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Research ArticleFinding Optimal Load Dispatch Solutions by Using a ProposedCuckoo Search Algorithm
Thang Trung Nguyen 1 Cong-Trang Nguyen1 Le Van Dai 2 and Nguyen Vu Quynh 3
1Power System Optimization Research Group Faculty of Electrical and Electronics Engineering Ton DucThang UniversityHo Chi Minh City Vietnam2Institute of Research and Development Duy Tan University Da Nang Vietnam3Department of Electromechanical and Electronic Faculty of Mechatronics and Electronics Lac Hong University Dong Nai Vietnam
Correspondence should be addressed to Le Van Dai levandaiduytaneduvn
Received 22 January 2019 Revised 16 April 2019 Accepted 16 May 2019 Published 28 May 2019
Academic Editor Changzhi Wu
Copyright copy 2019 Thang Trung Nguyen et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
Optimal load dispatch (OLD) is an important engineering problem in power system optimization field due to its significance ofreducing the amount of electric generation fuel and increasing benefit In the paper an improved cuckoo search algorithm (ICSA) isproposed for determining optimal generation of all available thermal generation units so that all constraints consisting of prohibitedpower zone (PPZ) real power balance (RPB) power generation limitations (PGL) ramp rate limits (RRL) and real power reserve(RPR) are completely satisfiedTheproposed ICSAmethodperformance ismore robust than conventional Cuckoo search algorithm(CCSA) by applying new modifications Compared to CCSA the proposed ICSA approach can obtain high quality solutionsand speed up the solution search ability The ICSA robustness is verified on different systems with diversification of objectivefunctions as well as the considered constraint set The results from the proposed ICSA method are compared to other algorithmsfor comparison The result comparison analysis indicates that the proposed ICSA approach is more robust than CCSA and otherexisting optimization approaches in finding solutions with significant quality and shortening simulation time Consequently itshould lead to a conclusion that the proposed ICSA approach deserves to be applied for finding solutions of OLD problem inpower system optimization field
1 Introduction
Over the past decades an enormous number of studieshave concerned and solved different optimization operationproblems in regard to electric grids by utilizing potentialsearch ability of optimization approaches Many concernedoperation problems regarding distribution power networktransmission network and different types of power plantas well as electric components in the network have beensuccessfully solved This study focuses on optimal loaddispatch (OLD) problem with the task of allocating thegenerated power of all considered thermal generation unitsto reduce the cost of burnt fossil fuels All physical andoperational constraints are required to be exactly satisfied Ifall generation units in each power plant and all power plants
are working under the most appropriate schedule total fuelcost of all units can be the smallest and consumers can getsignificant amount of revenue [1] The achievement is thanksto the meaning of OLD problem
A huge number of optimization approaches using math-ematical programming have been widely applied so farfor solving the considered OLD problem such as dynamicprogramming (DP) [2] lambda iterationmethod [3]NewtonRaphson and Lagrangian multiplier (NRLM) method [4]and linear programming (LP) [5 6] Conventional methodsfocused on the systems with simple constraints and convexobjective functionwhere nonlinear constraints and the effectsof valve loading process were not considered The complexlevel of the constraints and objective has been mentioned inmany articles For example the authors in [3] could only solve
HindawiMathematical Problems in EngineeringVolume 2019 Article ID 1564693 29 pageshttpsdoiorg10115520191564693
2 Mathematical Problems in Engineering
the OLD problem successfully by separating three-curveobjective function into three different single curve objectivefunctions In [7] amore realistic representation of the electricgeneration fuel function corresponding to multi fossil fuelsources was introduced in which the authors considered adiscontinuous cost function and the effects of valve loadingprocess (EoVLP) of thermal units More complicated modelswere also introduced For example some thermal generationunits were driven by burning multi fossil fuel sources (MFS)to generate electricity [8] or some operating conditions ofgenerators including upper generation boundary lower gen-eration boundary and prohibited power zones were added[9] However these methods have been only applied to thesystems where the generation power-fuel cost characteristicof thermal generation units was mathematically modelled asthe second order function and the effects of valve loadingprocess were ignored [10]
Other artificial intelligence-based advanced methodshave been recently applied for solving OLD problem Evolu-tionary programming-based approaches (EP) [1 11 12] andgenetic algorithm (GA) [13ndash15] were considered as fast algo-rithms because of their parallel search ability In addition GAand EP possessed good properties such as finding solutionsnearby global optimum capability of effectively handlingnonlinear constraints reliable search ability and not manyadjustment parameters [10] Thus it could become a suitablechoice for successfully solving OLD problem However GAwas prematurely convergent to local optimum solutions[16] In this regard simulated annealing approach (SA)[17] was a better probabilistic approach in finding solutionswith appropriate fitness but there was a high possibility ofeasily converging to local optimal zones when coping withcomplicatedly constrained problem It converges slower thanGA and EP though Differential evolution algorithm (DE)[10 18] also belongs to the same class as GA and EP Howeverit is more popular thanks to simple structure with severaladjustment parameters and high rate of success DE has beenmore widely and successfully applied than SA and GA [19]But its faster convergence manner led to the same drawbackas GA like high rate of falling into local optimum andhardly ever toward promising zones quickly In fact theseshortcomings could be tackled by setting population size tohigher value But high population size could suffer from longsimulation time to calculate fitness function and evaluatequality of solutions [20] Hopfield neural network (HNN)[21 22] focused on optimizing energy function and wasonly successfully applied for optimization problems whereobjective functions were differentiable HNN could be anappropriatemethod for large-scale systemswith highnumberof generation units but it needed long simulation time andmay also converge to local optimum solution zones [23]Particle swam optimization (PSO) [16 24] is a random searchapproach developed by behavior of a swarm or flock duringfood search process In comparison with GA PSO ownedmore advantages such as simpler implementation and fewparameters with easy selection However the success rateof PSO was highly influenced by adjustment parametersand it copes with high rate to be trapped in many zoneswith local optimum solutions [25] Harmony search (HS)
[26] is a metaheuristic-based method inspired from musicInstead of using gradient search HS has employed stochasticrandom search to exploit its potential ability Thus it tendedto converge to local optimal zones rather than global optimalzones [27] Biogeography-based optimization (BBO) [28]could compete with PSO and DE since its solutions weredirectly updated by migration from other existing solutionsand its solutions directly shared their attributes with othersolutions [29]
It is clear that each method has advantages as well asdisadvantages for different applications for finding OLDproblem solutions Hence another natural approach is tocombine different methods to exploit the advantages of eachmethod and enhance the overall searching capability Severalhybrid methods have been developed in such way includ-ing hybrid Genetic algorithm Pattern Search and Sequen-tial Quadratic Programming (GAndashPSndashSQP) method [30]hybrid Artificial Cooperative Search algorithm (HACSA)[31] hybrid PSO-SQP [32] and hybrid GA (HGA) [33]Basically these hybrid methods could deal with OLD prob-lem more effectively than each member method On theother hand they could suffer from the difficulty of selectingmany controllable parameters In addition to such popularoriginal algorithms and hybrid methods there are manyother original and improved methods that have been appliedfor solving the considered OLD problem These methods areSymbiotic organisms search algorithm (SOS) [34] and itsmodified version (MSOS) [34] teaching activity and learningactivity-based optimization (TLBO) [35] chemical reaction-based approach (CRBA) [36] enhanced particle swarmoptimization (EPSO) [37] sequential quadratic technique-based cross entropy approach (CEA-SQT) [38] traversesearch-based optimization approach (TSBO) [39] invasiveweed approach (IWA) [40] Improved Differential evolu-tion (IDE) [41] immune algorithm using power redistribu-tion IAPR [42] Colonial competitive differential evolution(CCDE) [43] Chaotic Bat algorithm (CBA) [44] Exchangemarket algorithm (EMA) [45] adaptive search techniquealgorithm and differential evolution (GRASP-DE) [46] 120579-Modified Bat Algorithm (120579-MBA) [47] Tournament-basedharmony search algorithm (TBHSA) [48] New Modified 120573-Hill Climbing Local Search Algorithm (M120573-HCLSA) [49]improved version of artificial bee colony algorithm (IABCA)[50] artificial cooperative search algorithm (ACSA) [51]and ameliorated greywolf optimization algorithm (AGWOA)[52] Among these methods ACSA and AGWOA werethe two latest methods which were applied for OLD andpublished in early 2019 However the demonstration of realperformance of the two methods is still questionable In factACSA has been tested only on systemswith small scale singlefuel and simple constraints such as generation limits andpower balance The largest scale system was considered in[51] to be 40-unit system Unlike [51] different types of fuelcost function complicated constraints and large scale systemwith 140 units have been taken into account in [52] Viacomparisons withmany existing methods AGWOAhas beenstated to be the best one with many surprising results Thusthe validation of reported solutions from the methodmust beverified and its strong search ability must be reevaluated In
Mathematical Problems in Engineering 3
the numerical results section we will report the verificationof the two questionable issues
In this paper we have proposed an improved cuckoosearch algorithm (ICSA) for dealing with large scale OLDproblem with the consideration of complicated constraintstogether with nondifferentiable fuel cost function In theproposed ICSA approach some newmodifications have beenperformed on conventional cuckoo search algorithm (CCSA)to improve the quality of CCSA The CCSA method wasfirst developed in 2009 [53] for solving a set of popularbenchmark functions and its highly superior performanceover PSO and GA has attracted a huge number of researchersin learning and applying for different optimization problemsin different fields Furthermore its improved variants are alsoan extremely vast number In relation toOLDproblemCCSAhas been applied and presented in [23 54ndash57] meanwhileits improved methods consisting of modified cuckoo searchalgorithm (MCSA) and improved cuckoo search algorithmwith one solution evaluation (OSE-CSA) have been respec-tively presented in [58 59] In [55] Basu has applied CCSAfor solving OLD problem with 40-unit system with singlefuel option and the effective of valve loading process 20-unitsystem with single fuel and quadratic fuel cost function and10-unit system with multiple fuel sources and without theeffects of valve loading process The author has made a bigeffort in demonstrating the high potential search of CCSAby comparing with many popular metaheuristic algorithmsbut the shortcoming of the study was neglecting complicatedconstraints and large scale systems
Studies in [56 57] have dealt with OLD problem withtwo power systems considering simple constraints and smallnumber of units Only three simplest constraints such aspower balance limitations of generation and prohibitedpower zones have been taken into account meanwhilethe largest system was solved to be 6-unit system Thusthere were few methods compared to CCSA and the realperformance of CCSA was not shown persuasively in thestudies In [23] authors have applied CCSA for solving differ-ent systems with very complicated constraints complicatedcharacteristics of thermal generating units and high numberof units Among the mentioned studies regarding CCSA forOLD problem authors in [23] could show the best view inevaluating the real performance of CCSA since there were sixcases that were carried out and a huge number of methodswere compared to CCSA In spite of the real potential searchability CCSA has been commented to be low convergence toglobal optimum and significantly improved better [58 59]MCSA in [58] has been proposed by using a new strategyfor the second generation technique Themutation operationin CCSA has been replaced with current-to-best1 model ofDE in [60] MCSA has been applied for solving four systemswith 3 6 15 and 40 units in which the most complicatedconstraint considered was prohibited power zone and onlysingle fuel source was taken into account MCSA methodhas been compared to other popular methods such as PSOGA and EP But the comparisons with CCSA have notbeen carried out Thus the improvement of such proposed
method in [58] was not proved persuasively OSE-CSA in[59] has canceled one evaluation time in case that OSE-CSAhas continued to improve solution quality The improvementseemed to be appropriate for CCSA in dealing with OLDproblem with complicated systems CCSA in [23 55] havebeen considered for comparison in [59] and they have beenproved to be less effective than OSE-CSA However OSE-CSA has used one more control parameter called one rankparameter and it needed to be tuned thoroughly for obtaininghigh performance In the proposed ICSA approach we havefocused on a new strategy of the second new solutiongeneration in CCSA method As shown in [58 59] CCSAhas become a strong search method thanks to the first newsolution generation which was performed by Levy flighttechnique while the second new solution generation couldnot take on local search function well In the second updateprogress via mutation operator two random old solutions areused to generate an increased step size However the mannercan lead to new low quality solutions because the increasedstep will be very small when iterative algorithm is carryingout at the last iterations In fact current solutions at finalseveral iterations tend to be close together and the differentvalues between each two ones are very small leading to a verysmall increased step In order to tackle the disadvantage of theCCSA we apply a new adaptive technique for improvementof solution quality Firstly we propose twoways for producingthe increased step including two-solution-based increasedstep and four-solution-based increased step The decisionwhen which step size will be used is dependent on theresult of comparison between fitness function ratio (FFR)and a predetermined parameter 119879119900119897 FFR is defined as aratio of deviation between fitness functions of the consideredsolution and the most promising solution to the fitness valueof the best one meanwhile 119879119900119897 is a boundary to give the finaldecision for the selection of a used step size At the beginning119879119900119897 is a fixed value for all solutions and then it will be adaptivebased on the comparison between it and FFR When FFR of asolution is less than 119879119900119897 119879119900119897 of the solution will be decreasedequally to ninety percent of the previous valueOtherwise thevalue of 119879119900119897 remains unchanged in case the FFR is equal toor higher than 119879119900119897 The adaptive technique has a significantlyimportant role in enhancing the potential search ability of theproposed method This proposed method is investigated onsix cases with different considered constraints different typesof fuel cost function and large scale systemsThe detail of thesix cases is as follows
Case 1 Four systems with single fuel source (SFS) and powerloss (PL) constraint
Case 2 A 110-unit system with SFS
Case 3 Four systems with SFS and the effects of valve loadingprocess (EoVLP)
Case 4 Two systems with SFS and PPZ and RPR constraints
Case 5 A 15-unit system with SFS and RRL PPZ and PLconstraints
4 Mathematical Problems in Engineering
Case 6 Three systems with multiple fuel sources (MFS) andEoVLP
The achieved results in terms of minimum fuel costaverage fuel cost maximum fuel cost and standard deviationfound by the proposed method compared to those obtainedby others reveal that the method is very efficient for theOLD problem In addition the performance improvementof the proposed method over CCSA is also investigatedvia the comparison of the best solution and all trial runsIn summary the main advantages of the proposed ICSAapproach over CCSA as well as the main contribution of thestudy are as follows
(i) Based on fitness function of each considered solutionlocal search or global search is decided to be appliedmore effectively
(ii) Find better solutions with smaller number of itera-tions and shorter execution time for each run
(iii) Shorten simulation time for the whole search of eachstudy case
However the proposed method also copes with the sameshortcomings as CSA Although the shortcomings do notcause bad results for the proposed method they make theproposed method be time consuming in tuning optimalparameter The shortcomings are analyzed as follows
(i) Control parameter probability of replacing controlvariables in each old solution must be tuned in rangebetween 0 and 1 There is no proper theory for deter-mining the most effective values of the parameterThus the performance of the proposed method mustbe tried by setting the parameter to values from 01 to1
(ii) The method uses more computation steps for searchprocessThus the proposedmethod uses higher num-ber of computation steps for each iteration Howeverdue to more effective search ability for each iterationthe proposed method can use smaller number ofiterations but it finds more effective solutions
The remaining parts of the paper are arranged as fol-lows Section 2 shows the objective and constraints of theconsidered OLD problem CCSA and the proposed methodare clearly explained in Section 3 Section 4 is in charge ofpresenting the implementation of ICSA method for the stud-ied problem The simulation results together with analysisand discussions are given in Section 5 Finally conclusion issummarized in Section 6 In addition appendix is also addedfor showing found solutions by the proposed ICSA approachfor test cases
2 Optimal Load DispatchProblem Description
21 Fuel Cost Function Forms with Single Fuel Source In theconsidered OLD problem the optimal operation of a set of
thermal generation units is concerned as the duty of reducingtotal cost of all the units which can be seen by the followingmodel
Reduce 119865 = 119873sum119894=1
119865119894 (119875119894) (1)
In traditional OLD problem fuel cost function of the119894119905ℎ generation unit 119865119894(119875119894) is represented as the second orderfunction with respect to real power output and coefficients asthe model below [2]
In addition for the case considering the effects ofvalve loading process on thermal generation units fuel costbecomes more complicated by adding sinusoidal term asbelow [12]
119865119894 (119875119894) = 1205721198941198752119894 + 120582119894119875119894 + 120575119894+ 1003816100381610038161003816120573119894 times sin (120574119894 times (119875119894min minus 119875119894))1003816100381610038161003816 (3)
Real Power Balance Constraint Total real power demand ofall loads in power system together with real power loss in allconductors must be equal to the generation from all availablethermal generation units The requirement is constrained bythe following equality
119873sum119894=1
119875119894 = 119875119863 + 119875119871 (4)
where total real power loss 119875119871 is determined by Kronrsquosequation below
119875119871 = 11986100 + 119873sum119895=1
1198610119895119875119895 + 119873sum119895=1
119873sum119894=1
119875119895119861119895119894119875119894 (5)
GenerationBoundaryConstraint For the purpose of economyand safe operation each thermal generation unit is con-strained by the lower generation bound and upper generationbound as the following model
119875119894min le 119875119894 le 119875119894max (6)
22 Fuel Cost Function Forms with Multi-Fuel Sources Inthis section fuel cost function of thermal generation units ismathematicallymodeled in terms of different forms from thatin the section above due to the consideration of multi-fuelsources Each type of fuel source is formed as each secondorder function and the fuel cost function form is the sumof different second order functions for the case of neglectingthe effects of valve loading progress But for the considerationcase of the effects the form ismore complexwith the presenceof sinusoidal terms [15] As a result the forms of cost functioncan be expressed in Equation (7) [21] and Equation (8) [15]
Mathematical Problems in Engineering 5
119865119894 (119875119894) =
1205751198941 + 1205821198941119875119894 + 12057211989411198752119894 fuel 1 119875119894min le 119875119894 le 11987511989411198981198861199091205751198942 + 1205821198942119875119894 + 12057211989421198752119894 fuel 2 1198751198942min le 119875119894 le 1198751198942max120575119894119895 + 120582119894119895119875119894 + 1205721198941198951198752119894 fuel 119895 119875119894119895min le 119875119894 le 119875119894119895max
(7)
119865119894 (119875119894) =
1205751198941 + 1205821198941119875119894 + 12057211989411198752119894 + 10038161003816100381610038161205731198941 times sin (1205741198941 times (119875119894min minus 119875119894))1003816100381610038161003816 for fuel 1 119875119894min le 119875119894 le 11987511989411198981198861199091205751198942 + 1205821198942119875119894 + 12057211989421198752119894 + 10038161003816100381610038161205731198942 times sin (1205741198942 times (119875119894min minus 119875119894))1003816100381610038161003816 for fuel 2 1198751198942min le 119875119894 le 1198751198942119898119886119909 119895 = 1 119898119894120575119894119895 + 120582119894119895119875119894 + 1205721198941198951198752119894 + 10038161003816100381610038161003816120573119894119895 times sin (120574119894119895 times (119875119894min minus 119875119894))10038161003816100381610038161003816 for fuel 119895 119875119894119895min le 119875119894 le 119875119894119895max
(8)
Cost function forms in Equations (7) and (8) are onlyincluded in objective function (1) meanwhile main con-straints in formulas (4) and (6) must be always satisfied
23 Prohibited Power Zone Real Power Reserve and RampRate Limit Constraints Prohibited power zones (PPZ) aredifferent ranges of power in fuel cost function that thermalgeneration units are not allowed to work due to operationprocess of steam or gas valves in their shaft bearing Thepower generation of units in the violated zones is harmfulto gas or steam turbines even destroyed shaft bearing Thusthe constraint is strictly observed In the fuel-power charac-teristic curve of generation units PPZ causes small violationzones and such curves become discontinuous As consideringPPZ constraint the determination of power generation ofunits is more complex and equal to either lower bound orupper bound Unlike PPZ constraint RPR constraint is notrelated to fuel-power feature curve but it causes difficulty foroptimization approaches in satisfying one more inequalityconstraint Each generation unit among the set of availablegeneration units must reserve real power so that the sumof real power from all generation units can be higher orequal to the requirement of power system for the purposeof stabilizing power system in case that there are some unitsstopping producing electricity On the contrary to PPZ con-straint ramp rate limit (RRL) constraint does not allowpoweroutput of thermal generating units outside a predeterminedrange The constraint considers maximum power change ofeach thermal generating unit as compared to the previouspower value Thus optimal generation must satisfy the RRLconstraint The PPZ constraint RPR constraint and RRLconstraint can be presented as follows
Prohibited Power Zones As considering PPZ constraint validworking zones of each thermal generating unit are notcontinuous and its generation must be outside the violatedzones as the following mathematical description
119875119894 isin
119875119894min le 119875119894 le 1198751198971198941119875119906119894119896minus1 le 119875119894 le 119875119897119894119896 k = 2 ni forall119894 isin Ω119875119906119894119899119894 le 119875119894 le 119875119894max
(9)
As observing Equation (9) generation units cannotbe operated within the violated zones except for startingpoint and end point Consequently the verification of PPZconstraint violation should be carried out first and thenthe correction should be done before dealing with otherconstraints such as real power reserve constraint and realpower balance Besides if power output of all units can satisfythe PPZ constraint generation limits in Equation (6) are alsoexactly met
Real Power Reserve Constraint Real power reserve in powersystem aims to enhance the ability of stability recovery ofpower system and avoid blackout In order to get high enoughpower for requirement all available units are constrained bythe following inequality
119873sum119894=1
119878119894 ge 119878119877 (10)
where 119878119894 is the real power reserve contribution of the 119894119905ℎthermal generation unit and the determination of 119878119894 can bedone by employing the two models below
119878119894 = 119875119894max minus 119875119894 119894119891 119878119894max gt (119875119894max minus 119875119894)119878119894max else
forall119894 notin Ω (11)
119878119894 = 0 forall119894 isin Ω (12)
Equation (10) shows that the constraint of prohibitedpower zones is not included in the real power reserveconstraints however prohibited power zones are alwaysstrictly considered and must be exactly satisfied
Ramp Rate Limit (RRL) Constraint In OLD problem allconsidered thermal generating units are supposed to be underworking status but previous active power of each thermalgenerating unit is not taken into account Thus increased ordecreased power is not constrained This assumption seemsto be not practical until RRL constraint is considered RRLconstraint considers initial power output and the power
6 Mathematical Problems in Engineering
change is supervised Regulated power can be higher or lowerthan the initial value as long as it is within a predeterminedrange Increased step size (ISS) and decreased step size (DSS)are given as input data and they are used to limit the change ofpower output of each thermal generating unit The constraintcan bemathematically expressed as the following formula [7]
1198751198940 minus 119863119878119878119894 le 119875119894 le 1198751198940 + 119868119878119878119894 (13)
where 1198751198940 is the initial power output of the 119894119905ℎ thermalgenerating unit before its power output is regulated 119868119878119878119894 and119863119878119878119894 are respectively maximum increased and decreasedstep sizes of the 119894119905ℎ thermal generating unit
3 The Proposed Cuckoo Search Algorithm
31 Classical Cuckoo Search Algorithm In search techniqueof CCSA [53] a set of solutions is randomly generated withina predetermined range in the first step and then the quality ofeach one is ranked by computing value of fitness functionThemost effective solution corresponding to the smallest valueof fitness function is determined and then search procedurecomes into a loop algorithm until the maximum iterationis reached In the loop algorithm two techniques updatingnew solutions two times (corresponding to two generations)are Levy flights and mutation technique which is calledstrange eggs identification technique The two generationscan produce promising quality solutions for CCSA Aftereach generation CCSA will carry out comparing fitness ofnewly updated solutions and initial solutions for keepingbetter ones and abandoning worse ones The most effectivesolution at last step of the loop search algorithm is determinedand it is restored as one candidate solution for a study caseThe detail of the two stages is as follows
311 Levy Flights Stage This is the first calculation step in theloop algorithm and it also produces new solutions in the firstgeneration for CCSA New solution 119878119900119897119899119890119908119909 is created by thefollowing model
119878119900119897119899119890119908119909 = 119878119900119897119909 + 120572 (119878119900119897119909 minus 119878119900119897119866119887119890119904119905) oplus Levy (120573) (14)
where 120572 is the positive scaling factor and it is nearly set todifferent values for different problems in the studies [53 62]In the work the most appropriate values for such factor canbe chosen to be 02505 for different systems
312 Discovery of Alien Eggs Stage The step plays a veryimportant role for updating new solutions 119878119900119897119899119890119908119909 of thewhole population However not every control variable ineach old solution is newly updated and the decision ofreplacement is dependent on comparison criteria as thefollowing equation
119878119900119897119899119890119908119909=
119878119900119897119909 + 1205761 (1198781199001198971199031198861198991198891 minus 1198781199001198971199031198861198991198892) if 1205762 lt 119875119886119878119900119897119909 119900119905ℎ119890119903119908119894119904119890
(15)
32 Proposed Algorithm In the part a new variant of CCSA(ICSA) is constructed by applying three effective changes onthe main functions of CCSA in order to shorten simula-tion time corresponding to reduction of iterations and findmore promising solutions The proposed amendments areexplained in detail as follows
(i) Suggest one more equation producing updated stepsize in addition to existing one in CCSA
(ii) Create a new selection standard by computing fitnessfunction ratio 119865119865119877119909 and comparing 119865119865119877119909 with apredetermined parameter 119879119900119897119909 Thus thanks to thestandard the existing updated step size and additionalupdate step size will be chosen more effectively
(iii) Automatically change value of 119879119900119897119909 for the xth solu-tion based on the result of comparing 119865119865119877119909 with theprevious 119879119900119897119909
Such three points are clarified by observing the followingsections
321 Strange Eggs Identification Technique (Mutation Tech-nique) The first proposed improvement in our proposedICSA approach is to select a more suitable formula forproducing new solutions with better fitness function valueIn CCSA Equation (16) below is used to produce a changingstep nearby old solutions for all current solutions
Δ1198781199001198971198991198901199081199091 = 1205763 (1198781199001198971199031198861198991198891 minus 1198781199001198971199031198861198991198892) (16)
The use of Equation (16) aims to produce a random walkaround old solutions in search zones with intent to findout promising solutions In order to reduce the possibilityof suffering the local trap and approach to other favorablezones for searching we propose a new Equation (17) Theformula is built by the idea of enlarging search zone withthe use of two more available solutions Obviously the largerchanging step can own higher performance in moving toother search spaces that the classical approach used in CCSAThe suggestion is mathematically expressed by the formulabelow
Δ1198781199001198971198991198901199081199092= 1205764 (1198781199001198971199031198861198991198891 minus 1198781199001198971199031198861198991198892 + 1198781199001198971199031198861198991198893 minus 1198781199001198971199031198861198991198894) (17)
The changing step obtained by using Eq (17) is namedfour-point changing step Now two solutions which arenewly formed by using two different changing steps shownin formulas (16) and (17) are found by the two followingmethods
It can be clearly observed that the distance between 119878119900119897119909(old solution) and 1198781199001198971198991198901199081199091 (new solution) is lower than thatbetween 119878119900119897119909 and 1198781199001198971198991198901199081199092 This difference can contribute ahighly efficient improvement to the proposed ICSA approachsearch ability
Mathematical Problems in Engineering 7
ΔSol2
Sol2
Sol3
Sol4
ΔSol1
Sol1
Solx
Solnew1
Solnew2
Figure 1 Simulation of solutions corresponding to the first itera-tions of the loop algorithm
For the CCSA case if two solutions 1198781199001198971199031198861198991198891 and 1198781199001198971199031198861198991198892are either slightly different or completely coincident suchnewly updated solution 1198781199001198971198991198901199081199091 does not have good chanceto leave the current zone and approach to more promisingzones In another word the new one is approximately coin-cident with the old one As the search task is taking place atsome last iterations this phenomenon becomes much worsebecause all current solutions are lumped in a small zone andthe capability of moving to other zones is impossible As aresult the CCSA approach will work ineffectively and searchstrategy is time consuming until other runs are started
Contrary to the two-point step size the new proposedformula may produce a large enough length to escape thelocal optimum zone and reach new favorable zones Itexplainswhy the four-point changing step has positive impacton the considered random walk rather than the two-pointchanging step
322 New Standard forChoosing theMostAppropriate Chang-ing Step In this section we extend our analysis to answer thequestionwhen to use the four-point step size FromEquations(18) and (19) two new solutions which are represented asthe results of the two-point-based factor and the four-pointstep size can be illustrated by using Figure 1 corresponding tothe search process at the first some iterations and Figure 2corresponding to the last some iterations For the sake ofsimplicity we rewrite the two equations as follows
Here we suppose that 1198781199001198971 and 1198781199001198972 are obtained byfour exact solutions 1198781199001198971 1198781199001198972 1198781199001198973 and 1198781199001198974 and calculatedas follows
ΔSol2
ΔSol1
Solx
Solnew1
Solnew2
Figure 2 Simulation of solutions corresponding to the last itera-tions of the loop algorithm
Δ1198781199001198971 = 1198781199001198971 minus 1198781199001198972 (22)
Δ1198781199001198972 = 1198781199001198973 minus 1198781199001198974 (23)
Asmentioned above the high changing step between newsolution and old solution can help to explore new favorablezones However in optimization algorithms searching stepscannot be arbitrarily large otherwise the algorithm maydiverge in particular for the cases that the consideredsolutions 119878119900119897119909 are not close together in solution search spaceFor example at the beginning of loop algorithm with thefirst iterations in Figure 1 1198781199001198971198991198901199081 is a better choice than1198781199001198971198991198901199082 because it is kept in a sufficient limit and does notlead to a risk of divergence In contrast as many of currentsolutions are in different positions but their distance is notvery short or approximately coincident such as at the lastiterations in Figure 2 1198781199001198971198991198901199081 and 119878119900119897119909 have a very shortdistance but 1198781199001198971198991198901199082 and 119878119900119897119909 have higher distance Accordingto the phenomenon in Figure 2 the proposed ICSA approachneeds to produce a high changing step to move solutions toother search zones without local optimum Hence 1198781199001198971198991198901199082would be preferred to 1198781199001198971198991198901199081
Based on the argument above the determination of thecondition for using either two-point changing step or four-point changing step is really crucial to the performance ofthe proposed ICSA approach in searching solutions of OLDproblem Here the ratio of 119865119865119877119909 which can be found byEquation (24) is suggested to be a suitable measurement forthe selection of two options
Δ119865119865119877119909 = 119865119865119909 minus 119865119865119887119890119904119905119865119865119887119890119904119905 (24)
For a particular set of the current solutions each individ-ual depending on its 119865119865119877119909 will create a corresponding newsolution by using either Equation (18) or (19) If the valueof one current solution is smaller than the predeterminedparameter 119879119900119897 Equation (19) is applied for updating suchconsidered solution 119909 Otherwise Equation (18) is a betteroption The steps of the modified algorithm are similar to the
8 Mathematical Problems in Engineering
If 1205765 lt 119875119886If FFRx lt Tolx119878119900119897119899119890119908119909 = 119878119900119897119909 + 1205764(1198781199001198971199031198861198991198891 minus 1198781199001198971199031198861198991198892 + 1198781199001198971199031198861198991198893 minus 1198781199001198971199031198861198991198894)else119878119900119897119899119890119908119909 = 119878119900119897119909 + 1205764(1198781199001198971199031198861198991198891 minus 1198781199001198971199031198861198991198892)end
Algorithm 1 New mutation technique applied in the proposed ICSA approach
conventional CSA except that an additional step should beadded at each iteration In this step the119865119865119889 of all individualsolutions should be calculated by utilizing Equation (24) andthen the result of comparing the ratio with 119879119900119897will be used todecidewhich updating formula should be selectedThewholedescription of the proposed standard and new mutationtechnique can be coded inMatlab program language by usingAlgorithm 1
323 Adjustment of Tolerance for Each Solution As pointedout above the proposed method needs assistances to deter-mine the most appropriate step size for finding out favorablesolution zones The given aim can be reached if the selectionof 119879119900119897119909 is reasonable however the range of this parameteris infinite and hard to select Thus the adaptation of tuningthe parameter is really necessary First of all the compari-son between 119879119900119897119909 and 119865119865119877119909 is carried out and then theadaptation will be determined based on the obtained resultfrom the comparison Results of comparison between the twoparameters can be either 119865119865119877119909 is less than 119879119900119897119909 or 119865119865119877119909is higher than 119879119900119897119909 The case that two parameters are equalhardly ever occurs
As the comer assumptionhappens (ie119865119865119877119909 is less than119879119900119897119909) at the considered time the four-point step size will beemployed for the 119909119905ℎ solution If 119879119900119897119909 remains unchanged atthe previous value the identification of improvement fromsuch four-point step size or two-point step size is vagueConsequently value of 119879119900119897119909 must be automatically reducedto a lower value in case that it has significant contribution tofound promising solution of previous iteration Clearly thedecrease of119879119900119897119909 can enable the proposedmethod to jump outlocal optimal zone and approachmore effective zones By trialand error method 119879119900119897119909 is selected to be a function of itselfthat is 09 of the previous value Finally the implementationof the proposed ICSA approach is presented in Algorithm 2
4 The Application of the ProposedICSA for OLD Problem
Thewhole computation steps of the proposed ICSA approachfor solving OLD problem are explained as follows
41 Handling Constraints and Randomly Producing InitialPopulation As shown in Section 2 the considered OLDproblem takes five following constraints into account
(i) Power balance constraint is shown in Equation (4)
(ii) Power output limitation constraint is shown in Equa-tion (6)
(iii) Prohibited power zone constraint is shown in Equa-tion (9)
(iv) Real power reserve constraint is shown in Equation(10)
(v) Ramp rate limit constraint is shown in Equation (13)
Among the five constraints ramp rate limit generationlimit and prohibited power zone seem to be more com-plicated than power balance and power reserve constraintsHowever the three constraints can be solved more easilybecause each unit is constrained independently in the threeconstraints whereas power balance constraint and powerreserve constraint consider all the thermal generating unitssimultaneously Power reserve constraint can be handledby penalizing the total generation of all units while powerbalance constraint can be solved by penalizing one violatedthermal generating unit The whole computation procedurefor solving all constraints and calculating fitness function ofsolutions is described in detail as follows
Step 1 Redefine maximum and minimum power output ofeach thermal generating unit as considering PPZ and RRLconstraints by using the following formulas
119875119894max = 119875119894max if 119875119894max le 119875i0 + 119868119878119878119894119875i0 + 119868119878119878119894 if 119875119894max gt 119875i0 + 119868119878119878119894
119894 = 1 119873(25)
119875119894min = 119875119894min if 119875119894min ge 119875i0 minus 119863119878119878119894119875i0 minus 119863119878119878119894 119890119897119904119890
119894 = 1 119873(26)
Mathematical Problems in Engineering 9
Produce initial population with119873119901119904 solutions (1198781199001198971 1198781199001198972 119878119900119897119909 119878119900119897119873119901119904)Calculate fitness function (1198651198651 1198651198652 119865119865119909 119865119865119873119901)Go to the loop algorithm by setting 119866 = 1
While (119866119898119886119909 gt 119866) (i) The first newly produced solutions119878119900119897119899119890119908119909 = 119878119900119897119909 + 120572(119878119900119897119909 minus 119878119900119897119866119887119890119904119905) oplus Levy(120573) (ii) Perform selection approach
119865119865119909 = 119865119865119909 if 119865119865119909 le 119865119865119899119890119908119909119865119865119899119890119908119909 119900119905ℎ119890119903119908119894119904119890 119909 = 1 119873119901119904
(v) Determine the most effective solution and its fitnessDetermine 119865119865119909 with the smallest value and assign 119878119900119897119909 to 119878119900119897119866119887119890119904119905If 119866119898119886119909 gt 119866 perform step (i) and increase 119866 to 119866 + 1 Otherwise stop the loop algorithm and report boththe smallest fitness together with 119878119900119897119866119887119890119904119905End while
Among the four Equations (25) and (26) are used firstin order to redefine upper bound and lower bound for allthermal generating units as considering RRL constraint Thethe redefined bounds continue to be redefined for the secondtime by using (27) and (28) as considering PPZ constraints
Step 2 (randomly produce initial population) For dealingwith the power balance constraint all available units areseparated into two groups in which the first group withdecision variables consists of the power output from thesecond unit to the last unit (P2 P3 PN) meanwhile onlythe power output of the first unit (1198751) belongs to the secondgroup with dependent variable So upper bound solution119878119900119897119898119886119909 and lower bound solution 119878119900119897119898119894119899 must be defined asfollows
Step 3 Handle prohibited power zone constraint for decisionvariables P2 P3 PN
After being randomly produced there is a high possi-bility that decision variables fall into PPZ and they violatePPZ constraint So the verification of falling into PPZ andcorrection of the violation should be accomplished by usingthe following formula
119875119894 =
119875119897119894119896 if 119875119897119894119896 lt 119875119894 le 119875119897119894119896 + 1198751198961198941198962119875119906119894119896 if (119875119894 gt 119875119897119894119896 + 1198751198961198941198962 ) amp (119875119894 lt 119875119906119894119896)119875119894 119890119897119904119890
119894 = 2 119873 amp 119896 = 1 119899119894
(31)
Step 4 Handle RPB constraint by calculating 1198751 and penaliz-ing 1198751 if it violates constraints
In this step power balance constraint is exactly handledby calculating and penalizing dependent variable (1198751) 1198751 isobtained by using formulas (4) and (5) as follows
1198751 = minus (11986101 minus 1 + 2sum119873119894=2 1198611119894119875119894) plusmn radicΔ211986111 (32)
where
Δ = (11986101 minus 1 + 2 119873sum119894=2
1198611119894119875119899)2 minus 411986111(119875119863 minus 119873sum
119894=2
119875119894+ 11986100 + 119873sum
119894=2
1198610119894119875119894 + 119873sum119894=2
119873sum119895=2
119875119894119861119894119895119875119895) amp Δ ge 0(33)
In Equation (32) 1198751 has been determined for the purposeof dealing with real power balance constraint However it isnot sure that 1198751 can satisfy upper bound and lower boundconstraints and prohibited power zone constraints So 1198751must be checked and penalized
Firstly 1198751 is checked and penalized for upper and lowerbound constraints by the following model
Δ1198751x =
0 if 1198751min le 1198751x le 1198751max
1198751min minus 1198751x if 1198751min gt 1198751x1198751x minus 1198751max if 1198751max lt 1198751x
(34)
In Equation (34) if the second case or the third caseoccurs it means P1 has violated either lower bound or upperbound and it would be penalized by using either (P1x= P1min-P1x) or (P1x =P1x -P1max) Otherwise ifP1 has not violatedthe bound constraints (ie the first case in (34) happened)
P1 would continue to be checked for PPZ constraint by thefollowing model
Δ1198751x
=
1198751 minus 1198751198971119896 if 1198751198971119896 lt 1198751 le 1198751198971119896 + 119875119896111989621198751199061119896 minus 1198751 if (1198751 gt 1198751198971119896 + 11987511989611198962 ) amp (1198751 lt 1198751199061119896)0 119890119897119904119890
(35)
Step 5 Handle real power reserve constraint (10)First of all 119878119894 is determined by using (11) and (12) and
then the 119909119905ℎ solution will be checked and penalized if poweroutput of all thermal generating units cannot satisfy RPRconstraint The penalty for violation of the constraint can becalculated by using equation (36)
Δ119878119877119909 =
0 if119873sum119894=1
119878119894119909 ge 119878119877119878119877 minus 119873sum119894=1
119878119894119909 119890119897119904119890 (36)
As a result real power reserve constraint can be solved byusing the penalty method
42 Calculate Fitness Function for Solutions Fitness functionof each solution is used to evaluate quality of solutionNormally the function is the sum of objective function andpenalty of violating constraints and is obtained by
43 The First Newly Updated Solutions by Levy Flights Tech-nique In this section the first newly updated solutionsare performed by employing Levy flights technique usingEquation (14) However each new solution can be out oftheir feasible operating zone such as PPZ and upper andlower limitations When the power output violates its PPZconstraints Equation (31) will be applied to tackle theconstraint Besides the following equation will be employedwhen power output is higher or lower than their limitations
119878119900119897119909 =
119878119900119897max if 119878119900119897max lt 119878119900119897119909119878119900119897min if 119878119900119897min gt 119878119900119897119909119878119900119897119909 Otherwise
119909 = 1 119873119901 (38)
After that Equations (32)-(37) are performed for deter-mining all variables and penalty terms Finally Equation (38)is employed to calculate fitness function
44 The Second Newly Updated Solutions by Using Muta-tion Technique The second newly updated solutions areaccomplished as presented in Section 3 above Similar to
Mathematical Problems in Engineering 11
the task after doing the first update each solution in thenew population must satisfy PPZ constraint and upper andlower boundaries by considering Equations (31) and (38)Then Equations (32)-(37) are performed for determiningall variables and penalty terms Finally Equation (38) isemployed to calculate fitness function and the solution withthe best value is assigned to the best one 11987811990011989711986611988711989011990411990545 Criterion of Stopping the Loop Algorithm In the loopalgorithm of using the proposed ICSA approach the solutionsearch work is stopped in case that the predeterminedmaximum iterations 119866119898119886119909 is reached For each search ter-mination the most effective solution is stored and anotherrun continues to be accomplished until the predeterminednumber of runs is reached After finishing the runs thebest one is found and reported In addition other valuessuch as the fitness of the worst solution and average fitnessof all solutions are also reported for comparing with othermethods
46 The Whole Iterative Process The whole iterative algo-rithm for implementing the proposed ICSA approach forcoping with OLD problem is described in detail in Figure 3
5 Results and Discussions
The proposed ICSA approach performance has been investi-gated on six cases with different fuel options different fuelcharacteristics and complicated constraints The details ofthe studied cases are presented as follows
Case 1 Four systems with single fuel source (SFS) and powerloss (PL) constraint
Subcase 11 A 3-unit system [57]Subcase 12 A 6-unit system [57]Subcase 13 A 3-unit system [56]Subcase 14 A 6-unit system [56]
Case 2 A 110-unit system with SFS [57]
Case 3 Four systems with SFS and the effects of valve loadingprocess (EoVLP)
Subcase 31 A 3-unit system supplying to a load of850MW [58]Subcase 32 A 13-unit system supplying to a load of1800MW [1]Subcase 33 A 13-unit system supplying to a load of2520MW [1]Subcase 34 A 40-unit system supplying to a load of2500MW [1]Subcase 35 An 80-unit system supplying to a load of4100MW [49]
Case 4 Two systems with SFS and PPZ and RPR constraints
Subcase 41 A 60-unit system supplying to a10600MW load [9]
Subcase 42 A 90-unit system supplying to a15900MW load [9]
Case 5 A 15-unit system with SFS and RRL PPZ and PLconstraints [61]
Case 6 Three systems with multiple fuel sources (MFS) andEoVLP
Subcase 61 An 80-unit system supplying to a21600MW load [15]Subcase 62 A 160-unit system supplying to a43200MW load [15]Subcase 63 A 320-unit system supplying to an86400MW load [54]
For each considered case with each load case the pro-posed ICSA approach is run 50 times on the programlanguage of Matlab and a PC with 4 GB of RAM and 24GHzprocessor The selection of adjustment parameters including119875119886 and 119879119900119897119909 is carefully considered to obtain the best optimalsolutions meanwhile two others such as 119873119901119904 and 119866119898119886119909 arechosen corresponding to the scale of particular test system9 values with the change of 01 in the range [01 09] are inturn selected for 119875119886 while 119879119900119897119909 is 001 at the beginning Theinformation including load demand119873119901119904 119866119898119886119909 and the best119875119886 is reported in Table 1
51 Obtained Results on Case 1 considering Four Systems withSFS and PL Constraint In this section we have implementedthe proposed ICSA approach for solving four systems dividedinto four subcases Tables 2 and 3 show the comparisons ofobtained results from Subcases 11 and 12 and Subcases 13and 14 respectively As listed in Table 2 the proposed ICSAmethod and CCSA can find equal fuel cost for Subcases 11whereas the reduction of fuel cost from the proposed ICSAmethod as compared to CCSA is clearer for Subcase 12 Asshown in Table 3 for comparing the proposed ICSA and threemethods consisting of CCSA ABC and FA the minimumfuel cost of the proposed ICSA is approximately equal to thatof these methods for Subcases 13 but much less than that ofthese methods for Subcase 14 Furthermore the proposedICSA has been run by setting 119873119901119904 and 119866119898119886119909 to 5 and 20but these values were much higher for CCSA ABC and FAThey are 20 and 5000 for CCSA 40 and 100 for ABC and 20and 5000 for FA Consequently the proposed method is veryefficient for Case 1 with four subcases
Optimal solutions obtained by ICSA for Case 1 are shownin Tables 16ndash18
52 Obtained Results on Case 2 considering 110-Unit Systemwith SFS In this section we have employed a very largescale system with 110 units but there were not challenges forobjective function and complex constraints since EoVLP andconstraints were not taken into account Both CCSA andthe proposed ICSA methods have been run for comparingwith BBO hybrid BBO and DE (DEBBO) and Opposi-tional real coded chemical reaction optimization algorithm(ORCCROA) in [36] IWA in [40] and AGWO in [52] As
12 Mathematical Problems in Engineering
Select parameters
- Correct solutions if violating PPZ constraint using Eq (31)- Determine using Eq (32)- Penalize for violating upper and lower bounds constraint using Eq (34)- Penalize for violating PPZ constraint using Eq (35)- Penalize the xth solution for violating RPR constraint using Eq (36)
- Determine quality of solutions employing Eq (37) - Select the most effective solution - Start the loop algorithm by selecting
- Perform the first solution update using Section 32 - Correct boundaries of solutions by using Eq (38)
- Correct solutions if violating PPZ constraint using Eq (31)- Determine using Eq (32)- Penalize for violating upper and lower bounds constraint using Eq (34)- Penalize for violating PPZ constraint using Eq (35)- Penalize the xth solution for violating RPR constraint using Eq(36)
- Determine fitness function using Eq (37)- Compare old solutions and new solutions to keep better ones
- Perform the second solution update using Algorithm 1 - Correct boundaries of solutions by using (38)
- Correct solutions if violating PPZ constraint using Eq (31)- Determine using Eq (32)- Penalize for violating upper and lower bounds constraint using Eq (34)- Penalize for violating PPZ constraint using Eq (35)- Penalize the xth solution for violating RPR constraint using Eq (36)
- Determine fitness function using Eq (37)- Compare old solutions and new solutions to keep better ones- Select the most effective solution
Stop
Start
- Redefine upper and lower bounds using Eqs (25)-(28)- Randomly generate initial population using Eq (30)
Nps Pa Gmax H> Tolx
P1xP1x
P1x
P1xP1x
P1x
P1xP1x
P1x
Sol<MN
Sol<MN
G = 1
G = Gmax G = G + 1
Figure 3 All computation steps for solving OLD problem by employing the proposed ICSA approach
shown in Table 4 AGWO [52] has reached less fuel cost thanICSA however the exact fuel cost which was recalculatedby using reported solution pointed out that the method hasreached a very high fuel cost of $2157404250 For comparisonwith other methods ICSA has found less fuel cost thanall these methods Particularly the reduction of generationfuel cost is significant as compared to BBO DEBBO andCCSA Execution time comparisons are also useful evidence
for indicating the high performance of ICSA Thus it canconclude that ICSA is a strong method for Case 2
Optimal solution obtained by ICSA for the case is shownin Table 19
53 Obtained Results on Case 3 considering Four Systems withSFS and EoVLP In this section the real performance of theproposed ICSA approach has been investigated based on five
Mathematical Problems in Engineering 13
Table 1 Information of considered cases and adjustment parameters
Case Fuel cost function Constraint Subcase No of units 119875119863 (MW) Nps Gmax Best 1198751198861 SFS PL
subcases with the gradual increase of number of units Thesmallest scale system considers 3 units but the largest scalesystem takes 80 units In addition to the implementation ofthe proposed ICSA we have also implemented CCSA forSubcase 31 and Subcase 35 for further comparison becauseCCSA has not been run for the two subcases so far
Comparison of obtained results from Subcase 31 shownin Table 5 indicates that the proposed ICSA is superior toCCSAwith lower fuel cost but it seems to be less effective thanMCSA [58] The minimum cost of MCSA reported in [58] isthe smallest fuel cost but the recalculated cost is much higherthan that of the proposed method Furthermore MCSA hasbeen implemented by setting very high values to 119873119901119904 and119866119898119886119909
Reports for Subcases 32 and 33 shown in Table 6are the comparisons of the proposed ICSA approach andother methods such as conventional Evolution programming(CEP) [1] Fast EP (FEP) [1] improved FEP (IFEP) [1] DE[12] multiplier Lagrange-based genetic algorithm with (GA-MU) [15] QPSO [16] GA-PS-SQP [30] PSO-SQP [32] M120573-HCLSA [49] IABCA [50] CCSA [59] OSE-CSA [59] SOS[34] MSOS [34] CEA-SQT [38] TSBO [39] IWA [40] andCBA [44] As observed from the table ICSAapproach obtainsbetter solutions than mostmethods excluding DE [10] CCSA
[59] OSE-CSA [59] SOS [34] MSOS [34] CEA-SQT [38]TSBO [39] IWA [40] and CBA [44] especially M120573-HCLSA[49] with lower cost $1796097 However recalculated costfrom reported solution of M120573-HCLSA is $179691 BesidesICSA is very fast as compared to most methods where twoother versions of Cuckoo search algorithm CCSA and OSE-CSA are also included except two methods in [34] Theprocessor of computer that all the methods run on is alsoreported in the final column Clearly ICSA approach is veryefficient for the case with the 13-unit system where effects ofvalve loading process are considered
In Subcase 34 the number of units is much larger thanthat of three subcases above up to 40 units [1] The obtainedresult comparisons with others are indicated in Table 7Clearly the minimum cost comparisons reveal that the pro-posedmethod is one of the leading methods due to the lowestcost except the comparison with CCSA [23] OSE-CSA [59]SOS [34] MSOS [34] EMA [45] 120579-MBA [47] and AGWOA[52] It is noted that AGWOA [52] has reported the bestminimum cost with $12140430 but recalculated minimumcost which was obtained by substituting reported optimalgeneration of all thermal generating units is $12141331 Theaverage and the maximum costs from the proposed methoddo not belong to the leading method group however the
Mathematical Problems in Engineering 15
Table 7 Result comparisons for Subcase 34
Approach Best cost Average cost Worst cost CPU time Computer($h) ($h) ($h) (s) (Processor-Ram)
execution time that ICSA approach takes is smaller thanmostones except IAPR [42] and 120579-MBA [47] which have been runon stronger computers Note that MSOS [34] has been fasterthan the proposed method for Subcases 31 and 32 abovebut it is too slower than the proposed method for the casenamely 1813 seconds compared to 146 seconds Comparedto two other versions of Cuckoo search CCSA [23] and OSE-CSA [59] the proposed method is also faster about threetimes although the processors are slightly different For thiscase 120579-MBA [47] shows a very good performance howeverthe method has not been tested onmore complicated systemsand larger scale and therefore more comparisons with themethod must end Clearly the proposed method is stillefficient for the case where large scale and effects of valveloading process are included
Subcase 35 is the largest scale system with 80 units Inaddition to the implementation of ICSA CCSA has beenalso run for the Subcase 35 for further investigation ofefficiency improvement of the proposed ICSA approach The
comparisons of minimum cost in Table 8 show that theproposed ICSA can find more optimal solution than M120573-HCLSA [49] AGWOA [52] and CCSA The proposed ICSAis also superior to CCSA in terms ofmore stable search abilityand lower fluctuation since its average cost and maximumcost are less than those of CCSA The outstanding figurecannot be reached as compared to AGWOA [52] howeverit is hard to conclude AGWOA [52] is superior to theproposed ICSA approach about more stable search abilityand lower fluctuation Actually comparison of the values ofpopulation and iterations as well as execution time cannot beaccomplished because the information was not reported in[52]Thus it can conclude that the proposed ICSA is effectivefor the subcase
Optimal solutions obtained by ICSA for the case areshown in Tables 20ndash23
54 Obtained Results on Case 4 with Two Systems consideringSFS and PPZ and RPR Constraints In this section two
16 Mathematical Problems in Engineering
Table 9 Result comparisons for Subcases 41 and 42
Subcase Approach Best cost ($h) Mean cost ($h) Worst cost ($h) Std dev ($h) CPU time (s) Computer (Processor-Ram)
test systems with SFS and PPZ and RPR constraints areconsidered The test system size is up to 60 and 90 units forSubcases 41 and 42 respectively Comparison for the casesis only performed with two Genetic algorithms consisting ofGA and IGA-MU in [9] and two other versions of Cuckoosearch algorithms including CCSA [23] and OSE-CSA [59]and presented in Table 9 Other studies have tended toignore such complicated constraints of PPZ and RPR Thecosts reported in Table 9 indicate that ICSA approach canobtain more effective solution than CCSA and OSE-CSAbecause it has reached lower minimum cost than the twoones Furthermore the proposed method also takes shortercomputation time for the two cases from about two timesto about three times although the processor of the proposedmethod is slightly strongerThemean costs of ICSA approachare much less than those from IGA-MU and GA and slightlyhigher than those from OSE-CSA but there is a trade-off between the proposed method and CCSA for the twosubcases In fact the proposed method obtains higher meancost for Subcase 41 but lower cost for Subcase 42 AlthoughGAmethods have been runon aweak computerwith 07 GHzof the processor compared to that with 24GHz in the studytheir execution times are significantly higher namely 56381seconds (GA) and 16258 seconds (IGA-MU) compared to09153 seconds of ICSA approach for Subcase 41 and 94093seconds (GA) and 25545 seconds (IGA-MU) compared to15892 seconds (the proposedmethod)The analysis can pointout that ICSA approach ismore efficient than these comparedmethods in terms of optimal solutions and execution time
Optimal solutions obtained by ICSA for Subcase 42 areshown in Table 24
55 Obtained Results on Case 5 with a 15-Unit System consid-ering SFS and RRL PPZ and PL Constraints In this section
a 15-unit system considering RRL PPZ and PL constraintsis considered to be solved for finding optimal solution Forefficiency investigation of the proposed ICSA we have alsoimplemented CCSA for comparison As listed in Table 10the proposed ICSA is the most effective method with thesmallest fuel cost The comparisons of control parameters aswell asCPU time are also good evidence to confirm the strongsearch of the proposed ICSA approach since it has been runby smaller values of control parameter and faster executiontime as compared to all methods excluding CCSA
Optimal solution obtained by ICSA for the case is shownin Table 25
56 Obtained Results on Case 6 withThree Systems consideringMFS and EoVLP In this section three test systems with thechallenge on objective function including multi-fossil fuelsources and effects of valve loading process are consideredThe scale is up to 80 units 160 units and 320 units forSubcases 61 62 and 63 respectively
Comparison for Subcase 61 reported in Table 11 revealsthat the proposed method is the best method in terms of thelowest best cost the lowestmean cost and the lowest standarddeviation and the fastest execution time The processor fromthis proposedmethod is about four times stronger thanCGA-MU and IGA-MU but the speed is from ten times to 35 timesfaster than these methods Compared to CCSA and OSE-CSA the proposed method is about two times faster but theprocessor is slightly stronger
Comparison for Subcase 62 is reported in Table 12Clearly the proposed ICSA approach obtains better values ofthe best mean and worst costs than most methods exceptMSOS [34] where the best cost difference is about $ 024However the proposed method is the second fastest onewith 1119 seconds where the first fastest one CBA [44] has
Mathematical Problems in Engineering 17
Table 11 Comparisons of found results for Subcase 61
Approach Best cost Mean cost Worst cost Std dev CPU time Computer($h) ($h) ($h) ($h) (s) (Processor-Ram)
spent 571 seconds Clearly MSOS is better than the proposedICSA approach with respect to slightly less best cost butis worse than the proposed method in terms of executiontime namely 2572 seconds compared to 1119 seconds whileCBA [44] is faster than the proposed method but obtainssignificantly worse costs The analysis can conclude that theproposed ICSA approach is very powerful for the subcasewith 160 units
Table 13 presents the comparison of three other methodsincludingCCSA [54] SOS [34] andMSOS [34] accompaniedwith the proposed method for Subcase 63 The obtainedresult comparisons imply that ICSA approach can obtainbetter values of the best and standard deviation costs thanCCSA and SOS but obtains slightly higher cost than MSOSby approximately $ 022 Besides the execution time fromICSA approach is much shorter than others especially it ishigher than five times faster than MSOS The four methodshave been run on approximately strong computers Brieflythe proposed ICSA approach can find and converge to morefavorable solution than other methods with shorter CPUtime except the comparison with MSOS which had bettersolution but spent higher than five times execution times
Consequently the proposed method is a very promisingoptimization algorithm for Subcase 63 a system up to 320units and with multi-fossil fuel sources and effects of valveloading process
Optimal solution obtained by ICSA for Subcase 63 isshown in Table 26
57 The Improvement of ICSA Approach Performance
571 The Outstanding Improvement over CCSA In this sec-tion the performance improvement of ICSA over CCSAhas been investigated by analyzing obtained results and setcontrol parameters Table 14 has been formed by addingreduction cost improvement level of the best cost executiontime and control parameters consisting of 119873119901119904 and 119866119898119886119909Among the compared factors reduction cost is the deviationof the cost of CCSA and that of ICSA whereas the improve-ment level is the ratio of the reduction cost to the cost ofCCSA The reduction costs indicate the proposed methodcould find either equal quality of solutions or higher qualityof solutions than CCSA for all study casesThe reduction costis from $0 to $42839 corresponding to the improvement level
18 Mathematical Problems in Engineering
Table 14 Summary of results obtained by CCSA and ICSA for all study cases
Study case Reduction cost ($) Improvement level () Execution time (s) Nps Gmax
Figure 4 The best run obtained by CCSA and ICSA for Case 2
that is from 0 to 08968The saving cost is not too much forone hour but the operation in one day onemonth or one yearis very high However it should be noted that CCSA has beenrun by setting much higher population size and iterations formany cases excluding study cases implemented in the studysuch as Case 2 Subcase 31 Subcase 35 and Case 5 Forinstance ICSA has used 119873119901119904 = 10 and 119866119898119886119909 = 15 for Subcase13 and119873119901119904 = 10 and119866119898119886119909 = 25 for Subcase 14 whereas CCSAhas been run by setting119873119901119904 = 20 and 119866119898119886119909 = 5000 for the twosubcases Similarly CCSA has been run for Subcases 32 33and 34 with much higher number of iterations For the lastsubcase CCSA has been run by setting 119873119901119904 = 320 and 119866119898119886119909
= 1200 but those of ICSA have been 10 and 9000 Due tothe higher value of control parameters CCSA has tended tospend more time in finding such high quality solutions foralmost all study cases Execution time of ICSA is less than18 seconds while that of CCSA is up to higher 75 secondsIt is clear that the proposed ICSA could find better optimalsolutions thanCCSA for such considered systems For furtherinvestigation of performance comparison the best runs over50 runs and fuel cost values of 50 runs obtained by CCSA andthe proposed ICSA for Case 2 Subcase 31 Subcase 35 andCase 5 have been plotted in from Figures 4ndash11 The best runcurves show the faster search of the proposed ICSA method
Mathematical Problems in Engineering 19
0 5 10 15 20 25 30 35 40 45 50Run
19795
198
19805
1981
19815
1982
19825
1983
19835
Fuel
cost
($)
CCSAICSA
times105
Figure 5The best fuel cost of 50 runs obtained by CCSA and ICSA for Case 2
0 2 4 6 8 10 12 14 16 18 20Iteration
8234
8236
8238
8240
8242
8244
8246
8248
8250
8252
Fitn
ess F
unct
ion
($)
CCSAICSA
Figure 6 The best run obtained by CCSA and ICSA for Subcase 31
whereas 50 values of fuel cost indicate that the proposedICSA can find many solutions with better quality Clearlythe proposed ICSA is outstanding in terms of stabilizationof solution search and faster convergence As a result it canconclude that the proposed ICSA approach is more effectivethanCCSA in solvingOLDproblemwith considered systems
572The Improvement of Results over Other Methods In thisarticle we have tested ICSA approach on 6 cases with 16
systems with different fuel cost forms different constraintsand different scale systems from 3 units to 320 units We havecompared the yielded results from ICSA approach and otherexisting ones for evaluating the efficiency of ICSA approachIn subsections above we have shown yielded results fromICSA approach and compared these results to those of otheronesHowever the demonstrationhad not been very good forobserving and comparing to lead to a conclusionThus in thesubsectionwehave summarized the result comparisons of the
20 Mathematical Problems in Engineering
0 5 10 15 20 25 30 35 40 45 50Run
8234
82342
82344
82346
82348
8235
82352
82354
82356
82358
8236
Fuel
cost
($)
CCSAICSA
Figure 7 The best fuel cost of 50 runs obtained by CCSA and ICSA for Subcase 31
0 1000 2000 3000 4000 5000 6000Iteration
242
244
246
248
25
252
254
256
Fitn
ess F
unct
ion
($)
CCSAICSA
times105
Figure 8 The best run obtained by CCSA and ICSA for Subcase 35
proposed and other ones Table 15 has reported the reductioncost (in $) of ICSA approach compared to other ones Inaddition we have converted the reduction cost into improve-ment level (in ) for better comparison The improvementhas been shown from the lowest level to the highest levelin terms of reduction cost and improvement percentageIn addition we have also given the slowest and the fastestexecution time of other compared methods together withthat of the proposed method The table implies that ICSA
approach can find better optimal solutions with less fuel costup to $052 for Subcase 13 $75229 for Subcase 14 $1775149for Case 2 $1181256 for Subcase 31 $8438 for Subcase 32$9113 for Subcase 33 $121181 for Subcase 34 $3412 forSubcase 35 $04329 for Subcase 41 $30227 for Subcase42 $407002 for Case 5 $18183 for Subcase 61 $13965 forSubcase 62 and $11817 for Subcase 63 These reductioncosts are equivalent to improvement level (IL) of 0032908968 82282 14143 047 038 099 001405
Mathematical Problems in Engineering 21
0 5 10 15 20 25 30 35 40 45 50Run
2428
243
2432
2434
2436
2438
244
2442
Fuel
cost
($)
CCSAICSA
times105
Figure 9 The best fuel cost of 50 runs obtained by CCSA and ICSA for Subcase 35
0 50 100 150 200 250 300 350 400Iteration
327
328
329
33
331
332
333
334
335
336
337
Fitn
ess F
unct
ion
($)
CCSAICSA
times104
Figure 10 The best run obtained by CCSA and ICSA for Case 5
00003 0002 12291 004 138 and 059 Thesequantitative comparisons reveal that larger scale systems canlead to better reduction cost but the improvement level isnot high because total cost of compared methods tends tobe large for large scale systems Furthermore very large scalesystems with nondifferentiable objective have been normallysolved by strong methods In fact systems in Case 3 havethe same characteristic with single fuel and effects of valveloading process but Subcase 34 is a larger scale system with
40 units while Subcases 31 32 and 33 are constructed by 3units 13 units and 13 units So the improvement percentageof Subcase 34 can be up to 099 whilst that of Subcases32 and 33 is 047 and 038 respectively Subcase 35 iswith the largest system 80 units but the reduction cost is notmuch only $3412 because compared methods with ICSA areeither state-of-the-art ones or improved ones Also Subcase41 and Subcase 42 have considered single fuel and PPZ andspinning reserve constraints but Subcase 42 is larger scale
22 Mathematical Problems in Engineering
Table 15 Performance improvement summary of the proposed method
Study cases Reduction cost ($) Improvement level () Execution time (s)From To From To Slowest method Fastest method Proposed method
Figure 11 The best fuel cost of 50 runs obtained by CCSA and ICSA for Case 5
system with 90 units and Subcase 41 is only with 60 unitsThus the improvement of Subcase 41 is lower with 00003but that of Subcase 42 is 0002 Similarly systems in Cases 6have the same featurewithmulti-fossil fuel sources and effectsof valve loading process but they are respectively constructedby 80 160 and 320 units As a result the improvements ofSubcase 62 138 and Subcase 63 059 are much higherthan Subcase 61 004 However Subcase 62 with smallernumber of units but getting higher improvement is easilyunderstood because there were nine compared methods butonly three compared methods are considered for Subcase 63In general the improvement is not high it is about under onedollar several dollars tens of dollars and over one thousand
dollars per hour however the saving cost will be significantif the operation is considered to be onemonth with 720 hoursor one year with 8760 hours
Execution time comparison can be evaluated by observ-ing the fastest and the slowest compared methods in Table 15These execution times of the proposed method are approxi-mately equal to that of the fastest methods for Subcases 3233 and 34 and much shorter than other fastest comparedmethods for other cases especially for Subcases 61 and 63The fastest method for Subcase 62 is CBA [44] showing 57seconds while that of the proposed method is 1119 secondsHowever it cannot conclude that CBA is more effective thanthe proposed method because the proposed method could
find better optimal solution with less fuel cost by $2130As considering execution time of the slowest comparedmethods it can point out that ICSA is a very fast optimizationtool since the execution time of these methods is 29496 and3425 seconds for Subcases 32 and 33 116735 seconds forSubcase 34 56381 seconds for Subcase 41 94093 secondsfor Subcase 42 12797 seconds for Case 5 30941 seconds forSubcase 61 7542 seconds for Subcase 62 and 9641 secondsfor Subcase 63 while the execution time of the proposedmethod for these cases is respectively 095 146 091 1589246 84828 1119 and 171384 seconds It is clearly shown thatICSA is very fast as compared to these methods
In summary the proposed method has found approx-imately high quality solutions with several standard state-of-the-art meta-heuristic algorithms and improved versionsof them together with other old methods In addition theproposed method could improve result better than approx-imately all methods with faster execution time Comparedto other methods with the fastest convergence speed andhigh quality solutions the proposed method has been as
fast as for some cases and much faster for other cases Thecomparison with the slowest methods could show that theproposed method was extremely powerful since it was up tonearly one thousand times faster Consequently the proposedICSA approach can be one of the strongest optimization toolsfor OLD problem
6 Conclusions
This paper has proposed a good ICSA method for solvingOLD problem in which many test systems with differentobjective functions and complicated constraints from simpleto complex have been used as studied cases The proposedICSA method has been developed by performing severalmodifications on the second solution update of CCSA whichcontained several drawbacks to global convergence and fastmanner The OLD problem has covered from single fuelto multi-fossil fuels from quadratic objective function tononconvex objective function in addition to PPZ RPR andRRL constraints Many existing optimization algorithms have
been concerned in aim to compare the performance and givethe final conclusion on the proposed method There have
been six main cases with sixteen subcases The evaluationshave been made at the end of each study case Clearly theproposed ICSA approach has yielded more effective optimalsolutions with faster execution time than almost all methodsConsequently it can be concluded that the proposed methodis much more superior to CCSA and is a very promisingmethod for solving OLD problem
Appendix
See Tables 16ndash26
Nomenclature
120575119894 120582119894 120572119894 120573119894 120574119894 Fuel cost functioncoefficients of the 119894119905ℎthermal generationunit11986100 1198610119895 119861119895119894 Power loss matrixcoefficients119865119865119909 119865119865119887119890119904119905 The values of fitness ofsolution 119909 and theso-far most effectivesolution among thecurrent set ofsolutions119898119894 Number of fuels burntin the 119894119905ℎ thermalgeneration unit
119873 Number of allavailable thermalgeneration units119899119894 Number of prohibitedpower zones of the 119894119905ℎthermal generationunit119875119906119894119896 119875119897119894119896 Upper and lowerlimits of the 119894119905ℎthermal generationunit corresponding tothe kth PPZ119875119886 Probability ofreplacing controlvariables in each oldsolution119875119863 Real power demand ofall loads in system119875119894119898119886119909 119875119894119898119894119899 The highest andlowest real poweroutputs of the iththermal generationunit119875119894119895119898119886119909 119875119894119895119898119894119899 The highest andlowest real poweroutputs of the 119894119905ℎthermal generationunit corresponding tothe 119895119905ℎ fuel type119878119894119898119886119909 Maximum real powerreserve contributionof the thermalgeneration unit 1198941198781199001198971199031198861198991198891 1198781199001198971199031198861198991198892 1198781199001198971199031198861198991198893 1198781199001198971199031198861198991198894 Randomly mixedsolutions from the setof current solutions119878119900119897119909 119878119900119897119866119887119890119904119905 The old solution x andthe most effectivesolution119878119877 Real power reserverequirement of system
120575119894119895 120582119894119895 120572119894119895 120573119894119895 120574119894119895 Fuel cost functioncoefficients of the 119894119905ℎthermal generationunit corresponding tothe 119895119905ℎ fuel type1205761 1205762 1205763 1205764 1205765 1205766 Random numbersbetween 0 and 1Ω Number of generationunits considering PPZconstraint
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare no conflicts of interest
References
[1] N Sinha R Chakrabarti and P K Chattopadhyay ldquoEvolution-ary programming techniques for economic load dispatchrdquo IEEETransactions on Evolutionary Computation vol 7 no 1 pp 83ndash94 2003
[2] P Hansen ldquoA separable approximation dynamic programmingalgorithm for economic dispatch with transmission lossesrdquoYugoslav Journal of Operations Research vol 12 no 2 2002
[3] N T Thang ldquoSolving economic dispatch problem with piece-wise quadratic cost functions using lagrange multiplier theoryrdquoin Proceedings of the 3rd International Conference on ComputerTechnology and Development (ICCTD rsquo11) pp 359ndash364 ASMEPress 2011
[4] S K Mishra and S K Mishra ldquoA comparative study of solutionof economic load dispatch problem in power systems in theenvironmental perspectiverdquoProcedia Computer Science vol 48pp 96ndash100 2015
[5] A A Al-Subhi and H K Alfares ldquoEconomic load dispatchusing linear programming a comparative studyrdquo InternationalJournal of Applied Industrial Engineering vol 3 no 1 pp 16ndash362016
28 Mathematical Problems in Engineering
[6] C Zhou G Huang and J Chen ldquoPlanning of electric powersystems considering virtual power plants with dispatchableloads included an inexact two-stage stochastic linear program-ming modelrdquoMathematical Problems in Engineering vol 2018Article ID 7049329 12 pages 2018
[7] T T Nguyen N V Quynh and L Van Dai ldquoImproved fireflyalgorithm a novel method for optimal operation of thermalgenerating unitsrdquo Complexity vol 2018 Article ID 7267593 23pages 2018
[8] S S Haroon S Hassan S Amin et al ldquoMultiple fuel machinespower economic dispatch using stud differential evolutionrdquoEnergies vol 11 no 6 pp 1ndash20 2018
[9] C-T Su and C-L Chiang ldquoNonconvex power economic dis-patch by improved genetic algorithm with multiplier updatingmethodrdquo Electric Power Components and Systems vol 32 no 3pp 257ndash273 2004
[10] NNoman andH Iba ldquoDifferential evolution for economic loaddispatch problemsrdquo Electric Power Systems Research vol 78 no8 pp 1322ndash1331 2008
[11] P Somasundaram and K Kuppusamy ldquoApplication of evolu-tionary programming to security constrained economic dis-patchrdquo International Journal of Electrical Power amp EnergySystems vol 27 no 5-6 pp 343ndash351 2005
[12] T Jayabarathi K Jayaprakash andD Jeyakumar ldquoEvolutionaryprogramming technique for different kinds of economic dis-patch problemsrdquo Electric Power Systems Research vol 73 no 2pp 169ndash176 2005
[13] P Subbaraj R Rengaraj and S Salivahanan ldquoEnhancementof combined heat and power economic dispatch using selfadaptive real-coded genetic algorithmrdquo Applied Energy vol 86no 6 pp 915ndash921 2009
[14] C-L Chiang and C-T Su ldquoAdaptive-improved genetic algo-rithm for the economic dispatch of units with multiple fueloptionsrdquo Cybernetics and Systems vol 36 no 7 pp 687ndash7042005
[15] C Chiang ldquoImproved genetic algorithm for power economicdispatch of units with valve-point effects and multiple fuelsrdquoIEEE Transactions on Power Systems vol 20 no 4 pp 1690ndash1699 2005
[16] K Meng H G Wang Z Y Dong and K P Wong ldquoQuantum-inspired particle swarm optimization for valve-point economicload dispatchrdquo IEEE Transactions on Power Systems vol 25 no1 pp 215ndash222 2010
[17] P Erdogmus A Ozturk and S Duman ldquoEnvironmen-taleconomic dispatch using genetic algorithm and simulatedannealingrdquo in Proceedings of the International Conference onElectric Power and Energy Conversion Systems EPECS rsquo09 pp1ndash4 IEEE 2009
[18] L D S Coelho and V C Mariani ldquoImproved differentialevolution algorithms for handling economic dispatch opti-mization with generator constraintsrdquo Energy Conversion andManagement vol 48 no 5 pp 1631ndash1639 2007
[19] H Dakuo W Fuli and M Zhizhong ldquoA hybrid genetic algo-rithm approach based on differential evolution for economicdispatch with valve-point effectsrdquo Electrical Power and EnergySystems vol 30 pp 31ndash38 2008
[20] C-F Chang J-J Wong J-P Chiou and C-T Su ldquoRobustsearching hybrid differential evolution method for optimalreactive power planning in large-scale distribution systemsrdquoElectric Power Systems Research vol 77 no 5-6 pp 430ndash4372007
[21] M R Farooqi P Jain and K R Niazi ldquoUsing Hopfield neuralnetwork for economic dispatch of power systemsrdquo in Proceed-ings of the National Power Engineering Conference PECon rsquo03pp 5ndash10 Bangi Malaysia 2003
[22] T Yalcinoz H Altun and U Hasan ldquoConstrained economicdispatch with prohibited operating zones a Hopfield neuralnetwork approachrdquo in Proceedings of the 10th MediterraneanElectrotechnical Conference Information Technology and Elec-trotechnology for the Mediterranean Countries ProceedingsMeleCon rsquo00 (Cat No 00CH37099) pp 570ndash573 LemesosCyprus 2000
[23] D N Vo P Schegner and W Ongsakul ldquoCuckoo searchalgorithm for non-convex economic dispatchrdquo IET GenerationTransmission amp Distribution vol 7 no 6 pp 645ndash654 2013
[24] J-B Park K-S Lee J-R Shin and K Y Lee ldquoA particleswarm optimization for economic dispatch with non-smoothcost functionsrdquo IEEETransactions on Power Systems vol 20 no1 pp 34ndash42 2005
[25] T Niknam H D Mojarrad and M Nayeripour ldquoA newfuzzy adaptive particle swarm optimization for non-smootheconomic dispatchrdquo Energy vol 35 no 4 pp 1764ndash1778 2010
[26] L D S Coelho and V C Mariani ldquoAn improved harmonysearch algorithm for power economic load dispatchrdquo EnergyConversion and Management vol 50 no 10 pp 2522ndash25262009
[27] M Mahdavi M Fesanghary and E Damangir ldquoAn improvedharmony search algorithm for solving optimization problemsrdquoAppliedMathematics and Computation vol 188 no 2 pp 1567ndash1579 2007
[28] A Bhattacharya and P K Chattopadhyay ldquoSolving complexeconomic load dispatch problem using biogeography-basedoptimizationrdquo Expert Systems with Applications vol 37 no 5pp 3605ndash3615 2010
[29] D Simon ldquoBiogeography-based optimizationrdquo IEEE Transac-tions on Evolutionary Computation vol 12 no 6 pp 702ndash7132008
[30] J S Alsumait J K Sykulski and A K Al-Othman ldquoAhybrid GA-PS-SQP method to solve power system valve-pointeconomic dispatch problemsrdquo Applied Energy vol 87 no 5 pp1773ndash1781 2010
[31] M S Turgut andGKDemir ldquoQuadratic approximationndashbasedhybrid Artificial Cooperative Search algorithm for economicemission load dispatchproblemsrdquo International Transactions onElectrical Energy Systems vol 27 no 4 pp 1ndash14 2017
[32] T A A Victoire and A E Jeyakumar ldquoHybrid PSO-SQPfor economic dispatch with valve-point effectrdquo Electric PowerSystems Research vol 71 no 1 pp 51ndash59 2004
[33] D-KHe F-LWang andZ-ZMao ldquoHybrid genetic algorithmfor economic dispatch with valve-point effectrdquo Electric PowerSystems Research vol 78 no 4 pp 626ndash633 2008
[34] D C Secui ldquoA modified symbiotic organisms search algorithmfor large scale economic dispatch problem with valve-pointloading effectsrdquo Energy vol 113 pp 366ndash384 2016
[35] K BhattacharjeeA Bhattacharya and S H N Dey ldquoTeaching-learning-based optimization for different economic dispatchproblemsrdquo Scientia Iranica vol 21 no 3 pp 870ndash884 2014
[36] K Bhattacharjee A Bhattacharya and S H N Dey ldquoOppo-sitional Real Coded Chemical Reaction Optimization for dif-ferent economic dispatch problemsrdquo International Journal ofElectrical Power amp Energy Systems vol 55 pp 378ndash391 2014
Mathematical Problems in Engineering 29
[37] V Hosseinnezhad M Rafiee M Ahmadian and M T AmelildquoSpecies-basedQuantum Particle SwarmOptimization for eco-nomic load dispatchrdquo International Journal of Electrical Poweramp Energy Systems vol 63 pp 311ndash322 2014
[38] M S P Subathra S E Selvan T A A Victoire A HChristinal andU Amato ldquoA hybrid with cross-entropymethodand sequential quadratic programming to solve economic loaddispatch problemrdquo IEEE Systems Journal vol 9 no 3 pp 1031ndash1044 2015
[39] J Zhan Q H Wu C Guo and X Zhou ldquoEconomic dispatchwith non-smooth objectives part I local minimum analysisrdquoIEEE Transactions on Power Systems vol 30 no 2 pp 710ndash7212015
[40] A K Barisal andR C Prusty ldquoLarge scale economic dispatch ofpower systems using oppositional invasive weed optimizationrdquoApplied Soft Computing vol 29 pp 122ndash137 2015
[41] M Basu ldquoImproved differential evolution for economic dis-patchrdquo International Journal of Electrical Power amp EnergySystems vol 63 pp 855ndash861 2014
[42] V S Aragon S C Esquivel and C C Coello ldquoAn immunealgorithm with power redistribution for solving economicdispatch problemsrdquo Information Sciences vol 295 pp 609ndash6322015
[43] M Ghasemi M Taghizadeh S Ghavidel and A AbbasianldquoColonial competitive differential evolution an experimentalstudy for optimal economic load dispatchrdquo Applied Soft Com-puting vol 40 pp 342ndash363 2016
[44] B R Adarsh T Raghunathan T Jayabarathi and X-S YangldquoEconomic dispatch using chaotic bat algorithmrdquo Energy vol96 pp 666ndash675 2016
[45] N Ghorbani and E Babaei ldquoExchange market algorithm foreconomic load dispatchrdquo International Journal of ElectricalPower amp Energy Systems vol 75 pp 19ndash27 2016
[46] J X V Neto G Reynoso-Meza T H Ruppel V C Marianiand L D S Coelho ldquoSolving non-smooth economic dispatchby a new combination of continuous GRASP algorithm anddifferential evolutionrdquo International Journal of Electrical Poweramp Energy Systems vol 84 pp 13ndash24 2017
[47] A Kavousi-Fard and A Khosravi ldquoAn intelligent 120579-ModifiedBat Algorithm to solve the non-convex economic dispatchproblem considering practical constraintsrdquoElectrical Power andEnergy Systems vol 82 pp 189ndash196 2016
[48] M A Al-Betar M A Awadallah A T Khader and A L BolajildquoTournament-based harmony search algorithm for non-convexeconomic load dispatch problemrdquo Applied Soft Computing vol47 pp 449ndash459 2016
[49] M A Al-Betar M A Awadallah I A Doush E Alsukhni andH ALkhraisat ldquoA non-convex economic dispatchproblemwithvalve loading effect using a new modified 120573-hill climbing localsearch algorithmrdquo Arabian Journal for Science and Engineeringpp 1ndash8 2018
[50] M A Awadallah M A Al-Betar A L Bolaji E M Alsukhniand H Al-Zoubi ldquoNatural selection methods for artificial beecolony with new versions of onlooker beerdquo Soft Computing pp1ndash40 2018
[51] S H Kaboli and A K Alqallaf ldquoSolving non-convex economicload dispatch problem via artificial cooperative search algo-rithmrdquo Expert Systems with Applications vol 128 pp 14ndash272019
[52] D Singh and J Dhillon ldquoAmeliorated greywolf optimization foreconomic load dispatch problemrdquo Energy vol 169 pp 398ndash4192019
[53] X S Yang and S Deb ldquoCuckoo search via Levy flightsrdquo inProceedings of the World Congress on Nature and BiologicallyInspired Computing (NaBIC rsquo09) pp 210ndash214 India 2009
[54] S Sahoo K M Dash R C Prusty and A K Barisal ldquoCom-parative analysis of optimal load dispatch through evolutionaryalgorithmsrdquo Ain Shams Engineering Journal vol 6 pp 107ndash1202015
[55] M Basu and A Chowdhury ldquoCuckoo search algorithm foreconomic dispatchrdquo Energy vol 60 pp 99ndash108 2013
[56] A B Serapiao ldquoCuckoo search for solving economic dispatchload problemrdquo Intelligent Control and Automation vol 04 no04 pp 385ndash390 2013
[57] A H Bindu and M D Reddy ldquoEconomic load dispatch usingcuckoo search algorithmrdquo International Journal of EngineeringResearch and Applications vol 3 pp 498ndash502 2013
[58] E Afzalan and M Joorabian ldquoAn improved cuckoo searchalgorithm for power economic load dispatchrdquo InternationalTransactions on Electrical Energy Systems vol 25 no 6 pp 958ndash975 2015
[59] T T Nguyen and D N Vo ldquoThe application of one rank cuckoosearch algorithm for solving economic load dispatch problemsrdquoApplied Soft Computing vol 37 pp 763ndash773 2015
[60] S M Islam S Das S Ghosh S Roy and P N Suganthan ldquoAnadaptive differential evolution algorithm with novel mutationand crossover strategies for global numerical optimizationrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 42 no 2 pp 482ndash499 2012
[61] Z L Gaing ldquoParticle swarm optimization to solving theeconomic dispatch considering the generator constraintsrdquo IEEETransactions on Power Systems vol 18 no 3 pp 1187ndash1195 2003
[62] X-S Yang and S Deb ldquoEngineering optimisation by Cuckoosearchrdquo International Journal of Mathematical Modelling andNumerical Optimisation vol 330 pp 43ndash51 2010
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2 Mathematical Problems in Engineering
the OLD problem successfully by separating three-curveobjective function into three different single curve objectivefunctions In [7] amore realistic representation of the electricgeneration fuel function corresponding to multi fossil fuelsources was introduced in which the authors considered adiscontinuous cost function and the effects of valve loadingprocess (EoVLP) of thermal units More complicated modelswere also introduced For example some thermal generationunits were driven by burning multi fossil fuel sources (MFS)to generate electricity [8] or some operating conditions ofgenerators including upper generation boundary lower gen-eration boundary and prohibited power zones were added[9] However these methods have been only applied to thesystems where the generation power-fuel cost characteristicof thermal generation units was mathematically modelled asthe second order function and the effects of valve loadingprocess were ignored [10]
Other artificial intelligence-based advanced methodshave been recently applied for solving OLD problem Evolu-tionary programming-based approaches (EP) [1 11 12] andgenetic algorithm (GA) [13ndash15] were considered as fast algo-rithms because of their parallel search ability In addition GAand EP possessed good properties such as finding solutionsnearby global optimum capability of effectively handlingnonlinear constraints reliable search ability and not manyadjustment parameters [10] Thus it could become a suitablechoice for successfully solving OLD problem However GAwas prematurely convergent to local optimum solutions[16] In this regard simulated annealing approach (SA)[17] was a better probabilistic approach in finding solutionswith appropriate fitness but there was a high possibility ofeasily converging to local optimal zones when coping withcomplicatedly constrained problem It converges slower thanGA and EP though Differential evolution algorithm (DE)[10 18] also belongs to the same class as GA and EP Howeverit is more popular thanks to simple structure with severaladjustment parameters and high rate of success DE has beenmore widely and successfully applied than SA and GA [19]But its faster convergence manner led to the same drawbackas GA like high rate of falling into local optimum andhardly ever toward promising zones quickly In fact theseshortcomings could be tackled by setting population size tohigher value But high population size could suffer from longsimulation time to calculate fitness function and evaluatequality of solutions [20] Hopfield neural network (HNN)[21 22] focused on optimizing energy function and wasonly successfully applied for optimization problems whereobjective functions were differentiable HNN could be anappropriatemethod for large-scale systemswith highnumberof generation units but it needed long simulation time andmay also converge to local optimum solution zones [23]Particle swam optimization (PSO) [16 24] is a random searchapproach developed by behavior of a swarm or flock duringfood search process In comparison with GA PSO ownedmore advantages such as simpler implementation and fewparameters with easy selection However the success rateof PSO was highly influenced by adjustment parametersand it copes with high rate to be trapped in many zoneswith local optimum solutions [25] Harmony search (HS)
[26] is a metaheuristic-based method inspired from musicInstead of using gradient search HS has employed stochasticrandom search to exploit its potential ability Thus it tendedto converge to local optimal zones rather than global optimalzones [27] Biogeography-based optimization (BBO) [28]could compete with PSO and DE since its solutions weredirectly updated by migration from other existing solutionsand its solutions directly shared their attributes with othersolutions [29]
It is clear that each method has advantages as well asdisadvantages for different applications for finding OLDproblem solutions Hence another natural approach is tocombine different methods to exploit the advantages of eachmethod and enhance the overall searching capability Severalhybrid methods have been developed in such way includ-ing hybrid Genetic algorithm Pattern Search and Sequen-tial Quadratic Programming (GAndashPSndashSQP) method [30]hybrid Artificial Cooperative Search algorithm (HACSA)[31] hybrid PSO-SQP [32] and hybrid GA (HGA) [33]Basically these hybrid methods could deal with OLD prob-lem more effectively than each member method On theother hand they could suffer from the difficulty of selectingmany controllable parameters In addition to such popularoriginal algorithms and hybrid methods there are manyother original and improved methods that have been appliedfor solving the considered OLD problem These methods areSymbiotic organisms search algorithm (SOS) [34] and itsmodified version (MSOS) [34] teaching activity and learningactivity-based optimization (TLBO) [35] chemical reaction-based approach (CRBA) [36] enhanced particle swarmoptimization (EPSO) [37] sequential quadratic technique-based cross entropy approach (CEA-SQT) [38] traversesearch-based optimization approach (TSBO) [39] invasiveweed approach (IWA) [40] Improved Differential evolu-tion (IDE) [41] immune algorithm using power redistribu-tion IAPR [42] Colonial competitive differential evolution(CCDE) [43] Chaotic Bat algorithm (CBA) [44] Exchangemarket algorithm (EMA) [45] adaptive search techniquealgorithm and differential evolution (GRASP-DE) [46] 120579-Modified Bat Algorithm (120579-MBA) [47] Tournament-basedharmony search algorithm (TBHSA) [48] New Modified 120573-Hill Climbing Local Search Algorithm (M120573-HCLSA) [49]improved version of artificial bee colony algorithm (IABCA)[50] artificial cooperative search algorithm (ACSA) [51]and ameliorated greywolf optimization algorithm (AGWOA)[52] Among these methods ACSA and AGWOA werethe two latest methods which were applied for OLD andpublished in early 2019 However the demonstration of realperformance of the two methods is still questionable In factACSA has been tested only on systemswith small scale singlefuel and simple constraints such as generation limits andpower balance The largest scale system was considered in[51] to be 40-unit system Unlike [51] different types of fuelcost function complicated constraints and large scale systemwith 140 units have been taken into account in [52] Viacomparisons withmany existing methods AGWOAhas beenstated to be the best one with many surprising results Thusthe validation of reported solutions from the methodmust beverified and its strong search ability must be reevaluated In
Mathematical Problems in Engineering 3
the numerical results section we will report the verificationof the two questionable issues
In this paper we have proposed an improved cuckoosearch algorithm (ICSA) for dealing with large scale OLDproblem with the consideration of complicated constraintstogether with nondifferentiable fuel cost function In theproposed ICSA approach some newmodifications have beenperformed on conventional cuckoo search algorithm (CCSA)to improve the quality of CCSA The CCSA method wasfirst developed in 2009 [53] for solving a set of popularbenchmark functions and its highly superior performanceover PSO and GA has attracted a huge number of researchersin learning and applying for different optimization problemsin different fields Furthermore its improved variants are alsoan extremely vast number In relation toOLDproblemCCSAhas been applied and presented in [23 54ndash57] meanwhileits improved methods consisting of modified cuckoo searchalgorithm (MCSA) and improved cuckoo search algorithmwith one solution evaluation (OSE-CSA) have been respec-tively presented in [58 59] In [55] Basu has applied CCSAfor solving OLD problem with 40-unit system with singlefuel option and the effective of valve loading process 20-unitsystem with single fuel and quadratic fuel cost function and10-unit system with multiple fuel sources and without theeffects of valve loading process The author has made a bigeffort in demonstrating the high potential search of CCSAby comparing with many popular metaheuristic algorithmsbut the shortcoming of the study was neglecting complicatedconstraints and large scale systems
Studies in [56 57] have dealt with OLD problem withtwo power systems considering simple constraints and smallnumber of units Only three simplest constraints such aspower balance limitations of generation and prohibitedpower zones have been taken into account meanwhilethe largest system was solved to be 6-unit system Thusthere were few methods compared to CCSA and the realperformance of CCSA was not shown persuasively in thestudies In [23] authors have applied CCSA for solving differ-ent systems with very complicated constraints complicatedcharacteristics of thermal generating units and high numberof units Among the mentioned studies regarding CCSA forOLD problem authors in [23] could show the best view inevaluating the real performance of CCSA since there were sixcases that were carried out and a huge number of methodswere compared to CCSA In spite of the real potential searchability CCSA has been commented to be low convergence toglobal optimum and significantly improved better [58 59]MCSA in [58] has been proposed by using a new strategyfor the second generation technique Themutation operationin CCSA has been replaced with current-to-best1 model ofDE in [60] MCSA has been applied for solving four systemswith 3 6 15 and 40 units in which the most complicatedconstraint considered was prohibited power zone and onlysingle fuel source was taken into account MCSA methodhas been compared to other popular methods such as PSOGA and EP But the comparisons with CCSA have notbeen carried out Thus the improvement of such proposed
method in [58] was not proved persuasively OSE-CSA in[59] has canceled one evaluation time in case that OSE-CSAhas continued to improve solution quality The improvementseemed to be appropriate for CCSA in dealing with OLDproblem with complicated systems CCSA in [23 55] havebeen considered for comparison in [59] and they have beenproved to be less effective than OSE-CSA However OSE-CSA has used one more control parameter called one rankparameter and it needed to be tuned thoroughly for obtaininghigh performance In the proposed ICSA approach we havefocused on a new strategy of the second new solutiongeneration in CCSA method As shown in [58 59] CCSAhas become a strong search method thanks to the first newsolution generation which was performed by Levy flighttechnique while the second new solution generation couldnot take on local search function well In the second updateprogress via mutation operator two random old solutions areused to generate an increased step size However the mannercan lead to new low quality solutions because the increasedstep will be very small when iterative algorithm is carryingout at the last iterations In fact current solutions at finalseveral iterations tend to be close together and the differentvalues between each two ones are very small leading to a verysmall increased step In order to tackle the disadvantage of theCCSA we apply a new adaptive technique for improvementof solution quality Firstly we propose twoways for producingthe increased step including two-solution-based increasedstep and four-solution-based increased step The decisionwhen which step size will be used is dependent on theresult of comparison between fitness function ratio (FFR)and a predetermined parameter 119879119900119897 FFR is defined as aratio of deviation between fitness functions of the consideredsolution and the most promising solution to the fitness valueof the best one meanwhile 119879119900119897 is a boundary to give the finaldecision for the selection of a used step size At the beginning119879119900119897 is a fixed value for all solutions and then it will be adaptivebased on the comparison between it and FFR When FFR of asolution is less than 119879119900119897 119879119900119897 of the solution will be decreasedequally to ninety percent of the previous valueOtherwise thevalue of 119879119900119897 remains unchanged in case the FFR is equal toor higher than 119879119900119897 The adaptive technique has a significantlyimportant role in enhancing the potential search ability of theproposed method This proposed method is investigated onsix cases with different considered constraints different typesof fuel cost function and large scale systemsThe detail of thesix cases is as follows
Case 1 Four systems with single fuel source (SFS) and powerloss (PL) constraint
Case 2 A 110-unit system with SFS
Case 3 Four systems with SFS and the effects of valve loadingprocess (EoVLP)
Case 4 Two systems with SFS and PPZ and RPR constraints
Case 5 A 15-unit system with SFS and RRL PPZ and PLconstraints
4 Mathematical Problems in Engineering
Case 6 Three systems with multiple fuel sources (MFS) andEoVLP
The achieved results in terms of minimum fuel costaverage fuel cost maximum fuel cost and standard deviationfound by the proposed method compared to those obtainedby others reveal that the method is very efficient for theOLD problem In addition the performance improvementof the proposed method over CCSA is also investigatedvia the comparison of the best solution and all trial runsIn summary the main advantages of the proposed ICSAapproach over CCSA as well as the main contribution of thestudy are as follows
(i) Based on fitness function of each considered solutionlocal search or global search is decided to be appliedmore effectively
(ii) Find better solutions with smaller number of itera-tions and shorter execution time for each run
(iii) Shorten simulation time for the whole search of eachstudy case
However the proposed method also copes with the sameshortcomings as CSA Although the shortcomings do notcause bad results for the proposed method they make theproposed method be time consuming in tuning optimalparameter The shortcomings are analyzed as follows
(i) Control parameter probability of replacing controlvariables in each old solution must be tuned in rangebetween 0 and 1 There is no proper theory for deter-mining the most effective values of the parameterThus the performance of the proposed method mustbe tried by setting the parameter to values from 01 to1
(ii) The method uses more computation steps for searchprocessThus the proposedmethod uses higher num-ber of computation steps for each iteration Howeverdue to more effective search ability for each iterationthe proposed method can use smaller number ofiterations but it finds more effective solutions
The remaining parts of the paper are arranged as fol-lows Section 2 shows the objective and constraints of theconsidered OLD problem CCSA and the proposed methodare clearly explained in Section 3 Section 4 is in charge ofpresenting the implementation of ICSA method for the stud-ied problem The simulation results together with analysisand discussions are given in Section 5 Finally conclusion issummarized in Section 6 In addition appendix is also addedfor showing found solutions by the proposed ICSA approachfor test cases
2 Optimal Load DispatchProblem Description
21 Fuel Cost Function Forms with Single Fuel Source In theconsidered OLD problem the optimal operation of a set of
thermal generation units is concerned as the duty of reducingtotal cost of all the units which can be seen by the followingmodel
Reduce 119865 = 119873sum119894=1
119865119894 (119875119894) (1)
In traditional OLD problem fuel cost function of the119894119905ℎ generation unit 119865119894(119875119894) is represented as the second orderfunction with respect to real power output and coefficients asthe model below [2]
In addition for the case considering the effects ofvalve loading process on thermal generation units fuel costbecomes more complicated by adding sinusoidal term asbelow [12]
119865119894 (119875119894) = 1205721198941198752119894 + 120582119894119875119894 + 120575119894+ 1003816100381610038161003816120573119894 times sin (120574119894 times (119875119894min minus 119875119894))1003816100381610038161003816 (3)
Real Power Balance Constraint Total real power demand ofall loads in power system together with real power loss in allconductors must be equal to the generation from all availablethermal generation units The requirement is constrained bythe following equality
119873sum119894=1
119875119894 = 119875119863 + 119875119871 (4)
where total real power loss 119875119871 is determined by Kronrsquosequation below
119875119871 = 11986100 + 119873sum119895=1
1198610119895119875119895 + 119873sum119895=1
119873sum119894=1
119875119895119861119895119894119875119894 (5)
GenerationBoundaryConstraint For the purpose of economyand safe operation each thermal generation unit is con-strained by the lower generation bound and upper generationbound as the following model
119875119894min le 119875119894 le 119875119894max (6)
22 Fuel Cost Function Forms with Multi-Fuel Sources Inthis section fuel cost function of thermal generation units ismathematicallymodeled in terms of different forms from thatin the section above due to the consideration of multi-fuelsources Each type of fuel source is formed as each secondorder function and the fuel cost function form is the sumof different second order functions for the case of neglectingthe effects of valve loading progress But for the considerationcase of the effects the form ismore complexwith the presenceof sinusoidal terms [15] As a result the forms of cost functioncan be expressed in Equation (7) [21] and Equation (8) [15]
Mathematical Problems in Engineering 5
119865119894 (119875119894) =
1205751198941 + 1205821198941119875119894 + 12057211989411198752119894 fuel 1 119875119894min le 119875119894 le 11987511989411198981198861199091205751198942 + 1205821198942119875119894 + 12057211989421198752119894 fuel 2 1198751198942min le 119875119894 le 1198751198942max120575119894119895 + 120582119894119895119875119894 + 1205721198941198951198752119894 fuel 119895 119875119894119895min le 119875119894 le 119875119894119895max
(7)
119865119894 (119875119894) =
1205751198941 + 1205821198941119875119894 + 12057211989411198752119894 + 10038161003816100381610038161205731198941 times sin (1205741198941 times (119875119894min minus 119875119894))1003816100381610038161003816 for fuel 1 119875119894min le 119875119894 le 11987511989411198981198861199091205751198942 + 1205821198942119875119894 + 12057211989421198752119894 + 10038161003816100381610038161205731198942 times sin (1205741198942 times (119875119894min minus 119875119894))1003816100381610038161003816 for fuel 2 1198751198942min le 119875119894 le 1198751198942119898119886119909 119895 = 1 119898119894120575119894119895 + 120582119894119895119875119894 + 1205721198941198951198752119894 + 10038161003816100381610038161003816120573119894119895 times sin (120574119894119895 times (119875119894min minus 119875119894))10038161003816100381610038161003816 for fuel 119895 119875119894119895min le 119875119894 le 119875119894119895max
(8)
Cost function forms in Equations (7) and (8) are onlyincluded in objective function (1) meanwhile main con-straints in formulas (4) and (6) must be always satisfied
23 Prohibited Power Zone Real Power Reserve and RampRate Limit Constraints Prohibited power zones (PPZ) aredifferent ranges of power in fuel cost function that thermalgeneration units are not allowed to work due to operationprocess of steam or gas valves in their shaft bearing Thepower generation of units in the violated zones is harmfulto gas or steam turbines even destroyed shaft bearing Thusthe constraint is strictly observed In the fuel-power charac-teristic curve of generation units PPZ causes small violationzones and such curves become discontinuous As consideringPPZ constraint the determination of power generation ofunits is more complex and equal to either lower bound orupper bound Unlike PPZ constraint RPR constraint is notrelated to fuel-power feature curve but it causes difficulty foroptimization approaches in satisfying one more inequalityconstraint Each generation unit among the set of availablegeneration units must reserve real power so that the sumof real power from all generation units can be higher orequal to the requirement of power system for the purposeof stabilizing power system in case that there are some unitsstopping producing electricity On the contrary to PPZ con-straint ramp rate limit (RRL) constraint does not allowpoweroutput of thermal generating units outside a predeterminedrange The constraint considers maximum power change ofeach thermal generating unit as compared to the previouspower value Thus optimal generation must satisfy the RRLconstraint The PPZ constraint RPR constraint and RRLconstraint can be presented as follows
Prohibited Power Zones As considering PPZ constraint validworking zones of each thermal generating unit are notcontinuous and its generation must be outside the violatedzones as the following mathematical description
119875119894 isin
119875119894min le 119875119894 le 1198751198971198941119875119906119894119896minus1 le 119875119894 le 119875119897119894119896 k = 2 ni forall119894 isin Ω119875119906119894119899119894 le 119875119894 le 119875119894max
(9)
As observing Equation (9) generation units cannotbe operated within the violated zones except for startingpoint and end point Consequently the verification of PPZconstraint violation should be carried out first and thenthe correction should be done before dealing with otherconstraints such as real power reserve constraint and realpower balance Besides if power output of all units can satisfythe PPZ constraint generation limits in Equation (6) are alsoexactly met
Real Power Reserve Constraint Real power reserve in powersystem aims to enhance the ability of stability recovery ofpower system and avoid blackout In order to get high enoughpower for requirement all available units are constrained bythe following inequality
119873sum119894=1
119878119894 ge 119878119877 (10)
where 119878119894 is the real power reserve contribution of the 119894119905ℎthermal generation unit and the determination of 119878119894 can bedone by employing the two models below
119878119894 = 119875119894max minus 119875119894 119894119891 119878119894max gt (119875119894max minus 119875119894)119878119894max else
forall119894 notin Ω (11)
119878119894 = 0 forall119894 isin Ω (12)
Equation (10) shows that the constraint of prohibitedpower zones is not included in the real power reserveconstraints however prohibited power zones are alwaysstrictly considered and must be exactly satisfied
Ramp Rate Limit (RRL) Constraint In OLD problem allconsidered thermal generating units are supposed to be underworking status but previous active power of each thermalgenerating unit is not taken into account Thus increased ordecreased power is not constrained This assumption seemsto be not practical until RRL constraint is considered RRLconstraint considers initial power output and the power
6 Mathematical Problems in Engineering
change is supervised Regulated power can be higher or lowerthan the initial value as long as it is within a predeterminedrange Increased step size (ISS) and decreased step size (DSS)are given as input data and they are used to limit the change ofpower output of each thermal generating unit The constraintcan bemathematically expressed as the following formula [7]
1198751198940 minus 119863119878119878119894 le 119875119894 le 1198751198940 + 119868119878119878119894 (13)
where 1198751198940 is the initial power output of the 119894119905ℎ thermalgenerating unit before its power output is regulated 119868119878119878119894 and119863119878119878119894 are respectively maximum increased and decreasedstep sizes of the 119894119905ℎ thermal generating unit
3 The Proposed Cuckoo Search Algorithm
31 Classical Cuckoo Search Algorithm In search techniqueof CCSA [53] a set of solutions is randomly generated withina predetermined range in the first step and then the quality ofeach one is ranked by computing value of fitness functionThemost effective solution corresponding to the smallest valueof fitness function is determined and then search procedurecomes into a loop algorithm until the maximum iterationis reached In the loop algorithm two techniques updatingnew solutions two times (corresponding to two generations)are Levy flights and mutation technique which is calledstrange eggs identification technique The two generationscan produce promising quality solutions for CCSA Aftereach generation CCSA will carry out comparing fitness ofnewly updated solutions and initial solutions for keepingbetter ones and abandoning worse ones The most effectivesolution at last step of the loop search algorithm is determinedand it is restored as one candidate solution for a study caseThe detail of the two stages is as follows
311 Levy Flights Stage This is the first calculation step in theloop algorithm and it also produces new solutions in the firstgeneration for CCSA New solution 119878119900119897119899119890119908119909 is created by thefollowing model
119878119900119897119899119890119908119909 = 119878119900119897119909 + 120572 (119878119900119897119909 minus 119878119900119897119866119887119890119904119905) oplus Levy (120573) (14)
where 120572 is the positive scaling factor and it is nearly set todifferent values for different problems in the studies [53 62]In the work the most appropriate values for such factor canbe chosen to be 02505 for different systems
312 Discovery of Alien Eggs Stage The step plays a veryimportant role for updating new solutions 119878119900119897119899119890119908119909 of thewhole population However not every control variable ineach old solution is newly updated and the decision ofreplacement is dependent on comparison criteria as thefollowing equation
119878119900119897119899119890119908119909=
119878119900119897119909 + 1205761 (1198781199001198971199031198861198991198891 minus 1198781199001198971199031198861198991198892) if 1205762 lt 119875119886119878119900119897119909 119900119905ℎ119890119903119908119894119904119890
(15)
32 Proposed Algorithm In the part a new variant of CCSA(ICSA) is constructed by applying three effective changes onthe main functions of CCSA in order to shorten simula-tion time corresponding to reduction of iterations and findmore promising solutions The proposed amendments areexplained in detail as follows
(i) Suggest one more equation producing updated stepsize in addition to existing one in CCSA
(ii) Create a new selection standard by computing fitnessfunction ratio 119865119865119877119909 and comparing 119865119865119877119909 with apredetermined parameter 119879119900119897119909 Thus thanks to thestandard the existing updated step size and additionalupdate step size will be chosen more effectively
(iii) Automatically change value of 119879119900119897119909 for the xth solu-tion based on the result of comparing 119865119865119877119909 with theprevious 119879119900119897119909
Such three points are clarified by observing the followingsections
321 Strange Eggs Identification Technique (Mutation Tech-nique) The first proposed improvement in our proposedICSA approach is to select a more suitable formula forproducing new solutions with better fitness function valueIn CCSA Equation (16) below is used to produce a changingstep nearby old solutions for all current solutions
Δ1198781199001198971198991198901199081199091 = 1205763 (1198781199001198971199031198861198991198891 minus 1198781199001198971199031198861198991198892) (16)
The use of Equation (16) aims to produce a random walkaround old solutions in search zones with intent to findout promising solutions In order to reduce the possibilityof suffering the local trap and approach to other favorablezones for searching we propose a new Equation (17) Theformula is built by the idea of enlarging search zone withthe use of two more available solutions Obviously the largerchanging step can own higher performance in moving toother search spaces that the classical approach used in CCSAThe suggestion is mathematically expressed by the formulabelow
Δ1198781199001198971198991198901199081199092= 1205764 (1198781199001198971199031198861198991198891 minus 1198781199001198971199031198861198991198892 + 1198781199001198971199031198861198991198893 minus 1198781199001198971199031198861198991198894) (17)
The changing step obtained by using Eq (17) is namedfour-point changing step Now two solutions which arenewly formed by using two different changing steps shownin formulas (16) and (17) are found by the two followingmethods
It can be clearly observed that the distance between 119878119900119897119909(old solution) and 1198781199001198971198991198901199081199091 (new solution) is lower than thatbetween 119878119900119897119909 and 1198781199001198971198991198901199081199092 This difference can contribute ahighly efficient improvement to the proposed ICSA approachsearch ability
Mathematical Problems in Engineering 7
ΔSol2
Sol2
Sol3
Sol4
ΔSol1
Sol1
Solx
Solnew1
Solnew2
Figure 1 Simulation of solutions corresponding to the first itera-tions of the loop algorithm
For the CCSA case if two solutions 1198781199001198971199031198861198991198891 and 1198781199001198971199031198861198991198892are either slightly different or completely coincident suchnewly updated solution 1198781199001198971198991198901199081199091 does not have good chanceto leave the current zone and approach to more promisingzones In another word the new one is approximately coin-cident with the old one As the search task is taking place atsome last iterations this phenomenon becomes much worsebecause all current solutions are lumped in a small zone andthe capability of moving to other zones is impossible As aresult the CCSA approach will work ineffectively and searchstrategy is time consuming until other runs are started
Contrary to the two-point step size the new proposedformula may produce a large enough length to escape thelocal optimum zone and reach new favorable zones Itexplainswhy the four-point changing step has positive impacton the considered random walk rather than the two-pointchanging step
322 New Standard forChoosing theMostAppropriate Chang-ing Step In this section we extend our analysis to answer thequestionwhen to use the four-point step size FromEquations(18) and (19) two new solutions which are represented asthe results of the two-point-based factor and the four-pointstep size can be illustrated by using Figure 1 corresponding tothe search process at the first some iterations and Figure 2corresponding to the last some iterations For the sake ofsimplicity we rewrite the two equations as follows
Here we suppose that 1198781199001198971 and 1198781199001198972 are obtained byfour exact solutions 1198781199001198971 1198781199001198972 1198781199001198973 and 1198781199001198974 and calculatedas follows
ΔSol2
ΔSol1
Solx
Solnew1
Solnew2
Figure 2 Simulation of solutions corresponding to the last itera-tions of the loop algorithm
Δ1198781199001198971 = 1198781199001198971 minus 1198781199001198972 (22)
Δ1198781199001198972 = 1198781199001198973 minus 1198781199001198974 (23)
Asmentioned above the high changing step between newsolution and old solution can help to explore new favorablezones However in optimization algorithms searching stepscannot be arbitrarily large otherwise the algorithm maydiverge in particular for the cases that the consideredsolutions 119878119900119897119909 are not close together in solution search spaceFor example at the beginning of loop algorithm with thefirst iterations in Figure 1 1198781199001198971198991198901199081 is a better choice than1198781199001198971198991198901199082 because it is kept in a sufficient limit and does notlead to a risk of divergence In contrast as many of currentsolutions are in different positions but their distance is notvery short or approximately coincident such as at the lastiterations in Figure 2 1198781199001198971198991198901199081 and 119878119900119897119909 have a very shortdistance but 1198781199001198971198991198901199082 and 119878119900119897119909 have higher distance Accordingto the phenomenon in Figure 2 the proposed ICSA approachneeds to produce a high changing step to move solutions toother search zones without local optimum Hence 1198781199001198971198991198901199082would be preferred to 1198781199001198971198991198901199081
Based on the argument above the determination of thecondition for using either two-point changing step or four-point changing step is really crucial to the performance ofthe proposed ICSA approach in searching solutions of OLDproblem Here the ratio of 119865119865119877119909 which can be found byEquation (24) is suggested to be a suitable measurement forthe selection of two options
Δ119865119865119877119909 = 119865119865119909 minus 119865119865119887119890119904119905119865119865119887119890119904119905 (24)
For a particular set of the current solutions each individ-ual depending on its 119865119865119877119909 will create a corresponding newsolution by using either Equation (18) or (19) If the valueof one current solution is smaller than the predeterminedparameter 119879119900119897 Equation (19) is applied for updating suchconsidered solution 119909 Otherwise Equation (18) is a betteroption The steps of the modified algorithm are similar to the
8 Mathematical Problems in Engineering
If 1205765 lt 119875119886If FFRx lt Tolx119878119900119897119899119890119908119909 = 119878119900119897119909 + 1205764(1198781199001198971199031198861198991198891 minus 1198781199001198971199031198861198991198892 + 1198781199001198971199031198861198991198893 minus 1198781199001198971199031198861198991198894)else119878119900119897119899119890119908119909 = 119878119900119897119909 + 1205764(1198781199001198971199031198861198991198891 minus 1198781199001198971199031198861198991198892)end
Algorithm 1 New mutation technique applied in the proposed ICSA approach
conventional CSA except that an additional step should beadded at each iteration In this step the119865119865119889 of all individualsolutions should be calculated by utilizing Equation (24) andthen the result of comparing the ratio with 119879119900119897will be used todecidewhich updating formula should be selectedThewholedescription of the proposed standard and new mutationtechnique can be coded inMatlab program language by usingAlgorithm 1
323 Adjustment of Tolerance for Each Solution As pointedout above the proposed method needs assistances to deter-mine the most appropriate step size for finding out favorablesolution zones The given aim can be reached if the selectionof 119879119900119897119909 is reasonable however the range of this parameteris infinite and hard to select Thus the adaptation of tuningthe parameter is really necessary First of all the compari-son between 119879119900119897119909 and 119865119865119877119909 is carried out and then theadaptation will be determined based on the obtained resultfrom the comparison Results of comparison between the twoparameters can be either 119865119865119877119909 is less than 119879119900119897119909 or 119865119865119877119909is higher than 119879119900119897119909 The case that two parameters are equalhardly ever occurs
As the comer assumptionhappens (ie119865119865119877119909 is less than119879119900119897119909) at the considered time the four-point step size will beemployed for the 119909119905ℎ solution If 119879119900119897119909 remains unchanged atthe previous value the identification of improvement fromsuch four-point step size or two-point step size is vagueConsequently value of 119879119900119897119909 must be automatically reducedto a lower value in case that it has significant contribution tofound promising solution of previous iteration Clearly thedecrease of119879119900119897119909 can enable the proposedmethod to jump outlocal optimal zone and approachmore effective zones By trialand error method 119879119900119897119909 is selected to be a function of itselfthat is 09 of the previous value Finally the implementationof the proposed ICSA approach is presented in Algorithm 2
4 The Application of the ProposedICSA for OLD Problem
Thewhole computation steps of the proposed ICSA approachfor solving OLD problem are explained as follows
41 Handling Constraints and Randomly Producing InitialPopulation As shown in Section 2 the considered OLDproblem takes five following constraints into account
(i) Power balance constraint is shown in Equation (4)
(ii) Power output limitation constraint is shown in Equa-tion (6)
(iii) Prohibited power zone constraint is shown in Equa-tion (9)
(iv) Real power reserve constraint is shown in Equation(10)
(v) Ramp rate limit constraint is shown in Equation (13)
Among the five constraints ramp rate limit generationlimit and prohibited power zone seem to be more com-plicated than power balance and power reserve constraintsHowever the three constraints can be solved more easilybecause each unit is constrained independently in the threeconstraints whereas power balance constraint and powerreserve constraint consider all the thermal generating unitssimultaneously Power reserve constraint can be handledby penalizing the total generation of all units while powerbalance constraint can be solved by penalizing one violatedthermal generating unit The whole computation procedurefor solving all constraints and calculating fitness function ofsolutions is described in detail as follows
Step 1 Redefine maximum and minimum power output ofeach thermal generating unit as considering PPZ and RRLconstraints by using the following formulas
119875119894max = 119875119894max if 119875119894max le 119875i0 + 119868119878119878119894119875i0 + 119868119878119878119894 if 119875119894max gt 119875i0 + 119868119878119878119894
119894 = 1 119873(25)
119875119894min = 119875119894min if 119875119894min ge 119875i0 minus 119863119878119878119894119875i0 minus 119863119878119878119894 119890119897119904119890
119894 = 1 119873(26)
Mathematical Problems in Engineering 9
Produce initial population with119873119901119904 solutions (1198781199001198971 1198781199001198972 119878119900119897119909 119878119900119897119873119901119904)Calculate fitness function (1198651198651 1198651198652 119865119865119909 119865119865119873119901)Go to the loop algorithm by setting 119866 = 1
While (119866119898119886119909 gt 119866) (i) The first newly produced solutions119878119900119897119899119890119908119909 = 119878119900119897119909 + 120572(119878119900119897119909 minus 119878119900119897119866119887119890119904119905) oplus Levy(120573) (ii) Perform selection approach
119865119865119909 = 119865119865119909 if 119865119865119909 le 119865119865119899119890119908119909119865119865119899119890119908119909 119900119905ℎ119890119903119908119894119904119890 119909 = 1 119873119901119904
(v) Determine the most effective solution and its fitnessDetermine 119865119865119909 with the smallest value and assign 119878119900119897119909 to 119878119900119897119866119887119890119904119905If 119866119898119886119909 gt 119866 perform step (i) and increase 119866 to 119866 + 1 Otherwise stop the loop algorithm and report boththe smallest fitness together with 119878119900119897119866119887119890119904119905End while
Among the four Equations (25) and (26) are used firstin order to redefine upper bound and lower bound for allthermal generating units as considering RRL constraint Thethe redefined bounds continue to be redefined for the secondtime by using (27) and (28) as considering PPZ constraints
Step 2 (randomly produce initial population) For dealingwith the power balance constraint all available units areseparated into two groups in which the first group withdecision variables consists of the power output from thesecond unit to the last unit (P2 P3 PN) meanwhile onlythe power output of the first unit (1198751) belongs to the secondgroup with dependent variable So upper bound solution119878119900119897119898119886119909 and lower bound solution 119878119900119897119898119894119899 must be defined asfollows
Step 3 Handle prohibited power zone constraint for decisionvariables P2 P3 PN
After being randomly produced there is a high possi-bility that decision variables fall into PPZ and they violatePPZ constraint So the verification of falling into PPZ andcorrection of the violation should be accomplished by usingthe following formula
119875119894 =
119875119897119894119896 if 119875119897119894119896 lt 119875119894 le 119875119897119894119896 + 1198751198961198941198962119875119906119894119896 if (119875119894 gt 119875119897119894119896 + 1198751198961198941198962 ) amp (119875119894 lt 119875119906119894119896)119875119894 119890119897119904119890
119894 = 2 119873 amp 119896 = 1 119899119894
(31)
Step 4 Handle RPB constraint by calculating 1198751 and penaliz-ing 1198751 if it violates constraints
In this step power balance constraint is exactly handledby calculating and penalizing dependent variable (1198751) 1198751 isobtained by using formulas (4) and (5) as follows
1198751 = minus (11986101 minus 1 + 2sum119873119894=2 1198611119894119875119894) plusmn radicΔ211986111 (32)
where
Δ = (11986101 minus 1 + 2 119873sum119894=2
1198611119894119875119899)2 minus 411986111(119875119863 minus 119873sum
119894=2
119875119894+ 11986100 + 119873sum
119894=2
1198610119894119875119894 + 119873sum119894=2
119873sum119895=2
119875119894119861119894119895119875119895) amp Δ ge 0(33)
In Equation (32) 1198751 has been determined for the purposeof dealing with real power balance constraint However it isnot sure that 1198751 can satisfy upper bound and lower boundconstraints and prohibited power zone constraints So 1198751must be checked and penalized
Firstly 1198751 is checked and penalized for upper and lowerbound constraints by the following model
Δ1198751x =
0 if 1198751min le 1198751x le 1198751max
1198751min minus 1198751x if 1198751min gt 1198751x1198751x minus 1198751max if 1198751max lt 1198751x
(34)
In Equation (34) if the second case or the third caseoccurs it means P1 has violated either lower bound or upperbound and it would be penalized by using either (P1x= P1min-P1x) or (P1x =P1x -P1max) Otherwise ifP1 has not violatedthe bound constraints (ie the first case in (34) happened)
P1 would continue to be checked for PPZ constraint by thefollowing model
Δ1198751x
=
1198751 minus 1198751198971119896 if 1198751198971119896 lt 1198751 le 1198751198971119896 + 119875119896111989621198751199061119896 minus 1198751 if (1198751 gt 1198751198971119896 + 11987511989611198962 ) amp (1198751 lt 1198751199061119896)0 119890119897119904119890
(35)
Step 5 Handle real power reserve constraint (10)First of all 119878119894 is determined by using (11) and (12) and
then the 119909119905ℎ solution will be checked and penalized if poweroutput of all thermal generating units cannot satisfy RPRconstraint The penalty for violation of the constraint can becalculated by using equation (36)
Δ119878119877119909 =
0 if119873sum119894=1
119878119894119909 ge 119878119877119878119877 minus 119873sum119894=1
119878119894119909 119890119897119904119890 (36)
As a result real power reserve constraint can be solved byusing the penalty method
42 Calculate Fitness Function for Solutions Fitness functionof each solution is used to evaluate quality of solutionNormally the function is the sum of objective function andpenalty of violating constraints and is obtained by
43 The First Newly Updated Solutions by Levy Flights Tech-nique In this section the first newly updated solutionsare performed by employing Levy flights technique usingEquation (14) However each new solution can be out oftheir feasible operating zone such as PPZ and upper andlower limitations When the power output violates its PPZconstraints Equation (31) will be applied to tackle theconstraint Besides the following equation will be employedwhen power output is higher or lower than their limitations
119878119900119897119909 =
119878119900119897max if 119878119900119897max lt 119878119900119897119909119878119900119897min if 119878119900119897min gt 119878119900119897119909119878119900119897119909 Otherwise
119909 = 1 119873119901 (38)
After that Equations (32)-(37) are performed for deter-mining all variables and penalty terms Finally Equation (38)is employed to calculate fitness function
44 The Second Newly Updated Solutions by Using Muta-tion Technique The second newly updated solutions areaccomplished as presented in Section 3 above Similar to
Mathematical Problems in Engineering 11
the task after doing the first update each solution in thenew population must satisfy PPZ constraint and upper andlower boundaries by considering Equations (31) and (38)Then Equations (32)-(37) are performed for determiningall variables and penalty terms Finally Equation (38) isemployed to calculate fitness function and the solution withthe best value is assigned to the best one 11987811990011989711986611988711989011990411990545 Criterion of Stopping the Loop Algorithm In the loopalgorithm of using the proposed ICSA approach the solutionsearch work is stopped in case that the predeterminedmaximum iterations 119866119898119886119909 is reached For each search ter-mination the most effective solution is stored and anotherrun continues to be accomplished until the predeterminednumber of runs is reached After finishing the runs thebest one is found and reported In addition other valuessuch as the fitness of the worst solution and average fitnessof all solutions are also reported for comparing with othermethods
46 The Whole Iterative Process The whole iterative algo-rithm for implementing the proposed ICSA approach forcoping with OLD problem is described in detail in Figure 3
5 Results and Discussions
The proposed ICSA approach performance has been investi-gated on six cases with different fuel options different fuelcharacteristics and complicated constraints The details ofthe studied cases are presented as follows
Case 1 Four systems with single fuel source (SFS) and powerloss (PL) constraint
Subcase 11 A 3-unit system [57]Subcase 12 A 6-unit system [57]Subcase 13 A 3-unit system [56]Subcase 14 A 6-unit system [56]
Case 2 A 110-unit system with SFS [57]
Case 3 Four systems with SFS and the effects of valve loadingprocess (EoVLP)
Subcase 31 A 3-unit system supplying to a load of850MW [58]Subcase 32 A 13-unit system supplying to a load of1800MW [1]Subcase 33 A 13-unit system supplying to a load of2520MW [1]Subcase 34 A 40-unit system supplying to a load of2500MW [1]Subcase 35 An 80-unit system supplying to a load of4100MW [49]
Case 4 Two systems with SFS and PPZ and RPR constraints
Subcase 41 A 60-unit system supplying to a10600MW load [9]
Subcase 42 A 90-unit system supplying to a15900MW load [9]
Case 5 A 15-unit system with SFS and RRL PPZ and PLconstraints [61]
Case 6 Three systems with multiple fuel sources (MFS) andEoVLP
Subcase 61 An 80-unit system supplying to a21600MW load [15]Subcase 62 A 160-unit system supplying to a43200MW load [15]Subcase 63 A 320-unit system supplying to an86400MW load [54]
For each considered case with each load case the pro-posed ICSA approach is run 50 times on the programlanguage of Matlab and a PC with 4 GB of RAM and 24GHzprocessor The selection of adjustment parameters including119875119886 and 119879119900119897119909 is carefully considered to obtain the best optimalsolutions meanwhile two others such as 119873119901119904 and 119866119898119886119909 arechosen corresponding to the scale of particular test system9 values with the change of 01 in the range [01 09] are inturn selected for 119875119886 while 119879119900119897119909 is 001 at the beginning Theinformation including load demand119873119901119904 119866119898119886119909 and the best119875119886 is reported in Table 1
51 Obtained Results on Case 1 considering Four Systems withSFS and PL Constraint In this section we have implementedthe proposed ICSA approach for solving four systems dividedinto four subcases Tables 2 and 3 show the comparisons ofobtained results from Subcases 11 and 12 and Subcases 13and 14 respectively As listed in Table 2 the proposed ICSAmethod and CCSA can find equal fuel cost for Subcases 11whereas the reduction of fuel cost from the proposed ICSAmethod as compared to CCSA is clearer for Subcase 12 Asshown in Table 3 for comparing the proposed ICSA and threemethods consisting of CCSA ABC and FA the minimumfuel cost of the proposed ICSA is approximately equal to thatof these methods for Subcases 13 but much less than that ofthese methods for Subcase 14 Furthermore the proposedICSA has been run by setting 119873119901119904 and 119866119898119886119909 to 5 and 20but these values were much higher for CCSA ABC and FAThey are 20 and 5000 for CCSA 40 and 100 for ABC and 20and 5000 for FA Consequently the proposed method is veryefficient for Case 1 with four subcases
Optimal solutions obtained by ICSA for Case 1 are shownin Tables 16ndash18
52 Obtained Results on Case 2 considering 110-Unit Systemwith SFS In this section we have employed a very largescale system with 110 units but there were not challenges forobjective function and complex constraints since EoVLP andconstraints were not taken into account Both CCSA andthe proposed ICSA methods have been run for comparingwith BBO hybrid BBO and DE (DEBBO) and Opposi-tional real coded chemical reaction optimization algorithm(ORCCROA) in [36] IWA in [40] and AGWO in [52] As
12 Mathematical Problems in Engineering
Select parameters
- Correct solutions if violating PPZ constraint using Eq (31)- Determine using Eq (32)- Penalize for violating upper and lower bounds constraint using Eq (34)- Penalize for violating PPZ constraint using Eq (35)- Penalize the xth solution for violating RPR constraint using Eq (36)
- Determine quality of solutions employing Eq (37) - Select the most effective solution - Start the loop algorithm by selecting
- Perform the first solution update using Section 32 - Correct boundaries of solutions by using Eq (38)
- Correct solutions if violating PPZ constraint using Eq (31)- Determine using Eq (32)- Penalize for violating upper and lower bounds constraint using Eq (34)- Penalize for violating PPZ constraint using Eq (35)- Penalize the xth solution for violating RPR constraint using Eq(36)
- Determine fitness function using Eq (37)- Compare old solutions and new solutions to keep better ones
- Perform the second solution update using Algorithm 1 - Correct boundaries of solutions by using (38)
- Correct solutions if violating PPZ constraint using Eq (31)- Determine using Eq (32)- Penalize for violating upper and lower bounds constraint using Eq (34)- Penalize for violating PPZ constraint using Eq (35)- Penalize the xth solution for violating RPR constraint using Eq (36)
- Determine fitness function using Eq (37)- Compare old solutions and new solutions to keep better ones- Select the most effective solution
Stop
Start
- Redefine upper and lower bounds using Eqs (25)-(28)- Randomly generate initial population using Eq (30)
Nps Pa Gmax H> Tolx
P1xP1x
P1x
P1xP1x
P1x
P1xP1x
P1x
Sol<MN
Sol<MN
G = 1
G = Gmax G = G + 1
Figure 3 All computation steps for solving OLD problem by employing the proposed ICSA approach
shown in Table 4 AGWO [52] has reached less fuel cost thanICSA however the exact fuel cost which was recalculatedby using reported solution pointed out that the method hasreached a very high fuel cost of $2157404250 For comparisonwith other methods ICSA has found less fuel cost thanall these methods Particularly the reduction of generationfuel cost is significant as compared to BBO DEBBO andCCSA Execution time comparisons are also useful evidence
for indicating the high performance of ICSA Thus it canconclude that ICSA is a strong method for Case 2
Optimal solution obtained by ICSA for the case is shownin Table 19
53 Obtained Results on Case 3 considering Four Systems withSFS and EoVLP In this section the real performance of theproposed ICSA approach has been investigated based on five
Mathematical Problems in Engineering 13
Table 1 Information of considered cases and adjustment parameters
Case Fuel cost function Constraint Subcase No of units 119875119863 (MW) Nps Gmax Best 1198751198861 SFS PL
subcases with the gradual increase of number of units Thesmallest scale system considers 3 units but the largest scalesystem takes 80 units In addition to the implementation ofthe proposed ICSA we have also implemented CCSA forSubcase 31 and Subcase 35 for further comparison becauseCCSA has not been run for the two subcases so far
Comparison of obtained results from Subcase 31 shownin Table 5 indicates that the proposed ICSA is superior toCCSAwith lower fuel cost but it seems to be less effective thanMCSA [58] The minimum cost of MCSA reported in [58] isthe smallest fuel cost but the recalculated cost is much higherthan that of the proposed method Furthermore MCSA hasbeen implemented by setting very high values to 119873119901119904 and119866119898119886119909
Reports for Subcases 32 and 33 shown in Table 6are the comparisons of the proposed ICSA approach andother methods such as conventional Evolution programming(CEP) [1] Fast EP (FEP) [1] improved FEP (IFEP) [1] DE[12] multiplier Lagrange-based genetic algorithm with (GA-MU) [15] QPSO [16] GA-PS-SQP [30] PSO-SQP [32] M120573-HCLSA [49] IABCA [50] CCSA [59] OSE-CSA [59] SOS[34] MSOS [34] CEA-SQT [38] TSBO [39] IWA [40] andCBA [44] As observed from the table ICSAapproach obtainsbetter solutions than mostmethods excluding DE [10] CCSA
[59] OSE-CSA [59] SOS [34] MSOS [34] CEA-SQT [38]TSBO [39] IWA [40] and CBA [44] especially M120573-HCLSA[49] with lower cost $1796097 However recalculated costfrom reported solution of M120573-HCLSA is $179691 BesidesICSA is very fast as compared to most methods where twoother versions of Cuckoo search algorithm CCSA and OSE-CSA are also included except two methods in [34] Theprocessor of computer that all the methods run on is alsoreported in the final column Clearly ICSA approach is veryefficient for the case with the 13-unit system where effects ofvalve loading process are considered
In Subcase 34 the number of units is much larger thanthat of three subcases above up to 40 units [1] The obtainedresult comparisons with others are indicated in Table 7Clearly the minimum cost comparisons reveal that the pro-posedmethod is one of the leading methods due to the lowestcost except the comparison with CCSA [23] OSE-CSA [59]SOS [34] MSOS [34] EMA [45] 120579-MBA [47] and AGWOA[52] It is noted that AGWOA [52] has reported the bestminimum cost with $12140430 but recalculated minimumcost which was obtained by substituting reported optimalgeneration of all thermal generating units is $12141331 Theaverage and the maximum costs from the proposed methoddo not belong to the leading method group however the
Mathematical Problems in Engineering 15
Table 7 Result comparisons for Subcase 34
Approach Best cost Average cost Worst cost CPU time Computer($h) ($h) ($h) (s) (Processor-Ram)
execution time that ICSA approach takes is smaller thanmostones except IAPR [42] and 120579-MBA [47] which have been runon stronger computers Note that MSOS [34] has been fasterthan the proposed method for Subcases 31 and 32 abovebut it is too slower than the proposed method for the casenamely 1813 seconds compared to 146 seconds Comparedto two other versions of Cuckoo search CCSA [23] and OSE-CSA [59] the proposed method is also faster about threetimes although the processors are slightly different For thiscase 120579-MBA [47] shows a very good performance howeverthe method has not been tested onmore complicated systemsand larger scale and therefore more comparisons with themethod must end Clearly the proposed method is stillefficient for the case where large scale and effects of valveloading process are included
Subcase 35 is the largest scale system with 80 units Inaddition to the implementation of ICSA CCSA has beenalso run for the Subcase 35 for further investigation ofefficiency improvement of the proposed ICSA approach The
comparisons of minimum cost in Table 8 show that theproposed ICSA can find more optimal solution than M120573-HCLSA [49] AGWOA [52] and CCSA The proposed ICSAis also superior to CCSA in terms ofmore stable search abilityand lower fluctuation since its average cost and maximumcost are less than those of CCSA The outstanding figurecannot be reached as compared to AGWOA [52] howeverit is hard to conclude AGWOA [52] is superior to theproposed ICSA approach about more stable search abilityand lower fluctuation Actually comparison of the values ofpopulation and iterations as well as execution time cannot beaccomplished because the information was not reported in[52]Thus it can conclude that the proposed ICSA is effectivefor the subcase
Optimal solutions obtained by ICSA for the case areshown in Tables 20ndash23
54 Obtained Results on Case 4 with Two Systems consideringSFS and PPZ and RPR Constraints In this section two
16 Mathematical Problems in Engineering
Table 9 Result comparisons for Subcases 41 and 42
Subcase Approach Best cost ($h) Mean cost ($h) Worst cost ($h) Std dev ($h) CPU time (s) Computer (Processor-Ram)
test systems with SFS and PPZ and RPR constraints areconsidered The test system size is up to 60 and 90 units forSubcases 41 and 42 respectively Comparison for the casesis only performed with two Genetic algorithms consisting ofGA and IGA-MU in [9] and two other versions of Cuckoosearch algorithms including CCSA [23] and OSE-CSA [59]and presented in Table 9 Other studies have tended toignore such complicated constraints of PPZ and RPR Thecosts reported in Table 9 indicate that ICSA approach canobtain more effective solution than CCSA and OSE-CSAbecause it has reached lower minimum cost than the twoones Furthermore the proposed method also takes shortercomputation time for the two cases from about two timesto about three times although the processor of the proposedmethod is slightly strongerThemean costs of ICSA approachare much less than those from IGA-MU and GA and slightlyhigher than those from OSE-CSA but there is a trade-off between the proposed method and CCSA for the twosubcases In fact the proposed method obtains higher meancost for Subcase 41 but lower cost for Subcase 42 AlthoughGAmethods have been runon aweak computerwith 07 GHzof the processor compared to that with 24GHz in the studytheir execution times are significantly higher namely 56381seconds (GA) and 16258 seconds (IGA-MU) compared to09153 seconds of ICSA approach for Subcase 41 and 94093seconds (GA) and 25545 seconds (IGA-MU) compared to15892 seconds (the proposedmethod)The analysis can pointout that ICSA approach ismore efficient than these comparedmethods in terms of optimal solutions and execution time
Optimal solutions obtained by ICSA for Subcase 42 areshown in Table 24
55 Obtained Results on Case 5 with a 15-Unit System consid-ering SFS and RRL PPZ and PL Constraints In this section
a 15-unit system considering RRL PPZ and PL constraintsis considered to be solved for finding optimal solution Forefficiency investigation of the proposed ICSA we have alsoimplemented CCSA for comparison As listed in Table 10the proposed ICSA is the most effective method with thesmallest fuel cost The comparisons of control parameters aswell asCPU time are also good evidence to confirm the strongsearch of the proposed ICSA approach since it has been runby smaller values of control parameter and faster executiontime as compared to all methods excluding CCSA
Optimal solution obtained by ICSA for the case is shownin Table 25
56 Obtained Results on Case 6 withThree Systems consideringMFS and EoVLP In this section three test systems with thechallenge on objective function including multi-fossil fuelsources and effects of valve loading process are consideredThe scale is up to 80 units 160 units and 320 units forSubcases 61 62 and 63 respectively
Comparison for Subcase 61 reported in Table 11 revealsthat the proposed method is the best method in terms of thelowest best cost the lowestmean cost and the lowest standarddeviation and the fastest execution time The processor fromthis proposedmethod is about four times stronger thanCGA-MU and IGA-MU but the speed is from ten times to 35 timesfaster than these methods Compared to CCSA and OSE-CSA the proposed method is about two times faster but theprocessor is slightly stronger
Comparison for Subcase 62 is reported in Table 12Clearly the proposed ICSA approach obtains better values ofthe best mean and worst costs than most methods exceptMSOS [34] where the best cost difference is about $ 024However the proposed method is the second fastest onewith 1119 seconds where the first fastest one CBA [44] has
Mathematical Problems in Engineering 17
Table 11 Comparisons of found results for Subcase 61
Approach Best cost Mean cost Worst cost Std dev CPU time Computer($h) ($h) ($h) ($h) (s) (Processor-Ram)
spent 571 seconds Clearly MSOS is better than the proposedICSA approach with respect to slightly less best cost butis worse than the proposed method in terms of executiontime namely 2572 seconds compared to 1119 seconds whileCBA [44] is faster than the proposed method but obtainssignificantly worse costs The analysis can conclude that theproposed ICSA approach is very powerful for the subcasewith 160 units
Table 13 presents the comparison of three other methodsincludingCCSA [54] SOS [34] andMSOS [34] accompaniedwith the proposed method for Subcase 63 The obtainedresult comparisons imply that ICSA approach can obtainbetter values of the best and standard deviation costs thanCCSA and SOS but obtains slightly higher cost than MSOSby approximately $ 022 Besides the execution time fromICSA approach is much shorter than others especially it ishigher than five times faster than MSOS The four methodshave been run on approximately strong computers Brieflythe proposed ICSA approach can find and converge to morefavorable solution than other methods with shorter CPUtime except the comparison with MSOS which had bettersolution but spent higher than five times execution times
Consequently the proposed method is a very promisingoptimization algorithm for Subcase 63 a system up to 320units and with multi-fossil fuel sources and effects of valveloading process
Optimal solution obtained by ICSA for Subcase 63 isshown in Table 26
57 The Improvement of ICSA Approach Performance
571 The Outstanding Improvement over CCSA In this sec-tion the performance improvement of ICSA over CCSAhas been investigated by analyzing obtained results and setcontrol parameters Table 14 has been formed by addingreduction cost improvement level of the best cost executiontime and control parameters consisting of 119873119901119904 and 119866119898119886119909Among the compared factors reduction cost is the deviationof the cost of CCSA and that of ICSA whereas the improve-ment level is the ratio of the reduction cost to the cost ofCCSA The reduction costs indicate the proposed methodcould find either equal quality of solutions or higher qualityof solutions than CCSA for all study casesThe reduction costis from $0 to $42839 corresponding to the improvement level
18 Mathematical Problems in Engineering
Table 14 Summary of results obtained by CCSA and ICSA for all study cases
Study case Reduction cost ($) Improvement level () Execution time (s) Nps Gmax
Figure 4 The best run obtained by CCSA and ICSA for Case 2
that is from 0 to 08968The saving cost is not too much forone hour but the operation in one day onemonth or one yearis very high However it should be noted that CCSA has beenrun by setting much higher population size and iterations formany cases excluding study cases implemented in the studysuch as Case 2 Subcase 31 Subcase 35 and Case 5 Forinstance ICSA has used 119873119901119904 = 10 and 119866119898119886119909 = 15 for Subcase13 and119873119901119904 = 10 and119866119898119886119909 = 25 for Subcase 14 whereas CCSAhas been run by setting119873119901119904 = 20 and 119866119898119886119909 = 5000 for the twosubcases Similarly CCSA has been run for Subcases 32 33and 34 with much higher number of iterations For the lastsubcase CCSA has been run by setting 119873119901119904 = 320 and 119866119898119886119909
= 1200 but those of ICSA have been 10 and 9000 Due tothe higher value of control parameters CCSA has tended tospend more time in finding such high quality solutions foralmost all study cases Execution time of ICSA is less than18 seconds while that of CCSA is up to higher 75 secondsIt is clear that the proposed ICSA could find better optimalsolutions thanCCSA for such considered systems For furtherinvestigation of performance comparison the best runs over50 runs and fuel cost values of 50 runs obtained by CCSA andthe proposed ICSA for Case 2 Subcase 31 Subcase 35 andCase 5 have been plotted in from Figures 4ndash11 The best runcurves show the faster search of the proposed ICSA method
Mathematical Problems in Engineering 19
0 5 10 15 20 25 30 35 40 45 50Run
19795
198
19805
1981
19815
1982
19825
1983
19835
Fuel
cost
($)
CCSAICSA
times105
Figure 5The best fuel cost of 50 runs obtained by CCSA and ICSA for Case 2
0 2 4 6 8 10 12 14 16 18 20Iteration
8234
8236
8238
8240
8242
8244
8246
8248
8250
8252
Fitn
ess F
unct
ion
($)
CCSAICSA
Figure 6 The best run obtained by CCSA and ICSA for Subcase 31
whereas 50 values of fuel cost indicate that the proposedICSA can find many solutions with better quality Clearlythe proposed ICSA is outstanding in terms of stabilizationof solution search and faster convergence As a result it canconclude that the proposed ICSA approach is more effectivethanCCSA in solvingOLDproblemwith considered systems
572The Improvement of Results over Other Methods In thisarticle we have tested ICSA approach on 6 cases with 16
systems with different fuel cost forms different constraintsand different scale systems from 3 units to 320 units We havecompared the yielded results from ICSA approach and otherexisting ones for evaluating the efficiency of ICSA approachIn subsections above we have shown yielded results fromICSA approach and compared these results to those of otheronesHowever the demonstrationhad not been very good forobserving and comparing to lead to a conclusionThus in thesubsectionwehave summarized the result comparisons of the
20 Mathematical Problems in Engineering
0 5 10 15 20 25 30 35 40 45 50Run
8234
82342
82344
82346
82348
8235
82352
82354
82356
82358
8236
Fuel
cost
($)
CCSAICSA
Figure 7 The best fuel cost of 50 runs obtained by CCSA and ICSA for Subcase 31
0 1000 2000 3000 4000 5000 6000Iteration
242
244
246
248
25
252
254
256
Fitn
ess F
unct
ion
($)
CCSAICSA
times105
Figure 8 The best run obtained by CCSA and ICSA for Subcase 35
proposed and other ones Table 15 has reported the reductioncost (in $) of ICSA approach compared to other ones Inaddition we have converted the reduction cost into improve-ment level (in ) for better comparison The improvementhas been shown from the lowest level to the highest levelin terms of reduction cost and improvement percentageIn addition we have also given the slowest and the fastestexecution time of other compared methods together withthat of the proposed method The table implies that ICSA
approach can find better optimal solutions with less fuel costup to $052 for Subcase 13 $75229 for Subcase 14 $1775149for Case 2 $1181256 for Subcase 31 $8438 for Subcase 32$9113 for Subcase 33 $121181 for Subcase 34 $3412 forSubcase 35 $04329 for Subcase 41 $30227 for Subcase42 $407002 for Case 5 $18183 for Subcase 61 $13965 forSubcase 62 and $11817 for Subcase 63 These reductioncosts are equivalent to improvement level (IL) of 0032908968 82282 14143 047 038 099 001405
Mathematical Problems in Engineering 21
0 5 10 15 20 25 30 35 40 45 50Run
2428
243
2432
2434
2436
2438
244
2442
Fuel
cost
($)
CCSAICSA
times105
Figure 9 The best fuel cost of 50 runs obtained by CCSA and ICSA for Subcase 35
0 50 100 150 200 250 300 350 400Iteration
327
328
329
33
331
332
333
334
335
336
337
Fitn
ess F
unct
ion
($)
CCSAICSA
times104
Figure 10 The best run obtained by CCSA and ICSA for Case 5
00003 0002 12291 004 138 and 059 Thesequantitative comparisons reveal that larger scale systems canlead to better reduction cost but the improvement level isnot high because total cost of compared methods tends tobe large for large scale systems Furthermore very large scalesystems with nondifferentiable objective have been normallysolved by strong methods In fact systems in Case 3 havethe same characteristic with single fuel and effects of valveloading process but Subcase 34 is a larger scale system with
40 units while Subcases 31 32 and 33 are constructed by 3units 13 units and 13 units So the improvement percentageof Subcase 34 can be up to 099 whilst that of Subcases32 and 33 is 047 and 038 respectively Subcase 35 iswith the largest system 80 units but the reduction cost is notmuch only $3412 because compared methods with ICSA areeither state-of-the-art ones or improved ones Also Subcase41 and Subcase 42 have considered single fuel and PPZ andspinning reserve constraints but Subcase 42 is larger scale
22 Mathematical Problems in Engineering
Table 15 Performance improvement summary of the proposed method
Study cases Reduction cost ($) Improvement level () Execution time (s)From To From To Slowest method Fastest method Proposed method
Figure 11 The best fuel cost of 50 runs obtained by CCSA and ICSA for Case 5
system with 90 units and Subcase 41 is only with 60 unitsThus the improvement of Subcase 41 is lower with 00003but that of Subcase 42 is 0002 Similarly systems in Cases 6have the same featurewithmulti-fossil fuel sources and effectsof valve loading process but they are respectively constructedby 80 160 and 320 units As a result the improvements ofSubcase 62 138 and Subcase 63 059 are much higherthan Subcase 61 004 However Subcase 62 with smallernumber of units but getting higher improvement is easilyunderstood because there were nine compared methods butonly three compared methods are considered for Subcase 63In general the improvement is not high it is about under onedollar several dollars tens of dollars and over one thousand
dollars per hour however the saving cost will be significantif the operation is considered to be onemonth with 720 hoursor one year with 8760 hours
Execution time comparison can be evaluated by observ-ing the fastest and the slowest compared methods in Table 15These execution times of the proposed method are approxi-mately equal to that of the fastest methods for Subcases 3233 and 34 and much shorter than other fastest comparedmethods for other cases especially for Subcases 61 and 63The fastest method for Subcase 62 is CBA [44] showing 57seconds while that of the proposed method is 1119 secondsHowever it cannot conclude that CBA is more effective thanthe proposed method because the proposed method could
find better optimal solution with less fuel cost by $2130As considering execution time of the slowest comparedmethods it can point out that ICSA is a very fast optimizationtool since the execution time of these methods is 29496 and3425 seconds for Subcases 32 and 33 116735 seconds forSubcase 34 56381 seconds for Subcase 41 94093 secondsfor Subcase 42 12797 seconds for Case 5 30941 seconds forSubcase 61 7542 seconds for Subcase 62 and 9641 secondsfor Subcase 63 while the execution time of the proposedmethod for these cases is respectively 095 146 091 1589246 84828 1119 and 171384 seconds It is clearly shown thatICSA is very fast as compared to these methods
In summary the proposed method has found approx-imately high quality solutions with several standard state-of-the-art meta-heuristic algorithms and improved versionsof them together with other old methods In addition theproposed method could improve result better than approx-imately all methods with faster execution time Comparedto other methods with the fastest convergence speed andhigh quality solutions the proposed method has been as
fast as for some cases and much faster for other cases Thecomparison with the slowest methods could show that theproposed method was extremely powerful since it was up tonearly one thousand times faster Consequently the proposedICSA approach can be one of the strongest optimization toolsfor OLD problem
6 Conclusions
This paper has proposed a good ICSA method for solvingOLD problem in which many test systems with differentobjective functions and complicated constraints from simpleto complex have been used as studied cases The proposedICSA method has been developed by performing severalmodifications on the second solution update of CCSA whichcontained several drawbacks to global convergence and fastmanner The OLD problem has covered from single fuelto multi-fossil fuels from quadratic objective function tononconvex objective function in addition to PPZ RPR andRRL constraints Many existing optimization algorithms have
been concerned in aim to compare the performance and givethe final conclusion on the proposed method There have
been six main cases with sixteen subcases The evaluationshave been made at the end of each study case Clearly theproposed ICSA approach has yielded more effective optimalsolutions with faster execution time than almost all methodsConsequently it can be concluded that the proposed methodis much more superior to CCSA and is a very promisingmethod for solving OLD problem
Appendix
See Tables 16ndash26
Nomenclature
120575119894 120582119894 120572119894 120573119894 120574119894 Fuel cost functioncoefficients of the 119894119905ℎthermal generationunit11986100 1198610119895 119861119895119894 Power loss matrixcoefficients119865119865119909 119865119865119887119890119904119905 The values of fitness ofsolution 119909 and theso-far most effectivesolution among thecurrent set ofsolutions119898119894 Number of fuels burntin the 119894119905ℎ thermalgeneration unit
119873 Number of allavailable thermalgeneration units119899119894 Number of prohibitedpower zones of the 119894119905ℎthermal generationunit119875119906119894119896 119875119897119894119896 Upper and lowerlimits of the 119894119905ℎthermal generationunit corresponding tothe kth PPZ119875119886 Probability ofreplacing controlvariables in each oldsolution119875119863 Real power demand ofall loads in system119875119894119898119886119909 119875119894119898119894119899 The highest andlowest real poweroutputs of the iththermal generationunit119875119894119895119898119886119909 119875119894119895119898119894119899 The highest andlowest real poweroutputs of the 119894119905ℎthermal generationunit corresponding tothe 119895119905ℎ fuel type119878119894119898119886119909 Maximum real powerreserve contributionof the thermalgeneration unit 1198941198781199001198971199031198861198991198891 1198781199001198971199031198861198991198892 1198781199001198971199031198861198991198893 1198781199001198971199031198861198991198894 Randomly mixedsolutions from the setof current solutions119878119900119897119909 119878119900119897119866119887119890119904119905 The old solution x andthe most effectivesolution119878119877 Real power reserverequirement of system
120575119894119895 120582119894119895 120572119894119895 120573119894119895 120574119894119895 Fuel cost functioncoefficients of the 119894119905ℎthermal generationunit corresponding tothe 119895119905ℎ fuel type1205761 1205762 1205763 1205764 1205765 1205766 Random numbersbetween 0 and 1Ω Number of generationunits considering PPZconstraint
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare no conflicts of interest
References
[1] N Sinha R Chakrabarti and P K Chattopadhyay ldquoEvolution-ary programming techniques for economic load dispatchrdquo IEEETransactions on Evolutionary Computation vol 7 no 1 pp 83ndash94 2003
[2] P Hansen ldquoA separable approximation dynamic programmingalgorithm for economic dispatch with transmission lossesrdquoYugoslav Journal of Operations Research vol 12 no 2 2002
[3] N T Thang ldquoSolving economic dispatch problem with piece-wise quadratic cost functions using lagrange multiplier theoryrdquoin Proceedings of the 3rd International Conference on ComputerTechnology and Development (ICCTD rsquo11) pp 359ndash364 ASMEPress 2011
[4] S K Mishra and S K Mishra ldquoA comparative study of solutionof economic load dispatch problem in power systems in theenvironmental perspectiverdquoProcedia Computer Science vol 48pp 96ndash100 2015
[5] A A Al-Subhi and H K Alfares ldquoEconomic load dispatchusing linear programming a comparative studyrdquo InternationalJournal of Applied Industrial Engineering vol 3 no 1 pp 16ndash362016
28 Mathematical Problems in Engineering
[6] C Zhou G Huang and J Chen ldquoPlanning of electric powersystems considering virtual power plants with dispatchableloads included an inexact two-stage stochastic linear program-ming modelrdquoMathematical Problems in Engineering vol 2018Article ID 7049329 12 pages 2018
[7] T T Nguyen N V Quynh and L Van Dai ldquoImproved fireflyalgorithm a novel method for optimal operation of thermalgenerating unitsrdquo Complexity vol 2018 Article ID 7267593 23pages 2018
[8] S S Haroon S Hassan S Amin et al ldquoMultiple fuel machinespower economic dispatch using stud differential evolutionrdquoEnergies vol 11 no 6 pp 1ndash20 2018
[9] C-T Su and C-L Chiang ldquoNonconvex power economic dis-patch by improved genetic algorithm with multiplier updatingmethodrdquo Electric Power Components and Systems vol 32 no 3pp 257ndash273 2004
[10] NNoman andH Iba ldquoDifferential evolution for economic loaddispatch problemsrdquo Electric Power Systems Research vol 78 no8 pp 1322ndash1331 2008
[11] P Somasundaram and K Kuppusamy ldquoApplication of evolu-tionary programming to security constrained economic dis-patchrdquo International Journal of Electrical Power amp EnergySystems vol 27 no 5-6 pp 343ndash351 2005
[12] T Jayabarathi K Jayaprakash andD Jeyakumar ldquoEvolutionaryprogramming technique for different kinds of economic dis-patch problemsrdquo Electric Power Systems Research vol 73 no 2pp 169ndash176 2005
[13] P Subbaraj R Rengaraj and S Salivahanan ldquoEnhancementof combined heat and power economic dispatch using selfadaptive real-coded genetic algorithmrdquo Applied Energy vol 86no 6 pp 915ndash921 2009
[14] C-L Chiang and C-T Su ldquoAdaptive-improved genetic algo-rithm for the economic dispatch of units with multiple fueloptionsrdquo Cybernetics and Systems vol 36 no 7 pp 687ndash7042005
[15] C Chiang ldquoImproved genetic algorithm for power economicdispatch of units with valve-point effects and multiple fuelsrdquoIEEE Transactions on Power Systems vol 20 no 4 pp 1690ndash1699 2005
[16] K Meng H G Wang Z Y Dong and K P Wong ldquoQuantum-inspired particle swarm optimization for valve-point economicload dispatchrdquo IEEE Transactions on Power Systems vol 25 no1 pp 215ndash222 2010
[17] P Erdogmus A Ozturk and S Duman ldquoEnvironmen-taleconomic dispatch using genetic algorithm and simulatedannealingrdquo in Proceedings of the International Conference onElectric Power and Energy Conversion Systems EPECS rsquo09 pp1ndash4 IEEE 2009
[18] L D S Coelho and V C Mariani ldquoImproved differentialevolution algorithms for handling economic dispatch opti-mization with generator constraintsrdquo Energy Conversion andManagement vol 48 no 5 pp 1631ndash1639 2007
[19] H Dakuo W Fuli and M Zhizhong ldquoA hybrid genetic algo-rithm approach based on differential evolution for economicdispatch with valve-point effectsrdquo Electrical Power and EnergySystems vol 30 pp 31ndash38 2008
[20] C-F Chang J-J Wong J-P Chiou and C-T Su ldquoRobustsearching hybrid differential evolution method for optimalreactive power planning in large-scale distribution systemsrdquoElectric Power Systems Research vol 77 no 5-6 pp 430ndash4372007
[21] M R Farooqi P Jain and K R Niazi ldquoUsing Hopfield neuralnetwork for economic dispatch of power systemsrdquo in Proceed-ings of the National Power Engineering Conference PECon rsquo03pp 5ndash10 Bangi Malaysia 2003
[22] T Yalcinoz H Altun and U Hasan ldquoConstrained economicdispatch with prohibited operating zones a Hopfield neuralnetwork approachrdquo in Proceedings of the 10th MediterraneanElectrotechnical Conference Information Technology and Elec-trotechnology for the Mediterranean Countries ProceedingsMeleCon rsquo00 (Cat No 00CH37099) pp 570ndash573 LemesosCyprus 2000
[23] D N Vo P Schegner and W Ongsakul ldquoCuckoo searchalgorithm for non-convex economic dispatchrdquo IET GenerationTransmission amp Distribution vol 7 no 6 pp 645ndash654 2013
[24] J-B Park K-S Lee J-R Shin and K Y Lee ldquoA particleswarm optimization for economic dispatch with non-smoothcost functionsrdquo IEEETransactions on Power Systems vol 20 no1 pp 34ndash42 2005
[25] T Niknam H D Mojarrad and M Nayeripour ldquoA newfuzzy adaptive particle swarm optimization for non-smootheconomic dispatchrdquo Energy vol 35 no 4 pp 1764ndash1778 2010
[26] L D S Coelho and V C Mariani ldquoAn improved harmonysearch algorithm for power economic load dispatchrdquo EnergyConversion and Management vol 50 no 10 pp 2522ndash25262009
[27] M Mahdavi M Fesanghary and E Damangir ldquoAn improvedharmony search algorithm for solving optimization problemsrdquoAppliedMathematics and Computation vol 188 no 2 pp 1567ndash1579 2007
[28] A Bhattacharya and P K Chattopadhyay ldquoSolving complexeconomic load dispatch problem using biogeography-basedoptimizationrdquo Expert Systems with Applications vol 37 no 5pp 3605ndash3615 2010
[29] D Simon ldquoBiogeography-based optimizationrdquo IEEE Transac-tions on Evolutionary Computation vol 12 no 6 pp 702ndash7132008
[30] J S Alsumait J K Sykulski and A K Al-Othman ldquoAhybrid GA-PS-SQP method to solve power system valve-pointeconomic dispatch problemsrdquo Applied Energy vol 87 no 5 pp1773ndash1781 2010
[31] M S Turgut andGKDemir ldquoQuadratic approximationndashbasedhybrid Artificial Cooperative Search algorithm for economicemission load dispatchproblemsrdquo International Transactions onElectrical Energy Systems vol 27 no 4 pp 1ndash14 2017
[32] T A A Victoire and A E Jeyakumar ldquoHybrid PSO-SQPfor economic dispatch with valve-point effectrdquo Electric PowerSystems Research vol 71 no 1 pp 51ndash59 2004
[33] D-KHe F-LWang andZ-ZMao ldquoHybrid genetic algorithmfor economic dispatch with valve-point effectrdquo Electric PowerSystems Research vol 78 no 4 pp 626ndash633 2008
[34] D C Secui ldquoA modified symbiotic organisms search algorithmfor large scale economic dispatch problem with valve-pointloading effectsrdquo Energy vol 113 pp 366ndash384 2016
[35] K BhattacharjeeA Bhattacharya and S H N Dey ldquoTeaching-learning-based optimization for different economic dispatchproblemsrdquo Scientia Iranica vol 21 no 3 pp 870ndash884 2014
[36] K Bhattacharjee A Bhattacharya and S H N Dey ldquoOppo-sitional Real Coded Chemical Reaction Optimization for dif-ferent economic dispatch problemsrdquo International Journal ofElectrical Power amp Energy Systems vol 55 pp 378ndash391 2014
Mathematical Problems in Engineering 29
[37] V Hosseinnezhad M Rafiee M Ahmadian and M T AmelildquoSpecies-basedQuantum Particle SwarmOptimization for eco-nomic load dispatchrdquo International Journal of Electrical Poweramp Energy Systems vol 63 pp 311ndash322 2014
[38] M S P Subathra S E Selvan T A A Victoire A HChristinal andU Amato ldquoA hybrid with cross-entropymethodand sequential quadratic programming to solve economic loaddispatch problemrdquo IEEE Systems Journal vol 9 no 3 pp 1031ndash1044 2015
[39] J Zhan Q H Wu C Guo and X Zhou ldquoEconomic dispatchwith non-smooth objectives part I local minimum analysisrdquoIEEE Transactions on Power Systems vol 30 no 2 pp 710ndash7212015
[40] A K Barisal andR C Prusty ldquoLarge scale economic dispatch ofpower systems using oppositional invasive weed optimizationrdquoApplied Soft Computing vol 29 pp 122ndash137 2015
[41] M Basu ldquoImproved differential evolution for economic dis-patchrdquo International Journal of Electrical Power amp EnergySystems vol 63 pp 855ndash861 2014
[42] V S Aragon S C Esquivel and C C Coello ldquoAn immunealgorithm with power redistribution for solving economicdispatch problemsrdquo Information Sciences vol 295 pp 609ndash6322015
[43] M Ghasemi M Taghizadeh S Ghavidel and A AbbasianldquoColonial competitive differential evolution an experimentalstudy for optimal economic load dispatchrdquo Applied Soft Com-puting vol 40 pp 342ndash363 2016
[44] B R Adarsh T Raghunathan T Jayabarathi and X-S YangldquoEconomic dispatch using chaotic bat algorithmrdquo Energy vol96 pp 666ndash675 2016
[45] N Ghorbani and E Babaei ldquoExchange market algorithm foreconomic load dispatchrdquo International Journal of ElectricalPower amp Energy Systems vol 75 pp 19ndash27 2016
[46] J X V Neto G Reynoso-Meza T H Ruppel V C Marianiand L D S Coelho ldquoSolving non-smooth economic dispatchby a new combination of continuous GRASP algorithm anddifferential evolutionrdquo International Journal of Electrical Poweramp Energy Systems vol 84 pp 13ndash24 2017
[47] A Kavousi-Fard and A Khosravi ldquoAn intelligent 120579-ModifiedBat Algorithm to solve the non-convex economic dispatchproblem considering practical constraintsrdquoElectrical Power andEnergy Systems vol 82 pp 189ndash196 2016
[48] M A Al-Betar M A Awadallah A T Khader and A L BolajildquoTournament-based harmony search algorithm for non-convexeconomic load dispatch problemrdquo Applied Soft Computing vol47 pp 449ndash459 2016
[49] M A Al-Betar M A Awadallah I A Doush E Alsukhni andH ALkhraisat ldquoA non-convex economic dispatchproblemwithvalve loading effect using a new modified 120573-hill climbing localsearch algorithmrdquo Arabian Journal for Science and Engineeringpp 1ndash8 2018
[50] M A Awadallah M A Al-Betar A L Bolaji E M Alsukhniand H Al-Zoubi ldquoNatural selection methods for artificial beecolony with new versions of onlooker beerdquo Soft Computing pp1ndash40 2018
[51] S H Kaboli and A K Alqallaf ldquoSolving non-convex economicload dispatch problem via artificial cooperative search algo-rithmrdquo Expert Systems with Applications vol 128 pp 14ndash272019
[52] D Singh and J Dhillon ldquoAmeliorated greywolf optimization foreconomic load dispatch problemrdquo Energy vol 169 pp 398ndash4192019
[53] X S Yang and S Deb ldquoCuckoo search via Levy flightsrdquo inProceedings of the World Congress on Nature and BiologicallyInspired Computing (NaBIC rsquo09) pp 210ndash214 India 2009
[54] S Sahoo K M Dash R C Prusty and A K Barisal ldquoCom-parative analysis of optimal load dispatch through evolutionaryalgorithmsrdquo Ain Shams Engineering Journal vol 6 pp 107ndash1202015
[55] M Basu and A Chowdhury ldquoCuckoo search algorithm foreconomic dispatchrdquo Energy vol 60 pp 99ndash108 2013
[56] A B Serapiao ldquoCuckoo search for solving economic dispatchload problemrdquo Intelligent Control and Automation vol 04 no04 pp 385ndash390 2013
[57] A H Bindu and M D Reddy ldquoEconomic load dispatch usingcuckoo search algorithmrdquo International Journal of EngineeringResearch and Applications vol 3 pp 498ndash502 2013
[58] E Afzalan and M Joorabian ldquoAn improved cuckoo searchalgorithm for power economic load dispatchrdquo InternationalTransactions on Electrical Energy Systems vol 25 no 6 pp 958ndash975 2015
[59] T T Nguyen and D N Vo ldquoThe application of one rank cuckoosearch algorithm for solving economic load dispatch problemsrdquoApplied Soft Computing vol 37 pp 763ndash773 2015
[60] S M Islam S Das S Ghosh S Roy and P N Suganthan ldquoAnadaptive differential evolution algorithm with novel mutationand crossover strategies for global numerical optimizationrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 42 no 2 pp 482ndash499 2012
[61] Z L Gaing ldquoParticle swarm optimization to solving theeconomic dispatch considering the generator constraintsrdquo IEEETransactions on Power Systems vol 18 no 3 pp 1187ndash1195 2003
[62] X-S Yang and S Deb ldquoEngineering optimisation by Cuckoosearchrdquo International Journal of Mathematical Modelling andNumerical Optimisation vol 330 pp 43ndash51 2010
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Mathematical Problems in Engineering 3
the numerical results section we will report the verificationof the two questionable issues
In this paper we have proposed an improved cuckoosearch algorithm (ICSA) for dealing with large scale OLDproblem with the consideration of complicated constraintstogether with nondifferentiable fuel cost function In theproposed ICSA approach some newmodifications have beenperformed on conventional cuckoo search algorithm (CCSA)to improve the quality of CCSA The CCSA method wasfirst developed in 2009 [53] for solving a set of popularbenchmark functions and its highly superior performanceover PSO and GA has attracted a huge number of researchersin learning and applying for different optimization problemsin different fields Furthermore its improved variants are alsoan extremely vast number In relation toOLDproblemCCSAhas been applied and presented in [23 54ndash57] meanwhileits improved methods consisting of modified cuckoo searchalgorithm (MCSA) and improved cuckoo search algorithmwith one solution evaluation (OSE-CSA) have been respec-tively presented in [58 59] In [55] Basu has applied CCSAfor solving OLD problem with 40-unit system with singlefuel option and the effective of valve loading process 20-unitsystem with single fuel and quadratic fuel cost function and10-unit system with multiple fuel sources and without theeffects of valve loading process The author has made a bigeffort in demonstrating the high potential search of CCSAby comparing with many popular metaheuristic algorithmsbut the shortcoming of the study was neglecting complicatedconstraints and large scale systems
Studies in [56 57] have dealt with OLD problem withtwo power systems considering simple constraints and smallnumber of units Only three simplest constraints such aspower balance limitations of generation and prohibitedpower zones have been taken into account meanwhilethe largest system was solved to be 6-unit system Thusthere were few methods compared to CCSA and the realperformance of CCSA was not shown persuasively in thestudies In [23] authors have applied CCSA for solving differ-ent systems with very complicated constraints complicatedcharacteristics of thermal generating units and high numberof units Among the mentioned studies regarding CCSA forOLD problem authors in [23] could show the best view inevaluating the real performance of CCSA since there were sixcases that were carried out and a huge number of methodswere compared to CCSA In spite of the real potential searchability CCSA has been commented to be low convergence toglobal optimum and significantly improved better [58 59]MCSA in [58] has been proposed by using a new strategyfor the second generation technique Themutation operationin CCSA has been replaced with current-to-best1 model ofDE in [60] MCSA has been applied for solving four systemswith 3 6 15 and 40 units in which the most complicatedconstraint considered was prohibited power zone and onlysingle fuel source was taken into account MCSA methodhas been compared to other popular methods such as PSOGA and EP But the comparisons with CCSA have notbeen carried out Thus the improvement of such proposed
method in [58] was not proved persuasively OSE-CSA in[59] has canceled one evaluation time in case that OSE-CSAhas continued to improve solution quality The improvementseemed to be appropriate for CCSA in dealing with OLDproblem with complicated systems CCSA in [23 55] havebeen considered for comparison in [59] and they have beenproved to be less effective than OSE-CSA However OSE-CSA has used one more control parameter called one rankparameter and it needed to be tuned thoroughly for obtaininghigh performance In the proposed ICSA approach we havefocused on a new strategy of the second new solutiongeneration in CCSA method As shown in [58 59] CCSAhas become a strong search method thanks to the first newsolution generation which was performed by Levy flighttechnique while the second new solution generation couldnot take on local search function well In the second updateprogress via mutation operator two random old solutions areused to generate an increased step size However the mannercan lead to new low quality solutions because the increasedstep will be very small when iterative algorithm is carryingout at the last iterations In fact current solutions at finalseveral iterations tend to be close together and the differentvalues between each two ones are very small leading to a verysmall increased step In order to tackle the disadvantage of theCCSA we apply a new adaptive technique for improvementof solution quality Firstly we propose twoways for producingthe increased step including two-solution-based increasedstep and four-solution-based increased step The decisionwhen which step size will be used is dependent on theresult of comparison between fitness function ratio (FFR)and a predetermined parameter 119879119900119897 FFR is defined as aratio of deviation between fitness functions of the consideredsolution and the most promising solution to the fitness valueof the best one meanwhile 119879119900119897 is a boundary to give the finaldecision for the selection of a used step size At the beginning119879119900119897 is a fixed value for all solutions and then it will be adaptivebased on the comparison between it and FFR When FFR of asolution is less than 119879119900119897 119879119900119897 of the solution will be decreasedequally to ninety percent of the previous valueOtherwise thevalue of 119879119900119897 remains unchanged in case the FFR is equal toor higher than 119879119900119897 The adaptive technique has a significantlyimportant role in enhancing the potential search ability of theproposed method This proposed method is investigated onsix cases with different considered constraints different typesof fuel cost function and large scale systemsThe detail of thesix cases is as follows
Case 1 Four systems with single fuel source (SFS) and powerloss (PL) constraint
Case 2 A 110-unit system with SFS
Case 3 Four systems with SFS and the effects of valve loadingprocess (EoVLP)
Case 4 Two systems with SFS and PPZ and RPR constraints
Case 5 A 15-unit system with SFS and RRL PPZ and PLconstraints
4 Mathematical Problems in Engineering
Case 6 Three systems with multiple fuel sources (MFS) andEoVLP
The achieved results in terms of minimum fuel costaverage fuel cost maximum fuel cost and standard deviationfound by the proposed method compared to those obtainedby others reveal that the method is very efficient for theOLD problem In addition the performance improvementof the proposed method over CCSA is also investigatedvia the comparison of the best solution and all trial runsIn summary the main advantages of the proposed ICSAapproach over CCSA as well as the main contribution of thestudy are as follows
(i) Based on fitness function of each considered solutionlocal search or global search is decided to be appliedmore effectively
(ii) Find better solutions with smaller number of itera-tions and shorter execution time for each run
(iii) Shorten simulation time for the whole search of eachstudy case
However the proposed method also copes with the sameshortcomings as CSA Although the shortcomings do notcause bad results for the proposed method they make theproposed method be time consuming in tuning optimalparameter The shortcomings are analyzed as follows
(i) Control parameter probability of replacing controlvariables in each old solution must be tuned in rangebetween 0 and 1 There is no proper theory for deter-mining the most effective values of the parameterThus the performance of the proposed method mustbe tried by setting the parameter to values from 01 to1
(ii) The method uses more computation steps for searchprocessThus the proposedmethod uses higher num-ber of computation steps for each iteration Howeverdue to more effective search ability for each iterationthe proposed method can use smaller number ofiterations but it finds more effective solutions
The remaining parts of the paper are arranged as fol-lows Section 2 shows the objective and constraints of theconsidered OLD problem CCSA and the proposed methodare clearly explained in Section 3 Section 4 is in charge ofpresenting the implementation of ICSA method for the stud-ied problem The simulation results together with analysisand discussions are given in Section 5 Finally conclusion issummarized in Section 6 In addition appendix is also addedfor showing found solutions by the proposed ICSA approachfor test cases
2 Optimal Load DispatchProblem Description
21 Fuel Cost Function Forms with Single Fuel Source In theconsidered OLD problem the optimal operation of a set of
thermal generation units is concerned as the duty of reducingtotal cost of all the units which can be seen by the followingmodel
Reduce 119865 = 119873sum119894=1
119865119894 (119875119894) (1)
In traditional OLD problem fuel cost function of the119894119905ℎ generation unit 119865119894(119875119894) is represented as the second orderfunction with respect to real power output and coefficients asthe model below [2]
In addition for the case considering the effects ofvalve loading process on thermal generation units fuel costbecomes more complicated by adding sinusoidal term asbelow [12]
119865119894 (119875119894) = 1205721198941198752119894 + 120582119894119875119894 + 120575119894+ 1003816100381610038161003816120573119894 times sin (120574119894 times (119875119894min minus 119875119894))1003816100381610038161003816 (3)
Real Power Balance Constraint Total real power demand ofall loads in power system together with real power loss in allconductors must be equal to the generation from all availablethermal generation units The requirement is constrained bythe following equality
119873sum119894=1
119875119894 = 119875119863 + 119875119871 (4)
where total real power loss 119875119871 is determined by Kronrsquosequation below
119875119871 = 11986100 + 119873sum119895=1
1198610119895119875119895 + 119873sum119895=1
119873sum119894=1
119875119895119861119895119894119875119894 (5)
GenerationBoundaryConstraint For the purpose of economyand safe operation each thermal generation unit is con-strained by the lower generation bound and upper generationbound as the following model
119875119894min le 119875119894 le 119875119894max (6)
22 Fuel Cost Function Forms with Multi-Fuel Sources Inthis section fuel cost function of thermal generation units ismathematicallymodeled in terms of different forms from thatin the section above due to the consideration of multi-fuelsources Each type of fuel source is formed as each secondorder function and the fuel cost function form is the sumof different second order functions for the case of neglectingthe effects of valve loading progress But for the considerationcase of the effects the form ismore complexwith the presenceof sinusoidal terms [15] As a result the forms of cost functioncan be expressed in Equation (7) [21] and Equation (8) [15]
Mathematical Problems in Engineering 5
119865119894 (119875119894) =
1205751198941 + 1205821198941119875119894 + 12057211989411198752119894 fuel 1 119875119894min le 119875119894 le 11987511989411198981198861199091205751198942 + 1205821198942119875119894 + 12057211989421198752119894 fuel 2 1198751198942min le 119875119894 le 1198751198942max120575119894119895 + 120582119894119895119875119894 + 1205721198941198951198752119894 fuel 119895 119875119894119895min le 119875119894 le 119875119894119895max
(7)
119865119894 (119875119894) =
1205751198941 + 1205821198941119875119894 + 12057211989411198752119894 + 10038161003816100381610038161205731198941 times sin (1205741198941 times (119875119894min minus 119875119894))1003816100381610038161003816 for fuel 1 119875119894min le 119875119894 le 11987511989411198981198861199091205751198942 + 1205821198942119875119894 + 12057211989421198752119894 + 10038161003816100381610038161205731198942 times sin (1205741198942 times (119875119894min minus 119875119894))1003816100381610038161003816 for fuel 2 1198751198942min le 119875119894 le 1198751198942119898119886119909 119895 = 1 119898119894120575119894119895 + 120582119894119895119875119894 + 1205721198941198951198752119894 + 10038161003816100381610038161003816120573119894119895 times sin (120574119894119895 times (119875119894min minus 119875119894))10038161003816100381610038161003816 for fuel 119895 119875119894119895min le 119875119894 le 119875119894119895max
(8)
Cost function forms in Equations (7) and (8) are onlyincluded in objective function (1) meanwhile main con-straints in formulas (4) and (6) must be always satisfied
23 Prohibited Power Zone Real Power Reserve and RampRate Limit Constraints Prohibited power zones (PPZ) aredifferent ranges of power in fuel cost function that thermalgeneration units are not allowed to work due to operationprocess of steam or gas valves in their shaft bearing Thepower generation of units in the violated zones is harmfulto gas or steam turbines even destroyed shaft bearing Thusthe constraint is strictly observed In the fuel-power charac-teristic curve of generation units PPZ causes small violationzones and such curves become discontinuous As consideringPPZ constraint the determination of power generation ofunits is more complex and equal to either lower bound orupper bound Unlike PPZ constraint RPR constraint is notrelated to fuel-power feature curve but it causes difficulty foroptimization approaches in satisfying one more inequalityconstraint Each generation unit among the set of availablegeneration units must reserve real power so that the sumof real power from all generation units can be higher orequal to the requirement of power system for the purposeof stabilizing power system in case that there are some unitsstopping producing electricity On the contrary to PPZ con-straint ramp rate limit (RRL) constraint does not allowpoweroutput of thermal generating units outside a predeterminedrange The constraint considers maximum power change ofeach thermal generating unit as compared to the previouspower value Thus optimal generation must satisfy the RRLconstraint The PPZ constraint RPR constraint and RRLconstraint can be presented as follows
Prohibited Power Zones As considering PPZ constraint validworking zones of each thermal generating unit are notcontinuous and its generation must be outside the violatedzones as the following mathematical description
119875119894 isin
119875119894min le 119875119894 le 1198751198971198941119875119906119894119896minus1 le 119875119894 le 119875119897119894119896 k = 2 ni forall119894 isin Ω119875119906119894119899119894 le 119875119894 le 119875119894max
(9)
As observing Equation (9) generation units cannotbe operated within the violated zones except for startingpoint and end point Consequently the verification of PPZconstraint violation should be carried out first and thenthe correction should be done before dealing with otherconstraints such as real power reserve constraint and realpower balance Besides if power output of all units can satisfythe PPZ constraint generation limits in Equation (6) are alsoexactly met
Real Power Reserve Constraint Real power reserve in powersystem aims to enhance the ability of stability recovery ofpower system and avoid blackout In order to get high enoughpower for requirement all available units are constrained bythe following inequality
119873sum119894=1
119878119894 ge 119878119877 (10)
where 119878119894 is the real power reserve contribution of the 119894119905ℎthermal generation unit and the determination of 119878119894 can bedone by employing the two models below
119878119894 = 119875119894max minus 119875119894 119894119891 119878119894max gt (119875119894max minus 119875119894)119878119894max else
forall119894 notin Ω (11)
119878119894 = 0 forall119894 isin Ω (12)
Equation (10) shows that the constraint of prohibitedpower zones is not included in the real power reserveconstraints however prohibited power zones are alwaysstrictly considered and must be exactly satisfied
Ramp Rate Limit (RRL) Constraint In OLD problem allconsidered thermal generating units are supposed to be underworking status but previous active power of each thermalgenerating unit is not taken into account Thus increased ordecreased power is not constrained This assumption seemsto be not practical until RRL constraint is considered RRLconstraint considers initial power output and the power
6 Mathematical Problems in Engineering
change is supervised Regulated power can be higher or lowerthan the initial value as long as it is within a predeterminedrange Increased step size (ISS) and decreased step size (DSS)are given as input data and they are used to limit the change ofpower output of each thermal generating unit The constraintcan bemathematically expressed as the following formula [7]
1198751198940 minus 119863119878119878119894 le 119875119894 le 1198751198940 + 119868119878119878119894 (13)
where 1198751198940 is the initial power output of the 119894119905ℎ thermalgenerating unit before its power output is regulated 119868119878119878119894 and119863119878119878119894 are respectively maximum increased and decreasedstep sizes of the 119894119905ℎ thermal generating unit
3 The Proposed Cuckoo Search Algorithm
31 Classical Cuckoo Search Algorithm In search techniqueof CCSA [53] a set of solutions is randomly generated withina predetermined range in the first step and then the quality ofeach one is ranked by computing value of fitness functionThemost effective solution corresponding to the smallest valueof fitness function is determined and then search procedurecomes into a loop algorithm until the maximum iterationis reached In the loop algorithm two techniques updatingnew solutions two times (corresponding to two generations)are Levy flights and mutation technique which is calledstrange eggs identification technique The two generationscan produce promising quality solutions for CCSA Aftereach generation CCSA will carry out comparing fitness ofnewly updated solutions and initial solutions for keepingbetter ones and abandoning worse ones The most effectivesolution at last step of the loop search algorithm is determinedand it is restored as one candidate solution for a study caseThe detail of the two stages is as follows
311 Levy Flights Stage This is the first calculation step in theloop algorithm and it also produces new solutions in the firstgeneration for CCSA New solution 119878119900119897119899119890119908119909 is created by thefollowing model
119878119900119897119899119890119908119909 = 119878119900119897119909 + 120572 (119878119900119897119909 minus 119878119900119897119866119887119890119904119905) oplus Levy (120573) (14)
where 120572 is the positive scaling factor and it is nearly set todifferent values for different problems in the studies [53 62]In the work the most appropriate values for such factor canbe chosen to be 02505 for different systems
312 Discovery of Alien Eggs Stage The step plays a veryimportant role for updating new solutions 119878119900119897119899119890119908119909 of thewhole population However not every control variable ineach old solution is newly updated and the decision ofreplacement is dependent on comparison criteria as thefollowing equation
119878119900119897119899119890119908119909=
119878119900119897119909 + 1205761 (1198781199001198971199031198861198991198891 minus 1198781199001198971199031198861198991198892) if 1205762 lt 119875119886119878119900119897119909 119900119905ℎ119890119903119908119894119904119890
(15)
32 Proposed Algorithm In the part a new variant of CCSA(ICSA) is constructed by applying three effective changes onthe main functions of CCSA in order to shorten simula-tion time corresponding to reduction of iterations and findmore promising solutions The proposed amendments areexplained in detail as follows
(i) Suggest one more equation producing updated stepsize in addition to existing one in CCSA
(ii) Create a new selection standard by computing fitnessfunction ratio 119865119865119877119909 and comparing 119865119865119877119909 with apredetermined parameter 119879119900119897119909 Thus thanks to thestandard the existing updated step size and additionalupdate step size will be chosen more effectively
(iii) Automatically change value of 119879119900119897119909 for the xth solu-tion based on the result of comparing 119865119865119877119909 with theprevious 119879119900119897119909
Such three points are clarified by observing the followingsections
321 Strange Eggs Identification Technique (Mutation Tech-nique) The first proposed improvement in our proposedICSA approach is to select a more suitable formula forproducing new solutions with better fitness function valueIn CCSA Equation (16) below is used to produce a changingstep nearby old solutions for all current solutions
Δ1198781199001198971198991198901199081199091 = 1205763 (1198781199001198971199031198861198991198891 minus 1198781199001198971199031198861198991198892) (16)
The use of Equation (16) aims to produce a random walkaround old solutions in search zones with intent to findout promising solutions In order to reduce the possibilityof suffering the local trap and approach to other favorablezones for searching we propose a new Equation (17) Theformula is built by the idea of enlarging search zone withthe use of two more available solutions Obviously the largerchanging step can own higher performance in moving toother search spaces that the classical approach used in CCSAThe suggestion is mathematically expressed by the formulabelow
Δ1198781199001198971198991198901199081199092= 1205764 (1198781199001198971199031198861198991198891 minus 1198781199001198971199031198861198991198892 + 1198781199001198971199031198861198991198893 minus 1198781199001198971199031198861198991198894) (17)
The changing step obtained by using Eq (17) is namedfour-point changing step Now two solutions which arenewly formed by using two different changing steps shownin formulas (16) and (17) are found by the two followingmethods
It can be clearly observed that the distance between 119878119900119897119909(old solution) and 1198781199001198971198991198901199081199091 (new solution) is lower than thatbetween 119878119900119897119909 and 1198781199001198971198991198901199081199092 This difference can contribute ahighly efficient improvement to the proposed ICSA approachsearch ability
Mathematical Problems in Engineering 7
ΔSol2
Sol2
Sol3
Sol4
ΔSol1
Sol1
Solx
Solnew1
Solnew2
Figure 1 Simulation of solutions corresponding to the first itera-tions of the loop algorithm
For the CCSA case if two solutions 1198781199001198971199031198861198991198891 and 1198781199001198971199031198861198991198892are either slightly different or completely coincident suchnewly updated solution 1198781199001198971198991198901199081199091 does not have good chanceto leave the current zone and approach to more promisingzones In another word the new one is approximately coin-cident with the old one As the search task is taking place atsome last iterations this phenomenon becomes much worsebecause all current solutions are lumped in a small zone andthe capability of moving to other zones is impossible As aresult the CCSA approach will work ineffectively and searchstrategy is time consuming until other runs are started
Contrary to the two-point step size the new proposedformula may produce a large enough length to escape thelocal optimum zone and reach new favorable zones Itexplainswhy the four-point changing step has positive impacton the considered random walk rather than the two-pointchanging step
322 New Standard forChoosing theMostAppropriate Chang-ing Step In this section we extend our analysis to answer thequestionwhen to use the four-point step size FromEquations(18) and (19) two new solutions which are represented asthe results of the two-point-based factor and the four-pointstep size can be illustrated by using Figure 1 corresponding tothe search process at the first some iterations and Figure 2corresponding to the last some iterations For the sake ofsimplicity we rewrite the two equations as follows
Here we suppose that 1198781199001198971 and 1198781199001198972 are obtained byfour exact solutions 1198781199001198971 1198781199001198972 1198781199001198973 and 1198781199001198974 and calculatedas follows
ΔSol2
ΔSol1
Solx
Solnew1
Solnew2
Figure 2 Simulation of solutions corresponding to the last itera-tions of the loop algorithm
Δ1198781199001198971 = 1198781199001198971 minus 1198781199001198972 (22)
Δ1198781199001198972 = 1198781199001198973 minus 1198781199001198974 (23)
Asmentioned above the high changing step between newsolution and old solution can help to explore new favorablezones However in optimization algorithms searching stepscannot be arbitrarily large otherwise the algorithm maydiverge in particular for the cases that the consideredsolutions 119878119900119897119909 are not close together in solution search spaceFor example at the beginning of loop algorithm with thefirst iterations in Figure 1 1198781199001198971198991198901199081 is a better choice than1198781199001198971198991198901199082 because it is kept in a sufficient limit and does notlead to a risk of divergence In contrast as many of currentsolutions are in different positions but their distance is notvery short or approximately coincident such as at the lastiterations in Figure 2 1198781199001198971198991198901199081 and 119878119900119897119909 have a very shortdistance but 1198781199001198971198991198901199082 and 119878119900119897119909 have higher distance Accordingto the phenomenon in Figure 2 the proposed ICSA approachneeds to produce a high changing step to move solutions toother search zones without local optimum Hence 1198781199001198971198991198901199082would be preferred to 1198781199001198971198991198901199081
Based on the argument above the determination of thecondition for using either two-point changing step or four-point changing step is really crucial to the performance ofthe proposed ICSA approach in searching solutions of OLDproblem Here the ratio of 119865119865119877119909 which can be found byEquation (24) is suggested to be a suitable measurement forthe selection of two options
Δ119865119865119877119909 = 119865119865119909 minus 119865119865119887119890119904119905119865119865119887119890119904119905 (24)
For a particular set of the current solutions each individ-ual depending on its 119865119865119877119909 will create a corresponding newsolution by using either Equation (18) or (19) If the valueof one current solution is smaller than the predeterminedparameter 119879119900119897 Equation (19) is applied for updating suchconsidered solution 119909 Otherwise Equation (18) is a betteroption The steps of the modified algorithm are similar to the
8 Mathematical Problems in Engineering
If 1205765 lt 119875119886If FFRx lt Tolx119878119900119897119899119890119908119909 = 119878119900119897119909 + 1205764(1198781199001198971199031198861198991198891 minus 1198781199001198971199031198861198991198892 + 1198781199001198971199031198861198991198893 minus 1198781199001198971199031198861198991198894)else119878119900119897119899119890119908119909 = 119878119900119897119909 + 1205764(1198781199001198971199031198861198991198891 minus 1198781199001198971199031198861198991198892)end
Algorithm 1 New mutation technique applied in the proposed ICSA approach
conventional CSA except that an additional step should beadded at each iteration In this step the119865119865119889 of all individualsolutions should be calculated by utilizing Equation (24) andthen the result of comparing the ratio with 119879119900119897will be used todecidewhich updating formula should be selectedThewholedescription of the proposed standard and new mutationtechnique can be coded inMatlab program language by usingAlgorithm 1
323 Adjustment of Tolerance for Each Solution As pointedout above the proposed method needs assistances to deter-mine the most appropriate step size for finding out favorablesolution zones The given aim can be reached if the selectionof 119879119900119897119909 is reasonable however the range of this parameteris infinite and hard to select Thus the adaptation of tuningthe parameter is really necessary First of all the compari-son between 119879119900119897119909 and 119865119865119877119909 is carried out and then theadaptation will be determined based on the obtained resultfrom the comparison Results of comparison between the twoparameters can be either 119865119865119877119909 is less than 119879119900119897119909 or 119865119865119877119909is higher than 119879119900119897119909 The case that two parameters are equalhardly ever occurs
As the comer assumptionhappens (ie119865119865119877119909 is less than119879119900119897119909) at the considered time the four-point step size will beemployed for the 119909119905ℎ solution If 119879119900119897119909 remains unchanged atthe previous value the identification of improvement fromsuch four-point step size or two-point step size is vagueConsequently value of 119879119900119897119909 must be automatically reducedto a lower value in case that it has significant contribution tofound promising solution of previous iteration Clearly thedecrease of119879119900119897119909 can enable the proposedmethod to jump outlocal optimal zone and approachmore effective zones By trialand error method 119879119900119897119909 is selected to be a function of itselfthat is 09 of the previous value Finally the implementationof the proposed ICSA approach is presented in Algorithm 2
4 The Application of the ProposedICSA for OLD Problem
Thewhole computation steps of the proposed ICSA approachfor solving OLD problem are explained as follows
41 Handling Constraints and Randomly Producing InitialPopulation As shown in Section 2 the considered OLDproblem takes five following constraints into account
(i) Power balance constraint is shown in Equation (4)
(ii) Power output limitation constraint is shown in Equa-tion (6)
(iii) Prohibited power zone constraint is shown in Equa-tion (9)
(iv) Real power reserve constraint is shown in Equation(10)
(v) Ramp rate limit constraint is shown in Equation (13)
Among the five constraints ramp rate limit generationlimit and prohibited power zone seem to be more com-plicated than power balance and power reserve constraintsHowever the three constraints can be solved more easilybecause each unit is constrained independently in the threeconstraints whereas power balance constraint and powerreserve constraint consider all the thermal generating unitssimultaneously Power reserve constraint can be handledby penalizing the total generation of all units while powerbalance constraint can be solved by penalizing one violatedthermal generating unit The whole computation procedurefor solving all constraints and calculating fitness function ofsolutions is described in detail as follows
Step 1 Redefine maximum and minimum power output ofeach thermal generating unit as considering PPZ and RRLconstraints by using the following formulas
119875119894max = 119875119894max if 119875119894max le 119875i0 + 119868119878119878119894119875i0 + 119868119878119878119894 if 119875119894max gt 119875i0 + 119868119878119878119894
119894 = 1 119873(25)
119875119894min = 119875119894min if 119875119894min ge 119875i0 minus 119863119878119878119894119875i0 minus 119863119878119878119894 119890119897119904119890
119894 = 1 119873(26)
Mathematical Problems in Engineering 9
Produce initial population with119873119901119904 solutions (1198781199001198971 1198781199001198972 119878119900119897119909 119878119900119897119873119901119904)Calculate fitness function (1198651198651 1198651198652 119865119865119909 119865119865119873119901)Go to the loop algorithm by setting 119866 = 1
While (119866119898119886119909 gt 119866) (i) The first newly produced solutions119878119900119897119899119890119908119909 = 119878119900119897119909 + 120572(119878119900119897119909 minus 119878119900119897119866119887119890119904119905) oplus Levy(120573) (ii) Perform selection approach
119865119865119909 = 119865119865119909 if 119865119865119909 le 119865119865119899119890119908119909119865119865119899119890119908119909 119900119905ℎ119890119903119908119894119904119890 119909 = 1 119873119901119904
(v) Determine the most effective solution and its fitnessDetermine 119865119865119909 with the smallest value and assign 119878119900119897119909 to 119878119900119897119866119887119890119904119905If 119866119898119886119909 gt 119866 perform step (i) and increase 119866 to 119866 + 1 Otherwise stop the loop algorithm and report boththe smallest fitness together with 119878119900119897119866119887119890119904119905End while
Among the four Equations (25) and (26) are used firstin order to redefine upper bound and lower bound for allthermal generating units as considering RRL constraint Thethe redefined bounds continue to be redefined for the secondtime by using (27) and (28) as considering PPZ constraints
Step 2 (randomly produce initial population) For dealingwith the power balance constraint all available units areseparated into two groups in which the first group withdecision variables consists of the power output from thesecond unit to the last unit (P2 P3 PN) meanwhile onlythe power output of the first unit (1198751) belongs to the secondgroup with dependent variable So upper bound solution119878119900119897119898119886119909 and lower bound solution 119878119900119897119898119894119899 must be defined asfollows
Step 3 Handle prohibited power zone constraint for decisionvariables P2 P3 PN
After being randomly produced there is a high possi-bility that decision variables fall into PPZ and they violatePPZ constraint So the verification of falling into PPZ andcorrection of the violation should be accomplished by usingthe following formula
119875119894 =
119875119897119894119896 if 119875119897119894119896 lt 119875119894 le 119875119897119894119896 + 1198751198961198941198962119875119906119894119896 if (119875119894 gt 119875119897119894119896 + 1198751198961198941198962 ) amp (119875119894 lt 119875119906119894119896)119875119894 119890119897119904119890
119894 = 2 119873 amp 119896 = 1 119899119894
(31)
Step 4 Handle RPB constraint by calculating 1198751 and penaliz-ing 1198751 if it violates constraints
In this step power balance constraint is exactly handledby calculating and penalizing dependent variable (1198751) 1198751 isobtained by using formulas (4) and (5) as follows
1198751 = minus (11986101 minus 1 + 2sum119873119894=2 1198611119894119875119894) plusmn radicΔ211986111 (32)
where
Δ = (11986101 minus 1 + 2 119873sum119894=2
1198611119894119875119899)2 minus 411986111(119875119863 minus 119873sum
119894=2
119875119894+ 11986100 + 119873sum
119894=2
1198610119894119875119894 + 119873sum119894=2
119873sum119895=2
119875119894119861119894119895119875119895) amp Δ ge 0(33)
In Equation (32) 1198751 has been determined for the purposeof dealing with real power balance constraint However it isnot sure that 1198751 can satisfy upper bound and lower boundconstraints and prohibited power zone constraints So 1198751must be checked and penalized
Firstly 1198751 is checked and penalized for upper and lowerbound constraints by the following model
Δ1198751x =
0 if 1198751min le 1198751x le 1198751max
1198751min minus 1198751x if 1198751min gt 1198751x1198751x minus 1198751max if 1198751max lt 1198751x
(34)
In Equation (34) if the second case or the third caseoccurs it means P1 has violated either lower bound or upperbound and it would be penalized by using either (P1x= P1min-P1x) or (P1x =P1x -P1max) Otherwise ifP1 has not violatedthe bound constraints (ie the first case in (34) happened)
P1 would continue to be checked for PPZ constraint by thefollowing model
Δ1198751x
=
1198751 minus 1198751198971119896 if 1198751198971119896 lt 1198751 le 1198751198971119896 + 119875119896111989621198751199061119896 minus 1198751 if (1198751 gt 1198751198971119896 + 11987511989611198962 ) amp (1198751 lt 1198751199061119896)0 119890119897119904119890
(35)
Step 5 Handle real power reserve constraint (10)First of all 119878119894 is determined by using (11) and (12) and
then the 119909119905ℎ solution will be checked and penalized if poweroutput of all thermal generating units cannot satisfy RPRconstraint The penalty for violation of the constraint can becalculated by using equation (36)
Δ119878119877119909 =
0 if119873sum119894=1
119878119894119909 ge 119878119877119878119877 minus 119873sum119894=1
119878119894119909 119890119897119904119890 (36)
As a result real power reserve constraint can be solved byusing the penalty method
42 Calculate Fitness Function for Solutions Fitness functionof each solution is used to evaluate quality of solutionNormally the function is the sum of objective function andpenalty of violating constraints and is obtained by
43 The First Newly Updated Solutions by Levy Flights Tech-nique In this section the first newly updated solutionsare performed by employing Levy flights technique usingEquation (14) However each new solution can be out oftheir feasible operating zone such as PPZ and upper andlower limitations When the power output violates its PPZconstraints Equation (31) will be applied to tackle theconstraint Besides the following equation will be employedwhen power output is higher or lower than their limitations
119878119900119897119909 =
119878119900119897max if 119878119900119897max lt 119878119900119897119909119878119900119897min if 119878119900119897min gt 119878119900119897119909119878119900119897119909 Otherwise
119909 = 1 119873119901 (38)
After that Equations (32)-(37) are performed for deter-mining all variables and penalty terms Finally Equation (38)is employed to calculate fitness function
44 The Second Newly Updated Solutions by Using Muta-tion Technique The second newly updated solutions areaccomplished as presented in Section 3 above Similar to
Mathematical Problems in Engineering 11
the task after doing the first update each solution in thenew population must satisfy PPZ constraint and upper andlower boundaries by considering Equations (31) and (38)Then Equations (32)-(37) are performed for determiningall variables and penalty terms Finally Equation (38) isemployed to calculate fitness function and the solution withthe best value is assigned to the best one 11987811990011989711986611988711989011990411990545 Criterion of Stopping the Loop Algorithm In the loopalgorithm of using the proposed ICSA approach the solutionsearch work is stopped in case that the predeterminedmaximum iterations 119866119898119886119909 is reached For each search ter-mination the most effective solution is stored and anotherrun continues to be accomplished until the predeterminednumber of runs is reached After finishing the runs thebest one is found and reported In addition other valuessuch as the fitness of the worst solution and average fitnessof all solutions are also reported for comparing with othermethods
46 The Whole Iterative Process The whole iterative algo-rithm for implementing the proposed ICSA approach forcoping with OLD problem is described in detail in Figure 3
5 Results and Discussions
The proposed ICSA approach performance has been investi-gated on six cases with different fuel options different fuelcharacteristics and complicated constraints The details ofthe studied cases are presented as follows
Case 1 Four systems with single fuel source (SFS) and powerloss (PL) constraint
Subcase 11 A 3-unit system [57]Subcase 12 A 6-unit system [57]Subcase 13 A 3-unit system [56]Subcase 14 A 6-unit system [56]
Case 2 A 110-unit system with SFS [57]
Case 3 Four systems with SFS and the effects of valve loadingprocess (EoVLP)
Subcase 31 A 3-unit system supplying to a load of850MW [58]Subcase 32 A 13-unit system supplying to a load of1800MW [1]Subcase 33 A 13-unit system supplying to a load of2520MW [1]Subcase 34 A 40-unit system supplying to a load of2500MW [1]Subcase 35 An 80-unit system supplying to a load of4100MW [49]
Case 4 Two systems with SFS and PPZ and RPR constraints
Subcase 41 A 60-unit system supplying to a10600MW load [9]
Subcase 42 A 90-unit system supplying to a15900MW load [9]
Case 5 A 15-unit system with SFS and RRL PPZ and PLconstraints [61]
Case 6 Three systems with multiple fuel sources (MFS) andEoVLP
Subcase 61 An 80-unit system supplying to a21600MW load [15]Subcase 62 A 160-unit system supplying to a43200MW load [15]Subcase 63 A 320-unit system supplying to an86400MW load [54]
For each considered case with each load case the pro-posed ICSA approach is run 50 times on the programlanguage of Matlab and a PC with 4 GB of RAM and 24GHzprocessor The selection of adjustment parameters including119875119886 and 119879119900119897119909 is carefully considered to obtain the best optimalsolutions meanwhile two others such as 119873119901119904 and 119866119898119886119909 arechosen corresponding to the scale of particular test system9 values with the change of 01 in the range [01 09] are inturn selected for 119875119886 while 119879119900119897119909 is 001 at the beginning Theinformation including load demand119873119901119904 119866119898119886119909 and the best119875119886 is reported in Table 1
51 Obtained Results on Case 1 considering Four Systems withSFS and PL Constraint In this section we have implementedthe proposed ICSA approach for solving four systems dividedinto four subcases Tables 2 and 3 show the comparisons ofobtained results from Subcases 11 and 12 and Subcases 13and 14 respectively As listed in Table 2 the proposed ICSAmethod and CCSA can find equal fuel cost for Subcases 11whereas the reduction of fuel cost from the proposed ICSAmethod as compared to CCSA is clearer for Subcase 12 Asshown in Table 3 for comparing the proposed ICSA and threemethods consisting of CCSA ABC and FA the minimumfuel cost of the proposed ICSA is approximately equal to thatof these methods for Subcases 13 but much less than that ofthese methods for Subcase 14 Furthermore the proposedICSA has been run by setting 119873119901119904 and 119866119898119886119909 to 5 and 20but these values were much higher for CCSA ABC and FAThey are 20 and 5000 for CCSA 40 and 100 for ABC and 20and 5000 for FA Consequently the proposed method is veryefficient for Case 1 with four subcases
Optimal solutions obtained by ICSA for Case 1 are shownin Tables 16ndash18
52 Obtained Results on Case 2 considering 110-Unit Systemwith SFS In this section we have employed a very largescale system with 110 units but there were not challenges forobjective function and complex constraints since EoVLP andconstraints were not taken into account Both CCSA andthe proposed ICSA methods have been run for comparingwith BBO hybrid BBO and DE (DEBBO) and Opposi-tional real coded chemical reaction optimization algorithm(ORCCROA) in [36] IWA in [40] and AGWO in [52] As
12 Mathematical Problems in Engineering
Select parameters
- Correct solutions if violating PPZ constraint using Eq (31)- Determine using Eq (32)- Penalize for violating upper and lower bounds constraint using Eq (34)- Penalize for violating PPZ constraint using Eq (35)- Penalize the xth solution for violating RPR constraint using Eq (36)
- Determine quality of solutions employing Eq (37) - Select the most effective solution - Start the loop algorithm by selecting
- Perform the first solution update using Section 32 - Correct boundaries of solutions by using Eq (38)
- Correct solutions if violating PPZ constraint using Eq (31)- Determine using Eq (32)- Penalize for violating upper and lower bounds constraint using Eq (34)- Penalize for violating PPZ constraint using Eq (35)- Penalize the xth solution for violating RPR constraint using Eq(36)
- Determine fitness function using Eq (37)- Compare old solutions and new solutions to keep better ones
- Perform the second solution update using Algorithm 1 - Correct boundaries of solutions by using (38)
- Correct solutions if violating PPZ constraint using Eq (31)- Determine using Eq (32)- Penalize for violating upper and lower bounds constraint using Eq (34)- Penalize for violating PPZ constraint using Eq (35)- Penalize the xth solution for violating RPR constraint using Eq (36)
- Determine fitness function using Eq (37)- Compare old solutions and new solutions to keep better ones- Select the most effective solution
Stop
Start
- Redefine upper and lower bounds using Eqs (25)-(28)- Randomly generate initial population using Eq (30)
Nps Pa Gmax H> Tolx
P1xP1x
P1x
P1xP1x
P1x
P1xP1x
P1x
Sol<MN
Sol<MN
G = 1
G = Gmax G = G + 1
Figure 3 All computation steps for solving OLD problem by employing the proposed ICSA approach
shown in Table 4 AGWO [52] has reached less fuel cost thanICSA however the exact fuel cost which was recalculatedby using reported solution pointed out that the method hasreached a very high fuel cost of $2157404250 For comparisonwith other methods ICSA has found less fuel cost thanall these methods Particularly the reduction of generationfuel cost is significant as compared to BBO DEBBO andCCSA Execution time comparisons are also useful evidence
for indicating the high performance of ICSA Thus it canconclude that ICSA is a strong method for Case 2
Optimal solution obtained by ICSA for the case is shownin Table 19
53 Obtained Results on Case 3 considering Four Systems withSFS and EoVLP In this section the real performance of theproposed ICSA approach has been investigated based on five
Mathematical Problems in Engineering 13
Table 1 Information of considered cases and adjustment parameters
Case Fuel cost function Constraint Subcase No of units 119875119863 (MW) Nps Gmax Best 1198751198861 SFS PL
subcases with the gradual increase of number of units Thesmallest scale system considers 3 units but the largest scalesystem takes 80 units In addition to the implementation ofthe proposed ICSA we have also implemented CCSA forSubcase 31 and Subcase 35 for further comparison becauseCCSA has not been run for the two subcases so far
Comparison of obtained results from Subcase 31 shownin Table 5 indicates that the proposed ICSA is superior toCCSAwith lower fuel cost but it seems to be less effective thanMCSA [58] The minimum cost of MCSA reported in [58] isthe smallest fuel cost but the recalculated cost is much higherthan that of the proposed method Furthermore MCSA hasbeen implemented by setting very high values to 119873119901119904 and119866119898119886119909
Reports for Subcases 32 and 33 shown in Table 6are the comparisons of the proposed ICSA approach andother methods such as conventional Evolution programming(CEP) [1] Fast EP (FEP) [1] improved FEP (IFEP) [1] DE[12] multiplier Lagrange-based genetic algorithm with (GA-MU) [15] QPSO [16] GA-PS-SQP [30] PSO-SQP [32] M120573-HCLSA [49] IABCA [50] CCSA [59] OSE-CSA [59] SOS[34] MSOS [34] CEA-SQT [38] TSBO [39] IWA [40] andCBA [44] As observed from the table ICSAapproach obtainsbetter solutions than mostmethods excluding DE [10] CCSA
[59] OSE-CSA [59] SOS [34] MSOS [34] CEA-SQT [38]TSBO [39] IWA [40] and CBA [44] especially M120573-HCLSA[49] with lower cost $1796097 However recalculated costfrom reported solution of M120573-HCLSA is $179691 BesidesICSA is very fast as compared to most methods where twoother versions of Cuckoo search algorithm CCSA and OSE-CSA are also included except two methods in [34] Theprocessor of computer that all the methods run on is alsoreported in the final column Clearly ICSA approach is veryefficient for the case with the 13-unit system where effects ofvalve loading process are considered
In Subcase 34 the number of units is much larger thanthat of three subcases above up to 40 units [1] The obtainedresult comparisons with others are indicated in Table 7Clearly the minimum cost comparisons reveal that the pro-posedmethod is one of the leading methods due to the lowestcost except the comparison with CCSA [23] OSE-CSA [59]SOS [34] MSOS [34] EMA [45] 120579-MBA [47] and AGWOA[52] It is noted that AGWOA [52] has reported the bestminimum cost with $12140430 but recalculated minimumcost which was obtained by substituting reported optimalgeneration of all thermal generating units is $12141331 Theaverage and the maximum costs from the proposed methoddo not belong to the leading method group however the
Mathematical Problems in Engineering 15
Table 7 Result comparisons for Subcase 34
Approach Best cost Average cost Worst cost CPU time Computer($h) ($h) ($h) (s) (Processor-Ram)
execution time that ICSA approach takes is smaller thanmostones except IAPR [42] and 120579-MBA [47] which have been runon stronger computers Note that MSOS [34] has been fasterthan the proposed method for Subcases 31 and 32 abovebut it is too slower than the proposed method for the casenamely 1813 seconds compared to 146 seconds Comparedto two other versions of Cuckoo search CCSA [23] and OSE-CSA [59] the proposed method is also faster about threetimes although the processors are slightly different For thiscase 120579-MBA [47] shows a very good performance howeverthe method has not been tested onmore complicated systemsand larger scale and therefore more comparisons with themethod must end Clearly the proposed method is stillefficient for the case where large scale and effects of valveloading process are included
Subcase 35 is the largest scale system with 80 units Inaddition to the implementation of ICSA CCSA has beenalso run for the Subcase 35 for further investigation ofefficiency improvement of the proposed ICSA approach The
comparisons of minimum cost in Table 8 show that theproposed ICSA can find more optimal solution than M120573-HCLSA [49] AGWOA [52] and CCSA The proposed ICSAis also superior to CCSA in terms ofmore stable search abilityand lower fluctuation since its average cost and maximumcost are less than those of CCSA The outstanding figurecannot be reached as compared to AGWOA [52] howeverit is hard to conclude AGWOA [52] is superior to theproposed ICSA approach about more stable search abilityand lower fluctuation Actually comparison of the values ofpopulation and iterations as well as execution time cannot beaccomplished because the information was not reported in[52]Thus it can conclude that the proposed ICSA is effectivefor the subcase
Optimal solutions obtained by ICSA for the case areshown in Tables 20ndash23
54 Obtained Results on Case 4 with Two Systems consideringSFS and PPZ and RPR Constraints In this section two
16 Mathematical Problems in Engineering
Table 9 Result comparisons for Subcases 41 and 42
Subcase Approach Best cost ($h) Mean cost ($h) Worst cost ($h) Std dev ($h) CPU time (s) Computer (Processor-Ram)
test systems with SFS and PPZ and RPR constraints areconsidered The test system size is up to 60 and 90 units forSubcases 41 and 42 respectively Comparison for the casesis only performed with two Genetic algorithms consisting ofGA and IGA-MU in [9] and two other versions of Cuckoosearch algorithms including CCSA [23] and OSE-CSA [59]and presented in Table 9 Other studies have tended toignore such complicated constraints of PPZ and RPR Thecosts reported in Table 9 indicate that ICSA approach canobtain more effective solution than CCSA and OSE-CSAbecause it has reached lower minimum cost than the twoones Furthermore the proposed method also takes shortercomputation time for the two cases from about two timesto about three times although the processor of the proposedmethod is slightly strongerThemean costs of ICSA approachare much less than those from IGA-MU and GA and slightlyhigher than those from OSE-CSA but there is a trade-off between the proposed method and CCSA for the twosubcases In fact the proposed method obtains higher meancost for Subcase 41 but lower cost for Subcase 42 AlthoughGAmethods have been runon aweak computerwith 07 GHzof the processor compared to that with 24GHz in the studytheir execution times are significantly higher namely 56381seconds (GA) and 16258 seconds (IGA-MU) compared to09153 seconds of ICSA approach for Subcase 41 and 94093seconds (GA) and 25545 seconds (IGA-MU) compared to15892 seconds (the proposedmethod)The analysis can pointout that ICSA approach ismore efficient than these comparedmethods in terms of optimal solutions and execution time
Optimal solutions obtained by ICSA for Subcase 42 areshown in Table 24
55 Obtained Results on Case 5 with a 15-Unit System consid-ering SFS and RRL PPZ and PL Constraints In this section
a 15-unit system considering RRL PPZ and PL constraintsis considered to be solved for finding optimal solution Forefficiency investigation of the proposed ICSA we have alsoimplemented CCSA for comparison As listed in Table 10the proposed ICSA is the most effective method with thesmallest fuel cost The comparisons of control parameters aswell asCPU time are also good evidence to confirm the strongsearch of the proposed ICSA approach since it has been runby smaller values of control parameter and faster executiontime as compared to all methods excluding CCSA
Optimal solution obtained by ICSA for the case is shownin Table 25
56 Obtained Results on Case 6 withThree Systems consideringMFS and EoVLP In this section three test systems with thechallenge on objective function including multi-fossil fuelsources and effects of valve loading process are consideredThe scale is up to 80 units 160 units and 320 units forSubcases 61 62 and 63 respectively
Comparison for Subcase 61 reported in Table 11 revealsthat the proposed method is the best method in terms of thelowest best cost the lowestmean cost and the lowest standarddeviation and the fastest execution time The processor fromthis proposedmethod is about four times stronger thanCGA-MU and IGA-MU but the speed is from ten times to 35 timesfaster than these methods Compared to CCSA and OSE-CSA the proposed method is about two times faster but theprocessor is slightly stronger
Comparison for Subcase 62 is reported in Table 12Clearly the proposed ICSA approach obtains better values ofthe best mean and worst costs than most methods exceptMSOS [34] where the best cost difference is about $ 024However the proposed method is the second fastest onewith 1119 seconds where the first fastest one CBA [44] has
Mathematical Problems in Engineering 17
Table 11 Comparisons of found results for Subcase 61
Approach Best cost Mean cost Worst cost Std dev CPU time Computer($h) ($h) ($h) ($h) (s) (Processor-Ram)
spent 571 seconds Clearly MSOS is better than the proposedICSA approach with respect to slightly less best cost butis worse than the proposed method in terms of executiontime namely 2572 seconds compared to 1119 seconds whileCBA [44] is faster than the proposed method but obtainssignificantly worse costs The analysis can conclude that theproposed ICSA approach is very powerful for the subcasewith 160 units
Table 13 presents the comparison of three other methodsincludingCCSA [54] SOS [34] andMSOS [34] accompaniedwith the proposed method for Subcase 63 The obtainedresult comparisons imply that ICSA approach can obtainbetter values of the best and standard deviation costs thanCCSA and SOS but obtains slightly higher cost than MSOSby approximately $ 022 Besides the execution time fromICSA approach is much shorter than others especially it ishigher than five times faster than MSOS The four methodshave been run on approximately strong computers Brieflythe proposed ICSA approach can find and converge to morefavorable solution than other methods with shorter CPUtime except the comparison with MSOS which had bettersolution but spent higher than five times execution times
Consequently the proposed method is a very promisingoptimization algorithm for Subcase 63 a system up to 320units and with multi-fossil fuel sources and effects of valveloading process
Optimal solution obtained by ICSA for Subcase 63 isshown in Table 26
57 The Improvement of ICSA Approach Performance
571 The Outstanding Improvement over CCSA In this sec-tion the performance improvement of ICSA over CCSAhas been investigated by analyzing obtained results and setcontrol parameters Table 14 has been formed by addingreduction cost improvement level of the best cost executiontime and control parameters consisting of 119873119901119904 and 119866119898119886119909Among the compared factors reduction cost is the deviationof the cost of CCSA and that of ICSA whereas the improve-ment level is the ratio of the reduction cost to the cost ofCCSA The reduction costs indicate the proposed methodcould find either equal quality of solutions or higher qualityof solutions than CCSA for all study casesThe reduction costis from $0 to $42839 corresponding to the improvement level
18 Mathematical Problems in Engineering
Table 14 Summary of results obtained by CCSA and ICSA for all study cases
Study case Reduction cost ($) Improvement level () Execution time (s) Nps Gmax
Figure 4 The best run obtained by CCSA and ICSA for Case 2
that is from 0 to 08968The saving cost is not too much forone hour but the operation in one day onemonth or one yearis very high However it should be noted that CCSA has beenrun by setting much higher population size and iterations formany cases excluding study cases implemented in the studysuch as Case 2 Subcase 31 Subcase 35 and Case 5 Forinstance ICSA has used 119873119901119904 = 10 and 119866119898119886119909 = 15 for Subcase13 and119873119901119904 = 10 and119866119898119886119909 = 25 for Subcase 14 whereas CCSAhas been run by setting119873119901119904 = 20 and 119866119898119886119909 = 5000 for the twosubcases Similarly CCSA has been run for Subcases 32 33and 34 with much higher number of iterations For the lastsubcase CCSA has been run by setting 119873119901119904 = 320 and 119866119898119886119909
= 1200 but those of ICSA have been 10 and 9000 Due tothe higher value of control parameters CCSA has tended tospend more time in finding such high quality solutions foralmost all study cases Execution time of ICSA is less than18 seconds while that of CCSA is up to higher 75 secondsIt is clear that the proposed ICSA could find better optimalsolutions thanCCSA for such considered systems For furtherinvestigation of performance comparison the best runs over50 runs and fuel cost values of 50 runs obtained by CCSA andthe proposed ICSA for Case 2 Subcase 31 Subcase 35 andCase 5 have been plotted in from Figures 4ndash11 The best runcurves show the faster search of the proposed ICSA method
Mathematical Problems in Engineering 19
0 5 10 15 20 25 30 35 40 45 50Run
19795
198
19805
1981
19815
1982
19825
1983
19835
Fuel
cost
($)
CCSAICSA
times105
Figure 5The best fuel cost of 50 runs obtained by CCSA and ICSA for Case 2
0 2 4 6 8 10 12 14 16 18 20Iteration
8234
8236
8238
8240
8242
8244
8246
8248
8250
8252
Fitn
ess F
unct
ion
($)
CCSAICSA
Figure 6 The best run obtained by CCSA and ICSA for Subcase 31
whereas 50 values of fuel cost indicate that the proposedICSA can find many solutions with better quality Clearlythe proposed ICSA is outstanding in terms of stabilizationof solution search and faster convergence As a result it canconclude that the proposed ICSA approach is more effectivethanCCSA in solvingOLDproblemwith considered systems
572The Improvement of Results over Other Methods In thisarticle we have tested ICSA approach on 6 cases with 16
systems with different fuel cost forms different constraintsand different scale systems from 3 units to 320 units We havecompared the yielded results from ICSA approach and otherexisting ones for evaluating the efficiency of ICSA approachIn subsections above we have shown yielded results fromICSA approach and compared these results to those of otheronesHowever the demonstrationhad not been very good forobserving and comparing to lead to a conclusionThus in thesubsectionwehave summarized the result comparisons of the
20 Mathematical Problems in Engineering
0 5 10 15 20 25 30 35 40 45 50Run
8234
82342
82344
82346
82348
8235
82352
82354
82356
82358
8236
Fuel
cost
($)
CCSAICSA
Figure 7 The best fuel cost of 50 runs obtained by CCSA and ICSA for Subcase 31
0 1000 2000 3000 4000 5000 6000Iteration
242
244
246
248
25
252
254
256
Fitn
ess F
unct
ion
($)
CCSAICSA
times105
Figure 8 The best run obtained by CCSA and ICSA for Subcase 35
proposed and other ones Table 15 has reported the reductioncost (in $) of ICSA approach compared to other ones Inaddition we have converted the reduction cost into improve-ment level (in ) for better comparison The improvementhas been shown from the lowest level to the highest levelin terms of reduction cost and improvement percentageIn addition we have also given the slowest and the fastestexecution time of other compared methods together withthat of the proposed method The table implies that ICSA
approach can find better optimal solutions with less fuel costup to $052 for Subcase 13 $75229 for Subcase 14 $1775149for Case 2 $1181256 for Subcase 31 $8438 for Subcase 32$9113 for Subcase 33 $121181 for Subcase 34 $3412 forSubcase 35 $04329 for Subcase 41 $30227 for Subcase42 $407002 for Case 5 $18183 for Subcase 61 $13965 forSubcase 62 and $11817 for Subcase 63 These reductioncosts are equivalent to improvement level (IL) of 0032908968 82282 14143 047 038 099 001405
Mathematical Problems in Engineering 21
0 5 10 15 20 25 30 35 40 45 50Run
2428
243
2432
2434
2436
2438
244
2442
Fuel
cost
($)
CCSAICSA
times105
Figure 9 The best fuel cost of 50 runs obtained by CCSA and ICSA for Subcase 35
0 50 100 150 200 250 300 350 400Iteration
327
328
329
33
331
332
333
334
335
336
337
Fitn
ess F
unct
ion
($)
CCSAICSA
times104
Figure 10 The best run obtained by CCSA and ICSA for Case 5
00003 0002 12291 004 138 and 059 Thesequantitative comparisons reveal that larger scale systems canlead to better reduction cost but the improvement level isnot high because total cost of compared methods tends tobe large for large scale systems Furthermore very large scalesystems with nondifferentiable objective have been normallysolved by strong methods In fact systems in Case 3 havethe same characteristic with single fuel and effects of valveloading process but Subcase 34 is a larger scale system with
40 units while Subcases 31 32 and 33 are constructed by 3units 13 units and 13 units So the improvement percentageof Subcase 34 can be up to 099 whilst that of Subcases32 and 33 is 047 and 038 respectively Subcase 35 iswith the largest system 80 units but the reduction cost is notmuch only $3412 because compared methods with ICSA areeither state-of-the-art ones or improved ones Also Subcase41 and Subcase 42 have considered single fuel and PPZ andspinning reserve constraints but Subcase 42 is larger scale
22 Mathematical Problems in Engineering
Table 15 Performance improvement summary of the proposed method
Study cases Reduction cost ($) Improvement level () Execution time (s)From To From To Slowest method Fastest method Proposed method
Figure 11 The best fuel cost of 50 runs obtained by CCSA and ICSA for Case 5
system with 90 units and Subcase 41 is only with 60 unitsThus the improvement of Subcase 41 is lower with 00003but that of Subcase 42 is 0002 Similarly systems in Cases 6have the same featurewithmulti-fossil fuel sources and effectsof valve loading process but they are respectively constructedby 80 160 and 320 units As a result the improvements ofSubcase 62 138 and Subcase 63 059 are much higherthan Subcase 61 004 However Subcase 62 with smallernumber of units but getting higher improvement is easilyunderstood because there were nine compared methods butonly three compared methods are considered for Subcase 63In general the improvement is not high it is about under onedollar several dollars tens of dollars and over one thousand
dollars per hour however the saving cost will be significantif the operation is considered to be onemonth with 720 hoursor one year with 8760 hours
Execution time comparison can be evaluated by observ-ing the fastest and the slowest compared methods in Table 15These execution times of the proposed method are approxi-mately equal to that of the fastest methods for Subcases 3233 and 34 and much shorter than other fastest comparedmethods for other cases especially for Subcases 61 and 63The fastest method for Subcase 62 is CBA [44] showing 57seconds while that of the proposed method is 1119 secondsHowever it cannot conclude that CBA is more effective thanthe proposed method because the proposed method could
find better optimal solution with less fuel cost by $2130As considering execution time of the slowest comparedmethods it can point out that ICSA is a very fast optimizationtool since the execution time of these methods is 29496 and3425 seconds for Subcases 32 and 33 116735 seconds forSubcase 34 56381 seconds for Subcase 41 94093 secondsfor Subcase 42 12797 seconds for Case 5 30941 seconds forSubcase 61 7542 seconds for Subcase 62 and 9641 secondsfor Subcase 63 while the execution time of the proposedmethod for these cases is respectively 095 146 091 1589246 84828 1119 and 171384 seconds It is clearly shown thatICSA is very fast as compared to these methods
In summary the proposed method has found approx-imately high quality solutions with several standard state-of-the-art meta-heuristic algorithms and improved versionsof them together with other old methods In addition theproposed method could improve result better than approx-imately all methods with faster execution time Comparedto other methods with the fastest convergence speed andhigh quality solutions the proposed method has been as
fast as for some cases and much faster for other cases Thecomparison with the slowest methods could show that theproposed method was extremely powerful since it was up tonearly one thousand times faster Consequently the proposedICSA approach can be one of the strongest optimization toolsfor OLD problem
6 Conclusions
This paper has proposed a good ICSA method for solvingOLD problem in which many test systems with differentobjective functions and complicated constraints from simpleto complex have been used as studied cases The proposedICSA method has been developed by performing severalmodifications on the second solution update of CCSA whichcontained several drawbacks to global convergence and fastmanner The OLD problem has covered from single fuelto multi-fossil fuels from quadratic objective function tononconvex objective function in addition to PPZ RPR andRRL constraints Many existing optimization algorithms have
been concerned in aim to compare the performance and givethe final conclusion on the proposed method There have
been six main cases with sixteen subcases The evaluationshave been made at the end of each study case Clearly theproposed ICSA approach has yielded more effective optimalsolutions with faster execution time than almost all methodsConsequently it can be concluded that the proposed methodis much more superior to CCSA and is a very promisingmethod for solving OLD problem
Appendix
See Tables 16ndash26
Nomenclature
120575119894 120582119894 120572119894 120573119894 120574119894 Fuel cost functioncoefficients of the 119894119905ℎthermal generationunit11986100 1198610119895 119861119895119894 Power loss matrixcoefficients119865119865119909 119865119865119887119890119904119905 The values of fitness ofsolution 119909 and theso-far most effectivesolution among thecurrent set ofsolutions119898119894 Number of fuels burntin the 119894119905ℎ thermalgeneration unit
119873 Number of allavailable thermalgeneration units119899119894 Number of prohibitedpower zones of the 119894119905ℎthermal generationunit119875119906119894119896 119875119897119894119896 Upper and lowerlimits of the 119894119905ℎthermal generationunit corresponding tothe kth PPZ119875119886 Probability ofreplacing controlvariables in each oldsolution119875119863 Real power demand ofall loads in system119875119894119898119886119909 119875119894119898119894119899 The highest andlowest real poweroutputs of the iththermal generationunit119875119894119895119898119886119909 119875119894119895119898119894119899 The highest andlowest real poweroutputs of the 119894119905ℎthermal generationunit corresponding tothe 119895119905ℎ fuel type119878119894119898119886119909 Maximum real powerreserve contributionof the thermalgeneration unit 1198941198781199001198971199031198861198991198891 1198781199001198971199031198861198991198892 1198781199001198971199031198861198991198893 1198781199001198971199031198861198991198894 Randomly mixedsolutions from the setof current solutions119878119900119897119909 119878119900119897119866119887119890119904119905 The old solution x andthe most effectivesolution119878119877 Real power reserverequirement of system
120575119894119895 120582119894119895 120572119894119895 120573119894119895 120574119894119895 Fuel cost functioncoefficients of the 119894119905ℎthermal generationunit corresponding tothe 119895119905ℎ fuel type1205761 1205762 1205763 1205764 1205765 1205766 Random numbersbetween 0 and 1Ω Number of generationunits considering PPZconstraint
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare no conflicts of interest
References
[1] N Sinha R Chakrabarti and P K Chattopadhyay ldquoEvolution-ary programming techniques for economic load dispatchrdquo IEEETransactions on Evolutionary Computation vol 7 no 1 pp 83ndash94 2003
[2] P Hansen ldquoA separable approximation dynamic programmingalgorithm for economic dispatch with transmission lossesrdquoYugoslav Journal of Operations Research vol 12 no 2 2002
[3] N T Thang ldquoSolving economic dispatch problem with piece-wise quadratic cost functions using lagrange multiplier theoryrdquoin Proceedings of the 3rd International Conference on ComputerTechnology and Development (ICCTD rsquo11) pp 359ndash364 ASMEPress 2011
[4] S K Mishra and S K Mishra ldquoA comparative study of solutionof economic load dispatch problem in power systems in theenvironmental perspectiverdquoProcedia Computer Science vol 48pp 96ndash100 2015
[5] A A Al-Subhi and H K Alfares ldquoEconomic load dispatchusing linear programming a comparative studyrdquo InternationalJournal of Applied Industrial Engineering vol 3 no 1 pp 16ndash362016
28 Mathematical Problems in Engineering
[6] C Zhou G Huang and J Chen ldquoPlanning of electric powersystems considering virtual power plants with dispatchableloads included an inexact two-stage stochastic linear program-ming modelrdquoMathematical Problems in Engineering vol 2018Article ID 7049329 12 pages 2018
[7] T T Nguyen N V Quynh and L Van Dai ldquoImproved fireflyalgorithm a novel method for optimal operation of thermalgenerating unitsrdquo Complexity vol 2018 Article ID 7267593 23pages 2018
[8] S S Haroon S Hassan S Amin et al ldquoMultiple fuel machinespower economic dispatch using stud differential evolutionrdquoEnergies vol 11 no 6 pp 1ndash20 2018
[9] C-T Su and C-L Chiang ldquoNonconvex power economic dis-patch by improved genetic algorithm with multiplier updatingmethodrdquo Electric Power Components and Systems vol 32 no 3pp 257ndash273 2004
[10] NNoman andH Iba ldquoDifferential evolution for economic loaddispatch problemsrdquo Electric Power Systems Research vol 78 no8 pp 1322ndash1331 2008
[11] P Somasundaram and K Kuppusamy ldquoApplication of evolu-tionary programming to security constrained economic dis-patchrdquo International Journal of Electrical Power amp EnergySystems vol 27 no 5-6 pp 343ndash351 2005
[12] T Jayabarathi K Jayaprakash andD Jeyakumar ldquoEvolutionaryprogramming technique for different kinds of economic dis-patch problemsrdquo Electric Power Systems Research vol 73 no 2pp 169ndash176 2005
[13] P Subbaraj R Rengaraj and S Salivahanan ldquoEnhancementof combined heat and power economic dispatch using selfadaptive real-coded genetic algorithmrdquo Applied Energy vol 86no 6 pp 915ndash921 2009
[14] C-L Chiang and C-T Su ldquoAdaptive-improved genetic algo-rithm for the economic dispatch of units with multiple fueloptionsrdquo Cybernetics and Systems vol 36 no 7 pp 687ndash7042005
[15] C Chiang ldquoImproved genetic algorithm for power economicdispatch of units with valve-point effects and multiple fuelsrdquoIEEE Transactions on Power Systems vol 20 no 4 pp 1690ndash1699 2005
[16] K Meng H G Wang Z Y Dong and K P Wong ldquoQuantum-inspired particle swarm optimization for valve-point economicload dispatchrdquo IEEE Transactions on Power Systems vol 25 no1 pp 215ndash222 2010
[17] P Erdogmus A Ozturk and S Duman ldquoEnvironmen-taleconomic dispatch using genetic algorithm and simulatedannealingrdquo in Proceedings of the International Conference onElectric Power and Energy Conversion Systems EPECS rsquo09 pp1ndash4 IEEE 2009
[18] L D S Coelho and V C Mariani ldquoImproved differentialevolution algorithms for handling economic dispatch opti-mization with generator constraintsrdquo Energy Conversion andManagement vol 48 no 5 pp 1631ndash1639 2007
[19] H Dakuo W Fuli and M Zhizhong ldquoA hybrid genetic algo-rithm approach based on differential evolution for economicdispatch with valve-point effectsrdquo Electrical Power and EnergySystems vol 30 pp 31ndash38 2008
[20] C-F Chang J-J Wong J-P Chiou and C-T Su ldquoRobustsearching hybrid differential evolution method for optimalreactive power planning in large-scale distribution systemsrdquoElectric Power Systems Research vol 77 no 5-6 pp 430ndash4372007
[21] M R Farooqi P Jain and K R Niazi ldquoUsing Hopfield neuralnetwork for economic dispatch of power systemsrdquo in Proceed-ings of the National Power Engineering Conference PECon rsquo03pp 5ndash10 Bangi Malaysia 2003
[22] T Yalcinoz H Altun and U Hasan ldquoConstrained economicdispatch with prohibited operating zones a Hopfield neuralnetwork approachrdquo in Proceedings of the 10th MediterraneanElectrotechnical Conference Information Technology and Elec-trotechnology for the Mediterranean Countries ProceedingsMeleCon rsquo00 (Cat No 00CH37099) pp 570ndash573 LemesosCyprus 2000
[23] D N Vo P Schegner and W Ongsakul ldquoCuckoo searchalgorithm for non-convex economic dispatchrdquo IET GenerationTransmission amp Distribution vol 7 no 6 pp 645ndash654 2013
[24] J-B Park K-S Lee J-R Shin and K Y Lee ldquoA particleswarm optimization for economic dispatch with non-smoothcost functionsrdquo IEEETransactions on Power Systems vol 20 no1 pp 34ndash42 2005
[25] T Niknam H D Mojarrad and M Nayeripour ldquoA newfuzzy adaptive particle swarm optimization for non-smootheconomic dispatchrdquo Energy vol 35 no 4 pp 1764ndash1778 2010
[26] L D S Coelho and V C Mariani ldquoAn improved harmonysearch algorithm for power economic load dispatchrdquo EnergyConversion and Management vol 50 no 10 pp 2522ndash25262009
[27] M Mahdavi M Fesanghary and E Damangir ldquoAn improvedharmony search algorithm for solving optimization problemsrdquoAppliedMathematics and Computation vol 188 no 2 pp 1567ndash1579 2007
[28] A Bhattacharya and P K Chattopadhyay ldquoSolving complexeconomic load dispatch problem using biogeography-basedoptimizationrdquo Expert Systems with Applications vol 37 no 5pp 3605ndash3615 2010
[29] D Simon ldquoBiogeography-based optimizationrdquo IEEE Transac-tions on Evolutionary Computation vol 12 no 6 pp 702ndash7132008
[30] J S Alsumait J K Sykulski and A K Al-Othman ldquoAhybrid GA-PS-SQP method to solve power system valve-pointeconomic dispatch problemsrdquo Applied Energy vol 87 no 5 pp1773ndash1781 2010
[31] M S Turgut andGKDemir ldquoQuadratic approximationndashbasedhybrid Artificial Cooperative Search algorithm for economicemission load dispatchproblemsrdquo International Transactions onElectrical Energy Systems vol 27 no 4 pp 1ndash14 2017
[32] T A A Victoire and A E Jeyakumar ldquoHybrid PSO-SQPfor economic dispatch with valve-point effectrdquo Electric PowerSystems Research vol 71 no 1 pp 51ndash59 2004
[33] D-KHe F-LWang andZ-ZMao ldquoHybrid genetic algorithmfor economic dispatch with valve-point effectrdquo Electric PowerSystems Research vol 78 no 4 pp 626ndash633 2008
[34] D C Secui ldquoA modified symbiotic organisms search algorithmfor large scale economic dispatch problem with valve-pointloading effectsrdquo Energy vol 113 pp 366ndash384 2016
[35] K BhattacharjeeA Bhattacharya and S H N Dey ldquoTeaching-learning-based optimization for different economic dispatchproblemsrdquo Scientia Iranica vol 21 no 3 pp 870ndash884 2014
[36] K Bhattacharjee A Bhattacharya and S H N Dey ldquoOppo-sitional Real Coded Chemical Reaction Optimization for dif-ferent economic dispatch problemsrdquo International Journal ofElectrical Power amp Energy Systems vol 55 pp 378ndash391 2014
Mathematical Problems in Engineering 29
[37] V Hosseinnezhad M Rafiee M Ahmadian and M T AmelildquoSpecies-basedQuantum Particle SwarmOptimization for eco-nomic load dispatchrdquo International Journal of Electrical Poweramp Energy Systems vol 63 pp 311ndash322 2014
[38] M S P Subathra S E Selvan T A A Victoire A HChristinal andU Amato ldquoA hybrid with cross-entropymethodand sequential quadratic programming to solve economic loaddispatch problemrdquo IEEE Systems Journal vol 9 no 3 pp 1031ndash1044 2015
[39] J Zhan Q H Wu C Guo and X Zhou ldquoEconomic dispatchwith non-smooth objectives part I local minimum analysisrdquoIEEE Transactions on Power Systems vol 30 no 2 pp 710ndash7212015
[40] A K Barisal andR C Prusty ldquoLarge scale economic dispatch ofpower systems using oppositional invasive weed optimizationrdquoApplied Soft Computing vol 29 pp 122ndash137 2015
[41] M Basu ldquoImproved differential evolution for economic dis-patchrdquo International Journal of Electrical Power amp EnergySystems vol 63 pp 855ndash861 2014
[42] V S Aragon S C Esquivel and C C Coello ldquoAn immunealgorithm with power redistribution for solving economicdispatch problemsrdquo Information Sciences vol 295 pp 609ndash6322015
[43] M Ghasemi M Taghizadeh S Ghavidel and A AbbasianldquoColonial competitive differential evolution an experimentalstudy for optimal economic load dispatchrdquo Applied Soft Com-puting vol 40 pp 342ndash363 2016
[44] B R Adarsh T Raghunathan T Jayabarathi and X-S YangldquoEconomic dispatch using chaotic bat algorithmrdquo Energy vol96 pp 666ndash675 2016
[45] N Ghorbani and E Babaei ldquoExchange market algorithm foreconomic load dispatchrdquo International Journal of ElectricalPower amp Energy Systems vol 75 pp 19ndash27 2016
[46] J X V Neto G Reynoso-Meza T H Ruppel V C Marianiand L D S Coelho ldquoSolving non-smooth economic dispatchby a new combination of continuous GRASP algorithm anddifferential evolutionrdquo International Journal of Electrical Poweramp Energy Systems vol 84 pp 13ndash24 2017
[47] A Kavousi-Fard and A Khosravi ldquoAn intelligent 120579-ModifiedBat Algorithm to solve the non-convex economic dispatchproblem considering practical constraintsrdquoElectrical Power andEnergy Systems vol 82 pp 189ndash196 2016
[48] M A Al-Betar M A Awadallah A T Khader and A L BolajildquoTournament-based harmony search algorithm for non-convexeconomic load dispatch problemrdquo Applied Soft Computing vol47 pp 449ndash459 2016
[49] M A Al-Betar M A Awadallah I A Doush E Alsukhni andH ALkhraisat ldquoA non-convex economic dispatchproblemwithvalve loading effect using a new modified 120573-hill climbing localsearch algorithmrdquo Arabian Journal for Science and Engineeringpp 1ndash8 2018
[50] M A Awadallah M A Al-Betar A L Bolaji E M Alsukhniand H Al-Zoubi ldquoNatural selection methods for artificial beecolony with new versions of onlooker beerdquo Soft Computing pp1ndash40 2018
[51] S H Kaboli and A K Alqallaf ldquoSolving non-convex economicload dispatch problem via artificial cooperative search algo-rithmrdquo Expert Systems with Applications vol 128 pp 14ndash272019
[52] D Singh and J Dhillon ldquoAmeliorated greywolf optimization foreconomic load dispatch problemrdquo Energy vol 169 pp 398ndash4192019
[53] X S Yang and S Deb ldquoCuckoo search via Levy flightsrdquo inProceedings of the World Congress on Nature and BiologicallyInspired Computing (NaBIC rsquo09) pp 210ndash214 India 2009
[54] S Sahoo K M Dash R C Prusty and A K Barisal ldquoCom-parative analysis of optimal load dispatch through evolutionaryalgorithmsrdquo Ain Shams Engineering Journal vol 6 pp 107ndash1202015
[55] M Basu and A Chowdhury ldquoCuckoo search algorithm foreconomic dispatchrdquo Energy vol 60 pp 99ndash108 2013
[56] A B Serapiao ldquoCuckoo search for solving economic dispatchload problemrdquo Intelligent Control and Automation vol 04 no04 pp 385ndash390 2013
[57] A H Bindu and M D Reddy ldquoEconomic load dispatch usingcuckoo search algorithmrdquo International Journal of EngineeringResearch and Applications vol 3 pp 498ndash502 2013
[58] E Afzalan and M Joorabian ldquoAn improved cuckoo searchalgorithm for power economic load dispatchrdquo InternationalTransactions on Electrical Energy Systems vol 25 no 6 pp 958ndash975 2015
[59] T T Nguyen and D N Vo ldquoThe application of one rank cuckoosearch algorithm for solving economic load dispatch problemsrdquoApplied Soft Computing vol 37 pp 763ndash773 2015
[60] S M Islam S Das S Ghosh S Roy and P N Suganthan ldquoAnadaptive differential evolution algorithm with novel mutationand crossover strategies for global numerical optimizationrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 42 no 2 pp 482ndash499 2012
[61] Z L Gaing ldquoParticle swarm optimization to solving theeconomic dispatch considering the generator constraintsrdquo IEEETransactions on Power Systems vol 18 no 3 pp 1187ndash1195 2003
[62] X-S Yang and S Deb ldquoEngineering optimisation by Cuckoosearchrdquo International Journal of Mathematical Modelling andNumerical Optimisation vol 330 pp 43ndash51 2010
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4 Mathematical Problems in Engineering
Case 6 Three systems with multiple fuel sources (MFS) andEoVLP
The achieved results in terms of minimum fuel costaverage fuel cost maximum fuel cost and standard deviationfound by the proposed method compared to those obtainedby others reveal that the method is very efficient for theOLD problem In addition the performance improvementof the proposed method over CCSA is also investigatedvia the comparison of the best solution and all trial runsIn summary the main advantages of the proposed ICSAapproach over CCSA as well as the main contribution of thestudy are as follows
(i) Based on fitness function of each considered solutionlocal search or global search is decided to be appliedmore effectively
(ii) Find better solutions with smaller number of itera-tions and shorter execution time for each run
(iii) Shorten simulation time for the whole search of eachstudy case
However the proposed method also copes with the sameshortcomings as CSA Although the shortcomings do notcause bad results for the proposed method they make theproposed method be time consuming in tuning optimalparameter The shortcomings are analyzed as follows
(i) Control parameter probability of replacing controlvariables in each old solution must be tuned in rangebetween 0 and 1 There is no proper theory for deter-mining the most effective values of the parameterThus the performance of the proposed method mustbe tried by setting the parameter to values from 01 to1
(ii) The method uses more computation steps for searchprocessThus the proposedmethod uses higher num-ber of computation steps for each iteration Howeverdue to more effective search ability for each iterationthe proposed method can use smaller number ofiterations but it finds more effective solutions
The remaining parts of the paper are arranged as fol-lows Section 2 shows the objective and constraints of theconsidered OLD problem CCSA and the proposed methodare clearly explained in Section 3 Section 4 is in charge ofpresenting the implementation of ICSA method for the stud-ied problem The simulation results together with analysisand discussions are given in Section 5 Finally conclusion issummarized in Section 6 In addition appendix is also addedfor showing found solutions by the proposed ICSA approachfor test cases
2 Optimal Load DispatchProblem Description
21 Fuel Cost Function Forms with Single Fuel Source In theconsidered OLD problem the optimal operation of a set of
thermal generation units is concerned as the duty of reducingtotal cost of all the units which can be seen by the followingmodel
Reduce 119865 = 119873sum119894=1
119865119894 (119875119894) (1)
In traditional OLD problem fuel cost function of the119894119905ℎ generation unit 119865119894(119875119894) is represented as the second orderfunction with respect to real power output and coefficients asthe model below [2]
In addition for the case considering the effects ofvalve loading process on thermal generation units fuel costbecomes more complicated by adding sinusoidal term asbelow [12]
119865119894 (119875119894) = 1205721198941198752119894 + 120582119894119875119894 + 120575119894+ 1003816100381610038161003816120573119894 times sin (120574119894 times (119875119894min minus 119875119894))1003816100381610038161003816 (3)
Real Power Balance Constraint Total real power demand ofall loads in power system together with real power loss in allconductors must be equal to the generation from all availablethermal generation units The requirement is constrained bythe following equality
119873sum119894=1
119875119894 = 119875119863 + 119875119871 (4)
where total real power loss 119875119871 is determined by Kronrsquosequation below
119875119871 = 11986100 + 119873sum119895=1
1198610119895119875119895 + 119873sum119895=1
119873sum119894=1
119875119895119861119895119894119875119894 (5)
GenerationBoundaryConstraint For the purpose of economyand safe operation each thermal generation unit is con-strained by the lower generation bound and upper generationbound as the following model
119875119894min le 119875119894 le 119875119894max (6)
22 Fuel Cost Function Forms with Multi-Fuel Sources Inthis section fuel cost function of thermal generation units ismathematicallymodeled in terms of different forms from thatin the section above due to the consideration of multi-fuelsources Each type of fuel source is formed as each secondorder function and the fuel cost function form is the sumof different second order functions for the case of neglectingthe effects of valve loading progress But for the considerationcase of the effects the form ismore complexwith the presenceof sinusoidal terms [15] As a result the forms of cost functioncan be expressed in Equation (7) [21] and Equation (8) [15]
Mathematical Problems in Engineering 5
119865119894 (119875119894) =
1205751198941 + 1205821198941119875119894 + 12057211989411198752119894 fuel 1 119875119894min le 119875119894 le 11987511989411198981198861199091205751198942 + 1205821198942119875119894 + 12057211989421198752119894 fuel 2 1198751198942min le 119875119894 le 1198751198942max120575119894119895 + 120582119894119895119875119894 + 1205721198941198951198752119894 fuel 119895 119875119894119895min le 119875119894 le 119875119894119895max
(7)
119865119894 (119875119894) =
1205751198941 + 1205821198941119875119894 + 12057211989411198752119894 + 10038161003816100381610038161205731198941 times sin (1205741198941 times (119875119894min minus 119875119894))1003816100381610038161003816 for fuel 1 119875119894min le 119875119894 le 11987511989411198981198861199091205751198942 + 1205821198942119875119894 + 12057211989421198752119894 + 10038161003816100381610038161205731198942 times sin (1205741198942 times (119875119894min minus 119875119894))1003816100381610038161003816 for fuel 2 1198751198942min le 119875119894 le 1198751198942119898119886119909 119895 = 1 119898119894120575119894119895 + 120582119894119895119875119894 + 1205721198941198951198752119894 + 10038161003816100381610038161003816120573119894119895 times sin (120574119894119895 times (119875119894min minus 119875119894))10038161003816100381610038161003816 for fuel 119895 119875119894119895min le 119875119894 le 119875119894119895max
(8)
Cost function forms in Equations (7) and (8) are onlyincluded in objective function (1) meanwhile main con-straints in formulas (4) and (6) must be always satisfied
23 Prohibited Power Zone Real Power Reserve and RampRate Limit Constraints Prohibited power zones (PPZ) aredifferent ranges of power in fuel cost function that thermalgeneration units are not allowed to work due to operationprocess of steam or gas valves in their shaft bearing Thepower generation of units in the violated zones is harmfulto gas or steam turbines even destroyed shaft bearing Thusthe constraint is strictly observed In the fuel-power charac-teristic curve of generation units PPZ causes small violationzones and such curves become discontinuous As consideringPPZ constraint the determination of power generation ofunits is more complex and equal to either lower bound orupper bound Unlike PPZ constraint RPR constraint is notrelated to fuel-power feature curve but it causes difficulty foroptimization approaches in satisfying one more inequalityconstraint Each generation unit among the set of availablegeneration units must reserve real power so that the sumof real power from all generation units can be higher orequal to the requirement of power system for the purposeof stabilizing power system in case that there are some unitsstopping producing electricity On the contrary to PPZ con-straint ramp rate limit (RRL) constraint does not allowpoweroutput of thermal generating units outside a predeterminedrange The constraint considers maximum power change ofeach thermal generating unit as compared to the previouspower value Thus optimal generation must satisfy the RRLconstraint The PPZ constraint RPR constraint and RRLconstraint can be presented as follows
Prohibited Power Zones As considering PPZ constraint validworking zones of each thermal generating unit are notcontinuous and its generation must be outside the violatedzones as the following mathematical description
119875119894 isin
119875119894min le 119875119894 le 1198751198971198941119875119906119894119896minus1 le 119875119894 le 119875119897119894119896 k = 2 ni forall119894 isin Ω119875119906119894119899119894 le 119875119894 le 119875119894max
(9)
As observing Equation (9) generation units cannotbe operated within the violated zones except for startingpoint and end point Consequently the verification of PPZconstraint violation should be carried out first and thenthe correction should be done before dealing with otherconstraints such as real power reserve constraint and realpower balance Besides if power output of all units can satisfythe PPZ constraint generation limits in Equation (6) are alsoexactly met
Real Power Reserve Constraint Real power reserve in powersystem aims to enhance the ability of stability recovery ofpower system and avoid blackout In order to get high enoughpower for requirement all available units are constrained bythe following inequality
119873sum119894=1
119878119894 ge 119878119877 (10)
where 119878119894 is the real power reserve contribution of the 119894119905ℎthermal generation unit and the determination of 119878119894 can bedone by employing the two models below
119878119894 = 119875119894max minus 119875119894 119894119891 119878119894max gt (119875119894max minus 119875119894)119878119894max else
forall119894 notin Ω (11)
119878119894 = 0 forall119894 isin Ω (12)
Equation (10) shows that the constraint of prohibitedpower zones is not included in the real power reserveconstraints however prohibited power zones are alwaysstrictly considered and must be exactly satisfied
Ramp Rate Limit (RRL) Constraint In OLD problem allconsidered thermal generating units are supposed to be underworking status but previous active power of each thermalgenerating unit is not taken into account Thus increased ordecreased power is not constrained This assumption seemsto be not practical until RRL constraint is considered RRLconstraint considers initial power output and the power
6 Mathematical Problems in Engineering
change is supervised Regulated power can be higher or lowerthan the initial value as long as it is within a predeterminedrange Increased step size (ISS) and decreased step size (DSS)are given as input data and they are used to limit the change ofpower output of each thermal generating unit The constraintcan bemathematically expressed as the following formula [7]
1198751198940 minus 119863119878119878119894 le 119875119894 le 1198751198940 + 119868119878119878119894 (13)
where 1198751198940 is the initial power output of the 119894119905ℎ thermalgenerating unit before its power output is regulated 119868119878119878119894 and119863119878119878119894 are respectively maximum increased and decreasedstep sizes of the 119894119905ℎ thermal generating unit
3 The Proposed Cuckoo Search Algorithm
31 Classical Cuckoo Search Algorithm In search techniqueof CCSA [53] a set of solutions is randomly generated withina predetermined range in the first step and then the quality ofeach one is ranked by computing value of fitness functionThemost effective solution corresponding to the smallest valueof fitness function is determined and then search procedurecomes into a loop algorithm until the maximum iterationis reached In the loop algorithm two techniques updatingnew solutions two times (corresponding to two generations)are Levy flights and mutation technique which is calledstrange eggs identification technique The two generationscan produce promising quality solutions for CCSA Aftereach generation CCSA will carry out comparing fitness ofnewly updated solutions and initial solutions for keepingbetter ones and abandoning worse ones The most effectivesolution at last step of the loop search algorithm is determinedand it is restored as one candidate solution for a study caseThe detail of the two stages is as follows
311 Levy Flights Stage This is the first calculation step in theloop algorithm and it also produces new solutions in the firstgeneration for CCSA New solution 119878119900119897119899119890119908119909 is created by thefollowing model
119878119900119897119899119890119908119909 = 119878119900119897119909 + 120572 (119878119900119897119909 minus 119878119900119897119866119887119890119904119905) oplus Levy (120573) (14)
where 120572 is the positive scaling factor and it is nearly set todifferent values for different problems in the studies [53 62]In the work the most appropriate values for such factor canbe chosen to be 02505 for different systems
312 Discovery of Alien Eggs Stage The step plays a veryimportant role for updating new solutions 119878119900119897119899119890119908119909 of thewhole population However not every control variable ineach old solution is newly updated and the decision ofreplacement is dependent on comparison criteria as thefollowing equation
119878119900119897119899119890119908119909=
119878119900119897119909 + 1205761 (1198781199001198971199031198861198991198891 minus 1198781199001198971199031198861198991198892) if 1205762 lt 119875119886119878119900119897119909 119900119905ℎ119890119903119908119894119904119890
(15)
32 Proposed Algorithm In the part a new variant of CCSA(ICSA) is constructed by applying three effective changes onthe main functions of CCSA in order to shorten simula-tion time corresponding to reduction of iterations and findmore promising solutions The proposed amendments areexplained in detail as follows
(i) Suggest one more equation producing updated stepsize in addition to existing one in CCSA
(ii) Create a new selection standard by computing fitnessfunction ratio 119865119865119877119909 and comparing 119865119865119877119909 with apredetermined parameter 119879119900119897119909 Thus thanks to thestandard the existing updated step size and additionalupdate step size will be chosen more effectively
(iii) Automatically change value of 119879119900119897119909 for the xth solu-tion based on the result of comparing 119865119865119877119909 with theprevious 119879119900119897119909
Such three points are clarified by observing the followingsections
321 Strange Eggs Identification Technique (Mutation Tech-nique) The first proposed improvement in our proposedICSA approach is to select a more suitable formula forproducing new solutions with better fitness function valueIn CCSA Equation (16) below is used to produce a changingstep nearby old solutions for all current solutions
Δ1198781199001198971198991198901199081199091 = 1205763 (1198781199001198971199031198861198991198891 minus 1198781199001198971199031198861198991198892) (16)
The use of Equation (16) aims to produce a random walkaround old solutions in search zones with intent to findout promising solutions In order to reduce the possibilityof suffering the local trap and approach to other favorablezones for searching we propose a new Equation (17) Theformula is built by the idea of enlarging search zone withthe use of two more available solutions Obviously the largerchanging step can own higher performance in moving toother search spaces that the classical approach used in CCSAThe suggestion is mathematically expressed by the formulabelow
Δ1198781199001198971198991198901199081199092= 1205764 (1198781199001198971199031198861198991198891 minus 1198781199001198971199031198861198991198892 + 1198781199001198971199031198861198991198893 minus 1198781199001198971199031198861198991198894) (17)
The changing step obtained by using Eq (17) is namedfour-point changing step Now two solutions which arenewly formed by using two different changing steps shownin formulas (16) and (17) are found by the two followingmethods
It can be clearly observed that the distance between 119878119900119897119909(old solution) and 1198781199001198971198991198901199081199091 (new solution) is lower than thatbetween 119878119900119897119909 and 1198781199001198971198991198901199081199092 This difference can contribute ahighly efficient improvement to the proposed ICSA approachsearch ability
Mathematical Problems in Engineering 7
ΔSol2
Sol2
Sol3
Sol4
ΔSol1
Sol1
Solx
Solnew1
Solnew2
Figure 1 Simulation of solutions corresponding to the first itera-tions of the loop algorithm
For the CCSA case if two solutions 1198781199001198971199031198861198991198891 and 1198781199001198971199031198861198991198892are either slightly different or completely coincident suchnewly updated solution 1198781199001198971198991198901199081199091 does not have good chanceto leave the current zone and approach to more promisingzones In another word the new one is approximately coin-cident with the old one As the search task is taking place atsome last iterations this phenomenon becomes much worsebecause all current solutions are lumped in a small zone andthe capability of moving to other zones is impossible As aresult the CCSA approach will work ineffectively and searchstrategy is time consuming until other runs are started
Contrary to the two-point step size the new proposedformula may produce a large enough length to escape thelocal optimum zone and reach new favorable zones Itexplainswhy the four-point changing step has positive impacton the considered random walk rather than the two-pointchanging step
322 New Standard forChoosing theMostAppropriate Chang-ing Step In this section we extend our analysis to answer thequestionwhen to use the four-point step size FromEquations(18) and (19) two new solutions which are represented asthe results of the two-point-based factor and the four-pointstep size can be illustrated by using Figure 1 corresponding tothe search process at the first some iterations and Figure 2corresponding to the last some iterations For the sake ofsimplicity we rewrite the two equations as follows
Here we suppose that 1198781199001198971 and 1198781199001198972 are obtained byfour exact solutions 1198781199001198971 1198781199001198972 1198781199001198973 and 1198781199001198974 and calculatedas follows
ΔSol2
ΔSol1
Solx
Solnew1
Solnew2
Figure 2 Simulation of solutions corresponding to the last itera-tions of the loop algorithm
Δ1198781199001198971 = 1198781199001198971 minus 1198781199001198972 (22)
Δ1198781199001198972 = 1198781199001198973 minus 1198781199001198974 (23)
Asmentioned above the high changing step between newsolution and old solution can help to explore new favorablezones However in optimization algorithms searching stepscannot be arbitrarily large otherwise the algorithm maydiverge in particular for the cases that the consideredsolutions 119878119900119897119909 are not close together in solution search spaceFor example at the beginning of loop algorithm with thefirst iterations in Figure 1 1198781199001198971198991198901199081 is a better choice than1198781199001198971198991198901199082 because it is kept in a sufficient limit and does notlead to a risk of divergence In contrast as many of currentsolutions are in different positions but their distance is notvery short or approximately coincident such as at the lastiterations in Figure 2 1198781199001198971198991198901199081 and 119878119900119897119909 have a very shortdistance but 1198781199001198971198991198901199082 and 119878119900119897119909 have higher distance Accordingto the phenomenon in Figure 2 the proposed ICSA approachneeds to produce a high changing step to move solutions toother search zones without local optimum Hence 1198781199001198971198991198901199082would be preferred to 1198781199001198971198991198901199081
Based on the argument above the determination of thecondition for using either two-point changing step or four-point changing step is really crucial to the performance ofthe proposed ICSA approach in searching solutions of OLDproblem Here the ratio of 119865119865119877119909 which can be found byEquation (24) is suggested to be a suitable measurement forthe selection of two options
Δ119865119865119877119909 = 119865119865119909 minus 119865119865119887119890119904119905119865119865119887119890119904119905 (24)
For a particular set of the current solutions each individ-ual depending on its 119865119865119877119909 will create a corresponding newsolution by using either Equation (18) or (19) If the valueof one current solution is smaller than the predeterminedparameter 119879119900119897 Equation (19) is applied for updating suchconsidered solution 119909 Otherwise Equation (18) is a betteroption The steps of the modified algorithm are similar to the
8 Mathematical Problems in Engineering
If 1205765 lt 119875119886If FFRx lt Tolx119878119900119897119899119890119908119909 = 119878119900119897119909 + 1205764(1198781199001198971199031198861198991198891 minus 1198781199001198971199031198861198991198892 + 1198781199001198971199031198861198991198893 minus 1198781199001198971199031198861198991198894)else119878119900119897119899119890119908119909 = 119878119900119897119909 + 1205764(1198781199001198971199031198861198991198891 minus 1198781199001198971199031198861198991198892)end
Algorithm 1 New mutation technique applied in the proposed ICSA approach
conventional CSA except that an additional step should beadded at each iteration In this step the119865119865119889 of all individualsolutions should be calculated by utilizing Equation (24) andthen the result of comparing the ratio with 119879119900119897will be used todecidewhich updating formula should be selectedThewholedescription of the proposed standard and new mutationtechnique can be coded inMatlab program language by usingAlgorithm 1
323 Adjustment of Tolerance for Each Solution As pointedout above the proposed method needs assistances to deter-mine the most appropriate step size for finding out favorablesolution zones The given aim can be reached if the selectionof 119879119900119897119909 is reasonable however the range of this parameteris infinite and hard to select Thus the adaptation of tuningthe parameter is really necessary First of all the compari-son between 119879119900119897119909 and 119865119865119877119909 is carried out and then theadaptation will be determined based on the obtained resultfrom the comparison Results of comparison between the twoparameters can be either 119865119865119877119909 is less than 119879119900119897119909 or 119865119865119877119909is higher than 119879119900119897119909 The case that two parameters are equalhardly ever occurs
As the comer assumptionhappens (ie119865119865119877119909 is less than119879119900119897119909) at the considered time the four-point step size will beemployed for the 119909119905ℎ solution If 119879119900119897119909 remains unchanged atthe previous value the identification of improvement fromsuch four-point step size or two-point step size is vagueConsequently value of 119879119900119897119909 must be automatically reducedto a lower value in case that it has significant contribution tofound promising solution of previous iteration Clearly thedecrease of119879119900119897119909 can enable the proposedmethod to jump outlocal optimal zone and approachmore effective zones By trialand error method 119879119900119897119909 is selected to be a function of itselfthat is 09 of the previous value Finally the implementationof the proposed ICSA approach is presented in Algorithm 2
4 The Application of the ProposedICSA for OLD Problem
Thewhole computation steps of the proposed ICSA approachfor solving OLD problem are explained as follows
41 Handling Constraints and Randomly Producing InitialPopulation As shown in Section 2 the considered OLDproblem takes five following constraints into account
(i) Power balance constraint is shown in Equation (4)
(ii) Power output limitation constraint is shown in Equa-tion (6)
(iii) Prohibited power zone constraint is shown in Equa-tion (9)
(iv) Real power reserve constraint is shown in Equation(10)
(v) Ramp rate limit constraint is shown in Equation (13)
Among the five constraints ramp rate limit generationlimit and prohibited power zone seem to be more com-plicated than power balance and power reserve constraintsHowever the three constraints can be solved more easilybecause each unit is constrained independently in the threeconstraints whereas power balance constraint and powerreserve constraint consider all the thermal generating unitssimultaneously Power reserve constraint can be handledby penalizing the total generation of all units while powerbalance constraint can be solved by penalizing one violatedthermal generating unit The whole computation procedurefor solving all constraints and calculating fitness function ofsolutions is described in detail as follows
Step 1 Redefine maximum and minimum power output ofeach thermal generating unit as considering PPZ and RRLconstraints by using the following formulas
119875119894max = 119875119894max if 119875119894max le 119875i0 + 119868119878119878119894119875i0 + 119868119878119878119894 if 119875119894max gt 119875i0 + 119868119878119878119894
119894 = 1 119873(25)
119875119894min = 119875119894min if 119875119894min ge 119875i0 minus 119863119878119878119894119875i0 minus 119863119878119878119894 119890119897119904119890
119894 = 1 119873(26)
Mathematical Problems in Engineering 9
Produce initial population with119873119901119904 solutions (1198781199001198971 1198781199001198972 119878119900119897119909 119878119900119897119873119901119904)Calculate fitness function (1198651198651 1198651198652 119865119865119909 119865119865119873119901)Go to the loop algorithm by setting 119866 = 1
While (119866119898119886119909 gt 119866) (i) The first newly produced solutions119878119900119897119899119890119908119909 = 119878119900119897119909 + 120572(119878119900119897119909 minus 119878119900119897119866119887119890119904119905) oplus Levy(120573) (ii) Perform selection approach
119865119865119909 = 119865119865119909 if 119865119865119909 le 119865119865119899119890119908119909119865119865119899119890119908119909 119900119905ℎ119890119903119908119894119904119890 119909 = 1 119873119901119904
(v) Determine the most effective solution and its fitnessDetermine 119865119865119909 with the smallest value and assign 119878119900119897119909 to 119878119900119897119866119887119890119904119905If 119866119898119886119909 gt 119866 perform step (i) and increase 119866 to 119866 + 1 Otherwise stop the loop algorithm and report boththe smallest fitness together with 119878119900119897119866119887119890119904119905End while
Among the four Equations (25) and (26) are used firstin order to redefine upper bound and lower bound for allthermal generating units as considering RRL constraint Thethe redefined bounds continue to be redefined for the secondtime by using (27) and (28) as considering PPZ constraints
Step 2 (randomly produce initial population) For dealingwith the power balance constraint all available units areseparated into two groups in which the first group withdecision variables consists of the power output from thesecond unit to the last unit (P2 P3 PN) meanwhile onlythe power output of the first unit (1198751) belongs to the secondgroup with dependent variable So upper bound solution119878119900119897119898119886119909 and lower bound solution 119878119900119897119898119894119899 must be defined asfollows
Step 3 Handle prohibited power zone constraint for decisionvariables P2 P3 PN
After being randomly produced there is a high possi-bility that decision variables fall into PPZ and they violatePPZ constraint So the verification of falling into PPZ andcorrection of the violation should be accomplished by usingthe following formula
119875119894 =
119875119897119894119896 if 119875119897119894119896 lt 119875119894 le 119875119897119894119896 + 1198751198961198941198962119875119906119894119896 if (119875119894 gt 119875119897119894119896 + 1198751198961198941198962 ) amp (119875119894 lt 119875119906119894119896)119875119894 119890119897119904119890
119894 = 2 119873 amp 119896 = 1 119899119894
(31)
Step 4 Handle RPB constraint by calculating 1198751 and penaliz-ing 1198751 if it violates constraints
In this step power balance constraint is exactly handledby calculating and penalizing dependent variable (1198751) 1198751 isobtained by using formulas (4) and (5) as follows
1198751 = minus (11986101 minus 1 + 2sum119873119894=2 1198611119894119875119894) plusmn radicΔ211986111 (32)
where
Δ = (11986101 minus 1 + 2 119873sum119894=2
1198611119894119875119899)2 minus 411986111(119875119863 minus 119873sum
119894=2
119875119894+ 11986100 + 119873sum
119894=2
1198610119894119875119894 + 119873sum119894=2
119873sum119895=2
119875119894119861119894119895119875119895) amp Δ ge 0(33)
In Equation (32) 1198751 has been determined for the purposeof dealing with real power balance constraint However it isnot sure that 1198751 can satisfy upper bound and lower boundconstraints and prohibited power zone constraints So 1198751must be checked and penalized
Firstly 1198751 is checked and penalized for upper and lowerbound constraints by the following model
Δ1198751x =
0 if 1198751min le 1198751x le 1198751max
1198751min minus 1198751x if 1198751min gt 1198751x1198751x minus 1198751max if 1198751max lt 1198751x
(34)
In Equation (34) if the second case or the third caseoccurs it means P1 has violated either lower bound or upperbound and it would be penalized by using either (P1x= P1min-P1x) or (P1x =P1x -P1max) Otherwise ifP1 has not violatedthe bound constraints (ie the first case in (34) happened)
P1 would continue to be checked for PPZ constraint by thefollowing model
Δ1198751x
=
1198751 minus 1198751198971119896 if 1198751198971119896 lt 1198751 le 1198751198971119896 + 119875119896111989621198751199061119896 minus 1198751 if (1198751 gt 1198751198971119896 + 11987511989611198962 ) amp (1198751 lt 1198751199061119896)0 119890119897119904119890
(35)
Step 5 Handle real power reserve constraint (10)First of all 119878119894 is determined by using (11) and (12) and
then the 119909119905ℎ solution will be checked and penalized if poweroutput of all thermal generating units cannot satisfy RPRconstraint The penalty for violation of the constraint can becalculated by using equation (36)
Δ119878119877119909 =
0 if119873sum119894=1
119878119894119909 ge 119878119877119878119877 minus 119873sum119894=1
119878119894119909 119890119897119904119890 (36)
As a result real power reserve constraint can be solved byusing the penalty method
42 Calculate Fitness Function for Solutions Fitness functionof each solution is used to evaluate quality of solutionNormally the function is the sum of objective function andpenalty of violating constraints and is obtained by
43 The First Newly Updated Solutions by Levy Flights Tech-nique In this section the first newly updated solutionsare performed by employing Levy flights technique usingEquation (14) However each new solution can be out oftheir feasible operating zone such as PPZ and upper andlower limitations When the power output violates its PPZconstraints Equation (31) will be applied to tackle theconstraint Besides the following equation will be employedwhen power output is higher or lower than their limitations
119878119900119897119909 =
119878119900119897max if 119878119900119897max lt 119878119900119897119909119878119900119897min if 119878119900119897min gt 119878119900119897119909119878119900119897119909 Otherwise
119909 = 1 119873119901 (38)
After that Equations (32)-(37) are performed for deter-mining all variables and penalty terms Finally Equation (38)is employed to calculate fitness function
44 The Second Newly Updated Solutions by Using Muta-tion Technique The second newly updated solutions areaccomplished as presented in Section 3 above Similar to
Mathematical Problems in Engineering 11
the task after doing the first update each solution in thenew population must satisfy PPZ constraint and upper andlower boundaries by considering Equations (31) and (38)Then Equations (32)-(37) are performed for determiningall variables and penalty terms Finally Equation (38) isemployed to calculate fitness function and the solution withthe best value is assigned to the best one 11987811990011989711986611988711989011990411990545 Criterion of Stopping the Loop Algorithm In the loopalgorithm of using the proposed ICSA approach the solutionsearch work is stopped in case that the predeterminedmaximum iterations 119866119898119886119909 is reached For each search ter-mination the most effective solution is stored and anotherrun continues to be accomplished until the predeterminednumber of runs is reached After finishing the runs thebest one is found and reported In addition other valuessuch as the fitness of the worst solution and average fitnessof all solutions are also reported for comparing with othermethods
46 The Whole Iterative Process The whole iterative algo-rithm for implementing the proposed ICSA approach forcoping with OLD problem is described in detail in Figure 3
5 Results and Discussions
The proposed ICSA approach performance has been investi-gated on six cases with different fuel options different fuelcharacteristics and complicated constraints The details ofthe studied cases are presented as follows
Case 1 Four systems with single fuel source (SFS) and powerloss (PL) constraint
Subcase 11 A 3-unit system [57]Subcase 12 A 6-unit system [57]Subcase 13 A 3-unit system [56]Subcase 14 A 6-unit system [56]
Case 2 A 110-unit system with SFS [57]
Case 3 Four systems with SFS and the effects of valve loadingprocess (EoVLP)
Subcase 31 A 3-unit system supplying to a load of850MW [58]Subcase 32 A 13-unit system supplying to a load of1800MW [1]Subcase 33 A 13-unit system supplying to a load of2520MW [1]Subcase 34 A 40-unit system supplying to a load of2500MW [1]Subcase 35 An 80-unit system supplying to a load of4100MW [49]
Case 4 Two systems with SFS and PPZ and RPR constraints
Subcase 41 A 60-unit system supplying to a10600MW load [9]
Subcase 42 A 90-unit system supplying to a15900MW load [9]
Case 5 A 15-unit system with SFS and RRL PPZ and PLconstraints [61]
Case 6 Three systems with multiple fuel sources (MFS) andEoVLP
Subcase 61 An 80-unit system supplying to a21600MW load [15]Subcase 62 A 160-unit system supplying to a43200MW load [15]Subcase 63 A 320-unit system supplying to an86400MW load [54]
For each considered case with each load case the pro-posed ICSA approach is run 50 times on the programlanguage of Matlab and a PC with 4 GB of RAM and 24GHzprocessor The selection of adjustment parameters including119875119886 and 119879119900119897119909 is carefully considered to obtain the best optimalsolutions meanwhile two others such as 119873119901119904 and 119866119898119886119909 arechosen corresponding to the scale of particular test system9 values with the change of 01 in the range [01 09] are inturn selected for 119875119886 while 119879119900119897119909 is 001 at the beginning Theinformation including load demand119873119901119904 119866119898119886119909 and the best119875119886 is reported in Table 1
51 Obtained Results on Case 1 considering Four Systems withSFS and PL Constraint In this section we have implementedthe proposed ICSA approach for solving four systems dividedinto four subcases Tables 2 and 3 show the comparisons ofobtained results from Subcases 11 and 12 and Subcases 13and 14 respectively As listed in Table 2 the proposed ICSAmethod and CCSA can find equal fuel cost for Subcases 11whereas the reduction of fuel cost from the proposed ICSAmethod as compared to CCSA is clearer for Subcase 12 Asshown in Table 3 for comparing the proposed ICSA and threemethods consisting of CCSA ABC and FA the minimumfuel cost of the proposed ICSA is approximately equal to thatof these methods for Subcases 13 but much less than that ofthese methods for Subcase 14 Furthermore the proposedICSA has been run by setting 119873119901119904 and 119866119898119886119909 to 5 and 20but these values were much higher for CCSA ABC and FAThey are 20 and 5000 for CCSA 40 and 100 for ABC and 20and 5000 for FA Consequently the proposed method is veryefficient for Case 1 with four subcases
Optimal solutions obtained by ICSA for Case 1 are shownin Tables 16ndash18
52 Obtained Results on Case 2 considering 110-Unit Systemwith SFS In this section we have employed a very largescale system with 110 units but there were not challenges forobjective function and complex constraints since EoVLP andconstraints were not taken into account Both CCSA andthe proposed ICSA methods have been run for comparingwith BBO hybrid BBO and DE (DEBBO) and Opposi-tional real coded chemical reaction optimization algorithm(ORCCROA) in [36] IWA in [40] and AGWO in [52] As
12 Mathematical Problems in Engineering
Select parameters
- Correct solutions if violating PPZ constraint using Eq (31)- Determine using Eq (32)- Penalize for violating upper and lower bounds constraint using Eq (34)- Penalize for violating PPZ constraint using Eq (35)- Penalize the xth solution for violating RPR constraint using Eq (36)
- Determine quality of solutions employing Eq (37) - Select the most effective solution - Start the loop algorithm by selecting
- Perform the first solution update using Section 32 - Correct boundaries of solutions by using Eq (38)
- Correct solutions if violating PPZ constraint using Eq (31)- Determine using Eq (32)- Penalize for violating upper and lower bounds constraint using Eq (34)- Penalize for violating PPZ constraint using Eq (35)- Penalize the xth solution for violating RPR constraint using Eq(36)
- Determine fitness function using Eq (37)- Compare old solutions and new solutions to keep better ones
- Perform the second solution update using Algorithm 1 - Correct boundaries of solutions by using (38)
- Correct solutions if violating PPZ constraint using Eq (31)- Determine using Eq (32)- Penalize for violating upper and lower bounds constraint using Eq (34)- Penalize for violating PPZ constraint using Eq (35)- Penalize the xth solution for violating RPR constraint using Eq (36)
- Determine fitness function using Eq (37)- Compare old solutions and new solutions to keep better ones- Select the most effective solution
Stop
Start
- Redefine upper and lower bounds using Eqs (25)-(28)- Randomly generate initial population using Eq (30)
Nps Pa Gmax H> Tolx
P1xP1x
P1x
P1xP1x
P1x
P1xP1x
P1x
Sol<MN
Sol<MN
G = 1
G = Gmax G = G + 1
Figure 3 All computation steps for solving OLD problem by employing the proposed ICSA approach
shown in Table 4 AGWO [52] has reached less fuel cost thanICSA however the exact fuel cost which was recalculatedby using reported solution pointed out that the method hasreached a very high fuel cost of $2157404250 For comparisonwith other methods ICSA has found less fuel cost thanall these methods Particularly the reduction of generationfuel cost is significant as compared to BBO DEBBO andCCSA Execution time comparisons are also useful evidence
for indicating the high performance of ICSA Thus it canconclude that ICSA is a strong method for Case 2
Optimal solution obtained by ICSA for the case is shownin Table 19
53 Obtained Results on Case 3 considering Four Systems withSFS and EoVLP In this section the real performance of theproposed ICSA approach has been investigated based on five
Mathematical Problems in Engineering 13
Table 1 Information of considered cases and adjustment parameters
Case Fuel cost function Constraint Subcase No of units 119875119863 (MW) Nps Gmax Best 1198751198861 SFS PL
subcases with the gradual increase of number of units Thesmallest scale system considers 3 units but the largest scalesystem takes 80 units In addition to the implementation ofthe proposed ICSA we have also implemented CCSA forSubcase 31 and Subcase 35 for further comparison becauseCCSA has not been run for the two subcases so far
Comparison of obtained results from Subcase 31 shownin Table 5 indicates that the proposed ICSA is superior toCCSAwith lower fuel cost but it seems to be less effective thanMCSA [58] The minimum cost of MCSA reported in [58] isthe smallest fuel cost but the recalculated cost is much higherthan that of the proposed method Furthermore MCSA hasbeen implemented by setting very high values to 119873119901119904 and119866119898119886119909
Reports for Subcases 32 and 33 shown in Table 6are the comparisons of the proposed ICSA approach andother methods such as conventional Evolution programming(CEP) [1] Fast EP (FEP) [1] improved FEP (IFEP) [1] DE[12] multiplier Lagrange-based genetic algorithm with (GA-MU) [15] QPSO [16] GA-PS-SQP [30] PSO-SQP [32] M120573-HCLSA [49] IABCA [50] CCSA [59] OSE-CSA [59] SOS[34] MSOS [34] CEA-SQT [38] TSBO [39] IWA [40] andCBA [44] As observed from the table ICSAapproach obtainsbetter solutions than mostmethods excluding DE [10] CCSA
[59] OSE-CSA [59] SOS [34] MSOS [34] CEA-SQT [38]TSBO [39] IWA [40] and CBA [44] especially M120573-HCLSA[49] with lower cost $1796097 However recalculated costfrom reported solution of M120573-HCLSA is $179691 BesidesICSA is very fast as compared to most methods where twoother versions of Cuckoo search algorithm CCSA and OSE-CSA are also included except two methods in [34] Theprocessor of computer that all the methods run on is alsoreported in the final column Clearly ICSA approach is veryefficient for the case with the 13-unit system where effects ofvalve loading process are considered
In Subcase 34 the number of units is much larger thanthat of three subcases above up to 40 units [1] The obtainedresult comparisons with others are indicated in Table 7Clearly the minimum cost comparisons reveal that the pro-posedmethod is one of the leading methods due to the lowestcost except the comparison with CCSA [23] OSE-CSA [59]SOS [34] MSOS [34] EMA [45] 120579-MBA [47] and AGWOA[52] It is noted that AGWOA [52] has reported the bestminimum cost with $12140430 but recalculated minimumcost which was obtained by substituting reported optimalgeneration of all thermal generating units is $12141331 Theaverage and the maximum costs from the proposed methoddo not belong to the leading method group however the
Mathematical Problems in Engineering 15
Table 7 Result comparisons for Subcase 34
Approach Best cost Average cost Worst cost CPU time Computer($h) ($h) ($h) (s) (Processor-Ram)
execution time that ICSA approach takes is smaller thanmostones except IAPR [42] and 120579-MBA [47] which have been runon stronger computers Note that MSOS [34] has been fasterthan the proposed method for Subcases 31 and 32 abovebut it is too slower than the proposed method for the casenamely 1813 seconds compared to 146 seconds Comparedto two other versions of Cuckoo search CCSA [23] and OSE-CSA [59] the proposed method is also faster about threetimes although the processors are slightly different For thiscase 120579-MBA [47] shows a very good performance howeverthe method has not been tested onmore complicated systemsand larger scale and therefore more comparisons with themethod must end Clearly the proposed method is stillefficient for the case where large scale and effects of valveloading process are included
Subcase 35 is the largest scale system with 80 units Inaddition to the implementation of ICSA CCSA has beenalso run for the Subcase 35 for further investigation ofefficiency improvement of the proposed ICSA approach The
comparisons of minimum cost in Table 8 show that theproposed ICSA can find more optimal solution than M120573-HCLSA [49] AGWOA [52] and CCSA The proposed ICSAis also superior to CCSA in terms ofmore stable search abilityand lower fluctuation since its average cost and maximumcost are less than those of CCSA The outstanding figurecannot be reached as compared to AGWOA [52] howeverit is hard to conclude AGWOA [52] is superior to theproposed ICSA approach about more stable search abilityand lower fluctuation Actually comparison of the values ofpopulation and iterations as well as execution time cannot beaccomplished because the information was not reported in[52]Thus it can conclude that the proposed ICSA is effectivefor the subcase
Optimal solutions obtained by ICSA for the case areshown in Tables 20ndash23
54 Obtained Results on Case 4 with Two Systems consideringSFS and PPZ and RPR Constraints In this section two
16 Mathematical Problems in Engineering
Table 9 Result comparisons for Subcases 41 and 42
Subcase Approach Best cost ($h) Mean cost ($h) Worst cost ($h) Std dev ($h) CPU time (s) Computer (Processor-Ram)
test systems with SFS and PPZ and RPR constraints areconsidered The test system size is up to 60 and 90 units forSubcases 41 and 42 respectively Comparison for the casesis only performed with two Genetic algorithms consisting ofGA and IGA-MU in [9] and two other versions of Cuckoosearch algorithms including CCSA [23] and OSE-CSA [59]and presented in Table 9 Other studies have tended toignore such complicated constraints of PPZ and RPR Thecosts reported in Table 9 indicate that ICSA approach canobtain more effective solution than CCSA and OSE-CSAbecause it has reached lower minimum cost than the twoones Furthermore the proposed method also takes shortercomputation time for the two cases from about two timesto about three times although the processor of the proposedmethod is slightly strongerThemean costs of ICSA approachare much less than those from IGA-MU and GA and slightlyhigher than those from OSE-CSA but there is a trade-off between the proposed method and CCSA for the twosubcases In fact the proposed method obtains higher meancost for Subcase 41 but lower cost for Subcase 42 AlthoughGAmethods have been runon aweak computerwith 07 GHzof the processor compared to that with 24GHz in the studytheir execution times are significantly higher namely 56381seconds (GA) and 16258 seconds (IGA-MU) compared to09153 seconds of ICSA approach for Subcase 41 and 94093seconds (GA) and 25545 seconds (IGA-MU) compared to15892 seconds (the proposedmethod)The analysis can pointout that ICSA approach ismore efficient than these comparedmethods in terms of optimal solutions and execution time
Optimal solutions obtained by ICSA for Subcase 42 areshown in Table 24
55 Obtained Results on Case 5 with a 15-Unit System consid-ering SFS and RRL PPZ and PL Constraints In this section
a 15-unit system considering RRL PPZ and PL constraintsis considered to be solved for finding optimal solution Forefficiency investigation of the proposed ICSA we have alsoimplemented CCSA for comparison As listed in Table 10the proposed ICSA is the most effective method with thesmallest fuel cost The comparisons of control parameters aswell asCPU time are also good evidence to confirm the strongsearch of the proposed ICSA approach since it has been runby smaller values of control parameter and faster executiontime as compared to all methods excluding CCSA
Optimal solution obtained by ICSA for the case is shownin Table 25
56 Obtained Results on Case 6 withThree Systems consideringMFS and EoVLP In this section three test systems with thechallenge on objective function including multi-fossil fuelsources and effects of valve loading process are consideredThe scale is up to 80 units 160 units and 320 units forSubcases 61 62 and 63 respectively
Comparison for Subcase 61 reported in Table 11 revealsthat the proposed method is the best method in terms of thelowest best cost the lowestmean cost and the lowest standarddeviation and the fastest execution time The processor fromthis proposedmethod is about four times stronger thanCGA-MU and IGA-MU but the speed is from ten times to 35 timesfaster than these methods Compared to CCSA and OSE-CSA the proposed method is about two times faster but theprocessor is slightly stronger
Comparison for Subcase 62 is reported in Table 12Clearly the proposed ICSA approach obtains better values ofthe best mean and worst costs than most methods exceptMSOS [34] where the best cost difference is about $ 024However the proposed method is the second fastest onewith 1119 seconds where the first fastest one CBA [44] has
Mathematical Problems in Engineering 17
Table 11 Comparisons of found results for Subcase 61
Approach Best cost Mean cost Worst cost Std dev CPU time Computer($h) ($h) ($h) ($h) (s) (Processor-Ram)
spent 571 seconds Clearly MSOS is better than the proposedICSA approach with respect to slightly less best cost butis worse than the proposed method in terms of executiontime namely 2572 seconds compared to 1119 seconds whileCBA [44] is faster than the proposed method but obtainssignificantly worse costs The analysis can conclude that theproposed ICSA approach is very powerful for the subcasewith 160 units
Table 13 presents the comparison of three other methodsincludingCCSA [54] SOS [34] andMSOS [34] accompaniedwith the proposed method for Subcase 63 The obtainedresult comparisons imply that ICSA approach can obtainbetter values of the best and standard deviation costs thanCCSA and SOS but obtains slightly higher cost than MSOSby approximately $ 022 Besides the execution time fromICSA approach is much shorter than others especially it ishigher than five times faster than MSOS The four methodshave been run on approximately strong computers Brieflythe proposed ICSA approach can find and converge to morefavorable solution than other methods with shorter CPUtime except the comparison with MSOS which had bettersolution but spent higher than five times execution times
Consequently the proposed method is a very promisingoptimization algorithm for Subcase 63 a system up to 320units and with multi-fossil fuel sources and effects of valveloading process
Optimal solution obtained by ICSA for Subcase 63 isshown in Table 26
57 The Improvement of ICSA Approach Performance
571 The Outstanding Improvement over CCSA In this sec-tion the performance improvement of ICSA over CCSAhas been investigated by analyzing obtained results and setcontrol parameters Table 14 has been formed by addingreduction cost improvement level of the best cost executiontime and control parameters consisting of 119873119901119904 and 119866119898119886119909Among the compared factors reduction cost is the deviationof the cost of CCSA and that of ICSA whereas the improve-ment level is the ratio of the reduction cost to the cost ofCCSA The reduction costs indicate the proposed methodcould find either equal quality of solutions or higher qualityof solutions than CCSA for all study casesThe reduction costis from $0 to $42839 corresponding to the improvement level
18 Mathematical Problems in Engineering
Table 14 Summary of results obtained by CCSA and ICSA for all study cases
Study case Reduction cost ($) Improvement level () Execution time (s) Nps Gmax
Figure 4 The best run obtained by CCSA and ICSA for Case 2
that is from 0 to 08968The saving cost is not too much forone hour but the operation in one day onemonth or one yearis very high However it should be noted that CCSA has beenrun by setting much higher population size and iterations formany cases excluding study cases implemented in the studysuch as Case 2 Subcase 31 Subcase 35 and Case 5 Forinstance ICSA has used 119873119901119904 = 10 and 119866119898119886119909 = 15 for Subcase13 and119873119901119904 = 10 and119866119898119886119909 = 25 for Subcase 14 whereas CCSAhas been run by setting119873119901119904 = 20 and 119866119898119886119909 = 5000 for the twosubcases Similarly CCSA has been run for Subcases 32 33and 34 with much higher number of iterations For the lastsubcase CCSA has been run by setting 119873119901119904 = 320 and 119866119898119886119909
= 1200 but those of ICSA have been 10 and 9000 Due tothe higher value of control parameters CCSA has tended tospend more time in finding such high quality solutions foralmost all study cases Execution time of ICSA is less than18 seconds while that of CCSA is up to higher 75 secondsIt is clear that the proposed ICSA could find better optimalsolutions thanCCSA for such considered systems For furtherinvestigation of performance comparison the best runs over50 runs and fuel cost values of 50 runs obtained by CCSA andthe proposed ICSA for Case 2 Subcase 31 Subcase 35 andCase 5 have been plotted in from Figures 4ndash11 The best runcurves show the faster search of the proposed ICSA method
Mathematical Problems in Engineering 19
0 5 10 15 20 25 30 35 40 45 50Run
19795
198
19805
1981
19815
1982
19825
1983
19835
Fuel
cost
($)
CCSAICSA
times105
Figure 5The best fuel cost of 50 runs obtained by CCSA and ICSA for Case 2
0 2 4 6 8 10 12 14 16 18 20Iteration
8234
8236
8238
8240
8242
8244
8246
8248
8250
8252
Fitn
ess F
unct
ion
($)
CCSAICSA
Figure 6 The best run obtained by CCSA and ICSA for Subcase 31
whereas 50 values of fuel cost indicate that the proposedICSA can find many solutions with better quality Clearlythe proposed ICSA is outstanding in terms of stabilizationof solution search and faster convergence As a result it canconclude that the proposed ICSA approach is more effectivethanCCSA in solvingOLDproblemwith considered systems
572The Improvement of Results over Other Methods In thisarticle we have tested ICSA approach on 6 cases with 16
systems with different fuel cost forms different constraintsand different scale systems from 3 units to 320 units We havecompared the yielded results from ICSA approach and otherexisting ones for evaluating the efficiency of ICSA approachIn subsections above we have shown yielded results fromICSA approach and compared these results to those of otheronesHowever the demonstrationhad not been very good forobserving and comparing to lead to a conclusionThus in thesubsectionwehave summarized the result comparisons of the
20 Mathematical Problems in Engineering
0 5 10 15 20 25 30 35 40 45 50Run
8234
82342
82344
82346
82348
8235
82352
82354
82356
82358
8236
Fuel
cost
($)
CCSAICSA
Figure 7 The best fuel cost of 50 runs obtained by CCSA and ICSA for Subcase 31
0 1000 2000 3000 4000 5000 6000Iteration
242
244
246
248
25
252
254
256
Fitn
ess F
unct
ion
($)
CCSAICSA
times105
Figure 8 The best run obtained by CCSA and ICSA for Subcase 35
proposed and other ones Table 15 has reported the reductioncost (in $) of ICSA approach compared to other ones Inaddition we have converted the reduction cost into improve-ment level (in ) for better comparison The improvementhas been shown from the lowest level to the highest levelin terms of reduction cost and improvement percentageIn addition we have also given the slowest and the fastestexecution time of other compared methods together withthat of the proposed method The table implies that ICSA
approach can find better optimal solutions with less fuel costup to $052 for Subcase 13 $75229 for Subcase 14 $1775149for Case 2 $1181256 for Subcase 31 $8438 for Subcase 32$9113 for Subcase 33 $121181 for Subcase 34 $3412 forSubcase 35 $04329 for Subcase 41 $30227 for Subcase42 $407002 for Case 5 $18183 for Subcase 61 $13965 forSubcase 62 and $11817 for Subcase 63 These reductioncosts are equivalent to improvement level (IL) of 0032908968 82282 14143 047 038 099 001405
Mathematical Problems in Engineering 21
0 5 10 15 20 25 30 35 40 45 50Run
2428
243
2432
2434
2436
2438
244
2442
Fuel
cost
($)
CCSAICSA
times105
Figure 9 The best fuel cost of 50 runs obtained by CCSA and ICSA for Subcase 35
0 50 100 150 200 250 300 350 400Iteration
327
328
329
33
331
332
333
334
335
336
337
Fitn
ess F
unct
ion
($)
CCSAICSA
times104
Figure 10 The best run obtained by CCSA and ICSA for Case 5
00003 0002 12291 004 138 and 059 Thesequantitative comparisons reveal that larger scale systems canlead to better reduction cost but the improvement level isnot high because total cost of compared methods tends tobe large for large scale systems Furthermore very large scalesystems with nondifferentiable objective have been normallysolved by strong methods In fact systems in Case 3 havethe same characteristic with single fuel and effects of valveloading process but Subcase 34 is a larger scale system with
40 units while Subcases 31 32 and 33 are constructed by 3units 13 units and 13 units So the improvement percentageof Subcase 34 can be up to 099 whilst that of Subcases32 and 33 is 047 and 038 respectively Subcase 35 iswith the largest system 80 units but the reduction cost is notmuch only $3412 because compared methods with ICSA areeither state-of-the-art ones or improved ones Also Subcase41 and Subcase 42 have considered single fuel and PPZ andspinning reserve constraints but Subcase 42 is larger scale
22 Mathematical Problems in Engineering
Table 15 Performance improvement summary of the proposed method
Study cases Reduction cost ($) Improvement level () Execution time (s)From To From To Slowest method Fastest method Proposed method
Figure 11 The best fuel cost of 50 runs obtained by CCSA and ICSA for Case 5
system with 90 units and Subcase 41 is only with 60 unitsThus the improvement of Subcase 41 is lower with 00003but that of Subcase 42 is 0002 Similarly systems in Cases 6have the same featurewithmulti-fossil fuel sources and effectsof valve loading process but they are respectively constructedby 80 160 and 320 units As a result the improvements ofSubcase 62 138 and Subcase 63 059 are much higherthan Subcase 61 004 However Subcase 62 with smallernumber of units but getting higher improvement is easilyunderstood because there were nine compared methods butonly three compared methods are considered for Subcase 63In general the improvement is not high it is about under onedollar several dollars tens of dollars and over one thousand
dollars per hour however the saving cost will be significantif the operation is considered to be onemonth with 720 hoursor one year with 8760 hours
Execution time comparison can be evaluated by observ-ing the fastest and the slowest compared methods in Table 15These execution times of the proposed method are approxi-mately equal to that of the fastest methods for Subcases 3233 and 34 and much shorter than other fastest comparedmethods for other cases especially for Subcases 61 and 63The fastest method for Subcase 62 is CBA [44] showing 57seconds while that of the proposed method is 1119 secondsHowever it cannot conclude that CBA is more effective thanthe proposed method because the proposed method could
find better optimal solution with less fuel cost by $2130As considering execution time of the slowest comparedmethods it can point out that ICSA is a very fast optimizationtool since the execution time of these methods is 29496 and3425 seconds for Subcases 32 and 33 116735 seconds forSubcase 34 56381 seconds for Subcase 41 94093 secondsfor Subcase 42 12797 seconds for Case 5 30941 seconds forSubcase 61 7542 seconds for Subcase 62 and 9641 secondsfor Subcase 63 while the execution time of the proposedmethod for these cases is respectively 095 146 091 1589246 84828 1119 and 171384 seconds It is clearly shown thatICSA is very fast as compared to these methods
In summary the proposed method has found approx-imately high quality solutions with several standard state-of-the-art meta-heuristic algorithms and improved versionsof them together with other old methods In addition theproposed method could improve result better than approx-imately all methods with faster execution time Comparedto other methods with the fastest convergence speed andhigh quality solutions the proposed method has been as
fast as for some cases and much faster for other cases Thecomparison with the slowest methods could show that theproposed method was extremely powerful since it was up tonearly one thousand times faster Consequently the proposedICSA approach can be one of the strongest optimization toolsfor OLD problem
6 Conclusions
This paper has proposed a good ICSA method for solvingOLD problem in which many test systems with differentobjective functions and complicated constraints from simpleto complex have been used as studied cases The proposedICSA method has been developed by performing severalmodifications on the second solution update of CCSA whichcontained several drawbacks to global convergence and fastmanner The OLD problem has covered from single fuelto multi-fossil fuels from quadratic objective function tononconvex objective function in addition to PPZ RPR andRRL constraints Many existing optimization algorithms have
been concerned in aim to compare the performance and givethe final conclusion on the proposed method There have
been six main cases with sixteen subcases The evaluationshave been made at the end of each study case Clearly theproposed ICSA approach has yielded more effective optimalsolutions with faster execution time than almost all methodsConsequently it can be concluded that the proposed methodis much more superior to CCSA and is a very promisingmethod for solving OLD problem
Appendix
See Tables 16ndash26
Nomenclature
120575119894 120582119894 120572119894 120573119894 120574119894 Fuel cost functioncoefficients of the 119894119905ℎthermal generationunit11986100 1198610119895 119861119895119894 Power loss matrixcoefficients119865119865119909 119865119865119887119890119904119905 The values of fitness ofsolution 119909 and theso-far most effectivesolution among thecurrent set ofsolutions119898119894 Number of fuels burntin the 119894119905ℎ thermalgeneration unit
119873 Number of allavailable thermalgeneration units119899119894 Number of prohibitedpower zones of the 119894119905ℎthermal generationunit119875119906119894119896 119875119897119894119896 Upper and lowerlimits of the 119894119905ℎthermal generationunit corresponding tothe kth PPZ119875119886 Probability ofreplacing controlvariables in each oldsolution119875119863 Real power demand ofall loads in system119875119894119898119886119909 119875119894119898119894119899 The highest andlowest real poweroutputs of the iththermal generationunit119875119894119895119898119886119909 119875119894119895119898119894119899 The highest andlowest real poweroutputs of the 119894119905ℎthermal generationunit corresponding tothe 119895119905ℎ fuel type119878119894119898119886119909 Maximum real powerreserve contributionof the thermalgeneration unit 1198941198781199001198971199031198861198991198891 1198781199001198971199031198861198991198892 1198781199001198971199031198861198991198893 1198781199001198971199031198861198991198894 Randomly mixedsolutions from the setof current solutions119878119900119897119909 119878119900119897119866119887119890119904119905 The old solution x andthe most effectivesolution119878119877 Real power reserverequirement of system
120575119894119895 120582119894119895 120572119894119895 120573119894119895 120574119894119895 Fuel cost functioncoefficients of the 119894119905ℎthermal generationunit corresponding tothe 119895119905ℎ fuel type1205761 1205762 1205763 1205764 1205765 1205766 Random numbersbetween 0 and 1Ω Number of generationunits considering PPZconstraint
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare no conflicts of interest
References
[1] N Sinha R Chakrabarti and P K Chattopadhyay ldquoEvolution-ary programming techniques for economic load dispatchrdquo IEEETransactions on Evolutionary Computation vol 7 no 1 pp 83ndash94 2003
[2] P Hansen ldquoA separable approximation dynamic programmingalgorithm for economic dispatch with transmission lossesrdquoYugoslav Journal of Operations Research vol 12 no 2 2002
[3] N T Thang ldquoSolving economic dispatch problem with piece-wise quadratic cost functions using lagrange multiplier theoryrdquoin Proceedings of the 3rd International Conference on ComputerTechnology and Development (ICCTD rsquo11) pp 359ndash364 ASMEPress 2011
[4] S K Mishra and S K Mishra ldquoA comparative study of solutionof economic load dispatch problem in power systems in theenvironmental perspectiverdquoProcedia Computer Science vol 48pp 96ndash100 2015
[5] A A Al-Subhi and H K Alfares ldquoEconomic load dispatchusing linear programming a comparative studyrdquo InternationalJournal of Applied Industrial Engineering vol 3 no 1 pp 16ndash362016
28 Mathematical Problems in Engineering
[6] C Zhou G Huang and J Chen ldquoPlanning of electric powersystems considering virtual power plants with dispatchableloads included an inexact two-stage stochastic linear program-ming modelrdquoMathematical Problems in Engineering vol 2018Article ID 7049329 12 pages 2018
[7] T T Nguyen N V Quynh and L Van Dai ldquoImproved fireflyalgorithm a novel method for optimal operation of thermalgenerating unitsrdquo Complexity vol 2018 Article ID 7267593 23pages 2018
[8] S S Haroon S Hassan S Amin et al ldquoMultiple fuel machinespower economic dispatch using stud differential evolutionrdquoEnergies vol 11 no 6 pp 1ndash20 2018
[9] C-T Su and C-L Chiang ldquoNonconvex power economic dis-patch by improved genetic algorithm with multiplier updatingmethodrdquo Electric Power Components and Systems vol 32 no 3pp 257ndash273 2004
[10] NNoman andH Iba ldquoDifferential evolution for economic loaddispatch problemsrdquo Electric Power Systems Research vol 78 no8 pp 1322ndash1331 2008
[11] P Somasundaram and K Kuppusamy ldquoApplication of evolu-tionary programming to security constrained economic dis-patchrdquo International Journal of Electrical Power amp EnergySystems vol 27 no 5-6 pp 343ndash351 2005
[12] T Jayabarathi K Jayaprakash andD Jeyakumar ldquoEvolutionaryprogramming technique for different kinds of economic dis-patch problemsrdquo Electric Power Systems Research vol 73 no 2pp 169ndash176 2005
[13] P Subbaraj R Rengaraj and S Salivahanan ldquoEnhancementof combined heat and power economic dispatch using selfadaptive real-coded genetic algorithmrdquo Applied Energy vol 86no 6 pp 915ndash921 2009
[14] C-L Chiang and C-T Su ldquoAdaptive-improved genetic algo-rithm for the economic dispatch of units with multiple fueloptionsrdquo Cybernetics and Systems vol 36 no 7 pp 687ndash7042005
[15] C Chiang ldquoImproved genetic algorithm for power economicdispatch of units with valve-point effects and multiple fuelsrdquoIEEE Transactions on Power Systems vol 20 no 4 pp 1690ndash1699 2005
[16] K Meng H G Wang Z Y Dong and K P Wong ldquoQuantum-inspired particle swarm optimization for valve-point economicload dispatchrdquo IEEE Transactions on Power Systems vol 25 no1 pp 215ndash222 2010
[17] P Erdogmus A Ozturk and S Duman ldquoEnvironmen-taleconomic dispatch using genetic algorithm and simulatedannealingrdquo in Proceedings of the International Conference onElectric Power and Energy Conversion Systems EPECS rsquo09 pp1ndash4 IEEE 2009
[18] L D S Coelho and V C Mariani ldquoImproved differentialevolution algorithms for handling economic dispatch opti-mization with generator constraintsrdquo Energy Conversion andManagement vol 48 no 5 pp 1631ndash1639 2007
[19] H Dakuo W Fuli and M Zhizhong ldquoA hybrid genetic algo-rithm approach based on differential evolution for economicdispatch with valve-point effectsrdquo Electrical Power and EnergySystems vol 30 pp 31ndash38 2008
[20] C-F Chang J-J Wong J-P Chiou and C-T Su ldquoRobustsearching hybrid differential evolution method for optimalreactive power planning in large-scale distribution systemsrdquoElectric Power Systems Research vol 77 no 5-6 pp 430ndash4372007
[21] M R Farooqi P Jain and K R Niazi ldquoUsing Hopfield neuralnetwork for economic dispatch of power systemsrdquo in Proceed-ings of the National Power Engineering Conference PECon rsquo03pp 5ndash10 Bangi Malaysia 2003
[22] T Yalcinoz H Altun and U Hasan ldquoConstrained economicdispatch with prohibited operating zones a Hopfield neuralnetwork approachrdquo in Proceedings of the 10th MediterraneanElectrotechnical Conference Information Technology and Elec-trotechnology for the Mediterranean Countries ProceedingsMeleCon rsquo00 (Cat No 00CH37099) pp 570ndash573 LemesosCyprus 2000
[23] D N Vo P Schegner and W Ongsakul ldquoCuckoo searchalgorithm for non-convex economic dispatchrdquo IET GenerationTransmission amp Distribution vol 7 no 6 pp 645ndash654 2013
[24] J-B Park K-S Lee J-R Shin and K Y Lee ldquoA particleswarm optimization for economic dispatch with non-smoothcost functionsrdquo IEEETransactions on Power Systems vol 20 no1 pp 34ndash42 2005
[25] T Niknam H D Mojarrad and M Nayeripour ldquoA newfuzzy adaptive particle swarm optimization for non-smootheconomic dispatchrdquo Energy vol 35 no 4 pp 1764ndash1778 2010
[26] L D S Coelho and V C Mariani ldquoAn improved harmonysearch algorithm for power economic load dispatchrdquo EnergyConversion and Management vol 50 no 10 pp 2522ndash25262009
[27] M Mahdavi M Fesanghary and E Damangir ldquoAn improvedharmony search algorithm for solving optimization problemsrdquoAppliedMathematics and Computation vol 188 no 2 pp 1567ndash1579 2007
[28] A Bhattacharya and P K Chattopadhyay ldquoSolving complexeconomic load dispatch problem using biogeography-basedoptimizationrdquo Expert Systems with Applications vol 37 no 5pp 3605ndash3615 2010
[29] D Simon ldquoBiogeography-based optimizationrdquo IEEE Transac-tions on Evolutionary Computation vol 12 no 6 pp 702ndash7132008
[30] J S Alsumait J K Sykulski and A K Al-Othman ldquoAhybrid GA-PS-SQP method to solve power system valve-pointeconomic dispatch problemsrdquo Applied Energy vol 87 no 5 pp1773ndash1781 2010
[31] M S Turgut andGKDemir ldquoQuadratic approximationndashbasedhybrid Artificial Cooperative Search algorithm for economicemission load dispatchproblemsrdquo International Transactions onElectrical Energy Systems vol 27 no 4 pp 1ndash14 2017
[32] T A A Victoire and A E Jeyakumar ldquoHybrid PSO-SQPfor economic dispatch with valve-point effectrdquo Electric PowerSystems Research vol 71 no 1 pp 51ndash59 2004
[33] D-KHe F-LWang andZ-ZMao ldquoHybrid genetic algorithmfor economic dispatch with valve-point effectrdquo Electric PowerSystems Research vol 78 no 4 pp 626ndash633 2008
[34] D C Secui ldquoA modified symbiotic organisms search algorithmfor large scale economic dispatch problem with valve-pointloading effectsrdquo Energy vol 113 pp 366ndash384 2016
[35] K BhattacharjeeA Bhattacharya and S H N Dey ldquoTeaching-learning-based optimization for different economic dispatchproblemsrdquo Scientia Iranica vol 21 no 3 pp 870ndash884 2014
[36] K Bhattacharjee A Bhattacharya and S H N Dey ldquoOppo-sitional Real Coded Chemical Reaction Optimization for dif-ferent economic dispatch problemsrdquo International Journal ofElectrical Power amp Energy Systems vol 55 pp 378ndash391 2014
Mathematical Problems in Engineering 29
[37] V Hosseinnezhad M Rafiee M Ahmadian and M T AmelildquoSpecies-basedQuantum Particle SwarmOptimization for eco-nomic load dispatchrdquo International Journal of Electrical Poweramp Energy Systems vol 63 pp 311ndash322 2014
[38] M S P Subathra S E Selvan T A A Victoire A HChristinal andU Amato ldquoA hybrid with cross-entropymethodand sequential quadratic programming to solve economic loaddispatch problemrdquo IEEE Systems Journal vol 9 no 3 pp 1031ndash1044 2015
[39] J Zhan Q H Wu C Guo and X Zhou ldquoEconomic dispatchwith non-smooth objectives part I local minimum analysisrdquoIEEE Transactions on Power Systems vol 30 no 2 pp 710ndash7212015
[40] A K Barisal andR C Prusty ldquoLarge scale economic dispatch ofpower systems using oppositional invasive weed optimizationrdquoApplied Soft Computing vol 29 pp 122ndash137 2015
[41] M Basu ldquoImproved differential evolution for economic dis-patchrdquo International Journal of Electrical Power amp EnergySystems vol 63 pp 855ndash861 2014
[42] V S Aragon S C Esquivel and C C Coello ldquoAn immunealgorithm with power redistribution for solving economicdispatch problemsrdquo Information Sciences vol 295 pp 609ndash6322015
[43] M Ghasemi M Taghizadeh S Ghavidel and A AbbasianldquoColonial competitive differential evolution an experimentalstudy for optimal economic load dispatchrdquo Applied Soft Com-puting vol 40 pp 342ndash363 2016
[44] B R Adarsh T Raghunathan T Jayabarathi and X-S YangldquoEconomic dispatch using chaotic bat algorithmrdquo Energy vol96 pp 666ndash675 2016
[45] N Ghorbani and E Babaei ldquoExchange market algorithm foreconomic load dispatchrdquo International Journal of ElectricalPower amp Energy Systems vol 75 pp 19ndash27 2016
[46] J X V Neto G Reynoso-Meza T H Ruppel V C Marianiand L D S Coelho ldquoSolving non-smooth economic dispatchby a new combination of continuous GRASP algorithm anddifferential evolutionrdquo International Journal of Electrical Poweramp Energy Systems vol 84 pp 13ndash24 2017
[47] A Kavousi-Fard and A Khosravi ldquoAn intelligent 120579-ModifiedBat Algorithm to solve the non-convex economic dispatchproblem considering practical constraintsrdquoElectrical Power andEnergy Systems vol 82 pp 189ndash196 2016
[48] M A Al-Betar M A Awadallah A T Khader and A L BolajildquoTournament-based harmony search algorithm for non-convexeconomic load dispatch problemrdquo Applied Soft Computing vol47 pp 449ndash459 2016
[49] M A Al-Betar M A Awadallah I A Doush E Alsukhni andH ALkhraisat ldquoA non-convex economic dispatchproblemwithvalve loading effect using a new modified 120573-hill climbing localsearch algorithmrdquo Arabian Journal for Science and Engineeringpp 1ndash8 2018
[50] M A Awadallah M A Al-Betar A L Bolaji E M Alsukhniand H Al-Zoubi ldquoNatural selection methods for artificial beecolony with new versions of onlooker beerdquo Soft Computing pp1ndash40 2018
[51] S H Kaboli and A K Alqallaf ldquoSolving non-convex economicload dispatch problem via artificial cooperative search algo-rithmrdquo Expert Systems with Applications vol 128 pp 14ndash272019
[52] D Singh and J Dhillon ldquoAmeliorated greywolf optimization foreconomic load dispatch problemrdquo Energy vol 169 pp 398ndash4192019
[53] X S Yang and S Deb ldquoCuckoo search via Levy flightsrdquo inProceedings of the World Congress on Nature and BiologicallyInspired Computing (NaBIC rsquo09) pp 210ndash214 India 2009
[54] S Sahoo K M Dash R C Prusty and A K Barisal ldquoCom-parative analysis of optimal load dispatch through evolutionaryalgorithmsrdquo Ain Shams Engineering Journal vol 6 pp 107ndash1202015
[55] M Basu and A Chowdhury ldquoCuckoo search algorithm foreconomic dispatchrdquo Energy vol 60 pp 99ndash108 2013
[56] A B Serapiao ldquoCuckoo search for solving economic dispatchload problemrdquo Intelligent Control and Automation vol 04 no04 pp 385ndash390 2013
[57] A H Bindu and M D Reddy ldquoEconomic load dispatch usingcuckoo search algorithmrdquo International Journal of EngineeringResearch and Applications vol 3 pp 498ndash502 2013
[58] E Afzalan and M Joorabian ldquoAn improved cuckoo searchalgorithm for power economic load dispatchrdquo InternationalTransactions on Electrical Energy Systems vol 25 no 6 pp 958ndash975 2015
[59] T T Nguyen and D N Vo ldquoThe application of one rank cuckoosearch algorithm for solving economic load dispatch problemsrdquoApplied Soft Computing vol 37 pp 763ndash773 2015
[60] S M Islam S Das S Ghosh S Roy and P N Suganthan ldquoAnadaptive differential evolution algorithm with novel mutationand crossover strategies for global numerical optimizationrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 42 no 2 pp 482ndash499 2012
[61] Z L Gaing ldquoParticle swarm optimization to solving theeconomic dispatch considering the generator constraintsrdquo IEEETransactions on Power Systems vol 18 no 3 pp 1187ndash1195 2003
[62] X-S Yang and S Deb ldquoEngineering optimisation by Cuckoosearchrdquo International Journal of Mathematical Modelling andNumerical Optimisation vol 330 pp 43ndash51 2010
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Mathematical Problems in Engineering 5
119865119894 (119875119894) =
1205751198941 + 1205821198941119875119894 + 12057211989411198752119894 fuel 1 119875119894min le 119875119894 le 11987511989411198981198861199091205751198942 + 1205821198942119875119894 + 12057211989421198752119894 fuel 2 1198751198942min le 119875119894 le 1198751198942max120575119894119895 + 120582119894119895119875119894 + 1205721198941198951198752119894 fuel 119895 119875119894119895min le 119875119894 le 119875119894119895max
(7)
119865119894 (119875119894) =
1205751198941 + 1205821198941119875119894 + 12057211989411198752119894 + 10038161003816100381610038161205731198941 times sin (1205741198941 times (119875119894min minus 119875119894))1003816100381610038161003816 for fuel 1 119875119894min le 119875119894 le 11987511989411198981198861199091205751198942 + 1205821198942119875119894 + 12057211989421198752119894 + 10038161003816100381610038161205731198942 times sin (1205741198942 times (119875119894min minus 119875119894))1003816100381610038161003816 for fuel 2 1198751198942min le 119875119894 le 1198751198942119898119886119909 119895 = 1 119898119894120575119894119895 + 120582119894119895119875119894 + 1205721198941198951198752119894 + 10038161003816100381610038161003816120573119894119895 times sin (120574119894119895 times (119875119894min minus 119875119894))10038161003816100381610038161003816 for fuel 119895 119875119894119895min le 119875119894 le 119875119894119895max
(8)
Cost function forms in Equations (7) and (8) are onlyincluded in objective function (1) meanwhile main con-straints in formulas (4) and (6) must be always satisfied
23 Prohibited Power Zone Real Power Reserve and RampRate Limit Constraints Prohibited power zones (PPZ) aredifferent ranges of power in fuel cost function that thermalgeneration units are not allowed to work due to operationprocess of steam or gas valves in their shaft bearing Thepower generation of units in the violated zones is harmfulto gas or steam turbines even destroyed shaft bearing Thusthe constraint is strictly observed In the fuel-power charac-teristic curve of generation units PPZ causes small violationzones and such curves become discontinuous As consideringPPZ constraint the determination of power generation ofunits is more complex and equal to either lower bound orupper bound Unlike PPZ constraint RPR constraint is notrelated to fuel-power feature curve but it causes difficulty foroptimization approaches in satisfying one more inequalityconstraint Each generation unit among the set of availablegeneration units must reserve real power so that the sumof real power from all generation units can be higher orequal to the requirement of power system for the purposeof stabilizing power system in case that there are some unitsstopping producing electricity On the contrary to PPZ con-straint ramp rate limit (RRL) constraint does not allowpoweroutput of thermal generating units outside a predeterminedrange The constraint considers maximum power change ofeach thermal generating unit as compared to the previouspower value Thus optimal generation must satisfy the RRLconstraint The PPZ constraint RPR constraint and RRLconstraint can be presented as follows
Prohibited Power Zones As considering PPZ constraint validworking zones of each thermal generating unit are notcontinuous and its generation must be outside the violatedzones as the following mathematical description
119875119894 isin
119875119894min le 119875119894 le 1198751198971198941119875119906119894119896minus1 le 119875119894 le 119875119897119894119896 k = 2 ni forall119894 isin Ω119875119906119894119899119894 le 119875119894 le 119875119894max
(9)
As observing Equation (9) generation units cannotbe operated within the violated zones except for startingpoint and end point Consequently the verification of PPZconstraint violation should be carried out first and thenthe correction should be done before dealing with otherconstraints such as real power reserve constraint and realpower balance Besides if power output of all units can satisfythe PPZ constraint generation limits in Equation (6) are alsoexactly met
Real Power Reserve Constraint Real power reserve in powersystem aims to enhance the ability of stability recovery ofpower system and avoid blackout In order to get high enoughpower for requirement all available units are constrained bythe following inequality
119873sum119894=1
119878119894 ge 119878119877 (10)
where 119878119894 is the real power reserve contribution of the 119894119905ℎthermal generation unit and the determination of 119878119894 can bedone by employing the two models below
119878119894 = 119875119894max minus 119875119894 119894119891 119878119894max gt (119875119894max minus 119875119894)119878119894max else
forall119894 notin Ω (11)
119878119894 = 0 forall119894 isin Ω (12)
Equation (10) shows that the constraint of prohibitedpower zones is not included in the real power reserveconstraints however prohibited power zones are alwaysstrictly considered and must be exactly satisfied
Ramp Rate Limit (RRL) Constraint In OLD problem allconsidered thermal generating units are supposed to be underworking status but previous active power of each thermalgenerating unit is not taken into account Thus increased ordecreased power is not constrained This assumption seemsto be not practical until RRL constraint is considered RRLconstraint considers initial power output and the power
6 Mathematical Problems in Engineering
change is supervised Regulated power can be higher or lowerthan the initial value as long as it is within a predeterminedrange Increased step size (ISS) and decreased step size (DSS)are given as input data and they are used to limit the change ofpower output of each thermal generating unit The constraintcan bemathematically expressed as the following formula [7]
1198751198940 minus 119863119878119878119894 le 119875119894 le 1198751198940 + 119868119878119878119894 (13)
where 1198751198940 is the initial power output of the 119894119905ℎ thermalgenerating unit before its power output is regulated 119868119878119878119894 and119863119878119878119894 are respectively maximum increased and decreasedstep sizes of the 119894119905ℎ thermal generating unit
3 The Proposed Cuckoo Search Algorithm
31 Classical Cuckoo Search Algorithm In search techniqueof CCSA [53] a set of solutions is randomly generated withina predetermined range in the first step and then the quality ofeach one is ranked by computing value of fitness functionThemost effective solution corresponding to the smallest valueof fitness function is determined and then search procedurecomes into a loop algorithm until the maximum iterationis reached In the loop algorithm two techniques updatingnew solutions two times (corresponding to two generations)are Levy flights and mutation technique which is calledstrange eggs identification technique The two generationscan produce promising quality solutions for CCSA Aftereach generation CCSA will carry out comparing fitness ofnewly updated solutions and initial solutions for keepingbetter ones and abandoning worse ones The most effectivesolution at last step of the loop search algorithm is determinedand it is restored as one candidate solution for a study caseThe detail of the two stages is as follows
311 Levy Flights Stage This is the first calculation step in theloop algorithm and it also produces new solutions in the firstgeneration for CCSA New solution 119878119900119897119899119890119908119909 is created by thefollowing model
119878119900119897119899119890119908119909 = 119878119900119897119909 + 120572 (119878119900119897119909 minus 119878119900119897119866119887119890119904119905) oplus Levy (120573) (14)
where 120572 is the positive scaling factor and it is nearly set todifferent values for different problems in the studies [53 62]In the work the most appropriate values for such factor canbe chosen to be 02505 for different systems
312 Discovery of Alien Eggs Stage The step plays a veryimportant role for updating new solutions 119878119900119897119899119890119908119909 of thewhole population However not every control variable ineach old solution is newly updated and the decision ofreplacement is dependent on comparison criteria as thefollowing equation
119878119900119897119899119890119908119909=
119878119900119897119909 + 1205761 (1198781199001198971199031198861198991198891 minus 1198781199001198971199031198861198991198892) if 1205762 lt 119875119886119878119900119897119909 119900119905ℎ119890119903119908119894119904119890
(15)
32 Proposed Algorithm In the part a new variant of CCSA(ICSA) is constructed by applying three effective changes onthe main functions of CCSA in order to shorten simula-tion time corresponding to reduction of iterations and findmore promising solutions The proposed amendments areexplained in detail as follows
(i) Suggest one more equation producing updated stepsize in addition to existing one in CCSA
(ii) Create a new selection standard by computing fitnessfunction ratio 119865119865119877119909 and comparing 119865119865119877119909 with apredetermined parameter 119879119900119897119909 Thus thanks to thestandard the existing updated step size and additionalupdate step size will be chosen more effectively
(iii) Automatically change value of 119879119900119897119909 for the xth solu-tion based on the result of comparing 119865119865119877119909 with theprevious 119879119900119897119909
Such three points are clarified by observing the followingsections
321 Strange Eggs Identification Technique (Mutation Tech-nique) The first proposed improvement in our proposedICSA approach is to select a more suitable formula forproducing new solutions with better fitness function valueIn CCSA Equation (16) below is used to produce a changingstep nearby old solutions for all current solutions
Δ1198781199001198971198991198901199081199091 = 1205763 (1198781199001198971199031198861198991198891 minus 1198781199001198971199031198861198991198892) (16)
The use of Equation (16) aims to produce a random walkaround old solutions in search zones with intent to findout promising solutions In order to reduce the possibilityof suffering the local trap and approach to other favorablezones for searching we propose a new Equation (17) Theformula is built by the idea of enlarging search zone withthe use of two more available solutions Obviously the largerchanging step can own higher performance in moving toother search spaces that the classical approach used in CCSAThe suggestion is mathematically expressed by the formulabelow
Δ1198781199001198971198991198901199081199092= 1205764 (1198781199001198971199031198861198991198891 minus 1198781199001198971199031198861198991198892 + 1198781199001198971199031198861198991198893 minus 1198781199001198971199031198861198991198894) (17)
The changing step obtained by using Eq (17) is namedfour-point changing step Now two solutions which arenewly formed by using two different changing steps shownin formulas (16) and (17) are found by the two followingmethods
It can be clearly observed that the distance between 119878119900119897119909(old solution) and 1198781199001198971198991198901199081199091 (new solution) is lower than thatbetween 119878119900119897119909 and 1198781199001198971198991198901199081199092 This difference can contribute ahighly efficient improvement to the proposed ICSA approachsearch ability
Mathematical Problems in Engineering 7
ΔSol2
Sol2
Sol3
Sol4
ΔSol1
Sol1
Solx
Solnew1
Solnew2
Figure 1 Simulation of solutions corresponding to the first itera-tions of the loop algorithm
For the CCSA case if two solutions 1198781199001198971199031198861198991198891 and 1198781199001198971199031198861198991198892are either slightly different or completely coincident suchnewly updated solution 1198781199001198971198991198901199081199091 does not have good chanceto leave the current zone and approach to more promisingzones In another word the new one is approximately coin-cident with the old one As the search task is taking place atsome last iterations this phenomenon becomes much worsebecause all current solutions are lumped in a small zone andthe capability of moving to other zones is impossible As aresult the CCSA approach will work ineffectively and searchstrategy is time consuming until other runs are started
Contrary to the two-point step size the new proposedformula may produce a large enough length to escape thelocal optimum zone and reach new favorable zones Itexplainswhy the four-point changing step has positive impacton the considered random walk rather than the two-pointchanging step
322 New Standard forChoosing theMostAppropriate Chang-ing Step In this section we extend our analysis to answer thequestionwhen to use the four-point step size FromEquations(18) and (19) two new solutions which are represented asthe results of the two-point-based factor and the four-pointstep size can be illustrated by using Figure 1 corresponding tothe search process at the first some iterations and Figure 2corresponding to the last some iterations For the sake ofsimplicity we rewrite the two equations as follows
Here we suppose that 1198781199001198971 and 1198781199001198972 are obtained byfour exact solutions 1198781199001198971 1198781199001198972 1198781199001198973 and 1198781199001198974 and calculatedas follows
ΔSol2
ΔSol1
Solx
Solnew1
Solnew2
Figure 2 Simulation of solutions corresponding to the last itera-tions of the loop algorithm
Δ1198781199001198971 = 1198781199001198971 minus 1198781199001198972 (22)
Δ1198781199001198972 = 1198781199001198973 minus 1198781199001198974 (23)
Asmentioned above the high changing step between newsolution and old solution can help to explore new favorablezones However in optimization algorithms searching stepscannot be arbitrarily large otherwise the algorithm maydiverge in particular for the cases that the consideredsolutions 119878119900119897119909 are not close together in solution search spaceFor example at the beginning of loop algorithm with thefirst iterations in Figure 1 1198781199001198971198991198901199081 is a better choice than1198781199001198971198991198901199082 because it is kept in a sufficient limit and does notlead to a risk of divergence In contrast as many of currentsolutions are in different positions but their distance is notvery short or approximately coincident such as at the lastiterations in Figure 2 1198781199001198971198991198901199081 and 119878119900119897119909 have a very shortdistance but 1198781199001198971198991198901199082 and 119878119900119897119909 have higher distance Accordingto the phenomenon in Figure 2 the proposed ICSA approachneeds to produce a high changing step to move solutions toother search zones without local optimum Hence 1198781199001198971198991198901199082would be preferred to 1198781199001198971198991198901199081
Based on the argument above the determination of thecondition for using either two-point changing step or four-point changing step is really crucial to the performance ofthe proposed ICSA approach in searching solutions of OLDproblem Here the ratio of 119865119865119877119909 which can be found byEquation (24) is suggested to be a suitable measurement forthe selection of two options
Δ119865119865119877119909 = 119865119865119909 minus 119865119865119887119890119904119905119865119865119887119890119904119905 (24)
For a particular set of the current solutions each individ-ual depending on its 119865119865119877119909 will create a corresponding newsolution by using either Equation (18) or (19) If the valueof one current solution is smaller than the predeterminedparameter 119879119900119897 Equation (19) is applied for updating suchconsidered solution 119909 Otherwise Equation (18) is a betteroption The steps of the modified algorithm are similar to the
8 Mathematical Problems in Engineering
If 1205765 lt 119875119886If FFRx lt Tolx119878119900119897119899119890119908119909 = 119878119900119897119909 + 1205764(1198781199001198971199031198861198991198891 minus 1198781199001198971199031198861198991198892 + 1198781199001198971199031198861198991198893 minus 1198781199001198971199031198861198991198894)else119878119900119897119899119890119908119909 = 119878119900119897119909 + 1205764(1198781199001198971199031198861198991198891 minus 1198781199001198971199031198861198991198892)end
Algorithm 1 New mutation technique applied in the proposed ICSA approach
conventional CSA except that an additional step should beadded at each iteration In this step the119865119865119889 of all individualsolutions should be calculated by utilizing Equation (24) andthen the result of comparing the ratio with 119879119900119897will be used todecidewhich updating formula should be selectedThewholedescription of the proposed standard and new mutationtechnique can be coded inMatlab program language by usingAlgorithm 1
323 Adjustment of Tolerance for Each Solution As pointedout above the proposed method needs assistances to deter-mine the most appropriate step size for finding out favorablesolution zones The given aim can be reached if the selectionof 119879119900119897119909 is reasonable however the range of this parameteris infinite and hard to select Thus the adaptation of tuningthe parameter is really necessary First of all the compari-son between 119879119900119897119909 and 119865119865119877119909 is carried out and then theadaptation will be determined based on the obtained resultfrom the comparison Results of comparison between the twoparameters can be either 119865119865119877119909 is less than 119879119900119897119909 or 119865119865119877119909is higher than 119879119900119897119909 The case that two parameters are equalhardly ever occurs
As the comer assumptionhappens (ie119865119865119877119909 is less than119879119900119897119909) at the considered time the four-point step size will beemployed for the 119909119905ℎ solution If 119879119900119897119909 remains unchanged atthe previous value the identification of improvement fromsuch four-point step size or two-point step size is vagueConsequently value of 119879119900119897119909 must be automatically reducedto a lower value in case that it has significant contribution tofound promising solution of previous iteration Clearly thedecrease of119879119900119897119909 can enable the proposedmethod to jump outlocal optimal zone and approachmore effective zones By trialand error method 119879119900119897119909 is selected to be a function of itselfthat is 09 of the previous value Finally the implementationof the proposed ICSA approach is presented in Algorithm 2
4 The Application of the ProposedICSA for OLD Problem
Thewhole computation steps of the proposed ICSA approachfor solving OLD problem are explained as follows
41 Handling Constraints and Randomly Producing InitialPopulation As shown in Section 2 the considered OLDproblem takes five following constraints into account
(i) Power balance constraint is shown in Equation (4)
(ii) Power output limitation constraint is shown in Equa-tion (6)
(iii) Prohibited power zone constraint is shown in Equa-tion (9)
(iv) Real power reserve constraint is shown in Equation(10)
(v) Ramp rate limit constraint is shown in Equation (13)
Among the five constraints ramp rate limit generationlimit and prohibited power zone seem to be more com-plicated than power balance and power reserve constraintsHowever the three constraints can be solved more easilybecause each unit is constrained independently in the threeconstraints whereas power balance constraint and powerreserve constraint consider all the thermal generating unitssimultaneously Power reserve constraint can be handledby penalizing the total generation of all units while powerbalance constraint can be solved by penalizing one violatedthermal generating unit The whole computation procedurefor solving all constraints and calculating fitness function ofsolutions is described in detail as follows
Step 1 Redefine maximum and minimum power output ofeach thermal generating unit as considering PPZ and RRLconstraints by using the following formulas
119875119894max = 119875119894max if 119875119894max le 119875i0 + 119868119878119878119894119875i0 + 119868119878119878119894 if 119875119894max gt 119875i0 + 119868119878119878119894
119894 = 1 119873(25)
119875119894min = 119875119894min if 119875119894min ge 119875i0 minus 119863119878119878119894119875i0 minus 119863119878119878119894 119890119897119904119890
119894 = 1 119873(26)
Mathematical Problems in Engineering 9
Produce initial population with119873119901119904 solutions (1198781199001198971 1198781199001198972 119878119900119897119909 119878119900119897119873119901119904)Calculate fitness function (1198651198651 1198651198652 119865119865119909 119865119865119873119901)Go to the loop algorithm by setting 119866 = 1
While (119866119898119886119909 gt 119866) (i) The first newly produced solutions119878119900119897119899119890119908119909 = 119878119900119897119909 + 120572(119878119900119897119909 minus 119878119900119897119866119887119890119904119905) oplus Levy(120573) (ii) Perform selection approach
119865119865119909 = 119865119865119909 if 119865119865119909 le 119865119865119899119890119908119909119865119865119899119890119908119909 119900119905ℎ119890119903119908119894119904119890 119909 = 1 119873119901119904
(v) Determine the most effective solution and its fitnessDetermine 119865119865119909 with the smallest value and assign 119878119900119897119909 to 119878119900119897119866119887119890119904119905If 119866119898119886119909 gt 119866 perform step (i) and increase 119866 to 119866 + 1 Otherwise stop the loop algorithm and report boththe smallest fitness together with 119878119900119897119866119887119890119904119905End while
Among the four Equations (25) and (26) are used firstin order to redefine upper bound and lower bound for allthermal generating units as considering RRL constraint Thethe redefined bounds continue to be redefined for the secondtime by using (27) and (28) as considering PPZ constraints
Step 2 (randomly produce initial population) For dealingwith the power balance constraint all available units areseparated into two groups in which the first group withdecision variables consists of the power output from thesecond unit to the last unit (P2 P3 PN) meanwhile onlythe power output of the first unit (1198751) belongs to the secondgroup with dependent variable So upper bound solution119878119900119897119898119886119909 and lower bound solution 119878119900119897119898119894119899 must be defined asfollows
Step 3 Handle prohibited power zone constraint for decisionvariables P2 P3 PN
After being randomly produced there is a high possi-bility that decision variables fall into PPZ and they violatePPZ constraint So the verification of falling into PPZ andcorrection of the violation should be accomplished by usingthe following formula
119875119894 =
119875119897119894119896 if 119875119897119894119896 lt 119875119894 le 119875119897119894119896 + 1198751198961198941198962119875119906119894119896 if (119875119894 gt 119875119897119894119896 + 1198751198961198941198962 ) amp (119875119894 lt 119875119906119894119896)119875119894 119890119897119904119890
119894 = 2 119873 amp 119896 = 1 119899119894
(31)
Step 4 Handle RPB constraint by calculating 1198751 and penaliz-ing 1198751 if it violates constraints
In this step power balance constraint is exactly handledby calculating and penalizing dependent variable (1198751) 1198751 isobtained by using formulas (4) and (5) as follows
1198751 = minus (11986101 minus 1 + 2sum119873119894=2 1198611119894119875119894) plusmn radicΔ211986111 (32)
where
Δ = (11986101 minus 1 + 2 119873sum119894=2
1198611119894119875119899)2 minus 411986111(119875119863 minus 119873sum
119894=2
119875119894+ 11986100 + 119873sum
119894=2
1198610119894119875119894 + 119873sum119894=2
119873sum119895=2
119875119894119861119894119895119875119895) amp Δ ge 0(33)
In Equation (32) 1198751 has been determined for the purposeof dealing with real power balance constraint However it isnot sure that 1198751 can satisfy upper bound and lower boundconstraints and prohibited power zone constraints So 1198751must be checked and penalized
Firstly 1198751 is checked and penalized for upper and lowerbound constraints by the following model
Δ1198751x =
0 if 1198751min le 1198751x le 1198751max
1198751min minus 1198751x if 1198751min gt 1198751x1198751x minus 1198751max if 1198751max lt 1198751x
(34)
In Equation (34) if the second case or the third caseoccurs it means P1 has violated either lower bound or upperbound and it would be penalized by using either (P1x= P1min-P1x) or (P1x =P1x -P1max) Otherwise ifP1 has not violatedthe bound constraints (ie the first case in (34) happened)
P1 would continue to be checked for PPZ constraint by thefollowing model
Δ1198751x
=
1198751 minus 1198751198971119896 if 1198751198971119896 lt 1198751 le 1198751198971119896 + 119875119896111989621198751199061119896 minus 1198751 if (1198751 gt 1198751198971119896 + 11987511989611198962 ) amp (1198751 lt 1198751199061119896)0 119890119897119904119890
(35)
Step 5 Handle real power reserve constraint (10)First of all 119878119894 is determined by using (11) and (12) and
then the 119909119905ℎ solution will be checked and penalized if poweroutput of all thermal generating units cannot satisfy RPRconstraint The penalty for violation of the constraint can becalculated by using equation (36)
Δ119878119877119909 =
0 if119873sum119894=1
119878119894119909 ge 119878119877119878119877 minus 119873sum119894=1
119878119894119909 119890119897119904119890 (36)
As a result real power reserve constraint can be solved byusing the penalty method
42 Calculate Fitness Function for Solutions Fitness functionof each solution is used to evaluate quality of solutionNormally the function is the sum of objective function andpenalty of violating constraints and is obtained by
43 The First Newly Updated Solutions by Levy Flights Tech-nique In this section the first newly updated solutionsare performed by employing Levy flights technique usingEquation (14) However each new solution can be out oftheir feasible operating zone such as PPZ and upper andlower limitations When the power output violates its PPZconstraints Equation (31) will be applied to tackle theconstraint Besides the following equation will be employedwhen power output is higher or lower than their limitations
119878119900119897119909 =
119878119900119897max if 119878119900119897max lt 119878119900119897119909119878119900119897min if 119878119900119897min gt 119878119900119897119909119878119900119897119909 Otherwise
119909 = 1 119873119901 (38)
After that Equations (32)-(37) are performed for deter-mining all variables and penalty terms Finally Equation (38)is employed to calculate fitness function
44 The Second Newly Updated Solutions by Using Muta-tion Technique The second newly updated solutions areaccomplished as presented in Section 3 above Similar to
Mathematical Problems in Engineering 11
the task after doing the first update each solution in thenew population must satisfy PPZ constraint and upper andlower boundaries by considering Equations (31) and (38)Then Equations (32)-(37) are performed for determiningall variables and penalty terms Finally Equation (38) isemployed to calculate fitness function and the solution withthe best value is assigned to the best one 11987811990011989711986611988711989011990411990545 Criterion of Stopping the Loop Algorithm In the loopalgorithm of using the proposed ICSA approach the solutionsearch work is stopped in case that the predeterminedmaximum iterations 119866119898119886119909 is reached For each search ter-mination the most effective solution is stored and anotherrun continues to be accomplished until the predeterminednumber of runs is reached After finishing the runs thebest one is found and reported In addition other valuessuch as the fitness of the worst solution and average fitnessof all solutions are also reported for comparing with othermethods
46 The Whole Iterative Process The whole iterative algo-rithm for implementing the proposed ICSA approach forcoping with OLD problem is described in detail in Figure 3
5 Results and Discussions
The proposed ICSA approach performance has been investi-gated on six cases with different fuel options different fuelcharacteristics and complicated constraints The details ofthe studied cases are presented as follows
Case 1 Four systems with single fuel source (SFS) and powerloss (PL) constraint
Subcase 11 A 3-unit system [57]Subcase 12 A 6-unit system [57]Subcase 13 A 3-unit system [56]Subcase 14 A 6-unit system [56]
Case 2 A 110-unit system with SFS [57]
Case 3 Four systems with SFS and the effects of valve loadingprocess (EoVLP)
Subcase 31 A 3-unit system supplying to a load of850MW [58]Subcase 32 A 13-unit system supplying to a load of1800MW [1]Subcase 33 A 13-unit system supplying to a load of2520MW [1]Subcase 34 A 40-unit system supplying to a load of2500MW [1]Subcase 35 An 80-unit system supplying to a load of4100MW [49]
Case 4 Two systems with SFS and PPZ and RPR constraints
Subcase 41 A 60-unit system supplying to a10600MW load [9]
Subcase 42 A 90-unit system supplying to a15900MW load [9]
Case 5 A 15-unit system with SFS and RRL PPZ and PLconstraints [61]
Case 6 Three systems with multiple fuel sources (MFS) andEoVLP
Subcase 61 An 80-unit system supplying to a21600MW load [15]Subcase 62 A 160-unit system supplying to a43200MW load [15]Subcase 63 A 320-unit system supplying to an86400MW load [54]
For each considered case with each load case the pro-posed ICSA approach is run 50 times on the programlanguage of Matlab and a PC with 4 GB of RAM and 24GHzprocessor The selection of adjustment parameters including119875119886 and 119879119900119897119909 is carefully considered to obtain the best optimalsolutions meanwhile two others such as 119873119901119904 and 119866119898119886119909 arechosen corresponding to the scale of particular test system9 values with the change of 01 in the range [01 09] are inturn selected for 119875119886 while 119879119900119897119909 is 001 at the beginning Theinformation including load demand119873119901119904 119866119898119886119909 and the best119875119886 is reported in Table 1
51 Obtained Results on Case 1 considering Four Systems withSFS and PL Constraint In this section we have implementedthe proposed ICSA approach for solving four systems dividedinto four subcases Tables 2 and 3 show the comparisons ofobtained results from Subcases 11 and 12 and Subcases 13and 14 respectively As listed in Table 2 the proposed ICSAmethod and CCSA can find equal fuel cost for Subcases 11whereas the reduction of fuel cost from the proposed ICSAmethod as compared to CCSA is clearer for Subcase 12 Asshown in Table 3 for comparing the proposed ICSA and threemethods consisting of CCSA ABC and FA the minimumfuel cost of the proposed ICSA is approximately equal to thatof these methods for Subcases 13 but much less than that ofthese methods for Subcase 14 Furthermore the proposedICSA has been run by setting 119873119901119904 and 119866119898119886119909 to 5 and 20but these values were much higher for CCSA ABC and FAThey are 20 and 5000 for CCSA 40 and 100 for ABC and 20and 5000 for FA Consequently the proposed method is veryefficient for Case 1 with four subcases
Optimal solutions obtained by ICSA for Case 1 are shownin Tables 16ndash18
52 Obtained Results on Case 2 considering 110-Unit Systemwith SFS In this section we have employed a very largescale system with 110 units but there were not challenges forobjective function and complex constraints since EoVLP andconstraints were not taken into account Both CCSA andthe proposed ICSA methods have been run for comparingwith BBO hybrid BBO and DE (DEBBO) and Opposi-tional real coded chemical reaction optimization algorithm(ORCCROA) in [36] IWA in [40] and AGWO in [52] As
12 Mathematical Problems in Engineering
Select parameters
- Correct solutions if violating PPZ constraint using Eq (31)- Determine using Eq (32)- Penalize for violating upper and lower bounds constraint using Eq (34)- Penalize for violating PPZ constraint using Eq (35)- Penalize the xth solution for violating RPR constraint using Eq (36)
- Determine quality of solutions employing Eq (37) - Select the most effective solution - Start the loop algorithm by selecting
- Perform the first solution update using Section 32 - Correct boundaries of solutions by using Eq (38)
- Correct solutions if violating PPZ constraint using Eq (31)- Determine using Eq (32)- Penalize for violating upper and lower bounds constraint using Eq (34)- Penalize for violating PPZ constraint using Eq (35)- Penalize the xth solution for violating RPR constraint using Eq(36)
- Determine fitness function using Eq (37)- Compare old solutions and new solutions to keep better ones
- Perform the second solution update using Algorithm 1 - Correct boundaries of solutions by using (38)
- Correct solutions if violating PPZ constraint using Eq (31)- Determine using Eq (32)- Penalize for violating upper and lower bounds constraint using Eq (34)- Penalize for violating PPZ constraint using Eq (35)- Penalize the xth solution for violating RPR constraint using Eq (36)
- Determine fitness function using Eq (37)- Compare old solutions and new solutions to keep better ones- Select the most effective solution
Stop
Start
- Redefine upper and lower bounds using Eqs (25)-(28)- Randomly generate initial population using Eq (30)
Nps Pa Gmax H> Tolx
P1xP1x
P1x
P1xP1x
P1x
P1xP1x
P1x
Sol<MN
Sol<MN
G = 1
G = Gmax G = G + 1
Figure 3 All computation steps for solving OLD problem by employing the proposed ICSA approach
shown in Table 4 AGWO [52] has reached less fuel cost thanICSA however the exact fuel cost which was recalculatedby using reported solution pointed out that the method hasreached a very high fuel cost of $2157404250 For comparisonwith other methods ICSA has found less fuel cost thanall these methods Particularly the reduction of generationfuel cost is significant as compared to BBO DEBBO andCCSA Execution time comparisons are also useful evidence
for indicating the high performance of ICSA Thus it canconclude that ICSA is a strong method for Case 2
Optimal solution obtained by ICSA for the case is shownin Table 19
53 Obtained Results on Case 3 considering Four Systems withSFS and EoVLP In this section the real performance of theproposed ICSA approach has been investigated based on five
Mathematical Problems in Engineering 13
Table 1 Information of considered cases and adjustment parameters
Case Fuel cost function Constraint Subcase No of units 119875119863 (MW) Nps Gmax Best 1198751198861 SFS PL
subcases with the gradual increase of number of units Thesmallest scale system considers 3 units but the largest scalesystem takes 80 units In addition to the implementation ofthe proposed ICSA we have also implemented CCSA forSubcase 31 and Subcase 35 for further comparison becauseCCSA has not been run for the two subcases so far
Comparison of obtained results from Subcase 31 shownin Table 5 indicates that the proposed ICSA is superior toCCSAwith lower fuel cost but it seems to be less effective thanMCSA [58] The minimum cost of MCSA reported in [58] isthe smallest fuel cost but the recalculated cost is much higherthan that of the proposed method Furthermore MCSA hasbeen implemented by setting very high values to 119873119901119904 and119866119898119886119909
Reports for Subcases 32 and 33 shown in Table 6are the comparisons of the proposed ICSA approach andother methods such as conventional Evolution programming(CEP) [1] Fast EP (FEP) [1] improved FEP (IFEP) [1] DE[12] multiplier Lagrange-based genetic algorithm with (GA-MU) [15] QPSO [16] GA-PS-SQP [30] PSO-SQP [32] M120573-HCLSA [49] IABCA [50] CCSA [59] OSE-CSA [59] SOS[34] MSOS [34] CEA-SQT [38] TSBO [39] IWA [40] andCBA [44] As observed from the table ICSAapproach obtainsbetter solutions than mostmethods excluding DE [10] CCSA
[59] OSE-CSA [59] SOS [34] MSOS [34] CEA-SQT [38]TSBO [39] IWA [40] and CBA [44] especially M120573-HCLSA[49] with lower cost $1796097 However recalculated costfrom reported solution of M120573-HCLSA is $179691 BesidesICSA is very fast as compared to most methods where twoother versions of Cuckoo search algorithm CCSA and OSE-CSA are also included except two methods in [34] Theprocessor of computer that all the methods run on is alsoreported in the final column Clearly ICSA approach is veryefficient for the case with the 13-unit system where effects ofvalve loading process are considered
In Subcase 34 the number of units is much larger thanthat of three subcases above up to 40 units [1] The obtainedresult comparisons with others are indicated in Table 7Clearly the minimum cost comparisons reveal that the pro-posedmethod is one of the leading methods due to the lowestcost except the comparison with CCSA [23] OSE-CSA [59]SOS [34] MSOS [34] EMA [45] 120579-MBA [47] and AGWOA[52] It is noted that AGWOA [52] has reported the bestminimum cost with $12140430 but recalculated minimumcost which was obtained by substituting reported optimalgeneration of all thermal generating units is $12141331 Theaverage and the maximum costs from the proposed methoddo not belong to the leading method group however the
Mathematical Problems in Engineering 15
Table 7 Result comparisons for Subcase 34
Approach Best cost Average cost Worst cost CPU time Computer($h) ($h) ($h) (s) (Processor-Ram)
execution time that ICSA approach takes is smaller thanmostones except IAPR [42] and 120579-MBA [47] which have been runon stronger computers Note that MSOS [34] has been fasterthan the proposed method for Subcases 31 and 32 abovebut it is too slower than the proposed method for the casenamely 1813 seconds compared to 146 seconds Comparedto two other versions of Cuckoo search CCSA [23] and OSE-CSA [59] the proposed method is also faster about threetimes although the processors are slightly different For thiscase 120579-MBA [47] shows a very good performance howeverthe method has not been tested onmore complicated systemsand larger scale and therefore more comparisons with themethod must end Clearly the proposed method is stillefficient for the case where large scale and effects of valveloading process are included
Subcase 35 is the largest scale system with 80 units Inaddition to the implementation of ICSA CCSA has beenalso run for the Subcase 35 for further investigation ofefficiency improvement of the proposed ICSA approach The
comparisons of minimum cost in Table 8 show that theproposed ICSA can find more optimal solution than M120573-HCLSA [49] AGWOA [52] and CCSA The proposed ICSAis also superior to CCSA in terms ofmore stable search abilityand lower fluctuation since its average cost and maximumcost are less than those of CCSA The outstanding figurecannot be reached as compared to AGWOA [52] howeverit is hard to conclude AGWOA [52] is superior to theproposed ICSA approach about more stable search abilityand lower fluctuation Actually comparison of the values ofpopulation and iterations as well as execution time cannot beaccomplished because the information was not reported in[52]Thus it can conclude that the proposed ICSA is effectivefor the subcase
Optimal solutions obtained by ICSA for the case areshown in Tables 20ndash23
54 Obtained Results on Case 4 with Two Systems consideringSFS and PPZ and RPR Constraints In this section two
16 Mathematical Problems in Engineering
Table 9 Result comparisons for Subcases 41 and 42
Subcase Approach Best cost ($h) Mean cost ($h) Worst cost ($h) Std dev ($h) CPU time (s) Computer (Processor-Ram)
test systems with SFS and PPZ and RPR constraints areconsidered The test system size is up to 60 and 90 units forSubcases 41 and 42 respectively Comparison for the casesis only performed with two Genetic algorithms consisting ofGA and IGA-MU in [9] and two other versions of Cuckoosearch algorithms including CCSA [23] and OSE-CSA [59]and presented in Table 9 Other studies have tended toignore such complicated constraints of PPZ and RPR Thecosts reported in Table 9 indicate that ICSA approach canobtain more effective solution than CCSA and OSE-CSAbecause it has reached lower minimum cost than the twoones Furthermore the proposed method also takes shortercomputation time for the two cases from about two timesto about three times although the processor of the proposedmethod is slightly strongerThemean costs of ICSA approachare much less than those from IGA-MU and GA and slightlyhigher than those from OSE-CSA but there is a trade-off between the proposed method and CCSA for the twosubcases In fact the proposed method obtains higher meancost for Subcase 41 but lower cost for Subcase 42 AlthoughGAmethods have been runon aweak computerwith 07 GHzof the processor compared to that with 24GHz in the studytheir execution times are significantly higher namely 56381seconds (GA) and 16258 seconds (IGA-MU) compared to09153 seconds of ICSA approach for Subcase 41 and 94093seconds (GA) and 25545 seconds (IGA-MU) compared to15892 seconds (the proposedmethod)The analysis can pointout that ICSA approach ismore efficient than these comparedmethods in terms of optimal solutions and execution time
Optimal solutions obtained by ICSA for Subcase 42 areshown in Table 24
55 Obtained Results on Case 5 with a 15-Unit System consid-ering SFS and RRL PPZ and PL Constraints In this section
a 15-unit system considering RRL PPZ and PL constraintsis considered to be solved for finding optimal solution Forefficiency investigation of the proposed ICSA we have alsoimplemented CCSA for comparison As listed in Table 10the proposed ICSA is the most effective method with thesmallest fuel cost The comparisons of control parameters aswell asCPU time are also good evidence to confirm the strongsearch of the proposed ICSA approach since it has been runby smaller values of control parameter and faster executiontime as compared to all methods excluding CCSA
Optimal solution obtained by ICSA for the case is shownin Table 25
56 Obtained Results on Case 6 withThree Systems consideringMFS and EoVLP In this section three test systems with thechallenge on objective function including multi-fossil fuelsources and effects of valve loading process are consideredThe scale is up to 80 units 160 units and 320 units forSubcases 61 62 and 63 respectively
Comparison for Subcase 61 reported in Table 11 revealsthat the proposed method is the best method in terms of thelowest best cost the lowestmean cost and the lowest standarddeviation and the fastest execution time The processor fromthis proposedmethod is about four times stronger thanCGA-MU and IGA-MU but the speed is from ten times to 35 timesfaster than these methods Compared to CCSA and OSE-CSA the proposed method is about two times faster but theprocessor is slightly stronger
Comparison for Subcase 62 is reported in Table 12Clearly the proposed ICSA approach obtains better values ofthe best mean and worst costs than most methods exceptMSOS [34] where the best cost difference is about $ 024However the proposed method is the second fastest onewith 1119 seconds where the first fastest one CBA [44] has
Mathematical Problems in Engineering 17
Table 11 Comparisons of found results for Subcase 61
Approach Best cost Mean cost Worst cost Std dev CPU time Computer($h) ($h) ($h) ($h) (s) (Processor-Ram)
spent 571 seconds Clearly MSOS is better than the proposedICSA approach with respect to slightly less best cost butis worse than the proposed method in terms of executiontime namely 2572 seconds compared to 1119 seconds whileCBA [44] is faster than the proposed method but obtainssignificantly worse costs The analysis can conclude that theproposed ICSA approach is very powerful for the subcasewith 160 units
Table 13 presents the comparison of three other methodsincludingCCSA [54] SOS [34] andMSOS [34] accompaniedwith the proposed method for Subcase 63 The obtainedresult comparisons imply that ICSA approach can obtainbetter values of the best and standard deviation costs thanCCSA and SOS but obtains slightly higher cost than MSOSby approximately $ 022 Besides the execution time fromICSA approach is much shorter than others especially it ishigher than five times faster than MSOS The four methodshave been run on approximately strong computers Brieflythe proposed ICSA approach can find and converge to morefavorable solution than other methods with shorter CPUtime except the comparison with MSOS which had bettersolution but spent higher than five times execution times
Consequently the proposed method is a very promisingoptimization algorithm for Subcase 63 a system up to 320units and with multi-fossil fuel sources and effects of valveloading process
Optimal solution obtained by ICSA for Subcase 63 isshown in Table 26
57 The Improvement of ICSA Approach Performance
571 The Outstanding Improvement over CCSA In this sec-tion the performance improvement of ICSA over CCSAhas been investigated by analyzing obtained results and setcontrol parameters Table 14 has been formed by addingreduction cost improvement level of the best cost executiontime and control parameters consisting of 119873119901119904 and 119866119898119886119909Among the compared factors reduction cost is the deviationof the cost of CCSA and that of ICSA whereas the improve-ment level is the ratio of the reduction cost to the cost ofCCSA The reduction costs indicate the proposed methodcould find either equal quality of solutions or higher qualityof solutions than CCSA for all study casesThe reduction costis from $0 to $42839 corresponding to the improvement level
18 Mathematical Problems in Engineering
Table 14 Summary of results obtained by CCSA and ICSA for all study cases
Study case Reduction cost ($) Improvement level () Execution time (s) Nps Gmax
Figure 4 The best run obtained by CCSA and ICSA for Case 2
that is from 0 to 08968The saving cost is not too much forone hour but the operation in one day onemonth or one yearis very high However it should be noted that CCSA has beenrun by setting much higher population size and iterations formany cases excluding study cases implemented in the studysuch as Case 2 Subcase 31 Subcase 35 and Case 5 Forinstance ICSA has used 119873119901119904 = 10 and 119866119898119886119909 = 15 for Subcase13 and119873119901119904 = 10 and119866119898119886119909 = 25 for Subcase 14 whereas CCSAhas been run by setting119873119901119904 = 20 and 119866119898119886119909 = 5000 for the twosubcases Similarly CCSA has been run for Subcases 32 33and 34 with much higher number of iterations For the lastsubcase CCSA has been run by setting 119873119901119904 = 320 and 119866119898119886119909
= 1200 but those of ICSA have been 10 and 9000 Due tothe higher value of control parameters CCSA has tended tospend more time in finding such high quality solutions foralmost all study cases Execution time of ICSA is less than18 seconds while that of CCSA is up to higher 75 secondsIt is clear that the proposed ICSA could find better optimalsolutions thanCCSA for such considered systems For furtherinvestigation of performance comparison the best runs over50 runs and fuel cost values of 50 runs obtained by CCSA andthe proposed ICSA for Case 2 Subcase 31 Subcase 35 andCase 5 have been plotted in from Figures 4ndash11 The best runcurves show the faster search of the proposed ICSA method
Mathematical Problems in Engineering 19
0 5 10 15 20 25 30 35 40 45 50Run
19795
198
19805
1981
19815
1982
19825
1983
19835
Fuel
cost
($)
CCSAICSA
times105
Figure 5The best fuel cost of 50 runs obtained by CCSA and ICSA for Case 2
0 2 4 6 8 10 12 14 16 18 20Iteration
8234
8236
8238
8240
8242
8244
8246
8248
8250
8252
Fitn
ess F
unct
ion
($)
CCSAICSA
Figure 6 The best run obtained by CCSA and ICSA for Subcase 31
whereas 50 values of fuel cost indicate that the proposedICSA can find many solutions with better quality Clearlythe proposed ICSA is outstanding in terms of stabilizationof solution search and faster convergence As a result it canconclude that the proposed ICSA approach is more effectivethanCCSA in solvingOLDproblemwith considered systems
572The Improvement of Results over Other Methods In thisarticle we have tested ICSA approach on 6 cases with 16
systems with different fuel cost forms different constraintsand different scale systems from 3 units to 320 units We havecompared the yielded results from ICSA approach and otherexisting ones for evaluating the efficiency of ICSA approachIn subsections above we have shown yielded results fromICSA approach and compared these results to those of otheronesHowever the demonstrationhad not been very good forobserving and comparing to lead to a conclusionThus in thesubsectionwehave summarized the result comparisons of the
20 Mathematical Problems in Engineering
0 5 10 15 20 25 30 35 40 45 50Run
8234
82342
82344
82346
82348
8235
82352
82354
82356
82358
8236
Fuel
cost
($)
CCSAICSA
Figure 7 The best fuel cost of 50 runs obtained by CCSA and ICSA for Subcase 31
0 1000 2000 3000 4000 5000 6000Iteration
242
244
246
248
25
252
254
256
Fitn
ess F
unct
ion
($)
CCSAICSA
times105
Figure 8 The best run obtained by CCSA and ICSA for Subcase 35
proposed and other ones Table 15 has reported the reductioncost (in $) of ICSA approach compared to other ones Inaddition we have converted the reduction cost into improve-ment level (in ) for better comparison The improvementhas been shown from the lowest level to the highest levelin terms of reduction cost and improvement percentageIn addition we have also given the slowest and the fastestexecution time of other compared methods together withthat of the proposed method The table implies that ICSA
approach can find better optimal solutions with less fuel costup to $052 for Subcase 13 $75229 for Subcase 14 $1775149for Case 2 $1181256 for Subcase 31 $8438 for Subcase 32$9113 for Subcase 33 $121181 for Subcase 34 $3412 forSubcase 35 $04329 for Subcase 41 $30227 for Subcase42 $407002 for Case 5 $18183 for Subcase 61 $13965 forSubcase 62 and $11817 for Subcase 63 These reductioncosts are equivalent to improvement level (IL) of 0032908968 82282 14143 047 038 099 001405
Mathematical Problems in Engineering 21
0 5 10 15 20 25 30 35 40 45 50Run
2428
243
2432
2434
2436
2438
244
2442
Fuel
cost
($)
CCSAICSA
times105
Figure 9 The best fuel cost of 50 runs obtained by CCSA and ICSA for Subcase 35
0 50 100 150 200 250 300 350 400Iteration
327
328
329
33
331
332
333
334
335
336
337
Fitn
ess F
unct
ion
($)
CCSAICSA
times104
Figure 10 The best run obtained by CCSA and ICSA for Case 5
00003 0002 12291 004 138 and 059 Thesequantitative comparisons reveal that larger scale systems canlead to better reduction cost but the improvement level isnot high because total cost of compared methods tends tobe large for large scale systems Furthermore very large scalesystems with nondifferentiable objective have been normallysolved by strong methods In fact systems in Case 3 havethe same characteristic with single fuel and effects of valveloading process but Subcase 34 is a larger scale system with
40 units while Subcases 31 32 and 33 are constructed by 3units 13 units and 13 units So the improvement percentageof Subcase 34 can be up to 099 whilst that of Subcases32 and 33 is 047 and 038 respectively Subcase 35 iswith the largest system 80 units but the reduction cost is notmuch only $3412 because compared methods with ICSA areeither state-of-the-art ones or improved ones Also Subcase41 and Subcase 42 have considered single fuel and PPZ andspinning reserve constraints but Subcase 42 is larger scale
22 Mathematical Problems in Engineering
Table 15 Performance improvement summary of the proposed method
Study cases Reduction cost ($) Improvement level () Execution time (s)From To From To Slowest method Fastest method Proposed method
Figure 11 The best fuel cost of 50 runs obtained by CCSA and ICSA for Case 5
system with 90 units and Subcase 41 is only with 60 unitsThus the improvement of Subcase 41 is lower with 00003but that of Subcase 42 is 0002 Similarly systems in Cases 6have the same featurewithmulti-fossil fuel sources and effectsof valve loading process but they are respectively constructedby 80 160 and 320 units As a result the improvements ofSubcase 62 138 and Subcase 63 059 are much higherthan Subcase 61 004 However Subcase 62 with smallernumber of units but getting higher improvement is easilyunderstood because there were nine compared methods butonly three compared methods are considered for Subcase 63In general the improvement is not high it is about under onedollar several dollars tens of dollars and over one thousand
dollars per hour however the saving cost will be significantif the operation is considered to be onemonth with 720 hoursor one year with 8760 hours
Execution time comparison can be evaluated by observ-ing the fastest and the slowest compared methods in Table 15These execution times of the proposed method are approxi-mately equal to that of the fastest methods for Subcases 3233 and 34 and much shorter than other fastest comparedmethods for other cases especially for Subcases 61 and 63The fastest method for Subcase 62 is CBA [44] showing 57seconds while that of the proposed method is 1119 secondsHowever it cannot conclude that CBA is more effective thanthe proposed method because the proposed method could
find better optimal solution with less fuel cost by $2130As considering execution time of the slowest comparedmethods it can point out that ICSA is a very fast optimizationtool since the execution time of these methods is 29496 and3425 seconds for Subcases 32 and 33 116735 seconds forSubcase 34 56381 seconds for Subcase 41 94093 secondsfor Subcase 42 12797 seconds for Case 5 30941 seconds forSubcase 61 7542 seconds for Subcase 62 and 9641 secondsfor Subcase 63 while the execution time of the proposedmethod for these cases is respectively 095 146 091 1589246 84828 1119 and 171384 seconds It is clearly shown thatICSA is very fast as compared to these methods
In summary the proposed method has found approx-imately high quality solutions with several standard state-of-the-art meta-heuristic algorithms and improved versionsof them together with other old methods In addition theproposed method could improve result better than approx-imately all methods with faster execution time Comparedto other methods with the fastest convergence speed andhigh quality solutions the proposed method has been as
fast as for some cases and much faster for other cases Thecomparison with the slowest methods could show that theproposed method was extremely powerful since it was up tonearly one thousand times faster Consequently the proposedICSA approach can be one of the strongest optimization toolsfor OLD problem
6 Conclusions
This paper has proposed a good ICSA method for solvingOLD problem in which many test systems with differentobjective functions and complicated constraints from simpleto complex have been used as studied cases The proposedICSA method has been developed by performing severalmodifications on the second solution update of CCSA whichcontained several drawbacks to global convergence and fastmanner The OLD problem has covered from single fuelto multi-fossil fuels from quadratic objective function tononconvex objective function in addition to PPZ RPR andRRL constraints Many existing optimization algorithms have
been concerned in aim to compare the performance and givethe final conclusion on the proposed method There have
been six main cases with sixteen subcases The evaluationshave been made at the end of each study case Clearly theproposed ICSA approach has yielded more effective optimalsolutions with faster execution time than almost all methodsConsequently it can be concluded that the proposed methodis much more superior to CCSA and is a very promisingmethod for solving OLD problem
Appendix
See Tables 16ndash26
Nomenclature
120575119894 120582119894 120572119894 120573119894 120574119894 Fuel cost functioncoefficients of the 119894119905ℎthermal generationunit11986100 1198610119895 119861119895119894 Power loss matrixcoefficients119865119865119909 119865119865119887119890119904119905 The values of fitness ofsolution 119909 and theso-far most effectivesolution among thecurrent set ofsolutions119898119894 Number of fuels burntin the 119894119905ℎ thermalgeneration unit
119873 Number of allavailable thermalgeneration units119899119894 Number of prohibitedpower zones of the 119894119905ℎthermal generationunit119875119906119894119896 119875119897119894119896 Upper and lowerlimits of the 119894119905ℎthermal generationunit corresponding tothe kth PPZ119875119886 Probability ofreplacing controlvariables in each oldsolution119875119863 Real power demand ofall loads in system119875119894119898119886119909 119875119894119898119894119899 The highest andlowest real poweroutputs of the iththermal generationunit119875119894119895119898119886119909 119875119894119895119898119894119899 The highest andlowest real poweroutputs of the 119894119905ℎthermal generationunit corresponding tothe 119895119905ℎ fuel type119878119894119898119886119909 Maximum real powerreserve contributionof the thermalgeneration unit 1198941198781199001198971199031198861198991198891 1198781199001198971199031198861198991198892 1198781199001198971199031198861198991198893 1198781199001198971199031198861198991198894 Randomly mixedsolutions from the setof current solutions119878119900119897119909 119878119900119897119866119887119890119904119905 The old solution x andthe most effectivesolution119878119877 Real power reserverequirement of system
120575119894119895 120582119894119895 120572119894119895 120573119894119895 120574119894119895 Fuel cost functioncoefficients of the 119894119905ℎthermal generationunit corresponding tothe 119895119905ℎ fuel type1205761 1205762 1205763 1205764 1205765 1205766 Random numbersbetween 0 and 1Ω Number of generationunits considering PPZconstraint
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare no conflicts of interest
References
[1] N Sinha R Chakrabarti and P K Chattopadhyay ldquoEvolution-ary programming techniques for economic load dispatchrdquo IEEETransactions on Evolutionary Computation vol 7 no 1 pp 83ndash94 2003
[2] P Hansen ldquoA separable approximation dynamic programmingalgorithm for economic dispatch with transmission lossesrdquoYugoslav Journal of Operations Research vol 12 no 2 2002
[3] N T Thang ldquoSolving economic dispatch problem with piece-wise quadratic cost functions using lagrange multiplier theoryrdquoin Proceedings of the 3rd International Conference on ComputerTechnology and Development (ICCTD rsquo11) pp 359ndash364 ASMEPress 2011
[4] S K Mishra and S K Mishra ldquoA comparative study of solutionof economic load dispatch problem in power systems in theenvironmental perspectiverdquoProcedia Computer Science vol 48pp 96ndash100 2015
[5] A A Al-Subhi and H K Alfares ldquoEconomic load dispatchusing linear programming a comparative studyrdquo InternationalJournal of Applied Industrial Engineering vol 3 no 1 pp 16ndash362016
28 Mathematical Problems in Engineering
[6] C Zhou G Huang and J Chen ldquoPlanning of electric powersystems considering virtual power plants with dispatchableloads included an inexact two-stage stochastic linear program-ming modelrdquoMathematical Problems in Engineering vol 2018Article ID 7049329 12 pages 2018
[7] T T Nguyen N V Quynh and L Van Dai ldquoImproved fireflyalgorithm a novel method for optimal operation of thermalgenerating unitsrdquo Complexity vol 2018 Article ID 7267593 23pages 2018
[8] S S Haroon S Hassan S Amin et al ldquoMultiple fuel machinespower economic dispatch using stud differential evolutionrdquoEnergies vol 11 no 6 pp 1ndash20 2018
[9] C-T Su and C-L Chiang ldquoNonconvex power economic dis-patch by improved genetic algorithm with multiplier updatingmethodrdquo Electric Power Components and Systems vol 32 no 3pp 257ndash273 2004
[10] NNoman andH Iba ldquoDifferential evolution for economic loaddispatch problemsrdquo Electric Power Systems Research vol 78 no8 pp 1322ndash1331 2008
[11] P Somasundaram and K Kuppusamy ldquoApplication of evolu-tionary programming to security constrained economic dis-patchrdquo International Journal of Electrical Power amp EnergySystems vol 27 no 5-6 pp 343ndash351 2005
[12] T Jayabarathi K Jayaprakash andD Jeyakumar ldquoEvolutionaryprogramming technique for different kinds of economic dis-patch problemsrdquo Electric Power Systems Research vol 73 no 2pp 169ndash176 2005
[13] P Subbaraj R Rengaraj and S Salivahanan ldquoEnhancementof combined heat and power economic dispatch using selfadaptive real-coded genetic algorithmrdquo Applied Energy vol 86no 6 pp 915ndash921 2009
[14] C-L Chiang and C-T Su ldquoAdaptive-improved genetic algo-rithm for the economic dispatch of units with multiple fueloptionsrdquo Cybernetics and Systems vol 36 no 7 pp 687ndash7042005
[15] C Chiang ldquoImproved genetic algorithm for power economicdispatch of units with valve-point effects and multiple fuelsrdquoIEEE Transactions on Power Systems vol 20 no 4 pp 1690ndash1699 2005
[16] K Meng H G Wang Z Y Dong and K P Wong ldquoQuantum-inspired particle swarm optimization for valve-point economicload dispatchrdquo IEEE Transactions on Power Systems vol 25 no1 pp 215ndash222 2010
[17] P Erdogmus A Ozturk and S Duman ldquoEnvironmen-taleconomic dispatch using genetic algorithm and simulatedannealingrdquo in Proceedings of the International Conference onElectric Power and Energy Conversion Systems EPECS rsquo09 pp1ndash4 IEEE 2009
[18] L D S Coelho and V C Mariani ldquoImproved differentialevolution algorithms for handling economic dispatch opti-mization with generator constraintsrdquo Energy Conversion andManagement vol 48 no 5 pp 1631ndash1639 2007
[19] H Dakuo W Fuli and M Zhizhong ldquoA hybrid genetic algo-rithm approach based on differential evolution for economicdispatch with valve-point effectsrdquo Electrical Power and EnergySystems vol 30 pp 31ndash38 2008
[20] C-F Chang J-J Wong J-P Chiou and C-T Su ldquoRobustsearching hybrid differential evolution method for optimalreactive power planning in large-scale distribution systemsrdquoElectric Power Systems Research vol 77 no 5-6 pp 430ndash4372007
[21] M R Farooqi P Jain and K R Niazi ldquoUsing Hopfield neuralnetwork for economic dispatch of power systemsrdquo in Proceed-ings of the National Power Engineering Conference PECon rsquo03pp 5ndash10 Bangi Malaysia 2003
[22] T Yalcinoz H Altun and U Hasan ldquoConstrained economicdispatch with prohibited operating zones a Hopfield neuralnetwork approachrdquo in Proceedings of the 10th MediterraneanElectrotechnical Conference Information Technology and Elec-trotechnology for the Mediterranean Countries ProceedingsMeleCon rsquo00 (Cat No 00CH37099) pp 570ndash573 LemesosCyprus 2000
[23] D N Vo P Schegner and W Ongsakul ldquoCuckoo searchalgorithm for non-convex economic dispatchrdquo IET GenerationTransmission amp Distribution vol 7 no 6 pp 645ndash654 2013
[24] J-B Park K-S Lee J-R Shin and K Y Lee ldquoA particleswarm optimization for economic dispatch with non-smoothcost functionsrdquo IEEETransactions on Power Systems vol 20 no1 pp 34ndash42 2005
[25] T Niknam H D Mojarrad and M Nayeripour ldquoA newfuzzy adaptive particle swarm optimization for non-smootheconomic dispatchrdquo Energy vol 35 no 4 pp 1764ndash1778 2010
[26] L D S Coelho and V C Mariani ldquoAn improved harmonysearch algorithm for power economic load dispatchrdquo EnergyConversion and Management vol 50 no 10 pp 2522ndash25262009
[27] M Mahdavi M Fesanghary and E Damangir ldquoAn improvedharmony search algorithm for solving optimization problemsrdquoAppliedMathematics and Computation vol 188 no 2 pp 1567ndash1579 2007
[28] A Bhattacharya and P K Chattopadhyay ldquoSolving complexeconomic load dispatch problem using biogeography-basedoptimizationrdquo Expert Systems with Applications vol 37 no 5pp 3605ndash3615 2010
[29] D Simon ldquoBiogeography-based optimizationrdquo IEEE Transac-tions on Evolutionary Computation vol 12 no 6 pp 702ndash7132008
[30] J S Alsumait J K Sykulski and A K Al-Othman ldquoAhybrid GA-PS-SQP method to solve power system valve-pointeconomic dispatch problemsrdquo Applied Energy vol 87 no 5 pp1773ndash1781 2010
[31] M S Turgut andGKDemir ldquoQuadratic approximationndashbasedhybrid Artificial Cooperative Search algorithm for economicemission load dispatchproblemsrdquo International Transactions onElectrical Energy Systems vol 27 no 4 pp 1ndash14 2017
[32] T A A Victoire and A E Jeyakumar ldquoHybrid PSO-SQPfor economic dispatch with valve-point effectrdquo Electric PowerSystems Research vol 71 no 1 pp 51ndash59 2004
[33] D-KHe F-LWang andZ-ZMao ldquoHybrid genetic algorithmfor economic dispatch with valve-point effectrdquo Electric PowerSystems Research vol 78 no 4 pp 626ndash633 2008
[34] D C Secui ldquoA modified symbiotic organisms search algorithmfor large scale economic dispatch problem with valve-pointloading effectsrdquo Energy vol 113 pp 366ndash384 2016
[35] K BhattacharjeeA Bhattacharya and S H N Dey ldquoTeaching-learning-based optimization for different economic dispatchproblemsrdquo Scientia Iranica vol 21 no 3 pp 870ndash884 2014
[36] K Bhattacharjee A Bhattacharya and S H N Dey ldquoOppo-sitional Real Coded Chemical Reaction Optimization for dif-ferent economic dispatch problemsrdquo International Journal ofElectrical Power amp Energy Systems vol 55 pp 378ndash391 2014
Mathematical Problems in Engineering 29
[37] V Hosseinnezhad M Rafiee M Ahmadian and M T AmelildquoSpecies-basedQuantum Particle SwarmOptimization for eco-nomic load dispatchrdquo International Journal of Electrical Poweramp Energy Systems vol 63 pp 311ndash322 2014
[38] M S P Subathra S E Selvan T A A Victoire A HChristinal andU Amato ldquoA hybrid with cross-entropymethodand sequential quadratic programming to solve economic loaddispatch problemrdquo IEEE Systems Journal vol 9 no 3 pp 1031ndash1044 2015
[39] J Zhan Q H Wu C Guo and X Zhou ldquoEconomic dispatchwith non-smooth objectives part I local minimum analysisrdquoIEEE Transactions on Power Systems vol 30 no 2 pp 710ndash7212015
[40] A K Barisal andR C Prusty ldquoLarge scale economic dispatch ofpower systems using oppositional invasive weed optimizationrdquoApplied Soft Computing vol 29 pp 122ndash137 2015
[41] M Basu ldquoImproved differential evolution for economic dis-patchrdquo International Journal of Electrical Power amp EnergySystems vol 63 pp 855ndash861 2014
[42] V S Aragon S C Esquivel and C C Coello ldquoAn immunealgorithm with power redistribution for solving economicdispatch problemsrdquo Information Sciences vol 295 pp 609ndash6322015
[43] M Ghasemi M Taghizadeh S Ghavidel and A AbbasianldquoColonial competitive differential evolution an experimentalstudy for optimal economic load dispatchrdquo Applied Soft Com-puting vol 40 pp 342ndash363 2016
[44] B R Adarsh T Raghunathan T Jayabarathi and X-S YangldquoEconomic dispatch using chaotic bat algorithmrdquo Energy vol96 pp 666ndash675 2016
[45] N Ghorbani and E Babaei ldquoExchange market algorithm foreconomic load dispatchrdquo International Journal of ElectricalPower amp Energy Systems vol 75 pp 19ndash27 2016
[46] J X V Neto G Reynoso-Meza T H Ruppel V C Marianiand L D S Coelho ldquoSolving non-smooth economic dispatchby a new combination of continuous GRASP algorithm anddifferential evolutionrdquo International Journal of Electrical Poweramp Energy Systems vol 84 pp 13ndash24 2017
[47] A Kavousi-Fard and A Khosravi ldquoAn intelligent 120579-ModifiedBat Algorithm to solve the non-convex economic dispatchproblem considering practical constraintsrdquoElectrical Power andEnergy Systems vol 82 pp 189ndash196 2016
[48] M A Al-Betar M A Awadallah A T Khader and A L BolajildquoTournament-based harmony search algorithm for non-convexeconomic load dispatch problemrdquo Applied Soft Computing vol47 pp 449ndash459 2016
[49] M A Al-Betar M A Awadallah I A Doush E Alsukhni andH ALkhraisat ldquoA non-convex economic dispatchproblemwithvalve loading effect using a new modified 120573-hill climbing localsearch algorithmrdquo Arabian Journal for Science and Engineeringpp 1ndash8 2018
[50] M A Awadallah M A Al-Betar A L Bolaji E M Alsukhniand H Al-Zoubi ldquoNatural selection methods for artificial beecolony with new versions of onlooker beerdquo Soft Computing pp1ndash40 2018
[51] S H Kaboli and A K Alqallaf ldquoSolving non-convex economicload dispatch problem via artificial cooperative search algo-rithmrdquo Expert Systems with Applications vol 128 pp 14ndash272019
[52] D Singh and J Dhillon ldquoAmeliorated greywolf optimization foreconomic load dispatch problemrdquo Energy vol 169 pp 398ndash4192019
[53] X S Yang and S Deb ldquoCuckoo search via Levy flightsrdquo inProceedings of the World Congress on Nature and BiologicallyInspired Computing (NaBIC rsquo09) pp 210ndash214 India 2009
[54] S Sahoo K M Dash R C Prusty and A K Barisal ldquoCom-parative analysis of optimal load dispatch through evolutionaryalgorithmsrdquo Ain Shams Engineering Journal vol 6 pp 107ndash1202015
[55] M Basu and A Chowdhury ldquoCuckoo search algorithm foreconomic dispatchrdquo Energy vol 60 pp 99ndash108 2013
[56] A B Serapiao ldquoCuckoo search for solving economic dispatchload problemrdquo Intelligent Control and Automation vol 04 no04 pp 385ndash390 2013
[57] A H Bindu and M D Reddy ldquoEconomic load dispatch usingcuckoo search algorithmrdquo International Journal of EngineeringResearch and Applications vol 3 pp 498ndash502 2013
[58] E Afzalan and M Joorabian ldquoAn improved cuckoo searchalgorithm for power economic load dispatchrdquo InternationalTransactions on Electrical Energy Systems vol 25 no 6 pp 958ndash975 2015
[59] T T Nguyen and D N Vo ldquoThe application of one rank cuckoosearch algorithm for solving economic load dispatch problemsrdquoApplied Soft Computing vol 37 pp 763ndash773 2015
[60] S M Islam S Das S Ghosh S Roy and P N Suganthan ldquoAnadaptive differential evolution algorithm with novel mutationand crossover strategies for global numerical optimizationrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 42 no 2 pp 482ndash499 2012
[61] Z L Gaing ldquoParticle swarm optimization to solving theeconomic dispatch considering the generator constraintsrdquo IEEETransactions on Power Systems vol 18 no 3 pp 1187ndash1195 2003
[62] X-S Yang and S Deb ldquoEngineering optimisation by Cuckoosearchrdquo International Journal of Mathematical Modelling andNumerical Optimisation vol 330 pp 43ndash51 2010
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6 Mathematical Problems in Engineering
change is supervised Regulated power can be higher or lowerthan the initial value as long as it is within a predeterminedrange Increased step size (ISS) and decreased step size (DSS)are given as input data and they are used to limit the change ofpower output of each thermal generating unit The constraintcan bemathematically expressed as the following formula [7]
1198751198940 minus 119863119878119878119894 le 119875119894 le 1198751198940 + 119868119878119878119894 (13)
where 1198751198940 is the initial power output of the 119894119905ℎ thermalgenerating unit before its power output is regulated 119868119878119878119894 and119863119878119878119894 are respectively maximum increased and decreasedstep sizes of the 119894119905ℎ thermal generating unit
3 The Proposed Cuckoo Search Algorithm
31 Classical Cuckoo Search Algorithm In search techniqueof CCSA [53] a set of solutions is randomly generated withina predetermined range in the first step and then the quality ofeach one is ranked by computing value of fitness functionThemost effective solution corresponding to the smallest valueof fitness function is determined and then search procedurecomes into a loop algorithm until the maximum iterationis reached In the loop algorithm two techniques updatingnew solutions two times (corresponding to two generations)are Levy flights and mutation technique which is calledstrange eggs identification technique The two generationscan produce promising quality solutions for CCSA Aftereach generation CCSA will carry out comparing fitness ofnewly updated solutions and initial solutions for keepingbetter ones and abandoning worse ones The most effectivesolution at last step of the loop search algorithm is determinedand it is restored as one candidate solution for a study caseThe detail of the two stages is as follows
311 Levy Flights Stage This is the first calculation step in theloop algorithm and it also produces new solutions in the firstgeneration for CCSA New solution 119878119900119897119899119890119908119909 is created by thefollowing model
119878119900119897119899119890119908119909 = 119878119900119897119909 + 120572 (119878119900119897119909 minus 119878119900119897119866119887119890119904119905) oplus Levy (120573) (14)
where 120572 is the positive scaling factor and it is nearly set todifferent values for different problems in the studies [53 62]In the work the most appropriate values for such factor canbe chosen to be 02505 for different systems
312 Discovery of Alien Eggs Stage The step plays a veryimportant role for updating new solutions 119878119900119897119899119890119908119909 of thewhole population However not every control variable ineach old solution is newly updated and the decision ofreplacement is dependent on comparison criteria as thefollowing equation
119878119900119897119899119890119908119909=
119878119900119897119909 + 1205761 (1198781199001198971199031198861198991198891 minus 1198781199001198971199031198861198991198892) if 1205762 lt 119875119886119878119900119897119909 119900119905ℎ119890119903119908119894119904119890
(15)
32 Proposed Algorithm In the part a new variant of CCSA(ICSA) is constructed by applying three effective changes onthe main functions of CCSA in order to shorten simula-tion time corresponding to reduction of iterations and findmore promising solutions The proposed amendments areexplained in detail as follows
(i) Suggest one more equation producing updated stepsize in addition to existing one in CCSA
(ii) Create a new selection standard by computing fitnessfunction ratio 119865119865119877119909 and comparing 119865119865119877119909 with apredetermined parameter 119879119900119897119909 Thus thanks to thestandard the existing updated step size and additionalupdate step size will be chosen more effectively
(iii) Automatically change value of 119879119900119897119909 for the xth solu-tion based on the result of comparing 119865119865119877119909 with theprevious 119879119900119897119909
Such three points are clarified by observing the followingsections
321 Strange Eggs Identification Technique (Mutation Tech-nique) The first proposed improvement in our proposedICSA approach is to select a more suitable formula forproducing new solutions with better fitness function valueIn CCSA Equation (16) below is used to produce a changingstep nearby old solutions for all current solutions
Δ1198781199001198971198991198901199081199091 = 1205763 (1198781199001198971199031198861198991198891 minus 1198781199001198971199031198861198991198892) (16)
The use of Equation (16) aims to produce a random walkaround old solutions in search zones with intent to findout promising solutions In order to reduce the possibilityof suffering the local trap and approach to other favorablezones for searching we propose a new Equation (17) Theformula is built by the idea of enlarging search zone withthe use of two more available solutions Obviously the largerchanging step can own higher performance in moving toother search spaces that the classical approach used in CCSAThe suggestion is mathematically expressed by the formulabelow
Δ1198781199001198971198991198901199081199092= 1205764 (1198781199001198971199031198861198991198891 minus 1198781199001198971199031198861198991198892 + 1198781199001198971199031198861198991198893 minus 1198781199001198971199031198861198991198894) (17)
The changing step obtained by using Eq (17) is namedfour-point changing step Now two solutions which arenewly formed by using two different changing steps shownin formulas (16) and (17) are found by the two followingmethods
It can be clearly observed that the distance between 119878119900119897119909(old solution) and 1198781199001198971198991198901199081199091 (new solution) is lower than thatbetween 119878119900119897119909 and 1198781199001198971198991198901199081199092 This difference can contribute ahighly efficient improvement to the proposed ICSA approachsearch ability
Mathematical Problems in Engineering 7
ΔSol2
Sol2
Sol3
Sol4
ΔSol1
Sol1
Solx
Solnew1
Solnew2
Figure 1 Simulation of solutions corresponding to the first itera-tions of the loop algorithm
For the CCSA case if two solutions 1198781199001198971199031198861198991198891 and 1198781199001198971199031198861198991198892are either slightly different or completely coincident suchnewly updated solution 1198781199001198971198991198901199081199091 does not have good chanceto leave the current zone and approach to more promisingzones In another word the new one is approximately coin-cident with the old one As the search task is taking place atsome last iterations this phenomenon becomes much worsebecause all current solutions are lumped in a small zone andthe capability of moving to other zones is impossible As aresult the CCSA approach will work ineffectively and searchstrategy is time consuming until other runs are started
Contrary to the two-point step size the new proposedformula may produce a large enough length to escape thelocal optimum zone and reach new favorable zones Itexplainswhy the four-point changing step has positive impacton the considered random walk rather than the two-pointchanging step
322 New Standard forChoosing theMostAppropriate Chang-ing Step In this section we extend our analysis to answer thequestionwhen to use the four-point step size FromEquations(18) and (19) two new solutions which are represented asthe results of the two-point-based factor and the four-pointstep size can be illustrated by using Figure 1 corresponding tothe search process at the first some iterations and Figure 2corresponding to the last some iterations For the sake ofsimplicity we rewrite the two equations as follows
Here we suppose that 1198781199001198971 and 1198781199001198972 are obtained byfour exact solutions 1198781199001198971 1198781199001198972 1198781199001198973 and 1198781199001198974 and calculatedas follows
ΔSol2
ΔSol1
Solx
Solnew1
Solnew2
Figure 2 Simulation of solutions corresponding to the last itera-tions of the loop algorithm
Δ1198781199001198971 = 1198781199001198971 minus 1198781199001198972 (22)
Δ1198781199001198972 = 1198781199001198973 minus 1198781199001198974 (23)
Asmentioned above the high changing step between newsolution and old solution can help to explore new favorablezones However in optimization algorithms searching stepscannot be arbitrarily large otherwise the algorithm maydiverge in particular for the cases that the consideredsolutions 119878119900119897119909 are not close together in solution search spaceFor example at the beginning of loop algorithm with thefirst iterations in Figure 1 1198781199001198971198991198901199081 is a better choice than1198781199001198971198991198901199082 because it is kept in a sufficient limit and does notlead to a risk of divergence In contrast as many of currentsolutions are in different positions but their distance is notvery short or approximately coincident such as at the lastiterations in Figure 2 1198781199001198971198991198901199081 and 119878119900119897119909 have a very shortdistance but 1198781199001198971198991198901199082 and 119878119900119897119909 have higher distance Accordingto the phenomenon in Figure 2 the proposed ICSA approachneeds to produce a high changing step to move solutions toother search zones without local optimum Hence 1198781199001198971198991198901199082would be preferred to 1198781199001198971198991198901199081
Based on the argument above the determination of thecondition for using either two-point changing step or four-point changing step is really crucial to the performance ofthe proposed ICSA approach in searching solutions of OLDproblem Here the ratio of 119865119865119877119909 which can be found byEquation (24) is suggested to be a suitable measurement forthe selection of two options
Δ119865119865119877119909 = 119865119865119909 minus 119865119865119887119890119904119905119865119865119887119890119904119905 (24)
For a particular set of the current solutions each individ-ual depending on its 119865119865119877119909 will create a corresponding newsolution by using either Equation (18) or (19) If the valueof one current solution is smaller than the predeterminedparameter 119879119900119897 Equation (19) is applied for updating suchconsidered solution 119909 Otherwise Equation (18) is a betteroption The steps of the modified algorithm are similar to the
8 Mathematical Problems in Engineering
If 1205765 lt 119875119886If FFRx lt Tolx119878119900119897119899119890119908119909 = 119878119900119897119909 + 1205764(1198781199001198971199031198861198991198891 minus 1198781199001198971199031198861198991198892 + 1198781199001198971199031198861198991198893 minus 1198781199001198971199031198861198991198894)else119878119900119897119899119890119908119909 = 119878119900119897119909 + 1205764(1198781199001198971199031198861198991198891 minus 1198781199001198971199031198861198991198892)end
Algorithm 1 New mutation technique applied in the proposed ICSA approach
conventional CSA except that an additional step should beadded at each iteration In this step the119865119865119889 of all individualsolutions should be calculated by utilizing Equation (24) andthen the result of comparing the ratio with 119879119900119897will be used todecidewhich updating formula should be selectedThewholedescription of the proposed standard and new mutationtechnique can be coded inMatlab program language by usingAlgorithm 1
323 Adjustment of Tolerance for Each Solution As pointedout above the proposed method needs assistances to deter-mine the most appropriate step size for finding out favorablesolution zones The given aim can be reached if the selectionof 119879119900119897119909 is reasonable however the range of this parameteris infinite and hard to select Thus the adaptation of tuningthe parameter is really necessary First of all the compari-son between 119879119900119897119909 and 119865119865119877119909 is carried out and then theadaptation will be determined based on the obtained resultfrom the comparison Results of comparison between the twoparameters can be either 119865119865119877119909 is less than 119879119900119897119909 or 119865119865119877119909is higher than 119879119900119897119909 The case that two parameters are equalhardly ever occurs
As the comer assumptionhappens (ie119865119865119877119909 is less than119879119900119897119909) at the considered time the four-point step size will beemployed for the 119909119905ℎ solution If 119879119900119897119909 remains unchanged atthe previous value the identification of improvement fromsuch four-point step size or two-point step size is vagueConsequently value of 119879119900119897119909 must be automatically reducedto a lower value in case that it has significant contribution tofound promising solution of previous iteration Clearly thedecrease of119879119900119897119909 can enable the proposedmethod to jump outlocal optimal zone and approachmore effective zones By trialand error method 119879119900119897119909 is selected to be a function of itselfthat is 09 of the previous value Finally the implementationof the proposed ICSA approach is presented in Algorithm 2
4 The Application of the ProposedICSA for OLD Problem
Thewhole computation steps of the proposed ICSA approachfor solving OLD problem are explained as follows
41 Handling Constraints and Randomly Producing InitialPopulation As shown in Section 2 the considered OLDproblem takes five following constraints into account
(i) Power balance constraint is shown in Equation (4)
(ii) Power output limitation constraint is shown in Equa-tion (6)
(iii) Prohibited power zone constraint is shown in Equa-tion (9)
(iv) Real power reserve constraint is shown in Equation(10)
(v) Ramp rate limit constraint is shown in Equation (13)
Among the five constraints ramp rate limit generationlimit and prohibited power zone seem to be more com-plicated than power balance and power reserve constraintsHowever the three constraints can be solved more easilybecause each unit is constrained independently in the threeconstraints whereas power balance constraint and powerreserve constraint consider all the thermal generating unitssimultaneously Power reserve constraint can be handledby penalizing the total generation of all units while powerbalance constraint can be solved by penalizing one violatedthermal generating unit The whole computation procedurefor solving all constraints and calculating fitness function ofsolutions is described in detail as follows
Step 1 Redefine maximum and minimum power output ofeach thermal generating unit as considering PPZ and RRLconstraints by using the following formulas
119875119894max = 119875119894max if 119875119894max le 119875i0 + 119868119878119878119894119875i0 + 119868119878119878119894 if 119875119894max gt 119875i0 + 119868119878119878119894
119894 = 1 119873(25)
119875119894min = 119875119894min if 119875119894min ge 119875i0 minus 119863119878119878119894119875i0 minus 119863119878119878119894 119890119897119904119890
119894 = 1 119873(26)
Mathematical Problems in Engineering 9
Produce initial population with119873119901119904 solutions (1198781199001198971 1198781199001198972 119878119900119897119909 119878119900119897119873119901119904)Calculate fitness function (1198651198651 1198651198652 119865119865119909 119865119865119873119901)Go to the loop algorithm by setting 119866 = 1
While (119866119898119886119909 gt 119866) (i) The first newly produced solutions119878119900119897119899119890119908119909 = 119878119900119897119909 + 120572(119878119900119897119909 minus 119878119900119897119866119887119890119904119905) oplus Levy(120573) (ii) Perform selection approach
119865119865119909 = 119865119865119909 if 119865119865119909 le 119865119865119899119890119908119909119865119865119899119890119908119909 119900119905ℎ119890119903119908119894119904119890 119909 = 1 119873119901119904
(v) Determine the most effective solution and its fitnessDetermine 119865119865119909 with the smallest value and assign 119878119900119897119909 to 119878119900119897119866119887119890119904119905If 119866119898119886119909 gt 119866 perform step (i) and increase 119866 to 119866 + 1 Otherwise stop the loop algorithm and report boththe smallest fitness together with 119878119900119897119866119887119890119904119905End while
Among the four Equations (25) and (26) are used firstin order to redefine upper bound and lower bound for allthermal generating units as considering RRL constraint Thethe redefined bounds continue to be redefined for the secondtime by using (27) and (28) as considering PPZ constraints
Step 2 (randomly produce initial population) For dealingwith the power balance constraint all available units areseparated into two groups in which the first group withdecision variables consists of the power output from thesecond unit to the last unit (P2 P3 PN) meanwhile onlythe power output of the first unit (1198751) belongs to the secondgroup with dependent variable So upper bound solution119878119900119897119898119886119909 and lower bound solution 119878119900119897119898119894119899 must be defined asfollows
Step 3 Handle prohibited power zone constraint for decisionvariables P2 P3 PN
After being randomly produced there is a high possi-bility that decision variables fall into PPZ and they violatePPZ constraint So the verification of falling into PPZ andcorrection of the violation should be accomplished by usingthe following formula
119875119894 =
119875119897119894119896 if 119875119897119894119896 lt 119875119894 le 119875119897119894119896 + 1198751198961198941198962119875119906119894119896 if (119875119894 gt 119875119897119894119896 + 1198751198961198941198962 ) amp (119875119894 lt 119875119906119894119896)119875119894 119890119897119904119890
119894 = 2 119873 amp 119896 = 1 119899119894
(31)
Step 4 Handle RPB constraint by calculating 1198751 and penaliz-ing 1198751 if it violates constraints
In this step power balance constraint is exactly handledby calculating and penalizing dependent variable (1198751) 1198751 isobtained by using formulas (4) and (5) as follows
1198751 = minus (11986101 minus 1 + 2sum119873119894=2 1198611119894119875119894) plusmn radicΔ211986111 (32)
where
Δ = (11986101 minus 1 + 2 119873sum119894=2
1198611119894119875119899)2 minus 411986111(119875119863 minus 119873sum
119894=2
119875119894+ 11986100 + 119873sum
119894=2
1198610119894119875119894 + 119873sum119894=2
119873sum119895=2
119875119894119861119894119895119875119895) amp Δ ge 0(33)
In Equation (32) 1198751 has been determined for the purposeof dealing with real power balance constraint However it isnot sure that 1198751 can satisfy upper bound and lower boundconstraints and prohibited power zone constraints So 1198751must be checked and penalized
Firstly 1198751 is checked and penalized for upper and lowerbound constraints by the following model
Δ1198751x =
0 if 1198751min le 1198751x le 1198751max
1198751min minus 1198751x if 1198751min gt 1198751x1198751x minus 1198751max if 1198751max lt 1198751x
(34)
In Equation (34) if the second case or the third caseoccurs it means P1 has violated either lower bound or upperbound and it would be penalized by using either (P1x= P1min-P1x) or (P1x =P1x -P1max) Otherwise ifP1 has not violatedthe bound constraints (ie the first case in (34) happened)
P1 would continue to be checked for PPZ constraint by thefollowing model
Δ1198751x
=
1198751 minus 1198751198971119896 if 1198751198971119896 lt 1198751 le 1198751198971119896 + 119875119896111989621198751199061119896 minus 1198751 if (1198751 gt 1198751198971119896 + 11987511989611198962 ) amp (1198751 lt 1198751199061119896)0 119890119897119904119890
(35)
Step 5 Handle real power reserve constraint (10)First of all 119878119894 is determined by using (11) and (12) and
then the 119909119905ℎ solution will be checked and penalized if poweroutput of all thermal generating units cannot satisfy RPRconstraint The penalty for violation of the constraint can becalculated by using equation (36)
Δ119878119877119909 =
0 if119873sum119894=1
119878119894119909 ge 119878119877119878119877 minus 119873sum119894=1
119878119894119909 119890119897119904119890 (36)
As a result real power reserve constraint can be solved byusing the penalty method
42 Calculate Fitness Function for Solutions Fitness functionof each solution is used to evaluate quality of solutionNormally the function is the sum of objective function andpenalty of violating constraints and is obtained by
43 The First Newly Updated Solutions by Levy Flights Tech-nique In this section the first newly updated solutionsare performed by employing Levy flights technique usingEquation (14) However each new solution can be out oftheir feasible operating zone such as PPZ and upper andlower limitations When the power output violates its PPZconstraints Equation (31) will be applied to tackle theconstraint Besides the following equation will be employedwhen power output is higher or lower than their limitations
119878119900119897119909 =
119878119900119897max if 119878119900119897max lt 119878119900119897119909119878119900119897min if 119878119900119897min gt 119878119900119897119909119878119900119897119909 Otherwise
119909 = 1 119873119901 (38)
After that Equations (32)-(37) are performed for deter-mining all variables and penalty terms Finally Equation (38)is employed to calculate fitness function
44 The Second Newly Updated Solutions by Using Muta-tion Technique The second newly updated solutions areaccomplished as presented in Section 3 above Similar to
Mathematical Problems in Engineering 11
the task after doing the first update each solution in thenew population must satisfy PPZ constraint and upper andlower boundaries by considering Equations (31) and (38)Then Equations (32)-(37) are performed for determiningall variables and penalty terms Finally Equation (38) isemployed to calculate fitness function and the solution withthe best value is assigned to the best one 11987811990011989711986611988711989011990411990545 Criterion of Stopping the Loop Algorithm In the loopalgorithm of using the proposed ICSA approach the solutionsearch work is stopped in case that the predeterminedmaximum iterations 119866119898119886119909 is reached For each search ter-mination the most effective solution is stored and anotherrun continues to be accomplished until the predeterminednumber of runs is reached After finishing the runs thebest one is found and reported In addition other valuessuch as the fitness of the worst solution and average fitnessof all solutions are also reported for comparing with othermethods
46 The Whole Iterative Process The whole iterative algo-rithm for implementing the proposed ICSA approach forcoping with OLD problem is described in detail in Figure 3
5 Results and Discussions
The proposed ICSA approach performance has been investi-gated on six cases with different fuel options different fuelcharacteristics and complicated constraints The details ofthe studied cases are presented as follows
Case 1 Four systems with single fuel source (SFS) and powerloss (PL) constraint
Subcase 11 A 3-unit system [57]Subcase 12 A 6-unit system [57]Subcase 13 A 3-unit system [56]Subcase 14 A 6-unit system [56]
Case 2 A 110-unit system with SFS [57]
Case 3 Four systems with SFS and the effects of valve loadingprocess (EoVLP)
Subcase 31 A 3-unit system supplying to a load of850MW [58]Subcase 32 A 13-unit system supplying to a load of1800MW [1]Subcase 33 A 13-unit system supplying to a load of2520MW [1]Subcase 34 A 40-unit system supplying to a load of2500MW [1]Subcase 35 An 80-unit system supplying to a load of4100MW [49]
Case 4 Two systems with SFS and PPZ and RPR constraints
Subcase 41 A 60-unit system supplying to a10600MW load [9]
Subcase 42 A 90-unit system supplying to a15900MW load [9]
Case 5 A 15-unit system with SFS and RRL PPZ and PLconstraints [61]
Case 6 Three systems with multiple fuel sources (MFS) andEoVLP
Subcase 61 An 80-unit system supplying to a21600MW load [15]Subcase 62 A 160-unit system supplying to a43200MW load [15]Subcase 63 A 320-unit system supplying to an86400MW load [54]
For each considered case with each load case the pro-posed ICSA approach is run 50 times on the programlanguage of Matlab and a PC with 4 GB of RAM and 24GHzprocessor The selection of adjustment parameters including119875119886 and 119879119900119897119909 is carefully considered to obtain the best optimalsolutions meanwhile two others such as 119873119901119904 and 119866119898119886119909 arechosen corresponding to the scale of particular test system9 values with the change of 01 in the range [01 09] are inturn selected for 119875119886 while 119879119900119897119909 is 001 at the beginning Theinformation including load demand119873119901119904 119866119898119886119909 and the best119875119886 is reported in Table 1
51 Obtained Results on Case 1 considering Four Systems withSFS and PL Constraint In this section we have implementedthe proposed ICSA approach for solving four systems dividedinto four subcases Tables 2 and 3 show the comparisons ofobtained results from Subcases 11 and 12 and Subcases 13and 14 respectively As listed in Table 2 the proposed ICSAmethod and CCSA can find equal fuel cost for Subcases 11whereas the reduction of fuel cost from the proposed ICSAmethod as compared to CCSA is clearer for Subcase 12 Asshown in Table 3 for comparing the proposed ICSA and threemethods consisting of CCSA ABC and FA the minimumfuel cost of the proposed ICSA is approximately equal to thatof these methods for Subcases 13 but much less than that ofthese methods for Subcase 14 Furthermore the proposedICSA has been run by setting 119873119901119904 and 119866119898119886119909 to 5 and 20but these values were much higher for CCSA ABC and FAThey are 20 and 5000 for CCSA 40 and 100 for ABC and 20and 5000 for FA Consequently the proposed method is veryefficient for Case 1 with four subcases
Optimal solutions obtained by ICSA for Case 1 are shownin Tables 16ndash18
52 Obtained Results on Case 2 considering 110-Unit Systemwith SFS In this section we have employed a very largescale system with 110 units but there were not challenges forobjective function and complex constraints since EoVLP andconstraints were not taken into account Both CCSA andthe proposed ICSA methods have been run for comparingwith BBO hybrid BBO and DE (DEBBO) and Opposi-tional real coded chemical reaction optimization algorithm(ORCCROA) in [36] IWA in [40] and AGWO in [52] As
12 Mathematical Problems in Engineering
Select parameters
- Correct solutions if violating PPZ constraint using Eq (31)- Determine using Eq (32)- Penalize for violating upper and lower bounds constraint using Eq (34)- Penalize for violating PPZ constraint using Eq (35)- Penalize the xth solution for violating RPR constraint using Eq (36)
- Determine quality of solutions employing Eq (37) - Select the most effective solution - Start the loop algorithm by selecting
- Perform the first solution update using Section 32 - Correct boundaries of solutions by using Eq (38)
- Correct solutions if violating PPZ constraint using Eq (31)- Determine using Eq (32)- Penalize for violating upper and lower bounds constraint using Eq (34)- Penalize for violating PPZ constraint using Eq (35)- Penalize the xth solution for violating RPR constraint using Eq(36)
- Determine fitness function using Eq (37)- Compare old solutions and new solutions to keep better ones
- Perform the second solution update using Algorithm 1 - Correct boundaries of solutions by using (38)
- Correct solutions if violating PPZ constraint using Eq (31)- Determine using Eq (32)- Penalize for violating upper and lower bounds constraint using Eq (34)- Penalize for violating PPZ constraint using Eq (35)- Penalize the xth solution for violating RPR constraint using Eq (36)
- Determine fitness function using Eq (37)- Compare old solutions and new solutions to keep better ones- Select the most effective solution
Stop
Start
- Redefine upper and lower bounds using Eqs (25)-(28)- Randomly generate initial population using Eq (30)
Nps Pa Gmax H> Tolx
P1xP1x
P1x
P1xP1x
P1x
P1xP1x
P1x
Sol<MN
Sol<MN
G = 1
G = Gmax G = G + 1
Figure 3 All computation steps for solving OLD problem by employing the proposed ICSA approach
shown in Table 4 AGWO [52] has reached less fuel cost thanICSA however the exact fuel cost which was recalculatedby using reported solution pointed out that the method hasreached a very high fuel cost of $2157404250 For comparisonwith other methods ICSA has found less fuel cost thanall these methods Particularly the reduction of generationfuel cost is significant as compared to BBO DEBBO andCCSA Execution time comparisons are also useful evidence
for indicating the high performance of ICSA Thus it canconclude that ICSA is a strong method for Case 2
Optimal solution obtained by ICSA for the case is shownin Table 19
53 Obtained Results on Case 3 considering Four Systems withSFS and EoVLP In this section the real performance of theproposed ICSA approach has been investigated based on five
Mathematical Problems in Engineering 13
Table 1 Information of considered cases and adjustment parameters
Case Fuel cost function Constraint Subcase No of units 119875119863 (MW) Nps Gmax Best 1198751198861 SFS PL
subcases with the gradual increase of number of units Thesmallest scale system considers 3 units but the largest scalesystem takes 80 units In addition to the implementation ofthe proposed ICSA we have also implemented CCSA forSubcase 31 and Subcase 35 for further comparison becauseCCSA has not been run for the two subcases so far
Comparison of obtained results from Subcase 31 shownin Table 5 indicates that the proposed ICSA is superior toCCSAwith lower fuel cost but it seems to be less effective thanMCSA [58] The minimum cost of MCSA reported in [58] isthe smallest fuel cost but the recalculated cost is much higherthan that of the proposed method Furthermore MCSA hasbeen implemented by setting very high values to 119873119901119904 and119866119898119886119909
Reports for Subcases 32 and 33 shown in Table 6are the comparisons of the proposed ICSA approach andother methods such as conventional Evolution programming(CEP) [1] Fast EP (FEP) [1] improved FEP (IFEP) [1] DE[12] multiplier Lagrange-based genetic algorithm with (GA-MU) [15] QPSO [16] GA-PS-SQP [30] PSO-SQP [32] M120573-HCLSA [49] IABCA [50] CCSA [59] OSE-CSA [59] SOS[34] MSOS [34] CEA-SQT [38] TSBO [39] IWA [40] andCBA [44] As observed from the table ICSAapproach obtainsbetter solutions than mostmethods excluding DE [10] CCSA
[59] OSE-CSA [59] SOS [34] MSOS [34] CEA-SQT [38]TSBO [39] IWA [40] and CBA [44] especially M120573-HCLSA[49] with lower cost $1796097 However recalculated costfrom reported solution of M120573-HCLSA is $179691 BesidesICSA is very fast as compared to most methods where twoother versions of Cuckoo search algorithm CCSA and OSE-CSA are also included except two methods in [34] Theprocessor of computer that all the methods run on is alsoreported in the final column Clearly ICSA approach is veryefficient for the case with the 13-unit system where effects ofvalve loading process are considered
In Subcase 34 the number of units is much larger thanthat of three subcases above up to 40 units [1] The obtainedresult comparisons with others are indicated in Table 7Clearly the minimum cost comparisons reveal that the pro-posedmethod is one of the leading methods due to the lowestcost except the comparison with CCSA [23] OSE-CSA [59]SOS [34] MSOS [34] EMA [45] 120579-MBA [47] and AGWOA[52] It is noted that AGWOA [52] has reported the bestminimum cost with $12140430 but recalculated minimumcost which was obtained by substituting reported optimalgeneration of all thermal generating units is $12141331 Theaverage and the maximum costs from the proposed methoddo not belong to the leading method group however the
Mathematical Problems in Engineering 15
Table 7 Result comparisons for Subcase 34
Approach Best cost Average cost Worst cost CPU time Computer($h) ($h) ($h) (s) (Processor-Ram)
execution time that ICSA approach takes is smaller thanmostones except IAPR [42] and 120579-MBA [47] which have been runon stronger computers Note that MSOS [34] has been fasterthan the proposed method for Subcases 31 and 32 abovebut it is too slower than the proposed method for the casenamely 1813 seconds compared to 146 seconds Comparedto two other versions of Cuckoo search CCSA [23] and OSE-CSA [59] the proposed method is also faster about threetimes although the processors are slightly different For thiscase 120579-MBA [47] shows a very good performance howeverthe method has not been tested onmore complicated systemsand larger scale and therefore more comparisons with themethod must end Clearly the proposed method is stillefficient for the case where large scale and effects of valveloading process are included
Subcase 35 is the largest scale system with 80 units Inaddition to the implementation of ICSA CCSA has beenalso run for the Subcase 35 for further investigation ofefficiency improvement of the proposed ICSA approach The
comparisons of minimum cost in Table 8 show that theproposed ICSA can find more optimal solution than M120573-HCLSA [49] AGWOA [52] and CCSA The proposed ICSAis also superior to CCSA in terms ofmore stable search abilityand lower fluctuation since its average cost and maximumcost are less than those of CCSA The outstanding figurecannot be reached as compared to AGWOA [52] howeverit is hard to conclude AGWOA [52] is superior to theproposed ICSA approach about more stable search abilityand lower fluctuation Actually comparison of the values ofpopulation and iterations as well as execution time cannot beaccomplished because the information was not reported in[52]Thus it can conclude that the proposed ICSA is effectivefor the subcase
Optimal solutions obtained by ICSA for the case areshown in Tables 20ndash23
54 Obtained Results on Case 4 with Two Systems consideringSFS and PPZ and RPR Constraints In this section two
16 Mathematical Problems in Engineering
Table 9 Result comparisons for Subcases 41 and 42
Subcase Approach Best cost ($h) Mean cost ($h) Worst cost ($h) Std dev ($h) CPU time (s) Computer (Processor-Ram)
test systems with SFS and PPZ and RPR constraints areconsidered The test system size is up to 60 and 90 units forSubcases 41 and 42 respectively Comparison for the casesis only performed with two Genetic algorithms consisting ofGA and IGA-MU in [9] and two other versions of Cuckoosearch algorithms including CCSA [23] and OSE-CSA [59]and presented in Table 9 Other studies have tended toignore such complicated constraints of PPZ and RPR Thecosts reported in Table 9 indicate that ICSA approach canobtain more effective solution than CCSA and OSE-CSAbecause it has reached lower minimum cost than the twoones Furthermore the proposed method also takes shortercomputation time for the two cases from about two timesto about three times although the processor of the proposedmethod is slightly strongerThemean costs of ICSA approachare much less than those from IGA-MU and GA and slightlyhigher than those from OSE-CSA but there is a trade-off between the proposed method and CCSA for the twosubcases In fact the proposed method obtains higher meancost for Subcase 41 but lower cost for Subcase 42 AlthoughGAmethods have been runon aweak computerwith 07 GHzof the processor compared to that with 24GHz in the studytheir execution times are significantly higher namely 56381seconds (GA) and 16258 seconds (IGA-MU) compared to09153 seconds of ICSA approach for Subcase 41 and 94093seconds (GA) and 25545 seconds (IGA-MU) compared to15892 seconds (the proposedmethod)The analysis can pointout that ICSA approach ismore efficient than these comparedmethods in terms of optimal solutions and execution time
Optimal solutions obtained by ICSA for Subcase 42 areshown in Table 24
55 Obtained Results on Case 5 with a 15-Unit System consid-ering SFS and RRL PPZ and PL Constraints In this section
a 15-unit system considering RRL PPZ and PL constraintsis considered to be solved for finding optimal solution Forefficiency investigation of the proposed ICSA we have alsoimplemented CCSA for comparison As listed in Table 10the proposed ICSA is the most effective method with thesmallest fuel cost The comparisons of control parameters aswell asCPU time are also good evidence to confirm the strongsearch of the proposed ICSA approach since it has been runby smaller values of control parameter and faster executiontime as compared to all methods excluding CCSA
Optimal solution obtained by ICSA for the case is shownin Table 25
56 Obtained Results on Case 6 withThree Systems consideringMFS and EoVLP In this section three test systems with thechallenge on objective function including multi-fossil fuelsources and effects of valve loading process are consideredThe scale is up to 80 units 160 units and 320 units forSubcases 61 62 and 63 respectively
Comparison for Subcase 61 reported in Table 11 revealsthat the proposed method is the best method in terms of thelowest best cost the lowestmean cost and the lowest standarddeviation and the fastest execution time The processor fromthis proposedmethod is about four times stronger thanCGA-MU and IGA-MU but the speed is from ten times to 35 timesfaster than these methods Compared to CCSA and OSE-CSA the proposed method is about two times faster but theprocessor is slightly stronger
Comparison for Subcase 62 is reported in Table 12Clearly the proposed ICSA approach obtains better values ofthe best mean and worst costs than most methods exceptMSOS [34] where the best cost difference is about $ 024However the proposed method is the second fastest onewith 1119 seconds where the first fastest one CBA [44] has
Mathematical Problems in Engineering 17
Table 11 Comparisons of found results for Subcase 61
Approach Best cost Mean cost Worst cost Std dev CPU time Computer($h) ($h) ($h) ($h) (s) (Processor-Ram)
spent 571 seconds Clearly MSOS is better than the proposedICSA approach with respect to slightly less best cost butis worse than the proposed method in terms of executiontime namely 2572 seconds compared to 1119 seconds whileCBA [44] is faster than the proposed method but obtainssignificantly worse costs The analysis can conclude that theproposed ICSA approach is very powerful for the subcasewith 160 units
Table 13 presents the comparison of three other methodsincludingCCSA [54] SOS [34] andMSOS [34] accompaniedwith the proposed method for Subcase 63 The obtainedresult comparisons imply that ICSA approach can obtainbetter values of the best and standard deviation costs thanCCSA and SOS but obtains slightly higher cost than MSOSby approximately $ 022 Besides the execution time fromICSA approach is much shorter than others especially it ishigher than five times faster than MSOS The four methodshave been run on approximately strong computers Brieflythe proposed ICSA approach can find and converge to morefavorable solution than other methods with shorter CPUtime except the comparison with MSOS which had bettersolution but spent higher than five times execution times
Consequently the proposed method is a very promisingoptimization algorithm for Subcase 63 a system up to 320units and with multi-fossil fuel sources and effects of valveloading process
Optimal solution obtained by ICSA for Subcase 63 isshown in Table 26
57 The Improvement of ICSA Approach Performance
571 The Outstanding Improvement over CCSA In this sec-tion the performance improvement of ICSA over CCSAhas been investigated by analyzing obtained results and setcontrol parameters Table 14 has been formed by addingreduction cost improvement level of the best cost executiontime and control parameters consisting of 119873119901119904 and 119866119898119886119909Among the compared factors reduction cost is the deviationof the cost of CCSA and that of ICSA whereas the improve-ment level is the ratio of the reduction cost to the cost ofCCSA The reduction costs indicate the proposed methodcould find either equal quality of solutions or higher qualityof solutions than CCSA for all study casesThe reduction costis from $0 to $42839 corresponding to the improvement level
18 Mathematical Problems in Engineering
Table 14 Summary of results obtained by CCSA and ICSA for all study cases
Study case Reduction cost ($) Improvement level () Execution time (s) Nps Gmax
Figure 4 The best run obtained by CCSA and ICSA for Case 2
that is from 0 to 08968The saving cost is not too much forone hour but the operation in one day onemonth or one yearis very high However it should be noted that CCSA has beenrun by setting much higher population size and iterations formany cases excluding study cases implemented in the studysuch as Case 2 Subcase 31 Subcase 35 and Case 5 Forinstance ICSA has used 119873119901119904 = 10 and 119866119898119886119909 = 15 for Subcase13 and119873119901119904 = 10 and119866119898119886119909 = 25 for Subcase 14 whereas CCSAhas been run by setting119873119901119904 = 20 and 119866119898119886119909 = 5000 for the twosubcases Similarly CCSA has been run for Subcases 32 33and 34 with much higher number of iterations For the lastsubcase CCSA has been run by setting 119873119901119904 = 320 and 119866119898119886119909
= 1200 but those of ICSA have been 10 and 9000 Due tothe higher value of control parameters CCSA has tended tospend more time in finding such high quality solutions foralmost all study cases Execution time of ICSA is less than18 seconds while that of CCSA is up to higher 75 secondsIt is clear that the proposed ICSA could find better optimalsolutions thanCCSA for such considered systems For furtherinvestigation of performance comparison the best runs over50 runs and fuel cost values of 50 runs obtained by CCSA andthe proposed ICSA for Case 2 Subcase 31 Subcase 35 andCase 5 have been plotted in from Figures 4ndash11 The best runcurves show the faster search of the proposed ICSA method
Mathematical Problems in Engineering 19
0 5 10 15 20 25 30 35 40 45 50Run
19795
198
19805
1981
19815
1982
19825
1983
19835
Fuel
cost
($)
CCSAICSA
times105
Figure 5The best fuel cost of 50 runs obtained by CCSA and ICSA for Case 2
0 2 4 6 8 10 12 14 16 18 20Iteration
8234
8236
8238
8240
8242
8244
8246
8248
8250
8252
Fitn
ess F
unct
ion
($)
CCSAICSA
Figure 6 The best run obtained by CCSA and ICSA for Subcase 31
whereas 50 values of fuel cost indicate that the proposedICSA can find many solutions with better quality Clearlythe proposed ICSA is outstanding in terms of stabilizationof solution search and faster convergence As a result it canconclude that the proposed ICSA approach is more effectivethanCCSA in solvingOLDproblemwith considered systems
572The Improvement of Results over Other Methods In thisarticle we have tested ICSA approach on 6 cases with 16
systems with different fuel cost forms different constraintsand different scale systems from 3 units to 320 units We havecompared the yielded results from ICSA approach and otherexisting ones for evaluating the efficiency of ICSA approachIn subsections above we have shown yielded results fromICSA approach and compared these results to those of otheronesHowever the demonstrationhad not been very good forobserving and comparing to lead to a conclusionThus in thesubsectionwehave summarized the result comparisons of the
20 Mathematical Problems in Engineering
0 5 10 15 20 25 30 35 40 45 50Run
8234
82342
82344
82346
82348
8235
82352
82354
82356
82358
8236
Fuel
cost
($)
CCSAICSA
Figure 7 The best fuel cost of 50 runs obtained by CCSA and ICSA for Subcase 31
0 1000 2000 3000 4000 5000 6000Iteration
242
244
246
248
25
252
254
256
Fitn
ess F
unct
ion
($)
CCSAICSA
times105
Figure 8 The best run obtained by CCSA and ICSA for Subcase 35
proposed and other ones Table 15 has reported the reductioncost (in $) of ICSA approach compared to other ones Inaddition we have converted the reduction cost into improve-ment level (in ) for better comparison The improvementhas been shown from the lowest level to the highest levelin terms of reduction cost and improvement percentageIn addition we have also given the slowest and the fastestexecution time of other compared methods together withthat of the proposed method The table implies that ICSA
approach can find better optimal solutions with less fuel costup to $052 for Subcase 13 $75229 for Subcase 14 $1775149for Case 2 $1181256 for Subcase 31 $8438 for Subcase 32$9113 for Subcase 33 $121181 for Subcase 34 $3412 forSubcase 35 $04329 for Subcase 41 $30227 for Subcase42 $407002 for Case 5 $18183 for Subcase 61 $13965 forSubcase 62 and $11817 for Subcase 63 These reductioncosts are equivalent to improvement level (IL) of 0032908968 82282 14143 047 038 099 001405
Mathematical Problems in Engineering 21
0 5 10 15 20 25 30 35 40 45 50Run
2428
243
2432
2434
2436
2438
244
2442
Fuel
cost
($)
CCSAICSA
times105
Figure 9 The best fuel cost of 50 runs obtained by CCSA and ICSA for Subcase 35
0 50 100 150 200 250 300 350 400Iteration
327
328
329
33
331
332
333
334
335
336
337
Fitn
ess F
unct
ion
($)
CCSAICSA
times104
Figure 10 The best run obtained by CCSA and ICSA for Case 5
00003 0002 12291 004 138 and 059 Thesequantitative comparisons reveal that larger scale systems canlead to better reduction cost but the improvement level isnot high because total cost of compared methods tends tobe large for large scale systems Furthermore very large scalesystems with nondifferentiable objective have been normallysolved by strong methods In fact systems in Case 3 havethe same characteristic with single fuel and effects of valveloading process but Subcase 34 is a larger scale system with
40 units while Subcases 31 32 and 33 are constructed by 3units 13 units and 13 units So the improvement percentageof Subcase 34 can be up to 099 whilst that of Subcases32 and 33 is 047 and 038 respectively Subcase 35 iswith the largest system 80 units but the reduction cost is notmuch only $3412 because compared methods with ICSA areeither state-of-the-art ones or improved ones Also Subcase41 and Subcase 42 have considered single fuel and PPZ andspinning reserve constraints but Subcase 42 is larger scale
22 Mathematical Problems in Engineering
Table 15 Performance improvement summary of the proposed method
Study cases Reduction cost ($) Improvement level () Execution time (s)From To From To Slowest method Fastest method Proposed method
Figure 11 The best fuel cost of 50 runs obtained by CCSA and ICSA for Case 5
system with 90 units and Subcase 41 is only with 60 unitsThus the improvement of Subcase 41 is lower with 00003but that of Subcase 42 is 0002 Similarly systems in Cases 6have the same featurewithmulti-fossil fuel sources and effectsof valve loading process but they are respectively constructedby 80 160 and 320 units As a result the improvements ofSubcase 62 138 and Subcase 63 059 are much higherthan Subcase 61 004 However Subcase 62 with smallernumber of units but getting higher improvement is easilyunderstood because there were nine compared methods butonly three compared methods are considered for Subcase 63In general the improvement is not high it is about under onedollar several dollars tens of dollars and over one thousand
dollars per hour however the saving cost will be significantif the operation is considered to be onemonth with 720 hoursor one year with 8760 hours
Execution time comparison can be evaluated by observ-ing the fastest and the slowest compared methods in Table 15These execution times of the proposed method are approxi-mately equal to that of the fastest methods for Subcases 3233 and 34 and much shorter than other fastest comparedmethods for other cases especially for Subcases 61 and 63The fastest method for Subcase 62 is CBA [44] showing 57seconds while that of the proposed method is 1119 secondsHowever it cannot conclude that CBA is more effective thanthe proposed method because the proposed method could
find better optimal solution with less fuel cost by $2130As considering execution time of the slowest comparedmethods it can point out that ICSA is a very fast optimizationtool since the execution time of these methods is 29496 and3425 seconds for Subcases 32 and 33 116735 seconds forSubcase 34 56381 seconds for Subcase 41 94093 secondsfor Subcase 42 12797 seconds for Case 5 30941 seconds forSubcase 61 7542 seconds for Subcase 62 and 9641 secondsfor Subcase 63 while the execution time of the proposedmethod for these cases is respectively 095 146 091 1589246 84828 1119 and 171384 seconds It is clearly shown thatICSA is very fast as compared to these methods
In summary the proposed method has found approx-imately high quality solutions with several standard state-of-the-art meta-heuristic algorithms and improved versionsof them together with other old methods In addition theproposed method could improve result better than approx-imately all methods with faster execution time Comparedto other methods with the fastest convergence speed andhigh quality solutions the proposed method has been as
fast as for some cases and much faster for other cases Thecomparison with the slowest methods could show that theproposed method was extremely powerful since it was up tonearly one thousand times faster Consequently the proposedICSA approach can be one of the strongest optimization toolsfor OLD problem
6 Conclusions
This paper has proposed a good ICSA method for solvingOLD problem in which many test systems with differentobjective functions and complicated constraints from simpleto complex have been used as studied cases The proposedICSA method has been developed by performing severalmodifications on the second solution update of CCSA whichcontained several drawbacks to global convergence and fastmanner The OLD problem has covered from single fuelto multi-fossil fuels from quadratic objective function tononconvex objective function in addition to PPZ RPR andRRL constraints Many existing optimization algorithms have
been concerned in aim to compare the performance and givethe final conclusion on the proposed method There have
been six main cases with sixteen subcases The evaluationshave been made at the end of each study case Clearly theproposed ICSA approach has yielded more effective optimalsolutions with faster execution time than almost all methodsConsequently it can be concluded that the proposed methodis much more superior to CCSA and is a very promisingmethod for solving OLD problem
Appendix
See Tables 16ndash26
Nomenclature
120575119894 120582119894 120572119894 120573119894 120574119894 Fuel cost functioncoefficients of the 119894119905ℎthermal generationunit11986100 1198610119895 119861119895119894 Power loss matrixcoefficients119865119865119909 119865119865119887119890119904119905 The values of fitness ofsolution 119909 and theso-far most effectivesolution among thecurrent set ofsolutions119898119894 Number of fuels burntin the 119894119905ℎ thermalgeneration unit
119873 Number of allavailable thermalgeneration units119899119894 Number of prohibitedpower zones of the 119894119905ℎthermal generationunit119875119906119894119896 119875119897119894119896 Upper and lowerlimits of the 119894119905ℎthermal generationunit corresponding tothe kth PPZ119875119886 Probability ofreplacing controlvariables in each oldsolution119875119863 Real power demand ofall loads in system119875119894119898119886119909 119875119894119898119894119899 The highest andlowest real poweroutputs of the iththermal generationunit119875119894119895119898119886119909 119875119894119895119898119894119899 The highest andlowest real poweroutputs of the 119894119905ℎthermal generationunit corresponding tothe 119895119905ℎ fuel type119878119894119898119886119909 Maximum real powerreserve contributionof the thermalgeneration unit 1198941198781199001198971199031198861198991198891 1198781199001198971199031198861198991198892 1198781199001198971199031198861198991198893 1198781199001198971199031198861198991198894 Randomly mixedsolutions from the setof current solutions119878119900119897119909 119878119900119897119866119887119890119904119905 The old solution x andthe most effectivesolution119878119877 Real power reserverequirement of system
120575119894119895 120582119894119895 120572119894119895 120573119894119895 120574119894119895 Fuel cost functioncoefficients of the 119894119905ℎthermal generationunit corresponding tothe 119895119905ℎ fuel type1205761 1205762 1205763 1205764 1205765 1205766 Random numbersbetween 0 and 1Ω Number of generationunits considering PPZconstraint
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare no conflicts of interest
References
[1] N Sinha R Chakrabarti and P K Chattopadhyay ldquoEvolution-ary programming techniques for economic load dispatchrdquo IEEETransactions on Evolutionary Computation vol 7 no 1 pp 83ndash94 2003
[2] P Hansen ldquoA separable approximation dynamic programmingalgorithm for economic dispatch with transmission lossesrdquoYugoslav Journal of Operations Research vol 12 no 2 2002
[3] N T Thang ldquoSolving economic dispatch problem with piece-wise quadratic cost functions using lagrange multiplier theoryrdquoin Proceedings of the 3rd International Conference on ComputerTechnology and Development (ICCTD rsquo11) pp 359ndash364 ASMEPress 2011
[4] S K Mishra and S K Mishra ldquoA comparative study of solutionof economic load dispatch problem in power systems in theenvironmental perspectiverdquoProcedia Computer Science vol 48pp 96ndash100 2015
[5] A A Al-Subhi and H K Alfares ldquoEconomic load dispatchusing linear programming a comparative studyrdquo InternationalJournal of Applied Industrial Engineering vol 3 no 1 pp 16ndash362016
28 Mathematical Problems in Engineering
[6] C Zhou G Huang and J Chen ldquoPlanning of electric powersystems considering virtual power plants with dispatchableloads included an inexact two-stage stochastic linear program-ming modelrdquoMathematical Problems in Engineering vol 2018Article ID 7049329 12 pages 2018
[7] T T Nguyen N V Quynh and L Van Dai ldquoImproved fireflyalgorithm a novel method for optimal operation of thermalgenerating unitsrdquo Complexity vol 2018 Article ID 7267593 23pages 2018
[8] S S Haroon S Hassan S Amin et al ldquoMultiple fuel machinespower economic dispatch using stud differential evolutionrdquoEnergies vol 11 no 6 pp 1ndash20 2018
[9] C-T Su and C-L Chiang ldquoNonconvex power economic dis-patch by improved genetic algorithm with multiplier updatingmethodrdquo Electric Power Components and Systems vol 32 no 3pp 257ndash273 2004
[10] NNoman andH Iba ldquoDifferential evolution for economic loaddispatch problemsrdquo Electric Power Systems Research vol 78 no8 pp 1322ndash1331 2008
[11] P Somasundaram and K Kuppusamy ldquoApplication of evolu-tionary programming to security constrained economic dis-patchrdquo International Journal of Electrical Power amp EnergySystems vol 27 no 5-6 pp 343ndash351 2005
[12] T Jayabarathi K Jayaprakash andD Jeyakumar ldquoEvolutionaryprogramming technique for different kinds of economic dis-patch problemsrdquo Electric Power Systems Research vol 73 no 2pp 169ndash176 2005
[13] P Subbaraj R Rengaraj and S Salivahanan ldquoEnhancementof combined heat and power economic dispatch using selfadaptive real-coded genetic algorithmrdquo Applied Energy vol 86no 6 pp 915ndash921 2009
[14] C-L Chiang and C-T Su ldquoAdaptive-improved genetic algo-rithm for the economic dispatch of units with multiple fueloptionsrdquo Cybernetics and Systems vol 36 no 7 pp 687ndash7042005
[15] C Chiang ldquoImproved genetic algorithm for power economicdispatch of units with valve-point effects and multiple fuelsrdquoIEEE Transactions on Power Systems vol 20 no 4 pp 1690ndash1699 2005
[16] K Meng H G Wang Z Y Dong and K P Wong ldquoQuantum-inspired particle swarm optimization for valve-point economicload dispatchrdquo IEEE Transactions on Power Systems vol 25 no1 pp 215ndash222 2010
[17] P Erdogmus A Ozturk and S Duman ldquoEnvironmen-taleconomic dispatch using genetic algorithm and simulatedannealingrdquo in Proceedings of the International Conference onElectric Power and Energy Conversion Systems EPECS rsquo09 pp1ndash4 IEEE 2009
[18] L D S Coelho and V C Mariani ldquoImproved differentialevolution algorithms for handling economic dispatch opti-mization with generator constraintsrdquo Energy Conversion andManagement vol 48 no 5 pp 1631ndash1639 2007
[19] H Dakuo W Fuli and M Zhizhong ldquoA hybrid genetic algo-rithm approach based on differential evolution for economicdispatch with valve-point effectsrdquo Electrical Power and EnergySystems vol 30 pp 31ndash38 2008
[20] C-F Chang J-J Wong J-P Chiou and C-T Su ldquoRobustsearching hybrid differential evolution method for optimalreactive power planning in large-scale distribution systemsrdquoElectric Power Systems Research vol 77 no 5-6 pp 430ndash4372007
[21] M R Farooqi P Jain and K R Niazi ldquoUsing Hopfield neuralnetwork for economic dispatch of power systemsrdquo in Proceed-ings of the National Power Engineering Conference PECon rsquo03pp 5ndash10 Bangi Malaysia 2003
[22] T Yalcinoz H Altun and U Hasan ldquoConstrained economicdispatch with prohibited operating zones a Hopfield neuralnetwork approachrdquo in Proceedings of the 10th MediterraneanElectrotechnical Conference Information Technology and Elec-trotechnology for the Mediterranean Countries ProceedingsMeleCon rsquo00 (Cat No 00CH37099) pp 570ndash573 LemesosCyprus 2000
[23] D N Vo P Schegner and W Ongsakul ldquoCuckoo searchalgorithm for non-convex economic dispatchrdquo IET GenerationTransmission amp Distribution vol 7 no 6 pp 645ndash654 2013
[24] J-B Park K-S Lee J-R Shin and K Y Lee ldquoA particleswarm optimization for economic dispatch with non-smoothcost functionsrdquo IEEETransactions on Power Systems vol 20 no1 pp 34ndash42 2005
[25] T Niknam H D Mojarrad and M Nayeripour ldquoA newfuzzy adaptive particle swarm optimization for non-smootheconomic dispatchrdquo Energy vol 35 no 4 pp 1764ndash1778 2010
[26] L D S Coelho and V C Mariani ldquoAn improved harmonysearch algorithm for power economic load dispatchrdquo EnergyConversion and Management vol 50 no 10 pp 2522ndash25262009
[27] M Mahdavi M Fesanghary and E Damangir ldquoAn improvedharmony search algorithm for solving optimization problemsrdquoAppliedMathematics and Computation vol 188 no 2 pp 1567ndash1579 2007
[28] A Bhattacharya and P K Chattopadhyay ldquoSolving complexeconomic load dispatch problem using biogeography-basedoptimizationrdquo Expert Systems with Applications vol 37 no 5pp 3605ndash3615 2010
[29] D Simon ldquoBiogeography-based optimizationrdquo IEEE Transac-tions on Evolutionary Computation vol 12 no 6 pp 702ndash7132008
[30] J S Alsumait J K Sykulski and A K Al-Othman ldquoAhybrid GA-PS-SQP method to solve power system valve-pointeconomic dispatch problemsrdquo Applied Energy vol 87 no 5 pp1773ndash1781 2010
[31] M S Turgut andGKDemir ldquoQuadratic approximationndashbasedhybrid Artificial Cooperative Search algorithm for economicemission load dispatchproblemsrdquo International Transactions onElectrical Energy Systems vol 27 no 4 pp 1ndash14 2017
[32] T A A Victoire and A E Jeyakumar ldquoHybrid PSO-SQPfor economic dispatch with valve-point effectrdquo Electric PowerSystems Research vol 71 no 1 pp 51ndash59 2004
[33] D-KHe F-LWang andZ-ZMao ldquoHybrid genetic algorithmfor economic dispatch with valve-point effectrdquo Electric PowerSystems Research vol 78 no 4 pp 626ndash633 2008
[34] D C Secui ldquoA modified symbiotic organisms search algorithmfor large scale economic dispatch problem with valve-pointloading effectsrdquo Energy vol 113 pp 366ndash384 2016
[35] K BhattacharjeeA Bhattacharya and S H N Dey ldquoTeaching-learning-based optimization for different economic dispatchproblemsrdquo Scientia Iranica vol 21 no 3 pp 870ndash884 2014
[36] K Bhattacharjee A Bhattacharya and S H N Dey ldquoOppo-sitional Real Coded Chemical Reaction Optimization for dif-ferent economic dispatch problemsrdquo International Journal ofElectrical Power amp Energy Systems vol 55 pp 378ndash391 2014
Mathematical Problems in Engineering 29
[37] V Hosseinnezhad M Rafiee M Ahmadian and M T AmelildquoSpecies-basedQuantum Particle SwarmOptimization for eco-nomic load dispatchrdquo International Journal of Electrical Poweramp Energy Systems vol 63 pp 311ndash322 2014
[38] M S P Subathra S E Selvan T A A Victoire A HChristinal andU Amato ldquoA hybrid with cross-entropymethodand sequential quadratic programming to solve economic loaddispatch problemrdquo IEEE Systems Journal vol 9 no 3 pp 1031ndash1044 2015
[39] J Zhan Q H Wu C Guo and X Zhou ldquoEconomic dispatchwith non-smooth objectives part I local minimum analysisrdquoIEEE Transactions on Power Systems vol 30 no 2 pp 710ndash7212015
[40] A K Barisal andR C Prusty ldquoLarge scale economic dispatch ofpower systems using oppositional invasive weed optimizationrdquoApplied Soft Computing vol 29 pp 122ndash137 2015
[41] M Basu ldquoImproved differential evolution for economic dis-patchrdquo International Journal of Electrical Power amp EnergySystems vol 63 pp 855ndash861 2014
[42] V S Aragon S C Esquivel and C C Coello ldquoAn immunealgorithm with power redistribution for solving economicdispatch problemsrdquo Information Sciences vol 295 pp 609ndash6322015
[43] M Ghasemi M Taghizadeh S Ghavidel and A AbbasianldquoColonial competitive differential evolution an experimentalstudy for optimal economic load dispatchrdquo Applied Soft Com-puting vol 40 pp 342ndash363 2016
[44] B R Adarsh T Raghunathan T Jayabarathi and X-S YangldquoEconomic dispatch using chaotic bat algorithmrdquo Energy vol96 pp 666ndash675 2016
[45] N Ghorbani and E Babaei ldquoExchange market algorithm foreconomic load dispatchrdquo International Journal of ElectricalPower amp Energy Systems vol 75 pp 19ndash27 2016
[46] J X V Neto G Reynoso-Meza T H Ruppel V C Marianiand L D S Coelho ldquoSolving non-smooth economic dispatchby a new combination of continuous GRASP algorithm anddifferential evolutionrdquo International Journal of Electrical Poweramp Energy Systems vol 84 pp 13ndash24 2017
[47] A Kavousi-Fard and A Khosravi ldquoAn intelligent 120579-ModifiedBat Algorithm to solve the non-convex economic dispatchproblem considering practical constraintsrdquoElectrical Power andEnergy Systems vol 82 pp 189ndash196 2016
[48] M A Al-Betar M A Awadallah A T Khader and A L BolajildquoTournament-based harmony search algorithm for non-convexeconomic load dispatch problemrdquo Applied Soft Computing vol47 pp 449ndash459 2016
[49] M A Al-Betar M A Awadallah I A Doush E Alsukhni andH ALkhraisat ldquoA non-convex economic dispatchproblemwithvalve loading effect using a new modified 120573-hill climbing localsearch algorithmrdquo Arabian Journal for Science and Engineeringpp 1ndash8 2018
[50] M A Awadallah M A Al-Betar A L Bolaji E M Alsukhniand H Al-Zoubi ldquoNatural selection methods for artificial beecolony with new versions of onlooker beerdquo Soft Computing pp1ndash40 2018
[51] S H Kaboli and A K Alqallaf ldquoSolving non-convex economicload dispatch problem via artificial cooperative search algo-rithmrdquo Expert Systems with Applications vol 128 pp 14ndash272019
[52] D Singh and J Dhillon ldquoAmeliorated greywolf optimization foreconomic load dispatch problemrdquo Energy vol 169 pp 398ndash4192019
[53] X S Yang and S Deb ldquoCuckoo search via Levy flightsrdquo inProceedings of the World Congress on Nature and BiologicallyInspired Computing (NaBIC rsquo09) pp 210ndash214 India 2009
[54] S Sahoo K M Dash R C Prusty and A K Barisal ldquoCom-parative analysis of optimal load dispatch through evolutionaryalgorithmsrdquo Ain Shams Engineering Journal vol 6 pp 107ndash1202015
[55] M Basu and A Chowdhury ldquoCuckoo search algorithm foreconomic dispatchrdquo Energy vol 60 pp 99ndash108 2013
[56] A B Serapiao ldquoCuckoo search for solving economic dispatchload problemrdquo Intelligent Control and Automation vol 04 no04 pp 385ndash390 2013
[57] A H Bindu and M D Reddy ldquoEconomic load dispatch usingcuckoo search algorithmrdquo International Journal of EngineeringResearch and Applications vol 3 pp 498ndash502 2013
[58] E Afzalan and M Joorabian ldquoAn improved cuckoo searchalgorithm for power economic load dispatchrdquo InternationalTransactions on Electrical Energy Systems vol 25 no 6 pp 958ndash975 2015
[59] T T Nguyen and D N Vo ldquoThe application of one rank cuckoosearch algorithm for solving economic load dispatch problemsrdquoApplied Soft Computing vol 37 pp 763ndash773 2015
[60] S M Islam S Das S Ghosh S Roy and P N Suganthan ldquoAnadaptive differential evolution algorithm with novel mutationand crossover strategies for global numerical optimizationrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 42 no 2 pp 482ndash499 2012
[61] Z L Gaing ldquoParticle swarm optimization to solving theeconomic dispatch considering the generator constraintsrdquo IEEETransactions on Power Systems vol 18 no 3 pp 1187ndash1195 2003
[62] X-S Yang and S Deb ldquoEngineering optimisation by Cuckoosearchrdquo International Journal of Mathematical Modelling andNumerical Optimisation vol 330 pp 43ndash51 2010
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
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Mathematical Problems in Engineering 7
ΔSol2
Sol2
Sol3
Sol4
ΔSol1
Sol1
Solx
Solnew1
Solnew2
Figure 1 Simulation of solutions corresponding to the first itera-tions of the loop algorithm
For the CCSA case if two solutions 1198781199001198971199031198861198991198891 and 1198781199001198971199031198861198991198892are either slightly different or completely coincident suchnewly updated solution 1198781199001198971198991198901199081199091 does not have good chanceto leave the current zone and approach to more promisingzones In another word the new one is approximately coin-cident with the old one As the search task is taking place atsome last iterations this phenomenon becomes much worsebecause all current solutions are lumped in a small zone andthe capability of moving to other zones is impossible As aresult the CCSA approach will work ineffectively and searchstrategy is time consuming until other runs are started
Contrary to the two-point step size the new proposedformula may produce a large enough length to escape thelocal optimum zone and reach new favorable zones Itexplainswhy the four-point changing step has positive impacton the considered random walk rather than the two-pointchanging step
322 New Standard forChoosing theMostAppropriate Chang-ing Step In this section we extend our analysis to answer thequestionwhen to use the four-point step size FromEquations(18) and (19) two new solutions which are represented asthe results of the two-point-based factor and the four-pointstep size can be illustrated by using Figure 1 corresponding tothe search process at the first some iterations and Figure 2corresponding to the last some iterations For the sake ofsimplicity we rewrite the two equations as follows
Here we suppose that 1198781199001198971 and 1198781199001198972 are obtained byfour exact solutions 1198781199001198971 1198781199001198972 1198781199001198973 and 1198781199001198974 and calculatedas follows
ΔSol2
ΔSol1
Solx
Solnew1
Solnew2
Figure 2 Simulation of solutions corresponding to the last itera-tions of the loop algorithm
Δ1198781199001198971 = 1198781199001198971 minus 1198781199001198972 (22)
Δ1198781199001198972 = 1198781199001198973 minus 1198781199001198974 (23)
Asmentioned above the high changing step between newsolution and old solution can help to explore new favorablezones However in optimization algorithms searching stepscannot be arbitrarily large otherwise the algorithm maydiverge in particular for the cases that the consideredsolutions 119878119900119897119909 are not close together in solution search spaceFor example at the beginning of loop algorithm with thefirst iterations in Figure 1 1198781199001198971198991198901199081 is a better choice than1198781199001198971198991198901199082 because it is kept in a sufficient limit and does notlead to a risk of divergence In contrast as many of currentsolutions are in different positions but their distance is notvery short or approximately coincident such as at the lastiterations in Figure 2 1198781199001198971198991198901199081 and 119878119900119897119909 have a very shortdistance but 1198781199001198971198991198901199082 and 119878119900119897119909 have higher distance Accordingto the phenomenon in Figure 2 the proposed ICSA approachneeds to produce a high changing step to move solutions toother search zones without local optimum Hence 1198781199001198971198991198901199082would be preferred to 1198781199001198971198991198901199081
Based on the argument above the determination of thecondition for using either two-point changing step or four-point changing step is really crucial to the performance ofthe proposed ICSA approach in searching solutions of OLDproblem Here the ratio of 119865119865119877119909 which can be found byEquation (24) is suggested to be a suitable measurement forthe selection of two options
Δ119865119865119877119909 = 119865119865119909 minus 119865119865119887119890119904119905119865119865119887119890119904119905 (24)
For a particular set of the current solutions each individ-ual depending on its 119865119865119877119909 will create a corresponding newsolution by using either Equation (18) or (19) If the valueof one current solution is smaller than the predeterminedparameter 119879119900119897 Equation (19) is applied for updating suchconsidered solution 119909 Otherwise Equation (18) is a betteroption The steps of the modified algorithm are similar to the
8 Mathematical Problems in Engineering
If 1205765 lt 119875119886If FFRx lt Tolx119878119900119897119899119890119908119909 = 119878119900119897119909 + 1205764(1198781199001198971199031198861198991198891 minus 1198781199001198971199031198861198991198892 + 1198781199001198971199031198861198991198893 minus 1198781199001198971199031198861198991198894)else119878119900119897119899119890119908119909 = 119878119900119897119909 + 1205764(1198781199001198971199031198861198991198891 minus 1198781199001198971199031198861198991198892)end
Algorithm 1 New mutation technique applied in the proposed ICSA approach
conventional CSA except that an additional step should beadded at each iteration In this step the119865119865119889 of all individualsolutions should be calculated by utilizing Equation (24) andthen the result of comparing the ratio with 119879119900119897will be used todecidewhich updating formula should be selectedThewholedescription of the proposed standard and new mutationtechnique can be coded inMatlab program language by usingAlgorithm 1
323 Adjustment of Tolerance for Each Solution As pointedout above the proposed method needs assistances to deter-mine the most appropriate step size for finding out favorablesolution zones The given aim can be reached if the selectionof 119879119900119897119909 is reasonable however the range of this parameteris infinite and hard to select Thus the adaptation of tuningthe parameter is really necessary First of all the compari-son between 119879119900119897119909 and 119865119865119877119909 is carried out and then theadaptation will be determined based on the obtained resultfrom the comparison Results of comparison between the twoparameters can be either 119865119865119877119909 is less than 119879119900119897119909 or 119865119865119877119909is higher than 119879119900119897119909 The case that two parameters are equalhardly ever occurs
As the comer assumptionhappens (ie119865119865119877119909 is less than119879119900119897119909) at the considered time the four-point step size will beemployed for the 119909119905ℎ solution If 119879119900119897119909 remains unchanged atthe previous value the identification of improvement fromsuch four-point step size or two-point step size is vagueConsequently value of 119879119900119897119909 must be automatically reducedto a lower value in case that it has significant contribution tofound promising solution of previous iteration Clearly thedecrease of119879119900119897119909 can enable the proposedmethod to jump outlocal optimal zone and approachmore effective zones By trialand error method 119879119900119897119909 is selected to be a function of itselfthat is 09 of the previous value Finally the implementationof the proposed ICSA approach is presented in Algorithm 2
4 The Application of the ProposedICSA for OLD Problem
Thewhole computation steps of the proposed ICSA approachfor solving OLD problem are explained as follows
41 Handling Constraints and Randomly Producing InitialPopulation As shown in Section 2 the considered OLDproblem takes five following constraints into account
(i) Power balance constraint is shown in Equation (4)
(ii) Power output limitation constraint is shown in Equa-tion (6)
(iii) Prohibited power zone constraint is shown in Equa-tion (9)
(iv) Real power reserve constraint is shown in Equation(10)
(v) Ramp rate limit constraint is shown in Equation (13)
Among the five constraints ramp rate limit generationlimit and prohibited power zone seem to be more com-plicated than power balance and power reserve constraintsHowever the three constraints can be solved more easilybecause each unit is constrained independently in the threeconstraints whereas power balance constraint and powerreserve constraint consider all the thermal generating unitssimultaneously Power reserve constraint can be handledby penalizing the total generation of all units while powerbalance constraint can be solved by penalizing one violatedthermal generating unit The whole computation procedurefor solving all constraints and calculating fitness function ofsolutions is described in detail as follows
Step 1 Redefine maximum and minimum power output ofeach thermal generating unit as considering PPZ and RRLconstraints by using the following formulas
119875119894max = 119875119894max if 119875119894max le 119875i0 + 119868119878119878119894119875i0 + 119868119878119878119894 if 119875119894max gt 119875i0 + 119868119878119878119894
119894 = 1 119873(25)
119875119894min = 119875119894min if 119875119894min ge 119875i0 minus 119863119878119878119894119875i0 minus 119863119878119878119894 119890119897119904119890
119894 = 1 119873(26)
Mathematical Problems in Engineering 9
Produce initial population with119873119901119904 solutions (1198781199001198971 1198781199001198972 119878119900119897119909 119878119900119897119873119901119904)Calculate fitness function (1198651198651 1198651198652 119865119865119909 119865119865119873119901)Go to the loop algorithm by setting 119866 = 1
While (119866119898119886119909 gt 119866) (i) The first newly produced solutions119878119900119897119899119890119908119909 = 119878119900119897119909 + 120572(119878119900119897119909 minus 119878119900119897119866119887119890119904119905) oplus Levy(120573) (ii) Perform selection approach
119865119865119909 = 119865119865119909 if 119865119865119909 le 119865119865119899119890119908119909119865119865119899119890119908119909 119900119905ℎ119890119903119908119894119904119890 119909 = 1 119873119901119904
(v) Determine the most effective solution and its fitnessDetermine 119865119865119909 with the smallest value and assign 119878119900119897119909 to 119878119900119897119866119887119890119904119905If 119866119898119886119909 gt 119866 perform step (i) and increase 119866 to 119866 + 1 Otherwise stop the loop algorithm and report boththe smallest fitness together with 119878119900119897119866119887119890119904119905End while
Among the four Equations (25) and (26) are used firstin order to redefine upper bound and lower bound for allthermal generating units as considering RRL constraint Thethe redefined bounds continue to be redefined for the secondtime by using (27) and (28) as considering PPZ constraints
Step 2 (randomly produce initial population) For dealingwith the power balance constraint all available units areseparated into two groups in which the first group withdecision variables consists of the power output from thesecond unit to the last unit (P2 P3 PN) meanwhile onlythe power output of the first unit (1198751) belongs to the secondgroup with dependent variable So upper bound solution119878119900119897119898119886119909 and lower bound solution 119878119900119897119898119894119899 must be defined asfollows
Step 3 Handle prohibited power zone constraint for decisionvariables P2 P3 PN
After being randomly produced there is a high possi-bility that decision variables fall into PPZ and they violatePPZ constraint So the verification of falling into PPZ andcorrection of the violation should be accomplished by usingthe following formula
119875119894 =
119875119897119894119896 if 119875119897119894119896 lt 119875119894 le 119875119897119894119896 + 1198751198961198941198962119875119906119894119896 if (119875119894 gt 119875119897119894119896 + 1198751198961198941198962 ) amp (119875119894 lt 119875119906119894119896)119875119894 119890119897119904119890
119894 = 2 119873 amp 119896 = 1 119899119894
(31)
Step 4 Handle RPB constraint by calculating 1198751 and penaliz-ing 1198751 if it violates constraints
In this step power balance constraint is exactly handledby calculating and penalizing dependent variable (1198751) 1198751 isobtained by using formulas (4) and (5) as follows
1198751 = minus (11986101 minus 1 + 2sum119873119894=2 1198611119894119875119894) plusmn radicΔ211986111 (32)
where
Δ = (11986101 minus 1 + 2 119873sum119894=2
1198611119894119875119899)2 minus 411986111(119875119863 minus 119873sum
119894=2
119875119894+ 11986100 + 119873sum
119894=2
1198610119894119875119894 + 119873sum119894=2
119873sum119895=2
119875119894119861119894119895119875119895) amp Δ ge 0(33)
In Equation (32) 1198751 has been determined for the purposeof dealing with real power balance constraint However it isnot sure that 1198751 can satisfy upper bound and lower boundconstraints and prohibited power zone constraints So 1198751must be checked and penalized
Firstly 1198751 is checked and penalized for upper and lowerbound constraints by the following model
Δ1198751x =
0 if 1198751min le 1198751x le 1198751max
1198751min minus 1198751x if 1198751min gt 1198751x1198751x minus 1198751max if 1198751max lt 1198751x
(34)
In Equation (34) if the second case or the third caseoccurs it means P1 has violated either lower bound or upperbound and it would be penalized by using either (P1x= P1min-P1x) or (P1x =P1x -P1max) Otherwise ifP1 has not violatedthe bound constraints (ie the first case in (34) happened)
P1 would continue to be checked for PPZ constraint by thefollowing model
Δ1198751x
=
1198751 minus 1198751198971119896 if 1198751198971119896 lt 1198751 le 1198751198971119896 + 119875119896111989621198751199061119896 minus 1198751 if (1198751 gt 1198751198971119896 + 11987511989611198962 ) amp (1198751 lt 1198751199061119896)0 119890119897119904119890
(35)
Step 5 Handle real power reserve constraint (10)First of all 119878119894 is determined by using (11) and (12) and
then the 119909119905ℎ solution will be checked and penalized if poweroutput of all thermal generating units cannot satisfy RPRconstraint The penalty for violation of the constraint can becalculated by using equation (36)
Δ119878119877119909 =
0 if119873sum119894=1
119878119894119909 ge 119878119877119878119877 minus 119873sum119894=1
119878119894119909 119890119897119904119890 (36)
As a result real power reserve constraint can be solved byusing the penalty method
42 Calculate Fitness Function for Solutions Fitness functionof each solution is used to evaluate quality of solutionNormally the function is the sum of objective function andpenalty of violating constraints and is obtained by
43 The First Newly Updated Solutions by Levy Flights Tech-nique In this section the first newly updated solutionsare performed by employing Levy flights technique usingEquation (14) However each new solution can be out oftheir feasible operating zone such as PPZ and upper andlower limitations When the power output violates its PPZconstraints Equation (31) will be applied to tackle theconstraint Besides the following equation will be employedwhen power output is higher or lower than their limitations
119878119900119897119909 =
119878119900119897max if 119878119900119897max lt 119878119900119897119909119878119900119897min if 119878119900119897min gt 119878119900119897119909119878119900119897119909 Otherwise
119909 = 1 119873119901 (38)
After that Equations (32)-(37) are performed for deter-mining all variables and penalty terms Finally Equation (38)is employed to calculate fitness function
44 The Second Newly Updated Solutions by Using Muta-tion Technique The second newly updated solutions areaccomplished as presented in Section 3 above Similar to
Mathematical Problems in Engineering 11
the task after doing the first update each solution in thenew population must satisfy PPZ constraint and upper andlower boundaries by considering Equations (31) and (38)Then Equations (32)-(37) are performed for determiningall variables and penalty terms Finally Equation (38) isemployed to calculate fitness function and the solution withthe best value is assigned to the best one 11987811990011989711986611988711989011990411990545 Criterion of Stopping the Loop Algorithm In the loopalgorithm of using the proposed ICSA approach the solutionsearch work is stopped in case that the predeterminedmaximum iterations 119866119898119886119909 is reached For each search ter-mination the most effective solution is stored and anotherrun continues to be accomplished until the predeterminednumber of runs is reached After finishing the runs thebest one is found and reported In addition other valuessuch as the fitness of the worst solution and average fitnessof all solutions are also reported for comparing with othermethods
46 The Whole Iterative Process The whole iterative algo-rithm for implementing the proposed ICSA approach forcoping with OLD problem is described in detail in Figure 3
5 Results and Discussions
The proposed ICSA approach performance has been investi-gated on six cases with different fuel options different fuelcharacteristics and complicated constraints The details ofthe studied cases are presented as follows
Case 1 Four systems with single fuel source (SFS) and powerloss (PL) constraint
Subcase 11 A 3-unit system [57]Subcase 12 A 6-unit system [57]Subcase 13 A 3-unit system [56]Subcase 14 A 6-unit system [56]
Case 2 A 110-unit system with SFS [57]
Case 3 Four systems with SFS and the effects of valve loadingprocess (EoVLP)
Subcase 31 A 3-unit system supplying to a load of850MW [58]Subcase 32 A 13-unit system supplying to a load of1800MW [1]Subcase 33 A 13-unit system supplying to a load of2520MW [1]Subcase 34 A 40-unit system supplying to a load of2500MW [1]Subcase 35 An 80-unit system supplying to a load of4100MW [49]
Case 4 Two systems with SFS and PPZ and RPR constraints
Subcase 41 A 60-unit system supplying to a10600MW load [9]
Subcase 42 A 90-unit system supplying to a15900MW load [9]
Case 5 A 15-unit system with SFS and RRL PPZ and PLconstraints [61]
Case 6 Three systems with multiple fuel sources (MFS) andEoVLP
Subcase 61 An 80-unit system supplying to a21600MW load [15]Subcase 62 A 160-unit system supplying to a43200MW load [15]Subcase 63 A 320-unit system supplying to an86400MW load [54]
For each considered case with each load case the pro-posed ICSA approach is run 50 times on the programlanguage of Matlab and a PC with 4 GB of RAM and 24GHzprocessor The selection of adjustment parameters including119875119886 and 119879119900119897119909 is carefully considered to obtain the best optimalsolutions meanwhile two others such as 119873119901119904 and 119866119898119886119909 arechosen corresponding to the scale of particular test system9 values with the change of 01 in the range [01 09] are inturn selected for 119875119886 while 119879119900119897119909 is 001 at the beginning Theinformation including load demand119873119901119904 119866119898119886119909 and the best119875119886 is reported in Table 1
51 Obtained Results on Case 1 considering Four Systems withSFS and PL Constraint In this section we have implementedthe proposed ICSA approach for solving four systems dividedinto four subcases Tables 2 and 3 show the comparisons ofobtained results from Subcases 11 and 12 and Subcases 13and 14 respectively As listed in Table 2 the proposed ICSAmethod and CCSA can find equal fuel cost for Subcases 11whereas the reduction of fuel cost from the proposed ICSAmethod as compared to CCSA is clearer for Subcase 12 Asshown in Table 3 for comparing the proposed ICSA and threemethods consisting of CCSA ABC and FA the minimumfuel cost of the proposed ICSA is approximately equal to thatof these methods for Subcases 13 but much less than that ofthese methods for Subcase 14 Furthermore the proposedICSA has been run by setting 119873119901119904 and 119866119898119886119909 to 5 and 20but these values were much higher for CCSA ABC and FAThey are 20 and 5000 for CCSA 40 and 100 for ABC and 20and 5000 for FA Consequently the proposed method is veryefficient for Case 1 with four subcases
Optimal solutions obtained by ICSA for Case 1 are shownin Tables 16ndash18
52 Obtained Results on Case 2 considering 110-Unit Systemwith SFS In this section we have employed a very largescale system with 110 units but there were not challenges forobjective function and complex constraints since EoVLP andconstraints were not taken into account Both CCSA andthe proposed ICSA methods have been run for comparingwith BBO hybrid BBO and DE (DEBBO) and Opposi-tional real coded chemical reaction optimization algorithm(ORCCROA) in [36] IWA in [40] and AGWO in [52] As
12 Mathematical Problems in Engineering
Select parameters
- Correct solutions if violating PPZ constraint using Eq (31)- Determine using Eq (32)- Penalize for violating upper and lower bounds constraint using Eq (34)- Penalize for violating PPZ constraint using Eq (35)- Penalize the xth solution for violating RPR constraint using Eq (36)
- Determine quality of solutions employing Eq (37) - Select the most effective solution - Start the loop algorithm by selecting
- Perform the first solution update using Section 32 - Correct boundaries of solutions by using Eq (38)
- Correct solutions if violating PPZ constraint using Eq (31)- Determine using Eq (32)- Penalize for violating upper and lower bounds constraint using Eq (34)- Penalize for violating PPZ constraint using Eq (35)- Penalize the xth solution for violating RPR constraint using Eq(36)
- Determine fitness function using Eq (37)- Compare old solutions and new solutions to keep better ones
- Perform the second solution update using Algorithm 1 - Correct boundaries of solutions by using (38)
- Correct solutions if violating PPZ constraint using Eq (31)- Determine using Eq (32)- Penalize for violating upper and lower bounds constraint using Eq (34)- Penalize for violating PPZ constraint using Eq (35)- Penalize the xth solution for violating RPR constraint using Eq (36)
- Determine fitness function using Eq (37)- Compare old solutions and new solutions to keep better ones- Select the most effective solution
Stop
Start
- Redefine upper and lower bounds using Eqs (25)-(28)- Randomly generate initial population using Eq (30)
Nps Pa Gmax H> Tolx
P1xP1x
P1x
P1xP1x
P1x
P1xP1x
P1x
Sol<MN
Sol<MN
G = 1
G = Gmax G = G + 1
Figure 3 All computation steps for solving OLD problem by employing the proposed ICSA approach
shown in Table 4 AGWO [52] has reached less fuel cost thanICSA however the exact fuel cost which was recalculatedby using reported solution pointed out that the method hasreached a very high fuel cost of $2157404250 For comparisonwith other methods ICSA has found less fuel cost thanall these methods Particularly the reduction of generationfuel cost is significant as compared to BBO DEBBO andCCSA Execution time comparisons are also useful evidence
for indicating the high performance of ICSA Thus it canconclude that ICSA is a strong method for Case 2
Optimal solution obtained by ICSA for the case is shownin Table 19
53 Obtained Results on Case 3 considering Four Systems withSFS and EoVLP In this section the real performance of theproposed ICSA approach has been investigated based on five
Mathematical Problems in Engineering 13
Table 1 Information of considered cases and adjustment parameters
Case Fuel cost function Constraint Subcase No of units 119875119863 (MW) Nps Gmax Best 1198751198861 SFS PL
subcases with the gradual increase of number of units Thesmallest scale system considers 3 units but the largest scalesystem takes 80 units In addition to the implementation ofthe proposed ICSA we have also implemented CCSA forSubcase 31 and Subcase 35 for further comparison becauseCCSA has not been run for the two subcases so far
Comparison of obtained results from Subcase 31 shownin Table 5 indicates that the proposed ICSA is superior toCCSAwith lower fuel cost but it seems to be less effective thanMCSA [58] The minimum cost of MCSA reported in [58] isthe smallest fuel cost but the recalculated cost is much higherthan that of the proposed method Furthermore MCSA hasbeen implemented by setting very high values to 119873119901119904 and119866119898119886119909
Reports for Subcases 32 and 33 shown in Table 6are the comparisons of the proposed ICSA approach andother methods such as conventional Evolution programming(CEP) [1] Fast EP (FEP) [1] improved FEP (IFEP) [1] DE[12] multiplier Lagrange-based genetic algorithm with (GA-MU) [15] QPSO [16] GA-PS-SQP [30] PSO-SQP [32] M120573-HCLSA [49] IABCA [50] CCSA [59] OSE-CSA [59] SOS[34] MSOS [34] CEA-SQT [38] TSBO [39] IWA [40] andCBA [44] As observed from the table ICSAapproach obtainsbetter solutions than mostmethods excluding DE [10] CCSA
[59] OSE-CSA [59] SOS [34] MSOS [34] CEA-SQT [38]TSBO [39] IWA [40] and CBA [44] especially M120573-HCLSA[49] with lower cost $1796097 However recalculated costfrom reported solution of M120573-HCLSA is $179691 BesidesICSA is very fast as compared to most methods where twoother versions of Cuckoo search algorithm CCSA and OSE-CSA are also included except two methods in [34] Theprocessor of computer that all the methods run on is alsoreported in the final column Clearly ICSA approach is veryefficient for the case with the 13-unit system where effects ofvalve loading process are considered
In Subcase 34 the number of units is much larger thanthat of three subcases above up to 40 units [1] The obtainedresult comparisons with others are indicated in Table 7Clearly the minimum cost comparisons reveal that the pro-posedmethod is one of the leading methods due to the lowestcost except the comparison with CCSA [23] OSE-CSA [59]SOS [34] MSOS [34] EMA [45] 120579-MBA [47] and AGWOA[52] It is noted that AGWOA [52] has reported the bestminimum cost with $12140430 but recalculated minimumcost which was obtained by substituting reported optimalgeneration of all thermal generating units is $12141331 Theaverage and the maximum costs from the proposed methoddo not belong to the leading method group however the
Mathematical Problems in Engineering 15
Table 7 Result comparisons for Subcase 34
Approach Best cost Average cost Worst cost CPU time Computer($h) ($h) ($h) (s) (Processor-Ram)
execution time that ICSA approach takes is smaller thanmostones except IAPR [42] and 120579-MBA [47] which have been runon stronger computers Note that MSOS [34] has been fasterthan the proposed method for Subcases 31 and 32 abovebut it is too slower than the proposed method for the casenamely 1813 seconds compared to 146 seconds Comparedto two other versions of Cuckoo search CCSA [23] and OSE-CSA [59] the proposed method is also faster about threetimes although the processors are slightly different For thiscase 120579-MBA [47] shows a very good performance howeverthe method has not been tested onmore complicated systemsand larger scale and therefore more comparisons with themethod must end Clearly the proposed method is stillefficient for the case where large scale and effects of valveloading process are included
Subcase 35 is the largest scale system with 80 units Inaddition to the implementation of ICSA CCSA has beenalso run for the Subcase 35 for further investigation ofefficiency improvement of the proposed ICSA approach The
comparisons of minimum cost in Table 8 show that theproposed ICSA can find more optimal solution than M120573-HCLSA [49] AGWOA [52] and CCSA The proposed ICSAis also superior to CCSA in terms ofmore stable search abilityand lower fluctuation since its average cost and maximumcost are less than those of CCSA The outstanding figurecannot be reached as compared to AGWOA [52] howeverit is hard to conclude AGWOA [52] is superior to theproposed ICSA approach about more stable search abilityand lower fluctuation Actually comparison of the values ofpopulation and iterations as well as execution time cannot beaccomplished because the information was not reported in[52]Thus it can conclude that the proposed ICSA is effectivefor the subcase
Optimal solutions obtained by ICSA for the case areshown in Tables 20ndash23
54 Obtained Results on Case 4 with Two Systems consideringSFS and PPZ and RPR Constraints In this section two
16 Mathematical Problems in Engineering
Table 9 Result comparisons for Subcases 41 and 42
Subcase Approach Best cost ($h) Mean cost ($h) Worst cost ($h) Std dev ($h) CPU time (s) Computer (Processor-Ram)
test systems with SFS and PPZ and RPR constraints areconsidered The test system size is up to 60 and 90 units forSubcases 41 and 42 respectively Comparison for the casesis only performed with two Genetic algorithms consisting ofGA and IGA-MU in [9] and two other versions of Cuckoosearch algorithms including CCSA [23] and OSE-CSA [59]and presented in Table 9 Other studies have tended toignore such complicated constraints of PPZ and RPR Thecosts reported in Table 9 indicate that ICSA approach canobtain more effective solution than CCSA and OSE-CSAbecause it has reached lower minimum cost than the twoones Furthermore the proposed method also takes shortercomputation time for the two cases from about two timesto about three times although the processor of the proposedmethod is slightly strongerThemean costs of ICSA approachare much less than those from IGA-MU and GA and slightlyhigher than those from OSE-CSA but there is a trade-off between the proposed method and CCSA for the twosubcases In fact the proposed method obtains higher meancost for Subcase 41 but lower cost for Subcase 42 AlthoughGAmethods have been runon aweak computerwith 07 GHzof the processor compared to that with 24GHz in the studytheir execution times are significantly higher namely 56381seconds (GA) and 16258 seconds (IGA-MU) compared to09153 seconds of ICSA approach for Subcase 41 and 94093seconds (GA) and 25545 seconds (IGA-MU) compared to15892 seconds (the proposedmethod)The analysis can pointout that ICSA approach ismore efficient than these comparedmethods in terms of optimal solutions and execution time
Optimal solutions obtained by ICSA for Subcase 42 areshown in Table 24
55 Obtained Results on Case 5 with a 15-Unit System consid-ering SFS and RRL PPZ and PL Constraints In this section
a 15-unit system considering RRL PPZ and PL constraintsis considered to be solved for finding optimal solution Forefficiency investigation of the proposed ICSA we have alsoimplemented CCSA for comparison As listed in Table 10the proposed ICSA is the most effective method with thesmallest fuel cost The comparisons of control parameters aswell asCPU time are also good evidence to confirm the strongsearch of the proposed ICSA approach since it has been runby smaller values of control parameter and faster executiontime as compared to all methods excluding CCSA
Optimal solution obtained by ICSA for the case is shownin Table 25
56 Obtained Results on Case 6 withThree Systems consideringMFS and EoVLP In this section three test systems with thechallenge on objective function including multi-fossil fuelsources and effects of valve loading process are consideredThe scale is up to 80 units 160 units and 320 units forSubcases 61 62 and 63 respectively
Comparison for Subcase 61 reported in Table 11 revealsthat the proposed method is the best method in terms of thelowest best cost the lowestmean cost and the lowest standarddeviation and the fastest execution time The processor fromthis proposedmethod is about four times stronger thanCGA-MU and IGA-MU but the speed is from ten times to 35 timesfaster than these methods Compared to CCSA and OSE-CSA the proposed method is about two times faster but theprocessor is slightly stronger
Comparison for Subcase 62 is reported in Table 12Clearly the proposed ICSA approach obtains better values ofthe best mean and worst costs than most methods exceptMSOS [34] where the best cost difference is about $ 024However the proposed method is the second fastest onewith 1119 seconds where the first fastest one CBA [44] has
Mathematical Problems in Engineering 17
Table 11 Comparisons of found results for Subcase 61
Approach Best cost Mean cost Worst cost Std dev CPU time Computer($h) ($h) ($h) ($h) (s) (Processor-Ram)
spent 571 seconds Clearly MSOS is better than the proposedICSA approach with respect to slightly less best cost butis worse than the proposed method in terms of executiontime namely 2572 seconds compared to 1119 seconds whileCBA [44] is faster than the proposed method but obtainssignificantly worse costs The analysis can conclude that theproposed ICSA approach is very powerful for the subcasewith 160 units
Table 13 presents the comparison of three other methodsincludingCCSA [54] SOS [34] andMSOS [34] accompaniedwith the proposed method for Subcase 63 The obtainedresult comparisons imply that ICSA approach can obtainbetter values of the best and standard deviation costs thanCCSA and SOS but obtains slightly higher cost than MSOSby approximately $ 022 Besides the execution time fromICSA approach is much shorter than others especially it ishigher than five times faster than MSOS The four methodshave been run on approximately strong computers Brieflythe proposed ICSA approach can find and converge to morefavorable solution than other methods with shorter CPUtime except the comparison with MSOS which had bettersolution but spent higher than five times execution times
Consequently the proposed method is a very promisingoptimization algorithm for Subcase 63 a system up to 320units and with multi-fossil fuel sources and effects of valveloading process
Optimal solution obtained by ICSA for Subcase 63 isshown in Table 26
57 The Improvement of ICSA Approach Performance
571 The Outstanding Improvement over CCSA In this sec-tion the performance improvement of ICSA over CCSAhas been investigated by analyzing obtained results and setcontrol parameters Table 14 has been formed by addingreduction cost improvement level of the best cost executiontime and control parameters consisting of 119873119901119904 and 119866119898119886119909Among the compared factors reduction cost is the deviationof the cost of CCSA and that of ICSA whereas the improve-ment level is the ratio of the reduction cost to the cost ofCCSA The reduction costs indicate the proposed methodcould find either equal quality of solutions or higher qualityof solutions than CCSA for all study casesThe reduction costis from $0 to $42839 corresponding to the improvement level
18 Mathematical Problems in Engineering
Table 14 Summary of results obtained by CCSA and ICSA for all study cases
Study case Reduction cost ($) Improvement level () Execution time (s) Nps Gmax
Figure 4 The best run obtained by CCSA and ICSA for Case 2
that is from 0 to 08968The saving cost is not too much forone hour but the operation in one day onemonth or one yearis very high However it should be noted that CCSA has beenrun by setting much higher population size and iterations formany cases excluding study cases implemented in the studysuch as Case 2 Subcase 31 Subcase 35 and Case 5 Forinstance ICSA has used 119873119901119904 = 10 and 119866119898119886119909 = 15 for Subcase13 and119873119901119904 = 10 and119866119898119886119909 = 25 for Subcase 14 whereas CCSAhas been run by setting119873119901119904 = 20 and 119866119898119886119909 = 5000 for the twosubcases Similarly CCSA has been run for Subcases 32 33and 34 with much higher number of iterations For the lastsubcase CCSA has been run by setting 119873119901119904 = 320 and 119866119898119886119909
= 1200 but those of ICSA have been 10 and 9000 Due tothe higher value of control parameters CCSA has tended tospend more time in finding such high quality solutions foralmost all study cases Execution time of ICSA is less than18 seconds while that of CCSA is up to higher 75 secondsIt is clear that the proposed ICSA could find better optimalsolutions thanCCSA for such considered systems For furtherinvestigation of performance comparison the best runs over50 runs and fuel cost values of 50 runs obtained by CCSA andthe proposed ICSA for Case 2 Subcase 31 Subcase 35 andCase 5 have been plotted in from Figures 4ndash11 The best runcurves show the faster search of the proposed ICSA method
Mathematical Problems in Engineering 19
0 5 10 15 20 25 30 35 40 45 50Run
19795
198
19805
1981
19815
1982
19825
1983
19835
Fuel
cost
($)
CCSAICSA
times105
Figure 5The best fuel cost of 50 runs obtained by CCSA and ICSA for Case 2
0 2 4 6 8 10 12 14 16 18 20Iteration
8234
8236
8238
8240
8242
8244
8246
8248
8250
8252
Fitn
ess F
unct
ion
($)
CCSAICSA
Figure 6 The best run obtained by CCSA and ICSA for Subcase 31
whereas 50 values of fuel cost indicate that the proposedICSA can find many solutions with better quality Clearlythe proposed ICSA is outstanding in terms of stabilizationof solution search and faster convergence As a result it canconclude that the proposed ICSA approach is more effectivethanCCSA in solvingOLDproblemwith considered systems
572The Improvement of Results over Other Methods In thisarticle we have tested ICSA approach on 6 cases with 16
systems with different fuel cost forms different constraintsand different scale systems from 3 units to 320 units We havecompared the yielded results from ICSA approach and otherexisting ones for evaluating the efficiency of ICSA approachIn subsections above we have shown yielded results fromICSA approach and compared these results to those of otheronesHowever the demonstrationhad not been very good forobserving and comparing to lead to a conclusionThus in thesubsectionwehave summarized the result comparisons of the
20 Mathematical Problems in Engineering
0 5 10 15 20 25 30 35 40 45 50Run
8234
82342
82344
82346
82348
8235
82352
82354
82356
82358
8236
Fuel
cost
($)
CCSAICSA
Figure 7 The best fuel cost of 50 runs obtained by CCSA and ICSA for Subcase 31
0 1000 2000 3000 4000 5000 6000Iteration
242
244
246
248
25
252
254
256
Fitn
ess F
unct
ion
($)
CCSAICSA
times105
Figure 8 The best run obtained by CCSA and ICSA for Subcase 35
proposed and other ones Table 15 has reported the reductioncost (in $) of ICSA approach compared to other ones Inaddition we have converted the reduction cost into improve-ment level (in ) for better comparison The improvementhas been shown from the lowest level to the highest levelin terms of reduction cost and improvement percentageIn addition we have also given the slowest and the fastestexecution time of other compared methods together withthat of the proposed method The table implies that ICSA
approach can find better optimal solutions with less fuel costup to $052 for Subcase 13 $75229 for Subcase 14 $1775149for Case 2 $1181256 for Subcase 31 $8438 for Subcase 32$9113 for Subcase 33 $121181 for Subcase 34 $3412 forSubcase 35 $04329 for Subcase 41 $30227 for Subcase42 $407002 for Case 5 $18183 for Subcase 61 $13965 forSubcase 62 and $11817 for Subcase 63 These reductioncosts are equivalent to improvement level (IL) of 0032908968 82282 14143 047 038 099 001405
Mathematical Problems in Engineering 21
0 5 10 15 20 25 30 35 40 45 50Run
2428
243
2432
2434
2436
2438
244
2442
Fuel
cost
($)
CCSAICSA
times105
Figure 9 The best fuel cost of 50 runs obtained by CCSA and ICSA for Subcase 35
0 50 100 150 200 250 300 350 400Iteration
327
328
329
33
331
332
333
334
335
336
337
Fitn
ess F
unct
ion
($)
CCSAICSA
times104
Figure 10 The best run obtained by CCSA and ICSA for Case 5
00003 0002 12291 004 138 and 059 Thesequantitative comparisons reveal that larger scale systems canlead to better reduction cost but the improvement level isnot high because total cost of compared methods tends tobe large for large scale systems Furthermore very large scalesystems with nondifferentiable objective have been normallysolved by strong methods In fact systems in Case 3 havethe same characteristic with single fuel and effects of valveloading process but Subcase 34 is a larger scale system with
40 units while Subcases 31 32 and 33 are constructed by 3units 13 units and 13 units So the improvement percentageof Subcase 34 can be up to 099 whilst that of Subcases32 and 33 is 047 and 038 respectively Subcase 35 iswith the largest system 80 units but the reduction cost is notmuch only $3412 because compared methods with ICSA areeither state-of-the-art ones or improved ones Also Subcase41 and Subcase 42 have considered single fuel and PPZ andspinning reserve constraints but Subcase 42 is larger scale
22 Mathematical Problems in Engineering
Table 15 Performance improvement summary of the proposed method
Study cases Reduction cost ($) Improvement level () Execution time (s)From To From To Slowest method Fastest method Proposed method
Figure 11 The best fuel cost of 50 runs obtained by CCSA and ICSA for Case 5
system with 90 units and Subcase 41 is only with 60 unitsThus the improvement of Subcase 41 is lower with 00003but that of Subcase 42 is 0002 Similarly systems in Cases 6have the same featurewithmulti-fossil fuel sources and effectsof valve loading process but they are respectively constructedby 80 160 and 320 units As a result the improvements ofSubcase 62 138 and Subcase 63 059 are much higherthan Subcase 61 004 However Subcase 62 with smallernumber of units but getting higher improvement is easilyunderstood because there were nine compared methods butonly three compared methods are considered for Subcase 63In general the improvement is not high it is about under onedollar several dollars tens of dollars and over one thousand
dollars per hour however the saving cost will be significantif the operation is considered to be onemonth with 720 hoursor one year with 8760 hours
Execution time comparison can be evaluated by observ-ing the fastest and the slowest compared methods in Table 15These execution times of the proposed method are approxi-mately equal to that of the fastest methods for Subcases 3233 and 34 and much shorter than other fastest comparedmethods for other cases especially for Subcases 61 and 63The fastest method for Subcase 62 is CBA [44] showing 57seconds while that of the proposed method is 1119 secondsHowever it cannot conclude that CBA is more effective thanthe proposed method because the proposed method could
find better optimal solution with less fuel cost by $2130As considering execution time of the slowest comparedmethods it can point out that ICSA is a very fast optimizationtool since the execution time of these methods is 29496 and3425 seconds for Subcases 32 and 33 116735 seconds forSubcase 34 56381 seconds for Subcase 41 94093 secondsfor Subcase 42 12797 seconds for Case 5 30941 seconds forSubcase 61 7542 seconds for Subcase 62 and 9641 secondsfor Subcase 63 while the execution time of the proposedmethod for these cases is respectively 095 146 091 1589246 84828 1119 and 171384 seconds It is clearly shown thatICSA is very fast as compared to these methods
In summary the proposed method has found approx-imately high quality solutions with several standard state-of-the-art meta-heuristic algorithms and improved versionsof them together with other old methods In addition theproposed method could improve result better than approx-imately all methods with faster execution time Comparedto other methods with the fastest convergence speed andhigh quality solutions the proposed method has been as
fast as for some cases and much faster for other cases Thecomparison with the slowest methods could show that theproposed method was extremely powerful since it was up tonearly one thousand times faster Consequently the proposedICSA approach can be one of the strongest optimization toolsfor OLD problem
6 Conclusions
This paper has proposed a good ICSA method for solvingOLD problem in which many test systems with differentobjective functions and complicated constraints from simpleto complex have been used as studied cases The proposedICSA method has been developed by performing severalmodifications on the second solution update of CCSA whichcontained several drawbacks to global convergence and fastmanner The OLD problem has covered from single fuelto multi-fossil fuels from quadratic objective function tononconvex objective function in addition to PPZ RPR andRRL constraints Many existing optimization algorithms have
been concerned in aim to compare the performance and givethe final conclusion on the proposed method There have
been six main cases with sixteen subcases The evaluationshave been made at the end of each study case Clearly theproposed ICSA approach has yielded more effective optimalsolutions with faster execution time than almost all methodsConsequently it can be concluded that the proposed methodis much more superior to CCSA and is a very promisingmethod for solving OLD problem
Appendix
See Tables 16ndash26
Nomenclature
120575119894 120582119894 120572119894 120573119894 120574119894 Fuel cost functioncoefficients of the 119894119905ℎthermal generationunit11986100 1198610119895 119861119895119894 Power loss matrixcoefficients119865119865119909 119865119865119887119890119904119905 The values of fitness ofsolution 119909 and theso-far most effectivesolution among thecurrent set ofsolutions119898119894 Number of fuels burntin the 119894119905ℎ thermalgeneration unit
119873 Number of allavailable thermalgeneration units119899119894 Number of prohibitedpower zones of the 119894119905ℎthermal generationunit119875119906119894119896 119875119897119894119896 Upper and lowerlimits of the 119894119905ℎthermal generationunit corresponding tothe kth PPZ119875119886 Probability ofreplacing controlvariables in each oldsolution119875119863 Real power demand ofall loads in system119875119894119898119886119909 119875119894119898119894119899 The highest andlowest real poweroutputs of the iththermal generationunit119875119894119895119898119886119909 119875119894119895119898119894119899 The highest andlowest real poweroutputs of the 119894119905ℎthermal generationunit corresponding tothe 119895119905ℎ fuel type119878119894119898119886119909 Maximum real powerreserve contributionof the thermalgeneration unit 1198941198781199001198971199031198861198991198891 1198781199001198971199031198861198991198892 1198781199001198971199031198861198991198893 1198781199001198971199031198861198991198894 Randomly mixedsolutions from the setof current solutions119878119900119897119909 119878119900119897119866119887119890119904119905 The old solution x andthe most effectivesolution119878119877 Real power reserverequirement of system
120575119894119895 120582119894119895 120572119894119895 120573119894119895 120574119894119895 Fuel cost functioncoefficients of the 119894119905ℎthermal generationunit corresponding tothe 119895119905ℎ fuel type1205761 1205762 1205763 1205764 1205765 1205766 Random numbersbetween 0 and 1Ω Number of generationunits considering PPZconstraint
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare no conflicts of interest
References
[1] N Sinha R Chakrabarti and P K Chattopadhyay ldquoEvolution-ary programming techniques for economic load dispatchrdquo IEEETransactions on Evolutionary Computation vol 7 no 1 pp 83ndash94 2003
[2] P Hansen ldquoA separable approximation dynamic programmingalgorithm for economic dispatch with transmission lossesrdquoYugoslav Journal of Operations Research vol 12 no 2 2002
[3] N T Thang ldquoSolving economic dispatch problem with piece-wise quadratic cost functions using lagrange multiplier theoryrdquoin Proceedings of the 3rd International Conference on ComputerTechnology and Development (ICCTD rsquo11) pp 359ndash364 ASMEPress 2011
[4] S K Mishra and S K Mishra ldquoA comparative study of solutionof economic load dispatch problem in power systems in theenvironmental perspectiverdquoProcedia Computer Science vol 48pp 96ndash100 2015
[5] A A Al-Subhi and H K Alfares ldquoEconomic load dispatchusing linear programming a comparative studyrdquo InternationalJournal of Applied Industrial Engineering vol 3 no 1 pp 16ndash362016
28 Mathematical Problems in Engineering
[6] C Zhou G Huang and J Chen ldquoPlanning of electric powersystems considering virtual power plants with dispatchableloads included an inexact two-stage stochastic linear program-ming modelrdquoMathematical Problems in Engineering vol 2018Article ID 7049329 12 pages 2018
[7] T T Nguyen N V Quynh and L Van Dai ldquoImproved fireflyalgorithm a novel method for optimal operation of thermalgenerating unitsrdquo Complexity vol 2018 Article ID 7267593 23pages 2018
[8] S S Haroon S Hassan S Amin et al ldquoMultiple fuel machinespower economic dispatch using stud differential evolutionrdquoEnergies vol 11 no 6 pp 1ndash20 2018
[9] C-T Su and C-L Chiang ldquoNonconvex power economic dis-patch by improved genetic algorithm with multiplier updatingmethodrdquo Electric Power Components and Systems vol 32 no 3pp 257ndash273 2004
[10] NNoman andH Iba ldquoDifferential evolution for economic loaddispatch problemsrdquo Electric Power Systems Research vol 78 no8 pp 1322ndash1331 2008
[11] P Somasundaram and K Kuppusamy ldquoApplication of evolu-tionary programming to security constrained economic dis-patchrdquo International Journal of Electrical Power amp EnergySystems vol 27 no 5-6 pp 343ndash351 2005
[12] T Jayabarathi K Jayaprakash andD Jeyakumar ldquoEvolutionaryprogramming technique for different kinds of economic dis-patch problemsrdquo Electric Power Systems Research vol 73 no 2pp 169ndash176 2005
[13] P Subbaraj R Rengaraj and S Salivahanan ldquoEnhancementof combined heat and power economic dispatch using selfadaptive real-coded genetic algorithmrdquo Applied Energy vol 86no 6 pp 915ndash921 2009
[14] C-L Chiang and C-T Su ldquoAdaptive-improved genetic algo-rithm for the economic dispatch of units with multiple fueloptionsrdquo Cybernetics and Systems vol 36 no 7 pp 687ndash7042005
[15] C Chiang ldquoImproved genetic algorithm for power economicdispatch of units with valve-point effects and multiple fuelsrdquoIEEE Transactions on Power Systems vol 20 no 4 pp 1690ndash1699 2005
[16] K Meng H G Wang Z Y Dong and K P Wong ldquoQuantum-inspired particle swarm optimization for valve-point economicload dispatchrdquo IEEE Transactions on Power Systems vol 25 no1 pp 215ndash222 2010
[17] P Erdogmus A Ozturk and S Duman ldquoEnvironmen-taleconomic dispatch using genetic algorithm and simulatedannealingrdquo in Proceedings of the International Conference onElectric Power and Energy Conversion Systems EPECS rsquo09 pp1ndash4 IEEE 2009
[18] L D S Coelho and V C Mariani ldquoImproved differentialevolution algorithms for handling economic dispatch opti-mization with generator constraintsrdquo Energy Conversion andManagement vol 48 no 5 pp 1631ndash1639 2007
[19] H Dakuo W Fuli and M Zhizhong ldquoA hybrid genetic algo-rithm approach based on differential evolution for economicdispatch with valve-point effectsrdquo Electrical Power and EnergySystems vol 30 pp 31ndash38 2008
[20] C-F Chang J-J Wong J-P Chiou and C-T Su ldquoRobustsearching hybrid differential evolution method for optimalreactive power planning in large-scale distribution systemsrdquoElectric Power Systems Research vol 77 no 5-6 pp 430ndash4372007
[21] M R Farooqi P Jain and K R Niazi ldquoUsing Hopfield neuralnetwork for economic dispatch of power systemsrdquo in Proceed-ings of the National Power Engineering Conference PECon rsquo03pp 5ndash10 Bangi Malaysia 2003
[22] T Yalcinoz H Altun and U Hasan ldquoConstrained economicdispatch with prohibited operating zones a Hopfield neuralnetwork approachrdquo in Proceedings of the 10th MediterraneanElectrotechnical Conference Information Technology and Elec-trotechnology for the Mediterranean Countries ProceedingsMeleCon rsquo00 (Cat No 00CH37099) pp 570ndash573 LemesosCyprus 2000
[23] D N Vo P Schegner and W Ongsakul ldquoCuckoo searchalgorithm for non-convex economic dispatchrdquo IET GenerationTransmission amp Distribution vol 7 no 6 pp 645ndash654 2013
[24] J-B Park K-S Lee J-R Shin and K Y Lee ldquoA particleswarm optimization for economic dispatch with non-smoothcost functionsrdquo IEEETransactions on Power Systems vol 20 no1 pp 34ndash42 2005
[25] T Niknam H D Mojarrad and M Nayeripour ldquoA newfuzzy adaptive particle swarm optimization for non-smootheconomic dispatchrdquo Energy vol 35 no 4 pp 1764ndash1778 2010
[26] L D S Coelho and V C Mariani ldquoAn improved harmonysearch algorithm for power economic load dispatchrdquo EnergyConversion and Management vol 50 no 10 pp 2522ndash25262009
[27] M Mahdavi M Fesanghary and E Damangir ldquoAn improvedharmony search algorithm for solving optimization problemsrdquoAppliedMathematics and Computation vol 188 no 2 pp 1567ndash1579 2007
[28] A Bhattacharya and P K Chattopadhyay ldquoSolving complexeconomic load dispatch problem using biogeography-basedoptimizationrdquo Expert Systems with Applications vol 37 no 5pp 3605ndash3615 2010
[29] D Simon ldquoBiogeography-based optimizationrdquo IEEE Transac-tions on Evolutionary Computation vol 12 no 6 pp 702ndash7132008
[30] J S Alsumait J K Sykulski and A K Al-Othman ldquoAhybrid GA-PS-SQP method to solve power system valve-pointeconomic dispatch problemsrdquo Applied Energy vol 87 no 5 pp1773ndash1781 2010
[31] M S Turgut andGKDemir ldquoQuadratic approximationndashbasedhybrid Artificial Cooperative Search algorithm for economicemission load dispatchproblemsrdquo International Transactions onElectrical Energy Systems vol 27 no 4 pp 1ndash14 2017
[32] T A A Victoire and A E Jeyakumar ldquoHybrid PSO-SQPfor economic dispatch with valve-point effectrdquo Electric PowerSystems Research vol 71 no 1 pp 51ndash59 2004
[33] D-KHe F-LWang andZ-ZMao ldquoHybrid genetic algorithmfor economic dispatch with valve-point effectrdquo Electric PowerSystems Research vol 78 no 4 pp 626ndash633 2008
[34] D C Secui ldquoA modified symbiotic organisms search algorithmfor large scale economic dispatch problem with valve-pointloading effectsrdquo Energy vol 113 pp 366ndash384 2016
[35] K BhattacharjeeA Bhattacharya and S H N Dey ldquoTeaching-learning-based optimization for different economic dispatchproblemsrdquo Scientia Iranica vol 21 no 3 pp 870ndash884 2014
[36] K Bhattacharjee A Bhattacharya and S H N Dey ldquoOppo-sitional Real Coded Chemical Reaction Optimization for dif-ferent economic dispatch problemsrdquo International Journal ofElectrical Power amp Energy Systems vol 55 pp 378ndash391 2014
Mathematical Problems in Engineering 29
[37] V Hosseinnezhad M Rafiee M Ahmadian and M T AmelildquoSpecies-basedQuantum Particle SwarmOptimization for eco-nomic load dispatchrdquo International Journal of Electrical Poweramp Energy Systems vol 63 pp 311ndash322 2014
[38] M S P Subathra S E Selvan T A A Victoire A HChristinal andU Amato ldquoA hybrid with cross-entropymethodand sequential quadratic programming to solve economic loaddispatch problemrdquo IEEE Systems Journal vol 9 no 3 pp 1031ndash1044 2015
[39] J Zhan Q H Wu C Guo and X Zhou ldquoEconomic dispatchwith non-smooth objectives part I local minimum analysisrdquoIEEE Transactions on Power Systems vol 30 no 2 pp 710ndash7212015
[40] A K Barisal andR C Prusty ldquoLarge scale economic dispatch ofpower systems using oppositional invasive weed optimizationrdquoApplied Soft Computing vol 29 pp 122ndash137 2015
[41] M Basu ldquoImproved differential evolution for economic dis-patchrdquo International Journal of Electrical Power amp EnergySystems vol 63 pp 855ndash861 2014
[42] V S Aragon S C Esquivel and C C Coello ldquoAn immunealgorithm with power redistribution for solving economicdispatch problemsrdquo Information Sciences vol 295 pp 609ndash6322015
[43] M Ghasemi M Taghizadeh S Ghavidel and A AbbasianldquoColonial competitive differential evolution an experimentalstudy for optimal economic load dispatchrdquo Applied Soft Com-puting vol 40 pp 342ndash363 2016
[44] B R Adarsh T Raghunathan T Jayabarathi and X-S YangldquoEconomic dispatch using chaotic bat algorithmrdquo Energy vol96 pp 666ndash675 2016
[45] N Ghorbani and E Babaei ldquoExchange market algorithm foreconomic load dispatchrdquo International Journal of ElectricalPower amp Energy Systems vol 75 pp 19ndash27 2016
[46] J X V Neto G Reynoso-Meza T H Ruppel V C Marianiand L D S Coelho ldquoSolving non-smooth economic dispatchby a new combination of continuous GRASP algorithm anddifferential evolutionrdquo International Journal of Electrical Poweramp Energy Systems vol 84 pp 13ndash24 2017
[47] A Kavousi-Fard and A Khosravi ldquoAn intelligent 120579-ModifiedBat Algorithm to solve the non-convex economic dispatchproblem considering practical constraintsrdquoElectrical Power andEnergy Systems vol 82 pp 189ndash196 2016
[48] M A Al-Betar M A Awadallah A T Khader and A L BolajildquoTournament-based harmony search algorithm for non-convexeconomic load dispatch problemrdquo Applied Soft Computing vol47 pp 449ndash459 2016
[49] M A Al-Betar M A Awadallah I A Doush E Alsukhni andH ALkhraisat ldquoA non-convex economic dispatchproblemwithvalve loading effect using a new modified 120573-hill climbing localsearch algorithmrdquo Arabian Journal for Science and Engineeringpp 1ndash8 2018
[50] M A Awadallah M A Al-Betar A L Bolaji E M Alsukhniand H Al-Zoubi ldquoNatural selection methods for artificial beecolony with new versions of onlooker beerdquo Soft Computing pp1ndash40 2018
[51] S H Kaboli and A K Alqallaf ldquoSolving non-convex economicload dispatch problem via artificial cooperative search algo-rithmrdquo Expert Systems with Applications vol 128 pp 14ndash272019
[52] D Singh and J Dhillon ldquoAmeliorated greywolf optimization foreconomic load dispatch problemrdquo Energy vol 169 pp 398ndash4192019
[53] X S Yang and S Deb ldquoCuckoo search via Levy flightsrdquo inProceedings of the World Congress on Nature and BiologicallyInspired Computing (NaBIC rsquo09) pp 210ndash214 India 2009
[54] S Sahoo K M Dash R C Prusty and A K Barisal ldquoCom-parative analysis of optimal load dispatch through evolutionaryalgorithmsrdquo Ain Shams Engineering Journal vol 6 pp 107ndash1202015
[55] M Basu and A Chowdhury ldquoCuckoo search algorithm foreconomic dispatchrdquo Energy vol 60 pp 99ndash108 2013
[56] A B Serapiao ldquoCuckoo search for solving economic dispatchload problemrdquo Intelligent Control and Automation vol 04 no04 pp 385ndash390 2013
[57] A H Bindu and M D Reddy ldquoEconomic load dispatch usingcuckoo search algorithmrdquo International Journal of EngineeringResearch and Applications vol 3 pp 498ndash502 2013
[58] E Afzalan and M Joorabian ldquoAn improved cuckoo searchalgorithm for power economic load dispatchrdquo InternationalTransactions on Electrical Energy Systems vol 25 no 6 pp 958ndash975 2015
[59] T T Nguyen and D N Vo ldquoThe application of one rank cuckoosearch algorithm for solving economic load dispatch problemsrdquoApplied Soft Computing vol 37 pp 763ndash773 2015
[60] S M Islam S Das S Ghosh S Roy and P N Suganthan ldquoAnadaptive differential evolution algorithm with novel mutationand crossover strategies for global numerical optimizationrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 42 no 2 pp 482ndash499 2012
[61] Z L Gaing ldquoParticle swarm optimization to solving theeconomic dispatch considering the generator constraintsrdquo IEEETransactions on Power Systems vol 18 no 3 pp 1187ndash1195 2003
[62] X-S Yang and S Deb ldquoEngineering optimisation by Cuckoosearchrdquo International Journal of Mathematical Modelling andNumerical Optimisation vol 330 pp 43ndash51 2010
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Algorithm 1 New mutation technique applied in the proposed ICSA approach
conventional CSA except that an additional step should beadded at each iteration In this step the119865119865119889 of all individualsolutions should be calculated by utilizing Equation (24) andthen the result of comparing the ratio with 119879119900119897will be used todecidewhich updating formula should be selectedThewholedescription of the proposed standard and new mutationtechnique can be coded inMatlab program language by usingAlgorithm 1
323 Adjustment of Tolerance for Each Solution As pointedout above the proposed method needs assistances to deter-mine the most appropriate step size for finding out favorablesolution zones The given aim can be reached if the selectionof 119879119900119897119909 is reasonable however the range of this parameteris infinite and hard to select Thus the adaptation of tuningthe parameter is really necessary First of all the compari-son between 119879119900119897119909 and 119865119865119877119909 is carried out and then theadaptation will be determined based on the obtained resultfrom the comparison Results of comparison between the twoparameters can be either 119865119865119877119909 is less than 119879119900119897119909 or 119865119865119877119909is higher than 119879119900119897119909 The case that two parameters are equalhardly ever occurs
As the comer assumptionhappens (ie119865119865119877119909 is less than119879119900119897119909) at the considered time the four-point step size will beemployed for the 119909119905ℎ solution If 119879119900119897119909 remains unchanged atthe previous value the identification of improvement fromsuch four-point step size or two-point step size is vagueConsequently value of 119879119900119897119909 must be automatically reducedto a lower value in case that it has significant contribution tofound promising solution of previous iteration Clearly thedecrease of119879119900119897119909 can enable the proposedmethod to jump outlocal optimal zone and approachmore effective zones By trialand error method 119879119900119897119909 is selected to be a function of itselfthat is 09 of the previous value Finally the implementationof the proposed ICSA approach is presented in Algorithm 2
4 The Application of the ProposedICSA for OLD Problem
Thewhole computation steps of the proposed ICSA approachfor solving OLD problem are explained as follows
41 Handling Constraints and Randomly Producing InitialPopulation As shown in Section 2 the considered OLDproblem takes five following constraints into account
(i) Power balance constraint is shown in Equation (4)
(ii) Power output limitation constraint is shown in Equa-tion (6)
(iii) Prohibited power zone constraint is shown in Equa-tion (9)
(iv) Real power reserve constraint is shown in Equation(10)
(v) Ramp rate limit constraint is shown in Equation (13)
Among the five constraints ramp rate limit generationlimit and prohibited power zone seem to be more com-plicated than power balance and power reserve constraintsHowever the three constraints can be solved more easilybecause each unit is constrained independently in the threeconstraints whereas power balance constraint and powerreserve constraint consider all the thermal generating unitssimultaneously Power reserve constraint can be handledby penalizing the total generation of all units while powerbalance constraint can be solved by penalizing one violatedthermal generating unit The whole computation procedurefor solving all constraints and calculating fitness function ofsolutions is described in detail as follows
Step 1 Redefine maximum and minimum power output ofeach thermal generating unit as considering PPZ and RRLconstraints by using the following formulas
119875119894max = 119875119894max if 119875119894max le 119875i0 + 119868119878119878119894119875i0 + 119868119878119878119894 if 119875119894max gt 119875i0 + 119868119878119878119894
119894 = 1 119873(25)
119875119894min = 119875119894min if 119875119894min ge 119875i0 minus 119863119878119878119894119875i0 minus 119863119878119878119894 119890119897119904119890
119894 = 1 119873(26)
Mathematical Problems in Engineering 9
Produce initial population with119873119901119904 solutions (1198781199001198971 1198781199001198972 119878119900119897119909 119878119900119897119873119901119904)Calculate fitness function (1198651198651 1198651198652 119865119865119909 119865119865119873119901)Go to the loop algorithm by setting 119866 = 1
While (119866119898119886119909 gt 119866) (i) The first newly produced solutions119878119900119897119899119890119908119909 = 119878119900119897119909 + 120572(119878119900119897119909 minus 119878119900119897119866119887119890119904119905) oplus Levy(120573) (ii) Perform selection approach
119865119865119909 = 119865119865119909 if 119865119865119909 le 119865119865119899119890119908119909119865119865119899119890119908119909 119900119905ℎ119890119903119908119894119904119890 119909 = 1 119873119901119904
(v) Determine the most effective solution and its fitnessDetermine 119865119865119909 with the smallest value and assign 119878119900119897119909 to 119878119900119897119866119887119890119904119905If 119866119898119886119909 gt 119866 perform step (i) and increase 119866 to 119866 + 1 Otherwise stop the loop algorithm and report boththe smallest fitness together with 119878119900119897119866119887119890119904119905End while
Among the four Equations (25) and (26) are used firstin order to redefine upper bound and lower bound for allthermal generating units as considering RRL constraint Thethe redefined bounds continue to be redefined for the secondtime by using (27) and (28) as considering PPZ constraints
Step 2 (randomly produce initial population) For dealingwith the power balance constraint all available units areseparated into two groups in which the first group withdecision variables consists of the power output from thesecond unit to the last unit (P2 P3 PN) meanwhile onlythe power output of the first unit (1198751) belongs to the secondgroup with dependent variable So upper bound solution119878119900119897119898119886119909 and lower bound solution 119878119900119897119898119894119899 must be defined asfollows
Step 3 Handle prohibited power zone constraint for decisionvariables P2 P3 PN
After being randomly produced there is a high possi-bility that decision variables fall into PPZ and they violatePPZ constraint So the verification of falling into PPZ andcorrection of the violation should be accomplished by usingthe following formula
119875119894 =
119875119897119894119896 if 119875119897119894119896 lt 119875119894 le 119875119897119894119896 + 1198751198961198941198962119875119906119894119896 if (119875119894 gt 119875119897119894119896 + 1198751198961198941198962 ) amp (119875119894 lt 119875119906119894119896)119875119894 119890119897119904119890
119894 = 2 119873 amp 119896 = 1 119899119894
(31)
Step 4 Handle RPB constraint by calculating 1198751 and penaliz-ing 1198751 if it violates constraints
In this step power balance constraint is exactly handledby calculating and penalizing dependent variable (1198751) 1198751 isobtained by using formulas (4) and (5) as follows
1198751 = minus (11986101 minus 1 + 2sum119873119894=2 1198611119894119875119894) plusmn radicΔ211986111 (32)
where
Δ = (11986101 minus 1 + 2 119873sum119894=2
1198611119894119875119899)2 minus 411986111(119875119863 minus 119873sum
119894=2
119875119894+ 11986100 + 119873sum
119894=2
1198610119894119875119894 + 119873sum119894=2
119873sum119895=2
119875119894119861119894119895119875119895) amp Δ ge 0(33)
In Equation (32) 1198751 has been determined for the purposeof dealing with real power balance constraint However it isnot sure that 1198751 can satisfy upper bound and lower boundconstraints and prohibited power zone constraints So 1198751must be checked and penalized
Firstly 1198751 is checked and penalized for upper and lowerbound constraints by the following model
Δ1198751x =
0 if 1198751min le 1198751x le 1198751max
1198751min minus 1198751x if 1198751min gt 1198751x1198751x minus 1198751max if 1198751max lt 1198751x
(34)
In Equation (34) if the second case or the third caseoccurs it means P1 has violated either lower bound or upperbound and it would be penalized by using either (P1x= P1min-P1x) or (P1x =P1x -P1max) Otherwise ifP1 has not violatedthe bound constraints (ie the first case in (34) happened)
P1 would continue to be checked for PPZ constraint by thefollowing model
Δ1198751x
=
1198751 minus 1198751198971119896 if 1198751198971119896 lt 1198751 le 1198751198971119896 + 119875119896111989621198751199061119896 minus 1198751 if (1198751 gt 1198751198971119896 + 11987511989611198962 ) amp (1198751 lt 1198751199061119896)0 119890119897119904119890
(35)
Step 5 Handle real power reserve constraint (10)First of all 119878119894 is determined by using (11) and (12) and
then the 119909119905ℎ solution will be checked and penalized if poweroutput of all thermal generating units cannot satisfy RPRconstraint The penalty for violation of the constraint can becalculated by using equation (36)
Δ119878119877119909 =
0 if119873sum119894=1
119878119894119909 ge 119878119877119878119877 minus 119873sum119894=1
119878119894119909 119890119897119904119890 (36)
As a result real power reserve constraint can be solved byusing the penalty method
42 Calculate Fitness Function for Solutions Fitness functionof each solution is used to evaluate quality of solutionNormally the function is the sum of objective function andpenalty of violating constraints and is obtained by
43 The First Newly Updated Solutions by Levy Flights Tech-nique In this section the first newly updated solutionsare performed by employing Levy flights technique usingEquation (14) However each new solution can be out oftheir feasible operating zone such as PPZ and upper andlower limitations When the power output violates its PPZconstraints Equation (31) will be applied to tackle theconstraint Besides the following equation will be employedwhen power output is higher or lower than their limitations
119878119900119897119909 =
119878119900119897max if 119878119900119897max lt 119878119900119897119909119878119900119897min if 119878119900119897min gt 119878119900119897119909119878119900119897119909 Otherwise
119909 = 1 119873119901 (38)
After that Equations (32)-(37) are performed for deter-mining all variables and penalty terms Finally Equation (38)is employed to calculate fitness function
44 The Second Newly Updated Solutions by Using Muta-tion Technique The second newly updated solutions areaccomplished as presented in Section 3 above Similar to
Mathematical Problems in Engineering 11
the task after doing the first update each solution in thenew population must satisfy PPZ constraint and upper andlower boundaries by considering Equations (31) and (38)Then Equations (32)-(37) are performed for determiningall variables and penalty terms Finally Equation (38) isemployed to calculate fitness function and the solution withthe best value is assigned to the best one 11987811990011989711986611988711989011990411990545 Criterion of Stopping the Loop Algorithm In the loopalgorithm of using the proposed ICSA approach the solutionsearch work is stopped in case that the predeterminedmaximum iterations 119866119898119886119909 is reached For each search ter-mination the most effective solution is stored and anotherrun continues to be accomplished until the predeterminednumber of runs is reached After finishing the runs thebest one is found and reported In addition other valuessuch as the fitness of the worst solution and average fitnessof all solutions are also reported for comparing with othermethods
46 The Whole Iterative Process The whole iterative algo-rithm for implementing the proposed ICSA approach forcoping with OLD problem is described in detail in Figure 3
5 Results and Discussions
The proposed ICSA approach performance has been investi-gated on six cases with different fuel options different fuelcharacteristics and complicated constraints The details ofthe studied cases are presented as follows
Case 1 Four systems with single fuel source (SFS) and powerloss (PL) constraint
Subcase 11 A 3-unit system [57]Subcase 12 A 6-unit system [57]Subcase 13 A 3-unit system [56]Subcase 14 A 6-unit system [56]
Case 2 A 110-unit system with SFS [57]
Case 3 Four systems with SFS and the effects of valve loadingprocess (EoVLP)
Subcase 31 A 3-unit system supplying to a load of850MW [58]Subcase 32 A 13-unit system supplying to a load of1800MW [1]Subcase 33 A 13-unit system supplying to a load of2520MW [1]Subcase 34 A 40-unit system supplying to a load of2500MW [1]Subcase 35 An 80-unit system supplying to a load of4100MW [49]
Case 4 Two systems with SFS and PPZ and RPR constraints
Subcase 41 A 60-unit system supplying to a10600MW load [9]
Subcase 42 A 90-unit system supplying to a15900MW load [9]
Case 5 A 15-unit system with SFS and RRL PPZ and PLconstraints [61]
Case 6 Three systems with multiple fuel sources (MFS) andEoVLP
Subcase 61 An 80-unit system supplying to a21600MW load [15]Subcase 62 A 160-unit system supplying to a43200MW load [15]Subcase 63 A 320-unit system supplying to an86400MW load [54]
For each considered case with each load case the pro-posed ICSA approach is run 50 times on the programlanguage of Matlab and a PC with 4 GB of RAM and 24GHzprocessor The selection of adjustment parameters including119875119886 and 119879119900119897119909 is carefully considered to obtain the best optimalsolutions meanwhile two others such as 119873119901119904 and 119866119898119886119909 arechosen corresponding to the scale of particular test system9 values with the change of 01 in the range [01 09] are inturn selected for 119875119886 while 119879119900119897119909 is 001 at the beginning Theinformation including load demand119873119901119904 119866119898119886119909 and the best119875119886 is reported in Table 1
51 Obtained Results on Case 1 considering Four Systems withSFS and PL Constraint In this section we have implementedthe proposed ICSA approach for solving four systems dividedinto four subcases Tables 2 and 3 show the comparisons ofobtained results from Subcases 11 and 12 and Subcases 13and 14 respectively As listed in Table 2 the proposed ICSAmethod and CCSA can find equal fuel cost for Subcases 11whereas the reduction of fuel cost from the proposed ICSAmethod as compared to CCSA is clearer for Subcase 12 Asshown in Table 3 for comparing the proposed ICSA and threemethods consisting of CCSA ABC and FA the minimumfuel cost of the proposed ICSA is approximately equal to thatof these methods for Subcases 13 but much less than that ofthese methods for Subcase 14 Furthermore the proposedICSA has been run by setting 119873119901119904 and 119866119898119886119909 to 5 and 20but these values were much higher for CCSA ABC and FAThey are 20 and 5000 for CCSA 40 and 100 for ABC and 20and 5000 for FA Consequently the proposed method is veryefficient for Case 1 with four subcases
Optimal solutions obtained by ICSA for Case 1 are shownin Tables 16ndash18
52 Obtained Results on Case 2 considering 110-Unit Systemwith SFS In this section we have employed a very largescale system with 110 units but there were not challenges forobjective function and complex constraints since EoVLP andconstraints were not taken into account Both CCSA andthe proposed ICSA methods have been run for comparingwith BBO hybrid BBO and DE (DEBBO) and Opposi-tional real coded chemical reaction optimization algorithm(ORCCROA) in [36] IWA in [40] and AGWO in [52] As
12 Mathematical Problems in Engineering
Select parameters
- Correct solutions if violating PPZ constraint using Eq (31)- Determine using Eq (32)- Penalize for violating upper and lower bounds constraint using Eq (34)- Penalize for violating PPZ constraint using Eq (35)- Penalize the xth solution for violating RPR constraint using Eq (36)
- Determine quality of solutions employing Eq (37) - Select the most effective solution - Start the loop algorithm by selecting
- Perform the first solution update using Section 32 - Correct boundaries of solutions by using Eq (38)
- Correct solutions if violating PPZ constraint using Eq (31)- Determine using Eq (32)- Penalize for violating upper and lower bounds constraint using Eq (34)- Penalize for violating PPZ constraint using Eq (35)- Penalize the xth solution for violating RPR constraint using Eq(36)
- Determine fitness function using Eq (37)- Compare old solutions and new solutions to keep better ones
- Perform the second solution update using Algorithm 1 - Correct boundaries of solutions by using (38)
- Correct solutions if violating PPZ constraint using Eq (31)- Determine using Eq (32)- Penalize for violating upper and lower bounds constraint using Eq (34)- Penalize for violating PPZ constraint using Eq (35)- Penalize the xth solution for violating RPR constraint using Eq (36)
- Determine fitness function using Eq (37)- Compare old solutions and new solutions to keep better ones- Select the most effective solution
Stop
Start
- Redefine upper and lower bounds using Eqs (25)-(28)- Randomly generate initial population using Eq (30)
Nps Pa Gmax H> Tolx
P1xP1x
P1x
P1xP1x
P1x
P1xP1x
P1x
Sol<MN
Sol<MN
G = 1
G = Gmax G = G + 1
Figure 3 All computation steps for solving OLD problem by employing the proposed ICSA approach
shown in Table 4 AGWO [52] has reached less fuel cost thanICSA however the exact fuel cost which was recalculatedby using reported solution pointed out that the method hasreached a very high fuel cost of $2157404250 For comparisonwith other methods ICSA has found less fuel cost thanall these methods Particularly the reduction of generationfuel cost is significant as compared to BBO DEBBO andCCSA Execution time comparisons are also useful evidence
for indicating the high performance of ICSA Thus it canconclude that ICSA is a strong method for Case 2
Optimal solution obtained by ICSA for the case is shownin Table 19
53 Obtained Results on Case 3 considering Four Systems withSFS and EoVLP In this section the real performance of theproposed ICSA approach has been investigated based on five
Mathematical Problems in Engineering 13
Table 1 Information of considered cases and adjustment parameters
Case Fuel cost function Constraint Subcase No of units 119875119863 (MW) Nps Gmax Best 1198751198861 SFS PL
subcases with the gradual increase of number of units Thesmallest scale system considers 3 units but the largest scalesystem takes 80 units In addition to the implementation ofthe proposed ICSA we have also implemented CCSA forSubcase 31 and Subcase 35 for further comparison becauseCCSA has not been run for the two subcases so far
Comparison of obtained results from Subcase 31 shownin Table 5 indicates that the proposed ICSA is superior toCCSAwith lower fuel cost but it seems to be less effective thanMCSA [58] The minimum cost of MCSA reported in [58] isthe smallest fuel cost but the recalculated cost is much higherthan that of the proposed method Furthermore MCSA hasbeen implemented by setting very high values to 119873119901119904 and119866119898119886119909
Reports for Subcases 32 and 33 shown in Table 6are the comparisons of the proposed ICSA approach andother methods such as conventional Evolution programming(CEP) [1] Fast EP (FEP) [1] improved FEP (IFEP) [1] DE[12] multiplier Lagrange-based genetic algorithm with (GA-MU) [15] QPSO [16] GA-PS-SQP [30] PSO-SQP [32] M120573-HCLSA [49] IABCA [50] CCSA [59] OSE-CSA [59] SOS[34] MSOS [34] CEA-SQT [38] TSBO [39] IWA [40] andCBA [44] As observed from the table ICSAapproach obtainsbetter solutions than mostmethods excluding DE [10] CCSA
[59] OSE-CSA [59] SOS [34] MSOS [34] CEA-SQT [38]TSBO [39] IWA [40] and CBA [44] especially M120573-HCLSA[49] with lower cost $1796097 However recalculated costfrom reported solution of M120573-HCLSA is $179691 BesidesICSA is very fast as compared to most methods where twoother versions of Cuckoo search algorithm CCSA and OSE-CSA are also included except two methods in [34] Theprocessor of computer that all the methods run on is alsoreported in the final column Clearly ICSA approach is veryefficient for the case with the 13-unit system where effects ofvalve loading process are considered
In Subcase 34 the number of units is much larger thanthat of three subcases above up to 40 units [1] The obtainedresult comparisons with others are indicated in Table 7Clearly the minimum cost comparisons reveal that the pro-posedmethod is one of the leading methods due to the lowestcost except the comparison with CCSA [23] OSE-CSA [59]SOS [34] MSOS [34] EMA [45] 120579-MBA [47] and AGWOA[52] It is noted that AGWOA [52] has reported the bestminimum cost with $12140430 but recalculated minimumcost which was obtained by substituting reported optimalgeneration of all thermal generating units is $12141331 Theaverage and the maximum costs from the proposed methoddo not belong to the leading method group however the
Mathematical Problems in Engineering 15
Table 7 Result comparisons for Subcase 34
Approach Best cost Average cost Worst cost CPU time Computer($h) ($h) ($h) (s) (Processor-Ram)
execution time that ICSA approach takes is smaller thanmostones except IAPR [42] and 120579-MBA [47] which have been runon stronger computers Note that MSOS [34] has been fasterthan the proposed method for Subcases 31 and 32 abovebut it is too slower than the proposed method for the casenamely 1813 seconds compared to 146 seconds Comparedto two other versions of Cuckoo search CCSA [23] and OSE-CSA [59] the proposed method is also faster about threetimes although the processors are slightly different For thiscase 120579-MBA [47] shows a very good performance howeverthe method has not been tested onmore complicated systemsand larger scale and therefore more comparisons with themethod must end Clearly the proposed method is stillefficient for the case where large scale and effects of valveloading process are included
Subcase 35 is the largest scale system with 80 units Inaddition to the implementation of ICSA CCSA has beenalso run for the Subcase 35 for further investigation ofefficiency improvement of the proposed ICSA approach The
comparisons of minimum cost in Table 8 show that theproposed ICSA can find more optimal solution than M120573-HCLSA [49] AGWOA [52] and CCSA The proposed ICSAis also superior to CCSA in terms ofmore stable search abilityand lower fluctuation since its average cost and maximumcost are less than those of CCSA The outstanding figurecannot be reached as compared to AGWOA [52] howeverit is hard to conclude AGWOA [52] is superior to theproposed ICSA approach about more stable search abilityand lower fluctuation Actually comparison of the values ofpopulation and iterations as well as execution time cannot beaccomplished because the information was not reported in[52]Thus it can conclude that the proposed ICSA is effectivefor the subcase
Optimal solutions obtained by ICSA for the case areshown in Tables 20ndash23
54 Obtained Results on Case 4 with Two Systems consideringSFS and PPZ and RPR Constraints In this section two
16 Mathematical Problems in Engineering
Table 9 Result comparisons for Subcases 41 and 42
Subcase Approach Best cost ($h) Mean cost ($h) Worst cost ($h) Std dev ($h) CPU time (s) Computer (Processor-Ram)
test systems with SFS and PPZ and RPR constraints areconsidered The test system size is up to 60 and 90 units forSubcases 41 and 42 respectively Comparison for the casesis only performed with two Genetic algorithms consisting ofGA and IGA-MU in [9] and two other versions of Cuckoosearch algorithms including CCSA [23] and OSE-CSA [59]and presented in Table 9 Other studies have tended toignore such complicated constraints of PPZ and RPR Thecosts reported in Table 9 indicate that ICSA approach canobtain more effective solution than CCSA and OSE-CSAbecause it has reached lower minimum cost than the twoones Furthermore the proposed method also takes shortercomputation time for the two cases from about two timesto about three times although the processor of the proposedmethod is slightly strongerThemean costs of ICSA approachare much less than those from IGA-MU and GA and slightlyhigher than those from OSE-CSA but there is a trade-off between the proposed method and CCSA for the twosubcases In fact the proposed method obtains higher meancost for Subcase 41 but lower cost for Subcase 42 AlthoughGAmethods have been runon aweak computerwith 07 GHzof the processor compared to that with 24GHz in the studytheir execution times are significantly higher namely 56381seconds (GA) and 16258 seconds (IGA-MU) compared to09153 seconds of ICSA approach for Subcase 41 and 94093seconds (GA) and 25545 seconds (IGA-MU) compared to15892 seconds (the proposedmethod)The analysis can pointout that ICSA approach ismore efficient than these comparedmethods in terms of optimal solutions and execution time
Optimal solutions obtained by ICSA for Subcase 42 areshown in Table 24
55 Obtained Results on Case 5 with a 15-Unit System consid-ering SFS and RRL PPZ and PL Constraints In this section
a 15-unit system considering RRL PPZ and PL constraintsis considered to be solved for finding optimal solution Forefficiency investigation of the proposed ICSA we have alsoimplemented CCSA for comparison As listed in Table 10the proposed ICSA is the most effective method with thesmallest fuel cost The comparisons of control parameters aswell asCPU time are also good evidence to confirm the strongsearch of the proposed ICSA approach since it has been runby smaller values of control parameter and faster executiontime as compared to all methods excluding CCSA
Optimal solution obtained by ICSA for the case is shownin Table 25
56 Obtained Results on Case 6 withThree Systems consideringMFS and EoVLP In this section three test systems with thechallenge on objective function including multi-fossil fuelsources and effects of valve loading process are consideredThe scale is up to 80 units 160 units and 320 units forSubcases 61 62 and 63 respectively
Comparison for Subcase 61 reported in Table 11 revealsthat the proposed method is the best method in terms of thelowest best cost the lowestmean cost and the lowest standarddeviation and the fastest execution time The processor fromthis proposedmethod is about four times stronger thanCGA-MU and IGA-MU but the speed is from ten times to 35 timesfaster than these methods Compared to CCSA and OSE-CSA the proposed method is about two times faster but theprocessor is slightly stronger
Comparison for Subcase 62 is reported in Table 12Clearly the proposed ICSA approach obtains better values ofthe best mean and worst costs than most methods exceptMSOS [34] where the best cost difference is about $ 024However the proposed method is the second fastest onewith 1119 seconds where the first fastest one CBA [44] has
Mathematical Problems in Engineering 17
Table 11 Comparisons of found results for Subcase 61
Approach Best cost Mean cost Worst cost Std dev CPU time Computer($h) ($h) ($h) ($h) (s) (Processor-Ram)
spent 571 seconds Clearly MSOS is better than the proposedICSA approach with respect to slightly less best cost butis worse than the proposed method in terms of executiontime namely 2572 seconds compared to 1119 seconds whileCBA [44] is faster than the proposed method but obtainssignificantly worse costs The analysis can conclude that theproposed ICSA approach is very powerful for the subcasewith 160 units
Table 13 presents the comparison of three other methodsincludingCCSA [54] SOS [34] andMSOS [34] accompaniedwith the proposed method for Subcase 63 The obtainedresult comparisons imply that ICSA approach can obtainbetter values of the best and standard deviation costs thanCCSA and SOS but obtains slightly higher cost than MSOSby approximately $ 022 Besides the execution time fromICSA approach is much shorter than others especially it ishigher than five times faster than MSOS The four methodshave been run on approximately strong computers Brieflythe proposed ICSA approach can find and converge to morefavorable solution than other methods with shorter CPUtime except the comparison with MSOS which had bettersolution but spent higher than five times execution times
Consequently the proposed method is a very promisingoptimization algorithm for Subcase 63 a system up to 320units and with multi-fossil fuel sources and effects of valveloading process
Optimal solution obtained by ICSA for Subcase 63 isshown in Table 26
57 The Improvement of ICSA Approach Performance
571 The Outstanding Improvement over CCSA In this sec-tion the performance improvement of ICSA over CCSAhas been investigated by analyzing obtained results and setcontrol parameters Table 14 has been formed by addingreduction cost improvement level of the best cost executiontime and control parameters consisting of 119873119901119904 and 119866119898119886119909Among the compared factors reduction cost is the deviationof the cost of CCSA and that of ICSA whereas the improve-ment level is the ratio of the reduction cost to the cost ofCCSA The reduction costs indicate the proposed methodcould find either equal quality of solutions or higher qualityof solutions than CCSA for all study casesThe reduction costis from $0 to $42839 corresponding to the improvement level
18 Mathematical Problems in Engineering
Table 14 Summary of results obtained by CCSA and ICSA for all study cases
Study case Reduction cost ($) Improvement level () Execution time (s) Nps Gmax
Figure 4 The best run obtained by CCSA and ICSA for Case 2
that is from 0 to 08968The saving cost is not too much forone hour but the operation in one day onemonth or one yearis very high However it should be noted that CCSA has beenrun by setting much higher population size and iterations formany cases excluding study cases implemented in the studysuch as Case 2 Subcase 31 Subcase 35 and Case 5 Forinstance ICSA has used 119873119901119904 = 10 and 119866119898119886119909 = 15 for Subcase13 and119873119901119904 = 10 and119866119898119886119909 = 25 for Subcase 14 whereas CCSAhas been run by setting119873119901119904 = 20 and 119866119898119886119909 = 5000 for the twosubcases Similarly CCSA has been run for Subcases 32 33and 34 with much higher number of iterations For the lastsubcase CCSA has been run by setting 119873119901119904 = 320 and 119866119898119886119909
= 1200 but those of ICSA have been 10 and 9000 Due tothe higher value of control parameters CCSA has tended tospend more time in finding such high quality solutions foralmost all study cases Execution time of ICSA is less than18 seconds while that of CCSA is up to higher 75 secondsIt is clear that the proposed ICSA could find better optimalsolutions thanCCSA for such considered systems For furtherinvestigation of performance comparison the best runs over50 runs and fuel cost values of 50 runs obtained by CCSA andthe proposed ICSA for Case 2 Subcase 31 Subcase 35 andCase 5 have been plotted in from Figures 4ndash11 The best runcurves show the faster search of the proposed ICSA method
Mathematical Problems in Engineering 19
0 5 10 15 20 25 30 35 40 45 50Run
19795
198
19805
1981
19815
1982
19825
1983
19835
Fuel
cost
($)
CCSAICSA
times105
Figure 5The best fuel cost of 50 runs obtained by CCSA and ICSA for Case 2
0 2 4 6 8 10 12 14 16 18 20Iteration
8234
8236
8238
8240
8242
8244
8246
8248
8250
8252
Fitn
ess F
unct
ion
($)
CCSAICSA
Figure 6 The best run obtained by CCSA and ICSA for Subcase 31
whereas 50 values of fuel cost indicate that the proposedICSA can find many solutions with better quality Clearlythe proposed ICSA is outstanding in terms of stabilizationof solution search and faster convergence As a result it canconclude that the proposed ICSA approach is more effectivethanCCSA in solvingOLDproblemwith considered systems
572The Improvement of Results over Other Methods In thisarticle we have tested ICSA approach on 6 cases with 16
systems with different fuel cost forms different constraintsand different scale systems from 3 units to 320 units We havecompared the yielded results from ICSA approach and otherexisting ones for evaluating the efficiency of ICSA approachIn subsections above we have shown yielded results fromICSA approach and compared these results to those of otheronesHowever the demonstrationhad not been very good forobserving and comparing to lead to a conclusionThus in thesubsectionwehave summarized the result comparisons of the
20 Mathematical Problems in Engineering
0 5 10 15 20 25 30 35 40 45 50Run
8234
82342
82344
82346
82348
8235
82352
82354
82356
82358
8236
Fuel
cost
($)
CCSAICSA
Figure 7 The best fuel cost of 50 runs obtained by CCSA and ICSA for Subcase 31
0 1000 2000 3000 4000 5000 6000Iteration
242
244
246
248
25
252
254
256
Fitn
ess F
unct
ion
($)
CCSAICSA
times105
Figure 8 The best run obtained by CCSA and ICSA for Subcase 35
proposed and other ones Table 15 has reported the reductioncost (in $) of ICSA approach compared to other ones Inaddition we have converted the reduction cost into improve-ment level (in ) for better comparison The improvementhas been shown from the lowest level to the highest levelin terms of reduction cost and improvement percentageIn addition we have also given the slowest and the fastestexecution time of other compared methods together withthat of the proposed method The table implies that ICSA
approach can find better optimal solutions with less fuel costup to $052 for Subcase 13 $75229 for Subcase 14 $1775149for Case 2 $1181256 for Subcase 31 $8438 for Subcase 32$9113 for Subcase 33 $121181 for Subcase 34 $3412 forSubcase 35 $04329 for Subcase 41 $30227 for Subcase42 $407002 for Case 5 $18183 for Subcase 61 $13965 forSubcase 62 and $11817 for Subcase 63 These reductioncosts are equivalent to improvement level (IL) of 0032908968 82282 14143 047 038 099 001405
Mathematical Problems in Engineering 21
0 5 10 15 20 25 30 35 40 45 50Run
2428
243
2432
2434
2436
2438
244
2442
Fuel
cost
($)
CCSAICSA
times105
Figure 9 The best fuel cost of 50 runs obtained by CCSA and ICSA for Subcase 35
0 50 100 150 200 250 300 350 400Iteration
327
328
329
33
331
332
333
334
335
336
337
Fitn
ess F
unct
ion
($)
CCSAICSA
times104
Figure 10 The best run obtained by CCSA and ICSA for Case 5
00003 0002 12291 004 138 and 059 Thesequantitative comparisons reveal that larger scale systems canlead to better reduction cost but the improvement level isnot high because total cost of compared methods tends tobe large for large scale systems Furthermore very large scalesystems with nondifferentiable objective have been normallysolved by strong methods In fact systems in Case 3 havethe same characteristic with single fuel and effects of valveloading process but Subcase 34 is a larger scale system with
40 units while Subcases 31 32 and 33 are constructed by 3units 13 units and 13 units So the improvement percentageof Subcase 34 can be up to 099 whilst that of Subcases32 and 33 is 047 and 038 respectively Subcase 35 iswith the largest system 80 units but the reduction cost is notmuch only $3412 because compared methods with ICSA areeither state-of-the-art ones or improved ones Also Subcase41 and Subcase 42 have considered single fuel and PPZ andspinning reserve constraints but Subcase 42 is larger scale
22 Mathematical Problems in Engineering
Table 15 Performance improvement summary of the proposed method
Study cases Reduction cost ($) Improvement level () Execution time (s)From To From To Slowest method Fastest method Proposed method
Figure 11 The best fuel cost of 50 runs obtained by CCSA and ICSA for Case 5
system with 90 units and Subcase 41 is only with 60 unitsThus the improvement of Subcase 41 is lower with 00003but that of Subcase 42 is 0002 Similarly systems in Cases 6have the same featurewithmulti-fossil fuel sources and effectsof valve loading process but they are respectively constructedby 80 160 and 320 units As a result the improvements ofSubcase 62 138 and Subcase 63 059 are much higherthan Subcase 61 004 However Subcase 62 with smallernumber of units but getting higher improvement is easilyunderstood because there were nine compared methods butonly three compared methods are considered for Subcase 63In general the improvement is not high it is about under onedollar several dollars tens of dollars and over one thousand
dollars per hour however the saving cost will be significantif the operation is considered to be onemonth with 720 hoursor one year with 8760 hours
Execution time comparison can be evaluated by observ-ing the fastest and the slowest compared methods in Table 15These execution times of the proposed method are approxi-mately equal to that of the fastest methods for Subcases 3233 and 34 and much shorter than other fastest comparedmethods for other cases especially for Subcases 61 and 63The fastest method for Subcase 62 is CBA [44] showing 57seconds while that of the proposed method is 1119 secondsHowever it cannot conclude that CBA is more effective thanthe proposed method because the proposed method could
find better optimal solution with less fuel cost by $2130As considering execution time of the slowest comparedmethods it can point out that ICSA is a very fast optimizationtool since the execution time of these methods is 29496 and3425 seconds for Subcases 32 and 33 116735 seconds forSubcase 34 56381 seconds for Subcase 41 94093 secondsfor Subcase 42 12797 seconds for Case 5 30941 seconds forSubcase 61 7542 seconds for Subcase 62 and 9641 secondsfor Subcase 63 while the execution time of the proposedmethod for these cases is respectively 095 146 091 1589246 84828 1119 and 171384 seconds It is clearly shown thatICSA is very fast as compared to these methods
In summary the proposed method has found approx-imately high quality solutions with several standard state-of-the-art meta-heuristic algorithms and improved versionsof them together with other old methods In addition theproposed method could improve result better than approx-imately all methods with faster execution time Comparedto other methods with the fastest convergence speed andhigh quality solutions the proposed method has been as
fast as for some cases and much faster for other cases Thecomparison with the slowest methods could show that theproposed method was extremely powerful since it was up tonearly one thousand times faster Consequently the proposedICSA approach can be one of the strongest optimization toolsfor OLD problem
6 Conclusions
This paper has proposed a good ICSA method for solvingOLD problem in which many test systems with differentobjective functions and complicated constraints from simpleto complex have been used as studied cases The proposedICSA method has been developed by performing severalmodifications on the second solution update of CCSA whichcontained several drawbacks to global convergence and fastmanner The OLD problem has covered from single fuelto multi-fossil fuels from quadratic objective function tononconvex objective function in addition to PPZ RPR andRRL constraints Many existing optimization algorithms have
been concerned in aim to compare the performance and givethe final conclusion on the proposed method There have
been six main cases with sixteen subcases The evaluationshave been made at the end of each study case Clearly theproposed ICSA approach has yielded more effective optimalsolutions with faster execution time than almost all methodsConsequently it can be concluded that the proposed methodis much more superior to CCSA and is a very promisingmethod for solving OLD problem
Appendix
See Tables 16ndash26
Nomenclature
120575119894 120582119894 120572119894 120573119894 120574119894 Fuel cost functioncoefficients of the 119894119905ℎthermal generationunit11986100 1198610119895 119861119895119894 Power loss matrixcoefficients119865119865119909 119865119865119887119890119904119905 The values of fitness ofsolution 119909 and theso-far most effectivesolution among thecurrent set ofsolutions119898119894 Number of fuels burntin the 119894119905ℎ thermalgeneration unit
119873 Number of allavailable thermalgeneration units119899119894 Number of prohibitedpower zones of the 119894119905ℎthermal generationunit119875119906119894119896 119875119897119894119896 Upper and lowerlimits of the 119894119905ℎthermal generationunit corresponding tothe kth PPZ119875119886 Probability ofreplacing controlvariables in each oldsolution119875119863 Real power demand ofall loads in system119875119894119898119886119909 119875119894119898119894119899 The highest andlowest real poweroutputs of the iththermal generationunit119875119894119895119898119886119909 119875119894119895119898119894119899 The highest andlowest real poweroutputs of the 119894119905ℎthermal generationunit corresponding tothe 119895119905ℎ fuel type119878119894119898119886119909 Maximum real powerreserve contributionof the thermalgeneration unit 1198941198781199001198971199031198861198991198891 1198781199001198971199031198861198991198892 1198781199001198971199031198861198991198893 1198781199001198971199031198861198991198894 Randomly mixedsolutions from the setof current solutions119878119900119897119909 119878119900119897119866119887119890119904119905 The old solution x andthe most effectivesolution119878119877 Real power reserverequirement of system
120575119894119895 120582119894119895 120572119894119895 120573119894119895 120574119894119895 Fuel cost functioncoefficients of the 119894119905ℎthermal generationunit corresponding tothe 119895119905ℎ fuel type1205761 1205762 1205763 1205764 1205765 1205766 Random numbersbetween 0 and 1Ω Number of generationunits considering PPZconstraint
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare no conflicts of interest
References
[1] N Sinha R Chakrabarti and P K Chattopadhyay ldquoEvolution-ary programming techniques for economic load dispatchrdquo IEEETransactions on Evolutionary Computation vol 7 no 1 pp 83ndash94 2003
[2] P Hansen ldquoA separable approximation dynamic programmingalgorithm for economic dispatch with transmission lossesrdquoYugoslav Journal of Operations Research vol 12 no 2 2002
[3] N T Thang ldquoSolving economic dispatch problem with piece-wise quadratic cost functions using lagrange multiplier theoryrdquoin Proceedings of the 3rd International Conference on ComputerTechnology and Development (ICCTD rsquo11) pp 359ndash364 ASMEPress 2011
[4] S K Mishra and S K Mishra ldquoA comparative study of solutionof economic load dispatch problem in power systems in theenvironmental perspectiverdquoProcedia Computer Science vol 48pp 96ndash100 2015
[5] A A Al-Subhi and H K Alfares ldquoEconomic load dispatchusing linear programming a comparative studyrdquo InternationalJournal of Applied Industrial Engineering vol 3 no 1 pp 16ndash362016
28 Mathematical Problems in Engineering
[6] C Zhou G Huang and J Chen ldquoPlanning of electric powersystems considering virtual power plants with dispatchableloads included an inexact two-stage stochastic linear program-ming modelrdquoMathematical Problems in Engineering vol 2018Article ID 7049329 12 pages 2018
[7] T T Nguyen N V Quynh and L Van Dai ldquoImproved fireflyalgorithm a novel method for optimal operation of thermalgenerating unitsrdquo Complexity vol 2018 Article ID 7267593 23pages 2018
[8] S S Haroon S Hassan S Amin et al ldquoMultiple fuel machinespower economic dispatch using stud differential evolutionrdquoEnergies vol 11 no 6 pp 1ndash20 2018
[9] C-T Su and C-L Chiang ldquoNonconvex power economic dis-patch by improved genetic algorithm with multiplier updatingmethodrdquo Electric Power Components and Systems vol 32 no 3pp 257ndash273 2004
[10] NNoman andH Iba ldquoDifferential evolution for economic loaddispatch problemsrdquo Electric Power Systems Research vol 78 no8 pp 1322ndash1331 2008
[11] P Somasundaram and K Kuppusamy ldquoApplication of evolu-tionary programming to security constrained economic dis-patchrdquo International Journal of Electrical Power amp EnergySystems vol 27 no 5-6 pp 343ndash351 2005
[12] T Jayabarathi K Jayaprakash andD Jeyakumar ldquoEvolutionaryprogramming technique for different kinds of economic dis-patch problemsrdquo Electric Power Systems Research vol 73 no 2pp 169ndash176 2005
[13] P Subbaraj R Rengaraj and S Salivahanan ldquoEnhancementof combined heat and power economic dispatch using selfadaptive real-coded genetic algorithmrdquo Applied Energy vol 86no 6 pp 915ndash921 2009
[14] C-L Chiang and C-T Su ldquoAdaptive-improved genetic algo-rithm for the economic dispatch of units with multiple fueloptionsrdquo Cybernetics and Systems vol 36 no 7 pp 687ndash7042005
[15] C Chiang ldquoImproved genetic algorithm for power economicdispatch of units with valve-point effects and multiple fuelsrdquoIEEE Transactions on Power Systems vol 20 no 4 pp 1690ndash1699 2005
[16] K Meng H G Wang Z Y Dong and K P Wong ldquoQuantum-inspired particle swarm optimization for valve-point economicload dispatchrdquo IEEE Transactions on Power Systems vol 25 no1 pp 215ndash222 2010
[17] P Erdogmus A Ozturk and S Duman ldquoEnvironmen-taleconomic dispatch using genetic algorithm and simulatedannealingrdquo in Proceedings of the International Conference onElectric Power and Energy Conversion Systems EPECS rsquo09 pp1ndash4 IEEE 2009
[18] L D S Coelho and V C Mariani ldquoImproved differentialevolution algorithms for handling economic dispatch opti-mization with generator constraintsrdquo Energy Conversion andManagement vol 48 no 5 pp 1631ndash1639 2007
[19] H Dakuo W Fuli and M Zhizhong ldquoA hybrid genetic algo-rithm approach based on differential evolution for economicdispatch with valve-point effectsrdquo Electrical Power and EnergySystems vol 30 pp 31ndash38 2008
[20] C-F Chang J-J Wong J-P Chiou and C-T Su ldquoRobustsearching hybrid differential evolution method for optimalreactive power planning in large-scale distribution systemsrdquoElectric Power Systems Research vol 77 no 5-6 pp 430ndash4372007
[21] M R Farooqi P Jain and K R Niazi ldquoUsing Hopfield neuralnetwork for economic dispatch of power systemsrdquo in Proceed-ings of the National Power Engineering Conference PECon rsquo03pp 5ndash10 Bangi Malaysia 2003
[22] T Yalcinoz H Altun and U Hasan ldquoConstrained economicdispatch with prohibited operating zones a Hopfield neuralnetwork approachrdquo in Proceedings of the 10th MediterraneanElectrotechnical Conference Information Technology and Elec-trotechnology for the Mediterranean Countries ProceedingsMeleCon rsquo00 (Cat No 00CH37099) pp 570ndash573 LemesosCyprus 2000
[23] D N Vo P Schegner and W Ongsakul ldquoCuckoo searchalgorithm for non-convex economic dispatchrdquo IET GenerationTransmission amp Distribution vol 7 no 6 pp 645ndash654 2013
[24] J-B Park K-S Lee J-R Shin and K Y Lee ldquoA particleswarm optimization for economic dispatch with non-smoothcost functionsrdquo IEEETransactions on Power Systems vol 20 no1 pp 34ndash42 2005
[25] T Niknam H D Mojarrad and M Nayeripour ldquoA newfuzzy adaptive particle swarm optimization for non-smootheconomic dispatchrdquo Energy vol 35 no 4 pp 1764ndash1778 2010
[26] L D S Coelho and V C Mariani ldquoAn improved harmonysearch algorithm for power economic load dispatchrdquo EnergyConversion and Management vol 50 no 10 pp 2522ndash25262009
[27] M Mahdavi M Fesanghary and E Damangir ldquoAn improvedharmony search algorithm for solving optimization problemsrdquoAppliedMathematics and Computation vol 188 no 2 pp 1567ndash1579 2007
[28] A Bhattacharya and P K Chattopadhyay ldquoSolving complexeconomic load dispatch problem using biogeography-basedoptimizationrdquo Expert Systems with Applications vol 37 no 5pp 3605ndash3615 2010
[29] D Simon ldquoBiogeography-based optimizationrdquo IEEE Transac-tions on Evolutionary Computation vol 12 no 6 pp 702ndash7132008
[30] J S Alsumait J K Sykulski and A K Al-Othman ldquoAhybrid GA-PS-SQP method to solve power system valve-pointeconomic dispatch problemsrdquo Applied Energy vol 87 no 5 pp1773ndash1781 2010
[31] M S Turgut andGKDemir ldquoQuadratic approximationndashbasedhybrid Artificial Cooperative Search algorithm for economicemission load dispatchproblemsrdquo International Transactions onElectrical Energy Systems vol 27 no 4 pp 1ndash14 2017
[32] T A A Victoire and A E Jeyakumar ldquoHybrid PSO-SQPfor economic dispatch with valve-point effectrdquo Electric PowerSystems Research vol 71 no 1 pp 51ndash59 2004
[33] D-KHe F-LWang andZ-ZMao ldquoHybrid genetic algorithmfor economic dispatch with valve-point effectrdquo Electric PowerSystems Research vol 78 no 4 pp 626ndash633 2008
[34] D C Secui ldquoA modified symbiotic organisms search algorithmfor large scale economic dispatch problem with valve-pointloading effectsrdquo Energy vol 113 pp 366ndash384 2016
[35] K BhattacharjeeA Bhattacharya and S H N Dey ldquoTeaching-learning-based optimization for different economic dispatchproblemsrdquo Scientia Iranica vol 21 no 3 pp 870ndash884 2014
[36] K Bhattacharjee A Bhattacharya and S H N Dey ldquoOppo-sitional Real Coded Chemical Reaction Optimization for dif-ferent economic dispatch problemsrdquo International Journal ofElectrical Power amp Energy Systems vol 55 pp 378ndash391 2014
Mathematical Problems in Engineering 29
[37] V Hosseinnezhad M Rafiee M Ahmadian and M T AmelildquoSpecies-basedQuantum Particle SwarmOptimization for eco-nomic load dispatchrdquo International Journal of Electrical Poweramp Energy Systems vol 63 pp 311ndash322 2014
[38] M S P Subathra S E Selvan T A A Victoire A HChristinal andU Amato ldquoA hybrid with cross-entropymethodand sequential quadratic programming to solve economic loaddispatch problemrdquo IEEE Systems Journal vol 9 no 3 pp 1031ndash1044 2015
[39] J Zhan Q H Wu C Guo and X Zhou ldquoEconomic dispatchwith non-smooth objectives part I local minimum analysisrdquoIEEE Transactions on Power Systems vol 30 no 2 pp 710ndash7212015
[40] A K Barisal andR C Prusty ldquoLarge scale economic dispatch ofpower systems using oppositional invasive weed optimizationrdquoApplied Soft Computing vol 29 pp 122ndash137 2015
[41] M Basu ldquoImproved differential evolution for economic dis-patchrdquo International Journal of Electrical Power amp EnergySystems vol 63 pp 855ndash861 2014
[42] V S Aragon S C Esquivel and C C Coello ldquoAn immunealgorithm with power redistribution for solving economicdispatch problemsrdquo Information Sciences vol 295 pp 609ndash6322015
[43] M Ghasemi M Taghizadeh S Ghavidel and A AbbasianldquoColonial competitive differential evolution an experimentalstudy for optimal economic load dispatchrdquo Applied Soft Com-puting vol 40 pp 342ndash363 2016
[44] B R Adarsh T Raghunathan T Jayabarathi and X-S YangldquoEconomic dispatch using chaotic bat algorithmrdquo Energy vol96 pp 666ndash675 2016
[45] N Ghorbani and E Babaei ldquoExchange market algorithm foreconomic load dispatchrdquo International Journal of ElectricalPower amp Energy Systems vol 75 pp 19ndash27 2016
[46] J X V Neto G Reynoso-Meza T H Ruppel V C Marianiand L D S Coelho ldquoSolving non-smooth economic dispatchby a new combination of continuous GRASP algorithm anddifferential evolutionrdquo International Journal of Electrical Poweramp Energy Systems vol 84 pp 13ndash24 2017
[47] A Kavousi-Fard and A Khosravi ldquoAn intelligent 120579-ModifiedBat Algorithm to solve the non-convex economic dispatchproblem considering practical constraintsrdquoElectrical Power andEnergy Systems vol 82 pp 189ndash196 2016
[48] M A Al-Betar M A Awadallah A T Khader and A L BolajildquoTournament-based harmony search algorithm for non-convexeconomic load dispatch problemrdquo Applied Soft Computing vol47 pp 449ndash459 2016
[49] M A Al-Betar M A Awadallah I A Doush E Alsukhni andH ALkhraisat ldquoA non-convex economic dispatchproblemwithvalve loading effect using a new modified 120573-hill climbing localsearch algorithmrdquo Arabian Journal for Science and Engineeringpp 1ndash8 2018
[50] M A Awadallah M A Al-Betar A L Bolaji E M Alsukhniand H Al-Zoubi ldquoNatural selection methods for artificial beecolony with new versions of onlooker beerdquo Soft Computing pp1ndash40 2018
[51] S H Kaboli and A K Alqallaf ldquoSolving non-convex economicload dispatch problem via artificial cooperative search algo-rithmrdquo Expert Systems with Applications vol 128 pp 14ndash272019
[52] D Singh and J Dhillon ldquoAmeliorated greywolf optimization foreconomic load dispatch problemrdquo Energy vol 169 pp 398ndash4192019
[53] X S Yang and S Deb ldquoCuckoo search via Levy flightsrdquo inProceedings of the World Congress on Nature and BiologicallyInspired Computing (NaBIC rsquo09) pp 210ndash214 India 2009
[54] S Sahoo K M Dash R C Prusty and A K Barisal ldquoCom-parative analysis of optimal load dispatch through evolutionaryalgorithmsrdquo Ain Shams Engineering Journal vol 6 pp 107ndash1202015
[55] M Basu and A Chowdhury ldquoCuckoo search algorithm foreconomic dispatchrdquo Energy vol 60 pp 99ndash108 2013
[56] A B Serapiao ldquoCuckoo search for solving economic dispatchload problemrdquo Intelligent Control and Automation vol 04 no04 pp 385ndash390 2013
[57] A H Bindu and M D Reddy ldquoEconomic load dispatch usingcuckoo search algorithmrdquo International Journal of EngineeringResearch and Applications vol 3 pp 498ndash502 2013
[58] E Afzalan and M Joorabian ldquoAn improved cuckoo searchalgorithm for power economic load dispatchrdquo InternationalTransactions on Electrical Energy Systems vol 25 no 6 pp 958ndash975 2015
[59] T T Nguyen and D N Vo ldquoThe application of one rank cuckoosearch algorithm for solving economic load dispatch problemsrdquoApplied Soft Computing vol 37 pp 763ndash773 2015
[60] S M Islam S Das S Ghosh S Roy and P N Suganthan ldquoAnadaptive differential evolution algorithm with novel mutationand crossover strategies for global numerical optimizationrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 42 no 2 pp 482ndash499 2012
[61] Z L Gaing ldquoParticle swarm optimization to solving theeconomic dispatch considering the generator constraintsrdquo IEEETransactions on Power Systems vol 18 no 3 pp 1187ndash1195 2003
[62] X-S Yang and S Deb ldquoEngineering optimisation by Cuckoosearchrdquo International Journal of Mathematical Modelling andNumerical Optimisation vol 330 pp 43ndash51 2010
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Mathematical Problems in Engineering 9
Produce initial population with119873119901119904 solutions (1198781199001198971 1198781199001198972 119878119900119897119909 119878119900119897119873119901119904)Calculate fitness function (1198651198651 1198651198652 119865119865119909 119865119865119873119901)Go to the loop algorithm by setting 119866 = 1
While (119866119898119886119909 gt 119866) (i) The first newly produced solutions119878119900119897119899119890119908119909 = 119878119900119897119909 + 120572(119878119900119897119909 minus 119878119900119897119866119887119890119904119905) oplus Levy(120573) (ii) Perform selection approach
119865119865119909 = 119865119865119909 if 119865119865119909 le 119865119865119899119890119908119909119865119865119899119890119908119909 119900119905ℎ119890119903119908119894119904119890 119909 = 1 119873119901119904
(v) Determine the most effective solution and its fitnessDetermine 119865119865119909 with the smallest value and assign 119878119900119897119909 to 119878119900119897119866119887119890119904119905If 119866119898119886119909 gt 119866 perform step (i) and increase 119866 to 119866 + 1 Otherwise stop the loop algorithm and report boththe smallest fitness together with 119878119900119897119866119887119890119904119905End while
Among the four Equations (25) and (26) are used firstin order to redefine upper bound and lower bound for allthermal generating units as considering RRL constraint Thethe redefined bounds continue to be redefined for the secondtime by using (27) and (28) as considering PPZ constraints
Step 2 (randomly produce initial population) For dealingwith the power balance constraint all available units areseparated into two groups in which the first group withdecision variables consists of the power output from thesecond unit to the last unit (P2 P3 PN) meanwhile onlythe power output of the first unit (1198751) belongs to the secondgroup with dependent variable So upper bound solution119878119900119897119898119886119909 and lower bound solution 119878119900119897119898119894119899 must be defined asfollows
Step 3 Handle prohibited power zone constraint for decisionvariables P2 P3 PN
After being randomly produced there is a high possi-bility that decision variables fall into PPZ and they violatePPZ constraint So the verification of falling into PPZ andcorrection of the violation should be accomplished by usingthe following formula
119875119894 =
119875119897119894119896 if 119875119897119894119896 lt 119875119894 le 119875119897119894119896 + 1198751198961198941198962119875119906119894119896 if (119875119894 gt 119875119897119894119896 + 1198751198961198941198962 ) amp (119875119894 lt 119875119906119894119896)119875119894 119890119897119904119890
119894 = 2 119873 amp 119896 = 1 119899119894
(31)
Step 4 Handle RPB constraint by calculating 1198751 and penaliz-ing 1198751 if it violates constraints
In this step power balance constraint is exactly handledby calculating and penalizing dependent variable (1198751) 1198751 isobtained by using formulas (4) and (5) as follows
1198751 = minus (11986101 minus 1 + 2sum119873119894=2 1198611119894119875119894) plusmn radicΔ211986111 (32)
where
Δ = (11986101 minus 1 + 2 119873sum119894=2
1198611119894119875119899)2 minus 411986111(119875119863 minus 119873sum
119894=2
119875119894+ 11986100 + 119873sum
119894=2
1198610119894119875119894 + 119873sum119894=2
119873sum119895=2
119875119894119861119894119895119875119895) amp Δ ge 0(33)
In Equation (32) 1198751 has been determined for the purposeof dealing with real power balance constraint However it isnot sure that 1198751 can satisfy upper bound and lower boundconstraints and prohibited power zone constraints So 1198751must be checked and penalized
Firstly 1198751 is checked and penalized for upper and lowerbound constraints by the following model
Δ1198751x =
0 if 1198751min le 1198751x le 1198751max
1198751min minus 1198751x if 1198751min gt 1198751x1198751x minus 1198751max if 1198751max lt 1198751x
(34)
In Equation (34) if the second case or the third caseoccurs it means P1 has violated either lower bound or upperbound and it would be penalized by using either (P1x= P1min-P1x) or (P1x =P1x -P1max) Otherwise ifP1 has not violatedthe bound constraints (ie the first case in (34) happened)
P1 would continue to be checked for PPZ constraint by thefollowing model
Δ1198751x
=
1198751 minus 1198751198971119896 if 1198751198971119896 lt 1198751 le 1198751198971119896 + 119875119896111989621198751199061119896 minus 1198751 if (1198751 gt 1198751198971119896 + 11987511989611198962 ) amp (1198751 lt 1198751199061119896)0 119890119897119904119890
(35)
Step 5 Handle real power reserve constraint (10)First of all 119878119894 is determined by using (11) and (12) and
then the 119909119905ℎ solution will be checked and penalized if poweroutput of all thermal generating units cannot satisfy RPRconstraint The penalty for violation of the constraint can becalculated by using equation (36)
Δ119878119877119909 =
0 if119873sum119894=1
119878119894119909 ge 119878119877119878119877 minus 119873sum119894=1
119878119894119909 119890119897119904119890 (36)
As a result real power reserve constraint can be solved byusing the penalty method
42 Calculate Fitness Function for Solutions Fitness functionof each solution is used to evaluate quality of solutionNormally the function is the sum of objective function andpenalty of violating constraints and is obtained by
43 The First Newly Updated Solutions by Levy Flights Tech-nique In this section the first newly updated solutionsare performed by employing Levy flights technique usingEquation (14) However each new solution can be out oftheir feasible operating zone such as PPZ and upper andlower limitations When the power output violates its PPZconstraints Equation (31) will be applied to tackle theconstraint Besides the following equation will be employedwhen power output is higher or lower than their limitations
119878119900119897119909 =
119878119900119897max if 119878119900119897max lt 119878119900119897119909119878119900119897min if 119878119900119897min gt 119878119900119897119909119878119900119897119909 Otherwise
119909 = 1 119873119901 (38)
After that Equations (32)-(37) are performed for deter-mining all variables and penalty terms Finally Equation (38)is employed to calculate fitness function
44 The Second Newly Updated Solutions by Using Muta-tion Technique The second newly updated solutions areaccomplished as presented in Section 3 above Similar to
Mathematical Problems in Engineering 11
the task after doing the first update each solution in thenew population must satisfy PPZ constraint and upper andlower boundaries by considering Equations (31) and (38)Then Equations (32)-(37) are performed for determiningall variables and penalty terms Finally Equation (38) isemployed to calculate fitness function and the solution withthe best value is assigned to the best one 11987811990011989711986611988711989011990411990545 Criterion of Stopping the Loop Algorithm In the loopalgorithm of using the proposed ICSA approach the solutionsearch work is stopped in case that the predeterminedmaximum iterations 119866119898119886119909 is reached For each search ter-mination the most effective solution is stored and anotherrun continues to be accomplished until the predeterminednumber of runs is reached After finishing the runs thebest one is found and reported In addition other valuessuch as the fitness of the worst solution and average fitnessof all solutions are also reported for comparing with othermethods
46 The Whole Iterative Process The whole iterative algo-rithm for implementing the proposed ICSA approach forcoping with OLD problem is described in detail in Figure 3
5 Results and Discussions
The proposed ICSA approach performance has been investi-gated on six cases with different fuel options different fuelcharacteristics and complicated constraints The details ofthe studied cases are presented as follows
Case 1 Four systems with single fuel source (SFS) and powerloss (PL) constraint
Subcase 11 A 3-unit system [57]Subcase 12 A 6-unit system [57]Subcase 13 A 3-unit system [56]Subcase 14 A 6-unit system [56]
Case 2 A 110-unit system with SFS [57]
Case 3 Four systems with SFS and the effects of valve loadingprocess (EoVLP)
Subcase 31 A 3-unit system supplying to a load of850MW [58]Subcase 32 A 13-unit system supplying to a load of1800MW [1]Subcase 33 A 13-unit system supplying to a load of2520MW [1]Subcase 34 A 40-unit system supplying to a load of2500MW [1]Subcase 35 An 80-unit system supplying to a load of4100MW [49]
Case 4 Two systems with SFS and PPZ and RPR constraints
Subcase 41 A 60-unit system supplying to a10600MW load [9]
Subcase 42 A 90-unit system supplying to a15900MW load [9]
Case 5 A 15-unit system with SFS and RRL PPZ and PLconstraints [61]
Case 6 Three systems with multiple fuel sources (MFS) andEoVLP
Subcase 61 An 80-unit system supplying to a21600MW load [15]Subcase 62 A 160-unit system supplying to a43200MW load [15]Subcase 63 A 320-unit system supplying to an86400MW load [54]
For each considered case with each load case the pro-posed ICSA approach is run 50 times on the programlanguage of Matlab and a PC with 4 GB of RAM and 24GHzprocessor The selection of adjustment parameters including119875119886 and 119879119900119897119909 is carefully considered to obtain the best optimalsolutions meanwhile two others such as 119873119901119904 and 119866119898119886119909 arechosen corresponding to the scale of particular test system9 values with the change of 01 in the range [01 09] are inturn selected for 119875119886 while 119879119900119897119909 is 001 at the beginning Theinformation including load demand119873119901119904 119866119898119886119909 and the best119875119886 is reported in Table 1
51 Obtained Results on Case 1 considering Four Systems withSFS and PL Constraint In this section we have implementedthe proposed ICSA approach for solving four systems dividedinto four subcases Tables 2 and 3 show the comparisons ofobtained results from Subcases 11 and 12 and Subcases 13and 14 respectively As listed in Table 2 the proposed ICSAmethod and CCSA can find equal fuel cost for Subcases 11whereas the reduction of fuel cost from the proposed ICSAmethod as compared to CCSA is clearer for Subcase 12 Asshown in Table 3 for comparing the proposed ICSA and threemethods consisting of CCSA ABC and FA the minimumfuel cost of the proposed ICSA is approximately equal to thatof these methods for Subcases 13 but much less than that ofthese methods for Subcase 14 Furthermore the proposedICSA has been run by setting 119873119901119904 and 119866119898119886119909 to 5 and 20but these values were much higher for CCSA ABC and FAThey are 20 and 5000 for CCSA 40 and 100 for ABC and 20and 5000 for FA Consequently the proposed method is veryefficient for Case 1 with four subcases
Optimal solutions obtained by ICSA for Case 1 are shownin Tables 16ndash18
52 Obtained Results on Case 2 considering 110-Unit Systemwith SFS In this section we have employed a very largescale system with 110 units but there were not challenges forobjective function and complex constraints since EoVLP andconstraints were not taken into account Both CCSA andthe proposed ICSA methods have been run for comparingwith BBO hybrid BBO and DE (DEBBO) and Opposi-tional real coded chemical reaction optimization algorithm(ORCCROA) in [36] IWA in [40] and AGWO in [52] As
12 Mathematical Problems in Engineering
Select parameters
- Correct solutions if violating PPZ constraint using Eq (31)- Determine using Eq (32)- Penalize for violating upper and lower bounds constraint using Eq (34)- Penalize for violating PPZ constraint using Eq (35)- Penalize the xth solution for violating RPR constraint using Eq (36)
- Determine quality of solutions employing Eq (37) - Select the most effective solution - Start the loop algorithm by selecting
- Perform the first solution update using Section 32 - Correct boundaries of solutions by using Eq (38)
- Correct solutions if violating PPZ constraint using Eq (31)- Determine using Eq (32)- Penalize for violating upper and lower bounds constraint using Eq (34)- Penalize for violating PPZ constraint using Eq (35)- Penalize the xth solution for violating RPR constraint using Eq(36)
- Determine fitness function using Eq (37)- Compare old solutions and new solutions to keep better ones
- Perform the second solution update using Algorithm 1 - Correct boundaries of solutions by using (38)
- Correct solutions if violating PPZ constraint using Eq (31)- Determine using Eq (32)- Penalize for violating upper and lower bounds constraint using Eq (34)- Penalize for violating PPZ constraint using Eq (35)- Penalize the xth solution for violating RPR constraint using Eq (36)
- Determine fitness function using Eq (37)- Compare old solutions and new solutions to keep better ones- Select the most effective solution
Stop
Start
- Redefine upper and lower bounds using Eqs (25)-(28)- Randomly generate initial population using Eq (30)
Nps Pa Gmax H> Tolx
P1xP1x
P1x
P1xP1x
P1x
P1xP1x
P1x
Sol<MN
Sol<MN
G = 1
G = Gmax G = G + 1
Figure 3 All computation steps for solving OLD problem by employing the proposed ICSA approach
shown in Table 4 AGWO [52] has reached less fuel cost thanICSA however the exact fuel cost which was recalculatedby using reported solution pointed out that the method hasreached a very high fuel cost of $2157404250 For comparisonwith other methods ICSA has found less fuel cost thanall these methods Particularly the reduction of generationfuel cost is significant as compared to BBO DEBBO andCCSA Execution time comparisons are also useful evidence
for indicating the high performance of ICSA Thus it canconclude that ICSA is a strong method for Case 2
Optimal solution obtained by ICSA for the case is shownin Table 19
53 Obtained Results on Case 3 considering Four Systems withSFS and EoVLP In this section the real performance of theproposed ICSA approach has been investigated based on five
Mathematical Problems in Engineering 13
Table 1 Information of considered cases and adjustment parameters
Case Fuel cost function Constraint Subcase No of units 119875119863 (MW) Nps Gmax Best 1198751198861 SFS PL
subcases with the gradual increase of number of units Thesmallest scale system considers 3 units but the largest scalesystem takes 80 units In addition to the implementation ofthe proposed ICSA we have also implemented CCSA forSubcase 31 and Subcase 35 for further comparison becauseCCSA has not been run for the two subcases so far
Comparison of obtained results from Subcase 31 shownin Table 5 indicates that the proposed ICSA is superior toCCSAwith lower fuel cost but it seems to be less effective thanMCSA [58] The minimum cost of MCSA reported in [58] isthe smallest fuel cost but the recalculated cost is much higherthan that of the proposed method Furthermore MCSA hasbeen implemented by setting very high values to 119873119901119904 and119866119898119886119909
Reports for Subcases 32 and 33 shown in Table 6are the comparisons of the proposed ICSA approach andother methods such as conventional Evolution programming(CEP) [1] Fast EP (FEP) [1] improved FEP (IFEP) [1] DE[12] multiplier Lagrange-based genetic algorithm with (GA-MU) [15] QPSO [16] GA-PS-SQP [30] PSO-SQP [32] M120573-HCLSA [49] IABCA [50] CCSA [59] OSE-CSA [59] SOS[34] MSOS [34] CEA-SQT [38] TSBO [39] IWA [40] andCBA [44] As observed from the table ICSAapproach obtainsbetter solutions than mostmethods excluding DE [10] CCSA
[59] OSE-CSA [59] SOS [34] MSOS [34] CEA-SQT [38]TSBO [39] IWA [40] and CBA [44] especially M120573-HCLSA[49] with lower cost $1796097 However recalculated costfrom reported solution of M120573-HCLSA is $179691 BesidesICSA is very fast as compared to most methods where twoother versions of Cuckoo search algorithm CCSA and OSE-CSA are also included except two methods in [34] Theprocessor of computer that all the methods run on is alsoreported in the final column Clearly ICSA approach is veryefficient for the case with the 13-unit system where effects ofvalve loading process are considered
In Subcase 34 the number of units is much larger thanthat of three subcases above up to 40 units [1] The obtainedresult comparisons with others are indicated in Table 7Clearly the minimum cost comparisons reveal that the pro-posedmethod is one of the leading methods due to the lowestcost except the comparison with CCSA [23] OSE-CSA [59]SOS [34] MSOS [34] EMA [45] 120579-MBA [47] and AGWOA[52] It is noted that AGWOA [52] has reported the bestminimum cost with $12140430 but recalculated minimumcost which was obtained by substituting reported optimalgeneration of all thermal generating units is $12141331 Theaverage and the maximum costs from the proposed methoddo not belong to the leading method group however the
Mathematical Problems in Engineering 15
Table 7 Result comparisons for Subcase 34
Approach Best cost Average cost Worst cost CPU time Computer($h) ($h) ($h) (s) (Processor-Ram)
execution time that ICSA approach takes is smaller thanmostones except IAPR [42] and 120579-MBA [47] which have been runon stronger computers Note that MSOS [34] has been fasterthan the proposed method for Subcases 31 and 32 abovebut it is too slower than the proposed method for the casenamely 1813 seconds compared to 146 seconds Comparedto two other versions of Cuckoo search CCSA [23] and OSE-CSA [59] the proposed method is also faster about threetimes although the processors are slightly different For thiscase 120579-MBA [47] shows a very good performance howeverthe method has not been tested onmore complicated systemsand larger scale and therefore more comparisons with themethod must end Clearly the proposed method is stillefficient for the case where large scale and effects of valveloading process are included
Subcase 35 is the largest scale system with 80 units Inaddition to the implementation of ICSA CCSA has beenalso run for the Subcase 35 for further investigation ofefficiency improvement of the proposed ICSA approach The
comparisons of minimum cost in Table 8 show that theproposed ICSA can find more optimal solution than M120573-HCLSA [49] AGWOA [52] and CCSA The proposed ICSAis also superior to CCSA in terms ofmore stable search abilityand lower fluctuation since its average cost and maximumcost are less than those of CCSA The outstanding figurecannot be reached as compared to AGWOA [52] howeverit is hard to conclude AGWOA [52] is superior to theproposed ICSA approach about more stable search abilityand lower fluctuation Actually comparison of the values ofpopulation and iterations as well as execution time cannot beaccomplished because the information was not reported in[52]Thus it can conclude that the proposed ICSA is effectivefor the subcase
Optimal solutions obtained by ICSA for the case areshown in Tables 20ndash23
54 Obtained Results on Case 4 with Two Systems consideringSFS and PPZ and RPR Constraints In this section two
16 Mathematical Problems in Engineering
Table 9 Result comparisons for Subcases 41 and 42
Subcase Approach Best cost ($h) Mean cost ($h) Worst cost ($h) Std dev ($h) CPU time (s) Computer (Processor-Ram)
test systems with SFS and PPZ and RPR constraints areconsidered The test system size is up to 60 and 90 units forSubcases 41 and 42 respectively Comparison for the casesis only performed with two Genetic algorithms consisting ofGA and IGA-MU in [9] and two other versions of Cuckoosearch algorithms including CCSA [23] and OSE-CSA [59]and presented in Table 9 Other studies have tended toignore such complicated constraints of PPZ and RPR Thecosts reported in Table 9 indicate that ICSA approach canobtain more effective solution than CCSA and OSE-CSAbecause it has reached lower minimum cost than the twoones Furthermore the proposed method also takes shortercomputation time for the two cases from about two timesto about three times although the processor of the proposedmethod is slightly strongerThemean costs of ICSA approachare much less than those from IGA-MU and GA and slightlyhigher than those from OSE-CSA but there is a trade-off between the proposed method and CCSA for the twosubcases In fact the proposed method obtains higher meancost for Subcase 41 but lower cost for Subcase 42 AlthoughGAmethods have been runon aweak computerwith 07 GHzof the processor compared to that with 24GHz in the studytheir execution times are significantly higher namely 56381seconds (GA) and 16258 seconds (IGA-MU) compared to09153 seconds of ICSA approach for Subcase 41 and 94093seconds (GA) and 25545 seconds (IGA-MU) compared to15892 seconds (the proposedmethod)The analysis can pointout that ICSA approach ismore efficient than these comparedmethods in terms of optimal solutions and execution time
Optimal solutions obtained by ICSA for Subcase 42 areshown in Table 24
55 Obtained Results on Case 5 with a 15-Unit System consid-ering SFS and RRL PPZ and PL Constraints In this section
a 15-unit system considering RRL PPZ and PL constraintsis considered to be solved for finding optimal solution Forefficiency investigation of the proposed ICSA we have alsoimplemented CCSA for comparison As listed in Table 10the proposed ICSA is the most effective method with thesmallest fuel cost The comparisons of control parameters aswell asCPU time are also good evidence to confirm the strongsearch of the proposed ICSA approach since it has been runby smaller values of control parameter and faster executiontime as compared to all methods excluding CCSA
Optimal solution obtained by ICSA for the case is shownin Table 25
56 Obtained Results on Case 6 withThree Systems consideringMFS and EoVLP In this section three test systems with thechallenge on objective function including multi-fossil fuelsources and effects of valve loading process are consideredThe scale is up to 80 units 160 units and 320 units forSubcases 61 62 and 63 respectively
Comparison for Subcase 61 reported in Table 11 revealsthat the proposed method is the best method in terms of thelowest best cost the lowestmean cost and the lowest standarddeviation and the fastest execution time The processor fromthis proposedmethod is about four times stronger thanCGA-MU and IGA-MU but the speed is from ten times to 35 timesfaster than these methods Compared to CCSA and OSE-CSA the proposed method is about two times faster but theprocessor is slightly stronger
Comparison for Subcase 62 is reported in Table 12Clearly the proposed ICSA approach obtains better values ofthe best mean and worst costs than most methods exceptMSOS [34] where the best cost difference is about $ 024However the proposed method is the second fastest onewith 1119 seconds where the first fastest one CBA [44] has
Mathematical Problems in Engineering 17
Table 11 Comparisons of found results for Subcase 61
Approach Best cost Mean cost Worst cost Std dev CPU time Computer($h) ($h) ($h) ($h) (s) (Processor-Ram)
spent 571 seconds Clearly MSOS is better than the proposedICSA approach with respect to slightly less best cost butis worse than the proposed method in terms of executiontime namely 2572 seconds compared to 1119 seconds whileCBA [44] is faster than the proposed method but obtainssignificantly worse costs The analysis can conclude that theproposed ICSA approach is very powerful for the subcasewith 160 units
Table 13 presents the comparison of three other methodsincludingCCSA [54] SOS [34] andMSOS [34] accompaniedwith the proposed method for Subcase 63 The obtainedresult comparisons imply that ICSA approach can obtainbetter values of the best and standard deviation costs thanCCSA and SOS but obtains slightly higher cost than MSOSby approximately $ 022 Besides the execution time fromICSA approach is much shorter than others especially it ishigher than five times faster than MSOS The four methodshave been run on approximately strong computers Brieflythe proposed ICSA approach can find and converge to morefavorable solution than other methods with shorter CPUtime except the comparison with MSOS which had bettersolution but spent higher than five times execution times
Consequently the proposed method is a very promisingoptimization algorithm for Subcase 63 a system up to 320units and with multi-fossil fuel sources and effects of valveloading process
Optimal solution obtained by ICSA for Subcase 63 isshown in Table 26
57 The Improvement of ICSA Approach Performance
571 The Outstanding Improvement over CCSA In this sec-tion the performance improvement of ICSA over CCSAhas been investigated by analyzing obtained results and setcontrol parameters Table 14 has been formed by addingreduction cost improvement level of the best cost executiontime and control parameters consisting of 119873119901119904 and 119866119898119886119909Among the compared factors reduction cost is the deviationof the cost of CCSA and that of ICSA whereas the improve-ment level is the ratio of the reduction cost to the cost ofCCSA The reduction costs indicate the proposed methodcould find either equal quality of solutions or higher qualityof solutions than CCSA for all study casesThe reduction costis from $0 to $42839 corresponding to the improvement level
18 Mathematical Problems in Engineering
Table 14 Summary of results obtained by CCSA and ICSA for all study cases
Study case Reduction cost ($) Improvement level () Execution time (s) Nps Gmax
Figure 4 The best run obtained by CCSA and ICSA for Case 2
that is from 0 to 08968The saving cost is not too much forone hour but the operation in one day onemonth or one yearis very high However it should be noted that CCSA has beenrun by setting much higher population size and iterations formany cases excluding study cases implemented in the studysuch as Case 2 Subcase 31 Subcase 35 and Case 5 Forinstance ICSA has used 119873119901119904 = 10 and 119866119898119886119909 = 15 for Subcase13 and119873119901119904 = 10 and119866119898119886119909 = 25 for Subcase 14 whereas CCSAhas been run by setting119873119901119904 = 20 and 119866119898119886119909 = 5000 for the twosubcases Similarly CCSA has been run for Subcases 32 33and 34 with much higher number of iterations For the lastsubcase CCSA has been run by setting 119873119901119904 = 320 and 119866119898119886119909
= 1200 but those of ICSA have been 10 and 9000 Due tothe higher value of control parameters CCSA has tended tospend more time in finding such high quality solutions foralmost all study cases Execution time of ICSA is less than18 seconds while that of CCSA is up to higher 75 secondsIt is clear that the proposed ICSA could find better optimalsolutions thanCCSA for such considered systems For furtherinvestigation of performance comparison the best runs over50 runs and fuel cost values of 50 runs obtained by CCSA andthe proposed ICSA for Case 2 Subcase 31 Subcase 35 andCase 5 have been plotted in from Figures 4ndash11 The best runcurves show the faster search of the proposed ICSA method
Mathematical Problems in Engineering 19
0 5 10 15 20 25 30 35 40 45 50Run
19795
198
19805
1981
19815
1982
19825
1983
19835
Fuel
cost
($)
CCSAICSA
times105
Figure 5The best fuel cost of 50 runs obtained by CCSA and ICSA for Case 2
0 2 4 6 8 10 12 14 16 18 20Iteration
8234
8236
8238
8240
8242
8244
8246
8248
8250
8252
Fitn
ess F
unct
ion
($)
CCSAICSA
Figure 6 The best run obtained by CCSA and ICSA for Subcase 31
whereas 50 values of fuel cost indicate that the proposedICSA can find many solutions with better quality Clearlythe proposed ICSA is outstanding in terms of stabilizationof solution search and faster convergence As a result it canconclude that the proposed ICSA approach is more effectivethanCCSA in solvingOLDproblemwith considered systems
572The Improvement of Results over Other Methods In thisarticle we have tested ICSA approach on 6 cases with 16
systems with different fuel cost forms different constraintsand different scale systems from 3 units to 320 units We havecompared the yielded results from ICSA approach and otherexisting ones for evaluating the efficiency of ICSA approachIn subsections above we have shown yielded results fromICSA approach and compared these results to those of otheronesHowever the demonstrationhad not been very good forobserving and comparing to lead to a conclusionThus in thesubsectionwehave summarized the result comparisons of the
20 Mathematical Problems in Engineering
0 5 10 15 20 25 30 35 40 45 50Run
8234
82342
82344
82346
82348
8235
82352
82354
82356
82358
8236
Fuel
cost
($)
CCSAICSA
Figure 7 The best fuel cost of 50 runs obtained by CCSA and ICSA for Subcase 31
0 1000 2000 3000 4000 5000 6000Iteration
242
244
246
248
25
252
254
256
Fitn
ess F
unct
ion
($)
CCSAICSA
times105
Figure 8 The best run obtained by CCSA and ICSA for Subcase 35
proposed and other ones Table 15 has reported the reductioncost (in $) of ICSA approach compared to other ones Inaddition we have converted the reduction cost into improve-ment level (in ) for better comparison The improvementhas been shown from the lowest level to the highest levelin terms of reduction cost and improvement percentageIn addition we have also given the slowest and the fastestexecution time of other compared methods together withthat of the proposed method The table implies that ICSA
approach can find better optimal solutions with less fuel costup to $052 for Subcase 13 $75229 for Subcase 14 $1775149for Case 2 $1181256 for Subcase 31 $8438 for Subcase 32$9113 for Subcase 33 $121181 for Subcase 34 $3412 forSubcase 35 $04329 for Subcase 41 $30227 for Subcase42 $407002 for Case 5 $18183 for Subcase 61 $13965 forSubcase 62 and $11817 for Subcase 63 These reductioncosts are equivalent to improvement level (IL) of 0032908968 82282 14143 047 038 099 001405
Mathematical Problems in Engineering 21
0 5 10 15 20 25 30 35 40 45 50Run
2428
243
2432
2434
2436
2438
244
2442
Fuel
cost
($)
CCSAICSA
times105
Figure 9 The best fuel cost of 50 runs obtained by CCSA and ICSA for Subcase 35
0 50 100 150 200 250 300 350 400Iteration
327
328
329
33
331
332
333
334
335
336
337
Fitn
ess F
unct
ion
($)
CCSAICSA
times104
Figure 10 The best run obtained by CCSA and ICSA for Case 5
00003 0002 12291 004 138 and 059 Thesequantitative comparisons reveal that larger scale systems canlead to better reduction cost but the improvement level isnot high because total cost of compared methods tends tobe large for large scale systems Furthermore very large scalesystems with nondifferentiable objective have been normallysolved by strong methods In fact systems in Case 3 havethe same characteristic with single fuel and effects of valveloading process but Subcase 34 is a larger scale system with
40 units while Subcases 31 32 and 33 are constructed by 3units 13 units and 13 units So the improvement percentageof Subcase 34 can be up to 099 whilst that of Subcases32 and 33 is 047 and 038 respectively Subcase 35 iswith the largest system 80 units but the reduction cost is notmuch only $3412 because compared methods with ICSA areeither state-of-the-art ones or improved ones Also Subcase41 and Subcase 42 have considered single fuel and PPZ andspinning reserve constraints but Subcase 42 is larger scale
22 Mathematical Problems in Engineering
Table 15 Performance improvement summary of the proposed method
Study cases Reduction cost ($) Improvement level () Execution time (s)From To From To Slowest method Fastest method Proposed method
Figure 11 The best fuel cost of 50 runs obtained by CCSA and ICSA for Case 5
system with 90 units and Subcase 41 is only with 60 unitsThus the improvement of Subcase 41 is lower with 00003but that of Subcase 42 is 0002 Similarly systems in Cases 6have the same featurewithmulti-fossil fuel sources and effectsof valve loading process but they are respectively constructedby 80 160 and 320 units As a result the improvements ofSubcase 62 138 and Subcase 63 059 are much higherthan Subcase 61 004 However Subcase 62 with smallernumber of units but getting higher improvement is easilyunderstood because there were nine compared methods butonly three compared methods are considered for Subcase 63In general the improvement is not high it is about under onedollar several dollars tens of dollars and over one thousand
dollars per hour however the saving cost will be significantif the operation is considered to be onemonth with 720 hoursor one year with 8760 hours
Execution time comparison can be evaluated by observ-ing the fastest and the slowest compared methods in Table 15These execution times of the proposed method are approxi-mately equal to that of the fastest methods for Subcases 3233 and 34 and much shorter than other fastest comparedmethods for other cases especially for Subcases 61 and 63The fastest method for Subcase 62 is CBA [44] showing 57seconds while that of the proposed method is 1119 secondsHowever it cannot conclude that CBA is more effective thanthe proposed method because the proposed method could
find better optimal solution with less fuel cost by $2130As considering execution time of the slowest comparedmethods it can point out that ICSA is a very fast optimizationtool since the execution time of these methods is 29496 and3425 seconds for Subcases 32 and 33 116735 seconds forSubcase 34 56381 seconds for Subcase 41 94093 secondsfor Subcase 42 12797 seconds for Case 5 30941 seconds forSubcase 61 7542 seconds for Subcase 62 and 9641 secondsfor Subcase 63 while the execution time of the proposedmethod for these cases is respectively 095 146 091 1589246 84828 1119 and 171384 seconds It is clearly shown thatICSA is very fast as compared to these methods
In summary the proposed method has found approx-imately high quality solutions with several standard state-of-the-art meta-heuristic algorithms and improved versionsof them together with other old methods In addition theproposed method could improve result better than approx-imately all methods with faster execution time Comparedto other methods with the fastest convergence speed andhigh quality solutions the proposed method has been as
fast as for some cases and much faster for other cases Thecomparison with the slowest methods could show that theproposed method was extremely powerful since it was up tonearly one thousand times faster Consequently the proposedICSA approach can be one of the strongest optimization toolsfor OLD problem
6 Conclusions
This paper has proposed a good ICSA method for solvingOLD problem in which many test systems with differentobjective functions and complicated constraints from simpleto complex have been used as studied cases The proposedICSA method has been developed by performing severalmodifications on the second solution update of CCSA whichcontained several drawbacks to global convergence and fastmanner The OLD problem has covered from single fuelto multi-fossil fuels from quadratic objective function tononconvex objective function in addition to PPZ RPR andRRL constraints Many existing optimization algorithms have
been concerned in aim to compare the performance and givethe final conclusion on the proposed method There have
been six main cases with sixteen subcases The evaluationshave been made at the end of each study case Clearly theproposed ICSA approach has yielded more effective optimalsolutions with faster execution time than almost all methodsConsequently it can be concluded that the proposed methodis much more superior to CCSA and is a very promisingmethod for solving OLD problem
Appendix
See Tables 16ndash26
Nomenclature
120575119894 120582119894 120572119894 120573119894 120574119894 Fuel cost functioncoefficients of the 119894119905ℎthermal generationunit11986100 1198610119895 119861119895119894 Power loss matrixcoefficients119865119865119909 119865119865119887119890119904119905 The values of fitness ofsolution 119909 and theso-far most effectivesolution among thecurrent set ofsolutions119898119894 Number of fuels burntin the 119894119905ℎ thermalgeneration unit
119873 Number of allavailable thermalgeneration units119899119894 Number of prohibitedpower zones of the 119894119905ℎthermal generationunit119875119906119894119896 119875119897119894119896 Upper and lowerlimits of the 119894119905ℎthermal generationunit corresponding tothe kth PPZ119875119886 Probability ofreplacing controlvariables in each oldsolution119875119863 Real power demand ofall loads in system119875119894119898119886119909 119875119894119898119894119899 The highest andlowest real poweroutputs of the iththermal generationunit119875119894119895119898119886119909 119875119894119895119898119894119899 The highest andlowest real poweroutputs of the 119894119905ℎthermal generationunit corresponding tothe 119895119905ℎ fuel type119878119894119898119886119909 Maximum real powerreserve contributionof the thermalgeneration unit 1198941198781199001198971199031198861198991198891 1198781199001198971199031198861198991198892 1198781199001198971199031198861198991198893 1198781199001198971199031198861198991198894 Randomly mixedsolutions from the setof current solutions119878119900119897119909 119878119900119897119866119887119890119904119905 The old solution x andthe most effectivesolution119878119877 Real power reserverequirement of system
120575119894119895 120582119894119895 120572119894119895 120573119894119895 120574119894119895 Fuel cost functioncoefficients of the 119894119905ℎthermal generationunit corresponding tothe 119895119905ℎ fuel type1205761 1205762 1205763 1205764 1205765 1205766 Random numbersbetween 0 and 1Ω Number of generationunits considering PPZconstraint
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare no conflicts of interest
References
[1] N Sinha R Chakrabarti and P K Chattopadhyay ldquoEvolution-ary programming techniques for economic load dispatchrdquo IEEETransactions on Evolutionary Computation vol 7 no 1 pp 83ndash94 2003
[2] P Hansen ldquoA separable approximation dynamic programmingalgorithm for economic dispatch with transmission lossesrdquoYugoslav Journal of Operations Research vol 12 no 2 2002
[3] N T Thang ldquoSolving economic dispatch problem with piece-wise quadratic cost functions using lagrange multiplier theoryrdquoin Proceedings of the 3rd International Conference on ComputerTechnology and Development (ICCTD rsquo11) pp 359ndash364 ASMEPress 2011
[4] S K Mishra and S K Mishra ldquoA comparative study of solutionof economic load dispatch problem in power systems in theenvironmental perspectiverdquoProcedia Computer Science vol 48pp 96ndash100 2015
[5] A A Al-Subhi and H K Alfares ldquoEconomic load dispatchusing linear programming a comparative studyrdquo InternationalJournal of Applied Industrial Engineering vol 3 no 1 pp 16ndash362016
28 Mathematical Problems in Engineering
[6] C Zhou G Huang and J Chen ldquoPlanning of electric powersystems considering virtual power plants with dispatchableloads included an inexact two-stage stochastic linear program-ming modelrdquoMathematical Problems in Engineering vol 2018Article ID 7049329 12 pages 2018
[7] T T Nguyen N V Quynh and L Van Dai ldquoImproved fireflyalgorithm a novel method for optimal operation of thermalgenerating unitsrdquo Complexity vol 2018 Article ID 7267593 23pages 2018
[8] S S Haroon S Hassan S Amin et al ldquoMultiple fuel machinespower economic dispatch using stud differential evolutionrdquoEnergies vol 11 no 6 pp 1ndash20 2018
[9] C-T Su and C-L Chiang ldquoNonconvex power economic dis-patch by improved genetic algorithm with multiplier updatingmethodrdquo Electric Power Components and Systems vol 32 no 3pp 257ndash273 2004
[10] NNoman andH Iba ldquoDifferential evolution for economic loaddispatch problemsrdquo Electric Power Systems Research vol 78 no8 pp 1322ndash1331 2008
[11] P Somasundaram and K Kuppusamy ldquoApplication of evolu-tionary programming to security constrained economic dis-patchrdquo International Journal of Electrical Power amp EnergySystems vol 27 no 5-6 pp 343ndash351 2005
[12] T Jayabarathi K Jayaprakash andD Jeyakumar ldquoEvolutionaryprogramming technique for different kinds of economic dis-patch problemsrdquo Electric Power Systems Research vol 73 no 2pp 169ndash176 2005
[13] P Subbaraj R Rengaraj and S Salivahanan ldquoEnhancementof combined heat and power economic dispatch using selfadaptive real-coded genetic algorithmrdquo Applied Energy vol 86no 6 pp 915ndash921 2009
[14] C-L Chiang and C-T Su ldquoAdaptive-improved genetic algo-rithm for the economic dispatch of units with multiple fueloptionsrdquo Cybernetics and Systems vol 36 no 7 pp 687ndash7042005
[15] C Chiang ldquoImproved genetic algorithm for power economicdispatch of units with valve-point effects and multiple fuelsrdquoIEEE Transactions on Power Systems vol 20 no 4 pp 1690ndash1699 2005
[16] K Meng H G Wang Z Y Dong and K P Wong ldquoQuantum-inspired particle swarm optimization for valve-point economicload dispatchrdquo IEEE Transactions on Power Systems vol 25 no1 pp 215ndash222 2010
[17] P Erdogmus A Ozturk and S Duman ldquoEnvironmen-taleconomic dispatch using genetic algorithm and simulatedannealingrdquo in Proceedings of the International Conference onElectric Power and Energy Conversion Systems EPECS rsquo09 pp1ndash4 IEEE 2009
[18] L D S Coelho and V C Mariani ldquoImproved differentialevolution algorithms for handling economic dispatch opti-mization with generator constraintsrdquo Energy Conversion andManagement vol 48 no 5 pp 1631ndash1639 2007
[19] H Dakuo W Fuli and M Zhizhong ldquoA hybrid genetic algo-rithm approach based on differential evolution for economicdispatch with valve-point effectsrdquo Electrical Power and EnergySystems vol 30 pp 31ndash38 2008
[20] C-F Chang J-J Wong J-P Chiou and C-T Su ldquoRobustsearching hybrid differential evolution method for optimalreactive power planning in large-scale distribution systemsrdquoElectric Power Systems Research vol 77 no 5-6 pp 430ndash4372007
[21] M R Farooqi P Jain and K R Niazi ldquoUsing Hopfield neuralnetwork for economic dispatch of power systemsrdquo in Proceed-ings of the National Power Engineering Conference PECon rsquo03pp 5ndash10 Bangi Malaysia 2003
[22] T Yalcinoz H Altun and U Hasan ldquoConstrained economicdispatch with prohibited operating zones a Hopfield neuralnetwork approachrdquo in Proceedings of the 10th MediterraneanElectrotechnical Conference Information Technology and Elec-trotechnology for the Mediterranean Countries ProceedingsMeleCon rsquo00 (Cat No 00CH37099) pp 570ndash573 LemesosCyprus 2000
[23] D N Vo P Schegner and W Ongsakul ldquoCuckoo searchalgorithm for non-convex economic dispatchrdquo IET GenerationTransmission amp Distribution vol 7 no 6 pp 645ndash654 2013
[24] J-B Park K-S Lee J-R Shin and K Y Lee ldquoA particleswarm optimization for economic dispatch with non-smoothcost functionsrdquo IEEETransactions on Power Systems vol 20 no1 pp 34ndash42 2005
[25] T Niknam H D Mojarrad and M Nayeripour ldquoA newfuzzy adaptive particle swarm optimization for non-smootheconomic dispatchrdquo Energy vol 35 no 4 pp 1764ndash1778 2010
[26] L D S Coelho and V C Mariani ldquoAn improved harmonysearch algorithm for power economic load dispatchrdquo EnergyConversion and Management vol 50 no 10 pp 2522ndash25262009
[27] M Mahdavi M Fesanghary and E Damangir ldquoAn improvedharmony search algorithm for solving optimization problemsrdquoAppliedMathematics and Computation vol 188 no 2 pp 1567ndash1579 2007
[28] A Bhattacharya and P K Chattopadhyay ldquoSolving complexeconomic load dispatch problem using biogeography-basedoptimizationrdquo Expert Systems with Applications vol 37 no 5pp 3605ndash3615 2010
[29] D Simon ldquoBiogeography-based optimizationrdquo IEEE Transac-tions on Evolutionary Computation vol 12 no 6 pp 702ndash7132008
[30] J S Alsumait J K Sykulski and A K Al-Othman ldquoAhybrid GA-PS-SQP method to solve power system valve-pointeconomic dispatch problemsrdquo Applied Energy vol 87 no 5 pp1773ndash1781 2010
[31] M S Turgut andGKDemir ldquoQuadratic approximationndashbasedhybrid Artificial Cooperative Search algorithm for economicemission load dispatchproblemsrdquo International Transactions onElectrical Energy Systems vol 27 no 4 pp 1ndash14 2017
[32] T A A Victoire and A E Jeyakumar ldquoHybrid PSO-SQPfor economic dispatch with valve-point effectrdquo Electric PowerSystems Research vol 71 no 1 pp 51ndash59 2004
[33] D-KHe F-LWang andZ-ZMao ldquoHybrid genetic algorithmfor economic dispatch with valve-point effectrdquo Electric PowerSystems Research vol 78 no 4 pp 626ndash633 2008
[34] D C Secui ldquoA modified symbiotic organisms search algorithmfor large scale economic dispatch problem with valve-pointloading effectsrdquo Energy vol 113 pp 366ndash384 2016
[35] K BhattacharjeeA Bhattacharya and S H N Dey ldquoTeaching-learning-based optimization for different economic dispatchproblemsrdquo Scientia Iranica vol 21 no 3 pp 870ndash884 2014
[36] K Bhattacharjee A Bhattacharya and S H N Dey ldquoOppo-sitional Real Coded Chemical Reaction Optimization for dif-ferent economic dispatch problemsrdquo International Journal ofElectrical Power amp Energy Systems vol 55 pp 378ndash391 2014
Mathematical Problems in Engineering 29
[37] V Hosseinnezhad M Rafiee M Ahmadian and M T AmelildquoSpecies-basedQuantum Particle SwarmOptimization for eco-nomic load dispatchrdquo International Journal of Electrical Poweramp Energy Systems vol 63 pp 311ndash322 2014
[38] M S P Subathra S E Selvan T A A Victoire A HChristinal andU Amato ldquoA hybrid with cross-entropymethodand sequential quadratic programming to solve economic loaddispatch problemrdquo IEEE Systems Journal vol 9 no 3 pp 1031ndash1044 2015
[39] J Zhan Q H Wu C Guo and X Zhou ldquoEconomic dispatchwith non-smooth objectives part I local minimum analysisrdquoIEEE Transactions on Power Systems vol 30 no 2 pp 710ndash7212015
[40] A K Barisal andR C Prusty ldquoLarge scale economic dispatch ofpower systems using oppositional invasive weed optimizationrdquoApplied Soft Computing vol 29 pp 122ndash137 2015
[41] M Basu ldquoImproved differential evolution for economic dis-patchrdquo International Journal of Electrical Power amp EnergySystems vol 63 pp 855ndash861 2014
[42] V S Aragon S C Esquivel and C C Coello ldquoAn immunealgorithm with power redistribution for solving economicdispatch problemsrdquo Information Sciences vol 295 pp 609ndash6322015
[43] M Ghasemi M Taghizadeh S Ghavidel and A AbbasianldquoColonial competitive differential evolution an experimentalstudy for optimal economic load dispatchrdquo Applied Soft Com-puting vol 40 pp 342ndash363 2016
[44] B R Adarsh T Raghunathan T Jayabarathi and X-S YangldquoEconomic dispatch using chaotic bat algorithmrdquo Energy vol96 pp 666ndash675 2016
[45] N Ghorbani and E Babaei ldquoExchange market algorithm foreconomic load dispatchrdquo International Journal of ElectricalPower amp Energy Systems vol 75 pp 19ndash27 2016
[46] J X V Neto G Reynoso-Meza T H Ruppel V C Marianiand L D S Coelho ldquoSolving non-smooth economic dispatchby a new combination of continuous GRASP algorithm anddifferential evolutionrdquo International Journal of Electrical Poweramp Energy Systems vol 84 pp 13ndash24 2017
[47] A Kavousi-Fard and A Khosravi ldquoAn intelligent 120579-ModifiedBat Algorithm to solve the non-convex economic dispatchproblem considering practical constraintsrdquoElectrical Power andEnergy Systems vol 82 pp 189ndash196 2016
[48] M A Al-Betar M A Awadallah A T Khader and A L BolajildquoTournament-based harmony search algorithm for non-convexeconomic load dispatch problemrdquo Applied Soft Computing vol47 pp 449ndash459 2016
[49] M A Al-Betar M A Awadallah I A Doush E Alsukhni andH ALkhraisat ldquoA non-convex economic dispatchproblemwithvalve loading effect using a new modified 120573-hill climbing localsearch algorithmrdquo Arabian Journal for Science and Engineeringpp 1ndash8 2018
[50] M A Awadallah M A Al-Betar A L Bolaji E M Alsukhniand H Al-Zoubi ldquoNatural selection methods for artificial beecolony with new versions of onlooker beerdquo Soft Computing pp1ndash40 2018
[51] S H Kaboli and A K Alqallaf ldquoSolving non-convex economicload dispatch problem via artificial cooperative search algo-rithmrdquo Expert Systems with Applications vol 128 pp 14ndash272019
[52] D Singh and J Dhillon ldquoAmeliorated greywolf optimization foreconomic load dispatch problemrdquo Energy vol 169 pp 398ndash4192019
[53] X S Yang and S Deb ldquoCuckoo search via Levy flightsrdquo inProceedings of the World Congress on Nature and BiologicallyInspired Computing (NaBIC rsquo09) pp 210ndash214 India 2009
[54] S Sahoo K M Dash R C Prusty and A K Barisal ldquoCom-parative analysis of optimal load dispatch through evolutionaryalgorithmsrdquo Ain Shams Engineering Journal vol 6 pp 107ndash1202015
[55] M Basu and A Chowdhury ldquoCuckoo search algorithm foreconomic dispatchrdquo Energy vol 60 pp 99ndash108 2013
[56] A B Serapiao ldquoCuckoo search for solving economic dispatchload problemrdquo Intelligent Control and Automation vol 04 no04 pp 385ndash390 2013
[57] A H Bindu and M D Reddy ldquoEconomic load dispatch usingcuckoo search algorithmrdquo International Journal of EngineeringResearch and Applications vol 3 pp 498ndash502 2013
[58] E Afzalan and M Joorabian ldquoAn improved cuckoo searchalgorithm for power economic load dispatchrdquo InternationalTransactions on Electrical Energy Systems vol 25 no 6 pp 958ndash975 2015
[59] T T Nguyen and D N Vo ldquoThe application of one rank cuckoosearch algorithm for solving economic load dispatch problemsrdquoApplied Soft Computing vol 37 pp 763ndash773 2015
[60] S M Islam S Das S Ghosh S Roy and P N Suganthan ldquoAnadaptive differential evolution algorithm with novel mutationand crossover strategies for global numerical optimizationrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 42 no 2 pp 482ndash499 2012
[61] Z L Gaing ldquoParticle swarm optimization to solving theeconomic dispatch considering the generator constraintsrdquo IEEETransactions on Power Systems vol 18 no 3 pp 1187ndash1195 2003
[62] X-S Yang and S Deb ldquoEngineering optimisation by Cuckoosearchrdquo International Journal of Mathematical Modelling andNumerical Optimisation vol 330 pp 43ndash51 2010
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10 Mathematical Problems in Engineering
Step 3 Handle prohibited power zone constraint for decisionvariables P2 P3 PN
After being randomly produced there is a high possi-bility that decision variables fall into PPZ and they violatePPZ constraint So the verification of falling into PPZ andcorrection of the violation should be accomplished by usingthe following formula
119875119894 =
119875119897119894119896 if 119875119897119894119896 lt 119875119894 le 119875119897119894119896 + 1198751198961198941198962119875119906119894119896 if (119875119894 gt 119875119897119894119896 + 1198751198961198941198962 ) amp (119875119894 lt 119875119906119894119896)119875119894 119890119897119904119890
119894 = 2 119873 amp 119896 = 1 119899119894
(31)
Step 4 Handle RPB constraint by calculating 1198751 and penaliz-ing 1198751 if it violates constraints
In this step power balance constraint is exactly handledby calculating and penalizing dependent variable (1198751) 1198751 isobtained by using formulas (4) and (5) as follows
1198751 = minus (11986101 minus 1 + 2sum119873119894=2 1198611119894119875119894) plusmn radicΔ211986111 (32)
where
Δ = (11986101 minus 1 + 2 119873sum119894=2
1198611119894119875119899)2 minus 411986111(119875119863 minus 119873sum
119894=2
119875119894+ 11986100 + 119873sum
119894=2
1198610119894119875119894 + 119873sum119894=2
119873sum119895=2
119875119894119861119894119895119875119895) amp Δ ge 0(33)
In Equation (32) 1198751 has been determined for the purposeof dealing with real power balance constraint However it isnot sure that 1198751 can satisfy upper bound and lower boundconstraints and prohibited power zone constraints So 1198751must be checked and penalized
Firstly 1198751 is checked and penalized for upper and lowerbound constraints by the following model
Δ1198751x =
0 if 1198751min le 1198751x le 1198751max
1198751min minus 1198751x if 1198751min gt 1198751x1198751x minus 1198751max if 1198751max lt 1198751x
(34)
In Equation (34) if the second case or the third caseoccurs it means P1 has violated either lower bound or upperbound and it would be penalized by using either (P1x= P1min-P1x) or (P1x =P1x -P1max) Otherwise ifP1 has not violatedthe bound constraints (ie the first case in (34) happened)
P1 would continue to be checked for PPZ constraint by thefollowing model
Δ1198751x
=
1198751 minus 1198751198971119896 if 1198751198971119896 lt 1198751 le 1198751198971119896 + 119875119896111989621198751199061119896 minus 1198751 if (1198751 gt 1198751198971119896 + 11987511989611198962 ) amp (1198751 lt 1198751199061119896)0 119890119897119904119890
(35)
Step 5 Handle real power reserve constraint (10)First of all 119878119894 is determined by using (11) and (12) and
then the 119909119905ℎ solution will be checked and penalized if poweroutput of all thermal generating units cannot satisfy RPRconstraint The penalty for violation of the constraint can becalculated by using equation (36)
Δ119878119877119909 =
0 if119873sum119894=1
119878119894119909 ge 119878119877119878119877 minus 119873sum119894=1
119878119894119909 119890119897119904119890 (36)
As a result real power reserve constraint can be solved byusing the penalty method
42 Calculate Fitness Function for Solutions Fitness functionof each solution is used to evaluate quality of solutionNormally the function is the sum of objective function andpenalty of violating constraints and is obtained by
43 The First Newly Updated Solutions by Levy Flights Tech-nique In this section the first newly updated solutionsare performed by employing Levy flights technique usingEquation (14) However each new solution can be out oftheir feasible operating zone such as PPZ and upper andlower limitations When the power output violates its PPZconstraints Equation (31) will be applied to tackle theconstraint Besides the following equation will be employedwhen power output is higher or lower than their limitations
119878119900119897119909 =
119878119900119897max if 119878119900119897max lt 119878119900119897119909119878119900119897min if 119878119900119897min gt 119878119900119897119909119878119900119897119909 Otherwise
119909 = 1 119873119901 (38)
After that Equations (32)-(37) are performed for deter-mining all variables and penalty terms Finally Equation (38)is employed to calculate fitness function
44 The Second Newly Updated Solutions by Using Muta-tion Technique The second newly updated solutions areaccomplished as presented in Section 3 above Similar to
Mathematical Problems in Engineering 11
the task after doing the first update each solution in thenew population must satisfy PPZ constraint and upper andlower boundaries by considering Equations (31) and (38)Then Equations (32)-(37) are performed for determiningall variables and penalty terms Finally Equation (38) isemployed to calculate fitness function and the solution withthe best value is assigned to the best one 11987811990011989711986611988711989011990411990545 Criterion of Stopping the Loop Algorithm In the loopalgorithm of using the proposed ICSA approach the solutionsearch work is stopped in case that the predeterminedmaximum iterations 119866119898119886119909 is reached For each search ter-mination the most effective solution is stored and anotherrun continues to be accomplished until the predeterminednumber of runs is reached After finishing the runs thebest one is found and reported In addition other valuessuch as the fitness of the worst solution and average fitnessof all solutions are also reported for comparing with othermethods
46 The Whole Iterative Process The whole iterative algo-rithm for implementing the proposed ICSA approach forcoping with OLD problem is described in detail in Figure 3
5 Results and Discussions
The proposed ICSA approach performance has been investi-gated on six cases with different fuel options different fuelcharacteristics and complicated constraints The details ofthe studied cases are presented as follows
Case 1 Four systems with single fuel source (SFS) and powerloss (PL) constraint
Subcase 11 A 3-unit system [57]Subcase 12 A 6-unit system [57]Subcase 13 A 3-unit system [56]Subcase 14 A 6-unit system [56]
Case 2 A 110-unit system with SFS [57]
Case 3 Four systems with SFS and the effects of valve loadingprocess (EoVLP)
Subcase 31 A 3-unit system supplying to a load of850MW [58]Subcase 32 A 13-unit system supplying to a load of1800MW [1]Subcase 33 A 13-unit system supplying to a load of2520MW [1]Subcase 34 A 40-unit system supplying to a load of2500MW [1]Subcase 35 An 80-unit system supplying to a load of4100MW [49]
Case 4 Two systems with SFS and PPZ and RPR constraints
Subcase 41 A 60-unit system supplying to a10600MW load [9]
Subcase 42 A 90-unit system supplying to a15900MW load [9]
Case 5 A 15-unit system with SFS and RRL PPZ and PLconstraints [61]
Case 6 Three systems with multiple fuel sources (MFS) andEoVLP
Subcase 61 An 80-unit system supplying to a21600MW load [15]Subcase 62 A 160-unit system supplying to a43200MW load [15]Subcase 63 A 320-unit system supplying to an86400MW load [54]
For each considered case with each load case the pro-posed ICSA approach is run 50 times on the programlanguage of Matlab and a PC with 4 GB of RAM and 24GHzprocessor The selection of adjustment parameters including119875119886 and 119879119900119897119909 is carefully considered to obtain the best optimalsolutions meanwhile two others such as 119873119901119904 and 119866119898119886119909 arechosen corresponding to the scale of particular test system9 values with the change of 01 in the range [01 09] are inturn selected for 119875119886 while 119879119900119897119909 is 001 at the beginning Theinformation including load demand119873119901119904 119866119898119886119909 and the best119875119886 is reported in Table 1
51 Obtained Results on Case 1 considering Four Systems withSFS and PL Constraint In this section we have implementedthe proposed ICSA approach for solving four systems dividedinto four subcases Tables 2 and 3 show the comparisons ofobtained results from Subcases 11 and 12 and Subcases 13and 14 respectively As listed in Table 2 the proposed ICSAmethod and CCSA can find equal fuel cost for Subcases 11whereas the reduction of fuel cost from the proposed ICSAmethod as compared to CCSA is clearer for Subcase 12 Asshown in Table 3 for comparing the proposed ICSA and threemethods consisting of CCSA ABC and FA the minimumfuel cost of the proposed ICSA is approximately equal to thatof these methods for Subcases 13 but much less than that ofthese methods for Subcase 14 Furthermore the proposedICSA has been run by setting 119873119901119904 and 119866119898119886119909 to 5 and 20but these values were much higher for CCSA ABC and FAThey are 20 and 5000 for CCSA 40 and 100 for ABC and 20and 5000 for FA Consequently the proposed method is veryefficient for Case 1 with four subcases
Optimal solutions obtained by ICSA for Case 1 are shownin Tables 16ndash18
52 Obtained Results on Case 2 considering 110-Unit Systemwith SFS In this section we have employed a very largescale system with 110 units but there were not challenges forobjective function and complex constraints since EoVLP andconstraints were not taken into account Both CCSA andthe proposed ICSA methods have been run for comparingwith BBO hybrid BBO and DE (DEBBO) and Opposi-tional real coded chemical reaction optimization algorithm(ORCCROA) in [36] IWA in [40] and AGWO in [52] As
12 Mathematical Problems in Engineering
Select parameters
- Correct solutions if violating PPZ constraint using Eq (31)- Determine using Eq (32)- Penalize for violating upper and lower bounds constraint using Eq (34)- Penalize for violating PPZ constraint using Eq (35)- Penalize the xth solution for violating RPR constraint using Eq (36)
- Determine quality of solutions employing Eq (37) - Select the most effective solution - Start the loop algorithm by selecting
- Perform the first solution update using Section 32 - Correct boundaries of solutions by using Eq (38)
- Correct solutions if violating PPZ constraint using Eq (31)- Determine using Eq (32)- Penalize for violating upper and lower bounds constraint using Eq (34)- Penalize for violating PPZ constraint using Eq (35)- Penalize the xth solution for violating RPR constraint using Eq(36)
- Determine fitness function using Eq (37)- Compare old solutions and new solutions to keep better ones
- Perform the second solution update using Algorithm 1 - Correct boundaries of solutions by using (38)
- Correct solutions if violating PPZ constraint using Eq (31)- Determine using Eq (32)- Penalize for violating upper and lower bounds constraint using Eq (34)- Penalize for violating PPZ constraint using Eq (35)- Penalize the xth solution for violating RPR constraint using Eq (36)
- Determine fitness function using Eq (37)- Compare old solutions and new solutions to keep better ones- Select the most effective solution
Stop
Start
- Redefine upper and lower bounds using Eqs (25)-(28)- Randomly generate initial population using Eq (30)
Nps Pa Gmax H> Tolx
P1xP1x
P1x
P1xP1x
P1x
P1xP1x
P1x
Sol<MN
Sol<MN
G = 1
G = Gmax G = G + 1
Figure 3 All computation steps for solving OLD problem by employing the proposed ICSA approach
shown in Table 4 AGWO [52] has reached less fuel cost thanICSA however the exact fuel cost which was recalculatedby using reported solution pointed out that the method hasreached a very high fuel cost of $2157404250 For comparisonwith other methods ICSA has found less fuel cost thanall these methods Particularly the reduction of generationfuel cost is significant as compared to BBO DEBBO andCCSA Execution time comparisons are also useful evidence
for indicating the high performance of ICSA Thus it canconclude that ICSA is a strong method for Case 2
Optimal solution obtained by ICSA for the case is shownin Table 19
53 Obtained Results on Case 3 considering Four Systems withSFS and EoVLP In this section the real performance of theproposed ICSA approach has been investigated based on five
Mathematical Problems in Engineering 13
Table 1 Information of considered cases and adjustment parameters
Case Fuel cost function Constraint Subcase No of units 119875119863 (MW) Nps Gmax Best 1198751198861 SFS PL
subcases with the gradual increase of number of units Thesmallest scale system considers 3 units but the largest scalesystem takes 80 units In addition to the implementation ofthe proposed ICSA we have also implemented CCSA forSubcase 31 and Subcase 35 for further comparison becauseCCSA has not been run for the two subcases so far
Comparison of obtained results from Subcase 31 shownin Table 5 indicates that the proposed ICSA is superior toCCSAwith lower fuel cost but it seems to be less effective thanMCSA [58] The minimum cost of MCSA reported in [58] isthe smallest fuel cost but the recalculated cost is much higherthan that of the proposed method Furthermore MCSA hasbeen implemented by setting very high values to 119873119901119904 and119866119898119886119909
Reports for Subcases 32 and 33 shown in Table 6are the comparisons of the proposed ICSA approach andother methods such as conventional Evolution programming(CEP) [1] Fast EP (FEP) [1] improved FEP (IFEP) [1] DE[12] multiplier Lagrange-based genetic algorithm with (GA-MU) [15] QPSO [16] GA-PS-SQP [30] PSO-SQP [32] M120573-HCLSA [49] IABCA [50] CCSA [59] OSE-CSA [59] SOS[34] MSOS [34] CEA-SQT [38] TSBO [39] IWA [40] andCBA [44] As observed from the table ICSAapproach obtainsbetter solutions than mostmethods excluding DE [10] CCSA
[59] OSE-CSA [59] SOS [34] MSOS [34] CEA-SQT [38]TSBO [39] IWA [40] and CBA [44] especially M120573-HCLSA[49] with lower cost $1796097 However recalculated costfrom reported solution of M120573-HCLSA is $179691 BesidesICSA is very fast as compared to most methods where twoother versions of Cuckoo search algorithm CCSA and OSE-CSA are also included except two methods in [34] Theprocessor of computer that all the methods run on is alsoreported in the final column Clearly ICSA approach is veryefficient for the case with the 13-unit system where effects ofvalve loading process are considered
In Subcase 34 the number of units is much larger thanthat of three subcases above up to 40 units [1] The obtainedresult comparisons with others are indicated in Table 7Clearly the minimum cost comparisons reveal that the pro-posedmethod is one of the leading methods due to the lowestcost except the comparison with CCSA [23] OSE-CSA [59]SOS [34] MSOS [34] EMA [45] 120579-MBA [47] and AGWOA[52] It is noted that AGWOA [52] has reported the bestminimum cost with $12140430 but recalculated minimumcost which was obtained by substituting reported optimalgeneration of all thermal generating units is $12141331 Theaverage and the maximum costs from the proposed methoddo not belong to the leading method group however the
Mathematical Problems in Engineering 15
Table 7 Result comparisons for Subcase 34
Approach Best cost Average cost Worst cost CPU time Computer($h) ($h) ($h) (s) (Processor-Ram)
execution time that ICSA approach takes is smaller thanmostones except IAPR [42] and 120579-MBA [47] which have been runon stronger computers Note that MSOS [34] has been fasterthan the proposed method for Subcases 31 and 32 abovebut it is too slower than the proposed method for the casenamely 1813 seconds compared to 146 seconds Comparedto two other versions of Cuckoo search CCSA [23] and OSE-CSA [59] the proposed method is also faster about threetimes although the processors are slightly different For thiscase 120579-MBA [47] shows a very good performance howeverthe method has not been tested onmore complicated systemsand larger scale and therefore more comparisons with themethod must end Clearly the proposed method is stillefficient for the case where large scale and effects of valveloading process are included
Subcase 35 is the largest scale system with 80 units Inaddition to the implementation of ICSA CCSA has beenalso run for the Subcase 35 for further investigation ofefficiency improvement of the proposed ICSA approach The
comparisons of minimum cost in Table 8 show that theproposed ICSA can find more optimal solution than M120573-HCLSA [49] AGWOA [52] and CCSA The proposed ICSAis also superior to CCSA in terms ofmore stable search abilityand lower fluctuation since its average cost and maximumcost are less than those of CCSA The outstanding figurecannot be reached as compared to AGWOA [52] howeverit is hard to conclude AGWOA [52] is superior to theproposed ICSA approach about more stable search abilityand lower fluctuation Actually comparison of the values ofpopulation and iterations as well as execution time cannot beaccomplished because the information was not reported in[52]Thus it can conclude that the proposed ICSA is effectivefor the subcase
Optimal solutions obtained by ICSA for the case areshown in Tables 20ndash23
54 Obtained Results on Case 4 with Two Systems consideringSFS and PPZ and RPR Constraints In this section two
16 Mathematical Problems in Engineering
Table 9 Result comparisons for Subcases 41 and 42
Subcase Approach Best cost ($h) Mean cost ($h) Worst cost ($h) Std dev ($h) CPU time (s) Computer (Processor-Ram)
test systems with SFS and PPZ and RPR constraints areconsidered The test system size is up to 60 and 90 units forSubcases 41 and 42 respectively Comparison for the casesis only performed with two Genetic algorithms consisting ofGA and IGA-MU in [9] and two other versions of Cuckoosearch algorithms including CCSA [23] and OSE-CSA [59]and presented in Table 9 Other studies have tended toignore such complicated constraints of PPZ and RPR Thecosts reported in Table 9 indicate that ICSA approach canobtain more effective solution than CCSA and OSE-CSAbecause it has reached lower minimum cost than the twoones Furthermore the proposed method also takes shortercomputation time for the two cases from about two timesto about three times although the processor of the proposedmethod is slightly strongerThemean costs of ICSA approachare much less than those from IGA-MU and GA and slightlyhigher than those from OSE-CSA but there is a trade-off between the proposed method and CCSA for the twosubcases In fact the proposed method obtains higher meancost for Subcase 41 but lower cost for Subcase 42 AlthoughGAmethods have been runon aweak computerwith 07 GHzof the processor compared to that with 24GHz in the studytheir execution times are significantly higher namely 56381seconds (GA) and 16258 seconds (IGA-MU) compared to09153 seconds of ICSA approach for Subcase 41 and 94093seconds (GA) and 25545 seconds (IGA-MU) compared to15892 seconds (the proposedmethod)The analysis can pointout that ICSA approach ismore efficient than these comparedmethods in terms of optimal solutions and execution time
Optimal solutions obtained by ICSA for Subcase 42 areshown in Table 24
55 Obtained Results on Case 5 with a 15-Unit System consid-ering SFS and RRL PPZ and PL Constraints In this section
a 15-unit system considering RRL PPZ and PL constraintsis considered to be solved for finding optimal solution Forefficiency investigation of the proposed ICSA we have alsoimplemented CCSA for comparison As listed in Table 10the proposed ICSA is the most effective method with thesmallest fuel cost The comparisons of control parameters aswell asCPU time are also good evidence to confirm the strongsearch of the proposed ICSA approach since it has been runby smaller values of control parameter and faster executiontime as compared to all methods excluding CCSA
Optimal solution obtained by ICSA for the case is shownin Table 25
56 Obtained Results on Case 6 withThree Systems consideringMFS and EoVLP In this section three test systems with thechallenge on objective function including multi-fossil fuelsources and effects of valve loading process are consideredThe scale is up to 80 units 160 units and 320 units forSubcases 61 62 and 63 respectively
Comparison for Subcase 61 reported in Table 11 revealsthat the proposed method is the best method in terms of thelowest best cost the lowestmean cost and the lowest standarddeviation and the fastest execution time The processor fromthis proposedmethod is about four times stronger thanCGA-MU and IGA-MU but the speed is from ten times to 35 timesfaster than these methods Compared to CCSA and OSE-CSA the proposed method is about two times faster but theprocessor is slightly stronger
Comparison for Subcase 62 is reported in Table 12Clearly the proposed ICSA approach obtains better values ofthe best mean and worst costs than most methods exceptMSOS [34] where the best cost difference is about $ 024However the proposed method is the second fastest onewith 1119 seconds where the first fastest one CBA [44] has
Mathematical Problems in Engineering 17
Table 11 Comparisons of found results for Subcase 61
Approach Best cost Mean cost Worst cost Std dev CPU time Computer($h) ($h) ($h) ($h) (s) (Processor-Ram)
spent 571 seconds Clearly MSOS is better than the proposedICSA approach with respect to slightly less best cost butis worse than the proposed method in terms of executiontime namely 2572 seconds compared to 1119 seconds whileCBA [44] is faster than the proposed method but obtainssignificantly worse costs The analysis can conclude that theproposed ICSA approach is very powerful for the subcasewith 160 units
Table 13 presents the comparison of three other methodsincludingCCSA [54] SOS [34] andMSOS [34] accompaniedwith the proposed method for Subcase 63 The obtainedresult comparisons imply that ICSA approach can obtainbetter values of the best and standard deviation costs thanCCSA and SOS but obtains slightly higher cost than MSOSby approximately $ 022 Besides the execution time fromICSA approach is much shorter than others especially it ishigher than five times faster than MSOS The four methodshave been run on approximately strong computers Brieflythe proposed ICSA approach can find and converge to morefavorable solution than other methods with shorter CPUtime except the comparison with MSOS which had bettersolution but spent higher than five times execution times
Consequently the proposed method is a very promisingoptimization algorithm for Subcase 63 a system up to 320units and with multi-fossil fuel sources and effects of valveloading process
Optimal solution obtained by ICSA for Subcase 63 isshown in Table 26
57 The Improvement of ICSA Approach Performance
571 The Outstanding Improvement over CCSA In this sec-tion the performance improvement of ICSA over CCSAhas been investigated by analyzing obtained results and setcontrol parameters Table 14 has been formed by addingreduction cost improvement level of the best cost executiontime and control parameters consisting of 119873119901119904 and 119866119898119886119909Among the compared factors reduction cost is the deviationof the cost of CCSA and that of ICSA whereas the improve-ment level is the ratio of the reduction cost to the cost ofCCSA The reduction costs indicate the proposed methodcould find either equal quality of solutions or higher qualityof solutions than CCSA for all study casesThe reduction costis from $0 to $42839 corresponding to the improvement level
18 Mathematical Problems in Engineering
Table 14 Summary of results obtained by CCSA and ICSA for all study cases
Study case Reduction cost ($) Improvement level () Execution time (s) Nps Gmax
Figure 4 The best run obtained by CCSA and ICSA for Case 2
that is from 0 to 08968The saving cost is not too much forone hour but the operation in one day onemonth or one yearis very high However it should be noted that CCSA has beenrun by setting much higher population size and iterations formany cases excluding study cases implemented in the studysuch as Case 2 Subcase 31 Subcase 35 and Case 5 Forinstance ICSA has used 119873119901119904 = 10 and 119866119898119886119909 = 15 for Subcase13 and119873119901119904 = 10 and119866119898119886119909 = 25 for Subcase 14 whereas CCSAhas been run by setting119873119901119904 = 20 and 119866119898119886119909 = 5000 for the twosubcases Similarly CCSA has been run for Subcases 32 33and 34 with much higher number of iterations For the lastsubcase CCSA has been run by setting 119873119901119904 = 320 and 119866119898119886119909
= 1200 but those of ICSA have been 10 and 9000 Due tothe higher value of control parameters CCSA has tended tospend more time in finding such high quality solutions foralmost all study cases Execution time of ICSA is less than18 seconds while that of CCSA is up to higher 75 secondsIt is clear that the proposed ICSA could find better optimalsolutions thanCCSA for such considered systems For furtherinvestigation of performance comparison the best runs over50 runs and fuel cost values of 50 runs obtained by CCSA andthe proposed ICSA for Case 2 Subcase 31 Subcase 35 andCase 5 have been plotted in from Figures 4ndash11 The best runcurves show the faster search of the proposed ICSA method
Mathematical Problems in Engineering 19
0 5 10 15 20 25 30 35 40 45 50Run
19795
198
19805
1981
19815
1982
19825
1983
19835
Fuel
cost
($)
CCSAICSA
times105
Figure 5The best fuel cost of 50 runs obtained by CCSA and ICSA for Case 2
0 2 4 6 8 10 12 14 16 18 20Iteration
8234
8236
8238
8240
8242
8244
8246
8248
8250
8252
Fitn
ess F
unct
ion
($)
CCSAICSA
Figure 6 The best run obtained by CCSA and ICSA for Subcase 31
whereas 50 values of fuel cost indicate that the proposedICSA can find many solutions with better quality Clearlythe proposed ICSA is outstanding in terms of stabilizationof solution search and faster convergence As a result it canconclude that the proposed ICSA approach is more effectivethanCCSA in solvingOLDproblemwith considered systems
572The Improvement of Results over Other Methods In thisarticle we have tested ICSA approach on 6 cases with 16
systems with different fuel cost forms different constraintsand different scale systems from 3 units to 320 units We havecompared the yielded results from ICSA approach and otherexisting ones for evaluating the efficiency of ICSA approachIn subsections above we have shown yielded results fromICSA approach and compared these results to those of otheronesHowever the demonstrationhad not been very good forobserving and comparing to lead to a conclusionThus in thesubsectionwehave summarized the result comparisons of the
20 Mathematical Problems in Engineering
0 5 10 15 20 25 30 35 40 45 50Run
8234
82342
82344
82346
82348
8235
82352
82354
82356
82358
8236
Fuel
cost
($)
CCSAICSA
Figure 7 The best fuel cost of 50 runs obtained by CCSA and ICSA for Subcase 31
0 1000 2000 3000 4000 5000 6000Iteration
242
244
246
248
25
252
254
256
Fitn
ess F
unct
ion
($)
CCSAICSA
times105
Figure 8 The best run obtained by CCSA and ICSA for Subcase 35
proposed and other ones Table 15 has reported the reductioncost (in $) of ICSA approach compared to other ones Inaddition we have converted the reduction cost into improve-ment level (in ) for better comparison The improvementhas been shown from the lowest level to the highest levelin terms of reduction cost and improvement percentageIn addition we have also given the slowest and the fastestexecution time of other compared methods together withthat of the proposed method The table implies that ICSA
approach can find better optimal solutions with less fuel costup to $052 for Subcase 13 $75229 for Subcase 14 $1775149for Case 2 $1181256 for Subcase 31 $8438 for Subcase 32$9113 for Subcase 33 $121181 for Subcase 34 $3412 forSubcase 35 $04329 for Subcase 41 $30227 for Subcase42 $407002 for Case 5 $18183 for Subcase 61 $13965 forSubcase 62 and $11817 for Subcase 63 These reductioncosts are equivalent to improvement level (IL) of 0032908968 82282 14143 047 038 099 001405
Mathematical Problems in Engineering 21
0 5 10 15 20 25 30 35 40 45 50Run
2428
243
2432
2434
2436
2438
244
2442
Fuel
cost
($)
CCSAICSA
times105
Figure 9 The best fuel cost of 50 runs obtained by CCSA and ICSA for Subcase 35
0 50 100 150 200 250 300 350 400Iteration
327
328
329
33
331
332
333
334
335
336
337
Fitn
ess F
unct
ion
($)
CCSAICSA
times104
Figure 10 The best run obtained by CCSA and ICSA for Case 5
00003 0002 12291 004 138 and 059 Thesequantitative comparisons reveal that larger scale systems canlead to better reduction cost but the improvement level isnot high because total cost of compared methods tends tobe large for large scale systems Furthermore very large scalesystems with nondifferentiable objective have been normallysolved by strong methods In fact systems in Case 3 havethe same characteristic with single fuel and effects of valveloading process but Subcase 34 is a larger scale system with
40 units while Subcases 31 32 and 33 are constructed by 3units 13 units and 13 units So the improvement percentageof Subcase 34 can be up to 099 whilst that of Subcases32 and 33 is 047 and 038 respectively Subcase 35 iswith the largest system 80 units but the reduction cost is notmuch only $3412 because compared methods with ICSA areeither state-of-the-art ones or improved ones Also Subcase41 and Subcase 42 have considered single fuel and PPZ andspinning reserve constraints but Subcase 42 is larger scale
22 Mathematical Problems in Engineering
Table 15 Performance improvement summary of the proposed method
Study cases Reduction cost ($) Improvement level () Execution time (s)From To From To Slowest method Fastest method Proposed method
Figure 11 The best fuel cost of 50 runs obtained by CCSA and ICSA for Case 5
system with 90 units and Subcase 41 is only with 60 unitsThus the improvement of Subcase 41 is lower with 00003but that of Subcase 42 is 0002 Similarly systems in Cases 6have the same featurewithmulti-fossil fuel sources and effectsof valve loading process but they are respectively constructedby 80 160 and 320 units As a result the improvements ofSubcase 62 138 and Subcase 63 059 are much higherthan Subcase 61 004 However Subcase 62 with smallernumber of units but getting higher improvement is easilyunderstood because there were nine compared methods butonly three compared methods are considered for Subcase 63In general the improvement is not high it is about under onedollar several dollars tens of dollars and over one thousand
dollars per hour however the saving cost will be significantif the operation is considered to be onemonth with 720 hoursor one year with 8760 hours
Execution time comparison can be evaluated by observ-ing the fastest and the slowest compared methods in Table 15These execution times of the proposed method are approxi-mately equal to that of the fastest methods for Subcases 3233 and 34 and much shorter than other fastest comparedmethods for other cases especially for Subcases 61 and 63The fastest method for Subcase 62 is CBA [44] showing 57seconds while that of the proposed method is 1119 secondsHowever it cannot conclude that CBA is more effective thanthe proposed method because the proposed method could
find better optimal solution with less fuel cost by $2130As considering execution time of the slowest comparedmethods it can point out that ICSA is a very fast optimizationtool since the execution time of these methods is 29496 and3425 seconds for Subcases 32 and 33 116735 seconds forSubcase 34 56381 seconds for Subcase 41 94093 secondsfor Subcase 42 12797 seconds for Case 5 30941 seconds forSubcase 61 7542 seconds for Subcase 62 and 9641 secondsfor Subcase 63 while the execution time of the proposedmethod for these cases is respectively 095 146 091 1589246 84828 1119 and 171384 seconds It is clearly shown thatICSA is very fast as compared to these methods
In summary the proposed method has found approx-imately high quality solutions with several standard state-of-the-art meta-heuristic algorithms and improved versionsof them together with other old methods In addition theproposed method could improve result better than approx-imately all methods with faster execution time Comparedto other methods with the fastest convergence speed andhigh quality solutions the proposed method has been as
fast as for some cases and much faster for other cases Thecomparison with the slowest methods could show that theproposed method was extremely powerful since it was up tonearly one thousand times faster Consequently the proposedICSA approach can be one of the strongest optimization toolsfor OLD problem
6 Conclusions
This paper has proposed a good ICSA method for solvingOLD problem in which many test systems with differentobjective functions and complicated constraints from simpleto complex have been used as studied cases The proposedICSA method has been developed by performing severalmodifications on the second solution update of CCSA whichcontained several drawbacks to global convergence and fastmanner The OLD problem has covered from single fuelto multi-fossil fuels from quadratic objective function tononconvex objective function in addition to PPZ RPR andRRL constraints Many existing optimization algorithms have
been concerned in aim to compare the performance and givethe final conclusion on the proposed method There have
been six main cases with sixteen subcases The evaluationshave been made at the end of each study case Clearly theproposed ICSA approach has yielded more effective optimalsolutions with faster execution time than almost all methodsConsequently it can be concluded that the proposed methodis much more superior to CCSA and is a very promisingmethod for solving OLD problem
Appendix
See Tables 16ndash26
Nomenclature
120575119894 120582119894 120572119894 120573119894 120574119894 Fuel cost functioncoefficients of the 119894119905ℎthermal generationunit11986100 1198610119895 119861119895119894 Power loss matrixcoefficients119865119865119909 119865119865119887119890119904119905 The values of fitness ofsolution 119909 and theso-far most effectivesolution among thecurrent set ofsolutions119898119894 Number of fuels burntin the 119894119905ℎ thermalgeneration unit
119873 Number of allavailable thermalgeneration units119899119894 Number of prohibitedpower zones of the 119894119905ℎthermal generationunit119875119906119894119896 119875119897119894119896 Upper and lowerlimits of the 119894119905ℎthermal generationunit corresponding tothe kth PPZ119875119886 Probability ofreplacing controlvariables in each oldsolution119875119863 Real power demand ofall loads in system119875119894119898119886119909 119875119894119898119894119899 The highest andlowest real poweroutputs of the iththermal generationunit119875119894119895119898119886119909 119875119894119895119898119894119899 The highest andlowest real poweroutputs of the 119894119905ℎthermal generationunit corresponding tothe 119895119905ℎ fuel type119878119894119898119886119909 Maximum real powerreserve contributionof the thermalgeneration unit 1198941198781199001198971199031198861198991198891 1198781199001198971199031198861198991198892 1198781199001198971199031198861198991198893 1198781199001198971199031198861198991198894 Randomly mixedsolutions from the setof current solutions119878119900119897119909 119878119900119897119866119887119890119904119905 The old solution x andthe most effectivesolution119878119877 Real power reserverequirement of system
120575119894119895 120582119894119895 120572119894119895 120573119894119895 120574119894119895 Fuel cost functioncoefficients of the 119894119905ℎthermal generationunit corresponding tothe 119895119905ℎ fuel type1205761 1205762 1205763 1205764 1205765 1205766 Random numbersbetween 0 and 1Ω Number of generationunits considering PPZconstraint
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare no conflicts of interest
References
[1] N Sinha R Chakrabarti and P K Chattopadhyay ldquoEvolution-ary programming techniques for economic load dispatchrdquo IEEETransactions on Evolutionary Computation vol 7 no 1 pp 83ndash94 2003
[2] P Hansen ldquoA separable approximation dynamic programmingalgorithm for economic dispatch with transmission lossesrdquoYugoslav Journal of Operations Research vol 12 no 2 2002
[3] N T Thang ldquoSolving economic dispatch problem with piece-wise quadratic cost functions using lagrange multiplier theoryrdquoin Proceedings of the 3rd International Conference on ComputerTechnology and Development (ICCTD rsquo11) pp 359ndash364 ASMEPress 2011
[4] S K Mishra and S K Mishra ldquoA comparative study of solutionof economic load dispatch problem in power systems in theenvironmental perspectiverdquoProcedia Computer Science vol 48pp 96ndash100 2015
[5] A A Al-Subhi and H K Alfares ldquoEconomic load dispatchusing linear programming a comparative studyrdquo InternationalJournal of Applied Industrial Engineering vol 3 no 1 pp 16ndash362016
28 Mathematical Problems in Engineering
[6] C Zhou G Huang and J Chen ldquoPlanning of electric powersystems considering virtual power plants with dispatchableloads included an inexact two-stage stochastic linear program-ming modelrdquoMathematical Problems in Engineering vol 2018Article ID 7049329 12 pages 2018
[7] T T Nguyen N V Quynh and L Van Dai ldquoImproved fireflyalgorithm a novel method for optimal operation of thermalgenerating unitsrdquo Complexity vol 2018 Article ID 7267593 23pages 2018
[8] S S Haroon S Hassan S Amin et al ldquoMultiple fuel machinespower economic dispatch using stud differential evolutionrdquoEnergies vol 11 no 6 pp 1ndash20 2018
[9] C-T Su and C-L Chiang ldquoNonconvex power economic dis-patch by improved genetic algorithm with multiplier updatingmethodrdquo Electric Power Components and Systems vol 32 no 3pp 257ndash273 2004
[10] NNoman andH Iba ldquoDifferential evolution for economic loaddispatch problemsrdquo Electric Power Systems Research vol 78 no8 pp 1322ndash1331 2008
[11] P Somasundaram and K Kuppusamy ldquoApplication of evolu-tionary programming to security constrained economic dis-patchrdquo International Journal of Electrical Power amp EnergySystems vol 27 no 5-6 pp 343ndash351 2005
[12] T Jayabarathi K Jayaprakash andD Jeyakumar ldquoEvolutionaryprogramming technique for different kinds of economic dis-patch problemsrdquo Electric Power Systems Research vol 73 no 2pp 169ndash176 2005
[13] P Subbaraj R Rengaraj and S Salivahanan ldquoEnhancementof combined heat and power economic dispatch using selfadaptive real-coded genetic algorithmrdquo Applied Energy vol 86no 6 pp 915ndash921 2009
[14] C-L Chiang and C-T Su ldquoAdaptive-improved genetic algo-rithm for the economic dispatch of units with multiple fueloptionsrdquo Cybernetics and Systems vol 36 no 7 pp 687ndash7042005
[15] C Chiang ldquoImproved genetic algorithm for power economicdispatch of units with valve-point effects and multiple fuelsrdquoIEEE Transactions on Power Systems vol 20 no 4 pp 1690ndash1699 2005
[16] K Meng H G Wang Z Y Dong and K P Wong ldquoQuantum-inspired particle swarm optimization for valve-point economicload dispatchrdquo IEEE Transactions on Power Systems vol 25 no1 pp 215ndash222 2010
[17] P Erdogmus A Ozturk and S Duman ldquoEnvironmen-taleconomic dispatch using genetic algorithm and simulatedannealingrdquo in Proceedings of the International Conference onElectric Power and Energy Conversion Systems EPECS rsquo09 pp1ndash4 IEEE 2009
[18] L D S Coelho and V C Mariani ldquoImproved differentialevolution algorithms for handling economic dispatch opti-mization with generator constraintsrdquo Energy Conversion andManagement vol 48 no 5 pp 1631ndash1639 2007
[19] H Dakuo W Fuli and M Zhizhong ldquoA hybrid genetic algo-rithm approach based on differential evolution for economicdispatch with valve-point effectsrdquo Electrical Power and EnergySystems vol 30 pp 31ndash38 2008
[20] C-F Chang J-J Wong J-P Chiou and C-T Su ldquoRobustsearching hybrid differential evolution method for optimalreactive power planning in large-scale distribution systemsrdquoElectric Power Systems Research vol 77 no 5-6 pp 430ndash4372007
[21] M R Farooqi P Jain and K R Niazi ldquoUsing Hopfield neuralnetwork for economic dispatch of power systemsrdquo in Proceed-ings of the National Power Engineering Conference PECon rsquo03pp 5ndash10 Bangi Malaysia 2003
[22] T Yalcinoz H Altun and U Hasan ldquoConstrained economicdispatch with prohibited operating zones a Hopfield neuralnetwork approachrdquo in Proceedings of the 10th MediterraneanElectrotechnical Conference Information Technology and Elec-trotechnology for the Mediterranean Countries ProceedingsMeleCon rsquo00 (Cat No 00CH37099) pp 570ndash573 LemesosCyprus 2000
[23] D N Vo P Schegner and W Ongsakul ldquoCuckoo searchalgorithm for non-convex economic dispatchrdquo IET GenerationTransmission amp Distribution vol 7 no 6 pp 645ndash654 2013
[24] J-B Park K-S Lee J-R Shin and K Y Lee ldquoA particleswarm optimization for economic dispatch with non-smoothcost functionsrdquo IEEETransactions on Power Systems vol 20 no1 pp 34ndash42 2005
[25] T Niknam H D Mojarrad and M Nayeripour ldquoA newfuzzy adaptive particle swarm optimization for non-smootheconomic dispatchrdquo Energy vol 35 no 4 pp 1764ndash1778 2010
[26] L D S Coelho and V C Mariani ldquoAn improved harmonysearch algorithm for power economic load dispatchrdquo EnergyConversion and Management vol 50 no 10 pp 2522ndash25262009
[27] M Mahdavi M Fesanghary and E Damangir ldquoAn improvedharmony search algorithm for solving optimization problemsrdquoAppliedMathematics and Computation vol 188 no 2 pp 1567ndash1579 2007
[28] A Bhattacharya and P K Chattopadhyay ldquoSolving complexeconomic load dispatch problem using biogeography-basedoptimizationrdquo Expert Systems with Applications vol 37 no 5pp 3605ndash3615 2010
[29] D Simon ldquoBiogeography-based optimizationrdquo IEEE Transac-tions on Evolutionary Computation vol 12 no 6 pp 702ndash7132008
[30] J S Alsumait J K Sykulski and A K Al-Othman ldquoAhybrid GA-PS-SQP method to solve power system valve-pointeconomic dispatch problemsrdquo Applied Energy vol 87 no 5 pp1773ndash1781 2010
[31] M S Turgut andGKDemir ldquoQuadratic approximationndashbasedhybrid Artificial Cooperative Search algorithm for economicemission load dispatchproblemsrdquo International Transactions onElectrical Energy Systems vol 27 no 4 pp 1ndash14 2017
[32] T A A Victoire and A E Jeyakumar ldquoHybrid PSO-SQPfor economic dispatch with valve-point effectrdquo Electric PowerSystems Research vol 71 no 1 pp 51ndash59 2004
[33] D-KHe F-LWang andZ-ZMao ldquoHybrid genetic algorithmfor economic dispatch with valve-point effectrdquo Electric PowerSystems Research vol 78 no 4 pp 626ndash633 2008
[34] D C Secui ldquoA modified symbiotic organisms search algorithmfor large scale economic dispatch problem with valve-pointloading effectsrdquo Energy vol 113 pp 366ndash384 2016
[35] K BhattacharjeeA Bhattacharya and S H N Dey ldquoTeaching-learning-based optimization for different economic dispatchproblemsrdquo Scientia Iranica vol 21 no 3 pp 870ndash884 2014
[36] K Bhattacharjee A Bhattacharya and S H N Dey ldquoOppo-sitional Real Coded Chemical Reaction Optimization for dif-ferent economic dispatch problemsrdquo International Journal ofElectrical Power amp Energy Systems vol 55 pp 378ndash391 2014
Mathematical Problems in Engineering 29
[37] V Hosseinnezhad M Rafiee M Ahmadian and M T AmelildquoSpecies-basedQuantum Particle SwarmOptimization for eco-nomic load dispatchrdquo International Journal of Electrical Poweramp Energy Systems vol 63 pp 311ndash322 2014
[38] M S P Subathra S E Selvan T A A Victoire A HChristinal andU Amato ldquoA hybrid with cross-entropymethodand sequential quadratic programming to solve economic loaddispatch problemrdquo IEEE Systems Journal vol 9 no 3 pp 1031ndash1044 2015
[39] J Zhan Q H Wu C Guo and X Zhou ldquoEconomic dispatchwith non-smooth objectives part I local minimum analysisrdquoIEEE Transactions on Power Systems vol 30 no 2 pp 710ndash7212015
[40] A K Barisal andR C Prusty ldquoLarge scale economic dispatch ofpower systems using oppositional invasive weed optimizationrdquoApplied Soft Computing vol 29 pp 122ndash137 2015
[41] M Basu ldquoImproved differential evolution for economic dis-patchrdquo International Journal of Electrical Power amp EnergySystems vol 63 pp 855ndash861 2014
[42] V S Aragon S C Esquivel and C C Coello ldquoAn immunealgorithm with power redistribution for solving economicdispatch problemsrdquo Information Sciences vol 295 pp 609ndash6322015
[43] M Ghasemi M Taghizadeh S Ghavidel and A AbbasianldquoColonial competitive differential evolution an experimentalstudy for optimal economic load dispatchrdquo Applied Soft Com-puting vol 40 pp 342ndash363 2016
[44] B R Adarsh T Raghunathan T Jayabarathi and X-S YangldquoEconomic dispatch using chaotic bat algorithmrdquo Energy vol96 pp 666ndash675 2016
[45] N Ghorbani and E Babaei ldquoExchange market algorithm foreconomic load dispatchrdquo International Journal of ElectricalPower amp Energy Systems vol 75 pp 19ndash27 2016
[46] J X V Neto G Reynoso-Meza T H Ruppel V C Marianiand L D S Coelho ldquoSolving non-smooth economic dispatchby a new combination of continuous GRASP algorithm anddifferential evolutionrdquo International Journal of Electrical Poweramp Energy Systems vol 84 pp 13ndash24 2017
[47] A Kavousi-Fard and A Khosravi ldquoAn intelligent 120579-ModifiedBat Algorithm to solve the non-convex economic dispatchproblem considering practical constraintsrdquoElectrical Power andEnergy Systems vol 82 pp 189ndash196 2016
[48] M A Al-Betar M A Awadallah A T Khader and A L BolajildquoTournament-based harmony search algorithm for non-convexeconomic load dispatch problemrdquo Applied Soft Computing vol47 pp 449ndash459 2016
[49] M A Al-Betar M A Awadallah I A Doush E Alsukhni andH ALkhraisat ldquoA non-convex economic dispatchproblemwithvalve loading effect using a new modified 120573-hill climbing localsearch algorithmrdquo Arabian Journal for Science and Engineeringpp 1ndash8 2018
[50] M A Awadallah M A Al-Betar A L Bolaji E M Alsukhniand H Al-Zoubi ldquoNatural selection methods for artificial beecolony with new versions of onlooker beerdquo Soft Computing pp1ndash40 2018
[51] S H Kaboli and A K Alqallaf ldquoSolving non-convex economicload dispatch problem via artificial cooperative search algo-rithmrdquo Expert Systems with Applications vol 128 pp 14ndash272019
[52] D Singh and J Dhillon ldquoAmeliorated greywolf optimization foreconomic load dispatch problemrdquo Energy vol 169 pp 398ndash4192019
[53] X S Yang and S Deb ldquoCuckoo search via Levy flightsrdquo inProceedings of the World Congress on Nature and BiologicallyInspired Computing (NaBIC rsquo09) pp 210ndash214 India 2009
[54] S Sahoo K M Dash R C Prusty and A K Barisal ldquoCom-parative analysis of optimal load dispatch through evolutionaryalgorithmsrdquo Ain Shams Engineering Journal vol 6 pp 107ndash1202015
[55] M Basu and A Chowdhury ldquoCuckoo search algorithm foreconomic dispatchrdquo Energy vol 60 pp 99ndash108 2013
[56] A B Serapiao ldquoCuckoo search for solving economic dispatchload problemrdquo Intelligent Control and Automation vol 04 no04 pp 385ndash390 2013
[57] A H Bindu and M D Reddy ldquoEconomic load dispatch usingcuckoo search algorithmrdquo International Journal of EngineeringResearch and Applications vol 3 pp 498ndash502 2013
[58] E Afzalan and M Joorabian ldquoAn improved cuckoo searchalgorithm for power economic load dispatchrdquo InternationalTransactions on Electrical Energy Systems vol 25 no 6 pp 958ndash975 2015
[59] T T Nguyen and D N Vo ldquoThe application of one rank cuckoosearch algorithm for solving economic load dispatch problemsrdquoApplied Soft Computing vol 37 pp 763ndash773 2015
[60] S M Islam S Das S Ghosh S Roy and P N Suganthan ldquoAnadaptive differential evolution algorithm with novel mutationand crossover strategies for global numerical optimizationrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 42 no 2 pp 482ndash499 2012
[61] Z L Gaing ldquoParticle swarm optimization to solving theeconomic dispatch considering the generator constraintsrdquo IEEETransactions on Power Systems vol 18 no 3 pp 1187ndash1195 2003
[62] X-S Yang and S Deb ldquoEngineering optimisation by Cuckoosearchrdquo International Journal of Mathematical Modelling andNumerical Optimisation vol 330 pp 43ndash51 2010
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Mathematical Problems in Engineering 11
the task after doing the first update each solution in thenew population must satisfy PPZ constraint and upper andlower boundaries by considering Equations (31) and (38)Then Equations (32)-(37) are performed for determiningall variables and penalty terms Finally Equation (38) isemployed to calculate fitness function and the solution withthe best value is assigned to the best one 11987811990011989711986611988711989011990411990545 Criterion of Stopping the Loop Algorithm In the loopalgorithm of using the proposed ICSA approach the solutionsearch work is stopped in case that the predeterminedmaximum iterations 119866119898119886119909 is reached For each search ter-mination the most effective solution is stored and anotherrun continues to be accomplished until the predeterminednumber of runs is reached After finishing the runs thebest one is found and reported In addition other valuessuch as the fitness of the worst solution and average fitnessof all solutions are also reported for comparing with othermethods
46 The Whole Iterative Process The whole iterative algo-rithm for implementing the proposed ICSA approach forcoping with OLD problem is described in detail in Figure 3
5 Results and Discussions
The proposed ICSA approach performance has been investi-gated on six cases with different fuel options different fuelcharacteristics and complicated constraints The details ofthe studied cases are presented as follows
Case 1 Four systems with single fuel source (SFS) and powerloss (PL) constraint
Subcase 11 A 3-unit system [57]Subcase 12 A 6-unit system [57]Subcase 13 A 3-unit system [56]Subcase 14 A 6-unit system [56]
Case 2 A 110-unit system with SFS [57]
Case 3 Four systems with SFS and the effects of valve loadingprocess (EoVLP)
Subcase 31 A 3-unit system supplying to a load of850MW [58]Subcase 32 A 13-unit system supplying to a load of1800MW [1]Subcase 33 A 13-unit system supplying to a load of2520MW [1]Subcase 34 A 40-unit system supplying to a load of2500MW [1]Subcase 35 An 80-unit system supplying to a load of4100MW [49]
Case 4 Two systems with SFS and PPZ and RPR constraints
Subcase 41 A 60-unit system supplying to a10600MW load [9]
Subcase 42 A 90-unit system supplying to a15900MW load [9]
Case 5 A 15-unit system with SFS and RRL PPZ and PLconstraints [61]
Case 6 Three systems with multiple fuel sources (MFS) andEoVLP
Subcase 61 An 80-unit system supplying to a21600MW load [15]Subcase 62 A 160-unit system supplying to a43200MW load [15]Subcase 63 A 320-unit system supplying to an86400MW load [54]
For each considered case with each load case the pro-posed ICSA approach is run 50 times on the programlanguage of Matlab and a PC with 4 GB of RAM and 24GHzprocessor The selection of adjustment parameters including119875119886 and 119879119900119897119909 is carefully considered to obtain the best optimalsolutions meanwhile two others such as 119873119901119904 and 119866119898119886119909 arechosen corresponding to the scale of particular test system9 values with the change of 01 in the range [01 09] are inturn selected for 119875119886 while 119879119900119897119909 is 001 at the beginning Theinformation including load demand119873119901119904 119866119898119886119909 and the best119875119886 is reported in Table 1
51 Obtained Results on Case 1 considering Four Systems withSFS and PL Constraint In this section we have implementedthe proposed ICSA approach for solving four systems dividedinto four subcases Tables 2 and 3 show the comparisons ofobtained results from Subcases 11 and 12 and Subcases 13and 14 respectively As listed in Table 2 the proposed ICSAmethod and CCSA can find equal fuel cost for Subcases 11whereas the reduction of fuel cost from the proposed ICSAmethod as compared to CCSA is clearer for Subcase 12 Asshown in Table 3 for comparing the proposed ICSA and threemethods consisting of CCSA ABC and FA the minimumfuel cost of the proposed ICSA is approximately equal to thatof these methods for Subcases 13 but much less than that ofthese methods for Subcase 14 Furthermore the proposedICSA has been run by setting 119873119901119904 and 119866119898119886119909 to 5 and 20but these values were much higher for CCSA ABC and FAThey are 20 and 5000 for CCSA 40 and 100 for ABC and 20and 5000 for FA Consequently the proposed method is veryefficient for Case 1 with four subcases
Optimal solutions obtained by ICSA for Case 1 are shownin Tables 16ndash18
52 Obtained Results on Case 2 considering 110-Unit Systemwith SFS In this section we have employed a very largescale system with 110 units but there were not challenges forobjective function and complex constraints since EoVLP andconstraints were not taken into account Both CCSA andthe proposed ICSA methods have been run for comparingwith BBO hybrid BBO and DE (DEBBO) and Opposi-tional real coded chemical reaction optimization algorithm(ORCCROA) in [36] IWA in [40] and AGWO in [52] As
12 Mathematical Problems in Engineering
Select parameters
- Correct solutions if violating PPZ constraint using Eq (31)- Determine using Eq (32)- Penalize for violating upper and lower bounds constraint using Eq (34)- Penalize for violating PPZ constraint using Eq (35)- Penalize the xth solution for violating RPR constraint using Eq (36)
- Determine quality of solutions employing Eq (37) - Select the most effective solution - Start the loop algorithm by selecting
- Perform the first solution update using Section 32 - Correct boundaries of solutions by using Eq (38)
- Correct solutions if violating PPZ constraint using Eq (31)- Determine using Eq (32)- Penalize for violating upper and lower bounds constraint using Eq (34)- Penalize for violating PPZ constraint using Eq (35)- Penalize the xth solution for violating RPR constraint using Eq(36)
- Determine fitness function using Eq (37)- Compare old solutions and new solutions to keep better ones
- Perform the second solution update using Algorithm 1 - Correct boundaries of solutions by using (38)
- Correct solutions if violating PPZ constraint using Eq (31)- Determine using Eq (32)- Penalize for violating upper and lower bounds constraint using Eq (34)- Penalize for violating PPZ constraint using Eq (35)- Penalize the xth solution for violating RPR constraint using Eq (36)
- Determine fitness function using Eq (37)- Compare old solutions and new solutions to keep better ones- Select the most effective solution
Stop
Start
- Redefine upper and lower bounds using Eqs (25)-(28)- Randomly generate initial population using Eq (30)
Nps Pa Gmax H> Tolx
P1xP1x
P1x
P1xP1x
P1x
P1xP1x
P1x
Sol<MN
Sol<MN
G = 1
G = Gmax G = G + 1
Figure 3 All computation steps for solving OLD problem by employing the proposed ICSA approach
shown in Table 4 AGWO [52] has reached less fuel cost thanICSA however the exact fuel cost which was recalculatedby using reported solution pointed out that the method hasreached a very high fuel cost of $2157404250 For comparisonwith other methods ICSA has found less fuel cost thanall these methods Particularly the reduction of generationfuel cost is significant as compared to BBO DEBBO andCCSA Execution time comparisons are also useful evidence
for indicating the high performance of ICSA Thus it canconclude that ICSA is a strong method for Case 2
Optimal solution obtained by ICSA for the case is shownin Table 19
53 Obtained Results on Case 3 considering Four Systems withSFS and EoVLP In this section the real performance of theproposed ICSA approach has been investigated based on five
Mathematical Problems in Engineering 13
Table 1 Information of considered cases and adjustment parameters
Case Fuel cost function Constraint Subcase No of units 119875119863 (MW) Nps Gmax Best 1198751198861 SFS PL
subcases with the gradual increase of number of units Thesmallest scale system considers 3 units but the largest scalesystem takes 80 units In addition to the implementation ofthe proposed ICSA we have also implemented CCSA forSubcase 31 and Subcase 35 for further comparison becauseCCSA has not been run for the two subcases so far
Comparison of obtained results from Subcase 31 shownin Table 5 indicates that the proposed ICSA is superior toCCSAwith lower fuel cost but it seems to be less effective thanMCSA [58] The minimum cost of MCSA reported in [58] isthe smallest fuel cost but the recalculated cost is much higherthan that of the proposed method Furthermore MCSA hasbeen implemented by setting very high values to 119873119901119904 and119866119898119886119909
Reports for Subcases 32 and 33 shown in Table 6are the comparisons of the proposed ICSA approach andother methods such as conventional Evolution programming(CEP) [1] Fast EP (FEP) [1] improved FEP (IFEP) [1] DE[12] multiplier Lagrange-based genetic algorithm with (GA-MU) [15] QPSO [16] GA-PS-SQP [30] PSO-SQP [32] M120573-HCLSA [49] IABCA [50] CCSA [59] OSE-CSA [59] SOS[34] MSOS [34] CEA-SQT [38] TSBO [39] IWA [40] andCBA [44] As observed from the table ICSAapproach obtainsbetter solutions than mostmethods excluding DE [10] CCSA
[59] OSE-CSA [59] SOS [34] MSOS [34] CEA-SQT [38]TSBO [39] IWA [40] and CBA [44] especially M120573-HCLSA[49] with lower cost $1796097 However recalculated costfrom reported solution of M120573-HCLSA is $179691 BesidesICSA is very fast as compared to most methods where twoother versions of Cuckoo search algorithm CCSA and OSE-CSA are also included except two methods in [34] Theprocessor of computer that all the methods run on is alsoreported in the final column Clearly ICSA approach is veryefficient for the case with the 13-unit system where effects ofvalve loading process are considered
In Subcase 34 the number of units is much larger thanthat of three subcases above up to 40 units [1] The obtainedresult comparisons with others are indicated in Table 7Clearly the minimum cost comparisons reveal that the pro-posedmethod is one of the leading methods due to the lowestcost except the comparison with CCSA [23] OSE-CSA [59]SOS [34] MSOS [34] EMA [45] 120579-MBA [47] and AGWOA[52] It is noted that AGWOA [52] has reported the bestminimum cost with $12140430 but recalculated minimumcost which was obtained by substituting reported optimalgeneration of all thermal generating units is $12141331 Theaverage and the maximum costs from the proposed methoddo not belong to the leading method group however the
Mathematical Problems in Engineering 15
Table 7 Result comparisons for Subcase 34
Approach Best cost Average cost Worst cost CPU time Computer($h) ($h) ($h) (s) (Processor-Ram)
execution time that ICSA approach takes is smaller thanmostones except IAPR [42] and 120579-MBA [47] which have been runon stronger computers Note that MSOS [34] has been fasterthan the proposed method for Subcases 31 and 32 abovebut it is too slower than the proposed method for the casenamely 1813 seconds compared to 146 seconds Comparedto two other versions of Cuckoo search CCSA [23] and OSE-CSA [59] the proposed method is also faster about threetimes although the processors are slightly different For thiscase 120579-MBA [47] shows a very good performance howeverthe method has not been tested onmore complicated systemsand larger scale and therefore more comparisons with themethod must end Clearly the proposed method is stillefficient for the case where large scale and effects of valveloading process are included
Subcase 35 is the largest scale system with 80 units Inaddition to the implementation of ICSA CCSA has beenalso run for the Subcase 35 for further investigation ofefficiency improvement of the proposed ICSA approach The
comparisons of minimum cost in Table 8 show that theproposed ICSA can find more optimal solution than M120573-HCLSA [49] AGWOA [52] and CCSA The proposed ICSAis also superior to CCSA in terms ofmore stable search abilityand lower fluctuation since its average cost and maximumcost are less than those of CCSA The outstanding figurecannot be reached as compared to AGWOA [52] howeverit is hard to conclude AGWOA [52] is superior to theproposed ICSA approach about more stable search abilityand lower fluctuation Actually comparison of the values ofpopulation and iterations as well as execution time cannot beaccomplished because the information was not reported in[52]Thus it can conclude that the proposed ICSA is effectivefor the subcase
Optimal solutions obtained by ICSA for the case areshown in Tables 20ndash23
54 Obtained Results on Case 4 with Two Systems consideringSFS and PPZ and RPR Constraints In this section two
16 Mathematical Problems in Engineering
Table 9 Result comparisons for Subcases 41 and 42
Subcase Approach Best cost ($h) Mean cost ($h) Worst cost ($h) Std dev ($h) CPU time (s) Computer (Processor-Ram)
test systems with SFS and PPZ and RPR constraints areconsidered The test system size is up to 60 and 90 units forSubcases 41 and 42 respectively Comparison for the casesis only performed with two Genetic algorithms consisting ofGA and IGA-MU in [9] and two other versions of Cuckoosearch algorithms including CCSA [23] and OSE-CSA [59]and presented in Table 9 Other studies have tended toignore such complicated constraints of PPZ and RPR Thecosts reported in Table 9 indicate that ICSA approach canobtain more effective solution than CCSA and OSE-CSAbecause it has reached lower minimum cost than the twoones Furthermore the proposed method also takes shortercomputation time for the two cases from about two timesto about three times although the processor of the proposedmethod is slightly strongerThemean costs of ICSA approachare much less than those from IGA-MU and GA and slightlyhigher than those from OSE-CSA but there is a trade-off between the proposed method and CCSA for the twosubcases In fact the proposed method obtains higher meancost for Subcase 41 but lower cost for Subcase 42 AlthoughGAmethods have been runon aweak computerwith 07 GHzof the processor compared to that with 24GHz in the studytheir execution times are significantly higher namely 56381seconds (GA) and 16258 seconds (IGA-MU) compared to09153 seconds of ICSA approach for Subcase 41 and 94093seconds (GA) and 25545 seconds (IGA-MU) compared to15892 seconds (the proposedmethod)The analysis can pointout that ICSA approach ismore efficient than these comparedmethods in terms of optimal solutions and execution time
Optimal solutions obtained by ICSA for Subcase 42 areshown in Table 24
55 Obtained Results on Case 5 with a 15-Unit System consid-ering SFS and RRL PPZ and PL Constraints In this section
a 15-unit system considering RRL PPZ and PL constraintsis considered to be solved for finding optimal solution Forefficiency investigation of the proposed ICSA we have alsoimplemented CCSA for comparison As listed in Table 10the proposed ICSA is the most effective method with thesmallest fuel cost The comparisons of control parameters aswell asCPU time are also good evidence to confirm the strongsearch of the proposed ICSA approach since it has been runby smaller values of control parameter and faster executiontime as compared to all methods excluding CCSA
Optimal solution obtained by ICSA for the case is shownin Table 25
56 Obtained Results on Case 6 withThree Systems consideringMFS and EoVLP In this section three test systems with thechallenge on objective function including multi-fossil fuelsources and effects of valve loading process are consideredThe scale is up to 80 units 160 units and 320 units forSubcases 61 62 and 63 respectively
Comparison for Subcase 61 reported in Table 11 revealsthat the proposed method is the best method in terms of thelowest best cost the lowestmean cost and the lowest standarddeviation and the fastest execution time The processor fromthis proposedmethod is about four times stronger thanCGA-MU and IGA-MU but the speed is from ten times to 35 timesfaster than these methods Compared to CCSA and OSE-CSA the proposed method is about two times faster but theprocessor is slightly stronger
Comparison for Subcase 62 is reported in Table 12Clearly the proposed ICSA approach obtains better values ofthe best mean and worst costs than most methods exceptMSOS [34] where the best cost difference is about $ 024However the proposed method is the second fastest onewith 1119 seconds where the first fastest one CBA [44] has
Mathematical Problems in Engineering 17
Table 11 Comparisons of found results for Subcase 61
Approach Best cost Mean cost Worst cost Std dev CPU time Computer($h) ($h) ($h) ($h) (s) (Processor-Ram)
spent 571 seconds Clearly MSOS is better than the proposedICSA approach with respect to slightly less best cost butis worse than the proposed method in terms of executiontime namely 2572 seconds compared to 1119 seconds whileCBA [44] is faster than the proposed method but obtainssignificantly worse costs The analysis can conclude that theproposed ICSA approach is very powerful for the subcasewith 160 units
Table 13 presents the comparison of three other methodsincludingCCSA [54] SOS [34] andMSOS [34] accompaniedwith the proposed method for Subcase 63 The obtainedresult comparisons imply that ICSA approach can obtainbetter values of the best and standard deviation costs thanCCSA and SOS but obtains slightly higher cost than MSOSby approximately $ 022 Besides the execution time fromICSA approach is much shorter than others especially it ishigher than five times faster than MSOS The four methodshave been run on approximately strong computers Brieflythe proposed ICSA approach can find and converge to morefavorable solution than other methods with shorter CPUtime except the comparison with MSOS which had bettersolution but spent higher than five times execution times
Consequently the proposed method is a very promisingoptimization algorithm for Subcase 63 a system up to 320units and with multi-fossil fuel sources and effects of valveloading process
Optimal solution obtained by ICSA for Subcase 63 isshown in Table 26
57 The Improvement of ICSA Approach Performance
571 The Outstanding Improvement over CCSA In this sec-tion the performance improvement of ICSA over CCSAhas been investigated by analyzing obtained results and setcontrol parameters Table 14 has been formed by addingreduction cost improvement level of the best cost executiontime and control parameters consisting of 119873119901119904 and 119866119898119886119909Among the compared factors reduction cost is the deviationof the cost of CCSA and that of ICSA whereas the improve-ment level is the ratio of the reduction cost to the cost ofCCSA The reduction costs indicate the proposed methodcould find either equal quality of solutions or higher qualityof solutions than CCSA for all study casesThe reduction costis from $0 to $42839 corresponding to the improvement level
18 Mathematical Problems in Engineering
Table 14 Summary of results obtained by CCSA and ICSA for all study cases
Study case Reduction cost ($) Improvement level () Execution time (s) Nps Gmax
Figure 4 The best run obtained by CCSA and ICSA for Case 2
that is from 0 to 08968The saving cost is not too much forone hour but the operation in one day onemonth or one yearis very high However it should be noted that CCSA has beenrun by setting much higher population size and iterations formany cases excluding study cases implemented in the studysuch as Case 2 Subcase 31 Subcase 35 and Case 5 Forinstance ICSA has used 119873119901119904 = 10 and 119866119898119886119909 = 15 for Subcase13 and119873119901119904 = 10 and119866119898119886119909 = 25 for Subcase 14 whereas CCSAhas been run by setting119873119901119904 = 20 and 119866119898119886119909 = 5000 for the twosubcases Similarly CCSA has been run for Subcases 32 33and 34 with much higher number of iterations For the lastsubcase CCSA has been run by setting 119873119901119904 = 320 and 119866119898119886119909
= 1200 but those of ICSA have been 10 and 9000 Due tothe higher value of control parameters CCSA has tended tospend more time in finding such high quality solutions foralmost all study cases Execution time of ICSA is less than18 seconds while that of CCSA is up to higher 75 secondsIt is clear that the proposed ICSA could find better optimalsolutions thanCCSA for such considered systems For furtherinvestigation of performance comparison the best runs over50 runs and fuel cost values of 50 runs obtained by CCSA andthe proposed ICSA for Case 2 Subcase 31 Subcase 35 andCase 5 have been plotted in from Figures 4ndash11 The best runcurves show the faster search of the proposed ICSA method
Mathematical Problems in Engineering 19
0 5 10 15 20 25 30 35 40 45 50Run
19795
198
19805
1981
19815
1982
19825
1983
19835
Fuel
cost
($)
CCSAICSA
times105
Figure 5The best fuel cost of 50 runs obtained by CCSA and ICSA for Case 2
0 2 4 6 8 10 12 14 16 18 20Iteration
8234
8236
8238
8240
8242
8244
8246
8248
8250
8252
Fitn
ess F
unct
ion
($)
CCSAICSA
Figure 6 The best run obtained by CCSA and ICSA for Subcase 31
whereas 50 values of fuel cost indicate that the proposedICSA can find many solutions with better quality Clearlythe proposed ICSA is outstanding in terms of stabilizationof solution search and faster convergence As a result it canconclude that the proposed ICSA approach is more effectivethanCCSA in solvingOLDproblemwith considered systems
572The Improvement of Results over Other Methods In thisarticle we have tested ICSA approach on 6 cases with 16
systems with different fuel cost forms different constraintsand different scale systems from 3 units to 320 units We havecompared the yielded results from ICSA approach and otherexisting ones for evaluating the efficiency of ICSA approachIn subsections above we have shown yielded results fromICSA approach and compared these results to those of otheronesHowever the demonstrationhad not been very good forobserving and comparing to lead to a conclusionThus in thesubsectionwehave summarized the result comparisons of the
20 Mathematical Problems in Engineering
0 5 10 15 20 25 30 35 40 45 50Run
8234
82342
82344
82346
82348
8235
82352
82354
82356
82358
8236
Fuel
cost
($)
CCSAICSA
Figure 7 The best fuel cost of 50 runs obtained by CCSA and ICSA for Subcase 31
0 1000 2000 3000 4000 5000 6000Iteration
242
244
246
248
25
252
254
256
Fitn
ess F
unct
ion
($)
CCSAICSA
times105
Figure 8 The best run obtained by CCSA and ICSA for Subcase 35
proposed and other ones Table 15 has reported the reductioncost (in $) of ICSA approach compared to other ones Inaddition we have converted the reduction cost into improve-ment level (in ) for better comparison The improvementhas been shown from the lowest level to the highest levelin terms of reduction cost and improvement percentageIn addition we have also given the slowest and the fastestexecution time of other compared methods together withthat of the proposed method The table implies that ICSA
approach can find better optimal solutions with less fuel costup to $052 for Subcase 13 $75229 for Subcase 14 $1775149for Case 2 $1181256 for Subcase 31 $8438 for Subcase 32$9113 for Subcase 33 $121181 for Subcase 34 $3412 forSubcase 35 $04329 for Subcase 41 $30227 for Subcase42 $407002 for Case 5 $18183 for Subcase 61 $13965 forSubcase 62 and $11817 for Subcase 63 These reductioncosts are equivalent to improvement level (IL) of 0032908968 82282 14143 047 038 099 001405
Mathematical Problems in Engineering 21
0 5 10 15 20 25 30 35 40 45 50Run
2428
243
2432
2434
2436
2438
244
2442
Fuel
cost
($)
CCSAICSA
times105
Figure 9 The best fuel cost of 50 runs obtained by CCSA and ICSA for Subcase 35
0 50 100 150 200 250 300 350 400Iteration
327
328
329
33
331
332
333
334
335
336
337
Fitn
ess F
unct
ion
($)
CCSAICSA
times104
Figure 10 The best run obtained by CCSA and ICSA for Case 5
00003 0002 12291 004 138 and 059 Thesequantitative comparisons reveal that larger scale systems canlead to better reduction cost but the improvement level isnot high because total cost of compared methods tends tobe large for large scale systems Furthermore very large scalesystems with nondifferentiable objective have been normallysolved by strong methods In fact systems in Case 3 havethe same characteristic with single fuel and effects of valveloading process but Subcase 34 is a larger scale system with
40 units while Subcases 31 32 and 33 are constructed by 3units 13 units and 13 units So the improvement percentageof Subcase 34 can be up to 099 whilst that of Subcases32 and 33 is 047 and 038 respectively Subcase 35 iswith the largest system 80 units but the reduction cost is notmuch only $3412 because compared methods with ICSA areeither state-of-the-art ones or improved ones Also Subcase41 and Subcase 42 have considered single fuel and PPZ andspinning reserve constraints but Subcase 42 is larger scale
22 Mathematical Problems in Engineering
Table 15 Performance improvement summary of the proposed method
Study cases Reduction cost ($) Improvement level () Execution time (s)From To From To Slowest method Fastest method Proposed method
Figure 11 The best fuel cost of 50 runs obtained by CCSA and ICSA for Case 5
system with 90 units and Subcase 41 is only with 60 unitsThus the improvement of Subcase 41 is lower with 00003but that of Subcase 42 is 0002 Similarly systems in Cases 6have the same featurewithmulti-fossil fuel sources and effectsof valve loading process but they are respectively constructedby 80 160 and 320 units As a result the improvements ofSubcase 62 138 and Subcase 63 059 are much higherthan Subcase 61 004 However Subcase 62 with smallernumber of units but getting higher improvement is easilyunderstood because there were nine compared methods butonly three compared methods are considered for Subcase 63In general the improvement is not high it is about under onedollar several dollars tens of dollars and over one thousand
dollars per hour however the saving cost will be significantif the operation is considered to be onemonth with 720 hoursor one year with 8760 hours
Execution time comparison can be evaluated by observ-ing the fastest and the slowest compared methods in Table 15These execution times of the proposed method are approxi-mately equal to that of the fastest methods for Subcases 3233 and 34 and much shorter than other fastest comparedmethods for other cases especially for Subcases 61 and 63The fastest method for Subcase 62 is CBA [44] showing 57seconds while that of the proposed method is 1119 secondsHowever it cannot conclude that CBA is more effective thanthe proposed method because the proposed method could
find better optimal solution with less fuel cost by $2130As considering execution time of the slowest comparedmethods it can point out that ICSA is a very fast optimizationtool since the execution time of these methods is 29496 and3425 seconds for Subcases 32 and 33 116735 seconds forSubcase 34 56381 seconds for Subcase 41 94093 secondsfor Subcase 42 12797 seconds for Case 5 30941 seconds forSubcase 61 7542 seconds for Subcase 62 and 9641 secondsfor Subcase 63 while the execution time of the proposedmethod for these cases is respectively 095 146 091 1589246 84828 1119 and 171384 seconds It is clearly shown thatICSA is very fast as compared to these methods
In summary the proposed method has found approx-imately high quality solutions with several standard state-of-the-art meta-heuristic algorithms and improved versionsof them together with other old methods In addition theproposed method could improve result better than approx-imately all methods with faster execution time Comparedto other methods with the fastest convergence speed andhigh quality solutions the proposed method has been as
fast as for some cases and much faster for other cases Thecomparison with the slowest methods could show that theproposed method was extremely powerful since it was up tonearly one thousand times faster Consequently the proposedICSA approach can be one of the strongest optimization toolsfor OLD problem
6 Conclusions
This paper has proposed a good ICSA method for solvingOLD problem in which many test systems with differentobjective functions and complicated constraints from simpleto complex have been used as studied cases The proposedICSA method has been developed by performing severalmodifications on the second solution update of CCSA whichcontained several drawbacks to global convergence and fastmanner The OLD problem has covered from single fuelto multi-fossil fuels from quadratic objective function tononconvex objective function in addition to PPZ RPR andRRL constraints Many existing optimization algorithms have
been concerned in aim to compare the performance and givethe final conclusion on the proposed method There have
been six main cases with sixteen subcases The evaluationshave been made at the end of each study case Clearly theproposed ICSA approach has yielded more effective optimalsolutions with faster execution time than almost all methodsConsequently it can be concluded that the proposed methodis much more superior to CCSA and is a very promisingmethod for solving OLD problem
Appendix
See Tables 16ndash26
Nomenclature
120575119894 120582119894 120572119894 120573119894 120574119894 Fuel cost functioncoefficients of the 119894119905ℎthermal generationunit11986100 1198610119895 119861119895119894 Power loss matrixcoefficients119865119865119909 119865119865119887119890119904119905 The values of fitness ofsolution 119909 and theso-far most effectivesolution among thecurrent set ofsolutions119898119894 Number of fuels burntin the 119894119905ℎ thermalgeneration unit
119873 Number of allavailable thermalgeneration units119899119894 Number of prohibitedpower zones of the 119894119905ℎthermal generationunit119875119906119894119896 119875119897119894119896 Upper and lowerlimits of the 119894119905ℎthermal generationunit corresponding tothe kth PPZ119875119886 Probability ofreplacing controlvariables in each oldsolution119875119863 Real power demand ofall loads in system119875119894119898119886119909 119875119894119898119894119899 The highest andlowest real poweroutputs of the iththermal generationunit119875119894119895119898119886119909 119875119894119895119898119894119899 The highest andlowest real poweroutputs of the 119894119905ℎthermal generationunit corresponding tothe 119895119905ℎ fuel type119878119894119898119886119909 Maximum real powerreserve contributionof the thermalgeneration unit 1198941198781199001198971199031198861198991198891 1198781199001198971199031198861198991198892 1198781199001198971199031198861198991198893 1198781199001198971199031198861198991198894 Randomly mixedsolutions from the setof current solutions119878119900119897119909 119878119900119897119866119887119890119904119905 The old solution x andthe most effectivesolution119878119877 Real power reserverequirement of system
120575119894119895 120582119894119895 120572119894119895 120573119894119895 120574119894119895 Fuel cost functioncoefficients of the 119894119905ℎthermal generationunit corresponding tothe 119895119905ℎ fuel type1205761 1205762 1205763 1205764 1205765 1205766 Random numbersbetween 0 and 1Ω Number of generationunits considering PPZconstraint
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare no conflicts of interest
References
[1] N Sinha R Chakrabarti and P K Chattopadhyay ldquoEvolution-ary programming techniques for economic load dispatchrdquo IEEETransactions on Evolutionary Computation vol 7 no 1 pp 83ndash94 2003
[2] P Hansen ldquoA separable approximation dynamic programmingalgorithm for economic dispatch with transmission lossesrdquoYugoslav Journal of Operations Research vol 12 no 2 2002
[3] N T Thang ldquoSolving economic dispatch problem with piece-wise quadratic cost functions using lagrange multiplier theoryrdquoin Proceedings of the 3rd International Conference on ComputerTechnology and Development (ICCTD rsquo11) pp 359ndash364 ASMEPress 2011
[4] S K Mishra and S K Mishra ldquoA comparative study of solutionof economic load dispatch problem in power systems in theenvironmental perspectiverdquoProcedia Computer Science vol 48pp 96ndash100 2015
[5] A A Al-Subhi and H K Alfares ldquoEconomic load dispatchusing linear programming a comparative studyrdquo InternationalJournal of Applied Industrial Engineering vol 3 no 1 pp 16ndash362016
28 Mathematical Problems in Engineering
[6] C Zhou G Huang and J Chen ldquoPlanning of electric powersystems considering virtual power plants with dispatchableloads included an inexact two-stage stochastic linear program-ming modelrdquoMathematical Problems in Engineering vol 2018Article ID 7049329 12 pages 2018
[7] T T Nguyen N V Quynh and L Van Dai ldquoImproved fireflyalgorithm a novel method for optimal operation of thermalgenerating unitsrdquo Complexity vol 2018 Article ID 7267593 23pages 2018
[8] S S Haroon S Hassan S Amin et al ldquoMultiple fuel machinespower economic dispatch using stud differential evolutionrdquoEnergies vol 11 no 6 pp 1ndash20 2018
[9] C-T Su and C-L Chiang ldquoNonconvex power economic dis-patch by improved genetic algorithm with multiplier updatingmethodrdquo Electric Power Components and Systems vol 32 no 3pp 257ndash273 2004
[10] NNoman andH Iba ldquoDifferential evolution for economic loaddispatch problemsrdquo Electric Power Systems Research vol 78 no8 pp 1322ndash1331 2008
[11] P Somasundaram and K Kuppusamy ldquoApplication of evolu-tionary programming to security constrained economic dis-patchrdquo International Journal of Electrical Power amp EnergySystems vol 27 no 5-6 pp 343ndash351 2005
[12] T Jayabarathi K Jayaprakash andD Jeyakumar ldquoEvolutionaryprogramming technique for different kinds of economic dis-patch problemsrdquo Electric Power Systems Research vol 73 no 2pp 169ndash176 2005
[13] P Subbaraj R Rengaraj and S Salivahanan ldquoEnhancementof combined heat and power economic dispatch using selfadaptive real-coded genetic algorithmrdquo Applied Energy vol 86no 6 pp 915ndash921 2009
[14] C-L Chiang and C-T Su ldquoAdaptive-improved genetic algo-rithm for the economic dispatch of units with multiple fueloptionsrdquo Cybernetics and Systems vol 36 no 7 pp 687ndash7042005
[15] C Chiang ldquoImproved genetic algorithm for power economicdispatch of units with valve-point effects and multiple fuelsrdquoIEEE Transactions on Power Systems vol 20 no 4 pp 1690ndash1699 2005
[16] K Meng H G Wang Z Y Dong and K P Wong ldquoQuantum-inspired particle swarm optimization for valve-point economicload dispatchrdquo IEEE Transactions on Power Systems vol 25 no1 pp 215ndash222 2010
[17] P Erdogmus A Ozturk and S Duman ldquoEnvironmen-taleconomic dispatch using genetic algorithm and simulatedannealingrdquo in Proceedings of the International Conference onElectric Power and Energy Conversion Systems EPECS rsquo09 pp1ndash4 IEEE 2009
[18] L D S Coelho and V C Mariani ldquoImproved differentialevolution algorithms for handling economic dispatch opti-mization with generator constraintsrdquo Energy Conversion andManagement vol 48 no 5 pp 1631ndash1639 2007
[19] H Dakuo W Fuli and M Zhizhong ldquoA hybrid genetic algo-rithm approach based on differential evolution for economicdispatch with valve-point effectsrdquo Electrical Power and EnergySystems vol 30 pp 31ndash38 2008
[20] C-F Chang J-J Wong J-P Chiou and C-T Su ldquoRobustsearching hybrid differential evolution method for optimalreactive power planning in large-scale distribution systemsrdquoElectric Power Systems Research vol 77 no 5-6 pp 430ndash4372007
[21] M R Farooqi P Jain and K R Niazi ldquoUsing Hopfield neuralnetwork for economic dispatch of power systemsrdquo in Proceed-ings of the National Power Engineering Conference PECon rsquo03pp 5ndash10 Bangi Malaysia 2003
[22] T Yalcinoz H Altun and U Hasan ldquoConstrained economicdispatch with prohibited operating zones a Hopfield neuralnetwork approachrdquo in Proceedings of the 10th MediterraneanElectrotechnical Conference Information Technology and Elec-trotechnology for the Mediterranean Countries ProceedingsMeleCon rsquo00 (Cat No 00CH37099) pp 570ndash573 LemesosCyprus 2000
[23] D N Vo P Schegner and W Ongsakul ldquoCuckoo searchalgorithm for non-convex economic dispatchrdquo IET GenerationTransmission amp Distribution vol 7 no 6 pp 645ndash654 2013
[24] J-B Park K-S Lee J-R Shin and K Y Lee ldquoA particleswarm optimization for economic dispatch with non-smoothcost functionsrdquo IEEETransactions on Power Systems vol 20 no1 pp 34ndash42 2005
[25] T Niknam H D Mojarrad and M Nayeripour ldquoA newfuzzy adaptive particle swarm optimization for non-smootheconomic dispatchrdquo Energy vol 35 no 4 pp 1764ndash1778 2010
[26] L D S Coelho and V C Mariani ldquoAn improved harmonysearch algorithm for power economic load dispatchrdquo EnergyConversion and Management vol 50 no 10 pp 2522ndash25262009
[27] M Mahdavi M Fesanghary and E Damangir ldquoAn improvedharmony search algorithm for solving optimization problemsrdquoAppliedMathematics and Computation vol 188 no 2 pp 1567ndash1579 2007
[28] A Bhattacharya and P K Chattopadhyay ldquoSolving complexeconomic load dispatch problem using biogeography-basedoptimizationrdquo Expert Systems with Applications vol 37 no 5pp 3605ndash3615 2010
[29] D Simon ldquoBiogeography-based optimizationrdquo IEEE Transac-tions on Evolutionary Computation vol 12 no 6 pp 702ndash7132008
[30] J S Alsumait J K Sykulski and A K Al-Othman ldquoAhybrid GA-PS-SQP method to solve power system valve-pointeconomic dispatch problemsrdquo Applied Energy vol 87 no 5 pp1773ndash1781 2010
[31] M S Turgut andGKDemir ldquoQuadratic approximationndashbasedhybrid Artificial Cooperative Search algorithm for economicemission load dispatchproblemsrdquo International Transactions onElectrical Energy Systems vol 27 no 4 pp 1ndash14 2017
[32] T A A Victoire and A E Jeyakumar ldquoHybrid PSO-SQPfor economic dispatch with valve-point effectrdquo Electric PowerSystems Research vol 71 no 1 pp 51ndash59 2004
[33] D-KHe F-LWang andZ-ZMao ldquoHybrid genetic algorithmfor economic dispatch with valve-point effectrdquo Electric PowerSystems Research vol 78 no 4 pp 626ndash633 2008
[34] D C Secui ldquoA modified symbiotic organisms search algorithmfor large scale economic dispatch problem with valve-pointloading effectsrdquo Energy vol 113 pp 366ndash384 2016
[35] K BhattacharjeeA Bhattacharya and S H N Dey ldquoTeaching-learning-based optimization for different economic dispatchproblemsrdquo Scientia Iranica vol 21 no 3 pp 870ndash884 2014
[36] K Bhattacharjee A Bhattacharya and S H N Dey ldquoOppo-sitional Real Coded Chemical Reaction Optimization for dif-ferent economic dispatch problemsrdquo International Journal ofElectrical Power amp Energy Systems vol 55 pp 378ndash391 2014
Mathematical Problems in Engineering 29
[37] V Hosseinnezhad M Rafiee M Ahmadian and M T AmelildquoSpecies-basedQuantum Particle SwarmOptimization for eco-nomic load dispatchrdquo International Journal of Electrical Poweramp Energy Systems vol 63 pp 311ndash322 2014
[38] M S P Subathra S E Selvan T A A Victoire A HChristinal andU Amato ldquoA hybrid with cross-entropymethodand sequential quadratic programming to solve economic loaddispatch problemrdquo IEEE Systems Journal vol 9 no 3 pp 1031ndash1044 2015
[39] J Zhan Q H Wu C Guo and X Zhou ldquoEconomic dispatchwith non-smooth objectives part I local minimum analysisrdquoIEEE Transactions on Power Systems vol 30 no 2 pp 710ndash7212015
[40] A K Barisal andR C Prusty ldquoLarge scale economic dispatch ofpower systems using oppositional invasive weed optimizationrdquoApplied Soft Computing vol 29 pp 122ndash137 2015
[41] M Basu ldquoImproved differential evolution for economic dis-patchrdquo International Journal of Electrical Power amp EnergySystems vol 63 pp 855ndash861 2014
[42] V S Aragon S C Esquivel and C C Coello ldquoAn immunealgorithm with power redistribution for solving economicdispatch problemsrdquo Information Sciences vol 295 pp 609ndash6322015
[43] M Ghasemi M Taghizadeh S Ghavidel and A AbbasianldquoColonial competitive differential evolution an experimentalstudy for optimal economic load dispatchrdquo Applied Soft Com-puting vol 40 pp 342ndash363 2016
[44] B R Adarsh T Raghunathan T Jayabarathi and X-S YangldquoEconomic dispatch using chaotic bat algorithmrdquo Energy vol96 pp 666ndash675 2016
[45] N Ghorbani and E Babaei ldquoExchange market algorithm foreconomic load dispatchrdquo International Journal of ElectricalPower amp Energy Systems vol 75 pp 19ndash27 2016
[46] J X V Neto G Reynoso-Meza T H Ruppel V C Marianiand L D S Coelho ldquoSolving non-smooth economic dispatchby a new combination of continuous GRASP algorithm anddifferential evolutionrdquo International Journal of Electrical Poweramp Energy Systems vol 84 pp 13ndash24 2017
[47] A Kavousi-Fard and A Khosravi ldquoAn intelligent 120579-ModifiedBat Algorithm to solve the non-convex economic dispatchproblem considering practical constraintsrdquoElectrical Power andEnergy Systems vol 82 pp 189ndash196 2016
[48] M A Al-Betar M A Awadallah A T Khader and A L BolajildquoTournament-based harmony search algorithm for non-convexeconomic load dispatch problemrdquo Applied Soft Computing vol47 pp 449ndash459 2016
[49] M A Al-Betar M A Awadallah I A Doush E Alsukhni andH ALkhraisat ldquoA non-convex economic dispatchproblemwithvalve loading effect using a new modified 120573-hill climbing localsearch algorithmrdquo Arabian Journal for Science and Engineeringpp 1ndash8 2018
[50] M A Awadallah M A Al-Betar A L Bolaji E M Alsukhniand H Al-Zoubi ldquoNatural selection methods for artificial beecolony with new versions of onlooker beerdquo Soft Computing pp1ndash40 2018
[51] S H Kaboli and A K Alqallaf ldquoSolving non-convex economicload dispatch problem via artificial cooperative search algo-rithmrdquo Expert Systems with Applications vol 128 pp 14ndash272019
[52] D Singh and J Dhillon ldquoAmeliorated greywolf optimization foreconomic load dispatch problemrdquo Energy vol 169 pp 398ndash4192019
[53] X S Yang and S Deb ldquoCuckoo search via Levy flightsrdquo inProceedings of the World Congress on Nature and BiologicallyInspired Computing (NaBIC rsquo09) pp 210ndash214 India 2009
[54] S Sahoo K M Dash R C Prusty and A K Barisal ldquoCom-parative analysis of optimal load dispatch through evolutionaryalgorithmsrdquo Ain Shams Engineering Journal vol 6 pp 107ndash1202015
[55] M Basu and A Chowdhury ldquoCuckoo search algorithm foreconomic dispatchrdquo Energy vol 60 pp 99ndash108 2013
[56] A B Serapiao ldquoCuckoo search for solving economic dispatchload problemrdquo Intelligent Control and Automation vol 04 no04 pp 385ndash390 2013
[57] A H Bindu and M D Reddy ldquoEconomic load dispatch usingcuckoo search algorithmrdquo International Journal of EngineeringResearch and Applications vol 3 pp 498ndash502 2013
[58] E Afzalan and M Joorabian ldquoAn improved cuckoo searchalgorithm for power economic load dispatchrdquo InternationalTransactions on Electrical Energy Systems vol 25 no 6 pp 958ndash975 2015
[59] T T Nguyen and D N Vo ldquoThe application of one rank cuckoosearch algorithm for solving economic load dispatch problemsrdquoApplied Soft Computing vol 37 pp 763ndash773 2015
[60] S M Islam S Das S Ghosh S Roy and P N Suganthan ldquoAnadaptive differential evolution algorithm with novel mutationand crossover strategies for global numerical optimizationrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 42 no 2 pp 482ndash499 2012
[61] Z L Gaing ldquoParticle swarm optimization to solving theeconomic dispatch considering the generator constraintsrdquo IEEETransactions on Power Systems vol 18 no 3 pp 1187ndash1195 2003
[62] X-S Yang and S Deb ldquoEngineering optimisation by Cuckoosearchrdquo International Journal of Mathematical Modelling andNumerical Optimisation vol 330 pp 43ndash51 2010
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12 Mathematical Problems in Engineering
Select parameters
- Correct solutions if violating PPZ constraint using Eq (31)- Determine using Eq (32)- Penalize for violating upper and lower bounds constraint using Eq (34)- Penalize for violating PPZ constraint using Eq (35)- Penalize the xth solution for violating RPR constraint using Eq (36)
- Determine quality of solutions employing Eq (37) - Select the most effective solution - Start the loop algorithm by selecting
- Perform the first solution update using Section 32 - Correct boundaries of solutions by using Eq (38)
- Correct solutions if violating PPZ constraint using Eq (31)- Determine using Eq (32)- Penalize for violating upper and lower bounds constraint using Eq (34)- Penalize for violating PPZ constraint using Eq (35)- Penalize the xth solution for violating RPR constraint using Eq(36)
- Determine fitness function using Eq (37)- Compare old solutions and new solutions to keep better ones
- Perform the second solution update using Algorithm 1 - Correct boundaries of solutions by using (38)
- Correct solutions if violating PPZ constraint using Eq (31)- Determine using Eq (32)- Penalize for violating upper and lower bounds constraint using Eq (34)- Penalize for violating PPZ constraint using Eq (35)- Penalize the xth solution for violating RPR constraint using Eq (36)
- Determine fitness function using Eq (37)- Compare old solutions and new solutions to keep better ones- Select the most effective solution
Stop
Start
- Redefine upper and lower bounds using Eqs (25)-(28)- Randomly generate initial population using Eq (30)
Nps Pa Gmax H> Tolx
P1xP1x
P1x
P1xP1x
P1x
P1xP1x
P1x
Sol<MN
Sol<MN
G = 1
G = Gmax G = G + 1
Figure 3 All computation steps for solving OLD problem by employing the proposed ICSA approach
shown in Table 4 AGWO [52] has reached less fuel cost thanICSA however the exact fuel cost which was recalculatedby using reported solution pointed out that the method hasreached a very high fuel cost of $2157404250 For comparisonwith other methods ICSA has found less fuel cost thanall these methods Particularly the reduction of generationfuel cost is significant as compared to BBO DEBBO andCCSA Execution time comparisons are also useful evidence
for indicating the high performance of ICSA Thus it canconclude that ICSA is a strong method for Case 2
Optimal solution obtained by ICSA for the case is shownin Table 19
53 Obtained Results on Case 3 considering Four Systems withSFS and EoVLP In this section the real performance of theproposed ICSA approach has been investigated based on five
Mathematical Problems in Engineering 13
Table 1 Information of considered cases and adjustment parameters
Case Fuel cost function Constraint Subcase No of units 119875119863 (MW) Nps Gmax Best 1198751198861 SFS PL
subcases with the gradual increase of number of units Thesmallest scale system considers 3 units but the largest scalesystem takes 80 units In addition to the implementation ofthe proposed ICSA we have also implemented CCSA forSubcase 31 and Subcase 35 for further comparison becauseCCSA has not been run for the two subcases so far
Comparison of obtained results from Subcase 31 shownin Table 5 indicates that the proposed ICSA is superior toCCSAwith lower fuel cost but it seems to be less effective thanMCSA [58] The minimum cost of MCSA reported in [58] isthe smallest fuel cost but the recalculated cost is much higherthan that of the proposed method Furthermore MCSA hasbeen implemented by setting very high values to 119873119901119904 and119866119898119886119909
Reports for Subcases 32 and 33 shown in Table 6are the comparisons of the proposed ICSA approach andother methods such as conventional Evolution programming(CEP) [1] Fast EP (FEP) [1] improved FEP (IFEP) [1] DE[12] multiplier Lagrange-based genetic algorithm with (GA-MU) [15] QPSO [16] GA-PS-SQP [30] PSO-SQP [32] M120573-HCLSA [49] IABCA [50] CCSA [59] OSE-CSA [59] SOS[34] MSOS [34] CEA-SQT [38] TSBO [39] IWA [40] andCBA [44] As observed from the table ICSAapproach obtainsbetter solutions than mostmethods excluding DE [10] CCSA
[59] OSE-CSA [59] SOS [34] MSOS [34] CEA-SQT [38]TSBO [39] IWA [40] and CBA [44] especially M120573-HCLSA[49] with lower cost $1796097 However recalculated costfrom reported solution of M120573-HCLSA is $179691 BesidesICSA is very fast as compared to most methods where twoother versions of Cuckoo search algorithm CCSA and OSE-CSA are also included except two methods in [34] Theprocessor of computer that all the methods run on is alsoreported in the final column Clearly ICSA approach is veryefficient for the case with the 13-unit system where effects ofvalve loading process are considered
In Subcase 34 the number of units is much larger thanthat of three subcases above up to 40 units [1] The obtainedresult comparisons with others are indicated in Table 7Clearly the minimum cost comparisons reveal that the pro-posedmethod is one of the leading methods due to the lowestcost except the comparison with CCSA [23] OSE-CSA [59]SOS [34] MSOS [34] EMA [45] 120579-MBA [47] and AGWOA[52] It is noted that AGWOA [52] has reported the bestminimum cost with $12140430 but recalculated minimumcost which was obtained by substituting reported optimalgeneration of all thermal generating units is $12141331 Theaverage and the maximum costs from the proposed methoddo not belong to the leading method group however the
Mathematical Problems in Engineering 15
Table 7 Result comparisons for Subcase 34
Approach Best cost Average cost Worst cost CPU time Computer($h) ($h) ($h) (s) (Processor-Ram)
execution time that ICSA approach takes is smaller thanmostones except IAPR [42] and 120579-MBA [47] which have been runon stronger computers Note that MSOS [34] has been fasterthan the proposed method for Subcases 31 and 32 abovebut it is too slower than the proposed method for the casenamely 1813 seconds compared to 146 seconds Comparedto two other versions of Cuckoo search CCSA [23] and OSE-CSA [59] the proposed method is also faster about threetimes although the processors are slightly different For thiscase 120579-MBA [47] shows a very good performance howeverthe method has not been tested onmore complicated systemsand larger scale and therefore more comparisons with themethod must end Clearly the proposed method is stillefficient for the case where large scale and effects of valveloading process are included
Subcase 35 is the largest scale system with 80 units Inaddition to the implementation of ICSA CCSA has beenalso run for the Subcase 35 for further investigation ofefficiency improvement of the proposed ICSA approach The
comparisons of minimum cost in Table 8 show that theproposed ICSA can find more optimal solution than M120573-HCLSA [49] AGWOA [52] and CCSA The proposed ICSAis also superior to CCSA in terms ofmore stable search abilityand lower fluctuation since its average cost and maximumcost are less than those of CCSA The outstanding figurecannot be reached as compared to AGWOA [52] howeverit is hard to conclude AGWOA [52] is superior to theproposed ICSA approach about more stable search abilityand lower fluctuation Actually comparison of the values ofpopulation and iterations as well as execution time cannot beaccomplished because the information was not reported in[52]Thus it can conclude that the proposed ICSA is effectivefor the subcase
Optimal solutions obtained by ICSA for the case areshown in Tables 20ndash23
54 Obtained Results on Case 4 with Two Systems consideringSFS and PPZ and RPR Constraints In this section two
16 Mathematical Problems in Engineering
Table 9 Result comparisons for Subcases 41 and 42
Subcase Approach Best cost ($h) Mean cost ($h) Worst cost ($h) Std dev ($h) CPU time (s) Computer (Processor-Ram)
test systems with SFS and PPZ and RPR constraints areconsidered The test system size is up to 60 and 90 units forSubcases 41 and 42 respectively Comparison for the casesis only performed with two Genetic algorithms consisting ofGA and IGA-MU in [9] and two other versions of Cuckoosearch algorithms including CCSA [23] and OSE-CSA [59]and presented in Table 9 Other studies have tended toignore such complicated constraints of PPZ and RPR Thecosts reported in Table 9 indicate that ICSA approach canobtain more effective solution than CCSA and OSE-CSAbecause it has reached lower minimum cost than the twoones Furthermore the proposed method also takes shortercomputation time for the two cases from about two timesto about three times although the processor of the proposedmethod is slightly strongerThemean costs of ICSA approachare much less than those from IGA-MU and GA and slightlyhigher than those from OSE-CSA but there is a trade-off between the proposed method and CCSA for the twosubcases In fact the proposed method obtains higher meancost for Subcase 41 but lower cost for Subcase 42 AlthoughGAmethods have been runon aweak computerwith 07 GHzof the processor compared to that with 24GHz in the studytheir execution times are significantly higher namely 56381seconds (GA) and 16258 seconds (IGA-MU) compared to09153 seconds of ICSA approach for Subcase 41 and 94093seconds (GA) and 25545 seconds (IGA-MU) compared to15892 seconds (the proposedmethod)The analysis can pointout that ICSA approach ismore efficient than these comparedmethods in terms of optimal solutions and execution time
Optimal solutions obtained by ICSA for Subcase 42 areshown in Table 24
55 Obtained Results on Case 5 with a 15-Unit System consid-ering SFS and RRL PPZ and PL Constraints In this section
a 15-unit system considering RRL PPZ and PL constraintsis considered to be solved for finding optimal solution Forefficiency investigation of the proposed ICSA we have alsoimplemented CCSA for comparison As listed in Table 10the proposed ICSA is the most effective method with thesmallest fuel cost The comparisons of control parameters aswell asCPU time are also good evidence to confirm the strongsearch of the proposed ICSA approach since it has been runby smaller values of control parameter and faster executiontime as compared to all methods excluding CCSA
Optimal solution obtained by ICSA for the case is shownin Table 25
56 Obtained Results on Case 6 withThree Systems consideringMFS and EoVLP In this section three test systems with thechallenge on objective function including multi-fossil fuelsources and effects of valve loading process are consideredThe scale is up to 80 units 160 units and 320 units forSubcases 61 62 and 63 respectively
Comparison for Subcase 61 reported in Table 11 revealsthat the proposed method is the best method in terms of thelowest best cost the lowestmean cost and the lowest standarddeviation and the fastest execution time The processor fromthis proposedmethod is about four times stronger thanCGA-MU and IGA-MU but the speed is from ten times to 35 timesfaster than these methods Compared to CCSA and OSE-CSA the proposed method is about two times faster but theprocessor is slightly stronger
Comparison for Subcase 62 is reported in Table 12Clearly the proposed ICSA approach obtains better values ofthe best mean and worst costs than most methods exceptMSOS [34] where the best cost difference is about $ 024However the proposed method is the second fastest onewith 1119 seconds where the first fastest one CBA [44] has
Mathematical Problems in Engineering 17
Table 11 Comparisons of found results for Subcase 61
Approach Best cost Mean cost Worst cost Std dev CPU time Computer($h) ($h) ($h) ($h) (s) (Processor-Ram)
spent 571 seconds Clearly MSOS is better than the proposedICSA approach with respect to slightly less best cost butis worse than the proposed method in terms of executiontime namely 2572 seconds compared to 1119 seconds whileCBA [44] is faster than the proposed method but obtainssignificantly worse costs The analysis can conclude that theproposed ICSA approach is very powerful for the subcasewith 160 units
Table 13 presents the comparison of three other methodsincludingCCSA [54] SOS [34] andMSOS [34] accompaniedwith the proposed method for Subcase 63 The obtainedresult comparisons imply that ICSA approach can obtainbetter values of the best and standard deviation costs thanCCSA and SOS but obtains slightly higher cost than MSOSby approximately $ 022 Besides the execution time fromICSA approach is much shorter than others especially it ishigher than five times faster than MSOS The four methodshave been run on approximately strong computers Brieflythe proposed ICSA approach can find and converge to morefavorable solution than other methods with shorter CPUtime except the comparison with MSOS which had bettersolution but spent higher than five times execution times
Consequently the proposed method is a very promisingoptimization algorithm for Subcase 63 a system up to 320units and with multi-fossil fuel sources and effects of valveloading process
Optimal solution obtained by ICSA for Subcase 63 isshown in Table 26
57 The Improvement of ICSA Approach Performance
571 The Outstanding Improvement over CCSA In this sec-tion the performance improvement of ICSA over CCSAhas been investigated by analyzing obtained results and setcontrol parameters Table 14 has been formed by addingreduction cost improvement level of the best cost executiontime and control parameters consisting of 119873119901119904 and 119866119898119886119909Among the compared factors reduction cost is the deviationof the cost of CCSA and that of ICSA whereas the improve-ment level is the ratio of the reduction cost to the cost ofCCSA The reduction costs indicate the proposed methodcould find either equal quality of solutions or higher qualityof solutions than CCSA for all study casesThe reduction costis from $0 to $42839 corresponding to the improvement level
18 Mathematical Problems in Engineering
Table 14 Summary of results obtained by CCSA and ICSA for all study cases
Study case Reduction cost ($) Improvement level () Execution time (s) Nps Gmax
Figure 4 The best run obtained by CCSA and ICSA for Case 2
that is from 0 to 08968The saving cost is not too much forone hour but the operation in one day onemonth or one yearis very high However it should be noted that CCSA has beenrun by setting much higher population size and iterations formany cases excluding study cases implemented in the studysuch as Case 2 Subcase 31 Subcase 35 and Case 5 Forinstance ICSA has used 119873119901119904 = 10 and 119866119898119886119909 = 15 for Subcase13 and119873119901119904 = 10 and119866119898119886119909 = 25 for Subcase 14 whereas CCSAhas been run by setting119873119901119904 = 20 and 119866119898119886119909 = 5000 for the twosubcases Similarly CCSA has been run for Subcases 32 33and 34 with much higher number of iterations For the lastsubcase CCSA has been run by setting 119873119901119904 = 320 and 119866119898119886119909
= 1200 but those of ICSA have been 10 and 9000 Due tothe higher value of control parameters CCSA has tended tospend more time in finding such high quality solutions foralmost all study cases Execution time of ICSA is less than18 seconds while that of CCSA is up to higher 75 secondsIt is clear that the proposed ICSA could find better optimalsolutions thanCCSA for such considered systems For furtherinvestigation of performance comparison the best runs over50 runs and fuel cost values of 50 runs obtained by CCSA andthe proposed ICSA for Case 2 Subcase 31 Subcase 35 andCase 5 have been plotted in from Figures 4ndash11 The best runcurves show the faster search of the proposed ICSA method
Mathematical Problems in Engineering 19
0 5 10 15 20 25 30 35 40 45 50Run
19795
198
19805
1981
19815
1982
19825
1983
19835
Fuel
cost
($)
CCSAICSA
times105
Figure 5The best fuel cost of 50 runs obtained by CCSA and ICSA for Case 2
0 2 4 6 8 10 12 14 16 18 20Iteration
8234
8236
8238
8240
8242
8244
8246
8248
8250
8252
Fitn
ess F
unct
ion
($)
CCSAICSA
Figure 6 The best run obtained by CCSA and ICSA for Subcase 31
whereas 50 values of fuel cost indicate that the proposedICSA can find many solutions with better quality Clearlythe proposed ICSA is outstanding in terms of stabilizationof solution search and faster convergence As a result it canconclude that the proposed ICSA approach is more effectivethanCCSA in solvingOLDproblemwith considered systems
572The Improvement of Results over Other Methods In thisarticle we have tested ICSA approach on 6 cases with 16
systems with different fuel cost forms different constraintsand different scale systems from 3 units to 320 units We havecompared the yielded results from ICSA approach and otherexisting ones for evaluating the efficiency of ICSA approachIn subsections above we have shown yielded results fromICSA approach and compared these results to those of otheronesHowever the demonstrationhad not been very good forobserving and comparing to lead to a conclusionThus in thesubsectionwehave summarized the result comparisons of the
20 Mathematical Problems in Engineering
0 5 10 15 20 25 30 35 40 45 50Run
8234
82342
82344
82346
82348
8235
82352
82354
82356
82358
8236
Fuel
cost
($)
CCSAICSA
Figure 7 The best fuel cost of 50 runs obtained by CCSA and ICSA for Subcase 31
0 1000 2000 3000 4000 5000 6000Iteration
242
244
246
248
25
252
254
256
Fitn
ess F
unct
ion
($)
CCSAICSA
times105
Figure 8 The best run obtained by CCSA and ICSA for Subcase 35
proposed and other ones Table 15 has reported the reductioncost (in $) of ICSA approach compared to other ones Inaddition we have converted the reduction cost into improve-ment level (in ) for better comparison The improvementhas been shown from the lowest level to the highest levelin terms of reduction cost and improvement percentageIn addition we have also given the slowest and the fastestexecution time of other compared methods together withthat of the proposed method The table implies that ICSA
approach can find better optimal solutions with less fuel costup to $052 for Subcase 13 $75229 for Subcase 14 $1775149for Case 2 $1181256 for Subcase 31 $8438 for Subcase 32$9113 for Subcase 33 $121181 for Subcase 34 $3412 forSubcase 35 $04329 for Subcase 41 $30227 for Subcase42 $407002 for Case 5 $18183 for Subcase 61 $13965 forSubcase 62 and $11817 for Subcase 63 These reductioncosts are equivalent to improvement level (IL) of 0032908968 82282 14143 047 038 099 001405
Mathematical Problems in Engineering 21
0 5 10 15 20 25 30 35 40 45 50Run
2428
243
2432
2434
2436
2438
244
2442
Fuel
cost
($)
CCSAICSA
times105
Figure 9 The best fuel cost of 50 runs obtained by CCSA and ICSA for Subcase 35
0 50 100 150 200 250 300 350 400Iteration
327
328
329
33
331
332
333
334
335
336
337
Fitn
ess F
unct
ion
($)
CCSAICSA
times104
Figure 10 The best run obtained by CCSA and ICSA for Case 5
00003 0002 12291 004 138 and 059 Thesequantitative comparisons reveal that larger scale systems canlead to better reduction cost but the improvement level isnot high because total cost of compared methods tends tobe large for large scale systems Furthermore very large scalesystems with nondifferentiable objective have been normallysolved by strong methods In fact systems in Case 3 havethe same characteristic with single fuel and effects of valveloading process but Subcase 34 is a larger scale system with
40 units while Subcases 31 32 and 33 are constructed by 3units 13 units and 13 units So the improvement percentageof Subcase 34 can be up to 099 whilst that of Subcases32 and 33 is 047 and 038 respectively Subcase 35 iswith the largest system 80 units but the reduction cost is notmuch only $3412 because compared methods with ICSA areeither state-of-the-art ones or improved ones Also Subcase41 and Subcase 42 have considered single fuel and PPZ andspinning reserve constraints but Subcase 42 is larger scale
22 Mathematical Problems in Engineering
Table 15 Performance improvement summary of the proposed method
Study cases Reduction cost ($) Improvement level () Execution time (s)From To From To Slowest method Fastest method Proposed method
Figure 11 The best fuel cost of 50 runs obtained by CCSA and ICSA for Case 5
system with 90 units and Subcase 41 is only with 60 unitsThus the improvement of Subcase 41 is lower with 00003but that of Subcase 42 is 0002 Similarly systems in Cases 6have the same featurewithmulti-fossil fuel sources and effectsof valve loading process but they are respectively constructedby 80 160 and 320 units As a result the improvements ofSubcase 62 138 and Subcase 63 059 are much higherthan Subcase 61 004 However Subcase 62 with smallernumber of units but getting higher improvement is easilyunderstood because there were nine compared methods butonly three compared methods are considered for Subcase 63In general the improvement is not high it is about under onedollar several dollars tens of dollars and over one thousand
dollars per hour however the saving cost will be significantif the operation is considered to be onemonth with 720 hoursor one year with 8760 hours
Execution time comparison can be evaluated by observ-ing the fastest and the slowest compared methods in Table 15These execution times of the proposed method are approxi-mately equal to that of the fastest methods for Subcases 3233 and 34 and much shorter than other fastest comparedmethods for other cases especially for Subcases 61 and 63The fastest method for Subcase 62 is CBA [44] showing 57seconds while that of the proposed method is 1119 secondsHowever it cannot conclude that CBA is more effective thanthe proposed method because the proposed method could
find better optimal solution with less fuel cost by $2130As considering execution time of the slowest comparedmethods it can point out that ICSA is a very fast optimizationtool since the execution time of these methods is 29496 and3425 seconds for Subcases 32 and 33 116735 seconds forSubcase 34 56381 seconds for Subcase 41 94093 secondsfor Subcase 42 12797 seconds for Case 5 30941 seconds forSubcase 61 7542 seconds for Subcase 62 and 9641 secondsfor Subcase 63 while the execution time of the proposedmethod for these cases is respectively 095 146 091 1589246 84828 1119 and 171384 seconds It is clearly shown thatICSA is very fast as compared to these methods
In summary the proposed method has found approx-imately high quality solutions with several standard state-of-the-art meta-heuristic algorithms and improved versionsof them together with other old methods In addition theproposed method could improve result better than approx-imately all methods with faster execution time Comparedto other methods with the fastest convergence speed andhigh quality solutions the proposed method has been as
fast as for some cases and much faster for other cases Thecomparison with the slowest methods could show that theproposed method was extremely powerful since it was up tonearly one thousand times faster Consequently the proposedICSA approach can be one of the strongest optimization toolsfor OLD problem
6 Conclusions
This paper has proposed a good ICSA method for solvingOLD problem in which many test systems with differentobjective functions and complicated constraints from simpleto complex have been used as studied cases The proposedICSA method has been developed by performing severalmodifications on the second solution update of CCSA whichcontained several drawbacks to global convergence and fastmanner The OLD problem has covered from single fuelto multi-fossil fuels from quadratic objective function tononconvex objective function in addition to PPZ RPR andRRL constraints Many existing optimization algorithms have
been concerned in aim to compare the performance and givethe final conclusion on the proposed method There have
been six main cases with sixteen subcases The evaluationshave been made at the end of each study case Clearly theproposed ICSA approach has yielded more effective optimalsolutions with faster execution time than almost all methodsConsequently it can be concluded that the proposed methodis much more superior to CCSA and is a very promisingmethod for solving OLD problem
Appendix
See Tables 16ndash26
Nomenclature
120575119894 120582119894 120572119894 120573119894 120574119894 Fuel cost functioncoefficients of the 119894119905ℎthermal generationunit11986100 1198610119895 119861119895119894 Power loss matrixcoefficients119865119865119909 119865119865119887119890119904119905 The values of fitness ofsolution 119909 and theso-far most effectivesolution among thecurrent set ofsolutions119898119894 Number of fuels burntin the 119894119905ℎ thermalgeneration unit
119873 Number of allavailable thermalgeneration units119899119894 Number of prohibitedpower zones of the 119894119905ℎthermal generationunit119875119906119894119896 119875119897119894119896 Upper and lowerlimits of the 119894119905ℎthermal generationunit corresponding tothe kth PPZ119875119886 Probability ofreplacing controlvariables in each oldsolution119875119863 Real power demand ofall loads in system119875119894119898119886119909 119875119894119898119894119899 The highest andlowest real poweroutputs of the iththermal generationunit119875119894119895119898119886119909 119875119894119895119898119894119899 The highest andlowest real poweroutputs of the 119894119905ℎthermal generationunit corresponding tothe 119895119905ℎ fuel type119878119894119898119886119909 Maximum real powerreserve contributionof the thermalgeneration unit 1198941198781199001198971199031198861198991198891 1198781199001198971199031198861198991198892 1198781199001198971199031198861198991198893 1198781199001198971199031198861198991198894 Randomly mixedsolutions from the setof current solutions119878119900119897119909 119878119900119897119866119887119890119904119905 The old solution x andthe most effectivesolution119878119877 Real power reserverequirement of system
120575119894119895 120582119894119895 120572119894119895 120573119894119895 120574119894119895 Fuel cost functioncoefficients of the 119894119905ℎthermal generationunit corresponding tothe 119895119905ℎ fuel type1205761 1205762 1205763 1205764 1205765 1205766 Random numbersbetween 0 and 1Ω Number of generationunits considering PPZconstraint
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare no conflicts of interest
References
[1] N Sinha R Chakrabarti and P K Chattopadhyay ldquoEvolution-ary programming techniques for economic load dispatchrdquo IEEETransactions on Evolutionary Computation vol 7 no 1 pp 83ndash94 2003
[2] P Hansen ldquoA separable approximation dynamic programmingalgorithm for economic dispatch with transmission lossesrdquoYugoslav Journal of Operations Research vol 12 no 2 2002
[3] N T Thang ldquoSolving economic dispatch problem with piece-wise quadratic cost functions using lagrange multiplier theoryrdquoin Proceedings of the 3rd International Conference on ComputerTechnology and Development (ICCTD rsquo11) pp 359ndash364 ASMEPress 2011
[4] S K Mishra and S K Mishra ldquoA comparative study of solutionof economic load dispatch problem in power systems in theenvironmental perspectiverdquoProcedia Computer Science vol 48pp 96ndash100 2015
[5] A A Al-Subhi and H K Alfares ldquoEconomic load dispatchusing linear programming a comparative studyrdquo InternationalJournal of Applied Industrial Engineering vol 3 no 1 pp 16ndash362016
28 Mathematical Problems in Engineering
[6] C Zhou G Huang and J Chen ldquoPlanning of electric powersystems considering virtual power plants with dispatchableloads included an inexact two-stage stochastic linear program-ming modelrdquoMathematical Problems in Engineering vol 2018Article ID 7049329 12 pages 2018
[7] T T Nguyen N V Quynh and L Van Dai ldquoImproved fireflyalgorithm a novel method for optimal operation of thermalgenerating unitsrdquo Complexity vol 2018 Article ID 7267593 23pages 2018
[8] S S Haroon S Hassan S Amin et al ldquoMultiple fuel machinespower economic dispatch using stud differential evolutionrdquoEnergies vol 11 no 6 pp 1ndash20 2018
[9] C-T Su and C-L Chiang ldquoNonconvex power economic dis-patch by improved genetic algorithm with multiplier updatingmethodrdquo Electric Power Components and Systems vol 32 no 3pp 257ndash273 2004
[10] NNoman andH Iba ldquoDifferential evolution for economic loaddispatch problemsrdquo Electric Power Systems Research vol 78 no8 pp 1322ndash1331 2008
[11] P Somasundaram and K Kuppusamy ldquoApplication of evolu-tionary programming to security constrained economic dis-patchrdquo International Journal of Electrical Power amp EnergySystems vol 27 no 5-6 pp 343ndash351 2005
[12] T Jayabarathi K Jayaprakash andD Jeyakumar ldquoEvolutionaryprogramming technique for different kinds of economic dis-patch problemsrdquo Electric Power Systems Research vol 73 no 2pp 169ndash176 2005
[13] P Subbaraj R Rengaraj and S Salivahanan ldquoEnhancementof combined heat and power economic dispatch using selfadaptive real-coded genetic algorithmrdquo Applied Energy vol 86no 6 pp 915ndash921 2009
[14] C-L Chiang and C-T Su ldquoAdaptive-improved genetic algo-rithm for the economic dispatch of units with multiple fueloptionsrdquo Cybernetics and Systems vol 36 no 7 pp 687ndash7042005
[15] C Chiang ldquoImproved genetic algorithm for power economicdispatch of units with valve-point effects and multiple fuelsrdquoIEEE Transactions on Power Systems vol 20 no 4 pp 1690ndash1699 2005
[16] K Meng H G Wang Z Y Dong and K P Wong ldquoQuantum-inspired particle swarm optimization for valve-point economicload dispatchrdquo IEEE Transactions on Power Systems vol 25 no1 pp 215ndash222 2010
[17] P Erdogmus A Ozturk and S Duman ldquoEnvironmen-taleconomic dispatch using genetic algorithm and simulatedannealingrdquo in Proceedings of the International Conference onElectric Power and Energy Conversion Systems EPECS rsquo09 pp1ndash4 IEEE 2009
[18] L D S Coelho and V C Mariani ldquoImproved differentialevolution algorithms for handling economic dispatch opti-mization with generator constraintsrdquo Energy Conversion andManagement vol 48 no 5 pp 1631ndash1639 2007
[19] H Dakuo W Fuli and M Zhizhong ldquoA hybrid genetic algo-rithm approach based on differential evolution for economicdispatch with valve-point effectsrdquo Electrical Power and EnergySystems vol 30 pp 31ndash38 2008
[20] C-F Chang J-J Wong J-P Chiou and C-T Su ldquoRobustsearching hybrid differential evolution method for optimalreactive power planning in large-scale distribution systemsrdquoElectric Power Systems Research vol 77 no 5-6 pp 430ndash4372007
[21] M R Farooqi P Jain and K R Niazi ldquoUsing Hopfield neuralnetwork for economic dispatch of power systemsrdquo in Proceed-ings of the National Power Engineering Conference PECon rsquo03pp 5ndash10 Bangi Malaysia 2003
[22] T Yalcinoz H Altun and U Hasan ldquoConstrained economicdispatch with prohibited operating zones a Hopfield neuralnetwork approachrdquo in Proceedings of the 10th MediterraneanElectrotechnical Conference Information Technology and Elec-trotechnology for the Mediterranean Countries ProceedingsMeleCon rsquo00 (Cat No 00CH37099) pp 570ndash573 LemesosCyprus 2000
[23] D N Vo P Schegner and W Ongsakul ldquoCuckoo searchalgorithm for non-convex economic dispatchrdquo IET GenerationTransmission amp Distribution vol 7 no 6 pp 645ndash654 2013
[24] J-B Park K-S Lee J-R Shin and K Y Lee ldquoA particleswarm optimization for economic dispatch with non-smoothcost functionsrdquo IEEETransactions on Power Systems vol 20 no1 pp 34ndash42 2005
[25] T Niknam H D Mojarrad and M Nayeripour ldquoA newfuzzy adaptive particle swarm optimization for non-smootheconomic dispatchrdquo Energy vol 35 no 4 pp 1764ndash1778 2010
[26] L D S Coelho and V C Mariani ldquoAn improved harmonysearch algorithm for power economic load dispatchrdquo EnergyConversion and Management vol 50 no 10 pp 2522ndash25262009
[27] M Mahdavi M Fesanghary and E Damangir ldquoAn improvedharmony search algorithm for solving optimization problemsrdquoAppliedMathematics and Computation vol 188 no 2 pp 1567ndash1579 2007
[28] A Bhattacharya and P K Chattopadhyay ldquoSolving complexeconomic load dispatch problem using biogeography-basedoptimizationrdquo Expert Systems with Applications vol 37 no 5pp 3605ndash3615 2010
[29] D Simon ldquoBiogeography-based optimizationrdquo IEEE Transac-tions on Evolutionary Computation vol 12 no 6 pp 702ndash7132008
[30] J S Alsumait J K Sykulski and A K Al-Othman ldquoAhybrid GA-PS-SQP method to solve power system valve-pointeconomic dispatch problemsrdquo Applied Energy vol 87 no 5 pp1773ndash1781 2010
[31] M S Turgut andGKDemir ldquoQuadratic approximationndashbasedhybrid Artificial Cooperative Search algorithm for economicemission load dispatchproblemsrdquo International Transactions onElectrical Energy Systems vol 27 no 4 pp 1ndash14 2017
[32] T A A Victoire and A E Jeyakumar ldquoHybrid PSO-SQPfor economic dispatch with valve-point effectrdquo Electric PowerSystems Research vol 71 no 1 pp 51ndash59 2004
[33] D-KHe F-LWang andZ-ZMao ldquoHybrid genetic algorithmfor economic dispatch with valve-point effectrdquo Electric PowerSystems Research vol 78 no 4 pp 626ndash633 2008
[34] D C Secui ldquoA modified symbiotic organisms search algorithmfor large scale economic dispatch problem with valve-pointloading effectsrdquo Energy vol 113 pp 366ndash384 2016
[35] K BhattacharjeeA Bhattacharya and S H N Dey ldquoTeaching-learning-based optimization for different economic dispatchproblemsrdquo Scientia Iranica vol 21 no 3 pp 870ndash884 2014
[36] K Bhattacharjee A Bhattacharya and S H N Dey ldquoOppo-sitional Real Coded Chemical Reaction Optimization for dif-ferent economic dispatch problemsrdquo International Journal ofElectrical Power amp Energy Systems vol 55 pp 378ndash391 2014
Mathematical Problems in Engineering 29
[37] V Hosseinnezhad M Rafiee M Ahmadian and M T AmelildquoSpecies-basedQuantum Particle SwarmOptimization for eco-nomic load dispatchrdquo International Journal of Electrical Poweramp Energy Systems vol 63 pp 311ndash322 2014
[38] M S P Subathra S E Selvan T A A Victoire A HChristinal andU Amato ldquoA hybrid with cross-entropymethodand sequential quadratic programming to solve economic loaddispatch problemrdquo IEEE Systems Journal vol 9 no 3 pp 1031ndash1044 2015
[39] J Zhan Q H Wu C Guo and X Zhou ldquoEconomic dispatchwith non-smooth objectives part I local minimum analysisrdquoIEEE Transactions on Power Systems vol 30 no 2 pp 710ndash7212015
[40] A K Barisal andR C Prusty ldquoLarge scale economic dispatch ofpower systems using oppositional invasive weed optimizationrdquoApplied Soft Computing vol 29 pp 122ndash137 2015
[41] M Basu ldquoImproved differential evolution for economic dis-patchrdquo International Journal of Electrical Power amp EnergySystems vol 63 pp 855ndash861 2014
[42] V S Aragon S C Esquivel and C C Coello ldquoAn immunealgorithm with power redistribution for solving economicdispatch problemsrdquo Information Sciences vol 295 pp 609ndash6322015
[43] M Ghasemi M Taghizadeh S Ghavidel and A AbbasianldquoColonial competitive differential evolution an experimentalstudy for optimal economic load dispatchrdquo Applied Soft Com-puting vol 40 pp 342ndash363 2016
[44] B R Adarsh T Raghunathan T Jayabarathi and X-S YangldquoEconomic dispatch using chaotic bat algorithmrdquo Energy vol96 pp 666ndash675 2016
[45] N Ghorbani and E Babaei ldquoExchange market algorithm foreconomic load dispatchrdquo International Journal of ElectricalPower amp Energy Systems vol 75 pp 19ndash27 2016
[46] J X V Neto G Reynoso-Meza T H Ruppel V C Marianiand L D S Coelho ldquoSolving non-smooth economic dispatchby a new combination of continuous GRASP algorithm anddifferential evolutionrdquo International Journal of Electrical Poweramp Energy Systems vol 84 pp 13ndash24 2017
[47] A Kavousi-Fard and A Khosravi ldquoAn intelligent 120579-ModifiedBat Algorithm to solve the non-convex economic dispatchproblem considering practical constraintsrdquoElectrical Power andEnergy Systems vol 82 pp 189ndash196 2016
[48] M A Al-Betar M A Awadallah A T Khader and A L BolajildquoTournament-based harmony search algorithm for non-convexeconomic load dispatch problemrdquo Applied Soft Computing vol47 pp 449ndash459 2016
[49] M A Al-Betar M A Awadallah I A Doush E Alsukhni andH ALkhraisat ldquoA non-convex economic dispatchproblemwithvalve loading effect using a new modified 120573-hill climbing localsearch algorithmrdquo Arabian Journal for Science and Engineeringpp 1ndash8 2018
[50] M A Awadallah M A Al-Betar A L Bolaji E M Alsukhniand H Al-Zoubi ldquoNatural selection methods for artificial beecolony with new versions of onlooker beerdquo Soft Computing pp1ndash40 2018
[51] S H Kaboli and A K Alqallaf ldquoSolving non-convex economicload dispatch problem via artificial cooperative search algo-rithmrdquo Expert Systems with Applications vol 128 pp 14ndash272019
[52] D Singh and J Dhillon ldquoAmeliorated greywolf optimization foreconomic load dispatch problemrdquo Energy vol 169 pp 398ndash4192019
[53] X S Yang and S Deb ldquoCuckoo search via Levy flightsrdquo inProceedings of the World Congress on Nature and BiologicallyInspired Computing (NaBIC rsquo09) pp 210ndash214 India 2009
[54] S Sahoo K M Dash R C Prusty and A K Barisal ldquoCom-parative analysis of optimal load dispatch through evolutionaryalgorithmsrdquo Ain Shams Engineering Journal vol 6 pp 107ndash1202015
[55] M Basu and A Chowdhury ldquoCuckoo search algorithm foreconomic dispatchrdquo Energy vol 60 pp 99ndash108 2013
[56] A B Serapiao ldquoCuckoo search for solving economic dispatchload problemrdquo Intelligent Control and Automation vol 04 no04 pp 385ndash390 2013
[57] A H Bindu and M D Reddy ldquoEconomic load dispatch usingcuckoo search algorithmrdquo International Journal of EngineeringResearch and Applications vol 3 pp 498ndash502 2013
[58] E Afzalan and M Joorabian ldquoAn improved cuckoo searchalgorithm for power economic load dispatchrdquo InternationalTransactions on Electrical Energy Systems vol 25 no 6 pp 958ndash975 2015
[59] T T Nguyen and D N Vo ldquoThe application of one rank cuckoosearch algorithm for solving economic load dispatch problemsrdquoApplied Soft Computing vol 37 pp 763ndash773 2015
[60] S M Islam S Das S Ghosh S Roy and P N Suganthan ldquoAnadaptive differential evolution algorithm with novel mutationand crossover strategies for global numerical optimizationrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 42 no 2 pp 482ndash499 2012
[61] Z L Gaing ldquoParticle swarm optimization to solving theeconomic dispatch considering the generator constraintsrdquo IEEETransactions on Power Systems vol 18 no 3 pp 1187ndash1195 2003
[62] X-S Yang and S Deb ldquoEngineering optimisation by Cuckoosearchrdquo International Journal of Mathematical Modelling andNumerical Optimisation vol 330 pp 43ndash51 2010
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
subcases with the gradual increase of number of units Thesmallest scale system considers 3 units but the largest scalesystem takes 80 units In addition to the implementation ofthe proposed ICSA we have also implemented CCSA forSubcase 31 and Subcase 35 for further comparison becauseCCSA has not been run for the two subcases so far
Comparison of obtained results from Subcase 31 shownin Table 5 indicates that the proposed ICSA is superior toCCSAwith lower fuel cost but it seems to be less effective thanMCSA [58] The minimum cost of MCSA reported in [58] isthe smallest fuel cost but the recalculated cost is much higherthan that of the proposed method Furthermore MCSA hasbeen implemented by setting very high values to 119873119901119904 and119866119898119886119909
Reports for Subcases 32 and 33 shown in Table 6are the comparisons of the proposed ICSA approach andother methods such as conventional Evolution programming(CEP) [1] Fast EP (FEP) [1] improved FEP (IFEP) [1] DE[12] multiplier Lagrange-based genetic algorithm with (GA-MU) [15] QPSO [16] GA-PS-SQP [30] PSO-SQP [32] M120573-HCLSA [49] IABCA [50] CCSA [59] OSE-CSA [59] SOS[34] MSOS [34] CEA-SQT [38] TSBO [39] IWA [40] andCBA [44] As observed from the table ICSAapproach obtainsbetter solutions than mostmethods excluding DE [10] CCSA
[59] OSE-CSA [59] SOS [34] MSOS [34] CEA-SQT [38]TSBO [39] IWA [40] and CBA [44] especially M120573-HCLSA[49] with lower cost $1796097 However recalculated costfrom reported solution of M120573-HCLSA is $179691 BesidesICSA is very fast as compared to most methods where twoother versions of Cuckoo search algorithm CCSA and OSE-CSA are also included except two methods in [34] Theprocessor of computer that all the methods run on is alsoreported in the final column Clearly ICSA approach is veryefficient for the case with the 13-unit system where effects ofvalve loading process are considered
In Subcase 34 the number of units is much larger thanthat of three subcases above up to 40 units [1] The obtainedresult comparisons with others are indicated in Table 7Clearly the minimum cost comparisons reveal that the pro-posedmethod is one of the leading methods due to the lowestcost except the comparison with CCSA [23] OSE-CSA [59]SOS [34] MSOS [34] EMA [45] 120579-MBA [47] and AGWOA[52] It is noted that AGWOA [52] has reported the bestminimum cost with $12140430 but recalculated minimumcost which was obtained by substituting reported optimalgeneration of all thermal generating units is $12141331 Theaverage and the maximum costs from the proposed methoddo not belong to the leading method group however the
Mathematical Problems in Engineering 15
Table 7 Result comparisons for Subcase 34
Approach Best cost Average cost Worst cost CPU time Computer($h) ($h) ($h) (s) (Processor-Ram)
execution time that ICSA approach takes is smaller thanmostones except IAPR [42] and 120579-MBA [47] which have been runon stronger computers Note that MSOS [34] has been fasterthan the proposed method for Subcases 31 and 32 abovebut it is too slower than the proposed method for the casenamely 1813 seconds compared to 146 seconds Comparedto two other versions of Cuckoo search CCSA [23] and OSE-CSA [59] the proposed method is also faster about threetimes although the processors are slightly different For thiscase 120579-MBA [47] shows a very good performance howeverthe method has not been tested onmore complicated systemsand larger scale and therefore more comparisons with themethod must end Clearly the proposed method is stillefficient for the case where large scale and effects of valveloading process are included
Subcase 35 is the largest scale system with 80 units Inaddition to the implementation of ICSA CCSA has beenalso run for the Subcase 35 for further investigation ofefficiency improvement of the proposed ICSA approach The
comparisons of minimum cost in Table 8 show that theproposed ICSA can find more optimal solution than M120573-HCLSA [49] AGWOA [52] and CCSA The proposed ICSAis also superior to CCSA in terms ofmore stable search abilityand lower fluctuation since its average cost and maximumcost are less than those of CCSA The outstanding figurecannot be reached as compared to AGWOA [52] howeverit is hard to conclude AGWOA [52] is superior to theproposed ICSA approach about more stable search abilityand lower fluctuation Actually comparison of the values ofpopulation and iterations as well as execution time cannot beaccomplished because the information was not reported in[52]Thus it can conclude that the proposed ICSA is effectivefor the subcase
Optimal solutions obtained by ICSA for the case areshown in Tables 20ndash23
54 Obtained Results on Case 4 with Two Systems consideringSFS and PPZ and RPR Constraints In this section two
16 Mathematical Problems in Engineering
Table 9 Result comparisons for Subcases 41 and 42
Subcase Approach Best cost ($h) Mean cost ($h) Worst cost ($h) Std dev ($h) CPU time (s) Computer (Processor-Ram)
test systems with SFS and PPZ and RPR constraints areconsidered The test system size is up to 60 and 90 units forSubcases 41 and 42 respectively Comparison for the casesis only performed with two Genetic algorithms consisting ofGA and IGA-MU in [9] and two other versions of Cuckoosearch algorithms including CCSA [23] and OSE-CSA [59]and presented in Table 9 Other studies have tended toignore such complicated constraints of PPZ and RPR Thecosts reported in Table 9 indicate that ICSA approach canobtain more effective solution than CCSA and OSE-CSAbecause it has reached lower minimum cost than the twoones Furthermore the proposed method also takes shortercomputation time for the two cases from about two timesto about three times although the processor of the proposedmethod is slightly strongerThemean costs of ICSA approachare much less than those from IGA-MU and GA and slightlyhigher than those from OSE-CSA but there is a trade-off between the proposed method and CCSA for the twosubcases In fact the proposed method obtains higher meancost for Subcase 41 but lower cost for Subcase 42 AlthoughGAmethods have been runon aweak computerwith 07 GHzof the processor compared to that with 24GHz in the studytheir execution times are significantly higher namely 56381seconds (GA) and 16258 seconds (IGA-MU) compared to09153 seconds of ICSA approach for Subcase 41 and 94093seconds (GA) and 25545 seconds (IGA-MU) compared to15892 seconds (the proposedmethod)The analysis can pointout that ICSA approach ismore efficient than these comparedmethods in terms of optimal solutions and execution time
Optimal solutions obtained by ICSA for Subcase 42 areshown in Table 24
55 Obtained Results on Case 5 with a 15-Unit System consid-ering SFS and RRL PPZ and PL Constraints In this section
a 15-unit system considering RRL PPZ and PL constraintsis considered to be solved for finding optimal solution Forefficiency investigation of the proposed ICSA we have alsoimplemented CCSA for comparison As listed in Table 10the proposed ICSA is the most effective method with thesmallest fuel cost The comparisons of control parameters aswell asCPU time are also good evidence to confirm the strongsearch of the proposed ICSA approach since it has been runby smaller values of control parameter and faster executiontime as compared to all methods excluding CCSA
Optimal solution obtained by ICSA for the case is shownin Table 25
56 Obtained Results on Case 6 withThree Systems consideringMFS and EoVLP In this section three test systems with thechallenge on objective function including multi-fossil fuelsources and effects of valve loading process are consideredThe scale is up to 80 units 160 units and 320 units forSubcases 61 62 and 63 respectively
Comparison for Subcase 61 reported in Table 11 revealsthat the proposed method is the best method in terms of thelowest best cost the lowestmean cost and the lowest standarddeviation and the fastest execution time The processor fromthis proposedmethod is about four times stronger thanCGA-MU and IGA-MU but the speed is from ten times to 35 timesfaster than these methods Compared to CCSA and OSE-CSA the proposed method is about two times faster but theprocessor is slightly stronger
Comparison for Subcase 62 is reported in Table 12Clearly the proposed ICSA approach obtains better values ofthe best mean and worst costs than most methods exceptMSOS [34] where the best cost difference is about $ 024However the proposed method is the second fastest onewith 1119 seconds where the first fastest one CBA [44] has
Mathematical Problems in Engineering 17
Table 11 Comparisons of found results for Subcase 61
Approach Best cost Mean cost Worst cost Std dev CPU time Computer($h) ($h) ($h) ($h) (s) (Processor-Ram)
spent 571 seconds Clearly MSOS is better than the proposedICSA approach with respect to slightly less best cost butis worse than the proposed method in terms of executiontime namely 2572 seconds compared to 1119 seconds whileCBA [44] is faster than the proposed method but obtainssignificantly worse costs The analysis can conclude that theproposed ICSA approach is very powerful for the subcasewith 160 units
Table 13 presents the comparison of three other methodsincludingCCSA [54] SOS [34] andMSOS [34] accompaniedwith the proposed method for Subcase 63 The obtainedresult comparisons imply that ICSA approach can obtainbetter values of the best and standard deviation costs thanCCSA and SOS but obtains slightly higher cost than MSOSby approximately $ 022 Besides the execution time fromICSA approach is much shorter than others especially it ishigher than five times faster than MSOS The four methodshave been run on approximately strong computers Brieflythe proposed ICSA approach can find and converge to morefavorable solution than other methods with shorter CPUtime except the comparison with MSOS which had bettersolution but spent higher than five times execution times
Consequently the proposed method is a very promisingoptimization algorithm for Subcase 63 a system up to 320units and with multi-fossil fuel sources and effects of valveloading process
Optimal solution obtained by ICSA for Subcase 63 isshown in Table 26
57 The Improvement of ICSA Approach Performance
571 The Outstanding Improvement over CCSA In this sec-tion the performance improvement of ICSA over CCSAhas been investigated by analyzing obtained results and setcontrol parameters Table 14 has been formed by addingreduction cost improvement level of the best cost executiontime and control parameters consisting of 119873119901119904 and 119866119898119886119909Among the compared factors reduction cost is the deviationof the cost of CCSA and that of ICSA whereas the improve-ment level is the ratio of the reduction cost to the cost ofCCSA The reduction costs indicate the proposed methodcould find either equal quality of solutions or higher qualityof solutions than CCSA for all study casesThe reduction costis from $0 to $42839 corresponding to the improvement level
18 Mathematical Problems in Engineering
Table 14 Summary of results obtained by CCSA and ICSA for all study cases
Study case Reduction cost ($) Improvement level () Execution time (s) Nps Gmax
Figure 4 The best run obtained by CCSA and ICSA for Case 2
that is from 0 to 08968The saving cost is not too much forone hour but the operation in one day onemonth or one yearis very high However it should be noted that CCSA has beenrun by setting much higher population size and iterations formany cases excluding study cases implemented in the studysuch as Case 2 Subcase 31 Subcase 35 and Case 5 Forinstance ICSA has used 119873119901119904 = 10 and 119866119898119886119909 = 15 for Subcase13 and119873119901119904 = 10 and119866119898119886119909 = 25 for Subcase 14 whereas CCSAhas been run by setting119873119901119904 = 20 and 119866119898119886119909 = 5000 for the twosubcases Similarly CCSA has been run for Subcases 32 33and 34 with much higher number of iterations For the lastsubcase CCSA has been run by setting 119873119901119904 = 320 and 119866119898119886119909
= 1200 but those of ICSA have been 10 and 9000 Due tothe higher value of control parameters CCSA has tended tospend more time in finding such high quality solutions foralmost all study cases Execution time of ICSA is less than18 seconds while that of CCSA is up to higher 75 secondsIt is clear that the proposed ICSA could find better optimalsolutions thanCCSA for such considered systems For furtherinvestigation of performance comparison the best runs over50 runs and fuel cost values of 50 runs obtained by CCSA andthe proposed ICSA for Case 2 Subcase 31 Subcase 35 andCase 5 have been plotted in from Figures 4ndash11 The best runcurves show the faster search of the proposed ICSA method
Mathematical Problems in Engineering 19
0 5 10 15 20 25 30 35 40 45 50Run
19795
198
19805
1981
19815
1982
19825
1983
19835
Fuel
cost
($)
CCSAICSA
times105
Figure 5The best fuel cost of 50 runs obtained by CCSA and ICSA for Case 2
0 2 4 6 8 10 12 14 16 18 20Iteration
8234
8236
8238
8240
8242
8244
8246
8248
8250
8252
Fitn
ess F
unct
ion
($)
CCSAICSA
Figure 6 The best run obtained by CCSA and ICSA for Subcase 31
whereas 50 values of fuel cost indicate that the proposedICSA can find many solutions with better quality Clearlythe proposed ICSA is outstanding in terms of stabilizationof solution search and faster convergence As a result it canconclude that the proposed ICSA approach is more effectivethanCCSA in solvingOLDproblemwith considered systems
572The Improvement of Results over Other Methods In thisarticle we have tested ICSA approach on 6 cases with 16
systems with different fuel cost forms different constraintsand different scale systems from 3 units to 320 units We havecompared the yielded results from ICSA approach and otherexisting ones for evaluating the efficiency of ICSA approachIn subsections above we have shown yielded results fromICSA approach and compared these results to those of otheronesHowever the demonstrationhad not been very good forobserving and comparing to lead to a conclusionThus in thesubsectionwehave summarized the result comparisons of the
20 Mathematical Problems in Engineering
0 5 10 15 20 25 30 35 40 45 50Run
8234
82342
82344
82346
82348
8235
82352
82354
82356
82358
8236
Fuel
cost
($)
CCSAICSA
Figure 7 The best fuel cost of 50 runs obtained by CCSA and ICSA for Subcase 31
0 1000 2000 3000 4000 5000 6000Iteration
242
244
246
248
25
252
254
256
Fitn
ess F
unct
ion
($)
CCSAICSA
times105
Figure 8 The best run obtained by CCSA and ICSA for Subcase 35
proposed and other ones Table 15 has reported the reductioncost (in $) of ICSA approach compared to other ones Inaddition we have converted the reduction cost into improve-ment level (in ) for better comparison The improvementhas been shown from the lowest level to the highest levelin terms of reduction cost and improvement percentageIn addition we have also given the slowest and the fastestexecution time of other compared methods together withthat of the proposed method The table implies that ICSA
approach can find better optimal solutions with less fuel costup to $052 for Subcase 13 $75229 for Subcase 14 $1775149for Case 2 $1181256 for Subcase 31 $8438 for Subcase 32$9113 for Subcase 33 $121181 for Subcase 34 $3412 forSubcase 35 $04329 for Subcase 41 $30227 for Subcase42 $407002 for Case 5 $18183 for Subcase 61 $13965 forSubcase 62 and $11817 for Subcase 63 These reductioncosts are equivalent to improvement level (IL) of 0032908968 82282 14143 047 038 099 001405
Mathematical Problems in Engineering 21
0 5 10 15 20 25 30 35 40 45 50Run
2428
243
2432
2434
2436
2438
244
2442
Fuel
cost
($)
CCSAICSA
times105
Figure 9 The best fuel cost of 50 runs obtained by CCSA and ICSA for Subcase 35
0 50 100 150 200 250 300 350 400Iteration
327
328
329
33
331
332
333
334
335
336
337
Fitn
ess F
unct
ion
($)
CCSAICSA
times104
Figure 10 The best run obtained by CCSA and ICSA for Case 5
00003 0002 12291 004 138 and 059 Thesequantitative comparisons reveal that larger scale systems canlead to better reduction cost but the improvement level isnot high because total cost of compared methods tends tobe large for large scale systems Furthermore very large scalesystems with nondifferentiable objective have been normallysolved by strong methods In fact systems in Case 3 havethe same characteristic with single fuel and effects of valveloading process but Subcase 34 is a larger scale system with
40 units while Subcases 31 32 and 33 are constructed by 3units 13 units and 13 units So the improvement percentageof Subcase 34 can be up to 099 whilst that of Subcases32 and 33 is 047 and 038 respectively Subcase 35 iswith the largest system 80 units but the reduction cost is notmuch only $3412 because compared methods with ICSA areeither state-of-the-art ones or improved ones Also Subcase41 and Subcase 42 have considered single fuel and PPZ andspinning reserve constraints but Subcase 42 is larger scale
22 Mathematical Problems in Engineering
Table 15 Performance improvement summary of the proposed method
Study cases Reduction cost ($) Improvement level () Execution time (s)From To From To Slowest method Fastest method Proposed method
Figure 11 The best fuel cost of 50 runs obtained by CCSA and ICSA for Case 5
system with 90 units and Subcase 41 is only with 60 unitsThus the improvement of Subcase 41 is lower with 00003but that of Subcase 42 is 0002 Similarly systems in Cases 6have the same featurewithmulti-fossil fuel sources and effectsof valve loading process but they are respectively constructedby 80 160 and 320 units As a result the improvements ofSubcase 62 138 and Subcase 63 059 are much higherthan Subcase 61 004 However Subcase 62 with smallernumber of units but getting higher improvement is easilyunderstood because there were nine compared methods butonly three compared methods are considered for Subcase 63In general the improvement is not high it is about under onedollar several dollars tens of dollars and over one thousand
dollars per hour however the saving cost will be significantif the operation is considered to be onemonth with 720 hoursor one year with 8760 hours
Execution time comparison can be evaluated by observ-ing the fastest and the slowest compared methods in Table 15These execution times of the proposed method are approxi-mately equal to that of the fastest methods for Subcases 3233 and 34 and much shorter than other fastest comparedmethods for other cases especially for Subcases 61 and 63The fastest method for Subcase 62 is CBA [44] showing 57seconds while that of the proposed method is 1119 secondsHowever it cannot conclude that CBA is more effective thanthe proposed method because the proposed method could
find better optimal solution with less fuel cost by $2130As considering execution time of the slowest comparedmethods it can point out that ICSA is a very fast optimizationtool since the execution time of these methods is 29496 and3425 seconds for Subcases 32 and 33 116735 seconds forSubcase 34 56381 seconds for Subcase 41 94093 secondsfor Subcase 42 12797 seconds for Case 5 30941 seconds forSubcase 61 7542 seconds for Subcase 62 and 9641 secondsfor Subcase 63 while the execution time of the proposedmethod for these cases is respectively 095 146 091 1589246 84828 1119 and 171384 seconds It is clearly shown thatICSA is very fast as compared to these methods
In summary the proposed method has found approx-imately high quality solutions with several standard state-of-the-art meta-heuristic algorithms and improved versionsof them together with other old methods In addition theproposed method could improve result better than approx-imately all methods with faster execution time Comparedto other methods with the fastest convergence speed andhigh quality solutions the proposed method has been as
fast as for some cases and much faster for other cases Thecomparison with the slowest methods could show that theproposed method was extremely powerful since it was up tonearly one thousand times faster Consequently the proposedICSA approach can be one of the strongest optimization toolsfor OLD problem
6 Conclusions
This paper has proposed a good ICSA method for solvingOLD problem in which many test systems with differentobjective functions and complicated constraints from simpleto complex have been used as studied cases The proposedICSA method has been developed by performing severalmodifications on the second solution update of CCSA whichcontained several drawbacks to global convergence and fastmanner The OLD problem has covered from single fuelto multi-fossil fuels from quadratic objective function tononconvex objective function in addition to PPZ RPR andRRL constraints Many existing optimization algorithms have
been concerned in aim to compare the performance and givethe final conclusion on the proposed method There have
been six main cases with sixteen subcases The evaluationshave been made at the end of each study case Clearly theproposed ICSA approach has yielded more effective optimalsolutions with faster execution time than almost all methodsConsequently it can be concluded that the proposed methodis much more superior to CCSA and is a very promisingmethod for solving OLD problem
Appendix
See Tables 16ndash26
Nomenclature
120575119894 120582119894 120572119894 120573119894 120574119894 Fuel cost functioncoefficients of the 119894119905ℎthermal generationunit11986100 1198610119895 119861119895119894 Power loss matrixcoefficients119865119865119909 119865119865119887119890119904119905 The values of fitness ofsolution 119909 and theso-far most effectivesolution among thecurrent set ofsolutions119898119894 Number of fuels burntin the 119894119905ℎ thermalgeneration unit
119873 Number of allavailable thermalgeneration units119899119894 Number of prohibitedpower zones of the 119894119905ℎthermal generationunit119875119906119894119896 119875119897119894119896 Upper and lowerlimits of the 119894119905ℎthermal generationunit corresponding tothe kth PPZ119875119886 Probability ofreplacing controlvariables in each oldsolution119875119863 Real power demand ofall loads in system119875119894119898119886119909 119875119894119898119894119899 The highest andlowest real poweroutputs of the iththermal generationunit119875119894119895119898119886119909 119875119894119895119898119894119899 The highest andlowest real poweroutputs of the 119894119905ℎthermal generationunit corresponding tothe 119895119905ℎ fuel type119878119894119898119886119909 Maximum real powerreserve contributionof the thermalgeneration unit 1198941198781199001198971199031198861198991198891 1198781199001198971199031198861198991198892 1198781199001198971199031198861198991198893 1198781199001198971199031198861198991198894 Randomly mixedsolutions from the setof current solutions119878119900119897119909 119878119900119897119866119887119890119904119905 The old solution x andthe most effectivesolution119878119877 Real power reserverequirement of system
120575119894119895 120582119894119895 120572119894119895 120573119894119895 120574119894119895 Fuel cost functioncoefficients of the 119894119905ℎthermal generationunit corresponding tothe 119895119905ℎ fuel type1205761 1205762 1205763 1205764 1205765 1205766 Random numbersbetween 0 and 1Ω Number of generationunits considering PPZconstraint
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare no conflicts of interest
References
[1] N Sinha R Chakrabarti and P K Chattopadhyay ldquoEvolution-ary programming techniques for economic load dispatchrdquo IEEETransactions on Evolutionary Computation vol 7 no 1 pp 83ndash94 2003
[2] P Hansen ldquoA separable approximation dynamic programmingalgorithm for economic dispatch with transmission lossesrdquoYugoslav Journal of Operations Research vol 12 no 2 2002
[3] N T Thang ldquoSolving economic dispatch problem with piece-wise quadratic cost functions using lagrange multiplier theoryrdquoin Proceedings of the 3rd International Conference on ComputerTechnology and Development (ICCTD rsquo11) pp 359ndash364 ASMEPress 2011
[4] S K Mishra and S K Mishra ldquoA comparative study of solutionof economic load dispatch problem in power systems in theenvironmental perspectiverdquoProcedia Computer Science vol 48pp 96ndash100 2015
[5] A A Al-Subhi and H K Alfares ldquoEconomic load dispatchusing linear programming a comparative studyrdquo InternationalJournal of Applied Industrial Engineering vol 3 no 1 pp 16ndash362016
28 Mathematical Problems in Engineering
[6] C Zhou G Huang and J Chen ldquoPlanning of electric powersystems considering virtual power plants with dispatchableloads included an inexact two-stage stochastic linear program-ming modelrdquoMathematical Problems in Engineering vol 2018Article ID 7049329 12 pages 2018
[7] T T Nguyen N V Quynh and L Van Dai ldquoImproved fireflyalgorithm a novel method for optimal operation of thermalgenerating unitsrdquo Complexity vol 2018 Article ID 7267593 23pages 2018
[8] S S Haroon S Hassan S Amin et al ldquoMultiple fuel machinespower economic dispatch using stud differential evolutionrdquoEnergies vol 11 no 6 pp 1ndash20 2018
[9] C-T Su and C-L Chiang ldquoNonconvex power economic dis-patch by improved genetic algorithm with multiplier updatingmethodrdquo Electric Power Components and Systems vol 32 no 3pp 257ndash273 2004
[10] NNoman andH Iba ldquoDifferential evolution for economic loaddispatch problemsrdquo Electric Power Systems Research vol 78 no8 pp 1322ndash1331 2008
[11] P Somasundaram and K Kuppusamy ldquoApplication of evolu-tionary programming to security constrained economic dis-patchrdquo International Journal of Electrical Power amp EnergySystems vol 27 no 5-6 pp 343ndash351 2005
[12] T Jayabarathi K Jayaprakash andD Jeyakumar ldquoEvolutionaryprogramming technique for different kinds of economic dis-patch problemsrdquo Electric Power Systems Research vol 73 no 2pp 169ndash176 2005
[13] P Subbaraj R Rengaraj and S Salivahanan ldquoEnhancementof combined heat and power economic dispatch using selfadaptive real-coded genetic algorithmrdquo Applied Energy vol 86no 6 pp 915ndash921 2009
[14] C-L Chiang and C-T Su ldquoAdaptive-improved genetic algo-rithm for the economic dispatch of units with multiple fueloptionsrdquo Cybernetics and Systems vol 36 no 7 pp 687ndash7042005
[15] C Chiang ldquoImproved genetic algorithm for power economicdispatch of units with valve-point effects and multiple fuelsrdquoIEEE Transactions on Power Systems vol 20 no 4 pp 1690ndash1699 2005
[16] K Meng H G Wang Z Y Dong and K P Wong ldquoQuantum-inspired particle swarm optimization for valve-point economicload dispatchrdquo IEEE Transactions on Power Systems vol 25 no1 pp 215ndash222 2010
[17] P Erdogmus A Ozturk and S Duman ldquoEnvironmen-taleconomic dispatch using genetic algorithm and simulatedannealingrdquo in Proceedings of the International Conference onElectric Power and Energy Conversion Systems EPECS rsquo09 pp1ndash4 IEEE 2009
[18] L D S Coelho and V C Mariani ldquoImproved differentialevolution algorithms for handling economic dispatch opti-mization with generator constraintsrdquo Energy Conversion andManagement vol 48 no 5 pp 1631ndash1639 2007
[19] H Dakuo W Fuli and M Zhizhong ldquoA hybrid genetic algo-rithm approach based on differential evolution for economicdispatch with valve-point effectsrdquo Electrical Power and EnergySystems vol 30 pp 31ndash38 2008
[20] C-F Chang J-J Wong J-P Chiou and C-T Su ldquoRobustsearching hybrid differential evolution method for optimalreactive power planning in large-scale distribution systemsrdquoElectric Power Systems Research vol 77 no 5-6 pp 430ndash4372007
[21] M R Farooqi P Jain and K R Niazi ldquoUsing Hopfield neuralnetwork for economic dispatch of power systemsrdquo in Proceed-ings of the National Power Engineering Conference PECon rsquo03pp 5ndash10 Bangi Malaysia 2003
[22] T Yalcinoz H Altun and U Hasan ldquoConstrained economicdispatch with prohibited operating zones a Hopfield neuralnetwork approachrdquo in Proceedings of the 10th MediterraneanElectrotechnical Conference Information Technology and Elec-trotechnology for the Mediterranean Countries ProceedingsMeleCon rsquo00 (Cat No 00CH37099) pp 570ndash573 LemesosCyprus 2000
[23] D N Vo P Schegner and W Ongsakul ldquoCuckoo searchalgorithm for non-convex economic dispatchrdquo IET GenerationTransmission amp Distribution vol 7 no 6 pp 645ndash654 2013
[24] J-B Park K-S Lee J-R Shin and K Y Lee ldquoA particleswarm optimization for economic dispatch with non-smoothcost functionsrdquo IEEETransactions on Power Systems vol 20 no1 pp 34ndash42 2005
[25] T Niknam H D Mojarrad and M Nayeripour ldquoA newfuzzy adaptive particle swarm optimization for non-smootheconomic dispatchrdquo Energy vol 35 no 4 pp 1764ndash1778 2010
[26] L D S Coelho and V C Mariani ldquoAn improved harmonysearch algorithm for power economic load dispatchrdquo EnergyConversion and Management vol 50 no 10 pp 2522ndash25262009
[27] M Mahdavi M Fesanghary and E Damangir ldquoAn improvedharmony search algorithm for solving optimization problemsrdquoAppliedMathematics and Computation vol 188 no 2 pp 1567ndash1579 2007
[28] A Bhattacharya and P K Chattopadhyay ldquoSolving complexeconomic load dispatch problem using biogeography-basedoptimizationrdquo Expert Systems with Applications vol 37 no 5pp 3605ndash3615 2010
[29] D Simon ldquoBiogeography-based optimizationrdquo IEEE Transac-tions on Evolutionary Computation vol 12 no 6 pp 702ndash7132008
[30] J S Alsumait J K Sykulski and A K Al-Othman ldquoAhybrid GA-PS-SQP method to solve power system valve-pointeconomic dispatch problemsrdquo Applied Energy vol 87 no 5 pp1773ndash1781 2010
[31] M S Turgut andGKDemir ldquoQuadratic approximationndashbasedhybrid Artificial Cooperative Search algorithm for economicemission load dispatchproblemsrdquo International Transactions onElectrical Energy Systems vol 27 no 4 pp 1ndash14 2017
[32] T A A Victoire and A E Jeyakumar ldquoHybrid PSO-SQPfor economic dispatch with valve-point effectrdquo Electric PowerSystems Research vol 71 no 1 pp 51ndash59 2004
[33] D-KHe F-LWang andZ-ZMao ldquoHybrid genetic algorithmfor economic dispatch with valve-point effectrdquo Electric PowerSystems Research vol 78 no 4 pp 626ndash633 2008
[34] D C Secui ldquoA modified symbiotic organisms search algorithmfor large scale economic dispatch problem with valve-pointloading effectsrdquo Energy vol 113 pp 366ndash384 2016
[35] K BhattacharjeeA Bhattacharya and S H N Dey ldquoTeaching-learning-based optimization for different economic dispatchproblemsrdquo Scientia Iranica vol 21 no 3 pp 870ndash884 2014
[36] K Bhattacharjee A Bhattacharya and S H N Dey ldquoOppo-sitional Real Coded Chemical Reaction Optimization for dif-ferent economic dispatch problemsrdquo International Journal ofElectrical Power amp Energy Systems vol 55 pp 378ndash391 2014
Mathematical Problems in Engineering 29
[37] V Hosseinnezhad M Rafiee M Ahmadian and M T AmelildquoSpecies-basedQuantum Particle SwarmOptimization for eco-nomic load dispatchrdquo International Journal of Electrical Poweramp Energy Systems vol 63 pp 311ndash322 2014
[38] M S P Subathra S E Selvan T A A Victoire A HChristinal andU Amato ldquoA hybrid with cross-entropymethodand sequential quadratic programming to solve economic loaddispatch problemrdquo IEEE Systems Journal vol 9 no 3 pp 1031ndash1044 2015
[39] J Zhan Q H Wu C Guo and X Zhou ldquoEconomic dispatchwith non-smooth objectives part I local minimum analysisrdquoIEEE Transactions on Power Systems vol 30 no 2 pp 710ndash7212015
[40] A K Barisal andR C Prusty ldquoLarge scale economic dispatch ofpower systems using oppositional invasive weed optimizationrdquoApplied Soft Computing vol 29 pp 122ndash137 2015
[41] M Basu ldquoImproved differential evolution for economic dis-patchrdquo International Journal of Electrical Power amp EnergySystems vol 63 pp 855ndash861 2014
[42] V S Aragon S C Esquivel and C C Coello ldquoAn immunealgorithm with power redistribution for solving economicdispatch problemsrdquo Information Sciences vol 295 pp 609ndash6322015
[43] M Ghasemi M Taghizadeh S Ghavidel and A AbbasianldquoColonial competitive differential evolution an experimentalstudy for optimal economic load dispatchrdquo Applied Soft Com-puting vol 40 pp 342ndash363 2016
[44] B R Adarsh T Raghunathan T Jayabarathi and X-S YangldquoEconomic dispatch using chaotic bat algorithmrdquo Energy vol96 pp 666ndash675 2016
[45] N Ghorbani and E Babaei ldquoExchange market algorithm foreconomic load dispatchrdquo International Journal of ElectricalPower amp Energy Systems vol 75 pp 19ndash27 2016
[46] J X V Neto G Reynoso-Meza T H Ruppel V C Marianiand L D S Coelho ldquoSolving non-smooth economic dispatchby a new combination of continuous GRASP algorithm anddifferential evolutionrdquo International Journal of Electrical Poweramp Energy Systems vol 84 pp 13ndash24 2017
[47] A Kavousi-Fard and A Khosravi ldquoAn intelligent 120579-ModifiedBat Algorithm to solve the non-convex economic dispatchproblem considering practical constraintsrdquoElectrical Power andEnergy Systems vol 82 pp 189ndash196 2016
[48] M A Al-Betar M A Awadallah A T Khader and A L BolajildquoTournament-based harmony search algorithm for non-convexeconomic load dispatch problemrdquo Applied Soft Computing vol47 pp 449ndash459 2016
[49] M A Al-Betar M A Awadallah I A Doush E Alsukhni andH ALkhraisat ldquoA non-convex economic dispatchproblemwithvalve loading effect using a new modified 120573-hill climbing localsearch algorithmrdquo Arabian Journal for Science and Engineeringpp 1ndash8 2018
[50] M A Awadallah M A Al-Betar A L Bolaji E M Alsukhniand H Al-Zoubi ldquoNatural selection methods for artificial beecolony with new versions of onlooker beerdquo Soft Computing pp1ndash40 2018
[51] S H Kaboli and A K Alqallaf ldquoSolving non-convex economicload dispatch problem via artificial cooperative search algo-rithmrdquo Expert Systems with Applications vol 128 pp 14ndash272019
[52] D Singh and J Dhillon ldquoAmeliorated greywolf optimization foreconomic load dispatch problemrdquo Energy vol 169 pp 398ndash4192019
[53] X S Yang and S Deb ldquoCuckoo search via Levy flightsrdquo inProceedings of the World Congress on Nature and BiologicallyInspired Computing (NaBIC rsquo09) pp 210ndash214 India 2009
[54] S Sahoo K M Dash R C Prusty and A K Barisal ldquoCom-parative analysis of optimal load dispatch through evolutionaryalgorithmsrdquo Ain Shams Engineering Journal vol 6 pp 107ndash1202015
[55] M Basu and A Chowdhury ldquoCuckoo search algorithm foreconomic dispatchrdquo Energy vol 60 pp 99ndash108 2013
[56] A B Serapiao ldquoCuckoo search for solving economic dispatchload problemrdquo Intelligent Control and Automation vol 04 no04 pp 385ndash390 2013
[57] A H Bindu and M D Reddy ldquoEconomic load dispatch usingcuckoo search algorithmrdquo International Journal of EngineeringResearch and Applications vol 3 pp 498ndash502 2013
[58] E Afzalan and M Joorabian ldquoAn improved cuckoo searchalgorithm for power economic load dispatchrdquo InternationalTransactions on Electrical Energy Systems vol 25 no 6 pp 958ndash975 2015
[59] T T Nguyen and D N Vo ldquoThe application of one rank cuckoosearch algorithm for solving economic load dispatch problemsrdquoApplied Soft Computing vol 37 pp 763ndash773 2015
[60] S M Islam S Das S Ghosh S Roy and P N Suganthan ldquoAnadaptive differential evolution algorithm with novel mutationand crossover strategies for global numerical optimizationrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 42 no 2 pp 482ndash499 2012
[61] Z L Gaing ldquoParticle swarm optimization to solving theeconomic dispatch considering the generator constraintsrdquo IEEETransactions on Power Systems vol 18 no 3 pp 1187ndash1195 2003
[62] X-S Yang and S Deb ldquoEngineering optimisation by Cuckoosearchrdquo International Journal of Mathematical Modelling andNumerical Optimisation vol 330 pp 43ndash51 2010
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
subcases with the gradual increase of number of units Thesmallest scale system considers 3 units but the largest scalesystem takes 80 units In addition to the implementation ofthe proposed ICSA we have also implemented CCSA forSubcase 31 and Subcase 35 for further comparison becauseCCSA has not been run for the two subcases so far
Comparison of obtained results from Subcase 31 shownin Table 5 indicates that the proposed ICSA is superior toCCSAwith lower fuel cost but it seems to be less effective thanMCSA [58] The minimum cost of MCSA reported in [58] isthe smallest fuel cost but the recalculated cost is much higherthan that of the proposed method Furthermore MCSA hasbeen implemented by setting very high values to 119873119901119904 and119866119898119886119909
Reports for Subcases 32 and 33 shown in Table 6are the comparisons of the proposed ICSA approach andother methods such as conventional Evolution programming(CEP) [1] Fast EP (FEP) [1] improved FEP (IFEP) [1] DE[12] multiplier Lagrange-based genetic algorithm with (GA-MU) [15] QPSO [16] GA-PS-SQP [30] PSO-SQP [32] M120573-HCLSA [49] IABCA [50] CCSA [59] OSE-CSA [59] SOS[34] MSOS [34] CEA-SQT [38] TSBO [39] IWA [40] andCBA [44] As observed from the table ICSAapproach obtainsbetter solutions than mostmethods excluding DE [10] CCSA
[59] OSE-CSA [59] SOS [34] MSOS [34] CEA-SQT [38]TSBO [39] IWA [40] and CBA [44] especially M120573-HCLSA[49] with lower cost $1796097 However recalculated costfrom reported solution of M120573-HCLSA is $179691 BesidesICSA is very fast as compared to most methods where twoother versions of Cuckoo search algorithm CCSA and OSE-CSA are also included except two methods in [34] Theprocessor of computer that all the methods run on is alsoreported in the final column Clearly ICSA approach is veryefficient for the case with the 13-unit system where effects ofvalve loading process are considered
In Subcase 34 the number of units is much larger thanthat of three subcases above up to 40 units [1] The obtainedresult comparisons with others are indicated in Table 7Clearly the minimum cost comparisons reveal that the pro-posedmethod is one of the leading methods due to the lowestcost except the comparison with CCSA [23] OSE-CSA [59]SOS [34] MSOS [34] EMA [45] 120579-MBA [47] and AGWOA[52] It is noted that AGWOA [52] has reported the bestminimum cost with $12140430 but recalculated minimumcost which was obtained by substituting reported optimalgeneration of all thermal generating units is $12141331 Theaverage and the maximum costs from the proposed methoddo not belong to the leading method group however the
Mathematical Problems in Engineering 15
Table 7 Result comparisons for Subcase 34
Approach Best cost Average cost Worst cost CPU time Computer($h) ($h) ($h) (s) (Processor-Ram)
execution time that ICSA approach takes is smaller thanmostones except IAPR [42] and 120579-MBA [47] which have been runon stronger computers Note that MSOS [34] has been fasterthan the proposed method for Subcases 31 and 32 abovebut it is too slower than the proposed method for the casenamely 1813 seconds compared to 146 seconds Comparedto two other versions of Cuckoo search CCSA [23] and OSE-CSA [59] the proposed method is also faster about threetimes although the processors are slightly different For thiscase 120579-MBA [47] shows a very good performance howeverthe method has not been tested onmore complicated systemsand larger scale and therefore more comparisons with themethod must end Clearly the proposed method is stillefficient for the case where large scale and effects of valveloading process are included
Subcase 35 is the largest scale system with 80 units Inaddition to the implementation of ICSA CCSA has beenalso run for the Subcase 35 for further investigation ofefficiency improvement of the proposed ICSA approach The
comparisons of minimum cost in Table 8 show that theproposed ICSA can find more optimal solution than M120573-HCLSA [49] AGWOA [52] and CCSA The proposed ICSAis also superior to CCSA in terms ofmore stable search abilityand lower fluctuation since its average cost and maximumcost are less than those of CCSA The outstanding figurecannot be reached as compared to AGWOA [52] howeverit is hard to conclude AGWOA [52] is superior to theproposed ICSA approach about more stable search abilityand lower fluctuation Actually comparison of the values ofpopulation and iterations as well as execution time cannot beaccomplished because the information was not reported in[52]Thus it can conclude that the proposed ICSA is effectivefor the subcase
Optimal solutions obtained by ICSA for the case areshown in Tables 20ndash23
54 Obtained Results on Case 4 with Two Systems consideringSFS and PPZ and RPR Constraints In this section two
16 Mathematical Problems in Engineering
Table 9 Result comparisons for Subcases 41 and 42
Subcase Approach Best cost ($h) Mean cost ($h) Worst cost ($h) Std dev ($h) CPU time (s) Computer (Processor-Ram)
test systems with SFS and PPZ and RPR constraints areconsidered The test system size is up to 60 and 90 units forSubcases 41 and 42 respectively Comparison for the casesis only performed with two Genetic algorithms consisting ofGA and IGA-MU in [9] and two other versions of Cuckoosearch algorithms including CCSA [23] and OSE-CSA [59]and presented in Table 9 Other studies have tended toignore such complicated constraints of PPZ and RPR Thecosts reported in Table 9 indicate that ICSA approach canobtain more effective solution than CCSA and OSE-CSAbecause it has reached lower minimum cost than the twoones Furthermore the proposed method also takes shortercomputation time for the two cases from about two timesto about three times although the processor of the proposedmethod is slightly strongerThemean costs of ICSA approachare much less than those from IGA-MU and GA and slightlyhigher than those from OSE-CSA but there is a trade-off between the proposed method and CCSA for the twosubcases In fact the proposed method obtains higher meancost for Subcase 41 but lower cost for Subcase 42 AlthoughGAmethods have been runon aweak computerwith 07 GHzof the processor compared to that with 24GHz in the studytheir execution times are significantly higher namely 56381seconds (GA) and 16258 seconds (IGA-MU) compared to09153 seconds of ICSA approach for Subcase 41 and 94093seconds (GA) and 25545 seconds (IGA-MU) compared to15892 seconds (the proposedmethod)The analysis can pointout that ICSA approach ismore efficient than these comparedmethods in terms of optimal solutions and execution time
Optimal solutions obtained by ICSA for Subcase 42 areshown in Table 24
55 Obtained Results on Case 5 with a 15-Unit System consid-ering SFS and RRL PPZ and PL Constraints In this section
a 15-unit system considering RRL PPZ and PL constraintsis considered to be solved for finding optimal solution Forefficiency investigation of the proposed ICSA we have alsoimplemented CCSA for comparison As listed in Table 10the proposed ICSA is the most effective method with thesmallest fuel cost The comparisons of control parameters aswell asCPU time are also good evidence to confirm the strongsearch of the proposed ICSA approach since it has been runby smaller values of control parameter and faster executiontime as compared to all methods excluding CCSA
Optimal solution obtained by ICSA for the case is shownin Table 25
56 Obtained Results on Case 6 withThree Systems consideringMFS and EoVLP In this section three test systems with thechallenge on objective function including multi-fossil fuelsources and effects of valve loading process are consideredThe scale is up to 80 units 160 units and 320 units forSubcases 61 62 and 63 respectively
Comparison for Subcase 61 reported in Table 11 revealsthat the proposed method is the best method in terms of thelowest best cost the lowestmean cost and the lowest standarddeviation and the fastest execution time The processor fromthis proposedmethod is about four times stronger thanCGA-MU and IGA-MU but the speed is from ten times to 35 timesfaster than these methods Compared to CCSA and OSE-CSA the proposed method is about two times faster but theprocessor is slightly stronger
Comparison for Subcase 62 is reported in Table 12Clearly the proposed ICSA approach obtains better values ofthe best mean and worst costs than most methods exceptMSOS [34] where the best cost difference is about $ 024However the proposed method is the second fastest onewith 1119 seconds where the first fastest one CBA [44] has
Mathematical Problems in Engineering 17
Table 11 Comparisons of found results for Subcase 61
Approach Best cost Mean cost Worst cost Std dev CPU time Computer($h) ($h) ($h) ($h) (s) (Processor-Ram)
spent 571 seconds Clearly MSOS is better than the proposedICSA approach with respect to slightly less best cost butis worse than the proposed method in terms of executiontime namely 2572 seconds compared to 1119 seconds whileCBA [44] is faster than the proposed method but obtainssignificantly worse costs The analysis can conclude that theproposed ICSA approach is very powerful for the subcasewith 160 units
Table 13 presents the comparison of three other methodsincludingCCSA [54] SOS [34] andMSOS [34] accompaniedwith the proposed method for Subcase 63 The obtainedresult comparisons imply that ICSA approach can obtainbetter values of the best and standard deviation costs thanCCSA and SOS but obtains slightly higher cost than MSOSby approximately $ 022 Besides the execution time fromICSA approach is much shorter than others especially it ishigher than five times faster than MSOS The four methodshave been run on approximately strong computers Brieflythe proposed ICSA approach can find and converge to morefavorable solution than other methods with shorter CPUtime except the comparison with MSOS which had bettersolution but spent higher than five times execution times
Consequently the proposed method is a very promisingoptimization algorithm for Subcase 63 a system up to 320units and with multi-fossil fuel sources and effects of valveloading process
Optimal solution obtained by ICSA for Subcase 63 isshown in Table 26
57 The Improvement of ICSA Approach Performance
571 The Outstanding Improvement over CCSA In this sec-tion the performance improvement of ICSA over CCSAhas been investigated by analyzing obtained results and setcontrol parameters Table 14 has been formed by addingreduction cost improvement level of the best cost executiontime and control parameters consisting of 119873119901119904 and 119866119898119886119909Among the compared factors reduction cost is the deviationof the cost of CCSA and that of ICSA whereas the improve-ment level is the ratio of the reduction cost to the cost ofCCSA The reduction costs indicate the proposed methodcould find either equal quality of solutions or higher qualityof solutions than CCSA for all study casesThe reduction costis from $0 to $42839 corresponding to the improvement level
18 Mathematical Problems in Engineering
Table 14 Summary of results obtained by CCSA and ICSA for all study cases
Study case Reduction cost ($) Improvement level () Execution time (s) Nps Gmax
Figure 4 The best run obtained by CCSA and ICSA for Case 2
that is from 0 to 08968The saving cost is not too much forone hour but the operation in one day onemonth or one yearis very high However it should be noted that CCSA has beenrun by setting much higher population size and iterations formany cases excluding study cases implemented in the studysuch as Case 2 Subcase 31 Subcase 35 and Case 5 Forinstance ICSA has used 119873119901119904 = 10 and 119866119898119886119909 = 15 for Subcase13 and119873119901119904 = 10 and119866119898119886119909 = 25 for Subcase 14 whereas CCSAhas been run by setting119873119901119904 = 20 and 119866119898119886119909 = 5000 for the twosubcases Similarly CCSA has been run for Subcases 32 33and 34 with much higher number of iterations For the lastsubcase CCSA has been run by setting 119873119901119904 = 320 and 119866119898119886119909
= 1200 but those of ICSA have been 10 and 9000 Due tothe higher value of control parameters CCSA has tended tospend more time in finding such high quality solutions foralmost all study cases Execution time of ICSA is less than18 seconds while that of CCSA is up to higher 75 secondsIt is clear that the proposed ICSA could find better optimalsolutions thanCCSA for such considered systems For furtherinvestigation of performance comparison the best runs over50 runs and fuel cost values of 50 runs obtained by CCSA andthe proposed ICSA for Case 2 Subcase 31 Subcase 35 andCase 5 have been plotted in from Figures 4ndash11 The best runcurves show the faster search of the proposed ICSA method
Mathematical Problems in Engineering 19
0 5 10 15 20 25 30 35 40 45 50Run
19795
198
19805
1981
19815
1982
19825
1983
19835
Fuel
cost
($)
CCSAICSA
times105
Figure 5The best fuel cost of 50 runs obtained by CCSA and ICSA for Case 2
0 2 4 6 8 10 12 14 16 18 20Iteration
8234
8236
8238
8240
8242
8244
8246
8248
8250
8252
Fitn
ess F
unct
ion
($)
CCSAICSA
Figure 6 The best run obtained by CCSA and ICSA for Subcase 31
whereas 50 values of fuel cost indicate that the proposedICSA can find many solutions with better quality Clearlythe proposed ICSA is outstanding in terms of stabilizationof solution search and faster convergence As a result it canconclude that the proposed ICSA approach is more effectivethanCCSA in solvingOLDproblemwith considered systems
572The Improvement of Results over Other Methods In thisarticle we have tested ICSA approach on 6 cases with 16
systems with different fuel cost forms different constraintsand different scale systems from 3 units to 320 units We havecompared the yielded results from ICSA approach and otherexisting ones for evaluating the efficiency of ICSA approachIn subsections above we have shown yielded results fromICSA approach and compared these results to those of otheronesHowever the demonstrationhad not been very good forobserving and comparing to lead to a conclusionThus in thesubsectionwehave summarized the result comparisons of the
20 Mathematical Problems in Engineering
0 5 10 15 20 25 30 35 40 45 50Run
8234
82342
82344
82346
82348
8235
82352
82354
82356
82358
8236
Fuel
cost
($)
CCSAICSA
Figure 7 The best fuel cost of 50 runs obtained by CCSA and ICSA for Subcase 31
0 1000 2000 3000 4000 5000 6000Iteration
242
244
246
248
25
252
254
256
Fitn
ess F
unct
ion
($)
CCSAICSA
times105
Figure 8 The best run obtained by CCSA and ICSA for Subcase 35
proposed and other ones Table 15 has reported the reductioncost (in $) of ICSA approach compared to other ones Inaddition we have converted the reduction cost into improve-ment level (in ) for better comparison The improvementhas been shown from the lowest level to the highest levelin terms of reduction cost and improvement percentageIn addition we have also given the slowest and the fastestexecution time of other compared methods together withthat of the proposed method The table implies that ICSA
approach can find better optimal solutions with less fuel costup to $052 for Subcase 13 $75229 for Subcase 14 $1775149for Case 2 $1181256 for Subcase 31 $8438 for Subcase 32$9113 for Subcase 33 $121181 for Subcase 34 $3412 forSubcase 35 $04329 for Subcase 41 $30227 for Subcase42 $407002 for Case 5 $18183 for Subcase 61 $13965 forSubcase 62 and $11817 for Subcase 63 These reductioncosts are equivalent to improvement level (IL) of 0032908968 82282 14143 047 038 099 001405
Mathematical Problems in Engineering 21
0 5 10 15 20 25 30 35 40 45 50Run
2428
243
2432
2434
2436
2438
244
2442
Fuel
cost
($)
CCSAICSA
times105
Figure 9 The best fuel cost of 50 runs obtained by CCSA and ICSA for Subcase 35
0 50 100 150 200 250 300 350 400Iteration
327
328
329
33
331
332
333
334
335
336
337
Fitn
ess F
unct
ion
($)
CCSAICSA
times104
Figure 10 The best run obtained by CCSA and ICSA for Case 5
00003 0002 12291 004 138 and 059 Thesequantitative comparisons reveal that larger scale systems canlead to better reduction cost but the improvement level isnot high because total cost of compared methods tends tobe large for large scale systems Furthermore very large scalesystems with nondifferentiable objective have been normallysolved by strong methods In fact systems in Case 3 havethe same characteristic with single fuel and effects of valveloading process but Subcase 34 is a larger scale system with
40 units while Subcases 31 32 and 33 are constructed by 3units 13 units and 13 units So the improvement percentageof Subcase 34 can be up to 099 whilst that of Subcases32 and 33 is 047 and 038 respectively Subcase 35 iswith the largest system 80 units but the reduction cost is notmuch only $3412 because compared methods with ICSA areeither state-of-the-art ones or improved ones Also Subcase41 and Subcase 42 have considered single fuel and PPZ andspinning reserve constraints but Subcase 42 is larger scale
22 Mathematical Problems in Engineering
Table 15 Performance improvement summary of the proposed method
Study cases Reduction cost ($) Improvement level () Execution time (s)From To From To Slowest method Fastest method Proposed method
Figure 11 The best fuel cost of 50 runs obtained by CCSA and ICSA for Case 5
system with 90 units and Subcase 41 is only with 60 unitsThus the improvement of Subcase 41 is lower with 00003but that of Subcase 42 is 0002 Similarly systems in Cases 6have the same featurewithmulti-fossil fuel sources and effectsof valve loading process but they are respectively constructedby 80 160 and 320 units As a result the improvements ofSubcase 62 138 and Subcase 63 059 are much higherthan Subcase 61 004 However Subcase 62 with smallernumber of units but getting higher improvement is easilyunderstood because there were nine compared methods butonly three compared methods are considered for Subcase 63In general the improvement is not high it is about under onedollar several dollars tens of dollars and over one thousand
dollars per hour however the saving cost will be significantif the operation is considered to be onemonth with 720 hoursor one year with 8760 hours
Execution time comparison can be evaluated by observ-ing the fastest and the slowest compared methods in Table 15These execution times of the proposed method are approxi-mately equal to that of the fastest methods for Subcases 3233 and 34 and much shorter than other fastest comparedmethods for other cases especially for Subcases 61 and 63The fastest method for Subcase 62 is CBA [44] showing 57seconds while that of the proposed method is 1119 secondsHowever it cannot conclude that CBA is more effective thanthe proposed method because the proposed method could
find better optimal solution with less fuel cost by $2130As considering execution time of the slowest comparedmethods it can point out that ICSA is a very fast optimizationtool since the execution time of these methods is 29496 and3425 seconds for Subcases 32 and 33 116735 seconds forSubcase 34 56381 seconds for Subcase 41 94093 secondsfor Subcase 42 12797 seconds for Case 5 30941 seconds forSubcase 61 7542 seconds for Subcase 62 and 9641 secondsfor Subcase 63 while the execution time of the proposedmethod for these cases is respectively 095 146 091 1589246 84828 1119 and 171384 seconds It is clearly shown thatICSA is very fast as compared to these methods
In summary the proposed method has found approx-imately high quality solutions with several standard state-of-the-art meta-heuristic algorithms and improved versionsof them together with other old methods In addition theproposed method could improve result better than approx-imately all methods with faster execution time Comparedto other methods with the fastest convergence speed andhigh quality solutions the proposed method has been as
fast as for some cases and much faster for other cases Thecomparison with the slowest methods could show that theproposed method was extremely powerful since it was up tonearly one thousand times faster Consequently the proposedICSA approach can be one of the strongest optimization toolsfor OLD problem
6 Conclusions
This paper has proposed a good ICSA method for solvingOLD problem in which many test systems with differentobjective functions and complicated constraints from simpleto complex have been used as studied cases The proposedICSA method has been developed by performing severalmodifications on the second solution update of CCSA whichcontained several drawbacks to global convergence and fastmanner The OLD problem has covered from single fuelto multi-fossil fuels from quadratic objective function tononconvex objective function in addition to PPZ RPR andRRL constraints Many existing optimization algorithms have
been concerned in aim to compare the performance and givethe final conclusion on the proposed method There have
been six main cases with sixteen subcases The evaluationshave been made at the end of each study case Clearly theproposed ICSA approach has yielded more effective optimalsolutions with faster execution time than almost all methodsConsequently it can be concluded that the proposed methodis much more superior to CCSA and is a very promisingmethod for solving OLD problem
Appendix
See Tables 16ndash26
Nomenclature
120575119894 120582119894 120572119894 120573119894 120574119894 Fuel cost functioncoefficients of the 119894119905ℎthermal generationunit11986100 1198610119895 119861119895119894 Power loss matrixcoefficients119865119865119909 119865119865119887119890119904119905 The values of fitness ofsolution 119909 and theso-far most effectivesolution among thecurrent set ofsolutions119898119894 Number of fuels burntin the 119894119905ℎ thermalgeneration unit
119873 Number of allavailable thermalgeneration units119899119894 Number of prohibitedpower zones of the 119894119905ℎthermal generationunit119875119906119894119896 119875119897119894119896 Upper and lowerlimits of the 119894119905ℎthermal generationunit corresponding tothe kth PPZ119875119886 Probability ofreplacing controlvariables in each oldsolution119875119863 Real power demand ofall loads in system119875119894119898119886119909 119875119894119898119894119899 The highest andlowest real poweroutputs of the iththermal generationunit119875119894119895119898119886119909 119875119894119895119898119894119899 The highest andlowest real poweroutputs of the 119894119905ℎthermal generationunit corresponding tothe 119895119905ℎ fuel type119878119894119898119886119909 Maximum real powerreserve contributionof the thermalgeneration unit 1198941198781199001198971199031198861198991198891 1198781199001198971199031198861198991198892 1198781199001198971199031198861198991198893 1198781199001198971199031198861198991198894 Randomly mixedsolutions from the setof current solutions119878119900119897119909 119878119900119897119866119887119890119904119905 The old solution x andthe most effectivesolution119878119877 Real power reserverequirement of system
120575119894119895 120582119894119895 120572119894119895 120573119894119895 120574119894119895 Fuel cost functioncoefficients of the 119894119905ℎthermal generationunit corresponding tothe 119895119905ℎ fuel type1205761 1205762 1205763 1205764 1205765 1205766 Random numbersbetween 0 and 1Ω Number of generationunits considering PPZconstraint
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare no conflicts of interest
References
[1] N Sinha R Chakrabarti and P K Chattopadhyay ldquoEvolution-ary programming techniques for economic load dispatchrdquo IEEETransactions on Evolutionary Computation vol 7 no 1 pp 83ndash94 2003
[2] P Hansen ldquoA separable approximation dynamic programmingalgorithm for economic dispatch with transmission lossesrdquoYugoslav Journal of Operations Research vol 12 no 2 2002
[3] N T Thang ldquoSolving economic dispatch problem with piece-wise quadratic cost functions using lagrange multiplier theoryrdquoin Proceedings of the 3rd International Conference on ComputerTechnology and Development (ICCTD rsquo11) pp 359ndash364 ASMEPress 2011
[4] S K Mishra and S K Mishra ldquoA comparative study of solutionof economic load dispatch problem in power systems in theenvironmental perspectiverdquoProcedia Computer Science vol 48pp 96ndash100 2015
[5] A A Al-Subhi and H K Alfares ldquoEconomic load dispatchusing linear programming a comparative studyrdquo InternationalJournal of Applied Industrial Engineering vol 3 no 1 pp 16ndash362016
28 Mathematical Problems in Engineering
[6] C Zhou G Huang and J Chen ldquoPlanning of electric powersystems considering virtual power plants with dispatchableloads included an inexact two-stage stochastic linear program-ming modelrdquoMathematical Problems in Engineering vol 2018Article ID 7049329 12 pages 2018
[7] T T Nguyen N V Quynh and L Van Dai ldquoImproved fireflyalgorithm a novel method for optimal operation of thermalgenerating unitsrdquo Complexity vol 2018 Article ID 7267593 23pages 2018
[8] S S Haroon S Hassan S Amin et al ldquoMultiple fuel machinespower economic dispatch using stud differential evolutionrdquoEnergies vol 11 no 6 pp 1ndash20 2018
[9] C-T Su and C-L Chiang ldquoNonconvex power economic dis-patch by improved genetic algorithm with multiplier updatingmethodrdquo Electric Power Components and Systems vol 32 no 3pp 257ndash273 2004
[10] NNoman andH Iba ldquoDifferential evolution for economic loaddispatch problemsrdquo Electric Power Systems Research vol 78 no8 pp 1322ndash1331 2008
[11] P Somasundaram and K Kuppusamy ldquoApplication of evolu-tionary programming to security constrained economic dis-patchrdquo International Journal of Electrical Power amp EnergySystems vol 27 no 5-6 pp 343ndash351 2005
[12] T Jayabarathi K Jayaprakash andD Jeyakumar ldquoEvolutionaryprogramming technique for different kinds of economic dis-patch problemsrdquo Electric Power Systems Research vol 73 no 2pp 169ndash176 2005
[13] P Subbaraj R Rengaraj and S Salivahanan ldquoEnhancementof combined heat and power economic dispatch using selfadaptive real-coded genetic algorithmrdquo Applied Energy vol 86no 6 pp 915ndash921 2009
[14] C-L Chiang and C-T Su ldquoAdaptive-improved genetic algo-rithm for the economic dispatch of units with multiple fueloptionsrdquo Cybernetics and Systems vol 36 no 7 pp 687ndash7042005
[15] C Chiang ldquoImproved genetic algorithm for power economicdispatch of units with valve-point effects and multiple fuelsrdquoIEEE Transactions on Power Systems vol 20 no 4 pp 1690ndash1699 2005
[16] K Meng H G Wang Z Y Dong and K P Wong ldquoQuantum-inspired particle swarm optimization for valve-point economicload dispatchrdquo IEEE Transactions on Power Systems vol 25 no1 pp 215ndash222 2010
[17] P Erdogmus A Ozturk and S Duman ldquoEnvironmen-taleconomic dispatch using genetic algorithm and simulatedannealingrdquo in Proceedings of the International Conference onElectric Power and Energy Conversion Systems EPECS rsquo09 pp1ndash4 IEEE 2009
[18] L D S Coelho and V C Mariani ldquoImproved differentialevolution algorithms for handling economic dispatch opti-mization with generator constraintsrdquo Energy Conversion andManagement vol 48 no 5 pp 1631ndash1639 2007
[19] H Dakuo W Fuli and M Zhizhong ldquoA hybrid genetic algo-rithm approach based on differential evolution for economicdispatch with valve-point effectsrdquo Electrical Power and EnergySystems vol 30 pp 31ndash38 2008
[20] C-F Chang J-J Wong J-P Chiou and C-T Su ldquoRobustsearching hybrid differential evolution method for optimalreactive power planning in large-scale distribution systemsrdquoElectric Power Systems Research vol 77 no 5-6 pp 430ndash4372007
[21] M R Farooqi P Jain and K R Niazi ldquoUsing Hopfield neuralnetwork for economic dispatch of power systemsrdquo in Proceed-ings of the National Power Engineering Conference PECon rsquo03pp 5ndash10 Bangi Malaysia 2003
[22] T Yalcinoz H Altun and U Hasan ldquoConstrained economicdispatch with prohibited operating zones a Hopfield neuralnetwork approachrdquo in Proceedings of the 10th MediterraneanElectrotechnical Conference Information Technology and Elec-trotechnology for the Mediterranean Countries ProceedingsMeleCon rsquo00 (Cat No 00CH37099) pp 570ndash573 LemesosCyprus 2000
[23] D N Vo P Schegner and W Ongsakul ldquoCuckoo searchalgorithm for non-convex economic dispatchrdquo IET GenerationTransmission amp Distribution vol 7 no 6 pp 645ndash654 2013
[24] J-B Park K-S Lee J-R Shin and K Y Lee ldquoA particleswarm optimization for economic dispatch with non-smoothcost functionsrdquo IEEETransactions on Power Systems vol 20 no1 pp 34ndash42 2005
[25] T Niknam H D Mojarrad and M Nayeripour ldquoA newfuzzy adaptive particle swarm optimization for non-smootheconomic dispatchrdquo Energy vol 35 no 4 pp 1764ndash1778 2010
[26] L D S Coelho and V C Mariani ldquoAn improved harmonysearch algorithm for power economic load dispatchrdquo EnergyConversion and Management vol 50 no 10 pp 2522ndash25262009
[27] M Mahdavi M Fesanghary and E Damangir ldquoAn improvedharmony search algorithm for solving optimization problemsrdquoAppliedMathematics and Computation vol 188 no 2 pp 1567ndash1579 2007
[28] A Bhattacharya and P K Chattopadhyay ldquoSolving complexeconomic load dispatch problem using biogeography-basedoptimizationrdquo Expert Systems with Applications vol 37 no 5pp 3605ndash3615 2010
[29] D Simon ldquoBiogeography-based optimizationrdquo IEEE Transac-tions on Evolutionary Computation vol 12 no 6 pp 702ndash7132008
[30] J S Alsumait J K Sykulski and A K Al-Othman ldquoAhybrid GA-PS-SQP method to solve power system valve-pointeconomic dispatch problemsrdquo Applied Energy vol 87 no 5 pp1773ndash1781 2010
[31] M S Turgut andGKDemir ldquoQuadratic approximationndashbasedhybrid Artificial Cooperative Search algorithm for economicemission load dispatchproblemsrdquo International Transactions onElectrical Energy Systems vol 27 no 4 pp 1ndash14 2017
[32] T A A Victoire and A E Jeyakumar ldquoHybrid PSO-SQPfor economic dispatch with valve-point effectrdquo Electric PowerSystems Research vol 71 no 1 pp 51ndash59 2004
[33] D-KHe F-LWang andZ-ZMao ldquoHybrid genetic algorithmfor economic dispatch with valve-point effectrdquo Electric PowerSystems Research vol 78 no 4 pp 626ndash633 2008
[34] D C Secui ldquoA modified symbiotic organisms search algorithmfor large scale economic dispatch problem with valve-pointloading effectsrdquo Energy vol 113 pp 366ndash384 2016
[35] K BhattacharjeeA Bhattacharya and S H N Dey ldquoTeaching-learning-based optimization for different economic dispatchproblemsrdquo Scientia Iranica vol 21 no 3 pp 870ndash884 2014
[36] K Bhattacharjee A Bhattacharya and S H N Dey ldquoOppo-sitional Real Coded Chemical Reaction Optimization for dif-ferent economic dispatch problemsrdquo International Journal ofElectrical Power amp Energy Systems vol 55 pp 378ndash391 2014
Mathematical Problems in Engineering 29
[37] V Hosseinnezhad M Rafiee M Ahmadian and M T AmelildquoSpecies-basedQuantum Particle SwarmOptimization for eco-nomic load dispatchrdquo International Journal of Electrical Poweramp Energy Systems vol 63 pp 311ndash322 2014
[38] M S P Subathra S E Selvan T A A Victoire A HChristinal andU Amato ldquoA hybrid with cross-entropymethodand sequential quadratic programming to solve economic loaddispatch problemrdquo IEEE Systems Journal vol 9 no 3 pp 1031ndash1044 2015
[39] J Zhan Q H Wu C Guo and X Zhou ldquoEconomic dispatchwith non-smooth objectives part I local minimum analysisrdquoIEEE Transactions on Power Systems vol 30 no 2 pp 710ndash7212015
[40] A K Barisal andR C Prusty ldquoLarge scale economic dispatch ofpower systems using oppositional invasive weed optimizationrdquoApplied Soft Computing vol 29 pp 122ndash137 2015
[41] M Basu ldquoImproved differential evolution for economic dis-patchrdquo International Journal of Electrical Power amp EnergySystems vol 63 pp 855ndash861 2014
[42] V S Aragon S C Esquivel and C C Coello ldquoAn immunealgorithm with power redistribution for solving economicdispatch problemsrdquo Information Sciences vol 295 pp 609ndash6322015
[43] M Ghasemi M Taghizadeh S Ghavidel and A AbbasianldquoColonial competitive differential evolution an experimentalstudy for optimal economic load dispatchrdquo Applied Soft Com-puting vol 40 pp 342ndash363 2016
[44] B R Adarsh T Raghunathan T Jayabarathi and X-S YangldquoEconomic dispatch using chaotic bat algorithmrdquo Energy vol96 pp 666ndash675 2016
[45] N Ghorbani and E Babaei ldquoExchange market algorithm foreconomic load dispatchrdquo International Journal of ElectricalPower amp Energy Systems vol 75 pp 19ndash27 2016
[46] J X V Neto G Reynoso-Meza T H Ruppel V C Marianiand L D S Coelho ldquoSolving non-smooth economic dispatchby a new combination of continuous GRASP algorithm anddifferential evolutionrdquo International Journal of Electrical Poweramp Energy Systems vol 84 pp 13ndash24 2017
[47] A Kavousi-Fard and A Khosravi ldquoAn intelligent 120579-ModifiedBat Algorithm to solve the non-convex economic dispatchproblem considering practical constraintsrdquoElectrical Power andEnergy Systems vol 82 pp 189ndash196 2016
[48] M A Al-Betar M A Awadallah A T Khader and A L BolajildquoTournament-based harmony search algorithm for non-convexeconomic load dispatch problemrdquo Applied Soft Computing vol47 pp 449ndash459 2016
[49] M A Al-Betar M A Awadallah I A Doush E Alsukhni andH ALkhraisat ldquoA non-convex economic dispatchproblemwithvalve loading effect using a new modified 120573-hill climbing localsearch algorithmrdquo Arabian Journal for Science and Engineeringpp 1ndash8 2018
[50] M A Awadallah M A Al-Betar A L Bolaji E M Alsukhniand H Al-Zoubi ldquoNatural selection methods for artificial beecolony with new versions of onlooker beerdquo Soft Computing pp1ndash40 2018
[51] S H Kaboli and A K Alqallaf ldquoSolving non-convex economicload dispatch problem via artificial cooperative search algo-rithmrdquo Expert Systems with Applications vol 128 pp 14ndash272019
[52] D Singh and J Dhillon ldquoAmeliorated greywolf optimization foreconomic load dispatch problemrdquo Energy vol 169 pp 398ndash4192019
[53] X S Yang and S Deb ldquoCuckoo search via Levy flightsrdquo inProceedings of the World Congress on Nature and BiologicallyInspired Computing (NaBIC rsquo09) pp 210ndash214 India 2009
[54] S Sahoo K M Dash R C Prusty and A K Barisal ldquoCom-parative analysis of optimal load dispatch through evolutionaryalgorithmsrdquo Ain Shams Engineering Journal vol 6 pp 107ndash1202015
[55] M Basu and A Chowdhury ldquoCuckoo search algorithm foreconomic dispatchrdquo Energy vol 60 pp 99ndash108 2013
[56] A B Serapiao ldquoCuckoo search for solving economic dispatchload problemrdquo Intelligent Control and Automation vol 04 no04 pp 385ndash390 2013
[57] A H Bindu and M D Reddy ldquoEconomic load dispatch usingcuckoo search algorithmrdquo International Journal of EngineeringResearch and Applications vol 3 pp 498ndash502 2013
[58] E Afzalan and M Joorabian ldquoAn improved cuckoo searchalgorithm for power economic load dispatchrdquo InternationalTransactions on Electrical Energy Systems vol 25 no 6 pp 958ndash975 2015
[59] T T Nguyen and D N Vo ldquoThe application of one rank cuckoosearch algorithm for solving economic load dispatch problemsrdquoApplied Soft Computing vol 37 pp 763ndash773 2015
[60] S M Islam S Das S Ghosh S Roy and P N Suganthan ldquoAnadaptive differential evolution algorithm with novel mutationand crossover strategies for global numerical optimizationrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 42 no 2 pp 482ndash499 2012
[61] Z L Gaing ldquoParticle swarm optimization to solving theeconomic dispatch considering the generator constraintsrdquo IEEETransactions on Power Systems vol 18 no 3 pp 1187ndash1195 2003
[62] X-S Yang and S Deb ldquoEngineering optimisation by Cuckoosearchrdquo International Journal of Mathematical Modelling andNumerical Optimisation vol 330 pp 43ndash51 2010
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
execution time that ICSA approach takes is smaller thanmostones except IAPR [42] and 120579-MBA [47] which have been runon stronger computers Note that MSOS [34] has been fasterthan the proposed method for Subcases 31 and 32 abovebut it is too slower than the proposed method for the casenamely 1813 seconds compared to 146 seconds Comparedto two other versions of Cuckoo search CCSA [23] and OSE-CSA [59] the proposed method is also faster about threetimes although the processors are slightly different For thiscase 120579-MBA [47] shows a very good performance howeverthe method has not been tested onmore complicated systemsand larger scale and therefore more comparisons with themethod must end Clearly the proposed method is stillefficient for the case where large scale and effects of valveloading process are included
Subcase 35 is the largest scale system with 80 units Inaddition to the implementation of ICSA CCSA has beenalso run for the Subcase 35 for further investigation ofefficiency improvement of the proposed ICSA approach The
comparisons of minimum cost in Table 8 show that theproposed ICSA can find more optimal solution than M120573-HCLSA [49] AGWOA [52] and CCSA The proposed ICSAis also superior to CCSA in terms ofmore stable search abilityand lower fluctuation since its average cost and maximumcost are less than those of CCSA The outstanding figurecannot be reached as compared to AGWOA [52] howeverit is hard to conclude AGWOA [52] is superior to theproposed ICSA approach about more stable search abilityand lower fluctuation Actually comparison of the values ofpopulation and iterations as well as execution time cannot beaccomplished because the information was not reported in[52]Thus it can conclude that the proposed ICSA is effectivefor the subcase
Optimal solutions obtained by ICSA for the case areshown in Tables 20ndash23
54 Obtained Results on Case 4 with Two Systems consideringSFS and PPZ and RPR Constraints In this section two
16 Mathematical Problems in Engineering
Table 9 Result comparisons for Subcases 41 and 42
Subcase Approach Best cost ($h) Mean cost ($h) Worst cost ($h) Std dev ($h) CPU time (s) Computer (Processor-Ram)
test systems with SFS and PPZ and RPR constraints areconsidered The test system size is up to 60 and 90 units forSubcases 41 and 42 respectively Comparison for the casesis only performed with two Genetic algorithms consisting ofGA and IGA-MU in [9] and two other versions of Cuckoosearch algorithms including CCSA [23] and OSE-CSA [59]and presented in Table 9 Other studies have tended toignore such complicated constraints of PPZ and RPR Thecosts reported in Table 9 indicate that ICSA approach canobtain more effective solution than CCSA and OSE-CSAbecause it has reached lower minimum cost than the twoones Furthermore the proposed method also takes shortercomputation time for the two cases from about two timesto about three times although the processor of the proposedmethod is slightly strongerThemean costs of ICSA approachare much less than those from IGA-MU and GA and slightlyhigher than those from OSE-CSA but there is a trade-off between the proposed method and CCSA for the twosubcases In fact the proposed method obtains higher meancost for Subcase 41 but lower cost for Subcase 42 AlthoughGAmethods have been runon aweak computerwith 07 GHzof the processor compared to that with 24GHz in the studytheir execution times are significantly higher namely 56381seconds (GA) and 16258 seconds (IGA-MU) compared to09153 seconds of ICSA approach for Subcase 41 and 94093seconds (GA) and 25545 seconds (IGA-MU) compared to15892 seconds (the proposedmethod)The analysis can pointout that ICSA approach ismore efficient than these comparedmethods in terms of optimal solutions and execution time
Optimal solutions obtained by ICSA for Subcase 42 areshown in Table 24
55 Obtained Results on Case 5 with a 15-Unit System consid-ering SFS and RRL PPZ and PL Constraints In this section
a 15-unit system considering RRL PPZ and PL constraintsis considered to be solved for finding optimal solution Forefficiency investigation of the proposed ICSA we have alsoimplemented CCSA for comparison As listed in Table 10the proposed ICSA is the most effective method with thesmallest fuel cost The comparisons of control parameters aswell asCPU time are also good evidence to confirm the strongsearch of the proposed ICSA approach since it has been runby smaller values of control parameter and faster executiontime as compared to all methods excluding CCSA
Optimal solution obtained by ICSA for the case is shownin Table 25
56 Obtained Results on Case 6 withThree Systems consideringMFS and EoVLP In this section three test systems with thechallenge on objective function including multi-fossil fuelsources and effects of valve loading process are consideredThe scale is up to 80 units 160 units and 320 units forSubcases 61 62 and 63 respectively
Comparison for Subcase 61 reported in Table 11 revealsthat the proposed method is the best method in terms of thelowest best cost the lowestmean cost and the lowest standarddeviation and the fastest execution time The processor fromthis proposedmethod is about four times stronger thanCGA-MU and IGA-MU but the speed is from ten times to 35 timesfaster than these methods Compared to CCSA and OSE-CSA the proposed method is about two times faster but theprocessor is slightly stronger
Comparison for Subcase 62 is reported in Table 12Clearly the proposed ICSA approach obtains better values ofthe best mean and worst costs than most methods exceptMSOS [34] where the best cost difference is about $ 024However the proposed method is the second fastest onewith 1119 seconds where the first fastest one CBA [44] has
Mathematical Problems in Engineering 17
Table 11 Comparisons of found results for Subcase 61
Approach Best cost Mean cost Worst cost Std dev CPU time Computer($h) ($h) ($h) ($h) (s) (Processor-Ram)
spent 571 seconds Clearly MSOS is better than the proposedICSA approach with respect to slightly less best cost butis worse than the proposed method in terms of executiontime namely 2572 seconds compared to 1119 seconds whileCBA [44] is faster than the proposed method but obtainssignificantly worse costs The analysis can conclude that theproposed ICSA approach is very powerful for the subcasewith 160 units
Table 13 presents the comparison of three other methodsincludingCCSA [54] SOS [34] andMSOS [34] accompaniedwith the proposed method for Subcase 63 The obtainedresult comparisons imply that ICSA approach can obtainbetter values of the best and standard deviation costs thanCCSA and SOS but obtains slightly higher cost than MSOSby approximately $ 022 Besides the execution time fromICSA approach is much shorter than others especially it ishigher than five times faster than MSOS The four methodshave been run on approximately strong computers Brieflythe proposed ICSA approach can find and converge to morefavorable solution than other methods with shorter CPUtime except the comparison with MSOS which had bettersolution but spent higher than five times execution times
Consequently the proposed method is a very promisingoptimization algorithm for Subcase 63 a system up to 320units and with multi-fossil fuel sources and effects of valveloading process
Optimal solution obtained by ICSA for Subcase 63 isshown in Table 26
57 The Improvement of ICSA Approach Performance
571 The Outstanding Improvement over CCSA In this sec-tion the performance improvement of ICSA over CCSAhas been investigated by analyzing obtained results and setcontrol parameters Table 14 has been formed by addingreduction cost improvement level of the best cost executiontime and control parameters consisting of 119873119901119904 and 119866119898119886119909Among the compared factors reduction cost is the deviationof the cost of CCSA and that of ICSA whereas the improve-ment level is the ratio of the reduction cost to the cost ofCCSA The reduction costs indicate the proposed methodcould find either equal quality of solutions or higher qualityof solutions than CCSA for all study casesThe reduction costis from $0 to $42839 corresponding to the improvement level
18 Mathematical Problems in Engineering
Table 14 Summary of results obtained by CCSA and ICSA for all study cases
Study case Reduction cost ($) Improvement level () Execution time (s) Nps Gmax
Figure 4 The best run obtained by CCSA and ICSA for Case 2
that is from 0 to 08968The saving cost is not too much forone hour but the operation in one day onemonth or one yearis very high However it should be noted that CCSA has beenrun by setting much higher population size and iterations formany cases excluding study cases implemented in the studysuch as Case 2 Subcase 31 Subcase 35 and Case 5 Forinstance ICSA has used 119873119901119904 = 10 and 119866119898119886119909 = 15 for Subcase13 and119873119901119904 = 10 and119866119898119886119909 = 25 for Subcase 14 whereas CCSAhas been run by setting119873119901119904 = 20 and 119866119898119886119909 = 5000 for the twosubcases Similarly CCSA has been run for Subcases 32 33and 34 with much higher number of iterations For the lastsubcase CCSA has been run by setting 119873119901119904 = 320 and 119866119898119886119909
= 1200 but those of ICSA have been 10 and 9000 Due tothe higher value of control parameters CCSA has tended tospend more time in finding such high quality solutions foralmost all study cases Execution time of ICSA is less than18 seconds while that of CCSA is up to higher 75 secondsIt is clear that the proposed ICSA could find better optimalsolutions thanCCSA for such considered systems For furtherinvestigation of performance comparison the best runs over50 runs and fuel cost values of 50 runs obtained by CCSA andthe proposed ICSA for Case 2 Subcase 31 Subcase 35 andCase 5 have been plotted in from Figures 4ndash11 The best runcurves show the faster search of the proposed ICSA method
Mathematical Problems in Engineering 19
0 5 10 15 20 25 30 35 40 45 50Run
19795
198
19805
1981
19815
1982
19825
1983
19835
Fuel
cost
($)
CCSAICSA
times105
Figure 5The best fuel cost of 50 runs obtained by CCSA and ICSA for Case 2
0 2 4 6 8 10 12 14 16 18 20Iteration
8234
8236
8238
8240
8242
8244
8246
8248
8250
8252
Fitn
ess F
unct
ion
($)
CCSAICSA
Figure 6 The best run obtained by CCSA and ICSA for Subcase 31
whereas 50 values of fuel cost indicate that the proposedICSA can find many solutions with better quality Clearlythe proposed ICSA is outstanding in terms of stabilizationof solution search and faster convergence As a result it canconclude that the proposed ICSA approach is more effectivethanCCSA in solvingOLDproblemwith considered systems
572The Improvement of Results over Other Methods In thisarticle we have tested ICSA approach on 6 cases with 16
systems with different fuel cost forms different constraintsand different scale systems from 3 units to 320 units We havecompared the yielded results from ICSA approach and otherexisting ones for evaluating the efficiency of ICSA approachIn subsections above we have shown yielded results fromICSA approach and compared these results to those of otheronesHowever the demonstrationhad not been very good forobserving and comparing to lead to a conclusionThus in thesubsectionwehave summarized the result comparisons of the
20 Mathematical Problems in Engineering
0 5 10 15 20 25 30 35 40 45 50Run
8234
82342
82344
82346
82348
8235
82352
82354
82356
82358
8236
Fuel
cost
($)
CCSAICSA
Figure 7 The best fuel cost of 50 runs obtained by CCSA and ICSA for Subcase 31
0 1000 2000 3000 4000 5000 6000Iteration
242
244
246
248
25
252
254
256
Fitn
ess F
unct
ion
($)
CCSAICSA
times105
Figure 8 The best run obtained by CCSA and ICSA for Subcase 35
proposed and other ones Table 15 has reported the reductioncost (in $) of ICSA approach compared to other ones Inaddition we have converted the reduction cost into improve-ment level (in ) for better comparison The improvementhas been shown from the lowest level to the highest levelin terms of reduction cost and improvement percentageIn addition we have also given the slowest and the fastestexecution time of other compared methods together withthat of the proposed method The table implies that ICSA
approach can find better optimal solutions with less fuel costup to $052 for Subcase 13 $75229 for Subcase 14 $1775149for Case 2 $1181256 for Subcase 31 $8438 for Subcase 32$9113 for Subcase 33 $121181 for Subcase 34 $3412 forSubcase 35 $04329 for Subcase 41 $30227 for Subcase42 $407002 for Case 5 $18183 for Subcase 61 $13965 forSubcase 62 and $11817 for Subcase 63 These reductioncosts are equivalent to improvement level (IL) of 0032908968 82282 14143 047 038 099 001405
Mathematical Problems in Engineering 21
0 5 10 15 20 25 30 35 40 45 50Run
2428
243
2432
2434
2436
2438
244
2442
Fuel
cost
($)
CCSAICSA
times105
Figure 9 The best fuel cost of 50 runs obtained by CCSA and ICSA for Subcase 35
0 50 100 150 200 250 300 350 400Iteration
327
328
329
33
331
332
333
334
335
336
337
Fitn
ess F
unct
ion
($)
CCSAICSA
times104
Figure 10 The best run obtained by CCSA and ICSA for Case 5
00003 0002 12291 004 138 and 059 Thesequantitative comparisons reveal that larger scale systems canlead to better reduction cost but the improvement level isnot high because total cost of compared methods tends tobe large for large scale systems Furthermore very large scalesystems with nondifferentiable objective have been normallysolved by strong methods In fact systems in Case 3 havethe same characteristic with single fuel and effects of valveloading process but Subcase 34 is a larger scale system with
40 units while Subcases 31 32 and 33 are constructed by 3units 13 units and 13 units So the improvement percentageof Subcase 34 can be up to 099 whilst that of Subcases32 and 33 is 047 and 038 respectively Subcase 35 iswith the largest system 80 units but the reduction cost is notmuch only $3412 because compared methods with ICSA areeither state-of-the-art ones or improved ones Also Subcase41 and Subcase 42 have considered single fuel and PPZ andspinning reserve constraints but Subcase 42 is larger scale
22 Mathematical Problems in Engineering
Table 15 Performance improvement summary of the proposed method
Study cases Reduction cost ($) Improvement level () Execution time (s)From To From To Slowest method Fastest method Proposed method
Figure 11 The best fuel cost of 50 runs obtained by CCSA and ICSA for Case 5
system with 90 units and Subcase 41 is only with 60 unitsThus the improvement of Subcase 41 is lower with 00003but that of Subcase 42 is 0002 Similarly systems in Cases 6have the same featurewithmulti-fossil fuel sources and effectsof valve loading process but they are respectively constructedby 80 160 and 320 units As a result the improvements ofSubcase 62 138 and Subcase 63 059 are much higherthan Subcase 61 004 However Subcase 62 with smallernumber of units but getting higher improvement is easilyunderstood because there were nine compared methods butonly three compared methods are considered for Subcase 63In general the improvement is not high it is about under onedollar several dollars tens of dollars and over one thousand
dollars per hour however the saving cost will be significantif the operation is considered to be onemonth with 720 hoursor one year with 8760 hours
Execution time comparison can be evaluated by observ-ing the fastest and the slowest compared methods in Table 15These execution times of the proposed method are approxi-mately equal to that of the fastest methods for Subcases 3233 and 34 and much shorter than other fastest comparedmethods for other cases especially for Subcases 61 and 63The fastest method for Subcase 62 is CBA [44] showing 57seconds while that of the proposed method is 1119 secondsHowever it cannot conclude that CBA is more effective thanthe proposed method because the proposed method could
find better optimal solution with less fuel cost by $2130As considering execution time of the slowest comparedmethods it can point out that ICSA is a very fast optimizationtool since the execution time of these methods is 29496 and3425 seconds for Subcases 32 and 33 116735 seconds forSubcase 34 56381 seconds for Subcase 41 94093 secondsfor Subcase 42 12797 seconds for Case 5 30941 seconds forSubcase 61 7542 seconds for Subcase 62 and 9641 secondsfor Subcase 63 while the execution time of the proposedmethod for these cases is respectively 095 146 091 1589246 84828 1119 and 171384 seconds It is clearly shown thatICSA is very fast as compared to these methods
In summary the proposed method has found approx-imately high quality solutions with several standard state-of-the-art meta-heuristic algorithms and improved versionsof them together with other old methods In addition theproposed method could improve result better than approx-imately all methods with faster execution time Comparedto other methods with the fastest convergence speed andhigh quality solutions the proposed method has been as
fast as for some cases and much faster for other cases Thecomparison with the slowest methods could show that theproposed method was extremely powerful since it was up tonearly one thousand times faster Consequently the proposedICSA approach can be one of the strongest optimization toolsfor OLD problem
6 Conclusions
This paper has proposed a good ICSA method for solvingOLD problem in which many test systems with differentobjective functions and complicated constraints from simpleto complex have been used as studied cases The proposedICSA method has been developed by performing severalmodifications on the second solution update of CCSA whichcontained several drawbacks to global convergence and fastmanner The OLD problem has covered from single fuelto multi-fossil fuels from quadratic objective function tononconvex objective function in addition to PPZ RPR andRRL constraints Many existing optimization algorithms have
been concerned in aim to compare the performance and givethe final conclusion on the proposed method There have
been six main cases with sixteen subcases The evaluationshave been made at the end of each study case Clearly theproposed ICSA approach has yielded more effective optimalsolutions with faster execution time than almost all methodsConsequently it can be concluded that the proposed methodis much more superior to CCSA and is a very promisingmethod for solving OLD problem
Appendix
See Tables 16ndash26
Nomenclature
120575119894 120582119894 120572119894 120573119894 120574119894 Fuel cost functioncoefficients of the 119894119905ℎthermal generationunit11986100 1198610119895 119861119895119894 Power loss matrixcoefficients119865119865119909 119865119865119887119890119904119905 The values of fitness ofsolution 119909 and theso-far most effectivesolution among thecurrent set ofsolutions119898119894 Number of fuels burntin the 119894119905ℎ thermalgeneration unit
119873 Number of allavailable thermalgeneration units119899119894 Number of prohibitedpower zones of the 119894119905ℎthermal generationunit119875119906119894119896 119875119897119894119896 Upper and lowerlimits of the 119894119905ℎthermal generationunit corresponding tothe kth PPZ119875119886 Probability ofreplacing controlvariables in each oldsolution119875119863 Real power demand ofall loads in system119875119894119898119886119909 119875119894119898119894119899 The highest andlowest real poweroutputs of the iththermal generationunit119875119894119895119898119886119909 119875119894119895119898119894119899 The highest andlowest real poweroutputs of the 119894119905ℎthermal generationunit corresponding tothe 119895119905ℎ fuel type119878119894119898119886119909 Maximum real powerreserve contributionof the thermalgeneration unit 1198941198781199001198971199031198861198991198891 1198781199001198971199031198861198991198892 1198781199001198971199031198861198991198893 1198781199001198971199031198861198991198894 Randomly mixedsolutions from the setof current solutions119878119900119897119909 119878119900119897119866119887119890119904119905 The old solution x andthe most effectivesolution119878119877 Real power reserverequirement of system
120575119894119895 120582119894119895 120572119894119895 120573119894119895 120574119894119895 Fuel cost functioncoefficients of the 119894119905ℎthermal generationunit corresponding tothe 119895119905ℎ fuel type1205761 1205762 1205763 1205764 1205765 1205766 Random numbersbetween 0 and 1Ω Number of generationunits considering PPZconstraint
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare no conflicts of interest
References
[1] N Sinha R Chakrabarti and P K Chattopadhyay ldquoEvolution-ary programming techniques for economic load dispatchrdquo IEEETransactions on Evolutionary Computation vol 7 no 1 pp 83ndash94 2003
[2] P Hansen ldquoA separable approximation dynamic programmingalgorithm for economic dispatch with transmission lossesrdquoYugoslav Journal of Operations Research vol 12 no 2 2002
[3] N T Thang ldquoSolving economic dispatch problem with piece-wise quadratic cost functions using lagrange multiplier theoryrdquoin Proceedings of the 3rd International Conference on ComputerTechnology and Development (ICCTD rsquo11) pp 359ndash364 ASMEPress 2011
[4] S K Mishra and S K Mishra ldquoA comparative study of solutionof economic load dispatch problem in power systems in theenvironmental perspectiverdquoProcedia Computer Science vol 48pp 96ndash100 2015
[5] A A Al-Subhi and H K Alfares ldquoEconomic load dispatchusing linear programming a comparative studyrdquo InternationalJournal of Applied Industrial Engineering vol 3 no 1 pp 16ndash362016
28 Mathematical Problems in Engineering
[6] C Zhou G Huang and J Chen ldquoPlanning of electric powersystems considering virtual power plants with dispatchableloads included an inexact two-stage stochastic linear program-ming modelrdquoMathematical Problems in Engineering vol 2018Article ID 7049329 12 pages 2018
[7] T T Nguyen N V Quynh and L Van Dai ldquoImproved fireflyalgorithm a novel method for optimal operation of thermalgenerating unitsrdquo Complexity vol 2018 Article ID 7267593 23pages 2018
[8] S S Haroon S Hassan S Amin et al ldquoMultiple fuel machinespower economic dispatch using stud differential evolutionrdquoEnergies vol 11 no 6 pp 1ndash20 2018
[9] C-T Su and C-L Chiang ldquoNonconvex power economic dis-patch by improved genetic algorithm with multiplier updatingmethodrdquo Electric Power Components and Systems vol 32 no 3pp 257ndash273 2004
[10] NNoman andH Iba ldquoDifferential evolution for economic loaddispatch problemsrdquo Electric Power Systems Research vol 78 no8 pp 1322ndash1331 2008
[11] P Somasundaram and K Kuppusamy ldquoApplication of evolu-tionary programming to security constrained economic dis-patchrdquo International Journal of Electrical Power amp EnergySystems vol 27 no 5-6 pp 343ndash351 2005
[12] T Jayabarathi K Jayaprakash andD Jeyakumar ldquoEvolutionaryprogramming technique for different kinds of economic dis-patch problemsrdquo Electric Power Systems Research vol 73 no 2pp 169ndash176 2005
[13] P Subbaraj R Rengaraj and S Salivahanan ldquoEnhancementof combined heat and power economic dispatch using selfadaptive real-coded genetic algorithmrdquo Applied Energy vol 86no 6 pp 915ndash921 2009
[14] C-L Chiang and C-T Su ldquoAdaptive-improved genetic algo-rithm for the economic dispatch of units with multiple fueloptionsrdquo Cybernetics and Systems vol 36 no 7 pp 687ndash7042005
[15] C Chiang ldquoImproved genetic algorithm for power economicdispatch of units with valve-point effects and multiple fuelsrdquoIEEE Transactions on Power Systems vol 20 no 4 pp 1690ndash1699 2005
[16] K Meng H G Wang Z Y Dong and K P Wong ldquoQuantum-inspired particle swarm optimization for valve-point economicload dispatchrdquo IEEE Transactions on Power Systems vol 25 no1 pp 215ndash222 2010
[17] P Erdogmus A Ozturk and S Duman ldquoEnvironmen-taleconomic dispatch using genetic algorithm and simulatedannealingrdquo in Proceedings of the International Conference onElectric Power and Energy Conversion Systems EPECS rsquo09 pp1ndash4 IEEE 2009
[18] L D S Coelho and V C Mariani ldquoImproved differentialevolution algorithms for handling economic dispatch opti-mization with generator constraintsrdquo Energy Conversion andManagement vol 48 no 5 pp 1631ndash1639 2007
[19] H Dakuo W Fuli and M Zhizhong ldquoA hybrid genetic algo-rithm approach based on differential evolution for economicdispatch with valve-point effectsrdquo Electrical Power and EnergySystems vol 30 pp 31ndash38 2008
[20] C-F Chang J-J Wong J-P Chiou and C-T Su ldquoRobustsearching hybrid differential evolution method for optimalreactive power planning in large-scale distribution systemsrdquoElectric Power Systems Research vol 77 no 5-6 pp 430ndash4372007
[21] M R Farooqi P Jain and K R Niazi ldquoUsing Hopfield neuralnetwork for economic dispatch of power systemsrdquo in Proceed-ings of the National Power Engineering Conference PECon rsquo03pp 5ndash10 Bangi Malaysia 2003
[22] T Yalcinoz H Altun and U Hasan ldquoConstrained economicdispatch with prohibited operating zones a Hopfield neuralnetwork approachrdquo in Proceedings of the 10th MediterraneanElectrotechnical Conference Information Technology and Elec-trotechnology for the Mediterranean Countries ProceedingsMeleCon rsquo00 (Cat No 00CH37099) pp 570ndash573 LemesosCyprus 2000
[23] D N Vo P Schegner and W Ongsakul ldquoCuckoo searchalgorithm for non-convex economic dispatchrdquo IET GenerationTransmission amp Distribution vol 7 no 6 pp 645ndash654 2013
[24] J-B Park K-S Lee J-R Shin and K Y Lee ldquoA particleswarm optimization for economic dispatch with non-smoothcost functionsrdquo IEEETransactions on Power Systems vol 20 no1 pp 34ndash42 2005
[25] T Niknam H D Mojarrad and M Nayeripour ldquoA newfuzzy adaptive particle swarm optimization for non-smootheconomic dispatchrdquo Energy vol 35 no 4 pp 1764ndash1778 2010
[26] L D S Coelho and V C Mariani ldquoAn improved harmonysearch algorithm for power economic load dispatchrdquo EnergyConversion and Management vol 50 no 10 pp 2522ndash25262009
[27] M Mahdavi M Fesanghary and E Damangir ldquoAn improvedharmony search algorithm for solving optimization problemsrdquoAppliedMathematics and Computation vol 188 no 2 pp 1567ndash1579 2007
[28] A Bhattacharya and P K Chattopadhyay ldquoSolving complexeconomic load dispatch problem using biogeography-basedoptimizationrdquo Expert Systems with Applications vol 37 no 5pp 3605ndash3615 2010
[29] D Simon ldquoBiogeography-based optimizationrdquo IEEE Transac-tions on Evolutionary Computation vol 12 no 6 pp 702ndash7132008
[30] J S Alsumait J K Sykulski and A K Al-Othman ldquoAhybrid GA-PS-SQP method to solve power system valve-pointeconomic dispatch problemsrdquo Applied Energy vol 87 no 5 pp1773ndash1781 2010
[31] M S Turgut andGKDemir ldquoQuadratic approximationndashbasedhybrid Artificial Cooperative Search algorithm for economicemission load dispatchproblemsrdquo International Transactions onElectrical Energy Systems vol 27 no 4 pp 1ndash14 2017
[32] T A A Victoire and A E Jeyakumar ldquoHybrid PSO-SQPfor economic dispatch with valve-point effectrdquo Electric PowerSystems Research vol 71 no 1 pp 51ndash59 2004
[33] D-KHe F-LWang andZ-ZMao ldquoHybrid genetic algorithmfor economic dispatch with valve-point effectrdquo Electric PowerSystems Research vol 78 no 4 pp 626ndash633 2008
[34] D C Secui ldquoA modified symbiotic organisms search algorithmfor large scale economic dispatch problem with valve-pointloading effectsrdquo Energy vol 113 pp 366ndash384 2016
[35] K BhattacharjeeA Bhattacharya and S H N Dey ldquoTeaching-learning-based optimization for different economic dispatchproblemsrdquo Scientia Iranica vol 21 no 3 pp 870ndash884 2014
[36] K Bhattacharjee A Bhattacharya and S H N Dey ldquoOppo-sitional Real Coded Chemical Reaction Optimization for dif-ferent economic dispatch problemsrdquo International Journal ofElectrical Power amp Energy Systems vol 55 pp 378ndash391 2014
Mathematical Problems in Engineering 29
[37] V Hosseinnezhad M Rafiee M Ahmadian and M T AmelildquoSpecies-basedQuantum Particle SwarmOptimization for eco-nomic load dispatchrdquo International Journal of Electrical Poweramp Energy Systems vol 63 pp 311ndash322 2014
[38] M S P Subathra S E Selvan T A A Victoire A HChristinal andU Amato ldquoA hybrid with cross-entropymethodand sequential quadratic programming to solve economic loaddispatch problemrdquo IEEE Systems Journal vol 9 no 3 pp 1031ndash1044 2015
[39] J Zhan Q H Wu C Guo and X Zhou ldquoEconomic dispatchwith non-smooth objectives part I local minimum analysisrdquoIEEE Transactions on Power Systems vol 30 no 2 pp 710ndash7212015
[40] A K Barisal andR C Prusty ldquoLarge scale economic dispatch ofpower systems using oppositional invasive weed optimizationrdquoApplied Soft Computing vol 29 pp 122ndash137 2015
[41] M Basu ldquoImproved differential evolution for economic dis-patchrdquo International Journal of Electrical Power amp EnergySystems vol 63 pp 855ndash861 2014
[42] V S Aragon S C Esquivel and C C Coello ldquoAn immunealgorithm with power redistribution for solving economicdispatch problemsrdquo Information Sciences vol 295 pp 609ndash6322015
[43] M Ghasemi M Taghizadeh S Ghavidel and A AbbasianldquoColonial competitive differential evolution an experimentalstudy for optimal economic load dispatchrdquo Applied Soft Com-puting vol 40 pp 342ndash363 2016
[44] B R Adarsh T Raghunathan T Jayabarathi and X-S YangldquoEconomic dispatch using chaotic bat algorithmrdquo Energy vol96 pp 666ndash675 2016
[45] N Ghorbani and E Babaei ldquoExchange market algorithm foreconomic load dispatchrdquo International Journal of ElectricalPower amp Energy Systems vol 75 pp 19ndash27 2016
[46] J X V Neto G Reynoso-Meza T H Ruppel V C Marianiand L D S Coelho ldquoSolving non-smooth economic dispatchby a new combination of continuous GRASP algorithm anddifferential evolutionrdquo International Journal of Electrical Poweramp Energy Systems vol 84 pp 13ndash24 2017
[47] A Kavousi-Fard and A Khosravi ldquoAn intelligent 120579-ModifiedBat Algorithm to solve the non-convex economic dispatchproblem considering practical constraintsrdquoElectrical Power andEnergy Systems vol 82 pp 189ndash196 2016
[48] M A Al-Betar M A Awadallah A T Khader and A L BolajildquoTournament-based harmony search algorithm for non-convexeconomic load dispatch problemrdquo Applied Soft Computing vol47 pp 449ndash459 2016
[49] M A Al-Betar M A Awadallah I A Doush E Alsukhni andH ALkhraisat ldquoA non-convex economic dispatchproblemwithvalve loading effect using a new modified 120573-hill climbing localsearch algorithmrdquo Arabian Journal for Science and Engineeringpp 1ndash8 2018
[50] M A Awadallah M A Al-Betar A L Bolaji E M Alsukhniand H Al-Zoubi ldquoNatural selection methods for artificial beecolony with new versions of onlooker beerdquo Soft Computing pp1ndash40 2018
[51] S H Kaboli and A K Alqallaf ldquoSolving non-convex economicload dispatch problem via artificial cooperative search algo-rithmrdquo Expert Systems with Applications vol 128 pp 14ndash272019
[52] D Singh and J Dhillon ldquoAmeliorated greywolf optimization foreconomic load dispatch problemrdquo Energy vol 169 pp 398ndash4192019
[53] X S Yang and S Deb ldquoCuckoo search via Levy flightsrdquo inProceedings of the World Congress on Nature and BiologicallyInspired Computing (NaBIC rsquo09) pp 210ndash214 India 2009
[54] S Sahoo K M Dash R C Prusty and A K Barisal ldquoCom-parative analysis of optimal load dispatch through evolutionaryalgorithmsrdquo Ain Shams Engineering Journal vol 6 pp 107ndash1202015
[55] M Basu and A Chowdhury ldquoCuckoo search algorithm foreconomic dispatchrdquo Energy vol 60 pp 99ndash108 2013
[56] A B Serapiao ldquoCuckoo search for solving economic dispatchload problemrdquo Intelligent Control and Automation vol 04 no04 pp 385ndash390 2013
[57] A H Bindu and M D Reddy ldquoEconomic load dispatch usingcuckoo search algorithmrdquo International Journal of EngineeringResearch and Applications vol 3 pp 498ndash502 2013
[58] E Afzalan and M Joorabian ldquoAn improved cuckoo searchalgorithm for power economic load dispatchrdquo InternationalTransactions on Electrical Energy Systems vol 25 no 6 pp 958ndash975 2015
[59] T T Nguyen and D N Vo ldquoThe application of one rank cuckoosearch algorithm for solving economic load dispatch problemsrdquoApplied Soft Computing vol 37 pp 763ndash773 2015
[60] S M Islam S Das S Ghosh S Roy and P N Suganthan ldquoAnadaptive differential evolution algorithm with novel mutationand crossover strategies for global numerical optimizationrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 42 no 2 pp 482ndash499 2012
[61] Z L Gaing ldquoParticle swarm optimization to solving theeconomic dispatch considering the generator constraintsrdquo IEEETransactions on Power Systems vol 18 no 3 pp 1187ndash1195 2003
[62] X-S Yang and S Deb ldquoEngineering optimisation by Cuckoosearchrdquo International Journal of Mathematical Modelling andNumerical Optimisation vol 330 pp 43ndash51 2010
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
test systems with SFS and PPZ and RPR constraints areconsidered The test system size is up to 60 and 90 units forSubcases 41 and 42 respectively Comparison for the casesis only performed with two Genetic algorithms consisting ofGA and IGA-MU in [9] and two other versions of Cuckoosearch algorithms including CCSA [23] and OSE-CSA [59]and presented in Table 9 Other studies have tended toignore such complicated constraints of PPZ and RPR Thecosts reported in Table 9 indicate that ICSA approach canobtain more effective solution than CCSA and OSE-CSAbecause it has reached lower minimum cost than the twoones Furthermore the proposed method also takes shortercomputation time for the two cases from about two timesto about three times although the processor of the proposedmethod is slightly strongerThemean costs of ICSA approachare much less than those from IGA-MU and GA and slightlyhigher than those from OSE-CSA but there is a trade-off between the proposed method and CCSA for the twosubcases In fact the proposed method obtains higher meancost for Subcase 41 but lower cost for Subcase 42 AlthoughGAmethods have been runon aweak computerwith 07 GHzof the processor compared to that with 24GHz in the studytheir execution times are significantly higher namely 56381seconds (GA) and 16258 seconds (IGA-MU) compared to09153 seconds of ICSA approach for Subcase 41 and 94093seconds (GA) and 25545 seconds (IGA-MU) compared to15892 seconds (the proposedmethod)The analysis can pointout that ICSA approach ismore efficient than these comparedmethods in terms of optimal solutions and execution time
Optimal solutions obtained by ICSA for Subcase 42 areshown in Table 24
55 Obtained Results on Case 5 with a 15-Unit System consid-ering SFS and RRL PPZ and PL Constraints In this section
a 15-unit system considering RRL PPZ and PL constraintsis considered to be solved for finding optimal solution Forefficiency investigation of the proposed ICSA we have alsoimplemented CCSA for comparison As listed in Table 10the proposed ICSA is the most effective method with thesmallest fuel cost The comparisons of control parameters aswell asCPU time are also good evidence to confirm the strongsearch of the proposed ICSA approach since it has been runby smaller values of control parameter and faster executiontime as compared to all methods excluding CCSA
Optimal solution obtained by ICSA for the case is shownin Table 25
56 Obtained Results on Case 6 withThree Systems consideringMFS and EoVLP In this section three test systems with thechallenge on objective function including multi-fossil fuelsources and effects of valve loading process are consideredThe scale is up to 80 units 160 units and 320 units forSubcases 61 62 and 63 respectively
Comparison for Subcase 61 reported in Table 11 revealsthat the proposed method is the best method in terms of thelowest best cost the lowestmean cost and the lowest standarddeviation and the fastest execution time The processor fromthis proposedmethod is about four times stronger thanCGA-MU and IGA-MU but the speed is from ten times to 35 timesfaster than these methods Compared to CCSA and OSE-CSA the proposed method is about two times faster but theprocessor is slightly stronger
Comparison for Subcase 62 is reported in Table 12Clearly the proposed ICSA approach obtains better values ofthe best mean and worst costs than most methods exceptMSOS [34] where the best cost difference is about $ 024However the proposed method is the second fastest onewith 1119 seconds where the first fastest one CBA [44] has
Mathematical Problems in Engineering 17
Table 11 Comparisons of found results for Subcase 61
Approach Best cost Mean cost Worst cost Std dev CPU time Computer($h) ($h) ($h) ($h) (s) (Processor-Ram)
spent 571 seconds Clearly MSOS is better than the proposedICSA approach with respect to slightly less best cost butis worse than the proposed method in terms of executiontime namely 2572 seconds compared to 1119 seconds whileCBA [44] is faster than the proposed method but obtainssignificantly worse costs The analysis can conclude that theproposed ICSA approach is very powerful for the subcasewith 160 units
Table 13 presents the comparison of three other methodsincludingCCSA [54] SOS [34] andMSOS [34] accompaniedwith the proposed method for Subcase 63 The obtainedresult comparisons imply that ICSA approach can obtainbetter values of the best and standard deviation costs thanCCSA and SOS but obtains slightly higher cost than MSOSby approximately $ 022 Besides the execution time fromICSA approach is much shorter than others especially it ishigher than five times faster than MSOS The four methodshave been run on approximately strong computers Brieflythe proposed ICSA approach can find and converge to morefavorable solution than other methods with shorter CPUtime except the comparison with MSOS which had bettersolution but spent higher than five times execution times
Consequently the proposed method is a very promisingoptimization algorithm for Subcase 63 a system up to 320units and with multi-fossil fuel sources and effects of valveloading process
Optimal solution obtained by ICSA for Subcase 63 isshown in Table 26
57 The Improvement of ICSA Approach Performance
571 The Outstanding Improvement over CCSA In this sec-tion the performance improvement of ICSA over CCSAhas been investigated by analyzing obtained results and setcontrol parameters Table 14 has been formed by addingreduction cost improvement level of the best cost executiontime and control parameters consisting of 119873119901119904 and 119866119898119886119909Among the compared factors reduction cost is the deviationof the cost of CCSA and that of ICSA whereas the improve-ment level is the ratio of the reduction cost to the cost ofCCSA The reduction costs indicate the proposed methodcould find either equal quality of solutions or higher qualityof solutions than CCSA for all study casesThe reduction costis from $0 to $42839 corresponding to the improvement level
18 Mathematical Problems in Engineering
Table 14 Summary of results obtained by CCSA and ICSA for all study cases
Study case Reduction cost ($) Improvement level () Execution time (s) Nps Gmax
Figure 4 The best run obtained by CCSA and ICSA for Case 2
that is from 0 to 08968The saving cost is not too much forone hour but the operation in one day onemonth or one yearis very high However it should be noted that CCSA has beenrun by setting much higher population size and iterations formany cases excluding study cases implemented in the studysuch as Case 2 Subcase 31 Subcase 35 and Case 5 Forinstance ICSA has used 119873119901119904 = 10 and 119866119898119886119909 = 15 for Subcase13 and119873119901119904 = 10 and119866119898119886119909 = 25 for Subcase 14 whereas CCSAhas been run by setting119873119901119904 = 20 and 119866119898119886119909 = 5000 for the twosubcases Similarly CCSA has been run for Subcases 32 33and 34 with much higher number of iterations For the lastsubcase CCSA has been run by setting 119873119901119904 = 320 and 119866119898119886119909
= 1200 but those of ICSA have been 10 and 9000 Due tothe higher value of control parameters CCSA has tended tospend more time in finding such high quality solutions foralmost all study cases Execution time of ICSA is less than18 seconds while that of CCSA is up to higher 75 secondsIt is clear that the proposed ICSA could find better optimalsolutions thanCCSA for such considered systems For furtherinvestigation of performance comparison the best runs over50 runs and fuel cost values of 50 runs obtained by CCSA andthe proposed ICSA for Case 2 Subcase 31 Subcase 35 andCase 5 have been plotted in from Figures 4ndash11 The best runcurves show the faster search of the proposed ICSA method
Mathematical Problems in Engineering 19
0 5 10 15 20 25 30 35 40 45 50Run
19795
198
19805
1981
19815
1982
19825
1983
19835
Fuel
cost
($)
CCSAICSA
times105
Figure 5The best fuel cost of 50 runs obtained by CCSA and ICSA for Case 2
0 2 4 6 8 10 12 14 16 18 20Iteration
8234
8236
8238
8240
8242
8244
8246
8248
8250
8252
Fitn
ess F
unct
ion
($)
CCSAICSA
Figure 6 The best run obtained by CCSA and ICSA for Subcase 31
whereas 50 values of fuel cost indicate that the proposedICSA can find many solutions with better quality Clearlythe proposed ICSA is outstanding in terms of stabilizationof solution search and faster convergence As a result it canconclude that the proposed ICSA approach is more effectivethanCCSA in solvingOLDproblemwith considered systems
572The Improvement of Results over Other Methods In thisarticle we have tested ICSA approach on 6 cases with 16
systems with different fuel cost forms different constraintsand different scale systems from 3 units to 320 units We havecompared the yielded results from ICSA approach and otherexisting ones for evaluating the efficiency of ICSA approachIn subsections above we have shown yielded results fromICSA approach and compared these results to those of otheronesHowever the demonstrationhad not been very good forobserving and comparing to lead to a conclusionThus in thesubsectionwehave summarized the result comparisons of the
20 Mathematical Problems in Engineering
0 5 10 15 20 25 30 35 40 45 50Run
8234
82342
82344
82346
82348
8235
82352
82354
82356
82358
8236
Fuel
cost
($)
CCSAICSA
Figure 7 The best fuel cost of 50 runs obtained by CCSA and ICSA for Subcase 31
0 1000 2000 3000 4000 5000 6000Iteration
242
244
246
248
25
252
254
256
Fitn
ess F
unct
ion
($)
CCSAICSA
times105
Figure 8 The best run obtained by CCSA and ICSA for Subcase 35
proposed and other ones Table 15 has reported the reductioncost (in $) of ICSA approach compared to other ones Inaddition we have converted the reduction cost into improve-ment level (in ) for better comparison The improvementhas been shown from the lowest level to the highest levelin terms of reduction cost and improvement percentageIn addition we have also given the slowest and the fastestexecution time of other compared methods together withthat of the proposed method The table implies that ICSA
approach can find better optimal solutions with less fuel costup to $052 for Subcase 13 $75229 for Subcase 14 $1775149for Case 2 $1181256 for Subcase 31 $8438 for Subcase 32$9113 for Subcase 33 $121181 for Subcase 34 $3412 forSubcase 35 $04329 for Subcase 41 $30227 for Subcase42 $407002 for Case 5 $18183 for Subcase 61 $13965 forSubcase 62 and $11817 for Subcase 63 These reductioncosts are equivalent to improvement level (IL) of 0032908968 82282 14143 047 038 099 001405
Mathematical Problems in Engineering 21
0 5 10 15 20 25 30 35 40 45 50Run
2428
243
2432
2434
2436
2438
244
2442
Fuel
cost
($)
CCSAICSA
times105
Figure 9 The best fuel cost of 50 runs obtained by CCSA and ICSA for Subcase 35
0 50 100 150 200 250 300 350 400Iteration
327
328
329
33
331
332
333
334
335
336
337
Fitn
ess F
unct
ion
($)
CCSAICSA
times104
Figure 10 The best run obtained by CCSA and ICSA for Case 5
00003 0002 12291 004 138 and 059 Thesequantitative comparisons reveal that larger scale systems canlead to better reduction cost but the improvement level isnot high because total cost of compared methods tends tobe large for large scale systems Furthermore very large scalesystems with nondifferentiable objective have been normallysolved by strong methods In fact systems in Case 3 havethe same characteristic with single fuel and effects of valveloading process but Subcase 34 is a larger scale system with
40 units while Subcases 31 32 and 33 are constructed by 3units 13 units and 13 units So the improvement percentageof Subcase 34 can be up to 099 whilst that of Subcases32 and 33 is 047 and 038 respectively Subcase 35 iswith the largest system 80 units but the reduction cost is notmuch only $3412 because compared methods with ICSA areeither state-of-the-art ones or improved ones Also Subcase41 and Subcase 42 have considered single fuel and PPZ andspinning reserve constraints but Subcase 42 is larger scale
22 Mathematical Problems in Engineering
Table 15 Performance improvement summary of the proposed method
Study cases Reduction cost ($) Improvement level () Execution time (s)From To From To Slowest method Fastest method Proposed method
Figure 11 The best fuel cost of 50 runs obtained by CCSA and ICSA for Case 5
system with 90 units and Subcase 41 is only with 60 unitsThus the improvement of Subcase 41 is lower with 00003but that of Subcase 42 is 0002 Similarly systems in Cases 6have the same featurewithmulti-fossil fuel sources and effectsof valve loading process but they are respectively constructedby 80 160 and 320 units As a result the improvements ofSubcase 62 138 and Subcase 63 059 are much higherthan Subcase 61 004 However Subcase 62 with smallernumber of units but getting higher improvement is easilyunderstood because there were nine compared methods butonly three compared methods are considered for Subcase 63In general the improvement is not high it is about under onedollar several dollars tens of dollars and over one thousand
dollars per hour however the saving cost will be significantif the operation is considered to be onemonth with 720 hoursor one year with 8760 hours
Execution time comparison can be evaluated by observ-ing the fastest and the slowest compared methods in Table 15These execution times of the proposed method are approxi-mately equal to that of the fastest methods for Subcases 3233 and 34 and much shorter than other fastest comparedmethods for other cases especially for Subcases 61 and 63The fastest method for Subcase 62 is CBA [44] showing 57seconds while that of the proposed method is 1119 secondsHowever it cannot conclude that CBA is more effective thanthe proposed method because the proposed method could
find better optimal solution with less fuel cost by $2130As considering execution time of the slowest comparedmethods it can point out that ICSA is a very fast optimizationtool since the execution time of these methods is 29496 and3425 seconds for Subcases 32 and 33 116735 seconds forSubcase 34 56381 seconds for Subcase 41 94093 secondsfor Subcase 42 12797 seconds for Case 5 30941 seconds forSubcase 61 7542 seconds for Subcase 62 and 9641 secondsfor Subcase 63 while the execution time of the proposedmethod for these cases is respectively 095 146 091 1589246 84828 1119 and 171384 seconds It is clearly shown thatICSA is very fast as compared to these methods
In summary the proposed method has found approx-imately high quality solutions with several standard state-of-the-art meta-heuristic algorithms and improved versionsof them together with other old methods In addition theproposed method could improve result better than approx-imately all methods with faster execution time Comparedto other methods with the fastest convergence speed andhigh quality solutions the proposed method has been as
fast as for some cases and much faster for other cases Thecomparison with the slowest methods could show that theproposed method was extremely powerful since it was up tonearly one thousand times faster Consequently the proposedICSA approach can be one of the strongest optimization toolsfor OLD problem
6 Conclusions
This paper has proposed a good ICSA method for solvingOLD problem in which many test systems with differentobjective functions and complicated constraints from simpleto complex have been used as studied cases The proposedICSA method has been developed by performing severalmodifications on the second solution update of CCSA whichcontained several drawbacks to global convergence and fastmanner The OLD problem has covered from single fuelto multi-fossil fuels from quadratic objective function tononconvex objective function in addition to PPZ RPR andRRL constraints Many existing optimization algorithms have
been concerned in aim to compare the performance and givethe final conclusion on the proposed method There have
been six main cases with sixteen subcases The evaluationshave been made at the end of each study case Clearly theproposed ICSA approach has yielded more effective optimalsolutions with faster execution time than almost all methodsConsequently it can be concluded that the proposed methodis much more superior to CCSA and is a very promisingmethod for solving OLD problem
Appendix
See Tables 16ndash26
Nomenclature
120575119894 120582119894 120572119894 120573119894 120574119894 Fuel cost functioncoefficients of the 119894119905ℎthermal generationunit11986100 1198610119895 119861119895119894 Power loss matrixcoefficients119865119865119909 119865119865119887119890119904119905 The values of fitness ofsolution 119909 and theso-far most effectivesolution among thecurrent set ofsolutions119898119894 Number of fuels burntin the 119894119905ℎ thermalgeneration unit
119873 Number of allavailable thermalgeneration units119899119894 Number of prohibitedpower zones of the 119894119905ℎthermal generationunit119875119906119894119896 119875119897119894119896 Upper and lowerlimits of the 119894119905ℎthermal generationunit corresponding tothe kth PPZ119875119886 Probability ofreplacing controlvariables in each oldsolution119875119863 Real power demand ofall loads in system119875119894119898119886119909 119875119894119898119894119899 The highest andlowest real poweroutputs of the iththermal generationunit119875119894119895119898119886119909 119875119894119895119898119894119899 The highest andlowest real poweroutputs of the 119894119905ℎthermal generationunit corresponding tothe 119895119905ℎ fuel type119878119894119898119886119909 Maximum real powerreserve contributionof the thermalgeneration unit 1198941198781199001198971199031198861198991198891 1198781199001198971199031198861198991198892 1198781199001198971199031198861198991198893 1198781199001198971199031198861198991198894 Randomly mixedsolutions from the setof current solutions119878119900119897119909 119878119900119897119866119887119890119904119905 The old solution x andthe most effectivesolution119878119877 Real power reserverequirement of system
120575119894119895 120582119894119895 120572119894119895 120573119894119895 120574119894119895 Fuel cost functioncoefficients of the 119894119905ℎthermal generationunit corresponding tothe 119895119905ℎ fuel type1205761 1205762 1205763 1205764 1205765 1205766 Random numbersbetween 0 and 1Ω Number of generationunits considering PPZconstraint
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare no conflicts of interest
References
[1] N Sinha R Chakrabarti and P K Chattopadhyay ldquoEvolution-ary programming techniques for economic load dispatchrdquo IEEETransactions on Evolutionary Computation vol 7 no 1 pp 83ndash94 2003
[2] P Hansen ldquoA separable approximation dynamic programmingalgorithm for economic dispatch with transmission lossesrdquoYugoslav Journal of Operations Research vol 12 no 2 2002
[3] N T Thang ldquoSolving economic dispatch problem with piece-wise quadratic cost functions using lagrange multiplier theoryrdquoin Proceedings of the 3rd International Conference on ComputerTechnology and Development (ICCTD rsquo11) pp 359ndash364 ASMEPress 2011
[4] S K Mishra and S K Mishra ldquoA comparative study of solutionof economic load dispatch problem in power systems in theenvironmental perspectiverdquoProcedia Computer Science vol 48pp 96ndash100 2015
[5] A A Al-Subhi and H K Alfares ldquoEconomic load dispatchusing linear programming a comparative studyrdquo InternationalJournal of Applied Industrial Engineering vol 3 no 1 pp 16ndash362016
28 Mathematical Problems in Engineering
[6] C Zhou G Huang and J Chen ldquoPlanning of electric powersystems considering virtual power plants with dispatchableloads included an inexact two-stage stochastic linear program-ming modelrdquoMathematical Problems in Engineering vol 2018Article ID 7049329 12 pages 2018
[7] T T Nguyen N V Quynh and L Van Dai ldquoImproved fireflyalgorithm a novel method for optimal operation of thermalgenerating unitsrdquo Complexity vol 2018 Article ID 7267593 23pages 2018
[8] S S Haroon S Hassan S Amin et al ldquoMultiple fuel machinespower economic dispatch using stud differential evolutionrdquoEnergies vol 11 no 6 pp 1ndash20 2018
[9] C-T Su and C-L Chiang ldquoNonconvex power economic dis-patch by improved genetic algorithm with multiplier updatingmethodrdquo Electric Power Components and Systems vol 32 no 3pp 257ndash273 2004
[10] NNoman andH Iba ldquoDifferential evolution for economic loaddispatch problemsrdquo Electric Power Systems Research vol 78 no8 pp 1322ndash1331 2008
[11] P Somasundaram and K Kuppusamy ldquoApplication of evolu-tionary programming to security constrained economic dis-patchrdquo International Journal of Electrical Power amp EnergySystems vol 27 no 5-6 pp 343ndash351 2005
[12] T Jayabarathi K Jayaprakash andD Jeyakumar ldquoEvolutionaryprogramming technique for different kinds of economic dis-patch problemsrdquo Electric Power Systems Research vol 73 no 2pp 169ndash176 2005
[13] P Subbaraj R Rengaraj and S Salivahanan ldquoEnhancementof combined heat and power economic dispatch using selfadaptive real-coded genetic algorithmrdquo Applied Energy vol 86no 6 pp 915ndash921 2009
[14] C-L Chiang and C-T Su ldquoAdaptive-improved genetic algo-rithm for the economic dispatch of units with multiple fueloptionsrdquo Cybernetics and Systems vol 36 no 7 pp 687ndash7042005
[15] C Chiang ldquoImproved genetic algorithm for power economicdispatch of units with valve-point effects and multiple fuelsrdquoIEEE Transactions on Power Systems vol 20 no 4 pp 1690ndash1699 2005
[16] K Meng H G Wang Z Y Dong and K P Wong ldquoQuantum-inspired particle swarm optimization for valve-point economicload dispatchrdquo IEEE Transactions on Power Systems vol 25 no1 pp 215ndash222 2010
[17] P Erdogmus A Ozturk and S Duman ldquoEnvironmen-taleconomic dispatch using genetic algorithm and simulatedannealingrdquo in Proceedings of the International Conference onElectric Power and Energy Conversion Systems EPECS rsquo09 pp1ndash4 IEEE 2009
[18] L D S Coelho and V C Mariani ldquoImproved differentialevolution algorithms for handling economic dispatch opti-mization with generator constraintsrdquo Energy Conversion andManagement vol 48 no 5 pp 1631ndash1639 2007
[19] H Dakuo W Fuli and M Zhizhong ldquoA hybrid genetic algo-rithm approach based on differential evolution for economicdispatch with valve-point effectsrdquo Electrical Power and EnergySystems vol 30 pp 31ndash38 2008
[20] C-F Chang J-J Wong J-P Chiou and C-T Su ldquoRobustsearching hybrid differential evolution method for optimalreactive power planning in large-scale distribution systemsrdquoElectric Power Systems Research vol 77 no 5-6 pp 430ndash4372007
[21] M R Farooqi P Jain and K R Niazi ldquoUsing Hopfield neuralnetwork for economic dispatch of power systemsrdquo in Proceed-ings of the National Power Engineering Conference PECon rsquo03pp 5ndash10 Bangi Malaysia 2003
[22] T Yalcinoz H Altun and U Hasan ldquoConstrained economicdispatch with prohibited operating zones a Hopfield neuralnetwork approachrdquo in Proceedings of the 10th MediterraneanElectrotechnical Conference Information Technology and Elec-trotechnology for the Mediterranean Countries ProceedingsMeleCon rsquo00 (Cat No 00CH37099) pp 570ndash573 LemesosCyprus 2000
[23] D N Vo P Schegner and W Ongsakul ldquoCuckoo searchalgorithm for non-convex economic dispatchrdquo IET GenerationTransmission amp Distribution vol 7 no 6 pp 645ndash654 2013
[24] J-B Park K-S Lee J-R Shin and K Y Lee ldquoA particleswarm optimization for economic dispatch with non-smoothcost functionsrdquo IEEETransactions on Power Systems vol 20 no1 pp 34ndash42 2005
[25] T Niknam H D Mojarrad and M Nayeripour ldquoA newfuzzy adaptive particle swarm optimization for non-smootheconomic dispatchrdquo Energy vol 35 no 4 pp 1764ndash1778 2010
[26] L D S Coelho and V C Mariani ldquoAn improved harmonysearch algorithm for power economic load dispatchrdquo EnergyConversion and Management vol 50 no 10 pp 2522ndash25262009
[27] M Mahdavi M Fesanghary and E Damangir ldquoAn improvedharmony search algorithm for solving optimization problemsrdquoAppliedMathematics and Computation vol 188 no 2 pp 1567ndash1579 2007
[28] A Bhattacharya and P K Chattopadhyay ldquoSolving complexeconomic load dispatch problem using biogeography-basedoptimizationrdquo Expert Systems with Applications vol 37 no 5pp 3605ndash3615 2010
[29] D Simon ldquoBiogeography-based optimizationrdquo IEEE Transac-tions on Evolutionary Computation vol 12 no 6 pp 702ndash7132008
[30] J S Alsumait J K Sykulski and A K Al-Othman ldquoAhybrid GA-PS-SQP method to solve power system valve-pointeconomic dispatch problemsrdquo Applied Energy vol 87 no 5 pp1773ndash1781 2010
[31] M S Turgut andGKDemir ldquoQuadratic approximationndashbasedhybrid Artificial Cooperative Search algorithm for economicemission load dispatchproblemsrdquo International Transactions onElectrical Energy Systems vol 27 no 4 pp 1ndash14 2017
[32] T A A Victoire and A E Jeyakumar ldquoHybrid PSO-SQPfor economic dispatch with valve-point effectrdquo Electric PowerSystems Research vol 71 no 1 pp 51ndash59 2004
[33] D-KHe F-LWang andZ-ZMao ldquoHybrid genetic algorithmfor economic dispatch with valve-point effectrdquo Electric PowerSystems Research vol 78 no 4 pp 626ndash633 2008
[34] D C Secui ldquoA modified symbiotic organisms search algorithmfor large scale economic dispatch problem with valve-pointloading effectsrdquo Energy vol 113 pp 366ndash384 2016
[35] K BhattacharjeeA Bhattacharya and S H N Dey ldquoTeaching-learning-based optimization for different economic dispatchproblemsrdquo Scientia Iranica vol 21 no 3 pp 870ndash884 2014
[36] K Bhattacharjee A Bhattacharya and S H N Dey ldquoOppo-sitional Real Coded Chemical Reaction Optimization for dif-ferent economic dispatch problemsrdquo International Journal ofElectrical Power amp Energy Systems vol 55 pp 378ndash391 2014
Mathematical Problems in Engineering 29
[37] V Hosseinnezhad M Rafiee M Ahmadian and M T AmelildquoSpecies-basedQuantum Particle SwarmOptimization for eco-nomic load dispatchrdquo International Journal of Electrical Poweramp Energy Systems vol 63 pp 311ndash322 2014
[38] M S P Subathra S E Selvan T A A Victoire A HChristinal andU Amato ldquoA hybrid with cross-entropymethodand sequential quadratic programming to solve economic loaddispatch problemrdquo IEEE Systems Journal vol 9 no 3 pp 1031ndash1044 2015
[39] J Zhan Q H Wu C Guo and X Zhou ldquoEconomic dispatchwith non-smooth objectives part I local minimum analysisrdquoIEEE Transactions on Power Systems vol 30 no 2 pp 710ndash7212015
[40] A K Barisal andR C Prusty ldquoLarge scale economic dispatch ofpower systems using oppositional invasive weed optimizationrdquoApplied Soft Computing vol 29 pp 122ndash137 2015
[41] M Basu ldquoImproved differential evolution for economic dis-patchrdquo International Journal of Electrical Power amp EnergySystems vol 63 pp 855ndash861 2014
[42] V S Aragon S C Esquivel and C C Coello ldquoAn immunealgorithm with power redistribution for solving economicdispatch problemsrdquo Information Sciences vol 295 pp 609ndash6322015
[43] M Ghasemi M Taghizadeh S Ghavidel and A AbbasianldquoColonial competitive differential evolution an experimentalstudy for optimal economic load dispatchrdquo Applied Soft Com-puting vol 40 pp 342ndash363 2016
[44] B R Adarsh T Raghunathan T Jayabarathi and X-S YangldquoEconomic dispatch using chaotic bat algorithmrdquo Energy vol96 pp 666ndash675 2016
[45] N Ghorbani and E Babaei ldquoExchange market algorithm foreconomic load dispatchrdquo International Journal of ElectricalPower amp Energy Systems vol 75 pp 19ndash27 2016
[46] J X V Neto G Reynoso-Meza T H Ruppel V C Marianiand L D S Coelho ldquoSolving non-smooth economic dispatchby a new combination of continuous GRASP algorithm anddifferential evolutionrdquo International Journal of Electrical Poweramp Energy Systems vol 84 pp 13ndash24 2017
[47] A Kavousi-Fard and A Khosravi ldquoAn intelligent 120579-ModifiedBat Algorithm to solve the non-convex economic dispatchproblem considering practical constraintsrdquoElectrical Power andEnergy Systems vol 82 pp 189ndash196 2016
[48] M A Al-Betar M A Awadallah A T Khader and A L BolajildquoTournament-based harmony search algorithm for non-convexeconomic load dispatch problemrdquo Applied Soft Computing vol47 pp 449ndash459 2016
[49] M A Al-Betar M A Awadallah I A Doush E Alsukhni andH ALkhraisat ldquoA non-convex economic dispatchproblemwithvalve loading effect using a new modified 120573-hill climbing localsearch algorithmrdquo Arabian Journal for Science and Engineeringpp 1ndash8 2018
[50] M A Awadallah M A Al-Betar A L Bolaji E M Alsukhniand H Al-Zoubi ldquoNatural selection methods for artificial beecolony with new versions of onlooker beerdquo Soft Computing pp1ndash40 2018
[51] S H Kaboli and A K Alqallaf ldquoSolving non-convex economicload dispatch problem via artificial cooperative search algo-rithmrdquo Expert Systems with Applications vol 128 pp 14ndash272019
[52] D Singh and J Dhillon ldquoAmeliorated greywolf optimization foreconomic load dispatch problemrdquo Energy vol 169 pp 398ndash4192019
[53] X S Yang and S Deb ldquoCuckoo search via Levy flightsrdquo inProceedings of the World Congress on Nature and BiologicallyInspired Computing (NaBIC rsquo09) pp 210ndash214 India 2009
[54] S Sahoo K M Dash R C Prusty and A K Barisal ldquoCom-parative analysis of optimal load dispatch through evolutionaryalgorithmsrdquo Ain Shams Engineering Journal vol 6 pp 107ndash1202015
[55] M Basu and A Chowdhury ldquoCuckoo search algorithm foreconomic dispatchrdquo Energy vol 60 pp 99ndash108 2013
[56] A B Serapiao ldquoCuckoo search for solving economic dispatchload problemrdquo Intelligent Control and Automation vol 04 no04 pp 385ndash390 2013
[57] A H Bindu and M D Reddy ldquoEconomic load dispatch usingcuckoo search algorithmrdquo International Journal of EngineeringResearch and Applications vol 3 pp 498ndash502 2013
[58] E Afzalan and M Joorabian ldquoAn improved cuckoo searchalgorithm for power economic load dispatchrdquo InternationalTransactions on Electrical Energy Systems vol 25 no 6 pp 958ndash975 2015
[59] T T Nguyen and D N Vo ldquoThe application of one rank cuckoosearch algorithm for solving economic load dispatch problemsrdquoApplied Soft Computing vol 37 pp 763ndash773 2015
[60] S M Islam S Das S Ghosh S Roy and P N Suganthan ldquoAnadaptive differential evolution algorithm with novel mutationand crossover strategies for global numerical optimizationrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 42 no 2 pp 482ndash499 2012
[61] Z L Gaing ldquoParticle swarm optimization to solving theeconomic dispatch considering the generator constraintsrdquo IEEETransactions on Power Systems vol 18 no 3 pp 1187ndash1195 2003
[62] X-S Yang and S Deb ldquoEngineering optimisation by Cuckoosearchrdquo International Journal of Mathematical Modelling andNumerical Optimisation vol 330 pp 43ndash51 2010
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
spent 571 seconds Clearly MSOS is better than the proposedICSA approach with respect to slightly less best cost butis worse than the proposed method in terms of executiontime namely 2572 seconds compared to 1119 seconds whileCBA [44] is faster than the proposed method but obtainssignificantly worse costs The analysis can conclude that theproposed ICSA approach is very powerful for the subcasewith 160 units
Table 13 presents the comparison of three other methodsincludingCCSA [54] SOS [34] andMSOS [34] accompaniedwith the proposed method for Subcase 63 The obtainedresult comparisons imply that ICSA approach can obtainbetter values of the best and standard deviation costs thanCCSA and SOS but obtains slightly higher cost than MSOSby approximately $ 022 Besides the execution time fromICSA approach is much shorter than others especially it ishigher than five times faster than MSOS The four methodshave been run on approximately strong computers Brieflythe proposed ICSA approach can find and converge to morefavorable solution than other methods with shorter CPUtime except the comparison with MSOS which had bettersolution but spent higher than five times execution times
Consequently the proposed method is a very promisingoptimization algorithm for Subcase 63 a system up to 320units and with multi-fossil fuel sources and effects of valveloading process
Optimal solution obtained by ICSA for Subcase 63 isshown in Table 26
57 The Improvement of ICSA Approach Performance
571 The Outstanding Improvement over CCSA In this sec-tion the performance improvement of ICSA over CCSAhas been investigated by analyzing obtained results and setcontrol parameters Table 14 has been formed by addingreduction cost improvement level of the best cost executiontime and control parameters consisting of 119873119901119904 and 119866119898119886119909Among the compared factors reduction cost is the deviationof the cost of CCSA and that of ICSA whereas the improve-ment level is the ratio of the reduction cost to the cost ofCCSA The reduction costs indicate the proposed methodcould find either equal quality of solutions or higher qualityof solutions than CCSA for all study casesThe reduction costis from $0 to $42839 corresponding to the improvement level
18 Mathematical Problems in Engineering
Table 14 Summary of results obtained by CCSA and ICSA for all study cases
Study case Reduction cost ($) Improvement level () Execution time (s) Nps Gmax
Figure 4 The best run obtained by CCSA and ICSA for Case 2
that is from 0 to 08968The saving cost is not too much forone hour but the operation in one day onemonth or one yearis very high However it should be noted that CCSA has beenrun by setting much higher population size and iterations formany cases excluding study cases implemented in the studysuch as Case 2 Subcase 31 Subcase 35 and Case 5 Forinstance ICSA has used 119873119901119904 = 10 and 119866119898119886119909 = 15 for Subcase13 and119873119901119904 = 10 and119866119898119886119909 = 25 for Subcase 14 whereas CCSAhas been run by setting119873119901119904 = 20 and 119866119898119886119909 = 5000 for the twosubcases Similarly CCSA has been run for Subcases 32 33and 34 with much higher number of iterations For the lastsubcase CCSA has been run by setting 119873119901119904 = 320 and 119866119898119886119909
= 1200 but those of ICSA have been 10 and 9000 Due tothe higher value of control parameters CCSA has tended tospend more time in finding such high quality solutions foralmost all study cases Execution time of ICSA is less than18 seconds while that of CCSA is up to higher 75 secondsIt is clear that the proposed ICSA could find better optimalsolutions thanCCSA for such considered systems For furtherinvestigation of performance comparison the best runs over50 runs and fuel cost values of 50 runs obtained by CCSA andthe proposed ICSA for Case 2 Subcase 31 Subcase 35 andCase 5 have been plotted in from Figures 4ndash11 The best runcurves show the faster search of the proposed ICSA method
Mathematical Problems in Engineering 19
0 5 10 15 20 25 30 35 40 45 50Run
19795
198
19805
1981
19815
1982
19825
1983
19835
Fuel
cost
($)
CCSAICSA
times105
Figure 5The best fuel cost of 50 runs obtained by CCSA and ICSA for Case 2
0 2 4 6 8 10 12 14 16 18 20Iteration
8234
8236
8238
8240
8242
8244
8246
8248
8250
8252
Fitn
ess F
unct
ion
($)
CCSAICSA
Figure 6 The best run obtained by CCSA and ICSA for Subcase 31
whereas 50 values of fuel cost indicate that the proposedICSA can find many solutions with better quality Clearlythe proposed ICSA is outstanding in terms of stabilizationof solution search and faster convergence As a result it canconclude that the proposed ICSA approach is more effectivethanCCSA in solvingOLDproblemwith considered systems
572The Improvement of Results over Other Methods In thisarticle we have tested ICSA approach on 6 cases with 16
systems with different fuel cost forms different constraintsand different scale systems from 3 units to 320 units We havecompared the yielded results from ICSA approach and otherexisting ones for evaluating the efficiency of ICSA approachIn subsections above we have shown yielded results fromICSA approach and compared these results to those of otheronesHowever the demonstrationhad not been very good forobserving and comparing to lead to a conclusionThus in thesubsectionwehave summarized the result comparisons of the
20 Mathematical Problems in Engineering
0 5 10 15 20 25 30 35 40 45 50Run
8234
82342
82344
82346
82348
8235
82352
82354
82356
82358
8236
Fuel
cost
($)
CCSAICSA
Figure 7 The best fuel cost of 50 runs obtained by CCSA and ICSA for Subcase 31
0 1000 2000 3000 4000 5000 6000Iteration
242
244
246
248
25
252
254
256
Fitn
ess F
unct
ion
($)
CCSAICSA
times105
Figure 8 The best run obtained by CCSA and ICSA for Subcase 35
proposed and other ones Table 15 has reported the reductioncost (in $) of ICSA approach compared to other ones Inaddition we have converted the reduction cost into improve-ment level (in ) for better comparison The improvementhas been shown from the lowest level to the highest levelin terms of reduction cost and improvement percentageIn addition we have also given the slowest and the fastestexecution time of other compared methods together withthat of the proposed method The table implies that ICSA
approach can find better optimal solutions with less fuel costup to $052 for Subcase 13 $75229 for Subcase 14 $1775149for Case 2 $1181256 for Subcase 31 $8438 for Subcase 32$9113 for Subcase 33 $121181 for Subcase 34 $3412 forSubcase 35 $04329 for Subcase 41 $30227 for Subcase42 $407002 for Case 5 $18183 for Subcase 61 $13965 forSubcase 62 and $11817 for Subcase 63 These reductioncosts are equivalent to improvement level (IL) of 0032908968 82282 14143 047 038 099 001405
Mathematical Problems in Engineering 21
0 5 10 15 20 25 30 35 40 45 50Run
2428
243
2432
2434
2436
2438
244
2442
Fuel
cost
($)
CCSAICSA
times105
Figure 9 The best fuel cost of 50 runs obtained by CCSA and ICSA for Subcase 35
0 50 100 150 200 250 300 350 400Iteration
327
328
329
33
331
332
333
334
335
336
337
Fitn
ess F
unct
ion
($)
CCSAICSA
times104
Figure 10 The best run obtained by CCSA and ICSA for Case 5
00003 0002 12291 004 138 and 059 Thesequantitative comparisons reveal that larger scale systems canlead to better reduction cost but the improvement level isnot high because total cost of compared methods tends tobe large for large scale systems Furthermore very large scalesystems with nondifferentiable objective have been normallysolved by strong methods In fact systems in Case 3 havethe same characteristic with single fuel and effects of valveloading process but Subcase 34 is a larger scale system with
40 units while Subcases 31 32 and 33 are constructed by 3units 13 units and 13 units So the improvement percentageof Subcase 34 can be up to 099 whilst that of Subcases32 and 33 is 047 and 038 respectively Subcase 35 iswith the largest system 80 units but the reduction cost is notmuch only $3412 because compared methods with ICSA areeither state-of-the-art ones or improved ones Also Subcase41 and Subcase 42 have considered single fuel and PPZ andspinning reserve constraints but Subcase 42 is larger scale
22 Mathematical Problems in Engineering
Table 15 Performance improvement summary of the proposed method
Study cases Reduction cost ($) Improvement level () Execution time (s)From To From To Slowest method Fastest method Proposed method
Figure 11 The best fuel cost of 50 runs obtained by CCSA and ICSA for Case 5
system with 90 units and Subcase 41 is only with 60 unitsThus the improvement of Subcase 41 is lower with 00003but that of Subcase 42 is 0002 Similarly systems in Cases 6have the same featurewithmulti-fossil fuel sources and effectsof valve loading process but they are respectively constructedby 80 160 and 320 units As a result the improvements ofSubcase 62 138 and Subcase 63 059 are much higherthan Subcase 61 004 However Subcase 62 with smallernumber of units but getting higher improvement is easilyunderstood because there were nine compared methods butonly three compared methods are considered for Subcase 63In general the improvement is not high it is about under onedollar several dollars tens of dollars and over one thousand
dollars per hour however the saving cost will be significantif the operation is considered to be onemonth with 720 hoursor one year with 8760 hours
Execution time comparison can be evaluated by observ-ing the fastest and the slowest compared methods in Table 15These execution times of the proposed method are approxi-mately equal to that of the fastest methods for Subcases 3233 and 34 and much shorter than other fastest comparedmethods for other cases especially for Subcases 61 and 63The fastest method for Subcase 62 is CBA [44] showing 57seconds while that of the proposed method is 1119 secondsHowever it cannot conclude that CBA is more effective thanthe proposed method because the proposed method could
find better optimal solution with less fuel cost by $2130As considering execution time of the slowest comparedmethods it can point out that ICSA is a very fast optimizationtool since the execution time of these methods is 29496 and3425 seconds for Subcases 32 and 33 116735 seconds forSubcase 34 56381 seconds for Subcase 41 94093 secondsfor Subcase 42 12797 seconds for Case 5 30941 seconds forSubcase 61 7542 seconds for Subcase 62 and 9641 secondsfor Subcase 63 while the execution time of the proposedmethod for these cases is respectively 095 146 091 1589246 84828 1119 and 171384 seconds It is clearly shown thatICSA is very fast as compared to these methods
In summary the proposed method has found approx-imately high quality solutions with several standard state-of-the-art meta-heuristic algorithms and improved versionsof them together with other old methods In addition theproposed method could improve result better than approx-imately all methods with faster execution time Comparedto other methods with the fastest convergence speed andhigh quality solutions the proposed method has been as
fast as for some cases and much faster for other cases Thecomparison with the slowest methods could show that theproposed method was extremely powerful since it was up tonearly one thousand times faster Consequently the proposedICSA approach can be one of the strongest optimization toolsfor OLD problem
6 Conclusions
This paper has proposed a good ICSA method for solvingOLD problem in which many test systems with differentobjective functions and complicated constraints from simpleto complex have been used as studied cases The proposedICSA method has been developed by performing severalmodifications on the second solution update of CCSA whichcontained several drawbacks to global convergence and fastmanner The OLD problem has covered from single fuelto multi-fossil fuels from quadratic objective function tononconvex objective function in addition to PPZ RPR andRRL constraints Many existing optimization algorithms have
been concerned in aim to compare the performance and givethe final conclusion on the proposed method There have
been six main cases with sixteen subcases The evaluationshave been made at the end of each study case Clearly theproposed ICSA approach has yielded more effective optimalsolutions with faster execution time than almost all methodsConsequently it can be concluded that the proposed methodis much more superior to CCSA and is a very promisingmethod for solving OLD problem
Appendix
See Tables 16ndash26
Nomenclature
120575119894 120582119894 120572119894 120573119894 120574119894 Fuel cost functioncoefficients of the 119894119905ℎthermal generationunit11986100 1198610119895 119861119895119894 Power loss matrixcoefficients119865119865119909 119865119865119887119890119904119905 The values of fitness ofsolution 119909 and theso-far most effectivesolution among thecurrent set ofsolutions119898119894 Number of fuels burntin the 119894119905ℎ thermalgeneration unit
119873 Number of allavailable thermalgeneration units119899119894 Number of prohibitedpower zones of the 119894119905ℎthermal generationunit119875119906119894119896 119875119897119894119896 Upper and lowerlimits of the 119894119905ℎthermal generationunit corresponding tothe kth PPZ119875119886 Probability ofreplacing controlvariables in each oldsolution119875119863 Real power demand ofall loads in system119875119894119898119886119909 119875119894119898119894119899 The highest andlowest real poweroutputs of the iththermal generationunit119875119894119895119898119886119909 119875119894119895119898119894119899 The highest andlowest real poweroutputs of the 119894119905ℎthermal generationunit corresponding tothe 119895119905ℎ fuel type119878119894119898119886119909 Maximum real powerreserve contributionof the thermalgeneration unit 1198941198781199001198971199031198861198991198891 1198781199001198971199031198861198991198892 1198781199001198971199031198861198991198893 1198781199001198971199031198861198991198894 Randomly mixedsolutions from the setof current solutions119878119900119897119909 119878119900119897119866119887119890119904119905 The old solution x andthe most effectivesolution119878119877 Real power reserverequirement of system
120575119894119895 120582119894119895 120572119894119895 120573119894119895 120574119894119895 Fuel cost functioncoefficients of the 119894119905ℎthermal generationunit corresponding tothe 119895119905ℎ fuel type1205761 1205762 1205763 1205764 1205765 1205766 Random numbersbetween 0 and 1Ω Number of generationunits considering PPZconstraint
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare no conflicts of interest
References
[1] N Sinha R Chakrabarti and P K Chattopadhyay ldquoEvolution-ary programming techniques for economic load dispatchrdquo IEEETransactions on Evolutionary Computation vol 7 no 1 pp 83ndash94 2003
[2] P Hansen ldquoA separable approximation dynamic programmingalgorithm for economic dispatch with transmission lossesrdquoYugoslav Journal of Operations Research vol 12 no 2 2002
[3] N T Thang ldquoSolving economic dispatch problem with piece-wise quadratic cost functions using lagrange multiplier theoryrdquoin Proceedings of the 3rd International Conference on ComputerTechnology and Development (ICCTD rsquo11) pp 359ndash364 ASMEPress 2011
[4] S K Mishra and S K Mishra ldquoA comparative study of solutionof economic load dispatch problem in power systems in theenvironmental perspectiverdquoProcedia Computer Science vol 48pp 96ndash100 2015
[5] A A Al-Subhi and H K Alfares ldquoEconomic load dispatchusing linear programming a comparative studyrdquo InternationalJournal of Applied Industrial Engineering vol 3 no 1 pp 16ndash362016
28 Mathematical Problems in Engineering
[6] C Zhou G Huang and J Chen ldquoPlanning of electric powersystems considering virtual power plants with dispatchableloads included an inexact two-stage stochastic linear program-ming modelrdquoMathematical Problems in Engineering vol 2018Article ID 7049329 12 pages 2018
[7] T T Nguyen N V Quynh and L Van Dai ldquoImproved fireflyalgorithm a novel method for optimal operation of thermalgenerating unitsrdquo Complexity vol 2018 Article ID 7267593 23pages 2018
[8] S S Haroon S Hassan S Amin et al ldquoMultiple fuel machinespower economic dispatch using stud differential evolutionrdquoEnergies vol 11 no 6 pp 1ndash20 2018
[9] C-T Su and C-L Chiang ldquoNonconvex power economic dis-patch by improved genetic algorithm with multiplier updatingmethodrdquo Electric Power Components and Systems vol 32 no 3pp 257ndash273 2004
[10] NNoman andH Iba ldquoDifferential evolution for economic loaddispatch problemsrdquo Electric Power Systems Research vol 78 no8 pp 1322ndash1331 2008
[11] P Somasundaram and K Kuppusamy ldquoApplication of evolu-tionary programming to security constrained economic dis-patchrdquo International Journal of Electrical Power amp EnergySystems vol 27 no 5-6 pp 343ndash351 2005
[12] T Jayabarathi K Jayaprakash andD Jeyakumar ldquoEvolutionaryprogramming technique for different kinds of economic dis-patch problemsrdquo Electric Power Systems Research vol 73 no 2pp 169ndash176 2005
[13] P Subbaraj R Rengaraj and S Salivahanan ldquoEnhancementof combined heat and power economic dispatch using selfadaptive real-coded genetic algorithmrdquo Applied Energy vol 86no 6 pp 915ndash921 2009
[14] C-L Chiang and C-T Su ldquoAdaptive-improved genetic algo-rithm for the economic dispatch of units with multiple fueloptionsrdquo Cybernetics and Systems vol 36 no 7 pp 687ndash7042005
[15] C Chiang ldquoImproved genetic algorithm for power economicdispatch of units with valve-point effects and multiple fuelsrdquoIEEE Transactions on Power Systems vol 20 no 4 pp 1690ndash1699 2005
[16] K Meng H G Wang Z Y Dong and K P Wong ldquoQuantum-inspired particle swarm optimization for valve-point economicload dispatchrdquo IEEE Transactions on Power Systems vol 25 no1 pp 215ndash222 2010
[17] P Erdogmus A Ozturk and S Duman ldquoEnvironmen-taleconomic dispatch using genetic algorithm and simulatedannealingrdquo in Proceedings of the International Conference onElectric Power and Energy Conversion Systems EPECS rsquo09 pp1ndash4 IEEE 2009
[18] L D S Coelho and V C Mariani ldquoImproved differentialevolution algorithms for handling economic dispatch opti-mization with generator constraintsrdquo Energy Conversion andManagement vol 48 no 5 pp 1631ndash1639 2007
[19] H Dakuo W Fuli and M Zhizhong ldquoA hybrid genetic algo-rithm approach based on differential evolution for economicdispatch with valve-point effectsrdquo Electrical Power and EnergySystems vol 30 pp 31ndash38 2008
[20] C-F Chang J-J Wong J-P Chiou and C-T Su ldquoRobustsearching hybrid differential evolution method for optimalreactive power planning in large-scale distribution systemsrdquoElectric Power Systems Research vol 77 no 5-6 pp 430ndash4372007
[21] M R Farooqi P Jain and K R Niazi ldquoUsing Hopfield neuralnetwork for economic dispatch of power systemsrdquo in Proceed-ings of the National Power Engineering Conference PECon rsquo03pp 5ndash10 Bangi Malaysia 2003
[22] T Yalcinoz H Altun and U Hasan ldquoConstrained economicdispatch with prohibited operating zones a Hopfield neuralnetwork approachrdquo in Proceedings of the 10th MediterraneanElectrotechnical Conference Information Technology and Elec-trotechnology for the Mediterranean Countries ProceedingsMeleCon rsquo00 (Cat No 00CH37099) pp 570ndash573 LemesosCyprus 2000
[23] D N Vo P Schegner and W Ongsakul ldquoCuckoo searchalgorithm for non-convex economic dispatchrdquo IET GenerationTransmission amp Distribution vol 7 no 6 pp 645ndash654 2013
[24] J-B Park K-S Lee J-R Shin and K Y Lee ldquoA particleswarm optimization for economic dispatch with non-smoothcost functionsrdquo IEEETransactions on Power Systems vol 20 no1 pp 34ndash42 2005
[25] T Niknam H D Mojarrad and M Nayeripour ldquoA newfuzzy adaptive particle swarm optimization for non-smootheconomic dispatchrdquo Energy vol 35 no 4 pp 1764ndash1778 2010
[26] L D S Coelho and V C Mariani ldquoAn improved harmonysearch algorithm for power economic load dispatchrdquo EnergyConversion and Management vol 50 no 10 pp 2522ndash25262009
[27] M Mahdavi M Fesanghary and E Damangir ldquoAn improvedharmony search algorithm for solving optimization problemsrdquoAppliedMathematics and Computation vol 188 no 2 pp 1567ndash1579 2007
[28] A Bhattacharya and P K Chattopadhyay ldquoSolving complexeconomic load dispatch problem using biogeography-basedoptimizationrdquo Expert Systems with Applications vol 37 no 5pp 3605ndash3615 2010
[29] D Simon ldquoBiogeography-based optimizationrdquo IEEE Transac-tions on Evolutionary Computation vol 12 no 6 pp 702ndash7132008
[30] J S Alsumait J K Sykulski and A K Al-Othman ldquoAhybrid GA-PS-SQP method to solve power system valve-pointeconomic dispatch problemsrdquo Applied Energy vol 87 no 5 pp1773ndash1781 2010
[31] M S Turgut andGKDemir ldquoQuadratic approximationndashbasedhybrid Artificial Cooperative Search algorithm for economicemission load dispatchproblemsrdquo International Transactions onElectrical Energy Systems vol 27 no 4 pp 1ndash14 2017
[32] T A A Victoire and A E Jeyakumar ldquoHybrid PSO-SQPfor economic dispatch with valve-point effectrdquo Electric PowerSystems Research vol 71 no 1 pp 51ndash59 2004
[33] D-KHe F-LWang andZ-ZMao ldquoHybrid genetic algorithmfor economic dispatch with valve-point effectrdquo Electric PowerSystems Research vol 78 no 4 pp 626ndash633 2008
[34] D C Secui ldquoA modified symbiotic organisms search algorithmfor large scale economic dispatch problem with valve-pointloading effectsrdquo Energy vol 113 pp 366ndash384 2016
[35] K BhattacharjeeA Bhattacharya and S H N Dey ldquoTeaching-learning-based optimization for different economic dispatchproblemsrdquo Scientia Iranica vol 21 no 3 pp 870ndash884 2014
[36] K Bhattacharjee A Bhattacharya and S H N Dey ldquoOppo-sitional Real Coded Chemical Reaction Optimization for dif-ferent economic dispatch problemsrdquo International Journal ofElectrical Power amp Energy Systems vol 55 pp 378ndash391 2014
Mathematical Problems in Engineering 29
[37] V Hosseinnezhad M Rafiee M Ahmadian and M T AmelildquoSpecies-basedQuantum Particle SwarmOptimization for eco-nomic load dispatchrdquo International Journal of Electrical Poweramp Energy Systems vol 63 pp 311ndash322 2014
[38] M S P Subathra S E Selvan T A A Victoire A HChristinal andU Amato ldquoA hybrid with cross-entropymethodand sequential quadratic programming to solve economic loaddispatch problemrdquo IEEE Systems Journal vol 9 no 3 pp 1031ndash1044 2015
[39] J Zhan Q H Wu C Guo and X Zhou ldquoEconomic dispatchwith non-smooth objectives part I local minimum analysisrdquoIEEE Transactions on Power Systems vol 30 no 2 pp 710ndash7212015
[40] A K Barisal andR C Prusty ldquoLarge scale economic dispatch ofpower systems using oppositional invasive weed optimizationrdquoApplied Soft Computing vol 29 pp 122ndash137 2015
[41] M Basu ldquoImproved differential evolution for economic dis-patchrdquo International Journal of Electrical Power amp EnergySystems vol 63 pp 855ndash861 2014
[42] V S Aragon S C Esquivel and C C Coello ldquoAn immunealgorithm with power redistribution for solving economicdispatch problemsrdquo Information Sciences vol 295 pp 609ndash6322015
[43] M Ghasemi M Taghizadeh S Ghavidel and A AbbasianldquoColonial competitive differential evolution an experimentalstudy for optimal economic load dispatchrdquo Applied Soft Com-puting vol 40 pp 342ndash363 2016
[44] B R Adarsh T Raghunathan T Jayabarathi and X-S YangldquoEconomic dispatch using chaotic bat algorithmrdquo Energy vol96 pp 666ndash675 2016
[45] N Ghorbani and E Babaei ldquoExchange market algorithm foreconomic load dispatchrdquo International Journal of ElectricalPower amp Energy Systems vol 75 pp 19ndash27 2016
[46] J X V Neto G Reynoso-Meza T H Ruppel V C Marianiand L D S Coelho ldquoSolving non-smooth economic dispatchby a new combination of continuous GRASP algorithm anddifferential evolutionrdquo International Journal of Electrical Poweramp Energy Systems vol 84 pp 13ndash24 2017
[47] A Kavousi-Fard and A Khosravi ldquoAn intelligent 120579-ModifiedBat Algorithm to solve the non-convex economic dispatchproblem considering practical constraintsrdquoElectrical Power andEnergy Systems vol 82 pp 189ndash196 2016
[48] M A Al-Betar M A Awadallah A T Khader and A L BolajildquoTournament-based harmony search algorithm for non-convexeconomic load dispatch problemrdquo Applied Soft Computing vol47 pp 449ndash459 2016
[49] M A Al-Betar M A Awadallah I A Doush E Alsukhni andH ALkhraisat ldquoA non-convex economic dispatchproblemwithvalve loading effect using a new modified 120573-hill climbing localsearch algorithmrdquo Arabian Journal for Science and Engineeringpp 1ndash8 2018
[50] M A Awadallah M A Al-Betar A L Bolaji E M Alsukhniand H Al-Zoubi ldquoNatural selection methods for artificial beecolony with new versions of onlooker beerdquo Soft Computing pp1ndash40 2018
[51] S H Kaboli and A K Alqallaf ldquoSolving non-convex economicload dispatch problem via artificial cooperative search algo-rithmrdquo Expert Systems with Applications vol 128 pp 14ndash272019
[52] D Singh and J Dhillon ldquoAmeliorated greywolf optimization foreconomic load dispatch problemrdquo Energy vol 169 pp 398ndash4192019
[53] X S Yang and S Deb ldquoCuckoo search via Levy flightsrdquo inProceedings of the World Congress on Nature and BiologicallyInspired Computing (NaBIC rsquo09) pp 210ndash214 India 2009
[54] S Sahoo K M Dash R C Prusty and A K Barisal ldquoCom-parative analysis of optimal load dispatch through evolutionaryalgorithmsrdquo Ain Shams Engineering Journal vol 6 pp 107ndash1202015
[55] M Basu and A Chowdhury ldquoCuckoo search algorithm foreconomic dispatchrdquo Energy vol 60 pp 99ndash108 2013
[56] A B Serapiao ldquoCuckoo search for solving economic dispatchload problemrdquo Intelligent Control and Automation vol 04 no04 pp 385ndash390 2013
[57] A H Bindu and M D Reddy ldquoEconomic load dispatch usingcuckoo search algorithmrdquo International Journal of EngineeringResearch and Applications vol 3 pp 498ndash502 2013
[58] E Afzalan and M Joorabian ldquoAn improved cuckoo searchalgorithm for power economic load dispatchrdquo InternationalTransactions on Electrical Energy Systems vol 25 no 6 pp 958ndash975 2015
[59] T T Nguyen and D N Vo ldquoThe application of one rank cuckoosearch algorithm for solving economic load dispatch problemsrdquoApplied Soft Computing vol 37 pp 763ndash773 2015
[60] S M Islam S Das S Ghosh S Roy and P N Suganthan ldquoAnadaptive differential evolution algorithm with novel mutationand crossover strategies for global numerical optimizationrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 42 no 2 pp 482ndash499 2012
[61] Z L Gaing ldquoParticle swarm optimization to solving theeconomic dispatch considering the generator constraintsrdquo IEEETransactions on Power Systems vol 18 no 3 pp 1187ndash1195 2003
[62] X-S Yang and S Deb ldquoEngineering optimisation by Cuckoosearchrdquo International Journal of Mathematical Modelling andNumerical Optimisation vol 330 pp 43ndash51 2010
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Figure 4 The best run obtained by CCSA and ICSA for Case 2
that is from 0 to 08968The saving cost is not too much forone hour but the operation in one day onemonth or one yearis very high However it should be noted that CCSA has beenrun by setting much higher population size and iterations formany cases excluding study cases implemented in the studysuch as Case 2 Subcase 31 Subcase 35 and Case 5 Forinstance ICSA has used 119873119901119904 = 10 and 119866119898119886119909 = 15 for Subcase13 and119873119901119904 = 10 and119866119898119886119909 = 25 for Subcase 14 whereas CCSAhas been run by setting119873119901119904 = 20 and 119866119898119886119909 = 5000 for the twosubcases Similarly CCSA has been run for Subcases 32 33and 34 with much higher number of iterations For the lastsubcase CCSA has been run by setting 119873119901119904 = 320 and 119866119898119886119909
= 1200 but those of ICSA have been 10 and 9000 Due tothe higher value of control parameters CCSA has tended tospend more time in finding such high quality solutions foralmost all study cases Execution time of ICSA is less than18 seconds while that of CCSA is up to higher 75 secondsIt is clear that the proposed ICSA could find better optimalsolutions thanCCSA for such considered systems For furtherinvestigation of performance comparison the best runs over50 runs and fuel cost values of 50 runs obtained by CCSA andthe proposed ICSA for Case 2 Subcase 31 Subcase 35 andCase 5 have been plotted in from Figures 4ndash11 The best runcurves show the faster search of the proposed ICSA method
Mathematical Problems in Engineering 19
0 5 10 15 20 25 30 35 40 45 50Run
19795
198
19805
1981
19815
1982
19825
1983
19835
Fuel
cost
($)
CCSAICSA
times105
Figure 5The best fuel cost of 50 runs obtained by CCSA and ICSA for Case 2
0 2 4 6 8 10 12 14 16 18 20Iteration
8234
8236
8238
8240
8242
8244
8246
8248
8250
8252
Fitn
ess F
unct
ion
($)
CCSAICSA
Figure 6 The best run obtained by CCSA and ICSA for Subcase 31
whereas 50 values of fuel cost indicate that the proposedICSA can find many solutions with better quality Clearlythe proposed ICSA is outstanding in terms of stabilizationof solution search and faster convergence As a result it canconclude that the proposed ICSA approach is more effectivethanCCSA in solvingOLDproblemwith considered systems
572The Improvement of Results over Other Methods In thisarticle we have tested ICSA approach on 6 cases with 16
systems with different fuel cost forms different constraintsand different scale systems from 3 units to 320 units We havecompared the yielded results from ICSA approach and otherexisting ones for evaluating the efficiency of ICSA approachIn subsections above we have shown yielded results fromICSA approach and compared these results to those of otheronesHowever the demonstrationhad not been very good forobserving and comparing to lead to a conclusionThus in thesubsectionwehave summarized the result comparisons of the
20 Mathematical Problems in Engineering
0 5 10 15 20 25 30 35 40 45 50Run
8234
82342
82344
82346
82348
8235
82352
82354
82356
82358
8236
Fuel
cost
($)
CCSAICSA
Figure 7 The best fuel cost of 50 runs obtained by CCSA and ICSA for Subcase 31
0 1000 2000 3000 4000 5000 6000Iteration
242
244
246
248
25
252
254
256
Fitn
ess F
unct
ion
($)
CCSAICSA
times105
Figure 8 The best run obtained by CCSA and ICSA for Subcase 35
proposed and other ones Table 15 has reported the reductioncost (in $) of ICSA approach compared to other ones Inaddition we have converted the reduction cost into improve-ment level (in ) for better comparison The improvementhas been shown from the lowest level to the highest levelin terms of reduction cost and improvement percentageIn addition we have also given the slowest and the fastestexecution time of other compared methods together withthat of the proposed method The table implies that ICSA
approach can find better optimal solutions with less fuel costup to $052 for Subcase 13 $75229 for Subcase 14 $1775149for Case 2 $1181256 for Subcase 31 $8438 for Subcase 32$9113 for Subcase 33 $121181 for Subcase 34 $3412 forSubcase 35 $04329 for Subcase 41 $30227 for Subcase42 $407002 for Case 5 $18183 for Subcase 61 $13965 forSubcase 62 and $11817 for Subcase 63 These reductioncosts are equivalent to improvement level (IL) of 0032908968 82282 14143 047 038 099 001405
Mathematical Problems in Engineering 21
0 5 10 15 20 25 30 35 40 45 50Run
2428
243
2432
2434
2436
2438
244
2442
Fuel
cost
($)
CCSAICSA
times105
Figure 9 The best fuel cost of 50 runs obtained by CCSA and ICSA for Subcase 35
0 50 100 150 200 250 300 350 400Iteration
327
328
329
33
331
332
333
334
335
336
337
Fitn
ess F
unct
ion
($)
CCSAICSA
times104
Figure 10 The best run obtained by CCSA and ICSA for Case 5
00003 0002 12291 004 138 and 059 Thesequantitative comparisons reveal that larger scale systems canlead to better reduction cost but the improvement level isnot high because total cost of compared methods tends tobe large for large scale systems Furthermore very large scalesystems with nondifferentiable objective have been normallysolved by strong methods In fact systems in Case 3 havethe same characteristic with single fuel and effects of valveloading process but Subcase 34 is a larger scale system with
40 units while Subcases 31 32 and 33 are constructed by 3units 13 units and 13 units So the improvement percentageof Subcase 34 can be up to 099 whilst that of Subcases32 and 33 is 047 and 038 respectively Subcase 35 iswith the largest system 80 units but the reduction cost is notmuch only $3412 because compared methods with ICSA areeither state-of-the-art ones or improved ones Also Subcase41 and Subcase 42 have considered single fuel and PPZ andspinning reserve constraints but Subcase 42 is larger scale
22 Mathematical Problems in Engineering
Table 15 Performance improvement summary of the proposed method
Study cases Reduction cost ($) Improvement level () Execution time (s)From To From To Slowest method Fastest method Proposed method
Figure 11 The best fuel cost of 50 runs obtained by CCSA and ICSA for Case 5
system with 90 units and Subcase 41 is only with 60 unitsThus the improvement of Subcase 41 is lower with 00003but that of Subcase 42 is 0002 Similarly systems in Cases 6have the same featurewithmulti-fossil fuel sources and effectsof valve loading process but they are respectively constructedby 80 160 and 320 units As a result the improvements ofSubcase 62 138 and Subcase 63 059 are much higherthan Subcase 61 004 However Subcase 62 with smallernumber of units but getting higher improvement is easilyunderstood because there were nine compared methods butonly three compared methods are considered for Subcase 63In general the improvement is not high it is about under onedollar several dollars tens of dollars and over one thousand
dollars per hour however the saving cost will be significantif the operation is considered to be onemonth with 720 hoursor one year with 8760 hours
Execution time comparison can be evaluated by observ-ing the fastest and the slowest compared methods in Table 15These execution times of the proposed method are approxi-mately equal to that of the fastest methods for Subcases 3233 and 34 and much shorter than other fastest comparedmethods for other cases especially for Subcases 61 and 63The fastest method for Subcase 62 is CBA [44] showing 57seconds while that of the proposed method is 1119 secondsHowever it cannot conclude that CBA is more effective thanthe proposed method because the proposed method could
find better optimal solution with less fuel cost by $2130As considering execution time of the slowest comparedmethods it can point out that ICSA is a very fast optimizationtool since the execution time of these methods is 29496 and3425 seconds for Subcases 32 and 33 116735 seconds forSubcase 34 56381 seconds for Subcase 41 94093 secondsfor Subcase 42 12797 seconds for Case 5 30941 seconds forSubcase 61 7542 seconds for Subcase 62 and 9641 secondsfor Subcase 63 while the execution time of the proposedmethod for these cases is respectively 095 146 091 1589246 84828 1119 and 171384 seconds It is clearly shown thatICSA is very fast as compared to these methods
In summary the proposed method has found approx-imately high quality solutions with several standard state-of-the-art meta-heuristic algorithms and improved versionsof them together with other old methods In addition theproposed method could improve result better than approx-imately all methods with faster execution time Comparedto other methods with the fastest convergence speed andhigh quality solutions the proposed method has been as
fast as for some cases and much faster for other cases Thecomparison with the slowest methods could show that theproposed method was extremely powerful since it was up tonearly one thousand times faster Consequently the proposedICSA approach can be one of the strongest optimization toolsfor OLD problem
6 Conclusions
This paper has proposed a good ICSA method for solvingOLD problem in which many test systems with differentobjective functions and complicated constraints from simpleto complex have been used as studied cases The proposedICSA method has been developed by performing severalmodifications on the second solution update of CCSA whichcontained several drawbacks to global convergence and fastmanner The OLD problem has covered from single fuelto multi-fossil fuels from quadratic objective function tononconvex objective function in addition to PPZ RPR andRRL constraints Many existing optimization algorithms have
been concerned in aim to compare the performance and givethe final conclusion on the proposed method There have
been six main cases with sixteen subcases The evaluationshave been made at the end of each study case Clearly theproposed ICSA approach has yielded more effective optimalsolutions with faster execution time than almost all methodsConsequently it can be concluded that the proposed methodis much more superior to CCSA and is a very promisingmethod for solving OLD problem
Appendix
See Tables 16ndash26
Nomenclature
120575119894 120582119894 120572119894 120573119894 120574119894 Fuel cost functioncoefficients of the 119894119905ℎthermal generationunit11986100 1198610119895 119861119895119894 Power loss matrixcoefficients119865119865119909 119865119865119887119890119904119905 The values of fitness ofsolution 119909 and theso-far most effectivesolution among thecurrent set ofsolutions119898119894 Number of fuels burntin the 119894119905ℎ thermalgeneration unit
119873 Number of allavailable thermalgeneration units119899119894 Number of prohibitedpower zones of the 119894119905ℎthermal generationunit119875119906119894119896 119875119897119894119896 Upper and lowerlimits of the 119894119905ℎthermal generationunit corresponding tothe kth PPZ119875119886 Probability ofreplacing controlvariables in each oldsolution119875119863 Real power demand ofall loads in system119875119894119898119886119909 119875119894119898119894119899 The highest andlowest real poweroutputs of the iththermal generationunit119875119894119895119898119886119909 119875119894119895119898119894119899 The highest andlowest real poweroutputs of the 119894119905ℎthermal generationunit corresponding tothe 119895119905ℎ fuel type119878119894119898119886119909 Maximum real powerreserve contributionof the thermalgeneration unit 1198941198781199001198971199031198861198991198891 1198781199001198971199031198861198991198892 1198781199001198971199031198861198991198893 1198781199001198971199031198861198991198894 Randomly mixedsolutions from the setof current solutions119878119900119897119909 119878119900119897119866119887119890119904119905 The old solution x andthe most effectivesolution119878119877 Real power reserverequirement of system
120575119894119895 120582119894119895 120572119894119895 120573119894119895 120574119894119895 Fuel cost functioncoefficients of the 119894119905ℎthermal generationunit corresponding tothe 119895119905ℎ fuel type1205761 1205762 1205763 1205764 1205765 1205766 Random numbersbetween 0 and 1Ω Number of generationunits considering PPZconstraint
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare no conflicts of interest
References
[1] N Sinha R Chakrabarti and P K Chattopadhyay ldquoEvolution-ary programming techniques for economic load dispatchrdquo IEEETransactions on Evolutionary Computation vol 7 no 1 pp 83ndash94 2003
[2] P Hansen ldquoA separable approximation dynamic programmingalgorithm for economic dispatch with transmission lossesrdquoYugoslav Journal of Operations Research vol 12 no 2 2002
[3] N T Thang ldquoSolving economic dispatch problem with piece-wise quadratic cost functions using lagrange multiplier theoryrdquoin Proceedings of the 3rd International Conference on ComputerTechnology and Development (ICCTD rsquo11) pp 359ndash364 ASMEPress 2011
[4] S K Mishra and S K Mishra ldquoA comparative study of solutionof economic load dispatch problem in power systems in theenvironmental perspectiverdquoProcedia Computer Science vol 48pp 96ndash100 2015
[5] A A Al-Subhi and H K Alfares ldquoEconomic load dispatchusing linear programming a comparative studyrdquo InternationalJournal of Applied Industrial Engineering vol 3 no 1 pp 16ndash362016
28 Mathematical Problems in Engineering
[6] C Zhou G Huang and J Chen ldquoPlanning of electric powersystems considering virtual power plants with dispatchableloads included an inexact two-stage stochastic linear program-ming modelrdquoMathematical Problems in Engineering vol 2018Article ID 7049329 12 pages 2018
[7] T T Nguyen N V Quynh and L Van Dai ldquoImproved fireflyalgorithm a novel method for optimal operation of thermalgenerating unitsrdquo Complexity vol 2018 Article ID 7267593 23pages 2018
[8] S S Haroon S Hassan S Amin et al ldquoMultiple fuel machinespower economic dispatch using stud differential evolutionrdquoEnergies vol 11 no 6 pp 1ndash20 2018
[9] C-T Su and C-L Chiang ldquoNonconvex power economic dis-patch by improved genetic algorithm with multiplier updatingmethodrdquo Electric Power Components and Systems vol 32 no 3pp 257ndash273 2004
[10] NNoman andH Iba ldquoDifferential evolution for economic loaddispatch problemsrdquo Electric Power Systems Research vol 78 no8 pp 1322ndash1331 2008
[11] P Somasundaram and K Kuppusamy ldquoApplication of evolu-tionary programming to security constrained economic dis-patchrdquo International Journal of Electrical Power amp EnergySystems vol 27 no 5-6 pp 343ndash351 2005
[12] T Jayabarathi K Jayaprakash andD Jeyakumar ldquoEvolutionaryprogramming technique for different kinds of economic dis-patch problemsrdquo Electric Power Systems Research vol 73 no 2pp 169ndash176 2005
[13] P Subbaraj R Rengaraj and S Salivahanan ldquoEnhancementof combined heat and power economic dispatch using selfadaptive real-coded genetic algorithmrdquo Applied Energy vol 86no 6 pp 915ndash921 2009
[14] C-L Chiang and C-T Su ldquoAdaptive-improved genetic algo-rithm for the economic dispatch of units with multiple fueloptionsrdquo Cybernetics and Systems vol 36 no 7 pp 687ndash7042005
[15] C Chiang ldquoImproved genetic algorithm for power economicdispatch of units with valve-point effects and multiple fuelsrdquoIEEE Transactions on Power Systems vol 20 no 4 pp 1690ndash1699 2005
[16] K Meng H G Wang Z Y Dong and K P Wong ldquoQuantum-inspired particle swarm optimization for valve-point economicload dispatchrdquo IEEE Transactions on Power Systems vol 25 no1 pp 215ndash222 2010
[17] P Erdogmus A Ozturk and S Duman ldquoEnvironmen-taleconomic dispatch using genetic algorithm and simulatedannealingrdquo in Proceedings of the International Conference onElectric Power and Energy Conversion Systems EPECS rsquo09 pp1ndash4 IEEE 2009
[18] L D S Coelho and V C Mariani ldquoImproved differentialevolution algorithms for handling economic dispatch opti-mization with generator constraintsrdquo Energy Conversion andManagement vol 48 no 5 pp 1631ndash1639 2007
[19] H Dakuo W Fuli and M Zhizhong ldquoA hybrid genetic algo-rithm approach based on differential evolution for economicdispatch with valve-point effectsrdquo Electrical Power and EnergySystems vol 30 pp 31ndash38 2008
[20] C-F Chang J-J Wong J-P Chiou and C-T Su ldquoRobustsearching hybrid differential evolution method for optimalreactive power planning in large-scale distribution systemsrdquoElectric Power Systems Research vol 77 no 5-6 pp 430ndash4372007
[21] M R Farooqi P Jain and K R Niazi ldquoUsing Hopfield neuralnetwork for economic dispatch of power systemsrdquo in Proceed-ings of the National Power Engineering Conference PECon rsquo03pp 5ndash10 Bangi Malaysia 2003
[22] T Yalcinoz H Altun and U Hasan ldquoConstrained economicdispatch with prohibited operating zones a Hopfield neuralnetwork approachrdquo in Proceedings of the 10th MediterraneanElectrotechnical Conference Information Technology and Elec-trotechnology for the Mediterranean Countries ProceedingsMeleCon rsquo00 (Cat No 00CH37099) pp 570ndash573 LemesosCyprus 2000
[23] D N Vo P Schegner and W Ongsakul ldquoCuckoo searchalgorithm for non-convex economic dispatchrdquo IET GenerationTransmission amp Distribution vol 7 no 6 pp 645ndash654 2013
[24] J-B Park K-S Lee J-R Shin and K Y Lee ldquoA particleswarm optimization for economic dispatch with non-smoothcost functionsrdquo IEEETransactions on Power Systems vol 20 no1 pp 34ndash42 2005
[25] T Niknam H D Mojarrad and M Nayeripour ldquoA newfuzzy adaptive particle swarm optimization for non-smootheconomic dispatchrdquo Energy vol 35 no 4 pp 1764ndash1778 2010
[26] L D S Coelho and V C Mariani ldquoAn improved harmonysearch algorithm for power economic load dispatchrdquo EnergyConversion and Management vol 50 no 10 pp 2522ndash25262009
[27] M Mahdavi M Fesanghary and E Damangir ldquoAn improvedharmony search algorithm for solving optimization problemsrdquoAppliedMathematics and Computation vol 188 no 2 pp 1567ndash1579 2007
[28] A Bhattacharya and P K Chattopadhyay ldquoSolving complexeconomic load dispatch problem using biogeography-basedoptimizationrdquo Expert Systems with Applications vol 37 no 5pp 3605ndash3615 2010
[29] D Simon ldquoBiogeography-based optimizationrdquo IEEE Transac-tions on Evolutionary Computation vol 12 no 6 pp 702ndash7132008
[30] J S Alsumait J K Sykulski and A K Al-Othman ldquoAhybrid GA-PS-SQP method to solve power system valve-pointeconomic dispatch problemsrdquo Applied Energy vol 87 no 5 pp1773ndash1781 2010
[31] M S Turgut andGKDemir ldquoQuadratic approximationndashbasedhybrid Artificial Cooperative Search algorithm for economicemission load dispatchproblemsrdquo International Transactions onElectrical Energy Systems vol 27 no 4 pp 1ndash14 2017
[32] T A A Victoire and A E Jeyakumar ldquoHybrid PSO-SQPfor economic dispatch with valve-point effectrdquo Electric PowerSystems Research vol 71 no 1 pp 51ndash59 2004
[33] D-KHe F-LWang andZ-ZMao ldquoHybrid genetic algorithmfor economic dispatch with valve-point effectrdquo Electric PowerSystems Research vol 78 no 4 pp 626ndash633 2008
[34] D C Secui ldquoA modified symbiotic organisms search algorithmfor large scale economic dispatch problem with valve-pointloading effectsrdquo Energy vol 113 pp 366ndash384 2016
[35] K BhattacharjeeA Bhattacharya and S H N Dey ldquoTeaching-learning-based optimization for different economic dispatchproblemsrdquo Scientia Iranica vol 21 no 3 pp 870ndash884 2014
[36] K Bhattacharjee A Bhattacharya and S H N Dey ldquoOppo-sitional Real Coded Chemical Reaction Optimization for dif-ferent economic dispatch problemsrdquo International Journal ofElectrical Power amp Energy Systems vol 55 pp 378ndash391 2014
Mathematical Problems in Engineering 29
[37] V Hosseinnezhad M Rafiee M Ahmadian and M T AmelildquoSpecies-basedQuantum Particle SwarmOptimization for eco-nomic load dispatchrdquo International Journal of Electrical Poweramp Energy Systems vol 63 pp 311ndash322 2014
[38] M S P Subathra S E Selvan T A A Victoire A HChristinal andU Amato ldquoA hybrid with cross-entropymethodand sequential quadratic programming to solve economic loaddispatch problemrdquo IEEE Systems Journal vol 9 no 3 pp 1031ndash1044 2015
[39] J Zhan Q H Wu C Guo and X Zhou ldquoEconomic dispatchwith non-smooth objectives part I local minimum analysisrdquoIEEE Transactions on Power Systems vol 30 no 2 pp 710ndash7212015
[40] A K Barisal andR C Prusty ldquoLarge scale economic dispatch ofpower systems using oppositional invasive weed optimizationrdquoApplied Soft Computing vol 29 pp 122ndash137 2015
[41] M Basu ldquoImproved differential evolution for economic dis-patchrdquo International Journal of Electrical Power amp EnergySystems vol 63 pp 855ndash861 2014
[42] V S Aragon S C Esquivel and C C Coello ldquoAn immunealgorithm with power redistribution for solving economicdispatch problemsrdquo Information Sciences vol 295 pp 609ndash6322015
[43] M Ghasemi M Taghizadeh S Ghavidel and A AbbasianldquoColonial competitive differential evolution an experimentalstudy for optimal economic load dispatchrdquo Applied Soft Com-puting vol 40 pp 342ndash363 2016
[44] B R Adarsh T Raghunathan T Jayabarathi and X-S YangldquoEconomic dispatch using chaotic bat algorithmrdquo Energy vol96 pp 666ndash675 2016
[45] N Ghorbani and E Babaei ldquoExchange market algorithm foreconomic load dispatchrdquo International Journal of ElectricalPower amp Energy Systems vol 75 pp 19ndash27 2016
[46] J X V Neto G Reynoso-Meza T H Ruppel V C Marianiand L D S Coelho ldquoSolving non-smooth economic dispatchby a new combination of continuous GRASP algorithm anddifferential evolutionrdquo International Journal of Electrical Poweramp Energy Systems vol 84 pp 13ndash24 2017
[47] A Kavousi-Fard and A Khosravi ldquoAn intelligent 120579-ModifiedBat Algorithm to solve the non-convex economic dispatchproblem considering practical constraintsrdquoElectrical Power andEnergy Systems vol 82 pp 189ndash196 2016
[48] M A Al-Betar M A Awadallah A T Khader and A L BolajildquoTournament-based harmony search algorithm for non-convexeconomic load dispatch problemrdquo Applied Soft Computing vol47 pp 449ndash459 2016
[49] M A Al-Betar M A Awadallah I A Doush E Alsukhni andH ALkhraisat ldquoA non-convex economic dispatchproblemwithvalve loading effect using a new modified 120573-hill climbing localsearch algorithmrdquo Arabian Journal for Science and Engineeringpp 1ndash8 2018
[50] M A Awadallah M A Al-Betar A L Bolaji E M Alsukhniand H Al-Zoubi ldquoNatural selection methods for artificial beecolony with new versions of onlooker beerdquo Soft Computing pp1ndash40 2018
[51] S H Kaboli and A K Alqallaf ldquoSolving non-convex economicload dispatch problem via artificial cooperative search algo-rithmrdquo Expert Systems with Applications vol 128 pp 14ndash272019
[52] D Singh and J Dhillon ldquoAmeliorated greywolf optimization foreconomic load dispatch problemrdquo Energy vol 169 pp 398ndash4192019
[53] X S Yang and S Deb ldquoCuckoo search via Levy flightsrdquo inProceedings of the World Congress on Nature and BiologicallyInspired Computing (NaBIC rsquo09) pp 210ndash214 India 2009
[54] S Sahoo K M Dash R C Prusty and A K Barisal ldquoCom-parative analysis of optimal load dispatch through evolutionaryalgorithmsrdquo Ain Shams Engineering Journal vol 6 pp 107ndash1202015
[55] M Basu and A Chowdhury ldquoCuckoo search algorithm foreconomic dispatchrdquo Energy vol 60 pp 99ndash108 2013
[56] A B Serapiao ldquoCuckoo search for solving economic dispatchload problemrdquo Intelligent Control and Automation vol 04 no04 pp 385ndash390 2013
[57] A H Bindu and M D Reddy ldquoEconomic load dispatch usingcuckoo search algorithmrdquo International Journal of EngineeringResearch and Applications vol 3 pp 498ndash502 2013
[58] E Afzalan and M Joorabian ldquoAn improved cuckoo searchalgorithm for power economic load dispatchrdquo InternationalTransactions on Electrical Energy Systems vol 25 no 6 pp 958ndash975 2015
[59] T T Nguyen and D N Vo ldquoThe application of one rank cuckoosearch algorithm for solving economic load dispatch problemsrdquoApplied Soft Computing vol 37 pp 763ndash773 2015
[60] S M Islam S Das S Ghosh S Roy and P N Suganthan ldquoAnadaptive differential evolution algorithm with novel mutationand crossover strategies for global numerical optimizationrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 42 no 2 pp 482ndash499 2012
[61] Z L Gaing ldquoParticle swarm optimization to solving theeconomic dispatch considering the generator constraintsrdquo IEEETransactions on Power Systems vol 18 no 3 pp 1187ndash1195 2003
[62] X-S Yang and S Deb ldquoEngineering optimisation by Cuckoosearchrdquo International Journal of Mathematical Modelling andNumerical Optimisation vol 330 pp 43ndash51 2010
Hindawiwwwhindawicom Volume 2018
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Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
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Submit your manuscripts atwwwhindawicom
Mathematical Problems in Engineering 19
0 5 10 15 20 25 30 35 40 45 50Run
19795
198
19805
1981
19815
1982
19825
1983
19835
Fuel
cost
($)
CCSAICSA
times105
Figure 5The best fuel cost of 50 runs obtained by CCSA and ICSA for Case 2
0 2 4 6 8 10 12 14 16 18 20Iteration
8234
8236
8238
8240
8242
8244
8246
8248
8250
8252
Fitn
ess F
unct
ion
($)
CCSAICSA
Figure 6 The best run obtained by CCSA and ICSA for Subcase 31
whereas 50 values of fuel cost indicate that the proposedICSA can find many solutions with better quality Clearlythe proposed ICSA is outstanding in terms of stabilizationof solution search and faster convergence As a result it canconclude that the proposed ICSA approach is more effectivethanCCSA in solvingOLDproblemwith considered systems
572The Improvement of Results over Other Methods In thisarticle we have tested ICSA approach on 6 cases with 16
systems with different fuel cost forms different constraintsand different scale systems from 3 units to 320 units We havecompared the yielded results from ICSA approach and otherexisting ones for evaluating the efficiency of ICSA approachIn subsections above we have shown yielded results fromICSA approach and compared these results to those of otheronesHowever the demonstrationhad not been very good forobserving and comparing to lead to a conclusionThus in thesubsectionwehave summarized the result comparisons of the
20 Mathematical Problems in Engineering
0 5 10 15 20 25 30 35 40 45 50Run
8234
82342
82344
82346
82348
8235
82352
82354
82356
82358
8236
Fuel
cost
($)
CCSAICSA
Figure 7 The best fuel cost of 50 runs obtained by CCSA and ICSA for Subcase 31
0 1000 2000 3000 4000 5000 6000Iteration
242
244
246
248
25
252
254
256
Fitn
ess F
unct
ion
($)
CCSAICSA
times105
Figure 8 The best run obtained by CCSA and ICSA for Subcase 35
proposed and other ones Table 15 has reported the reductioncost (in $) of ICSA approach compared to other ones Inaddition we have converted the reduction cost into improve-ment level (in ) for better comparison The improvementhas been shown from the lowest level to the highest levelin terms of reduction cost and improvement percentageIn addition we have also given the slowest and the fastestexecution time of other compared methods together withthat of the proposed method The table implies that ICSA
approach can find better optimal solutions with less fuel costup to $052 for Subcase 13 $75229 for Subcase 14 $1775149for Case 2 $1181256 for Subcase 31 $8438 for Subcase 32$9113 for Subcase 33 $121181 for Subcase 34 $3412 forSubcase 35 $04329 for Subcase 41 $30227 for Subcase42 $407002 for Case 5 $18183 for Subcase 61 $13965 forSubcase 62 and $11817 for Subcase 63 These reductioncosts are equivalent to improvement level (IL) of 0032908968 82282 14143 047 038 099 001405
Mathematical Problems in Engineering 21
0 5 10 15 20 25 30 35 40 45 50Run
2428
243
2432
2434
2436
2438
244
2442
Fuel
cost
($)
CCSAICSA
times105
Figure 9 The best fuel cost of 50 runs obtained by CCSA and ICSA for Subcase 35
0 50 100 150 200 250 300 350 400Iteration
327
328
329
33
331
332
333
334
335
336
337
Fitn
ess F
unct
ion
($)
CCSAICSA
times104
Figure 10 The best run obtained by CCSA and ICSA for Case 5
00003 0002 12291 004 138 and 059 Thesequantitative comparisons reveal that larger scale systems canlead to better reduction cost but the improvement level isnot high because total cost of compared methods tends tobe large for large scale systems Furthermore very large scalesystems with nondifferentiable objective have been normallysolved by strong methods In fact systems in Case 3 havethe same characteristic with single fuel and effects of valveloading process but Subcase 34 is a larger scale system with
40 units while Subcases 31 32 and 33 are constructed by 3units 13 units and 13 units So the improvement percentageof Subcase 34 can be up to 099 whilst that of Subcases32 and 33 is 047 and 038 respectively Subcase 35 iswith the largest system 80 units but the reduction cost is notmuch only $3412 because compared methods with ICSA areeither state-of-the-art ones or improved ones Also Subcase41 and Subcase 42 have considered single fuel and PPZ andspinning reserve constraints but Subcase 42 is larger scale
22 Mathematical Problems in Engineering
Table 15 Performance improvement summary of the proposed method
Study cases Reduction cost ($) Improvement level () Execution time (s)From To From To Slowest method Fastest method Proposed method
Figure 11 The best fuel cost of 50 runs obtained by CCSA and ICSA for Case 5
system with 90 units and Subcase 41 is only with 60 unitsThus the improvement of Subcase 41 is lower with 00003but that of Subcase 42 is 0002 Similarly systems in Cases 6have the same featurewithmulti-fossil fuel sources and effectsof valve loading process but they are respectively constructedby 80 160 and 320 units As a result the improvements ofSubcase 62 138 and Subcase 63 059 are much higherthan Subcase 61 004 However Subcase 62 with smallernumber of units but getting higher improvement is easilyunderstood because there were nine compared methods butonly three compared methods are considered for Subcase 63In general the improvement is not high it is about under onedollar several dollars tens of dollars and over one thousand
dollars per hour however the saving cost will be significantif the operation is considered to be onemonth with 720 hoursor one year with 8760 hours
Execution time comparison can be evaluated by observ-ing the fastest and the slowest compared methods in Table 15These execution times of the proposed method are approxi-mately equal to that of the fastest methods for Subcases 3233 and 34 and much shorter than other fastest comparedmethods for other cases especially for Subcases 61 and 63The fastest method for Subcase 62 is CBA [44] showing 57seconds while that of the proposed method is 1119 secondsHowever it cannot conclude that CBA is more effective thanthe proposed method because the proposed method could
find better optimal solution with less fuel cost by $2130As considering execution time of the slowest comparedmethods it can point out that ICSA is a very fast optimizationtool since the execution time of these methods is 29496 and3425 seconds for Subcases 32 and 33 116735 seconds forSubcase 34 56381 seconds for Subcase 41 94093 secondsfor Subcase 42 12797 seconds for Case 5 30941 seconds forSubcase 61 7542 seconds for Subcase 62 and 9641 secondsfor Subcase 63 while the execution time of the proposedmethod for these cases is respectively 095 146 091 1589246 84828 1119 and 171384 seconds It is clearly shown thatICSA is very fast as compared to these methods
In summary the proposed method has found approx-imately high quality solutions with several standard state-of-the-art meta-heuristic algorithms and improved versionsof them together with other old methods In addition theproposed method could improve result better than approx-imately all methods with faster execution time Comparedto other methods with the fastest convergence speed andhigh quality solutions the proposed method has been as
fast as for some cases and much faster for other cases Thecomparison with the slowest methods could show that theproposed method was extremely powerful since it was up tonearly one thousand times faster Consequently the proposedICSA approach can be one of the strongest optimization toolsfor OLD problem
6 Conclusions
This paper has proposed a good ICSA method for solvingOLD problem in which many test systems with differentobjective functions and complicated constraints from simpleto complex have been used as studied cases The proposedICSA method has been developed by performing severalmodifications on the second solution update of CCSA whichcontained several drawbacks to global convergence and fastmanner The OLD problem has covered from single fuelto multi-fossil fuels from quadratic objective function tononconvex objective function in addition to PPZ RPR andRRL constraints Many existing optimization algorithms have
been concerned in aim to compare the performance and givethe final conclusion on the proposed method There have
been six main cases with sixteen subcases The evaluationshave been made at the end of each study case Clearly theproposed ICSA approach has yielded more effective optimalsolutions with faster execution time than almost all methodsConsequently it can be concluded that the proposed methodis much more superior to CCSA and is a very promisingmethod for solving OLD problem
Appendix
See Tables 16ndash26
Nomenclature
120575119894 120582119894 120572119894 120573119894 120574119894 Fuel cost functioncoefficients of the 119894119905ℎthermal generationunit11986100 1198610119895 119861119895119894 Power loss matrixcoefficients119865119865119909 119865119865119887119890119904119905 The values of fitness ofsolution 119909 and theso-far most effectivesolution among thecurrent set ofsolutions119898119894 Number of fuels burntin the 119894119905ℎ thermalgeneration unit
119873 Number of allavailable thermalgeneration units119899119894 Number of prohibitedpower zones of the 119894119905ℎthermal generationunit119875119906119894119896 119875119897119894119896 Upper and lowerlimits of the 119894119905ℎthermal generationunit corresponding tothe kth PPZ119875119886 Probability ofreplacing controlvariables in each oldsolution119875119863 Real power demand ofall loads in system119875119894119898119886119909 119875119894119898119894119899 The highest andlowest real poweroutputs of the iththermal generationunit119875119894119895119898119886119909 119875119894119895119898119894119899 The highest andlowest real poweroutputs of the 119894119905ℎthermal generationunit corresponding tothe 119895119905ℎ fuel type119878119894119898119886119909 Maximum real powerreserve contributionof the thermalgeneration unit 1198941198781199001198971199031198861198991198891 1198781199001198971199031198861198991198892 1198781199001198971199031198861198991198893 1198781199001198971199031198861198991198894 Randomly mixedsolutions from the setof current solutions119878119900119897119909 119878119900119897119866119887119890119904119905 The old solution x andthe most effectivesolution119878119877 Real power reserverequirement of system
120575119894119895 120582119894119895 120572119894119895 120573119894119895 120574119894119895 Fuel cost functioncoefficients of the 119894119905ℎthermal generationunit corresponding tothe 119895119905ℎ fuel type1205761 1205762 1205763 1205764 1205765 1205766 Random numbersbetween 0 and 1Ω Number of generationunits considering PPZconstraint
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare no conflicts of interest
References
[1] N Sinha R Chakrabarti and P K Chattopadhyay ldquoEvolution-ary programming techniques for economic load dispatchrdquo IEEETransactions on Evolutionary Computation vol 7 no 1 pp 83ndash94 2003
[2] P Hansen ldquoA separable approximation dynamic programmingalgorithm for economic dispatch with transmission lossesrdquoYugoslav Journal of Operations Research vol 12 no 2 2002
[3] N T Thang ldquoSolving economic dispatch problem with piece-wise quadratic cost functions using lagrange multiplier theoryrdquoin Proceedings of the 3rd International Conference on ComputerTechnology and Development (ICCTD rsquo11) pp 359ndash364 ASMEPress 2011
[4] S K Mishra and S K Mishra ldquoA comparative study of solutionof economic load dispatch problem in power systems in theenvironmental perspectiverdquoProcedia Computer Science vol 48pp 96ndash100 2015
[5] A A Al-Subhi and H K Alfares ldquoEconomic load dispatchusing linear programming a comparative studyrdquo InternationalJournal of Applied Industrial Engineering vol 3 no 1 pp 16ndash362016
28 Mathematical Problems in Engineering
[6] C Zhou G Huang and J Chen ldquoPlanning of electric powersystems considering virtual power plants with dispatchableloads included an inexact two-stage stochastic linear program-ming modelrdquoMathematical Problems in Engineering vol 2018Article ID 7049329 12 pages 2018
[7] T T Nguyen N V Quynh and L Van Dai ldquoImproved fireflyalgorithm a novel method for optimal operation of thermalgenerating unitsrdquo Complexity vol 2018 Article ID 7267593 23pages 2018
[8] S S Haroon S Hassan S Amin et al ldquoMultiple fuel machinespower economic dispatch using stud differential evolutionrdquoEnergies vol 11 no 6 pp 1ndash20 2018
[9] C-T Su and C-L Chiang ldquoNonconvex power economic dis-patch by improved genetic algorithm with multiplier updatingmethodrdquo Electric Power Components and Systems vol 32 no 3pp 257ndash273 2004
[10] NNoman andH Iba ldquoDifferential evolution for economic loaddispatch problemsrdquo Electric Power Systems Research vol 78 no8 pp 1322ndash1331 2008
[11] P Somasundaram and K Kuppusamy ldquoApplication of evolu-tionary programming to security constrained economic dis-patchrdquo International Journal of Electrical Power amp EnergySystems vol 27 no 5-6 pp 343ndash351 2005
[12] T Jayabarathi K Jayaprakash andD Jeyakumar ldquoEvolutionaryprogramming technique for different kinds of economic dis-patch problemsrdquo Electric Power Systems Research vol 73 no 2pp 169ndash176 2005
[13] P Subbaraj R Rengaraj and S Salivahanan ldquoEnhancementof combined heat and power economic dispatch using selfadaptive real-coded genetic algorithmrdquo Applied Energy vol 86no 6 pp 915ndash921 2009
[14] C-L Chiang and C-T Su ldquoAdaptive-improved genetic algo-rithm for the economic dispatch of units with multiple fueloptionsrdquo Cybernetics and Systems vol 36 no 7 pp 687ndash7042005
[15] C Chiang ldquoImproved genetic algorithm for power economicdispatch of units with valve-point effects and multiple fuelsrdquoIEEE Transactions on Power Systems vol 20 no 4 pp 1690ndash1699 2005
[16] K Meng H G Wang Z Y Dong and K P Wong ldquoQuantum-inspired particle swarm optimization for valve-point economicload dispatchrdquo IEEE Transactions on Power Systems vol 25 no1 pp 215ndash222 2010
[17] P Erdogmus A Ozturk and S Duman ldquoEnvironmen-taleconomic dispatch using genetic algorithm and simulatedannealingrdquo in Proceedings of the International Conference onElectric Power and Energy Conversion Systems EPECS rsquo09 pp1ndash4 IEEE 2009
[18] L D S Coelho and V C Mariani ldquoImproved differentialevolution algorithms for handling economic dispatch opti-mization with generator constraintsrdquo Energy Conversion andManagement vol 48 no 5 pp 1631ndash1639 2007
[19] H Dakuo W Fuli and M Zhizhong ldquoA hybrid genetic algo-rithm approach based on differential evolution for economicdispatch with valve-point effectsrdquo Electrical Power and EnergySystems vol 30 pp 31ndash38 2008
[20] C-F Chang J-J Wong J-P Chiou and C-T Su ldquoRobustsearching hybrid differential evolution method for optimalreactive power planning in large-scale distribution systemsrdquoElectric Power Systems Research vol 77 no 5-6 pp 430ndash4372007
[21] M R Farooqi P Jain and K R Niazi ldquoUsing Hopfield neuralnetwork for economic dispatch of power systemsrdquo in Proceed-ings of the National Power Engineering Conference PECon rsquo03pp 5ndash10 Bangi Malaysia 2003
[22] T Yalcinoz H Altun and U Hasan ldquoConstrained economicdispatch with prohibited operating zones a Hopfield neuralnetwork approachrdquo in Proceedings of the 10th MediterraneanElectrotechnical Conference Information Technology and Elec-trotechnology for the Mediterranean Countries ProceedingsMeleCon rsquo00 (Cat No 00CH37099) pp 570ndash573 LemesosCyprus 2000
[23] D N Vo P Schegner and W Ongsakul ldquoCuckoo searchalgorithm for non-convex economic dispatchrdquo IET GenerationTransmission amp Distribution vol 7 no 6 pp 645ndash654 2013
[24] J-B Park K-S Lee J-R Shin and K Y Lee ldquoA particleswarm optimization for economic dispatch with non-smoothcost functionsrdquo IEEETransactions on Power Systems vol 20 no1 pp 34ndash42 2005
[25] T Niknam H D Mojarrad and M Nayeripour ldquoA newfuzzy adaptive particle swarm optimization for non-smootheconomic dispatchrdquo Energy vol 35 no 4 pp 1764ndash1778 2010
[26] L D S Coelho and V C Mariani ldquoAn improved harmonysearch algorithm for power economic load dispatchrdquo EnergyConversion and Management vol 50 no 10 pp 2522ndash25262009
[27] M Mahdavi M Fesanghary and E Damangir ldquoAn improvedharmony search algorithm for solving optimization problemsrdquoAppliedMathematics and Computation vol 188 no 2 pp 1567ndash1579 2007
[28] A Bhattacharya and P K Chattopadhyay ldquoSolving complexeconomic load dispatch problem using biogeography-basedoptimizationrdquo Expert Systems with Applications vol 37 no 5pp 3605ndash3615 2010
[29] D Simon ldquoBiogeography-based optimizationrdquo IEEE Transac-tions on Evolutionary Computation vol 12 no 6 pp 702ndash7132008
[30] J S Alsumait J K Sykulski and A K Al-Othman ldquoAhybrid GA-PS-SQP method to solve power system valve-pointeconomic dispatch problemsrdquo Applied Energy vol 87 no 5 pp1773ndash1781 2010
[31] M S Turgut andGKDemir ldquoQuadratic approximationndashbasedhybrid Artificial Cooperative Search algorithm for economicemission load dispatchproblemsrdquo International Transactions onElectrical Energy Systems vol 27 no 4 pp 1ndash14 2017
[32] T A A Victoire and A E Jeyakumar ldquoHybrid PSO-SQPfor economic dispatch with valve-point effectrdquo Electric PowerSystems Research vol 71 no 1 pp 51ndash59 2004
[33] D-KHe F-LWang andZ-ZMao ldquoHybrid genetic algorithmfor economic dispatch with valve-point effectrdquo Electric PowerSystems Research vol 78 no 4 pp 626ndash633 2008
[34] D C Secui ldquoA modified symbiotic organisms search algorithmfor large scale economic dispatch problem with valve-pointloading effectsrdquo Energy vol 113 pp 366ndash384 2016
[35] K BhattacharjeeA Bhattacharya and S H N Dey ldquoTeaching-learning-based optimization for different economic dispatchproblemsrdquo Scientia Iranica vol 21 no 3 pp 870ndash884 2014
[36] K Bhattacharjee A Bhattacharya and S H N Dey ldquoOppo-sitional Real Coded Chemical Reaction Optimization for dif-ferent economic dispatch problemsrdquo International Journal ofElectrical Power amp Energy Systems vol 55 pp 378ndash391 2014
Mathematical Problems in Engineering 29
[37] V Hosseinnezhad M Rafiee M Ahmadian and M T AmelildquoSpecies-basedQuantum Particle SwarmOptimization for eco-nomic load dispatchrdquo International Journal of Electrical Poweramp Energy Systems vol 63 pp 311ndash322 2014
[38] M S P Subathra S E Selvan T A A Victoire A HChristinal andU Amato ldquoA hybrid with cross-entropymethodand sequential quadratic programming to solve economic loaddispatch problemrdquo IEEE Systems Journal vol 9 no 3 pp 1031ndash1044 2015
[39] J Zhan Q H Wu C Guo and X Zhou ldquoEconomic dispatchwith non-smooth objectives part I local minimum analysisrdquoIEEE Transactions on Power Systems vol 30 no 2 pp 710ndash7212015
[40] A K Barisal andR C Prusty ldquoLarge scale economic dispatch ofpower systems using oppositional invasive weed optimizationrdquoApplied Soft Computing vol 29 pp 122ndash137 2015
[41] M Basu ldquoImproved differential evolution for economic dis-patchrdquo International Journal of Electrical Power amp EnergySystems vol 63 pp 855ndash861 2014
[42] V S Aragon S C Esquivel and C C Coello ldquoAn immunealgorithm with power redistribution for solving economicdispatch problemsrdquo Information Sciences vol 295 pp 609ndash6322015
[43] M Ghasemi M Taghizadeh S Ghavidel and A AbbasianldquoColonial competitive differential evolution an experimentalstudy for optimal economic load dispatchrdquo Applied Soft Com-puting vol 40 pp 342ndash363 2016
[44] B R Adarsh T Raghunathan T Jayabarathi and X-S YangldquoEconomic dispatch using chaotic bat algorithmrdquo Energy vol96 pp 666ndash675 2016
[45] N Ghorbani and E Babaei ldquoExchange market algorithm foreconomic load dispatchrdquo International Journal of ElectricalPower amp Energy Systems vol 75 pp 19ndash27 2016
[46] J X V Neto G Reynoso-Meza T H Ruppel V C Marianiand L D S Coelho ldquoSolving non-smooth economic dispatchby a new combination of continuous GRASP algorithm anddifferential evolutionrdquo International Journal of Electrical Poweramp Energy Systems vol 84 pp 13ndash24 2017
[47] A Kavousi-Fard and A Khosravi ldquoAn intelligent 120579-ModifiedBat Algorithm to solve the non-convex economic dispatchproblem considering practical constraintsrdquoElectrical Power andEnergy Systems vol 82 pp 189ndash196 2016
[48] M A Al-Betar M A Awadallah A T Khader and A L BolajildquoTournament-based harmony search algorithm for non-convexeconomic load dispatch problemrdquo Applied Soft Computing vol47 pp 449ndash459 2016
[49] M A Al-Betar M A Awadallah I A Doush E Alsukhni andH ALkhraisat ldquoA non-convex economic dispatchproblemwithvalve loading effect using a new modified 120573-hill climbing localsearch algorithmrdquo Arabian Journal for Science and Engineeringpp 1ndash8 2018
[50] M A Awadallah M A Al-Betar A L Bolaji E M Alsukhniand H Al-Zoubi ldquoNatural selection methods for artificial beecolony with new versions of onlooker beerdquo Soft Computing pp1ndash40 2018
[51] S H Kaboli and A K Alqallaf ldquoSolving non-convex economicload dispatch problem via artificial cooperative search algo-rithmrdquo Expert Systems with Applications vol 128 pp 14ndash272019
[52] D Singh and J Dhillon ldquoAmeliorated greywolf optimization foreconomic load dispatch problemrdquo Energy vol 169 pp 398ndash4192019
[53] X S Yang and S Deb ldquoCuckoo search via Levy flightsrdquo inProceedings of the World Congress on Nature and BiologicallyInspired Computing (NaBIC rsquo09) pp 210ndash214 India 2009
[54] S Sahoo K M Dash R C Prusty and A K Barisal ldquoCom-parative analysis of optimal load dispatch through evolutionaryalgorithmsrdquo Ain Shams Engineering Journal vol 6 pp 107ndash1202015
[55] M Basu and A Chowdhury ldquoCuckoo search algorithm foreconomic dispatchrdquo Energy vol 60 pp 99ndash108 2013
[56] A B Serapiao ldquoCuckoo search for solving economic dispatchload problemrdquo Intelligent Control and Automation vol 04 no04 pp 385ndash390 2013
[57] A H Bindu and M D Reddy ldquoEconomic load dispatch usingcuckoo search algorithmrdquo International Journal of EngineeringResearch and Applications vol 3 pp 498ndash502 2013
[58] E Afzalan and M Joorabian ldquoAn improved cuckoo searchalgorithm for power economic load dispatchrdquo InternationalTransactions on Electrical Energy Systems vol 25 no 6 pp 958ndash975 2015
[59] T T Nguyen and D N Vo ldquoThe application of one rank cuckoosearch algorithm for solving economic load dispatch problemsrdquoApplied Soft Computing vol 37 pp 763ndash773 2015
[60] S M Islam S Das S Ghosh S Roy and P N Suganthan ldquoAnadaptive differential evolution algorithm with novel mutationand crossover strategies for global numerical optimizationrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 42 no 2 pp 482ndash499 2012
[61] Z L Gaing ldquoParticle swarm optimization to solving theeconomic dispatch considering the generator constraintsrdquo IEEETransactions on Power Systems vol 18 no 3 pp 1187ndash1195 2003
[62] X-S Yang and S Deb ldquoEngineering optimisation by Cuckoosearchrdquo International Journal of Mathematical Modelling andNumerical Optimisation vol 330 pp 43ndash51 2010
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
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Submit your manuscripts atwwwhindawicom
20 Mathematical Problems in Engineering
0 5 10 15 20 25 30 35 40 45 50Run
8234
82342
82344
82346
82348
8235
82352
82354
82356
82358
8236
Fuel
cost
($)
CCSAICSA
Figure 7 The best fuel cost of 50 runs obtained by CCSA and ICSA for Subcase 31
0 1000 2000 3000 4000 5000 6000Iteration
242
244
246
248
25
252
254
256
Fitn
ess F
unct
ion
($)
CCSAICSA
times105
Figure 8 The best run obtained by CCSA and ICSA for Subcase 35
proposed and other ones Table 15 has reported the reductioncost (in $) of ICSA approach compared to other ones Inaddition we have converted the reduction cost into improve-ment level (in ) for better comparison The improvementhas been shown from the lowest level to the highest levelin terms of reduction cost and improvement percentageIn addition we have also given the slowest and the fastestexecution time of other compared methods together withthat of the proposed method The table implies that ICSA
approach can find better optimal solutions with less fuel costup to $052 for Subcase 13 $75229 for Subcase 14 $1775149for Case 2 $1181256 for Subcase 31 $8438 for Subcase 32$9113 for Subcase 33 $121181 for Subcase 34 $3412 forSubcase 35 $04329 for Subcase 41 $30227 for Subcase42 $407002 for Case 5 $18183 for Subcase 61 $13965 forSubcase 62 and $11817 for Subcase 63 These reductioncosts are equivalent to improvement level (IL) of 0032908968 82282 14143 047 038 099 001405
Mathematical Problems in Engineering 21
0 5 10 15 20 25 30 35 40 45 50Run
2428
243
2432
2434
2436
2438
244
2442
Fuel
cost
($)
CCSAICSA
times105
Figure 9 The best fuel cost of 50 runs obtained by CCSA and ICSA for Subcase 35
0 50 100 150 200 250 300 350 400Iteration
327
328
329
33
331
332
333
334
335
336
337
Fitn
ess F
unct
ion
($)
CCSAICSA
times104
Figure 10 The best run obtained by CCSA and ICSA for Case 5
00003 0002 12291 004 138 and 059 Thesequantitative comparisons reveal that larger scale systems canlead to better reduction cost but the improvement level isnot high because total cost of compared methods tends tobe large for large scale systems Furthermore very large scalesystems with nondifferentiable objective have been normallysolved by strong methods In fact systems in Case 3 havethe same characteristic with single fuel and effects of valveloading process but Subcase 34 is a larger scale system with
40 units while Subcases 31 32 and 33 are constructed by 3units 13 units and 13 units So the improvement percentageof Subcase 34 can be up to 099 whilst that of Subcases32 and 33 is 047 and 038 respectively Subcase 35 iswith the largest system 80 units but the reduction cost is notmuch only $3412 because compared methods with ICSA areeither state-of-the-art ones or improved ones Also Subcase41 and Subcase 42 have considered single fuel and PPZ andspinning reserve constraints but Subcase 42 is larger scale
22 Mathematical Problems in Engineering
Table 15 Performance improvement summary of the proposed method
Study cases Reduction cost ($) Improvement level () Execution time (s)From To From To Slowest method Fastest method Proposed method
Figure 11 The best fuel cost of 50 runs obtained by CCSA and ICSA for Case 5
system with 90 units and Subcase 41 is only with 60 unitsThus the improvement of Subcase 41 is lower with 00003but that of Subcase 42 is 0002 Similarly systems in Cases 6have the same featurewithmulti-fossil fuel sources and effectsof valve loading process but they are respectively constructedby 80 160 and 320 units As a result the improvements ofSubcase 62 138 and Subcase 63 059 are much higherthan Subcase 61 004 However Subcase 62 with smallernumber of units but getting higher improvement is easilyunderstood because there were nine compared methods butonly three compared methods are considered for Subcase 63In general the improvement is not high it is about under onedollar several dollars tens of dollars and over one thousand
dollars per hour however the saving cost will be significantif the operation is considered to be onemonth with 720 hoursor one year with 8760 hours
Execution time comparison can be evaluated by observ-ing the fastest and the slowest compared methods in Table 15These execution times of the proposed method are approxi-mately equal to that of the fastest methods for Subcases 3233 and 34 and much shorter than other fastest comparedmethods for other cases especially for Subcases 61 and 63The fastest method for Subcase 62 is CBA [44] showing 57seconds while that of the proposed method is 1119 secondsHowever it cannot conclude that CBA is more effective thanthe proposed method because the proposed method could
find better optimal solution with less fuel cost by $2130As considering execution time of the slowest comparedmethods it can point out that ICSA is a very fast optimizationtool since the execution time of these methods is 29496 and3425 seconds for Subcases 32 and 33 116735 seconds forSubcase 34 56381 seconds for Subcase 41 94093 secondsfor Subcase 42 12797 seconds for Case 5 30941 seconds forSubcase 61 7542 seconds for Subcase 62 and 9641 secondsfor Subcase 63 while the execution time of the proposedmethod for these cases is respectively 095 146 091 1589246 84828 1119 and 171384 seconds It is clearly shown thatICSA is very fast as compared to these methods
In summary the proposed method has found approx-imately high quality solutions with several standard state-of-the-art meta-heuristic algorithms and improved versionsof them together with other old methods In addition theproposed method could improve result better than approx-imately all methods with faster execution time Comparedto other methods with the fastest convergence speed andhigh quality solutions the proposed method has been as
fast as for some cases and much faster for other cases Thecomparison with the slowest methods could show that theproposed method was extremely powerful since it was up tonearly one thousand times faster Consequently the proposedICSA approach can be one of the strongest optimization toolsfor OLD problem
6 Conclusions
This paper has proposed a good ICSA method for solvingOLD problem in which many test systems with differentobjective functions and complicated constraints from simpleto complex have been used as studied cases The proposedICSA method has been developed by performing severalmodifications on the second solution update of CCSA whichcontained several drawbacks to global convergence and fastmanner The OLD problem has covered from single fuelto multi-fossil fuels from quadratic objective function tononconvex objective function in addition to PPZ RPR andRRL constraints Many existing optimization algorithms have
been concerned in aim to compare the performance and givethe final conclusion on the proposed method There have
been six main cases with sixteen subcases The evaluationshave been made at the end of each study case Clearly theproposed ICSA approach has yielded more effective optimalsolutions with faster execution time than almost all methodsConsequently it can be concluded that the proposed methodis much more superior to CCSA and is a very promisingmethod for solving OLD problem
Appendix
See Tables 16ndash26
Nomenclature
120575119894 120582119894 120572119894 120573119894 120574119894 Fuel cost functioncoefficients of the 119894119905ℎthermal generationunit11986100 1198610119895 119861119895119894 Power loss matrixcoefficients119865119865119909 119865119865119887119890119904119905 The values of fitness ofsolution 119909 and theso-far most effectivesolution among thecurrent set ofsolutions119898119894 Number of fuels burntin the 119894119905ℎ thermalgeneration unit
119873 Number of allavailable thermalgeneration units119899119894 Number of prohibitedpower zones of the 119894119905ℎthermal generationunit119875119906119894119896 119875119897119894119896 Upper and lowerlimits of the 119894119905ℎthermal generationunit corresponding tothe kth PPZ119875119886 Probability ofreplacing controlvariables in each oldsolution119875119863 Real power demand ofall loads in system119875119894119898119886119909 119875119894119898119894119899 The highest andlowest real poweroutputs of the iththermal generationunit119875119894119895119898119886119909 119875119894119895119898119894119899 The highest andlowest real poweroutputs of the 119894119905ℎthermal generationunit corresponding tothe 119895119905ℎ fuel type119878119894119898119886119909 Maximum real powerreserve contributionof the thermalgeneration unit 1198941198781199001198971199031198861198991198891 1198781199001198971199031198861198991198892 1198781199001198971199031198861198991198893 1198781199001198971199031198861198991198894 Randomly mixedsolutions from the setof current solutions119878119900119897119909 119878119900119897119866119887119890119904119905 The old solution x andthe most effectivesolution119878119877 Real power reserverequirement of system
120575119894119895 120582119894119895 120572119894119895 120573119894119895 120574119894119895 Fuel cost functioncoefficients of the 119894119905ℎthermal generationunit corresponding tothe 119895119905ℎ fuel type1205761 1205762 1205763 1205764 1205765 1205766 Random numbersbetween 0 and 1Ω Number of generationunits considering PPZconstraint
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare no conflicts of interest
References
[1] N Sinha R Chakrabarti and P K Chattopadhyay ldquoEvolution-ary programming techniques for economic load dispatchrdquo IEEETransactions on Evolutionary Computation vol 7 no 1 pp 83ndash94 2003
[2] P Hansen ldquoA separable approximation dynamic programmingalgorithm for economic dispatch with transmission lossesrdquoYugoslav Journal of Operations Research vol 12 no 2 2002
[3] N T Thang ldquoSolving economic dispatch problem with piece-wise quadratic cost functions using lagrange multiplier theoryrdquoin Proceedings of the 3rd International Conference on ComputerTechnology and Development (ICCTD rsquo11) pp 359ndash364 ASMEPress 2011
[4] S K Mishra and S K Mishra ldquoA comparative study of solutionof economic load dispatch problem in power systems in theenvironmental perspectiverdquoProcedia Computer Science vol 48pp 96ndash100 2015
[5] A A Al-Subhi and H K Alfares ldquoEconomic load dispatchusing linear programming a comparative studyrdquo InternationalJournal of Applied Industrial Engineering vol 3 no 1 pp 16ndash362016
28 Mathematical Problems in Engineering
[6] C Zhou G Huang and J Chen ldquoPlanning of electric powersystems considering virtual power plants with dispatchableloads included an inexact two-stage stochastic linear program-ming modelrdquoMathematical Problems in Engineering vol 2018Article ID 7049329 12 pages 2018
[7] T T Nguyen N V Quynh and L Van Dai ldquoImproved fireflyalgorithm a novel method for optimal operation of thermalgenerating unitsrdquo Complexity vol 2018 Article ID 7267593 23pages 2018
[8] S S Haroon S Hassan S Amin et al ldquoMultiple fuel machinespower economic dispatch using stud differential evolutionrdquoEnergies vol 11 no 6 pp 1ndash20 2018
[9] C-T Su and C-L Chiang ldquoNonconvex power economic dis-patch by improved genetic algorithm with multiplier updatingmethodrdquo Electric Power Components and Systems vol 32 no 3pp 257ndash273 2004
[10] NNoman andH Iba ldquoDifferential evolution for economic loaddispatch problemsrdquo Electric Power Systems Research vol 78 no8 pp 1322ndash1331 2008
[11] P Somasundaram and K Kuppusamy ldquoApplication of evolu-tionary programming to security constrained economic dis-patchrdquo International Journal of Electrical Power amp EnergySystems vol 27 no 5-6 pp 343ndash351 2005
[12] T Jayabarathi K Jayaprakash andD Jeyakumar ldquoEvolutionaryprogramming technique for different kinds of economic dis-patch problemsrdquo Electric Power Systems Research vol 73 no 2pp 169ndash176 2005
[13] P Subbaraj R Rengaraj and S Salivahanan ldquoEnhancementof combined heat and power economic dispatch using selfadaptive real-coded genetic algorithmrdquo Applied Energy vol 86no 6 pp 915ndash921 2009
[14] C-L Chiang and C-T Su ldquoAdaptive-improved genetic algo-rithm for the economic dispatch of units with multiple fueloptionsrdquo Cybernetics and Systems vol 36 no 7 pp 687ndash7042005
[15] C Chiang ldquoImproved genetic algorithm for power economicdispatch of units with valve-point effects and multiple fuelsrdquoIEEE Transactions on Power Systems vol 20 no 4 pp 1690ndash1699 2005
[16] K Meng H G Wang Z Y Dong and K P Wong ldquoQuantum-inspired particle swarm optimization for valve-point economicload dispatchrdquo IEEE Transactions on Power Systems vol 25 no1 pp 215ndash222 2010
[17] P Erdogmus A Ozturk and S Duman ldquoEnvironmen-taleconomic dispatch using genetic algorithm and simulatedannealingrdquo in Proceedings of the International Conference onElectric Power and Energy Conversion Systems EPECS rsquo09 pp1ndash4 IEEE 2009
[18] L D S Coelho and V C Mariani ldquoImproved differentialevolution algorithms for handling economic dispatch opti-mization with generator constraintsrdquo Energy Conversion andManagement vol 48 no 5 pp 1631ndash1639 2007
[19] H Dakuo W Fuli and M Zhizhong ldquoA hybrid genetic algo-rithm approach based on differential evolution for economicdispatch with valve-point effectsrdquo Electrical Power and EnergySystems vol 30 pp 31ndash38 2008
[20] C-F Chang J-J Wong J-P Chiou and C-T Su ldquoRobustsearching hybrid differential evolution method for optimalreactive power planning in large-scale distribution systemsrdquoElectric Power Systems Research vol 77 no 5-6 pp 430ndash4372007
[21] M R Farooqi P Jain and K R Niazi ldquoUsing Hopfield neuralnetwork for economic dispatch of power systemsrdquo in Proceed-ings of the National Power Engineering Conference PECon rsquo03pp 5ndash10 Bangi Malaysia 2003
[22] T Yalcinoz H Altun and U Hasan ldquoConstrained economicdispatch with prohibited operating zones a Hopfield neuralnetwork approachrdquo in Proceedings of the 10th MediterraneanElectrotechnical Conference Information Technology and Elec-trotechnology for the Mediterranean Countries ProceedingsMeleCon rsquo00 (Cat No 00CH37099) pp 570ndash573 LemesosCyprus 2000
[23] D N Vo P Schegner and W Ongsakul ldquoCuckoo searchalgorithm for non-convex economic dispatchrdquo IET GenerationTransmission amp Distribution vol 7 no 6 pp 645ndash654 2013
[24] J-B Park K-S Lee J-R Shin and K Y Lee ldquoA particleswarm optimization for economic dispatch with non-smoothcost functionsrdquo IEEETransactions on Power Systems vol 20 no1 pp 34ndash42 2005
[25] T Niknam H D Mojarrad and M Nayeripour ldquoA newfuzzy adaptive particle swarm optimization for non-smootheconomic dispatchrdquo Energy vol 35 no 4 pp 1764ndash1778 2010
[26] L D S Coelho and V C Mariani ldquoAn improved harmonysearch algorithm for power economic load dispatchrdquo EnergyConversion and Management vol 50 no 10 pp 2522ndash25262009
[27] M Mahdavi M Fesanghary and E Damangir ldquoAn improvedharmony search algorithm for solving optimization problemsrdquoAppliedMathematics and Computation vol 188 no 2 pp 1567ndash1579 2007
[28] A Bhattacharya and P K Chattopadhyay ldquoSolving complexeconomic load dispatch problem using biogeography-basedoptimizationrdquo Expert Systems with Applications vol 37 no 5pp 3605ndash3615 2010
[29] D Simon ldquoBiogeography-based optimizationrdquo IEEE Transac-tions on Evolutionary Computation vol 12 no 6 pp 702ndash7132008
[30] J S Alsumait J K Sykulski and A K Al-Othman ldquoAhybrid GA-PS-SQP method to solve power system valve-pointeconomic dispatch problemsrdquo Applied Energy vol 87 no 5 pp1773ndash1781 2010
[31] M S Turgut andGKDemir ldquoQuadratic approximationndashbasedhybrid Artificial Cooperative Search algorithm for economicemission load dispatchproblemsrdquo International Transactions onElectrical Energy Systems vol 27 no 4 pp 1ndash14 2017
[32] T A A Victoire and A E Jeyakumar ldquoHybrid PSO-SQPfor economic dispatch with valve-point effectrdquo Electric PowerSystems Research vol 71 no 1 pp 51ndash59 2004
[33] D-KHe F-LWang andZ-ZMao ldquoHybrid genetic algorithmfor economic dispatch with valve-point effectrdquo Electric PowerSystems Research vol 78 no 4 pp 626ndash633 2008
[34] D C Secui ldquoA modified symbiotic organisms search algorithmfor large scale economic dispatch problem with valve-pointloading effectsrdquo Energy vol 113 pp 366ndash384 2016
[35] K BhattacharjeeA Bhattacharya and S H N Dey ldquoTeaching-learning-based optimization for different economic dispatchproblemsrdquo Scientia Iranica vol 21 no 3 pp 870ndash884 2014
[36] K Bhattacharjee A Bhattacharya and S H N Dey ldquoOppo-sitional Real Coded Chemical Reaction Optimization for dif-ferent economic dispatch problemsrdquo International Journal ofElectrical Power amp Energy Systems vol 55 pp 378ndash391 2014
Mathematical Problems in Engineering 29
[37] V Hosseinnezhad M Rafiee M Ahmadian and M T AmelildquoSpecies-basedQuantum Particle SwarmOptimization for eco-nomic load dispatchrdquo International Journal of Electrical Poweramp Energy Systems vol 63 pp 311ndash322 2014
[38] M S P Subathra S E Selvan T A A Victoire A HChristinal andU Amato ldquoA hybrid with cross-entropymethodand sequential quadratic programming to solve economic loaddispatch problemrdquo IEEE Systems Journal vol 9 no 3 pp 1031ndash1044 2015
[39] J Zhan Q H Wu C Guo and X Zhou ldquoEconomic dispatchwith non-smooth objectives part I local minimum analysisrdquoIEEE Transactions on Power Systems vol 30 no 2 pp 710ndash7212015
[40] A K Barisal andR C Prusty ldquoLarge scale economic dispatch ofpower systems using oppositional invasive weed optimizationrdquoApplied Soft Computing vol 29 pp 122ndash137 2015
[41] M Basu ldquoImproved differential evolution for economic dis-patchrdquo International Journal of Electrical Power amp EnergySystems vol 63 pp 855ndash861 2014
[42] V S Aragon S C Esquivel and C C Coello ldquoAn immunealgorithm with power redistribution for solving economicdispatch problemsrdquo Information Sciences vol 295 pp 609ndash6322015
[43] M Ghasemi M Taghizadeh S Ghavidel and A AbbasianldquoColonial competitive differential evolution an experimentalstudy for optimal economic load dispatchrdquo Applied Soft Com-puting vol 40 pp 342ndash363 2016
[44] B R Adarsh T Raghunathan T Jayabarathi and X-S YangldquoEconomic dispatch using chaotic bat algorithmrdquo Energy vol96 pp 666ndash675 2016
[45] N Ghorbani and E Babaei ldquoExchange market algorithm foreconomic load dispatchrdquo International Journal of ElectricalPower amp Energy Systems vol 75 pp 19ndash27 2016
[46] J X V Neto G Reynoso-Meza T H Ruppel V C Marianiand L D S Coelho ldquoSolving non-smooth economic dispatchby a new combination of continuous GRASP algorithm anddifferential evolutionrdquo International Journal of Electrical Poweramp Energy Systems vol 84 pp 13ndash24 2017
[47] A Kavousi-Fard and A Khosravi ldquoAn intelligent 120579-ModifiedBat Algorithm to solve the non-convex economic dispatchproblem considering practical constraintsrdquoElectrical Power andEnergy Systems vol 82 pp 189ndash196 2016
[48] M A Al-Betar M A Awadallah A T Khader and A L BolajildquoTournament-based harmony search algorithm for non-convexeconomic load dispatch problemrdquo Applied Soft Computing vol47 pp 449ndash459 2016
[49] M A Al-Betar M A Awadallah I A Doush E Alsukhni andH ALkhraisat ldquoA non-convex economic dispatchproblemwithvalve loading effect using a new modified 120573-hill climbing localsearch algorithmrdquo Arabian Journal for Science and Engineeringpp 1ndash8 2018
[50] M A Awadallah M A Al-Betar A L Bolaji E M Alsukhniand H Al-Zoubi ldquoNatural selection methods for artificial beecolony with new versions of onlooker beerdquo Soft Computing pp1ndash40 2018
[51] S H Kaboli and A K Alqallaf ldquoSolving non-convex economicload dispatch problem via artificial cooperative search algo-rithmrdquo Expert Systems with Applications vol 128 pp 14ndash272019
[52] D Singh and J Dhillon ldquoAmeliorated greywolf optimization foreconomic load dispatch problemrdquo Energy vol 169 pp 398ndash4192019
[53] X S Yang and S Deb ldquoCuckoo search via Levy flightsrdquo inProceedings of the World Congress on Nature and BiologicallyInspired Computing (NaBIC rsquo09) pp 210ndash214 India 2009
[54] S Sahoo K M Dash R C Prusty and A K Barisal ldquoCom-parative analysis of optimal load dispatch through evolutionaryalgorithmsrdquo Ain Shams Engineering Journal vol 6 pp 107ndash1202015
[55] M Basu and A Chowdhury ldquoCuckoo search algorithm foreconomic dispatchrdquo Energy vol 60 pp 99ndash108 2013
[56] A B Serapiao ldquoCuckoo search for solving economic dispatchload problemrdquo Intelligent Control and Automation vol 04 no04 pp 385ndash390 2013
[57] A H Bindu and M D Reddy ldquoEconomic load dispatch usingcuckoo search algorithmrdquo International Journal of EngineeringResearch and Applications vol 3 pp 498ndash502 2013
[58] E Afzalan and M Joorabian ldquoAn improved cuckoo searchalgorithm for power economic load dispatchrdquo InternationalTransactions on Electrical Energy Systems vol 25 no 6 pp 958ndash975 2015
[59] T T Nguyen and D N Vo ldquoThe application of one rank cuckoosearch algorithm for solving economic load dispatch problemsrdquoApplied Soft Computing vol 37 pp 763ndash773 2015
[60] S M Islam S Das S Ghosh S Roy and P N Suganthan ldquoAnadaptive differential evolution algorithm with novel mutationand crossover strategies for global numerical optimizationrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 42 no 2 pp 482ndash499 2012
[61] Z L Gaing ldquoParticle swarm optimization to solving theeconomic dispatch considering the generator constraintsrdquo IEEETransactions on Power Systems vol 18 no 3 pp 1187ndash1195 2003
[62] X-S Yang and S Deb ldquoEngineering optimisation by Cuckoosearchrdquo International Journal of Mathematical Modelling andNumerical Optimisation vol 330 pp 43ndash51 2010
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
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AnalysisInternational Journal of
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Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Mathematical Problems in Engineering 21
0 5 10 15 20 25 30 35 40 45 50Run
2428
243
2432
2434
2436
2438
244
2442
Fuel
cost
($)
CCSAICSA
times105
Figure 9 The best fuel cost of 50 runs obtained by CCSA and ICSA for Subcase 35
0 50 100 150 200 250 300 350 400Iteration
327
328
329
33
331
332
333
334
335
336
337
Fitn
ess F
unct
ion
($)
CCSAICSA
times104
Figure 10 The best run obtained by CCSA and ICSA for Case 5
00003 0002 12291 004 138 and 059 Thesequantitative comparisons reveal that larger scale systems canlead to better reduction cost but the improvement level isnot high because total cost of compared methods tends tobe large for large scale systems Furthermore very large scalesystems with nondifferentiable objective have been normallysolved by strong methods In fact systems in Case 3 havethe same characteristic with single fuel and effects of valveloading process but Subcase 34 is a larger scale system with
40 units while Subcases 31 32 and 33 are constructed by 3units 13 units and 13 units So the improvement percentageof Subcase 34 can be up to 099 whilst that of Subcases32 and 33 is 047 and 038 respectively Subcase 35 iswith the largest system 80 units but the reduction cost is notmuch only $3412 because compared methods with ICSA areeither state-of-the-art ones or improved ones Also Subcase41 and Subcase 42 have considered single fuel and PPZ andspinning reserve constraints but Subcase 42 is larger scale
22 Mathematical Problems in Engineering
Table 15 Performance improvement summary of the proposed method
Study cases Reduction cost ($) Improvement level () Execution time (s)From To From To Slowest method Fastest method Proposed method
Figure 11 The best fuel cost of 50 runs obtained by CCSA and ICSA for Case 5
system with 90 units and Subcase 41 is only with 60 unitsThus the improvement of Subcase 41 is lower with 00003but that of Subcase 42 is 0002 Similarly systems in Cases 6have the same featurewithmulti-fossil fuel sources and effectsof valve loading process but they are respectively constructedby 80 160 and 320 units As a result the improvements ofSubcase 62 138 and Subcase 63 059 are much higherthan Subcase 61 004 However Subcase 62 with smallernumber of units but getting higher improvement is easilyunderstood because there were nine compared methods butonly three compared methods are considered for Subcase 63In general the improvement is not high it is about under onedollar several dollars tens of dollars and over one thousand
dollars per hour however the saving cost will be significantif the operation is considered to be onemonth with 720 hoursor one year with 8760 hours
Execution time comparison can be evaluated by observ-ing the fastest and the slowest compared methods in Table 15These execution times of the proposed method are approxi-mately equal to that of the fastest methods for Subcases 3233 and 34 and much shorter than other fastest comparedmethods for other cases especially for Subcases 61 and 63The fastest method for Subcase 62 is CBA [44] showing 57seconds while that of the proposed method is 1119 secondsHowever it cannot conclude that CBA is more effective thanthe proposed method because the proposed method could
find better optimal solution with less fuel cost by $2130As considering execution time of the slowest comparedmethods it can point out that ICSA is a very fast optimizationtool since the execution time of these methods is 29496 and3425 seconds for Subcases 32 and 33 116735 seconds forSubcase 34 56381 seconds for Subcase 41 94093 secondsfor Subcase 42 12797 seconds for Case 5 30941 seconds forSubcase 61 7542 seconds for Subcase 62 and 9641 secondsfor Subcase 63 while the execution time of the proposedmethod for these cases is respectively 095 146 091 1589246 84828 1119 and 171384 seconds It is clearly shown thatICSA is very fast as compared to these methods
In summary the proposed method has found approx-imately high quality solutions with several standard state-of-the-art meta-heuristic algorithms and improved versionsof them together with other old methods In addition theproposed method could improve result better than approx-imately all methods with faster execution time Comparedto other methods with the fastest convergence speed andhigh quality solutions the proposed method has been as
fast as for some cases and much faster for other cases Thecomparison with the slowest methods could show that theproposed method was extremely powerful since it was up tonearly one thousand times faster Consequently the proposedICSA approach can be one of the strongest optimization toolsfor OLD problem
6 Conclusions
This paper has proposed a good ICSA method for solvingOLD problem in which many test systems with differentobjective functions and complicated constraints from simpleto complex have been used as studied cases The proposedICSA method has been developed by performing severalmodifications on the second solution update of CCSA whichcontained several drawbacks to global convergence and fastmanner The OLD problem has covered from single fuelto multi-fossil fuels from quadratic objective function tononconvex objective function in addition to PPZ RPR andRRL constraints Many existing optimization algorithms have
been concerned in aim to compare the performance and givethe final conclusion on the proposed method There have
been six main cases with sixteen subcases The evaluationshave been made at the end of each study case Clearly theproposed ICSA approach has yielded more effective optimalsolutions with faster execution time than almost all methodsConsequently it can be concluded that the proposed methodis much more superior to CCSA and is a very promisingmethod for solving OLD problem
Appendix
See Tables 16ndash26
Nomenclature
120575119894 120582119894 120572119894 120573119894 120574119894 Fuel cost functioncoefficients of the 119894119905ℎthermal generationunit11986100 1198610119895 119861119895119894 Power loss matrixcoefficients119865119865119909 119865119865119887119890119904119905 The values of fitness ofsolution 119909 and theso-far most effectivesolution among thecurrent set ofsolutions119898119894 Number of fuels burntin the 119894119905ℎ thermalgeneration unit
119873 Number of allavailable thermalgeneration units119899119894 Number of prohibitedpower zones of the 119894119905ℎthermal generationunit119875119906119894119896 119875119897119894119896 Upper and lowerlimits of the 119894119905ℎthermal generationunit corresponding tothe kth PPZ119875119886 Probability ofreplacing controlvariables in each oldsolution119875119863 Real power demand ofall loads in system119875119894119898119886119909 119875119894119898119894119899 The highest andlowest real poweroutputs of the iththermal generationunit119875119894119895119898119886119909 119875119894119895119898119894119899 The highest andlowest real poweroutputs of the 119894119905ℎthermal generationunit corresponding tothe 119895119905ℎ fuel type119878119894119898119886119909 Maximum real powerreserve contributionof the thermalgeneration unit 1198941198781199001198971199031198861198991198891 1198781199001198971199031198861198991198892 1198781199001198971199031198861198991198893 1198781199001198971199031198861198991198894 Randomly mixedsolutions from the setof current solutions119878119900119897119909 119878119900119897119866119887119890119904119905 The old solution x andthe most effectivesolution119878119877 Real power reserverequirement of system
120575119894119895 120582119894119895 120572119894119895 120573119894119895 120574119894119895 Fuel cost functioncoefficients of the 119894119905ℎthermal generationunit corresponding tothe 119895119905ℎ fuel type1205761 1205762 1205763 1205764 1205765 1205766 Random numbersbetween 0 and 1Ω Number of generationunits considering PPZconstraint
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare no conflicts of interest
References
[1] N Sinha R Chakrabarti and P K Chattopadhyay ldquoEvolution-ary programming techniques for economic load dispatchrdquo IEEETransactions on Evolutionary Computation vol 7 no 1 pp 83ndash94 2003
[2] P Hansen ldquoA separable approximation dynamic programmingalgorithm for economic dispatch with transmission lossesrdquoYugoslav Journal of Operations Research vol 12 no 2 2002
[3] N T Thang ldquoSolving economic dispatch problem with piece-wise quadratic cost functions using lagrange multiplier theoryrdquoin Proceedings of the 3rd International Conference on ComputerTechnology and Development (ICCTD rsquo11) pp 359ndash364 ASMEPress 2011
[4] S K Mishra and S K Mishra ldquoA comparative study of solutionof economic load dispatch problem in power systems in theenvironmental perspectiverdquoProcedia Computer Science vol 48pp 96ndash100 2015
[5] A A Al-Subhi and H K Alfares ldquoEconomic load dispatchusing linear programming a comparative studyrdquo InternationalJournal of Applied Industrial Engineering vol 3 no 1 pp 16ndash362016
28 Mathematical Problems in Engineering
[6] C Zhou G Huang and J Chen ldquoPlanning of electric powersystems considering virtual power plants with dispatchableloads included an inexact two-stage stochastic linear program-ming modelrdquoMathematical Problems in Engineering vol 2018Article ID 7049329 12 pages 2018
[7] T T Nguyen N V Quynh and L Van Dai ldquoImproved fireflyalgorithm a novel method for optimal operation of thermalgenerating unitsrdquo Complexity vol 2018 Article ID 7267593 23pages 2018
[8] S S Haroon S Hassan S Amin et al ldquoMultiple fuel machinespower economic dispatch using stud differential evolutionrdquoEnergies vol 11 no 6 pp 1ndash20 2018
[9] C-T Su and C-L Chiang ldquoNonconvex power economic dis-patch by improved genetic algorithm with multiplier updatingmethodrdquo Electric Power Components and Systems vol 32 no 3pp 257ndash273 2004
[10] NNoman andH Iba ldquoDifferential evolution for economic loaddispatch problemsrdquo Electric Power Systems Research vol 78 no8 pp 1322ndash1331 2008
[11] P Somasundaram and K Kuppusamy ldquoApplication of evolu-tionary programming to security constrained economic dis-patchrdquo International Journal of Electrical Power amp EnergySystems vol 27 no 5-6 pp 343ndash351 2005
[12] T Jayabarathi K Jayaprakash andD Jeyakumar ldquoEvolutionaryprogramming technique for different kinds of economic dis-patch problemsrdquo Electric Power Systems Research vol 73 no 2pp 169ndash176 2005
[13] P Subbaraj R Rengaraj and S Salivahanan ldquoEnhancementof combined heat and power economic dispatch using selfadaptive real-coded genetic algorithmrdquo Applied Energy vol 86no 6 pp 915ndash921 2009
[14] C-L Chiang and C-T Su ldquoAdaptive-improved genetic algo-rithm for the economic dispatch of units with multiple fueloptionsrdquo Cybernetics and Systems vol 36 no 7 pp 687ndash7042005
[15] C Chiang ldquoImproved genetic algorithm for power economicdispatch of units with valve-point effects and multiple fuelsrdquoIEEE Transactions on Power Systems vol 20 no 4 pp 1690ndash1699 2005
[16] K Meng H G Wang Z Y Dong and K P Wong ldquoQuantum-inspired particle swarm optimization for valve-point economicload dispatchrdquo IEEE Transactions on Power Systems vol 25 no1 pp 215ndash222 2010
[17] P Erdogmus A Ozturk and S Duman ldquoEnvironmen-taleconomic dispatch using genetic algorithm and simulatedannealingrdquo in Proceedings of the International Conference onElectric Power and Energy Conversion Systems EPECS rsquo09 pp1ndash4 IEEE 2009
[18] L D S Coelho and V C Mariani ldquoImproved differentialevolution algorithms for handling economic dispatch opti-mization with generator constraintsrdquo Energy Conversion andManagement vol 48 no 5 pp 1631ndash1639 2007
[19] H Dakuo W Fuli and M Zhizhong ldquoA hybrid genetic algo-rithm approach based on differential evolution for economicdispatch with valve-point effectsrdquo Electrical Power and EnergySystems vol 30 pp 31ndash38 2008
[20] C-F Chang J-J Wong J-P Chiou and C-T Su ldquoRobustsearching hybrid differential evolution method for optimalreactive power planning in large-scale distribution systemsrdquoElectric Power Systems Research vol 77 no 5-6 pp 430ndash4372007
[21] M R Farooqi P Jain and K R Niazi ldquoUsing Hopfield neuralnetwork for economic dispatch of power systemsrdquo in Proceed-ings of the National Power Engineering Conference PECon rsquo03pp 5ndash10 Bangi Malaysia 2003
[22] T Yalcinoz H Altun and U Hasan ldquoConstrained economicdispatch with prohibited operating zones a Hopfield neuralnetwork approachrdquo in Proceedings of the 10th MediterraneanElectrotechnical Conference Information Technology and Elec-trotechnology for the Mediterranean Countries ProceedingsMeleCon rsquo00 (Cat No 00CH37099) pp 570ndash573 LemesosCyprus 2000
[23] D N Vo P Schegner and W Ongsakul ldquoCuckoo searchalgorithm for non-convex economic dispatchrdquo IET GenerationTransmission amp Distribution vol 7 no 6 pp 645ndash654 2013
[24] J-B Park K-S Lee J-R Shin and K Y Lee ldquoA particleswarm optimization for economic dispatch with non-smoothcost functionsrdquo IEEETransactions on Power Systems vol 20 no1 pp 34ndash42 2005
[25] T Niknam H D Mojarrad and M Nayeripour ldquoA newfuzzy adaptive particle swarm optimization for non-smootheconomic dispatchrdquo Energy vol 35 no 4 pp 1764ndash1778 2010
[26] L D S Coelho and V C Mariani ldquoAn improved harmonysearch algorithm for power economic load dispatchrdquo EnergyConversion and Management vol 50 no 10 pp 2522ndash25262009
[27] M Mahdavi M Fesanghary and E Damangir ldquoAn improvedharmony search algorithm for solving optimization problemsrdquoAppliedMathematics and Computation vol 188 no 2 pp 1567ndash1579 2007
[28] A Bhattacharya and P K Chattopadhyay ldquoSolving complexeconomic load dispatch problem using biogeography-basedoptimizationrdquo Expert Systems with Applications vol 37 no 5pp 3605ndash3615 2010
[29] D Simon ldquoBiogeography-based optimizationrdquo IEEE Transac-tions on Evolutionary Computation vol 12 no 6 pp 702ndash7132008
[30] J S Alsumait J K Sykulski and A K Al-Othman ldquoAhybrid GA-PS-SQP method to solve power system valve-pointeconomic dispatch problemsrdquo Applied Energy vol 87 no 5 pp1773ndash1781 2010
[31] M S Turgut andGKDemir ldquoQuadratic approximationndashbasedhybrid Artificial Cooperative Search algorithm for economicemission load dispatchproblemsrdquo International Transactions onElectrical Energy Systems vol 27 no 4 pp 1ndash14 2017
[32] T A A Victoire and A E Jeyakumar ldquoHybrid PSO-SQPfor economic dispatch with valve-point effectrdquo Electric PowerSystems Research vol 71 no 1 pp 51ndash59 2004
[33] D-KHe F-LWang andZ-ZMao ldquoHybrid genetic algorithmfor economic dispatch with valve-point effectrdquo Electric PowerSystems Research vol 78 no 4 pp 626ndash633 2008
[34] D C Secui ldquoA modified symbiotic organisms search algorithmfor large scale economic dispatch problem with valve-pointloading effectsrdquo Energy vol 113 pp 366ndash384 2016
[35] K BhattacharjeeA Bhattacharya and S H N Dey ldquoTeaching-learning-based optimization for different economic dispatchproblemsrdquo Scientia Iranica vol 21 no 3 pp 870ndash884 2014
[36] K Bhattacharjee A Bhattacharya and S H N Dey ldquoOppo-sitional Real Coded Chemical Reaction Optimization for dif-ferent economic dispatch problemsrdquo International Journal ofElectrical Power amp Energy Systems vol 55 pp 378ndash391 2014
Mathematical Problems in Engineering 29
[37] V Hosseinnezhad M Rafiee M Ahmadian and M T AmelildquoSpecies-basedQuantum Particle SwarmOptimization for eco-nomic load dispatchrdquo International Journal of Electrical Poweramp Energy Systems vol 63 pp 311ndash322 2014
[38] M S P Subathra S E Selvan T A A Victoire A HChristinal andU Amato ldquoA hybrid with cross-entropymethodand sequential quadratic programming to solve economic loaddispatch problemrdquo IEEE Systems Journal vol 9 no 3 pp 1031ndash1044 2015
[39] J Zhan Q H Wu C Guo and X Zhou ldquoEconomic dispatchwith non-smooth objectives part I local minimum analysisrdquoIEEE Transactions on Power Systems vol 30 no 2 pp 710ndash7212015
[40] A K Barisal andR C Prusty ldquoLarge scale economic dispatch ofpower systems using oppositional invasive weed optimizationrdquoApplied Soft Computing vol 29 pp 122ndash137 2015
[41] M Basu ldquoImproved differential evolution for economic dis-patchrdquo International Journal of Electrical Power amp EnergySystems vol 63 pp 855ndash861 2014
[42] V S Aragon S C Esquivel and C C Coello ldquoAn immunealgorithm with power redistribution for solving economicdispatch problemsrdquo Information Sciences vol 295 pp 609ndash6322015
[43] M Ghasemi M Taghizadeh S Ghavidel and A AbbasianldquoColonial competitive differential evolution an experimentalstudy for optimal economic load dispatchrdquo Applied Soft Com-puting vol 40 pp 342ndash363 2016
[44] B R Adarsh T Raghunathan T Jayabarathi and X-S YangldquoEconomic dispatch using chaotic bat algorithmrdquo Energy vol96 pp 666ndash675 2016
[45] N Ghorbani and E Babaei ldquoExchange market algorithm foreconomic load dispatchrdquo International Journal of ElectricalPower amp Energy Systems vol 75 pp 19ndash27 2016
[46] J X V Neto G Reynoso-Meza T H Ruppel V C Marianiand L D S Coelho ldquoSolving non-smooth economic dispatchby a new combination of continuous GRASP algorithm anddifferential evolutionrdquo International Journal of Electrical Poweramp Energy Systems vol 84 pp 13ndash24 2017
[47] A Kavousi-Fard and A Khosravi ldquoAn intelligent 120579-ModifiedBat Algorithm to solve the non-convex economic dispatchproblem considering practical constraintsrdquoElectrical Power andEnergy Systems vol 82 pp 189ndash196 2016
[48] M A Al-Betar M A Awadallah A T Khader and A L BolajildquoTournament-based harmony search algorithm for non-convexeconomic load dispatch problemrdquo Applied Soft Computing vol47 pp 449ndash459 2016
[49] M A Al-Betar M A Awadallah I A Doush E Alsukhni andH ALkhraisat ldquoA non-convex economic dispatchproblemwithvalve loading effect using a new modified 120573-hill climbing localsearch algorithmrdquo Arabian Journal for Science and Engineeringpp 1ndash8 2018
[50] M A Awadallah M A Al-Betar A L Bolaji E M Alsukhniand H Al-Zoubi ldquoNatural selection methods for artificial beecolony with new versions of onlooker beerdquo Soft Computing pp1ndash40 2018
[51] S H Kaboli and A K Alqallaf ldquoSolving non-convex economicload dispatch problem via artificial cooperative search algo-rithmrdquo Expert Systems with Applications vol 128 pp 14ndash272019
[52] D Singh and J Dhillon ldquoAmeliorated greywolf optimization foreconomic load dispatch problemrdquo Energy vol 169 pp 398ndash4192019
[53] X S Yang and S Deb ldquoCuckoo search via Levy flightsrdquo inProceedings of the World Congress on Nature and BiologicallyInspired Computing (NaBIC rsquo09) pp 210ndash214 India 2009
[54] S Sahoo K M Dash R C Prusty and A K Barisal ldquoCom-parative analysis of optimal load dispatch through evolutionaryalgorithmsrdquo Ain Shams Engineering Journal vol 6 pp 107ndash1202015
[55] M Basu and A Chowdhury ldquoCuckoo search algorithm foreconomic dispatchrdquo Energy vol 60 pp 99ndash108 2013
[56] A B Serapiao ldquoCuckoo search for solving economic dispatchload problemrdquo Intelligent Control and Automation vol 04 no04 pp 385ndash390 2013
[57] A H Bindu and M D Reddy ldquoEconomic load dispatch usingcuckoo search algorithmrdquo International Journal of EngineeringResearch and Applications vol 3 pp 498ndash502 2013
[58] E Afzalan and M Joorabian ldquoAn improved cuckoo searchalgorithm for power economic load dispatchrdquo InternationalTransactions on Electrical Energy Systems vol 25 no 6 pp 958ndash975 2015
[59] T T Nguyen and D N Vo ldquoThe application of one rank cuckoosearch algorithm for solving economic load dispatch problemsrdquoApplied Soft Computing vol 37 pp 763ndash773 2015
[60] S M Islam S Das S Ghosh S Roy and P N Suganthan ldquoAnadaptive differential evolution algorithm with novel mutationand crossover strategies for global numerical optimizationrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 42 no 2 pp 482ndash499 2012
[61] Z L Gaing ldquoParticle swarm optimization to solving theeconomic dispatch considering the generator constraintsrdquo IEEETransactions on Power Systems vol 18 no 3 pp 1187ndash1195 2003
[62] X-S Yang and S Deb ldquoEngineering optimisation by Cuckoosearchrdquo International Journal of Mathematical Modelling andNumerical Optimisation vol 330 pp 43ndash51 2010
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Figure 11 The best fuel cost of 50 runs obtained by CCSA and ICSA for Case 5
system with 90 units and Subcase 41 is only with 60 unitsThus the improvement of Subcase 41 is lower with 00003but that of Subcase 42 is 0002 Similarly systems in Cases 6have the same featurewithmulti-fossil fuel sources and effectsof valve loading process but they are respectively constructedby 80 160 and 320 units As a result the improvements ofSubcase 62 138 and Subcase 63 059 are much higherthan Subcase 61 004 However Subcase 62 with smallernumber of units but getting higher improvement is easilyunderstood because there were nine compared methods butonly three compared methods are considered for Subcase 63In general the improvement is not high it is about under onedollar several dollars tens of dollars and over one thousand
dollars per hour however the saving cost will be significantif the operation is considered to be onemonth with 720 hoursor one year with 8760 hours
Execution time comparison can be evaluated by observ-ing the fastest and the slowest compared methods in Table 15These execution times of the proposed method are approxi-mately equal to that of the fastest methods for Subcases 3233 and 34 and much shorter than other fastest comparedmethods for other cases especially for Subcases 61 and 63The fastest method for Subcase 62 is CBA [44] showing 57seconds while that of the proposed method is 1119 secondsHowever it cannot conclude that CBA is more effective thanthe proposed method because the proposed method could
find better optimal solution with less fuel cost by $2130As considering execution time of the slowest comparedmethods it can point out that ICSA is a very fast optimizationtool since the execution time of these methods is 29496 and3425 seconds for Subcases 32 and 33 116735 seconds forSubcase 34 56381 seconds for Subcase 41 94093 secondsfor Subcase 42 12797 seconds for Case 5 30941 seconds forSubcase 61 7542 seconds for Subcase 62 and 9641 secondsfor Subcase 63 while the execution time of the proposedmethod for these cases is respectively 095 146 091 1589246 84828 1119 and 171384 seconds It is clearly shown thatICSA is very fast as compared to these methods
In summary the proposed method has found approx-imately high quality solutions with several standard state-of-the-art meta-heuristic algorithms and improved versionsof them together with other old methods In addition theproposed method could improve result better than approx-imately all methods with faster execution time Comparedto other methods with the fastest convergence speed andhigh quality solutions the proposed method has been as
fast as for some cases and much faster for other cases Thecomparison with the slowest methods could show that theproposed method was extremely powerful since it was up tonearly one thousand times faster Consequently the proposedICSA approach can be one of the strongest optimization toolsfor OLD problem
6 Conclusions
This paper has proposed a good ICSA method for solvingOLD problem in which many test systems with differentobjective functions and complicated constraints from simpleto complex have been used as studied cases The proposedICSA method has been developed by performing severalmodifications on the second solution update of CCSA whichcontained several drawbacks to global convergence and fastmanner The OLD problem has covered from single fuelto multi-fossil fuels from quadratic objective function tononconvex objective function in addition to PPZ RPR andRRL constraints Many existing optimization algorithms have
been concerned in aim to compare the performance and givethe final conclusion on the proposed method There have
been six main cases with sixteen subcases The evaluationshave been made at the end of each study case Clearly theproposed ICSA approach has yielded more effective optimalsolutions with faster execution time than almost all methodsConsequently it can be concluded that the proposed methodis much more superior to CCSA and is a very promisingmethod for solving OLD problem
Appendix
See Tables 16ndash26
Nomenclature
120575119894 120582119894 120572119894 120573119894 120574119894 Fuel cost functioncoefficients of the 119894119905ℎthermal generationunit11986100 1198610119895 119861119895119894 Power loss matrixcoefficients119865119865119909 119865119865119887119890119904119905 The values of fitness ofsolution 119909 and theso-far most effectivesolution among thecurrent set ofsolutions119898119894 Number of fuels burntin the 119894119905ℎ thermalgeneration unit
119873 Number of allavailable thermalgeneration units119899119894 Number of prohibitedpower zones of the 119894119905ℎthermal generationunit119875119906119894119896 119875119897119894119896 Upper and lowerlimits of the 119894119905ℎthermal generationunit corresponding tothe kth PPZ119875119886 Probability ofreplacing controlvariables in each oldsolution119875119863 Real power demand ofall loads in system119875119894119898119886119909 119875119894119898119894119899 The highest andlowest real poweroutputs of the iththermal generationunit119875119894119895119898119886119909 119875119894119895119898119894119899 The highest andlowest real poweroutputs of the 119894119905ℎthermal generationunit corresponding tothe 119895119905ℎ fuel type119878119894119898119886119909 Maximum real powerreserve contributionof the thermalgeneration unit 1198941198781199001198971199031198861198991198891 1198781199001198971199031198861198991198892 1198781199001198971199031198861198991198893 1198781199001198971199031198861198991198894 Randomly mixedsolutions from the setof current solutions119878119900119897119909 119878119900119897119866119887119890119904119905 The old solution x andthe most effectivesolution119878119877 Real power reserverequirement of system
120575119894119895 120582119894119895 120572119894119895 120573119894119895 120574119894119895 Fuel cost functioncoefficients of the 119894119905ℎthermal generationunit corresponding tothe 119895119905ℎ fuel type1205761 1205762 1205763 1205764 1205765 1205766 Random numbersbetween 0 and 1Ω Number of generationunits considering PPZconstraint
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare no conflicts of interest
References
[1] N Sinha R Chakrabarti and P K Chattopadhyay ldquoEvolution-ary programming techniques for economic load dispatchrdquo IEEETransactions on Evolutionary Computation vol 7 no 1 pp 83ndash94 2003
[2] P Hansen ldquoA separable approximation dynamic programmingalgorithm for economic dispatch with transmission lossesrdquoYugoslav Journal of Operations Research vol 12 no 2 2002
[3] N T Thang ldquoSolving economic dispatch problem with piece-wise quadratic cost functions using lagrange multiplier theoryrdquoin Proceedings of the 3rd International Conference on ComputerTechnology and Development (ICCTD rsquo11) pp 359ndash364 ASMEPress 2011
[4] S K Mishra and S K Mishra ldquoA comparative study of solutionof economic load dispatch problem in power systems in theenvironmental perspectiverdquoProcedia Computer Science vol 48pp 96ndash100 2015
[5] A A Al-Subhi and H K Alfares ldquoEconomic load dispatchusing linear programming a comparative studyrdquo InternationalJournal of Applied Industrial Engineering vol 3 no 1 pp 16ndash362016
28 Mathematical Problems in Engineering
[6] C Zhou G Huang and J Chen ldquoPlanning of electric powersystems considering virtual power plants with dispatchableloads included an inexact two-stage stochastic linear program-ming modelrdquoMathematical Problems in Engineering vol 2018Article ID 7049329 12 pages 2018
[7] T T Nguyen N V Quynh and L Van Dai ldquoImproved fireflyalgorithm a novel method for optimal operation of thermalgenerating unitsrdquo Complexity vol 2018 Article ID 7267593 23pages 2018
[8] S S Haroon S Hassan S Amin et al ldquoMultiple fuel machinespower economic dispatch using stud differential evolutionrdquoEnergies vol 11 no 6 pp 1ndash20 2018
[9] C-T Su and C-L Chiang ldquoNonconvex power economic dis-patch by improved genetic algorithm with multiplier updatingmethodrdquo Electric Power Components and Systems vol 32 no 3pp 257ndash273 2004
[10] NNoman andH Iba ldquoDifferential evolution for economic loaddispatch problemsrdquo Electric Power Systems Research vol 78 no8 pp 1322ndash1331 2008
[11] P Somasundaram and K Kuppusamy ldquoApplication of evolu-tionary programming to security constrained economic dis-patchrdquo International Journal of Electrical Power amp EnergySystems vol 27 no 5-6 pp 343ndash351 2005
[12] T Jayabarathi K Jayaprakash andD Jeyakumar ldquoEvolutionaryprogramming technique for different kinds of economic dis-patch problemsrdquo Electric Power Systems Research vol 73 no 2pp 169ndash176 2005
[13] P Subbaraj R Rengaraj and S Salivahanan ldquoEnhancementof combined heat and power economic dispatch using selfadaptive real-coded genetic algorithmrdquo Applied Energy vol 86no 6 pp 915ndash921 2009
[14] C-L Chiang and C-T Su ldquoAdaptive-improved genetic algo-rithm for the economic dispatch of units with multiple fueloptionsrdquo Cybernetics and Systems vol 36 no 7 pp 687ndash7042005
[15] C Chiang ldquoImproved genetic algorithm for power economicdispatch of units with valve-point effects and multiple fuelsrdquoIEEE Transactions on Power Systems vol 20 no 4 pp 1690ndash1699 2005
[16] K Meng H G Wang Z Y Dong and K P Wong ldquoQuantum-inspired particle swarm optimization for valve-point economicload dispatchrdquo IEEE Transactions on Power Systems vol 25 no1 pp 215ndash222 2010
[17] P Erdogmus A Ozturk and S Duman ldquoEnvironmen-taleconomic dispatch using genetic algorithm and simulatedannealingrdquo in Proceedings of the International Conference onElectric Power and Energy Conversion Systems EPECS rsquo09 pp1ndash4 IEEE 2009
[18] L D S Coelho and V C Mariani ldquoImproved differentialevolution algorithms for handling economic dispatch opti-mization with generator constraintsrdquo Energy Conversion andManagement vol 48 no 5 pp 1631ndash1639 2007
[19] H Dakuo W Fuli and M Zhizhong ldquoA hybrid genetic algo-rithm approach based on differential evolution for economicdispatch with valve-point effectsrdquo Electrical Power and EnergySystems vol 30 pp 31ndash38 2008
[20] C-F Chang J-J Wong J-P Chiou and C-T Su ldquoRobustsearching hybrid differential evolution method for optimalreactive power planning in large-scale distribution systemsrdquoElectric Power Systems Research vol 77 no 5-6 pp 430ndash4372007
[21] M R Farooqi P Jain and K R Niazi ldquoUsing Hopfield neuralnetwork for economic dispatch of power systemsrdquo in Proceed-ings of the National Power Engineering Conference PECon rsquo03pp 5ndash10 Bangi Malaysia 2003
[22] T Yalcinoz H Altun and U Hasan ldquoConstrained economicdispatch with prohibited operating zones a Hopfield neuralnetwork approachrdquo in Proceedings of the 10th MediterraneanElectrotechnical Conference Information Technology and Elec-trotechnology for the Mediterranean Countries ProceedingsMeleCon rsquo00 (Cat No 00CH37099) pp 570ndash573 LemesosCyprus 2000
[23] D N Vo P Schegner and W Ongsakul ldquoCuckoo searchalgorithm for non-convex economic dispatchrdquo IET GenerationTransmission amp Distribution vol 7 no 6 pp 645ndash654 2013
[24] J-B Park K-S Lee J-R Shin and K Y Lee ldquoA particleswarm optimization for economic dispatch with non-smoothcost functionsrdquo IEEETransactions on Power Systems vol 20 no1 pp 34ndash42 2005
[25] T Niknam H D Mojarrad and M Nayeripour ldquoA newfuzzy adaptive particle swarm optimization for non-smootheconomic dispatchrdquo Energy vol 35 no 4 pp 1764ndash1778 2010
[26] L D S Coelho and V C Mariani ldquoAn improved harmonysearch algorithm for power economic load dispatchrdquo EnergyConversion and Management vol 50 no 10 pp 2522ndash25262009
[27] M Mahdavi M Fesanghary and E Damangir ldquoAn improvedharmony search algorithm for solving optimization problemsrdquoAppliedMathematics and Computation vol 188 no 2 pp 1567ndash1579 2007
[28] A Bhattacharya and P K Chattopadhyay ldquoSolving complexeconomic load dispatch problem using biogeography-basedoptimizationrdquo Expert Systems with Applications vol 37 no 5pp 3605ndash3615 2010
[29] D Simon ldquoBiogeography-based optimizationrdquo IEEE Transac-tions on Evolutionary Computation vol 12 no 6 pp 702ndash7132008
[30] J S Alsumait J K Sykulski and A K Al-Othman ldquoAhybrid GA-PS-SQP method to solve power system valve-pointeconomic dispatch problemsrdquo Applied Energy vol 87 no 5 pp1773ndash1781 2010
[31] M S Turgut andGKDemir ldquoQuadratic approximationndashbasedhybrid Artificial Cooperative Search algorithm for economicemission load dispatchproblemsrdquo International Transactions onElectrical Energy Systems vol 27 no 4 pp 1ndash14 2017
[32] T A A Victoire and A E Jeyakumar ldquoHybrid PSO-SQPfor economic dispatch with valve-point effectrdquo Electric PowerSystems Research vol 71 no 1 pp 51ndash59 2004
[33] D-KHe F-LWang andZ-ZMao ldquoHybrid genetic algorithmfor economic dispatch with valve-point effectrdquo Electric PowerSystems Research vol 78 no 4 pp 626ndash633 2008
[34] D C Secui ldquoA modified symbiotic organisms search algorithmfor large scale economic dispatch problem with valve-pointloading effectsrdquo Energy vol 113 pp 366ndash384 2016
[35] K BhattacharjeeA Bhattacharya and S H N Dey ldquoTeaching-learning-based optimization for different economic dispatchproblemsrdquo Scientia Iranica vol 21 no 3 pp 870ndash884 2014
[36] K Bhattacharjee A Bhattacharya and S H N Dey ldquoOppo-sitional Real Coded Chemical Reaction Optimization for dif-ferent economic dispatch problemsrdquo International Journal ofElectrical Power amp Energy Systems vol 55 pp 378ndash391 2014
Mathematical Problems in Engineering 29
[37] V Hosseinnezhad M Rafiee M Ahmadian and M T AmelildquoSpecies-basedQuantum Particle SwarmOptimization for eco-nomic load dispatchrdquo International Journal of Electrical Poweramp Energy Systems vol 63 pp 311ndash322 2014
[38] M S P Subathra S E Selvan T A A Victoire A HChristinal andU Amato ldquoA hybrid with cross-entropymethodand sequential quadratic programming to solve economic loaddispatch problemrdquo IEEE Systems Journal vol 9 no 3 pp 1031ndash1044 2015
[39] J Zhan Q H Wu C Guo and X Zhou ldquoEconomic dispatchwith non-smooth objectives part I local minimum analysisrdquoIEEE Transactions on Power Systems vol 30 no 2 pp 710ndash7212015
[40] A K Barisal andR C Prusty ldquoLarge scale economic dispatch ofpower systems using oppositional invasive weed optimizationrdquoApplied Soft Computing vol 29 pp 122ndash137 2015
[41] M Basu ldquoImproved differential evolution for economic dis-patchrdquo International Journal of Electrical Power amp EnergySystems vol 63 pp 855ndash861 2014
[42] V S Aragon S C Esquivel and C C Coello ldquoAn immunealgorithm with power redistribution for solving economicdispatch problemsrdquo Information Sciences vol 295 pp 609ndash6322015
[43] M Ghasemi M Taghizadeh S Ghavidel and A AbbasianldquoColonial competitive differential evolution an experimentalstudy for optimal economic load dispatchrdquo Applied Soft Com-puting vol 40 pp 342ndash363 2016
[44] B R Adarsh T Raghunathan T Jayabarathi and X-S YangldquoEconomic dispatch using chaotic bat algorithmrdquo Energy vol96 pp 666ndash675 2016
[45] N Ghorbani and E Babaei ldquoExchange market algorithm foreconomic load dispatchrdquo International Journal of ElectricalPower amp Energy Systems vol 75 pp 19ndash27 2016
[46] J X V Neto G Reynoso-Meza T H Ruppel V C Marianiand L D S Coelho ldquoSolving non-smooth economic dispatchby a new combination of continuous GRASP algorithm anddifferential evolutionrdquo International Journal of Electrical Poweramp Energy Systems vol 84 pp 13ndash24 2017
[47] A Kavousi-Fard and A Khosravi ldquoAn intelligent 120579-ModifiedBat Algorithm to solve the non-convex economic dispatchproblem considering practical constraintsrdquoElectrical Power andEnergy Systems vol 82 pp 189ndash196 2016
[48] M A Al-Betar M A Awadallah A T Khader and A L BolajildquoTournament-based harmony search algorithm for non-convexeconomic load dispatch problemrdquo Applied Soft Computing vol47 pp 449ndash459 2016
[49] M A Al-Betar M A Awadallah I A Doush E Alsukhni andH ALkhraisat ldquoA non-convex economic dispatchproblemwithvalve loading effect using a new modified 120573-hill climbing localsearch algorithmrdquo Arabian Journal for Science and Engineeringpp 1ndash8 2018
[50] M A Awadallah M A Al-Betar A L Bolaji E M Alsukhniand H Al-Zoubi ldquoNatural selection methods for artificial beecolony with new versions of onlooker beerdquo Soft Computing pp1ndash40 2018
[51] S H Kaboli and A K Alqallaf ldquoSolving non-convex economicload dispatch problem via artificial cooperative search algo-rithmrdquo Expert Systems with Applications vol 128 pp 14ndash272019
[52] D Singh and J Dhillon ldquoAmeliorated greywolf optimization foreconomic load dispatch problemrdquo Energy vol 169 pp 398ndash4192019
[53] X S Yang and S Deb ldquoCuckoo search via Levy flightsrdquo inProceedings of the World Congress on Nature and BiologicallyInspired Computing (NaBIC rsquo09) pp 210ndash214 India 2009
[54] S Sahoo K M Dash R C Prusty and A K Barisal ldquoCom-parative analysis of optimal load dispatch through evolutionaryalgorithmsrdquo Ain Shams Engineering Journal vol 6 pp 107ndash1202015
[55] M Basu and A Chowdhury ldquoCuckoo search algorithm foreconomic dispatchrdquo Energy vol 60 pp 99ndash108 2013
[56] A B Serapiao ldquoCuckoo search for solving economic dispatchload problemrdquo Intelligent Control and Automation vol 04 no04 pp 385ndash390 2013
[57] A H Bindu and M D Reddy ldquoEconomic load dispatch usingcuckoo search algorithmrdquo International Journal of EngineeringResearch and Applications vol 3 pp 498ndash502 2013
[58] E Afzalan and M Joorabian ldquoAn improved cuckoo searchalgorithm for power economic load dispatchrdquo InternationalTransactions on Electrical Energy Systems vol 25 no 6 pp 958ndash975 2015
[59] T T Nguyen and D N Vo ldquoThe application of one rank cuckoosearch algorithm for solving economic load dispatch problemsrdquoApplied Soft Computing vol 37 pp 763ndash773 2015
[60] S M Islam S Das S Ghosh S Roy and P N Suganthan ldquoAnadaptive differential evolution algorithm with novel mutationand crossover strategies for global numerical optimizationrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 42 no 2 pp 482ndash499 2012
[61] Z L Gaing ldquoParticle swarm optimization to solving theeconomic dispatch considering the generator constraintsrdquo IEEETransactions on Power Systems vol 18 no 3 pp 1187ndash1195 2003
[62] X-S Yang and S Deb ldquoEngineering optimisation by Cuckoosearchrdquo International Journal of Mathematical Modelling andNumerical Optimisation vol 330 pp 43ndash51 2010
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
find better optimal solution with less fuel cost by $2130As considering execution time of the slowest comparedmethods it can point out that ICSA is a very fast optimizationtool since the execution time of these methods is 29496 and3425 seconds for Subcases 32 and 33 116735 seconds forSubcase 34 56381 seconds for Subcase 41 94093 secondsfor Subcase 42 12797 seconds for Case 5 30941 seconds forSubcase 61 7542 seconds for Subcase 62 and 9641 secondsfor Subcase 63 while the execution time of the proposedmethod for these cases is respectively 095 146 091 1589246 84828 1119 and 171384 seconds It is clearly shown thatICSA is very fast as compared to these methods
In summary the proposed method has found approx-imately high quality solutions with several standard state-of-the-art meta-heuristic algorithms and improved versionsof them together with other old methods In addition theproposed method could improve result better than approx-imately all methods with faster execution time Comparedto other methods with the fastest convergence speed andhigh quality solutions the proposed method has been as
fast as for some cases and much faster for other cases Thecomparison with the slowest methods could show that theproposed method was extremely powerful since it was up tonearly one thousand times faster Consequently the proposedICSA approach can be one of the strongest optimization toolsfor OLD problem
6 Conclusions
This paper has proposed a good ICSA method for solvingOLD problem in which many test systems with differentobjective functions and complicated constraints from simpleto complex have been used as studied cases The proposedICSA method has been developed by performing severalmodifications on the second solution update of CCSA whichcontained several drawbacks to global convergence and fastmanner The OLD problem has covered from single fuelto multi-fossil fuels from quadratic objective function tononconvex objective function in addition to PPZ RPR andRRL constraints Many existing optimization algorithms have
been concerned in aim to compare the performance and givethe final conclusion on the proposed method There have
been six main cases with sixteen subcases The evaluationshave been made at the end of each study case Clearly theproposed ICSA approach has yielded more effective optimalsolutions with faster execution time than almost all methodsConsequently it can be concluded that the proposed methodis much more superior to CCSA and is a very promisingmethod for solving OLD problem
Appendix
See Tables 16ndash26
Nomenclature
120575119894 120582119894 120572119894 120573119894 120574119894 Fuel cost functioncoefficients of the 119894119905ℎthermal generationunit11986100 1198610119895 119861119895119894 Power loss matrixcoefficients119865119865119909 119865119865119887119890119904119905 The values of fitness ofsolution 119909 and theso-far most effectivesolution among thecurrent set ofsolutions119898119894 Number of fuels burntin the 119894119905ℎ thermalgeneration unit
119873 Number of allavailable thermalgeneration units119899119894 Number of prohibitedpower zones of the 119894119905ℎthermal generationunit119875119906119894119896 119875119897119894119896 Upper and lowerlimits of the 119894119905ℎthermal generationunit corresponding tothe kth PPZ119875119886 Probability ofreplacing controlvariables in each oldsolution119875119863 Real power demand ofall loads in system119875119894119898119886119909 119875119894119898119894119899 The highest andlowest real poweroutputs of the iththermal generationunit119875119894119895119898119886119909 119875119894119895119898119894119899 The highest andlowest real poweroutputs of the 119894119905ℎthermal generationunit corresponding tothe 119895119905ℎ fuel type119878119894119898119886119909 Maximum real powerreserve contributionof the thermalgeneration unit 1198941198781199001198971199031198861198991198891 1198781199001198971199031198861198991198892 1198781199001198971199031198861198991198893 1198781199001198971199031198861198991198894 Randomly mixedsolutions from the setof current solutions119878119900119897119909 119878119900119897119866119887119890119904119905 The old solution x andthe most effectivesolution119878119877 Real power reserverequirement of system
120575119894119895 120582119894119895 120572119894119895 120573119894119895 120574119894119895 Fuel cost functioncoefficients of the 119894119905ℎthermal generationunit corresponding tothe 119895119905ℎ fuel type1205761 1205762 1205763 1205764 1205765 1205766 Random numbersbetween 0 and 1Ω Number of generationunits considering PPZconstraint
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare no conflicts of interest
References
[1] N Sinha R Chakrabarti and P K Chattopadhyay ldquoEvolution-ary programming techniques for economic load dispatchrdquo IEEETransactions on Evolutionary Computation vol 7 no 1 pp 83ndash94 2003
[2] P Hansen ldquoA separable approximation dynamic programmingalgorithm for economic dispatch with transmission lossesrdquoYugoslav Journal of Operations Research vol 12 no 2 2002
[3] N T Thang ldquoSolving economic dispatch problem with piece-wise quadratic cost functions using lagrange multiplier theoryrdquoin Proceedings of the 3rd International Conference on ComputerTechnology and Development (ICCTD rsquo11) pp 359ndash364 ASMEPress 2011
[4] S K Mishra and S K Mishra ldquoA comparative study of solutionof economic load dispatch problem in power systems in theenvironmental perspectiverdquoProcedia Computer Science vol 48pp 96ndash100 2015
[5] A A Al-Subhi and H K Alfares ldquoEconomic load dispatchusing linear programming a comparative studyrdquo InternationalJournal of Applied Industrial Engineering vol 3 no 1 pp 16ndash362016
28 Mathematical Problems in Engineering
[6] C Zhou G Huang and J Chen ldquoPlanning of electric powersystems considering virtual power plants with dispatchableloads included an inexact two-stage stochastic linear program-ming modelrdquoMathematical Problems in Engineering vol 2018Article ID 7049329 12 pages 2018
[7] T T Nguyen N V Quynh and L Van Dai ldquoImproved fireflyalgorithm a novel method for optimal operation of thermalgenerating unitsrdquo Complexity vol 2018 Article ID 7267593 23pages 2018
[8] S S Haroon S Hassan S Amin et al ldquoMultiple fuel machinespower economic dispatch using stud differential evolutionrdquoEnergies vol 11 no 6 pp 1ndash20 2018
[9] C-T Su and C-L Chiang ldquoNonconvex power economic dis-patch by improved genetic algorithm with multiplier updatingmethodrdquo Electric Power Components and Systems vol 32 no 3pp 257ndash273 2004
[10] NNoman andH Iba ldquoDifferential evolution for economic loaddispatch problemsrdquo Electric Power Systems Research vol 78 no8 pp 1322ndash1331 2008
[11] P Somasundaram and K Kuppusamy ldquoApplication of evolu-tionary programming to security constrained economic dis-patchrdquo International Journal of Electrical Power amp EnergySystems vol 27 no 5-6 pp 343ndash351 2005
[12] T Jayabarathi K Jayaprakash andD Jeyakumar ldquoEvolutionaryprogramming technique for different kinds of economic dis-patch problemsrdquo Electric Power Systems Research vol 73 no 2pp 169ndash176 2005
[13] P Subbaraj R Rengaraj and S Salivahanan ldquoEnhancementof combined heat and power economic dispatch using selfadaptive real-coded genetic algorithmrdquo Applied Energy vol 86no 6 pp 915ndash921 2009
[14] C-L Chiang and C-T Su ldquoAdaptive-improved genetic algo-rithm for the economic dispatch of units with multiple fueloptionsrdquo Cybernetics and Systems vol 36 no 7 pp 687ndash7042005
[15] C Chiang ldquoImproved genetic algorithm for power economicdispatch of units with valve-point effects and multiple fuelsrdquoIEEE Transactions on Power Systems vol 20 no 4 pp 1690ndash1699 2005
[16] K Meng H G Wang Z Y Dong and K P Wong ldquoQuantum-inspired particle swarm optimization for valve-point economicload dispatchrdquo IEEE Transactions on Power Systems vol 25 no1 pp 215ndash222 2010
[17] P Erdogmus A Ozturk and S Duman ldquoEnvironmen-taleconomic dispatch using genetic algorithm and simulatedannealingrdquo in Proceedings of the International Conference onElectric Power and Energy Conversion Systems EPECS rsquo09 pp1ndash4 IEEE 2009
[18] L D S Coelho and V C Mariani ldquoImproved differentialevolution algorithms for handling economic dispatch opti-mization with generator constraintsrdquo Energy Conversion andManagement vol 48 no 5 pp 1631ndash1639 2007
[19] H Dakuo W Fuli and M Zhizhong ldquoA hybrid genetic algo-rithm approach based on differential evolution for economicdispatch with valve-point effectsrdquo Electrical Power and EnergySystems vol 30 pp 31ndash38 2008
[20] C-F Chang J-J Wong J-P Chiou and C-T Su ldquoRobustsearching hybrid differential evolution method for optimalreactive power planning in large-scale distribution systemsrdquoElectric Power Systems Research vol 77 no 5-6 pp 430ndash4372007
[21] M R Farooqi P Jain and K R Niazi ldquoUsing Hopfield neuralnetwork for economic dispatch of power systemsrdquo in Proceed-ings of the National Power Engineering Conference PECon rsquo03pp 5ndash10 Bangi Malaysia 2003
[22] T Yalcinoz H Altun and U Hasan ldquoConstrained economicdispatch with prohibited operating zones a Hopfield neuralnetwork approachrdquo in Proceedings of the 10th MediterraneanElectrotechnical Conference Information Technology and Elec-trotechnology for the Mediterranean Countries ProceedingsMeleCon rsquo00 (Cat No 00CH37099) pp 570ndash573 LemesosCyprus 2000
[23] D N Vo P Schegner and W Ongsakul ldquoCuckoo searchalgorithm for non-convex economic dispatchrdquo IET GenerationTransmission amp Distribution vol 7 no 6 pp 645ndash654 2013
[24] J-B Park K-S Lee J-R Shin and K Y Lee ldquoA particleswarm optimization for economic dispatch with non-smoothcost functionsrdquo IEEETransactions on Power Systems vol 20 no1 pp 34ndash42 2005
[25] T Niknam H D Mojarrad and M Nayeripour ldquoA newfuzzy adaptive particle swarm optimization for non-smootheconomic dispatchrdquo Energy vol 35 no 4 pp 1764ndash1778 2010
[26] L D S Coelho and V C Mariani ldquoAn improved harmonysearch algorithm for power economic load dispatchrdquo EnergyConversion and Management vol 50 no 10 pp 2522ndash25262009
[27] M Mahdavi M Fesanghary and E Damangir ldquoAn improvedharmony search algorithm for solving optimization problemsrdquoAppliedMathematics and Computation vol 188 no 2 pp 1567ndash1579 2007
[28] A Bhattacharya and P K Chattopadhyay ldquoSolving complexeconomic load dispatch problem using biogeography-basedoptimizationrdquo Expert Systems with Applications vol 37 no 5pp 3605ndash3615 2010
[29] D Simon ldquoBiogeography-based optimizationrdquo IEEE Transac-tions on Evolutionary Computation vol 12 no 6 pp 702ndash7132008
[30] J S Alsumait J K Sykulski and A K Al-Othman ldquoAhybrid GA-PS-SQP method to solve power system valve-pointeconomic dispatch problemsrdquo Applied Energy vol 87 no 5 pp1773ndash1781 2010
[31] M S Turgut andGKDemir ldquoQuadratic approximationndashbasedhybrid Artificial Cooperative Search algorithm for economicemission load dispatchproblemsrdquo International Transactions onElectrical Energy Systems vol 27 no 4 pp 1ndash14 2017
[32] T A A Victoire and A E Jeyakumar ldquoHybrid PSO-SQPfor economic dispatch with valve-point effectrdquo Electric PowerSystems Research vol 71 no 1 pp 51ndash59 2004
[33] D-KHe F-LWang andZ-ZMao ldquoHybrid genetic algorithmfor economic dispatch with valve-point effectrdquo Electric PowerSystems Research vol 78 no 4 pp 626ndash633 2008
[34] D C Secui ldquoA modified symbiotic organisms search algorithmfor large scale economic dispatch problem with valve-pointloading effectsrdquo Energy vol 113 pp 366ndash384 2016
[35] K BhattacharjeeA Bhattacharya and S H N Dey ldquoTeaching-learning-based optimization for different economic dispatchproblemsrdquo Scientia Iranica vol 21 no 3 pp 870ndash884 2014
[36] K Bhattacharjee A Bhattacharya and S H N Dey ldquoOppo-sitional Real Coded Chemical Reaction Optimization for dif-ferent economic dispatch problemsrdquo International Journal ofElectrical Power amp Energy Systems vol 55 pp 378ndash391 2014
Mathematical Problems in Engineering 29
[37] V Hosseinnezhad M Rafiee M Ahmadian and M T AmelildquoSpecies-basedQuantum Particle SwarmOptimization for eco-nomic load dispatchrdquo International Journal of Electrical Poweramp Energy Systems vol 63 pp 311ndash322 2014
[38] M S P Subathra S E Selvan T A A Victoire A HChristinal andU Amato ldquoA hybrid with cross-entropymethodand sequential quadratic programming to solve economic loaddispatch problemrdquo IEEE Systems Journal vol 9 no 3 pp 1031ndash1044 2015
[39] J Zhan Q H Wu C Guo and X Zhou ldquoEconomic dispatchwith non-smooth objectives part I local minimum analysisrdquoIEEE Transactions on Power Systems vol 30 no 2 pp 710ndash7212015
[40] A K Barisal andR C Prusty ldquoLarge scale economic dispatch ofpower systems using oppositional invasive weed optimizationrdquoApplied Soft Computing vol 29 pp 122ndash137 2015
[41] M Basu ldquoImproved differential evolution for economic dis-patchrdquo International Journal of Electrical Power amp EnergySystems vol 63 pp 855ndash861 2014
[42] V S Aragon S C Esquivel and C C Coello ldquoAn immunealgorithm with power redistribution for solving economicdispatch problemsrdquo Information Sciences vol 295 pp 609ndash6322015
[43] M Ghasemi M Taghizadeh S Ghavidel and A AbbasianldquoColonial competitive differential evolution an experimentalstudy for optimal economic load dispatchrdquo Applied Soft Com-puting vol 40 pp 342ndash363 2016
[44] B R Adarsh T Raghunathan T Jayabarathi and X-S YangldquoEconomic dispatch using chaotic bat algorithmrdquo Energy vol96 pp 666ndash675 2016
[45] N Ghorbani and E Babaei ldquoExchange market algorithm foreconomic load dispatchrdquo International Journal of ElectricalPower amp Energy Systems vol 75 pp 19ndash27 2016
[46] J X V Neto G Reynoso-Meza T H Ruppel V C Marianiand L D S Coelho ldquoSolving non-smooth economic dispatchby a new combination of continuous GRASP algorithm anddifferential evolutionrdquo International Journal of Electrical Poweramp Energy Systems vol 84 pp 13ndash24 2017
[47] A Kavousi-Fard and A Khosravi ldquoAn intelligent 120579-ModifiedBat Algorithm to solve the non-convex economic dispatchproblem considering practical constraintsrdquoElectrical Power andEnergy Systems vol 82 pp 189ndash196 2016
[48] M A Al-Betar M A Awadallah A T Khader and A L BolajildquoTournament-based harmony search algorithm for non-convexeconomic load dispatch problemrdquo Applied Soft Computing vol47 pp 449ndash459 2016
[49] M A Al-Betar M A Awadallah I A Doush E Alsukhni andH ALkhraisat ldquoA non-convex economic dispatchproblemwithvalve loading effect using a new modified 120573-hill climbing localsearch algorithmrdquo Arabian Journal for Science and Engineeringpp 1ndash8 2018
[50] M A Awadallah M A Al-Betar A L Bolaji E M Alsukhniand H Al-Zoubi ldquoNatural selection methods for artificial beecolony with new versions of onlooker beerdquo Soft Computing pp1ndash40 2018
[51] S H Kaboli and A K Alqallaf ldquoSolving non-convex economicload dispatch problem via artificial cooperative search algo-rithmrdquo Expert Systems with Applications vol 128 pp 14ndash272019
[52] D Singh and J Dhillon ldquoAmeliorated greywolf optimization foreconomic load dispatch problemrdquo Energy vol 169 pp 398ndash4192019
[53] X S Yang and S Deb ldquoCuckoo search via Levy flightsrdquo inProceedings of the World Congress on Nature and BiologicallyInspired Computing (NaBIC rsquo09) pp 210ndash214 India 2009
[54] S Sahoo K M Dash R C Prusty and A K Barisal ldquoCom-parative analysis of optimal load dispatch through evolutionaryalgorithmsrdquo Ain Shams Engineering Journal vol 6 pp 107ndash1202015
[55] M Basu and A Chowdhury ldquoCuckoo search algorithm foreconomic dispatchrdquo Energy vol 60 pp 99ndash108 2013
[56] A B Serapiao ldquoCuckoo search for solving economic dispatchload problemrdquo Intelligent Control and Automation vol 04 no04 pp 385ndash390 2013
[57] A H Bindu and M D Reddy ldquoEconomic load dispatch usingcuckoo search algorithmrdquo International Journal of EngineeringResearch and Applications vol 3 pp 498ndash502 2013
[58] E Afzalan and M Joorabian ldquoAn improved cuckoo searchalgorithm for power economic load dispatchrdquo InternationalTransactions on Electrical Energy Systems vol 25 no 6 pp 958ndash975 2015
[59] T T Nguyen and D N Vo ldquoThe application of one rank cuckoosearch algorithm for solving economic load dispatch problemsrdquoApplied Soft Computing vol 37 pp 763ndash773 2015
[60] S M Islam S Das S Ghosh S Roy and P N Suganthan ldquoAnadaptive differential evolution algorithm with novel mutationand crossover strategies for global numerical optimizationrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 42 no 2 pp 482ndash499 2012
[61] Z L Gaing ldquoParticle swarm optimization to solving theeconomic dispatch considering the generator constraintsrdquo IEEETransactions on Power Systems vol 18 no 3 pp 1187ndash1195 2003
[62] X-S Yang and S Deb ldquoEngineering optimisation by Cuckoosearchrdquo International Journal of Mathematical Modelling andNumerical Optimisation vol 330 pp 43ndash51 2010
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
been concerned in aim to compare the performance and givethe final conclusion on the proposed method There have
been six main cases with sixteen subcases The evaluationshave been made at the end of each study case Clearly theproposed ICSA approach has yielded more effective optimalsolutions with faster execution time than almost all methodsConsequently it can be concluded that the proposed methodis much more superior to CCSA and is a very promisingmethod for solving OLD problem
Appendix
See Tables 16ndash26
Nomenclature
120575119894 120582119894 120572119894 120573119894 120574119894 Fuel cost functioncoefficients of the 119894119905ℎthermal generationunit11986100 1198610119895 119861119895119894 Power loss matrixcoefficients119865119865119909 119865119865119887119890119904119905 The values of fitness ofsolution 119909 and theso-far most effectivesolution among thecurrent set ofsolutions119898119894 Number of fuels burntin the 119894119905ℎ thermalgeneration unit
119873 Number of allavailable thermalgeneration units119899119894 Number of prohibitedpower zones of the 119894119905ℎthermal generationunit119875119906119894119896 119875119897119894119896 Upper and lowerlimits of the 119894119905ℎthermal generationunit corresponding tothe kth PPZ119875119886 Probability ofreplacing controlvariables in each oldsolution119875119863 Real power demand ofall loads in system119875119894119898119886119909 119875119894119898119894119899 The highest andlowest real poweroutputs of the iththermal generationunit119875119894119895119898119886119909 119875119894119895119898119894119899 The highest andlowest real poweroutputs of the 119894119905ℎthermal generationunit corresponding tothe 119895119905ℎ fuel type119878119894119898119886119909 Maximum real powerreserve contributionof the thermalgeneration unit 1198941198781199001198971199031198861198991198891 1198781199001198971199031198861198991198892 1198781199001198971199031198861198991198893 1198781199001198971199031198861198991198894 Randomly mixedsolutions from the setof current solutions119878119900119897119909 119878119900119897119866119887119890119904119905 The old solution x andthe most effectivesolution119878119877 Real power reserverequirement of system
120575119894119895 120582119894119895 120572119894119895 120573119894119895 120574119894119895 Fuel cost functioncoefficients of the 119894119905ℎthermal generationunit corresponding tothe 119895119905ℎ fuel type1205761 1205762 1205763 1205764 1205765 1205766 Random numbersbetween 0 and 1Ω Number of generationunits considering PPZconstraint
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare no conflicts of interest
References
[1] N Sinha R Chakrabarti and P K Chattopadhyay ldquoEvolution-ary programming techniques for economic load dispatchrdquo IEEETransactions on Evolutionary Computation vol 7 no 1 pp 83ndash94 2003
[2] P Hansen ldquoA separable approximation dynamic programmingalgorithm for economic dispatch with transmission lossesrdquoYugoslav Journal of Operations Research vol 12 no 2 2002
[3] N T Thang ldquoSolving economic dispatch problem with piece-wise quadratic cost functions using lagrange multiplier theoryrdquoin Proceedings of the 3rd International Conference on ComputerTechnology and Development (ICCTD rsquo11) pp 359ndash364 ASMEPress 2011
[4] S K Mishra and S K Mishra ldquoA comparative study of solutionof economic load dispatch problem in power systems in theenvironmental perspectiverdquoProcedia Computer Science vol 48pp 96ndash100 2015
[5] A A Al-Subhi and H K Alfares ldquoEconomic load dispatchusing linear programming a comparative studyrdquo InternationalJournal of Applied Industrial Engineering vol 3 no 1 pp 16ndash362016
28 Mathematical Problems in Engineering
[6] C Zhou G Huang and J Chen ldquoPlanning of electric powersystems considering virtual power plants with dispatchableloads included an inexact two-stage stochastic linear program-ming modelrdquoMathematical Problems in Engineering vol 2018Article ID 7049329 12 pages 2018
[7] T T Nguyen N V Quynh and L Van Dai ldquoImproved fireflyalgorithm a novel method for optimal operation of thermalgenerating unitsrdquo Complexity vol 2018 Article ID 7267593 23pages 2018
[8] S S Haroon S Hassan S Amin et al ldquoMultiple fuel machinespower economic dispatch using stud differential evolutionrdquoEnergies vol 11 no 6 pp 1ndash20 2018
[9] C-T Su and C-L Chiang ldquoNonconvex power economic dis-patch by improved genetic algorithm with multiplier updatingmethodrdquo Electric Power Components and Systems vol 32 no 3pp 257ndash273 2004
[10] NNoman andH Iba ldquoDifferential evolution for economic loaddispatch problemsrdquo Electric Power Systems Research vol 78 no8 pp 1322ndash1331 2008
[11] P Somasundaram and K Kuppusamy ldquoApplication of evolu-tionary programming to security constrained economic dis-patchrdquo International Journal of Electrical Power amp EnergySystems vol 27 no 5-6 pp 343ndash351 2005
[12] T Jayabarathi K Jayaprakash andD Jeyakumar ldquoEvolutionaryprogramming technique for different kinds of economic dis-patch problemsrdquo Electric Power Systems Research vol 73 no 2pp 169ndash176 2005
[13] P Subbaraj R Rengaraj and S Salivahanan ldquoEnhancementof combined heat and power economic dispatch using selfadaptive real-coded genetic algorithmrdquo Applied Energy vol 86no 6 pp 915ndash921 2009
[14] C-L Chiang and C-T Su ldquoAdaptive-improved genetic algo-rithm for the economic dispatch of units with multiple fueloptionsrdquo Cybernetics and Systems vol 36 no 7 pp 687ndash7042005
[15] C Chiang ldquoImproved genetic algorithm for power economicdispatch of units with valve-point effects and multiple fuelsrdquoIEEE Transactions on Power Systems vol 20 no 4 pp 1690ndash1699 2005
[16] K Meng H G Wang Z Y Dong and K P Wong ldquoQuantum-inspired particle swarm optimization for valve-point economicload dispatchrdquo IEEE Transactions on Power Systems vol 25 no1 pp 215ndash222 2010
[17] P Erdogmus A Ozturk and S Duman ldquoEnvironmen-taleconomic dispatch using genetic algorithm and simulatedannealingrdquo in Proceedings of the International Conference onElectric Power and Energy Conversion Systems EPECS rsquo09 pp1ndash4 IEEE 2009
[18] L D S Coelho and V C Mariani ldquoImproved differentialevolution algorithms for handling economic dispatch opti-mization with generator constraintsrdquo Energy Conversion andManagement vol 48 no 5 pp 1631ndash1639 2007
[19] H Dakuo W Fuli and M Zhizhong ldquoA hybrid genetic algo-rithm approach based on differential evolution for economicdispatch with valve-point effectsrdquo Electrical Power and EnergySystems vol 30 pp 31ndash38 2008
[20] C-F Chang J-J Wong J-P Chiou and C-T Su ldquoRobustsearching hybrid differential evolution method for optimalreactive power planning in large-scale distribution systemsrdquoElectric Power Systems Research vol 77 no 5-6 pp 430ndash4372007
[21] M R Farooqi P Jain and K R Niazi ldquoUsing Hopfield neuralnetwork for economic dispatch of power systemsrdquo in Proceed-ings of the National Power Engineering Conference PECon rsquo03pp 5ndash10 Bangi Malaysia 2003
[22] T Yalcinoz H Altun and U Hasan ldquoConstrained economicdispatch with prohibited operating zones a Hopfield neuralnetwork approachrdquo in Proceedings of the 10th MediterraneanElectrotechnical Conference Information Technology and Elec-trotechnology for the Mediterranean Countries ProceedingsMeleCon rsquo00 (Cat No 00CH37099) pp 570ndash573 LemesosCyprus 2000
[23] D N Vo P Schegner and W Ongsakul ldquoCuckoo searchalgorithm for non-convex economic dispatchrdquo IET GenerationTransmission amp Distribution vol 7 no 6 pp 645ndash654 2013
[24] J-B Park K-S Lee J-R Shin and K Y Lee ldquoA particleswarm optimization for economic dispatch with non-smoothcost functionsrdquo IEEETransactions on Power Systems vol 20 no1 pp 34ndash42 2005
[25] T Niknam H D Mojarrad and M Nayeripour ldquoA newfuzzy adaptive particle swarm optimization for non-smootheconomic dispatchrdquo Energy vol 35 no 4 pp 1764ndash1778 2010
[26] L D S Coelho and V C Mariani ldquoAn improved harmonysearch algorithm for power economic load dispatchrdquo EnergyConversion and Management vol 50 no 10 pp 2522ndash25262009
[27] M Mahdavi M Fesanghary and E Damangir ldquoAn improvedharmony search algorithm for solving optimization problemsrdquoAppliedMathematics and Computation vol 188 no 2 pp 1567ndash1579 2007
[28] A Bhattacharya and P K Chattopadhyay ldquoSolving complexeconomic load dispatch problem using biogeography-basedoptimizationrdquo Expert Systems with Applications vol 37 no 5pp 3605ndash3615 2010
[29] D Simon ldquoBiogeography-based optimizationrdquo IEEE Transac-tions on Evolutionary Computation vol 12 no 6 pp 702ndash7132008
[30] J S Alsumait J K Sykulski and A K Al-Othman ldquoAhybrid GA-PS-SQP method to solve power system valve-pointeconomic dispatch problemsrdquo Applied Energy vol 87 no 5 pp1773ndash1781 2010
[31] M S Turgut andGKDemir ldquoQuadratic approximationndashbasedhybrid Artificial Cooperative Search algorithm for economicemission load dispatchproblemsrdquo International Transactions onElectrical Energy Systems vol 27 no 4 pp 1ndash14 2017
[32] T A A Victoire and A E Jeyakumar ldquoHybrid PSO-SQPfor economic dispatch with valve-point effectrdquo Electric PowerSystems Research vol 71 no 1 pp 51ndash59 2004
[33] D-KHe F-LWang andZ-ZMao ldquoHybrid genetic algorithmfor economic dispatch with valve-point effectrdquo Electric PowerSystems Research vol 78 no 4 pp 626ndash633 2008
[34] D C Secui ldquoA modified symbiotic organisms search algorithmfor large scale economic dispatch problem with valve-pointloading effectsrdquo Energy vol 113 pp 366ndash384 2016
[35] K BhattacharjeeA Bhattacharya and S H N Dey ldquoTeaching-learning-based optimization for different economic dispatchproblemsrdquo Scientia Iranica vol 21 no 3 pp 870ndash884 2014
[36] K Bhattacharjee A Bhattacharya and S H N Dey ldquoOppo-sitional Real Coded Chemical Reaction Optimization for dif-ferent economic dispatch problemsrdquo International Journal ofElectrical Power amp Energy Systems vol 55 pp 378ndash391 2014
Mathematical Problems in Engineering 29
[37] V Hosseinnezhad M Rafiee M Ahmadian and M T AmelildquoSpecies-basedQuantum Particle SwarmOptimization for eco-nomic load dispatchrdquo International Journal of Electrical Poweramp Energy Systems vol 63 pp 311ndash322 2014
[38] M S P Subathra S E Selvan T A A Victoire A HChristinal andU Amato ldquoA hybrid with cross-entropymethodand sequential quadratic programming to solve economic loaddispatch problemrdquo IEEE Systems Journal vol 9 no 3 pp 1031ndash1044 2015
[39] J Zhan Q H Wu C Guo and X Zhou ldquoEconomic dispatchwith non-smooth objectives part I local minimum analysisrdquoIEEE Transactions on Power Systems vol 30 no 2 pp 710ndash7212015
[40] A K Barisal andR C Prusty ldquoLarge scale economic dispatch ofpower systems using oppositional invasive weed optimizationrdquoApplied Soft Computing vol 29 pp 122ndash137 2015
[41] M Basu ldquoImproved differential evolution for economic dis-patchrdquo International Journal of Electrical Power amp EnergySystems vol 63 pp 855ndash861 2014
[42] V S Aragon S C Esquivel and C C Coello ldquoAn immunealgorithm with power redistribution for solving economicdispatch problemsrdquo Information Sciences vol 295 pp 609ndash6322015
[43] M Ghasemi M Taghizadeh S Ghavidel and A AbbasianldquoColonial competitive differential evolution an experimentalstudy for optimal economic load dispatchrdquo Applied Soft Com-puting vol 40 pp 342ndash363 2016
[44] B R Adarsh T Raghunathan T Jayabarathi and X-S YangldquoEconomic dispatch using chaotic bat algorithmrdquo Energy vol96 pp 666ndash675 2016
[45] N Ghorbani and E Babaei ldquoExchange market algorithm foreconomic load dispatchrdquo International Journal of ElectricalPower amp Energy Systems vol 75 pp 19ndash27 2016
[46] J X V Neto G Reynoso-Meza T H Ruppel V C Marianiand L D S Coelho ldquoSolving non-smooth economic dispatchby a new combination of continuous GRASP algorithm anddifferential evolutionrdquo International Journal of Electrical Poweramp Energy Systems vol 84 pp 13ndash24 2017
[47] A Kavousi-Fard and A Khosravi ldquoAn intelligent 120579-ModifiedBat Algorithm to solve the non-convex economic dispatchproblem considering practical constraintsrdquoElectrical Power andEnergy Systems vol 82 pp 189ndash196 2016
[48] M A Al-Betar M A Awadallah A T Khader and A L BolajildquoTournament-based harmony search algorithm for non-convexeconomic load dispatch problemrdquo Applied Soft Computing vol47 pp 449ndash459 2016
[49] M A Al-Betar M A Awadallah I A Doush E Alsukhni andH ALkhraisat ldquoA non-convex economic dispatchproblemwithvalve loading effect using a new modified 120573-hill climbing localsearch algorithmrdquo Arabian Journal for Science and Engineeringpp 1ndash8 2018
[50] M A Awadallah M A Al-Betar A L Bolaji E M Alsukhniand H Al-Zoubi ldquoNatural selection methods for artificial beecolony with new versions of onlooker beerdquo Soft Computing pp1ndash40 2018
[51] S H Kaboli and A K Alqallaf ldquoSolving non-convex economicload dispatch problem via artificial cooperative search algo-rithmrdquo Expert Systems with Applications vol 128 pp 14ndash272019
[52] D Singh and J Dhillon ldquoAmeliorated greywolf optimization foreconomic load dispatch problemrdquo Energy vol 169 pp 398ndash4192019
[53] X S Yang and S Deb ldquoCuckoo search via Levy flightsrdquo inProceedings of the World Congress on Nature and BiologicallyInspired Computing (NaBIC rsquo09) pp 210ndash214 India 2009
[54] S Sahoo K M Dash R C Prusty and A K Barisal ldquoCom-parative analysis of optimal load dispatch through evolutionaryalgorithmsrdquo Ain Shams Engineering Journal vol 6 pp 107ndash1202015
[55] M Basu and A Chowdhury ldquoCuckoo search algorithm foreconomic dispatchrdquo Energy vol 60 pp 99ndash108 2013
[56] A B Serapiao ldquoCuckoo search for solving economic dispatchload problemrdquo Intelligent Control and Automation vol 04 no04 pp 385ndash390 2013
[57] A H Bindu and M D Reddy ldquoEconomic load dispatch usingcuckoo search algorithmrdquo International Journal of EngineeringResearch and Applications vol 3 pp 498ndash502 2013
[58] E Afzalan and M Joorabian ldquoAn improved cuckoo searchalgorithm for power economic load dispatchrdquo InternationalTransactions on Electrical Energy Systems vol 25 no 6 pp 958ndash975 2015
[59] T T Nguyen and D N Vo ldquoThe application of one rank cuckoosearch algorithm for solving economic load dispatch problemsrdquoApplied Soft Computing vol 37 pp 763ndash773 2015
[60] S M Islam S Das S Ghosh S Roy and P N Suganthan ldquoAnadaptive differential evolution algorithm with novel mutationand crossover strategies for global numerical optimizationrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 42 no 2 pp 482ndash499 2012
[61] Z L Gaing ldquoParticle swarm optimization to solving theeconomic dispatch considering the generator constraintsrdquo IEEETransactions on Power Systems vol 18 no 3 pp 1187ndash1195 2003
[62] X-S Yang and S Deb ldquoEngineering optimisation by Cuckoosearchrdquo International Journal of Mathematical Modelling andNumerical Optimisation vol 330 pp 43ndash51 2010
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
119873 Number of allavailable thermalgeneration units119899119894 Number of prohibitedpower zones of the 119894119905ℎthermal generationunit119875119906119894119896 119875119897119894119896 Upper and lowerlimits of the 119894119905ℎthermal generationunit corresponding tothe kth PPZ119875119886 Probability ofreplacing controlvariables in each oldsolution119875119863 Real power demand ofall loads in system119875119894119898119886119909 119875119894119898119894119899 The highest andlowest real poweroutputs of the iththermal generationunit119875119894119895119898119886119909 119875119894119895119898119894119899 The highest andlowest real poweroutputs of the 119894119905ℎthermal generationunit corresponding tothe 119895119905ℎ fuel type119878119894119898119886119909 Maximum real powerreserve contributionof the thermalgeneration unit 1198941198781199001198971199031198861198991198891 1198781199001198971199031198861198991198892 1198781199001198971199031198861198991198893 1198781199001198971199031198861198991198894 Randomly mixedsolutions from the setof current solutions119878119900119897119909 119878119900119897119866119887119890119904119905 The old solution x andthe most effectivesolution119878119877 Real power reserverequirement of system
120575119894119895 120582119894119895 120572119894119895 120573119894119895 120574119894119895 Fuel cost functioncoefficients of the 119894119905ℎthermal generationunit corresponding tothe 119895119905ℎ fuel type1205761 1205762 1205763 1205764 1205765 1205766 Random numbersbetween 0 and 1Ω Number of generationunits considering PPZconstraint
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare no conflicts of interest
References
[1] N Sinha R Chakrabarti and P K Chattopadhyay ldquoEvolution-ary programming techniques for economic load dispatchrdquo IEEETransactions on Evolutionary Computation vol 7 no 1 pp 83ndash94 2003
[2] P Hansen ldquoA separable approximation dynamic programmingalgorithm for economic dispatch with transmission lossesrdquoYugoslav Journal of Operations Research vol 12 no 2 2002
[3] N T Thang ldquoSolving economic dispatch problem with piece-wise quadratic cost functions using lagrange multiplier theoryrdquoin Proceedings of the 3rd International Conference on ComputerTechnology and Development (ICCTD rsquo11) pp 359ndash364 ASMEPress 2011
[4] S K Mishra and S K Mishra ldquoA comparative study of solutionof economic load dispatch problem in power systems in theenvironmental perspectiverdquoProcedia Computer Science vol 48pp 96ndash100 2015
[5] A A Al-Subhi and H K Alfares ldquoEconomic load dispatchusing linear programming a comparative studyrdquo InternationalJournal of Applied Industrial Engineering vol 3 no 1 pp 16ndash362016
28 Mathematical Problems in Engineering
[6] C Zhou G Huang and J Chen ldquoPlanning of electric powersystems considering virtual power plants with dispatchableloads included an inexact two-stage stochastic linear program-ming modelrdquoMathematical Problems in Engineering vol 2018Article ID 7049329 12 pages 2018
[7] T T Nguyen N V Quynh and L Van Dai ldquoImproved fireflyalgorithm a novel method for optimal operation of thermalgenerating unitsrdquo Complexity vol 2018 Article ID 7267593 23pages 2018
[8] S S Haroon S Hassan S Amin et al ldquoMultiple fuel machinespower economic dispatch using stud differential evolutionrdquoEnergies vol 11 no 6 pp 1ndash20 2018
[9] C-T Su and C-L Chiang ldquoNonconvex power economic dis-patch by improved genetic algorithm with multiplier updatingmethodrdquo Electric Power Components and Systems vol 32 no 3pp 257ndash273 2004
[10] NNoman andH Iba ldquoDifferential evolution for economic loaddispatch problemsrdquo Electric Power Systems Research vol 78 no8 pp 1322ndash1331 2008
[11] P Somasundaram and K Kuppusamy ldquoApplication of evolu-tionary programming to security constrained economic dis-patchrdquo International Journal of Electrical Power amp EnergySystems vol 27 no 5-6 pp 343ndash351 2005
[12] T Jayabarathi K Jayaprakash andD Jeyakumar ldquoEvolutionaryprogramming technique for different kinds of economic dis-patch problemsrdquo Electric Power Systems Research vol 73 no 2pp 169ndash176 2005
[13] P Subbaraj R Rengaraj and S Salivahanan ldquoEnhancementof combined heat and power economic dispatch using selfadaptive real-coded genetic algorithmrdquo Applied Energy vol 86no 6 pp 915ndash921 2009
[14] C-L Chiang and C-T Su ldquoAdaptive-improved genetic algo-rithm for the economic dispatch of units with multiple fueloptionsrdquo Cybernetics and Systems vol 36 no 7 pp 687ndash7042005
[15] C Chiang ldquoImproved genetic algorithm for power economicdispatch of units with valve-point effects and multiple fuelsrdquoIEEE Transactions on Power Systems vol 20 no 4 pp 1690ndash1699 2005
[16] K Meng H G Wang Z Y Dong and K P Wong ldquoQuantum-inspired particle swarm optimization for valve-point economicload dispatchrdquo IEEE Transactions on Power Systems vol 25 no1 pp 215ndash222 2010
[17] P Erdogmus A Ozturk and S Duman ldquoEnvironmen-taleconomic dispatch using genetic algorithm and simulatedannealingrdquo in Proceedings of the International Conference onElectric Power and Energy Conversion Systems EPECS rsquo09 pp1ndash4 IEEE 2009
[18] L D S Coelho and V C Mariani ldquoImproved differentialevolution algorithms for handling economic dispatch opti-mization with generator constraintsrdquo Energy Conversion andManagement vol 48 no 5 pp 1631ndash1639 2007
[19] H Dakuo W Fuli and M Zhizhong ldquoA hybrid genetic algo-rithm approach based on differential evolution for economicdispatch with valve-point effectsrdquo Electrical Power and EnergySystems vol 30 pp 31ndash38 2008
[20] C-F Chang J-J Wong J-P Chiou and C-T Su ldquoRobustsearching hybrid differential evolution method for optimalreactive power planning in large-scale distribution systemsrdquoElectric Power Systems Research vol 77 no 5-6 pp 430ndash4372007
[21] M R Farooqi P Jain and K R Niazi ldquoUsing Hopfield neuralnetwork for economic dispatch of power systemsrdquo in Proceed-ings of the National Power Engineering Conference PECon rsquo03pp 5ndash10 Bangi Malaysia 2003
[22] T Yalcinoz H Altun and U Hasan ldquoConstrained economicdispatch with prohibited operating zones a Hopfield neuralnetwork approachrdquo in Proceedings of the 10th MediterraneanElectrotechnical Conference Information Technology and Elec-trotechnology for the Mediterranean Countries ProceedingsMeleCon rsquo00 (Cat No 00CH37099) pp 570ndash573 LemesosCyprus 2000
[23] D N Vo P Schegner and W Ongsakul ldquoCuckoo searchalgorithm for non-convex economic dispatchrdquo IET GenerationTransmission amp Distribution vol 7 no 6 pp 645ndash654 2013
[24] J-B Park K-S Lee J-R Shin and K Y Lee ldquoA particleswarm optimization for economic dispatch with non-smoothcost functionsrdquo IEEETransactions on Power Systems vol 20 no1 pp 34ndash42 2005
[25] T Niknam H D Mojarrad and M Nayeripour ldquoA newfuzzy adaptive particle swarm optimization for non-smootheconomic dispatchrdquo Energy vol 35 no 4 pp 1764ndash1778 2010
[26] L D S Coelho and V C Mariani ldquoAn improved harmonysearch algorithm for power economic load dispatchrdquo EnergyConversion and Management vol 50 no 10 pp 2522ndash25262009
[27] M Mahdavi M Fesanghary and E Damangir ldquoAn improvedharmony search algorithm for solving optimization problemsrdquoAppliedMathematics and Computation vol 188 no 2 pp 1567ndash1579 2007
[28] A Bhattacharya and P K Chattopadhyay ldquoSolving complexeconomic load dispatch problem using biogeography-basedoptimizationrdquo Expert Systems with Applications vol 37 no 5pp 3605ndash3615 2010
[29] D Simon ldquoBiogeography-based optimizationrdquo IEEE Transac-tions on Evolutionary Computation vol 12 no 6 pp 702ndash7132008
[30] J S Alsumait J K Sykulski and A K Al-Othman ldquoAhybrid GA-PS-SQP method to solve power system valve-pointeconomic dispatch problemsrdquo Applied Energy vol 87 no 5 pp1773ndash1781 2010
[31] M S Turgut andGKDemir ldquoQuadratic approximationndashbasedhybrid Artificial Cooperative Search algorithm for economicemission load dispatchproblemsrdquo International Transactions onElectrical Energy Systems vol 27 no 4 pp 1ndash14 2017
[32] T A A Victoire and A E Jeyakumar ldquoHybrid PSO-SQPfor economic dispatch with valve-point effectrdquo Electric PowerSystems Research vol 71 no 1 pp 51ndash59 2004
[33] D-KHe F-LWang andZ-ZMao ldquoHybrid genetic algorithmfor economic dispatch with valve-point effectrdquo Electric PowerSystems Research vol 78 no 4 pp 626ndash633 2008
[34] D C Secui ldquoA modified symbiotic organisms search algorithmfor large scale economic dispatch problem with valve-pointloading effectsrdquo Energy vol 113 pp 366ndash384 2016
[35] K BhattacharjeeA Bhattacharya and S H N Dey ldquoTeaching-learning-based optimization for different economic dispatchproblemsrdquo Scientia Iranica vol 21 no 3 pp 870ndash884 2014
[36] K Bhattacharjee A Bhattacharya and S H N Dey ldquoOppo-sitional Real Coded Chemical Reaction Optimization for dif-ferent economic dispatch problemsrdquo International Journal ofElectrical Power amp Energy Systems vol 55 pp 378ndash391 2014
Mathematical Problems in Engineering 29
[37] V Hosseinnezhad M Rafiee M Ahmadian and M T AmelildquoSpecies-basedQuantum Particle SwarmOptimization for eco-nomic load dispatchrdquo International Journal of Electrical Poweramp Energy Systems vol 63 pp 311ndash322 2014
[38] M S P Subathra S E Selvan T A A Victoire A HChristinal andU Amato ldquoA hybrid with cross-entropymethodand sequential quadratic programming to solve economic loaddispatch problemrdquo IEEE Systems Journal vol 9 no 3 pp 1031ndash1044 2015
[39] J Zhan Q H Wu C Guo and X Zhou ldquoEconomic dispatchwith non-smooth objectives part I local minimum analysisrdquoIEEE Transactions on Power Systems vol 30 no 2 pp 710ndash7212015
[40] A K Barisal andR C Prusty ldquoLarge scale economic dispatch ofpower systems using oppositional invasive weed optimizationrdquoApplied Soft Computing vol 29 pp 122ndash137 2015
[41] M Basu ldquoImproved differential evolution for economic dis-patchrdquo International Journal of Electrical Power amp EnergySystems vol 63 pp 855ndash861 2014
[42] V S Aragon S C Esquivel and C C Coello ldquoAn immunealgorithm with power redistribution for solving economicdispatch problemsrdquo Information Sciences vol 295 pp 609ndash6322015
[43] M Ghasemi M Taghizadeh S Ghavidel and A AbbasianldquoColonial competitive differential evolution an experimentalstudy for optimal economic load dispatchrdquo Applied Soft Com-puting vol 40 pp 342ndash363 2016
[44] B R Adarsh T Raghunathan T Jayabarathi and X-S YangldquoEconomic dispatch using chaotic bat algorithmrdquo Energy vol96 pp 666ndash675 2016
[45] N Ghorbani and E Babaei ldquoExchange market algorithm foreconomic load dispatchrdquo International Journal of ElectricalPower amp Energy Systems vol 75 pp 19ndash27 2016
[46] J X V Neto G Reynoso-Meza T H Ruppel V C Marianiand L D S Coelho ldquoSolving non-smooth economic dispatchby a new combination of continuous GRASP algorithm anddifferential evolutionrdquo International Journal of Electrical Poweramp Energy Systems vol 84 pp 13ndash24 2017
[47] A Kavousi-Fard and A Khosravi ldquoAn intelligent 120579-ModifiedBat Algorithm to solve the non-convex economic dispatchproblem considering practical constraintsrdquoElectrical Power andEnergy Systems vol 82 pp 189ndash196 2016
[48] M A Al-Betar M A Awadallah A T Khader and A L BolajildquoTournament-based harmony search algorithm for non-convexeconomic load dispatch problemrdquo Applied Soft Computing vol47 pp 449ndash459 2016
[49] M A Al-Betar M A Awadallah I A Doush E Alsukhni andH ALkhraisat ldquoA non-convex economic dispatchproblemwithvalve loading effect using a new modified 120573-hill climbing localsearch algorithmrdquo Arabian Journal for Science and Engineeringpp 1ndash8 2018
[50] M A Awadallah M A Al-Betar A L Bolaji E M Alsukhniand H Al-Zoubi ldquoNatural selection methods for artificial beecolony with new versions of onlooker beerdquo Soft Computing pp1ndash40 2018
[51] S H Kaboli and A K Alqallaf ldquoSolving non-convex economicload dispatch problem via artificial cooperative search algo-rithmrdquo Expert Systems with Applications vol 128 pp 14ndash272019
[52] D Singh and J Dhillon ldquoAmeliorated greywolf optimization foreconomic load dispatch problemrdquo Energy vol 169 pp 398ndash4192019
[53] X S Yang and S Deb ldquoCuckoo search via Levy flightsrdquo inProceedings of the World Congress on Nature and BiologicallyInspired Computing (NaBIC rsquo09) pp 210ndash214 India 2009
[54] S Sahoo K M Dash R C Prusty and A K Barisal ldquoCom-parative analysis of optimal load dispatch through evolutionaryalgorithmsrdquo Ain Shams Engineering Journal vol 6 pp 107ndash1202015
[55] M Basu and A Chowdhury ldquoCuckoo search algorithm foreconomic dispatchrdquo Energy vol 60 pp 99ndash108 2013
[56] A B Serapiao ldquoCuckoo search for solving economic dispatchload problemrdquo Intelligent Control and Automation vol 04 no04 pp 385ndash390 2013
[57] A H Bindu and M D Reddy ldquoEconomic load dispatch usingcuckoo search algorithmrdquo International Journal of EngineeringResearch and Applications vol 3 pp 498ndash502 2013
[58] E Afzalan and M Joorabian ldquoAn improved cuckoo searchalgorithm for power economic load dispatchrdquo InternationalTransactions on Electrical Energy Systems vol 25 no 6 pp 958ndash975 2015
[59] T T Nguyen and D N Vo ldquoThe application of one rank cuckoosearch algorithm for solving economic load dispatch problemsrdquoApplied Soft Computing vol 37 pp 763ndash773 2015
[60] S M Islam S Das S Ghosh S Roy and P N Suganthan ldquoAnadaptive differential evolution algorithm with novel mutationand crossover strategies for global numerical optimizationrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 42 no 2 pp 482ndash499 2012
[61] Z L Gaing ldquoParticle swarm optimization to solving theeconomic dispatch considering the generator constraintsrdquo IEEETransactions on Power Systems vol 18 no 3 pp 1187ndash1195 2003
[62] X-S Yang and S Deb ldquoEngineering optimisation by Cuckoosearchrdquo International Journal of Mathematical Modelling andNumerical Optimisation vol 330 pp 43ndash51 2010
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
119873 Number of allavailable thermalgeneration units119899119894 Number of prohibitedpower zones of the 119894119905ℎthermal generationunit119875119906119894119896 119875119897119894119896 Upper and lowerlimits of the 119894119905ℎthermal generationunit corresponding tothe kth PPZ119875119886 Probability ofreplacing controlvariables in each oldsolution119875119863 Real power demand ofall loads in system119875119894119898119886119909 119875119894119898119894119899 The highest andlowest real poweroutputs of the iththermal generationunit119875119894119895119898119886119909 119875119894119895119898119894119899 The highest andlowest real poweroutputs of the 119894119905ℎthermal generationunit corresponding tothe 119895119905ℎ fuel type119878119894119898119886119909 Maximum real powerreserve contributionof the thermalgeneration unit 1198941198781199001198971199031198861198991198891 1198781199001198971199031198861198991198892 1198781199001198971199031198861198991198893 1198781199001198971199031198861198991198894 Randomly mixedsolutions from the setof current solutions119878119900119897119909 119878119900119897119866119887119890119904119905 The old solution x andthe most effectivesolution119878119877 Real power reserverequirement of system
120575119894119895 120582119894119895 120572119894119895 120573119894119895 120574119894119895 Fuel cost functioncoefficients of the 119894119905ℎthermal generationunit corresponding tothe 119895119905ℎ fuel type1205761 1205762 1205763 1205764 1205765 1205766 Random numbersbetween 0 and 1Ω Number of generationunits considering PPZconstraint
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare no conflicts of interest
References
[1] N Sinha R Chakrabarti and P K Chattopadhyay ldquoEvolution-ary programming techniques for economic load dispatchrdquo IEEETransactions on Evolutionary Computation vol 7 no 1 pp 83ndash94 2003
[2] P Hansen ldquoA separable approximation dynamic programmingalgorithm for economic dispatch with transmission lossesrdquoYugoslav Journal of Operations Research vol 12 no 2 2002
[3] N T Thang ldquoSolving economic dispatch problem with piece-wise quadratic cost functions using lagrange multiplier theoryrdquoin Proceedings of the 3rd International Conference on ComputerTechnology and Development (ICCTD rsquo11) pp 359ndash364 ASMEPress 2011
[4] S K Mishra and S K Mishra ldquoA comparative study of solutionof economic load dispatch problem in power systems in theenvironmental perspectiverdquoProcedia Computer Science vol 48pp 96ndash100 2015
[5] A A Al-Subhi and H K Alfares ldquoEconomic load dispatchusing linear programming a comparative studyrdquo InternationalJournal of Applied Industrial Engineering vol 3 no 1 pp 16ndash362016
28 Mathematical Problems in Engineering
[6] C Zhou G Huang and J Chen ldquoPlanning of electric powersystems considering virtual power plants with dispatchableloads included an inexact two-stage stochastic linear program-ming modelrdquoMathematical Problems in Engineering vol 2018Article ID 7049329 12 pages 2018
[7] T T Nguyen N V Quynh and L Van Dai ldquoImproved fireflyalgorithm a novel method for optimal operation of thermalgenerating unitsrdquo Complexity vol 2018 Article ID 7267593 23pages 2018
[8] S S Haroon S Hassan S Amin et al ldquoMultiple fuel machinespower economic dispatch using stud differential evolutionrdquoEnergies vol 11 no 6 pp 1ndash20 2018
[9] C-T Su and C-L Chiang ldquoNonconvex power economic dis-patch by improved genetic algorithm with multiplier updatingmethodrdquo Electric Power Components and Systems vol 32 no 3pp 257ndash273 2004
[10] NNoman andH Iba ldquoDifferential evolution for economic loaddispatch problemsrdquo Electric Power Systems Research vol 78 no8 pp 1322ndash1331 2008
[11] P Somasundaram and K Kuppusamy ldquoApplication of evolu-tionary programming to security constrained economic dis-patchrdquo International Journal of Electrical Power amp EnergySystems vol 27 no 5-6 pp 343ndash351 2005
[12] T Jayabarathi K Jayaprakash andD Jeyakumar ldquoEvolutionaryprogramming technique for different kinds of economic dis-patch problemsrdquo Electric Power Systems Research vol 73 no 2pp 169ndash176 2005
[13] P Subbaraj R Rengaraj and S Salivahanan ldquoEnhancementof combined heat and power economic dispatch using selfadaptive real-coded genetic algorithmrdquo Applied Energy vol 86no 6 pp 915ndash921 2009
[14] C-L Chiang and C-T Su ldquoAdaptive-improved genetic algo-rithm for the economic dispatch of units with multiple fueloptionsrdquo Cybernetics and Systems vol 36 no 7 pp 687ndash7042005
[15] C Chiang ldquoImproved genetic algorithm for power economicdispatch of units with valve-point effects and multiple fuelsrdquoIEEE Transactions on Power Systems vol 20 no 4 pp 1690ndash1699 2005
[16] K Meng H G Wang Z Y Dong and K P Wong ldquoQuantum-inspired particle swarm optimization for valve-point economicload dispatchrdquo IEEE Transactions on Power Systems vol 25 no1 pp 215ndash222 2010
[17] P Erdogmus A Ozturk and S Duman ldquoEnvironmen-taleconomic dispatch using genetic algorithm and simulatedannealingrdquo in Proceedings of the International Conference onElectric Power and Energy Conversion Systems EPECS rsquo09 pp1ndash4 IEEE 2009
[18] L D S Coelho and V C Mariani ldquoImproved differentialevolution algorithms for handling economic dispatch opti-mization with generator constraintsrdquo Energy Conversion andManagement vol 48 no 5 pp 1631ndash1639 2007
[19] H Dakuo W Fuli and M Zhizhong ldquoA hybrid genetic algo-rithm approach based on differential evolution for economicdispatch with valve-point effectsrdquo Electrical Power and EnergySystems vol 30 pp 31ndash38 2008
[20] C-F Chang J-J Wong J-P Chiou and C-T Su ldquoRobustsearching hybrid differential evolution method for optimalreactive power planning in large-scale distribution systemsrdquoElectric Power Systems Research vol 77 no 5-6 pp 430ndash4372007
[21] M R Farooqi P Jain and K R Niazi ldquoUsing Hopfield neuralnetwork for economic dispatch of power systemsrdquo in Proceed-ings of the National Power Engineering Conference PECon rsquo03pp 5ndash10 Bangi Malaysia 2003
[22] T Yalcinoz H Altun and U Hasan ldquoConstrained economicdispatch with prohibited operating zones a Hopfield neuralnetwork approachrdquo in Proceedings of the 10th MediterraneanElectrotechnical Conference Information Technology and Elec-trotechnology for the Mediterranean Countries ProceedingsMeleCon rsquo00 (Cat No 00CH37099) pp 570ndash573 LemesosCyprus 2000
[23] D N Vo P Schegner and W Ongsakul ldquoCuckoo searchalgorithm for non-convex economic dispatchrdquo IET GenerationTransmission amp Distribution vol 7 no 6 pp 645ndash654 2013
[24] J-B Park K-S Lee J-R Shin and K Y Lee ldquoA particleswarm optimization for economic dispatch with non-smoothcost functionsrdquo IEEETransactions on Power Systems vol 20 no1 pp 34ndash42 2005
[25] T Niknam H D Mojarrad and M Nayeripour ldquoA newfuzzy adaptive particle swarm optimization for non-smootheconomic dispatchrdquo Energy vol 35 no 4 pp 1764ndash1778 2010
[26] L D S Coelho and V C Mariani ldquoAn improved harmonysearch algorithm for power economic load dispatchrdquo EnergyConversion and Management vol 50 no 10 pp 2522ndash25262009
[27] M Mahdavi M Fesanghary and E Damangir ldquoAn improvedharmony search algorithm for solving optimization problemsrdquoAppliedMathematics and Computation vol 188 no 2 pp 1567ndash1579 2007
[28] A Bhattacharya and P K Chattopadhyay ldquoSolving complexeconomic load dispatch problem using biogeography-basedoptimizationrdquo Expert Systems with Applications vol 37 no 5pp 3605ndash3615 2010
[29] D Simon ldquoBiogeography-based optimizationrdquo IEEE Transac-tions on Evolutionary Computation vol 12 no 6 pp 702ndash7132008
[30] J S Alsumait J K Sykulski and A K Al-Othman ldquoAhybrid GA-PS-SQP method to solve power system valve-pointeconomic dispatch problemsrdquo Applied Energy vol 87 no 5 pp1773ndash1781 2010
[31] M S Turgut andGKDemir ldquoQuadratic approximationndashbasedhybrid Artificial Cooperative Search algorithm for economicemission load dispatchproblemsrdquo International Transactions onElectrical Energy Systems vol 27 no 4 pp 1ndash14 2017
[32] T A A Victoire and A E Jeyakumar ldquoHybrid PSO-SQPfor economic dispatch with valve-point effectrdquo Electric PowerSystems Research vol 71 no 1 pp 51ndash59 2004
[33] D-KHe F-LWang andZ-ZMao ldquoHybrid genetic algorithmfor economic dispatch with valve-point effectrdquo Electric PowerSystems Research vol 78 no 4 pp 626ndash633 2008
[34] D C Secui ldquoA modified symbiotic organisms search algorithmfor large scale economic dispatch problem with valve-pointloading effectsrdquo Energy vol 113 pp 366ndash384 2016
[35] K BhattacharjeeA Bhattacharya and S H N Dey ldquoTeaching-learning-based optimization for different economic dispatchproblemsrdquo Scientia Iranica vol 21 no 3 pp 870ndash884 2014
[36] K Bhattacharjee A Bhattacharya and S H N Dey ldquoOppo-sitional Real Coded Chemical Reaction Optimization for dif-ferent economic dispatch problemsrdquo International Journal ofElectrical Power amp Energy Systems vol 55 pp 378ndash391 2014
Mathematical Problems in Engineering 29
[37] V Hosseinnezhad M Rafiee M Ahmadian and M T AmelildquoSpecies-basedQuantum Particle SwarmOptimization for eco-nomic load dispatchrdquo International Journal of Electrical Poweramp Energy Systems vol 63 pp 311ndash322 2014
[38] M S P Subathra S E Selvan T A A Victoire A HChristinal andU Amato ldquoA hybrid with cross-entropymethodand sequential quadratic programming to solve economic loaddispatch problemrdquo IEEE Systems Journal vol 9 no 3 pp 1031ndash1044 2015
[39] J Zhan Q H Wu C Guo and X Zhou ldquoEconomic dispatchwith non-smooth objectives part I local minimum analysisrdquoIEEE Transactions on Power Systems vol 30 no 2 pp 710ndash7212015
[40] A K Barisal andR C Prusty ldquoLarge scale economic dispatch ofpower systems using oppositional invasive weed optimizationrdquoApplied Soft Computing vol 29 pp 122ndash137 2015
[41] M Basu ldquoImproved differential evolution for economic dis-patchrdquo International Journal of Electrical Power amp EnergySystems vol 63 pp 855ndash861 2014
[42] V S Aragon S C Esquivel and C C Coello ldquoAn immunealgorithm with power redistribution for solving economicdispatch problemsrdquo Information Sciences vol 295 pp 609ndash6322015
[43] M Ghasemi M Taghizadeh S Ghavidel and A AbbasianldquoColonial competitive differential evolution an experimentalstudy for optimal economic load dispatchrdquo Applied Soft Com-puting vol 40 pp 342ndash363 2016
[44] B R Adarsh T Raghunathan T Jayabarathi and X-S YangldquoEconomic dispatch using chaotic bat algorithmrdquo Energy vol96 pp 666ndash675 2016
[45] N Ghorbani and E Babaei ldquoExchange market algorithm foreconomic load dispatchrdquo International Journal of ElectricalPower amp Energy Systems vol 75 pp 19ndash27 2016
[46] J X V Neto G Reynoso-Meza T H Ruppel V C Marianiand L D S Coelho ldquoSolving non-smooth economic dispatchby a new combination of continuous GRASP algorithm anddifferential evolutionrdquo International Journal of Electrical Poweramp Energy Systems vol 84 pp 13ndash24 2017
[47] A Kavousi-Fard and A Khosravi ldquoAn intelligent 120579-ModifiedBat Algorithm to solve the non-convex economic dispatchproblem considering practical constraintsrdquoElectrical Power andEnergy Systems vol 82 pp 189ndash196 2016
[48] M A Al-Betar M A Awadallah A T Khader and A L BolajildquoTournament-based harmony search algorithm for non-convexeconomic load dispatch problemrdquo Applied Soft Computing vol47 pp 449ndash459 2016
[49] M A Al-Betar M A Awadallah I A Doush E Alsukhni andH ALkhraisat ldquoA non-convex economic dispatchproblemwithvalve loading effect using a new modified 120573-hill climbing localsearch algorithmrdquo Arabian Journal for Science and Engineeringpp 1ndash8 2018
[50] M A Awadallah M A Al-Betar A L Bolaji E M Alsukhniand H Al-Zoubi ldquoNatural selection methods for artificial beecolony with new versions of onlooker beerdquo Soft Computing pp1ndash40 2018
[51] S H Kaboli and A K Alqallaf ldquoSolving non-convex economicload dispatch problem via artificial cooperative search algo-rithmrdquo Expert Systems with Applications vol 128 pp 14ndash272019
[52] D Singh and J Dhillon ldquoAmeliorated greywolf optimization foreconomic load dispatch problemrdquo Energy vol 169 pp 398ndash4192019
[53] X S Yang and S Deb ldquoCuckoo search via Levy flightsrdquo inProceedings of the World Congress on Nature and BiologicallyInspired Computing (NaBIC rsquo09) pp 210ndash214 India 2009
[54] S Sahoo K M Dash R C Prusty and A K Barisal ldquoCom-parative analysis of optimal load dispatch through evolutionaryalgorithmsrdquo Ain Shams Engineering Journal vol 6 pp 107ndash1202015
[55] M Basu and A Chowdhury ldquoCuckoo search algorithm foreconomic dispatchrdquo Energy vol 60 pp 99ndash108 2013
[56] A B Serapiao ldquoCuckoo search for solving economic dispatchload problemrdquo Intelligent Control and Automation vol 04 no04 pp 385ndash390 2013
[57] A H Bindu and M D Reddy ldquoEconomic load dispatch usingcuckoo search algorithmrdquo International Journal of EngineeringResearch and Applications vol 3 pp 498ndash502 2013
[58] E Afzalan and M Joorabian ldquoAn improved cuckoo searchalgorithm for power economic load dispatchrdquo InternationalTransactions on Electrical Energy Systems vol 25 no 6 pp 958ndash975 2015
[59] T T Nguyen and D N Vo ldquoThe application of one rank cuckoosearch algorithm for solving economic load dispatch problemsrdquoApplied Soft Computing vol 37 pp 763ndash773 2015
[60] S M Islam S Das S Ghosh S Roy and P N Suganthan ldquoAnadaptive differential evolution algorithm with novel mutationand crossover strategies for global numerical optimizationrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 42 no 2 pp 482ndash499 2012
[61] Z L Gaing ldquoParticle swarm optimization to solving theeconomic dispatch considering the generator constraintsrdquo IEEETransactions on Power Systems vol 18 no 3 pp 1187ndash1195 2003
[62] X-S Yang and S Deb ldquoEngineering optimisation by Cuckoosearchrdquo International Journal of Mathematical Modelling andNumerical Optimisation vol 330 pp 43ndash51 2010
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
119873 Number of allavailable thermalgeneration units119899119894 Number of prohibitedpower zones of the 119894119905ℎthermal generationunit119875119906119894119896 119875119897119894119896 Upper and lowerlimits of the 119894119905ℎthermal generationunit corresponding tothe kth PPZ119875119886 Probability ofreplacing controlvariables in each oldsolution119875119863 Real power demand ofall loads in system119875119894119898119886119909 119875119894119898119894119899 The highest andlowest real poweroutputs of the iththermal generationunit119875119894119895119898119886119909 119875119894119895119898119894119899 The highest andlowest real poweroutputs of the 119894119905ℎthermal generationunit corresponding tothe 119895119905ℎ fuel type119878119894119898119886119909 Maximum real powerreserve contributionof the thermalgeneration unit 1198941198781199001198971199031198861198991198891 1198781199001198971199031198861198991198892 1198781199001198971199031198861198991198893 1198781199001198971199031198861198991198894 Randomly mixedsolutions from the setof current solutions119878119900119897119909 119878119900119897119866119887119890119904119905 The old solution x andthe most effectivesolution119878119877 Real power reserverequirement of system
120575119894119895 120582119894119895 120572119894119895 120573119894119895 120574119894119895 Fuel cost functioncoefficients of the 119894119905ℎthermal generationunit corresponding tothe 119895119905ℎ fuel type1205761 1205762 1205763 1205764 1205765 1205766 Random numbersbetween 0 and 1Ω Number of generationunits considering PPZconstraint
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare no conflicts of interest
References
[1] N Sinha R Chakrabarti and P K Chattopadhyay ldquoEvolution-ary programming techniques for economic load dispatchrdquo IEEETransactions on Evolutionary Computation vol 7 no 1 pp 83ndash94 2003
[2] P Hansen ldquoA separable approximation dynamic programmingalgorithm for economic dispatch with transmission lossesrdquoYugoslav Journal of Operations Research vol 12 no 2 2002
[3] N T Thang ldquoSolving economic dispatch problem with piece-wise quadratic cost functions using lagrange multiplier theoryrdquoin Proceedings of the 3rd International Conference on ComputerTechnology and Development (ICCTD rsquo11) pp 359ndash364 ASMEPress 2011
[4] S K Mishra and S K Mishra ldquoA comparative study of solutionof economic load dispatch problem in power systems in theenvironmental perspectiverdquoProcedia Computer Science vol 48pp 96ndash100 2015
[5] A A Al-Subhi and H K Alfares ldquoEconomic load dispatchusing linear programming a comparative studyrdquo InternationalJournal of Applied Industrial Engineering vol 3 no 1 pp 16ndash362016
28 Mathematical Problems in Engineering
[6] C Zhou G Huang and J Chen ldquoPlanning of electric powersystems considering virtual power plants with dispatchableloads included an inexact two-stage stochastic linear program-ming modelrdquoMathematical Problems in Engineering vol 2018Article ID 7049329 12 pages 2018
[7] T T Nguyen N V Quynh and L Van Dai ldquoImproved fireflyalgorithm a novel method for optimal operation of thermalgenerating unitsrdquo Complexity vol 2018 Article ID 7267593 23pages 2018
[8] S S Haroon S Hassan S Amin et al ldquoMultiple fuel machinespower economic dispatch using stud differential evolutionrdquoEnergies vol 11 no 6 pp 1ndash20 2018
[9] C-T Su and C-L Chiang ldquoNonconvex power economic dis-patch by improved genetic algorithm with multiplier updatingmethodrdquo Electric Power Components and Systems vol 32 no 3pp 257ndash273 2004
[10] NNoman andH Iba ldquoDifferential evolution for economic loaddispatch problemsrdquo Electric Power Systems Research vol 78 no8 pp 1322ndash1331 2008
[11] P Somasundaram and K Kuppusamy ldquoApplication of evolu-tionary programming to security constrained economic dis-patchrdquo International Journal of Electrical Power amp EnergySystems vol 27 no 5-6 pp 343ndash351 2005
[12] T Jayabarathi K Jayaprakash andD Jeyakumar ldquoEvolutionaryprogramming technique for different kinds of economic dis-patch problemsrdquo Electric Power Systems Research vol 73 no 2pp 169ndash176 2005
[13] P Subbaraj R Rengaraj and S Salivahanan ldquoEnhancementof combined heat and power economic dispatch using selfadaptive real-coded genetic algorithmrdquo Applied Energy vol 86no 6 pp 915ndash921 2009
[14] C-L Chiang and C-T Su ldquoAdaptive-improved genetic algo-rithm for the economic dispatch of units with multiple fueloptionsrdquo Cybernetics and Systems vol 36 no 7 pp 687ndash7042005
[15] C Chiang ldquoImproved genetic algorithm for power economicdispatch of units with valve-point effects and multiple fuelsrdquoIEEE Transactions on Power Systems vol 20 no 4 pp 1690ndash1699 2005
[16] K Meng H G Wang Z Y Dong and K P Wong ldquoQuantum-inspired particle swarm optimization for valve-point economicload dispatchrdquo IEEE Transactions on Power Systems vol 25 no1 pp 215ndash222 2010
[17] P Erdogmus A Ozturk and S Duman ldquoEnvironmen-taleconomic dispatch using genetic algorithm and simulatedannealingrdquo in Proceedings of the International Conference onElectric Power and Energy Conversion Systems EPECS rsquo09 pp1ndash4 IEEE 2009
[18] L D S Coelho and V C Mariani ldquoImproved differentialevolution algorithms for handling economic dispatch opti-mization with generator constraintsrdquo Energy Conversion andManagement vol 48 no 5 pp 1631ndash1639 2007
[19] H Dakuo W Fuli and M Zhizhong ldquoA hybrid genetic algo-rithm approach based on differential evolution for economicdispatch with valve-point effectsrdquo Electrical Power and EnergySystems vol 30 pp 31ndash38 2008
[20] C-F Chang J-J Wong J-P Chiou and C-T Su ldquoRobustsearching hybrid differential evolution method for optimalreactive power planning in large-scale distribution systemsrdquoElectric Power Systems Research vol 77 no 5-6 pp 430ndash4372007
[21] M R Farooqi P Jain and K R Niazi ldquoUsing Hopfield neuralnetwork for economic dispatch of power systemsrdquo in Proceed-ings of the National Power Engineering Conference PECon rsquo03pp 5ndash10 Bangi Malaysia 2003
[22] T Yalcinoz H Altun and U Hasan ldquoConstrained economicdispatch with prohibited operating zones a Hopfield neuralnetwork approachrdquo in Proceedings of the 10th MediterraneanElectrotechnical Conference Information Technology and Elec-trotechnology for the Mediterranean Countries ProceedingsMeleCon rsquo00 (Cat No 00CH37099) pp 570ndash573 LemesosCyprus 2000
[23] D N Vo P Schegner and W Ongsakul ldquoCuckoo searchalgorithm for non-convex economic dispatchrdquo IET GenerationTransmission amp Distribution vol 7 no 6 pp 645ndash654 2013
[24] J-B Park K-S Lee J-R Shin and K Y Lee ldquoA particleswarm optimization for economic dispatch with non-smoothcost functionsrdquo IEEETransactions on Power Systems vol 20 no1 pp 34ndash42 2005
[25] T Niknam H D Mojarrad and M Nayeripour ldquoA newfuzzy adaptive particle swarm optimization for non-smootheconomic dispatchrdquo Energy vol 35 no 4 pp 1764ndash1778 2010
[26] L D S Coelho and V C Mariani ldquoAn improved harmonysearch algorithm for power economic load dispatchrdquo EnergyConversion and Management vol 50 no 10 pp 2522ndash25262009
[27] M Mahdavi M Fesanghary and E Damangir ldquoAn improvedharmony search algorithm for solving optimization problemsrdquoAppliedMathematics and Computation vol 188 no 2 pp 1567ndash1579 2007
[28] A Bhattacharya and P K Chattopadhyay ldquoSolving complexeconomic load dispatch problem using biogeography-basedoptimizationrdquo Expert Systems with Applications vol 37 no 5pp 3605ndash3615 2010
[29] D Simon ldquoBiogeography-based optimizationrdquo IEEE Transac-tions on Evolutionary Computation vol 12 no 6 pp 702ndash7132008
[30] J S Alsumait J K Sykulski and A K Al-Othman ldquoAhybrid GA-PS-SQP method to solve power system valve-pointeconomic dispatch problemsrdquo Applied Energy vol 87 no 5 pp1773ndash1781 2010
[31] M S Turgut andGKDemir ldquoQuadratic approximationndashbasedhybrid Artificial Cooperative Search algorithm for economicemission load dispatchproblemsrdquo International Transactions onElectrical Energy Systems vol 27 no 4 pp 1ndash14 2017
[32] T A A Victoire and A E Jeyakumar ldquoHybrid PSO-SQPfor economic dispatch with valve-point effectrdquo Electric PowerSystems Research vol 71 no 1 pp 51ndash59 2004
[33] D-KHe F-LWang andZ-ZMao ldquoHybrid genetic algorithmfor economic dispatch with valve-point effectrdquo Electric PowerSystems Research vol 78 no 4 pp 626ndash633 2008
[34] D C Secui ldquoA modified symbiotic organisms search algorithmfor large scale economic dispatch problem with valve-pointloading effectsrdquo Energy vol 113 pp 366ndash384 2016
[35] K BhattacharjeeA Bhattacharya and S H N Dey ldquoTeaching-learning-based optimization for different economic dispatchproblemsrdquo Scientia Iranica vol 21 no 3 pp 870ndash884 2014
[36] K Bhattacharjee A Bhattacharya and S H N Dey ldquoOppo-sitional Real Coded Chemical Reaction Optimization for dif-ferent economic dispatch problemsrdquo International Journal ofElectrical Power amp Energy Systems vol 55 pp 378ndash391 2014
Mathematical Problems in Engineering 29
[37] V Hosseinnezhad M Rafiee M Ahmadian and M T AmelildquoSpecies-basedQuantum Particle SwarmOptimization for eco-nomic load dispatchrdquo International Journal of Electrical Poweramp Energy Systems vol 63 pp 311ndash322 2014
[38] M S P Subathra S E Selvan T A A Victoire A HChristinal andU Amato ldquoA hybrid with cross-entropymethodand sequential quadratic programming to solve economic loaddispatch problemrdquo IEEE Systems Journal vol 9 no 3 pp 1031ndash1044 2015
[39] J Zhan Q H Wu C Guo and X Zhou ldquoEconomic dispatchwith non-smooth objectives part I local minimum analysisrdquoIEEE Transactions on Power Systems vol 30 no 2 pp 710ndash7212015
[40] A K Barisal andR C Prusty ldquoLarge scale economic dispatch ofpower systems using oppositional invasive weed optimizationrdquoApplied Soft Computing vol 29 pp 122ndash137 2015
[41] M Basu ldquoImproved differential evolution for economic dis-patchrdquo International Journal of Electrical Power amp EnergySystems vol 63 pp 855ndash861 2014
[42] V S Aragon S C Esquivel and C C Coello ldquoAn immunealgorithm with power redistribution for solving economicdispatch problemsrdquo Information Sciences vol 295 pp 609ndash6322015
[43] M Ghasemi M Taghizadeh S Ghavidel and A AbbasianldquoColonial competitive differential evolution an experimentalstudy for optimal economic load dispatchrdquo Applied Soft Com-puting vol 40 pp 342ndash363 2016
[44] B R Adarsh T Raghunathan T Jayabarathi and X-S YangldquoEconomic dispatch using chaotic bat algorithmrdquo Energy vol96 pp 666ndash675 2016
[45] N Ghorbani and E Babaei ldquoExchange market algorithm foreconomic load dispatchrdquo International Journal of ElectricalPower amp Energy Systems vol 75 pp 19ndash27 2016
[46] J X V Neto G Reynoso-Meza T H Ruppel V C Marianiand L D S Coelho ldquoSolving non-smooth economic dispatchby a new combination of continuous GRASP algorithm anddifferential evolutionrdquo International Journal of Electrical Poweramp Energy Systems vol 84 pp 13ndash24 2017
[47] A Kavousi-Fard and A Khosravi ldquoAn intelligent 120579-ModifiedBat Algorithm to solve the non-convex economic dispatchproblem considering practical constraintsrdquoElectrical Power andEnergy Systems vol 82 pp 189ndash196 2016
[48] M A Al-Betar M A Awadallah A T Khader and A L BolajildquoTournament-based harmony search algorithm for non-convexeconomic load dispatch problemrdquo Applied Soft Computing vol47 pp 449ndash459 2016
[49] M A Al-Betar M A Awadallah I A Doush E Alsukhni andH ALkhraisat ldquoA non-convex economic dispatchproblemwithvalve loading effect using a new modified 120573-hill climbing localsearch algorithmrdquo Arabian Journal for Science and Engineeringpp 1ndash8 2018
[50] M A Awadallah M A Al-Betar A L Bolaji E M Alsukhniand H Al-Zoubi ldquoNatural selection methods for artificial beecolony with new versions of onlooker beerdquo Soft Computing pp1ndash40 2018
[51] S H Kaboli and A K Alqallaf ldquoSolving non-convex economicload dispatch problem via artificial cooperative search algo-rithmrdquo Expert Systems with Applications vol 128 pp 14ndash272019
[52] D Singh and J Dhillon ldquoAmeliorated greywolf optimization foreconomic load dispatch problemrdquo Energy vol 169 pp 398ndash4192019
[53] X S Yang and S Deb ldquoCuckoo search via Levy flightsrdquo inProceedings of the World Congress on Nature and BiologicallyInspired Computing (NaBIC rsquo09) pp 210ndash214 India 2009
[54] S Sahoo K M Dash R C Prusty and A K Barisal ldquoCom-parative analysis of optimal load dispatch through evolutionaryalgorithmsrdquo Ain Shams Engineering Journal vol 6 pp 107ndash1202015
[55] M Basu and A Chowdhury ldquoCuckoo search algorithm foreconomic dispatchrdquo Energy vol 60 pp 99ndash108 2013
[56] A B Serapiao ldquoCuckoo search for solving economic dispatchload problemrdquo Intelligent Control and Automation vol 04 no04 pp 385ndash390 2013
[57] A H Bindu and M D Reddy ldquoEconomic load dispatch usingcuckoo search algorithmrdquo International Journal of EngineeringResearch and Applications vol 3 pp 498ndash502 2013
[58] E Afzalan and M Joorabian ldquoAn improved cuckoo searchalgorithm for power economic load dispatchrdquo InternationalTransactions on Electrical Energy Systems vol 25 no 6 pp 958ndash975 2015
[59] T T Nguyen and D N Vo ldquoThe application of one rank cuckoosearch algorithm for solving economic load dispatch problemsrdquoApplied Soft Computing vol 37 pp 763ndash773 2015
[60] S M Islam S Das S Ghosh S Roy and P N Suganthan ldquoAnadaptive differential evolution algorithm with novel mutationand crossover strategies for global numerical optimizationrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 42 no 2 pp 482ndash499 2012
[61] Z L Gaing ldquoParticle swarm optimization to solving theeconomic dispatch considering the generator constraintsrdquo IEEETransactions on Power Systems vol 18 no 3 pp 1187ndash1195 2003
[62] X-S Yang and S Deb ldquoEngineering optimisation by Cuckoosearchrdquo International Journal of Mathematical Modelling andNumerical Optimisation vol 330 pp 43ndash51 2010
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
28 Mathematical Problems in Engineering
[6] C Zhou G Huang and J Chen ldquoPlanning of electric powersystems considering virtual power plants with dispatchableloads included an inexact two-stage stochastic linear program-ming modelrdquoMathematical Problems in Engineering vol 2018Article ID 7049329 12 pages 2018
[7] T T Nguyen N V Quynh and L Van Dai ldquoImproved fireflyalgorithm a novel method for optimal operation of thermalgenerating unitsrdquo Complexity vol 2018 Article ID 7267593 23pages 2018
[8] S S Haroon S Hassan S Amin et al ldquoMultiple fuel machinespower economic dispatch using stud differential evolutionrdquoEnergies vol 11 no 6 pp 1ndash20 2018
[9] C-T Su and C-L Chiang ldquoNonconvex power economic dis-patch by improved genetic algorithm with multiplier updatingmethodrdquo Electric Power Components and Systems vol 32 no 3pp 257ndash273 2004
[10] NNoman andH Iba ldquoDifferential evolution for economic loaddispatch problemsrdquo Electric Power Systems Research vol 78 no8 pp 1322ndash1331 2008
[11] P Somasundaram and K Kuppusamy ldquoApplication of evolu-tionary programming to security constrained economic dis-patchrdquo International Journal of Electrical Power amp EnergySystems vol 27 no 5-6 pp 343ndash351 2005
[12] T Jayabarathi K Jayaprakash andD Jeyakumar ldquoEvolutionaryprogramming technique for different kinds of economic dis-patch problemsrdquo Electric Power Systems Research vol 73 no 2pp 169ndash176 2005
[13] P Subbaraj R Rengaraj and S Salivahanan ldquoEnhancementof combined heat and power economic dispatch using selfadaptive real-coded genetic algorithmrdquo Applied Energy vol 86no 6 pp 915ndash921 2009
[14] C-L Chiang and C-T Su ldquoAdaptive-improved genetic algo-rithm for the economic dispatch of units with multiple fueloptionsrdquo Cybernetics and Systems vol 36 no 7 pp 687ndash7042005
[15] C Chiang ldquoImproved genetic algorithm for power economicdispatch of units with valve-point effects and multiple fuelsrdquoIEEE Transactions on Power Systems vol 20 no 4 pp 1690ndash1699 2005
[16] K Meng H G Wang Z Y Dong and K P Wong ldquoQuantum-inspired particle swarm optimization for valve-point economicload dispatchrdquo IEEE Transactions on Power Systems vol 25 no1 pp 215ndash222 2010
[17] P Erdogmus A Ozturk and S Duman ldquoEnvironmen-taleconomic dispatch using genetic algorithm and simulatedannealingrdquo in Proceedings of the International Conference onElectric Power and Energy Conversion Systems EPECS rsquo09 pp1ndash4 IEEE 2009
[18] L D S Coelho and V C Mariani ldquoImproved differentialevolution algorithms for handling economic dispatch opti-mization with generator constraintsrdquo Energy Conversion andManagement vol 48 no 5 pp 1631ndash1639 2007
[19] H Dakuo W Fuli and M Zhizhong ldquoA hybrid genetic algo-rithm approach based on differential evolution for economicdispatch with valve-point effectsrdquo Electrical Power and EnergySystems vol 30 pp 31ndash38 2008
[20] C-F Chang J-J Wong J-P Chiou and C-T Su ldquoRobustsearching hybrid differential evolution method for optimalreactive power planning in large-scale distribution systemsrdquoElectric Power Systems Research vol 77 no 5-6 pp 430ndash4372007
[21] M R Farooqi P Jain and K R Niazi ldquoUsing Hopfield neuralnetwork for economic dispatch of power systemsrdquo in Proceed-ings of the National Power Engineering Conference PECon rsquo03pp 5ndash10 Bangi Malaysia 2003
[22] T Yalcinoz H Altun and U Hasan ldquoConstrained economicdispatch with prohibited operating zones a Hopfield neuralnetwork approachrdquo in Proceedings of the 10th MediterraneanElectrotechnical Conference Information Technology and Elec-trotechnology for the Mediterranean Countries ProceedingsMeleCon rsquo00 (Cat No 00CH37099) pp 570ndash573 LemesosCyprus 2000
[23] D N Vo P Schegner and W Ongsakul ldquoCuckoo searchalgorithm for non-convex economic dispatchrdquo IET GenerationTransmission amp Distribution vol 7 no 6 pp 645ndash654 2013
[24] J-B Park K-S Lee J-R Shin and K Y Lee ldquoA particleswarm optimization for economic dispatch with non-smoothcost functionsrdquo IEEETransactions on Power Systems vol 20 no1 pp 34ndash42 2005
[25] T Niknam H D Mojarrad and M Nayeripour ldquoA newfuzzy adaptive particle swarm optimization for non-smootheconomic dispatchrdquo Energy vol 35 no 4 pp 1764ndash1778 2010
[26] L D S Coelho and V C Mariani ldquoAn improved harmonysearch algorithm for power economic load dispatchrdquo EnergyConversion and Management vol 50 no 10 pp 2522ndash25262009
[27] M Mahdavi M Fesanghary and E Damangir ldquoAn improvedharmony search algorithm for solving optimization problemsrdquoAppliedMathematics and Computation vol 188 no 2 pp 1567ndash1579 2007
[28] A Bhattacharya and P K Chattopadhyay ldquoSolving complexeconomic load dispatch problem using biogeography-basedoptimizationrdquo Expert Systems with Applications vol 37 no 5pp 3605ndash3615 2010
[29] D Simon ldquoBiogeography-based optimizationrdquo IEEE Transac-tions on Evolutionary Computation vol 12 no 6 pp 702ndash7132008
[30] J S Alsumait J K Sykulski and A K Al-Othman ldquoAhybrid GA-PS-SQP method to solve power system valve-pointeconomic dispatch problemsrdquo Applied Energy vol 87 no 5 pp1773ndash1781 2010
[31] M S Turgut andGKDemir ldquoQuadratic approximationndashbasedhybrid Artificial Cooperative Search algorithm for economicemission load dispatchproblemsrdquo International Transactions onElectrical Energy Systems vol 27 no 4 pp 1ndash14 2017
[32] T A A Victoire and A E Jeyakumar ldquoHybrid PSO-SQPfor economic dispatch with valve-point effectrdquo Electric PowerSystems Research vol 71 no 1 pp 51ndash59 2004
[33] D-KHe F-LWang andZ-ZMao ldquoHybrid genetic algorithmfor economic dispatch with valve-point effectrdquo Electric PowerSystems Research vol 78 no 4 pp 626ndash633 2008
[34] D C Secui ldquoA modified symbiotic organisms search algorithmfor large scale economic dispatch problem with valve-pointloading effectsrdquo Energy vol 113 pp 366ndash384 2016
[35] K BhattacharjeeA Bhattacharya and S H N Dey ldquoTeaching-learning-based optimization for different economic dispatchproblemsrdquo Scientia Iranica vol 21 no 3 pp 870ndash884 2014
[36] K Bhattacharjee A Bhattacharya and S H N Dey ldquoOppo-sitional Real Coded Chemical Reaction Optimization for dif-ferent economic dispatch problemsrdquo International Journal ofElectrical Power amp Energy Systems vol 55 pp 378ndash391 2014
Mathematical Problems in Engineering 29
[37] V Hosseinnezhad M Rafiee M Ahmadian and M T AmelildquoSpecies-basedQuantum Particle SwarmOptimization for eco-nomic load dispatchrdquo International Journal of Electrical Poweramp Energy Systems vol 63 pp 311ndash322 2014
[38] M S P Subathra S E Selvan T A A Victoire A HChristinal andU Amato ldquoA hybrid with cross-entropymethodand sequential quadratic programming to solve economic loaddispatch problemrdquo IEEE Systems Journal vol 9 no 3 pp 1031ndash1044 2015
[39] J Zhan Q H Wu C Guo and X Zhou ldquoEconomic dispatchwith non-smooth objectives part I local minimum analysisrdquoIEEE Transactions on Power Systems vol 30 no 2 pp 710ndash7212015
[40] A K Barisal andR C Prusty ldquoLarge scale economic dispatch ofpower systems using oppositional invasive weed optimizationrdquoApplied Soft Computing vol 29 pp 122ndash137 2015
[41] M Basu ldquoImproved differential evolution for economic dis-patchrdquo International Journal of Electrical Power amp EnergySystems vol 63 pp 855ndash861 2014
[42] V S Aragon S C Esquivel and C C Coello ldquoAn immunealgorithm with power redistribution for solving economicdispatch problemsrdquo Information Sciences vol 295 pp 609ndash6322015
[43] M Ghasemi M Taghizadeh S Ghavidel and A AbbasianldquoColonial competitive differential evolution an experimentalstudy for optimal economic load dispatchrdquo Applied Soft Com-puting vol 40 pp 342ndash363 2016
[44] B R Adarsh T Raghunathan T Jayabarathi and X-S YangldquoEconomic dispatch using chaotic bat algorithmrdquo Energy vol96 pp 666ndash675 2016
[45] N Ghorbani and E Babaei ldquoExchange market algorithm foreconomic load dispatchrdquo International Journal of ElectricalPower amp Energy Systems vol 75 pp 19ndash27 2016
[46] J X V Neto G Reynoso-Meza T H Ruppel V C Marianiand L D S Coelho ldquoSolving non-smooth economic dispatchby a new combination of continuous GRASP algorithm anddifferential evolutionrdquo International Journal of Electrical Poweramp Energy Systems vol 84 pp 13ndash24 2017
[47] A Kavousi-Fard and A Khosravi ldquoAn intelligent 120579-ModifiedBat Algorithm to solve the non-convex economic dispatchproblem considering practical constraintsrdquoElectrical Power andEnergy Systems vol 82 pp 189ndash196 2016
[48] M A Al-Betar M A Awadallah A T Khader and A L BolajildquoTournament-based harmony search algorithm for non-convexeconomic load dispatch problemrdquo Applied Soft Computing vol47 pp 449ndash459 2016
[49] M A Al-Betar M A Awadallah I A Doush E Alsukhni andH ALkhraisat ldquoA non-convex economic dispatchproblemwithvalve loading effect using a new modified 120573-hill climbing localsearch algorithmrdquo Arabian Journal for Science and Engineeringpp 1ndash8 2018
[50] M A Awadallah M A Al-Betar A L Bolaji E M Alsukhniand H Al-Zoubi ldquoNatural selection methods for artificial beecolony with new versions of onlooker beerdquo Soft Computing pp1ndash40 2018
[51] S H Kaboli and A K Alqallaf ldquoSolving non-convex economicload dispatch problem via artificial cooperative search algo-rithmrdquo Expert Systems with Applications vol 128 pp 14ndash272019
[52] D Singh and J Dhillon ldquoAmeliorated greywolf optimization foreconomic load dispatch problemrdquo Energy vol 169 pp 398ndash4192019
[53] X S Yang and S Deb ldquoCuckoo search via Levy flightsrdquo inProceedings of the World Congress on Nature and BiologicallyInspired Computing (NaBIC rsquo09) pp 210ndash214 India 2009
[54] S Sahoo K M Dash R C Prusty and A K Barisal ldquoCom-parative analysis of optimal load dispatch through evolutionaryalgorithmsrdquo Ain Shams Engineering Journal vol 6 pp 107ndash1202015
[55] M Basu and A Chowdhury ldquoCuckoo search algorithm foreconomic dispatchrdquo Energy vol 60 pp 99ndash108 2013
[56] A B Serapiao ldquoCuckoo search for solving economic dispatchload problemrdquo Intelligent Control and Automation vol 04 no04 pp 385ndash390 2013
[57] A H Bindu and M D Reddy ldquoEconomic load dispatch usingcuckoo search algorithmrdquo International Journal of EngineeringResearch and Applications vol 3 pp 498ndash502 2013
[58] E Afzalan and M Joorabian ldquoAn improved cuckoo searchalgorithm for power economic load dispatchrdquo InternationalTransactions on Electrical Energy Systems vol 25 no 6 pp 958ndash975 2015
[59] T T Nguyen and D N Vo ldquoThe application of one rank cuckoosearch algorithm for solving economic load dispatch problemsrdquoApplied Soft Computing vol 37 pp 763ndash773 2015
[60] S M Islam S Das S Ghosh S Roy and P N Suganthan ldquoAnadaptive differential evolution algorithm with novel mutationand crossover strategies for global numerical optimizationrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 42 no 2 pp 482ndash499 2012
[61] Z L Gaing ldquoParticle swarm optimization to solving theeconomic dispatch considering the generator constraintsrdquo IEEETransactions on Power Systems vol 18 no 3 pp 1187ndash1195 2003
[62] X-S Yang and S Deb ldquoEngineering optimisation by Cuckoosearchrdquo International Journal of Mathematical Modelling andNumerical Optimisation vol 330 pp 43ndash51 2010
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Mathematical Problems in Engineering 29
[37] V Hosseinnezhad M Rafiee M Ahmadian and M T AmelildquoSpecies-basedQuantum Particle SwarmOptimization for eco-nomic load dispatchrdquo International Journal of Electrical Poweramp Energy Systems vol 63 pp 311ndash322 2014
[38] M S P Subathra S E Selvan T A A Victoire A HChristinal andU Amato ldquoA hybrid with cross-entropymethodand sequential quadratic programming to solve economic loaddispatch problemrdquo IEEE Systems Journal vol 9 no 3 pp 1031ndash1044 2015
[39] J Zhan Q H Wu C Guo and X Zhou ldquoEconomic dispatchwith non-smooth objectives part I local minimum analysisrdquoIEEE Transactions on Power Systems vol 30 no 2 pp 710ndash7212015
[40] A K Barisal andR C Prusty ldquoLarge scale economic dispatch ofpower systems using oppositional invasive weed optimizationrdquoApplied Soft Computing vol 29 pp 122ndash137 2015
[41] M Basu ldquoImproved differential evolution for economic dis-patchrdquo International Journal of Electrical Power amp EnergySystems vol 63 pp 855ndash861 2014
[42] V S Aragon S C Esquivel and C C Coello ldquoAn immunealgorithm with power redistribution for solving economicdispatch problemsrdquo Information Sciences vol 295 pp 609ndash6322015
[43] M Ghasemi M Taghizadeh S Ghavidel and A AbbasianldquoColonial competitive differential evolution an experimentalstudy for optimal economic load dispatchrdquo Applied Soft Com-puting vol 40 pp 342ndash363 2016
[44] B R Adarsh T Raghunathan T Jayabarathi and X-S YangldquoEconomic dispatch using chaotic bat algorithmrdquo Energy vol96 pp 666ndash675 2016
[45] N Ghorbani and E Babaei ldquoExchange market algorithm foreconomic load dispatchrdquo International Journal of ElectricalPower amp Energy Systems vol 75 pp 19ndash27 2016
[46] J X V Neto G Reynoso-Meza T H Ruppel V C Marianiand L D S Coelho ldquoSolving non-smooth economic dispatchby a new combination of continuous GRASP algorithm anddifferential evolutionrdquo International Journal of Electrical Poweramp Energy Systems vol 84 pp 13ndash24 2017
[47] A Kavousi-Fard and A Khosravi ldquoAn intelligent 120579-ModifiedBat Algorithm to solve the non-convex economic dispatchproblem considering practical constraintsrdquoElectrical Power andEnergy Systems vol 82 pp 189ndash196 2016
[48] M A Al-Betar M A Awadallah A T Khader and A L BolajildquoTournament-based harmony search algorithm for non-convexeconomic load dispatch problemrdquo Applied Soft Computing vol47 pp 449ndash459 2016
[49] M A Al-Betar M A Awadallah I A Doush E Alsukhni andH ALkhraisat ldquoA non-convex economic dispatchproblemwithvalve loading effect using a new modified 120573-hill climbing localsearch algorithmrdquo Arabian Journal for Science and Engineeringpp 1ndash8 2018
[50] M A Awadallah M A Al-Betar A L Bolaji E M Alsukhniand H Al-Zoubi ldquoNatural selection methods for artificial beecolony with new versions of onlooker beerdquo Soft Computing pp1ndash40 2018
[51] S H Kaboli and A K Alqallaf ldquoSolving non-convex economicload dispatch problem via artificial cooperative search algo-rithmrdquo Expert Systems with Applications vol 128 pp 14ndash272019
[52] D Singh and J Dhillon ldquoAmeliorated greywolf optimization foreconomic load dispatch problemrdquo Energy vol 169 pp 398ndash4192019
[53] X S Yang and S Deb ldquoCuckoo search via Levy flightsrdquo inProceedings of the World Congress on Nature and BiologicallyInspired Computing (NaBIC rsquo09) pp 210ndash214 India 2009
[54] S Sahoo K M Dash R C Prusty and A K Barisal ldquoCom-parative analysis of optimal load dispatch through evolutionaryalgorithmsrdquo Ain Shams Engineering Journal vol 6 pp 107ndash1202015
[55] M Basu and A Chowdhury ldquoCuckoo search algorithm foreconomic dispatchrdquo Energy vol 60 pp 99ndash108 2013
[56] A B Serapiao ldquoCuckoo search for solving economic dispatchload problemrdquo Intelligent Control and Automation vol 04 no04 pp 385ndash390 2013
[57] A H Bindu and M D Reddy ldquoEconomic load dispatch usingcuckoo search algorithmrdquo International Journal of EngineeringResearch and Applications vol 3 pp 498ndash502 2013
[58] E Afzalan and M Joorabian ldquoAn improved cuckoo searchalgorithm for power economic load dispatchrdquo InternationalTransactions on Electrical Energy Systems vol 25 no 6 pp 958ndash975 2015
[59] T T Nguyen and D N Vo ldquoThe application of one rank cuckoosearch algorithm for solving economic load dispatch problemsrdquoApplied Soft Computing vol 37 pp 763ndash773 2015
[60] S M Islam S Das S Ghosh S Roy and P N Suganthan ldquoAnadaptive differential evolution algorithm with novel mutationand crossover strategies for global numerical optimizationrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 42 no 2 pp 482ndash499 2012
[61] Z L Gaing ldquoParticle swarm optimization to solving theeconomic dispatch considering the generator constraintsrdquo IEEETransactions on Power Systems vol 18 no 3 pp 1187ndash1195 2003
[62] X-S Yang and S Deb ldquoEngineering optimisation by Cuckoosearchrdquo International Journal of Mathematical Modelling andNumerical Optimisation vol 330 pp 43ndash51 2010
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences