Imperialist competitive algorithm for solving non-convex dynamic economic power dispatch Behnam Mohammadi-ivatloo a , Abbas Rabiee b , Alireza Soroudi c, * , Mehdi Ehsan a a Center of Excellence in Power System Management and Control, Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran b Department of Electrical Engineering, Abhar Branch, Islamic Azad University, Abhar, Iran c Department of Electrical Engineering, Damavand Branch, Islamic Azad University, Damavand, Iran article info Article history: Received 8 September 2011 Received in revised form 11 June 2012 Accepted 12 June 2012 Available online 15 July 2012 Keywords: Dynamic economic dispatch Imperialist competitive algorithm Prohibited operation zone Valve-point effect Ramp-rate limits Optimization abstract Dynamic economic dispatch (DED) aims to schedule the committed generating units’ output active power economically over a certain period of time, satisfying operating constraints and load demand in each interval. Valve-point effect, the ramp rate limits, prohibited operation zones (POZs), and transmission losses make the DED a complicated, non-linear constrained problem. Hence, in this paper, imperialist competitive algorithm (ICA) is proposed to solve such complicated problem. The feasibility of the proposed method is validated on five and ten units test system for a 24 h time interval. The results obtained by the ICA are compared with other techniques of the literature. These results substantiate the applicability of the proposed method for solving the constrained DED with non-smooth cost functions. Besides, to examine the applicability of the proposed ICA on large power systems, a test case with 54 units is studied. The results confirm the suitability of the ICA for large-scale DED problem. Ó 2012 Elsevier Ltd. All rights reserved. 1. Introduction A power utility needs to ensure that the electrical power is generated with minimum cost. Hence, for economic operation of the system, the total demand must be appropriately shared among the generating units with an objective to minimize the total generation cost of the system. Thus, economic dispatch (ED) is one of the important problems of power system operation and control. Tradi- tional ED problem, attempts to minimize the cost of supplying energy subject to constraints on static behavior of the generating units. It is assumed that the amount of power to be supplied by a given set of committed units is constant for a given interval of time. However, to avoid shortening of the life of their equipment, plant operators, try to keep thermal gradients inside the turbine within safe limits. This mechanical constraint is usually translated into a limit on the rate of increase of the electrical output. Such ramp-rate constraints lead to the construction of dynamic economic dispatch (DED) problem, which is an extension of conventional ED problem. DED refers to the problem of determining minimum cost of dispatch of generators for a given horizon of time, taking into consideration the constraints imposed on system operation by the generator ramp-rate limitations. To solve DED problem, generators are modeled using inputeoutput curves in most of the power system operation studies. Traditionally an approximate quadratic function used to model the generator inputeoutput curves [1,2]. But, the generating units with multi-valve steam turbines exhibit a greater variation in the fuel-cost functions; and thus the natural inputeoutput curve is non-linear and non-smooth due to the effect of multiple steam admission valves (known as valve-points effect) [3,4]. Besides, generating units may have certain prohibited oper- ation zones (POZs) due to limitations of machine components or instability concerns. Hence, considering the effect of valve-points and POZs in generators’ cost function, makes the DED a non- convex optimization problem. Lots of optimization methods including classical and heuristic algorithms were applied to solve DED problem. Due to non- convexity of the DED problem, application of classical methods like Lagrangian relaxation [5] and dynamic programming [6] are restricted. In recent years, Maclaurin series approximation has been applied to model the valve-point effects [7,8] but it has been shown that this method leads to non-optimal solution. * Corresponding author. Tel.: þ98 (0)2166165954; fax: þ98 (0)216602326. E-mail addresses: [email protected](B. Mohammadi-ivatloo), a_rabiee@ ee.sharif.edu (A. Rabiee), [email protected](A. Soroudi), ehsan@ sharif.edu (M. Ehsan). Contents lists available at SciVerse ScienceDirect Energy journal homepage: www.elsevier.com/locate/energy 0360-5442/$ e see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.energy.2012.06.034 Energy 44 (2012) 228e240
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Energy 44 (2012) 228e240
Contents lists available
Energy
journal homepage: www.elsevier .com/locate/energy
Imperialist competitive algorithm for solving non-convex dynamic economicpower dispatch
Behnam Mohammadi-ivatloo a, Abbas Rabiee b, Alireza Soroudi c,*, Mehdi Ehsan a
aCenter of Excellence in Power System Management and Control, Department of Electrical Engineering, Sharif University of Technology, Tehran, IranbDepartment of Electrical Engineering, Abhar Branch, Islamic Azad University, Abhar, IrancDepartment of Electrical Engineering, Damavand Branch, Islamic Azad University, Damavand, Iran
a r t i c l e i n f o
Article history:Received 8 September 2011Received in revised form11 June 2012Accepted 12 June 2012Available online 15 July 2012
0360-5442/$ e see front matter � 2012 Elsevier Ltd.http://dx.doi.org/10.1016/j.energy.2012.06.034
a b s t r a c t
Dynamic economic dispatch (DED) aims to schedule the committed generating units’ output activepower economically over a certain period of time, satisfying operating constraints and load demand ineach interval. Valve-point effect, the ramp rate limits, prohibited operation zones (POZs), andtransmission losses make the DED a complicated, non-linear constrained problem. Hence, in this paper,imperialist competitive algorithm (ICA) is proposed to solve such complicated problem. The feasibility ofthe proposed method is validated on five and ten units test system for a 24 h time interval. The resultsobtained by the ICA are compared with other techniques of the literature. These results substantiate theapplicability of the proposed method for solving the constrained DED with non-smooth cost functions.Besides, to examine the applicability of the proposed ICA on large power systems, a test case with 54units is studied. The results confirm the suitability of the ICA for large-scale DED problem.
� 2012 Elsevier Ltd. All rights reserved.
1. Introduction
A power utility needs to ensure that the electrical power isgeneratedwithminimumcost. Hence, for economic operation of thesystem, the total demand must be appropriately shared among thegenerating units with an objective to minimize the total generationcost of the system. Thus, economic dispatch (ED) is one of theimportant problems of power system operation and control. Tradi-tional ED problem, attempts to minimize the cost of supplyingenergy subject to constraints on static behavior of the generatingunits. It is assumed that the amount of power to be supplied bya given set of committed units is constant for a given interval of time.However, to avoid shortening of the life of their equipment, plantoperators, try to keep thermal gradients inside the turbine withinsafe limits. This mechanical constraint is usually translated intoa limit on the rate of increase of the electrical output. Such ramp-rateconstraints lead to the construction of dynamic economic dispatch(DED) problem, which is an extension of conventional ED problem.
DED refers to the problem of determining minimum cost ofdispatch of generators for a given horizon of time, taking intoconsideration the constraints imposed on system operation by thegenerator ramp-rate limitations. To solve DED problem, generatorsare modeled using inputeoutput curves in most of the powersystem operation studies. Traditionally an approximate quadraticfunction used to model the generator inputeoutput curves [1,2].But, the generating units with multi-valve steam turbines exhibita greater variation in the fuel-cost functions; and thus the naturalinputeoutput curve is non-linear and non-smooth due to the effectof multiple steam admission valves (known as valve-points effect)[3,4]. Besides, generating units may have certain prohibited oper-ation zones (POZs) due to limitations of machine components orinstability concerns. Hence, considering the effect of valve-pointsand POZs in generators’ cost function, makes the DED a non-convex optimization problem.
Lots of optimization methods including classical and heuristicalgorithms were applied to solve DED problem. Due to non-convexity of the DED problem, application of classical methodslike Lagrangian relaxation [5] and dynamic programming [6] arerestricted. In recent years, Maclaurin series approximation has beenapplied to model the valve-point effects [7,8] but it has been shownthat this method leads to non-optimal solution.
B. Mohammadi-ivatloo et al. / Energy 44 (2012) 228e240 229
More recent works have been around artificial intelligence (AI)methods, such as artificial neural networks (ANN), simulatedannealing (SA), genetic algorithms (GA), differential evolution (DE),particle swarm optimization (PSO), evolutionary programming(EP), tabu search (TS), and hybrid methods. Optimization methodsbased on AI have shown better performance in solving the DEDproblem with capability of modeling more realistic objectivefunctions and constraints. In [9] hybrid EP and sequential quadraticprogramming (SQP) method has been proposed to solve non-convex DED problem. Chaotic quantum genetic algorithm (CQGA)is used in [10] for solving DED problem considering the effect ofwind generation. DE algorithms have received attention in solvingDED problems [11e17]. Other heuristic search methods have beenapplied to solve DED problems in the past decade. These include GA[1], quantum GA (QGA) [18], artificial immune system method [19],artificial bee colony algorithm (ABC) [20], PSO [15,16,21,22],multiple TS (MTS) algorithm [23], enhanced cross-entropy method[24], and SA algorithm [25]. Hybrid methods such as hybrid artifi-cial immune systems and SQP [26], hybrid EP and SQP method[9,22], hybrid swarm intelligence based harmony search algorithm[4], hybrid seeker optimization algorithm (SOA) and SQP [27],hybrid Hopfield neural network (HNN) and quadratic programming(QP) [28,29], adaptive hybrid DE algorithm [30], hybrid PSO andSQP [31], and artificial immune system (AIS) [32] are found to beeffective in solving complex optimization problems such as DEDproblem. Table 1 summarizes and compares different proposedalgorithms for solution of the DED problem.
In this paper, an imperialist competitive algorithm (ICA) isproposed to solve constrained non-convex DED problems. ICA isrecently proposed by Atashpaz-Gargari and Lucas [38]. This algo-rithm is inspired by the imperialistic competitive. ICA has shown
Table 1Summary of the proposed algorithms for solution of DED problem in literature.
Reference Algorithm POZ Valve
[2] Quadratic programming No No[4] Hybrid harmony search algorithm No Yes[6] Constructive dynamic programming No No[7] Maclaurin series-based Lagrangian method No Yes[9] Hybrid EP and SQP No Yes[10] Chaotic quantum GA No Yes[11] Chaotic sequence based DE No Yes[13] DE No Yes[14] Improved DE No Yes[15] Improved PSO No Yes[16] Modified DE No Yes[17] Chaotic DE No Yes[18] Quantum GA No Yes[19] Artificial immune system No Yes[20] Artificial bee colony No Yes[21] Adaptive PSO No Yes[22] Deterministically guided PSO No Yes[23] Multiple tabu search Yes No[24] Enhanced cross-entropy method No Yes[25] Simulated annealing No Yes[26] Hybrid AIS and SQP No Yes[27] Hybrid SOA and SQP No Yes[28] Hybrid HNN and QP No No[29] Hybrid HNN and QP No No[30] Adaptive hybrid DE No Yes[31] Hybrid PSO and SQP No Yes[32] Artificial immune system No Yes[33] Maclaurin series-based Lagrangian method No Yes[34] Modified hybrid EP and SQP No Yes[35] Hybrid DE No Yes[36] Covariance matrix adapted evolution strategy No Yes[37] Improved chaotic PSO No YesProposed Imperialist competitive algorithm Yes Yes
good performance in solving optimization problems in differentareas such as template matching [39], DG planning [40], optimaldesign of plate-fin heat exchangers [41] and electromagneticproblems [42]. This algorithm also has been successfully applied topower systemproblems like as PSS (power system stabilizer) design[43], linear induction motor design [44], unit commitment [45] andmodel reduction of a detailed transformer model [46]. The chaoticversion of the ICA is presented in [47] for global optimization.
Application of ICA to benchmark and large scale DED test casesshow that ICA is capable to find better results comparingwith otherheuristic algorithms. The rest of the paper is organized as follows:
In Section 2 the mathematical formulation of the DED problemis given, considering POZs, ramp-rate limits, valve-point effects andtransmission losses. Section 3 proposes the ICA and describes itsimplementation on DED problems. Section 4 is devoted to casestudies and numerical results. In this section, four application casesare studied, and the corresponding comparisons with the recentlyapplied methods are presented. Conclusions are finally outlined inSection 5.
2. Dynamic economic dispatch problem formulation
The objective function of DED problem is to minimize thetotal production cost over the operating horizon, which can bewritten as:
minTC ¼XTt¼1
XNi¼1
CitðPitÞ (1)
where Cit (in $/h) is the production cost of unit i at time t, N is thenumber of dispatchable power generation units and Pit (in MW) is
point Transmission loss Test cases and time horizon
B. Mohammadi-ivatloo et al. / Energy 44 (2012) 228e240230
the power output of ith unit at time t. T is the total number of hoursin the operating horizon. The production cost of a generation unitconsidering valve-point effects is defined as:
CitðPitÞ ¼ aiP2it þ biPit þ ci þ
���eisin�fi�Pmini � Pit
����� (2)
where ai, bi and ci are the fuel cost coefficients of the ith unit, ei andfi are the valve-point coefficients of the ith unit. The units of theabove coefficients are ($/MW2 h), ($/MWh), ($/h), ($/h) and(1/MW), respectively. Pmin
i (in MW) is the minimum capacity limitof unit i. The added sinusoidal term in the production cost functionreflects the effect of valve-points. The DED problem is non-convexand non-differentiable considering valve-point effects [48].
The objective function of the DED problem (1) should be mini-mized subject to the following constraints:
1. Real power balance
Hourly power balance considering network transmission lossesis written as:
XNi¼1
Pit ¼ PDðtÞ þ PlossðtÞ t ¼ 1;2;.; T (3)
where Ploss(t) and PD(t) (both inMW) are total transmission loss andtotal load demand of the system at time t, respectively. System lossis a function of units power production and the topology of thenetwork which can be calculated using the results of load flowproblem [31] or Kron’s loss formula known as B� matrix coeffi-cients [28]. In this work, B� matrix coefficients method is used tocalculate system loss, as follows:
PlossðtÞ ¼XNi¼1
XNj¼1
PitBijPjt þXNi¼1
Bi0Pit þ B00 t ¼ 1;2;.; T (4)
2. Generation limits of units:
Pmini � Pit � Pmax
i i ¼ 1;.;N; t ¼ 1;2;.; T (5)
where Pmaxi (in MW) is the maximum power outputs of ith unit.
3. Rampup and ramp down constraints: The output power changerate of the thermal unit must be in an acceptable range to avoidundue stresses on the boiler and combustion equipments [49].The ramp rate limits of generation units are stated as follows:
Pit � Pit�1 � URi i ¼ 1;.;N; t ¼ 1;2;.; T (6)
Pit�1 � Pit � DRi i ¼ 1;.;N; t ¼ 1;2;.; T (7)
where URi is the ramp up limit of the ith generator (MW/h) and DRiis the ramp down limit of the ith generator (MW/h). Consideringramp rate limits of unit, generator capacity limit Eq. (5) can berewritten as follows:
max�Pmini ; Pit�1 � DRi
�� Pit � min
�Pmaxi ; Pit�1 þ URi
�i ¼ 1;.;N; t ¼ 1;2;.; T
(8)
4. Prohibited operation zones limits (POZs):
Generating units may have certain restricted operation zonedue to limitations of machine components or instabilityconcerns. The allowable operation zones of generation unit canbe defined as:
Pit˛
8><>:
Pmini � Pit � Pli;1
Pui;j�1 � Pit � Pli;j j ¼ 2;3;.;Mi i ¼ 1;.;N t ¼ 1;2;.;TPui;Mi
� Pit � Pmaxi
(9)
where Pli;j and Pui;j are the lower and upper limits of the jth pro-hibited zone of unit i, respectively. Mi is the number of prohibitedoperation zones of unit i.
3. Imperialist competitive algorithm
The ICA was first proposed in [38]. It is inspired by the imperi-alistic competition. It starts with an initial population called colo-nies. The colonies are then categorized into two groups namely,imperialists (best solutions) and colonies (rest of the solutions). Theimperialists try to absorb more colonies to their empire. The colo-nies will change according to the policies of imperialists. Thecolonies may take the place of their imperialist if they becomestronger than it (propose a better solution). The flowchart ofproposed algorithm which is the same as [38] for solving the DEDproblem is depicted in Fig. 1. The imperialist competitive algorithmis very strong in pattern recognition. This aspect is used in this
B. Mohammadi-ivatloo et al. / Energy 44 (2012) 228e240232
Step 6. Exchange the position of a colony and the imperialist if itis stronger (CPc > IPi). If there are several colonies better thanthe imperialist, then the imperialist will be replaced by the bestof them.Step 7. Compute the empire’s power, i.e. EPi for all empires asfollows:
EPi ¼ w1 � IPi þw2 �Xc˛Ei
CPc (15)
where,w1 andw2 are weighting factors which are selected in a waythat the algorithm will not be trapped into a local Minima. For thisreason, the value of w1 is selected as a number about 10e20% andw2 ¼ 1 � w1.
Step 8. Pick the weakest colony and give it to one of the bestempires (select the destination empire probabilistically basedon its power (EPi)).
Table 4Comparison of optimization results for 5-unit test system (case 1).
Method Minimumcost ($)
Averagecost ($)
Maximumcost ($)
Number oftrial runs
SA [25] 47,356 NA NA NAAPSO [21] 44,678 NA NA NAAIS [19] 44,385.43 44,758.8363 45,553.7707 30GA [20] 44,862.42 44,921.76 45,893.95 30PSO [20] 44,253.24 45,657.06 46,402.52 30ABC [20] 44,045.83 44,064.73 44,218.64 30MSL [7] 49,216.81 NA NA NAHS [4] 44,367.23 NA NA 25HHS [4] 43,154.8554 NA NA 25DE [13] 45,372 43,739 44,214 20DE [14] 45,800 NA NA 10Proposed (ICA) 43,117.055 43,144.472 43,209.533 100
NA denotes that the value was not available in the literature.
Step 9. Eliminate the empires that have no colony.Step 10. If more than one empire remained andIteration� Itermax then Iteration¼ Iterationþ 1 and go to Step. 5Step 11. End.
It should be noted that the Nc and Nimp are given constants andare determined by the expert who uses the algorithm. Typically10e20% of Nc would be a good choice for Nimp. The steps of thealgorithm is shown in Fig. 1.
The operating schedule of each country (for all operatingperiods) is binary coded. The generation value in time t of unit iis calculated as follows: suppose that the row t in the columncorresponding to unit i is vec ¼ [string of binary values].
0 50 100 150 200
4.2
4.4
4.6
4.8
5
5.2
5.4
5.6
Iteration
To
tal C
ost ($)
Fig. 2. Convergence characteristics of the ICA algorithm for 5-unit test system.
Table 5Analysis of objective function for different number of trial runs for 5-unit testsystem.
Fig. 3. Distribution of the objective function for 100 trial runs for 5-unit test system.
B. Mohammadi-ivatloo et al. / Energy 44 (2012) 228e240 233
Pit ¼ ðPmaxi � Pmin
i Þ�½vec0*½2n�1;n ¼ N : 1��=2N þ Pmini . Where N is
the number of generating units. The binary coding ofeach country (which may become an imperialist or not) canbe helpful in easily using the crossover and mutation operatorsof GA.
Table 6Optimal 24-h schedule of ten-unit test system (case 2).
In this section, the proposed ICA is applied to four test systemswith different number of generating units. All the programs aredeveloped using MATLAB 7.1 on a Pentium IV personal computerwith 3.6 GHz speed processor and 2 GB RAM.
4.1. Determination of ICA parameters
The number of colonies is an important factor in determina-tion of optimal solution. If the number of colonies is increasedthen it can better explore the solution space but it wouldaugment the computation burden. In this paper, the number ofcolonies are assumed to be Nc ¼ 100 like as most of the ICApapers [38e40]. The following procedure has been adopted tocalculate optimum value of the mutation and crossover proba-bilities and w1, w2 ¼ 1 � w1. The value of w1 is varied from 0.05 to0.9 the problem is solved for various combinations of mutationand crossover probabilities. There is a unique set of w1 andprobability values the total cost is minimum. The results foroptimal value of w1 for 10-unit test case, which is equal to 0.15are given in Table 2. It should be noted that due to lack of spaceonly the table corresponding to the optimal values are givenhere.
For all cases, the dispatch horizon is selected as one day with24 dispatch periods where each period is assumed to be 1 h. Inthis paper the stopping criteria is defined as reaching to themaximum number of iterations (Itermax ¼ 200 for cases 1e3 and800 iterations for case 4) and when no more than one imperialistexists in the search space. For getting better starting point, firstDED problem is solved without considering valve-point effects,losses, and POZs, which results in a convex quadratic program-ming problem. The initial sets of colonies are generated byrandomly perturbing the results of quadratic programmingproblem.
NA denotes that the value was not available in the literature.
B. Mohammadi-ivatloo et al. / Energy 44 (2012) 228e240234
4.2. Case 1: five unit system
The first test system is a 5-unit test system. The data for thissystem are adapted from [25] and provided in B. In this test system,transmission losses and ramp rate constraints are considered. Thehourly load profile for this case is presented in last column ofTable 3.
The DED problem of 5-unit system is solved using the proposedalgorithm. The valve-point effects, transmission losses, ramp rateconstraints and generation limits are considered in this system. Theprohibited operating zones are not considered in this test case forthe sake of comparison of results with those reported in literatureusing different methods. Table 3 shows the obtained results for thissystem.
0 50 100 150 200
1
1.05
1.1
1.15
1.2
1.25
1.3
1.35
x 106
Iteration
To
ta
l C
os
t ($
)
Fig. 4. Convergence characteristics of the ICA algorithm for 10-unit test system.
These results are compared with several methods presented inrecent literature in terms of minimum cost, mean cost, andmaximum cost over 100 runs in Table 4. The results of the proposedalgorithm are in bold. Themaximum iteration number is selected tobe 200. The convergence characteristic of the proposed algorithm isdepicted in Fig. 2. By investigating the results presented in Table 4,it can be observed that the obtained results outperform the othercited methods for 5-unit test case. In order to analysis thecomputational efficiency of the proposed algorithm, differentsimulations are done considering number of trial runs and itsdistribution. Table 5 show the minimum, average, maximum andstandard deviation (SD) of objective function for different numbersof trial runs. Fig. 3 shows the distribution of the objective functionfor trial run number of 100.
4.3. Case 2: ten unit system without transmission loss
The second test system is ten-unit test system. In this case,generators capacity limits, ramp rate constraint and valve-pointeffects are considered. The transmission losses are ignored in thiscase for sake of comparison. The data for this system is adaptedfrom [25] and provided in B. The hourly load profile for this case ispresented in last column of Table 6.
Table 6 shows the obtained results for 10-unit system withoutconsidering transmission losses. Theminimum cost, mean cost, andmaximum cost of obtained optimal results are compared withresults of previously developed algorithms such as differentialevolution (DE) [13], hybrid EP and SQP [9], Hybrid PSO-SQP [31],deterministically guided PSO (DGPSO) [22], modified hybrid EP-SQP(MHEP-SQP) [34], improved PSO (IPSO) [15], hybrid DE (HDE) [35],improved DE (IDE) [14], artificial bee colony algorithm (ABC) [20],modified differential evolution (MDE) [16], covariance matrixadapted evolution strategy (CMAES) [36], artificial immune system
10 20 30 40 50 60 70 80 90 100
1.018
1.0185
1.019
1.0195
1.02
1.0205
1.021
1.0215
1.022x 10
6
Run Number
To
tal C
ost ($)
Cost
Minimum
Average
Maximum
Fig. 5. Distribution of the objective function for 100 trial runs for 10-unit test systemwithout losses (case 2).
Table 9Optimal 24-h schedule of ten-unit test system (case 3).
B. Mohammadi-ivatloo et al. / Energy 44 (2012) 228e240 235
(AIS) [19], hybrid swarm intelligence based harmony search algo-rithm (HHS) [4], improved chaotic particle swarm optimizationalgorithm (ICPSO) [37], hybrid artificial immune systems andsequential quadratic programming (AIS-SQP) [26], hybrid SOA-SQPalgorithm [27], chaotic sequence based differential evolution algo-rithm (CS-DE) [11], chaotic differential evolution (CDE)method [17],adaptive hybrid differential evolution algorithm (AHDE) [30], andenhanced cross-entropy method (ECE) [24], harmony search (HS)[4], DE [11] and Improved DE [14] in Table 7. Results of the proposedmethod are in bold. Themaximum iteration number and number oftrails are selected to be 200 and 100, respectively. The convergencecharacteristic of the proposed algorithm is depicted in Fig. 4. It canbe observed that the obtained resultswith ICA algorithm is less thanthose of reported in literature. Table 8 show the minimum, average,maximum and standard deviation (SD) of objective function fordifferent numbers of trial runs. Fig. 5 shows the distribution of theobjective function for trial run number of 100.
4.4. Case 3: ten unit system with transmission loss
The data for this case is similar to case 2. In this case, thetransmission losses also considered. The B� matrix coefficients of
Table 10Comparison of optimization results for case 3.
Method Minimum cost ($) Average cost ($) Maximum cost ($)
NA denotes that the value was not available in the literature.
this system can be found in [25] which is given in per unit(100 MW base). The proposed algorithm applied to ten-unit testcase with taking into account the transmission losses. The corre-sponding generation dispatch is presented in Table 9. Theminimum cost, mean cost, and maximum cost of obtained optimalresults over 100 runs are compared with the results of evolu-tionary programming (EP) [34], hybrid EP-SQP (EP-SQP) [34],modified hybrid EP-SQP (MHEP-SQP) [34], genetic algorithm (GA)[20], particle swarm optimization (PSO) [20], improved PSO (IPSO)[15], enhanced cross-entropy method (ECE) [24], artificial beecolony algorithm (ABC) [20] and artificial immune system (AIS)[19] in Table 10. Results of the proposed method are in bold.
0 50 100 150 200
1
1.05
1.1
1.15
1.2
1.25
Iteration
To
tal C
ost ($)
Fig. 6. Convergence characteristics of the ICA algorithm for 10-unit test system withloss.
Table 11Analysis of objective function for different number of trial runs for 10-unit testsystem with considering losses (case 3).
Fig. 7. Distribution of the objective function for 100 trial runs for 10-unit test systemwith losses (case 3).
0 100 200 300 400 500 600 700 800
1.8
1.9
2
2.1
2.2
2.3
2.4
x 106
Iteration
To
tal C
ost ($)
ICA
GA
PSO
Fig. 8. Convergence characteristics of the ICA algorithm compared with PSO and GAfor case 4.
B. Mohammadi-ivatloo et al. / Energy 44 (2012) 228e240236
The convergence characteristic of the proposed algorithm isdepicted in Fig. 6. The minimum, average, maximum and standarddeviation (SD) of objective function for different numbers of trialruns are shown in Table 11. Distribution of the objective function for100 trial runs are depicted in Fig. 7.
4.5. Case 4: 54 unit system
In this case, a 54-unit test system is employed. The data of thissystem is adopted from [50]. The valve-point effects and POZs areconsidered here. Hence this is a large non-convex test case. Theresults obtained using the ICA are presented in Table A.1 for theload demand which is also given in Table A.1. Beside the ICA, twodifferent algorithms (GA [1] and PSO [51]) are used for optimaldispatch of this system. For GA algorithm, mutation and selectionrates are 0.2 and 0.5, respectively. For PSO algorithm cognitive andsocial parameters are equal to 1 and 2.5, respectively. Themaximum iteration number for PSO and GA are same as ICA. Theobtained results over 25 trial runs are compared in Table 12.Results of the proposed method are in bold. The minimum costobtained using ICA is 1,807,081.174 $/day, whereas for the case ofGA and PSO algorithms theminimum costs are 1,834,373.494 $/day
Table 12Comparison of optimization results for case 4.
Method Minimum cost ($) Average cost ($) Maximum cost ($)
and 1,832,121.861 $/day, respectively. With assumption that thedaily load profile is same as studied day during the entire year, itmeans that using ICA will result in 9,139,850.75 $ annual savingcomparing to PSO and 9,961,696.80 $ annual saving comparing toGA. It should be mentioned that in a practical power system thedaily load profile is changing and DED problem should be solvedfor each day separately and the numbers are provided just forillustration of the economic effect of better solution. It is observedthat the performance of the proposed method is better for largescale test cases too, and the proposed method can be used forscheduling of practical large power systems. The convergencecharacteristics of the ICA algorithm compared with PSO and GA forthis case are given in Fig. 8. The maximum iteration number forthis case is selected to be 800.
5. Conclusion
In this paper, the imperialist competitive algorithm (ICA) hasbeen applied to solve the DED problem of generating unitsconsidering the valve-point effects, prohibited operation zones(POZs), ramp rate limits and transmission losses. The effectivenessof the proposed algorithm has been examined by comprehensivestudies on DED problems of different dimensions and complex-ities. At the first, the ICA is tested on five and ten units test systemfor a 24 h time interval. The results justify the applicability of theproposed method for solving the constrained DED with non-smooth cost functions. Also the proposed algorithm is imple-mented on a 54 units test system and the ICA is compared withtwo well-known heuristic algorithm, i.e. GA and PSO. Numericalexperiments on 4 test systems show that the proposed methodcan obtain lower total generation cost, so it provides a newand efficient approach to solve large-scale constrained DEDproblem.
Appendix A. Hourly optimum dispatch of 54-unit test system
Optimal 24-h schedule of 54-unit test system is provided inTable A.1.
B. Mohammadi-ivatloo et al. / Energy 44 (2012) 228e240 237
Appendix B. Test system’s data
Generating units’ characteristics of 5-unit test system areprovided in Table B.1. The B� matrix coefficients of this test systemare as follows.
Generating units’ characteristics of 10-unit test system areprovided in Table B.2. The B�matrix coefficients of this test systemin per-unit in 100 MW base are as follows.
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