Research Article Energy Optimization for Distributed ...downloads.hindawi.com/journals/mpe/2016/6379253.pdf · Energy Optimization for Distributed Energy Resources Scheduling with

Research ArticleEnergy Optimization for Distributed Energy ResourcesScheduling with Enhancements in Voltage Stability Margin

Hugo Morais1 Tiago Sousa2 Angel Perez1 Hjoumlrtur Joacutehannsson1 and Zita Vale2

1Automation and Control Group Department of Electrical Engineering Denmark Technical University (DTU) ElektrovejBuilding 326 2800 Lyngby Denmark2GECAD Knowledge Engineering and Decision Support Research Center Polytechnic Institute of Porto (IPP)R Dr Antonio Bernardino de Almeida 431 4200-072 Porto Portugal

Correspondence should be addressed to Hugo Morais hmmhugogmailcom

Received 30 July 2015 Revised 3 March 2016 Accepted 5 April 2016

Academic Editor Xavier Delorme

Copyright copy 2016 Hugo Morais et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The need for developing new methodologies in order to improve power system stability has increased due to the recent growth ofdistributed energy resources In this paper the inclusion of a voltage stability index in distributed energy resources scheduling isproposed Two techniques were used to evaluate the resulting multiobjective optimization problem the sum-weighted Pareto frontand an adapted goal programmingmethodologyWith this newmethodology the system operators can consider both the costs andvoltage stability Priority can be assigned to one objective function according to the operating scenario Additionally it is possibleto evaluate the impact of the distributed generation and the electric vehicles in the management of voltage stability in the futureelectric networks One detailed case study considering a distribution network with high penetration of distributed energy resourcesis presented to analyse the proposed methodology Additionally the methodology is tested in a real distribution network

1 Introduction

The growing use of distributed generation (DG) in differentvoltage levels has been changing the power systems operationconcept To support the network operation and also to takeadvantage of the distribution energy resources it is importantto develop new operation and management methodologiesOne key aspect to guarantee adequate service levels is thepower system stability margin that is ensured by adequateancillary services These services are traditionally providedby centralized power plants with high power capacity andcoordinated by the system operators in the transmissionlevel However in a near future and considering the growingpenetration of distributed energy resources in medium andlow voltage distribution network the system stability shouldalso be ensured by the system operator in the distributionlevel as well as by the aggregators (eg virtual power plants)that manage the distributed energy resources [1 2]

Behind the DG units the consumers storage systemsand electric vehicles (EVs) are also important to support the

power system stability [3] At the distribution level the dis-tribution system operator and the aggregators can participatein several ancillary services [1 4] such as primary secondarytertiary frequency and voltage control fault-ride-throughcapability the congestion management the power lossesminimization in distribution networks the monitoring thewaveform quality and the islanded operation of networks

Extensive reviews on voltage stability indexes can befound in the literature [5ndash9] with special focus on onlineassessment methods for voltage stability Nevertheless usingthis index to enhance scheduling reconfiguration and dis-patch solutions has shown its potential to improve thesolutions regarding the voltage stability limitation [10ndash12]These approaches make the 119871-index a possible option toenhance the solutions which was initially proposed in [13]based on the power flow equations however recent attentionto its use application and possible improvements has beenreported in [14] A thorough comparison with other indexescan be found in [8] as well as a discussion on 119871-indexlimitations [9]

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 6379253 20 pageshttpdxdoiorg10115520166379253

2 Mathematical Problems in Engineering

The line stability indexes like the 119871119898119899

[15] or the VCPI[16] are more accurate than the 119871-index to predict the voltagecollapse proximity in a real time operation However inthe day-ahead scheduling optimization the main goal ofthe voltage stability index is not based on determining theproximity to the collapse but to influence the distributedresources scheduling for contributing to the system stabilityAs stated in [16] the evolution of the 119871-index is similar asthe other suggested indexes (when the 119871-index increases the119871119898119899

and VCPI also increase) meaning if we optimize the119871-index we are also improving the 119871

119898119899and VCPI indexes

The 119871-index is used in the present study because it is easierto integrate in the optimization problem than the othersuggested indexes With the 119871-index the objective functiondoes not depend on the use of 119876 and consequently not on 119875due to the load consumption in each bus (also depends onthe resources scheduling such as distributed generation andelectric vehicles)

The paper proposes an energy resource schedulingproblem with a multiobjective function incorporating theoperation cost and the voltage stability This multiobjectiveoptimization problem will be applied for a scenario in adistribution network with a high penetration of distributedenergy resources mainly an intensive EVs penetration In theoperation cost different technologies of DG and the use ofEVs with griddable capability are considered also known asvehicle-to-grid (V2G)The use of 119871-index is proposed to dealwith the voltage stability in the joint optimization problemThe 119871-index was initially proposed in [13] based on thepower flow equations Two optimization techniques are pro-posed in this paper for solving the proposed multiobjectiveenergy resource scheduling problem These techniques willdetermine the nondominated solutions of the multiobjectiveoptimization problem namely the weighted-sum methodand an adapted goal programming methodology Thus thenondominated solutions represent the Pareto front that wasproposed in [17] yet the application in engineering andscience fields only began in the end of the seventies [18]Furthermore the goal programming methodology can bevery useful in real application due to the complex character-istics of the objective functions In a regular power systemoperation the system operators can establish a predefinedrange in the operation cost objective function and in thecritical situations (operation near from boundaries) thesystem operators can limit the objective function concerningthe voltage stability index

To demonstrate the effectiveness of the proposedmethodologies concerning voltage stability two studies wereincluded in the first one the sensitivity analysis is performedconsidering variations in the power demand in the voltageangle and in the voltage magnitude on the slack bus (fromthe distribution networkrsquos point of view the reference busis the connection in an upstream network) In the secondanalysis the loadability limit is determined for an hourconsidering three different scheduling objective functions(operation cost 119871-index and multiobjective) allowing thedetermination of the maximum load that can be supplied(voltage stability boundary) considering the voltage controlconstraints This approach is equivalent to the bifurcations

determined with continuation power flow algorithms thatallow to calculate the loadability limit for the power system[19ndash21] Both analyses show the improvements in the energyresource scheduling problem through the incorporation of119871-index as another objective function In addition bothmethodologies are tested in a distribution network with highpenetration of distributed energy resources consideringthe use of electric vehicles allowing the 119871-index and theoperation cost evaluation The weighted-sum method isalso applied to a real distribution network to evaluate itsperformance in a large network

After the Introduction Section 2 presents an overviewconcerning the energy resource scheduling problem Sec-tion 3 focuses on the mathematical formulation and on theimplementation of the proposed methodologies Section 4shows the case study considering a 33-bus distributionnetwork and finally the most important conclusions arepresented in Section 5

2 Energy Resource SchedulingOverview and Contributions

The development of energy resources scheduling methodsconsidering the distributed resources in different voltage lev-els of power systems is an important research topic Typicallythe energy resource scheduling consists in an optimizationproblem to determine the best scheduling to minimize theoperation cost of the available resources [22] However in asmart grid context it is also important to take into accountother aspects than just the economic one such as powerquality voltage stability environmental aspects or the loaddiagramprofileTherefore all these aspects can be included inthe energy resource scheduling providing different solutionsto help the system operators in the decision making process

Several authors have proposed different methodologiesto deal with the energy resource scheduling considering dis-tributed energy resources such as DG and active consumerswith demand response programs and the network operationIn [23] it is described a framework for aggregators to deter-mine the energy resource scheduling based on the concept ofquality-of-service in power system A more complex negoti-ation perspective is presented in [24] considering multilevelnegotiation layers between aggregators and electricitymarketparticipation For amicrogrid level perspective [25] proposesa multiagent base platform allowing the scheduling of thedistributed energy resources

Other works deal with the energy resource scheduling tointegrate the electric vehicles with V2G capability A compre-hensive and exhaustive review is presented in [26] concerningthe impact of EVs in the distribution network In [27] theauthors proved that EVs can improve the management ofintermittent renewable resources such as wind farms andin [28] it is shown that EVs can be used to level the dailyload diagram Wu et al [29] claim that the charging controlin EVs is required for a well accommodation in the powersystem To handle the large number of electric vehiclesseveral artificial intelligence algorithms have been proposed[30ndash32] to provide the scheduling of charge and dischargeenergy from EVs batteries Another innovative perspective

Mathematical Problems in Engineering 3

is proposed in [33] considering a hierarchical model tocoordinate the energy resource scheduling in smart grid withelectric vehicles The integration of plug-in hybrid electricvehicles in microgrids resource scheduling is proposed [34]

The use of multiobjective functions in the energyresources scheduling problems is an important challengeto improve the quality of the obtained solutions Someapproaches are proposed considering the environmentaspects [30 35] or to levelling the load diagram [28 36]in the energy resource scheduling problem However as ispossible to see in [26] few work was developed consideringthe contribution of the distributed energy resources andmainly the electric vehicles to the ancillary services like thevoltage stability The inclusion of a voltage stability indexin the energy resource scheduling problem turns into amultiobjective function because it is a competing objectivewith the operation cost The main contributions of this workare as follows

(1) To propose a multiobjective model to deal with theoperation cost and voltage stability in the energyresource scheduling problem

(2) To use distributed energy resources namely dis-tributed generation and electric vehicles for con-tributing to the power system voltage stability

(3) To apply theweighted-summethodology and to adaptthe goal programmingmethodology to determine thePareto front of the proposed multiobjective energyresource scheduling problem

(4) Test the proposed multiobjective approach in a realdistribution network

3 Energy Resource Scheduling inDistribution Network

The energy resource scheduling is an important task inthe present and the future power systems operation Thegrowing penetration of distributed generation and otherenergy resources such as the electric vehicles increases sig-nificantly the problem complexity [37] The energy resourcescheduling can consider several objective functions most ofthem based on the energy costs or on the entities profitsHowever technical aspects such as the system stabilityare becoming more important in new operation paradigmof the future distribution networks In this paper it isproposed a multiobjective energy resource scheduling forthe distributed energy resources considering two objectivefunctions namely the operation cost and the voltage stabilityusing two different methodologies The first methodologycalled weighted-sum is one of the most popular methodsto solve multiobjectives problems The second implementedapproach is the modified weighted goal programming whichis also used in several real applications These methodologiescan be used by an aggregator with the responsibility tocontrol different distributed resources as well as part of thedistribution network

The goal programming approach can result in non-Paretooptimal solutions [38] and the execution time for each

simulation should be higher due to the increased number ofconstraints (one of the objective function is formulated asconstraint) [39] On the other hand it is possible to obtain anapproach of Pareto front with few simulationsTherefore theuse of goal programming approach was selected consideringthe characteristics of the objective functions In fact whenthe system is operating normally the system operators canestablish a predefined range in the operation cost objectivefunction and in critical situations (operation near to bound-aries) the system operators can define the objective functionconcerning the voltage stability index

31 Operation Cost Objective Function (1198651) The operation

cost function 1198651is composed by several terms concerning

different distributed energy resources useoperation coststhat are given by

min 1198651=

119879

sum

119905=1

[

119873DG

sum

DG=1119888119860(DG119905)119883DG(DG119905) + 119888119861(DG119905)

sdot 119875DG(DG119905) + 119888119862(DG119905)1198752

DG(DG119905)

+

119873SP

sum

SP=1119888SP(SP119905)119875SP(SP119905)

+

119873EV

sum

EV=1(119888Dch(EV119905) + 119888Deg(EV))

sdot 119875Dch(EV119905) minus 119888Ch(EV119905)

sdot 119875Ch(EV119905) +

119873119871

sum

119871=1

119888NSD(119871119905)119875NSD(119871119905)

+

119873DG

sum

DG=1119888GCP(DG119905)119875GCP(DG119905)]

(1)

For the DG units a quadratic function is used which iscommonly employed for fossil fuel units [40] In DG unitsbased on renewable sources (eg wind or solar) the linearterm (119888

119861) of the quadratic function is the only one considered

The cost with energy acquisition to external suppliers is alsoconsidered (119875SP) that allows the balance between the DGEVs and demand in the distribution network In this formu-lation the cost with EVs discharge (119888Dch(EV119905)) is consideredand also the benefit to the aggregator from charging EVs(119888Ch(EV119905)) In addition the battery degradation cost (119888Deg(EV))is considered during the EVs discharging process [41 42]Finally two penalization costs are considered The first one119875NSD(119871119905) penalizes the aggregator when nonsupplied demandsituations occur The second one 119875GCP(DG119905) refers to ldquotake-or-payrdquo contracts violation These contracts are consideredmainly for wind and solar units and the penalization occurswhen generation curtailment is necessary The penalizationterms are important to make a robust mathematical for-mulation in order to handle with critical situations fromhigh consumer demands or high power generation from DGunits


32 Voltage Stability Objective Function (1198652) In the proposed

mathematical formulation the voltage stability is achievedconsidering the load index (119871-index) minimization In [13]the following expression is proposed that determines the 119871-index (119871

119895) considering bus 119894 as a generation bus and bus 119895 as

the load bus

119871119895=

1003816100381610038161003816100381610038161003816100381610038161003816

1 +119880119894

119880119895

1003816100381610038161003816100381610038161003816100381610038161003816

(2)

In [43] and most recently in [44] a new expression isproposed to determine the 119871-index using measurements ofvoltage phasors at the bus and is defined as

119871119895=

4 [119881119894119881119895cos (120579

119894minus 120579119895) minus 1198812

119895cos2 (120579

119894minus 120579119895)]

1198812

119894

(3)

The 119871-index value is between 0 and 1 and the optimalvalue is close to 0 If the maximum 119871-index in the system isless than 1 the system is stable in terms of voltage level Thesystem is unstable if the 119871-index value is above 1 [13] Fromthe optimization point of view the goal is to minimize themaximum value of 119871-index in all buses Basically the 119871-indexminimization involves taking into account the bus far fromthe stressed condition boundaries

The evaluation of 119871-index implies the use of expression(3) in all consumption buses However in the future distribu-tion networks there will be generation connected in severalbuses changing the consumption buses to the generationbuses in someperiods of the day depending on the distributedenergy resources installed and on the generation and loadforecast in each one Therefore all buses are comparedwith the bus connected to the high voltage level in orderto determine the 119871-index where the minimization of thefunction 119865

2 which is the maximum 119871-index in each period

119905 is formulatedmin 119865

2= max (119871 index

(bus119905)) (4)Function 119865

2is a nonconvex function which requires

more time to find the optimal solutionThe epigraph variable119865Aux is used to turn the 119865

2function into a convex one

min 1198652= 119865Aux(119905)

subjected to 119865Aux(119905) ge (119871 index(bus119905))

(5)

where the epigraph variable 119865Aux removes the nonconvexityof function 119865

2(the maximum 119871-index) turning the optimiza-

tion problem simpler to be solved The use of the epigraphvariables is detailed explained and illustrated in [45] turninga nonlinear optimization problem into a linear optimizationproblem

33 Multiobjective Function Weighted-Sum Approach (119865119882)

The weighted-sum method [46] transforms the multiobjec-tive function 119865 into a single one by summing all functions(1198651and 119865

2) where each function is multiplied by a different

weight (120573 and 120575) as it is formulatedmin 119865 = 120573119865

1+ 1205751198652SF

120573 + 120575 = 1

(6)

where the weight factors are between 0 and 1 for giving moreor less relevance to each objective function Additionallythe sum of the two weight factors must be equal to 1 Touniform the objective functions the voltage stability pricefactor (SF) is includedThe voltage stability can be quantifiedas a price signal meaning that themultiobjective function canbe treated as a single objective function to optimize the costIn the present paper the value of SF is equal to the energy costof the most expensive distributed resource as given by

SF = max (119888Res Sche) (7)

where the 119888Res Sche contains the prices of all resourcesscheduled (DG external suppliers and EVs) solving theoptimization problem with just the operation cost function1198651 For the DG units that use a quadratic function it is

considered an average price determined by themultiplicationof the DG maximum generation power and the coefficientsof the quadratic function and then divided by the samemaximum generation power Typically the price selectedwill be the most expensive resource scheduled in the peakperiods because in those periods it has the highest consump-tion power However different expression can be also useddepending on the aggregatorrsquos strategies and on the normalnetwork operation conditions The weighted-sum method isthe most traditional and popular method that parametricallychanges the weights among objective functions to obtain thePareto front [47]

34 Multiobjective Function Goal Programming Consideringthe Utopia Point Approach (119865

119866) The goal programming

was firstly proposed in [48 49] and it is used in a largerange of problems in different areas [50] Several variationsof the original method have been proposed such as thereference goal programming [51] or the Archimedean goalprogramming (also known as weighted goal programming)[52]Thegoal programming consists in the definition of a goalfor the objective function converting the original objectivefunction into a constraint as it is described

min119909isin119883119889

minus119889+

119896

sum

119894=1

(119889+

119894+ 119889minus

119894)

subjected to 119865119895(119909) + 119889

+

119895+ 119889minus

119895= 119887119895

119889+

119895 119889minus

119895ge 0

119895 = 1 2 119896

(8)

In order to copewith variations in the initial goal positive(119889+

119894) and negative (119889minus

119894) deviation variables for each objective

function should be added to the new constraintThe objectivein (8) is to minimize the positive and negative deviationvariables [39] Additionally a weight factor can be multipliedin each deviation variable turning themethod into aweightedgoal programming The Pareto front can be also obtained bythis method through changing the weights of the positive andnegative deviation in each simulation [39 53]

The proposed methodology is based on the goal pro-gramming method with additional changes in order to


Start

(1) (5)

Utopia point (Up)definition

Objective functionsrange definition

Objective functionssteps and fix values

definition

(10) (11)

Pareto front solution

Up

Pareto front

UpStart

F1 optimization (F1p) F2 optimization (F2p)

(F1n) and (F2n)

F1

fix v

alue

sF1

fix v

alue

s

F2 fix values

F2 fix values

F1n

F1n

F2n

F2n

F1p

F1p

F2p

F2p

Point obtained with F1


F1 optimization considering F1 optimization consideringfix values to F1fix values to F2

Figure 1 Flowchart of the proposed goal programming methodology

adapt this approach to the envisaged problem The proposedmethodology uses some principles of the normal-boundaryintersection proposed in [54] The main idea is to establishonly one objective function as constraint trying to optimizethe other objective functionThemain advantage is the abilityto execute simulations only in the predefined ranges for eachparameter Figure 1 presents the flowchart of the proposedgoal programming methodology

In the first step the optimal solution for each objectivefunction is determined using the expressions (1) and (5)respectively The utopia point (Up) is determined usingthe obtained individual points (119865

1119901) and (119865

2119901) The utopia

point also known as ideal point can be described as thepoint with the best result for both objective functions ofthe multiobjective optimization problem [55] Consideringthe utopia point (Up) and the results obtained in the single


objective functions the users can reduce function limits inorder to guarantee more adequate solution that results in thepoints 119865

1119899and 119865

2119899 These values can also be determined by

functions according to the operation contextsAfterwards the fixed values for functions 119865

1and 119865

2are

obtained using functions steps (1198651 step) and (119865

2 step) that arecalculated as

1198651 step =

(1198651119899minus Up)

num steps (9)

1198652 step =

(1198652119899minus Up)

num steps (10)

where num steps is the total number of steps that is definedby the users and they can be different for each objectivefunction In this paper a total number of 19 steps areconsidered for each objective function

Finally the fixed values for each objective function areused to determine new solutions (119865

1198921) and (119865

1198922)

1198651198921= min (119865

1+ 10119865

2119889+ + 01119865

2119889minus) (11)

1198651198922= min (119865

2+ 10119865

1119889+ + 01119865

1119889minus) (12)

In (11) the objective is to minimize the function 1198651

obtained with (1) plus the positive (1198652119889+) and negative (119865

2119889minus)

deviations in objective function 1198652 The negative deviation is

multiplied by 01 which means an incentive to the 119871-indexreductionThe positive deviation is multiplied by 10 to penal-ize the increase of the 119871-index because the positive deviationis only used to guarantee the problem feasibility Additionallyit is necessary to transform the objective function 119865

2(5) into

a constraint

1198652= 119865Aux + 1198652119889+ minus 1198652119889minus 119865

2119889+ 1198652119889minus ge 0 (13)

The same approach is used to obtain a new solution (1198651198922)

optimizing the objective function 1198652(12) considering the

expression (5) for function 1198652plus the positive (119865

1119889+) and

negative (1198651119889minus) deviations in objective function 119865

1 Then the

objective function (1) is converted into a constraint

1198651=

119879

sum

119905=1

[

119873DG

sum

DG=1119888119860(DG119905)119883DG(DG119905) + 119888119861(DG119905)119875DG(DG119905)

+ 119888119862(DG119905)119875

2

DG(DG119905) +

119873SP

sum

SP=1119888SP(SP119905)119875SP(SP119905)

+

119873EV

sum

EV=1(119888Dch(EV119905) + 119888Deg(EV)) 119875Dch(EV119905)

minus 119888Ch(EV119905)119875Ch(EV119905) +

119873119871

sum

119871=1

119888NSD(119871119905)119875NSD(119871119905)

+

119873DG

sum

DG=1119888GCP(DG119905)119875GCP(DG119905)] + 1198651119889+ minus 1198651119889minus

1198651119889+ 1198651119889minus ge 0

(14)

35 Problem Constraints The energy resource schedulingshould use an accurate model of the network to achievescheduling results that are feasible in the electric network(avoiding lines congestion and bus voltage violations) Forthis purpose an AC power flow is included as constraint inthe energy resource scheduling problem The active powerbalance equation establishes that the active power injected ineach bus 119894 is equal to the active power generation minus theactive power demand in the same bus

119873119894

DG

sum

DG=1(119875119894

DG(DG119905) minus 119875119894

GCP(DG119905)) +

119873119894

SP

sum

SP=1119875119894

SP(SP119905)

+

119873119894

EV

sum

EV=1119875119894

Dch(EV119905) minus

119873119894

119871

sum

119871=1

(119875119894

Load(119871119905) minus 119875119894

NSD(119871119905))

minus

119873119894

EV

sum

EV=1119875119894

Ch(EV119905)

= 1198661198941198941198812

119894(119905)

+ 119881119894(119905)sum

119895isin119879119871119894

119881119895(119905)(119866119894119895cos 120579119894119895(119905)

+ 119861119894119895sin 120579119894119895(119905))

(15)

The active power injected in bus 119894 is defined as the sumof all power flow through the lines that connect to this busthe power generation is the sum of the power generationfromDG and external suppliers and EVs discharge the activepower demand is equal to the sum of the EVs charge plusthe forecast consumers demand in each bus less than thenonsupplied energy A more detailed loads model should beused for dynamic evaluation of voltage stability Howeverin the scheduling process the use of forecast values allowsdetermining the impact of the power demand in the voltagestability indexes [5ndash9]

In addition the reactive power injected in bus 119894 is alsoequal to the reactive power generationminus the active powerdemand

119873119894

DG

sum

DG=1119876119894

DG(DG119905) +

119873119894

SP

sum

SP=1119876119894

SP(SP119905)

minus

119873119894

119871

sum

119871=1

(119876119894

Load(119871119905) minus 119876119894

NSD(119871119905))

= 119881119894(119905)sum

119895isin119879119871119894

119881119895(119905)(119866119894119895sin 120579119894119895(119905)

minus 119861119894119895cos 120579119894119895(119905))

minus 1198611198941198941198812

119894(119905)

forall119905 isin 1 119879 forall119894 isin 1 119873Bus 120579119894119895(119905) = 120579119894(119905) minus 120579119895(119905)

(16)

where 120579119894119895(119905)

is the voltage angle difference between bus 119894 and119895 119879119871119894 contains the set of all lines that are connected to thebus 119894 119866

119894119895and 119861

119894119895represent the real and imaginary part of the

admittance matrix corresponding to the 119894 row and 119895 columnrespectively


For the AC power flow model it is also to establish themaximum and minimum limits for the voltage magnitudeand angle respectively

119881119894

Min le 119881119894(119905) le 119881119894

Max forall119905 isin 1 119879

120579119894

Min le 120579119894(119905) le 120579119894


(17)

Before solving the energy resource scheduling problem aslack bus is necessarily selected in the distribution networkFor this slack bus a fixed value for the voltage magnitude andangle is specified

Finally the line thermal limit (upper limit) is establishedfor the power flow from bus 119894 to bus 119895 and vice versa as isdefined

100381610038161003816100381610038161003816119881119894(119905)[119910119894119895119881119894119895(119905)

+ 119910sh 119894119881119894(119905)]lowast100381610038161003816100381610038161003816le 119878

Max119879119871 119894 to 119895

100381610038161003816100381610038161003816119881119895(119905)[119910119894119895119881119895119894(119905)

+ 119910sh 119895119881119895(119905)]lowast100381610038161003816100381610038161003816le 119878

Max119879119871 119895 to 119894

forall119905 isin 1 119879 forall119894 119895 isin 1 119873Bus forall119879119871 isin 1 119873119879119871 119894 = 119895 119881119894119895(119905)

= 119881119894(119905)minus 119881119895(119905)

(18)

The distribution network can be connected to upstreamnetworks by transformers that change the voltage level fromhigh voltage (HV) to medium voltage (MV) These HVMVtransformers have an upper limit (maximum capacity)

radic(

119873119894

SP

sum

SP=1119875119894

SP(SP119905))

2

+ (

119873119894

SP

sum

SP=1119876119894

SP(SP119905))

2

le 119878MaxTFR HV MV(119894)

forall119905 isin 1 119879 forall119894 isin 1 119873Bus

(19)

Similarly the buses in the distribution network can havetransformers from MV to low voltage (LV) connecting smalldistributed energy resources such as photovoltaic units andEVs to the MV side of bus 119894

radic1198752

TFR MV LV(119894119905) + 1198762

TFR MV LV(119894119905) le 119878MaxTFR MV LV(119894)


(20)

The 119875TFR MV LV and 119876TFR MV LV are determined by

119875TFR MV LV(119894119905) =

119873119894

DG

sum

DG=1(119875119894

DG(DG119905) minus 119875119894

GCP(DG119905))

+

119873119894

EV

sum

EV=1(119875119894

Dch(EV119905) minus 119875119894

Ch(EV119905))

minus

119873119894

119871

sum

119871=1

(119875119894

Load(119871119905) minus 119875119894

NSD(119871119905))

119876TFR MV LV(119894119905) =

119873119894

DG

sum

DG=1119876119894

DG(DG119905)

minus

119873119894

119871

sum

119871=1

(119876119894

Load(119871119905) minus 119876119894

NSD(119871119905))

(21)

Regarding the DG units the minimum and maximumlimits for active reactive and apparent power generation areconsidered

119875Min(DG119905)119883DG(DG119905) le 119875DG(DG119905)

le 119875Max(DG119905)119883DG(DG119905)


le 119876Max(DG119905)119883DG(DG119905)

radic(119875DG(DG119905))2

+ (119876DG(DG119905))2

le 119878Max(DG119905)

forall119905 isin 1 119879 forallDG isin 1 119873DG

(22)

where 119883DG(DG119905) is used to control if the DG unit will beturned on or off considering the optimized solution

Regarding DG units with ldquotake-or-payrdquo contracts withthe system operator mainly renewable sources the systemoperator is mandatory to dispatch all the forecasted powerthat is given by

119875DG(DG119905) + 119875GCP(DG119905) = 119875DGForecast(DG119905)

forall119905 isin 1 119879 forallDG isin 1 119873DG (23)

In the case of external supplier a maximum limit for theactive and reactive generation respectively is defined as

119875SP(SP119905) le 119875Max(SP119905)

119876SP(SP119905) le 119876Max(SP119905)

forall119905 isin 1 119879 forallSP isin 1 119873SP

(24)


Regarding EVs the amount of energy stored at the endof period 119905 is determined

119864Stored(EV119905) = 119864Stored(EV119905minus1) minus 119864Trip(EV119905) + 120578119888(EV)119875Ch(EV119905) minus1

120578119889(EV)

119875Dch(EV119905)

forall119905 isin 1 119879 forallEV isin 1 119873EV Δ119905 = 1 119905 = 1 997888rarr 119864Stored(EV119905minus1) = 119864Initial(EV)

(25)

where 119864Trip(EV119905) corresponds to the typical daily travel profilefor reducing the energy stored in the battery when theEV is in travel Thus the system operator must ensurethe energy required for the EV user to travel in the timehorizon of the energy resource scheduling problem A tripforecast can be considered in the 119864Trip(EV119905) according to thehistory consumption profile for each EV [56] The chargeand discharge efficiency(120578

119888(EV) and 120578119889(EV)) are included inthe batteries balance equation (25)

The energy resource scheduling problem also considersthe maximum and minimum energy stored in the EVsbatteries

119864BatMin(EV119905) le 119864Stored(EV119905) le 119864BatMax(EV119905)

forall119905 isin 1 119879 forallEV isin 1 119873EV (26)

where 119864BatMax(EV119905) is related to the batteryrsquos capacity and119864BatMin(EV119905) corresponds to the minimum amount of energystored in the battery and in which period that energy must

be guaranteed EV users and system operator must use anadequate communication system to exchange informationabout several parameters such as the 119864BatMin(EV119905) [57]

The chargedischarge rates have their own upper limitsthat are formulated as

119875Ch(EV119905) le 119875Max(EV119905)119883Ch(EV119905)

119875Dch(EV119905) le 119875Max(EV119905)119883Dch(EV119905)

forall119905 isin 1 119879 forallEV isin 1 119873EV

(27)

where the maximum chargedischarge limits depend on theconnection points (normal charge or fast charge points) Forinstance EVs can be connected in a single phase (eg athome) therefore the chargedischarge limit is lower thanwhen EVs are connected in three-phasemode (eg a parkinglot at the work)

Finally the sum of the two binary variables for charge anddischarge must be lower or equal to 1 to avoid a simultaneouscharge and discharge in the same period 119905

119883Ch(EV119905) + 119883Dch(EV119905) le 1 forall119905 isin 1 119879 forallEV isin 1 119873EV 119883Ch(EV119905) 119883Dch(EV119905) isin 0 1 (28)

The on-load tap changer (OLTC) in the HVMV powertransformer ismodeled (29) considering different steps (STP)and a correspondent binary variable (119883TFR(STP119905)) for eachone Equation (30) assures that only one step is used and thepower voltage in the power transformer (119881

0(119905)) is defined in

(31) Consider the following

Δ119881TFR(119905) = 119881STPTFR(119905)119883TFR(STP119905) (29)

119873STP

sum

STP=1119883TFR(STP119905) = 1 (30)

119881119894(119905)= 119881

Base119894(119905)

+

119873STP

sum

STP=1Δ119881119894

TFR(STP119905)


(31)

36 Software and Solvers Used Both methodologies havebeen implemented inMATLAB software interconnectedwiththe general algebraic modeling system (GAMS) [58] MAT-LAB is used to process all the data regarding the resourcescharacteristics and contracts and afterwards to organize all

the results GAMS is used to run the optimization algorithmsGAMS offers a large set of solvers in the same platform Theproposed energy resource scheduling is classified as a mixed-integer nonlinear programming (MINLP) problem

In GAMS software the DICOPT solver was used [59]because it solves the MINLP problems by splitting them intomixed-integer programming (MIP) and nonlinear program-ming (NLP) subproblems The coordination between thesetwo subproblems is important to obtain the optimal solutionof a MINLP problem and DICOPT coordinates MIP andNLP solutions through ldquoOuter approximationrdquo ldquoEqualityrelaxationrdquo and ldquoaugmented penaltyrdquo These coordinationmechanisms will create and handle relaxed problems tobe solved by the MIP and NLP solvers afterwards theobtained solutions are penalized then the relaxed problemsare decreased until the stopping criteria of DICOPT isreached

The two solvers used to solve the two subproblems areCPLEX for the MIP subproblems and the CONOPT for theNLP subproblems DICOPT uses an iterative process thatstops when the MIP and NLP subproblems return solutionswith a difference less than a predefined error that has beenfixed at 001 The local optima solutions are the main


Table 1 Distributed energy resources information

Resource Units Total installed power (kW) Price scheme (mukWh)Max Mean Min

PV 32 1320 0254 0187 0110Wind 5 505 0136 0091 0060Small hydro 2 80 0145 0117 0089CHP 15 725 0105 0075 0057Biomass 3 350 0226 0201 0186WTE 1 10 0056 0056 0056Fuel cell 8 440 0200 0055 0010Total DG 66 3430 mdash mdash mdashExternal suppliers 10 2900 0150 0105 0060EVs charge 2000 mdash 0EVs discharge 0040

Table 2 Electric vehicle technical information [61ndash67]

EV model Battery Charge rate (kW) Discharge rate (kW)Capacity (kWh) Range (km) Slow Fast

MiEV 160 160 3 50 3C-Zero 160 150 22 275 22Fluence ZE 220 185 3 43 3Leaf 240 160 66 50 66Kangoo ZE 220 170 3 43 3Zoe 220 150 3 43 3Prius 44 20 3 mdash 3

obstacle to overcome by DICOPT due to the nonconvexitiesthat characterize a MINLP problem Therefore the solverdoes not guarantee the global optimum even incorporatingalgorithms to handle this kind of obstacle

4 Case Study

The present section shows the main results of the proposedmethodologies and it is divided into five subsections Sec-tion 41 presents the information and input data used in thefirst case study of this paper Section 42 is related to thePareto front results of the two proposed methodologies tosolve themultiobjective energy resource scheduling problemSection 43 presents an evaluation concerning the voltagestability margin of different Pareto front solutions deter-mined by the adapted goal programming methodology InSection 44 the behaviour of the adapted goal programmingmethodology in different operation scenarios is evaluatedsuch as voltage angle variation voltage magnitude variationand load consumption variation Section 45 presents thePareto front results of the proposed weighted-sum method-ology for a real scenario

41 33-Bus Distribution Network Description In the firstcase study a 33-bus distribution network in [60] is usedThe network supplies 218 consumers including domestic

commercial and industrial consumers The network has 66DG units spread over the buses 32 photovoltaic (PV) 15combined heat and power (CHP) 8 fuel cell 5 wind 3biomass 2 small hydro and 1 waste-to-energy (WTE) unitThe network is connected to aHVupstreamnetwork throughbus 0The distribution system operator or the aggregator cannegotiate energy with external suppliers in bilateral negotia-tions andor electricity markets The negotiated energy flowsto distribution network through bus 0 Figure 2 shows the33-bus distribution network In Table 1 the power capacityand the energy cost of each generation technology of externalsuppliers and of EVs are presented

Regarding the EVs the management of 2000 EVs isconsidered that can come and go from the network Table 2presents the seven EVsmodels that are used in this case study[61ndash67] A simulation tool [56] is used to generate the dailytraveling profiles for the 2000 EVs This simulator obtainsthe bus location that each EV will have to connect in thedistribution network In terms of the batteriesrsquo degradationcost a cost of 003mukWh considering the work proposedin [41 42] has been defined In addition the EVs dischargecost was established at 004mukWh This value is basedon the profits of the EVrsquos owner however an extra incentiveestablished in the contracts should be considered to stimulatethe participation in these events


0

11819

20

21

2

3

4

567

8

11

910

1213

1415

16

17

2223

24

2526

27

28

29

30

31

32

50kW

30kW

30kW 9

kW 9kW

9kW

30kW

9kW

9kW

30kW

30kW

30kW

30kW

9kW

30kW

30kW

9kW9

kW3

kW

30kW9

kW

3kW

9kW

9kW

30kW

30kW

9kW

3kW

9kW

9kW

3kW

30kW

30kW

10kW40

kW50kW

100kW

50kW

10kW

100kW

50kW

10kW 200

kW

10kW

25kW 25

kW 100kW

150kW

100kW

100kW 100

kW

50kW

50kW

50kW50

kW

50kW

100kW

25kW200

kW10

kW

50kW

10kW

30kW

200kW

25kW25

kW

PV panel

Wind

WTE

CHP

Fuel cell

Biomass

Hydro

EV with V2G sim

sim

sim

simsim

sim

sim

Figure 2 33-bus distribution network configuration in 2040 [60]

42 33-Bus Distribution Network Results Considering thedescribed scenario two proposed methodologies were per-formed namely the weighted-sum and the adapted goalprogramming methodologies In the weighted-summethod-ology 500 runs have beenmade and in each run the weightedfactors changed from 0 to 1 in steps of 0002 (considering thatin each run the sum of the two weights must be equal to 1)For the adapted goal programming methodology 40 pointswere tested The simulations were performed on a computerwith two processors Intel Xeon E5645 240GHz each onewith two cores 24GB of random-access-memory

Figure 3 shows the Pareto front obtained by the weighted-sum methodology

The weighted-sum method has been able to find 360nondominated solutions most of themwith very close valuesin both objective functions Considering these results it ispossible to conclude that 500 weights is an excessive numberthat influences the execution time of this methodology Theoperation cost changed between 6933 and 9054mu andthe 119871-index changed between 00305 and 01613 In Figure 4

L-in

dex

All solutionsWeighted-sum Pareto (500 points)

002004006008

01012014016018

7100 7300 7500 7700 7900 8100 8300 8500 8700 8900 91006900Operation cost (mu)

Figure 3 Pareto front for weighted-sum method considering 500different weights

the Pareto front comparison between the two implementedmethodologies is presented


Weighted-sum Pareto (500 points)Objective function 1

Objective function 2

ldquoBestrdquo solution

Weighted-sum Pareto (40 points)

L-in

dex

002004006008

01012014016018


Proposed method (40 points)

Figure 4 Pareto front for weighted-summethod and proposed goalprogramming

Simulations

L-index 00642Best solution050

100

150

200

250

Nor

mal

ized

dist

ance

Operation cost 7236mu

mdash

Figure 5 Normalized distance to the utopia point

Considering the excessive number of weights tested inthe first simulation (Figure 3) 40 weights were tested for theweighted-sum method Using the same number of simula-tions in both methods allows getting a better comparisonAs shown in Figure 4 the curves are overlapped and it isvery hard to see differences between the two methodologiesThe overlapping of Pareto front curves allows to concludethat both proposed methodologies are suitable to solvethe multiobjective energy resource scheduling consideringthe operation cost and the voltage stability In the goalprogramming methodology the reduction of the solutionsranges (119865

1119901= 1198651119899

and 1198652119901

= 1198652119899) was not considered

However in a real application it is possible to reduce the rangeof the operation costs due to the low variation presented by119871-index In this paper the normalized distance to the utopiapoint was used The normalized distance of each obtainedsolution to the utopia point is presented in Figure 5

The Pareto front gives useful multiple choices to thedecisionmaker (ie distribution system operator or resourceaggregator) However the decision maker must choose theldquobestrdquo nondominated solution that satisfies its requirementsThe choice should be made according to a specific strategydepending of each decision maker Some methods have beenproposed to select the best choice of the Pareto front [68] Inthis method the ldquobestrdquo solution corresponds to an operation

0100200300400500600700

1 3 5 7 9 11 13 15 17 19 21 23

Ope

ratio

n co

st (m

u)

Time (h)

Objective function 1Objective function 2ldquoBestrdquo solution

Figure 6 Operation cost in each period


0002004006008

01012014016018

L-in

dex

3 5 7 9 11 13 15 17 19 21 231Time (h)

Figure 7 119871-index in each period

cost of 7236mu and to an 119871-index of 00642 as is shown byFigure 5 In Figures 6 and 7 the comparisons of the operationcost and the 119871-index for the ldquobestrdquo solution and for thesolutions considering each objective function are presented

Regarding to the operation cost a high variation occursin successive periods when the minimization of the 119871-indexobjective function is only consideredHowever the 119871-index isconstant throughout the daywith a very small value of 00305The 119871-index value represented in Figure 7 corresponds tothe maximum value in each hour and it can be obtainedin different buses in each period Results of the ldquobestrdquosolution also present a constant behaviour of 119871-index andthe operation cost is according to the expected values for thisparameter

Looking more carefully at Figure 6 it is possible to seethat in some periods the operation cost obtained in the ldquobestrdquosolution is higher than the cost obtained by the optimal solu-tion in the minimization of 119871-index This happens becausethe optimization process considers the 24 periods and notan optimization for each period In fact one of the mostdifficult aspects in the optimization is to schedule the chargeand discharge periods of EVs For instance the operationcost in period 16 is higher in the ldquobestrdquo solution than in


302826242210 12 14 16 18 20 3286420Bus

1101102103104105106107

Volta

ge m

agni

tude

(pu)


Figure 8 Voltage magnitude in hour 20

0500

100015002000250030003500400045005000

Pow

er (k

W)

Vehicle dischargeExternal supplierFuel cellSmall hydroMSWBiomass

CHPWindPhotovoltaicLoad consumptionTotal consumption

3 5 7 9 11 13 15 17 19 21 231Time (h)

Figure 9 Energy resource scheduling considering the ldquobestrdquo solu-tion

the optimal solution for the individual 119871-index optimizationdue to the higher amount of scheduled EVs charging in theldquobestrdquo solution which increase the energy stored in the EVsbatteries This energy will be used in the remaining periodsavoiding the use of more expensive resources The voltagemagnitude in each bus is presented in Figure 8

As expected the use of 119871-index function results in amore stable voltage profile in each bus Figure 9 presents theenergy resource scheduling by each technology for the ldquobestrdquosolution

Figure 9 shows the high impact of the PV generationduring the hourswith a high solar radiation (aroundmidday)Another important aspect is the use of electric vehiclesdischarge at the end of the day (ie peak periods) The useof EVs discharge in these peak periods can be restrictedby the use of these vehicles because approximately 96 ofthe time cars are parked and in only 4 of the time carsare used for transportation [69] In the peak periods manyvehicles are traveling However many of them are parkedwith the possibility of being connected in the electric network

1 3 5 7 9 11 13 15 17 19 21 23Time (h)

Vehicle chargeVehicle dischargeVehicle energy state

minus1500

minus1000

minus500

0

500

1000

1500

2000

Pow

er (k

W)

minus15000

minus10000

minus5000

0

5000

10000

15000

20000

Ener

gy (k

Wh)

Figure 10 Electric vehicles charge and discharge considering theldquobestrdquo solution

and then discharging the batteries energy if necessary Inthe present simulation most of the EVs have trips betweenperiods 18 to 22 with an average time of one hourThismeansthere are lots of vehicles connected to the network in eachperiod Furthermore most of the EVs have high amountsof energy storage in their batteries due to the previousscheduling allowing the use of EVs discharge when EVs areconnected to the network Figure 10 shows the EVs chargeand discharge and energy stored in the battery for the ldquobestrdquosolution

In Figure 10 one can see that EVs charge their batteryduring the night (ie off-peak periods) but also in hourswith high PV generation because this case study consideredthat PV has ldquotake-or-payrdquo contracts so the aggregator ordistribution system operator must fully dispatch the energygenerated by PV Another important aspect is the use ofcharge and discharge processes in the same periods In facteach EV only charges or discharges in each period Howeverthe optimization schedules the charge of some EVs and thedischarge of other EVs in order to guarantee lower 119871-indexvalues This means that EVs can be used as a resource toimprove the voltage stabilitymargin in the future distributionnetworks Some detailed information concerning the perfor-mance and execution time of the proposed methodologies isshown in Table 3

The high number of variables including the discrete onesleads to a very complex problem with high execution timeThe execution time for one run (a simulation with specificweights) is in average of around 25 minutes However theexecution time of a simulation can change between 13 and58 minutes The total execution time is higher than 27 hoursfor the weighted-sum method with 500 different weights (orsimulations) When 40 different weights are used the timebecomes more acceptable to a little bit more than 2 hoursThese execution times are only possible due to the use ofparallel processing (8 cores in this case) In the adaptedgoal programming method the time is higher (33 hours)because it is necessary to determine the results for eachobjective function regardless of defining the simulation stepsand continuing with the rest of the methodrsquos process (see


Table 3 Performance and execution time

Performance indicators Weighted-sum method Proposed goal programming500 weights 40 weights

Average execution time for one run (in minutes) 241867 234567 261942Total simulation time (in hours) 277693 22725 32931Memory used for one run Around 60MBNumber of variables 36796Number of discrete variables 11424

Table 4 Voltage stability margin in period 21

Pareto front point 120582

Minimum 1198651

1297Minimum 119865

22115

ldquoBestrdquo solution 2107

Figure 1) In this case the parallel processing only can beused after determining the simulation steps and fixed valuesThe memory space is not a critical aspect in the optimizationprocess when a single simulation is considered However itcan be a constraint when the parallel processing is used

43 Voltage StabilityMargin To determine the voltage stabil-itymargin it is necessary to use the optimization formulationgiven in [19 21]

max 120582119905

subjected to 119875load = 120582119905119875load1015840(119871119905)

119876load = 120582119905119876load1015840(119871119905)

forall119905 isin 1 119879

forall119871 isin 1 119873119871

(32)

The maximum loadability will be given by the loadincrement parameter 120582

119905for each hour and it is a parameter of

interest for assessing voltage stability [19] and as described in(32) the loads are considered with a constant powermodel forthe calculations In (32) the limits in the generators capacityand the network power balance equations are included asproblem constraints these are presented in (15) (16) and (22)to (24)The calculation of 120582was performed for hour 21 whichis the peak hour and the hourwith higher differences betweenthe solutions with different objective functions

Three scenarios were studied considering the use of singleobjective functions (operation cost and 119871-index optimiza-tion) and the ldquobestrdquo solution for the multiobjective functionTable 4 shows the obtained solutions in period 21

These results show that the proposed multiobjectiveenergy resource scheduling methodology allows increasingthe voltage stability margin effectively and thus including the119871-index helps improving the solution when compared to thescheduling obtained only with theminimumoperational cost(minimum 119865

1)

Distribution network

Slackgenerator

Bus 0

Systemslack bus slack bus

ldquoVirtualrdquo

ZTH

Figure 11 Transmission system equivalent

44 Sensitivity Analysis In order to evaluate the proposedmethodology behaviour in different operation scenariosthree different sensitivities analyses are performed One ofthe key points of these analyses is the assumption that dis-tribution network is connected to the transmission networkThe transmission network can be represented by itsTheveninequivalent that is shown in Figure 11 in which 33-bussdistribution network HVMV connects to the transmissionnetwork from bus 0

In the distribution network scheduling process the slackbus is represented by the connection to the transmissionnetwork However this connection can be seen as a ldquovirtualrdquoslack bus due to the existence of a system slack bus In prac-tice the ldquovirtualrdquo will impose the operation conditions of dis-tribution network but depends on the Thevenin impedanceof all system Taking this aspect into account the ldquovirtualrdquoslack bus can have different values of voltage magnitude andvoltage angle according to the system operation scenario

The analyses consist in the variation of the differentparameters In the first one (Figures 12 and 13) the ldquovirtualrdquoslack bus voltage angle is changed In the second analysis theldquovirtualrdquo slack bus voltage magnitude is changed (Figures 14and 15) In the third analysis the consumers power demandis changed (Figures 16 and 17)

To perform these analyses some assumptions are consid-ered

(i) The 119871-index is computed considering the powersystem slack bus as reference

(ii) The generation capacity has been increased two times

(iii) The load consumption has been increased two timesin analyses 1 and 2


0

02

04

06

08

1

12

L-in

dex


5 10 15 20 25 30 35 40 45 50 55 600ldquoVirtualrdquo slack bus voltage angle (∘)

Figure 12 119871-index sensitivity considering the ldquovirtualrdquo slack busvoltage angle variation

0 5 10 15 20 25 30 35 40 45 50 55 6088008850890089509000905091009150920092509300

Ope

ratio

n co

st (m

u)

ldquoVirtualrdquo slack bus voltage angle (∘)


Figure 13 Operation cost sensitivity considering the ldquovirtualrdquo slackbus voltage angle variation

(iv) The ldquovirtualrdquo slack bus voltage and angle in analysis 2is of 50∘

(v) The lines thermal limits are increased to avoid viola-tions

By analysing Figures 12ndash17 it is possible to conclude thatin general theminimization of the119871-index objective functionprovides better results concerning the 119871-index value andthe minimization of the operation cost function providesbetter results in terms of operation cost This is a logicaland expected conclusion Most interesting is the analysisof the ldquobestrdquo solution results In fact the results obtainedusing the multiobjective function give a solution for 119871-indexnearby the 119871-index objective function and solutions nearbythe operation cost objective function in terms of operationcost

In Figures 12 and 13 it is possible to see that the 119871-indexincreases with the voltage angle increasing and the differ-ences between the objective functions decrease Howeverthe use of the operation cost objective function (119865

1) results

in 119871-index values higher than 1 for voltage angles higherthan 55∘ whereas the use of 119871-index objective function (119865

2)

or the ldquobestrdquo solution objective function keeps the values


09

095

1

105

11

115

L-in

dex

104

103

102

101

100

099

098

097

096

095

094

093

092

105

ldquoVirtualrdquo slack bus voltage magnitude (pu)

Figure 14 119871-index sensitivity considering the ldquovirtualrdquo slack busvoltage magnitude variation

ldquoVirtualrdquo slack bus voltage magnitude (pu)Objective function 1Objective function 2ldquoBestrdquo solution

105

104

103

102

101

100

099

098

097

096

095

094

093

092

71007110712071307140715071607170718071907200

Ope

ratio

n co

st (m

u)

Figure 15 Operation cost sensitivity considering the ldquovirtualrdquo slackbus voltage magnitude variation

of 119871-index below 1 It is important to mention that in realoperation values of 119871-index higher than 1 are not possiblebecause the system would collapse However in the presentanalysis considering the day-ahead horizon this type of valuescan be obtained mainly because the operation conditions aresignificantly increased to analyse their limits

Figure 13 shows that the operation cost is more or lessconstant in the simulations This means that the operationcost is independent from the voltage angle

Regarding the voltage magnitude analysis from Figures14 and 15 it is possible to see that the 119871-index is higherthan 1 in three situations when the objective function 1 isused In fact the evolution is not constant This happensbecause the objective function only considers the costs andin some cases it can schedule resources at the same priceyet with different impact on the 119871-index as in this caseWhen the 119871-index is included in the objective function thevalues obtained for the 119871-index have a constant evolutionremaining below the limit As in the voltage angle evaluationthe operation costs are more or less constant

Analysing Figures 16 and 17 regarding the load sensitivityit is possible to see that the use of objective function 2 limitsthe impact of the load increase in the 119871-index In this case


100

120

140

160

180

200

220

240

260

280

300

320

340

360

380

400

Load ()

094

0945

095

0955

096

0965

097

L-in

dex


Figure 16 119871-index sensitivity considering the total load consump-tion variation

550060006500700075008000850090009500

1000010500

Ope

ratio

n co

st (m

u)

120

140

160

180

200

220

240

260

280

300

320

340

360

380

400

100

Load ()Objective function 1Objective function 2ldquoBestrdquo solution

Figure 17 Operation cost sensitivity considering the total loadconsumption variation

the electric vehicles charge and discharge are scheduled indifferent periods trying to minimize the impact of the loadincrease in 119871-index When the operation costs are includedin the objective function the 119871-index increases remainingbelow the limit As expected the cost increases with the loadincreaseThe load increases four times and the cost increasesaround two times This fact can seem odd but it can bejustified by the total system demand which is composed ofthe load consumption and the electric vehicles charge Onthe other hand the PV panels have ldquotake-or-payrdquo contractswith a price higher than the average This means that theload increase is supplied by the generation units or externalsuppliers with a lower generation cost reducing the impactin the operation cost

45 Real Scenario Analysis Another important aspect toevaluate the proposedmethodology is its application in a realscenario The proposed methodology assumes a higher pen-etration of distributed generation and electric vehicles Thisrepresents a future vision of power systems operation In thisway themethodologywas tested in a real network concerningthe network characteristics considering a scenario of DGand EVs penetration according to [70 71] The considerednetwork (Figure 18) is a real 30 kV distribution network

supplied by one high voltage substation (6030 kV) with 90MVA of maximum power capacity distributed by 6 feederswith a total number of 937 buses and 464 MVLV powertransformers [72]This distribution network has already beenused for many years and it has suffered many reformulationsIt is partly composed of aluminium conductors and copperconductors and the distribution is made by power lines andunderground cables The study results in 548 DG units mostof them using photovoltaic panels and 464 aggregated loadsin MVLV power transformers

Figure 19 shows the obtained results using the weighted-sum method considering 200 simulations with differentweights

The weighted-sum method found 70 nondominatedsolutions The operation cost changes between 93131 and94161mu and the 119871-index between 001293 and 09512muThis means that the utopia point is defined by an operationcost of 93131mu and an 119871-index of 001293mu The ldquobestrdquosolution (Figure 20) is given by the weight factor (120573) of 069resulting in an operation cost of 93168mu and an 119871-indexof 003506 Figure 21 shows the differences in the hourlyoperation costs

The cost increases 187 with objective function 2 andincreases only 004 when the ldquobestrdquo solution objectivefunction is used Regarding 119871-index the value obtained withobjective function 2 is constant during the day and equal to001242 When the objective function 1 is used the 119871-indexis of 009512 imposed in the peak hour (period 12) In theldquobestrdquo solution objective function the 119871-index is 003506 Byanalysing the differences in operation cost and in119871-index (seeFigure 22) it is possible to see higher differences in the 119871-index than in the operation costsThedifferences in operationcosts are mainly due to the electric vehicles schedulingIn fact in some periods the EVs discharge happens tosupply the charge of other EVs (see Figure 23) Howeverthe charge and discharge allows reducing the power flow insome lines of the network reducing the differences in thevoltage magnitude and angle and consequently reducing the119871-index Furthermore the systemuses a better reactive powerscheduling to reduce the voltage differences between busesreducing the 119871-index values The 119871-index value is the samein most of periods (9 to 21) in the case of ldquobestrdquo solutionHowever this value is not constant in the buses In fact the 119871-index is imposed in each period by different buses accordingto the resources scheduling

By analysing all the presented resources it is possible toconclude that the proposed multiobjective energy resourcescheduling and the two proposed weighted-sum and adaptedgoal programming methodologies can be successfully usedin real networks with large penetration of distributed energyresources The inclusion of 119871-index objective function in thedistribution networks can support significantly the manage-ment of all power system

5 Conclusions

The future power systems will be operated considering a largerange of different distributed energy resources connected in


765 768

780

803

812

862

877

937

903

21536

68 94

106

171 179174

127

95

181

214 215

219

222

264

306

275

237

261

235

234

328338372

341

345

357

360

473496 563

397

564 590 592

598618

624

623 664

595

665 681 685

689

690

700

730

736737735

764

SupplierConnections between VPPsConnections between VPPs and supplier

middot middot middot



middot middot middot middot middot middot

middot middot middotmiddot middot middot


middot middot middot middot middot middot middot middot middot






Figure 18 937 bus distribution network configuration [72]

All solutionsNondominated solutions

0001002003004005006007008009

01

L-in

dex

9320

0

9340

0

9480

0

9380

0

9400

0

9420

0

9440

0

9500

0

9300

0

9360

0

9460

0

Operation cost (mu)

Figure 19 Pareto front for weighted-sum method

different voltage levels This new reality involves a bettermanagement of the distributed energy resources in order toimprove the system stability

A methodology to integrate the voltage stability in thescheduling of distributed energy resources in a distribu-tion level was presented in this paper The proposed for-mulation results in a multiobjective optimization problemconsidering the operation costs and the voltage stabilityThe voltage stability is ldquomeasuredrdquo by the load index value(119871-index)

Two case studies are presented considering a 33-bus dis-tribution network and a real Portuguese distribution networkwith 937 buses In both cases the obtained results show theadvantage of the proposed methodologies mainly when acombination of the two objective functions is used By usingthe ldquobestrdquo solution objective function it is possible obtain asignificantly better 119871-index values with a short operation costincrease

The voltage stability margin is evaluated in a peak period(in the 33-bus distribution network case study) consider-ing the three objective functions showing the effectivenessof the method Additionally a sensitivity analysis is pre-sented considering extreme cases allowing evaluating thebehaviour of the proposed method in complex operationscenarios


Simulations0

1

2

3

4

5N

orm

aliz

ed d

istan

ce

L-index 003506Operation cost 93168mu

Best solution


200025003000350040004500500055006000

Ope

ratio

n co

st (m

u)

3 5 7 9 11 13 15 17 19 21 231Time (h)



1 3 5 7 9 11 13 15 17 19 21 23Time (h)


0001002003004005006007008009

01

L-in

dex


Nomenclature

Parameters

120573 Function 1198651weight


120582119905 Load increment parameter in period 119905

120578119888 Grid-to-vehicle efficiency

120578119889 Vehicle-to-grid efficiency

3 5 7 9 11 13 15 17 19 21 231Time (h) minus15000

minus10000

minus5000

0

5000

10000

15000

20000

Ener

gy (k

Wh)

minus2000minus1000

010002000300040005000600070008000

Pow

er (k

W)



119861 Imaginary part in admittance matrix [S]119888119860 Fixed component of cost function [muh]

119888119861 Linear component of cost function

[mukW]119888119888 Quadratic component of cost function

[mukW2]119888 Resource cost in period 119905 [mukW]119864 Stored energy in the battery of vehicle at the

end of period 119905 [kWh]119864Initial Energy stored in the battery of vehicle at the

beginning of period 1 [kWh]119864Trip Energy consumption in the battery during a

trip that occurs in period 119905 [kWh]119865 Objective function119866 Real part in admittance matrix [S]119873 Total number of resourcesnum steps Number of steps119871 Load index value in the bus119878 Maximum apparent [kVA]SF Voltage stability price factor119879 Total number of periods119879119871 Set of lines connected to a certain bus119881 Complex amplitude of voltage [V]Up Utopia point119910 Series admittance of line that connects two

buses [S]119910sh Shunt admittance of line that connects two

buses [S]

Variables

Δ119881 Voltage difference120579 Voltage angle119865Aux Epigraph variable to handle with the load index119875 Active power [kW]119876 Reactive power [kVAr]119878 Apparent power [kVA]119881 Voltage magnitude [V]119883 Binary variable


Indices

1 Operation cost function2 Load index functionBase Base valueBatMax Battery energy capacityBatMin Minimum stored energy to be guaranteed

at the end of period 119905Bus BusCh Charge process of the electric vehicle119889+ Positive deviation119889minus Negative deviation

Dch Discharge process of the electric vehicleDG Distributed generation unitDGForecast Forecast power of distributed generation

unit in period 119905EV Electric vehicle119892 Objective function solution for the goal

programmingGCP Generation curtailment power119894 119895 Bus 119894 and bus 119895119871 Load LoadMax Upper bound limitMin Lower bound limitNSD Nonsupplied demandRes Sche Resource scheduledSP External supplierStored Stored energy in the battery of the vehicleSTP Tap stepTFR Power transformerTFR HV MV Transformer that connects from high

voltage to medium voltageTFR MV LV Transformer that connects from medium

voltage to low voltage119879119871 Line

Competing Interests

The authors declare that they have no competing interests

References

[1] M Braun ldquoTechnological control capabilities ofDER to providefuture ancillary servicesrdquo International Journal of DistributedEnergy Resources vol 3 no 3 pp 191ndash206 2007

[2] P Joskow ldquoIncentive regulation in theory and practice elec-tricity distribution and transmission networksrdquo in EconomicRegulation and Its Reform What Have We Learned N L RoseEd University of Chicago Press 2013

[3] J C Ferreira V Monteiro and J L Afonso ldquoVehicle-to-anything application (V2Anything App) for electric vehiclesrdquoIEEE Transactions on Industrial Informatics vol 10 no 3 pp1927ndash1937 2014

[4] J Oyarzabal J Marti A Ilo M Sebastian D Alvira and KJohansen ldquoIntegration of DER into power system operationthrough virtual power plant concept applied for voltage regu-lationrdquo in Proceedings of the CIGREIEEE PES Joint Symposiumon Integration ofWide-Scale Renewable Resources into the PowerDelivery System pp 1ndash7 Calgary Canada July 2009

[5] X Wu J Liu and S Yan ldquoStudy on voltage stability ofdistribution networksrdquo in Proceedings of theWorkshop on PowerElectronics and Intelligent Transportation System (PEITS rsquo08)pp 399ndash403 Guangzhou China August 2008

[6] Y Zhichun L Le Jian W Zilin and G Hanyang ldquoAnalyticalmethod of the impact of distributed generation on static voltagestability of distribution network and its developmentrdquo Indone-sian Journal of Electrical Engineering and Computer Science vol11 no 9 pp 5018ndash5029 2013

[7] J Zhao Y Yang and Z Gao ldquoA review on on-line voltagestability monitoring indices andmethods based on local phasormeasurementrdquo in Proceedings of the 17th Power Systems Compu-tation Conference Stockholm Sweden August 2011

[8] A K Sinha and D Hazarika ldquoA comparative study of voltagestability indices in a power systemrdquo International Journal ofElectrical Power amp Energy Systems vol 22 no 8 pp 589ndash5962000

[9] M Glavic and T Van Cutsem ldquoA short survey of methods forvoltage instability detectionrdquo in Proceedings of the IEEE Powerand Energy Society General Meeting pp 1ndash8 San Diego CalifUSA July 2011

[10] M Arun and P Aravindhababu ldquoA new reconfiguration schemefor voltage stability enhancement of radial distribution sys-temsrdquo Energy Conversion and Management vol 50 no 9 pp2148ndash2151 2009

[11] A Rabiee M Vanouni and M Parniani ldquoOptimal reactivepower dispatch for improving voltage stability margin using alocal voltage stability indexrdquo Energy Conversion and Manage-ment vol 59 pp 66ndash73 2012

[12] M H Moradi S M Reza Tousi and M Abedini ldquoMulti-objective PFDE algorithm for solving the optimal siting andsizing problem ofmultiple DG sourcesrdquo International Journal ofElectrical Power and Energy Systems vol 56 pp 117ndash126 2014

[13] P Kessel and H Glavitsch ldquoEstimating the voltage stability of apower systemrdquo IEEE Transactions on Power Delivery vol 1 no3 pp 346ndash354 1985

[14] Y Wang C Wang F Lin W Li L Y Wang and J ZhaoldquoIncorporating generator equivalentmodel into voltage stabilityanalysisrdquo IEEE Transactions on Power Systems vol 28 no 4 pp4857ndash4866 2013

[15] M Moghavvemi and M O Faruque ldquoPower system securityand voltage collapse a line outage based indicator for pre-dictionrdquo International Journal of Electrical Power and EnergySystem vol 21 no 6 pp 455ndash461 1999

[16] M Moghavvemi and O Faruque ldquoReal-time contingency eval-uation and ranking techniquerdquo IEE ProceedingsmdashGenerationTransmission and Distribution vol 145 no 5 pp 517ndash524 1998

[17] V Pareto Manuale di Economia Politica Societa EditriceLibraria 1906

[18] W StadlerMulticriteria Optimization in Engineering and in theSciences Springer US 1988

[19] T Van Cutsem ldquoVoltage instability phenomena countermea-sures and analysis methodsrdquo Proceedings of the IEEE vol 88no 2 pp 208ndash227 2000

[20] R Avalos and C Canizares ldquoEquivalency of continuation andoptimization methods to determine saddle-node and limit-induced bifurcations in power systemsrdquo IEEE Transactions onCircuits and Systems I Regular Papers vol 56 no 1 pp 210ndash223 2009

[21] G D Irisarri X Wang J Tong and S Mokhtari ldquoMaximumloadability of power systems using interior point non-linear


optimizationmethodrdquo IEEETransactions on Power Systems vol12 no 1 pp 162ndash172 1997

[22] G Binetti A Davoudi D Naso B Turchiano and F L LewisldquoA distributed auction-based algorithm for the nonconvexeconomic dispatch problemrdquo IEEE Transactions on IndustrialInformatics vol 10 no 2 pp 1124ndash1132 2014

[23] Y Huang S Mao and R M Nelms ldquoAdaptive electricityscheduling in microgridsrdquo IEEE Transactions on Smart Gridvol 5 no 1 pp 270ndash281 2014

[24] HMorais T Pinto Z Vale and I Praca ldquoMultilevel negotiationin smart grids for VPP management of distributed resourcesrdquoIEEE Intelligent Systems vol 27 no 6 pp 8ndash16 2012

[25] H S V S Kumar Nunna and S Doolla ldquoMultiagent-baseddistributed-energy-resourcemanagement for intelligent micro-gridsrdquo IEEE Transactions on Industrial Electronics vol 60 no 4pp 1678ndash1687 2013

[26] M Yilmaz and P T Krein ldquoReview of the impact of vehicle-to-grid technologies on distribution systems and utility interfacesrdquoIEEETransactions on Power Electronics vol 28 no 12 pp 5673ndash5689 2013

[27] S Bellekom R Benders S Pelgrom and H Moll ldquoElectriccars and wind energy two problems one solution A studyto combine wind energy and electric cars in 2020 in TheNetherlandsrdquo Energy vol 45 no 1 pp 859ndash866 2012

[28] H Morais T Sousa Z Vale and P Faria ldquoEvaluation of theelectric vehicle impact in the power demand curve in a smartgrid environmentrdquoEnergy Conversion andManagement vol 82pp 268ndash282 2014

[29] D Wu D C Aliprantis and L Ying ldquoLoad scheduling anddispatch for aggregators of plug-in electric vehiclesrdquo IEEETransactions on Smart Grid vol 3 no 1 pp 368ndash376 2012

[30] A Y Saber and G K Venayagamoorthy ldquoPlug-in vehicles andrenewable energy sources for cost and emission reductionsrdquoIEEE Transactions on Industrial Electronics vol 58 no 4 pp1229ndash1238 2011

[31] J H Zhao FWen Z Y Dong Y Xue andK PWong ldquoOptimaldispatch of electric vehicles and wind power using enhancedparticle swarm optimizationrdquo IEEE Transactions on IndustrialInformatics vol 8 no 4 pp 889ndash899 2012

[32] T Sousa Z Vale J P Carvalho T Pinto and H Morais ldquoAhybrid simulated annealing approach to handle energy resourcemanagement considering an intensive use of electric vehiclesrdquoEnergy vol 67 pp 81ndash96 2014

[33] F Kennel D Gorges and S Liu ldquoEnergymanagement for smartgrids with electric vehicles based on hierarchical MPCrdquo IEEETransactions on Industrial Informatics vol 9 no 3 pp 1528ndash1537 2013

[34] C Chen and S Duan ldquoOptimal integration of plug-in hybridelectric vehicles in microgridsrdquo IEEE Transactions on IndustrialInformatics vol 10 no 3 pp 1917ndash1926 2014

[35] Y A Katsigiannis P S Georgilakis and E S Karapidakis ldquoMul-tiobjective genetic algorithm solution to the optimumeconomicand environmental performance problem of small autonomoushybrid power systems with renewablesrdquo IET Renewable PowerGeneration vol 4 no 5 pp 404ndash419 2010

[36] J Druitt and W-G Fruh ldquoSimulation of demand managementand grid balancing with electric vehiclesrdquo Journal of PowerSources vol 216 pp 104ndash116 2012

[37] W Su H Eichi W Zeng and M-Y Chow ldquoA survey on theelectrification of transportation in a smart grid environmentrdquoIEEE Transactions on Industrial Informatics vol 8 no 1 pp 1ndash10 2012

[38] M Zeleny and J Cochrane Multiple Criteria Decision Making1982

[39] R T Marler and J S Arora ldquoSurvey of multi-objective opti-mization methods for engineeringrdquo Structural and Multidisci-plinary Optimization vol 26 no 6 pp 369ndash395 2004

[40] J Grainger and W Stevenson Jr Power System AnalysisMcGraw-Hill ScienceEngineeringMath 1994

[41] S B Peterson J Apt and J F Whitacre ldquoLithium-ion batterycell degradation resulting from realistic vehicle and vehicle-to-grid utilizationrdquo Journal of Power Sources vol 195 no 8 pp2385ndash2392 2010

[42] S B Peterson J FWhitacre and J Apt ldquoThe economics of usingplug-in hybrid electric vehicle battery packs for grid storagerdquoJournal of Power Sources vol 195 no 8 pp 2377ndash2384 2010

[43] T K A Rahman and G B Jasmon ldquoA new technique forvoltage stability analysis in a power system and improvedloadflow algorithm for distribution networkrdquo in Proceedings ofthe International Conference on Energy Management and PowerDelivery (EMPD rsquo95) vol 2 pp 714ndash719 Singapore November1995

[44] K JMakasa andGKVenayagamoorthy ldquoEstimation of voltagestability index in a power systemwith Plug-in Electric Vehiclesrdquoin Proceedings of the 8th iREP Symposium Bulk Power SystemDynamics and Control (iREP rsquo10) pp 1ndash7 Rio de Janeiro BrazilAugust 2010

[45] R Horst P Pardalos and N Van Thoai Introduction to GlobalOptimization (Nonconvex Optimization and Its Applications)Springer Berlin Germany 2000

[46] L A Zadeh ldquoOptimality and non-scalar-valued performancecriteriardquo IEEE Transactions on Automatic Control vol 8 no 1pp 59ndash60 1963

[47] I Y Kim and O L De Weck ldquoAdaptive weighted-sum methodfor bi-objective optimization pareto front generationrdquo Struc-tural andMultidisciplinary Optimization vol 29 no 2 pp 149ndash158 2005

[48] A Charnes W W Cooper and R O Ferguson ldquoOptimalestimation of executive compensation by linear programmingrdquoManagement Science vol 1 no 2 pp 138ndash151 1955

[49] A Charnes andW CooperManagement Models and IndustrialApplications of Linear Programming John Wiley amp Sons NewYork NY USA 1961

[50] O P Dubey R K Dwivedi and S N Singh ldquoGoal program-ming a survey (1960ndash2000)rdquo The IUP Journal of OperationsManagement vol 11 no 2 pp 29ndash53 2012

[51] W Ogryczak ldquoA goal programming model of the referencepoint methodrdquo Annals of Operations Research vol 51 pp 33ndash44 1994

[52] A Charnes and W Cooper ldquoGoal programming and multipleobjective optimizations part 1rdquoEuropean Journal of OperationalResearch vol 1 no 1 pp 39ndash54 1977

[53] K Miettinen Nonlinear Multiobjective Optimization 1999[54] I Das and J Dennis ldquoNormal-boundary intersection a new

method for generating the pareto surface in nonlinear multi-criteria optimization problemsrdquo SIAM Journal on Optimizationvol 8 no 3 pp 631ndash657 1996

[55] T Vincent andW GranthamOptimality in Parametric SystemsJohn Wiley amp Sons New York NY USA 1981

[56] J Soares B Canizes C Lobo Z Vale and H Morais ldquoElectricvehicle scenario simulator tool for smart grid operatorsrdquo Ener-gies vol 5 no 6 pp 1881ndash1899 2012


[57] H Liu H Ning Y Zhang and M Guizani ldquoBattery status-aware authentication scheme for V2G networks in smart gridrdquoIEEE Transactions on Smart Grid vol 4 no 1 pp 99ndash110 2013

[58] R E RosenthalGAMSmdashAUserrsquos GuideWashingtonDCUSA2014

[59] I E Grossmann J Viswanathan A Vecchietti R Raman andE Kalvelagen DICOPTmdashA Userrsquos Guide 2008

[60] M Silva H Morais and Z Vale ldquoAn integrated approach fordistributed energy resource short-term scheduling in smartgrids considering realistic power system simulationrdquo EnergyConversion and Management vol 64 pp 273ndash288 2012

[61] Renault ldquoRenault ZOE ZE Technical Specificationrdquo 2013httpwwwrenaultcomeninnovationl-univers-du-designpagesshow-car-zoe-previewaspx

[62] Renault ldquoRenault Kangoo ZE Technical Specificationrdquo 2013httpwwwrenaultcomenvehiculesrenaultpageskangoo-express-zeaspx

[63] Renault ldquoRenault Fluence ZE Technical Specificationsrdquo 2013httpwwwrenault-zecomze-rangefluence-zepresentation-1935html

[64] Nissan Nissan Leaf Technical Specifications 2013 httpwwwnissancoukvehicleselectricvehiclesleafhtmvehicleselectricvehiclesleafleaf-enginespecifications

[65] Toyota Prius Plug-In Technical Specifications 2013 httpwwwtoyotacomprius-plug-in

[66] Citroen ldquoCitroen C-Zero Technical Specificationsrdquo 2013httpwwwcitroencoukhomenew-carscar-rangecitroen-c-zerofull-electric

[67] Mitsubishi ldquoMitsubishi i-MiEV Technical Specificationsrdquo 2013httpwwwmitsubishi-motorscomspecialevwhatisindexhtml

[68] Y Liut M Yoshioka K Homma and T Shibuya ldquoEfficientlyfinding the rsquoBestrsquo solution with multi-objectives from multipletopologies in topology library of analog circuitrdquo in Proceedingsof the Asia and South Pacific Design Automation Conference pp498ndash503 IEEE Yokohama Japan January 2009

[69] W Kempton and J Tomic ldquoVehicle-to-grid power fundamen-tals calculating capacity and net revenuerdquo Journal of PowerSources vol 144 no 1 pp 268ndash279 2005

[70] EREC Renewable Energy Scenario to 2040 2004[71] EREC Energy [R]evolutionmdashA Sustainable World Energy Out-

look EREC 2007[72] S Goncalves H Morais T Sousa and Z Vale ldquoEnergy

resource scheduling in a real distribution network managed byseveral virtual power playersrdquo in Proceedings of the IEEE PESTransmission and Distribution Conference and Exposition (TampDrsquo12) pp 1ndash8 Orlando Fla USA May 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


The line stability indexes like the 119871119898119899

[15] or the VCPI[16] are more accurate than the 119871-index to predict the voltagecollapse proximity in a real time operation However inthe day-ahead scheduling optimization the main goal ofthe voltage stability index is not based on determining theproximity to the collapse but to influence the distributedresources scheduling for contributing to the system stabilityAs stated in [16] the evolution of the 119871-index is similar asthe other suggested indexes (when the 119871-index increases the119871119898119899

and VCPI also increase) meaning if we optimize the119871-index we are also improving the 119871

119898119899and VCPI indexes

The 119871-index is used in the present study because it is easierto integrate in the optimization problem than the othersuggested indexes With the 119871-index the objective functiondoes not depend on the use of 119876 and consequently not on 119875due to the load consumption in each bus (also depends onthe resources scheduling such as distributed generation andelectric vehicles)

The paper proposes an energy resource schedulingproblem with a multiobjective function incorporating theoperation cost and the voltage stability This multiobjectiveoptimization problem will be applied for a scenario in adistribution network with a high penetration of distributedenergy resources mainly an intensive EVs penetration In theoperation cost different technologies of DG and the use ofEVs with griddable capability are considered also known asvehicle-to-grid (V2G)The use of 119871-index is proposed to dealwith the voltage stability in the joint optimization problemThe 119871-index was initially proposed in [13] based on thepower flow equations Two optimization techniques are pro-posed in this paper for solving the proposed multiobjectiveenergy resource scheduling problem These techniques willdetermine the nondominated solutions of the multiobjectiveoptimization problem namely the weighted-sum methodand an adapted goal programming methodology Thus thenondominated solutions represent the Pareto front that wasproposed in [17] yet the application in engineering andscience fields only began in the end of the seventies [18]Furthermore the goal programming methodology can bevery useful in real application due to the complex character-istics of the objective functions In a regular power systemoperation the system operators can establish a predefinedrange in the operation cost objective function and in thecritical situations (operation near from boundaries) thesystem operators can limit the objective function concerningthe voltage stability index

To demonstrate the effectiveness of the proposedmethodologies concerning voltage stability two studies wereincluded in the first one the sensitivity analysis is performedconsidering variations in the power demand in the voltageangle and in the voltage magnitude on the slack bus (fromthe distribution networkrsquos point of view the reference busis the connection in an upstream network) In the secondanalysis the loadability limit is determined for an hourconsidering three different scheduling objective functions(operation cost 119871-index and multiobjective) allowing thedetermination of the maximum load that can be supplied(voltage stability boundary) considering the voltage controlconstraints This approach is equivalent to the bifurcations

determined with continuation power flow algorithms thatallow to calculate the loadability limit for the power system[19ndash21] Both analyses show the improvements in the energyresource scheduling problem through the incorporation of119871-index as another objective function In addition bothmethodologies are tested in a distribution network with highpenetration of distributed energy resources consideringthe use of electric vehicles allowing the 119871-index and theoperation cost evaluation The weighted-sum method isalso applied to a real distribution network to evaluate itsperformance in a large network

After the Introduction Section 2 presents an overviewconcerning the energy resource scheduling problem Sec-tion 3 focuses on the mathematical formulation and on theimplementation of the proposed methodologies Section 4shows the case study considering a 33-bus distributionnetwork and finally the most important conclusions arepresented in Section 5

2 Energy Resource SchedulingOverview and Contributions

The development of energy resources scheduling methodsconsidering the distributed resources in different voltage lev-els of power systems is an important research topic Typicallythe energy resource scheduling consists in an optimizationproblem to determine the best scheduling to minimize theoperation cost of the available resources [22] However in asmart grid context it is also important to take into accountother aspects than just the economic one such as powerquality voltage stability environmental aspects or the loaddiagramprofileTherefore all these aspects can be included inthe energy resource scheduling providing different solutionsto help the system operators in the decision making process

Several authors have proposed different methodologiesto deal with the energy resource scheduling considering dis-tributed energy resources such as DG and active consumerswith demand response programs and the network operationIn [23] it is described a framework for aggregators to deter-mine the energy resource scheduling based on the concept ofquality-of-service in power system A more complex negoti-ation perspective is presented in [24] considering multilevelnegotiation layers between aggregators and electricitymarketparticipation For amicrogrid level perspective [25] proposesa multiagent base platform allowing the scheduling of thedistributed energy resources

Other works deal with the energy resource scheduling tointegrate the electric vehicles with V2G capability A compre-hensive and exhaustive review is presented in [26] concerningthe impact of EVs in the distribution network In [27] theauthors proved that EVs can improve the management ofintermittent renewable resources such as wind farms andin [28] it is shown that EVs can be used to level the dailyload diagram Wu et al [29] claim that the charging controlin EVs is required for a well accommodation in the powersystem To handle the large number of electric vehiclesseveral artificial intelligence algorithms have been proposed[30ndash32] to provide the scheduling of charge and dischargeenergy from EVs batteries Another innovative perspective















min 1198651=

119879

sum

119905=1

[

119873DG

sum

DG=1119888119860(DG119905)119883DG(DG119905) + 119888119861(DG119905)

sdot 119875DG(DG119905) + 119888119862(DG119905)1198752

DG(DG119905)

+

119873SP

sum

SP=1119888SP(SP119905)119875SP(SP119905)

+

119873EV

sum

EV=1(119888Dch(EV119905) + 119888Deg(EV))


sdot 119875Ch(EV119905) +

119873119871

sum

119871=1

119888NSD(119871119905)119875NSD(119871119905)

+

119873DG

sum

DG=1119888GCP(DG119905)119875GCP(DG119905)]

(1)








the load bus

119871119895=

1003816100381610038161003816100381610038161003816100381610038161003816

1 +119880119894

119880119895

1003816100381610038161003816100381610038161003816100381610038161003816

(2)


119871119895=

4 [119881119894119881119895cos (120579

119894minus 120579119895) minus 1198812

119895cos2 (120579

119894minus 120579119895)]

1198812

119894

(3)






(bus119905)) (4)Function 119865




min 1198652= 119865Aux(119905)


(5)








1+ 1205751198652SF

120573 + 120575 = 1

(6)








min119909isin119883119889

minus119889+

119896

sum

119894=1

(119889+

119894+ 119889minus

119894)

subjected to 119865119895(119909) + 119889

+

119895+ 119889minus

119895= 119887119895

119889+

119895 119889minus

119895ge 0

119895 = 1 2 119896

(8)







Start

(1) (5)




definition

(10) (11)


Up

Pareto front

UpStart


(F1n) and (F2n)

F1

fix v

alue

sF1

fix v

alue

s

F2 fix values

F2 fix values

F1n

F1n

F2n

F2n

F1p

F1p

F2p

F2p







1119901) and (119865

2119901) The utopia




1119899and 119865



1and 119865

2are



1198651 step =

(1198651119899minus Up)

num steps (9)

1198652 step =

(1198652119899minus Up)

num steps (10)



1198921) and (119865

1198922)

1198651198921= min (119865

1+ 10119865

2119889+ + 01119865

2119889minus) (11)

1198651198922= min (119865

2+ 10119865

1119889+ + 01119865

1119889minus) (12)



2119889minus)



2(5) into

a constraint

1198652= 119865Aux + 1198652119889+ minus 1198652119889minus 119865

2119889+ 1198652119889minus ge 0 (13)




1119889+) and


1 Then the


1198651=

119879

sum

119905=1

[

119873DG

sum

DG=1119888119860(DG119905)119883DG(DG119905) + 119888119861(DG119905)119875DG(DG119905)

+ 119888119862(DG119905)119875

2

DG(DG119905) +

119873SP

sum

SP=1119888SP(SP119905)119875SP(SP119905)

+

119873EV

sum


minus 119888Ch(EV119905)119875Ch(EV119905) +

119873119871

sum

119871=1

119888NSD(119871119905)119875NSD(119871119905)

+

119873DG

sum


1198651119889+ 1198651119889minus ge 0

(14)


119873119894

DG

sum

DG=1(119875119894

DG(DG119905) minus 119875119894

GCP(DG119905)) +

119873119894

SP

sum

SP=1119875119894

SP(SP119905)

+

119873119894

EV

sum

EV=1119875119894

Dch(EV119905) minus

119873119894

119871

sum

119871=1

(119875119894

Load(119871119905) minus 119875119894

NSD(119871119905))

minus

119873119894

EV

sum

EV=1119875119894

Ch(EV119905)

= 1198661198941198941198812

119894(119905)

+ 119881119894(119905)sum

119895isin119879119871119894

119881119895(119905)(119866119894119895cos 120579119894119895(119905)

+ 119861119894119895sin 120579119894119895(119905))

(15)



119873119894

DG

sum

DG=1119876119894

DG(DG119905) +

119873119894

SP

sum

SP=1119876119894

SP(SP119905)

minus

119873119894

119871

sum

119871=1

(119876119894

Load(119871119905) minus 119876119894

NSD(119871119905))

= 119881119894(119905)sum

119895isin119879119871119894

119881119895(119905)(119866119894119895sin 120579119894119895(119905)

minus 119861119894119895cos 120579119894119895(119905))

minus 1198611198941198941198812

119894(119905)


(16)

where 120579119894119895(119905)


119894119895and 119861





119881119894

Min le 119881119894(119905) le 119881119894


120579119894

Min le 120579119894(119905) le 120579119894


(17)



100381610038161003816100381610038161003816119881119894(119905)[119910119894119895119881119894119895(119905)

+ 119910sh 119894119881119894(119905)]lowast100381610038161003816100381610038161003816le 119878

Max119879119871 119894 to 119895

100381610038161003816100381610038161003816119881119895(119905)[119910119894119895119881119895119894(119905)

+ 119910sh 119895119881119895(119905)]lowast100381610038161003816100381610038161003816le 119878

Max119879119871 119895 to 119894


= 119881119894(119905)minus 119881119895(119905)

(18)


radic(

119873119894

SP

sum

SP=1119875119894

SP(SP119905))

2

+ (

119873119894

SP

sum

SP=1119876119894

SP(SP119905))

2



(19)


radic1198752

TFR MV LV(119894119905) + 1198762



(20)


119875TFR MV LV(119894119905) =

119873119894

DG

sum

DG=1(119875119894

DG(DG119905) minus 119875119894

GCP(DG119905))

+

119873119894

EV

sum

EV=1(119875119894

Dch(EV119905) minus 119875119894

Ch(EV119905))

minus

119873119894

119871

sum

119871=1

(119875119894

Load(119871119905) minus 119875119894

NSD(119871119905))

119876TFR MV LV(119894119905) =

119873119894

DG

sum

DG=1119876119894

DG(DG119905)

minus

119873119894

119871

sum

119871=1

(119876119894

Load(119871119905) minus 119876119894

NSD(119871119905))

(21)



le 119875Max(DG119905)119883DG(DG119905)


le 119876Max(DG119905)119883DG(DG119905)

radic(119875DG(DG119905))2

+ (119876DG(DG119905))2

le 119878Max(DG119905)


(22)






119875SP(SP119905) le 119875Max(SP119905)

119876SP(SP119905) le 119876Max(SP119905)


(24)




120578119889(EV)

119875Dch(EV119905)


(25)












(27)








119873STP

sum

STP=1119883TFR(STP119905) = 1 (30)

119881119894(119905)= 119881

Base119894(119905)

+

119873STP

sum

STP=1Δ119881119894

TFR(STP119905)


(31)













4 Case Study






0

11819

20

21

2

3

4

567

8

11

910

1213

1415

16

17

2223

24

2526

27

28

29

30

31

32

50kW

30kW

30kW 9

kW 9kW

9kW

30kW

9kW

9kW

30kW

30kW

30kW

30kW

9kW

30kW

30kW

9kW9

kW3

kW

30kW9

kW

3kW

9kW

9kW

30kW

30kW

9kW

3kW

9kW

9kW

3kW

30kW

30kW

10kW40

kW50kW

100kW

50kW

10kW

100kW

50kW

10kW 200

kW

10kW

25kW 25

kW 100kW

150kW

100kW

100kW 100

kW

50kW

50kW

50kW50

kW

50kW

100kW

25kW200

kW10

kW

50kW

10kW

30kW

200kW

25kW25

kW

PV panel

Wind

WTE

CHP

Fuel cell

Biomass

Hydro

EV with V2G sim

sim

sim

simsim

sim

sim





L-in

dex


002004006008

01012014016018









L-in

dex

002004006008

01012014016018




Simulations


100

150

200

250

Nor

mal

ized

dist

ance


mdash



1119901= 1198651119899

and 1198652119901




0100200300400500600700

1 3 5 7 9 11 13 15 17 19 21 23

Ope

ratio

n co

st (m

u)

Time (h)




0002004006008

01012014016018

L-in

dex

3 5 7 9 11 13 15 17 19 21 231Time (h)






302826242210 12 14 16 18 20 3286420Bus

1101102103104105106107

Volta

ge m

agni

tude

(pu)



0500

100015002000250030003500400045005000

Pow

er (k

W)



3 5 7 9 11 13 15 17 19 21 231Time (h)





1 3 5 7 9 11 13 15 17 19 21 23Time (h)


minus1500

minus1000

minus500

0

500

1000

1500

2000

Pow

er (k

W)

minus15000

minus10000

minus5000

0

5000

10000

15000

20000

Ener

gy (k

Wh)











Minimum 1198651

1297Minimum 119865

22115




max 120582119905


119876load = 120582119905119876load1015840(119871119905)

forall119905 isin 1 119879

forall119871 isin 1 119873119871

(32)






1)


Slackgenerator

Bus 0


ldquoVirtualrdquo

ZTH










0

02

04

06

08

1

12

L-in

dex




0 5 10 15 20 25 30 35 40 45 50 55 6088008850890089509000905091009150920092509300

Ope

ratio

n co

st (m

u)








1) results


2)



09

095

1

105

11

115

L-in

dex

104

103

102

101

100

099

098

097

096

095

094

093

092

105




105

104

103

102

101

100

099

098

097

096

095

094

093

092

71007110712071307140715071607170718071907200

Ope

ratio

n co

st (m

u)







100

120

140

160

180

200

220

240

260

280

300

320

340

360

380

400

Load ()

094

0945

095

0955

096

0965

097

L-in

dex



550060006500700075008000850090009500

1000010500

Ope

ratio

n co

st (m

u)

120

140

160

180

200

220

240

260

280

300

320

340

360

380

400

100










5 Conclusions



765 768

780

803

812

862

877

937

903

21536

68 94

106

171 179174

127

95

181

214 215

219

222

264

306

275

237

261

235

234

328338372

341

345

357

360

473496 563

397

564 590 592

598618

624

623 664

595

665 681 685

689

690

700

730

736737735

764
















0001002003004005006007008009

01

L-in

dex

9320

0

9340

0

9480

0

9380

0

9400

0

9420

0

9440

0

9500

0

9300

0

9360

0

9460

0

Operation cost (mu)







Simulations0

1

2

3

4

5N

orm

aliz

ed d

istan

ce


Best solution


200025003000350040004500500055006000

Ope

ratio

n co

st (m

u)

3 5 7 9 11 13 15 17 19 21 231Time (h)



1 3 5 7 9 11 13 15 17 19 21 23Time (h)


0001002003004005006007008009

01

L-in

dex


Nomenclature

Parameters






3 5 7 9 11 13 15 17 19 21 231Time (h) minus15000

minus10000

minus5000

0

5000

10000

15000

20000

Ener

gy (k

Wh)

minus2000minus1000

010002000300040005000600070008000

Pow

er (k

W)











buses [S]

Variables



Indices








Competing Interests


References



















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra























min 1198651=

119879

sum

119905=1

[

119873DG

sum

DG=1119888119860(DG119905)119883DG(DG119905) + 119888119861(DG119905)

sdot 119875DG(DG119905) + 119888119862(DG119905)1198752

DG(DG119905)

+

119873SP

sum

SP=1119888SP(SP119905)119875SP(SP119905)

+

119873EV

sum

EV=1(119888Dch(EV119905) + 119888Deg(EV))


sdot 119875Ch(EV119905) +

119873119871

sum

119871=1

119888NSD(119871119905)119875NSD(119871119905)

+

119873DG

sum

DG=1119888GCP(DG119905)119875GCP(DG119905)]

(1)








the load bus

119871119895=

1003816100381610038161003816100381610038161003816100381610038161003816

1 +119880119894

119880119895

1003816100381610038161003816100381610038161003816100381610038161003816

(2)


119871119895=

4 [119881119894119881119895cos (120579

119894minus 120579119895) minus 1198812

119895cos2 (120579

119894minus 120579119895)]

1198812

119894

(3)






(bus119905)) (4)Function 119865




min 1198652= 119865Aux(119905)


(5)








1+ 1205751198652SF

120573 + 120575 = 1

(6)








min119909isin119883119889

minus119889+

119896

sum

119894=1

(119889+

119894+ 119889minus

119894)

subjected to 119865119895(119909) + 119889

+

119895+ 119889minus

119895= 119887119895

119889+

119895 119889minus

119895ge 0

119895 = 1 2 119896

(8)







Start

(1) (5)




definition

(10) (11)


Up

Pareto front

UpStart


(F1n) and (F2n)

F1

fix v

alue

sF1

fix v

alue

s

F2 fix values

F2 fix values

F1n

F1n

F2n

F2n

F1p

F1p

F2p

F2p







1119901) and (119865

2119901) The utopia




1119899and 119865



1and 119865

2are



1198651 step =

(1198651119899minus Up)

num steps (9)

1198652 step =

(1198652119899minus Up)

num steps (10)



1198921) and (119865

1198922)

1198651198921= min (119865

1+ 10119865

2119889+ + 01119865

2119889minus) (11)

1198651198922= min (119865

2+ 10119865

1119889+ + 01119865

1119889minus) (12)



2119889minus)



2(5) into

a constraint

1198652= 119865Aux + 1198652119889+ minus 1198652119889minus 119865

2119889+ 1198652119889minus ge 0 (13)




1119889+) and


1 Then the


1198651=

119879

sum

119905=1

[

119873DG

sum

DG=1119888119860(DG119905)119883DG(DG119905) + 119888119861(DG119905)119875DG(DG119905)

+ 119888119862(DG119905)119875

2

DG(DG119905) +

119873SP

sum

SP=1119888SP(SP119905)119875SP(SP119905)

+

119873EV

sum


minus 119888Ch(EV119905)119875Ch(EV119905) +

119873119871

sum

119871=1

119888NSD(119871119905)119875NSD(119871119905)

+

119873DG

sum


1198651119889+ 1198651119889minus ge 0

(14)


119873119894

DG

sum

DG=1(119875119894

DG(DG119905) minus 119875119894

GCP(DG119905)) +

119873119894

SP

sum

SP=1119875119894

SP(SP119905)

+

119873119894

EV

sum

EV=1119875119894

Dch(EV119905) minus

119873119894

119871

sum

119871=1

(119875119894

Load(119871119905) minus 119875119894

NSD(119871119905))

minus

119873119894

EV

sum

EV=1119875119894

Ch(EV119905)

= 1198661198941198941198812

119894(119905)

+ 119881119894(119905)sum

119895isin119879119871119894

119881119895(119905)(119866119894119895cos 120579119894119895(119905)

+ 119861119894119895sin 120579119894119895(119905))

(15)



119873119894

DG

sum

DG=1119876119894

DG(DG119905) +

119873119894

SP

sum

SP=1119876119894

SP(SP119905)

minus

119873119894

119871

sum

119871=1

(119876119894

Load(119871119905) minus 119876119894

NSD(119871119905))

= 119881119894(119905)sum

119895isin119879119871119894

119881119895(119905)(119866119894119895sin 120579119894119895(119905)

minus 119861119894119895cos 120579119894119895(119905))

minus 1198611198941198941198812

119894(119905)


(16)

where 120579119894119895(119905)


119894119895and 119861





119881119894

Min le 119881119894(119905) le 119881119894


120579119894

Min le 120579119894(119905) le 120579119894


(17)



100381610038161003816100381610038161003816119881119894(119905)[119910119894119895119881119894119895(119905)

+ 119910sh 119894119881119894(119905)]lowast100381610038161003816100381610038161003816le 119878

Max119879119871 119894 to 119895

100381610038161003816100381610038161003816119881119895(119905)[119910119894119895119881119895119894(119905)

+ 119910sh 119895119881119895(119905)]lowast100381610038161003816100381610038161003816le 119878

Max119879119871 119895 to 119894


= 119881119894(119905)minus 119881119895(119905)

(18)


radic(

119873119894

SP

sum

SP=1119875119894

SP(SP119905))

2

+ (

119873119894

SP

sum

SP=1119876119894

SP(SP119905))

2



(19)


radic1198752

TFR MV LV(119894119905) + 1198762



(20)


119875TFR MV LV(119894119905) =

119873119894

DG

sum

DG=1(119875119894

DG(DG119905) minus 119875119894

GCP(DG119905))

+

119873119894

EV

sum

EV=1(119875119894

Dch(EV119905) minus 119875119894

Ch(EV119905))

minus

119873119894

119871

sum

119871=1

(119875119894

Load(119871119905) minus 119875119894

NSD(119871119905))

119876TFR MV LV(119894119905) =

119873119894

DG

sum

DG=1119876119894

DG(DG119905)

minus

119873119894

119871

sum

119871=1

(119876119894

Load(119871119905) minus 119876119894

NSD(119871119905))

(21)



le 119875Max(DG119905)119883DG(DG119905)


le 119876Max(DG119905)119883DG(DG119905)

radic(119875DG(DG119905))2

+ (119876DG(DG119905))2

le 119878Max(DG119905)


(22)






119875SP(SP119905) le 119875Max(SP119905)

119876SP(SP119905) le 119876Max(SP119905)


(24)




120578119889(EV)

119875Dch(EV119905)


(25)












(27)








119873STP

sum

STP=1119883TFR(STP119905) = 1 (30)

119881119894(119905)= 119881

Base119894(119905)

+

119873STP

sum

STP=1Δ119881119894

TFR(STP119905)


(31)













4 Case Study






0

11819

20

21

2

3

4

567

8

11

910

1213

1415

16

17

2223

24

2526

27

28

29

30

31

32

50kW

30kW

30kW 9

kW 9kW

9kW

30kW

9kW

9kW

30kW

30kW

30kW

30kW

9kW

30kW

30kW

9kW9

kW3

kW

30kW9

kW

3kW

9kW

9kW

30kW

30kW

9kW

3kW

9kW

9kW

3kW

30kW

30kW

10kW40

kW50kW

100kW

50kW

10kW

100kW

50kW

10kW 200

kW

10kW

25kW 25

kW 100kW

150kW

100kW

100kW 100

kW

50kW

50kW

50kW50

kW

50kW

100kW

25kW200

kW10

kW

50kW

10kW

30kW

200kW

25kW25

kW

PV panel

Wind

WTE

CHP

Fuel cell

Biomass

Hydro

EV with V2G sim

sim

sim

simsim

sim

sim





L-in

dex


002004006008

01012014016018









L-in

dex

002004006008

01012014016018




Simulations


100

150

200

250

Nor

mal

ized

dist

ance


mdash



1119901= 1198651119899

and 1198652119901




0100200300400500600700

1 3 5 7 9 11 13 15 17 19 21 23

Ope

ratio

n co

st (m

u)

Time (h)




0002004006008

01012014016018

L-in

dex

3 5 7 9 11 13 15 17 19 21 231Time (h)






302826242210 12 14 16 18 20 3286420Bus

1101102103104105106107

Volta

ge m

agni

tude

(pu)



0500

100015002000250030003500400045005000

Pow

er (k

W)



3 5 7 9 11 13 15 17 19 21 231Time (h)





1 3 5 7 9 11 13 15 17 19 21 23Time (h)


minus1500

minus1000

minus500

0

500

1000

1500

2000

Pow

er (k

W)

minus15000

minus10000

minus5000

0

5000

10000

15000

20000

Ener

gy (k

Wh)











Minimum 1198651

1297Minimum 119865

22115




max 120582119905


119876load = 120582119905119876load1015840(119871119905)

forall119905 isin 1 119879

forall119871 isin 1 119873119871

(32)






1)


Slackgenerator

Bus 0


ldquoVirtualrdquo

ZTH










0

02

04

06

08

1

12

L-in

dex




0 5 10 15 20 25 30 35 40 45 50 55 6088008850890089509000905091009150920092509300

Ope

ratio

n co

st (m

u)








1) results


2)



09

095

1

105

11

115

L-in

dex

104

103

102

101

100

099

098

097

096

095

094

093

092

105




105

104

103

102

101

100

099

098

097

096

095

094

093

092

71007110712071307140715071607170718071907200

Ope

ratio

n co

st (m

u)







100

120

140

160

180

200

220

240

260

280

300

320

340

360

380

400

Load ()

094

0945

095

0955

096

0965

097

L-in

dex



550060006500700075008000850090009500

1000010500

Ope

ratio

n co

st (m

u)

120

140

160

180

200

220

240

260

280

300

320

340

360

380

400

100










5 Conclusions



765 768

780

803

812

862

877

937

903

21536

68 94

106

171 179174

127

95

181

214 215

219

222

264

306

275

237

261

235

234

328338372

341

345

357

360

473496 563

397

564 590 592

598618

624

623 664

595

665 681 685

689

690

700

730

736737735

764
















0001002003004005006007008009

01

L-in

dex

9320

0

9340

0

9480

0

9380

0

9400

0

9420

0

9440

0

9500

0

9300

0

9360

0

9460

0

Operation cost (mu)







Simulations0

1

2

3

4

5N

orm

aliz

ed d

istan

ce


Best solution


200025003000350040004500500055006000

Ope

ratio

n co

st (m

u)

3 5 7 9 11 13 15 17 19 21 231Time (h)



1 3 5 7 9 11 13 15 17 19 21 23Time (h)


0001002003004005006007008009

01

L-in

dex


Nomenclature

Parameters






3 5 7 9 11 13 15 17 19 21 231Time (h) minus15000

minus10000

minus5000

0

5000

10000

15000

20000

Ener

gy (k

Wh)

minus2000minus1000

010002000300040005000600070008000

Pow

er (k

W)











buses [S]

Variables



Indices








Competing Interests


References



















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













the load bus

119871119895=

1003816100381610038161003816100381610038161003816100381610038161003816

1 +119880119894

119880119895

1003816100381610038161003816100381610038161003816100381610038161003816

(2)


119871119895=

4 [119881119894119881119895cos (120579

119894minus 120579119895) minus 1198812

119895cos2 (120579

119894minus 120579119895)]

1198812

119894

(3)






(bus119905)) (4)Function 119865




min 1198652= 119865Aux(119905)


(5)








1+ 1205751198652SF

120573 + 120575 = 1

(6)








min119909isin119883119889

minus119889+

119896

sum

119894=1

(119889+

119894+ 119889minus

119894)

subjected to 119865119895(119909) + 119889

+

119895+ 119889minus

119895= 119887119895

119889+

119895 119889minus

119895ge 0

119895 = 1 2 119896

(8)







Start

(1) (5)




definition

(10) (11)


Up

Pareto front

UpStart


(F1n) and (F2n)

F1

fix v

alue

sF1

fix v

alue

s

F2 fix values

F2 fix values

F1n

F1n

F2n

F2n

F1p

F1p

F2p

F2p







1119901) and (119865

2119901) The utopia




1119899and 119865



1and 119865

2are



1198651 step =

(1198651119899minus Up)

num steps (9)

1198652 step =

(1198652119899minus Up)

num steps (10)



1198921) and (119865

1198922)

1198651198921= min (119865

1+ 10119865

2119889+ + 01119865

2119889minus) (11)

1198651198922= min (119865

2+ 10119865

1119889+ + 01119865

1119889minus) (12)



2119889minus)



2(5) into

a constraint

1198652= 119865Aux + 1198652119889+ minus 1198652119889minus 119865

2119889+ 1198652119889minus ge 0 (13)




1119889+) and


1 Then the


1198651=

119879

sum

119905=1

[

119873DG

sum

DG=1119888119860(DG119905)119883DG(DG119905) + 119888119861(DG119905)119875DG(DG119905)

+ 119888119862(DG119905)119875

2

DG(DG119905) +

119873SP

sum

SP=1119888SP(SP119905)119875SP(SP119905)

+

119873EV

sum


minus 119888Ch(EV119905)119875Ch(EV119905) +

119873119871

sum

119871=1

119888NSD(119871119905)119875NSD(119871119905)

+

119873DG

sum


1198651119889+ 1198651119889minus ge 0

(14)


119873119894

DG

sum

DG=1(119875119894

DG(DG119905) minus 119875119894

GCP(DG119905)) +

119873119894

SP

sum

SP=1119875119894

SP(SP119905)

+

119873119894

EV

sum

EV=1119875119894

Dch(EV119905) minus

119873119894

119871

sum

119871=1

(119875119894

Load(119871119905) minus 119875119894

NSD(119871119905))

minus

119873119894

EV

sum

EV=1119875119894

Ch(EV119905)

= 1198661198941198941198812

119894(119905)

+ 119881119894(119905)sum

119895isin119879119871119894

119881119895(119905)(119866119894119895cos 120579119894119895(119905)

+ 119861119894119895sin 120579119894119895(119905))

(15)



119873119894

DG

sum

DG=1119876119894

DG(DG119905) +

119873119894

SP

sum

SP=1119876119894

SP(SP119905)

minus

119873119894

119871

sum

119871=1

(119876119894

Load(119871119905) minus 119876119894

NSD(119871119905))

= 119881119894(119905)sum

119895isin119879119871119894

119881119895(119905)(119866119894119895sin 120579119894119895(119905)

minus 119861119894119895cos 120579119894119895(119905))

minus 1198611198941198941198812

119894(119905)


(16)

where 120579119894119895(119905)


119894119895and 119861





119881119894

Min le 119881119894(119905) le 119881119894


120579119894

Min le 120579119894(119905) le 120579119894


(17)



100381610038161003816100381610038161003816119881119894(119905)[119910119894119895119881119894119895(119905)

+ 119910sh 119894119881119894(119905)]lowast100381610038161003816100381610038161003816le 119878

Max119879119871 119894 to 119895

100381610038161003816100381610038161003816119881119895(119905)[119910119894119895119881119895119894(119905)

+ 119910sh 119895119881119895(119905)]lowast100381610038161003816100381610038161003816le 119878

Max119879119871 119895 to 119894


= 119881119894(119905)minus 119881119895(119905)

(18)


radic(

119873119894

SP

sum

SP=1119875119894

SP(SP119905))

2

+ (

119873119894

SP

sum

SP=1119876119894

SP(SP119905))

2



(19)


radic1198752

TFR MV LV(119894119905) + 1198762



(20)


119875TFR MV LV(119894119905) =

119873119894

DG

sum

DG=1(119875119894

DG(DG119905) minus 119875119894

GCP(DG119905))

+

119873119894

EV

sum

EV=1(119875119894

Dch(EV119905) minus 119875119894

Ch(EV119905))

minus

119873119894

119871

sum

119871=1

(119875119894

Load(119871119905) minus 119875119894

NSD(119871119905))

119876TFR MV LV(119894119905) =

119873119894

DG

sum

DG=1119876119894

DG(DG119905)

minus

119873119894

119871

sum

119871=1

(119876119894

Load(119871119905) minus 119876119894

NSD(119871119905))

(21)



le 119875Max(DG119905)119883DG(DG119905)


le 119876Max(DG119905)119883DG(DG119905)

radic(119875DG(DG119905))2

+ (119876DG(DG119905))2

le 119878Max(DG119905)


(22)






119875SP(SP119905) le 119875Max(SP119905)

119876SP(SP119905) le 119876Max(SP119905)


(24)




120578119889(EV)

119875Dch(EV119905)


(25)












(27)








119873STP

sum

STP=1119883TFR(STP119905) = 1 (30)

119881119894(119905)= 119881

Base119894(119905)

+

119873STP

sum

STP=1Δ119881119894

TFR(STP119905)


(31)













4 Case Study






0

11819

20

21

2

3

4

567

8

11

910

1213

1415

16

17

2223

24

2526

27

28

29

30

31

32

50kW

30kW

30kW 9

kW 9kW

9kW

30kW

9kW

9kW

30kW

30kW

30kW

30kW

9kW

30kW

30kW

9kW9

kW3

kW

30kW9

kW

3kW

9kW

9kW

30kW

30kW

9kW

3kW

9kW

9kW

3kW

30kW

30kW

10kW40

kW50kW

100kW

50kW

10kW

100kW

50kW

10kW 200

kW

10kW

25kW 25

kW 100kW

150kW

100kW

100kW 100

kW

50kW

50kW

50kW50

kW

50kW

100kW

25kW200

kW10

kW

50kW

10kW

30kW

200kW

25kW25

kW

PV panel

Wind

WTE

CHP

Fuel cell

Biomass

Hydro

EV with V2G sim

sim

sim

simsim

sim

sim





L-in

dex


002004006008

01012014016018









L-in

dex

002004006008

01012014016018




Simulations


100

150

200

250

Nor

mal

ized

dist

ance


mdash



1119901= 1198651119899

and 1198652119901




0100200300400500600700

1 3 5 7 9 11 13 15 17 19 21 23

Ope

ratio

n co

st (m

u)

Time (h)




0002004006008

01012014016018

L-in

dex

3 5 7 9 11 13 15 17 19 21 231Time (h)






302826242210 12 14 16 18 20 3286420Bus

1101102103104105106107

Volta

ge m

agni

tude

(pu)



0500

100015002000250030003500400045005000

Pow

er (k

W)



3 5 7 9 11 13 15 17 19 21 231Time (h)





1 3 5 7 9 11 13 15 17 19 21 23Time (h)


minus1500

minus1000

minus500

0

500

1000

1500

2000

Pow

er (k

W)

minus15000

minus10000

minus5000

0

5000

10000

15000

20000

Ener

gy (k

Wh)











Minimum 1198651

1297Minimum 119865

22115




max 120582119905


119876load = 120582119905119876load1015840(119871119905)

forall119905 isin 1 119879

forall119871 isin 1 119873119871

(32)






1)


Slackgenerator

Bus 0


ldquoVirtualrdquo

ZTH










0

02

04

06

08

1

12

L-in

dex




0 5 10 15 20 25 30 35 40 45 50 55 6088008850890089509000905091009150920092509300

Ope

ratio

n co

st (m

u)








1) results


2)



09

095

1

105

11

115

L-in

dex

104

103

102

101

100

099

098

097

096

095

094

093

092

105




105

104

103

102

101

100

099

098

097

096

095

094

093

092

71007110712071307140715071607170718071907200

Ope

ratio

n co

st (m

u)







100

120

140

160

180

200

220

240

260

280

300

320

340

360

380

400

Load ()

094

0945

095

0955

096

0965

097

L-in

dex



550060006500700075008000850090009500

1000010500

Ope

ratio

n co

st (m

u)

120

140

160

180

200

220

240

260

280

300

320

340

360

380

400

100










5 Conclusions



765 768

780

803

812

862

877

937

903

21536

68 94

106

171 179174

127

95

181

214 215

219

222

264

306

275

237

261

235

234

328338372

341

345

357

360

473496 563

397

564 590 592

598618

624

623 664

595

665 681 685

689

690

700

730

736737735

764
















0001002003004005006007008009

01

L-in

dex

9320

0

9340

0

9480

0

9380

0

9400

0

9420

0

9440

0

9500

0

9300

0

9360

0

9460

0

Operation cost (mu)







Simulations0

1

2

3

4

5N

orm

aliz

ed d

istan

ce


Best solution


200025003000350040004500500055006000

Ope

ratio

n co

st (m

u)

3 5 7 9 11 13 15 17 19 21 231Time (h)



1 3 5 7 9 11 13 15 17 19 21 23Time (h)


0001002003004005006007008009

01

L-in

dex


Nomenclature

Parameters






3 5 7 9 11 13 15 17 19 21 231Time (h) minus15000

minus10000

minus5000

0

5000

10000

15000

20000

Ener

gy (k

Wh)

minus2000minus1000

010002000300040005000600070008000

Pow

er (k

W)











buses [S]

Variables



Indices








Competing Interests


References



















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Start

(1) (5)




definition

(10) (11)


Up

Pareto front

UpStart


(F1n) and (F2n)

F1

fix v

alue

sF1

fix v

alue

s

F2 fix values

F2 fix values

F1n

F1n

F2n

F2n

F1p

F1p

F2p

F2p







1119901) and (119865

2119901) The utopia




1119899and 119865



1and 119865

2are



1198651 step =

(1198651119899minus Up)

num steps (9)

1198652 step =

(1198652119899minus Up)

num steps (10)



1198921) and (119865

1198922)

1198651198921= min (119865

1+ 10119865

2119889+ + 01119865

2119889minus) (11)

1198651198922= min (119865

2+ 10119865

1119889+ + 01119865

1119889minus) (12)



2119889minus)



2(5) into

a constraint

1198652= 119865Aux + 1198652119889+ minus 1198652119889minus 119865

2119889+ 1198652119889minus ge 0 (13)




1119889+) and


1 Then the


1198651=

119879

sum

119905=1

[

119873DG

sum

DG=1119888119860(DG119905)119883DG(DG119905) + 119888119861(DG119905)119875DG(DG119905)

+ 119888119862(DG119905)119875

2

DG(DG119905) +

119873SP

sum

SP=1119888SP(SP119905)119875SP(SP119905)

+

119873EV

sum


minus 119888Ch(EV119905)119875Ch(EV119905) +

119873119871

sum

119871=1

119888NSD(119871119905)119875NSD(119871119905)

+

119873DG

sum


1198651119889+ 1198651119889minus ge 0

(14)


119873119894

DG

sum

DG=1(119875119894

DG(DG119905) minus 119875119894

GCP(DG119905)) +

119873119894

SP

sum

SP=1119875119894

SP(SP119905)

+

119873119894

EV

sum

EV=1119875119894

Dch(EV119905) minus

119873119894

119871

sum

119871=1

(119875119894

Load(119871119905) minus 119875119894

NSD(119871119905))

minus

119873119894

EV

sum

EV=1119875119894

Ch(EV119905)

= 1198661198941198941198812

119894(119905)

+ 119881119894(119905)sum

119895isin119879119871119894

119881119895(119905)(119866119894119895cos 120579119894119895(119905)

+ 119861119894119895sin 120579119894119895(119905))

(15)



119873119894

DG

sum

DG=1119876119894

DG(DG119905) +

119873119894

SP

sum

SP=1119876119894

SP(SP119905)

minus

119873119894

119871

sum

119871=1

(119876119894

Load(119871119905) minus 119876119894

NSD(119871119905))

= 119881119894(119905)sum

119895isin119879119871119894

119881119895(119905)(119866119894119895sin 120579119894119895(119905)

minus 119861119894119895cos 120579119894119895(119905))

minus 1198611198941198941198812

119894(119905)


(16)

where 120579119894119895(119905)


119894119895and 119861





119881119894

Min le 119881119894(119905) le 119881119894


120579119894

Min le 120579119894(119905) le 120579119894


(17)



100381610038161003816100381610038161003816119881119894(119905)[119910119894119895119881119894119895(119905)

+ 119910sh 119894119881119894(119905)]lowast100381610038161003816100381610038161003816le 119878

Max119879119871 119894 to 119895

100381610038161003816100381610038161003816119881119895(119905)[119910119894119895119881119895119894(119905)

+ 119910sh 119895119881119895(119905)]lowast100381610038161003816100381610038161003816le 119878

Max119879119871 119895 to 119894


= 119881119894(119905)minus 119881119895(119905)

(18)


radic(

119873119894

SP

sum

SP=1119875119894

SP(SP119905))

2

+ (

119873119894

SP

sum

SP=1119876119894

SP(SP119905))

2



(19)


radic1198752

TFR MV LV(119894119905) + 1198762



(20)


119875TFR MV LV(119894119905) =

119873119894

DG

sum

DG=1(119875119894

DG(DG119905) minus 119875119894

GCP(DG119905))

+

119873119894

EV

sum

EV=1(119875119894

Dch(EV119905) minus 119875119894

Ch(EV119905))

minus

119873119894

119871

sum

119871=1

(119875119894

Load(119871119905) minus 119875119894

NSD(119871119905))

119876TFR MV LV(119894119905) =

119873119894

DG

sum

DG=1119876119894

DG(DG119905)

minus

119873119894

119871

sum

119871=1

(119876119894

Load(119871119905) minus 119876119894

NSD(119871119905))

(21)



le 119875Max(DG119905)119883DG(DG119905)


le 119876Max(DG119905)119883DG(DG119905)

radic(119875DG(DG119905))2

+ (119876DG(DG119905))2

le 119878Max(DG119905)


(22)






119875SP(SP119905) le 119875Max(SP119905)

119876SP(SP119905) le 119876Max(SP119905)


(24)




120578119889(EV)

119875Dch(EV119905)


(25)












(27)








119873STP

sum

STP=1119883TFR(STP119905) = 1 (30)

119881119894(119905)= 119881

Base119894(119905)

+

119873STP

sum

STP=1Δ119881119894

TFR(STP119905)


(31)













4 Case Study






0

11819

20

21

2

3

4

567

8

11

910

1213

1415

16

17

2223

24

2526

27

28

29

30

31

32

50kW

30kW

30kW 9

kW 9kW

9kW

30kW

9kW

9kW

30kW

30kW

30kW

30kW

9kW

30kW

30kW

9kW9

kW3

kW

30kW9

kW

3kW

9kW

9kW

30kW

30kW

9kW

3kW

9kW

9kW

3kW

30kW

30kW

10kW40

kW50kW

100kW

50kW

10kW

100kW

50kW

10kW 200

kW

10kW

25kW 25

kW 100kW

150kW

100kW

100kW 100

kW

50kW

50kW

50kW50

kW

50kW

100kW

25kW200

kW10

kW

50kW

10kW

30kW

200kW

25kW25

kW

PV panel

Wind

WTE

CHP

Fuel cell

Biomass

Hydro

EV with V2G sim

sim

sim

simsim

sim

sim





L-in

dex


002004006008

01012014016018









L-in

dex

002004006008

01012014016018




Simulations


100

150

200

250

Nor

mal

ized

dist

ance


mdash



1119901= 1198651119899

and 1198652119901




0100200300400500600700

1 3 5 7 9 11 13 15 17 19 21 23

Ope

ratio

n co

st (m

u)

Time (h)




0002004006008

01012014016018

L-in

dex

3 5 7 9 11 13 15 17 19 21 231Time (h)






302826242210 12 14 16 18 20 3286420Bus

1101102103104105106107

Volta

ge m

agni

tude

(pu)



0500

100015002000250030003500400045005000

Pow

er (k

W)



3 5 7 9 11 13 15 17 19 21 231Time (h)





1 3 5 7 9 11 13 15 17 19 21 23Time (h)


minus1500

minus1000

minus500

0

500

1000

1500

2000

Pow

er (k

W)

minus15000

minus10000

minus5000

0

5000

10000

15000

20000

Ener

gy (k

Wh)











Minimum 1198651

1297Minimum 119865

22115




max 120582119905


119876load = 120582119905119876load1015840(119871119905)

forall119905 isin 1 119879

forall119871 isin 1 119873119871

(32)






1)


Slackgenerator

Bus 0


ldquoVirtualrdquo

ZTH










0

02

04

06

08

1

12

L-in

dex




0 5 10 15 20 25 30 35 40 45 50 55 6088008850890089509000905091009150920092509300

Ope

ratio

n co

st (m

u)








1) results


2)



09

095

1

105

11

115

L-in

dex

104

103

102

101

100

099

098

097

096

095

094

093

092

105




105

104

103

102

101

100

099

098

097

096

095

094

093

092

71007110712071307140715071607170718071907200

Ope

ratio

n co

st (m

u)







100

120

140

160

180

200

220

240

260

280

300

320

340

360

380

400

Load ()

094

0945

095

0955

096

0965

097

L-in

dex



550060006500700075008000850090009500

1000010500

Ope

ratio

n co

st (m

u)

120

140

160

180

200

220

240

260

280

300

320

340

360

380

400

100










5 Conclusions



765 768

780

803

812

862

877

937

903

21536

68 94

106

171 179174

127

95

181

214 215

219

222

264

306

275

237

261

235

234

328338372

341

345

357

360

473496 563

397

564 590 592

598618

624

623 664

595

665 681 685

689

690

700

730

736737735

764
















0001002003004005006007008009

01

L-in

dex

9320

0

9340

0

9480

0

9380

0

9400

0

9420

0

9440

0

9500

0

9300

0

9360

0

9460

0

Operation cost (mu)







Simulations0

1

2

3

4

5N

orm

aliz

ed d

istan

ce


Best solution


200025003000350040004500500055006000

Ope

ratio

n co

st (m

u)

3 5 7 9 11 13 15 17 19 21 231Time (h)



1 3 5 7 9 11 13 15 17 19 21 23Time (h)


0001002003004005006007008009

01

L-in

dex


Nomenclature

Parameters






3 5 7 9 11 13 15 17 19 21 231Time (h) minus15000

minus10000

minus5000

0

5000

10000

15000

20000

Ener

gy (k

Wh)

minus2000minus1000

010002000300040005000600070008000

Pow

er (k

W)











buses [S]

Variables



Indices








Competing Interests


References



















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1119899and 119865



1and 119865

2are



1198651 step =

(1198651119899minus Up)

num steps (9)

1198652 step =

(1198652119899minus Up)

num steps (10)



1198921) and (119865

1198922)

1198651198921= min (119865

1+ 10119865

2119889+ + 01119865

2119889minus) (11)

1198651198922= min (119865

2+ 10119865

1119889+ + 01119865

1119889minus) (12)



2119889minus)



2(5) into

a constraint

1198652= 119865Aux + 1198652119889+ minus 1198652119889minus 119865

2119889+ 1198652119889minus ge 0 (13)




1119889+) and


1 Then the


1198651=

119879

sum

119905=1

[

119873DG

sum

DG=1119888119860(DG119905)119883DG(DG119905) + 119888119861(DG119905)119875DG(DG119905)

+ 119888119862(DG119905)119875

2

DG(DG119905) +

119873SP

sum

SP=1119888SP(SP119905)119875SP(SP119905)

+

119873EV

sum


minus 119888Ch(EV119905)119875Ch(EV119905) +

119873119871

sum

119871=1

119888NSD(119871119905)119875NSD(119871119905)

+

119873DG

sum


1198651119889+ 1198651119889minus ge 0

(14)


119873119894

DG

sum

DG=1(119875119894

DG(DG119905) minus 119875119894

GCP(DG119905)) +

119873119894

SP

sum

SP=1119875119894

SP(SP119905)

+

119873119894

EV

sum

EV=1119875119894

Dch(EV119905) minus

119873119894

119871

sum

119871=1

(119875119894

Load(119871119905) minus 119875119894

NSD(119871119905))

minus

119873119894

EV

sum

EV=1119875119894

Ch(EV119905)

= 1198661198941198941198812

119894(119905)

+ 119881119894(119905)sum

119895isin119879119871119894

119881119895(119905)(119866119894119895cos 120579119894119895(119905)

+ 119861119894119895sin 120579119894119895(119905))

(15)



119873119894

DG

sum

DG=1119876119894

DG(DG119905) +

119873119894

SP

sum

SP=1119876119894

SP(SP119905)

minus

119873119894

119871

sum

119871=1

(119876119894

Load(119871119905) minus 119876119894

NSD(119871119905))

= 119881119894(119905)sum

119895isin119879119871119894

119881119895(119905)(119866119894119895sin 120579119894119895(119905)

minus 119861119894119895cos 120579119894119895(119905))

minus 1198611198941198941198812

119894(119905)


(16)

where 120579119894119895(119905)


119894119895and 119861





119881119894

Min le 119881119894(119905) le 119881119894


120579119894

Min le 120579119894(119905) le 120579119894


(17)



100381610038161003816100381610038161003816119881119894(119905)[119910119894119895119881119894119895(119905)

+ 119910sh 119894119881119894(119905)]lowast100381610038161003816100381610038161003816le 119878

Max119879119871 119894 to 119895

100381610038161003816100381610038161003816119881119895(119905)[119910119894119895119881119895119894(119905)

+ 119910sh 119895119881119895(119905)]lowast100381610038161003816100381610038161003816le 119878

Max119879119871 119895 to 119894


= 119881119894(119905)minus 119881119895(119905)

(18)


radic(

119873119894

SP

sum

SP=1119875119894

SP(SP119905))

2

+ (

119873119894

SP

sum

SP=1119876119894

SP(SP119905))

2



(19)


radic1198752

TFR MV LV(119894119905) + 1198762



(20)


119875TFR MV LV(119894119905) =

119873119894

DG

sum

DG=1(119875119894

DG(DG119905) minus 119875119894

GCP(DG119905))

+

119873119894

EV

sum

EV=1(119875119894

Dch(EV119905) minus 119875119894

Ch(EV119905))

minus

119873119894

119871

sum

119871=1

(119875119894

Load(119871119905) minus 119875119894

NSD(119871119905))

119876TFR MV LV(119894119905) =

119873119894

DG

sum

DG=1119876119894

DG(DG119905)

minus

119873119894

119871

sum

119871=1

(119876119894

Load(119871119905) minus 119876119894

NSD(119871119905))

(21)



le 119875Max(DG119905)119883DG(DG119905)


le 119876Max(DG119905)119883DG(DG119905)

radic(119875DG(DG119905))2

+ (119876DG(DG119905))2

le 119878Max(DG119905)


(22)






119875SP(SP119905) le 119875Max(SP119905)

119876SP(SP119905) le 119876Max(SP119905)


(24)




120578119889(EV)

119875Dch(EV119905)


(25)












(27)








119873STP

sum

STP=1119883TFR(STP119905) = 1 (30)

119881119894(119905)= 119881

Base119894(119905)

+

119873STP

sum

STP=1Δ119881119894

TFR(STP119905)


(31)













4 Case Study






0

11819

20

21

2

3

4

567

8

11

910

1213

1415

16

17

2223

24

2526

27

28

29

30

31

32

50kW

30kW

30kW 9

kW 9kW

9kW

30kW

9kW

9kW

30kW

30kW

30kW

30kW

9kW

30kW

30kW

9kW9

kW3

kW

30kW9

kW

3kW

9kW

9kW

30kW

30kW

9kW

3kW

9kW

9kW

3kW

30kW

30kW

10kW40

kW50kW

100kW

50kW

10kW

100kW

50kW

10kW 200

kW

10kW

25kW 25

kW 100kW

150kW

100kW

100kW 100

kW

50kW

50kW

50kW50

kW

50kW

100kW

25kW200

kW10

kW

50kW

10kW

30kW

200kW

25kW25

kW

PV panel

Wind

WTE

CHP

Fuel cell

Biomass

Hydro

EV with V2G sim

sim

sim

simsim

sim

sim





L-in

dex


002004006008

01012014016018









L-in

dex

002004006008

01012014016018




Simulations


100

150

200

250

Nor

mal

ized

dist

ance


mdash



1119901= 1198651119899

and 1198652119901




0100200300400500600700

1 3 5 7 9 11 13 15 17 19 21 23

Ope

ratio

n co

st (m

u)

Time (h)




0002004006008

01012014016018

L-in

dex

3 5 7 9 11 13 15 17 19 21 231Time (h)






302826242210 12 14 16 18 20 3286420Bus

1101102103104105106107

Volta

ge m

agni

tude

(pu)



0500

100015002000250030003500400045005000

Pow

er (k

W)



3 5 7 9 11 13 15 17 19 21 231Time (h)





1 3 5 7 9 11 13 15 17 19 21 23Time (h)


minus1500

minus1000

minus500

0

500

1000

1500

2000

Pow

er (k

W)

minus15000

minus10000

minus5000

0

5000

10000

15000

20000

Ener

gy (k

Wh)











Minimum 1198651

1297Minimum 119865

22115




max 120582119905


119876load = 120582119905119876load1015840(119871119905)

forall119905 isin 1 119879

forall119871 isin 1 119873119871

(32)






1)


Slackgenerator

Bus 0


ldquoVirtualrdquo

ZTH










0

02

04

06

08

1

12

L-in

dex




0 5 10 15 20 25 30 35 40 45 50 55 6088008850890089509000905091009150920092509300

Ope

ratio

n co

st (m

u)








1) results


2)



09

095

1

105

11

115

L-in

dex

104

103

102

101

100

099

098

097

096

095

094

093

092

105




105

104

103

102

101

100

099

098

097

096

095

094

093

092

71007110712071307140715071607170718071907200

Ope

ratio

n co

st (m

u)







100

120

140

160

180

200

220

240

260

280

300

320

340

360

380

400

Load ()

094

0945

095

0955

096

0965

097

L-in

dex



550060006500700075008000850090009500

1000010500

Ope

ratio

n co

st (m

u)

120

140

160

180

200

220

240

260

280

300

320

340

360

380

400

100










5 Conclusions



765 768

780

803

812

862

877

937

903

21536

68 94

106

171 179174

127

95

181

214 215

219

222

264

306

275

237

261

235

234

328338372

341

345

357

360

473496 563

397

564 590 592

598618

624

623 664

595

665 681 685

689

690

700

730

736737735

764
















0001002003004005006007008009

01

L-in

dex

9320

0

9340

0

9480

0

9380

0

9400

0

9420

0

9440

0

9500

0

9300

0

9360

0

9460

0

Operation cost (mu)







Simulations0

1

2

3

4

5N

orm

aliz

ed d

istan

ce


Best solution


200025003000350040004500500055006000

Ope

ratio

n co

st (m

u)

3 5 7 9 11 13 15 17 19 21 231Time (h)



1 3 5 7 9 11 13 15 17 19 21 23Time (h)


0001002003004005006007008009

01

L-in

dex


Nomenclature

Parameters






3 5 7 9 11 13 15 17 19 21 231Time (h) minus15000

minus10000

minus5000

0

5000

10000

15000

20000

Ener

gy (k

Wh)

minus2000minus1000

010002000300040005000600070008000

Pow

er (k

W)











buses [S]

Variables



Indices








Competing Interests


References



















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119881119894

Min le 119881119894(119905) le 119881119894


120579119894

Min le 120579119894(119905) le 120579119894


(17)



100381610038161003816100381610038161003816119881119894(119905)[119910119894119895119881119894119895(119905)

+ 119910sh 119894119881119894(119905)]lowast100381610038161003816100381610038161003816le 119878

Max119879119871 119894 to 119895

100381610038161003816100381610038161003816119881119895(119905)[119910119894119895119881119895119894(119905)

+ 119910sh 119895119881119895(119905)]lowast100381610038161003816100381610038161003816le 119878

Max119879119871 119895 to 119894


= 119881119894(119905)minus 119881119895(119905)

(18)


radic(

119873119894

SP

sum

SP=1119875119894

SP(SP119905))

2

+ (

119873119894

SP

sum

SP=1119876119894

SP(SP119905))

2



(19)


radic1198752

TFR MV LV(119894119905) + 1198762



(20)


119875TFR MV LV(119894119905) =

119873119894

DG

sum

DG=1(119875119894

DG(DG119905) minus 119875119894

GCP(DG119905))

+

119873119894

EV

sum

EV=1(119875119894

Dch(EV119905) minus 119875119894

Ch(EV119905))

minus

119873119894

119871

sum

119871=1

(119875119894

Load(119871119905) minus 119875119894

NSD(119871119905))

119876TFR MV LV(119894119905) =

119873119894

DG

sum

DG=1119876119894

DG(DG119905)

minus

119873119894

119871

sum

119871=1

(119876119894

Load(119871119905) minus 119876119894

NSD(119871119905))

(21)



le 119875Max(DG119905)119883DG(DG119905)


le 119876Max(DG119905)119883DG(DG119905)

radic(119875DG(DG119905))2

+ (119876DG(DG119905))2

le 119878Max(DG119905)


(22)






119875SP(SP119905) le 119875Max(SP119905)

119876SP(SP119905) le 119876Max(SP119905)


(24)




120578119889(EV)

119875Dch(EV119905)


(25)












(27)








119873STP

sum

STP=1119883TFR(STP119905) = 1 (30)

119881119894(119905)= 119881

Base119894(119905)

+

119873STP

sum

STP=1Δ119881119894

TFR(STP119905)


(31)













4 Case Study






0

11819

20

21

2

3

4

567

8

11

910

1213

1415

16

17

2223

24

2526

27

28

29

30

31

32

50kW

30kW

30kW 9

kW 9kW

9kW

30kW

9kW

9kW

30kW

30kW

30kW

30kW

9kW

30kW

30kW

9kW9

kW3

kW

30kW9

kW

3kW

9kW

9kW

30kW

30kW

9kW

3kW

9kW

9kW

3kW

30kW

30kW

10kW40

kW50kW

100kW

50kW

10kW

100kW

50kW

10kW 200

kW

10kW

25kW 25

kW 100kW

150kW

100kW

100kW 100

kW

50kW

50kW

50kW50

kW

50kW

100kW

25kW200

kW10

kW

50kW

10kW

30kW

200kW

25kW25

kW

PV panel

Wind

WTE

CHP

Fuel cell

Biomass

Hydro

EV with V2G sim

sim

sim

simsim

sim

sim





L-in

dex


002004006008

01012014016018









L-in

dex

002004006008

01012014016018




Simulations


100

150

200

250

Nor

mal

ized

dist

ance


mdash



1119901= 1198651119899

and 1198652119901




0100200300400500600700

1 3 5 7 9 11 13 15 17 19 21 23

Ope

ratio

n co

st (m

u)

Time (h)




0002004006008

01012014016018

L-in

dex

3 5 7 9 11 13 15 17 19 21 231Time (h)






302826242210 12 14 16 18 20 3286420Bus

1101102103104105106107

Volta

ge m

agni

tude

(pu)



0500

100015002000250030003500400045005000

Pow

er (k

W)



3 5 7 9 11 13 15 17 19 21 231Time (h)





1 3 5 7 9 11 13 15 17 19 21 23Time (h)


minus1500

minus1000

minus500

0

500

1000

1500

2000

Pow

er (k

W)

minus15000

minus10000

minus5000

0

5000

10000

15000

20000

Ener

gy (k

Wh)











Minimum 1198651

1297Minimum 119865

22115




max 120582119905


119876load = 120582119905119876load1015840(119871119905)

forall119905 isin 1 119879

forall119871 isin 1 119873119871

(32)






1)


Slackgenerator

Bus 0


ldquoVirtualrdquo

ZTH










0

02

04

06

08

1

12

L-in

dex




0 5 10 15 20 25 30 35 40 45 50 55 6088008850890089509000905091009150920092509300

Ope

ratio

n co

st (m

u)








1) results


2)



09

095

1

105

11

115

L-in

dex

104

103

102

101

100

099

098

097

096

095

094

093

092

105




105

104

103

102

101

100

099

098

097

096

095

094

093

092

71007110712071307140715071607170718071907200

Ope

ratio

n co

st (m

u)







100

120

140

160

180

200

220

240

260

280

300

320

340

360

380

400

Load ()

094

0945

095

0955

096

0965

097

L-in

dex



550060006500700075008000850090009500

1000010500

Ope

ratio

n co

st (m

u)

120

140

160

180

200

220

240

260

280

300

320

340

360

380

400

100










5 Conclusions



765 768

780

803

812

862

877

937

903

21536

68 94

106

171 179174

127

95

181

214 215

219

222

264

306

275

237

261

235

234

328338372

341

345

357

360

473496 563

397

564 590 592

598618

624

623 664

595

665 681 685

689

690

700

730

736737735

764
















0001002003004005006007008009

01

L-in

dex

9320

0

9340

0

9480

0

9380

0

9400

0

9420

0

9440

0

9500

0

9300

0

9360

0

9460

0

Operation cost (mu)







Simulations0

1

2

3

4

5N

orm

aliz

ed d

istan

ce


Best solution


200025003000350040004500500055006000

Ope

ratio

n co

st (m

u)

3 5 7 9 11 13 15 17 19 21 231Time (h)



1 3 5 7 9 11 13 15 17 19 21 23Time (h)


0001002003004005006007008009

01

L-in

dex


Nomenclature

Parameters






3 5 7 9 11 13 15 17 19 21 231Time (h) minus15000

minus10000

minus5000

0

5000

10000

15000

20000

Ener

gy (k

Wh)

minus2000minus1000

010002000300040005000600070008000

Pow

er (k

W)











buses [S]

Variables



Indices








Competing Interests


References



















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












120578119889(EV)

119875Dch(EV119905)


(25)












(27)








119873STP

sum

STP=1119883TFR(STP119905) = 1 (30)

119881119894(119905)= 119881

Base119894(119905)

+

119873STP

sum

STP=1Δ119881119894

TFR(STP119905)


(31)













4 Case Study






0

11819

20

21

2

3

4

567

8

11

910

1213

1415

16

17

2223

24

2526

27

28

29

30

31

32

50kW

30kW

30kW 9

kW 9kW

9kW

30kW

9kW

9kW

30kW

30kW

30kW

30kW

9kW

30kW

30kW

9kW9

kW3

kW

30kW9

kW

3kW

9kW

9kW

30kW

30kW

9kW

3kW

9kW

9kW

3kW

30kW

30kW

10kW40

kW50kW

100kW

50kW

10kW

100kW

50kW

10kW 200

kW

10kW

25kW 25

kW 100kW

150kW

100kW

100kW 100

kW

50kW

50kW

50kW50

kW

50kW

100kW

25kW200

kW10

kW

50kW

10kW

30kW

200kW

25kW25

kW

PV panel

Wind

WTE

CHP

Fuel cell

Biomass

Hydro

EV with V2G sim

sim

sim

simsim

sim

sim





L-in

dex


002004006008

01012014016018









L-in

dex

002004006008

01012014016018




Simulations


100

150

200

250

Nor

mal

ized

dist

ance


mdash



1119901= 1198651119899

and 1198652119901




0100200300400500600700

1 3 5 7 9 11 13 15 17 19 21 23

Ope

ratio

n co

st (m

u)

Time (h)




0002004006008

01012014016018

L-in

dex

3 5 7 9 11 13 15 17 19 21 231Time (h)






302826242210 12 14 16 18 20 3286420Bus

1101102103104105106107

Volta

ge m

agni

tude

(pu)



0500

100015002000250030003500400045005000

Pow

er (k

W)



3 5 7 9 11 13 15 17 19 21 231Time (h)





1 3 5 7 9 11 13 15 17 19 21 23Time (h)


minus1500

minus1000

minus500

0

500

1000

1500

2000

Pow

er (k

W)

minus15000

minus10000

minus5000

0

5000

10000

15000

20000

Ener

gy (k

Wh)











Minimum 1198651

1297Minimum 119865

22115




max 120582119905


119876load = 120582119905119876load1015840(119871119905)

forall119905 isin 1 119879

forall119871 isin 1 119873119871

(32)






1)


Slackgenerator

Bus 0


ldquoVirtualrdquo

ZTH










0

02

04

06

08

1

12

L-in

dex




0 5 10 15 20 25 30 35 40 45 50 55 6088008850890089509000905091009150920092509300

Ope

ratio

n co

st (m

u)








1) results


2)



09

095

1

105

11

115

L-in

dex

104

103

102

101

100

099

098

097

096

095

094

093

092

105




105

104

103

102

101

100

099

098

097

096

095

094

093

092

71007110712071307140715071607170718071907200

Ope

ratio

n co

st (m

u)







100

120

140

160

180

200

220

240

260

280

300

320

340

360

380

400

Load ()

094

0945

095

0955

096

0965

097

L-in

dex



550060006500700075008000850090009500

1000010500

Ope

ratio

n co

st (m

u)

120

140

160

180

200

220

240

260

280

300

320

340

360

380

400

100










5 Conclusions



765 768

780

803

812

862

877

937

903

21536

68 94

106

171 179174

127

95

181

214 215

219

222

264

306

275

237

261

235

234

328338372

341

345

357

360

473496 563

397

564 590 592

598618

624

623 664

595

665 681 685

689

690

700

730

736737735

764
















0001002003004005006007008009

01

L-in

dex

9320

0

9340

0

9480

0

9380

0

9400

0

9420

0

9440

0

9500

0

9300

0

9360

0

9460

0

Operation cost (mu)







Simulations0

1

2

3

4

5N

orm

aliz

ed d

istan

ce


Best solution


200025003000350040004500500055006000

Ope

ratio

n co

st (m

u)

3 5 7 9 11 13 15 17 19 21 231Time (h)



1 3 5 7 9 11 13 15 17 19 21 23Time (h)


0001002003004005006007008009

01

L-in

dex


Nomenclature

Parameters






3 5 7 9 11 13 15 17 19 21 231Time (h) minus15000

minus10000

minus5000

0

5000

10000

15000

20000

Ener

gy (k

Wh)

minus2000minus1000

010002000300040005000600070008000

Pow

er (k

W)











buses [S]

Variables



Indices








Competing Interests


References



















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















4 Case Study






0

11819

20

21

2

3

4

567

8

11

910

1213

1415

16

17

2223

24

2526

27

28

29

30

31

32

50kW

30kW

30kW 9

kW 9kW

9kW

30kW

9kW

9kW

30kW

30kW

30kW

30kW

9kW

30kW

30kW

9kW9

kW3

kW

30kW9

kW

3kW

9kW

9kW

30kW

30kW

9kW

3kW

9kW

9kW

3kW

30kW

30kW

10kW40

kW50kW

100kW

50kW

10kW

100kW

50kW

10kW 200

kW

10kW

25kW 25

kW 100kW

150kW

100kW

100kW 100

kW

50kW

50kW

50kW50

kW

50kW

100kW

25kW200

kW10

kW

50kW

10kW

30kW

200kW

25kW25

kW

PV panel

Wind

WTE

CHP

Fuel cell

Biomass

Hydro

EV with V2G sim

sim

sim

simsim

sim

sim





L-in

dex


002004006008

01012014016018









L-in

dex

002004006008

01012014016018




Simulations


100

150

200

250

Nor

mal

ized

dist

ance


mdash



1119901= 1198651119899

and 1198652119901




0100200300400500600700

1 3 5 7 9 11 13 15 17 19 21 23

Ope

ratio

n co

st (m

u)

Time (h)




0002004006008

01012014016018

L-in

dex

3 5 7 9 11 13 15 17 19 21 231Time (h)






302826242210 12 14 16 18 20 3286420Bus

1101102103104105106107

Volta

ge m

agni

tude

(pu)



0500

100015002000250030003500400045005000

Pow

er (k

W)



3 5 7 9 11 13 15 17 19 21 231Time (h)





1 3 5 7 9 11 13 15 17 19 21 23Time (h)


minus1500

minus1000

minus500

0

500

1000

1500

2000

Pow

er (k

W)

minus15000

minus10000

minus5000

0

5000

10000

15000

20000

Ener

gy (k

Wh)











Minimum 1198651

1297Minimum 119865

22115




max 120582119905


119876load = 120582119905119876load1015840(119871119905)

forall119905 isin 1 119879

forall119871 isin 1 119873119871

(32)






1)


Slackgenerator

Bus 0


ldquoVirtualrdquo

ZTH










0

02

04

06

08

1

12

L-in

dex




0 5 10 15 20 25 30 35 40 45 50 55 6088008850890089509000905091009150920092509300

Ope

ratio

n co

st (m

u)








1) results


2)



09

095

1

105

11

115

L-in

dex

104

103

102

101

100

099

098

097

096

095

094

093

092

105




105

104

103

102

101

100

099

098

097

096

095

094

093

092

71007110712071307140715071607170718071907200

Ope

ratio

n co

st (m

u)







100

120

140

160

180

200

220

240

260

280

300

320

340

360

380

400

Load ()

094

0945

095

0955

096

0965

097

L-in

dex



550060006500700075008000850090009500

1000010500

Ope

ratio

n co

st (m

u)

120

140

160

180

200

220

240

260

280

300

320

340

360

380

400

100










5 Conclusions



765 768

780

803

812

862

877

937

903

21536

68 94

106

171 179174

127

95

181

214 215

219

222

264

306

275

237

261

235

234

328338372

341

345

357

360

473496 563

397

564 590 592

598618

624

623 664

595

665 681 685

689

690

700

730

736737735

764
















0001002003004005006007008009

01

L-in

dex

9320

0

9340

0

9480

0

9380

0

9400

0

9420

0

9440

0

9500

0

9300

0

9360

0

9460

0

Operation cost (mu)







Simulations0

1

2

3

4

5N

orm

aliz

ed d

istan

ce


Best solution


200025003000350040004500500055006000

Ope

ratio

n co

st (m

u)

3 5 7 9 11 13 15 17 19 21 231Time (h)



1 3 5 7 9 11 13 15 17 19 21 23Time (h)


0001002003004005006007008009

01

L-in

dex


Nomenclature

Parameters






3 5 7 9 11 13 15 17 19 21 231Time (h) minus15000

minus10000

minus5000

0

5000

10000

15000

20000

Ener

gy (k

Wh)

minus2000minus1000

010002000300040005000600070008000

Pow

er (k

W)











buses [S]

Variables



Indices








Competing Interests


References



















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

11819

20

21

2

3

4

567

8

11

910

1213

1415

16

17

2223

24

2526

27

28

29

30

31

32

50kW

30kW

30kW 9

kW 9kW

9kW

30kW

9kW

9kW

30kW

30kW

30kW

30kW

9kW

30kW

30kW

9kW9

kW3

kW

30kW9

kW

3kW

9kW

9kW

30kW

30kW

9kW

3kW

9kW

9kW

3kW

30kW

30kW

10kW40

kW50kW

100kW

50kW

10kW

100kW

50kW

10kW 200

kW

10kW

25kW 25

kW 100kW

150kW

100kW

100kW 100

kW

50kW

50kW

50kW50

kW

50kW

100kW

25kW200

kW10

kW

50kW

10kW

30kW

200kW

25kW25

kW

PV panel

Wind

WTE

CHP

Fuel cell

Biomass

Hydro

EV with V2G sim

sim

sim

simsim

sim

sim





L-in

dex


002004006008

01012014016018









L-in

dex

002004006008

01012014016018




Simulations


100

150

200

250

Nor

mal

ized

dist

ance


mdash



1119901= 1198651119899

and 1198652119901




0100200300400500600700

1 3 5 7 9 11 13 15 17 19 21 23

Ope

ratio

n co

st (m

u)

Time (h)




0002004006008

01012014016018

L-in

dex

3 5 7 9 11 13 15 17 19 21 231Time (h)






302826242210 12 14 16 18 20 3286420Bus

1101102103104105106107

Volta

ge m

agni

tude

(pu)



0500

100015002000250030003500400045005000

Pow

er (k

W)



3 5 7 9 11 13 15 17 19 21 231Time (h)





1 3 5 7 9 11 13 15 17 19 21 23Time (h)


minus1500

minus1000

minus500

0

500

1000

1500

2000

Pow

er (k

W)

minus15000

minus10000

minus5000

0

5000

10000

15000

20000

Ener

gy (k

Wh)











Minimum 1198651

1297Minimum 119865

22115




max 120582119905


119876load = 120582119905119876load1015840(119871119905)

forall119905 isin 1 119879

forall119871 isin 1 119873119871

(32)






1)


Slackgenerator

Bus 0


ldquoVirtualrdquo

ZTH










0

02

04

06

08

1

12

L-in

dex




0 5 10 15 20 25 30 35 40 45 50 55 6088008850890089509000905091009150920092509300

Ope

ratio

n co

st (m

u)








1) results


2)



09

095

1

105

11

115

L-in

dex

104

103

102

101

100

099

098

097

096

095

094

093

092

105




105

104

103

102

101

100

099

098

097

096

095

094

093

092

71007110712071307140715071607170718071907200

Ope

ratio

n co

st (m

u)







100

120

140

160

180

200

220

240

260

280

300

320

340

360

380

400

Load ()

094

0945

095

0955

096

0965

097

L-in

dex



550060006500700075008000850090009500

1000010500

Ope

ratio

n co

st (m

u)

120

140

160

180

200

220

240

260

280

300

320

340

360

380

400

100










5 Conclusions



765 768

780

803

812

862

877

937

903

21536

68 94

106

171 179174

127

95

181

214 215

219

222

264

306

275

237

261

235

234

328338372

341

345

357

360

473496 563

397

564 590 592

598618

624

623 664

595

665 681 685

689

690

700

730

736737735

764
















0001002003004005006007008009

01

L-in

dex

9320

0

9340

0

9480

0

9380

0

9400

0

9420

0

9440

0

9500

0

9300

0

9360

0

9460

0

Operation cost (mu)







Simulations0

1

2

3

4

5N

orm

aliz

ed d

istan

ce


Best solution


200025003000350040004500500055006000

Ope

ratio

n co

st (m

u)

3 5 7 9 11 13 15 17 19 21 231Time (h)



1 3 5 7 9 11 13 15 17 19 21 23Time (h)


0001002003004005006007008009

01

L-in

dex


Nomenclature

Parameters






3 5 7 9 11 13 15 17 19 21 231Time (h) minus15000

minus10000

minus5000

0

5000

10000

15000

20000

Ener

gy (k

Wh)

minus2000minus1000

010002000300040005000600070008000

Pow

er (k

W)











buses [S]

Variables



Indices








Competing Interests


References



















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














L-in

dex

002004006008

01012014016018




Simulations


100

150

200

250

Nor

mal

ized

dist

ance


mdash



1119901= 1198651119899

and 1198652119901




0100200300400500600700

1 3 5 7 9 11 13 15 17 19 21 23

Ope

ratio

n co

st (m

u)

Time (h)




0002004006008

01012014016018

L-in

dex

3 5 7 9 11 13 15 17 19 21 231Time (h)






302826242210 12 14 16 18 20 3286420Bus

1101102103104105106107

Volta

ge m

agni

tude

(pu)



0500

100015002000250030003500400045005000

Pow

er (k

W)



3 5 7 9 11 13 15 17 19 21 231Time (h)





1 3 5 7 9 11 13 15 17 19 21 23Time (h)


minus1500

minus1000

minus500

0

500

1000

1500

2000

Pow

er (k

W)

minus15000

minus10000

minus5000

0

5000

10000

15000

20000

Ener

gy (k

Wh)











Minimum 1198651

1297Minimum 119865

22115




max 120582119905


119876load = 120582119905119876load1015840(119871119905)

forall119905 isin 1 119879

forall119871 isin 1 119873119871

(32)






1)


Slackgenerator

Bus 0


ldquoVirtualrdquo

ZTH










0

02

04

06

08

1

12

L-in

dex




0 5 10 15 20 25 30 35 40 45 50 55 6088008850890089509000905091009150920092509300

Ope

ratio

n co

st (m

u)








1) results


2)



09

095

1

105

11

115

L-in

dex

104

103

102

101

100

099

098

097

096

095

094

093

092

105




105

104

103

102

101

100

099

098

097

096

095

094

093

092

71007110712071307140715071607170718071907200

Ope

ratio

n co

st (m

u)







100

120

140

160

180

200

220

240

260

280

300

320

340

360

380

400

Load ()

094

0945

095

0955

096

0965

097

L-in

dex



550060006500700075008000850090009500

1000010500

Ope

ratio

n co

st (m

u)

120

140

160

180

200

220

240

260

280

300

320

340

360

380

400

100










5 Conclusions



765 768

780

803

812

862

877

937

903

21536

68 94

106

171 179174

127

95

181

214 215

219

222

264

306

275

237

261

235

234

328338372

341

345

357

360

473496 563

397

564 590 592

598618

624

623 664

595

665 681 685

689

690

700

730

736737735

764
















0001002003004005006007008009

01

L-in

dex

9320

0

9340

0

9480

0

9380

0

9400

0

9420

0

9440

0

9500

0

9300

0

9360

0

9460

0

Operation cost (mu)







Simulations0

1

2

3

4

5N

orm

aliz

ed d

istan

ce


Best solution


200025003000350040004500500055006000

Ope

ratio

n co

st (m

u)

3 5 7 9 11 13 15 17 19 21 231Time (h)



1 3 5 7 9 11 13 15 17 19 21 23Time (h)


0001002003004005006007008009

01

L-in

dex


Nomenclature

Parameters






3 5 7 9 11 13 15 17 19 21 231Time (h) minus15000

minus10000

minus5000

0

5000

10000

15000

20000

Ener

gy (k

Wh)

minus2000minus1000

010002000300040005000600070008000

Pow

er (k

W)











buses [S]

Variables



Indices








Competing Interests


References



















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










302826242210 12 14 16 18 20 3286420Bus

1101102103104105106107

Volta

ge m

agni

tude

(pu)



0500

100015002000250030003500400045005000

Pow

er (k

W)



3 5 7 9 11 13 15 17 19 21 231Time (h)





1 3 5 7 9 11 13 15 17 19 21 23Time (h)


minus1500

minus1000

minus500

0

500

1000

1500

2000

Pow

er (k

W)

minus15000

minus10000

minus5000

0

5000

10000

15000

20000

Ener

gy (k

Wh)











Minimum 1198651

1297Minimum 119865

22115




max 120582119905


119876load = 120582119905119876load1015840(119871119905)

forall119905 isin 1 119879

forall119871 isin 1 119873119871

(32)






1)


Slackgenerator

Bus 0


ldquoVirtualrdquo

ZTH










0

02

04

06

08

1

12

L-in

dex




0 5 10 15 20 25 30 35 40 45 50 55 6088008850890089509000905091009150920092509300

Ope

ratio

n co

st (m

u)








1) results


2)



09

095

1

105

11

115

L-in

dex

104

103

102

101

100

099

098

097

096

095

094

093

092

105




105

104

103

102

101

100

099

098

097

096

095

094

093

092

71007110712071307140715071607170718071907200

Ope

ratio

n co

st (m

u)







100

120

140

160

180

200

220

240

260

280

300

320

340

360

380

400

Load ()

094

0945

095

0955

096

0965

097

L-in

dex



550060006500700075008000850090009500

1000010500

Ope

ratio

n co

st (m

u)

120

140

160

180

200

220

240

260

280

300

320

340

360

380

400

100










5 Conclusions



765 768

780

803

812

862

877

937

903

21536

68 94

106

171 179174

127

95

181

214 215

219

222

264

306

275

237

261

235

234

328338372

341

345

357

360

473496 563

397

564 590 592

598618

624

623 664

595

665 681 685

689

690

700

730

736737735

764
















0001002003004005006007008009

01

L-in

dex

9320

0

9340

0

9480

0

9380

0

9400

0

9420

0

9440

0

9500

0

9300

0

9360

0

9460

0

Operation cost (mu)







Simulations0

1

2

3

4

5N

orm

aliz

ed d

istan

ce


Best solution


200025003000350040004500500055006000

Ope

ratio

n co

st (m

u)

3 5 7 9 11 13 15 17 19 21 231Time (h)



1 3 5 7 9 11 13 15 17 19 21 23Time (h)


0001002003004005006007008009

01

L-in

dex


Nomenclature

Parameters






3 5 7 9 11 13 15 17 19 21 231Time (h) minus15000

minus10000

minus5000

0

5000

10000

15000

20000

Ener

gy (k

Wh)

minus2000minus1000

010002000300040005000600070008000

Pow

er (k

W)











buses [S]

Variables



Indices








Competing Interests


References



















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















Minimum 1198651

1297Minimum 119865

22115




max 120582119905


119876load = 120582119905119876load1015840(119871119905)

forall119905 isin 1 119879

forall119871 isin 1 119873119871

(32)






1)


Slackgenerator

Bus 0


ldquoVirtualrdquo

ZTH










0

02

04

06

08

1

12

L-in

dex




0 5 10 15 20 25 30 35 40 45 50 55 6088008850890089509000905091009150920092509300

Ope

ratio

n co

st (m

u)








1) results


2)



09

095

1

105

11

115

L-in

dex

104

103

102

101

100

099

098

097

096

095

094

093

092

105




105

104

103

102

101

100

099

098

097

096

095

094

093

092

71007110712071307140715071607170718071907200

Ope

ratio

n co

st (m

u)







100

120

140

160

180

200

220

240

260

280

300

320

340

360

380

400

Load ()

094

0945

095

0955

096

0965

097

L-in

dex



550060006500700075008000850090009500

1000010500

Ope

ratio

n co

st (m

u)

120

140

160

180

200

220

240

260

280

300

320

340

360

380

400

100










5 Conclusions



765 768

780

803

812

862

877

937

903

21536

68 94

106

171 179174

127

95

181

214 215

219

222

264

306

275

237

261

235

234

328338372

341

345

357

360

473496 563

397

564 590 592

598618

624

623 664

595

665 681 685

689

690

700

730

736737735

764
















0001002003004005006007008009

01

L-in

dex

9320

0

9340

0

9480

0

9380

0

9400

0

9420

0

9440

0

9500

0

9300

0

9360

0

9460

0

Operation cost (mu)







Simulations0

1

2

3

4

5N

orm

aliz

ed d

istan

ce


Best solution


200025003000350040004500500055006000

Ope

ratio

n co

st (m

u)

3 5 7 9 11 13 15 17 19 21 231Time (h)



1 3 5 7 9 11 13 15 17 19 21 23Time (h)


0001002003004005006007008009

01

L-in

dex


Nomenclature

Parameters






3 5 7 9 11 13 15 17 19 21 231Time (h) minus15000

minus10000

minus5000

0

5000

10000

15000

20000

Ener

gy (k

Wh)

minus2000minus1000

010002000300040005000600070008000

Pow

er (k

W)











buses [S]

Variables



Indices








Competing Interests


References



















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

02

04

06

08

1

12

L-in

dex




0 5 10 15 20 25 30 35 40 45 50 55 6088008850890089509000905091009150920092509300

Ope

ratio

n co

st (m

u)








1) results


2)



09

095

1

105

11

115

L-in

dex

104

103

102

101

100

099

098

097

096

095

094

093

092

105




105

104

103

102

101

100

099

098

097

096

095

094

093

092

71007110712071307140715071607170718071907200

Ope

ratio

n co

st (m

u)







100

120

140

160

180

200

220

240

260

280

300

320

340

360

380

400

Load ()

094

0945

095

0955

096

0965

097

L-in

dex



550060006500700075008000850090009500

1000010500

Ope

ratio

n co

st (m

u)

120

140

160

180

200

220

240

260

280

300

320

340

360

380

400

100










5 Conclusions



765 768

780

803

812

862

877

937

903

21536

68 94

106

171 179174

127

95

181

214 215

219

222

264

306

275

237

261

235

234

328338372

341

345

357

360

473496 563

397

564 590 592

598618

624

623 664

595

665 681 685

689

690

700

730

736737735

764
















0001002003004005006007008009

01

L-in

dex

9320

0

9340

0

9480

0

9380

0

9400

0

9420

0

9440

0

9500

0

9300

0

9360

0

9460

0

Operation cost (mu)







Simulations0

1

2

3

4

5N

orm

aliz

ed d

istan

ce


Best solution


200025003000350040004500500055006000

Ope

ratio

n co

st (m

u)

3 5 7 9 11 13 15 17 19 21 231Time (h)



1 3 5 7 9 11 13 15 17 19 21 23Time (h)


0001002003004005006007008009

01

L-in

dex


Nomenclature

Parameters






3 5 7 9 11 13 15 17 19 21 231Time (h) minus15000

minus10000

minus5000

0

5000

10000

15000

20000

Ener

gy (k

Wh)

minus2000minus1000

010002000300040005000600070008000

Pow

er (k

W)











buses [S]

Variables



Indices








Competing Interests


References



















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










100

120

140

160

180

200

220

240

260

280

300

320

340

360

380

400

Load ()

094

0945

095

0955

096

0965

097

L-in

dex



550060006500700075008000850090009500

1000010500

Ope

ratio

n co

st (m

u)

120

140

160

180

200

220

240

260

280

300

320

340

360

380

400

100










5 Conclusions



765 768

780

803

812

862

877

937

903

21536

68 94

106

171 179174

127

95

181

214 215

219

222

264

306

275

237

261

235

234

328338372

341

345

357

360

473496 563

397

564 590 592

598618

624

623 664

595

665 681 685

689

690

700

730

736737735

764
















0001002003004005006007008009

01

L-in

dex

9320

0

9340

0

9480

0

9380

0

9400

0

9420

0

9440

0

9500

0

9300

0

9360

0

9460

0

Operation cost (mu)







Simulations0

1

2

3

4

5N

orm

aliz

ed d

istan

ce


Best solution


200025003000350040004500500055006000

Ope

ratio

n co

st (m

u)

3 5 7 9 11 13 15 17 19 21 231Time (h)



1 3 5 7 9 11 13 15 17 19 21 23Time (h)


0001002003004005006007008009

01

L-in

dex


Nomenclature

Parameters






3 5 7 9 11 13 15 17 19 21 231Time (h) minus15000

minus10000

minus5000

0

5000

10000

15000

20000

Ener

gy (k

Wh)

minus2000minus1000

010002000300040005000600070008000

Pow

er (k

W)











buses [S]

Variables



Indices








Competing Interests


References



















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










765 768

780

803

812

862

877

937

903

21536

68 94

106

171 179174

127

95

181

214 215

219

222

264

306

275

237

261

235

234

328338372

341

345

357

360

473496 563

397

564 590 592

598618

624

623 664

595

665 681 685

689

690

700

730

736737735

764
















0001002003004005006007008009

01

L-in

dex

9320

0

9340

0

9480

0

9380

0

9400

0

9420

0

9440

0

9500

0

9300

0

9360

0

9460

0

Operation cost (mu)







Simulations0

1

2

3

4

5N

orm

aliz

ed d

istan

ce


Best solution


200025003000350040004500500055006000

Ope

ratio

n co

st (m

u)

3 5 7 9 11 13 15 17 19 21 231Time (h)



1 3 5 7 9 11 13 15 17 19 21 23Time (h)


0001002003004005006007008009

01

L-in

dex


Nomenclature

Parameters






3 5 7 9 11 13 15 17 19 21 231Time (h) minus15000

minus10000

minus5000

0

5000

10000

15000

20000

Ener

gy (k

Wh)

minus2000minus1000

010002000300040005000600070008000

Pow

er (k

W)











buses [S]

Variables



Indices








Competing Interests


References



















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Simulations0

1

2

3

4

5N

orm

aliz

ed d

istan

ce


Best solution


200025003000350040004500500055006000

Ope

ratio

n co

st (m

u)

3 5 7 9 11 13 15 17 19 21 231Time (h)



1 3 5 7 9 11 13 15 17 19 21 23Time (h)


0001002003004005006007008009

01

L-in

dex


Nomenclature

Parameters






3 5 7 9 11 13 15 17 19 21 231Time (h) minus15000

minus10000

minus5000

0

5000

10000

15000

20000

Ener

gy (k

Wh)

minus2000minus1000

010002000300040005000600070008000

Pow

er (k

W)











buses [S]

Variables



Indices








Competing Interests


References



















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Indices








Competing Interests


References



















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Energy Optimization for Distributed ...downloads.hindawi.com/journals/mpe/2016/6379253.pdf · Energy Optimization for Distributed Energy Resources Scheduling with

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