EPSO approach A multiyear dynamic transmission expansion planning model using a discretebased Electric Power Systems Research

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A multiyear dynamic transmission expansion planning model using a discrete based EPSO approach

Manuel Costeira da Rocha

, João Tomé Saraiva

aECEDepartment,FacultyofEngineering,UniversityofPorto,RuaDr.RobertoFrias,4200-465Porto,Portugal

bINESCTECandECEDepartment,FacultyofEngineering,UniversityofPorto,CampusdaFEUP,RuaDr.RobertoFrias,4200-465Porto,Portugal

a r t i c l e i n f o

Articlehistory:

Received24January2012

Receivedinrevisedform11July2012 Accepted15July2012

Available online 9 August 2012

Keywords:

Transmissionexpansionplanning Investments

Dynamicmultiyearmodel Discreteevolutionaryparticleswarm optimization

a b s t r a c t

Thispaperpresentsamultiyeardynamictransmissionexpansionplanning,TEP,modelaimingatmin- imizingoperationandinvestmentcostsalongtheentireplanninghorizonwhileensuringanadequate qualityofserviceandenforcingconstraintsmodelingtheoperationofthenetworkalongtheplanning horizon.Thedevelopedmodelprofitsfromtheexperienceofplannerswhenpreparingalistofpossi- blebranch(linesandtransformers)additionseachofthemassociatedtothecorrespondinginvestment cost.TheobjectiveofsolvingaTEPproblemistoselectanumberofelementsofthislistandprovideits schedulingalongtheplanninghorizonsuchthatoneisfacingamixedintegeroptimizationproblem.In thiscase,thisproblemwassolvedusingadiscreteevolutionaryparticleswarmoptimizationalgorithm, DEPSO,basedonalreadyreportedEPSOapproachesbutparticularlysuitedtotreatdiscreteproblems.

ApartfromdetailingthedevelopedDEPSO,thispaperdescribesthemathematicalformulationoftheTEP problemandtheadoptedsolutionalgorithm.ItalsoincludesresultsoftheapplicationoftheDEPSOto theTEPproblemusingtwotestnetworkswidelyusedbyotherresearchersonthisarea.

© 2012 Elsevier B.V. All rights reserved.

1. Introduction

Inthelasttwodecadespowersystemswentthrougharestruc- turingprocessaimingatintroducingmarketmechanismstolink thegenerationandthedemandandliberalizingprogressivelythe accesstothenetworks.Asthisprocessdeveloped,newchallenges wereintroducednamelyintermsofdecouplingdistributionnet- workactivitiesfromretailing.Asaresult,mostcountriesthatwent throughrestructuringtypicallyimplementedcompetitivemecha- nismsintheextremeactivitiesofthevaluechain,generationand retailing,whilekeepingnetworkactivitiesasregulatedmonopo- lies.Inthefirstyearsafterthismove,alargeraccentwasputon shorttermactivities,forinstanceillustratedbytheimplementation ofday-aheadmarkets.Astimepassed,longtermexpansionplan- ningactivitiesregainedinterestandcontinuetobeamajorconcern ofgenerationand transmissioncompanies.Regardingtransmis- sion,expansionplansmustnowbepreparedinadecoupledway fromgenerationandfromdistribution.Transmissionnetworkswill nowhavetofollowandinmostcasestoanticipatetherequests bothfromnewgenerationandnewdemandintroducinganew levelofuncertaintyregardingthelocationofconnectionpoints.

∗Correspondingauthor.Tel.:+351222094230;fax:+351222094150.

E-mailaddresses:[email protected](M.C.d.Rocha), [email protected](J.T.Saraiva).

Theincreasingnumberofwindparkstogetherwiththeirincreas- inginstalledcapacityleadstopowersurplusinsomedistribution networksthatnowstarttoinjectintransmission.Astheinstalled capacityofwindparksincrease,connectionpointsarealsomoving fromdistributiontotransmissioncreatingnewchallengestotrans- missionplanners.Furthermore,inEuropelongtermplansareunder developmentinordertobuildasupertransmissiongridenabling movinglargeamountsofenergyatlongerdistances,forinstance usinghydroresourcesinScandinaviaandsolarandwindresources insoutherncountries,asSpainandPortugal.Consideringallthese aspects,transmissionexpansionplanningproblemsareevenmore complexthaninthepastduetoanumberofaspects:

-theyhaveamultiperioddynamicnatureanditshouldbemain- tained an holistic view over the entire horizon, eventually discretized in a number of annual periods.This holistic view meansthatrunninganexpansionexerciseovernpperiodsisnot thesameasrunningnpindependentexercisesin asequential way.Treatingthewholehorizoninthesameproblemmeansthat whencommissioningaprojectforaperiodwearetakingnotonly intoaccounttherequirementsofthatperiodbutalsoitsimpact inthefuture;

-theyexhibitageographiccouplinginthesensethatnewinstal- lations will not be selected as answers to local problems. In meshednetworksastransmissionones,solvingaproblemcan 0378-7796/$–seefrontmatter© 2012 Elsevier B.V. All rights reserved.

http://dx.doi.org/10.1016/j.epsr.2012.07.012

alsoaddressandhavepositiveimpactsonbottlenecksinother locations,sothatthereisaglobalviewthatshouldbemaintained;

-theyarediscreteproblemsduetothenatureofinvestmentalter- natives;

-theyareaffectedbyloaduncertaintyoverthehorizon.Aplan shouldbeadequatenotonlyforagivenloadevolutionbutthe decisionmakershouldnotfeelanyrelevantregretwhateverthe future.Thisimmediatelyleadstoriskanalysisunderwhichflex- iblesolutionsaremostwelcome.

Soaccordingwiththeseindications,theTEPproblemcanbe definedasacomplexoptimizationprobleminvolvingalargenum- berofvariablesandconstraintsthataimsatschedulinganumber ofinvestmentsontransmissionassetsalonganextendedhorizon.

Thisproblemhasamixedintegernonlinearprogrammingnature, which,infact,correspondstoacombinatorialproblemanditcan beformulatedconsideringseveralobjectives,thatareusuallycon- tradictory.

Theliteratureonthistopicincludesalargenumberofpublica- tionsthatcanbegatheredintwolargegroups.Ononeside,there areapplicationsdesignedtoanalyzepre-preparedexpansionplans.

Mostoftheseformulationscorrespondtosoftwarepackagesdevel- opedbyutilitiesorbyresearchcentersthatwererelatedwiththem.

PackagesasTRELSSandCREAMdevelopedbyEPRIandseveraloth- ersimplementedbyCEPELinBrazil,ENELinItalyandEDFinFrance areexamplesoftheseapproaches.Ontheotherhand,thereareopti- mizationmodelsdesignedtobuildexpansionplansaccordingto somecriteria.Thenumberofpublicationsonthistopicisverylarge [1]andthereisnotacommonandgeneraltransmissionexpansion formulationadoptedbyallresearchers.Traditionally,theexpan- sionformulationsincludedcontinuousvariablestorepresentthe capacityofnewbranchesthusrequiringapproximationstoobtain afinaltechnicallyfeasiblesolution.Forinstance,Refs.[2,3]describe linearandnon-linearapproachestotheTEPproblem.Someother papersas[4–7]describemixedintegerformulationsandadopt,for instance,Branch&BoundandBendersDecompositionbasedmeth- odsinawaytopreservethediscretenatureofinvestments.Some othersselectinvestmentsaccordingtoaMeritIndexortoatrade- offrelationbetweentheinvestmentcostandtheresultingbenefit [8–11].

More recently several emergent techniques as Simulated Annealing, Genetic Algorithms, Tabu Search and Game Theory startedtobeappliedtothisproblem. Refs.[12,13]describethe applicationofGeneticAlgorithmstothetransmissionexpansion problem[14],detailstheuseofTabuSearch[15],adoptsSimu- latedAnnealingand[16]usesGrasp.Finally,ref.[17]describesa multiagentimplementationbasedoncooperativegames.These authorsmentiontheadvantagesoftheseapproachestoaddressthis complexcombinatorialproblemintermsofidentifyingafeasible solutioninamanageablecomputationtime.

InviewofthecharacteristicsoftheTEPproblem,this paper describesamultiyeardynamicapproachthataimsatminimizing theinvestmentand operationcostsalong theplanninghorizon whileenforcinglimitsforreliability indices,namelyconsidering N−1contingencies,and forthemaximuminvestmentcost and forthenumberofon-going projectsineachperiod ofthehori- zon.Given the mixed integer natureof theresulting problem, weadoptedadiscreteevolutionaryparticleswarmoptimization (DEPSO)approach tosolvethis problemthat correspondstoan enhancementoforiginalPSOalgorithmsbothintermsofintroduc- inganevolutionaryflavortotheruleofgenerationofnewparticles andtreatingdiscretevariablesinamoreadequateway.Accord- ingly,theproposeddynamicTEPmodelbasedonDEPSOisthemain contributionofthispaperandallowsforeseeingtheapplicationof DEPSOtoothercomplexcombinatorialproblems.

Asaresultoftheseideas,thispaperisstructuredasfollows.After thisSection1,Section2detailsthediscreteevolutionaryparticle swarmoptimization,DEPSO,andSection3detailstheTEPmathe- maticalformulationandtheapplicationoftheimplementedDEPSO tothisproblem.Section4presentsresultsfortwoCaseStudies, namelyprovidingcomparisonswithresultsreportedintheliter- aturebyotherresearchersandfinallySection5drawsthemost relevantconclusionsofthisresearch.

2. Discreteevolutionaryparticleswarmoptimization

2.1. Heuristictools

Heuristicmethodsgostep-by-stepgenerating,evaluating,and selectingsolutions,withorwithoutinteractingwiththeplanner.

Takingadvantagefromplanersexperienceinputs,thecomputa- tionalperformanceofheuristicmethodsisusuallybetterthanthat ofclassicalmathematicalmethods.Insomecaseslocalsearches are also performed. The solutions are evaluated and classified accordingtothefitnessfunctionthatconsiderstechnical,finan- cialandservicecriteriaanddata.HeuristicToolsinclude,among others,evolutionaryalgorithmsandparticleswarmoptimization.

Inparticular,evolutionaryalgorithmsareusuallyorganizedinthe followingsteps:

initializearandompopulationPofnptelements;

repeat–reproduction(byrecombinationand/ormutation),evaluation, selectionandtest;

untiltestispositive(forterminationcriteriabasedonfitness,onnumber ofgenerationsorothercriteria).

Evolutionary computation offers several advantages when facingdifficultoptimizationproblems[18]:asitsconceptualsim- plicityandbroadapplicability,itoutperformsclassicmethodson realproblemsandithasalargepotentialtouseknowledgeand hybridizewithothermethods.Theclassicalparticleswarmopti- mization,PSO, wasproposedbyKennedy in1995,basedonthe parallelexplorationofthesearchspacebyasetof“particles”or alternativesthataresuccessivelytransformedalongtheprocess [19].ThebasicPSOmodelisdefinedaccordingto(1)and(2).Under thisscheme,themovementruleofaparticleptisgivenby(2)and itisusedtochangethepositionoftheparticleX_ptⁱ obtainedfor iterationitoitsnewpositionX_ptⁱ⁺¹inthefollowingiterationi+1.

X_ptⁱ⁺¹=X_ptⁱ +V_ptⁱ⁺¹ (1) V_ptⁱ⁺¹=V_ptⁱ +Rndⁱ_mem,pt⁺¹ ·Wmem·(bpt−X_ptⁱ )

+Rndⁱ⁺¹_coop,pt·Wcoop·(b_G−X_ptⁱ ) (2)

Thismovementruleincludesthreetermsasfollows:

-theInertiaTermgivenbyV_ptⁱ.Thistermindicatesthatthemove- mentoftheparticleptisinfluencedbythemovementithadin thepreviousiteration;

-theMemoryTermindicatingthatthemovementoftheparticle isattractedbythebestofitsancestors,bpt,or,inotherwords, bythebestparticlethatwasobtainedinpreviousiterationsin thispositionofthepopulation.Thistermisdeterminedbythe memoryweightWmemsetinthebeginningoftheprocessandby arandomnumberRndⁱ_mem,pt⁺¹ sampledineachiterationfroman uniformdistributionin[0.0;1.0];

-similarly,theCooperationTermincludesinformationaboutthe bestglobalparticlesofaridentifiedintheentirepopulationinall previousiterations,bG.Thistermisalsodeterminedbythecoop- erationweightWcooptypicallysetinthebeginningoftheprocess andbyarandomnumber Rndⁱ⁺¹_coop,pt sampledfromanuniform distributionin[0.0;1.0]ineachiteration.

Fig.1.MovementruleintheEPSOalgorithm.

Some refinements were introduced in this scheme namely becauseseveraltestsindicatedthatthismovementrulewasade- quatetomaketheswarmconvergetothezonewheretheoptimum wasbutfailedtofinetunetheconvergencetotheaccurateoptimum position.

2.2. DiscreteevolutionaryPSO,EPSO

In2002MirandaandFonseca[20]introducedtheevolutionary particleswarmoptimization(EPSO)thatjointsthemostinteresting featuresofparticleswarmmethodologiesandevolutionaryalgo- rithms.ThestructureoftheEPSOalgorithmissummarizedbelow [21].

InitializearandompopulationPofnptparticles Repeat

Replication,mutation,reproduction,selection,test Untiltestispositive

EPSOfocusesinregionsofthesearchspacewherebettercontri- butionsforthesolutioncanbefound,insteadofconductingablind samplingofthespace.Inthiscase,themovementruleofparticlept alsoincludestheInertia,theMemoryandtheCooperationTerms mentionedaboveandthevelocityV_ptⁱ⁺¹determiningthenewposi- tionofthisparticle,X_ptⁱ⁺¹,isnowgivenby(3)andthisschemeis illustratedinFig.1.

V_ptⁱ⁺¹=W_ine,pt^i+1∗ ·V_ptⁱ +W_mem,pt^i+1∗ ·(b_pt−X_ptⁱ )+W_coop,pt^i+1∗ ·(b^∗_G−X_ptⁱ )·P (3)

In expression(3) W_ine,pt^i+1∗,W_mem,pt^i+1∗ and W_coop,pt^i+1∗ representthe Inertia,theMemoryandtheCooperationweights.Inthefirstplace, thealgorithm samplestheweightsW_ine,ptⁱ⁺¹ ,W_mem,ptⁱ⁺¹ andW_coop,ptⁱ⁺¹ thatarethenmutatedinW_ine,pt^i+1∗,W_mem,pt^i+1∗ andW_coop,pt^i+1∗ using(4)in whichrepresentsalearningparameterfixedexternally.Inthese expressionsthesign*denotesmutatedvalues.Thebestglobalpar- ticle,bG,alsoundergoesmutationaccordingto(5)sothatalocal searcharoundthecurrentbestglobalparticleisperformed.

W_pt^i+1∗=W_ptⁱ⁺¹·[logN(0,1)] (4) b^∗_G=b_G+W_bG^i+1∗·N(0,1) (5) Finally,theCooperationTermisalsoinfluencedbyacommu- nicationfactorP.Whentheparticleshaveseveraldimensionsand arerepresentedbyavector,thecommunicationfactorPismod- eledbyadiagonalmatrixhaving0and1valuesonthediagonal.

The1valuesareusedtocommunicate theinformationin some dimensionsofb^∗_Gtothenewparticle.The1valuesinthismatrix haveprobabilitypandthe0valueshaveprobability1−p.Several reportedtestsshowedthatpvaluessetat0.2or0.3leadtobetter resultswhencomparedwithmoreclassicalandmoredeterministic schemeswherep=1.

Theapproachintroducedinthispaperis basedonadiscrete modelingofEPSO,DEPSO.Possible solutionsarerepresentedby vectorsofintegersinthesensethatboththevelocityvectorVand theparticlesXareforcedtobeintegers,usingaroundingprocess oftheoutputof(3).Thismodelisboostedbylocalsearchnearby thebestglobalsolutionseverfounded.Thesefeaturesenhancethe successfulprogressionofDEPSOtowardthemostpromisingsolu- tions.FurtherdetailsregardingtheapplicationoftheDEPSOtothe TEPproblemareprovidedinSection3.3.

3. Transmissionexpansionplanningmodel

3.1. Generalmodelingissues

AsindicatedinSection1,theTEPproblemexhibitsanumberof featuresthatcontributetoturnitintoacomplexproblemnamely in terms of the holistic viewthat should bemaintained along theentirehorizonandregardingthediscretenatureofexpansion projects.Asaresultoftheseconcerns,inthedevelopedformu- lationweexplicitlyconsideraplanninghorizonstructuredinnp periodsthat willberepresentedat a time inthemathematical problemandwealsoconsideraprojectlistbasedonnprojprojects includingnewlinesandtransformersthataretechnicallyimple- mentable.Eachof theseprojectsischaracterizedbythepairof nodesbetweenwhichitcanbebuilt,bythetechnicaldataofthe lineorthetransformerandbytheinvestmentcost.Theobjectiveof theTEPproblemisthereforetoidentifythebestpossiblesolution thatintegratesanumberofelementsofthisprojectlist,adequately scheduled along the years of theplanning horizon, sothat we minimizeacriteriumtobespecifiedandanumberofconstraints areenforced.

ThesearchspacethatwillbeanalyzedintheTEPproblemisdis- creteandittypicallyincludesalargenumberofpossiblealternative plansgivenbynp^nproj,wherenpisthenumberofperiodsinthe planninghorizonandnprojisthenumberofinvestmentprojectsin theprojectlistdefinedbytheplanner.AsolutionXptcorresponds toanexpansionplananditincludesanumberofprojectsselected amongthislist.ThefirstpopulationXoftheDEPSOincludingnpt particlesisrandomlygeneratedanditcorrespondstoamatrixof npt×nproj.Eachlineofthismatrixcorrespondstoaparticleinthe population,thatis,toapossiblesolutiontotheTEPproblem.Each elementofthislineisassociatedwitheachprojectintheproject listanditcontainsinformationabout:

-theyear inwhich it willbecommissioned, thatis an integer between1andnp;

-or0ifthisparticularprojectwasnotselectedtothisparticular particle;

-ornp+1meaningitwaspostponed.

Thisdesignofaparticlewithintegersfrom0tonp+1,andin particularwithapossiblestatesbelow1andabovenp,isrelevant whenusingDEPSObecauseitshouldbepossibletoevolvewiththe samedifficultyfromanystateto0,nonselectingtheproject,orto np+1,postponingtheprojectbeyondthepossibledefinedstates.

Ifthe0andnp+1stateswherenotallowed,theroundingswithbe limitedto1,inthelowerlevel,andtonpatthehigherleveland sothe1andthenpstateswouldbeartificiallyfavoredeventually leadingtoinadequateplans.

3.2. Mathematicalformulation

TheformulationoftheTEPmodelisgivenby(6)–(9).Forapar- ticularparticleptiniterationi,theobjectivefunctionisgivenby (6)anditcorrespondstotheadditionoftheinvestmentandthe operationcostsineachperiodadequatelytransferredtotheinitial

periodusinganinterestraterr.Inthisformulation,ptistheindex ofaparticleinthepopulation,iistheiterationcounter,pandjare integersrepresentingaperiodintheplanninghorizonandaproject intheprojectlist.Ontheotherhand,OC_pt,pⁱ representstheoper- ationcostofthisparticleinperiodp,IC_jistheinvestmentcostof projectj,K_pt,p,jⁱ isabinaryvariableindicatingthatprojectjissched- uledtostartoperationatperiodpintheexpansionplanassociated toparticlept.

min Costⁱ_pt=

p=1

j=1 IC_j·K_pt,p,jⁱ ) (1+rr)^p

(6)

Subjectedto:

Powerflowandgeneratorlimitsforeachperiod (7) Financialconstraints,globalorforeachperiod (8)

Reliabilityconstraints (9)

Theoperationcostassociatedwitheachparticleptiniteration iisestimatedsolvingaDCOPFalgorithmforeachperiodpinthe planninghorizonaccordingto(10)–(14).Inthisformulationc_kis theoperatingcostofthegeneratorconnectedtobusk,in$/MWh, Pg_kistheoutputofthegeneratorconnectedtobusk,PNS_kmodels aficticiousgeneratortorepresentthePowerNotSuppliedinbus k,havingalargepenalizationcostgivenbyG,Pd_krepresentsthe loadinbusk,Pg_k^minandPg_k^maxaretheminimumandthemaximum generationsinbusk,P_b^maxisthemaximumflowinbranchbofthe transmissionnetworkanda_bkistheDCsensitivitycoefficientrelat- ingtheinjectedpowerinbuskwiththeactivepowerflowinbranch b.Foreachbranchbwithextremenodesmandn,thesecoefficients arederiveddividingthedifferenceofthephasesbetweennodesm andnbythebranchreactance.Then,thephaseinnodemandthe phaseinnodenaresustitutedbyexpressionsestablishedusingthe elementsinlinesmandnoftheinversetheadmittancematrixof theDCmodelmultipliedbytheinjectedpowerineachnodekof thesystem.Makingthissubstitution,finallyyeldsforbranchban expressionwritteninfunctionoftheinjectedpowersineachnode k.Thecoefficientsofthisexpressionaretheso-calledsensitivity coefficientsoftheflowinbranchbregardingtheinjectedpowers inthesystemnodes,thatisthecoefficientsa_bk.Inthiscase,given thatweareconsideringfictitiousgeneratorstorepresentPower NotSupplied,theinjectedpowerinnodekisgivenbytheaddition ofthegenerationPg_kwithPNS_kandsubtractedfromthedemand, Pd_k.Theflowinbranchbisfinallyconstrainedbyaminimumand amaximumvaluethatareusuallysymmetricalofeachother,that is,P_b^min=−P_b^max,indicatingthattheflowcanrangefrom−P^max_b toP_b^max,thusleadingtoconstraints(14).Foreachparticleinthe population,thisproblemissolvedforeveryperiodp=1,...,npcon- sideringtheequipments(linesandtransformers)thatareincluded inthatparticletostartoperationinperiodp.

minOC=

c_k·Pg_k+G

PNS_k (10)

Subjectto:

PNS_k=

Pd_k (11)

Pg^min_k ≤Pgg≤Pg_k^max (12)

PNS_k≤Pd_k (13)

−P^max_b ≤

a_bk·(Pg_k+PNS_k−Pd_k)≤P_b^max (14) Theobjectivefunctionofthisproblemminimizesthegeneration cost,subjectedtoaglobalbalanceEq.(11),tothegenerationlimits (12),tonodallimitsonthePowerNotSuppliedineachbus(13) andtothebranchflowlimits(14).NonzerovaluesofPowerNot

Suppliedareundisirableandsothevariablesassociatedwiththese ficticiousgeneratorshavealargepenalizationgivenbyGin(10).On theotherhand,itisclearthatifthegenerationplustransmission systemisunabletosupplythedemand,nonzerovaluesofPower NotSuppliedwillbeobtainedinsomenodes.Inanycase,giventhat PowerNotSuppliedmodelsademandreduction,itbecomesclear thatPNSinnodekcannotexceedthedemandinitiallyspecified forthatnode,thusjustifyingtheintegrationofconstraints(13).

Regardingthismodel,theDC-OPFissolvedusingasimplexcode programmedinMatlab,andincludedintheMatpowerlibrary.

Whilesolvingthisproblem,networkandgeneratorlimitcon- straintsareenforcedbut iftransmissioncapacity isunsufficient thenPNSwillbenonzerothusincreasingthevalueoftheobjective function(10).Thisformulationassumesthatthenetworkisloss- less.Inordertoincreasetherealismofthemodel,thisDC-OPFcan bemodifiedtoincludeanestimateoftransmissionlossesaccord- ingtothefollowingschemethattypicallyconvergesinlessthan 5iterations.Theuseoftheoptimizationproblem(10)–(14)with theestimateofthetransmissionlossesisillustratedin[22]con- sideringthePortuguesetransmissionsystem.Itprovidedaccurate resultsin4–5iterations,namelyconsideringthatinwell-developed transmissionsystemstheleveloftransmissionlossesistypically reducedanditvariesintherangefrom1to1.5%oftheload.

Algorithmtoincludeanestimateofbranchactivelosses (i)run aninitialDC-OPFusing(10)–(14) andcomputevoltage

phasesusingtheDCmodel;

(ii)estimateactivelossesinbranchm-nusing(15).Inthisexpres- sion,gmnistheconductanceofbranchm-nandmnisthephase differenceacrossthisbranch;

Lossmn≈2·gmn·(1−cosmn) (15) (iii)addhalfofthelossesinbranchm-ntotheoriginalloadsin nodesmandn.Runanewdispatchusing(10)–(14)andupdate voltagephases;

(iv)endifthedifferenceofvoltagephasesinallnodesissmaller thanaspecifiedthreshold.Ifnot,returnto(ii).

Inordertocharacterizeeachparticleinthepopulation,weuse afitnessfunctionbasedon(6)plusanumberofpenaltytermsas follows.Thesepenaltytermsaresetatpredefinedlargepositive numbers,whichareaddedtothefitnessfunctionwhenanythresh- oldisviolated.Ifforagivenparticle,anythresholdisviolated,the correspondingfitnessefunctionassumesalargepositivevalueand soit getspenalized, giventhatwe areaddressingaminimizing problemwhensolvingtheTEPproblem.

Inthefirstplace,aftersolvingthepreviousDCdispatchproblem, wegetthegenerationcost,theleveloflossesandtheeventualnon zerovalueofPNSfortheentiresystem,PNS(N).Iftheleveloflosses exceedsamaximumpercentageregardingthetotalgenerationin thesystem,thenthisparticleispenalizedwithaterm˛1 inthe fitnessfunctionandifPNS(N)isnotzero,thenthepenaltyterm˛₂ isalsointroducedinthefitnessfunction.Ontheotherhand,each oftheprojectsischaracterizedbyitsinvestment.Whenaproject isselectedforaparticularperiod,itsinvestmentisreferredtothe initialperiod,usingtheinterestraterralreadyusedin(6)andin linewiththelevelofrisk ofthis typeofinvestment.Regarding thesecosts,wecanestablishtwo kindsofconstraints.Thefirst onecorrespondstothemaximumnumberofprojectsthatcanbe implemented per period or themaximum investment cost per period.Thislimitationarisesduetofinancialoroperationalreasons andifitisviolatedweconsiderapenaltyterm˛₃ inthefitness function.Thesecondonecorrespondstothemaximumnumberof projectsorthemaximuminvestmentovertheentirehorizonandit modelsaglobalfinancialconstraint.Ifitisviolated,thenapenalty term ˛4 is included in the fitness function. Finally, regarding

reliability aspects, the fitness function also penalizes plans in whichthePNSisnon-zerofornetworkconfigurationsassociated toN−1contingencies.Itisalsopossibletoincludepenaltiesfora selectednumberofN−2contingenciesgiventhattheGridCodes ofseveralcountriestypicallydetailalistofcontingencies,namely N−2, regarding which thesystem is supposed tosurvive. The penaltyover PNS(N−1)and eventually regarding otherhigher ordercontingenciesismadeusingthepenaltyterm˛5.

As a result of all these considerations, the fitness function, Ffunction,pt,characterizingthequalityofeachparticleptinthepopu- lationinaparticulariterationofthealgorithmisgivenby(16)that correspondsto(6)plusthesefivepenaltyterms.

Ffunction,pt=

p=1

j=1 IC_j·K_pt,p,jⁱ ) (1+rr)^p

+

y=1

˛y (16)

Using thisexpression,we evaluateeach particleinthepop- ulationsothatwecanapplythemovementrulesoftheDEPSO algorithmthatwillbedetailedinthenextsection.Atthispoint itisimportanttomentionthatwhencalculatingthevalueofthe fitnessfunctionforaparticleptwithexpression(16),theOCval- uesineachperiodarenotrecalculated.Infact,foreachperiodand foreachparticlewealreadyhaveinformationabouttheprojects tobeimplemented.Thesenewprojectstogetherwiththeinitial transmissionnetworkdefinethenetworktoanalyzeinthatperiod.

Usingthisinformation,theDC-OPFdetailedabovewasruntoesti- matetheoperationcostsfor everyperiod. Thesecostsarethen usedin(16)toobtainthefitnessfunction oftheparticleunder analysis..

3.3. Solutionalgorithm

ThediscretenatureoftheTEPprobleminspiredtheadoption ofasetofchangesintheEPSOalgorithm.Asmentionedabove,a populationisamatrixwithnpt×nprojpositionsthathasaninte- gerfrom0tonp+1ineachposition.Tointroducemorediversity onthesearch,theDEPSOworkswithtwopopulationsthatcorre- spondtoclonesofthebestpopulation, obtainedattheendofa particulariteration.SimilarlytotheEPSO,theDEPSOisinitialized byrandomlysamplinganinitialpopulation,thatis,bysampling integersfrom0tonp+1foreachpositionofthematrixmentioned before. Afterthis initializationprocedure, theDEPSOevolvesas follows:

-Replication – the best populationobtained at the end of the previousiterationisclonedtwice,sothatineachiterationthe algorithmisactuallyworkingwithtwopopulations;

-Mutationofweights–theInertia,theMemoryandtheCooper- ationweightsmentionedinSection2.2aremutatedusing(17).

Thisexpressionusestheweightfromthepreviousiterationand itincludesthelogisticfunctiontoinduceachaoticsearchpattern.

Duetothenon-repetitioncharacteristicofchaoticfunctions,their adoptionallowshigherspeedsonoverallsearchesthanstochas- ticergodicsearchesthatdependonprobabilities[23].Itshould benoticedthattheseweightsarecomputedforeverypositionof everyparticle;

W_pt,j^i+1∗=

0.5+ 1

1+exp^−w

i∗ pt,j

(17) -Mutationofthebestglobal–thebestglobalparticleisavector withasmanypositionsastheelementsoftheprojectlist.Itsnproj positionsarealsomutatedincaserandomlygeneratednumbers N(0,1)takevalueslessthanaparameterk_bg∈[0,1].Thisallows introducingchangesinthecurrentbestglobalparticle,sothatwe makealocalsearcharoundthecurrentbestglobal.Todothis,the

correspondingweightismutatedinthefirstplaceusing(17)and theneachpositionjofthebestglobalundergoesmutationusing (18);

b^∗_Gj=b_Gj+round(2·W_bGj^i+1∗−1) (18)

-Recombination–afterhavingmutatedtheInertia,theMemory and theCooperation weightsand havingmutatedsome posi- tionsofthebestglobalsofaridentifiedparticle,weusethesame recombinationruleoftheEPSO(3)tocomputethemovement fromiterationitoi+1.IneachiterationoftheDEPSOwearecon- cernedinobtainingtechnicallyfeasiblesolutions,sothatateach stepweroundthevalueobtainedfrom(3)tothenearestinte- ger.Thenewparticleptisthentheresultoftheadditionofthe particleptinthepreviousiterationwiththecomputedvelocity vector.Itisalsoimportanttonoticethatasaresultofthismove- mentrule,eachpositionofeachparticleisfilledwithaninteger thatcaneventuallyexceedthesearchspace,namelynp+1.Ifthat isthecase, theparticleliesoutsidethesearchspaceand itis thenreturnedbacktothesearchspacebyplacingiteitheronthe edgeofthatspace,thatisintegerslargerthannp+1,arereplaced bynp+1;

-RecombinationbyLamarkianevolution–whenapplyingthemove- ment andtherecombinationrules (3),itis possibletoobtain a zero velocity vector.If that is thecase, that particlewould notmovewhengoingfromiterationitoiterationi+1.Inthese cases,toinducesomeextradiversity,wealsousedtheideasof JeanBaptisteLamarke,abiologistthatlivedintheXVIII/XIXcen- turies.Lamarkedevelopedwhatcanbecalledaprotoevolution theorythatprivilegedchangesatthemacroscopicorfenotype levelaccordingtowhichlivingbeings wouldbetransmutated along time inorder togeneratemorecomplex entities.Using thismacroscopicidea,ifaparticlehasazerovelocity,thensome ofitspositionsaremutated,namelytheonesregardingwhich randomly generated numbers N(0,1) takevalues less than a parameterk_Lam∈[0,1].Themutatedelementinpositionjofsuch aparticleiscomputedusinganexpressionsimilarto(18)also includingthelogistic functionin orderto introducea chaotic searchpattern;

-Selection–atthispoint,allparticlesinthetwopopulationswere mutatedandsowecanevaluatethemcomputingthefitnessfunc- tiongivenby(16).Then,wegoalongthetwopopulations,wetake theparticleptfrompopulation1andtheparticleptfrompopula- tion2anditsurvivestheonehavingbetterfitness,amorereduced valuefor(16)inthiscase.Thistournamentschemeyieldsthenew population,whichcorrespondstotheoutputofiterationi+1.At theendofthisstep,thebestparticleinthenewpopulationis comparedwiththecurrentbestglobalparticletoupdatethebest globalsofaridentified;

-Termination step –finally, itis checkedwhetherthis iterative schemeends.Thiscorrespondstocheckifthemaximumnumber ofiterationswasalreadycompletedorifaconvergencecriterium isvalid.Inthefirstcase,thealgorithmstopswithouthavingcon- vergedeventuallysuggestingthatalargernumberofiterations shouldbecompleted.Inthesecondcase,thealgorithmconverges if,forinstance,thebestglobalparticlewasnotupdatedforapre- specifiednumberofiterationsorifthevalueofthefitnessfunction ofthebestglobalparticledidnotchangemorethanathreshold forapre-specifiednumberofiterations.

Theoutputofthealgorithmisapopulationofsolutions,among whichisthebestparticle.Theplannercanthensimplychoosethe bestglobalsolutionorconducta finaldecision steptakinginto accountatrade-offanalysis.

Fig.2.Singlelinediagramofthe6busGarvernetwork.

4. CaseStudies

4.1. Garvernetwork 4.1.1. Data

TheGarvernetworkwasdetailedintheseminalpaper[24]and sincethenithasbeenusedbyseveralotherresearchersnamely tocomparedifferentTEPapproaches.Thisnetworkispresented inFig.2 andit includes6 busesand 6 lines.Intheinitialcon- figutaion,bus 6is notconnected totherestof thesystemand theexistingdemandinbuses1–5is760MWwhiletheinstalled generationcapacityintheseinterconnectednodesisjust510MW.

Asaresult,theexpansionplanningexercisewillobviouslyhave topromotetheinterconnectionof bus6 totherestofthesys- teminordertopreventPowerNotSupplied.Table1detailsthe busdata,namelytheinstalledgenerationcapacityandtheactive powerdemand.Regardingthegeneratoroperatingcostsusedin theobjectivefunction(10)oftheDC-OPFmodeldescribedinSec- tion3.2weused25$/MWhforthegeneratorsconnectedtobus1 and40$/MWhforthegeneratorsconnectedtobuses4–6.Onthe otherhand,thecoefficientGthatpenalizesnonzeroPNSvalues wassetat10,000$/MWh.Thepenaltycoefficientsusedin(16)in casesomeestablishedthresholdisviolatedweresetat100,000so thatanyviolationstronglypenalizestheassociatedparticle.Table2 indicatesthebranchdataandfinallyTable3detailstheprojectlist thatincludes17possibleaddtions.

4.1.2. Singleperiodanalysis

UsingthedataabovewesolvedasingleperiodTEPexercise.The DEPSOalgorithmwasruninseveraltestsnamelytoevaluatesev- eraldesignoptionsasthesizeofthepopulations,theintroduction oftheLamarkianoperatorincaseanullvelocityvectorisobtained andthemutationofthebestglobaleverfoundparticleasaway Table1

Busdata–installedgenerationcapacity(MW)andactivepowerdemand(MW).

Busnumber Installed

capacity(MW)

Activepower demand(MW)

1 150 80

2 0 240

3 360 40

4 0 160

5 0 240

6 600 0

Table2

Branchdata,resistance(pu),reactance(pu)andcapacity(MW).

Frombus Tobus Resistance(pu) Reactance(pu) Capacity(MW)

1 2 0.10 0.40 100

1 4 0.15 0.60 80

1 5 0.05 0.20 100

2 3 0.05 0.20 100

2 4 0.10 0.40 100

3 5 0.05 0.20 100

Fig.3. InfluenceofseveralchoicesoftheparameterkbGassociatedwiththemuta- tionofthebestglobalparticleonthefrequencyoftheidentificationoftheoptimal solution.

toinducesomelocalsearch.Inthispaperandduetospacelimi- tationswewillsummarizesomeoftheresultsthatwereobtained alongthisresearch.Inthisscope,Fig.3illustratestheinfluenceof severalchoicesoftheparameterk_bGassociatedwiththemutation ofthebestglobalparticleandFig.4presentstheinfluenceofthe parameterkLam determiningtheuseoftheLamarkianevolution whenthevelocityofaparticleiszero.

Finally,regardingthissingleperiodanalysis,itisimportantto stressthattheoptimalidentifiedsolutionisthesameasthesolu- tionmentionedintheliteraturealthoughthefinalpopulationsmay includeothersolutionshavingthesametotalinvestmentcostand havingzeroPowerNotSupplied.Thissolutionincludesonenew branchbetweennodes3and5and3newbranchesbetweennodes 4and6.Thisfinalsolutionwasobtainedinthescopeofanumber oftestsinwhichtheDEPSOran100timesfor10,20,30and50 particlesineachpopulationandFig.5illustratestheconvergence characteristcsofthealgorithm.TheperformanceoftheDEPSOis verygoodbecauseitprovidedverylargeratesofidentificationof theoptimalsolutionina reducednumberofiterationsevenfor smallpopulationsof10particles.Forexample,forapopulationwith 20particlesafter10iterationsthebestsolutionwasidentifiedin 100%oftheruns,thatisinallthe100TEPexercices.

Fig.4.InfluenceoftheparameterkLamdeterminingtheuseoftheLamarkianevo- lutionwhenthevelocityofaparticleiszeroonthefrequencyoftheidentification oftheoptimalsolution.

Table3

Projectlistincludingextremenodesofpossiblenewbranchesandinvestmentcost.

Branchnumber Frombus Tobus Resistance

(pu)

Reactance (pu)

Capacity (MW)

Inv.cost (10⁶$)

1 2 6 0.08 0.03 100 30

2 2 6 0.08 0.03 100 30

3 2 6 0.08 0.03 100 30

4 2 4 0.10 0.40 100 40

5 5 6 0.1476 0.61 78 61

6 3 5 0.05 0.20 100 20

7 3 5 0.05 0.20 100 20

8 3 5 0.05 0.20 100 20

9 4 6 0.08 0.30 100 30

10 4 6 0.08 0.30 100 30

11 4 6 0.08 0.30 100 30

12 4 6 0.08 0.30 100 30

13 4 6 0.08 0.30 100 30

14 1 4 0.15 0.60 80 60

15 1 5 0.05 0.20 100 20

16 1 2 0.10 0.40 100 40

17 2 3 0.05 0.20 100 20

Fig.5. PerformanceoftheDEPSOalgorithmfor10,20,30and50particlesinterms ofthefrequencyoftheidentificationofthebestsolution.

4.1.3. Multiperiodanalysis

Inasecondstep,weusedagaintheGarvernetworkpresented inFig.2torunafourperiodtest.Inthiscase,theprojectliststill containsthe17projectsinTable3butthesearchspaceismuch largerthaninthesingleperiodtestanditincludes6¹⁷possible combinations.Recallthateachpositionjofeachparticlecanbe

filledwithanintegerfrom0tonp+1,thatis,from0to5inthiscase.

Regardingtheresults,thebestsolutionincludestheconstruction of6linesspreadbythe4periodsasfollows:

- inperiod1– 1line1–5,1line2–6,1line3–5and1line4–6;

-inperiod2thereisnonewaddition;

- inperiod3– 1line4–6;

-inperiod4–1line3–5.

Thissolutionhasaninvestmentcostof150M$.Itwasobtained notconsideringthepenaltytermin(16)overthelossesanditwas admittedthatthedemandincreasedatarateof5%perperiod.The evolutionofthenetworksincetheinitialconfigurationinFig.2to period1,andfinallytoperiod4isdisplayedinFig.6.

TheperformanceoftheDEPSOwasevaluatedrunningtheTEP exercise100timesforpopulationswith10,20,30,50and80par- ticles.Fig.7illustratestheresultsintermsofthefrequencyofthe identificationofthebestsolution.Thesegraphsshowthatpopula- tionswith30particlesyieldverygoodresultssincethebestsolution isidentifiedin95%ofthecasesafter40iterations.Ifthenumberof particlesislarger,80inthiscase,asimilarpercentageisobtained after20iterations.

Fig.6.Evolutionofthenetworkfromperiods1to4–period1ontheleftsideandperiod4ontheright.

Fig.7. Frequencyofidentificationofthebestsolutionfor10,20,30,50and80 particles.

Finally,theperformanceoftheDEPSOwascomparedwiththe oneoftheEPSOasreportedin[21].Fig.8showstherateofconver- genceoftheEPSOandoftheDEPSOalgorithmsforpopulationswith 10,30,50,80and100particlesconsideringthatineachcasethe algorithmwasrun100times.AsshowninthisFigure,theDEPSO performsbetterthantheEPSOforthesamenumberofparticles anditerations.ThisindicatesthattheDEPSOwasabletoidentify thebestsolutionmoretimesthantheEPSOunderthesamecon- ditions,intermsofthenumberofparticlesanditerations.Onthe otherhand,forpopulationswith50particlesormoretheDEPSO identifiedthebestsolutioninallthe100runsofthealgorithm.

4.2. IEEEreliabilitytestsystem 4.2.1. Data

TheIEEEReliabilityTestSystem,RTS,wasoriginallydescribedin [25].Since1979ithasbeenusedbyseveralresearchersfordifferent purposesanditwassubjectedtoanumberofadaptationsinorder toturnitmoresuitabletotestparticularapplications.Theoriginal

Fig.8. ComparisonbetweentheEPSOandtheDEPSOfordifferentpopulationsizes anditerations.

systemincludes24nodesand38branches(linesandtransformers) atthevoltagelevelsof138and230kV.Theoriginaltotaldemandis 2850MWandtheinstalledgenerationcapacityis3405MW.Sev- eraltestsconductedinthescopeofreliabilitycalculationsshowed thatthetransmissionsystemoftheRTSnetworkwaslighlyloaded andsointhispaperweincreasedthedemandandtheinstalled generationcapacitytothetripleoftheiroriginalvalues,thatis,to 8550MWandto10,215MWwhilemaintainingthecharateristics ofthetransmissionsystem,namelyintermsofthebranchcapaci- tiesreportedin[25].Thisreferencealsocontainsthecompletedata forthissystemnamelythebranchresistanceandreactancevalues andthegeneratoroperatingcostvalues.

Ontheotherhand,when doing themultiperiod analysiswe useda10%valuefortherrrateandweadmitedthatthedemand increasedinallbusesby5%perperiod.Giventhisdemandincrease, we admitted that the transmission system alsohad toaccom- modatethe connectionof two new generators as follows: one connectedtobus4witha capacityof300MWand anotherone tobus19withacapacityof591MW.Finally,apartfromtheoper- ationandtheinvestmentcosts,thefitnessfunctionalsoincluded Table4

Projectlistincludingextremenodesofpossiblenewbranches,typeandinvestmentcost.

Branchnumber Frombus Tobus Type Resistance(pu) Reactance(pu) Capacity(MW) Inv.cost(10⁶$)

1 3 24 Transf. 0.0 0.4195 400 500

2 9 11 Transf. 0.0 0.4195 400 500

3 10 11 Transf. 0.0 0.4195 400 500

4 10 12 Transf. 0.0 0.4195 400 500

5 1 5 Line 0.1090 0.4225 175 220

6 1 5 Line 0.1090 0.4225 175 220

7 2 4 Line 0.1640 0.6335 175 330

8 2 4 Line 0.1640 0.6335 175 330

9 2 6 Line 0.2485 0.9600 175 500

10 2 6 Line 0.2485 0.9600 175 500

11 6 10 Line 0.0695 0.3025 175 160

12 7 8 Line 0.0795 0.0032 175 160

13 7 8 Line 0.0795 0.0032 175 160

14 8 10 Line 0.2135 0.8255 175 430

15 11 13 Line 0.0305 0.2389 500 660

16 12 13 Line 0.0305 0.2380 500 660

17 14 16 Line 0.0250 0.1945 500 540

18 15 21 Line 0.0315 0.2450 500 680

19 15 24 Line 0.0335 0.2595 500 720

20 16 17 Line 0.0165 0.1295 500 360

21 16 17 Line 0.0165 0.1295 500 360

22 16 19 Line 0.0150 0.1150 500 320

23 17 18 Line 0.0090 0.0720 500 200

24 20 23 Line 0.0140 0.1080 500 300

25 11 13 Line 0.0305 0.2389 500 660

26 12 13 Line 0.0305 0.2380 500 660

27 11 14 Line 0.0305 0.2380 500 580

28 14 16 Line 0.0250 0.1945 500 540

Table5

Bestidentifiedexpansionplansobtainedinseveralruns.

Test npt Inv.cost(M$) Expansionprojectsincludedinthebestidentifiedplans

3–24 10–12 1–5 1–5 2–6 6–10 7–8 7–8 11–13 11–13 16–17

4periods 30 2599.16 – 1 1 – 2 1 1 2 1 3 –

4periods 100 2527.44 – 1 1 – 4 1 1 2 1 3 –

4periods 150 2427.72 4 1 1 – – 1 1 2 1 – 4

1period 100 1280.00 – – 1 1 – 1 1 1 – – 1

Fig.9.Evolutionofthefitnessfunctionfor30,100and150particlesforthefour periodtest.

penaltytermsifthenumberofprojectsperperiodexceeded6and ifPNSwasnotzerofortheNsystemandforN−1contingencies.In thiscase,weusedthesamevaluesforthepenaltytermsthatwere mentionedinSection4.1.1fortheGarvernetwork.

RegardingthestoppingcriteriaoftheDEPSOalgorithm,wecon- ductednumerousrunsconsideringamaximumof5000iterations.

Thealgorithmwouldalsostopif1000iterationswerecompleted withoutimprovingthecurrentbestglobalsolution.Inordertocon- ductthesetests,weadmittedthattheplannerspecifiedalistof possibleinvestmentprojectscontaining28newbranches,linesand transformers.Themaincharacteristicsoftheseprojectsaredetailed inTable4.

4.2.2. Singleperiodanalysis

Inthiscase,theDEPSOalgorithmwastestedwithpopulations including10,30and100particlesanditwaspossibletoconclude that10particleswastooshorttoensureanadequatefrequency ofidentificationofthebestsolution,evenafterrunning1000iter- ations.Ontheotherhand,forapopulationof30particlesa95%

frequencyofthebestsolutionwasachievedafter900iterations while for a population of 100particles a 100% frequency was obtainedafter590iterations.Thebestsolutionthatwasidenti- fiedhasaninvestmentcostof1280M$anditincludesbuildingthe following6branches:twonewbranchesconnectingnodes1and 5,onenewbranchconnectingnodes6and10,twonewbranches connectingnodes7and8andonenewbranchconnectingnodes 16and17.Forapopulationof30particles,thealgorithmanalyzed (30particles×2 populations×900iterations)=54×10³ possible investmentplanswhilethetotalnumberofparticlesinthesearch spacewas3²⁸≈22.9×10¹².

4.2.3. Multiperiodperiodanalysis

Finally,thedevelopedDEPSOwasalsotestedconsideringafour periodhorizon.In thiscase,we usedthesameprojectlistwith 28newpossibleequipments,meaningthatthesearchspacenow includes6²⁸≈6.1×10²¹potentialsolutions.Inthiscase,wecon- ductedseveraltestsnamelyusingpopulationswith30,100and 150particlesandFig.9 displays3examplesofthefitnessevo- lution.Table5characterizesthesolutionsthatwereobtainedin thesethreerunstogetherwiththesolutionfromthesingleperiod study.Foreachrun,thistableindicatestheperiodinwhicheach

equipmentshouldstartoperation.Theprojectsnotlistedarenot consideredinanyofthesesolutions.Theseresultsdeservesome comments:

- therearesignificantdifferencesamongthesingleperiodandthe multiperiodsolutions.Forinstance,buildingabranchconnecting nodes16and17isincludedinthesingleperiodsolutionwhileit isonlyincludedinperiod4ofthemultiperiodsolutionobtained withapopulationof150particles;

-ontheotherhand,oneoftheprojects7–8isdelayedfromperiod 1toperiod2whengoingtothemultiperiodanalysis;

-several projects not elected in thesingle period solutionare includedinthemultiperiodsolutionsthatwereobtained.These includeinstallingthetransformers3–24and10–12andthelines 2–6and11–13;

-finally,theseresultsandthedifferencesbetweenthesolution obtainedforthesingleperiodanalysisandforthemultiperiod testsconfirmthatamultiperiodstudyisnotnecessarilyacombi- nationofstaticschedulesobtainedinsequenceforeachplanning period.Infact,thesetofprojectselectedforthesingleperiod solutionisnotequaltotheprojectsincludedinthefirstperiod ofanyofthethreereportedmultiperiodsolutions.Thismeans thatexplicitlymodelingintheproblemalltheperiodsalongthe planninghorizonimposesanewdynamictothesolutionidenti- ficationthatisnotadequatelycapturedbysolvingindependently asequenceofyearlyplanningexercises.

5. Conclusions

Thispaperdescribesadiscreteapproachoftheevolutionarypar- ticleswarmoptimization,DEPSO,thatwasusedtosolveamultiyear transmissionexpansionplanningproblem.Thisproblemisvery complexduetoitsmathematicalcharacteristicsanditstypically largecombinatorialnature.Thismeansitisimportanttodevelop modelsthatadequatelyaddressthisproblem,namelyinviewofthe involvedinvesmentcoststhatareultimatelyreflectedinthetar- iffspaidbynewtorkusers.Theresultsthatwereobtainedandthat arepartiallyreportedinthispaperdemonstratethatthedeveloped DEPSOapproachisaccurateandthatitallowstheidentificationof goodqualitysolutionswithlessparticlesandlessiterationswhen comparedwithclassicalparticleswarmoptimizationalgorithms.

Ontheotherhand,theresultsthatwereobtainednamelyforthe IEEERTSsystemalsocompareinafavorablewaytootherresults reportedbyotherresearchteams.

Finally,thisresearchlinewillbefollowedinfuturepublications namely to enhance the TEP model so that it addresses uncer- taintiesthataretypicallypresentinthistypeofexercices.These uncertainties can affecttheevolution of thedemand along the planninghorizonandwillleadtotheinclusionof riskconcepts inthisapproach.Anotherpointofpossibledevelopmentconsists ofenhancingthedevelopedmodelsothatitcanaccommodateDC transmissionsystemsgiventhegrowinginterestonthem.Regard- ingthisissue,itisimportanttonoticethatthegeneralformulation oftheTEPproblemisstillvalidincaseDCtransmissionsystems areconsidered.Theincorporationofthesesystemswouldrequire extendingthelistofcandidateprojectstobesuppliedbytheuser