Convergence-Analysis-of-the-Incremental-Cost-Conse

IEEETRANSACTIONSONPOWERSYSTEMS,VOL.27,NO.4,NOVEMBER20121761ConvergenceAnalysisoftheIncrementalCostConsensusAlgorithmUnderDifferentCommunicationNetworkTopologiesinaSmartGridZiangZhang,StudentMember,IEEE,andMo-YuenChow,Fellow,IEEEAbstract—Inasmartgrid,effectivedistributedcontrolalgo-rithmscouldbeembeddedindistributedcontrollerstoproperlyallocateelectricalpoweramongconnectedbusesautonomously.Byselectingtheincrementalcostofeachgenerationunitastheconsensusvariable,theincrementalcostconsensus(ICC)algorithmisabletosolvetheconventionalcentralizedeconomicdispatchprobleminadistributedmanner.Themathematicalformulationofthealgorithmhasbeenpresentedinthispaper.TheresultsofseveralcasestudieshavealsobeenpresentedtoshowthatthedifferencebetweennetworktopologieswillinfluencetheconvergencerateoftheICCalgorithm.IndexTerms—Distributedcontrol,economicdispatch,leader-followerconsensus,multi-agentsystem,smartgrid.I.INTRODUCTIONECONOMICdispatchisoneofthefundamentalprob-lemsinpowersystems.Itisessentiallyanoptimizationproblemwiththeobjectiveofreducingthetotalgenerationcost,subjecttoseveralconstraints.Previouseffortstosolvetheeconomicdispatchproblem(EDP)havebeenmadebyimplementingvariousnumericalmethodsandoptimizationtechniques.Theconventionalmethodsincludethelambda-it-erationmethod[1]andthegradientsearchmethod[2].Thesemethodsprovideafeasiblesolutionwhenthefuelcostfunctionisconvex.Moresophisticatedtechniqueshavebeenemployedtosolvethenonconvexcostfunction,suchasgeneticalgorithms(GA)[3],particleswarmoptimization(PSO)[4],andmorerecentlytheimprovedparticleswarmoptimization(IPSO)[5].Theperformanceandapplicabilityofeconomicdispatchhasbeenimprovedbytheseoptimizationtechniques.However,itisessentialtohaveasinglecontrolcenterthatcanaccessthestateoftheentiresystem.Thiscentrallycontrolledframeworkmaycausesomeperformancelimitationsinthefuturepowergrid.Inthispaper,weexplorethesolutionoftheEDPfromanotherangle.WeproposeusingadistributedalgorithmtosolvetheEDPinadistributedfashion,ratherthantheconventionalcentralcontroller.ManuscriptreceivedFebruary17,2011;revisedJuly12,2011,October20,2011,January30,2012;acceptedFebruary14,2012.DateofpublicationApril13,2012;dateofcurrentversionOctober17,2012.ThisworkwassupportedinpartbytheNationalScienceFoundation(NSF)underAwardNumberEEC-08212121.Paperno.TPWRS-00140-2011.TheauthorsarewiththeDepartmentofElectricalandComputerEngi-neering,NorthCarolinaStateUniversity,Raleigh,NC27606USA(e-mail:[e-mail];[e-mail]).Colorversionsofoneormoreofthefiguresinthispaperareavailableonlineat(NCS).First,inordertoachievetheoptimaloperatingconditions,thecentralcontrollerwillberequiredtohaveahighlevelofcon-nectivity,whichwillimposeasubstantialcomputationalburdenandistypicallymoresensitivetofailuresandmodelingerrorsthandistributedcontrolschemes[7].Anotherchallengeisthatthetopologyofthesmartgridisunknown,notonlybecauseofthevarietyofconfigurationsofthepowergridandcommu-nicationnetworktopologies,butalsobecause“plug-and-play”technologieswillmakethetopologytime-varying[8].Thus,inordertocontrolthiskindofNCS,or,specificallyspeaking,thesmartgrid,arobustalgorithmshouldbeabletooperatecorrectlyinthepresenceoflimitedandunreliablecommunicationcapa-bilities,andoftenintheabsenceofacentralcontrolmechanism[9].Thus,distributedcontrolalgorithmsaremoresuitableforsolvingthisproblem.Afundamentalproblemindistributedcontrolsystemsistheneedforallthenodestoreachacon-sensus.Theconsensusproblemhasbeenstudiedinseveralareas,suchassocialscience,animalscience[10],andcom-puterscience[11].Consensusalgorithmsandtheirapplicationhavebeenstudiedextensivelyinthesystemandcontrolarea[12]–[18].Oneofthepotentialalgorithmsthatcanbeusedtoimprovethepowersystemistheconsensusalgorithmofamulti-vehiclesystemintroducedin[12].Themulti-vehiclesystemperformscooperativetasks,andthevehiclesareabletocommunicatewithsomesetofothervehiclesthroughthecommunicationnetwork.Similarly,thefuturepowergridcanbeconsideredasanaggregationofcontrollablepowerelec-tronicsdeviceswhichcontrolledbydistributedagentsthroughthecommunicationnetwork[8].Thisfuturepowersystemiscomposedoftwonetworks:acommunicationnetworkandapowernetwork.Weproposedanincrementalcostconsensus(ICC)algorithmasanexampletoillustratetheuseofdistributedcontrolonasmartgridin[19].Severalpreliminarysimulationswithoutgen-eratordynamicsalsohavebeenshownin[19]todemonstratethefeasibilityoftheICCalgorithm.Inthispaper,wefocusonconvergenceanalyses,suchastherelationshipbetweencom-municationtopologyandconvergencerate.WeextendedtheICCalgorithmbyaddingthegeneratorcapacityintoconsider-ation.Wealsosuggestusingthenodecentralitymeasurementtohelpthe“leaderelection”intheleader-followerconsensusalgorithm.InordertostudythetransientresponseoftheICCalgorithm,wealsoextendthesimulationmodelbyincludinggeneratordynamics.TheperformanceofICCalgorithmunderU.S.GovernmentworknotprotectedbyU.S.copyright.1