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首页 Individual Branch and Path Necessary Conditions …

Individual Branch and Path Necessary Conditions for Saddle-Node Bifurcation Voltage Collapse.pdf

Individual Branch and Path Nece…

cwhawk
2012-02-05 0人阅读 举报 0 0 暂无简介

简介:本文档为《Individual Branch and Path Necessary Conditions for Saddle-Node Bifurcation Voltage Collapsepdf》,可适用于工程科技领域

IEEETRANSACTIONSONPOWERSYSTEMS,VOL,NO,FEBRUARYIndividualBranchandPathNecessaryConditionsforSaddleNodeBifurcationVoltageCollapseSantiagoGrijalva,SeniorMember,IEEEAbstractThispaperexplorestherelationbetweensaddlenodebifurcationvoltagecollapseandcomplexflowlimitsofindividualtransmissionelementsTwonecessaryconditionsforpowersystemvoltagecollapsearepresentedFirst,whenatransferofpowertakesplaceinapowersystem,atleastonelinemustreachitsstatictransferstabilitylimit(STSL)atorbeforethepointofvoltagecollapseSecond,forapointtopointtransfer,apathfromthesourcetothesink,formedbylinesallofwhichhavereachedtheirSTSLlimits,mustbeformedinthenetworkbeforethepointofcollapseisreachedWepresentnumericalexamplesconfirmingthesetwonecessaryconditionsIndexTermsCollapse,distributionfactor,loadingmargin,powersystemvoltagestabilityIINTRODUCTIONANIMPORTANTpropertyofsecurepowersystemsistheabilitytomaintainreasonablylargemarginstothepointsofdynamicandstaticvoltagecollapseInthelastdecade,duetoincreaseddemand,reducedinvestmentintransmission,largermarkettransfers,andrenewableenergyvariability,electricgridsinmanyregionsoftheworldarebeingoperatedwithlessmargintovoltagestabilitylimits,Hence,theabilitytomonitordistancetocollapseinrealtimesecurityapplicationssuchassecurityanalysis(SA),securityconstrainedoptimalpowerflow(SCOPF),andnonlinearavailabletransfercapability(ATC),isessentialtoensurethatthepowergridcontinuestobeoperatedinasecuremannerTheexactdeterminationofthesystemmargintovoltagecollapserequirescontinuationpowerflowsolutions,fornormaloperation,foralargesetofcontingencies,andformultipleplausiblesystemscenariosSeveralindiceshavebeenproposedasindicatorsofproximitytocollapseTheseindicesarebasedontheoverallconditionsofthegridApowersystemwilleventuallyreachitspointofstaticvoltagecollapseunderasufficientlylargetransferofactivepowerfromasourcetoasinkAsthetransferincreases,voltagecollapsecanoccurduetotwodifferentmechanisms:)thepowerflowJacobianmatrixbecomessingular(saddlenodebifurcation),,or)acontrollimit(reactivepowergeneratorlimit,tapchangerlimit,etc)isencountered,causingashifttoanewsystemofalgebraicequations,whichdoesnothaveasolution(limitinducedbifurcation),Inthispaper,weManuscriptreceivedJanuary,revisedMay,,August,,November,,andFebruary,acceptedMarch,DateofpublicationAugust,dateofcurrentversionJanuary,PapernoTPWRSTheauthoriswiththeGeorgiaInstituteofTechnology,Atlanta,GAUSA(email:sgrijalvaecegatechedu)DigitalObjectIdentifierTPWRSwillfocusexclusivelyoncollapseduetopowerflowJacobiansaddlenodebifurcationWhilecollapsephenomenadependontheconditionsoftheentirenetwork,ithasbeenshowninthatforthecaseofasystemwithperfectvoltagecontrol(anidealsysteminwhichallthebusesareassumedtobePVbuses),voltagecollapseduetosaddlenodebifurcationcanbemonitoredbylookingatthebehavioroftheflowofactiveandreactivepowerinindividualtransmissionelementsThatresultisextendedinthispaper)bydemonstratinganindividualbranchnecessaryconditionforvoltagecollapseforthecaseofgenericsystems(systemsthatincludeslack,PV,andPQbuses),and)bydemonstratinganecessaryconditionforvoltagecollapserelatedtoasinglenetworktransferpathThecentraltopicofthispaperisthereforetherelationbetweensystemwidevoltagestabilitylimitsandthelimitsofindividualtransmissiondevicesIIBACKGROUNDInthissection,wepresenttheformaldefinitionsofpowertransfer,staticcollapse,branchcomplexflowtrajectory,andpowertransferdistributionfactorsThesedefinitionswereoriginallywritteninTheyareincludedhereforclarityandasbackgroundforthenewresultsandproofspresentedinSectionsIII–Vofthispaper,andforthenumericalexamplesinSectionVIWewillassumethroughoutthepaperthatthepowersystemconsistsofasingleelectricislandDefinition:PowerTransfer:Atransferofactivepoweracrossaconnectedpowernetworkwithbusesisabalancedvariationofthebusnetactivepowerinjections,andisdenotedby,whereisascalarparameterthatrepresentsthesizeofthetransferinmegawattsorinperunit,andisavectorofsizeformedbysellingandbuyingbusparticipationfactorsThevectordescribesthefractionofactivepowerthatisbeinginjectedorextractedateachsellingorbuyingbusThesevectorsmeettheconditionthatandThedefinitionofatransferassumesapreestablishedlossallocationschemeoralosslesspowersystemThevectorspecifiesthedirectionofatransferinapplicationssuchasavailabletransfercapability,theshiftofgeneratorcontrolsintheinnerloopiterationsoftheoptimalpowerflow(OPF),orchangesintotalsystemloadingrepresentedbyconstantpowerAnincreaseintotalpowersystemdemandcanbesimulatedbymodelingatransferfromdispatchedunitstotheconsumptionpointsusingparticipationfactorsproportionaltothedemandincreaseDefiningatransferinthisgenericmanneraccommodatesalargevarietyofpossiblechangesofactivepowerinasystem$©IEEEGRIJALVA:INDIVIDUALBRANCHANDPATHNECESSARYCONDITIONSDefinition:StaticCollapseDuetoSaddleNodeBifurcation:LetbethesizeofapowertransferLetbethevalueofatthepretransferornormaloperationsystemconditionWeassume(throughoutthispaper)thatthereisanACpowerflowsolutionforthisinitialoperatingconditionThetransferwillincreasefromuptoapointwherethesystemreachesasaddlenodebifurcationandtheJacobianmatrixbecomessingularTherearepowerflowsolutionsforallintheintervalandnosolutionifDefinition:BranchComplexFlowTrajectory:ThestateofthepowersystemisgivenbyavectorcontainingthebusvoltagemagnitudesandanglesIfweconsideratransmissionlineconnectingbusesand,theflowsofactiveandreactivepowerinthelinearecompletelydeterminedbyknowingtheparametersofthelineandtheanglesandvoltagemagnitudesatthetwoterminalbuses:()()wheretheflowsandthestatevariablesarefunctionsofthetransferparameteristhemagnitudeandtheangleofthebranchadmittanceobtainedfromthetransmissionlinemodelThefollowingexpressioncanbeobtainedbyrearranging()and()byleavingthesineandcosineintherighthandside,takingthesquare,andadding()Wenotethat()isindependentofthevoltageanglesAsthetransferincreases,thepreviousequationwillresultinatrajectoryofthecomplexflowoflineintheplane,asaparametricfunctionofthetransfersizeIfthevoltagemagnitudesandarekeptconstant,then()wouldrepresenttheequationofaconstant“operatingcircle”,withcenterandradiusgivenby()()wherethesubscriptindicatesacircleparameterIngeneral,thevoltagemagnitudesattheterminalsofalinewillchangeasafunctionofWenotethatifthevoltagemagnitudesdecrease,thecenterofthecirclewillmoveclosertotheoriginofcoordinatesanditsradiuswillbecomesmallerEquation()willbeusedlaterontodemonstratethatthecomplexflows()and()ofatransmissionlineareboundedDefinition:PowerTransferDistributionFactor:Thepowertransferdistributionfactor(PTDF)ofalineisthesensitivityofthetransmissionlineactivepowerflowwithrespecttotheparameterassociatedwiththetransferBecausewearenotconsideringcontrollimitsorlimitinducedbifurcation,wecanassumethatiscontinuousanddifferentiableinThedistributionfactoristhereforedeterminedas()Ifthesystemisassumedtobelossless,then,andalsoStartingataninitialoperatingpoint,wherethelineflowisequalto,theposttransferflowinthelinecanbeobtainedas()TheactivepowerflowsinallthetransmissionelementscanbegroupedinavectorThisvectormayincludeflowsatbothendsofeachtransmissionlineAvectorofdistributionfactorscanalsobeformedThisvectorofdistributionfactorscanbeobtainedusingthesolvedpowerflowJacobianmatrixas()IIIBOUNDEDNESSOFTHEBRANCHCOMPLEXFLOWInthissection,wedemonstratethatthecomplexflowofatransmissionlineisboundedTheactivepowerlimitassociatedwiththisboundisdefinedasthestatictransferstabilitylimit(STSL)ProvidedinthissectionisasufficientconditionforcollapsedirectlyrelatedtotheSTSLforthecaseoftransmissionelementsthatareradiallyconnectedtothegridLemma:ThelinecomplexflowisboundedProof:Thevoltagemagnitudeattheterminalofageneratoristheresultofexciteraction,whichisboundedbythemagneticfieldandexcitationcurrentThus,voltagemagnitudesinpowersystemsarepositiveandfiniteAllthetransmissionlineparameters,andarefiniteSincethesineandcosinefunctionshaverange,thisimpliesthatthecomplexflowisboundedTogetanexplicitboundonthecomplexpowerflow,wenotethatthequantities,andin()and()arefiniteInaddition,itfollowsfrom()thatisstrictlypositiveUsingthecircledefinitionsin()and(),wewrite()as()Thisequationimpliesthat()Sinceandarefinite,andisstrictlypositive,itfollowsthat()Thus,theactiveandreactivepowerflowsofatransmissionlineareboundedbythesequantitiesAsthetransferincreasesinthesystem,theflowsofactivepowerinthetransmissionelementswillchangeLargerflowsIEEETRANSACTIONSONPOWERSYSTEMS,VOL,NO,FEBRUARYofactivepowerintransmissionlinesusuallyrequirelargerflowsofreactivepowerAfter(),ifasaresultofatransfertheactivepowerflowinalineincreasessufficiently,thecomplexflowtrajectorywouldeventuallyreachapointwhereIftheterminalvoltagesareassumedtobeconstant,thiscanbeeasilyseenbynotingthatwithlargeactivepowerflow,thereactiveflowwillchangefollowingacircleshapeuntilthetrajectoryreachesaverticalpointThisisalsotrueiftheterminalvoltagesofthelinevarywiththetransferThisverticalityconditioninthecomplexflowplanerepresentsahardlimitoftheflowofactivepowerinthelineDefinition:LineStaticTransferStabilityLimit(STSL):Givenapowertransfer,thestatictransferstabilitylimit(STSL)forthesendingendofatransmissionlineisdefinedasthemaximumamountofactivepowerflowthatcanbesentthroughthelinewhentheconditionisreachedbythelinecomplexflowtrajectoryWedenotethislimitbyTheSTSLlimitisnotreachableforeverytransmissionline,sincethepointofvoltagecollapsemayoccurbeforethecomplexflowtrajectoryofagivenlinereachestheverticalityconditionIfthelimitisreachedforagivenline,itwillcorrespondtoacertainvalueofthetransferparameterIthasbeenobservedexperimentallythatforcertainlines,itispossiblethatifthetransfercontinuestoincreasebeyond,thereactivepowerofthelinewillcontinuetoincreasewhiletheactivepowerwillstarttodecreaseTheSTSLlimitofalinecanbereachedintwoways:)Ifthederivativeofthereactivepowerwithrespectto(thereactivepowerdistributionfactor)isfinite,thentheactivepowerdistributionfactormustbeequaltozeroatthatpoint:())Ifthederivativeofthereactivepowerwithrespecttotendstoinfinity,thentheactivepowerdistributionfactormustbefiniteandpositive:()Both()and()implythatTheSTSLofalinehasbeenreachedifeithercondition()or()aremetbythecomplexflowofalineDefinition:RadialLine:Givenapowersystemandapowertransfer,alineorasetofidenticallinesconnectingtwobusesandissaidtoberadialfortransferifopeningthelineorlinespreventsthetransferfromtakingplaceWhiletheconditionprovidedby()appearsinlinesthatarepartofmeshedtopologies,theconditionsin()appearinlinesthatareradialLetusprovideinsightofwhythisisthecaseConsiderabusthatisasourcepointofatransferwithparticipationfactor(entryinpositionofthetransfervector)LetusassumethatthisbusisconnectedtotherestofthesystemthroughlineIflineisopen,itisnotpossibletotransferpowerfrombusHence,lineisaradiallineforthistransferIfthesystemislossless,thedistributionfactorofthislinewillbeexactlyanditwillbeconstantforanyvalueofthetransferparameterInthecaseofasystemwithlosses,thisdistributionfactorwillbecloseto,butconstrainedbythefactthataboutofthepowerbeingtransferredneedstoflowthroughthatparticularlineOtherexamplesofradiallineswouldbeasetoftwoormoreidenticalcircuitsconnectingtwobuses,oralinebetweentworegionsortwocontrolareasthatistheonlypathforagiventransferInsummary,whenlinesareradial,thetopologyofnetworkimposesaconstraintonthedistributionfactorbeingnonzero,andhence,theSTSLoftheselinesisreachedundertheconditionsin()Ontheotherhand,whenlinesarepartofmeshedtopologies,thedistributionfactorscanvaryaschangesandtheSTSLmaybereachedundertheconditionsgivenby()WenowproveformallythatifatransferincreasescausingaradiallinetoreachitsSTSL,thencollapsemustoccurimmediatelyifthetransferistocontinuetoincreasebeyondthispointLemma:ASufficientConditionforCollapse:LetaninitialpowersystemsolutionbeavailableforaparameterassociatedwithatransferLetusassumethatincreasestoapoint,wheretheactivepowerofalinereachesitsSTSLlimit,andthedistributionfactorisstrictlypositive,eg,aradiallineThenthesystemceasestohaveapowerflowsolutionforProof:WeprovethisbycontradictionContradictionhypothesis(CH):Thereexistsapowerflowsolutionforavalueofthetransfer,whereispositiveandarbitrarilysmallFor,theactivepowerflowmusthavechangedto()Sincethedistributionfactorandarestrictlypositive,thetermcorrespondingtotheintegralin()isstrictlypositive,andhence,ThisisnotpossiblebecausetheactivepowerflowofthelineisboundedbyitsSTSLlimitWehavereachedacontradictionThus,therecannotbeapowerflowsolutionforLetusnowturnourattentiontolinesthatarenotradial,ie,linesthatarepartofmeshedsystemsInthiscase,howtheflowsaredistributeddependsontheentirestructureofthegridanditsparametersDistributionfactorscanvarysignificantlyasafunctionofthetransferInparticular,closertothepointofcollapse,theymayvanishatthepointofSTSLreachingtheconditionstatedin()Weareinterestedinestablishingarelationbetweenthepointwherecondition()mayholdforalineandinmeshednetworksThus,wewillnowfocusonmeshedtopologiesanddistributionfactorsthatmaychangetheirsignsduringthetransferGRIJALVA:INDIVIDUALBRANCHANDPATHNECESSARYCONDITIONSIVBRANCHNECESSARYCONDITIONFORCOLLAPSELemma:ThederivativesofthelineactivepowerflowwithrespecttothestatevariablesareboundedProof:Thederivativesofthetransmissionlineactivepowerflowwithrespecttothestatevariablesattheterminalbusescanbeobtaineddirectlyfrom()Thetermsintheresultingexpressionscanbereplacedbytheexpressionsin(),(),(),and(),toobtain()()()()AfterLemma,thelineactiveandreactiveflowsaswellastheparametersandareallboundedThus,alltheexpressionsin()–()abovearealsoboundedThiscompletestheproofCorollary:ThematrixofderivativesoflineactiveflowswithrespecttostatevariablesisboundedProof:Sinceeachentryinthematrixisbounded,thematrixisalsoboundedDefinition:NetworkCutforaPointtoPointTransfer:Anetworkcutforatransferfromasourcebustoasinkbusisagroupoftransmissionlinesthatseparatethesetofbusesofthenetworkintotwomutuallyexclusiveandcomprehensivepartitionsand,suchthatandDefinition:PositiveandNegativeCuts:Forapointtopointtransfer,acutisdefinedtobepositivewithrespecttothetransferiftheflowthroughthecutincreasesasthetransferincreasesItisnegativeiftheflowthroughthecutdecreasesasthetransferincreasesLemma:Inalosslesspowersystem,forapointtopointtransfer,thesumofthedistributionfactorsofthebranchesinanydirectedpositivecutseparatingthesourcebusformthesinkbusisequaltooneForanegativecut,thesumisequaltominusoneProof:Duetoconservationofactivepowerandsincethesystemislossless,alltheincreaseinpowerinjectionatthesourcemustbeextractedfromthesinkThus,theresultingchangeinthetotalflowinthelinesofanypositivecutseparatingthesourcefromthesinkmustbeequaltothetransferredamount:()Takingthederivativewithrespectto,wehavethat()whereisdefinedasthedistributionfactorofthepositivecutInthecaseofanegativecut,thetransferredamountis,whichyieldsforthesumofdistributionfactorsInthelossycase,thetransfermayincreaseordecreasethetotallossesSincelossesrepresentarathersmallpercentageofthetotalpowerinthesystem,theexpressioncanbeappliedtolossysystemsWewilluseLemmasandinordertointroducetheindividualbranchnecessaryconditionforcollapseWewillconsiderapointtopointtransfer,Wewillassumewithoutlossofgeneralitythatfortheinitialornormaloperatingconditionat,allactivepowerdistributionfactorsarepositive,eg,alinethathasanegativedistributionfactorcanberedefinedtobefrombustobussothatitsdistributionfactorispositiveTheorem:IndividualBranchNecessaryConditionforCollapse:LetusconsiderapointtopointtransferIfvariesuntilthepointofcollapseat,thenthereisavalue,with,suchthatnolinehasreacheditsSTSLforvaluesandatleastonelinehasreacheditsSTSLforvaluesInotherwords,atleastonelinemustreachitsSTSLbeforeoratthepointofcollapseProof:Toprovethistheorem,weneedtoprovetwocases:STLSinradiallinesandSTSLoflinesinmeshedtopologiesWehaveproventhefirstpartinLemmawhereitwasestablishedthatwhenthelinethatreachestheSTSLisradialandparticipatesinthetransfer,thenitsdistributionfactorisconstant,condition()holds,andiftheSTSLisreached,itdoessoatthepointofsystemcollapseItislefttoaddressthesecondcase:linesthatparticipateinthetransferandarepartofmeshedtopologies,wherecondition()willholdWedemonstratethisbycontradictionContradictionhypothesis(CH):NolineSTSLisreachedbeforeoratthepointofcollapseAftercondition(),thisisequivalenttostatingthatstartingwithalllinesparticipatinginthetransf

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Individual Branch and Path Necessary Conditions for Saddle-Node Bifurcation Voltage Collapse

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