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Equilibrium bidding in joint transmission and energy markets

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Equilibrium bidding in joint transmission and energy markets
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Babayğit, Cihan
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Deregulated electricity markets
Financial transmission rights
Nash equilibrium
FTR and energy settlement
Matrix game
Reinforcement learning
Dissertations, Academic -- Industrial Engineering -- Doctoral -- USF   ( lcsh )
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ABSTRACT: Participants in deregulated electric power markets compete for financial transmission rights (FTRs) to hedge against losses due to transmission congestion by submitting bids to the independent system operator (ISO). The ISO obtains an FTR allocation, that maximizes sales revenue while satisfying simultaneous feasibility. This FTR allocation remains in place for a length of time during which the participants compete in the energy market to maximize their total payoff from both FTR and energy markets. Energy markets (bi-lateral, day ahead, real time) continue until the the end of the current FTR period, at which time the participants can choose to modify their FTR holdings for the next FTR period. As in any noncooperative game, finding Nash equilibrium bidding strategies is of critical importance to the participants in both FTR and energy markets.In this research, a two-tier matrix game theoretic modeling approach is developed that can be used to obtain equilibrium bidding behavior of the participants in both FTR and energy markets considering the total payoff from FTR and energy. The matrix game model presents a significant deviation from the bilevel optimization approach commonly used to model FTR and energy allocation problems. A reinforcement learning (RL) algorithm is also developed which uses a simulation model and a value maximization approach to obtain the equilibrium bidding strategies in each market. The model and the RL based solution approach allow consideration of multi-dimensional bids (for both FTR and energy markets), network contingencies, varying demands, and many participants. The value iteration based RL algorithm obtains pure strategy Nash equilibrium for FTR and energy allocation.A sample network with three buses and four participants is considered for demonstrating the viability of the game theoretic model for FTR market. A PJM network example with five buses, five generators and three loads is also considered to analyze equilibrium bidding behavior in joint FTR and energy markets. Several numerical experiments on the sample networks are conducted using the approach of statistical design of experiments (DOE) to assess impacts of variations of bid and network parameters on the market outcomes like participant payoffs and equilibrium strategies.
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Dissertation (Ph.D.)--University of South Florida, 2007.
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EquilibriumBiddinginJointTransmissionandEnergyMarketsbyCihanBabayigitAdissertationsubmittedinpartialfulllmentoftherequirementsforthedegreeofDoctorofPhilosophyDepartmentofIndustrialandManagementSystemsEngineeringCollegeofEngineeringUniversityofSouthFloridaMajorProfessor:TapasK.Das,Ph.D.RalphFehr,Ph.D.,P.E.KandethodyRamachandran,Ph.D.AlexSavachkin,Ph.D.JoseZayas-Castro,Ph.D.DateofApproval:November8,2007Keywords:DeregulatedElectricityMarkets,FinancialTransmissionRights,NashEquilibrium,FTRandEnergySettlement,MatrixGame,ReinforcementLearningcCopyright2007,CihanBabayigit

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DEDICATIONTomynewbornsonLuisCan,mywifeandfamily

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ACKNOWLEDGEMENTSFirst,Iwouldliketothankmyadvisor,Dr.TapasK.Dasforhisguidanceandsupportthroughoutthecourseofthisdissertation.Thankyouforshowingnewwaysofapproachingandsolvingaproblem.IwishtothankDr.RalphFehr,Dr.KandethodyRamachandran,Dr.AlexSavachkin,andDr.JoseZayas-Castroforservinginmycommitteeandfortheirsuggestions;andDr.AydinSunolforchairingmydefense.IwouldliketothankIMSEdepartmentadministrationandstafortheirassistanceandtryingtoimprovetheworkenvironment.Lastbutbynomeansleast,Iwouldliketothankmyfamilyandfriendsfortheircondence,supportandcompanionship.Youwillcontinuetobepartofmystudies.

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TABLEOFCONTENTSLISTOFTABLESiiiLISTOFFIGURESivABSTRACTvCHAPTER1INTRODUCTION1CHAPTER2LITERATUREREVIEW62.1FTR:FinancialTransmissionRights72.1.1Point-to-PointFTRandFlowgateFTR72.1.2TypesofFTR:OptionsandObligations92.1.3SimultaneousFeasibilityandRevenueAdequacy102.2SettlementApproachesinFTRMarket112.2.1BiddingStrategies122.2.2FTRAuctions132.2.3OptimalPowerFlow142.2.4PerformanceMeasures162.2.5FTRMarketPower17CHAPTER3AMATRIXGAMEMODELFORSETTLEMENTOFFTRMARKET193.1AMatrixGameModelFormulationforFTRAllocation193.1.1ComputationofPayoMatrixElements213.1.1.1ISO'sFTRRevenueMaximizationModel213.1.1.2ExpectedFTRRevenueofaBidder233.1.1.3FTRUtilityofaBidder233.2SolutionofMatrixGameforEquilibriumFTRBiddingStrategy243.2.1AValueIterationAlgorithmforN-PlayerMatrixGames25CHAPTER4JOINTFTRANDENERGYBIDDINGMODEL284.1AMatrixGameModelforJointFTRandEnergyMarketSettle-ment294.2AMatrixGameModelFormulationforJointMarketSettlement314.2.1FTRAllocationModelFormulation31i

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4.2.2EnergyAllocationModelFormulation33CHAPTER5NUMERICALEXAMPLE:FTRMARKETSETTLEMENT395.1TheSampleNetwork395.2EquilibriumBiddingStrategiesforDierentNetworkScenarios405.3ImpactofBidParameterDiscretization435.4ImpactofBidParameterVariations445.5ImpactoftheNetworkParameterVariations48CHAPTER6NUMERICALEXAMPLE:JOINTFTRANDENERGYMAR-KETSETTLEMENTS516.1TheSampleNetwork516.2ImpactofFTRsinMarketSettlement526.3ImpactofContingencyandDemandScenarioVariability566.4ImpactofGeneratorCostFunctionVariations60CHAPTER7CONCLUSIONS65REFERENCES68ABOUTTHEAUTHOREndPageii

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LISTOFTABLESTable5.1NetworkandBidValues41Table5.2EquilibriumBiddingStrategiesforSixteenNetworkScenarios42Table5.3StrategieswithHigherPayosthanNashEquilibrium42Table5.4ImpactofBidParameterDiscretization44Table5.5ImpactofTypeMixParameter49Table5.6ANOVAwithBidder2'sPayos50Table5.7ANOVAwithBidder1'sPayos50Table6.1EquilibriumBiddingStrategyofGenerator1andBusLMPs53Table6.2BiddingParametersandFactorsofGenerators53Table6.3EquilibriumPayoswithoutFTRandwithFTRs55Table6.4EquilibriumPayosforContingency-DemandProbabilityMatri-ces57Table6.5ContingencyScenarios57Table6.6DemandScenarios58Table6.7FTRMarketDataforContingency-DemandVariations58Table6.8EquilibriumPayosforDierentCostFunctionofGenerator5and261Table6.9JointContingency-DemandProbabilityMatrix62Table6.10FTRMarketDataforCostFunctionVariationsVariations63iii

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LISTOFFIGURESFigure4.1FTRandEnergyMarketOperationCycle29Figure4.2MatrixGameModelSolutionStepsforJointMarket38Figure5.1FTRBiddersina3-BusPowerNetwork40Figure5.2PriceEectonBidder1'sAverageUtility45Figure5.3PriceEectonBidder2'sAverageUtility46Figure5.4QuantityEectonBidder1'sAverageUtility47Figure5.5QuantityEectonBidder2'sAverageUtility47Figure5.6StrategicImpactofQuantityParameter48Figure5.7TypeMixEectonBidder1'sAverageUtility48Figure5.8TypeMixEectonBidder2'sAverageUtility49Figure6.1PJM5-BusPowerNetwork52Figure6.2GeneratorFactorEectsforContingency-DemandVariations60Figure6.3LoadFactorEectsforContingency-DemandVariations61Figure6.4GeneratorFactorEectsforGenerationCostVariations64Figure6.5LoadFactorEectsforGenerationCostVariations64iv

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EQUILIBRIUMBIDDINGINJOINTTRANSMISSIONANDENERGYMARKETSCihanBabayigitABSTRACTParticipantsinderegulatedelectricpowermarketscompetefornancialtrans-missionrightsFTRstohedgeagainstlossesduetotransmissioncongestionbysubmittingbidstotheindependentsystemoperatorISO.TheISOobtainsanFTRallocation,thatmaximizessalesrevenuewhilesatisfyingsimultaneousfeasibility.ThisFTRallocationremainsinplaceforalengthoftimeduringwhichtheparticipantscompeteintheenergymarkettomaximizetheirtotalpayofrombothFTRanden-ergymarkets.Energymarketsbi-lateral,dayahead,realtimecontinueuntilthetheendofthecurrentFTRperiod,atwhichtimetheparticipantscanchoosetomodifytheirFTRholdingsforthenextFTRperiod.Asinanynoncooperativegame,nd-ingNashequilibriumbiddingstrategiesisofcriticalimportancetotheparticipantsinbothFTRandenergymarkets.Inthisresearch,atwo-tiermatrixgametheo-reticmodelingapproachisdevelopedthatcanbeusedtoobtainequilibriumbiddingbehavioroftheparticipantsinbothFTRandenergymarketsconsideringthetotalpayofromFTRandenergy.ThematrixgamemodelpresentsasignicantdeviationfromthebileveloptimizationapproachcommonlyusedtomodelFTRandenergyal-locationproblems.AreinforcementlearningRLalgorithmisalsodevelopedwhichv

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usesasimulationmodelandavaluemaximizationapproachtoobtaintheequilibriumbiddingstrategiesineachmarket.ThemodelandtheRLbasedsolutionapproachallowconsiderationofmulti-dimensionalbidsforbothFTRandenergymarkets,networkcontingencies,varyingdemands,andmanyparticipants.ThevalueiterationbasedRLalgorithmobtainspurestrategyNashequilibriumforFTRandenergyallocation.AsamplenetworkwiththreebusesandfourparticipantsisconsideredfordemonstratingtheviabilityofthegametheoreticmodelforFTRmarket.APJMnetworkexamplewithvebuses,vegeneratorsandthreeloadsisalsoconsideredtoanalyzeequilibriumbiddingbehaviorinjointFTRandenergymarkets.SeveralnumericalexperimentsonthesamplenetworksareconductedusingtheapproachofstatisticaldesignofexperimentsDOEtoassessimpactsofvariationsofbidandnetworkparametersonthemarketoutcomeslikeparticipantpayosandequilibriumstrategies.vi

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CHAPTER1INTRODUCTIONCapacitylimitationsinthetransmissiongridconstrainmovementofpoweracrossthegridandtherebyimposedierentiallocationalmarginalpricesLMPs.Thisphenomenon,whichmayexposemarketparticipantstovolatileenergypricesisde-scribedastransmissioncongestion.TransmissioncongestioncreatesadilemmaforthesystemoperatorISOastherevenuescollectedfromthecustomersretailersexceedthepaymentstogenerators/suppliers.ElectricitymarketsuseinstrumentslikenancialtransmissionrightsFTRstohedgethemarketparticipantsfromthevolatilityofthecongestioncharges.ThisisaccomplishedthrougharedistributionoftheexcessrevenueamongtheFTRholdersHogan[1].Hence,FTRisanancialriskinstrumentintendedtoosettransmissionusers'congestioncharges.AnFTRisrepresentedasMWamountbetweentwopointsinthetransmissionnetworkandisvalidoveradenedperiodoftime.ThedenitionofFTRalsodependsonhowthetransmissioncapacityinMWisspeciedandmeasured.FTRsdenedbetweenanytwoparticularbusesinthesystemareknownaspoint-to-pointnancialtransmissionrightPTP-FTR.HolderofPTP-FTRisentitledtobepaidifthedierenceinthelocationalmarginalpricesbetweenthespeciedpointsofwithdrawalandinjectionLMPispositive.AmuchlesscommonFTRtypeisreferredtoasowgatenancialtransmissionrightFGR-FTR,whichgrantstheholderacapacityreservationorschedulingpriorityforusingspecictransmissionlinksorowgates.FGR-FTRbetweenanytwonodescanbeobtainedbycombining1

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thecapacityreservationsFGRsofthelinesconnectingthenodes.Inthisresearch,onlyPTPnancialtransmissionrightsareconsidered.PTP-FTRscanbefurtherclassiedasobligationsoroptionsdependingonthenancialsettlementstrategy.AnFTRobligationisbi-directionalandcanhaveanegativeorapositivepricedierenceLMP.IncaseofanegativevalueofLMP,theholderoftheFTRmakesapaymentequaltotheproductoftheFTRquantitytimestheLMP.AnFTRoptionisuni-directionalandcanonlyhaveanonnegativevalue.Thatis,theholderofanFTRoptionisnotexpectedtomakeapaymenttotheISOiftheLMPisnegative,butwillbepaidbytheISOwhentheLMPispositive.Inthisresearch,aPTP-FTRbidderhasachoicetobidforanycombinationofobligationandoption.AfterthePTP-FTRbidsaresubmitted,twoimportantconditionsthataretakenintoaccountinallocatingtheavailableFTRsare:1.Simultaneousfeasibility2.RevenueadequacySimultaneousfeasibilityreferstotheconditionwheretheallocatedFTRsarewithinthecapabilityoftheexistingtransmissionsystem.Thatis,testedbycheckingifthepowerowsthatoccurduetotheallocatedFTRsfallwithintheconstraintsofthesystem.Revenueadequacyisaconditionwherebythenetpayments,collectedbythesystemoperatorthroughtheactualdispatchofenergy,shouldbegreaterorequaltothepaymentsmadetotheFTRholders.Ithasbeenshownthatrevenueadequacyfollowsfromsimultaneousfeasibility[1].ItisconsideredthattoacquireFTRonapath,marketparticipantssubmitstrate-gicbidstoISOcomprisedoffourparameters:obligationprice,optionprice,obligationquantity,andoptionquantity.FTRbiddersareassumedtosubmitdierentbidsfor2

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theFTRpaths.Basedonthebids,theFTRsareallocatedsuchthatISO'srevenuefromFTRsalesismaximizedwhilesatisfyingtheconstraintsofsimultaneousfeasi-bilitycondition.FTRbiddersattempttomaximizetheirexpectedutilityforholdinganFTR.TwodierentapproachesforobtainingequilibriumFTRbiddingstrategiesareexamined:1.ConsideringbiddingintheFTRmarketaloneassumingthattheestimatesofLMPsareavailable,and2.ConsideringbiddingintheFTRmarketinconjunctionwithbiddingintheenergymarkettoderivetheactualLMPs.ThesolutionofjointFTRandenergymarketsisnotfoundintheopenliterature.InaderegulatedelectricpowermarketwithmultiplebiddersforFTRs,itises-sentialtounderstandtheequilibriumbiddingbehaviorandtheresultingpayostotheparticipants.Itisalsoimperativetounderstandtheimpactofvariousnetworkparametervaluesontheequilibriumoutcomes.Acommonapproachadoptedintheelectricmarketliteraturetoobtainequilibriumbiddingbehaviorisabi-levelop-timizationmethod.ThisapproachhasalsobeenusedintheFTRmarket.Inthebi-levelmethod,theupperlevelproblemobtainstheequilibriumbidsofthepar-ticipants,whilethelowerlevelproblemndsthecorrespondingFTRallocationviaISO'srevenuemaximizationsubjecttotheSFTconstraints.Theupperlevelproblemattemptstondanequilibriumstrategybyrepeatedlyupdatingbidders'strategiesoneatatimewhileassumingstrategiesofotherbiddersxeduntilnofurtherchangeinthestrategiesispossible.Suchanapproachcanbefoundinarecentpaper[2],whichconsiderspriceastheonlyPTP-FTRbidparameter,andassumesthateachbiddersubmitsbidforasingleFTRpath.ItisalsoassumedinthepaperthattheLMPestimateisknowntoabidder,andthebidderutilityisafunctionoftherisk3

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coecientandthevarianceofLMPoftheFTRpath.VariousmethodsthathavebeenusedtoforecastLMPsarepricesimulationmethods[3],statisticalmethodslikearticialneuralnetworkmethod[3],[4],andtimeseriesmethod[5].Inthisresearch,amatrixgametheoreticapproachtoexaminingequilibriumFTRandenergybiddingbehavioroftheparticipantsinaderegulatedpowermarketispresented.ThebidsareconsideredtobeadiscretevaluedvectorofFTRpricesandquantities.Also,abidderisallowedtobidonanysubsetoftheavailableFTRpathsinanetwork.Thematrixgametheoreticmodelallowssimultaneoussolutionoftheequilibriumbiddingbehavioroftheparticipants.ArecentlydevelopedvalueiterationbasedreinforcementlearningRLapproachisusedinsolvingforequilibriumFTRbidingstrategies[6].AsampleFTRnetwork,thatwasstudiedin[2],isusedinnumericallydemonstratingthematrixgametheoreticmodelingapproachforFTRallocationunderLMPassumptions.APJM-5busexampleisusedtostudythejointFTRandenergyallocationprocess.Experimentsareconductedtodeterminetheimpactsofthebidandnetworkparametersthroughsinglefactoranalysisandmulti-factoranalysisusingfactorialexperimentaldesignandconsequentanalysisofvariance.Thisdissertationisorganizedasfollows.Chapter2providestheliteratureaboutthetransmissionrightsinderegulatedelectricmarket.Chapter3presentstheformu-lationstomodeltheFTRallocationproblemasamatrixgameandvalueiterationbasedRLalgorithmtosolvetheequilibriumbiddingstrategies.Chapter4presentstheformulationstointegratetheFTRandenergymarketsandndtheequilibriumstrategieswiththisjointmodel.Chapter5presentsthenumericalexperimentsandtheirresultswhenFTRallocationsaredoneusingestimationofLMPs.Chapter6providesthenumericalexperimentswithresultstorevealthesignicantfactors4

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whentheFTRandenergymarketsaresettledjointly.ConclusionsaresummarizedinChapter7.5

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CHAPTER2LITERATUREREVIEWThemovetowardsaderegulatedelectricpowerindustryhasraisedtheawarenessofthecriticalimpactoftransmissioncongestiononpowernetworks.Limitationsinthetransmissiongridconstrainlong-distancemovementofpower,whichresultsinhigherpricesincertainlocationsofthenetwork.Thisphenomenonistermedastransmissioncongestion,andthedierenceinthelocationalmarginalpricesLMPsbetweenanytwobussesiscalledcongestioncosttransmissionchargetotheparticipantsatthosebusses.Electricitymarketsusevarioustransmissionrightmechanismstohedgethemarketparticipantsfromthevolatilityofthetransmissioncharges.Transmissionrightsallowtheirholderstoderivebenetsfromtheuseofthetransmissioncapacityasfollows:1.Financialbenetsresultingfromtheuseofthecapacity2.Righttousethetransmissioncapacity.Hence,thetransmissionrightscouldbenancialand/orphysicalinnature.Apurelynancialapproach,knownasnancialtransmissionrightsFTR,providemarkettradersandothermarketparticipantsaninstrumentforconstructingnancialhedges.TheFederalEnergyRegulatoryCommissionFERC,inanoticeofpublicrulingNOPRin2002,proposedlocation-basedmarginalpricingtogetherwithFTRsasthemechanismtobuildecientenergymarkets[7].Theaboveruling,accordingtoHogan[8],setstherightincentivestothemarketparticipants.LMP-basedFTR6

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marketshavebeeninoperationinNewYorkandPJMforafewyears.NewEnglandhasrecentlyadoptedthis,whiletheMidwestandCaliforniamarketsarescheduledtoimplementthisstructuresoon[9].TheabovemarketstogetherrepresentasignicantportionofUnitedState'selectricitymarket.2.1FTR:FinancialTransmissionRightsTransmissioncongestioncreatesadilemmaforthesystemoperatorastherevenuescollectedfromthecustomersexceedthepaymentstogenerators.Theadditionalrevenueisreferredtoasthecongestionrevenue.Hogan[10]suggeststhataconvenientsolutiontothisdilemmawouldbetore-distributethecongestionrevenuethroughasystemoflong-runnancialtransmissionrightsFTRs.FTRisdenedasanancialriskinstrumentintendedtoosetthecongestionchargesincurredinatransmissionnetwork.FTRservestonotonlyprotectamarketparticipantfromthelosseslinkedwithcongestionbutalsoasameansofgeneratingrevenueinaderegulatedmarket,inawaysimilartothestocksinthenancialsector.ItisalsoreferredtointheliteratureastransmissioncongestioncontractTCCorcongestionrevenuerightCRR.VarioustypesofexistingFTRcontractsarediscussedbelow.2.1.1Point-to-PointFTRandFlowgateFTRThedenitionofFTRfurtherdependsonhowthetransmissioncapacityisspec-ied.FTRbetweenanytwoparticularbusesinasystemisknownaspoint-to-pointPTPnancialtransmissionright,whichisalsocalledrmtransmissionrightorjustFTR.OwnerofPTPFTRisentitledtobepaidifthedierenceinthelocationalpricesbetweenthespeciedpointofwithdrawalandinjectionispositive.TheotherlessfrequentlyusedFTRtypeistheowgatereferredtoasFGR.Inaconstraineddispatch,aowgatehasashadowprice,whichdenestheowgate'smarketclear-7

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ingprice.FlowgateFTRisacontracttocollecttheshadowpricefromtherealizeddispatchforaspeciedquantityoftheconstraint.ThisapproachcreatestheideaofsellingthelinelimitsorresourcesbesidestheelectricowonthelinesTrHoganF02.Generatorsandloadsareinterestedinbeingabletotransferpowerbetweentwospeciclocationsinthenetwork.O'neillet.al[11]statethatPTPFTRsarewellsuitedforhedgingcongestioncostforsuchcasesinthelongrun.However,existenceofalargenumberofpossiblePTPcombinationsmakesitdiculttochooseinthedynamicenvironmentofelectrictrade.Asaresult,resellersofPTPcontractsfaceathinmarket.Furthermore,anychangeincongurationofPTPrightsrequiresimulta-neousfeasibilitywhichhastobesolvedbytheregionaltransmissionoperatorRTOwithallotherrights.Thiscentralizedoptimizationprocedurelimitsthedevelopmentofo-RTOsecondarymarketsforPTPrights[12].Adamson[13]presentsanewmethodoftradingPTPtransmissionrightswhichisclaimedtodecreasethelevelofcentraloptimization.Thismethodisbasedontheprinciplesofunequalexchangeratesbetweendierentpoint-to-pointtransmissionrights.ThoughFERChasrecentlyendorsedthemeritofhavingowgatetransmissionrights,thereisalotofdebatesurroundingthesubject.Chaoetal.[14]proposeow-basedsystemsasapotentiallyecientmethodoftradingrightsthatdoesnotrequirecentralizedoptimization.Moreover,electricitytradersoftenarguethatsincetherewillbefewercongestedowgatescomparedtonumberofnodecombinationsinPTPrights,ow-basedrightswillyieldamoreliquidforwardmarketforenergyandtransmission.However,thenumberoftransmissionrightsthatmustbedenedtoaccountforalloftheactualorpossibleconstraintsonanactualnetworkmaybelargeandimpractical[15].Andrew[16]suggeststhatFGRsystemswhenimplementedinlargegridsmaynotcaptureallthecongestioncostsinthesystem,forcingtheoperatortoprovidecontinuousupdateofthenewcommerciallysignicantowgates.8

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2.1.2TypesofFTR:OptionsandObligationsFinancialtransmissionrightscanbedenedasobligationsoroptionsdependingonthenancialtreatmenttotheholderoftheright.Obligation:AnFTRobligationisbi-directionalandcanhaveanegativevalueorapositivevalue.IncaseofanegativevaluetheholderoftheFTRmakesapaymentequaltotheproductoftheFTRquantitytimesthepricedierence.Asimilarpay-mentisreceivedbytheholder,fromtheISO,iftheFTRhasapositivevalue.ThistypeofFTRiscommonlyusedduetoitseaseofimplementationthoughitisnotaspracticalfromaneconomicstandpoint.Option:AnFTRoptionisuni-directionalandcanonlyhaveapositivevalue.Thatis,theholderoftheFTRisnotexpectedtomakeapaymenttotheISOiftheFTRisnegativebutwillalwaysbepaidbytheISOwhentheFTRispositive.ItisnaturalthatthistypeofFTRwouldbemoreappealingtomarketparticipantshoweveranFTRoptiontendstocostmoreatauctionthanitsequivalentobligation.Moreover,practicalimplementationofthistypeofFTRisacomplextask.Forexample,inapowernetworkof3000buses,therewouldbeupwardsof100,000constraints.SolvingforanFTRoptionwouldrequireevaluationofeachoftheseindividualconstraints,ataskwhichisdaunting.AnFTRauctionforbothpoint-to-pointobligationsandoptionsissimilartoaneconomicdispatchproblem[1].ButinFTRauctionforoptions,evaluatingacontingencyandconstraintconditionrequiressolvinganunconstrainedoptimalpowerowfortheworst-caseimpact.Thisincreasesthecomplexityofthemodelandhencesuccesswiththisauctionmodelhasnotyetbeendemonstrated.InthecaseofNYISOworkingwithanACAlternatingCurrentpowernetworkfurthercomplicatesitseortsatimplementingFTRoptions.Ontheotherhand,anFTRauctionforowgate9

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obligationsoroptionsismorecomplicatedanditisnotlikeaneconomicdispatchproblem.Inthismodel,therecouldbealargenumberofowgatesintherealgridthatgreatlycomplicatesanyconstructionofhedges.Alsotherequiredowgateamountstohedgeanytransactionchangefrequentlywithchangingdispatchrestrictions.Ahybridmodelwithpoint-to-pointandowgateobligationsandoptionscanproduceacomputationalchallengeanditisnotclearwhetherthisauctionmodelcouldbesolvedforarealisticgrid.O'neilletal.[11]proposeanauction-basedprocessthatallowsthemarketparticipantstoacquireandrecongurethenancialtransmissionrights.Thepapershowsthatbyallowingowgateandpoint-to-pointobligationsandoptionstobereconguredandexchanged,themarketcandecidewhatcombinationofnancialrightsaremostusefultothemarketparticipants.2.1.3SimultaneousFeasibilityandRevenueAdequacyTwoimportantaspects,whichareconsideredtoassessviabilityoftheFTRsaresimultaneousfeasibilityandrevenueadequacy.Inelectricpowernetworks,conditionssuchasthermallimits,powerlimits,generatingcapacity,anddemandvarywithtime.Asaresult,FTRallocationmayneedtobevariedtomaintainelectricalandeconomicconstraints.Theseconstraintsareknownassimultaneousfeasibilityandrevenueadequacy.SimultaneousfeasibilityreferstotestingwhethertheallocatedFTRsarewithinthecapabilityoftheexistingtransmissionsystem.Thatis,thepowerowsthatoccurduetotheallocatedFTRmustfallwithintheconstraintsofthesystem.Revenueadequacyisaconditionwherebythenetpayments,collectedbythesystemoperatorthroughtheactualdispatchofenergy,shouldbegreaterthanthepaymentstotheFTRholders.TheISOconstantlychecksfortheviabilityoftheseallocatedFTRsbyperformingasimultaneousfeasibilitytestSFT.10

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TheSFTisdonebymodelingtheFTRsasgenerationatpointofinjectionsourcepointandloadatpointofwithdrawalsinkpoint.AnACpower-owanalysisiscarriedouttoevaluateifthesystemwillremainwithinallpermissiblelimits,includingsinglecontingencysituationslikethelossofalineetc..Thispowerowanalysisemploysanoptimizationprogramthatisusedtosimulatetheworkingoftheactualpowernetwork.Thisprogramiscalledtheeconomicdispatchprogramoroptimalpowerow.Oncetheparametersofthepowernetworkhavebeenmodeledintotheprogram,itsimulatestheactualoperationofthenetworkundervaryingconditionsandparameters.IftheoutstandingFTRviolateanyofthesystemlimitsthentheyareconsideredtobeunviable.ThiswillusuallyresultintheISOrunningtheallocationprogramagaintoobtainare-allocationorintheworstcaseaskingthemarketparticipantstobidagain.TheallocatedFTRmustalsoensurethattheexcessrevenuecongestionchargescollectedbytheISOisadequatetocoverthepaymentsrequiredundertheFTRs.Fortunatelyhowever,bothoftheaboveconditionsneednotbeindividuallyveried.Ithasbeenproventhatrevenueadequacyfollowsfromsimultaneouslyfeasibility[1].2.2SettlementApproachesinFTRMarketIdeally,anoptimalsecurityconstraineddispatchisthedispatchthatgivessimilarresultsliketheunconstraineddispatchwithequalLMPsatallsystembuses.However,constrainedtransmissionconsiderablyimpactsenergypricesasindicatedbylargeuctuationsinLMPs.ThisforcesparticipantstoplaystrategicallywiththemarkettoolslikeauctionsandFTRs.Intheend,thesystemsettlesdownbysolvingasocial-welfareproblem.In[2],LiandShahidehpourformulateFTRauctionproblemasalinearprogramwiththeobjectiveofmaximizingtherevenuecollectedfromtheFTR11

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auctionmarket.AnFTRbidder'sobjectiveistomaximizeitsexpectedutilityforholdinganFTR.2.2.1BiddingStrategiesTheauctionisthecentralmechanismofanFTRmarket.TobuyorsellFTR,marketparticipantssubmitquantity,costinformationandpointsofinjectionanddeliveryintheformofbids,totheISO.TheISObeingtheneutralpartyarbiterrunsanFTRauctiontoallocatetheFTRs.ThebidsareallocatedsuchthattheymaximizetherevenuefromFTRsaleswhilesatisfyingthesimultaneousfeasibilitycondition.Asaresult,user'sbiddingbehaviorsbecomesignicant.BiddersmaketheirdecisionsbasedonanticipatedsystemoperatingconditionswhileholdingFTRs.Specically,theyneedtoestimateLMPdierencesbetweensinkandsourcepointsonacertainFTRpathandidentifypotentialcompetitorsandtheircorrespondingbiddinginformation[2].Themethodsthatforecasttheelectricitypriceincludepricesimulationmethods[3],statisticalmethodsuchasarticialneuralnetwork[3],[4],andtimeseries[5].EachFTRbidder'sobjectiveistomaximizeitsexpectedutilityforholdinganFTR.Understandingthisbehaviorisaworkinprogress.TheproblemistodeterminethecompetitiveequilibriumbiddingstrategiesinanFTRauctionbyconsideringtheequilibriumprotsrealizedintheenergymarketastheperformancemeasure.AsoundbiddingstrategyisneededforpurchasingFTRs,forthefollowingreasons.Ifthebidpriceistoolow,sucientFTRsmaynotbeallocatedresultinginthesupplierpayinghighcongestioncosts.Ontheotherhand,biddingtoohighwilllikelywinownershipoftheauctionedFTR,butmaymeanlossofprot.MarketparticipantsintheireortstobidforFTRsshouldtakeintoaccounttwoimportantfactors.Oneofthesefactorsistheestimatedpricedierentialbetweenthesupplynodeandtheloadnode,forwhichanaccuratepredictionoffuturelocationalprices12

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isimportant.Theotherfactoristheanticipatedtotalsupplyquantityasitideallyrepresentstheamountofpowerthatneedstobehedgedagainstthecongestioncosts.LiandShahidehpour[2]modeltheFTRbiddingproblemasabileveloptimizationwiththeuppersubproblemrepresentingbiddersandthelowersubproblemrepresent-ingthesolutiontotheISO'sFTRmarketclearingproblem.TheirresultspresentedbiddingdierencesbetweenFTRobligationsandoptions.Furthermore,theresultsshowedthatforecastinganaccurateLMPdierencesandproperriskpreferenceswerethecriticalpointsinFTRbiddingandbidders'payos.2.2.2FTRAuctionsAnauctionisdenedasthemethodofallocatinggoodsundercompetition.Auc-tionsareessentiallypricingmechanisms.Increasingly,inaderegulatedpowermarketenvironment,marketparticipantswhomayormaynothavetheabilitytoactuallyschedulepowersupplyinthenetwork,areattemptingtobuyandsellFTRinordertoprotfromtherevenuesandpaymentswarrantedundertheFTRsystem.Asthenumberofparticipantsincreases,thecompetitiontobuyorsellFTRsalsoincreases.ThisrisingtrendincompetitionhasensuredthatatanytimetherearenumerousparticipantsvyingforthesameFTR.Suchasituationfosterstheneedtoutilizewell-establishedmethodsofsellingorbuyingFTRtomultiplemarketparticipants,whileincreasingtherevenuegeneratedfortheISO.Worldwide,avarietyofauctionstrate-giesareutilizedinmanymarketscenariostocarryouttradingonsimilargrounds.Ideally,theFTRauctionmechanismshouldincreasepricecertaintyoftheenergymarketandimprovemarketeciencyinthelongrun.Thechoiceofanauctionmechanismaectsthepricesinthemarket.Itishenceworthwhiletoinvestigatethevariousauctionrulesandstrategies.13

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Auctionsusedinelectricitymarketsarecalledmulti-unitauctionssincemorethanoneunitofthesametypeisauctioned.Twoformsofmulti-unitelectricityauctionsarecommonlyusedinpresentdayrestructuredelectricitymarkets:uniformpriceauctionanddiscriminatoryauction.Theuniformpriceauctionisfurtherdierentiatedasrstpriceuniformauctionandsecondpriceuniformauction.AreviewoftheliteraturesuggeststhatdierentauctionruleshavenotbeencomparedfortheireectivenessintradingFTR.TheinitialandmostimportantrequirementtotestvariousauctionrulesonthesellingandbuyingofFTRisacomputationalframeworkabletoevaluatetheeectsofdierentauctionstrategiesonFTRallocation.Thatis,ameansofallocatingtheFTRisrequiredinwhichanauctionframeworkcanbeembedded.CommonlypracticedmethodsofFTRallocationuseDCDirectCurrentapproximationstoanACnetworkscenariotodetermineloadowsandguaranteesimultaneousfeasibilityandrevenueadequacy.Inrecentyears,systemoperatorshavebegunusingnon-linearoptimizationalgorithmsthatmodelanACnetworkwithoutanyapproximations.TheoptimalpowerowOPFisapopularalgorithmthatisutilizedforthispurpose.2.2.3OptimalPowerFlowOPFisagenerictermthatdescribesabroadclassofproblemswhichtrytooptimizeaspecicobjectivefunctionwhilesatisfyingconstraintsdictatedbytheop-erationalandphysicalcharacteristicsoftheelectricnetwork[17].TheOPFalgorithmisconcernedwiththephysicalallocationofgenerationcapacitiesandtheobjectivefunctionisgenerallytominimizecostofsupplyingpower.ItisthereforenecessarytotransformitsstructuretomakeitsuitableforFTRallocationassuggestedbyHogan[1].14

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OPFisinvolvedinthesolutionofalarge-scalenon-linearmathematicalpro-grammingproblemandithastakenmathematiciansandscientistsmanydecadesofresearchtodevelopecientalgorithmstosolveit.TheuniquefeatureofOPFisthatthecostofoperatingthenetworkcanbeminimizedwhilemaintainingthefunc-tionalconstraints.Signicantprogresshasbeenachievedinthisarea[18],[19],[20].ManyoptimizationtechniqueshavebeenemployedtosolvetheOPFproblem.Thetechniquesareclassiedasfollows[17]:1.Nonlinearprogramming2.Quadraticprogramming3.Newtonbasedsolutionsofoptimalityconditions4.Linearprogramming5.HybridversionsofLPandIP6.InteriorpointmethodsThederegulatedpowermarkethasincreasedtheneedforfastreliableandaccurateOPFscapableofsimulatingtherealtimespotmarket.Variationsoccurringinrealtimehavetobemodeledwithgreateraccuracy.RecentyearshaveseenthefocusshiftingtoadaptingtheformulationofOPFtoworkinaderegulatedpowermarketenvironment[21],[22],[23],[24],[25].IncreaseincomputationalcapabilitiesandimprovedmathematicalalgorithmshaveensuredthatOPFisrobustenoughtobeemployedinaderegulatedenvironment.TheriseintheuseoftheWorldWideWebduetoincreaseinbandwidthanddatatransfercapabilities[22]broughttheoptimalpowerowtotheforefrontofrealtime,onlinemultipleparticipantinteractionsimulationsofthederegulatedmarket.In[21],YongandLassatertakeintoaccount15

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thepresenceofmultiplemarketparticipantswhohavetheabilitytoselecttheirownenergysupplier.TheformulationforOPFcanbecategorizedbasedonthetypeofpowernetwork:alternatingcurrentoptimalpowerowAC-OPFanddirectcurrentoptimalpowerowDC-OPF.Intheliterature,DC-OPFiswidelyusedtoformulatetheOPFprobleminACnetworks,sinceDC-OPFisfastercomparedtotheAC-OPF.TheDC-OPFformulationisobtainedbyignoringreactivepowerbalanceequations,linelossesandtapdependenceintransformerreactanceandassumingthatallvoltagemagnitudesareidenticallyoneperunit.TheDC-OPFhenceconvertsthepowerowproblemtoalinearproblemandsolvesalinearsetofequations.ThedisadvantageofDC-OPFisthatallnon-linearsystemparametersareconvertedtolinearformtherebycompromisingontheabilityoftheoptimizationprogramtoaccuratelymodelthesystem.TheformulationforAC-OPFiscomplexandnonlinearinnature.Thiscomplexitythough,isosetbythebenetsthatitoers.ThemajorbenetofAC-OPFisthatitinternalizeslosses,i.e.,duringtheeconomicdispatchprocessthesupplygeneratorsaresetatahigherleveltocompensateforboththeactualloadandthelossesthatoccurfromthedispatchforthatload.TheLMPsthatresultfromthisdispatchwillreectthecostofgenerationtocompensatefortheselosses.ItalsomodelsreactivepowerandvoltageconstraintsinthesystemmakingtheAC-OPFdispatchamuchmoreaccurateandrealisticrepresentationofthepowersystem.AsaresulttheAC-OPFprovidesuswithacomprehensiveframeworkfortheFTRauctionprocess.2.2.4PerformanceMeasuresElectricdispatchproblemcanhavedierentsolutions.Whichdispatchalternativeisoptimaldependsontheperformancemeasures.In[26],Alomoushintroducessome16

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performanceindicestoquantifyseverityofcongestion,degreeofsystemutilizationanduniformityofenergyprices.Totalcongestioncharge,totalsystemgeneration,indexoftotalgenerationcharge,indexoflocationalmarginalprices,averagelocationalmarginalpriceofgeneration/loadandsystemutilizationaretheindicesthattheauthorproposes.Basedontheseindicesdierentdispatchscenarioscanbecomparedandoptimaldispatchcanbedetermined.Someindiceswillbemoreimportantthantheothersdependingontheperformancepriorities.Powersystemstabilityisanotherperformancemeasurethathastobetakenintoconsiderationwhilechoosingtheappropriatedispatchscenario.Somedispatchesmayyieldmorepreferredoutcomes;however,itmaycreatealesssecurepowersystem.Powersystemswithveryclosetothevoltagecollapsepoint,whichcanbemeasuredbyanappropriatesteady-statevoltagestabilityindicator[27],[28],shouldbeavoided.2.2.5FTRMarketPowerInFTRmarkets,itisimportanttomonitorsituationsinwhichholderscanin-creasetheirFTRpayoutsbyincreasingtherelevantcongestionintentionally.Becauseofparticularcharacteristicsofpowermarkets,itispossibletoexercisemarketpowerasinnootherkindofmarket.In[29],[30],[31],strategicbehaviorofFTRholdershasbeenanalyzedinatwo-nodenetwork.IthasbeenshownthatifageneratorintheimportingnodeholdsanFTR,itincreasesitspowermarket;however,iftheFTRisheldbyageneratorintheexportingnode,ithasnoeectinthemarketpowerofthegenerator.Oren[32]arguesthatcentralizedimplementationofFTRswillresultininecientdispatchandmarketpowerforthegenerators.Stoft[29]providesacounter-argumenttoOren'sconclusionandprovesthatundercertainex-cesscapacity"conditions,nancialtransmissionrightscurbmarketpower.Joskowetal.[33],[34]studytheimpactofallocationofnancialandphysicaltransmissionrights17

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onmarketpower.[33],[34],[35]concludethattheeectofrightsdependuponnu-merousfactorsincludingthecongurationoftheunderlyingmarketpowerproblemslocationofbuyerandsellerandthemicrostructureofthemarketfortransmissionrights.In[36],BautistaandQuintanaproposeamethodbasedonrelativehedgingpositionratiostoscreenanddiscriminateFTRswithmarketpowerpotential.18

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CHAPTER3AMATRIXGAMEMODELFORSETTLEMENTOFFTRMARKET3.1AMatrixGameModelFormulationforFTRAllocationLetI=f1;2;;IgdenotethesetofpathsofsourceandsinklocationsforwhichFTRscanbeobtained.Also,letN=f1;2;;NgdenotethesetofparticipantsbiddingfortheavailableFTRs.Abiddern2NisconsideredtobidonasubsetofpathsInIwitha4-dimensionalbidvectorforeachpathconsistingofpriceasafractionoftheLMPestimateforbothobligationandoption,quantityasafractionofthemaximumowlimits,andFTRtypemixfactorindicatingproportionofthequantitythatiscategorizedasobligation,andtherestofthequantityasoption.Abidvectorforbiddernforpathi2Incanbedenotedasani=kn;obiLMPni;kn;opiLMPni;lniQi;mni;19

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wherekn;obinthbidder'sobligationpricebidfactorvaluedbetween0and1,kn;opinthbidder'soptionpricebidfactorvaluedbetween0and1,LMPninthbidder'sforecastofcumulativeLMPdierencebetweensinkandsourcebusesofpathioverallFTRholdingperiodsPi.e.,LMPni=PPp=1LMPni;p.lninthbidder'squantitybidfactorvaluedbetween0and1,Qiapproximatemaximumowofelectricityonpathi,andmninthbidder'sbidforFTRtypemixfactorvaluedbetween0and1onpathi,where:0indicatesallFTRquantitybidasoptions,and0.5indicatesequaldivisionofFTRquantitybetweenoptionsandobligations.Thebidvectorofparticipantncanbegivenasan=ani;8i2In.Thenthecompletebidvectorofalltheplayerscanbedenotedasa=a1;a2;;aN.SincejInjdenotesthenumberofpathsonwhichbidsaresubmittedbybiddern,thedimensionofthebidvectorforthebiddernis4jInj.SincetheNbiddersareincompetitiontomaximizetheirFTRbenets,thenon-cooperativebiddingprocesscanbemodeledasanN-playermatrixgame,ifthecontinuousbidvectorelementsaresuitablydiscretized.Ahigherlevelofdiscretizationgivesbetterapproximationtotherealscenario,thoughatahighercostofcomputation.If,forexample,thebidvectorelementskn;obi,kn;opi,lni,andmniarediscretizedatelevenlevelseachstartingat0with0.1incrementsupto1,thentheactionspaceofbidderncanbegivenas1111111jInj.Hence,thematrixgameconsistsofNpayomatriceseachofsize1111111jI1j1111111jI2j1111111jINj.20

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Formulationofamatrixgamewouldrequirecomputationofthepayomatrixelements.TheseelementsrepresentutilitiesofthebiddersobtainedfromadetailedconsiderationofvarianceandriskassociatedwithLMPestimates,networkcon-straintsandcontingencies,andISO'ssettlement.Inwhatfollows,thedetailsforcomputingtheelementsofthepayomatricesareprovided.3.1.1ComputationofPayoMatrixElementsForanygivenbidcombination,theISO'srevenuemaximizationmodelissolvedrsttodeterminetheFTRallocationsforthebiddersforeachpathincludedintheirbids.TheFTRquantitiesarethenusedtocalculaterevenuesfortheplayers,whicharethenconvertedtoutilityvaluesconsideringthevariabilitiesintheLMPestimatesoftheplayersandtheirriskcoecients.3.1.1.1ISO'sFTRRevenueMaximizationModelAdcmodelisadoptedforFTRallocation.Adaptationofthedcmodelissolelyforsimplicationofthecomputationalneedsofthisresearch.Atrueacmodelcanbesubstitutedforreallifeimplementationsrequiringhighercomputingpower.TheoptimizationmodelmaximizesISO'srevenuefromFTRallocationswhileconsideringsimultaneousfeasibilityforthenetwork.Themodelcanbegivenasfollows.maxNXn=1Xi2Inn;obiFTRn;obi+n;opiFTRn;opi.1s.t.NXn=1Xi2In[Dn;ci;lFTRn;obi+max;Dn;ci;lFTRn;opi]Bcl8l;c.221

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NXn=1Xi2In[)]TJ/F20 11.955 Tf 9.298 0 Td[(Dn;ci;lFTRn;obi+max;)]TJ/F20 11.955 Tf 9.299 0 Td[(Dn;ci;lFTRn;opi]Bcl8l;c.3FTRn;obimniQni8n;i.4FTRn;opi)]TJ/F20 11.955 Tf 11.955 0 Td[(mniQni8n;i.5whereFTRn;obiquantityofobligationFTRallocatedtonthbidderonpathidecisionvariableFTRn;opiquantityofoptionFTRallocatedtonthbidderonpathidecisionvariablen;obiobligationbidpriceofnthbidderonpathin;opioptionbidpriceofnthbidderonpathiDn;ci;lPTDFofthenthbidder'sithpathonlinelundercontin-gencycBclcapacitylimitoflinelundercontingencycQniupperbiddingquantityofbiddernforpathii.e.,QilniTherelationshipbetweenatransactionpowerinjectionatonebustobewith-drawnatanotherbusandhowmuchofthattransmissionowonalineiscalledthepowertransferdistributionfactorPTDF.PTDFscanbeusedinISO'sFTRsettlementmodeltocheckthelinelimits.Thatis,linecapacitiesbecomeresourcesinFTRsettlementmodel.Whentheshadowpricesareaddedoveralllines,marginalclearingpriceforanFTRisfound.22

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3.1.1.2ExpectedFTRRevenueofaBidderBasedonISO'sFTRallocation,theexpectedrevenueforbiddern,Rn,canbeobtainedasfollows:Rn=Xi2In[LMPniFTRn;obi+maxLMPni;0FTRn;opi)]TJ/F15 11.955 Tf -220.838 -31.394 Td[(MCPn;obiFTRn;obi+MCPn;opiFTRn;opi].6whereRnnthbidder'sexpectedrevenueMCPn;obimarketclearingpriceforobligationFTRofbidderninpathiMCPn;opimarketclearingpriceforoptionFTRofbidderninpathiRnshowstheexpectedrevenue$overallpathsthattheparticipantnhassubmittedbidsforThisisapeculiarsituationsincetheparticipantcanplaywiththebidsonthepathsthatheisbidding.Atthesametime,participantcompetesbothwithhisrivalsandhimselftomaximizethetotalFTRrevenue.DiscriminatorypriceauctionisusedfortheISO'sFTRsettlementmodel.Asaresult,MCPn;obiandMCPn;opiaresimplybiddern'sobligationandoptionpricebidrespectivelyforpathi.Thatis,marginalclearingpriceforanFTRpathisdeterminedbythewinningbidder.3.1.1.3FTRUtilityofaBidderRevenuesforthebiddersarecalculatedattheenergymarketwiththeactualLMPs.However,sincethenodalenergypricesarevolatile,ameanvalueandavarianceareconsideredbythebiddersforeachLMPestimate.TohedgeagainstthevariabilityofLMPs,thebiddersconsiderariskfactorincomputationoftheir23

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actualpayoutilityasfollows[2]:Un=Rn)]TJ/F20 11.955 Tf 13.948 0 Td[(rnvarRn;.7whereUnutilityofbiddern,rnriskcoecientofbiddern,andvarRnvarianceofbiddern'srevenueestimate.Thelevelofriskdependsonthebidderbehaviorneutral,risk-averse,orrisk-taker,whichiscapturedbythesignandmagnitudeofrvalue.Biddersarechosentoberiskaverseinthisstudy.ThevariancecanbeobtainedfromthecovarianceestimatesoftheLMPsasshownin[2].Inthefollowingsection,analgorithmispresentedthatcanbeusedtoobtainaNashequilibriumFTRbiddingstrategyforthebidders.3.2SolutionofMatrixGameforEquilibriumFTRBiddingStrategyInthissectionarecentlydevelopedapproachisdiscussedtoobtainNashequilib-riumofN-playermatrixgames[6].LetVnadenotethepayomatrixofthenthplayerofwhichrna1;;aNarethematrixelements.DenethevalueofanactionantoplayernasVnan=Xfa1;;aNnangpan;a)]TJ/F21 7.97 Tf 6.587 0 Td[(nrna1;;an;;aN;.8where:pan;a)]TJ/F21 7.97 Tf 6.587 0 Td[(ndenotestheprobabilityofchoiceofanactioncombinationa)]TJ/F21 7.97 Tf 6.587 0 Td[(nbyalltheotherplayerswhileplayernchoseactionan.IndecisionmakingproblemswithasingleplayerthosearemodeledasMDPsandSMDPs,thereexistoptimalvaluesforeachstate-actionpair,andthehighestvaluedeterminestheoptimalactionineach24

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state[37].DrawingananalogyfromMDPs,formatrixgamesthathavemultipleplayersandasinglestate,itisconjecturedthatthereexistoptimalvaluesoverallactionsoftheplayersthatcanyieldpureNEstrategies.However,theprobabilitiespan;a)]TJ/F21 7.97 Tf 6.587 0 Td[(nneededtocomputethesevaluesareimpossibletoobtainforreallifeprob-lemswithoutpriorknowledgeofbidders'behavior.Therefore,alearningapproachisemployedtoestimatethevaluesoftheactionsasfollows..8isrewrittenasVnt+1an=)]TJ/F20 11.955 Tf 11.955 0 Td[(t[Vntan]+trna1;;an;;aN;.9where:tdenotestheiterationcount.Thealgorithmpresentedbelowutilizesthevaluelearningscheme.9toderivepureNEstrategiesforN-playermatrixgames.3.2.1AValueIterationAlgorithmforN-PlayerMatrixGamesItisassumedthatthegamehasN-playersandeachplayernhasasetofAnpossibleactionstochoosefrom.Hence,NdierentrewardmatricesofsizejA1jjA2jjANjareavailable.TheAlgorithm:1.Eliminaterowsandcolumnsofthematricesassociatedwiththedominatedactions.Adominatedactionisonethatwillneverbeadoptedbyarationalplayerirrespectiveofthechoicesofotherplayers.Anactionan2Anforplayernissaidtobedominatedifrnan;a)]TJ/F21 7.97 Tf 6.587 0 Td[(nrn^an;a)]TJ/F21 7.97 Tf 6.587 0 Td[(n,where:^an2Annananda)]TJ/F21 7.97 Tf 6.587 0 Td[(ndenotestheactionsofallotherplayers.2.Letiterationcountt=0.InitializethevaluesforallactionsoftheplayerVnantozero.Alsoinitializethelearningparameter0,explorationparameter25

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0,andparameters,neededtoobtainsuitabledecayratesoflearningandexploration,respectively.LetMaxstepsdenotethemaximumiterationcount.3.IftMaxsteps,continuelearningofthevaluesthroughthefollowingsteps.aActionSelection:GreedyactionselectionforpurestrategyNashequilibrium:Eachplayern,withprobability)]TJ/F20 11.955 Tf 12.567 0 Td[(t,choosesagreedyaction^anforwhichVn^anVna;8a2Ann^an.Atieisbrokenarbitrarily.Withprobabilityt,theplayerchoosesanexploratoryactionfromtheremain-ingelementsofAnexcludingthegreedyaction,whereeachexploratoryactionischosenwithequalprobability.bValueUpdating:Updatethespecicvaluesforeachplayerncorrespondingtothechosenactionanusingthelearningschemegivenbelow.Vnt+1an)]TJ/F20 11.955 Tf 11.955 0 Td[(tVntan+t)]TJ/F20 11.955 Tf 5.48 -9.684 Td[(rnan;a)]TJ/F21 7.97 Tf 6.587 0 Td[(n:.10cSettt+1.dUpdatethelearningparameterstandexplorationparametertfollowingtheDCMschemegivenbelow[38]:t=0 1+u;whereu=t2 +t;.11where:0denotestheinitialvalueofalearning/explorationrate,andisalargevaluee.g.,109chosentoobtainasuitabledecayrateforthelearning/explorationparameters.Explorationrategenerallyhasalarge26

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startingvaluee.g.,0.8andaquickerdecay,whereaslearningratehasasmallstartingvaluee.g.,0.1andveryslowdecayrate.Exactchoiceofthesevaluesdependsontheapplication[38,39].eIft
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CHAPTER4JOINTFTRANDENERGYBIDDINGMODELTherehavebeenmanystudiesinliteratureabouteithertheFTRmarketortheenergymarket,however,thereisonlyalimitednumberofstudiesthatexaminethebothmarketstogether[11]isoneexample.FTRandenergymarketsaecteachotherdirectly,infact,revenuesforholdinganFTRisdeterminedatenergymarketandoneofthemainmotivationofhavinganFTRistohedgeagainstthevolatileenergymarketLMPs.Therefore,integratingthesetwomarketswillnotonlyreectthereallifescenariobutalsomakeitpossibletoanalyzedierentaspectsofthejointmarketthatwouldbebasedonassumptionsotherwise.SuchajointmodelcanbeusedtoinvestigatetheeectsofFTRsonparticipants'biddingstrategiesinenergymarket.Eectsofdynamicenvironmentoftheenergymarketsuchasvaryingcontingencyanddemandscenariosontheequilibriumsettlementscanalsobestudiedthroughsuchanapproach.MarketpowerduetoFTRs,eectsofsuppliers'generationcostfunctionsaresomeothertopicsthatcanbeanalyzedwithajointmodel.Inshort,anintegratedmodelofbothFTRandenergymarketsisamorerealisticrepresentationofreallifescenarioandwillallowamoredetailedanalysisaboutthepowermarket.AjointtransmissionandenergymarketimplementationinreallifeisgiveninFigure4.1.Asseenfromthegure,rstparticipantscompeteforFTRs.AfterISOclearstheFTRmarket,participantsholdtheallocatedFTRsuntiltheendofFTRauctionhorizonduringwhichenergymarketsettlementstakeplaceregularly.28

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Figure4.1FTRandEnergyMarketOperationCycle4.1AMatrixGameModelforJointFTRandEnergyMarketSettlementLetI=f1;2;;IgdenotethesetofpathsofinjectionandwithdrawallocationsforwhichFTRscanbeobtained.LetN=f1;2;;NgdenotethesetofbiddersparticipatinginthejointmarketsomeofwhomaregeneratorsdenotedbysetG,andloadsdenotedbysetLwhere:GNandLN.Also,letSbethesetofactionsin29

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thejointmarket.Abiddern2NisconsideredtobidonasubsetofpathsInIinFTRmarketwithanactionain2Sinforeachpathwhere:Sinisthesetofstrategiesavailabletobiddernonpathi.Itisassumedthatloadshaveinelasticconstantdemands,therefore,onlygeneratorscompeteinenergymarket.Ageneratorg2Gisconsideredtobidwithastrategyag2Sgwhere:Sgdenotesthesetofavailableactionsforgeneratorginenergymarket.ThecycleofFTRandenergymarketoperationsisgiveninFigure4.1.Amatrixgamemodelisusedtosolvethenon-cooperativecompetitionamongbidders.AschematicdiagramofthestepsofthejointmatrixgamemodelispresentedinFigure4.2.Initialstepstep0istodenethenetworkparameterssuchasthesetofFTRpathsofeachbidder,generationcostfunctionsofsuppliers,contingencyanddemandscenarioprobabilitiesandstrategyspaceofthebidders.Atstep1,weinitializethestrategiesofallbidderstotherststrategyintheirstrategyset.Step2-4and14-17istocoverallstrategycombinationsbytheparticipantsonalltheirpathsinFTRmarket.Attheendofeachstrategycombination,FTRmarketissettledbytheISOatstep5.AfterallocatingFTRstotheparticipantsinstep5,algorithmcontinueswiththeenergymarketoperationsstartingwithinitializingthestrategiesofallgeneratorstotherststrategyintheirenergystrategysetstep6.Steps7-8and11-12aretoexploreallthepossibleenergystrategycombinationsbythegenerators.Attheendofeachstrategycombination,ISOsettlestheenergymarketanddeterminesthegenerationquantitiesbythegeneratorstogetherwiththebusLMPsstep9.Atstep10,themodelisreadytocalculatethepayosofthegeneratorsforthecorrespondingstrategycombination.Aftercomputingthepayosofgenerators,itistransferredtoenergypayomatrixwhichiscompletedattheendofstep12.Atstep13,equilibriumstrategyintheenergymarketisfoundbyRLalgorithmforthecurrentFTRallocation.Thisequilibriumpointisusednotonlytocalculatethe30

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payosofthegeneratorsbutalsoloadswhocompeteinFTRmarketstep14.ThiscyclecontinuesuntilalltheFTRstrategiesarevisitedandcorrespondingpayosaretransferredtoFTRpayomatrixendofstep17.Atstep18,RLalgorithmisusedtondtheequilibriumstrategyintheFTRmarket.ThevalueiterationbasedreinforcementlearningalgorithmtondtheequilibriumstrategyinbothmarketsisexplainedinSection3.2.Detailsaboutthestrategyvectorsofthebidders,ISO'ssettlementmodelsandcalculationofthepayosinbothFTRandenergymarketisdiscussedinSection4.2.4.2AMatrixGameModelFormulationforJointMarketSettlementInthissection,bidvectorofparticipantsarepresentedforbothFTRandenergymarkets.ParticipantscompetewitheachotherbysubmittingbidstoISOwhichsettlesthemarketforthegivenbids.ISO'smarketsettlementmodelsarealsogiventogetherwithpayocalculationsofthebidders.4.2.1FTRAllocationModelFormulationAbiddernsubmitsherFTRbenetfunctionwhichisanon-decreasingquadraticconcavefunctiondenedasfnX=nXn)]TJ/F20 11.955 Tf 12.642 0 Td[(nX2nwhere:XnisthequantityofFTReitherinformofobligationoroption.Therefore,thebidderisrequiredtosubmitalinearparameterandaquadraticparameterforherbenetfunction,andthetypeofFTRtoISOforFTRauction.Thus,abidvectorforbiddernonpathi2Incanbedenotedasain=)]TJ/F20 11.955 Tf 5.48 -9.684 Td[(in;in;kin;31

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whereinnthbidder'slinearpricebidonFTRpathi,innthbidder'squadraticpricebidonpathi,kinnthbidder'sFTRtypeselectiononpathivalued1forobligationand0foroptiontype.AfterparticipantssubmittheirbidsforFTRauction,ISOallocatestheFTRsbasedonanoptimizationmodelwithanobjectiveofrevenuemaximizationStep5inFigure4.2.Thisdcmodelcanbegivenasfollows.maxNXn=1Xi2IninFTRi;obn+FTRi;opn)]TJ/F20 11.955 Tf 11.956 0 Td[(inFTRi;obn+FTRi;opn2.1s.t.NXn=1Xi2In[Di;cn;lFTRi;obn+max;Di;cn;lFTRi;opn]Bcl8l;c.2NXn=1Xi2In[)]TJ/F20 11.955 Tf 9.298 0 Td[(Di;cn;lFTRi;obn+max;)]TJ/F20 11.955 Tf 9.299 0 Td[(Di;cn;lFTRi;opn]Bcl8l;c.3FTRi;obnkniM8n;i.4FTRi;opn)]TJ/F20 11.955 Tf 11.955 0 Td[(kniM8n;i.532

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whereFTRi;obnquantityofobligationFTRallocatedtonthbidderonpathidecisionvariableFTRi;opnquantityofoptionFTRallocatedtonthbidderonpathidecisionvariableDi;cn;lPTDFofthenthbidder'sithpathonlinelundercontin-gencycBclcapacitylimitoflinelundercontingencycMbig-MvalueusedtoconstrainallocationofFTRonlytotheselectedtypeISO'sFTRrevenuemaximizationmodeldeterminestheFTRallocationandcor-respondingcostsforallparticipants.FTRcostofabiddernwhoisbiddingwithin;in;kinstrategyvectoronherFTRpathsIniscalculatedasFCn=Xi2IninFTRi;obn+FTRi;opn)]TJ/F20 11.955 Tf 11.956 0 Td[(inFTRi;obn+FTRi;opn2:AlthoughFTRcostsarecalculatedattheendofFTRauction,ISOhastocleartheenergymarketbeforeFTRrevenuescanbecalculated,i.e.,energysettlementdataareneededtocomputeFTRrevenue.Therefore,computationofFTRrevenueandprotareexplainedinthefollowingsection.4.2.2EnergyAllocationModelFormulationGeneratorshavemarginalrealcostfunctionswhichareassumedtobequadraticconvexfunctionsdenedashgZg=ogZg+ogZg2where:Zgisthequantityofelectricitysuppliedbygeneratorg.However,generatorg2GsubmitsherenergycostfunctionashgZg=gZg+gZg2wherelinearandquadraticcostcoecientsarepartofherstrategytomaximizeherpayo.Thegeneratoralsohastosubmitthe33

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lowerandupperboundcapacityofherpowergeneration.Therefore,ageneratorisrequiredtosubmitalinearcostparameter,aquadraticcostparameter,lowerproductionlimit,andupperproductionlimittoISOforenergyauction.Thus,abidvectorforgeneratorgcanbedenotedasag=g;g;p g; pg;whereggthgenerator'slinearcostbid,ggthgenerator'squadraticcostbid,p ggthgenerator'slowergenerationlimit, pggthgenerator'sgenerationcapacity.Afterparticipantssubmittheirbidsforenergyauction,ISOdeterminesthepowersupplyamongthegeneratorsbasedonanoptimizationmodelwithanobjectiveofcostminimizationofsupplyingpowertoconsumersStep9inFigure4.2.TherearetworandomfactorsintheISO'smodelthatisascertainedatthetimeofenergyauction:1.Contingencysituation2.DemandsituationISOclearstheenergymarketbasedonthecurrentcontingencysituationandconsumerdemands.ItisassumedthatthereareCcontingencyscenariosoflinesandUdemandscenariosofloads.Letc2f1;2;;Cgdenotethecurrentcontingencyscenarioandu2f1;2;;Ugdenotethecurrentdemandscenario.Also,letb2Bdenotethebuses,ij2Adenotethearcsdirectedlines,andm2Rdenotethetransmissionlineloopspresentinthenetwork.AllthearcsthatareinloopmaredenotedbythesetAm.Similarly,allthegeneratorsloadsthatarelocatedatbusbaredenotedby34

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GbLb.ThenISO'senergysettlementmodelcanbegivenasfollows.minXg2GgZg+gZ2g.6s.t.Xg2GbZg)]TJ/F20 11.955 Tf 13.948 0 Td[(Qub+Xb:ib2ATib)]TJ/F25 11.955 Tf 18.393 11.357 Td[(Xj:bj2ATbj=08b.7Xij2AmsijmTij=08m.8Tij Tcij8ij.9Zg pg8g.10Zgp g8g.11Zg0;Tij08g;8ij.1235

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whereQubtotalquantityofdemandatbusbunderdemandscenarioui.e.,Qub=Pl2LbqulZgquantityofelectricitysuppliedbygeneratorgdecisionvari-ableTijamountofelectricowonarcijdecisionvariablesijmKirchhovoltagecoecientforarcijinloopm,equalsto1ifijinthesamedirectionwiththeloopmand-1ifintheoppositedirectionwiththeloopm TcijElectricowcapacityofarcijunderthecontingencysce-nariocISO'senergycostminimizationmodeldeterminestheallocationofpowergen-erationbysupplierstomeetthedemand.EnergyprotofageneratorgforthecontingencyscenariocanddemandscenariouiscalculatedasEPc;ug=LMPc;ubgZc;ug)]TJ/F15 11.955 Tf 13.948 0 Td[([ogZc;ug+ogZc;ug2];where:LMPc;ubgdenotestheLMPatthebuswheregeneratorgislocated.Loadsdonotcompeteinenergymarket,however,theymakepaymentstotheISObasedontheenergysettlementdata.Therefore,energyprotofaloadforthecontingencyscenariocanddemandscenariodisbasicallythecostofherdemandwhichcanbestatedasEPc;ul=)]TJ/F20 11.955 Tf 11.291 0 Td[(LMPc;ublqc;ul;where:LMPc;ubldenotestheLMPatthebuswhereloadlislocated.Asstatedintheprevioussection,FTRcostsFCarecomputedattheendoftheFTRsettlement,however,energysettlementdataareneededtocalculatetheFTRrevenueofapar-36

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ticipant.Therefore,FTRprotscanbecalculatedatthisstagetogetherwithenergyprots.FTRprotofabiddernwhohasasetofFTRpathsInwillbecalculatedforthecontingencyscenariocanddemandscenariouasFPc;un=Xi2In[LMPinFTRi;obn+maxLMPin;0FTRi;opn)]TJ/F20 11.955 Tf 13.947 0 Td[(FCin]:Intheequationabove,LMP'sandFTRquantitiesareforcontingencyscenariocanddemandscenariou.Itisassumedthateachofthecontingency-demandscenariohasaprobabilitytooccurdenedwithjointprobabilitymatrixc;u.Sincethecontingencyanddemandscenariothatwillhappenduringtheenergyauctionisun-knowntothemarketparticipants,theaveragepayovalueoverallcontingencyanddemandscenariosissignicant.ExpectedpayoofbiddernforanFTRandenergysettlementwithdierentcontingencyanddemandscenariosisdPOn=CXc=1UXu=1c;uFPc;un+EPc;un:37

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Figure4.2MatrixGameModelSolutionStepsforJointMarket38

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CHAPTER5NUMERICALEXAMPLE:FTRMARKETSETTLEMENTInordertodemonstratethematrixgametheoreticapproachtoobtainequilibriumbiddingstrategiesforanFTRmarket,asamplepowernetwork,asstudiedin[2],wasadopted.ByvaryingthenetworkparameterslikecontingenciesandLMPdierencesbetweenthenodes,sixteendierentnetworkscenariosarecreatedforwhichequilib-riumFTRbiddingstrategiesarepresented.Sinceinthematrixgameformulationthecontinuousbidparametersobligationprice,optionprice,quantity,andtypemixarediscretized,theeectoftheextentofdiscretizationisexaminednext.Thereafter,theimpactofindividualbidparametersofthebiddersundertheassumptionthattheotherbidderschoosetheiractionsuniformlyfromtheavailablesetsisstudied.Finally,theimpactofthenetworkparametersontheequilibriumFTRbiddingstrategiesisinvestigatedthroughananalysisofvarianceANOVAviaa24factorialexperiment.Thecolumnsofthetablesthatdonothaveunitsareingenericunits.5.1TheSampleNetworkThesamplenetworkconsistingofthreebusesandfourbiddersisdepictedinFigure5.1.TheBidders3and4areconsiderednon-strategic,henceonlybidders1and2areconsideredstrategicbiddersinthematrixgame.ThepathsbetweensourceandsinkbusesonwhichthebiddersbidareshownintheFigure5.1,whichalsoindicatesthereactancevaluesandowlimitsofeachline.39

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Figure5.1FTRBiddersina3-BusPowerNetwork5.2EquilibriumBiddingStrategiesforDierentNetworkScenariosFourkeynetworkrelatedparametersthatwereconsideredinthisstudyarecontin-gencyc,LMPsl,variancesoftheLMPestimatesv,andtheriskcoecientr.Sixteendierentnetworkscenarioswerecreatedbyvaryingeachofthefournet-workparametersattwolevels.Theparametersl;v;andrwhichcouldbevariedforbothstrategicbidderswerevariedonlyforbidder2.Inordertosimplifythenumeri-calexposition,theobligationandtheoptionpricebidsareconsideredtobeidentical,whichreducedthesizeofthebidvectorfromfourtothreedimensions.Itisnotedhowever,thatobligationFTRmaybecomealiability,whereastheoptionFTRdoesnothavesucharisk,andhencethebidpricescouldbedierent.Ourmodelisgeneralandaccommodatesthischaracteristic.Foreachofthesixteenscenarios,thepossiblenumberofbidchoicesofthetwoplayerswaskeptconstantat125withvelevelsofdiscretizationsforeachofprice,quantity,andthetypemix.Table5.1showsthe40

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Table5.1NetworkandBidValues valuesofthenetworkandthebidparameters.Foreachscenario,thepayomatriceswereconstructedandthevalueiterationbasedlearningalgorithmwasimplemented.ThenetworkscenariosandthecorrespondingpurestrategyequilibriumasobtainedbytheRLalgorithmarepresentedinTable5.2.AsindicatedinthelastcolumnofTable5.2,intenoutofthethirteenscenarioshavingpurestrategyNashequilibria,theRLalgorithmconvergedtoaNashequilib-riumpoint.AmongthemultipleNashequilibriathatexistforscenariosvrandclvr,thestrategiesthattheRLalgorithmconvergedtohavehigherpayosforbothbidderscomparedtotheotherNashequilibriumpoints.Inthreeoftheremainingscenarioswith'No'inthelastcolumn,theRLalgorithmconvergedtonon-NEstrategiesyieldinghigherpayosforbothofthebidderscomparedtotheNEpayos.Recog-nizingthesesolutionsiscriticalsinceallbiddersmustaccepttostayatthesepointsinordertogainthebenetsofthesehigherthanNashequilibriumpayos.Forthesescenarios,Table5.3showsacomparisonofthepayosfromtheNashequilibriumstrategiesandthecorrespondingnon-NEstrategiesobtainedbytheRLalgorithm.Theremainingthreescenarioswitha'-'inthelastcolumndonothaveapure41

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Table5.2EquilibriumBiddingStrategiesforSixteenNetworkScenarios Table5.3StrategieswithHigherPayosthanNashEquilibrium 42

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strategyNashequilibrium.TheRLalgorithmconvergedtostrategieswithahighpayodistributionforthebidders.5.3ImpactofBidParameterDiscretizationAsdiscussedearlier,discretizationofthebidparametersisessentialtoformulatingthenon-cooperativebehaviorofthebiddersasamatrixgame.Anerdiscretizationofthecontinuousparametersisrequiredtominimizethedeviationfromtheactualproblemscenarioandthetrueequilibrium.Atthesametime,nerdiscretizationoftheparametersofamultidimensionalbidvectorexpandstheactionspace,whichincreasesthedimensionsofthepayomatricesandtheresultingcomputationalre-quirements.Inordertoexposethesignicanceofdiscretization,theimpactofpriceparameterdiscretizationontheequilibriumbiddingstrategiesisstudied.Fivedierentlevelsofdiscretizationofthepriceparameter;5;10;15;and20wereconsideredwhilethediscretizationofquantityandtypemixparameterswerekeptconstantat5levelseach.Thisresultedinpayomatrixsizesvaryingfrom7575355to500500055.TheequilibriumpayosoftheplayersaregiveninTable5.4.Asevidentfromthepayos,theequilibriumstrategiesvariedquitesignicantlywiththelevelofdiscretization.Italsoappearsthatwithnerpricediscretizationthepayosofthebiddersincreased.Thisisduetothefactthatthealgorithmalwayslooksforanequilibriumwithhighvalues,andasdiscretizationincreases,thealgorithmhasmorecandidatestochoosefrom.43

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Table5.4ImpactofBidParameterDiscretization 5.4ImpactofBidParameterVariationsTheequilibriumoutcomeofamatrixgameisaresultantoftheparameterval-uesoftheparticipants'bidvectors.Thoughitisdicult,itisdesirabletoextractinsightintotheimpactoftheindividualbidparameterontheequilibriumpayos.Therefore,anexperimentwhereimpactofeachbidparameterwasgraphicallyana-lyzedisconductedasfollows.Itisacknowledgedthattheobservationsmadeinthissectionhaveproblemspecicinterpretationswithsomepotentialforgeneralization.Intheexperiment,thenetworkparametervaluesweremaintainedatthefollowing.Forbidder1:LMP=$20,variance=0.2,riskcoecient=0.003,andforbidder2:LMP=$10.5,variance=0.2,riskcoecient=0.002.MaximumquantityQwasconsideredtobe300,andthenetworkwasassumedtohavenocontingency.Thepricefactorofbidders1and2werevariedintenstepsbetween0.1and0.95instepsof0.1.Figure5.2showstheimpactofpricevariationsbybidder2onbidder1pay-os.Thepayosofbidder1,asplotted,wereaveragedoverallpossiblecombinations080ofquantityandtypemixparametersofthetwobidders,whereeachbidderhas108possiblebidchoices.Forallbidder1pricefactorvaluesupto0.7,thepayowaszero.Forbidpricefactorbeyond0.7,bidder1'spayoswereidentical44

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forallbidpricefactorslessthanorequalto0.7bybidder2.Hence,onlythebidpricefactorscenarioswithbothbidsgreaterthanorequalto0.7arecriticalasshowninFigure5.2.Asbidder2changesitspricefactor,theoptimalpricebidforbidder1alsochanges.Forexample,asbidder2changespricefactorfrom0.7to0.8,theoptimalpricebidforbidder1changesfrom0.8to0.9.Similarly,Figure5.3showstheimpactofpricebidvariationsofbidder1onthebidder2'spayosutility.Ageneralconclusionthatcanbedrawnfromtheaboveisthatasignicantinteractionexistsbetweenthebidderpricesinhowtheyimpactthebidderutilities.Theexactlevelofinteractionswilldependonthenetworkparametervalues. Figure5.2PriceEectonBidder1'sAverageUtilityAnalyses,similartothatofprice,werealsoconductedwithquantityandtypemixparameters.TheresultsfromtheinvestigationofthequantityparameterarepresentedinFigures5.4and5.5.Forbothbidders,thequantityeectappearstobesomewhatidentical.Thebidderpayosincreasewithincreaseinthequantitybid,andtheyleveloafter0.5forbidder1and0.7forbidder2irrespectiveofthecompetitor'sbid.Thisindicatesthatforthegivenproblemparameters,thequantitybidshouldbekeptatthemaximumpossiblevalue.However,itwasourconjecturethatinthe45

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Figure5.3PriceEectonBidder2'sAverageUtilitypresenceofhighvaluesofvarianceand/orriskcoecient,thechoiceofthequantityparametercouldbecomestrategic.Totestthisconjecture,thesamplenetworkwasstudiedunderanewscenariowiththefollowingnetworkparameters.Forbidder1:LMP=$20,variance=0.2,riskcoecient=0.003,andforbidder2:LMP=$13,variance=2,riskcoecient=0.01.Thestrategicimpactofbidder2'squantitybidonherpayo,whichstartstodeclinebeyondacertainvalueofquantitybid,isshowninFigure5.6.Thisisinclearcontrasttothehigherthebetterbehaviorseenearlier.AgeneralconclusioncanbestatedthatFTRquantitycouldbeasignicantparameterandshouldbeconsideredinthebiddingprocess.TheresultsoftheinvestigationontheimpactoftypemixparameteronthebidderpayosaregiveninFigures5.7and5.8.ItappearsfromFigure5.7thatbidder1'spayoisnotaectedbyitschoiceofthetypemixparameter,andisonlyminimallyaectedbythechoiceofbidder2'stypemixparameter.Ontheotherhand,bidder2spayoiscompletelyindependentofbidder1'sstrategy,asevidentfromtheoverlappingcurvesinFigure5.8.Bidder2suersasignicantdecreaseinutilitywiththechoiceofhighervaluesofthetypemixfactori.e.,higherproportionof46

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Figure5.4QuantityEectonBidder1'sAverageUtility Figure5.5QuantityEectonBidder2'sAverageUtilityobligation.Table5.5depicts,forasamplescenario,howthetotalFTRallocationaswellasitsobligationandoptioncomponentschangeforbidder2,asthebiddervariesitstypemixbid.ThissupportsthetrendobservedinFigure5.8,sincebidder2winsthemostFTRwhenthetypemixfactorissetatzeroi.e.,alloption,andtheFTRallocationdecreasesasmoreobligationsareaddedtothemix.Itisconcludedthat47

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Figure5.6StrategicImpactofQuantityParametertypemixparametercouldplayasignicantroleinamulti-bidderFTRsettlementprocessandthusshouldbeadequatelyinvestigated. Figure5.7TypeMixEectonBidder1'sAverageUtility5.5ImpactoftheNetworkParameterVariationsTheimpactofthenetworkparametersontheequilibriumpayosofthebidderswasstudiedthroughananalysisofvarianceANOVAviaa4-factordesignedexper-iment.Thefactors,theirlevels,andthesixteen4experimentswerepresentedin48

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Figure5.8TypeMixEectonBidder2'sAverageUtilityTable5.5ImpactofTypeMixParameter Table5.1and5.2.TwosetsofANOVAwereperformedusingpayosofbidder1andbidder2giveninTable5.2asexperimentaloutcomes.Sinceeachoutcomeisasinglereplicate,normalprobabilityplotsofthefactorandinteractioneectswereconstructedtoobtainerrorsumofsquareSSestimates.TheANOVAresultsaregiveninTables5.6and5.7.ItappearsfromTable5.6thatbidder2'spayoisaectedbyallfourofthefactorsandisinsensitivetoanyofthefactorinteractions.Amongthesignicantfactors,theLMPappearstobethemostcriticalwithap-valueof49

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Table5.6ANOVAwithBidder2'sPayos Table5.7ANOVAwithBidder1'sPayos 0.0001.Table5.7showsthat,forthegivennetwork,bidder1'spayoisaectedonlybytheLMPestimateofbidder2andthecontingencyinthenetwork.Asexpected,varianceandriskcoecientparametersofbidder2whicharetheothertwofactorsconsideredintheexperimenthavenosignicantimpactonthepayoofbidder1.50

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CHAPTER6NUMERICALEXAMPLE:JOINTFTRANDENERGYMARKETSETTLEMENTSInordertodemonstratethematrixgametheoreticapproachtoobtainequilibriumbiddingstrategiesforjointFTRandenergymarkets,asamplepowernetworkisadopted.First,impactoftheFTRsonthestrategiesofabidderisinvestigatedbyassigningadierentFTRpathtothebidder.FTReectshavealsobeenanalyzedbycomparingthepayosofparticipantswithandwithoutFTRs.Thereafter,theimpactofthecontingencyanddemandvariationsintheelectricmarkettotheequilibriummarketpointhasbeenexaminedbychangingthefrequenciesofcontingencyanddemandscenarios.Finally,theimpactofgenerators'marginalcostfunctionontheequilibriumbiddingpayoshasbeenstudiedbyvaryingthecostfunctionsforsomegenerators.Thecolumnsofthetablesthatdonothaveunitsareingenericunits.6.1TheSampleNetworkThesamplenetworkchosentoexaminethejointmarketisPJM5-busexampleinwhichtherearevegeneratorsandthreeloads.ThelocationofeachgeneratorandloadtogetherwiththetransmissionlinesandtheirreactancevaluesaredepictedinFigure6.1.51

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Figure6.1PJM5-BusPowerNetwork6.2ImpactofFTRsinMarketSettlementTheimpactofFTRsonthebiddingstrategiesofparticipantshasbeenstudiedbyassigningcertainquantitiesofFTRondierentpaths.OnlyonebidderisselectedandcertainFTRquantitiesareallocatedtothisbidderondierentFTRpathsoneatatime.TosimplifytheexperimentandsingleouttheFTReect,otherparticipantsarenotallocatedanyFTRs.GeneratorscompetewitheachotherbychangingtheirstrategiesandnallyanequilibriumpointisreachedthroughRLalgorithm.Inthisexperimentation,generator1ischosentoholdtheFTRonvaryingpathscenarioswhicharegivenin6.1.ThebiddingparametersandfactorsofgeneratorsforenergymarketisshowninTable6.2.Forexample,ifgenerator4selectssecondlinearstrategyandsecondquadraticstrategytobidthengenerator4'scostfunctionwillbehZ4=1:430Z4+1:20:025Z24where0Z4200.AsseeninTable6.2,anygeneratorwhobidswiththerstlinearandquadraticstrategyinfactbidslessthanhermarginalcostfunction.Thischaracteristicisintegratedinthebiddingmodeltoenlargebiddingspaceofabidder.Forexample,abiddermayaccepttoloseenergyrevenueinreturnofhighFTRrevenue.Biddingparametersinthetableare52

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Table6.1EquilibriumBiddingStrategyofGenerator1andBusLMPs Table6.2BiddingParametersandFactorsofGenerators similartoPJM5-Busexampleandbasicallyexhibitstheideaofhavingloadswithexpensivelocalgenerationandcheapergenerationinotherfurtherbuses.Ageneratorhastotalofsixstrategycombinationslinear2quadraticfactors.ForeachFTRpathscenario,theequilibriumstrategyofgenerator1andthecorrespondingLMPsateachbusarepresentedinTable6.1includingthenoFTRcase.Thecostof150MWFTRisassumedtobe$2400.AlltheequilibriumpointsreachedfordierentFTRpathscenarioshavethepropertyofpureNashequilibrium.AsseeninTable6.1,generator1haslittleeectontheLMPswhichchangeminimallybetweengenerator1'sminimumpricestrategy,0:9;0:8,onpath1)]TJ/F15 11.955 Tf -422.701 -23.907 Td[(2,andhermaximumpricestrategy,;1:2,onpath2)]TJ/F15 11.955 Tf 12.916 0 Td[(1.WhenthedierentFTRscenariosatTable6.1areexamined,itisobservedthatgenerator1chooses53

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thestrategycombinationthatmaximizesitssumofenergyprotandFTRprot.Forexample,whengenerator1competeswithoutFTR,electricmarketsettlesdownwithgenerator1having:9;1:2strategyforherlinearandquadraticpricefactorsandLMPof$59:9atherbus,bus1.However,whengenerator1isallocatedanobligationFTRof150MWonpath1)]TJ/F15 11.955 Tf 13.217 0 Td[(2,themarketsettlesonanequilibriumpointwithgenerator1having:9;0:8strategy.Bychangingherstrategyfrom:9;1:2to:9;0:8,generator1doesnotincreaseherenergyrevenuesinceLMPatbus1remainssameat$59:9,however,generator1'sFTRrevenueincreasesduetoLMP1)]TJ/F18 7.97 Tf 6.586 0 Td[(2increase.Ontheotherhand,ifthedirectionofFTRisreversed,i.e.,selectingpath2-1,obligationFTRbecomesaliability.Generator1adoptstothisconditionbyswitchingherequilibriumstrategyfactorto;1:2.Asaresult,notonlyLMP2)]TJ/F18 7.97 Tf 6.586 0 Td[(1improvesbutalsoherenergyrevenueincreasesbyariseatthebus1LMP.AspresentedinTable6.1,whenFTRpath5)]TJ/F15 11.955 Tf 11.817 0 Td[(4isallocatedtogenerator1,shestaysatthesamestrategyof;1:2.Asaresult,LMPatbus1continuestobe$60:1andLMP5)]TJ/F18 7.97 Tf 6.587 0 Td[(4is$4:2.Whereas,ifFTRpath4)]TJ/F15 11.955 Tf 11.409 0 Td[(5isallocatedtogenerator1,generator1hasaconictingstrategyoutcomes.Ifgenerator1choosesthestrategyof:9;0:8,shereceivesasmallerFTRliabilityLMP4)]TJ/F18 7.97 Tf 6.587 0 Td[(5=$)]TJ/F15 11.955 Tf 11.834 0 Td[(3:9butherenergyrevenuedecreaseswithLMPof$59:9atbus1.Ifgenerator1attainsthestrategyof2;1:2,shegetsahigherenergyrevenuewithLMPof$60:1butFTRliabilityincreaseswithaLMP4)]TJ/F18 7.97 Tf 6.586 0 Td[(5of$)]TJ/F15 11.955 Tf 12.866 0 Td[(4:2.Asstatedearlier,generator1choosesthestrategywhichmaximizesheroverallpayo.Expectedquantityofelectricitythatgenerator2suppliesis109:5MWwhengenerator1selectsstrategy:9;0:8,and102:6MWwhengenerator1chooses2;1:2.Asaresult,heroverallpayoforthe:9;0:8strategyis109:559:9+150)]TJ/F15 11.955 Tf 9.299 0 Td[(3:9=$5975:3andforthe;1:2strategyis102:660:1+150)]TJ/F15 11.955 Tf 9.299 0 Td[(4:2=$5533:3.Therefore,generator1choosesthestrategyof0:9;0:8.54

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Table6.3EquilibriumPayoswithoutFTRandwithFTRs InordertoassesstheoverallimpactofFTRs,FTRsaremadeavailabletosomeoftheparticipantssimultaneouslyandcorrespondingequilibriumpayoshavebeencomparedwiththepayosofnoFTRcase.ResultsaregiveninTable6.3.TheshadedcellsinthetableindicatestheFTRpaththatthecorrespondingparticipantbidstoacquireFTR.Toincreasethecomputationalperformance,participantshavediscretizedpricebidstrategiesstartingataminimumvalue.Table6.3showsthatgenerator2,3andload1raisetheirpayoswhentheyholdFTRs.TheirequilibriumFTRbiddingstrategiesalsoindicatethattheyarewillingtobuythecorrespondingFTRs,i.e.,theirpricebidsarehigherthantheminimumpricebidstrategy.Load2'sequilibriumpricebidisalsohigherthantheminimumpricestrategy.Thus,load2iswillingtoattaintheFTRonpath3-5,however,ISOwhichhasanobjectiveofmaximizingFTRsalesrevenue,doesnotallocateanyFTRtoload2withthegivenbid.Ontheotherhand,load2doesnotnditprotabletobidmorethanhercurrentbidwhichislessthanhermaximumpricebid.Generator1istheonlyparticipantwhobidsminimumpriceforherFTR.ThereasoncanbeseenatherdecreasingequilibriumpayowhileholdingFTR,i.e.,generator1doesnotnditbenecialtopossesthisFTRpathevenwiththeminimumpricebid.Finally,thechangeinthepayosoftheparticipantsthatdonotholdanyFTRisinthemixed55

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direction.Whilegenerator4and5seeadecreaseintheirpayo,load2and3seeasmallincrease.ItappearsthatexistenceofFTRsmakethegeneratorsbidmorecompetitivelyintheenergymarketwhichdecreasetherevenueforgeneratorsandcostforloads.6.3ImpactofContingencyandDemandScenarioVariabilityTostudytheeectofcontingencyanddemandscenariosvariabilityonthemarketequilibriumpayos,foursetsofjointprobabilitymatrixc;uaregenerated.Thesematriceswhicharecreatedas22factorialdesign,areshowninTable6.4.Bothfactorscontingencyvariability,demandvariabilityhavelowlevelandhighlevel.Notationsusedinthistableareasfollows1.Lowlevelofcontingencyvariabilityandlowleveldemandvariability2.Lowlevelofcontingencyvariabilityandhighlevelofdemandvariabilityd3.Highlevelofcontingencyvariabilityandlowlevelofdemandvariabilityc4.Highlevelofcontingencyvariabilityandhighlevelofdemandvariabilitydc5.EectofdemandvariabilityD6.EectofcontingencyvariabilityC7.JointeectofcontingencyanddemandvariabilityDCTherearefourcontingencyscenariosandthreedemandscenarios.ContingencyscenariosandcorrespondinglinelimitsaregiveninTable6.5.Contingencyscenario1isbasicallythereisnocontingencyandallthelinelimitsareattheirnormallevels.Contingencyscenario2;3and4correspondsadecreaseinthelinelimitsof12;45and34,respectively.Lowlevelforcontingencyvariabilityfactormeansthatmajorityof56

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Table6.4EquilibriumPayosforContingency-DemandProbabilityMatrices Table6.5ContingencyScenarios thetimeitwillremaininthenocontingencyscenario,1.Highlevelforcontingencyvariabilityfactormeansthatthereisahigherprobabilityforthecontingencyscenariotobeotherthannocontingencyscenario,i.e.,frequencyofline12;45or34beingdownishigher.TherearethreedemandscenariosofloadswhicharepresentedinTable6.6.Demandscenario1iscreatedtoreectamediumdemandbyallloads.Similarly,2representsahighdemand,and3representsalowdemandatthenetwork.Lowlevelofdemandvariabilityfactormeansthatmajorityofthetimethedemandwillremaininthemediumdemandscenario,1.Similarly,highlevelofdemandvariabilityfactormeansthatprobabilityofhavingunusualdemandssuchasloworhighquantitieswillbebigger.FTRmarketdataareshowninTable6.7.Asseenfromthetable,generators1,2,and3,andloads1and2competeintheFTRmarket.TheirFTRpathsinterms57

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Table6.6DemandScenarios Table6.7FTRMarketDataforContingency-DemandVariations ofsourceanddestinationbuses,biddingparametersandfactorsofbothlinearandquadraticcomponentsofthebenetfunctionarealsogiveninTable6.7.Itisnotgiveninthetable,however,bothobligationandoptiontypeFTRisavailabletothebiddersaspartoftheirFTRbiddingstrategy.Ifload1bidder6choosestobidforFTRonpath3)]TJ/F15 11.955 Tf 12.103 0 Td[(4withthesecondlinearfactorandtherstquadraticfactorthenherbenetfunctionwillbefX6=810X6)]TJ/F15 11.955 Tf 14.645 0 Td[(30:002X26.Lowandhighlevelsoflinearandquadraticcoecientsofthebenetfunctionallowthebiddertoadjustthepriceonherpath.IfabiddervaluesherFTRpathlowthenthebiddercanchooserstlinearfactorandsecondquadraticfactorwhichwillkeepthepriceforthatFTRlow.Generators'biddingparametersandfactorsforenergymarketsettlementremainsameasshowninTable6.2.58

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EquilibriumpayosandfactoreectsoftheparticipantsforeachcontingencyanddemandvariabilitylevelisgiveninTable6.4.ThechangesinthefactoreectsbasedonsourceofvariationsarepresentedinFigure6.2forgeneratorsandinFigure6.3forloads.ItshouldbenotedthatthesepayosareobtainedafterseriesofenergymarketsettlementswhichisfollowedbytheFTRmarketsettlement.EachsettlementissubjecttoISO'sallocationoptimizationmodelandparticipantsbidstrategicallytomaximizetheiroverallpayos.Therefore,theyareresultofthecomplexrelationsamongmarketdynamics.Figure6.2showsthatincreaseinthedemandvariabilityreducesthepayosofgenerator1,2,3,and4whereasimprovesthepayoofgenerator5.Whendemandvariabilityincreases,averagedemanddecreaseswhichpullstheaverageLMPsatthebusesdownresultinginlowerenergyrevenueforgenerators.However,ISOallocatesmorepowergenerationtogenerator5inthepresenceofhighdemandvariabilitywhichosetsthelowerenergyrevenueduetolowerLMPandevenincreasesheroverallprot.Contingencyvariabilityincreasehassimilareectonthepayoofgenerator3,4,and5,however,hasapositiveeectonthepayosofgenerator1and2.TheLMPatbus1wheregenerator1and2arelocated,isstilllow,howeverLMPsoftheFTRpathsofgenerator1and2)]TJ/F15 11.955 Tf 13.101 0 Td[(5and1)]TJ/F15 11.955 Tf 13.102 0 Td[(2,respectivelyincreaseswhichincreasestheiroverallpayos.Whentheloadpayosareanalyzedunderincreaseddemandvariability,itappearsthatallofthemareaectedpositively.ThisispartlyduetothefactthattheiraveragedemanddecreasesandpartlybecauseofthedecreaseattheLMPsoftheirbuses.Itisalsoobservedthatimprovementinload1'spayoissmallerthanload2and3.ThiscanbeexplainedbytheFTRthatload1isholding,3)]TJ/F15 11.955 Tf 13.04 0 Td[(4.Whereas,thisFTRpathhelpsload1toincreaseherpayowhenthecontingencyvariabilityincreasessinceLMPincreasesonpath3)]TJ/F15 11.955 Tf 13.196 0 Td[(4.Load2and3whodonotholdanyFTRseeadecreaseintheirpayos59

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becauseofincreasedLMPsattheirbuses.Thejointeectofcontingencyanddemandvariabilityonthepayosoftheparticipantsisnegligible.Inshort,contingencyanddemandvariabilityhavesignicanteectsonthepayosofthemarketparticipants.ThedirectionoftheseeectschangefrombiddertobidderandfromcasetocaseshowingthecomplexrelationsbetweenthemarketdynamicssuchasequilibriumLMPs,whetherthebidderholdsanyFTR,generationquantities. Figure6.2GeneratorFactorEectsforContingency-DemandVariations6.4ImpactofGeneratorCostFunctionVariationsInordertoanalyzetheeectofgeneratorcostfunctionvariabilityonthemarketequilibriumpayos,linearcostcoecientofgenerator2and4isvariedattwolevelslowandhighforminga22factorialdesign.ThesecoecientlevelsareshowninTable6.8.Notationsusedinthistableareasfollows1.Lowmarginalcostforgenerator2andlowmarginalcostforgenerator52.Lowmarginalcostforgenerator2andhighmarginalcostforgenerator5g53.Highmarginalcostforgenerator2andlowmarginalcostforgenerator5g260

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Figure6.3LoadFactorEectsforContingency-DemandVariationsTable6.8EquilibriumPayosforDierentCostFunctionofGenerator5and2 4.Highmarginalcostforgenerator2andhighmarginalcostforgenerator5g5g25.Eectofgenerator5marginalcostfunctionG56.Eectofgenerator2marginalcostfunctionG27.Jointeectofgenerator5and2marginalcostfunctionsG5G2FourcontingencyandthreedemandscenariospresentedintheprevioussectionseeTable6.5and6.6staysameforthisexperimentation.Anewjointcontingency61

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Table6.9JointContingency-DemandProbabilityMatrix -demandprobabilitymatrixwhichiscomposedofmediumcontingencyanddemandvariability,isgeneratedforthisproblemunlikethelow-highcontingency-demandvariabilitycombinationsintheprevioussection.ThisnewjointprobabilitymatrixisgiveninTable6.9.AbiggerFTRmarketwhosedataareshowninTable6.10,iscreatedforthisproblem.Asseenfromthetable,generators1,2,and3,andloads1and2con-tinuetobethecompetitorswiththesameFTRpathsintheFTRmarket,however,generator2hasanextraFTRpath,3)]TJ/F15 11.955 Tf 12.436 0 Td[(2.ThebiddingparametersandfactorsofbothlinearandquadraticcomponentsofthebenetfunctionarealsogiveninTable6.10.ObligationandoptiontypeFTRsarestillpartofFTRbiddingstrategyofthebidders.Generators'biddingparametersandfactorsforenergymarketsettlementremainsameasshowninTable6.2.EquilibriumpayosandfactoreectsoftheparticipantsforthemarginalcostfunctioncombinationsaregiveninTable6.8.ThechangesinthefactoreectsbasedonsourceofvariationsarepresentedinFigure6.4forgeneratorsandinFigure6.5forloads.WhenthebusLMPsareexamined,itisseenthatgenerator5'sbiddingpriceaectstheLMPsoverthewholenetworkpricesetterwhereasgenerator2'sbiddingpricehasaminimalaectoverthebusLMPsincludingbus1whereshe62

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Table6.10FTRMarketDataforCostFunctionVariationsVariations islocatedpricetaker.Thisobservationexplainsthetrendinthegures.AsseenfromFigure6.4,allgenerators'payosarehigherwhengenerator5hasahighermarginalcost.Ahighermarginalcostforgenerator5makeshertobidhigherintheenergymarketresultinginhigherLMPsandincreasedenergyprotsforthegenerators.HigherLMPsmeanincreasedenergycostsfortheloadsasseenin6.5.Whiletheincreaseinthepayoofgenerator4ismaximum,generator5hasalowerincreaseinherpayo.Itisobservedthatinthisnetworkgenerator4and5actlikesubstitutes.Thus,whengenerator5bidshigherpriceforpowergeneration,ISOwhohasagoalofminimizingtotalcostofpowergeneration,allocatesmoreMWtobeproducedbygenerator4andlessbygenerator5creatingmoreprotforgenerator4andlessforgenerator5.Sincegenerator2ispricetaker,theLMPsonthebusesdonotdiermuchwhenshebidshigherpriceduetohighermarginalcost.Theonlyparticipantwhosepayoissignicantlyaectedbygenerator2'shigherpricebiddingisherselfwhichisobviouslybecauseofherhighermarginalcost.Tosummarize,impactofagenerator'sproductioncostfunctionisaproductofcomplexrelationsamongthemarketdynamics.Ifageneratorhasahighercostfunction,thismay63

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decreaseherequilibriumpayo,mayormaynotaecttheequilibriumpayosoftheotherparticipantsdependingonwhethersheisapricemakerorapricetaker. Figure6.4GeneratorFactorEectsforGenerationCostVariations Figure6.5LoadFactorEectsforGenerationCostVariations64

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CHAPTER7CONCLUSIONSFinancialtransmissionrightisconsideredanimportantmechanismforpowermar-ketparticipantstohedgeagainstpriceuncertaintiesresultingfromtransmissioncon-gestion.FTRalsoservesasameansofgeneratingrevenueinaderegulatedmarket,inawaysimilartothestocksinthenancialsector.AframeworkforFTRallocationwasoriginallyintroducedin[10].ThoughbiddingstrategiesinanFTRmarketishighlyinuencedbythebiddingstrategiesintheenergymarketandviceversa,toourknowledge,noattempthasbeenmadepriortothisresearchtojointlymodelandexamineequilibriumbiddingbehaviorsinFTRandenergymarkets.Inthisdissertation,agametheoreticmodelforexaminingnon-cooperativebid-dingstrategiesforacquiringFTRsinaderegulatedpowermarketispresented.Thematrixgametheoreticmodelpresentsasignicantdeparturefromthecommonlyusedbi-leveloptimizationapproachfoundintheliterature,anditallowsconsiderationofmultidimensionalbidswithmanybidders,multipleFTRpaths,dierentobligationandoptionprices,andcontingenciesandvaryingdemands.AvalueiterationbasedRLalgorithmisusedasasolutiontoolforthematrixgamemodel.AsamplepowernetworkisusedinelaboratedemonstrationofthematrixgamemodelforanalyzingFTRbiddingstrategies.SixteendierentnumericalscenariosareconstructedfromthesamplenetworkforwhichequilibriumFTRbiddingsolutionsarepresented.ThequalityofthesolutionsintermsoftheirNashpropertyandbidderpayosaredis-cussed.ItisshownthatthevalueiterationbasedRLalgorithmisabletondNash65

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equilibriumsolutioninmajorityoutof13oftheproblemscenariosforwhichpurestrategyNashequilibriumexist.Additionalexperimentationswerealsoconductedtostudytheimpactofbidpa-rametersonequilibriumsolution.Thenumericalresultsshowthatpriceisanimpor-tantfactoranditsvaluecouldsignicantlyaltertheFTRallocationoutcome.TheFTRquantitybidisshowntobeafunctionofriskandvarianceparametersofthenetwork.Withouthighvaluesofriskandvariance,quantitybidcouldbehaveinanonstrategicmannerhigherthebetter.Thecombinationofobligationandoptiontypemixbidmayhavesignicantimpactonthepayosofthebidders,andhenceshouldbeconsideredwhilebidding.Astatisticallydesigned2-levelfactorialexperimentprovidedanidealmeansforin-vestigatingimpactsoffourdierentnetworkrelatedparameterscontingency,LMP,varianceofLMPestimates,andriskcoecientofthebiddersontheequilibriumoutcome.Theresultsshowthatallfourofthefactorssignicantlyimpactequilib-riumFTRsettlement,buttheirinteractionswerenotsignicant.Itwasfoundthatsomecontingenciesinthenetworkcancreatefavorablebiddingpositionsforsomeofthebidders.Theresultsindicatethatanaccurateconsiderationofthenetworkparametersiscrucialindetermininganequilibriumbiddingstrategy.InthejointFTRandenergymarketmodel,LMPsaredirectlyattainedfromtheenergymarketwheregeneratorscompetetomaximizetheirpayos.IntegrationofFTRandenergymarketsrevealsthecomplexrelationsamongthemarketdynamics.Italsoallowstoincorporatedetailcharacteristicsofpowermarketsuchasvaryingcontingencyanddemandscenarios.Generatorsconsiderallcontingencyanddemandscenariosandtrytomaximizetheirexpectedpayos.ExperimentationswiththejointFTRandenergymarketsviaaPJM-5busnetworkexampleshowedthatFTRholdingshaveasignicantimpactonboththeenergymarketstrategiesandthejoint66

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equilibriumpayos.Whenthecontingencyanddemandscenariovariabilitieswerechanged,payosoftheparticipantswereaected.Marginalcostfunctionsofthegeneratorswerealsofoundtohaveinuenceontheequilibriummarketsettlement.Ithasbeenobservedthatdependingontheproductioncapacityandnetworklocation,somegeneratorshaveinuenceoverallnetworkLMPspricesetterswhileothersdonotpricetakers.Themodelandthesolutionapproachpresentedherewillhelpthemarketpar-ticipantstobetterevaluatetheirFTRandenergybiddingstrategies,andthusaidthemarketstoreachanequilibrium,reducinguncertaintyfortheparticipants.Theresearchoutcomeswillalsoserveasvaluabletoolsforthedesignersoftherestruc-turedpowermarkets.Itisalsoexpectedthatrestructuredmarketsdesignedusingtheapproachdevelopedherewillprovideahigherlevelofmarketreliabilitythanwhathasbeenexperimentedsofar.67

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REFERENCES[1]W.W.Hogan.Financialtransmissionrightsformulations.Technicalreport,HarvardElectricityPolicyGroup,2002.[2]T.LiandM.Shahidehpour.Risk-constrainedftrbiddingstrategyintransmissionmarkets.IEEETransactionsonPowerSystems,202:1014{1021,2005.[3]M.Shahidehpour,H.Yamin,andZ.Li.MarketOperationsinElectricPowerSystems.2002.[4]Y.HongandC.Hsiao.Locationalmarginalpriceforecastinginderegulatedelectricmarketsusingarecurrentneuralnetwork.InPowerEng.Soc.WinterMeeting,2001.[5]J.Contreras,R.Espinola,F.J.Nogales,andA.J.Conejo.Arimamodelstopredictnext-dayelectricityprices.IEEETrans.PowerSyst.,18:1014{1020,2003.[6]V.NanduriandT.K.Das.Areinforcementlearningapproachtondingnashequilibriumofmulti-playermatrixgame.Technicalreport,IMSE,UniversityofSouthFlorida,2007.[7]FERC,NOPR.www.ferc.gov.[8]W.W.Hogan.Transmissionmarketdesign.Technicalreport,HarvardElectricityPolicyGroup,2003.[9]O.Alsac,J.M.Bright,S.Brignone,M.Prais,C.Silva,B.Stott,andN.Vempati.Therightstoghtpricevolatility.IEEEPowerandEnergyMagazine,2:47{57,2004.[10]W.W.Hogan.Competitiveelectricitymarketdesign:Awholesaleprimer.Tech-nicalreport,HarvardElectricityPolicyGroup,1998.[11]R.P.Oneill,U.Helman,B.F.Hobbs,W.R.Stewart,andM.H.Rothkopf.Ajointenergyandtransmissionrightsauction:Proposalandproperties.IEEETransactionsonPowerSystems,174:1058{1067,2002.[12]R.D.Tabors.Forwardmarketsfortransmissionthatclearatlmp:Ahybridproposal.InHawaiiInternationalConferenceOnSystemSciences,2001.68

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[13]S.Adamson.Designofatransmissionrightsexchange.Technicalreport,FrontierEconomicsInc.,TwoBrattleSquare,Cambridge,MA02138,2001.[14]C.Hung-Po,S.Peck,S.Oren,andR.Wilson.Flow-basedtransmissionrightsandcongestionmanagement.ElectricityJournal,138:38{58,2000.[15]W.W.Hogan.Flowgatesrightsandwrongs.Technicalreport,HarvardElectricityPolicyGroup,2000.[16]L.Andrew.Canowgatesreallywork?ananalysisoftransmissionconges-tioninpjmmarketfromapril1,1998-april30,2000.Technicalreport,MarketDevelopmentDepartment,PJMInterconnection,2000.[17]J.A.Momoh,M.E.El-Hawary,andR.Adapa.Areviewofselectedoptimalpowerowliteratureto1993.IEEEtransactionsonpowersystems,14:96{111,1999.[18]J.Carpentier.Towardsasecureandoptimalautomaticoperationofpowersystems.BulletindelaSocieteFrancaisedesElectriciens,3:431{447,1962.[19]O.AlsacandB.Stott.Optimalloadowwithsteadystatesecurity.IEEEtransactionsonpowerapparatusandsystems,pages745{754,1974.[20]J.A.Momoh,R.J.Koessler,M.S.Bond,andB.Stott.Challengestooptimalpowerow.IEEETransactionsonPowerSystems,121:444{447,1997.[21]T.YongandR.Lasseter.Optimalpowerowformulationinmarketofretailwheeling.Technicalreport,PSERC-CornellUniversity,2003.[22]R.D.Zimmerman,R.J.Thomas,D.Gan,andC.Murillo-Sanchez.Aninternet-basedplatformfortestinggenerationschedulingauctions.InHawaiiinterna-tionalconferenceonsystemsciences.IEEE,1998.[23]G.P.HarrisonandA.R.Wallace.Maximizingdistributedgenerationcapacityinderegulatedmarkets.IEEE/PESTransmissionandDistributionConferenceandExposition,2003.[24]J.Contreras,A.Losi,M.Russo,andF.F.Wu.Simulationandevaluationofoptimizationproblemsolutionsindistributedenergymanagementsystems.IEEETransactionsonPowerSystems,171:57{62,2002.[25]A.P.S.Meliopoulos,S.W.Kang,andG.Cokkinides.Probabilistictransfercapa-bilityassessmentinaderegulatedenvironment.InHawaiiInternationalConfer-enceOnSystemSciences,2000.[26]M.I.Alomoush.Performanceindicestomeasureandcomparesystemutilizationandcongestionseverityofdierentdispatchscenarios.ElectricPowerSystemsResearch,74:223{230,2005.69

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[27]P.KesselandH.Glavitsch.Estimatingthevoltagestabilityofapowersystem.IEEETransactionsPowerDeliv.,1:346{354,1986.[28]G.HuangandN.Nair.Voltagestabilityconstrainedloadcurtailmentproceduretoevaluatepowersystemreliabilitymeasures.InProceedingsof2000IEEE-PowerEng.Soc.WinterMeeting,2002.[29]StevenStoft.Financialtransmissionrightsmeetcournot:Howtcc'scurbmarketpower.EnergyJournal,1999.[30]J.Bushnell.Transmissionrightsandmarketpower.ElectricityJournal,12:77{85,1999.[31]A.PhilpottandG.Pritchard.Ontransmissionrightsinelectricitypoolmarkets.Technicalreport,UniversityAucklandPress,2001.[32]S.S.Oren.Passivetransmissionrightswillnotdothejob.ElectricityJournal,10:22{33,1997.[33]P.JoskowandP.Tirole.Transmissionrightsandmarketpoweronelectricpowernetworks.ii:Physicalrights.mimeo,MITandIDEI,1998.[34]P.JoskowandP.Tirole.Transmissionrightsandmarketpoweronelectricpowernetworks.i:Financialrights.mimeo,MITandIDEI,1998.[35]J.Bushnell.Transmissionrightsandmarketpower.mimeo,UniversityofCali-forniaEnergyInstitute,1998.[36]G.BautistaandV.Quintana.Screeningandmitigationofexacerbatedmar-ketpowerduetonancialtransmissionrights.IEEETransactionsonPowerSystems,20:213{222,2005.[37]M.L.Puterman.MarkovDecisionProcesses.JohnWileyandSons,NewYorkChichesterBrisbaneTorontoSingapore,1994.[38]T.K.Das,A.Gosavi,S.Mahadevan,andN.Marchalleck.Solvingsemi-Markovdecisionproblemsusingaveragerewardreinforcementlearning.ManagementScience,454,1999.[39]A.Gosavi,N.Bandla,andT.K.Das.Areinforcementlearningapproachtoairlineseatallocationformultiplefareclasseswithoverbooking.IIEtransactions,Specialissueonadvancesonlarge-scaleoptimizationforlogistics,productionandmanufacturingsystems,2002.70

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ABOUTTHEAUTHORCihanBabayigitreceivedhisB.S.degreeinIndustrialEngineeringfromBogaziciUniversity,_Istanbul,Turkey,in2000.HereceivedhisM.S.degreeinIndustrialandManufacturingSystemsEngineeringfromOhioUniversity,Athens,Ohio,in2003.Histhesistopicwasapplicationofnewgeneticalgorithmmodelsintheareaofcellularmanufacturing.HeiscurrentlyaPh.D.studentattheIndustrialandManagementSystemsEngineeringdepartmentatUniveristyofSouthFlorida,Tampa,Florida.HisresearchisintheeldofstochasticgametheoreticmodelingofderegulatedelectricitymarketsandrelatedRLbasedsolutionapproaches.HeisastudentmemberofTheInstituteforOperationsResearchandManagementSciencesINFORMS.


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Equilibrium bidding in joint transmission and energy markets
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ABSTRACT: Participants in deregulated electric power markets compete for financial transmission rights (FTRs) to hedge against losses due to transmission congestion by submitting bids to the independent system operator (ISO). The ISO obtains an FTR allocation, that maximizes sales revenue while satisfying simultaneous feasibility. This FTR allocation remains in place for a length of time during which the participants compete in the energy market to maximize their total payoff from both FTR and energy markets. Energy markets (bi-lateral, day ahead, real time) continue until the the end of the current FTR period, at which time the participants can choose to modify their FTR holdings for the next FTR period. As in any noncooperative game, finding Nash equilibrium bidding strategies is of critical importance to the participants in both FTR and energy markets.In this research, a two-tier matrix game theoretic modeling approach is developed that can be used to obtain equilibrium bidding behavior of the participants in both FTR and energy markets considering the total payoff from FTR and energy. The matrix game model presents a significant deviation from the bilevel optimization approach commonly used to model FTR and energy allocation problems. A reinforcement learning (RL) algorithm is also developed which uses a simulation model and a value maximization approach to obtain the equilibrium bidding strategies in each market. The model and the RL based solution approach allow consideration of multi-dimensional bids (for both FTR and energy markets), network contingencies, varying demands, and many participants. The value iteration based RL algorithm obtains pure strategy Nash equilibrium for FTR and energy allocation.A sample network with three buses and four participants is considered for demonstrating the viability of the game theoretic model for FTR market. A PJM network example with five buses, five generators and three loads is also considered to analyze equilibrium bidding behavior in joint FTR and energy markets. Several numerical experiments on the sample networks are conducted using the approach of statistical design of experiments (DOE) to assess impacts of variations of bid and network parameters on the market outcomes like participant payoffs and equilibrium strategies.
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Dissertation (Ph.D.)--University of South Florida, 2007.
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Adviser: Tapas K. Das, Ph.D.
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Deregulated electricity markets.
Financial transmission rights.
Nash equilibrium.
FTR and energy settlement.
Matrix game.
Reinforcement learning.
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