Two-Stage Chance and Expected Value Constrained Stochastic Unit Commitment

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Two-Stage Chance and Expected Value Constrained Stochastic Unit Commitment Formulations, Algorithms and Case Studies
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Wang, Qianfan
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Doctorate ( Ph.D.)
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University of Florida
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Industrial and Systems Engineering
Committee Chair:
Guan, Yongpei
Committee Members:
Hartman, Joseph C
Lan, Guanghui
Hager, William Ward
Wang, Jianhiu

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algorithms -- optimization -- stochastic
Industrial and Systems Engineering -- Dissertations, Academic -- UF
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Abstract:
Stochastic programming is a common approach to solving decision-making problems under uncertainty in power systems optimization. In this dissertation, we first present a unit commitment model with uncertain wind power output from the perspective of independent system operators (ISOs). The problem is formulated as a chance-constrained two-stage (CCTS) stochastic program. Our model ensures that, with high probability, a large portion of the wind power output at each operating hour will be utilized. Second, CCTS is proposed to study the optimal {bidding} strategy for independent power producers in the deregulated electricity market. Third, an expected value constrained stochastic unit commitment formulation is considered to handle integrating wind power. Finally, we study the stochastic unit commitment problem with uncertain demand response to enhance the reliability unit commitment process for ISOs. Accordingly, Sample Average Approximation (SAA) algorithms are developed for these stochastic unit commitment models. The computational results indicate the proposed algorithms can solve large-scale power grid optimization problems.
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by Qianfan Wang.
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Thesis (Ph.D.)--University of Florida, 2013.
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Adviser: Guan, Yongpei.
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TWO-STAGECHANCEANDEXPECTEDVALUECONSTRAINEDSTOCHASTICUNITCOMMITMENT:FORMULATIONS,ALGORITHMSANDCASESTUDIESByQIANFANWANGADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2013

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c2013QianfanWang 2

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Tomyparents 3

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ACKNOWLEDGMENTS IwouldliketoexpressmyearnestappreciationtomyadvisorDr.YongpeiGuanforhisgeneroussupport,advice,andencouragementduringmydoctoralstudies.IthankDr.Guanforofferingmetheinvaluableopportunitytojoinhisresearchgrouptopursuemyresearchinterests.Withouthisinspiration,guidanceandsupervisionthroughoutmyresearch,thisdissertationworkwouldneverbenished.ManythankstoDr.JianhuiWangforhisinsightfuldiscussiontodeveloptheresearchideainthisworkandforhishelpfulsuggestionstoimprovethequalityofourthreeresearchpapers.Iamindebtedtotherestofmycommitteemembers,Dr.JosephC.Hartman,Dr.WilliamW.HagerandDr.GuanghuiLan,whomadevaluablecommentsandsuggestionsonthisdissertation.IamgratefultothestaffatDepartmentofIndustrialandSystemsEngineering,Ms.CynthiaBlunt,Ms.LeslieSuzanneRedding,fortheiradministrativesupports.IthankmyfriendsatUniversityofFloridafortheirfriendship,mymentorsandmanagersoftheinternshipsatArgonneNationalLaboratory,SASInstituteandAlstomGridforsharingtheirindustryexperienceswithme.Finally,IappreciatemygirlfriendXiaofeiYueandmyfamilyfortheirencouragement,supportandlove. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 8 LISTOFFIGURES ..................................... 10 ABSTRACT ......................................... 11 CHAPTER 1INTRODUCTION ................................... 12 ChallengesofWindPowerIntegration ....................... 13 EffectofWindPoweronISO ......................... 13 EffectofWindPoweronGENCO ....................... 15 StochasticProgramming ............................... 16 Chance-ConstrainedStochasticProgram .................. 17 ExpectedValueConstrainedStochasticProgram .............. 17 DissertationOutline ................................. 18 2STOCHASTICUCWITHUNCERTAINWINDPOWEROUTPUT ........ 21 Nomenclature ..................................... 21 BackgroundandLiteratureReview ......................... 22 MathematicalFormulation .............................. 24 ProblemFormulation .............................. 24 ChanceConstraintDescription ........................ 27 Policy1 ................................. 27 Policy2 ................................. 27 Policy3 ................................. 27 SampleAverageApproximation ........................... 28 ScenarioGeneration .............................. 28 SolutionValidation ............................... 30 Upperbound .............................. 30 Lowerbound .............................. 31 SummaryoftheCombinedSAAAlgorithm .................. 31 MethodsToSolveTheSAAProblem ........................ 32 SortingApproach ................................ 32 StrongFormulation ............................... 32 ComputationalResult ................................ 34 Six-BusSystem ................................. 35 A5-scenarioexample ......................... 36 Experimentsatdifferentrisklevels .................. 37 Experimentsatdifferentscenariosizes ................ 38 5

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Modied118-BusSystem ........................... 39 ConcludingRemarks ................................. 40 3STOCHASTICPRICE-BASEDUC ......................... 42 Nomenclature ..................................... 42 BackgroundandLiteratureReview ......................... 44 MathematicalFormulation .............................. 45 MarketFramework ............................... 45 ProblemFormulation .............................. 46 SampleAverageApproximation ........................... 49 SAAProblem .................................. 50 ConvergenceAnalysisandSolutionValidation ............... 51 SAAAlgorithmFramework ........................... 52 HeuristicsforSolvingEachSAAProblem .................. 53 Innerupperbound ........................... 54 Innerlowerbound ........................... 54 ComputationalResults ................................ 54 ScenarioGenerationforUncertainWindPowerandPrice ......... 55 Three-GeneratorSystem ........................... 56 Optimalsolutionwithtenscenarios .................. 57 Sensitivityanalysisfordifferentrisklevelsandscenariosizes ... 58 ComputationalResultsforaComplicatedSystem .............. 58 Multi-BusSystem ................................ 59 ConcludingRemarks ................................. 61 4STOCHASTICEXPECTEDVALUECONSTRAINEDUC ............. 63 Nomenclature ..................................... 63 Motivation ....................................... 65 MathematicalFormulation .............................. 66 SolutionMethodology ................................ 68 ScenarioGeneration .............................. 68 MILPreformulationofchanceconstraint ............... 69 ReformulationofSAAproblem .................... 70 SolutionValidation ............................... 70 Upperbound .............................. 71 Lowerbound .............................. 72 SummaryoftheCombinedSAAAlgorithm .................. 74 ComputationalResults ................................ 75 Six-BusSystem ................................. 77 Revised118-BusSystems ........................... 78 SAAalgorithmanalysisfor118SW .................. 78 Distributedwindpowersystem118DW ................ 79 ConcludingRemarks ................................. 80 6

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5STOCHASTICUCWITHUNCERTAINDEMANDRESPONSE ......... 82 SolutionMethodologyandCaseStudy ....................... 83 SolutionMethodology ............................. 83 CaseStudy ................................... 84 DeterministicDRvs.stochasticDR .................. 85 Risklevelsanddemandresponseeffect ............... 86 ConcludingRemarks ................................. 87 6CONCLUSIONSANDSUGGESTIONSFORFUTURERESEARCH ...... 88 UnitCommitmentwithUncertainContingency ................... 88 OtherRecoursestoHedgeWindPowerUncertainty ............... 89 APPENDIX:PROOFOFPROPOSITION2.1 ...................... 90 REFERENCES ....................................... 95 BIOGRAPHICALSKETCH ................................ 101 7

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LISTOFTABLES Table page 2-1Busdata ....................................... 35 2-2Generatordata .................................... 36 2-3Fueldata ....................................... 36 2-4Transmissionlinedata ................................ 36 2-5Optimalunitcommitment .............................. 37 2-6Computationalresultsforthesix-bussystemwithdifferentrisklevels-usingpolicy1 ........................................ 38 2-7Computationalresultsforthesix-bussystemwithdifferentrisklevels-usingpolicy2 ........................................ 38 2-8Computationalresultsforthesix-bussystemwithdifferentrisklevels-usingpolicy3 ........................................ 38 2-9Computationalresultsforthe118-bussystem ................... 40 2-10Computationaltimeforthe118-bussystem:MILPandstrongformulation ... 41 3-1Generatordata .................................... 56 3-2Fueldata ....................................... 57 3-3Pumped-storage ................................... 57 3-4Optimalunitcommitment .............................. 57 3-5ComputationalresultsforacomplicatedsystemforeachSAAproblem-heuristicmethod(risklevel:10%) ............................... 60 3-6Resultsofsolutionvalidationforacomplicatedsystem(risklevel:10%) .... 60 3-7Bussettings ...................................... 61 3-8Computationalresultsfordistributedsystem .................... 61 4-1Computationalresultsforthesix-bussystemwithdifferentutilizationrates ... 77 4-2Computationalresultsforthesix-bussystemwithdifferentrisklevels ...... 77 4-3Computationalresultsforthe118-bussystemwithdifferentcombinationsofiterationsandsamplesizes ............................. 79 4-4Computationalresultsfor118DW-withoutexpectedvalueconstraint ...... 80 8

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4-5Computationalresultsfor118DW-withexpectedvalueconstraint ....... 80 5-1Deterministicvs.stochastic ............................. 85 5-2Computationalresultsfordifferentrisks ...................... 86 5-3Computationalresultsfordifferentelasticities ................... 87 9

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LISTOFFIGURES Figure page 2-1ProposedcombinedSAAalgorithm ......................... 33 2-2Six-bussystem .................................... 35 2-3Windutilizationineachoperatinghourforthe5scenariocase ......... 37 2-4PlottingthesolutionsofSAAwithdifferentscenariosizes ............ 39 3-1ProposedSAAalgorithm .............................. 53 3-2Obj.($)oftheSAAproblemwithdifferentrisklevels ................ 58 3-3Obj.($)oftheSAAproblemwithdifferentscenariosizes ............. 59 4-1ProposedSAAalgorithm .............................. 76 5-1Step-wiseapproximationofprice-elasticdemandcurve ............. 85 10

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyTWO-STAGECHANCEANDEXPECTEDVALUECONSTRAINEDSTOCHASTICUNITCOMMITMENT:FORMULATIONS,ALGORITHMSANDCASESTUDIESByQianfanWangMay2013Chair:YongpeiGuanMajor:IndustrialandSystemsEngineering Stochasticprogrammingisacommonapproachtosolvingdecision-makingproblemsunderuncertaintyinpowersystemsoptimization.Inthisdissertation,werstpresentaunitcommitmentmodelwithuncertainwindpoweroutputfromtheperspectiveofindependentsystemoperators(ISOs).Theproblemisformulatedasachance-constrainedtwo-stage(CCTS)stochasticprogram.Ourmodelensuresthat,withhighprobability,alargeportionofthewindpoweroutputateachoperatinghourwillbeutilized.Second,CCTSisproposedtostudytheoptimalbiddingstrategyforindependentpowerproducersinthederegulatedelectricitymarket.Third,anexpectedvalueconstrainedstochasticunitcommitmentformulationisconsideredtohandleintegratingwindpower.Finally,westudythestochasticunitcommitmentproblemwithuncertaindemandresponsetoenhancethereliabilityunitcommitmentprocessforISOs.Accordingly,SampleAverageApproximation(SAA)algorithmsaredevelopedforthesestochasticunitcommitmentmodels.Thecomputationalresultsindicatetheproposedalgorithmscansolvelarge-scalepowergridoptimizationproblems. 11

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CHAPTER1INTRODUCTION Mathematicaloptimizationhasseenabroadrangeofapplicationsintheelectricpowergridoperationandplanning.Applicationsincludedoptimalpowerow,unitcommitment,security-constrainedeconomicdispatch,powersystemexpansion,transmissionplanning,contingencyanalysis,etc.[ 72 ],[ 76 ].Recently,rapiddevelopmentsinrenewableenergyandsmartgridencouragenewresearchofadvancedformulationsandalgorithmsfromtheoptimizationsociety.Inparticular,asthemostcrucialdecisionsforpowersystemoperators,unitcommitment(UC)isofgreatconcerntothepracticeinpowerindustry. TheUCproblemaimstominimizethepowergenerationcostsubjecttoasetofphysicalconstraintsinpowersystemoperations.ThesystemoperatorrunstheUCoptimizationmodeltoobtaintheoptimalscheduling(commitmentandpowergeneration)solutionforeachgeneratortosatisfyelectricitydemand.Mixedintegerprogramming(MIP)isappliedtomodelandsolvetheUCprobleminmostU.S.electricitymarketssuchasPJM,MidwestISO(MISO)andISONewEngland(ISO-NE)[ 25 ]fortheirmarketclearings.Thereaderisreferredto[ 70 ]forcomprehensiveintroductionsofMIP.ThedeterministicUCproblemisformulatedasfollows: minx,ybTx+cTy (1) s.t.Fxf, (1) Gyg, (1) Ax+Byh, (1) Intheaboveformulation,xrepresentstheunitcommitmentdecisionandyrepresentstheeconomicdispatchdecision.Constraints( 1 )describetheunitphysicalconstraints c[2012]IEEE.REPRINTED,WITHPERMISSION,FROM[ 64 66 ] 12

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(e.g.,start-up/shut-down,minup/down-timeconstraints).Constraints( 1 )representthedispatchconstraints(e.g.,reserverequirements,transmissionlimits,demandbalanceconstraints).Finally,constraints( 1 )describethecouplingconstraintsfordecisionsxandy(e.g.generationupper/lowerlimits,ramping-up/downlimits).ChallengesofWindPowerIntegration Anumberofinitiativeshavebeenlaunchedtoincreasethe utilization ofwindpowerindifferentcountriesandregions(e.g.,[ 23 ]and[ 1 ]).Withtheexplosivegrowthofinstalledwindpowercapacity,powersystemoperationparadigmshavefacedprofoundchallenges. Astheelectricitymarketmonitorandcoordinator,theIndependentSystemOperator(ISO)usuallyestablisheseffectivewindpowerintegrationthroughmarketregulation.Amarketparticipant,e.g.,IndependentPowerProducer(IPP)orGenerationCompany(Genco),canactivelycontributetowindpowerintegrationusingtheirgenerationportfoliocombinations.Inthisdissertation,wefocusontheuncertainwindpowerintegrationfromtheperspectivesofISOandIPP. EffectofWindPoweronISO Highpenetrationofwindpowerhasgreatlychallengedthewaythepowersystemhasbeenoperated.Ononehand,windpowerissustainableandhaszerocarbonemissions.Ontheotherhand,windpowerisintermittentandverydifculttopredict.Theuctuationinwindpoweroutputrequiressufcientrampingcapabilityavailableinthesystemtoaddresstheinherentvariabilityanduncertainty.Thetraditionalpowersystemoperationmethods,whichweredesignedtoaddresslimiteduncertaintyinthesystemsuchasloadvariation,havefailedtoconsiderthevariationfromtheunprecedentedscaleofwindpowerutilization.Hence,large-scaleuseofwindpowerproductioncallsforadvancedpowersystemoperationmethodstomaintainthesecurityofsystemoperationsbybetterschedulinggenerationsources. Researchhasbeendonetoimprovepowersystemoperationmethodssuchasunitcommitmenttoaccommodatelargeamountsofwindpower.Ashort-termgeneration 13

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schedulingmodelforWindPowerIntegrationintheLiberalisedElectricityMarkets(WILMAR)wasproposedin[ 11 ],[ 59 ],and[ 58 ].WILMARisastochasticrollingunitcommitmentmodelwithwindpowerscenarios.Themodelhasbeensuccessfullyusedinseveralwindintegrationstudies.Ummelsetal.[ 60 ]analyzedtheimpactsofwindpoweronthermalgenerationunitcommitmentanddispatchintheDutchsystem,whichhasasignicantshareofcombinedheatandpower(CHP)units.BouffardandGaliana[ 17 ]usedastochasticunitcommitmentmodeltocalculatethereserverequirementsbysimulatingthewindpowerrealizationinthescenariosincomparisonwiththetraditionalpre-denedreserverequirements.Ruizetal.[ 50 ]proposedastochasticformulationtomanageuncertaintyintheunitcommitmentproblemandextendedthemodeltoconsideruncertaintyandvariabilityinwindpowerbyusingthesamestochasticframework[ 49 ].Wangetal.[ 63 ]presentedasecurity-constrainedunitcommitment(SCUC)algorithmthattakesintoaccounttheintermittencyandvariabilityofwindpowergeneration.Benders'decompositionwasusedtodecreasethecomputationalrequirementsbroughtbyalargenumberofwindpowerscenarios.Astochasticunitcommitmentmodelwasproposedin[ 61 ].Variouswindpowerforecastsanddifferentlevelsofreserverequirementsweresimulated.Itwasfoundthatwindpowerforecasterrorshavesignicantimpactonunitcommitmentanddispatch. Morerecently,theintervaloptimizationapproachandthescenario-basedmethodarecomparedtosolvestochasticSCUCwithuncertainwindpoweroutputin[ 73 ]. In [ 45 ],bothloadandwindpoweruncertainties areconsidered ,andtheadaptiveparticleswarmoptimizationalgorithm isapplied tosolvethestochasticunitcommitmentproblem.In[ 54 ],aquantile-basedscenariotreeisdiscussedandcomparedwithotherscenariotree formulationsthrough thestochasticunitcommitment framework withhighwindpenetration.In[ 22 ],acomputationalframeworkincorporatingtheWeatherResearchandForecast(WRF)model ispresented .Windpoweruncertaintyquanticationis 14

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addressedbyWRFandthentakenintoaccountinthestochasticoptimizationasscenarios. Mostofthemodelspresentedsofaraimtominimizetheoveralloperatingcost,whichallowsthecurtailmentofwindpower.Thewindpowercurtailmentoccurswhentransmissioncongestionexistsorthereisoversupplyofwindpowerduetothetechnicalconstraintsoftheotherconventionalunitssuchascapacityorminimumon/offtimeconstraints.Inthiscase,theunusedwindpowerbecomesawaste.Thewindpowercurtailmentwilldampentheincentiveofwindpowerinvestmentinthelongrunandmaycausemoreemissionsfromthealternativeenergysources. Forthesereasons, itisdesirablethatthesystemoperatorsareabletoutilizeasmuchwindpoweraspossible.Inpractice,thesystemoperatorsinsomeregionslikeGermanyarerequiredtouserenewableenergysuchaswindpowerasapriorityovertheotherconventionalgenerationsources[ 26 ].Hence,becauseofthewindpoweruncertaintyandvariability,thesystemoperatorshavereliabilityconcernsindispatchingtheirsystemswithlargeamountsofwindpowerwhilewantingtoutilizewindtothelargestpossibleextentatthesametime.Therefore,thesystemoperatorsneedtodetermineaproperunitcommitmentstrategywhichcanbalancetheneedtospillwindpowerduetoreliabilityandotherreasonswhilestilltakingthemostadvantageofwindpower.EffectofWindPoweronGENCO Insteadofincorporatingwindpowerintotheunitcommitmentattheelectricitymarketclearing,analternativewayforanISOistodistributethewindpowerintegrationtomarketparticipants. One common approachistointroducegreencerticatestoensureutilizationofrenewableenergyaseffectivelyaspossible[ 32 ].Thisapproachisbasically imposing thenationaltargetforrenewableenergy utilizationon eitherthedemandsideincludingconsumersordistributioncompanies(e.g.,DenmarkandGermany)[ 41 ],orthegenerationside(e.g.,Italy)[ 41 ].Iftheregulationisapplied to thedemandside,consumersordistributorswillberequiredtoprovethattheyconsume 15

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atleastthespeciedamountofrenewableenergyby submitting certicatestotheauthoritiesatagiventime.Iftheregulationisapplied to thegenerationside,everysupplier,exceptrenewableenergyproducersorimporters,isrequiredtoensurethat a certainpercentageoftheenergy producedbythem ,isrenewableenergy. Thisregulationhasexerteda large impactonelectricitymarketeconomicsandoperations,inparticularmarketparticipantssuchasindependentpowerproducers(IPPs)thatownthermalunitsaswellasrenewablegenerationresources likewindpower Underthisregulation ,eachproducer has to utilize asmuchrenewableenergyaspossibleforpossibleextraprotobtainedfromthegreencerticatemarket.Ontheotherside,anIPPowningtraditionalthermalunitsandwindpowerturbineshastoface two-fold uncertaintiespriceandwindpower output uncertainties when submitting bidstothemarket.Ifthereisamismatchbetween theamountsubmittedin day-aheadand the real-timeoutputs [ 28 ],apenaltywillbeimposed(e.g.,[ 40 ]and[ 46 ]).Dueto the intermittentnatureofwindpower, signicant penalties canbegenerated.To avoidsuch signicant penalties ,anefcientapproachtohandletheuncertaintiesisbasedonthemixedutilizationofwindpowerandpumped-storageunits[ 28 ].StochasticProgramming Stochasticprogrammingisawell-knownmethodtotackletheuncertaintyinthedecision-makingprocess.Inthestochasticprogramming,theuncertainparameterisusuallyassociatedwithaprobabilitydistribution.Asanexample,considertheclassicaltwo-stagestochasticprogram[ 14 ].Therststage(here-and-now)decisionsaremadebeforetheuncertaintyoccurs;thesecondstage(wait-and-see)decisionsvaryaccordingtothescenariorealization.Thegeneralformulationofthetwo-stagestochasticprogramwithxedrecoursecanbedescribedasfollows: mincTx+E[Q(x,)]s.t.Ax=b,x0,(1) where 16

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Q(x,)=minfqy () jWy () =h)]TJ /F6 11.955 Tf 11.96 0 Td[(Tx,y()0g.(1) Here,xdenotestherststagedecisionvariable,y () denotesthesecond-stagedecisionvariable, and isarandomvector.Thisformulationminimizestheobjectivefunctionwhichcontainstheexpectedvalueofrecoursecostonsecond-stagevariables. Inthefollowingsections,weintroducetheothertwobranchesofstochasticprogramming:chance-constrained(orprobabilityconstrained)stochasticprogramandexpectedvalueconstrainedstochasticprogram.Chance-ConstrainedStochasticProgram Pioneeringworkinchance-constrainedoptimizationwasdonebyCharnes,CooperandSymonds[ 20 ].Chanceconstraintshavebeenappliedtoalargeclassofoptimizationproblemsincludingfacilitylocation[ 12 ],callcenterstafngservice[ 30 ],nancialportfoliooptimization[ 27 ],andoptimalpowerow[ 75 ]. Thegeneralchance-constrainedstochasticprogramcanbedescribedasfollows: minx2Xf(x)s.t.PrfG(x,)0g1)]TJ /F8 11.955 Tf 11.95 0 Td[(,(1) whereX
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thattheCVaRconstrainedproblemisaspecialcaseoftheexpectedvalueconstrainedproblem.Thegeneralexpectedvalueconstrainedstochasticprogramproposeanadditionalconstraintinto( 1 )asfollows: E[G(x,y(),)] ` (1) Expectedvalueconstrainedstochasticprogramhasbeenappliedtosupplychainnetworkdesignandportfoliooptimization[ 68 ].DissertationOutline InChapter 2 ,wepresentaunitcommitmentproblemwithuncertainwindpoweroutput.Theproblemisformulatedasachance-constrainedtwo-stage(CCTS)stochasticprogram.Ourmodelensuresthat,withhighprobability,alargeportionofthewindpoweroutputateachoperatinghourwillbeutilized.Theproposedmodelincludesboththetwo-stagestochasticprogramandthechance-constrainedstochasticprogramfeatures.Thesetypesofproblemsarechallengingandhaveneverbeenstudiedtogetherbefore,eventhoughthealgorithmsforthetwo-stagestochasticprogramandthechance-constrainedstochasticprogramhavebeenrecentlydevelopedseparately.Inthischapter,acombinedsampleaverageapproximation(SAA)algorithmisdevelopedtosolvethemodeleffectively.TheconvergencepropertyandthesolutionvalidationprocessofourproposedcombinedSAAalgorithmisdiscussedandpresentedinthischapter.Finally,computationalresultsindicatethatincreasingtheutilizationofwindpoweroutputmightincreasethetotalpowergenerationcost,andourexperimentsalsoverifythattheproposedalgorithmcansolvelarge-scalepowergridoptimizationproblems. InChapter 3 ,weproposeanoptimal bidding strategyforIPPsinthederegulatedelectricitymarket.TheIPPsareassumedtobepricetakers,whoseobjectivesaretomaximizetheirprotsconsidering priceandwindpoweroutputuncertainties,whileensuringhighwindpowerutilization .Theproblemisformulatedasatwo-stage 18

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stochasticprice-basedunitcommitment problem withchanceconstraintstoensurewindpowerutilization.Inourmodel,therststagedecisionincludesunitcommitmentand quantityofelectricity submitted to theday-aheadmarket.Thesecondstagedecisionincludesgenerationdispatch, actualusageof windpower,and amountof energyimbalance betweentheday-aheadand real-timemarkets.Thechanceconstraintisappliedtoensure acertainpercentageof windpowerutilization soasto complywith renewableenergyutilizationregulations Finally,theSAAapproach isappliedtosolvetheproblem,and the computationalresultsarereportedfortheproposedSAAalgorithmshowingthesensitivityofthetotalprotastherequirementofwindpowerutilizationchanges. Chapter 4 proposes an expectedvalueandchanceconstrainedstochasticoptimizationapproachfortheunitcommitmentproblemwithuncertainwindpoweroutput.Inourmodel,theutilizationofwindpowercanbeadjustedbychangingtheutilizationrateintheproposedexpectedvalueconstraint.Meanwhile,thechanceconstraintisusedtorestricttheprobabilityofloadimbalance.WeusetheSAAmethodtotransformtheexpectedobjectivefunction,theexpectedvalueconstraintandthechanceconstraintintosampleaveragereformulations.Furthermore,wediscussacombinedSAAframeworkthatconsidersboththeexpectedvalueandthechanceconstraintstoconstructstatisticalupperandlowerboundsfortheoptimizationproblem.Finally,wetesttheperformanceoftheproposedalgorithmwithdifferentutilizationratesanddifferentrisklevelsforasix-bussystem.WealsostudyarevisedIEEE118-bussystemtoshowthescalabilityofourmodelandalgorithm. Chapter 5 studiesthestochasticunitcommitmentproblemwithuncertaindemandresponse toenhancethereliabilityunitcommitmentprocessforISOs .Althoughdemandresponse(DR)encouragescustomerstovoluntarilyscheduleelectricityconsumptionbasedonpricesignals,theresponsefromtheconsumersidecouldbeuncertainduetoavarietyofreasons.Inthischapter,weuseastochasticrepresentationofDRby 19

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scenario,andeachscenario correspondsto aprice-elasticdemandcurve.Contingencyconstraintsareconsideredandinaddition,achanceconstraintisappliedtoensurethelossofloadprobability(LOLP)lowerthanapre-denedrisklevel.Then,theSAAmethodisappliedtosolvetheproblem. Finally,Chapter 6 concludesthedissertationandprovidesgeneralsuggestionsforfutureresearch. 20

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CHAPTER2STOCHASTICUCWITHUNCERTAINWINDPOWEROUTPUTNomenclature A.SetsandIndices BG Setofbuseswiththermalgenerationunits. BW Setofbuseswithwindfarms. B Setofallbuses. E Setoftransmissionlineslinking buspairs b Setofgeneratorsatbusb. T Timehorizon(e.g.,24hours). B.Parameters kbi,j Lineowdistributionfactorfortransmissionlinelinkingbusiandbusjdue tothenetinjectionatbusb. Ui,j Transmissionowlimitontransmissionlinewhichlinksbusiandbusj. Dbt Demandatbusbintimeperiodt. bi Start-upcostforgeneratoriatbusb. bi Shut-downcostforgeneratoriatbusb. bi Costofgeneratingminimumpoweroutputforgeneratoriifitisturnedonat busb. Fc( qbit ) Fuelcostforgeneratoriatbusbintimeperiodtwhenitsgenerationis qbit t Penaltycost perunitofenergy shortage intimeperiodt Gbi Minimum-uptimeforgeneratoriatbusb. Hbi Minimum-downtimeforgeneratoriatbusb. c[2012]IEEE.REPRINTED,WITHPERMISSION,FROM[ 64 ] 21

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Rt Amountofspinningreserveneeded forthewholepowersystem intime periodt. URbi Ramp-upratelimitofgeneratoriatbusb. DRbi Ramp-downratelimitofgeneratoriatbusb. LBbi Lowerboundofelectricitygeneratedbygeneratoriatbusb. UBbi Upperboundofelectricitygeneratedbygeneratoriatbusb. wbt() Arandomparameterindicatingthe windpoweroutputoravailablecapacity atbusbintimeperiodt. C.DecisionVariables QGt Totalamountofelectricitygeneratedbythermalunitsintimeperiodt. QWt Totalamountofwindpowercommittedtobe utilized(delivered) intime periodt. ^qbt Amountofwindpowercommittedtobeutilized(delivered) atbusbintime periodt. qbit Amountofelectricitygeneratedbygeneratoriatbusbintimeperiodt. obit Binaryvariabletoindicateifgeneratoriatbusbisonintimeperiodt. ubit Binaryvariabletoindicateifgeneratoriatbusbisstartedupintimeperiod t. vbit Binaryvariabletoindicateifgeneratoriatbusbisshutdownintimeperiod t. Sbt() Amountofenergy shortage atbusbintimeperiodt( second-stagedecision variable).BackgroundandLiteratureReview Inthischapter,wepresentanovelunitcommitmentmodelthatcantakeintoaccountwindpowerforecastingerrorswhilemaintainingthesystemreliabilityincase 22

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ofsuddenuctuationsinwindpoweroutput. Inourmodel,thesystemoperatorscanrequestaportionofthewindpoweroutputtobeutilizedatacertainprobability.Inthisway,theriskofalargeamountofwindbeingcurtailedwillbeadjustablebytheoperators. Weusethechance-constrainedoptimizationtechniquetoformulatetheproblemto ensurethat,withhighprobability,alargeportionofthewindpoweroutputateachoperatinghourwillbeutilized. Sincethewindpoweruncertaintyiscapturedbyanumberofwindpowerscenariosinourapproach, alargepartofthewindpoweroutput,denedbythesystemoperators,willbeutilizedinalargeportionofscenarios. Thesystemoperatorscanrenetheunitcommitmentsolutionbydeningappropriateattitudestoward risk andcost[ 39 ].Somesystemoperatorsmaypreferalowerriskofcurtailingthewindpowerwhiletheothersmaybepronetospillwindpowerwhensystemconstraintstakeeffect.Theriskpreferencereectsthevarioustreatmentsofwindpowerinreality[ 15 ]. Chance-constrainedoptimizationhasbeen previouslystudiedtosolve thestochasticunitcommitmentproblemwithuncertainloadin[ 42 ]andtransmissionplanningproblemin[ 74 ].In[ 42 ],withtheconsiderationofthehourlyloaduncertaintyanditscorrelationstructure,theunitcommitmentproblemisinitiallyformulatedasachanceconstrainedoptimizationprobleminwhichtheloadisrequiredtobemetwithaspeciedhighprobabilityovertheentiretimehorizon.Inthesolutionapproach,theprobabilityconstraintisreplacedbyasetofseparateprobabilityconstraintseachofwhichcouldbeinvertedtoobtainasetofequivalentdeterministiclinearinequalities.Finally,thedeterministicformofthestochasticconstraintisusedinsolvingtheproblemiteratively.In[ 74 ],thechanceconstraintisappliedtotransmissionplanning,anditisintheformthatthenot-overload-probabilityforthetransmissionlineisrequiredtobemorethanaspeciedprobability.Two-stepoptimizationprocesswithageneticalgorithmisappliedtosolvetheproblem. 23

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Inthischapter,thechanceconstraintisappliedtodescribepoliciestoensuretheutilizationofwindpoweroutput.Inourapproach,differentpoliciesleadtodifferenttypesofchanceconstraints. Someofthesetypesofconstraintscouldnotbeinvertedtoobtainequivalentdeterministiclinearinequalities.Thus,thealgorithmdevelopedin[ 42 ]couldnotbedirectlyappliedheretosolveourproblem.Inaddition,thealgorithmproposedin[ 74 ]wasnotdesignedtosolvetwostagestochasticprograms,anditalsocouldnotbedirectlyappliedheretosolveourproblem. Therefore,weproposetostudyasampleaverageapproximation(SAA)algorithmtosolvetheproblem.Ourapproachcanprovideasolutionthatconvergestotheoptimaloneasthenumberofsamplesincreases.MathematicalFormulation ProblemFormulation Ourmodelcontainsbothchance-constrainedandtwo-stagestochasticprogramfeatures.Thegeneraltwo-stageandchance-constrainedstochasticprogramsaredescribedinSection 1 Inthischapter,wedevelopachance-constrainedtwo-stagestochasticunitcommitmentformulation,combining( 1 ),( 1 ),and( 1 ),toaddressuncertainwindpoweroutput.Wecallit a CCTSprogram,whichcontainsbothchance-constrainedandtwo-stagestochasticprogramfeatures. Inourmodel,theonlyuncertaintyconsideredisthewindpoweravailability.Therststageofthestochasticprogramconsistsofthetraditionalunitcommitmentproblemwithtransmissionconstraintsandthedecisiononthetotalamountofwindpowercommittedtobeutilized(delivered),formedinlightofaprobabilisticwindpowerforecast.Thesecondstagerepresentsthepenaltycostduetoenergyshortageoncetheactualwindpoweroutputisknown.Thechanceconstraintensurestheutilizationofwindpoweroutput. Thedetailedformulationisdescribedasfollows(denotedasthetrueproblem). 24

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minXb2BGTXt=1Xi2b(biubit+bivbit+biobit+Fc( qbit ))+TXt=1 t E"Xb2BWSbt()# (2) s.t.LBbiobit qbit UBbiobit,8i2b,8b,8t (2) )]TJ /F6 11.955 Tf 9.3 0 Td[(obi(t)]TJ /F5 7.97 Tf 6.59 0 Td[(1)+obit)]TJ /F6 11.955 Tf 11.96 0 Td[(obik0, (2) 1k)]TJ /F3 11.955 Tf 11.96 0 Td[((t)]TJ /F3 11.955 Tf 11.95 0 Td[(1)Gbi,8i2b,8b,8tobi(t)]TJ /F5 7.97 Tf 6.59 0 Td[(1))]TJ /F6 11.955 Tf 11.96 0 Td[(obit+obik1, (2) 1k)]TJ /F3 11.955 Tf 11.96 0 Td[((t)]TJ /F3 11.955 Tf 11.95 0 Td[(1)Hbi,8i2b,8b,8t)]TJ /F6 11.955 Tf 9.3 0 Td[(obi(t)]TJ /F5 7.97 Tf 6.59 0 Td[(1)+obit)]TJ /F6 11.955 Tf 11.96 0 Td[(ubit0,8i2b,8b,8t (2) obi(t)]TJ /F5 7.97 Tf 6.59 0 Td[(1))]TJ /F6 11.955 Tf 11.96 0 Td[(obit)]TJ /F6 11.955 Tf 11.95 0 Td[(vbit0,8i2b,8b,8t (2) qbit )]TJ ET BT /F6 11.955 Tf 96.9 -267.47 Td[(qbi(t)]TJ /F5 7.97 Tf 6.59 0 Td[(1) (2)]TJ /F6 11.955 Tf 11.96 0 Td[(obi(t)]TJ /F5 7.97 Tf 6.58 0 Td[(1))]TJ /F6 11.955 Tf 11.95 0 Td[(obit)LBbi+(1+obi(t)]TJ /F5 7.97 Tf 6.59 0 Td[(1))]TJ /F6 11.955 Tf 11.96 0 Td[(obit)URbi, (2) 8i2b,8b,8t qbi(t)]TJ /F5 7.97 Tf 6.59 0 Td[(1) )]TJ ET BT /F6 11.955 Tf 115.58 -321.26 Td[(qbit (2)]TJ /F6 11.955 Tf 11.96 0 Td[(obi(t)]TJ /F5 7.97 Tf 6.58 0 Td[(1))]TJ /F6 11.955 Tf 11.95 0 Td[(obit)LBbi+(1)]TJ /F6 11.955 Tf 11.95 0 Td[(obi(t)]TJ /F5 7.97 Tf 6.59 0 Td[(1)+obit)DRbi, (2) 8i2b,8b,8tQGt+QWt=Xb2BDbt,8t (2) Xb2BW^qbt=QWt,8t (2) Xb2 BG Xi2b qbit =QGt,8t (2) Xb2BGXi2bUBbiobit Rt +Xb2BDbt,8t (2) )]TJ /F6 11.955 Tf 9.3 0 Td[(UijXb2Bkbij(^qbt+Xn2bqbnt)]TJ /F6 11.955 Tf 11.96 0 Td[(Dbt)Uij,8(i,j)2E,8t (2) Pr( G(x,)0 )1)]TJ /F8 11.955 Tf 11.95 0 Td[( (2) Sbt( )=maxf0,^qbt)]TJ /F6 11.955 Tf 11.96 0 Td[(wbt()g ,8b,8t, 2
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Intheaboveformulation, thedecisionvariablesSbt()aresecond-stagevariables,correspondingtoyin( 1 )( 1 ),andothersarerststagevariables,correspondingtoxin( 1 )( 1 ).WedeneFc( qbit )=cbi( qbit )2+bbi qbit .Noteherebecauseofconstraints( 2 ),wehavebiobit+Fc( qbit )=(cbi( qbit )2+bbi qbit +bi)obit .Theobjectivefunction( 2 )iscomposedof powergenerationcostsintherststageandpenaltycostduetoenergyshortageinthesecondstage.Inourmodel,foreachtimeperiodinthesecondstage,ifthewindpoweroutputislargerthantheamountofwindpowercommittedtobeutilized,theexcesswindpowercanbecurtailedwithoutpenalty,becausetheutilizationofwindpowerusageisguaranteedbythechanceconstraint( 2 ).Excesswindpowerwillnotbesoldtoathirdpartywiththeconsiderationofpotentialtransmissioncongestionandotherconstraints.Ifthewindpoweroutputislessthantheamountofwindpowercommittedtobeutilized,penaltycostwillbetriggeredduetoenergyshortage. ThehourlyUCconstraintslistedaboveincludetheunitgenerationcapacityconstraints( 2 ),unitminimum-uptimeconstraints( 2 ) (e.g.,whenobi(t)]TJ /F5 7.97 Tf 6.59 0 Td[(1)=0andobit=1,itmeansthattheunitisturnedonintimeperiodt.TheninthefollowingGbitimeperiods,itshouldbeon,becausewehave)]TJ /F3 11.955 Tf 9.3 0 Td[(0+1)]TJ /F6 11.955 Tf 12.12 0 Td[(obik0atthismoment,andobik=1canbeguaranteedforatleastGbitimeperiods) ,unitminimum-downtimeconstraints( 2 ) (theexplanationissimilartotheonefor(6)) ,unitstart-upconstraints( 2 ),unitshut-downconstraints( 2 ),unitrampingupconstraints( 2 ),unitrampingdownconstraints( 2 ),systempowerbalanceconstraints( 2 2 2 ),systemspinningreserverequirements( 2 ),andtransmissioncapacityconstraint s ( 2 )(Notethatthecalculationofkbijisthesameastheonedescribedin[ 67 ]).Constraints( 2 )-( 2 )arerststageconstraints;constraint ( 2 )indicatesthatG(x,)0(fornotationbrevity,weusextorepresentalltherststagedecisionvariables)shouldbesatisedwithaprobabilityofatleast1)]TJ /F8 11.955 Tf 12.09 0 Td[(;constraint( 2 )isdescribedindetailinthefollowingpartB; constraints( 2 ) are thesecond-stageconstraint s which indicatetheamountof 26

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energyshortage,incasethewindpoweroutputislessthantheamountofwindpowercommittedtobeutilized. ChanceConstraintDescription Weapplythreepoliciestoguaranteetheutilizationofwindpoweranddevelopthecorrespondingthreetypesofchanceconstraints withconstraintmappingsG1,G2andG3,respectively. Policy1 ConstraintmappingG1denes thatfor theentire time planning horizon(24h),thereisatleast1)]TJ /F8 11.955 Tf 12.55 0 Td[(chancetheusageofthetotalwindpowergenerationislargerthanorequalto, 0<<100% Pr( TXt=1Xb2BWwbt())]TJ /F4 7.97 Tf 17 14.94 Td[(TXt=1QWt0)1)]TJ /F8 11.955 Tf 11.96 0 Td[(.(2)Policy2 ConstraintmappingG2denes thatfor eachparticularoperating houronthetime planning horizon,thereisatleast1)]TJ /F8 11.955 Tf 12.58 0 Td[(chancetheusageofthewindpowerislargerthanorequalto 0<<100% Pr( Xb2BWwbt())]TJ /F6 11.955 Tf 11.95 0 Td[(QWt0)1)]TJ /F8 11.955 Tf 11.95 0 Td[( 8t. (2) Policy3 ConstraintmappingG3 considersthejointprobabilitywhichisatleast1)]TJ /F8 11.955 Tf 12.14 0 Td[(chancetheusageofwindpowerislargerthanorequalto 0<<100% forevery operatinghour Pr( Xb2BWwb())]TJ /F6 11.955 Tf 11.96 0 Td[(QW0)1)]TJ /F8 11.955 Tf 11.95 0 Td[( (2) wherewb()=[wb1(),wb2(),...,wbT()]T,QW=[QW1,QW2,...,QWT]T,and0isaTdimensionalvectorofzeros. 27

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Fromabove,itcanbeobservedthatpolicy3ismost restrictive ,whilepolicy1istheleast restrictive one.Inpolicy3,werequirethatatleastofwindpowerisutilizedduringeachofthe24operatinghourstomakeanoutcomeoftherandomwindgenerationamountaqualiedone(i.e.,satisfyingthechanceconstraint).Thus,policy3ismore restrictive thanpolicy2,whichismore restrictive thanpolicy1.SampleAverageApproximation Sampleaverageapproximation(SAA)isaneffectivemethodtosolvechance-constrainedandtwo-stage stochastic problems.ThebasicideaofSAAistoapproximatethetruedistributionofrandomvariableswithanempiricaldistributionbyMonteCarlosamplingtechnology.AnumberoftheoreticalresearchandcomputationalstudiesofSAAhavebeendevelopedforchance-constrainedstochasticproblems( e.g., [ 43 ]and[ 37 ])andtwo-stagestochasticproblems( e.g., [ 34 ]and[ 5 ]). However,thereisnoexistingSAAmethodtosolvethemodelthatcontainsbothchance-constrainedandtwo-stagestochasticprogramfeatures. Inthissection,wedevelopacombinedSAAalgorithmtosolvetheCCTSprogram.ThecombinedSAAframeworkcontainsthreeparts:scenariogeneration,convergenceanalysis,andsolutionvalidation.ForeachSAAproblem,wesolvethecorresponding m ixedi ntegerl inear p rogram(MILP)efcientlybydevelopingastrongformulation.Thedetailsareshowninthefollowingsubsections.ScenarioGeneration InSAA,the truedistributionofwindpowergenerationisreplacedbyanempiricaldistributionusingcomputersimulation. WeuseMonteCarlosimulationtogeneratescenarios. AssumethewindpowerissubjecttoamultivariatenormaldistributionN(,)( oneofmanypossibledistributionsofwindpower )foreverytimeperiodt,where vector ischosenastheforecastedwindpowerand matrix describesthevolatility.TheMonteCarlosimulation generates alargenumberofscenarios,eachwiththesameprobability1=N.Ineachscenario,thereare24hourly,randomwindpower outputs basedontheforecastedgeneration.TodecreasethevarianceofsimpleMonte 28

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Carlosimulation,theLatinhypercubesampling(LHS)isemployedtomakethestatisticaldistributionttherealdistributionbetter[ 63 ]. Afterthescenariosaregenerated (e.g.,Nscenarios) ,theexpectedvaluefunctionE[Pb2BWSbt( )]isestimatedbythesampleaveragefunctionN)]TJ /F5 7.97 Tf 6.59 0 Td[(1PNj=1Pb2BWSbt( j)(see,e.g.,[ 34 ]).Ontheotherhand, ingeneral, thechanceconstraintcanbeestimatedbyanindicatorfunctionN)]TJ /F5 7.97 Tf 6.59 0 Td[(1PNj=11(0,1)(G(x, j)) (see,e.g.,[ 43 ]),whichrequiresthatacertainpercentageofthesamplessatisfythechanceconstraint. Thevalueoftheindicatorfunction1(0,1)(G(x, j))isequaltoonewhenG(x, j)2(0,1)orzerowhenG(x, j)0. Thecorrespondingformulationisshownasfollows (denotedastheSAAproblem) : minXb2BGTXt=1Xi2b(biubit+bivbit+biobit+Fc( qbit ))+N)]TJ /F5 7.97 Tf 6.58 0 Td[(1TXt=1NXj=1 t Xb2BWSbt( j)s.t.( 2)-221()]TJ /F3 11.955 Tf 21.25 0 Td[(2 ))]TJ /F3 11.955 Tf 11.96 0 Td[(( 2)-222()]TJ /F3 11.955 Tf 21.26 0 Td[(13 )N)]TJ /F5 7.97 Tf 6.59 0 Td[(1NXj=11(0,1)(G(x, j)) ^qbtwbt( j)+Sbt( j)(8t;8b2BW;j=1,2,...,N) qbit ,^qbt,Sbt( j)0;obit,ubit,vbit2f0,1g,8t,8i,8b. First,as thesamplesizeNgoestoinnity,wecanprovethatthe objective oftheaboveformulation converges tothatofthetrueproblem asshowninthefollowingproposition. Proposition2.1. Let^NrepresenttheobjectivevalueoftheSAAproblem,andrepresenttheobjectivevalueofthetrueprogram.Wehave^N!andD(^xN,x)!0w.p.1asN!1,whereD(^xN,x)representsthedistancebetweentheoptimalsolution^xNfortheSAAproblemandtheoptimalsolutionxforthetrueproblem. Proof. Thedetailsoftheconvergenceproofaregivenin AppendixA Next,wediscussthesolutionvalidationprocessinthefollowingsubsection. 29

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SolutionValidation Solutionvalidation forthetwo-stageandchance-constrained problems havebeenwellstudied in [ 43 ] and [ 5 ], respectively .Inthissection,wedevelopacombinedalgorithm that embedsthesolutionvalidationofthechance-constrainedproblemintothatofthetwo-stageproblem. Assumethatxisanoptimalsolution fortheSAAproblem ,andvisthecorrespondingobjectivevalue. ForagivencandidatesolutionfortheSAAproblem,solutionvalidationprovidesaschemetovalidateitsqualitybyobtainingupperandlowerboundsforthecorresponding optimalobjectivevalue .Weconstructtheupperandlowerboundsasfollows: Upperbound SinceCCTScontainsachanceconstraint,westartwiththevericationoffeasibilityofthegivensolutionx.Todo this ,we rst estimatethe trueprobabilityfunctionof the chanceconstraint q(x)=PrfG(x,)>0g. (2) Followingthemethod described in[ 6 ]and[ 43 ],weconstruct a (1)]TJ /F8 11.955 Tf 12.06 0 Td[()-condenceupperboundonq(x): U(x)=^qN0(x)+zp ^qN0(x)(1)]TJ /F3 11.955 Tf 12.2 0 Td[(^qN0(x))=N0,(2) whereN0isthesamplesizeforthevalidationofthechanceconstraint,and^qN0(x)istheestimatedvalueofq(x)forthegivensamplesizeN0. Ifthisupperboundofq(x)islessthantherisklevel ,thenxisfeasiblewith condence level(1)]TJ /F8 11.955 Tf 12.21 0 Td[(). Then ,wecanevaluatethecorrespondingupperboundof the optimalvalueforthe second -stagepartinCCTS, thesameasthevalidationprocessforthenormaltwo-stagestochasticproblemasdescribedin [ 5 ] : U(v)=cTx+1 N0N0Xn=1Q(x,n).(2) 30

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ItiseasytoseethatU(v)istheupperboundforCCTS. Lowerbound Togetthelowerboundfortheobjectivevaluev,wetake ^S iterations.Foreachiteration1s ^S ,weruntheNscenarioSAAproblemMtimes.FortheseMruns,wefollowthesameschemeastheonedescribedin[ 6 ]and[ 43 ]topickthe Lth smallestoptimalvalue,denotedasvLs,astheapproximatedlowerboundforthechance-constrainedpartwith condence level(1)]TJ /F8 11.955 Tf 12.81 0 Td[(),whereLiscalculatedasdescribedin[ 43 ].Finally,takingtheaverageoffvLs,1s ^S gprovidesthelowerboundforCCTS. SummaryoftheCombinedSAAAlgorithm Inthealgorithm,weputthecalculationoftheupperboundforCCTSintheloopofthecalculationofthelowerboundforthechance-constrainedpartinordertospeedupthealgorithm.The proposedcombinedSAAalgorithm issummarized inthefollowingsteps(alsoseeowchartinFig. 2-1 ). 1. For s =1,2,..., ^S ,repeatthefollowingsteps: (a) Form=1,2,...,M,repeatthefollowingsteps: i. Solve theassociatedSAAwithNscenarios.Denotethesolutionasxmandtheoptimalvalueasvm; ii. Generatescenarios1,2,...,N0.Estimateq( xm )by^qN0(xm)anduse( 2 )togetU(xm); iii. IfU(xm),goto(d);else,skip(d)andgotonextiteration; iv. Estimatethecorrespondingupperbound forCCTS using( 2 ), basedontheN0scenariosgeneratedin(b) ; (b) Pickthesmallestupperboundin Step (1)astheapproximatedupperbound ^gs ; (c) SorttheMoptimalvalues obtainedinStep (1)innondecreasingorder,e.g.,v1v2...vM.PicktheLthoptimalvaluevLanddenoteitasvLs. 2. TakingtheaverageofvL1,vL2,...,vL ^S ,wegetthelowerboundv=1 ^S P ^S s=1vLs. 31

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3. Takingtheminimumof^g1,^g2,...,^g ^S ,wegettheupperbound^g=min1s ^S ^gs. 4. Estimatetheoptimalitygapgivenby (^g)]TJ /F3 11.955 Tf 12.24 0 Td[(v)=v100% .MethodsToSolveTheSAAProblem TheSAAproblemisanMILP. Wecanusethestandardbranchandboundalgorithmwhichisimplementedinmostcommercialsolvers.Themainproblemishowtosolvethe SAAproblem effectivelywiththechanceconstraintsunderdifferentpolicies.Inthissection,weintroduce a sortingapproachfor policies 1and2. Forpolicy3,wederive a strongformulationasstudiedin[ 37 ]tospeedupthealgorithm. SortingApproach Aftertaking samples ,wecan simplify ( 2 )bysortingtheright-hand-sidevaluesoftheconstraintsforeachsample,(i.e. ,windpowerin eachday) andpicking thed(1)]TJ /F8 11.955 Tf 12.39 0 Td[()Neth right-hand-sidevalue toconstructadeterministicconstraint.Similarly,aftertakingsamples,wecansimplify( 2 )bysortingtheright-hand-sidevaluesoftheconstraintsforeachsample(i.e. ,windpowerin eachhour)andpickingthed(1)]TJ /F8 11.955 Tf 10.09 0 Td[()Neth right-hand-sidevalue toconstructadeterministicconstraint.StrongFormulation Itcanbeobservedthat thesortingmethoddoesnot workfor ( 2 )becausethesortingalgorithmcannothandlethejointprobability case describedin( 2 ).Instead, reformulatingasanMILPallowssolutionoftheproblemincorporating( 2 ). ForagivensamplesizeN, constraint( 2 )canbereformulatedasfollows: Xb2BWwbt(j))]TJ /F6 11.955 Tf 11.96 0 Td[(QWtMzj (2) (t=1,2,...,T;j=1,2,...,N) (2) NXj=1zjN (2) zj2f0,1g (2) (j=1,2,...,N). (2) 32

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. Initialization SolveSAAproblem Estimateq(x) m=m+1 U(x)? EstimateUBusing( 2 ) m==M? Pick^gs SortandpickvLs s== ^S ? s=s+1 TaketheaverageoffvLsgtogetlowerboundv ^g=min1s ^S ^gs Estimatetheoptimalitygap no yes no yes no yes Figure2-1. ProposedcombinedSAAalgorithm 33

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Thestar-inequalitiescan speedup thecomputationoftheaboveMILPmodel(see,e.g.,[ 43 ],[ 37 ],and[ 29 ]).Moreover,ithasbeenprovedthattheMILPmodelcanbetransformedintoastrongformulationafteraddingthestar-inequalities[ 37 ].Todothis,weintroduceanewsetofbinaryvariablesfrtj:j=1,2,...,q;t=1,2,...,Tganddenehtj=Pb2BWwbt(j).Withoutlossofgenerality,weassumethatht1ht2htN.The strong formulationisdescribedasfollows: rtj)]TJ /F6 11.955 Tf 11.96 0 Td[(rt(j+1)0 (2) (j=1,2,...,q;t=1,2,...,T) (2) z[j])]TJ /F6 11.955 Tf 11.96 0 Td[(rtj0(j=1,2,...,q;t=1,2,...,T) (2) QWt+qXj=1(htj)]TJ /F6 11.955 Tf 11.95 0 Td[(ht(j+1))rtjht1 (2) (t=1,2,...,T) (2) NXj=1zjN (2) rtj,zj2f0,1g, (2) where[j]representsthescenarioindexcorrespondingtothejthlargesthvalueintimeperiodt, andinequalities( 2 )arethestarinequalities. ComputationalResult Inthissection,asix-bussystemandarevised118-bussystemarestudiedtoillustratetheproposedalgorithms.Inthesix-bussystem,werun the computationalexperimentsatdifferentrisklevelstocomparetheresultsofdifferentpolicies.Thesolutionvalidationisneglectedforsimplicity,and computational experimentsatdifferentsamplesizesaretestedto verify theconvergencepropertyofthecombinedSAAalgorithm.Inthe revised 118-bussystem,werunthecomputationalexperimentstotesttheentirecombinedSAAalgorithm described inSectionsIIIandIV.Thealgorithm 34

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Table2-1. Busdata BusIDTypeUnitWindFarmHourlyLoad(MW) B1ThermalG1--B2ThermalG2--B3Thermal--300B4Wind-W1300B5Thermal--200B6ThermalG3-iscodedinC++usingCPLEX12.1.All experiments are implemented onacomputerworkstationwithIntelQuadCore2.40GHzand8GBmemory.Six-BusSystem Thesix-bussystemincludesthreegenerators,onewindfarm, threeloads ,andsixtransmissionlines.ThelayoutofthesystemisdepictedinFig. 2-2 ,andthecharacteristicsofthebuses,thermalunits,andtransmissionlinesaredescribedinTables 2-1 2-4 B4 B5 B6 B1 B2 B3 G1 G2 W1 G3 Figure2-2. Six-bussystem WeassumethewindfarmislocatedatbusB4,=85%,andt=600,8t.Thewindpowerisassumedmultivariatenormaldistributed,withthehourlymeanforecastedoutputsrangingbetween10)]TJ /F3 11.955 Tf 12.73 0 Td[(100andastandarddeviationof45%oftheexpectedvalues.TorunthemodelinCPLEXeffectively,weusetheinterpolationmethod[ 57 ]to 35

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Table2-2. Generatordata UnitLowerUpperMin-downMin-upRamp(MW)(MW)(h)(h)(MW/h) G11003002450G2802003340G31503503215 Table2-3. Fueldata Unitb(MBtu/c(MBtu/Start-upFuelFuelPrice(MBtu)MWh)MW2h)(MBtu)($/MBtu) G15060.00041001.2469G2405.50.00013001.2461G3604.50.00501.2462 approximatethefuelcostfunction.Apiecewiselinearfunctionreplacesthefuelcostfunctionin( 2 ).A5-scenarioexample Werstgiveanexamplewith5scenariostoillustratePolicy3.BasedonthechanceconstraintdescriptioninPolicy3with=20%,thereisonlyonescenarioinwhichtherearetimeperiodswhosewindutilizationsarelessthan85%.FromtheresultsshowninFigure 2-3 ,wecanobservethatthechanceconstraintissatisedandonlyscenario3canhaveutilizationbelow85%. TheunitcommitmentresultsintheoptimalsolutionarelistedinTable 2-5 Table2-4. Transmissionlinedata LineIDFromToXFlowLimit(MW) L1B1B20.170200L2B1B40.150200L3B2B30.258300L4B2B40.197200L5B3B60.140100L6B4B50.150200L7B5B60.160400 36

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Figure2-3. Windutilizationineachoperatinghourforthe5scenariocase Table2-5. Optimalunitcommitment Hours(1-24) G1111111111111111111111111G2000000011111111111111111G3111100011111111100000000 Experimentsatdifferentrisklevels Next,weuse200scenariostorunexperimentsondifferentrisklevels.TheresultsarereportedinTables 2-6 2-7 ,and 2-8 forcomparison. Itcanbeobservedthatthe total cost(columnobj.inTable 2-8 )isreducedastherisklevelincreasesfrom15%to100%.Thisisreasonable because the fuel costforthermalplantsmightbehigherifthepolicyonwindpowergenerationismore restrictive andlesswindpowerwillbecurtailed.Anextremecaseis=100%inwhichthechanceconstraintcanbeneglected.Insuchacase,theoptimalcostissmallerthanthatatanyotherrisklevel.Meanwhile,thewindutilizationisatitslowestvalueaswell(below50%). Herethewindutilizationismeasuredastheaveragewindusageunderallscenarios,whichisequaltotheratiobetweenPTt=1P200j=1Pb2BWminf^qbt,wbt(j)gandPTt=1P200j=1Pb2BWwbt(j) .FromTables 2-6 2-8 ,wecanalsoobservethatPolicy3is 37

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themost restrictive one.Forthesamegivenrisklevel,thewindpowerutilizationisthehighestamongallthreepolices. Table2-6. Computationalresultsforthesix-bussystemwithdifferentrisklevels-usingpolicy1 RiskLevelObj.($)UtilizationCPUTime(sec) 15% 53038.384.01%0.2920%52541.383.04%0.3140%51473.381.92%0.2570%48235.575.01%0.29100%46213.848.9%0.11 Table2-7. Computationalresultsforthesix-bussystemwithdifferentrisklevels-usingpolicy2 RiskLevelObj.($)UtilizationCPUTime(sec) 15% 63315.590.00%0.0220%60462.588.75%0.0740%53782.183.57%0.0870%48699.175.09%0.1100%46213.848.9%0.11 Table2-8. Computationalresultsforthesix-bussystemwithdifferentrisklevels-usingpolicy3 RiskLevelObj.($)UtilizationCPUTime(sec) 15% 79372.593.86%1.4720%76850.593.5%2.0740%69268.392.2%3.8570%62789.590.14%53.28100%46213.848.9%0.11 Experimentsatdifferentscenariosizes Wehave shown theconvergencepropertyofthecombinedSAAalgorithminSectionIII.Here,wesetdifferentsamplesizesfortheSAAalgorithmtoverifythattheoptimalsolutionindeedconvergesasthescenariosizeincreases(e.g.,seeFig. 2-4 ).Policy3isappliedandtherisklevelissetto=10%forcomparison. 38

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Figure2-4. PlottingthesolutionsofSAAwithdifferentscenariosizes Modied118-BusSystem AmodiedIEEE118-bussystem,basedontheonegivenonlineatmotor.ece.iit.edu/data,isusedtotesttheSAAalgorithm. Weselectallthegeneratorsthatusecoalasthefuel.Intotal,thereare 33thermalgenerators.Wetakeall186transmissionlinesand91loads.Sincethereareonly33generators,wereducethevalueoftheloadateachbustoensuretheexistenceofasolution. Additionally,weconsiderawindfarmatBus3,therisklevel=10%,and=85%. Thedetailedrevised118-bussystemdataisgivenonlineat cso.ise.ufl.edu/data_r118.xls Wethenapplypolicy3,andthecomputationalresultsarereportedinTable 2-9 Therstcolumnrepresentsthecombinationofiterationnumbers( ^S ,M)(i.e.,iterationnumbersforobtainingthelowerboundandforthechanceconstraintpart),andvalidation'sscenariosize(N0). ThesecondcolumnrepresentsthescenariosizeoftheSAAproblem.ThethirdcolumnrepresentsthelowerboundobtainedbytheSAAalgorithm.ThefourthcolumnrepresentstheupperboundobtainedbytheSAA 39

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algorithm.ThefthcolumnrepresentsthegapwhichiscalculatedbyUB)]TJ /F4 7.97 Tf 6.59 0 Td[(LB LB100%.Finally,thesixthcolumnrepresentstheCPUtimeofthealgorithm. Table2-9. Computationalresultsforthe118-bussystem ( ^S M,N0)NLBUBGapTime(sec) (55,1000)104702904771841.46%3.6504728394772100.9%6.751004735784778570.9%10.2(520,2000)104719944752330.6%15.8504742714763250.4%25.41004745604775620.6%40.3(2020,3000)104723564739950.34%64.5504746924762300.32%103.81004747014761250.30%145.8 FromTable 2-9 ,wecanobservethatastheiterationnumberandthesamplesizeincrease,theoptimalitygapdecreases.Forthelastcaseinwhichthesamplesizeis100,theiterationnumbersare ^S =M=20,andthevalidationscenariosizeN0=3000,theoptimalitygapisaround0.30%.Thatis,theproposedalgorithmconvergesfastandcansolvetheproblemeffectively. Finally,wecomparetheperformancebetweenthedefaultMILPandthe strong formulationapproaches,andreporttheresultsinTable 2-10 .Itcanbeobservedfromthetablethatthe strong formulationapproachtakesmuchlesstimethanthedefaultMILPapproachwhentherisklevelisnottrivial(e.g.,0<<100%).Theresultsshowthescalabilityofthe strong formulationapproachtosolvelarge-scaleproblems.Itcanalsobeobservedfromthetablethattheoptimalobjectivevaluedecreasesastherisklevelincreases. ConcludingRemarks Inthischapter,achanceconstrainedtwo-stagestochasticprogramconsideringtheuncertainwindpoweroutputwasstudied.Inourapproach,thechanceconstraintguaranteestheminimumusageofthewindpowerbysettingarisklevel,whichlimits 40

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Table2-10. Computationaltimeforthe118-bussystem:MILPandstrongformulation NObj.($)MILP(sec)Strong(sec) 0%1004774630.160.161504819470.20.182004820590.230.2120%1004727598.080.315047283354.160.49200473072154.510.57100%1004687120.190.211504689240.210.242004688830.250.29 thechancethatalargeamountofwindpowermightbecurtailed.Westudiedthreedifferenttypesofpolicesandcomparedthewindutilizationsbythesepolicies.TheresultsveriedthatPolicy3 isthemostrestrictiveone. Then,westudiedacombinedSAAalgorithmthatcanderiveanoptimalsolutionwhenthesamplesizeincreases.ThenalcomputationalresultsverifytheeffectivenessoftheproposedSAAalgorithmandtherelatedsolutionvalidationprocess,andshowthattheproposedmodelcanhelpincreasetheusageofwindpower. 41

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CHAPTER3STOCHASTICPRICE-BASEDUCNomenclature A.SetsandIndices N Numberofscenarios T Timehorizon(24hour s ). B Numberofbuses. b,b Setsofthermalgeneratorsandhydrounitsinbusb,respectively. B.Parameters SU bi Start-upcostofthermalgenerator i inbusb. SD bi Shut-downcost of thermalgenerator i inbusb. Fc( p bit) F uelcost of thermalgenerator i intimetatbusbwhenitsgenerationis p bit MU bi Minimum-uptimeforthermalgenerator i inbusb. MD bi Minimum-downtimeforthermalgenerator i inbusb. URbi Ramp-upratelimitforthermalgenerator i inbusb. DRbi Ramp-downratelimitforthermalgenerator i inbusb. Lbi Lower limit ofelectricitygeneratedbythermalgenerator i inbusb. Ubi Upper limit ofelectricitygeneratedbythermalgenerator i inbusb. Sbeginib Waterreserve levelofpumped-storage unit iinbusbinthersttimeperiod. S endib Waterreserve levelofpumped-storage unit iinbusbinthelasttimeperiod. L Hib Lower limit of power pumpedin/outbypumped-storage unit iinbusbinone timeperiod. U Hib Upper limit of power pumpedin/outbypumped-storage unit iinbusbinone c[2012]IEEE.REPRINTED,WITHPERMISSION,FROM[ 66 ] 42

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timeperiod. bt Penaltycostper MW forenergyimbalance in time t inbusb. 1 Efciencyofgeneratingpowerbythepumped-storageunit. 2 Efciencyof absorbing powerbythepumped-storageunit. C.DecisionVariables p bit Electricitygenerationamountbythermalgenerator i intime t inbusb. ybit Binary decision variable:ifthermalgenerator i isonintime t inbusb; otherwise. obit Binary decision variable:ifthermalgenerator i isstartedupintime t in busb;otherwise. vbit Binary decision variable:ifthermalgenerator i isshutdownintime t in busb;otherwise. hbit Binary decision variabletoindicatewhetherpumped-storageuniti generates(e.g.,1)orconsumes(e.g.,0)power intime t inbusb. s bit Waterreservelevelofpumped-storageunitiintimetinbusb. q Btb Electricity bid intotheday-aheadmarketintime t inbusb. q Gtb Totalamountofthermalunitgeneration intime t inbusb. q Wtb Windpower sold inthe real-timemarket intime t inbusb. q H+itb Powergeneratedfrompumped-storageunit iintime t inbusb. q H)]TJ /F4 7.97 Tf -6.59 -10.26 Td[(itb Powerconsumedbypumped-storageunitiintime t inbusb. q imbtb Powerimbalanceintime t inbusb. Note: Someofthesedecisionvariablesaresecondstagevariableswhenthey are followedby() ,whererepresentsarandomvectorfollowingacertain probabilisticdistribution. D.RandomNumbers 43

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RDAtb() Day-ahead marketpriceofelectricityintime t inbusb. RRTtb() Real-time marketpriceofelectricityintime t inbusb. W tb() Windpower output intime t inbusb.BackgroundandLiteratureReview Inthischapter,weproposeto studytheoptimalbiddingstrategyforanIPPwhosegenerationportfoliomayconsistofthermal,hydroandwindpowerunits. TheobjectiveofanIPP'sself-schedulingproblemistomaximizeitsprotwhileensuringhighutilizationofwindpoweroutputto complywithrelatedregulations .Inourapproach, the IPPisassumedtobeapricetakerinthemarket,whichmeanstheIPPdoesnothavecontroloverthemarketpricesbybiddingstrategically.OneofthereasonsfortheIPPtobeapricetakerisitsrelativelysmallshareofgenerationinthetotalgenerationcapacity inthemarket .SincetheIPPisaprice-taker,themarketpricesarepurelyinputtotheIPP'sownprotmaximizationproblem.Thus,theobjectiveoftheIPPistooptimizeitsowngenerationportfoliobasedontheforecastedday-aheadandreal-timepriceinformation.Withthisprice-baseddecisionmaking,theIPPtriestocomeupwithitsbestgenerationschedulethatcanbe bid intothemarketsubjecttophysicalconstraintsofthegeneratorssuchasminon/off,capacitylimits,etc.Alongwiththegenerationquantityobtainedfromtheoptimization,theIPPcanbidalowpricetoensuretheacceptanceofitsbidsinthemarket.ThisproblemistypicallydenedasthePrice-basedUnitCommitment(PBUC)problemasshownintheliterature(see,e.g.,[ 9 35 36 52 ]). Inthischapter, theIPPisconsideredto operate andscheduleafewnumberofthermalgenerators,severalwindfarmsandpumped-storage units .We propose anovel bidding strategythatcanmaximizetheexpectedprotwiththeconsiderationofwindpowerforecastingerrorsandensurehighutilizationofwindpoweroutput forIPPs The chanceconstraintrequiresacertainprobabilityatwhich agivenportionof windpower must beutilized.Forexample,wecandeneaprobabilityof95%atwhichthe utilizationof windpowerisnosmallerthanacertain number(e.g.,85%) .Applications 44

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ofchance-constrainedoptimizationinpowersystemhavebeenstudiedrecently[ 42 ],[ 74 ],[ 64 ]and[ 65 ].However,noneofthepreviousstudies has investigated windpower bidding strategies withchance constraints ,whichisthemainfocusofthischapter. Inthischapter,westudyasampleaverageapproximation(SAA)methodtosolvethechanceconstrained powerproducerbidding problem.AscomparedtotheSAAalgorithmsrecentlydevelopedforthetwo-stagestochasticprogramdescribedin [ 64 ] ,[ 34 ]and[ 5 ],andforthechance-constrainedsingle-stagestochasticprogramdescribedin[ 43 ]and[ 37 ],wedevelopadifferentSAAalgorithm duetothebinarydecisionvariablesforthehydrounitsinthesecondstage The validationprocessandconvergenceproof oftheproposedapproach are also investigated. Thecasestudiesshow theproposedalgorithminthischapterprovidestightlowerandupperbounds,which illustrate theeffectivenessofourapproach. Thecontributionsofthischapteraresummarizedasfollows: 1. MuchofthepreviousresearchstudiedthestochasticunitcommitmentproblemfromanISO'sperspective.Inthischapter,ourcontributionsfocusontheprice-basedunitcommitment(PBUC)fromaprice-takingIPP'sperspective. Atwo-stagestochasticprogrammingmodelisstudiedtosolvetheproblem. 2. ComparedwithotherworksinPBUC[ 35 36 ], thisis therstworktoapplythechance-constrainedstochasticprogrammingtoaddressbiddingstrategiesonathermal-wind-hydrogenerationportfolio. 3. TheSAAframeworkisadjustedspecicallyforourchance-constrainedstochasticprogrammingmodel.Wealsoproposeaheuristic-basedSAAalgorithmforthisspecicproblemstructure. MathematicalFormulation MarketFramework Inaderegulatedelectricitymarket ,theIPPssubmitbidseachday,andthemarketoperatorprovidesthemarketclearingpricesofelectricity. TheIPPswithboththermalandwindpowerunitsinthegenerationportfoliofaceatleasttwomainsourcesofuncertainties:marketpricesandvariablewindpoweroutput.IPPshavetoconsider 45

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theuncertaintyofwindpowerintheirbiddingastheirwindpowerforecastswillnotbeperfectlyaccurateanderrorsalwaysexist.Inaddition,someoftheelectricitymarkets(suchasPJM[ 4 ])areenforcingpenaltiesonthemismatchbetweenIPPs'day-aheadbidsandtheiractualgenerationinthereal-timemarket.Onecaneasilyobservethat uncertainwindpoweroutputcontributessignicantlytothemismatch .Inthemeantime,IPPscannotchoosetoabandontoomuchwindpowertoavoiditsuctuationastheymaybesubjecttocertainregulationsthatrequirerenewableenergylikewindpower accounting foracertainshareoftheirtotalgenerationoutput[ 41 ].Hence,IPPsneedtooptimizetheirbiddingstrategiesthatcanbalancetheobjectiveofprotmaximizationandtherisksassociatedwithwindpowerrealizations,whilemakingsurethewindpowerisutilizedtothegreatestdegree.Basedontheabovediscussion,theproposedmethodinthischaptermodelssuchabiddingprocessforIPPsbyformulatingtheproblemasatwo-stagestochasticprogram.Tomaximizethetotalexpectedprot,theIPPsdecidethequantitytobesubmittedtothemarketintherststage(day-aheadmarket),consideringthepossiblerealizationsofuncertainmarketpricesandwindpoweroutputinthesecondstage(real-timemarket).Chanceconstraintsareusedtomodeltheleastpercentageofwindpowerutilization.Ourmodelingframeworkalsocapturesthetwo-settlement(day-aheadandreal-time)marketprocedureasinmostoftheU.S.marketsbymodelingthetwomarketsintheobjectivefunction(e.g.,PJM[ 4 ]). ProblemFormulation ThePBUCproblemisformulatedasatwo-stagechanceconstrainedstochasticprogrammingproblem.Therststageofthemodelisaunitcommitmentproblemwithdecisionsoncommitmentandquantityoftheelectricityofferedtotheday-aheadmarket.Thesecondstageofthemodelisaneconomicdispatchproblemwithdecisionsonthermalandhydrounitdispatch,theactualusageofwindpower,andenergyimbalance.Weconsiderthechanceconstraintatthesecondstage,inwhichtheactualwindpower 46

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utilizedcouldbedifferentfromthewindpoweroutput. Wedescribethenalformulationoftheproblem(denotedasthetrueproblem)asfollows. max)]TJ /F4 7.97 Tf 16.33 14.94 Td[(TXt=1 BXb=1 Xi2 b ( SUbiobit+SDbivbit )+E[Q(y,o,v, qB ,)] (3) s.t.)]TJ ET BT /F6 11.955 Tf 136.66 -153.69 Td[(ybi(t)]TJ /F5 7.97 Tf 6.59 0 Td[(1) + ybit )]TJ ET BT /F6 11.955 Tf 208.19 -153.69 Td[(ybik 0,1k)]TJ /F3 11.955 Tf 11.96 0 Td[((t)]TJ /F3 11.955 Tf 11.96 0 Td[(1) MU bi,8i2 b,8b,8t (3) ybi(t)]TJ /F5 7.97 Tf 6.58 0 Td[(1) )]TJ ET BT /F6 11.955 Tf 172.35 -207.48 Td[(ybit + ybik 1,1k)]TJ /F3 11.955 Tf 11.96 0 Td[((t)]TJ /F3 11.955 Tf 11.96 0 Td[(1) MD bi,8i2 b,8b,8t (3) )]TJ ET BT /F6 11.955 Tf 136.66 -261.28 Td[(ybi(t)]TJ /F5 7.97 Tf 6.59 0 Td[(1) + ybit )]TJ ET BT /F6 11.955 Tf 208.19 -261.28 Td[(obit 0,8i2 b,8b,8t (3) ybi(t)]TJ /F5 7.97 Tf 6.58 0 Td[(1) )]TJ ET BT /F6 11.955 Tf 172.35 -315.07 Td[(ybit )]TJ ET BT /F6 11.955 Tf 198.89 -315.07 Td[(vbit 0,8i2 b,8b,8t (3) ybit obit vbit 2f0,1g,8i2 b,8b,8t, (3) whereQ(y,o,v, qB ,)isequalto maxTXt=1 BXb=1 ( RDAtb()qBtb+RRTtb()qimbtb()) )]TJ ET BT /F4 7.97 Tf 140.29 -468.02 Td[(TXt=1BXb=1Xi2b Fc(pbit()))]TJ ET BT /F4 7.97 Tf 260.79 -468.02 Td[(TXt=1BXb=1 btjqimbtb()j (3) s.t.Lbiybit p b it()Ubiybit,8i2 b,8b,8t (3) p b it())]TJ ET BT /F6 11.955 Tf 168.8 -541.02 Td[(p b i(t)]TJ /F5 7.97 Tf 6.58 0 Td[(1)()(2)]TJ /F6 11.955 Tf 11.96 0 Td[(y b i(t)]TJ /F5 7.97 Tf 6.58 0 Td[(1))]TJ /F6 11.955 Tf 11.96 0 Td[(y b it) L b i +(1+y b i(t)]TJ /F5 7.97 Tf 6.59 0 Td[(1))]TJ /F6 11.955 Tf 11.96 0 Td[(y b it)UR b i,8i2 b,8b,8t (3) p b i(t)]TJ /F5 7.97 Tf 6.59 0 Td[(1)())]TJ ET BT /F6 11.955 Tf 187.49 -594.82 Td[(p b it()(2)]TJ /F6 11.955 Tf 11.96 0 Td[(y b i(t)]TJ /F5 7.97 Tf 6.58 0 Td[(1))]TJ /F6 11.955 Tf 11.96 0 Td[(y b it)L b i+(1)]TJ /F6 11.955 Tf 11.95 0 Td[(y b i(t)]TJ /F5 7.97 Tf 6.59 0 Td[(1)+y b it)DR b i,8i2 b,8b,8t (3) 47

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Xi2 b p bit()=qGtb(),8b,8t (3) qWt b ()+qGt b ()+ Xi2b( qH+ itb ())]TJ /F6 11.955 Tf 11.96 0 Td[(qH)]TJ ET BT /F4 7.97 Tf 297.45 -48.75 Td[(itb () ) =qBt b +qimbt b (),8b,8t (3) s b it()=s b i(t)]TJ /F5 7.97 Tf 6.58 0 Td[(1)()+ 2 qH)]TJ /F4 7.97 Tf -7.17 -10.71 Td[(itb())]TJ /F6 11.955 Tf 13.15 8.09 Td[(qH+itb() 1 8i2b,8b,8t (3) h b it()LHibqH+itb()h b it()UHib,8i2b,8b,8t (3) (1)]TJ /F6 11.955 Tf 11.96 0 Td[(h b it())LHibqH)]TJ /F4 7.97 Tf -7.16 -10.71 Td[(itb()(1)]TJ /F6 11.955 Tf 11.96 0 Td[(h b it())UHib, 8i2b,8b,8t (3) s b iT()=Sendib, sbi0()=Sbeginib ,8i2b,8b (3) Pr(BXb=1Wtb()BXb=1qWtb(),8t)1)]TJ /F8 11.955 Tf 11.95 0 Td[( (3) p b it(),qWt b (),qGt b (),qH+it b (),qH)]TJ /F4 7.97 Tf -7.17 -10.7 Td[(it b (),s b it()0,h b it()2f0,1g, qBtbfree, 8i,8b,8t. (3) Theobjectivefunction( 3 )istomaximizetheexpectedtotalprot.Itisequaltotheexpectedrevenue E[PTt=1PBb=1( RDAtb()qBtb+RRTtb()qimbtb() )] whichfollowsthetwo -settlement marketprocedureinmostU.S.electricitymarkets, minustheexpectedpowergenerationcost PTt=1PBb=1Pi2b(SUbiobit+SDbivbit)+E[PTt=1PBb=1Pi2bFc(pbit())] ,andtheexpectedpenaltycost E[PTt=1PBb=1btjqimbtb()j] Itshouldbenotedherethatqimbtb()capturestheamountofimbalancebetweentheday-aheadbidamountandthereal-timegenerationoutput.Asthisimbalanceismainlycausedbythevariablewindpowerandinaccuratewindpowerforecasting,theadditionalpenaltybtwhichisimposedbymarketoperatorscanreducetheuncertaintyofthemarketrelatedtouncertainwindgeneration. Theunitcommitmentconstraintsattherststagelisted above includeconstraints( 3 )to( 3 ),representingtheunitminimum-uptimerequirementwhentheunit isturnedon (e.g.,constraints( 3 )),theunitminimum-downtimerequirementwhentheunit isturnedoff(e.g.,constraints( 3 )) ,theunitstart-upcondition(e.g.,constraints 48

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( 3 )),andtheunitshut-downcondition(e.g.,constraints( 3 )).Thehourlyeconomicdispatchconstraintsatthesecondstageincludeunitgenerationupperandlower limit constraints( 3 ),unitrampingupconstraints( 3 ),unitrampingdownconstraints( 3 ), totalthermalgenerationoutput ( 3 )( itsumsupallthegenerationbythermalunits) ,powerbalanceconstraints( 3 ) (thetotalsystemgenerationshouldbeequalto the amountofenergy offeredintheday-aheadmarket plustheimbalance) hydrowaterinventorybalance constraints( 3 ), hydrounitpumpin/out limit constraints(( 3 )and( 3 )) andrst/lastperiod water reservationamount constraints( 3 ). Thechanceconstraint( 3 )isassociatedwitharisklevel (e.g.,=10%), whichmeansthetotalutilizationofwindpowerhastobelargerthanorequalto(e.g.,=85%)foratleast100(1)]TJ /F8 11.955 Tf 12.74 0 Td[()percentofchance.Ascanbeseen,addingthisconstraintcanhelpIPPscomplywithregulationswhichrequireacertainpercentageofwindpowerutilizationatahighprobability. Inaddition, y b it,o b itandv b itarerst-stagedecisionvariables,andothersaresecond-stagedecisionvariables. Intheobjectivefunction ofourmodel ,thereexistsanabsolutevaluewhichindicatestheimbalancepenalty.Itcanbereformulated by usinglinearprogramming asshownin [ 18 ] .Forinstance,thefollowingminimizationproblem min b tjqimbt b j,subjecttoAx=b(3) canbereformulatedasfollows: minf b tdt b j)]TJ /F6 11.955 Tf 17.93 0 Td[(dt b qimbt b dt b andAx=bg(3) a fterintroduc ingan auxiliaryvariabledt b .Thus,wecanreplacetheabsolute value partin( 3 )withalinear function ( 3 ).SampleAverageApproximation Inthissection,weapply a sampleaverageapproximation(SAA)methodtosolvethestochasticprogramshownabove.SAAiscomposedofthreesteps:1)scenario 49

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generationtoapproximatethetruedistribution,2)convergenceanalysistoshowtheconvergencepropertyofthealgorithm,and3)solutionvalidationtoverifythatthesolutionconvergestotheoptimalone. Thereadersarereferredto[ 5 ]formoredetailsregardingthetraditionalSAAmethoddescription Fornotationbrevity,theproposedmathematicalmodelcanbeabstractedasfollows: minxf2Xf(xf)+E[Q(xf,)] where Q(xf,)=mincxs()s.t.Axs()=g)]TJ /F6 11.955 Tf 11.95 0 Td[(Dxf,xs()0PrfH(xf,xs(),)0g1)]TJ /F8 11.955 Tf 11.96 0 Td[(.Intheaboveformulation,xfandxsrepresenttherstandsecondstagedecisionvariables,andf(xf)andQ(xf,)representtherstandsecondstageobjectivefunctions.Inaddition,Xrepresentsthefeasibleregionofxf,c,g,A,Darevectors/matricesofparameters,andHistheconstraintmapping. SAAProblem Inourapproach,theSAAproblemisgeneratedsimilarlytotheonedescribedin[ 64 ]and[ 5 ].Forinstance,aMonteCarlosimulationmethodisutilizedforscenariogenerationpurposes.Afterthescenariosaregenerated(e.g. ,N scenarios),theobjectivefunctionE[Q(xf,)]canbelinearizedandreplacedbythesampleaveragefunctionN)]TJ /F5 7.97 Tf 6.59 0 Td[(1PNj=1Q(xf,j)[ 34 ].Meanwhile,anindicatorfunction1(0,1)(H(xf,xs(j),j)isintroducedtoestimatethechanceconstraintasdescribedin[ 43 ].Wehave 1(0,1)(H(xf,xs(j),j))=8>><>>:1,ifH(xf,xs(j),j)2(0,1);0,ifH(xf,xs(j),j)=2(0,1). Byintroducingbinarydecisionvariablesztoindicateifaconstraintissatised,whenasamplesizeNisgiven,wecanlinearizethechanceconstraintasthefollowing 50

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constraints( 3 )( 3 ),andtheSAAproblemcanbedescribedasfollows: max)]TJ /F4 7.97 Tf 16.33 14.94 Td[(TXt=1 BXb=1 Xi2 b ( SUbiobit+SDbivbit )+N)]TJ /F5 7.97 Tf 6.59 0 Td[(1 NXj=1 TXt=1BXb=1 ( RDAtb(j)qBtb+RRTtb(j)qimbtb(j) ))]TJ ET BT /F4 7.97 Tf 102.95 -117.45 Td[(TXt=1BXb=1Xi2b Fc(pbit(j)))]TJ ET BT /F4 7.97 Tf 228.06 -117.45 Td[(TXt=1BXb=1 btjqimbtb(j)j# (3) s.t.( 3)-222()]TJ /F3 11.955 Tf 21.25 0 Td[(2 ))]TJ /F3 11.955 Tf 11.95 0 Td[(( 3)-221()]TJ /F3 11.955 Tf 21.25 0 Td[(6 ),( 3)-222()]TJ /F3 11.955 Tf 21.26 0 Td[(8 ))]TJ /F3 11.955 Tf 11.96 0 Td[(( 3)-222()]TJ /F3 11.955 Tf 21.26 0 Td[(16 ),and( 3)-222()]TJ /F3 11.955 Tf 21.25 0 Td[(18 ) BXb=1 Wt b (j))]TJ ET BT /F4 7.97 Tf 172.47 -175.42 Td[(BXb=1 qWt b (j)Mzj,8t,8j (3) NXj=1zjN (3) zj2f0,1g,j=1,2,...,N. (3) ConvergenceAnalysisandSolutionValidation Weusestatisticalmethodstoanalyzethesolutionof the SAAproblemandprovideconvergenceanalysisandsolutionvalidation. AsthesamplesizeNgoestoinnity,weclaimthattheobjectiveoftheSAAproblemconvergestothatofthetrueproblem.Toprovetheconvergenceproperty,weneedtorstprovetheconvergenceofthechanceconstrainedpart.Thisresultcanbeachievedusinga similarapproachasdescribed in[ 43 ].Secondly,afterconvertingthechanceconstraintintotheMILPformulation,weshouldnoticethattherst-stageprobleminthewholeSAAproblemofthetrueproblemisapureintegerprogram,andthesecond-stageproblemisamixedintegerlinearprogram.Accordingto[ 5 ],thesolutionofsuchanSAAproblemwillconvergetothatofthetrueproblem. In[ 5 ]and[ 43 ],theproceduresforsolutionvalidationofSAAproblemshavebeendevelopedforthetwo-stageproblemandforthechance-constrainedproblem,respectively.LetxandvbeanoptimalsolutionandthecorrespondingoptimalobjectivevaluefortheSAAproblem,respectively.Tovalidatethequalityofx,thevalidation 51

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processobtainsupperandlowerboundsforvofthetrueproblem.Usually,thesolutionvalidationneedstoconsiderthefeasibilitywhendealingwithchanceconstraints(e.g.,chanceconstraintscontainbothrstandsecondstagedecisionvariables),becauseitisnotguaranteedthatthesolutionoftheSAAproblemalwayssatisesthechanceconstraintswithalargescenariosize.However,inthischapter,thechanceconstraintsareonlyconsideredinthesecondstage.Thesecond-stagedecisionismadeafterthescenariosarerealized.Thus,thechanceconstraintscanalwaysbesatisedbytuningthesecondstagedecisionvariables.Weapplydirectlythevalidationprocessin[ 5 ]toconstructstatisticalboundsfortheobjectivevalueofourSAAproblem. SAAAlgorithmFramework TodescribetheSAAalgorithm,werstintroduce additionalnotation .Forinstance,weletNbethescenariosizeoftheSAAproblem, K betheiterationnumber,N0bethescenariosizeof the validationprocess toobtaina lowerbound,bgbethelowerboundofthetrueproblem,xkandvkbetheoptimalsolutionandoptimalobjectivevalueiniteration k ,andvbetheupperboundofthetrueproblem.TheSAAalgorithmcanbesummarizedasfollows withaowchartinFig. 3-1 : 1. Set k=1,2,...,K andrepeatthefollowingstepsforeachk: (a) ForagivensamplesizeN,generateacorrespondingSAAproblemandsolvetheSAA problem toobtainxkandvk; (b) ForagivensamplesizeN0forthevalidationprocess,generateindependentscenarios1,2,...,N0,andestimatethelowerboundoftheproblemusingthefollowingformula: bgk= f(xfk) +1 N0N0Xn=1Q( xfk ,n),(3) whereQ(xfk,n)isthesecond-stageproblemdenedin( 3 )-( 3 )withxfxedasxfk. 2. Taketheaverageofv1,v2,...,vK .T heupperbound canbeobtainedas v=1 KPKk=1vk followingTheorem1in[ 38 ] 52

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3. Takethemaximumofbg1,bg2,...,bgK .T helowerbound canbeobtainedas bg=max1kKbgk 4. Estimate theoptimalitygap: ( v )]TJ /F10 11.955 Tf 12.02 .5 Td[(bg)=bg100% Figure3-1. ProposedSAAalgorithm HeuristicsforSolvingEachSAAProblem AsshowninSection 3 eachSAAproblemcontainsmanyintegervariableswhichmaketheproblemhardtosolveby commercialsolverslike CPLEX[ 2 ]underdefaultsettings.Inaddition,therearemanybinarydecisionvariablesinthesecondstageduetohydrooperations.Whenthesystembecomeslarger,thenumberofintegervariableswillincreasesignicantly.Toreducethecomputationalcomplexityoftheproblem,weapplyheuristicmethodstoobtaingoodfeasiblesolutions,and tightinner 53

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upperandlowerboundsoftheoptimalobjectivevalueforeachSAAproblem. Basedonoptimizationtheory,inourheuristicapproach,wesolvearelaxationproblembyrelaxingthesecondstagebinarydecisionvariablestobefractional,whichprovidesanupperboundforourmaximizationproblem.Inaddition,anyfeasiblesolutionleadstoacorrespondinglowerbound. Innerupperbound Thebasicideaistousetherelaxationtogetaninnerupperboundforamaximizationproblem.SincetheintegervariablesleadtothedifcultyofsolvingeachSAAproblem,werelaxtheintegralityforconstraints( 3 )and( 3 ) whilemaintainingtheintegralityofthezvariablesin( 3 ) togetaninnerupperboundforthegivenSAAproblem. Noteherethatthisapproachismoreeffectiveforsmallsamplesizes. Innerlowerbound WecreateafeasiblesolutiontogetaninnerlowerboundforthegivenSAAprobleminthispart.Notethattherst-stagesolutionfromobtaining the innerupperboundin 3 shouldalsosatisfytherst-stageconstraintsintheSAAproblem.Therefore,wecanxtherst-stagesolutionobtainedfromtheabovepartin 3 ,andsolvethesecond-stagesub-problemtoobtainafeasiblesolutionandacorrespondinginnerlowerboundfortheSAAproblem. Moreover,whenweusetheinnerupperbound andcorrespondingsolution derivedin 3 to replacevkandxkinStep1.1in 3 inthecalculationofobtaining theupperboundinthevalidationprocess,westillobtaintheupperboundfortheoriginaltrueproblem.Similarly,usingtheinnerlowerboundand correspondingsolution obtainedin 3 to replacevkandxkinStep1.1in 3 inthecalculationofobtaining thelowerboundinthevalidationprocessprovidesthelowerboundfortheoriginaltrueproblem. ComputationalResults Inthissection,werststudy athree -generatorsystem inasinglebus toillustratetheproposedalgorithm. Second,weconsideramorecomplicatedlargesystem ina 54

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singlebustoevaluatetheperformanceoftheheuristicapproach.Finally,weevaluatetheperformance ofouralgorithmona generalizedmulti-bussystem(e.g.,thermal,windandhydrounitsarelocatedindifferentbuses),bycomparingitwiththecaseinwhicheachbusisconsideredseparately ItshouldbenotedthatourSAAsolutionframeworkcanbeappliedtolargersystemsaswellalthoughthesizesofthetestsystemsusedinthischapteraremoderate.ThereasonforustouseamoderatesizeinstanceisthatanIPPwithanexcessivelylargegenerationportfoliomaymostlikelybeabletoinuencethemarketprices,whichviolatesourassumptionofaprice-taker. For the three -generatorsystem,werunthecomputationalexperiments on the SAAproblems atdifferentrisklevelsanddifferentsamplesizesforcomparison.TheSAAalgorithm describedin Section 3 isalsotestedforthissystem. For thecomplicatedsystem, weconsider the heuristicmethod describedinSection 3 tosolvetheSAAproblemandrunthecomputationalexperimentstotesttheheuristic-basedSAAalgorithm. Thecodes are writteninC++andtheproblemissolved withCPLEX12.1.Alltheexperimentsareimplementedonacomputerworkstationwith4IntelCoresand8GBRAM.ScenarioGenerationforUncertainWindPowerandPrice IntheSAAframework,weneedtogeneratescenariosbyMonteCarlosimulation.Inourapproach,MISOwindpowerandpricehistoricaldataare utilizedforour casestudies.Thewindpowerdataisavailablein the NationalRenewableEnergyLaboratory(NREL)2006easternwinddataset.TheLocationalMarginalPrice(LMP)historicaldataisprovidedbyMISO.Thestate-of-the-arttimeseriesmodelsforwindpowergenerationareintwocategories:windspeed-basedapproachesandwindpower-basedapproaches.Thewindspeed-basedapproaches(see,e.g.,[ 19 44 ])applythetimeseriesmodeltogeneratewindspeedscenariosandconvertthemintowindpoweroutput.Thewindpower-basedapproachesconsiderwindpowertimeseriesdirectly(see,e.g.,[ 21 ]).Tocapturethewindpoweruncertainty,wenowrstapplythetimeseriesmodeltoanalyzethehistoricalwinddataavailablefromNREL[ 3 ].Themethod 55

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in[ 21 ]isappliedtoconstructtheARIMA-basedmodel.AMonteCarlosimulationisthenperformedonrandomnoisewhichissubjecttoanormaldistributionintheARIMAmodeltogeneratescenarios.AsLMPsareverydifculttoforecastthemselvesduetoavarietyoffactorssuchasstrategicbiddingortransmissioncongestion,weassumetheuncertaintiesofday-aheadandreal-timeLMPstofollow aGaussiandistribution andfollowthemethoddescribedin[ 16 ]togeneratepricescenarios. Thatis,weusethehistoricaldatatogetthemeanandvariancefortheGaussiandistributionfortheday-aheadandreal-timeLMPs,respectively.Then,theiidsamplesaregeneratedfromtheGaussiandistributionwiththeestimatedmeanandvariance,plusthewhitenoisefollowingthestandardnormaldistribution. Notethattheproposedmodelinthischaptercanbeappliedtootherscenariogenerationapproacheswithoutlossofgeneralitybychangingthescenariogenerationapproachaccordingly.Forexample,asdescribedin[ 63 ],thewindpowercanbeassumedtofollowamultivariateGaussiandistributioninMonteCarlosimulation. Three-GeneratorSystem Inthis subsection ,westudyasimplecaseinwhich an IPPownsandoperatesthreethermalgenerators,onewindfarm,andonepumped-storageunit. Forthissmallinstance,eachSAAproblemcanbesolvedbyCPLEXwithdefaultsettingsdirectly.Therefore,theheuristicmethodinSection 3 isnotconsidered. We report the computational results at differentrisklevelsandscenariosizes.Thecharacteristicsofthermal and pumped-storageunitsaredescribedin Tables 3-1 3-3 Table3-1. Generatordata UnitLowerUpperMin-downMin-upRamp (MW) (MW) (h) (h) (MW/h) G1501002440G21001503330G320503215 56

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Table3-2. Fueldata Unitab( MMBtu c( MMBtu Start-upFuelPrice (MMBtu) /MWh) /MW2h) ( MMBtu )($/ MMBtu ) G15060.00041001.246G2405.50.00013001.246G3604.50.00501.246 Table3-3. Pumped-storage UpperLowerStartLevelEndLevel12 (MW) (MW) ( MWh )( MWh ) 2005100.90.9 InordertorunthemodelinCPLEXeffectively,welinearizethefuelcostfunctionbyusingtheinterpolationmethod[ 57 ].Accordingly,thefuelcostfunctionin( 3 )isreplacedbyapiecewiselinearfunction.Optimalsolutionwithtenscenarios Wereporttheoptimalsolutionof the SAAalgorithmwithtenscenariosand risklevel =10%inthissubsection.Table 3-4 reports theunitcommitment statusforeachgenerator .ItcanbeobservedthatG1iscommittedmostly,ThereasonisthatG1hasmoreexiblelower/upperboundsandramplimitsthanG2andG3.TheexibilityofthesecharacteristicsallowsG1toaccommodatethewindpowerbetter. Table3-4. Optimalunitcommitment Hours(1-24) G1000000000001111111111111G2000000000001111111110000G3000000000000000000000000 Toshowtheeffectivenessofthepumped-storageunit, wecomparetheaverageimbalanceqimbtbvaluewithandwithoutthepumped-storageunit.Theaverageimbalancedecreasesfrom6.8to2.5whenwehavethepumped-storageunitinthesystem. 57

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Sensitivityanalysisfordifferentrisklevelsandscenariosizes Thenumerical resultsondifferentrisklevelsanddifferentscenariosizesarereportedinthefollowing Figs. 3-2 and 3-3 .FromFig. 3-2 ,itcanbeobservedthatthetotalprotincreasesastherisklevelincreases.Thisisreasonablebecausethetotalprotwillbeloweriftheutilizationrequirementofwindpoweroutputismorerestrictive. ToverifytheconvergencepropertyoftheSAAalgorithmshowninSection 3 ,theSAAalgorithmistestednumericallybysettingdifferentsamplesizes.TheresultsshowninFig. 3-3 ( withtheriskleveltobe10% )indicatethattheobjectivefunctionoscillatesatthebeginningwhenthesamplesizeissmallandthenconvergesslowlytotheoptimalobjectivevalue. Figure3-2. Obj.($)oftheSAAproblemwithdifferentrisklevels ComputationalResultsforaComplicatedSystem Inthissubsection,wereportthecasestudyresultofamorecomplicatedsystem. WeassumetheIPPowns vegenerators,vewindfarms,andtwohydrounits G1andG2usedinthethree-generatorsystemareduplicatedinthiscasestudysetting 58

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Figure3-3. Obj.($)oftheSAAproblemwithdifferentscenariosizes Underthissetting,eachSAAproblemcannotbesolvedtooptimalitywithin thetwo-hourtimelimit whenthescenariosizereaches50.Thereasonisthatthewindpowerscenariosvaryextensivelysuchthatthecomputationalcomplexityisdominatedbythechanceconstraint.However,ourheuristicapproachcanstillprovidetightinnerlowerandupperbounds.AsshowninTable 3-5 ,thelowerboundmatchestheupperboundwhenthesamplesizeisnolargerthan50.Whenthesamplesizeincreases,wecanstopourheuristicalgorithmifthetimelimitisreached.Accordingly,wecanstillreportthecorrespondinglowerandupperboundsforeachSAA.ThedrawbackforthisapproachisthatitpotentiallyincreasestheoptimalitygapforeachSAAproblem.However,asweincreasetheiterationnumberKandthevalidationprocesssamplesizeN0,thenalestimatedoptimalitygapfortheSAAframeworkcanstillbereducedtoasmallnumber,asshowninTable 3-6 .Multi-BusSystem IntheabovelargersysteminSection 3 ,weassumealltheIPP'spowergenerationresourcesare inasinglebus,oraggregated .ItiscommoninpracticethattheIPP'spowergenerationresourcesaredistributedatdifferentbuses asdescribedinthe 59

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Table3-5. ComputationalresultsforacomplicatedsystemforeachSAAproblem-heuristicmethod(risklevel:10%) N Inner LB Inner UBCPUTime(s) 101286901286903.830123820123820605.6501246881246882476.9 Table3-6. Resultsofsolutionvalidationforacomplicatedsystem(risklevel:10%) (K,N0)LBUBGapCPUTime(s) (10,200)1224101296725.9%1334.5(30,500)1210401287656.3%3243.8(50,800)1228181266373.1%5453.2 modelinSection 4 Insuchacase,theelectricitypricesatdifferentbusesmightbedifferent. One canseparatelysolvetheproblemforeachbus, whichcansavethecomputationaltime .Inthissubsection,we useour modeltoconsiderdifferentbusessimultaneouslysincetheobjectiveoftheIPPshouldbetomaximizetheprotofitsentiregenerationportfoliolocatedatdifferentbuses.Whileeachbuscontainsitsownpriceinformationandpowerbiddingbalanceconstraints,onechanceconstraint isapplied onthetotalwindutilizationforthewhole multi-bus generationportfolio.Thechanceconstraint is thecouplingconstraintforallbuses. Inourexperiment,weassumetheIPP'spowergenerationresourcesaredistributedinvedifferentbuses. G1usedinthethree-generatorsystemisduplicatedasthefourthgeneratorinthiscasestudy .ThedetailedsettingsaresummarizedinTable 3-7 ThecomputationalresultsarereportedinTable 3-8 Itcanbeobservedthatthe proposedmethodwhichconsiders the couplingchanceconstraintprovidesalargertotalprot. Thismatchesthetheoreticalresult.Thatis,anysolutionoftheseparatedchanceconstrainedproblemmustbeafeasiblesolutionofthecouplingchanceconstrainedproblem,whichleadstothefactthatthecouplingchanceconstrainedproblemprovides 60

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Table3-7. Bussettings BusNo.WindThermalHydro 11112120311142125100 abettersolutionwithahighertotalprot.Wethereforeconcludethatsharingresourcesinsidethemulti-bussystemcantackletheuncertaintiesbetterandofferahigherprotingeneral. Table3-8. Computationalresultsfordistributedsystem RiskLevelObj.withCouplingObj.withSeparatingChanceConstraintChanceConstraints 0%798736296810%922667645240%113419104145100%153631153631 ConcludingRemarks Inthischapter ,astochasticprogrammingmodelisproposedtoaddresstheprice-basedunitcommitmentproblemwithwindpowerutilization constraints .Ourmodelincorporatesday-aheadprice,real-timeprice,andwindpoweroutputuncertainties. In therststage, an IPPmakesdecisionsonunitcommitmentand theamountofenergyoffered fortheday-aheadmarket. T heeconomicdispatchofgeneratorsismadeinthesecondstage.Achanceconstraintisconsideredtoensuretheutilizationofthevolatilewindpowertoalargeextent.Inotherwords,thereisagreatchancetheusageofthewindpowersatisesapre-dened percentage .Thechanceconstraintallowsthepowerproducertoadjusttheutilizationofwindpowerbasedondifferentregulations. 61

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Ourmodel maximizes theprotandaccommodatestherequiredusageofwindpoweroutput. AnSAAalgorithmisdevelopedtosolvetheproblem ,andthe objectivevalueoftheSAAproblemconvergestotheoptimaloneasthescenariosizeincreases. For morecomplicatedsystems, weproposeaheuristicapproachtoacceleratetheSAAalgorithm. Ourimplementationprovidesthe overall upperandlowerboundsforthetrueproblem.The reasonable estimated optimalitygapandmoderatecomputationaltimeverifythatourapproachiseffectiveinsolvingthisproblem. 62

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CHAPTER4STOCHASTICEXPECTEDVALUECONSTRAINEDUCNomenclature A.IndicesandParameters B Indexsetofallbuses. E Indexsetoftransmissionlineslinkingtwobuses. BG Setofbuseswiththermalgenerationunits. BW Setofbuseswithwindfarms. b Setofthermalgeneratorsatbusb. T Timehorizon(e.g.,24hours). SU bi Start-upcostofthermalgenerator i at busb. SD bi Shut-downcost of thermalgenerator i at busb. Fc( q bit()) F uelcost of thermalgenerator i atbusbintimetwhenitsgeneration amount is q bit() MU bi Minimumup timeforthermalgenerator i atbusb. MD bi Minimumdown timeforthermalgenerator i atbusb. URbi Ramp-upratelimitforthermalgenerator i atbusb. DRbi Ramp-downratelimitforthermalgenerator i atbusb. Lbi Lowerboundofelectricitygeneratedbythermalgenerator i atbusb. Ubi Upperboundofelectricitygeneratedbythermalgenerator i atbusb. Cij Transmission capacityfor the transmission line linking busiandbusj. Kbij Lineow distributionfactorforthetransmission line linking busi andbusj, duetothenetinjectionatbusb +t Penaltycostper MW forloadcurtailment in time t )]TJ /F4 7.97 Tf -.65 -7.58 Td[(t Penaltycostper MW forgenerationcurtailment in time t 63

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The utilization rateofthewind power output. Risklevelassociatedwiththechanceconstrainton loadimbalance Thetoleranceofloadimbalanceinthechanceconstraint. B.DecisionVariables obit Binary decision variable:1ifthermalgenerator i atbusbisonintimet; 0otherwise. ubit Binary decision variable:1ifthermalgenerator i atbusbisstartedupin timet;0otherwise. vbit Binary decision variable:1ifthermalgenerator i atbusbisshutdownin timet;0otherwise. qbit() Electricitygenerationamountbythermalgenerator i atbusbintimet correspondingtoscenario ^qbt() Amountofwindpowerutilized (delivered) atbusbintimet corresponding toscenario QGt() Totalgenerationamountbythermal units intime t correspondingto scenario QWt() Totalwindpowercommittedtobeutilized(delivered)intime t corresponding toscenario Qimb+t() Totalloadcurtailmentintime t correspondingtoscenario Qimb)]TJ /F4 7.97 Tf -14.2 -7.58 Td[(t() Totalgenerationcurtailmentintime t correspondingtoscenario C.RandomNumbers Wbt() Arandomparameterindicatingthe uncertain windpower output(decidedby theweather)bywindfarmatbusbintimet correspondingtoscenario Dbt() Arandomparameterindicatingthe uncertain loadatbusbintimet correspondingtoscenario 64

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Motivation Inthischapter, besidesthechanceconstraint enforcinga lowenergyimbalanceprobability,ageneralexpectedvalueconstraintisintroducedinthestochasticprogrammingframeworktoensuretheoverallexpectedamountofwindpowerusage. Arelated concept istherecentlydevelopedconditional value -atrisk(CVaR)decisioncriterionapproach. CVaRisariskcontrolmetric,whichisknowntopossessbetterpropertiesthanvalue-at-risk(VaR)[ 48 ]. In[ 16 ],CVaR hasbeenapplied to optimalwindpowertradingstrategiesinLMPmarkets.In[ 7 ],amoregeneralmodelconsideringtheconventionaluniton/offoperationsandrampingconstraints is developed,inwhichCVaRisappliedtostudythetradingofthermalenergyanduncertainwindpoweroutput.Boththewindandthermalunitsareassumedtobeconnectedtothesamebusownedbyanindependentpowerproducer.Ithasbeenshownrecently in [ 69 ]thattheCVaRconstrainedproblem is aspecialcaseoftheexpectedvalueconstrainedproblem.Inthischapter,westudythegeneralexpectedvalueconstrainedproblem.Inourapproach, an SAA methodwillbe used to solvesuchastochasticoptimizationprobleminvolving boththechanceand theexpectedvalueconstraints. Thedeveloped SAAalgorithmcombinesthestatisticalanalysisofboththeexpectedvalueconstraintandthechanceconstraint.Themaincontributionsof this chapterare listed asfollows: 1. Weproposeastochasticoptimizationmodelthataddressesbothwindpoweroutputandloaduncertainties. 2. Weintroducetheexpectedvalueconstraintintheproposedstochasticoptimizationmodeltoensurewindpowerutilization.Tothebestofourknowledge,thisistherststudyontheexpectedvalueconstrainedstochasticunitcommitmentproblem. 3. WedevelopanewcombinedSAAalgorithmtosolvetheexpectedvalueandchanceconstrainedstochasticunitcommitmentproblem,whichhasneverbeenstudiedbefore. 4. Ourproposedapproachwillhelpenhancetheunitcommitment procedure byISOs/RTOstoensuretheutilizationofwindpoweroutput,whilemaintainingthesystemreliability. 65

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MathematicalFormulation Inthis section ,wedevelopatwo-stagestochasticunitcommitmentformulationconsideringboththeexpectedvalueconstraintandthechanceconstrainttoaddressuncertainwindpoweroutput.Therststageistodetermine theday-aheadunitcommitmentdecisionsthatinclude turn-on / turn-off decisions of thermal powergeneratingunitsbysatisfyingunitcommitment physical constraints.Thesecondstagecontainsthedecisionson thereal-timedispatchofthermalunitsandtheactualamountofwindpowerusage Thepenaltycostisintroducedinthesecondstagetocontroltheloadimbalance. Thedetailedformulationisdescribedasfollows: minTXt=1 X b2B Xi2 b ( SUbiubit+SDbivbit )+E[Q(o,u,v,)] (4) s.t.)]TJ ET BT /F6 11.955 Tf 112.68 -294.55 Td[(obi(t)]TJ /F5 7.97 Tf 6.59 0 Td[(1) + obit )]TJ ET BT /F6 11.955 Tf 185.56 -294.55 Td[(obik 0, (4) 8k :1k)]TJ /F3 11.955 Tf 11.95 0 Td[((t)]TJ /F3 11.955 Tf 11.96 0 Td[(1)MUbi,8i2b, 8b2B ,8t obi(t)]TJ /F5 7.97 Tf 6.59 0 Td[(1) )]TJ ET BT /F6 11.955 Tf 149.16 -348.34 Td[(obit + obik 1, (4) 8k :1k)]TJ /F3 11.955 Tf 11.95 0 Td[((t)]TJ /F3 11.955 Tf 11.96 0 Td[(1) MD bi,8i2 b,8b2B,8t )]TJ ET BT /F6 11.955 Tf 112.68 -402.14 Td[(obi(t)]TJ /F5 7.97 Tf 6.59 0 Td[(1) + obit )]TJ ET BT /F6 11.955 Tf 185.56 -402.14 Td[(ubit 0, 8i2b,8b2B,8t (4) obi(t)]TJ /F5 7.97 Tf 6.59 0 Td[(1) )]TJ ET BT /F6 11.955 Tf 149.16 -429.03 Td[(obit )]TJ ET BT /F6 11.955 Tf 176.26 -429.03 Td[(vbit 0, 8i2b,8b2B,8t (4) obit ubit vbit 2f0,1g, 8i2b,8b2B,8t, (4) where Q( o u v ,)=min TXt=1Xb2BXi2b Fc(qbit())+TXt=1+tQimb+t()+TXt=1)]TJ /F4 7.97 Tf -.65 -7.89 Td[(tQimb)]TJ /F4 7.97 Tf -14.2 -7.89 Td[(t() (4) Lbiobit q b it()Ubiobit,8i2 b,8b2B,8t (4) q b it())]TJ ET BT /F6 11.955 Tf 152.58 -625.13 Td[(q b i(t)]TJ /F5 7.97 Tf 6.58 0 Td[(1)()(2)]TJ /F6 11.955 Tf 11.96 0 Td[(o b i(t)]TJ /F5 7.97 Tf 6.59 0 Td[(1))]TJ /F6 11.955 Tf 11.95 0 Td[(o b it) L b i + 66

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(1+o b i(t)]TJ /F5 7.97 Tf 6.59 0 Td[(1))]TJ /F6 11.955 Tf 11.96 0 Td[(o b it)UR b i,8i2 b,8b2B,8t (4) q b i(t)]TJ /F5 7.97 Tf 6.58 0 Td[(1)())]TJ ET BT /F6 11.955 Tf 171.27 -38.85 Td[(q b it()(2)]TJ /F6 11.955 Tf 11.96 0 Td[(o b i(t)]TJ /F5 7.97 Tf 6.59 0 Td[(1))]TJ /F6 11.955 Tf 11.95 0 Td[(o b it)L b i+(1)]TJ /F6 11.955 Tf 11.95 0 Td[(o b i(t)]TJ /F5 7.97 Tf 6.59 0 Td[(1)+o b it)DR b i,8i2 b,8b2B,8t (4) QGt()=Xb2BGXi2bqbit(), 8t (4) QWt()=Xb2BW^qbt(), 8t (4) Qimb+t())]TJ /F6 11.955 Tf 11.95 0 Td[(Qimb)]TJ /F4 7.97 Tf -14.2 -7.89 Td[(t() =Xb2BDbt())]TJ /F3 11.955 Tf -177.73 -30.25 Td[((QWt()+QGt()),8t (4) )]TJ /F6 11.955 Tf 9.3 0 Td[(CijXb2BKbij(^qbt()+X r 2bqbrt())]TJ /F6 11.955 Tf 11.96 0 Td[(Dbt())Cij,8(i,j)2E, 8t (4) ^qbt()Wbt(),8b2B,8t (4) Pr( )]TJ /F3 11.955 Tf 9.3 0 Td[(QWt()+QGt())]TJ /F10 11.955 Tf 11.95 11.36 Td[(Xb2BDbt(), 8t )1)]TJ /F8 11.955 Tf 11.95 0 Td[(, (4) E"TXt=1QWt()#E"TXt=1Xb2BWWbt()#, (4) qbit(),^qbt(),QGt(),QWt(),Qimbt()0. (4) Intheaboveformulation,wedenoteFc(.)as the fuelcostfunction. The objectivefunction( 4 )iscomposedoftheunitcommitmentcostsintherststage, and thefuelcostaswellas the penaltycostduetoloadimbalanceinthesecondstage. Minimumup/down-timeconstraints( 4 )and( 4 ) meanthestatus(onoroff)of eachunit shouldlastforaminimumtimeonceitisstarteduporshutdown.Constraints( 4 )and( 4 )indicatethestart-up and shut -downoperationsforeachunit. Constraints ( 4 ) describe theupperandlowerboundsofpoweroutputofeachunit andrampingconstraints( 4 ) and ( 4 )limitthemaximum incrementordecrement ofpower generation ofeachunit betweentwoadjacentperiods.Constraints( 4 )and( 4 )representthetotalthermalgenerationand theactual windpower utilized Constraints( 4 )describethepossible 67

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loadimbalancewhichispenalizedintheobjectivefunction.Constraints( 4 )indicatethetransmissioncapacityconstraintsandconstraints( 4 )describethatthewindpowerutilizedshouldbenomorethanthemaximumavailablewindpower.Thechanceconstraint( 4 )requiresthatthechanceofloadimbalance beyondthetolerancelevel shouldbebelowapre-denedrisklevel.Finally,theexpectedvalueconstraint( 4 )canguaranteetheusageofwindpoweroutputisnolessthanacertainratioofthemaximumavailablewindpower. SolutionMethodology Inthissection,we develop acombinedSAA algorithm tosolvetheexpectedvalueconstrainedandchanceconstrainedtwo-stagestochasticprogram.Essentially,SAAisusedtoapproximatetheactualdistributionwithanempiricaldistributioncorrespondingtoarandomsample[ 43 ].Thebasicframework of thecombinedSAAframeworkcontainsthreeparts:scenariogeneration,convergenceanalysis,andsolutionvalidation.Forconvergenceproperties,theproofsfortheexpectedvalueconstrainedstochasticprogramandthechanceconstrainedstochasticprogramareprovidedin[ 69 ]and[ 43 ]respectively, whichcanbedirectlyappliedheretoshowtheconvergencepropertyofthecombinedSAAalgorithm. Thereforewefocusonthescenariogenerationandsolutionvalidationinthefollowinganalysis.ScenarioGeneration We useMonteCarlosimulationtogeneratescenariosfor the windpoweroutput andload Assumingthe windpoweroutputfollowsamultivariatenormaldistributionN(,) [ 63 ] ,whereisthepredictedvalueof the windpoweroutputandmatrixdenotesitsvolatility ,wecanrun MonteCarlo simulation togenerateNscenariosandeachscenariohasthesameprobability1 N.Nowwecanreplacethesecond-stageobjectivefunctionby 1 NNXn=1TXt=1Xb2BXi2bFc(qbit(n))+TXt=1+tQimb+t(n) 68

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+TXt=1)]TJ /F4 7.97 Tf -.65 -7.89 Td[(tQimb)]TJ /F4 7.97 Tf -14.2 -7.89 Td[(t(n). (4) Fortheexpectedvalueconstraint,weuse the MonteCarlo simulation togenerateNscenariostoestimateconstraint( 4 )by: 1 NNXn=1TXt=1QWt(n) NNXn=1TXt=1Xb2BWWbt(n). (4) Meanwhile,thechanceconstraint( 4 )canbeestimatedas follows[ 6 ]: 1 NNXn=11[)]TJ /F5 7.97 Tf 6.59 0 Td[(,](QWt()+QGt())]TJ /F10 11.955 Tf 11.95 11.35 Td[(Xb2BDbt(),8t)1)]TJ /F8 11.955 Tf 11.95 0 Td[(, (4) where1[)]TJ /F5 7.97 Tf 6.59 0 Td[(,](.)isanindicatorfunction,i.e., 1[)]TJ /F5 7.97 Tf 6.59 0 Td[(,](x)=8>><>>:1,ifx2[)]TJ /F3 11.955 Tf 9.3 0 Td[(,];0,ifx=2[)]TJ /F3 11.955 Tf 9.3 0 Td[(,]. Similarly,constraints( 4 )-( 4 )canbereplacedrespectivelybyscenario-basedconstraints, following thetraditionalSAAmethod forthetwo-stagestochasticprogram [ 5 ]. TheultimateSAAproblemisprovidedinthefollowing Subsection 4 afterreformulating the indicatorconstraint( 4 ). MILPreformulationofchanceconstraint Afterwegeneratethescenarios,thechanceconstraint isconvertedto anindicatorfunction( 4 )intheSAAproblem.WeuseanMILPmodeltoreformulatethissampledchanceconstraint. Foragiven samplesizeN,weintroduceabinarydecisionvariablez indicating whetherthechanceconstraintissatisedinthecorrespondingscenario. Thenthechanceconstraintcanberepresentedasfollows: )]TJ /F3 11.955 Tf 9.3 0 Td[()]TJ /F6 11.955 Tf 11.95 0 Td[(MznQWt()+QGt())]TJ /F10 11.955 Tf 11.96 11.36 Td[(Xb2BDbt()+Mzn, 8t,8n (4) NXn=1znN,zn2f0,1g,8n. (4) 69

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ReformulationofSAAproblem TheultimateSAAproblemformulationisasfollows: minTXt=1 Xb2B Xi2 b ( SUbiubit+SDbivbit )+1 NNXn=1TXt=1Xb2BXi2bFc(qbit(n))+TXt=1+tQimb+t(n)+TXt=1)]TJ /F4 7.97 Tf -.65 -7.89 Td[(tQimb)]TJ /F4 7.97 Tf -14.2 -7.89 Td[(t(n) (4) s.t.( 4)-222()]TJ /F3 11.955 Tf 21.26 0 Td[(2 ))]TJ /F3 11.955 Tf 11.96 0 Td[(( 4)-222()]TJ /F3 11.955 Tf 21.26 0 Td[(6 ),( 4)-222()]TJ /F3 11.955 Tf 21.26 0 Td[(8 ))]TJ /F3 11.955 Tf 11.96 0 Td[(( 4)-222()]TJ /F3 11.955 Tf 21.26 0 Td[(15 ),( 4)-222()]TJ /F3 11.955 Tf 21.26 0 Td[(18 ),82f1,...,Ng,( 4)-222()]TJ /F3 11.955 Tf 21.26 0 Td[(20 ),( 4)-221()]TJ /F3 11.955 Tf 21.25 0 Td[(22 ),( 4)-222()]TJ /F3 11.955 Tf 21.25 0 Td[(23 ).SolutionValidation Thebasicideaforthevalidationprocessisto applystatisticaltechnique s toapproximatetheupperandlowerbounds of theoptimalobjectivevalue of theSAAproblem.Theoptimalitygapcanbeobtainedfromthevalidationprocess withacondencelevel. Fornotationbrevitytointroduceourproposedalgorithm,themathematicalmodelcanbeabstractedasfollows: mincTx+E[Q(x,)] (4) s.t.Axb, (4) E[G(x,y(),)] ` (4) Pr(H(x,y(),)0), (4) x0, (4) whereQ(x,)=mind()Ty() (4) s.t.T()x+Wy()=h(),y()0. (4) 70

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Upperbound First,we noticethatthesecondstageproblemmightnothave the completerecourseproperty[ 5 ]. Thatis,thesecondstageisnotguaranteedtobefeasibleforagivenunitcommitmentdecision. Ifthisisthecase, wecanenlargetheinthechanceconstrainttoallowmoreimbalance .Sucha settingcan guaranteethatforany rststagesolution x (e.g.unitcommitmentdecision) andanyscenario,therealwaysexistsafeasiblesolutiony()forthesecondstage, i.e., Q(x,)<1forallxand. Itcanbealsoobservedthat Q(x,)0. Then wecanassumethattheexpectedvalueE[Q(x,)]iswell-denedandnitevaluedforagivendistributionof. Weknow that anyfeasiblesolutioncanprovide an upperboundoftheoptimal objective value.Let f1,2,,Ngbeasampleofsize N and xbetheoptimalrststagesolution of theSAAproblem.Weknowitispossiblethatxisinfeasibletotheoriginalproblem.Denep(x):=PrfH(x,y(),)gandq(x):=E[G(x,y(),)], anduse thefollowingtwoconstraints intheSAAproblem : p(x)~, (4) q(x)~`, (4) where~and~`` Constraints( 4 )and( 4 ) areequivalenttothechanceconstraint( 4 )andtheexpectedvalueconstraint( 4 )when~=and~`=`. NotehereifthesamplesizeN islargeenough, when ~and~``,thefeasiblesolutionoftheSAAproblemismorelikelytobefeasibletotheoriginalproblem. Now,foragivensizeN,rstbasedonthemethoddescribed in[ 6 ],weconstructthe(1)]TJ /F8 11.955 Tf 12.3 0 Td[()condenceupperboundforthechanceconstraint( 4 ): Uc(x)=pN(x)+zr pN(x)(1)]TJ /F6 11.955 Tf 11.95 0 Td[(pN(x)) N, (4) 71

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wherepN(x)=1 NPNn=11[0,1)(H(x,y(n),n)).IfUc(x)islessthan orequalto therisklevel,thenxisfeasiblefor the chanceconstraintwith the condencelevel(1)]TJ /F8 11.955 Tf 12.47 0 Td[().Ifnot,wewilldecreasethevalueof~andsolvetheproblemagaintocheckwhetherthe updated optimalsolution leadstoanewvalueUc(x),whichislessthanorequalto. Secondly ,wecomputethe(1)]TJ /F8 11.955 Tf 12.29 0 Td[()condenceupperboundfortheexpectedvalueconstraint: Ue(x)=qN(x)+zs PNn=1[G(x,y(n),n))]TJ /F6 11.955 Tf 11.96 0 Td[(qN(x)]2 N(N)]TJ /F3 11.955 Tf 11.95 0 Td[(1), (4) whereqN(x)=1 NPNn=1G(x,y(n),n).IfUe(x)islessthanorequalto`,weclaimxisfeasiblefortheexpectedconstraintwith the condencelevel(1)]TJ /F8 11.955 Tf 12.5 0 Td[().Ifnot,wecandecreasethevalueof~`toimprovethechanceof obtaininga feasiblesolution. Sinceitiswell-knownthatPr(ATB)Pr(A)Pr(B),weconcludethatxisfeasiblefortheoriginalproblemwithacondencelevel,where(1)]TJ /F8 11.955 Tf 11.92 0 Td[()2.Thecorrespondingupperboundofthetrueproblemisgivenasfollows: U(x)=cTx+1 NNXn=1Q(x,n). (4) Lowerbound Toobtainalowerbound,we rstconsiderthefollowing Lagrangerelaxationformulation(denotedasP0)oftheoriginalproblem : minfcTx+E[Q(x,)]+(E[G(x,y(),)])]TJ /F8 11.955 Tf 11.95 0 Td[(`)gs.t.Constraints( 4 )and( 4 ). (4) Denotetheoptimal objective valueofP0asv0,andtheoptimal objective valueof the originalproblemasv.FromLagrangianduality,wehave v0v (4) 72

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forany0.Especially,the equation holdswhenistheoptimalLagrangemultiplierandthereisnodualitygap. Now,werst generateNscenarios,andestimatethechanceconstraint( 4 )as1 NPNn=11[0,1)(H(x,y(n),n)).TheLagrange relaxation problemcanbeapproximatedasfollows(denotedasP1): mincTx+E[Q(x,)]+(E[G(x,y(),)])]TJ /F8 11.955 Tf 11.95 0 Td[(`)s.t.Constraint( 4 ),1 NNXn=11[0,1)(H(x,y(n),n)). (4) Then ,afterthescenariosaregenerated(e.g.,Nscenarios),theexpectedobjectivevalueintheLagrangian relaxation objectivefunctionisestimatedbythesampleaverage approximation function: 1 NNXn=1(Q(x,n)+(G(x,y(n),n))]TJ /F8 11.955 Tf 11.95 0 Td[(`)). NowtheLagrange relaxation problemisapproximatedasthefollowingSAAproblem(denotedasP2): minnfN(x)=cTx+1 NNXn=1(Q(x,n)+(G(x,y(n),n))]TJ /F8 11.955 Tf 11.96 0 Td[(`))os.t.Constraint( 4 ),1 NNXn=11[0,1)(H(x,y(n),n)). (4) WesolvetheSAAproblemwithNscenariosMtimes.FortheseMruns,wedenotetheoptimalvalueofP1problemtobe^vP11,^vP12,,^vP1M,andtheoptimalvalueofP2problemtobe^vP21,^vP22,,^vP2M.Wefollowthesameschemeastheonedescribedin[ 6 ]and[ 43 ],andpicktheLthsmallestoptimalvalueofP1.Withoutlossof generality ,we 73

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canassumethevalueis^vP1L.Fromtheconclusionsin[ 6 ]and[ 43 ],wehave: Pr(^vP1Lv0)=1)]TJ /F8 11.955 Tf 11.96 0 Td[(. (4) Atthesametime ,foreachruni=1,2,,M,wehaveE[^vP2i]^vP1i[ 51 ]. Inparticular ,wehave E[^vP2L]^vP1L. (4) NowweapproximateE[^vP2L]bystatisticaltechnique s .WesetSasthenumberoftotaliterations.Ineachiterations,1sS,wesolvetheSAAwithNscenariosMtimestoobtain^vP2L,andre-denoteitasvP2Ls.Let vL=vP2L1++vP2LS S, (4) ~v=vL)]TJ /F6 11.955 Tf 11.96 0 Td[(zs PSs=1[vP2Ls)]TJ /F3 11.955 Tf 12.25 0 Td[(vL]2 S(S)]TJ /F3 11.955 Tf 11.95 0 Td[(1). (4) Wecanobtainthat: Pr)]TJ /F3 11.955 Tf 5.77 -9.68 Td[(~vE[^vP2L]=1)]TJ /F8 11.955 Tf 11.95 0 Td[(. (4) Frominequalities( 4 ),( 4 ),( 4 )and( 4 ),wecanseethatPr(~vv)(1)]TJ /F8 11.955 Tf 11.96 0 Td[()2,i.e.,~visthelowerboundofvwithacondencelevel,where(1)]TJ /F8 11.955 Tf 11.95 0 Td[()2.SummaryoftheCombinedSAAAlgorithm Inthissubsection,wesummarizethecombinedSAAalgorithmasfollows (theowchartisshowninFig. 2-1 ) : 1. Fors=1,2,,S,repeatthefollowingsteps: (a) Form=1,2,,M,repeatthefollowingsteps: i. Setc>0,e>0,2(0,1),~=, and ~`=`. 74

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ii. SolvetheSAAproblemwithNscenarios: min^fN(x)=cTx+1 NNXn=1Q(x,n)s.t.Axb,x0,1 NNXn=11(0,1)(H(x,y(n),n))~,1 NNXn=1G(x,y(n),n)~`, where thesecondstageisthesameas( 4 ).Let(xm,ym)betheoptimalsolutiontotheSAAproblem,andmbetheoptimalLagrangemultiplier. iii. Generatescenarios1,2,,N0 foralargenumberN0 anduse( 4 )and( 4 )toobtain Uc(xm)andUe(xm) .CheckwhetherUc(xm)andUe(xm)`hold.Iftheformerfails,reduce~byc.Ifthelatterfails,decrease~`to~`)]TJ /F8 11.955 Tf 11.96 0 Td[(e.Returntob). iv. Estimatethecorrespondingupperboundusing( 4 ),basedontheN0scenariosgeneratedinc). (b) Pickthesmallestupperboundin1)astheapproximatedupperbound^gs. (c) ByusingtheLagrangemultiplierobtainedinb),sorttheMoptimalvaluesofP2in a nondecreasingorder,e.g.,^vP21^vP22^vP2M.PicktheLth smallest optimalvalue^vP2LanddenoteitasvP2Ls. 2. Takingtheminimumof^g1,^g2,,^gS,wegettheupperbound^g=min1sS^gs. 3. Compute vLbasedon( 4 )and theestimatedlowerbound~v basedon( 4 ) 4. Estimatetheoptimalitygapusing(^g)]TJ /F3 11.955 Tf 12.74 0 Td[(~v)=~v100% withthecondencelevelatleast(1)]TJ /F8 11.955 Tf 11.96 0 Td[()2 .ComputationalResults Inthissection,weperformcasestudiesforasix-bussystemandtworevisedIEEE118-bussystemstoshowtheeffectivenessoftheproposedapproach.Werstperformsensitivityanalysis of theSAAproblem on thesix-bussystem.ThenthecombinedSAAalgorithmdescribedinSection 4 isappliedtosolvethetworevisedIEEE-118bussystems. WeuseC++withCPLEX12.1toimplementtheproposedformulationsand 75

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Figure4-1. ProposedSAAalgorithm 76

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algorithms.Alltheexperimentsareconductedonacomputerworkstationwith4IntelCoresand8GBRAM.Six-BusSystem Thesix-bussystemcontainsthreethermalgeneratorsandonewindfarm.Thesettingsofthewindfarm,thethermalgenerators,theloadforecasts,andthetransmissionlinesarethesameastheonesdescribedin[ 64 ],exceptthewindfarmislocated atbusB2 .Weprovide a sensitivityanalysison the utilizationrateintheexpectedvalueconstraintand the risklevelinthechanceconstraint. Wesetthesamplesize N tobe50.Wealsorequirezero probabilityofloadimbalance .Thatis,therisklevelof the chanceconstraintiszero. Thecomputational results arereported inTable 4-1 .Itcanbeobservedthatthe total cost(columnobj.inTable 4-1 )increasesastheutilizationrateincreasesfrom60%to90%. Thisisbecause theutilizationpolicybecomesmorerestrictiveasincreases. Table4-1. Computationalresultsforthesix-bussystemwithdifferentutilizationrates UtilizationObj.($)CPUTime(sec) 60%26977.52.770%27646.220.580%32991.910.285%39603.99.590%46215.99.1 Toinvestigatethesignicanceof thesettingsonthechanceconstraint ,wereportthe results inTable 4-2 withdifferentrisklevels.Thesamplesizeissettobe50andissettobe70%.Itcanbeobservedthatthe total costdecreasesastherisklevel Table4-2. Computationalresultsforthesix-bussystemwithdifferentrisklevels RiskLevelObj.($)CPUTime(sec) 0.0527235.577.90.1026649.0161.40.1526284.3160.20.2025928.5821.6 77

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increasesfrom0.05to0.2,becausetheload imbalance policybecomemorerelaxedastherisklevelincreases. Finally,weruntheexperimentswithdifferentscenariosizesandobservethattheoptimalobjectivevalue converges asthesamplesizeincreasesto70(therisklevelissettobe0.1andissettobe70%).Revised118-BusSystems TworevisedIEEE118-bussystems(named118SWand118DW)basedontheonegivenonlineathttp://motor.ece.iit.edu/dataarestudiedinthissection.Apartfrom the 33thermalgeneratorsintheoriginalIEEE118-bussystem,118SWiscreatedbyaddingasinglecentralizedwindfarmand118DWiscreatedbyadding10windfarmsat10differentbuses.Allthe186transmissionlinesintheoriginalsystemareselectedforbothrevisedsystems.SAAalgorithmanalysisfor118SW Thecomputationalresultsofouralgorithmfor118SW are reportedinTable 4-3 .Intherstcolumn,weindicatethatdifferentcombinationsofvalidationsettings(i.e. iterationnumbers(S,M)andvalidationscenarionumber(N0))areconsidered in theexperiments.ThescenariosizeoftheSAAproblemisgiveninthesecondcolumn.ThelowerboundandupperboundobtainedbytheSAAalgorithm arereported inthethirdandfourthcolumns.ThefthcolumnrepresentsthegapwhichiscalculatedbyUB)]TJ /F4 7.97 Tf 6.59 0 Td[(LB LB100%.Finally,theCPUtimeofthealgorithmisreportedinthesixthcolumn. Fromthetable,wehavethefollowingobservations:First,when thescenariosizeof the SAAproblemisN=10,wealreadyobtainasmalloptimalitygapat0.44%.Suchanobservationsuggeststhatthesolutionof the SAAproblemwith the scenariosizeN=10isalreadyveryclosetothesolutionofthetrueproblem. Secondly ,theoptimalitygap decreases as the scenariosizeof the SAAproblemand the validation problem increases. Finally,i nthelastrow,whenthescenariosizeof the SAAproblemisN=50 78

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Table4-3. Computationalresultsforthe118-bussystemwithdifferentcombinationsofiterationsandsamplesizes (SM,N0)NLBUBGapTime(sec) (55,100)105855785881780.44%127205858735881780.39%402505866515882060.27%1379(105,500)105861335877000.27%316205868185877000.15%783505871445877340.10%2319(2010,1000)105868295877580.15%1364205872195877580.09%3325505873805877930.07%9005 withthe iterationnumbersandthevalidation scenariosizebeing(2010,1000),thesmallestoptimalitygapisachievedat0.07%.Distributedwindpowersystem118DW Weruntheexperimentson118DWinthissectionandanalyzetheimpactoftheexpectedvalueconstraintondistributedwindpowerresources.Intotal,thereare10windfarmsat10differentbuses(busnumbersareshowninTable 4-4 ).Undersuchasetting,theassociatedwindpoweruncertaintiesaffectthepowergridoperationsdramatically. Werstrelaxtheexpectedvalueconstraintandsolvethestochasticoptimizationproblem.TheutilizationratesofwindpoweratspecicbusesaresummarizedinTable 4-4 .ThelowestutilizationandhighestutilizationareobservedinB36andB96,respectively.ThereasonoflowutilizationisthatthereisonethermalgeneratorinB36andthecapacityofthetransmissionlineconnectedwiththisbusisrelativelylowercomparedwithothers.Thewindpowerhastobecurtailedduetothe limited transmissionlinecapacity.Conversely,thetotalcapacityofthetransmissionlinesconnectedwithB96ismuchlarger(therearevetransmissionlinesconnectedwithB96)andnogeneratorislocatedthere. 79

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Table4-4. Computationalresultsfor118DW-withoutexpectedvalueconstraint BusNO.UtilizationBusNO.Utilization B690%B5665%B1678%B6676%B2673%B7674%B3640%B8671%B4690%B9696% Thenwesolvetheproblemwiththeexpectedvalueconstraintenforced.Theutilizationrateissettobe85%.ThecorrespondingutilizationratesateachstudiedbusarereportedinTable 4-5 .Itcanbeobservedthat the utilizationratesatallbusesareincreasedcomparedwiththoseinTable 4-4 .Ontheotherhand,wecanalsoobservethat the utilizationincrementsatdifferentbusesarenotthesame as thelargestincrement is 22%(B26)andthesmallestincrement is 2%(B96),whichshowstheadvantageofapplying theproposed solutionapproachtominimizethetotalcostwhileensuringtheoverallhighwindpowerutilizationrate. Inotherwords, ourmodelcanhelpcoordinategeneration injections atdifferentbusesandprovidetheoptimal generation strategyateachbus. Table4-5. Computationalresultsfor118DW-withexpectedvalueconstraint BusNO.UtilizationBusNO.Utilization B697%B5669%B1684%B6690%B2695%B7685%B3655%B8678%B4693%B9698% ConcludingRemarks Inthischapter,weproposedtheexpectedvalueconstrainedstochasticprogramtostudytheuncertainwindpower generation .Byadjustingtheutilizationrateofwindresources,thesystemoperatorcanchangethe utilizationrates ofwindresources 80

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through the expectedvalueconstraint. Inaddition ,thechanceconstraintisappliedtomodelthe loadimbalance withasmall probability Thus,our modelincorporate s both the expectedvalueconstraintandchanceconstraint. Accordingly ,we proposed a combinedSAA algorithmtosolvetheproblem .We alsoperformed thesensitivityanalysison differentrisklevelsandwindutilizationratesfor thesix-bussystem.Finally,thecomputation al resultsof the revisedIEEE118-bussystemsexhibitthescalabilityofour proposed modelsandalgorithms. Itisworthwhiletonotethatbypreviousstudy[ 64 ],ISOscanaddressahighwindpowerutilizationwithachance-constrainedstochasticprogram.Theexpectedvalueconstraint,fromanotherperspective,allowstheISOsenforceanoverallhighwindpowerusage.TheISOscanaccordinglychoosethedifferentpoliciesandcorrespondingconstraintstoenhancethewindpowerpenetrationlevel.Insomeextremecase,theISOsmightwanttohavebothchanceconstraintandexpectedvalueconstraintonthewindpowerutilization.TheproposedcombinedSAAalgorithminthisresearchcanbeappliedtosuchacasewithoutlossofgenerality. 81

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CHAPTER5STOCHASTICUCWITHUNCERTAINDEMANDRESPONSE TheobjectiveofanIndependentSystemOperator(ISO)istomaximizethesocialwelfareforelectricityproducersandcustomers.CustomersparticipatingintheDemandResponse(DR)programcanexpectsavingsbyreducingtheirelectricityusageduringpeakperiods[ 8 ].Intheliterature,DRwasmostlymodeledasaxeddemandcurve.However,duetoavarietyofreasonssuchaslackofattention,latencyincommunication,andchangeinconsumptionbehavior,theactualresponsefromtheconsumerstoapricesignalisuncertaininnature.Hence,thecustomerbehaviorisexplicitlymodeledbyanuncertaindemandelasticityinthischapter,whichmeanscustomershavedifferentresponsestotheelectricitypricesunderdifferentscenarios.Wealsoconsidergeneratoroutages andtransmissionlinecontingencies whichcanbeaddressedbyDRprogramstoavoidorreduceforcedloadcurtailment. OurproposedapproachcanbeappliedtoenhancethereliabilityunitcommitmentprocessforISOs. Weconsideratwo-stagestochasticprogrammingformulationwithunitcommitmentdecisionsattherststageandreal-timegenerationandloadamountdecisionsatthesecondstage.Theobjectiveistomaximizethesocialwelfare: max)]TJ /F3 11.955 Tf 9.3 0 Td[(( cgs +PTt=1PiE[fi(xit())])+PTt=1PbE[Ft,b,(dbt())] )]TJ /F10 11.955 Tf 11.29 8.97 Td[(PTt=1E[twt()] (5) wherecgsdenotesgeneratorstart-upandshut-downcosts[ 64 ],fi(xit())representsthefuelcostforgeneratoriattimetwhenthegenerationamountisxit(),Ft,b,()representstheconsumerbenetatbusbattimet withdbt()representingtheamountofelasticloadatbusbattimet(notehereeachbusloadincludesbothinelasticandelasticloads,andtheconsumerbenetfortheinelasticloadiszero,cf.[ 55 ].Therefore, c[2012]IEEE.REPRINTED,WITHPERMISSION,FROM[ 65 ] 82

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theobjectivefunctiononlyincludeselasticloads), andwt()andtrepresentthetotalamountofloadcurtailmentandunitpenaltycost attimet,respectively. Ourmodelincludes generationupper/lowerboundconstraints,minup/-down time constraints,start-up/shut-downconstraints,ramp-up/-downconstraints, spinningreserveconstraints, and transmissioncapacityconstraints (cf.[ 64 ]) Boththegenerationupper/lowerboundandtransmissioncapacityconstraintsconsidercontingencies.Forinstance,thegenerationboundconstraintswithuncertaingeneratorcontingencyconsideration aremodeledasfollows: Liyit(1)]TJ /F6 11.955 Tf 11.96 0 Td[(Ci())xit()Uiyit(1)]TJ /F6 11.955 Tf 11.95 0 Td[(Ci())8i,8t, (5) Pr(Ci()=1)=i 8i, (5) whereLiandUiarelowerandupperboundsofgeneratori,yitisabinaryvariabletoindicateifgeneratoriison during timeperiodt,Ci()isarandombinaryparameterindicatingthecontingencyofgeneratori,andiisthegivenprobabilityvaluethatthecontingencyhappensforgeneratori.Constraints( 5 )enforcethegenerationoutputtobezeroduringthecontingency.Finally,achanceconstraintisintroducedtoformulatethelossofloadprobability(LOLP)asfollows: Pr Xb(dbt()+^dbt())Xixit(),8t!1)]TJ /F8 11.955 Tf 11.95 0 Td[(,(5) whereisdenedasriskleveland^dbt()representstheamountofinelasticloadatbusbattimet. SolutionMethodologyandCaseStudy SolutionMethodology AnSAAmethodisutilizedtosolvetheproblem.Inourapproach,aMonteCarlomethodisrstappliedtogeneratescenarios(e.g.,Nscenarios).Then,theexpectedvaluefunctionisreplacedwiththesampleaveragefunction,andaccordinglythechanceconstraintisreplacedwithanMILPreformulationasin[ 43 ]. Theprice-elastic 83

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demandcurveforeachISOcouldbedifferentwiththecommonpartthatthedemandisanon-increasingfunctionofprice(cf.[ 55 ]and[ 33 ]).Thiscurvecanbeobtainedbysimulationandhistoricaldataanalysis.Withoutlossofgenerality ,inthischapter,theprice-elasticdemandcurveisdescribedasd b t()=A b t p b t ()t ,b (cf.[ 56 ])withthepurposetoillustrateourproposedsolutionapproach. Foragivenelasticity b t (),A b t canbedecidedbythegivenreferencepoint(Dreft, b ,Preft, b ).Then,astep-wisefunctionisappliedtoapproximatethisdemand-pricefunction asdescribed in[ 56 ]: Ft,b,(dbt())=PKk=1pkt,brkt,b() (5) dbt()=PKk=1rkt,b(), 0rkt,b()lkt,b (5) whereKisthenumberofsteps(seeFig. 5-1 ),pkt,bandlkt,baregivenforeachk,andrkt,b()isanauxiliarydecisionvariable. Basedon( 5 )and( 5 ),andthemaxobjective,wehaverkt,b()=lkt,binthesolutionifdbt()Pku=1lut,b.Thus,thediscrepancybetweentheapproximationandtheintegralofthecurve(consumerbenet)isequaltothedifferencebetweentheshadedareaabovethecurveandtheonebelowthecurve,andthisdiscrepancyconvergestozeroasK!+1. Inaddition,bt()isassumedtofollowanormaldistribution( ourmethodologycanalsobeappliedtootherdistributions )andaMonteCarlomethodisappliedtogeneratebt()forobtainingFt,b,underdifferentscenarios. Fortheprobabilityconstraints,duringthescenariogenerationprocess,werandomlysetgeneratoriundercontingencystatusforiNscenarios,and transmissionline(m,n)undercontingencystatusformnNscenarios,wheremnisthegivenprobabilityvaluethatthecontingencyhappensfortransmissionline(m,n)andNrepresentsthetotalnumberofscenarios. CaseStudy WestudytherevisedIEEE118-bussystem(onlineatece.iit.edu/data) with33generators toillustratetheresults. 84

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Figure5-1. Step-wiseapproximationofprice-elasticdemandcurve DeterministicDRvs.stochasticDR WerstsettheriskleveltobezeroandcomparetheresultsbasedonthedeterministicDRandstochasticDRrepresentations,toshowhowstochasticDRworksbetter. Weassumepossiblecontingenciesoccurontwotransmissionlinesandtwogeneratorsinthissubsection. ForthedeterministicDR,theprice-elasticdemandcurveiscertaininwhichthemeanvalueoftheelasticityistakentogeneratethecurve.SeveralindicatorsaregiveninTable 5-1 forcomparisonpurposes.Itcan Table5-1. Deterministicvs.stochastic DeterministicStochastic Number ofStart-ups 82 144 ExpectedReserve Amount (MW) 1191 2100 ExpectedLoadLoss(MW) 833 120 SolutionTime(sec.) 35.2 50.6 85

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beobservedthatthestochasticformulationapproachputsmoregeneratorsonlinetoprovideadditionalcapacityforunexpectedconsumptionbehaviors.Thisapproachprovidesmorereserves( foreachscenario,itismeasuredasthedifferencebetweenthetotalgenerationcapacityofonlinegeneratorsandtheload )whichleadtolessloadcurtailment.Risklevelsanddemandresponseeffect TheoptimalobjectivevaluescorrespondingtodifferentrisklevelsarereportedinTable 5-2 .Therisklevelisrepresentedbytheprobabilitydenedinthechanceconstraint( 5 ),whichindicates thepossibilityoftheloadbeing curtailed.Itcanbeobservedthatthesocialwelfareincreaseswhentherisklevelincreases, becauseallowingloadcurtailmentprovidesmoreexibilityforgenerationscheduling. Table5-2. Computationalresultsfordifferentrisks Obj.($)Time(sec.) 0 1252530 24.5 10% 1323200 57.8 30% 1464530 131.9 ToshowtheeffectivenessofDR,weassumebtthesameforeachbandtandcomparetheoptimalsocialwelfareusingagroupofelasticitieswithdifferentmeanandstandarddeviationvalues(e.g.,and). ItcanbeobservedinTable 5-3 thatthetotalsocialwelfarehasatendencytoincreaseasthedemandelasticityincreases.Butitisnotguaranteedthatthereisalwaysapositivecorrelationbetweenelasticityandwelfare.Itdependsoneachspecicprice-elasticdemandcurve.Also,ourconclusionisbasedonthereferencepointmodelingapproachweused.Itmaynotbegeneralizedtoothermodelingmethods.However,thegeneralmodelingframeworkwedescribedinthischaptercanaccommodateotherdemandsidemodelingapproaches.Ourproposedsolutionapproachcansolvethesesmodelsefcientlyandnumerically. 86

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Table5-3. Computationalresultsfordifferentelasticities (,)Obj.($)Time(sec.) (-0.8,0.2) 1252530 24.90 (-2,1) 1494270 30.28 (-3,2) 1738670 14.57 ConcludingRemarks Inthischapter,weprovidedageneralmodelingframeworkthatconsiderstheuncertaindemand-sideresponseinwhichprice-elasticdemandcurvesvarybyscenario.Thisframeworkcanaccommodatedifferentdemandsidemodelingapproaches. Inaddition,theproposedchanceconstraintcontrolstheLOLPandthesampleaverageapproximationmethodcansolvetheIEEE118-bussystemefciently.Finalcasestudiesindicatethatthestochasticrepresentationofuncertaindemandresponsecanleadtomoreavailablegenerationcapacity,ascomparedtoitsdeterministiccounterpart. 87

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CHAPTER6CONCLUSIONSANDSUGGESTIONSFORFUTURERESEARCH Inthisdissertation,severalstochasticprogrammingformulationsareproposedtocapturetheuncertaintyintheunitcommitmentproblem.Inparticular,achance-constrainedtwo-stagestochasticprogramisdevelopedtostudythestochasticunitcommitmentwithuncertainwindpoweroutputforISO.Thisideaisextendedtostudytheprice-basedunitcommitmentforelectricitymarketparticipantIPP.Then,anexpectedvalueconstrainedstochasticprogramisconsideredtoguaranteehighutilizationofwindpowerinISO'sunitcommitmentproblemfromadifferentperspective.Finally,thechance-constrainedtwo-stagestochasticprogramisappliedtostudytheuncertaindemandresponse.WedevelopcorrespondingSAAframeworksfortheabovehybridmodels.ThecasestudiesindicatethesignicanceofthenewstochasticunitcommitmentmodelsandeffectivenessofthenewSAAalgorithms.Next,webrieydiscusssomefutureresearchdirections.UnitCommitmentwithUncertainContingency Unexpectedcontingenciesofpowergridelements,suchastransmissionlinesandgenerators,canresultindramaticelectricityshortagesorevenlarge-scaleblackouts(see,e.g.,[ 10 24 47 71 ]).Giventhetopologyofapowergridstructure,unitcommitmentisamongthemostcrucialdecisionsforasystemoperatortohandlepost-contingency.Thewell-knownN-1andN-2securitycriteriaareimplementedinindustrypracticewithstochasticunitcommitmentmodels(see,e.g.,[ 50 62 ]).Thesecriteriahavealsobeengeneralizedtoconsidermultiplecontingencycases(e.g.,N-kcriterion[ 13 ]).WiththeN-krule,apowergridwiththeNcomponentswillcontinuetomeetdemandwheneveranykorfewercomponentssufferacontingency.However,verylimitedresearch[ 31 53 ]hasbeendonetoincorporatethegeneralN-kruleintheunitcommitmentusingthestochasticprogrammingorrobustoptimizationduetothecomputationalintractability. 88

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Robustoptimizationisanotherstate-of-the-artapproachtosolvedecision-makingunderuncertainty.Comparedwithstochasticprogramming,robustoptimizationdoesnotrequireanexplicitprobabilitydistributionoftheuncertainparameter.Theuncertaintyrepresentationisadeterministicsetintherobustoptimization.Robustoptimizationprovidestheoptimalsolutionundertheworst-casescenario,thusitssolutionimmunizesallthepossiblescenariorealizations(i.e.alwaysbefeasible).Itisanimportantandinterestingtopictoformulateandsolvethecontingency-constrainedunitcommitmentwithstochasticprogramming/robustoptimizationandcomparetheirperformances.OtherRecoursestoHedgeWindPowerUncertainty Sincedemandresponsecanbeconsideredasanadditionalreservecapacity,itwillbeinterestingtousethedemandresponsetobalancetheuncertainwindpowergeneration.Recently,somepromisingenergystoragetechniquessuchasplug-inhybridelectricvehicle(PHEV)inthesmartgridcanaccommodatethewindpowerinasignicantmanner.Asafuturesubject,weareinterestedinmodelingandsolvingmoreadvancedstochasticunitcommitmentwithdiversiedresourcestohedgethewindpoweruncertainty. 89

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APPENDIX:PROOFOFPROPOSITION2.1 Convergenceproofsforthechance-constrainedandthetwo-stagestochasticprogramshavebeenstudiedin[ 43 ]and[ 5 ],respectively.Thisappendix,however,providestherstproofforthecasethatcontainsbothchance-constrainedandtwo-stagestochasticprogramfeatures. InourCCTSprogram,thesamplesoftherandomvariable(windgeneration) are usedintheapproximations forboththesecond-stagevalue andthechanceconstraintpart. Oncethesamplesaregenerated,t hestochasticproblemwillbecomeadeterministicproblem.Weshowinthispartthatthe objective ofthedeterministicproblemisconvergenttothatofthestochasticproblemasthescenariosizegoestoinnity. Recallthat our CCTSprogram(e.g.,thetrueproblem)canbeexpressedasfollows: minXb2BGTXt=1Xi2b(biubit+bivbit+biobit+Fc( qbit ))+TXt=1 t E[Xb2BWSbt( )] (A) s.t.( 2)-222()]TJ /F3 11.955 Tf 21.26 0 Td[(2 ))]TJ /F3 11.955 Tf 11.96 0 Td[(( 2)-222()]TJ /F3 11.955 Tf 21.26 0 Td[(13 )PrfG(x,)0g1)]TJ /F8 11.955 Tf 11.95 0 Td[(Sbt( )=maxf0,^qbt)]TJ /F6 11.955 Tf 11.96 0 Td[(wbt()g,8t,8,8b2BW qbit ,^qbt0;obit,ubit,vbit2f0,1g,8t,8i,8b. Beforestartingtheproofoftheconvergence,weintroducethefollowingtwoapproximated models : (i)Replacethechance-constrainedpartbythesampleapproximation. minXb2BGTXt=1Xi2b(biubit+bivbit+biobit+Fc( qbit )) 90

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+TXt=1 t E[Xb2BWSbt( )] (A) s.t.( 2)-222()]TJ /F3 11.955 Tf 21.26 0 Td[(2 ))]TJ /F3 11.955 Tf 11.96 0 Td[(( 2)-222()]TJ /F3 11.955 Tf 21.26 0 Td[(13 )N)]TJ /F5 7.97 Tf 6.59 0 Td[(1NXj=11(0,1)(G(x, j)) Sbt( )=maxf0,^qbt)]TJ /F6 11.955 Tf 11.96 0 Td[(wbt()g,8t,8,8b2BW qbit ,^qbt0;obit,ubit,vbit2f0,1g,8t,8i,8b. (ii) Replace boththechance-constrainedand thesecond -stagepartsbythesampleapproximation. minXb2BGTXt=1Xi2b(biubit+bivbit+biobit+Fc( qbit ))+N)]TJ /F5 7.97 Tf 6.58 0 Td[(1TXt=1NXj=1 t Xb2BWSbt( j) (A) s.t.( 2)-222()]TJ /F3 11.955 Tf 21.26 0 Td[(2 ))]TJ /F3 11.955 Tf 11.96 0 Td[(( 2)-222()]TJ /F3 11.955 Tf 21.25 0 Td[(13 )N)]TJ /F5 7.97 Tf 6.59 0 Td[(1NXj=11(0,1)(G(x, j)) ^qbtwbt( j)+Sbt( j)(8t;8b2BW;j=1,2,...,N) qbit ,^qbt,Sbt( j)0;obit,ubit,vbit2f0,1g,8t,8i,8b. Beforewegivethedetailedproofoftheproposition,weshowacorollarybasedonanassumptionandProposition2in[ 43 ]. Assumption1. Thereisanoptimalsolutionxofthetrueproblem ( 1 ) suchthatforany >0thereisx2X,whereXisthefeasibleregionfortheproblem,withkx)]TJ /F3 11.955 Tf 11.65 0 Td[(xk and q(x) PropositionA.1. SupposethatthesignicancelevelsofthetrueandSAAproblemsarethesame(i.e. ,=N ),thesetXiscompact,thefunctionf(x) anddecisionvariablesare 91

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continuous,G(x,)isaCaratheodoryfunction,andtheaboveassumptionholds,then^N!andD(^xN,x)!0w.p.1asN!1. InProposition2,theconvergencepropertyholdsforthecontinuouscase.Nowweshowtheconvergencepropertyholdsforthemixedintegercaseinthefollowingcorollary. Corollary1. Forthemix-integercase,supposetheobjectivefunctionis f(x,y) ,wherex2Xisthesetofbinaryvariables,andy2Yisthesetofcontinuousvariables.IfXisnite,andtheotherassumptionsintheabovepropositionhold,then,westillhave^N!andD(^xN,x)!0w.p.1asN!1. Proof. LetjXj=)]TJ /F1 11.955 Tf 20.59 0 Td[(.WedenotetheelementsofsetXinorder: x1,x2,...,x)]TJ ET BT /F1 11.955 Tf 396.95 -233.1 Td[(.Foreachxedxi,wecanapplyProposition2forcontinuousvariableyandgetacorrespondingconvergentsolutionbysolvingtheSAAproblem,i.e., minffN(xj,y)g!minff(xj,y)gg(xj),(A) wherefN(.,.)representstheobjectivefunctionfortheSAAproblemwhenthesamplesizeisN. Withoutlossofgenerality,weassume =min1i,j)]TJ /F2 11.955 Tf 8.14 7.53 Td[(kg(xi))]TJ /F6 11.955 Tf 11.96 0 Td[(g(xj)k>0.(A) Nowletxbetheintegerpartintheoptimalsolutionofthetrueproblem,and^xNbetheintegerpartintheoptimalsolutionoftheSAAproblemwhenthesamplesizeisN.Basedon( A ),thereexistsalargeconstantnumberN0suchthat kfN(^xN,^yN))]TJ /F6 11.955 Tf 11.95 0 Td[(g(^xN)k< 2(A) and kfN(x,^yN))]TJ /F6 11.955 Tf 11.95 0 Td[(g(x)k< 2,(A) 92

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whenN>N0.Meanwhile,weshouldhave fN(x,^yN)fN(^xN,^yN),(A) since(^xN,^yN)istheoptimalsolutionfortheSAAproblemwhenthesamplesizeisN. Ontheotherhand,itisobviousthat g(^xN)g(x),(A) basedonthedenitionofx.If^xN6=x,thenbasedon( A )and( A ), 0g(^xN))]TJ /F6 11.955 Tf 11.95 0 Td[(g(x)g(^xN))]TJ /F6 11.955 Tf 11.95 0 Td[(g(x)+fN(x,^yN))]TJ /F6 11.955 Tf 11.96 0 Td[(fN(^xN,^yN). Thus kg(^xN))]TJ /F6 11.955 Tf 11.95 0 Td[(g(x)kkg(^xN))]TJ /F6 11.955 Tf 11.95 0 Td[(g(x)+fN(x,^yN))]TJ /F6 11.955 Tf 11.96 0 Td[(fN(^xN,^yN)kkg(^xN))]TJ /F6 11.955 Tf 11.95 0 Td[(fN(^xN,^yN)k+kg(x))]TJ /F6 11.955 Tf 11.96 0 Td[(fN(x,^yN)k<, wherethethirdinequalityfollowsfrom( A )and( A ).Thiscontradictswith( A )andtheoriginalconclusionholds. Nowweproveourpropositionintwosteps. First ,weprovethatthesolutionof( A ) convergesto thatof( A ).Noticethat( A )isapuretwo-stage stochastic program,wheretherststage decision variablesarecontinuousordiscreteandtheexpectationfunctionintheobjectivefunctioniscontinuous.Basedontheconclusionin[ 5 ], theSAAofthisproblemconvergestothetruevalueofthetwo-stagestochasticprogram 93

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Second ,weprovethatthesolutionof( A )convergestothatof( A ).Itiseasytoseethatourmodelsatisesallconditionsintheabovecorollary.Then,accordingly,thesolutionof( A )convergestothatof( A ). Therefore,thesolutionofourSAAproblem( A )convergestothatofthetrueproblem( A ).Theconclusionholds. 94

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BIOGRAPHICALSKETCH QianfanWangreceivedhisB.S.incomputerscienceandB.S.ineconomicsfromthePekingUniversity,Beijing,Chinain2009.HereceivedhisPh.D.inindustrialandsystemsengineeringfromtheUniversityofFlorida,Gainesville,FLinspring,2013.HewasavisitingstudentattheArgonneNationalLaboratoryinfall,2010andspring,2011.HealsointernedwithSASInstitute,Cary,NCandAlstomGrid,Redmond,WAinsummer,2011andsummer,2012. 101