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Data-Driven Risk-Averse Stochastic Program and Renewable Energy Integration

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Title:
Data-Driven Risk-Averse Stochastic Program and Renewable Energy Integration
Creator:
Zhao, Chaoyue
Place of Publication:
[Gainesville, Fla.]
Florida
Publisher:
University of Florida
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english
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1 online resource (123 p.)

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Industrial and Systems Engineering
Committee Chair:
GUAN,YONGPEI
Committee Co-Chair:
GEUNES,JOSEPH PATRICK
Committee Members:
RICHARD,JEAN-PHILIPPE P
HAGER,WILLIAM WARD
Graduation Date:
8/9/2014

Subjects

Subjects / Keywords:
Demand curves ( jstor )
Distance functions ( jstor )
Electric generators ( jstor )
Electricity ( jstor )
Optimal solutions ( jstor )
Probability distributions ( jstor )
Robust optimization ( jstor )
Total costs ( jstor )
Unit costs ( jstor )
Wind power ( jstor )
Industrial and Systems Engineering -- Dissertations, Academic -- UF
data-driven -- power -- renewable -- robust -- stochastic
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bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Industrial and Systems Engineering thesis, Ph.D.

Notes

Abstract:
With increasing penetration of renewable energy into the power grid and its intermittent nature, it is crucial and challenging for system operators to provide reliable and cost effective daily electricity generation scheduling. In this dissertation, we present our recently developed innovative modeling and solution approaches to address this challenging problem. We start with developing several optimization-under-uncertainty models, including both stochastic and robust optimization ones, to solve reliability unit commitment problems for ISOs so as to ensure power system cost efficiency while ensuring high utilization of renewable energy. Then, we extend our research to study data-driven risk-averse two-stage stochastic program, for which the distribution of the random variable is within a given confidence set. By introducing a new class of probability metrics, we construct the confidence set based on historical data, and provide a framework to solve the problem for both discrete and continuous distribution cases. Our approach is guaranteed to obtain an optimal solution and in addition, we prove that our risk-averse stochastic program converges to the risk-neutral case as the size of historical data increases to infinity. Moreover, we show the "value of data" by analyzing the convergence rate of our solution approach. Finally, we illustrate examples of using this framework and discuss its application on renewable energy integration. ( en )
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (Ph.D.)--University of Florida, 2014.
Local:
Adviser: GUAN,YONGPEI.
Local:
Co-adviser: GEUNES,JOSEPH PATRICK.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2015-08-31
Statement of Responsibility:
by Chaoyue Zhao.

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UFRGP
Rights Management:
Applicable rights reserved.
Embargo Date:
8/31/2015
Resource Identifier:
969977049 ( OCLC )
Classification:
LD1780 2014 ( lcc )

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DATA-DRIVENRISK-AVERSESTOCHASTICPROGRAMANDRENEWABLEENERGYINTEGRATIONByCHAOYUEZHAOADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2014

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c2014ChaoyueZhao 2

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TomyparentsYongchunZhaoandYajingRen 3

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ACKNOWLEDGMENTS IwouldliketoexpressmydeepestgratitudetomyadvisorDr.YongpeiGuan,forhisconstantsupport,guidance,encouragementandfriendshipforthepastfouryears.Iammorethangratefultohavetheinvaluableopportunitytoworkwithhim,andtolearnfromhim.Hisknowledge,passion,wisdomandenthusiasmhelpmoldmeintotheresearcherIamtodayandinspiremetohopefullybecomeaprofessorlikehiminfuture.Withouthisguidanceandsupervision,thisdissertationworkwouldneverbepossible.IamverygratefultoDr.JosephGeunes,Dr.Jean-PhilippeRichardandDr.WilliamHagerforbeingonmyPhDdissertationcommittee,fortheirvaluablecommentsandsuggestionsonthisdissertation,andfortheirsinceresupport,suggestionsandguideonme.Ialsoappreciatetheirresponsivenessandexibilitytosupportmycompressedtimeline.IamalsogratefultothestaffatDepartmentofIndustrialandSystemsEngineering,Ms.CynthiaBlunt,Ms.SaraPons,fortheiradministrativesupports.Inaddition,Iwouldliketothankmycollaborators:JianhuiWangandJean-PaulWatson,fortheiradvicesandsuggestions.Ithasbeenagreatexperiencetoworkwiththem.ManythankstomyamazinggraduatestudentcolleaguesandfriendsatUniversityofFlorida,itisyourfriendshipthatmademygraduatestudyoneofthebestexperiencesofmylife.Finally,IdeeplyappreciatemyparentsYongchunZhao,YajingRen,andmyboyfriendYuanxiangWang,fortheirencouragement,supportandlove. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 8 LISTOFFIGURES ..................................... 9 ABSTRACT ......................................... 10 CHAPTER 1INTRODUCTION ................................... 11 2MULTI-STAGEROBUSTUNITCOMMITMENTCONSIDERINGWINDANDDEMANDRESPONSEUNCERTAINTIES ..................... 18 2.1ProblemDescriptionandLiteratureReview ................. 18 2.2Nomenclature .................................. 22 2.3MathematicalFormulation ........................... 24 2.3.1DeterministicModel .......................... 25 2.3.2LinearizingtheObjectiveFunction .................. 26 2.3.2.1Linearizingrbt(dbt) ...................... 26 2.3.2.2Linearizingfbi(xbit) ...................... 28 2.3.3UncertainWindPowerOutputFormulation .............. 28 2.3.4UncertainDemandResponseCurveFormulation .......... 29 2.3.5RobustOptimizationFormulation ................... 31 2.4SolutionMethodology ............................. 32 2.4.1ProblemReformulation ......................... 33 2.4.2Benders'Decomposition ........................ 34 2.4.2.1Feasibilitycuts ........................ 35 2.4.2.2Optimalitycuts ........................ 36 2.5CaseStudy ................................... 36 2.5.1DifferentDemandResponseScenarios ............... 37 2.5.2WindPowerOutputUncertainty .................... 38 2.5.3WindPowerOutputandDemandResponseUncertainties ..... 39 2.5.4118TWSystem ............................. 40 2.6Summary .................................... 41 3UNIFIEDSTOCHASTICANDROBUSTUNITCOMMITMENT ......... 42 3.1ProblemDescriptionandLiteratureReview ................. 42 3.2Nomenclature .................................. 44 3.3MathematicalFormulation ........................... 46 3.4DecompositionAlgorithmsandSolutionFramework ............ 49 3.4.1ScenarioGeneration .......................... 49 3.4.2LinearizingFi(.) ............................. 49 5

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3.4.3TheUncertaintySetoftheLoad .................... 49 3.4.4AbstractFormulation .......................... 50 3.4.5Benders'DecompositionAlgorithm .................. 51 3.4.6Benders'CutsfortheStochasticOptimizationPart ......... 52 3.4.6.1Feasibilitycuts ........................ 52 3.4.6.2Optimalitycuts ........................ 53 3.4.7Benders'CutsfortheRobustOptimizationPart ........... 53 3.4.7.1Feasibilitycuts ........................ 54 3.4.7.2Optimalitycuts ........................ 55 3.4.8SpecialCasesandDiscussions .................... 55 3.5ComputationalResults ............................. 57 3.5.1SensitivityAnalysis ........................... 58 3.5.1.1Effectofuncertaintyset ................... 58 3.5.1.2Sensitivityanalysisofobjectiveweight ......... 59 3.5.2ProposedApproachvsStochasticOptimizationApproach ..... 59 3.5.3ProposedApproachvsRobustOptimizationApproach ....... 62 3.6Summary .................................... 62 4DATA-DRIVENRISK-AVERSETWO-STAGESTOCHASTICPROGRAM .... 64 4.1ProblemDescriptionandLiteratureReview ................. 64 4.2-StructureProbabilityMetrics ........................ 67 4.2.1Denition ................................ 68 4.2.2RelationshipsamongMetrics ..................... 69 4.3SolutionMethodology ............................. 73 4.3.1DiscreteCase .............................. 73 4.3.2ContinuousCase ............................ 79 4.4NumericalExperiments ............................ 87 4.4.1NewsvendorProblem .......................... 87 4.4.2FacilityLocationProblem ....................... 90 4.5Summary .................................... 93 5DATA-DRIVENUNITCOMMITMENTPROBLEM ................. 94 5.1ProblemDescriptionandLiteratureReview ................. 94 5.2MathematicalFormulations .......................... 95 5.2.1StochasticUnitCommitmentProblem ................ 95 5.2.2Data-DrivenUnitCommitmentFormulation ............. 100 5.2.2.1Wassersteinmetric ..................... 100 5.2.2.2Referencedistribution .................... 101 5.2.2.3Condencesetconstruction ................ 102 5.3SolutionMethodology ............................. 104 5.3.1ExactSeparationApproach ...................... 107 5.3.2BilinearSeparationApproach ..................... 109 5.4ConvergenceAnalysis ............................. 110 5.5CaseStudy ................................... 113 6

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5.6Summary .................................... 115 6CONCLUSIONS ................................... 116 REFERENCES ....................................... 117 BIOGRAPHICALSKETCH ................................ 123 7

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LISTOFTABLES Table page 2-1Differentprice-elasticdemandcurvescenarios .................. 38 2-2Thecomparisonoftwosettings ........................... 39 2-3Theuncertaindemandresponsecase ....................... 40 2-4Thecomparisonoftwosystemswithmultiplewindsources ........... 40 3-1ResultsunderdifferentRatio%andBudget%settings .............. 58 3-2ComparisonbetweenSOandSRapproaches .................. 61 3-3ComparisonbetweenROandSRapproaches .................. 63 4-1Facilitiesthatarenotopen .............................. 92 5-1ComparisonbetweenSOandDD-SUCapproaches ............... 114 8

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LISTOFFIGURES Figure page 2-1Anexampleofprice-elasticdemandcurve ..................... 24 2-2Step-wisefunctionapproximationoftheprice-elasticdemandcurve ...... 27 2-3Theuncertaintyofprice-elasticdemandcurve .................. 30 2-4Windpoweroutputevolutionovertime ....................... 37 3-1Flowchartoftheproposedalgorithm ........................ 56 3-2Therelationshipbetweentheobjectivevalueandtheobjectiveweight ..... 59 4-1Relationshipsamongmembersof-structureprobabilitymetricsclass ..... 71 4-2Effectsofhistoricaldata ............................... 88 4-3Effectsofcondencelevel .............................. 89 4-4Effectsofhistoricaldata ............................... 91 4-5Effectsofsamples .................................. 92 9

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyDATA-DRIVENRISK-AVERSESTOCHASTICPROGRAMANDRENEWABLEENERGYINTEGRATIONByChaoyueZhaoAugust2014Chair:YongpeiGuanMajor:IndustrialandSystemsEngineeringWithincreasingpenetrationofrenewableenergyintothepowergridanditsintermittentnature,itiscrucialandchallengingforsystemoperatorstoprovidereliableandcosteffectivedailyelectricitygenerationscheduling.Inthisdissertation,wepresentourrecentlydevelopedinnovativemodelingandsolutionapproachestoaddressthischallengingproblem.Westartwithdevelopingseveraloptimization-under-uncertaintymodels,includingbothstochasticandrobustoptimizationones,tosolvereliabilityunitcommitmentproblemsforIndependentSystemOperators(ISOs)soastoensurepowersystemcostefciencywhilemaintainingahighutilizationofrenewableenergy.Then,weextendourresearchtostudydata-drivenrisk-aversetwo-stagestochasticprogram,forwhichthedistributionoftherandomvariableiswithinagivencondenceset.Byintroducinganewclassofprobabilitymetrics,weconstructthecondencesetbasedonhistoricaldata,andprovideaframeworktosolvetheproblemforbothdiscreteandcontinuousdistributioncases.Ourapproachisguaranteedtoobtainanoptimalsolutionandinaddition,weprovethatourrisk-aversestochasticprogramconvergestotherisk-neutralcaseasthesizeofhistoricaldataincreasestoinnity.Moreover,weshowthe“valueofdata”byanalyzingtheconvergencerateofoursolutionapproach.Finally,weillustrateexamplesofusingthisframeworkanddiscussitsapplicationonrenewableenergyintegration. 10

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CHAPTER1INTRODUCTIONTheUnitCommitment(UC)problemisoneofthemostimportantproblemsinpowersystems,whichevolvesdeterminingeffectiveon/offdailyschedulingofgenerationunits,tosatisfytheforecastedelectricityconsumptionoveragiventimehorizon,whileadheringtotheunits'physicalandtransmissionconstraintsatminimumoperationcost.The“unitcommitment”decisionistodeterminethestart-upandshut-downscheduleofgenerationunitsofeachhourovertheplanningtimehorizonwhilesatisfyingthestart-up/shut-downandon/offstatusconstraints.The“economicdispatch”decisionistheonethatcanmaketheelectricitysupplymeettheelectricityloadsimultaneously,withconsiderationofgenerationandtransmissionconstraints.Inthewholesaleelectricitymarkets,anIndependentSystemOperator(ISO)acceptsbidsfrombothsupplyandloadsides,andrunstheunitcommitmentproblemtoobtainthegenerationschedule,tokeepthepowersystembalanced.However,theincreasingintegrationofintermittentrenewableenergyintothepowersystembringsahighdegreeofuncertaintytoboththesupplysideandtheloadside,andconsequentlybringsprofoundchallengestotheISOinordertomaintainastableandreliablepowergrid.Ononehand,governmentpoliciesencouragethelargescalepenetrationofrenewableenergy,whichleadstopositiveimpactsongreenhousegasreduction,waterconservation,andenergysecurity.Forinstance,thesystemoperatorsinsomeregions,suchasGermany,considerrenewableenergyasahigherpriorityoverotherconventionalgenerationsources[ 23 ].Besides,theCaliforniaPublicUtilitiesCommissionenforcesatleast33percentagesoftheelectricityretailsalesfromtherenewableenergyby2020[ 39 ].Ontheotherhand,duetotheintermittentnatureofrenewableenergy,itis verydifcult toaccuratelypredictitsoutput.Theinaccurateforecastofrenewableenergyoutputintroducesconsiderableuncertaintiesintothesystemoperator'sdecision-makingprocess.Iftheuncertaintiescannotbe 11

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handledwell,transmissionviolations,loadblackoutsandevencascadingfailuresmayoccurinrealtime,whichcouldcausehigh-pricedremedyactionstocorrectpotentialsystemimbalance.Overtheyears,industryputssignicanteffortsondealingwithuncertaintiesinthepowersystemandhasdevelopedmanyeffectivewaystohandletheuncertainties.Onewaytopreventthesystemimbalancecausedbytheuncertainsystemenvironmentistoreservegenerationcapacity,e.g.,theoperatingreserves.Forexample,atMISO,theoperatingreservesincluderegulatingreserve,spinningreserve,andsupplementalreserve[ 15 ].Althoughoperatingreservesareregardedaseffectivewaystoaccommodateuncertainties,theyarenotsufcienttocoverlargeruncertainties,suchasrenewableenergyoutputuncertainties.Recently,two-stagestochasticandrobustoptimizationapproacheshavebeenstudiedextensivelytoaccommodateuncertainties.Forthetwo-stagestochasticoptimizationapproach,theday-aheadunitcommitmentdecisionismadeintherststagebeforetheuncertainproblemparameterrepresentingtherealtimeinformationisrealizedandtheeconomicdispatchamountismadeinthesecondstageaftertheuncertainparameterisrealized.Theobjectiveistominimizethetotalexpectedcostandtheuncertainproblemparameter(e.g.,windpower)iscapturedbyanumberofscenarios.Inrecentworks,signicantcontributionhasbeenmadebyusingthestochasticoptimizationmodelstosolvetheunitcommitmentproblemunderuncertainties,inparticular,underwindpoweroutputuncertainty.Forinstance,recentlyin[ 5 ],[ 64 ],and[ 63 ],astochasticunitcommitmentmodelisintroducedforshort-termoperationstointegratewindpowerintheLiberalisedElectricityMarkets.Thismodelhasbeensuccessfullyimplementedandusedinseveralwindpowerintegrationstudies.Inaddition,astochasticformulation,whichallowstheexplicitmodelingofthesourcesofuncertaintyintheunitcommitmentproblem,isproposedin[ 55 ],atwo-stagesecurity-constrainedunitcommitment(SCUC)algorithmthatconsiderstheunitcommitmentdecisionintherststageandtakesintoaccountthe 12

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intermittencyandvolatilityofwindpowergenerationinthesecondstageisintroducedin[ 70 ],andastochasticunitcommitmentmodel,consideringvariouswindpowerforecastsandtheirimpactsonunitcommitmentanddispatchdecisions,isproposedin[ 69 ].Signicantresearchprogresshasalsobeenmadetosolvesecurity-constrainedstochasticunitcommitmentmodels.Forinstance,security-constrainedstochasticunitcommitmentformulationsaddressingmarket-clearingaredescribedin[ 13 ]andthecorrespondingcasestudiesareperformedin[ 14 ].In[ 75 ],ascenario-treebasedstochasticsecurity-constrainedunitcommitmentisstudied.Mostrecently,two-stagestochasticprogrammingapproacheshavebeenstudiedtoconsiderbothslow-startandfast-startgeneratorsin[ 49 ],inwhichtheslow-startgeneratorsarecommittedintherststageandfast-startgeneratorsarecommittedinthesecondstage.Theseapproacheshavealsobeenstudiedtoestimatethecontributionofdemandexibilityinreplacingoperatingreservesin[ 48 ]andensurehighutilizationofwindpoweroutputbyaddingadditionalchanceconstraintsin[ 71 ].Inpractice,asignicantamountofavailabledataforISOs/RTOsmakesitpossibletotakesamplesandgeneratescenariosforthestochasticoptimizationapproach.However,itisalwayschallengingforthestochasticoptimizationapproachtodealwithlarge-sizedinstanceswhenthescenariosizeincreasessignicantly.Therefore,differentscenarioreductionapproacheshavebeenproposedtoselectimportantscenarios.Inthisway,thesmallsamplesizemayleadtothefeasibilityissues.Thatis,theday-aheadunitcommitmentdecisionmightnotbefeasibleforsomescenarioswhicharenotselected.Recently,robustoptimizationapproacheshavebeenproposedtoensuretherobustnessandmaketheday-aheadunitcommitmentfeasibleformostoutcomesoftherealtimeuncertaininputparameter.Fortherobustoptimizationapproach,theuncertainparameterisdescribedwithinagivendeterministicuncertaintysetandtheobjectiveistominimizetheworst-casecostthatincludestherst-stageunitcommitmentandthesecond-stageeconomicdispatchcosts.Recentresearchworksincludetwo-stage 13

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robustunitcommitmentmodelandBenders'decompositionalgorithmdevelopmentstoensuresystemrobustnessunderloaduncertaintiesintroducedin[ 32 ]and[ 7 ],two-stagerobustoptimizationmodelswithslightlydifferentuncertaintysetstoproviderobustday-aheadunitcommitmentdecisionsunderwindpoweroutputuncertaintiespresentedin[ 78 ]and[ 30 ],arobustbiddingstrategyinapool-basedmarketbysolvingarobustmixed-integerlinearprogramandgeneratingabiddingcurvedescribedin[ 5 ],arobustoptimizationmodeltointegratePHEVsintotheelectricgridtohandlethemostrelevantplanninguncertaintiesproposedin[ 28 ],andarobustoptimizationapproachtosolvecontingency-constrainedunitcommitmentwithN-ksecuritycriterionanddecidereserveamountstoensuresystemrobustnessintroducedin[ 59 ].Theadvantageoftherobustoptimizationapproachisthatitrequiresminimalinformationoftheinputuncertainparameter(aslongastheinformationissufcienttogeneratethedeterministicuncertaintyset)andensurestherobustnessoftheobtainedunitcommitmentdecision,i.e.,theday-aheadunitcommitmentdecisionisfeasibleformostoutcomesoftherealtimeuncertainproblemparameter.However,thisapproachalwaysfacesthechallengesonitsoverconservatism,duetoitsobjectivefunctionofminimizingtheworst-casecost,becausetheworstcasehappensrarely.Inthisdissertation,wepresentourrecentlydevelopedinnovativemodelingandsolutionapproachestoaddresstheunitcommitmentproblem.Westartwithdevelopingseveraloptimization-under-uncertaintymodels,includingbothstochasticandrobustoptimizationones,tosolvereliabilityunitcommitmentproblemsforISOssoastoensurepowersystemcostefciencywhilemaintainingahighutilizationofrenewableenergy.Inaddition,toaddresstheshortagesofstochasticoptimizationandrobustoptimizationapproaches,weproposeaninnovativeuniedstochasticandrobustunitcommitmentmodeltotakeadvantageofbothstochasticandrobustoptimizationapproaches.Then,weextendourresearchtostudythedata-drivenrisk-aversetwo-stagestochasticprogram,inwhichthedistributionoftherandomvariableiswithinagivencondence 14

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set.Finally,weapplytheproposeddata-drivenrisk-aversetwo-stagestochasticoptimizationframeworktopowersystemproblems.Inthisdissertation,weunfoldthediscussionasfollows.InChapter 2 ,wedescribeamulti-stagerobustunitcommitmentproblemconsideringwindanddemandresponse(DR)uncertainties.Inthischapter,DRprogramsareintroducedasreserveresourcetomitigatewindpoweroutputuncertainty.However,theprice-elasticdemandcurveisnotexactlyknowninadvance,whichprovidesanotherdimensionofuncertainty.ToaccommodatethecombineduncertaintiesfromwindpowerandDR,weallowthewindpoweroutputtovarywithinagivenintervalwiththeprice-elasticdemandcurvealsovaryinginthisdissertation.WedeveloparobustoptimizationapproachtoderiveanoptimalunitcommitmentdecisionforthereliabilityunitcommitmentrunsbyISOs/RTOs,withtheobjectiveofmaximizingtotalsocialwelfareunderthejointworst-casewindpoweroutputanddemandresponsescenario.Theproblemisformulatedasamulti-stagerobustmixed-integerprogrammingproblem.AnexactsolutionapproachleveragingBenders'decompositionisdevelopedtoobtaintheoptimalrobustunitcommitmentschedulefortheproblem.Additionalvariablesareintroducedtoparameterizetheconservatismofourmodelandavoidover-protection.Finally,wetesttheperformanceoftheproposedapproachusingacasestudybasedontheIEEE118-bussystem.Theresultsverifythatourproposedapproachcanaccommodatebothwindpoweranddemandresponseuncertainties,anddemandresponsecanhelpaccommodatewindpoweroutputuncertaintybyloweringtheunitloadcost.InChapter 3 ,weproposeanoveluniedstochasticandrobustunitcommitmentmodelthattakesadvantageofbothstochasticandrobustoptimizationapproaches,thatis,thisinnovativemodelcanachievealowexpectedtotalcostwhileensuringthesystemrobustness.Byintroducingweightsforthecomponentsforthestochasticandrobustpartsintheobjectivefunction,systemoperatorscanadjusttheweightsbased 15

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ontheirpreferences.Finally,aBenders'decompositionalgorithmisdevelopedtosolvethemodelefciently.Thecomputationalresultsindicatethatthisapproachprovidesamorerobustandcomputationallytrackableframeworkascomparedtothestochasticoptimizationapproachandamorecost-effectiveunitcommitmentdecisionascomparedtotherobustoptimizationapproach.Chapter 4 presentsadata-drivenrisk-aversetwo-stagestochasticoptimizationframework.Inmostpractice,theactualdistributionofarandomparameterisunknown.Instead,onlyaseriesofhistoricdataareavailable.Inthischapter,wedevelopadata-drivenstochasticoptimizationapproachtoprovidearisk-aversedecisionmakingunderuncertainty.Inourapproach,startingfromagivensetofhistoricaldata,werstconstructacondencesetfortheunknownprobabilitydistributionutilizingaclassof-structureprobabilitymetrics.Then,wedescribethereferencedistributionsandsolutionapproachestosolvethedevelopedtwo-stagerisk-aversestochasticprogram,correspondingtothegivensetofhistoricaldata,forthecasesinwhichthetrueprobabilitydistributionsarediscreteandcontinuous,respectively.Morespecically,forthecaseinwhichthetrueprobabilitydistributionisdiscrete,wereformulatetherisk-averseproblemtoatraditionaltwo-stagerobustoptimizationproblem.Forthecaseinwhichthetrueprobabilitydistributioniscontinuous,wedevelopasamplingapproachtoobtaintheupperandlowerboundsfortherisk-averseproblem,andprovethatthesetwoboundsconvergetotheoptimalobjectivevalueexponentiallyfastasthesamplesizeincreases.Furthermore,weprovethat,forbothcases,asmoredatasamplesareobserved,therisk-averseproblemconvergestotherisk-neutraloneexponentiallyfastaswell.Finally,theexperimentalresultsonnewsvendorandfacilitylocationproblemsshowhownumericallytheoptimalobjectivevalueoftherisk-aversestochasticprogramconvergestotherisk-neutralone,whichindicatesthevalueofdata.Chapter 5 utilizesthedata-drivenrisk-aversetwo-stagestochasticoptimizationtechniquestosolvetheunitcommitmentproblemunderrenewableenergyoutput 16

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uncertainty.Thetraditionalwaytodescribetherenewableenergyoutputuncertaintyistoassumetheoutputfollowsacertaindistribution,e.g.,NormaldistributionandWeibulldistribution.However,inpractice,theinformationabouttherenewableenergyoutputisusuallyincomplete,andtheinaccuratedistributionassumptionmayleadtobiasedUCandEDsolutions.Inthischapter,insteadofassumingthedistributionofrenewableenergyoutputisknown,aseriesofhistoricaldata,whicharedrawnfromthetruedistribution,areobserved.Weconstructthecondencesetoftherenewableenergyoutputdistribution,andproposearisk-averseapproachtoaddresstheunitcommitmentproblemunderrenewableenergyoutputuncertainty.Thatis,weconsidertheworstcasedistributioninthecondenceset.WeintroducetheWassersteindistributionmetricandproposethesolutionapproachestotacklethedata-drivenrisk-aversestochasticoptimizationframework.Weprovethatasthenumberofhistoricaldatagoestoinnity,therisk-aversesolutionconvergestotherisk-neutralsolution.OurcasestudyontheIEEE118-bussystemveriestheeffectivenessofourproposedsolutionapproach.Finally,Chapter 6 concludesthedissertationandprovidesgeneralsuggestionsforfutureresearch. 17

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CHAPTER2MULTI-STAGEROBUSTUNITCOMMITMENTCONSIDERINGWINDANDDEMANDRESPONSEUNCERTAINTIES 2.1ProblemDescriptionandLiteratureReviewInrecentyears,windenergy penetrationhasincreasedsubstantially andisexpectedtocontinuegrowinginthefuture .Forexample,the U.S.DepartmentofEnergydescribedascenario thatwindenergycouldgenerate20%of nation'selectricityby 2030[ 44 ].However,duetoitsintermittentnature,windpoweris inherently verydifcult topredict.Moreover,themagnitudeofwindpoweroutputvarianceismuchlargerthanthatofthetraditionalloadvariance. Asaresult, traditionalpowersystemoperationmethods areinsufcientto maintainsystemreliability. Duetophysicalconstraintsofthepowersystem(e.g.,rampinglimitsofconventionalgeneratorsandtransmissionlinecapacities),windpowercurtailmentoccursfrequently,whichconsequentlyleadstolowutilizationofwindpoweranddampenstheincentiveofwindpowerinvestmentinthelongrun.Therefore,thesystemoperatorsinsomeregions,suchasGermany,considerrenewableenergyahigherpriorityovertheotherconventionalgenerationsources[ 23 ]. Recentstudieshavefocusedondevelopingstochasticoptimizationmodelswiththeobjectiveofminimizingthetotalexpectedcost,includingashort-termstochasticrollingunitcommitmentmodel [ 5 , 63 ] ,astochasticunitcommitmentmodeltocalculatereserverequirementsbysimulatingthewindpowerrealization s andcomparingitsperformance withthetraditionalpre-denedreserverequirements[ 12 ],andastudyontheimpactsofwindpoweronthermalgenerationunitcommitment[ 65 ].Relatedresearchcanalsobefoundin[ 55 ],[ 70 ],and[ 69 ].All ofthis research indicates thatwindpoweroutputuncertaintyandwindpowerforecasterrorshaveasignicantimpactonunitcommitment c[2013]IEEE.REPRINTED,WITHPERMISSION,FROM[ 77 ] 18

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anddispatch,and moreadvancedpowersystemoperationmethodsare requiredtomakethesystemreliable. Morerecently, toensure high utilizationofwindpower,achanceconstrainedoptimizationmodel[ 71 ]andarobustoptimizationmodel[ 31 ]havebeendevelopedtosolvetheproblem.Intheformercase,achanceconstraintisdevelopedtoensurethataportionofthewindpoweroutput(e.g.,90%)beutilizedatacertainprobability.Inthisway,theriskofalargeamountofwind power beingcurtailed canbeadjusted bythe system operators.Inthelattercase,windpoweroutputisassumedto lie withinanintervaldenedbyquantiles . Allofthe windpoweroutputwithinthisintervalwillbeutilized.Unitcommitmentdecisionsare thenmadebyconsideringtheworst-case windpoweroutput scenario . Whilethesetwoapproachesensurehighutilizationofwindpoweroutput,bothapproachestendtocommitmoreconventionalgeneratorstoaccommodatethewindpoweroutputuncertainty. Asanalternative,apumpedstoragehydrounitisconsideredin[ 31 ].Pumpedstoragehydroisexibleandeasyto operate ,andcanreducethetotalcostundertheworst-casescenariosignicantly.However,duetolocationalandgeographicallimitations,pumpedstorage hydro units cannot bewidely adopted . Incomparison, demandresponse(DR)hasbeenshown tobean efcientapproachtoreducepeakload[ 60 ][ 35 ]. Italsohaspotentialtoaccommodatewindpoweroutputuncertainty .Forinstance,whenthewindpoweroutputishigherthanexpected,DRprogramscanhelpabsorbtheextrawindpower.Ontheother hand ,DRprogramscanhelpdecreasetheloadwhenthewindpoweroutputislow.Moreimportantly,thisapproachcanbewidelyapplied,ascomparedtopumpedstoragehydro.Ingeneral, DR aimstomanage end-useconsumers' electricityconsumption patterns viatime-varying prices ,or by offeringnancialincentivestoreducetheconsumptionofelectricityattimesofhighelectricitypricesorwhensystem reliability isjeopardized[ 43 ]. DR canbenetload-servingentities,consumers,andIndependent 19

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SystemOperators(ISOs)[ 60 ][ 35 ][ 43 ].Inparticular,forISOs(thefocusofthischapter),DRcanhelpbalanceelectricityconsumptionandgeneration,and can thereforeensureamorestable,reliableandcontrollable powergrid . TheU.S.DepartmentofEnergypredictedthatby2019,thetotalU.S.peakdemandcouldbereduced20%byDRwithfullparticipation[ 41 ].Inorderto“ensurethatdemandresponseistreatedcomparablytootherresources,”theFederalEnergyRegulatoryCommission(FERC)requiresthatISOsandRegionalTransmissionOrganizations(RTOs)“acceptbidsfromdemandresponseresourcesintheirmarketsforcertainancillaryservices,comparabletootherresources”[ 40 ].Severalregionalgridoperators(e.g.,NYISO,PJM,ISO-NE,andERCOT)haveprovidedopportunitiesforconsumerstoparticipateinDRprogramsinordertointegrateDRresourcesintothewholesaleenergymarketstepbystep. Inmost research ,theprice-elasticdemandcurveis characterized bypriceelasticity,whichrepresentsthesensitivityofelectricitydemand(Q) withrespect tothechangeofprice(P)[ 2 , 61 ].Forasmallchangeinprice(P),priceelasticityisdenedas =Q=Q P=P.(2)In[ 60 ],theelasticityvalueissimpliedas=Q=P, resultinginalinearizedprice-elasticdemandcurve .In[ 34 ],theprice-elasticdemandcurve isapproximated asastepwiselinearcurve.In[ 36 ],providedthatthechangeinpriceofonecommoditywillnotonlyaffectitsdemand,butalsomayaffectthedemandofanothercommodity,theconceptof“self-elasticity”and“cross-elasticity”isdeveloped.Thechapteralsoanalyzeshowtheseelasticitiescanmodel consumers' behaviorsandthesetofspotprice s .Intheaboveresearch,DRwasmostlymodeledasaxedprice-elasticdemandcurve.However,duetoavarietyofreasonsincludinglackofattention,latencyincommunication,andchangeinconsumptionbehavior,theactualprice-elasticdemandcurveisuncertaininnature [ 47 ] .Inotherwords,theactualresponsefrom 20

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theconsumersinrealtimecouldbedifferentfromtheforecastedvalues. Therefore ,theconsumerbehaviorshouldbemodeledbyanuncertainprice-elasticdemandcurve,whichmeansconsumershavedifferentresponsepatternstotheelectricitypricesunderdifferentscenarios.Inthiscase,theprice-elasticdemandcurvecan vary withinacertainrange.Hence,weproposeanefcientrobustunitcommitmentapproachthatcanconsiderwindpoweroutputuncertaintyandinexactDRinformationinthischapter.Weassumewind power outputiswithinagivenintervalandtheprice-elasticdemandcurveisalsovaryingwithina given range.Theobjectiveistomaximizethesocial welfare (denedinSection 2.3 ) undertheworst-case joint windpoweroutputandprice-elasticdemandcurvescenario.Ourrststagevariablesareunitcommitmentdecisions;thesecondstageconsiders economicdispatchforeachthermalgeneratoraftertheworst-casewindpoweroutputscenarioisrealized ;inthethirdstage,weconsidertheworst-caseprice-elasticdemandcurve. Byusingthisrobustoptimizationapproach, the reliabilityunitcommitmentrunprocess(e.g.,referredasreliabilityunitcommitmentatERCOTandreliabilityassessmentcommitmentatMidwestISO)ateachISOcanbestrengthened . Ascomparedtotherecentworksonrobustoptimizationtosolvepowersystemoptimizationproblems[ 4 , 28 , 31 , 59 ],thecontributionsofthischaptercanbesummarizedasfollows: 1. Bothwindpoweroutputanddemandresponseuncertaintiesareconsideredintheunitcommitmentproblem. 2. Amulti-stagerobustoptimizationmodelisdevelopedtoformulatetheproblem,ascomparedtopreviouslystudiedtwo-stagerobustoptimizationmodels. 3. Atractablesolutionapproachisproposedtosolvethemulti-stagerobustoptimizationproblemandthecomputationalresultsverifytheeffectivenessofourproposedapproach. The remainderofthe chapterisorganizedasfollows . InSection 2.3 ,wedescribehowtoformulatetheuncertaintysets describing theuncertain windpoweroutputandtheuncertainregionoftheprice-elasticdemandcurve. Wethen deriveamulti-stage 21

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robustoptimizationmodeltodecidetheoptimal robust unitcommitmentschedule.InSection 2.4 ,wetakeadvantageoftheproblemstructureandtransformthemulti-stagerobustoptimizationproblem intoatwo-stageproblem .Then,wedevelopaBenders'decompositionalgorithmtosolvetheproblem.InSection 2.5 ,weprovidecasestudiesand examineassociated computationalresults. Weconcludewithasummaryofourcontributionsanddiscussionsin Section 2.6 . 2.2Nomenclature A. SetsandParameters T Setoftimeperiods. B Setofbuses. K Setofstepsintroduced toapproximatetheprice-elasticdemandcurve . Gb Setofgeneratorsatbusb. Setoftransmissionlineslinkingtwobuses. SUbi Start-upcostforgeneratoriatbusb. SDbi Shut-downcostforgeneratoriatbusb. MUbi Minimumup timeforgeneratoriatbusb. MDbi Minimumdown timeforgeneratoriatbusb. Lbi Lowerboundofelectricitygeneratedbythermalgenerator i atbusb. Ubi Upperboundofelectricitygeneratedbythermalgenerator i atbusb. RUbi Ramp-up rate limitforgeneratoriatbusb. RDbi Ramp-down rate limitforgeneratoriatbusb. Cij Transmission capacityfor thetransimissionlinelinking busiandbusj. Kbij Lineowdistributionfactorforthetransimissionlinelinkingbusiandbusj,duetothenetinjectionatbusb, asdescribedin[ 73 ]. D0tb Theinelasticpartofdemand atbusbintimeperiodt . 22

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DMtb Themaximumdemand atbusbintimeperiodt . Wbt Theforecastedwindpoweroutput atbusbintimeperiodt . Wb+t Theupper deviation ofthecondenceintervalforthewindpoweroutput atbusbintimeperiodt . Wb)]TJ /F4 7.97 Tf -6.51 -7.58 Td[(t Thelower deviation ofthecondenceintervalforthewindpoweroutput atbusbintimeperiodt . $bt The upperboundforthetotaldeviationsoftherealprice-elasticdemandcurvefromtheforecastedcurveatbusbintimeperiodt . b The cardinalitybudgettorestrictthenumberoftimeperiodsinwhichthewindpoweroutputisfarawayfromitsforecastedvalueatbusb . `bkt Thekthsteplengthintheprice-elasticdemandcurveatbusbintimeperiodt. pbkt Thepriceatstepk intheprice-elasticdemandcurve atbusbintimeperiodt. bt Thegivenpriceelasticityatbusbintimeperiodt. B. DecisionVariables ybit Binaryvariabletoindicateifgeneratoriison atbusbintimeperiodt . ubit Binaryvariabletoindicateifgeneratoriisstartedup atbusbintimeperiodt . vbit Binaryvariabletoindicateifgeneratoriisshutdown atbusbintimeperiodt . wbt Windpoweroutput atbusbintimeperiodt . dbt Actual electricitydemand atbusbintimeperiodt . xbit Amountofelectricity produced bygeneratori atbusbintimeperiodt . hbkt The auxillary variableintroducedfordemandatstepk intheprice-elasticdemandcurve atbusbintimeperiodt. rbt(.) Theintegraloftheprice-elasticdemandcurveatbusbintimeperiodt. fbi(.) Thefuelcostfunctionofgeneratori atbusb . C. RandomParameters pbt Electricityprice atbusbintimeperiodt . 23

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2.3MathematicalFormulationInthissection,werstdescribethe deterministic modelforISO s /RTOsto determine unitcommitmentdecisionswiththeobjectiveofmaximizingtotalsocialwelfare. Inthismodel ,thewindpoweroutputwbtisassumeddeterministicandtheprice-elasticdemandcurveisalsocertain.Ingeneral,demandwilldecreasewhenelectricitypriceincreases.However,someelectricityconsumptionwillnotbeaffectedbyelectricityprices,suchascriticalloadslikehospitalsandairports.Wedenethispartofdemand as “inelasticdemand”.Accordingly,theotherpartofdemandvaryingwithelectricitypricesisreferredtoas“elasticdemand” [ 60 ] . . . Quantity(MW) . Price($/MW) . Demandcurve . Suppliersurplus . Consumersurplus . Supplycurve . D0tb . DMtb . inelastic . demand . elastic . demand . dbt . pbt Figure2-1. Anexampleofprice-elasticdemandcurve Inthischapter,weassumeloadateachbusincludesbothinelasticandelastic components . WecanmodelthedemandcurveandsupplycurveasshowninFig. 2-1 .Theelectricitysupplyanddemandreachanequilibriumattheintersectionpoint(dbt,pbt),correspondingtotheelectricitydemanddbtleadingtothemaximumsocialwelfare,whichisdenedasthesummationofconsumersurplusandsuppliersurplusasshowninFig. 2-1 .Sincetheinelasticdemandparthasaninnitemarginalvalue,weassumethattheconsumersurplusfortheinelasticdemandpartisaconstantasdenedin[ 60 ].Inaddition,forourprice-elasticdemandcurve,weassumetheelasticloadisanon-increasingfunctionoftheelectricityprice(cf.[ 60 ],[ 36 ],and[ 42 ]).LetD0tbbe 24

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theinelasticdemand atbusbintimeperiodt .Becausethedemandhastheinelasticcomponent,wehavedbtD0tb. WefurtherimposeanupperlimitDMtbondbt,yielding:D0tbdbtDMtb. Accordingly,thesocialwelfareisequalto thedifferencebetweentheintegralofthedemandcurvefromD0tbtodbt(denotedasrbt(dbt)inourmodel)plusaconstant(i.e.,theintegralofthedemandcurvefrom0toD0tb) andtheintegralofthesupplycurvefrom0todbt. Inourmodel,forthecalculationconvenience,weomittheconstantpartinourobjectivefunction,whichwillprovidethesameoptimalsolution. Finally,weletfbi(xbit)representthefuelcostforgeneratoriatbusbintimeperiodt.The deterministic modelcanbedescribedasfollows(denotedasD-UC): 2.3.1DeterministicModel maxXt2TXb2Brbt(dbt))]TJ /F10 11.955 Tf 10.51 11.36 Td[(Xt2TXb2BXi2Gb(fbi(xbit)+SUbiubit+SDbivbit) (2) s.t.)]TJ /F3 11.955 Tf 9.3 0 Td[(ybi(t)]TJ /F8 7.97 Tf 6.58 0 Td[(1)+ybit)]TJ /F3 11.955 Tf 11.96 0 Td[(ybik0, (2) 8k:1k)]TJ /F5 11.955 Tf 11.96 0 Td[((t)]TJ /F5 11.955 Tf 11.95 0 Td[(1)MUbi, 8t2T,8b2B,8i2Gb ybi(t)]TJ /F8 7.97 Tf 6.59 0 Td[(1))]TJ /F3 11.955 Tf 11.96 0 Td[(ybit+ybik1, (2) 8k:1k)]TJ /F5 11.955 Tf 11.96 0 Td[((t)]TJ /F5 11.955 Tf 11.95 0 Td[(1)MDbi,8t2T,8b2B,8i2Gb)]TJ /F3 11.955 Tf 9.3 0 Td[(ybi(t)]TJ /F8 7.97 Tf 6.58 0 Td[(1)+ybit)]TJ /F3 11.955 Tf 11.96 0 Td[(ubit0,8t2T,8b2B,8i2Gb (2) ybi(t)]TJ /F8 7.97 Tf 6.59 0 Td[(1))]TJ /F3 11.955 Tf 11.96 0 Td[(ybit)]TJ /F3 11.955 Tf 11.95 0 Td[(vbit0,8t2T,8b2B,8i2Gb (2) LbiybitxbitUbiybit,8t2T,8b2B,8i2Gb (2) xbit)]TJ /F3 11.955 Tf 11.95 0 Td[(xbi(t)]TJ /F8 7.97 Tf 6.59 0 Td[(1)ybi(t)]TJ /F8 7.97 Tf 6.59 0 Td[(1)RUbi+(1)]TJ /F3 11.955 Tf 11.96 0 Td[(ybi(t)]TJ /F8 7.97 Tf 6.59 0 Td[(1))MAbi, (2) 8t2T,8b2B,8i2Gbxbi(t)]TJ /F8 7.97 Tf 6.59 0 Td[(1))]TJ /F3 11.955 Tf 11.96 0 Td[(xbit ybitRDbi+(1)]TJ /F3 11.955 Tf 11.96 0 Td[(ybit)MAbi , (2) 8t2T,8b2B,8i2Gb 25

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Xb2B(Xi2Gbxbit+wbt)=Xb2Bdbt,8t2T (2) )]TJ /F3 11.955 Tf 9.3 0 Td[(CijXb2BKbij(Xq2Gbxbqt+wbt)]TJ /F3 11.955 Tf 11.96 0 Td[(dbt)Cij, (2) 8(i,j)2,8t2TD0tbdbtDMtb,8t2T,8b2B (2) ybit,ubit,vbit2f0,1g,xbit,dbt0,8t2T,8b2B,8i2Gb. (2) Intheaboveformulation,theobjectivefunctionin( 2 )istomaximizethesocialwelfare(withouttheconstantpart). Constraints( 2 )and( 2 )representtheminimumup-timeandtheminimumdown-timerestrictionsrespectively.Constraints( 2 )and( 2 )computetheunitstart-upandshut-downstatevariables. Constraints( 2 ) enforce theupperandlowerlimitsofpoweroutputofeachunit.Constraints( 2 )and( 2 ) enforce therampingratelimitsofeachunit.Constraints( 2 )ensureloadbalanceandrequirethesupplytomeetthedemand.Constraints( 2 )aretransmissionlinecapacitylimits.Finally,constraints( 2 ) enforce thelowerandupperlimitsforactualdemand,duetothecontributionofelasticdemand [ 61 ] .Inthefollowingsubsections,wedescribehowtolinearizetheobjectivefunction,generatetheuncertaintysetsforthewindpoweroutputandprice-elasticdemandcurve,andobtainanalrobustoptimizationformulation. 2.3.2LinearizingtheObjectiveFunctionTherearetwononlineartermsintheobjectivefunctionfor(D-UC):rbt(dbt)andfbi(xbit).Nowwedescribehowtolinearizethesetwononlinearterms. 2.3.2.1Linearizingrbt(dbt)Theelasticityvaluedeterminestheexibilityofthedemand.Wemodelthe consumer demandresponseofthe“elasticdemand”partasaprice-elasticdemandcurve.Ifpriceelasticityisconstant,wecanrepresenttheprice-elasticdemandcurveas: dbt=Abt(pbt)bt, (2) 26

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wherebtisthegivenpriceelasticityatbusbintimeperiodt,andAbtisaparameterthatcanbedecidedbyagivenreferencepoint(dbt,pbt)[ 61 ].Finally,noteherethatourproposedsolutionmethodcanbeappliedtoothermodeling approaches of demandelasticitywithoutlossofgenerality. Forinstance,forsomeISOs/RTOs(e.g.,MISO),theprice-elasticdemandcurveitselfisastep-wisefunction. AsshowninFig. 2-2 ,forageneralprice-elasticdemandcurverepresentedas( 2 ),astep-wisefunctioncanbeappliedtoapproximatethisprice-elasticdemandcurve.Wecanapproximaterbt(dbt)as: rbt(dbt)=Xk2Kpbkthbkt, (2) dbt=Xk2Khbkt, (2) 0hbktlbkt,8k2K (2) wherelbktisthekthsteplengthofthestep-wisefunction,pbktisthecorrespondingpriceatstepk, hktistheauxiliaryvariableintroducedfordemandatstepk,andKisthesetofsteps. . . Demand(MW) . Price($/MW) . dbt . D0tb . DMtb . pb1t . pb2t . pbKt . lb1t . lb2t . lbKt . dbt . pbt Figure2-2. Step-wisefunctionapproximationoftheprice-elasticdemandcurve Noticethatpbktisstrictlydecreasingwithk.Sincewearemaximizingrbt(dbt), accordingtoequations( 2 )and( 2 ) ,wehave: 27

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hbst=8>>>>>><>>>>>>:lbst,ifss0,whenPs0)]TJ /F8 7.97 Tf 6.59 0 Td[(1k=1lbkt
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forecastedvalueatbusb.Thebudgetbcanbeadjustedbythesystemoperators.Forexample,ifbissettobe0,thewindoutputuctuationateachbusisassumedtobesmallandcanbeapproximatedbyitsforecastedvalue.Ifb=6,signicantuctuationsofwindpoweroutputareassumedtooccurinnomorethansixtimeintervals.Itcanbeobservedthatthis“budgetparameter”bcanbeusedtoadjusttheconservatismofthesystem.Forinstance,ourproposedapproachismoreconservatismasbincreases.Meanwhile,itisprovedin[ 8 ]that,foranygivenbudgetblessthan24,therobustoptimalunitcommitmentsolutionobtainedbasedonthisuncertaintysetisstillfeasibleforanypossiblewindpoweroutputbetweenitsgivenlowerandupperboundswithahighprobability(e.g.,when8,therobustoptimalunitcommitmentsolutionwillbefeasiblewithaprobabilityhigherthan95%asshownin[ 31 ]).Underthissetting,ateachbusb,theworst-casewindpoweroutputscenariohappenswhenwindpoweroutputreachesitslowerbound,upperboundorforecastedvalue,andthetotalnumberofperiodsinwhichwindpoweroutputisnotatitsforecastedvalueshouldbenomorethanthebudgetb. Accordingly, theuncertaintysetcanbedescribedasfollows: W:=(w2RjBjjTj:wbt=Wbt+zb+tWb+t)]TJ /F3 11.955 Tf 11.96 0 Td[(zb)]TJ /F4 7.97 Tf -5.76 -7.89 Td[(tWb)]TJ /F4 7.97 Tf -6.51 -7.89 Td[(t,Xt2T(zb+t+zb)]TJ /F4 7.97 Tf -5.76 -7.89 Td[(t)b,8t2T,8b2B),wherezb+tandzb)]TJ /F4 7.97 Tf -5.76 -7.58 Td[(tarebinaryvariables.Ifzb+t=1,thewindpoweroutputwillreachitsupperlimit,andifzb)]TJ /F4 7.97 Tf -5.76 -7.59 Td[(t=1,thewindpoweroutputwillreachitslowerbound,andifbothofthemare0,theforecastedvalueisachieved. 2.3.4UncertainDemandResponseCurveFormulationInSubsection 2.3.2.1 ,weprovide a descriptionoftheprice-elasticdemandcurveandhowtouselinearfunctionstoapproximateit.Inpractice,however,asdescribedinSection 2.1 ,the actual price-elasticdemandcurveis uncertain .WhenISOs/RTOsmakeunitcommitmentdecisions, itisnecessarytoallow theprice-elasticdemandcurve 29

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tovarywithinacertainrange. As showninFig. 2-3 ,foragivencertainpricep0,thecorrespondingdemandisuncertain (e.g.,therangeford0asindicatedinthegure) .Similarly,foragivendemandd0,the corresponding pricecanvarywithinacertainrange (e.g.,therangeforp0asindicatedinthegure) .Accordingly,wecanformulatethe . . Demand(MW) . Price($/MW) . Differentscenariosof . price-elasticdemand . curve . Predictedcurve . d0 . p0 . Therange . forp0 . Therange . ford0 Figure2-3. Theuncertaintyofprice-elasticdemandcurve price-elasticdemandcurveasdbt=Abt(pbt)bt+btordbt=Abt(pbt+bt)bt,wherebt representsa deviation used todescribetheuncertaintyoftheprice-elasticdemandcurve.Inourmodel,weassumedbt=Abt(pbt+bt)btforcomputationalconvenience.Followingthepreviousstepsonapproximatingtheprice-elasticdemandcurveasastep-wisecurve,foreachdbtintheprice-elasticdemandcurve,thecorrespondingpbktisallowedto varywithin therangepbkt2[pbkt)]TJ /F5 11.955 Tf 12.29 0 Td[(^bkt,pbkt+^bkt],wherepbkt represents the estimated valueofpbkt,bktisthedeviationofpbktand^bktistheupperlimitofbkt. Toadjustthe conservativeness ,weintroducetheparameter$bttorestrictthetotalamountofdeviations,i.e.,Pk2Kbkt.Wecanadjusttheconservativenessofourproposedapproachthroughchangingthevalueof$bt.Thesmallerthevalue$btis,thelessuncertaintythedemandresponsecurvehas.Theuncertaintysetofthedemand 30

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responsecurvecanbedescribedasfollows: =n:)]TJ /F5 11.955 Tf 10.67 0 Td[(^tbkbkt^tbk, (2) )]TJ /F6 11.955 Tf 9.29 0 Td[($btXk2Kbkt$bt, (2) 8t2T,8b2B,8k2Ko. (2) 2.3.5RobustOptimizationFormulationWiththeconsiderationofbothwindanddemandresponseuncertainties,wedevelopathree-stagerobustoptimizationformulationto determinerobustday-aheadreliabilityunitcommitmentdecisionsforISOs/RTOs .Intherststage , we include the unitcommitment decisions for each generation unit whileconsidering allunitcommitmentconstraintswithunknownwindpoweroutputanddemandresponsepatterns.Afterrealizingtheworst-casewindpoweroutput, wedecideinthesecondstagethedispatchlevelforeachunit tomaximize thetotal socialwelfarewiththeconsiderationoftheworst-caseprice-elasticdemandcurve.Finally, in thethirdstage, weconsider theuncertaintyoftheprice-elasticdemandcurvethatminimizestheexpectedtotalsocialwelfare.Thederivedrobustoptimizationformulation is shownasfollows: maxy,u,vn)]TJ /F10 11.955 Tf 11.95 11.36 Td[(Xt2TXb2BXi2Gb(SUbiubit+SDbivbit)+minw2Wmaxx,h2(min2Xt2TXb2BXk2K(pbkt+bkt)hbkt)]TJ /F10 11.955 Tf 11.29 11.35 Td[(Xt2TXb2BXi2Gbbit)o (2) s.t.Constraints( 2 ))]TJ /F1 11.955 Tf 11.95 0 Td[(( 2 ),ybit,ubit,vbit2f0,1g,8t2T,8b2B,8i2Gb (2) whereX=n Constraints( 2 ))]TJ /F1 11.955 Tf 11.96 0 Td[(( 2 ),( 2 ), 31

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Xb2B Xi2Gbxbit+wbt!=Xb2BXk2Khbkt,8t2T (2) hbktlbkt,8t2T,8b2B,8k2K (2) )]TJ /F3 11.955 Tf 9.3 0 Td[(CijXb2BKbij Xq2Gbxbqt+wbt)]TJ /F10 11.955 Tf 12.01 11.36 Td[(Xk2Khbkt!Cij,8(i,j)2,8t2T (2) D0tbXk2KhbktDMtb,8t2T,8b2B (2) xbit0,hbkt0, (2) 8t2T,8b2B,8i2Gb,8k2Ko. Noteherethatintheaboveformulation,constraints( 2 )arethereformulationsofconstraints( 2 ),constraints( 2 )arederivedfromconstraints( 2 ),constraints( 2 )arederivedfromconstraints( 2 ),andconstraints( 2 )arethereformulationsofconstraints( 2 ). 2.4SolutionMethodologyFornotationbrevity,weusematricesandvectorstorepresenttheconstraintsandvariables.Theaboveformulationcanberepresentedasfollows: maxy,u,v(^aTu+^bTv)+minw2Wmaxx,h,2(min2QTh+^cTh)]TJ /F5 11.955 Tf 12 0 Td[(^eT) (2) s.t.^Ay+^Bu+^Cv0, (2) y,u,v2f0,1g, (2) where =n^Dx^Fy+^g, (2) ^Px)]TJ /F5 11.955 Tf 12.18 2.66 Td[(^J^Ky, (2) ^Rh^q, (2) ^Tx+^Sh^s+^Ow, (2) x,h0o, (2) 32

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=n^U^mo. (2) Constraint( 2 )representsconstraints( 2 )-( 2 );constraint( 2 )representsconstraints( 2 )-( 2 );constraint( 2 )representsconstraints( 2 );constraint( 2 )representsconstraints( 2 )and( 2 );constraint( 2 )representsconstraints( 2 )and( 2 ); constraint( 2 )representsconstraints( 2 );constraint( 2 )representsconstraints( 2 )and( 2 ). 2.4.1ProblemReformulationTosolvetheaboveformulation,werstdualizetheconstraintsin( 2 )andcombineitwiththesecondstagedecisionvariablesandconstraints.Weobtainthefollowing two-stageformulation : maxy,u,v(^aTu+^bTv)+minw2Wmaxx,h,,2(^mT+^cTh)]TJ /F5 11.955 Tf 12 0 Td[(^eT) (2) s.t.^Ay+^Bu+^Cv0, (2) y,u,v2f0,1g, (2) where =\n^UT)]TJ /F3 11.955 Tf 11.96 0 Td[(h0, (2) 0o. (2) Duetothespecialproblemstructure(forinstance,thethirdstageuncertaintyisonlyinvolvedintheobjectivefunction),takingthedualformulationdoesnotgeneratenonlineartermsfortheresultingtwo-stagerobustoptimizationproblem.Then,wecandualizetheremainingconstraintsin, and transformthesecond-stageproblemasfollows: !(y)=minw2W,,,,,(^Fy+^g)T+(^Ky)T+^qT+(^s+^Ow)T (2) 33

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s.t.^DT+^PT+^TT0, (2) ^RT+^ST)]TJ /F6 11.955 Tf 11.96 0 Td[(^c, (2) ^JT=^e, (2) ^U^m, (2) ,,,,0, (2) where,,,,aredualvariablesforconstraints( 2 ),( 2 ),( 2 ),( 2 ),and( 2 )respectively.Intheaboveformulation,wehavethebilinearterm:wT^OT.Let^OT=,andbyusingtheuncertaintysetW,wehave wT=Xt2TXb2Bbtwbt (2) =Xt2TXb2Bbt(Wbt+zb+tWb+t)]TJ /F3 11.955 Tf 11.95 0 Td[(zb)]TJ /F4 7.97 Tf -5.76 -7.9 Td[(tWb)]TJ /F4 7.97 Tf -6.51 -7.9 Td[(t) (2) =Xt2TXb2B(btWbt+b+tWb+t+b)]TJ /F4 7.97 Tf -5 -7.9 Td[(tWb)]TJ /F4 7.97 Tf -6.51 -7.9 Td[(t) (2) s.t.bt=(^OT)bt,8t2T,8b2B (2) b+t)]TJ /F3 11.955 Tf 21.92 0 Td[(Mzb+t,8t2T,8b2B (2) b+tbt)]TJ /F3 11.955 Tf 11.95 0 Td[(M(1)]TJ /F3 11.955 Tf 11.96 0 Td[(zb+t),8t2T,8b2B (2) b)]TJ /F4 7.97 Tf -5 -7.89 Td[(t)]TJ /F3 11.955 Tf 21.92 0 Td[(Mzb)]TJ /F4 7.97 Tf -5.76 -7.89 Td[(t,8t2T,8b2B (2) b)]TJ /F4 7.97 Tf -5 -7.89 Td[(t)]TJ /F6 11.955 Tf 21.92 0 Td[(bt)]TJ /F3 11.955 Tf 11.95 0 Td[(M(1)]TJ /F3 11.955 Tf 11.96 0 Td[(zb)]TJ /F4 7.97 Tf -5.77 -7.89 Td[(t),8t2T,8b2B (2) Xt2T(zb+t+zb)]TJ /F4 7.97 Tf -5.76 -7.89 Td[(t)b,8b2B. (2) NowwecanreplacethebilineartermwT^OTby( 2 )andaddconstraints( 2 )to( 2 )toremovethebilinearterm. 2.4.2Benders'DecompositionWecanusetheBenders'decompositionalgorithmtosolvetheabovethree-stagerobustoptimizationproblem.Wedenote#asthesecond-stageoptimalobjectivevalue 34

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andthenconsiderthefollowingmasterproblem.Byaddingfeasibilityandoptimalitycuts,wecansolvethisproblemiteratively: maxy,u,v^aTu+^bTv+#s.t.Constraints( 2 )and( 2 ),Feasibilitycuts,Optimalitycuts. 2.4.2.1FeasibilitycutsWeuse the L-shapedmethodtogeneratefeasibilitycuts.Inthiscase,whenwecheckthefeasibilityin,wedonotneedtoconsiderconstraints( 2 ),( 2 )and( 2 )sincetheywillnotaffectthefeasibility.Thefeasibilitycheckproblemisshownasfollows: maxx,h,1,2)]TJ /F8 7.97 Tf 17.71 14.95 Td[(3Xi=1^eTi1)]TJ /F8 7.97 Tf 18.37 14.95 Td[(3Xi=1^eTi2 (2) s.t.^Dx+11)]TJ /F6 11.955 Tf 11.95 0 Td[(12^Fy+^g, (2) ^Rh+21)]TJ /F6 11.955 Tf 11.95 0 Td[(22^q, (2) ^Tx+^Sh+31)]TJ /F6 11.955 Tf 11.95 0 Td[(32^s+^Ow, (2) 10,20, (2) where^erepresentsthevectorwithallcomponents1.NowwetakethedualoftheaboveformulationandreplacethenonlineartermwT^OTbyusingthesamescheme: !L(y)=minw2W,^,^,^,^(^Fy+^g)T^+^qT^+^sT^+(W)T^+(W+)T^++(W)]TJ /F5 11.955 Tf 7.08 -4.94 Td[()T^)]TJ ET BT /F1 11.955 Tf 433.45 -553.86 Td[((2) s.t.^DT^+^TT^0, (2) ^RT^+^ST^0, (2) Constraints( 2 )to( 2 ), 35

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^,^,^,^,^+,^)]TJ /F2 11.955 Tf 10.41 -4.94 Td[(2[0,1], (2) where^,^,and^aredualvariablesforconstraints( 2 ),( 2 ),and( 2 ).Decisionvariables^,^+,^)]TJ /F1 11.955 Tf 10.41 -4.34 Td[(aredenedas( 2 )to( 2 ).Thenwecanperformthefollowingstepstocheckfeasibility: (1) If!L(y)=0,yisfeasible; (2) If!L(y)<0,generatea corresponding feasibilitycut!L(y)0. 2.4.2.2OptimalitycutsAssumingintheithiteration,wesolvethemasterproblemandobtain#iandyi.Since#isthesecond-stageoptimalobjectivevalue,ifwesubstituteyiintothesubproblemandget!(yi),weshouldhave!(yi)#i.If!(yi)<#i,wecanclaimthatyiisnotanoptimalsolutionandwecangenerateacorrespondingoptimalitycutasfollows: !(y)#. 2.5CaseStudyWeevaluatetheperformanceofourproposedapproachbytestingarevisedIEEE118-bussystemavailableonlineat http: //motor.ece.iit.edu/data.Thesystemcontains118buses,33generators,and186transmissionlines. ApartfromtheoriginalIEEE118-bussystem,wecreatethe118SWsystembyaddingasinglewindfarmatbus10andcreatethe118TWbyaddingthreewindfarmsatthreedifferentbuses. The operationaltime horizonis24hours, andeachtimeperiodissettobeonehour.Weuseave-piecepiecewiselinearfunctiontoapproximatethegenerationcostfunction[ 31 ]anduseten-piecepiece-wisesegments[ 72 ]toapproximatetheprice-elasticdemandcurve.Thereferencepointforthedemandresponsecurveis(80,25)[ 61 ]. ThepatternofthewindpoweroutputincludingitsupperandlowerboundsisillustratedinFig. 2-4 ,basedonthestatisticsfromNationalRenewableEnergyLaboratory(NREL). 36

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Figure2-4. Windpoweroutputevolutionovertime Finally,alltheexperimentsareimplementedusingCPLEX12.1,atIntelQuadCore2.40GHzwith8GBmemory. 2.5.1DifferentDemandResponseScenariosWerstconsidertheeffectsofdifferentdemandresponsescenarios.Toshowtheeffects,weassumethedemandresponsecurveiscertainandelasticitybtisthesameforeachbandt(denoteditas).Wesettheinelasticdemandequalto80%oftheforecasteddemandandthedemand'supperlimitequalto120%oftheforecasteddemand.Inthiscase,wetestthreedifferentelasticityvalues[ 72 ],e.g., =)]TJ /F5 11.955 Tf 9.3 0 Td[(0.5,=)]TJ /F5 11.955 Tf 9.29 0 Td[(1,and=)]TJ /F5 11.955 Tf 9.3 0 Td[(2 .Wetestthe118SWsystemwithfourdifferentcardinalitybudgets,e.g., =2,=4,=6,and=8 .Tocomparetheirperformance,foreachsystem,wecomputetheunitloadcost(ULC),whichisequaltothetotalcost(i.e.,unitcommitmentcostplusfuelcost)dividedbythetotaldemand.Weintroducealinearpenaltycostfunctionfortheunsatiseddemandortransmissioncapacity/rampratelimitviolations,andtheunitpenaltycostissettobe7947/MWh[ 31 ].WereporttheresultsofULC,thesocialwelfare,andtheCPUtimeinTable 2-1 . FromTable 2-1 wecanobservethatULChasatendencytodecreaseastheelasticityvalueincreases,andthesocialwelfarehasatendencytoincreaseastheelasticityvalueincreases.Thisindicatesthat,forthiscasestudy,introducingdemandresponsecanhelpreducetheunitloadcostandincreasethesocialwelfare . 37

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Table2-1. Differentprice-elasticdemandcurvescenarios ULCSocialWelfareTime(s) 2-0.512.2051.169e+07601-112.2397.032e+06579-212.2703.276e+065984-0.512.3391.167e+07729-112.3687.018e+06713-212.4093.261e+067066-0.512.4161.166e+07991-112.4457.010e+061005-212.4853.253e+069858-0.512.5351.165e+071373-112.5536.998e+061318-212.6033.239e+061296 2.5.2WindPowerOutputUncertaintyInthissubsection,wetestthe118SWsystemundervariouswindcardinalitybudgets(e.g.,differentvalues).Wetestthesystemundertwodifferentsettings.Undertherstsetting(denotedasCD),wederiverobustunitcommitmentdecisionswiththeconsiderationofwindpoweroutputuncertainty,butwithoutdemandresponse.Wesetthedemandtobetheforecasteddemand.Underthesecondsetting(denotedasDR),wederiverobustunitcommitmentdecisionswiththeconsiderationofwindpoweroutputuncertaintyanddeterministicdemandresponse.Thatis,toshowtheeffectivenessofthedemandresponse,weconsiderthecaseinwhichonlywindpoweroutputuncertaintyisconsideredandthedemandresponsecurveisassumedtobecertain. Forthedemandresponsecurve,wesettheelasticityvalue=)]TJ /F5 11.955 Tf 9.3 0 Td[(1 .WereporttheresultsofULC,theCPUtime,and thetotalnumberofBenders'cuts(alsoreferredtothenumberofiterations) inTable 2-2 .FromtheresultsreportedinTable 2-2 wecanobservethat,undertheworst-casewindpoweroutputscenario,thecasewithdemandresponsehaslessunitloadcostthanthecasewithoutdemandresponsegiventhesamewindcardinalitybudget. 38

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Table2-2. Thecomparisonoftwosettings SystemULCTime(s)Cuts 2CD12.51832011DR12.239579104CD12.62441315DR12.368713116CD12.72289314DR12.4551005138CD12.814112614DR12.553131814 2.5.3WindPowerOutputandDemandResponseUncertaintiesInthissubsection,weanalyzethecase,denotedasUDR,consideringbothwindpoweroutputanddemandresponseuncertainties.BesidestheuncertaintysettingsforthewindpoweroutputdescribedinSubsection 2.5.2 ,wetesttwodifferentdeviationvaluesfortheprice-elasticdemandcurve,e.g.,bkt=5and10,8t2T,8b2B,8k2K.Inaddition,weset$bt=0.5Pk2Kbkt,8t2T,8b2B. Wealsocomparethesesettingswiththecaseinwhichbkt=0,8t2T,8b2B,8k2K,i.e.,thedeterministicDRcase. TheresultsofULC,thesocialwelfare(denotedasS.W.), andtheconsumersurplusandthesuppliersurplus asdenedinSection 2.3 areshowninTable 2-3 . FromTable 2-3 ,wecanobservethefollowing: 1. Given adeviation butdifferentwindcardinalitybudgets,undertheworst-casescenariowhenweincreasethevalueof,ULCincreasesandthesocialwelfare,bothconsumersurplusandsuppliersurplus,decreases.Thisisduetotheneedtoprovidemoregenerationtoguaranteethatthesupplymeetsthedemand. 2. Foraxedwindcardinalitybudgetbutwithdifferentdeviations,thesocialwelfaredecreaseswhenthesystemhasmoredemandresponseuncertainty.Thisisbecauseweconsiderthesocialwelfareundertheworst-casescenario.Whenwehavemoredemandresponseuncertainty,thealgorithmgivesusamoreconservativesolution.WealsoobservethatULCandthesuppliersurplusdecreaseswhenthedeviationvalueincreases.Buttheconsumersurplusdoesn'tchangetoomuchasthedeviationvalueincreases.Thereasonforthisisthatwhenthedemandresponsecurvehasmoreuncertainty,thecurveshiftstoachievea 39

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Table2-3. Theuncertaindemandresponsecase ULCS.W.(e+6)Consumer(e+6)Supplier(e+6) 2012.2397.032464.099432.93303512.2056.518344.106542.41181012.0026.013484.113731.899754012.3687.017734.086262.93147512.3396.503674.092722.410951012.1425.998974.100041.898936012.4457.009674.078262.93141512.4166.494784.084752.410031012.2215.991134.092291.898848012.5536.997544.066652.93089512.5356.48244.07252.40991012.3495.978544.079751.89879 smallerdemandequilibriumpoint,whichcorrespondstosmallertotalsystemcostandULC. 2.5.4118TWSystem Wereporttheexperimentresultsforthe118TWsysteminthissubsectionandanalyzetheimpactofthedemandresponseondistributedwindresources.Intotal,therearethreewindfarmsatthreedifferentbuses:buses10,26,and32.TheoutputofeachwindfarmfollowsthedistributiongivenbyFig. 2-4 .WetestsystemCDandsystemDRasdenedinSectionIV-Bunderdifferentwindcardinalitybudget,andreportULC,theCPUtime,andthetotalnumberofBenders'cutsinTable 2-4 : Table2-4. Thecomparisonoftwosystemswithmultiplewindsources SystemULCTime(s)Cuts 1CD12.990699312DR12.8021127292CD13.0588153314DR12.8335208694CD13.1336287513DR12.904534087 40

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FromtheresultsinTable 2-4 ,wecanobservethat,undertheworst-casewindpoweroutputscenario,withmultiplewindresources,thecasewithdemandresponsestillhassmallerunitloadcostthanthecasewithoutdemandresponsegiventhesamewindcardinalitybudget.However,theCPUtimeincreasesdramaticallyasthenumberofwindfarmsincreases. 2.6SummaryInthischapter,wedeveloped arobustoptimizationapproachforunitcommitmenttomaximizesocialwelfareunderworst-casewindpoweroutputanddemandresponsescenarios .Anuncertainprice-elasticdemandcurveisusedtomodelconsumers'responsetopricesignals.Benders'decompositionisappliedtosolvetheproblem. OurnalcomputationalresultsonanIEEE118-bussystemverifythatourproposedapproachcanaccommodatebothwindpoweranddemandresponseuncertainties,anddemandresponsecanhelpaccommodatewindpoweroutputuncertaintybyloweringtheunitloadcost. Noteherethatinthischapter,weonlyconsiderthewindpoweroutputanddemandresponseuncertainties.However,ourapproachcanbeappliedtoothersystemuncertainties.Forinstance,inourmodelsetting,weassumetheinelasticdemandiscertain.Inpractice,theinelasticdemandcanbeuncertain.Forthiscase,wecanregardthispartofinelasticdemandasnegativesupplyandcombineitwithwindpoweroutputuncertaintytoconstructouruncertaintyset.WecanalsoseparateitfromwindpoweroutputuncertaintyandbuildaseparateuncertaintysetaswedidforthewindpoweroutputuncertaintycaseinSection 2.3 .Thesamedecompositionapproachdescribedinthischaptercanbeappliedtosolvetheproblem.Meanwhile,forthecurrentelectricitymarkets,ancillaryservice(regulation-up,regulation-down,spinandnon-spin)isregardedasamorecommonwaytoaccommodatewinduncertainty,ascomparedtodemandresponse. 41

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CHAPTER3UNIFIEDSTOCHASTICANDROBUSTUNITCOMMITMENT 3.1ProblemDescriptionandLiteratureReviewAsmentionedinSection 1 ,therearebroadlytwobranchesoftwo-stageoptimizationapproachestodealwiththeunitcommitmentproblemunderuncertainties,namelytwo-stagestochasticoptimizationapproachandtwo-stagerobustoptimizationapproach.Forthetwo-stagestochasticoptimizationapproach,theuncertainparameterisgenerallycapturedbyaseriesofscenarios,andtheobjectiveistominimizetheexpectedtotalcostormaximizetheexpectedtotalsocialwelfareoverallthescenarios.Recently,signicantworkhasbeendonebyusingthestochasticoptimizationmodelstosolvetheunitcommitmentproblemunderuncertainties,frombothloadsideandsupplyside[ 70 – 72 ].Althoughthetwo-stagestochasticoptimizationapproachisaneffectiveapproachtosolvetheUCproblem,ithassomepracticallimitations.First,sincetheuncertainparameterisoftenassumedtofollowaprobabilitydistribution,whichisestimatedbythehistoricalrealizationsoftheuncertainparameter,theaccuracyofprobabilitydistributionestimationiscriticaltodeterminetheoptimalUCdecisions.However,inpractice,itisusuallydifculttoestimatethedistributionoftheuncertainparameterprecisely.Second,toguaranteeareliabilityunitcommitmentrun,alargenumberofscenariosshouldbegenerated,whichalwaysbringschallengesforthestochasticoptimizationapproachtodealwithlarge-scaledinstances.However,ifthenumberofselectedsceneriesisreduced,thefeasibilityissuesmayoccur.Thatis,theday-aheadunitcommitmentdecisionmightnotbefeasibleforsomescenarioswhicharenotconsidered.Therefore,ontheotherhand,robustoptimizationapproacheshavedrawnmoreattentionrecentlyforsystemoperatorstoensurethesystemrobustness c[2013]IEEE.REPRINTED,WITHPERMISSION,FROM[ 76 ] 42

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andmaketheday-aheadunitcommitmentfeasibleformostoutcomesoftherealtimeuncertainparameter.Fortherobustoptimizationapproach,theuncertainparameterisdescribedwithinagivendeterministicuncertaintysetandtheobjectiveistominimizethetotalcostundertheworst-casescenario.Therst-stagedecisionistheunitcommitmentdecisionthatismadebeforetherealizationoftheworst-casescenario,andthesecond-stagedecisionistheeconomicdispatchdecisionwhichismadeaftertheworst-casescenarioisrealized.Theadvantageoftherobustoptimizationapproachisthatitdoesnotrequirethedistributioninformationoftheinputuncertainparameter,asthestochasticoptimizationapproachesdo.Instead,aslongastheinformationissufcienttogeneratethedeterministicuncertaintyset,theday-aheadunitcommitmentdecisioncanbeobtainedtobefeasibleformostoutcomesoftherealtimeuncertainproblemparameter.Recently,two-stagerobustunitcommitmentmodelsareproposedandBenders'decompositionalgorithmisutilizedtoensuresystemrobustnessunderuncertainties[ 7 , 32 , 78 ].Sincerobustoptimizationachievestheoptimalsolutionwiththeconsiderationofworst-casescenario,thisapproachalwaysfacesthechallengesonitsoverconservatism,duetoitsobjectivefunctionofminimizingtheworst-casecost,becausetheworstcasehappensrarely.Toaddresstheshortagesofstochasticoptimizationapproachandrobustoptimizationapproach,inthischapter,weproposeaninnovativeuniedstochasticandrobustunitcommitmentmodeltotakeadvantageofbothstochasticandrobustoptimizationapproaches.Inourmodel,weputtheweightsfortheexpectedtotalcostandtheworst-casecostrespectivelyintheobjectivefunction.Thisapproachallowsthesystemoperatorstodecidetheweightforeachobjectivebasedontheirpreferences.Inaddition,iftheweightintheobjectivefunctionfortherobustoptimizationpartiszero,thenourmodelturnsouttobeatwo-stagestochasticoptimizationproblemwithadditional 43

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constraintsgeneratedbasedontherobustoptimizationapproach.Themaincontributionofourproposedinnovativeapproachcanbesummarizedasfollows: (1) Ourproposedapproachtakestheadvantagesofboththestochasticandrobustoptimizationapproaches.Ourapproachcanprovideaday-aheadunitcommitmentdecisionthatcanleadtoaminimumexpectedtotalcostwhileensuringthesystemrobustness. (2) Ourproposedapproachcangeneratealessconservativesolutionascomparedtothetwo-stagerobustoptimizationapproach,andamorerobustsolutionascomparedtothetwo-stagestochasticoptimizationapproach. (3) OurproposedapproachcanbeimplementedinasingleBenders'decompositionframework.Thecomputationaltimecanbecontrolledbythesystemoperators.Meanwhile,thesystemoperatorscanalsoadjusttheweightsintheobjectivefunction,basedontheirpreferencesonstochasticand/orrobustoptimizationapproaches.Theremainderofthischapterisorganizedasfollows.Section 3.3 describesouruniedstochasticandrobustunitcommitmentmodel.InSection 3.4 ,wedevelopaBenders'decompositionalgorithmtosolvetheproblem.Intheproposedalgorithm,weapplybothfeasibilityandoptimalitycutsforthestochasticandrobustoptimizationpartsrespectively.Allthesecutsareaddedintothemasterproblem,whichprovidestheday-aheadunitcommitmentdecision.Section 3.5 providesandanalyzesthecomputationalexperimentsthroughseveralcasestudies.Finally,Section 3.6 summarizesourresearch. 3.2Nomenclature A. SetsandParameters B Indexsetofallbuses. Indexsetoftransmissionlineslinkingtwobuses. Gb Setofthermalgeneratorsatbusb. T Timehorizon(e.g.,24hours). SU bi Start-upcostofthermalgenerator i at busb. 44

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SD bi Shut-downcost of thermalgenerator i at busb. Fi(.) Fuelcost of thermalgenerator i . MU bi Minimumup timeforthermalgenerator i atbusb. MD bi Minimumdown timeforthermalgenerator i atbusb. RUbi Ramp-upratelimitforthermalgenerator i atbusb. RDbi Ramp-downratelimitforthermalgenerator i atbusb. Lbi Lowerboundofelectricitygeneratedbythermalgenerator i atbusb. Ubi Upperboundofelectricitygeneratedbythermalgenerator i atbusb. Cij Capacityfor the transmission line linking busiandbusj. Kbij Lineow distributionfactorforthetransmission line linking busi andbusj,duetothenetinjectionatbusb . bt Theweightoftheloadatbusbintimet . t Thebudgetparametertodescribetheuncertaintysetforthetotalloadintimet. Thebudgetparametertodescribetheuncertaintysetforthetotalloadforthewholeoperationalhorizon. Theweightfortheexpectedtotalgenerationcostintheobjectivefunction. Db+t Theupper bound oftheload atbusbintimet . Db)]TJ /F4 7.97 Tf -5.46 -7.59 Td[(t Thelower bound oftheload atbusbintimet . Dbt Theforecastedloadatbusbintimet. dbt Arandomparameterrepresentingtheload atbusbintimet fortherobustoptimizationpart. dbt() Theload atbusbintimet correspondingtoscenarioforthestochasticoptimizationpart. jbit Theinterceptofthejthsegmentlinefor thegenerationcostfor generatoriatbusbintimet. jbit Theslopeofthejthsegmentlinefor thegenerationcostfor generator i atbusbintimet. 45

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B. First-stageVariables ybit Binary decision variable:“1”ifthermalgenerator i atbusbisonintimet;“0”otherwise. ubit Binary decision variable:“1”ifthermalgenerator i atbusbisstartedupintimet;“0”otherwise. vbit Binary decision variable:“1”ifthermalgenerator i atbusbisshutdownintimet;“0”otherwise. C. Second-stageVariables qbit() Electricitygenerationamountbythermalgenerator i atbusbintimet correspondingtoscenario forthestochasticoptimizationpart. xbit Electricitygenerationamountbythermalgenerator i atbusbintimet fortherobustoptimizationpart. bit() Auxiliaryvariableforthestochasticoptimizationpartrepresentingthefuelcostofthermalgenerator i atbusbintimetcorrespondingtoscenario. bit Auxiliaryvariablefortherobustoptimizationpartrepresentingthefuelcostofthermalgenerator i atbusbintimet. 3.3MathematicalFormulationInthissection,wedevelopatwo-stageunitcommitmentformulationconsideringboththeexpectedtotalgenerationcostandtheworst-casescenariogenerationcost.Therststageistodetermine theday-aheadunitcommitmentdecisionthatincludes turn-on / turn-off decisions of thermal generatorsbysatisfyingunitcommitment physical constraints.Thesecondstagecontainsthedecisionson theeconomicdispatchforthethermalgeneratorsundereachscenarioforthestochasticoptimizationpartandtheworst-casescenariofortherobustoptimizationpart .Inourmodel,aparameter2[0,1]isintroducedtorepresenttheweightoftheexpectedtotalgenerationcostandaccordingly1)]TJ /F6 11.955 Tf 12.38 0 Td[(representstheweightoftheworst-casegenerationcost.Especially,forthecase=1,onlytheexpectedtotalgenerationcostisconsideredintheobjectivefunctionandforthecase=0,onlytheworst-casegenerationcostisconsidered.The 46

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detailedformulationisdescribedasfollows: minTXt=1 X b2B Xi2 Gb ( SUbiubit+SDbivbit )+E[Q(y,u,v,)]+(1)]TJ /F6 11.955 Tf 11.96 0 Td[()maxd2Dminx2(y,u,v,d)TXt=1 X b2B Xi2 Gb Fi(xbit) (3) s.t.)]TJ ET BT /F3 11.955 Tf 101.41 -145.03 Td[(ybi(t)]TJ /F8 7.97 Tf 6.58 0 Td[(1) + ybit )]TJ ET BT /F3 11.955 Tf 172.93 -145.03 Td[(ybik 0, 8k :1k)]TJ /F5 11.955 Tf 11.96 0 Td[((t)]TJ /F5 11.955 Tf 11.96 0 Td[(1)MUbi,8i2Gb, 8b2B ,8t (3) ybi(t)]TJ /F8 7.97 Tf 6.59 0 Td[(1) )]TJ ET BT /F3 11.955 Tf 137.1 -198.82 Td[(ybit + ybik 1, 8k :1k)]TJ /F5 11.955 Tf 11.96 0 Td[((t)]TJ /F5 11.955 Tf 11.96 0 Td[(1) MD bi,8i2 Gb,8b2B,8t (3) )]TJ ET BT /F3 11.955 Tf 101.41 -252.61 Td[(ybi(t)]TJ /F8 7.97 Tf 6.58 0 Td[(1) + ybit )]TJ ET BT /F3 11.955 Tf 172.93 -252.61 Td[(ubit 0, 8i2Gb,8b2B,8t (3) ybi(t)]TJ /F8 7.97 Tf 6.59 0 Td[(1) )]TJ ET BT /F3 11.955 Tf 137.1 -279.51 Td[(ybit )]TJ ET BT /F3 11.955 Tf 163.63 -279.51 Td[(vbit 0, 8i2Gb,8b2B,8t (3) ybit , ubit , vbit 2f0,1g, 8i2Gb,8b2B,8t, (3) whereQ( y , u , v ,)isequalto min TXt=1Xb2BXi2Gb Fi(qbit()) (3) s.t.Lbiybit q b it()Ubiybit,8i2 Gb,8b2B,8t (3) q b it())]TJ ET BT /F3 11.955 Tf 158.77 -461.29 Td[(q b i(t)]TJ /F8 7.97 Tf 6.59 0 Td[(1)()(2)]TJ /F3 11.955 Tf 11.95 0 Td[(y b i(t)]TJ /F8 7.97 Tf 6.59 0 Td[(1))]TJ /F3 11.955 Tf 11.95 0 Td[(y b it) L b i +(1+y b i(t)]TJ /F8 7.97 Tf 6.59 0 Td[(1))]TJ /F3 11.955 Tf 11.95 0 Td[(y b it)UR b i,8i2 Gb,8b2B,8t (3) q b i(t)]TJ /F8 7.97 Tf 6.59 0 Td[(1)())]TJ ET BT /F3 11.955 Tf 177.45 -515.09 Td[(q b it()(2)]TJ /F3 11.955 Tf 11.95 0 Td[(y b i(t)]TJ /F8 7.97 Tf 6.59 0 Td[(1))]TJ /F3 11.955 Tf 11.95 0 Td[(y b it)L b i+(1)]TJ /F3 11.955 Tf 11.95 0 Td[(y b i(t)]TJ /F8 7.97 Tf 6.59 0 Td[(1)+y b it)DR b i,8i2 Gb,8b2B,8t (3) X b2B Xi2 Gb qbit()=Xb2Bdbt(),8t (3) )]TJ /F3 11.955 Tf 9.3 0 Td[(CijXb2BKbij(Xr2Gbqbrt())]TJ /F3 11.955 Tf 11.95 0 Td[(dbt())Cij, 8(i,j)2,8t (3) 47

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and (y,u,v,d)=nx:Lbiybitx b itUbiybit,8i2 Gb,8b2B,8t (3) x b it)]TJ /F3 11.955 Tf 11.95 0 Td[(x b i(t)]TJ /F8 7.97 Tf 6.59 0 Td[(1)(2)]TJ /F3 11.955 Tf 11.95 0 Td[(y b i(t)]TJ /F8 7.97 Tf 6.59 0 Td[(1))]TJ /F3 11.955 Tf 11.96 0 Td[(y b it) L b i +(1+y b i(t)]TJ /F8 7.97 Tf 6.59 0 Td[(1))]TJ /F3 11.955 Tf 11.95 0 Td[(y b it)UR b i,8i2 Gb,8b2B,8t (3) x b i(t)]TJ /F8 7.97 Tf 6.59 0 Td[(1))]TJ /F3 11.955 Tf 11.96 0 Td[(x b it(2)]TJ /F3 11.955 Tf 11.95 0 Td[(y b i(t)]TJ /F8 7.97 Tf 6.59 0 Td[(1))]TJ /F3 11.955 Tf 11.96 0 Td[(y b it)L b i+(1)]TJ /F3 11.955 Tf 11.95 0 Td[(y b i(t)]TJ /F8 7.97 Tf 6.59 0 Td[(1)+y b it)DR b i,8i2 Gb,8b2B,8t (3) X b2B Xi2 Gb xbit=Xb2Bdbt,8t (3) )]TJ /F3 11.955 Tf 9.3 0 Td[(CijXb2BKbij(Xr2Gbxbrt)]TJ /F3 11.955 Tf 11.96 0 Td[(dbt)Cij, 8(i,j)2,8t o. (3) Intheaboveformulation,wedenoteFi(.)as the generationcostfunctionofgeneratori. The objectivefunction( 3 )iscomposedoftheunitcommitmentcostintherststage, and boththeexpectedeconomicdispatchcostandtheworst-caseeconomicdispatchcostinthesecondstage. Constraints( 3 )and( 3 ) representeachunit'sminimumup-timeandminimumdown-timerestrictionsrespectively.Constraints( 3 )and( 3 )indicatethestart-up and shut -downoperationsforeachunit. Constraints ( 3 )and( 3 ) enforce theupperandlowerlimitsofthepowergenerationamountofeachunit . Therampingupconstraints( 3 )and( 3 )requiretherst-hourminimumgenerationrestriction(e.g.,Lbi)asdescribedin[ 27 ]andlimitthemaximum increment ofthepower generation amount ofeachunit betweentwoadjacentperiodswhenthegeneratorison.Similarly,rampingdownconstraints( 3 )and( 3 )requirethelast-hourminimumgenerationrestriction(e.g.,Lbi)andenforcethemaximumdecrementofthepowergenerationamountofeachunitbetweentwoadjacentperiodswhenthegeneratorison.Constraints( 3 )and( 3 )ensureloadbalanceandconstraints( 3 )and( 3 )representthetransmissioncapacityconstraints. 48

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3.4DecompositionAlgorithmsandSolutionFramework 3.4.1ScenarioGeneration We useMonteCarlosimulationtogeneratescenariosfor the uncertainload.WeassumethattheloadfollowsamultivariatenormaldistributionN(D,)withitspredictedvalueDandvolatilitymatrix .Wecanrun MonteCarlo simulation togenerateNscenarioseachwiththesameprobability1=N.Aftergeneratingscenarios,wecanreplacethesecond-stageexpectedtotalcostobjectivetermby: 1 NNXn=1TXt=1Xb2BXi2GbFi(qbit(n)). (3) 3.4.2LinearizingFi(.)ThegenerationcostFi(.)is usually expressedasaquadraticfunction, for whichweuse a J-piecepiecewiselinearfunctiontoapproximate. Forinstance,wehave bit(n)jbitybit+jbitqbit(n), (3) 8t2T,8b2B,8i2Gb,8j=1,,J,8n=1,,N forthestochasticoptimizationpart,andsimilarly,wehave bitjbitybit+jbitxbit, (3) 8t2T,8b2B,8i2Gb,8j=1,,J fortherobustoptimizationpart. 3.4.3TheUncertaintySetoftheLoad Togeneratetheuncertaintysetfortherobustoptimizationpart ,we assume theloadforeachtimeperiodtateachbusbisbetweenalowerboundDb)]TJ /F4 7.97 Tf -5.46 -7.58 Td[(tandanupperboundDb+t, whichcanbedecidedbythe5thand95thpercentilesoftherandomloadoutput. Inaddition,weassumeforeachgiventimeperiodt,thesummationoftheweightedloadsatallbusesisboundedabovebyt,andthesummationoftheweightedloadswithinthe 49

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wholeoperationalhorizonisboundedaboveby. Accordingly, theuncertaintysetcanbedescribedasfollows: D:=(d2RjBjjTj:Db)]TJ /F4 7.97 Tf -5.46 -7.89 Td[(tdbtDb+t,8t,8b (3) Xb2Btbdtbt,8t (3) TXt=1Xb2Btbdtb). (3) 3.4.4AbstractFormulationFornotationbrevity,weusematricesandvectorstorepresenttheconstraintsandvariables.Forexample,weuseetorepresentthevectorwithallcomponentsequalto1.Themathematicalmodelcanbeabstractedasfollows: miny,u,v(aTu+bTv)+1 NNXn=1eT(n)+(1)]TJ /F6 11.955 Tf 11.95 0 Td[()maxd2Dminx,eTs.t.Ay+Bu+Cvr, (3) Fy)]TJ /F7 11.955 Tf 11.96 0 Td[(Dq(n)g,n=1,,N (3) Ky)]TJ /F7 11.955 Tf 11.96 0 Td[(Pq(n))]TJ /F7 11.955 Tf 11.95 0 Td[(J(n)0,n=1,,N (3) Tq(n)Sd(n)+s,n=1,,N (3) Fy)]TJ /F7 11.955 Tf 11.96 0 Td[(Dx (d) g, (3) Ky)]TJ /F7 11.955 Tf 11.96 0 Td[(Px (d) )]TJ /F7 11.955 Tf 11.96 0 Td[(J (d) 0, (3) Tx (d) Sd+s, (3) y,u,v2f0,1g,x (d) ,q(n)0, (d) ,(n)free ,8n (3) where D=nd2RjBjjTj:d)]TJ /F2 11.955 Tf 10.41 -5.41 Td[(dd+,UTdzo. (3) Constraint( 3 )representsconstraints( 3 )-( 3 );constraint( 3 )representsconstraints( 3 )-( 3 );constraint( 3 )representsconstraint( 3 );constraint 50

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( 3 )representsconstraints( 3 )and( 3 ).Constraint( 3 )representsconstraints( 3 )-( 3 );constraint( 3 )representsconstraint( 3 );constraint( 3 )representsconstraints( 3 )and( 3 ). 3.4.5Benders'DecompositionAlgorithmWecanusetheBenders'decompositionalgorithmtosolvetheaboveproblem.First,foreachscenarion,n=1,,N,wedualizetheconstraints( 3 )to( 3 )andobtainthefollowingdualformulation forthesecond-stageeconomicdispatchforthestochasticpart : Sn(y)=maxn,n,n(Fy)]TJ /F7 11.955 Tf 11.96 0 Td[(g)Tn+(Ky)Tn+(s+Sd(n))Tns.t.DTn+PTn+TTn0, (3) JTn=e, (3) n,n,n0, (3) wheren,n,naredualvariablescorrespondingtothescenarionforconstraints( 3 ),( 3 )and( 3 )respectively.Similarly,wedualizetheconstraints( 3 )to( 3 )andobtainthefollowingdualformulation forthesecond-stageeconomicdispatchfortherobustpart : R(y)=maxd2D,,,(Fy)]TJ /F7 11.955 Tf 11.95 0 Td[(g)T+(Ky)T+(s+Sd)Ts.t.DT+PT+TT0, (3) JT=e, (3) ,,0, (3) where,,aredualvariablesforconstraints( 3 ),( 3 )and( 3 )respectively.Wedenotenasthesecond-stageoptimal economicdispatch costcorrespondingtoscenarion,n=1,,Nandasthesecond-stageoptimal economicdispatch costundertheworst-casescenario.Thenthemasterproblem canbedescribedasfollows, 51

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andtheproblemcanbesolved byaddingfeasibilityandoptimalitycutsiteratively: miny,u,v2f0,1g(aTu+bTv)+1 NNXn=1n+(1)]TJ /F6 11.955 Tf 11.95 0 Td[()s.t.Ay+Bu+Cvr,Feasibilitycuts,Optimalitycuts. 3.4.6Benders'CutsfortheStochasticOptimizationPart 3.4.6.1FeasibilitycutsWeuse the L-shapedmethodtogeneratefeasibilitycuts.Inthiscase,wedon'tneedtoconsiderconstraints( 3 ) sincewecanalwayschoose(n)tomake theseconstraints satised. Thus ,theywillnotaffectthefeasibility .Forconstraint s ( 3 )and( 3 )inscenarion,n=1,,N,thefeasibilitycheckproblemisshownasfollows: minq(n),4Xj=1eTj (3) s.t.Dq(n)+1)]TJ /F6 11.955 Tf 11.96 0 Td[(2Fy)]TJ /F7 11.955 Tf 11.96 0 Td[(g, (3) Tq(n)+3)]TJ /F6 11.955 Tf 11.95 0 Td[(4Sd(n)+s, (3) q(n)0,j0,j=1,,4. (3) T hedualoftheaboveformulation canbedescribedasfollows : !Sn(y)=max^n,^n(Fy)]TJ /F7 11.955 Tf 11.95 0 Td[(g)T^n+(Sd(n)+s)T^n (3) s.t.DT^n+TT^n0, (3) ^n,^n2[0,1], (3) where^nand^naredualvariables correspondingtothenthscenario forconstraints( 3 )and( 3 )respectively.Thenwecanperformthefollowingstepstocheckfeasibility: (1) If!Sn(y)=0,yisfeasible forthenthscenario ; 52

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(2) If!Sn(y)>0,generatea corresponding feasibilitycut!Sn(y)0. 3.4.6.2Optimalitycuts Ateach iteration, after solvingthemasterproblem,weobtainn,n=1,,N,andy.Ifwesubstituteyintothesubproblemandget Sn(y),weshouldhave Sn(y)n.If Sn(y)>n,wecanclaimthatyisnotanoptimalsolutionandwecangenerateacorrespondingoptimalitycutasfollows: Sn(y)n. 3.4.7Benders'CutsfortheRobustOptimizationPart Aftersolvingthemasterproblem, weapplythebilinearapproachtoget R(y). Thebilinearapproachisprovedtoconvergetooptimalitywithasmallgap(lessthan0.05%)inareasonableandmuchshortertimethantheexactseparationalgorithmdoes[ 32 ]. Wesolvethefollowingtwolinearprograms iteratively : R1(y,d)=max,,(Fy)]TJ /F7 11.955 Tf 11.96 0 Td[(g)T+(Ky)T+(s+Sd)T(SUB1)s.t.DT+PT+TT0, (3) JT=e, (3) ,,0; (3) R2(y,,,)=maxdTSd+(Fy)]TJ /F7 11.955 Tf 11.96 0 Td[(g)T+(Ky)T+sT(SUB2)s.t.d)]TJ /F2 11.955 Tf 10.4 -5.41 Td[(dd+, (3) UTdz. (3) Next,wediscussthefeasibilityandoptimalitycutsbasedon theproposedapproach . 53

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3.4.7.1FeasibilitycutsIftherst-stagesolutionyisinfeasiblethenweshouldhave R(y) infeasibleor unbounded.Notethat R2(y,,,)is feasibleand boundedfromabovebecausethefeasibleregionofdisapolyhedra(e.g.,bounded),andtheobjectivefunctioniscontinuous .Therefore,weonlyneedtocheckthefeasibilityof(SUB1).Thefeasibilitycheck foragivend isthesameas thatforthe stochasticcase: !R1(y,d)=max^,^(Fy)]TJ /F7 11.955 Tf 11.95 0 Td[(g)T^+(s+Sd)T^ (3) (FEA1)s.t.DT^+TT^0, (3) ^,^2[0,1]. (3) Noteherethatinorder to check thefeasibilityundertheworst-casescenario,weneedtosolve(FEA1)andthefollowing(FEA2)iteratively tondtheworst-caseloadd . !R2(y,^,^)=maxd^TSd+(Fy)]TJ /F7 11.955 Tf 11.96 0 Td[(g)T^+sT^ (3) (FEA2)s.t.Constraints( 3 )and( 3 ). Thisprocedurestopswhenwehave !R2(y,^,^)!R1(y,d). Thedetailedalgorithmisshownasfollows : (1) Pickanextremepointd in D ; (2) Solve(FEA1),andstoretheoptimalobjectivevalue!R1(y,d)andtheoptimalsolution^and^; (3) Solve(FEA2),andstoretheoptimalobjectivevalue!R2(y,^,^)andtheoptimalsolutiond; (4) If!R2(y,^,^)>!R1(y,d),letd=dandgotoStep(2).OtherwisegotoStep(5); (5) If!R1(y,d)=0, terminatethefeasibilitycheckfortherobustpart,andgotothefeasibilitycheckforthestochasticpart.Otherwise,add thefeasibilitycut!R1(y,d)0tothemasterproblem. 54

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3.4.7.2OptimalitycutsAteachiteration,wesolvethemasterproblemandobtainandy. Similarly,if R(y)>,wecangenerateacorrespondingoptimalitycutasfollows: R(y). Thedetailedalgorithmtoobtainoptimalitycutsisasfollows : (1) Pickanextremepointd in D; (2) Solve(SUB1),andstoretheoptimalobjectivevalue R1(y,d)andtheoptimalsolutionand; (3) Solve(SUB2),andstoretheoptimalobjectivevalue R2(y,,,)andtheoptimalsolutiond; (4) If R2(y,,,)> R1(y,d),letd=dandgotoStep(2).OtherwisegotoStep(5); (5) If R2(y,,,)>,generatethecorrespondingoptimalitycut R2(y,,,)tothemasterproblem.Otherwise, terminatetheoptimalitycheckfortherobustpartandgototheoptimalitycheck forthestochasticpart.Finally,theowchartofourproposedalgorithmisshowninFig. 3-1 . 3.4.8SpecialCasesandDiscussionsTherobustoptimizationapproach helpsensuretherobustness of therst-stageunitcommitmentdecision .Wecan consideraspecialcaseinwhichwe only usetheconstraints providedbythe robustoptimization approach toguaranteethesolutionrobustnesswithoutconsideringtheworst-case economicdispatchcost intheobjectivefunction(i.e.,=1).Inaddition,ifwe only takeasmallnumberofscenarios, then wedon'tneedtousetheBenders'decomposition approach togeneratecutsforthestochasticpart. Instead ,wecan put thestochasticpartinthemasterproblem,andaddfeasibilitycutsbyconsideringtheconstraintsintherobustpart.Therefore,the special casecanbereformulatedasfollows: miny,u,v(aTu+bTv)+1 NNXn=1eT(n) 55

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. . PickanextremepointdinD . n=1,solvethemasterproblemandgety,u,v,,Sn,n=1,,N . Get!R1(y,d),^and^ . Get!R2(y,^,^)andd . d=d . !R2(y,^,^)>!R1(y,d)? . !R1(y,d)>0? . Generateafeasibilitycut!R1(y,d)0 . Get!Sn(y) . Generateafeasibilitycut!Sn(y)0 . !Sn(y)>0? . n=N? . n=n+1 . n=1,get R1(y,d)and, . Get R2(y,,,)andd . d=d . R2(y,,,)> R1(y,d)? . R2(y,,,)>? . Generateanoptimalitycut R2(y,,,) . Get Sn(y) . Generateanoptimalitycut Sn(y)n . Sn(y)>n . n=N? . n=n+1 . Stopandoutputthesolution . yes . no . yes . no . yes . no . no . yes . yes . no . yes . no . no . yes . no . yes Figure3-1. Flowchartoftheproposedalgorithm 56

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s.t.Constraints( 3 )-( 3 ),Fy)]TJ /F7 11.955 Tf 11.95 0 Td[(Dx(d)g,Tx(d)Sd+s,8d2Dy,u,v2f0,1g,x(d)0,(n)free.Fortheaboveformulation, thedecisionvariablex(d)istheauxiliaryvariablerepresentingthegenerationamountcorrespondingtothecaseinwhichtheloadisd. Constraints( 3 )-( 3 ) areput inthemasterproblem.WeapplytherobustfeasibilitycheckdescribedinSection 4 tondtheworst-caseloadd.ThenwecanaddtheBenders'feasibilitycutsintheform!R1(y,d)0tothemasterproblem.Noteherethatinthissection,weprovideageneralframeworktosolvetheuniedstochasticandrobustunitcommitmentproblem.Thereareseveraldeviationswecanexploreforeachpartinthealgorithm.Forinstance,forthestochasticpart,wecanchoosetheBenders'decompositionapproachorsolvethedeterministicequivalentformulation(asaboveforthespecialcase);fortherobust optimality part,wecanconsiderincludingornotincludingtheworst-casecostintheobjectivefunction;fortherobustfeasibilitypart,oncetheworst-caseloaddisdetected,wecanaddthedualcutintheform!R1(y,d)0,oragroupofprimalcutsintheformFy)]TJ /F7 11.955 Tf 12.79 0 Td[(Dx(d)g,Tx(d)Sd+s,similartotheonedescribedin[ 78 ]. 3.5ComputationalResultsInthissection,wereportexperimentalresultsforamodiedIEEE118-bussystem,basedontheonegivenonlineathttp://motor.ece.iit.edu/data,toshowtheeffectivenessoftheproposedapproach.Thesystemcontains118buses,33generators,and186transmissionlines.Theoperationaltimeintervalis24hours. Inourexperiments,wesetthefeasibilitytolerancegaptobe10)]TJ /F8 7.97 Tf 6.58 0 Td[(6andtheoptimalitytolerancegap ( R2)]TJ /F5 11.955 Tf 12.68 2.66 Td[()= tobe10)]TJ /F8 7.97 Tf 6.59 0 Td[(4.TheMIPgaptoleranceforthemasterproblemistheCPLEXdefaultgap. WeuseC++withCPLEX12.1toimplementtheproposedformulationsandalgorithms.Allexperimentsareexecutedonacomputerworkstationwith4IntelCoresand8GBRAM. 57

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Inourexperiment,werstperformsensitivityanalysisofourproposedapproachintermsoftheeffectsoftheuncertaintysetandtheobjectiveweight.Then,wecomparetheperformancesofourproposedapproachwiththestochasticandrobustoptimizationapproaches. 3.5.1SensitivityAnalysis 3.5.1.1EffectofuncertaintysetForconvenience,wenormalizetheweightparameterbt=1.WerstletDb+t=(1+ R atio%)DbtandDb)]TJ /F4 7.97 Tf -5.45 -7.58 Td[(t=(1)]TJ ET BT /F1 11.955 Tf 152.11 -180.56 Td[(R atio%)Dbt,8t,8b.Then,foreachtimet,weletthebudgett=(1+ B udget%)Pb2BDbt.Finally,welettheoverallbudget=0.9PTt=1Pb2BDb+t.Inourexperiment,weallowthesensitivityanalysisparameters R atio%and B udget%tovaryfrom0to25%.Inaddition,weuseave-piecepiecewiselinearfunctiontoapproximatethe generation costfunction.WetesttheperformanceofourproposedapproachundervariousRatio%andBudget% settings . NoteherethatwhenBudget%>Ratio%,constraint( 3 )isredundantandaccordinglythecomputationalresultsforthesecasesarethesameastheoneinwhichBudget%=Ratio% .Inthisexperiment,wesetthesamplesizeNtobe5,andreporttheoptimalobjectivevalue,thenumberofstart-ups,andthecomputationaltimeforeachsettinginTable 3-1 . Table3-1. ResultsunderdifferentRatio%andBudget%settings Ratio%Budget%5%15%25% 5%Obj.Val.($)737,134737,134737,134#ofStart-ups101010Time(sec)48484815%Obj.Val.($)737,439738,372738,272#ofStart-ups101111Time(sec)6116316325%Obj.Val.($)738,561739,374740,175#ofStart-ups121213Time(sec)1063921193 58

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FromresultsshowninTable 3-1 ,wecanobserve that (1) givenanidenticalBudget%,theobjectivevalueandthenumberofstart-upsincreaseas Ratio%increases,because theproblembecomesmoreconservative,andsimilarly, (2) givenanidenticalRatio%,theobjectivevalueandthenumberofstart-upsincrease asBudget%increases, becausethesystem allows more loaductuations . 3.5.1.2Sensitivityanalysisofobjectiveweight Inthissubsection,wetesttheperformanceofourproposedapproachundervariousobjectiveweight settings .Note here thatif=0,onlytheworst-casetotalcostisconsideredandif=1,onlytheexpectedtotalcostisincluded.Forthisexperiment,wesetRatio%=10%andBudget%=10%.Thecomputationalresultsareshown in Fig. 4-2 . Figure3-2. Therelationshipbetweentheobjectivevalueandtheobjectiveweight FromFig. 4-2 wecanobservethat,asincreases,theoptimalobjectivevaluedecreases,becausetheproblembecomes less conservativeastheworst-casecostcomponenthas asmaller weight. 3.5.2ProposedApproachvsStochasticOptimizationApproachInthissubsection,wesetRatio%=Budget%and compare theperformance s ofour proposed approachwiththetraditionaltwo-stagestochastic optimizationapproach underdifferentBudget%andscenario-size settings .Inthisexperiment,wedonot 59

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includethe worst-case generationcostintheobjectivefunction(i.e.,=1). The totalcostscorrespondingto the traditionaltwo-stagestochasticoptimizationapproach(SO)andourproposeduniedstochasticandrobustoptimizationapproach(SR)areobtainedbythefollowing two steps: (1) Solvetheproblem byusingthe SOandSRapproachesrespectively,andobtainthecorresponding rst-stageunitcommitment solutions . (2) Fix theunitcommitment solutions obtainedin(1)andsolvethesecond-stageproblemrepeatedlyfor50randomly generatedinstancesto obtainthetotalcostforeachapproach. Tocomparetheperformancesbetweenthese two approaches,weintroduceapenaltycost attherateof $ 5,000 =MWh [ 7 ] ,forany powerimbalance ortransmissioncapacity/ramp-ratelimitviolation . Wersttesttheperformancesof theSOandSR approachesundervariousBudget% settings . Wesetthescenariosizetobe5andreportthecomputationalresultsinTable 3-2 .TheBudget%scenarios are givenintherstcolumn.Thetotalcosts (T.C.) obtainedbyeachapproacharereportedinthethirdcolumn.TheUCcosts (UC.C.) foreachapproacharegiveninthefourthcolumnandthenumbersofstart-upsaregiveninthefthcolumn.Finally,theCPUtimesarerecordedinthesixthcolumn.FromTable 3-2 ,wehavethefollowingobservations: (1) First,weobservethatwhenBudget%5,theunitcommitmentdecisionsobtainedfromthe SOandSR approachesarethesame. This isbecausetheuncertaintysetissosmallthattheunitcommitmentdecisionsobtainedfromtheSOapproacharerobustenoughtoaccommodatetheuncertainty,andthereisnoneedtogeneratefeasibilitycutsintheSRapproach.However,whenBudget%10,theUCdecisionsobtainedfromtheSOapproacharenotfeasibletosomesimulatedloadscenarios. But theUCdecisionsobtainedfromtheSRapproach arealwaysfeasible underdifferentbudgetlevels.Therefore, dueto thepenaltycostfortheviolationinthe power balanceortransmissioncapacityconstraints,theSRapproachincursa smaller totalcostthantheSOapproachwhenBudget%10%. Moreover, whenBudget%increases,thetotalcostgapbetween these twoapproachesincreases.ThisresultveriesthattheproposedapproachcanprovideamorerobustsolutionascomparedtotheSOapproach,especiallywhenthesystemhasmoreuncertainties. 60

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Table3-2. ComparisonbetweenSOandSRapproaches Budget%ModelT.C.($)UC.C.($)#ofStart-upsTime(sec) 3%SO737,79949,5001062SR737,79949,50010505%SO738,11249,5001059SR738,11249,500104710%SO 740,878 49,50010 62 SR739,91152,5001112615%SO 752,575 49,50010 63 SR741,17051,0001116720%SO 782,361 49,50010 63 SR742,86654,00012222 (2) Second,weobservethat, as comparedtotheSO approach ,theSR approach requires moregenerators committed to providesufcient generationcapacitytoguarantee thesupplymeetingtheload .Asaresult,theSRapproachhas alarger UCcostthantheSOapproach.This resultalsoveries thattheproposedSRapproachcanprovideamorerobustsolutionascomparedtotheSOapproach.Next,wetestthesystemperformancesoftheSOandSRapproachesundervariousscenario-size settings .WesetBudget%=20%andtestthescenariosizeN=1,5,10,and20,respectively.First,weobservethatfortheSOapproach,thenumberofstart-upsincreaseswhenthenumberofscenariosincreases(e.g.,thenumberofstart-upsis10 whenN=1 andthisnumberincreasesto11 whenN=20 ).Ontheotherhand,inourproposedSRapproach,thenumberofstart-upsremainsthesame.Therefore,theproposedSRapproachismorerobustthanthetraditionalSOapproach.Second,weobservethatasthescenariosizeincreases,thetotalcostoftheSOapproachandthetotalcostgapbetweentheSOandSRapproachesdecrease(e.g.,thetotalcostofSOis$ 782,361 andthetotalcostofSRis$742,866whenN=1;thetotalcostofSOis$ 758,496 andthetotalcostofSRis$742,866whenN=20). Thisisbecauseasthescenariosizeincreases,therst-stageunitcommitmentsolutionobtainedbytheSOapproachbecomesmorerobustsothatthesystemincurslesspenaltycost 61

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inthesecondstage. However,thereisstillabiggapbetweenthetwowhenN=20,whichveriesthatitisnecessarytoadd the robustuncertaintysettoensuresystemrobustnessoftheday-aheadunitcommitmentdecision. 3.5.3ProposedApproachvsRobustOptimizationApproach Wealsocompare theperformancesofourproposed SRapproachwith thetraditionaltwo-stagerobust optimization (RO) approach undervariousBudget%settings.Weset the scenariosizefor the SRapproachtobe5andsummarizethecomputationalresultsinTable 3-3 .TheBudget%settings,thetotalcosts,theUCcosts,thenumbersofstart-ups,andtheCPUtimesare reported intherst,third,fourth,fthandsixthcolumnsrespectively. First,fromourcomputationalresults,weobservethattherearenopenaltycostsincurredforboththeSRandROapproaches,whichmeansthattheunitcommitmentdecisionsforeachapproacharefeasibleforallgeneratedscenarios. Second,fromTable 3-3 weobservethat ourproposedSRapproachcommitslessnumberofunitsintherststagethan the ROapproach. Thatis, as comparedtotheRO approach ,SR leadstosmallerunitcommitmentandtotalcosts. ThisresultindicatesthatourproposedSRapproachcangeneratealessconservativesolutionascomparedtotheROapproachwhilemaintainingsystemrobustness.Finally,weobserve that ,theCPUtimesfortheROapproacharelargerthanourproposedSRapproachforeachtestedinstance, becausetheinitialsolutionfortheROapproachisworsethanthatofourproposedSRapproach,andmoreoptimalitycutsaregeneratedtomakethealgorithmconverge. 3.6SummaryInthischapter,wedevelopedauniedstochasticandrobustunitcommitmentmodelforISOs/RTOstoperformreliabilityunitcommitmentruns, soas toachievearobustandcost-effectiveunitcommitmentsolution.Ourproposedapproachtakestheadvantagesofboththestochasticandrobustoptimizationapproaches.Theuncertaintysetprovidedbytherobustoptimizationapproachensurestherobustnessoftheunit 62

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Table3-3. ComparisonbetweenROandSRapproaches Budget%ModelT.C.($)UC.C.($)#ofStart-upsTime(sec) 3%RO737,29449,50010375SR737,27549,50010495%RO738,51451,00011292SR737,27549,500104810%RO739,51554,00012375SR738,19052,5001112715%RO739,86854,00012339SR738,50651,0001116820%RO749,06463,30015303SR739,32054,00012223 commitmentdecision.Meanwhile,theexpectedtotalcostintheobjectivefunctionprovidessystemoperatorsexibilitytoadjustthecost-effectivenessoftheproposedapproach.TheproposedmodelcanbesolvedefcientlybyourproposedBenders'decompositionframework,whichincludesbothfeasibilityandoptimalitycuts.Finally,computationalexperimentsshowtheeffectivenessofourproposedapproach. 63

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CHAPTER4DATA-DRIVENRISK-AVERSETWO-STAGESTOCHASTICPROGRAM 4.1ProblemDescriptionandLiteratureReviewStochasticprogrammingcaneffectivelyhelpdecisionmakinginanuncertainenvironment.Particularly,aspecicclassofstochasticprogramminghasbeenstudiedextensively,whichiscalledtwo-stagestochasticprogram.Typically,atwo-stagestochasticprogram(denotedasSP)canbeformulatedasfollows: minxcTx+E[Q(x,)] (4) (SP)s.t.x2X,wherexistherst-stagedecisionvariable,inaconvexsetX,andQ(x,)=miny()2YfH(y()):G(x,y())0,2grepresentingthesecond-stageprobleminwhichtherandomvariableisonaprobabilityspace(,(),P),whereisthesamplespacefor,()isthe-algebraof,andPistheassociateprobabilitydistribution.Inthisformulation,theprobabilitydistributionisassumedknownandsignicantresearchprogresshasbeenmadeinthetheoreticalanalysisanddevelopingefcientalgorithms[ 9 ].However,inpractice,duetolimitedavailableinformationontherandomparameters,itisgenerallydifculttopreciselyestimatetheprobabilitydistributionP.Toaddressthisissue,distributionallyrobuststochasticprogramhasbeenproposed.Inthisapproach,bydeninganuncertaintysetDofthetrueprobabilitydistributionP,adistributionallyrobusttwo-stagestochasticprogramcanbereformulatedasfollows(denotedasRA-SP),withtheobjectiveofminimizingtheexpectedcostundertheworst-casedistributioninD: minxcTx+maxP2DEP[Q(x,)] (4) (RA-SP)s.t.x2X. 64

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OneimportantfactorforsolvingthedistributionallyrobuststochasticprogramistheconstructionofD.Severalapproacheshavebeenproposedintheliterature.Forinstance,in[ 25 ],theconicsetsforthemeanvectorofobserveddataareconsidered;in[ 58 ],byrestrictingthedensityfunctionfwithl1norm,thedistributionallyrobuststochasticprogramcanbereformulatedasaCVaRproblemandsolvedbyasampleaverageapproximationapproach.Morerecently,thedatadrivenapproachhasbeenappliedtoconstructthecondencesetD.Thatis,insteadofknowingthetrueprobabilitydistributionP,aseriesofhistoricaldatathataretakenfromPcanbeobserved.WithareferencedistributionP0determinedbythehistoricaldataandapredenedmetricd(P0,P)tomeasurethedistancebetweenP0andP,thecondencesetDcanberepresentedas D=fP:d(P,P0)g, (4) wherethetoleranceisdependentonthenumberofhistoricaldataobserved.Intuitively,DgetstighteraroundthetruedistributionPwithmorehistoricaldataobserved.Thatis,becomessmaller,andaccordinglyRA-SPbecomeslessconservative.TherehasbeenresearchprogressmadebyusingthedatadrivenapproachtoconstructthedistributioncondenceuncertaintysetD.Forinstance,in[ 18 ],Disconstructedbasedonthemeanandcovariancematricesoftherandomvariableslearnedfromthehistoricaldata.Inaddition,Dcanbeconstructeddifferentlybydeningdifferentmetrics.Forexample,-divergences,whicharedenedintheformd(PjP0):=R(f() f0())f0()dwithreferencedensityfunctionf0correspondingtoP0andtruedensityfunctionfcorrespondingtoP,areusuallyusedformeasuringthedistancebetweenthereferencedistributionandthetruedistribution.In[ 6 ],-divergencesareusedforthemetricdandthesingle-stagerobustcounterpartproblemcanbereformulatedasatractableproblem.Thisapproachhasbeenextendedtothetwo-stagestochasticprogramcasein[ 38 ]andthechanceconstrainedcasein[ 29 ].Besides,itisalsowell-knownthat 65

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-divergences,especiallyKL-divergencedKL(PjP0):=Rlog(f() f0())f0()d,hasbeencommonlyappliedinbothmachinelearningandinformationtheory,e.g.,[ 46 ]and[ 62 ].However,-divergencesareingeneralnotmetricssincemost-divergencesdonotsatisfythetriangleinequalityandeventhesymmetricpropertyd(PjP0)=d(P0jP).Also,asindicatedin[ 21 ]andthemetrics'relationshipsin[ 24 ],therearenoconvergenceresultsforgeneral-divergenceswithempiricalprobabilitydistributionasP0,forthecaseinwhichthetruedensityfunctionfiscontinuous.Inthischapter,weintroduceanewclassofmetrics,whichiscalled-structureprobabilitymetricsclass.GiventwoprobabilitydistributionsPandQ,the-structureprobabilitymetricsaredenedasfollows: d(P,Q)=suph2HZhdP)]TJ /F10 11.955 Tf 11.96 16.27 Td[(ZhdQ, (4) whereHisaclassofreal-valuedboundedmeasurablefunctionson.-structureprobabilitymetricsclassisrstintroducedin[ 79 ],andithasbeenappliedininformationtheory[ 26 ],mathematicalstatistics[ 57 ],masstransportationproblems[ 52 ],andseveralareasincomputerscience,includingprobabilisticconcurrency,imageretrieval,datamining,andbioinformatics[ 20 ].Themembersin-structureprobabilitymetricsclassareKantorovichmetric,Fortet-Mouriermetric,TotalVariationmetric,BoundedLipschitzmetric,andKolmogorov/Uniformmetric.WewillintroducethesemetricsindetailsinSection 4.2 .Inourchapter,weutilize-structureprobabilitymetricstohelpconstructthecondenceuncertaintysetfortheprobabilitydistributioninRA-SP(i.e.,Problem( 4 )).Then,wedevelopalgorithmstosolveRA-SPforbothdiscreteandcontinuoustruedistributioncases.Finally,weshowthatRA-SPconvergestoSP(i.e.,Problem( 4 ))asthenumberofhistoricaldataincreasestoinnityandfurtherexplorethevalueofdatabyinvestigatingtheconvergencerateandshowingnumericalexperimentresults.Ourcontributionscanbesummarizedasfollows: 66

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1. Weapplyanewclassofmetrics,-structureprobabilitymetrics,toconstructthecondencesetDfortheambiguousdistributionsandexploretherelationshipsamongthemembersinthe-structureprobabilitymetricsclass. 2. WeprovideframeworkstosolveRA-SPforbothdiscreteandcontinuousdistributioncases.ForthecaseinwhichthetruedistributionPisdiscrete,wecanreformulateRA-SPasatraditionaltwo-stagerobustoptimizationproblem.ForthecaseinwhichPiscontinuous,weproposeasamplingapproachandobtainthelowerandupperboundsoftheoptimalobjectivevalueofRA-SP.WefurtherprovethatbothlowerandupperboundsconvergetotheoptimalobjectivevalueofRA-SPasthesamplesizegoestoinnity. 3. Weexploretheconvergenceratesforbothouterandinnerloops.Thatis,fortheouterloop,weprovethatasthenumberofhistoricaldatagoestoinnity,therisk-averseproblemRA-SPconvergestoSPexponentiallyfast.Inaddition,fortheinnerloopforwhichthetruedistributioniscontinuous,asthesamplesizegoestoinnity,weprovethatbothlowerandupperboundsconvergetotheoptimalobjectivevalueofRA-SPexponentiallyfast.Theremainingpartofthischapterisorganizedasfollows:InSection 4.2 ,weintroducethe-structureprobabilitymetricsclassandstudytherelationshipsamongthemembersinthisclass.InSection 4.3 ,wedescribethesolutionapproachestosolveRA-SPforbothdiscreteandcontinuoustruedistributioncases,andexploretheconvergencerateforeachcase.InSection 4.4 ,weperformnumericalstudiesondata-drivenrisk-aversenewsvendorandfacilitylocationproblems.Finally,inSection 4.5 ,weconcludeourresearch. 4.2-StructureProbabilityMetricsInthissection,weintroduceanewclassofmetrics:-structureprobabilitymetrics.Werstintroducethedenitionsofthemetricmembersofthatclass,e.g.,Kantorovichmetric,Fortet-Mouriermetric,TotalVariationmetric,BoundedLipschitzmetric,andKolmogorov/Uniformmetric.Then,weinvestigatetherelationshipsamongthesemetrics.Weshowthatifthesupportingspaceisbounded,theTotalVariationmetricisadominatingmetricintheclass,i.e.,theconvergencewiththeTotalVariationmetriccanguaranteetheconvergencewithothermetricsintheclass. 67

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4.2.1DenitionAsdescribedin Section 4.1 ,foranytwoprobabilitydistributionsPandQ,the-structureprobabilitymetricsaredenedbyd(P,Q)=suph2HjRhdP)]TJ /F10 11.955 Tf 12.77 9.63 Td[(RhdQj,whereHisaclassofreal-valuedboundedmeasurablefunctionson. Ingeneral,the-structuremetricssatisfythepropertiesofmetrics,i.e.,d(P,Q)=0ifandonlyifP=Q,d(P,Q)=d(Q,P)(symmetricproperty)andd(P,Q)d(P,O)+d(O,Q)foranyprobabilitydistributionO(triangleinequality).Inthefollowing,wedene(x,y)asthedistancebetweenrandomvariablesxandyanddenedasthedimensionof.IfrandomvariablexfollowsdistributionP,wedenoteitasP=L(x). Then,wederivedifferenttypesofmetricsbasedondifferentdenitionsofH. Kantorovichmetric:ForKantorovichmetric(denotedasdK(P,Q)), H =fh:khkL1g,wherekhkL:=supfh(x))]TJ /F4 7.97 Tf 6.58 0 Td[(h(y) (x,y):x6=ying.Manymetricsknowninstatistics,measuretheory,ergodictheory,andfunctionalanalysis,arespecialcasesoftheKantorovichmetric[ 67 ].Kantorovichmetricalsohasmanyapplicationsintransportationtheory[ 52 ],andincomputerscienceincludingprobabilisticconcurrency,imageretrieval,datamining,andbioinformatics[ 20 ]. Fortet-Mouriermetric:ForFortet-Mouriermetric(denotedasdFM(P,Q)), H =fh:khkC1g,wherekhkC:=supfh(x))]TJ /F4 7.97 Tf 6.59 0 Td[(h(y) c(x,y):x6=yingandc(x,y)=(x,y)maxf1,(x,a)p)]TJ /F8 7.97 Tf 6.58 0 Td[(1,(y,a)p)]TJ /F8 7.97 Tf 6.59 0 Td[(1gforsomep1anda2.Noteherethatwhenp=1,Fortet-MouriermetricisthesameasKantorovichmetric.Therefore,Fortet-mouriermetricisusuallyusedasageneralizationofKantorovichmetric,withtheapplicationsonmasstransportationproblems[ 52 ]. TotalVariationmetric:ForTotalVariationmetric(denotedasdTV(P,Q)), H =fh:khk11g,wherekhk1:=supx2jh(x)j.AnotherdenitionoftheTotalVariationmetricisdTV(P,Q):=2supB2()jP(B))]TJ /F16 11.955 Tf 12.3 0 Td[(Q(B)j.Thetotalvariationmetrichasacouplingcharacterization (detailedproofsareshownin[ 37 ]) : dTV(P,Q)=2inffPr(X6=Y):P=L(X),Q=L(Y)g. (4) Thetotalvariancemetriccanbeappliedininformationtheory[ 16 ]andinstudyingtheergodicityofMarkovChains[ 45 ].Lateron,wewillprovethattheconvergencewithrespecttotheTotalVariationmetricimpliestheconvergencewithrespecttoothermetricsinthegeneral-structureprobabilitymetricsclasswhenisbounded. 68

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BoundedLipschitzmetric:ForBoundedLipschitzmetric(denotedasdBL(P,Q)), H =fh:khkBL1g,wherekhkBL:=khkL+khk1.BoundedLipschitzmetricisusuallyusedtoprovetheconvergenceofprobabilitymeasuresintheweaktopology[ 22 ]. Uniform/Kolmogorovmetric:ForUniformmetric(alsocalledKolmogorovmetric,denotedasdU(P,Q)), H =fI(,t],t2Rdg.TheUniformmetricisoftenusedinprovingtheclassicalcentrallimittheorem,and commonlyutilizedin theKolmogorov-Smirnovstatistic for hypothesistesting[ 56 ]. Accordingtothedenition,wehavedU(P,Q)=suptjP(xt))]TJ /F16 11.955 Tf 11.96 0 Td[(Q(xt)j. Wassersteinmetric:WassersteinmetricisdenedasdW(P,Q):=inffE[(X,Y)]:P=L(X),Q=L(Y)g,wheretheinmumistakenoveralljointdistributionswithmarginalsPandQ.AlthoughWassersteinmetricisnotamemberinthegeneral-structureprobabilitymetricsclass,bytheKantorovich-Rubinsteintheorem [ 33 ] ,theKantorovichmetricis equivalent totheWassersteinmetric. Inparticular,when =R, dW(P,Q)=Z+1jF(x))]TJ /F3 11.955 Tf 11.95 0 Td[(G(x)jdx, (4) whereF(x)andG(x)arethedistributionfunctionsderivedfromPandQrespectively. ThisconclusionholdsfollowingtheargumentthatinffE[(X,Y)]:P=L(X),Q=L(Y)g=R10jF)]TJ /F8 7.97 Tf 6.59 0 Td[(1(t))]TJ /F3 11.955 Tf 12.13 0 Td[(G)]TJ /F8 7.97 Tf 6.59 0 Td[(1(t)jdt,asstatedinTheorem6.0.2in[ 3 ]andR10jF)]TJ /F8 7.97 Tf 6.58 0 Td[(1(t))]TJ /F3 11.955 Tf 12.23 0 Td[(G)]TJ /F8 7.97 Tf 6.59 0 Td[(1(t)jdt=R+1jF(x))]TJ /F3 11.955 Tf 12.23 0 Td[(G(x)jdx. Wassersteinmetricalsohaswideapplicationsontransportation problems [ 52 ]. 4.2.2RelationshipsamongMetricsInthissubsection,weexploretherelationshipsamongthemembersof-structureprobabilitymetrics.Basedontherelationships,wedemonstratethattheTotalVariationmetricisthedominatingmetricifthesupportingspaceisbounded.ByprovingthattheTotalVariationmetricisequaltotheL1metricwhentheprobabilitydensityexists,wecanalsoclaimthattheL1metricisthedominatingmetricofthe-structureprobabilitymetricsclass.First,weexploretherelationshipsbetweentheTotalVariationmetricandtheWassersteinmetric.WedenoteBasthediameterof,andwehavethefollowinglemma. 69

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Lemma1. TherelationshipsbetweentheTotalVariationmetricandtheWassersteinmetricareasfollows: Ifisbounded,2dW(P,Q)BdTV(P,Q); Ifisaniteset,dmindTV(P,Q)2dW(P,Q),wheredmin=minx6=y(x,y). Proof. SeethedetailedproofinTheorem4in[ 24 ]. Next,westudytherelationshipsamongtheBoundedLipschitzmetric,theWassersteinmetricandtheTotalVariationmetric.SincekhkBL:=khkL+khk1,thefeasibleregionofhfortheBoundedLipschitzmetricismorerestrictivethantheonefortheKantorovichmetricandtheonefortheTotalVariationmetric.Thereforewehavethefollowinglemma: Lemma2. TherelationshipsamongtheBoundedLipschitz,Wasserstein,andTotalVariationmetricsareasfollows: dBL(P,Q)dW(P,Q), dBL(P,Q)dTV(P,Q).Moreover,therelationshipbetweentheTotalVariationmetricandtheUniformmetriccanbeobtainedbythedenitionsofeachmetric.TheTotalVariationmetricistondasetBamongalltheBorelsetof(),tominimizethevalueof2jP(B))]TJ /F16 11.955 Tf 12.19 0 Td[(Q(B)j.ButtheUniformmetricistondasetBamongallthesetsintheform(,x],tominimizethevalueofjP(B))]TJ /F16 11.955 Tf 11.96 0 Td[(Q(B)j.Sinceset(,x]isaborelset,wehave: Lemma3. TherelationshipbetweentheTotalVariationmetricandtheUniformmetricis:2dU(P,Q)dTV(P,Q).Finally,weexploretherelationshipsbetweentheWassersteinmetricandtheFortet-Mouriermetric.Fornotationbrevity,weletL=maxf1,Bp)]TJ /F8 7.97 Tf 6.59 0 Td[(1g.Wehavethefollowinglemma: Lemma4. TherelationshipsbetweentheWassersteinmetricandtheFortet-Mouriermetricareasfollows: 70

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dW(P,Q)dFM(P,Q), dFM(P,Q)LdW(P,Q). Proof. TherstinequalityisobviousbyfollowingthedenitionsoftheWassersteinmetricandtheFortet-Mouriermetric.Forthesecondinequality, fortheFortet-Mouriermetric, wehavejh(x))]TJ /F3 11.955 Tf 11.34 0 Td[(h(y)jc(x,y)=(x,y)maxf1,(x,a)p)]TJ /F8 7.97 Tf 6.58 0 Td[(1,(y,a)p)]TJ /F8 7.97 Tf 6.58 0 Td[(1g(x,y)L, wheretheequationholdsfollowingthedenitionofc(x,y) . Now wecanobserve dFM(P,Q)suph:jh(x))]TJ /F4 7.97 Tf 6.59 0 Td[(h(y)jL(x,y)jZhdP)]TJ /F10 11.955 Tf 11.95 16.28 Td[(ZhdQj=Lsuph:jh(x))]TJ /F4 7.97 Tf 6.59 0 Td[(h(y)jL(x,y)jZh LdP)]TJ /F10 11.955 Tf 11.95 16.27 Td[(Zh LdQj=Lsupg:jg(x))]TJ /F4 7.97 Tf 6.58 0 Td[(g(y)j(x,y)jZgdP)]TJ /F10 11.955 Tf 11.95 16.27 Td[(ZgdQj=LdW(P,Q).Thenthesecondinequalityholds. Tosummarize,therelationshipsamongthemembersof-structureprobabilitymetricsclassareshowninFigure 4-1 . . . K . W . BL . TV . U . FM . 1 . 1 . B=2 . 2=dmin . 1 . 1 . 2 . L . 1 Figure4-1. Relationshipsamongmembersof-structureprobabilitymetricsclass BasedonLemmas 1 to 4 ,weconcludethefollowingpropositionwithoutproof. 71

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Proposition4.1. Ifthesupportisbounded,theKantorovichmetric,Fortet-Mouriermetric,BoundedLipschitzmetricandKolmogorovmetricaredominatedbytheTotalVariationmetric.Moreover,ifthedensityfunctionscorrespond ing toPandQexist(denoteasf(x)andg(x)),thenwecandenetheL1metricofthedensitiesf,gasdL1(f,g):=RRdjf(x))]TJ /F3 11.955 Tf 12.52 0 Td[(g(x)jdx.Inthefollowinglemma,weshowtherelationshipbetweentheL1metricandtheTotalVariationmetric. Lemma5. dL1(f,g)=dTV(P,Q). Proof. First,lettingA:=fx2:f(x)>g(x)g,wehave RRdjf(x))]TJ /F3 11.955 Tf 11.95 0 Td[(g(x)jdx=ZAjf(x))]TJ /F3 11.955 Tf 11.95 0 Td[(g(x)jdx+ZAcjf(x))]TJ /F3 11.955 Tf 11.96 0 Td[(g(x)jdx=ZA(f(x))]TJ /F3 11.955 Tf 11.96 0 Td[(g(x))dx+ZAc(g(x))]TJ /F3 11.955 Tf 11.96 0 Td[(f(x))dx=2ZA(f(x))]TJ /F3 11.955 Tf 11.96 0 Td[(g(x))dx=2(P(A))]TJ /F16 11.955 Tf 11.96 0 Td[(Q(A)).ThelaststepisbecauseofRRd(f(x))]TJ /F3 11.955 Tf 11.95 0 Td[(g(x))dx=0.Second,foranyB2(), jP(B))]TJ /F16 11.955 Tf 11.96 0 Td[(Q(B)j=jZBf(x)dx)]TJ /F10 11.955 Tf 11.96 16.28 Td[(ZBg(x)dxj=jZB\A(f(x))]TJ /F3 11.955 Tf 11.96 0 Td[(g(x))dx+ZB\Ac(f(x))]TJ /F3 11.955 Tf 11.95 0 Td[(g(x))dxjmaxfZB\A(f(x))]TJ /F3 11.955 Tf 11.95 0 Td[(g(x))dx,ZB\Ac(g(x))]TJ /F3 11.955 Tf 11.96 0 Td[(f(x))dxgmaxfZA(f(x))]TJ /F3 11.955 Tf 11.96 0 Td[(g(x))dx,ZAc(g(x))]TJ /F3 11.955 Tf 11.95 0 Td[(f(x))dxg=1 2Zjf(x))]TJ /F3 11.955 Tf 11.96 0 Td[(g(x)jdx.Therefore,wehavedTV(P,Q)=dL1(f,g). Withtheabovelemmasandproposition,weexplorethemethodologiestosolveRA-SP(i.e.,Problem( 4 ))andstudytheconvergenceratesinthenextsection. 72

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4.3SolutionMethodologyInthissection,weinvestigatethesolutionmethodologies tosolve thedata-drivenrisk-aversetwo-stagestochasticprogram(i.e.,RA-SP asdescribedin( 4 ) ).Fortheoreticalconvenience,weassumethesupportingspaceisbounded,whichisnotarestrictiveassumptioninpractice.Thesectionisorganizedbyansweringthefollowingimportantquestions: 1. HowtodeterminethereferencedistributionP0? 2. Howtorepresentthevalueofdependingontheamountofhistoricaldata,i.e.,theconvergencerate? 3. Howtosolvetheproblemwithrespecttodifferent-structureprobabilitymetrics?Weanswerthesequestionsbyconsideringtwocases:(i)thetrueprobabilitydistributionisdiscreteand(ii)thetrueprobabilitydistributioniscontinuous. 4.3.1DiscreteCaseForthecaseinwhichthetruedistributionisdiscrete,werstconsiderthecasethatthesupportingspaceisnite,e.g.,f1,2,,Ng.Forthecasethatthesupportingspaceisinnite,wecanusetheframeworkofcontinuousprobabilitydistributioncase,whichwewilldiscusslater.GivenMhistoricaldata10,20,,M0,toestimatethereferencedistributionP0,weconsidertheempiricaldistributionofhistoricaldata,i.e.,thecumulativedistributionfunction(CDF)thatputsmass1=Mateachdatapointi0,i=1,M.Formally,theempiricaldistributionisdenedas P0(x)=1 MMXi=1i0(x), (4) wherei0(x)isanindicatorvariableequalto1ifi0xand0otherwise.Inthiscase,sincethesupportingspaceisdiscrete,thereferencedistributionP0canberepresentedbyitsmassprobabilityp10,p20,,pN0,wherepi0isequaltothenumberofhistoricaldatamatchingidividedbyM. 73

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AfteridentifyingthereferencedistributionP0,wediscussthevalueof,i.e.,theconvergencerateoftheempiricaldistributiontothetruedistributionofthediscretecase.WerststudytheconvergencerateoftheTotalVariationmetricforthediscretedistribution,andthenweexploretheconvergenceratesofothermetricsintheclassforamoregeneralcase,i.e.,thetruedistributionisnotlimitedtobediscrete(theseresultswillbeusedforlateroncontinuouscaseconvergencerateanalysis).FortheTotalVariationmetric,ifthetruedistributionisdiscrete,Pinsker'sinequality[ 17 ]showsdTV(P0,P)p dKL(PjP0),wheredKL(PjP0)isthediscretecaseKL-divergencedenedasPiln(p0i pi)p0i.Since[ 50 ]showsthatdKL(PjP0)convergesinmeasuretoa-squaredistributedrandomvariableasnumberofhistoricaldataMgoestoinnity,wecanclaimthattheconvergencerateofdTV(P0,P)isboundedbythatofadistributedrandomvariableasMgoestoinnity.Meanwhile,noteherethatforthecaseinwhichthetruedistributioniscontinuous,thetotalvariationmetricdoesnotconvergeasdescribedin[ 21 ],whichisalsothereasonthattheKLdivergencedoesnotconvergeasindicatedinSection1.Next,weexploretheconvergencerateoftheKolmogorovmetricforthegeneraldistributioncase(eitherdiscreteorcontinuous).TheconvergencepropertyandtheconvergenceratecanbeobtainedbyutilizingthefollowingDvoretzky-Kiefer-Wolfowitzinequality. Proposition4.2(Dvoretzky-Kiefer-Wolfowitz). Forthed=1case,Pr(dU(P0,P))1)]TJ /F5 11.955 Tf 12.18 0 Td[(2e)]TJ /F8 7.97 Tf 6.59 0 Td[(2M2.Forthed>1case,forany>0,thereexistsaconstantnumberCsuchthatPr(dU(P0,P))1)]TJ /F3 11.955 Tf 11.95 0 Td[(Ce)]TJ /F8 7.97 Tf 6.58 0 Td[((2)]TJ /F12 7.97 Tf 6.59 0 Td[()M2.Remark:Atighterratehasbeenobtainedin[ 1 ]forthed>1caseinwhichPr(dU(P0,P))1)]TJ /F3 11.955 Tf 11.96 0 Td[(C2(d)]TJ /F8 7.97 Tf 6.59 0 Td[(1)Md)]TJ /F8 7.97 Tf 6.58 0 Td[(1e)]TJ /F8 7.97 Tf 6.59 0 Td[(2M2,forsomeC0.Finally,weexploretheconvergencerateoftheWassersteinmetric(noteheretheratealsoappliestotheKantorovichmetric)forthegeneraldistributioncase,and 74

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thereforewecanobtaintheconvergenceratesoftheFortet-MourierandBoundedLipschitzmetricsaccordingly. Proposition4.3. Ifd=1,thenPr(dW(P0,P))1)]TJ /F5 11.955 Tf 11.96 0 Td[(2e)]TJ /F8 7.97 Tf 6.59 0 Td[(2M2. Proof. Ford=1,basedonthedenitionoftheWassersteinmetric,dW(P0,P)=R+1jF(x))]TJ /F3 11.955 Tf 12.42 0 Td[(G(x)jdx,whereFandGaredistributionfunctionsderivedfromP0andPrespectively.AccordingtotheHolder'sinequalitykfgk1kfkpkgkqwith1=p+1=q=1,lettingf=F(x))]TJ /F3 11.955 Tf 11.95 0 Td[(G(x),g=1,p=1,andq=1,wehave Z+1jF(x))]TJ /F3 11.955 Tf 11.95 0 Td[(G(x)jdxsupxjF(x))]TJ /F3 11.955 Tf 11.95 0 Td[(G(x)j.Moreover,accordingtotheDvoretzky-Kiefer-WolfowitzinequalitydescribedinProposition 4.2 ,wehavePr(supxjF(x))]TJ /F3 11.955 Tf 12.14 0 Td[(G(x)j)1)]TJ /F5 11.955 Tf 12.14 0 Td[(2e)]TJ /F8 7.97 Tf 6.59 0 Td[(2M2.Therefore,Pr(dW(P0,P))1)]TJ /F5 11.955 Tf 11.96 0 Td[(2e)]TJ /F8 7.97 Tf 6.59 0 Td[(2M2. WethencanimmediatelyobtaintheconvergenceratesfortheFortet-MourierandBoundedLipschitzmetricsforthed=1case,followingtherelationshipsamongthe-structureprobabilitymetricsasdescribedinSubsection 4.2.2 . Corollary1. Ifd=1,thenPr(dFM(P0,P))1)]TJ /F5 11.955 Tf 12.26 0 Td[(2e)]TJ /F13 5.978 Tf 7.78 3.26 Td[(2M L22andPr(dBL(P0,P))1)]TJ /F5 11.955 Tf 11.96 0 Td[(2e)]TJ /F8 7.97 Tf 6.59 0 Td[(2M2.Next,weprovethatforageneraldimensiond,P0alsoconvergestoPexponentiallyfastwithrespecttotheWassersteinmetric.In[ 10 ],ithasshownfor=RK,theconvergencerateoftheempiricaldistributiontothetruedistributionis P(dW(P0,P))1)]TJ /F5 11.955 Tf 11.96 0 Td[(exp()]TJ /F6 11.955 Tf 10.5 8.08 Td[( 2M2),forsome0.Motivatedbythis,wenowexploreanexactconvergencerateforaboundedspace.Itisshownin“Particularcase5”in[ 11 ]that dW(,P0)diam()p 2dKL(jP0) (4) 75

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holdsfor82P(),whereP()isthesetofallprobabilitymeasuresdenedon.Withinequality( 4 ),wehavethefollowingproposition. Proposition4.4. Foragenerald1,wehave P(dW(P0,P))1)]TJ /F5 11.955 Tf 11.96 0 Td[(exp()]TJ /F6 11.955 Tf 15.87 8.08 Td[(2 2B2M). Proof. Letset B:=f2P():dW(,P)g(4)andC()bethecollectionofallboundedcontinuousfunctions:!R,withthesupremumnorm,i.e.,kk1=supxj(x)j.Therefore,wehave Pr(dW(P,P0))=Pr(P02B)PrZdP0inf2BZdexp)]TJ /F3 11.955 Tf 9.3 0 Td[(Minf2BZdEeMRdP0 (4) =exp)]TJ /F3 11.955 Tf 9.3 0 Td[(Minf2BZd)]TJ /F5 11.955 Tf 15.76 8.09 Td[(1 MlogEeMRdP0=exp)]TJ /F3 11.955 Tf 9.3 0 Td[(Minf2BZd)]TJ /F5 11.955 Tf 15.76 8.09 Td[(1 MlogEePMi=1(i) (4) =exp)]TJ /F3 11.955 Tf 9.3 0 Td[(Minf2BZd)]TJ /F5 11.955 Tf 11.95 0 Td[(logZedP0, (4) for82C(),whereinequality( 4 )followsfromtheChebyshev'sexponentialinequality,equation( 4 )followsfromthedenitionofP0,andequation( 4 )followsfromtheassumptionthatthehistoricaldataarei.i.d.drawn.Considering():=sup2C()Rd)]TJ /F5 11.955 Tf 12.87 0 Td[(logRedP0,followingthecontinuityandboundednessfromthedenitionofC(),thereexistaseriesnsuchthatlimn!1Rnd)]TJ /F5 11.955 Tf 12.58 0 Td[(logRendP0=().Therefore,foranysmallpositivenumber0>0,thereexistsanN0suchthat())]TJ /F10 11.955 Tf 12.35 9.63 Td[(Rnd)]TJ /F5 11.955 Tf 12.36 0 Td[(logRendP00foranynN0.Therefore,accordingto( 4 )bysubstitutingwithn,wehave Pr(P02B)exp)]TJ /F3 11.955 Tf 9.3 0 Td[(Minf2BZnd)]TJ /F5 11.955 Tf 11.96 0 Td[(logZendP0 76

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exp)]TJ /F3 11.955 Tf 9.3 0 Td[(Minf2B(())]TJ /F6 11.955 Tf 11.96 0 Td[(0).AccordingtoLemma6.2.13in[ 19 ],weknowthat()=dKL(jP0).Forthecase2B,followingthedenitionofsetBin( 4 ),wehavedW(,P0).Inaddition,following( 4 )wehavedKL(jP0)2=(2B2).Therefore, Pr(P02B)exp)]TJ /F3 11.955 Tf 9.3 0 Td[(M2 2B2)]TJ /F6 11.955 Tf 11.95 0 Td[(0.Let0==Mforanyarbitrarysmallpositive,then Pr(dW(P,P0))=Pr(P02B)exp)]TJ /F6 11.955 Tf 15.86 8.08 Td[(2 2B2M+.Sinceisarbitrarilysmall,wehaveP(dW(P0,P))1)]TJ /F5 11.955 Tf 11.96 0 Td[(exp()]TJ /F12 7.97 Tf 14.23 4.71 Td[(2 2B2M). Similarly,theconvergenceratealsoappliesfortheKantorovichmetric,andmoreover,wecanobtaintheconvergenceratesfortheFortet-MourierandBoundedLipschitzmetricsforthegeneraldimensioncaseasfollows. Corollary2. Foragenerald1,wehaveP(dFM(P0,P))1)]TJ /F5 11.955 Tf 12.19 0 Td[(exp()]TJ /F12 7.97 Tf 18.55 4.7 Td[(2 2B2L2M)andPr(dBL(P0,P))1)]TJ /F5 11.955 Tf 11.96 0 Td[(exp()]TJ /F12 7.97 Tf 14.23 4.7 Td[(2 2B2M).Withtheconvergencerates,wecancalculatethevalueofaccordingly.TaketheconvergenceratefortheWassersteinmetricobtainedbyProposition 4.4 forexample.Assumingthecondencelevelissettobe,thatis,P(dW(P0,P))1)]TJ /F5 11.955 Tf 12.86 0 Td[(exp()]TJ /F12 7.97 Tf 13.84 4.7 Td[(2 2B2M)=,thenwecanobtain=q 2Blog(1 1)]TJ /F12 7.97 Tf 6.59 0 Td[()=M.Similarly,wecancalculatethevalueoffordifferentmetricsbasedonCorollary 2 .Next,weexplorethemethodologytosolvetheproblem.Assuming=f1,2,,Ng,RA-SP(i.e.,Problem( 4 ))canbereformulatedas: minxcTx+maxpiNXi=1piQ(x,i)s.t.Xipi=1, (4) 77

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maxhiNXi=1hip0i)]TJ /F4 7.97 Tf 17.29 14.94 Td[(NXi=1hipi,8hi:khk1. (4) Forkhk1,accordingly,wehave Kantorovich:jhi)]TJ /F3 11.955 Tf 11.96 0 Td[(hjj(i,j),8i,j, Fortet-Mourier:jhi)]TJ /F3 11.955 Tf 11.96 0 Td[(hjj(i,j)maxf1,(i,a)p)]TJ /F8 7.97 Tf 6.59 0 Td[(1,(j,a)p)]TJ /F8 7.97 Tf 6.59 0 Td[(1g,8i,j, Bounded-Lipschitz:jhi)]TJ /F3 11.955 Tf 11.96 0 Td[(hjj(i,j),jfij1,8i,j, TotalVariation:jhij1,8i.Fortheabovefourmetrics,theconstraintscanbesummarizedasPiaijhibj,j=1,,J.First,wedevelopthereformulationofconstraint( 4 ).Consideringtheproblem maxhiNXi=1hip0i)]TJ /F4 7.97 Tf 17.3 14.94 Td[(NXi=1hipis.t.NXiaijhibj,j=1,,J,wecangetitsdualformulationasfollows: minJXj=1bjujs.t.JXj=1aijujp0i)]TJ /F3 11.955 Tf 11.96 0 Td[(pi,8i=1,,N,whereuisthedualvariable.Therefore,forthediscretedistributioncase,RA-SPcanbereformulatedasfollows(denotedbyFR-M): minxcTx+maxpiNXi=1piG(x,i)s.t.NXi=1pi=1,JXj=1bjuj, 78

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JXj=1aijujp0i)]TJ /F3 11.955 Tf 11.95 0 Td[(pi,8i=1,,N.Fortheuniformmetric,wecanobtainthereformulationdirectlyfromthedenitionofUniformmetric(denotedbyFR-U): mincTx+maxpiNXi=1piG(x,i)s.t.Xipi=1,jjXi=1p0i)]TJ /F3 11.955 Tf 11.95 0 Td[(pij,8j=1,,J.Next,wesummarizethealgorithmtosolvethecaseinwhichthetruedistributionisdiscrete. Algorithm1Algorithmforthediscretecase Input:Historicaldata1,2,,Ndrawni.i.d.fromthetruedistributionandcondencelevel.Output:Theobjectivevalueoftherisk-averseproblemRA-SP.Step1:ObtainthereferencedistributionP0(x)=1 NPNi=1i(x)andthevalueofbasedonthehistoricaldata.Step2:UsethereformulationFR-MorFR-Utosolvetheproblem.Step3:Outputthesolution. 4.3.2ContinuousCaseInthissection,wediscussthecaseinwhichthetruedistributioniscontinuous.Similarly,wedevelopthemethodologiestosolvethecontinuouscasebyansweringtheabovethreequestionsrespectively.Forthecontinuouscase,weconsiderthereferenceprobabilitydensityfunction(pdf)insteadofthereferencecumulativedistributionfunction(cdf),andweuseKernelDensityEstimationtoconstructthereferencepdf.Assumingweobservensamples1,2,,n,whicharedrawnfromthetruedistribution,thekerneldensityfunctionisdenedas fn(x)=1 nhdnnXi=1K(x)]TJ /F6 11.955 Tf 11.96 0 Td[(i hn), (4) 79

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whereK(x)=Qdk=1k(xk),inwhichkisaBorelmeasurablefunction(kernel)withdimension1,satisfyingk0,RRk(x)dx=1.NoteherethatK(x)alsosatisesK0,RRdK(x)dx=1.Byusingthekerneldensityestimation,wecanseethatthereferenceprobabilitydistributionP0derivedfromthedensityfunctionfn(x)isacontinuousdistribution.Next,weanalyzetheconvergencepropertiesandtheconvergenceratesofthereferencedistributiontothetruedistribution,correspondingtodifferentmetricsin-structureprobabilitymetricsclass,byusingkerneldensityestimation.Undermildassumptions,wecanprovethatthereferencedensitydistributionP0derivedfromfnconvergestothetrueprobabilitydistributionexponentiallyfast,underdifferentmetrics. Proposition4.5. Iflimn!1hn=0andlimn!1nh2dn=1,thenforall>0,Pr(jfn(x))]TJ /F3 11.955 Tf -446.23 -23.91 Td[(E[fn(x)]j1>)2expf)]TJ /F3 11.955 Tf 15.28 0 Td[(nh2dn2=(2V2)gforeveryx,whereV=supyK(y). Proof. WeletZi=K(x)]TJ /F12 7.97 Tf 6.59 0 Td[(i hn)=hdn,thenbasedonthedenitionofkerneldensityfunction( 4 ),wehaveE[fn(x)]=E[1 nPni=1Zi]=E[Zi]sincethati,i=1,,Naredrawni.i.d..Besides,itiseasytoobservethat jZij=jK(x)]TJ /F6 11.955 Tf 11.95 0 Td[(i hn)=hdnjV hdn. (4) AccordingtotheHoeffding'sinequality,wehave Pr(jfn(x))]TJ /F3 11.955 Tf 11.96 0 Td[(E[fn(x)]j>)=Pr(j1 nnXi=1Zi)]TJ /F3 11.955 Tf 11.95 0 Td[(E[Zi]j>)2expf)]TJ /F5 11.955 Tf 15.28 0 Td[(2n2=(2V hdn)2g=2expf)]TJ /F3 11.955 Tf 15.28 0 Td[(nh2dn2=(2V2)g.Thusweconcludetheproof. Proposition4.6. Ifthetruedensityfunctionf(x)isboundedandcontinuousalmosteverywhere,thenlimn!1jE[fn(x)])]TJ /F3 11.955 Tf 11.96 0 Td[(f(x)j1=0. 80

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Proof. Ononehand,wehaveE(fn(x))=RRdh)]TJ /F4 7.97 Tf 6.58 0 Td[(dK(x)]TJ /F4 7.97 Tf 6.58 0 Td[(y h)f(y)dy=RRdK(u)f(x)]TJ /F3 11.955 Tf 12.13 0 Td[(uh)du.Ontheotherhand,basedonthedenitionofK(u),wehaveRRdK(u)du=1,andthereforef(x)=RRdK(u)f(x)du.Then, jE[fn(x)])]TJ /F3 11.955 Tf 11.96 0 Td[(f(x)j=ZRdK(u)(f(x)]TJ /F3 11.955 Tf 11.95 0 Td[(uh))]TJ /F3 11.955 Tf 11.95 0 Td[(f(x))duZkukAK(u)(f(x)]TJ /F3 11.955 Tf 11.96 0 Td[(uh))]TJ /F3 11.955 Tf 11.96 0 Td[(f(x))du+ZkukAK(u)(f(x)]TJ /F3 11.955 Tf 11.95 0 Td[(uh))]TJ /F3 11.955 Tf 11.95 0 Td[(f(x))du, (4) foranyvalueA0.Fortheseconditemin( 4 ),wecanobservethat ZkukAK(u)(f(x)]TJ /F3 11.955 Tf 11.95 0 Td[(uh))]TJ /F3 11.955 Tf 11.95 0 Td[(f(x))du2supx2Rdf(x)ZkukAK(u)du.Inaddition,sinceK(u)0andRu2RdK(u)du=1,wecanalwayschooseAlargeenoughtomakesupx2Rdf(x)jRkukAK(u)dujlessthananyarbitrary.Fortherstitemof( 4 ),wehave ZkukAK(u)(f(x)]TJ /F3 11.955 Tf 11.96 0 Td[(uh))]TJ /F3 11.955 Tf 11.96 0 Td[(f(x))dusupx2RdjK(x)jZkukA(f(x)]TJ /F3 11.955 Tf 11.95 0 Td[(uh))]TJ /F3 11.955 Tf 11.96 0 Td[(f(x))du.Sincefisboundedandcontinuousalmosteverywhere,thereexistsaconstantCsuchthatjf(x)]TJ /F3 11.955 Tf 12.43 0 Td[(uh))]TJ /F3 11.955 Tf 12.43 0 Td[(f(x)jCkuhkalmosteverywhere.Withoutconsideringthesetwithmeasure0,wehavejRkukAK(u)(f(x)]TJ /F3 11.955 Tf 12.19 0 Td[(uh))]TJ /F3 11.955 Tf 12.18 0 Td[(f(x))dujCAhsupx2RdjK(x)j.Similarly,foragivenA,CAhsupx2RdjK(x)jcanbelessthananyarbitrarywhenhisverysmall.SojE[fn(x)])]TJ /F3 11.955 Tf 11.95 0 Td[(f(x)j1!0whenhgoestozero,soasngoestoinnity. Withtheabovetwopropositions,wecanestimatetheconvergenceratefortheTotalVariationmetricasfollows: Theorem4.1. Ifthetruedensityfunctionf(x)isboundedandcontinuousalmosteverywhere,limn!1hn=0,andlimn!1nh2dn=1,thenforall>0,thereexistsn0suchthat Pr(dTV(P0,P)>)2expf)]TJ /F3 11.955 Tf 15.27 0 Td[(nh2dn2=(2V2)g (4) 81

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fornn0,whereV=supyK(y)andP0,Pareprobabilitydistributionsderivedfromfnandfrespectively. Proof. AccordingtotheHolder'sinequality,wehavedL1(fn,f)supxjfn(x))]TJ /F3 11.955 Tf 12.69 0 Td[(f(x)j.AccordingtoLemma 5 ,wehavedL1(fn,f)=dTV(P0,P).BasedonPropositions 4.5 and 4.6 ,weprovethetheorem. WithTheorem 4.1 ,itiseasytoderivetheconvergenceratesforothermetricsbasedonLemmas 1 to 4 . Corollary3. Ifthetruedensityfunctionf(x)isboundedandcontinuousalmostevery-where,limn!1hn=0,andlimn!1nhdn=1,thenforall>0,thereexistsann0suchthat Pr(dW(P0,P)>)2expf)]TJ /F3 11.955 Tf 15.27 0 Td[(nh2dn2=(2B2V2)g, (4) Pr(dBL(P0,P)>)2expf)]TJ /F3 11.955 Tf 15.28 0 Td[(nh2dn2=(2V2)g, (4) Pr(dFM(P0,P)>)2expf)]TJ /F3 11.955 Tf 15.28 0 Td[(nh2dn2=(2(BLV)2)g, (4) Pr(dU(P0,P)>)2expf)]TJ /F5 11.955 Tf 15.28 0 Td[(2nh2dn2=V2g, (4) fornn0,whereV=supyK(y)andP0,Pareprobabilitydistributionsderivedfromfnandfrespectively.Giventhatthereferencedistributionconvergestothetruedistributionexponentiallyfast,next,weexploretheconvergencepropertyoftheoptimalsolutionandtheoptimalobjectivevalueofRA-SPtotheoptimalsolutionandtheoptimalobjectivevalueofSP,i.e.,Problem( 4 ).Wehavethefollowingtheorem. Theorem4.2. TheoptimalsolutionandtheoptimalobjectivevalueofRA-SP,i.e.,Problem( 4 ),convergetotheoptimalsolutionandtheoptimalobjectivevalueofSP,i.e.,Problem( 4 ),respectively. Proof. AccordingtoProposition 4.4 andCorollary 2 ,weknowasthehistoricaldatasizeMgoestoinnity,thetruedistributionPconvergestoP0inprobability.Sinceas 82

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Mgoestoinnity,goestozero,sotheworst-caseprobabilitydistribution,denotedasP0,convergestoP0.Therefore,theworst-casedistributionP0convergestothetruedistributionPinprobability.Sinceisbounded,thenforanygivenx2X,thefunctionQ(x,)isboundedandcontinuouswith.AccordingtotheHelly-Braytheorem,wecanclaimthatforanygivenx2X, limM!1supP:d(P,P0)EP[Q(x,)]=limM!1EP0[Q(x,)]=EP[Q(x,)]. (4) Withequation( 4 ),werstexploretheconvergencepropertyofoptimalvalues.Werepresent^vastheoptimalvalueand^xastheoptimalsolutionofthefollowingproblem: minxcTx+limM!1supP:d(P,P0)EP[Q(x,)] (4) s.t.x2X.Besides,werepresentvastheoptimalvalueandxastheoptimalsolutionofproblemSP(i.e.,problemasdescribedin( 4 )).Thenweneedtoprovethat^v=v.Duetothefactthat^vv,iftheequation^v=vdoesnothold,wehave^v>v.AccordingtoEquation( 4 ),wecanobservethat cTx+limM!1supP:d(P,P0)EP[Q(x,)]=cTx+EP[Q(x,)]. (4) Therefore, cT^x+limM!1supP:d(P,P0)EP[Q(^x,)]=^v>v=cTx+EP[Q(x,)]=cTx+limM!1supP:d(P,P0)EP[Q(x,)], (4) 83

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whichviolatesthat^xistheoptimalsolutiontoproblem( 4 ).Consequently,wehave^v=v,whichindicatestheconvergencepropertyofoptimalvalues.Besides,sincetheoptimalsolutionofproblem( 4 )convergesto^x,and^xisalsoanoptimalsolutionofproblem( 4 )dueto^v=v,wecanclaimthattheoptimalsolutionofproblem( 4 )convergestotheoptimalsolutionofproblem( 4 ). Next,wederiveasamplingapproachtosolveRA-SP(i.e.,Problem( 4 ))forthecaseinwhichthetrueprobabilitydistributioniscontinuous,withanymetricFin-structureprobabilitymetricsclass.Wedenote f(x)=cTx+supP:dF(P,P0)EP[Q(x,)], (4) whereP0istheprobabilitydistributionderivedfromfnasindicatedin( 4 ),andcanbecalculatedbyusingTheorem 4.1 andCorollary 3 .Weconsiderthefollowingtwoproblems: H+(x,)=cTx+supFN:dF(FN,F0N)+EFN[Q(x,)], (4) H)]TJ /F5 11.955 Tf 7.08 -4.94 Td[((x,)=cTx+supFN:dF(FN,F0N))]TJ /F12 7.97 Tf 6.59 0 Td[(EFN[Q(x,)], (4) whereF0NistheempiricaldistributionofNsamples10,20,,N0drawnfromthereferencedistributionP0,andFNisanydiscretedistributionof10,20,,N0.Fornotationbrevity,letF(,N)representtheconvergenceratecorrespondstodifferentmetricFobtainedbyProposition 4.4 andCorollary 2 .Wehavethefollowingtheorem: Proposition4.7. Foragivenrststagedecisionx,H)]TJ /F5 11.955 Tf 7.08 -4.34 Td[((x,)isalowerboundoff(x)withprobability1)]TJ /F6 11.955 Tf 12.78 0 Td[(F(,N)andH+(x,)isanupperboundoff(x)withprobability1)]TJ /F6 11.955 Tf 11.96 0 Td[(F(,N). Proof. Forthelowerboundpart,werstassumethatFNistheworst-casedistributionofProblem( 4 ).ThenFNshouldsatisfytheconstraintdF(FN,F0N))]TJ /F6 11.955 Tf 12.77 0 Td[(.Also,basedonProposition 4.4 andCorollary 2 ,wehavedF(F0N,P0)withprobability 84

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1)]TJ /F6 11.955 Tf 12.47 0 Td[(F(,N).SincedFisametric,itsatisesthetriangleinequality,i.e.,dF(FN,P0)dF(FN,F0N)+dF(F0N,P0).ThuswehavedF(FN,P0)withprobability1)]TJ /F6 11.955 Tf 12.55 0 Td[(F(,N).Therefore, PrfH)]TJ /F5 11.955 Tf 7.09 -4.93 Td[((x,)f(x)g=PrfcTx+EFN[Q(x,)]cTx+supP:dF(P,P0)EP[Q(x,)]g=PrfEFN[Q(x,)]supP:dF(P,P0)EP[Q(x,)]gPrfFN2fP:dF(P,P0)gg1)]TJ /F6 11.955 Tf 11.96 0 Td[(F(,N).Byusingthesamelogic,wecanprovetheupperboundpart.Thedetailedproofisomittedhere. Fromthepropositionwecansee,ononehand,asdecreases,thelowerboundtendstoincrease,andtheupperboundtendstodecrease,andtheupperboundandlowerboundconvergeasgoesto0.Butnoticeherewhenisverysmall,weneedtondalargeNtoguaranteeahighprobability.Ontheotherhand,considertheproblem H(x)=cTx+supFN:dF(FN,F0N)EFN[Q(x,)] (4) andweknowH+(x,)H(x)H)]TJ /F5 11.955 Tf 7.08 -4.34 Td[((x,).Therefore,H(x)alsoconvergestotheobjectiveofRA-SPexponentiallyfast.ThenwecanalsosolveRA-SPbysolvingtheminimizationproblemof( 4 ).NowweanalyzetheupperboundandlowerboundofproblemRA-SP.Welet v+=minx2XH+(x,), (4) v)]TJ /F12 7.97 Tf -1.37 -7.89 Td[(=minx2XH)]TJ /F5 11.955 Tf 7.09 -4.94 Td[((x,), (4) andletvastheoptimalvalueofRA-SP.Wehavethefollowingtheorem: 85

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Theorem4.3. Prfv)]TJ /F12 7.97 Tf -1.38 -7.89 Td[(vg1)]TJ /F6 11.955 Tf 11.95 0 Td[(F(,N), (4) Prfvv+g1)]TJ /F6 11.955 Tf 11.95 0 Td[(F(,N). (4) Proof. Wejustverifythestatement( 4 ),andthenthestatement( 4 )holdsfollowingthesamelogic.Denotexandx)]TJ /F12 7.97 Tf -1.17 -7.29 Td[(aretheoptimalsolutionstotheproblemRA-SPandproblem( 4 ),respectively. Prfv)]TJ /F12 7.97 Tf -1.37 -7.89 Td[(vg=PrfH)]TJ /F5 11.955 Tf 7.09 -4.93 Td[((x)]TJ /F12 7.97 Tf -1.16 -7.89 Td[(,)f(x)gPrfH)]TJ /F5 11.955 Tf 7.09 -4.94 Td[((x,)f(x)g (4) 1)]TJ /F6 11.955 Tf 11.95 0 Td[(F(,N), (4) whereinequality( 4 )holdssincex)]TJ /F12 7.97 Tf -1.17 -7.29 Td[(istheoptimalsolutiontotheproblem( 4 )sothatH)]TJ /F5 11.955 Tf 7.08 -4.34 Td[((x)]TJ /F12 7.97 Tf -1.17 -7.29 Td[(,)H)]TJ /F5 11.955 Tf 7.09 -4.34 Td[((x,).Theinequality( 4 )isderivedbyusingProposition 4.7 . Nowwediscussthecalculationsofvaluesforthecontinuouscase.WecanfollowthesimilarlogicasdiscussedinSubsection 4.3.1 andrepresentasafunctionofcondencelevelandthenumberofhistoricaldataM,basedonTheorem 4.1 andCorollary 3 .Inaddition,iftheprobabilitiesfor( 4 )tobetheupperboundof( 4 )and( 4 )tobethelowerboundof( 4 )aregiven,wecanalsocalculatethenumberof,basedonProposition 4.4 andCorollary 2 .Next,weconcludethealgorithmtosolvethecaseinwhichthetruedistributioniscontinuousinAlgorithm2.Noteherethatforthecasethatthetruedistributionisdiscretebutthesupportingspaceisinnite,wecanemploythesamealgorithm,exceptusingtheempiricaldistributionasthereferencedistribution.Inthatcase,theconvergencerate(i.e.,thevalueof)canbeobtainedbyusingProposition 4.4 andCorollary 2 . 86

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Algorithm2Algorithmforthecontinuouscase Input:Historicaldata1,2,,Mdrawni.i.d.fromthetruedistributionandcondenceleveland.Output:Theestimatedobjectivevalueoftherisk-averseproblem.Step1:ObtainthekerneldensityfunctionfM(x)asshownin( 4 )andbasedonthehistoricaldata.Step2:Setgap=1000,N=0.Step3:Whilegap>,do1)N=N+N.2)SimulateNsamplesfromfM(x),andobtainbasedonProposition 4.4 andCorollary 2 .3)Obtaintheupperboundbysolving( 4 ).4)Obtainthelowerboundbysolving( 4 ).5)Obtaintheoptimalitygapbetweenthelowerandupperbounds.Step4:Outputthesolution. 4.4NumericalExperimentsInthissection,byusingthe-structureprobabilitymetricstoconstructthecondenceset,weexploretheperformancesforbothdiscreteandcontinuouscases.Weshowthesystemperformancebytwoexamples:NewsvendorproblemandFacilityLocationproblem. 4.4.1NewsvendorProblemInthissubsection,weuseatwo-stageNewsvendorproblemasanexampletoconductournumericalexperiments.Anewsvendorplacesanorderyofnewspapersonedayaheadataunitcostofc1.Onthesecondday,sheobservestherealdemanddandplacesasupplementalorderxataunitcostofc2.Ifc1c2,thenthenewsvendorcanputallherpurchaseordersonthesecondday,basedontheknowninformationofdemand.Forthiscase,thetwo-stagenewsvendorproblemcanbereducedtobeasingle-stageproblem.Ifc2p,thenthenewsvendorwillnotputanyorderontheseconddaytoavoidlossofmoney,andsimilarly,thetwo-stagenewsvendorproblemcanbereducedtobeasingle-stageproblem.Therefore,withoutlossofgeneralization,weassumec1
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problemcanbeformulatedasfollows: maxy0)]TJ /F3 11.955 Tf 9.3 0 Td[(c1y+minP:d(P,P0)EPmaxx0[pminfx+y,d()g)]TJ /F3 11.955 Tf 20.59 0 Td[(c2x], (4) wherecanbeanymetricinthe-structureprobabilitymetricsclass,andcanbecalculatedbasedonProposition 4.1 andCorollary 2 ifthetruedistributionisdiscreteandTheorem 4.4 andCorollary 3 ifthetruedistributioniscontinuous.Inthefollowing,byusingthenewsvendorproblemasanexample,weexploretheperformanceofourapproachforthediscretecase.Wesetc1=3,c2=4,andp=5.Weassumethedemandfollowsadiscretedistributionwithtwoscenarios:10and20eachwithprobabilities0.4and0.6respectively.Werststudytheeffectsofthenumberofobservedhistoricaldata,bysettingthecondenceleveltobe0.99andvaryingthenumberofhistoricaldatafrom10to10,000. Figure4-2. Effectsofhistoricaldata TheresultsareshowninFigure 4-2 .Fromthegure,wecanobservethat,nomatterwhatkindofmetricsweuse,asthenumberofhistoricaldataincreases,theobjectivevalueofproblemRA-SPtendstoincrease.Thisresultconformstothe 88

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intuition,becausethevalueofdecreasesasthenumberofhistoricaldataincreases.Therefore,accordingly,therisk-averseproblemRA-SPbecomeslessconservative.Wecanalsoobservethat,whenthenumberofhistoricaldataexceeds2,000,thegapsbetweentherisk-averseproblemRA-SPandrisk-neutralproblemSPareverysmall(lessthan0.2)undertheWassersteinandFortet-Mouriermetrics.Furthermore,whenthenumberofhistoricaldataexceeds5,000,thegapsbetweentherisk-averseproblemRA-SPandtherisk-neutralproblemSParesmallunderallmetrics.Next,weexploretheeffectsofcondencelevelontheobjectivevalueofRA-SP.Wesetthenumberofhistoricaldatatobe5,000andtestvedifferentcondencelevels:0.7,0.8,0.9,0.95,and0.99.WereporttheresultsinFigure 4-3 . Figure4-3. Effectsofcondencelevel Fromthegure,wecanobservethatasthecondencelevelincreases,thegapsbetweentherisk-averseproblemRA-SPandrisk-neutralproblemSPalsoincrease.Thisisduetothefactthat,asthecondencelevelincreases,thevalueofincreases.Thus,theproblembecomesmoreconservativeandthetrueprobabilitydistributionismorelikelytobeinthecondencesetD. 89

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4.4.2FacilityLocationProblemInthissubsection,toshowthesystemperformanceforthecontinuousdistributioncase,weusethefacilitylocationproblemasanexample,e.g.,acompanythatsuppliesindependentgrocerystoresdecideswhatfacilities,amonglocationsi=1,,M,toopen.EachfacilityiassociateswithaxedcostFiandacapacityCiifitisopen.Assumethatthecompanyhasastablesetofdemandsitesj=1,,N,whosedemandsareindependentlydistributed.Thereisashipmentcosttobringaunitproductfromfacilityitodemandsitej.Attherststage,thecompanyneedstodecidewhichfacilitiestoopen,andatthesecondstage,afterrealizingthedemanddjatsitej,thecompanyneedstodecidetheamountofproductstobeshippedfromfacilityitositej.Accordingly,thedata-drivenrisk-aversetwo-stagestochasticfacilitylocationproblemcanbeformulatedasfollows: minyMXi=1Fiyi+maxP2DEP"MXi=1NXj=1Tijxij()#s.t.NXj=1xijCiyi,i=1,,M,MXi=1xij=dj(),j=1,,N,yi2f0,1g,xij0,i=1,,M,j=1,,N.Weassumethereare10facilitylocationsand10demandsites.Foreachlocationi,thecapacityis15+iandthexedcostis100+i.Meanwhile,theunitshippingcostfromlocationitodemandsitejis5+0.008i.Inthisexperiment,wetestthecasesinwhichtherandomvariablefollowsaUniformdistribution,Normaldistribution,GammadistributionandWeibulldistribution,respectively.Similartothediscretecase,westudytheeffectofhistoricaldatarst.Wesetthenumberofsampleswetakefromthereferencedistributionas100,andcomputethe 90

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estimatedvalueoftheriskaverseproblem( 4 )describedas( 4 ).WereporttheobjectivevaluescorrespondingtovariousnumbersofhistoricaldatainFigure 4-5 . Figure4-4. Effectsofhistoricaldata FromFigure 4-4 wecanobservethat,foraxednumberofsamples,asthenumberofhistoricaldataincreases,theobjectivevaluestendtodecrease.Thatisbecauseasthenumberofhistoricaldataincreases,thevalueofdecreasesandtheproblembecomeslessconservative.Inaddition,wereportthefacilitiesthatarenotopenintherststageinTable 4.4.2 fortheWeibullmetricforcomparison.Fromthetablewecanobservethat,fordifferentmetricsanddifferentnumberofhistoricaldata,wehavedifferentrst-stagedecisions.Next,wesetthenumberofhistoricaldatatobe5000,andtesttheeffectsofthenumberofsamplesfortheproposedsamplingapproach.Weobtaintheupperand 91

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Metrics1010050010002000400060008000 K4,101,103,103,103,106,103,106,10BL4,101,103,103,103,107,93,106,10FM4,101,103,103,103,106,103,106,10U4,101,103,104,94,97,93,107,9TV5,91,104,94,93,106,103,107,9 Table4-1. Facilitiesthatarenotopen lowerbounds,andtheestimatedvaluescorrespondingtodifferentsamplenumbersrespectivelyandshowtheresultsinFigure 4-5 . Figure4-5. Effectsofsamples FromFigure 4-5 wecanobserve,nomatterwhatthetruedistributionis,asthenumberofsamplesincreases,thegapbetweentheupperboundandthelowerboundtendstodecrease. 92

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4.5SummaryInthischapter,weapplyanewclassofprobabilitymetrics,-structureprobabilitymetrics,toconstructthecondencesetofambiguousdistributions,bylearningfromthehistoricaldata.Basedonthis,wefurtherdevelopaframeworktosolvetherisk-aversetwo-stagestochasticprogramforbothcasesinwhichthetruedistributionisdiscreteandcontinuous,respectively.Wereformulatetherisk-averseproblemasatraditionalrobustoptimizationproblemforthediscretecase.Forthecontinuouscase,weproposeare-samplingapproachtoprovidethestatisticalupperandlowerboundsfortheoptimalobjectivevalueoftherisk-averseproblem.Inaddition,theseboundsareprovedtoconvergetotheoptimalobjectivevalueasthesamplesizeincreasestoinnity.Wealsoprovethatunder-structureprobabilitymetrics,therisk-averseproblemconvergestotherisk-neutraloneexponentiallyfastasthenumberofhistoricaldataincreasestoinnity.Theexperimentalresultsofnewsvendorandfacilitylocationproblemsshowtheeffectivenessoftheproposedapproachandnumericallyshowthevalueofdata. 93

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CHAPTER5DATA-DRIVENUNITCOMMITMENTPROBLEM 5.1ProblemDescriptionandLiteratureReviewInthischapter,wepresentadata-drivenrisk-aversestochasticoptimizationapproachtosolvetheunitcommitmentproblem.Thatis,insteadofknowingtheexactdistributionoftheuncertainparameter,wecanobserveaseriesofhistoricaldatawhicharedrawnfromthetruedistribution.Bylearningfromthehistoricaldata,weconstructareferencedistributionoftheuncertainparameteraswellasacondencesetofthetruedistribution,andmakedecisionsinconsiderationofthecasethatthetruedistributioncanvarywithinthecondenceset.Then,wedevelopadata-drivenrisk-aversetwo-stagestochasticunitcommitmentframeworkandproposethesolutionapproachestosolvethedevelopedframework.Ourcontributionscanbedescribedasfollows: 1. Unlikethetraditionaltwo-stagestochasticunitcommitmentproblem,wedonotassumethattheprobabilitydistributionsoftheuncertainparameters,e.g.,electricityloadandwindpoweroutput,areknown.Instead,weconstructcondencesetsoftheprobabilitydistributionsandproposethesolutionmethodologytosolvearisk-aversestochasticunitcommitmentproblem. 2. Weexaminetheconvergencepropertiesofthedata-drivenstochasticunitcommitmentframework,andwedemonstratethatasthenumberofhistoricaldataincreases,therisk-averseproblemconvergestotherisk-neutralproblem,i.e.,thetraditionaltwo-stagestochasticunitcommitmentproblem.Inthatcase,ourproblembecomeslessconservativeasmorehistoricaldataareobserved.Theremainderofthischapterisorganizedasfollows.InSection 5.2 ,wedescribethemathematicalformulationsofthetraditionalstochasticunitcommitmentproblemandthedata-drivenstochasticunitcommitmentproblemrespectively.InSection 5.3 ,wediscussthesolutionmethodologiestosolvethedata-drivenstochasticunitcommitmentproblem.InSection 5.4 ,weprovethatasthenumberofhistoricaldataincreases,thedata-drivenstochasticunitcommitmentproblemconvergestothetraditionalstochasticunitcommitmentproblem.InSection5,weconcludeourwork. 94

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5.2MathematicalFormulationsInthissection,wedenethenotationsandintroducetheformulationofthetraditionaltwo-stagestochasticunitcommitmentproblemrst.Thenbasedontheobservedhistoricaldata,weconstructthereferencedistributionandthecondencesetofthetrueprobabilitydistribution,bybringingtheconceptof“Wassersteinmetric”.Basedonthereferencedistributionandthecondenceset,wethenbuildtheformulationofthedata-drivenrisk-aversetwo-stagestochasticunitcommitmentproblem. 5.2.1StochasticUnitCommitmentProblemInthischapter,wehavetwodimensionsofuncertainties,i.e.,windpoweroutputandelectricityload.However,sincethewindpoweroutputcanberegardedasanegativeload,withoutlossofgenerality,weonlyconsidertheelectricityloadastheuncertainparameter.Wesummarizethenotationsbysets,parameters,rst-stagevariablesandsecond-stagevariableslistedasfollows. A. SetsandParameters B Indexsetofallbuses. E Indexsetoftransmissionlineslinkingtwobuses. Gb Setofthermalgeneratorsatbusb. T Timehorizon(e.g.,24hours). SU bi Start-upcostofthermalgenerator i at busb. SD bi Shut-downcost of thermalgenerator i at busb. Fi(.) Fuelcost of thermalgenerator i . MU bi Minimumup timeforthermalgenerator i atbusb. MD bi Minimumdown timeforthermalgenerator i atbusb. RUbi Ramp-upratelimitforthermalgenerator i atbusb. RDbi Ramp-downratelimitforthermalgenerator i atbusb. Lbi Lowerboundofelectricitygeneratedbythermalgenerator i atbusb. 95

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Ubi Upperboundofelectricitygeneratedbythermalgenerator i atbusb. Cij Capacityfor the transmission line linking busiandbusj. Kbij Lineow distributionfactorforthetransmission line linking busi andbusj,duetothenetinjectionatbusb . Db+t Theupper bound oftheload atbusbintimet . Db)]TJ /F4 7.97 Tf -5.46 -7.58 Td[(t Thelower bound oftheload atbusbintimet . dtb() Theload atbusbintimet correspondingtoscenario. jbit Theinterceptofthejthsegmentlineforthegenerationcostforgeneratoriatbusbintimet. jbit Theslopeofthejthsegmentlineforthegenerationcostforgenerator i atbusbintimet. B. First-stageVariables ybit Binary decision variable:“1”ifthermalgenerator i atbusbisonintimet;“0”otherwise. ubit Binary decision variable:“1”ifthermalgenerator i atbusbisstartedupintimet;“0”otherwise. vbit Binary decision variable:“1”ifthermalgenerator i atbusbisshutdownintimet;“0”otherwise. C. Second-stageVariables qbit() Electricitygenerationamountbythermalgenerator i atbusbintimet correspondingtoscenario . bit() Auxiliaryvariablerepresentingthefuelcostofthermalgenerator i atbusbintimetcorrespondingtoscenario.Withthenotations,wedevelopatwo-stagestochasticunitcommitmentformulationbyminimizingtheexpectedtotalgenerationcost.Therststageistoobtaintheday-aheadon/offschedule(i.e.,unitcommitmentdecision)forthethermalgeneratorsadheringtotheoperationalconstraints.Aftertheuncertainloadisrealized,thesecondstageistoobtaintherealtimeeconomicdispatchdecisionforthethermalgeneratorswhilesatisfyingthephysicalandtransmissionconstraints.Thedetailedformulation(denoted 96

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asSUC)canbedescribedasfollows: minTXt=1 X b2B Xi2 Gb ( SUbiubit+SDbivbit )+E[Q(y,u,v,)] (5) s.t.)]TJ ET BT /F3 11.955 Tf 103.94 -103.92 Td[(ybi(t)]TJ /F8 7.97 Tf 6.58 0 Td[(1) + ybit )]TJ ET BT /F3 11.955 Tf 175.46 -103.92 Td[(ybik 0, 8k :1k)]TJ /F5 11.955 Tf 11.96 0 Td[((t)]TJ /F5 11.955 Tf 11.95 0 Td[(1)MUbi,8i2Gb, 8b2B ,8t, (5) ybi(t)]TJ /F8 7.97 Tf 6.58 0 Td[(1) )]TJ ET BT /F3 11.955 Tf 139.62 -157.71 Td[(ybit + ybik 1, 8k :1k)]TJ /F5 11.955 Tf 11.96 0 Td[((t)]TJ /F5 11.955 Tf 11.95 0 Td[(1) MD bi,8i2 Gb,8b2B,8t , (5) )]TJ ET BT /F3 11.955 Tf 103.94 -211.5 Td[(ybi(t)]TJ /F8 7.97 Tf 6.58 0 Td[(1) + ybit )]TJ ET BT /F3 11.955 Tf 175.46 -211.5 Td[(ubit 0, 8i2Gb,8b2B,8t, (5) ybi(t)]TJ /F8 7.97 Tf 6.58 0 Td[(1) )]TJ ET BT /F3 11.955 Tf 139.62 -238.4 Td[(ybit )]TJ ET BT /F3 11.955 Tf 166.16 -238.4 Td[(vbit 0, 8i2Gb,8b2B,8t, (5) ybit , ubit , vbit 2f0,1g, 8i2Gb,8b2B,8t, (5) whereQ( y , u , v ,)isequalto min TXt=1Xb2BXi2Gb Fi(qbit()) (5) s.t.Lbiybit q b it()Ubiybit,8i2 Gb,8b2B,8t, (5) q b it())]TJ ET BT /F3 11.955 Tf 131.23 -420.18 Td[(q b i(t)]TJ /F8 7.97 Tf 6.59 0 Td[(1)()(2)]TJ /F3 11.955 Tf 11.96 0 Td[(y b i(t)]TJ /F8 7.97 Tf 6.59 0 Td[(1))]TJ /F3 11.955 Tf 11.96 0 Td[(y b it) L b i +(1+y b i(t)]TJ /F8 7.97 Tf 6.59 0 Td[(1))]TJ /F3 11.955 Tf 11.96 0 Td[(y b it)RU b i,8i2 Gb,8b2B,8t, (5) q b i(t)]TJ /F8 7.97 Tf 6.59 0 Td[(1)())]TJ ET BT /F3 11.955 Tf 149.91 -473.98 Td[(q b it()(2)]TJ /F3 11.955 Tf 11.96 0 Td[(y b i(t)]TJ /F8 7.97 Tf 6.59 0 Td[(1))]TJ /F3 11.955 Tf 11.96 0 Td[(y b it)L b i+(1)]TJ /F3 11.955 Tf 11.96 0 Td[(y b i(t)]TJ /F8 7.97 Tf 6.59 0 Td[(1)+y b it)RD b i,8i2 Gb,8b2B,8t, (5) )]TJ /F3 11.955 Tf 9.29 0 Td[(CijXb2BKbij(Xr2Gbqbrt())]TJ /F3 11.955 Tf 11.96 0 Td[(dtb())Cij, 8(i,j)2E,8t, (5) X b2B Xi2 Gb qbit()=Xb2Bdtb(),8t. (5) Intheaboveformulation, constraints( 5 )and( 5 ) indicatethataminimumup-timeandaminimumdown-timeareneededinordertostartupandshutdownthethermalunit.Constraints( 5 )and( 5 )arethestart-up and shut -downoperationalconstraints 97

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foreachthermalunit. Constraint ( 5 ) enforces theupperandlowerboundsoftheelectricitygenerationamountofeachthermalunit . Therampingupconstraint( 5 )andrampingdownconstraint( 5 )indicatethemaximum increment anddecrementofthepower generation amount ofeachunit betweentwoadjacentperiodswhenthethermalunitison.Constraint( 5 )representsthetransmissioncapacityconstraintsandconstraint( 5 )ensuresloadbalance.Next,weintroducethetraditionalsolutionapproachtoaddressthestochasticunitcommitmentproblem.First,notethatthefuelcostfunctionFi(.)isnormallyaquadraticfunction.Thetraditionalwaytolinearizethequadraticfunctionistousemultiplepiecesofpiecewiselinearfunctiontoapproximateit,andwealsoutilizethistechniqueinthischapter.Forinstance,weuseaK-piecepiecewiselinearfunctionasfollowstoapproximateFi(.): bitjbitybit+jbitxbit, (5) 8t=1,,T,8b2B,8i2Gb,8k=1,,K.Inordertokeepnotationsbrevity,weusetheabstractformulationtorepresentSUC.Forinstance,weusevectorstorepresentvariablesandmatrixtorepresentconstraints,thenSUCcanbeabstractedasfollows: miny,u,v(aTu+bTv)+E[Q(y,u,v,)]s.t.Ay+Bu+Cvr, (5) whereQ(y,u,v,)isequalto mineT() (5) s.t.Dx()f+Fy, (5) Gx()g+Hd(), (5) Jx()+K()h+Ly. (5) 98

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Noteherethatconstraint( 5 )representsconstraints( 5 )-( 5 );constraint( 5 )representsconstraints( 5 )-( 5 );constraint( 5 )representsconstraint( 5 )andconstraint( 5 )representsconstraint( 5 ).TosolveSUC,scenario-basedapproachesarecommonlyused.Thekeyideaofscenario-basedapproachesistoassumethattherandomparameterfollowsagivendistribution,forexample,multivariatenormaldistributionN(D,)withitspredictedvalueDandvolatilitymatrix.Inaddition,simulationtechniquessuchasMontoCarlosimulationarecommonlyusedtogeneratescenariosfortherandomparameter.Therefore,theexpectedcostintheobjectivefunctioncanbeestimatedastheaveragevalueofcostscorrespondingtodifferentgeneratedscenarios.Forexample,ifweuseMontoCarlosimulationtogenerateNscenariosfortheelectricityload,i.e.,dtb(1),dtb(2),,dtb(N),wecanapproximatetheobjectivefunctionofSUCasfollows: miny,u,v(aTu+bTv)+1 NNXn=1Q(y,u,v,n). (5) Accordingly,thestochasticunitcommitmentproblemistransformedintoadeterministicMIPtoalargerextent.Basedonthelagrangerelaxationtechnique,theproblemcanbedecomposedintoscenario-basedsubproblems[ 53 ],andthencanbeaddressedbysolvingeachsubproblem.However,duetotheincompleteinformationoftheelectricityload,itisverydifculttoaccuratelyestimateitsprobabilitydistribution.Consequently,theobtainedunitcommitmentdecisioncanbebiased.Forexample,iftheonlineunitsarenotcapableofprovidingenoughelectricitytobalancetheload,electricityshortageorblackoutmayoccur.Ontheotherside,ifthegeneratedelectricityexceedstheelectricityconsumption,thewindpoweroutputneedstobecurtailedtokeepthesystembalanced.Totacklethesechallengingproblems,theconceptofdata-drivenrisk-aversestochasticoptimizationhasbeenproposed.Inthenextsection,weintroducethedata-driven 99

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risk-aversetwo-stagestochasticunitcommitmentproblemanddiscussthesolutionapproachestoaddresstheproblem. 5.2.2Data-DrivenUnitCommitmentFormulationInthissection,wedevelopadata-drivenrisk-aversetwo-stagestochasticunitcommitmentmodel,inwhichinsteadofknowingtheexactdistributionoftheelectricityload,aseriesofhistoricaldataofloadareobserved.Basedontheobservedhistoricaldata,webuildthecondencesetfortheprobabilitydistributionoftheelectricityload,byusingtheWassersteinmetric.Thenweproposethesolutionmethodologytosolvethedata-drivenrisk-aversetwo-stagestochasticunitcommitmentproblem.Finally,weanalyzetheconvergencepropertiesofthemodelandshowthattherisk-aversenessofthemodeldecreasesasmorehistoricaldataareobservedandeventuallydoesnotexistasthenumberofhistoricaldatagoestoinnity. 5.2.2.1WassersteinmetricTheWassersteinmetricisdenedasadistancefunctionbetweentwoprobabilitydistributionsonagivensupportingspace.Morespecically,giventwoprobabilitydistributionsPandQonthesupportingspace,WassersteinmetricisdenedasdW(P,Q):=inffE[(X,Y)]:P=L(X),Q=L(Y)g,where(X,Y)isdenedasthedistancebetweenrandomvariablesXandY,andtheinmumistakenoveralljointdistributionswithmarginalsPandQ.First,asindicatedin[ 68 ],theWassersteinmetricisindeedametricsinceitsatisesthepropertiesofmetrics.Thatis,dW(P,Q)=0ifandonlyifP=Q,dW(P,Q)=dW(Q,P)(symmetricproperty)anddW(P,Q)dW(P,O)+dW(O,Q)foranyprobabilitydistributionO(triangleequality).Inaddition,bytheKantorovich-Rubinsteintheorem[ 33 ],theWassersteinmetricisequivalenttotheKantorovichmetric,whichisdenedas dK(P,Q)=suph2HjZhdP)]TJ /F10 11.955 Tf 11.95 16.27 Td[(ZhdQj, (5) 100

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whereH=fh:khkL1g,andkhkL:=supfh(x))]TJ /F4 7.97 Tf 6.59 0 Td[(h(y) (x,y):x6=ying.Inparticular,when=R, dW(P,Q)=Z+1jF(x))]TJ /F3 11.955 Tf 11.96 0 Td[(G(x)jdx, (5) whereF(x)andG(x)arethedistributionfunctionsderivedbyPandQrespectively.ThisconclusionholdsfollowingtheargumentthatinffE[(X,Y)]:P=L(X),Q=L(Y)g=R10jF)]TJ /F8 7.97 Tf 6.59 0 Td[(1(t))]TJ /F3 11.955 Tf 12.22 0 Td[(G)]TJ /F8 7.97 Tf 6.58 0 Td[(1(t)jdt,asstatedinTheorem6.0.2in[ 3 ]andR10jF)]TJ /F8 7.97 Tf 6.59 0 Td[(1(t))]TJ /F3 11.955 Tf 12.23 0 Td[(G)]TJ /F8 7.97 Tf 6.59 0 Td[(1(t)jdt=R+1jF(x))]TJ /F3 11.955 Tf 12.42 0 Td[(G(x)jdx.TheconceptofWassersteinmetricisrstintroducedbyLeonidVasershtein,andcommonlyappliedinmanyareas.Forexample,manymetricsknowninstatistics,measuretheory,ergodictheory,functionalanalysis,etc.,arespecialcasesoftheWasserstein/Kantorovichmetric[ 67 ].Wasserstein/Kantorovichalsohasmanyapplicationsintransportationtheory[ 52 ],andsomeapplicationsincomputersciencelikeprobabilisticconcurrency,imageretrieval,datamining,andbioinformatics,etc[ 20 ].Inthischapter,weuseWassersteinmetrictoconstructthecondencesetfortheprobabilitydistribution. 5.2.2.2ReferencedistributionAfterobservingaseriesofhistoricaldata,wecanhaveanestimationofthetrueprobabilitydistribution,whichiscalledreferenceprobabilitydistribution.Intuitively,themorehistoricaldatawehave,themoreaccurateestimationofthetrueprobabilitydistributionwecanobtain.Manysignicantworkshavebeenmadetogetthereferencedistribution,withbothparametricapproachesandnonparametricapproaches.Forinstance,fortheparametricapproaches,thetrueprobabilitydistributionisusuallyestimatedasaparticulardistributionfunction,e.g.,normaldistribution,andtheparameters(e.g.,meanandvariance)areestimatedbylearningfromthehistoricaldata.Ontheotherhand,thenonparametricestimations,suchaskerneldensityestimation,arealsoprovedtobeeffectiveapproachestoobtainthereferenceprobabilitydistribution(e.g.,[ 54 ]and[ 51 ]).Inthischapter,weuseanonparametricapproach,morespecically, 101

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theempiricaldistributionfunctiontoestimatethetrueprobabilitydistribution.Theempiricaldistributionfunctionisastepfunctionthatjumpsupby1=NateachoftheNindependentandidentically-distributeddatapoints.Thatis,givenNi.i.d.historicaldatapoints10,20,,N0,theempiricaldistributionisdenedas P0(x)=1 NNXi=1i0(x), (5) wherei0(x)isoneifxi0andzeroelsewhere.Basedonthestronglawoflargenumbers,itcanbeprovedthatthereferencedistributionP0pointwiseconvergestothetrueprobabilitydistributionPalmostsurely[ 66 ].ByGlivenko-Cantellitheorem,thisresultcanbestrengthenedbyprovingtheuniformconvergenceofPtoP0[ 74 ].InChapter 4 ,wehaveprovedthatundertheWassersteinmetric,theempiricaldistributionP0exponentiallyconvergestothetruedistributionP. 5.2.2.3CondencesetconstructionWiththepreviouslydenedprobabilitymetricandreferenceprobabilitydistribution,wenowconstructthecondencesetofthetrueprobabilitydistribution.Intuitively,themorehistoricaldatathatcanbeobserved,the“closer”thereferencedistributionistothetruedistribution.Ifweusetorepresentthedistancebetweenthereferencedistributionandthetruedistribution,thenthemorehistoricaldatawehave,thesmallerthevalueofis,andthetighterthecondencesetbecomes.Therefore,thecondencesetDcanberepresentedasfollows: D=fP:d(P,P0)g, (5) wherethevalueofdependsonthenumberofhistoricaldata.Morespecically,fromChapter 4 ,wehaveshowntheexactrelationshipbetweenthenumberofhistoricaldataandthevalueof,inthefollowingproposition: 102

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Proposition5.1. Forageneraldimensiondofthesupportingspace,wehave P(dW(P0,P))1)]TJ /F5 11.955 Tf 11.95 0 Td[(exp()]TJ /F6 11.955 Tf 15.34 8.09 Td[(2 2B2N),whereNisthenumberofhistoricaldata,andBisthediameterof.Specically,ifd=1,wehave Pr(dW(P0,P))1)]TJ /F5 11.955 Tf 11.96 0 Td[(2exp()]TJ /F5 11.955 Tf 9.3 0 Td[(2N2).Moreover,weconsideraspecialcasethatP0andPareproductmeasures.Thatis,P0=di=1P(i)0andP=di=1P(i),whereP(i)0andP(i)aretheprobabilitydistributionsdenedontheBorel-algebraofR.Besides,weconsiderthecasethat(x,y)iswithL1norm(i.e.,(x,y)=Pdi=1jxi)]TJ /F3 11.955 Tf 12.63 0 Td[(yij).Then,basedonthedenitionofWassersteinmetric,wehave dW(P0,P)=dXi=1dW(P(i)0,P(i)),wheredW(P(i)0,P(i))=RRjP(i)0(x))]TJ /F16 11.955 Tf 12.95 0 Td[(P(i)(x)jdx.Furthermore,accordingtoHolder'sinequality,wehaveRRjP(i)0(x))]TJ /F16 11.955 Tf 13.44 0 Td[(P(i)(x)jdxkP(i)0(x))]TJ /F16 11.955 Tf 13.44 0 Td[(P(i)(x)k1.Finally,theDvoretzky-Kiefer-Wolfowitztheoremleadstothefollowinginequalities: Pr(dW(P0,P))dXi=1Pr(dW(Pi0,Pi) d)2dexp()]TJ /F5 11.955 Tf 10.49 8.09 Td[(2n2 d2).Inconclusion,forthespecialcasethatP0andPareproductmeasuresand(x,y)=Pdi=1jxi)]TJ /F3 11.955 Tf 11.96 0 Td[(yij,wehave Pr(dW(P0,P))1)]TJ /F5 11.955 Tf 11.96 0 Td[(2dexp()]TJ /F5 11.955 Tf 10.49 8.08 Td[(2n2 d2),foragenerald1.InsteadofknowingthetrueprobabilitydistributionP,weassumePcanvarywithinthecondencesetDandweconsidertheworst-caseexpectedvalueE[Q(y,u,v,)].Inthatcase,theproposedapproachismoreconservativethanthetraditionalstochastic 103

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optimizationapproach.Thatis,theproposedapproachisarisk-averseapproach.Therefore,thedata-drivenrisk-aversetwo-stagestochasticunitcommitmentproblemcanbeformulatedasfollows: miny,u,v(aTu+bTv)+maxP2DEP[Q(y,u,v,)](DD)]TJ /F3 11.955 Tf 11.96 0 Td[(SUC)s.t.Ay+Bu+Cvr. (5) 5.3SolutionMethodologyInthissection,weproposethesolutionapproachtoaddresstheproblem(DD-SUC).WeassumetheelectricityloadforeachtimeperiodtateachbusbisbetweenalowerboundDb)]TJ /F4 7.97 Tf -5.46 -7.59 Td[(tandanupperboundDb+t,andwithoutlossofgenerality,weletd()=.Accordingly,thesupportingspace(oruncertaintyset)fortherandomelectricityloadcanbedescribedasfollows: :=(2RjBjjTj:Db)]TJ /F4 7.97 Tf -5.45 -7.9 Td[(ttbDb+t,8t,8b). (5) AsindicatedinSubsection 5.2.2.2 ,ifwehaveNsamplesofhistoricaldata1,2,,N,thereferencedistributionP0canbedenedastheempiricaldistribution,i.e.,P0=1 NPNi=1i(x).Ontheotherhand,accordingtothedenitionoftheWassersteinmetric,thecondencesetDcanbereformulatedasfollows: D=nP:inffE[(Z,W)]:P=L(Z),P0=L(W)go. (5) Therefore,basedonthedenitionofE[(Z,W)]andpropertiesofconditionaldensity,wecanobtainthefollowingreformulationofE[(Z,W)]: E[(Z,W)]=1 NNXi=1Zw2fwji(wjz=i)(w,i)dw, (5) wherefwji(wjz=i)istheconditionaldensityfunctionwhenz=i.Fornotationbrevity,weletfi(w)=fwji(wjz=i)andi(w)=(w,i).Thenthesecondstageproblemof 104

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(DD-SUC)canbereformulatedas: maxfY(w)0Zw2Q(y,u,v,w)fY(w)dws.t.fY(w)=1 NNXi=1fi(w), (5) Zw2fi(w)dw=1,8i, (5) 1 NNXi=1Zw2fi(w)i(w)dw, (5) wherefY(w)isthedensityfunctionofP.Theconstraints( 5 )and( 5 )arebasedonthepropertiesofconditionaldensityfunctionandtheconstraint( 5 )istherepresentationofthecondencesetD.Bysubstitutingconstraint( 5 )intotheobjectivefunction,wecanobtainitsequivalentformulationasfollows: maxfi(w)01 NNXi=1Zw2Q(y,u,v,w)fi(w)dws.t.Zw2fi(w)dw=1,8i, (5) 1 NNXi=1Zw2fi(w)i(w)dw. (5) Sincetheaboveproblemisasemi-inniteproblem,thereisnodualitygap.ThenwecanconsidertheLagrangiandualproblemthatcanbewrittenas: L(i,)=maxfi01 NNXi=1Z(Q(y,u,v,w))]TJ /F3 11.955 Tf 11.96 0 Td[(Ni)]TJ /F6 11.955 Tf 11.96 0 Td[(i(w))fi(w)dw+NXi=1i+, (5) whereiandaredualvariablesofconstraints( 5 )and( 5 )respectively.ThedualproblemisthentominimizeL(i,)withrespecttoi,,andissubjecttobenonnegative.Next,wearguethat8w2,Q(y,u,v,w))]TJ /F3 11.955 Tf 12.31 0 Td[(Ni)]TJ /F6 11.955 Tf 12.31 0 Td[(i(w)0.Ifthisargumentdoesnothold,thenthereexistsw0suchthatQ(y,u,v,w0))]TJ /F3 11.955 Tf 10.37 0 Td[(Ni)]TJ /F6 11.955 Tf 10.37 0 Td[(i(w0)>0.Itmeansthereexistsasmallnumber,suchthatQ(y,u,v,w0))]TJ /F3 11.955 Tf 12.41 0 Td[(Ni)]TJ /F6 11.955 Tf 12.41 0 Td[(i(w0)>.SincethefunctionQ(y,u,v,w))]TJ /F3 11.955 Tf 12.23 0 Td[(Ni)]TJ /F6 11.955 Tf 12.22 0 Td[(i(w)iscontinuouswithw,ifQ(y,u,v,w0))]TJ ET BT /F1 11.955 Tf 224.03 -687.85 Td[(105

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Ni)]TJ /F6 11.955 Tf 12.51 0 Td[(i(w0)>,thereexistsasmallballB(w0,),suchthatQ(y,u,v,w))]TJ /F3 11.955 Tf 12.51 0 Td[(Ni)]TJ /F6 11.955 Tf -439.26 -23.9 Td[(i(w)>for8w2B(w0,).Therefore,wecanletfi(w)tobearbitrarylargewhenw2B(w0,),thenL(i,)isarbitrarylargeaswell,whichleadstoacontradiction.Hence,theargumentQ(y,u,v,w0))]TJ /F3 11.955 Tf 11.96 0 Td[(Ni)]TJ /F6 11.955 Tf 11.96 0 Td[(i(w0)0forallw2holds.Inthatcase,maxfi01 NNXi=1Z(Q(y,u,v,w0))]TJ /F3 11.955 Tf 11.95 0 Td[(Ni)]TJ /F6 11.955 Tf 11.95 0 Td[(i(w0))fi(w)dw+NXi=1i+=NXi=1i+,withtheoptimalsolutionfi=0,i=1,,N.Then,thedualformulationisreformulatedas:mini,0NXi=1i+s.t.Q(y,u,v,w))]TJ /F3 11.955 Tf 11.96 0 Td[(Ni)]TJ /F6 11.955 Tf 11.95 0 Td[(i(w)0,8w2,8i=1,,N.Fromtheaboveformulation,itiseasytoobservethattheoptimaliisequalto1 Nmaxw2fQ(y,u,v,w))]TJ /F6 11.955 Tf 13.07 0 Td[(i(w)gandthesecond-stagerisk-averseoptimizationproblemisequivalentto min0(1 NNXi=1maxw2fQ(y,u,v,w))]TJ /F6 11.955 Tf 11.95 0 Td[(i(w)g+). (5) Therefore,wehavethefollowingtheorem: Theorem1. Theproblem(DD-SUC)isequivalenttothefollowingtwo-stagerobustoptimizationproblem: miny,u,v,0(aTu+bTv)++1 NNXi=1maxw2fQ(y,u,v,w))]TJ /F6 11.955 Tf 11.95 0 Td[(i(w)g (5) (R)]TJ /F3 11.955 Tf 11.96 0 Td[(SUC)s.t.Ay+Bu+Cvr, (5) whereQ(y,u,v,w)isequalto mineT(w) (5) s.t.Dx(w)f+Fy, (5) Gx(w)g+Hw, (5) 106

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Jx(w)+K(w)h+Ly. (5) Next,wediscussthesolutionapproachtoaddressthe(R-SUC)problem.Inthischapter,weutilizetheBenders'decompositionalgorithmtosolvetheproblem.First,wedualizetheconstraints( 5 )to( 5 )andobtainthefollowingdualformulationandcombineitwiththesecondstagetogetthesubproblem(denotedasSUB-SUC): i(y)=maxw,,,(f+Fy)T+(g+Hw)T+(h+Ly)T)]TJ /F6 11.955 Tf 11.96 0 Td[(i(w)s.t.DT+GT+JT0, (5) KT=e, (5) ,,0, (5) where,,aredualvariablesforconstraints( 5 ),( 5 )and( 5 )respectively.Letidenotetheoptimalvalueofthesubproblem,wecanobtainthefollowingmasterproblem:miny,u,v(aTu+bTv)+1 NNXi=1is.t.Ay+Bu+Cvr,Feasibilitycuts,Optimalitycuts.Theproblemcanbesolvedbyaddingfeasibilityandoptimalitycutsiteratively.Noticehereinthesubproblem(SUB-SUC),wehaveabilineartermwTHT.Inthefollowingpart,weproposetwoseparationapproachestoaddressthebilinearterm. 5.3.1ExactSeparationApproachSincethesupportingspaceoftherandomelectricityloadisdescribedasapolytopeasshownin( 5 ),itcanbeveriedthattheoptimalsolutionwtothesubproblem(SUB-SUC)shouldsatisfythefollowingproposition. 107

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Proposition5.2. Thereexistsanoptimalsolution(w,,,)tothesubproblem(SUB-SUC)suchthatwtb=Db)]TJ /F4 7.97 Tf -5.45 -7.59 Td[(torwtb=Db+tforeachbusbineachtimeperiodt. Proof. Foraxedsolution(,,),theproblemistomaximizeaconcavefunctionwithapolyhedralfeasibleregion,soatleastoneextremepointwofistheoptimalsolutiontothesubproblem,i.e.,wtb=Db)]TJ /F4 7.97 Tf -5.45 -7.58 Td[(torwtb=Db+t. Weletbinaryvariablezb+t=1ifwtb=Db+tandbinaryvariablezb)]TJ /F4 7.97 Tf -5.76 -7.58 Td[(t=1ifwtb=Db)]TJ /F4 7.97 Tf -5.46 -7.58 Td[(t,andthenbasedonProposition 5.2 ,wehavethefollowingconstraintshold: zb+t+zb)]TJ /F4 7.97 Tf -5.76 -7.9 Td[(t=1, (5) wtb=Db+tzb+t+Db)]TJ /F4 7.97 Tf -5.46 -7.89 Td[(tzb)]TJ /F4 7.97 Tf -5.76 -7.89 Td[(t, (5) whereconstraint( 5 )indicatestheoptimalloadwilleitherachieveitslowerboundorupperbound.Withconstraints( 5 )and( 5 ),wenowaddressthebilineartermwTHT.LetHT=,wehave wTHT=Xt2TXb2B(Db+tzb+t+Db)]TJ /F4 7.97 Tf -5.46 -7.89 Td[(tzb)]TJ /F4 7.97 Tf -5.76 -7.89 Td[(t)bt=Xt2TXb2B(Db+tzb+tbt+Db)]TJ /F4 7.97 Tf -5.46 -7.89 Td[(tzb)]TJ /F4 7.97 Tf -5.76 -7.89 Td[(tbt)=Xt2TXb2B(Db+tb+t+Db)]TJ /F4 7.97 Tf -5.46 -7.89 Td[(tb)]TJ /F4 7.97 Tf -4.57 -7.89 Td[(t) (5) s.t.bt=(HT)bt,8t2T,8b2B (5) b+t)]TJ /F3 11.955 Tf 21.92 0 Td[(Mzb+t,8t2T,8b2B (5) b+tbt)]TJ /F3 11.955 Tf 11.95 0 Td[(M(1)]TJ /F3 11.955 Tf 11.96 0 Td[(zb+t),8t2T,8b2B (5) b)]TJ /F4 7.97 Tf -4.57 -7.89 Td[(t)]TJ /F3 11.955 Tf 21.92 0 Td[(Mzb)]TJ /F4 7.97 Tf -5.76 -7.89 Td[(t,8t2T,8b2B (5) b)]TJ /F4 7.97 Tf -4.57 -7.9 Td[(t)]TJ /F6 11.955 Tf 21.92 0 Td[(bt)]TJ /F3 11.955 Tf 11.95 0 Td[(M(1)]TJ /F3 11.955 Tf 11.96 0 Td[(zb)]TJ /F4 7.97 Tf -5.76 -7.9 Td[(t),8t2T,8b2B. (5) NowwecanreplacethebilineartermwTHTwith( 5 )andaddconstraints( 5 )to( 5 )tothesubproblem. 108

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Next,weinvestigatehowtogenerateBenders'feasibilitycutsandoptimalitycuts.Forthefeasibilitycuts,weonlyconsiderconstraints( 5 )and( 5 )sinceonlytheseconstraintsaffectthefeasibility.WeuseL-shapemethodtogeneratethefeasibilitycutswiththefollowingfeasibilitycheckproblem: $(y)=max^,^(f+Fy)T^+g^+(D+)T^++(D)]TJ /F5 11.955 Tf 7.09 -4.93 Td[()T^)]TJ /F3 11.955 Tf -260.86 -31.83 Td[(s.t.DT^+GT^0,Constraints( 5 )to( 5 ),^,^2[0,1].If$(y)=0,yisafeasiblesolution.If$(y)>0,wecangenerateafeasibilitycut$(y)0.Fortheoptimalitycuts,afterwesolvethemasterproblemandobtainyandifori=1,,N,wesubstituteyintothesubproblemandgeti(y).Foreachi,ifi(y)>i,thenyisnotanoptimalsolutionandwecangenerateacorrespondingoptimalitycuti(y)i. 5.3.2BilinearSeparationApproachInthissubsection,wediscussthebilinearapproachtogenerateBenders'feasibilitycuts.Similarly,itisnotnecessarytoconsiderconstraint( 5 )sinceitdoesnotaffectthefeasibility.Therefore,thefeasibilitycheckproblemfortheseconstraintsisshownasfollows(denotedasFEA): (y)=max^,^(f+Fy)T^+(g+Hw)T^ (5) s.t.DT^+GT^0, (5) ^,^2[0,1]. (5) Next,weinitiatethevalueofwasoneoftheextremepoints,andwiththexedw,wesolve(FEA)toobtaintheoptimalvalue(y)(denotedas1(y,w))andoptimalsolution 109

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^and^.Then,byxing^=^and^=^,wemaximizetheobjective( 5 )withrespecttow2.Inthatcase,wecanobtaintheoptimalvalue(denotedas2(y,^,^))andtheoptimalvaluew.If2(y,^,^)>1(y,w),letw=wandprocessititeratively.Otherwise,checkwhether1(y,w)=0,ifso,wecanterminatethefeasibilitycheck;ifnot,addthefeasibilitycut1(y,w)0tothemasterproblem.Then,wegeneratetheBenders'optimalitycutsbyusingthebilinearapproach.Similarly,weinitiatethevalueofwasoneextremepointof,andsolvetheproblem(DR-SUC)toobtaintheoptimalvalueforeachi=1,,Nwiththexedw(denotedas1i(y,w))andoptimalsolution,and.Then,byxing=,=and=,weobtaintheoptimalvalue(denotedas2i(y,,,))andtheoptimalvaluewofthefollowingproblem2i(y,,,)=maxw2THw)]TJ /F6 11.955 Tf 11.95 0 Td[(i(w)+(f+Fy)T+gT+(h+Ly)T.If2i(y,,,)>1i(y,w),letw=wandprocessititerativelyuntil2i(y,,,)1i(y,w).Thencheckwhether2i(y,,,)>i,ifso,generatethecorrespondingoptimalitycut2i(y,,,)itothemasterproblem;otherwise,outputthesolutions. 5.4ConvergenceAnalysisInthissection,weexaminetheconvergencepropertiesoftheDD-SUCtoSUCasthenumberofhistoricaldataincreases.WedemonstratethatasthecondencesetDshrinkswithmoreobservedhistoricaldata,therisk-averseproblemDD-SUCconvergestotherisk-neutralproblemSUC.Werstanalyzetheconvergencepropertyofthesecondstageobjectivevalue. Proposition5.3. limN!1min0n+1 NPNi=1maxw2fQ(y,u,v,w))]TJ /F6 11.955 Tf 11.96 0 Td[(i(w)go=EP0[Q(y,u,v,w)],thatis,lim!0supP2DEP[Q(y,u,v,w)]=EP0[Q(y,u,v,w)]. Proof. First,sincesupP2DEP[Q(y,u,v,w)]isequivalentto( 5 ),weknowthat( 5 )existsforanyN.Ontheotherside,accordingtotheLawofLargeNumbers 110

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andHelly-Braytheorem,wehave limN!1(+1 NNXi=1maxw2fQ(y,u,v,w))]TJ /F6 11.955 Tf 11.96 0 Td[((w,i)g)=EP0[maxw2fQ(y,u,v,w))]TJ /F6 11.955 Tf 11.96 0 Td[((w,)g].Therefore, limN!1min0(+1 NNXi=1maxw2fQ(y,u,v,w))]TJ /F6 11.955 Tf 11.95 0 Td[((w,i)g)min0limN!1(+1 NNXi=1maxw2fQ(y,u,v,w))]TJ /F6 11.955 Tf 11.95 0 Td[((w,i)g) (5) =min0EP0[maxw2fQ(y,u,v,w))]TJ /F6 11.955 Tf 11.95 0 Td[((w,)g] (5) Nextweshowthatforagiveny,u,v, min0EP0[maxw2fQ(y,u,v,w))]TJ /F6 11.955 Tf 11.96 0 Td[((w,)g]=EP0[Q(y,u,v,)]. (5) Foragiven,weassumetheoptimalsolutiontomaxw2fQ(y,u,v,w))]TJ /F6 11.955 Tf 12.52 0 Td[((w,)gisw(),then EP0[maxw2fQ(y,u,v,w))]TJ /F6 11.955 Tf 11.95 0 Td[((w,)g]=EP0[Q(y,u,v,w())])]TJ /F6 11.955 Tf 11.96 0 Td[(EP0[(w(),)g].Sinceforanyw2,Q(y,u,v,w())isbounded,wedenotetheupperboundofQ(y,u,v,w())asM,then EP0[Q(y,u,v,w())])]TJ /F6 11.955 Tf 11.95 0 Td[(EP0[(w(),)g]M)]TJ /F6 11.955 Tf 11.95 0 Td[(EP0[(w(),)g].First,weargueEP0[(w(),)g]=0.Ifnot,wecanlettobepositiveinnity,thenmin0EP0[maxw2fQ(y,u,v,w))]TJ /F6 11.955 Tf 13.08 0 Td[((w,)g]willbeunbounded.Therefore,EP0[(w(),)g]=0.Sinceforany,(w(),)0,thenwehave(w(),)=0,forany2.Itmeansw()=,8.Inthatcase, min0EP0[maxw2fQ(y,u,v,w))]TJ /F6 11.955 Tf 11.95 0 Td[((w,)g]=min0EP0[fQ(y,u,v,w()))]TJ /F6 11.955 Tf 11.96 0 Td[((w(),)g] 111

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=min0EP0[fQ(y,u,v,))]TJ /F6 11.955 Tf 11.96 0 Td[((,)g]=EP0[Q(y,u,v,)].From( 5 )and( 5 ),wehave lim!0supP2DEP[Q(y,u,v,w)]EP0[Q(y,u,v,w)].Ontheotherhand,sinceP02D,wehave supP2DEP[Q(y,u,v,w)]EP0[Q(y,u,v,w)],whichimmediatelyyields lim!0supP2DEP[Q(y,u,v,w)]EP0[Q(y,u,v,w)].Therefore,wehavethetheoremholds. Wedenotetheoptimalvaluetoproblem(DD-SUC)withNnumberofhistoricaldataas (N)andtheoptimalvaluetoproblem(SUC)as (0),wehavethefollowingtheorem: Proposition5.4. limN!1 (N)= (0). Proof. First,noticethatN!1isequivalentto!0.Therefore,toprovelimN!1 (N)= (0)isequivalenttoprovelim!0 ()= (0),where ()isthesameas (N).DenoteV(y,u,v)astheobjectivevalueofproblem(DD-SUC)andW(y,u,v)astheobjectivevalueofproblem(SUC),withrespecttoanygivensolutiony,u,v.AccordingtoProposition 5.3 ,foranyarbitrarysmallnumber,andanygivensolutiony,u,v,thereexists>0,suchthat jV(y,u,v))]TJ /F3 11.955 Tf 11.96 0 Td[(W(y,u,v)j,8. (5) 112

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Thenforany,denotetheoptimalsolutionto(DD-SUC)asy,u,v,andtheoptimalsolutionto(SUC)as^y,^u,^v,then j ())]TJ /F6 11.955 Tf 11.96 0 Td[( (0)j= ())]TJ /F6 11.955 Tf 11.95 0 Td[( (0)=V(y,u,v))]TJ /F3 11.955 Tf 11.95 0 Td[(W(^y,^u,^v)V(^y,^u,^v))]TJ /F3 11.955 Tf 11.95 0 Td[(W(^y,^u,^v)jV(^y,^u,^v))]TJ /F3 11.955 Tf 11.95 0 Td[(W(^y,^u,^v)jTherefore,wecanprovetheproposition. 5.5CaseStudyInthissection,wetestthesystemperformancewithamodiedIEEE118-bussystem,basedontheonegivenonlineathttp://motor.ece.iit.edu/data.Thesystemcontains118buses,33generators,and186transmissionlines.Theoperationaltimeintervalis24hours.Inourexperiments,wesetthefeasibilitytolerancegaptobe10)]TJ /F8 7.97 Tf 6.59 0 Td[(6andtheoptimalitytolerancegaptobe10)]TJ /F8 7.97 Tf 6.58 0 Td[(4.TheMIPgaptoleranceforthemasterproblemistheCPLEXdefaultgap.WeuseC++withCPLEX12.1toimplementtheproposedformulationsandalgorithms.Allexperimentsareexecutedonacomputerworkstationwith4IntelCoresand8GBRAM.Inourexperiment,wecomparetheperformancesofourproposedapproachwiththestochasticoptimizationapproach.Weintroduceapenaltycostwiththevalue$ 5,000 =MWh [ 7 ] ,foranypowerimbalanceortransmissioncapacity/ramp-ratelimitviolation.Besides,weassumetheuncertainloadfollowsamultivariatenormaldistribution,andwegeneratesamplesastheobservedhistoricaldata.Inaddition,wesetthecondencelevelas0.99.Thenwecompareourproposedmodelwiththetraditionaltwo-stagestochasticunitcommitmentmodel.Weobtainthetotalcostscorrespondingtoourproposeddata-drivenrisk-aversestochasticoptimizationapproach(DD-SUC)andthetraditionaltwo-stagestochastic 113

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optimizationapproach(SO)throughthefollowingsteps:1)ObtaintheunitcommitmentdecisionsbyusingtheDD-SUCapproachandSOapproachrespectively.2)FixtheobtainedUCdecisionsandsolvethesecond-stageproblemrepeatedlyfor50randomlygeneratedsamplestoobtainthetotalcostsforeachapproachrespectively.WereporttheresultsinTable 5-1 .Therstcolumnrepresentsthenumberofsamples.ThethirdcolumncomputesthevalueofbasedonProposition 5.1 .Thenumbersofstart-upsaregiveninthefourthcolumnandunitcommitmentcostsaregiveninthefthcolumn.Finally,thesixthcolumngivesthetotalcosts. Table5-1. ComparisonbetweenSOandDD-SUCapproaches #ofsamplesModel#ofstart-upsUC.C.($)T.C.($) 1SO26.626215440602828DD-SUC26.62629590560442510SO8.420215440602823DD-SUC8.42029586554446550SO3.765225495602884DD-SUC3.765285805544394100SO2.663235555602960DD-SUC2.663285805543703 FromTable 3-2 ,wehavethefollowingobservations: (1) Ononehand,asthenumberofsamplesincreases,thenumberofstart-upsobtainedbytheproposedDD-SUCapproachdecreases,andtheunitcommitmentcostdecreases,aswellasthetotalcost.Thatisbecausemorehistoricaldataareobserved,sothattheuncertaintysetshrinks,whichleadstolessconservativesolutions.Ontheotherhand,asthenumberofsamplesincreases,thenumberofstart-upsobtainedbytheSOapproachincreases,andtheunitcommitmentcostincreases,sincemorehistoricaldataleadstomorerobustsolutionsfortheSOapproach. (2) Second,forthetestednumberofsamples,theUCdecisionsderivedbytheSOapproacharenotfeasibletosomesimulatedloadscenarios.However,theUCdecisionsobtainedfromtheproposedapproachhavenofeasibilityissue.Therefore,theSOapproachhasmoretotalcoststhantheproposedDD-SUCapproach,sincethepenaltycostsoccurbyusingtheSOapproach. 114

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(2) Third,ascomparedtotheSO approach ,theDD-SUC approach bringsmoregeneratorsonlineto providesufcient generationcapacitytomaintainthesystembalance.Asaresult,theDD-SUCapproachhas alarger UCcostthantheSOapproach.This resultveries thattheproposedDD-SUCapproachismorerobustthantheSOapproach. 5.6SummaryInthischapter,weproposeadata-drivenrisk-aversetwo-stagestochasticoptimizationapproachtocopewiththeunitcommitmentproblem.Bylearningfromthehistoricaldatathataredrawnfromanambiguousprobabilitydistributionoftheloadandrenewableenergyoutput,weconstructacondencesetoftheprobabilitydistributionwithWassersteinmetricanddevelopadata-drivenrisk-aversetwo-stagestochasticunitcommitmentmodel.Inaddition,wereformulatethemodelintoatraditionaltwo-stagerobustoptimizationproblemanddemonstratethatasthenumberofhistoricaldataincreases,thedata-drivenrisk-aversetwo-stagestochasticunitcommitmentmodelconvergestothetraditionaltwo-stagestochasticunitcommitmentmodel.Finally,arevisedIEEE118-bussystemisexaminedtoshowtheefciencyoftheproposedmodel. 115

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CHAPTER6CONCLUSIONSInthisdissertation,severaloptimization-under-uncertaintyformulationsareproposedtocapturetheuncertaintyintheunitcommitmentproblem.First,weproposearobustapproachtocopewithtwodimensionaluncertainties,i.e.,electricityloadandwindpoweroutput.Then,wedevelopauniedstochasticandrobustunitcommitmentproblem,whichcantakeadvantageofbothrobustoptimizationapproachandstochasticoptimizationapproachandmeanwhileovercometheirdisadvantages.Then,weextendourresearchtothefundamentaldata-drivenrisk-aversetwo-stagestochasticoptimization,forwhichthedistributionoftherandomvariablevarieswithinagivencondenceset.Basedonthehistoricaldata,weconstructthecondenceset,withanewclassofprobabilitymetrics.Inaddition,weprovidethesolutionmethodologytodealwiththeproblemforbothdiscreteandcontinuousdistributioncases.Moreover,weprovethatourrisk-aversestochasticprogramconvergestotherisk-neutralcaseasthesizeofhistoricaldataincreasestoinnity.Finally,weapplytheproposeddata-drivenrisk-aversetwo-stagestochasticoptimizationmodeltopowersystemoperationalproblems.Possiblefuturedirectionsinclude:1)extendingcurrentresearchtothechance-constrainedandmulti-stagecases,andanalyzingtheirapplicationsonpowersystemsoperationsandsupplychainmanagement.2)developingexactapproachestosolvelarge-sizedproblemsinpowersystems. 116

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BIOGRAPHICALSKETCH Ms.ChaoyueZhaoreceivedherPh.D.fromtheUniversityofFloridainsummer2014inIndustrialandSystemsEngineering.Beforethat,sheobtainedherB.S.degreeinInformationandComputingSciencesfromtheFudanUniversity,China,in2010.Herresearchinterestsincludedata-drivenstochasticoptimizationandstochasticintegerprogramwiththeirapplicationsonsmartgrid,energysystems,andsupplychainmanagement.ShehascollaboratedwithSandiaandArgonneNationalLabsforherresearchandworkedatPacicGas&ElectricCompany,oneofthelargestutilitiesinUS.HerresearchhasledtopaperspublishedontheagshipjournalforpowersystemsandabeststudentpaperawardintheCIStrackofISERCconference2012.ShewasawardedGraduateStudentCouncilOutstandingResearchAwardin2014,andwasalsooneoftherecipientsoftheOfceofResearchTravelGrants,theGraduateStudentCouncilTravelGrants,NSFStudentTravelAward,andthe2013MixedIntegerProgrammingWorkshopStudentTravelAward. 123