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A Unified State-Space and Scenario Tree Framework for Multi-Stage Stochastic Optimization

Permanent Link: http://ufdc.ufl.edu/UFE0042028/00001

Material Information

Title: A Unified State-Space and Scenario Tree Framework for Multi-Stage Stochastic Optimization an Application to Emission-Constrained Hydro-Thermal Scheduling
Physical Description: 1 online resource (166 p.)
Language: english
Creator: Rebennack, Steffen
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: emissions, hydrothermal, midterm, optimization, sampling, scenarios, sddp, stochastic, uncertainty
Industrial and Systems Engineering -- Dissertations, Academic -- UF
Genre: Industrial and Systems Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: In the hydro-thermal scheduling problem, one is interested in determining the optimal operating policy for the use of hydro and thermal resources in order to minimize total expected costs of fulfilling the demand for electricity over a given time horizon. Originally proposed in 1991 by Pereira and Pinto, Stochastic Dual Dynamic Programming (SDDP) remains to date the most efficient algorithm which is able to cope with inflow uncertainty and a detailed representation of a system's characteristics. In this dissertation, we propose several extensions of the SDDP methodology: We embed the SDDP algorithm into a scenario tree framework, incorporate CO2 emission allowance constraints, and supplement the profit maximization models to account for CO2 emission allowance markets. These extensions allows us to additionally deal with uncertainties related to the evolution of demand and fuel prices. From a practical standpoint, this is an innovation as fuel price and electricity demand uncertainty could not be taken into account efficiently in hydro-thermal power systems so far, and from a technical standpoint, this is a new approach unifying the state-space and scenario tree framework. The importance of such an approach was made evident by the global economic crisis of 2008 when several countries experienced huge variations in demand and faced sudden and sharp increases in fuel costs due to oil price swings, with implications not only on total incurred costs but also regarding security of supply. Despite the uncertainty surrounding the design of a mechanism which is ultimately accepted by nations worldwide, the necessity to implement measures to curb emissions of greenhouse gases on a global scale is consensual. The electricity sector plays a fundamental role in this puzzle and countries may soon have to revise their operating policy directives in order to make them compatible with additional constraints imposed by such regulations. Managing an annual emission allowance is somewhat similar to managing water reservoirs since one must determine the optimal trade-off between consuming parts of the limited amount of a resource in the present moment or saving it for future use. The decision to deplete the CO2 stock on hand may only be assessed in terms of its expected future costs, which depend on the evolution of hydrological conditions. Thus, a reservoir model for the CO2 emission quota has been proposed, respecting the stage decomposition framework of stochastic dynamic programming methods. This reservoir model allows for CO2 allowances to expire at given times. This is practically of high importance as this model reflects the currently implemented policy of the EU Emission Trading Scheme. The deregulation of the electricity markets made it necessary to incorporate uncertain electricity prices into the optimization models. Those models are typically solved using a hybrid method of the stochastic dynamic programming and stochastic dual dynamic programming. We extend those methods by incorporating stochastic CO2 emission allowance prices and stochastic fuel prices. The input data are derived by a fundamental model which allows us to capture the joint correlation of electricity market prices, CO2 emission allowance prices, fuel prices and hydro inflows.
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.
Statement of Responsibility: by Steffen Rebennack.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Pardalos, Panagote M.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2011-08-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0042028:00001

Permanent Link: http://ufdc.ufl.edu/UFE0042028/00001

Material Information

Title: A Unified State-Space and Scenario Tree Framework for Multi-Stage Stochastic Optimization an Application to Emission-Constrained Hydro-Thermal Scheduling
Physical Description: 1 online resource (166 p.)
Language: english
Creator: Rebennack, Steffen
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: emissions, hydrothermal, midterm, optimization, sampling, scenarios, sddp, stochastic, uncertainty
Industrial and Systems Engineering -- Dissertations, Academic -- UF
Genre: Industrial and Systems Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: In the hydro-thermal scheduling problem, one is interested in determining the optimal operating policy for the use of hydro and thermal resources in order to minimize total expected costs of fulfilling the demand for electricity over a given time horizon. Originally proposed in 1991 by Pereira and Pinto, Stochastic Dual Dynamic Programming (SDDP) remains to date the most efficient algorithm which is able to cope with inflow uncertainty and a detailed representation of a system's characteristics. In this dissertation, we propose several extensions of the SDDP methodology: We embed the SDDP algorithm into a scenario tree framework, incorporate CO2 emission allowance constraints, and supplement the profit maximization models to account for CO2 emission allowance markets. These extensions allows us to additionally deal with uncertainties related to the evolution of demand and fuel prices. From a practical standpoint, this is an innovation as fuel price and electricity demand uncertainty could not be taken into account efficiently in hydro-thermal power systems so far, and from a technical standpoint, this is a new approach unifying the state-space and scenario tree framework. The importance of such an approach was made evident by the global economic crisis of 2008 when several countries experienced huge variations in demand and faced sudden and sharp increases in fuel costs due to oil price swings, with implications not only on total incurred costs but also regarding security of supply. Despite the uncertainty surrounding the design of a mechanism which is ultimately accepted by nations worldwide, the necessity to implement measures to curb emissions of greenhouse gases on a global scale is consensual. The electricity sector plays a fundamental role in this puzzle and countries may soon have to revise their operating policy directives in order to make them compatible with additional constraints imposed by such regulations. Managing an annual emission allowance is somewhat similar to managing water reservoirs since one must determine the optimal trade-off between consuming parts of the limited amount of a resource in the present moment or saving it for future use. The decision to deplete the CO2 stock on hand may only be assessed in terms of its expected future costs, which depend on the evolution of hydrological conditions. Thus, a reservoir model for the CO2 emission quota has been proposed, respecting the stage decomposition framework of stochastic dynamic programming methods. This reservoir model allows for CO2 allowances to expire at given times. This is practically of high importance as this model reflects the currently implemented policy of the EU Emission Trading Scheme. The deregulation of the electricity markets made it necessary to incorporate uncertain electricity prices into the optimization models. Those models are typically solved using a hybrid method of the stochastic dynamic programming and stochastic dual dynamic programming. We extend those methods by incorporating stochastic CO2 emission allowance prices and stochastic fuel prices. The input data are derived by a fundamental model which allows us to capture the joint correlation of electricity market prices, CO2 emission allowance prices, fuel prices and hydro inflows.
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.
Statement of Responsibility: by Steffen Rebennack.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Pardalos, Panagote M.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2011-08-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0042028:00001


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PanagoteM.PardaloswasnotonlymyPhDadvisorbutalsoamentorforlife.Hewasabletondaperfectbalancebetweenclosesupervisionandacademicfreedom,allowingmetoexploredifferentresearchdirections.Itwashisforesighttointroducemetotheenergyapplicationeld.Furthermore,hewasabletoopenseeminglylockeddoorsformeandthatmorethanjustonce.MarioV.F.PereirainvitedmetohiscompanyPSRwhereIspendaveryproductiveandenjoyabletimeoveraperiodofmorethanthreemonths.TheinsightIhavegainedduringthesevisitsonthepowersystemsoptimizationpracticeingeneralandonSDDPinparticular,areofunmeasurablewealthtome.HisscienticdiscussionswereofgreatvalueandIthankhimforco-advisingthisdissertation.NikoA.IliadisallowedmetogainrsthandresearchexperienceduringmytimeasaninternathiscompanyEnerCoRD.Thiswasthestartofmyexposuretorealworldenergysystemoptimizationproblems,leadingtothisdissertationtopic.BrunoFlachwasthesafeportformyresearch.Heansweredallmytechnicalquestionsonstochasticprogramming,hydro-thermalschedulingandSDDPwithgreatskill,knowledge,andpatience;shapingthisdissertationinauniqueway.JosefKallrathisasteady,candidandsinceresupporterofmyprofessionalcareereversinceIwashisinternattheBASFcooperationin2004.Iwanttothankhimnotonlyforhisnumerousscienticcollaborations,butalsoforhistrustandfaithinme.LotharKountzwasmymathteacherinhighschoolduringmy9th,10th,and11thgrade.Itwashisdedicationtomath,hisskillsinteaching,hisabilitiestomotivatemeandhisfaithinmewhichleadmedownthiswonderfulpath.Icannotexpresshowmuchthismeanstome.TimothyJ.Anderson,JeanPhilippeRichardandJ.ColeSmithshapedthisresearchthroughvariouscommentsandsuggestionsastheirroleofmydissertationcommitteemembers. 4

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page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 8 LISTOFFIGURES ..................................... 9 LISTOFALGORITHMS .................................. 10 LISTOFABBREVIATIONS ................................ 11 ABSTRACT ......................................... 13 CHAPTER 1INTRODUCTION ................................... 15 1.1ElectricityMarketDeregulation ........................ 18 1.2CO2Emissions ................................. 20 1.3TypesofUncertainties ............................. 24 1.4DissertationStructure ............................. 26 1.5DissertationContributions ........................... 27 2MULTI-STAGESTOCHASTICLINEARPROGRAMMINGAPPLIEDTOHYDRO-THERMALSCHEDULING ......................... 29 2.1Multi-stageStochasticProgrammingFormulation .............. 29 2.1.1Assumptions .............................. 31 2.1.2DeterministicEquivalentProgramming ................ 32 2.2Application:Hydro-ThermalSchedulingProblem .............. 33 2.2.1AdditionalConstraints ......................... 39 2.2.2IstheHydro-thermalSchedulingWorldLinear? .......... 42 3SOLUTIONTECHNIQUES:ASURVEY ...................... 44 3.1DeterministicModels .............................. 45 3.2Scenario-BasedMethods ........................... 46 3.2.1ScenarioGenerationandReduction ................. 47 3.2.2ApplicationstoReservoirManagement ................ 48 3.3Sampling-BasedMethods ........................... 49 3.3.1StochasticDynamicProgramming:ExpectedFutureCostInterpolation ............................... 50 3.3.1.1Methodology ......................... 51 3.3.1.2Extensions,variationsandrelatedmethods ........ 61 3.3.2StochasticDualDynamicProgramming:ExpectedFutureCostExtrapolation .............................. 63 3.3.2.1Methodology ......................... 63 6

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.................... 69 3.3.2.3Extensions,variationsandrelatedmethods ........ 71 4SDDPT:FUELCOSTANDELECTRICITYDEMANDUNCERTAINTYINTHEFRAMEWORKOFSDDP .............................. 77 4.1UncertaintiesviaScenarioTrees ....................... 80 4.1.1FutureCostFunctionCutsforSDDPT ................ 83 4.1.2SDDPT:SDDPwithScenarioTree .................. 86 4.1.3MultivariateScenarioTrees ...................... 90 4.1.4ScenarioTreevs.MarkovChain ................... 91 4.2CaseStudyforCentralAmerica ........................ 92 4.2.1Panama ................................. 94 4.2.2CostaRica ............................... 99 4.3Discussion ................................... 100 5CO2EMISSIONCONSTRAINEDSDDP ...................... 104 5.1CO2EmissionConstrainedStochasticHydro-ThermalScheduling .... 106 5.2CO2EmissionQuotaModelingviaReservoirs ................ 110 5.2.1One-StageDispatchProgramming .................. 111 5.2.2FutureCostFunctionCutsforCO2ReservoirsinSDDP ...... 114 5.2.3SDDPwithCO2Reservoir ....................... 117 5.2.4SDDPTwithCO2Reservoir ...................... 120 5.2.5MultipleGHGReservoirs ........................ 120 5.3CaseStudyforGuatemala .......................... 121 5.4Discussion ................................... 131 6PROFITMAXIMIZATIONINDEREGULATEDELECTRICITYMARKETSANDCO2EMISSIONMARKETS ............................. 133 6.1Multi-StageStochasticProgrammingFormulation .............. 137 6.1.1One-StageDispatchProgramming .................. 140 6.1.2FutureBenetFunctionCutsforCO2ReservoirsandScenarioTree ................................... 143 6.2Discussion ................................... 144 7CONCLUSIONS ................................... 145 APPENDIX:NOMENCLATURE .............................. 146 REFERENCES ....................................... 152 BIOGRAPHICALSKETCH ................................ 166 7

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Table page 4-1ThermalplantsconsideredforthePanamapowersystem ............ 102 4-2ComputationalresultsforthepowersystemofPanamawithdifferentelectricitydemandscenarios .................................. 102 4-3ThermalplantsconsideredfortheCostaRicapowersystem ........... 103 4-4ComputationalresultsforthepowersystemofCostaRicawithdifferentelectricitydemandscenarios .................................. 103 5-1CO2emissionfactorsfordifferenttypesofthermalplants ............ 121 5-2ThermalplantsconsideredfortheGuatemalapowersystem ........... 132 5-3MonthlyelectricitydemandfortheGuatemalapowersystem .......... 132 5-4ComputationalresultsfordifferentquotalevelsontheGuatemalapowersystem ........................................ 132 A-1Indicesandsets ................................... 146 A-2Objectivefunctions .................................. 147 A-3Decisionvariablesandvaluesobtainedthroughoptimization .......... 148 A-4Inputdata ....................................... 149 8

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Figure page 1-1CO2emissionssharesfordifferentsectorsintheUSAin2007 ......... 22 1-2CO2emissionssharesfordifferentsectorsfortheUSAstatesin2007 ..... 23 1-3CO2emissionsperUSstatein2007inmillionmetrictons ............ 24 4-1Scenariotreewith4stages ............................. 81 4-2ElectricitydemandscenariosforthepowersystemsofPanamaandCostaRica.A)Panama.B)CostaRica .......................... 95 4-3Hydro-electricsystemofPanama .......................... 96 4-4Hydro-electricreservoirsystemofCostaRica ................... 99 5-1CO2emissionreservoirs ............................... 111 5-2Hydro-electricreservoirsystemofGuatemala ................... 122 5-3AnnualCO2emissionsandoperationalcostfortheGuatemalapowersystematdifferentquotalevels ............................... 124 5-4YearlygenerationmixforGuatemalapowersystemwithdifferentquotalevels 125 5-5MonthlydispatchingdecisionsfortheGuatemalapowersystemforthequotafreecase,whichisthenusedasthebasecaseshowingthemonthlydifferenceinelectricityproduction ............................... 128 5-6Averageelectricitymarginal(=spot)pricesfortheGuatemalapowersystematdifferentquotalevels ................................. 130 5-7AverageCO2emissionallowancemarginal(=spot)pricesfortheGuatemalapowersystematdifferentquotalevels ....................... 130 9

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Algorithm page 3-1SDP:StochasticDynamicProgramming(Generic) ................ 57 3-2InowScenarioGeneration ............................. 59 3-3SDDP:StochasticDualDynamicProgramming(Generic) ............ 70 4-1SDDPT:StochasticDualDynamicProgrammingwithScenarioTree ...... 89 5-1StochasticDualDynamicProgrammingwithCO2Reservoir ........... 119 10

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ADPApproximateDynamicProgrammingcf.conferCO2CarbondioxideCVaRConditionalValueatRiskDPDynamicProgramminge.g.exempligratiaetal.etaliiETSEmissionTradingSchemeEVSExpectedValueSolutionGHGGreenHouseGasGWhGigawatthouri.e.idestLPLinearProgrammingMILPMixedIntegerLinearProgrammingMWMegawattMWhMegawatthourPAR(k)PeriodicAutoregressiveModeloflag-kRHSRightHandSides.t.subjecttoSDDPStochasticDualDynamicProgrammingSDDPTStochasticDualDynamicProgrammingwithscenarioTree 11

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62 ].Manycountriesaroundtheworldrelyonhydro-electricpowerasoneoftheirelectricitysources.Hydro-electricpowercanbealargeelectricitysourcewithbasicallyzeromarginalelectricityproductioncost.Furthermore,hydro-electricpowerisacleanenergysourceemittingnegligibleamountsofgreenhousegases.However,theinvestmentcostaretypicallyimmense.Inconventionalpowersystems,thegoalistominimizeexpectedelectricitygenerationcost,whilemaintaininganadequatesecurityofsupply[ 124 ].Inthislight,ifwearegivenapurelyhydro-electricpowersystem,thenthesecurityofsupplyistheonlyconcern.However,iftherearethermalgenerationunitsnexttothehydro-electricpowerstations,thenthereistrade-offbetweenshort-termcosts(e.g.withinamonth)andlong-termeconomicandsecurityconsiderations(e.g.years).Thatis,thereisatrade-offbetweenusingthewaterforelectricitygenerationinthecurrentperiod(leadingtolowergenerationcostinthisperiodandmaybeapowershortageinthefutureifadraughtoccurs)orsavingthewaterforfutureperiods(leadingtohigherimmediatecostandmaybespillageinthefutureifaverywetseasonoccurs).Asthewaterinowsarenotknownwithafairamountofcertaintyoveralongertimeperiod(e.g.years),thoseinowshavetobeconsidereduncertain.Hydro-thermalpowersystemoperatorsfacedifferentuncertaintieswhendecisionshavetobemade.Thewaterinowsarethemostprominentamongthoseuncertainties-especiallywhenplanninginamid-termtolong-termhorizon.Untiltoday,therearenomodelsthatcanpredictthehydroinowswithagoodenoughprecisionforsuchatimeperiodtheoptimizationmodelsdesire;e.g.,oneyear.However,thoseinowsarenottotallyrandom,astherearesomeseasonalpatternsandastheinowstendtodepend 15

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14 15 ]istoensurethatanoptimalsolutiontoa(convex)optimizationproblemremainsfeasible,eventhoughtheremightbesomenoiseinthedata;i.e.,thedataintheoptimizationmodelmightnotbecorrectasitissubjecttouncertainty.Hence,thefeasibilityisthemainconcern.Mostpopulararethefollowingtwomethodstreatinguncertaintyinthewayrobustoptimizationisunderstood.Therstmethodassumesthatthedatavaryinacertain,knowninterval.Thisleadstoveryconservativesolutions.Thesecondmethoddoesalsonotassumethatthedistributionoftheuncertaintyisknown,butinstead,themeanandthestandarddeviationareavailable.Then,arobustcounterparttotheoriginalmodelisformed,whichignoressomerareeventswiththehelpofsafetyparameters.Again,theresultingsolutionsarerobustagainstcertainchangesintheuncertainparameter(s).Inbothcases,therobustcounterpartsremainconvexproblems[ 18 ].Floudasandco-workers[ 91 103 ]considertherobustnessofschedulingproblemsunderuncertainty.Specically,twocasesareconsideredwhichtintotheclassicationabove:boundeduncertaintyanduncertaintywithknownprobabilitydistribution.Theirrobustoptimizationframeworkisappliedtomixed-integerlinearprogrammingtechniquesforshort-termschedulingproblemstoderivesolutionswhichare,inasense,immuneagainstdatauncertainty.Similarly,VerderameandFloudas[ 171 ]proposearobustoptimizationapproachtowardsdemanduncertaintyfortheoperationalplanningofbatchplants.Theauthorsarguethatthetimehorizonofseveralmonthsmakeitmandatorytoconsiderdemanduncertainties.TheintegrationofboththeoperationalplanningandschedulingforbatchplantsunderdemandandprocessingtimeuncertaintyhasbeenproposedbyVerderameandFloudas[ 172 ]aswell. 16

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151 ].Especiallyinapplicationswherearobustsolutionisessential(suchassystemdesigns),stochasticprogrammingmethodsmightfail[ 6 ].However,whenthedistributionsofthedatacanbeestimatedfairlywell,thenthestochasticprogrammingapproachmightbefavorable;especiallywhendecisionsaremadeonaroutinelybasis,arobustoptimizationapproachmightbetooconservative.Onemightalsowanttocombinetheadvantagesofboththestochasticandtherobustapproachwhenitcomestodesignquestions[ 142 ].Despitetheadvantagesofrobustoptimization,stochasticoptimizationisthestandardinhydro-thermalscheduling.Thestochasticprogrammingmethodologyhasbeenusedformorethan30yearsagoandisacceptedasthetoolofchoiceinthehydro-thermalschedulingcommunity.Theprimaryreasonisthatthewaterinowsseemtobeindeedstochastic;thatis,theinowsfollowadistributionwhichcanbeestimatedfairlywellwiththedataavailable.Usingananalyticapproach,investmentdecisions(e.g.,innewpowerplants)havetotakeintoaccountthewholelifetimeoftheassetsathand.Thisleadstolong-termoptimizationproblems,havingatypicalhorizonof15-20years.Reservoirmanagementofhydro-thermalenergyplantsisatypicalmid-termhorizonproblemwithalengthof1to3years.Incontrast,short-termoptimizationproblemscovertypicallyuptooneweek(sometimesalsodays).Examplesareunitcommitment[ 152 ],economicdispatch 17

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173 ],andoptimalpowerowproblems[ 132 ].Asmentionedbefore,hydroinowsarenotknowinadvanceoverthehorizonofmid-termorlong-termoptimizationproblems.Therefore,thoseproblemsaretypicallytreatedwithstochasticapproachedwhereastheshort-termoptimizationproblemsaretypicallydeterministic. 18

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85 ].Forelectricutilities,thegenerationandhedgingdecisionsshouldbemadejointly[ 58 113 ]astheyarecorrelatedand,hence,theseparationtheorembyHolthausen[ 80 ]doesnotapply.Variousdifferentriskmeasuresforpowerutilitieshavebeenintroduced.ValueatRisk(VaR)beingtheestablishedstandardinthebankingindustryandothermajorapplicationareas[ 92 ],isalsoanimportantriskmeasureinthepowerindustry.EversinceRockafellarandUryasev[ 145 ]introducedtheirvariantcalledConditionalValueatRisk(CVaR),itgainedincreasingpopularityintheriskmanagementcommunityandrecentlyalsointheenergyindustry[ 180 ].Itsconvexityandcoherence[ 5 ]makeCVaRanattractiveriskmeasureforenergyapplications,especiallyinthecontextofhydro-thermalscheduling[ 82 85 ].Energysystemsaroundtheworldareindifferentstatesofthederegulationprocess.Therearestillcountrieswhicharecentrallydispatched,e.g.,CentralandSouthAmericancountries,whileothershavefacetsofboththeregulatedandthederegulatedmarkets.InGermany,forinstance,somepublicutilitiesareconfrontedwiththesituationthattheyhavetoservetheelectricitydemandoftheircustomerswhileusingtheirownassetsaswellasdifferenttypesofenergycontracts.However,theyarenotallowedtosellelectricityinthespot-market.SuchasituationisdescribedbyRebennacketal.[ 141 ],whereashort-termportfoliooptimizationproblemwithaquarterhourresolution 19

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23 ]modellibraryunderthenamepoutil(PortfolioOptimizationforelectricUTILities)[ 61 ].ThismodelwasthenextendedbyRebennacketal.[ 139 ]tothetradinginthebalancingmarketwheretheimportantfeatureofthemodelremainsthedispatchofthepowerplantindiscretesteps.Thederegulationofenergymarketshasagreatimpactontheenergysystemandisnotwithoutfailures[ 97 ].Duetomarketpowerabusebydominantplayers,poormarketdesign,andthintradingofforwardandfuturescontracts,deregulatedelectricitymarketsmayfailtosupplyelectricityreliablyandcheaply.ExamplesofsuchfailureswerereportedinUK,Norway,Alberta(Canada)andCalifornia(USA)[ 176 ].Hence,thesemarketshavetobedesignedproperly,monitoredandanalyzedwithgreatcare.Insums,inaderegulatedelectricitymarket,theclassicalleast-costminimizationproblemforhydro-thermalpowersuppliersisreplacedbyrevenuemaximizationmodels.Inotherwords,theleastcostoperationofthesystem,oncetheconcernofthecentraldispatcher,isreplacedbythechallengeofoptimallybiddingintheelectricitymarket.However,underthehypothesisofabsenceofmarketpower,thesystemoperationwhereagentsarefreetosubmitpriceandquantitybidsisshowntobeequivalenttothatwhichresultsfromacentralizedleast-costscheduling.Hence,asapricetaker,theoptimalbiddingstrategyisgivenbythemarginalsystemcost,whichcanbederivedthroughacostminimizationmodel[ 36 69 ].Furthermore,throughafundamentalapproach,theelectricityspotpricescanbeestimatedthroughthecentrallydispatchedleast-costschedulingproblem;again,assumingtheabsenceofmarketpower[ 166 ].WewilldiscussthatmoredetailedinChapter 6 ofthisdissertation. 20

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1 ].TheKyotoProtocolaimstoreduceemissionsofsixgreenhousegases:CO2,methane,nitrousoxide,hydrouorocarbonsperuorocarbons,andsulphurhexauoride[ 2 ].WefocusinthisdissertationonCO2emissions;thoughtheconceptsdiscussedapplyingeneral.Themainquestion,however,isonlyansweredpartlybytheKyotoProtocol:howtoachievethesereductiontargetsfortheglobalgreenhousegasemissionseconomicallyandecologicallyworthwhile?Therearebasicallytwoalternativetoachieveanemissionreductiononacountrylevel:emissiontaxesorCap-and-Tradesystems.Ataxsystemdoesexactlydowhatitsnamesuggests:enforcingataxonemissions.Suchataxcanbeputintoplacewithasmallamountofbureaucracy,itiseasilycontrollableandthepriceisxed.However,thistaxseemspoliticallyinfeasibleandhasthedisadvantageofnotlimitingtheemissionsdirectly.Incontrast,aCap-and-Trademechanismputsacapontheemissionsofawholesystemwhereastheemissionpriceisdeterminedbyamarketmechanism[ 45 ].ThemainmotivationbehindacarbontaxoraCap-and-Trademechanismcomesfromthefactthatclimatechangeisrecognizedasaglobalproblemandcanonlybesolvedglobally.Hence,itisnotimportantatwhichexactlocationthegreenhousegasemissionsreductionisachieved,butmoreimportantly,thatthereductionisdoneinacostefcientway.WilliSutton,aninfamousUSbankrobber,oncerepliedtothequestionwhyherobbedbanks:Becausethat'swherethemoneyis.ExactlythisisthequestionforCO2emissions,too.Wheredohuman-madeCO2emissionscomefrom?TheU.S. 21

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48 ].Figure 1-1 showstheCO2emissionsinpercentageforvedifferentsectorsforthewholeUSin2007.ThegurerevealsthattheelectricityproductiontakesthelargestshareoftheCO2emissionswith38%followedbythetransportationsectorwith36%.Hence,aimingforCO2emissionreductionsinthepowerindustryisanaturalchoicenotonlybecauseitisthelargestCO2emitterbutalsobecausetherearecleantechnologiesreadilyavailable,e.g.,throughsustainableenergysourcessuchaswind,solar,biomassornuclear. Figure1-1. CO2emissionssharesfordifferentsectorsintheUSAin2007;datasource[ 48 ] TheenergysystemsintheUSvarysignicantlybetweenthedifferentstateswhichexplainsFigure 1-2 .ThestateofVermont(VL)hasthehighestrateofnuclear-generatedpowerintheUSwhilehavingnocoal-redpowerplant.Thismakestheirenergysystemclean.OntheotherspectrumistheStateofWestVirginiawhichgeneratesmostofitselectricitywithcoal-redpowerplants.TheUSemittedin2007approximately6,017millionmetrictonscarbondioxide[ 48 ].TheemittedmetrictonsCO2varywidelyamongthedifferentstatesoftheUS,asshowninFigure 1-3 .InthelightofFigure 1-2 ,thiscanpartlybeexplainedbythe 22

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CO2emissionssharesfordifferentsectorsfortheUSAstatesin2007;datasource[ 48 ] differentenergymixbutalsobythedifferentenergydemandsandeconomiesamongtheUSstates.ThelargestmultinationalemissiontradingschemeintheworldistheEuropeanUnionEmissionTreadingScheme(EUETS)forCO2[ 51 ].ThegovernmentsoftheEUmemberstatesagreedonnationalemissioncapsandallocatetheallowancestotheirindustrialoperatorsviaso-callednationalallocationplans.PlantoperatorshavetomonitorandannuallyreporttheirCO2emissionsandtheyhavetoreturntheusedemissionallowancesofCO2ineachyear;althoughtheCO2emissionsaregivenforseveralyearsinadvanceinordertoavoidannualanomalies.Thoseinstallationswhichhaveallowancesleftovercanselltheminthemarketorsavethemforfutureuse.Thosethatexceedtheirtotalemissionshavetopayaneof100epertonofCO2emissionsandtheirnamesarepublished. 23

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CO2emissionsperUSstatein2007inmillionmetrictons;datasource[ 48 ] SettingtheCO2emissionallowancesinaCap-and-Tradesystematanappropriatelevelisanon-trivialtaskastheexperiencewiththeEuropeanETSshows[ 157 ].ItturnsoutthatthetradingsystemisverysensitivetochangesintheamountofCO2emissionallowancesissued.Ifnotdoneproperly,thereareeithertoomanyallowancesavailable,leadingtoamarketpriceofzerofortheCO2emissionallowancesortoofewandthemarketpriceshitsthelevelofthene.Hence,appropriatetoolsandmechanismsarerequiredforsettingthoselevelsmeaningful.ElectricitycompanieswithinanemissiontradingschemeforCO2allowancesfacenewchallenges,managingtheirportfolioofassetsinanoptimalway.TheneedforoptimizationtoolswasalsoempiricallyshownbyEhrhartetal.[ 46 ],whereasimulationofCO2priceswasperformed. 24

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1. withinastageandindependentofpreviousstages; 2. dependentonpreviousstages;seasonaldependencies; 3. dependentonlyonpreviousstage;varyeverystate; 4. political/macro-economical.Examplesforeachuncertaintyare 1. outageofplants,loaductuations; 2. hydroinow; 3. electricityspotprices,CO2spotprices; 4. fuelprices,electricitydemand.Thesesourcesofuncertaintycanbetreatedasfollows: 1. withineachstage,separateofeachothertypicallytreatedviaMonteCarlosamplingthisisnotthefocusofthisdissertationandwerefertoCosta[ 31 ]; 2. viamultivariateperiodicautoregressivemodels,duetotheirlinearity,thisisthestandardapproachusedwithinthestochasticdualdynamicprogrammingscheme,discussedinChapter 3 ; 3. viaMarkov-Chains,transitionmatricesprovidetheconditionalprobability,usedinChapter 6 ; 4. viascenariotrees,topicofChapter 4 25

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2 ,webrieyreviewtheconceptsofmulti-stagestochastic(linear)programmingandapplyittohydro-thermalscheduling.Theintentionistoembedthehydro-thermalschedulingproblemsofthefollowingchaptersinawelldenedframework.Chapter 3 reviewsthesolutiontechniquescommonlyusedandexistingintheliterature.Especiallytheconceptsofstochasticdynamicprogramming(SDP)andstochasticdualdynamicprogramming(SDDP)arediscussedindetails.Thelattermethodisadoptedtosolveproblemsdescribedinthelaterchapters.Theincorporationoffuelcostuncertaintyandelectricitydemanduncertaintyinthecontextofhydro-thermalschedulingandtheirincorporationinthedynamicprogrammingframeworkisdiscussedinChapter 4 .Theproposedmethodcombinesatreeapproachforhandlingcertaintypesofuncertaintywiththeclassicalautoregressivemodelsforthewaterinows.ComputationaltestsareperformedontherealpowersystemsofPanamaandCostaRica.CO2emissioncapsonahydro-thermalenergysystemisthesubjectofChapter 5 .Thosequotasaremodeledusingreservoirs,allowingtheincorporationofemissionsystemcapsinthedynamicprogrammingframework,regardlessofthelengthofthestagesinthemodel,thetimehorizonoftheCO2emissionallowancesissuedandtheirexpirationdate.AcasestudyontherealpowersystemofGuatemalashowsthe 26

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6 .TheCO2emissionquotapricesareforecastedusingafundamentalapproach,wherethewholeelectricitypowersystemisdispatchedcentrallyinanoptimalway.Thisisdonebyusingacostminimizationmodel,incorporatingthetreeframeworkforfuelpriceandelectricitypriceuncertainty(developedinChapter 4 )aswellastheCO2emissionconstraints(developedinChapter 5 )intothestandardleast-costhydro-thermalschedulingmodel.Thismodelallowscapturingthejointcorrelationsofthedifferentstochasticitiesintheenergysystem:inow,electricityprice,fuelcost,electricitydemandandCO2emissionallowanceprice.ThisdissertationisconcludedinChapter 7 71 94 ].Inparticular,thedissertationhasthefollowingmajorcontributions: 1. Incorporationofthescenariotreeapproachtowardsthemodelingofuncertaintyintothedynamicprogrammingframework.Thisisnovelandcanbeseenasaunicationofthediscrete,tree-communityandthecontinuous,MarkovChain-community;cf.Chapter 4 .ThisextendstheresearchofPereiraandPinto[ 125 ]toanewalgorithmwecallSDDPTandisanimprovementoftheresultsofPereiraetal.[ 121 ]. 27

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ModelingofCO2emissionquotasonacentrallydispatchedhydro-thermalpowersystem.TheproposedoptimizationmodelallowsthecalculationofmarginalCO2emissionpricesforapowersystemviathedualmultipliersofthereservoirconstraints.ThisprovidesinsightsontheeffectsofCO2capsontheoperationalpart;cf.Chapter 5 .Theproposedreservoirmodeliscapableofaccommodatingadetailedrepresentationofemissionsandrelatedconstraints.ItisthusanalternativeformulationtotheworkofBelsnesetal.[ 13 ]. 3. Jointmodelingofseveraluncertaintiesalongwiththeircorrelationinaderegulatedelectricitymarket.Asub-modelisusedtogeneratethosepriceforecastsandastochasticoptimizationmodelisproposed;cf.Chapter 6 .ThisextendstheworkofBelsnesetal.[ 13 ]andMoetal.[ 112 ]. 28

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35 ].Thissentencecapturestheveryideaofstochasticprogrammingthatdecisionshavetobemadeunderuncertain/unknowndata.Throughoutthefollowing55years,arichtheoryandapplicationsofstochasticprogramminghavebeenestablished.WebegininSection 2.1 withagenericmulti-stagestochasticprogrammingformulationanddiscusstheunderlyingassumptionsoftheproblemandtheirformulations.ThissectionisbasedontheworkofBirgeandLouveaux[ 21 ]aswellasKallandWallace[ 93 ].Afterwards,thehydro-thermalschedulingproblemisintroducedandframedinthepreviouslypresentedcontextofmulti-stagestochasticprogramminginSection 2.2 .Wediscussextensionsandthevalidityofthemodelpresented.Thesediscussionsarethebasisfortheproceedingchapters. 29

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havetobemetbythestochasticdecisionvariablesxt(!).Then,themulti-stagestochasticprogramcanbewrittenasmincx+minE!22[q2(!2)x2(!2)+...++minE!t2j!t1,...,!2[qt(!t)xt(!t)]+...++minE!T2j!T1,...,!2[qT(!T)xT(!T)]... 30

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2.2 ontheexampleofhydro-thermalscheduling.Constraints( 2 )-( 2 )ofthemultistageprogramhaveaspecialstructureknownasnonanticipativityconstraints.Theideaisthatfordecisionsatstaget,therandomevents!t+1,...,!Tarenotknown,meaningthatthefuturestageshavenotbeenobserved.Inotherwords,thedecisionsintherststage,x,areindependentoftherandomevents!2,...,!Tandhavetobemadeaprioribeforetheyareobserved. 1. xedrecourse;i.e.,theconstraintmatrixWisxedandnotrandom,whichisshowninformulation( 2 )-( 2 )asWdoesnotdependontherandomoutcomes!; 2. deterministicstochasticconstraintmatrixTt(!t);i.e.,justlikeW,matrixTt(!t)isactuallyxedandreducestoTt; 3. continuousdecisionvariablesinlinearexpressions;i.e.,onlynon-negativecontinuousvariablesareincludedinthemodel,representedbyconstraints( 2 ). 4. discrete,niteandknowndistributionoftherandomoutcomes;asthisisaverystrongassumption,itissufcienttoassumethatwecanapproximatethe(discreteorcontinuous)distributionofthedatabyadiscretedistributionfairlywell; 5. tomeetconstraints( 2 )-( 2 )inaprobabilisticsense;i.e.,theyhavetobemetalmostsurely;asweassumeanitely,discretedistribution,thisassumptionisobsolete; 31

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thatthedistributionoftherandomoutcomeofstagetdependsontheoutcomeofallpreviousstages.Lateron,wewillassumetheMarkovproperty;i.e,thedistributionofstagetdependsonlyontheoutcomeofthepreviousstaget1.Forthehydro-thermalschedulingproblemsconsideredinthisdissertation,thelinearityofalldecisionvariablesisassumed.Theresultingmulti-stagestochasticlinearprogramsaremucheasertosolvecomparedtostochasticintegerprogramswithintegervariablesoccurringnotonlyintherststagedecisions.Oneexplanationofthischallengeisthefactthattheexpectedtthstagevaluefunctions(seebelow)arediscontinuousandnon-convex;cf.toBlairandJeroslow[ 22 ].Theresearchonsolutionmethodsformulti-stagestochasticintegerprogrammingproblemshavebeendrivenbytheresearchgroupsofSchultzandSen.Forreviews,werefertoSchultzetal.[ 154 ],Schultz[ 153 ]andSen[ 156 ].Thevalidityofthoseassumptionsforourapplicationtohydro-thermalschedulingisdiscussedlateroninSection 2.2.2 2 )-( 2 )becomequicklyverylarge,andhence,theyaredifculttosolve.Oneideatotacklethisproblemandtoexploititsblockdiagonalstructureistowritethemulti-stagestochasticprogramasaone-stageprograminthefollowingwaymincx+Q2(x) 32

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2 )fort=T1,...,1,leadingtotheexpectedt+1thstagevaluefunction Now,denethetthstagevaluefunctionfort=2,3,...,T1Qt(xt1,!t):=minxtfqt(!t)xt+Qt+1(xt)jWtxt=ht(!t)Tt1(!t)xt1,xt0g. 2 )-( 2 )isdeterministic.Hence,iftheexpectedsecond-stagevaluefunctionQ2(x)isanalyticallygiven,thenthemulti-stagestochasticprogram( 2 )-( 2 )reducestoadeterministic(non)linearprogram,becauseQ2(x)mightbeanonlinearfunctioninx.Formulation( 2 )-( 2 )isthereforealsocalleddeterministicequivalentprogramto( 2 )-( 2 ).IfweaddtotheassumptionslistedinSection 2.1.1 alsothatthecostcoefcientsqt(!t)arexed,thenQ2(x)isaconvexfunctioninvariablex.Thereasonisthatthestochasticpartonlyaffectstherighthandsideofconstraints( 2 )( 2 ).Thisyieldstopiecewiselinearexpectedfuturecostfunctions. 33

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34

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2 )onlydependsindirectlyonut.Constraints( 2 )ensurethattheremainingelectricitydemandissatisedthroughthethermalpowerplantsproduction.Weassumethatthethermalplantshaveonlythecapacityconstraints( 2 )restrictingtheiroperations.Asthevariableoperationcostsaswellastheelectricitydemandareassumedtobeknown,thethermalcomplementfunctionisadeterministicoptimizationproblem;thoughdependentonthestochasticvariablesutfort2.Toensurefeasibilityforanygivenhydrogenerationlevelsut,onecanreplacetheequality( 2 )bytheinequality.Wecannaturallyassumethattheoperationcostctjperunitenergyarepositive.Thisimpliesthatct(ut)isanon-negative,convexfunctioninut.Moreprecisely,ct()isapiecewiselinearfunction.Withoutlossofgenerality,wecanfurtherassumethateachhydroplantihasahydroreservoirwhichissubjecttoalowerboundv 35

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2 )and( 2 ),modelingthewaterbalanceconstraints.Forthescheduledhydro-electricgenerationu1andut(!t),respectively,thethermalcomplementfunctionsc1(u1)andctut(!t)determinethecostofmeetingtheelectricitydemandduringtheplanninghorizon. 36

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2 )-( 2 )doesnot(directly)includeanythermalgenerationdecisions.Thismakesthisformulationverygeneric.Inprincipal,anycomplementgenerationfunctioncouldreplacefunctionsct()in( 2 ).Insertingthethermalcomplementfunction( 2 )-( 2 )into( 2 )-( 2 ),leadstothefollowingmulti-stagestochasticprogrammingproblemz:=minXj2Jc1jg1j+1+minE!222"Xj2Jc2jg2j(!2)+2(!2)+...++minE!t2t"Xj2Jctjgtj(!t)+t(!t)#+...++minE!T2T"Xj2JcTjgTj(!T)+T(!T)#...# 37

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2 ).Theobjectivefunction( 2 )isthesumoftheminimumexpectedoperationcostsperstage.Lookingattherststage,werealizethatthewaterinowisassumedtobeknown,makingtherststagedecisionsdeterministic.Thesameholdstrueatanystaget.Thiswayofmodelingismotivatedbytheideathatthewaterinowsthroughoutthisperiod(remember,wearelookingattimehorizonsofoneweektoonemonth)canbeobservedandthedecisionsonthehydrooperationcanbeadopted[ 162 ].ThisisknownintheliteratureasWait-and-Seemodel.Wewaitforthestochasticevent(e.g.inow)tobeseenandmakethedecisionthen.Incontrast,theHere-and-Nowmodelsneedadecisionbeforetheoutcomeoftherandomeventisknown,[ 54 ].WewillseeinChapter 3 howaHere-and-Nowapproachchangesthemodelformulation.Incontrasttodeterministicmathematicalprogramming,problem( 2 )-( 2 )doesnotprovideasinglesolutionoftheproblem.Forstages2toT,awholesetofsolutionsispresented,dependentontherandomeventsoccurring.Hence,thissolutionsetisnaturallycalledapolicy.Inpractice,oneismostinterestedintherststagesolutions.Lookingataweeklyormonthlyresolutionofproblem( 2 )-( 2 )withatimehorizonofafewyears,therststagesolutioncarriestheinformationofthefuturestages;e.g.,balancingtherisksofrationing,draughtsorspillage.Thisrststagesolutionisthenusedastargets(inoneortheothersense)fortheshort-termoptimizationproblems.Hence,onetypicallyrunsoptimizationproblem( 2 )-( 2 )everyweekormonthtoexploitthenewinformationavailableonthestochasticcomponents.Wenowdiscusshow( 2 )-( 2 )tsintotheframeworkofmulti-stagestochasticprogrammingasgivenbyformulation( 2 )-( 2 ).Thedecisionvariablesxaregivenasvectorsing1,u1,v2,s1,and1;similarlyforvariablesxt(!t).Withthis 38

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2 )aregiventhroughtherststageelectricitydemandconstraints( 2 ),therststagewaterbalanceconstraints( 2 )andtherststagelimitsonthedecisionvariables( 2 ),deningconstraintmatrixAandtherighthandsidevectorb.Foragivent2T1,thetthstagestochasticconstraints( 2 )aregivenbythecorrespondingelectricitydemandconstraints( 2 ),waterbalanceconstraints( 2 )andvariablesbounds( 2 ).TherecoursematrixWtcorrespondstothecoefcientsgivenbythevariablesgt,ut,vt+1,st,andt,whilematrixTtisgivenbythecoefcientsofvt.Hence,bothmatrixesarex;i.e.,theydonotchangewiththerandomwaterinow.Incontrast,therighthandsidevalueht(!t)isindeedstochastic,representingtheuncertainhydroinowsat(!t).Thisistheonlystochasticcomponentamongtheinputdata.Withthenotationsabove,theobjectivefunctiontsnaturallyintothegenericstochasticframework.Hence,theprogramhasxedrecourseandaxstochasticmatrixTt.Thereasonisthatthetechnologydoesnotchangeinthesystem;e.g.,thethermalandhydrogenerationefciencyremainsthesamethroughouttheoptimizationhorizon.ThelinearityassumptionisdiscussedinSection 2.2.2 whiletheassumptionsonthedistributionsoftheuncertaininowsarepartofChapter 3 .AsurveyofgeneralstochasticprogrammingmodelsforenergyproblemsisgivenbyWallaceandFleten[ 173 ].Labadie[ 99 ]surveysvariousdifferentapproacheshowtoattackuncertainhydroinowsinmulti-reservoirsystems. 2 )-( 2 )morepractical,additionalconstraintshavebeenproposed[ 130 131 ]: 39

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40

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2 )-( 2 )remainspreserved;i.e.,theresultingstochasticprogrammingformulationrespectsthestructureof( 2 )-( 2 ).Thisisveryimportant,assolutiontechniqueslikeDynamicProgramming(DP)exploitthisstructure.WewillcomebacktothisinChapter 5 wherewediscussthemodelingofCO2emissionconstraints,spanningmultiplestages.Sofar,werestrictedourselvestolinearrelationshipsbetween,andfor,thevariables.Sometimes,thestructuremayallowtoincludeadiscretestructure(mostlikelyinatwo-stageproblem)whichreducestoalinearprogram.Examplesforsuchstructuresarenetworktypeconstraints.Moregenerally,suchareductiontolinearprogrammingisalwayspossible,iftheconstraintsetoftheprobleminextensiveformistotallyunimodular[ 3 26 143 ]. 41

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182 ].Thedailyexperiencetellsusthattheworldofdecisionsisnotlinear.However,evencomputerscientistshavetoadmitthattheworldisnotblackandwhite.Thesameholdstrueforhydro-thermalscheduling.Theproblemofinterestmightnotbelinear, 3 .Thesecutsprovidetheexpectedwatervalueforallfuturestagesconsidered. 42

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175 ],forinstance,introducedapiecewiselinearapproximationofthenonlinearWeymouthpanhandleequationstomodelandsolveagasnetworkoptimizationproblem.ThecomputationalresultsontheBelgiumGasnetworkdemonstratedtheeffectivenessofthismodel.ThecommonlinearizationoftheactiveandreactivepowerowequationsarediscussedbyRubio-Barrosetal.[ 150 ].AcomparisonofdifferentpowerowmodelsispresentedbyOverbyeetal.[ 120 ].Eventhoughtheserelationsarenon-convexandnon-smooth,theselinearizationsworkverywellinpractice,whenthepowersystemisinsteadystateasthenthethreemainassumptionsonthislinearizationaresatised:lossesareneglectable,phaseangledifferencesatadjacentbusesaresmall,andthereactanceismuchgreaterthantheresistance.Despitetheargumentabove,onecanpragmaticallyclaimthatthereasonismainlydrivenbytheoperationsresearchscience:Largescale,non-linear,non-convex,non-smoothmulti-stagestochasticprogrammingproblemsareforthefuturegeneration,optimistically. 43

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181 ].Sincethistime,signicantalgorithmicimprovementshavebeenmade.Wesurveythehighlightsofthelatestdevelopmentsinthisareatosolvethehydro-thermalschedulingproblemsdiscussedinChapter 2 .ThesurveybyLabadie[ 99 ]in2004coversahugevarietyofdifferentmodelsformulti-reservoiroptimizationproblemsandreviewstheirsolutionmethods.Discrete,non-linearandmulti-objectiveproblemsarediscussednexttolinearmodels;astonishingly,SDDPhasnotbeenmentioned.Hence,thischapterisagoodcomplementtothemainfocusisgivenonDPmethodssuchasSDDP.Afundamentaldifferenceisthewayinowuncertaintyistreated.Weclassifythesolutionmethodsintothreedifferentgroups: 1. deterministicmodels, 2. scenario-basedmethods, 3. sampling-basedmethods.Deterministicmodelstreatthehydroinows(andotherpossibleuncertainties)asknown.Asthisisnotthemainfocusofthisdissertation,weonlybrieydiscussdifferentdeterministicapproachesinSection 3.1 whichhavebeenappliedtothehydro-thermalschedulingproblems.Scenario-basedmethodsgenerateup-frontasetofrealizationsoftherandomspace.Therealizationsarethenusedtogeneratetheextensiveformofthestochasticprogram,yieldingforourapplicationtoa(very)large-scale(linear)programmingprograms.Thosemathematicalprogramsarethentypicallysolvedexactly;i.e,toglobal(=local)optimality.Hence,thesolutionqualitydependsontheapproximationoftherealizationstotheoriginal,stochasticprogram.Fortwo-stagestochasticprograms,thenumberofrealizationscanbefairlylargeleadingtogoodsolutions.However,for 44

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3.2 ,wereviewscenario-basedmethodswhichhavebeensuccessfullyappliedtothehydro-thermschedulingproblems.GeneralsolutionmethodsincludetheL-shapedmethod[ 160 ],thediagonalquadraticapproximationmethod[ 115 ]andtheaugmentedLagrangiandecompositionmethod[ 149 ].Incontrast,sampling-basedmethodsgeneratesamplesoftherandomspaceon-the-yandsolvetheresultingproblemsapproximately.Often,probabilisticconvergenceresultsareestablishedfortheproposedsolutionmethodswhilestatisticalmethodsstopthealgorithmsafteranitenumberofsteps.Thereisarichclassofsuchmethodsavailableintheliterature.Themostrelevantmethodsappliedtohydro-thermalschedulingarereviewedinSection 3.3 .Othermethodsincludethestochasticlinearizationmethod[ 53 ],theauxiliaryfunctionmethod[ 33 ],thestochasticdecompositionmethod[ 78 ],thesamplepathoptimization[ 144 ],ortheseparableapproximationmethod[ 128 ].Someauthorsclassifythesolutionmethodsforstochastichydro-thermalschedulingproblemsintoLinearProgramming(LP)andDPmethods.Theclassicationintoscenario-basedandsampling-basedmethodsisslightlybroader,asLPmethodsaretypicallyscenario-basedmethodsandDPmethodsaresampling-basesmethods. 2 )canbetakenintoaccount.However,thosemodelsaretypicallyrestrictedtoMixedIntegerLinearProgramming(MILP)problemsorconvexquadraticprogrammingproblems,aslarge-scalegeneralnon-convexmodelsarecurrentlynotcomputationaltractable.Nevertheless,unitcommitment,start-upcostsaswellasminimumupanddowntimeconstraintstintotheframeworkofMILP.Hence,thisapproachallowstheincorporation 45

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30 ]usedinterior-pointmethodstosolveaschedulingproblemforthehydro-dominatedenergysystemoftheSwissRailways.Medinaetal.[ 111 ]presentedaclipping-offinteriorpointalgorithmforsolvingadeterministic,large-scale,MILPmid-termhydro-thermalschedulingproblem.ResultsontheSpanishpowersystemrevealthesuperiorityofthealgorithmoverstandardinteriorpointmethods.Asemideniteprogrammingapproachtowardsaconvexquadratichydro-thermalschedulingproblemwasdevelopedbyFuentesandQuintana[ 59 ].Theirapproachrunsinpolynomialtimeduetoaconvexquadraticmodelingofthecostfunctionandconstraints,thus,avoidingintegervariables.Pereiraetal.[ 123 ]successfullycombinedanalyticalmodelsandMonteCarlosimulationtoovercomecomputationalchallengesposedontheanalyticalmodel.Theideaistosolveeasierproblemsanalyticallyandleavethedetailedmodeltothesimulationscheme.Testonamulti-reservoirhydro-electricpowersystemillustratethemethod.TheideaofMonteCarlosimulation,asusedbyBaslisetal.[ 8 ],istogenerateafairlylargenumberofscenarioswhicharethenusedasinputdatainadeterministicmodel.TheauthorsusedeterministicMILPproblemsandapplyittotheGreekpowersystem.Computationaltestswereperformedwith100MonteCarloscenariosusinghourlytimesteps,providingadistributionoftheoptimalsolution. 174 ].Thecomputedscenarioshavetypicallytheshapeofatreebranchingateachstageorafansetofindividualscenarios. 46

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146 ]pioneeredtheworkonscenarioanalysisandaggregationwiththeirworkin1991.Givenasetofscenarios,ratherthansolvingthedeterministicequivalentintheextensiveform,theyproposedaprogressivehedgingalgorithm,whichiterativelygeneratespoliciesbymodiedscenariosubproblems.Specicconditionsarederivedforwhichtheobtainedpolicy(solutiontothestochasticoptimizationproblem)convergestotheoptimalpolicycorrespondingtothedeterministicequivalent. 43 ].Intheirwork,differentpossibilitiesforscenario-treegenerationarestressed,i.e.,usingclusteranalysisincaseofexternalscenariopathgenerator,orimportancesamplingtobuildawholetreefromscratch.Theirmethodiswidelyapplicableasconvexitywithrespecttotherandomparametersisnotrequired.HylandandWallace[ 84 ]presentascenariogenerationmethodformulti-stageproblems.Thelimitednumberofdiscreteoutcomessatisfystatisticalpropertieswhichhavetobegivenasinput.Usingnon-linearprogramming,theappealingideaistominimizesomemeasureofdistancebetweenthespecicationsandthestatisticalpropertiesofthediscreteapproximation.Hylandetal.[ 83 ]presentaheuristicmethodforthegenerationofscenariosformulti-stagestochasticoptimizationproblems.Asthecriticalpropertiesofthegenerated 47

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147 ]aswellasHeitschetal.[ 77 ]forstochasticprogrammingproblemsallowtheanalysisofthegeneratedscenariotreesbymeasuringtheclosenessoftheobtainedsolutiontothe(unknown)solutionoftheoriginalstochasticprogram.Basedonthisstabilitytheory,Dupacovaetal.[ 44 ]aswellasHeitschandRomisch[ 74 ]developedaframeworkforscenariogenerationandreduction,approximatingtheunderlyingdiscreteprobabilitydistributionwhichreplacedthe(continuous)stochasticprocesses.ThesescenariotreeapproximationschemeswerefurtherdevelopedbyHeitschandRomisch[ 76 ].ThedevelopedscenarioreductionalgorithmshavebeenimplementedinGAMS[ 23 ]havingthesolvernameSCENRED[ 60 ]. HylandandWallace hasbeenappliedtotheNordicelectricitymarketbyFletenetal.[ 58 ].Theauthorsusedascenariotreeapproachtowardsanportfoliomanagementprobleminthederegulatedhydro-powerelectricitymarket.Numericalresultsarepresentedforave-stagescenariotreemodelwith256scenariosspanningatwoyeartimehorizon.Similarly,Shresthaetal.[ 159 ]usedascenariotreewith243scenariosoversixperiodstomodelandsolveahydro-powerprotmaximizationproblem.HeitschandRomisch[ 75 ]discussastochasticpowermanagementproblemviamultivariatescenariotrees.Thescenarioscarrytheinformationofuncertaintiesfromelectricalload,streamowstohydrounits,marketpricesoffuelandelectricity.Inordertoreducethedeterministicequivalentofthemulti-stagestochasticprogram,ascenarioreductiontechniquebasedonrecursivedeletionandbundlingofscenarioswasused. 48

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70 ]discussscenariogenerationandscenarioreductiononahydropowerprotmaximizationproblem.Again,amultivariatescenariotreeisused,capturinguncertaintiesinelectricalload,spinningreserve,inows,fuel/electricityprices.ThesoftwareSCENREDwasusedforthescenariotreereduction. Eichhornetal. [ 49 50 ]walkthereadernicelythroughtheprocessofscenarioreduction,scenariotreegeneration(approximation)andmulti-stagestochasticprogrammingmodelingviaariskmanagementproblemforelectricityportfolios.Differentriskmeasures(linearandmixedinteger)arediscussed,allowingthemaximizationofexpectedrevenuesandtheminimizationofrisksimultaneously.Theresultingmathematicalprogramsarelargescalelinearormixedintegerprograms.Inordertocapturetheinowuncertainty,alargescenariotreemayberequired,leadingtoverylargescaledeterministicequivalentprograms;cf.[ 57 159 ].Especiallyifthenumberofhydroreservoirsislarge,thecorrelationamongthehydroinowsrequiresafairlylargenumberofscenarios.Thus,sampling-basedmethodsreceivedgreatattentionintheliteratureaswellasinpracticetosolvemid-termandlong-termhydro-thermschedulingproblems. 2 )( 2 ),seeBellman[ 11 ].Thisso-calledBellmanrecursionisapowerfuldecompositionoftheproblemandisthebasisformany(mainlysampling-based)solutionalgorithmsofstochasticprograms.AsurveyofDPtechniquesforwaterreservoirmanagementwaspresentedbyLamondandBoukhtouta[ 100 ]in 1996 49

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2 .Theliteratureuntilthe1980'sfocusedmainlyonstochasticdynamicprogrammingmethods,interpolatingthetth-stagevaluefunctionwhilediscretizingthestatespace.However,thisleadstoanexponentialincreaseinthesizeoftheproblemtosolve,knownasthecurseofdimensionality;theseconceptswillbecarefullyaddressedinSection 3.3.1 .Inordertoovercomethisproblem,aremarkableseriesofresearchfollowedbasedonnestedBenders'decompositionmethod[ 16 ].AmongthemarethepapersbyPereiraandPinto[ 124 ],Jacobsetal.[ 89 ]andMorton[ 114 ],whileVelasquezetal.[ 170 ]presentedamodicationofthedualdynamicprogrammingalgorithmof PereiraandPinto forthemulti-stagecase.ThecurseofdimensionalitywasthennallybrokenbythestochasticdualdynamicprogrammingmethodofPereira[ 122 ]andPereiraandPinto[ 125 ].ThisisthesubjectofSection 3.3.2 .Theresearchafter1991onhydro-thermalschedulingproblemswasverymuchdrivenbythisSDDPmethod.Recently,theso-calledApproximateDynamicProgramming(ADP)algorithmshavebeenappliedtoavarietyofmulti-stagestochasticoptimizationproblemswithadynamicstructure.Similartostochasticdualdynamicprogrammingmethod,thecurseofdimensionalitycanbebrokenbyADPapproaches;cf.Powell[ 129 ].ADPhasbeensuccessfullyappliedtovariousrealworldoptimizationproblemsincludingproblemsarisingfromenergyapplications;cf.toEndersetal.[ 52 ]andLohndorfandMinner[ 107 ]. 50

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2 intheformofequations( 2 )-( 2 ).LetusnowhaveacloserlookatthemethodologyofSDP.Thefollowingismainlybasedon[ 122 130 131 ]. 2 )-( 2 )intheone-stagerecursionframework,consideraspecicstaget.Then,thetaskistondoptimalhydro-thermalschedulingdecisionsforthisstageatanexpectedminimalcost;i.e.,minimalexpectedoperationcostforstagetplusminimalexpectedfuturecosts.Perassumption,atthisstaget,thepasthasoccurredwhichmeansthatthepreviousinowsaswellasthehydroreservoirlevelsvtareknown;withvtbeingavectorinthehydroreservoirsi2I.Then,themulti-stagestochasticprogram( 2 )-( 2 )canbedecomposedintoso-calledone-stagedispatchproblemsasfollowszt(vt):=minE!2t"Xj2Jctjgtj(!)+t(!)+zt+1vt+1(!)# 3 )ensuresthattheelectricitydemandcorrespondingtoperiodtissatised,withthepossibilityofrationing.Thewaterbalanceequationsaremodeledthrough( 3 )wherevtiaretheinitialwaterreservoirlevelsforstaget.Theboundsanddomainforeachvariablearegivenin( 3 ).Theobjectivefunction( 3 )isthesum 51

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2 )-( 2 )canbeobtainedinthesensethatzz1(v1)withtheinitialwaterreservoirlevelv1.However,thetrickypartishowtocorrectlyguesstherighthydroreservoirlevelsvtforeachone-stagedispatchproblem( 3 )-( 3 ).Asthoseone-stagedispatchproblemscannotbesolvedcomputationallyforthewholecontinuumofreservoirlevelsvt,adiscretizationintoNvaluesvnt,n2N=f1,...,Ng,maybechosen.Problem( 3 )-( 3 )isthensolvedforthoseNvaluesvnt.Typically,SDPalgorithmsinterpolatethesolutionvaluescorrespondingtotheNreservoirlevelstoobtainafuturecostfunctionforthepreviousstage.Thesameconceptappliestothepastinows.AsmentionedinChapter 2 ,wedenotebytthesetofpossibleoutcomes,conditionedonthepastoutcomes.Forthesakeofthisdissertation,werestrictourselvestorandomprocessesdependentexclusivelyontheoutcomeofthepreviousstage.Suchmodelsarecalledmodelsoflag-1.Hence,tisnowthesetofallpossibleoutcomes!t2,conditionedonlyonthepreviousoutcome!t1;i.e.,t:=j!t1.Inthiscase,theone-stagedispatchproblem( 3 )-( 3 )dependsonanotherstatevariable!t1as 52

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2 ,weassumethatwecanapproximatethedistributionwithdiscreteandnitesamplesofaknowndistribution.InthecontextofSDP(lateronalsoforSDDP),theseinowscanbemodeledasalinearautoregressivemodelviaacontinuousMarkovprocess(incontrasttoaMarkovchain),takingintoconsiderationthecorrelationtotheinowsofthepreviousstage(s).Asweassumealag-1model,theinowsinstagetdependonlyonthepreviousinowsinstaget1.Asimpleversionofthisinowmodelisthengiventhrough withinowmeant,standarddeviation&t,modelparameters1and2,andindependentrandomvariablessampledfromanappropriatedistribution;typicallyastandardnormaldistribution.WediscussthisinowmodelingreaterdetailinSection 3.3.1.1 53

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3 )-( 3 )reducesthentoaone-stagedeterministicoptimizationproblemzt(vt,at1):=minXl2Lpl"Xj2Jctjgltj+lt+zt+1vlt+1,alt# 3 )-( 3 )decomposesintoLindependentproblems,oneforeachwaterinowsamplealti.Inordertosimulatethestochasticinowthroughouttheplanninghorizon,asampleofMso-calledforwardinowsamt,m2M=f1,...,Mg,isderivedfromthelinearautoregressivemodel( 3 ).Theforwardinowsaretypicallyequallyprobable;i.e.,pm=1=M.Thoseforwardinowsamplesarethenthepastinows.Hence,wehaveforeachstoragediscretizationn2N,forwardinowm2Mandbackwardopening 54

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3 )-( 3 )bytheaverageoperationalcost.Hence,thisisnotnecessaryfortherststageandweobtainfortheone-stagedispatch 55

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2 )-( 2 )astheaveragevalueovertheMinowscenarios;i.e.,z1:=Xm2Mpmzm1(v1,am0)z. 2 )-( 2 )forhydro-thermalschedulingintroducedinChapter 2 assumesadeterministicrststagedecision.Hence,thewaterinowsa1ofstageonearedeterministicandnotstochastic.However,inordertocalculatez1(asanestimateofz),Mone-stagedispatchproblems( 3 )-( 3 )aresolved.Again,theconceptofWait-and-Seecomeshereintoplaywhereweassumetoknowtheinowsoftherststagewhenthedecisionismade.Onecanfurtherobservethattherststagedoesnotdependonl,nobackwardopeningsarepresent.Thesebackwardopeningsreectthestochasticwaterinowinthefuturestages.TheSDPissummarizedinAlgorithm 3-1 .Therunningtimeisdominatedbythenumberoflinearprogrammingproblemssolved;whichisMNL(T1)+M.Thatdoesnotlooktoobad.However,theproblemisthatthestatespace(givenbyvtand 56

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3.3.1.1 3 )-( 3 ) 3 ) 3 )-( 3 ) 3 ) 12 101 ].Therefore,mostapplicationsofSDPtohydro-thermalsystemsarerestrictedtosinglereservoirsystemsorsystemswithafewreservoirs. 135 ].However,asymptotically,thegrowthremainsexponential 57

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3-1 isastaticalgorithminthesensethatthediscretizationofthestatespacehastobechoseninadvance.Asthequalityoftheobtainedsolutiondependscruciallyonthechoiceofthisdiscretization,thereiscertainlyatrade-offbetweencomputationaltimeandsolutionquality.Furthermore,thereisnomechanismavailablemeasuringthequalityoftheobtainedsolution.Insum,therearethreemainchallengesforSDPalgorithms:curseofdimensionality,staticdiscretizationofstatespaceandlackofsolutionqualitymeasure. 3 ).Letushaveacloserlookhowthatworks.Beforehand,aseriesofinowscenariosaregeneratedforeachstaget.Thiscanbedonebyapplying( 3 )anappropriatenumberoftimeswhilesamplingfromthedistributionof.However,weactuallywanttheinowdecomposedintothereservoirsi2I.Thiscanbedonebyapplying( 3 )foreachoftheIreservoirs,takingintoaccountthattherandomvariablesarecorrelated.Foreachforwardscenariomandstaget,wesampleIindependentrandomvariablesmtiN(0,1) 3 )togeneratecorrelatedinowsforeachofthereservoirs.ThesamecanbedonetoobtaintheLbackwardopeningsforeachforwardinowscenario.Algorithm 3-2 summarizestheseideas.Theinputdatat,&t,1and2areestimatedusinghistoricaldata.Typically,suchdataareavailableoverarelativelylargetimeperiod,letussay30to50years.However, 58

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3 ) 3 ) 3 ) 3 ) 3 ) 3 ) 59

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3 )-( 3 ).Theunderlyingassumptionwasthatthehydroinowisstochastic,butthattheuncertaininowrevealsitselfduringthestagethedecisionhastobemade.Therefore,weassumethatinstagetatthetimewhenwemakethedecisionofthewaterreleasesandthethermalproduction,theactualhydroinowisknown.Thisisthereasonwhyallthedecisionvariableshavethescenarioindexl=1,...,L.Inotherwords,anoptimaldecisionforstagetcanbemade.RecognizethattherearestillLinowscenariosaswedonotknowtheinowinadvance(beforestaget).ThisisknownintheliteratureasWait-and-Seemodel.Wewaitforthestochasticeventtooccurandmakethedecisionafterwards.Thiswayofmodelingthehydro-thermalschedulingproblemsismotivatedbytheideathatthewaterinowsthroughoutthisstage(remember,wearelookingattimehorizonsofoneweektoonemonthforeachstage)canbeobservedandthedecisionsontheoperationofthehydro-thermalsystemcanbeadjusted.Incontrast,assumingthatthewaterinowsforstagetareindeedstochasticduringstaget,thentheone-stagedispatchproblems( 3 )-( 3 )changeasfollowszt(vt,at1):=minXj2Jctjgtj+t+Xl2Lplzt+1vlt+1,alt 60

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3 ),thedecisionvariablesonthespillagesltiandthewaterreservoirlevelsvlt+1iattheendofstagetarestochastic(theyhavescenarioindexl).Recognizethatproblem( 3 )-( 3 )doesnotdecomposeintoLindependentlinearprogrammingproblemsincontrasttotheWait-and-Seeformulation.Modelsofthetype( 3 )-( 3 ),requiringadecisionbeforetheoutcomeoftherandomeventisknown,arecalledHere-and-Nowmodels;cf.FaberandStedinger[ 54 ]aswellasStedingeretal.[ 162 ]forfurtherdiscussiononthistopic.Fortheremainderofthisdissertation,werestrictourselvestotheWait-and-Seemodels. 72 ],Buras[ 24 ],and Turgeon [ 167 168 ].SurveysaregivenbyYakowitz[ 177 ],Yeh[ 181 ],andStedinger[ 161 ].AmorerecentreviewofSDPmethods,theirapplicationsandcomparisonstootherstochasticapproacheswasgivenbySahinidis[ 151 ].OneofthestrengthsofSDPmethodsisthattheycanhandlenon-convex,non-smoothone-stageprograms;i.e.,theone-stagedispatchproblems( 3 )-( 3 )canbasicallybeofanytype.Thoseone-stageproblemscouldevenbesolvedapproximately,forinstance,withheuristicmethods.AthoroughdiscussionofcomputationalaspectsofSDPmethodswasgivenbyHanson[ 73 ].Specialfocusisgivenoncomputationalwaystoovercomethecurseofdimensionality;i.e,withgridcomputing,parallelizationandbetterdatastructures. 61

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55 ]foralong-termhydro-thermalschedulingproblem.TheproposedalgorithmovercomesseveraldrawbacksofDPmethods;e.g.,thediscretizationofthestateandcontrolvariablesisnotrequired.AvariantofDParethedifferentialdynamicprogrammingmethodsoriginallyintroducedbyMayne[ 110 ]fordeterministicoptimalcontrolproblems.Thebasicideaisaquadraticapproximation,usingsuccessiveapproximations.ExtensionstothestochasticcasearegivenbyJacobsonandMayne[ 90 ].WerefertoYakowitz[ 178 ]formoredetailsondifferentialdynamicprogrammingmethods.ChaerandMonzon[ 28 ]discussthestabilityregionforSDPalgorithms.Morespecically,theyderivearelationshipbetweenthetimeintegrationstep,thespacediscretizationstepandthemaximalincomingandoutcomingows.TheUruguayanhydro-thermalsystemischosenforthecomputationaltests.AcomputationalcomparisonofnestedBendersdecompositionandDPmethodsforhydro-electricoptimizationproblemswasperformedbyArchibalsetal.[ 4 ].Thetestedproblemsareofmid-size(havingupto17reservoirs)witharelativeshorttimehorizonofveperiods.TheresultsofDPalgorithmswherewithin3.2%oftheoptimallycomputedsolutionsusingthesimplexmethod.Asurveyofstochastic(anddeterministic)dynamicprogrammingalgorithmsisgivenbyYakowitz[ 177 ]in1982.Hediscusseddifferentoptimizationproblemsrelatedtowaterresourcesamongwhicharealsoreservoirmanagementproblemsthoseareveryclosetoourhydro-thermalschedulingproblems.Weclosethissectionwitharemarkablecitationfromthe1982article:Thesituationwithrespecttostochasticdynamicprogrammingisthatthereare,asyet,nowidelyapplicablecomputationaldevisesotherthandiscretedynamicprogramming(DDP).Becauseoftheircurseofdimensionality,...DDPisnotadequateforsolvingmanywaterresourceproblemsof 62

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122 ]andPereiraandPinto[ 125 ]wasnallyabletoovercomethebiggestdrawbackofthepreviouslyusedmethods(likeSDP):thecurseofdimensionality.SDDPusesdualinformationtounderestimatethefuturecostfunctionviaextrapolationtechniques,notonlytoreducethesizeofthestatespace,butalsotomakethediscretizationprocessdynamicandtogenerateboundsontheoptimalsolutionatthesametime,thuseliminatingallmajordrawbacksoftheSDPmethods.SDDPisnowanestablishedmethodanditisstillstate-of-the-artinsolvinghydro-thermalschedulingproblems.Itisextensivelyusedinoperationsstudiesanddispatchcentersinmorethan30countriesacrosstheworldspanningallvecontinents,includingSouth,CentralandNorthAmerica,Austria,Spain,Norway,Turkey,NewZealandandChina.In2008,a10yearsuccessstoryofSDDPfortheBraziliansystemwaspresentedbyMaceiraetal.[ 109 ].ThereareseveralexcellentdescriptionsoftheSDDPalgorithmavailableinliteraturenexttotheoriginalpublicationsofPereira[ 122 ]andPereiraandPinto[ 125 ].AmongthemarethearticlesbydeOliveiraetal.[ 37 ],TilmantandKelman[ 165 ],Gjelsviketal.[ 65 ],aswellasBezerraetal.[ 19 ].ThedocumentsofthecompanyPSRfortheirhydro-thermaloptimizationtools,namedafterthealgorithmSDDP,areespeciallydetailedonthemodelingaspects[ 130 131 ]. 63

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3 )-( 3 )isjointlyconvexinvtandat1.Therefore,rstignorethefunctionszt+1(,)in( 3 ).Then,zt(,)isaconvexfunctionjointlyinvtandat1,astheybothappearintheRHSofalinearprogram.Forthelaststage,perconstruction,wehavezT+10.Withthelatterargument,zT(,)isconvex,jointlyinvtandat1.Asthesumofconvexfunctionsisconvex(plt0),thefunctionzT(,)isconvex.Iteratingthisargumentprovesthatzt(,)isaconvexfunctioninthereservoirlevelvtandthepastinowsat1.TheconvexityofthefuturecostfunctionisimportantasthisallowsSDDPtounderestimatethefuturecostfunctionviaextrapolation.Thisworksasfollows.Evaluatingfunctionzt(,)atthespecicpointsvntandamt1foralln2Nandm2M,leadstothefunctionvalueszt(vnt,amt1).Ifwecanalsoobtaintheslopestmnandatmnofthefunctionzt(vnt,amt1)atthispoint(recall,thesearebothvectorsinthereservoirsi),thenwecanextrapolatethewholefunctionzt(,),duetoitsconvexity.Inotherwords,wecanunderestimatethefunctionzts(,)viathe(linear)slopesoftheplanesatthepointsvntandamt1.Hence,weobtainthefollowinglinearprogram,deningalowerboundonthetruefunctionzt(,)z 64

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3 )-( 3 )intoLindependentproblems( 3 )-( 3 ).Theslopestmnandatmnarethengivenbytmn:=@zt(vt,) 3 )=====@PLl=1plzlt(vt,) 3 )=====@PLl=1plzlt(,at1) 3 )====LXl=1pl@zlt(,at1) 3 )asarowvectorofthewaterreservoirsi2I,foragivenstoragevaluevntandpreviouswaterinowsamt1.Thenwederive forallt2T1,m2M,andn2N.Fornotationalconvenience,wedene consistentwiththezerofuturecostforthelaststageT.Asfortheevaluationpointsvntandamt1thelinearplaneandthefunctionzt(,)toucheachother,oneobtainszt(vnt,amt1)=tmnvnt+atmnamt1+ctmn, 65

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3 )isalsoanapproximationofthetrueexpectedoperationalcostforandafterstaget.Evenattheevaluationpointofvntandamt1,functionzt(vnt,amt1)maynotreectthetruecost,ingeneral.Thepiecewiselinearapproximationofthefuturecostfunctionvia( 3 )-( 3 )allowsus,basedonformulation( 3 )-( 3 ),toderivethefollowinglinearprogrammingformulationfortheone-stagedispatchproblemzlmnt(vnt,amt1):=minXj2Jctjgtj+t+t+1 3 ),settingtheslopesofthepiecewiselinearapproximationofthefuturecostfunctionforstageTaszero,theminimizationin( 3 )forcesdecisionvariableT+1tovaluezeroinanyoptimalsolution. 66

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3 )-( 3 )viarelation( 3 )byusingthepiecewiselinearapproximationofthefuturefunctioncuts,leadingtozm1(v1):=minXj2Jc1jgm1j+m1+2 67

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3 )doesnothaveabackwardopeningscenarioindexl,asthosewhereusedtoconstructthefuturecostfunctionsinthebackwardpass.Furthermore,( 3 )isnolongerdependentonthepreviouswaterinow(s).Inthebackwardpass,thosepreviousinowsareimportant,astheycarryinformationonthestochasticwaterinowinthecurrentstage.Intheforwardsimulationphase,thisisencodedindirectlyintheMforwardinowscenarios.Trackingtheencounteredoperationalcostforagiveninowscenariomwhilerunningthroughthestagesprovidestheactualcostofoperationforthisinowscenario.andcalculatesas wheregmtjandmtareoptimalsolutionsofthecorrespondingvariablesinproblem( 3 )-( 3 ).Hence,theaveragecostoverallMinowscenariosisthengivenby 68

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Wecanstopthebackward-forwarditerations,oncetheencounteredoperationcost^zliesinsidethecondenceintervalz 3-3 .Oneobservesthatthecomputationalcomplexityofthealgorithmisdominated(similartoSDP)bythenumberofLPproblemstobesolved.Ineachmainiteration,thereareM(1+NLjT1j)LPproblemsforeachbackwardpassandMTLPproblemsforeachforwardsimulation.IncontrasttotheSDPalgorithm,whereNgrowsexponentiallyinthereservoirsize(seeSection 3.3.1 ),atthebeginning,onetypicallyassignsinSDDPonestoragevaluediscretizationforeachforwardinowscenario;i.e.,N=1.DuringtheMonteCarlosimulationphase,newstoragevaluesaregeneratedwhichdynamicallyupdatethediscretizationsetNtobeusedinthenextbackwardpass,ifthealgorithmdoesnotconvergebefore.Thisleadstoalineargrowsofthesizeofthestoragediscretizationinthenumberofmainiterations.PracticalexperienceshowsthattheSDDPalgorithmconvergesinafewiterations(lessthan20)evenforlarge-scalepowersystemswithmorethan30hydroreservoirs.Thus,theSDDPalgorithmisabletobreakthecurseofdimensionality. 106 ]presenttherstconvergenceanalysisofsampling-basedmultistagestochasticlinearBendersdecompositionmethodswheretheuncertainparametersoccuronlyintheconstraintrighthandsides(neededforconvexityreasons). 69

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3.3.1.1 3 )-( 3 ) 3 )and( 3 ) 3 )-( 3 ) 4 ) 4 )-( 4 ) 3 ) 3 ) 3 ) 3 ) 70

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127 ]provedtheconvergenceofsampling-basedmulti-stagestochasticlinearprogramsundersomemilderassumptionswhicharesatisedforSDDPalgorithms.Linderothetal.[ 104 ]computationallytesttheconvergencebehaviorofsampleaverageapproximationsmethodsfortwo-stagestochasticlinearprogramswithrecourse.Thereportedresultsareveryencouragingasgoodsolutionswerecomputedforalltestedproblemswithreasonablecomputationaleffort.AverythoroughdiscussionontheconvergencebehavioroftheSDDPalgorithmisgivenbyShapiro[ 158 ].Again,theanalysisisrestrictedtothecaseoftwo-stages.However,theauthorreportsatheoreticallyveryslowconvergence;cf.alsoLemarechaletal.[ 102 ].Eventhoughnoanalyticalresultsforthemulti-stagecasearepresented,theauthorsconjecturethatforthemulti-stagecase,itiscomputationallyimpossibletocomputetheexactsolutions,ingeneral.Forpracticalproblems,however,Shapiroadmits:withareasonablecomputationalefforttheSDDPalgorithmcouldproduceapracticallyacceptableandimplementablepolicy.Asapracticalrecommendation,astrongerstoppingcriteriaasthecondenceinterval( 3 )fortheSDDPalgorithmissuggested;i.e.,theapplicationofat-testontheexpectedvaluecorrespondingtodifferentpolicies. 71

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PereiraandPinto inthefollowingastheclassicalSDDPalgorithm.DinizanddosSantos[ 39 ]aswellasdosSantosandDiniz[ 42 ]proposeanextensionoftheclassicalSDDPalgorithmbyincorporatingtheinformationofseveralfuturestagesintoonestage.Thisisdonebyinclusionofadditionalvariablesandconstraintsofthestagesahead.Theprocessiscontrolledbytheso-calledtimestepaggregationfactor.Theaimofthistechniqueistoreducethenumberoffuturefunctioncutstobegenerated,yieldingaspeed-upofcomputationaltime.However,duetotheincreaseoftheproblemsizes,thereisatrade-offbetweenthetimestepaggregationfactorandtheconvergenceincrease.Itisworthmentioningthatthisapproachhasthesameaccuracy(withrespecttothetimediscretization)asthetheclassicalSDDPalgorithm.NumericalresultsfortheBraziliansystemarepresented,demonstratingasignicantspeed-upcomparedtotheclassicalSDDPalgorithm.Inordertoenhancetheinowmodels,InfangerandMorton[ 88 ]proposecutsharingtechniquesamongdifferentscenariosubproblemsatthesamestage.ThisallowssamplingbasedmethodsliketheclassicalSDDPtohaveserialdependenciesoftherandomoutcomes.MaceiraandDamazio[ 108 ]presentaparadoxcasefortheBrazilianhydro-thermalsystemwherecounter-intuitiveresultswhereobtainedbyusingaPeriodicAutoregressiveModeloflag-k;PAR(k).Thekeyobservationsarethenegativecoefcientsinthestreamowmodelandtheproposedsolutionistoreducetheorderofthemodelinthosecases.SimilarobservationshavebeenmadebyNoakesetal.[ 116 ].ForthecaseofthepowersystemofBrazil,Granvilleetal.[ 68 ]suggestedanelectricitytransmissionconstrainedhydro-thermalstochasticoptimizationmodel.TheFistandSecondKirchoff'slawsdrivingthepowerowhavebeenlinearized.Thenaturalgassupply,demandandtransportationnetworkhasbeenincorporatedintotheframeworkofSDDPbyBezerraetal.[ 20 ].Thegaspressuredifferencebetween 72

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164 ]considerastaticandadynamichydropower-irrigationmanagementproblem.Inthestaticproblem,axedamountofwaterisassignedtotheirrigationsystemswhileinthedynamicapproach,thewaterirrigationispartoftheoptimizationprocess.TheresultingproblemswheresolvedusingSDDP.ComputationalstudyshowedthebenetsofthedynamicmodeloverthestaticapproachontheexampleoftheEuphratesriverinTurkeyandSyriawithanexpectedprotincreaseof6%.Donohue[ 40 ]developedtheso-calledAbridgedNestedDecomposition(AND)methodoriginallydesignedtosolvethedynamicvehicleallocationproblem.ThismethodissimilartotheSDDPalgorithmthoughrequiringasmallersamplesizeforconvergenceasdescribedbyDonohueandBirge[ 41 ].BirgeappliedthismethodtotherealpowersystemofColombia;cf.[ 126 ].ChenandPowell[ 29 ]introducedaCUtting-PlaneandPartial-Sampling(CUPPS)algorithm.Thebasicideaistouseforwardsamplingpassestogeneratevalidcutswhichsupportthefuturecostfunction(theauthorscallitexpectedrecursefunctionwhichistheestablishedterminthestochasticprogrammingcommunity)frombelow(forminimizationproblems).IncontrasttoSDDP,thereisnobackwardpassandthecutsaregeneratedintheforwardsimulationphase.Theproposedmethodisconvergentwithprobabilityone.TheconceptofGeneralizedDualDynamicProgramming(GDDP)wasintroducedbyVelasquez[ 169 ].SimilartoSDDP,theGDDPapproachusesBenderscuts.Controltheoryaspectsallowthealgorithmtodistinguishbetweenstatevariablesandcontrol 73

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134 ]andRead[ 133 ]in1984whichiscalledConstructiveDualDynamicProgramming(CDDP).Thebasicideaistosolvethedualofthedynamicprogrammingformulationdirectly.Thisallowstoconstructthemarginalvaluesurfaceexactly,deninganoperatingpolicyoverthewholestate-space.ThisisthemajordifferencetotheclassicalSDDPalgorithmwherealocallyaccurateapproximationiscomputedinstead.AverydetaileddescriptionoftheCDDPalgorithmdiscussingalsoefcientimplementationtechniquesisgivenbyReadandHindsberger[ 135 ].CDDPalgorithmsarelimitedtolower/mediumdimensionalproblemsasthecurseofdimensionalityisnotbroken.Thus,ifthenumberofhydroreservoirsinthesystemislarge(e.g.,asfortheBrazilianpowersystem),thenCDDPmethodsarenotcomputationalfeasibleandSDDPmethodsarepreferable.However,CDDPalgorithmsposesseveraldesirablefeatures.First,byconstructingthemarginalvaluesurfaceexactly,simulationstudiescanbeperformed.Second,linearityisnotrequired,allowingnonlinearriskmeasureorgame-theoreticcomponents.SeveralextensionstotheoriginalCDDPhavebeenproposedintheliterature.Todealwithinowcorrelations,YangandRead[ 179 ]suggestedtheuseofanadditionaldimension.Again,withanincreaseinthedimension,Kerretal.[ 96 ]embeddedriskmeasuresintotheframeworkofCDDP.GamingcomponentsweresuccessfullyaddedbyScottandRead[ 155 ]whichwasthenfollowedbyaseriesofpublications,namelybyBatstoneandScott[ 10 ],Stewartetal.[ 163 ],aswellasReadetal.[ 136 ].ACDDPalgorithmhasbeenappliedtotheNewZealandhydro-powersystembyCulyetal.[ 34 ]andlaterbyCraddocketal.[ 32 ].CDDPhasalsobeenappliedtotheNordicpowersystems,[ 135 ]. 74

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1 ,thederegulationoftheelectricitymarketaddsanotherstochasticcomponenttothehydro-thermalschedulingproblem:Electricityspotprices.ThisstochasticcomponentistypicallymodeledasaMarkovProcess,makingthespotprice(s)additionalstatevariable(s)nexttothereservoirlevelsandpastinows.Intheprotmaximizationmodel,thefuturecostfunctionbecomesafuturebenetfunctionwhichisaconcavefunction,jointlyinthereservoirlevelsandpastinows.Recallthatthereasonwastheconcavityoflinearmaximizationproblemsinrighthandsidevariations.However,withrespecttothespotprices,thefuturebenetfunctionbecomesconvex.Thereasonisthatthespotpricesappearintheobjectivefunction.Thus,thefuturebenetfunctionissaddle-shaped,whichmakesitimpossibletoapplytheSDDPalgorithmdirectly.Thiscomesfromtheconcavityrequirementonthefuturebenetfunctioninordertoapproximateitbyapiecewiselinearfunction.Inordertoovercomethisdifculty,GjelsvikandWallace[ 66 ]introducedahybridSDP/SDDPmethod.TheappealingideaistocombinetheindividualstrengthoftheSDPandtheSDDPmethodthelackofconcavityassumptionandthebreakofthecurseofdimensionality,respectivelyinordertoovercometheirindividualweaknesses.Inthisscheme,thespotpriceforecastsaretreatedviaMarkovChainsinadiscretemanner(intheSDPframework)whilethereservoirlevelsandwaterinowsaremodeledbycontinuousapproximations(intheSDDPframework).AdescriptionofthemethodisgivenbyGjelsviketal.[ 64 ]andbyGjelsviketal.[ 63 ]aswellasinthetechnicalreportbyPereiraetal.[ 121 ].AdetaileddescriptionofthehybridSDP/SDDPmethodwithapplicationtotheNordiccountriesisgivenbyGjelsviketal.[ 65 ].Apenaltyfunctionapproachtowardsriskmanagementforhydro-thermalprotmaximizationwasintroducedbyKristiansen[ 98 ].TheresultingmodelswheresolvedusingahybridSDP/SDDPapproach.ComputationalresultsforoneofNorway'spower 75

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87 ]alsousedahybridSDP/SDDPapproachtosolveahydro-thermalprotmaximizationproblemsubjecttoriskconstraints.Usingthesamemethodology,Iliadisetal.[ 86 ]benchmarkeddifferentriskmeasuresforhydro-electricagentsinaderegulatedmarketwithfocusontheCVaRriskmeasure. 76

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PereiraandPinto ,thestochasticdualdynamicprogrammingalgorithmdealswithstochasticmulti-stagelinearprogramsinwhichtheuncertainparameterslieontheRHSoftheconstraintmatrix.Whileoriginallyappliedinthecontextoftheleast-costhydro-thermalschedulingproblem,ithassincebeenextendedtoadiversesetofapplications;cf.Iliadis[ 85 ],Flachetal.[ 56 ],Chabaretal.[ 27 ],aswellasCosta[ 31 ]andChapter 3 .Uncertaintyininowswasattheheartoftheoriginalapplicationandsomeoftheaforementionedextensionstothealgorithmweredevisedinordertocopewithuncertaintiesofadifferentnaturewhichcouldnotbeincorporatedintotheoriginalframeworkinastraightforwardmanner.ThischapterintendstocontributetothisbodyofworkbyfurtherextendingtheSDDPalgorithminordertotakeintoaccountadditionalsourcesofstochasticity.Thecentralissuesregardinguncertaintyanditseffectsonthedecisionstobetakenmayvarydependingonthetimehorizonandcharacteristicsofthesystemunderconsideration.Predominantlyhydrosystemsaremoreconcernedwithinowuncertainty,sincethatdirectlyaffectsthesystem'scapacityofsustainedenergyproduction.Thermalsystems,ontheotherhand,areusuallyfocusedonguaranteeingreliabilityattimesofpeakdemand,thusmakingunitoutagesanimportantissue.WefollowthespiritbyZimmermann[ 183 ]andChapter 1 toclassifytheuncertaintiesforourrealworldoptimizationproblemwithrespecttoitscontext.Ingeneral,uncertaintiesrelatedtothehydro-thermalschedulingproblemmaybebroadlyclassiedintofourgroups.Foreachoneofthesegroups,thewaytomathematicallyrepresenttheseuncertaintieshasanimmediateimpactonthemethodologiestoefcientlysolvetheresultingproblems.Therstgroupmayinclude,forexample,theavailabilityofeachgeneratingunit.Inthiscase,thebestapproachisusuallytoperformaprobabilistic 77

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31 ].Thesecondgroupincludessourcesofuncertaintytowhichatimeseriesmodelmaybeaccuratelyttedandexpectedtoprovidereasonableforecasts-uncertaintyininowsisanexamplethatliesonthiscategory.ThethirdgrouprelatestorandomvariableswhoseevolutionintimeisbetterrepresentedbyMarkovChains.Thatissometimesthecaseofelectricityspotprices[ 27 112 ].Finally,thefourthgroupdealswithsourcesofuncertaintythataremorecloselyrelatedtostructural,politicalormacro-economicalconditionsandcanonlybecharacterizedintheformofascenariotree,particularlywhenoneisinterestedinlong-termprojectionsratherthanshort-termforecasts.Growthinelectricitydemandortheevolutionoffuelpricesarethemostprominentexamplesinthisgroup,andareexactlythemotivationforourworkinthischapter.Giventherecentglobaleconomiccrisisandhugeswingsinoilprices,itbecameevidentthatrelyingonpointestimatesforkeyvariablessuchasdemandinfuturetimestagesandfuelcostsforthermalplantsmayresultinbiasedandriskydecisions.Pereiraetal.[ 121 ]proposedthemodelingoftheelectricitydemanduncertaintyasalinearauto-regressiveprocess.ThisistheoreticallypossibleandamenabletotheapplicationoftheSDDPalgorithmsince,aswillbeshowninSection 4.1 ,demandappearsattheRHSoftheconstraintmatrixand,hence,thefuturecostfunctionisaconvexfunctionindemand.Inpractice,however,alinearauto-regressivemodelseemsnottobeagoodpredictorfordemandsincethesearemean-revertingprocessesandarenotabletocapturethepossibilitythatfuturedemandmayfollowstructuralregimeswhicharecompletelydifferentfromthatofthepresent.Asthefuelpriceappearintheobjectivefunction,anauto-regressiveprocessmodelleadstoafuturecostfunctionhavingasaddleshape,destroyingthenecessaryconvexityoftheproblemwhichallowsittobesolvedwithSDDP.Hence,aMarkovChainapproachseemstobethenaturalwayandwasproposedby Pereiraetal. aswell,leadingtofuelpriceclusterswithtransitionprobabilities.Wewilldiscussthedifferencestoourproposedapproachin 78

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4.1.4 .Again,suchamodelisdifculttocalibrateanditseemsnottocapturethefuelpricedevelopmentcompletely.Morecomplexmodelsas,forinstance,proposedbyBatlleandBarqun[ 9 ]seemtobemoreappropriatetocapturethefuelpriceuncertainty,whichtheninturncanbetransformedintoascenariotree.Intheliterature,thereisawiderangeofpublicationssuggestingscenariotreeapproachesforthestochasticloadprocessandthestochasticfuelprices;cf.NowakandRomisch[ 117 ].Therearedifferentefcientmethodologiesforthegenerationofscenariosandthereductionofthesizeofthetreeiscomputationallyveryimportant;cf.Chapter 3 .Alagrangianrelaxationtechniqueforashort-termhydro-thermalschedulingproblemunderdemanduncertaintywasdevelopedbyDentchevaandRomisch[ 38 ].ThemaincontributionofthischapteristheextensionofDPbasedalgorithms,likeSDDP,byembeddingitintoascenariotreeframework,thuscapturingadditionalsourcesofuncertaintywhichcannotcurrentlybedealtwithinanefcientandmeaningfulmanner.AmongtheDPmethods,SDDPisthemethodofchoiceifitcomestohydro-thermalpowersystemswithasignicantshareofhydropowerandalargenumberofhydroreservoirs;cf.Chapter 3 .Thus,wediscussthejointhandlingofscenario-basedandsampling-basedstochasticmodelingontheSDDPmethod.Inotherwords,weembedthescenariotreeframeworkintheclassicalSDDPframeworkandcallthisextensionSDDPT(SDDPwithscenarioTree).Apaperbasedonthischapterhasbeensubmittedforpublicationtoascienticjournal[ 138 ].Theremainderofthischapterisorganizedasfollows.Anextensionoftheclassicalmulti-stagestochastichydro-thermaloptimizationproblemtouncertaintiesmodeledviascenariotreesispresentedinSection 4.1 alongwiththenecessaryalgorithmicmodicationsinordertosolvethismodelwithSDDPapproaches.CasestudiesfortherealpowersystemsofPanamaandCostaRicaareperformedinSection 4.2 .WeconcludethischapterwithSection 4.3 79

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2 )-( 2 ),wheretheelectricitydemandandfuelpriceswereconsideredtobedeterministic,seeChapter 2 .Thisproblemessentiallyconsistsofdeterminingtheoptimaloperatingpolicyfortheuseofhydroandthermalresourcessoastominimizetotalexpectedcostsinordertofullltheknowndemand.Weassumethatwearegivenahydro-thermalpowersystemwhichhastocentrallydispatchthegeneration.Theresultingproblemhasstillitsjusticationalsointhecontextofderegulatedelectricitymarkets,asitisthecoreofmanyoptimizationproblemssuchastheprotmaximizationproblembyMoetal.[ 112 ]ortheoptimalexpansionproblembyGorenstinetal.[ 67 ].Furthermore,manyhydro-thermalsystemsarestillcentrallydispatched;e.g.,CentralandSouthAmericancountries.Costminimizationmodelsforacentrallydispatchedpowersystemmightalsobeusedinafundamentalmodelinordertoforecastelectricityprices.ThisisdiscussedinChapter 6 .Letusdiscussnowhowtoincludeuncertaintiesintothehydro-thermalschedulingproblem( 2 )-( 2 ),whicharebestcapturedviascenariotrees.Candidatesforsuchuncertaintiesarefuelpriceuncertaintyandelectricitydemanduncertainty;i.e.,thedatactjordtarenowstochastic:ctj()ordt(),respectively,with2t.FollowingthenotationforthehydroinowuncertaintyofChapter 2 ,wedenotebytthesetofpossiblefuelpriceoutcomesconditionedonallfuelpriceoutcomesprevioustostaget.Weassumethatthehydroinowandtheuncertaintytreatedhereviaatreeareindependent;i.e.,randomoutcome!2and2tarestatisticallyindependent.Thisisjustiedinthecontextofourhydro-thermalapplicationasthehydroinowsshouldhavenoinuenceonthefuelpricesandtheelectricitydemand.Withoutlossofgenerality,inthissection,wediscussthisconceptofscenario-basedmodelingofuncertaintyontheexampleoffuelprices.Theideageneralizesnaturallyforother 80

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Scenariotreewith4stagesS1=fs11g,S2=fs12g,S3=fs13,s23g,S4=fs14,s24,s34,s44g,s112=fs11g,s123=fs13,s23g,s134=fs14,s24,s34g,s234=fs44g,ps11s121=ps23s443=1,ps12s132+ps12s232=1,ps13s143+ps13s243+ps13s343=1 4.1.3 .InChapter 2 ,wedenectjastheoperationalcostforelectricitygenerationofoneMWhforthermalplantj2Jinstaget.Theseoperationalcostaretypicallycalculatedastheproductoffuelpriceandfuelconsumption1.Whilethefuelconsumptionisatechnologicalparameterforeachthermalplantremainingbasicallyconstantthroughoutthelifecycleoftheplant,thefuelpriceissubjecttouncertainty.Thus,onetypicallyconsidersfuelcostuncertaintyinordertocapturethethermalplants'generationcostuncertainty.However,tokeepthenotationsimpler,weassumethethermalgenerationcostctjtobestochastic. 81

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4-1 .Then,atstagetforascenarios2St1,weobservethefuelpricecstjcorrespondingtoscenariosandtheone-stagedispatchproblem( 3 )-( 3 )isgivenbyzts(vt):=minE!2t24Xj2Jcstjgstj(!)+st(!)+X2st+1pst+1zt+1vst+1(!)35 4 )82

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4 )isoftypeWait-and-Seeforthefuelpriceuncertainty;i.e.,weassumetoknowtherandomfuelpriceduringstagetwhenwehavetomakethedecision;thisissimilartothehydroinowmodelingasdiscussedinChapter 2 andChapter 3 .Foragiveninowscenariosl2Landinitialwaterreservoirlevelvt,theexpectedcostatstagetandforascenarios=st2Stcanthenbecalculatedfromzlts(vnt,amt1):=minXj2Jcstjglstj+lst+X2st+1pst+1zt+1lst+1,alt 4 ).Toderivehosefuturefunctioncuts,wefollowthecentralthemeofChapter 3 4 )-( 4 ),foragivenscenarios2St,wenowhavest+1futurecostfunctions:oneforeachfuturescenarioproceedingfrom 83

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4 ).Then,zlts(,)isaconvexfunctionjointlyinvtandat1,astheybothappearintheRHSofanLPproblem(thesameargumentholdstruefortheelectricitydemanddt).Forthelaststage,zT+10.Withthelatterargument,zlTs(,)isconvexinvtandat1.Asthesumofconvexfunctionsisconvex(plt0),thefunctionzTs(,)isconvex.Iteratingthisargumentandexploitingthatpst+10aswell,wediscoverthatzts(,)isaconvexfunctioninthereservoirlevelvtandthepastinowsat1.Thisleadstothefollowing 4 )isaconvexfunction,jointlyinthereservoirlevelsvtandthepreviousinowsat1.Evaluatingthisfunctionzts(,)atthespecicpointsvntandamt1foralln2Nandm2M,leadstothefunctionvalueszts(vnt,amt1).Ifwehavealsotheslopestmnsandatmnsofthefunctionzts(vnt,amt1)atthispoint,thenwecanextrapolatethewholefunctionzts(,),duetoitsconvexity.Inotherwords,wecanunderestimatethefunctionzts(,)viathe(linear)slopesoftheplanesatthepointsvntandamt1.Hence,weobtainthefollowinglinearprogram,deningalowerboundonthetruefunctionzts(,)z 84

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4 )====@PLl=1plzlts(vt,) 4 )====@PLl=1plzlts(,at1) 3 )====LXl=1pl@zlts(,at1) 4 )asarowvectorofthewaterreservoirsi2I,foragivenstoragevaluevntandpreviouswaterinowsamt1.Thenweobtain Asfortheevaluationpointsvntandamt1thelinearplaneandthefunctionzts(,)toucheachother,oneobtainszts(vnt,amt1)=tmnsvnt+atmnsamt1+ctmns, 85

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4 )-( 4 )withtheoptimizationproblem( 4 )-( 4 )leadstothelinearprogrammingproblemzlts(vnt,amt1):=minXj2Jcstjglstj+lst+X2st+1pst+1t+1 4.1.1 .Thisworksasfollows.Intherststage,noobjectivefunctioncutshavetobecalculated;rememberthattheobtainedfuturefunctioncutsinstagetshowupintheformulationofstaget1.Hence,nobackwardopeningsareneededfortherststage(noindexlanymore).Furthermore,theinitialwaterreservoirlevelsrepresentthestateofthecurrenthydro-systemandareassumedtobeknown.AssumingaWait-and-Seemodelfor 86

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4 )-( 4 )reducestozm1s1(v1):=minXj2Jc1jgm1j+m1+X2s12ps122 4 ),weobtainalowerboundontheoveralloperationcostthroughz 87

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wheregmstjandmstareoptimalsolutionsofthecorrespondingvariablesinproblem( 4 )-( 4 ).Hence,theaveragecostoverallMinowscenariosisthengivenby whichisanestimatorofthesamplemeanoftherealoperationcostswhentheactualstochasticinowsoccur.Thestandarddeviationiscalculatedviarelation( 3 )andtheSDDPTalgorithmcanstopthebackward-forwarditerations,oncetheencounteredoperationcost^zliesinsidethecondenceinterval( 3 ).Apseudo-codeoftheSDDPTmethodisgiveninAlgorithm 4-1 .OneobservesthatthecomputationalcomplexityoftheSDDPTalgorithmisdominatedbythenumberofLPproblemstobesolved.Ineachmainiteration,thereareM1+NLPt2T1StLPproblemsforeachbackwardpassandMPt2TStLPproblemsforeachforward 88

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3 4 )-( 4 ) 4 )and( 4 ) 4 )-( 4 ) 4 ) 4 )-( 4 ) 4 ) 4 ) 3 ) 3 ) 89

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4 )-( 4 )areindependentforeachinowscenariosm2M,l2Landscenarioss2St.(Duetothestructureofthedecomposition,theLPproblemsaretimedependent.)ThissuggestsanaturalwaytoparallelizetheSDDPTAlgorithm 4-1 .However,onehastorecognizethataperfectspeed-upcannotbeexpected,asonetypicallyexploitsthesimilarityoftheLPproblemstobesolved;i.e.,forgivenstaget,theLPproblemsvaryonlyintheRHSvalues(observethatat+1~m~naltisaconstantforstagetaswell).Hence,warm-startsusingdualsimplexalgorithmsaretypicallyusedtoexploitthatstructure. 3 .RecallthatweusedtheargumentofLPproblemsbeingconvexinvariationsintheRHStoprovetheconvexity.Hence,addingastatevariablecorrespondingtosomedataotherthantheRHSmaydestroytheconvexityproperty;e.g.,astatevariableaffectingtheobjectivefunctionleadstoaconcavefuturecostfunctioninthatvariable. 90

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4.1.1 again,itisapparentthattheuncertaintycapturedviathetreedoesnotaffecttheconvexityofthefutureconstfunctionsinthestatevariablesvtandat1.Thisimpliesthattheuncertaintycapturedbythetreecanaffectanycoefcientoftheone-stagedispatchproblem( 4 )-( 4 )(e.g.,thermalgenerationcost,powercoefcientforhydroplants,electricitydemand,lowerandupperboundsonthermalgenerationdecisions,waterrelease,waterspillagereservoirlevelandspillage,aswellascoefcientsofalltheconstraintsdiscussedinChapter 2 ).TheonlychangeinLPproblems( 4 )-( 4 )isthattheappropriatecoefcientshaveascenarioindexs.Furthermore,thoseuncertaintiescanbehandledjointly.Thus,anymultivariatescenariotreecouldbeusedtomodelvariousuncertaintiessimultaneously. 121 ]proposedaMarkovChaintocopewithfuelpriceuncertainty.ThisMarkovChainhasastatespacesizeofKandistime-homogeneous.Hence,thetransitionprobabilitydistributiondoesnotdependonthestagesandcanberepresentedbyarightstochasticmatrix.ThisleadstoKpriceclustersforeachstagetwithstageindependenttransitionprobability.TheincorporationwiththeclassicalSDDPworksasfollows.Eachforwardinowm2Misassignedexactlyonesuchpricecluster(hence,MK)andthefuturecostfunctionin( 3 )issubstitutedbytheexpectedvalueofthefuturecostfunctionsforeachpricecluster.ThisleadstotheverynicepropertythatthenumberofLPproblemstobesolvedremainsthesameasintheclassicalSDDP.However,themaindrawbackofthismethodisthatitispracticallyverytrickytodenetheinitialvaluesforthecostclustersandtoderivemeaningfultransitionprobabilities.Furthermore,itisquestionableifthefuelpricesreallyevolveaccordingtoatime-homogeneousMarkovChain. 91

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4.1.1 .Hence,onewantstomakesuretouseatreeassmallaspracticallyfeasible.However,asdescribedinSection 4.1.2 ,theimplementationofaparallelizationschemewouldbesimpleandpossiblycapableofachievingcomputationaltimescomparabletothoseinwhichuncertaintyindemandorfuelpricesisnotconsidered.Furthermore,allthetechniquesforscenariogenerationandscenarioreductionreadilyavailableintheliterature(cf.Chapter 3 )canbeusedtogeneratethinscenariotrees,keepingtherunningtimeoftheSDDPTalgorithmcomputationallyfeasible. 92

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173 ].Weusethosecutsinourstudiestoobtainsolutionsfortherststage.Thisallowsustostudythedemandeffectsonanannualbasisfordifferentrststagedecisions.Inourcomputationalresults,weconsider7differentelectricitydemandscenarios.TheelectricitydemandforJanuaryisthesameforeachscenariowhilethedemandforallothermonthsaregivenbythecumulativepercentagechangerelativetothereferencescenarioat1.75%,1.00%and0.50%forthePanamasystemand1.00%,0.50%and0.25%fortheCostaRicasystem,respectively;i.e.,instaget,1
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4-2 (a)andforCostaRicainFigure 4-2 (b),respectively.ThepresentedSDDPTalgorithmhasbeenimplementedinMosel(Version3.0.0),analgebraicmodelinglanguagedevelopedbyY.ComlombaniandS.HeipckeforDashOptimization(nowFICO).TheresultingLPproblemsfromthedecompositionaresolvedusingXpressOptimizer(Version20.00.05).ThetimehorizonofchoiceisoneyearwithmonthlystageswheretherstmonthisJanuaryandthelastmonthconsideredisDecember.Toachieveaccuratecomputationalresultsandtoreducenoise,weuseM=100forwardinowscenariosandL=50backwardopeningsfortheSDDPalgorithm.Theinowsareforecastedviaalinearautoregressivemodeloflag-1;thecorrespondingparametersofequation( 3 )areestimatesusingrealdata. 4-1 .WecanseethatthegenerationcostperMWhrangedfrom$71.3to$313.4fortherstmonth.TheACP1toACP4fuelsareaspecialmixofdifferentfuelsforthoseparticularthermalplantsonly.Fortheconsecutivemonths,weassumethesamefuelprices,whileanannualdiscountrateof10%applies.Fortherstmonthconsidered,thexedthermalgenerationis82.4GWhwithagenerationcostofmillion$10.3andthethermalcapacityis426.4GWh.Overaoneyearhorizon,thexedgenerationcostaccumulatestomillion$115.5.AschematicdiagramofthehydrosystemofPanamaisshowninFigure 4-3 .Theinstalledhydro-electriccapacityforJanuarywas529.4GWh.TheelectricitydemandforJanuaryisassumedtobeknownat537.3GWh.Further,weassumeanelectricitydemandpresentingseasonaleffectswithadifferenceof2.1%betweenJanuaryandDecember.ThispatterncanbeseeninFigure 4-2 94

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B ElectricitydemandscenariosforthepowersystemsofPanamaandCostaRica.A)Panama.B)CostaRica 95

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Hydro-electricsystemofPanama

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4-2 .Computationalresultsareshownfor7differentelectricitydemandscenarios,anaverageofthoseandastochasticscenariocorrespondingtoFigure 4-2 .Thesecondrowgivesthethermalgenerationfortherststageforthebestcomputedsolution,whiletherowlabeledHydroindicatesthehydro-electricgenerationinGWhforJanuary.IntherowDemand,theyearlyelectricitydemandisgiven,whileinrowlabeledCosttheyearlygenerationcostareprovided(excludingthexedgenerationcost).EVSstandsforExpectedValueSolutionandthecorrespondingrowindicatesthecorrespondingyearlygenerationcost.Therowsprovidethepercentagechangewithrespecttothereferencescenario.TheresultsinTable 4-2 revealthattherststagedecisionsareverysensitivetochangesinelectricitydemand;recallthatthedemandoftherststageisthesameineachscenario.Thisisexplainedbytheveryideaofhydro-thermalelectricitysystems,whereonewantstohedgeagainstdryseasonswheretheinstalledthermalcapacitymightnotbesufcienttomeettheelectricitydemand(orveryexpensivethermalgenerationunitsmightbeneeded).Relativelyfullhydro-reservoirscanpreventelectricityshortagesduringthoseseasons.However,thiscomeswiththeriskthatsomewatermighthavetobespilledifawetseasonoccurs.Thisexplainsthetrendsinthehigher(lower)thermalelectricitygenerationfordemandincreases(decreases).However,thethermalelectricitygenerationdoesnotincrease(withademandincrease)withthesamerateasitdecreases(withademanddecrease);thesameholdstruefortheannualcost.Therstreasonisgivenbythehedgingagainstdryseasons;i.e.,theincreaseinfuturedemandleadstoanproportionalhigherincreaseofthehydroelectricitythanadecreaseforthecaseofdemanddecreases.Thesecondreasonisthatadecreaseindemandmayallowtousesomehydro-electricpowerintherststagetoavoidtheproductionusingthemostexpensivethermalplants.Thethirdreasonexplainingtherelativesimilarityinelectricityproductionofthecaseof+1.00%and 97

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4-2 ,thethermalandhydroelectricgenerationdecisionsfortherststagediffersignicantlyfortheaverageandstochasticcase.Thereasonisonceagainthattheextremecasesofelectricitydemandincreasesmayleadtoelectricityshortagesinfuturestageswhicharepenalizedheavily.Hence,thehigherreservoirlevelsinthestochasticcasecomparedtotheaveragecaseisahedgingagainstfutureelectricityshortages.Usingtheexpectedelectricitydemandasthesinglescenarioofchoiceleadsinourcasetothesamerststagedecisionsasthereferencescenario.Now,usingthisrststagesolutioninanyofthe7demandscenariosleadstotheso-calledEVS.Recognizethatweusethistermwithrespecttothetwo-stageuncertaintyintheelectricitydemandembeddedinamulti-stagestochasticoptimizationcontext.Hence,thisiscanbeseenasanadoptionofthisrecognizedterminology[ 21 ].Theincreaseinannualoperationcostsbyignoringtherandomvariationsintheelectricitydemandcomparedtothestochasticapproachiscalledthevalueofthestochasticsolution(VSS).Forourdata,wehavethattheVSSismillion$1.279,correspondingtoa0.81%costdecreasecomparedtotheEVS. 98

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Hydro-electricreservoirsystemofCostaRica;excludingadditional24run-of-theriverplants 4-3 .ForJanuary,thethermalcapacityamountedto490.4GWhwhilethexedgenerationcostperannumismillion$13.6.ThehydrosystemofCostaRicaconsistsof3reservoirswithelectricgeneratorsand26run-of-the-riverplants.Thehydro-systemforthe3hydro-reservoirsisshowninFigure 4-4 .Theinstalledhydro-electriccapacitywas2,438MW. 99

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4-4 showaverysimilarpatterntotheresultsobtainedforthePanamasystem.FollowingtheconventionsforthePanamacasestudy,theexpectedvalueofperfectinformationis$915,870whichis0.92%ofthecostincurredunderperfectinformation.TheVSSamountstomillion$1.142,ora1.12%costsavingcomparedtotheEVS. 3 intotwobroadclasses:scenario-basedandsampling-based.Thosetwoapproachestowardsuncertaintymodelingseemtobecontrarytoeachotheranddividetheresearchcommunity.However,thehybridmethodproposedinthisChapteremploysthebestfeaturesofbothmethodswhichallowsefcientlysolvinglarge-scalerealworldproblems.Morespecically,theadvantagesofscenario-basedmodeling(e.g.,copingwithcomplexforecastingtools)andsampling-basedmodeling(e.g.,copingwithlargenumberofinowsampleswithadetailedreservoirssystemwhileavoidingthecurseofdimensionality)aremergedtoa 100

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101

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ThermalplantsconsideredforthePanamapowersystem Min.Generation[MW]15151540201212010000Capacity[MW]40404015842.844963053.571218FuelType111233141567Cost[$/MWh]109.6109.6109.6122.0230.3152.474.596.271.3160.7313.4171.2 FuelType1:Bunker,2:DieselM.,3:DieselL.,4:ACP1,5:ACP2,6:ACP3,7:ACP4 Table4-2. ComputationalresultsforthepowersystemofPanamawithdifferentelectricitydemandscenarios.ThegeneratedelectricityisgiveninGWhandthecostaregivenin$1000 Scenario-1.75%-1.00%-0.50%Reference+0.50%+1.00%+1.75%AverageStochasticThermal257.8257.5264.1276.1268.2340.4340.2286.3302.0-6.63%-6.74%-4.34%-2.85%23.31%23.23%3.71%9.38%Hydro279.5279.8273.2261.2197.1196.9269.1251.0235.37.01%7.12%4.58%3.02%-24.64%-24.55%-3.92%-9.92%Demand5,866.56,110.66,280.26,455.66,636.96,824.37,117.56,470.2-9.12%-5.34%-2.72%2.81%5.71%10.25%0.23%Cost109,702128,330141,329151,664162,295189,495214,229156,721157,374-27.67%-15.39%-6.81%7.01%24.94%41.25%EVS109,794128,371141,333151,664163,198193,536222,675158,6530.08%0.03%0.00%0.00%0.56%2.13%3.94%

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ThermalplantsconsideredfortheCostaRicapowersystem Min.Generation[MW]000097.50019.601000Capacity[MW]14343626130130.57826.14210662425FuelType1771877887918Cost[$/MWh]103.3223.3223.384.52.7198.1186.72.72.7155.113.6868.72.7 FuelType1:Bunker,7:Diesel,8:Geoice,9:GeoCR Table4-4. ComputationalresultsforthepowersystemofCostaRicawithdifferentelectricitydemandscenarios.ThegeneratedelectricityisgiveninGWhandthecostaregivenin$1000 Scenario-1.00%-0.50%-0.25%Reference+0.25%+0.25%+1.00%AverageStochastic Thermal187.0195.8207.0215.8229.7236.9318.1227.2232.9-13.37%-9.30%-4.12%9.75%47.40%23.23%5.26%7.89%Hydro614.5605.7594.5585.7571.8564.6483.4574.3568.64.93%3.43%1.52%-2.37%-3.59%-17.47%-1.94%-2.91%Demand9,052.99,303.79,432.39,563.19,696.19,831.310,108.59,569.7-5.34%-2.71%-1.37%1.39%2.80%5.70%0.07%Cost70,57478,81584,19991,002103,768113,684158,176100,032100,947-22.45%-13.39%-7.48%14.03%24.92%73.82%EVS71,57579,68984,47891,002105,685116,616165,576102,0891.42%1.11%0.33%0.00%1.85%2.58%4.68%

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1 thattheelectricitysectorcontributesasignicantshareoftheoverallCO2emissionsandthus,isanaturalchoiceforCO2emissionregulations.HavinganannuallimitonthetotalCO2emissionallowancesonapowersystemdirectlyaffectsthewaysystemoperatorsdenetheoperatingscheduleofeachplantsinceanewelementmustnowbefactoredintotheequationontopoftheusualsourcesofuncertaintysuchasdemandandinows.Whileitisdesirablethatthegenerationof 104

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13 ]modelCO2reservoirsinSDDPviaahydroreservoir,wherethermalplantsareinterpretedashydro-electricstationsusingCO2allowances(insteadofwater)andtransmissionlinecostareusedtomodelthethermalgenerationcost.Inthispaper,weintroduceanalternativereservoirmodel,allowingcomplicatingconstraints(e.g.,fuelavailability)onthethermalplantsandallowingCO2emissionstoexpireasdeterminedbytheEUETSregulations.Otherthanthearticleby Belsnesetal. ,theliteratureonCO2emissionconstrainedhydro-thermalcostminimizationproblemsisverythin.ThereareafewarticlesforthecaseofliberalizedelectricitymarketsandCO2emissiontrading.However,thisissubjectofChapter 6 wherewereviewthosemethods.Thecontributionofthischapterisonthemodelingaspectoftheemission-constrainedhydro-thermalschedulingproblem.WeproposearepresentationofGHGemissionquotasasreservoirs,thusallowingittobereadilyembeddedintothestochasticdualdynamicprogrammingalgorithmwiththeadditionofstatevariablesintothefuturecostfunctions.Theproposedapproachisexibleandcapableofhandlingtherepresentationofemissionconstraintsbothatasystem-widelevelandinamoredetailedviewthat 105

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137 ].Althoughweadopttheroleofasystemoperatorthatdeterminesthedispatchofbothhydroandthermalplantsinacentralizedfashion,theutilizedmodelmayalsoprovideinsightsofthepotentialconsequencesofimposingemissionquotasinliberalizedmarkets.Underthehypothesisofabsenceofmarketpower,thesystemoperationwhereagentsarefreetosubmitpriceandquantitybidsisshowntobeequivalenttothatwhichresultsfromacentralizedleast-costscheduling;cf.Chapter 6 .Theremainderofthischapterisorganizedasfollows.InSection 5.1 ,weformulatetheproblemofinterestasastochastichydro-thermalschedulingproblem.Inordertosolvethismodel,aCO2reservoirispresentedinSection 5.2 alongwiththederivationofthefuturefunctioncuts,necessarytoincorporatethismethodologyintotheframeworkofDP.AcasestudyfortherealpowersystemofGuatemalaispresentedinSection 5.3 .WeconcludewithadiscussioninSection 5.4 2 whichissubjecttoCO2emissioncaps.Givenisahydro-thermalsystemwithIhydropowerplantsi2I=f1,...,IgandJthermalplantsj2J=fj,...,Jg.Decisionscanbemadeatdiscretestagest2T=f1,...,Tg(e.g.,monthly)ontheelectricitygenerationgtjofthethermalplantsj2Jandtheelectricitygenerationiutiofthehydropowerplantsi2I.Theobjectiveistheminimizationoftheexpectedoperationalcostzofthesysteminthelong-termhorizon,particularlytakingintoaccountCO2emissionquotas.Theoperationcostconsistsofthevariablecostfortheelectricityproductionofthethermal 106

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2 .Theinowsatiperhydroreservoiri2Iandstaget2Tareassumedtobestochasticwhileallothertechnicalspecicationsofthesystemareknown;inparticular,thegenerationcostctjperthermalplantj2Jandstaget2Taswellastheelectricitydemanddtperstaget2Taregivenasanaveragevalueoverthetimediscretizationlength;i.e.,monthly.Wenextformalizethisproblemasamathematicalprogram. 2 .Eachrandomoutcomerevealsacertainhydroinowforthecorrespondingstage.Then,thedescribedmid-termhydro-thermalschedulingproblemcanbemodeledasthefollowingmulti-stagestochasticlinearprogrammingproblem 107

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108

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2 ),thehydroinowsa1iareknownwithcertaintyintherststagewhenthegenerationdecisionsaremade.Thedecisionsinallotherstagesarestochastic.Letusnowhavealookatthefuturestagest>1.There,theknownelectricitydemanddthastobemetbythermaland/orhydroelectricityproductionasstatedinconstrains( 5 )wherearationingoft(!t)ispossiblebutpenalizedintheobjective.Thewaterbalanceequations( 5 )ensurethatthereservoirlevelvt+1i(!t)forreservoiriattheendofstagetequalsthereservoirlevelvti(!t1)atthebeginningofthestageplusthestochasticwaterinowati(!t)minustheturbinedwateruti(!t)minusthespilledwaterstiplustheinowsfromtheplantsimmediatelyupstreamofplanti,eitherfromupstreamhydropowergenerationorspillage;withthenotationv2i(!1)v2i.Constraints( 5 )modeltheemissionallowancesperhorizon,wheretheemittedtonsCO2duetothermalgenerationhavetobelessthenorequaltotheCO2quotaECO2yplustheadditionalCO2allowancesboughtvianesfy.Inthisformulation,inordertoreducethenotationalburden,weassumethattheemissionallowancesexpirerightbeforenewonceareissued.However,thegeneralizationisstraightforwardandthereservoirmodelproposedinSection 5.2 ismorepreciseinthisregard.Recognizethatconstraints( 5 )mayspanmultiplestages.Thisisparticularlyproblematicforthedecompositionmethodsavailableintheliterature.WediscussthisingreaterdetailbelowwhereweprovideanalternativemodeloftheCO2emissionconstraints( 5 ).Theobjectivefunction( 5 )isthengivenasthesumoftherststagegenerationcostplustheexpectedcostofthermalpowergenerationinthefuturestages,includingpossiblefeesforrationingandCO2emissionquotaviolations.Additionallinearoperationalconstraintscanbeaddedtothemodel( 5 )-( 5 )inordertomakeitmorepractical;e.g.,linearizedelectricityandgasnetworkconstraints,multipleloadblocks,andsub-systems.Foracomprehensivelistofconstraintsproposedintheliterature,refertoChapter 2 109

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5 )mayspanmultiplestagesand,hence,destroystheblockdiagonalstructureoftheoriginalhydro-thermalschedulingproblem( 2 )-( 2 ).Dynamicprogrammingmethods(andotherdecompositiontechniques)exploitthisstructureandcannotdealwiththisconstraintinitspresentform.Hence,wesuggestaformulationoftheCO2emissionquotasrespectingthestagedecompositionframeworkofdynamicprogrammingmethodslikeSDPandSDDP.ThequotaontheCO2emission,modeledviaconstraint( 5 ),canbeinterpretedasareservoirasfollows:Atanygiventimey2Yg(e.g.,atthebeginningoftheyear)itrainsCO2emissionrights,llingtheemissionsreservoir,seeFigure 5-1 .Ateachtimestaget2T,thereisabalanceequationfortheCO2emissionsasfollowset+1=etXj2JBjgtj+ft,t2TnYg 5 ),theemissionsallowancesdonotexpireattheendofthestaget,et+1,aretheCO2emissionsallowancesatthebeginningofstaget,et,minustheemissionallowancesusedviathermalelectricitygeneration,Pj2JBjgtj,plustheemissionrightsned,ftplustheemissionallowancesissued,ECO2t.Noticethatweneedtohavea(non-negative)variableftfortheemissionsexceedingthequotaforeachstaget2T(incontrasttoy2Yg)inordertoensurethattheemissionreservoirlevelet+1isnonnegativeattheendofeachstage.Inparticular,variablesftensurefeasibilityoftheCO2reservoirconstraintsintheone-stagedispatchproblemfortheSDDPalgorithm,derivedinSection 5.2.1 110

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CO2emissionreservoirs Withequation( 5 ),weareabletomodelthetwocases(i)theemissionallowancesexpireatstagesy2Yeor(ii)thereisnoexpirationdate.Thisisrealizedviadenition (5) forallt2Tande0=e1beingtheinitialCO2emissionallowances.Thismodelallowsthatemissionallowancesexpirewhenevernewallowancesareissuedornot,asitisthecaseintheEUETS;i.e.,YeYg. 5 )-( 5 )allowsastagedecompositionofthehydro-thermalschedulingproblem;cf.Chapter 3 .Beingatstaget,thecouplingbetweenthepreviousandfuturestagesisthengiventhroughthehydroreservoirlevelsvtastheinitialreservoirlevelforstaget,theCO2emissionslevelet,andthepreviousinowsat1theso-calledstatevariables.FollowingthelogicoftheLPproblems( 3 )-( 3 )and( 3 )-( 3 ),decomposingtheproblemintostagesandapplyingbackwardopeninginowscenariosl2Leachwith(conditional)probabilitypl,thefollowingdeterministicone-stagedispatchproblemis 111

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3 )-( 3 )asweaddtheCO2emissionreservoirconstraints( 5 )-( 5 )andthestatevariableset.Likeproblem( 3 )-( 3 ),theaboveLPproblemdecomposesintoLindependentproblems,oneforeachwaterinowsamplealt.AlongthelinesofLPproblems( 3 )-( 3 ),MforwardinowsareusedtosimulatethestochastichydroinowswhilethereservoirstoragevaluesvtandtheCO2

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3 andChapter 4 .Formulation( 3 )113

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3 )changesinthiscontextthentozm1(v1,e1,am0):=minXj2Jc1jg1j+1+CCO2f1+z2v2,am1 5 )-( 5 ).FollowingtheprinciplesofChapter 4 ,evaluatingthisfunctionztataspecicpointvnt,entandant1leadstoafunctionvaluezt(vnt,ent,amt1)2R.Ifweknowalsotheslopestmn,etmnandatmnofztatthispointvnt,entandant1,thenwecanextrapolatethefunctionztjustlikeintheclassicalSDDP.Inotherwords,wecanunderestimatethefunctionztviathe(linear)slopesofthepointsvnt,entandamt1.Hence, 114

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3 )and( 3 ).Letusdenoteby~lmntthedualmultipliersofconstraints( 5 ),thenweobtain forallt2T1,m2M,andn2N.Similarly,etmncanbederivedfort2T1,m2M,andn2Nthroughetmn=@zt(,et,) 5 )fort2TnYgandthedualmultipliersofconstraint( 5 )fort2Yg,respectively,forgivenemissionreservoirlevelent.Then,weobtainetmn=Xl2Lpllmnt,t2TnYg,m2Mn2N 115

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5 ),yieldingtotheLPproblemszlmnt(vnt,ent,amt1):=minXj2Jctjgtj+t+CCO2ft+t+1 116

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117

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perforwardinowscenariom2M.Theaveragecostisthenobtainedbytakingtheexpectationovertheforwardinowscenariosasinequation( 3 ).JustlikeintheclassicalSDDPalgorithm,thestandarddeviationofthoseMcostsarecalculatedviarelation( 3 )andthealgorithmcanstop,assoonastheestimatedoperationcost^zlieinsidethecondenceinterval( 3 ). 118

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3 5 )-( 5 ) 5 ),( 5 )or( 5 ),and( 5 ) 5 )-( 5 ) 5 ) 5 )-( 5 ) 5 )-( 5 ) 5 ) 3 ) 3 ) 3 ) 119

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5-1 .Analytically,therunningtimeinthenumberofLPproblemstobesolvedineachmainiterationofAlgorithm 5-1 remainsthesameasthegenericSDDPAlgorithm 3-3 :M(1+NLjT1j)+MT.However,thenumberofmainiterationsmayincrease,asthestatevariableoftheCO2reservoirhastobediscretized,too.ThismayalsoleadtoanincreaseinthesizeofthediscretizationsetN. 4 ,weembedthescenariotreemodelingofuncertaintyintotheframeworkofSDDPandobtainedtheSDDPTalgorithm.ThisalgorithmcanalsobecombinedwiththeCO2reservoirconstraints( 5 )-( 5 ).Thestate-spaceoftheone-stagedispatchproblemssolvedbytheSDDPTalgorithmisthenincreasedbytheCO2reservoirlevelsetalongwiththecorrespondingCO2emissionconstraints.Consistentwiththeincreaseinthestate-space,thefuturefunctioncutse.g.,inconstraints( 4 )areextendedbytheadditionalvariable,combiningtheCO2reservoirconstraintsthroughthevariousstages.ThefuturecostfunctioncutscorrespondingtotheCO2reservoirdependthenalsooneachscenarios2St.Inotherwords,et+1mnhasanadditionalscenarioindexs:et+1mns. 6 .Typically,thoseprot 120

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CO2emissionfactorsfordifferenttypesofthermalplants.Allotherplantsareassumedtohavezeroemissions typeCO2factorunitsource coal2.86[tonCO2/ton][ 81 ]diesel22.38[poundsCO2/gallon][ 47 ]bunker78.8[kgCO2/MMBtu][ 47 ] maximizationmodelsincludeonlya(small)subsetofpowerplantsofthewholepowersystem;e.g.,theplantsbeingownedbyoneoftheseveralpowerproducersinacountry.InordertocaptheGHGemissionsonacountrylevel,GHGemissionquotasmightbeassignedtoeachproducerseparatelyasitisdoneintheEUETS.Hence,inordertoobtainanaccuratefundamentalmodel,thecostminimizationmodelshouldbedividedintodifferentregions(e.g.,oneforeachpowersupplier)subjecttocertainquota.Thiscanbemodeledbyassigningeachregionitsownreservoir,whereallowancesareuseduponlybythecorrespondingpowerplants. 5-2 .Weassumeaconstantfuelpricethroughouttheplanninghorizon,leadingtoaconstantproductionpriceforeachplant.Guatemalahadtwohydro-reservoirsplants,schematicallyshowninFigure 5-2 ,aswellastenrun-of-the-riverplants.Therun-of-the-riverplants'totalinstalledcapacitywas354.9MW.TheCO2emissionsfactorspertypeofplantusedtocalculatetheCO2emissionsperplantoftheGuatemalapowersystemareprovidedinTable 5-1 .TheSDDPalgorithmwiththeCO2emissionreservoirconstraintshasbeenimplementedusingthemodelinglanguageMosel(Version3.0.0).TheresultinglinearprogrammingproblemsfromthedecompositionaresolvedusingXpressOptimizer(Version20.00.05).ThetimehorizonofchoiceisoneyearwithmonthlystageswheretherstmonthisJanuaryandthelastmonthconsideredisDecember.Weapply 121

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Hydro-electricreservoirsystemofGuatemala anannualdiscountrateof10%.TheinowuncertaintyiscapturedwithinSDDPviascenarioswhicharegeneratedbylinearautoregressivemodelswithcorrelatedinnovationsestimationoftheparameterswascarriedoutbyttingalag-1modeltorealinowdataofthepast38years.WeuseM=100inowscenariosforoursimulationsduringtheSDDPsimulationphasewhileL=25inowscenariosareusedforthebackwardpass.Hence,thepresentedresultsareaveragesoverthose100forwardinowscenarios.Theelectricitydemandfortheplanninghorizonofoneyearisassumedtobeknownandfollowsseasonalpattern;seeTable 5-3 .Boththeemissionneforexceedingagivenannualquotaandthepenaltyforrationingwerearbitrarilyset,theformerbeingxedto$100andthelattersettotentimesthelargestgenerationcost.OperatingtheGuatemalapowersystemwithoutCO2emissionquotaleadsto(average)annualoperationcostofmillion$348and(average)emissionsof3.635million 122

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5-3 .OneobservesinFigure 5-3A thattheaverageCO2emissionsdecreaseataslowerratethanthequotalevelsaredecreased.Figure 5-3A alsoshowstheCO2emissionsoftheworstcasescenariooccurred;i.e.,thecasewiththelowestinowsobserved.Interestingly,thequotalevelof$100doesnotpushtheworstcaseemissionsbelowthequotalevel.Hence,theprobabilityofexceedingthequotalimitisabovezeroforalltestedquotalevelslessthan3.95;thevaluesareshowninTable 5-4 inrowfour.Givenanacceptableriskofexceedingtheyearlyallowance,onecouldeasilyidentifytheoptimalpenaltylevelbyrunningthemodelseveraltimes.Table 5-4 showsalsotheaverageannualoperationcostfordifferentquotalevelsinrowtwo,includingtheeventuallyincurrednes.RowthreeprovidestheaverageamountofCO2emissionsexceedingtheallocatedquota.Inrowveandsix,theriskofnotmeetingtheelectricitydemandareshown,givenasthemaximumrationingpercentagewithrespecttothetotalelectricitydemandamongthe100scenariosandasthenumberofscenarioswithdemandrationing,respectively;fortheparticularoptimaloperationalpolicycalculated.Reviewingthenumbers,therationingriskisnegligible.Theaverageannualoperationcostsoverall100scenariosareshowninFigure 5-3B .Theimportantinformationinthisgureisgivenbytheslopeoftheoperationalcostcurve:Theslopecanbeinterpretedastheincremental/marginalcostforCO2reduction.WhenexcludingCO2emissionnes,onegainsagoodapproximationofincrementalCO2emissionreductioncostwhicharefortherst17,000tonsCO2roughly$52pertonandforthelast10,000tonsover$150perton.Asthesecostareoperationalcost,theyareshort-termCO2reductioncost.Again,theseresultscouldbeusedbypolicymakersasguidelinesandcomparedtosociety'swillingnesstoreduceemissionsinordertoreachaplausiblecompromise.Figure 5-4 showsthegenerationmixoverthespanofthewholeyearforthedifferentquotalevelsimposed.WhiletheproductionoftheCO2emissionfreesources 123

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B AnnualCO2emissionsandoperationalcostfordifferentquotalevels.A)AverageandmaximumCO2emissions. 124

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YearlygenerationmixforGuatemalapowersystemwithdifferentquotalevels. (geo,co-generationandhydro)remainbasicallyunchanged,thedirtycoalisreplacedsteadilybythemoreexpensivebutcleanerbunker. 125

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B

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D

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F Monthlydispatchingdecisionsforthequotafreecase,whichisthenusedasthebasecaseshowingthemonthlydifferenceinelectricityproduction.A)Quota:None.B)Quota:3.4Milliontons.C)Quota:3.6Milliontons.D)Quota:3.7Milliontons.E)Quota:3.8Milliontons.F)Quota:3.95Milliontons.

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5-5A to 5-5F showthegenerationmixforeachmonthovertheplanninghorizonofoneyearfordifferentquotalevels.Mostobviousarethechangesintheelectricityproductionwithcoalredplantsintherststageswhenimplyinghighquotas.Itcanbeclearlyseenthatthecoalgenerationisreplacedbybunkerinmostofthecases.However,forthecasesofrelativemoderateemissionquotalevelsof3.9,3.95or4.0milliontonsofCO2,thehydroresourcesareoperatedmoreaggressively,leadingtoincreasesintherationingrisk;cf.Table 5-4 .Oneobservesanincreaseinthehydrogenerationfortherststages(andhenceadecreaseinthewaterreservoirlevels)leadingtothishigherrisk.Controversy,forlowerquotalevels,lesswaterisusedintherststagesbutinsteadthecapacityofthebunkerplantsisusedtoproduceelectricitywhichinlaterstagescouldreplacedirtierplants,suchascoalordiesel.Themoretheinowuncertaintyunfolds,themorecoalcanbeusedwhenexpectedthatenoughCO2emissionsareavailable.ThemarginalCO2emissionpricesandelectricitypricesareshownforthedifferentquotalevelsandthe12stagesinFigure 5-6 andFigure 5-7 ,respectively.ThetrendofdecreasingpricesisexplainedbytheexpirationoftheCO2emissionsattheendoftheplanninghorizonofoneyear;i.e.,theCO2emissionrightshavenofuturevalueabovetheplanninghorizon.Thestochasticwaterinowdrivesthistrendfurther.Whileatthebeginning,onemighthavetobeveryconservativewithrespecttoCO2emissionsformost(orevenall)oftheinowscenarios,whileattheendofthehorizon,theemissionsquotamightonlyaffectafewofthescenarios.Possibleendeffectsofthehydro-thermalgenerationplanwherethewaterhaslessfuturevaluethanatthebeginningareveryminorforthisdatasetanddonotaltertheresults.Stage6,7and8(June,JulyandAugust)arethecrucialstagesforthisdataset.Outofthe100inowscenarios,severaldroughtsareoccurringduringthesestages;leadingtoashortageinwater,increasingthethermalproductionuptotheircapacitylimitsandwiththat,increasingtheCO2emissionsandtheelectricityproductioncost.Oncethese 129

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Averageelectricitymarginal(=spot)pricesfordifferentquotalevels Figure5-7. AverageCO2emissionallowancemarginal(=spot)pricesfordifferentquotalevels 130

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131

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ThermalplantsconsideredfortheGuatemalapowersystem NumberofPlants13118311310CumulativeCapacity[MW]24.0120.441.4729.891.5132.413.058.0227.0FuelType112223345Cost[$/MWh]129.9132.061.667.168.741.245.92.71.0CO2Emission[kg/MWh]625.0635.2544.1593.5607.31001.01115.400 FuelType1:Diesel,2:Bunker,3:Coal,4:GEO,5:Co-generation Table5-3. MonthlyelectricitydemandfortheGuatemalapowersystem JanuaryFebruaryMarchAprilMayJuneJulyAugustSeptemberOctoberNovemberDecember GWh801738828771844805840847806850820838 Table5-4. ComputationalresultsfordifferentquotalevelsontheGuatemalapowersystem (Average)Cost[million$]370.1367.0364.5362.4359.5357.6355.8353.8351.6351.1350.8349.3349.3348.4CO2EmissionsAboveQuota[tons]491434262425156911096344272741625723240-CO2EmissionsAboveQuota[%]4636282519181410104110-Max.DemandRationing[MWh%]000000.020.020.020.010.210.380.380.380DemandRationing[%]00000111136670

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1. uncertaintyofspotprices,and 2. riskappetite1havetobeincludedintheoptimizationmodels.ThepresenceofaCO2market,tradingCO2emissionallowances,makestheprotmaximizationmodelsevenmoreinvolved.CO2emissionallowancepricesaredeterminedviamarketmechanisms,addinganadditionalsourceofuncertaintyintothemodels:CO2emissionallowanceprices.Thus,theoptimizationmodelofthischapterdiffersfromthemodelsdiscussedinthepreviouschaptersinthefollowingthreemainaspects: 133

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3 .Contrary,allotherthreeuncertaintiesaremodeledisadiscreteway.Thefuelpriceandelectricitydemanduncertaintiesarecapturedbyamultivariatescenariotree,thus,employingthemethodologyofChapter 4 .AMarkovChainapproachisusedtowardstheelectricityspotpriceandCO2emissionallowancepriceuncertainty,deningpriceclusterswithtransitionprobabilities.Inorderfortheproposedmodeltobepracticallyrelevant,wehavetomakethefollowingtwoassumptions: 1. Theconsideredpowersub-systemaswellasthewholepowersystemhaveasignicantshareofinstalledhydro-electriccapacity;i.e.,thesystemsarehydro-dominated. 2. Allplayerscorrespondingtothesub-systemsintheelectricitymarketarepricetakers.Asweconsiderstochastichydroinowsinthemodel,therstassumptionisapparent.ThepriceclustersfortheelectricityspotpricesaswellastheCO2emissionallowancepricesaremodeledbeingstaticinthesensethattheycannotbeinuencedbythe 134

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118 ]andOtero-Novasetal.[ 119 ],complementarity-basedmodelsbyBushnell[ 25 ]andHobbsandHelman[ 79 ],aswellasSDPtechniquesbyKelmanetal.[ 95 ]andScottandRead[ 155 ].Inpresenceofmarketpower,theleast-costsolution(centrallydispatchmodel)isexpectedtodifferfromtheCournot-Nashequilibriumwhichmoreaccuratelymodelthereality.ThiswasempiricallyshownfortheBrazilianpowersystembyBarrosoetal.[ 7 ].Incontrast,inthecaseofperfectcompetitioni.e.,absenceofmarketpowerthecentrallydispatchedsolutionisthesameasthesolutionofthemarket-baseddispatchasempiricallyshownbyLinoetal.[ 105 ].Hence,asapricetaker,theoptimalbiddingstrategyisgivenbythemarginalsystemcost,whichcanbederivedthroughacostminimizationmodelasarguedbyGrossandFinlay[ 69 ].Thisleadstotheconceptoffundamentalmodeling.Throughaleast-costmodelofthewholesystem,marginalelectricitypricesaswellasmarginalCO2emissionallowancepricescanbederived.ThesepricesaregivenasthedualmultipliersfromanoptimalsolutionpolicyofthecorrespondingelectricitydemandconstraintsaswellasCO2emissionreservoirconstraints.Giventheabsenceofmarketpowerinthesystem,thesepricescanthenbeusedaspriceforecastsinthesub-system'smodel. 135

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13 ].Theauthorssolvedahydro-thermalprotmaximizationprobleminthemid-termtolong-termhorizoninthepresenceofCO2emissionallowancemarkets.Similartotheapproachproposedinthischapter,thepricesarealsoforecastedusingfundamentalmodeling.Inthischapter,weconsideradditionaluncertaintiesviatheformofscenariotreesandallowadetailedmodelingoftheCO2emissionconstraints;cf.Chapter 5 .Incontrasttoafundamentalmodel,RongandLahdelma[ 148 ]useascenariotreeapproachtomodeljointlytheuncertaintiesofheatdemand,electricityspotpricesandCO2emissionallowanceprices.Theirmulti-stagestochasticoptimizationmodelfocusesontheCO2emissionstradingofacombinedheatandpowerproducer.Nohydro-plantsareconsideredintheirpaper.Nevertheless,thispaperisrelevanttoourworkastheoptimizationoftheoperationandtheemissiontradingareconsideredjointlyinonemodel;cf.Section 6.1 .BenzandTrueck[ 17 ]considerdifferentstochasticmodelstocaptureandpredictthespotpricedynamicsofCO2emissionallowancesintheshort-term.Short-termmodelshavetobemuchmoredetailedthanthemodelsrequiredforourpurposes,asweassumeaveragepricesforeachstage;i.e.,averagemonthlyprices. BenzandTrueck suggestMarkovswitchingandAR-GARCHmodelstocapturethecharacteristicsofthe 136

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17 ].Themaincontributionofthischapteristhejointmodelingofthreedifferenttypesofuncertaintiesinasingleprotmaximizationproblemwhilecapturingtheircorrelation.Thosethreetypesofuncertaintiesarecapturedby(linear)timeseriesmodels(e.g.,hydroinows),scenariotrees(e.g.,fuelpricesandelectricitydemand)andMarkovChains(e.g.,electricityspotpricesandCO2emissionallowanceprices).Theestimationofthemarketbaseddataarederivedviaafundamentalmodel.Further,theresultingprotmaximizationmodelissolvedusingahybridSDP/SDDPTmethod.Partsofthischapterhavebeenpublishedin[ 140 ].Theremainderofthischapterisorganizedasfollows.Theprotmaximizationproblemisformulatedasamulti-stagestochasticprogrammingprobleminSection 6.1 .AsolutiontechniquetosolvethestochasticmodelisderivedinSection 6.1.1 .WeconcludewithadiscussioninSection 6.2 137

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6 )and( 6 ).Furthermore,thermalplantscanbeusedtogenerateelectricityaswellthroughvariablesgtj(t)atoperationalcostsofctj(t),whereeachMWhproducedreleasesBjtonsofCO2emissions.InordertomeettheCO2emissionallowancequotaECO2y,additionalallowancescanbeboughteCBt(t)intheCO2emissionallowancemarketforthepriceofPCBt(t).Ifthereareenoughallowancesavailable,thenthoseallowancescanalsobesoldintheCO2marketviavariableseCSt(t)foragainofPCSt(t).Thegeneratedelectricityisusedtomeetthegivenquantitycontracts'demanddCt.AccesselectricitycanbesoldintheelectricityspotmarketforPEST().Theoverallaimisthentomaximize 139

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6 )-( 6 )doesonlyallowthesalesofelectricityinthespotmarketbutnottobuy.Incasethatthepurchaseofelectricityinthespotmarketispossible,thethermalgenerationdecisionsandthehydro-electricitygenerationdecisionsbecomedecoupled!Thereasonisthatinthiscase,thethermalplantsareusedwhenevertheelectricitygenerationcost(includingtheCO2emissioncost)arelessthantheelectricityspot(sales)price.However,whenincludingriskconstraints,boundsontheelectricityspotpricetrading,and/orderivativecontracts,thentheprotmaximizationproblemhastobesolvedjointlyforthethermalandthehydrogeneration.NexttoCO2emissionallowancemarkets,theKyotoprotocolofferstwoadditionalmechanisms:JointImplementation(JI)andCleanDevelopmentMechanism(CDM).Theideaisthatpartoftheemissionreductioncanbeachievedbyconductingemission-reducingprojectsinotherindustrializedcountrieswithKyototargets(JI)andincountrieswithouttargets(CDM).Thepossibilityofsuchprojectscanbeincorporatedintheoptimizationmodel( 6 )-( 6 )viadeterministicvariables;cf.Rebennacketal.[ 140 ].Inaddition,alltheconstraintsdiscussedinChapter 2 canbeappliedaswell. 6 )-( 6 )canbesolved.Inordertoexploittheblockdiagonalstructureofproblem,weusedynamicprogrammingtechniquestodecomposetheproblemintoone-stagedispatchproblems. 140

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3 ,SDDPmethodsrelyontheconvexityofthefuturecostfunctioninitsstatevariables.Formaximizationproblems,thistranslatestotheconcavityrequirementsofthefuturebenetfunctioninitsstatevariables.RecallthatthereasonfortheconcavityofthefuturebenetfunctionliesintheoccurrenceoftheuncertaintyastheRHSvaluesoftheLPproblem(andthelinearmodelforthewaterinows).However,introducingastatevariableforanobjectivefunctioncoefcientleadstoasaddleshapefuturebenetfunction.ThisproblemhasbeensolvedbyGjelsvikandWallace[ 66 ]bytreatingtheuncertaintywithrespecttotheelectricitypricesasMarkovChains.Byintroducingpriceclustersassociatedwithacertainhydroinow,thestatespaceisdiscretized;cf.Chapter 3 ,thus,keepingtheconcavityofthefuturebenetfunctioninthestatevariables.TheelectricitypriceuncertaintyaswellastheCO2emissionallowancemarketpriceuncertaintyaretreatedviaMarkovChains.Incontrast,theMarkovprocessandMarkovChainapproachtowardsinowandmarketpriceuncertainty,respectively,fuelpriceuncertaintyistreatedthoughascenariotree.ConsistentwithChapter 4 ,weassumethatthefuelpricesareindependentfromthehydroinows.Though,ourmodelcapturesthecorrelationamongelectricityprices,CO2emissionallowanceprices,hydroinowsandfuelprices.ThestochasticspotmarketelectricitypriceandthestochasticCO2emissionspotpricesaremodeledasdiscretecost.Thesecost/pricesarehandledjointlyandgroupedintoclusterssktwithk2Kst=f1,...,Kstg,oneforeachscenarios2St;weallowtohavedifferentpriceclustersforeachstagetandfuelpricescenarios.ThescenarioindexallowsustocapturethecorrelationofthefuelpricesandthemarketpricesforelectricityaswellasCO2emissionallowances.Hence,foreachclusterk2Kstforscenarios2St,wehavethetupleskt=[Psk,ESt,Psk,CSt],

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3 )-( 3 ),( 4 )-( 4 )and( 5 )-( 5 ),weobtainaone-stagedispatchproblemforeachfuelprice 142

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5 )-( 5 ),alldecisionvariablescarryinowindexl,decomposingtheproblemintoLindependentones.Thefuturebenetfunctionszkt+1s(,,)canbeoverestimatedusingtheinformationofthedualmultipliers. 6 )and( 6 )andnotintheRHSofthecorrespondingmathematicalprograms.Inordertopreservetheconcavityofthefuturebenetfunctioninthestatevariables,wetreatedthoseuncertaintiesasadiscreteMarkovChainandscenariotree,respectively. 143

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144

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125 ]inseveralways.First,ascenariotreeframeworktocaptureuncertaintiesdrivenbypoliticalandmacro-economicalforcesisembeddedintotheSDDPalgorithm.Examplesforthistypeofuncertaintiesarefuelpricesandelectricitydemand.Thefeasibilityandefciencyoftheso-calledSDDPTalgorithmisdemonstratedbytworealcasestudiesforthepowersystemsofPanamaandCostaRica.Second,amodelforCO2emissioncapsonhydro-thermalpowersystemsisproposed.Themulti-stageconstraintsoftheCO2emissionallowancesarere-formulatedasareservoirconstraint,respectingthestagedecompositionframeworkofSDDP.ThedualmultiplierscorrespondingtotheCO2emissionreservoirconstraintsprovidemarginalCO2emissionallowanceprices.ThosepricesareoperationalsystempricesforCO2emissionreductioninthemid-term.ComputationalresultsareperformedfortherealpowersystemofGuatemala.Third,aprotmaximizationmodelforelectricutilitiesinaderegulatedelectricitymarkethasbeenproposed.TheutilityissubjecttoCO2emissionquotaswhereallowancesaretradedinamarketenvironment.Theproposedmodelincludesstochasticparametersinthehydroinows,fuelprices,electricitymarketpricesandCO2emissionallowanceprices.ThepresentedscenariotreemodelaswellasthereservoirmodelfortheCO2emissionallowancescanbeappliedtohydro-thermalexpansionplanningproblems.Inparticular,long-termmodelssubjecttoCO2emissionallowancecapsareofpracticalinterestforregulatorsandsociety,asthemarginalCO2pricesrepresenttheinvestmentcostforCO2emissionreductions. 145

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ThenomenclaturethroughoutthisarticleissummarizedinTables A-1 A-4 .Aindicatesanoptimalsolutionvalue. TableA-1. Indicesandsets SymbolSizeMeaning

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Objectivefunctions SymbolUnitMeaning

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Decisionvariablesandvaluesobtainedthroughoptimization SymbolUnitMeaning 148

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Inputdata SymbolUnitMeaning 4 )1modelparameterforthelinearautoregressiveinowmodel2m3modelparameterforthelinearautoregressiveinowmodel'tdenedas&t1=&t1g 149

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A-4 .Continued SymbolUnitMeaning 150

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A-4 .Continued SymbolUnitMeaning 151

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SteffenRebennackobtainedhisdiplomainmathematicsattheRuprecht-KarlsUniversitatHeidelberginHeidelberg,Germany,underthesupervisionofProf.Dr.GerhardReinelt,inJuly2006.InDecember2007,heobtainedaMasterofScienceoftheIndustrialandSystemsEngineeringDepartmentoftheUniversityofFloridainGainesville,USA.InAugust2010,SteffenRebennackobtainedtheMasterofScienceinManagementfromtheWarringtonCollegeofBusinessAdministrationattheUniversityofFlorida.SteffenRebennackgraduatedinAugust2010fromtheUniversityofFloridawithaPhDdegreeoftheIndustrialandSystemsEngineeringDepartmentunderthesupervisionofProf.Dr.Dr.h.c.mult.PanosM.Pardalos.SteffenRebennack'sresearchinterestsareinpowersystemsmodeling,powersystemsoptimization,stochasticoptimization,decompositionmethods,globaloptimization,integerprogrammingandcombinatorialoptimization. 166