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Stochastic Integer Optimization and Applications in Energy Systems

Permanent Link: http://ufdc.ufl.edu/UFE0042029/00001

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Title: Stochastic Integer Optimization and Applications in Energy Systems
Physical Description: 1 online resource (117 p.)
Language: english
Creator: Zheng, Qipeng
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: decomposition, electricity, expansion, gas, integer, location, optimization, planning, risk, stochastic, system, transmission
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

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Abstract: Everyday, we are faced with a lot of uncertainties and discrete decisions. Stochastic mixed integer programming is well suited to help us handle this situation. However, this type of optimization problems are not easy to solve. The first half this dissertation gives a brief review of stochastic programming and stochastic mixed integer programming, and proposes a solution method, embedded Benders' decomposition. Of all these difficult problems, those arising from energy systems are very urgent and important, since in the modern age, instead of human force, people rely more on other energy sources to keep the whole society running. The second half of this dissertation is about stochastic integer optimization applications in energy systems. Firstly, this dissertation studies the stochastic security constrained unit commitment problem, which includes both day-ahead and real time unit commitment, making it a very typical stochastic mixed integer program. Numerical results show that embedded Benders decomposition method suits well this problem, especially when it has a large number of scenarios. Secondly, this dissertation discusses optimization models and algorithms in the natural gas industry, and proposes natural gas transmission system expansion planning models which include both natural gas transmission network expansion and LNG (Liquified Natural Gas) terminals location planning. These models take into account the uncertainties of demands and supplies in the future, which make the models stochastic integer programs with discrete subproblems. In addition, this dissertation considers risk control in these models by including probabilistic constraints, such as a limit on CVaR (Conditional Value at Risk). In order to solve the large-scale problems, especially those with large numbers of scenarios, the embedded Benders decomposition algorithm is applied to tackle the discrete subproblems. Numerical results show that this algorithm is efficient for solving large scale stochastic natural gas transportation system expansion planning problems.
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 Qipeng Zheng.
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 2011-02-28

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System ID: UFE0042029:00001

Permanent Link: http://ufdc.ufl.edu/UFE0042029/00001

Material Information

Title: Stochastic Integer Optimization and Applications in Energy Systems
Physical Description: 1 online resource (117 p.)
Language: english
Creator: Zheng, Qipeng
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: decomposition, electricity, expansion, gas, integer, location, optimization, planning, risk, stochastic, system, transmission
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: Everyday, we are faced with a lot of uncertainties and discrete decisions. Stochastic mixed integer programming is well suited to help us handle this situation. However, this type of optimization problems are not easy to solve. The first half this dissertation gives a brief review of stochastic programming and stochastic mixed integer programming, and proposes a solution method, embedded Benders' decomposition. Of all these difficult problems, those arising from energy systems are very urgent and important, since in the modern age, instead of human force, people rely more on other energy sources to keep the whole society running. The second half of this dissertation is about stochastic integer optimization applications in energy systems. Firstly, this dissertation studies the stochastic security constrained unit commitment problem, which includes both day-ahead and real time unit commitment, making it a very typical stochastic mixed integer program. Numerical results show that embedded Benders decomposition method suits well this problem, especially when it has a large number of scenarios. Secondly, this dissertation discusses optimization models and algorithms in the natural gas industry, and proposes natural gas transmission system expansion planning models which include both natural gas transmission network expansion and LNG (Liquified Natural Gas) terminals location planning. These models take into account the uncertainties of demands and supplies in the future, which make the models stochastic integer programs with discrete subproblems. In addition, this dissertation considers risk control in these models by including probabilistic constraints, such as a limit on CVaR (Conditional Value at Risk). In order to solve the large-scale problems, especially those with large numbers of scenarios, the embedded Benders decomposition algorithm is applied to tackle the discrete subproblems. Numerical results show that this algorithm is efficient for solving large scale stochastic natural gas transportation system expansion planning problems.
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 Qipeng Zheng.
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 2011-02-28

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0042029:00001


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STOCHASTICINTEGEROPTIMIZATIONANDAPPLICATIONSINENERG YSYSTEMS By QIPENGZHENG ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2010

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c r 2010QipengZheng 2

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Tomylovelydaughter,DeHaoZheng,mywifeandmyparents 3

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ACKNOWLEDGMENTS First,Iwouldliketothanksmyadvisor,Dr.PanagoteM.Pard alos,forhisconstant supports,encouragementandmotivation.Heisagreatmento r,friendandcollaborator. Heisalwayswillingtohelpandgiveinsightfulguidance.Ia mverygratefultobeone ofhisstudents.Hetaughtmemanydifferentwaystoapproach aresearchproblem andthepersistencetoaccomplishanygoal.Hewasresponsib lefortransformingme toaresearcherfromastudent.Withouthishelp,thisdisser tationwouldnothavebeen possible. IamverygratefultoDr.JonathanColeSmith,Dr.WilliamHag er,Dr.Yongpei GuanandDr.VladimirBoginskiforbeingonmyPhDdissertati oncommittee,andtheir support,suggestions,guideandexperiencesharedwithme. AlsoIwouldliketothank Dr.SiriphongLawphongpanichforhissupportsandsuggesti onsonmyresearch. Ialsowouldliketothankallmycollaborators:AshwinAruls elvan,JianhuiWang, NengFan,SteffenRebennack,AlexeySorokin,NikoA.Iliadi s,HongshengXu,Siqian Shen,YingyanLou,TaoZhang.Iamgratefulfortheiradvices andsuggestions.They taughtmehowtobeagoodresearcherandithasbeenagreatexp erienceworkingwith them.IwanttothankallthefellowstudentsinDepartmentof IndustrialandSystems EngineeringatUniversityofFloridaforthediscussionsan dtheirfriendship.Ialsowould liketothankallothersofmyfriendswhoarenotdirectlyass ociatedtomydissertation butarealwayssupportingme. Finally,Iwouldliketothankmyfather,DongxiangZheng;my mother,ZhengjunLi; mywife,YoushanZhao;andmylovelydaughter,DeHaoZheng,f ortheirimmenselove, support,andmoralencouragement. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ..................................4 LISTOFTABLES ......................................7 LISTOFFIGURES .....................................8 ABSTRACT .........................................9 CHAPTER 1INTRODUCTION ...................................11 1.1StochasticProgramming ............................12 1.1.1GeneralFormulation ..........................12 1.1.2SolutionMethods ............................14 1.2StochasticMixedIntegerProgramming ....................18 1.2.1FormulationandPreviousApproaches ................18 1.2.2GeneratingValidBendersCutsfromDiscreteSubprobl ems ....20 1.3OutlineofthisDissertation ...........................21 2EMBEDDEDBENDERS'DECOMPOSITION ...................23 2.1Two-stageEmbeddedBenders'Decomposition ...............23 2.2StochasticTwo-stageEmbeddedBenders'Decomposition .........28 2.3DeterministicMultistageEmbeddedBendersDecomposit ion .......28 2.4TheStochasticMultistageAlgorithm .....................31 3STOCHASTICSECURITYCONSTRAINEDUNITCOMMITMENTMODELS .34 3.1Introduction ...................................34 3.2ProblemFormulation ..............................35 3.3ProblemDecomposition ............................39 3.4SolutionAlgorithm ...............................47 3.5NumericalExamples ..............................49 4OPTIMIZATIONMODELSINNATURALGASINDUSTRY ............53 4.1Introduction ...................................53 4.2OptimizationinGasProduction(Recovery) .................55 4.2.1ProductionSchedulingConsideringWellPlacement ........55 4.2.1.1MixedIntegerLinearProgrammingFormulation .....56 4.2.1.2NonlinearProgrammingFormulation ............58 4.2.2TotalGasRecoveryMaximization ...................59 4.3NaturalGasPipelineNetworkOptimization .................62 4.3.1CompressorStationAllocationProblem ...............63 4.3.2LeastGasPurchaseProblemandOptimalDimensioning ......65 5

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4.3.3MinimumFuelConsumptionProblem .................69 4.4NaturalGasMarketModels ..........................71 4.4.1ReallocationProbleminaRegulatedNaturalGasMarke t .....71 4.4.2DeregulatedNaturalGasMarketModels ...............74 4.4.3CombiningNaturalGasSystemandElectricitySystem .......78 4.4.3.1ElectricitySystemReliabilityStudy .............78 4.4.3.2OptimizationinNaturalGasContracts ...........80 4.5Conclusion ...................................83 5NATURALGASNETWORKEXPANSIONPLANNING ..............84 5.1Introduction ...................................84 5.2ExpansionPlanningModels ..........................86 5.2.1TheStochasticPlanningModel ....................88 5.2.2ThePlanningModelwithRiskConstraints ..............94 5.3EmbeddedBendersDecomposition .....................96 5.4NumericalExamples ..............................106 6CONCLUSIONS ...................................110 REFERENCES .......................................112 BIOGRAPHICALSKETCH ................................117 6

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LISTOFTABLES Table page 3-1SCUCSetsandIndices ...............................35 3-2SCUCParameters ..................................35 3-3SCUCDecisionVariables ..............................36 3-4GeneratorsData ...................................50 3-5Loadforecastofasimpleexample .........................50 3-6Solutionofthe4-unitSCUCwith3scenarios ...................50 3-7ComputationalResultsofSCUC ..........................52 4-1Maintenancecyclelength ..............................81 5-1EXPNSetsandIndices ...............................89 5-2Parameters ......................................90 5-3DecisionVariables ..................................91 5-4GroupsofInstancesEXPN .............................107 5-5Computationaltimesforinstancesgroup1 .....................107 5-6Computationaltimesforinstancesgroup2 .....................108 5-7Computationaltimesforinstancesgroup3 .....................108 7

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LISTOFFIGURES Figure page 3-1LinearApproximationoftheFuelCostFunction ..................38 4-1WorldGasConsumptioninBillionCubicFeet ...................54 4-2Agaspipelinenetworkcongurationproblemwiththreeb ranches. .......64 4-3Leastcostproblemnetwork. ............................66 4-4Agaspipelinenetwork. ...............................69 4-5ParticipantsRelationshipinRegulatedGasMarket. ...............74 4-6RelationshipbetweenGasNetworkandElectricityNetwo rk. ...........78 4-7GasContractsModeledbyReservoirs. .......................81 5-1USnaturalgastransmissioncorridorfromEIA2008 ...............85 5-2ExistingandProposedNorthAmericanLNGTerminals .............86 5-3Naturalgaslongtermconsumptionexpectation ..................87 5-4Anaturalgastransmissionnetworkexample ...................88 5-5DiscreteExpansionCosts ..............................89 5-6SolutionofaSimpleEXPNExample ........................93 5-7ValueatRiskv.s.ConditionalValueatRisk ....................94 5-8EmbeddedBendersDecompositionAlgorithmforEXPN .............105 5-9MinimalCostV.S.LimitofCVaR ..........................109 8

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy STOCHASTICINTEGEROPTIMIZATIONANDAPPLICATIONSINENERG YSYSTEMS By QipengZheng August2010 Chair:PanagoteM.PardalosMajor:IndustrialandSystemsEngineering Everyday,wearefacedwithalotofuncertaintiesanddiscre tedecisions.Stochastic mixedintegerprogrammingiswellsuitedtohelpushandleth issituation.However,this typeofoptimizationproblemsarenoteasytosolve.Therst halfthisdissertationgives abriefreviewofstochasticprogrammingandstochasticmix edintegerprogramming, andproposesasolutionmethod,embeddedBenders'decompos ition.Ofallthese difcultproblems,thosearisingfromenergysystemsareve ryurgentandimportant, sinceinthemodernage,insteadofhumanforce,peoplerelym oreonotherenergy sourcestokeepthewholesocietyrunning.Thesecondhalfof thisdissertationisabout stochasticintegeroptimizationapplicationsinenergysy stems.Firstly,thisdissertation studiesthestochasticsecurityconstrainedunitcommitme ntproblem,whichincludes bothday-aheadandrealtimeunitcommitment,makingitaver ytypicalstochastic mixedintegerprogram.Numericalresultsshowthatembedde dBendersdecomposition methodsuitswellthisproblem,especiallywhenithasalarg enumberofscenarios. Secondly,thisdissertationdiscussesoptimizationmodel sandalgorithmsinthenatural gasindustry,andproposesnaturalgastransmissionsystem expansionplanning modelswhichincludebothnaturalgastransmissionnetwork expansionandLNG (LiquiedNaturalGas)terminalslocationplanning.These modelstakeintoaccountthe uncertaintiesofdemandsandsuppliesinthefuture,whichm akethemodelsstochastic integerprogramswithdiscretesubproblems.Inaddition,t hisdissertationconsidersrisk 9

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controlinthesemodelsbyincludingprobabilisticconstra ints,suchasalimitonCVaR (ConditionalValueatRisk).Inordertosolvethelarge-sca leproblems,especiallythose withlargenumbersofscenarios,theembeddedBendersdecom positionalgorithmis appliedtotacklethediscretesubproblems.Numericalresu ltsshowthatthisalgorithmis efcientforsolvinglargescalestochasticnaturalgastra nsportationsystemexpansion planningproblems. 10

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CHAPTER1 INTRODUCTION Optimizationormathematicalprogrammingisaveryvibrant andimportantbranch ofmathematics,whichhasabroadareaofapplicationsandgr eatimplicationsinourreal life.Amathematicalprogrammingproblemistominimizeano bjectivefunctioneitheror notwithrespecttoasetofconstraints,whichisshownasfol lows, Min f ( x ) s.t. g ( x ) 0, (1–1) where x isthedecisionvariablevectorin R n ,and f ( x ) and g ( x ) arevectorsoffunctions of x in R l and R m respectively.When l =1 ,wehaveauniqueobjectivefunction,which makesthetypeofmoststudiedmathematicalprogrammingpro blems.However,there arealotofproblemswith l > 1 ,whicharecalledmultiobjectiveoptimization.Depending onthepropertiesoftheobjectivefunction f ( x ) ,constraints g ( x ) andrestrictionsonthe decisionvariables,manydifferenttypesofmathematicalp rogrammingproblemsare dened,andaccordinglydifferentsolutiontechniquesare developed. FollowingthepioneeringresearchbyDantzig,vonNeumann, Kuhn,Tucker,etc,in the1940sandthe1950s,mathematicalprogramminghasbeeng ainingmoreandmore attentions,whileinuencingthereallifemorebroadlyand deeply.Ofallresearchelds ofmathematicalprogramming,linearprogrammingisaveryb asictype,whichisrichina lotapplicationareas,andistherstmathematicalprogram mingexperienceorclassfor mostoftheresearchersinoptimizationandoperationsrese arch.Inlinearprogramming problems,both f ( x ) and g ( x ) arelinearfunctionsof x ,shownasfollows, Min c T x s.t. Ax b x 0, (1–2) 11

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where c and b arecolumnvectorsgivenin R n and R m respectively,and A isaxed matrixin R m n Becauseoftheextensivestudies,welldevelopedtheories, andsolutiontechniques onlinearprogramming,suchassimplexmethod,interiorpoi ntmethod,etc,manyother optimizationstudies,e.g.,stochasticprogramming,mixe dintegerprogramming,etc,are basedonlinearprogramming. 1.1StochasticProgramming Inreality,thereisagreatneedforustoincorporatethefut ureuncertaintieswhen wetrytomakesomedecisionsrightnow,suchasapplications inenergy,nance, economics,business,transportation,etc.Stochasticpro grammingoroptimizationhas beencontinuouslygainingmoreandmorepopularitysinceit sbirthin1950's.Stochastic programmingtakesintoaccountallpossiblefutureoutcome s,andassumesthatwe performoptimallyunderanysituationwhentheuncertainti esunfold,andminimizesthe summationofthecurrentcostandthefutureexpectedcost.I nlongrun,thisactuallycan helpustoachieveabettercurrentdecisionthanitwouldhav ebeenifweonlyconsider somescenario(s)oreventheexpectedoutcome.1.1.1GeneralFormulation Themostextensivelystudiedstochasticprogrammingprobl emsarethestochastic linearprograms,whichonlyinvolvelinearconstraintsand continuousvariables.Inthe twostagestochasticprograms,therandomnessisonlyobser vedonce.Decisionsneed tobemadebothbeforeandtheaftertheuncertaintiesunfold .Thegeneralformulationof thistypeofproblemsisshownasfollows, Min c T x + E [ Q ( x w ) ] s.t. Ax b (1–3) where w isanrandomvector,and c and x arerespectivelyagivenvectorofcostsanda decisionvectorin R n ,and E [ Q ( x w ) ] iscalledthevaluefunctionorrecoursefunction, 12

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whichistheexpectedfuturecostof Q ( x w ) ,whichiscostofthedecisionsmadeafter theuncertaintiesunfoldandisshownasfollows, Q ( x w )= Min [ d ( w )] T y s.t. F ( w ) y g ( w ) T ( w ) x y 0, (1–4) where y isthedecisionvectoraftertheoutcomeoftherandomvariab le w isobserved. F ( w ) iscalledtherecoursematrix,and T ( w ) isthetechnologymatrix.Butinmostof theliterature,insteadof F ( w ) ,axedrecoursematrix F isused,whichisconsidered tobeindependentofthescenarios.( 1–12 )and( 1–13 )areusuallyreferredtoasthe rstandsecondstageproblemsrespectively.Whentherando mvariableisdiscretely distributedandhasanitenumberofoutcomes,thestochast icprogramiscompletelya deterministiclinearprogrammingproblemasfollows, Min c T x + X 2 Prob ( )[ d ( )] T y ( ) s.t. Ax b F ( ) y ( )+ T ( ) x g ( ), 8 2 y ( ) 0, 8 2 (1–5) where y ( ) isthedecisionvectorforscenario withcorrespondingprobability Prob ( ) and isthesetofallpossibleoutcomesofrandomvariable w .Duetothespecial structureoftheaboveproblem,decompositionalgorithmsa reveryusefulwhen experiencingabignumberofscenarios.Benders'decomposi tion[ 9 ]iswellsuitedto handlethissituation.AbriefintroductionoftheBenders' decompositionalgorithmis presentedinthefollowingsection. Inthetwo-stagestochasticprogrammingproblems,weassum ethattherandom variablewillberealizedonlyonce,anddecisionsaremadeb othbeforeandafterthat event.However,inreality,wemayneedtomakeaseriesofdec isionsalongatime 13

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sequence,withinwhichasequenceofrandomeventshappenal ternatelywiththe decisions.Multistagestochasticprogrammingisaverygoo dmodelingtoolforthistype ofproblems.Itgeneralizestheconceptofthetwo-stagesto chasticprogramming,and canbeseenasasequenceoftwo-stagestochasticprograms.I tsformulationisshown asfollows, Q t ( x t 1 ( ), )= Min [ d t ( )] T x t ( )+ E [ Q t +1 ( x t ( ) )] s.t. F t ( ) x t ( ) g t ( ) T t ( ) x t 1 ( ), x t ( ) 0, (1–6) wherethesubscript t and t 1 denotethestages, and denotethescenariosofstage t and t 1 respectively.Multistagestochasticprogrammingproblem saremuchmore difculttosolveduetothecurseofdimensionalityexplosi on. 1.1.2SolutionMethods Thecomputationaldifcultyofstochasticprogramminglie sinthefactthatitinvolves toomanydecisionvariablesandconstraintsbecauseforeac hscenarioawholesecond stageformulation,( 1–13 ),isrequired.Iftherststagedecisionisgiven,thesecon d stageproblemcanbedecomposedtomanysmallerproblemswhi chcanbesolved separately.Thenwecanprovidesomefeedbacktotherststa getotellwhether thegivenrststagesolutionisgoodornot.Benders'cutisa verygoodmediathat coordinatesthisback-and-forthcommunication.Bendersd ecompositionwasproposed byBenders[ 9 ]in1962,whichisexplainedbrieyasfollows.Supposewear edealing withthefollowingoptimizationproblem[P], [P]:Min c T x + d T y s.t. x 2 X Ex + Fy g y 0. (1–7) 14

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Givenasolution ^ x ,theaboveprogramreducestoalinearprogrammingproblema s follows,called[SP], [SP]:Min d T y s.t. Fy g E ^ x u (1–8) y 0. Itscorrespondingdualprogramisshownasfollows,called[ DSP], [DSP]:Min ( g E ^ x ) T u s.t. F T u d u 0. (1–9) [DSP]and[SP]sharethesameoptimalobjectivevaluesincet heyarebothlinear programs.Theoptimalobjectivevalueof[SP]or[DSP]isapi ecewiselinearfunction withrespecttomasterproblemdecisionvariable, x .In[DSP],thefeasibleregionisnot relatedtothemasterproblemdecision, x .[DSP]isalinearprogram,andthenitsoptimal solutionisonthevertexofthefeasibleregion.Thismeansi fwecangetalltheextreme pointsofthe[DSP],wecandenethevaluefunctionofthesub problem.Uptonow,we assumethe[DSP]isfeasibleandbounded.Ifthe[DSP]isinfe asible,whichmeansthe [SP]isunbounded,thentheoriginalproblem[P]isunbounde d.If[DSP]isunbounded, whichmeansthe[SP]isinfeasible,thenthegivenrststage decision, ^ x ,isnotafeasible solutiontotheoriginalproblem[P].Hence,inordertoprev entunboundednessorthis kindofrststagesolution,weneed ( g Ex ) T v 0 ,where v isanextremerayofthe unbounded[DSP].Thentheoriginalproblemcanberedeneda sfollows, [MP]:Min c T x + s.t. x 2 X (1–10) ( g Ex ) T ^ u i 8 i 2I 15

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( g Ex ) T ^ v j 0, 8 j 2J where I and J aretheextreme-pointsetandextreme-raysetof[DSP]respe ctively. However,includingallextremepointsandextremerayswoul dnotmakeitaveryefcient formulation.Ifweonlyincludesubsetsofallextremepoint sandrays,wewouldbeable togetanlowerboundoftheproblem.Thisproblemwithsubset sofextremepointsand raysiscalledrestrictedmasterproblem,[RMP],whichissh ownasfollows, [RMP]:Min c T x + s.t. x 2 X ( g Ex ) T ^ u i 8 i 2I r ( g Ex ) T ^ v j 0, 8 j 2J r (1–11) where I r and J r aretheextreme-pointsubsetandextreme-raysubsetof[DSP ] respectively. Ancombinationoffeasiblesolutionsofbothmasterandsubp roblemsyieldsan upperboundoftheoriginalproblem.Sowecaniterativelyso lvethe[RMP]and[DSP] toupdatethelowerboundandupperbounduntiltheymatcheac hother.Givenavery smallvalue, ,theBenders'decompositionalgorithmisshownasfollows, Step0.SetUB= 1 ,LB= 1 I r = J r = ; ; Step1.Solve[RMP],andoptimalsolutionandobjectivevalu eare ^ x and w respectively;LB max ( LB w ) ; Step2.Solve[DSP],andoptimalsolutionis ^ u orextremeray ^ v UB min ( UB c T ^ x +( g E ^ x ) T ^ u ) ; I r I r [f ^ u g or J r J r [f ^ v g ; Step3.IfUB LB ,stop;O/Wgotostep1. Whenextendedtostochasticprogramming,informationfrom thesolutionofevery subproblemneedtobeconsidered.Generallytherearetwowa ysoffeedingbackthe futureinformation,byeithertheaggregatedcutsordisagg regatedcuts.VanSlyke 16

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andWets[ 63 ]proposedtheL-shapedmethod,whichaddsasingleaggregat edcutat eachiteration,whileBirgeandLouveaux[ 11 ]proposedamulticutmethod.Detailed discussionofadvantagesanddisadvantagesofbothalgorit hmscanbefoundin[ 10 ]. AlsowecanapplyBendersdecompositionsequentiallytodea lwiththemultistage stochasticlinearprograms.Themostdifcultparttosolve amultistagestochastic programmingproblemisthedimensionalityexplosion.Thev ariablesandscenarios growsexponentiallyasthenumberofstagesgoesup.Pereira andPinto(1991)[ 50 ] proposedStochasticDualDynamicProgramming(SDDP),whic hcandealwiththe curseofdimensionality,tosolveamultistagehydropowerp lantplanningproblem.Their methoditerativelyuseBender'scuttoapproximatetheexpe cted-cost-to-gofunction, anduseMonteCarloforwardsimulationtoavoidtheexplicit enumerationofallpossible scenarios. SDDPisthestochasticversionofDynamicDualProgramming( DDP),which appliesBenders'decompositioninthemultistageproblem. Supposewehavemultistage problemwhichhasfollowingformat, Min c T 1 x 1 + c T 2 x 2 + c T 3 x 3 + s.t. A 1 x 1 b 1 E 1 x 1 + A 2 x 2 b 2 E 2 x 2 + A 3 x 3 b 3 Sincethesubprobleminstage n isonlyrelatedtothesubprobleminstage n 1 ,Benders'cutscanstillbeappliedtoachievethecommunica tionbetweentwo consecutivestages.Givenasolutionofstage n 1 ,i.e., x n 1 ,wecanndaBenders' cutforthestage n 1 bysolvingthestage n problemwith x n 1 beingxed,where thedualoptimalsolutionisthecoefcientsoftheBenders' cut.TheDDPalgorithmis composedoftwomajorprocedures,i.e.,theforwardandback warditerations,where 17

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forwarditerationsndafeasiblesolutionofeachstageand backwarditerationsndthe Benders'cutsforthepreviousstage.Wecangettheupperbou ndbycalculatingthe costofafeasiblesolution,thelowerboundbycalculatingt heapproximatedobjective function. Intraditionalstochasticprogramming,allvariablesdeno tingeveryscenarioare includedintheformulation.Forthemultistageproblem,th evariablesizecouldbe easilyincreasedtobillions,trillions,evenmore.SDDPis analgorithmthatnicelyavoids thedimensionalityexplosionbytheMonteCarloforwardsim ulation.Inthebackward iterationofSDDP,Benders'cutsforstage n 1 areobtainedbyusingtheaverageof theoptimaldualsolutionscorrespondingtodifferentscen ariosofstage n .Forexample, n 1 = P mj =1 p j n j n 1 ,where j n 1 istheoptimaldualsolutionofstage n subproblemunder scenario j ,whichisassociatedwithprobability p j n .Ateachforwarditeration,instead ofenumeratingallpossiblescenarios,SDDPonlyndsevera lsamplepathsbyMonte Carlosimulation,andsolvethecorrespondingproblemsalo ngthesamplepathsto obtainafeasiblesolution. 1.2StochasticMixedIntegerProgramming Inreality,wealsoneedtomakealotofdiscretedecisionsun deruncertainties, whichneedtoincludeintegervariablesintheoptimization problems,begettingthe stochasticmixedintegerprograms.Theformulationoftwostagestochasticmixed integerprogramsareverysimilartothetwo-stagestochast iclinearprograms,except thatithasintegerrestrictionsonthedecisionvariablese itherintherststage( 1–12 ),or thesecondstage( 1–13 ),orboth. 1.2.1FormulationandPreviousApproaches Thegeneralformulationofstochasticmixedintegerprogra msisasfollows, Min c T x + E [ Q ( x w ) ] (1–12) s.t. Ax b 18

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x 2 R n 1 + Z m 1 + where Q ( x w ) istherecoursefunctionshownasfollows, Q ( x w )= Min [ d ( w )] T y (1–13) s.t. F ( w ) y g ( w ) T ( w ) x y 2 R n 2 + Z m 2 + Stochasticmixedintegerprogramming(SMIP)hasbeendrawi ngalotofattention recently.Whenintegervariablesexistonlyintherststag e,theproblemisrelatively easiertosolve,sincegenerallyL-shapedmethod([ 63 ])orBendersdecomposition([ 9 ]) wouldwork.Thisisbecausethevaluefunctionforthesecond stageisconvexwith respecttotherststagevariables.However,thesecondsta gevaluefunctionbecomes non-convexanddiscontinuous,ingeneral,whentherearein tegervariableswithin thesecondstageasdiscussedin[ 12 ].ThismakesBendersdecomposition([ 9 ])or generalizedBendersdecomposition([ 31 ])notreadilyapplicablebecauseoftheduality gapofintegerprograms.Withinthelasttwodecades,alotof researchhasbeendoneto solveSIMPproblemswithintegervariablesinthesecondsta ge.LaporteandLouveaux [ 38 ]proposedadecomposition-basedbranch-and-cutmethod,w herebothfeasibility andoptimalitycutsareapplied,forSMIPwithpurebinaryva riablesintherststage. CareandTind[ 18 ]proposedageneralizedL-shapemethodbygeneralizedBend ers decomposition([ 31 ]),wherebothGomorycutsandbranch-and-boundalgorithma re applied.SheraliandFraticelli[ 60 ]andSheraliandZhu[ 61 ]developedmodiedBenders decompositionmethodsbysequentiallyconvexifyingthedi scretesubproblemusing reformulation-linearizationtechnique([ 59 ]).NtaimoandSen[ 48 ],SenandHigle[ 57 ] andNtaimo[ 47 ]proposeddecompositionmethodsforSMIPwithrandomrecou rse anddiscretesecondstagebasedondisjunctiveprogramming ([ 5 ]).Ahmedetal.[ 1 ] 19

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developedanitebranch-and-cutsolutionalgorithmforSM IPwithasecondstage programofpureintegervariables.1.2.2GeneratingValidBendersCutsfromDiscreteSubprobl ems Inordertotacklethisdifculty,wemayneedtoconvexifyth esecondstagemixed integerprogramstogetvalidandeffectiveBenderscutsfor therststageormaster problem.TheapproachmadebySheraliandFraticelli[ 60 ]isinspiring.Theyanalyzethe followingtwostagemixedintegerprogram, [ P ] : Min c T x + d T y s.t. Ax + Dy b x 2 X x 2f 0,1 g n y 2 Y where y isavectorincludingintegervariables,and X isanonemptypolytope.Ifwecan ndtheconvexhullofthefollowingregion, f Ax + Dy b y 2 Y g (1–14) foranygiven x ,thenBenders'decompositioncanbeappliedbecausethelin ear relaxationofthesubproblem(convexhullformulation)wil lhavethesameoptimal solutionasthediscretesubproblem,andthenthesubproble mcansimplybetreatedas alinearprogrammingproblem.Toachievethisgoal,Reformu lation-Linearization-Technique orLift-and-Projectcutsareiterativelyaddedtothesubpr oblemin[ 60 ].Thesearecalled globalcuts,whichmeansthattheyarevalidfortheoriginal problem[ P ]butfocuson cuttingtheregionof( 1–14 ),whichhasthefollowingformat, T k y + T k x k k =1,..., K (1–15) where k denotesthe k th cut.Therearealsoothercuts([ 6 ],[ 58 ],etc)whichpossessthe sameproperties( 1–15 )has.Withthesecutsadded,therelaxedsubproblemwillbea s 20

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follows, Min d T y s.t. Dy b Ax y r T k y k T k ^ x k =1,..., K where y r isthelinearrelaxationoftheset Y .Thenwejustneedtosolvethe abovelinearproblemtoproduceBenderscutfortherststag e.Forconvenience,the subproblemisassumedtobefeasiblegivenanyrststagesol ution, ^ x ,sincewecan alwaysaddanarticialvariableandassignabigpenaltyoni ttomaketheproblem feasible.Supposetheoptimaldualsolutionsare 1 2 ,and 3 correspondingtothe abovethreeconstraintsrespectively.ThenavalidBenders 'cutcanbeobtainedas follows, z ( b Ax ) T 1 + r T 2 + K X k =1 ( k T k x ) 3 k Evenwhentheconvexhullofthesubproblemisnotcompletely obtained,theBenders cutsarestillvalidtotherststageproblem.Thisisbecaus ethatthetherelaxed subproblemalwaysprovidesalowerboundtothesubproblem, whichmeansitalso providesavalidlowerboundfor z 1.3OutlineofthisDissertation Thedissertationisorganizedinsuchawaythatwerstintro duceourproposed methodsforstochasticprogrammingproblems,andthendesc ribesomestochastic optimizationmodels,especiallyintheenergysystemsarea ,andnallydiscusshowto applyouralgorithms.InChapter 2 ,weintroducetheEmbeddedBendersdecomposition method,whichalsoexploitsBenderscutsforthesecondstag esubproblems,andthese cutsarereusablegivenanyrststagesolution.Alsoourmet hodgeneratesmultiplecuts 21

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whilesolvingthesubproblemofonlyonescenario,bytaking advantageofthespecial structureofthemodels. Thesecondhalfofthisdissertationisaboutstochasticopt imizationapplicationsin theareaofenergysystems,suchasstochasticunitcommitme ntproblems,optimization modelsinnaturalgasindustry.Chapter 3 introducesthestochasticsecurityconstrained unitcommitmentmodel,whichtriestosolvebothday-aheadc ommitmentschedule andreal-timecommitmentschedule,whileconsideringmini mumupanddowntimes, spinningreservesandnonspiningreserves,unitcapacitie s,andpiecewiselinearfuel costfunctions.WeapplytheEmbeddedBendersdecompositio nonthisproblem,and thenumericalresultsshowthatEBDalgorithmoutperformst hedefaultCPLEXMIP solverforproblemswithlargenumbersofscenarios.Thecom putationaltimealmost increaselinearlywhenweincreasethesizeoftheproblem,w hichmakeEBDavery reliablemethodforsolvingstochasticsecurityconstrain edunitcommitmentproblems. Chapter 4 givesadetailedsurveyofoptimizationmodelsinthenatura lgasindustryby focusingonthenaturalgasproduction,transportation,an dmarket.Chapter 5 proposes expansionplanningmodelswhichincludebothnaturalgastr ansmissionnetwork expansionandLNG(LiquiedNaturalGas)terminalslocatio nplanning.Thesemodels takeintoaccounttheuncertaintiesofdemandsandsupplies inthefuture,whichmake themodelsstochasticintegerprogramswithdiscretesubpr oblems.Alsoweconsider riskcontrolinourmodelsbyincludingprobabilisticconst raints,suchasalimitonCVaR (ConditionalValueatRisk).Inordertosolvethelarge-sca leproblems,especiallywith alargenumberofscenarios,wealsoapplytheembeddedBende rsdecomposition algorithm.Numericalresultsshowthatouralgorithmisef cientforlargescalestochastic naturalgastransportationsystemexpansionplanningprob lems.Chapter 6 concludes thisdissertation,whilealsotalkingaboutfutureresearc h. 22

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CHAPTER2 EMBEDDEDBENDERS'DECOMPOSITION ThemethodproposedbySheraliandFraticelli(2002)[ 60 ]modiestheBenders' decompositionfordiscretesubproblembysequentiallycon vexifyingthemusing Reformulation-LinearizationTechniques(RLT)orlift-an d-projectmethod.However, ittakesalotofeffortstondaglobalcutforeachsubproble mbyusingRLTor lift-and-projectmethod.Attheeachiteration,theiralgo rithmneedstoaddaglobal cuttogetabetterconvexhullforthediscretesubproblem.I nthisdissertation,amethod calledembeddedBenders'decompositionwhichapplyBender 'scuttohelpapproximate theconvexhullofthediscretesubproblemisproposed.TheB enders'cutaddedtothe subproblemisvalidforanyrststagedecisionandthenthey arereusablealongthe iterativecomputations,whichsavealottimetogeneratedi fferentnewconvexication cutsateachiteration.SeveralvariantsofEmbeddedBender s'Decompositionalgorithm tosolvedifferenttypesofproblemsareexplainedindetail sinthefollowingsections. 2.1Two-stageEmbeddedBenders'Decomposition Atypicaldeterministicformulationoftwostagemixedinte gerprogramisasfollows, P 0 : Min c T 1 x 1 + d T 1 z 1 + c T 2 x 2 + d T 2 z 2 s.t. A 1 x 1 + B 1 z 1 b 1 1 E 1 x 1 + F 1 z 1 + G 1 x 2 h 1 1 A 2 x 2 + B 2 z 2 b 2 2 where x i isavectorofcontinuousvariablesand z i isavectorofintegervariables,for i =1,2 .Forconvenience,werstdiscusshowtosolveatwo-stagede terministic problem,whereeachstagecontainsbothcontinuousandinte gervariables. When x 1 and z 1 aregiven,wehavethefollowingproblem, P 1 : Min c T 2 x 2 + d T 2 z 2 23

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s.t. G 1 x 1 h 1 E 1 x 1 F 1 z 1 1 A 2 x 2 + B 2 z 2 b 2 2 BysolvingP 1 ,wecangettheoptimalsolution,( z 2 x 2 ).With x 1 z 1 z 2 ,wecansolve thefollowingdualproblem,i.e.,DP 1 DP 1 : Max ( h 1 E 1 x 1 F 1 z 1 ) T 1 +( b 2 B 2 z 2 ) T 2 s.t. G T 1 1 + A T2 2 c 2 x 2 SupposetheoptimalsolutiontoDP 1 is( 1 2 ).Thenthenewcut, ( h 1 E 1 x 1 F 1 z 1 ) T 1 +( b 2 B 2 z 2 ) T 2 ,constructedbyusing( 1 2 ),isaglobalcutsincethe feasibleregionofproblemDP 1 doesnotdependon( x 1 z 1 ,and z 2 ).So,byaddingthese embeddedBender'scuts,wecanhelpapproximatetheconvexh ullofthediscrete subproblem,andformulateanewsubproblem,RLP 1 ,asfollows, RLP 1 : Min d T 2 z 2 + s.t. ( h 1 E 1 x 1 F 1 z 1 ) T k1 +( b 2 B 2 z 2 ) T k 2 k 2 K k z 2 1, where K isasubsetofthevertexsetofproblemDP 1 ,andintegervariable z 2 isrelaxed tocontinuousvariable.Denotetheoptimaldualsolutionof constraint k ofRLP 1 by k .A validBenders'cutfortherststageproblemisasfollows, e T + X k 2 K ( h 1 E 1 x 1 F 1 z 1 ) T k1 + b T 2 k 2 jk (2–1) Therestrictedmasterproblemisasfollows, RMP 0 : Min c T 1 x 1 + d T 1 z 1 + s.t. A 1 x 1 + B 1 z 1 b 1 e T + Xk 2 K ( h 1 E 1 x 1 F 1 z 1 ) T k1 + b T 2 k 2 jk j 2 J 24

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where J isasubsetofthevertexsetofdualproblemtoRLP 1 Thealgorithmforsolvingthetwostageproblemisshownbelo w, Step0.SetUB= 1 ,LB= 1 K = J = ; Step1.SolveRMP 0 ,andoptimalsolutionandobjectivevalueare( x 1 z 1 )and w respectively.LB max ( LB w ) Step2.SolveP 1 ,andoptimalsolutionis( x 2 z 2 ). UB min ( UB c T 1 x 1 + d T 1 z 1 + c T 2 x 2 + d T 2 z 2 ) Step3.IfUB LB ,stop.O/Wgotostep4. Step4.Ifnecessary,solveDP 1 ,andoptimalsolutionis( 1 2 ). K K [ ( 1 2 ) Step5.SolveRLP 1 ,andoptimalsolutionis =( k k 2 K ). J J [ Gotostep1. P 1 isanintegerprogrammingproblemwithasmallernumberofin tegervariables,which needlittleefforttosolve.Thisalgorithmdecomposeamixe dintegerprogramming problemintomultiplesmallerMIPs,whichmakethetotalcom putationaltimeless.We cangeneralizethisalgorithmtomultistageMIPsasDDPisge neralizationofBenders' decomposition.Eventually,thegeneralizedembeddedBend ers'decompositioncan exploittheSDDP'sideastoavoidthedimensionalityexplos ion. BelowisasimpleexampleofimplementingEmbeddedBendersD ecomposition: P 0 : Min 2 x 1 +3 x 2 + z 1 +6 z 2 +4 x 3 +3 x 4 + z 3 + z 4 s.t. x 1 +2 x 2 +4 z 1 +3 z 3 4, 2 x 1 + x 2 + z 1 + z 2 2, x 2 +2 z 1 + z 2 +3 x 3 + x 4 3, 2 x 3 +3 x 4 + z 3 +2 z 4 5, x 3 + x 4 +2 z 3 + z 4 2, 25

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x 0, z 2f 0,1 g 4 Firstweneedtosolvetherststagesubproblemwithoutcons ideringanyfuture information. DP 0 : Min 2 x 1 +3 x 2 + z 1 +6 z 2 s.t. x 1 +2 x 2 +4 z 1 +3 z 2 4, 2 x 1 + x 2 + z 1 + z 2 2, x 0, z 2f 0,1 g 2 Theoptimalobjectivevalueis w =2 ,withoptimalsolutionbeing( x 1 x 2 z 1 z 2 )= (0.5,0,1,0) .Thenwecanset LB =2 .Byxingtherststagevariableas( x 1 x 2 z 1 z 2 ) wesolvethesecondstagemixedintegersubproblemtogetafe asiblesolutionforthe wholeproblem,whichprovidesanupperbound. P 1 : Min 2 x 1 +3 x 2 + z 1 +6 z 2 +4 x 3 +3 x 4 + z 3 + z 4 s.t. x 2 +2 z 1 + z 2 +3 x 3 + x 4 3, 3 2 x 3 +3 x 4 + z 3 +2 z 4 5, 4 x 3 + x 4 +2 z 3 + z 4 2, 5 x 0, z 2f 0,1 g 2 TheoptimalsolutionofP 1 is( x 3 x 4 z 3 z 4 )= (0,1,0,1) .Thentheupperboundcanbe calculatedasfollows UB = c 1 x 1 + c 2 x 2 + d 1 z 1 + d 2 z 2 + c 3 x 3 + c 4 x 4 + d 3 z 3 + d 4 z 4 =6 26

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Byxing ( z 3 z 4 ) ,wecansolveDP 1 asalinearprogram,ofwhichtheoptimaldual solutionis ( 3 4 5 ) = (0,1,0) .Withtheoptimaldualsolution,wecanconstructa inner/convexicationcutasinthefollowinglinearprogra m, RLP 1 : Min z 3 + z 4 + s.t. (5 z 3 z 4 )( 4 ), z 3 1, 3 z 4 1 4 Thenwesolvetheaboverelaxed/convexiedlinearprogram, andtheoptimalprimaland dualsolutionare, ( ^ ,^ z 3 ,^ z 4 ) = (3,0,1) and( ^ ,^ 3 ,^ 4 )=( 1,0, 1 )respectively.Thenewcut forrststageisconstructedasfollows, 5 4 ^ +^ 3 +^ 3 =4 Thenwesolvetherststagemixedintegerprogramagainwith thenewcutfromthe secondstage, RMP 0 : Min 2 x 1 +3 x 2 + z 1 +6 z 2 + s.t. x 1 +2 x 2 +4 z 1 +3 z 2 4, 2 x 1 + x 2 + z 1 + z 2 2, 4, x 0, z 2f 0,1 g 2 TheoptimalobjectiveoftheaboverststageMIPwithnewout er/feedbackcutis w =6 ,andthenwecanupdateLB= 6 .NowwehaveUB=LB,andoptimalsolutionis ( x 1 x 2 z 1 z 2 x 3 x 4 z 3 z 4 )=( .5,0,1,0,0,1,0,1 ). 27

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2.2StochasticTwo-stageEmbeddedBenders'Decomposition WecanextendthedeterministicversionofEmbeddedBenders 'Decomposition tosolvethestochastictwo-stagemixedintegerprograms.T hedifferencehereisthat weneedtodosimulationstogettheapproximatedexpectedup perbound,whichisthe sampleaverageofsampledsubproblemsplustherststageso lution.Or,weneedto solveallsubproblemsforallscenariostogettheexactexpe ctedcostgivenarststage solution.Thealgorithm(withtheapproximatedupperbound )isshownasfollows, Step1.SetUB= 1 ,LB= 1 K = J = ; Step2.SolveRMP 0 ,andoptimalsolutionandobjectivevalueare( x 1 z 1 )and w respectively.LB max ( LB w ) Step3.For i =1: n Getasample, ,fromallscenarios SolveP 1 ( ) ,andoptimalsolutionis( x 2 ( ) z 2 ( ) ). Z UB ( )= c T 1 x 1 + d T 1 z 1 + c T 2 x 2 ( )+ d T 2 z 2 ( )) SolveDP 1 ( ) ,andoptimalsolutionis( 1 ( ) 2 ( ) ). K K [ ( 1 ( ) 2 ( )) SolveRLP 1 ( ) ,andoptimalsolutionis ( )=( k ( ), k 2 K ( ) ). J ( ) J ( ) [ ( ) Step4.UpdateUBbythesampleaverageof Z UB ( ) 's Step5.IfUB-LB < ,stop;o/w,gotostep1. Usuallythismethodisusedwhenthereexistahugenumberofs cenarios,and n isasmallernumberascomparedtothecardinalityofthescen arioset.Whenthetotal numberofscenariosisnotsobig,wecouldsolveallthesubpr oblemscorrespondingto thescenariostogettheexactexpectedfuturecost. 2.3DeterministicMultistageEmbeddedBendersDecomposit ion Ascanbeseeninthetwostagealgorithm,therearetwosetofc uts: Outer/Feedback(OF)Cuts J i :thecutswhichprovideinformationfromthefuture stages. 28

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Inner/Convexication(IC)Cuts K i :thecutswhichconvexifythemixedinteger subproblem. ItiseasytonotethattherststagedoesnothaveICcuts,and laststagedoesnothave anyOFcuts. Foranystage i ,weneedtosolveamixedintegerprogramtogetafeasiblesol ution, MIP i Min c T i x i + d T i z i + i s.t. A i x i + B i z i b i E i 1 ^ x i 1 F i 1 ^ z i 1 ( ji ) T E i x i +( ji ) T F i z i + i w j i 8 j 2J i x i 0, z i 2f 0,1 g n i Afterwegetasolutionofstage i ,wecanx z i inMIP i andobtainalinearprogram, whichisdenotedbyLP i ,shownasfollows, LP i Min c T i x i + i s.t. A i x i b i E i 1 ^ x i 1 F i 1 ^ z i 1 B i ^ z i i ( ji ) T E i x i + i w j i ( ji ) T F i ^ z i 8 j 2J i r j i x i 0. InordertogetanICcut,whichtriestoconvexifythesubprob lemMIP i ,weneedto solveeitherthelinearprogramLP i oritsdualDLP i togetthedualoptimalsolution. DLP i Max ( b i E i 1 ^ x i 1 F i 1 ^ z i 1 B i ^ z i ) T i + X j 2J i w j i ( ji ) T F i ^ z i r j i s.t. A Ti i + X j 2J i ( ji ) T E i T r j i c i X j 2J i r j i =1, 29

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0, r j i 0 2J i Supposetheoptimaldualsolutionis ^ new i and ^ r j new i 8 j 2J i .ThenanewICcutis constructedasfollows, ( new i ) T B i + X j 2J i ( ji ) T F i ^ r j new i # z i + i ( new i ) T b i + X j 2J i w j i ^ r j new i ( new i ) T E i 1 ^ x i 1 ( new i ) T F i 1 ^ z i 1 (2–2) AfteraddingthethenewlyconstructedICcut,wecangetther elaxed/convexied subproblemRLP i asfollows, RLP i Min d T i z i + i s.t. ( k i ) T z i + i v k i ( k i ) T E i 1 ^ x i 1 ( k i ) T F i 1 ^ z i 1 8 k 2K i ki 0 z i 1, i where k i =( k i ) T B i + X j 2 J i ( ji ) T F i ^ r j k i v k i =( k i ) T b i + X j 2J i w j i ^ r j k i SinceRLP i isarelaxedlinearprogramoftheMIP i ,avalidBenders(OF)cutfor stage i 1 canbeconstructedbyusingtheoptimaldualsolutionofRLP i .Supposethe newlyobtainedoptimaldualsolutionis ^ newi ^ k new i 8 k 2K i .ThenewOFcutforstage i 1 isasfollows, i 1 e T ^ newi + X k 2K i ^ k new i v k i ( k i ) T E i 1 x i 1 ( k i ) T F i 1 z i 1 30

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Ifwelet newi 1 = X k 2K i ^ k new i ^ k i w new i 1 = e T ^ newi + X k 2K i ^ k new i v k i ThenthenewOFcutwillbelike, ( newi 1 ) T E i 1 x i 1 +( newi 1 ) T F i 1 z i 1 + i 1 w new i 1 (2–3) Tosummarize,thedeterministicmultistageembeddedBende rsdecompositionalgorithm isshownasfollows, Step0.UB= 1 ,LB= 1 K i = J i = ; forallstages. Step1.Forstage i =1,2,..., T (a)SolveMIP i withxedvaluesof ^ x i 1 and ^ z i 1 ,andsupposeoptimal solutionis( ^ x i ^ z i ). If i =1 ,LB max ( LB c T 1 x 1 + d T 1 z 1 + 1 ) If i = T ,UB min UB P Ti =1 ( c T i x i + d T i z i ) (b)If i > 1 ,solveLP i ,andsupposetheoptimalsolutionis( ^ i i )and dualoptimalsolutionis( ^ new i ^ r j new i 8 j 2J i ).ConstructanewICcut asin( 2–2 ),andadditinto K i (c)If i > 1 ,solveRLP i ,andsupposeoptimaldualsolutionis ( ^ k new i 8 k 2K i ^ newi ). ConstructanewOFcutasin( 2–3 ),andadditinto J i 1 Step2.IfUB LB ,stop;otherwisegotoStep1. 2.4TheStochasticMultistageAlgorithm Inthestochasticcase,weassumetheuncertaintiesamongst agesareindependent ofeachother.Thisimpliesthatforanystagewithanypossib leoutcome,therewillbe onlyonefuturebenetfunction.Thismakestheproblemeasi erwithoutlosinggenerality becausethereisnoneedtogeneratethewholescenariotree. Sincewehavealready proposedamethodtoobtainvalidfeedbackcutsevenwhenthe subproblemsaremixed integerprogram,wecanstilltakeadvantageofthismethodt ohandlethedimensionality 31

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explosioninSDDP.AlsoSDDPonlyallowstherandomvariable sontherighthandside, becauseitonlyconstructanaggregatedcutforeachstageif atrialvectorofprevious stageisgiven.Thenewapproachinthisproposalistoconstr uctmultiplecutsatatime, whichcancopewithrandomvariablesatanywhereinthesubpr oblem.However,we focusondealingwithproblemwhererandomvariablesappear intheobjectivefunction andrighthandside. Asisshowninthedeterministicalgorithm,wecalculatethe thepresentstage problembasedonthexedvaluesofpreviousstagedecision. Ofcoursewecan calculatethepreviousstagesolutionforeachnodeofthede cisiontreeinstochastic case.Doingthiswillleadtothedimensionalityexplosiond isaster.Forexample,ifwe aregoingtosolveamultistagestochasticmixedintegerpro gram,eachstageofwhich onlyhas3scenarios,therewillbe 3 12 =531441 leafnodes,andthenumberofinteger variablesinonlytheleafnodeswillbe 3 3 12 1.6 million.Thatisadifcultproblem foranystateoftheartMIPsolver.However,thisissueactua llycanbecircumventedby usingforwardsimulation,whichisshownasfollows, Step0.SolveMIP 1 ,andsupposethesolutionis ^ x 1 and ^ z 1 .Let x s 1 =^ x 1 z s 1 =^ z 1 s =1,2,..., N Step1.Forstage i =1,..., T For s =1,..., S Samplea( c s i b s i )from n ( c i b i ), 2 i o ; SolveMIP i with( c s i b s i ); Savethesolution( ^ x s i ^ z s i ). Thisactuallyprovideanestimateoftheupperbound, ^ UB = 1 S S X s =1 T X i =1 ( c s i ) T ^ x s i +( d s i ) T ^ z s i Alsoweneedtocopewiththefuturebenetfunctions(OFcuts ).Thebackward recursionshownbelowismainlytoconstructtheOFcutstore ecttheinformationof futurestages. 32

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For i = T T 1, T 2,...,2 Foreach ^ x s i s =1,2,..., S Foreachscenario 2 i SolveLP i with ^ x s i ConstructanICcutforRLP i asin( 2–2 ) SolveRLP i ConstructanOFcutforMIP i 1 asin( 2–3 ) SolveMIP 1 ,whichprovidealowerbound, LB = c T 1 x 1 + d T 1 z 1 + P 2 2 1 Thestochasticmultistagealgorithmactuallykeepsrunnin gthebackwardrecursion andforwardsimulationuntilUBandLBaresufcientlyclose toeachother.The multistagestochasticalgorithmisshownasfollows, Step0.RunForwardSimulationandcalculatethe ^ UB Step1.RunBackwardRecursionandcalculatethe LB Step2.If ^ UB LB ,stop.OtherwisegotoStep0. Itisinterestingtonotethat ^ UB isanestimationofminimummeanvalueoftotal cost, ^ TC = ^ UB = 1 S S X s =1 T X i =1 ( c s i ) T ^ x s i +( d s i ) T ^ z s i Soitwouldbenicewealsocancalculatetheestimationofits standarddeviationwhich isasfollows, ^ TC = vuut 1 S S X s =1 [ ( c s i ) T ^ x s i +( d s i ) T ^ z s i ^ TC ] 2 Hencewemaystopwhen LB fallsintotherange h ^ UB ^ TC ^ UB +^ TC i 33

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CHAPTER3 STOCHASTICSECURITYCONSTRAINEDUNITCOMMITMENTMODELS Unitcommitmenthasbeenaveryimportantprobleminthepowe rsystem,because itistoreducetheproductioncostofelectricitybyoptimal lyschedulingthecommitments ofgenerationunits.Thisisalsoachallengingproblemsinc eitinvolvesagreatamount integervariables.Traditionally,themostlyusedmethodi sLagrangianrelaxation.Inthis chapter,weintroduceanewtypeofunitcommitmentmodelwhi chtakesintoaccount theuncertaintiesofdemandsandthesecurityconstraints, e.g.,spinningreserves,and non-spinningreserves,toincreasethesystemrobustnessd uringcontingencies.Also, weapplytheEBDmethodpreviouslyproposedinChapter2toth esemodels. 3.1Introduction Sincethe1980s,theenergysectorhasbeenexperiencingadr amaticchange fromregulatedmarkettoderegulatedmarket.Thisintroduc esalotofuncertaintiesto theelectricityproducers,suchasprices,demands,etc.Re cently,inordertocounter thetrendofglobalclimatechange,moreandmorerenewablee nergysourcesare introducedintotheenergymarket.Thisalsobringsuncerta inties,suchaswindpower, solarpower,becauseoftheweather.Thismakesstochasticp rogrammingmodelsvery necessaryforproductioncompaniestoachieveprotmaximi zationorcostminimization. Therearegenerallytwotypesoffossilfuelgenerators,qui ck-startgeneratorsand traditionalgenerators,inuseinmostoftheelectricityco mpanies.Thetraditional generatorsareusuallyusingcoalandtakesalongtimetoget started,i.e.,acouple ofhours,whichhastobescheduledadayahead.Thequick-sta rtgenerators,instead, aretransferringgasoroilenergytoelectricity,andcange tstartedalmostimmediately, say,inlessthan10minutes.Thenquick-startgeneratorsar eusuallyusedasremedies tomeetthehighdemandsinrealtime.Becausebinaryvariabl esareusedtomodel whetherageneratorisonoroff,thefuelcostandstartupcos tminimizationproblem isastochasticmixedintegerprogramwithdiscreteseconds tage.Thisisverydifcult 34

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Table3-1.SCUCSetsandIndices N c numberofcoalpowerunits N g numberofgaspowerunits thesetofallpossiblescenarios T lengthofplanninghorizon i j indicesofgenerators t timeperiod indicesofscenarios Table3-2.SCUCParameters SU it start-upcostofunit i inperiod t SD g it shut-downcostofunit i inperiod t l i minimumdowntimeofunit i L i minimumuptimeofunit i P min i minimumamountofpowergeneratedbyunit i P max i maximumamountofpowergeneratedbyunit i U i rampinguplimitofunit i D i rampingdownlimitofunit i S max i maximumspinningreserveofunit i RS t spinningreserverequirementattime t ofscenario RO t operatingreserverequirementattime t ofscenario PD t real-timesystemdemandattime t ofscenario PL t real-timesystemlossesattime t ofscenario tosolvedirectlybyanystateoftheartcommercialoptimiza tionsoftwarewhenwe experienceabignumberofscenarios.Followingthissectio n,werstintroducethe model,andthenapplytheEBDalgorithmtosolveit,andnall yshowthenumerical resultsfordifferentsettings. 3.2ProblemFormulation Inthestochasticsecurityconstrainedunitcommitmentpro blem,wehaveboth day-aheadandreal-timeunitcommitmentscheduling.Inthe day-aheadscheduling, weneedtomakecommitmentplansforallgeneratingunits,in cludebothnon-fast-start generatorsandfast-startgenerators.However,inthereal timescheduling,onlyfast-start unitscanberescheduled.Alsothepowergeneratedbyoneuni tcanbeadjustedinreal timeifitsstatusis“on”atthattimeperiod.Inordertofaci litatethedescriptionofour model,welistthesetsandindicesusedinthischapterinTab le 3-1 ,parametersinTable 35

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Table3-3.SCUCDecisionVariables it commitmentdecisionofunit i inperiod t r it startupactionofunit i atperiod t it shutdownactionofunit i atperiod t p it amountofpowergeneratedbyunit i inperiod t ofscenario s it spinningreserveofunit i inperiod t ofscenario q it operatingreserveofunit i inperiod t ofscenario r it real-timestartupactionofgasunit i inperiod t ofscenario it real-timeshutdownactionofgasunit i inperiod t ofscenario jt commitmentdecisionofgasunit i inperiod t ofscenario y it startuprescheduleindicatorofgasunit i inperiod t ofscenario z it shutdownrescheduleindicatorofgasunit i inperiod t ofscenario 3-2 ,anddecisionvariablesinTable 3-3 it denotesthecommitmentstatusofunit i at timeperiod t ,with“ 0 ”meaning“off”and“ 1 ”viceversa. r it isthestart-upactionindicator, ofwhich“ 1 ”meansthereisastart-upactionand“ 0 ”viceversa,and it istheshut-down actionindicator. jt istherescheduledcommitmentstatusvariableoffast-star tunit j at time t inscenario .Sodothestart-upactionindicatorvariable, r jt ,andtheshut-down actionindicatorvariable, jt y jt isthestart-upreschedulingindicator,ofwhich“ 1 ”means astart-upactionhappensinrealtimebutnotintheday-ahea dschedule,and“ 0 ”means real-timescheduleisassameasday-aheadone,and“ 1 ”meansthereisastart-up actioninday-aheadschedulebutnotintherealtime.Theext ensiveformulationis shownasthefollowing,[ ESCUC ]: min T X t =1 X i 2f N c [ N g g ( SU it r it + SD it it ) (3–1) + T X 2 Prob T X t =1 24 X i 2f N c [ N g g F i p it + X j 2 N g SU jt y jt + SD jt z jt 35 (3–2) s.t. it i ( t 1) i 8 i 2 N c = t ,...,min f t + L i 1, T g t =2,..., T (3–3) i ( t 1) it 1 i 8 i 2 N c = t ,...,min f t + l i 1, T g t =2,..., T (3–4) r it it i ( t 1) 8 i 2f N c [ N g g t =1,..., T (3–5) 36

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it it + i ( t 1) 8 i 2f N c [ N g g t =1,..., T (3–6) X i 2f N c [ N g g p it PD t + PL t t =1,..., T 8 2 (3–7) X i 2f N c [ N g g s it RS t t =1,..., T 8 2 (3–8) X j 2 N g q jt RO t t =1,..., T 8 2 (3–9) y jt r jt r jt 8 j 2 N g t =1,..., T 8 2 (3–10) z jt jt jt 8 j 2 N g t =1,..., T 8 2 (3–11) r jt jt j ( t 1) 8 j 2 N g t =1,..., T 8 2 (3–12) jt jt + j ( t 1) 8 j 2 N g t =1,..., T 8 2 (3–13) p it P min i it 8 i 2 N c t =1,..., T 8 2 (3–14) p it + s it P max i it 8 i 2 N c t =1,..., T 8 2 (3–15) p jt P min j jt 8 j 2 N g t =1,..., T 8 2 (3–16) p jt + s jt P max j jt 8 j 2 N g t =1,..., T 8 2 (3–17) D i p it p i ( t 1) U i 8 i 2f N c [ N g g t =1,..., T 8 2 (3–18) s it S max i 8 i 2f N c [ N g g t =1,..., T 2 (3–19) q jt (1 jt ) P max j 8 j 2 N g t =1,..., T 8 2 (3–20) jt r jt jt 2f 0,1 g 8 j 2 N g t =1,..., T 2 (3–21) it r it it 2f 0,1 g 8 i 2f N c [ N g g t =1,..., T (3–22) y jt z jt 2f 1,0,1 g 8 j 2 N g t =1,..., T 8 2 (3–23) p s q 0, 8 2 (3–24) wherewecanjusttreatboth r it and it aspositivecontinuousvariablessincethereare positivecostsrelatedtothemintheobjectivefunction.Fo rconvenience,let p bea vectorcomposedofall p it i =1,..., N c t =1,..., T .Sodo s q r r g y and z throughtherestofthischapter. 37

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( : L ; L s r ? s ? t ? u ? v ? w t u v w Figure3-1.LinearApproximationoftheFuelCostFunction Becausethecostfunctionitselfisconvex,usuallyaquadra ticfunctionwitha positivesecondorderderivative,itspiecewiselinearapp roximationfunctionisstill convex.Hencewecanusethefollowingfunctionandconstrai ntstoapproximatethe originalfunction F ( p ) intheobjectivefunction,asisshowninFigure 3-1 F ( p )= K X k =1 C k k Andweneedtoaddthefollowingsintotheconstraints, p = K X k =1 k k K X k =1 k =1 k 0, k =1,..., K 38

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3.3ProblemDecomposition Itisnontrivialtohaveasecondstagewithdiscretevariabl esespeciallywhenthere areahugenumberofscenarios.Inthischapter,wearegoingt oadoptthesamemethod proposedby[ 67 ].Theextensiveformulation,[ESCUC],istoodifculttoso lvebecause ofthehugenumberofscenarios.Sowewouldliketodecompose theproblemintotwo stages:themasterproblemandsubproblems. Therestrictedmasterproblemisshownasfollows, [ RMP ]: min T X t =1 X i 2f N c [ N g g ( SU it r it + SD it it ) + X 2 Prob (3–25) s.t. it i ( t 1) i 8 i 2 N c = t ,...,min f t + L i 1, T g t =2,..., T (3–26) i ( t 1) it 1 i 8 i 2 N c = t ,...,min f t + l i 1, T g t =2,..., T (3–27) r it it i ( t 1) 8 i 2f N c [ N g g t =1,..., T (3–28) it it + i ( t 1) 8 i 2f N c [ N g g t =1,..., T (3–29) it 2f 0,1 g 8 i 2f N c [ N g g t =1,..., T (3–30) r it it 0, 8 i 2f N c [ N g g t =1,..., T (3–31) X i 2 N c ^ x n it it + X ji 2 N g ^ d n jt r jt + ^ e n jt jt + b n 8 n 2J 2 (3–32) where r it and it arerelaxedtononnegativecontinuousvariables,becauset heyare relatedtopositivecostsandaredeterminedbybinaryvaria bles it and i ( t 1) isa upperboundvariablefortherecoursefunctionofscenario ,and ^ x n it ^ d n jt ^ e n jt andb n arethecoefcientsofcut n ,whichwillbeexplainedindetailslater.Whentherststag e decisionvariablesarexed,thesecondstagewouldbedecom posedto j j separate subproblemssinceonlyonescenariowillhappeninreality. Theonlydifferencebetween twosubproblemsarethedemandsasshowninthefollowingfor mulation.Foreach 39

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scenario 2 ,thesubproblem,withxedvaluesofrststagedecisionvar iables,isas follows,[ SP ]: min T X t =1 24 X i 2f N c [ N G g X k 2 K i C i k it k + X j 2f N g g SU jt y jt + SD jt z jt 35 s.t. p it = X k 2 K i i k it k 8 i 2f N c [ N g g t =1,..., T X k 2 K i it k =1, 8 i 2f N c [ N g g t =1,..., T X i 2f N c [ N g g p it PD t + PL t t =1,..., T X i 2f N c [ N g g s it RS t t =1,..., T X j 2 N g q jt RO t t =1,..., T y jt r jt ^ r jt 8 j 2 N g t =1,..., T z jt jt ^ jt 8 j 2 N g t =1,..., T r g jt jt j ( t 1) 8 j 2 N g t =1,..., T g jt jt + j ( t 1) 8 j 2 N g t =1,..., T P min i ^ it p it + s it P max i ^ it 8 i 2 N c t =1,..., T P min j jt p jt + s jt P max j jt 8 j 2 N g t =1,..., T D i p it p i ( t 1) U i 8 i 2f N c [ N g g t =1,..., T s it S max i 8 i 2f N c [ N g g t =1,..., T q jt (1 jt ) P max j 8 j 2 N g t =1,..., T jt r jt jt 2f 0,1 g 8 j 2 N g t =1,..., T y jt z jt 2f 1,0,1 g 8 j 2 N g t =1,..., T p s q 0, 40

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Foranygiven ^ ^ r and ^ ,SP mightbeinfeasible,whichmeanswemayneed tosolveitsdualinordertoconstructacutfor[RMP].Howeve r,wecanrelaxthe subproblemtomakeitalwaysfeasiblebyintroducingadummy generatorwithhigher costs.Wecallthenewproblemtherelaxedsubproblem,which isshownasfollows, [ RSP ]: min T X t =1 24 X i 2f N c [ N G [ d g X k 2 K i C i k it k + X j 2f N g [ d g SU j y jt + SD j z jt 35 (3–33) s.t. p it = X k 2 K i i k it k 8 i 2f N c [ N g [ d g t =1,..., T (3–34) X k 2 K i it k =1, 8 i 2f N c [ N g [ d g t =1,..., T (3–35) X i 2f N c [ N g [ d g p it PD t + PL t t =1,..., T (3–36) X i 2f N c [ N g [ d g s it RS t t =1,..., T (3–37) X j 2f N g [ d g q jt RO t t =1,..., T (3–38) y jt r jt ^ r jt 8 j 2f N g [ d g t =1,..., T (3–39) z jt jt ^ jt 8 j 2f N g [ d g t =1,..., T (3–40) r g jt jt j ( t 1) 8 j 2f N g [ d g t =1,..., T (3–41) g jt jt + j ( t 1) 8 j 2f N g [ d g t =1,..., T (3–42) p it P min i ^ it 8 i 2 N c t =1,..., T (3–43) p it + s it P max i ^ it 8 i 2 N c t =1,..., T (3–44) p jt P min j jt 8 j 2f N g [ d g t =1,..., T (3–45) p jt + s jt P max j jt 8 j 2f N g [ d g t =1,..., T (3–46) D i p it p i ( t 1) U i 8 i 2f N c [ N g [ d g t =1,..., T (3–47) s it S max i 8 i 2f N c [ N g [ d g t =1,..., T (3–48) 41

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q jt (1 jt ) P max j 8 j 2f N g [ d g t =1,..., T (3–49) jt r jt jt 2f 0,1 g 8 j 2f N g [ d g t =1,..., T (3–50) y jt z jt 2f 1,0,1 g 8 j 2f N g [ d g t =1,..., T (3–51) p s q 0, (3–52) whereweassume ^ r dt = ^ dt =0 t =1,..., T .Becausetherelaxedsubproblemisstill amixedintegerprogram,weneedtofurtherdecomposeitinor dertondalegitimate dualoptimalsolutiontoconstructacuttingplaneforthe[R MP].Ourstrategyistokeep convexifyingthesubproblemwhilereturningcutstothe[RM P]alongtheiterations. Whenthebinaryvector isxed,thesubproblemassociatedwithscenario reduceto alinearprogram,[LP ],shownasfollows, [ LP ]: min T X t =1 X i 2f N c [ N g [ d g X k 2 K i C i k it k (3–53) s.t. p it = X k 2 K i i k it k 8 i 2f N c [ N g [ d g t =1,..., T (3–54) X k 2 K i it k =1, 8 i 2f N c [ N g [ d g t =1,..., T l it (3–55) X i 2f N c [ N g [ d g p it PD t + PL t t =1,..., T h I t (3–56) X i 2f N c [ N g [ d g s it RS t t =1,..., T h II t (3–57) X j 2f N g [ d g q jt RO t t =1,..., T h III t (3–58) P min i ^ it p it + s it P max i ^ it 8 i 2 N c t =1,..., T u it (3–59) P min j ^ jt p jt + s it P max j ^ jt 8 j 2f N g [ d g t =1,..., T v jt (3–60) D i p it p i ( t 1) U i 8 i 2f N c [ N g [ d g t =1,..., T w it (3–61) s it S max i 8 i 2f N c [ N g [ d g t =1,..., T r I it (3–62) 42

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q jt (1 ^ jt ) P max j 8 j 2f N g [ d g t =1,..., T r II jt (3–63) p s q 0. (3–64) Solvingtheabove[LP ]cangenerateaglobalcut,whichisshownin( 3–65 ),to convexifythefollowing[RLP ]whichcapturetheintegerpartof[RSP]. T X t =1 X i 2 N c ^ it P min i ^ u it + P max i ^ u + it + T X t =1 X j 2f N g [ d g jt h P min j ^ v jt + P max j ^ v + jt r II jt i + T X t =1 h ^ h I t PD t + PL t + ^ h II t RS t + ^ h III t RO t i + T X t =1 24 X i 2f N c [ N g [ d g ^ l it D i ^ w it + U i ^ w + it + S max i ^ r I it + X j 2f N g [ d g P max j ^ r II it 35 (3–65) where ^ l ^ h I ^ h II ^ h III ^ u ^ v ^ w ^ r I ^ r II jt aretheoptimaldualsolutioncorresponding toconstraints( 3–55 )–( 3–63 )respectively.By“global”,itmeansthatthecutisvalidfo r [RLP ]givenanyrststagesolution,i.e.,thesolutionfromthem asterproblem, ^ ^ r and ^ .Forconvenience,werewrite( 3–65 )invectorformatasfollows, b + f + a ^ Thenwecanincludetheseglobalcutstoconstructarelaxedv ersionofthesubproblems asfollows,[ RLP ]: min T X t =1 24 X j 2f N g [ d g SU jt y jt + SD jt z jt 35 + T X t =1 X j 2f N g [ d g ( SU jt + SD jt ) (3–66) s.t. y jt r jt 1 ^ r jt 8 j 2f N g [ d g t =1,..., T jt (3–67) z jt jt 1 ^ jt 8 j 2f N g [ d g t =1,..., T jt (3–68) r jt jt + j ( t 1) 0, 8 j 2f N g [ d g t =1,..., T (3–69) 43

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jt + jt j ( t 1) 0, 8 j 2f N g [ d g t =1,..., T (3–70) b k + f k + a k ^ 8 k 2K k (3–71) jt 1, 8 j 2f N g [ d g t =1,..., T jt (3–72) y jt 2, 8 j 2f N g [ d g t =1,..., T jt (3–73) z jt 2, 8 j 2f N g [ d g t =1,..., T jt (3–74) y z r 0, (3–75) where K isthesetcontainingallglobalcutsforscenario Proposition3.1. Thecut,( 3–65 ),obtainedbysolving[LP ]isaglobalcutfor[RLP ] givenanyrststagesolution, ^ ^ r and ^ Sincetheabove[RLP ]isalinearprogram,whosefeasibleregionisanapproximat ion oftheconvexhullof[RSP ]sfeasibleregion,wecanderiveavalidBenderscutfor [RMP],shownin( 3–76 ),bysolvingitsdualproblemoptimally. T X t =1 X i 2 N c X k 2 K ^ k a k it it + T X t =1 X j 2 N g ^ jt r jt +^ jt jt + X k 2K f k ^ k + X j 2f N g [ d g ^ jt +^ jt + T X t =1 X i 2f N g [ d g ^ jt +2 ^ jt +2 ^ jt (3–76) where ^ k istheoptimaldualsolutioncorrespondingtothe k th globalcutsin[RLP ],and ^ ^ ^ ^ and ^ ,aretheoptimaldualsolutionscorrespondingtoconstrain ts( 3–67 ), ( 3–68 ),( 3–72 ),( 3–73 )and( 3–74 )respectively. Proposition3.2. Thecutfrom[RLP ]isavalidBenderscutfor[RMP]. Itisinterestingtonotethatall[LP ]ssharethesamedualspace(dualfeasible region)sincethecostsintheobjectivefunctionsandlefthand-sidecoefcientsarethe same.Hencethedualsolutionobtainedbysolvingaspecic[ LP ]couldbeusedto constructtheglobalcutsforotherscenarios.Inconstrain t( 3–71 ),thesetofcuts, K isdesignatedtoonlyonesinglescenario, .However,wecangeneralizethissettoall scenarios,whichissupportedbythefollowingcorollary. 44

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Corollary1. Theglobalcutofscenario ,( 3–65 ),withrighthandsidebeingchangedto f a ^ ,isalsovalidfor[RLP ]givenanyrststagesolution,forall 2 Forallscenarios,thisconstraint,( 3–65 ),isalmostthesameexcept f k ,whichis calculatedasfollows, f k = T X t =1 h ^ h I k t PD t + PL t + ^ h II k t RS t + ^ h III k t RO t i + f c k wherethe ( k ) inthesuperscriptsdenotes k th optimaldualsolutionin K ,and f c k = T X t =1 24 X i 2f N c [ N g [ d g ^ l k it D i ^ w k it + U i ^ w + k it + S max i ^ r I k it + X j 2f N g [ d g P max j ^ r II k it 35 whichisindependentofscenario .So f k canbeconsideredasanafnefunctionof PD t PL t RS t and RO t ,inwhich f c k istheconstant.Thenall[RLP ]scansharethe sameglobalcutsetexceptfortherighthandsides.Insteado fmultipleglobalcutsetsfor allthescenarios,weonlyneedtomaintainasingleglobalcu tsetasfollows, b k + f k + a k ^ 8 k 2K (3–77) wheretheonlydifferencesbetweenthescenariosarethenam esofvariablesandright handsides. Afterwereplace( 3–71 )in[RLP ]by( 3–77 ),theleft-hand-sidecoefcientsof[RLP ] arenotdependentofthescenariosanymorebecause b k isthesameforallscenarios. Thisalsomeansthatall[RLP ]shavethesamedualfeasibleregionbecausetheyare alllinearprograms.Soanoptimaldualsolutiontoonescena rioisalsoafeasibledual solutiontoanotherscenario.Hencethedualoptimalsoluti ons, ^ ^ ^ ^ ^ ,and ^ to[RLP ]canhelpconstructvalidBenderscutsfromallotherscenar ios,butwith different f k 's,whichisstatedinthefollowingtheorem. Proposition3.3. Forall 2 P Tt =1 P i 2 N c P k 2 K ^ k a k it it + P Tt =1 P j 2 N g ^ jt r jt +^ jt jt + (3–78) 45

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P k 2K f k ^ k + P j 2f N g [ d g ^ jt +^ jt + P Tt =1 P i 2f N g [ d g ^ jt +2 ^ jt +2 ^ jt isavalidBenderscutfor[RMP]givenanyrststagesolution ,where ^ ^ ^ ^ ^ and ^ arethedualoptimalsolutionsto[RLP ]. Withthesedisaggregatedcutsbeingaddedintothe[RMP],we needtoinclude j j recoursevariables, s.Inthecaseofabignumberofscenarios,thiscouldincreas e thecomputationalburdenofsolvingtherestrictedmasterp roblem.However,allthe cutsgeneratedbythesamedualsolutionof[RLP ]canbeaggregatedtoonesingle cutbyaddingthemtogetherwhilemultiplyingeachofthemby theprobabilityofits correspondingscenario.Theaggregatedcutisshownasfoll ows, T X t =1 X i 2 N c X k 2 K ^ k a k it it + T X t =1 X j 2 N g ^ jt r jt +^ jt jt + T X t =1 h h I t PD t + PD t + h II t RS t + h III t RO t i + X k 2K ^ k f c k + X j 2f N g [ d g ^ jt +^ jt + T X t =1 X i 2f N g [ d g ^ jt +2 ^ jt +2 ^ jt (3–79) where P 2 isreplacedby ,and RS RO PD and PL aretheexpectationsof therandomspinningreserve,operatingreserveanddemand. h I t h II t and h III t are aggregatedoptimaldualsolutionsasfollows, h I t = X k 2K ^ k ^ h I k t h II t = X k 2K ^ k ^ h II k t h III t = X k 2K ^ k ^ h III k t Ifwechoosetoaddtheaggregatedcutstotherelaxedmasterp roblem,RMP,the term, P 2 ,initsobjectivefunctioncanbethensimplyreplacedby .According tothenumberofscenarios,wecouldchoosedifferentstrate giestoaddvalidBenders cuts.Asdiscussedin[ 10 ],thedisaggregatedschemeischoseninthecaseofasmall 46

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numberofscenarios,andviceversa.Weprefertheaggregate dschemebecausethe aggregatedcutscontainmoreinformationfromallscenario sandareveryeasyto generateduetothesharingofsamedualspace. 3.4SolutionAlgorithm IfthereisnoBenders'cutbeingaddedinthe[RMP],withouti ncludingrecourse variable intheobjectivefunction,itsoptimalsolutionis (^ ^ ,^ r ^ )=0 .Itisbecause thatallthevariablesarenonnegativeandthecostsofstart upandshutdownarepositive. Hence0isthebestobjectivevaluethat[RMP]canachieve.Th enwecanusethisas theinitialsolutionfortheEmbeddedBendersDecompositio n.Ateachiteration,afterwe solvethe[RMP],itsoptimalobjectivevalueisusedasalowe rboundof[ESCUC].An upperboundcanbeobtainedasfollows, Z UB =Z RMP ^ + X 2 Prob Z RSP (3–80) where Z RSP Z RMP and ^ aretheoptimalobjectivevaluesof[RSP ]and[RMP],andthe solutionof respectively.Thisactuallyrepresentsthecostofafeasib lesolutiontothe relaxed[ESCUC]withadummycostlygeneratorbeingadded. Thelowerboundbasedonthesolutionof[RMP]couldimprovev eryslowlyin practicewhenthe UB and LB areveryclosetoeachother.Oneofthemethodstoavoid slowconvergenceorevenstallingistoapplytheintegerL-s hapedcutsinceitisan optimalitycutwhichensurestoimprovethelowerboundifth ereexistsasolutionwitha higherobjectivevalue.AnintegerL-shaped“optimality”c utisasfollows, z ( Q (^ x ) L ) X j 2 T x j X j 2 F x j j T j +1 + L where Q (^ x ) istherecoursefunctionof ^ x ,therststagesolution,and L isalowerbound forthesecondstageproblem,and T = f j j ^ x j =1 g and F = f j j ^ x j =0 g ,if x isthe rststagedecisionvariableand ^ x isthecurrentsolution.Thisfollowsfromthefact thattherighthandsidewillbeequalto Q (^ x ) if x =^ x ,andlessthan L otherwisesince 47

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P j 2 T x j P j 2 F x j j T j +1 0 if x 6 =^ x .Wereferinterestedreadersto[ 38 ]forthe detailedproof.Withouthavingtodenethetwosets, T and F ,afterrearrangingterms thecutcanbeexpressedbythefollowingequivalentinequal ity, ( Q (^ x ) L ) X j (1 2^ x j ) x j + z Q (^ x ) ( Q (^ x ) L ) X j ^ x j Therststageproblem,[RMP],isapureintegerprogramwith only0-1variables,and thenwecanapplytheintegerL-shaped“optimality”cutin[R MP],whichisshownas follows, [ Q (^ ) L ] 24 T X t =1 X i 2f N c [ N g g (1 2^ it ) it 35 + Q (^ ) [ Q (^ ) L ] 24 T X t =1 X i 2f N c [ N g g ^ it 35 (3–81) wheretherecourse Q ( ) isafunctionofonly ,thecommitmentstatusofbothcoaland gaspowergenerators,sincethebestoptimalobjectivevalu eofthesecondstageis uniquelydenedoncetheyaredetermined.Ifallgenerators remain“on”ateachtime periods,therewillbenostartupandshutdowncost,andthen theoptimalobjective costcanbeusedasanlowerboundofthesecondstage, L .TheembeddedBenders decompositionalgorithmisshownasfollows, Step0.Set UB= 1 LB=0 K = ; (^ ,^ r ^ )=0 Z UB =0 Z RMP =0 ,and ^ =0 ; Step1.Solve[RSP ], 8 2 ,andsupposethatoptimalsolutionandobjective valueare( ^ ^ p ^ q ^ ^ y ^ z ^ r g ^ g ),and Z RSP 8 2 ; Update Z UB ; UB min(UB,Z UB ) Step2.Solve[LP ],anddualoptimalsolutionsare( ^ w ^ h ^ u and ^ v ); Addthisnewdualsolutiontotheset, K ; Repeatthisforall 2 ; Step3.Solve[RLP ],andsupposetheoptimaldualsolutionis( ^ ^ ^ ^ l ^ r and ^ ); Addanewaggregatedcut,asin( 3–79 ),into[RMP]; Repeatthisforall 2 ; 48

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Step4.AddanintegerL-shaped“optimality”cut,asin( 3–81 ),into[RMP]; Step5.Solve[RMP],andsupposethatoptimalsolutionandob jectivevalueare ( ^ ,^ r ^ ,^ )and Z RMP respectively; LB max( LB ,Z RMP ) ; Step6.If UB LB ,stop;Otherwise,gotoStep1. where isasmallvalueforthegaptolerance.Asisshownabove,were peatedlysolve [LP ]and[RLP ]forallscenariosinstep2and3respectively.However,ino rderto improvecomputationalefciencywedonotneedtorepeatfor allscenariossinceall [LP ]sand[RLP ]sarecorrespondingtothesamerststagedecision.Oneway isto samplefromallthescenariosandonlysolvealimitedamount of[LP ]sand[RLP ]s. Inthisalgorithm,wemaintaintwosetsofcuts:oneforconve xifyingthemixed integersubproblems,andoneforconstructingthefuturebe netfunctions.Therstset ofcutsarecalledinnerconvexication(IC)cuts,andthese condsetofcutsarereferred toasouterfeedback(OF)cuts.Becausebothtypesofcutsare actuallyBenderscuts, andICcutsareembeddedinthesubproblemstoprovidevalidO Fcuts,wecallthis algorithmEmbeddedBendersDecompositionalgorithm.When thealgorithmactually terminate,wemayneedtocheckthesolutioninordertodeter mineiftheoriginal [ESCUC]isfeasibleornot.Anyvariablerelatedtothedummy costlygeneratorshould beequaltozero.Otherwise,[ESCUC]isinfeasiblebecausee venthealltheavailable generatorsareturnedon,someofrequirementconstraints( 3–7 ),( 3–8 )or( 3–9 )cannot besatised,whichmeansthedemandsareactuallygreaterth anthetotalgeneration capacityofallunits. 3.5NumericalExamples Inthissection,wepresentnumericalresultsofouralgorit hmonservalproblems withdifferentsizesandsettings.WecodeourembeddedBend ersdecomposition algorithminMicrosoftVisualC++whilecallingCPLEX10(Co ncertTechnology)tosolve thedecomposedproblems.AllprogramsareruninMicrosoftW indowsXPProfessional 2002SP2onaDellDesktopwithIntelPentium4CPU3.40GHzand 2GBofRAM. 49

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Table3-4.GeneratorsData StartupGenerationPower10Minutes UnitCostCost(MW)Spinning (MBtu)(MBtu/MW)MaxMin(MW) G110081201010G280101002010G315012502010G418015601010 Table3-5.Loadforecastofasimpleexample Loads(MW) Prob ( ) t =1 t =2 10.222026020.525028030.3270300 Asecurityconstrainedunitcommitmentproblemwithfourge nerators,ofwhichG3 andG4arefast-startgenerators,isdiscussedbelow.Thege neratordataareshown inTable 3-4 .Forconvenience,wesolveaproblemwithtwotimeperiodand three scenarios,withdatashowninTable 3-5 ByapplyingEmbeddedBendersComposition,after5iteratio nswith10cutsadded intherststage,thealgorithmreachestheoptimalityandr eturnsthesameoptimal solutionasthecompletemodelsolvedbyCPLEX,whichtakes2 3interationsandadds 8cuts.TheresultsareshowninTable 3-6 .Computationaltimes(inmilliseconds)of moreexamplesareshowninTable 3-7 ,inwhichwelistthetotalcomputationaltimes, andcomputingtimesfor[RMP],[SP],[LP]and[RSP].Ascanbe seeninTable 3-7 computingtimesalmostincreaselinearlywithrespecttoth enumberofscenarios,which Table3-6.Solutionofthe4-unitSCUCwith3scenarios CostGeneration(MW) (MBtu) t G1G2G3G4 112010000 14730 21201004001120100300 25540 212010050101120100500 36080 21201005030 50

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meansourEBDmethodiswellsuitedtoproblemswithalargenu mberofscenarios. Also,theEBDalgorithmspendsabigportionoftimetosolve[ SP]and[LP].Henceit ispossibletofurtherreducecomputingtimeifwedonotcalc ulatenewICandOFcuts foreachscenarioinStep1becauseloopingthroughallscena riostakesalotoftime, especiallywhenwehaveahugenumberofscenarios.Moreadva ncedimplementation couldhelptoachievethisandimprovetheoverallperforman ce. 51

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Table3-7.ComputationalResultsofSCUC InstanceGroup j j ESCUCRMPRSPLPRLPTTLItCntTTL/ESCUCRMP/ESCUC 1472201876261530511.284.68 j N c j =2 3622345176231844513.613.77 j N g j =3 879232135677471712521.672.94 j K j =4 20141203367479634019528.501.44 10078120218005803218319523.460.26 100024922187148439624714873545.970.01 14720318712677593512.624.32 j N c j =2 3622055159363876514.133.31 j N g j =3 8792181464126631871523.682.76 j K j =6 201562183708124644114526.371.40 1001000236185341126318945518.950.24 1000504851871845021264818486343.660.00 162203220788058159.373.27 j N c j =2 37863249311635924.600.81 j N g j =3 8125172137547481642513.141.38 j K j =8 2040615631544831338948.350.38 10076111561680245481705142.240.02 1000152526125133855772113407830.880.00 1632332657779654510.383.70 j N c j =2 37847238623137824.850.60 j N g j =3 81411721628109471956413.871.22 j K j =11 2042217232817747357748.480.41 10088901401643679461670141.880.02 1000227877126133635483113384030.590.00 52

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CHAPTER4 OPTIMIZATIONMODELSINNATURALGASINDUSTRY Withthesurgeoftheglobalenergydemand,naturalgasplays anincreasingly importantroleintheglobalenergymarket.Tomeetthedeman d,optimizationtechniques havebeenwidelyusedinthenaturalgasindustry,andyielde dalotofpromisingresults. Inthischapter,wegiveadetaileddiscussionofoptimizati onmodelsinthenaturalgas industrywiththefocusonthenaturalgasproduction,trans portation,andmarket. 4.1Introduction Concernedaboutglobalwarmingandshortageofcrudeoil,pe oplebecomemore interestedinnaturalgaswhichisarelativelycleanenergy sourceandabundantin manyplaces.Naturalgasmainlyconsistsofmethane,andwhe nburnt,itreleasesa fairamountofenergyandlessgreenhousegases( e.g. ,CO 2 )thanoilandcoal.As wecanseefromFig. 4-1 ,theworldgasconsumption/productionislinearlygrowing since1980fromapproximately52,890billioncubicfeettoa pproximately104,424 billioncubicfeetin2006,accordingtotheInternationalE nergyAnnual2006from USDepartmentofEnergy,EnergyInformationAdministratio n(EIA).Moreover,the naturalgasconsumptionisexpectedtocontinuetogrowline arlytoapproximately153 trillioncubicfeetin2030,whichisanaveragegrowthrateo fabout1.6percentperyear accordingtotheInternationalEnergyOutlook2009fromEIA In2008,theresidentialuseofnaturalgasaccountedfor21% ,thecommercialuse for13%,theindustrialusefor34%,thetransportationfor3 %andtheelectricpower productionfor29%theAnnualEnergyReview2009fromEIA.Th eindustrialsector isexpectedtoremainthelargestend-usesectorfornatural gasthrough2030withan expectedshareof40%accordingtotheInternationalEnergy Outlook2009fromEIA. Theelectricpowergenerationfromnaturalgaswasthesecon dlargestconsumerof naturalgasaftertheindustrialsectorin2006.Theelectri citygenerationaccounted in2006for32%oftheworld'stotalnaturalgasconsumption. Duetotheworldwide 53

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n n n Figure4-1.WorldGasConsumptioninBillionCubicFeetdiscussions/attemptstoreducegreenhousegasemissions, theelectricitygeneration vianaturalgasisexpectedtobecomeevenmoreimportantand itsshareoftheworld's totalnaturalgasconsumptionisexpectedtoincreaseto35% in2030accordingtothe InternationalEnergyOutlook2009fromEIA.Hence,natural gasremainsanimportant sourceofenergyforboththeindustrialandtheelectricity sectors. Thischapterdiscussesdifferentoptimizationmodelsinth enaturalgasindustry. Wefocusonthreekeyapplications:thenaturalgasproducti on,thenaturalgas transportation,andthenaturalgasmarket.Thischapteris organizedinsuchaway thatwestartwiththeintroductionoftheproblemitself,an dthendiscussamathematical formulationoftheproblemandnallyreviewsolutiontechn iquestosolvethesemodels. However,whenwellknownalgorithms,suchasBranch&Cut,ar eusedtosolvethe mathematicalprograms,wedonotgointodetailsbutreferto theliteratureinstead. Section 4.2 discussestheoptimizationapplicationsingasrecoveryan dproduction. Wefocusontheproductionschedulingproblemandthemaxima lrecoveryproblem. Section 4.3 focusesongastransportation,wherethenetworkdesignpro blemsand theoptimalfuelcostproblemarediscussed.Thenaturalgas marketisdiscussedin Section 4.4 .WeconcludewithSection 4.5 54

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4.2OptimizationinGasProduction(Recovery) Thereisstillahugeamountofgasnaturalgasreservesinthe world:in2009, thereserveswereestimatedat6,254trillioncubicfeet;69 trillioncubicfeetabovethe estimatefor2008.Thisfollowsthegeneralupwardstrendof theworldnaturalgas reservesoveryears.Withashareofapproximately40.7%,th eMiddleEasthasthe largestnaturalgasreservesoftheworld,followedbyEuras iawith32.2%andAfrica with7.8%.Onthecountrylevel,Russiahasapproximately26 .9%oftheworldsnatural gasreservesandholdstogetherwithIran(15.9%)andQatar( 14.3%)approximately 57%oftheworld'snaturalgasreserveswhilethetop20count riesholdtogether90.7%. Interestingly,formostregions,thereserves-to-product ionratesaresubstantial,with anworldwideestimateof63yearsaccordingtoBP2008report .Hence,naturalgas productionandrecoverywillcontinuetobeanimportanttas kinthefuture. Optimizationmodelsandtechniquesareappliedextensivel yinnaturalgasrecovery processes,suchasproductionscheduling,placementofwel lhead,gasrecovery systemsorfacilitiesdesigns.Forasurveyongasandoilrec overyandproduction,we referthereadertoHorne[ 34 ].Theseoptimizationproblemsarecomputationallydifcu lt tosolve.Onereasonisthatahugenumberofparametersaresu bjecttouncertainties. Anotherreasonarethenonlinear/nonsmooth/nonconvexfun ctionsandconstraints,due tothepropertiesofgasproductionoperationsasexplained in[ 8 ].Inthefollowing,we discusssomespecicoptimizationproblemsoccurringinth egasproduction. 4.2.1ProductionSchedulingConsideringWellPlacement Usually,agasreservoirisaccessedbydrillingmultiplewe llsonitssurface.Also gaswithdrawalfromanyofthewellswillleadtopressurered uctionsatallwellsdrilled onthesamereservoir.Thenthepressurereductionswillcom ebacktodecreasethe withdrawalrateateverywellforthenextperiod.Theoptima lproductionscheduling problemistondtheoptimalwithdrawalrateateverydrille dwellateachtimeperiod whiledeterminingthewelllocationatthesametime. 55

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4.2.1.1MixedIntegerLinearProgrammingFormulation MurrayandEdgar[ 44 ]formulatethisproblemasamixedintegerlinearprogrammi ng (MILP)problem.Theytrytodeterminetheoptimalwellcong uration(withdrawalrates) whilesatisfyingthedemandschedulewithoutexceedingit. Drillingornotataparticular location, i ,canbedenotedbyabinaryvariable,say, y i .Hence,thedrillingdecisioncan onlybemadeatparticularlocations i whichhavetobeidentiedbeforehand.Also,use q k i todenotethewithdrawalratefromwell i attimeperiod k .Theinteractionbetween withdrawalratesandpressuresatallthewellscanbedeline atedbythefollowinggas owequation, r k g r + q = c t @ @ t (4–1) where =2 R p 0 z ( ) ( ) d .Includingthisconstraintinamathematicalprogramming formulationleadstohugecomputationaldifculties.Howe ver,asstatedin[ 44 ],this nonlinearconstrainthasaverygoodlinearizationsubstit ute,calledinuenceequations [ 2 64 ].Intheseequations,thepressuredropatwell i isalinearfunctionofwithdrawal owratesfromalldrilledwells.Thisisdenedbyinuencef unctionmatrices, k k =1,..., m ,where ij denotesthepressuredropatwell i foraunitowatwell j during timeperiod k .Themaximalprotproblemcanbeformulatedasfollows, max m X k =1 n X i =1 b k i q k i (4–2) s.t. n X j =1 kij q k j = p k i i =1,..., n k =1,..., m (4–3) n X j =1 kij q k j p k i i =1,..., n k =1,..., m (4–4) l X k =1 n X j =1 kij q k j ^ p l i i =1,..., n l =1,..., m (4–5) n X j =1 q k j d k k =1,..., m (4–6) q k i M i y i i =1,..., n (4–7) 56

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q k i 0, i =1,..., n k =1,..., m (4–8) y i 2f 0,1 g i =1,..., n (4–9) where,fromwell i duringtimeperiod k b k i isthebenetofoneunitgasow, p k i isthe pressurereduction,and p k i isthemaximalpressurereductionatperiod k ^ p l i isthe maximaltotalpressuredropallowedfromtheinitialtimepo inttotimeperiod l and d k is thedemandattime k M i isabignumbertoboundthewithdrawalowrateif y i =1 Itsobjectivefunctionisthetotalbenetfromthewithdraw alofgas.Constraints( 4–3 ) computethepressuredropateachwelllocationduringevery timeperiod.Constraints ( 4–4 )specifytheupperboundbywhichthepressurecandropdurin gaspecicsingle periodforeachwelllocation.Alsothereisanupperboundby whichthepressurecan dropduringtheperiodbetweentheinitialtimepointandthe currenttimeperiod,which isstatedinconstraints( 4–5 ).Constraints( 4–6 )ensurethatthetotalgaswithdrawal fromallwellsdoesnotexceedthedemandateachtimeperiod. Constraints( 4–7 )show thatonlydrilledwellscanhaveapositivewithdrawalowra te.Thisresultsinamixed integerprogramming(MIP)problem,whichcanbesolvedbywe llBranch&Boundor Branch&Cuttechniques.Wereferthereaderto[ 35 37 39 46 65 ]forcomprehensive discussionsofthesetechniques. Letusdiscussnowthedrawbacksoftheproposedmodel( 4–2 )-( 4–9 ).Themodel doesnotincludeanyothercostsuchaswelldrillingcost,it doesnottakeintoaccount therelationshipbetweentheprotcoefcient b k i andthedemand d k ,anditassumes thattheoperatorcanchooseanyowratewithoutconsiderin gtheconcurrentwellhead pressure.Also,afterthederegulationofthenaturalgasma rket,theconstraint( 4–6 )is notnecessaryandcanbeincorporatedintotheobjectivefun ctioninstead.Furthermore, thedifferentperiodsareintercorrelatedtoeachother.Fo rinstance,thepriceofgasat timeperiod t willaffectthedemandatthenexttimeperiod t +1 andviceversa.By 57

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incorporatingallthesefactors,anonlinearmixedinteger programmingproblemcanbe formulated.4.2.1.2NonlinearProgrammingFormulation Amultiple-stagenonlinearoptimizationproblemisalsopr oposedbyMurrayand Edgarin[ 44 ].Theyformulateanonlinearproblemforeachtimeperiodta kinginto accounttheinteractionsbetweentwoconsecutivestages.T heobjectivefunction foreachtimeperiod k incorporatesmorefactorssuchasthewellplacementcost, compressoroperatingcost,compressorsetupcost,andthep riceofgas,whichisshown asfollows, f k = n X j =1 Aq k j C w q k j q k j + U k j q k j C k j q k j + D k j (4–10) where A isthepriceperunitgasow,and C w isthesetupcostofanywellplacement. Insteadofusingthebinaryvariables y i todenotewhetherawellisdrillornot,this nonlinearprogrammingformulationusestheterm q k j q k j + toapproximate y i ,where isa smallconstantcomparedtothemagnitudeofgaswithdrawal owrates q k j j =1,..., n k =1,..., m .Tobeabletousethisapproximation,themagnitudeoftheo wratesare assumedtobeknown. U k j istheoperatingcostofthecompressorsforaunitowof q k j C k j q k j and D k j approximatethesetupcostofacompressoratthislocationb efore timeperiod k .Setting D k j = C k j q k 1 j makesthesummationofthesetwotermsequal to0,whichensuresthatthecompressorsetupcostonlyoccur once.Forthenonlinear formulation,thedeliverabilityequationsareconsidered besidestheconstraintsinthe MIPformulation.Thedeliverabilityconstraintsspecifyt herelationshipbetweenthe withdrawalrateandwellheadpressure,whichisalsoapprox imatedbylinearfunctions andshownasfollows, q k j e 1 j + e 2 j kj j =1,..., n k =1,..., m (4–11) where e 1 j and e 2 j arethelinearcoefcientsand kj isthebottom-holepressureatwellsite j aftertimeperiod k 58

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Alsoamulti-stagebasedalgorithmisproposedin[ 44 ],inwhichallstages(time periods)aresolvedinansequentialorderfrom1to m .Wedescribethisalgorithmas follows: Step1:Setuptheproblem:obtainparameters, e 1 j and e 2 j ,bysomeregression techniques;assumethatnocompressorisneededinitiallya ndset U 1 j = C k j = D k j =0 ;startfromtherstperiodproblem. Step2:Solvetheperiod k problemwithanappropriatenonlinearprogramming algorithm,suchasthegradientprojectionmethod[ 55 ]. Step3:Examinethedualvariablesofthedeliverabilitycon straints.Ifnoneis positive,anoptimalsolutionhasbeenfoundfortimeperiod k ,thengoto Step6.Otherwise,gotoStep4. Step4:Ifallpositivedualvariablesareassociatedwithde liverabilityconstraints ofthelowestfeasibledeliverypressure,anoptimalsoluti onisfoundfor timeperiod k ,thengotoStep6.Otherwise,gotoStep5. Step5:Selectthedeliverabilityconstraintwiththelarge stassociateddual variable,andthenrelaxthisconstrainttothenextlowestd elivery pressure.GotoStep2. Step6:Byusingthecurrentperiodoptimalsolution,update theparameters inthenextperiodproblem.If k = m ,terminatethewholeprogram. Otherwise,set k = k +1 ,andgotoStep2. Thedrawbackoftheproposedmodelisthatitdoesnotconside ralltimeperiods togetherbutconsidersthemseparately.Obviously,withth isapproach,anoptimal solutiontothepracticalproblemcannotbeobtained,asthe interactionsamongalltime periodsarenottakenintoaccount.4.2.2TotalGasRecoveryMaximization Inordertowithdrawasmuchnaturalgasfromareservoiraspo ssible,oneoptionis tousewaterooding.Thisleadstothefollowingimmediateq uestion.Whatisanoptimal waterinjectionratewithrespecttodifferentobjectives, suchasthemaximalultimate recovery,orthetotalrevenues?Alotofmodelshavebeenpro posedforthisproblem. MantiniandBeyer[ 41 ]proposedoptimalcontrolmodelstothissystemanddened severalobjectivefunctionsduetodifferentaspectsofthe problem. 59

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Now,supposetherearetwowellsdrilledonthesurfaceofthe gasreservoir,onefor gasrecoveryandoneforwaterinjection.Therefore,let r ( t ) denotethewithdrawalrate ofgaswhichisboundedbythemaximumrateofgasextraction r m ( t ) .Throughthewater injection,wellwaterisinjectedintothereservoiratthen onnegativerate s ( t ) .Thismodel assumesaconstant g whichistheratioofgasentrappedbehindtheinjectedwater to thevolumeofwateratanytime.Themodeltomaximizetheulti mategasrecoverycan thenbestatedas max Z 1 0 r ( t ) dt (4–12) s.t. PV = NRT (4–13) dV dt = s ( t ) gs ( t ), (4–14) dN dt = r ( t ) gs ( t ) P ( t ) RT (4–15) r m ( t ) r ( t ) 0, s ( t ) 0, where P ( t ) V ( t ) arethepressureandvolumeofthegasreservoir, N ( t ) istheamount ofgaswhichisnotentrappedattime t R istheuniversalconstantofgas,and T is thetemperature.Constraint( 4–13 )istheidealgaslaw,constraint( 4–14 )showsthe entrappedgasequalstoconstant g timesthevolumeofthewaterwhileconstraint ( 4–15 )statesthatgasisentrappedatthecurrentpressureinther eservoirandremains atthesamepressureandhasnoeffectonthereservoir.Byint roducinganothervariable Q = P = RT andpluggingconstraint( 4–13 )intoconstraint( 4–15 ),amoreconcisemodel canbeobtainedasfollows, max Z 1 0 r ( t ) dt s.t. dV dt = (1+ g ) s ( t ), dQ dt = r ( t )+ P ( t ) s ( t ) V ( t ) 60

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r m ( t ) r ( t ) 0, s ( t ) 0. MantiniandBeyer[ 41 ]alsodiscussseveralotherobjectivefunctions.Forexamp le, theobjectivefunctiontomaximizethepresentworthvalueo fthenetrevenuesfor internalrateofreturn, ,notequalto0,is Z 1 0 e t [ r ( t ) s ( t )] dt where istheratioofthewaterprice(percubicmeter)tothegaspri ce(permole), and isthegaspricepermole.Duetothepresentsofthedifferent ialequations,these problemsaregenerallycomputationallydifculttosolve. However,MantiniandBeyer establishedaveryinterestingtheorem,characterizingth epropertiesof(some)optimal solutionsofthecontrolvariable r ( t ) and s ( t ) .Letusre-statethistheoremhere. Theorem4.1. [ 41 ]Theobjectivefunction R 1 0 r ( t ) dt ismaximizedbyanyfunctions ^ r and ^ s suchthat, Z t 1 0 ^ r ( t )= V 0 ( P 0 P c ), (4–16) Z t 2 0 ^ r ( t )= P c ( V 0 V c ) 1+ g (4–17) ^ r ( t )=0, 8 t > t 2 (4–18) ^ s ( t )= 8>>>>>><>>>>>>: 0,0 t < t 1 ^ r ( t ) P c t 1 t t 2 0, t > t 2 (4–19) for t 1 and t 2 areanynumberswith 0 < t 1 < t 2 ,where P 0 and V 0 aretheinitialpressure andvolumerespectivelyandgasrecoverystopswhen P P c or V V c Thistheoremleadstotheinterestingstatementthatitisop timaltostartthe wateroodingwhenthersttime P islowerthan P c ;thatis,theentrappedgasisat 61

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thelowestpossiblepressure.Although,inpractice,thism aynotbevalidforsome specicgaswellsduetodiscrepanciesbetweenmodelingand reality. 4.3NaturalGasPipelineNetworkOptimization Originallynaturalgaswastreatedasabyproductofcrudeoi lorcoalminingand wasspared.Thearesintheminingeldwereusuallynatural gas[ 62 ].Notuntilthe introductionofpipelinesdidthenaturalgasbecomeoneoft hemajorsourcesofenergy. Theearliestgaspipelineswereconstructedinthe1890'san dtheywerenotasefcient asthosethatweareusingnowadays.Themoderngaspipelines didnotcomeintobeing untilthesecondquarteroftwentiethcentury.Becauseofth epropertiesofnaturalgas, pipelinesweretheonlywaytotransportitfromtheproducti onsitestothedemanding places,beforetheconceptofLiqueedNaturalGas(LNG).Th etransportationofnatural gasviapipelinesremainsstillveryeconomical,butitishi ghlyimpracticalacrossoceans. AlthoughLNGmarketisburgeoninginhighspeednow,pipelin enetworkremainsthe maintransportationsystemfornaturalgas. Gaspipelinesplayamajorroleinenergysupplyandsecurity .TheNordStream GasPipeline(NSGP)project,transportingRussiangastoGe rmany,isoneofthe recentlargescalepipelineprojects.TheNSGPisplannedas atwin-pipelinewitha totalcapacityof55billioncubicmetersperannum.Theesti matedinvestmentcost are4billioneuros,nancedbyajointventureofthethreeco mpaniesJSCGazprom, BASFAGandE.ONAG.Notleast,thedecisiontobuildthemarin epipelinewasdriven politically,passingbyPoland,Lithuania,Estonia,Belar usandUkraine,inorderto increasethenaturalgassupplysecurityforGermany,mainl y. Afterthepostwargaspipelineboom,alotofresearchhasbee ndoneinoptimization applicationstopipelinenetworks;forinstance,howtoset upthepipelinenetwork,howto determinetheoptimaldiameterofthepipelines,howtoallo catecompressorstationsin thepipelinenetwork,andwhatistheminimalfuelconsumpti onofthenetwork.Typically, themathematicalprogrammingformulationsofthepipeline optimizationproblems 62

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containalotofnonlinear/nonconvex/nonsmoothconstrain tsandfunctions.Themost commonconstraintsaretheso-calledWeymouthpanhandleeq uations,whichrelatethe pressureandowratethroughasegmentofpipeline( i j ).Theyreadasfollows sign ( f ij ) f 2 ij = p 2 i p 2 j ,( i j ) 2 A p (4–20) where f ij istheowrateofpipeline( i j ), p i and p j arethepressuresatnode i and j respectively.Hence,thedirectionofthegasowdependson thepressuredifferenceof thetwonodes i and j .Therefore,thenonsmoothfunctionsign ( f ij ) isneeded. Recently,moreresearchisrelatedtothenetworkoptimizat ionofgastransmission; giventhenetworkstructureotherthanthedesignofthenetw orktopology.Oneofthe fewpapersdealingwiththedesignofnetworktopologyisthe onebyRothfarbetal.[ 56 ], wheretheauthorsproposeatreegeneratingalgorithmtodes ignthenetworktopology. 4.3.1CompressorStationAllocationProblem Onceanetworktopologyischosen,oneproblemistodetermin etheoptimal congurationofthepipelinesandthelocationofthecompre ssorstationsinthisnetwork. Becauseofthehighsetupcostandhighmaintenancecost,iti sdesirabletohavethe bestnetworkdesignwiththelowestcost.Thisproblemconce rnsalotofvariables:the numberofcompressorstationswhichisanintegervariable, thepipelinelengthbetween twocompressorstations,thediametersofthepipelines,an dthesuctionanddischarge gaspressuresatcompressorstations.Thisproblemiscompu tationallyverychallenging sinceitincludesnotonlynonlinearfunctionsinbothobjec tiveandconstraintsbut,in addition,alsointegervariables. Asimpleandtypicalnetworkforthistypeofproblemisshown inFig. 4-2 .Node s isthesupplynodewherethegasisproduced.Nodes a and b arethedemand nodeswherethegasisconsumed.Thetrapezoids 1 through 6 denotethecompressor stations.Therearethreebranches: s to3istherstbranch,3to a isthesecondbranch, and3to b isthethirdbranch. 63

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s 1 2 3 4 5 6 a b Figure4-2.Agaspipelinenetworkcongurationproblemwit hthreebranches. Supposethereareatmost n compressorstationstobesetup,and n 1 n 2 ,and n 3 denotethenumberofcompressorstationsonbranch1,2,and3 respectively.For eachpipelinesegment i ,thereareveassociatedparameters:theowrate f i ,the dischargepressure(fromtheupstreamcompressor) p d i ,thesuctionpressure(fromthe downstreamcompressor) p s i ,thediameter d i ,andthelength l i TheformulationforthethreebranchesproblembyEdgaretal .[ 26 27 ]readsas, Min n X i =1 ( O y + C c ) T s s r r 1 1 p d i p s i z ( r 1) r + n +1 X i =1 C l l i d i (4–21) s.t. p d i p s i i =1,..., n (4–22) p d i K i p s i i =1,..., n (4–23) p di p d i p d i i =1,..., n (4–24) p si p s i p s i i =1,..., n (4–25) l i l i l i i =1,..., n (4–26) d i p d i p d i i =1,..., n (4–27) f i = Ad 8 3 i ( p d i ) 2 ( p s i ) 2 l i 1 2 i =1,..., n (4–28) n 1 X i =1 l i + n 1 + n 2 X i = n 1 +1 l i = L 1 (4–29) n 1 X i =1 l i + n 1 + n 3 X i = n 1 +1 l i = L 2 (4–30) 64

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where r istheratioofspecicheats, T s isthesuctiontemperature, z isthegas compressibilityfactor, s istheefciencyfactor, O y and C c arecostfunctionswith respecttohorsepower.Theobjectivefunction( 4–21 )containstwoparts,ofwhich therstisthecompressorstationcostsandthesecondisthe maintenancecostsof thepipelinesegments.Constraints( 4–22 )-( 4–27 )aretheupperandlowerboundson pressures,pipelinelengthsanddiameters. L 1 and L 2 arethedistancesbetweenthe supplynodeandtwodemandnodes. Model( 4–21 )-( 4–30 )canbesolvedbyapplyingBranchandBoundtechniques usingreducedgradientnonlinearoptimizationmethodtoso lvethesubproblemateach nodeintheBranchandBoundtree[ 26 27 ].Thedrawbackofthismodelisthatithighly dependsonthetopologyofthenetwork.4.3.2LeastGasPurchaseProblemandOptimalDimensioning Inthemodernnaturalgasindustry,thegasproductioncompa niesarerarely afliatedwiththegastransmissionanddistributioncompa nies.Thus,forgasdistribution companies,oneproblemistodeterminethebestowrateandg aspressuresineach pipelinebywhichtheleastcostonpurchasinggasfromprodu cersisachieved.This problemcanbeformulatedasaoptimizationproblemwithlin earobjectivefunctionand nonlinear/noconvexconstraints. ConsidernowFig. 4-3 s 1 and s 2 arethesuppliesforsourcenodes 1 and 2 ,the setofwhichisdenotedby N s .Nodes6to9aredemandnodeswithdemands s i i =6,7,8,9 .Inthismodel,therearetwokindsofarcs:thosewithcompre ssorstations suchas (1,4) and (2,4) ,whichisdenotedby A c ;andthosewithout,whicharealso calledpipelinearcsanddenotedby A p .Flowsonarcswithcompressorsaredirected suchthat f ij 0 8 ( i j ) 2 A c ,andowsonpipelinearcsareundirectedandthedirection dependsonthepressuresofbothendsofthisarc. 65

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n n n n n n n n n n n n n n n n n n n n J J J J J J J J J J J J J J J J J J J J J J J J^ n n n n ? ? ? ? s 6 s 7 s 8 s 9 r r r r 6789 r r r 345 r r 12 ? ? s 1 s 2 Figure4-3.Leastcostproblemnetwork. Amathematicalprogrammingformulationcanbestatedas min X i 2 N s c i s i (4–31) s.t. X j 2 A +i f ij X j 2 A i f ji = s i 8 i 2 N (4–32) sign ( f ij ) f 2 ij = C ij ( p 2 i p 2 j ), 8 ( i j ) 2 A p (4–33) f 2 ij C ij ( p 2 i p 2 j ), 8 ( i j ) 2 A c (4–34) s i s i s i 8 i 2 N (4–35) p i p i p i 8 i 2 N (4–36) f ij 0, 8 ( i j ) 2 A c (4–37) where p i isthegaspressureatnode i c i isthepurchasecostperunitgasfromsupplier i ,and C ij ancoefcientforarc ( i j ) ,whichisdeterminedbythelength,diameterandso on. A +i denotesthesetofarcswhichareemanatingfromnode i ,while A i denotethe oneofincomingarcstonode i Thenonlinearconstraintsofthemodelabovecanbesimplie dbyletting i substitute p 2 i .Then,constraints( 4–33 ),( 4–34 ),and( 4–36 )canbereplacedby 66

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sign ( f ij ) f 2 ij = C ij ( i j ), 8 ( i j ) 2 A p f 2 ij = C ij ( i j ), 8 ( i j ) 2 A p i i i 8 i 2 N Withthissubstitution,the`only'nonlinearfunctionslef taresign ( f ij ) and f 2 ij DeWolfandSmeers[ 25 ]proposeapiecewiselinearprogrammingalgorithmto solvethisproblem,inwhichtheyconstructapiecewiseline arapproximationtothe nonlinearconstraintsandsolvetherelaxedproblembysimp lexalgorithmextensions [ 22 ].Theperformanceofthealgorithmdependshighlyonthecho iceoftheinitialpoint. Itiscrucialtohaveagoodstartingsolution,whichcanbeob tainedbysolvingthe followingproblem: Min X ( i j ) 2 A j f ij j f 2 ij 3 C 2 ij (4–38) s.t. X j 2 A +i f ij X j 2 A i f ji = s i 8 i 2 N s i s i s i 8 i 2 N Theobjectivefunction( 4–38 )inthisproblemistheamountofmechanicalenergy consumedinthegaspipelineperunittime.ItsKKTnecessary conditions(see[ 7 ])is equivalenttotheconstraints( 4–32 ),( 4–33 ),and( 4–35 ).TheKKTnecessarypointisa goodapproximationstartingpointwhichdoesnottakeintoa ccountpressures'bounds andtheexistenceofcompressors.Thealgorithmproposedby [ 25 ]isasfollows: (o) Initialization: Let( f 0 p 0 s 0 )beavectorofows,pressures,andnet suppliesthatsatisfyconstraints( 4–32 ),( 4–33 ),( 4–34 ),( 4–35 ),and ( 4–37 ).Replacethenonlinearfunctionsign ( f ij ) f 2 ij byapiecewiselinear approximationincluding f 0 ij asabreakpoint.Use f 0 ij asstartingpointfor thepiecewiselinearprogrammingapproach.Alsoset k =1 (i) Iteration k : Solvetheapproximationproblembythepiecewiselinear programmingapproach.Let( f k p k s k )bethesolution. 67

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(ii) Stoppingrule: Compute f k ij bythefollowingequation: f k ij = sign ( p k i p k j ) C ij j ( p k i ) 2 ( p k j ) 2 j 1 2 Iftheerror e k ij = f k ij f k ij isgreaterthanagiventolerance,forexample, 10 5 ,thenadd f k ij asanewdiscretizationpointandreturntostep(i). Otherwisestopandtheincumbentsolutionisoptimal. Itcanbenoticedthattheoptimalobjectivefunctionvalueo fproblem( 4–31 )isa functionofthediametersofthepipelines,say, Q ( D ) ,becausetheparameter C ij of pipeline ( i j )isafunctionofthediameter,where C ij = K ij D ij and K ij isacoefcient.If thenetworkstructureandthelengthofeachpipelinearexe d,theinvestmentproblem istondthebestpipelinediameterswhichachievethelowes tinvestmentcostincluding boththegaspurchasecost Q ( D ) andthepipelineconstructioncost C ( D ) .Theyare givenas C ( D )= X ( i j ) 2 A =( k G D 2 ij + k I G D ij + k II G ) l ij where l ij isthelengthofpipeline ( i j ) .Thentheinvestmentproblembecomes Min C ( D )+ Q ( D ) (4–39) s.t. D ij 0, 8 ( i j ) 2 A whichisabilevelprogrammingproblem.Thesecondpartofth ecostfunction, Q ( D ) ,is nonconvex/nodifferentialandhasanimplicitdomain.DeWo lfandSmeers[ 23 ]propose howtogetonegeneralizedsubgradient,asinthenextpropos ition. Proposition4.1. Denoteby f s anoptimalsolutionoftheoperationsproblem ( 4–31 ).Let w ij beanoptimalvalueofthedualvariableassociatedtoconstr aint( 4–33 ). Then (..., w ij 5 K 2 ij D 4 ij ,...) 2 @ Q ( D ), (4–40) where @ Q ( D ) isthegeneralizedsubdifferential. Theinvestmentproblem( 4–39 )canbesolvedbyabundlemethodwhichperforms wellfornondifferentialoptimizationproblems.Byusinga bundlemethod,wedonot 68

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i j s k p q t Figure4-4.Agaspipelinenetwork.needtoknowtheexplicitdomainoftheobjectivefunction.H enceitisagoodtforthe investmentproblembecausetheobjectivefunctiondomaini simplicit.Ateachstep,it onlyneedsthevalueoftheobjectivefunctionandoneoftheg eneralizedsubgradient, whichcanbecomputedby( 4–40 ).Thedualvariables, w ij ,canbeobtainedwhilesolving theoperationsproblemsbyusingsimplexalgorithmextensi ons.Readersmayndmore comprehensivediscussionsofthebundlemethodin[ 33 ]. 4.3.3MinimumFuelConsumptionProblem Tolettheconsumerreceiveanacceptablewithdrawalrateof gas,thepipeline needstomaintainacertainpressure.Thisisachievedbyadd ingcompressorstations inthenetwork.Onewellknownproblemistheminimalfuelcos tproblemduetothe fuelconsumptionofcompressorstations,whichareusually consideredasspecialarcs inthenetworkofthistypeofmodels.Theminimalfuelcostpr oblemhasbeenwidely discussedintheliterature;seeforinstance[ 20 32 51 – 53 66 ]. AntypicalgaspipelinenetworkisshowninFig. 4-4 .Node s isthesourcenode, and t p ,and q arethedemandnodes.Arc ( j t ) isanordinarypipelinearc,arcs ( i j ) ( k p ) ( s q ) arecompressorstationarcs.Ineachcompressorstation ( i j ) ,thereare C ij compressors,andthepressuresat i and j aredenotedby p i and p j respectively.Let A I denotethesetofcompressorstationarcs, A II denotethesetofordinarypipearcs, V 69

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denotethenodeset.Then,theminimalfuelcostproblemcanb estatedas Min X ( i j ) 2 A I g ij ( x ij p i p j )= X ( i j ) 2 A I x ij Z i RT i [( p j p i ) 1] ij (4–41) s.t. X j 2 A +i x ij X j 2 A i x ji = b j 8 i 2 V (4–42) p 2 i p 2 j = R ij x 2 ij 8 ( i j ) 2 A II (4–43) 0 x ij u ij 8 i j 2 A (4–44) p L i p i p U i 8 i 2 V (4–45) ( x ij n ij p i p j ) 2D ij 8 ( i j ) 2 A I (4–46) n ij 2 0,1,2,..., N ij 8 ( i j ) 2 A I (4–47) where p L i and p U i arethelowerandupperboundsonthepressureofnode i .Ateach compressorstation ( i j ) u ij isthecapacity, N i j isthetotalnumberofcompressor, and x ij n ij arethegasowrateandnumberofcompressorinuserespectiv ely.Also thereareseveralotherrelatedparametersfor ( i j ) : z i isthegascompressibilityfactor, T i isthegastemperature, ij isthecompressoradiabaticefciency,and R ij isagas constant.Themostcomplicatedconstraintis( 4–46 )inwhich D ij isthefeasibledomain ofcompressorstation ( i j ) asforvariabletriplet ( x ij n ij p i p j ) .Thefeasibledomainisstated belowbythesetofequations, h ij s 2 ij = A H + B H ( q ij s ij )+ C H ( q ij s ij ) 2 + D H ( q ij s ij ) 3 (4–48) ij = C E ( q ij s ij ) 2 + B E ( q ij s ij )+ A E 100 (4–49) S min s ij S max (4–50) Surge q ij s ij Stonewall (4–51) h ij = Z i RT i [( p j p i ) 1] (4–52) q ij = Z i RT i x ij p i n ij (4–53) 70

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Intheaboveequations, q ij denotetheowthroughthecompressorunit, s ij denotethe speedofthecompressor(s),and A H B H C H D H C E B E A E arethecompressorunit's constants. Thisproblemisverydifculttosolve,anditssolutionalgo rithmsarehighly dependentonthetopologyofunderlyingnetwork.Mostofthe algorithmsforthis problemarebasedondynamicprogramming[ 51 – 53 ]andgradientsearchapproaches [ 66 ].Alsometaheuristicapproacheshavebeenconducted,such asantcolony optimization[ 20 ]orgeneticalgorithms[ 32 ]. 4.4NaturalGasMarketModels Governmentregulationoverthegasindustrydatesbacktoth eearlydaysofnatural gasusage.Attherstglance,thisseamstobereasonable,as governmentandthe publicarethemainusersofnaturalgasandinvestmentsinth enaturalgasindustry aretremendous.Notuntilthe1980sbeganthederegulationo fthisindustrytoimprove bothequityandefciencyofthenaturalgasmarket.Between theoriginalproducers andendusers,thereexistsavarietyofparticipants,eacho fwhichactstooptimizeits ownbenets.Underdifferentgovernmentpolicies,alotofn aturalgasmarketmodels areproposed.Inthissectionwediscussoptimizationmodel sofbotharegulatedanda deregulatedgasmarket.4.4.1ReallocationProbleminaRegulatedNaturalGasMarke t O'Neiletal.[ 49 ]proposeamodelonhowtoallocategastouserswithdifferen t prioritiesunderthegovernmentregulationswhenencounte redagasshortage emergency.Inthismodeltherearemultiplegastransmissio nsystemsamongwhich anytwosystemsarenotnecessarilyconnectedphysically.A llusersaredivideinto9 categorieswithpriorities1through9.Thetransportation networkiscomposedoftwo typesofarcsandnodes:thephysicalarcsandnodeswhichrea llyexistinpracticedenotedby A phy and N phy ,respectively-andthepseudocounterpartswhicharefor convenienceofmodeling-denotedby A pseudo and N pseudo ,respectively.Let K w betheset 71

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ofuserswhowithdrawgasfromgassystem w .Thismodelalsoincludesthepanhandle constraints( 4–20 )foreachofthepipelinearcs.However,insteadofusingthe actual nonlinearconstraints,thismodelincorporatestwolinear izedapproximationconstraints ineachiteration,whichreadas ij f ij + i p i j p j ij 8 ( i j ), 1ij p i p j 2ij 8 ( i j ), where 1ij ,and 2ij areparametersdeterminedateachiterationthrough ij = 1 j f new ij j 1ij =(1 1 )( p new i p new j ), 2ij =(1+ 1 )( p new i p new j ), 1 = max f ( r 2 ) m 1 2 2 g withthepositiveconstants r 2 2 TheallocationalgorithmproposedbyO'Neiletal.[ 49 ]isasfollows: Step0:Allocatetheminimumamountsthatallusersmustrece ive.Ifnofeasible solutionexists,thenstop;noallocationexistsunderthes pecied parameters. Step1:Allocategasaccordingtotheprioritieswithineach transporter'ssystem, startingwithpriority1andproceedinginascendingordero fpriority. Step2:Determineifpriorities1through5aresatised.Ifs o,gotostep4. Otherwise,xthelower(6through9)priorityusers,inpipe lineswitha shortageinanyhigherpriority,attheirlowerbounds. Step3:Allocategasaccordingtotheprioritieswithinthee ntiresystem. Step4:Incorporatethelinearizednonlinearconstraintsa ndndtheoptimal solutionminimizingtheamounttransferredbetweensystem s,asinthe optimizationproblem( 4–54 )-( 4–64 ). Thelinearprogrammingformulationusedintheallocationp roblem[ 49 ]canbe statedasfollows, 72

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Min X ( i j ) 2 I j f ij j + X ( i j ) 2 S f ij (4–54) s.t. X j 2 A +i f ij X j 2 A i f ji = s i Xk 2 K 9 X l =0 d ikl 8 i 2 N (4–55) X i 2 N X k 2 K d ikl + u l = d l l =0,...,9, (4–56) 5 X l =1 X k 2 K w X i 2 N d ikl + r w = g w 8 w 2 W (4–57) ij f ij + i p i j p j ij 8 ( i j ) 2 A ps (4–58) 1ij p i p j 2ij 8 ( i j ) 2 A vc (4–59) 0 s i s i 8 i 2 N (4–60) d ikl d ikl d ikl 8 i 2 N k 2 K l =0,...,9 (4–61) p i p i p i 8 i 2 N (4–62) u l 0, l =0,1,...,9, (4–63) r w 0, 8 w 2 W (4–64) where s isthesupply, d isthedemand, u istheslackvariableforthedemandofeach priority,and r istheslackvariableforthedemandofpriority1through5.I nconstraints ( 4–58 ), f ij + i p i j p j isthelinearizedversionofthepanhandleequation,where i and j arethecoefcientsoftherstorderTaylorseriesexpansio n. A ps and A vc denote thepipelinearcsetandthecompressorarcset,respectivel y.Theobjectivefunctionis theamountofgastransferredbetweentwosystems, I isthesetofphysicalarcsthat connecttwosystems,and S isthesetofpseudoarcsthatrealizeswappingbyallowing owintoredistributionnode.Thisisoneofearliestmathem aticalmodelsdescribingthe naturalgasmarketunderregulation. 73

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Producers Pipeline Companies Local Distribution Companies Residential, Commercial, Industrial Customers Figure4-5.ParticipantsRelationshipinRegulatedGasMar ket. 4.4.2DeregulatedNaturalGasMarketModels InNorthAmerica,beforethe1980's,thenaturalgasmarketh adbeengreatly regulatedbythegovernmentsincethe1930's.Intheregulat edmarket,therewere primarilyfourparticipants:thegasproducers,thegaspip elinecompanies,localgas distributioncompanies,andcustomers.Therelationshipo ftheseparticipantisshown inFig. 4-5 ,whereproducerssoldgastopipelinecompanies,andpipeli necompanies soldthegastolocalgasdistributioncompanies,andthenlo caldistributioncompanies soldthegastovariouscustomers,suchasindustrial,comme rcial,andresidential customers.Inthisregulatedmarket,gaspricesineachofth eabovetransactionsare tightlyregulatedbyFederalandStategovernmentsaspipel inecompaniesandlocal distributioncompanieshadmonopoliesinthegasmarket.Si ncethemid1980's,a seriesofderegulationpolicieshavebeenannounced.These policesencouragepipeline companiestoswitchfromtheirtraditionalroleasownersof naturalgasbyallowing producersandbuyerstobypassthepipelinecompaniesintha tthebuyerscantransport theirowngasthroughthepipelinesystembypayingsomefees Thederegulationofthegasmarketnotonlychangedtheroles oftheformer participantsbutalsohelpedtocreatemoreparticipants,s uchasthegasmarketing companies.Manymodelshavebeenproposedforthederegulat edgasmarket, 74

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especiallyforNorthAmericaandEurope.Optimalpurchasin gstrategiesconsidering storage,contract,spotprices,peakdaydemandslocaldist ributioncompaniesunder NorthAmericagasmarketconditionshavebeenstudiedbyAve ryetal.[ 4 ].Amodel basedongeneralizednetworktoprovideoptimalstrategies forthemarketingcompanies andlocaldistributioncompanies,andasystem,GRIDNET,to storeallthedealed informationwereproposedbyBrooksandNeill[ 17 ]andBrooks[ 15 ].TheNatural GasTransmissionandDistributionModule(NGTDM)isanimpo rtantmodelofthe NorthAmericangasmarket,whichisasubmoduleoftheU.S.De partmentofEnergy's NationalEnergyModelingSystem(NEMS).TheGasSystemAnal ysisModel(GSAM) isanotherNorthAmericangasmarketmodel,whichtriestoma ximizethesocial welfarefunctiontogettheequilibrium,seeforinstanceGa brieletal.[ 30 ].Oneofthe mostrecentNorthAmericangasmarketmodelsistheMixedCom plementarity-Based EquilibriumModelofNaturalGasMarkets;seeGabrieletal. [ 29 ]. Gabrieletal.[ 29 ]considersixtypesofparticipants:thepipelineoperator s,the productionoperators,themarketers/shippers,thestorag ereservoiroperators,the peakgasoperators,andthecustomers.Eachparticipantist ryingtominimizecost ormaximizeprotforitself.Forthesakeofsimplicity,thi smodelassumesonlylinear relationshipwithineachproblemfacedbyaparticipant.He nce,everyparticipantfaces alinearprogrammingproblem.Becausenaturalgasisahighl yseasonalproduct,the modelspeciesthreeseasonsineachyear,whicharedenoted by s =1,2,3 .Everyyear hasindex y 2 Y s =1 :lowdemandseason,Apr.-Oct.; s =2 :highdemandseason,Nov.,Dec.,Feb.,Mar.; s =3 :peakdemandseason,Jan. Inthisformulation,pipelinegasisavailableforallthree seasons,andgasisinjected tostoragereservoirinseason1andextractedinseason2and 3,andpeakgasisonly usedinthepeakseason. 75

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Theoperatorofpipeline a istryingtomaximizeitsownprotbysolvingthefollowing problem, Max X y 2 Y 3 X s =1 days s asy f asy (4–65) s.t. f asy f a 8 s y (4–66) f asy 0, 8 s y (4–67) where days s isthenumberofdaysinseason s asy and f asy arethepricesandow ratesrespectivelyofpipeline a inseason s ofyear y .Constraints( 4–66 )aretheupper boundconstraintsoftheows. asy aretheequilibriumshowpricesdeterminedbythe optimizationproblemsoftheotherparticipants.Othertha n asy ,therearesomeother conditionsrelatingthispipelineoperatorproblemtotheo therpipelinesandotherkinds ofparticipants.TheseconditionsareusuallycalledMarke t-Clearingconditions.The correspondingMarket-Clearingconditionsforthegaspipe lineoperatorproblemreads days 1 f a 1 y = X r 2 R ( n 1 ( a )) days 1 g ary + X m 2 M ( n 1 ( a )) days 1 h am 1 y a 1 y free 8 y 2 Y (4–68) days s f asy = X m 2 M ( n s ( a )) days s h amsy asy free s =2,3, 8 y 2 Y (4–69) ThesetwoMarket-Clearingconditionsstatethatallthesup pliesequalallthedemands. g ary istheowrateofgastostorageoperator r fromtheproducersofseason 1 through arc a ,and h amsy isthegasowratefromproducersofseason s tomarketer m through arc a Theproductionoperator'sproblem,forproductioncompany c 2 C atnode n 2 N ,is tomaximizeitsprotbysolvingthefollowingproblem, Max X y 2 Y 3 X s =1 days s nsy q csy c pr c q csy (4–70) s.t. q csy q c 8 s y (4–71) 76

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X y 2 Y 3 X s =1 days s q csy prod c (4–72) q csy 0, 8 s y (4–73) where nsy and c pr c arethepriceofgassoldbytheproductioncompanyandcostto produceoneunitofgas,respectively,forcompany c ,and q csy istheproductionrateof thecompanyinseason s ofyear y .Constraints( 4–71 )specifytheupperboundsofthe productionrateineachperiod,andconstraints( 4–72 )givethetotalproductioncapacity forthewholeplanninghorizon.Exceptthisoptimizationpr oblem,thecouplingconditions fortheproductioncompany c atnode n areasfollows, X c 2 C ( n ) days 1 q c 1 y = X a 2 A +n X r 2 R ( n 1 ( a )) days 1 g ary + X m 2 M ( n 1 ( a )) days 1 h am 1 y n 1 y free 8 y 2 Y (4–74) X c 2 C ( n ) days s q csy = X a 2 A +n X m 2 M ( n s ( a )) days s h asy asy free s =2,3, 8 y 2 Y (4–75) Thestoragereservoiroperator'sproblem,themarketer'sp roblem,andthepeak gasoperator'sproblemarealldescribedinthesameway,rs tthelinearprogramming problemandthenthemarket-clearingconditions.Sinceall operator'sproblemsare linearprogrammingproblems,theKKTconditionsarenecess aryandsufcient. CombiningalltheKKTconditionsandmarket-clearingcondi tionsofeveryoperator's problem,wethengetaLinearComplementarityProblem(LCP) ,whichisaspecialcase ofnonlinearcomplementarityproblem(NCP)orvariational inequalityproblem(VI). Gabrieletal.[ 29 ]provedthatthereexistsasolutionofthesystemandthepri cesare uniqueinthiscase.FormoredetailsaboutLCP,NCP,andVI,w ereferthereader,for instance,to[ 21 28 40 45 ]. AlsoalotofmodelsfortheEuropeangasmarketshavebeenpro posed.A stochasticStackelberg-Nash-Cournotequilibriummodelf ornaturalgasproducers areproposedbyDeWolfandSmeers[ 24 ].BretonandZaccour[ 14 ]proposeaduopoly 77

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Gas Supply ElectricityNetwork Combined-CycleElectricity Production GasNetwork NonGasElectricity NonElectricityGas Demand ElectricityDemand Figure4-6.RelationshipbetweenGasNetworkandElectrici tyNetwork. producermodel.ArecentEuropeangasmarketmodelsimilart othemodelin[ 29 ]is GASTALEproposedbyBootsetal.[ 13 ]. 4.4.3CombiningNaturalGasSystemandElectricitySystem Naturalgasiswidelyusedinelectricityproduction.Becau secombined-cycleplants arehighlyefcientandhavelessdamagetotheenvironment, moreandmorepower plantsofthistypearebuildaroundtheworld.Hencetheelec tricityandthegassystem arenowhighlycorrelated.Herewediscusssomerelatedopti mizationapplications regardingthisrelationship.4.4.3.1ElectricitySystemReliabilityStudy Duetotheincreasingnumberofcombined-cyclepowerplants beingbuilt,electricity productionreliesmoreandmoreontheamountofgasthepower plantscanget. However,theelectricityplantsarenottheonlyusersofnat uralgas;seeSec. 4.1 .In ordertoperformareliabilityanalysisoftheelectricitys ystem,itisimportanttostudy themaximalamountofgaswhichthegasnetworkcansupplytot heelectricityplants. Therelationbetweengasnetworkandelectricitynetworkis showninFig. 4-6 .Munoz etal.[ 43 ]studiedtheproblemofthemaximalgassupplytheelectrici tysystemcan receive,takingintoaccounttheothergasusers,thepipeli necapacityandtheproduction capacity.Theformulationisverysimilartothegaspipelin eoperationsproblem( 4–31 ). 78

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Insteadofminimizingthegaspurchasecostasin( 4–31 ),thisproblemmaximizesthe totalelectricitywhichcanbeproducedbyusinggasfromthe gassystem.Itcanbe formulatedas Max X i 2 N e A i e i + B i e 2 i + C i e 3 i (4–76) s.t. X j 2 A +i f ij X j 2 A i f ji = s i d i e i 8 i 2 N (4–77) sign ( f ij ) f 2 ij = C ij ( p 2 i p 2 j ), 8 ( i j ) 2 A p (4–78) f 2 ij C ij ( p 2 i p 2 j ), 8 ( i j ) 2 A c (4–79) s i s i s i 8 i 2 N (4–80) p i p i p i 8 i 2 N (4–81) d i d i d i 8 i 2 N (4–82) e i e i e i 8 i 2 N (4–83) f ij 0, 8 ( i j ) 2 A c (4–84) wheretheobjectivefunctionisapolynomialfunctionofwit hdrawalofgasfromthegas network. e i isthegaswithdrawaltoproduceelectricity. d i isthedemandnotrelatedto electricityproduction. A +i denotesthesetofarcswhichareemanatingfromnode i ,while A i denotesthesetofincomingarcstonode i Munozetal.[ 43 ]solvetheaboveproblemintwophases.First,bydroppingal l nonlinearconstraints,amixedintegerlinearprogramming problemisobtainedand thensolved,wheretheintegervariablesdenotethedirecti onsofowsinthepipeline segments.Second,byknowingthedirectionsofowsfromthe phaseIproblem, anonlinearproblemissolved.However,twotheoreticalque stionsstillremainin thecorrectnessofoptimalityobtainedbythemethod.First ,itremainsunanswered whetherthesolutionfromphaseIwillensurethephaseIIpro blemtobefeasible. Second,itisnottruethatthesecondphaseproblemisaconve xproblemforwhicha 79

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simplecounterexamplecaneasilybeconstructed,suchasth esinglepipelinesegment problem.4.4.3.2OptimizationinNaturalGasContracts Manyelectricityproductionplantsusealotofsourcesamon gwhichnaturalgas isaveryreliablealternativetomeetthehighelectricityd emand.Theoptimizationof fuelcontractsforahydro-basedpowersystemisaverygoode xample.Inhydropower systems,precipitationvariesfromseasontoseason.Forth elowprecipitationseasons, theplantsneedtobuygastogenerateelectricity. Letusnowdiscussamodelwhichdealswiththeoptimaldispat chstrategywhile consideringtheparticularspecicationsofgassupplycon tractsasinChabaretal. [ 19 ].Thismodelassumesatake-or-paycontract,whichiswidel yadopted,especially inEurope.Ifatake-or-paycontractissigned,specifyinga monthlyamountandatotal annualamount,thenatleast X % ofthemonthlyamounthastobeboughteverymonth andatleast Y % ofthecontractedannualamountfortheyearhastobebought. Hence, theremightbesomegasexcessbasedoncontractsofthistype .Tworeservoirsare addedintothismodeltoaccommodatethesituationswherega sexcessexists.All excessesofgasnotconsumedmonthlyarestoredinthegasres ervoirA,thedifference betweentheannualtake-or-payamountandthesumofallmont hlytake-or-payamounts oftheyearisstoredinreservoirB.Also,oneofthegascontr actprovisionsstatethatthe gaspurchasedatanytimepointcannot“stayinthereservoir ”,oractuallyholdbythe gasproviderbymorethan N timeperiods,whichmeansthatifanyamountofgasstays inthereservoirmorethan N timeperiods,itwillhavetobediscarded. GD t isusedto denotetheamountofgasdiscardedattime t .Figure 4-7 showshowthemodel,based onreservoirs,dealswiththecontractprovisions. Alsothemaintenancescheduleismodeledbyreservoirs.Ac titiousremaining-hours reservoirisassignedtoeverypowerunitforeachmaintenan cecycle.Fora3power units3cyclesproblem,therewillbe9reservoirs.Thelengt hofeachkindofcycleis 80

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Reservoir A Reservoir B Power Plant GTR ARM GToP GD (Y%-X%)M GS Figure4-7.GasContractsModeledbyReservoirs.Table4-1.Maintenancecyclelength CycleFrequencyAverageDurationCost(MMR$) Combustor8000hours7days3.5 Hotpathcircuit24000hours14days10 Majormaintenance48000hours21days20 showninTable 4-1 .Foreachpowerunit,thereservoirsarelledwiththeamoun tof remaininghoursofoperationuntilnextmaintenance.Theca pacityofeachreservoir isthelengthofthecycle.Astheunitoperates,allreservoi rsforthatunitarereduced bythequantityoftheelapsedhours.Aftermaintenance,the ctitiousmaintenance reservoirislledtoitscapacity. Consideringalsothemaintenanceschedulingofthethermal plant,adynamic programmingformulationoftheproblem,foragivenstagean dprice,isproposedby Chabaretal.[ 19 ]: FBF k t ( VA t VB t f VH i j t 8 i j g k t ) (4–85) = Max RI t + S X s =1 p t +1 ( k s ) FBF s t +1 ( VA t +1 VB t +1 f VH ij t +1 8 i j g k t +1 ) (4–86) s.t. VA t +1 = VA t + ARM t GToP t + GTR t GD t (4–87) VB t +1 = VB t GTR t (4–88) VH i j t +1 = VH i j t (1 x i j t )+ VH j x i j t r EG i t 8 i j (4–89) n X i =1 i t EG i t = H c ( CToP t + GToP t + r G t ), (4–90) 81

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where VA t and VB t arethevolumeofgasinreservoirsAandB,respectively,and VH i j t isthe“volume”ofremaininghoursofoperationthattheunit i hasuntilthenext maintenanceofcycle j GToP t istheamountofgasactuallyusedtogenerateelectricity, and GS t istheamountofgaspurchasedorsoldtothegasspotmarket. ARM t isthe amountofgaspurchasedfromthegasdistributor,andshould beboundedbelow by X % M GTR t istheamountofgastransferfromAtoB,and GD t istheamountof gasdiscardedwhenitisinthereservoirmorethanthemaximu mstoragetime, N n and m arethetotalnumberofpowerunitsandtotalnumberofmainte nancecycles, respectively. RI t istheimmediaterevenueinstage t s t isthespotpriceinstage t ofscenario s p t +1 ( k s ) isthetransitionprobabilityofthespotpriceofscenario k in stage t tothespotpriceofscenario s instage t +1 x i j t isthebinarydecisionvariable associatedwiththescheduleofmaintenanceofcycle j forunit i atstage t VH j is themaximumcapacityofthereservoirofremaininghoursofo perationuntilthenext maintenanceofcycle j EG j t istheenergygeneratedbyunit i atstage t r isaninverse coefcientofthepowerunit,and i t isconversionfactorfromMMBTUtoMWhofunit i at stage t ,and H c istheheatrateofthegas. Constraints( 4–87 )-( 4–89 )arethectitiousreservoirbalanceconstraintsand( 4–90 ) isthetransformationfromgastoelectricity.Exceptconst raints( 4–87 )-( 4–90 ),thereare alsoalotofotherconstraints,suchasgasconsumptionprio rityconstraints,maximum andminimumgasconsumptionconstraints,maintenancecons traints,constraintsrelated tothemechanismimplementedforthemodelingofthecontrac tsandsoon.Forthis problem,eachstageisamixedintegerlinearprogrammingpr oblem.Andthewhole problemissolvedbyusingstochasticdualdynamicprogramm ing,rstproposedby PereiraandPinto[ 50 ]. Alsothenaturalgasmarketcanbemodeledasanaturalgasval uechain.The primarycomponentisnaturalgasinthischains.Variousmar ketmodelsareproposed andutilizedinrealityatdifferentstagesalongthisvalue chain,e.g.,production, 82

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transportationandprocessing,storage,importterminals andmarkets,wholesale andretailmarkets.Pleasereferto[ 42 ]formoredetailsaboutmarketmodelswithinthe naturalgasvaluechain. 4.5Conclusion Thischapterdiscussvariousoptimizationmodelsoccurrin ginthenaturalgas industry;focusingonthreeaspects:production,transpor tation,andmarket.Aswecan see,thenaturalgasindustryisacomplexsystemandingreat needofoptimization techniquestoimproveperformance.Especiallythenonline arandnonconvexnatureof theproblemsmakesitcomputationallychallengingtondgo odsolutions.Weobserve thatlinearizationtechniquesareacommonmethodtotackle thesenonconvexfunctions, oftenreducingtheproblemtoa(series)oflinearormixedin tegerlinerprogramming problems.Withthecomputationalpowerofcomputersincrea singoverthelastdecade, theuseofmeta-heuristicsisbecomemoreandmorepopular;e speciallyforproblems whichcannotbehandledwiththecurrentMINLPsolverseithe rduetothesizeofthe problemorduetothedegeneracy. Thederegulationofthegasmarketintroducedadditionalmo delingaspectsand computationalchallenges:various(additional)stochast icelementshavebeenaddedto the`classical'problems.Thisunderlyingstructureofthe problemscannotbeignoredby anyseriousmodelandweexpectthatfutureresearchwillfoc usonstochasticmodels and,especially,onnewtechniqueshowtosolvethese(large -scale)practicalproblems whenalsointegerandnonconvex,nonlinearfunctionsarepr esent. 83

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CHAPTER5 NATURALGASNETWORKEXPANSIONPLANNING Duetotheincreasingdemandsfornaturalgas,itisplayinga moreimportantrole intheenergysystem,anditssystemexpansionplanningisdr awingmoreattentions. Inthischapter,weproposeexpansionplanningmodelswhich includebothnatural gastransmissionnetworkexpansionandLNG(LiquiedNatur alGas)terminals locationplanning.Thesemodelstakeintoaccounttheuncer taintiesofdemandsand suppliesinthefuture,whichmakethemodelsstochasticint egerprogramswithdiscrete subproblems.Alsoweconsiderriskcontrolinourmodelsbyi ncludingprobabilistic constraints,suchasalimitonCVaR(ConditionalValueatRi sk).Inordertosolvethe large-scaleproblems,especiallywithalargenumberofsce narios,weproposethe embeddedBendersdecompositionalgorithm,whichappliesB enderscutsinbothrst andsecondstages,totacklethediscretesubproblems.Nume ricalresultsshowthat ouralgorithmisefcientforlargescalestochasticnatura lgastransportationsystem expansionplanningproblems. 5.1Introduction Naturalgas,whichoncewasconsideredthebyproductorspar egasofoiland coalmining,hasbecomeaverypreciousandimportantenergy sourceintheworld's energysystem.Itisarelativelycleanerenergysourcecomp aredtocoalandoilbecause itreleaseslessgreenhousegas.Especiallyaftertheintro ductionofcombinedcycle powerturbines,whichismuchmoreefcientthanthetraditi onalelectricitypower generatorsusingcoal,naturalgasisplayingamoreimporta ntroleintheworld'senergy supply.From1986to2006,theannualworld'sconsumptionof naturalgashasbeen doubledto102.2trillioncubicfeetfrom52.9trillioncubi cfeetaccordingtoEIA(Energy InformationAdministration)2009annualreport.Itsannua ldemandisforecastedto increaseby50%in2030.Becauseoftheincreasingdemandsfo rnaturalgas,itis veryimportanttostudythenaturalgassystemexpansionpla nning,especiallyits 84

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Figure5-1.USnaturalgastransmissioncorridorfromEIA20 08 transportationsystem.Traditionally,itstransportatio nsystemismainlycomposedof transmissionpipelines.(TheUStransmissioncorridorofn aturalgasisshowninFigure 5-1 accordingtoEIA2008report.)Withtheincreasingintercon tinentalLNG(Liquied NaturalGas)shipment,theexpansionplanningonselecting thelocationsandsizesof theLNGterminalsshouldalsobeconsideredanimportantpar tofthetransportation system.TheproposedandaccomplishedLNGterminalsofUSis showninFigure 5-2 accordingtoFederalEnergyRegulationCommission. Asisdiscussedin[ 68 ],mathematicalmodelingandoptimizationhavebeen extensivelyappliedinnaturalgasindustryandyieldedalo tofgreatresults.Inthis paper,wetrytocomeupwithastochasticexpansionplanning modelwhichconsiders bothtransmissionpipelinenetworkexpansionplanningand LNGterminallocation planning.Wemodeltheexpansionofapipelineandthesetupo fanewsizeofLNG terminalbybinaryvariables.Themodelistryingtominimiz ethetotalexpansioncost andtransmissioncostwhileconsideringthewholetranspor tationsystem.Alsothis modelassumesgeneralizednetworkowsasin[ 16 ].Inaddition,wealsoproposearisk 85

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Figure5-2.ExistingandProposedNorthAmericanLNGTermin als managementmodelwhichincludesCVaR(ConditionalValueat Risk)riskconstraints,in ordertobalancebetweentheminimalcostandtheriskoflosi ngdemands.Duetothe existenceofintegervariableswithinbothstages,thestoc hasticprogramisnotatrivial problemtosolve,especiallywhenthenumberofscenariosis big. Theremainingpartofthischapterisorganizedasfollows.F irst,insection 5.2 ,we proposethestochasticmodelandtheriskmanagementmodelw ithCVaRconstraints.In section 5.3 weexplainourEmbeddedBendersDecompositionprocessandp roposethe algorithm.Section 5.4 showsthecomputationalresultsandcomparesthesolutions with differentCVaRconstraints. 5.2ExpansionPlanningModels Wehavealreadyknownthattheworld'snaturalgassupplywil lnotlastforever becausethereservesaredwindlingandwillnotgrowbythems elves.Soavery interestingandimportantquestiontoaskiswhetherweneed toexpandourgassystem. 86

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Figure5-3.NaturalgaslongtermconsumptionexpectationTheup-shootingtrendwillnotalwaysholdinthefuture,and sometimethedemand willdropforsure.However,itseemsthegasreservescansti llsupportusformultiple decadesorevenahundredyears.(Anaturalgaslongtermcons umptionexpectingis showninFigure 5-3 )Hence,itisimportanttodotheexpansionplanningeconomi cally andreliably,whichcancopewithdifferentfuturesituatio ns.Thegasreservesarequite differentindifferentregionsoftheworld.AccordingtoEI AInternationalEnergyOutlook 2009,theworldaverageRTP(Reserves-To-Production)rati oisabout63years;Central andSouthAmericaRTPisabout48years;RussiaandAfricaRTP are78and79years respectively;RTPofmiddleeastismorethan100years;USpr oductionrateisabout 20TCF(TrillionCubicFeet)peryearanditsestimatedreser vesareabout1747.47TCF, whichmakeitsRTP87years.Theimbalanceofnaturalgasrese rvesandeconomic growthindifferentregionsmakeintercontinentaltranspo rtationnecessary.Themain intercontinentaltransportationisLNGshipment.Inthena tionallevel,itisimportant toanalyzethewholenaturalgassystembyconsideringpipel inenetworksandLNG locationstogether.Anetworkexamplewhichconsidersboth ofthemisshowninFigure 5-4 ,inwhichalltransmissionlinesareexpandableandthoseda shedlinesdenote possiblenewtransmissionlines,andnodesassociatedwith “LNG”arepossibleLNG terminals. 87

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10 1 2 3 4 5 6 7 8 9 11 LNG LNG LNG LNG LNG LNG LNG Figure5-4.Anaturalgastransmissionnetworkexample5.2.1TheStochasticPlanningModel Ourmodelingaimstoformulatetheprobleminthesystemleve lwhileconsidering uncertaintiesoffuturesuppliesanddemands.Theobjectiv eistominimizethetotalcost, whichincludesthecostsofbotharcexpansionsandLNGtermi nalexpansions,andthe transmissioncosts,whilesatisfyingalldemands.Inthism odel,weassumediscrete expansions,whichareactuallywhatishappeningnow.Forex ample,thediametersof thegaspipelinesandsizesofLNGcontainersareusuallydis cretewhenyoutrytobuy themfromthemanufacturers.Figure 5-5 showsthediscretizedexpansioncostsofthe gaspipeline.Weuse0-1integervariable, k ij ,todenotewhetheranexpansionofsize kij ismadeforarc ( i j ) ,andthentotalcostofpipelineexpansionis Cost ARC = X ( i j ) 2 A X k 2 K ij c k ij k ij SodoestheLNGterminalopeningcostasfollows, Cost LNG = X i 2 N LNG X k 2 K i c k i k i 88

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s t u v %s r n r %v %u %t Figure5-5.DiscreteExpansionCostsTable5-1.EXPNSetsandIndices N Thesetofallnodesinthenetwork A Thesetofallarcsinthenetwork A +i Thesetofoutgoingarcsfromnode i A i Thesetofincomingarcstonode i K ij Thesetofallpossibleexpansionsizesonarc ( i j ) K i Thesetofallpossibleexpansionsizesonarc i N LNG Possible(approved)LNGterminals Thesetofallscenariosofthedemandpatterns Denoteaspecicscenario Fortheconvenience,wemakethefollowingthreeassumption s,(A.1-A.3),which alwaysholdthroughoutthewholepaper.A.1 Assumediscretedistributionofuncertainties, = f 1 2 ,..., r g ,where r isanite positiveinteger; A.2 (1 l ) j A j P i 2 N SF 0 P i 2 N d 0 ,and (1 l ) j A j P i 2 N SF 1 ( ) P i 2 N d 1 ( ), 8 2 where l =max ( i j ) 2 A l ij ; A.3 Makingthemaximumexpansiononeverynodeandarcisenought osatisfyall demands. 89

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Table5-2.Parameters kij The k th expansionsizeofarc ( i j ) ki The k th expansionsizeofLNGport i c 0, k ij c 1, k ij Currentandfuturecostsoftheexpansionofsize kij onarc ( i j ) c 0, k i c 1, k i Costoftheexpansionofsize ki ofLNGport i h 0 ij h 1 ij Unittransportationcostofarc ( i j ) d 0 i Currentdemandofthenode i d 1 i ( ) Futuredemandofnode i underscenario l ij Transmissionlossrateonarc ( i j ) SF 0 i Currentselfsupplylimitofnode i SL 0i CurrentLNGsupplylimitofnode i SF 1 i ( ) Futureselfsupplylimitofnode i underscenario SL 1i ( ) FutureLNGsupplylimitofnode i underscenario u ij Thepreviouscapacityofarc ( i j ) v i ThepreviouscapacityofLNGport i Pr ( ) Probabilityofscenario Inordertofacilitatethedescriptionofourmodels,Table 5-1 denesallthesetsof arcs,nodes,scenarios,etc,andTable 5-2 denesallcoefcientsandparameters,while Table 5-3 denesalldecisionvariablesofbothrstandsecondstages Ourstochasticplanningmodelistominimizethecurrentcos tplustheexpected futurecostwhichareshownin( 5–1 )and( 5–2 )respectively.Withineachofthem, therearethreeparts:arc(pipeline)expansioncost,LNGte rminalexpansioncost andtransportationcost.( 5–3 )denestheowbalanceconstraints,wheregasloss isconsideredbymultiplyingdifferentfactorsonallthein comingows,sinceinreality therearealwaysleakingproblemsandcompressorstationsn eedtousesomegas tomaintainpressureofpipelines.Weassumebidirectional owsoneacharcandthe capacityconstraintofeacharcisdenedin( 5–4 ).Sinceweincludetransportation costintheobjectivefunction,foreacharctheoptimalsolu tionwillonlyhavenonzero owatmostinonedirection.( 5–5 )isthearcexpansionconstraints.Constraint( 5–6 ) requiresthatLNGsupplyatanyLNGportnodecannotexceedit sthroughputcapacity, whileconstraint( 5–7 )requiresthatLNGsupplyatanyLNGportnodealsocannot exceeditsLNGsupplylimit.Constraint( 5–8 )statesthatthetotalsupplyofeveryLNG 90

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Table5-3.DecisionVariables 0, k ij Binaryvariabletodenotewhethera kij expansionismaderightnowforarc ( i j ) 1, k ij ( ) Binaryvariabletodenotewhethera kij expansionismadeinthefutureforarc ( i j ) underscenario 0, k i Binaryvariabletodenotewhethera ki expansionismaderightnowforLNGport i 1, k i ( ) Binaryvariabletodenotewhethera ki expansionismadeinthefutureforLNGport i underscenario f 0 ij Flowofarc ( i j ) rightnow f 1 ij ( ) Flowofarc ( i j ) inthefutureunderscenario s 0 i TotalSupplyfromnode i rightnow s 1 i ( ) TotalSupplyfromnode i inthefutureunderscenario g 0 i LNGsupplyfromnode i rightnow g 1 i ( ) FutureLNGsupplyfromnode i inthefuture u 0 ij ThecurrentCapacityofarc ( i j ) u 1 ij ( ) ThefutureCapacityofarc ( i j ) underscenario v 0 i ThecurrentcapacityofLNGport i v 1 i ( ) ThefuturecapacityofLNGport i underscenario nodeshouldbelessthanitsselfsupplylimitplusitsLNGsup ply.( 5–9 )istheLNG throughputcapacityexpansionconstraints.AtallthenonLNGnodes,thesupplyis boundedbyitsselfsupplylimit,whichisshownin( 5–10 ).Constraints( 5–11 ),( 5–12 ) and( 5–13 )denethenonnegativecontinuousow,capacityandsupply variables,and binaryexpansionvariables.Constraint( 5–14 )-( 5–24 )denethesecondstagefeasible region,whichalmostreplicatestherststage j j timesforallscenarioswithdifferent demands,supplies,and,mostimportantly,thedecisionvar iables.Thewholeextensive formulationofthestochasticplanningproblemisshownint hefollowingmixedinteger linearminimizationprogram,[EXPN]. [ EXPN ]: Min X ( i j ) 2 A X k 2 K ij c 0, k ij 0, k ij + X i 2 N LNG X k 2 K i c 0, k i 0, k i + X ( i j ) 2 A h 0 ij f 0 ij (5–1) + X 2 Pr ( ) 24 X ( i j ) 2 A X k 2 K ij c 1, k ij 1, k ij ( )+ X i 2 N LNG X k 2 K i c 1, k i 1, k i ( )+ X ( i j ) 2 A h 1 ij f 1 ij ( ) 35 (5–2) 91

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s.t. X ( i j ) 2 A +i f 0 ij X ( j i ) 2 A i (1 l ji ) f 0 ji = s 0 i d 0 i 8 i 2 N (5–3) f 0 ij + f 0 ji u 0 ij 8 ( i j ) 2 A (5–4) u 0 ij = u ij + X k 2 K ij kij 0, k ij 8 ( i j ) 2 A (5–5) g 0 i v 0 i 8 i 2 N LNG (5–6) g 0 i SL 0i 8 i 2 N LNG (5–7) s 0 i g 0 i + SF 0 i 8 i 2 N LNG (5–8) v 0 i = v i + X k 2 K i ki 0, k i 8 i 2 N LNG (5–9) s 0 i SF 0 i 8 i 2 N n N LNG (5–10) s 0 i f 0 ij u 0 ij v 0 i g 0 i ( ) 0, 8 ( i j ) 2 A i 2 N (5–11) 0, k ij 2f 0,1 g 8 k 2 K ij ,( i j ) 2 A (5–12) 0, k i 2f 0,1 g 8 k 2 K i i 2 N LNG (5–13) ( constraintstobecontinued ) [EXPN]isatwostagemixedintegerprogram,wherebinaryvar iablesarepresent inbothstages.Becausethesecondstagealsoincludesbinar yvariables,wecan notgenerateBenderscutsdirectlyasintheL-Shapedmethod .Onemethodof gettingvalidBenderscutsfromthesecondstagefortherst stageistheEmbedded BendersDecompositionwhichrelaxesthesecondstagetoali nearprogramandalso useBenderscutstoapproximatetheconvexhullofthesecond stageprogram.To enhancetheconvergence,wealsoaddintegerL-shapedoptim alitycuts[ 38 ]when necessary.ThesolutiontothesimpleexampleisshowninFig ure 5-6 ,whichindicates onlyLNGterminalexpansionsareneededatthecurrentpoint ,withsize2LNGterminal expansionsatnode1and10,andsize3LNGterminalexpansion satnode2,3and11. 92

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10 1 2 3 4 5 6 7 8 9 11 LNG-3 LNG-2 LNG LNG LNG-2 LNG-3 LNG-3 Figure5-6.SolutionofaSimpleEXPNExample Theconstraintsofsecondstageareasfollows, ( constraintscontinued ) X ( i j ) 2 A +i f 1 ij ( ) X ( j i ) 2 A i (1 l ji ) f 1 ji ( )= s 1 i ( ) d 1 i ( ), 8 i 2 N 2 (5–14) f 1 ij ( )+ f 1 ji ( ) u 1 ij ( ), 8 ( i j ) 2 A 2 (5–15) u 1 ij ( )= u 0 ij + X k 2 K ij ki j 1, k ij ( ), 8 ( i j ) 2 A 2 (5–16) g 1 i ( ) v 1 i ( ), 8 i 2 N LNG 2 (5–17) g 1 i ( ) SL 1i ( ), 8 i 2 N LNG 2 (5–18) s 1 i ( ) g 1 i ( )+ SF 1 i ( ), 8 i 2 N LNG 2 (5–19) v 1 i ( )= v 0 i + X k 2 K i ki 1, k i ( ), 8 i 2 N LNG 2 (5–20) s 1 i ( ) SF 1 i ( ), 8 i 2 N n N LNG 2 (5–21) s 1 i ( ), f 1 ij ( ), u 1 ij ( ), v 1 i ( ), g 1 i ( ) 0, 8 ( i j ) 2 A i 2 N 2 (5–22) 1, k ij ( ) 2f 0,1 g 8 k 2 K ij ,( i j ) 2 A 2 (5–23) 1, k i ( ) 2f 0,1 g 8 k 2 K i i 2 N LNG 2 (5–24) 93

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( :H; ‹ ƒ ƒ ƒš s H Figure5-7.ValueatRiskv.s.ConditionalValueatRisk5.2.2ThePlanningModelwithRiskConstraints Thesolutionto[EXPN]mighthavetosacricealottosatisfy anextremescenario whichhasbigdemands,whichmeanswemaypayforextremely“b ad”thingswhich areveryunlikelytohappen.Inordertondsuchkindofscena riosandtoleraterisks toacertainextent,wemayneedtoincludeameasurewhichcan helpuslocatethose scenariosandtellhow“bad”theyareandhow“unlikely”they are.Ariskmanagement modelwithchanceconstraintswouldtakecareofthissituat ionwhilecontrollingtherisk inanacceptablemanner. ValueatRisk(VaR)andConditionalValueatRisk(CVaR)aret wogenerallyused riskmeasuresintheliteratureduetotheirstructuralandc omputationaleasiness comparedtovariance.Asisstatedin[ 36 ],VaRhasbeenwidelyusedinnancial areasandisalsothestandardriskmeasureofBankforIntern ationalSettlements. Mathematicallyspeaking,VaRistheminimumvalue,suchtha ttheprobabilityofrandom lossisgreaterthanorequaltothisvalueislessthanasmall predenedapercentage 94

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(theriskyarea),whichisdenedasfollows, VaR = inf f l 2 R : P ( L ( X y ) l ) 1 g where L ( X y ) isthelossfunctionofrandomvariable X anddecisionvariable y isthecondencelevel.However,modelingVaRconstraintsn eedstouseinteger variables,whichmakessomelinearprogrammingmodelsdif culttosolve.Instead CVaRconstraintsdoesnotneedtointroduceintegervariabl esandonlyinvolveslinear constraints.Ingeneral,CVaRistheexpectedlossgiventhe factthatthelossisgreater thanorequaltoVaR,whichisshownasfollows, CVaR = E f L ( X y ) j L ( X y ) VaR g Asdiscussedin[ 3 ],CVaRconstraintsaretighterthanVaRconstraintssincet herisk constraintsaregenerallyoftheformatasVaR orCVaR ,andCVaR is abiggervaluethanVaR ,asshowninFigure 5-7 ,inwhich F L ( l ) isthecumulative distributionfunctionofrandomloss L ,and L min and L max arerespectivelytheminimum andmaximumvaluesthat L cantake. Inourriskmanagementmodel,wewouldliketouseCVaRasther iskmeasure becauseitnotonlyprovidesatighterboundbutalsoconsist sofonlylinearconstraints andcontinuousvariables.Theriskinthisexpansionproble mistheshortageofgas supplytothecustomers.Soweintroduceanewvariable, i ( ) ,todenotetheshortage inthefutureatnode i underscenario .Sotheoriginalowbalanceconstraint( 5–14 ) inthesecondstage(stochasticpart)ischangedtoconstrai nt( 5–25 ),whichincludes theshortagevariable i ( ) .Also,inordertodifferentiatenodesbypriorities,weonl y allowacertainamountofshortage, i ,forthenodesin N R ,asubsetofallnodes.Thisis realizedbyconstraints( 5–26 )and( 5–27 ). 95

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TheexpansionplanningmodelwithCVaRconstraints,whichs haresthesame objectivefunction( 5–1 )+( 5–2 ),samerststageconstraints( 5–3 )-( 5–13 )and ( 5–15 )-( 5–24 )with[EXPN],isformulatedasfollows, [ EXPN-R ]: Min ( 5–1 )+( 5–2 ) s.t. ( 5–3 ) ( 5–13 ), X ( i j ) 2 A +i f 1 ij ( ) X ( j i ) 2 A i (1 l ji ) f 1 ji ( ) = s 1 i ( ) d 1 i ( )+ i ( ), 8 i 2 N 2 (5–25) ( 5–15 ) ( 5–24 ), i ( )=0, 8 i 2 N n N R 2 (5–26) i i ( ) 0, 8 i 2 N R 2 (5–27) X i 2 N i ( ) + w ( ), 8 2 (5–28) + X 2 Pr ( ) 1 w ( ) (5–29) w ( ) 0, 8 2 (5–30) whereconstraints( 5–28 )-( 5–30 )aretheriskconstraintswhichareequivalentto CVaR .Constraint( 5–30 )introducesthenonnegativecontinuousvariable, w ( ) todenotetheamountoflossgreaterVaR forscenario .Constraint( 5–28 )denesthe boundsforthetotallossesofallscenarios,wherethesolut ionof ,aftersolvingthe problem,isactuallyVaR formostofthetimeasdiscussedin[ 54 ].Constraint( 5–29 ) thennallydenestheboundonCVaR 5.3EmbeddedBendersDecomposition ThetwoproblemsproposedinSection 5.2 arebothstochasticmixedintegerlinear programs,whichalsoincludeintegervariablesinthesecon dstage.Whenthenumberof scenariosarebig,theseproblemswillincludeahugenumber ofintegervariables,which 96

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maketheproblemdifculttosolveasawhole.Henceitisnece ssarytodecompose theproblemsandsolvethesecondstageproblemsseparately .L-shapedmethod[ 63 ] andBendersDecomposition[ 9 ]arewidelyusedtosolvethetwostagestochastic linearprogramsasdescribedin[ 10 ].However,theyarenotdirectlyapplicabletoour problemsbecausethesecondstageincludesintegervariabl esandsolvingmixed integerprogramsdoesnotgenerallyproduceusefuldualsol utions. Inthischapter,weproposeadifferentkindofvalidglobalc utstoapproximatethe convexhullofthesecondstagediscreteproblem.Inconstra ints( 5–14 )-( 5–24 ),thereare bothlinearandintegerparts.Henceitispossibletofurthe rdecomposethesubproblem itselfbyBendersdecomposition,andtheBenderscutswithi nthesubproblemcan helpcutoffthecombinationoftherststageandsecondstag eintegersolution.Our approachtriestoalsouseBenderscutstoconvexifythesubp roblem.Sowecallour algorithmEmbeddedBendersDecompositioninwhichBenders cutsaregenerated forbothmasterandsubproblems.Inthissection,wewilldis cusshowtoimplement ourdecompositionschemetosolvethestochasticexpansion planningproblem.The restrictedmasterproblem,[RMP],isasfollows, [ RMP ]: Min X ( i j ) 2 A X k 2 K ij c 0, k ij 0, k ij + X i 2 N LNG X k 2 K i c 0, k i 0, k i + X ( i j ) 2 A h 0 ij f 0 ij + X 2 Pr ( ) z ( ) (5–31) s.t. ( 5–3 ) ( 5–12 ), z ( ) X ( i j ) 2 A ^ x t ij ( ) u 0 ij + X i 2 N LNG ^ y t i ( ) v 0 i + r t ( ), 8 t 2T ( ), 2 (5–32) where t denotesthe t th cutofscenario ,and ^ x t ij ( ) ^ y t i ( ) and r t ( ) arerespectively theoptimaldualmultipliersandsumproductofmultipliers andrighthandsidesofthe relaxedsubproblemofscenario ,whichwillbeshownlater.Thisprogramisalways feasiblebecauseof(A.2)and(A.3).Also,intheabove[RMP] formulation,weshowhow thedisaggregatedcutsareadded,andwealsowilltalkabout theaggregatedcutsand comparethesetwokindsofcutsaddingschemesafterwenish thediscussionofhow 97

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thevalidBenderscutsfortherststagearegenerated.(Not e[RMP]itselfcanbefurther decomposed,sincethenetworkconstraintsarenotrelatedt otheexpansiondecisions directly.) Aftersolvethe[RMP]andobtainitssolution,wecansolveth esecondstage problemorthesubproblembyxingtherststageexpansiond ecisions, ^ u 0 and ^ v 0 .Becausewealsohaveexpansiondecisionvariablesinthese condstage,the subproblemsarealwaysfeasibleduetotheassumptions,(A. 2)and(A.3).Thisallowsus tosolvethesubproblemsofdifferentscenariosseparately becausethereisnocoupling betweenscenariosexcepttherststagedecisions.Thesubp roblemcorrespondingto scenario isshownasfollows, [ SP( ) ]: Min X ( i j ) 2 A X k 2 K ij c 1, k ij 1, k ij ( )+ X i 2 N LNG X k 2 K i c 1, k i 1, k i ( )+ h 1 ij f 1 ij ( ) s.t. X ( i j ) 2 A +i f 1 ij ( )+ X ( j i ) 2 A i (1 l ji ) f 1 ji ( )+ s 1 i ( )= d 1 i ( ), 8 i 2 N f 1 ij ( ) f 1 ji ( )+ u 1 ij ( ) 0, 8 ( i j ) 2 A u 1 ij ( ) X k 2 K ij kij 1, k ij ( )=^ u 0 ij 8 ( i j ) 2 A g 1 i ( ) v 1 i ( ) 0, 8 i 2 N LNG g 1 i ( ) SL 1i ( ), 8 i 2 N LNG s 1 i ( ) g 1 i ( ) SF 1 i ( ), 8 i 2 N LNG v 1 i ( ) X k 2 K i ki 1, k i ( )=^ v 0 i 8 i 2 N LNG s 1 i ( ) SF 1 i ( ), 8 i 2 N n N LNG s 1 i ( ), f 1 ij ( ), u 1 ij ( ), v 1 i ( ), g 1 i ( ) 0, 8 ( i j ) 2 A i 2 N 1, k ij ( ) 2f 0,1 g 8 k 2 K ij ,( i j ) 2 A 1, k i ( ) 2f 0,1 g 8 k 2 K i i 2 N LNG 98

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Thisisamixedintegerlinearprogramandwillnotdirectlyp rovideusefulBenderscuts for[RMP]ingeneral.However,comparedtothewholeproblem ,thisprogramismuch easiertosolvesinceitonlyinvolvesonescenarioandhasmu chlessbothintegerand continuousvariables.Aftersolve[SP ( ) ]andxtheintegervariableatitsoptima, ^ 1 ( ) and ^ 1 ( ) ,thenwecansolvethefollowing[LP ( ) ]togenerateembeddedBenderscuts, theglobalcuts,forthesubproblemofscenario .Again,theprogram[LP ( ) ]isalways feasiblesince[SP( )]isalwaysfeasibleandthebinaryvariablesarexedasthe optimal (feasible)solutionof[SP( )]. [ LP( ) ]: Min X ( i j ) 2 A h 1 ij f 1 ij ( ) (5–33) s.t. X ( i j ) 2 A +i f 1 ij ( )+ X ( j i ) 2 A i (1 l ji ) f 1 ji ( )+ s 1 i ( )= d 1 i ( ), 8 i 2 N (5–34) f 1 ij ( ) f 1 ji ( )+ u 1 ij ( ) 0, 8 ( i j ) 2 A (5–35) u 1 ij ( )= X k 2 K ij kij ^ 1, k ij ( )+^ u 0 ij 8 ( i j ) 2 A (5–36) g 1 i ( ) v 1 i ( ) 0, 8 i 2 N LNG (5–37) g 1 i ( ) SL 1i ( ), 8 i 2 N LNG (5–38) s 1 i ( ) g 1 i ( ) SF 1 i ( ), 8 i 2 N LNG (5–39) v 1 i ( )= X k 2 K i ki ^ 1, k i ( )+^ v 0 i 8 i 2 N LNG (5–40) s 1 i ( ) SF 1 i ( ), 8 i 2 N n N LNG (5–41) s 1 i ( ), f 1 ij ( ), u 1 ij ( ), v 1 i ( ), g 1 i ( ) 0, 8 ( i j ) 2 A i 2 N (5–42) Nowwehaveapurelinearprogramwithoutexpansiondecision s,andthensolvingit canhelpgeneratethefollowingglobalcutwhichisvalidgiv enanyrststageexpansion decisionstatus, ^ u 0 and ^ v 0 X ( i j ) 2 A ^ p u ij 24 0@ X k 2 K ij kij 1, k ij ( ) 1A +^ u 0 ij 35 99

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+ X i 2 N LNG ^ p v i X k 2 K i ki 1, k i ( ) +^ v 0 i # + X i 2 N ^ q d i d 1 i ( )+^ q SF i SF 1 i ( ) + X i 2 N LNG ^ q SL i SL 1i ( ) (5–43) where ^ p u ij ^ p v ij ^ q d i ^ q SL i ^ q SF i aretheoptimaldualmultiplierscorrespondingto( 5–36 ), ( 5–40 ),( 5–37 ),( 5–38 ),( 5–39 )and( 5–41 )respectively.Forconvenience,thiscutcanbe rewritteninvectorformatasfollows, a T l 1 ( )+ b T l 1 ( )+( P u l ) T ^ u 0 +( P v l ) T ^ v 0 +( Q d l ) T d 1 ( )+( Q SL l ) T SL 1 ( )+( Q SF l ) T SF 1 ( ) (5–44) where l denotethe l th cut.Thenwecanincludetheseglobalcutstoconstructarela xed versionofthesubproblemsasfollows, [ RSP( ) ]: Min X ( i j ) 2 A X k 2 K ij c 1, k ij 1, k ij ( )+ X i 2 N LNG X k 2 K i c 1, k i 1, k i ( )+ (5–45) s.t. a T l 1 ( )+ b T l 1 ( )+( P u l ) T ^ u 0 +( P v l ) T ^ v 0 +( Q d l ) T d 1 ( )+( Q SL l ) T SL 1 ( )+( Q SF l ) T SF 1 ( ), 8 l 2L ( ), (5–46) 0 1, k ij ( ) 1, 8 k 2 K ij ,( i j ) 2 A (5–47) 0 1, k i ( ) 1, 8 k 2 K i i 2 N LNG (5–48) wherethesecondstageexpansiondecisionvariablesarerel axedtobecontinuouswhile beingboundedwithin [0,1] Proposition5.1. ( 5–43 )isvalidBenderscutsfor[RSP( )]givenanyrststagesolution ^ u 0 and ^ v 0 Proof. In[LP ( ) ],rststagesolutions ^ u 0 and ^ v 0 onlyexistonthetherighthandsides ofconstraints( 5–36 )and( 5–40 ).[LP( )]swithdifferentrststagesolutionssharethe samedualspaceeventheyhavedifferentprimalfeasiblereg ions.Thisisbecausethe objectivecoefcientsandlefthandsidecoefcientsareth esamefordifferentproblems. 100

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Thecut,( 5–43 ),isconstructedbyusingthedualsolutionto[LP ( ) ],andthenitisvalid foranygivenrststagedecision. Thismeansthatwestillcanusethepreviouslygeneratedemb eddedBenders cuts,( 5–43 ),forthecurrentrststagedecision.Sowecouldsequentia llyaddthe embeddedBenderscutsto[RSP ( ) ].Butweneedtochangetherighthandsidesofthe embeddedBenderscutsbyplugginginthecurrentrststages olution.Subproblemsof allscenarioshavethesamelefthandsidesbecausetheuncer taintiesintheexpansion planningmodellieinthedemandsandsupplies,whichareall ontherighthandsides oftheconstraints.Soall[LP( )]ssharethesamedualspace,andthenthedualoptimal solutiontoany[LP ( ) ], ( a b P u P v Q d Q SL Q SF ) ,canbeusedtoconstructembedded Benderscutsfor[LP ( ) ], 8 2 ,asfollows, a T 1 ( )+ b T 1 ( )+( P u ) T ^ u 0 +( P v ) T ^ v 0 +( Q d ) T d 1 ( )+( Q SL ) T SL 1 ( )+( Q SF ) T SF 1 ( ). Hencewedonotneedtomaintainaindividualsetofdualoptim alsolutions, L ( ) ,for everyscenario,butonlyneedtomaintainoneset L forallscenarios,becausethe dualsolutionscanbeusedforallscenarios.Thenwecansolv ethe[RSP( )]toderive Benderscutsfortherststage,whichisshownasfollows, z ( ) X l 2L ^ r t l ( ) ( P u l ) T u 0 +( P v l ) T v 0 +( Q d l ) T d 1 ( )+( Q SL l ) T SL 1 ( )+( Q SF l ) T SF 1 ( ) + X ( i j ) 2 A X k 2 K ij ^ k t ij ( )+ X i 2 N LNG X k 2 K i ^ k t i ( ) (5–49) where ^ r t l ( ) istheoptimaldualmultipliercorrespondingtothe l th embeddedBenders cutin L ,and t denotesthe t th Benderscutfor[RMP]. ^ tij ( ) and ^ t i ( ) areoptimaldual multiplierscorrespondingto( 5–47 )and( 5–48 )respectively. Proposition5.2. ( 5–49 )isavalidBenderscutfor[RMP]. 101

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Proof. Supposethat Z SP ( ) and Z RSP ( ) aretheoptimalobjectivevaluesof[SP ( ) ]and [RSP ( ) ]respectively.Givenanyrststagesolution, Z SP ( ) Z RSP ( ) ,because( 5–43 )is aglobalcutfor[RSP ( ) ]andthefeasibleregionof[RSP ( ) ]contains[SP ( ) ]'s.Also,we haveZ RSP ( ) X l 2L ^ r t l ( ) ( P u l ) T u 0 +( P v l ) T v 0 +( Q d l ) T d 1 ( )+( Q SL l ) T SL 1 ( )+( Q SF l ) T SF 1 ( ) + X ( i j ) 2 A X k 2 K ij ^ k t ij ( )+ X i 2 N LNG X k 2 K i ^ k t i ( ). Hence,( 5–49 )isavalidBenderscutfor[RMP]. Ifwemoveallitemsinvolvingvariablestothelefthandside forconstraint( 5–46 ),we willget a T l 1 ( ) b T l 1 ( )+ ( P u l ) T ^ u 0 +( P v l ) T ^ v 0 +( Q d l ) T d 1 ( )+( Q SL l ) T SL 1 ( )+( Q SF l ) T SF 1 ( ), 8 l 2L where L ( ) isreplacedby L ,sinceanyembeddedBenderscutisvalidforallrestricted subproblems.Itisinterestingtonotethatthecoefcients of 1 ( ) and 1 ( ) are independentofscenarios.Soall[RSP ( ) ]shavethesamelefthandsidecoefcients andobjectivefunction,whichmeanstheysharethesamedual space(dualfeasible region).Hencesolvingone[RSP( )]meansthatweobtainmultiplecutsforallscenarios, andthenwehavefollowingproposition.Proposition5.3. Forall 2 z ( ) X l 2L ^ r t l ( ) ( P u l ) T u 0 +( P v l ) T v 0 +( Q d l ) T d 1 ( )+( Q SL l ) T SL 1 ( )+( Q SF l ) T SF 1 ( ) + X ( i j ) 2 A ^ X k 2 K ij k t ij ( )+ X i 2 N LNG X k 2 K i ^ k t i ( ) (5–50) isavalidBenderscutfor[RMP],where ^ r t l ( ),^ tij ( ),^ t i ( ) arethedualoptimalsolution to[RSP( )]. 102

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BecauseofProposition 5.3 ,inconstraint( 5–32 )weonlyneedtomaintainasingle setofdualsolutions, T ,insteadofmultiplesetsforeachindividualscenario, T ( ) .In the[RMP],however,westillneed j j recoursevariables, z ( ) s,and jTjj j Benders cuts,whichwerefertoasdisaggregatedcuts.Actuallyeach dualsolutionin T is correspondingtomultipledisaggregatedcutsin( 5–32 ),whichcanbeaggregatedtoone cut,( 5–51 ).Theaggregatedcutisobtainedbyaddingtheweighteddisa ggregatedcuts together,wheretheweighttoacutisitscorrespondingprob ability.Itisshownasfollows, z X l 2L ^ r t l ( ) h ( P u l ) T u 0 +( P v l ) T v 0 +( Q d l ) T d 1 +( Q SL l ) T SL 1 +( Q SF l ) T SF 1 i + X ( i j ) 2 A X k 2 K ij ^ k t ij ( )+ X i 2 N LNG X k 2 K i ^ k t i ( ), (5–51) where P 2 Pr ( ) z ( ) isreplacedby z ,and d 1 SL 1 and SF 1 aretheexpecteddemand andsupplyvectors,whichareequalto P 2 Pr ( ) d 1 ( ) P 2 Pr ( ) SL 1 ( ) and P 2 Pr ( ) SF 1 ( ) respectively.Aggregationcanreducethenumbersofrecour se variablesandBenderscutsgreatlyifthereareahugeamount ofscenarios.Ifthe aggregatedcuts( 5–51 )areusedin[RMP],itonlyneedsonerecoursevariable, z ,and itsobjectivefunctionneedstobemodiedaccordinglyasfo llows, X ( i j ) 2 A X k 2 K ij c 0, k ij 0, k ij + X i 2 N LNG X k 2 K i c 0, k i 0, k i + X ( i j ) 2 A h 0 ij f 0 ij + z (5–52) Accordingtothenumberofscenarios,wecouldchoosediffer entstrategiestoaddvalid Benderscutsin[RMP].Asisdiscussedin[ 10 ],thedisaggregatedschemeischosen inthecaseofasmallnumberofscenarios,andviceversa.For themodel[EXPN], weprefertheaggregatedschemebecausetheaggregatedcuts needlessvariables, containmoreinformationfromallscenarios,andareveryea sytogenerateduetothe sharingofsamedualspaceamong[RSP ( ) ]s.Forthe[EXPN-R]model,wecandothe samedecompositionbyseparatingtheriskconstraints,and changingtheowbalance constraintas( 5–25 ).Becauseconstraint( 5–29 )bundlesallthevariablestogether, 103

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itisincludedthe[RMP].Sodothevariables, w ( ) s,relatedtothisconstraint.The remainingriskconstraints,( 5–26 5–27 5–28 ),areaddedto[SP ( ) ]s.Thenthesame decompositionstrategyapplies. Asisdescribedinsection 5.3 ,Benderscutsareusedinboththemasterproblem andthesubproblems,andBenderscutsinthesubproblemshel ptoconstructthe Benderscutsforthemasterproblem,whichisthereasonthat wenamethisalgorithm EmbeddedBendersDecomposition.Inpractice,Bendersdeco mpositioncouldconverge veryslowly.Then,inordertospeedupconvergence,inthema sterproblemweadd anothertypeofcuts,integerL-shaped“optimality”cuts,w hichisproposedby[ 38 ].An integerL-shaped“optimality”cutisasfollows, z ( Q (^ x ) L ) X j 2 T x j X j 2 F x j j T j +1 + L where Q ( x ) istherecoursefunctionand L isalowerboundforthesecondstage problem,and T = f j j ^ x j =1 g and F = f j j ^ x j =0 g ,if x istherststagedecisionvariable and ^ x isthecurrentsolution.Thisfollowsfromthefactthatther ighthandsidewillbe equalto Q (^ x ) if x =^ x ,andlessthan L otherwisesince P j 2 T x j P j 2 F x j j T j +1 0 if x 6 =^ x .Wereferinterestedreadersto[ 38 ]forthedetailedproof.Withouthavingto denethetwosets, T and F ,afterrearrangingtermsthecutcanbeexpressedbythe followingequivalentinequality, ( Q (^ x ) L ) X j (1 2^ x j ) x j + z Q (^ x ) ( Q (^ x ) L ) X j ^ x j Becausethesecondstageproblems,SP ( ) s,areactuallyaffectedonlybythe binarysolutions, ^ 0 and ^ 0 because ^ u 0 and ^ v 0 aredeterminedifthesebinaryvariables arechosen.ThismeanswecanusetheintegerL-shaped“optim ality”cutsforsolving ourmodelsbecauserststagecontinuousdecisions(ows)d oesnotaffectthesecond 104

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Step0. Set UB= 1 LB=0 L = ; T = ; ^ z =0 ; Solve[RMP]withouttakingintoaccountrecoursevariable, z ; Gettheoptimalobjective, Z RMP ,andsolution( ^ u 0 ,^ v 0 ); Step1. For 2 Solve[SP ( ) ]with( ^ u 0 ,^ v 0 ); Gettheoptimalobjective, Z SP ( ) ,andsolution( ^ 1 ( ), ^ 1 ( ) ); Solve[LP( )]with( ^ u 0 ,^ v 0 )and( ^ 1 ( ), ^ 1 ( ) ); Compute( a b Q d Q SL Q SF P u P v )aftersolvingLP( ); L L[ ( a b Q d Q SL Q SF P u P v ) ; Solve[RSP( )]with( ^ u 0 ,^ v 0 ),obtainoptimaldualsolution (^ r ,^ ,^ ) ; Constructanewaggregatedcutasin( 5–51 )andadditto T ; EndFor U=Z RMP ^ z + P 2 Pr ( )Z SP ( ); UB min (UB,U) ; Step2. AddanewintegerL-shapedcutto[RMP]asin( 5–53 ); Step3. Solve[RMP]andobtainoptimalobjective, Z RMP ,andsolution( ^ z ,^ u 0 ,^ v 0 ); LB max (LB,Z RMP ) ; Step4. If UB LB ,thenstop;OtherwisegotoStep1. Figure5-8.EmbeddedBendersDecompositionAlgorithmforE XPN stage.AnintegerL-shaped“optimality”cutfor[RMP]isasf ollows, h Q (^ 0 ^ 0 ) L i 24 X ( i j ) 2 A X k 2 K ij 1 2^ 0, k ij 0, k ij + X i 2 N LNG X k 2 K i 1 2 ^ 0, k i 0, k i 35 + z Q (^ 0 ^ 0 ) h Q (^ 0 ^ 0 ) L i 24 X ( i j ) 2 A X k 2 K ij ^ 0, k ij + X i 2 N LNG X k 2 K i 0, k i 35 (5–53) Thebestthecase,inthesenseoflowestcost,fortheseconds tageproblemsisthat noexpansionisneeded.Eventhereisnoexpansioncost,ther eisalwaystransportation cost.Hence,alowerbound L istheminimaltransportationcostofthesecondstage, whichcanbecalculatedasfollows, L = X 2 Pr ( ) h T f where f istheoptimalsolutionofthenetworkowproblemofscenari o Aninitialsolutionoftherststagedecisionscanbeobtain edbysolvingthe[RMP] withoutincludingtherecoursevariable z andanyBenderscut.Thelowerboundis 105

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actuallytheoptimalobjectivevalueof[RMP].Theupperlow erboundcanbeobtainedby addingupthetotalcostofafeasiblesolution.Theembedded Bendersdecomposition algorithmisshowninFigure 5-8 5.4NumericalExamples Inthissection,wepresentnumericalresultsofouralgorit hmonservalproblems withdifferentsizes.WecodeourembeddedBendersdecompos itionalgorithmin MicrosoftVisualC++whilecallingCPLEX10(ConcertTechno logy)tosolvethe decomposedproblems.AllprogramsareruninMicrosoftWind owsXPProfessional 2002SP2onaDellDesktopwithIntelPentium4CPU3.40GHzand 2GBofRAM. Wetestthreegroupsofinstances,eachofwhichhasdifferen tnumbersofLNG nodes,totalnodes,arcs,arcexpansionandLNGterminalexp ansionsizes.Also,we assumeallarcsareexpandable,andsamepossiblearcandnod eexpansioncapacities atdifferentarcsandLNGnodes.Solet j K ij j denotesthenumberofpossibleexpansion sizesofallarcs,and j K i j denotesthenumberofpossibleexpansionsizesofallLNG nodes.Thenthenumberofbinaryvariablesintheextensivef ormulation,[EXPN], is ( j K i jj N LNG j + j K ij jj A j ) ( 1+ j j ) ,whichmeans ( j K i jj N LNG j + j K ij jj A j ) binaryvariablesineachdecomposedproblem.Ineachgroup, werandomlygenerate differentamountsofscenariosforaspecicinstance.Then umbersofscenarios rangefrom 10 to 10 thousand.Whenthenumberofscenariosisbig,thisextensiv e formulationisnotaneasyproblem.Whiledealingwiththeex tensiveformulation, [EXPN],directly,CPLEX-MIPsolverdoesnotefcientlysol vetheinstanceswithabig numberofscenarios,e.g., 10 kscenarios,witheitherexceedingthe2hourcomputational timelimitorrunningoutofcomputermemory.However,ourEB Dalgorithmcansolve theseinstanceswithabignumberofscenariosinatimelyman ner.InTable1,we denethreegroupsofinstances.Andthencomputationalres ultsareshowninTable 5-5 5-6 and 5-7 ,wheretimesarecountedinseconds.Inthethreetables,wel istthat totalcomputationaltimes,andcomputingtimesforRMP,SP, LPandRSP.Ascanbe 106

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Table5-4.GroupsofInstancesEXPN GroupNo. j N LNG jj N jj A jj K ij jj K i j 124522237123337112055 Table5-5.Computationaltimesforinstancesgroup1 j j TotalRMPSPLPRSP 105.1570.5493.2030.9710.4342011.6990.5797.7652.1821.173502.8760.0312.0450.4990.253 1005.7810.0473.8891.0290.56620012.5740.0358.2732.6101.18930018.6520.03111.7173.0481.61350029.1550.04618.4775.9232.62670036.1980.3224.4216.5114.045 1k58.6600.04734.6329.3435.4552k115.2440.04768.91618.67213.059 10k419.7150.31286.709125.6293.209 seeninthetables,computingtimealmostincreaselinearly withrespecttothenumber ofscenarios,whichmeansourEBDmethodiswellsuitedtopro blemswithahuge numberofscenarios.Also,theEBDalgorithmspendsabigpor tionoftimetosolve SPandLP.Henceitispossibletofurtherreducecomputingti meifwedonotcalculate newICandOFcutsforeachscenarioinStep1becauseloopingt hroughallscenarios takesalotoftime,especiallywhenwehaveahugenumberofsc enarios,e.g.10kor more.Moreadvancedimplementationcouldhelptoachieveth isandimprovetheoverall performance. Inadditiontotestingthecomputationaltimeofouralgorit hm,wealsoconduct numericalexamplesontheriskmanagementmodel,[ESPN-R], tondouthowthe riskconstraintsaffecttheperformance(optimalobjectiv evalue)ofthemodel.Figure 5-9 showshowtheoptimalobjectivefunctionchangeswhenwevar ythecondence level, ,andtheupperboundoftheriskmeasure,ConditionalValuea tRisk.Asis seenfromthegure,theoptimalobjectivevaluedecreasesa stheupperbound, ,on CVaRincreases.Fourlinesaredrawnaccordingtodifferent condencelevels,and 107

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Table5-6.Computationaltimesforinstancesgroup2 j j TotalRMPSPLPRSP 101.6080.1560.9040.3450.203202.6390.1251.7480.5040.262506.1130.1093.8741.3830.745 10011.6960.1267.8152.4221.33320023.2210.11015.3054.9742.83230037.4900.10925.6337.0024.74650052.6500.10838.80910.1523.58170078.3420.9451.56718.2228.459 1k100.6560.10968.51323.7128.3222k190.8510.093134.32645.3238.179 10k934.0130.109685.224240.2018.479 Table5-7.Computationaltimesforinstancesgroup3 j j TotalRMPSPLPRSP 3.5770.1240.2800.1090.64 101.6090.1091.0330.3010.156203.4490.1252.1720.6990.453507.2220.0944.8631.4400.825 10015.2160.09310.5172.8481.75820027.9320.10918.7675.8443.21230041.8370.09428.9968.7224.02550073.9500.09350.58414.9778.296700127.5780.09585.58326.90214.998 1k145.0970.09498.22626.96319.8142k295.8850.094215.37460.72419.693 10k1290.8800.94996.116276.14318.526 thelinewithhighercondencelevelboundsfrombelowtheon ewithlowercondence level,whichsaysthatweneedtopaymoreifwantwewanttobem oresecure(higher condencelevel).Anotherinterestingfacttonotefromthe gureisthatthefourlines becomemoredeviatedfromeachotherwhentheupperbound increases.Thisismay beexplainedbyasimpleexample.If =0 ,thentheoptimalobjectivevalueofany condencelevelshouldbethesame,becausetherighthandsi deofconstraint( 5–29 )is 0 ,andthen willnotbeabletoaffecttheoptimalsolutionof and w i s.With gradually increasing,theeffectivenessof ontheoptimalobjectivevaluekeepsincreasing.It, directlyreadingfromthegure,lookslikethattheoptimal objectivevalueisaconvex 108

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! # $ # $ # $ $ # % & ( r$ Figure5-9.MinimalCostV.S.LimitofCVaRfunctionof .Thisistrueifwearedealingwithpurelinearmodels.Howev er,thismay notholdforourmodelssincewealsohavediscretedecisionv ariables. 109

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CHAPTER6 CONCLUSIONS Thisdissertationdiscussesstochasticintegerprogrammi nganditsapplications intheenergysystems.Firstthisdissertationgivesabrief reviewaboutstochastic programming,stochasticintegerprogramming,andsolutio nmethods.Thenthis dissertationproposestheEmbeddedBenders'Decompositio n(EBD)methodfor bothdeterministicandstochasticmixedintegerprogrammi ng,whichincludeinteger decisionvariablesinthesecondstage.Thismethodtakesad vantageofBenders' decompositionandappliestheBenders'cutsinbothrstand secondstages.All Benders'cutsgeneratedforthesecondstagearereusableal ongtheiterationsofthe algorithmsincethesecutsarevalidforthesecondstagegiv enanyrststagedecision. Next,thisdissertationstudiesthetwo-stagestochastics ecurityconstrainedunit commitment(SSCUC)problembyapplyingtheEBDmethod.TheS SCUCproblem includesbothday-aheadschedulingofcoalredgenerators andreal-timequick-start generatorsscheduling.ComputationalresultsshowthatEB Disverywellsuitedfor theSSCUCproblems,especiallywhendealingwithalotofsce narios.Afterthat,a detailedreviewofoptimizationmodelsandtechniquesappl iedinnaturalgasindustry ispresented.Also,thisdissertationproposesamathemati calprogrammingmodelfor naturalgastransmissionsystemexpansionplanning,which ,toourbestknowledge,is therstmodelthatcombinestransmissionlineexpansionan dLNGterminalexpansion together,andconsidersuncertaintiesindemandsandsuppl ies,andriskcontrolling. NumericalresultsindicatethatCVaRisaverygoodriskmeas ureinthesenseoftaking risktoreducecost.Becauseexpansionsaremodeledbyinteg ervariables,themodel isatwostagestochasticmixedintegerprogramswhereinteg ervariablesexistinboth stages.SMIPofthistypeisaverychallengingproblem.EBDa lgorithmisalsoapplied tosolvetheexpansionplanningproblem.Benderscutsareim plementedinbothstages, buttheyactasdifferentroles.TheBenderscutsfortheseco ndstagearereusablegiven 110

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anyrststagesolution,andsolvingthesubproblemsofones cenariocanhelpgenerate multiplecutsforallscenariosduetotheirsharingofdualf easiblespace.Thenumerical resultsshowthattheEBDalgorithmisalsoverywellsuitedf ortheexpansionplanning problems,especiallywhenthereareahugenumberofscenari os.Futureresearch wouldfocusonhowtofurtherimprovetheEBDalgorithmbyred ucingthecomputational timeonRSP,LPandRSP.Also,moresophisticatedtechniques togeneratestrong Benderscutscouldbeusedtoimprovetheconvergencerateof thealgorithm. Therearetwotypesoffutureresearchwewouldliketopursue ,theoriesand applications.Onthetheoreticalside,wewouldliketonda nitelyconvergentmethod forgeneralstochasticmixedintegerprogramsbasedonther esultsofEmbedded BendersDecomposition.Also,thedevelopmentoffastconve rgentmethodsfor multistagestochasticintegerprogramsbasedonBendersde compositionandpolyhedral theoryisfascinating.Ontheapplicationside,networkbas edstochasticunitcommitment problemsareveryimportantinreality,sincegeneratorsar elocatedinadecentralized powergrid.Network-basedmodelscanprovidemoreinsights abouthowtocoordinate andintegrateallresourceswithinthepowergrid.Thenatur algascontractoptimization problemisanotherveryinterestingapplicationproblemon whichwecanapplythe multistageEBDifwecanenhancetheconvergence.Moreandad vancedstudiesof polyhedralpropertiesofsubproblemineachstagearecruci altodevelopadvanced algorithmsforthemultistageproblems. 111

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BIOGRAPHICALSKETCH QipengZhengwasborninaheavyindustrytown,whichisamajo rmanufacturer ofChina'saluminumproduction,inZibo,Shandong,in1979. Uponhisgraduationfrom thelocalhighschoolin1997,hewenttoBeijingforcollegei nNorthChinaUniversityof Technology,majoringinIndustrialAutomation.Afterobta ininghisbachelor'sdegreein 2001,hedevoted8monthsinastartupcompanytodevelopings oftwareforembedded systems.In2002,hestartedpursuinghismaster'sdegreein theDepartmentof AutomationatTsinghuaUniversity,China.Withhismaster' sdegree,hecameto Gainesville,Florida,topursuehisdoctorateintheDepart mentofIndustrialandSystems EngineeringatUniversityofFlorida,insummer2005. 117