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Mixed Integer Programming Approaches to 0-1 Knapsack Problems and Unified Stochastic and Robust Optimization on Wind Power Investment

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Title:
Mixed Integer Programming Approaches to 0-1 Knapsack Problems and Unified Stochastic and Robust Optimization on Wind Power Investment
Creator:
Zhao, Kun
Place of Publication:
[Gainesville, Fla.]
Florida
Publisher:
University of Florida
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Language:
english
Physical Description:
1 online resource (102 p.)

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Industrial and Systems Engineering
Committee Chair:
HARTMAN,JOSEPH C
Committee Co-Chair:
GUAN,YONGPEI
Committee Members:
RICHARD,JEAN-PHILIPPE P
AYTUG,HALDUN
Graduation Date:
8/8/2015

Subjects

Subjects / Keywords:
Algorithms ( jstor )
Financial investments ( jstor )
Investment decisions ( jstor )
Linear programming ( jstor )
Operations research ( jstor )
Optimal solutions ( jstor )
Robust optimization ( jstor )
Run time ( jstor )
Transmission lines ( jstor )
Wind power ( jstor )
0-1kp -- dynamic-programming -- mixed-integer -- optimization -- robust -- stochastic -- wind-power
Industrial and Systems Engineering -- Dissertations, Academic -- UF
Genre:
bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Industrial and Systems Engineering thesis, Ph.D.

Notes

Abstract:
This dissertation covers a theoretical study of combining dynamic programming approach with cutting planes to solve the binary knapsack problems. In addition, motivated by the theoretical research, a further study is conducted on applications in wind power investment with or without transmission expansions. In the theoretical study, forward dynamic programming is used to generate cutting planes for an equivalent integer programming formulation of the binary knapsack problem. Upper bounds from intermediate stage solutions are used to derive cutting planes while solutions determined from exploring ``infeasible states" generate traditional cover inequalities. The effectiveness is illustrated at solving binary knapsack problems through a branch-and-cut framework. In the application part, short term wind power investment problem and long term simultaneous wind power and transmission investment problem are studied, where the 0-1 knapsack is involved as the budget constraint in the mixed-integer program formulation. Unified stochastic and robust optimization framework is proposed to solve the problems, and Bender's decomposition algorithms are utilized to generate optimality cuts. Computational experiments verify the effectiveness of the solution approaches. ( en )
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In the series University of Florida Digital Collections.
General Note:
Includes vita.
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Includes bibliographical references.
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Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (Ph.D.)--University of Florida, 2015.
Local:
Adviser: HARTMAN,JOSEPH C.
Local:
Co-adviser: GUAN,YONGPEI.
Statement of Responsibility:
by Kun Zhao.

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UFRGP
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Copyright Zhao, Kun. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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LD1780 2015 ( lcc )

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MIXEDINTEGERPROGRAMMINGAPPROACHESTO0-1KNAPSACKPROBLEMSANDUNIFIEDSTOCHASTICANDROBUSTOPTIMIZATIONONWINDPOWERINVESTMENTByKUNZHAOADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2015

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c2015KunZhao

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ToQinfangXueandYunshengZhao-MomandDad,Iloveyou.

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ACKNOWLEDGMENTSIwouldliketoacknowledgeallthepeoplewhohavehelpedmeandmademyPh.D.possible.Firstly,theunconditionaltrustandencouragementfrommyparentshavealwaysbeenmygreatestassets.IwouldliketothankDr.I-lokChangforalwaysbeingpatientandsupportivetome.IwouldliketothankmyacademicadvisorDr.JosephC.HartmanandDr.YongpeiGuan,forgivingmetheopportunitiestoworkoninterestingresearchtopics,andforthevaluablediscussions.IwouldliketothankDr.Jean-PhilippeRichardandDr.Aytug,Haldunforservinginmycommittee.IwouldalsoliketothankDr.ColeSmithandourDepartmentofIndustrialandSystemsEngineeringforadmittingmetothedoctoralprogram.Lastbutnotleast,IwouldliketothankallmyfriendsatUF,andextendmythankstoXiaojing,Yuan,Xuan,andVijay. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ................................... 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 9 ABSTRACT ......................................... 10 CHAPTER 1INTRODUCTION ................................... 11 1.1TheKnapsackProblems ............................. 11 1.2TheWindPowerInvestmentProblem ...................... 12 2DYNAMICPROGRAMMIGBASEDINEQUALITIESFORTHE0-1KNAPSACKPROBLEM ....................................... 14 2.1MotivationandLiteratureReview ........................ 14 2.2TheoryReview .................................. 16 2.2.1IPSeparationAlgorithm ......................... 16 2.2.2DynamicProgrammingApproach .................... 19 2.3InequalityDerivation ............................... 21 2.3.1CoverInequalitiesfromDPInfeasibleStates ............... 21 2.3.2ProtInequalities ............................. 26 2.3.3ANumericalExample .......................... 28 2.4ComputationalResults .............................. 31 2.4.1Implementation .............................. 31 2.4.2ExperimentalDesign ........................... 33 2.4.3ResultsSummary ............................. 33 2.5ConcludingRemarks ............................... 38 3DYNAMICPROGRAMMINGBASEDINEQUALITIESFORTHEMULTIDIMENSIONALKNAPSACKPROBLEM ................................ 39 3.1MotivationandLiteratureReview ........................ 39 3.2TheoryReview .................................. 41 3.2.1CoverInequality ............................. 41 3.2.2DynamicProgrammingApproach .................... 42 3.3Inequalityderivation ............................... 43 3.3.1CoverInequalitiesandProtInequalitiesforMKP ........... 43 3.3.2ModiedProtInequality ........................ 45 3.3.3ANumericalExample .......................... 47 3.4ComputationalResults .............................. 48 3.4.1Implementation .............................. 48 5

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3.4.2ExperimentalDesign ........................... 49 3.4.3ResultsandDiscussions ......................... 49 3.5ConcludingRemarks ............................... 51 4RISK-AVERSEWINDPOWERINVESTMENTTHROUGHUNIFIEDSTOCHASTICANDROBUSTOPTIMIZATION ........................... 52 4.1MotivationandLiteratureReview ........................ 52 4.2NotationandMathematicalFormulation .................... 55 4.2.1Nomenclature ............................... 55 4.2.2MathematicalModel ........................... 56 4.3LinearizationandDecomposition ........................ 60 4.3.1StochasticOptimization ......................... 60 4.3.2BilinearTermintheObjective ...................... 61 4.3.3RobustOptimization ........................... 61 4.3.4Algorithm ................................. 63 4.4ComputationalResults .............................. 65 4.4.1A6-BusCase ............................... 66 4.4.2IEEE118-BusCase ............................ 70 4.5ConcludingRemarks ............................... 73 5SIMULTANEOUSWINDPOWERANDTRANSMISSIONINVESTMENTFORVERTICALLYINTEGRATEDUTILITIES ............................... 74 5.1MotivationandLiteratureReview ........................ 74 5.2NotationandMathematicalFormulation .................... 76 5.2.1Nomenclature ............................... 76 5.2.2MathematicalModel ........................... 78 5.3LinearizationandDecomposition ........................ 82 5.3.1StochasticOptimization ......................... 82 5.3.2RobustOptimization ........................... 85 5.3.3Algorithm ................................. 86 5.4ComputationalResults .............................. 86 5.4.1A6-BusCase ............................... 86 5.4.2IEEE118-BusCase ............................ 91 5.5ConcludingRemarks ............................... 93 6CONCLUSION ..................................... 94 REFERENCES ........................................ 95 BIOGRAPHICALSKETCH ................................. 102 6

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LISTOFTABLES Table page 2-1Runtimesummary(millisec)fork=5 ......................... 34 2-2Runtimesummary(millisec)fork=10 ........................ 35 2-3Runtime(millisec)forcorrelatedcases,n=500,k=5 ................. 36 2-4Runtime(millisec)forcorrelatedcases,n=500,k=10 ................ 36 2-5Runtime(millisec)breakdown,stronglycorrelated .................. 37 2-6Runtime(millisec)breakdown,inversestronglycorrelated .............. 37 2-7Runtimebreakdown,almoststronglycorrelated ................... 38 3-1Runtime(millisec)summaryfortwodimensionalMKP ............... 50 4-1Transmissionlinecharacters .............................. 66 4-2Generatoroers .................................... 67 4-3Busdata ........................................ 67 4-4Sensitivityanalysis-budget(WIMP) ......................... 68 4-5Sensitivityanalysis-budget(WIMP+WRP) ..................... 68 4-6Sensitivityanalysis-installment(WIMP) ....................... 69 4-7Sensitivityanalysis-installment(WIMP+WRP) ................... 69 4-8SensitivityAnalysis-uncertaintyset ......................... 69 4-9118-busdata ...................................... 70 4-10118-busbasecase ................................... 71 4-11Protchangesunderworst-casescenario:4-candidate ................ 71 4-12Protchangesunderworst-casescenario:6-candidate ................ 72 5-1Busdata ........................................ 88 5-2Expecteddemandlevelandintensity ......................... 89 5-3Transmissionlinedata ................................. 89 5-4Generatorcapacityfactor ............................... 89 5-56-buscaseweightchange ............................... 90 7

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5-66-buscasewindblocksize ............................... 91 5-76-buscaseuncertaintyset ............................... 91 5-8118-busdata ...................................... 92 5-9118-busexpecteddemandlevelandintensity ..................... 92 5-10118-buscaseweightchange .............................. 92 5-11118-buswindblocksize ................................ 93 5-12118-buscaseuncertaintyset .............................. 93 8

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LISTOFFIGURES Figure page 2-1LinearsolutiontoKP ................................. 18 2-2UpdatedsolutiontoKP ................................ 18 2-3DPillustration ..................................... 20 2-4AllfeasiblesolutionsfromforwardDP ......................... 28 2-5GenerateCIsfromDP ................................. 29 4-1Windpowerinvestmentmodel:algorithmowchart ................. 65 4-2The6bussystem ................................... 66 5-1Windpowerandtransmissioninvestmentmodel:algorithmowchart ........ 87 5-2Themodied6bussystem .............................. 88 9

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyMIXEDINTEGERPROGRAMMINGAPPROACHESTO0-1KNAPSACKPROBLEMSANDUNIFIEDSTOCHASTICANDROBUSTOPTIMIZATIONONWINDPOWERINVESTMENTByKunZhaoAugust2015Chair:JosephHartmanCochair:YongpeiGuanMajor:IndustrialandSystemsEngineeringThisdissertationcoversatheoreticalstudyofcombiningdynamicprogrammingapproachwithcuttingplanestosolvethebinaryknapsackproblems.Inaddition,motivatedbythetheoreticalresearch,afurtherstudyisconductedonapplicationsinwindpowerinvestmentwithorwithouttransmissionexpansions.Inthetheoreticalstudy,forwarddynamicprogrammingisusedtogeneratecuttingplanesforanequivalentintegerprogrammingformulationofthebinaryknapsackproblem.Upperboundsfromintermediatestagesolutionsareusedtoderivecuttingplaneswhilesolutionsdeterminedfromexploring\infeasiblestates"generatetraditionalcoverinequalities.Theeectivenessisillustratedatsolvingbinaryknapsackproblemsthroughabranch-and-cutframework.Intheapplicationpart,shorttermwindpowerinvestmentproblemandlongtermsimultaneouswindpowerandtransmissioninvestmentproblemarestudied,wherethe0-1knapsackisinvolvedasthebudgetconstraintinthemixed-integerprogramformulation.Uniedstochasticandrobustoptimizationframeworkisproposedtosolvetheproblems,andBender'sdecompositionalgorithmsareutilizedtogenerateoptimalitycuts.Computationalexperimentsverifytheeectivenessofthesolutionapproaches. 10

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CHAPTER1INTRODUCTION 1.1TheKnapsackProblemsTheknapsackproblemisoneofthemostprominentfamiliesincombinatorialoptimization.Itiscommonlyusedinpracticalproblems,suchascapitalbudgeting,nancialmanagement,resourcesallocation,etc.Allproblemvariationsintheknapsackfamilyinvolvethedecisionofchoosingasubsetofdierentitemstobettedinoneormore\knapsacks".Thetypicalgoalistomaximizetheaggregatebenetgeneratedwithoutviolatingthespeciedcapacities.Withintheknapsackfamily,variousproblemdesignshavebeendevelopedbasedondierentapplications,amongwhichthesimplestyetthemostimportantproblemisthe0-1knapsackproblem.Inthe0-1knapsackproblem,itisassumedthatasetofitemsisgiven,eachwithaprotandaweight.Abasicbinarydecisionneedstobemadeoneachitemintheset,altogetherinducingacombineddecisionthatmaximizesthetotalprotwhilesatisfyingasinglecapacityconstraint.Extensionsofthe0-1knapsackproblemincludetheboundedknapsackproblem,whichincreasestheavailabilityofeachitemtoaboundedpositiveinteger,theunboundedknapsackproblem,whichallowstheintegerupperboundintheboundedknapsackproblemtogotoinnity,andthemultipleknapsackproblem,wherebinaryitemsarepackedintodisjointknapsacks,themultiple-choiceknapsackproblem,wheretheitemsarecategorizedintoseparatedsub-classes.Asoneofthedirectextensions,themulti-dimensionalknapsackproblem,whichsometimesistreatedasaspecialcaseofageneralintegerlinearprogram[ 1 ],isstudiedinthisdissertation.Thesinglecapacityconstraintisexpandedtomorethanonelevel,andthedicultyofsolvingtheproblemtooptimalityisincreasedsignicantly.ItiswellknownthattheknapsackproblemsareNP-hard[ 1 ].Totackletheseproblems,theexactapproachesoftenincludedynamicprogrammingorbranchandbound,eachhavingitsownstrengthsandlimitations.Studiesontheknapsackpolytope,relaxations,heuristics,preprocessingmethods,amongothers,shedlightsondeeperunderstandingoftheproblems,leadingtomoreecientapproaches. 11

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Inthisdissertation,thesingleconstraintandmulti-dimensionalbinaryknapsackproblemsarestudied.Chapter 2 studiesthe0-1knapsackproblem,whereanewapproachisproposed,togeneratethetraditionalcoverinequalitiesusingdynamicprogramming,andupperboundsaredenedbyintermediatestageoptimalsolutions.InChapter 3 ,thestudyisextendedtothemulti-dimensionalknapsackproblems. 1.2TheWindPowerInvestmentProblemInspiredbythecuttingplanestudiesonknapsackproblems,twoapplicationsonwindpowerinvestmentarepresented,whereBender'sdecompositionmethodisutilizedtogeneratecuttingplanes,andthe0-1knapsackproblemisincludedasasubproblemofmakinginvestmentdecisions.Targetingtoreducetheemissionofgreenhousegasesandpollutant,renewableenergysourceshavereceivedincreasingsupportworldwide.Havingtheworld'ssecondlargesttotalamountofrenewableelectricitygeneration,13%ofelectricityintheU.S.isgeneratedbyrenewableenergysourcesinyear2014,amongwhich,windpoweraccountedforthesecondlargestshareof34%,following48%ofhydropower.U.S.governmenthasestablishedvariousnancialincentives,andenforcedtargetstandardstopromoterenewableenergygeneration[ 2 ].Thoughthecostsofinstallingrenewableenergyhavedecreasedsubstantiallywiththetechnologydevelopmentovertheyears,renewableenergyinstallmentsarestillcapitalintensiveinvestmentsingeneral.Inaddition,theavailabilityofrenewableenergysourcesmayvaryonanannual,seasonal,orevendailybasis[ 2 ].Evenwithsubsidiesfromthepublicfunds,investinginrenewableenergycanberisky.Moreover,relativeissues,suchastransmissionexpansion,alsoneedtobeconsideredalong.Stochasticoptimizationisacommonapproachinmodelinguncertaintyintheelectricitymarket.Robustoptimization,ontheotherhand,hedgestheworst-casescenarioinanuncertaintyset.Theformermethodneedsalargenumberofscenariostocovertheexceptionalcases,whilethelattertendstoresultinrobust,yetconservativedecisions.Uniedmodels,therefore,areproposedtoobtainarobustdecisionwithscenariossimulated. 12

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Alongthislineofresearch,inthisdissertation,auniedstochasticandrobustoptimizationmodelonwindpowerinvestmentisproposedinChapter 4 .Chapter 5 extendstheassumptionstosimultaneousinvestmentinwindpowerandtransmissionexpansioninthelongterm. 13

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CHAPTER2DYNAMICPROGRAMMIGBASEDINEQUALITIESFORTHE0-1KNAPSACKPROBLEM 2.1MotivationandLiteratureReviewWeconsidera0-1knapsackproblem(KP),whereasetN=f1,2,...,ngofitemsarepackedintoaknapsackofcapacityc.Eachitemjhasprotpjandweightwj.Theobjectiveistomaximizethetotalprotoftheitemsincludedintheknapsack.Inmathematicalform: maxnXj=1pjxj:nXj=1wjxjc,,xj2f0,1g, (2{1) wherepj,wj,c2Zforallj2N,xj=1ifitemjisincludedandxj=0otherwise.Toeliminatetrivialcases,itisalsoassumedthateachpj>0,0c.KPisaclassicintegerlinearproblem,oftenincludedasasubproblemofmorecomplexproblems,suchasthemultipleknapsackproblem,generalassignmentproblem,orgeneralmixedintegerprogrammingproblems.KPisdenedasNP-hard,butcanbesolvedinpseudo-polynomialtimeusingdynamicprogramming(DP)[ 3 ].Inmanyrealworldapplications,KPisoftenstudiedasacapitalbudgetingproblem,packingproblem,orcuttingproblem,etc.[ 1 ].DierentapproachesforsolvingKPhavebeenstudiedovertheyears.Followingthegreedyalgorithm,whereitemsareaddedtotheknapsackindescendingorderofprotperunitweightratiountilthecapacityislled,thesolutiontothelinearprogrammingrelaxationoftheoriginalknapsackproblem(LKP)denesanupperboundontheoptimalintegersolution[ 4 ].DP,rstintroducedbyBellman[ 5 ],solvesKPexactlyinO(nc)time.However,whenproblemsizes,denedbynandc,andcoecients,pjorwj,arelarge,solutiontimeanddatastoragespacecangrowsignicantly.ComputationalimprovementshavebeendevelopedusingDPwithlists[ 6 ]anddivideandconquerforDP[ 1 ].Branchandboundalgorithms[ 7 ]alsondtheuniversaloptimalsolution,where\feasiblesolutionswereenumeratedimplicitly"[ 1 ].While 14

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theworstcaseruntimeforbranchandboundisO(2n),relaxationandmodicationsontheproblemcanlargelyimproveeciency.Theconceptofa\coreproblem"wasintroducedbyBalasandZemel[ 8 ].Basedontheknownoptimalsolutionx,thecoreoftheKPisdenedasasubsetCore=fa,...,bgN,wherea=minfj2Njxj=0gandb=maxfj2Njxj=1g,(assuminga
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Combinationsandextensionsofbasicalgorithmshavebeenshowntobeecient,suchasbranchandboundcombinedwithanupperboundbyMartelloandToth[ 7 , 24 ],abranchandboundcombinedwiththecorealgorithmbyMartelloandToth[ 25 ]andPisinger[ 26 ],DPcombinedwiththecorealgorithmbyPisinger[ 27 ],andthecombinationofDP,coreandboundsfrombranchandboundbyMartelloetal.[ 28 ].However,wearenotawareofanyresearchcombiningDPwiththetraditionalseparationalgorithm.Inthischapter,westudythesharedpropertiesofthesetwobasicapproaches.InspiredbyHartmanetal.[ 29 ],wepresentanewwayofgeneratingtraditionalcoverinequalitiesfromstagesofanequivalentDP,andderiveanewsetofvalidinequalities.Theseareimplementedinacut-and-branchsolutionprocedureforKP.Themethodistestedonuncorrelated,weaklycorrelated,stronglycorrelated,inverselystronglycorrelated,andalmoststronglycorrelatedprobleminstances.Computationalresultsshowthattheapproachiseective.Thechapterisorganizedasfollows.InSection 2.2 ,wereviewtherelateddenitionofcoverinequalities,IPseparationalgorithm,anddynamicprogrammingapproaches.InequalitiesandalgorithmsarederivedinSection 2.3 andSection 2.4 showsthecomputationalresults. 2.2TheoryReview 2.2.1IPSeparationAlgorithmConsiderKP( 2{1 ).AsetCNisacoverfor( 2{1 )ifitstotalweightexceedsknapsackcapacity,i.e.,Pj2Cwj>c.GivenanycoverC,thecoverinequality(CI)isdenedas Xj2CxjjCj)]TJ /F5 11.955 Tf 17.94 0 Td[(1, (2{2) andisvalidforallfeasibleintegersolutionsoftheKP[ 10 ].Acover,C0,isaminimalcoverifnoneofitspropersubsetisacover,or: Xj2C0nfkgwjc,8k2C0. (2{3) 16

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Theinequalitygeneratedfromacover,C,isgenerallyweakerthanthatfromaminimalcover,C0C,for: nx2f0,1gn:Xj2C0xjjC0j)]TJ /F5 11.955 Tf 17.93 0 Td[(1onx2f0,1gn:Xj2CxjjCj)]TJ /F5 11.955 Tf 17.93 0 Td[(1o. (2{4) TheclassicseparationalgorithmbeginswithsolvingtheLPrelaxation(LKP)of( 2{1 ).IftheLPsolutionisintegral,itisalsothesolutionto( 2{1 ).Inmostcases,theLPsolutionisfractional.Crowderetal.[ 14 ]denedtheseparationproblemasaminimizationproblemoftheform: =minXj2N(1)]TJ /F4 11.955 Tf 11.96 0 Td[(xj)zj:Xj2Nwjzj>c,zj2f0,1g, (2{5) whichisalsoaknapsackproblem.DPecientlysolves( 2{5 )ifthecoecientswjarenottoobig.Iftheresultingoptimalobjective<1,somezj=1hasacoecient(1)]TJ /F4 11.955 Tf 12.1 0 Td[(xj)<1,andthefollowinginequalityisthemostviolatedCIatthecurrentsuboptimalsolution: Xj2CzjxjjCj)]TJ /F5 11.955 Tf 17.94 0 Td[(1,whereC=fj2N:zj=1g. (2{6) Ateachiteration,themostviolatedCIisaddedtothelinearrelaxationof( 2{1 ).TheupdatedLPisthenre-solved,andthesolutionisimprovedby1)]TJ /F10 11.955 Tf 12.3 0 Td[(.If1,thecurrentsuboptimalsolutionxsatisesallcoverinequalitiesandreachesthebestsolutionpossibleusingtheCIseparation. Example Considerasimpletwovariableproblem: maxf15x1+7x2:5x1+3x26,xi2f0,1ggTheinitialsolutiontotheLKPisx1=1andx2=1=3,asshowninFigure 2-1 .Aftertherst(andonly)iteration,thecoverinequalityx1+x21isadded.Theupdatedsolutionthenbecomesx1=1andx2=0.Figure 2-2 showsthefeasibleregionafteraddingtherst(andonly)coverinequality. 17

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Figure2-1. LinearsolutiontoKP Figure2-2. UpdatedsolutiontoKP 18

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SincesolvingforexactCIcanbecostly,aheuristiccanbeusedtodeneabaseCIandperformliftingtechniquestoimproveitsstrength. 2.2.2DynamicProgrammingApproachTheDPapproachtosolving( 2{1 )isthroughtheBellmanrecursion[ 5 ].Thestateisdenedasutilizedcapacityintheknapsack,=0,...,c,withitemsr2Nasstages.Theintermediatestager,stateproblemisdenedasfollows: fr()=maxrXj=1pjxj,s.t.rXj=1wjxj,xj2f0,1g,j2N (2{7) Theintermediateoptimalobjectivevaluescanbeobtainedbyrecursion: fr()=maxffr)]TJ /F8 7.97 Tf 6.58 0 Td[(1(),cr+fr)]TJ /F8 7.97 Tf 6.58 0 Td[(1()]TJ /F4 11.955 Tf 11.96 0 Td[(wr)g,forr=1,...,n. (2{8) wheref0()=0,8=0,...,c.Sincecandrn,thereareanitenumberofoperations,andDPsolvesexactlyinO(nc).AsolutionS1NisdominatedbyanothersolutionS2N,ifPj2S1pjPj2S2wj.AsisstatedbyKellar,etal.[ 1 ],adominatedsolutionwillnotresultinanyoptimalsolutionsofKP.Atsomestagerandstate,ifthereexiststwosetsS1andS2,withmaxfS1g=maxfS2g=r,also=Pj2S1wj=Pj2S2wj,andPj2S1pjfr)]TJ /F8 7.97 Tf 6.59 0 Td[(1(),xr=1and0otherwise.Wethencontinuetondtheoptimalsolutiontofr)]TJ /F8 7.97 Tf 6.58 0 Td[(1()]TJ /F4 11.955 Tf 12.66 0 Td[(xrwr).Thisprocessisrepeateduntiltheresidualweightequals0.Forinstance,considerasimplethree-variableproblem: maxf2x1+x2+3x3,s.t.2x1+2x2+x33g. 19

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Figure2-3. DPillustration Figure 2-3 plotsitems(orstages)onthehorizontalaxisagainstweights(orstates)ontheverticalaxis.Startingfromtheorigin,whichrepresentstheemptyknapsack,item1iseitherplacedintheknapsack(weightincreasedtow1=2)ornot(weightstaysat0),sopoints(1,2)and(1,0)indicatethesetwocases.Similarly,ateachstager3,startingfromeachexistingpositions(r)]TJ /F5 11.955 Tf 12.8 0 Td[(1,),twopointsarecreated,indicatingwhetheritemrisselectedontopofstager)]TJ /F5 11.955 Tf 13.14 0 Td[(1positionornot.Linesegmentsconnecting(r)]TJ /F5 11.955 Tf 13.15 0 Td[(1,)andpoints(r,+wr)and(r,)trackthepathofhowstagerpositionsareachieved.Inthisway,wecangraphallpossiblesolutionpathstothisproblem.Theobjectivefvalues,calculatedfromBellmanrecursion,aremarkedbesideeachpoint.Atpoint(2,2),bothsetsfx1gandfx2ghavethesameweight,butfx1ghashighervalue,sosetfx2ghereisdominatedbyfx1g,andintermediateoptimalobjectiveis2.Ahorizontallineisdrawnattheknapsackcapacity.Ifapoint,forinstancepoint(2,4),isabovethehorizontalline,thepositionisinfeasible.Atthenalstage,thehighestfvalueofallpointsonorbelowthecapacitylinegivestheoptimalobjectivevalue.Inthiscase,theoptimalvalueisobtainedatpoint(3,3)withf=5.Wethencantrackfromthispointbackandndtheoptimalsolutionfx1,x3g. 20

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2.3InequalityDerivation 2.3.1CoverInequalitiesfromDPInfeasibleStatesWeobservethatCIscanalsobegeneratedbysolvingtheDPofKPforstatesbiggerthanknapsackcapacity.IftheoptimalsolutionxatstateisinfeasibletoKP,i.e.: Xj2Cwjxj=>c,C=fj2N:xj=1g,avalidCItoKPcanbedenedas Xi2CxijCj)]TJ /F5 11.955 Tf 17.93 0 Td[(1.Thus,wecangeneratemultipleCIsbysolvingthroughoneDPstage. Lemma1. Solving( 2{8 )tooptimality,forallstates=1,...,Pnj=1wj,andallstagesr=1,...,n,denesallcoverinequalitiestoKP. Proof. DPtracksintermediatestageoptimalsolutionsforeachstate=1,...,Pnj=1wjineachstager2N.Ateachstate>cinanystage,iftheoptimalsolutionL0Nisnottheemptyset,Pj2L0wj=>c.Bydenition,L0denesacoverofcapacity.Firstassumethatallintermediateoptimalsolutionsareunique,i.e.atanystate, if9L0,~L0Ns.t.Xj2L0wj=Xj2~L0wj=,thenL0=~L0. (2{9) Bygoingthrough( 2{8 ),wehaveessentiallyrecordedallpossiblesubsetsofN,andtherefore,denedallcovers.However,dominancerelations,asdescribedinSection 2.2 ,occurofteninlargeKPproblems.Inthiscase,thedominatedsolutionscanbekeptinaretrievallist.InadditiontoBellmanrecursiondeningCIsfromintermediateoptimalsolutions,wecanloopthroughtheretrievallistandalsodeneCIsfromthedominatedsolutions.Inthisway,allitemcombinationswillalsohavebeenloopedthroughattheendofDP.Thatis,wecanalsodeneallpossibleCIs. 21

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ThesetofallcoversforagivenKPincludeallminimalandnon-minimalcovers.Foreciency,weareonlyconcernedwithminimalcovers.Thisisaddressedinthefollowingtheorem. Theorem2.1. SolvingDPthroughstate(c+wr)ateachstager2NdenesallminimalcoverstoKP. Proof. FromLemma 1 ,weconcludethatwecanidentifyallcoversthroughsolvingDPforallstatesinallstagesandkeepingaretrievallistfordominatedsolutionsifnecessary.Itisthensucienttoshowthat,ifacover,Cnew,isgeneratedinstageratastate>c+wr,itisnotaminimalcover.IfcoverCnewisdenedatstager,thenr2Cnew.Anycoverinf1,2,...,rginwhichrisnotanelementwouldhavebeendenedinapreviousstage.ConsiderthesubsetCnewnfrg,whichsatises: Xj2Cnewnfrgwj=)]TJ /F4 11.955 Tf 11.95 0 Td[(wr>c.Thatis,thesubset,CnewnfrgCnew,isalsoacover.Thus,Cnewisnotaminimalcover.Therefore,nostate>c+wrinstagercangenerateaminimalcover. Corollary2.1.1. Sortingallitemsbydescendingprot-weightratio,i.e.: p1 w1p2 w2pn wn. (2{10) DPdenestheCIgeneratedfromtherstiterationoftheseparationproblem( 2{5 )inthestageofthesplititem. Proof. ThesolutiontotheLKPrelaxationisgreedy.IftheLKPsolutionisnotintegral,asin[ 1 ],denetheindexofthesplititemas: m=maxk2N:kXj=1wj
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TheoptimalLKPsolutionhastheform: x=(1,...,1,d,0,...,0),where d=(c)]TJ /F6 7.97 Tf 11.95 14.94 Td[(m)]TJ /F8 7.97 Tf 6.59 0 Td[(1Xj=1wj)=wm2(0,1).Tosolvetheseparationproblem( 2{5 ),itisobviousthateachxj=1,costs1)]TJ /F4 11.955 Tf 12.09 0 Td[(xj=0forzj,sotheoptimalsolutionwillassignzj=1.However,onlyincludingtheseitemsdoesnotdeneacover,forPm)]TJ /F8 7.97 Tf 6.59 0 Td[(1j=1wjc.Since1)]TJ /F4 11.955 Tf 11.45 0 Td[(xm=1)]TJ /F4 11.955 Tf 11.45 0 Td[(d<1,while1)]TJ /F4 11.955 Tf 11.44 0 Td[(xj=1,foralljm+1,and: mXj=1wj>m)]TJ /F8 7.97 Tf 6.58 0 Td[(1Xj=1wj+dwm=m)]TJ /F8 7.97 Tf 6.59 0 Td[(1Xj=1wj+c)]TJ /F7 11.955 Tf 11.95 8.96 Td[(Pm)]TJ /F8 7.97 Tf 6.59 0 Td[(1j=1wj wmwm=c,theoptimalsolutionwillassignzm=1.Theseparationproblemhasoptimalobjectivevalue=1)]TJ /F4 11.955 Tf 11.95 0 Td[(d<1.Therefore,therstCIfromsolving( 2{5 )is: mXj=1xjm)]TJ /F5 11.955 Tf 11.95 0 Td[(1. (2{11) Ontheotherhand,DPstartstogenerateCIsatstagem,whentheknapsackislledforthersttime: m)]TJ /F8 7.97 Tf 6.59 0 Td[(1Xj=1wjc.Atstagem,Pmj=1wj
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Also,theremainingstatesPmj=1wj+1cannotbereachedatstagem,sonocoverisgeneratedafter( 2{11 ).Therefore,( 2{11 )istheonlyCIgeneratedbyDPthroughstagem. Corollary2.1.3. AteachstagersuchthatPrj=1wj>c,denethesetofallcoversthatcanbegeneratedatstagerasCr=fCiN:Pj2Ciwj>cg.TheCIdenedfromtherstcover,Cm,whichsatises: Xj2CmwjXj2Ciwj,forallCi2Cr (2{12) isaminimalcover. Proof. Therstcover,Cm,generatedatstagerwillsatisfy( 2{12 )andoccurinstateminf:=Pj2Ciwj>c,Ci2Crg.SinceCmNisnite,thereexistsq2Cmsuchthatwqwjforallj2Cm.DeletingelementqfromCm,theremainingsetofCmisthesubsetwithlargesttotalweight.However: Xj2Cmnfqgwj=Xj2Cmwj)]TJ /F4 11.955 Tf 11.96 0 Td[(wqc,and Xj2C0wj=minXj2CNwj>c. 24

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SincePm)]TJ /F8 7.97 Tf 6.58 0 Td[(1j=1wjc,maxfC0g=m,andsoC0f1,2,...,mg.Asf1,2,...,mgisnotaminimalcover,thereexistsk2f1,2,...,m)]TJ /F5 11.955 Tf 11.96 0 Td[(1g,suchthat Xj2f1,...,mgnfkgwjXj2C0wj>c,andsoC0isapropersubsetoff1,...,mg.Corollary 2.1.1 hasshownthattheCI( 2{11 )inducedfromthecoverf1,...,mgisalsodenedinstagem.Thatis,DPgeneratesbothCIsfromC0andf1,...,mg.Thatis,atleasttwoCIsaregeneratedinstagem.Notethat1)]TJ /F4 11.955 Tf 12.31 0 Td[(xj=0forallf1,...,m)]TJ /F5 11.955 Tf 12.31 0 Td[(1g,theresultingobjectivevalueoftheseparationproblem =Xj2C0(1)]TJ /F4 11.955 Tf 11.95 0 Td[(xj)=1)]TJ /F4 11.955 Tf 11.96 0 Td[(d<1.Thus,theCIdenedfromC0isviolatedbythesameamountof( 2{11 ),buthasamathematicallystrongerform. ConcludingfromCorollary 2.1.1 to 2.1.4 ,therstCIfromDPisthesameorstrongerthantherstCIfrom( 2{5 ). Corollary2.1.5. Supposetheintermediateoptimalsolutionx=(x1,...,xr)atstagerandstate=maxf=Pj2f1,...,rgwjxj;
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Proof. IftheremainingitemssatisfyPnj=r+1wjcr,thetotalweightofitemsincludedintheknapsackwillnotexceedcbysettingxj=1forallj=r+1,...,n,i.e., rXj=1wjxj+nXj=r+1wjrXj=1wjxj+cr=cClearly,nomoreminimalCIswillbegeneratedinthefuturestages. 2.3.2ProtInequalitiesWecandirectlygenerateanothersetofinequalitiesusingintermediatestageoptimalsolutionsfromearlyDPstages. Theorem2.2. SupposetheDPintermediateoptimalobjectivevalueatstagerisfr.Thefollowinginequality: rXj=1pjxjfr (2{13) isvalidforallfeasiblesolutionsofKP. Proof. AnyfeasiblesolutiontoKP,xf,satisesthefollowinginequality: rXj=1wjxfjnXj=1wjxfjc.Therefore,ther-vectorfxf1,...,xfrgisfeasibletothestagersubproblem: maxrXj=1pjxj:rXj=1wjxjc,xj2f0,1g.Sincefristheoptimalobjectivevalueatstager,itfollowsthat: rXj=1pjxfjfr. 26

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Theorem2.3. SupposetheDPintermediateoptimalobjectivevalueatstagerisfr,thefollowinginequality: nXj=1pjxjfr (2{14) isvalidforalloptimalsolutionsofKP. Proof. WeknowthattheoptimalobjectiveforKP,f,isanupperboundforallintermediateoptimalobjectivevalues,i.e., ffr,8r2N (2{15) Consideringanyoptimalsolutionx=fx1,x2,...,xng,itsatises: nXj=1pjxj=f (2{16) Therefore,combining( 2{15 )and( 2{16 ),inequality( 2{14 )holdsforalloptimalsolutions. Corollary2.3.1. Afeasiblesolutionx02f0,1gnsatisesallinequalities( 2{13 )and( 2{14 )forallstagesr2N,ifandonlyifx0isintheoptimalitysetofKP. Proof. Supposex0=fx01,x02,...,x0ngisintheoptimalityset.Sincex0isafeasiblesolution,fromTheorem2,x0satises( 2{13 ).Sincex0isanoptimalsolution,fromTheorem3,x0satises( 2{14 ).Thatis,x0satisesallproposedinequalities.Nowsupposex0=fx01,x02,...,x0ngsatisesinequalities( 2{13 )and( 2{14 )forallstages,thenitsatises: nXj=1pjx0jfr,andrXj=1pjx0jf,8r2NSincetheintermediatestageoptimalobjectivevaluesfromforwardDP,frvaluesarenondecreasingandconvergetofasrgoeston,i.e., 9k2N,s.t.8krn,fr=f. 27

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Thatis, nXj=1pjx0jfandnXj=1pjx0jfTherefore,Pnj=1pjx0j=f,i.e.,x0isintheoptimalitysetofKP. 2.3.3ANumericalExampleConsiderthefollowingfour-variableproblem: maxf15x1+7x2+10x3+4x4:5x1+3x2+6x3+4x49,xi2f0,1gg.ThesolutiontotheLKPisx1=x2=1,x3=1=6,x4=0,withobjectivevalue232=3.ForwardDPgeneratesCIandprotinequalities( 2{13 )and( 2{14 )atstage3,wherethestage3objectiveis22,withx1=x2=1,x3=0.Since22<32(=15+7+10),theprotinequalitiesaredenedas: 15x1+7x2+10x322and15x1+7x2+10x3+4x422Figure 2-4 showspathsforallfeasiblesolutionsoftheKP.Thecircledpointistheoptimalatstage3,fromwhichtheprotinequalitiesweredened. Figure2-4. AllfeasiblesolutionsfromforwardDP 28

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Figure2-5. GenerateCIsfromDP TherstcoverdenedbyextendingtheDPalgorithmbeyondtheknapsackcapacityisfx1,x3g,with=5+6=11>9,sothattherstCIis: x1+x31.ThisCIisvoilatedby: 1)]TJ /F7 11.955 Tf 17.5 11.36 Td[(Xj2f1,3g(1)]TJ /F4 11.955 Tf 11.96 0 Td[(xj)=1 6.Continuinginstage3,thesetfx1,x2,x3gwithatotalweightof14>9isalsorecognizedbyDPasacover,andthesecondCIisdenedas: x1+x2+x32.Figure 2-5 showsthepathsdeningcoversthatcanbegenerateduptostatec+wrateachstager.ThecircledpointsdenethetwoCIsatstage3.Addingtheseinequalitiestothe 29

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LKP,theupdatedsystemusingallstage3informationoftheDPbecomes: max15x1+7x2+10x3+4x4s.t.5x1+3x2+6x3+4x4915x1+7x2+10x32215x1+7x2+10x3+4x422x1+x31x1+x2+x32xi2[0,1],i=1,2,3,4Theupdatedsolutionis: x1=x2=1,x3=0,x4=0.25,withobjectivevalue23.Nowconsidertheseparationalgorithm( 2{5 ).Therstseparationproblemisdenedas: =min4Xj=1(1)]TJ /F4 11.955 Tf 11.96 0 Td[(xj)zj,s.t.5z1+3z2+6z3+4z4>9,zj2f0,1g (2{17) Thesolutiontotherstiterationis: z1=z2=z3=1,z4=0,with1=5 6.ThentherstCIisdenedasx1+x2+x32,anditisviolatedby1)]TJ /F10 11.955 Tf 12.08 0 Td[(1=1=6.NotethattherstCIsgeneratedusingSAandDPbothimprovethesolutionby1/6,whiletheCIfromDPhasamathematicallystrongerform.AddingtherstCItotheLKP,theupdatedlinearsolutionis: x1=1,x2=2 3,x3=1 3,x4=0, 30

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withobjectivevalue23.Solvingtheseparationproblem( 2{17 )again,thesolutiontotheseconditerationis: z1=z3=1,z2=z4=0,with2=2 3.ThenthesecondCIisdenedasx1+x31,anditisviolatedby1)]TJ /F10 11.955 Tf 12.8 0 Td[(2=1=3.Notethatseparationalgorithm( 2{17 )takestwoiterationstogeneratetwoCIs,whilebothCIsaredenedbysolvingthroughstage3ofDP. 2.4ComputationalResultsThissectionpresentsthecomputationalperformanceofthetechniquesdescribedinSection 2.3 usingrandomlygeneratedinstances.AllexperimentswerecodedinJavawithCPLEX12.5usingConcert2.9technologyonanIBMSystemx3650withtwoIntelE5640Xeonprocessorsand24GBmemory.Atimelimitof5minuteswassetforallexperiments. 2.4.1ImplementationIftheoptimalsolutiontotheLKPwasfractional,wegeneratedprotandcoverinequalitiesthroughpartiallysolvingtheforwardDP( 2{8 ).AllinequalitieswereaddedtotheoriginalKPandthesystemwassolvedusingCPLEXdefaultIPsolver.TodeneCIs,weareonlyinterestedinminimalcovers.Oncewedenetheminimalcovers,theCIsfromnon-minimalcoversareredundant.FromTheorem1,allminimalcoverswillbedenedbystatec+wrineachstager2N.Insteadofsolvingthroughallstates=1,...,Pnj=1wj,westoppedatstatec+wrofeachstager.Instates>cwhereCIsweredened,thesolutionsnolongercontributedtooptimality,sowedidnotrecordtheoptimalobjectivevaluebeyondtheknapsackcapacityc.Notethattheprotinequalitiesprovidedusefulrestrictionswhentheknapsackcouldnotincludealltheitemsthroughstager,sotheinequalities( 2{13 )and( 2{14 )wereaddedtothesystematstagesrm,wheremwasdenedastheindexofthesplititem.Moreover,sincetheintermediatestageoptimalsolutionswerenotupdatedineverystage,toavoidredundancy,inequalities( 2{13 )weregeneratedinthestageswheretheintermediatestageoptimalobjective 31

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valueincreased,i.e.,iffrwasupdatedinstager,thefollowinginequalitywasdened: Xjr)]TJ /F8 7.97 Tf 6.59 0 Td[(1pjxjfr)]TJ /F8 7.97 Tf 6.58 0 Td[(1.WeexpectthesefeasibilitycutstobemorerestrictiveinlaterDPstages.Inthestoppingstage,wegeneratedthelasttwoprotinequalities( 2{13 )and( 2{14 ),andthiswastheonlytimeinequality( 2{14 )wasadded.TwosetsofstoppingcriteriawereconsideredforDP.OnesetofexperimentsranDPthroughaconstantfractionofallstages,bn=3corbn=2c.Fractionofstagesgetslargerasthetotalnumberofstagesdoes.Theothersetofexperimentsrecordedthestageofthesplititemm,andstoppedatstagesb4m=3candb3m=2c,forusefulinequalitiesweregeneratedwhentheknapsackcapacitycouldbeexceeded.ThesmallernumberofDPstagestraversed,thefewernumberofinequalitiesweregenerated,buttheshortertheDPruntime.Ingeneral,thereisatradeobetweentheruntimeandtheamountofinformationweget.InSection 2.3 ,itwasshownthatwewereabletodeneallminimalcovers.However,intheexperiment,wefoundthatinordertodeneallminimalCIsfromeachstage,wehadtodenealotredundantconstraintswhichlargelyincreasedsolverpreprocessingtime.ItwasmoreecienttoaddtherstCIsfromeachstage,whichwereguaranteedtobeminimal.Therefore,ateachstager,westoppedatthestatewheretherstCIwasdenedorstatec+wjwithoutdeninganyCI.Inaddition,keepingretrievallistsforalldominatedsolutionswastimeconsuming,especiallyforlargeproblemsizes,sowedidnotkeeparetrievallistintheexperiments.Theresultsfromthefollowingmodelswererecorded: Default:CPLEXIPdefaultsolver; U:onlyinequalities( 2{13 )wereadded; L:onlyinequalities( 2{14 )wereadded; Cover:onlyrstCIswereadded; 32

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UCvr:inequalities( 2{13 )andCIswereadded; LCvr:inequalities( 2{14 )andCIswereadded; UL:inequalities( 2{13 )and( 2{14 )wereadded; All:CIs,inequalities( 2{13 )and( 2{14 )werealladdedtoKP.WerecordedtheDPtimeofgeneratinginequalities,andtheCPLEXdefaultIPsolvertimeforallmodels.Sincemaxj2Nfwjgc,theworstcasecomplexityfortheDPpartofourmodelsisO(n(c+maxj2Nfwjg))=O(nc). 2.4.2ExperimentalDesignOurexperimentsfollowedtheinstancechoicesdescribedinMartelloetal.[ 28 ].Fiveproblemcaseswereconsideredinourstudies,eachtestedwithfourdierentproblemsizes,n=50,100,200,and500,andtwolevelsofcapacity,overdatarangeofR=1000. Uncorrelated:bothpjandwjindependentlyanduniformlygeneratedin[1,R]; Weaklycorrelated:wjrandomlygeneratedin[1,R],andpj2[wj)]TJ /F4 11.955 Tf 12.28 0 Td[(R=10,wj+R=10]whilepj1; Stronglycorrelated:wjrandomlygeneratedin[1,R],andpj=wj+R=10; Inversestronglycorrelated:pjrandomlygeneratedin[1,R],andwj=pj+R=10; Almoststronglycorrelated:wjrandomlygeneratedin[1,R],andpj2[wj+R=10)]TJ /F4 11.955 Tf 11.95 0 Td[(R=500,wj+R=10)]TJ /F4 11.955 Tf 11.95 0 Td[(R=500].Foreachcasewithproblemsizen,teninstancesweretestedoneachofthetwolevelsofcapacity(roundeddowntotheclosestinteger),c=1 knXj=1wj,wherek=5,10Inall,wehadatotalof400instances. 2.4.3ResultsSummaryAsPisinger[ 30 ]described,themorecorrelatedinstancesandbiggerproblemsweregenerallymorediculttosolve.CPLEXIPsolverwasecientinsolvingfairsizedKPs.Theruntimesforallmodelswerelongerforlargerproblems.Inourexperiments,atimelimitof 33

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5minuteswasimposedasanupperboundforallinstances.Usingthedefaultsolver,9outof400instanceshitthetimelimit,whileallotherinstancesweresolvedtooptimality.All9instanceshadinverselystronglycorrelateddata,amongwhichoneinstancehadn=200variables,andtheothereightinstanceshadn=500variables. Table2-1. Runtimesummary(millisec)fork=5 CasesnDefaultBestDP+IP%ImproveBestModelEqual Uncorrelated5025.39.960.87%LUL,LCvr,Cvr10028.215.843.97%UAll,L,LCvr20033.138.9-17.52%Default-50089.9113.5-26.25%Default-Weakly5029.916.644.48%All-Correlated1003534.61.14%LCvrL200102.964.737.12%ULU500106.5104.81.60%U-Strongly5029.613.454.73%Cvr-Correlated10051.712.675.63%UUCvr20064.528.755.50%UAll,UL500228.892.459.62%UL,ULInverse5053.8688.85%L-Strongly100201.331.184.55%AllULCorrelated2001519.914.199.07%ULLCvr500120024.659.399.95%ULLAlmost504222.646.19%ULUStrongly100115.621.181.75%L-Correlated200136.12879.43%U-500226.7128.443.36%UUL Table 2-1 and 2-2 presenttheruntimesummaryforallproblemsizesandallcasesunderbothcapacitylevels.ColumnDefaultrepresentsthedefaultCPLEXIPsolvertime.ColumnBestDP+IPrepresentthetotalruntime,i.e.,theDPtimeforgeneratinginequalitiesplustheIPsolvertime,ofthebestperformingDPmodelsforeachcase.Ourbestperforminginequalitiesaddedmodelshaveshorteraverageruntimesthanthedefaultsolverin38outof40cases.Column%Improvecalculatesthepercentageofimprovementinruntime,i.e.: %Improve=(Default)-(BestDP+IP) (Default) 34

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Ourmodelshadmoderateperformanceinlargeruncorrelatedorsomeweaklycorrelatedcases.However,inthemorecorrelatedcases,i.e.stronglycorrelated,inversestronglycorrelatedandalmoststronglycorrelatedcases,thepercentageimprovementinruntimeareallwellabove40%,underbothcapacitylevelsandforallproblemsizes.Whencapacitydecreasesfrom1=5to1=10ofthetotalweights,problemsizesaresmallerandDPrunsfasterforcalculatingthesamenumberofstages,sotheimprovementbyaddingDPbasedinequalitiesismoreobviousinmostcases.Forinstance,inthen=500stronglycorrelatedcase,the%Improveincreasesfrom59.62%to71.87%,whenkincreasesfrom5to10.Similarly,%Improveincreasesfrom43.36%to71.89%whencapacitydecreasesforthen=500almoststronglycorrelatedcase. Table2-2. Runtimesummary(millisec)fork=10 CasesnDefaultBestDP+IP%ImproveBestModelEqual Uncorrelated50227.565.91%UL-10017.31042.20%ULAll,L,LCvr,U20027.527.30.73%UAll,UL50065.563.82.60%U-Weakly5024.412.349.59%AllULCorrelated10036.230.914.64%AllUL,LCvr20053.944.517.44%U-500120.668.942.87%U-Strongly5026.98.668.03%All-Correlated10061.11182.00%UUL200109.914.686.72%ULU,L500196.655.371.87%UULInverse5023.79.460.34%ULAllStrongly100249.78.696.56%L-Correlated2003017262.599.79%LLCvr500120547.652.299.96%UL-Almost5052.827.448.11%AllULStrongly10085.531.563.16%U-Correlated20091.226.970.50%U-50020156.571.89%ULU,Cvr ColumnBestModelidentiesthebestperformingmodels,i.e.,modelswiththeshortestruntimes,forallcase.ColumnEqualprovidesmodelshavingruntimeswithinminf10%,10millisecondsgabovetheshortestruntime.Sincetheruntimedierencesfromthebest 35

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performingmodelsaresmall,thesemodelsareconsideredasgoodasthebestperformingmodels. Table2-3. Runtime(millisec)forcorrelatedcases,n=500,k=5 CasesDefaultDPAllCvrLLCvrUUCvrUL Strongly228.8186.3107108.995.110792.4106.195.1Inverse120024.6277.469.6270076.959.767.3150280150319.659.3Almost226.7175.1160.1154.6144.8157.3128.4158.5136.8 Fromtheresults,addingupperandlowerboundstogetherisoneofthemostecientmodelsin21outof40cases,and20casesindicatethataddingupperboundsaloneisecient.Thesetwomodelsarethemostfrequentbestmodels.Addingallthreetypesofinequalitiesoraddinglowerboundsaloneoccursasoneofthefastestmodelsin11cases.AddingCIsandlowerboundstogetherappearstobeecientin6cases,andaddingCIsalonein3cases.OnecaseindicatesthataddingCIsandupperboundstogetherisecient,whileinthatcase,thefastestmodelistoaddupperboundsalone.SinceweonlyaddtherstCIfromeachstage,andCPLEXdefaultsolveralsogeneratesminimumCIs,thebestmodels,inmostcases,haveCIsaddedtogetherwithprotinequalities.TheperformanceofmodelswithCIsaddedaloneisexpectedtohighlyreecttheperformanceofthedefaultsolver. Table2-4. Runtime(millisec)forcorrelatedcases,n=500,k=10 CasesDefaultDPAllCvrLLCvrUUCvrUL Strongly196.693.763.575.564.266.555.365.560.3Inverse120547.6138.658.3120547.6575.1579.160148.760152.152.2Almost20189.765.361.964.363.557.264.356.5 Table 2-3 andTable 2-4 comparetheruntimesfromallmodelsforthestrongly,inversestrongly,andalmoststronglycorrelatedn=500casesunderbothcapacitylevels.RuntimesfromCPLEXdefaultsolverandDPapproachareconsiderasbenchmarks.Defaultsolverruntimes,shownincolumnDefault,dependonthegeneraldicultyoftheproblems,whileDPruntimes,shownincolumnDP,aremostlyinuencedbytheproblemsizes.Inthecasesshown,runtimesfromDPapproachareshorterthanCPLEXdefaultsolver,whileruntimesfrommodelsAllandULareshorterthanDPinallthreecasesforbothcapacitylevels,with 36

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ULperformingslightlybetterthanAll.Forstronglycorrelatedandalmoststronglycorrelatedcases,modelUprovidesthemostimprovementinruntimesunderbothcapacitylevels,butintheinversestronglycorrelatedcase,theupperboundsneedtopairwiththelowerboundsforbetterperformance.TheDPsolverreachesthetimelimitinmanycases. Table2-5. Runtime(millisec)breakdown,stronglycorrelated CapacityModelStopFractionDPTimeIPTime k=5LFracN1/371.223.9UFracN1/371.520.9ULFracN1/371.523.6k=10UFracN1/346.88.5ULFracN1/350.210.1 Table 2-5 , 2-6 ,and 2-7 providefurtherdetailsofthebestperformingmodelsfromTable 2-3 and 2-4 .ColumnStopprovidesthestoppingcriteriaforDPthathavementionedbefore.Thestoppingcriteriavaryindierentcases.FracNstopsDPataconstantfractionofallstages,andStage+stopsDPrelativetothestageofsplititem.ColumnsDPTimeandIPTimeprovidetheruntimesofDPandIPsolverparts,respectively. Table2-6. Runtime(millisec)breakdown,inversestronglycorrelated CapacityModelStopFractionDPTimeIPTime k=5AllStage+1/358.411.2LStage+1/348.511.2LCvrStage+1/358.39ULStage+1/3509.3k=10AllStage+1/215.442.9ULStage+1/212.439.8 Forstronglycorrelatedcases,asshowninTable 2-5 ,DPtakesthemajorityoftheruntimeforbothcapacitylevels.Allmodelsrunthough1=3ofthetotalnumberofstages,insteadof1=2ofallstagestosametimeontheDPparts.Foralmoststronglycorrelatedcases,DPpartstakemuchlongerthanIPpartinallcases.Forcapacitylevelk=5,theDPtimesforrunningthrough4m=3stagesor1=2allstagesareclose.EachstoppingcriterionperformsslightlybetterwithmodelUandUL,respectively.Forcapacitylevelk=10,modelU 37

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indicatesthatbothStage+andFracNresultinshortestruntime.Thatis,4m=3stagesand1=3allstagesareclose. Table2-7. Runtimebreakdown,almoststronglycorrelated CapacityModelStopFractionDPTimeIPTime k=5UStage+1/3117.111.3ULFracN1/2125.711.1k=10CvrFracN1/352.59.4UStage+/FracN1/348.29ULFracN1/346.410.1 Forinversestronglycorrelatedcases,DPtakesmajorityofruntimefork=5cases,whileIPtakesmajorityofruntimefork=10cases.Allmodelsusethesplititemasastoppingindicator.Allk=5modelsrunthrough4m=3stagesofDP,whereenoughinequalitieshavebeendenedforreducingIPsolvertime,anditisbettertostopearlyandsavetimeonDPparts.Thebestmodelsfork=10casesrunthrough1=2ofallstagestodenemoreinequalities,forDPpartisfastenoughunderthesmallercapacity.Ingeneral,thesuitablestoppingcriteriavaryfordierentcases.Whilethestageofthesplititemcanbeagoodindicator,thebalancebetweenlongerDPtimeandmoreinequalitiesalwaysneedstobeconsidered. 2.5ConcludingRemarksInthischapter,weintroducedanewwayofgeneratingcoverinequalitiesforthe0-1knapsackproblemfromsolvingtheforwarddynamicprogrammingthrough\infeasiblestates".TwosetsofprotinequalitiesweredenedfromtheintermediatestageoptimalsolutionsofDP.ThebalancebetweensolvingthroughlongerDPsforgeneratingmoreusefulinequalitiestoshortenIPsolvertimeandsolvingshorterDPsbutlongerIPsolvertimevariedindierentcases.Ourmodelswereeectiveinmosttestinstances,andthemostecientinmorecorrelatedcases.Resultdetailswereshownforn=500problemswithstrongly,inverselystrongly,andalmoststronglycorrelateddata. 38

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CHAPTER3DYNAMICPROGRAMMINGBASEDINEQUALITIESFORTHEMULTIDIMENSIONALKNAPSACKPROBLEM 3.1MotivationandLiteratureReviewThemultidimensionalknapsackproblem(MKP)isoneofthedirectgeneralizationsofthe0-1knapsackproblem(0-1KP).Weconsiderasetofitems,N=f1,...,ng,andaknapsackofdimensionald2Z+.Eachitemj=1,...,nischaracterizedbyitsprot,pj,andasetofweights,fw1j,...,wdjg,whichrepresentstheresourceseachitemtakesineachdimensionoftheknapsack.Wewanttopickasubsetofitemsthatyieldsthemostprot,whilenotexceedinganyoftheknapsackconstraints.Whend=1,thisproblemisreducedtotheonedimensionalKP.ForgeneralMKP,weconsidercaseswithd2.Inmathematicalform,aMKPisdenedasfollows: maxnXj=1pjxjs.t.nXj=1wijxjci,i=1,...,d,xj2f0,1g,j=1,...,n, (3{1) whereallconstantspj,wij,ci,d,fori=1,...,d,j=1,...,n,areintegers.FollowingKelleretal.[ 31 ],toavoidtriviality,wefurtherassumethateachitemj2Nhasstrictlypositiveprotandweights,i.e.,pj>0andwij0,whilePdi=1wij1.Inorderforanyitemtopossiblybechosen,weassumeeachknapsackcapacityci>0,andwijciforallj2Nandi=1,...,d.Toensurewehavetomakeachoice,weassumePnj=1wijci,fori=1,...,d.Otherwise,theproblemcanbeeasilyreplacedbyaformulationwithlessitemsorlessknapsackconstraints.MKPisoneofthemostwellknownstronglyNP-hardintegerlinearproblems[ 31 , 32 ].SurveyshavebeenconductedtostudydierentperspectivesofMKP.Herewenamesomeofthekeyndings.Kelleretal.[ 31 ]providedacomprehensivedescriptiononitshistoricaldevelopment,exactandheuristicalgorithms.Freville[ 33 ]surveyedontheexactand 39

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approximationalgorithms,aswellastheoreticalpropertiesofMKP.FrevilleandHana[ 34 ]alsoreviewedvariousalgorithmsandavailablecommercialsoftwareforsolvingMKP,withthefocusonsurrogateandcompositedualityrelatedndingsandalgorithms.ExactsolutionofMKPcanbeobtainedusingdynamicprogramming(DP)orbranchandbound[ 31 ].DirectextensionofDPrecursionfrom0-1KPisvalidtoMKP.However,comparingto0-1KP,dominancerelationsinMKPareseldom[ 31 ].DuetothespacerequirementofDP,basicorenhancedDPonlysolvessmallinstances[ 33 ].GilmoreandGomory[ 35 ]presentedtheDPalgorithmintwo-dimensionalMKPasoneoftheearlierreferencesonsolvingMKPbyDP.Boyeretal.[ 36 ]usedDPwithsurrogaterelaxationtoderivevalidupperandlowerbounds,andappliedbranchandboundtondexactsolutions.BecauseofthedicultiesinsolvingMKPtooptimality,therearemorestudiesonrelaxationsandpreprocessingreductionmethods.Lagrangianandsurrogate[ 37 , 38 ]relaxationarethetwopopulardirections.Accordingto[ 31 ],boundsfromsurrogaterelaxationsaremoreecientthanLagrangianrelaxation,andthecompositeofthetwobringsmodestimprovementstosurrogaterelaxation.BasedonLagrangianandsurrogateconstraints,FrevilleandPlateau[ 39 ]presentedapreprocessingprocedureforlargeMKPinstances.Osorioetal.[ 40 ]generatedcutsandusedconstraintpairingbasedonsurrogateanalysis.Balevetal.[ 41 ]providedaDPbasedpreprocessingreductionalgorithmusingtwosetsofbounds:asequenceofnon-increasingupperboundsandasequenceofnon-decreasinglowerboundsobtainedbyxingasubsetofthevariablestotheopositevalueofafeasiblesolution.Besidesthetwodirectionsmentionedabove,therearestudiesonotherstrongvalidinequalities.BektasandOguz[ 42 ]extendedtheseparationalgorithmforcoverinequalitieson0-1KPtoMKP.KaparisandLetchford[ 43 ]extendedtheliftedcoverinequalitiestoMKP,andshowedthattheirenhancedgloballiftedcoverinequalitiesdominatealltheotheravailableupperboundsforMKP.Inaddition,variousheuristicalgorithmshavebeendevelopedalongtheway.ChuandBeasley[ 44 ]providedadetailedsurveyofexistingheuristicsandpresentedanewgenetic 40

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algorithm.FleszarandHindi[ 45 ]comparedseveralLP-relaxationbasedheuristicsandpresentedimprovedprocedures.WeingartnerandNess[ 46 ]developedupperandlowerbounds,basedondynamicprogrammingframeworkandtestedtheheuristicalgorithmontwo-dimensionalMKP.MarstenandMorin[ 47 ]presentedaheuristicofahybridapproachthatcombinedDPandbranchandboundalgorithms.Boyeretal.[ 48 ]usedDPtosolvethesurrogaterelaxationofMKP,whileasecondarylistiskeptfordominatedstates.TheysuggestedbranchandcutproceduretoexplorethesecondarylistinordertoimprovethegapbetweentheheuristicssolutionandtheoptimalsolutionofMKP.Puchingeretal.[ 32 ]extendedthecoreconceptfrom0/1KPtoMKP,andsolvedexactandapproximatecoresizesfordierentbenchmarkinstances.TheyalsocombinedanintegerLPbasedandamemetricalgorithmsoncoreproblems.BertsimasandDemir[ 49 ]presentedanapproximateDPapproachtoMKPandshowedthattheirheuristicwasecient.Moreover,MKPisoftentreatedasaspecialcaseorasubproblemofageneral0/1integerlinearprogram,andithasbeencommonlyusedindierentapplications,suchasthecapitalbudgetingproblems[ 46 , 50 ],thecuttingstockproblem[ 35 ],theportfolioselectionproblem[ 33 ],etc.Accordingto[ 31 ],practicalMKPproblemsusuallyhavearelativelysmallnumberofconstraintsbutalargenumberofvariables.InthisChapter,weextendthediscussionsfromChapter 2 andgenerateDPbasedinequalitiesforMKP.Section 3.2 reviewstherelevanttheories;Section 3.3 discussestheinequalityderivations;andSection 3.4 providesthecomputationalresults. 3.2TheoryReview 3.2.1CoverInequalityRecallfromChapter 2 ,for0-1KP,ifasetCNexceedstheknapsackcapacity,c,i.e.,Pj2Cwj>c,thenitisacovertoKP,andwecandenethefollowingcoverinequality(CI)fromthecoverC: Xj2CxjjCj)]TJ /F5 11.955 Tf 17.93 0 Td[(1, (3{2) 41

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whichisvalidforallfeasiblesolutions.Acoverisminimalifeliminatinganyitemfromthesetresultsinafeasiblesolution.Ifacovercontainsoneormoreminimalcoversasitspropersubset,andCIdenedfromthatcoverisgenerallyweakerthanCIsdenedfromitsminimalcoversubsets.Givenanon-integraloptimalsolution,^x=f^x1,...,^xng,tothelinearrelaxationof0-1KP,theseparationproblemtondthemostviolatedCIcanbedenedasaminimizationversionofthe0-1KP[ 14 ],i.e., =minnXj2N(1)]TJ /F4 11.955 Tf 11.95 0 Td[(xj)zj:Xj2Nwjzj>c,zj2f0,1go (3{3) If( 3{3 )hasanoptimalobjectivevalue<1withtheoptimalsolutionfzjgj=1,...,n,thecurrentmostviolatedCIcanbedenedas: Xj2CzjxjjCj)]TJ /F5 11.955 Tf 17.93 0 Td[(1,whereC=fj2N:zj=1g. (3{4) TheCIdenedfromacoveroraminimalcoverisnotguaranteedtobefacet-dening[ 43 ].LiftingtechniquesareusuallyappliedonaCItodeneastrongerorfacet-deninginequality. 3.2.2DynamicProgrammingApproachTheDPapproachon0-1KPcanbeeasilyextendedtoMKP:eachitemisregardedasastage,andeachsetofutilizedcapacitiesinalldimensionsoftheknapsackisregardedasastate.AnintermediatestagekproblemforMKPcanbedenedasfollows: maxkXj=1pjxjs.t.kXj=1wijxji,fori=1,...,dxj2f0,1g,forj=1,...,k. (3{5) 42

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AstraightforwardextensionoftheBellmanrecursionfor0-1KPndstheoptimalobjectivevaluetothestagek,state(1,2,...,d)problem: fk(1,2,...,d)=8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:fk)]TJ /F8 7.97 Tf 6.58 0 Td[(1(1,2,...,d),ifi><>>:1iffk(1,2,...,d)>fk)]TJ /F8 7.97 Tf 6.58 0 Td[(1(1,2,...,d)0otherwise, (3{7) followedbyndingtheoptimalsolutiontofk)]TJ /F8 7.97 Tf 6.58 0 Td[(1(1)]TJ /F4 11.955 Tf 11.96 0 Td[(w1kxk,2)]TJ /F4 11.955 Tf 11.96 0 Td[(w2kxk,...,d)]TJ /F4 11.955 Tf 11.95 0 Td[(wdkxk). 3.3Inequalityderivation 3.3.1CoverInequalitiesandProtInequalitiesforMKPDPbasedinequalitiesfor0-1KPderivedinChapter2canbeextendedtoMKP. Lemma2. SolvingDPthroughstatefci+wijgi=1,...,dateachstagejdenesallminimalcoverinequalitiesforMKP. Proof. Whensolvingthroughastatef1,...,dg,wherei>ciforsomei2f1,...,dg,theoptimalsolution,x,obtainedatthisstatesatises: Xj2Cwijxj=i>ci,C=fj2N,xj=1g. (3{8) Thatis,theithknapsackconstraintisviolated,soavalidCIcanbedenedas: Xj2CxjjCj)]TJ /F5 11.955 Tf 17.93 0 Td[(1 (3{9) 43

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FromTheorem 2.1 ofChapter 2 ,weknowthatsolvingDPthroughstatec+wjateachstagejdenesallminimalcoverstothesingleknapsackconstraint.ForMKP,eachconstraintcanberegardedasasingle-levelKP,i.e.,whenasetviolatesoneoftheconstraints,itisacovertoMKP.Therefore,solvingDPthroughallconstraintsdenesallminimalcoverstoMKP. Lemma3. ProtinequalitiesarevalidforMKP. Proof. InthesamemannerastheprotinequalitiesdenedforKP,letfkbetheintermediatestageoptimalobjectivevalueatstagek,i.e.,theoptimalvaluefor( 3{5 ),thentheprotinequalitiesaredenedasfollows: kXj=1pjxjfk, (3{10) nXj=1pjxjfk. (3{11) Theproofthatshows( 3{10 )and( 3{11 )arevalidforalloptimalsolutionstoMKPfollowtheexactproofsofTheorem 2.2 and 2.3 fromChapter 2 ,soweonlyillustratethebasicideashere.Toshow( 3{10 )isvalid,givenanyfeasiblesolution,xf,thek-vectorofitsrstkentries,fxf1,...,xfkg,isafeasiblesolutiontothestagekproblem( 3{5 ),for: kXj=1wijxfjnXj=1wijxfjci,8i=1,...,d (3{12) Sincefkisthestagekoptimalobjective,inequality( 3{10 )isvalidforxf.Inwords,( 3{10 )isavalidinequalityforallfeasiblesolutionstoMKP.Toshow( 3{11 )isvalid,sincetheoptimalobjectivevaluesforDPrecursionsarenon-decreasing,denotetheoptimalobjectivevalueforMKPasf,thenfkf.ConsideranyoptimalsolutiontoMKP,x,itsatises: nXj=1pjxj=fTherefore,( 3{11 )isvalidforx.Thatis,( 3{11 )isvalidforalloptimalsolutionstoMKP. 44

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3.3.2ModiedProtInequality Theorem3.1. Let~fkjxk=1betheoptimalobjectivevaluetothestagekproblemxingxk=1.Thenthefollowingmodiedinequalityisvalidforallfeasiblesolutions: kXj=1pjxj(1)]TJ /F4 11.955 Tf 11.96 0 Td[(xk)fk)]TJ /F8 7.97 Tf 6.59 0 Td[(1+xk~fkjxk=1 (3{13) Proof. Takinganyfeasiblesolution,xf,ofMKP,itsatises( 3{12 ).Therefore,thek-vectorfxf1,...,xfkgisafeasiblesolutiontothestageksubproblem.SinceMKPisabinaryproblem,xfk=0or1.Ifxfk=1,therighthandsideof( 3.1 )canbereducedto~fkjxk=1.Weknowthatxfisinthesetoffeasiblesolutionsxingxk=1.Bydenition,~fkjxk=1istheoptimalobjectivevalueatstagekxingxk=1.Thefollowinginequalityisvalid: kXj=1pjxfj~fkjxk=1 (3{14) Nowifxfk=0,therighthandsideof( 3.1 )canbesimpliedtofk)]TJ /F8 7.97 Tf 6.58 0 Td[(1.Forthesamereasonasabove,the(k)]TJ /F5 11.955 Tf 10.36 0 Td[(1)-vector,fxf1,...,xfk)]TJ /F8 7.97 Tf 6.58 0 Td[(1gisafeasiblesolutiontothestagek)]TJ /F5 11.955 Tf 10.36 0 Td[(1subproblem.Bydenition,fk)]TJ /F8 7.97 Tf 6.59 0 Td[(1istheoptimalobjectiveforstagek)]TJ /F5 11.955 Tf 11.96 0 Td[(1subproblem,soitfollowsthat: kXj=1pjxj=k)]TJ /F8 7.97 Tf 6.59 0 Td[(1Xj=1pjxjfk)]TJ /F8 7.97 Tf 6.59 0 Td[(1. (3{15) Therefore,inequality( 3.1 )isvalidforallfeasiblesolutions. Corollary3.1.1. Iftheintermediateoptimalobjectiveatstagekisthesamevalueforxingxk=1orxk=0,thenthemodiedprotinequalitygeneratedfromstagekistheprotinequality. Proof. If~fkjxk=1=~fkjxk=0,wehavefoundalternativeintermediateoptimalsolutionsforthestageksubproblem,andfk=~fkjxk=1.Ontheotherhand,theoptimalobjectivevalueatstagekxingxk=0isalsotheoptimalobjectivevalueforstagek)]TJ /F5 11.955 Tf 12 0 Td[(1problem,i.e.,~fkjxk=0=fk)]TJ /F8 7.97 Tf 6.59 0 Td[(1. 45

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Therighthandsideof( 3.1 )thenbecomes:(1)]TJ /F4 11.955 Tf 11.96 0 Td[(xk)fk)]TJ /F8 7.97 Tf 6.59 0 Td[(1+xk~fkjxk=1=(1)]TJ /F4 11.955 Tf 11.96 0 Td[(xk)fk+xkfk=fkThatis,( 3.1 )isreducedtotheprotinequalityatstagek. Corollary3.1.2. Ifxingxk=1atstagekgivesanoptimalobjectivevaluedierentfromfk)]TJ /F8 7.97 Tf 6.59 0 Td[(1,themodiedprotinequalityisstrongerthantheprotinequalitygeneratedatstagek. Proof. Recallthattheprotinequalitydenedfromstagekis: kXj=1pjxj=k)]TJ /F8 7.97 Tf 6.59 0 Td[(1Xj=1pjxj+pkxkfk (3{16) Inequality( 3.1 )canbere-writtenas: k)]TJ /F8 7.97 Tf 6.58 0 Td[(1Xj=1pjxj+)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(pk+fk)]TJ /F8 7.97 Tf 6.59 0 Td[(1)]TJ /F5 11.955 Tf 12.05 2.66 Td[(~fkjxk=1xkfk)]TJ /F8 7.97 Tf 6.59 0 Td[(1. (3{17) Weconsiderthefollowingtwocasesseparately: Case1:~fkjxk=10,pk+fk)]TJ /F8 7.97 Tf 6.59 0 Td[(1)]TJ /F5 11.955 Tf 12.16 2.66 Td[(~fkjxk=1>pk.Thus,( 3{17 )isstrongerthan( 3{16 )inthiscase. Case2:~fkjxk=1>fk)]TJ /F8 7.97 Tf 6.59 0 Td[(1.Weknowthatxk=1instagekoptimalsolution,andfk=~fkjxk=1.Bydenition,xk1,then )]TJ /F5 11.955 Tf 5.57 -7.03 Td[(~fkjxk=1)]TJ /F4 11.955 Tf 11.96 0 Td[(fk)]TJ /F8 7.97 Tf 6.59 0 Td[(1xk~fkjxk=1)]TJ /F4 11.955 Tf 11.95 0 Td[(fk)]TJ /F8 7.97 Tf 6.58 0 Td[(1=fk)]TJ /F4 11.955 Tf 11.95 0 Td[(fk)]TJ /F8 7.97 Tf 6.58 0 Td[(1 (3{18) Adding( 3{18 )to( 3{17 ),wegetexactly( 3{16 ).Therefore,( 3{17 )isstrongerthan( 3{16 )inthiscase. 46

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3.3.3ANumericalExampleConsiderthefollowingfour-variabletwo-constraintMKPexample: max5x1+6x2+10x3+8x4s.t.4x1+5x2+6x3+7x4106x1+5x2+7x3+6x412xj2f0,1g,j=1,2,3,4 (3{19) WestarttogenerateDP-basedinequalitiesinstage3.Theoptimalobjectivevalueatstage3is11,sotheprotinequalitiesaredenedas: 5x1+6x2+10x311 (3{20) 5x1+6x2+10x3+8x411. (3{21) Fixingx3,theoptimalobjectivevalue~f3jx3=1=10,andwehavef3=f2=11.Thus,themodiedprotinequalityisdenedas: 5x1+6x2+10x311(1)]TJ /F4 11.955 Tf 11.96 0 Td[(x3)+10x3=11)]TJ /F4 11.955 Tf 11.95 0 Td[(x3or5x1+6x2+11x311 (3{22) Noticethatthecoecientforx3isbigerin( 3{22 )than( 3{20 ),so( 3{22 )dominates( 3{20 ).Inaddition,theCIsaregeneratedinstage3 x2+x31 (3{23) x1+x31 (3{24) x1+x2+x32 (3{25) Afterstage3,weadd( 3{22 )-( 3{25 )to( 3{19 ). 47

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3.4ComputationalResultsThissectionpresentsthecomputationalperformanceofthetechniquesdescribedinSection 3.3 usingrandomlygeneratedinstances.AllexperimentswerecodedinJavawithCPLEX12.5usingConcert2.9technologyonanIBMSystemx3650withtwoIntelE5640Xeonprocessorsand24GBmemory. 3.4.1ImplementationIftheoptimalsolutiontotheLKPwasfractional,wegeneratedprotandcoverinequalitiesthroughpartiallysolvingtheforwarddynamicprogramming(DP)problem.AllinequalitieswereaddedtotheoriginalKPandthesystemwassolvedusingCPLEXdefaultIPsolver.FromSection 3.3 ,weknowthatsolvingDPthroughstateci+wijateachstagejforeachknapsackconstraintiwillndallminimalCIs.AretrievallistwasneededinordertodeneCIsfromthedominatedstatesaswasin0-1KP.SincedominatedstatesarerareinMKP,wedidnotkeeptheretrievallistintheexperiments.Also,inordertodeneallminimalCIs,wehadtodenealotredundantconstraints,whichincreasethepreprocessingtimeofthesolver.ItwasmoreecienttoaddtherstCIsfromeachstageandeachconstraint,whichareguaranteedtobeminimal.In0-1KP,oneofthecrucialassumptionsforourDP-basedinequalitiestobepreferableisthatallitemsaresortedbydescendingprot-weightratio.InMKP,however,eachitemischaracterizedwithmultipleweights.Kelleretal.[ 31 ]summarizedsomechoicesforeciencyofanitemforMKPinplaceoftheprot-weightratioforKP.Thegeneralnotionoftheeciencyrjforeachitemjcanbedenedas rj=pj Pdi=1iwij (3{26) 48

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wherefigi=1,...,disasetofmultipliers.Lettingalli=1,wegetthedirectgeneralizationofonedimensionalprot-weightratio: r1j=pj Pdi=1wij (3{27) Totakethesizesoftheknapsacksintoconsiderandleti=1=ci,wegetascaledratio: r2j=pj Pdi=1(wij=ci) (3{28) Therearealsootherchoicesofrj.Weused( 3{28 )topre-sortalltheitems. 3.4.2ExperimentalDesignWefollowedtheexperimentaldesigndescribedin[ 49 ].Similartotheproblemdesignforthe0-1KP,weconsideredrandomgeneratedcasesinthreebasicproblemtypes.Foreachi=1,...,d,j2N,anddatarangeR,theproblemtypesweredenedasfollows: Uncorrelateddata:eachpj,wijrandomlygeneratedin[1,R]; Weaklycorrelateddata:eachwijrandomlygeneratedin[1,R],andpj2[Pdi=1wij=d)]TJ /F4 11.955 Tf 11.96 0 Td[(R=10,Pdi=1wij=d+R=10],whilepj1; Stronglycorrelateddata:eachwijrandomlygeneratedin[1,R],andpj=Pdi=1wij=d+R=10Foreachinstancewithproblemsizen=jNj,eachcapacitywascalculatedasafractionofthetotalweight,i.e.: ci=j1 knXj=1wijk 3.4.3ResultsandDiscussionsAsKelleretal.[ 31 ]pointedout,puredynamicprogrammingapproach,withcomplexityO(ncdmax),hadlimitedsuccessinsolvingsmallsizeMKPs.SincewepartiallysolvetheforwardDPtogeneratetheproposedinequalities,thetimeandspacerequirementsareexpectedtobeproportionaltotherequirementsforsolvingthecorrespondingfullDPrecursion. 49

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WetesteddierentcombinationsofproposedinequalitiesonrandomlygeneratedMKPswithd=2constraints.Foreachproblemtype,weconsideredproblemsizesn=50,100,200,and400,overdatarangeofR=500,andcapacitylevelsk=5,10,20,and40.Foreachcase,twentyinstancesweregenerated.Table 3-1 presentstheaverageruntimeofCPLEXIPsolverwithorwithoutDPbasedinequalitiesadded.Theperformancedependsoninstances,andwepresenttheselectedcaseswheretheruntimewithinequalitiesaddedshowsclearimprovement.ColumnfracNshowedthefractionoftotalnumberofstagesDPwentthrough.ColumnnumInstcountedthenumberofinstances,outoftwenty,thathadimprovementafterDPbasedinequalitiesadded.ColumndefaultTlistedtheaveragedefaultsolvertime.ColumnaddedTpresenttheaverageIPsolvertimefromthebestperforminginequality-addedmodels.Column%impAvgcalculatestheaveragepercentageimprovementsoftheseselectedinstances. Table3-1. Runtime(millisec)summaryfortwodimensionalMKP nkfracNTypenumInstdefaultTaddedT%impAvg 5051/5uncor1656.8129.3147.52%weak559.5430.6212.56%strong1074.4447.8939.51%100101/5uncor1377.5061.2541.56%weak6180.25153.2519.75%strong8182.17157.8329.83%200201/5uncor9221.25184.2530.86%weak5149.7592.5019.65%400401/8uncor5798.80378.1027.35%weak5729.50468.6335.71% AsisshowninTable 3-1 ,theaveragepercentageimprovementonsolverruntimeofthelistedcasesrangedfromover10%toover40%.About90%ofthebestmodelswerecombinationsofupperandlowerbounds.Fromthetable,inequalitiesweremoreeectiveinsmallerinstances.Forlargerinstances,ingeneral,solvingthroughalargerfractionofallstageswouldresultinbetterresults.Inordertogeneratemoreusefulinequalities,weneededtosolvethroughenoughDPstages.However,asmoreDPstagesweresolved,thetimeandspace 50

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requiredbyDPincreased.Therefore,therewasatrade-obetweentheresourcestakenbyDPandtheamountofinformationgeneratedfromDP. 3.5ConcludingRemarksThischapterextendedtheworkofChapter 2 tothemultidimensionalknapsackproblem(MKP).Traditionalcoverinequalitiesandprotinequalities,generatedthroughpartiallysolvingforwardDP,werealsovalidforMKP.DuetothetimeandspacerequirementsbyDP,eectivenesswasshowninselectedinstances. 51

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CHAPTER4RISK-AVERSEWINDPOWERINVESTMENTTHROUGHUNIFIEDSTOCHASTICANDROBUSTOPTIMIZATION 4.1MotivationandLiteratureReviewOverthepasttwodecades,windpowerhasbeenpromotedtobeoneofthemainstreampowersourcesinmanycountries[ 51 , 52 ].Withthetechnologicaldevelopmentandtheworldwiderapidgrowthofwindpowercapacity,windenergyhasbecomethemostaordableandoneofthemostpopularrenewableenergysources.From2004to2013,globalwindinvestmentincreasedwiththecompoundannualgrowthrateof21%[ 52 ].By2014,windpowersuppliesaround4%ofworld'selectricitydemand[ 53 ].Unliketheconventionalelectricitygenerations,thereisnoCO2orotherpollutionemissionassociatedwithwindpowerproduction[ 54 ].FollowingthetraditionalwindenergymarketsinEurope,andmorerecentlyinNorthAmericaandChina,moreandmorecountrieshaveestablishedpublicpoliciesandregulations,suchastaxexemption,subsidies,quota,feed-intari,toencouragewindpowergeneration,forbothenvironmentalandeconomicalconcerns[ 55 ].Unsurprisingly,thecostandbenetanalysisofwindpowerinvestmentstarttoattractattentions.Ononehand,investinginwindpoweravoidscostsoffuelandwaterconsumptions,aswellaschargesongreenhousegasandpollutantemissions.Onetheotherhand,windpowerinvestmentrequireslargecapitalcosts,plusadditionalconcernsregardingpublicacceptanceofnewinfrastructure[ 52 ].Afterinitialinstallation,thereoccuradditionaloperatingcosts,includingexpensesonregularmonitoringandmaintenancetosustainhighperformancethroughoutitsoperationallifespan.Additionally,withnewwindpowerinstallments,possibleadjustmentsintheoverallelectricitynetworkalsoneedtobeconsidered[ 56 ].Thoughthecapitalandoperatingcostsarefairlypredictable,thewindintensityfactor,i.e.thestrengthandqualityofwindresourcewhichdeterminetheactualcapabilityofwindpowerproduction,ishighlyunpredictable,especiallyinshortterm(dailyandseasonal)[ 54 ].Moreover,sincewindpowerisnon-dispatchable,theintensityfactoructuation,togetherwithdemandvariability, 52

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directlycausesthevolatilityoftheelectricityprice,leavingtheprotabilityofwindpowerinvestmentquestionable.Studieshavebeenconductedtoevaluatewindpenetrationscenariosandinvestmentpossibilitiesinvariouselectricitymarketsindierentregions.First,publicandprivateinvestmentannualsummaries,includingthetrendsofwindinvestmentsandrelatedissuessuchasenvironmentalimpactsandopportunities,arereportedbygovernments/associationsaroundtheworld.Forinstance,annualreportbyU.S.DepartmentofEnergy[ 52 ]describesthecomprehensiveimageofthecurrentwindpowermarketintheU.S.andfuturetargets;annualreportbyEuropeanWindEnergyAssociation[ 51 ]providesthesummaryofmarketenvironmentforthewholeEuropeandabriefdescriptionofthewindinvestmentstatusinitsmembercountries.Meanwhile,researchershavealsoperformedextensivestudiesinprovidingecientpoliciesandassociatedsimulationvalidationsfordierentcountries.SnyderandKaiser[ 57 ]comparetheoshorewindpowerdevelopmentbetweenEuropeandUS.Sinden[ 58 ]presentstrendsandavailabilitiesoftheUKwindresources,andexaminedtheoccurrenceofextremelowandhighwindspeedeventsbasedontheUKwindrecord.Strbacetal.[ 59 ]proposeageneticmodeltoestimatemaximumlossfortheUKelectricitysystem.Theyconcludedthatthesystemwouldbeabletoaccommodatesignicantincrementsinwindpowergeneration,butitwasalsonecessarytoretainalargeportionofconventionalplantstoensurethesecurityofsupply.AkdagandGuler[ 60 ]giveareviewandcostanalysisofwindpowerinvestmentinTurkey.KongnamandNuchprayoon[ 61 ]provideafeed-intarischemetopromotewindenergygenerationundertheregulatedenvironmentinThailand.Kaldellis[ 62 ]maximizeswindpenetrationinanautonomouselectricitygridbasedonthedatafortheAegeanArchipelagoarea,inwhichWeibulldistributionwasusedtosimulatewindpotential.Kaldellisetal.[ 63 ]estimatethemaximumwindenergyinautonomouselectricalislandnetworkswithtimeperiodconcerns,usinganintegratednumericalalgorithm.BurkeandO'Malley[ 64 ]maximizewindpowerpenetrationinanexistingtransmissionnetworkwhilepreservingthetraditionalnetworksecuritystandards,usingatimeseriesmodel.Jongheetal.[ 65 ]useastaticlinear 53

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programmingmodeltondanoptimaltechnologymixofconventionalpowergenerationwithhighlevelwindpowerpenetration.Thesestudiesarefurtherextendedtoincorporateuncertaintiesindemandandpowergenerationusingstochasticoptimizationapproaches.Therearemoreliteratureoninvestmentproblemsinconventionalgenerations[ 66 { 68 ]toaccommodatedemandandpowergenerationuncertainties,butlimitedliteratureonwindpowergenerations.DennyandO'Malley[ 69 ]useadispatchmodeltodeterminethenetbenetsofwindintegrationinIrelandundervariousassumptions,andndthecriticalvalueofwindpenetrationfordierentscenarios.Pousinhoetal.[ 70 ]maximizetheprotofwindproduction,withthecasestudyforPortugal.Theuncertainmarketpricesarerepresentedbyscenarios,andaconditionalvalue-at-risk(CVaR)componentisaddedtotheobjectivefunctionforrisk-aversedecisions.BaringoandConejo[ 71 { 73 ]buildtwo-stagemodelsandrepresentloadandwindintensityasscenariosinthestochasticoptimizationframework,amongwhich[ 71 ]reformulatesthebilevelproblemintoamixed-integerlinearprogram(MILP),[ 72 ]solvesthesecond-stagestochasticproblemusingBender'sDecomposition,and[ 73 ]addsaCVaRcomponenttotheobjectivefunctionandobtainsarisk-aversesolution.Weproposeatwo-stagestochasticoptimizationmodelforthewindpowerinvestmentproblem,withtheobjectiveofmaximizingtheexpectedtotalprotforoneyear.Inaddition,weobtainamorerobustdecisionbyaddingtherobustoptimizationparttotheoriginalstochasticmodel.Therestofthepaperisorganizedasfollows.InSectionII,wepresentthemodelformulation.SectionIIIdiscussesthelinearizationanddecompositionprocedure:thestochasticoptimizationpartistransformedintoanequivalentMILPandintegratedintothemasterproblem,andfortherobustoptimizationpart,weprovideaBender'sdecompositionalgorithmtosolvetheproblem.Finally,SectionIVprovidesthecomputationalresults. 54

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4.2NotationandMathematicalFormulation 4.2.1NomenclatureInthissubsection,weintroducetheindexsets,constants,anddecisionvariablesusedinthemathematicalformulation. Indices T Indexsetofdemandblocks N Indexsetofallbuses Wt Indexsetofscenariosinthetthdemandblock G Indexsetofallgenerationunits Gn Indexsetofgenerationunitsatbusn Bi Indexsetofblocksoftheithgenerationunit Dn Indexsetofdemandslocatedatbusn L Indexsetoftransmissionlines Constants Ht Durationofdemandblockt MNn Maximumwindpowerthatcanbeinstalledatbusn wt,! Weightofthe!thscenarioindemandblockt Fn Amortizedxedinvestmentcostatbusn Vn Amortizedvariableinvestmentcostatbusn ^Vn Annualinvestmentcostatbusn Cwindmax Annualbudgetforwindpowerinvestment In,t,! Windintensityinscenario!ofdemandblocktatbusn o(k) Sending-endbusoflinek r(k) Receiving-endbusoflinek Bk Susceptanceoflinek 55

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dj,t,! Thejthdemandinscenario!ofdemandblockt dj,t Theexpectedjthdemandofdemandblockt pgi,b Oerpriceofthebthblockofithgenerationunit gmaxi,b Generationcapacityofthebthblockoftheithgenerationunit fmaxk Flowcapacityoflinek Variables Yn Binaryindicatorforsetupatbusn Xn Installedwindpowercapacityatbusn n,t,! Thelocalmarginalpriceinscenario!ofdemandblocktatbusn Pn,t,! Windpowerproducedinscenario!anddemandblocktatbusn gi,b,t,! Powerproducedbythebthblockoftheithgeneratorinscenario!ofdemandblockt ~gi,b,t Powerproducedbythebthblockoftheithgeneratordemandblockt n,t,! Voltageangleatbusninscenario!ofdemandblocktforstochasticoptimization ~n,t Voltageangleatbusnindemandblocktforrobustoptimization fk,t,! Powerowthroughlinekininscenario!ofdemandblocktforstochasticoptimization ~fk,t Powerowthroughlinekindemandblocktforrobustoptimization ~d+,)]TJ /F6 7.97 Tf -10.2 -7.64 Td[(n,t Auxiliaryvariablesforsupplyshortageandexcessforrobustoptimization n,t Windintensityatbusnindemandblocktforrobustoptimization 4.2.2MathematicalModelWefollowthedirectionasdescribedin[ 71 ].Theproblemisdesignedonbehalfofaprivatewindpowerinvestor,consideringwindplantinvestmentsandoperationsinanexistingtransmissionnetwork.Theinvestmentdecisionismadeunderapool-basedelectricitymarket,withinwhichthewindpowerissoldatthelocalmarginalpriceatthebuswheretheplantislocated.Assumingnosubsidiesfrompublicfunds,theinvestorneedstorecoverherinvestmentfromsellingelectricitytothemarket,whileaimstomaximizetheprot.Moreover,perfect 56

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competitionisassumedsothatpowerproducedbyconventionalproducersistradedatthemarginalcosts.Intheproblemsetting,weconsiderashort-termwindpowerinvestmentplanforoneyear.Therststageofthetwo-stagemodelmakesinvestmentdecisions,whilethesecondstageensurestheowandloadbalanceofthenetwork.Thesecondstageconsistsofbothstochasticandrobustoptimizationapproaches,inwhichthestochasticpartmodelsuncertaintyinscenariosandtherobustpartconsiderstheworst-casescenariowithinanuncertaintyset.Thederivingmathematicalformulation,denotedasRWI,canbedescribedasfollows: max)]TJ /F7 11.955 Tf 11.29 11.35 Td[(Xn2N(FnYn+VnXn)+Xt2THtX!2Wtwt!Xn2Nn,t,!Pn,t,!+Q (4{1) s.t.XnMNnYn,8n2N (4{2) Xn2N^VnXnCwindmax (4{3) (RWI)Pn,t,!In,t,!Xn,8n2N,8!2Wt,8t2T (4{4) Yn2f0,1g,Xn0,8n2N (4{5) n,t,!=n,t,!,8n2N,8!2Wt,8t2T, (4{6) wheren,t,!2argminn Xi2GnXb2Bici,bgi,b,t,! (4{7) s.t.Xi2GnXb2Bigi,b,t,!)]TJ /F7 11.955 Tf 20.91 11.35 Td[(Xk:o(k)=nfk,t,!+Xk:r(k)=nfk,t,!+Pn,t,!=Xj2Dndj,t,!:n,t,!,8n2N (4{8) fk,t,!=Bko(k),t,!)]TJ /F10 11.955 Tf 11.95 0 Td[(r(k),t,!:k,t,!,8k2L (4{9) )]TJ /F4 11.955 Tf 9.3 0 Td[(fmaxkfk,t,!fmaxk:mink,t,!,maxk,t,!,8k2L (4{10) 0gi,b,t,!gmaxi,b:'mini,b,t,!,'maxi,b,t,!,8b2Bi,8i2G (4{11) )]TJ /F10 11.955 Tf 9.3 0 Td[(n,t,!:minn,t,!,maxn,t,!,8n2Nnfn:ref.g (4{12) 57

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n,t,!=0:n,t,!,forn:ref. (4{13) o8!2Wt,8t2Tand Q=min2maxf,g,,d+,d)]TJ /F11 7.97 Tf 6.26 -2.27 Td[(2Xn2NXt2T()]TJ /F4 11.955 Tf 9.3 0 Td[(Md)]TJ /F6 7.97 Tf -1.15 -7.89 Td[(n,t) (4{14) :=nXi2GnXb2Bi~gi,b,t)]TJ /F7 11.955 Tf 20.9 11.35 Td[(Xk:o(k)=n~fk,t+Xk:r(k)=n~fk,t+Xn+d+n,t)]TJ /F4 11.955 Tf 11.96 0 Td[(d)]TJ /F6 7.97 Tf -1.15 -7.89 Td[(n,t=Xj2Dndj,t:~n,t,8n2N (4{15) ~fk,t=Bk~o(k),t)]TJ /F5 11.955 Tf 12.68 2.65 Td[(~r(k),t:~k,t,8k2L (4{16) )]TJ /F4 11.955 Tf 9.3 0 Td[(fmaxk~fk,tfmaxk:~mink,t,~maxk,t,8k2L (4{17) 0~gi,b,tgmaxi,b:~'mini,b,t,~'maxi,b,t,8b2Bi,8i2G (4{18) )]TJ /F10 11.955 Tf 9.3 0 Td[(~n,t:~minn,t,~maxn,t,8n2Nnfn:ref.g (4{19) ~n,t=0:~n,t,forn:ref. (4{20) ~d+n,t0,~d)]TJ /F6 7.97 Tf -1.15 -7.89 Td[(n,t0 (4{21) o8t2T,2,wheretheuncertaintysetisdeneas =n2RjNjjTj:n,t2[I 2n,t, I2n,t],Xn2Nn,t2[I 1t, I1t],Xt2TXn2Nn,t2[I , I]o. (4{22) Intheaboveformulation,( 4{1 )-( 4{6 )describestherst-stagedecisionproblem,i.e.,thewindpowerinstalledateachcandidatelocation,( 4{7 )-( 4{13 )isthesecond-stagestochasticoptimization,and( 4{14 )-( 4{21 )isthesecond-stagerobustoptimization.Morespecically,objectivefunction( 4{1 )maximizestheoverallprotoftheinvestment:theexpectedrevenue,minustheexpectedannualinvestmentcostandthepenaltyofsupply 58

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excessundertheworst-casescenario.Theinitialwindpowerinvestmentcostisassumedtobeamortizedovertheoperationallifespan.Forsimplicity,itisassumedthatthereareaxedcostandavariablecostateachbusifwindpowerisinstalled,andthecostsatallbusessumuptothetotalinvestmentcostinoneyear.Theexpectedrevenuecomesfromtheweightedaveragerevenueofallscenariosfromthestochasticpart.Sinceanexcessinpowersupplydrivesdownthemarketprice,apenaltytermisaddedforsupplyexcessfortherobustpart.Constraint( 4{2 )setsanupperboundforthewindpowerinstalledateachbus,andconstraint( 4{3 )setstheinvestmentbudgetlimit.Constraint( 4{4 )statesthatthewindpowerproducedateachbusisthewindpowerinstalledatthatbusmultipliedbyawindintensityfactorundereachscenario.Constraint( 4{6 )presentsthatthesellingpricesaredeterminedbythelocalmarginalprices(LMP),whicharethedualvaluescorrespondingtoconstraints( 4{8 )bysolvingthesecond-stagestochasticoptimizationproblem( 4{7 )-( 4{13 )undereachdemandblockandeachscenario.Theobjective( 4{7 )forthesecond-stagestochasticoptimizationminimizestheconventionalgenerationcost.Theobjective( 4{14 )forthesecond-stagerobustoptimizationndstheworstcasesupplyexcess.Constraints( 4{8 )and( 4{15 )ensuretheloadbalanceateachbus.Inconstraint( 4{15 ),theauxiliaryvariablesd+n,tandd)]TJ /F6 7.97 Tf -1.15 -7.58 Td[(n,tcorrespondtosupplyshortageandexcess,respectively.Constraint( 4{9 )and( 4{16 )establishthepowerowthrougheachline.Constraint( 4{10 )and( 4{17 )ensurethatthepowerowiswithincapacity,andconstraint( 4{11 )and( 4{18 )guaranteethatthepowergenerationbyconventionalgeneratoriswithincapacity.Thelowerboundforallconventionalpowergenerationsissettobezero,whichcanbeeasilyraisedtoapositivevalue.Constraint( 4{12 )and( 4{19 )indicatetherangeforthevoltageangle.Herewelettheboundstobe)]TJ /F10 11.955 Tf 9.3 0 Td[(to,while,again,itiseasytoreplacetherangeswithsmallerones.Constraint( 4{13 )and( 4{20 )imposethevoltageangleatthereferencebustobezeroineachscenarioandeachdemandblock.Finally,theuncertaintysetforrobustoptimizationisdenedin( 4{22 ).Theupperandlowerboundscanbeobtainedthroughintervalforecasting. 59

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4.3LinearizationandDecompositionInthissection,wederivesolutionalgorithmstosolveRWI. 4.3.1StochasticOptimizationAsisdescribedin[ 71 ],weuseKKTequilibriumconstraintstotransformthestochasticpartintoamixedintegerlinearprogram(MILP).Forillustration,thesecondstagestochasticoptimizationproblem( 4{7 )-( 4{13 )canbeabstractedasfollows: =argminfDz,s.t.Ez=H)]TJ /F4 11.955 Tf 11.96 0 Td[(Gp (4{23) JzKg, (4{24) wherezrepresentsthestochasticvariablesfg,f,g,andprepresentsthewindproductionPfromthemasterproblem.Sincethestochasticconstraintsarealllinear,constraints( 4{23 )-( 4{24 )canbereplacedbyitsequivalentKKToptimalityconditions: ( 4{23 )-( 4{24 )D+E+J=0 (4{25) (K)]TJ /F4 11.955 Tf 11.96 0 Td[(Jz)=0 (4{26) 0,(K)]TJ /F4 11.955 Tf 11.96 0 Td[(Jz)0, (4{27) whereandrepresentthedualvariablesfor( 4{23 )and( 4{24 ),respectively.Thebilinearcomplimentaryconstraints( 4{26 )canthenbefurtherlinearizedbyintroducingbinaryvariables,andwehavethefollowingequivalentconstraints: ( 4{23 )-( 4{25 ),( 4{27 ) (4{28) Mu (4{29) K)]TJ /F4 11.955 Tf 11.95 0 Td[(JzM(1)]TJ /F4 11.955 Tf 11.96 0 Td[(u) (4{30) u2f0,1g. (4{31) 60

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Wecanthenintegrate( 4{28 )-( 4{31 )tothemasterproblemtoformaMILP. 4.3.2BilinearTermintheObjectiveThesecondtermin( 4{1 )isabilinearterm.Togetanequivalentlinearform,wereformulatethedualobjectiveofthesecond-stagestochasticoptimizationfollowingasimilarapproachasdescribedin[ 71 ].Sinceallconstraints( 4{23 )-( 4{24 )arelinear,byStrongDualityTheorem,theprimalanddualoptimalobjectivevaluesareequal.Thatis, Dz=(H)]TJ /F4 11.955 Tf 11.95 0 Td[(Gp)+K. (4{32) Notethatthedetailedexpressionof(H)]TJ /F4 11.955 Tf 11.95 0 Td[(Gp)is Xn2Nn,t,!(Xj2Dndj,t,!)]TJ /F4 11.955 Tf 11.95 0 Td[(Pn,t,!).(4{33)Combining( 4{32 )and( 4{33 ),wecanrewriteequation( 4{32 )as Xn2Nn,t,!Pn,t,!=Xn2Nn,t,!Xj2Dndj,t,!+K)]TJ /F4 11.955 Tf 11.95 0 Td[(Dz. (4{34) Inaddition,from( 4{6 ),wehaven,t,!=n,t,!,sothebilineartermPn2Nn,t,!Pn,t,!intheobjectivefunction( 4{1 )canbereplacedbytheright-handsideformulaof( 4{34 ). 4.3.3RobustOptimizationTherobustoptimizationpartcanbeabstractedasfollows: Q=min2max~z2()]TJ /F4 11.955 Tf 9.29 0 Td[(Md)]TJ /F5 11.955 Tf 7.09 -4.93 Td[() (4{35) =nE~z+d+)]TJ /F4 11.955 Tf 11.96 0 Td[(d)]TJ /F5 11.955 Tf 10.41 -4.94 Td[(=H)]TJ /F4 11.955 Tf 11.95 0 Td[(Gx (4{36) J~zKo (4{37) wheredrepresentstheauxiliaryvariablesd+n,tandd)]TJ /F6 7.97 Tf -1.15 -7.58 Td[(n,t,xrepresentswindpowerinstalled,Xn,fromthemasterproblem,and~zrepresentsvariablesf~g,~f,~gintherobustpart.Forxedx, 61

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wedualizeconstraints( 4{36 )-( 4{37 )togetthefollowingdualformulation: Q(x,,~,~):=min2,~z(H)]TJ /F4 11.955 Tf 11.95 0 Td[(Gx)~+K~ (4{38) s.t.nE~+J~=0 (4{39) 0~M (4{40) ~0o, (4{41) where~and~aredualvariablesfor( 4{36 )and( 4{37 ),respectively,and~2~representsthedualvariablesforconstraints( 4{15 ).Theobjective( 4{38 )containsabilinearterm~.HereweapplyabilinearapproachinaBender'sdecompositionmannertotheproblem.Accordingto[ 74 ],thisapproachconvergesinashortertimethanexactseparationalgorithm.Forxedvaluesofand~,wedenethefollowingtwolinearsubproblems: SUB1:Q1(x,):=Q(x,,~,~)s.t.( 4{39 )-( 4{41 )SUB2:Q2(x,~,~):=Q(x,,~,~)s.t.( 4{39 )-( 4{41 )TraditionalBender'sdecompositionaddsbothfeasibilitycutsandoptimalitycuts.Inthiscase,sincetherststagesolutionsxonlyappearsintheobjectivefunction( 4{38 ),feasibilityisguaranteed.Wesolvethetwosubproblemsandupdate~,~andvaluesiteratively.IfQ2(x,~,~)
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4.3.4AlgorithmFromSection 5.2 and 5.3 ,theintegratedmasterproblem,denotedasWIMP,becomes max)]TJ /F7 11.955 Tf 11.29 11.36 Td[(Xn2B(FnYn+VnXn)+Xt2THtX!2Wtwt,!hXn2Nn,t,!Xj2Dndj,t,!)]TJ /F7 11.955 Tf 11.29 11.36 Td[(Xi2GXb2Bici,bgi,b,t,!)]TJ /F7 11.955 Tf 11.95 11.36 Td[(Xk2Lmaxk,t,!+mink,t,!fmaxk)]TJ /F7 11.955 Tf 11.95 11.36 Td[(Xi2GXb2Bi'maxi,b,t,!gmaxi,b)]TJ /F7 11.955 Tf 27.84 11.36 Td[(Xn2Nnfn:ref.gmaxn,t,!+minn,t,!i+Q (4{43) s.t.Constraints( 4{2 )-( 4{5 ) (4{44) Constraints( 4{8 )-( 4{13 ) (4{45) nci,b)]TJ /F10 11.955 Tf 11.95 0 Td[(n(i),t,!+'maxi,b,t,!)]TJ /F10 11.955 Tf 11.95 0 Td[('mini,b,t,!=0,8i2G,8b2Bi (4{46) o(k),t(!))]TJ /F10 11.955 Tf 11.96 0 Td[(r(k),t,!)]TJ /F10 11.955 Tf 11.96 0 Td[(k,t,!+maxk,t,!)]TJ /F10 11.955 Tf 11.96 0 Td[(mink,t,!=0,8k2L (4{47) )]TJ /F7 11.955 Tf 20.18 11.36 Td[(Xkjo(k)=nBkk,t,!+Xkjr(k)=nBkk,t,!+maxn,t,!)]TJ /F10 11.955 Tf 11.96 0 Td[(minn,t,!=0,8nnfn:ref.g (4{48) )]TJ /F7 11.955 Tf 20.18 11.36 Td[(Xkjo(k)=nBkk,t,!+Xkjr(k)=nBkk,t,!)]TJ /F10 11.955 Tf 11.96 0 Td[(n,t,!=0,forn:ref. (4{49) maxk,t,!Mumaxk,t,!,8k2L (4{50) (WIMP)fmaxk)]TJ /F4 11.955 Tf 11.95 0 Td[(fk,t,!M(1)]TJ /F4 11.955 Tf 11.95 0 Td[(umaxk,t,!),8k2L (4{51) mink,t,!Mumink,t,!,8k2L (4{52) fk,t,!+fmaxkM(1)]TJ /F4 11.955 Tf 11.95 0 Td[(umink,t,!),8k2L (4{53) 'maxi,b,t,!Mumaxi,b,t,!,8i2G,8b2Bi (4{54) gmaxi,b)]TJ /F4 11.955 Tf 11.96 0 Td[(gi,b,t,!M(1)]TJ /F4 11.955 Tf 11.96 0 Td[(umaxi,b,t,!),8i2G,8b2Bi (4{55) 'minib,t(!)Mumini,b,t!,8i2G,8b2Bi (4{56) gib,t(!)M(1)]TJ /F4 11.955 Tf 11.96 0 Td[(umini,b,t!),8i2G,8b2Bi (4{57) maxn,t,!Mumaxn,t,!,8n2Nnfn:ref.g (4{58) )]TJ /F10 11.955 Tf 11.95 0 Td[(n,t,!M(1)]TJ /F4 11.955 Tf 11.96 0 Td[(umaxn,t,!),8n2Nnfn:ref.g (4{59) minn,t,!Muminn,t,!,8n2Nnfn:ref.g (4{60) +n,t,!M(1)]TJ /F4 11.955 Tf 11.96 0 Td[(uminn,t,!),8n2Nnfn:ref.g (4{61) 63

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umaxk,t,!,umink,t,!2f0,1g,8k2L (4{62) umaxi,b,t,!,umini,b,t,!2f0,1g,8i2G,8b2Bi (4{63) umaxn,t,!,uminn,t,!2f0,1g,8n2Nnfn:ref.g (4{64) o8t2T,8!2Wt.Therobustpart,denotedasWRP,becomes Q=min2,~,~,~',~,~Xt2TXn2N~n,thXj2Dndj,t)]TJ /F10 11.955 Tf 11.95 0 Td[(Xni+Xk2Lfmaxk~maxk,t+~mink,t+Xi2GXb2Bi~'maxib,tgmaxib+Xn2N~maxn,t+~minn,t (4{65) s.t.n~n(i),t+~'maxib,t)]TJ /F5 11.955 Tf 13.63 0 Td[(~'minib,t=0,8i2G,8b2Bi (4{66) )]TJ /F5 11.955 Tf 9.57 2.65 Td[(~o(k),t+~r(k),t+~k,t+~maxk,t)]TJ /F5 11.955 Tf 13.25 2.65 Td[(~mink,t=0,8k2L (4{67) )]TJ /F7 11.955 Tf 20.18 11.36 Td[(Xkjo(k)=nBk~k,t+Xkjr(k)=nBk~k,t+~maxn,t)]TJ /F5 11.955 Tf 12.97 2.66 Td[(~minn,t=0,8n2Nnfn:ref.g (4{68) )]TJ /F7 11.955 Tf 20.18 11.36 Td[(Xkjo(k)=nBk~k,t+Xkjr(k)=nBk~k,t)]TJ /F5 11.955 Tf 13.13 0 Td[(~n,t=0,n:ref. (4{69) 0~n,tM,8n2N (4{70) ~n,t,~k,tfree,~maxk,t,~mink,t,~'maxib,t,~'minib,t,~maxn,t,~minn,t0o,8t2T,2Ateachiterationwegothroughthefollowingprocedure(alsoseeowchart 4-1 ): 1) Initialization 2) SolveWIMPtoobtaintheoptimalQandXn; 3) Solve(SUB1);storetheoptimalobjectiveQ1(x,),andoptimalsolutions~,~,~max,~min,~'max,~'min,~max,~min,~; 4) Solve(SUB2);storetheoptimalobjectiveQ2(x,~,~)),andoptimalsolutions; 5) IfQ2
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Figure4-1. Windpowerinvestmentmodel:algorithmowchart 4.4ComputationalResultsThissectionpresentsthecomputationalresultsoftheproposedalgorithmona6-busandtheIEEE118-buscases.AllexperimentswerecodedinC++withCPLEX12.5usingConcert2.9technologyonanIBMSystemx3650withtwoIntelE5640Xeonprocessorsand24GBmemory. 65

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Figure4-2. The6bussystem 4.4.1A6-BusCaseWerstconsidera6-bussystem,basedonthedataprovidedin[ 75 ].Thenetwork,showninFigure 4-2 ,consistsofsixbuses,seventransmissionlines,threegeneratorsatbus1,2,and6,andthreedemandslocatedatbus3,4,and5.ThedetailedinformationofthesystemispresentedinTable 4-1 4-3 .Table 4-1 providesthesend-endbus,receive-endbus,owcapacityandsusceptanceofeachline.Table2presentstheoersizesandpricesateachgenerator,andthesevaluesareassumedtobeconstantthroughouttheplanningyear.Inthissystem,weconsiderthreecandidatesforinstallingwindpower,i.e.,bus1,2,and4.Itisassumedthatthexedandvariablecostsateachbusare$1millionand$8000/MW,respectively. Table4-1. Transmissionlinecharacters linesendBusreceiveBusmaxFlowSusceptance 1122005.88242142006.66673233003.87604242005.07615361007.14296452006.66677564006.2500 Weconsideredtwodemandblocks.Table 4-3 providestheoperatinghoursineachdemandblock,thepeakload,maximumwindpowerinstallment,theexpecteddemand 66

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Table4-2. Generatoroers GeneratorOersizeOerprice 180,80,70,7065,75,85,95265,55,50,4561,71,81,91385,85,65,6563,80,88,98 levelsandintensitylevels.Ateachbuswherethereisademand,ademandlevelisrandomlygeneratedin[0.75,1.0],andthedemandistheproductofpeakloadandthedemandlevel.Similarly,ateachbuswherewindpowerisavailable,awindintensitylevelisrandomgeneratedin[0.35,0.55].Torepresentuncertaintyinscenarios,weassumedcorrelatedscenariossimilarto[ 76 ]:threedemandlevels,i.e.5%higherandlowerthantheexpectedvalue,andthreeintensitylevels,i.e.20%higherandlowerthantheexpectedvalue.Thatis,atotalof9scenarioswithineachdemandblockisconsidered,andweassumedthatallscenariosareequallyweighted. Table4-3. Busdata dmdBlockhoursBuspeakLoaddmdLevelmaxInstallwindIntensity 110001003000.392003000.41633000.920043000.893000.4652000.870060000215001003000.4322003000.3733000.820043000.783000.4252000.810060000 Wersttestedabasecase,whichallowsfullbudget,i.e.sothebudgetconstraintisnotrestrictive,andthelimitsforuncertaintysetaretakenfromthe95%condenceinterval.Theoptimalsolutionistoinstall300MWofwindatallthreecandidatebuses,andobtainaprotof$6.24108.Wethenperformedsensitivityanalysisonbudget,maximumpowerinstallmentatcandidatebuses,andsizeofuncertaintyset.Foreachcase,wepresenttheexpectedprot, 67

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andthepercentagechangerelativetothebasecase,runtimewithandwithoutrobustoptimization.Addingrobustoptimizationresultsinlongerruntimeasexpected,becausethemasterproblemisrevisitedafteraddingeachcut.Wealsoreportthenumberofcutsaddedbytherobustoptimization.Tables 4-4 and 4-5 presenttheresultswhenbudgetgoesfrom100%(basecase),90%,downto60%offullbudget.Ingeneral,theexpectedprotgoesdownasbudgetshrinks.Sinceaddingcutsfromrobustoptimizationreducethefeasibleregion,therisk-averseoptimalsolutionforbasecasesis0.5414%lowerthantheoriginaloptimalsolution.Asbudgetdecreaseto90%and80%,therisk-neutralmaximumprotdecreasesalong,buttherisk-averseoptimalsolutionstaysthesame.At70%and60%budget,addingtherobustoptimizationpartornotcometothesameoptimalsolution,duetothetightnessofthebudgetconstraint.Thepercentagechangesinprotsaresmallerwithrobustoptimizationadded,i.e.robustoptimizationgivesamoresecureinvestmentdecisionasbudgetchanges. Table4-4. Sensitivityanalysis-budget(WIMP) BudgetProt($)%changeTime(s)Bus1Bus2Bus4 Base6.2435E+08-3300.00300.00300.0090%6.2425E+08-0.016%3300.00300.00180.9180%6.2342E+08-0.148%3300.00171.68248.3270%6.1886E+08-0.879%4233.7596.25300.0060%6.1332E+08-1.766%3143.7596.25300.00 Table4-5. Sensitivityanalysis-budget(WIMP+WRP) BudgetProt($)%changeTime(s)Bus1Bus2Bus4Cuts Base6.2097E+08-18255.79221.83213.96490%6.2097E+080.000%14255.79221.83213.96380%6.2097E+080.000%13255.79221.83213.96370%6.1886E+08-0.339%6233.7596.25300.00160%6.1332E+08-1.232%4143.7596.25300.001 Tables 4-6 and 4-7 providetheresultswhenmaximumwindinstallmentatcandidatebusesincreaseordecrease10%or20%fromthebasecaseof300MW.Itisassumedthatfullbudgetisprovidedinallcases.Again,itisshownthataddingtherobustoptimizationprovidesmorestableandconservativesolutionwithrespecttochangesinmaximuminstallment.Comparing 68

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Table4-6. Sensitivityanalysis-installment(WIMP) maxInstallProt%changeTimeBus1Bus2Bus4 3606.2730E+080.473%3360.00221.92360.003306.2590E+080.248%4330.00221.83282.363006.2435E+08-3300.00221.83300.002706.2250E+08-0.296%4270.00221.92238.202406.2045E+08-0.624%3240.00240.00240.00 thepercentagechangesinexpectedprot,robustoptimizationensurestheoptimalobjectivevaluestayingthesamewhenmaximuminstallmentincreaseby20%ordecreaseby10%,whilewithoutit,theobjectivevaluedieredbyalmost$5million.Whenmaximuminstallmentdecreaseby20%,robustoptimizationdoesnotaddadditionalrestrictiveconstraint. Table4-7. Sensitivityanalysis-installment(WIMP+WRP) maxInstallProt%changeTimeBus1Bus2Bus4cuts 3606.2097E+080.000%23255.75360.00213.9053306.2097E+080.000%17255.79330.00213.9633006.2097E+08-18255.79300.00213.9642706.2097E+080.000%14255.75270.00213.9042406.2045E+08-0.083%3240.00240.00240.000 Table 4-8 presentstheresultswhensizeofuncertaintysetstakevaluesfrom85%upto99%condenceintervals(CI),andthepercentagechangeoftheobjectivevalueswithrespecttotheresultfromtheintegratedmasterproblem.Asisshown,thewiderthecondenceintervalis,themoreconservativetheresultsare,becausetheworstcasescenarioistakenfromawiderrange.Whileuncertaintysetfrom85%CIistoonarrowtogenerateadditionalrestrictions,uncertaintysetfrom99%CIshrinkstheexpectedprotbylessthan1%. Table4-8. SensitivityAnalysis-uncertaintyset SizeProt($)%changeBus1Bus2Bus4Time(s)cuts Master6.2435E+08-300.00300.00300.003-85%6.2435E+080.000%300.00300.00300.003090%6.2379E+08-0.090%300.00212.82232.8112395%6.2097E+08-0.542%255.79221.83213.9618499%6.1898E+08-0.860%226.01161.25255.47153 69

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4.4.2IEEE118-BusCaseThebasicdataoftheIEEE118-Bustestsystemisprovidedin[ 77 ].Thesystemconsistsof186transmissionlines,54generators,and91demands.Weassumethateachgeneratorprovidesthreeblocksizes,i.e.,50%,25%,and25%ofitsmaximumcapacity.Asismentionedbefore,theconventionalgeneratorsareassumedtooerelectricityattheirmarginalprices,oerpricesareassignedasthemeanvaluesofthequadraticcostfunctionfromthecorrespondingintervalsofoersizes.Thexedandvariablecostsatallcandidatebusesareassumedtobe$1millionand$0.1million,respectively.Withthesamesetupasthe6-buscase,weconsiderninescenarioswithineachofthetwodemandblocks.TheexpecteddemandlevelsforalldemandsandexpectedintensitiesatcandidatebusesareprovidedinTable 4-9 . Table4-9. 118-busdata dmdBlockdmdLevelBus4-candidateIntensity6-candidateIntensity 10.92070.3900.390330.4160.41654-0.450730.4600.460890.4000.400100-0.47020.81070.4320.432330.3700.37054-0.510730.4200.420890.5300.530100-0.440 Inthissystem,wersttestthebasecaseon4-candidatecase,i.e.,bus7,33,73,89,and6-candidatecase,i.e.,bus54and100additionaltopreviouscandidates.Thebasecases,again,allowfullbudget,withuncertaintysettakenfromthe95%CI.AsisshowninTable 4-10 ,theoptimalsolutionsfromintegratedmasterproblemforboth4-candidateand6-candidatecasesaretoinstallfullcapacityof300MWatallcandidatebuses.Withtherobustoptimizationapplied,thewindpowerinstallmentatbus73waslimitedto200MWin4-candidatecase,and180MWin6-candidatescase.Whennumberofcandidatebusesincreasesfromfourtosix, 70

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morewindcapacityisavailableintheentirenetwork,addingtherobustoptimizationbecomesmoresensitivetolesspreferredLMPprices,andresultsinlowerwindpowerinstallmentatbus73. Table4-10. 118-busbasecase ModelProt($)Time(s)Bus7Bus33Bus54Bus73Bus89Bus100 4-candidateWIMP1.21E+0926300300-300300-WIMP+WRP1.12E+0986300300-200300-6-candidateWIMP1.86E+094930300300300300300300WIMP+WRP1.77E+0919175300300300180300300 Wethencomparetheinvestmentcostandprotdierencesunderworst-casescenariosbetweenrisk-neutral(WIMP)andrisk-averse(WIMP+WRP)decisions,forboth4-candidateand6-candidatecases.Thatis,weusetheoptimalsolutionsfromtheintegratedmasterproblem,tocalculatetheinvestmentcostandprotunderworst-casescenarios,andcomparethesevalueswiththeoptimalsolutionsfromtherobustoptimization. Table4-11. Protchangesunderworst-casescenario:4-candidate CasesProt($)extraInv($)Prot($)Time(s)CutsWIMP+WRPunderWorstWIMP+WRP Base1.215E+091.12E+091.00E+071.112E+0926862Budget90%1.111E+091.103E+090.00E+001.103E+092154280%9.958E+089.877E+080.00E+009.877E+0816522maxInstall2701.106E+091.031E+097.00E+061.024E+09146022409.842E+089.406E+084.00E+069.366E+0820622Uncertaintyset97%1.098E+091.22E+071.086E+0945199%1.096E+091.24E+071.084E+09501 Tables 4-11 and 4-12 exhibittheresultsforthebasecase,andcasewithchangesinbudget(10%or20%decreaseinbudget),maximumwindpowerinstallment(10%or20%decreaseinmaximuminstallment),andsizeofuncertaintyset(use97%or99%CIs).ThesecondandthirdcolumnspresenttheprotfromWIMPandtheproblemwithrobust 71

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optimizationadded.ColumnextraInvgivestheextrainvestmentcostofrisk-neutraldecisionsiftheworst-casescenariohappens,andthefollowingcolumnProtunderWorstgivestheprotofrisk-neutraldecisionsundertheworst-casescenario.Theruntimeoftheproblemwithorwithoutrobustoptimization,andthenumberofcutsaddedfromtherobustoptimizationarereportedinthelastthreecolumns. Table4-12. Protchangesunderworst-casescenario:6-candidate CasesProt($)extraInv($)Prot($)Time(s)CutsWIMP+WRPunderWorstWIMP+WRP Base1.859E+091.766E+091.20E+071.754E+094930191751Budget90%1.731E+091.723E+090.00E+001.723E+0915082473180%1.597E+091.588E+090.00E+001.588E+091512191maxInstall2701.715E+091.643E+099.00E+061.634E+0960169412401.557E+091.498E+096.00E+061.492E+09951691Uncertaintyset97%1.765E+091.22E+071.752E+0921531199%1.763E+091.24E+071.751E+09101161 Itisshown,fromTable 4-11 and 4-12 ,thatalthoughtheexpectedprotofWIMPishigherthanitwithWRP,underworst-casescenario,theprotgeneratedbytheserisk-neutraldecisionarelowerthanprotgeneratedbyrisk-aversedecisionsinthebasecases,andcaseswhenmaximuminstallmentorsizeofuncertaintysetchanges.Whenbudgetdecreases,thewindpowerinstallmentatdierentbusesareadjustedbytherobustoptimization,butthetotalinstallmentremainsthesame.Sinceweassumedequalinvestmentcostatallcandidatebuses,theprotgeneratedbyrisk-neutraldecisionsisthesameasrisk-aversedecisionsunderworst-casescenarios.WecanalsocomparethepercentagechangesinprotbyWIMPanditwithWRP.In4-candidatecases,whenbudgetshrinksby20%,protfromWIMPdecreasesby18.01%,andby11.99%withWRPadded;whenmaximuminstallmentreducesby20%,protfromWIMPdecreasesby18.97%and16.18%withWRPadded.In6-candidatecases,20%decreaseofbudgetresultsin14.09%protdecreasesbyWIMPandby10.10%whenWRPisadded; 72

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20%decreaseinmaximuminstallmentresultsin16.21%decreaseinprotfromWIMPand15.18%whenWRPisadded.Thatis,addingWRPoccursmorerobustresultsinallcases.Intheproposedrobustmodel,thereareatotalof41312variables,amongwhich11206binaryvariables,and30106continuousvariables.Allprogramsarecompletedwithinreasonabletime. 4.5ConcludingRemarksInthischapter,weproposedauniedstochasticandrobustoptimizationbilevelmodeltosolveawindpowerinvestmentproblem,forpotentialinvestorsinanexistingtransmissionnetwork.Weusedthestochasticoptimizationapproachtosimulateuncertainty,andtherobustoptimizationapproachtocapturetheworst-casescenarioswithinanuncertaintyset.Stochasticoptimizationalonerequiresafairnumberofscenariostomodeluncertainty,whilerobustoptimizationaloneresultsinextremeconservativedecisions.Theuniedmodeltakestheadvantageofbothmethodstoreacharisk-aversedecisionforthewindpowerinvestors,whilestayingawayfromtheextremesofeachoneofthem. 73

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CHAPTER5SIMULTANEOUSWINDPOWERANDTRANSMISSIONINVESTMENTFORVERTICALLYINTEGRATEDUTILITIES 5.1MotivationandLiteratureReviewWiththeconcernsofglobalwarmingandpollutionoverthepast20years,peoplebegantolookforpossibilitiesofintegratingrenewableenergyintotheelectricitymarket.Amongthevariousrenewableenergysources,windpowerisoneofthemostaordableandcontrollablesources[ 51 , 52 ].In2014,79.1%ofnewpowerinstallationintheEuropeanUnion(EU)wasrenewable.Whilewindpower,beingthetechnologywiththehighestinstallationrate,accountedfor43.7%oftotalnewenergyinstallationsinEUincludingbothconventionalandrenewableenergysources[ 78 ].Governmentsaroundtheworldhaveestablishedpublicpoliciesandregulationstopromotethegrowthofwindcapacity[ 55 ].AswindpowerpenetrationincreasesrapidlyintheU.S.,thederegulatedpowerpoolsareformedinsomestates,replacingthetraditionalverticallyintegratedutilities.Windpowerinvestorsneedtorecovertheirinvestmentfromthemarket,buttheirprojectsareusuallypartiallysubsidizedbypublicfunds[ 52 , 71 ].Therefore,itisalsointhecentralplanner'sinteresttominimizethecostsofthewindproject[ 79 ].Meanwhile,morethanhalfofthestatesarestillservedbytheverticallyintegratedutilities.Toevaluatetheprotabilityofpotentialwindprojects,rstly,windprojectsarecapitalintensiveinvestments.Thedevelopmentoftechnology,alongwiththemarketgrowth,hasdramaticallydecreasedtheinvestmentcostsovertheyears,butalsoencouragedlargenumberofnewprojectsandnewplayerstojointhemoreandmorematuremarket.Besides,theessentialfeaturesofthewindsource,suchasintermittentoutput,non-dispatchableproduction,anduncertainintensity,bringindeterminacy.Moreover,theuctuationofmarketdemand,togetherwiththeaforementionedcharacteristicsofwind,createvolatilityintheenergymarketprice,andfurtherdenetheriskassociatedwithwindpowerinvestment[ 51 ].Alongwiththedevelopmentofrenewableenergycomesdiscussionsontransmissionnetworkexpansionandsystemreconstruction.Theoriginallyelectricitynetworkwas 74

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designedforconventionalgenerators.Inadditiontothesystemreliability,thecongestionofexistingnetwork,therateofinvestmentrecovery,andotherrelatedissues,suchaslandandweatherconditions,etc.,thetransmissionnetworkinvestmentnowadaysneedtoconsidernewrequirementsraisedbytheemergeofrenewableenergysourcesinthelongrun.Consequently,networkexpansionplanningandrenewableenergyinvestmentarenaturallyboundedtogether[ 80 , 81 ].Studieshavebeenconductedontransmissionplanningandconventionalgenerationexpansions[ 82 { 86 ].Rohetal.[ 85 ]simulateajointauctionmarketwithcompetitiveresources.Lopezetal.[ 82 ]proposeastochasticmodelwithuncertaindemand.Astandarddeviationtermisaddedtotheobjectivefunctionasforriskmeasure,andamixedintegernonlinearprogramissolved.Wangetal.[ 86 ]formulateabi-levelmodel,wheretheupperlevelsolvestheoptimalplanningstrategiesandthelowerlevelensuresmarketclearance.MurphyandSmeers[ 84 ]proposedthreesequentialmodelsthatreectperfectcompetition,longtermoligopoly,andtwostageCournotgamewithrststageinvestmentdecisionsandsecondstagesalesdecisions.Jenabietal.[ 83 ]alsopresentabi-levelmodelforsimultaneousdecisionongeneratorandtransmissioninvestments.Firstlevelplansthetransmissionnetworksandthesecondleveldecidesongenerationexpansion.Theycomparetheresultsbetweendierentincentivesonobjectivesofbothlevels,maximizingsocialwelfare,orinvestorprot.Inmorerecentliteratures,discussionshavebeencarriedoutonintegratingthewindpowerinvestmentintothetransmissionnetworkexpansion.Siano[ 87 ]provideprobabilisticevaluationsonwindpenetration,wherewindintensityandloaduncertaintyaresimulatedbyMonteCarlosimulations.ParkandBaldick[ 88 ]presentatwo-stagestochasticmodelminimizingtotalcostanduseasequentialapproximationapproach.Windintensityanddemandarerepresentedbydependentrandomvariables.Maurovich-Horvatetal.[ 89 ]proposedbi-levelstochasticmodelstocomparetheinvestmentdecisionsbymerchantinvestor(protmaximizing),transmissionsystemoperator(TSO)orcentralplanner(socialwelfaremaximizing).Theupperlevelplansthetransmissioninvestment,andtheconventionalandwindproducersmakeinvestment 75

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decisioninthelowerlevelbasedonthegiventransmissionnetwork.BaringoandConejo[ 79 ]designatwostagestochasticmodelforwindplantinvestmentandtransmissionnetworkreinforcementdecisions,aimingtomaximizesocialwelfare.Themodelisthenreformulatedwithequilibriumconstraints,anduncertaintyinwindintensityanddemandarepresentedinscenarios.Inthischapter,ascomparedtotheexistingstudiesdescribedabovemostlyforthederegulatedmarkets,weproposeatwo-stagestochasticoptimizationmodelforaverticallyintegratedutilityinaregulatedmarketenvironment.Weconsiderbothwindpowerandtransmissionlineinvestmentswiththeobjectiveofminimizingtotalcostoveragivenplanninghorizon.Thisstudyextendstheworkdescribedin[ 79 ]byconsideringtheinvestmentplanforaverticallyintegratedutilityinsteadofamarketparticipantinaderegulatedmarketenvironment,plustherobustoptimizationapproachtohedgeagainsttheworst-casescenarios,soastoprovidearisk-aversesolution.Meanwhile,thisstudyextendstheworkdescribedin[ 90 ]byconsideringwindpowerinvestmentinsteadofconventionalgenerationcapacityexpansion,andusingauniedstochasticandrobustoptimizationapproachascomparedtorobustoptimizationapproaches.Therestofthechapterisorganizedasfollows.InSection 5.2 ,wepresentthenotationandmathematicalformulation.Ourmodelcapturesbothstochasticandrobustoptimizationfeatures.Section 5.3 discussesthesolutiontechniquestosolvetheproposedmodel,whichincludesthedevelopmentofadecompositionprocedure.Inparticular,thestochasticoptimizationpartistransformedintoanequivalentMILPandintegratedintothemasterproblem.Fortherobustoptimizationpart,weprovideaBender'sdecompositionalgorithmtosolvetheproblem.Finally,inSection 5.4 ,weapplyourproposedapproachtosolvethesix-busandIEEE118-bussystems,andcomputationalresultsshowtheeectivenessofourapproach. 5.2NotationandMathematicalFormulation 5.2.1NomenclatureThissubsectionliststheindices,constants,andvariablesneededforourproposedmodel. 76

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Indices DB Indexsetofdemandblocks N Indexsetofbuses T Indexsetofyearsintheplanninghorizon L Indexsetofexistingtransmissionlines LN Indexsetofcandidatetransmissionlines Bi Indexsetofblocksoftheithgenerationunit G Indexsetofallgenerationunits Gn Indexsetofgenerationunitsatbusn Indexsetofscenarios Kn Indexsetofwindpowerblocksatbusn Constants w! Weightofscenario! FWn Fixedwindpowerinvestmentcostatbusn VWn Variablewindpowerinvestmentcostatbusn CLl Costofbuildingtransmissionlinel Pwindn,k Uptokblocksofwindpowercapacitytobeinstalledatbusn CRn,t Costofloadcurtailmentatbusninyeart CGn,t Powerproductioncostofconventionalgeneratoratbusninyeart o(l) Sending-endbusoflinel r(l) Receiving-endbusoflinel Bl Susceptanceoflinel fmaxl Transmissioncapacityonlinel dn,t,b,! Powerloadatbusnindemandblockbofyeartandscenario! 77

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dn,t,b Expectedpowerloadatbusnindemandblockbofyeart Gft,b Capacityfactorofgenerationunitindemandblockbofyeart Pn,t Productioncapacityofgenerationunitatbusninyeart In,t,b,! Windintensityatbusnindemandblockbofyeartandscenario! min Lowerlimitforthevoltageangles max Upperlimitforthevoltageangles Decisionvariables Yn Binaryindicatorforset-upwindpoweratbusn Xwindn Installedwindpowercapacityatbusn xwindn,k Binaryindicatorthatequals1ifkblocksofwindpowercapacityisinstalledatbusn Xlinel Binaryindicatorforbuildingtransmissionlinel pn,t,b,! Windpowerproductionatbusnindemandblockbofyeartinscenario! gn,t,b,! Powerproducedbygeneratoratbusnindemandblockbofyeartinscenario! ~gn,t,b Powerproducedbygeneratoratbusnindemandblockbofyeart rn,t,b,! Loadsheddingatbusnindemandblockbofyeartinscenario! ~rn,t,b Loadsheddingatbusnindemandblockbofyeart fl,t,b,! Powerowthroughlinelindemandblockbofyeartinscenario! ~fl,t,b Powerowthroughlinelindemandblockbofyeart n,t,b,! Voltageangleatbusnindemandblockbofyeartinscenario! ~n,t,b Voltageangleatbusnindemandblockbofyeart Windintensityfactoratbusnforrobustoptimization 5.2.2MathematicalModelInthisproblem,weconsideralongtermwindpowerandtransmissionlineinvestmentplanbyaverticallyintegratedutility.Atwo-stagemodelisconstructed,withtherststagemakinginvestmentdecisions,andthesecondstagemakingoperationaldecisionsgivenresults 78

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fromtherststage.Uncertaintyofwindintensityiscapturedbysecond-stagestochasticandrobustoptimization:thestochasticoptimizationmodelswindintensityanddemandlevelinscenarios,andtherobustoptimizationhedgestheworst-casescenariousinganuncertaintyset.Themathematicalformulation,denotedasRWTI,isdesignedasfollows: minXn2N(FWnYn+VWnXwindn)+Xl2LNCLlXlinel+minp,g,r,f,Q1+(1)]TJ /F10 11.955 Tf 11.96 0 Td[()max2Imin~g,~r,~f,~Q2 (5{1) s.t.XwindnMYn,8n2N (5{2) Xwindn=Xk2Knxwindn,kPwindn,k,8n2N (5{3) Xk2Knxn,k=1,8n2N (5{4) xwindn,k,Ywindn,Xlinel,2f0,1g (5{5) whereminp,g,r,f,Q1=Xn,t,b,!w!(CGn,tgn,t,b,!+CRn,trn,t,b,!) (5{6) s.t.npn,t,b,!+gn,t,b,!+Xl:r(l)=nfl,t,b,!)]TJ /F7 11.955 Tf 18.75 11.36 Td[(Xl:o(l)=nfl,t,b,!=dn,t,b,!)]TJ /F4 11.955 Tf 11.95 0 Td[(rn,t,b,!:n,t,b,!,8n2N (5{7) 0pn,t,b,!In,t,b,!Xwindn:n,t,b,!,8n2N (5{8) fl,t,b,!=Bl(o(l),t,b,!)]TJ /F10 11.955 Tf 11.96 0 Td[(r(l),t,b,!):l,t,b,!,8l2L (5{9) fl,t,b,!)]TJ /F4 11.955 Tf 11.95 0 Td[(Bl(o(l),t,b,!)]TJ /F10 11.955 Tf 11.96 0 Td[(r(l),t,b,!)+(1)]TJ /F4 11.955 Tf 11.95 0 Td[(Xlinel)M0:al,t,b,!,8l2LN (5{10) fl,t,b,!)]TJ /F4 11.955 Tf 11.95 0 Td[(Bl(o(l),t,b,!)]TJ /F10 11.955 Tf 11.96 0 Td[(r(l,t,b,!))]TJ /F5 11.955 Tf 11.95 0 Td[((1)]TJ /F4 11.955 Tf 11.95 0 Td[(Xlinel)M0:bl,t,b,!,8l2LN (5{11) )]TJ /F4 11.955 Tf 9.3 0 Td[(fmaxlfl,t,b,!fmaxl:al,t,b,!,bl,t,b,!,8l2L (5{12) )]TJ /F4 11.955 Tf 9.3 0 Td[(fmaxlXlinelfl,t,b,!fmaxlXlinel:al,t,b,!,bl,t,b,!,8l2LN (5{13) 0gn,t,b,!Gft,bPn,t:n,t,b,!,8n2N (5{14) minn,t,b,!max: an,t,b!, bn,t,b,!,8n2N (5{15) rn,t,b,!0,8n2N (5{16) 79

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o8t2T,b2B,!2andmax2Imin~g,~r,~f,~Q2=Xn,t,b(CGn,t~gn,t,b+CRn,t~rn,t,b) (5{17) s.t.nn,t,bXwindn+~gn,t,b+Xk:r(k)=n~fl,t,b)]TJ /F7 11.955 Tf 20.9 11.36 Td[(Xk:o(k)=n~fl,t,b=dn,t,b)]TJ /F5 11.955 Tf 11.57 0 Td[(~rn,t,b:~n,t,b,8n2N (5{18) ~fl,t,b=Bl(~o(l),t,b)]TJ /F5 11.955 Tf 12.68 2.65 Td[(~r(l),t,b):~l,t,b,8l2L (5{19) ~fl,t,b)]TJ /F4 11.955 Tf 11.96 0 Td[(Bl(~o(l),t,b)]TJ /F5 11.955 Tf 12.69 2.66 Td[(~r(l),t,b)+(1)]TJ /F4 11.955 Tf 11.96 0 Td[(Xlinel)M0:~al,t,b,8l2LN (5{20) ~fl,t,b)]TJ /F4 11.955 Tf 11.96 0 Td[(Bl(~o(l),t,b)]TJ /F5 11.955 Tf 12.69 2.66 Td[(~r(l),t,b))]TJ /F5 11.955 Tf 11.96 0 Td[((1)]TJ /F4 11.955 Tf 11.96 0 Td[(Xlinel)M0:~bl,t,b,8l2LN (5{21) )]TJ /F4 11.955 Tf 9.3 0 Td[(fmaxl~fl,t,bfmaxl:~al,t,b,~bl,t,b,8l2L (5{22) )]TJ /F4 11.955 Tf 9.3 0 Td[(fmaxlXlinel~fl,t,bfmaxlXlinel:~al,t,b,~bl,t,b,8l2LN (5{23) 0~gn,t,bGft,bPn,t:~n,t,b,8n2N (5{24) min~n,t,bmax:~ an,t,b,~ bn,t,b,8n2N (5{25) ~rn,t,b0,8n (5{26) g8t2T,b2BTheuncertaintysetIisdenedas: I=f2RjNjjTjjDBj,Iminn,t,bn,t,bImaxn,t,b,IminXn,t,bn,t,bImaxg (5{27) Intheaboveformulation,( 5{2 )-( 5{5 )describetherst-stageproblem.Theobjective( 5{1 )minimizesthetotalcostovertheplanninghorizon:initialinvestmentcostofwindpowerandtransmissionlinefromtherststage,andtheexpectedcostfromconventionalpowerproductionandloadsheddingfromthesecondstage.Sinceweconsideralongtermplan,theinvestmentcostsofwindpowerinstallmentandtransmissionlinesarethesumofthepresentvaluesofinitialinvestmentcostsandannualcostsincurredovertheplanningyears.Thecostsofconventionalproductionandloadsheddingcomefrombothstochasticand 80

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robustoptimization,andtheexpectedcostistheweightedaverageoftwoobjectives.Constraint( 5{2 )indicateswhetherwindpowerisinstalledatbusn,andconstraint( 5{3 )and( 5{4 )decidewindcapacity,i.e.,thenumberofthewindpowerblocksinstalledateachbus.Forexample,supposethatwindpowerisavailablein50MW-block,andatmost4blockscanbeinstallatbusn,thenPwindn,1=50,Pwindn,2=100,Pwindn,3=150,Pwindn,4=200.Iftheinvestmentdecisionistoinstallupto3blocksatthebus,wehavexn,3=1,andxn,k=0fork6=3.Second-stagestochasticoptimizationproblemis( 5{6 )-( 5{16 ),andsecond-stagerobustoptimizationproblemis( 5{17 )-( 5{26 ).Thestochasticobjective( 5{6 )takestheweightedaverageoftheconventionalproductioncostandloadsheddingfromallscenarios,andtherobustobjective( 5{17 )establishthecostunderworst-casescenariooftheuncertaintyset.Weassumethatallscenariosareequallyweighted.Constraints( 5{7 )and( 5{18 )presentstheloadbalanceateachbus.Constraint( 5{8 )imposestheupperboundofwindpowerproductiontobetheproductofwindintensityandthepowercapacityinstalledatthatbus.Constraints( 5{9 )and( 5{19 )denethepowerowontheexistingtransmissionlines,andsimilarly,constraints( 5{10 )-( 5{11 )and( 5{20 )-( 5{21 )denethepowerowonthenewtransmissionlinesifrst-stageproblemdecidestobuildthem.Constraints 5{12 -( 5{13 )and( 5{22 )-( 5{23 )ensurethatthepowerowsonbothexistingandcandidatetransmissionlinesarewithincapacity.Constraints( 5{14 )and( 5{24 )indicatethattheproductionupperboundofaconventionalgeneratoristheproductofafactorandtheoriginalcapacityonthatunit.Forsimplicity,thelowerboundsforconventionalproductionaredenedtobezeros,andthesevaluescaneasilybereplacedbypositivelevels.Constraints( 5{15 )and( 5{25 )settheupperandlowerboundsofthevoltageangleateachbus.Finally,theuncertaintysetforrobustoptimizationisdenedas( 5{27 ).Thelowerandupperboundscanbeobtainedfromancondenceinterval(CI)ofhistoricaldata. 81

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5.3LinearizationandDecompositionThissectionpresentsthesolutionprocedureforsolvingRWTI.Thestochasticoptimizationproblemislinearizedandthenintegratedintothemasterproblem,whiletherobustoptimizationproblemisdualizedandoptimalitycutsaregenerated. 5.3.1StochasticOptimizationSimilarto[ 79 ],wetaketheprimal-dualapproachtothestochasticoptimizationpart.Inadditiontotheprimalconstraints( 5{7 )-( 5{16 ),weadditsdualconstraintsandenforcethestrongdualityequality.Thedualconstraintsareasfollows: nn,t,b,!)]TJ /F10 11.955 Tf 11.96 0 Td[(n,t,b,!w!CGn,t,8n2N (5{28) n,t,bw!CRn,t,8n2N (5{29) n,t,b,!)]TJ /F10 11.955 Tf 11.96 0 Td[(n,t,b,!0,8n2N (5{30) r(l),t,b,!)]TJ /F10 11.955 Tf 11.96 0 Td[(o(l),t,b,!+l,t,b,!+al,t,b,!)]TJ /F10 11.955 Tf 11.96 0 Td[(bl,t,b,!=0,8l2L (5{31) r(l),t,b,!)]TJ /F10 11.955 Tf 11.96 0 Td[(o(l),t,b,!+al,t,b,!)]TJ /F10 11.955 Tf 11.96 0 Td[(bl,t,b,!+al,t,b,!)]TJ /F10 11.955 Tf 11.96 0 Td[(bl,t,b,!=0,8l2LN (5{32) )]TJ /F7 11.955 Tf 23.31 11.36 Td[(Xo(l)=n,l2LBll,t,b,!+Xr(l)=n,l2LBll,t,b,!)]TJ /F7 11.955 Tf 26.69 11.36 Td[(Xo(l)=n,l2LNBl(al,t,b,!)]TJ /F10 11.955 Tf 11.96 0 Td[(bl,t,b,!)+Xr(l)=n,l2LNBl(al,t,b,!)]TJ /F10 11.955 Tf 11.96 0 Td[(bl,t,b,!)+ an,t,b,!)]TJ /F10 11.955 Tf 11.96 0 Td[( bn,t,b,!=0,8n2N (5{33) n,t,b,!,l,t,b,!,free,n,t,b,!,al,t,b,!,bl,t,b,!,al,t,b,!,bl,t,b,!,n,t,b,!, an,t,b,!, bn,t,b,!0 (5{34) o8t2T,b2DB,!2wheren,t,b,!,l,t,b,!,n,t,b,!,al,t,b,!,bl,t,b,!,al,t,b,!,bl,t,b,!,n,t,b,!, an,t,b,!,and bn,t,b,!aredualvariables.Sinceallconstraints( 5{7 )-( 5{16 )arelinear,strongdualityholds.AccordingtotheStrongDualityTheorem,theprimalanddualoptimalobjectivevaluesareequal,sowe 82

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havethefollowingstrongdualityequality: Xn,t,b,!w!(CGn,tgn,t,b,!+CRn,trn,t,b,!)=Xn,t,b,!n,t,b,!dn,t,b,!)]TJ /F7 11.955 Tf 15.6 11.36 Td[(Xn,t,b,!n,t,b,!In,t,b,!Xwindn)]TJ /F7 11.955 Tf 23.26 11.36 Td[(Xl2LN,t,b,!(al,t,b,!+bl,t,b,!)(1)]TJ /F4 11.955 Tf 11.95 0 Td[(Xlinel)M)]TJ /F7 11.955 Tf 19.88 11.36 Td[(Xl2L,t,b,!(al,t,b,!+bl,t,b,!)fmaxl)]TJ /F7 11.955 Tf 23.26 11.36 Td[(Xl2LN,t,b,!(al,t,b,!+bl,t,b,!)fmaxlXlinel)]TJ /F7 11.955 Tf 15.6 11.35 Td[(Xn,t,b,!n,t,b,!Gft,bPn,t+Xn,t,b,! an,t,b,!min)]TJ /F7 11.955 Tf 16.26 11.35 Td[(Xn,t,b,! bn,t,b,!max (5{35) Therearetwocasesofbilineartermsintheaboveequation( 5{35 ).Intherstcase,thebilineartermshavetheformsmax(1)]TJ /F4 11.955 Tf 12.52 0 Td[(X)andmaxX,where,0,and1)]TJ /F4 11.955 Tf 12.51 0 Td[(X,X2f0,1g.Lety=(1)]TJ /F4 11.955 Tf 12.14 0 Td[(X)and~y=X,theneachtermcanbereplacedbythefollowingbig-Mformulation: maxy (5{36) s.t.y+XM (5{37) y(1)]TJ /F4 11.955 Tf 11.96 0 Td[(X)M (5{38) max~y (5{39) s.t.~y+(1)]TJ /F4 11.955 Tf 11.96 0 Td[(X)M (5{40) ~yXM. (5{41) Intheothercase,thebilineartermshavetheformX.Bydenition,similarto[ 79 ],0,X=PxP,andP0,x2f0,1g,sowecanrewritethetermasX=PxP.Letz=xP,theneachxPtermcanbereplacebythefollowingsetofconstraints: z=P)]TJ /F10 11.955 Tf 11.96 0 Td[( (5{42) xP zxP (5{43) (1)]TJ /F4 11.955 Tf 11.95 0 Td[(x)P (1)]TJ /F4 11.955 Tf 11.95 0 Td[(x)P, (5{44) whereand aretheupperandlowerboundsfor. 83

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Consequently,constraint( 5{35 )canbereplacedbythefollowingconstraints: Xn,t,b,!w!(CGn,tgn,t,b,!+CRn,trn,t,b,!)=Xn,t,b,!n,t,b,!dn,t,b,!)]TJ /F7 11.955 Tf 20.55 11.36 Td[(Xl2L,t,b,!(al,t,b,!+bl,t,b,!)fmaxl)]TJ /F7 11.955 Tf 22.6 11.36 Td[(Xl2LN,t,b,!(yal,t,b,!+ybl,t,b,!)M)]TJ /F7 11.955 Tf 23.26 11.36 Td[(Xl2LN,t,b,!(~yal,t,b,!+~ybl,t,b,!)fmaxl)]TJ /F7 11.955 Tf 16.27 11.36 Td[(Xn,t,b,!In,t,b,!zn,t,b,!,k)]TJ /F7 11.955 Tf 15.6 11.36 Td[(Xn,t,b,!n,t,b,!Gft,bPn,t+Xn,t,b,! an,t,b,!min)]TJ /F7 11.955 Tf 16.26 11.36 Td[(Xn,t,b,! bn,t,b,!max (5{45) nyal,t,b,!al,t,b,!+XlinelM,8l2LN (5{46) yal,t,b,!(1)]TJ /F4 11.955 Tf 11.95 0 Td[(Xlinel)M,8l2LN (5{47) ybl,t,b,!bl,t,b,!+XlinelM,8l2LN (5{48) ybl,t,b,!(1)]TJ /F4 11.955 Tf 11.95 0 Td[(Xlinel)M,8l2LN (5{49) ~yal,t,b,!al,t,b,!+(1)]TJ /F4 11.955 Tf 11.96 0 Td[(Xlinel)M,8l2LN (5{50) ~yal,t,b,!XlinelM,8l2LN (5{51) ~ybl,t,b,!bl,t,b,!+(1)]TJ /F4 11.955 Tf 11.96 0 Td[(Xlinel)M,8l2LN (5{52) ~ybl,t,b,!XlinelM,8l2LN (5{53) zn,t,b,!,k=n,t,b,!Pwindn,k)]TJ /F10 11.955 Tf 11.96 0 Td[(n,t,b,!,k,8n2N,k2Kn (5{54) xwindn,kPwindn,k zn,t,b,!,kxwindn,kPwindn,k,8n2N,k2Kn (5{55) (1)]TJ /F4 11.955 Tf 11.95 0 Td[(xwindn,k)Pwindn,k n,t,b,!,k(1)]TJ /F4 11.955 Tf 11.95 0 Td[(xwindn,k)Pwindn,k,8n2N,k2Kn (5{56) yal,t,b,!,ybl,t,b,!,~yal,t,b,!,~ybl,t,b,!,zn,t,b,!,k,n,t,b,!,k0 (5{57) ot2T,b2DB,!2Fromhere,wecanintegratethestochasticdualconstraintsandequivalentlinearconstraintsforstrongdualityequalitytotherst-stageproblemandformthefollowingintegratedmasterproblem,denotedasWTIMP: minXn2N(FWnYn+VWnXwindn)+Xl2LNCLlXlinel+Xn,t,b,!w!(CGn,tgn,t,b,!+CRn,trn,t,b,!)+(1)]TJ /F10 11.955 Tf 11.96 0 Td[()Q2s.t.( 5{2 )-( 5{5 ),( 5{7 )-( 5{16 ),( 5{28 )-( 5{34 ),and( 5{45 )-( 5{57 ). 84

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5.3.2RobustOptimizationForthesecond-stagerobustoptimization,werstdualizetherobustconstraints( 5{18 )-( 5{26 )andgetthefollowingdualproblem: max2Imax~,~,~,~,~'Q=Xn,t,b~n,t,b(dn,t,b)]TJ /F10 11.955 Tf 11.96 0 Td[(n,t,bXwindn))]TJ /F7 11.955 Tf 19.26 11.36 Td[(Xl2LN,t,b(~al,t,b+~bl,t,b)(1)]TJ /F4 11.955 Tf 11.95 0 Td[(Xlinel)M)]TJ /F7 11.955 Tf 15.88 11.35 Td[(Xl2L,t,b(~al,t,b+~bl,t,b)fmaxl)]TJ /F7 11.955 Tf 19.26 11.35 Td[(Xl2LN,t,b(~al,t,b+~bl,t,b)fmaxlXlinel)]TJ /F7 11.955 Tf 11.59 11.36 Td[(Xn,t,b~n,t,bGft,bPn,t+Xn,t,b~ an,t,bmin)]TJ /F7 11.955 Tf 12.26 11.36 Td[(Xn,t,b~ bn,t,bmax (5{58) s.t.n~n,t,bCRn,t,8n2N (5{59) ~n,t,b)]TJ /F5 11.955 Tf 12.8 2.65 Td[(~n,t,bCGn,t,8n2N (5{60) ~r(l),t,b)]TJ /F5 11.955 Tf 13.43 2.66 Td[(~o(l),t,b+~l,t,b+~al,t,b)]TJ /F5 11.955 Tf 12.96 2.66 Td[(~bl,t,b=0,8l2L (5{61) ~r(l),t,b)]TJ /F5 11.955 Tf 13.43 2.66 Td[(~o(l),t,b+~al,t,b)]TJ /F5 11.955 Tf 13.25 2.66 Td[(~bl,t,b+~al,t,b)]TJ /F5 11.955 Tf 12.96 2.66 Td[(~bl,t,b=0,8l2LN (5{62) )]TJ /F7 11.955 Tf 23.31 11.35 Td[(Xo(l)=n,l2LBl~l,t,b+Xr(l)=n,l2LBl~l,t,b)]TJ /F7 11.955 Tf 26.69 11.35 Td[(Xo(l)=n,l2LNBl(~al,t,b)]TJ /F5 11.955 Tf 13.26 2.65 Td[(~bl,t,b)+Xr(l)=n,l2LNBl(~al,t,b)]TJ /F5 11.955 Tf 13.26 2.66 Td[(~bl,t,b)+~ an,t,b)]TJ /F5 11.955 Tf 14.14 2.66 Td[(~ bn,t,b=0,8l2L[LN (5{63) ~n,t,b,~l,t,bfree,~al,t,b,~bl,t,b,~al,t,b,~bl,t,b,~n,t,b,~ an,t,b,~ bn,t,b0 (5{64) o8t2T,b2DB.Thereisabilinearterminthedualobjective( 5{58 ),whereandarebothcontinuousvariables.SimilartoChapter 4 ,weuseaBender'sdecompositionmethodtosolvethebilinearissue,butwiththeprimalapproach.LetXrepresenttherst-stageinvestmentdecisionvariables,XwindnandXlinel,andlet~representdualvariablesf~,~,~a,~b,~a,~b,~,~'a,~'bg.Forxedvaluesofand~,wedenethefollowingtwosubproblems: SUB1:max2Imax~Q1(X,)=Q(X,,~)s.t.( 5{59 )-( 5{64 )SUB2:max2Imax~Q2(X,~)=Q(X,,~)s.t.( 5{59 )-( 5{64 ) 85

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Similartotheprimalapproachdescribedin[ 91 ],aftersolvingmaxmaxQbysolvingsubproblemsSUB1andSUB2iteratively,weobtaintheoptimalsolutions,i.e.,theworst-casescenarioforthecurrentrst-stagesolutions.IfQ2>Q2,thefollowingprimalconstraintsareaddedtotheintegratedmasterproblem: Q2Q2(,~g,~r,~f,~) (5{65) (~g,~r,~f,~)2f( 5{18 )-( 5{26 )g. (5{66) 5.3.3AlgorithmThecompletealgorithmprocedureisasfollows(alsoseeowchat 5-1 ): 1) Initialization; 2) SolveWTIMPtoobtaintheoptimalsolutionsXwindn,Xlinel,andoptimalvalueQ2; 3) Solve(SUB1);storetheoptimalobjectiveQ1(X,),andoptimalsolution~,~,~,~,~ ; 4) Solve(SUB2);storetheoptimalobjectiveQ2(X,~,~,~,~,~ ),andoptimalsolution; 5) IfQ2>Q1,let=andrepeat3) 6) IfQ2>Q2,addtheprimalinequalities( 5{65 )-( 5{66 )toWTIMP.Otherwise,stopandexporttheoptimalsolution. 5.4ComputationalResultsTheproposedalgorithmistestedona6-busandtheIEEE118-buscases,andthecomputationalresultsareprovidedinthissection.ExperimentswerecodedinC++withCPLEX12.5usingConcert2.9technologyonanIBMSystemx3650withtwoIntelE5640Xeonprocessorsand24GBmemory. 5.4.1A6-BusCaseWersttestasix-busexampleforillustration,modiedfromthesix-bussystemprovidedin[ 75 ].AsshowninFigure 5-2 ,itisassumedthattheexistingnetworkhavesixbuses,sixtransmissionlines,threegeneratorsand3demands.ThedetailedinformationispresentedinTable 5-1 5-4 .Itisassumedthatthexedandvariablecostsofwindinstallmentare 86

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Figure5-1. Windpowerandtransmissioninvestmentmodel:algorithmowchart $3000and$200/MW,respectively;thecostofbuildingatransmissionlineis$5000,costofconventionalgenerationis$100/MW,andthepenaltycostofloadsheddingis$2000/MW.Table 5-1 providesdataateachbus:maxInstallliststhemaximumwindcapacitythatcanbeinstalled,genCapacityliststhecapacityofconventionalgenerators,andpeakLoadliststhepeakdemandateachbus.Weconsiderave-yearplan,andtwodemandblocksineachyear.Table 5-2 presenttheexpecteddemandlevelandintensityateachbusineachdemandblockofeachyear.Theactualdemandistheproductofpeakloadanddemandlevel.Theexpected 87

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Figure5-2. Themodied6bussystem Table5-1. Busdata BusmaxInstallgenCapacitypeakLoad 130030002300200030030043000300500200604000 demandlevelsarerandomlygeneratedfrom[0.8,1.0],andtheexpectedintensitylevelsarerandomlygeneratedfrom[0.35,0.6].Weusecorrelatedscenarios,asisdescribedin[ 76 ].Giventheexpecteddemandlevel,weassumetwomoredemandlevelsthatare5%higherorlowerthanexpected,andtwomoreintensitylevelsthatare20%higherorlowerthanexpected.Thatisatotalofninescenarios,andallscenariosareassumedtobeequallyweighted.Table 5-3 providesthelinecharacteristics.TheexistingtransmissionlinesareLine1-Line6,andthetwocandidatelinesareLine7andLine8.Thetableprovidesthesend-endbus,receive-endbus,owcapacityandsusceptanceofeachline.Table 5-4 presentthecapacityfactorsofconventionalgeneratorsineachdemandblockofeachyear.Thesecapacityfactorsarerandomlygeneratedfrom[0.8,1.0].Whenconsideringwindpowerandtransmissionlineinvestmenttogether,therecanbeasubstitutioneect,e.g.,whenthedemandishigherthanpoweravailableatsomebus,wecaneitherbuildawindpowerplantatthelocationtoboosttheself-supply,orbuildanewtransmissionlinetoreceivepowerfromlocationswherethereisasupplyexcess.Therecan 88

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Table5-2. Expecteddemandlevelandintensity dmdBlkBusdmdLevelIntensityY1Y2Y3Y4Y5Y1Y2Y3Y4Y5 11000000.4050.5410.450.3630.4972000000.5650.5090.4820.3150.430.8320.9460.7840.7780.8270000040.9470.9060.8310.8460.8140.4530.4130.3160.3660.32150.8650.9080.8940.90.892000006000000000021000000.3770.3360.3890.5860.3292000000.4920.5630.4220.5920.34330.7370.7270.7880.7420.7390000040.7080.8770.7470.7630.7980.3870.5070.3880.5650.47550.7880.950.9020.8810.8650000060000000000 Table5-3. Transmissionlinedata LinesendBusrecBusmaxFlowSusceptance 1122005.8822142006.6673233003.8764361007.1435452006.6676564006.2507152005.0008242005.000 Table5-4. Generatorcapacityfactor YeardmdBlockgenFactor 110.80220.897210.94620.952310.95920.950410.90720.823510.88820.846 89

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alsobeacomplimentaryeect,e.g.,whenawindpowerplantisinstalledatsomebus,buttheowcapacityoftheexistingtransmissionlinesconnectedtothisbusislow,thesystemneedstoconsiderbuildingnewtransmissionlinesinordertodistributetheincreasedsupply.Sinceinvestmentbudgetisnotaconstraintinourproposedmodel,weexpecttoseemoreofthecomplimentaryeect.Wersttestabasecase.Recallthatthesecond-stagecostofconventionalgenerationsandloadsheddingistheweightedaverageoftheobjectivevaluesfromstochasticandrobustoptimization.Thebasecaseassignsequalweightstothetwoparts.Itisassumedthatwindcapacityisavailablein100MWblocksforinstallation,anduptothreeblockscanbeinstalledateachcandidatebus.Finally,theuncertaintysetforwindintensityisobtainedfroma95%condenceinterval.Thebasecaseoptimalsolutionistoinstall300MWwindcapacityatbus2and4,andbuildanewtransmissionlineconnectingthetwobuses,whichincursatotalcostof$718385.Wethenmoveontothesensitivityanalysisonweightsassignedtostochasticandrobustoptimization,sizeofwindblocks,andsizeoftheuncertaintyset. Table5-5. 6-buscaseweightchange Weight()Cost($)Time(s)CutsBus1Bus2Bus4Line7Line8 080348532000010.180500773000010.37758713103000010.5718385320300300010.762524321300300300010.9529749113003003000114819991-30030030010 Table 5-5 presentstheresultswithdierentweightsassignedtothetwopartsofthesecond-stage.Itreportstheoptimalinvestmentsolutions,theoptimalobjective,i.e.,thetotalcost,runtime,andthenumberofsetsofprimalconstraintsadded.TherstcolumnWeight()liststheweightsassignedtothestochasticpart,sotheweightsassignedtotherobustpartis(1)]TJ /F10 11.955 Tf 12.12 0 Td[().Weconsidertherobustoptimizationalonewhen=0,andstochasticoptimizationalonewhen=1.UsingrobustoptimizationaloneresultsininvestingLine8 90

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only,withoutanynewwindpowerinstallment,whichisfairlyconservative.Usingstochasticoptimizationaloneresultsininvestmentinallthreecandidatebuses,togetherwithLine7.Asweightsonrobustpartdecreases,theexpectedtotalcostdecreases,andtheoptimalsolutionsmovefrommorerisk-aversetomorerisk-neutraldecisions. Table5-6. 6-buscasewindblocksize wBlk(MW)Cost($)Time(s)CutsBus1Bus2Bus4Line7Line8 50737207210150150011007183853203003000115070035842045045001 Table 5-6 presentstheresultswhenblocksizesofwindpowerchange.Inthiscase,theoptimalsolutionsaretoinstallfullwindcapacityatbus2and4,andalsoinvestinLine8.Theexpectedtotalcostdecreasesasmorewindpowerisavailableateachcandidatebuses.Sincethexedcostofbuildingwindpowerplantstaysthesameandthevariablecostisrelativelylow,itispreferredtoinstallmorewindcapacityifavailable. Table5-7. 6-buscaseuncertaintyset SetCost($)Time(s)CutsBus1Bus2Bus4Line7Line8 90%693733323003003000195%7183853203003000199%733264320300001 Table 5-7 showstheeectofchangingthesizeofuncertaintysetforwindintensity.Whentheuncertaintysetisdenedfrom90%,95%,upto99%CI,theoptimalinvestmentdecisiongoesfrominvestinginwindpoweratallthreecandidatebuses,toinvestingattwooutofthreecandidates,andfurtherdowntoonlyinvestingwindpoweratbus2.AstheuncertaintysetisdenedfromawiderCI,theexpectedcostishigher,andtheoptimalinvestmentdecisionismoreconservative. 5.4.2IEEE118-BusCaseDataofIEEE118-Bussystemisprovidedin[ 77 ].Thetestsystemconsistsof186transmissionlines,54generators,and91demands.Inourexperiments,alldemandsaredoubledforillustrationpurpose.Therelativetolerancegapissettobe10)]TJ /F8 7.97 Tf 6.59 0 Td[(4. 91

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Table5-8. 118-busdata LinesendBusrecBusowMaxSusceptance 1712200302121320030312592003041315200305138020030 Table 5-8 presenttheinformationofcandidatetransmissionline.Weconsidervecandidatetransmissionlines,andtheirsend-endandreceive-endbuses,owcapacityandsusceptanceareasprovided. Table5-9. 118-busexpecteddemandlevelandintensity YeardmdBlkdmdLevelIntensity 110.9200.40520.8100.377210.9390.54120.8250.336310.8850.45020.7420.389410.8840.36320.8160.586510.7860.49720.8960.329 Table5-10. 118-buscaseweightchange Weight()Cost($)Time(s)CutsBus7Bus12Bus13L1L2L3L4L5 01.127E+086951000000100.31.122E+087221300300300001000.51.102E+087681300300300001000.71.082E+087701300300300001000.91.063E+0893413003003000011011.053E+08343-30030030000100 Weconsider3candidatelocationsforwindinstallment,atbus7,12,and13.Thexedandvariablecostsofwindinstallmentsare$100000and$3000/MW,respectively.Withthesamesetupastheprevious6-buscase,weconsiderthreedemandlevelsandthreeintensitylevelswithineachtwodemandblocksofeachyearovera5-yearhorizon.Theexpecteddemandlevelsarerandomlygeneratedfrom[0.8,1],andintensityisrandomlygeneratedfrom[0.3,0.6]. 92

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Table5-11. 118-buswindblocksize wBlk(MW)Cost($)Time(s)CutsBus7Bus12Bus13L1L2L3L4L5 501.116E+086351150150150001001001.102E+087681300300300001001501.089E+08775145045045000110 Table5-12. 118-buscaseuncertaintyset SetCost($)Time(s)CutsBus7Bus12Bus13L1L2L3L4L5 95%1.102E+0876813003003000010097%1.103E+0876813003003000011099%1.105E+08813130030030000100 Table 5-9 providestheexpecteddemandlevelsandwindintensitiesineachdemandblockofeachyear,andthesevaluesareusedforallbuses.Table 5-10 , 5-11 ,and 5-12 presentthechangesintheoptimalsolutionsalongwiththechangesinweights(),sizeofavailablewindblocks,andsizeoftheuncertaintyset.Inthissetting,windpowerandLine3arealwaysinvested,whileLine1,2,and5areneverchosen.Again,theexpectedtotalcostsdecreaseswhentheweightsassignedtostochasticoptimizationincreases,morewindcapacityareavailableforinstallation,ortheuncertaintysetisdenedfromasmallerCI.Thereareatotalof197553variables,amongwhich44870arebinary,and152683arecontinuous.Allexperimentsarecompletedwithinareasonabletime. 5.5ConcludingRemarksWeproposedauniedstochasticandrobustoptimizationtwo-stagemodelonalongtermwindpowerandtransmissionlineinvestmentproblem,foraverticallyintegratedutility.Stochasticoptimizationwasusedtosimulateuncertainty,androbustoptimizationwasusedtohedgetheworst-casescenarioswithinanuncertaintyset.Whilestochasticoptimizationcoversordinarysituations,andmakesrisk-neutraldecisions,robustoptimizationusuallyresultsinconservativesolutions.Theuniedmodelproducearisk-aversedecision,whilethelevelofconservativenesscanbeadjustedbyalteringtheweightsassignedtostochasticandrobustparts,orthesizeoftheuncertaintysetfortherobustoptimization. 93

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CHAPTER6CONCLUSIONInthisdissertation,atheoreticalstudyofcombiningdynamicprogrammingapproachwithcuttingplaneswasconductedtosolvethebinaryknapsackproblems.Motivatedbythetheoreticalresearch,twoapplicationsonwindpowerinvestmentwerefurtherstudied.InChapter 2 ,forwarddynamicprogrammingwaspartiallysolvedtogeneratecoverinequalitiesandprotinequalitiesforthe0-1knapsackproblem.Therewasatrade-obetweenshorterruntimeandmoreusefulinequalitiesgeneratedfromDP.Experimentswereconductedonproblemswithuncorrelated,weaklycorrelated,stronglycorrelated,inversestronglycorrelated,andalmoststronglycorrelateddata.ComparedtoCPLEXIPsolver,theproposedapproachwaseectiveinsolvingmosttestinstances,whilelargerimprovementwasshowninmorecorrelatedcases.ThisstudywasextendedtothemultidimensionalknapsackprobleminChapter 3 .InChapter 4 ,auniedstochasticandrobustoptimizationtwo-stagemodelwasproposedtosolveashorttermwindpowerinvestmentprobleminanexistingtransmissionnetwork,wherestochasticoptimizationapproachwasusedtosimulateuncertainty,andtherobustoptimizationapproachwasusedtocapturetheworst-casescenarioswithinanuncertaintyset.InChapter 5 ,atwo-stagemodelwasproposedforlongtermwindpowerandtransmissioninvestmentproblem.Bender'sdecompositionalgorithms,withprimalordualconstraints,wereutilizedtogenerateoptimalitycutsforbothmodels. 94

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BIOGRAPHICALSKETCH Kun Zhao was born in Xi'an, and raised in Wuhan, China. She received her bachelor's in mathematics and economics from American University (AU), Washington, DC in 2009, and master's in mathematics from AU in 2010. She joined the Industrial and Systems Engineering (ISE) doctoral program at University of Florida (UF) in 2010. She received a aster of cience in ISE in 2013 and a master of science in management from Warrington School of Business at UF in 2014. She received Ph.D. from ISE in 2015. 102