Benefits of Decomposition Methods to Speed-up Energy ... · level (decoposition in modelnodes or...

a project by

ManuelWetzel,FriederBorggrefe21.09.2016

DLR

BenefitsofDecompositionMethodstoSpeed-upEnergySystemModellingandApplicationtoStochasticOptimization

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1. Introduction• TheproblemofuncertaintyinEnergySystemModelling

2. Currentimplementationofstochasticoptimization• Fromdeterministictostochasticmodelling• ImprovingconvergencebyEnhancedBendersapproaches

3. Challengesandpossibleimprovements• Currentchallenges• Application topathoptimization

Content

M.Wetzeland F.Borggrefe(DLR) Application of Stochastic Optimization inESM 21.09.2016

TheproblemofuncertaintyinEnergySystemModelling

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Theproblemofuncertaintyinenergysystemmodelling

• Longterm capacity expansion planning requires making decisions now,which have animpact onthe energy system for several decades.

• Context scenarios can give ageneral direction of development,however alargenumber of consistent sub-scenariosare possible.Thecurrentapproach usually considers only the main scenario and evaluatesrobustness by sensitivity analysis of sub scenarios.

Robust scenario High

Robust scenario Base

Robust scenario Low

Sensitivity analysis






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Model result Low

Model result Base

Model result High

Pathdevelopment of anuncertain parameter


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Bendersdecompositionseparatingcapacityexpansionplanning andeconomicdispatch

Bendersdecompositionfortwo-stagestochasticoptimization(L-shapedMethod)[Benders1962andVanSlyke 1969]

Masterproblem:• Decidenowwhichcapacities

shouldbeexpanded

Subproblem:• Decidelateroneconomic

dispatchtosatisfyelectricaldemandwithcapacitiesgivenbymasterproblem

1stStage:Capacity expansion –Installed capacities

2ndStage:Dispatch –Generationvariables

Linkingvariables:Installed capacities,connecting the capacityexpansion problem and the dispatch problem

Mathematicalformulation inLPtable


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Bendersdecompositionseparatingcapacityexpansionplanning andeconomicdispatch

1.Two-stagedecisiontree

Stage1(Masterproblem)Investmentdecision

Stage2(Subproblem)Stochasticeconomicdispatchscenarios

2.Multi-stagedecisiontree

Stage2Dispatch&Investmentdecision2020

2016 2050

…

20252020

Stage1Investmentdecision2016

Dispatch

Investment 2045

Stage3Dispatch&Investmentdecision2025


Current implementation of stochasticprogramming

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REMix SIMPLE

TheREMix Model

• Capacity expansion planning andeconomic dispatch

• High spatial,temporalandtechnological resolution

• Modularapproach to include heatsector,electromobility,DSMandothers

• Simplified version of REMix,usedas adevelopment platform

• Scalability of model dimensionsby generic data generation

Ø Modified for demonstratingimplementation approaches forstochastic optimization


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Implementationof EnhancedBendersdecompositionapproaches

GAMSSimpleModel:• Capacity expansionplanning

and economic dispatch

1.Implementationof aBendersdecomposition approachSeperating capacity expansionand economic dispatch

Masterproblem:Investmentdecision

Subproblem:Economicdispatch

Suggested solutionCapacities for installed plantcapacities,electric storages and linksfor each region

Optimality CutInformation about the marginals of the firststage decision variables


DeterministicversionofTheGAMSSIMPLEmodel

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• Each subproblem is weighted by aprobability and represents aspecific setof assumptions for uncertain parameters


1.Adaptationof the Regions:• Seperate regions and type1.Implementationof aBendersdecomposition approachSeperate regions and type

2.Stochasticmodel withscenarios• Each scenario has a

probability• Stochasticmodel is stillsingle

threat

Masterproblem:Investmentdecision

Subproblem:Economicdispatchscenarios


Setof uncertain parameters• Electrical demand• Fuelprices• Wind timeseries• Solartimeseries• E-mobility usage

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Types of optimality cuts

Singlecut• One optimality cut periteration

is submitted to master problem

Multicut• Each subscenario submits an

optimality cut each iteration

Clustercut• Each cluster submits an

optimality cut each iteration

Subproblem:Dispatch

Master:Investment

Weightedbyscenario probability

Weightedbyscenario probability

• Multicutsgive detailed information butincrease complexity [Birge 1988]• Singlecutsaggregate information thereforemore iterations are necessary• Dynamicswitching between cut-types is possible [Skar 2014]


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• SmallESMwith 3powerplants,3storages,2node linksand 8760h• Formulation as Deterministic Equivalent for small problems faster than

Bendersdecomposition,butleads tomemory restrictions inlargemodels

1.Adaptationof the Regions:• Seperate regions and type

2.Stochasticmodel withscenarios

4.Grid computing• Scenariosand parts of the

modell can be placed inseveral threats

5.DeterministicModel• One modell formaster and

subproblem

3.SimpleComputing• Scenariosare processedone

afteranother

Testing withModelnodes:3Scenarios:100=40GBRAM

Testing withModelnodes:3Scenarios:100=3-4GBRAM

Testing withModelnodes:3Scenarios:100<<3-4GBRAM


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BendersDecompositionMaster Subproblem

• Subproblems canbesolvedinparallel• Memorydemandscales withnumber

ofparallelsolveprocesses


DeterministicEquivalent

• SizeofLPincreaseswithnumberofscenarios,solvedbySIMPLEX/Barrier

• OutofmemoryfortypicalREMixproblemswithscenariodimension


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Improvement to achieve asychronity

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Synchronous Model Asynchronous Model

tIteration1 Iteration2 Iteration3 Iteration4Iteration1 Iteration2

• NewMastercan be startedas soon as afraction of scenarios are solved,new iteration has only partial information therefore the number ofiterations increases [Linderoth2003]

• Trade-off betweenmore iterations and parallelisation


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Algorithm parameters

Number ofclusters

Number ofscenarios

Number ofconcurrentmasters

Number ofnodes

Number oftimesteps

Deletingoptimalitycuts?

Clustersubmissionby GUSS

Multiplecuts perscenario?

Number oftechnologies

Typeofoptimality

cuts

Fractionofsolved

scenarios

• Largenumber of parameters to optimize performance of EnhancedBendersAlgorithm making systematic testingnecessary

SIMPLEModel

Optimality Cuts

GUSSusage

Asynchronity

Solvelinkand Solver

Solveroptions

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Method Clustercut Multicut Multicut +TR MC+ TR+GUSS

Iterations 138 77 54 61

Timeto solve 5:54h 4:42h 3:14h 2:51h

CPUload max 300% 449% 466% 134%

CPUload avg. 66.9% 70.8% 56.2% 18.6%

RAMusage max 0.58GB 0.58GB 0.48GB 0.67GB

RAMusage avg. 0.11GB 0.16 GB 0.11GB 0.26GB

Computational test results


• Modelbuilding timelimiting constraint,models are solved faster thangenerated therefore low average CPUload (100%equals 1core outof 32)

• Exchangeof information between GAMSand solver can be accelerated byusing shared memory and running GAMSand CPLEXindifferentthreads,this comes atthe cost of increased memory demand

Challenges and possible improvements

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• Reducing computational overhead by using methods which simplify orreduce model building time(Usage of GUSS,externalizing model building)

• Balancing CPUload for migration to highperformance computing

• Generatingbetter cuts - insufficient electrical generation willcause allplanttypes inaregion to be built

• Deleting unused cuts inorder to keep the masterproblem as small aspossible while retaining important information

• Improving the starting point for the Trust-Regionapproach

• Combining stochastic optimization with decomposition onthe scenariolevel (decoposition inmodel nodes or timesteps)inaNested Bendersapproach

Challenges


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Challenges

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Summevon%CPU

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%CPU

%MEM


• Largescale scenario with typical synchronous behavior• Gapsindicate solving master while peaks represent parallelsubproblems• Highcorrelation between CPUand memory load indicating scalability with

number of parallelprocesses

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Application to path optimization

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Motivating examplefrom introduction

Result:3Context scenariopathways including3subscenarios each


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Application to path optimization

2016 205020302020 2040 205020302020 2040

• Application of Bendersdecomposition incapacity expansion planning andeconomic dispatch leads to alargenumber of subproblems which can besolved inparallel

Masterproblem Subproblems


ProjectBEAM-ME

a project by

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• J.Benders,"Partitioningproceduresforsolvingmixedvariablesprogramming problems,"NumerischeMathematik,vol.4,pp.238–252,1962.

• R.M.VanSlyke andR.Wets,"L-shapedlinearprogramswithapplicationstooptimalcontrolandstochasticprogramming," SIAMJournalonAppliedMathematics,vol.17,no.4,pp.638–663,1969.

• J.R.Birge andF.V.Louveaux,"Amulticut algorithmfortwo-stagestochasticlinearprograms,"EuropeanJournalofOperationalResearch,vol.34,no.3,pp.384–392,1988.

• C.Skar,G.Doorman and A.Tomasgard,"Large-scale powersystem planning using enhancedBendersdecomposition,"PowerSystemsComputation Conference(PSCC),pp.1-7,2014,,doi:10.1109/PSCC.2014.7038297

• J.T.Linderoth andS.J.Wrigth,"Implementingadecompositionalgorithmforstochasticprogrammingonacomputationalgrid," ComputationalOptimizationandApplications,vol.24,pp.207–250,2003.

References

Backup

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GAMSGUSS

7.GUSSimplementation

1.Adaptationof the Regions:• Seperate regions and type

2.Stochasticmodel withscenarios

4.Grid computing• Scenariosand parts of the

modell can be placed inseveral threats

GUSS

• Gathersdatafromdifferentsources/symbolsincollectionofmodels

• Updatesabasemodelinstancewiththisscenariodata

• Solvestheupdatedmodelinstanceand

• Scattersthescenario resultsto


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GUSS

• Gathersdatafromdifferentsources/symbolsincollectionofmodels

• Updatesabasemodelinstancewiththisscenariodata

• Solvestheupdatedmodelinstanceand

• Scattersthescenario resultsto

UsingGUSS

• GUSSisaGAMSfacilitythatpermitssolutionofasetofscenariosforaGAMSmodelmodifyingdatatoruneachscenario.

• GUSSisnotreallyasolver• organizesandpassesdata to

theothergamssolvers• Collectionofmodelsaresolved

inasinglepasswithoutneedingLOOPovermultiplesolves.

• fasterthanloopbecauseonlyonemodelisbuild


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Challenges

• Increasenumberofcutshanded fromonescenarioinoneiteration

• Furtherimplementationsnecessarytoevaluateexpectedperformanceofstochasticoptimization inREMix andaccelerateconvergenceo Deletionofunnecessaryoptimalitycuts,o Differentparametrisationsofalgorithm,o Dynamicadaptationoftrustregion,o Improvementofstartingpoint

• FurtherdevelopmenttowardsCPUloadbalancing necessaryformigration tohighperformancecomputing

• UsingREMix:ModularityofREMix environmentcomplicatestheimplementationofBendersdecompositionInformationaboutallactivevariables,equationsandmarginals necessary


Summary

• ModelledinGAMSandsolvedinCPLEXtypicallywithBarrierAlgorithm• Formulationasdeterministicequivalentimpossibleduetomemory

restrictions

• Staringpointbyregional presolve withmeanoverstochasticparameters• UtilisationoftheGAMSGridComputingenvironmentincombinationwith

GUSSinordertosubmitclustersofsubscenarios tobesolvedsimultanously• Optimalitycutsareaddedtothemasterproblemeitherassinglecuts,

clustercuts orfullmulticut• Currentlynocut-deletionprocedureimplemented

29M.Wetzeland F.Borggrefe(DLR) Application of Stochastic Optimization inESM 21.09.2016

• ModelledinGAMSandsolvedinCPLEXtypicallywithBarrierAlgorithm• Formulationasdeterministicequivalentimpossibleduetomemory

restrictions

• Staringpointbyregional presolve withmeanoverstochasticparameters• UtilisationoftheGAMSGridComputingenvironmentincombinationwith

GUSSinordertosubmitclustersofsubscenarios tobesolvedsimultanously• Optimalitycutsareaddedtothemasterproblemeitherassinglecuts,

clustercutsorfullmulticut• Currentlynocut-deletionprocedureimplemented

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• WelchenEinflusshatdieWahleinesDekomponierungsverfahrens aufdieWahldesSolvers?MachtesdabeieinenUnterschiedobnachz.B.Zeit,KnotenoderErzeugungundZubaudekomponiertwird?

• SindSlackvariablen wieBalanceEnsure undLoadEnsure fürDekomponierungsverfahren immernoch sinnvoll?OderkannalternativfrüherabgebrochenundeineneueIterationgestartetwerden(branch andcut,analog zuMIP)?


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• BenötigteZeitfürdenBaudesGAMSModells– mussreduziertwerden,– bei100SzenarienundmehrerenSekundenModellbauzeitgehteinGroßteilderZeitin

denOverhead.– GUSSscheintinderHinsichtgutanwendbardaModellnureinmalgebautwerdenmuss.– FürIterativeVerfahrenwäreoptimalwennsicheinvorhergebautesGUSSModellmit

neuenParameternupdatenließe(spartModellbauzeitinspäterenIterationen

• CPUsmüssensich(nahezu)vollständigauslastenlassen,– d.h.wennTeiledesClustersfertigsinddürfendiesenichtaufneueAufgabenwarten

müssen.– ->AsynchronesVerfahrennötig!!!– Problem:EssolltentrotzdemkeineneuenModellegebautwerden.


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• HöhereGranularitätmusserreichtwerden.Wenn80kCoresgleichzeitiganeinemProblemrechnensollen,dieAnzahlsinnvollerSzenarienjedocheherimBereichvon30-100liegt(ohneZufallsvariablen/vollständigeKombinatorik)müssendieSubproblemeweiterunterteiltwerden(z.B.nachKnoten/Zeitschritten).

⇒ ZusammenspielderKopplung von1.)Dekomponierung einesstochastischenProblems und2.)Dekomponierung innerhalb einesSubproblems


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• HoherAnpassungsgradderstochastischenElemente:verschiedeneProjektewerdenunterschiedlichestochastischeParameterverwenden,wiekanndasGanzebenutzerfreundlichimplementiertwerden?

• HoheModularitätvonREMix erfordertautomatischesAnpassenderBendersDekomposition (WelcheVariablenwerdenwannentschieden,welcheGleichungensindaktiv,welcheMarginals müssenfürdieOptimalitätscutszurückgeführtwerden)

• Modellsollweiterhindeterminitisch verwendetwerdenkönnenohneinallenFälleneinestochastischeDimensionberücksichtigenzumüssen(Optionalität führtzuKomplexitätdesGAMSCodes)


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References


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