Modeling uncertainty in reservoir loss functions ... - Wiley Online Library

WATER RESOURCES RESEARCH,

VOL. 35, NO. 9, PAGES 2815-2823, SEPTEMBER,

1999

Modeling uncertainty in reservoir loss functions using fuzzy sets Ram½shS. V. Tecgavarapu Departmentof Civil and GeologicalEngineering,Universityof Manitoba,Winnipeg,Canada

Slobodan

P. Simonovic

Natural ResourcesInstitute,Universityof Manitoba, Winnipeg,Canada

Abstract. Imprecisioninvolvedin the definition of reservoirlossfunctionsis addressed usingfuzzy set theory concepts.A reservoiroperationproblem is solvedusingthe conceptsof fuzzy mathematicalprogramming.Membershipfunctionsfrom fuzzy set theory are usedto representthe decisionmaker'spreferencesin the definition of shapeof loss curves. These functions

are assumed to be known

and are used to model

the uncertainties.

Linear and nonlinearoptimizationmodelsare developedunder fuzzy environment.A new approachis presentedthat involvesdevelopmentof compromisereservoiroperating policiesbasedon the rules from the traditional optimizationmodelsand their fuzzy equivalentswhile consideringthe preferencesof the decisionmaker. The imprecision associated with the definitionof penaltyand storagezonesand uncertaintyin the penalty coefficientsare the main issuesaddressedthroughthis study.The modelsdevelopedare appliedto the Green Reservoir,Kentucky.Simulationsare performedto evaluatethe operatingrules generatedby the modelsconsideringthe uncertaintiesin the loss functions.Resultsindicatethat the reservoiroperatingpoliciesare sensitiveto changein the shapesof lossfunctions. 1.

Introduction

In the past few decadesa variety of optimizationmodels have been developedfor long- and short-term and real-time operationof single-andmultiple-reservoir systems. Deterministic and stochasticapproacheswere used to handle various issuesarisingout of the modelingprocess.Problemsin this area have been addressedby many researchersusing a wide variety of optimizationtools. These tools range from simple simulationapproachesto complexoptimizationmodels.Yeh [1985] providesan excellentstate-of-the-artreviewof optimization modelsusedfor reservoiroperation.Most recent studies [e.g., Simonovic,1991; Hipel, 1992] have addressedthe multiobjectivenature of reservoiroperation problems.With increasingattention toward the development of operation modelsbasedon economicobjectives(handledthroughlossor penaltyfunctions),issuesrelated to uncertaintyand imprecisionin the definitionof theseobjectiveshavebecomea priority. Recentstudies[e.g.,Lund and Ferreira,1996]haveemphasizedthe need to developrealisticlossfunctionsfor inclusion in the optimizationmodels.The emphasisof the presentstudy is on short-termreservoiroperationsconsideringthe imprecision in the definition

of conventional

loss functions.

Optimizationof the short-termreservoiroperationsis generally achievedby formulatinga model to minimize the economic losesincurred from deviationsin operation from the releaseand storagevolumevaluesset as a target for the planningperiod.Theselossesare usuallyrepresentedby lossfunctions [Datta and Burges,1984] which reflect the penalty incurred for a specificdeviation from the target. Reservoir Copyright1999by the American GeophysicalUnion. Paper number 1999WR900165. 0043-1397/99/1999WR900165509.00

operationproblemsconcernedwith the minimizationof shortterm economiclosseshave been addressedby many researchers. Can and Houck [1984] addressedthe operationproblem usinggoal programming.They usedreservoirlossfunctionsto minimizethe penaltiesassociated with the deviationsfrom the targets.In an anotherwork [Can and Houck, 1985] they discussedthe problemsassociated with real-time reservoiroperations. Simonovicand Burn [1989] presented an improved methodologyfor derivingshort-termoperatingpolicieswhile at the same time obtaining the optimal operating horizon. Followingthiswork,Rezniceket al. [1991]workedon the same problem but used goal programminginstead of linear programming.All of the aboveworksinvolvedformulationswhich usedpiecewiselinearizedlossfunctionsof storageand release. One of the difficult aspectsin thesemodelsis the quantification of lossfunctions.These functionsare usuallyderived or• the basisof the experienceof reservoiroperatorsand therefore are highlysubjective.The valueswhichmake up the lossfunctions are penalty coefficients,and their selectionis ultimately the reservoiroperator'spreference.Penaltycoefficientsare the pointson the lossfunctionswhich definethe penaltyin monetary units correspondingto the penalty zones.These values are usuallyderivedfrom economicinformationconsideringthe impactsof reservoiroperation.Despitethe utility of lossfunctions in variousreservoiroperationproblems,there still exist unresolvedquestionsabouttheir derivation,shapes,and associatedpenaltycoefficients.In a recentstudy,Lund and Ferreira [1996] state that the most difficult and expensivepart of any practicalreservoiroperationmodelis usuallythe development of penaltyfunctions. The imprecisenature of lossfunctionsassociatedwith the difficultiesin determiningthe shapeand penalty coefficients makes the reservoir operation problem difficult to handle.

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MODELING

UNCERTAINTY

Also, the operatingpoliciesentirelydependon the exactdef- given by Zimmermann [1987]. Fuzzy linear programming inition of thesefunctions.In practice,the penaltycoefficients (FLP) utilizes the Bellman-Zadehcriterion for solutionof are not crispnumbersbut are certainaspirationlevelswhich many problemswhere the goals and constraintsare fuzzy. are not well defined. Also, it can be observed that the loss Many applicationsand variationsof FLP can be found in functionvaluesare, in turn, the decisionmaker's degreesof literature [e.g.,Kikuchi et al., 1991; Cui and Blockley,1990]. importance attached to violation of various target values. The presentstudyusesbothlinearandnonlinearprogramming Therefore the decision-making processinvolvesdealingwith underfuzzyenvironment. Theproblemdealtis nonsymmetric the problemin an environmentwhere the objectivesand con- in a sensein that the objectivefunctionis crispor well defined straintsimposedare vague.Fuzzy set theoryconceptscan be and the constraints are fuzzy.A fuzzymathematicalprogramusefulin this context,as they can provide an alternativeap- ming problem is consideredto be symmetricwhen both the proachto dealwith thoseproblemsin whichthe objectives and objectivefunction and the constraintsare vague. constraints are not well defined or information about them is To solvethe problemsthat are nonsymmetric, a procedure by Zimmermann[1987] is used.The procedureinnot precise.A wealthof literaturerelatedto conceptsof fuzzy suggested setsandtheir applications is availableelsewhere[Zadeh,1965; cludesthe followingsteps:(1) the mathematicalprogramming Zimmermann,1987;Shrestha et al., 1996].Recentstudiespro- model is solved,and the objectivefunctionvalue is obtained; which vide examplesof usefulapplicationof fuzzyset theoryto res- (2) the model is againsolvedwith modifiedconstraints ervoiroperationproblems.Kindler[1992]usedfuzzysettheory are consideredfuzzy; and (3) the model is solved,with the in order to developa water allocationmodelwherethe water objectivefunctionandconstraints (whichwere earlierassumed requirementsare assumedto be fuzzy quantities.Reservoir asfuzzy)replacedby their fuzzyequivalents usingmembership operationruleswere derivedby Russelland Campbell[1996] functions.The objectivefunctionin step1 or 2 becomesa fuzzy usingfuzzy logic programming.They indicatethat their ap- constraintin step3. The fuzzyconstraintsin the presentstudy proachcanbe appliedonly to problemswith few controlvari- are relatedto the penaltyzonesand coefficients, whereasthe ables.Shresthaet al. [1996]developeda fuzzy-rule-based res- objectivefunctionis the penaltyvalue in monetaryunits.The ervoir operation model. The rule-base is developed proceduredescribed in steps1-3 canbe represented in a mathconsideringdifferentoperationaldecisions takenby the reser- ematicalform. For a minimizationproblemthe stepsare given voir managers.A recentstudyby Fontaneet al. [1997]provides below: a usefulapplicationof fuzzy setsin planningreservoiroperaStep 1: Minimize CX subjectto tionswith impreciseobjectives.Membershipfunctionsare obAX_> b (1) tained on the basisof the actual surveyswhich are used in a dynamicprogrammingmodel. They emphasizethe practical whereCX is theobjective function, X = Ix 1,x2, "'It is the valueof membershipfunctionswhen includedin optimization matrix of decisionvariables, and AX is the constraint matrix. frameworkfor reservoiroperationproblems.Lossfunctions Let the objectivefunctionvalue obtainedby solvingthe above bear a closeresemblance to the membershipfunctionsusedin problembe fo. the fuzzy set theory. This similaritycan be a motivationfor Step 2: Minimize CX subjectto replacingthe former with the latter. However,in the present studythis problemis not handled.It is assumedthat the loss AX -> b + to (2) functionsare availableand the decisionmaker'spreferences Here the tolerance interval to, by which the b value can are modeledusinglinear membershipfunctions. change, is addedto the right-handsideof (2). Let the objective The present study concentrateson the developmentof a methodologyto handlethe imprecisioninvolvedin the defini- functionvalue obtainedfrom the solutionof step2 be flStep 3: Maximize X subjectto tion of lossfunctionswhile at the same time addressingthe short-termreservoiroperationproblem. The paper is orgaAX- X to-> b (3) nizedasfollows.Decisionmakingundera fuzzyenvironmentis discussed first. Formulationsaddressingthe problemsof imCX + X(fl -- f0) -