1. WATER SYSTEM OPTIMIZATION: CONCEPTS AND METHODS

1 1. WATER SYSTEM OPTIMIZATION: CONCEPTS AND DEFINITIONSE ngineering project design and optimization can be effectively approached using CONCEPTS ofsystems analysis. A SYSTEM can be thought of as a set of components or processes that transformresource inputs into product (goods and services) outputs. The basic concept of a SYSTEM is rep-resented in Figure (a)I(x)O(y)12345678(b)2quantityqualityti me (c)Figure : Representation of a SystemIn Figure , the SYSTEM is defined by a boundary which separates those components that arean interrelated part of the SYSTEM from those outside which are part of the "environment".Determining the boundary depends on the physical SYSTEM , the technological and spatial ele-ments and the assumptions and the purposes for which the analysis is being conducted (seeFigure ).

2 For example, in a WATER resources SYSTEM , the analyst must decide which hydro-logic basin and WATER sources, dams, reservoir, and conveyance systems, and service areas andwater uses to include in the SYSTEM .The inputs define the flow of resource into the SYSTEM and the outputs and products from thesystem. A SYSTEM often has several subsystems. In the more detailed representation of , the inputs include controllable or decision variables, which represent design choices that areopen to the engineer. Assigning values to controllable variables establishes an outputs describe the performance of the SYSTEM or its consequences upon the indicate the effects of applying design and planning decisions via the input variables andare evaluated against SYSTEM objectives and criteria in order to assess the worth of the respectivealternatives in terms of time, reliability, costs or other appropriate.

3 Detailed Representation of a RESOURCES SYSTEMS DESCRIPTIONSW ater resources systems modeling may be treated at various levels of specificity as illustrated byFigure If the design is concerned with local WATER supply planning, then the SYSTEM bound-ary would include the key elements shown by Problem 1 in Figure If basin-wide multipur-pose planning or operation is of concern, the SYSTEM boundary must be expanded to include thekinds of elements shown in Problem 2. The engineer might be interested in statewide allocationof WATER among basins and uses as illustrated by Problem 3. As a further example, the specificelements and interconnections of a multipurpose basin are further depicted in the SYSTEM blockdiagram of Figure This type of diagram is useful in constructing the mathematical optimi-zation or simulation models for the SYSTEM .

4 Table summarizes many of the relevant input,outputs, decision variables, and SYSTEM constraints and components of WATER resources MODELS OF SYSTEMS:OVERVIEW AND CONCEPTSF igure is a representation of a modeling space , with each face of the cube representing animportant dimension of quantitative models. Depending on whether variable relationships areprobabilistic or deterministic, static or dynamic, and linear or nonlinear (as represented by thefaces of the cube) various analytical techniques (the corners) are required to handle AREAINDUSTRYCITYBIRD REFUGEIRRIGATEDAGRICULTUREPROBLEM 2: Multipurpose SystemRIVERSPRINGTREATMENTPLANTSERVICE AREAWELL FIELDPROBLEM 2: LOCAL WATERSUPPLY SYSTEMBearRiverUintahGreat SaltLakeWeberRiverJordanRiverSevier RiverWestWestColoradoSoutheastColoradoCe darBeaverLowerColoradoPROBLEM 3: Statewide WATER ResourceAllocation PlanFigure.

5 Levels of Specificity in WATER Resources Systems Modeling5 Output --BenefitFunctionsCity AWater SupplyFlow AugmentationFlood ControlWaste DisposalWatershedPowerPlantIrri-gationDo wnstreamEffectsCost --InputFunctionsRes.#1 OperatingPoliciesRes.#2 ObjectivesEfficiency;Redistribution;Envi ronmental Quality; , Legislation, DecisionsB2B1B3B4-7B8B9B10C1C2C3C4-7C8C9 C10(t)QEconomic-PoliticalInterfacePhysic al-EconomicInterface(acres)(tons/yr)Figu re : Hierarchy of Systems and Systems Functions6 Table : Elements of WATER Resources SystemsInputs to WATER Resources sources: for example, surface WATER flow, sedimentation, or salt load, sources: for example, desalting WATER , imported and recycling: for example, treated WATER from treatment plant, recycling WATER in natural , resourcesOutputs of WATER Resources allocation to user and wildlife and quality of the WATER resource of the of streamSystem Decision and use coordination and allocation and location of treatment of treatments and treatment level allocation to various of investment: for example, stages of development, interest and subsidy strategiesConstraints on Systems constraints: for example, budget, B/C constraints: for example, tradeoff between : for example, WATER and technology constraints: for example, probability of WATER : SYSTEM output may have to meet certain standards.

6 For example, effluent standards from wastewater treatmentplantsSystem Physical and Engineering and management SYSTEM and control and other protecting or collection systems comprised of (a) canals, (b) pipes, (c) pumping stations and other control SYSTEM properties of stream: for example, roughness, properties of stream: for example, rate of aeration, rate of properties of stream: for example, hardness, pH7 Deterministic/ProbabilisticNumberofVaria blesStatic/DynamicProbabilisticDynamicSt aticDeterministicFigure : Modeling Space (Cube)Broadly speaking the purpose of modeling may be either predictive or prescriptive. Predictivemodels of systems are constructed to clarify the internal structure of a SYSTEM and predict itsbehavior or response to input variables.

7 On the other hand, prescriptive models strive not only toreproduce the behavior of the SYSTEM itself, but also to evaluate the consequences of design alter-natives according to predetermined measures of the model structure for predictive or prescriptive models must be based either onformal theory or some very strong plausibility arguments. Systems models cannot be devised bysimply using statistical manipulations of data and information to determine variable , all the information relevant to the SYSTEM may not be quantifiable as numerical , systems modeling techniques may be quantitative and nonquantitative or provides a general classification of modeling METHODS and techniques useful in systemsanalysis. The entries in the Table are classified under the heading of predictive or prescriptivemodels according to the theoretical basis for model construction.

8 Table relates the models tothe modeling cubic dimensions shown in Figure overview of course cannot provide detailed descriptions of the various modelingapproaches. Whole textbooks are devoted to these subjects. Instead, this discussion simply triesto provide a basic classification of the techniques in order to understand where optimization fitamong the various : An Overview of Approaches to Systems ModelingApplication or UseProblem TypePredictivePrescriptiveQuantitativeDe terministicSystems Transformationsalgebraic equations,differential equations, statevariable formulations, input-output analysisOptimization ProceduresClassical Optimization Theory;differential calculus, lagrangians,optimal control theoryNetworksgraph theoryNetworksCPM and PERTS tochasticStochastic Processesinventory theory, queuingtheory, Markov processesDecision AnalysisStatistical (Bayesian) decisiontheory, game theoryStatistical Modelsregression analysis,component and factoranalysis, stepwise multipleregression, discriminantanalysis, econometricanalysisSimulationDeterminist ic and stochasticmodel components and modelsMonte Carlo METHODS , searchtechniques for dominant solutionsNon-quantitativeVerbal Modelsscenarios, survey researchVerbal ModelsDelphi inquiriesPeople Modelsrole playingPeople Modelsoperational gaming9 Table.

9 Types of Models in Modeling Space TimeRandomState ofDomainFunctionalVarianceVarianceVarian ceVarianceType of Model or SolutionApproachStaticDynamicDeterminist icProbabilisticSingle VariateMultivariateLinearNonlinearSystem s TransformationLinear, nonlinear systems1st order diff. equationsXXXstate variablesXXXI nput-output analysisXXXO ptimizationClassical OptimizationDifferential CalculusXXXXL agrange MultipliersXXXXO ptimal Control TheoryMathematical Programminglinear programmingXXXX nonlinear programmingXXXX integer programmingXXXdynamic programmingXXXX stochastic programmingXXXX genetic programmingXXXXXS tochasticInventoryXQueuingXXMarkovXXMult ivariateXXNetworksGraph TheoryCPM and GENERAL MODEL OF SYSTEM Design and OptimizationEngineering design problems can be mathematically described by three functions associated witheach of the design factors.

10 The physical processes (design or production function), the resourcefunction's costs, and the output's or product's values (benefit functions) (see Figure ). Thedefinition of each function is derived from different sources. SYSTEM Design Variables(X , X , .. , X )12n(b , b , .. , b )12m(z , z , .. , z )12kPhysicalResourcesOutput: Goodsand ServicesResource CostFunctionsOutput BenefitFunctionsInputCostUtilityOutputBe nefitUtilityg (x) <=> bmmg*(x) = ZDesign(Production)FunctionC = c(x)V = B - CB = h(z) = h'(x)EvaluationOptimizationMax VFigure : Model of Systems Design and OptimizationThe design function, based on the physical nature of the SYSTEM without regard to value,describes the maximum product that can be obtained from the input of any given set ofresources.