STOCHASTIC DYNAMIC PROGRAMMING FOR OPTIMUM RESERVOIR OPERATION1

ABSTRACT. For a multipurpose single reservoir a deterministic optimal operating policy can be readily devised by the dynamic programming method. However, this method can only be applied to sets of deterministic stream flows as might be used repetitively in a Monte Carlo study or possibly in a historical study. This paper reports a study in which an optimal operating policy for a multipurpose reservoir was determined, where the optimal operating policy is stated in terms of the state of the reservoir indicated by the storage volume and the river flow in the preceding month and uses a stochastic dynamic programming approach. Such a policy could be implemented in real time operation on a monthly basis or it could be used in a design study. As contrasted with deterministic dynamic programming, this method avoids the artificiality of using a single set of stream flows. The data for this study are the conditional probabilities of the stream flow in successive months, the physical features of the reservoir in question, and the return functions and constraints under which the system operates.     

PRACTICAL IMPLICATIONS IN THE USE OF STOCHASTIC DYNAMIC PROGRAMMING FOR RESERVOIR OPERATION1

ABSTRACT: A stochastic dynamic programming model is applied to a small hydroelectric system. The variation in number of stage iterations and the computer time required to reach steady state conditions with changes in the number of storage states is investigated. The increase in computer time required to develop the storage probability distributions with increase in the number of storage states is reviewed. It is found that for an average of seven inflow states, the largest number of storage states for which it is computationally feasible to develop the storage probability distributions is nine. It is shown that use of the dynamic program results based on a small number of storage states results in unrealistically skewed storage probability distributions. These skewed distributions are attributed to “trapping” states at the low end of the storage range.     

THE OPTIMALITY OF CAPACITY SHARING IN STOCHASTIC DYNAMIC PROGRAMMING PROBLEMS OF SHARED RESERVOIR OPERATION1

ABSTRACT: This paper is concerned with finding an optimal allocation of water entitlements for each of two users of water who share a reservoir. Two instruments of allocation are considered. The first, release sharing, involves sharing the releases from the reservoir; the second, capacity sharing, is concerned with allocating to each user of water a share of inflows, reservoir capacity and leakage and evaporation losses. Stochastic dynamic programming problems of reservoir operation under each type of sharing arrangement are formulated. It is shown that the maximum discounted expected profit from reservoir operation over the life of the storage using capacity sharing is at least as large as that obtained using release sharing and that release sharing is not Pareto efficient.     

COMPARISON OF RESERVOIR LINEAR OPERATION RULES USING LINEAR AND DYNAMIC PROGRAMMING1

ABSTRACT: Mathematical optimization techniques are used to study the operation and design of a single, multi-purpose reservoir system. Optimal monthly release policies are derived for Hoover Reservoir, located in Central Ohio, using chance-constrained linear programming and dynamic programming-regression methodologies. Important characteristics of the former approach are derived, discussed, and graphically illustrated using Hoover Reservoir as a case example. Simulation procedures are used to examine and compare the overall performance of the optimal monthly reservoir release policies derived under the two approaches. Results indicate that, for the mean detention time and the corresponding safe yield target water supply release under existing design of Hoover Reservoir, the dynamic programming policies produce lower average annual losses (as defined by a two-sided quadratic loss function) while achieving at least as high reliability levels when compared to policies derived under the chance-constrained linear programming method. In making this comparison, the reservoir release policies, although not identical, are assumed to be linear. This restricted form of the release policy is necessary to make the chance-constrained programming method mathematically tractable.     

A STOCHASTIC DYNAMIC PROGRAMMING BASED APPROACH TO THE OPERATION OF A MULTI-RESERVOIR SYSTEM1

ABSTRACT: An heuristic iterative technique based upon stochastic dynamic programming is presented for the analysis of the operation of a three reservoir ‘Y’ shaped hydroelectric system. The technique is initiated using historical inflow data for the downstream reservoir. At each iteration the optimal policies for the downstream hydroelectric generating unit are used to provide relative weightings or targets for operation of upstream reservoirs. New input inflows to the downstream reservoir are then obtained by running the historical streamflow record through the optimal policies for the upstream reservoirs. These flows are then used to develop a new operating policy for the downstream reservoir and hence new targets for the upstream reservoirs. The process is continued until the operating policies for each reservoir provide the same overall system benefit for two successive iterations. Results obtained from the procedure are compared to the results obtained by historical operation of the system. The procedure is shown to develop operating policies which give benefits which are as close to the historical benefits as can be expected given the choice of the number of storage state variables.     

COMPARISON OF STOCHASTIC AND DETERMINISTIC DYNAMIC PROGRAMMING FOR RESERVOIR OPERATING RULE GENERATION1

ABSTRACT: Two dynamic programming models — one deterministic and one stochastic — that may be used to generate reservoir operating rules are compared. The deterministic model (DPR) consists of an algorithm that cycles through three components: a dynamic program, a regression analysis, and a simulation. In this model, the correlation between the general operating rules, defined by the regression analysis and evaluated in the simulation, and the optimal deterministic operation defined by the dynamic program is increased through an iterative process. The stochastic dynamic program (SDP) describes streamflows with a discrete lag-one Markov process. To test the usefulness of both models in generating reservoir operating rules, real-time reservoir operation simulation models are constructed for three hydrologically different sites. The rules generated by DPR and SDP are then applied in the operation simulation model and their performance is evaluated. For the test cases, the DPR generated rules are more effective in the operation of medium to very large reservoirs and the SDP generated rules are more effective for the operation of small reservoirs.     

IMPLEMENTATION OF AN OPTIMIZATION MODEL FOR OPERATION OF A METROPOLITAN RESERVOIR SYSTEM1

ABSTRACT: A deterministic dynamic programming optimization model with a refining sectioning search procedure is developed and implemented to find least cost withdrawal and release patterns for water supple from a multiple reservoir system serving a metropolitan area. Applications are made to teh four reservoir system operated by the city of Dallas, Texas. A realistic cost structure, including nonlinear power consumption, block rate unit power costs, and flow dependent power consumption for intracity water distribution, is utilized. Applications are made to find least cost operating patterns and, as well, by inclusion of a water loss penalty function, supply patterns which will reduce evaporation water losses for the Dallas system.     

COMPUTATIONAL IMPROVEMENT FOR STOCHASTIC DYNAMIC PROGRAMMING MODELS OF URBAN WATER SUPPLY RESERVOIRS1

ABSTRACT: This paper describes two methods that are introduced to improve the computational effort of stochastic dynamic programming (SDP) as applicable to the operation of multiple urban water supply reservoir systems. The stochastic nature of streamflow is incorporated explicitly by considering it in the form of a multivariate probability distribution. The computationally efficient Gaussian Legendre quadrature method is employed to compute the conditional probabilities of streamflow, which accounts for the serial correlation of streamflow into each storage and the cross correlation between the streamflow into various storages. A realistic assumption of cross correlation of streamflow is introduced to eliminate the need to consider the streamflow combinations which are unlikely to occur in the SDP formulation. A “corridor” approach is devised to eliminate the need to consider the infeasible and/or inferior storage volume combinations in the preceding stage in computing the objective function in the recursive relation. These methods are verified in terms of computational efficiency and accuracy by using a hypothetical example of three interconnected urban water supply reservoirs. Therefore, it can be concluded that these methods allow SDP to be more attractive for deriving optimal operating rules for multiple urban water supply reservoir systems.     

A COMPARISON OF SOME DYNAMIC,LINEAR AND POLICY ITERATION METHODS FOR RESERVOIR OPERATION1

Within the past few years, a number of papers have been published in which stochastic mathematical programming models, incorporating first order Markov chains, have been used to derive alternative sequential operating policies for a multiple purpose reservoir. This paper attempts to review and compare three such mathematical modeling and solution techniques, namely dynamic programming, policy iteration, and linear programming. It is assumed that the flows into the reservoir are serially correlated stochastic quantities. The design parameters are assumed fixed, i.e., the reservoir capacity and the storage and release targets, if any, are predetermined. The models are discrete since the continuous variables of time, volume, and flow are approximated by discrete units. The problem is to derive an optimal operating policy. Such a policy defines the reservoir release as a function of the current storage volume and inflow. The form of the solution and some of the advantages, limitations and computational efficiencies of each of the models and their algorithms are compared using a simplified numerical example.     

A COMPARISON OF TWO METHODS FOR MULTIOBJECTIVE OPTIMIZATION FOR RESERVOIR OPERATION1

ABSTRACT: Two major objectives in operating the multireservoir system of the Upper Colorado River basin are maximization of hydroelectric power production and maximization of the reliability of annual water supply. These two objectives conflict. Optimal operation of the reservoir system to achieve both is unattainable. This paper seeks the best compromise solution for an aggregated reservoir as a surrogate of the multireservoir system by using two methods: the constraint method and the method of combined stochastic and deterministic modeling. Both methods are used to derive the stationary optimal operating policy for the aggregated reservoir by using stochastic dynamic programming but with different objective functions and minimum monthly release constraints. The resulting operating policies are then used in simulated operation of the reservoir with historical inflow records to evaluate their relative effectiveness. The results show that the policy obtained from the combination method would yield more hydropower production and higher reliability of annual water supply than that from the constraint-method policy.     

A RISK-BASED APPROACH FOR FLOOD CONTROL OPERATION OF A MULTIPURPOSE RESERVOIR1

ABSTRACT: Many approaches are available for operation of a multipurpose reservoir during flood season; one of them is allocation of storage space for flood control. A methodology to determine a reservoir operation policy based on explicit risk consideration is presented. The objective of the formulation is to maximize the reservoir storage at the end of a flood season while ensuring that the risk of an overflow is within acceptable limits. The Dynamic Programming technique has been used to solve the problem. This approach has been applied to develop operation policies for an existing reservoir. The performance of the policy was evaluated through simulation and was found to be satisfactory.     

APPLICATION OF THE GENETIC ALGORITHM FOR OPTIMIZING OPERATION RULES OF THE LiYuTan RESERVOIR IN TAIWAN1

ABSTRACT: A procedure to apply genetic algorithm to optimize operation rules is proposed and applied to the LiYuTan Reservoir in Taiwan. The designed operation rules are operation zones with discount rates of water supply. The first step of the procedure is to predefine the shape of boundary curves of operation zones according to reservoir storage routing. Then, relatively fewer variables are used to describe the curves, and a last genetic algorithm (GA) is applied to optimize the curves. The procedure is applied to the newly built LiYuTan Reservoir for increasing domestic water demands. Shortage index is used to evaluate the performance of operation zones. A year is divided into 36 operational periods, with each month containing three operational periods. The shortage indexes calculated in operational periods are 9.81, 8.27, and 7.13, respectively, for the reservoir without operation rules, applying operation zones optimized by GA with encoding 36 storage levels for each curve, and adopting operation zones optimized by GA with encoding the curves with predefined shape. The average deficits for the three cases are 77.2, 43.6, and 33.3 (10⁴ m³/day), respectively. The results indicate that operation zones optimized by the proposed procedure have smaller shortage indexes and lower average deficits. In addition, the optimized operation zones have less variation and thus are more practical for operation. Conclusively, the proposed procedure utilizing GA to optimize operation zones with predefined shape can provide better and realistic outcomes through limited iterations.     

A RESERVOIR OPERATING MODEL WITH INORGANIC QUALITY CONSTRAINTS FOR THE TRUCKEE RIVER1

ABSTRACT: This study investigated low flow augmentation as a means of meeting inorganic water quality standards for the Truckee River at the California-Nevada state line. A digital inorganic water quality model was combined with a deterministic dynamic reservoir operating model in an iterative process which allowed the optimization of releases subject to selected inorganic water quality constraints as well as downstream demands. Results from model runs with varied flow and river loading data indicate that flow augmentation may be a feasible and relatively inexpensive way of meeting standards for this system except in time of severe drought.     

A MATHEMATICAL MODEL FOR OPTIMUM DESIGN AND CONTROL OF METROPOLITAN WASTEWATER MANAGEMENT SYSTEMS1

ABSTRACT: A comprehensive mathematical model (Urban Wastewater Management Model) has been developed to continuously simulate time-varying wastewater flows and qualities in complex metropolitan combined sewerage systems. The model serves three functions: (1) assessment of existing or planned system performance in relation to other wastewater discharges in either a metropolitan or river basin area; (2) determination of the optium operation or automatic control of existing or planned systems during rainstorms; and (3) determination of the most economically feasible combination of design alternatives for improving or expanding existing systems to meet specified performance criteria. The model provides an efficient engineering tool for evaluating and controlling pollutant discharges from combined sewerage systems (including treatment plants) to receiving waters, while considering the time and spacial variations of rainfall and dry-weather flows and qualities as well as economic constraints.     

A MATHEMATICAL MODEL FOR EVALUATING THE POTENTIAL OF DESALTING1

ABSTRACT The Office of Saline Water, which has federal responsibility for developing low-cost, saline sources of fresh water, has recognized the need for an improved method of forecasting the future potential of desalting in this country. The magnitude of the role of desalting will influence the plans of federal, state, and local water resource agencies and the research and development programs of manufacturers. A dynamic simulation model has been developed by Arthur D. Little, Inc. under contract by OSW to translate relevant factors of water supply and demand into a forecast of desalting potential. The model projects the needs for desalting in 20 hydrologic regions of the U.S. Model performance has thus far been demonstrated by the development of a forecast and a battery of related sensitivity tests. Current results indicate the following potential desalting capacities: 225 MGD in 1980; 2,250 MGD in 2000; and 7,000 MGD in 2020. Significant improvements in desalting economics promise to increase these potentials by a factor of four or five by 2000-2020. Model inputs and results are continuing to be refined. When completed, OSW will have a dynamic tool with which to guide its R&D program.     

THE OPTIMAL TIME TO START THE OPERATION OF A DESALTING PLANT IN ISRAEL1

The study contains the methodology of finding the optimal time to start the operation of a desalting plant or any other non conventional source of water for agricultural use in Israel. The essence of the methodology is the moving equilibrium price and quantity of agricultural products. The market of agricultural products is represented by a single demand curve and a supply curve which is derived from the agricultural sector production function. Both the demand for and the supply of agricultural products functions are shifting over time. The marginal value product of water is the critical value that determines the optimal time of a desalting plant. The study is static in the sense that dynamic aspects per se are not dealt with-e.g. learning by doing of building and operating desalting plants. The empirical results although used for illustrative purposes can be used as reference points for other detailed and more ‘exact’ studies on water desalination.     

EVALUATION OF METHODS FOR ESTIMATING STREAM WATER QUALITY PARAMETERS IN A TRANSIENT MODEL FROM STOCHASTIC DATA1

ABSTRACT: In this study the estimation of parameters in water quality models represented by linear first order partial differential equations is investigated. Two sets of simulated input-output data, one with input noise and the other with output measurement error, were used. The parameters were estimated by a gradient technique (Bard's method) and a pattern search technique. The results indicate that the output measurement error significantly affects the values of parameter estimates as compared to the noise added to the input. Bard's method consistently gave results with a smaller sum of square value.     

A VERIFICATION OF THE QUAL-1 WATER QUALITY MODEL FOR THE LOWER MISSISSIPPI RIVER1

ABSTRACT: This paper describes the verification of the QUAL-1 mass transport model for the lower Mississippi River between St. Francisville and Point a la Hache using dye studies conducted by the U. S. Geological Survey. QUAL-1 is a one-dimensional steady-state model for rivers and is capable of predicting longitudinal profiles of soluble materials entering rivers from point sources. Both conservative and nonconservative parameters of water quality can be considered. The major problems surmounted were the determination of a diffusion coefficient and the use of transient data to verify a steady state model.     

APPLICATION OF A LOW-FLOW ASSESSMENT MODEL FOR THE MONONGAHELA RIVER BASIN1

ABSTRACT: The application of a low-flow assessment model is illustrated for the Monogahela River Basin. The model simulates the impact of reservoir operating rules and consumptive use limitation policies on low-flow frequency at downstream locations in the basin. Policies are evaluated using an observed flow sequence and synthetic flow inputs. The paper reviews the historical development of flow management on the Monogahela to provide background for the current study.     

A DECISION SUPPORT TOOL FOR THE MANAGEMENT OF MULTI‐RESERVOIR SYSTEMS1

ABSTRACT: A decision support tool is developed for the management of water resources, focusing on multipurpose reservoir systems. This software tool has been designed in such a way that it can be suitable to hydrosystems with multiple water uses and operating goals, calculating complex multi‐reservoir systems as a whole. The mathematical framework is based on the parameterization‐simulation‐optimization scheme. The main idea consists of a parametric formulation of the operating rules for reservoirs and other projects (i.e., hydropower plants). This methodology enables the radical decrease of the number of decision variables, making feasible the location of the optimal management policy, which maximizes the system yield and the overall operational benefit and minimizes the risk for the management decisions. The program was developed using advanced software engineering techniques. It is adaptable in a wide range of water resources systems, and its purpose is to support water and power supply companies and related authorities. It already has been applied to two of the most complicated hydrosystems of Greece, the first time as a planning tool and the second time as a management tool.