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.     

OPTIMAL RESERVOIR MANAGEMENT AND CROP PLANNING UNDER DETERMINISTIC AND STOCHASTIC INFLOWS1

ABSTRACT: This study analyzes planning under deterministic and stochastic inflows for the Mayurakshi project in India. Models are developed to indicate the optimal storage of reservoir water, the transfer of water to the producing regions, and the spillage of water from the reservoir, if needed. A deterministic programming model was first formulated to represent the existing situation. A chance-constrained model then was constructed to evaluate potential violations of the deterministic model. Both models were quantified for the command area. Data were collected from surveys of the area and from government agencies. Both the deterministic and change-constrained models suggest a more intensive cropping program in the region. Both emphasize more dependence on rabi and less on kharif crops. The chance-constrained especially suggests use of more water in the rabi season. Important chances in cropping programs and labor use take place under the chance-constrained model.     

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.     

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.     

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 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 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.     

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.     

IRRIGATION WATER DELIVERY SYSTEM OPERATION VIA AGGREGATE STATE DYNAMIC PROGRAMMING1

ABSTRACT: A modified dynamic programming (DP) approach that is called aggregate state dynamic programming (ASDP) is presented to optimally operate irrigation water delivery systems. ASDP can be applied to multiple reservoir systems without encountering dimensionality problems. In addition, the random nature of water supply and consumptive crop demands can be incorporated into the technique. A case study is presented to display the application of ASDP. Using a sum-of-squared shortages objective, ASDP out performs a traditional separation technique and approaches the theoretical (ideal) optimum. Problem settings that are conducive to the use of ASDP and limitations of the technique are presented.     

A COMPARISON OF STREAMFLOW GENERATION MODELS FOR RESERVOIR CAPACITY-YIELD ANALYSIS1

ABSTRACT: An analysis of four streamflow generation schemes for the use in the estimation of the required conservation storage for a single reservoir is presented. The comparison of the generating schemes should aid in the selection of an appropriate model type for the reservoir sizing problem. The streamflow generation models are compared using two criteria. The first comparison is between the statistics of the generated streamflow sequences and the corresponding statistics from the historical record. The second evaluation compares the median reservoir size determined by each model with the required storage based on the historical flow sequence. The results of a comparative analysis for monthly streamflow data for the Rzav River in Yugoslavia are presented and discussed. The results indicate that both evaluation criteria are required to discriminate between the various options.     

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.     

RESERVOIR OPERATION ANI EVALUATION OF DOWNSTREAM FLOW AUGMENTATION1

ABSTRACT: Operation of a storage‐based reservoir modifies the downstream flow usually to a value higher than that of natural flow in dry season. This could be important for irrigation, water supply, or power production as it is like an additional downstream benefit without any additional investment. This study addresses the operation of two proposed reservoirs and the downstream flow augmentation at an irrigation project located at the outlet of the Gandaki River basin in Nepal. The optimal operating policies of the reservoirs were determined using a Stochastic Dynamic Programming (SDP) model considering the maximization of power production. The modified flows downstream of the reservoirs were simulated by a simulation model using the optimal operating policy (for power maximization) and a synthetic long‐term inflow series. Comparing the existing flow (flow in river without reservoir operation) and the modified flow (flow after reservoir operation) at the irrigation project, the additional amount of flow was calculated. The reliability analysis indicated that the supply of irrigation could be increased by 25 to 100 percent of the existing supply over the dry season (January to April) with a reliability of more than 80 percent.     

MULTIOBJECTWE OPTIMIZATION OF RESERVOIR SYSTEMS OPERATION1

ABSTRACT: Development of optimal operational policies for large-scale reservoir systems is often complicated by a multiplicity of conflicting project uses and purposes. A wide range of multiobjective optimization methods are available for appraising tradeoffs between conificting objectives. The purpose of this study is to provide guidance as to those methods which are best suited to dealing with the challenging large-scale, nonlinear, dynamic, and stochastic characteristics of multireservoir system operations. As a case study, the selected methodologies are applied to the Han River Reservoir System in Korea for four principal project objectives: water supply and low flow augmentation; annual hydropower production, reliable energy generation, and minimization of risk of violating firm water supply requirements. Additional objectives such as flood control are also considered, but are imposed as fixed constraints.     

OPTIMAL ALLOCATION OF ARTIFICIAL AERATION ALONG A POLLUTED STREAM USING DYNAMIC PROGRAMMING1

ABSTRACT: When a series of aerators are used to raise the level of dissolved oxygen in a polluted stream through instream artificial aeration augmentation, the system is governed by the basic dissolved oxygen mass balance equation with the existence of artificial aeration as its boundary conditions. A mathematical model is formulated for the optimization of the allocation of aeration capacity to each of the series of aerators subject to a limitation on total available aeration capacity. The objective function is the minimization of the sum of the squares of the aeration costs and the costs incurred by damaging or unnecessarily improving the system. The original constrained allocation problem is simplified by converting it to an unconstrained one via the use of Lagrange multiplier. A discretized dynamic programming algorithm is formulated for finding the optimal allocation policy. A typical optimal aeration capacity allocation policy and its corresponding dissolved oxygen sag profile for the illustrated numerical example is presented, and the relationship between the total available aeration capacity and Lagrange multiplier is also developed treating weighting factors as parameters.     

A STOCHASTIC DYNAMIC PROGRAMMING MODEL FOR THE OPTIMUM OPERATION OF A MULTI-PURPOSE RESERVOIR1

ABSTRACT: The main objective of this paper is to present a stockastic dynamic programming model useful in determining the optimal operating policy of a single multipurpose surface reservoir. It is the unreliability of forecasting the amount of future streamflow which makes the problem of a reservoir operation a stochastic process. In this paper the stochastic nature of the streamflow is taken into account by considering the correlation between the streamflows of each pair of consecutive time intervals. This interdependence is used to calculate the probability of transition from a given state and stage to its succeeding ones. A dynamic programming model with a physical equation and a stochastic recursive equation is developed to find the optimum operational policy. For illustrative purposes, the model is applied to a real surface water reservoir system.     

A NONLINEAR PROGRAMMING ALGORITHM FOR REAL-TIME HOURLY RESERVOIR OPERATIONS1

ABSTRACT: The optimization of real-time operations for a single reservoir system is studied. The objective is to maximize the sum of hourly power generation over a period of one day subject to constraints of hourly power schedules, daily flow requirement for water supply and other purposes, and the limitations of the facilities. The problem has a nonlinear concave objective function with nonlinear concave and linear constraints. Nonlinear Duality Theorems and Lagrangian Procedures are applied to solve the problem where the minimization of the Lagrangian is carried out by a modified gradient projection technique along with an optimal stepsize determination routine. The dimension of the problem in terms of the number of variables and constraints is reduced by eliminating the 24 continuity equations with a special implicit routine. A numerical example is presented using data provided by the Bureau of Reclamation, Sacramento, California.     

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.     

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 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 A SINGLE RESERVOIR OPERATION POLICY TO A RAIN WATER CISTERN SYSTEM1

The operation policy for a single reservoir is applied to a rain water cistern system because the functions of a cistern are similar to a simple single reservoir. Since the cistern is a closed system, water loss is negligible. In this study, a dynamic programming analysis has been made to study the effects of the probable weekly rainfall and the water storage in the cistern towards the water consumption policy. The result of this study indicates that the water consumption rate should be adjusted into a lower rate when the water storage in the cistern is low and/or when the expected probable weekly rainfall is low if the owner of the cistern does not want to risk the chance of an empty cistern. The demand for a reliable method for forecasting weekly rainfall is evident in this study.