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.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

COST EFFECTIVE OPERATION OF URBAN WATER SUPPLY SYSTEM USING DYNAMIC PROGRAMMING1

ABSTRACT: Efficient operation of a city water supply system is an important goal of all municipalities. Efficient operation should result in minimum operation cost through reduction in total energy use and/ or reduction in on-peak energy consumption. An optimization model was designed for operating the water supply systems of cities using groundwater. The Newton-Raphson pipe network was used for network analysis and a dynamic programming optimization algorithm was used for determining a schedule for pump operation in the pipe network system. The model is most suitable for use in small cities with up to 45,000 in population, but with large-scale disintegration techniques may also be used for larger cities. The savings in operation costs are a function of energy cost and energy use pattern and water use pattern in the area.     

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 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.     

MULTI-OBJECTIVE RESERVOIR OPERATION INCLUDING TURBIDITY CONTROL1

ABSTRACT: The operational problems of a reservoir are expressed by three coordinates: space, time stage, and objective. The operational procedure is formulated using dynamic programming as a multi-objective problem. After comparing the scalar and the vector optimization, the scalar optimization technique is applied to turbidity analysis in a reservoir.     

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.     

THE VALUE OF STREAMFLOW FORECASTING IN RESERVOIR OPERATION1

ABSTRACT: The value of streamflow forecasts in reservoir operation depends on a number of factors and may vary considerably. Assessment of forecast benefits is presented here for three specific systems. Statistical streamflow models of increasing forecasting ability are coupled with a recently developed stochastic control method in extensive simulation experiments. The performance of the system is statisticafly evaluated with regard to energy generation and flood and drought prevention. The results indicate that forecast benefits are system specific and may range from quite substantial to fairly minimal.     

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 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.     

AN APPLICATION OF MATHEMATICAL PROGRAMMING IN PLANNING SURFACE WATER STORAGE1

Designing a surface reservoir involves the concept of reservoir yield. This concept embodies three basic information items: hydrologic regime, active storage volume, and reservoir release policy. In the actual case presented below, the magnitude of the active storage was prescribed by a legal procedure, so that the planning issue became that of determining the reservoir yield given the hydrological information. A stochastic dynamic programming model was formulated to derive a schedule of seasonal optimal reservoir releases and their respective probabilities of occurrence. This schedule is the reservoir yield. The yearly cycle was divided into three seasons representing the actual climatic conditions, and conditional probabilities linking streamflows in consecutive seasons were estimated. An operating policy was postulated, based on the same set of legal decisions that prescribed the active storage volume, and target reservoir releases were assumed. Similarly, target storages at the end of each season were set up. The optimizing/ minimizing criterion in the dynamic programming formulation was the sum of squares of deviations of actual releases and final storage volumes from their respective targets.     

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.     

APPLICATION OF CROP YIELD FUNCTIONS IN RESERVOIR OPERATION1

ABSTRACT: A model is developed for real-time operation of an irrigation reservoir with the objective of maximizing the value of multiple crop yields during a growing season. The model employs monthly additive and product forms of crop yield functions for dry matter and grain crops, respectively. The resulting nonlinear optimization model uses a log transform to reduce nonlinearities in the model. An application of the proposed model is compared to a common operating rule used in simulation models. The proposed model results were better in terms of net benefits from crop yields. The model uses GAMS (General Algebraic Modeling System) language. It requires an IBM-compatible microcomputer and is suitable for use by a reservoir manager.     

MULTIVARIATE CLASSIFICATION OF SMALL ORDER WATERSHEDS IN THE QUABBIN RESERVOIR BASIN,MASSACHUSEVFS1

ABSTRACT: A multivariate approach was used to analyze hydrologic, geologic, geographic, and water-chemistry data from small order watersheds in the Quabbin Reservoir Basin in central Massachusetts. Eighty three small order watersheds were delineated and landscape attributes defining hydrologic, geologic, and geographic features of the watersheds were compiled from geographic information system data layers. Principal components analysis was used to evaluate 11 chemical constituents collected bi-weekly for 1 year at 15 surface-water stations in order to subdivide the basin into subbasins comprised of watersheds with similar water quality characteristics. Three principal components accounted for about 90 percent of the variance in water chemistry data. The principal components were defined as a biogeochemical variable related to wet. land density, an acid-neutralization variable, and a road-salt variable related to density of primary roads. Three subbasins were identified. Analysis of variance and multiple comparisons of means were used to identify significant differences in stream water chemistry and landscape attributes among subbasins. All stream water constituents were significantly different among subbasins. Multiple regression techniques were used to relate stream water chemistry to landscape attributes. Important differences in landscape attributes were related to wetlands, slope, and soil type.