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.     

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.     

FLOOD CONTROL RESERVOIR OPERATIONS UNDER ENVIRONMENTAL RESTRAINTS1

ABSTRACT: A flood control reservoir protects valuable developments on the downstream flood plain by storing flood waters and releasing them at a rate that will reduce the downstream damage. The water surface level of the flood pool behind the dam can fluctuate considerably during the occurrence of a large magnitude flood causing the inundation of trees, low vegetation, and water based recreation facilities located in those areas of the flood pool area that are normally well above the water level. The amount of damage that will occur in the upper levels of the flood storage area will depend on the depth and duration of the inundation that occurs. This, in turn, is directly related to the operating policy for the reservoir. A dynamic programming optimization model of flood control reservoir operation is presented. This model determines the reservoir operating schedule that minimizes downstream flood damages. Various constraints are added to the model to account for the environmental impacts of long periods of flood storage.     

RESERVOIR RELEASES TO USES WITH DIFFERENT RELIABILITY REQUIREMENTS1

ABSTRACT: Releases from a reservoir may be allocated to a number of uses, each of which may require a given volume of water at a different reliability. The paper provides a method that can be used to estimate the volume of water associated with a given reliability for each use of water when the proportion of releases allocated to each use is known. These results can be used to evaluate the meeting of specified objectives under a published release policy derived by stationary stochastic dynamic programming. The results can also be used to solve water allocation problems when the probability distribution of available water is known (or can be estimated) and water has multiple uses, each of which has different volume and reliability requirements.     

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.     

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.     

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.     

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.     

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 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 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.     

OPTIMAL CONTROL OF RESERVOIR RELEASES TO MINIMIZE SEDIMENTATION IN RIVERS AND RESERVOIRS1

ABSTRACT: An optimal control methodology and computational model are developed to evaluate multi‐reservoir release schedules that minimize sediment scour and deposition in rivers and reservoirs. The sedimentation problem is formulated within a discrete‐time optimal control framework in which reservoir releases represent control variables and reservoir bed elevations, storage levels, and river bed elevations represent state variables. Constraints imposed on reservoir storage levels and releases are accommodated using a penalty function method. The optimal control model consists of two interfaced components: a one‐dimensional finite‐difference simulation module used to evaluate flow hydraulics and sediment transport dynamics, and a successive approximation linear quadratic regulator (SALQR) optimization algorithm used to update reservoir release policies and solve the augmented control problem. Hypothetical two‐reservoir and five‐reservoir networks are used to demonstrate the methodology and its capabilities, which is a vital phase towards the development of a more robust optimal control model and application to an existing multiple‐reservoir river network.     

DYNAMIC PROGRAMMING FOR OPTIMAL CAPACITY EXPANSION OF WASTEWATER TREATMENT PLANTS1

ABSTRACT: A dynamic programming procedure for the planning and operation of a wastewater treatment plant over a long period of time is presented. In order to meet increased demands for wastewater treatment in the future, the expansion of existing plants must be considered. Dynamic programming is employed to determine the optimal schedule of expansion at each plant, simultaneously determining an optimal operating policy (treatment level). The optimal schedule of expansion at each plant depends on the following: (1) the shape of the projected wastewater demand function; (2) the interest rate used; (3) the locations and capacities of the facilities available; and (4) the rates of increase of the costs of construction, labor, chemicals, and electric power. An example illustrating the use of the procedure is presented.     

Multireservoir real-time operations for flood control using balanced water level index method

This paper presents a real-time simulation-optimization operation procedure for determining the reservoir releases at each time step during a flood. The proposed procedure involves two models, i.e., a hydrological forecasting model and a reservoir operation model. In the reservoir operation model, this paper compares two flood-control operation strategies for a multipurpose multireservoir system. While Strategy 1 is the real-time joint reservoir operations without using the balanced water level index (BWLI) method, Strategy 2 involves real-time joint reservoir operations using the BWLI method. The two strategies presented are formulated as mixed-integer linear programming (MILP) problems. The idea of using the BWLI method is derived from the HEC-5 program developed by the US Army Corps of Engineers. The proposed procedure has been applied to the Tanshui River Basin system in Taiwan using the 6h ahead forecast data of six typhoons. A comparison of the results obtained from the two strategies reveals that Strategy 2 performs much better than Strategy 1 in determining the reservoir real-time releases throughout the system during flood emergencies in order to minimize flooding, while maintaining all reservoirs in the system in balance if possible. Consequently, the proposed model using the BWLI method demonstrates its effectiveness in estimating real-time releases.     

Developing a New Operations Plan for the Bow River Basin Using Collaborative Modeling for Decision Support

The Bow River Basin is a cornerstone of Alberta's development. In 2010, stakeholders representing interests from agriculture, municipalities, environment, and more formed the Bow River Project Research Consortium to help determine the potential for improving the operations in the basin. At present, upstream reservoirs are operated primarily for hydropower, whereas downstream reservoirs are operated for irrigation. Through Collaborative Modeling for Decision Support the stakeholders were able to develop a new method for operating the system that would dramatically improve environmental performance. The main components of the new operating strategy called for: purchase or setting aside of a small amount of storage volume in the power reservoirs; a set of rules for releases from that storage; an agreement by the major irrigation districts with the largest water licenses to utilize their ability to shift deliveries to and from their large offstream storage reservoirs to allow for increased instream flows, and to allow junior water license holders (mainly municipal and industrial supplies) an uninterrupted water supply; limitations of reservoir fluctuations to improve inreservoir habitat for fisheries; and increased minimum flows throughout the system leading to improved environmental outcomes. Costs of this strategy were minimal, impacts on power revenue were estimated at <US$2 million/yr on average. Compensatory arrangements should be possible.     

A PROPOSED MODIFICATION TO REGULATION OF LAKE OKEECHOBEE1

ABSTRACT: The current Lake Okeechobee regulation schedule is two feet higher than previous schedules that were in operation during the early 1970's. Its implementation was in response to prolonged periods of drought that occurred during the 1960's and early 1970's and the large increases in consumptive uses that were projected, and are presently occurring in south Florida. The additional storage provided by the schedule undoubtedly helped prevent more severe water shortages during the record setting 1980–1982 drought. However, two environmental concerns associated with the present schedule surfaced in recent years with the return to more normal rainfall conditions. First, the present schedule allows frequent high water conditions to exist in the lake that appear to be stressful to the unique littoral zone habitat of the lake. Second, the allowable buildup of storage prior to the dry season, combined with the large required decrease in storage prior to the hurricane (wet) season, contribute to the need for large regulation releases to tidewater. These large discharges have undesirable impacts on ecosystems of the downstream estuaries. This paper presents an alternative schedule that better meets the needs of the estuarine habitats without negatively impacting the other objectives of managing the lake.     

OPTIMIZATION OF MULTIPURPOSE RESERVOIR SYSTEM OPERATION1

ABSTRACT: The goal programming approach for multipurpose reservoir operation has been proposed and applied to the Bhadra reservoir system, having irrigation and hydropower production as dual purposes, in India. The objective of the model is to satisfy sequentially a series of operating criteria. Two goal programming models, one with the objective function as minimizing the deviations from storage targets and the other with the objective function as minimizing the deviations from release targets, have been formulated and applied to the reservoir system under study. The results proved that the model with release targets is preferred over the model with storage targets for determining operational policies for multipurpose reservoir system.     

EFFECTS OF CLIMATE CHANGE ON WATER RESOURCES IN THE DELAWARE RWER BASIN1

ABSTRACT: The effects of potential climate change on water resources in the Delaware River basin were determined. The study focused on two important water-resource components in the basin: (1) storage in the reservoirs that supply New York City, and (2) the position of the salt front in the Delaware River estuary. Current reservoir operating procedures provide for releases from the New York City reservoirs to maintain the position of the salt front in the estuary downstream from freshwater intakes and ground-water recharge zones in the Philadelphia metropolitan area. A hydrologic model of the basin was developed to simulate changes in New York City reservoir storage and the position of the salt front in the Delaware River estuary given changes in temperature and precipitation. Results of simulations indicated that storage depletion in the New York City reservoirs is a more likely effect of changes in temperature and precipitation than is the upstream movement of the salt front in the Delaware River estuary. In contrast, the results indicated that a rise in sea level would have a greater effect on movement of the salt front than on storage in the New York City reservoirs. The model simulations also projected that, by decreasing current mandated reservoir releases, a balance can be reached wherein the negative effects of climate change on storage in the New York City reservoirs and the position of the salt front in the Delaware River estuary are minimized. Finally, the results indicated that natural variability in climate is of such magnitude that its effects on water resources could overwhelm the effects of long-term trends in precipitation and temperature.     

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.     

CLIMATE CHANGE IMPACTS UNCERTAINTY FOR WATER RESOURCES IN THE SAN JOAQUIN RIVER BASIN,CALIFORNIA1

ABSTRACT: A climate change impacts assessment for water resources in the San Joaquin River region of California is presented. Regional climate projections are based on a 1 percent per year CO₂ increase relative to late 20th Century CO₂ conditions. Two global projections of this CO₂ increase scenario are considered (HadCM2 and PCM) during two future periods (2010 to 2039 and 2050 to 2079). HadCM2 projects faster warming than PCM. HadCM2 and PCM project wetter and drier conditions, respectively, relative to present climate. In the HadCM2 case, there would be increased reservoir inflows, increased storage limited by existing capacity, and increased releases for deliveries and river flows. In the PCM case, there would be decreased reservoir inflows, decreased storage and releases, and decreased deliveries. Impacts under either projection case cannot be regarded as more likely than the other. Most of the impacts uncertainty is attributable to the divergence in the precipitation projections. The range of assessed impacts is too broad to guide selection of mitigation projects. Regional planning agencies can respond by developing contingency strategies for these cases and applying the methodology herein to evaluate a broader set of CO₂ scenarios, land use projections, and operational assumptions. Improved agency access to climate projection information is necessary to support this effort.