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.     

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.     

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.     

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.     

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.     

OPTIMIZING FLOOD CONTROL ALLOCATION FOR A MULTIPURPOSE RESERVOIR1

ABSTRACT. In the last decade much research has been devoted to applying the systems analysis approach to water resources problems. A popular research goal has been determination of the “best” method of operating a multipurpose reservoir. The goal of this study was to derive the economically optimum flood control diagram for a multipurpose reservoir by systems analysis. The technique employed to optimize the flood control diagram was programmed so that the optimization process could be applied to other multipurpose reservoirs. Two computer programs developed at the U.S. Army Corps of Engineers' Hydrologic Engineering Center were utilized with modifications to simulate the operation of Folsom Reservoir in central California. Economic analyses were incorporated along with an optimization technique into the reservoir operations program; and the resultant program was capable of routing a sequence of monthly reservoir inflows, computing benefits for various flood control diagrams (as dictated by the optimization procedure), and selecting the economically optimum flood control diagram. The univariate gradient technique was the optimization procedure employed. The two computer programs are on file at the Hydrologic Engineering Center in Davis, California.     

DERIVING THE NONLINEAR RISK-BENEFIT ALGORITHM FOR RESERVOIRS1

ABSTRACT: The Nonlinear Risk-Benefit (NRB) Algorithm includes risk as one of the objectives in a multiple-objective optimization problem. The NRB Algorithm is derived by extending the Surrogate Worth Trade-Off method to quadratic programming. This category of problem is common in water resources planning and design, especially multipurpose reservoir systems. Consequently, an example is given using the algorithm for optimally operating a multipurpose reservoir.     

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.     

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.     

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.     

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 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.     

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.     

REASONABLY FULL HYDROLOGIC DEVELOPMENT OF RESERVOIR SITES1

The size of multipurpose reservoir development is usually determined by an economic analysis of reservoir capabilities and the present and projected water resources needs which can be satisfied. This analysis is referred to as project formulation, wherein optimum conditions are sought. In responding to multiple objectives, i.e., national economic development, regional development and environmental quality, which are being considered in river basin planning in recent years, reservoirs should provide for reasonably full hydrologic development. Additional storage will be needed to provide opportunities for economic development, as well as meet unexpected development. Also, it provides more flow regulation capability for quality of environment considerations. An analysis has been made on twelve reservoir sites in the New York State portion of the Susquehanna River Basin to determine the so-called “reasonably full hydrologic development of reservoir sites.” Hydrologic, economic, environmental and physical characteristics of the sites are taken into consideration. For normal conditions, it can be concluded that a yield equivalent to about 80 percent of the average discharge (runoff) can be considered as reasonably full hydrologic development for reservoir sites in the Susquehanna River Basin in New York. The same technique can be applied elsewhere to determine reasonably full hydrologic development of reservoir sites.     

MANAGEMENT MODEL FOR CONTROLLING NITRATE CONTAMINATION IN THE NEW JERSEY PINE BARRENS AQUIFER1

ABSTRACT: Land use planning in rapidly developing areas can serve as an effective tool for minimizing water quality impacts on ground water supplies. A land use management model applied to Jackson Township of the New Jersey Pine Barrens was developed. The management model consisted of a simulation model for the transport of nitrates from septic tank systems through the aquifer and a multiobjective, goal programming optimization model to determine population density restrictions using 208 areawide planning population projections. Results showed that growth may have to be curtailed in several areas of Jackson Township and that current population projections over the next 30 years may result in unacceptably high nitrate concentrations downgradient of Jackson Township. The management framework provides a flexible approach to land use planning.     

Scenario‐Based and Scenario‐Neutral Assessment of Climate Change Impacts on Operational Performance of a Multipurpose Reservoir

Scenario‐based and scenario‐neutral impacts assessment approaches provide complementary information about how climate change‐driven effects on streamflow may change the operational performance of multipurpose dams. Examining a case study of Cougar Dam in Oregon, United States, we simulated current reservoir operations under scenarios of plausible future hydrology. Streamflow projections from the CGCM3.1 general circulation model for the A1B emission scenario were used to generate stochastic reservoir inflows that were then further perturbed to simulate a potentially drier future. These were then used to drive a simple reservoir model. In the scenario‐based analysis, we found reservoir operations are vulnerable to climate change. Increases in fall and winter inflow could lead to more frequent flood storage, reducing flexibility to store incoming flood flows. Uncertainty in spring inflow volume complicates projection of future filling performance. The reservoir may fill more or less often, depending on whether springs are wetter or drier. In the summer, drawdown may occur earlier to meet conservation objectives. From the scenario‐neutral analysis, we identified thresholds of streamflow magnitude that can predict climate change impacts for a wide range of scenarios. Our results highlight projected operational challenges for Cougar Dam and provide an example of how scenario‐based and scenario‐neutral approaches may be applied concurrently to assess climate change impacts.     

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.     

Interactive two-stage stochastic fuzzy programming for water resources management

In this study, an interactive two-stage stochastic fuzzy programming (ITSFP) approach has been developed through incorporating an interactive fuzzy resolution (IFR) method within an inexact two-stage stochastic programming (ITSP) framework. ITSFP can not only tackle dual uncertainties presented as fuzzy boundary intervals that exist in the objective function and the left- and right-hand sides of constraints, but also permit in-depth analyses of various policy scenarios that are associated with different levels of economic penalties when the promised policy targets are violated. A management problem in terms of water resources allocation has been studied to illustrate applicability of the proposed approach. The results indicate that a set of solutions under different feasibility degrees has been generated for planning the water resources allocation. They can help the decision makers (DMs) to conduct in-depth analyses of tradeoffs between economic efficiency and constraint-violation risk, as well as enable them to identify, in an interactive way, a desired compromise between satisfaction degree of the goal and feasibility of the constraints (i.e., risk of constraint violation).     

Comparison of Two Spatial Optimization Techniques: A Framework to Solve Multiobjective Land Use Distribution Problems

Two spatial optimization approaches, developed from the opposing perspectives of ecological economics and landscape planning and aimed at the definition of new distributions of farming systems and of land use elements, are compared and integrated into a general framework. The first approach, applied to a small river catchment in southwestern France, uses SWAT (Soil and Water Assessment Tool) and a weighted goal programming model in combination with a geographical information system (GIS) for the determination of optimal farming system patterns, based on selected objective functions to minimize deviations from the goals of reducing nitrogen and maintaining income. The second approach, demonstrated in a suburban landscape near Leipzig, Germany, defines a GIS-based predictive habitat model for the search of unfragmented regions suitable for hare populations (Lepus europaeus), followed by compromise optimization with the aim of planning a new habitat structure distribution for the hare. The multifunctional problem is solved by the integration of the three landscape functions (“production of cereals,” “resistance to soil erosion by water,” and “landscape water retention”). Through the comparison, we propose a framework for the definition of optimal land use patterns based on optimization techniques. The framework includes the main aspects to solve land use distribution problems with the aim of finding the optimal or best land use decisions. It integrates indicators, goals of spatial developments and stakeholders, including weighting, and model tools for the prediction of objective functions and risk assessments. Methodological limits of the uncertainty of data and model outcomes are stressed. The framework clarifies the use of optimization techniques in spatial planning.