بهره برداری بهینه از مخازن چند منظوره با استفاده از برنامه ریزی پویا تصادفی انعطاف پذیر

This paper presents a fuzzy stochastic dynamic programming (FSDP) approach to derive steady-state multipurpose reservoir operating policies. The vagueness associated with some operating objectives as well as with the decision-making process is apprehended through fuzzy set theory. Operating objectives are considered as fuzzy sets and their membership functions represent decision maker’s preferences and satisfaction associated with particular states of the system. At each stage of the FSDP algorithm, current operating objectives are considered as flexible constraints while the fuzzy goal is obtained from the set of maximizing decisions calculated at the previous stage. The decision function results from the aggregation of the flexible constraints and the fuzzy goal. Continuous re-optimization models, with discrete FSDP-derived membership functions approximated by cubic splines, are used for implementing FSDP-derived results in real-time or simulated operation. The model can also employ different hydrologic state variables to describe the temporal persistence of the streamflow process. The proposed approach is implemented to derive operating policies for the Mansour Eddahbi reservoir (Morocco) with current inflow as the hydrologic state variable.

Long-term optimal management of multipurpose reservoirs is a sequential, stochastic and often fuzzy optimization problem. Sequential decisions have to be taken so that time-dependent and imprecise demands can be fulfilled under constraints imposed by the economical and physical limitations of the system, as well as by the stochastic nature of inflows. As pointed out by Yeh [1], there is no general method for optimizing reservoir operation; it ranges from simulation to optimization models. If early studies focused on hydropower reservoirs, a recent shift has been observed toward multipurpose systems. This movement can be explained by the raising concerns about environmental quality and by our ability to cope with evolving objectives in the management of water resources systems (Loucks [2]). The multiobjectivity issue becomes even more complex when some of the objectives are difficult-to-quantify, i.e. they cannot or should not be converted to monetary units. For example, such objectives can be environmental quality, recreation, flood control, subsistence agriculture in developing countries. To deal with this issue, traditional operation models based on economic objectives must be adapted so that the inherent vagueness can be captured in the definition of the loss (or penalty) functions (Teegavarapu and Simonovic [3]). An alternative to this approach consists in eliciting decision makers’ satisfaction associated with particular states of the system and then to integrate it into an optimization model (Fontane et al. [4]). This paper presents a general optimization approach for deriving efficient reservoir operating rules while considering: (1) the operating objectives as flexible constraints, (2) the temporal persistence of the hydrologic conditions, and (3) the unboundedness of the planning period. The flexibility of the constraints allows us to capture decision makers’ preferences on the solution and to relax the set of possible solutions so that partially feasible solutions can also be examined. Stochastic dynamic programming (SDP) is a powerful technique for optimizing reservoir operation problems in which the stochastic nature of the inflows plays a key role and has a deep impact on system performance. In the implicit SDP formulation, many sequences of streamflows are used as inputs to deterministic DP models with well-defined termination time. Then, the optimal policies are traditionally determined by regression analysis. In the explicit SDP formulation, the temporal persistence of the streamflow process is directly incorporated in the optimization algorithm by means of transition probabilities. This second alternative better exploits the information found in most hydrologic time series, but it is also more computationally demanding since it requires an additional state variable (the hydrologic state variable). Recent applications of SDP can be found in Tejada-Guibert et al. [5] and [6], Liang et al. [7], Kim and Palmer [8]. Like other optimization techniques, SDP can be fuzzified so as to capture the imprecise nature of the constraints and/or the objectives. Fuzzy set theory and fuzzy logic provide mathematical frameworks for dealing with vague objects and approximate reasoning. There are basically two ways to implement those concepts in reservoir operation problems. The first utilizes a fuzzy-rule based scheme to derive operating rules using an “IF-THEN” principle to emulate reservoir operators’ knowledge (Shrestha et al. [9]). The second relies on fuzzified traditional optimization techniques such as linear programming (LP) and dynamic programming (DP). Teegavarapu and Simonovic [3] optimize short-term reservoir operation by minimizing economic losses with LP and non-LP formulations in which both the penalty coefficients and zones are considered fuzzy. Fontane et al. [4] use a deterministic fuzzy DP (FDP) algorithm to determine optimal monthly release decisions based on Bellman and Zadeh’s framework [10]. This process is repeated a large number of times using synthetically generated equally likely streamflows so that the release rule can be obtained from regression analyses. This approach presents some interesting features but also has some limitations. For example, it implicitly assumes that current decisions are independent of future events and decisions beyond the planning horizon. So, the distant future is ignored. The maximal solutions are supplied by the max–min formulation, which only depends on the most pessimistic objective. Due to the nature of DP, this lack of compensation deeply affects the overall sequence of decisions; one unambitious objective is enough to make a trajectory uninteresting. In addition, the implementation of FDP-derived results is very sensitive to the hydrological regime, so that the effectiveness of the release rule can easily become questionable because of the low coefficients of determination for the regressions. Finally, the implementation of an FDP approach in a Monte Carlo simulation framework only considers the stochastic variability of the hydrologic inputs, ignoring the temporal persistence found in most hydrologic time series. The proposed FSDP model addresses these issues. In the present study, operating objectives are considered as flexible constraints of a stochastic optimization problem over an unbounded planning horizon. The recursive fuzzy DP equation is thus generalized by (1) directly incorporating the probability distributions of the hydrologic inputs, (2) explicitly considering the unboundedness of the planning horizon, (3) modeling multiobjective decision-making by allowing compensatory connectives. Consequently, the reservoir operation problem is analyzed as a never-ending sequence of decisions, in which current and future decisions may influence each other. In this paper, the objectives faced by the decision maker are the immediate and the future consequences associated with a release decision. The immediate consequences are the degrees indicating how well the various current operating objectives are satisfied. The future consequences, on the other hand, are numbers indicative of how well the current release decision affects future operations. Important features of the FSDP model are: (1) it emphasizes multipurpose use; (2) it accommodates multiparticipatory decision-making framework; and (3) it is not restricted to conventional (economic) objectives. This optimization model can therefore be used to derive sustainable operating rules for multipurpose multireservoir systems since it promotes both economic and equity efficiencies as recommended by the World Commission on Dams (WCD [11]). As pointed out by Tilmant et al. [12], the main difference between the FSDP and the classical SDP approaches is to be found in the way the vagueness of the reservoir operation problem is handled, rather than in the optimization results, and thus, in the performance of the system. Further applications of the FSDP model to reservoir operation problems can be found in Tilmant et al. [13] and [14]. This paper also presents a methodology for identifying stationary policies for FSDP algorithms. Further, this paper vividly demonstrates that careful attention should be paid to the determination of the level of compensation between the satisfaction degrees of the objectives. To illustrate this, we implement the FSDP model to derive steady-state operating policies for the Mansour Eddahbi reservoir in Morocco. The main services are: irrigation, flood control and hydropower. Several FSDP models are formulated, each with a specific level of compensation. Their performances are then compared in terms of reliability and resiliency of system operation with different satisfactory states. Those results are obtained from simulation using continuous re-optimization models. The paper is organized as follows. In the second section, flexible stochastic dynamic programming is presented and the aggregation issues are discussed. The third section is devoted to the case study and the application of the optimization model. Finally, conclusions are given in section four.

The paper presents a fuzzy optimization approach to derive steady-state operating policy for multipurpose reservoirs. The methodology can be summarized in four steps: (1) construct membership functions from water users’ preferences; (2) run several FSDP models with different compensation parameters γ; (3) identify selection criteria; and (4) perform a sensitivity analysis to determine the “best” compensation parameter by implementing continuous re-optimization models with membership functions generated by the FSDP algorithm developed in step (2). The case study of the Mansour Eddahbi reservoir in Morocco illustrates the methodology. The explicit fuzzy SDP approach presented in this paper handles the vagueness associated with certain reservoir operating objectives as well as the randomness of the hydrologic conditions. By considering the optimization problem as a flexible constraint satisfaction problem, we are able to bring into the picture the decision-maker preferences on the solution. This allows us to go around the economic estimation of the consequences associated with a decision, while taking into account a variety of intangible operating objectives and their relative importance. Therefore, the approach is well suited for situations where classical optimization techniques cannot be implemented due to the absence of information and/or methods for assessing the economic benefits (or losses) of system operation. The model also explicitly considers the impact of current decisions on the future and the influence of future decisions on the best course of action now. This paper also demonstrates that the traditional framework for fuzzy decision-making based on the max–min optimization problem yields unsatisfactory system performance and, due to the lack of convergence, is computationally demanding. Although the aggregation of the objectives to form the decision function is close to the “hard” intersection, some level of compensation must be taken into account when implementing the FSDP algorithm. Nevertheless, the questions of how much compensation is required, at what time, and for what system’s status are left open. We addressed these issues by assuming that the level was both independent of the system’s status and time-invariant. Then, the level itself was determined by a sensitivity analysis in which we compared the performance of the system using four criteria, such as the reliability and the resiliency of simulated system operation.