Elsevier

Advances in Water Resources

Volume 32, Issue 9, September 2009, Pages 1429-1443
Advances in Water Resources

Evaluation of stochastic reservoir operation optimization models

https://doi.org/10.1016/j.advwatres.2009.06.008Get rights and content

Abstract

This paper investigates the performance of seven stochastic models used to define optimal reservoir operating policies. The models are based on implicit (ISO) and explicit stochastic optimization (ESO) as well as on the parameterization–simulation–optimization (PSO) approach. The ISO models include multiple regression, two-dimensional surface modeling and a neuro-fuzzy strategy. The ESO model is the well-known and widely used stochastic dynamic programming (SDP) technique. The PSO models comprise a variant of the standard operating policy (SOP), reservoir zoning, and a two-dimensional hedging rule. The models are applied to the operation of a single reservoir damming an intermittent river in northeastern Brazil. The standard operating policy is also included in the comparison and operational results provided by deterministic optimization based on perfect forecasts are used as a benchmark. In general, the ISO and PSO models performed better than SDP and the SOP. In addition, the proposed ISO-based surface modeling procedure and the PSO-based two-dimensional hedging rule showed superior overall performance as compared with the neuro-fuzzy approach.

Introduction

The major task of reservoir operation is to decide how much water should be released now and how much should be retained for future use given some available and/or forecasted information at the beginning of the current time period. In practice, reservoir operators usually follow rule curves, which stipulate the actions that should be taken conditioned on the current state of the system.

Rule curves are typically constructed from simulation models that provide reservoir responses to predefined operating policies. Since there are often a large number of feasible policies, mathematical optimization techniques may assist in identifying the best one as they implicitly look at all possible alternatives [48]. The application of optimization to solve reservoir operation problems has been a topic extensively studied during the last few decades [19], [47], [48]. Various of these studies deal with deterministic optimization models, which do not consider the uncertainties of some variables such as future reservoir inflows. Uncertainty becomes important when expected values of inflows cannot appropriately represent highly variable hydrologic conditions or when the inflows cannot be reliably forecasted for a relatively long period. In such cases, the problem is typically addressed by stochastic dynamic programming (SDP). SDP is the most popular type of explicit stochastic optimization (ESO), an approach that incorporates probabilistic inflow methods directly into the optimization problem. However, several authors have pointed out that, despite the continuing research, there is still a gap between theoretical developments and real-world implementations [19], [40], [47], [34], [48]. Reservoir operators are usually skeptical to replace simulation with sophisticated optimization models since the latter are mathematically more complex, especially when stochasticity is explicitly included.

In order to cope with this issue, the present paper investigates two techniques, namely, implicit stochastic optimization (ISO) and parameterization–simulation–optimization (PSO), which are both able to incorporate inflow uncertainties and to provide rule curves in an arguably simpler way than ESO. In brief, ISO uses deterministic optimization to operate the reservoir under several equally likely inflow scenarios and then examines the resulting set of optimal operating data to develop the rule curves. In a different way, PSO first predefines a shape for the rule curve based on some parameters and then applies heuristic strategies to look for the combination of parameters that provides the best reservoir operating performance under possible inflow scenarios. In this way, most stochastic aspects of the problem, including spatial and temporal correlations of unregulated inflows, are implicitly included [19].

The utilization of ISO for finding reservoir operating policies was first exploited by Young [50] in a study that utilized dynamic programming applied to annual operations. The optimal releases found by the dynamic programming model were regressed on the current reservoir storage and the projected inflow for the year. The regression equation could be thus used to obtain the reservoir release at any time given the present storage and inflow conditions. Karamouz and Houck [15] extended Young’s procedure by adding one extra constraint to the optimization model specifying that the release must be within a given percentage of the release defined by the previously found operating policy. The releases were again established by least squares multiple regression on the current-period inflow and initial storage. Kim and Heo [16] used ISO combined with two types of linear equations for the regression analysis to define monthly operating rules for a multipurpose reservoir. Willis et al. [46] devised a different approach that utilized the probability mass function of the optimal releases, conditioned on reservoir storage and inflow. Modern alternatives to the classical regression analysis are the application of artificial neural networks (ANN) [10], [4], [13], [30], [3] and fuzzy rule-based modeling (FRB) [35], [24], [28], [31], [33] to inferring the operating rules. In theory, ANN and FRB methods may improve the rule curves since they can ‘learn’ the operating policies from the data and are able to handle more efficiently the complex nonlinear relationships among the involved variables. An additional advantage of fuzzy logic is that it is more flexible and allows incorporation of expert opinions, which could make it more acceptable to operators [31]. Most of the published studies show that these two techniques outperform regression-based ISO and SDP. Recent works [20], [23] have been also successfully applying an approach that combines both ANN and FRB called adaptive neuro-fuzzy inference system (ANFIS), created by Jang [14].

The PSO technique is formally presented in the work of Koutsoyiannis and Economou [17] and has been utilized in several other studies [18], [25], [27]. A number of authors successfully applied the simulation–optimization principle of PSO to derive reservoir rule curves [22], [1], [29], [5], [6], [44], [26]. Since genetic algorithms and other direct search methods are commonly used for the parameter calibrations, this technique is sometimes called direct search approach.

Unlike most previous applications of ISO and PSO to reservoirs, this study explores their use in a system under semiarid conditions with the objective of mitigating hydrological droughts. In addition, the ISO models employed here use quadratic programming (QP) rather than dynamic programming (DP) in their deterministic optimization process. QP does not require discretization of the state variables. Apart from this, one ISO- and another PSO-based model proposed in this study showed best overall performance among all investigated models.

This manuscript is divided into six sections including this introduction. The next three sections present, respectively, the ISO, PSO and SDP models utilized in the study. Section 5 describes the case study and shows the application of the rule curves generated by all models to its operation. The last section concludes the study.

Section snippets

Overview of the ISO approach

Implicit stochastic optimization, also referred to as Monte Carlo optimization, uses a deterministic optimization model to find optimal reservoir releases under several different inflow ensembles. For each inflow sequence realization, a different operating policy is found. The set of all operating policies is then examined in order to construct reservoir release rules (Fig. 1). Classically, multiple regression analysis is applied to the optimization results in order to develop operating rules

Overview of the PSO approach

The ISO technique introduced above tries to find a shape for a reservoir rule curve by exploiting operating data generated by optimization. In a rather different way, parameterization–simulation–optimization starts with the shape of the rule already established and defined by some few parameters. A set of initial parameter values is chosen and the reservoir is operated using the predefined rule under a number of inflow scenarios or a long inflow series. The parameters are then changed and the

SDP model

The recursive function F of the employed stochastic dynamic programming model is:Ftn(S(t-1),I(t))=minimizefeasibleS(t)Z(t)+I(t+1)PI(t+1)Ft+1n-1(S(t),I(t+1))where t is the current month and n is the total number of remaining months. Initial storage S(t-1) and current inflow I(t) are the state variables, while final storage S(t) is the decision variable. Z(t) is the objective function in Eq. (1) and PI(t+1) is the unconditional inflow transition probability (no correlation between consecutive

Case study

The state of Paraíba (Fig. 10) is one of the poorest of Brazil and has 98% of its surface within the so-called Polygon of Droughts, an area of over one million square kilometers comprising most of Brazil’s northeast region. In the period from 1997 to 1999, an extreme drought fell upon this region aggravating the critical water storage condition of the Epitácio Pessoa reservoir that supplies Campina Grande, the state’s second largest city (370 000 inhabitants). This caused the reservoir to reach

Conclusions

This paper evaluated the applicability of several reservoir operation optimization models. Rule curves developed by ISO and PSO models, that implicitly deal with inflow uncertainties, were compared to those obtained by SDP, in which the stochasticity is explicitly incorporated. In principle, the investigated models aimed to build reservoir operating policies that estimated the amount of release based on the reservoir’s current status, defined by its initial storage level and the projected

Acknowledgements

The first author greatly acknowledges the Alexander von Humboldt Foundation and its Georg Forster Research Fellowship program for the financial support received in order to carry out this research in Germany.

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