Evaluation of stochastic reservoir operation optimization models
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:where t is the current month and n is the total number of remaining months. Initial storage and current inflow are the state variables, while final storage is the decision variable. is the objective function in Eq. (1) and 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.
References (50)
- et al.
Inferring operating rules for reservoir operations using fuzzy regression and ANFIS
Fuzzy Set Syst
(2007) - et al.
Stochastic generation of monthly flows for ephemeral streams
J Hydrol
(1980) - Bravo JM. Otimização da operação de um reservatório para controle de cheias com base na previsão de vazão. Master’s...
- Celeste AB, Billib M. The role of spill and evaporation in reservoir optimization models. Water Resour Manage 2009....
- et al.
Neural network based decision support model for optimal reservoir operation
Water Resour Manage
(2005) - et al.
Multireservoir modeling with dynamic programming and neural networks
J Water Resour Plan Manage
(2001) - et al.
Optimizing the reservoir operating rule curves by genetic algorithms
Hydrol Process
(2005) Real coded genetic algorithm optimization of long term reservoir operation
J Am Water Resour Assoc
(2003)- et al.
A globally convergent Lagrangian barrier algorithm for optimization with general inequality constraints and simple bounds
Math Comput
(1997) - D’Errico J. Surface fitting using gridfit; 2005....
Optimal hedging and carryover storage value
J Water Resour Plan Manage
Use of Monte Carlo optimization and artificial neural networks for deriving reservoir operating rules
Annu J Hydraul Eng JSCE
Practical optimization
Reliability, resiliency, and vulnerability criteria for water resource system performance evaluation
Water Resour Res
Application of ANN for reservoir inflow prediction and operation
J Water Resour Plan Manage
ANFIS: adaptive-network-based fuzzy inference system
IEEE Trans Syst Man Cybern
Annual and monthly reservoir operating rules generated by deterministic optimization
Water Resour Res
Evaluation of the parameterization–simulation–optimization approach for the control of reservoir systems
Water Resour Res
A decision support tool for the management of multi-reservoir systems
J Am Water Resour Assoc
Optimal operation of multireservoir systems: state-of-the-art review
J Water Resour Plan Manage
Optimal operation of a multi-purpose reservoir using neuro-fuzzy technique
Water Resour Manage
Direct search approaches using genetic algorithms for optimization of water reservoir operating policies
J Water Resour Plan Manage
Reservoir operation using a dynamic programming fuzzy rule-based approach
Water Resour Manage
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