Solving nonlinear water management models using a combined genetic algorithm and linear programming approach

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Abstract

Gradient-based nonlinear programming (NLP) methods can solve problems with smooth nonlinear objectives and constraints. However, in large and highly nonlinear models, these algorithms can fail to find feasible solutions, or converge to local solutions which are not global. Evolutionary search procedures in general, and genetic algorithms (GAs) specifically, are less susceptible to the presence of local solutions. However, they often exhibit slow convergence, especially when there are many variables, and have problems finding feasible solutions in constrained problems with “narrow” feasible regions. In this paper, we describe strategies for solving large nonlinear water resources models management, which combine GAs with linear programming. The key idea is to identify a set of complicating variables in the model which, when fixed, render the problem linear in the remaining variables. The complicating variables are then varied by a GA. This GA&LP approach is applied to two nonlinear models: a reservoir operation model with nonlinear hydropower generation equations and nonlinear reservoir topologic equations, and a long-term dynamic river basin planning model with a large number of nonlinear relationships. For smaller instances of the reservoir model, the CONOPT2 nonlinear solver is more accurate and faster, but for larger instances, the GA&LP approach finds solutions with significantly better objective values. The multiperiod river basin model is much too large to be solved in its entirety. The complicating variables are chosen here so that, when they are fixed, each period's model is linear, and these models can be solved sequentially. This approach allows sufficient model detail to be retained so that long-term sustainability issues can be explored.

Introduction

Gradient-based nonlinear programming (NLP) algorithms are widely available, and can be applied to problems with smooth nonlinear objectives and constraints. Sparsity-exploiting implementations like MINOS [15] and CONOPT2 [4] have solved many problems with thousands of constraints and variables. However, these NLP solvers generally converge to the local solution nearest to the starting point, and their theoretical convergence properties do not hold for nonsmooth problems. Their speed and reliability also decrease as problem size and complexity increase.

Because of these limitations, other methods have been applied to solve large-scale nonlinear reservoir management models in recent years, including evolutionary search methods. These include genetic algorithms (GAs), which often find good approximate solutions even when the model functions are multimodal, discontinuous, or nondifferentiable [17]. Recently, there has been a significant growth of interest in using GAs for water resources planning and design. McKinney and Lin [11] and Huang and Mayer [9] applied a binary-coded GA to models of pump-and-treat groundwater remediation. Savic and Walters [18] applied a GA to least-cost design of water distribution networks, and Halhal et al. [8] studied water network rehabilitation, replacement, and expansion by using a GA. Oliveira and Loucks [16] developed a GA-based approach to search for effective operating policies for dynamic models of multipurpose multi-reservoir systems.

In this paper, we present a combined GA and linear programming (LP) strategy, which is used to solve large nonlinear water resources management models that are difficult, if not impossible, to solve using currently available NLP solvers. The strategy begins by choosing a set of complicating variables in the original model. When these variables are fixed, the model becomes much easier to solve (in our examples, it is linear). The complicating variables are varied by a GA, and a linear program is solved to compute the optimal objective value for each set of fixed values suggested by the GA. In principle, another class of search algorithms could replace the GA, for example simulated annealing or tabu search [6].

This strategy has much in common with the widely used benders and generalized benders decomposition (GBD) procedures [5], which vary the complicating variables by solving the benders “master problem”. For an application of GBD to multiperiod reservoir models of the form used in this paper, see [3]. However, for the GBD master problem to be tractable, the original model must have special structure (it is sufficient for the complicating and noncomplicating variables to appear separably everywhere). Using a search algorithm to vary the complicating variables, there are no restrictions on how the complicating variables can appear in the model. Of course, this generality is achieved at a price: convergence is not guaranteed, and is often slow in later iterations.

We apply the GA&LP approach to two models: a multiperiod reservoir operation model with nonlinear hydropower generation and reservoir topologic equations, and a long-term dynamic river basin planning model with a large number of nonlinear relationships.

Section snippets

Basics of GAs

Detailed descriptions of GAs in water resources literature can be found in McKinney and Lin [11], Huang and Mayer [9], Savic and Walters [18], and Oliveira and Loucks [16]. Only a brief introduction is given here, in order to help readers understand the proposed methodology in this paper.

GAs are a subclass of general artificial-evolution search methods based on natural selection and the mechanisms of population genetics [12]. They belong to a family of optimization techniques in which the

Elastic formulation

Most search methods, including GAs, find feasible solutions to constrained problems by penalizing infeasibilities. This requires that constraints which involve complicating variables be formulated in an “elastic” way, by including deviation variables which measure the amount of violation. These are penalized in the objective. The general mathematical statement of the elasticized problem follows. Let x and y be the vectors of noncomplicating and complicating variables, respectively, let p and n

Implementation of the GA

The GA used here discretizes each component of y into 2L equally spaced values, and represents the discretized variable as a binary string containing L bits, as described in [14]. If we assume that the ith component of y has finite lower and upper bounds li and ui, and that bj is the value of the jth bit in the string, then the discretized component, denoted by yi, is given byyi=li+[(ui−li)/(2L−1)]j=1Lbj2j−1.If y has m components, each individual in the GA population is represented by a bit

Application to a reservoir operation model

Fig. 2 shows a network diagram of a five-reservoir system, where the reservoirs are capable of hydropower generation, as well as water supply, flood control, and flow augmentation. An optimization model is developed to maximize the production of energy, while satisfying constraints arising from flow augmentation and flood control. Such models are solved by GBD in [3].

The objective is to maximize the sum of the ratios of energy generated to energy demand over all time periods,maxz=t∈Tn∈pwstPW

Application to a river basin planning model

The GA&LP approach has also been applied to a very large and complex multiperiod model of long-term irrigation planning and water allocation in the Syr Darya River basin in Central Asia. A more extensive description of the model is in [2]. As illustrated in Fig. 9, the long-term modeling framework is composed of an inter-year control program (IYCP) and a series of yearly models (YM). The modeling framework is designed to connect intra-year short-term decisions dynamically with inter-year

Discussion and conclusion

This GA&LP approach combines a modeling strategy and a choice of algorithms. The modeling strategy requires careful selection of the complicating variables. In the reservoir model, these are chosen to render the model linear in the remaining variables. In the long-term planning situation, the complicating variables are chosen to yield both linearity and the ability to solve each yearly model sequentially. If these choices are made properly, computational results show that a fairly standard GA

Acknowledgements

The authors are grateful to Professor Daniel P. Loucks and another anonymous reviewer for their very helpful comments and suggestions.

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