Research papers
Optimal operation of multi-reservoir hydropower systems using enhanced comprehensive learning particle swarm optimization

https://doi.org/10.1016/j.jher.2015.06.003Get rights and content

Highlights

  • ECLPSO is applied to the optimal operation of multi-reservoir hydropower systems.

  • The outflow and storage volume constraints are appropriately enforced.

  • The penalty factor is dynamically adjusted.

  • The optimization framework helps to derive feasible high quality operation policies robustly.

Abstract

Metaheuristics are promising optimization algorithms for tackling reservoir-system operation. Comprehensive learning particle swarm optimization (CLPSO) is a state-of-the-art metaheuristic that is strong in exploration. Recently we have proposed enhanced CLPSO (ECLPSO) to improve the exploitation performance of CLPSO. In this paper, we apply ECLPSO to the optimal operation of multi-reservoir hydropower systems. Two novel strategies are proposed to handle the various physical and operational constraints. First, the outflow and storage volume constraints are appropriately enforced to achieve a tradeoff between preserving diversity and facilitating convergence. Second, with the penalty function technique adopted to penalize the constraint violations and convert the original constrained problem into an unconstrained one, the penalty factor is dynamically adjusted in order to encourage exploration of the search space in the beginning and gradually guide the search to concentrate in the feasible region. The short-term scheduling of a 4-reservoir hydrothermal power system and the long-term planning of China's Xiluodu–Xiangjiaba–Threegorges 3-reservoir hydropower system are studied. Experimental results demonstrate that ECLPSO helps to robustly derive feasible high quality solutions for the two cases studied. The contribution to performance improvement by ECLPSO as well as the constraint enforcement and penalty factor adjustment strategies are analyzed.

Introduction

Hydropower has become a critical source of electricity as it is renewable, clean, and cheap. Most hydropower is generated from plants constructed within reservoirs. A reservoir impounds water to serve various purposes such as hydropower generation, flood control, navigation, and/or water provisioning. A multi-reservoir system consists of multiple reservoirs sited within the same river basin. The cascaded reservoirs are hydraulically coupled as the outflow of a reservoir constitutes part of the inflow into its immediate downstream reservoir (Orero and Irving, 1998). The optimal operation of the cascaded reservoirs is beneficial for the overall development of the river basin.

The optimal operation of multi-reservoir hydropower systems is challenging to solve because of the following factors: (1) the input (e.g. inflow) imprecision and uncertainties need to be addressed (Simonovic, 1987); (2) the decision-making process is multi-stage and dynamic (Simonovic, 1987); (3) the resulting optimization problem is often large-scale with a lot of decision variables and constraints (Labadie, 2004); (4) the hydraulic coupling among the reservoirs complicates the problem (Orero and Irving, 1998); (5) because of the various physical and operational constraints, it isn't easy to find a feasible solution that satisfies all the constraints (Simonovic, 1987); (6) the hydropower performance model is usually nonlinear (El-Hawary and Christensen, 1979), could be nonconvex (Tauxe et al., 1980), discontinuous and non-differentiable (Lyra and Ferreira, 1995), and even mixed-integer (Wang and Zhang, 2012); and (7) sometimes multiple conflicting objectives are considered (Lyra, Ferreira, 1995, Tauxe et al, 1980). The input imprecision and uncertainties can be addressed implicitly through forecast data, long/representative historical records, and/or synthetically generated data (Labadie, 2004). The multiple objectives can be transformed into a single objective through using techniques such as the weighting and ε-constraint (Miettinen, 1999).

Various optimization algorithms have been applied to solve the optimization problems related to the operation of multi-reservoir hydropower systems. The algorithms are usually classified into three general categories: traditional optimizers, modern metaheuristics, and hybrid approaches. Traditional optimizers, including linear programming, nonlinear programming, dynamic programming, and optimal control theory, have rigorous mathematical foundations (Labadie, 2004). A metaheuristic is essentially a high level intelligent strategy to guide the search toward the promising region. A lot of different metaheuristics have been proposed, including simulated annealing, tabu search, genetic algorithm, differential evolution, and particle swarm optimization, just to name a few (Boussaïd et al., 2013). The metaheuristics are inspired by different nature principles from biology, ethology, or physics. Compared with traditional optimizers, modern metaheuristics are significantly more flexible as they don't require the objective and constraints to be continuous, differentiable, linear, or convex, and they usually can efficiently solve large-scale problems. Although a metaheuristic might not find the exact optimum, it often can find a near-optimal solution. In addition, metaheuristics can be directly linked with simulation models (e.g. flood inundation and water quality) without requiring simplifying any assumptions in the models. Hybrid approaches try to combine the strengths of different optimization algorithms. For example, Li et al. (2012) applied a hybrid approach to the optimal operation of China's Three Gorges cascaded hydropower system; the approach first exploits incremental dynamic programming to narrow down the search space, and then takes advantage of genetic algorithm to solve the nonlinear and nonconvex problem.

Introduced in 1995 (Eberhart, Kennedy, 1995, Kennedy, Eberhart, 1995), particle swarm optimization (PSO) is a swarm-intelligence inspired metaheuristic simulating the movements of organisms in a bird flock or fish school. PSO solves an optimization problem using a swarm of particles, with each particle representing a candidate solution. Metaheuristics (including PSO) need to achieve a balance between exploration and exploitation. Exploration is the ability to search different regions for locating a good solution, while exploitation is the ability to concentrate the search around a small region for refining a hopeful solution (Boussaïd et al., 2013). Comprehensive learning PSO (CLPSO) (Liang et al., 2006) is state-of-the-art PSO variant that is strong in exploration but weak in exploitation. We have recently proposed enhanced CLPSO (ECLPSO) (Yu and Zhang, 2014) to improve the exploitation performance of CLPSO. In this paper, we apply ECLPSO to the optimal operation of multi-reservoir hydropower systems. Two novel strategies are proposed to handle the various physical and operational constraints. First, the outflow and storage volume constraints are appropriately enforced to achieve a tradeoff between preserving diversity and facilitating convergence. Second, with the penalty function technique adopted to penalize the constraint violations and convert the original constrained problem into an unconstrained one, the penalty factor is dynamically adjusted in order to encourage exploration of the search space in the beginning and gradually guide the search to concentrate in the feasible region. The short-term scheduling of a 4-reservoir hydrothermal power system and the long-term planning of China's Xiluodu–Xiangjiaba–Threegorges 3-reservoir hydropower system are studied.

The rest of this paper is organized as follows. In Section 2, a generalized problem formulation is presented. Section 3 reviews the working principle of ECLPSO. Related works on constraint handling mechanisms and the application of PSO to optimal reservoir-system operation are discussed in Section 4. Section 5 details the application implementation of ECLPSO. In Section 6, the performance of ECLPSO is evaluated on the selected cases. Section 7 concludes the paper.

Section snippets

Generalized problem formulation

For the optimal operation of multi-reservoir hydropower systems, the short/medium/long-term optimization problems (Chen et al, 2013, Ji et al, 2014, Li et al, 2012, Li et al, 2014a, Li et al, 2014b, Orero, Irving, 1998) have a similar generalized formulation. A single-reservoir system could be regarded as the special case of a multi-reservoir system when the number of reservoirs degenerates to one. Before putting the generalized problem formulation into perspective, decision variables, state

Enhanced comprehensive learning particle swarm optimization

In PSO, all the particles “fly” in the search space. Let there be D decision variables, the swarm of particles move in D-dimensional space. Each particle, denoted as p, is associated with a position Posp = (Posp,1, Posp,2, …, Posp,D), a flight velocity Velp = (Velp 1, Velp,2, …, Velp,D), and a fitness that indicates its performance. PSO relies on iterative learning to find the optimum. In each iteration, p adjusts its velocity according to its previous velocity, its historical best position

Constraint handling mechanisms

The most popular technique to handle constraints is the use of penalty functions. The penalty function technique worsens the fitness of infeasible solutions to favor the evolution of feasible solutions. The constraint violations are incorporated into the objective function so that the original constrained problem is converted into an unconstrained problem. Although being simple, a penalty function requires an appropriate selection of the penalty factor which is dependent on the specific problem

Representation and initialization of each particle

For each particle p, its position Posp, as shown in Eq. (20), is a N×T matrix representing candidate reservoir outflow rates. Each dimension of the position is randomly initialized from [Oi,t¯,Oi,t¯].Posp=[O1,1,O1,2,,O1,TO2,1,O2,2,,O2,TON,1,ON,2,,ON,T]

The power discharge rate Qi,t and the spillage rate Si,t are determined from the outflow rate Oi,t as follows. If Oi,tQi,t¯, then Qi,t = Oi,t and Si,t = 0; otherwise, Qi,t=Qi,t¯ and Si,t = Oi,t - Qi,t.

Enforcement of the outflow and storage volume constraints

The dimensional

Experimental studies

Two representative cases are studied, with case 1 being the short-term scheduling of a 4-reservoir hydrothermal power system introduced in Orero and Irving (1998) and case 2 being the long-term planning of China's Xiluodu–Xiangjiaba–Threegorges 3-reservoir hydropower system.

The following performance issues are investigated: (1) how the PbE and ALPs enhancements as well as the constraint enforcement and penalty factor adjustment strategies help improve the algorithm performance; and (2) how

Conclusions and future work

An optimization framework for the operation of multi-reservoir hydropower systems has been proposed in this paper. The framework adopts our recently proposed algorithm ECLPSO as the optimization tool because ECLPSO is well balanced in exploration and exploitation. The outflow and storage volume constraints are appropriately enforced to achieve a tradeoff between preserving diversity and facilitating convergence. The penalty function technique is employed to convert the original constrained

Acknowledgement

This work was supported by the Public Benefit Special Research Fund of the Ministry of Water Resources of the People's Republic of China (201201017). The work of Hui Qin was supported by the National Natural Science Foundation of China (51209008).

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