An improved particle swarm optimization algorithm for unit commitment

https://doi.org/10.1016/j.ijepes.2006.02.011Get rights and content

Abstract

This paper presents an improved particle swarm optimization algorithm (IPSO) for power system unit commitment. IPSO is an extension of the standard particle swarm optimization algorithm (PSO) which uses more particles’ information to control the mutation operation, and is similar to the social society in that a group of leaders could make better decisions. The convergence property of the proposed IPSO method is analyzed using standard results from the dynamic system theory and some guidelines are derived for proper algorithm parameter selection. A new adaptive strategy for choosing parameters is also proposed to assure convergence of IPSO method, and the proposed algorithm adopts the orthogonal design to generate initial population that are scattered uniformly over feasible solution space. Furthermore, this method combines relaxation technique to zero-one variable and penalty function method to transform the problem to a nonlinear continuous variable optimization one by taking into account more constraints. The feasibility of the proposed method is demonstrated from 10 to 100 unit systems, and the test results are compared with those obtained by Evolutionary Programming (EP) and Genetic Algorithm (GA) in terms of solution quality and convergence properties. The simulation results show that the proposed method is capable of obtaining higher quality solutions.

Introduction

The task of unit commitment (UC) in power system refers to the optimization problem for determining the on/off states of generating units that minimize the operating cost for a given time horizon [1]. The committed units must meet the system forecasted demand and spinning reserve requirement at minimum operating cost, subject to a large set of operating constraints. Since improved UC schedule may save the electric utilities millions of dollars per year in production costs. UC is an important optimization task in the daily operation planning of modern power system.

The unit commitment has commonly been formulated as a high-dimensional, nonlinear, mixed-integer combinatorial optimization problem. Many methods have been developed for solving the UC problem. The exact solution to the problem can be obtained only by complete enumeration, often at the cost of a prohibitive computation time requirement for realistic power system [1]. Research endeavors, therefore, have been focused on efficient, near-optimal UC algorithms which can be applied to large-scale power systems and have reasonable storage and computation time requirements. A survey of literature on the UC methods reveals that various numerical optimization techniques have been employed to approach the UC problem. Specifically, there are priority list methods [2], [3], dynamic programming [4], [5], [6], Lagrangian relaxation methods [7], [8], [9], branch-and-bound methods [10], integer programming [11] and mixed-integer programming [12]. Among these methods, the priority list method is one of the earliest and simplest approaches to address the UC problem. Most priority list schemes are built around a shut-down algorithm based on the relative efficiency of each unit. Priority listing is often used in the dynamic programming approach. The dynamic programming algorithm is a useful technique that provides optimal solutions in small systems of generators. However, the consideration of all combinations of units, as implemented in this technique, has proven impractical even for systems of moderate size. Hence, most applications incorporate some heuristics. These heuristics however, may produce suboptimal solutions. The Lagrangian relaxation method is a fair agreement between solution quality and computation requirements for large-scale UC problems. Nevertheless, the application of this dual optimization technique to the nonconvex UC problem gives rise to some technical problems such as “relaxation of constraints” or “duality gap”. The branch-and-bound method adopts a linear function to represent the fuel consumption and time-dependent start-up cost and obtains the required lower and upper bounds. The shortcoming of branch-and-bound is the exponential growth in the execution time with the size of the UC problem. The mixed-integer programming method uses linear programming technique to solve and check for an integer solution. The method has only been applied to small UC problem and has required major assumptions which limit the solution space.

In the last decade, many new stochastic search methods have been developed for the global optimization problems, such as genetic algorithms [13], [14], [15] and evolutionary programming [16], [17]. They are attracting much attention, because of their great potential for modeling engineering problems in environments which have been resistant to solution by classic techniques. Having in common processes of natural evolution, these algorithms share many similarities: each maintains a population of solutions which are evolved through random alterations and selection. The differences between these procedures lie in the representation technique they utilize to encode candidates, the type of alterations they use to create new solutions, and the mechanism they employ for selecting new “parents”. In power systems these algorithms have yielded satisfactory results across a great variety of problem types [18].

Particle swarm optimization (PSO) is one of the stochastic search techniques developed by Kennedy and Eberhart [19]. It was developed through simulation of a simplified social system, and has been found to be robust in solving continuous nonlinear optimization problems. The PSO technique can generate high-quality solutions within shorter calculation time and stable convergence characteristic than other stochastic methods [19]. Although the PSO seems to be sensitive to the tuning of some weights or parameters, many researches are still in progress for proving its potential in solving complex power system problems [20], [21]. Kassabalidi et al. [22] introduced dynamic security border identification using enhanced particle swarm optimization. Naka et al. [23] proposed a hybrid particle swarm optimization for distribution state estimation. Abido proposed an efficient and reliable particle swarm based approach to solve the optimal power flow problems [24]. It has been found that the PSO method quickly finds the high-quality optimal solution for many power system optimization problems. But, PSO has a more global searching ability at the beginning of the run and a local search near the end of the run. Therefore, while solving problems with more local optima, there are more possibilities for the PSO to explore local optima at the end of run. However, the UC problem does have these properties in itself. For these reasons, a reliable global approach to power system optimization problems would be of considerable value to power engineering community.

In this paper, we propose an improved particle swarm optimization algorithm (IPSO) to solve the UC problem. The basic idea of the IPSO method is to use more particles’ information to control the mutation operation and is similar to the social society in that a group of leaders could make better decisions. The convergence property of the proposed IPSO is analyzed using standard results from the dynamic system theory and some guidelines are derived for proper algorithm parameter selection. A new adaptive strategy for choosing parameters is also proposed to assure convergence of IPSO method, and the proposed algorithm adopts the orthogonal design to generate initial population that are scattered uniformly over feasible solution space. Furthermore, this method combines relaxation technique to zero-one variable and penalty function method to transform the UC problem to a nonlinear continuous variable optimization one by taking into account more constraints. The algorithm is successfully tested on a modeled UC problem previously addressed by some existing techniques. Test results show that IPSO has a good convergent property and can obtain high quality solutions. Compared with PSO, IPSO exhibits significant improvement in computational efficiency.

Section snippets

Problem formulation

In this section, we first formulate the UC problem. The objective of the UC problem is minimization of the total operating costs over the scheduling horizon. Therefore, the objective function is expressed as the sum of fuel and start-up costs of the generating units. Mathematically, the function is as follows:minF=t=1Ti=1ICipit·uit+Sit·uit·1-uit-1,where Ci(pit) fuel cost function of the ith unit with generation output, pit, at the tth hour. Usually, it is a quadratic polynomial with

Particle swarm optimization

Particle swarm optimization (PSO) is one of the modern heuristic algorithms developed by Kennedy and Eberhart [19], [25]. It is a multi-agent search technique that traces its evolution to the emergent motion of a flock of birds searching for food. It uses a number of particles that constitute a swarm. Each particle traverses the search space looking for the global minimum (or maximum). In a PSO system, particles fly around in a multidimensional search space. During flight, each particle adjusts

Numerical results

The efficiency of the proposed method was verified by solving a reported UC problem [14]. A base set of 10 units was initially chosen along with a 24-h demand. This set as well as the demand schedule is shown in Table 1. The spinning reserve was assumed to be 10% of the load demand in each study system.

The proposed IPSO method has been implemented in Matlab 6.5 programming language and numerical tests are carried on a Pentium IV 2.0G computer. Some parameters must be assigned before IPSO is

Conclusion

An improved particle swarm optimization (IPSO) has been developed for determination of the global or near-global optimum solution for UC problem. The improved particle swarm optimization uses more particles’ information to control the mutation operation. A new adaptive strategy for choosing parameters is also proposed to assure convergence of IPSO method. To further enhance convergence of IPSO, the proposed method adopts the orthogonal design method to generate initial population that are

Acknowledgements

This work is supported by the Outstanding Young Research Investigator Fund (No. 60225006) and Innovative Research Group Fund (No. 60421002) of Natural Science Foundation of China.

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