Solution of non-convex economic dispatch problem considering valve loading effect by a new Modified Differential Evolution algorithm

Abstract

This paper presents Economic Dispatch (ED) solution considering valve loading effect by a new Modified Differential Evolution (MDE) algorithm. Considering valve loading effect changes ED into a non-convex optimization problem. This non-convexity challenges analytical and heuristic methods in finding optimal solution in reasonable time. Differential Evolution (DE) is one of evolutionary algorithms, which has been used in many optimization problems due to its simplicity and efficiency. The proposed MDE is in the framework of differential evolution owning new mutation operator and selection mechanism inspired from Genetic Algorithm (GA), Particle Swarm Optimization (PSO) and Simulated Annealing (SA), respectively. In other words, positive characteristics of DE, GA, PSO and SA are combined to create a new efficient stochastic search technique. In the solution of the non-convex ED by the proposed stochastic search technique, an effective constraint handling method is also presented. The proposed MDE is examined on three ED test systems and compared with some of the most recently published ED solution methods. These comparisons reveal the efficiency and robustness of the proposed MDE.

Introduction

Economic Dispatch (ED) is defined as the process of allocating generation levels to the generating units in the mix, so that the system load is supplied entirely and most economically [1], [2]. ED is sub problem of unit commitment and determines the final generation level of each committed generator. Early methods were (a) the base load method where the next most efficient unit is loaded to its maximum capability, then the second most efficient unit is loaded, etc., (b) “best point loading”, where units are successively loaded to their lowest heat rate point, beginning with the most efficient unit and working down to the least efficient unit, etc. Then, it was recognized that the incremental method, later known as the equal incremental method, yielded the most economic results [2]. After that, several classical optimization techniques, such as gradient method [3], lambda iteration method [4], linear programming [5], quadratic programming [6], non-linear programming [7], Lagrangian relaxation algorithm [8] and dynamic programming [9] were proposed to solve ED problem.

By perfect modeling of the final cost of generation and taking valve loading effect into account, the cost function of generators take non-convex form [10]. The theoretical assumptions behind previous algorithms (except dynamic programming) that may not be suitable for the ED formulation are convexity and differentiability. Furthermore, they are local optimizers in nature, i.e., they might converge to local solutions instead of global ones if the initial guess happens to be in the neighborhood of a local solution. Dynamic programming method may cause the dimensions of the ED problem to become extremely large, thus requiring enormous computational efforts.

For overcoming these deficiencies, Artificial Intelligence Methods have been used to solve ED problem, such as Genetic Algorithm (GA) [11], Tabu Search (TS) [12], Hopfield neural network [13], ant colony optimization [14], different types of Evolutionary Programming (EP) [15], Evolutionary Strategy (ES) [16], Particle Swarm Optimization (PSO) [17], [18], [19], [20] and Bacterial Foraging (BF) [21]. Moreover, for reinforcement of these stochastic search algorithms, hybrid methods like combination of evolutionary programming with Sequential Quadratic Programming (SQP) [22] and combination of differential evolution with SQP [23] are also proposed. Differential Evolution (DE) is a type of evolutionary algorithm originally proposed by Price and Storn for optimization problems over a continuous domain [24]. DE is exceptionally simple and significantly fast and robust. The basic idea of DE is to adapt the search during the evolutionary process. At the start of the evolution, the perturbations are large since parent populations are far away from each other. As the evolutionary process matures, the population converges to a small region and the perturbations adaptively become small. As a result, the evolutionary algorithm performs a global exploratory search during the early stages of the evolutionary process and local exploitation during the mature stage of the search [25]. In this paper a new Modified Differential Evolution (MDE) algorithm is proposed to solve non-convex economic dispatch. Structure of this algorithm is based on DE. However, it has a new mutation operator inspired from PSO [26] and GA plus a new selection mechanism inspired from SA [27]. In other words, the positive characteristics of DE, PSO, GA and SA are combined to create a new hybrid stochastic search technique. Also, an equality constraint handling method is proposed, which enhances the performance of the proposed MDE to solve the non-convex ED.

This paper is organized as follows. Section 2 describes ED problem formulation considering valve loading effect, prohibited operating zone (POZ) constraints and ramp rate limits. Moreover, the proposed method for constraints handling is presented in this section. In Section 3, the proposed solution method is introduced. For this purpose, at first a brief description of DE technique is presented. Then the proposed MDE is described. Obtained results from the MDE to solve the non-convex ED problem are presented in Section 4. Besides, the MDE is compared with some of the most recently published ED solution methods. Section 5 concludes the paper.