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Abstract：A phase-based adaptive differential evolution (PADE) algorithm is proposed to solve the economic load dispatch (ELD) considering valve-point effects (VPE) and transmission losses. To a great extent, PADE makes up for the drawbacks of the traditional differential evolution (DE) through three improvements. First, we establish an archive of storing successful individuals to improve the quality of offspring. Second, to balance the exploring and exploiting ability of the algorithm, a phase-based mutation operation is carried out. Third, two control parameters are adaptively adjusted, which is helpful for enhancing the robustness of the algorithm. In addition, two types of repair methods of constraint handling are employed for the ELD without or with transmission losses to help PADE find feasible solutions more efficiently. A performance comparison between PADE and other DE approaches from the literature was carried out on six ELD test cases which consider a set of operating constraints including the VPE and transmission losses. Results show a competitive PADE performance in all test cases regarding the compared DE approaches. Compared to methods from the literature, the costs obtained by PADE are lower in most cases while the corresponding constraint violations reach a lower level.
1. Introduction
In power systems, economic load dispatch (ELD) [1] is an important, complex optimization problem that allocates the required generation among the available generators. The main purpose of ELD is to find the optimal distribution of the generating units to make the fuel cost minimize while meeting all inequality and equality constraints in practice. Therefore, an accurate mathematical model is needed for adapting the ELD problems with different characteristics. So far, the quadratic function has been widely used as the classical mathematical model for solving the cost of the ELD problems. Moreover, addition conditions are often considered in the actual system, which constitute various versions for the ELD problems such as the valve-point effects (VPE) [2], transmission losses [3], and prohibited operating zones (POZs) [4]. Many classical methods, such as linear programming [5], quadratic programming [6], lambda iteration method [7], and gradient method [1], have been applied to solve the conventional ELD problems with the quadratic function. The above classical methods require that the generators exhibit convex and smooth characteristics. However, the input-output characteristics of generators are generally nonconvex and non-smooth in practical thermal generation plants. The reasons are caused by the VPE, POZs and so on, which may make the cost function produce a number of local optima. Therefore, these classical methods can hardly find the global optima for such types of problems. Dynamic programming [8] is different from the above classical methods. It can handle nonconvex and nonsmooth ELD problems, but it is restricted by the curse of dimensionality that inevitably leads to extra computational cost. Thus it is not also a suitable one for practical ELD problems. In order to seek the optimal solution of ELD, more and more scholars tend to use modern intelligent algorithms. These modern intelligent algorithms have huge advantage over classical methods, especially for the ELD problems with various operation constraints. They are essentially approximation methods, which do not need the derivative information of the ELD problems. The practical ELD problems are quite complex, so most of scholars are committed to continuously improving or hybrid modern intelligent algorithms and developing new constraint handling mechanisms at present. Qin and Cheng et al. [9] adopted an orthogonal designed method and proposed auxiliary vector generation based on multiple strategies to enhance the effectiveness of orthogonal designed operation. Based on the adaptive adjustment of acceleration coefficients, the algorithm’s robustness and global search capability can be improved by employing a tent chaotic map. Furthermore, a repair method is designed to handle the practical constraints. Coelho and Mariani [10] put forward an improved harmony search (IHS) algorithm based on exponential distribution for the ELD problems. In IHS, a repair process is adopted to find feasible solutions. Al-Betar and Awadallah et al. [11] developed a modified version of harmony search algorithm that is called tournament-based harmony search (THS) algorithm, where the tournament selection process replaces the random selection process in the memory consideration operator...... 
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