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Top1. Introduction
The optimal power flow (OPF) proposed by “Carpentier (1979)”, is an optimization tool through which the electric utilities strive to determine secure operating conditions for a power system. The OPF solution aims to optimize a selected objective function via optimal adjustment of the power system control variables, while satisfying various equality and inequality constraints. OPF problem has been rigorously studied over the past few decades. Many optimization techniques have emerged so far and have been applied to solve the problem. In the past, a wide variety of conventional optimization techniques such as Newton method “Monticelli and Liu (1992)”, linear programming “Stott, Marinho and Alsac (1979)”, dynamic programming “Xie and Song (2001)”, and interior point methods “Wei, Sasaki, Kubokawa and Yokoyama (1998)” have been developed to solve OPF problem. Generally, these techniques suffer due to algorithmic complexity, insecure convergence, and sensitiveness to initial search point, etc. Usually, these methods rely on the assumption that the fuel cost characteristic of a generating unit is a smooth, convex function. However, in practical system, it is not possible, or appropriate, to represent the unit’s fuel cost characteristics as convex function. This situation arises when valve-points non-linearities effect of thermal generating units and prohibited operating zones constraints are considered. Therefore, new optimization methods are required to deal with these difficulties.
In recent years, many heuristic algorithms, such as Tabu search (TS) “Abido (2002)”, genetic algorithms (GA) “Devaraj and Yegnanarayana (2005)”, evolutionary programming (EP) “Sood (2007)”, particle swarm optimization (PSO) “Roy, Ghoshal and Thakur (2009)”, differential evolution (DE) “Liu and Lampinen (2005); Varadarajan and Swarup (2008)”, artificial bee colony (ABC) “Kwannetr, Leeton and Kulworawanichpong (2010)”, bacteria foraging optimization (BFO) “Tang, Li and Wu (2008)”, biogeography-based optimization (BBO) “Roy, Ghoshal and Thakur (2010a); Roy, Ghoshal and Thakur (2010b)” and gravitational search algorithm (GSA) “Bhattacharya and Roy (2012); Roy, Mandal and Bhattacharya (2012)” have successfully proved their capabilities in solving the OPF problem, without any restrictions on the shape of the cost curves. The results reported are promising and encouraging for further research in this direction. Moreover, many hybrid algorithms “Ongsakul and Bhasaputra (2002); Chunjie, Huiru and Chen (2010)” have been introduced to enhance the search efficiency.
DE proposed by R. Storn and K. Price “Storn and Price (1995)” is one of the latest evolutionary optimization algorithms. It is a simple but powerful population-based stochastic search technique for solving global optimization problems. Its effectiveness and efficiency have been successfully demonstrated in many application fields. Like other evolutionary algorithms, two fundamental processes drive the evolution of a DE population: exploration of different regions of the search space, and the selection process, which ensures the exploitation of previous knowledge about the fitness landscape. Practical experience, however, shows that DE may occasionally stop proceeding toward the global optimum level and may converge to a local optimum or any other point. This situation is usually referred to as premature convergence, where the population converges to some local optima of a multimodal objective function, losing its diversity. This is due to the lack of exploitation ability of the original DE algorithm.