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Top1. Introduction
In the power system operation, the OPF problem is a very popular one. The objective of an optimal power flow (OPF) problem is to find the steady state operation point of generators in the system so as their total generation cost is minimized while satisfying various generator and system constraints such as real and reactive power of generators, bus voltages, transformer taps, switchable capacitor banks, and transmission line capacity limits (Carpentier, 1979). In the OPF problem, the controllable variables usually determined: 1) real power output of generators, 2) voltage magnitude at generation buses, 3) injected reactive power at compensation buses, 4) and transformer tap settings. This is a classic and large-scale problem, but it is extensively studied due to its significance in the power system operation. Traditionally, mathematical programming techniques can effectively deal with the problem and the OPF problem has been widely studied in the literature (Happ and Wirgau, 1981; Huneault and Galiana, 1991; Momoh, Adapa, and El-Hawary, 1999; Pandya and Joshi, 2008). However, due to the incorporation of FACTS devices to systems, valve-point effects or multiple fuels to generators recently, the OPF problem becomes more complicated and the mathematical programming techniques are not a proper selection. Therefore, it requires more powerful search methods for a better implementation.
Several methods have been applied for solving this problem. The purpose of the solution methods applied to this problem is to find the optimal solution, so that the effectiveness of these methods can be evaluated. The better method is the one which can find better optimal solution than other methods for the problem in terms of the minimum objective function. However, not all methods can be successfully applied to solve this problem due to handling several variables and constraints. Therefore, the effective methods for successfully solving the OPF problem are usually the powerful method for solving optimization problems. The OPF problem has been solved by several conventional methods such as gradient-based method (Wood and Wollenberg, 1996), linear programming (LP) (Abou El-Ela and Abido, 1992; Mota-Palomino and Quintana, 1986), non-linear programming (NLP) (Dommel and Tinny, 1968; Pudjianto, Ahmed, and Strbac, 2002), quadratic programming (QP) (Burchett, Happ, and Vierath, 1984; Granelli and Montagna, 2000), Newton-based methods (Sun et al., 1984; Santos and da Costa, 1995; Lo and Meng, 2004), semidefinite programming (Bai et al., 2008), and interior point method (IPM) (Yan and Quintana, 1999; Wang and Liu, 2005; Capitanescu et al., 2007). Generally, the conventional methods can find the optimal solution for an optimization problem with a very short time. However, the main drawback of these methods is that they are difficult to deal with non-convex optimization problems with a non-differentiable objective. Moreover, these methods are also very difficult for dealing with large-scale problems due to large search space. In addition, meta-heuristic search methods recently developed have shown that they have the capability to deal with this complicated problem. Several meta-heuristic search methods have been also widely applied for solving the OPF problem such as genetic algorithm (GA) (Lai and Ma, 1997; Wu, Cao, and Wen, 1998; Osman, Abo-Sinna, and Mousa, 2004); simulated annealing (SA) (Roa-Sepulveda and Pavez-Lazo, 2003), tabu search (TS) (Abido, 2002), evolutionary programming (EP) (Wu and Ma, 1995; Yuryevich and Wong, 1999); particle swarm optimization (PSO) (Abido, 2001), and differential evolution (DE) (Cai, Chung, and Wong, 2008). These meta-heuristic search methods can overcome the main drawback of the conventional methods with the problem not required to be differentiable. However, the optimal solutions obtained by these methods for optimization problems are near optimum and the quality of the solutions is not high when they deal with large-scale problems; that is the obtained solutions may be local optimums with long computational time.