Evolving ant direction differential evolution for OPF with non-smooth cost functions

Abstract

In this paper, an effective and reliable algorithm, termed as evolving ant direction differential evolution (EADDE) algorithm, for solving the optimal power flow problem with non-smooth and non-convex generator fuel cost characteristics is presented. In this method, suitable mutation operator for differential evolution (DE) is found by ant colony search. The genetic algorithm evolves the ant colony parameters and the Newton–Raphson method solves the power flow problem. The proposed algorithm has been examined on the standard IEEE 30-bus and IEEE 57-bus systems with three different objective functions. Different cases were considered to investigate the robustness of the proposed method in finding the global solution. The EADDE provides better results compared to classical DE and other methods recently reported in the literature as demonstrated by simulation results.

Introduction

The optimal power flow problem (OPF) problem is an optimization tool through which the electric utilities strive to determine secure and 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. The OPF problem, in general, is a large-scale highly constrained nonlinear non-convex optimization problem. OPF problem has been solved by conventional and evolutionary based algorithms. Many mathematical programming techniques such as linear programming (LP), nonlinear programming (NLP), quadratic programming (QP), Newton method and interior point methods (IPM) have been applied to solve the OPF problem successfully (Momoh et al., 1999, Habiabollahzadeh and Luo, 1989, Burchet et al., 1984, Mota-Palomino and Quintana, 1986, Sun et al., 1984, Yan and Quintana, 1999). Usually, these methods rely on the assumption that the fuel cost characteristic of a generating unit is a smooth, convex function. However, there are situations where it is not possible, or even appropriate, to represent the unit’s fuel cost characteristics as a convex function. This situation arises when valve-points’ units prohibited operating zones and piecewise quadratic cost characteristics are present (Sayah and Zehar, 2008).

In recent years, many heuristic algorithms, such as genetic algorithms (GA) (Lai and Ma, 1997), evolutionary programming (Yuryevich and Wong, 1999), simulated annealing (Roa-Sepulveda and Pavez-Lazo, 2003), tabu search (Abido, 2002) and particle swarm optimization (Abido, 2002) have been proposed for solving the OPF problem, without any restrictions on the shape of the cost curves. Moreover, many hybrid algorithms have been introduced to enhance the search efficiency. For instance, a hybrid tabu search and simulated annealing (TS/TA) (Ongsakul and Bhasaputra, 2002) was applied to solve the OPF with flexible alternating current transmission systems (FACTS) device problem; hybrid evolutionary programming and tabu search or improved tabu search (ITS) (Lin et al., 2002) was used to solve the economic dispatch problem with non-smooth cost functions. Meanwhile, improved evolutionary programming (IEP) (Wang et al., 2002) was successfully used to solve combinatorial optimization problems.

In the recent past, Storn and Price introduced a powerful evolutionary algorithm called differential evolution (DE) to solve numerical global optimization problems, like the OPF problem (Storn and Price, 1997). DE is a numerical optimization approach that is simple, easy to implement, faster than many evolutionary algorithms and robust. DE combines simple arithmetic operators with the classical operators of crossover, mutation and selection to evolve from a randomly generated starting population to a final solution. Each parent competes one-on-one against its corresponding offspring and the fitter of the two becomes a member of the next generation.

The DE algorithm has been successfully applied to various power system optimization problems such as generation expansion planning (Kannan et al., 2005) and hydrothermal scheduling (Lakshminarasimman and Subramanian, 2006). Figueroa and Cederio (2004) applied DE for power system state estimation. Coelho and Mariani (2006) used this algorithm for economic dispatch with valve-point effect. Basu (2008) applied DE for solving the OPF problem incorporating FACTS devices. The hybrid differential evolution (HDE) has been employed for the solution of a large capacitor placement problem (Chiou et al., 2004). The mixed integer hybrid differential evolution (MIHDE) has been employed for hydrothermal coordination (Lakshminarasimman and Subramanian, 2007a), hydrothermal optimal power flow (Lakshminarasimman and Subramanian, 2007b) and the network reconfiguration problem (Su and Lee, 2003). The variable scaling hybrid differential evolution algorithm has been used for solving network reconfiguration of distribution systems (Chiou et al., 2005).

Colorni and Dorigo Maniezzo (1992) proposed the concept of ant system (AS) and applied it to the traveling salesman problem (TSP) (Dorigo and Gambardella, 1997). The ant algorithm has been inspired by the behavior of real ant colonies, in particular, by their foraging behavior. Recently, the ant algorithm has been applied to various optimization problems, such as the short-term generation scheduling problem (Yu et al., 1998), unit commitment (Sisworahardio and El-Keib, 2002) and hydro-electric generation scheduling (Huang, 2001).

In this paper, an efficient evolving ant direction DE based approach is proposed to solve the OPF problem with non-smooth cost functions. Evolving ant direction mutation operator selection is suggested to the original DE algorithm. Though there are five mutation operations stated in this paper, the EADDE uses only one mutation operator in every generation during the solution process. The proposed EADDE method embedded with the ant colony search is able to constantly choose different but most appropriate mutation operators during the solution process to accelerate the search for the global optimum solution. In the proposed algorithm, power flow problem is solved by Newton Raphson method; the control variables for power flow are optimized (overall optimization) by the differential evolution method. The mutation operator (out of five operators) for the differential evolution is selected by ant direction search and the ant parameters are evolved by genetic algorithm method. The proposed approach has been examined and tested on IEEE 30-bus and IEEE 57-bus standard test systems with three different objective functions. Simulation results demonstrate that the EADDE algorithm is superior to the original DE algorithm and provides significantly better results compared to those reported in the literature.

This paper introduces the combined application of the evolving concept and the ant direction hybrid differential evolution for solving the OPF problem, which was not applied to OPF in the literature so far. The remainder of the paper is organized as follows: Section 2 describes the formulation of an optimal power flow problem, while Section 3 explains the standard DE approach. Section 4 then details the procedure of the proposed evolving and direction DE and Section 5 presents the results of the optimization and compares the methods used to solve the case studies of optimal power flow problems with IEEE 30-bus and IEEE 57-bus systems. Lastly, Section 6 outlines the conclusion.