Quasi-oppositional differential evolution for optimal reactive power dispatch
Introduction
The reactive power dispatch (RPD) plays an important role for improving economy and security of power system operation. Although the reactive power generation has no production cost, however it affects the overall generation cost by the way of the active power loss. The RPD is a nonlinear, non-convex and non-differentiable optimization problem. It minimizes active power loss and improves voltage profile and voltage stability by adjusting control variables such as generator voltages, transformer tap settings, and reactive power output of shunt VAR compensators in a power system while satisfying several equality and inequality constraints.
Several classical mathematical methods [1], [2], [3], [4], [5], [6], [7], [8] such as linear programming, quadratic programming, gradient projection method, interior point method, reduced gradient method and Newton method have been applied to solve RPD problem of power system. These methods are computationally fast but these methods optimize the objective function by linearizing it. The RPD is a non-linear multimodal optimization problem with a mixture of discrete and continuous variables. It has multiple local optima. Hence, it is so hard to find the global optimum of reactive power dispatch problem by using classical mathematical methods. For these reasons, researchers have developed computational intelligence-based techniques to solve the RPD problem.
In recent years, computational intelligence-based techniques, such as evolutionary programming [9], adaptive genetic algorithm [10], particle swarm optimization [11], hybrid stochastic search technique [12], hybrid particle swarm optimization [13], multiagent-based particle swarm optimization [14], bacterial foraging based optimization [15], differential evolution [16], [21], quantum-inspired evolutionary algorithm [17], self adaptive real coded genetic algorithm [18], seeker optimization algorithm [19], comprehensive learning particle swarm optimization (CLPSO) [20], biogeography-based optimization [22], hybrid shuffled frog leaping algorithm and Nelder–Mead simplex search [23], gravitational search algorithm [24], quasi-oppositional teaching learning based optimization [25], and opposition-based gravitational search algorithm [26] have been applied to solve RPD problem. These techniques have shown effectiveness in overcoming the disadvantages of classical methods.
Since the mid 1990s, many techniques originated from Darwin’s natural evolution theory have emerged. These techniques are usually termed by “evolutionary computation methods” including evolutionary algorithms (EAs), swarm intelligence and artificial immune system. Differential evolution (DE) [27], [28], [29], a relatively new member in the family of evolutionary algorithms, first proposed over 1995–1997 by Storn and Price at Berkeley is a novel approach to numerical optimization. It is a population-based stochastic parallel search evolutionary algorithm which is very simple yet powerful. The main advantages of DE are its capability of solving optimization problems which require minimization process with nonlinear, non-differentiable and multi-modal objective functions.
The basic concept of opposition-based learning (OBL) [31], [32], [33] was originally introduced by Tizhoosh. The main idea behind OBL is for finding a better candidate solution and the simultaneous consideration of an estimate and its corresponding opposite estimate (i.e., guess and opposite guess) which is closer to the global optimum. OBL was first utilized to improve learning and back propagation in neural networks by Ventresca and Tizhoosh [34], and since then, it has been applied to many EAs, such as differential evolution [35], particle swarm optimization [36] and ant colony optimization [37].
Quasi-oppositional based learning (QOBL) is implemented on differential evolution (DE). The proposed quasi-oppositional differential evolution (QODE) along with basic differential evolution (DE) is applied to solve the RPD problem. The RPD is a combinatorial optimization problem involving nonlinear functions having multiple local optima and nonlinear and discontinuous constraints. In order to evaluate the proposed method, the proposed QODE is tested on IEEE 30-bus, 57-bus and 118-bus test systems with different objective functions that reflect active power loss minimization, voltage profile improvement and voltage stability enhancement. Test results obtained from QODE have been compared with those obtained by other evolutionary methods reported in the literature. From numerical results, it is found that the proposed QODE based approach provides better solution.
Section snippets
Problem formulation
The objective of the RPD is to minimize the active power loss and to improve voltage profile and voltage stability while satisfying equality and inequality constraints. Three objective functions and constraints are formulated as follows.
A brief description of differential evolution
Differential evolution (DE) is a type of evolutionary algorithm originally proposed by Price and Storn [29] for optimization problems over a continuous domain. DE is exceptionally simple, significantly faster 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
Application of the proposed method
The proposed QODE and DE have been applied to solve RPD problems. Three different test systems with three different objective functions have been studied to verify its applicability. Programs have been written in MATLAB-7 language and executed on a 3.0 GHz Pentium-IV personal computer. In order to demonstrate the effectiveness of the proposed QODE for solution of three different RPD problems, IEEE 30-bus, 57-bus and 118-bus test systems have been considered. The results obtained from proposed
Conclusion
In this paper, QODE is demonstrated and successfully applied to solve RPD problem. The RPD problem is formulated as a nonlinear optimization problem with equality and inequality constraints of power system. In this study, different objective functions such as minimization of active power loss and enhancement of voltage profile and voltage stability are considered. The proposed QODE approach is tested on IEEE 30-bus, 57-bus and 118-bus test systems to demonstrate its effectiveness. Due to
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