Decomposition-based modern metaheuristic algorithms for multi-objective optimal power flow – A comparative study

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Abstract

This article presents multi-objective variants of two popular metaheuristics, namely, the artificial bee colony algorithm (ABC) and the teaching learning based optimization algorithm (TLBO). Both of them are used to solve an optimal power flow problem. The proposed multi-objective variants are based on a decomposition approach, where the multi-objective optimization problem is decomposed into a number of scalar optimization sub-problems which are simultaneously optimized. The proposed algorithms are tested on the IEEE 30-bus system with different objectives. In addition, an algorithm based on fuzzy set theory is used to select the best committed solution. The proposed approaches are compared with others metaheuristic algorithms available in the specialized literature. Results indicate that the proposed approaches are highly competitive and also able to generate a well-distributed set of non-dominated solutions for the optimal power flow problem.

Introduction

The optimal power flow (OPF) problem has a significant importance in the power system׳s operation, planning, economic scheduling, and security. It is a non-linear con strained optimization problem, where the solution attains the control variables optimal adjustment, while at the same time satisfying equality and inequality constraints related to the equipments׳ rating, in order to optimize a certain objective function.

In general, the optimal power flow problem may include several objective functions, possibly in conflict with each other. Such kind of optimization problem has a set of possible solutions (named Pareto optimal set), which represent the best commitment among the objectives (Stadler, 1988). Two major solution approaches may be identified:

  • (1)

    The first approach is based on conventional methods. Such as Gradient-based Methods, Non-Linear Programming (NLP), Quadratic Programming (QP), Linear Programming (LP) and Interior Point Methods (Lee et al., 1985, Momoh et al., 1999a, Momoh et al., 1999b), the Weighting Method (Kuo et al., 2005), and the ε-Constraint Method (Hsiao et al., 1994).

  • (2)

    The second approach is based on the use of metaheuristic algorithms such as the Differential Evolution (DE) (Abido and Al-Ali., 2012), the Non-dominated Sorting Genetic Algorithm II (NSGA-II) (Jeyadevi et al., 2011, Deb et al., 2002), Particle Swarm Optimization (PSO) (Abido, 2011), Harmony search algorithm (Sivasubramani and Swarup, 2011), and the Hybrid Evolutionary Programming Technique (Alawode et al., 2010).

Conventional methods are based on an estimation of the global minimum. However, due to difficulties of differentiability, non-linearity, and non-convexity, these methods may not guarantee to reach the global optimum (Yamille et al., 2008). Moreover, these methods exhibit some limitations, depending upon the type of problem, e.g., when the objective function is not available in algebraic form. Thus, metaheuristics (from which evolutionary algorithms is a particular subclass) have become a popular choice for solving complex optimization problems, due to their flexibility, generality, and ease of use. Additionally, most metaheuristics require little or no specific domain knowledge.

Modern multi-objective evolutionary algorithms (MOEAs) aim at generating a number of Pareto-optimal solutions as diverse as possible. Indeed, MOEAs need a density estimator that distributes solutions along the Pareto front (e.g., crowding distance, fitness sharing, niching). However, there is evidence that these methods cannot always provide good results, especially when dealing with complex multi-objective problems (MOP) (Zhang and Li, 2007, Li and Zhang, 2009).

Recently, a novel MOEA framework called the multi-objective evolutionary algorithm based on decomposition (MOEA/D) (Zhang and Li, 2007), has been proposed. MOEA/D decomposes a MOP into several single-objective optimization sub-problems with neighborhood relationship. In this way, a set of optimal solutions is achieved by minimizing each sub-problem instead of using the traditional Pareto ranking methods. This has given rise to a new generation of MOEAs. Nevertheless, the performance of MOEA/D in power system applications has not been fully investigated.

This paper proposes a modified artificial bee colony algorithm and a teaching-learning algorithm in the MOEA/D framework. The proposed approaches are used to solve an optimal power flow problem, with competing objectives.

In order to minimize the total fuel cost, the active power losses and a voltage stability index (Kessel and Glavitsch, 1986), the proposed algorithms estimate the following optimal values: (i) the generators׳ voltage magnitudes; (ii) generators׳ active power outputs, (iii) transformers׳ tap settings; (iv) the compensating value for shunt elements (reactors/capacitors). In addition, an algorithm based on fuzzy set theory is used to select the best committed solution.

The effectiveness of the proposed approaches is demonstrated and compared with respect to a MOEA based on decomposition, which is representative of the state-of-the-art in the area: MOEA/D-DRA (Zhang et al., 2009). Results are also compared with respect to the NSGA-II (Deb et al., 2002), which remains as the most popular Pareto-based MOEA. The methods are applied on an IEEE 30-bus test system. Additionally, results reported in the open research (Abido and Al-Ali, 2012, Sivasubramani and Swarup, 2011) are also included for a comparative study.

The rest of the paper is organized as follows. Section 2 presents some basic background. In Section 3, the general framework of the proposed approaches is summarized. Section 4 presents the problem formulation and the method based on fuzzy theory for choosing the best committed solution. Simulation results and a comparative study are presented in Section 5. Finally, our conclusions are provided in Section 6.

Section snippets

Multi-objective optimization

A multi-objective optimization problem (MOP) is formulated as follows:MinF(x)={f1(x),...,fm(x)}SubjecttoxΩwhere x is the vector of decision variables, and Ω is the feasible region within the decision space. F:Ωmis defined as the m objective functions mapping.

In multi-objective optimization, the goal is to find the best possible trade off among the objectives since, frequently, one objective can be improved only at the expense of worsening another. To describe the concept of optimality for

Multi-objective artificial bee colony

The proposed Multi-Objective Artificial Bee Colony Algorithm based on Decomposition (MOABC/D) utilizes the Tchebycheff approach to decompose the MOP into N scalar optimization sub-problems by choosing N weighting vectors: wi={w1i,...,wmi}, where i=1,…, N and m=number of objective functions. For two objective functions, i.e., m=2, wi=(w1i,w2i) can be set asw1i=(j1)/(N1),w2i=1w1iThis method for generating weighting vectors works well for the formulation in this paper. However other methods may

Problem statement

In this paper, an optimal power flow problem is formulated as a multi-objective optimization problem where three objective functions are taken into account for minimization, while satisfying a number of equality and inequality constraints. The problem is formulated in the sequel.

Simulation results and comparison

In order to assess the effectiveness of the proposed algorithms, the MOABC/D and MOTLA/D have been compared with respect to the MOEA/D-DRA (Zhang et al., 2009) and NSGA-II (Deb et al., 2002). The algorithms have been tested in the IEEE 30 bus system. This system consists of six generators, four transformers with off-nominal tap ratio, and nine reactive power injection sources. The complete system data are given in (Lee et al., 1985). The optimization problem has 24 parameters.

The parameters

Conclusions

This paper presented two multi-objective optimization methods based on decomposition for solving a multi-objective optimal power flow problem. The first method is based on intelligent behavior of honey bees and the second one is based on the teaching-learning strategy.

In order to validate the effectiveness and performance of the proposed methods, MOABC/D and MOTLA/D were applied to the IEEE 30 bus test system and compare with respect to MOEA/D-DRA and the NSGA-II in two different

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    1

    Carlos A. Coello Coello acknowledges support from CONACyT project no. 103570.

    2

    Juan M. Ramírez acknowledges support from CONACyT project nos. 167933 and 188167.

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