Decomposition-based modern metaheuristic algorithms for multi-objective optimal power flow – A comparative study
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:where x is the vector of decision variables, and Ω is the feasible region within the decision space. is 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: , where i=1,…, N and m=number of objective functions. For two objective functions, i.e., , can be set asThis 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
References (25)
- et al.
A modified artificial bee colony for real-parameters optimization
Inf. Sci.
(2012) - et al.
Solving multiobjective optimal reactive power dispatch using modified NSGA-II
Int. J. Electr. Power Energy Syst.
(2011) - et al.
Teaching-learning based optimization: a novel method for constrained mechanical design optimization problems
Comput.-Aided Des.
(2011) - et al.
Multi-objective harmony search algorithm for optimal power flow problem
Int. J. Electr. Power .Energy Syst.
(2011) Multiobjective particle swarm optimization for optimal power flow problem
- et al.
Multi-objective optimal power flow using differential evolution
Arab. J. Sci. Eng.
(2012) - et al.
Multiobjective optimal power flow using hybrid evolutionary algorithm
Int. J. Electr. Electron. Eng.
(2010) - et al.
A fast and elitist multiobjective genetic algorithm: NSGA-II
IEEE Trans. Evol. Comput.
(2002) - et al.
A computer package for optimal multiobjective VAR planning in large scale power systems
IEEE Trans. Power Syst.
(1994) - Karaboga, D., 2005. An Idea Based on a Honey Bee Swarm for Numerical Optimization, Technical-Report TR06. Erciyes...
Estimating the voltage stability of a power system
IEEE Trans. Power Deliv.
Estimating the voltage stability of a power system
IEEE Trans. Power Deliv.
Cited by (57)
A new algorithm based on gray wolf optimizer and shuffled frog leaping algorithm to solve the multi-objective optimization problems
2020, Applied Soft Computing JournalParticle filter and Levy flight-based decomposed multi-objective evolution hybridized particle swarm for flexible job shop greening scheduling with crane transportation
2020, Applied Soft Computing JournalCitation Excerpt :Especially with the scheduling problems becoming more and more complicated, a number of scholars are now more inclined to resort to hybrid intelligent methods for the awareness of such combinations usually perform better than a single technique [15]. The combination of tabu search (TS) and PSO [16], MOEA/D and artificial bee colony [17], simulated annealing (SA) and genetic algorithm (GA) [18], have been proposed in recent years to solve optimal scheduling problems more effectively. Motivated by the advantages of hybrid intelligent optimization methods, in this paper, we proposed an efficient hybrid algorithm, namely PLMEAPS, where we hybridize MOEA/D with PSO, and PSO has positive utility to overcome the weakness of the MOEA/D in the ability to perform local search ability and improve the search accuracy of the algorithm.
A multiple search strategies based grey wolf optimizer for solving multi-objective optimization problems
2020, Expert Systems with ApplicationsCitation Excerpt :To enhance the search ability, a polynomial-based mutation (PBM) was designed (Deb & Agarwal, 1999). Their proposed algorithm was validated on DTLZ (DTLZ1-DTLZ7) and UF (UF1-UF10) test functions, and compared with five well-known MOEAs: MOGWO, MOEA/D, MOPSO based on Decomposition (MOPSO/D) (Peng & Zhang, 2008), Multi-objective Artificial Bee Colony Algorithm based on Decomposition (MOABC/D) (Medina, Das, Coello Coello & Ramírez, 2014), Multi-objective Teaching-Learning Algorithm based on Decomposition (MOTLA/D) (Medina et al., 2014) and decomposition-based Multi-objective Particle Swarm Optimizer (dMOPSO) (Zapotecas-Martínez & Coello, 2011) by considering two performance metrics: Normalized Hypervolume (Hn) (Zitzler & Thiele, 1998, 2003) and Inverted Generational Distance plus (IGD+) (Ishibuchi, Masuda, Tanigaki & Nojima, 2015). Apart from the aforementioned studies, other researchers also have published prominent works (Emary, Yamany, Hassanien & Snasel, 2015; Li, Wang & Chen, 2019; Nguyen, Thom & Dao, 2017; Nuaekaew, Artrit, Pholdee & Bureerat, 2017; Sahoo & Chandra, 2017; Tsai, Nguyen & Dao, 2017; Xia et al., 2019).
- 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.