Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 7410293, 7 pages

http://dx.doi.org/10.1155/2016/7410293

## A Hybrid Algorithm of Particle Swarm Optimization and Tabu Search for Distribution Network Reconfiguration

^{1}School of Electronic Information and Electrical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China^{2}Department of Computer Science and Electrical Engineering, Georgia Institute of Technology, Atlanta, GA 30314, USA

Received 15 January 2016; Accepted 19 July 2016

Academic Editor: Mauro Pontani

Copyright © 2016 Sidun Fang and Xiaochen Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper deals with the distribution network reconfiguration problem. A hybrid algorithm of particle swarm optimization (PSO) and tabu search (TS) is proposed as the searching algorithm. The new algorithm shares the advantages of PSO and TS, which has a fast computation speed and a strong ability to avoid local optimal solution. After a thorough comparison, network random key (NRK) is introduced as the corresponding coding strategy among various tree representation strategies. NRK could completely avoid the generation of infeasible solutions during the searching process and has a good locality property, which allows the new hybrid algorithm to perform to its fullest potential. The proposed algorithm has been validated through an IEEE 33 bus test case. Compared with other algorithms, the proposed method is both accurate and computationally efficient. Furthermore, a test to solve another problem also proves the robustness of the proposed algorithm for a different problem.

#### 1. Introduction

Distribution system is usually designed with loops while running in a radial structure. Distribution network reconfiguration (DNR) is a process of altering the topological structure of distribution feeders by changing the open/closed status of the sectionalizing and tie switches [1]. DNR is not limited to fault isolation; from time to time network reconfiguration is performed to achieve various goals, such as system loss reduction, overloads relieving [2], load aggregation [3, 4], and system reliability improvement [5].

The performance and efficiency of any DNR algorithm largely rely on a wise combination of a smart topological coding strategy and an efficient searching algorithm. As a nondeterministic polynomial hard (NP-hard) problem, DNR has been heavily studied with various searching algorithms, from step-by-step heuristics, such as branch-exchange method [6], to metaheuristics based algorithms, such as tabu search (TS) [4], simulated annealing [7], genetic algorithm (GA) [8–10], and particle swarm optimization (PSO) [11]. Apart from the choice of searching algorithms, the distribution network representation or coding strategy is equally important due to the topological nature of DNR. Various coding strategies have been studied in DNR problem, including binary string representation [5] and Prüfer number representation [3, 12].

After a brief comparison of different coding strategies and existing searching algorithms, this paper proposes a new DNR algorithm. The new method adopts a hybrid optimization of PSO and TS as the searching algorithm and network random keys (NRK) as the corresponding coding strategy. To boost up the overall searching efficiency, a direct method for distribution system power flow analysis [13] is introduced, which has been proved to be both robust and time-efficient.

In recent years, PSO has been successfully applied to solving different kinds of problems, ranging from multimodal and topological mathematical problems [14, 15] to aerospace [11, 16–20] and chemical engineering [21, 22]. It is famous for its easy realization and fast convergence, while suffering from the possibility of early convergence to local optimums. In the proposed hybrid algorithm, whenever early convergence occurs, the original particle swarm would be separated into three groups of swarms. Swarm 1 continues performing the basic PSO algorithm; swarm 2 is replaced with newly generated random particles; and swarm 3 begins to perform TS on each particle. With the integration of TS, the hybrid PSO algorithm could effectively avoid local optimum by accepting worse solution under certain condition during the searching process.

NRK, which is originally used for GAs, in this paper, has been applied to PSO algorithm. As a topological coding strategy, NRK could completely avoid the possibility of generating unfeasible solutions when using heuristic algorithms in graph optimization problems. It also transforms the original discrete DNR problem into a continuous optimization problem. When applied in GAs, NRK is no more than a coding strategy, which possesses little physical meaning during the solution searching process. However, when used in PSO, the coding strategy has a physical meaning. The value of the “key” in the NRK can be interpreted as an importance index for each branch in the graph. The whole searching process could be interpreted as a process of adjusting the importance index of each branch and choosing the most important branches to form the optimal tree structure.

The remainder of this paper is structured as follows.

In Section 2, the DNR problem is formulated as an optimization problem. In Section 3, different network topology representation schemes are discussed, and NRK is introduced. In Section 4, the hybrid algorithm of PSO and TS is proposed and explained in detail. In Section 5, the new algorithm is tested on an IEEE 33 bus system with numerical results. The conclusion is drawn in Section 6.

#### 2. Problem Representation

DNR is originally used in planned outages for maintenance purpose or fault isolation to restore service. A. Merlin and H. Back [14] were the first to come to the idea that reconfiguration may lead to a system total loss reduction and they tried to search for such an optimal configuration using the branch-and-bound method. Since then, loss reduction has been considered as a common objective for the study of new DNR optimization algorithms.

A noticeable characteristic of DNR is the repeated analysis of power flow during the solution searching process. In order to improve searching efficiency, several refined or approximate algorithms for power flow analysis have been studied, such as decoupled method [15], hashing table method [16], and perturbation method [2]. A direct method [13] is adopted in this paper, which has been proved to be highly efficient in distribution network power flow analysis.

Another characteristic of DNR is the topological constraints, which means any feasible solution of a DNR problem should represent a tree structure with every node being connected. Configuration space is the set of allowed system configurations over which the optimal system configuration is to be searched for [7]. In DNR, only solutions that belong to the configuration space are considered feasible.

Assume that a distribution network has branches. A DNR problem for system losses reduction can be formulated aswhere is an -dimensional vector. If branch is closed, ; otherwise . and represent the active and reactive power flow on branch . and are penalty factors, while and are penalty functions for node voltage constraint and branch current constraint.

#### 3. Network Topology Representation

The process of searching the optimal DNR solution involves the graph theory of optimal spanning tree. Let graph represent the topology of a distribution network, where stands for vertices and stands for edges. Each potential solution is a spanning tree of . All spanning trees of the graph make up the configuration space. A good network topology representation strategy should have four characteristics:(1)*Being Easy to Encode and Decode*. A less complicated coding strategy would cost less time to encode and decode, thus leading to a boost in computational efficiency.(2)*Being Compatible with Other Optimization Algorithms*. Many metaheuristic algorithms have their own limitations in dealing with different types of optimization problems. For example, GAs require a binary string representation, and PSO requires continuous variables. A good coding strategy should be compatible with corresponding searching algorithms.(3)*Avoiding Infeasible Solutions*. Topological constraint is one of the thorniest issues in DNR, especially when it comes to the utilization of metaheuristic algorithms. Whenever an infeasible solution is generated, the original searching process will be interrupted. A good coding strategy should effectively rule out the possibility of generating infeasible solutions, which would greatly improve the computational efficiency and avoid the tedious topological checking process.(4)*Having a Good Locality Property*. A good locality property means that the objective function value is relatively continuous and smooth, rather than irregular jumps within a local area in the searching space. Most metaheuristic algorithms determine the best searching direction based on current objective function values. Then the algorithms will lead the searching process towards the most promising direction. In other words, a coding strategy with bad locality properties will greatly restrict the effectiveness of the searching algorithm. The configuration space generated by a good coding strategy should always keep a high locality.

There are many different ways to represent the distribution network topology, and each of them has its merits and flaws.

Binary string representation is the most intuitive and straightforward way to represent the network topology by assigning a binary string . The dimension of is the total number of switches. The elements in are set to be 0 or 1, representing the open and closed status of each switch. However, binary string representation is usually blamed for the high probability of generating infeasible solutions when applied by many searching algorithms such as SA or PSO. Genetic operators such as crossover or mutation almost always generate infeasible solution, which forces the algorithm to stop.

In order to reduce the probability of generating infeasible solutions, homeomorphism [12] and fundamental loop [17] representation method are widely adopted. The graph theory of homeomorphism simplifies the original graph by smoothing out unnecessary vertices from the original graph. After the simplification, each branch in the new graph represents a group of branches in the original graph. According to the graph theory, one and only one branch could be opened in each branch group in order to form a tree structure. Similarly, fundamental loop representation avoids infeasible solutions by introducing fundamental loop tables. Only one branch should be opened in each fundamental loop. These two methods help to reduce the probability of generating infeasible solutions and keep the searching process from interruption. However, none of the methods above could completely avoid infeasible solutions, and additional checking rules are still necessary.

Random key (RK) is an efficient method for encoding and scheduling problems. Rothlauf et al. [9] proposes a tree representation for GAs using RK, by the name of network random keys. Queiroz and Lyra [3] are the first to introduce the combination of NRK and GAs in the DNR problem.

Taking a 5-node system as an example, see Figure 1. The NRK coding and decoding process goes as follows.