Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 527128, 13 pages

http://dx.doi.org/10.1155/2015/527128

## Enhanced GSA-Based Optimization for Minimization of Power Losses in Power System

^{1}Key Laboratory of Industrial Internet of Things and Networked Control, Ministry of Education, Chongqing University of Posts and Telecommunications, Chongqing 400065, China^{2}Research Center on Complex Power System Analysis and Control, Chongqing University of Posts and Telecommunications, Chongqing 400065, China^{3}Department of Electrical Engineering, Hubei Minzu University, Enshi 445000, China

Received 14 July 2015; Revised 24 November 2015; Accepted 1 December 2015

Academic Editor: Luis J. Yebra

Copyright © 2015 Gonggui Chen et al. 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

Gravitational Search Algorithm (GSA) is a heuristic method based on Newton’s law of gravitational attraction and law of motion. In this paper, to further improve the optimization performance of GSA, the memory characteristic of Particle Swarm Optimization (PSO) is employed in GSAPSO for searching a better solution. Besides, to testify the prominent strength of GSAPSO, GSA, PSO, and GSAPSO are applied for the solution of optimal reactive power dispatch (ORPD) of power system. Conventionally, ORPD is defined as a problem of minimizing the total active power transmission losses by setting control variables while satisfying numerous constraints. Therefore ORPD is a complicated mixed integer nonlinear optimization problem including many constraints. IEEE14-bus, IEEE30-bus, and IEEE57-bus test power systems are used to implement this study, respectively. The obtained results of simulation experiments using GSAPSO method, especially the power loss reduction rates, are compared to those yielded by the other modern artificial intelligence-based techniques including the conventional GSA and PSO methods. The results presented in this paper reveal the potential and effectiveness of the proposed method for solving ORPD problem of power system.

#### 1. Introduction

Optimal reactive power dispatch (ORPD), as one of the significant optimization problems in power system operation, is to minimize the given objective function such as total active power transmission losses by optimizing settings of control variables while satisfying a set of constraints during the entire dispatch period. Control variables contain discrete variables such as tap positions of transformers and amount of reactive compensation and continuous variables like generator voltages. Constraints consist of a series of equality constraints and inequality constraints [1]. Besides, it is worth noting that the ORPD problem in this paper is a single-objective optimization problem different from the study in [1] which researches a multiobjective optimization problem. In [1], the multiobjective ORPD problem seeks for a compromise solution for minimization of power losses and index simultaneously, but this paper is required to find out a global optimal solution for minimization of . Thus, ORPD problem in this paper is a complex mixed integer nonlinear optimization problem with a number of constraints and has the challenge of searching for the global optimal solution.

Numerous classical approaches including gradient-based optimization algorithms and many mathematical programming methods [2–5] have been developed and applied for solving the ORPD problem in the past. However, these traditional techniques almost only optimize the differentiable objective functions and they have difficulties in dealing with the nonconvex, nonlinear, discontinuous functions with constraints [6, 7]. But now a number of modern artificial intelligence-based techniques with stochastic optimization such as Genetic Algorithm (GA) [8], Differential Evolution (DE) [9], Particle Swarm Optimization (PSO) [10], and Gravitational Search Algorithm (GSA) have been applied to solve different ORPD problems efficiently without the abovementioned restraints, which overcomes the defects of conventional techniques. Furthermore, different method has its peculiar strength; for instance, the process of variation and hybridization in GA increases the diversity of population, which contributes to obtain the better solutions; DE uses the differences between individuals to change the individual itself, and this operation utilizes the distribution features of population to improve the search capacity effectively; PSO gets highlighted by virtue of the memory characteristic from imitating animals’ predation process containing social and individual behaviors; the pattern of movement of GSA contributes significantly to the high efficiency of the search process. Conversely, every method yet has its own weakness: GA is apt to converge prematurely and the long and complicated evolution procedures of it increase the running time; in DE, the individuals’ differences decrease with the increase of number of iterations, which impacts the increase of diversity of population directly. PSO tends to be trapped into prematurity in the latter period of searching, which directly lessens the possibility of acquiring the better solution; different from the algorithms based on biology, GSA is a memoryless algorithm, which is adverse to recording of the optimal value during the process of searching. According to these features, the exertion of merging individual superiorities into a new algorithm has become wide now in various engineering fields, which avoids their own disadvantages by benefiting from each other’s advantages; for example, a method composed of chaotic embedded Backtracking Search Optimization Algorithm (BSA) and Binary Charged System Search (BCSS) algorithm is proposed for solving Short-Term Hydrothermal Generation Scheduling (SHTGS) in [11]; in [12], the authors present a new hybrid evolutionary algorithm based on new fuzzy adaptive PSO algorithm and Nelder-Mead simplex search method to solve distribution feeder reconfiguration problem; the combination of the vertical search algorithm and presented lateral search algorithm is used to solve the midterm schedule for thermal power plants problem in [13]. But this paper highly favors the superiorities of PSO and GSA. On the basis of GSA, PSO is merged with it as fine tuning to improve the quality of solutions, which forms GSAPSO method for solving ORPD problem.

Gravitational Search Algorithm (GSA), a heuristic evolutionary optimization algorithm, was proposed by Rashedi et al. in [14]. GSA deriving from the thought of Newton’s law stands out depending on the flexible and efficient movement characteristic, and naturally, its application is wide. GSA-based Photovoltaic (PV) excitation control strategy is used for single-phase operation of three-phase wind-turbine coupled induction generator [15]. GSA is applied to coordinate Power System Stabilizers (PSSs) and Thyristor Controlled Series Capacitor (TCSC) controllers simultaneously, which is demonstrated to achieve good robust performance for damping the low frequency interarea oscillations [16]. GSA is proposed to find the optimal solution for optimal power flow problem in a power system [17].

Particle Swarm Optimization (PSO) proposed by Kennedy and Eberhart [18] owns numerous absorbing aspects: simple thought, convenient implementation, high efficiency, powerful search ability, and so on. But what the most prominent feature is belongs to the memory peculiarity during the search process, the memory peculiarity contributing particles to record the global optimum and individual optimum in every generation. The applications of PSO are far more than the authors care to mention; for instance, enhanced PSO approach is applied for optimal scheduling of hydrosystem [19]; robust PID controller tuning based on the constrained PSO is researched in [20]; PSO is used for optimization of acoustic filters [21]; a hybrid Particle Swarm Optimization algorithm is proposed for solving the problem of optimal reactive power dispatch within a wind farm [10].

In this paper, GSAPSO is the combination of GSA and PSO, which not only retains movement characteristic in the search process of GSA but also increases capability of sharing information and memory ability. In this work, GSA, PSO, and GSAPSO have been examined and tested in IEEE14-bus, IEEE30-bus, and IEEE57-bus test systems for the solution of ORPD problem of power system with the objective minimizing total active power transmission losses . The obtained performances of GSA, PSO, and GSAPSO are compared. And the power loss reduction rates of the proposed GSAPSO algorithm are also compared to those of other optimization methods. The simulation results reveal that the proposed GSAPSO approach can obtain a better optimum effect than these compared algorithms and the results’ distribution of it is more concentrated than conventional GSA and PSO methods; besides, the proposed GSAPSO can avoid falling into the local optimum.

The rest of this paper is organized as follows: Section 2 introduces the mathematical modeling of ORPD problem. GSAPSO algorithm is described in detail in Section 3. Section 4 presents the calculation process of GSAPSO algorithm for ORPD problem. Some simulation experiments are shown in Section 5, and Section 6 gives the conclusions.

#### 2. Mathematical Modeling

The mathematical modeling of ORPD problem is composed of two parts: objectives and constraints. The former provided in the paper are to minimize , and the latter contain equality constraints and inequality constraints [1].

##### 2.1. Objective Functions

The objective to minimize the total active power transmission losses in reactive power optimization is expressed as follows:where represents the total active power losses in transmission lines; is the number of network branches; is the conductance of the th branch which connects bus and bus ; and , respectively, denote the voltage magnitude of the th and th bus; is the voltage phase between buses and .

##### 2.2. System Constraints

The aforementioned objective function is subject to the system constraints which include the equality and inequality constraints.

###### 2.2.1. Equality Constraints

There are two equality constraints describing the active and reactive power balance, which are expressed as follows:where is the number of the buses connecting with the th bus; is the number of total buses except for swing bus; is the number of PQ buses; and are the active power of the th generator bus and the th load bus, respectively; and are, respectively, the reactive power at the th generator bus and the th load bus; and , respectively, represent the real part and imaginary part of which is the element of the -bus matrix (bus admittance matrix) at the th row and the th column.

Equations (2) are considered as the termination conditions of calculating the Jacobian matrix in Newton-Raphson load flow calculation.

###### 2.2.2. Inequality Constraints

The descriptions of inequality constraints are given based on the state variables and control variables, respectively.

*(i) Inequality Constraints of Control Variables*(a)The limit for generator bus voltages:where denotes the number of PV buses and is the voltages at the th generator bus.(b)The limit for tap positions of transformers:where is the number of transformers; is the tap positions of the th transformer, which is a discrete variable.(c)The limit for amount of reactive compensation:where is the number of the banks of capacitor or inductor; denotes the reactive compensation capacity of the th bank of capacitor or inductor.

*(ii) Inequality Constraints of State Variables*(a)The limit for voltages at PQ bus:where is the voltage at the th load bus.(b)The limit for reactive compensation capacity at PV bus:where is the reactive compensation capacity at the generator .(c)The limit for apparent power of transmission line:where is the apparent power of the transmission line between buses and .

##### 2.3. Handling of Constraints

What is worth mentioning is that, during the process of optimization, equality constraints and inequality constraints of control and state variables are satisfied as the following explanations, respectively.(i)The two equality constraints are satisfied by Newton-Raphson power flow algorithm in load flow calculation.(ii)The generator bus voltages (), tap positions of transformers (), and amount of reactive compensation () are the control variables which can be self-restricted according to their limits by the algorithm.(iii)The limits on active power generation at the swing bus , voltages at PQ bus (), reactive compensation capacity at PV bus (), and apparent power of transmission line () are state variables which are restricted by the objective function combining the penalty function.

##### 2.4. Formulation

In short, the ORPD problem can be formulated as a complex nonlinear constrained optimization mathematical model, and a compact expression is given inwhere and denote the vector of control variables and the vector of state variables, respectively; and , respectively, represent the equality constraints and inequality constraints of system.

In this paper, and are expressed as follows:where “” denotes transposition.

#### 3. Description of GSAPSO Algorithm

##### 3.1. Brief Introduction of GSA

GSA proposed by Rashedi et al. in 2009 is a newly developed stochastic search algorithm which is inspired by the law of gravity and law of motion [14]. In GSA, a series of agents are considered as objects and their performances are measured by their masses, and all these objects attract each other by the gravity force, while this force causes a global movement of all objects towards the objects with heavier masses [22]. The description of GSA about how to solve the problem is as follows.

Assume there are agents distributed in space and the position of the th agent is defined as inwhere denotes the position of the th agent in the th dimension and is the dimension of the search space.

The mass of every agent is computed based on the current agents’ fitness as follows:where and represent the mass and fitness value of the th agent at iteration ; and , respectively, denote the best and worst fitness value among the agents at iteration , which is defined as follows:

In accordance with the law of gravity, the force acting on agent by agent is computed as follows:where is the gravitational constant at iteration ; represents a small constant which can avoid the denominator equal to zero; , defined as , denotes the Euclidian distance between agent and agent .

We always use rather than in (15) because of the better performance of in most cases based on many simulation experiments. The better performance refers to the lower power losses in this paper. And is reduced from an initial value with iteration as follows:where is the initial gravitational constant; is a constant greater than zero; is the current number of iterations; and represents the maximum number of iterations.

On the basis of (15), the total force acting on agent can be given aswhere represents random number drawn uniformly on ; denotes the set of the first agents with the best fitness value and biggest mass, which is a function and reduced with time from the initial value .

Based on the law of motion, the acceleration of the th agent is computed as follows:

The updates of velocity and position of agent at the next iteration are computed as follows:where and are the velocity and position of agent at iteration in the th dimension.

##### 3.2. Memory Characteristic of PSO

Schools of fish and flocks of birds always find foods, which is attributed to the communication among individuals and the memory ability for individual best direction and global best direction. PSO algorithm simulates the behaviors of animals, whose update of velocity is defined as follows:where is the inertia weight; and represent the acceleration factors; and , respectively, denote the best position of particle and the best position in swarm in the th dimension at iteration .

##### 3.3. GSAPSO

Different from PSO, every agent in GSA determines the direction by the total force from other agents but lacks the communication with others so as to miss the memory ability. In this paper, the proposed GSAPSO is an enhanced GSA-based optimization algorithm, combining the memory characteristic of PSO based on the GSA, which is helpful to the agents to move to the global best position. The difference between GSA and GSAPSO is the update modes of velocity and position which are crucial for the artificial intelligence-based algorithms. And the updates of velocity and position in GSAPSO combining the law of gravity and memory characteristic of PSO are defined as follows:It is worth pointing out that the inertia weight impacting velocity in (20) is not introduced in (21), because GSA can determine the direction by the total force from other agents and only need to reinforce the impact of the memory for the best position of every agent and the best position in all the agents. And the steps of GSAPSO are depicted as follows.

*Step 1. *Generate the initial population.

*Step 2. *Compute the fitness of every agent, and record and .

*Step 3. *Update , , , and of the population based on (13), (14), and (16).

*Step 4. *Compute and of every agent according to (15), (17), and (18).

*Step 5. *Update the velocity and position by using (21) and (22).

*Step 6 (check stop criterion). *Go to the next step if the number of iterations reaches the maximum number of iterations; otherwise go back and continue Step .

*Step 7. *Select the solution with the best fitness as the global best solution.

And the computational flow of the GSAPSO algorithm is shown in Figure 1.