AMENDED PARTICLE SWARM OPTIMIZATION ALGORITHM FOR REAL POWER LOSS REDUCTION AND STATIC VOLTAGE STABILITY MARGIN INDEX ENHANCEMENT

In this paper, Amended Particle Swarm Optimization Algorithm (APSOA) is proposed with the combination of Particle Swarm Optimization (PSO) and Gravitational Search Algorithm (GSA) for solving the optimal reactive power dispatch Problem. PSO is one of the most widely used evolutionary algorithms in hybrid methods due to its simplicity, convergence speed, an ability of searching Global optimum. GSA has many advantages such as, adaptive learning rate, memory-less algorithm and, good and fast convergence. Proposed hybridized algorithm is aimed at reduce the probability of trapping in local optimum. In order to assess the efficiency of proposed algorithm, it has been tested on Standard IEEE 30 system and compared to other standard algorithms. The simulation results demonstrate worthy performance of the Amended Particle Swarm Optimization Algorithm (APSOA) in solving optimal reactive power dispatch problem


Introduction
In recent years the optimal reactive power dispatch (ORPD) problem has received great attention as a result of the improvement on economy and security of power system operation. Gradient method [1,2] Newton method [3] and linear programming [4][5][6] like various mathematical techniques have been adopted to solve the optimal reactive power dispatch problem. But they have difficulty in handling inequality constraints. Many Evolutionary algorithms such as have been proposed to solve the reactive power dispatch problem [7][8][9][10]. In this paper, Amended Particle Swarm Optimization Algorithm (APSOA) is proposed with the combination of Particle Swarm Optimization (PSO) and Gravitational Search Algorithm (GSA) for solving the optimal reactive power dispatch Problem. Merging the optimization algorithms is a way to balance the overall exploration and Exploitation ability. PSO is one of the most widely used evolutionary algorithms in hybrid methods due to its simplicity, convergence speed, an ability of searching Global optimum. GSA has many advantages [11] such as, adaptive learning rate, memory-less algorithm and, good and fast convergence. There are some studies in the literature [12][13][14][15][16][17][18][19] which have been done to synthesize PSO with other algorithms. These hybrid algorithms are aimed at reducing the probability of trapping in local optimum. PSO is one of the most widely used evolutionary algorithms in hybrid methods due to its simplicity, convergence speed, an ability of searching Global optimum. GSA has many advantages such as, adaptive learning rate, memoryless algorithm and, good and fast convergence. This hybridized algorithm is aimed at reduce the probability of trapping in local optimum. In order to assess the efficiency of proposed algorithm, it has been tested on Standard IEEE 30 system and compared to other standard algorithms. The simulation results demonstrate worthy performance of the Amended Particle Swarm Optimization Algorithm (APSOA) in solving optimal reactive power dispatch problem.

Modal Analysis for Voltage Stability Evaluation
Power flow equations of the steady state system is given by, Where ΔP = bus real power change incrementally. ΔQ = bus reactive Power injection change incrementally. Δθ = bus voltage angle change incrementally. ΔV = bus voltage Magnitude change incrementally. Jpθ , JPV , JQθ , JQV are sub-matrixes of the System voltage stability in jacobian matrix and both P and Q get affected by this.

Modes of Voltage Instability
Voltage Stability characteristics of the system have been identified through computation of the Eigen values and Eigen vectors. Where, ξ denote the right eigenvector matrix of JR, η denote the left eigenvector matrix of JR, ∧ denote the diagonal eigenvalue matrix of JR.
From the equations (5) and (6), Or ξi denote the ith column right eigenvector & η is the ith row left eigenvector of JR. λi indicate the ith Eigen value of JR.
reactive power variation of the ith modal is given by, where, Where ξji is the jth element of ξi ith modal voltage variation is mathematically given by, When the value of | λi | =0 then the ith modal voltage will get collapsed.
In equation (8), when ΔQ = ek is assumed , then ek has all its elements zero except the kth one being 1. Then ∆V can be formulated as follows, ƞ 1k is k th element of ƞ 1 At bus k V -Q sensitivity is given by,

Problem Formulation
To minimize the real power loss and also to maximize the static voltage stability margin (SVSM) is the key objectives of the reactive power dispatch problem.

Minimization of Real Power Loss
Real power loss (Ploss) minimization in transmission lines is mathematically given as, Where n is the number of transmission lines, gk is the conductance of branch k, Vi and Vj are voltage magnitude at bus i and bus j, and θij is the voltage angle difference between bus i and bus j.

Minimization of Voltage Deviation
At load buses minimization of the voltage deviation magnitudes (VD) is stated as follows, Where nl is the number of load busses and Vk is the voltage magnitude at bus k.

System Constraints
These are the following constraints subjected to objective function as given below, Load flow equality constraints: where, nb is the number of buses, P G and Q G are the real and reactive power of the generator, P D and Q D are the real and reactive load of the generator, and G ij and B ij are the mutual conductance and susceptance between bus i and bus j.
Generator bus voltage (V Gi ) inequality constraint: Load bus voltage (V Li ) inequality constraint: Switchable reactive power compensations (Q Ci ) inequality constraint: Reactive power generation (Q Gi ) inequality constraint: Transformers tap setting (T i ) inequality constraint: Transmission line flow (S Li ) inequality constraint: Where, nc, ng and nt are numbers of the switchable reactive power sources, generators and transformers.

Particle Swarm Optimization
Particle Swarm Optimization (PSO) is an evolutionary computation technique which is inspired from social behavior of bird flocking. It uses a number of particles (candidate solutions) which fly around in the search space to find best solution. Meanwhile, they all look at the best particle (best solution) in their paths. In other words, particles consider their own best solutions as well as the best solution has found so far. Each particle in PSO should consider the current position, the current velocity, the distance to pbest, and the distance to gbest to modify its position. PSO was mathematically modeled as follow: The PSO starts with randomly placing the particles in a problem space. In each iteration, the velocities of particles are calculated using equation (24). After defining the velocities, the position of masses can be calculated by equation (25). The process of changing particles' position will continue until meeting an end criterion.

Gravitational Search Algorithm
Gravitational Search Algorithm (GSA) is a novel heuristic optimization method which is inspired from is the Newton's theory that states: Every particle in the universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. GSA can be considered as a collection of agents (candidate solutions) whose have masses proportional to their value of fitness function. During generations, all masses attract each other by the gravity forces between them. A heavier mass has the bigger attraction force. Therefore, the heavier masses which are probably close to the global Optimum attract the other masses proportional to their Distances. The GSA was mathematically modeled as follow. Suppose a system with N agents. The algorithm starts with randomly placing all agents in search space. During all epochs, the gravitational forces from agent j on agent i at a Specific time t is defined as follows; The G(t) is calculated as (27): In a problem space with the dimension d, the total force that acts on agent i is calculated as the following equation: According to the law of motion, the acceleration of an agent is proportional to the result force and inverse of its mass, so the acceleration of all agents should be calculated as follow: The velocity and position of agents are calculated as follows: In GSA, at first all masses are initialized with random Values. Each mass is a candidate solution. After Initialization, velocities for all masses are defined using (30). Meanwhile the gravitational constant, total forces, and accelerations are calculated as (27), (28), and (29) respectively. The positions of masses are calculated using (31). Finally GSA will be stopped by meeting an end criterion.

Amended Particle Swarm Optimization Algorithm (APSOA)
Two algorithms can be hybridized in high-level or low-level with relay or co evolutionary method as homogeneous or heterogeneous. In this paper, we hybridize PSO with GSA using low-level co evolutionary heterogeneous hybrid. The hybrid is low-level because we combine the functionality of both algorithms. It is co-evolutionary because we do not use both algorithm one after another. In other words, they run in parallel. It is heterogeneous because there are two different algorithms that are involved to produce final results. The basic idea of proposed algorithm is to combine the ability of social thinking (gbest) in PSO with the local search Capability of GSA. In order to combine these algorithms, (32) is proposed as follow: In each iteration, the positions of particles are updated as follow: In the proposed algorithm, at first, all agents are randomly initialized. Each agent is considered as a candidate solution. After initialization, Gravitational force, gravitational constant, and resultant forces among agents are calculated using (26), (27), and (28) respectively. After that, the accelerations of particles are defined as (29). In each iteration, the best solution so far should be updated. After calculating the accelerations and with updating the best solution so far, the velocities of all agents can be calculated using (32). Finally, the positions of agents are defined as (33). The process of updating velocities and positions will be stopped by meeting an end criterion. The steps of hybridized algorithm are represented in fig.1. In Amended Particle Swarm Optimization Algorithm (APSOA), the quality of solutions (fitness) is considered in the updating procedure. The agents near good solutions try to attract the other agents which are exploring the search space. When all agents are near a good solution, they move very slowly. In this case, the gBest help them to exploit the global best. Amended Particle Swarm Optimization Algorithm (APSOA) use a memory (gBest) to save the best solution has found so far, so it is accessible anytime.

Simulation Results
The efficiency of the proposed Amended Particle Swarm Optimization Algorithm (APSOA) is demonstrated by testing it on standard IEEE-30 bus system. 6 generator buses, 24 load buses and 41 transmission lines of which four branches are (6-9), (6-10) , (4)(5)(6)(7)(8)(9)(10)(11)(12) and ( Table 5. In Table 1 optimal values of the control variables are given.  Table 2 indicates the optimal values of the control variables & there is no limit violations in state variables. Mainly static voltage stability margin (SVSM) has increased from 0.2486 to 0.2498. contingency analysis was conducted using the control variable setting obtained in case 1 and case 2 to determine the voltage security of the system. In Table 3 the Eigen values equivalents to the four critical contingencies are given. Result reveal about the Eigen value has been improved considerably for all contingencies in the second case.     Minimum loss Evolutionary programming [20] 5.0159 Genetic algorithm [21] 4.665 Real coded GA with Lindex as SVSM [22] 4.568 Real coded genetic algorithm [23] 4.5015 Proposed APSOA method 4.2006

Conclusion
In this paper, Amended Particle Swarm Optimization Algorithm (APSOA) has been successfully solved the optimal reactive power dispatch problem. Proposed hybridized algorithm is aimed at reduce the probability of trapping in local optimum. In order to assess the efficiency of proposed algorithm, it has been tested on Standard IEEE 30 system and compared to other standard algorithms. The simulation results demonstrate worthy performance of the Amended Particle Swarm Optimization Algorithm (APSOA) in solving optimal reactive power dispatch problem. Real power loss has been reduced & static voltage stability margin index has been enhanced.