Solving Economic Load Dispatch Problem UsingParticle Swarm Optimization Technique

Economic load dispatch (ELD)is one of the important problems ofpower system operation. Conventional methods like Lambda iteration methodare not efficientfor complex ELD problems. Particle swarm optimization is preferred in ELD problem due to its high performance.The Inertia Weight PSO and Constriction Factor PSO algorithms are performed on threeunit and sixunit systems. The analysis of ELD problem is performed by Conventional method and PSO method. In this paper,losses are neglected in the ELD problem. PSO algorithm obtains the best solution forELD problem.


Introduction
The economic load dispatch problemisone of theoptimization problems. The mainaim of thisproblem is to minimize the total cost of generation.The conventional methods require more computation time in the ELD problem. PSO and Genetic Algorithm methodsare mainly used in ELD problems. The conventional method cannot obtain the best solution tothe ELD problem.PSO is the most efficient methodfor economic load dispatch problem.Various PSO algorithms used in this paper are Inertia weight PSO (IPSO) and Constriction factor PSO (CPSO) algorithm [1].
Various optimization problems are not solved by the single optimization. Different optimization methods are available for various optimization problems. The PSO algorithm offers the best solution for optimization problems. Modern optimization methods like PSO areeffective for engineering problems. The main aspect of PSO algorithm is its simplicity and a relatively less number of parameters.

Problem Formulation
The main focusof the ELD problem is to generate power at the minimum cost while satisfying certain constraints.Economic load dispatch determines the optimum share of the power demand subject to various system constraints. The ELD problem is expressed as (1).
F i (P i ) = (a i + b i *P i + c i *P i 2 ) ii) Inequality constraints [2] The Output power of each generator should be between its lower limit and upper limit.

Pimin≤ Pi≤ Pimax (4)
Pimin -Minimum output power of ith generator Pimax -Maximum output power of ith generator Pi -Output power of ith generator

Particle Swarm Optimization
The concept of PSO was developed by Kennedy and Eberhart. In PSO, a swarm consists of several particles. The PSO technique is based on the representation of social psychology. Each particle in PSO has a random position and random velocity. Each particle searches for a speed-adjusted position based on their flying and neighbourhood flying experience. If one particle first finds the best path to food then other particles will quickly follow the best path.The parameters that affect the performance of PSOare Swarm size, Number of iterations and Acceleration coefficients.

A. Computational Procedure of PSO [3]
The various steps of the PSO algorithm are as follows.

Parameter selection
Selection of PSO parameters such as population size, maximum iteration and acceleration constant.

Initialization of Population
In PSO,the particles are initialized with position p within the generator limits and velocity V.

Evaluation of Objective function
Calculate the fuel cost of the plant for each particle with the help of generated output power.

Selection of Previous best and Global best position
Set the initial output power for every particle to its previous best and set the best of the previous best to the global best.

Velocity and Position updation
Calculate the updated Position and Velocity of the particle.

Termination step
The PSO algorithm stops after a sufficient best fitness or maximum iterations are reached. Shi and Eberhart [4] developed Inertia weight PSO to enhancethe performance of PSO. This algorithm is known as Inertia weight PSO (IPSO). The inertia weight w is used to limit the velocity below its maximum value. This method enables the faster convergence of the swarm. The equations used in this algorithm are (5) and (6) respectively.

B. Constriction factor Particle swarm optimization
After the standard PSO algorithm, various PSO algorithms were introduced. Clerc [5] developed a Constriction factor PSO (CPSO) algorithm to improve the PSO performance. Constriction factor k is included to increase the convergence rate of PSO. The equations used in this algorithm are (8) and (9) respectively.

A. Case Study 1: 3 UNIT SYSTEM
In this case study three unit thermal system is considered. All units have the minimum and maximum generation limits. The generation data of three unit thermal system without lossis given in Table I

B. Case Study 2: 6 UNIT SYSTEM
In this case study six unit thermal system is considered. All units have the minimum and maximum generation limits. The generation data of six unit thermal system without lossis given in Table II[7].

CONCLUSION
In this paper, the ELD problem is solved using the conventional method and various PSO algorithms. The two test systems taken for the ELD problem are threeunit system and the sixunit system.The total cost is minimum in the PSO algorithm as compared to Conventional lambda iteration method.CPSO algorithm provides a faster convergence rate compared to the IPSO algorithm.It is concluded that both IPSO and CPSO algorithm gives the best solution for ELD problem.The total cost reductionis more inthe six-unit system than the three-unit system.