Particle Swarm Optimization Based Economic Load Dispatch with Valve Point Loading

Particle Swarm Optimization Based Economic Load Dispatch with Valve Point Loading

Parmvir Singh Bhullar1, Jaspreet Kaur Dhami2 Department of Electrical Engineering B.B.S.B.E.College, Fatehgarh Sahib

Abstract- Economic load dispatch (ELD) problem is a quite important task in the operational planning of a power system, where fuel cost is minimized by allocating generation to the committed units subject to the constraints imposed. A small change in the formulation of objective function will come into picture with the introduction of valve point loading effects. In this paper an effective and reliable particle swarm optimization (PSO) technique is proposed for the economic load dispatch problem. The results have been demonstrated for Economic Load Dispatch of standard 3-generator and 10-generator systems with and without consideration of the transmission losses.

Keywords- Economic Load dispatch, Particle Swarm Optimization, Valve Point Loading Effects.

1. INTRODUCTION

   The classical economic dispatch (ED) problem is most important, very well known and heavily researched problem in the power system. The operating cost is reduced by proper allocation of the amount of power to the committed units subject to the equality and inequality constraints [1].

   Traditionally, economic dispatch (ED) problems are solved using Lagrangian multipliers [2] and require the units with monotonically increasing piecewise linear cost functions. Unfortunately, this assumption is not applicable practically since modern units input-output characteristics are inherently highly non-linear due to valve- point loading, multiple fuel- effects and other type of constraints. It can be solved using either deterministic or stochastic methods. However, deterministic methods such as lambda iteration, base-point participation, hill climbing, gradient methods and others depend on the simplicity-based concept that assumes the ED problems generator cost curves to be convex. Hence, using quadratic or piecewise quadratic, monotonically increasing cost functions, the solutions of the ED problem were used to

   be estimated. However, with the addition of the practical system constraints, such as the valve-point effects, the generator cost curves are no longer convex and the ED problem becomes non-convex constrained optimization problem. Thus, there is a wide trend of adopting stochastic algorithms which are able to effectively solve the economic dispatch problem. Propitious results have been reported during the past few years and several methods like genetic algorithm (GA) [3], evolutionary programming (EP) [4], tabu search (TS) [5], simulated annealing (SA) [6], differential evolution (DE) [7] and many other methods were successfully implemented in practical ED problems significantly improving the existing results of the problem.

   In this paper, the PSO method is used in solving the non- convex economic dispatch problem. The results of experiments performed on standard practical test systems of 3 and 10 generators are reported and discussed. It is also important to mention that undoubtedly many established research papers have properly applied PSO in ED problem with non-convex characteristic. But many of these research works have not properly studied the effect of transmission losses in the load dispatch problems. Moreover parameter tuning is the most important factor in this algorithm that is applied up to the best level in this paper. In next section formulation of economic load dispatch problem is discussed and in third section PSO algorithm and its implementation is discussed. In ensuing section, test problems as well as the experimental results are reported. The paper is summarized in Section V.

2. ED PROBLEM FORMULATION

The classic economic dispatch problem aims to supply the required quantity of power at the lowest possible cost [8]. The dispatch problem can be stated mathematically as follows:

To minimize the total fuel cost at thermal plants: III. PROPOSED ALGORITHMS

Minimize

(1)

1. Introduction

   This section describes the proposed Particle Swarm Optimization method. It is an optimization and search

   (2) subject to the equality real power balance constraints:

   (3)

   technique based on the principles of social behavior of animals. The method was developed in 1995 by James Kennedy and Russell Eberhart [9]. PSO is very good at finding good enough solutions for a large range of problems, such as constrained optimization problems, multi-objective

   2 (4)

   optimization problems, etc. It is a simple and powerful optimization tool which scatters random particles, i.e.,

   the inequality constraint of limits on the generator outputs is:

   (5)

   solutions in the problem space. These particles, called swarms collect information from each array constructed by their respective positions. The particles update their positions using the velocity of particles. Position and velocity are both

   where ai, bi and ci are the cost coefficients of the i-th generator and NG is the number of generators committed to the operating system. Pgi is the power output of the i-th generator, PD is the load demand, PL represents the transmission losses.

   Bij =ijth element of loss coefficient symmetric matrix B, Bi0 =ith element of the loss coefficient vector and B00 =loss coefficient constant

   However, the cost function of a generator is not always differentiable due to valve-point effects and/or change of

   updated in a heuristic manner using guidance from particles own experience and the experience of its neighbors. The PSO algorithm requires less computation time and less memory because of its inherent simplicity.

   Most important factor in PSO algorithm is the setting or adjusting the value of parameters given in equation (7):

   (7)

   fuels. The valve-point effects introduce ripples in the heat-rate

   (8)

   curve. The fuel cost function with valve-point loadings of the generators is as shown in Fig 1:

   Fig. 1. The Valve-Point Effect

   Here w describes inertia weight that controls the momentum of the particle by weighing the contribution of the previous velocitybasically controlling how much memory of the previous flight direction will influence the new velocity. C1, C2 are the acceleration coefficients. In a classical PSO, Kennedy and Eberhart [10] described that a relatively high value of the cognitive component, compared with the social component, will result in excessive wandering of individuals through the search space. In contrast, a relatively high value of the social component may lead particles to rush prematurely toward a local optimum. Moreover, they suggested setting either of the acceleration coefficients at 2, in order to make the mean of both stochastic factors unity, so that particles would overfly only half the time of search. Since then, this suggestion has been extensively used for most studies. Suganthan [11] tested a method of linearly decreasing both

   (6)

   acceleration coefficients with time, but observed that the fixed acceleration coefficients at 2 generate better solutions. and

   where ai, bi, ci, ei and fi are the fuel cost coefficients of

   generator i with valve-point loading effects.

   are the particles personal and global best values respectively. A flowchart showing general algorithm of PSO is drawn below:

   Fig. 2. Flowchart showing PSO Algorithm

2. Implementation of PSO to ELD Problem

In this paper, an algorithm is developed to solve a constrained nd unconstrained ED problem using PSO to obtain a high quality solution. The PSO algorithm is utilized mainly to determine the optimal allocation of power among the committed units, thus minimizing the total generation cost. To implement the PSO algorithm to solve the ED problem, mentioned steps should be followed:

Step I: Initialize randomly the velocity of particles, generation of each generator according to the limit of each unit. These initial individuals must be feasible candidate solutions that satisfy the practical operation constraints.

Step 2: Each set of solution in the space should satisfy the following equation (3). If any combination doesnt satisfy the constraints then they are set according to the power balance equation:

V min = -0.5*Pmin V max = +0.5*Pmax

Step 7: Modify the member position x of each individual Pg according to equation (8).

Step 8: If the fitness value of each individual is better than the previous pbest value, the current value is set to be pbest. If the best pbest is better than gbest, the value is set to be gbest.

Step 9: If the number of iterations reaches the maximum, then go to step 10.Otherwise, go to step 3.

Step 10: The individual that generates the final gbest is the optimal generation power of each unit with the minimum total generation cost.

IV. EXPERIMENTAL RESULTS

In this section, the results of ELD after the implementation of proposed algorithm based on PSO are discussed. The programme is implemented in MATLAB 7.5.0. The developed algorithm for ELD problem based on PSO, have been discussed in Section III. The main objective is to minimize the cost of generation of plants considering valve point loading effects. The performance is evaluated with and without considering transmission losses using two generator test systems, i.e. three generator test system and ten generator test system and results are compared for with and without valve loading effects and for with and without transmission losses.

1. Case I- 3 Generator test system

   Electric power system of three generators has been studied. In this case, the load demand, PD is taken as 850MW. Operating cost coefficients for each generator are given in Table I

   TABLE I. Cost Coefficients of 3-Unit System

   Unit	1	2	3
   	0.001562	0.00194	0.00482
   	7.92	7.85	7.97
   	561	310	78
   	300	200	150
   	0.0315	0.042	0.063
   	600	400	200
   	100	100	50

   (9)

   B-Coefficient Matrix:

   Step 3: The objective is to be evaluated for each individual i.e. fitness of each individual is to be calculated. B=

   Step 4: Compare fitness of each individual with its pbest. The

   best fitness value among the pbests is denoted as gbest.

   Step 5: Modify the member velocity v of each individual Pg, according to equation (7).

   Step 6: The velocity components constraint occurring in the limits from the following conditions are checked.

   The Table II and Table III compare the results giving minimum costs with and without transmission losses. In Run 1 and Run 2, valve point loading effects are not considered. In Run 3 and 4, valve loading effects are also considered. Here in

   PSO parameters C1 and C2 are set at 2 and w is taken as 0.7. However the no. of particles and maximum iterations are set at 100.

   Table II. Comparison of Results using PSO without Valve Point Loading Effects

   Unit No.	Without Losses Run 1	With Losses Run 2
   1	423	376.1
   2	313	376.1
   3	114	104
   Total Generation (MW)	850	856.3144
   Minimum Cost ($/hr.)	8196.9	8257.9

   Table III. Comparison of Results using PSO with Valve Point Loading Effects

   Unit No.	Without Losses Run 3	With Losses Run 4
   1	481	401
   2	279	276
   3	90	181
   Total Generation (MW)	850	857.9054
   Minimum Cost ($/hr.)	8217	8290

   Figure3 and 4 shows the convergence characteristics of the cost function for with and without transmission losses, without valve loading effects. Figure 5 and 6 are the convergence characteristics with valve loading effects.

   Fig. 3. Convergence characteristics for 3-unit system without losses and without valve loading.

   Fig. 4. Convergence characteristics for 3-unit system with losses and without valve loading.

   Fig. 5. Convergence characteristics for 3-unit system without losses and with valve loading.

   Fig. 6. Convergence characteristics for 3-unit system with losses and with valve loading.

2. Case II- 10 Generator System

Electric power system of ten generators has been studied in this case. Here, the load demand, PD is taken as 2000 MW. Operating cost coefficients for each generator are given in Table IV.

Table IV. Cost Coefficients of 10- unit system

0.014

1

B- coefficient matrix is shown in nest column for considering the transmission losses.

The Table V and Table VI compare the results giving minimum costs with and without transmission losses. In Run 1 and Run 2, valve point loading effects are not considered. In Run 3 and 4, valve loading effects are also considered. Here in PSO parameters C1 and C2 are set at 2 and w is taken as 0.7. However the no. of particles and maximum iterations are set at 100.

Figure 7 and 8 shows the convergence characteristics of the cost function of 10- generator system for with and without transmission losses, without valve loading effects. Figure 9 and 10 are the convergence characteristics with valve loading effects.

B- coefficient matrix = * B

Table V. Comparison of Results using PSO without Valve Point Loading Effects

Table VI. Comparison of Results using PSO with Valve Point Loading Effects

It is clear from the Table V and VI that cost is significantly improved with the consideration of transmission losses and with valve loading effects.

Fig. 7. Convergence characteristics for 10-unit system without losses and without valve loading.

Fig. 8. Convergence characteristics for 10-unit system with losses and without valve loading.

Fig. 9. Convergence characteristics for 10-unit system without losses and with valve loading.

Fig. 10. Convergence characteristics for 10-unit system with losses and with valve loading.

The non-convex economic problem of power dispatch is solved using PSO strategy. These results are compared with the results available in literature for 3-generator system [12, 13] and for 10-generator system [14] and it is found that results are significantly improved by the proposed algorithm. Tuning of various parameters of PSO is important and it is found that the values of parameters in this paper are perfect for the improvement of results. The results demonstrate that PSO out performs other methods, particularly for non-convex cases, in terms of solution quality, dynamic convergence, computational efficiency, robustness and stability. The proposed algorithm can be applied to other non-convex, and non-smooth cost function having different constraints like prohibited operating zones, ramp rates and multi-fuel options. The proposed algorithm can also be applied to other power system optimization problems like dynamic economic dispatch and reactive power dispatch.

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