PSO-ANN For Economic Load Dispatch With Valve Point Loading Effects

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International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (ISSN 2250-2459, Volume 2, Issue 5, May 2012)

137

PSO-ANN For Economic Load Dispatch With Valve Point

Loading Effects

Tripti Gupta

1

, Dr. Manjaree Pandit

2

1,2_{Department of Electrical Engineering, Madhav Institute of Technology and Science Gwalior, INDIA}

1

[email protected]

2_{[email protected]}

Abstract— This paper presents a new efficient approach to Economic Load Dispatch (ELD) problems with non-convex cost functions using PSO-ANN. The practical ELD problems have non-smooth cost functions considering valve point loading effects with equality and inequality constraints that make the problem of finding the global optimum difficult using any traditional mathematical approach. Therefore, Particle Swarm Optimization (PSO) technique is used for generating training data for the neural network. This paper proposes a PSO-ANN for the fast on-line implementation of the economic load dispatch problem. Test results of system with 3 generating units are given to illustrate the effectiveness of the proposed method.

Keywords— Economic load Dispatch, Particle Swarm optimization, Artificial Neural Network, valve point loading effects.

I. INTRODUCTION

Economic load dispatch problem is one of the fundamental issues in power system operation. Its objective is to allocate the power demand among committed generators in the most economical manner, while all physical and operational constraints are satisfied. This is a real time problem for properly allocating the real power output among the online generating units so as the total production cost of thermal units is minimized.

Many researchers have been done for the problem as reported in the literature [1]. At the early time, the objective function of the ED problem was approximately represented by a single quadratic function so that mathematical programming techniques could be implemented to solve it [2].

This simplified model has been used since the Mathematical methods require the fuel cost function to be differential. However, the input-output characteristics of thermal generating units are actually more complicated due to the effects of valve point loadings [3]. The practical ED with nonlinearities translates into a complicated optimization problem having complex and Non convex characteristics, with multiple minima, making the challenge of obtaining the global minima, very difficult.

Conventional gradient based optimization methods fail to model these discontinuities and usually result in inaccurate dispatches causing loss of revenue. Dynamic programming [4] has no restriction on cost curve, but this method is computationally extensive, and suffers from the problem of dimensionality.

Methods like dynamic programming [4], genetic algorithm [5], [6], [7], [8], evolutionary programming [9], [10]–[12], artificial intelligence [13], and particle swarm optimization [14]–[24] solve nonconvex optimization problems efficiently and often achieve a fast and near global optimal solution.

Recently, Eberhart and Kennedy suggested a Particle Swarm Optimization (PSO) based on the analogy of swarm of bird and school offish. In PSO, each individual makes its decision based on its own experience together with other individual's experiences [25]. The individual particles are drawn stochastically towards the position of present velocity of each individual, their own previous best performance, and the best previous performance of their neighbors. The main advantages of the PSO algorithm are summarized as: simple concept, easy implementation, and computational efficiency when compared with mathematical Algorithm and other heuristic optimization techniques.

In this paper, data is generated by using Particle Swarm Optimization and implemented with artificial neural network for three unit generating system with valve point loading effects. The proposed PSO has been tested on 3-unit systems and the obtained various loads on three generators respectively. And best total costs for each optimised combination are trained with the neural network with different hidden neurons and learning rate. The optimized output, cost are target and loads are treated as input values for ANN.

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International Journal of Emerging Technology and Advanced Engineering

138 II. PROBLEM FORMULATION

A. Economic load dispatch

1) ED Problem with Smooth Cost Functions

The ED problem is to find the optimal combination of power generations that minimizes the total generation cost while satisfying an equality constraint and inequality constraints. The most simplified cost function of each generator can be represented as a quadratic function as given in whose solution can be obtained by the conventional mathematical methods.

(1)

(2)

Where

C total generation cost Fj cost function of generator j aj,bj,cj cost coefficients of generator j Pj electrical output of generator j J set of all generators

2)

Problem constraints

Power balance constraints: While minimizing the total generation cost, the total generation should be equal to the total system demand .The equality constraint for the ELD problem can be given as

(3)

Generation capacity constraints: For stable operation, real power output of each generator is restricted by lower and upper limits as follows:

P

j

P

j

P

j

1 ,

2 ,...,

J

max

min





(4)

Where Pjmin, Pjmax are the minimum, maximum outputs of generator respectively.

3) Non-smooth Cost Function with Valve-Point Effects

In reality, the objective function of an ELD problem has non differentiable points according to valve point loading effects. Therefore, the objective function should be composed of a set of non-smooth cost functions .The generator with multi-valve steam turbines has very different input-output curvecompared with the smooth cost function [6]. Typically, the valve point results in, as each steam valve starts to open, the ripples like in to take account for the valve-point effects, sinusoidal functions are added to the quadratic cost functions as follows:

(5)

Where , are the coefficients of generator j reflecting valve point effects

0 100 200 300 400 500 600

0 1000 2000 3000 4000 5000 6000

Unit Power output

Fu

el

C

os

t

P1 Quad P1 VPL P2 Quad P2 VPL P3 Quad P3 VPL

Fig.1. Cost curves of 3-unit system with and without valve point Loading effects

B. Particle swarm optimization

PSO is one of the modern heuristic algorithms developed byKennedy and Eberhart [6]. The advantages of PSO are ease of implementation and only few parameters to adjust [7], [8]. The PSO algorithm searches in parallel using a group of individuals, in a physical n dimensional search space, the Position and velocity of individual i are represented as the Vectors and

respectively, in the PSO algorithm.

Let

and

respectively, be the position of the individual i and its neighbours ' best position so far[2].Using the information, the updated velocity of individual i is modified under the following equation in the PSO algorithm

(6) Where

Vik velocityof individual at iteration W weight parameter

c1, c2 weight factors

rand1, rand2 random numbers between 0 and1 Xik position of individual i iteration k

Pbestik best position of individual until iteration k

Gbestik best position of the group until iteration k

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International Journal of Emerging Technology and Advanced Engineering

139

(7)

The search mechanism of the PSO using the modified velocity and position of the individual i based on (6) and (7) is illustrated in fig 2

Fig.2 search mechanism of PSO

C. Artificial neural networks

Artificial Neural Networks, heron referred to as ANN, are an attempt at modeling the processing power of the human brain. Humans are able to adapt to new situations and learn quickly when given the correct context. Computers are relatively slow at performing simple human tasks such as recognizing a lizard in a painting of the jungle. ANN work by simulating the structure of the human brain. At their basic level they consist of a network of neurons connected by synapses.

Neurons are the basic elements of an ANN. Neurons accept inputs from other connections and produce an output by firing their synapse. Neurons typically perform a weighted sum on all of their input connections and then pass it through a transfer function to produce its output. A simple block diagram showing the process of a neuron applying the transfer function to its inputs before emitting its output may be seen in Fig 1. The traditional ANN is a binary network in which a synapse either fires or doesn’t fire. This type of transfer function is a step function in which the neuron compares its weighted sum to a threshold and then either emits a 1 or a 0 (fires or doesn’t fire its synapse). While binary networks have their uses, most engineering applications involve the real number system. ANN has thus been adapted to use real numbers. The principles are the same, but rather than only outputting a 1 or 0, a neuron can output a real number on any range, typically [0, 1].

Fig.3. neuron block diagram

III. PSO-ANNMODEL

The PSO and ANN have different approaches for solving the problem depending on their inherent properties. both the methods have different approaches to a minimization problem having one global minimum and a number of local minima in the performance surface (fig ). It is an herent property of ANN to search in the local region whereas PSO explores the entire search space for the optimization of the concerned variable. It is often seen that while training, the network get stuck in a local minimum of the network error but PSO finds a global minimum and does not stop at it .

Fig.4.search space in neural network

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International Journal of Emerging Technology and Advanced Engineering

140 PSO has a good ability to explore the whole search space for the minimization of the cost function in a minimization problem but the problem with PSO is that all the particles may not rest at the global minimum and may fall short of target therefore, a hybrid PSO-ANN model having a well balanced search (exploration and exploitation) property of both ANN and PSO can be used for a minimization problem. The hybrid model possesses well –balanced search ability for the global minimum in the search space .PSO searches for all the minima in the search space i.e. network error ,then ANN does a fine search around them to get global minimum.

The raw data was then transferred to a suitable form depending upon the activation function of the neuron, which in the present case is the sigmoid function (fig.) the input and target values were linearly normalized in order to train the network in the non-linear region of the activation function avoiding the saturation region .

Sigmoid function: f(x)=1/1+exp(-x)

Thereafter, the transformed data set was divided into two parts, first part containing 100 data points was used for training the network and the other part containing 30 data points was used for testing the robustness of the network in predicting with input values which were not used in training.

Training Neural Networks with PSO

The PSO algorithm is vastly different than any of the traditional methods of training. PSO does not just train one network, but rather trains a network of networks. PSO builds a set number of ANN and initializes all network weights to random values and starts training each one. On each pass through a data set, PSO compares each network’s fitness.

The network with the highest fitness is considered the global best. The other networks are updated based on the global best network rather than on their personal error or fitness. Each neuron contains a position and velocity. The position corresponds to the weight of a neuron. The velocity is used to update the weight. The velocity is used to control how much the position is updated. If a neuron is further away (the position is further from the global best position) then it will adjust its weight more than a neuron that is closer to the global best.

Algorithm for the proposed PSO-ANN

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International Journal of Emerging Technology and Advanced Engineering

141 IV. RESULTS AND DISCUSSION

Usually, training in a neural network is a slow intelligence process. So to speed it up optimization is required at every point, i.e., in deciding the neural network architecture, the number and type of training/ testing patterns, the learning rate, the error goal etc. The training patterns consist of input/output pairs. The proposed approach is tested on a standard test system taken from[9]. This system has a non-convex cost characteristic as shown in Fig.1.

A. Data Generation

The cost and generation limits of three unit system are given in Table. I. The load patterns are generated by varying the total load demand within the limits of each generator. The cost function for non-convex problem for each load is calculated with the help of cost coefficients and generator coefficient reflecting the valve point effects. The PSO is applied for each load pattern to obtain the optimum value of cost and real power sharing at different generating units.

TABLE I

DATAGENERATION

The PSO will optimize the cost for each load varying from 766MW to 943MW. This table wills shows that by varying the population size the cost can be optimize.

TABLE II

EFFECT OF POPULATION SIZE

Pop size

Min. cost

Max. cost Mean S D

25 8235.4944 8796.6658 8574.6585 113.7768

50 8242.1601 8870.4137 88704.9876 110.9763 100 8234.0715 8768.4520 85423.1324 88.9853

In table II effect of different population size is shown and Best cost is found as 8234.0715 at pop size =100, trial=10.

1 2 3 4 5 6 7 8 9 10

8200 8300 8400 8500

trials

co

st

popsize 100 popsize 50 popsize 25

Fig.7. Convergence characteristics for different pop size

B. Performance of PSO-ANN

Power Systems are complex systems with a large number of control variables. ANN has proven to be excellent tools for mapping complex systems to a known output. The proposed solution is to construct and train a neural network to predict economic load dispatch for a given power system.

TABLE III.

EFFECT OF HIDDEN NEURONS LEARNING RATE =0.005

The best performance is obtained for 50 hidden layers for which the mean square error is 0.003 in 86 iterations with CPU time 47 sec time. The various characteristic curves during implementation of ANN for 50 hidden neurons are shown in figures.

Generator a b c e f Pmin

(MW) Pmax

(MW)

Unit 1 0.00156 7.92 561 300 0.031 100 600 Unit 2 0.00194 7.85 310 200 0.042 100 400

Unit 3 0.00482 7.97 78 150 0.063 50 150

s.no Hidden neurons

Iterations MSE CPU time

1 10 2000 0.0115 0:01:40

2 20 318 0.009 0:01:40

3 30 188 0.007 0:01:40

4 40 190 0.004 0:01:40

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International Journal of Emerging Technology and Advanced Engineering

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Fig.8. Performance characteristic of ANN for 50 HLN

Mean Squared Error gives average squared difference between outputs and targets. Lower values are better. Zero means no error.

This fig shows the mean square error continuously getting reduced meeting the goal at 86 epochs.

Fig.9.Training state of ANN

Fig.10.Regression plot: Relationship between network target and output

Regression R Values measure the correlation between outputs and targets. An R value of 1 means a close relationship, 0 a random relationship.

In the proposed algorithm R value is 0.9091 this shows that a very close relation between the output and target values.

Finally training results of economic load dispatch with 3 generating unit system by PSO-ANN is accomplished in the following figures.

0 10 20 30 40 50 60 70 80 90 100

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Training patterns

P4

target ANN

Fig.11.Result of optimal cost computation by ANN

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International Journal of Emerging Technology and Advanced Engineering

143

[image:7.612.71.272.142.399.2]

Fig 13.Result of Generator 2(P2) obtained by ANN

Fig.14.Result of Generator 3(P3) obtained by ANN

V. CONCLUSION

In this paper PSO-ANN for classical economic load dispatch problem with valve point loading is proposed.For this, optimized data is generated by using PSO with different loads and then trained for different load patterns by using proposed algorithm. Finally The target values are met within the permissible time duration. The results show the effectiveness of the proposed method for obtaining the target values with minimum mean square error 0.003.

VI. ACKNOWLEDGMENT

The authors sincerely acknowledge the financial support provided by UGC, New Delhi under major research project entitled Power System Optimization and Security Assessment Using Soft Computing Techniques, vide F No.34-399/2008 (SR) dated, 24th December 2008. The second author acknowledges UGC research award for post doctoral work sanctioned by UGC, New Delhi vide letter no. F-30-120 (SC)/2009 (SA-II). The authors also thank the Director, M.I.T.S. Gwalior for providing facilities for carrying out this work.

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BIOGRAPHICAL NOTES

Tripti Gupta obtained her B.E. degree from Institute of information technology and management Gwalior (India) in 2010. She is pursuing her M.E. degree In Electrical engineering from Madhav Institute of Technology & Science Gwalior (India). Her area of interest are Power System and Evolutionary computation application to power system.