Elsevier

Applied Soft Computing

Volume 2, Issue 4, February 2003, Pages 297-305
Applied Soft Computing

On-line emission and economic load dispatch using adaptive Hopfield neural network

https://doi.org/10.1016/S1568-4946(02)00062-5Get rights and content

Abstract

This paper presents an adaptive Hopfield neural network (AHNN) based methodology, where the slope of the activation function is adjusted, for finding approximate Pareto solutions for the multi-objective optimization problem of emission and economic load dispatch (EELD). We have placed emphasis on finding solutions quickly, rather than the global Pareto solutions, so that our algorithm can be employed in large on-line power systems where variations in load are quite frequent. To enable faster convergence, adaptive learning rates have been developed by using energy function and applied to the slope adjustment method. The proposed algorithm has been tested on selected IEEE bus benchmark systems. The convergence of AHNN is found to be nearly 50% faster than the non-adaptive version.

Introduction

The operation planning of a power system is characterized by maintaining a high degree of economy and reliability. Traditionally, to solve the emission and economic dispatch problem, a Lagrangian augmented function is first formulated, and the optimal conditions are obtained by partial derivation of this function [1], [2]. In traditional method, calculation of the penalty factor as well as the incremental loss is always the key point in the solution algorithm. The problem can be solved using the lambda-iteration method, Newton–Raphson method, gradient method, genetic algorithms (GAs) or fuzzy based algorithms [3].

Among these methods, the lambda-iteration method [4] has been applied in many software packages due to its ease of implementation and used by power utilities for economic load dispatch (ELD). However, this method is not directly applicable for multi-objective emission and economic load dispatch (EELD) problem. Further, the experimental results have shown that the lambda-iteration method has oscillatory problems in large-scale systems [5], resulting in slower solution time. The genetic algorithm based approaches have shown better results for larger systems [6] than the lambda-iteration method, but their usage for the on-line ELD and EELD problems involve larger time. Neural networks have been used to solve this ELD problem for on-line dispatch, as the convergence of neural networks is much faster than the methods discussed earlier [5].

The Hopfield neural network (HNN) has been applied in various fields of optimization since Hopfield proposed the model in 1982. HNN based approaches have been proposed for solving the ELD [5], [7] and Emission problems [10] separately. In this paper, we propose a HNN based approach for finding Pareto solutions for the multi-objective EELD problem. Such a combined optimization problem has been studied before [8], [9], [11], [13], but they have not considered Pareto solutions. Like in the case of [5], [10], we have employed slope adjustment technique for faster convergence of the neural network. Sample IEEE benchmark cases of varying number of generators and loads with varying number of bus bars have been used to test our algorithm [8], [9].

Section snippets

Economic dispatch formulation

Consider a power generation system with i generators. The ELD problem is to find the optimal combination of power generation that minimizes the total cost while satisfying the total demand. The cost function of ELD problem is defined as follows:F=ifi(P)In Eq. (1), the generation cost function fi(Pi) in US$/h is usually expressed as a quadratic polynomial [7]fi(Pi)=(aiPi2+biPi+ci)where fi (Pi) is the cost of the ith generator in US$/h; Pi the power output of generator i in MW; ai, bi and ci the

The Hopfield neural network

The Hopfield network consists of a single layer of neurons, where the output of each of the neuron is given as an input to all the other neurons. The output of each of the neuron is a sigmoidal function of the weighted net input of that neuron.

The continuous neuron model is a generalized Hopfield network in which the computational energy decreases continuously in time [16]. For a very high gain parameter (U0) of the neurons, continuous model perform similar to the discrete model. Since the

Adaptive Hopfield network

The traditional Hopfield neural network discussed in the previous section requires a large number of iterations to solve the emission and economic load dispatch problem and it has problem in convergence during the transient. In order to increase the speed of convergence, the slope adjustment method is employed in this paper as in [5].

Since energy is to be minimized and its convergence depends on the gain parameter U0, the gradient–descent method can be applied to adjust the gain parameter asU

Simulation results and discussions

The algorithm presented in this paper for emission and economic load dispatch has been tested on IEEE14 bus, IEEE30 bus and IEEE57 [14] bus systems. The test system was provided with generators P1P5 on IEEE14 bus system, P6P11 on IEEE30 bus system and P12P15 on IEEE57 bus system. The generator details are given in Table 1. The results are obtained for minimum fuel cost, minimum emission while meeting the transmission losses and also satisfying the power generating limits of the generators.

Conclusion

In this paper, we have presented an approach using the adaptive Hopfield neural network (AHNN) for emission and economic load dispatch multi-objective optimization problem. The proposed algorithm employs slope adjustment technique for faster convergence to find approximate Pareto solutions. Because of its faster convergence, this method can be used for on-line problems effectively.

Acknowledgements

The authors are grateful to the anonymous referees for their meticulous corrections and valuable comments without which the paper would not be in its present form. The authors are also thankful to their respective institutes for providing necessary facilities for carrying out this research.

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