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

1. 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 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 Among these methods, the lambda-iteration method 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 , resulting in slower solution time. The genetic algorithm based approaches have shown better results for larger systems 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 . 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 and Emission problems 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 , but they have not considered Pareto solutions. Like in the case of , 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
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