Designing a Modified Hopfield Network to Solve an Economic Dispatch Problem with Nonlinear Cost Function Ivan Nunes da Silva, Leonardo Nepomuceno, Thiago Masson Bastos State University of São Paulo – UNESP UNESP/FE/DEE, CP 473, CEP 17033-360 Bauru – SP, Brazil e-mail: ivan@feb.unesp.br Abstract - Economic Dispatch (ED) problems have recently been solved by artificial neural networks approaches. In most of these dispatch models, the cost function must be linear or quadratic. Therefore, functions that have several minimum points represent a problem to the simulation since these approaches have not accepted nonlinear cost function. Another drawback pointed out in the literature is that some of these neural approaches fail to converge efficiently towards feasible equilibrium points. This paper discusses the application of a modified Hopfield architecture for solving ED problems defined by nonlinear cost function. The internal parameters of the neural network adopted here are computed using the valid-subspace technique, which guarantees convergence to equilibrium points that represent a solution for the ED problem. Simulation results and a comparative analysis involving a 3-bus test system are presented to illustrate efficiency of the proposed approach. 1. INTRODUCTION In power systems, the Economic Dispatch (ED) can be defined as the process of allocating generation levels to the generating units, so that the system load is supplied entirely and most economically. Many ED approaches have been proposed in the literature to formulate and solve this problem. In [1] it is provided a review of the advances in such field. The economic dispatch definition above is quite large so that many specific optimization models applied to power system, such as optimal power flow, unit commitment, generation scheduling, etc., may be faced as economic dispatch models. It must be clear however that these models vary in complexity and have different scopes of application. The Classic Economic Dispatch (CED) discussed in [1] and [2] is the starting point of the ED problem. CED is concerned with minimization of total operating costs while supplying entirely the system demand and enforcing the limits on generation levels. In CED procedures, the transmission network representation is totally neglected. Some effective applications of artificial neural networks to the ED problem have recently been presented in the literature. In [3] an approach trying to unify unit commitment and generation dispatch functions is described. A hybrid Hopfield network is adopted such that the energy function of the Hopfield network is able to deal with discrete and continuous terms. In [4] a Hopfield neural network is proposed to solve CED problem with general non-convex cost functions. The computation effort for solving the problem is high due to large number of iterations to obtain the optimality. In [5] an analytic Hopfield method reducing considerably this computation effort is proposed. However the method is not applied to non-convex cost functions. In [6] a Hopfield model for ED problem considering prohibited zones was developed. In [7] a neural network approach for solving CED with transmission capacity constraints was proposed. In [8] a restructuring of the approach described in [7] was proposed for solving the unconstrained ED, i.e., the ED with transmission capacity constraints and also the multi- area ED problems. A solution to ED problems using decision trees and considering nonlinear cost function is presented in [9]. Most of the neural network applications described above fail to converge efficiently towards the equilibrium points representing the dispatch problem solutions. A careful analysis of the results presented in some of these papers reveals that infeasible solutions are sometimes obtained. In the modified Hopfield approach proposed in [10] to solve CED problem, the optimization and constraint terms involved with problem mapping (Section 3) are treated in different stages. The modified Hopfield approach guarantees the network convergence to a feasible optimal solution [11]. The problems associated with speed of convergence, depicted in [3] and [4], were also satisfactorily handled in [10]. As demonstrated in [11], the modified Hopfield approach is also applicable to general non-convex cost functions; and this includes CED problems with non-monotonically increasing incremental cost units, such as that studied in [12]. This paper applies the modified Hopfield approach described in [10] to solve a more representative dispatch problem. In the problem being dealt with in this work, the cost function includes nonlinear terms. A 3-bus test system is used to validate the methodology discussed in this paper. The results have pointed out that the modified Hopfield approach is robust enough to deal also with nonlinear cost functions. The paper is organized as follows. In Section 2, the formulation of the dispatch problem is introduced. In Section 3, the modified Hopfield network is presented, and valid- subspace technique, used to design the network parameters, is described. A mapping of the economic dispatch problem with nonlinear cost functions using the modified Hopfield network is formulated in Section 4. In Section 5, simulation results are presented to validate the developed approach. In Section 6, the main conclusions about the paper are presented. 0-7803-7278-6/02/$10.00 ©2002 IEEE