Applied Soft Computing 13 (2013) 292–301 Contents lists available at SciVerse ScienceDirect Applied Soft Computing j ourna l ho me p age: www.elsevier.com/l ocate/asoc Augmented Lagrange Hopfield network initialized by quadratic programming for economic dispatch with piecewise quadratic cost functions and prohibited zones Vo Ngoc Dieu a,∗ , Peter Schegner b a Department of Power Systems, Faculty of Electrical and Electronic Engineering, Ho Chi Minh City University of Technology, Ho Chi Minh City, Viet Nam b Institute of Electrical Power Systems and High Voltage Engineering, Faculty of Electrical and Computer Engineering, Technische Universität Dresden, 01069 Dresden, Germany a r t i c l e i n f o Article history: Received 9 September 2011 Received in revised form 2 July 2012 Accepted 11 August 2012 Available online 21 August 2012 Keywords: Augmented Lagrange Hopfield network Economic dispatch Piecewise quadratic fuel cost function Prohibited zones Quadratic programming a b s t r a c t This paper proposes a method based on quadratic programming (QP) and augmented Lagrange Hopfield network (ALHN) for solving economic dispatch (ED) problem with piecewise quadratic cost functions and prohibited zones. The ALHN method is a continuous Hopfield neural network with its energy function based on augmented Lagrange function which can properly deal with constrained optimization problems. In the proposed method, the QP method is firstly used to determine the fuel cost curve for each unit and initialize for the ALHN method, then a heuristic search is used for repairing prohibited zone violations, and the ALHN method is finally applied for solving the problem if any violations found. The proposed method has been tested on different systems and the obtained results are compared to those from many other methods in the literature. The result comparison has indicated that the proposed method has obtained better solution quality than many other methods. Therefore, the proposed QP-ALHN method could be a favorable method for solving the ED problem with piecewise quadratic cost functions and prohibited zones. © 2012 Elsevier B.V. All rights reserved. 1. Introduction The operation cost in power systems needs to be minimized at each time satisfying operating constraints via economic dispatch (ED) problem. In practical power system operation conditions, many thermal generating units, especially those units which are supplied with multiple fuel sources like coal, natural gas, and oil require that their fuel cost functions may be segmented as piece- wise quadratic cost functions to represent for different types of fuel. In addition, thermal generating units may have prohibited oper- ating zones due to physical limitations on components of units. Consequently, a unit with prohibited zones, its whole operating region will be broken into several isolated feasible sub-regions. Therefore, the ED problem with piecewise quadratic cost func- tions and prohibited zones is to minimize the total fuel cost among the available fuels for each thermal unit subject to load demand, generation limits, and prohibited zone constraints. This is a non- convex and complicated optimization problem since it contains the discontinuous values at operating characteristics of generat- ing units forming multiple local optima. Therefore, the classical solution methods are difficult to deal with this type of problem. One approach for solving the problem with such units having mul- tiple fuel options is to linearize the segments and solve them by ∗ Corresponding author. E-mail address: vndieu@gmail.com (V.N. Dieu). traditional methods [1]. A better approach is to retain the assump- tion of piecewise quadratic cost functions instead of linearized segments and proceed to solve them. A hierarchical approach based on the numerical method (HNUM) has been proposed in [2] as one way to approach to the problem. However, the major prob- lem for the numerical methods is their exponentially growing time complexities for larger systems with non-convex constraints. Since the first implementation for solving benchmark problem of Travel- ing Salesman [3], Hopfield neural network (HNN) has become very popular for solving constrained optimization problems. The basic structure of the HNN is based on the highly connected networks of nonlinear analog neurons. The advantage of the HNN is that it can solve complex optimization problems in fast manner since all of the neurons simultaneously and continuously change their analog state in parallel. Moreover, the HNN also has simple technical imple- mentation and properly handles the variables’ upper and lower limits using its sigmoid function. However, the HNN only guar- antees convergence to a local minimum of optimization problems. Therefore, the HNN should be further improved for obtaining global optimal solution of practical problems. The application of the HNN in [4] has created difficulties in handling some kinds of inequal- ity constraints and dealing with large-scale problems with many constraints. Moreover, the final solution of the HNN method is also sensitive to the choice of penalty factors associated with constraints in its energy function. For solving the problem by the enhanced Lagrangian neural network (ELANN) [5] method, the dynamics of Lagrange multipliers associated with equality and inequality 1568-4946/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.asoc.2012.08.026