Economic Load Dispatch With Multiple Fuel Options and Valve Point Effect Using Cuckoo Search Algorithm With Different Distributions

Nomenclature

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  a_ij, b_ij, c_ij fuel cost coefficients for fuel type j of unit i

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  e_ij, f_ij fuel cost coefficients for fuel type j of unit i reflecting valve-point effects

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  N total number of generating units

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  m_i number of fuel types of unit i

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  P_i power output of unit i

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  P_i,max maximum power output of unit i

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  P_i,min minimum power output of unit i

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  P_ij,min minimum power output for fuel j of unit i

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  P_D total system load demand

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  P_L total transmission loss

Introduction

The objective of economic load dispatch (ELD) is to minimize total fuel cost of thermal units while satisfying both equality and inequality constraints including load balance constraint, upper and lower generation limit on thermal units (Lee & Kim, 2002). Traditionally, the fuel cost function of thermal unit is approximately represented as one single quadratic curve because each generating unit used only one fossil fuel to produces electricity. However, it is more realistic to represent the fuel function as a segmented piece-wise quadratic functions because several fuels are burned (Dieu et al., 2013).

Several methods have been applied for solving ELD problem with multiple fuel options so far. The lamda-iteration has been valued as a simple and effective one (Thang 2011). However, the disadvantages of the method are that the values of lamda and updated step size are randomly chosen initially. This can lead to a non-optimal solution or non-convergence. The best solution has been found after the method has been performed 93 independent runs with various values of lamda and fuel type. The computational time for each trial is short but total time for whole is long. Enhanced Augmented Lagrange Hopfield Network (ALHN) (Dieu & Ongsakul, 2008) solves ELD problem in two phases and gains good solutions and short simulation time. However, the gained simulation results depend on setting a large number of parameters. The Differential Evolution (DE) (Nasimul & Hitoshi, 2008) algorithm is found to be a powerful evolutionary algorithm for global optimization in many real problems. Self-Adaptive Differential Evolution (SDE) (Balamurugan & Subramanian, 2007) is a good method to solve ELD problem with valve point effects. The application of Hopfield neural network (HNN) (Park et al., 1993) with merit of simplicity created difficulties in handling some kinds of inequality constraints. For solving the problem by the enhanced Lagrangian neural network (ELANN) (Lee & Kim, 2002) method, the dynamics of Lagrange multipliers including equality and inequality constraints were improved to guarantee its convergence to the optimal solutions, and the momentum technique was also employed in its learning algorithm to achieve fast computational time. Both HNN (Park et al., 1993) and ELANN (Lee & Kim, 2002) were involved a large number of iterations for convergence.