Solution of nonconvex and nonsmooth economic dispatch by a new Adaptive Real Coded Genetic Algorithm
Introduction
Power systems should be operated under a high degree of economy. Economic dispatch is an important optimization task addressing this vital concern for power system operations. ED is defined as the process of allocating generation levels to the generating units, so that the system load is supplied entirely and most economically (Wood & Wollenberg, 1996). ED is subproblem of unit commitment (UC) (Patra et al., 2009, Yuan et al., 2009) and determines the generation level of each committed unit. Practical economic dispatch has complex and nonlinear characteristics with many equality and inequality constraints (Li, 1998, Li and Aggarwal, 2000). Traditionally, in the ED problem, the cost function of each generator has been approximated by a single quadratic function, and the valve-point effects and multi-fuel source options were ignored. This would often introduce inaccuracy into the resulting dispatch. Large modern generating units with multi-valve steam turbines have a number of steam admission valves that are opened sequentially to obtain ever-increasing output of units and the valve-point effects produce a ripple-like heat rate curve. However, since the cost curve of a generator is highly nonlinear, containing discontinuities owing to valve-point loadings, the cost function is more realistically denoted as a segmented piecewise nonlinear function (Sinha, Chakrabarti, & Chattopadhyay, 2003) rather than a single quadratic function. Moreover, many generating units, specifically those which are supplied with multi-fuel source lead to the problem of determining the most economic fuel to burn (Lio & Cai, 2005). Solution spaces of these problems have multiple minima due to valve-point effects and multi-fuel options.
Recently several stochastic techniques are proposed to solve ED problem (Altun and Yalcinoz, 2008, Kuo, 2008, Yuan et al., 2009). Some methods such as hybrid differential evolution (HDE) (Yuan et al., 2009), genetic algorithm (GA) (Chiang, 2007, Ling and Leung, 2007), modified PSO (MPSO) with a dynamic search space reduction strategy (Park, Lee, Shin, & Lee, 2005), evolutionary strategy optimization (ESO) (Pereira-Neto, Unsihuay, & Saavedra, 2005), quantum evolutionary algorithm (Babu, Das, & Patvardhan, 2008), partition approach algorithm (PAA) (Lin, Gow, & Tsay, 2007), pattern search (PS) method (Al-Sumait, AL-Othman, & Sykulski, 2007), self-tuning hybrid differential evolutionary (ST-HDE) (Wang, Chiou, & Liu, 2007), hybrid genetic algorithm (HGA) (He, Wang, & Mao, 2008), hybrid bacterial foraging (BF) technique (Panigrahi & Pandi, 2008), self-organizing hierarchical particle swarm optimization (SOH-PSO) (Chaturvedi, Pandit, & Srivastava, 2008) and society-civilization algorithm (SCA) combined with particle swarm optimization (PSO) called CSO (Selvakumar & Thanushkodi, 2009) have been proposed to solve ED problem in light of the valve-point effects. Moreover, a number of heuristic techniques such as Taguchi method (TM) (Yuan et al., 2009), Hopfield neural network (HNN) (Park, Kim, Eom, & Lee, 1993), adaptive Hopfield neural network (AHNN) (Lee, Sode-Yome, & Park, 1998), and evolutionary programming (EP) (Sadasivam & Sadasivam, 2000) have been applied to solve ED problem with the consideration of multiple fuel source options. Recently, a few modern approaches such as improved genetic algorithm with multiplier updating (IGA-MU) (Chiang, 2005), new particle swarm optimization (NPSO) with local random search (NPSO-LRS) (Selvakumar & Thanushkodi, 2007), anti-predatory particle swarm optimization (APSO) (Selvakumar & Thanushkodi, 2008), and various evolutionary algorithms (EA) (Manoharan, Kannan, Baskar, & Iruthayarajan, 2008) proposed to solve ED considering both valve loading effects and multi-fuel options together.
To obtain an accurate and practical economic dispatch solution, the realistic operation of the ED problem should take both valve-point effects and multiple fuels into account, which usually are found in practical power systems simultaneously. To solve this ED problem, a new Adaptive Real Coded Genetic Algorithm (ARCGA) with new genetic operators including arithmetic-average-bound crossover (AABX) and B-Spline wavelet mutation (BWM) is proposed. The proposed ARCGA is executed adaptively and can provide a more diverse search of the solution space. So, better optimum solutions with lower computation burden can be found compared with the previous stochastic search techniques proposed to solve the ED problem.
The remaining parts of the paper are organized as follows. In Section 2, the formulation of the practical ED problem including valve loading effect and multiple fuel option is presented. The proposed solution method, i.e. ARCGA, is introduced in the third section. Application of the proposed ARCGA to solve the ED problem is described in Section 4. Obtained numerical results are presented and discussed in Section 5. Section 6 concludes the paper.
Section snippets
Problem formulation
ED problem can be formulated as follows (Wood & Wollenberg, 1996):where is the total generation cost ($/hr), n is the number of dispatchable units, is the generation output of ith dispatchable unit, and is the fuel cost function of ith dispatchable unit ($/hr).
The objective function of (1) is subject to power balance and generating capacity constraints:where is the total active power demand, is the total
Proposed Adaptive Real Coded Genetic Algorithm solution
GAs are search and optimization procedures that inspired by the natural genetics. GA begins with a population of randomly generated chromosomes, which each one represents a potential solution for the optimization problem. The evolution of a GA population is generally based on the selection of the parents according to their fitness values, generation of the offspring chromosomes (e.g. by crossover and mutation operators) and survival of the fittest. For more details about GAs, the interested
Application of the proposed ARCGA to solve the ED problem
Application of proposed ARCGA to solve the ED problem can be summarized as the following step by step algorithm:
- (1)
At first the initial population of the ARCGA is randomly produced where each gene of the chromosomes represents output of a dispatchable generating unit. In the production of the initial population, the output of (n − 1) dispatchable units can be chosen arbitrary within their respective generating capacity constraints (3) while the output of reference unit is constrained by the system
Numerical results and analysis
The proposed algorithm was implemented in the MATLAB software package on a simple Pentium IV personal computer with 1.8 GHz CPU with 256 MB RAM. Owing to the randomness of the stochastic algorithms, their performance cannot be judged by the result of a single run. Many trials with independent population initializations should be made to acquire a useful conclusion of the performance of the algorithm. In this paper, the statistical indices such as the best, mean and worst results (generation
Conclusion
In this paper, a new stochastic search technique is proposed to solve the nonconvex and nonsmooth ED problem including both valve loading effects and multiple fuel source options of units. The proposed ARCGA has new genetic operators including AABX crossover and BWM mutation. By the proposed operators, the ARCGA can benefit from both global and local search abilities. Also, to enhance the convergence behavior of the ARCGA, an adaptation process is incorporated in its evolution. To show the
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