تکامل دیفرانسیلی صحیح ترکیبی کدگذاری شده - روش برنامه ریزی پویا برای توزیع بار اقتصادی با گزینه های سوخت چندگانه

This paper presents a novel and efficient approach through a hybrid integer coded differential evolution – dynamic programming (ICDEDP) scheme to solve the economic dispatch (ED) problem with multiple fuel options. A dynamic programming (DP) based simplified recursive algorithm is developed for optimal scheduling of the generating units in the ED problem. The proposed hybrid scheme is developed in such a way that an integer coded differential evolution (ICDE) is acting as a main optimizer to identify the optimal fuel options, and the DP is used to find the fitness of each agent in the population of the ICDE, which makes a quick decision to direct the search towards the optimal region. The hybrid ICDEDP decision vector consists of a sequence of integer numbers representing the fuel options of each unit to optimize quality of search and computation time. A gene swap operator is introduced in the proposed algorithm to improve its convergence characteristics. In order to show the efficiency and effectiveness, the proposed hybrid ICDEDP approach has been examined and tested with numerical results using the ten generation unit economic dispatch problem with multiple fuel options. The test result shows that the proposed hybrid ICDEDP algorithm has high quality solution, superior convergence characteristics and shorter computation time.

The main objective of the economic dispatch problem is to determine the optimal combination of power outputs for all generating units that minimizes the total fuel cost while satisfying load demand and operating constraints. This makes the ED problem a large scale nonlinear constrained optimization problem. Conventional techniques offer good results, but when the search space is nonlinear and has discontinuities, they become very complicated with a slow convergence ratio and do not always seek the optimal solution. New numerical methods are needed to cope with these difficulties, especially those with high speed search for the optimal and not being trapped in local minima. The stochastic search algorithms such as genetic algorithm (GA) [1], evolutionary programming (EP) [2] and [3], simulated annealing (SA) [4] and PSO [5] and [6], may prove to be very effective in solving nonlinear ED problems without any restriction on the shape of the cost curves. Although these heuristic methods do not always guarantee discovering the globally optimal solution in finite time, they often provide a fast and reasonable solution (sub-optimal, nearly global optimal). SA is applied in many power system problems, but setting the control parameters of the SA algorithm is a difficult task, and the convergence speed is slow when applied to a real system [7]. Though GA methods have been employed successfully to solve complex optimization problems, recent research has identified some deficiencies in GA performance. This degradation in efficiency is apparent in applications with highly epistatic objective functions. Moreover, the premature convergence of GA degrades its performance and reduces its search capability that leads to a higher probability of obtaining a local optimum [8]. EP seems to be a good method to solve optimization problems. When applied to problems consisting of more numbers of local optima, the solutions obtained from the EP method is just near the global optimum one. Also, the GA and EP take long simulation times in order to obtain a solution for such problems. Therefore, hybrid methods combining two or more optimization methods were introduced [9] and [10]. The generation cost function for fossil fired plants can be represented as a segmented piecewise quadratic function. The generating units, particularly those that can be supplied with multi-fuel sources (coal, natural gas, or oil), lead to the problem of determining the most economic fuel to burn. Lin and Vivani [11] have discussed such a problem using the hierarchical approach of Lagrangian multipliers (λ) method to find the incremental fuel cost for subsystems comprising groups of units. The solution searches for the optimal λ for various choices of fuel and the generation range of the units iteratively. Park et al. [12] proposed to apply a Hopfield neural network (HNN) to the ED problem for a piecewise quadratic cost function. Lee et al. [13] presented an improved adaptive Hopfield neural network approach for finding the solution for ED with multiple fuel options. It is well known that the HNN converges very slowly and normally takes a large number of iterations. Baskar et al. [14] discussed a hybrid real coded GA method for solving the ED problem with multiple fuel options. In this paper, a hybrid ICDEDP algorithm that combines a heuristic optimization technique and a mathematical algorithm is proposed to solve the ED problem with multiple fuel options. Differential evolution (DE), developed by Storn and Price, is one of the most excellent evolutionary algorithms. DE is a robust statistical method for cost minimization, which does not make use of a single nominal parameter vector but instead uses a population of equally important vectors. The fittest of an offspring competes one to one with of the corresponding parent, which is different from the other evolutionary algorithms [15]. By using a simple and direct evolution process, the convergence speed of DE becomes very fast. However, the faster convergence may lead to a higher probability of obtaining a local optimum. Generally, this drawback can be overcome by using a large population size but which leads to increased computation time in evaluation of the fitness function. In order to overcome this drawback a hybrid method is presented to solve the ED problem with multiple fuel options by integrating the integer coded DE with the dynamic programming (DP) method. A DP [16] based recursive algorithm that minimizes fuel cost has been developed in a simple form to evaluate the fitness of each individual in the population of the hybrid ICDEDP algorithm. The main advantages of the proposed method for solving the ED problem with multiple fuel options are its simple concept, great robustness, high quality solution and better computation efficiency.

A new hybrid technique based on an integer coded differential evolution and dynamic programming approach for solving the economic dispatch problem with multiple fuel options is presented in this paper. The developed DP based recursive algorithm for the ED problem is a non-iterative direct method, which gives the optimal generation dispatch to minimize the total fuel cost for the given choice of fuel options of each unit in a power system. The proposed algorithm has been tested on a ten units system with multiple fuel options to demonstrate its effectiveness and robustness. The ICDE is applied as a base level search to direct the search towards the optimal fuel options, and the DP based recursive approach is used as fine tuning to determine the optimal generation schedule and, thereby, the fitness function of each decision vector in the hybrid ICDEDP algorithm. The developed hybrid ICDEDP algorithm narrows the solution space, which accelerates the convergence of the algorithm. The comparison of the results with those of other methods reported in the literature shows the superiority of the proposed method and its potential for solving the ED problem with multiple fuel options.