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IPASJ International Journal of Electrical Engineering (IIJEE) Web Site: http://www.ipasj.org/IIJEE/IIJEE.htm A Publisher for Research Motivation........ Email: editoriijee@ipasj.org Volume 1, Issue 6, December 2013 ISSN 2321-600X A Thermal Economic Dispatch Problem Solving By Particle Swarm Optimization Manjeet Singh1, Divesh Thareja2 1 Department of Electrical and Electronics Engineering, Assistant Professor, HCTM Technical Campus, Kaithal, India. 2 Department of Electrical Engineering, Assistant Professor, BGIET, Sangrur, India. ABSTRACT The primary objective of the ELD is to minimize the total cost of generation while maintaining the operational costs of the available generation resources. This paper consist a new approach to solve thermal units economic dispatch (ED) problems. Economic dispatch is a highly constrained optimization problem in power system encompassing interaction among decision variables. The particle swarm optimization (PSO) technique is demonstrated. This paper presents a Particle Swarm Optimization (PSO) based solution for optimal flow with generating units having fuel costs curves while satisfying the constraints.PSO has been examined and tested for standard 10 generating units. Index Terms: Particle swarm optimization, non-smooth fuel cost functions, valve point effects. 1. INTRODUCTION Economic dispatch is generation allocation problem and defined as the process of calculating the generation of the generating units so that the system load is meet while satisfying all the constraints. Historically economic dispatch is being carried out since 1920. It was the time when engineers were concerned with the problem of economic allocation of generation or the proper division of the load among the generating units available. The methods for solving this kind of problem include traditional operational research algorithms (such as linear programming, quadratic programming, dynamic programming, gradient methods and Lagrange relaxation approaches) and modern methods (such as simulated annealing and evolutionary algorithms). Some of these methods are successful in locating the optimal solution, but they are usually slow in convergence and require heavy computational cost [2]. Economic load dispatch problem plays an important role in the operation of power systems. It is a method to determine the most efficient, low cost and reliable operation of a power system by dispatching the available electricity generation resources to supply the load on the system. The primary objective of the ELD is to minimize the total cost of generation while maintaining the operational costs of the available generation resources. The economic load dispatch is very important optimized problem solution in power system operations for allocating generation among the committed units such that the system constraints imposed are satisfied and energy requirements in terms of Rupees per hour ( Rs/hr) or Dollars per hour($/hr)are minimized.Eberhart and Kennedy suggested a particle swarm optimization (PSO) based on the analogy of swarm of bird and school of fish [4]. The PSO mimics the behaviour of individuals in a swarm to maximize the survival of the species. In PSO, each individual makes his decision using his own experience together with other individuals’ experiences [3]. Partical swarm optimization (PSO) is one of the modern heuristic algorithms, which can be used to solve non linear optimization problems [3]. It is a population-based search algorithm and searches in parallel using a group of particles similar to Genetic Algorithms (GA). The original PSO suggested by Kennedy and Eberhart is based on the analogy of swarm of bird and school of fish [4]. Each particle in PSO makes its decision using its own experience and its neighbor’s experiences for evolution. That is, particles approach to the optimum through its present velocity, previous experience, and the best experience of its neighbors [2].In this paper, an alternative approach is proposed to the non-smooth ED problem using a PSO which focuses on equality and inequality constraints when modifying each individual’s search. The equality constraint (i.e., the supply/demand balance) is easily satisfied by specifying a variable (i.e., a generator output) at random in each iteration as a slag generator whose value is determined by the difference between the total system demand and the total generation excluding the slag generator. However, the inequality constraints in the next position of an individual produced by the PSO algorithm can violate the inequality constraints. In this case, the position of any individual violating the constraints is set to its minimum or maximum position depending on the velocity evaluated. Volume 1, Issue 6, December 2013 Page 5 IPASJ International Journal of Electrical Engineering (IIJEE) Web Site: http://www.ipasj.org/IIJEE/IIJEE.htm A Publisher for Research Motivation........ Email: editoriijee@ipasj.org Volume 1, Issue 6, December 2013 ISSN 2321-600X The main objective of this study is to introduce the use of Particle Swarm Optimization (PSO) technique to the subject of power system economic load dispatch. In this paper, the PSO method has been employed to solve economic dispatch problem with a valve point effects. 2. PROBLEM FORMULATION 2.1Objective Function. The objective of the economic dispatch problem is to minimize the total fuel cost of thermal power plants subjected to the operating constraints of a power system. The simplified cost function of each generator can be represented as a quadratic function as described in (1). = ) (1) = + + (2) Where, Total generation cost. Cost function of generator i. , ,cost coefficients of generator i. Power of generator i. N number of generators. 2.2 Cost Function Considering Valve-Point Effects. The generator with multi-valve steam turbines has very different input-output curve compared with the smooth cost function. Typically, the valve point results in, as each steam valve starts to open, the ripples like in Fig. 1. To take account for the valve-point effects, sinusoidal functions are added to the quadratic cost functions as follows: (3) Where and are the coefficients of unit i reflecting valve point effects. Figure1.Fuel cost versus power output for four valve steam turbine unit. 2.3 System Constraints. 2.3.1 Equality /Active power balance equation. For power balance, an equality constraint should be satisfied. The total generated power should be the same as total load demand plus the total line loss. = + (4) Where the total system demand and is the total line loss. 2.3.2 Inequality Constraints. Generation output of each generator should be laid between maximum and minimum limits. The corresponding inequality constraints for each generator are: Volume 1, Issue 6, December 2013 Page 6 IPASJ International Journal of Electrical Engineering (IIJEE) Web Site: http://www.ipasj.org/IIJEE/IIJEE.htm A Publisher for Research Motivation........ Email: editoriijee@ipasj.org Volume 1, Issue 6, December 2013 ISSN 2321-600X (5) Where and are the minimum and maximum output of generator i, respectively. 3. IMPLEMENTATION OF PSO FOR ED PROBLEMS 3.1. Particle Swarm Optimization. Particle swarm optimization (PSO) is a population-based optimization method first proposed by Kennedy and Eberhart in 1995, inspired by social behaviour of bird flocking or fish schooling. The PSO as an optimization tool provides a population-based search procedure in which individuals called particles change their position (state) with time. In a PSO system, particles fly around in a multidimensional search space. During flight, each particle adjusts its position according to its own experience (This value is called pbest), and according to the experience of a neighboring particle (This value is called gbest), made use of the best position encountered by itself and its neighbor (Fig. 2). Figure2. Concept of a searching point by PSO. This modification can be represented by the concept of velocity. Velocity of each agent can be modified by the following equation: (6) Using the above equation, a certain velocity, which gradually gets close to pbest and gbest can be calculated .The current position (searching point in the solution space) can be modified by the following equation. (7) NP is the number of particles in a group. NG is the number of members in particles. is the pointer of iterations (generations). is the inertia weight factor C1 and C2 are the acceleration constant. R1 and R2 are uniform random values in the range [0, 1]. is the velocity of member of particle at iteration, is the current position of member of particle at iteration. Suitable selection of inertia weight w provides a balance between global and local explorations,thus requiring less iteration on average to find a sufficiently optimal solution. As originally developed, often decreases linearly from about 0.9 to 0.4 during a run. In general, the inertia weight is set according to the following equation [4]. (8) Where, Volume 1, Issue 6, December 2013 Page 7 IPASJ International Journal of Electrical Engineering (IIJEE) Web Site: http://www.ipasj.org/IIJEE/IIJEE.htm A Publisher for Research Motivation........ Email: editoriijee@ipasj.org Volume 1, Issue 6, December 2013 ISSN 2321-600X is the maximum number of iterations (generations). is the current number of iterations. 4. SIMULATION RESULTS AND DISCUSSION The proposed PSO algorithm is tested on standard 10 generating units. The test system consists of 10 thermal units without losses (Table1), 24 hour load. The optimal setting of the PSO control parameters are: c1=2, c2=2, numbers of particles is 10 and number of iterations is 100. The Inertia weight was kept between 0.4 and 0.9 4.1The OPF with quadratic fuel cost functions. The proposed PSO is applied to standard 10 generating units. The obtained results using PSO are given in Tables 1. Fig.3. shows the cost convergence of PSO for various numbers of generations. It was clearly shown that there is no rapid change in the fuel cost function value after 40 iterations. Hence it is clears from the Fig3. Table.1 Ten-unit generator characteristics. Hours P(MW) Hours P(MW) Hours P(MW) Hours P(MW) 1 1036 7 1702 13 2072 19 1776 2 1110 8 1776 14 1924 20 2072 3 1258 9 1924 15 1776 21 1924 4 1406 10 2072 16 1554 22 1628 5 1480 11 2146 17 1480 23 1332 6 1628 12 2220 18 1628 24 1184 Total 2114300 cost($/hr) CPT(sec) 7.691574 6 x 10 2.24 2.22 2.2 2.18 Cost($/hr.) 2.16 2.14 2.12 2.1 0 10 20 30 40 50 60 70 80 90 100 No of Iterations. Figure3. Convergence characteristic of the 10 generating units Volume 1, Issue 6, December 2013 Page 8 IPASJ International Journal of Electrical Engineering (IIJEE) Web Site: http://www.ipasj.org/IIJEE/IIJEE.htm A Publisher for Research Motivation........ Email: editoriijee@ipasj.org Volume 1, Issue 6, December 2013 ISSN 2321-600X 4 x 10 13 12 11 10 Cost($/hr.) 9 8 7 6 5 0 5 10 15 20 25 No of Hours. Figure4.Cost-Load curve for Test problem. 5. CONCLUSION The proposed algorithm has been tested on the 10 generating units. Particle Swarm Optimization has implemented on ten generating units having twenty four hours load with non-smooth fuel cost functions. It was clearly shown that there is no rapid change in the fuel cost function value after 40 iterations The PSO is successfully and effectively used for minimization the overall cost functions. REFERENCE [1] A. J. Wood, and B. F. Wollenbergy, “Power Generation, Operation, and Control”, New York, NY, John Wiley & Sons, Inc., 1984. [2] J. Kennedy and R. C. Eberhart, “Swarm Intelligence, San Francisco”, CA, Morgan Kaufmann Publishers, 2001. [3] K. Y. Lee and M. A. El-Sharkawi (Editors), “Modern Heuristic Optimization Techniques with Applications to Power Systems”, IEEE Power Engineering Society (02TP160), 2002. [4] J. Kennedy and R. C. Eberhart, “Particle swarm optimization,” Proceedings of IEEE International Conference on Neural Networks (ICNN’95), Vol. IV, pp. 1942-1948, Perth, Australia, 1995. [5]H. Yoshida, K. Kawata, Y. Fukuyama, S. Takayama, and Y. Nakanishi, “A particle swarm optimization for reactive power and voltage control considering voltage security assessment”, IEEE Trans. Power Syst., vol. 15, pp. 1232–1239, Nov. 2000. [6] M. Clerc and J. Kennedy, “The particle swarm-explosion, stability, and convergence in a multidimensional Complex space”, IEEE Trans. Evol.Comput., vol. 6, no. 1, pp. 58–73, Feb. 2002. [7] M. A. Abido, “Optimal design of power-system stabilizers using particle swarm optimization” , IEEE Trans. Energy Conv., vol. 17, no. 3, pp. 406–413, Sept. 2002. [8] S. Pothiya, I. Ngamroo, W. Kongprawechnon, "Application of multiple tabu search algorithm to solve dynamic economic dispatch considering generator constraints", in 1November 2007. [9] Z. X. Liang and J. D. Glover, “A zoom feature for a dynamic programming solution to economic dispatch including transmission losses”, IEEE Trans. on Power Systems, Vol. 7. No. 2, pp. 544-550, May 1992. Volume 1, Issue 6, December 2013 Page 9

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IPASJ International Journal of Electrical Engineering (IIJEE)
Web Site: http://www.ipasj.org/IIJEE/IIJEE.htm
A Publisher for Research Motivation........ Email: editoriijee@ipasj.org
Volume 1, Issue 6, December 2013 ISSN 2321-600X

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