Solving Economic Load Dispatch Problems in Power Systems using Genetic Algorithm and Particle Swarm Optimization

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1 Solving Economic Load Dispatch Problems in Power Systems using Genetic Algorithm and Particle Swarm Optimization Loveleen Kaur 1, Aashish Ranjan 2, S.Chatterji 3, and Amod Kumar 4 1 Asst. Professor, PEC University of Technology, Chandigarh, India 2 Student, PEC University of Technology, Chandigarh, India 3 Director, UCER, Gr Noida, Uttar Pradesh, India 4 Chief Scientist, CSIO, Chandigarh, India Abstract In this paper, comparative study of two methods, Particle Swarm Optimization (PSO) and Genetic Algorithm (GA) is used to solve Economic dispatch problem in power systems. The feasibility of both the methods is demonstrated for a six-generator system and a fifteen- generator system. The results from the experiments show that the PSO method gives a better quality solution to the Economic Dispatch problem than the Genetic Algorithm technique. Index Terms Power systems, Particle Swarm Optimization, Genetic Algorithm, Economic Dispatch. I. INTRODUCTION ECONOMIC Dispatch (ED) problem is one of the fundamental issues in the operation of Power Systems. In an interconnected Power System, the objective is to minimize the operating cost of the system without violating the system constraints. This means that the active and reactive power of the individual units is varied in such a manner that the total fuel cost of the system is minimum while meeting the total load demand on the system. This is known as the Optimal Power Flow (OPF) problem. In order to find the optimal solution of the OPF problem, the selected objective function is optimized (maximized or minimized). The objective function could be of different forms like fuel cost, transmission cost, and the system security. Usually, the total production cost is taken as an objective function. For the simplification of an OPF problem, the transmission losses are not considered. In actual practice, the transmission losses are taken into consideration. But, this makes the ED problem more complicated and a different solution procedure needs to be used to find the solution. Several factors like operating efficiencies of generators, fuel cost, and transmission losses, influence power generation at minimum cost. It is not necessary that the generator with the highest efficiency guarantees least cost of operation as it may be located at a place where the cost of the fuel is high. Likewise, if the load center is far-off from the plant, transmission losses would be quite high and hence the cost of operation would be high. Hence, the ED problem becomes the determination of the generation of individual plants such that the total cost of operation is least. There are a number of mathematical programming methods and optimization problem techniques for the determination of an optimal solution of an Economic Dispatch problem [1], [2], [18], [19]. Some of the Artificial Intelligence techniques like Hopfield Neural Networks are used to solve economic dispatch problem with transmission capacity constraints [3], [4]. Genetic Algorithms are adaptive metaheuristic search algorithm. Its source of inspiration is the process of natural selection, according to which, the competition among the individuals over limited resources results in fitter ones dominating over the weaker ones. It is based on bio-inspired operators like mutation, crossover, and selection. As compared to another popular method, called Simulated Annealing (SA), GA gives faster results due to its parallel search techniques. In GAs, a population of points at each iteration is generated. Among all these points, the best point converges towards the optimal solution. GA has been used for various types of optimization problems of power systems like feeder reconfiguration and placement of capacitors in the distribution system [1], [5]-[7]. 279 Loveleen Kaur, Aashish Ranjan, S.Chatterji, and Amod Kumar

2 In [5], Walters and Sheble make use of a GA-based algorithm to solve an ED problem for valve point discontinuities. Yalcinoz et al. use a neural network based approach for solving the ED problem under constrained transmission conditions [4]. Chen and Chang present a GA based approach for solving the ED problem in a large-scale system [8]. Through numerical results, they show that their proposed method is faster than the lambda-iteration method in a large-scale system. Fung et al. make use of Simulated Annealing (SA) and Tabu Search (TS) techniques in an integrated parallel Genetic Algorithm to solve the ED problem [9]. Yalcinoz et al. propose a new genetic approach based on arithmetic crossover for solving the problem of economic dispatch. They use elitism, arithmetic crossover and mutation in GA to find the solution of the ED problem more efficiently [10]. Although GA is widely used for solving the optimization problems, it suffers from various deficiencies. In GA, the crossover and mutation operations do not promise better fitness of successive population [11], [16]. Also, the performance of GA is further degraded by its premature convergence. Particle Swarm Optimization (PSO) is a computational technique which is used for solving the optimization problem. It was developed by Eberhart and Kennedy [17], [18]. This method is quite powerful in solving continuous nonlinear optimization problems [13]-[16], [19]. It generates solutions of higher quality with stable convergence characteristics. In [20], the authors present a PSO for the control of voltage and reactive power. Their method takes care of the voltage security by the use of continuation power flow and a contingency analysis technique. Neka et al. present the application of a hybrid PSO method for the solution of a practical distribution state estimation problem [21]. II. OVERVIEW OF PARTICLE SWARM OPTIMIZATION Particle Swarm optimization (PSO) is originally attributed to Kennedy and Eberhart. Its source of inspiration was the social behavior of a bird flock. In a swarm of birds looking for food, the birds adjust their position as per its own experience and the experience of its neighboring birds. This simulation is used for optimization of nonlinear multi-dimensional functions. Let x denotes the position of a particle and v denotes its flight velocity. In a d- dimensional space, pbest and gbest denote the previous best position of the particle and the best position obtained by all particles so far, respectively. The updated position and velocity of each particle can be calculated using its current velocity and its distance from the gbest as: New Velocity, v i (k+1) = w * v i (k) + C 1 * (pbest i x i (k)) + C 2 * (gbest x i (k)) New Position, x i (k + 1) = x i (k) + v i (k + 1) i particle index k discrete time index w inertia weight factor v i velocity of i th particle x i position of i th particle pbest i best position found by i th particle (personal best) gbest best position found by swarm (global best, best of personal bests) C 1, C 2 acceleration constant. With the addition of w, known as inertia weight factor, the PSO is modified. This inertia weight factor is added to provide the balance between global and local explorations. It helps in reducing the number of iterations. The constants, C 1 and C 2, known as acceleration constants, pull each particle towards the particle and global best positions. Lower values of C 1 and C 2 let the particles roam away from the target region while the higher values leads to their abrupt movement towards the target region. Typically, both C 1 and C 2 are given a value of 2. III. OVERVIEW OF GENETIC ALGORITHM Genetic Algorithm (GA) is an adaptive heuristic search algorithm. It is based on the evolutionary ideas of natural selection and genetics. According to the concept of natural selection, given by Charles Darwin, the competition among the individuals for limited resources leads to the domination of the fittest individuals over the weaker ones. GA is used for solving both constrained and unconstrained optimization problems. In GA, from a population of candidate solutions, individuals are 280 Loveleen Kaur, Aashish Ranjan, S.Chatterji, and Amod Kumar

3 randomly selected to be parents and they are used to produce children for next generation. Only the fittest among all the individuals will survive and reproduce, producing the next generation. In this way, the future generations will keep getting better than the previous generations. So, in an optimization problem, the objective function improves with each iteration. The nature of the problem decides the population size. Usually, the initial population is randomly generated, allowing the entire range of possible solutions. During each generation, individual solutions are chosen through a fitness-based process. In this process, the fitter solutions have a higher probability of getting selected (by the fitness function). The fitness function is problem dependent. It measures the quality of the represented solution. Sometimes, in some cases, it is difficult or impossible to define the fitness function. In that case, we can use simulation to determine the fitness function. After the selection of fitter solutions, a second generation population of solutions is generated. The second generation consists of three types of children: elite children, crossover children, and mutation children. Elite children are the ones with the best fitness level. Crossover children (solution) are produced by taking more than one parent solution. The mutation children (solution) are produced by changing one or more genes of a single parent. By producing a child solution using crossover and mutation, the new solution generated has several characteristics of its parents. Then again new parents are selected for producing new children and the process continues until a termination condition is reached. Some of the common terminating conditions are: 1) a solution satisfying minimum criteria is found 2) fixed number of generations reached 3) allocated computation time or money is reached 4) highest ranking solution s fitness is reached 5) A combination of above. If the stopping criteria are modified, the solution may change as per the new conditions. obtained are discussed and compared. Two different systems, having six and fifteen generators, are taken as the test system. For simplicity, the losses, ramp rate limits and the prohibited zones are not considered. Under the same evaluation function and individual definition, we perform 20 trials using the two methods and the quality of the solution is observed. 1) PSO Method: Population size = 100 Generations = 200 Inertia weight factor: w max = 0.9 and w min = 0.4 Acceleration constants: C 1 =2 and C 2 =2. 2) GA Method Population size = 100 Generations = 200 Crossover rate, P c =0.8 Mute rate, P m = 0.01 Case Study Example 1: Six-Unit system: The system consists of six generators and the total load demand on the system is 1263 MW. The characteristics of the six thermal units are given in table Ⅰ. The power outputs of the six generators are represented by P 1, P 2, P 3, P 4, P 5 and P 6. These values are randomly generated. The best solution obtained by applying these methods is shown in table Ⅱ. TABLE 1 GENERATING UNIT CAPACITY AND COEFFICIENTS IV. NUMERICAL EXAMPLES AND RESULTS The PSO and the GA method are used to solve an Economic Dispatch(ED) problem and the results 281 Loveleen Kaur, Aashish Ranjan, S.Chatterji, and Amod Kumar

4 TABLE II BEST SOLUTION OF 6-UNIT SYSTEM (BEST INDIVIDUAL) Unit Power PSO method GA method Output P 1 (MW) P 2 (MW) P 3 (MW) P 4 (MW) P 5 (MW) P 6 (MW) Total Power output (MW) Total Generation cost ($/hr) 15,276 15,276 TABLE Ⅲ COMPARISON BETWEEN BOTH METHODS (20 TRIALS) Generation cost ($) Max Min PSO GA Example 2: Fifteen-Unit System: The system consists of fifteen generators and the total load demand on the system is 2630 MW. The characteristics of the fifteen generators are given in table Ⅳ. The experimental results are shown in Tables V and VI. These results are found to satisfy the system constraints. TABLE Ⅳ GENERATING UNIT DATA TABLE Ⅴ BEST SOLUTION OF 15-UNIT SYSTEM (BEST INDIVIDUAL) Unit Power Output PSO method GA method P 1 (MW) P 2 (MW) P 3 (MW) P 4 (MW) P 5 (MW) P 6 (MW) P 7 (MW) P 8 (MW) P 9 (MW) P 10 (MW) P 11 (MW) P 12 (MW) P 13 (MW) P 14 (MW) P 15 (MW) Total Power output (MW) Total Generation cost($/hr) ,604 32,844 TABLE Ⅵ COMPARISON BETWEEN BOTH METHODS (20 TRIALS) Generation cost ($) Max Min PSO GA Loveleen Kaur, Aashish Ranjan, S.Chatterji, and Amod Kumar V. DISCUSSION AND CONCLUSION In this paper, Particle Swarm Optimization (PSO) and Genetic Algorithm (GA) have been successfully implemented to solve the Economic Dispatch problem with generator constraints. For simplicity, some of the nonlinear characteristics of the generator like ramp rate limits and valve point zones have not been considered. For the testing of the above two methods, six generator and a fifteen

5 generator systems have been used. The results obtained show that PSO gives higher quality solution than the GA. REFERENCES [1] A. Bakirtzis, V. Petridis, and S. Kazarlis, Genetic algorithm solution to the economic dispatch problem, Proc. Inst. Elect. Eng. Gen., Transm. Dist., vol. 141, no. 4, pp , July [2] F. N. Lee and A. M. Breipohl, Reserve constrained economic dispatch with prohibited operating zones, IEEE Trans. Power Syst., vol. 8, pp , Feb [3] C.-T. Su and G.-J. Chiou, A fast-computation hopfield method to eco-nomic dispatch of power systems, IEEE Trans. Power Syst., vol. 12, pp , Nov [4] T. Yalcinoz and M. J. Short, Neural networks approach for solving economic dispatch problem with transmission capacity constraints, IEEE Trans. Power Syst., vol. 13, pp , May [5] D. C. Walters and G. B. Sheble, Genetic algorithm solution of economic dispatch with valve point loading, IEEE Trans. Power Syst., vol. 8, pp , Aug [6] K. P. Wong and Y. W. Wong, Genetic and genetic/simulated Annealing approaches to economic dispatch, Proc. Inst. Elect. Eng., pt. C, vol. 141, no. 5, pp , Sept [7] G. B. Sheble and K. Brittig, Refined genetic algorithm Economic dispatch example, IEEE Trans. Power Syst., vol. 10, pp , Feb [8] P.-H. Chen and H.-C. Chang, "Large-Scale economic dispatch by genetic algorithm," IEEE Trans. Power Syst., vol. 10, pp , Nov [9] C. C. Fung, S. Y. Chow, and K. P. Wong, Solving the economic dis-patch problem with an integrated parallel genetic algorithm, in Proc. PowerCon Int. Conf., vol. 3, 2000, pp [10] T. Yalcionoz, H. Altun, and M. Uzam, Economic dispatch solution using a genetic algorithm based on arithmetic crossover, in Proc. IEEE Proto Power Tech. Conf., Proto, Portugal, Sept [11] D. B. Fogel, Evolutionary Computation: Toward a New Philosophy of Machine Intelligence, 2 ed. Piscataway, NJ: IEEE Press, [12] K. S. Swarup and S. Yamashiro, Unit commitment solution method-ology using genetic algorithm, IEEE Trans. Power Syst., vol. 17, pp , Feb [13] J. Kennedy and R. Eberhart, Particle swarm optimization, Proc. IEEE Int. Conf. Neural Networks, vol. IV, pp , [14] Y. Shi and R. Eberhart, A modified particle swarm optimizer, Proc. IEEE Int. Conf. Evol. Comput., pp , May [15] Y. Shi and R. C. Eberhart, "Empirical study of particle swarm optimization," in Proc. Congr. Evol. Comput., NJ, 1999, pp [16] R. C. Eberhart and Y. Shi, Comparison between genetic algorithms and particle swarm optimization, Proc. IEEE Int. Conf. Evol. Comput., pp , May [17] Eberhart, R. C., Dobbins, R. W., and Simpson, P. K. (1996), Computational Intelligence PC Tools, Boston: Academic Press. [18] Eberhart, R. C., and Kennedy, J. (1995), A new optimizer using particle swarm theory, Proc. Sixth International Symposium on Micro Machine and Human Science (Nagoya, Japan), IEEE Service Center, Piscataway, NJ, [19] P. J. Angeline, "Using selection to improve particle swarm optimization," Proc. IEEE Int. Conf. Evol. Comput., pp , May [20] 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 , Nov [21] S. Naka, T. Genji, T. Yura, and Y. Fukuyama, Practical distribution state estimation using hybrid particle swarm optimization, Proc. IEEE Power Eng. Soc. Winter Meeting, vol. 2, pp , [22] H. Saadat, Power System Analysis. New York: McGraw-Hill, Loveleen Kaur, Aashish Ranjan, S.Chatterji, and Amod Kumar