A Fast Computational Genetic Algorithm for Economic Load Dispatch

Transcription

1 A Fast Computational Genetic Algorithm for Economic Load Dispatch M.Sailaja Kumari 1, M.Sydulu Sailaja_matam@Yahoo.com 1, 2 Department of Electrical Engineering National Institute of Technology, Warangal Andhra Pradesh, India, 2 sydulumaheswarapu@yahoo.co.in Abstract This paper presents a Fast Genetic Algorithm (FGA) approach for solving Economic Load Dispatch (ELD) problem. GAs perform powerful global searches, but their long computation times limit them when solving large scale optimization problems. The present paper describes a method to overcome this limitation by starting with random solutions within the search space and narrowing down the search space by considering the minimum and maximum errors of the population members. Since the search space is restricted to a small region within the available search space, the algorithm works very fast. This feature of the algorithm is attractive when applied to ELD of large systems. The convergence of the algorithm can be expected with in one or two generations independent of size of the system. The results have been demonstrated for ELD of standard 3 generator, 6 generator, 20 generator and 38 generator systems with and without consideration of transmission system losses. In all the cases the Fast GA shows reliable convergence. The final results obtained using Fast GA are compared with conventional GA and found to be encouraging. Index Terms Economic dispatch, Genetic algorithms, Incremental fuel cost of generators, Optimal dispatch. G I. INTRODUCTION ENETIC Algorithms (GAs) are optimization algorithms based on the mechanics of natural selection and natural genetics. GAs have been developed by John Holland in 1960s. Some fundamental ideas have been borrowed from genetics and are artificially used to construct more robust algorithms requiring minimal problem information [1]. Unlike classical search and optimization methods GA starts its search with a random set of solutions, instead of single solution. The Holland s GA that uses simple genetic operators is known as Simple Genetic Algorithm (SGA). SGA starts with a random population of solutions (chromosomes in GA terminology). Each population member is then evaluated for the given objective function and is assigned fitness. The algorithm then checks for the stopping condition. The condition can be either all population members assume equal 349 fitness values or when a maximum number of generations are reached. If the condition is not satisfied, the population members are arranged in descending order of their fitness value and genetic operators are applied to produce new and better fit population from old population while applying genetic operators like recombination, elitism, crossover and mutation. This completes one iteration and in GA terminology one generation. The generation count is updated and the process is repeated. SGA is capable of locating the near-optimal solutions, but requires a large number of generations to converge. In addition to SGA, there exists a number of advanced GAs, designed to have advanced features. Suitable combination of GA operators also are used [2] to enhance the performance of SGA. Economic Load Dispatch (ELD) is one of the important optimization problems in modern Energy Management Systems (EMS). ELD determines the optimal real power settings of generating units in order to minimize total fuel cost of thermal plants. Various mathematical programming methods and optimization techniques have previously been applied for solution of ELD. These include Lambda iteration method, participation factors method and gradient methods. ELD problems in practice are usually hard for traditional mathematical programming methodologies because of the equality and inequality constraints. GAs have also been applied for solution of ELD. [3, 4] present ELD using GA. A unit based encoding scheme is used, which restricts the applicability of GA to large-scale systems. In unit based encoding the chromosome length increase with number of units in the system. [5] presented a lambda-based GA approach for solving ED problem. The solution time grows approximately linearly with problem size rather than geometrically. An Extended compact Genetic Algorithm (ECGA) and an adaptive discretization technique named split-on-demand (SOD) were proposed [6] to solve ELD problem. A repair operator is defined for making the

2 infeasible solutions to satisfy the equality constraint. [7] proposed a combined economic and emission dispatch by considering both the economy and emission objectives. A comparison of GA, Micro GA and Evolutionary Programming is brought out for different size of systems. [8] presents Economic power Dispatch using a hybrid technique, where GA parameters are controlled using Fuzzy logic technique. In the present paper a fast computation genetic algorithm is proposed based on lambda-based GA approach. The paper presents a methodology to overcome long computation times involved with simple Genetic Algorithms. The fast computation feature of the developed algorithm is advantageous and can be used in on line power system studies. II. ECONOMIC LOAD DISPATCH A. Problem Formulation The objective of Economic Load Dispatch (ELD) for power system consisting of thermal generating units is to find the optimal combination of power generations that minimizes the total generation cost while satisfying the specified equality and inequality constraints. The fuel cost function of the generator is represented as a quadratic function of generator active powers. The minimization function J can be obtained as sum of the fuel costs F i of all the generating units. Min J= Subjected to NG F (1) i i=1 power outputs must be equal to the total power demand by the load plus losses, must be satisfied. If transmission system losses are neglected, the equality constraint becomes, the sum of the power outputs must be equal to the total power demand by the load. Also, the power output of each unit must be greater than or equal to the minimum power permitted and less than or equal to the maximum power permitted on that unit. B. Transmission system Losses To achieve true ELD transmission system losses must be taken into account, as power generating units are spread over large areas. Using B-coefficients method, the network losses are expressed as a quadratic function of unit generations as NG NG loss Gi ij Gj i=1 j=1 P = P B P (5) In (5) B ij are called as B-coefficients or loss coefficients. III. Simple Genetic Algorithm (SGA) A. Representing a solution in GA To find the optimal decision variables, to optimize an objective function and to satisfy the constraints, the variables are represented in binary strings. Eqn.(6) gives a mapping function defined to [1] take care of variable bounds. In (6) li is the string length used to code i th variable, DV(s i ) is the min max decoded value of the string s i. X i and X i are lower and upper bounds of variable x i. x -x x=x + DV 2-1 max min min i i i i li (s i ) (6) NG P =P +P i=1 Gi D loss P P P (3) Gimin Gi Gimax In (1) fuel cost of generating unit is given by 2 F = a + bp + cp i i i i Gi Gi Where a i, b i, c i are cost coefficients of unit i, P Gi is real power generation of unit i. This is a constrained optimization problem, that may be solved using [9] advanced calculus methods that involve Legrange function. The necessary condition for the existence of minimum cost operating condition for thermal power system is that the incremental cost rates of all the units be equal to some undetermined value Lambda (λ). Along with the necessary condition, the equality constraint, the sum of the (2) (4) 350 Variable bounds are taken care by the mapping function given by (6). After choosing a string representation, a random population of solutions is generated. Then fitness is assigned to each population member, which represents the goodness of each solution. The string is evaluated in the context of objective function and constraints. In the absence of constraints, the objective function is treated as fitness function. In case of a minimization problem, the solution with smaller fitness is better. Genetic Operators: The SGA uses reproduction operator, elitism operator, crossover operator and mutation operator. B. Reproduction operator The main goal of reproduction operator is to make duplicates of the best fit solutions in the population and eliminate least fit solutions, while keeping the population size constant. Some commonly used methods include tournament selection, proportionate selection or Roulette Wheel Selection

3 (RWS), and ranking selection. In tournament selection tournaments are played between two solutions and the better solution is chosen and placed in the mating pool. Two other solutions are selected again randomly and another slot in the mating pool is filled with a better solution. Each solution can take part in the tournament twice. The process is repeated until the mating pool is filled with new good quality solutions. In proportionate selection, solutions are assigned copies proportional to their fitness values. It can be thought of as a roulette-wheel mechanism, where the wheel is divided into N (population size) divisions, where the size of each is marked in proportion to the fitness of each population member. Thereafter the wheel is spun N times, each time selecting the solution indicated by the pointer. In RWS a string with a higher fitness has a higher probability of being copied into the mating pool. In RWS the number of copies a population member gets in the next population is equal to f i /f avg. This selection operator is slow compared to tournament selection, as it merely calculates the average fitness all population members. Fitness scaling techniques are used to enhance the performance of RWS operator. Ranking selection ranks the population members according to their fitness, from worst (rank1) to best (rank N), where N is the population size. Each member in the sorted list is assigned fitness equal to the rank of the solution in the list. Thereafter the proportional selection operator is applied with the ranked fitness values and N solutions are chosen from mating pool. C. Crossover operator The reproduction operator makes copies of the best fit solutions in the next generation, but it cannot create any diversity in the population members. It only makes more copies of best fit solutions at the cost of least fit solutions. Crossover and mutation operators create new solutions in the population. There exist a number of crossover operators in the literature, but in all the operators, two solutions are randomly selected from the mating pool. These are called parent chromosomes. Then a crossover site is randomly selected and a portion of the string is exchanged between the two parent chromosomes. These are called child chromosomes. The process is repeated until the population is filled with new chromosomes. This is the concept behind single point and multi point crossover operators. Another type of crossover known as uniform crossover creates a random mask and generates the child chromosomes from parents using the bit value in the mask. D. Elitism Operator In order not to loose the good solutions from the population elitism operator is used. This operator prevents all population members from undergoing crossover operation. If a crossover probability of P c is used, 100P c % of population undergoes crossover, whereas 100(1-Pc)% of population is copied to 351 next population without any modification. This operator preserves the best fit chromosomes in the current population, and prevents them from being modified during the crossover operation. E. Mutation Operator The crossover operator is mainly responsible for bringing diversity in the population; mutation operator is also used for bringing further diversity in the population. The bit wise mutation operator flips the selected bit from 1 to 0 and vice versa, with a mutation probability P m. SGA is very simple and straightforward. Reproduction operator selects good strings; crossover operator recombines them to form two better strings. The mutation operator alters the strings locally. If bad strings are created in crossover they are eliminated by the reproduction operator. GAs with these simple operators, constitute a potential search and optimization algorithms. IV. ELD using SGA and FGA For solution of ELD using SGA, incremental fuel cost of the generators i.e. lambda is encoded in the chromosome. The algorithm for implementing ELD without losses using Simple GA is as follows. 1. Read population size, chromosome length, unit data, Pdemand, Probability of Elitism, Crossover and mutation 2. Randomly generate population of chromosomes. 3. Decode the chromosomes using (6). 4. Lambda_act = Lambda_min + (Lambda_max - Lambda_min)* Decoded lambda 5. Use the Lambda_act and cost coefficients of the generators, and calculate real power output of the generators (Pgen). 6. Calculate the error of each chromosome as (Sum of Pgen) Pdemand. 7. Fitness(i) of each chromosome is calculated as 1/(1+error(i)/Pdemand). 8. Arrange the chromosomes in the descending order of their fitness. 9. Check if error(1) *Pdemand 10. If yes STOP and calculate Optimal fuel cost and Pgen of units 11. Check if Fitness(1)=Fitness(last chromosome) 12. If yes print All chromosomes have equal values, calculate Optimal fuel cost and Pgen of units and STOP. 13. Apply elitism, Reproduction (RWS), crossover and mutation and generate new population from old one. 14. Update generation count. 15. Check if Generation count > maximum generations?

4 16. If yes, print Problem not converged in maximum number of generations, STOP. 17. Repeat from step 3. For ELD using Fast GA incremental fuel cost of the generators i.e. Lambda is encoded in the chromosome. The algorithm for implementing ELD without losses using Fast GA is as follows. 1. Read population size, chromosome length, unit data, Pdemand, Probability of Elitism, Crossover and mutation 2. Randomly generate population of chromosomes. 3. Decode the chromosomes using eqn. (6). 4. Lambda_act = Lambda_min + (Lambda_max - Lambda_min) * Decoded lambda 5. Use the Lambda_act and cost coefficients of the generators, and calculate real power output of the generators (Pgen). 6. Calculate the error of each chromosome as (Sum of Pgen) Pdemand. 7. Identify Minimum positive error and corresponding chromosome no.(i1) 8. Identify Minimum negative error and corresponding chromosome no.(i2) 9. Set Lambda_max= Lambda_act(I1) 10. Set Lambda_min=Lambda_act(I2) 11. For all chromosomes calculate Lambda_act(chrom) = Lambda_min+(Lambda_max-Lambda_min)*Decoded lambda 12. Calculate Pgen from Lambda_act(chrom) 13. Error(chrom)=(Sum of Pgen)- Pdemand 14. Fitness(i) of each chromosome is calculated as 1/(1+Error(i)/Pdemand). 15. Arrange the chromosomes in the descending order of their fitness. 16. Check if error(1) *Pdemand 17. If yes STOP and calculate Optimal fuel cost and Pgen of units 18. Check if Fitness(1)=Fitness(last chromosome) 19. If yes print All chromosomes have equal values, calculate Optimal fuel cost and Pgen of units and STOP. 20. Apply elitism, Reproduction (RWS), crossover and mutation and generate new population from old one. 21. Update generation count. 22. Check if Generation count > maximum generations? 23. If yes, print Problem not converged in maximum number of generations, STOP. Repeat from step Upto step 6 both algorithms are same. In the fast GA, after the errors of all chromosomes are evaluated, the search space is restricted. This is done by identifying the chromosome which has minimum positive error, and setting the lambda_act of this chromosome to be Lambda_max. Then, identify the chromosome with minimum negative error, and set the lambda_act of this chromosome to be Lambda_min. This will largely reduce the search space from wide Lamdba_max, Lambda_min to small region. The remaining steps are just same as for simple GA. Since the search space is now a very narrow region between the maximum and minimum lambda values, the algorithm converges in two or three generations. For ELD considering transmission losses, in step 5 and 6 of the algorithms the losses are accounted for while calculating the real power output of generators and then error of each chromosome is evaluated as [(sum of Pgen)- Pdemand-Ploss]. The other steps remain same. The present paper uses loss coefficients for calculation of losses. V. RESULTS AND DISCUSSION The developed algorithm is tested on standard 3 generator, 6generator, 20 generator and 38generator systems. For every case the chromosome length, population size, Probability of crossover, Mutation and elitism considered with Simple GA and Fast GA are same. Roulette Wheel Parent selection technique, Uniform crossover and bit-wise mutation are used in all cases. Data for 38 generator system is presented in the appendix. Tables I to IV present the test system results without consideration of system losses. Tables V to VIII provide the test system results while taking system losses into consideration. Tables I presents the results of standard 3 generator system [9] for a power demand of 850MW without considering transmission losses. The results obtained are in agreement with the results shown in [9]. Further, for the same GA parameters FGA locates the optimal solution in just one iteration. The GA parameters considered for this case are Population size: 30, Chromosome length: 16 bits, max no of generations: 100, Elitism probability: 0.15, Crossover probability: 0.9, Mutation probability: TABLE I 3 GENERATOR SYSTEM WITHOUT LOSS Simple GA Fast GA Pdemand (MW) 850 Pg1 (MW) Pg2 (MW) Pg3 (MW) FC ($/hr) λ($/mwh)

5 2 1 Generations Time (Sec.) TABLE II 6GENERATOR SYSTEM WITHOUT LOSS Simple GA Fast GA Pdemand (MW) Pg1 (MW) Pg2 (MW) Pg3 (MW) Pg4 (MW) Pg5 (MW) Pg6 (MW) FC ($/hr) λ($/mwh) Generations Time (Sec.) Tables II presents the results of standard 6 generator system [10] for a power demand of 283.4MW. The GA parameters considered for this case are population size: 40, string length: 16, max no of generations: 100, elitism probability: 0.15, crossover probability: 0.95, mutation probability: Tables III presents the results of 20 generator system [11] for power demands of 1200, 2500 and 3600MW. The GA parameters for this test system (without losses) are population size: 40, string length: 16, max no of iterations: 100, elitism probability: 0.12, crossover probability: 0.98, mutation probability : In all the cases the FGA shows reliable convergence in just 1 iteration. TABLE III 20 GENERATOR SYSTEM WITHOUT LOSSES SGA FGA SGA FGA SGA FGA Pd Pg Pg Pg Pg Pg Pg Pg Pg Pg Pg Pg Pg Pg Pg Pg Pg Pg Pg Pg Pg FC ($/hr) λ($/mwh ) generatio ns 7 1 Time Table IV presents results obtained for 38 generator system for power demands of 7500, 8600 MW without losses. The population size: 60, string length: 16, max no of generations: 100, elitism probability: 0.12, crossover probability: 0.92, mutation probability: For the two cases considered FGA provides better optimal solution in just one generation. TABLE IV 38 GENERATOR SYSTEM WITHOUT LOSSES Method Simple GA Fast GA Simple GA Fast GA Pdemand Pg Pg Pg Pg Pg Pg Pg Pg Pg Pg Pg Pg Pg Pg Pg Pg Pg Pg Pg Pg Pg Pg Pg Pg Pg Pg Pg Pg Pg Pg Pg Pg Pg Pg Pg Pg Pg Pg FC($/hr) λ($/mwh) generations 12 Time (Sec)

6 Tables V presents the results of standard 3 generator system [9] for a power demand of 850MW while considering transmission losses. The GA parameters considered for this system are population size: 40, string length: 16, max no of generations:100, elitism probability: 0.15, crossover probability:0.95, mutation probability: The results obtained are in agreement with the results shown in [9]. TABLE V 3GENERATOR SYSTEM WITH LOSSES Simple GA Fast GA Pdemand (MW) 850 Pg1 (MW) Pg2 (MW) Pg3 (MW) FC ($/hr) λ($/mwh) Generations Loss Time (Sec.) respect to number of generations. From the figure it is obvious that, in case of FGA algorithm the error is considerably reduced in the first generation itself, so this leads to very fast convergence to optimal solution. TABLE VII 6GENERATOR SYSTEM WITH LOSSES Simple GA Fast GA Pdemand (MW) Pg1 (MW) Pg2 (MW) Pg3 (MW) Pg4 (MW) Pg5 (MW) Pg6 (MW) FC ($/hr) λ($/mwh) Generations Loss(MW) Time (Sec.) FGA SGA FGA SGA Error Error generations 5 Fig. 1. Variation of error of best fit chromosome with number of generations for 3 generator system with loss generations Further, for the same GA parameters FGA locates better optimal solution in just 2 generations compared to 10 generations with SGA. Fig.1 shows the variation of error of the best fit chromosome with number of generations. The figure clearly demonstrates the superiority of the developed FGA over SGA. Table VII presents the generator power outputs, optimal fuel cost and loss obtained for 6 generator system. The GA parameters considered are population size: 40, string length: 16, max no of generations: 100, Elitism probability: 0.15, crossover probability: 0.95, mutation probability: From the table it is clear that, for the same GA parameters the proposed FGA presents the optimal solution in just 2 generations compared to 8 generations with SGA. Fig. 2. shows the variation of error of the best fit chromosome with 354 Fig. 2. Variation of error of best fit chromosome with number of generations for 6generator system with loss Table VII provides the optimal generator outputs and optimal fuel cost with losses for 20 generator system [11]. The considered GA parameters are population size: 40, string length: 16, max no of generations: 100, elitism probability: 0.15, crossover probability: 0.95 (Pd=1200), crossover probability: 0.9 (Pd=2500), crossover probability: 0.92 (Pd=3600), mutation probability: From the table it is very clear that, for all the cases considered, FGA provides the optimal solution in minimum time. Further, for a power demand of 2500MW, SGA and

7 FGA provide a Fuel cost of ($/hr) and ($/hr) against ($/hr) as reported in [11] FGA SGA 10 TABLE VII 20 GENERATOR SYSTEM WITH LOSSES Gen in MW SGA FGA SGA FGA SGA FGA Pd 1200 MW 2500MW 3600MW Pg Pg Pg Pg Pg Pg Pg Pg Pg Pg Pg Pg Pg Pg Pg Pg Pg Pg Pg Pg FC ($/hr) λ($/mwh) generations Loss (MW) Time Error generations Fig. 4. Variation of error of best fit chromosome with number of generations for 20 generator system with loss, Pd=3600 VI. CONCLUSION The paper presented a Fast computation algorithm for Economic Load dispatch problem. For the same Genetic operators and parameters, the proposed algorithm shows better and faster convergence over simple GA. Large computational times involved with GAs can be overcome by assessing the variation of error with the solution at hand. A number of test cases have been studied and the algorithm showed reliable convergence. The developed algorithm can be used for real time ELD problems, as the solution accuracy is achieved in minimum time. The only limitation of the algorithm is, if the solution represents a single variable (lambda in this case), the error variation can be easily assessed with respect to this single variable. But if the chromosome represents a number of variables, the error variation with respect to all the variables cannot be predicted and it becomes difficult to restrict the search space to small regions FGA SGA Error generations Fig. 3. Variation of error of best fit chromosome with number of generations for 20 generator system with loss, Pd=1200 Fig. 3 and 4 display the variation of error of the best fit chromosome with number of generations for a power demand of 1200MW and 2500Mw respectively. Even here the FGA shows its superiority over SGA. 355

8 APPENDIX 38 GENERATOR SYSTEM DATA Gen.No. a b c Pmin Pmax REFERENCES [1] Kalyanmoy Deb, Multi-objective Optimization using Evolutionary algorithms, John Wiley and Sons, [2] Anastasios G.Bakirtzis, Pandel N.Biskas, Christoforos E.Zoumas, Vasilios Petridis, Optimal Power Flow by Enhanced Genetic Algorithm, IEEE Transactions on Power Systems, Vol.17, No.2, May 2002, pp [3] Sheble, G.B. and K. Brittig, Refined genetic algorithm-economic dispatch example, IEEE paper 94 WM PWRS, presented at the IEEE/PES 1994 winter meeting. [4] Walters, D.C. and G.B.Sheble, Genetic algorithm solution of Economic dispatch with valve point loading, IEEE Trans. on Power Systems, Vol. 8, No. 3, pp , August [5] Po-Hung Chen and Hong-Chan Chang, Large-Scale Economic Dispatch by Genetic Algorithm, IEEE Trans. On Power Systems, Vol. 10, No.4, November [6] Chao-Hong Chen and Ying-ping Chen, Real Coded ECGA for Economic Dispatch, GECCO 07, July 7-11, 2007, London, England, United Kingdom. [7] P.Venkatesh, R.Gnanadass and Narayana Prasad Padhy, Comparison and Application of Evolutionary Programming Techniques to combined Economic Emission Dispatch with Line flow Constraints, IEEE Transactions on Power Systems, Vol 18, No.2, May [8] A.Laoufi, A.Hazzab and M.Rahli, Economic Dispatch using Fuzzy- Genetic Algorithm, International Journal of Applied Engineering Research, ISSN Vol.1, No.3 (2006). [9] Allen J.Wood and Bruce F.Wollenberg, Power Generation Operation and control, A wiley-interscience Publication, John Wiley & Sons, INC, [10] Alsac,O. and Stott.B, Optimal Load Flow with Steady State Security, IEEE Transactions on Power Apparats and Systems, Vol.93, No.3, [11] Ching-Tzong Su and Chien-Tung Lin, New approach with a Hopfield Modeling Framework to Economic Dispatch, IEEE Transactions on Power Systems, Vol. 15, No. 2, May