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  1. 1. K. Sudhakara Reddy, Dr. M. Damodar Reddy/ International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue4, July-August 2012, pp.2325-2330 Economic Load Dispatch Using Firefly Algorithm K. Sudhakara Reddy1, Dr. M. Damodar Reddy2 1-(M.tech Student, Department of EEE, SV University, Tirupati 2- (Associate professor, Department of EEE, SV University, Tirupati)ABSTRACT Economic Load Dispatch (ELD) problem problem solving algorithms. In the Fireflyin power systems has been solved by various algorithm[14][15][16], the objective function of aoptimization methods in the recent years, for given optimization problem is associated with theefficient and reliable power production. This flashing light or light intensity which helps thepaper introduces a solution to ELD problem using swarm of fireflies to move to brighter and morea new metaheuristic nature-inspired algorithm attractive locations in order to obtain efficientcalled Firefly Algorithm (FFA). The proposed optimal solutions.approach has been applied to various test systems. In this research paper we will show how theThe results proved the efficiency and robustness firefly algorithm can be used to solve the economicof the proposed method when compared with the load dispatch optimization problem. A briefother optimization algorithms. description and mathematical formulation of ELD problem has been discussed in the following section.Keywords - Economic Load Dispatch, Firefly The concept of Firefly Algorithm is discussed inAlgorithm. section 3. The respective algorithm and parameter setting of FFA has been provided in section 4.1.Introduction Simulation studies are discussed in section 5 and Economic Load Dispatch (ELD) seeks the conclusion is drawn in section 6.best generation schedule for the generating plants tosupply the required demand plus transmission loss 2. Formulation of the Economic Loadwith the minimum generation cost. Significant Dispatch Problemeconomical benefits can be achieved by finding a 2.1. Economic Dispatchbetter solution to the ELD problem. So, a lot of The objective of economic load dispatch ofresearches have been done in this area. Previously a electric power generation is to schedule thenumber of calculus-based approaches including committed generating unit outputs so as to meet theLagrangian Multiplier method [1] have been applied load demand at minimum operating cost whileto solve ELD problems. These methods require satisfying all units and operational constraints of theincremental cost curves to be monotonically power system.increasing/piece-wise linear in nature. But the input- The economic dispatch problem is aoutput characteristics of modern generating units are constrained optimization problem and it can behighly non-linear in nature, so some approximation is mathematically expressed as follows:required to meet the requirements of classicaldispatch algorithms. Therefore more interests havebeen focused on the application of artificial (2.1)intelligence (AI) technology for solution of these Where, FT : total generation cost (Rs/hr)problems. Several AI methods, such as Genetic n : number of generatorsAlgorithm[2] Artificial Neural Networks[3], Pn : real power generation of n thSimulated Annealing[4], Tabu Search[5], generator (MW)Evolutionary Programming[6], Particle Swarm Fn(Pn) : generation cost for PnOptimization[7], Ant Colony Optimization[8], Subject to a number of power systemsDifferential Evolution[9], Harmony search network equality and inequality constraints. TheseAlgorithm[10], Dynamic Programming[11], Bio- constraints include:geography based optimization[12], Intelligent waterdrop Algorithm[13] have been developed and applied 2.2. System Active Power Balancesuccessfully to small and large systems to solve ELD For power balance, an equality constraintproblems in order to find much better results. Very should be satisfied. The total power generated shouldrecently, in the study of social insects behavior, be the same as total load demand plus the total linecomputer scientists have found a source of inspiration lossesfor the design and implementation of optimizationalgorithms. Particularly, the study of fireflies’behavior turned out to be very attractive to develop 2325 | P a g e
  2. 2. K. Sudhakara Reddy, Dr. M. Damodar Reddy/ International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue4, July-August 2012, pp.2325-2330 Where, PD : total system demand (MW) The firefly algorithm has three particular PL: transmission loss of the system idealized rules which are based on some of the major(MW) flashing characteristics of real fireflies. The characteristics are as follows:2.3. Generation Limits i) All fireflies are unisex and they will move towards Generation output of each generator should more attractive and brighter ones regardless their sex.be laid between maximum and minimum limits. The ii) The degree of attractiveness of a firefly iscorresponding inequality constraints for each proportional to its brightness which decreases as thegenerator are distance from the other firefly increases. This is due to the fact that the air absorbs light. If there is not a brighter or more attractive firefly than a particular (2.3) one, it will then move randomly. Where, Pn,min : minimum power output limit iii) The brightness or light intensity of a firefly isof nth generator (MW) determined by the value of the objective function of a Pn,max : maximum power output limit of nth given problem. For maximization problems, the lightgenerator (MW) intensity is proportional to the value of the objectiveThe generation cost function Fn(Pn) is usually function.expressed as a quadratic polynomial: 3.2. Attractiveness In the firefly algorithm, the form of attractiveness function of a firefly is given by the (2.4) following monotonically decreasing function: Where, an, bn and cn are fuel costcoefficients. with m≥12.4. Network Losses (3.1) Since the power stations are usually spread Where, r is the distance between any twoout geographically, the transmission network losses fireflies,must be taken into account to achieve true economic β0 is the initial attractiveness at rdispatch. Network loss is a function of unit =0, andgeneration. To calculate network losses, two methods γ is an absorption coefficientare in general use. One is the penalty factors method which controls the decrease of the light intensity.and the other is the B coefficients method. The latteris commonly used by the power utility industry. In 3.3. Distancethe B coefficients method, network losses are The distance between any two fireflies i and j atexpressed as a quadratic function: positions xi and xj respectively can be defined as : (2.5) (3.2) Where Bmn are constants called B where xi,k is the kth component of the spatialcoefficients or loss coefficients coordinate xi of the ith firefly and d is the number of dimensions.3. The Firefly Algorithm3.1. Description 3.4. Movement The Firefly Algorithm is a metaheuristic, The movement of a firefly i which isnature-inspired optimization algorithm which is attracted by a more attractive i.e., brighter firefly j isbased on the social flashing behavior of fireflies. It is given by :based on the swarm behavior such as fish, insects orbird schooling in nature. Although the fireflyalgorithm has many similarities with other algorithms (3.3)which are based on the so-called swarm intelligence, Where the first term is the current positionsuch as the famous Particle Swarm Optimization of a firefly, the second term is used for considering a(PSO), Artificial Bee Colony optimization (ABC) firefly’s attractiveness to light intensity seen byand Bacterial Foraging (BFA) algorithms, it is indeed adjacent fireflies and the third term is used for themuch simpler both in concept and implementation. random movement of a firefly in case there are noIts main advantage is that it uses mainly real random brighter ones. The coefficient α is a randomizationnumbers, and it is based on the global communication parameter determined by the problem of interest,among the swarming particles called as fireflies. rand is a random number generator uniformly distributed in the space [0,1]. 2326 | P a g e
  3. 3. K. Sudhakara Reddy, Dr. M. Damodar Reddy/ International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue4, July-August 2012, pp.2325-2330 4. Algorithm A reasonable loss coefficient matrix of Step 1: Read the system data such as cost power system network was employed to draw the coefficients, minimum and maximum power transmission line loss and satisfy the transmission limits of all generator units, power demand capacity constraints. The program is written in and B-coefficients. MATLAB software package. Step 2: Initialize the parameters and constants of Firefly Algorithm. They are noff, αmax, αmin, 5.1 Three-Unit System β0, γmin, γmax and itermax (maximum number The generator cost coefficients, generation of iterations). limits and B-coefficient matrix of three unit system Step 3: Generate noff number of fireflies (xi) are given below. Economic Load Dispatch solution randomly between λmin and λmax . for three unit system is solved using conventional Step 4: Set iteration count to 1. Lambda iteration method and Firefly algorithm Step 5: Calculate the fitness values corresponding to method. Test results of Firefly method are given in noff number of fireflies. table 5.2. Comparison of test results of Lambda- Step 6: Obtain the best fitness value GbestFV by iteration method and Firefly algorithm are shown in comparing all the fitness values and also table 5.3. obtain the best firefly values GbestFF corresponding to the best fitness value Table 5.1 Cost coefficients and power limits of 3- GbestFV. Unit system. Step 7: Determine alpha(α) value of currentUnit iteration using the following equation: an bn cn Pn,min Pn,max α (iter)= αmax -(( αmax - αmin) (current iteration number )/ itermax) 1 1243.5311 38.30553 0.03546 35 210 Step 8: Determine the rij values of each firefly using the following equation: 2 1658.5696 36.32782 0.02111 130 325 rij= GbestFV -FV rij is obtained by finding the difference 3 1356.6592 38.27041 0.01799 125 315 between the best fitness value GbestFV (GbestFV is the best fitness value i.e., jth firefly) and fitness value FV of the ith firefly. The loss coefficient matrix of 3-Unit system Step 9: New xi values are calculated for all the Bij= fireflies using the following equation: Table 5.2 Test results of Firefly Algorithm for 3-Unit (4.1) System Where, β0 is the initial attractiveness Pow γ is the absorption co-efficient Sl er P1 P2 P3 Ploss Fuel rij is the difference between the best fitness value Dem Cost . (M (M (M (M GbestFV and fitness value FV of the ith firefly. and λ (Rs/ N W) W) W) W) α (iter) is the randomization parameter ( In this o. (M hr) present work, α (iter) value is varied between 0.2 W) and 0.01) rand is the random number between 0 and 44.4 70.3 156. 129. 5.77 185 1 350 1. 387 012 267 208 698 64.5 In this present work, x → λ 45.4 82.0 174. 150. 7.56 208 2 400 Step 10: Iteration count is incremented and if 762 784 994 496 813 12.3 iteration count is not reached maximum then 46.5 93.9 193. 171. 9.61 231 3 450 go to step 5 291 374 814 862 271 12.4 Step 11: GbestFF gives the optimal solution of the 47.5 105. 212. 193. 11.9 254 Economic Load Dispatch problem and the 4 500 977 88 728 306 144 65.5 results are printed. 48.6 117. 231. 214. 14.4 278 5 550 824 907 738 831 769 72.4 5. Simulation Results 49.7 130. 250. 236. 17.3 303 The effectiveness of the proposed firefly 6 600 836 021 846 437 040 34.0 algorithm is tested with three and six generating unit 50.9 142. 270. 258. 20.3 328 systems. Firstly, the problem is solved by 7 650 017 223 053 124 997 51.0 conventional Lambda iterative method and then 52.0 154. 289. 279. 23.7 354 Firefly algorithm optimization method is used to 8 700 371 514 360 894 680 24.4 solve the problem. 2327 | P a g e
  4. 4. K. Sudhakara Reddy, Dr. M. Damodar Reddy/ International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue4, July-August 2012, pp.2325-2330 Table 5.3 Comparison of test results of Lambda- Bij= iteration method and Firefly Algorithm for 3-Unit System Fule Cost (Rs/hr) Lambda Sl. Power Firefly iteration No. Demand(MW) Algorithm method 1 350 18570.7 18564.5 2 400 20817.4 20812.3 Table 5.5 Test results of Firefly method for 6-Unit System 3 450 23146.8 23112.4 Po 4 500 25495.2 25465.5 w F 5 550 27899.3 27872.4 er ue S D Pl l P1 P2 P3 P4 P5 P6 6 600 30359.3 30334.0 l e oss C ( ( ( ( ( ( . m ( os M M M M M M 7 650 32875.0 32851.0 N an λ M t W W W W W W o d W (R ) ) ) ) ) ) 8 700 35446.3 35424.4 . ( ) s/ M hr W ) ) 5.2 Six-Unit System 47 23 95 10 20 18 14 32 The generator cost coefficients, generation 60 .3 .8 .6 0. 2. 1. .2 09 limits and B-coefficient matrix of six unit system are 1 10 0 41 60 38 70 83 19 37 4. given below. Economic Load Dispatch solution for 9 3 9 8 2 8 4 7 six unit system is solved using conventional Lambda 48 26 10 10 21 19 16 34 iteration method and Firefly Algorithm method. Test 65 .1 .0 7. 9. 6. 6. .7 48 results of Firefly method are given in table 5.5. 2 10 0 73 67 26 66 77 95 28 2. Comparison of test results of Lambda-iteration 1 9 4 8 5 4 1 6 method and Firefly Algorithm are shown in table 5.6. 49 28 11 11 23 21 19 36 70 .0 .2 8. 8. 0. 2. .4 91 3 10 0 14 90 95 67 76 74 31 2. Table 5.4 Cost coefficients and power limits of 6- 6 8 8 5 3 5 9 2 Unit system 49 30 11 13 12 24 22 22 39Unit an Bn Cn Pn,min Pn,max 75 .8 .4 .2 0. 7. 4. 8. .3 38 4 0 45 75 26 44 51 46 18 10 4.1 756.79886 38.53 0.15240 10 125 1 6 5 6 5 6 2 8 02 451.32513 46.15916 0.10587 10 150 50 32 14 14 13 25 24 25 41 80 .6 .5 .4 1. 6. 7. 3. .3 89 53 1049.9977 40.39655 0.02803 35 225 0 61 86 84 54 04 66 00 31 6. 3 3 3 8 5 4 9 2 94 1243.5311 38.30553 0.03546 35 210 51 34 17 15 14 27 25 44 28 85 .4 .7 .7 2. 4. 0. 7. 45 6 .55 1658.5696 36.32782 0.02111 130 325 0 83 10 67 70 61 89 85 0. 56 9 2 5 8 4 7 9 36 1356.6592 38.27041 0.01799 125 315 52 36 21 15 27 31 47 16 28 90 .3 .8 .0 3. 2. .9 04 7 3. 4. 0 16 48 77 22 73 88 5. The loss co-efficient matrix of 6-Unit system 93 17 2 1 5 7 7 1 3 53 38 24 17 16 29 35 49 28 95 .1 .9 .4 5. 1. 7. .6 68 8 7. 0 58 99 14 21 88 48 29 2. 64 5 8 5 4 2 1 5 1 2328 | P a g e
  5. 5. K. Sudhakara Reddy, Dr. M. Damodar Reddy/ International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue4, July-August 2012, pp.2325-2330Table 5.6 Comparison of test results of Lambda- Economic Dispatch,” IEEE Transaction oniteration method Power System, Vol. 15, No. 2, 2000, pp.and Firefly Algorithm for 6-Unit System 541-545. [4]. Basu M. “A simulated annealing based goal Fule Cost (Rs/hr) attainment method for economic emissionSl Power Lambda load dispatch of fixed head hydrothermal Firefly power systems” International Journal ofno. Demand(MW) iteration Algorithm Electrical Power and Energy Systems method 2005;27(2):147–53.1 600 32129.8 32094.7 [5]. W. M. Lin, F. S. Cheng and M. T. Tsay, “An Improved Tabu Search for Economic2 650 34531.7 34482.6 Dispatch with Multiple Minima,” IEEE Transaction on Power Systems, Vol. 17, No.3 700 36946.4 36912.2 1, 2002, pp. 108-112. [6]. T. Jayabarathi, G. Sadasivam and V.4 750 39422.1 39384.0 Ramachandran, “Evolutionary Programming based Economic dispatch of5 800 41959.0 41896.9 Generators with Prohibited Operating Zones,” Electrical Power System Research,6 850 44508.1 44450.3 Vol. 52, No. 3, 1999, pp. 261- 266. [7]. J.B. Park, K. S. Lee, J. R. Shin and K. Y.7 900 47118.2 47045.3 Lee, “A Particle Swarm Optimization for Economic Dispatch with Non Smooth Cost8 950 49747.4 49682.1 Functions,” IEEE Transaction on Power Systems, Vol. 8, No. 3, 1993, pp. 1325-1332. [8]. Huang JS. “Enhancement of hydroelectric From the tabulated results, we can observe Generation scheduling using ant colonythat the Firefly Algorithm optimization method can system based optimization approache”.obtain lower fuel cost than conventional Lambda IEEE Transactions on Energy Conversioniteration method. 2001;16(3):296–301. [9]. N. Nomana and H. Iba, “Differential6. Conclusions Evolution for Economic Load Dispatch Economic Load Dispatch problem is solved Problems,” Electric Power Systemsby using Lambda iteration method and Firefly Research, Vol. 78, No. 8, 2008, pp. 1322-Algorithm. The programs are written in MATLAB 1331.software package. The solution algorithm has been [10]. Z.W. Geem, J.H. Kim, and G.V.tested for two test systems with three and six Loganathan, “A new heuristic optimizationgenerating units. The results obtained from Firefly algorithm: harmony search,” Simulation,Algorithm are compared with the results of Lambda Vol. 76, No. 2, pp. 60-68, 2001.iteration method. Comparison of test results of both [11].Z. X. Liang and J. D. Glover, “A Zoommethods reveals that Firefly Algorithm is able to give Feature for a Dynamic Programmingmore optimal solution than Lambda iteration method. Solution to Economic Dispatch includingThus, it develops a simple tool to meet the load Transmission Losses,” IEEE Transactionsdemand at minimum operating cost while satisfying on Power Systems, Vol. 7, No. 2, 1992, pp.all units and operational constraints of the power 544-550.system. [12]. A. Bhattacharya and P. K. Chattopadhyay, “Biogeography- Based Optimization forReferences Different Economic Load Dispatch [1]. Tkayuki S, Kamu W. “Lagrangian Problems,” IEEE Transactions on Power relaxation method for price based unit Systems, Vol. 25, No. 2, 2010, pp. 1064- commitment problem” Engineering 1077. Optimization -Taylor & Francis Journal, [13]. S. R. Rayapudi, “An Intelligent Water Drop 36(6): 705-19. Algorithm for Solving Economic Load [2]. P. H. Chen and H. C. Chang, “Large-Scale Dispatch Problem,” International Journal Economic Dispatch by Genetic Algorithm,” of Electrical and Electronics Engineering, IEEE Transactions on Power System, Vol. Vol. 5, No. 2, 2011, pp. 43-49. 10, No. 4, 1995, pp. 1919-1926. [14] Yang, X. S., “Firefly algorithm for [3]. C.-T. Su and C.-T. Lin, “New Approach with multimodal optimization”, in Stochastic a Hopfield Modeling Framework to Algorithms Foundations and Appplications 2329 | P a g e
  6. 6. K. Sudhakara Reddy, Dr. M. Damodar Reddy/ International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue4, July-August 2012, pp.2325-2330 (Eds O. Watanabe and T. eugmann), SAGA [15] X.-S. Yang, “Firefly Algorithm, Levy Flights 2009, LectureNotes in Computer Science, and Global Optimization”, Research and 5792, Springer-Verlag,Berlin,2009,pp.169- Development in Intelligent Systems XXVI 178. (Eds M. Bramer, R. Ellis, Petridis), Springer London, 2010, pp. 209-218.[16] N. Chai-ead, P. Aungkulanon and P. Luangpaiboon “Bees and Firefly Algorithms for Noisy Non-Linear Optimisation Problems”, proceedings of international multi conference of engineers and computer scientits 2011 Vol II IMECS 2011 March 16-18, Hong Kong. 2330 | P a g e

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