An Effective Biogeography Based Optimization Algorithm to Slove Economic Load Dispatch Problem

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1 Joural of Computer Sciece 8 (9): , 202 ISSN Sciece Publicatios A Effective Biogeography Based Optimizatio Algorithm to Slove Ecoomic Load Dispatch Problem Vaitha, M. ad 2 K. Thaushkodi Departmet of Electrical ad Electroics Egieerig, Coimbatore Istitute of Egieerig ad Techology, Coimbatore, Idia 2 Departmet of Electrical ad Electroics Egieerig, Akshaya College of Egieerig ad Techology, Kiathukadavu, Coimbatore, Idia Abstract: Problem statemet: Implemetatio of a Effective Biogeography Based Algorithm (EBBO) for Ecoomic Load Dispatch (ELD) problems i power system i order to obtai optimal ecoomic dispatch with miimum geeratio cost. Approach: A viable methodology has bee implemeted for a 20 uit geerator system to miimize the fuel cost fuctio cosiderig the trasmissio loss ad system operatig limit costraits ad is compared with other approaches such as BBO, Lambda Iteratio ad Hopfield Model. Results: Proposed algorithm has bee applied to ELD problems for verifyig its feasibility ad the compariso of results are tabulated ad pictorial visualizatio for covergece of EBBO is represeted. Coclusio: Comparig with the other existig techiques, the EBBO gives better result by cosiderig the quality of the solutio obtaied. This method could be a alterative approach for solvig the ELD problems i practical power system. Key words: Ecoomic Load Dispatch (ELD), Effective Biogeography Based Algorithm (EBBO), low geeratio cost, quadratic cost fuctio, lambda iteratio, hopfield model INTRODUCTION The most sigificat crisis i the plaig ad operatio of electric power geeratio system is the effective schedulig of all geerators i a system to meet the required demad. The Ecoomic Load Dispatch (ELD) is the importat optimizatio problem to schedule the geeratio amog geeratig uits i power system. The mai aim of ELD problem is to miimize the operatio cost by satisfyig the various operatioal costraits i order met the load demad. May traditioal algorithms (Wood ad Wolleberg, 996) like lambda iteratio, Gradiet search, Newto method are applied to optimize ELD problems however i these methods it is assumed that the icremetal cost curves of the uits are mootoically icreasig piecewise liear fuctios, but the practical systems are oliear. I the past years may optimizatio algorithms are beig developed to solve the ELD problems such as Geetic Algorithms (GA) (Che ad Chag, 995; Orero ad Irvig, 996) Particle Swarm Optimizatio (PSO) (Gaig, 2003; Selvakumar ad Thaushkodi, 2007; Kuo, 2008; Khamsawag et al., 2009), Simulated Aealig (SA) (Wog ad Fug, 993; Wog, 995), Differetial Evolutio (DE) (Das et al., 2008; Biogeography Based Optimizatio (BBO) (Bhattacharya ad Chattopadhyay, 200a). GA is ispired by the study of geetics ad coceptually based o atural evolutio mechaisms. PSO is a robust stochastic optimizatio techique based o the movemet ad itelligece of swarms. SA is a stochastic optimizatio techique which is based o the priciples of statistical egieerig. DE is techically populatio based Evolutioary Algorithm. Biogeography is the ature s way of distributig species. The migratio of species from oe islad to aother, evolutio of ew species ad extictio of species are expressed by the mathematical models of biogeography. This study describes a ew optimizatio algorithm, a Effective Biogeography Based Optimizatio Algorithm. This algorithm is validated by applyig it to 20 uits system with geerator costraits, power balace costraits ad trasmissio loss. The total geeratio cost ad the computatioal time obtaied by this method is better or comparable whe compared to other methods. MATERIALS AND METHODS The ELD problem havig a objective fuctio miimizes the total geeratio cost, F T, while fulfillig Khamsawag ad Jiriwibhakor, 2009) ad Correspodig Author: Vaitha, M., Departmet of Electrical ad Electroics Egieerig, Coimbatore Istitute of Egieerig ad Techology, Coimbatore, Idia 482

2 various costraits whe supplyig the required load demad of a power system. The objective fuctio is give by Eq. : 2 T = i Gi = i Gi + i Gi + i i= i= () mi F mi F (P ) mi A P B P C where, P Gi is the output power geerated by the ith geerator, F i (P Gi ) is the Geeratio cost fuctio of ith geerator ad A i, B i, C i are the Cost coefficiets of ith geerator, is the umber of geerators. Two costraits are cosidered i this problem, i.e., the geeratio capacity of each geerator ad the power balace of the etire power system. Costrait : This costrait is a iequality costrait for each geerator. For ormal system operatios, real power output of each geerator is withi its lower ad upper bouds ad is kow as geeratio capacity costrait give by Eq. 2: P P P (2) mi max Gi Gi Gi mi max where, PGi ad PGi are the lower ad upper limit of the power geerated by ith geerator. Costrait 2: This costrait is a equality costrait. I which the equilibrium is met whe the total power geeratio must equals the total demad P D ad the real power loss i trasmissio lies P L. This is kow as power balace costrait ca be expressed as give i Eq. 3: PGi = PD + PL (3) i= The trasmissio losses are cosidered as a fuctio of the geerators output, ca be expressed as give i Eq. 4: (4) P = P B P + B P + B L Gi ij Gj oi Gi oo i= j= i= where, B ij, B oi, B oo are the trasmissio power loss B- coefficiets, which are assumed to be costat. I the summary, the objective of ecoomic power dispatch optimizatio is to miimize F T subject to the costraits give by the Eq J. Computer Sci., 8 (9): , 202 Particle Swarm Optimizatio: PSO is a populatio based optimizatio techique, motivated by biological cocepts like swarmig ad flockig. PSO is iitialized velocity ad positio. 483 with the populatio, which is radomly geerated ad it always coducts a search i the populatio of particles. Every particle i the populatio represets a possible cadidate solutio (i.e. fitess) to the give problem. I a PSO system, the search towards optima is carried out i a multidimesioal search space. Every particle memorizes its best solutio i additio to its positio achieved so far is kow as Pbest, the Persoal best. It also kows the best value alog with its positio foud i the group amog Pbest, kow as Gbest, the Global best. The basic theory of PSO isists o acceleratig each particle towards its Pbest ad the Gbest locatios as show i Fig.. Differetial evolutio: DE is also a populatio based optimizatio algorithm. The optimizatio process i DE is carried with four basic operatios amely, Iitializatio, Mutatio, Crossover ad Selectio. Through iitializatio operatio ew populatio is created ad the idividuals are kow as target vectors. New parameters are itroduced by the mutatio operatio ito the populatio ad geerate a mutat vector. The crossover operatio geerates trial vectors by combiig the parameter of the mutat vectors with the target vectors. Selectio is the process through which the ext geeratio populatio vector is created by comparig the fitess of target vector ad trial vector. Biogeography: BBO is based o the cocept of Biogeography, two differet processes idetified as Migratio ad Mutatio are carried out. The populatio of idividuals or cadidate solutios ca be represeted as a solutio vector havig itegers. Every iteger i the solutio vector is equal to oe SIV. The quality of the solutios is evaluated by SIVs. The good solutios are cosidered as high HSI habitats where as others are kow as low HSI habitats. The habitats HSI is the fitess fuctio to a give problem. By usig the migratio operatio the iformatio is shared betwee habitats probabilistically. A sudde chage ca occur i the HSI of a atural habitat due to some atural calamities or other evets kow as mutatio. The diversity of the populatios is icreased by the mutatio Proposed EBBO approach: A popular research tred is to merge or combie the PSO with the other techiques, especially the other evolutioary computatio techiques. Evolutioary operators like selectio, crossover ad mutatio have bee applied ito the PSO. I EBBO approach, first the PSO cocept is used to iitialize the populatio of particle with its

3 J. Computer Sci., 8 (9): , 202 Fig. : Cocept of modificatio of a searchig poit by PSO The velocity of the particle is updated if the calculated velocity is out of boudary or closely to zero (rad (0, )), a mutatio operator of the DE is activated; recalculate the velocity of this particle by usig DE mutatio operator. If the calculated velocity usig DE is less tha crossover rate (CR), calculate the immigratio rate λ ad the emigratio rate for each idividual Xi ad Modify the populatio with migratio operator ad update the positio accordig to the ew velocity. Usig this, agai calculate the Gbest value. Step 3: Each pbest values are compared with the other pbest values i the populatio. The best evaluatio value amog the pbest is deoted as gbest. Step 4: The member velocity V of each idividual i the populatio is updated accordig to the velocity update Equatio V = w V + c r (pbest x ) (t+ ) (t) (t) i i i i + c r (gbest x ) (t) 2 2 i EBBO Algorithm: Step : The idividuals of the populatio are radomly iitialized. The velocities of the differet particles are also radomly geerated keepig the velocity withi the maximum ad miimum value [0.5 to -0.5]. These iitial idividuals must be feasible cadidate solutios that satisfy the practical operatio costraits (both the liear ad o-liear costraits) of the give problem. Choose CR, F values. Step 2: The cost fuctio of each idividual is calculated i the populatio usig the evaluatio fuctio F T. The preset value is set as the pbest value. 484 Step 5: The member velocity, V of each idividual i the populatio is checked. If the calculated velocity is out of boudary or closely to zero (rad (0, )), a mutatio operator of the DE is activated, recalculate the velocity of this particle by usig mutatio operator V = F ((x x ) (x x )) (t+ ) (t) (t) (t) (t) i k i q i Else go to step 8, without activatig DE mutatio operator. Step 6: If the calculated velocity usig DE is less tha CR value, Calculate the immigratio rate λ ad the emigratio rate for each idividual Xi Else go to step 8

4 J. Computer Sci., 8 (9): , 202 Step 7: Modify the populatio with migratio operator ad go to step 2 Step 8: The positio of each idividual is modified accordig to the positio update equatio New positio=old positio + updated velocity Go to step 2 Step 9: Cotiue the process, util a maximum iteratio is obtaied. RESULTS The performace of the proposed algorithm was tested o a 20-uit system with a demad of 2500 MW. The software was writte i matlab-7 ad executed. The results of fuel cost ad cpu time obtaied by the proposed EBBO algorithm are compared with other methods such as BBO (Bhattacharya ad Chattopadhyay, 200b), Lambda Iteratio ad Hopfield Model (Su ad Li, 2000) to evaluate the performace of the proposed method. The iput data ad trasmissio loss coefficiets for 20 uits system is take from (Su ad Li, 2000). Table provides the statistic results that ivolved the geeratio cost, evaluatio value ad CPU time. Table 2 provides the parameter settig for the proposed method. Figure 2 provides the characteristics graph betwee iteratio ad the total geeratio cost. Fig. 2: Covergece characteristic of 20-geerator system Table : Best Power Output for 20-Geerator System Uit output EBBO BBO Lambda iteratio Hopfield model P (MW) P2 (MW) P3 (MW) P4 (MW) P5 (MW) P6 (MW) P7 (MW) P8 (MW) P9 (MW) P0 (MW) P (MW) P2 (MW) P3 (MW) P4 (MW) P5 (MW) P6 (MW) P7 (MW) P8 (MW) P9 (MW) P20 (MW) Total Power Output (MW) Total Geeratio Cost ($/h) Power Loss) (MW) CPU time/ iteratio(sec)

5 J. Computer Sci., 8 (9): , 202 Table 2: Parameter Settigs C C2 Iteratio ω mi ω max CR F µ λ DISCUSSION The previous method results take from the literature is compared with the proposed method. The compariso proves that all the four algorithms have the potetial to fid the global solutio, but the miimum geeratio costs achieved by EBBO is less tha those reported i recet literature. It is also clear that the EBBO algorithm is efficiet ad require less computatioal time. As a whole it ca be said that the EBBO algorithm is computatioally efficiet tha earlier metioed methods. Thus, the EBBO algorithm is more reliable to fid out the miimum fuel cost i this example. CONCLUSION This study presets a ovel codig scheme for EBBO algorithm to solve practical ELD issues ad cofirmed by a simulatio process. The proposed combied method uses the mutatio property of DE ad migratio property of BBO i PSO, which ca provide a good optimal solutio eve whe the problem begis with optimal solutio. The performace of proposed codig scheme used i the case study of 20-uits system with trasmissio loss, proved to have saliet features icludig better quality solutio, stable covergece characteristics ad good computatioal efficiecy whe compared with the results obtaied from other heuristic methods. REFERENCES Bhattacharya, A. ad P.K. Chattopadhyay, 200a. Biogeography-based optimizatio for differet ecoomic load dispatch problems. IEEE Tras. Power Syst., 25: DOI: 0.09/TPWRS Bhattacharya, A. ad P.K. Chattopadhyay, 200b. Hybrid Differetial evolutio with biogeographybased optimizatio for solutio of ecoomic load dispatch. IEEE Tras. Power Syst., 25: DOI: 0.09/TPWRS Che, P.H. ad H.C. Chag, 995. Large-scale ecoomic dispatch by geetic algorithm. IEEE Tras. Power Syst., 0: DOI: 0.09/ Das, S., A. Abraham ad A. Koar, Particle Swarm Optimizatio ad Differetial Evolutio Algorithms: Techical Aalysis, Applicatios ad Hybridizatio Perspectives. st Ed., Spriger- Verlag Berli Heidelberg. Gaig, Z.L., Particle Swarm optimizatio to solvig the ecoomic dispatch cosiderig the geerator costraits. IEEE Tras. Power Syst., 8: DOI: 0.09/TPWRS Khamsawag, S. ad S. Jiriwibhakor, Solvig the ecoomic dispatch problem usig ovel particle swarm optimizatio. It. J. Elec., Comput. Syst. Eg., 3: Khamsawag, S., P. Waakar, S. Pothiya ad S. Jiriwibhakor, Solvig the ecoomic dispatch problem by usig differetial evolutio. Proceedigs of the 6th Iteratioal Coferece o Electrical Egieerig/Electroics, Computer, Telecommuicatios ad Iformatio Techology, May 6-9, IEEE Xplore Press, Choburi, Thailad, pp: DOI: 0.09/ECTICON Kuo, C.C., A ovel codig scheme for practical ecoomic dispatch by modified particle swarm approach. IEEE Tras. Power Syst., 23: DOI: 0.09/TPWRS Orero, S.O. ad M.R. Irvig, 996. Ecoomic dispatch of geerators with prohibited operatig zoes: A geetic algorithm approach. IEE Proce. Geer. Tras. Distrib., 43: DOI: 0.049/ipgtd: Selvakumar, I.A. ad K. Thaushkodi, A ew particle swarm optimizatio solutio to ocovex ecoomic dispatch problems. IEEE Tras. Power Syst., 22: DOI: 0.09/TPWRS Su, C.T. ad C.T. Li, New approach with a Hopfield modelig framework to ecoomic dispatch. IEEE Tras. Power Syst., 5: DOI: 0.09/ Wog, K.P. ad C.C. Fug, 993. Simulated aealig based ecoomic dispatch algorithm. IEE Proce. Ge. Tras. ad Distrib., 40: Wog, K.P., 995. Solvig power system optimizatio problems usig simulated aealig. Egi. Appli. Artificial Itel., 8: DOI: 0.06/ (95) Wood, A.J. ad B.F. Wolleberg, 996. Power Geeratio, Operatio ad Cotrol. 2d Ed., Joh Wiley ad Sos, New York, ISBN-0: , pp: