Application of Imperialist Competitive Algorithm to Solve Constrained Economic Dispatch

Transcription

1 Iteratioal Joural o Electrical Egieerig ad Iformatics Volume 4, Number 4, December 202 Applicatio of Imperialist Competitive Algorithm to Solve Costraied Ecoomic Dispatch Ghasem Mokhtari, Ahmad Javid Ghaizadeh *, Esmaeil Ebrahimi Electrical Egieerig Departmet, Amirkabir Uiversity of Techology, Tehra, Ira ghaizadeh@aut.ac.ir Abstract: I the costraied dyamic ED problem, cosiderig some practical operatio costraits of geerators, such as ramp rate limits ad prohibited operatig zoes, electric power geeratio of uits are scheduled. So, this paper, cosiderig these costraits, presets a ew optimizatio techique based o Imperialist Competitive Algorithm (ICA to solve the Ecoomic Dispatch (ED problem i power systems. To show the efficiecy of the proposed method, this algorithm is applied to solve costraied dyamic ED problem of two power systems with 6 ad 5 uits. The results are compared to those achieved from covetioal methods such as Simulated Aealig (SA, Geetic Algorithm (GA ad Particle Swarm Optimizatio (PSO. The experimetal results show that the proposed method is capable to determie the solutios of ED problem more fast ad accurate tha other covetioal approaches. Keywords: Ecoomic Dispatch, Optimizatio, Imperialist Competitive Algorithm (ICA. Itroductio Oe of the fudametal cases i power system operatio is Ecoomic Dispatch (ED problem. It should be a optimizatio problem to reduce the operatio cost of uits while satisfyig costraits. Two types of methods are used to optimize the geeratio output of uits. The first type of methods employed various mathematical programmig techiques. These covetioal methods iclude the lambdaiteratio method, the base poit ad participatio factors method, the gradiet method ad etc. [3]. To use these kids of methods for solvig ED problem, the mai assumptio is that the icremetal cost curves of the uits are icreased liearly. However, i practical systems cosiderig the oliear characteristics of geerators, this assumptio is ot acceptable. I additio, i a large system the methods based o mathematical programmig have oscillatory problem [4]. Noted methods (mathematical methods usually used to solve oe type of ED problem kow as static ED problem. The secod type of ED problem is kow as dyamic ED problem. The objective fuctio of dyamic ED is to determie geerator output cosiderig power balace, geeratig uit capacity limits, prohibited operatig zoes, valve poit effect ad multiple fuel cost. To solve the secod type of ED problem (dyamic ED problem, radom search optimizatio methods, such as Evolutioary Programmig (EP [5], Hybrid Immue Geetic Algorithm (HIGA [6], Tabu Search (TS [7] ad Particle Swarm Optimizatio (PSO [4] have bee used. Most of the radom search optimizatio methods have two importat problems:. Trappig i local miima. 2. Slow rate of covergece. Received: Jauary st, 202. Accepted: November 29 th, 202 3

2 Ghasem Mokhtari, et al. To prevet these kids of problem, this paper proposes the use of a ew evolutioary algorithm kow as Imperialist Competitive Algorithm (ICA to solve dyamic ED problem. The ICA is a metaheuristic optimizatio method that is based o modelig of the attempts of coutries to domiate other courtiers. Like other evolutioary oes, this algorithm starts with a iitial populatio. Populatios are i two types: coloies ad imperialists that all together form some empires. Imperialistic competitio amog these empires forms the basis of this algorithm. Durig this competitio, weak empires collapse ad powerful oes take possessio of their coloies. Imperialistic competitio coverges to a state i which there exists oly oe empire ad its coloies are the same positio ad have the same cost as the imperialist [8]. I [9] ICA is used for static ED problem. I this paper, this algorithm is applied to solve the costraied ED problem. The rest of this paper orgaized as follow: The mathematic formulatio of the dyamic ED problem is preseted i sectio II. The ICA is itroduced i sectio III ad the implemetatio of this algorithm for ED problem is explaied i sectio IV. Fially, i sectio V, the proposed method is verified by usig two differet power systems with 6 ad 5 uits. 2. Problem Formulatio The dyamic ED problem is reviewed here for completeess ad to clarify the costraits that are observed. The objective fuctio of the dyamic ED problem is to schedule the output of geeratio uits to meet the required demad at miimum operatig cost while satisfyig all uits' costraits. So, it ca be said as follow: The objective fuctio of the ED problem ca be expressed as [0]: 2 mi Ft = Fi( Pi = ( ai + bp i i + cp i i i= i= ( where F t is the total operatio cost; F i is the geeratio cost fuctio of the ith geerator, a i, b i ad c i are the coefficiets of cost fuctio of the ith geerator; P i is the output power of the ith geerator ad is the umber of geerators. The mai equality costrait is the costrait of power balacig which ca be expressed as follow: P = P + P i D L i= (2 where PD is the load demad ad PL is the total trasmissio losses, which is a fuctio of the uit power outputs that ca be represeted as follow: P = PB P + B P + B L i ij j 0i i 00 i= j= i= (3 where B is matrix that depeds o trasmissio lies parameters. To achieve the actual ecoomic operatio, two iequality costraits of geerator are take ito accout [0]. Firstly, the ramp rate restrictios of geerators are caused i view of the fact that the thermal geeratig outputs caot be adjusted quickly. To cosider the operatig process, the ramp rate limits are icluded i the ED problem to make sure the feasibility of the solutios. As a result, the iequality costraits because of ramp rate limits for uit geeratio chages are preseted as follow: 4

3 Applicatio of Imperialist Competitive Algorithm to Solve Costraied Ecoomic Dispatch mi 0 max 0 max( Pi, Pi DRi Pi mi( Pi, Pi + URi (4 where P i 0 is the previous output power, P i mi ad P i max are the miimum ad maximum outputs of the ith geerator, respectively. URi, DRi are the up ramp ad dow ramp limits of the ith geerator. Secodly, the prohibited operatig zoes i the performace curve of a typical thermal uit are because of steam valve operatio or fluctuatio i a shaft bearig []. Practically, the shape of the iput output curve i the eighborhood of the prohibited zoe is difficult to be determied by actual performace testig or operatig records. I the actual operatio, adjustig the geeratio output P i of a uit must avoid the uit operatig i the prohibited zoes. The feasible operatig zoes of uit i ca be preseted as follows: mi l Pi Pi Pi, P P P P j = 2,3,..., u max Pi, P i i Pi u l i i, j i i, j i (5 where i is the umber of prohibited zoes of the ith geerator. P i,j l ; P i,j u are the lower ad upper power output of the prohibited zoes j of the ith geerator, respectively. 3. Imperialist Competitive Algorithm Imperialism is the policy of spreadig the power of a imperial beyod its ow boudaries. A imperialist domiate other coutries by direct rule or by less obvious meas such as cotrol of market for goods or raw materials. The Imperialist Competitive Algorithm (ICA was proposed by Esmaeil ad Caro [8], this method is a ew sociopolitically motivated global search strategy that has recetly bee itroduced for dealig with differet optimizatio tasks. I [0], this algorithm is used to fid the optimized weights of artificial eural etwork. Moreover, [2, 3] desig a optimal cotroller usig this algorithm. Figure shows the flow chart of the ICA [8]. Like the same evolutioary algorithms, ICA commeces with a iitial populatio of P coutries which are geerated radomly withi the feasible space. Each coutry is show by coutry i =[p, p 2,, p Nvar ] which N var is dimesio of optimizatio problem. The best coutries i the iitial populatio, cosiderig the cost fuctio of them, are selected as the imperialists ad other couties kow as the coloies of these imperialists. To build iitial empires, coloies are divided amog imperialists Based o imperialist s power. To divide the coloies amog imperialists proportioally, the followig ormalized cost of a imperialist is defied: C = c max( c i (6 Where c is the cost of th imperialist ad C is the ormalized cost. As a result the ormalized power of each imperialist ca be defied: p C = N imp i = C i (7 5

4 Ghasem Mokhtari, et al. After dividig coloies amog imperialists, these coloies start closig to their empire. The total power of a empire ca be determied by the power of imperialist coutry plus a percetage of power of its coloies as follow: power = cos t ( imperialists +ε mea{(cos t ( coloies of empires } (8 The way by which the coloies move toward empires ca be foud i [8]. Figure. Flow chart of the ICA 6

5 Applicatio of Imperialist Competitive Algorithm to Solve Costraied Ecoomic Dispatch Where power is the total power (cost of the th empire ad ξ is a positive umber less tha. Cosiderig [8], its amout is set to 0.. After startig competitio, ay empire that caot succeed i this competitio ad caot icrease its power will be elimiated from the competitio. The results of competitio will be a icrease i the power of strog empires ad a decrease i the power of weak oes. With passig computatio, Weak empires will lose their power ad they will collapse. At the ed, by usig the movemet of coloies toward their relevat imperialist ad also the collapse mechaism, oe imperial will exit which is the aswer of optimizatio problem. 4. Applicatio Of ICA To ED Problem I this sectio the implemetatio of the ICA for solvig dyamic ED problem is described. To apply this algorithm to ED problem, two mechaisms should be specified. A. Iitializatio ad structure of solutios B. Costraits hadlig These two mechaisms ca be used as follow i ICA to determie optimized output of geeratig uits i ED problem. A. Iitializatio ad structure of solutios I the iitializatio process, several iitial solutios are created radomly for the ICA. If the iitial populatio of the ICA (coutries cosidered as the output geeratio value of uits (P j 0, P j2 0,, P j 0. j=,.., m. where m is the umber of coutries i ICA ad is the umber of geerators. To create a set of solutios, equality ad iequality costraits should be satisfied. So, the procedure as follow is applied to ICA. Step ( Step (2 Step (3 read system data the boudary of feasible solutio is defied as follow: mi 0 Pi_ lower= max( Pi, Pi DRi max 0 Pi _ upper = mi( Pi, Pi + URi for i=: P = P + rad( ( P P i i _ lower i _ upper i _ lower Ed where rad( is a uiform radom value i the rage [0, ]. B. Costraits hadlig with ICA I evolutioary optimizatio algorithm, hadlig the existig equality ad iequality costraits is very importat. I literatures, differet methods for hadlig costraits i evolutioary computatio optimizatio algorithms are preseted. The most commoly used costraits hadlig methods amog them are: preservig feasible solutio method, ifeasible solutio rejectio method, pealty fuctio method ad solutio repair method [3]. I this paper the last two methods are used as follow to hadle costrait i ICA. To hadle equality costraits i equatio (2, pealty fuctio method used as [4]. The evaluatio fuctio is adopted as (6, it is reciprocal of the geeratio cost fuctio F cost ad power balace costraits P pbc as i (7 ad (8. (6 f = Fcost + Ppbc where, 7

6 Ghasem Mokhtari, et al. F ( Fi( Pi Fmi i= = + abs( ( F F cost max mi 2 pbc = + ( i D L i= P P P P F max maximum geeratio cost amog all idividuals i the iitial populatio; F mi miimum geeratio cost amog all idividuals i the iitial populatio. Moreover, to satisfy the iequality costrait of prohibited zoe, solutio repair method is used. To prevet solutio to fall i the prohibited zoe we do as follow: Step (: for all uits specify prohibited zoes (suppose [a k i,b k i ] as the kth prohibited zoe of uit i Step (2: for each uit k k If k ( ai + bi ai < Pi < 2 k k k Pi = ai + ( ai bi rad( k k Else if ( ai + bi < Pi < bi 2 k k+ k Pi = ai + ( ai bi rad( Ed 5. Case Studies To verify the effectiveess of the ICA for ED problem, two differet power systems with 6 ad 5 uits were tested. Moreover, a compariso is made with SA, GA, HIGA, PSO, TS ad MTS that are listed i [4, 6, 0]. The software like i [0] is implemeted by the MATLAB laguage o a o a laptop computer with Core2Duo 2.5 GHz CPU ad 2 GByte RAM. Accordig to the experiece, the followig parameters are used i the ICA. Number of coutries= Number of imperialist= 8 Maximum iteratio= 30 Revolutio Rate=0.2 Damp Ratio = 0.99 Uitig Threshold=0.02 Moreover, the parameters of the other optimizatio methods are listed i [23]. A. 6 geeratig uit system The system cotais 6 uits, 26 buses ad 46 lies [4]. The characteristics of the uits are listed i Table I. The load demad is 263 MW. I ormal operatio of the system, the loss coefficiets B matrix with 00 MVA base capacity is show i [4]. Table. Geeratio uit data of six uits system 0 P i (MW mi P i (MW max P i (MW a i ($ b i ($/MW c i ($/MW 2 UR i (MW/h DR i (MW/h Prohibited zoes (MW [20,240] [350,3] [90,0] [40,60] [50,70] [20,240] [,90] [0,20] [90,0] [40,50] [75,85] [00,05] (7 (8 8

7 Applicatio of Imperialist Competitive Algorithm to Solve Costraied Ecoomic Dispatch I [6, 0] six methods (MTS, PSO, HIGA TS, GA ad SA were employed to test o this system. Cosiderig these results ad the result of proposed method, Table II elaborates the best solutios of all methods, which satisfy the practical system costraits such as the ramp rate limits ad prohibited zoes of uits. It ca be clearly see i Table II that the solutio of the proposed method is the secod oe which has the miimum cost amog all methods. With schedulig the output power of uits correctly by the proposed method, the losses of system is decreased; therefore, total cost of system is reduced. I additio, i Table 3, the statistical results of all methods such as maximum ad miimum geeratio cost, best geeratio cost ad stadard deviatio are show. These results are determied after 30 trials. Table 2. Best solutio of six uits system Uit power output HIGA [6] SA [0] GA [0] TS [0] PSO [0] MTS [0] ICA P (MW P 2 (MW P 3 (MW P 4 (MW P 5 (MW P 6 (MW Total output (MW P loss (MW Total cost ($/h Table 3. Compariso of ICA performace with other methods for 6 uits system Methods Maximum cost ($/h Average cost ($/h Miimum cost ($/h Stadard deviatio SA [0] GA [0] TS [0] PSO [0] MTS [0] ICA B. 5geeratig uit system The secod system to verify the proposed method cotais 5 uits. The characteristics of uits are listed i Table 4. I this system, the load demad is fixed at 2630 MW. Moreover, the loss coefficiet B matrices are show i [4]. Table 4. Geeratio uit data of 5 uits system Uit P 0 i (MW P mi i (MW P max i (MW a i ($ b i ($/MW c i ($/MW 2 UR i (MW/h DR i (MW/h Prohibited zoes (MW [, 225] [305, 335] [420, 450] [, 200] [305, 335] [390, 420] [230, 2] [365, 395] [430, 4] [30, 40] [,65] 9

8 Ghasem Mokhtari, et al. After simulatig this case, the best solutios obtaied from all methods that satisfy the system costraits are show i Table 5. It ca be clearly see that the proposed method has the cheaper solutio amog the all methods. I additio, after performig 30 trials, the statistical results of all method are listed i Table 6. The results of Table 6 show that i spite of icreasig the size of system, the proposed method is acceptable ad truthful. Table 5. Best solutio of six uits system Uit power output SA [0] GA [0] TS [0] PSO [0] MTS [0] ICA P (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 0 (MW P (MW P 2 (MW P 3 (MW P 4 (MW P 5 (MW Total output (MW P loss (MW Total cost ($/h Table 6. Compariso of ICA performace with other methods for 5 uits system Methods Maximum cost ($/h Average cost ($/h Miimum cost ($/h Stadard deviatio Average CPU time (s SA GA TS PSO MTS ICA C. Computatio efficiecy Table 7 shows the computatioal time of the proposed method for both case studies. It ca be see that proposed method has eough efficiecy for solvig ED problem. I additio, Figure 2, shows the covergece characteristic of the proposed method for 5 geeratig uit system. It ca be see that the proposed method after 9 iteratios reaches to the optimum poit. The best method amog other methods, as show i [0], it reaches to the optimum poit after aroud 85 iteratios. So, the covergece of the proposed method to the optimum solutio is faster tha the other methods Table 7. Computatioal time of the proposed method Average CPU time (s 6 geeratig uit system geeratig uit system

9 Applicatio of Imperialist Competitive Algorithm to Solve Costraied Ecoomic Dispatch 2 x 04 Fiitess Fuctio Iteratio Figure 2. Covergece characteristic of the proposed method 6. Coclusio I this paper Imperialist Competitive Algorithm (ICA for solvig the Ecoomic Dispatch (ED problem was preseted. Two appropriate strategies for iitializatio ad hadlig the equality ad iequality costraits were utilized which always provide feasible solutios. To show the ability of the proposed method for ED problem, two differet test systems are used. Comparig the results of the proposed method with other covetioal methods that is listed i [0], it ca be derived that the proposed method ca fid the best solutio better tha other methods. Moreover, the effectiveess of the proposed method is show by low computatioal time ad fast covergece. Refereces [] A. J. Wood, B. F. Wolleberg, "Power geeratio, operatio ad cotrol," New York: Joh Wiley & Sos, 994. [2] R. A. Jabr, A. H. Cooick ad B. J. Cory, "A homogeeous liear programmig algorithm for the security costraied ecoomic dispatch problem," IEEE Trasactios o Power Systems, vol. 5, o. 3, pp , Aug [3] S. D. Che ad J. F. Che, "A direct NewtoRaphso ecoomic emissio dispatch," Iteratioal Joural of Electrical Power & Eergy Systems, vol. 25, o. 5, pp. 447, Ju [4] T. A. A. Victoire ad A. E. Jeyakumar, "Discussio of particle swarm optimizatio to solvig the ecoomic dispatch cosiderig the geerator costraits," IEEE Trasactios o Power Systems, vol. 9, o. 4, pp , Nov [5] M. A. Azad ad E. Ferades, "A modified differetial evolutio based solutio techique for ecoomic dispatch problems," Joural of Idustrial ad Maagemet Optimizatio, vol. 8, o. 4, pp , Nov 202. [6] A. Rabii, S. Mobaiee, B. Mohamady ad A. Suroody, "A ew heuristic algorithm for solvig ocovex ecoomic power dispatch," Joural of Applied Scieces, vol., o. 23, pp , 20. [7] W. Ogsakul, S. Dechaupaprittha, I. Ngamroo, "Parallel tabu search algorithm for costraied ecoomic dispatch," IEE Proceedigs Geeratio, Trasmissio ad Distributio, vol. 5, o. 2, pp. 5766, [8] E. AtashpazGargari ad C. Lucas, "Imperialist competitive algorithm: A algorithm for optimizatio ispired by imperialistic competitio," IEEE Cogress o Evolutioary Computatio, CEC 2007, pp , 2528 Sept [9] H. Chahkadi Nejad ad R. Jahai, "A ew approach to ecoomic load dispatch of power system usig Imperialist Competitive Algorithm," Australia Joural of Basic ad Applied Scieces, vol. 5, o. 9, pp ,

10 Ghasem Mokhtari, et al. [0] S. Pothiya, I. Ngamroo, W. Kogprawecho, "Applicatio of multiple tabu search algorithm to solve dyamic ecoomic dispatch cosiderig geerator costraits," Eergy Coversio ad Maagemet, vol. 49, pp , Apr [] P. H. Che ad H. C. Chag, "Largescale ecoomic dispatch by geetic algorithm," IEEEE Trasactioss o Power Systems, vol. 0, o.4, pp , Nov [2] M. Abdechiri, K. Faez, H. Bahrami, "Neural Network Learig Based o Chaotic Imperialist Competitive Algorithm," 2d Iteratioal Workshop o Itelliget Systemss ad Applicatios ( ISA, pp. 5, 2223 May 200. [3] E. AtashpazGargari, F. Hashemzadeh, C. Lucas, "Desigig MIMO PIID cotrollerr usig coloial competitive algorithm: Applied to distillatio colum process," IEEEE Cogress o Evolutioary Computatio, CEC 2008, (IEEE World Cogress o Computatioal Itelligece, pp , Ghasem Mokhtari was bor i Mashhad. He received the B.S. degree i electrical egieerig from Ferdowsi Uiversity of Mashhad i He is curretly pursuig the M.S. degreee i the electrical egieerigg departmet at the Amirkabir Uiversity of Techology (AUT, Tehra, Ira. His research iterest iclude TEP problem, power market ad power system optimizatio. Ahmad Javid Ghaizadeh was bor i Herat, Afghaista, i 98. He received his B.Sc. ad M.Sc. i electrical egieerig from Sahad Uiversity of Techology, Tabriz, Ira ad Ferdowsi Uiversity of Mashhad, Mashhad, Ira, i 2004 ad 2009, respectively. He is curretly pursuig the Ph.D. degree i the electrical egieerig departmet at the Amirkabirr Uiversity of Techology (AUT, Tehra, Ira. His research iterests iclude Power Quality, power system optimizatio ad operatio ad trasformerss trasiets. Esmaeil Ebrahimi was bor i Maku, Ira, i 987. He received the B. S. degree i electricall egieerig from Tabriz Uiversity, Tabriz, Ira, i He is curretly pursuig the M. S. degree i the Electrical Egieerigg departmet, Amirkabir Uiversity of Techology (AUT, Tehra, Ira. His research iterests iclude distributed geeratio, power protectio, ad power electroics. 562