Economic dispatch solution using efficient heuristic search approach

Transcription

1 Leonardo Journal of Scences Economc dspatch soluton usng effcent heurstc search approach Samr SAYAH QUERE Laboratory, Faculty of Technology, Electrcal Engneerng Department, Ferhat Abbas Unversty, Setf 1, Algera E-mal: Correspondng author, phone: Abstract A new effcent hybrd dfferental evoluton algorthm wth harmony search (DE-HS) s proposed n ths paper to solve the economc load dspatch (ELD) problem. The fresh ndvdual generaton mechansm of harmony search (HS) s utlzed to enhance the local search capablty of the orgnal dfferental evoluton (DE) method. The proposed hybrd algorthm s valdated usng the IEEE 30 bus test system and the practcal Algeran 59 bus system. The results confrm the sutablty of the proposed strategy to fnd accurate and feasble optmal solutons for ELD problems. Keywords Economc power dspatch (EPD); Dfferental Evoluton (DE); Harmony Search (HS) Introducton Economc load dspatch s a fundamental functon n modern power system operaton and control. The conventonal economc load dspatch (ELD) problem of power generaton nvolves allocaton of power output to dfferent thermal unts to mnmze the operatng cost subect to dverse equalty and nequalty constrants of the power system. Ths makes the ELD problem a large-scale hghly nonlnear constraned optmzaton problem. 38

2 Leonardo Journal of Scences It s therefore of great mportance to solve ths problem as quckly and accurately as possble. Unfortunately, the nput-output characterstcs of modern unts are nherently nonlnear and non-conve because of valve-pont loadngs, prohbted operatng zones and multple fuels, and furthermore they may generate multple local mnmum ponts n the cost functon. Conventonal technques offer good results, but when the search space s nonlnear and has dscontnutes, these technques become dffcult to solve wth a slow convergence rato and not always seekng to the global optmal soluton. New numercal methods are then needed to cope wth these dffcultes, specally, those wth hgh speed search to the optmal and not beng trapped n local mnma. The ELD problem has been solved va several tradtonal optmzaton methods ncludng gradent-based technques, newton methods, lnear programmng, and quadratc programmng [1]. Most of these technques are not capable to solve effcently the optmzaton problems wth a non-conve, non-contnuous, and hghly non-lnear soluton space. Several heurstc global search tools have evolved n the last decades that have facltated solvng the ELD problems wth non-smooth cost functons. Among these methods could be named the genetc algorthms [1, 2], evolutonary programmng [3], neural networks [4], dfferental evoluton [5], smulated annealng [6], tabu search [7], partcle swarm optmzaton [8], and ant colony optmzaton [9]. Also the hybrdzaton and combnaton of these methods have been used to solve more effectvely ELD problems [10-12]. Dfferental evoluton and harmony search algorthms are two of these global optmzaton technques. Dfferental evoluton (DE) algorthm whch s nspred by bologcal and socologcal motvatons has been developed and ntroduced by Storn and Prce [13] n DE algorthm mproves a populaton of canddate solutons over several generatons usng the mutaton, crossover and selecton operators n order to reach an optmal soluton [14]. In DE, the ftness of an offsprng s one-to-one competed wth that of the correspondng parent. Ths one-to-one competton makes convergence speed of DE faster than other evolutonary algorthms. Nevertheless, ths faster convergence property yelds n a hgher probablty of searchng toward a local optmum or gettng premature convergence [15-16]. Proposed by Geem et al. [17] n 2001, the harmony search (HS) method s nspred by the underlyng prncples of the muscans mprovsaton of the harmony. In the HS algorthm, muscan mprovses the ptches of hs/her nstrument to obtan a better state of 39

3 Economc dspatch soluton usng effcent heurstc search approach Samr SAYAH harmony. The ptch of each muscal nstrument determnes the aesthetc qualty, ust as the obectve functon value s determned by the set of values assgned to each decson varable [18]. On the other hand, HS s good at dentfyng the hgh performance regons of the soluton space at a reasonable tme. However, several studes [19-21] have shown that the orgnal HS method has a lmtaton n dealng wth the multmodal and constraned optmzaton problems. To overcome the drawbacks of DE and HS, an effcent combnaton strategy of DE and HS (termed DE-HS) s ntroduced n ths study. The proposed DE-HS conssts essentally n a strong co-operaton of DE and HS, snce the DE-based update polcy of estng ndvduals and HS-based generaton strategy of new ndvduals are merged together [22]. In ths study, a new effcent combned dfferental evoluton and harmony search (DE-HS) algorthm s proposed for solvng practcal ELD problems. The performance of the proposed method s tested on two case studes usng the IEEE 30 bus test system and the practcal Algeran 59 bus system. Numercal results obtaned by the proposed approach were compared wth other optmzaton results reported n the lterature recently. Economc load dspatch problem formulaton The man obectve of ELD s to mnmze the total generaton cost of the power system wthn a defned nterval (typcally 1h). Obectve functon The obectve functon for the ELD reflects the cost assocated wth generatng power n the system. The obectve functon F for the entre power system can then be wrtten as the sum of the fuel cost model for each generator: ng F = 1 F (1) where F s the fuel cost functon of the th generator n ($/h) and ng s the number of onlne generatng unts to be dspatched. Tradtonally, the fuel cost curve s appromated usng a smple smooth quadratc functon [5] gven by: 40

4 Leonardo Journal of Scences F (P ) a P b P c, (2) 2 where P s the power of generator. a, b and c are the cost coeffcents of generator. Power balance constrant Ths constrant s based on the prncple of equlbrum between total system generaton and total system loads P D and transmsson losses P L [5]. That s where P L ng 1 P P D P L can be obtaned usng the B matr loss formula [5], gven by: (3) ng 1 ng 1 ng P P B P P B B (4) L where B, B 0 and B 00 are the loss coeffcents. Generator capacty constrants For stable operaton, the real power generaton of each generator should be restrcted between ts lower and upper lmts of generaton. The generator capacty constrants are epressed as a par of nequalty constrants, as follows: P mn P P (5) ma Overvew of dfferental evoluton algorthm The dfferental evoluton algorthm (DE) s a populaton based algorthm lke genetc algorthm usng the smlar operators; crossover, mutaton and selecton. In DE, each decson varable s represented n the chromosome by a real number. As n any other evolutonary algorthm, the ntal populaton of DE s randomly generated, and then evaluated. After that, the selecton process takes place. Durng the selecton stage, three parents are chosen and they generate a sngle offsprng whch competes wth a parent to determne whch one passes to the followng generaton. DE generates a sngle offsprng (nstead of two lke n the genetc algorthm) by addng the weghted dfference vector between two parents to a thrd parent. If 41

5 Economc dspatch soluton usng effcent heurstc search approach Samr SAYAH the resultng vector yelds a lower obectve functon value than a predetermned populaton member, the newly generated vector replaces the vector to whch t was compared. An optmzaton task consstng of D parameters can be presented by a D-dmensonal vector. In DE, a populaton of NP soluton vectors s randomly created at the start. Ths populaton s successfully mproved over G generatons by applyng mutaton, crossover and selecton operators, to reach an optmal soluton [14, 15]. The man steps of the DE algorthm are gven bellow: Intalzaton Evaluaton Repeat Mutaton Crossover Evaluaton Selecton Untl (Termnaton crtera are met) Intalzaton Typcally, each decson parameter n every vector of the ntal populaton s assgned a randomly chosen value from wthn ts correspondng feasble bounds: (0), mn ma mn, = 1,, NP, = 1,, D, (6) where denotes a unformly dstrbuted random number wthn the range [0, 1], generated anew for each value of. ma and mn are the upper and lower bounds of the th decson parameter, respectvely. vector Mutaton The mutaton operator creates mutant vectors a wth the dfference of two other randomly vectors followng equaton [17]: by perturbng a randomly selected and, accordng to the (G) (G) (G) (G) ( ), = 1,, NP, (7) a b c b c 42

6 Leonardo Journal of Scences where a, b and c are randomly chosen vectors among the NP populaton, and a b c. a, and are selected anew for each parent vector. The scalng constant s an algorthm b c control parameter used to adust the perturbaton sze n the mutaton operator and mprove algorthm convergence. Crossover The crossover operaton generates tral vectors mutant vectors wth ts target or parent vectors dstrbuton of the followng form [17]: by mng the parameters of the accordng to a selected probablty (G), (G), (G), f C R otherwse or q, = 1,, NP, = 1,, D (8) where denotes a unformly dstrbuted random number wthn [0, 1], generated anew for each value of. q s a randomly chosen nde 1,...,N P that guarantees that the tral vector gets at least one parameter from the mutant vector. The crossover constant CR s an algorthm parameter that controls the dversty of the populaton and ads the algorthm to escape from local mnma. Selecton The selecton operator forms the populaton by choosng between the tral vectors and ther predecessors (target vectors) those ndvduals that present a better ftness or are more optmal accordng to (9) [17]. (G) (G) (G), f f ( ) f ( ) (G1), = 1,, NP (9) (G), otherwse Ths optmzaton process s repeated for several generatons, allowng ndvduals to mprove ther ftness as they eplore the soluton space n search of optmal values. DE has three essental control parameters whch are: the scalng factor, the crossover constant CR and the populaton sze NP. The scalng factor s a value n the range [0, 2] that controls the amount of perturbaton n the mutaton process. The crossover constant s 43

7 Economc dspatch soluton usng effcent heurstc search approach Samr SAYAH a value n the range [0, 1] that controls the dversty of the populaton. The populaton sze determnes the number of ndvduals n the populaton and provdes the algorthm enough dversty to search the soluton space. Overvew of harmony search algorthm The harmony search (HS) s a musc-nspred evolutonary algorthm, mmckng the mprovsaton process of musc players searchng for a perfect state of harmony [17-18]. The harmony n musc s analogous to the soluton vector and the behavor of muscan s mprovsaton s analogous to local and global search schemes n optmzaton technques. In the HS algorthm, muscal performances seek a perfect state of harmony determned by aesthetc estmaton, as the optmzaton algorthms seek a best state (.e. global optmum) determned by obectve functon value. The optmzaton procedure of the HS algorthm conssts of the followng steps: 1. Intalze algorthm parameters and harmony memory (HM). 2. Improvse a new harmony from HM. 3. Update harmony memory. 4. Repeat Steps 3 and 4 untl the stoppng crteron s met. The detaled descrpton of these steps s summarzed n the followng [18]: Step 1. Intalze the HS Memory (HM). The ntal HM conssts of a certan number of randomly generated solutons for the optmzaton problem under consderaton. For a D- dmenson problem, an HM wth the sze of HMS can be represented as follows: D D HM (10) HMS HMS HMS D where... ( = 1, 2,, HMS ) s a soluton canddate. 1 2 D Step 2. Improvse a new soluton... from the HM. Each component of ths 1 2 D soluton,, s obtaned based on the harmony memory consderng rate (HMCR). The HMCR s defned as the probablty of selectng a component from the present HM 44

8 Leonardo Journal of Scences members, and 1 HMCR s, therefore, the probablty of generatng t randomly: X HMS wth probablty HMCR wth probablty (1 HMCR) (11) where =1, 2,, D X s the set of the possble range of values for the th decson parameter. After, every component obtaned from the HM s eamned to determne whether t should be ptch-adusted. Ths operaton uses the PAR parameter, whch s the ptch adustment rate as follows: value of Yes, then Yes wth probablty PAR Ptch adustng decson for No wth probablty (1 PAR) The PAR determnes the probablty of a canddate from the HM to be mutated. The ( 1 PAR) sets the rate of dong nothng. If the ptch adustment decson for s replaced as follows: bw (13) where bw s an arbtrary dstance bandwdth and s a random number generated usng unform dstrbuton between 0 and 1. Step 3. Update the HM. Frst, the new soluton from Step 2 s evaluated. If t yelds a better ftness than that of the worst member n the HM, t wll replace that one. Otherwse, t s elmnated. Step 4. Repeat Step 2 to Step 3 untl a termnaton crteron (e.g., mamal number of teratons) s met. HS method s a random search technque whch t does not requre any pror doman nformaton, such as the gradent of the obectve functons. However, dfferent from those populaton-based evolutonary approaches, t only utlzes a sngle search memory to evolve. Therefore, the HS method has the nterestng characterstcs of algorthm smplcty. (12) s Hybrdzaton of dfferental evoluton and harmony search Durng the past decade, hybrdzaton of evolutonary algorthms has ganed growng popularty, whch can overcome ther ndvdual drawbacks whle beneft from each other s 45

9 Economc dspatch soluton usng effcent heurstc search approach Samr SAYAH strengths [15, 22]. In ths contet, Gao et al. [22] have proposed a new hybrd optmzaton technque, namely DE-HS, by embeddng the HS nto the DE method. Both the DE and HS have ther advantages and weakness. Especally, the DE updates the current ndvduals based on only the dfferences among certan randomly selected ones. Unfortunately, t does not have an eplct mechansm for generatng new chromosomes. On the other hand, the HS method has the dstngushng feature of usng the combnaton of all the estng ones to obtan fresh ndvduals. Therefore, a natural dea s to embed ths ndvdual generaton approach nto the orgnal DE so that t can converge n a faster way [22]. The teraton procedure of the DE-HS can be eplaned as follows [22]. The chromosomes n the DE populaton are frst updated accordng to the DE prncple n (6), (7) and (8). Net, these chromosomes are consdered as the harmony memory members of the HS method. Note that the sze of the HM s the same as that of the DE populaton,.e., HMS = NP. Thus, a new ndvdual s produced ether based on the HM members (DE populaton) or n a random manner as n Step 2 of HS algorthm. The key HS parameters, HMCR and PAR, are also used here. Ths fresh canddate s then compared wth the worst DE ndvdual for possble replacement as n Step 3 of the HS method. Fnally, the whole populaton s updated, and the DE evoluton contnues to the net teraton. It can be seen from the above descrpton that the DE and HS are complementary rather than compettve wth each other. Incorporatng the HS-based new ndvdual generaton strategy nto the DE can ndeed enhance ts local eploraton n the search space and result n a superor optmzaton performance. The stoppng crtera are the condtons under whch the search process wll stop. In ths work, the search procedure wll termnate whenever the predetermned mamum number of teratons G ma s reached, or whenever the global best soluton does not mprove over a predetermned number of teratons. Constrants handlng It s worth mentonng that the control varables are self-constraned. In order to keep the tral vectors wthn ther bounds, the control parameter that eceeds a feasble bound s adusted to the correspondng volated bound. 46

10 Leonardo Journal of Scences A penalty functon approach s used to handle the power balance constrant and nequalty constrants. The etended obectve functon F ~ F F K ng 1 P P D P L 2 nc 1 PF ~ (or ftness functon) s defned by: where K denotes the penalty factor of the equalty constrant; nc represents the number of nequalty constrants and PF (14) s the penalty functon for the th nequalty constrant; gven as: lm 2 K (U U ) f volated PF (15) 0 otherwse K s the penalty factor for the th nequalty constrant; varable U. lm U s the lmt value of the Results and Dscusson The proposed DE-HS approach was tested usng the IEEE 30 bus benchmark system [16] and the practcal Algeran 59-bus system [24]. In order to compare the results obtaned from DE-HS, optmzaton problems are also solved usng the conventonal DE and HS methods. For all optmzaton methods (DE-HS, DE and HS), 50 ndependent runs were made n each case study usng the same control parameter settngs, as follows: populaton sze (or HM sze) NP = HMS = 20, mutaton factor = 0.50, crossover constant CR = 0.99, HM consderng rate HMCR = 0.99, ptch adustment rate PAR = 0.10, dstance bandwdth bw = 0.05 and mamum number of teratons G ma = All the programs were mplemented under the MATLAB computatonal envronment, on an Intel Core 2 Duo 2.10 GHz Laptop computer wth 3 GB RAM under Wndows 7. Case 1: Results for the IEEE 30 bus system The IEEE 30 bus 6 generators test system s descrbed n [23], where the total load was set to MW. Table 1 shows the data for the s generators. Over 50 repeated trals, the average soluton cost acheved by DE-HS was $/h wth a mnmum cost of $/h, and a mamum cost of $/h (0.0099% dfference). The average eecuton tme 47

11 Economc dspatch soluton usng effcent heurstc search approach Samr SAYAH was 0.15 second. Table 2 gves the soluton detals for a mnmum cost, whch converged after 40 teratons and 0.14 second. It s mportant to pont out that all varables were able to stay wthn ther permssble lmts sown n Table 1. The convergence of DE-HS for the tral run wth the mnmum fuel cost s shown n Fg. 1. Table 1. Generators data of the IEEE 30 bus system Parameter Gen. 1 Gen. 2 Gen. 3 Gen. 4 Gen. 5 Gen. 6 a ($/Mw 2 h) b ($/MWh) c ($/h) mn P ma P (MW) (MW) a, b and c = cost coeffcents Table 2. The best economc dspatch soluton found by DE-HS for case 1 Parameter Value P1 (MW) P2 (MW) P3 (MW) P4 (MW) P5 (MW) P6 (MW) Total power generaton (MW) Total power demand (MW) Fuel cost ($/h) Loss (MW) 9.79 Eecuton tme (s) 0.14 Table 3 presents the comparson of the mnmum, mamum, mean cost, mean teraton and average computaton tme for the smulaton over 50 tral runs obtaned by the proposed hybrd DE-HS wth the DE and HS algorthms. As seen from Table 3, the best, worst and average solutons obtaned wth the proposed DE-HS algorthm are better than the best, worst and average results of all other methods. It s observed that the best soluton found by DE-HS s dentcal to that found by DE. Besdes, the mean cost gven by DE-HS s better 48

12 Leonardo Journal of Scences compared to other methods. From Table 3, t s also clear that the suggested method converges n less teratons (43 teratons) compared to the conventonal DE (63 teratons) and HS (578 teratons). The mnmum cost and mamum cost are very close (0.0099% dfference) ndcatng a good convergence characterstc and a great robustness of the proposed hybrd DE-HS. Table 3. Comparson results of the three methods (DE-HS, DE and HS) for case 1 Mnmum cost Mamum cost Mean cost Mean Average CPU Method ($/h) ($/h) ($/h) teraton Tme (s) HS DE DE-HS HS = Harmony Search; DE = Dfferental Evoluton Fg. 1 represents the convergence curves of the best solutons found by DE-HS, DE and HS. Ths fgure shows that DE-HS converges well and has the better convergence behavor than the DE and HS. Ths faster convergence behavor of the proposed DE-HS s due to the ncorporaton of the HS ndvdual generaton strategy nto the orgnal DE whch enhances the local search capablty of the orgnal DE method. Fgure 1. Comparson of the convergence curves of DE-HS, DE (Dfferental Evoluton), and HS (Harmony Search) (case 1) The results of the proposed approach were compared n Table 4 to those reported usng Gradent Proecton Method (GPM) [23], Nonlnear Programmng (NLP) [25], 49

13 Economc dspatch soluton usng effcent heurstc search approach Samr SAYAH Evolutonary Programmng (EP) [26], Improved Partcle Swarm optmzaton (IPSO) [27], and Improved Evolutonary Programmng (IEP) [28]. It can be seen that the results gven by DE-HS are close to those reported n the lterature. As a concluson, we can say that the suggested DE-HS has the ablty to fnd acceptable solutons n a reasonable tme compared to other approaches whch allow to an onlne mplementaton. Table 4. Comparson of smulaton results for the IEEE 30 bus system GPM NLP EP IPSO IEP Parameters HS DE DE-HS [23] [25] [26] [27] [28] P1 (MW) P2 (MW) P3 (MW) P4 (MW) P5 (MW) P6 (MW) Total Gen (MW) Cost ($/h) Loss (MW) CPU tme (s) GPM = Gradent Proecton Method; NLP = Nonlnear Programmng; EP = Evolutonary Programmng; IPSO = Improved Partcle Swarm optmzaton; IEP = Improved Evolutonary Programmng; HS = Harmony Search; DE = Dfferental Evoluton Case 2: Results for the practcal Algeran 59 bus system The Algeran network conssts of 59 buses and 10 generators supplyng a total load of MW [24]. Table 5 provdes the parameters assocated wth the generatng unts (t s notng that the generator no. 5 at bus no. 13 s not operatonal). The results found by DE-HS were compared n Table 6 wth the results of DE and HS methods. From the comparatve results of Table 6, t s evdent that the hybrd DE-HS outperforms all other methods n terms of achevng successfully the best mean cost of $/h wth less teratons and less computaton tme than the orgnal DE. It s also observed that the mnmum and mamum costs acheved by DE-HS are very close, whch gves a clear ndcaton of stablty of the results and robustness of the proposed strategy. 50

14 Leonardo Journal of Scences Table 5. Generator parameters of the Algeran network Bus Gen. a b c (MW) (MW) ($/MW 2 h) ($/MWh) ($/h) = Mnmum power generaton; = Mamum power generaton; a, b and c = cost coeffcents mn P mn P ma P ma P Table 6. Comparson results of the three methods (DE-HS, DE and HS) for case 2 Mnmum cost Mamum cost Mean cost Mean Average CPU Method ($/h) ($/h) ($/h) teraton Tme (s) HS DE DE-HS HS = Harmony Search; DE = Dfferental Evoluton The convergence characterstcs of DE-HS, DE and HS are depcted n Fg. 2. It s qute clear that the proposed DE-HS approach has a faster convergence rate. Fgure 2. Convergence behavour of DE-HS, DE (Dfferental Evoluton) and HS (Harmony Search) algorthms (case 2) 51

15 Economc dspatch soluton usng effcent heurstc search approach Samr SAYAH The results of the proposed approach were compared to those reported usng genetc algorthm (GA) [24] and ant colony optmzaton method (ACO) [29]. The comparson results are gven n Table 7. From ths table, t can be seen that the proposed approach gves better results wth a very short computaton tme and the savng n fuel cost s sgnfcant. Ths demonstrates the potental and effectveness of the proposed method to solve the nonlnear optmzaton problems. Table 7. Comparson of smulaton results for the Algeran network Parameters GA ACO [13] [14] HS DE DE-HS P1 (MW) P2 (MW) P3 (MW) P4 (MW) P5 (MW) P6 (MW) P7 (MW) P8 (MW) P9 (MW) P10 (MW) Total Generaton (MW) Cost ($/h) Loss (MW) CPU tme (s) GA = Genetc Algorthm; ACO = Ant Colony Optmzaton; HS = Harmony Search; DE = Dfferental Evoluton Conclusons Incorporatng the HS-based new ndvdual generaton strategy nto the DE can ndeed enhance ts local eploraton n the search space and result n a superor optmzaton performance. Computatonal results show that the proposed hybrd approach has the ablty to converge to hgh qualty solutons wth stable convergence characterstc and good computaton effcency. Consderng the cases and comparatve study presented n ths paper, DE-HS algorthm appears to be very effcent n partcular for ts fast convergence to the global 52

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