An Improved Shuffled Frog Leaping Algorithm Approach for Unit Commitment Problem

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1 Australan Journal of Basc and Appled Scences, 5(6): , 2011 ISSN An Improved Shuffled Frog Leapng Algorthm Approach for Unt Commtment Problem 1 R. Jahan, 2 H. Chahkand Nejad, 1 O. Khayat, 1 M. Mohammad Abad, 1 H. has Zadeh 1 Islamc Azad Unversty, Brjand Branch, Brjand, Iran. 2 Electrcal Engneerng Department, Islamc Azad Unversty, Brjand Branch, Brjand, Iran. Abstract: Solvng unt commtment problem (UCP) s a complex optmzaton process. The exact soluton of the UCP s acheved through a complete enumeraton of all possble combnatons of generatng unts, whch could be a great number. enerally behavor of Unt commtment s formulated as optmzaton problems whch are nonlnear, mxed-nteger combnatons and studed n large-scale. The am s to mnmze the total operatng cost through a generaton schedulng, Whle relatng to a varety of constrants. Ths means that t s desrable to fnd the optmal generatng unt commtment n the power system for the next hours. A new approach va an mproved verson of Shuffled Frog Leapng Algorthm (ISFLA) s ntroduced for solvng optmal unt commtment (UC) problem. We have appled ISFLA to a ten-unt test system and ts multples. To compare obtaned results wth those of many other UC methods,t s clear that not only the ISFLA procedure satsfy constrants very well, t also has some advantages over the others, Such as a good convergence, fast calculatng speed and hgh accuracy. Key words: Improved Shuffled Frog Leapng Algorthm (ISFLA), Power System, Unt Commtment, System Constrants. INTRODUCTION The objectve of the economc schedulng of generators s to guarantee the optmum combnaton of generators connected to the system to provde the load demand. The economc dspatch problem nvolves two separate steps namely unt commtment and on-lne economc load dspatch. The unt commtment nvolves the selecton of unts that wll supply the antcpated load of the system at mnmum cost over a requred perod of tme as well as provdng a specfed margn of the operatng reserve, known as the spnnng reserve. The on-lne economc dspatch dstrbutes the load among those operatng unts that are paralleled wth the system n such a manner so as to mnmze the total cost of supplyng the mnute to mnute requrements of the system. It s qute expensve to run too many generatng unts. A great deal of money can be saved by turnng unts off (decommtng them) when they are not needed. The generc UC can be formulated as to mnmze the operatonal cost subject to mnmum up-tme and down-tme constrants, crew constrants, ramp constrants, unt capablty lmts, duraton of unts, unt status, generaton constrants and reserve constrants. Several soluton technques have been appled to ths problem, ether by usng determnstc, meta heurstc, and hybrd approaches. Determnstc approaches nclude prorty lst (PL) (A.J. Wood & B.F. Wollenberg, 1976), dynamc programmng (DP) (C.K. Pang & H.C. Chen, 1976), Lagrangan Relaxaton (LR) (S.J. Wang, et al. 1995), nteger mxed-nteger programmng (J.A. Muckstadt & R.C. Wlson, 1968), and the branch-andbound methods The prorty lst s the smplest and fastest but acheves poor fnal soluton (J.A. Muckstadt & R.C. Wlson, 1983). Meta-heurstc approach, such as genetc algorthm (A) (S.A. Kazarls, et al. 1996), (H. Ma, A.A, et al. 1994), (Damouss, I., et al. 2004), evolutonary programmng (EP) (K.A. Juste, et al. 1999), smulated annealng (SA) (F. Zhuang & F.D. alana, 1990), tabu search (TS) (A.H. Mantawy, et al. 1999), partcle swarm optmzaton (PSO) (Lee, T., & Chen, C., 2007), (Tng, T., et al. 2003), greedy random adaptve search procedure (RASP) (Vana, A., et al. 2003) are also beng wdely nvestgated to solve the UC problem. These meta-heurstc methods optmzaton methods attract much attenton, because of ther ablty to search not only local optmal soluton but also global optmal soluton and can easly deal wth varous dffcult nonlnear constrants. However, these meta-heurstc methods requre a consderable amount of computatonal tme to fnd the near-global mnmum especally for a large-scale UCP. Correspondng Author: R. Jahan, Wth Islamc Azad Unversty, Brjand Branch, Brjand, Iran. E-mal: Rjahanh@gmal.com 1379

2 Aust. J. Basc & Appl. Sc., 5(6): , 2011 Shuffled frog leapng algorthm (SFLA) s a memetc meta-heurstc that s based on evoluton of memes carred by nteractve ndvduals and a global exchange of nformaton among the frog populaton. It combnes the advantages of the genetc-based memetc algorthm (MA) and the socal behavor-based PSO algorthm wth such characterstcs as smple concept, fewer parameters adjustment, prompt formaton, great capablty n global search and easy mplementaton. Whle proposed prmarly for solvng the multobjectve engneerng problems such as water resource dstrbuton (Eusuff, M.M., and Lansey, K.E., 2003), brdge deck repars (Hatem Elbehary, et al. 2006) and job-shop schedulng arrangement (Alreza Rahm-Vahed,&Al Hossen Mrzae, 2007). The ISFLA s appled to the wdely used ten-unt test system and ts multples. Comparng our results wth those of many UC solvng methods presented n relevant publcatons reveals that the SFLA method s a more effectve technque among the varous methods from both the operaton cost and executon tme aspects. Ths paper s organzed as follows. Secton 2 provdes the mathematcal formulaton of the UCP. Secton 3 ntroduces the bascs of SFLA. Secton 4 proposes an Improved SFLA (ISFLA) method for solvng UCP. Secton 5 gves the numercal example. Secton 6 outlnes the conclusons Nomenclature: t U t P t Index of unts Index of tme-steps Number of generatng unts On/Off status of unt at tme-step t eneraton output of unt at tme-step t F ( P ) a t b c Runnng cost of the unt at tme-step t,, Runnng cost coeffcents of unt S S0 P Dt λ P Rt P max P mn M TO TS S Start-up cost of unt, 1, Start-up cost coeffcents of unt System load demand at tme-step t Equal loss ncremental rate Spnnng reserve Unt maxmum generaton output lmt Unt mnmum generaton output lmt Start up and down tmes lmtaton Mnmum up tme of unt th Mnmum down tme of unt th TO TS Duraton durng whch unt s contnuously on Duraton durng whch unt s contnuously off 1380

3 Aust. J. Basc & Appl. Sc., 5(6): , 2011 UC Problem Formulaton: 2.1. Objectve Functon: It s assumed that the schedule perods are wthn 24 hours and are dvded nto 24 tme-steps. Total cost Is the sum of runnng and start up cost for all unts over the whole schedulng perods. Overall objectve functon of the UC problem s: 24 MnF( U, P ) [ U F ( P ) U (1 U ) S ] t t t t t t t 1 t1 1 enerally, runnng cost, per unt wthn tme nterval s a functon of the generator power output. Cost functon s usually as follows : (1) F ( P ) ap bp c 2 t t t (2) Start up cost of the generator reles on the tme the unt has been off pror to the start up. And ts represented as follows: T S S (1 ) 0 S1 e (3) Shut down cost s gven as each unt s constant value. The shut down cost s consdered 0 for each unt System Constrants: Many constrants are to be satsfed n a unt commtment problem. Power system ndvdually, power pool, relablty councl and so on. may be mposed dfferent rules on the schedulng the unts, dependng on the generaton makeup, load-curve characterstcs and.... Spnnng reserve descrbes total amount of generaton avalable from all unts synchronzed n a system, regardless of present suppled load and losses beng ncurred n a system. Spnnng reserve must be carred out n such a way that one or more unts loss doesn t cause too far a drop n the system frequency. Spnnng reserve must obey certan rules whch state that reserve must be capable of makng loss of most heavly loaded unt up n a gven perod of tme. Reserve requrement also calculated as a functon of the probablty of not havng suffcent generaton to meet the load, by makng people. 1. Power balance constrant 1 UP P, t1,2,3,... t t Dt P t s calculated by the runnng unts at tme-step t accordng to equal loss ncremental rate prncple and met: df1 t df2t dft... (5) dp dp dp 1t 2t t t 1,2,...,24, 1,2,..., (4) 2. Spnnng Reserve If spnnng reserve s be more than 7% of the total load at each tme nterval, t must satsty: 1 UP 1.07 P, t1,2,3,...24 t max Dt (6) 3. Unt eneraton Output Lmtaton P P P 1,2,...,24, 1,2,..., mn max t (7) 4. Start Uage tu zabap- and Down Tmes Lmtaton 1381

4 Aust. J. Basc & Appl. Sc., 5(6): , t 1 U U M t t 1 (8) 5. Mnmum Up and Down-Tme Constrants TO TS TO TS (9) (10) III. Revew of SFLA and MSFLA: III.1. Shuffled Frog Leapng Algorthm: Shuffled Frog Leapng Algorthm (SFLA) s a heurstc search algorthm presented for the frst tme by Eusuff and Lansey n The man purpose of ths algorthm was achevng a method to solve complcated optmzaton problems wthout any use of tradtonal mathematcal optmzaton tools. In fact, the SFL algorthm s combnaton of meme-based genetc algorthm or Memetc Algorthm and Partcle Swarm Optmzaton (PSO). Ths algorthm has been nspred from memetc evoluton of a group of frogs when seekng for food. In ths method, a soluton to a gven problem s presented n the form of a strng, called frog whch has been consdered as a control vector n ths paper as follows n (11). The ntal populaton of frogs s parttoned nto groups or subsets called memeplexes and the number of frogs n each subset s equal. The SFL algorthm s based on two search technques: local search and global nformaton exchange technques. Based on local search, the frogs n each subset mprove ther postons to have more foods (to reach the best soluton). In second technque, obtaned nformaton between subsets s compared to each other (after each local search n subsets). The procedure of SFL algorthm wll be as follows: 1) An ntal populaton of P frogs (P solutons) created randomly whch consdered n ths paper as follows: (11) X1. Populaton (11). X P ( p) (2 N te ) X [ Te, Te,..., Te, Sw, Sw,..., Sw ] 1 2 Nte 1 2 Nte 2) The entre populaton s dvded nto m subsets (m memeplexes), each contanng n frogs (.e., P = m n), n such a way that the frst frog of sorted populaton goes to the frst memeplex, the second frog goes to the second memeplex, frog m goes to m memeplex, and frog m+1 goes to the frst memeplex agan, etc. therefore, n each memeplex, there wll be n frogs. 3) Ths step s based on local search. Wthn each local memeplex, the frogs wth the best and the worst ftness are dentfed as and, respectvely. Also, the frog wth the global best ftness (the best soluton) s dentfed as. Then, the poston of the worst frog s updated (based on frog leapng rule) as follows: D rand ( X X ) b w X ( new) X ( old) D w w ( D D D ) mn max (12) Where rand s a random number between 0 and 1; D max s the maxmum allowed change n frog s poston. If ths process produces a better soluton the worst frog s poston ( X ( old)). w ( X ( new)) w, new poston of the worst frog), t replaces 1382

5 Aust. J. Basc & Appl. Sc., 5(6): , 2011 Fg. 1: The orgnal frog leapng rule. Otherwse, the calculatons n equatons 1 and 2 are repeated wth respect to the global best frog (.e. replaces). If no mprovement becomes possble n ths case, then a new soluton s randomly generated to replace the worst frog (X w ). Because of all arrays n X are ntegers, obtaned solutons from equatons 1 and 2 must be rounded after each teraton. 4) Contnue of prevous step for a number of predefned teratons. 5) After mprovement n frog s postons, new populaton s sorted n a descendng order accordng to ther ftness. 6) If the convergence crtera are satsfed, stop. Otherwse, go to step 2 and repeat agan. III.2. Modfed Shuffled Frog Leapng Algorthm: Accordng to prevous secton, the worst frog n each memeplex mproves ts poston toward the best frog s poston or the global best poston n the same memeplex. But accordng to equatons 1 and 2 and Fg. (1), the possble new poston of the worst frog s restrcted n the lne segment between ts current poston (X w ) and the best frog s poston (X b ), and the worst frog wll never jump over the best one (see Fg. (2)). these lmtatons not only slow down the convergence speed, but also cause premature convergence. Hence, the equatons 1 and 2 must be replaced by new equatons as follows: D rand C( X X ) W b w W [ rw, r w,..., r w ] T 1 1,max 2 2,max Nte Nte,max Xw D f D Dmax Xw( new) D Xw Dmax f D D T DD max (13) (14) Fg. 2: The new frog leapng rule. Where rand s a random number between 0 and 1; C s a constant n the range between 1 and 2; r are random numbers between -1 and 1; w, max are the maxmum allowed percepton and acton uncertantes n the th dmenson of the search space; D max s the maxmum allowed change n frog s poston. Because of all arrays n X are ntegers, obtaned solutons from equatons 13 and 14 must be rounded after each teraton. By applyng equatons 13-14, local search space n each memeplex ncreases. Therefore, the convergence speed ncreases and convergence probablty to acheve the best soluton wll ncrease. For applyng MSFL algorthm to a UCP problem, followng steps must be taken: 1383

6 Aust. J. Basc & Appl. Sc., 5(6): , ) In ths step, requred parameters and nformaton such as number of memeplexes, number of frogs and etc, are defned and determned. 2) The constraned objectve functon s converted to an unconstraned objectve functon accordng to: F( x) f( x) k( ( h ( x)) ) 2 Nueq j 1 Neq j 1 k ( ( Max[0 g ( x)]) ) j 2 2 (15) Above formula s objectve functon of OPUC problem where N eq and N ueq are the number of equal and unequal constrants, respectvely. Also, g (x) and h (x) are equal and unequal constrants, respectvely. k 1, k 2 (k 1, k 2 > 0) are penalty factors whch must have a large value. IV. Optmzaton Strategy by MSFLA: On/Off statue can be easly represented by bnary codng: 1 s on statue and 0 s off statue. If the schedulng perod s dvded nto 24 tme-steps and there are total unts. Then each unt has 24 bts (Fg. 3)..e. 2nd bt of unt 1 represents the on/off statue of unt 1 at 2nd tme-step. One bnary codng ndvdual can be combned accordng to the order of unts and each ndvdual has total 24 bts. Per bt of each ndvdual n one populaton s produced randomly. Ths paper transforms the orgnal constraned UC problem nto unconstraned one by usng penalty functon. n C MnF u jr j 1 j Where, F s orgnal objectve functon; n c s the number of volaton constrants; R j and u j are the volaton value and penalty coeffcent of jth constrant, respectvely. Equaton (16) only ncludes spnnng reserve, start up and down tmes, mnmum up- and down-tme constrants. The power balance and unt generaton output lmtaton s consdered n the load dspatch. The ftness functon s: K TF n C F ujr j1 j Constant K s proportonal coeffcent. The value of K and u j should be selected accordng to the specfc problem. The values should let the ftness value of feasble soluton be around 1 to prevent computer treatng too large or small value. (16) (17) Fg. 3: The bnary representaton of unt commtment. The proposed optmzaton problem can be summarzed as: Step 1) Intalze the parameters of the unt commtment problem. 1384

7 Aust. J. Basc & Appl. Sc., 5(6): , 2011 Step 2) Set the chromosomes as the 24-hours bts unts. Create an ntal populaton of k frogs generated randomly. Step 3) Sort the populaton ncreasngly and dvde the frogs nto p memplexes each holdng q frogs such that k = p q.. The dvson s done wth the frst frog gong to the frst memplex, second one gong to the second memplex, the pth frog to the pth memplex and the p + lth frog back to the frst memplex. Step 4) For each memplex, Regardng to the problem constrants, set the cost functon whch s to be mnmzed (Equaton (11)). Step 4-1:Set p 1 = 0 where p 1 counts the number of memeplexes and wll be compared wth the total number of memeplexes p. Set y 1 = 0 where y 1 = 0 counts the number of evolutonary steps and wll be compared wth the maxmum number of steps (y max ), to be completed wth n each memeplex. Step 4-2:Set p 1 = p Step 4-3:Set y 1 = y Step 4-4:For each memplex, the frogs wth the best ftness and worst ftness are dentfed as X w and X b, respectvely. Also the frog wth the global best ftness X g s dentfed. Then the poston of the worst frog X w for the memplex s adjusted as follows: B rand(.) ( X b X w) (18) new X old X B ( B B B ) w w max max Where rand (.) s a random number between1 and 0 and Bmax s the maxmum allowed change n the frogs poston. If the evolutons produce a better frog (soluton), t replaces the older frog. Otherwse, X b s replaced by X g n (18) and the process s repeated. If non mprovement becomes possble n ths case a random frog s generated whch replaces the old frog. Step 4-5:If P 1 < P, return to step5-2. If y 1 < y, return to step 5-3. Other wse go to step 4. Step 5: Check the convergence. Stop n the case that convergence crtera s satsfed. Otherwse, consder the new populaton as the ntal populaton and return to the step4. The best soluton found n the search process s consdered as the output results of the algorthm. Step 6: Prnt out the optmal soluton to the target problem. ncludng poston and optmal unt commtment soluton, and the correspondng ftness value represent the mnmum Cost Functon. V. Numercal Results: Fast Messy genetc algorthm program usng vsual C++ s presented n ths paper. 10 generaton unt system and ts multples (10-100). s tested and the results are compared to other algorthms. Table 1 gves the 24-h unts outputs for the ten-unt case. Test results and Comparson wth s shown n Table 2 and Table 3, respectvely. Obtaned results represents a better and optmzed global optmal soluton to the problem and verfes the correctness of the proposed algorthm. Table 1: Optmal Unt Commtment Result. Unt On/Off Statue of Per Tme-Step Number Table 2: Total costs of the FMA method for test systems. No. of unts Best cost ($) Average cost ($) Worst cost ($) , , , ,122,766 1,125,109 1,125, ,242,153 2,248,267 2,248, ,363,430 3,370,828 3,371, ,485,583 4,486,233 4,487, ,605,009 5,606,992 5,608,

8 Aust. J. Basc & Appl. Sc., 5(6): , 2011 Table 3: Total Cost Comparson of Several Methods. NO. of Unts SPL EP LR A PSO MA , , , , , , ,123,938 1,127,256 1,128,362 1,126,243 1,125,983 1,128, ,248,645 2,252,612 2,250,223 2,251,911 2,250,012 2,249, ,371,178 3,376,255 3,374,994 3,376,625 3,374,174 3,370, ,492,909 4,505,536 4,496,729 4,504,933 4,501,538 4,494, ,615,530 5,633,800 5,620,305 5,627,437 5,625,376 5,616,314 NO.of Unts DP LRA MILP ALR ACSA SFLA n ths paper , , , , , ,122,622-1,126,720-1,122, ,242,178-2,249,790-2,242, ,371,079-3,371,188-3,363, ,501,844-4,494,487-4,485, ,613,127 5,605,189 5,615,893-5,605,009 VI. Concluson: SFLA s effcently mplemented to solve the UC problem. SFLA total producton costs over the scheduled tme horzon are less expensve than other methods on the large number of generatng unts. The proposed algorthm consder varous constrants successfully and the genetc operatons are mproved based on the power system characterstc. The test results represent the SFLA effects on searchng global or near global optmal soluton n the UC problem. Also the results demonstrate a better convergence and they are hghly mproved. REFERENCES Alreza Rahm-Vahed and Al Hossen Mrzae, Solvng a b-crtera permutaton flow-shop problem usng shuffled frog-leapng algorthm, Soft Computng, Sprnger-Verlag. Cohen, A.I. and M. Yoshmura, A Branch-and-Bound Algorthm for Unt Commtment, IEEE Transacton on Power Systems, PAS-102(2): Damouss, I., A. Bakrtzs and P. Dokopoulos, A soluton to the untcommtment problem usng nteger-coded genetc algorthm. IEEE Transactons on Power Systems, 19(2): Dllon, T.S. and K. Wedwn, Integer programmng approach to the problem of optmal unt commtment wth probablstc reserve determnaton, IEEE Transacton on Power Systems, PAS-97. Eusuff, M.M. and K.E. Lansey, Optmzaton of Water Dstrbuton Network Desgn Usng the Shuffled Frog Leapng Algorthm, J Water Resour Plan Manage, 129(3): Hatem Elbehary, Emad Elbeltag and Tarek Hegazy, Comparson of Two Evolutonary Algorthms for Optmzaton of Brdge Deck Repars, Computer-Aded Cvl and Infrastructure Engneerng, 21: Juste, K.A., H. Kta, E. Tanaka and J. Hasegawa, An evolutonary programmng soluton to the unt commtment problem, IEEE Transactons on Pover Systems, 14(4): Kazarls, S.A., A.. Bakrtzs and V. Petrds, A genetc algorthm soluton to the unt commtment problem, IEEE Trunsactons on Power Systems, 11(1): Kazarls, S. and A. Bakrtzs, A genetc algorthm soluton to the unt commtment problem. IEEE Transactons on Power Systems, 11(1): Lee, T. and C. Chen, Unt commtment wth probablstc reserve An IPSO approach. Energy Converson and Management, 48: Ma, H., A.A. El-Keb and R.E. Smth, A genetc algorthm-based approach to economc dspatch of power systems, IEEE Proceedngs, Southeastcon 94. Creatve Technology Transfer - A lobal Affar, pp: Mantawy, A.H., Y.L. Abdel-Magd and S.Z. Selm, Integratng genetc algorthms, tabu search, and smulated annealng for the unt commtment problem, IEEE Transactons on Power Systems, 14(3): Muckstadt, J.A. and R.C. Wlson, An Applcaton of Mxed-Integer Programmng Dualty to Schedulng Thermal eneratng Systems, IEEE Transacton on Power Systems, pp: Pang, C.K. and H.C. Chen, Optmal short-term thermal unt commtment, IEEE Transacton on Pover Systems, PAS-95, pp: Tng, T., M. Rao and C. Loo, Solvng unt commtment problem usng hybrd partcle swarm optmzaton. Journal of Heurstcs, 9: Vana, A., J. Sausa and M. Matos, Usng RASP to solve the unt commtment problem. Annals of Operatons Research, 120(1):

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