Generation Maintenance Scheduling for Generation Expansion Planning Using a Multi-Objective Binary. Gravitational Search Algorithm

Transcription

1 Internatonal Transacton of Electrcal an Computer Engneers System, 2014, Vol. 2, No. 2, Avalable onlne at Scence an Eucaton Publshng DOI: /teces Generaton Mantenance Scheulng for Generaton Expanson Plannng Usng a Mult-Objectve Bnary Gravtatonal Search Algorthm Iman Gorooh Sarou *, Mohamma Tagh Amel, Mehra Setayesh nazar Faculty of Electrcal an Computer Engneerng, Shah Behesht Unversty, Tehran, Iran *Corresponng author: _gorooh@sbu.ac.r Receve December 22, 2013; Revse February 18, 2014; Accepte February 23, 2014 Abstract Generaton mantenance scheulng (GMS) s an mportant an effectve part of Generaton expanson plannng (GEP) problem. Preventve-mantenance scheules nee to be optmze to trae-off between two conflctng objectves, reucng the overall cost an mprovng the relablty. Ths paper presents a mult-objectve bnary gravtatonal search algorthm (BGSA) for solvng GMS problem of generaton systems as a sub-problem of the man GEP problem. In the propose metho, a fuzzy membershp functon s efne for each term n the objectve functon. There are three objectve functons n ths problem. The frst objectve functon s levelng reserve capacty when unt mantenance outages are consere. The secon an the thr objectves whch are also objectves of the man GEP problem, are to mnmze the operaton an mantenance (O&M) cost an the relablty nex of Expecte Energy Not Supple (EENS). As GMS problem s a sub-problem of the man GEP problem, t s solve for a typcal soluton of the man GEP problem. The propose metho s apple to solve GMS problem for 4-bus test system from Granger & Stevenson an IEEE-RTS 24-bus test system for a plannng horzon of one year an two years, respectvely. To verfy the capablty of the propose BGSA base metho, a bnary genetc algorthm (BGA) metho s also mplemente to solve GMS problem an then the results are compare. Keywors: generaton mantenance scheulng, generaton expanson plannng, levelze reserve, Gravtatonal Search Algorthm, Genetc Algorthm, mult-objectve Cte Ths Artcle: Iman Gorooh Sarou, Mohamma Tagh Amel, an Mehra Setayesh nazar, Generaton Mantenance Scheulng for Generaton Expanson Plannng Usng a Mult-Objectve Bnary Gravtatonal Search Algorthm. Internatonal Transacton of Electrcal an Computer Engneers System, vol. 2, no. 2 (2014): o: /teces Introucton Generaton Expanson Plannng (GEP) s the frst key step n long-term plannng ssues, after correctly forecastng the loa for a specfe future pero. In general terms, GEP s an optmzaton problem n whch the am s to etermne the new generaton plants n terms of when to be nveste, what type an capacty they shoul be an where they shoul be nstalle, so that the cost functon s mnmze an varous constrants are satsfe. An essental feature means that the relablty requrement wthn the power nustry s very hgh. One of the man subjects that s of great mportance to the relablty of power systems s to mantan the avalablty of generatng unts n an acceptable level. On the other han, the mantenance scheules of power generatng unts play a crtcal role n both plannng an operaton management sectons of power system. The number of generators s ncrease an the number of reserve margns are ecrease by the electrcty eman growng n moern power systems. Consequently, GMS has become a complex, mult-object-constrane optmzaton problem for power systems [1,2,3,4]. unt commtment, generaton spatch, mport/export of generatng unts, generaton expanson plannng an such other plannng actvtes are rectly affecte by GMS problem ecsons. For ths reason, GMS problem s essental to solve for the plannng of the certan an relable power system operaton n a moern power system. The purpose of mantenance scheulng s to fn the sequence of scheule outages of generatng unts over a gven pero of tme such that the level of energy reserve s mantane. A lot of research has been carre on GMS for several ecaes. A fuzzy system approach wth genetc enhancement s apple to GMS n [1]. In ths reference, fuzzy sets are use to optmally ajust membershp functons parameters. [2-7] have ntrouce meta-heurstc algorthms base approaches such as mofe screte partcle swarm optmzaton (MDPSO) an smulate annealng to form robust methos for solvng the GMS problem. In these references, economc benefts an relable operaton of a power system are mprove, subject to satsfyng system loa eman, allowable mantenance wnow, crew an resource constrants. Securty-constrane GMS problem are solve n [8,9,10,11] usng an evolutonary programmng-base soluton technque an general

2 45 Internatonal Transacton of Electrcal an Computer Engneers System algebrac moelng system, conserng uncertantes n the loa, fuel an mantenance costs. In [12,13], formulatons are presente that enable Ant Colony Optmzaton (ACO) for contnuous omans to seek the optmal soluton of GMS problem. The objectve functons of these formulatons conser the effect of economy as well as relablty. In these references, varous constrants such as spnnng reserve, uraton of mantenance crew are taken nto account. [14,15,16] have propose a ecson tree an mxe nteger lnear formaton for solvng the GMS problem that clearly conser the thermal unts agng momentum. Ths allows the system operators to etermne the thermal unts mantenance outage scheule base on the cost analyss of mantenance tasks. Ths cost analyzng shoul be one by optmzng the tme nterval between consecutve mantenance tasks. In the present paper, GMS problem s solve as a subproblem of the man GEP problem to optmze an reajust the mantenance scheme of generatng unts accorng to the economcal nces an relablty nces of the system operaton n fferent tmes. A multobjectve fuzzy approach s use to solve GMS problem. Two objectves of mnmzng O&M cost an mnmzng the EENS nex are common to the GMS an GEP problems. Therefore, a probablstc proucton smulaton s propose to obtan EENS nex an O&M cost of the generatng unts accorng to ther mantenance scheules. The rsk of the system's supply beng not suffcent may be ncrease urng the scheule mantenance outages. Therefore, levelng reserve capacty (the system's nstalle capacty mnus the mum loa an the mantenance capacty urng the pero uner examnaton) s consere as the another objectve functon to guarantee the electrcty supply relablty of the power system. Gravtatonal Search Algorthm (GSA) [17,18] s base on the law of gravty an mass nteractons. The hgher effcency of GSA n compare to some other heurstc optmzaton methos such as Partcle Swarm Optmzaton (PSO), Central Force Optmzaton (CFO) an Real Genetc Algorthm (RGA) n solvng varous nonlnear functons, are emonstrate n [17]. [17] has examne GSA on a set of varous stanar bench mark functons. The results obtane by GSA n most cases have prove superor results an n all cases were comparable wth PSO, RGA an CFO. Consequently, Bnary Gravtatonal Search Algorthm (BGSA) s use as the optmzaton tool to solve GMS problem n the present paper. Beses, a fuzzy membershp functon s efne for each term n the objectve functon. GMS problem s solve for the best soluton of the man GEP problem that s evaluate wthout conserng mantenance scheulng of generatng unts. 4-bus test system from Granger & Stevenson for a plannng horzon of one year an IEEE- RTS 24-bus test system for a plannng horzon of two years are use as test systems. BGA s also mplemente to solve GMS problem to satsfy the effectveness of the propose BGSA, an then the results are compare an nterprete. The results ncate that the BGSA s an effectve alternatve for the soluton of the propose GMS problem. The man contrbuton of ths paper s summarze at the followng: -Proposng a new methoology for solvng GMS problem as a sub-problem of the man GEP problem takng nto account the economcal nces an relablty nces -Proposng new formulaton for computng O&M cost an EENS -Applyng mult-objectve BGSA to solve the propose GMS problem an verfyng ts compatblty n comparson wth BGA. 2. Problem Formulaton In ths secton, confgure formulaton for GMS problem has been evelope. The problem s to etermne the set of generatng unts nvolvng n mantenance n each tme nterval. In fact, GMS varables are tme ntervals an ay number of them, when each generatng unt s allowe to be scheule for mantenance. In ong so, beses satsfyng varous constrants, such as the mantenance crew constrant an transmsson state constrant, the present value of O&M costs an the nex of EENS shoul be mnmze. In aton the reserve capacty s levelze urng the tme ntervals. The propose objectve functons as well as the varous constrants are scusse at the followng subsectons Fuzzy Mult-objectve Formulaton A fuzzy optmzaton proceure s use n ths paper for hanlng the multple objectves. In ths approach, objectve functon terms can be summe up by transformng elements of objectve functons nto fuzzy oman wth a sutable membershp functon. Membershp functons ncate the egree of satsfacton of the objectves. A hgher membershp value mples greater satsfacton wth the soluton. The shape of membershp functon epens on the nature of the relate objectve an t may be ece by the ecson maker [19,20,21]. Overall satsfacton of plan s obtane from Eq. (1). Mn μ = k1. μreserve + k2. μcos t + k3. μeens (1) Where, k1, k2, k3 are the non-negatve weghtng coeffcents satsfyng the conton: k 1 +k 2 +k 3 =1, whch can be tune usng the expert knowlege. Depenng on the purpose, the optmzaton proceure can be realy aapte smply by changng the values of the weghtng coeffcents an ths s an nterestng feature for the multobjectve formulaton. μ reserve, μ cost an μeens are respectvely the membershp functon values for three objectve functons, levelzng the reserve capacty, mnmzng O&M cost an mnmzng the relablty nex of EENS, whch are escrbe at the followng. In the crsp oman, ether the objectve s satsfe or t s volate, respectvely, membershp values of unty or zero are mple. Fuzzy set membershp may have a value from zero to unty. Lower an upper boun values of an objectve wth a strctly monotoncally ecreasng an contnuous functon, create a membershp functon. A commonly use membershp functon s use for both objectve functons employe n ths problem, as shown n Fgure 1 [21,22]. Ths relatonshp s expresse mathematcally n Eq. (2). Inex of f ( X ) n ths fgure an equaton can be replace by nces M t, EENS an I resv.

3 Internatonal Transacton of Electrcal an Computer Engneers System 46 1 f ( ) mn X < f f f ( X) µ = f mn f ( X) f f ( X) mn < (2) f f 0 f < f( X) As can be seen from Fgure 1, a soluton wth a lower f ( X ) has a greater membershp value, an vce versa. In aton, mn f an f can be calculate by solvng the sngle objectve problem for mnmzng an mzng the objectve functon, separately [23]. µ f ( X ) 1 0 mn f f f ( X ) Fgure 1. Membershp functon use for all objectve functons ( f ( X ) can be replace by any of nces M t, EENS or I resv) Loa Capacty (MW) Peak loa: 800 MW Mantenance:300MW Mantenance:200MW Mantenance:100MW... Average N nt Tme ntervals (ays) 2.2. Levelze Reserve Capacty For convenence n scheulng, the whole mantenance pero s ve nto N st stages that each stage s ether one week, ten ays or one month. The levelze reserve metho s a straght an nstnctve metho that raws up a mantenance scheule by levelng the system's net reserve capacty throughout the tme ntervals. Actually, sum of the system loa an capacty of generatng unt's mantenance outages throughout the tme subntervals are Fgure 2. Metho of reserve capacty levelng usng n Eq. (3) mae unform n ths metho. Not that, the tme subntervals are specfe urng the tme ntervals, conserng the startng ay an the length of generatng unts mantenance outages. Ths s a metho conventonally use n GMS problem an t has foun we applcaton n electrc utltes owng to ts easy an unerstanable concept. As aforementone, reserve capacty levelng s one of the objectve functon of the propose metho. In ong so, an nex s efne to evaluate ths objectve functon. Fgure 2 an Eq. (3) are

4 47 Internatonal Transacton of Electrcal an Computer Engneers System use to calculate ths nex, takng nto account the generatng unt's mantenance outages. 12 area uner mum capacty th m= 1l ne n m tme nt erval Iresv = 1 area uner average capacty lne (3) As can be seen, Iresv mn s equal to 0, as long as the mum capacty s the same throughout the tme ntervals. N nt s total number of tme subntervals, that s epenent on GMS problem soluton O&M Cost O&M cost s one of the objectves use n propose fuzzy mult-objectve approach. O&M costs n each year are compose of fuel costs an non-fuel O&M costs of generaton, whch can be evaluate as: mn base HRGk EG peak + HRG N m k EG per Nb N k k, b, f var tot Mt = CFGkb, Comkb, EG + f = 1 b= 1k= 1 = 1 fx k + Comkb, PG (4) base EG, peak EG an tot EG,respectvely, enote the energy generate n base an peak capacty, an total energy generate by the type-k th plant unt locate at bus b, n pero f of year t of the stuy pero. In ths stuy, the energy generate by each unt n the system s calculate by probablstc proucton smulaton. In ths approach the force outages of thermal unts are convolve wth the nverte equvalent loa uraton curve (ELDC) an, consequently, the effect of unexpecte outages of thermal unts upon other unts s accounte n a probablstc way. The net effect s that the expecte generatng costs of the system s ncrease by an ncrease of peakng unts generaton n orer to make up the reucton of base unts generaton ue to scheule outages for mantenance an unt falures [24]. Inces of base kb,, f, t peak kb,, f, t tot kb,, f, t EG, EG an EG are calculate usng probablstc proucton smulaton conserng an ELDC. Eq. (5) shows the equaton of the ELDC for fxe an canate unts: ( ) ( 1) ( 1) f ( x) = pf ( x) + qf ( x c) (5) Where P s the operaton rate of generatng unt an q =1-p s equal to the unt s Force Outage Rate (FOR); c s generatng capacty for unt n p.u; energy generate by each unt s etermne usng Eq. (6) Where x s equal to x Eg = Tp ( 1) f ( x) x (6) x 1 C j. j= Expecte Energy not Supple As alreay mentone, EENS s another objectve functon of the propose metho. A generaton unt maybe trppe at a rate gven by ts FOR, whch represents the percentage of tme, when unt maybe unavalable ue to unexpecte outages. Some porton of the energy eman may not be serve owng to generatng unt's FORs an mantenance outages. Probablstc proucton smulaton nvolvng ELDC s also mplemente to compute EENS as n Eq. (7). x + C t ( EENS = T f n) ( x) x (7) C t where C t s the total capacty of all of the generatng unts urng the tme ntervals Constrants of the Propose Metho In the propose GMS problem, two types of constrants are mpose: techncal constrants comprsng of GEP constrants an basc constrants consstng of fuel constrants, mantenance constrants, mantenance crew constrants an transmsson state constrants. The transmsson state constrants are some specal lmts mpose on the generatng unt mantenance plan n a certan area by the transmsson network operaton state. Some of these constrants are escrbe at the followng: Techncal Constrants Generaton capacty shoul be suffcent n meetng the loa requrements, though there are uncertantes that may cause generaton unts to trp unexpectely at any tme as n Eq. (8). Eq. (8) smply states that the nstalle capacty n the crtcal pero must le between the gven mum an mnmum reserve margns a t an b t, respectvely above the peak eman (D t,c ) urng the crtcal pero of the year. ( ) ( ) ( ) 1+ at Dtc, P Ktc, 1+ bt Dtc, t = 1,..., T (8) Basc Constrants a) The contnuty of mantenance actvty: Mantenance must be complete n a contnuous tme nterval once starte as n Eq. (9) stk+ Sk 1 mk, st = Sk k S (9) st= st k b) The mnmum tme nterval between two mantenances to each generatng unt: Suppose that the tme nterval s consere as N stages; then ths constrant for unt k s presente as follows: ( ) st st + S N k S (10) k2 k1 k1 c) Mantenance crew constrant: The mantenance crew constrant means that there are lmtatons ue to the mantenance manpower avalablty; Normally, two generatng unts cannot be scheule for mantenance together n the same power plant an at the same tme,.e. V r,st =1. Only a few power plants wth conserable

5 Internatonal Transacton of Electrcal an Computer Engneers System 48 mantenance resources can allow V r,st >1. However, the followng equaton must stll be met [24]: mk, st Vr, st k V r st = 1,..., Nst (11) Fgure 3. The flowchart of BGSA 3. Soluton Methoology As aforementone, the purpose of ths problem s to fn the optmum mantenance scheule of generatng unts. The objectves are escrbe n Secton 2. In ths secton, frstly, BGSA s escrbe, an then t s employe to the propose metho Bnary Gravtatonal Search Algorthm (BGSA) In the BGSA [17,18], agents are consere as objects an ther performance s measure by ther masses. All these objects attract each other by the gravty force, an a global movement of all objects towars the objects wth heaver masses, s generate by ths force. As a result,

6 49 Internatonal Transacton of Electrcal an Computer Engneers System masses cooperate usng a rect form of communcaton, through gravtatonal force. The heavy masses whch correspon to goo solutons guarantees the explotaton step of the algorthm, owng to ther more slowly movement than lghter ones. The BGSA coul be consere as a small artfcal worl of masses obeyng the Newtonan laws of gravtaton an moton system of masses. The flow agram of BGSA s epcte n Fgure 3. worst(t) an best(t) n ths fgure are efne as follows (for a mnmzaton problem): best() t = mn ft j () t (12) j { 1,... N} worst() t = ft j () t (13) j { 1,... N} Where, ft (t) represent the ftness value of the agent at tme t. In BGSA a system wth N agents (masses) s consere. The poston of the th agent s efne by: 1 n X = ( x,..., x,..., x ) = 1, 2,..., N (14) The step of Ftness evaluaton s performe usng followng equatons: ft () t worst() t m ( t) = = 1,2,..., N best() t worst() t m () t M () t = N mj () t j= 1 To upate agents poston Eqs. (17-24) are use: M() t M j() t Fj () t = G() t ( xj () t x ()) t Rj () t + ε (15) (16) (17) Rj () t = X (), t X j () t (18) 2 The total force that acts on agent n a menson ( F () t ) s suppose to be a ranomly weghte sum of th components of the forces exerte from other agents, gvng a stochastc characterstc to the algorthm: N () = j j () (19) j= 1, j F t ran F t Where r an j s a ranom number n the nterval [0,1]. F () t a () t = M () t (20) v ( t + 1) = ran v () t + a () t (21) v s lmte to an approprate nterval for a better convergence as Eq. (22). v < v (22) In BGSA a probablty functon s efne for each agent as: S( v ( t)) = tanh( v ( t)) (23) Poston change of each agent n a menson of bnary space wll cause ts value change from 0 to 1 or vce versa. Followng the above calculaton of probablty functon each agent moves n each menson as Eq. (24). f ran < S( v ( t + 1)) then x ( t + 1) = complement( x ( t)) else x ( t+ 1) = x ( t) (24) t=1 t=t t=t st=1 st=2 st=3 st=st st= N st t= number of years st=number of stages u=1 u=2 b b b b b b b b b b b 1 u=u b 3 b 2 b 4 b 5 u= N u b b b b b u=number of generatng unt Fgure 4. The ata structure of each agent poston of propose BGSA Table 1. Integer structure an corresponng escrpton for the poston of each agent n each menson Bnary coe Bnary to ecmal converson Descrpton There s no mantenance scheule n ths tme stage The startng ay of the tme stage when generatng unt allowe to be scheule for mantenance There s no mantenance scheule n ths tme stage 3.2. Applyng BGSA to the Propose Metho To solve GMS problem usng BGSA, problem varables are combne an represente as a mxe nteger cong n each agent poston. The ata structure of each agent poston s epcte n Fgure 4. As can be seen from ths fgure, the bnary to ecmal converson of each fve bts of poston of each agent n each menson refers to the startng ay of the tme stage st when generatng unt u s

7 Internatonal Transacton of Electrcal an Computer Engneers System 50 allowe to be scheule for mantenance n year t of the stuy pero, as shown n Table 1. In the propose BGSA, canate solutons of ntal populaton are ranomly selecte between all solutons whch satsfy the problem constrants an also new solutons obtane through the BGSA s proceure are checke to sustan feasblty. Flowchart of the man GEP problem along wth the propose metho to solve GMS problem s presente n Fgure 5. As can be seen from ths fgure GMS problem s solve as a sub-problem of the man GEP problem. Generaton nvestment cost an transmsson enhancement cost presente n ths fgure are other objectve functons of the man GEP problem. To hanle the problem complexty, t s assume that, the transmsson enhancement requrements are approxmately proportonal to the length-base overloas calculatng by runnng an AC power flow urng crtcal operatng pero. It s worthwhle to note that, as GEP problem s not the man subject of ths paper an GMS problem s solve for a typcal soluton of the man GEP problem, so objectves of generaton an transmsson enhancement costs are not consere n secton of smulaton results of ths paper. Intal nformaton GEP Problem constrants Generate ntal ranom soluton (expanson plans) GMS subproblem GMS subproblem constrants Objectve functon Generate ntal ranom soluton (mantenance plans) Stop Yes Converge? Investment cost Select the crtcal pero No Probablstc proucton smulaton levelze reserve metho Algorthm operators AC Power flow O&M cost expecte Energy not supple (EENS) Inex of reserve capacty levelzng Multobjectve Fuzzy metho Operaton cost Cost of expecte Energy not supple Cost of transmsson enhancement Objectve functon Stop Yes Converge? No Algorthm operators 4. Smulaton Fgure 5. Flowchart of the man GEP problem along wth the propose metho to solve GMS problem In orer to emonstrate the effectveness of the propose approach, t s apple to the 4-bus test system from Granger & Stevenson for a plannng horzon of one year an an IEEE-RTS 24-bus test system for a plannng horzon of two years wth growng complexty. The best soluton of the man GEP problem applyng to case stues wthout conserng mantenance scheulng of generatng unts, whch GMS problem s solve for, s presente n Table 2. In ths stuy, for convenence n scheulng, the whole mantenance year s ve nto 12 stages (N st =12);.e. every stage s one month. Canate plants for GEP problem are selecte from four fferent types of natural gas unts, coal unts, ol unts an nuclear unts.

8 51 Internatonal Transacton of Electrcal an Computer Engneers System Table 2. The best soluton of the man GEP problem applyng to both case stues Number of unts to be ae Unt Descrpton Case stuy 1 Case stuy 2 Unt name Unt type Capacty (MW) Mantenance uraton (ays) 1st year 1st year 2n year U12 Fxe U20 Fxe U50 Fxe U76 Fxe U100 Fxe U155 Fxe U197 Fxe U318 Fxe U350 Fxe U400 Fxe FCOA Canate FCC Canate FOIL Canate NUCL Canate Two case stues an ther results are escrbe at the followng. In the frst case, the problem s apple to a 4- bus, 1-generator case from Granger & Stevenson [25]. The fgure of ths system s presente n Fgure 6. As can be seen, there s only one fxe nuclear unt of 318 MW nomnal capacty, an 8% FOR; total network loa s 500 MW. For ths case stuy, GMS problem s solve for a plannng horzon of one year. An IEEE-RTS 24-bus test GMS problem s apple for a plannng horzon of two years wth growng complexty [26]. There are 32 fxe unts of 3,405 MW total nomnal generatng capacty; Total network loa s 2,850 MW. The propose BGSA that s valate for case stuy 1 s also employe to solve the GMS problem for ths case stuy. The mum loa urng the stages for both Case stues are presente n Table 3. system s selecte as the secon case stuy to whch the Fgure 6. Case stuy 1: 4 bus test system from Granger & Stevenson Table 3. Maxmum loa (MW) urng the stages for both case stues (Y (Year), M (Month)) Case Stues 1 st M 2 n M 3 r M 4 th M 5 th M 6 th M 7 th M 8 th M 9 th M 10 th M 11 th M 12 th M Case Stuy 1 Y Case Stuy 2 Y Y Frst, BGSA s use as the optmzaton tool, then BGA [22] s use to solve the problem for feasble solutons, satsfyng the problem constrants. As the performance of heurstc search methos reles on ther parameters, so the algorthm parameters are tune for both BGSA an BGA to obtan a better soluton. In both cases, populaton sze s set to 100 (N= 100) an mum teraton s 60. In ths αt stuy, equaton ( Gt () = Ge T 0 ) s use to set parameter G of BGSA, where G 0 s set to 100 an α s set to 12, an T s the total number of teratons. Total objectve functon s evaluate for ths case stuy as escrbe n Secton 2. A probablstc proucton smulaton s also use to

9 Internatonal Transacton of Electrcal an Computer Engneers System 52 calculate the energy generate by each unt an also the expecte energy not serve n each pero for these case stues, takng nto account the generatng unts mantenance scheule Smulaton Results Fgure 7 shows the ELDC of the probablstc proucton smulaton for the propose mantenance plan by the BGSA for case stuy 1. As can be seen, the equvalent loa uraton curve s compute for sx fxe an canate unts wth respect to ther FOR. Table 4 shows propose mantenance plan by BGSA for both case stues. The mantenance scheule for each generatng unt s shown n ths table by enotng month an ay of the mantenance startng urng n each year of the scheulng horzon. Two scheulng turns are consere for unts whch are fxe or be ae at the frst year of plannng horzon. It s noteworthy that the propose mantenance plan by the BGSA an BGA applyng to case stuy 1 are the same. Total generate energy an O&M costs of generaton of each unt urng the frst stage for mantenance plan of case stuy 1 are presente n Table 5. The value of three objectve functons, ther membershp functon values an total objectve functon for two case stues are presente n Table 6. As can be seen from ths table, scounte values of total operatng costs are respectvely equal to K$ an K$ for case stues 1 an 2. In calculatng the total objectve functon nexes k 1, k 2 an k 3 are assume to be equal (.e. k 1 = k 2 = k 3 =1/3). Member shp functon values are compute from the value of objectve functons usng Eq. (2) an Fgure 1. Eq. (1) s use for calculatng the total objectve functon. Usng reserve capacty levelzng metho shown n Fgure 2 an Eq. (3), nex of I resv s specfe by consstng the capacty fgure. Dong so, sum of areas uner mum capacty lne of tme ntervals an also the average capacty an the area uner average capacty lne s calculate as shown n Table 6. Table 4. Propose mantenance plan by BGSA for case stues (Y (Year), M (Month), D (Day), FT (Frst Turn), ST (Secon Turn)) Case Stues Unts Frst Unt Secon Unt Thr Unt 4 th Unt 5 th Unt 6 th Unt Y M D Y M D Y M D Y M D Y M D Y M D U Case stuy 1 FCC FOIL U12 FT ST U20 FT ST U50 FT ST U76 FT ST U100 FT ST U155 FT ST Case Stuy 2 U197 FT ST U350 FT ST U400 FT ST FCOA FT ST FCC FT ST FOIL FT ST NUCL FT ST Table 5. Generate Energy an O&M costs of generaton of each unt urng frst stage for propose mantenance plan by BGSA for Case stuy 1 Fuel costs (K$) Non-fuel O&M costs (K$) O&M cost nvolvng fuel costs Unt number Unt Type Total Generate Energy (GWH) (K$) 1 U F-CC F-CC FOIL FOIL FOIL FOIL 5.09 Total M t

10 53 Internatonal Transacton of Electrcal an Computer Engneers System Case stuy1 Case stuy2 Average capacty (MW) Table 6. Smulaton results of propose BGSA-base metho for both case stues Area uner Sum of areas uner EENS average capacty mum capacty lne of I resv Mt ( K $ ) µ (GWH) reserve µ cost µ EENS µ lne (MW ays) tme ntervals (MW ays) BGSA Versus BGA In orer to evaluate the performance of propose BGSA-base metho, a stochastc heurstc search, BGA, s also mplemente to solve GMS problem [20]. The convergence rate of search algorthm an the fnal results of both metho are compare n ths subsecton. In BGA, the tune values of C r an C m are 0.92 an The results show that, propose mantenance scheules by BGSA an BGA are the same for case stuy 1, but have some fferences for case stuy 2. As can be seen from Fgure 7. ELDC for case stuy 1 Fgure 8, BGSA has prove a better soluton n comparson wth BGA for case stuy 2. Table 7 shows the average of best ftness, fnal best ftness an number of teratons to convergence for BGSA an BGA for both case stues. As ths table llustrates BGSA proves better results an has a hgher convergence rate than BGA. In aton, the goo convergence rate of BGSA for both case stues coul be conclue from Fgure 8. Accorng to ths fgure, BGSA tens to fn the global optmum faster than BGA. Fgure 8. BGSA an BGA Convergence manner for case stues

11 Internatonal Transacton of Electrcal an Computer Engneers System 54 Table 7. Comparson of BGSA an BGA performance for both case stues Case stues Algorthm Average best ftness value Fnal best ftness value Number of teratons to convergence Case stuy 1 Case stuy 2 5. Conclusons The power system's electrcty supply relablty s closely relate to ts reserve capacty, nclung spnnng reserve an col reserve, an the capacty reserve s nfluence by loa varaton an generatng unt mantenance outage. Both research an practce have shown that GMS s a constrane optmzaton problem. The mantenance scheule that satsfes all the constrants s calle a 'feasble' scheule. The optmal soluton among many feasble scheules s a scheule that has an optmal (usually a mnmum) objectve functon value. In ths stuy, a gravtatonal base mult-objectve fuzzy approach s use to solve GMS problem. There are three objectve functons. The frst objectve functon s reserve capacty levelzng. O&M costs an the nex of EENS are efne as the secon an thr objectve functons. A novel metho s evelope for calculatng the reserve capacty levelzng objectve functon. Power system probablstc proucton smulaton s use to calculate the energy generate by each unt, O&M cost as well as the nex of EENS n each pero. The 4-bus test system from Granger & Stevenson an the IEEE-RTS 24-bus test system are use as test systems to numercally evaluate the effcency of the propose metho. Smulaton results are prove for the test systems for a plannng horzon of one year an a plannng horzon of two years wth growng complexty, respectvely. The results of the BGSA are compare an valate aganst BGA n solvng the GMS problem. The results ncate that the BGSA s an effectve alternatve to the soluton of the propose GMS problem. Nomenclature BGSA BGA BGSA BGA var Comkb, Varable O&M cost ($/MWH) of the k th type plant unt locate at bus b fx Comkb, Fxe O&M cost ($/MW) of the k th type plant unt locate at bus b M k,b,h Number of type-k plant unts locate at bus b, n pero f of year t of the stuy pero c The crtcal pero as the pero of the year for whch the fference between the corresponng avalable generatng capacty an the peak eman has the smallest value M k,st Mantenance state, where 0 = no mantenance n tme stage st (operaton) an 1 = uner mantenance n tme stage st T Stuy pero (n years) M t Dscounte value of O&M cost I resv Inex of reserve capacty levelzng mn f Mnmum value of f ( X ) f Maxmum value of f ( X ) N k Number of fferent types of unts PG k Maxmum capacty (MW) of a k th type unt N per Number of peros consere n each year M t O&M costs n year t mn HRG Heat rate at the mnmum operatng k level (kcal/mwh) of the k th type plant HRG Heat rate at the mum operatng k level (kcal/mwh) of the k th type plant CFG Fuel cost ($/kcal) of the k th type plant kb, unt locate at bus b St k S k Startng stage for mantenance (the week or month) Number of stages for mantenance (the number of weeks or months) S { 1, 2,..., k },The set of generatng unts nvolve n mantenance n the pero St k1 an St k2 V r,st N u N st G(t) M (t) uner examnaton The startng tmes of the frst an secon mantenances, respectvely Maxmum number of generatng unts that the mantenance crew V r can work on smultaneously n stage st Number of generatng unt Number of tme stages n each year Gravtatonal constant at tme t The gravtatonal mass relate to agent at tme t x The poston of th agent n the th menson F () t j The force actng on mass from mass j R () t j Euclan stance between two agents an j a () t The acceleraton of agent at tme t, an n recton th v () t The velocty of an agent C r Coeffcent of cross over operator of BGA C m Coeffcent of mutaton operator of BGA References [1] Hossen Sef, Mohamma Saegh Sepasan, Electrc Power System Plannng, Issues, Algorthms an Solutons, Sprnger- Verlag, Berln, 2011.

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