A Profit-Based Unit Commitment using Different Hybrid Particle Swarm Optimization for Competitive Market

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1 A.A. Abou El Ela, e al./ Inernaonal Energy Journal 9 (2008) A rof-based Un Commmen usng Dfferen Hybrd arcle Swarm Opmzaon for Compeve Marke A. A. Abou El Ela*, G.E. Al + and H.S. Abd El-Ghany + Absrac Two proposed approaches are presened for opmal schedulng of un commmen (UC) n compeve marke. The parcle swarm opmzaon (SO) echnque s used o fnd ou he soluon of boh opmal UC schedulng and power generaon dspachng problems, smulaneously. These approaches depend on wo sgmod funcons o oban he bnary values for he SO echnque. The frs approach consders he fuzzfcaon of power generaon coss as a sgmod funcon, whle he second approach consders he fuzzfcaon of power generaon as a sgmod funcon. An exponenal funcon s proposed o mnmze power generaon coss as well as maxmze her own prof, whle all load demand and he power generaon consrans are sasfed. Therefore, he generaons companes (GECO) can schedule her oupu power accordng o a maxmum own prof. Ths means ha, he GECO mus ake a decson, how much power and reserve generaons should be sold n he markes o oban a maxmum own prof. Dfferen applcaons are carred ou usng varous sandard es sysems o show he capably of he proposed approaches he compeve marke. Keywords Bddng sraeges, compeve aucon markes, hybrd parcle swarm opmzaon (HSO), opmzaon mehods, power generaon dspach, un commmen.. ITRODUCTIO Elecrcy raders make bds and ers ha are mached subjec o he approval of an ndependen conrac admnsraor (ICA) who ensures ha he sysem s operang safely whn lms. Tradonal power sysem operaon, plannng, and conrol need changes. In he pas, ules had o produce power o sasfy her cusomers wh he mnmum producon cos. Tha means ules run UC wh he condon ha all demand and reserve mus be me. Afer he srucure changed; however, hey are more compeve under deregulaon. The objecve of UC s no o mnmze coss as before, bu o make he maxmum prof for company. A survey of leraure on UC mehods reveals ha varous numercal opmzaon echnques have been employed o address he UC problems. Specfcally, here are: prory ls mehods [], neger programmng [2], dynamc programmng [3], mxed-neger programmng [4], branch-and-bound mehods [5], and Lagrangan relaxaon mehods [6]. There are anoher classes of numercal echnques appled o he UC problem, whch are: Mea-heursc approaches nclude exper sysems (ES) [7], fuzzy logc (FL) [8], arfcal neural neworks (As) [9], genec algorhm (GA) [0], evoluonary programmng (E) [], smulaed annealng (SA) [2], and abu search (TS) [3]. These mehods can accommodae more complcaed consrans and are clamed o have beer soluon qualy. * Elecrcal Engneerng Deparmen, Faculy of Engneerng, Shebn El Kom, Mnoufya Unversy, Egyp. + Elecrcal Engneerng Deparmen, Faculy of Engneerng, Tana Tana Unversy, Egyp. Correspondng auhor; Tel: E-mal: draaa50@homal.com Dynamc programmng mehod [3] has many advanages such as s ably o manan soluon feasbly. everheless, hs mehod has dmensonal problem wh a large power sysem because he problem sze ncreases rapdly wh he number of generang uns o be commed, whch resuls n an unaccepable soluon me. Branch-and-bound adops a lnear funcon o represen he fuel consumpon and me-dependen sar cos and obans he requred lower and upper bounds. The dsadvanage of he branch-and-bound mehod s he exponenal growh n he execuon me wh he sze of he UC problem. The neger and mxed-neger mehods adop lnear programmng echnque o solve and check for an neger soluon. These mehods have only been appled o small UC problems and have requred major assumpons ha lm he soluon space. The Lagrangan relaxaon mehod provdes a fas soluon, bu may suffer from numercal convergence and soluon qualy problems. SA s a powerful, general-purpose sochasc opmzaon echnque, whch can heorecally converge asympocally o a global opmum soluon. One man drawback of SA s ha, akes a large compuaonal me o fnd he near-global mnmum soluon. GAs s a general-purpose sochasc and parallel search mehods based on he mechancs of naural selecon and naural genecs. The SO approach s movaed from he socal behavor of brd flockng and fsh schoolng. Kennedy and Eberhar nroduced SO n 995 n erms of socal and cognve behavor. The SO has been wdely used as a problem-solvng mehod n engneerng and compuer scence. The SO has been used o solve he opmal power flow problem [4], he reacve power and volage conrol problem [5], and he dsrbuon sae esmaon problem [6]. In solvng he un commmen problem, generally wo basc decsons are nvolved, namely he un commmen (UC) decson and he economc dspach

2 282 (ED) decson. The UC decson nvolves he deermnaon of he generang uns o be runnng durng each hour of he operaon and plannng horzon, consderng sysem capacy requremens, ncludng he reserve, and he consrans on he sar up and shu down of he generaon uns. The ED decson nvolves he allocaon of he sysem demand and spnnng reserve capacy among he operang uns durng each specfc hour of he operaon. Ths paper proposes wo approaches based on hybrd parcle swarm opmzaon (HSO) approaches n solvng he UC problem. A proposed exponenal objecve funcon s presened whch leads o fas convergence of SO soluon. 2. ARTICLE SWARM OTIMIZATIO arcle swarm opmzaon s a compung echnque nroduced by Kennedy and Eberhar n 995, whch was nspred by he socal behavor of brd flockng or fsh schoolng [7]. SO s nspred by parcles movng around n he search space. The ndvduals n a SO hus have her own posons and veloces. These ndvduals are denoed as parcles. Tradonally, SO has no crossover beween ndvduals, has no muaon, and parcles are never subsued by oher ndvduals durng he run [7]. The updae of he parcles s accomplshed o calculae a new velocy for each parcle (poenal soluon) based on s prevous velocy ( v d ), he parcle's locaon a whch he bes fness so far has been acheved ( pbes d ), and he populaon global locaon ( gbes d ) a whch he bes fness so far has been acheved. Then, each parcle s poson n he soluon hyperspace s updaed. The modfed velocy and poson of each parcle can be calculaed usng he curren velocy and dsance from pbes d o gbes as shown n d he followng equaons [8]: v ( n+ ) ( n) ( n) d = w. vd + c. rand (.).( pbes d xd ( n) + c2. rand 2 (.).( gbes d x d ) ( n+ ) ( n) ( n+ ) d = xd + vd x Velocy of parcle a eraon ; n d-dmensonal ( n) space s lmed by: v d, mn < vd < vd,max. Approprae selecon of nera wegh facor (w) n Equaon provdes a balance beween global and local exploraons. In general, he nera wegh facor (w) s se o he followng equaon: ) A.A. Abou El Ela, e al./ Inernaonal Energy Journal 9 (2008) () (2) wmax wmn w = wmax er (3) er max The velocy of parcle n d-dmensonal space s lmed by some maxmum value, v d,max. Ths lm enhances he local exploraon of he problem space and realscally smulaes he ncremenal changes of human learnng. To ensure unform velocy hrough all dmensons, he maxmum velocy n he d-dmenson s presened as: v d,max x x d,max d,mn = (4) 3. ROFIT-BASED UC ROBLEM FORMULATIO A prof-based UC (BUC) problem under compeve envronmen s presened as an opmzaon procedure. The objecve funcon of BUC s o maxmze he prof subjec o all prevalng consrans [0]: Max F = RV TC (5) In a resrucured sysem, GECO sell he power generaon n energy marke and sell reserve n he reserve (ancllary) marke. When he power reserve s used, GECO receves he spo prce for he reserve ha s generaed. In hs case, reserve prce s much lower han he spo prce. Revenue and coss n Equaon 5 can be calculaed from [9]: RV = S. + (( ) R + (6) TC = T T T [( r ). ) + r. + R r r. S ) R. U )].U + SUC.( U ).U + SDC.( U ). U,(-) (7) The generaor fuel-cos funcon can be expressed as: 2 ( F ) = a + b. + c. (8) where, a, b and c are he un cos coeffcens. Subjec o:. Demand consran: ' U D =,,T (9) = 2. Reserve consran: ' RU SR =,, T (0) = 3. ower generaon and reserve lms: mn R (, ) max 0 (, ) max mn 4. Mnmum up and downme consrans: on Sar-up cos s calculaed from: on [ X (, -) - T ][ U (, - ) U =,, () =,, (2) ] 0 [ X (, -) - T ][ U - U (, -) ] 0 SUC HSC, = CSC, X X (, -) (, -) T > T + CH + CH (3) (4) (5)

3 A.A. Abou El Ela, e al./ Inernaonal Energy Journal 9 (2008) OTIMAL UC SCHEMES Two hybrd parcle swarm opmzaon (HSO) approaches n solvng he UC problem are proposed. Based HSO Mehod The orgnal verson of SO operaes on real values. The erm hybrd parcle swarm opmzaon was frs menoned n [20], whereby he erm hybrd mean he combnaon of SO and GA. However, n hs approach, hybrd s mean o hghlgh he concep of blendng real valued SO (solvng economc load dspach (ELD)) wh bnary valued SO (solvng UC) runnng ndependenly and smulaneously. The bnary SO (BSO) s made possble wh a smple modfcaon o he parcle swarm algorhm. Ths BSO solves bnary problems smlar o hose radonally opmzed by GAs. Kennedy and Eberhar [2] showed ha he bnary parcle swarm was able o successfully opmze he De Jong [22] sue of es funcons. Furher, Kennedy and Spears [23] compared he bnary parcle swarm algorhm o GAs comprsng crossover only, muaon only, and boh crossover and muaon, n Spears mulmodal random problem generaor. I was seen ha he parcle swarm found global opma faser han any of he hree knds of GAs n all condons excep for problems feaurng low dmensonaly. In bnary parcle swarm, X and bes can ake values of 0 or only. The V velocy wll deermne a probably hreshold. If V s hgher, he ndvdual s more lkely o choose, and lower values favor he 0 choce. Such a hreshold needs o say n he range [0.0,.0]. One sraghforward funcon for accomplshng hs s common n neural neworks. The funcon s called he sgmod funcon and s defned as follows: μ ( V ) = (6) + exp( V ) Random number (drawn from a unform dsrbuon beween 0.0 and.0) s hen generaed, whereby X s se o f he random number s less han he value from he sgmod funcon as llusraed n he followng equaon: If Rand() < μ( V ), hen U =, else U = 0 (7) In he UC problem, of generaor. U Frs roposed HSO Approach represens he on or sae Ths approach s dependen on a suggesed fuzzy membershp funcon, as shown n Fgure ha can be expressed as: Cmax C μ ( c) = (8) C C max mn Fg.. Membershp funcon of he frs proposed HSO approach. Second roposed HSO Approach Ths approach s dependen on a suggesed fuzzy membershp funcon, as shown n Fgure 2 ha can be expressed as: mn μ ( ) = (9) max mn Fg. 2. Membershp funcon of he second proposed HSO approach. 5. ROOSED FEATURE OF FITESS FUCTIO Recenly, several mehods use he cos funcon x) o evaluae a feasble soluon, for he opmzaon problems as [0], [8]: Φ ( x) x) (20) f = nd he consran volaon measure consrans were defned n [8]. Φ u (x) for he r + m Then, he oal evaluaon of an ndvdual, whch can be nerpreed as he error (for a mnmzaon problem) of an ndvdual x, s obaned as: Φ ( x) = Φ ( x) + Φ ( x) (2) f u In hs paper, a new approach of he consran volaon measure Φ u (x) s proposed, whch resuls n reducng formulaon and compuaon requremen for he r + m consrans, as: r + m = + Φ ( x) = exp( g ( x)) (22) u + where, g ( x) max{ 0, g ( x) } =. In oher words, g + (x) s h he magnude of he volaon of he equaly and nequaly consran, r + m. Where, r s he number of nequaly consrans, and m s he number of equaly consrans.

4 284 ower Demand, Reserve and ower Generaon Consrans The objecve of he UC problem s formulaed as a combnaon of oal producon cos (as he man objecve) wh power balance (as equaly consrans) and spnnng reserve as well as generaon lms (as nequaly consrans), whereby TC n (7) and Φ u (x) are equvalen o he blend of power balance and spnnng reserve consrans, respecvely. Consequenly, he formulaon of he proposed fness funcon can be expressed as: Φ x) = Φ f ( x) + w.exp( c. Φ d ( x)) +. exp( c2. R ( x)) + w3.exp( c3. Φ g ( x)) ( w2 Φ (23) Where, w s se o f a volaon of (9) s occurred n, whle w 2 = 0 and w 3 =0 whenever (9) s no volaed. Lkewse, w 2 s also se o whenever a volaon of (0) s deeced, oherwse remans equal o 0. The choce of c and c 2 are dependng on he accuracy and he speed of he convergence requremens whch may be equal o 2. The frs erm n he penaly facor s he power balance consran. Ths erm s formulaed, for prof-based un commmen (BUC), as: Φ d ' ( x ) = max 0, ( U D ) (24) = The second erm n he penaly facor s he reserve consran, where R s 0% of power demand D '. Ths erm s formulaed for secury-consran un commmen (SCUC) and BUC as: Φ R ( x) = max 0, ( = ' ', maxu ( D + R )) A.A. Abou El Ela, e al./ Inernaonal Energy Journal 9 (2008) (25) The hrd erm n he penaly facor s he generaon consrans. Ths erm s formulaed for SCUC and BUC as: Φ g (26) where, he maxmum power generaon lm s defned as: Φ and he mnmum power generaon lm s defned as: ( x) = Φ g max ( x) + Φ g mn ( x) g max ( x) = max 0, = (,max ). U ) (27) Φ( x) = {[( r). ) + r. + R )].U + SUC.( U )]U } + w.exp( c. Φ d ( x)) + w exp( c. Φ ( x)) + w.exp( c. Φ ( ))) (29) 2. 2 R 3 3 g x Whle, he fness funcon for evaluang every parcle n he populaon of SO for ceran hours can be expressed as: {[( r). ) + r. + R )] T Φ( x ) =.U + SUC.( U )]U } + w (30).exp( c. Φ d ( x)) + w2.exp( c2. Φ R ( x)) + w3.exp( c3. Φ g ( x)) The fness funcons for evaluang every parcle n he populaon of SO for ceran hours for BUC can be expressed as: Φ (x) T = {[( r ). ) + r. + R )].U + SUC.( U )]U } { S. + (( r) (3). R + + Φ + r. S ) R. U } w.exp( c. d ( x)) w 2.exp( c2. Φ R ( x)) + w3.exp( c3. Φ g ( x)) Sasfyng he Mnmum Up and Down Tme Consrans The mn-up (MU) and mn-down (MD) me consrans are checked a every eraon. The resulng advanages for hs ype of represenaon are:. Reduced problem formulaon complexy. 2. The populaon pool consss of a se of feasble soluons, hus ncreasng he accuracy of he obaned soluons. 3. Wh he se of feasble soluons, he power sysem operaor can rade beween he soluon whn hs se accordng o he performance ndex TC. 4. An urgen propery of he suggesed approach s he fas convergence compared o ha obaned from [8]. Fgure 3 shows he flowchar of he wo proposed approaches for he opmal schedulng of UC. Φ g mn ( x) = max 0, = (,mn ). U ) (28) By subsung Equaon 7 no Equaon 23, he fness funcon for evaluang every parcle n he populaon of SO for an hour can be defned as:

5 A.A. Abou El Ela, e al./ Inernaonal Energy Journal 9 (2008) SIMULATIO RESULTS o Sar Read npu daa Inal populaon random (ndxg*) bes= and Ubes=U For each hour () Check mn up/down me Call objecve funcon Las hour done? Gen= Yes Rankng (&U) as objec fun se gbes Calculae velocy and new poson (v and ) o Un saus decson from SO approaches HSO, Approach and Approach 2 For each hour () Check mn up/down me call objecve funcon Las hour done? Yes Gen+ In hs secon, wo case sudes are used o llusrae he effecveness of he proposed approaches n erms of s soluon qualy. Smulaons are carred ou usng wo es sysems adaped from [0] and [9]. The frs sysem consss of hree generang uns, 2-hour schedulng perods. The second sysem consss of en generang uns, 24-hour schedulng perods. The effec of r and he reserve prce on he prof of GECO are smulaed usng he frs es sysem. The second es sysem s used o show he capably of he proposed approaches. All smulaon resuls are compared wh he resuls obaned usng he radonal UC and LR- E mehods [9]. The SO echnque seems o be sensve o he unng of some weghs or parameers, accordng o he experences of many references [9]. The parameers of he proposed approaches are gven as: - opulaon sze = 00; - Inal nera wegh (wmax) = 0.9; - Fnal nera wegh (wmn ) = 0.4; - Acceleraon consan c = 2 and c 2 = 2; Fgure 3 shows he flow char of he proposed procedure for he wo proposed approaches. Frs Tes Sysem: Table shows he power generaon and reserve schedulng usng he frs approach a r equals o and reserve prce equals o 4% of he spo prce. A prof-based UC, he un s a all schedulng perods o sell power and reserve generaon o oban he hghes prof. Fgure 4 shows he dfferen values of he revenue, cos and prof a he varous operang hours. In hs fgure, he prof of GECO, whch s he dfferen beween he revenue and generaon coss, has he hghes value a hour 7 because he load demand s aken from only wo uns (see Table ) ha have low sar-up coss, whle he generaon coss are remaned fxed and he spo prce s ncreased. Compare bes and and se bes Is ermnaon crera sasfed? Yes o Rankng ( and U) se gbes Calculae and prn (cos, prof, U,, Xonof, and CU me) Sop Fg. 3. Flowchar of he wo proposed approaches.

6 286 A.A. Abou El Ela, e al./ Inernaonal Energy Journal 9 (2008) Fg. 4. Revenue, generaon coss and prof of GECO for 3-un sysem. Table. ower and reserve generaon for 3-un es sysem (r = 0.005, reserve prce= 4% of spo prce) usng he frs proposed approach. Tradonal Un Commmen rof-based Un Commmen Hour Un Un 2 Un 3 Cos rof Un Reserve Cos rof Un 2 Un 3 ($) ($) (MW) ($) ($) 0 00/0 70/ / /0 50/ / /40 200/ / /55 200/ / /70 400/0 200/ /70 200/ /95 400/0 200/ /0 200/ /00 400/0 200/ /0 200/ /80 400/0 200/ /0 200/ /5 350/50 200/ / / /0 00/0 30/ /35 200/ /0 00/40 200/ /40 200/ / /50 200/ Toal Table 2. Comparson beween he dfferen approaches for he oal producon cos, prof of GECO and CU for 3-un es sysem. BUC SCUC Approach Cos ($) rof ($) CU (sec) Cos ($) rof ($) CU (sec) HSO Approach Approach LR-E [36] Tradonal Table 2 shows a comparson beween he dfferen approaches for he oal producon coss, prof of GECO and he compuaonal me (CU) a r = and reserve prce= 0.04 of spo prce. The frs proposed approach gves he bes values for he generaon coss, prof of GECO and compuaonal me compared wh he second proposed approach and HSO approach. Fgure 5 shows he effec of probably r, ha power reserve s called and generaed, on he prof of GECO usng he radonal prof and he prof-based mehods. The power reserve paymen prce s fxed a 4% of spo prce whle r s changed from o Fgure 6 shows he effec of reserve prce on he prof of GECO usng he radonal prof and he prof-based mehods, when he probably r s fxed a From Fgures 5 and 6, he prof of GECO s ncreased usng he proposed prof-based mehod compared wh he radonal prof mehod because he power demand and power reserve. Fgure 7 shows he fness shapes of he proposed approaches compared o he based HSO mehod. In hs fgure, he fness of he frs proposed approach has he fases convergence compared wh oher approaches.

7 A.A. Abou El Ela, e al./ Inernaonal Energy Journal 9 (2008) Fg. 5. Effec of r on prof of GECO. Fg. 6. Effec of reserve prce on prof of GECO. Fg. 7. Fness funcon of hree mehod wh he generaon of SO echnques. Fg. 8. Revenue, generaon cos and prof of GECO for 0- un sysem. 60 CU (sec) HOS Approach 2 Approach o. of power generaon un Fg. 9. CU agans he number of power generaon uns for dfferen approaches. Fgure 7 shows he fness shapes of he proposed approaches compared o he based HSO mehod. In hs fgure, he fness of he frs proposed approach has he fases. Second Tes Sysem: Tables 3 and 4 nclude he same resuls of Tables and 2. Fgure 8 presens he same resuls of Fgure 4 bu for he second es sysem. Fgure 9 shows he compuaonal me (CU) agans he number of power generaon uns for he dfferen approaches. The dfferen approaches are appled o 2-un [0], 3-un [9], 4-un [8] and 0-un [9]. From hs fgure, he frs proposed approach has a mnmum compuaonal me compared o oher approaches.

8 288 A.A. Abou El Ela, e al./ Inernaonal Energy Journal 9 (2008) Table 3. Comparson beween he dfferen approaches for prof of GECO and CU for 0-un es sysem usng dfferen approaches. BUC Mehod rof ($) CU (sec) HSO Approach Approach LR-E [36] Table 4. ower and reserve generaon for 0-un es sysem (r = 0.005, reserve prce= % of spo prce) usng he frs proposed approach. ower (MW) / Reserve (MW) /0 245/70 0/0 0/0 0/0 0/0 0/ /0 295/75 0/0 0/0 0/0 0/0 0/ /0 395/60 0/0 0/0 0/0 0/0 0/ /0 455/0 0/0 0/0 0/0 0/0 0/ /0 455/0 0/0 0/0 0/0 0/0 0/ /0 455/0 0/0 30/0 0/0 0/0 0/ /0 455/0 0/0 30/0 0/0 0/0 0/ /0 455/0 0/0 30/0 0/0 0/0 0/ /0 455/0 30/0 30/0 0/0 0/0 0/ /0 455/0 30/0 30/0 62/0 68/0 0/0 455/0 455/0 30/0 30/0 62/0 80/0 0/ /0 455/0 30/0 30/0 62/0 80/0 0/ /0 455/0 30/0 30/0 62/0 0/0 0/ /0 455/0 30/0 30/0 30/32 0/0 0/ /0 455/0 0/0 30/0 60/2 0/0 0/ /0 455/0 0/0 30/0 0/0 0/0 0/ /0 455/0 0/0 30/0 0/0 0/0 0/ /0 455/0 0/0 30/0 0/0 0/0 0/ /0 455/0 0/0 30/0 0/0 0/0 0/ /0 455/0 0/0 30/0 0/0 0/0 0/ /0 455/0 0/0 30/0 0/0 0/0 0/ /0 455/0 0/0 30/0 0/0 0/0 0/ /0 455/0 0/0 0/0 0/0 0/0 0/ /0 345/80 0/0 0/0 0/0 0/0 0/0 Toal prof: 0966 $ 7. COCLUSIO Two effcen and accurae approaches for opmal schedulng of un commmen (UC) n compeve marke have been presened n hs paper. The opmal soluon of of UC power generaons schedulng problems has been obaned, smulaneously. These approaches depend on wo fuzzy sgmod funcons o accelerae he soluon convergence compared wh he HSO and LR- E echnques. oban he bnary values for SO echnque. These approaches have been used for maxmzng he prof of GECO by consderng he power and reserve generaon, smulaneously. A proposed exponenal objecve funcon has been successfully appled whch leads o fas convergence of he SO echnque. The resuls he frs approach have he lowes generaon coss, hghes prof of GECO and lowes CU compared wh he oher approaches. OMECLATURE F ( ) roducon cos of un n me perod ($). F rof of GECO ($). RV Revenue of GECO ($). SUC Sar-up cos for un n me perod ($). TC Toal cos of GECO ($). CH The cold sar hour (hr) a un. CSC The un's cold sar-up cos a un ($). HSC The un's ho sar-up cos a un ($). HSO Hybrd parcle swarm opmzaon. ' D Forecased demand a hour (MW). umber of generaor uns. A chosen number of nervals. mn Mnmum lm of generaor (MW). ower generaon of un a hour (MW).

9 A.A. Abou El Ela, e al./ Inernaonal Energy Journal 9 (2008) max Maxmum lm of generaor (MW). R Reserve generaon of un a hour (MW). SDC Shu-down cos of un a me perod ($). S Forecased spo prce a hour ($). Forecased reserve a hour (MW). ' SR T umber of hours. T Mnmum -me of un (hr). on T Mnmum-on me of un (hr). U On/ saus of generaor a hour. on (, - ) X Tme duraon for whch un has been on-me a hour (hr). X Tme duraon for whch un has been (, -) -me a hour (hr). R Forecased reserve prce a hour. r robably ha he reserve s called and generaed. (n) v Velocy of parcle a eraon n. d (n) x Curren poson of parcle a eraon d n. w Inera wegh facor. n umber of eraons. n umber of parcles n a group. m umber of members n a parcle. c and c 2 Acceleraon consan of SO. rand ( ) and Random numbers beween 0 and. rand 2 ( ) ermax and er SCUC BUC Maxmum and he curren number of eraons. Secury-consran un commmen. rof-based un commmen. REFERECES [] Senjyu, T., Shmabukuro, K., Uezao, K., and Funabash, T A fas echnque for un commmen problem by exended prory ls. IEEE Transacons on ower Sysem 8(2): [2] Dllon, T.S., Edwn, K.W., Kochs, H.D., and Taud, R. J Ineger programmng approach o he problem of opmal un commmen wh probablsc reserve deermnaon. IEEE Transacons on ower Applcaon Sysems 97(6): [3] Ouyang, Z., and Shahdehpour, S.M. 99. An nellgen dynamc programmng for un commmen applcaon. IEEE Transacons on ower Sysem 6(3): [4] Mucksad, J.A., and Wlson, R.C An applcaon of mxed-neger programmng dualy o schedulng hermal generang sysems. IEEE Transacons on ower Applcaon Sysem [5] Chen, C., and Wang, S Branch-and-bound Schedulng for hermal generang uns. IEEE Transacons on Energy Converson 8(2): [6] Redondo,.J., and Conejo, A.J., 999. Shor-erm hydro-hermal coordnaon by Lagrangan relaxaon soluon of he dual problem. IEEE Transacons on ower Sysem 4(): [7] Kohar, D.., and Ahmed, A An Exper Sysem Approach o Un Commmen roblem. In IEEE Tencon roceedngs on Communcaon, Conrol, and ower Engneerng. 9-2 Ocober. pp [8] El-Saadawa, M.M., Tanawa, M.A., and Tawfkb, E A fuzzy opmzaon-based approach o large-scale hermal un commmen. Elecrc ower Sysems Research 72: [9] ayak, R., and Sharma, J.D Hybrd neural nework and smulaed annealng approach o he un commmen problem. Compuers and Elecrcal Engneerng 26(6): [0] Rcher, C.W., and Sheblé, G.B., 2000 A prof-based un commmen GA for he compeve envronmen. IEEE Transacons on ower Sysem 5(2): [] Rajan, C.C., and Mohan, M.R An evoluonary programmng-based Tabu search mehod for solvng he un commmen problem. IEEE Transacons on ower Sysem 9(): [2] Smopoulos, D.., Kavaza, S.D., and Vournas, C.D Relably consraned un commmen usng smulaed annealng. IEEE Transacons on ower Sysems 2(4): [3] Manawy, A.H., Abdel-Magd, Y.L., and Selm, S.Z., 998. Un commmen by abu search. IEE roceedngs on General Transmsson Dsrbuon 45(): [4] Abdo, M.A Opmal power flow usng parcle swarm opmzaon. Inernaonal Journal Elecrc ower Energy Sysem 24(7): [5] Yoshda, H., Kawaa, K., Fukuyama, Y., and akansh, Y A parcle swarm opmzaon for reacve power and volage conrol consderng volage secury assessmen. IEEE Transacons on ower Sysem 5(4): [6] aka, S., Genj, T., Yura, T., and Fukuyama, Y., A hybrd parcle swarm opmzaon for dsrbuon sae esmaon. IEEE Transacons on ower Sysem. 8(): [7] Balc, H.H., and Valenzuela, J.F., Schedulng elecrc power generaors usng parcle swarm opmzaon combned wh he Lagrangan relaxaon mehod. Inernaonal Journal Appled Mahemacs and Compuer Scence 4(3): [8] Tng, T.O., Rao, M.V.C., and Loo, C.K A novel approach for un commmen problem va an effecve hybrd parcle swarm opmzaon. IEEE Transacons on ower Sysem 2(): [9] Aavryanupap,., Ka, H., Tanaka, E., and Hasegawa, J A hybrd LR E for solvng new prof-based UC problem under compeve envronmen. IEEE Transacons on ower Sysem 8(): [20] aka, S., Genj, T., Yura, T., and Fukuyama, Y., A hybrd parcle swarm opmzaon for dsrbuon sae esmaon. IEEE Transacons on ower Sysem 8(): [2] Kennedy J., and Eberhar, R.C., 997. A dscree bnary verson of he parcle swarm algorhm. In roceedngs of he Inernaonal Conference on Sysems, Man, Cybernecs. scaaway, ew Jersey,

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