Particle Swarm Optimization

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1 Paricle Swarm Opimizaion Edied by Alesandar Lazinica In-Tech inechweb.org

2 VII Conens Preface V 1. Novel Binary Paricle Swarm Opimizaion 001 Mojaba Ahmadieh Khanesar, Hassan Tavaoli, Mohammad Teshnehlab and Mahdi Aliyari Shoorehdeli 2. Swarm Inelligence Applicaions in Elecric Machines 011 Amr M. Amin and Omar T. Hegazy 3. Paricle Swarm Opimizaion for HW/SW Pariioning 049 M. B. Abdelhalim and S. E. D Habib 4. Paricle Swarms in Saisical Physics 077 Andrei Bauu and Elena Bauu 5. Individual Parameer Selecion Sraegy for Paricle Swarm Opimizaion 089 Xingjuan Cai, Zhihua Cui, Jianchao Zeng and Ying Tan 6. Personal Bes Oriened Paricle Swarm Opimizer 113 Chang-Huang Chen, Jong-Chin Hwang and Sheng-Nian Yeh 7. Paricle Swarm Opimizaion for Power Dispach wih Pumped Hydro 131 Po-Hung Chen 8. Searching for he bes Poins of inerpolaion using swarm inelligence echniques Djerou L., Khelil N., Zerara A. and Baouche M. 9. Paricle Swarm Opimizaion and Oher Meaheurisic Mehods in Hybrid Flow Shop Scheduling Problem M. Fire Ercan 10. A Paricle Swarm Opimizaion echnique used for he improvemen of analogue circui performances Mourad Fahfah, Yann Cooren, Mourad Loulou and Paric Siarry Paricle Swarm Opimizaion Applied for Locaing an Inruder by an Ulra-Wideband Radar Newor Rodrigo M. S. de Oliveira, Carlos L. S. S. Sobrinho, Josivaldo S. Araújo and Rubem Farias 183

3 Paricle Swarm Opimizaion for Power Dispach wih Pumped Hydro 7 Po-Hung Chen Deparmen of Elecrical Engineering, S. John s Universiy Taiwan 1. Inroducion Recenly, a new evoluionary compuaion echnique, nown as paricle swarm opimizaion (PSO), has become a candidae for many opimizaion applicaions due o is high-performance and flexibiliy. The PSO echnique was developed based on he social behavior of flocing birds and schooling fish when searching for food (Kennedy & Eberhar, 1995). The PSO echnique simulaes he behavior of individuals in a group o maximize he species survival. Each paricle flies in a direcion ha is based on is experience and ha of he whole group. Individual paricles move sochasically oward he posiion affeced by he presen velociy, previous bes performance, and he bes previous performance of he group. The PSO approach is simple in concep and easily implemened wih few coding lines, meaning ha many can ae advanage of i. Compared wih oher evoluionary algorihms, he main advanages of PSO are is robusness in conrolling parameers and is high compuaional efficiency (Kennedy & Eberhar, 2001). The PSO echnique has been successfully applied in areas such as disribuion sae esimaion (Naa e al., 2003), reacive power dispach (Zhao e al., 2005), and elecromagneic devices design (Ho e al., 2006). In he previous effor, a PSO approach was developed o solve he capacior allocaion and dispaching problem (Kuo e al., 2005). This chaper inroduces a PSO approach for solving he power dispach wih pumped hydro (PDWPH) problem. The PDWPH has been reconed as a difficul as wihin he operaion planning of a power sysem. I aims o minimize oal fuel coss for a power sysem while saisfying hydro and hermal consrains (Wood & Wollenberg, 1996). The opimal soluion o a PDWPH problem can be obained via exhausive enumeraion of all pumped hydro and hermal uni combinaions a each ime period. However, due o he compuaional burden, he exhausive enumeraion approach is infeasible in real applicaions. Convenional mehods (El-Hawary & Ravindranah, 1992; Jeng e al., 1996; Allan & Roman, 1991; Al- Agash, 2001) for solving such a non-linear, mix-ineger, combinaorial opimizaion problem are generally based on decomposiion mehods ha involve a hydro and a hermal sub-problem. These wo sub-problems are usually coordinaed by LaGrange mulipliers. The opimal generaion schedules for pumped hydro and hermal unis are hen sequenially obained via repeiive hydro-hermal ieraions. A well-recognized difficuly is ha soluions o hese wo sub-problems can oscillae beween maximum and minimum generaions wih sligh changes of mulipliers (Guan e al., 1994; Chen, 1989). Consequenly,

4 132 Paricle Swarm Opimizaion soluion cos frequenly ges suc a a local opimum raher han a he global opimum. However, obaining an opimal soluion is of prioriy concern o an elecric uiliy. Even small percenage reducion in producion coss ypically leads o considerable savings. Obviously, a comprehensive and efficien algorihm for solving he PDWPH problem is sill in demand. In he previous effors, a dynamic programming (DP) approach (Chen, 1989) and a geneic algorihm (GA) echnique (Chen & Chang, 1999) have been adoped o solve he PDWPH problem. Alhough he GA has been successfully applied o solve he PDWPH problem, recen sudies have idenified some deficiencies in GA performance. This decreased efficiency is apparen in applicaions in which parameers being opimized are highly correlaed (Eberhar & Shi, 1998; Boeringer & Werner, 2004). Moreover, premaure convergence of he GA reduces is performance and search capabiliy (Angeline, 1998; Juang, 2004). This wor presens new soluion algorihms based on a PSO echnique for solving he PDWPH problem. The proposed approach combines a binary version of he PSO echnique wih a muaion operaion. Kennedy and Eberhar firs inroduced he concep of binary PSO and demonsraed ha a binary PSO was successfully applied o solve a discree binary problem (Kennedy & Eberhar, 1997). In his wor, since all Taipower s pumped hydro unis are designed for consan power pumping, novel binary encoding/decoding echniques are judiciously devised o model he discree characerisic in pumping mode as well as he coninuous characerisic in generaing mode. Moreover, since he basic PSO approach converges fas during he iniial period and slows down in he subsequen period and someimes lands in a local opimum, his wor employs a muaion operaion ha speeds up convergence and escapes local opimums. Represenaive es resuls based on he acual Taipower sysem are presened and analyzed, illusraing he capabiliy of he proposed PSO approach in pracical applicaions. 2. Modeling and Formulaion 2.1 Lis of symbols DR i Down ramp rae limis of hermal uni i. Fi ( Psi ) Producion coss for P si. I j Naural inflow ino he upper reservoir of pumped hydro plan j in hour. N h Number of pumped hydro unis. N s Number of hermal unis. P hj Power generaion (posiive) or pumping (negaive) of pumped hydro plan j in hour. P L Sysem load demand in hour. P si Power generaion of hermal uni i in hour. Q j Waer discharge of pumped hydro plan j in hour. Q j, p Waer pumping of pumped hydro plan j in hour.

5 Paricle Swarm Opimizaion for Power Dispach wih Pumped Hydro 133 Rhj ( Phj ) Spinning reserve conribuion of pumped hydro plan j for R req Sysem s spinning reserve requiremens in hour. Rsi ( Psi ) Spinning reserve conribuion of hermal uni i for S j Waer spillage of pumped hydro plan j in hour. P si. P hj. T Number of scheduling hours. UR i Up ramp rae limis of hermal uni i. V j Waer volume of he upper reservoirs of plan j a he end of hour. V j, l Waer volume of he lower reservoirs of plan j a he end of hour. v i Velociy of paricle i a ieraion. x i Posiion (coordinae) of paricle i a ieraion. 2.2 Modeling a pumped hydro plan A pumped hydro plan, which consiss of an upper and a lower reservoir, is designed o save fuel coss by generaing during pea load hours wih waer in he upper reservoir, which is pumped up from he lower reservoir o he upper reservoir during ligh load hours (Fig. 1). P G Figure 1. Pumped hydro plan In generaing mode, he equivalen-plan model can be derived using an off-line mahemaical procedure ha maximizes oal plan generaion oupu under differen waer discharge raes (Wood & Wollenberg, 1996). The generaion oupu of an equivalen pumped hydro plan is a funcion of waer discharged hrough urbines and he conen (or he ne head) of he upper reservoir. The general form is expressed as Phj 1 = f ( Q j, V j ) (1) The quadraic discharge-generaion funcion, considering he ne head effec, uilized in his wor as a good approximaion of pumped hydro plan generaion characerisics is given as Phj = α j Q j + β j Q j + γ j (2)

6 134 Paricle Swarm Opimizaion where coefficiens j 1 α, 1 β j, and j 1 γ depend on he conen of he upper reservoir a he end of hour -1. In his wor, he read-in daa includes five groups of α, β, γ coefficiens ha are associaed wih differen sorage volume, from minimum o maximum, for he upper reservoir (firs quadran in Fig. 2). Then, he corresponding coefficiens for any reservoir volume are calculaed using a linear inerpolaion (Chen, 1989) beween he wo closes volume. In pumping mode, since all Taipower s pumped hydro unis are designed for consan power pumping, he characerisic funcion of a pumped hydro plan is a discree disribuion (hird quadran in Fig. 2). Oupu (MW) Vmax Pmax Vmin Pumping (cubic meer per second) Discharge (cubic meer per second) Inpu (MW) Figure 2. Typical inpu-oupu characerisic for a pumped hydro plan 2.3 Objecive funcion and consrains The pumped hydro scheduling aemps seeing he opimal generaion schedules for boh pumped hydro and hermal unis while saisfying various hydro and hermal consrains. Wih division of he oal scheduling ime ino a se of shor ime inervals, say, one hour as one ime inerval, he pumped hydro scheduling can be mahemaically formulaed as a consrained nonlinear opimizaion problem as follows: Problem: T Ns Minimize F ( P ) (3) = 1i= 1 Subjec o he following consrains: Sysem power balance Ns Nh Psi + Phj PL = 0 i= 1 j= 1 i si (4) Waer dynamic balance V j 1 = V j + I j Q j + Q j, p S j (5)

7 Paricle Swarm Opimizaion for Power Dispach wih Pumped Hydro V j, l = V j, l + Q j Q j, p + S j (6) Thermal generaion and ramp rae limis 1 1 Max( Psi, Psi DRi ) Psi Min( Psi, Psi + URi ) (7) Waer discharge limis Waer pumping limis Reservoir limis Q Q j Q j Q j (8) j,p Q j, p Q j,p (9) V j V j V j (10) V j,l V j, l V j,l (11) Sysem s spinning reserve requiremens N s N + h R si( Psi ) Rhj ( Phj ) Rreq (12) i= 1 j= 1 3. Refined PSO Soluion Mehodology 3.1 Basic PSO echnique Consider an opimizaion problem of D variables. A swarm of N paricles is iniialized in which each paricle is assigned a random posiion in he D-dimensional hyperspace such ha each paricle s posiion corresponds o a candidae soluion for he opimizaion problem. Le x denoe a paricle s posiion (coordinae) and v denoe he paricle s fligh velociy over a soluion space. Each individual x in he swarm is scored using a scoring funcion ha obains a score (finess value) represening how good i solves he problem. The bes previous posiion of a paricle is Pbes. The index of he bes paricle among all paricles in he swarm is Gbes. Each paricle records is own personal bes posiion (Pbes), and nows he bes posiions found by all paricles in he swarm (Gbes). Then, all paricles ha fly over he D-dimensional soluion space are subjec o updaed rules for new posiions, unil he global opimal posiion is found. Velociy and posiion of a paricle are updaed by he following sochasic and deerminisic updae rules: + 1 vi = wvi + c1rand( ) ( Pbesi xi ) + c2rand( ) ( Gbes xi ) (13) x i = xi + vi (14) where w is an ineria weigh, c 1 and c 2 are acceleraion consans, and Rand( ) is a random number beween 0 and 1.

8 136 Paricle Swarm Opimizaion Equaion (13) indicaes ha he velociy of a paricle is modified according o hree componens. The firs componen is is previous velociy, v i, scaled by an ineria, w. This componen is ofen nown as habiual behavior. The second componen is a linear aracion oward is previous bes posiion, Pbes i, scaled by he produc of an acceleraion consan, c 1, and a random number. Noe ha a differen random number is assigned for each dimension. This componen is ofen nown as memory or self-nowledge. The hird componen is a linear aracion oward he global bes posiion, Gbes, scaled by he produc of an acceleraion consan, c 2, and a random number. This componen is ofen nown as eam wor or social nowledge. Fig. 3 illusraes a search mechanism of a PSO echnique using he velociy updae rule (13) and he posiion updae rule (14). x i v i Pbes i +1 v i +1 x i Gbes Pbes j +1 v j x j v j +1 x j Figure 3. Searching mechanism of a PSO Acceleraion consans c 1 and c 2 represen he weighs of he sochasic acceleraion erms ha push a paricle oward Pbes and Gbes, respecively. Small values allow a paricle o roam far from arge regions. Conversely, large values resul in he abrup movemen of paricles oward arge regions. In his wor, consans c 1 and c 2 are boh se a 2.0, following he ypical pracice in (Eberhar & Shi, 2001). Suiable correcion of ineria w in (13) provides a balance beween global and local exploraions, hereby reducing he number of ieraions when finding a sufficienly opimal soluion. An ineria correcion funcion called ineria weigh approach (IWA) (Kennedy & Eberhar, 2001) is uilized in his wor. During he IWA, he ineria weigh w is modified according o he following equaion: wmax wmin w = wmax Ir (15) Ir where w max and w min are he iniial and final ineria weighs, Ir max is he maximum number of ieraion, and Ir is he curren number of ieraion. max

9 Paricle Swarm Opimizaion for Power Dispach wih Pumped Hydro Binary encoding For exposiion ease, consider a pumped hydro plan wih four unis. Fig. 4 presens a paricle sring ha ranslaes he encoded parameer-waer discharges of each plan ino heir binary represenaions. Hour Figure 4. Paricle sring for a pumped hydro plan wih four unis Using a plan s waer discharge insead of he plan s generaion oupu, he encoded parameer is more beneficial when dealing wih difficul waer balance consrains. Each paricle sring conains 24 sub-srings ha represen he soluion for hourly discharge/pumping schedules of he pumped hydro plan during a 24-hour period. Each sub-sring is assigned he same number of five bis. The firs bi is used o idenify wheher he plan is in generaing or pumping mode. The remaining four bis are used o represen a normalized waer discharge, q j, in generaing mode, or he number of pumping unis in pumping mode. In generaing mode, he resoluion equals 1/2 4 of he discharge difference from minimum o maximum. 3.3 Decoding of paricle sring A paricle wihin a binary PSO approach is evaluaed hrough decoding he encoded paricle sring and compuing he sring s scoring funcion using he decoded parameer. The following seps summarize he deailed decoding procedure. Sep 1. Decode he firs bi o deermine wheher he plan is in generaing or pumping mode: Hour b 1 b 2 b 3 b 4 b 5 b 2 =b 3 =b 4 =b 5 = 0 => idle mode b 1 = 0 => pumping mode b 1 = 1 => generaing mode Sep 2. If in idle mode, P hj =0, go o Sep 10; if in pumping mode, go o Sep 3; if in generaing mode, go o Sep 6. Sep 3. Decode he remaining four bis of he sub-sring o calculae he number of pumping unis, N p, and he oal volume of pumped waer, Q j, p : Hour 0 b 2 b 3 b 4 b 5 N p 5 = ( b i = 2 i ) b i { 0,1} (16) where Q j, p = Q j,sp N p (17) Q j, sp is he consan volume for pumping per uni.

10 138 Paricle Swarm Opimizaion Sep 4. Calculae he upper boundary of pumped waer: 1 Q j, p = Min[ Q j, p, (V j, l V j, l )] (18) If he oal volume of pumped waer exceed he upper boundary, hen decrease he number of pumping unis unil he upper boundary is saisfied. Sep 5. Calculae he MW power for pumping: ( P N ) P hj = j,sp p (19) where P j, sp is he consan power for pumping per uni. Then go o sep 10. Sep 6. Decode he remaining four bis of he sub-sring o obain a normalized discharge, q j, in decimal values: Hour 1 b 2 b 3 b 4 b q 5 ( ( i 1 = b 2 ) ) b { 0,1} (20) j i i= 2 Sep 7. Calculae he upper boundary of discharge: i Q j 1 = Min[Q j, (V j,l V j, l )] (21) Sep 8. Translae he normalized value, q j, o he acual value, Q j : Sep 9. Calculae he generaion oupu, P hj, using (2). Sep 10. Calculae he remaining hermal loads, P rm : Q j = Q j + q j Q j Q j (22) rm P L hj = P P (23) Sep 11. Coninue wih compuaions of he 10 seps from hour 1 o hour 24. Sep 12. Perform he uni commimen (UC) for he remaining hermal load profile, and reurn he corresponding hermal cos. In his wor, a UC pacage based on he neural newor (Chen & Chen, 2006) is used o perform he UC as aing ino accoun fuel coss, sar-up coss, ramp rae limis, and minimal upime/downime consrains. Sep 13. Translae he corresponding hermal cos ino he score of he i-h paricle using a scoring funcion (deails are found in he nex Secion). Sep 14. Repea hese 13 seps for each paricle from he firs o las paricle.

11 Paricle Swarm Opimizaion for Power Dispach wih Pumped Hydro Scoring funcion The scoring funcion adoped is based on he corresponding hermal producion cos. To emphasize he bes paricles and speed up convergence of he evoluionary process, he scoring funcion is normalized ino a range of 0 1. The scoring funcion for he i-h paricle in he swarm is defined as SCORE( i ) = 1 + i 1 cos( i ) cos( Gbes 1 ) (24) where SCORE(i) is he score (finess value) of he i-h paricle; cos(i) is he corresponding hermal cos of he i-h paricle; cos(gbes) is he cos of he highes raning paricle sring, namely, he curren bes paricle; and, i is a scaling consan ( i =100 in his sudy). 3.5 Muaion operaion The basic PSO approach ypically converges rapidly during he iniial search period and hen slows. Then, checing he posiions of paricles showed ha he paricles were very ighly clusered around a local opimum, and he paricle velociies were almos zero. This phenomenon resuled in a slow convergence and rapped he whole swarm a a local opimum. Muaion operaion is capable of overcoming his shorcoming. Muaion operaion is an occasional (wih a small probabiliy) random alernaion of he Gbes sring, as shown in Fig. 5. This wor inegraes a PSO echnique wih a muaion operaion providing bacground variaion and occasionally inroduces beneficial maerials ino he swarm o speed up convergence and escape local opimums. Gbes: New Paricle: Figure 5. Muaion operaion The soluion mehodology for solving he pumped hydro scheduling problem using he proposed approach is oulined in he general flow char (Fig. 6). 4. Tes Resuls The proposed approach was implemened on a MATLAB sofware and execued on a Penium IV 3.0GHz personal compuer. Then, he proposed approach was esed for a porion of he Taipower sysem, which consiss of 34 hermal unis and he Ming-Hu pumped hydro plan wih four unis. In addiion o he ypical consrains lised in Secion 2, he Taipower sysem has hree addiional feaures ha increase problem difficuly. a. The Taipower sysem is an isolaed sysem. Thus i is self-sufficien a all imes. The 300MW sysem s spinning reserve requiremen mus be saisfied each hour. b. Thermal unis, due o heir ramp rae limis, have difficuly handling large load flucuaions, especially a noon lunch brea. c. The lower reservoir of Ming-Hu pumped hydro plan has only a small sorage volume. Table 1 presens deailed daa for he Ming-Hu pumped hydro plan. The hermal sysem consiss of 34 hermal unis: six large coal-fired unis, eigh small coal-fired unis, seven oil-

12 140 Paricle Swarm Opimizaion fired unis, en gas urbine unis, and hree combined cycle unis. For daa on he characerisics of he 34-uni hermal sysem, please refer o (Chen & Chang, 1995). Read in daa and define consrains. 1. Iniialize swarm: (a) Randomize each paricle ino a binary sring. (b) Randomize velociy of each paricle. 2. Evaluae paricles: (a) Decode each paricle o obain a MW schedule of P/S unis. (deail in Secion 3.3) (b) Do hermal uni commimen for he remaining hermal loads o obain a producion cos. (c) Score each paricle using (24). (d) Iniialize each Pbes o equal he curren posiion of each paricle. (e) Gbes equals he bes one among all Pbes. 3. Updae velociy of paricle using (13). 4. Updae posiion of paricle using (14). 5. Decode and score he new paricle posiion. 6. Updae Pbes if he new posiion is beer han ha of Pbes. 7. Updae Gbes if he new posiion is beer han ha of Gbes. Repea for each paricle. 8. Muaion operaion: Perform muaion if Gbes remains unchanged wihin he laes 50 ieraions. Repea for each ieraion. Gbes is he opimal soluion. Figure 6. General flow char of he proposed approach

13 Paricle Swarm Opimizaion for Power Dispach wih Pumped Hydro 141 Insalled Capaciy Maximal Maximal Discharge Pumping (m 3 /s) (m 3 /s) Lower Reservoir Maximal Minimal Sorage Sorage (10 3 m 3 ) (10 3 m 3 ) Efficiency 250MW ,756 1, Table 1. Characerisics of he Ming-Hu pumped hydro plan The proposed approach was esed on a summer weeday whose load profile (Fig. 7) was obained by subracing expeced generaion oupu of oher hydro plans and nuclear unis from he acual sysem load profile. Fig. 8 and 9 presen schemaics of es resuls. Fig. 8 shows he oal generaion/pumping schedules creaed by he proposed approach. Fig. 9 shows he remaining hermal load profiles. The opimal schedules for pumped hydro unis and hermal unis are obained wihin 3 minues, saisfying Taipower s ime requiremen. To invesigae furher how he proposed approach and exising mehods differ in performance, his wor adops a DP mehod (Chen, 1989) and a GA mehod (Chen & Chang, 1999) as he benchmar for comparison. Table 2 summarizes he es resuls obained using hese hree mehods MW Load Facor= HOUR Figure 7. Summer weeday load profile Several ineresing and imporan observaions are derived from his sudy and are summarized as follows: a. The generaion/pumping profile generally follows he load flucuaion, a finding ha is consisen wih economic expecaions. The Ming-Hu pumped hydro plan generaes 3,893 MWh power during pea load hours and pumps up 5,250 MWh power during ligh load hours, resuling in a cos saving of NT$5.91 million in one day, where cos saving = (cos wihou pumped hydro) - (cos wih pumped hydro). b. The pumped hydro unis are he primary source of sysem spinning reserve due o heir fas response characerisics. The sysem s spinning reserve requiremen accouns for he fac ha pumped hydro unis do no generae power a heir maximum during pea load hours. c. Variaion of waer sorage in he small lower reservoir is always reained wihin he maximum and minimum boundaries. The final volume reurns o he same as he iniial volume. d. The load facor is improved from 0.82 o 0.88 due o he conribuion of he four pumped hydro unis.

14 142 Paricle Swarm Opimizaion e. Noably, boh cos saving and execuion ime for he proposed approach are superior o eiher a DP or a GA mehod. Toal Generaion: 3,893 (MW*Hr) Toal Pumping: 5,250 (MW*Hr) MW HOUR Figure 8. Hourly generaion/pumping schedules MW Wihou pumped hydro (Load Facor=0.82) Wih pumped hydro (Load Facor=0.88) HOUR Figure 9. Conras of wo remaining hermal load profiles Cos Saving (10 3 NT$) Execuion Time (second) Load Mehod Facor DP , GA , RPSO , Table 2. Performance comparison wih exising mehods

15 Paricle Swarm Opimizaion for Power Dispach wih Pumped Hydro Conclusion This wor presens a novel mehodology based on a refined PSO approach for solving he power dispach wih pumped hydro problem. An advanage of he proposed echnique is he flexibiliy of PSO for modeling various consrains. The difficul waer dynamic balance consrains are embedded and saisfied hroughou he proposed encoding/decoding algorihms. The effec of ne head, consan power pumping characerisic, hermal ramp rae limis, minimal upime/downime consrains, and sysem s spinning reserve requiremens are all considered in his wor o mae he scheduling more pracical. Numerical resuls for an acual uiliy sysem indicae ha he proposed approach has highly aracive properies, a highly opimal soluion and robus convergence behavior for pracical applicaions. 6. References Al-Agash, S. (2001). Hydrohermal scheduling by augmened Lagrangian: consideraion of ransmission consrains and pumped-sorage unis, IEEE Transacions on Power Sysem, Vol. 16, pp Allan, R. N. & Roman, J. (1991). Reliabiliy assessmen of hydrohermal generaion sysems conaining pumped sorage plan, IEE Proceedings-C, Generaion, Transmission, and Disribuion, Vol. 138, pp Angeline, P. J. (1998). Evoluionary opimizaion versus paricle swarm opimizaion: philosophy and performance differences, Lecure Noes in Compuer Science, Vol. 1447, pp Boeringer, D. W. & Werner, D. H. (2004). Paricle swarm opimizaion versus geneic algorihms for phased array synhesis, IEEE Transacions on Anennas Propagaion, Vol. 52, pp Chen, P. H. (1989). Generaion scheduling of hydraulically coupled plans, Maser s hesis, Dep. Elec. Eng., Naional Tsing-Hua Universiy, Taiwan Chen, P. H. & Chang, H. C. (1995). Large-scale economic dispach by geneic algorihm, IEEE Transacions on Power Sysem, Vol. 10, pp Chen, P. H. & Chang, H. C. (1999). Pumped-sorage scheduling using a geneic algorihm, Proceedings of he Third IASTED Inernaional Conference on Power and Energy Sysems, pp , USA Chen, P. H. & Chen, H. C. (2006). Applicaion of evoluionary neural newor o power sysem uni commimen, Lecure Noes in Compuer Science, Vol. 3972, pp Eberhar, R. & Shi, Y. (1998). Comparison beween geneic algorihms and paricle swarm opimizaion, Lecure Noes in Compuer Science, vol. 1447, pp Eberhar, R. & Shi, Y. (2001). Paricle swarm opimizaion: developmens, applicaions and resources, Proceedings of IEEE Inernaional Congress on Evoluionary Compuaion, Vol. 1, pp El-Hawary, M. E. & Ravindranah, K. M. (1992). Hydro-hermal power flow scheduling accouning for head variaions, IEEE Transacions on Power Sysem, Vol. 7, pp Guan, X.; Luh, P. B.; Yen, H. & Rogan, P. (1994). Opimizaion-based scheduling of hydrohermal power sysems wih pumped-sorage unis, IEEE Transacions on Power Sysem, Vol. 9, pp

16 144 Paricle Swarm Opimizaion Ho, S. L.; Yang, S.; Ni, G. & Wong, H. C. (2006). A paricle swarm opimizaion mehod wih enhanced global search abiliy for design opimizaions of elecromagneic devices, IEEE Transacions on Magneics, Vol. 42, pp Jeng, L. H.; Hsu, Y. Y.; Chang, B. S. & Chen, K. K. (1996). A linear programming mehod for he scheduling of pumped-sorage unis wih oscillaory sabiliy consrains, IEEE Transacions on Power Sysem, Vol. 11, pp Juang, C. F. (2004). A hybrid of geneic algorihm and paricle swarm opimizaion for recurren newor design, IEEE Transacions on Sysems, Man, and Cyberneics, Vol. 34, pp Kennedy, J. & Eberhar, R. (1995). Paricle swarm opimizaion, Proceedings of IEEE Inernaional Conference on Neural Newors, Vol. 4, pp Kennedy, J. & Eberhar, R. (1997). A discree binary version of he paricle swarm algorihm, Proceedings of Inernaional Conference on Sysems, Man, and Cyberneics, pp , Piscaaway, NJ Kennedy, J. & Eberhar, R. (2001). Swarm Inelligence, Morgan Kaufmann, San Francisco Kuo, C. C.; Chen, P. H.; Hsu, C. L. & Lin, S. S. (2005). Paricle swarm opimizaion for capacior allocaion and dispaching problem using ineracive bes-compromise approach, WSEAS Transacions on Sysem, Vol. 4, pp Naa, S.; Genji, T.; Yura, T. & Fuuyama, Y. (2003). A hybrid paricle swarm opimizaion for disribuion sae esimaion, IEEE Transacions on Power Sysem, Vol. 18, pp Wood, A. J. & Wollenberg, B. F. (1996). Power Generaion, Operaion, and Conrol, 2 nd ed., John Wiley & Sons, New Yor Zhao, B.; Guo, C. X. & Cao, Y. J. (2005). A muliagen-based paricle swarm opimizaion approach for opimal reacive power dispach, IEEE Transacions on Power Sysem, Vol. 20, pp