A Novel Solution Based on Differential Evolution for Short-Term Combined Economic Emission Hydrothermal Scheduling

Transcription

1 Engineering, 2009, 1, 1-54 ublished Online June 2009 in SciRes (hp:// A Noel Soluion Based on Differenial Eoluion for Shor-Term Combined Economic Emission Hydrohermal Scheduling Chengfu Sun 1, Songfeng Lu 2 School of Compuer Science and Technology, Huazhong Uniersiy of Science and Technology, Wuhan, China 1 aason509@smail.hus.edu.cn, 2 lusongfeng@sina.com Receied April 17, 2009; reised May 13, 2009; acceped May 18, 2009 Absrac This paper presens a noel approach based on differenial eoluion for shor-erm combined economic emission hydrohermal scheduling, which is formulaed as a bi-obecie problem: 1) imizing fuel cos and 2) imizing emission cos. A penaly facor approach is employed o coner he bi-obecie problem ino a single obecie one. In he proposed approach, heurisic rules are proposed o handle waer dynamic balance consrains and heurisic sraegies based on prioriy lis are employed o repair acie power balance consrains iolaions. A feasibiliy-based selecion echnique is also deised o handle he reseroir sorage olumes consrains. The feasibiliy and effecieness of he proposed approach are demonsraed and he es resuls are compared wih hose of oher mehods repored in he lieraure. Numerical experimens show ha he proposed mehod can obain beer-qualiy soluions wih higher precision han any oher opimizaion mehods. Hence, he proposed mehod can well be exended for soling he large-scale hydrohermal scheduling. Keywords: Hydrohermal ower Sysems, Economic Load Scheduling, Combined Economic Emission Scheduling, Differenial Eoluion 1. Inroducion One of he maor problems exising oday on elecric power sysems is he opimum scheduling of hydrohermal plans. Shor-erm hydrohermal scheduling is a daily planning ask in power sysems and is main obecie is o imize he oal operaional cos subeced o a ariey of consrains of hydraulic and power sysem nework. As he source for hydropower is he naural waer resources, he operaional cos of hydroelecric plans is insignifican. Thus, he obecie of imizing he operaional cos of a hydrohermal sysem essenially reduces o imize he fuel cos of hermal plans oer a scheduling horizon while saisfying arious consrains. Due o increasing concern oer amospheric polluion, harmful emission produced by he hermal unis mus be imized simulaneously. So a reised economic power dispach program considering boh he fuel cos and emission is required. Bu imizing polluion may lead o an increase in generaion cos and ice ersa. The imporance of he generaion scheduling problem of hydrohermal sysems is well recognized. Therefore, many mehods hae been deised o sole his difficul opimizaion problem for seeral decades. Some of hese mehods are dynamic programg mehodology [1], linear programg [2], and decomposiion echniques [3]. Recenly, aside from he aboe mehods, opimal hydrohermal scheduling problems hae been soled by mea-heurisic approaches such as geneic algorihm [4-6], culural algorihm [7] and paricle swarm opimizaion [8] ec. Various heurisic mehods such as heurisic search echnique [9], fuzzy saisfying eoluionary programg procedures [10] and fuzzy decision-making sochasic echnique [11] hae been applied o sole Copyrigh 2009 SciRes. Engineering, 2009, 1, 1-54

2 C. F. SUN ET AL. 47 muli-obecie shor-erm hydrohermal scheduling problems. Because hese mea-heurisic opimizaion mehods are able o proide higher qualiy soluions, hey hae receied more ineres. One of hese mea-heurisic opimizaion mehods is differenial eoluion (DE) [13]. A new opimizaion mehod known as DE, which is a sochasic search algorihm based on populaion cooperaion and compeiion of indiiduals, has gradually become more popular and has been successfully applied o sole opimizaion problems paricularly inoling non-smooh obecie funcion. DE combines he simple arihmeic operaors wih he classical eoluion operaors of crossoer, muaion and selecion o eole from a randomly generaed populaion o a final soluion. The DE algorihm has been applied o arious fields of power sysem opimizaion such as dynamic economic dispach wih ale-poin effecs [14], hydrohermal scheduling [15], economic dispach wih non-smooh and non-conex cos funcions [16], opimal reacie power planning in large-scale disribuion sysem [17], and economic dispach problem [18]. This work presens a noel approach based on differenial eoluion o sole shor-erm combined economic emission scheduling of cascaded hydrohermal sysems. Moreoer, heurisic rules are proposed o handle he waer dynamic balance consrains and heurisic sraegies based on prioriy lis are employed o handle acie power balance consrains. A he same ime, a feasibiliy-based selecion echnique is deised o handle he reseroir sorage olumes consrains. The resuls obained wih he proposed approach were analyzed and compared wih he resuls of he differenial eoluion [12] and ineracie fuzzy saisfying mehod based on eoluionary programg [10] repored in he lieraure. The remainder of he paper is organized as follows. The formulaion of he shor-erm combined economic emission scheduling of hydrohermal power sysems wih cascaded reseroirs is inroduced in Secion 2, while Secion 3 explains he classical DE. Secion 4 describes he implemenaion of he proposed mehod for soling he shor-erm hydrohermal scheduling and oulines heurisic sraegies o handle waer dynamic balance consrains and acie power balance consrains. Secion 5 presens he opimizaion resuls for he shor-erm hydrohermal power sysems scheduling. Lasly, secion 6 draws he conclusions. 2. roblem Formulaion The hydrohermal scheduling problem combined economic emission scheduling is formulaed as a bi-obecie opimizaion problem. I is concerned wih he aemp o imize he fuel cos and as well as he emission of hermal unis, while making full use of he aailabiliy of hydro-resources as much as possible. In he formulaion of he hydrohermal scheduling problem, he following obecies and consrains mus be aken ino accoun and he equaliy and inequaliy consrains mus simulaneously be saisfied Noaions In order o formulae he hydrohermal scheduling problem mahemaically, he following noaions is inroduced firs: fi si fuel cos of hermal plan i including ale poin loading ei si emission of hermal plani including ale poin loading asi, bsi, c si esi, f si coefficiens of hermal generaing plan i si, si, si, si, si emission coefficiens of hermal plan i T oal ime inerals oer scheduling horizon N s, N h number of hermal and hydro plans respeciely h power generaion of hydro generaing plan a ime ineral si power generaion of hermal generaing uni i a ime ineral D power demand a ime ineral L oal ransmission loss a ime ineral C1, C2, C3, C4, C5, C6 power generaion coefficiens of hydro plan V h sorage olume of reseroir a ime ineral Q h waer discharge rae of he h reseroir a ime. si si imum and imum power generaion by hermal plan i si si imum and imum power generaion by hydro plan V h, V h imum and imum sorage olumes of reseroir Ih inflow of hydro reseroir a ime ineral S h spillage discharge rae of hydro plan a ime ineral m waer ranspor delay from reseroir m o R u number of upsream hydro generaing plans direcly aboe reseroir curren ieraion generaion N number of he parameer ecors p 2.2. Obecie Funcions Copyrigh 2009 SciRes. Engineering, 2009, 1, 1-54

3 48 C. F. SUN ET AL Economic Scheduling In his paper, non-smooh fuel cos funcion of hermal generaing uni wih ale-poin effecs is considered. 2 sin f a b c e f i si si si si si si si si si si (1) For a gien hydrohermal sysem, he problem may be described as imizaion o f oal fuel cos associaed o he on-line N unis for T inerals in he gien ime horizon as defined by Equaion (2) under a se of operaing consrains as follows: [ f ] (2) Emission Scheduling In his sudy, he amoun of emission from each generaor can be described as he sum of a quadraic and an exponenial funcion. The economic emission scheduling problem can be expressed as he imizaion of oal amoun of emission release defined by Equaion(4) as 2.3. Consrains F T While imizing he aboe wo obecies, he following consrains mus be saisfied simulaneously. Acie power balance consrain The hydroelecric generaion is a funcion of waer discharge rae and reseroir waer head, which can be expressed as follows: N s 1 i1 2 exp e i si si si si si si si si si (3) Ns T N s E [ e ] 1 i1 Nh i i si si 0 si h D L i1 1 (4) (5) 2 2 h C1 Vh C2 Qh C3 Vh Qh C4 Vh C5 Qh C 6 (6) eneraion limis consrains si si si 1) Reseroir sorage olumes consrains 2) Discharge raes limi (7) h h h Vh Vh V h Q Q Q h h h 3) Waer dynamic balance consrains (8) (9) (10) Ru V V I Q S Q S h h, 1 h h h hm, m hm, m m1 3. Oeriew of Differenial Eoluion Algorihm (11) As a populaion-based and sochasic global opimizer, differenial eoluion (DE) is one of he laes eoluionary opimizaion mehods proposed by Sorn and rice [13]. In a DE algorihm, candidae soluions are randomly generaed and eoled o final indiidual soluion by simple echnique combining simple arihmeic operaors wih he classical eens of muaion, crossoer and selecion. One of he mos frequenly used muaion sraegies, named DE/rand/1/bin, will be employed in his paper Muaion Operaion The essenial ingredien in he muaion operaion is he ecor difference. For each arge ecor Xi i1, 2,, Np, he weighed difference beween wo randomly seleced ecors X l and X m is added o a hird randomly seleced ecor X k o generae a muaed ecor V i using he following equaion. i k l m V X F X X (12) where X k, X l and X m are randomly seleced ecors and i k l m; The muaion facor F 0 is a user chosen parameer o conrol he amplificaion of he difference beween wo indiiduals so as o aoid search sagnaion. 3.2 Crossoer Operaion Following he muaion phase, he crossoer operaion is performed in order o increase he diersiy in he searching process. U i, Vi, if CR or q Xi, oherwise where 0,1 (13), generaed anew for each alue of, is a uniformly disribued random number. The crossoer facor CR 0,1 conrols he diersiy of he populaion. X i,, Vi, and Ui, are he h parameer of he i h arge ecor, muan ecor and rial ecor a generaion, respeciely Selecion Operaion Copyrigh 2009 SciRes. Engineering, 2009, 1, 1-54

4 Thereafer, a selecion operaor is applied o compare he finess funcion alue of wo compeing ecors, namely, arge and rial ecors o deere who can surie for he nex generaion. f X 1 i i Ui if f U f X Xi oherwise i C. F. SUN ET AL. 49 (14) where denoes he finess funcion under opimizaion (imizaion). 4. Implemenaion of he roposed Mehod for Soling he Shor-Term Hydrohermal Scheduling In his secion, he procedures for soling shor-erm scheduling problem of hydrohermal power sysem are described in deails. Especially, heurisic sraegies will be gien o handle consrains of hydrohermal scheduling problem. The process of he proposed mehod for soling hydrohermal scheduling can be summarized as follows Srucure of arameer Soluion Vecor The srucure of a soluion for hydrohermal scheduling problem is composed of a se of decision ariables which represen he discharge rae of he each hydro plan and he power generaed by each hermal uni oer he scheduling horizon. Qh11 Qh21 QhN 1 h s11 s21 sns1 Qh12 Qh22 QhNh2 s 12 s22 sns2 k QhT 1 Qh2T QhNT h st 1 s2t snt s (15) The elemens Qh and si si ( 1, 2, N h ; i 1, 2,, N s ) are subeced o he waer discharge rae and he hermal generaing capaciy consrains as depiced in Equaion. (10) and (7), respeciely. The waer discharge rae of he h hydro plan in he dependen ineral mus saisfy he waer dynamic balance consrains in Equaion (11) Iniializaion arameer Vecors of each parameer ecor k p r Q Q r Q Q (16) h h q h h r (17) si si s si si r where q and s are he random numbers uniformly disribued in 0, Combined Economic and Emission Scheduling The shor-erm combined economic emission scheduling of hydrohermal power sysems wih cascaded reseroirs is a bi-obecie problem wih he aemp o imize simulaneously fuel cos and emission of hermal plans. The bi-obecie opimizaion problem can be ransformed ino a single obecie one by inroducing price penaly facors h. For more deails, see Ref. [12] Soluion Modificaion New alues of waer discharge rae Qh, 1 and power generaion are generaed hrough muaion and si, 1 crossoer operaion according o Equaions (12) and (13), respeciely. The new alues are no always guaraneed o saisfy he consrains Equaions (10) and (7), respeciely. If any alue iolaing is consrain is modified in he following way: Q if Q Q h h, 1 h h, 1 h, 1 h h, 1 h Qh if Qh, 1 Qh Q Q if Q Q Q si if si, 1 si, 1, 1 if, 1 si if si, 1 si si si si si si (18) (19) 4.5. Heurisic Sraegies o Handle Equaliy Consrains Handling Waer Dynamic Balance Consrains To mee exacly he resricions on he iniial and final reseroir sorage, he waer discharge rae of he h hydro plan in he dependen ineral d is hen calculaed using Equaion(21). The dependen waer discharge rae mus saisfy he consrains in Equaion (10). Assug he spillage in Equaion (11) o be zero for simpliciy, he waer dynamic balance consrains are During he iniializaion process, he candidae soluion X k 1, 2,, N is ran- Ru domly iniialized wihin he feasible range as follows: 0, m T T T (20) V V Q Q I h ht h hm h 1 1 m1 1 Copyrigh 2009 SciRes. Engineering, 2009, 1, 1-54

5 50 C. F. SUN ET AL. where V is he iniial sorage olume of reseroir ; VhT h0 is he final sorage olume of reseroir. The procedures for repairing he waer dynamic balance iolaions in hydrohermal scheduling are as follows: Sep 1: Se 1. Sep 2: Randomly choose a ime ineral d as a dependen ineral and se coun 1. Sep 3: In order o mee equaliy consrain in Equaion (11), he waer discharge rae of he h hydro plan in he dependen ineral d is hen calculaed by Qhd T T Ru T Q V V Q Q I hd ho ht h hm, m h 1 1 m1 1 d (21) If he compued Q hd doesn iolae he consrains in Equaion (10) hen go o sep 7; oherwise go o he nex sep Sep 4: Change Q hd using Equaion(18). Sep 5: A new random ime ineral d is chosen ensuring ha i is no repeaedly seleced and coun coun 1. Sep 6: If cou n T, hen go o sep 3; oherwise go o nex sep. Sep 7: 1, if Nh, hen go o sep 2; oherwise go o nex sep. Sep 8: The modificaion process is eraed Handling Acie ower Balance Consrains The power balance equaliy consrains in Equaion(5) sill remain o be resoled afer he waer dynamic balance consrains are presered. The heurisic sraegy based on prioriy lis is proposed for handling he power balance consrains. In his paper, prioriy lis is creaed according o each hermal plan parameer. When he hermal plan is a is imum oupu power, he aerage full-load cos i of hermal plan i a ime ineral is defined by 1 fi si 2h ei si i (22) si where h is price penaly facor a ime ineral, 1 and 2 are he weigh facors. The deail procedures for handling acie power balance consrains are as follows: Sep 1: Calculae he aerage full-load cos i using Equaion(22) a ime ineral. Arrange hem in ascending order of i o obain a prioriy lis L. Sep 2: Se 1. Sep 3: Se emp _ L L. Sep 4: The amoun of acie power balance iolaion a ime ineral is calculaed Ns Nh si h D i1 1 by 1 ( ). In his paper he power loss is no considered for simpliciy. Sep 5: If 0, go o Sep 14; if 0, go o Sep 6; if 0, go o Sep 10. Sep 6: Se m 1. Sep 7: Se power of he generaor uni k wih highes i in emp _ L o be k sk.then delee hermal uni k from emp _ L. Sep 8: Calculae he oal power generaed by all hermal unis a ime ineral. If N ( h k si h D su m 1 sum N h sum h D 1, se ) and go o sep 14; oherwise se k sk. Sep 9 : m m 1. If m Ns, hen go o Sep 7; oherwise go o Sep 14 Sep 10: Se m 1. Sep 11: Se power of he generaor uni k wih low- in emp _ L o be k sk.then delee es i hermal uni k from emp _ L. Sep 12: Calculae he oal power generaed by all hermal unis a ime ineral. If N ( h k si h D su m 1 sum N h sum h D 1, se ) and go o sep 14; oherwise se k sk. Sep 13: m m 1.If m Ns, hen go o Sep 11; oherwise go o Sep 14. Sep 14: 1.If T, hen go o Sep 3; oherwise go o Sep 15. Sep 15: The modificaion process is eraed Selecion Based Technique for Handling Reseroir Sorage Volumes Consrains In his work, he feasibiliy-based selecion rules are applied o he proposed approach for handling he inequaliy consrains of reseroir sorage olumes consrains. The procedures for repairing he reseroir sorage olumes consrains are as follows: Sep 1: The oerall reseroir sorage olumes consrains iolaion of soluion x is CV x, which is defined as Copyrigh 2009 SciRes. Engineering, 2009, 1, 1-54

6 C. F. SUN ET AL. 51 N h T CV x 0, V, h Vh Vh Vh (23) 1 1 Sep 2: (1) If boh parameer ecors are feasible, hen he one wih he beer finess alue wins. (2) Oherwise, if boh parameer ecors are infeasible, hen he one wih he less alue of CV x wins. (3) Oherwise, he feasible parameer ecors always wins. 5. Simulaion Resuls In his secion, a es sysem consising of a muli-chain cascade of four hydro unis and hree hermal unis is sudied o demonsrae he feasibiliy and effecieness of he proposed mehod for soling shor-erm hydrohermal scheduling wih cascaded reseroirs. The enire scheduling period is chosen as one day wih 24 inerals of 1 hour each. The load demand of he sysem, hydro and hermal uni coefficiens, reseroir inflows and reseroir limis are aken from he lieraure [10]. In order o compare wih Ref. [12], he parameers for populaion size and imum number of generaions allowed are se as follows: N p 70, imum number of ieraions Maxier 400, respeciely. Before proceeding o he simulaed calculaion, careful selecion of muaion and crossoer facor is imporan o produce a compeen resul. The following alues for muaion and crossoer facor were seleced by parameer seing hrough rial and error for he presen es sysem: muaion facor F 0.44, crossoer facor CR Under he chosen parameers, i has been found o proide opimum resuls. The proposed approach is performed 10 rials for differen cases of hydrohermal scheduling. According o [12], he oal cos can be presened as follow s for a rade off beween fuel cos and emission cos. 1 si 2 si TC F h E (24) where 1 and 2 are he weigh facors. The resuls of proposed mehod for obaining com- emission scheduling (CEES, bined economic 1 1 and 2 1 ) soluion are illusraed as follows. In his case, he aluesi of hermal uni 1, 2 and 3 a ime inerals 1, 2, 3, 4, 5, 6, 24 are , and , while a oher ime inerals hey are , and Bu he prioriy lis is1, 2, 3 oer he enire scheduling horizon. The hermal uni 1 wih he lowes i will hae he highes prioriy o be dispached more generaion power. The opimal hydrohermal generaion schedule for CEES is shown in Figure 1 and he opimal hourly waer discharge rae obained by he proposed mehod for CEES is presened in Figure 2. Figure 1. Hydrohermal generaion (MW) schedule for CEES. Copyrigh 2009 SciRes. Engineering, 2009, 1, 1-54

7 52 C. F. SUN ET AL. Figure 2. Hourly hydro plan discharge ( m 3 ) for CEES. Figure 3. Reseroir sorage olumes for CEES. Copyrigh 2009 SciRes. Engineering, 2009, 1, 1-54

8 C. F. SUN ET AL. 53 The raecories of reseroir sorage olumes for CEES are shown in Figure 3. Table 1 shows ha using he proposed mehod opimal fuel cos is found o be $ , while amoun emission is found o be lb. In Table 1, he opimal soluions of he fuel cos and emission cos for economic load scheduling (ELS, 1 1 and 2 0 ), economic emission scheduling (EES, 1 0 and 2 1 h ) and CEES obained from he proposed approach hae been compared wih hose of DE [12]. From he resuls i is quie eiden ha he proposed mehod proides beer soluions for shor-erm combined economic emission hydrohermal scheduling wih cascaded hydro reseroirs. Table 2 presens he bes, wors and mean alue of fuel cos and emission of CEES obained by differenial eoluion wihou prioriy lis, paricle swarm opimizaion wihou prioriy lis and he proposed approach. From he analysis of resuls in Table 2, i can be seen ha he proposed approach can produce aluable rade off soluions for CEES. I also shows ha he wo obecies of imizing he fuel cos and emission cos are of conflicing naure, ha is o say, imizing polluion increases fuel cos and ice ersa. From he resuls of CEES, i clearly sees ha wih some compromise in fuel cos, i is possible o obain huge reducion in emission. I can be seen clearly from Table 1 ha he proposed mehod yields much beer resuls in erms of fuel cos, he amoun of emission han known opimizaion mehods repored in he lieraure. I is also ery imporan o noe ha compared wih he resuls of fuzzy saisfying [10] he beer resuls from [12] are obained based on iolaing he consrains of he es sysem, such as he resuls of Table 1, Table 3 and Table 5 in Ref. [12], from which i is clearly shown ha he power generaion of hermal uni s 1 iolaes is consrain which is 20 s a some ime inerals. Howeer, in his sudy we obain een beer resuls while sricly saisfying all consrains of he es sysem. 6. Conclusions In his paper, a noel approach in combinaion wih noel equaliy consrain handling echniques has been successfully inroduced o sole hydrohermal scheduling wih non-smooh fuel and emission cos funcions. The maor adanages of his noel mehod are as follows: 1) In order o handle consrains effeciely, heurisic rules are proposed o handle waer dynamic balance consrains and heurisic sraegies based on prioriy lis are employed o handle acie power balance consrains; 2) The feasibiliy-based selecion rules are deeloped o handle he reseroir sorage olumes consrains. Addiionally, he improed heurisic sraegies can be simply incorporaed ino differenial eoluion. Hence he proposed mehod does no require he use of penaly funcions and explores he opimum soluion a a relaiely lesser compuaional effor. Numerical experimens show ha he proposed mehod can obain beer-qualiy soluions wih higher precision han any oher opimizaion mehods repored in he lieraure. Hence, he proposed mehod can well be exended for soling he large-scale hydrohermal scheduling. 7. Acknowledgemens The auhors graefully acknowledge he financial suppors from Naional Naural Science Foundaion of China under ran no The auhors hank he anonymous Reiewers and Ediors for consrucie and deailed commens. Table 1. Comparison of cos for ELS, EES and CEES by proposed mehod and DE. The proposed mehod Differenial eoluion (DE) The proposed mehod Fuel cos ($) Emission (lb) ELS EES CEES ELS EES CEES Table 2. Comparison of cos of CEES by proposed mehod, DE and SO. Bes Value Wors Value Mean Value Fuel cos($) Emission(lb) Differenial Fuel cos($) eoluion wihou prioriy lis Emission(lb) SO wihou prioriy lis Fuel cos($) Emission(lb) Copyrigh 2009 SciRes. Engineering, 2009, 1, 1-54

9 54 C. F. SUN ET AL. 8. References [1] J. Tang and B. eer, Hydrohermal scheduling ia exended differenial dynamic programg and mixed coordinaion, IEEE Transacions on ower Sysem, Vol. 10, pp , [2] M. iekuowski, Opimal shor-erm scheduling for a large-scale cascaded hydro sysem, IEEE Transacions on ower Sysem, Vol. 9, pp , [3] M. V. F. ereira and L. M. V.. ino, A decomposiion approach o he economic dispach of he hydrohermal sysems, IEEE Transacions on ower Sysems, Vol. 101, pp , [4] M. Ramirez and. E. Onae, The shor-erm hydrohermal coordinaion ia geneic algorihms, Elecric ower Componens and Sysems, Vol. 34, pp. 1-19, [5] E. il, J. Busos, and H. Rudnick, Shor-erm hydrohermal generaion scheduling model using a geneic algorihm, IEEE Transacions on ower Sysem, Vol. 18, pp , [6] X. Yuan and Y. Yuan, A hybrid chaoic geneic algorihm for shor-erm hydro sysem scheduling, Mahemaics and Compuers in Simulaion, Vol. 59, pp , [7] X. Yuan and Y. Yuan, Applicaion of culural algorihm o generaion scheduling of hydrohermal sysems, Energy Conersion and Managemen, Vol. 47, pp , [8] B. Yu, X. Yuan, and J. Wang, Shor-erm hydro-hermal scheduling using paricle swarm opimizaion mehod, Energy Conersion and Managemen, Vol. 48, pp , [9] J. S. Dhillon and D.. Kohari, Muli-obecie shorerm hydrohermal scheduling based on heurisic search echnique, Asian Journal of Informaion Technology, Vol. 6, pp , [10] M. Basu, An ineracie fuzzy saisfying mehod based on eoluionary programg echnique for muli-obecie shor-erm hydrohermal scheduling, Elecric ower Sysems Research, Vol. 69, pp , [11] J. S. Dhillon, S. C. ari, and D.. Kohari, Fuzzy decision-making in sochasic muli-obecie shor-erm hydrohermal scheduling, IEE roceedings of eneraion Transmission and Disribuion, Vol. 149, pp , [12] K. K. Mandal and N. Chakrabory, Shor-erm combined economic emission scheduling of hydrohermal power sysems wih cascaded reseroirs using differenial eoluion, Energy Conersion and Managemen, Vol. 50, pp , [13] R. Sorn and K. rice, Differenial eoluion a simple and efficien heurisic for global opimizaion oer coninuous spaces, Journal of lobal Opimizaion, Vol. 11, pp , [14] X. H. Yuan, L. Wang, Y. C. Zhang, and Y. B. Yuan, A hybrid differenial eoluion mehod for dynamic economic dispach wih ale-poin effecs, Exper Sysem wih Applicaion, Vol. 36, pp , [15] X. H. Yuan, B. Cao, B. Yang, and Y. B. Yuan. Hydrohermal scheduling using chaoic hybrid differenial eoluion, Energy Conersion and Managemen, Vol. 49, pp , [16] S. K. Wang, J.. Chiou, and C. W. Liu, Non-smooh/ non-conex economic dispach by a noel hybrid differenial eoluion algorihm, IEE roceeding eneraion Transmission and Disribuion, Vol. 1, pp , [17] C. F. Changa, J. J. Wong, J.. Chiou, and C. T. Su, Robus searching hybrid differenial eoluion mehod for opimal reacie power planning in large-scale disribuion syems, Elecric ower Sysems Research, Vol. 77, pp , [18] J.. Chiou, Variable scaling hybrid differenial eoluion for large-scale economic dispach problems, Elecric ower Sysems Research, Vol. 77, pp , Copyrigh 2009 SciRes. Engineering, 2009, 1, 1-54