Application of Particle Swarm Optimization to Economic Dispatch Problem: Advantages and Disadvantages

Transcription

1 Appcaton of Partce Swarm Optmzaton to Economc Dspatch Probem: Advantages and Dsadvantages Kwang Y. Lee, Feow, IEEE, and Jong-Bae Par, Member, IEEE Abstract--Ths paper summarzes the state-of-art partce swarm optmzaton (PSO) appcatons for resovng the economc dspatch (ED) probem, whch s consdered as one of the compex probems to be taced. The PSO technques have drawn much attenton from the power system communty and been successfuy apped n many compex optmzaton probems n power systems. Ths paper focuses on the appcaton of PSO technques to the ED probems and descrbes ther advantages and dsadvantages n resovng the ED probems. Index Terms Partce swarm optmzaton, economc dspatch, advantages and dsadvantages of PSO. I. ITRODUCTIO ARTICLE swarm optmzaton (PSO) s one of the Pmodern heurstc agorthms, whch can be used to sove nonnear and non-contnuous optmzaton probems []. It s a popuaton-based search agorthm and searches n parae usng a group of partces smar to other AI-based heurstc optmzaton technques. The orgna PSO suggested by Kennedy and Eberhart s based on the anaogy of swarm of brd and schoo of fsh []. Each partce n PSO maes ts decson usng ts own experence and ts neghbor s experences for evouton. That s, partces approach to the optmum through ts present veocty, prevous experence, and the best experence of ts neghbors [3]. The man advantages of the PSO agorthm are summarzed as: smpe concept, easy mpementaton, robustness to contro parameters, and computatona effcency when compared wth mathematca agorthm and other heurstc optmzaton technques. In a physca n-dmensona search space, the poston and veocty of partce- are represented as the vectors X x,, x V v v n the PSO agorthm. and n n, Gbest Gbest Let x,, x and Gbest x,, x n be the best poston of partce and ts neghbors best poston so far, respectvey. The modfed veocty and poston of each partce can be cacuated usng the current veocty and the K. Y. Lee s wth the Department of Eectrca Engneerng, The Pennsyvana State Unversty, Unversty Par, PA 680, USA. (wangee@psu.edu) J.-B. Par s wth the Department of Eectrca Engneerng, Konu Unversty, Seou, 43-70, Korea. (jbaepar@onu.ac.r) n dstance from V to Gbest as foows: ( V c r X ) c r ( Gbest X ) () X X V () V veocty of partce at teraton, nerta weght factor, c,c acceeraton coeffcents, r,r random numbers between 0 and, X poston of partce at teraton, best poston of partce unt teraton, Gbest best poston of the group unt teraton. In ths veocty updatng process, the vaues of parameters such as, c, and c shoud be determned n advance. In genera, the weght s set accordng to the foowng equaton []: mn Iter Iter (3), mn nta, fna weghts, Iter mum teraton number, Iter current teraton number. Recenty, PSO has been successfuy apped to varous feds of power system optmzaton. Most of power system optmzaton probems ncudng economc dspatch (ED) have compex and nonnear characterstcs wth heavy equaty and nequaty constrants. The prmary objectve of the ED probem s to determne the optma combnaton of power outputs of a generatng unts so as to meet the requred oad demand at mnmum operatng cost whe satsfyng system equaty and nequaty constrants. In the tradtona ED probem, the cost functon for each generator has been approxmatey represented by a snge quadratc functon and s soved usng mathematca programmng based on the optmzaton technques such as ambda-teraton method, gradent-based method, etc. [4]. These mathematca methods requre ncrementa or margna fue cost curves whch shoud be monotoncay ncreasng to fnd goba optma souton. Unfortunatey, the nput-output X/06/$ IEEE 88 PSCE 006

2 characterstcs of generatng unts are nherenty hghy nonnear because of prohbted operatng zones, vave-pont oadngs, and mutpe effects, etc. Thus, the practca ED probem s represented as a non-smooth optmzaton probem wth equaty and nequaty constrants, whch cannot be soved by the tradtona mathematca methods. Dynamc programmng (DP) method [5] can sove such types of probems, but t suffers from so-caed the curse of dmensonaty. Over the past few years, n order to sove these probems, many saent methods have been deveoped such as genetc agorthm (GA) [6], evoutonary programmng (EP) [7], [8], Tabu search [9], neura networ approaches [0], and PSO based approaches []-[5]. Among these methods, the focus of ths paper s to survey and summarze PSO appcatons to the ED probems. We hope that ths paper w serve as a good startng pont for those nterested n earnng about the deveopment of PSO and ts appcatons n ED probems. II. ED PROBLEM FORMULATIOS A. Objectve Functon The objectve of the economc dspatch probem s to mnmze the tota fue cost of therma power pants subjected to the operatng constrants of a power system. The smpfed cost functon of each generator can be represented as a quadratc functon as descrbed n (5). F T F ( P ) (4) F ( P ) a b P cp (5) F T tota generaton cost, F cost functon of generator, a, b, c cost coeffcents of generator, P power of generator, number of generators. ) Cost Functon Consderng Vave-Pont Effects The generatng unts wth mut-vave steam turbnes exhbt a greater varaton n the fue-cost functons. Snce the vave pont resuts n the rppes as shown n Fg., a cost functon contans hgher order nonnearty []. To tae account for the vave-pont effect, snusoda functons are added to the quadratc const functons. Therefore, equaton (5) shoud be repaced as (6) as foows: F ( P ) a b P cp e sn( f ( P, mn P )) (6) where e and f are the coeffcents of unt refectng vavepont effects. Fg.. Exampe of cost functon wth 5 vaves. ) Cost Functon wth Mutpe Fues Snce the dspatchng unts are practcay supped wth mut-fue sources, each unt shoud be represented wth severa pecewse quadratc functons refectng the effects of fue type changes [] as shown n Fg.. In genera, a pecewse quadratc functon can be used to represent the nput-output curve of a generator wth mutpe fues and descrbed as (7). a b P c P f P mn P P a b P cp f P P P F ( P ) (7) an bnp cnp f Pn P P where a p, bp, cp are the cost coeffcents of generator for the p-th power eve. $ $/MW Incrementa cost FUEL FUEL FUEL3 Mn P P Max Power[MW] Fg.. Pecewse quadratc and ncrementa cost functons of a generator. B. Equaty and Inequaty Constrants ) Actve power baance equaton For power baance, an equaty constrant shoud be satsfed. The tota generated power shoud be the same as tota oad demand pus the tota ne oss where oss. P PD P Loss P D s the tota system demand and Loss (8) P s the tota ne 89

3 ) Mnmum and mum power mts Generaton output of each generator shoud be ad between mum and mnmum mts. The correspondng nequaty constrants for each generator are P P P (9), mn,, mn and P, are the mnmum and mum output of generator, respectvey. 3) Generator ramp rate mts The generator output cannot be rased or owered to any vaue nstantaneousy. The operatng range of a onne unts s restrcted by ther ramp rate mts as foows: P P 0 DR and P P 0 UR (0) 0 s the prevous output power of unt. DR and UR are the down ramp and up ramp mts, respectvey. 4) System spnnng reserve requrements The system spnnng reserve constrant for securng power system securty s summarzed as foows; MnP P UR,, SR () where S R s the system spnnng reserve requrement n MW. 5) Generator prohbted operatng zones The operatng zone of a generatng unt may not be avaabe aways for power generaton due to mtatons n practca operatng constrants as shown n Fg. 3. The constrant s descrbed n (); $ PZ : Prohbted Zone PZ PZ Mn Max MW Fg. 3. Exampe of cost functon wth two prohbted operatng zones. P u P P, u P,,mn PZ, P P P P P P,,,,,3,, () PZ, u, and P, are the ower and upper boundary of prohbted operatng zone of unt, respectvey. PZ, s the number of prohbted zones of unt. 6) Lne fow constrants PLf, PLf,,,,, L (3) Lf, s the rea power of ne and L s the number of transmsson nes. III. SURVEY OF PSO APPLICATIOS TO ED PROBLEMS The practca ED probem must consder not ony the cost functon wth vave-pont and mut-fue effects but aso equaty and nequaty constrants such as power baance, power generaton mts, system spnnng reserve, generator ramp rate mts, and generator prohbted operatng zones, etc. Par et a. [] suggested a modfed PSO technque to sove the ED probems wth non-smooth cost functons ncorporatng dynamc search space reducton strategy. In ths paper, the objectve functon s formuated as a combnaton of pecewse quadratc cost functons nstead of havng a snge convex functon for each generatng unt n order to consder practca operatng condtons e vave-pont and mut-fue effects as descrbed n (6) and (7), respectvey. In order to mpement the PSO agorthm for sovng the ED wth vavepont effect, the parameters were set as foows; the popuaton sze s set as 0, nta weght (.e., ) s.0, fna weght (.e., mn ) s 0.5, and acceeraton coeffcents are set as. And PSO agorthm for sovng the ED wth mutpe fues assgned the parameters as foows; the popuaton sze s 30, nta weght (.e., ) s 0.5, fna weght (.e., mn ) s 0., and acceeraton coeffcents are set as. Aso, n the paper, a technque to treat equaty and nequaty constrants n a PSO mechansm s suggested. E-Gaad et a. [] added new constrants to the ED probems by consderng system spnnng reserve and generator prohbted operatng zones as gven n () and (), respectvey. In ths paper, the conventona PSO technque s apped to sove non-smooth ED probems. Gang [3] consdered the generator ramp rate mts (0) n the same probem treated n reference []. Aso, the paper adopts the orgna PSO technque whe the ftness functon s mapped nto [0,]. In the paper, the parameters were used as foows; the popuaton sze s 00, the mum teraton number s 00, fna weght (.e., ) s 0.9, nta weght (.e., mn ) s 0.4, and acceeraton coeffcents are. Gang [4] carred out the dynamc ED that must satsfy not ony the system oad demand and the spnnng reserve capacty but aso some practca operaton constrants that ncude the ramp rate mts (0), the prohbted operatng zones (), and the ne fow mts (3). Vctore and Jeyaumar [5] aso mpemented the dynamc ED probem usng PSO combned wth sequenta quadratc programmng (SQP). Ths hybrd method ntegrates PSO agorthm as the man optmzer wth SQP as the oca optmzer to fne-tune the souton regon. In order to mpement the PSO agorthm, the parameters were set as foows: the popuaton sze s 00, the mum teraton number s 30000, nta weght (.e., ) s.3, fna weght (.e., mn ) s 0.7, and acceeraton coeffcents are. IV. ADVATAGES AD DISADVATAGES OF PSO A PSO s consdered as one of the most powerfu methods for resovng the non-smooth goba optmzaton probems 90

4 and has many ey advantages as foows: PSO s a dervatve-free technque just e as other heurstc optmzaton technques. PSO s easy n ts concept and codng mpementaton compared to other heurstc optmzaton technques. PSO s ess senstvty to the nature of the objectve functon compared to the conventona mathematca approaches and other heurstc methods. PSO has mted number of parameters ncudng ony nerta weght factor and two acceeraton coeffcents n comparson wth other competng heurstc optmzaton methods. Aso, the mpact of parameters to the soutons s consdered to be ess senstve compared to other heurstc agorthms [6]. PSO seems to be somewhat ess dependent of a set of nta ponts compared to other evoutonary methods, mpyng that convergence agorthm s robust. PSO technques can generate hgh-quaty soutons wthn shorter cacuaton tme and stabe convergence characterstcs than other stochastc methods [3]. The major drawbac of PSO, e n other heurstc optmzaton technques, s that t acs somewhat a sod mathematca foundaton for anayss to be overcome n the future deveopment of reevant theores. Aso, t can have some mtatons for rea-tme ED appcatons such as 5- mnute dspatch consderng networ constrants snce the PSO s aso a varant of stochastc optmzaton technques requrng reatvey a onger computaton tme than mathematca approaches. However, t s beeved that the PSO-based approach can be apped n the off-ne rea-word ED probems such as day-ahead eectrcty marets. Aso, the PSO-based approach s beeved that t has ess negatve mpact on the soutons than other heurstc-based approaches. However, t st has the probems of dependency on nta pont and parameters, dffcuty n fndng ther optma desgn parameters, and the stochastc characterstc of the fna outputs. V. COCLUSIO The man focus of ths dscusson s to survey and summarze the appcatons of PSO for sovng the ED probems ncudng the advantages and dsadvantages of PSObased approaches. The PSO agorthm has been gettng much attenton n varous appcatons ncudng power system optmzaton probems. Especay, n sovng the ED probems, t s beeved that the PSO-based appcaton to nonsmooth ED probems outperforms other state-of-the-art heurstc or mathematca agorthms. However, PSO agorthms st need further research and deveopment to mprove ts performance and to obtan the robustness. Aso, for the appcaton n the rea-word ED probems, t s necessary to combne the conventona mathematca approach wth the PSO methods based on ther own merts. VI. REFERECES [] K. Y. Lee and M. A. E-Sharaw (Edtors), Modern Heurstc Optmzaton Technques wth Appcatons to Power Systems, IEEE Power Engneerng Socety (0TP60), 00. [] J. Kennedy and R. C. Eberhart, Partce swarm optmzaton, Proceedngs of IEEE Internatona Conference on eura etwors (IC 95), Vo. IV, pp , Perth, Austraa, 995. [3] J. Kennedy and R. C. Eberhart, Swarm Integence, San Francsco, CA: Morgan Kaufmann Pubshers, 00. [4] A. J. Wood, and B. F. Woenbergy, Power Generaton, Operaton, and Contro, ew Yor, Y: John Wey & Sons, Inc., 984. [5] Z. X. Lang and J. D. Gover, A zoom feature for a dynamc programmng souton to economc dspatch ncudng transmsson osses, IEEE Trans. on Power Systems, Vo. 7. o., pp , May 99. [6] D. C. Waters and G. B. Shebe, Genetc agorthm souton of economc dspatch wth the vave pont oadng, IEEE Trans. on Power Systems, Vo. 8, o. 3, pp , Aug [7] H. T. Yang, P. C. Yang, and C. L. Huang, Evoutonary programmng based economc dspatch for unts wth non-smooth fue cost functons, IEEE Trans. on Power Systems, Vo., o., pp. -8, Feb [8]. Snha, R. Charabart, and P. K. Chattopadhyay, Evoutonary programmng technques for economc oad dspatch, IEEE Trans. on Evoutonary Computatons, Vo. 7, o., pp , Feb [9] W. M. Ln, F. S. Cheng, and M. T. Tsay, An mproved Tabu search for economc dspatch wth mutpe mnma, IEEE Trans. on Power Systems, Vo. 7, o., pp. 08-, Feb. 00. [0] K. Y. Lee, A. Sode-Yome, and J. H. Par, Adaptve Hopfed neura networ for economc oad dspatch, IEEE Trans. on Power Systems, Vo. 3, o., pp , May 998. [] J. B. Par, K. S. Lee, J. R. Shn, and K. Y. Lee, A partce swarm optmzaton for economc dspatch wth nonsmooth cost functons, IEEE Trans. on Power Systems, Vo. 0, o., pp. 34-4, Feb [] A. E-Gaad, M. E-Hawary, A. Saam, and A. Kaas, Partce swarm optmzer for constraned economc dspatch wth prohbted operatng zones, Canadan Conference on Eectrca and Computer Engneerng, Vo., pp. 78-8, 00. [3] Z. L. Gang, Partce swarm optmzaton to sovng the economc dspatch consderng the generator constrants, IEEE Trans. on Power Systems, Vo. 8, o. 3, pp , Aug [4] Z. L. Gang, Constraned dynamc economc dspatch souton usng partce swarm optmzaton, IEEE Power Engneerng Socety Genera Meetng, pp , 004. [5] T. A. A. Vctore and A. E. Jeyaumar, Reserve constraned dynamc dspatch of unts wth vave-pont effects, IEEE Trans. on Power Systems, Vo. 0, o. 3, pp. 73-8, Aug [6] R. C. Eberhart and Y. Sh, Comparson between genetc agorthms and partce swarm optmzaton, Proc. IEEE Int. Conf. Evo. Compu., pp. 6-66, May 998. VII. BIOGRAPHIES Kwang Y. Lee receved the B.S. degree n Eectrca Engneerng from Seou atona Unversty, Seou, Korea, n 964, the M.S. degree n Eectrca Engneerng from orth Daota State Unversty, Fargo, D, n 967, and the Ph. D, degree n Systems Scence from Mchgan State Unversty, East Lansng, n 97. He s currenty a Professor of Eectrca Engneerng at the Pennsyvana State Unversty, Unversty Par, PA. Hs research nterests ncude contro theory, computatona ntegence and ther appcaton to power systems. Dr. Lee s currenty the Drector of Power Systems Contro Laboratory at Penn State. He s a Feow of IEEE, an Assocate Edtor of IEEE Transactons on eura etwors, and Edtor of IEEE Transactons on Energy Converson. 9

5 Jong-Bae Par receved B.S., M.S., and Ph.D. degrees from Seou atona Unversty n 987, 989, and 998, respectvey. For , he was wth Korea Eectrc Power Corporaton, and for he was an Assstant Professor at Anyang Unversty, Korea. Currenty, he s an Assstant Professor of Eectrca Engneerng at Konu Unversty, Seou, Korea. Hs major research topcs ncude power system operaton, pannng, economcs, and marets. 9