A New Approach for Solving the Unit Commitment Problem by Adaptive Particle Swarm Optimization

Transcription

1 A New Aroach for Solvn he Un Commmen Problem by Adave Parcle Swarm Omzaon V.S. Paala, Suden Member, IEEE, and I. Erlch, Senor Member, IEEE Absrac Ths aer resens a new aroach for formulan he un commmen roblem whch resuls n a conserable reducon n he number of decson varables. The scheduln varables are coded as neers reresenn he oeraon erods of a eneran un. The un commmen roblem s solved usn a new arameer free adave arcle swarm omzaon (APSO) aroach. Ths alorhm roves soluons o he major demers of PSO such as arameer unn, selecon of omal swarm sze and roblem deenden enaly funcons. The consraned omzaon roblem s solved usn adave enaly funcon aroach. The enaly erms ada o he erformance of he swarm. So no addonal enaly coeffcen unn s requred. Ths aer descrbes he roosed alorhm wh he new varable formulaon and resens es resuls on a en un es sysem. The resuls demonsrae he robusness of he new alorhm n solvn he un commmen roblem. Index Terms Bnary and Ineer rorammn, Parcle swarm omzaon, Penaly funcon, Un commmen. I. INTRODUCTION NIT commmen (UC) s a nonlnear mxed neer Uomzaon roblem o schedule he oeraon of he eneran uns a mnmum oeran cos whle sasfyn he demand and reserve requremens. The UC roblem has o deermne he on/off sae of he eneran uns a each hour of he lannn erod and omally dsach he load and reserve amon he commed uns. UC s he mos snfcan omzaon as n he oeraon of he ower sysems. Solvn he UC roblem for lare ower sysems s comuaonally exensve. The comlexy of he UC roblems rows exonenally o he number of eneran uns. Several soluon sraees have been roosed o rove qualy soluons o he UC roblem and ncrease he oenal savns of he ower sysem oeraor. These nclude deermnsc and sochasc search aroaches. Deermnsc aroaches nclude he rory ls mehod [], dynamc rorammn [2], Laranan Relaxaon [3] and he branchand-bound mehods [4]. Alhouh hese mehods are smle and fas, hey suffer from numercal converence and soluon qualy roblems. The sochasc search alorhms such as Ths Projec s suored by he German Federal Mnsry of Educaon and Research (BMBF) under he ran 03SF032A. Paala Venaa Swaroo (venaa.aala@un-due.de) s wh he nsue of elecrcal ower sysems, Unversä Dusbur-Essen. Isvan Erlch (svan.erlch@un-due.de) s wh he nsue of elecrcal ower sysems, Unversä Dusbur-Essen, Dusbur, Germany. arcle swarm omzaon [5]-[8], enec alorhms [9], evoluonary rorammn [0], smulaed annealn [], an colony omzaon [2] and abu search [3] are able o overcome he shorcomns of radonal omzaon echnques. These mehods can handle comlex nonlnear consrans and rove hh qualy soluons. However, all hese alorhms suffer from he curse of dmensonaly. The ncreased roblem sze adversely effecs he comuaonal me and he qualy of he soluons. Ths aer addresses a new aroach o handle he un commmen varables. The scheduln of a un s exressed as oeraon erods or duy cycles. Hence he un commmen varables are coded as neers. Ths formulaon drascally reduces he number of decson varables and hence can overcome he shorcomns of sochasc search alorhms for UC roblems. Due o smlcy and less arameer unn, adave arcle swarm omzaon s used for solvn he roblem. II. PROBLEM FORMULATION The objecve of he UC roblem s o mnmze he oal oeran coss subjeced o a se of sysem and un consrans over he scheduln horzon. s assumed ha he roducon cos, PC for un a any ven me nerval s a quadrac funcon of he eneraor ower ouu,. 2 PC = a + b + c () Where a, b, c are he un cos coeffcens. The eneraor sar-u cos deends on he me he un has been swched off ror o he sar u, T off. The sar-u cos SC a any ven me s assumed o be an exonenal cos curve. T off, SC = σ + δ ex (2) τ Where σ s he ho sar-u cos, δ he cold sar-u cos and τ s he cooln me consan. The oal oeran coss, OC T for he scheduln erod T s he sum of he roducon coss and he sar-u coss. T N OC T PC, U, + SC,, = = ( U, ) U = (3) Where U, s he bnary varable o ndcae he on/off sae of he un a me. U, = f un s commed a me, oherwse U, =0. The overall objecve s o mnmze OC T subjec o a number of sysem and un consrans. All he eneraors are 2008 IEEE.

2 2 assumed o be conneced o he same bus sulyn he oal sysem demand. Therefore, he newor consrans are no aen no accoun. Power balance consran The oal eneraed ower a each hour mus be equal o he load of he corresondn hour, D. N U, = =, D (4) Snnn reserve consran For relable oeraon, he ower sysem has o manan a ceran meawa caacy as snnn reserve, R. N = max U D + R, Generaon lm consran mn, max (5) (6) Mnmum u/down me consrans The mnmum u/down me consrans ndcae ha a un mus be on/off for a ceran number of hours before can be shu off or brouh onlne, resecvely. T T on, MUT (7) off, MDT (8) The nal un saes a he sar of he scheduln erod mus be aen no accoun. Where mn & max are he mnmum and maxmum eneraon lm of he h un, T on & T off reresens he duraon durn whch he h un s connuously on and off resecvely and MUT & MDT are he mnmum u-me and down-me resecvely. III. PARTICLE SWARM OPTIMIZATION Parcle swarm omzaon (PSO) was frs roosed by Kennedy and Eberhar [4] n 995. PSO s a oulaon based searchn alorhm. Ths aroach smulaes he smlfed socal sysem such as fsh schooln and brds flocn. PSO s nalzed by a oulaon of oenal soluons called arcles. Each arcle fles n he search sace wh a ceran velocy. The arcle s flh s nfluenced by conve and socal nformaon aaned durn s exloraon. I has very few unable arameers and he evoluonary rocess s very smle. I has been successfully aled o solve nonlnear, combnaoral, mulmodal and mul-objecve roblems. s caable of rovn qualy soluons o many comlex ower sysem roblems. In hs sudy adave arcle swarm omzaon (APSO) [5], [6] s used for solvn he UC roblem. APSO s a arameer free echnque. The alorhm s able o adjus he flh of he arcles based on her erformances. So no secal arameer unn s requred. s also caable of fndn he mos arorae swarm sze. In hs alorhm, dfferen szed rous of arcles called Trbes move n he search sace o fnd he omal soluon. All he arcles n a rou share her flyn exerences n order o fnd a local mnmum. The rbes exlore several romsn areas smulaneously and assocae amon each oher o dece on he lobal mnmum. IV. PARTICLE FORMULATION All alorhms solve he UC roblem by bnary rorammn. So he arcle s coded as a bnary srn reresenn he on/off saus of he eneraors for each hour of he scheduln erod T. Bu n he new aroach resened here, he bnary UC varables are formulaed as neers as shown below F.. The new UC varable formulaon The un commmen schedule n f. s reresened by fve neers. Each neer reresens he connuous on/off erod of a eneran un. Neave neer ndcaes he erod when he un s swched off and osve neer ndcaes he erod when he un s n oeraon. The number of neers requred o reresen he scheduln decsons should be defned before he sar of he omzaon. For a sysem wh N eneran uns and P neers reresenn he UC schedule of each un, he arcle consss of N*T connuous varables reresenn he ower eneraon levels of uns a each hour and N*P neer varables reresenn UC schedule of he uns a each hour. Ths formulaon would reduce he number of UC decson varables comared o he bnary codn by (T-P)/T%. V. APSO APPROACH TO UC The major drawbacs of evoluonary alorhms are arameer unn, omal oulaon/swarm sze, remaure converence and roblem deenden enaly coeffcens. These demers can be successfully overcome by usn he adave arcle swarm omzaon alorhm. APSO can be used as a blac box. s ndeenden of he roblem ben solved. The neers reresenn oeran schedule of a un s deced as I, =,2,..., P. The omzaon rocess can be exlaned by he follown ses. Se : Swarm nalzaon The evoluonary rocess sars wh a snle arcle reresenn a snle rbe. The arcle s randomly nalzed whn he secfed lms. The UC neer varables are eneraed as shown below: T T rand_neer n,n, =,...P - P P I = (9) P- T - = I, P = A fness value s calculaed usn (3) and s assned o he

3 3 arcle. The consrans (4)-(8) are verfed and a enaly erm s evaluaed for each volaon. Ths enaly s hen added o he arcles fness value. The arcle memorzes s revous wo erformances. Inally hese varables are nalzed o he arcle s curren oson. Se 2: Inalze nformaon rous. Each arcle has a rou of nformers o asss n s search rocess. The sze of hs rou s adave. Inally hs rou consss of he arcle self and he bes arcle n he rbe. Se 3: Udae he curren oson of he arcles. The erformance of he arcles can be an mrovemen (+), saus quo (=) or a deeroraon (-). The arcles are raded based on her revous wo erformances. A arcle s consered excellen f s revous wo erformances are mrovemens (++), ood f has jus mroved s erformance (=+)/(+=), neural f has he follown confuraon (-+)/(+-)/(= =) and bad f s revous erformances are deeroraons (--). If he arcle haens o be bad or neural, hen vo mehod s used as udae sraey whereas f he arcles are excellen or ood, hen Gaussan udae sraey s used. These sraees are exlaned below. A. Pvo sraey Idenfy he ndvual bes erformance of he arcle, and bes nformer of he arcle,. Defne wo hyersheres, H and H wh and as ceners and radus equal o he dsance beween hem. A on s randomly chosen n each of hese hyersheres. A wehed combnaon of hese ons ves he new oson vecor, x + of he arcle. + x = crandom H 2 ( ) + c random( H ) (0) c = f ( ) f ( ) + f ( ) () c2 = f ( ) f ( ) + f ( ) (2) Where c and c 2 are he weh facors. These values are calculaed usn he objecve funcon (4), f corresondn o and. B. Gaussan Udae sraey Ths udae sraey s very smlar o sandard PSO udae equaons. The velocy of he arcle a eraon + deend on s revous velocy (v ), s revous bes erformance ( ) and erformance of he bes nformer ( d ) as shown below. δ = x (3) d δ = x (4) + v = v + auss_ rand ( v ) + = x + + ( δ, δ 2) + auss_ rand( δ, δ 2) (5) x χ (6) Where auss_rand(µ, σ) eneraes a normal dsrbued numbers wh mean µ and sandard devaon σ. The Gaussan dsrbuon wll rove boh lobal and local exloraon around and. Ths aroach elmnaes he use of acceleraon coeffcens. Hence he udae sraey s oally ndeenden of he unn arameers. The velocy of neer UC varables s bounded o v mn and v max. The udae rocedure for he neer UC varables can be exlaned by he f.2. On Off F. 2. Udae rocedure for UC neer varables of a snle eneraor wh fve oeran erods To each neer or duy cycle a ceran velocy s added. The arrows ndcae he drecon of velocy. The lef arrow mlcaes a neave velocy whch causes a reducon n he corresondn neer or oeraon erod. Where as a rh arrow ndcaes a osve velocy whch hels o ncrease he duy cycle or he corresondn neer. There s no need o assn a smo ranson robably or muaon robably for alern he on/off saus of he uns. The UC neer varables are reaed le any oher decson varables. Afer udan he neer varables hey are celed o he neares neers. A fness value s calculaed usn (3) and s assned o he arcle. The consrans (4)-(8) are verfed and a enaly erm s evaluaed for each volaon. Ths enaly s hen added o he arcles fness value. The udae rocedure for he arcles of all rbes consue one cycle. The rbes are allowed o exlore and evolve for a ceran number of cycles. Se 4: chec he bounds on all varables The Generaon lm consran s checed for all connuous varables. If he varables cross he bounds, hey are made o say on he boundary. The sum of he neer UC varables of each un mus be equal o he scheduln erod T. P = 2 {,...,N} 3 I = T (7) Ths consran s mlemened by usn he follown fuzzy rules. P = ± f I < T Rule : I T I (8) = = ( I > T) Rule2: = ± T 2 2 ( I + I > T) Rule3: I = ± ( T I ) f f f I > T.... (9) =..... > P = ± f I T Rulem: I T I = = Se 5: Chec for he ermnaon creron Chec f he curren eraon number has reached he ermnaon lm. If yes, ex oherwse chec f he number of 4 5 T

4 4 cycles s less han he oal sze of curren nformaon rous. Ths condon wll ensure ha all he rbes have enouh me o exlore, evolve and share nformaon wh s nformers. If he second condon s obeyed o o se 3 or else roceed o se 6. Se 6: Adaaons Evaluae he erformance of he rbes. A rbe s consered ood/bad f has more number of ood/bad arcles. Snce he bad rbe do no have he caably o convere, needs more nformaon or arcles o fnd an omal soluon. The bes arcle of he bad rbe wll enerae a new arcle for he new rbe. Each bad rbe conrbues a arcle each o he new rbe. The new arcles wll be n conac o her arens by addn hem o her resecve nformaon rous. On he oher hand, he ood rbe s caable of convern o an omal soluon. I may no need all s arcles for furher exloraon. Therefore he wors arcle s evenually removed from he rbe. The nformaon rous of all he arcles are udaed by relacn hs deleed arcle wh he bes arcle of ha rbe. Ths wll ensure ha he search rocess do no send me on unwaned arcles. Hence he alorhm can fnd he omum wh mnmum funcon evaluaons. Afer he adaaons on he rbes, connue o se 3. VI. CONSTRAINTS HANDLING Consraned omzaon roblems are usually solved usn enaly funcon aroach. The nfeasble arcles wll be enalzed for he consran volaon by addn a enaly erm o he fness value. Many sac and dynamc enaly funcons [7] have been successfully esed on a number of roblems. Bu all hese enaly funcons requre he bes enaly coeffcens whch can be deermned only hrouh reeaed rals. Hence a self adave enaly funcon aroach [8] s used. Two enaly erms, d(x) and (x) are added o each arcle. These erms deend on he erformance of he whole swarm. The new fness value s he sum of he dsance value, d(x) and he enaly value, (x). The dsance value s defned as follows: b'( x) f r f =0 d( x) = (20) ' 2 2 oherwse OCT + b'( x) Where number of feasble arcle r f = (2) swarm sze ' OC T s he normalzed oeraon cos and b ' (x) s he sum of he normalzed volaon of each consran dved by he oal number of consrans. For a swarm wh no feasble arcles, he frs and foremos objecve of he omzaon s o fnd a feasble arcle.e. reduce b ' (x). When here are feasble arcles n he swarm hen he alorhm res o reduce he ' roo mean square sum of he OC T and b ' (x). The nformaon carred by he nfeasble arcle can be very helful n fndn he omal soluon. For nsance, when here are few feasble arcles, nfeasble arcles wh low consran volaon can hel n fndn more feasble arcles. On he oher hand when here are more feasble arcles, nfeasble arcles wh low objecve value can ve more nformaon abou he lobal omum. So a every me sae, he search rocess requres an arorae se of nfeasble arcles o fnd he omal soluon. Ths henomenon s mlemened by he enaly value, (x). In bnary rorammn, secal soluon echnques have o be aled o sasfy he mnmum u-me and down-me consrans. The arcles whch volae consrans (7) and (8) should be correced by usn a rear or muaon oeraor [9]. Some alorhms rove arbrary feasble arcles a he sar of he search rocess. Bu he APSO aroach wh he new UC varable formulaon can handle he MUT, MDT consrans whou any correcon oeraors. VII. NUMERICAL EXAMPLE The APSO alorhm was esed on a en eneraor sysem consern a me horzon of 24 hours. The dae s ven n Table I and II. The snnn reserve s assumed o be 5% of he load demand. The APSO roram was mlemened n vsual c lanuae. Tes resuls are comared aans he bnary rorammn resuls n [6]. In bnary PSO he arcle consss of 0*24 connuous varables reresenn he eneraon levels of he en uns and 0*24 bnary varables reresenn he oeraon schedule of he uns a each hour. Whereas, he new aroach, he arcle consss of 0*24 connuous varables and 0*5 neer varables. Hence here s aroxmaely 80% reducon n he number of UC decson varables comared o he bnary rorammn aroach. Un max mn TABLE I GENERATING UNIT DATA Fuel cos Sar-u cos a b c σ δ τ MUT (hrs) MDT (hrs) TABLE II HOURLY LOAD DEMAND INS (hrs) Hour Demand Hour Demand The oal oeran coss usn bnary rorammn are $565,804 accordn o [6] whereas for he new aroach, he coss are $56,586. The UC schedules of he wo aroaches

5 5 are resened n Tables III and IV. The resuls mly ha he new aroach exlos he search sace far beer han he bnary PSO aroach. TABLE III OPTIMAL UC SCHEDULE USING BINARY PROGRAMMING Un Un Schedule OC($) 565,450 TABLE IV OPTIMAL UC SCHEDULE USING NEW APPROACH UNIT UNIT SCHEDULE OC($) 56,586 The qualy of he soluon obaned by he new aroach can also be examned usn he reserve allocaon shown n fure 3. In bnary mehod, here s hue unwaned reserve allocaon durn he low demand hours. In he new aroach here s omal reserve allocaon. The reserve allocaon by he new aroach s very close o he requred reserve caacy. F. 4 shows he converence endency of APSO wh he new UC neer varable formulaon. Reserve reserve caacy bnary mehod new mehod Tme (h) F. 3. Reserve allocaon by bnary rorammn and he new aroach oeraon cos (000$) funcon evaluaons F. 4. Converence endency of he new aroach VIII. CONCLUSION The new UC varable formulaon causes conserable reducon n he number of decson varables and hus he sze of omzaon roblem. I sll requres he defnon of oeraon erods or neers requred for he UC varable formulaon whch can no be deermned omally a ror. However, n he near fuure, auhors are lannn o resen an exenson of he roosed alorhm ha allows o overcome hs drawbac by usn adave arcle sze for neer coded UC varables. Also he converence behavor could be made faser by usn secal oeraors ha can asss he arcles o sasfy he equaly demand consran and o remove he excess reserve allocaon. The roosed adave arcle swarm omzaon maes raccal mlemenaons much easer due o he fac ha APSO s oally free from arameer unn. The alorhm s self adave and s herefore ndeenden of he roblem o be solved. The adave enaly funcon aroach nroduced reduces he burden of selecn he bes enaly coeffcens. The arcles do no requre any rear sraees for sasfyn he consrans. As a resul, he alorhm s caable of effcenly exlorn he search sace and eneran qualy soluons. Ths research s a conrbuon o he develomen of new robus arameer free evoluonary alorhms for solvn he un commmen roblem. REFERENCES [] Johnson, R.C.; Ha, H.H.; Wrh, W.J.; Lare Scale Hydro-Thermal Un Commmen Mehod and Resuls, IEEE Transacons on Power Aaraus and Sysems Volume PAS-90, Issue 3,May 97 Pae(s): [2] Lowery, P.G.; Generaon un commmen by dynamc rorammn, IEEE Trans. Power A. Sys., vol. PAS-02, , 983. [3] Zhuan, F. and Galana, F. D. Toward a more rorous and raccal un commmen by Laranan relaxaon, IEEE Trans. Power Sys., vol. 3, no. 2, , May 988. [4] Cohen, A. I. and Yoshmura, M. A branch-and-bound alorhm for un commmen, IEEE Trans. Power A. Sys., vol. PAS-02, no. 2, , 983. [5] Tn, T.O.; Rao, M.V.C.; Loo, C.K.; A novel aroach for un commmen roblem va an effecve hybr arcle swarm omzaon, IEEE Transacons on Power Sysems, Volume 2, Issue, Feb Pae(s):4-48. [6] Zwe-Lee Gan, Dscree arcle swarm omzaon alorhm for un commmen, IEEE Power Enneern Socey General Meen, 2003, Volume, 3-7 July 2003.

6 6 [7] Tn, T.O; Rao, M.V.C.; Loo, C.K.; and Nu, S.S.; Solvn Un Commmen Problem Usn Hybr. Parcle Swarm Omzaon, Journal of Heurscs, 9: , [8] Saber, A.Y.; Senjyu, T.; Yona, A.; Funabash, T.; Un commmen comuaon by fuzzy adave arcle swarm omzaon, IET Generaon, Transmsson & Dsrbuon, Volume, Issue 3, May 2007 Pae(s): [9] Kazarls, S.A.; Barzs, A.G.; Pers, V.; A enec alorhm soluon o he un commmen roblem IEEE Transacons on Power Sysems, Volume, Issue, Feb. 996 Pae(s):83-92 [0] Juse, K.A.; Ka, H.; Tanaa, E.; Haseawa, J.; An evoluonary rorammn soluon o he un commmen roblem, IEEE Transacons on Power Sysems, Volume 4, Issue 4, Nov. 999 Pae(s): [] Smooulos, D.N.; Kavaza, S.D.; Vournas, C.D.; Un commmen by an enhanced smulaed annealn alorhm, IEEE Transacons on Power Sysems, Volume 2, Issue, Feb Pae(s):68-76 [2] Ssworahardjo, N.S.; El-Keb, A.A.; Un commmen usn he an colony search alorhm, Lare Enneern Sysems Conference on Power Enneern 2002, LESCOPE 02 Pae(s):2 6. [3] Manawy, A.H.; Abdel-Ma, Y.L.; Selm, S.Z.; Un commmen by abu search, IEE Proceedns Generaon, Transmsson and Dsrbuon, Volume 45, Issue, Jan. 998 Pae(s): [4] Kennedy, J.; Eberhar, R.; Parcle swarm omzaon, IEEE Inernaonal Conference on Neural Newors, 995, Volume 4, 27 Nov.- Dec. 995 Pae(s): [5] Maurce Clerc, TRIBES, a Parameer Free Parcle Swarm Omzer, French verson: Presened a OEP'03, Pars, France. [6] Paala, V.S; Wlch, M.; Snh, S.N.; Erlch, I.; Reacve ower manaemen n offshore wnd farms by adave PSO, Inernaonal conference on nellen sysem alcaon o ower sysem 2007.Ths aer won he bes aer award. [7] Gen, M.; Chen, R.; A survey of enaly echnques n enec alorhms, Proceedns of IEEE Inernaonal Conference on Evoluonary Comuaon, 996, May 996 Pae(s): [8] Tessema, B.; Yen, G.G.; A Self Adave Penaly Funcon Based Alorhm for Consraned Omzaon, IEEE Conress on Evoluonary Comuaon, July 2006 Pae(s): [9] Yuan Xaohu; Yuan Yanbn; Wan Chen; Zhan Xaoan, An Imroved PSO Aroach for Prof-based Un Commmen n Elecrcy Mare, IEEE/PES Transmsson and Dsrbuon Conference and Exhbon: Asa and Pacfc, 2005 Pae(s):-4. BIOGRAPHIES Venaa Swaroo Paala receved he B.E deree n elecrcal enneern from Faculy of Elecrcal enneern, Naarjuna Unversy, Inda n 2002, and M.Sc. deree n elecrcal enneern wh emhass on ower and auomaon from Unversy Dusbur- Essen, Germany n He s currenly a Ph.D. suden a he Unversy of Dusbur-Essen, Germany. Hs research neress nclude sochasc omzaon under uncerany usn evoluonary alorhms. Isvan Erlch (953) receved hs Dl.-In. deree n elecrcal enneern from he Unversy of Dresden/Germany n 976. Afer hs sudes, he wored n Hunary n he feld of elecrcal dsrbuon newors. From 979 o 99, he joned he Dearmen of Elecrcal Power Sysems of he Unversy of Dresden aan, where he receved hs PhD deree n 983. In he erod of 99 o 998, he wored wh he consuln comany EAB n Berln and he Fraunhofer Insue IITB Dresden resecvely. Durn hs me, he also had a eachn assnmen a he Unversy of Dresden. Snce 998, he s Professor and head of he Insue of Elecrcal Power Sysems a he Unversy of Dusbur- Essen/Germany. Hs major scenfc neres s focused on ower sysem sably and conrol, modeln and smulaon of ower sysem dynamcs ncludn nellen sysem alcaons. He s a member of VDE and senor member of IEEE.