Gravitational Search Algorithm for Optimal Economic Dispatch R.K.Swain a*, N.C.Sahu b, P.K.Hota c

Transcription

1 Avalable onlne a Procea Technology 6 (2012 ) n Inernaonal Conference on Councaon, Copung & Secury [ICCCS-2012] Gravaonal Search Algorh for Opal Econoc Dspach R.K.Swan a*, N.C.Sahu b, P.K.Hoa c a* Deparen of Elecrclal Engneerng, KIST, Bhubaneswar b Deparen, Bhubaneswar C Deparen of Elecrclal Engneerng, VSSUT, Burla Absrac Ths paper presens a novel opzaon approach o consrane econoc loa spach (ELD) probles usng gravaonal search algorh (GSA). Econoc spach eernes he elecrcal power o be generae by he coe generang uns n a power syse so ha he generaon cos o be nze whle sasfyng he consrans. Ths paper presens a new algorh base on law of gravy an ass neracon o solve econoc loa spach proble (ELD) by a new opzaon algorh calle as Gravaonal Search Algorh (GSA) The sulaon resuls reveal ha he evelope echnque s easy o pleen, converge wh less execuon e an hghly opal soluon for econoc spach wh nu generaon cos can be acheve. Sulaons resuls were perfore over varous syses wh fferen nuber of generang uns an coparsons are perfore wh oher prevalen approaches. The fnngs affre he robusness, fas convergence an profcency of propose ehoology over oher exsng echnque The Publshe Auhors. Publshe by Elsever by Elsever L. Selecon L. Selecon an/or an/or peer-revew uner responsbly of he of Deparen of Copuer of Scence Copuer & Engneerng, Scence & Naonal Engneerng, Insue of Naonal Technology Insue Rourkela of Technology Rourkela Keywor:Gravaonal search algorh, opal power spach,valve pon loang 1. Inroucon The econoc spach proble (EDP) s relae o he opu generaon scheulng of avalable generaors n a power syse o nze oal fuel cos whle sasfyng he loa ean an operaonal consrans. EDP plays an poran role n operaon plannng an conrol of oern power syses [1]. Over he pas few years, a nuber of approaches have been evelope for solvng EDP usng classcal aheacal prograng ehos [2-4].Meanwhle, classcal opzaon ehos are hghly sensve o sarng pon an frequenly converge o local opzaon soluon or verge alogeher. Lnear prograng ehos are fas an relable bu an savanages assocae wh he pecewse lnear cos approxaon. Nonlnear prograng ehos have a proble of convergence an algorhc coplexy Recenly, n orer o ake nuercal ehos ore convenen for solvng EDPs oern opzaon echnques have been successfully eploye o solve he EDPs as a non sooh The Auhors. Publshe by Elsever L. Selecon an/or peer-revew uner responsbly of he Deparen of Copuer Scence & Engneerng, Naonal Insue of Technology Rourkela o: /.procy

2 412 R.K.Swan e al. / Procea Technology 6 ( 2012 ) opzaon proble. A nuber of convenonal approaches have been evelope for solvng EDPs such as graen eho, lnear prograng algorh[5],labe eraon eho,quarac prograng, non-lnear prograng algorh[6], lagrangan relaxaon algorh [7], an he arfcal nellgence echnology has been successfully use o solve EDPs such as genec algorh [8-9],neural neworks[10], sulae annealng an abu search[11],evoluonary prograng[12-13],parcle swar opzaon [14],an colony opzaon [15] an so on. Recenly a new heursc search algorh, naely gravaonal search algorh (GSA) ovae by gravaonal law an law of oon has been propose by Rashe e al [16-18].They have been apple successfully n solvng varous non lnear funcons. The obane resuls confr he hgh perforance an effcen of he propose ehe.gsa has a flexble an well- balance echans o enhance exploraon ably. Man obecve of hs suy o presen he use of GSA opzaon echnque for econoc operaon of power syse. In hs paper GSA eho has been propose o solve econoc spach proble wh valve pon effec for 3 an 13 un es syse. The resuls obane wh he propose GSA approach were analyze an copare wh oher opzaon resuls repore n he leraure [17].Ths paper s organze as fellows. In secon 2, he proble forulaon s presene.in secon 3 he concep an applcaon of GSA s explane. The paraeer sengs for he es syse o evaluae he perforance of GSA an resuls are scusse n secon 4.The concluson are gven n secon 5 2. Proble forulaon he econoc loa spach proble can be escrbe as an opzaon (nzaon) process wh he followng obecve funcon Mn n 1 FC P Where FC ( P ) s he oal cos funcon of he h un an P s he power generae by he h un. (1) Subec no power balance equaon: D P P L Where D 1 s he syse ean an P L s he ranssson loss, an generang capacy consrans: (2) P P P for =1, 2,... n (3) n ax Where P n an ax P are he nu an axu power oupu of h un. The fuel cos funcon whou valve- pon loang of he generang un by f (4) 2 ( P ) a b P c P An he fuel cos funcon conserng valve pon loang of he generang un are gven as 2 f ( P ) a b P c P e sn( f ( P n P )) (5)

3 R.K.Swan e al. / Procea Technology 6 ( 2012 ) Where a, b, an c are he fuel cos coeffcens of he h un an e an f are he fuel cos coeffcen of he h un wh valve-pon effecs. The generang uns wh ul-valve sea urbne exhb a greaer varaon n he fuel cos funcons. The valve-pon effecs nrouce rpples n he hea rae curves 3. Gravaonal search Algorh The gravaonal search algorh (GSA), s one of he newes heursc search algorh was evelope by Rashe e al. n 2009[16]. GSA s followe he physcal law of gravy an he law of oon. The gravaonal force beween wo parcles s recly proporonal o he prouc of her asses an nversely proporonal o he square of he sance beween he. In he propose algorh, agens are consere as obecs an her perforance s easure by her asses. n P p, p,, p, 1,2 (6), p s he poson of he h where F h ass n he enson an n s he enson of he search space. ac on ass an s efne as follows. M p M G p p R M s he ass of he of he obec, M s he ass of he obec, G s he gravaonal R s he Euclan sance beween he wo obecs an, an s a sall (7) where consan a e, consan. R ( ) p ( ), p ( ) 2 The oal force acng on he agen n he enson s calculae as follows. F (8) ran F (9) where ran s a rano nuber n he nerval [0, 1]. Accorng o he law of oon, he acceleraon of he agen, a e, n he gven as follows; F M n (10) h enson, s To fn he velocy of a parcle s a funcon of s curren velocy ae o s curren acceleraon. Therefore, he nex poson an nex velocy of an agen can be calculae as follows. v ( 1) (11) p ran pv 1 p v 1 where ran s a unfor rano varable n he nerval [0, 1]. The gravaonal consan, G, s nalze a he begnnng an wll be ecrease wh he e o conrol he search accuracy. In oher wors, G s funcon of he nal value (G 0 ) an e (): G G G, 0 (12) (13)

4 414 R.K.Swan e al. / Procea Technology 6 ( 2012 ) G G e 0 1 T The asses of he agens are calculae usng fness evaluaon. A heaver ass eans a ore effcen agen. Ths eans ha beer agens have hgher aracons an oves ore slowly. Supposng he equaly of he gravaonal an nera ass, he values of asses s calculae usng he ap of fness. The gravaonal an neral asses are upang by he followng equaons. (14) f bes wors wors (15) M 1 f represens he fness value of he agen a e, an he bes an where wors n he populaon respecvely ncae he sronges an he weakness agen accorng o her fness value. For a nzaon proble: bes wors n f (17) l, ax f (18) l, For a Maxzaon proble: bes wors ax f (19) l, n f (20) l, 4. Gravaonal search algorh base econoc loa spach In orer o hanle he consrans convenenly, he srucure of soluons for ED proble has solve by he propose eho s copose of a se of real power oupu ecson varables for each un n all over he scheulng peros. The secon proves he soluon ehoology o he above-enone econoc spach probles hrough gravaonal search algorh Inalzaon In he nalzaon proceure, he canae solu s ranoly nalze whn he feasble range n such a way ha shoul sasfy he consran gven by Eq. (4). A coponen of a canae s nalze as P ~U ( P n, P ax ),where U s he unfor srbuon of he varables rangng n he nerval of P P ). ( n, ax (16)

5 R.K.Swan e al. / Procea Technology 6 ( 2012 ) Fness evaluaon(obecve funcon) The fness evaluaon n each agen n he populaon se s evaluae usng he equaon (4). Ieraon coun fro hs sep, =1. Upae G (), bes (), wors () an M () for =1, Agen force calculaon The oal force acng on he agen n he enson s calculae n equaon (7) Evaluaon of acceleraon of an agen The acceleraon of an agen n h enson over T spach pero has evaluae usng equaon (10) The nex velocy of an agen s calculae by ang he acceleraon of an agen o he curren velocy an also poson of an agen wll upae Soppng creron Repea he sep fro 4.3 o 4.7 unl he soppng crera s reache. There are varous crera avalable o sop a sochasc opzaon algorh. In hs paper, o copare wh he prevous resuls, axu nuber of eraons s chosen as he soppng creron. If he soppng creron s no sasfe, he above proceure s repeae fro fness evaluaon wh ncreene eraon. 5. Copuaonal proceure The purpose GSA approach for econoc loa spach proble wh valve- pon effec can be suarze as follows. The copuaonal ehoology of GS algorh has gven n fgure 1. Sep 1. Search space enfcaon Sep 2. Generae nal populaon beween nu an axu values. Sep 3. Fness evaluaon of agens. Sep 4. Upae gravaonal consan G (), pbes () an wors () n he populaon an upae he ass of he obec M (). Sep 5. Force calculaon n fferen recon. Sep 6. Calculaon of acceleraon an velocy of an agen. Sep 7. Upang he poson of an agen. Sep 8. Repea sep 3 o sep 7 unl he sop crera s sasfe

6 416 R.K.Swan e al. / Procea Technology 6 ( 2012 ) Sep 9. Sop. Generae Inal Populaon Evaluae he Fness Funcon Upae he G (), bes (), wors () of he populaon each agen Upae velocy an poson Meeng en of creron 6. Sulaon resuls Fg. 1. Flow char for GSA algorh Each propose gravaonal search algorh has been pleene n coan lne MATLAB for soluon of wo es cases of econoc loa spach. In hs paper, o access he effcency of he propose algorh has been apple o 3 an 13 heral uns of ED probles n whch obecve funcon s non sooh because he valve pon effec are aken no accoun. The enre progra s run Penu-IV, 2.80GHZ wh 506,604KB RAM PC Tes case 1 Reurn bes soluon The npu aa for hree-generaor syse are gven n [13] an es aa has gven n able I.The axu oal power oupu of he generaor s 850 MW. These resuls gve he nu generaon cos for each approach afer 50 runs. The seup for he propose algorh s execue wh he followng paraeer M=100, Where G 0 s se o 100, s se o 8, Maxu eraon nubers are 100 for hs case suy. The oal generaon cos obane by hs propose eho s $/h. The

7 R.K.Swan e al. / Procea Technology 6 ( 2012 ) execuon e for hs case s 0.45s.. However, he pre ephass n hs work o have coparave perforance of EP wh respec o GSA. The resul of hs es case s shown n Table II. The convergence characersc has been shown n Fg. 2. Table 1. Un aa for es case I (hree-un syses) Generaors P n P ax a b c e f Table.2 Resul of es case I Evoluon Meho Mean Te. Bes Te Mean Cos Max. Cos ($) Mn Cos n Secs In Sec. ($) ($) ( $) CEP [13] FEP [13] CEP [13] FEP [13] GSA x Cos $/h Cos $/h Nuber of eraons Nuber of eraons Fg.1 Convergence characerscs of es case I Fg.1 Convergence characerscs of es case II 6.2. Tes case II The un aa of 13 generang uns has aken fro reference [13 ] whch has shown n Table 3. In hs case he loa ean s 1800MW. The resuls fro he sulaon are suarze n Table 4. For obanng he nu generaon cos, ean e an bes e s acheve by he propose eho. The sulaon paraeers has been seup for he propose algorh s M (ass) =100, Where G 0 s se

8 418 R.K.Swan e al. / Procea Technology 6 ( 2012 ) o 100, s se o 8, Maxu eraon nubers are The nu generaon cos obane by he propose eho s $/h.the convergence graph of he GSA eho s shown n fgure 3. Table 3. Un aa for es case II (hreen-un syses) Generaors P n P ax a b c e f Table 4. Sulaon resul II (hreen-un syses) Evoluon Meho Mean Bes Mean Cos Maxu Cos Mnu Cos Te Te ($) ($) ($) CEP [13] FEP [13] MFEP[13] IFEP [13] GSA [13] Concluson New approaches of usng gravaonal search algorh base o solve for econoc loa spach s presene. The presene evaluaon funcon oel an opally selece he ass of an agens have enhance he perforance of he gravaonal search algorh. A coparave suy was carre ou beween he propose EP an GSA echnque. The GSA eho gves beer resuls wh reuce copuaonal e. Hence, he suy shows ha GSA coul be a prosng echnque for solvng coplcae opzaon probles n power syses. The GSA has prove he global opal soluon

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