Gravitational Search Algorithm for Solving Combined Economic and Emission Dispatch

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1 IFOTEH-JAHORIA Vol. 4 March 205. ravtatoal Search Algorthm for Solvg Combed Ecoomc ad Emsso Dspatch Jorda Radosavlevć Faculty of Techcal Sceces Uversty of ršta Kosovsa Mtrovca Kosovsa Mtrovca Serba orda.radosavlevc@pr.ac.rs Abstract Ths paper presets a gravtatoal search algorthm (SA) for solvg the combed ecoomc ad emsso dspatch (CEED) problem power systems. umercal results for the stadard IEEE 30-bus sx-geerator test system have bee preseted to llustrate the performace ad applcablty of the proposed approach. The results obtaed are compared to those reported the recet lterature. Those results show that the proposed algorthm provdes effectve ad robust hgh-qualty soluto of the CEED problem. Key words- ecoomc dspatch; emsso; gravtatoal search algorthm; I. ITRODUCTIO Ecoomc dspatch (ED) problem has a sgfcat mportace the power system s operato ecoomc schedulg ad securty. The ED problem soluto ams to mmze the cost of geerato of electrc power though optmal adustmet of the commtted geeratg ut outputs whle at the same tme satsfyg all ut ad system costrats. It s a large-scale o-lear costraed optmzato problem. Wth the creased publc awareess of the evrometal polluto the tradtoal ED whch gores the pollutat emssos of the fossl fuels used by the thermal plats o loger satsfes the eeds []. Whe the evrometal cocers are combed wth the ED the the problem becomes CEED problem. Ths problem cosders two obectves such as mmzato of the fuel cost ad emsso from the thermal power plats wth both equalty ad equalty costrats. So the CEED problem s a multobectve mathematcal problem whch coflctg obectves are optmzed smultaeously. Evrometal aspect adds complexty to the soluto of the ecoomc dspatch problem due to the olear characterstcs of the mathematcal models used to represet emssos. I addto the CEED problem ca be complcated eve further f o-smooth ad o-covex fuel cost fuctos are used to model geerators such as valve pot loadg effects. All these cosderatos mae the CEED problem a hghly olear ad a multmodal optmzato problem [2]. eerally three approaches to hadle the CEED problem have bee reported [3]. I the frst approach the emsso s treats as a costrat wth a permssble lmt. However ths formulato has a severe dffculty gettg the trade-off relatos betwee cost ad emsso. The secod approach treats the emsso as aother obectve addto to the cost obectve. I ths case the CEED problem s coverted to a sgle obectve optmzato problem ether by lear combato of both obectves or by cosderg oe obectve at a tme for optmzato. I the thrd approach smultaeously coflctg obectves are evaluated together the soluto of the CEED problem. Both the fuel cost ad the emsso are mmzed together. I practce the ecoomc dspatch problem has bee solved by usg determstc (classcal) ad populato-based optmzato methods. I the past few decades may classcal optmzato methods such as gradet method ewto s method lear programmg o-lear programmg dyamc programmg goal programmg techque ad Lagraga relaxato algorthm have bee appled to varous ED problems. However most of them have dffcultes to solve ED problems due to o-learty ad o-covexty fuel cost ad emsso characterstcs. The covetoal optmzato methods are hghly sestve to the startg pot ad frequetly coverge to local optmum soluto. Moreover these methods are ot able to fd a soluto wth a sgfcat computatoal tme for medum or large-scale CEED problem [2]. Recetly may populato-based methods have bee used to solve complex costraed optmzato problems. eerally achevg optmal or ear optmal soluto for a specfc problem wll requre multple trals as well as approprate tug of assocated parameters [4]. A wde varety of populato-based techques such as artfcal bee coloy algorthm (ABC) [] spral optmzato algorthm (SOA) [2] geetc algorthm (A) o-domated sortg geetc algorthm (SA) ched areto geetc algorthm (A) [5] o-domated sortg geetc algorthm (SA II) [67] dfferetal evoluto (DE) [89] mult obectve dfferetal evoluto (MODE) [0] partcle swarm optmzato (SO) [] mult-obectve partcle swarm optmzato (MOSO) [3] modfed bacteral foragg algorthm (MBFA) [2] etc. have bee appled solvg the o-lear CEED problems wth dfferet obectve fuctos. Ths paper proposes a SA algorthm to solve the CEED problem. The performace of the proposed algorthm s tested

2 o the stadard IEEE 30-bus sx-geerator test system. umercal results obtaed by the proposed approach were compared wth other optmzato results reported the lterature recetly. II. ROBLEM FORMULATIO The soluto of the combed ecoomc ad emsso dspatch problem s acheved by mmzg the obectve fucto (OF) combed wth the weghted sum method uder the system costrats []. OF = M w F ( ) + ( w) γ E ( ) () I Eq. () the fuel cost rate s show wth F ( ) ad emsso rate (to/h) wth E ( ). Scalg factor weght factor ad the set of all the thermal geerato uts are deoted as γ w ( 0 w ) ad respectvely. w = correspods to the mmzato of total fuel cost oly lewse w = 0 correspods to the mmzato of total emsso oly. A. fucto fucto of each geerator the system may be represeted as a quadratc fucto of real power geerato: 2 ( ) a + b c F = + (2) where a b ad c are the cost coeffcets. B. Emsso fucto Fossl-fueled thermal uts cause atmospherc waste emsso composed of gases ad partcles such as carbo doxde (CO 2 ) sulfur doxde (SO 2 ) troge oxde (O x ). Dfferet mathematcal models were proposed to represet the emsso fucto of thermal geeratg uts [6]. I ths paper the emsso fucto of each thermal ut s defed as the sum of a quadratc fucto ad a expoetal fucto []: 2 ( ) α + β + η + ξ ( ) E exp λ = (to/h) (3) where α β η ξ ad λ are coeffcets of the th geerator emsso characterstcs. I the Eqs. (2)-(3) the s MW. C. Costrats Durg the mmzato process some equalty ad equalty costrats must be satsfed. I ths process a equalty costrat s called a power balace ad a equalty costrat s called a geerato capacty costrat. C. ower balace costrat The total power geerato must cover the total load demad load ad the real power trasmsso les. Accordgly the power balace costrat ca be represeted as follows: load = 0 (4) The trasmsso es of the system are represeted by coeffcets (B ) ormally referred to as B- matrces. The B- matrces approxmate the system es as a quadratc fucto of the geerator real powers: = (5) B + B0 + B00 where B B 0 ad B 00 are the coeffcets of the B- matrces. C2. eerato capacty costrat For stable operato real power output of each geerator s restrcted by mmum as follows: m ad mum m power lmts ( ) (6) C3. Slac geerator calculato To eforce actve power balace costrat gve Eq. (4) a depedet geerator (slac geerator) should be selected. As the slac geerator the geerator whch dexed old wth s adopted. The value of geerato power s calculated by usg Eq. (7) where the tal value of power old frst = = 0 []. s set to zero ( ) After obtag old = old = (7) from Eq. (5). Accordg to ths followg equato: load ew power ew ew s determed s calculated usg the = ew ew = + (8) load The result of ths equato s cotrolled Eq. (9) ad f the error value (ε) s below error tolerace value TOL ε (e.g. TOL ε =0-6 ) the equato satsfes the power balace costrat. ε = ε TOLε (9) ew old

3 The obtaed s checed whether t satsfes the costrat defed Eq. (6) or ot. Cosequetly the varable lm s defed as: f > lm m m = f < (0) m f Iequalty costrat of the depedet varable that s s added to the obectve fucto as a quadratc pealty terms. The ew expaded obectve fucto to be mmzed becomes: OF where λ p s the pealty factor. p lm ( ) 2 = OF + λ () p III. OVERVIEW OF SA The gravtatoal search algorthm (SA) s a ewly stochastc search algorthm developed by Rashed et al. [3]. I ths algorthm agets are cosdered as obects ad ther performaces are measured by ther masses. The SA could be cosdered as a solated system of masses. It s le a small artfcal world of masses obeyg the ewtoa laws of gravtato ad moto. By lapse of tme the masses wll be attracted by the heavest mass whch t represets a optmum soluto the search space. SA has bee verfed as havg hgh-qualty performace solvg dfferet optmzato problems [7-20]. I a system wth agets (masses) the posto of the th aget s defed by: [ x... x x ] X =... for = 2... (2) where s the search spase dmeso of the problem.e. the umber of cotrol varables ad x defes the posto of the th aget the th dmeso. After evaluatg the curret populato ftess the mass of each aget s calculated as follows: m M = m m = worst() t worst() t (3) ft = (4) best where ft (t) represet the ftess value of the aget at tme (terato) t. best(t) ad worst(t) s the best ad worst ftess of all agets respectvely ad defed as follows (for a mmzato problem): best worst ( t) ( t) m ft { }... = (5) ft {... } = (6) Accordg to ewto gravtato theory the total force that acts o the th aget the th dmeso at t tme s specfed as follows: F = r() t M R Kbest ε M ( x x ) + (7) where r s a radom umber the terval [0 ] (t) s gravtatoal costat at tme t M (t) ad M (t) are masses of agets ad ɛ s a small costat ad R (t) s the Euclda dstace betwee the two agets ad gve by the followg equato: ( t) X ( t) X ( t) 2 R = (8) Kbest s the set of frst K agets wth the best ftess value ad bggest mass whch s a fucto of tme talzed to K 0 at the begg ad decreased wth tme. I such a way at the begg all agets apply the force ad as tme passes Kbest s decreased learly ad at the ed there wll be ust oe aget applyg force to the others. By the law of moto the accelerato of the th aget at t tme the th dmeso s gve by followg equato: ( t) F ( t) M ( t) a = (9) The searchg strategy o ths oto ca be defed to fd the ext velocty ad ext posto of a aget. ext velocty of a aget s defed as a fucto of ts curret velocty added to ts curret accelerato. Hece the ext posto ad ext velocty of a aget ca be computed as follows: v ( t ) = r v ( t) a ( t) ( t + ) = x ( t) + v ( t +) + (20) + x (2) where r s a uform radom varable the terval [0 ]. Ths radom umber s utlzed to gve a radomzed characterstc to the search. x represets the posto of aget dmeso v s the velocty ad a s the accelerato. It must be poted out that the gravtatoal costat (t) s mportat determg the performace of SA. It s talzed at the begg ad wll be reduced wth tme to cotrol the search accuracy. I other words the gravtatoal costat s a fucto of the tal value 0 ad tme t: ( t) = exp( α t T ) 0 (22)

4 where α s a user specfed costat t the curret terato ad T s the mum terato umber. The parameters of mum terato T populato sze tal gravtatoal costat 0 ad costat α cotrol the performace of SA. A. SA mplemetato roposed SA approach has bee appled to solve the CEED problem. The cotrol varables of the CEED problem costtute the dvdual posto of several agets that represet a complete soluto set. I a system wth agets the posto of the th aget s defed by: [ x... x x ] X =... for = 2... ad (23) = The elemets of aget X are real power outputs of all geerato uts except the slac geerator. Dfferet steps to solve the CEED problem usg SA are lsted as follow: Step Search space detfcato. Italze SA parameters le: T 0 ad α. Step 2 Italzato: geerate radom populato of agets. The tal postos of each aget are radomly selected betwee mmum ad mum values of the cotrol varables (.e. real power outputs of the geerato uts). Step 3 Calculate the real power output of slac geerator for each aget curret populato. Step 4 Calculate the ftess value for each aget usg (). Step 5 Update the (t) (22) best(t) (5) worst(t) (6) ad M (t) (3) for =2... Step 6 Calculato of the total force dfferet drectos usg (7). Step 7 Calculato of accelerato of each aget usg (9). Step 8 Calculato of velocty of all agets usg (20). Step 9 Update each aget s posto usg (2). Step 0 Repeat Steps 3-9 utl the stop crtera s reached. That s a predefed umber of terato T. Step Retur best soluto. Stop. IV. SIMULATIO RESULTS The proposed SA algorthm s tested o the stadard IEEE 30-bus sx-geerator test system for load = MW. Ths test system s wdely used as bechmar the power system feld for solvg the CEED problem [2]. The fuel cost coeffcets ad the O x emsso coeffcets cludg the lmts of geerato for the geerators of the test system are lsted Table I. I ths study the scalg factor () s tae as γ = 000 ($/to) ad the error tolerace value (9) s O x TOL =0 6 ε MW. The B- matrx values are show as follows: B = B 0 = [ ] B 00 = [ ] The algorthm have bee mplemeted MATLAB 20b computg evromet ad ru o a 2.20 Hz C wth 3.0 B RAM. Twety cosecutve test rus have bee performed for each case examed. The results show are the best values obtaed over these 20 rus. The SA parameters used for the smulato are adopted as follow: α s set to 0 ad 0 s set to. The populato sze ad mum terato umber T are set to 50 ad 200 respectvely for all case studes. For the purpose of comparso wth the reported results the test system s cosdered for two cases as follows: : Wth cosderg ; : Wth eglectg. Table II shows the optmum soluto values of SA for the weght factor: w = (fuel cost mmzato) w = 0 (O x emsso mmzato) ad w = 0.5 (combed fuel cost ad O x emsso mmzato CEED mmzato). Uder the same system data cotrol varable lmts ad costrats the results for Cases A ad B obtaed usg the SA approach are compared to some other algorthms reported the lterature as show Tables III ad IV respectvely. From these tables t ca be see that the proposed approach outperforms may techques used to solve CEED problems because the results obtaed usg SA are ether better or comparable to those obtaed usg other techques. Ths hghlghts ts ablty to fd better qualty soluto. Fgs. -3 llustrates the covergece characterstcs of SA for the fuel cost O x emsso ad combed fuel cost ad O x emsso mmzato respectvely. As ca bee see the proposed SA algorthm s coverge to ts global optmal soluto very small umber of terato for all cases. TABLE I. EERATIO LIMITS FUEL COST AD EMISSIO COEFFICIETS OF THE TEST SYSTEM. Ut a m b c α e e e-2 2.0e e e e-2 5.0e e e e-2.0e e e e-2 2.0e e e e-2.0e e e-2 5.5e-2.0e β η ξ λ - 5 -

5 TABLE II. THE BEST SOLUTIO FOR FUEL COST AD O X EMISSIO. w eerato (MW) O x emsso (to/h) (MW) TABLE III. COMARISO OF BEST SOLUTIO FOR CASE A. Methods mmzato (w=) O x emsso mmzato (w=0) CEED mmzato (w=0.5) O x emsso (to/h) O x emsso (to/h) O x emsso (to/h) ABC [] MOSO [3] A [5] SA [5] A [5] SA II [6] SA II [7] DE [8] DE [9] MODE [0] SO [] MBFA [2] MODE/SO [4] MA θ-so [5] SA TABLE IV. COMARISO OF BEST SOLUTIO FOR CASE B. Methods mmzato (w=) O x emsso mmzato (w=0) CEED mmzato (w=0.5) O x emsso (to/h) O x emsso (to/h) O x emsso (to/h) SOA [2] MOSO [3] A [5] SA [5] A [5] SA II [6] SA II [7] DE [8] DE [9] SO [] MBFA [2] MODE/SO [4] MA θ-so [5] SA w= 206 w= OF OF Iterato Fg.. Covergece characterstcs of SA case fuel cost m. (w=) Iterato Fg. 2. Covergece characterstcs of SA case O x emsso m. (w=0)

6 OF w= Iterato Fg. 3. Covergece characterstcs of SA case combe fuel cost ad O x emsso mmzato - CEED (w=0.5). V. COCLUSIO I ths paper a SA optmzato algorthm has bee proposed ad successfully appled to solve the CEED problem. Smulato results show that the SA approach provdes effectve ad robust hgh-qualty soluto. Moreover the results obtaed usg SA are ether better or comparable to those obtaed usg other techques reported the lterature. ACKOWLEDEMET Ths wor was supported by the Mstry of Educato Scece ad Techologcal Developmet of the Republc of Serba uder research grat TR REFERECES [] D. Ayd S. Ozyo C. Yasar ad T. Lao Artfcal bee coloy algorthm wth dyamc populato sze to combed ecoomc ad emsso dspatch problem Iteratoal Joural of Electrcal ower ad Eergy Systems vol. 54 pp [2] L. Beasla A. Belmada ad M. Rahl Spral optmzato algorthm for solvg combed ecoomc ad emsso dspatch Iteratoal Joural of Electrcal ower ad Eergy Systems vol. 62 pp [3] M. A. Abdo Multobectve partcle swarm optmzato for evrometal/ecoomc dspatch problem Electrc ower Systems Research vol. 79 pp [4] J. Radosavlevć. Arsć ad M. Jevtć Optmal power flow usg hybrd SOSA algorthm 55th Iteratoal Scetfc Coferece of Rga Techcal Uversty o ower ad Electrcal Egeerg (RTUCO204) October Rga Latva pp [5] M. A. Abdo Multobectve evolutoary algorthm for electrc power dspatch problem IEEE Trasactos o Evolutoary Computato vol. 0 (3) pp [6] R. T. F. Ah Kg H.C.S. Rughooputh ad K. Deb Evolutoary multobectve evrometal/ecoomc dspatch: stochastc vs. determstc approaches Evolutoary Mult-Crtero Optmzato Lecture otes Computer Scece vol. 34 (0) pp [7] K. O. Alawode A. M. Jubrl ad O. A. Komolafe Multobectve optmal power flow usg hybrd evolutoary algorthm Iteratoal Joural of Electrcal ower ad Eergy Systems vol. 4 (7) pp [8] A. A. Abou El M. A. Abdo ad S. R. Spea Dfferetal evoluto algorthm for emsso costraed ecoomc power dspatch problem Electrc ower Systems Research vol. 80 (0) pp [9] R. E. erez-uerrero ad J. R. Cedeo-Maldoado Dfferetal evoluto based ecoomc evrometal power dspatch IEEE ower Symposum roceedgs of the 37th Aual orth Amerca 2005 pp [0] L. H. Wu Y.. Wag X. F. Yua ad S. W. Zhou Evrometal/ecoomc power dspatch problem usg mult-obectve dfferetal evoluto algorthm Electrc ower Systems Research vol. 80 (9) pp [] J. Hazra ad A. K. Sha A mult-obectve optmal power flow usg partcle swarm optmzato Europea Trasactos o Electrcal ower vol. 2 () pp [2]. K. Hota A. K. Barsal ad R. Charabart Ecoomc emsso load dspatch through fuzzy based bacteral foragg algorthm Iteratoal Joural of Electrcal ower ad Eergy Systems vol. 32 (7) pp [3] E. Rashed H. ezamabad-pour ad S. Saryazd SA: A gravtatoal search algorthm Iformato Sceces vol. 79 pp [4] D. W. og Y. Zhag ad C. L. Q Evrometal/ecoomc power dspatch usg a hybrd mult-obectve optmzato algorthm Iteratoal Joural of Electrcal ower ad Eergy Systems vol. 32 pp [5] T. am ad H. Doagou-Moarrad Multobectve ecoomc/emsso dspatch by multobectve θ-partcle swarm optmsato IET eerato Trasmso & Dstrbuto vol. 6 (5) pp [6] A. Bhattacharya ad. K. Chattopadhyay Solvg ecoomc emsso load dspatch problems usg hybrd dfferetal evoluto Appled Soft Computg vol. pp [7] E. Rashed. ezamabad-pour S. Saryazd Flter modelg usg gravtatoal search algorthm Egeerg Applcatos of Artfcal Itellgece vol. 24 () pp [8] S. Duma Y. Somez U. uvec et al Optmal reactve power dspatch usg a gravtatoal search algorthm IET eer Trasm Dstrb vol. 6 (6) pp [9] S. Duma U. uvec Y. Somez. Yoruere Optmal power flow usg gravtatoal search algorthm Eergy Cov. Maag. vol. 59 pp [20] J. Radosavlevć M. Jevtć. Arsć D. Klmeta Optmal power flow for dstrbuto etwors usg gravtatoal search algorthm Electrcal Egeerg vol. 96 (4) pp