Non-Smooth Economic Dispatch Solution by Using Enhanced Bat-Inspired Optimization Algorithm

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1 World Academy of Scece, Egeerg ad Techology Iteratoal Joural of Mathematcal ad Computatoal Sceces Vol:8, No:4, 014 No-Smooth Ecoomc Dspatch Soluto by Usg Ehaced Bat-Ispred Optmzato Algorthm Farhad Namdar, Reza Sedaghat Iteratoal Scece Idex, Mathematcal ad Computatoal Sceces Vol:8, No:4, 014 waset.org/publcato/ Abstract Ecoomc dspatch (ED) has bee cosdered to be oe of the key fuctos electrc power system operato whch ca help to buld up effectve geeratg maagemet plas. The practcal ED problem has o-smooth cost fucto wth olear costrats whch make t dffcult to be effectvely solved. Ths paper presets a ovel heurstc ad effcet optmzato approach based o the ew Bat algorthm (BA) to solve the practcal o-smooth ecoomc dspatch problem. The proposed algorthm easly takes care of dfferet costrats. I addto, two ewly troduced modfcatos method s developed to mprove the varety of the bat populato whe creasg the covergece speed smultaeously. The smulato results obtaed by the proposed algorthms are compared wth the results obtaed usg other recetly develop methods avalable the lterature. Keywords No-smooth, ecoomc dspatch, bat-spred, olear practcal costrats, modfed bat algorthm. A I. INTRODUCTION S power demad creases ad sce the fuel cost of the power geerato s exorbtat, reducg the operato costs of power systems becomes a mportat topc. Ecoomc Dspatch (ED) s oe of the most mportat problems to be solved for smooth ad ecoomc operato of a power system. A good load dspatch reduces the producto cost, creases the system relablty ad mzes the eergy capablty of thermal uts [1]. ED s a process for sharg the total load o a power system amog varous geeratg plats to acheve greatest ecoomy of operato. The ED, a olear optmzato problem s bascally solved to geerate optmal amout of geeratg power from the fossl fuel based geeratg uts the system by mmzg the fuel cost ad satsfyg all system costrats of power system[]. Over the past few years, a umber of dervatve-based approaches such as gradet method [3], lambda terato method (LIM) [4], [5], lear programmg (LP) [6], quadratc programmg (QP) [7], Lagraga multpler method [8], classcal techque based o co-ordato equatos [9] were appled to solve ELD problems. There are complex ad olear characterstcs wth equalty ad equalty costrats assocated to the Practcal ED. These characterstcs may be mposed to the problem to esure the system operator of system relablty durg dsturbaces ad a secure operato. These assumptos proved to be feasble for practcal mplemetato because of ther olear characterstcs of prohbted operatg zoes (POZs) Farhad Namdar ad Reza Sedaghat are wth Departmet of Electrcal ad Electroc Egeerg, Egeerg Faculty, Loresta Uversty, Loresta, Ira (e-mal: reza_sedaghat@yahoo.com). or valve-pot effects real geerators. Practcal ED tself has complex ad olear characterstcs wth may equalty ad equalty costrats, whch are brefly descrbed the followg. I practce due to physcal operato lmtato such as faults the maches themselves or the assocated auxlares, such as boler ad feed pumps uts ca have prohbted operatg zoes ad geerators that operate these zoes may experece amplfcato of vbratos ther shaft beargs, whch should be avoded practcal applcatos. I addto, large moder geeratg uts wth mult-valve steam turbes have a umber of steam admsso valves that are opeed sequetally to obta ever-creasg output of the uts ad the valve-pot effects produce a rpple-lke heat rate curve. Moreover may geeratg uts, specfcally those whch are suppled wth mult-fuel source lead to the problem of determg the most ecoomc fuel to bur. Cosequetly, mult-fuel uts, prohbted operatg zoes ad valve loadg effects should be cosdered to solve a realstc ED problem, whch makes hard the fdg of the optmum soluto [10], [11]. Recetly, moder heurstc optmzato techques have bee appled to solve ED problem due to ther abltes of fdg a almost global optmal soluto, such as geetc algorthms (GA) [1], [13], Tabu search (TS) [14], smulated aealg (SA) [15], dfferetal evoluto (DE) [16] ad Partcle Swarm Optmzato (PSO) [17], [18] are cosdered as realstc ad powerful soluto schemes to obta the global optmums power system optmzato problems ad due to ther ablty to fd a almost global optmal soluto for ED problems wth operatg costrats. I ths respect, a modfed verso of Bat Algorthm (BA) as a evolutoary meta-heurstc algorthm s employed to solve the proposed realstc ED problem. BA tres to formulate ad smulate the jourey of bats search of utrtous resource or chasg preys. The algorthm s smple cocept thus easy to mplemet sce there are ot may adjustg parameter cluded the formulato. However, the orgal algorthm suffers low covergece rate ad t s dested to get trapped local optma due to the lack of dversty the populato. Two modfcato stages were emplaced the orgal algorthm to help crease the covergece rate of the algorthm ad dversfy the populato. Iterspersg the populato to the etre search space mproves the odds of fdg the global optma. The robustess ad capablty of the propose methodology s demostrated by applyg the procedure to oe IEEE stadard test system. I all, the ma cotrbutos of ths study ca be summarzed as follows: Iteratoal Scholarly ad Scetfc Research & Iovato 8(4) scholar.waset.org/ /

2 World Academy of Scece, Egeerg ad Techology Iteratoal Joural of Mathematcal ad Computatoal Sceces Vol:8, No:4, 014 Iteratoal Scece Idex, Mathematcal ad Computatoal Sceces Vol:8, No:4, 014 waset.org/publcato/ Proposg a comprehesve model for ED to cosder practcal costrat real systems Modfyg the orgal BA to eable t of seekg the search space faster ad more precsely II. PRACTICAL ED MATHEMATICAL DESCRIPTION The objectve of the ED s to mmze the total geerato cost of a power system over some approprate perod whle satsfyg varous costrats. The power system balace of codtos for system demad, power losses ad etre geerator power. The mathematcal represetato of the classcal ED problem ad the proposed practcal ED are descrbed ths secto. A. Classc ED ED ts classcal formulato s to mmze the summg costs of thermal geeratg uts whch are geerally cosdered as a secod order polyomal fucto of the geerato [], the form of (1): f( X) Cost( X ) = F( P ) = (a ) +bp +cp = 1 =1 = I whch P deotes output power of the th ut ad stads for the umber of geerators the etwork. The polyomal coeffcets of cost for th ut are represeted by a, b ad c as well. The covetoal ED optmzato problem s subjected to the followg costrats forcg geerators to produce power wth specfc lmts so that ther total geerato equals total power demad the etwork (D) plus the trasmsso etwork loss. However, the etwork loss s ot cosdered ths paper for smplcty. P=D = 1 m P P P I the above formulato, the lower ad upper bouds of power geerato for th ut s deoted by P m ad P respectvely. B. Proposed Practcal Ed Formulato 1. Valve-Pot Loadgs Effects However, t s more practcal to cosder the effect of valve pot loadg for thermal power plats [19]. These effects, whch occur as each steam admsso valve a turbe, create a rpplg fluece o the ut s cost curve. Cosderg the valve-pot effects, the fuel cost fucto of the th thermal geeratg ut s expressed as the sum of a quadratc ad a susodal fucto the followg form: F P =a + e f m ( ) bp+ cp + s( ( P P)) The costs of valve loadg effect are represeted by the coeffcets e ad f the susodal term. (1) () (3) (4). Multple Fuels Sce the dspatchg uts are practcally provded wth mult-fuel sources, each ut should really be represeted wth several pecewse quadratc fuctos reflectg the effects of fuel type chages, ad the geerator must detfy the ecoomc fuel to bur [0]. Thus, sce dfferet fuels possess varous costs, the fal geerato cost of the uts wll be depedet o ther choce of fuel leadg to dvded cost fucto for geerators as follows: a +b P +c P ; Fuel 1: P P P a +b P +c P F( P ) =... m a j +bjp +cjp ; Fuel j: Pj P P m m ; Fuel : P P P ad by the addto of the term related to valve loadg effect the above formulato turs to the followg form: F ( P ) = a j +bj P + m Fuel j: P j P j = 1,,.., f m cj P + ej s( f j( Pj P)) ; P The umber of dfferet fuel types that are provded s deoted by f. Apart from the two prevously covetoal costrats, the secure operato of etwork madates respectg to the followg costrats: 3. Ramp Rate Lmts P (7) P0 UR; If geerato creases P0 P DR; If geerato decreases Ramp-up ad ramp-dow rate lmts of th ut are deoted by UR ad DR respectvely. Also P 0 s the actve power output of th ut the prevous hour. It s of sgfcat matter that due to the cosderato of ram rates, the output power of each ut s ow bouded by ew lmt as follows: m ( P, P0 ) m(, 0 ); DR P P P + UR = 1,,.., 4. Prohbted Operatg Zoes (POZs) Each geerator has ts geerato capacty lmtato, whch ca ot be exceeded [1]. The prohbted operatg zoes the put-output performace curve due to steam valve operatg shaft bearg are beg cosdered determg the optmum ED ths paper. I practce, whe adjustg the geerato output of a ut oe must avod operato the prohbted zoes. Thus, the shape of the put-output curve the eghborhood of the prohbted zoes s dffcult to determed, the best ecoomcal approach s acheved by avodg the operato these areas. Oe mght represet the POZ restrctos for th as: (5) (6) (8) Iteratoal Scholarly ad Scetfc Research & Iovato 8(4) scholar.waset.org/ /

3 World Academy of Scece, Egeerg ad Techology Iteratoal Joural of Mathematcal ad Computatoal Sceces Vol:8, No:4, 014 Iteratoal Scece Idex, Mathematcal ad Computatoal Sceces Vol:8, No:4, 014 waset.org/publcato/ m LB P P P,1 UB LB P, j 1 P P, j; j =,3,.., NP UB P, j P P For = 1,,.., N GP (9) N GP s the umber of geerators corporatg POZ ad NP s the umber of POZs of th ut. Besdes, P LB,j ad P UB,j refer to the lower ad upper boudary of j th POZ of th geerator respectvely. 5. Spg Reserve Due to the cluso of POZ the formulato of ED problem, spg reserve of the system should be wrtte the followg form: m{( P ), } ( ) (10) P S Ω Θ S = 0 Θ (11) Sr= S = 1 S S (1) r R I whch S ad S refer to the spg reserve of th ut ad ts mum value respectvely. The set of operatg uts s deoted by Ω ad the set of operatg uts wth POZ s represeted by Θ. S R refers to the total spg reserve requred by the system. III. OPTIMIZATION TECHNIQUE A. Orgal Bat Algorthm (BA) Bats are fascatg amals. They are the oly mammals wth wgs, ad they also have advaced capablty of echolocato. It s estmated that there are about 1000 dfferet bat speces, whch accouts for up to 0% of all mammal speces. Ther sze rages from the ty bumblebee bats (of about 1.5 to g) to gat bats wth a wgspa of about m ad weght of up to about 1 kg. Mcrobats typcally have a forearm legth of about. to 11 cm. Most bats use echolocato to a certa degree; amog all the speces, mcrobats are a famous example because they use echolocato extesvely, whereas megabats do ot. Most mcrobats are sectvores. Mcrobats use a type of soar, called echolocato, to detect prey, avod obstacles, ad locate ther roostg crevces the dark. These bats emt a very loud soud pulse ad lste for the echo that bouces back from the surroudg objects. Ther pulses vary propertes ad ca be correlated wth ther hutg strateges, depedg o the speces. Most bats use short, frequecy-modulated sgals to sweep through about a octave; others more ofte use costat-frequecy sgals for echolocato. Ther sgal badwdth vares wth speces ad ofte creases by usg more harmocs. Studes show that mcrobats use the tme delay from the emsso ad detecto of the echo, the tme dfferece betwee ther two ears, ad the loudess varatos of the echoes to buld up a three-dmesoal scearo of the surroudg. They ca detect the dstace ad oretato of the target, the type of prey, ad eve the movg speed of the prey, such as small sects. Ideed, studes suggest that bats seem to be able to dscrmate targets by the varatos of the Doppler effect duced by the wg-flutter rates of the target sects. Bat Algorthm (BA) s a metaheurstc populato based optmzato algorthm whch was frst spred from the search of bats to fd ther food []. Bats sed some sgals to the evromet ad te lste to ts echo whch s called echolocato process. BA s maly costructed by the use of 4 ma deas []: 1) the dfferece betwee the prey ad food s dstgushed through the use of echolocato process; ) Each bat the posto X fles wth the velocty of V producg a especal pulse wth the frequecy ad loudess of f ad A respectvely; 3) the loudess of A chages dfferet ways such as reducg from a large value to a low value; ad 4) the frequecy f ad rate r of each pulse s regulated automatcally. Itally, all bats fly radomly the search space producg radom pulses. After each fly, the posto of each bat s updated as follows: V = V + f ( X X ); = 1,, N X X N f f f f N ew old G Bat ew old ew = + V ; = 1,..., Bat m m = + ϕ1( ) ; = 1,..., Bat (13) where X G dcates the best global soluto. The upper ad lower frequecy lmts of th bat are represeted by f ad f m respectvely. The populato sze s equal to the total umber of bats deoted by N Bat ad φ 1 s a radomly geerated umber betwee 0 ad 1. The secod movemet the bat posto s smulated as follows: ew old old X = X + ε A ; = 1,, N mea Bat (14) where ε s a radom umber the rage of [-1,1] ad A mea old s the mea value of ampltude of all bats. Oce the posto of bats s mproved by the above adjustmets a ew radom dvdual X ew s geerated case the rate of ts sgal r s greater tha a radom value β. Ths ew soluto wll be serted to the populato case the followg costrat s respected: [ β < A]&[ f( X ) < f( Gbest)] (15) As metoed formerly the value of sgal ampltudes geerated by bats has a gradual decrease formulated by: ew old A = αa Iter + 1 r = r 0 [1 exp( γ t)] (16) where t represets terato umber. α ad γ are costat parameters as well. The ma steps of proposed BA are as follows: Iteratoal Scholarly ad Scetfc Research & Iovato 8(4) scholar.waset.org/ /

4 World Academy of Scece, Egeerg ad Techology Iteratoal Joural of Mathematcal ad Computatoal Sceces Vol:8, No:4, 014 Iteratoal Scece Idex, Mathematcal ad Computatoal Sceces Vol:8, No:4, 014 waset.org/publcato/ Step 1. Italze the bat populato or ther posto X old ad ther veloctes V old. Defe pulse frequecy f at X old. Italze pulse rates r ad the loudess A. Step. Geerate ew solutos by adjustg frequecy, ad updatg veloctes ad locatos/solutos (Equato (13]). Step 3. f (rad > r ) Select a soluto amog the best solutos. Geerate a local soluto aroud the selected best soluto. Step 4. Else geerate a ew soluto by flyg radomly. Step 5. If ( [ β < A]&[ f( X) < f( Gbest)] ) Accept the ew solutos, crease r ad reduce A. Step 6. Rak the bats ad fd the curret best X ew. Step 7. whle (terato < Max umber of teratos) Post process results ad vsualzato. The algorthm stops wth the total-best soluto. B. Modfed Bat Algorthm The orgal BA suffers some drawbacks such as possblty of gettg trapped local optma ad low rate of covergece to the optmal soluto. Two modfcatos are devsed ad added to the algorthm order to mprove ts covergece rate ad dversty as follows: 1. Modfcato Method 1 I the frst modfcato step, t s attempted to dversfy the bat populato usg Lévy flght whch s defed as a radom walk wth regular ad dspersed step legths accordg to heavy-taled probablty dstrbuto [3]. The mathematcal represetato the Lévy flght s formulates as: Le vy( ω) ~ τ = t ω ; (1< ω 3) (17) Ths dea s borrowed to geerate a ew dvdual each terato as follows: ew old X = X + ϕ1 Levy ( ω) (18) Ths ew soluto mght replace the th bat the populato case t excels the objectve fucto.. Modfcato Method The secod modfcato step s devsed to tersperse radomly geerated solutos the populato based o covetoal GA operators of crossover ad mutato. To do so, three bats X b1, X b ad X b3 are chose radomly such that b1 b b3 for th bat the populato ad two test solutos wll be geerated as follows: XTest,1 = Xb + ϕ 1 1 ( Xb X ) b3 XTest, = ϕ XG + ϕ3 ( XG X ) (19) (0) The above dvduals are compared to the th bat ad the oe whch ehaces the objectve fucto replaces X. IV. SOLUTION PROCEDURE I order to apply MBA to the ED problem, the followg steps should be mplemeted: Step 1. Defg the put data. Here all data cludg the etwork data, algorthm data (such as umber of bats, tal postos, costat coeffcets ad etc.), objectve fucto parameters, costrats parameters ad etc. are defed completely. Step. Formato of the ftess fucto. It s oted that the ftess fucto clude the objectve fucto ad the pealty values related to the problem costrats. Step 3. Geerato of the tal populato based o the formato gve prevous secto. Step 4. Evaluato of the objectve fuctos for each bat separately ad fdg the best soluto. Step 5. Movemet of the bat populato to the ew mproved postos. Step 6. Applcato of the proposed modfcato methods accordg to (17)-(0). Step 7. Updatg the value of the best dvdual. Step 8. Check the termato crtero. If the termato crtero s satsfed, the fsh the algorthm ad prt the results else retur to step 5 ad repeat the steps. V. SIMULATION RESULTS Effectveess of the proposed approach to solve proposed realstc ED problem s llustrated by applyg the method o a test system. For the sake of better demostratg the robustess of the proposed methodology, the procedure s appled to the 40-ut IEEE power system as well secod test case s fully troduced [6]. The optmzato problem s solved 100 tmes to geerate the results, have bee doe to the IEEE-40 ut system ad the optmzato results ad the ut allotted outputs were llustrated Tables I ad II respectvely. TABLE I COST FUNCTION OPTIMIZATION USING THE PROPOSED METHOD ON THE IEEE 40-UNIT TEST SYSTEM Method Best Average Worst IFEP [4] MPSO [5] ESO [6] PSO-LRS [5] Improved GA [7] HPSOWM[8] IGAMU[9] NPSO [5] HDE[30] NPSO-LRS [5] BA Proposed MBA 1, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , Probg to the results reveals the domace of the proposed MBA fdg the optmal soluto compared to other methods. The mprovemets are obvous all average, worst ad the best solutos. The covergece behavor of the proposed MBA s depcted Fg. 1 ad t s see that the Iteratoal Scholarly ad Scetfc Research & Iovato 8(4) scholar.waset.org/ /

5 World Academy of Scece, Egeerg ad Techology Iteratoal Joural of Mathematcal ad Computatoal Sceces Vol:8, No:4, 014 modfcatos have ehaced covergece rate of the algorthm. MBA optmzato s a promsg techque for solvg complcated problems power system. Iteratoal Scece Idex, Mathematcal ad Computatoal Sceces Vol:8, No:4, 014 waset.org/publcato/ TABLE II OPTIMAL OPERATING POINT OF THE GENERATORS ON THE IEEE 40-UNIT TEST SYSTEM Ut Optmal Geerato Ut Optmal Geerato Fg. 1 The covergece speed of the proposed MBA o the IEEE 40-ut test system VI. CONCLUSION Ths paper proposed a ew ature spred bat algorthm for solvg the ecoomc load dspatch problem wth o-smooth cost fuctos ad t has bee compared wth dfferet PSO ad IWD techques wth multple fuel opto, valve loadg effects, power geerato lmts, spg reserve, ramp rate lmts ad POZs. To mprove the search process, two techques,.e. Lévy flght ad tersperse radomly geerated solutos the populato based o covetoal GA operator's varable reducto, ad zoom feature are added to the Bat Algorthm (BA). Studed results cofrm that the proposed MBA s much superor to other covetoal methods terms of hgh-qualty soluto, stable covergece characterstc, ad good computato effcecy. Therefore, ths results shows that REFERENCES [1] J.J. Grager ad W.D. Steveso, Jr., Power System Aalyss. New York: McGraw-Hll, [] B.H. Chowdhury, S. Rahma, A revew of recet advaces ecoomc dspatch, IEEE Tras Power Syst 1990;5(4): [3] J.C. Dodu, P. Mart, A. Merl, J. Pouget, A optmal formulato ad soluto of short-rage operatg problems for a power system wth flow costrats, IEEE Proc; 60(1):54 63, 197. [4] C.L. Che, S.C. Wag, Brach ad boud schedulg for thermal geeratg uts, IEEE Tras Eergy Covers;8():184 9, [5] P. Aravdhababu, K.R. Nayar, Ecoomc dspatch based o optmal lambda usg radal bass fucto etwork, It J Electr Power Eergy Syst;4(7):551 6, 00. [6] J. Parkh, D. Chattopadhyay, A mult-area lear programmg approach for aalyss of ecoomc operato of the Ida power system, IEEE Tras Power Syst;11(1):5 8, [7] J.Y. Fa, L. Zhag, Real-tme ecoomc dspatch wth le flow ad emsso costrats usg quadratc programmg, IEEE Tras Power Syst;13():30 5, [8] J. Nada, L. Har, M.L. Kothar, Ecoomc emsso dspatch wth le flow costrats usg a classcal techque, IEE Proc Geer Tras Dstrb;141(1):1 10, [9] A.A. El-Keb, H. Ma, J.L. Hart, Evrometally costraed ecoomc dspatch usg the Lagraga relaxato method, IEEE Tras Power Syst;9(4):173 9, [10] C.T. Su, C.T. L, New approach wth a Hopfeld modelg framework to ecoomc dspatch, IEEE Trasactos o Power System, 15(), , 000. [11] T. Nkam, H. Doagou Mojarrad, M. Nayerpour, A ew fuzzy adaptve partcle swarm optmzato for o-smooth ecoomc dspatch, Eergy. do: /j.eergy , 009. [1] D.C. Walters, G.B. Sheble, Geetc algorthm soluto of ecoomc dspatch wth valve pot loadg, IEEE Tras Power Syst ;8:135 3, [13] C.L. Chag, Improved geetc algorthm for power ecoomc dspatch of uts wth valve-pot effects ad multple fuels, IEEE Tras Power Syst 005;0(4):1690 9, 005. [14] W. Ogsakul, S. Dechaupaprtttha, I. Ngamroo, Parallel tabu search algorthm for costraed ecoomc dspatch, IEE Proc Geer Trasm Dstrb ;151():157 66, 004. [15] K.P. Wag, C.C. Fug, Smulate aealg base ecoomc dspatch algorthm, IEE Proc C; 140(6):507 13, 006. [16] N. Noma ad H. Iba, Dfferetal evoluto for ecoomc load dspatch problems, Electrcal Power Systems Research, vol.78, o.8, pp , 008. [17] J. B. Park, K. S. Lee, J. R. Sh ad K. Y. Lee, A partcle swarm optmzato for ecoomc dspatch wth o-smooth cost fuctos, IEEE Trasactos o Power Systems, vol.0, o.1, pp.34-4, 005. [18] Z. L. Gag, Partcle swarm optmzato to solvg the ecoomc dspatch cosderg the geerator costrats, IEEE Trasactos o Power Systems, vol.18, o.3, pp , 003. [19] T. Nkam, R. Azzpaah-Abarghooee, R. Sedaghat, A. Kavous-Fard, A Ehaced hybrd partcle swarm optmzato ad smulated aealg for practcal ecoomc dspatch, Eergy Educato Scece ad Techology Part A: Eergy Scece ad Research, 30(1): , 01. [0] J.B. Park, K.S. Lee, J.R. Sh, K.Y. Lee, A partcle swarm optmsato for ecoomc dspatch wth osmooth cost fucto, IEEE Tras Power Syst ;0(1):34 4, 005. [1] A.J. Wood, B.F. Wolleberg, Power Geerato Operato ad Cotrol, Joh Wley & Sos: New York, [] J.C. Dodu, P. Mart, A. Merl, J. Pouget, A optmal formulato ad soluto of short-rage operatg problems for a power system wth flow costrats, IEEE Proc; 60(1):54 63, 197. [3] C.L. Che, S.C. Wag, Brach ad boud schedulg for thermal geeratg uts, IEEE Tras Eergy Covers;8():184 9, [4] P. Aravdhababu, K.R. Nayar, Ecoomc dspatch based o optmal lambda usg radal bass fucto etwork, It J Electr Power Eergy Syst;4(7):551 6, 00. Iteratoal Scholarly ad Scetfc Research & Iovato 8(4) scholar.waset.org/ /

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