An Efficient Meta Heuristic Algorithm to Solve Economic Load Dispatch Problems

Transcription

1 A Effcet Meta Heurstc Algorthm to Solve Ecoomc Load Dspatch Problems R. Subramaa* (C.A.), K. Thaushkod* ad A. Prakash* Abstract: The Ecoomc Load Dspatch (ELD) problems power geerato systems are to reduce the fuel cost by reducg the total cost for the geerato of electrc power. Ths paper presets a effcet Modfed Frefly Algorthm (MFA), for solvg ELD Problem. The ma objectve of the problems s to mmze the total fuel cost of the geeratg uts havg quadratc cost fuctos subjected to lmts o geerator true power output ad trasmsso losses. The MFA s a stochastc, Meta heurstc approach based o the dealzed behavour of the flashg characterstcs of frefles. Ths paper presets a applcato of MFA to ELD for 3,6,13 ad 15 geerator test case systems. MFA s appled to ELD problem ad compared ts soluto qualty ad computato effcecy to Geetc Algorthm (GA), Dfferetal Evoluto (DE), Partcle Swarm Optmzato (PSO), Artfcal Bee Coloy optmzato (ABC), Bogeography-Based Optmzato (BBO), Bacteral Foragg Optmzato (BFO), Frefly Algorthm (FA) techques. The smulato result shows that the proposed algorthm outperforms prevous optmzato methods. Keywords: Artfcal Bee Coloy Optmzato, Bogeography-Based Optmzato, Ecoomc Load Dspatch, Frefly Algorthm, Geetc Algorthm, Partcle Swarm Optmzato. 1 Itroducto1 Electrcal power dustry restructurg has created hghly vbrat ad compettve market that altered may aspects of the power dustry. I ths chaged scearo, scarcty of eergy resources, creasg power geerato cost, evromet cocer, ever growg demad for electrcal eergy ecesstate optmal dspatch. Ecoomc Load Dspatch (ELD) s oe of the mportat optmzato problems power systems that have the objectve of dvdg the power demad amog the ole geerators ecoomcally whle satsfyg varous costrats. Sce the cost of the power geerato s exorbtat, a optmum dspatch saves a cosderable amout of moey. Optmal geerato dspatch s oe of the most mportat problems power system egeerg, beg a techque commoly used by operators every day system operatos. Optmal geerato seeks to allocate the real ad reactve power throughout power system obtag optmal operatg state that reduces cost ad Iraa Joural of Electrcal & Electroc Egeerg, 013. Paper frst receved 11 Mar. 013 ad revsed form 19 Oct * The Authors are wth the Akshaya College of Egeerg ad Techology, Combatore, Ida. E-mals: ssst.m1mm3@gmal.com, thaush1@gmal.com ad prksh830@gmal.com. mproves overall system effcecy. The ELD problem reduces the system cost by allocatg the real power amog ole geeratg uts. I the ELD problem the classcal formulato presets defceces due to smplcty of models. Here, the power system modelled through the power balace equato ad geerators are modelled wth smooth quadratc cost fuctos ad geerator output costrats, trasmsso loss costrats [1] ad securty costrats []. To mprove power system studes, ew models are cotuously beg developed that result a more effcet system operatos. Cost fuctos that cosder valve pot loadgs, fuel swtchg, ad prohbted operatg zoes as well as costrats that provde more accurate represetato of system such as: emsso, ramp rate lmts, le flow lmts, spg reserve requremet ad system voltage profle. The mproved models geerally crease the level of complexty of the optmzato problem due to the o-learty assocated wth them. Tradtoal algorthms lke lambda terato, base pot partcpato factor, gradet method, ad Newto method ca solve the ELD problems effectvely f ad oly f the fuel-cost curves of the geeratg uts are pece-wse lear ad mootocally creasg. The basc ELD cosders the power balace costrat apart 46 Iraa Joural of Electrcal & Electroc Egeerg, Vol. 9, No. 4, Dec. 013

2 from the geeratg capacty lmts. However, a practcal ELD must take ramp rate lmts, prohbted operatg zoes, valve pot effects, ad mult fuel optos to cosderato to provde the completeess for the ELD formulato. The resultg ELD s a ocovex optmzato problem, whch s a challegg oe ad caot be solved by the tradtoal methods. Practcal ELD problems have olear, o-covex type objectve fucto wth tese equalty ad equalty costrats. Recet advaces computato ad the search for better results of complex optmzato problems have fometed the developmet of techques kow as Evolutoary Algorthms. The methods for solvg these kds of problems clude tradtoal mathematcal programmg such as lear programmg, quadratc programmg, dyamc programmg, gradet methods, Lagraga relaxato ad covetoal methods lke Taguch Method (TM) [3] ad Gravtatoal Search Algorthm (GSA) [4] approaches ad moder meta-heurstc methods such as Geetc Algorthms (GA) [5-7], Evolutoary Programmg [8], Hop Feld Neural Network (HNN) [9], Partcle Swarm Optmzato (PSO) [10-13], Bacteral Foragg Optmzato (BFO) [14], Artfcal Bee Coloy (ABC) [15], Dfferetal Evoluto (DE) [16] ad Bogeography-Based Optmzato (BBO) [17] are some of these methods whch are successful locatg the optmal soluto but they are usually slow covergece. Problem Formulato The classcal ELD problem s a optmzato problem that determes the power output of each ole geerator that wll result a least cost system operatg state. The objectve of the classcal ecoomc dspatch s to mmze the total system cost where the total system cost s a fucto composed by the sum of the cost fuctos of each geerator. Ths power allocato s doe cosderg system balace betwee geerato ad loads, ad feasble regos of operato for each geeratg ut. The Ecoomc dspatch problem s a fuel cost mmzato of problem whe several geerators are operated to meet the requred power demad. The objectve fucto s gve by: MmzeFt = F( P) (1) = 1 where F t s total fuel cost $/h, P s the power output of th Geerator MW ad F ( P ) s the fuel cost equato of the th plat expressed as follows. F ( P ) = P a + b P + c () = 1 where a, b ad c are the fuel cost coeffcets of th Geerator $/MW h, $/MWh, ad $/h respectvely. The total fuel cost to be mmzed s subject to the followg costrats. P = Pd + Pl = 1 (3) where P d ad P l are the system power demad ad power loss MW respectvely. The system power loss s gve by the relato: PL = P Bj Pj + B0 P + B00 = 1 j= 1 = 1 (4) where B ad B o are the loss coeffcet matrces ad B oo s the loss coeffcet costat. The equalty costrat s gve by m max P P P (5) m max where P ad P are the mmum ad maxmum geerato lmt of th Geerator MW respectvely..1 Valve Pot Loadg Effects The valve-pot loadg effect has bee modelled as a recurrg rectfed susodal fucto, such as the oe show Fg. 1 ad Eq. (6) represets fuel cost cludg valve pot loadg. m F (P ) = a P + b P + c + e s( f ( P P )) (6). Ramp Rate Lmt Costrats I ELD studes, the ut geerato output s usually assumed to be adjusted stataeously. Eve though ths assumpto smplfes the problem, t does ot reflect the actual operatg processes of the geeratg ut. Therefore, practcal stuatos, the operatg rage of all ole uts s restrcted by ther ramp rate lmts, for forcg the uts operato cotually betwee two adjacet specfc perods. The equalty costrats due to ramp rate lmts are gve by: Fg. 1 operatg cost characterstcs wth valve pot loadg. Subramaa et al: A Effcet Meta Heurstc Algorthm to Solve Ecoomc Load Dspatch Problems 47

3 f geerato creases, P P 0 UR (7) f geerato decreases, 0 P (8) P DR 0 where P ad P are the curret ad prevous power output of ut, respectvely. UR ad DR are the up ad dow ramp rate lmt of the -th geeratg ut ( MW/h), respectvely..3 Prohbted Operatg Zoes (POZ) Prohbted Operatg Zoes (POZ) the putoutput curve of geerator are due to steam valve operato or vbrato ts shaft bearg. I practce, t s dffcult to determe the prohbted zoe by actual performace testg or operatg records. I actual operato, the best ecoomy s acheved by avodg operato these areas. Hece, the feasble operato zoe of ut ca be gve as follows: m L P P P,1 U L P,k-1 P P, k k = 1... z (9) U max P,z P P 3 The Frefly Algorthm The Frefly Algorthm (FA) [18-0], s a Meta heurstc, ature-spred, optmzato algorthm whch s based o the socal flashg behavor of frefles, or lghtg bugs, the summer sky the tropcal temperature regos. It was developed by Dr. X-She Yag at Cambrdge Uversty 007, ad t s based o the swarm behavor such as fsh, sects, or brd schoolg ature. I partcular, although the frefly algorthm has may smlartes wth other algorthms whch are based o the so-called swarm tellgece, such as the famous Partcle Swarm Optmzato, ad Artfcal Bee Coloy optmzato algorthms t s deed much smpler both cocept ad mplemetato. The ma advatage s that t uses maly real radom umbers, ad t s based o the global commucato amog the swarm partcles [.e., the frefles], ad as a result, t seems more effectve optmzato such as the ELD problem our case. The FA has three partcular dealzed rules. They are All frefles are usex, ad they wll move towards more attractve ad brghter oes regardless ther sex. The degree of attractveess of a frefly s proportoal to ts brghtess whch decreases as the dstace from the other frefly creases due to the fact that the ar absorbs lght. If there s ot a brghter or more attractve frefly tha a partcular oe, t wll the move radomly. The brghtess or lght testy of a frefly s determed by the value of the objectve fucto of a gve problem. For maxmzato problems, the lght testy s proportoal to the value of the objectve fucto. 3.1 Algorthm Step 1: Read the system data such as cost coeffcets, mmum ad maxmum power lmts of all geerator uts, power demad ad B-coeffcets. Step : Italze the parameters ad costats of Frefly Algorthm. They are off, α max, α m, β 0, γ m, γ max ad termax (maxmum umber of teratos). Step 3: Geerate off umber of frefles radomly betwee λ m ad λ max. Step 4: Set terato cout to 1. Step 5: Calculate the ftess values correspodg to off umber of frefles. Step 6: Obta the best ftess value GbestFV by comparg all the ftess values ad also obta the best frefly values GbestFF correspodg to the best ftess value GbestFV. Step 7: Determe alpha (α) value of curret terato usg the followg equato: α (ter) = αmax - ((αmax - αm) (curret Iterato umber)/ termax) Step 8: Determe the r j values of each frefly usg the followg equato: r j = GbestFV -FV r j s obtaed by fdg the dfferece betwee the best ftess value GbestFV (GbestFV s the best ftess value.e., j th frefly) ad ftess value FV of the th frefly. Step 9: New x values are calculated for all the frefles usg the followg equato: X ew = Xold + β0 *exp( γ. rj *( xj x ) 1 (10) α ( ter)*( rad ) I Eq. (10), β 0 s the tal attractveess γ s the absorpto co-effcet r j s the dfferece betwee the best ftess value GbestFV ad ftess value FV of the th frefly. α(ter) s the radomzato parameter ( I ths work, α (ter) s set to 0.) rad s the radom umber betwee 0 ad 1.I ths work, x λ. Step 10: Iterato cout s cremeted ad f terato cout s ot reached maxmum the go to step 5. Step 11: GbestFF gves the optmal soluto of the Ecoomc Load Dspatch problem ad the results are prted. The basc steps of the FA ca be summarzed as the pseudo code for Frefly Algorthm as follows. 3. Pseudo Code for Frefly Algorthm Objectve fucto f(x), x = (x 1,,x d )T Geerate tal populato of frefles x (=1,.., ) Lght testy I at x s determed by f(x ) Defe lght absorpto coeffcet γ whle (t < MaxGeerato) for = 1: all frefles for j = 1: all frefles f (Ij > I), More frefly towards j d-dmeso; ed f 48 Iraa Joural of Electrcal & Electroc Egeerg, Vol. 9, No. 4, Dec. 013

4 Attractveess vares wth dstace r va exp [-γ r ] Evaluate ew solutos ad update lght testy. ed for j ed for Rak the frefles ad fd the curret best. ed whle Post process results ad vsualzato. 4 Modfed Frefly Algorthm The Modfed Frefly Algorthm (MFA) was proposed ths paper to mprove the explorato of the searchg optmum soluto. Two modfcatos have bee doe. Frstly, stead of usg Cartesa dstace of r j, the modfcato was doe by fdg the mmum varato dstace betwee frefles ad secodly, to mprove the explorato or dversty of the caddate of soluto, the smple mutato correspods to α s adopted the FA process. Thus t wll ehace the optmum results solvg ELD. The proposed modfcatos ca be summarzed as the pseudo code gve below. 4.1 Costrat Hadlg MFA A sgfcat factor the applcato of optmzato techques s how the algorthm hadles the costrats cocerg the problem. The POZ costrats Eq. (9) are utlzed as follows. If the geerato of th ut s settled ts j th POZ,.e.: LB P,j UB P P,j (11) The the amout of geerato s cut to the earest boudary of the j th POZ as follows: LB UB P,j + P LB,j P = (1),j P = P fp P P (13) P fp P P LB LB ave, j, j, j UB ave UB, j, j, j For a olear optmzato problem wth equalty ad equalty costrats, a commo method s the pealty method. The dea s to defe a pealty fucto so that the costraed problem s trasformed to a ucostraed problem. Now we ca defe: M N ν jψ j ( x ) = 1 = 1 ( x,, ν j ) = f ( x ) + μφ ( x ) + μ (14) where μ j 1 ad v j 0 whch should be large eough, depedg o the soluto qualty eeded. As we ca see, whe a equalty costrat t met, ts effect or cotrbuto to Π s zero. However, whe t s volated, t s pealzed heavly as t creases Π sgfcatly. Smlarly, t s true whe equalty costrats becomes tght or exactly. It should be metoed that geerato ad ramp rate lmts are smlar type of costrats. These costrats together state the overall upper/lower geerato lmts of the uts. 4. Pseudo Code for MFA Objectve fucto f(x) x = (x 1,,x d ) T Geerate tal populato of frefles x (=1,, ) Lght testy I at x s determed by f (x ) Defe lght absorpto coeffcet γ whle (t < MaxGeerato) for = 1: all frefles for j = 1: all frefles f (I j > I ), More frefly towards j d-dmeso; ed f Fd the mmum varato dstace of all frefles r = m((frefly frefly j)) Attractveess vares wth dstace r va exp[-γ r ] Evaluate ew solutos ad update lght testy ed for j ed for radum Mutato f radum < probablty of mutato Rak the frefles ad fd the curret best ed whle Post process results ad vsualzato. 5 Smulato Results To solve the ELD problem, the MFA s coded wth MATLAB programmg ad t was ru o a computer wth a Itel Core Duo processor, wdows operatg system. Mathematcal calculatos ad comparsos ca be doe very quckly ad effectvely wth MATLAB. Sce the performace of the proposed algorthm sometmes depeds o put parameters, they should be carefully chose. After several rus, the followg put cotrol parameters are foud to be best for optmal performace of the proposed algorthm. I ths proposed method, we represet ad assocate each frefly wth a vald power output (.e., potetal soluto) ecoded as a real umber for each power geerator ut, whle the fuel cost objectve.e., the objectve fucto of the problem s assocated ad represeted by the lght testy of the frefles. I ths smulato, the values of the cotrol parameters are: α = 0., γ =1.0, β 0 = 1.0 ad =1, ad the maxmum geerato of frefles (teratos) s 100. The values of the fuel cost, the power lmts of each geerator, the power loss coeffcets, ad the total power load demad are suppled as puts to the frefly algorthm. The power output of each geerator, the total system power, the fuel cost wth trasmsso losses are cosdered as outputs of the proposed MFA algorthm. Itally, the objectve fucto of the gve problem s formulated ad t s assocated wth the lght testy of the swarm of the frefles. Subramaa et al: A Effcet Meta Heurstc Algorthm to Solve Ecoomc Load Dspatch Problems 49

5 The MFA has bee proposed for 3, 6, 13 ad 15 mache IEEE stadard test systems. The proposed MFA method has bee compared wth varous optmzato methods ad s tabulated from Tables 1 to 4. Accordg to the result obtaed, the MFA for ELD s more advatageous tha all other Algorthms. From the smulato results of 3, 6, 13 ad 15 geerator test system for ELD usg MFA method, the total fuel cost ad total le losses are decreased tha all other algorthms. 6 Cocluso The proposed MFA to solve ELD problem by cosderg the practcal costrats has bee preseted ths paper. From the comparso table t s observed that the proposed algorthm exhbts a better performace wth respect to all other techques. The effectveess of MFA was demostrated ad tested ths research. From the smulatos, t ca be see that MFA gave the best result of total cost mmzato compared to all other optmzato methods. I future, the proposed MFA ca be used to solve ELD cosderg the valve pot loadg effects. Table 1 Comparso table for 3- ut system (Pd=850 MW) wth valve pot loadg effects. S. GA PSO Descrpto No [5] [11] MFA 1. P 1 (MW) P (MW) P 3 (MW) Power Output(MW) Fuel cost ($/h) Table Comparso table for IEEE 13-ut test system (Pd=1800 MW) wth valve pot effect. Ut Power (MW) MFA ICA-PSO [10] P P P P P P P P P P P P P Total geerato Geerato cost ($/h) Table 3 Comparso table for IEEE 15- ut test system (Pd=630 MW) wth trasmsso loss. Ut Power FA PSO GA MFA (MW) [19] [11] [7] P P P P P P P P P P P P P P P Total power (MW) Losses (MW) Geerato cost ($/h) 3,697 3,704 3,858 33,113 Refereces [1] A. Badr, S. Jadd ad M. Parsa-Moghaddam, Impact of partcpats market power ad trasmsso costrats o GeCos Nash equlbrum pot, Iraa Joural of Electrcal ad Electroc Egeerg, Vol. 3, Nos. 1 &, pp.1-9, Ja [] M. R. Aghamohammad, Statc securty costraed geerato schedulg usg sestvty characterstcs of eural etwork, Iraa Joural of Electrcal ad ElectrocEgeerg, Vol. 4, No. 3, pp , Jul [3] L. Derog ad C. Yg, Taguch Method for Solvg the Ecoomc Dspatch Problem Wth No smooth Cost Fuctos, IEEE trasactos o power systems, Vol. 0, No. 4, pp , Nov [4] S. Duma, U. Güveç ad N. Yörükere, Gravtatoal Search Algorthm for Ecoomc Dspatch wth Valve-Pot Effects, Iteratoal Revew of Electrcal Egeerg, Vol. 5, No. 6, pp , Dec [5] D. C. Walters ad G. B. Sheble, Geetc Algorthm Soluto of Ecoomc Dspatch wth Valve Pot Loadg, IEEE Trasactos o Power Systems,Vol. 8, No. 3, pp , Aug [6] A. Bakrtzs, V. Petrds ad S. Kazarls, Geetc Algorthm Soluto to the Ecoomc Dspatch Problem, Proceedgs. Ist. Elect. Eg. Geerato, Trasmsso Dstrbuto, Vol. 141, No. 4, pp , July Iraa Joural of Electrcal & Electroc Egeerg, Vol. 9, No. 4, Dec. 013

6 Table 4 Comparso table Showg Smulato Result of 6-geerator ut (P d =163 MW) wth varous optmzato methods. S. No Descrpto GA PSO DE ABC BBO BFO [7] [11] [17] [15] [17] [14] FA MFA 1. P 1 (MW) P (MW) P 3 (MW) P 4 (MW) P 5 (MW) P 6 (MW) Power Output MW) 8. P loss (MW) Fuel cost ($/h) Executo tme (sec.) [7] G. B. Sheble ad K. Brttg, Refed geetc algorthm- ecoomc dspatch example, IEEE Tras. o Power Systems, Vol. 10, pp , Feb [8] N. Sha, R. Chakrabart ad P. K. Chattopadhyay, Evolutoary programmg techques for ecoomc load dspatch, IEEE Trasactos o Evolutoary Computato, Vol. 7, No. 1, pp , Feb [9] C. T. Su ad C. T. L, New approach wth a Hopfeld modelg framework to ecoomc dspatch, IEEE Trasactos o Power Systems, Vol. 15, No., pp , 000. [10] J. G. Vlachogas ad K. Y. Lee, Ecoomc Load Dspatch - A Comparatve Study o Heurstc Optmzato Techques wth a Improved Coordated Aggregato-Based PSO, IEEE Trasactos o Power Systems, Vol. 4, No., pp , May 009. [11] J. B. Park, K. S. Lee, J. R. Sh ad K. Y. Lee, A Partcle Swarm Optmzato for Ecoomc Dspatch wth Nosmooth Cost Fuctos, IEEE Trasactos o Power Systems, Vol. 0, No. 1, pp. 34-4, Feb [1] J. B. Park, Y. W. Jeog, J. R. Sh ad K. Y. Lee, A Improved Partcle Swarm Optmzato for Nocovex Ecoomc Dspatch Problems, IEEE Trasactos o Power Systems, Vol. 5, No. 1, pp , Feb [13] I. A. Selvakumar ad K. Thaushkod, A ew partcle swarm optmzato soluto to o covex ecoomc load dspatch problems, IEEE Trasactos o Power Systems, Vol., No. 1, pp. 4-51, Feb [14] B. K. Pagrah ad V. R. Pad, Bacteral foragg optmzato: Nelder-Mead hybrd algorthm for ecoomc load dspatch. IET Geer. Trasm, Dstrb. Vol., No. 4. pp , 008. [15] D. Karaboga ad B. Basturk, Artfcal Bee Coloy (ABC) Optmzato Algorthm for Solvg Costraed Optmzato Problems, Sprger-Verlag, IFSA, LNAI, Vol. 459, pp , 007. [16] R. Stor ad K. Prce, Dfferetal Evoluto-A Smple ad Effcet Adaptve Scheme for Global Optmzato Over Cotuous Spaces, Iteratoal Computer Scece Isttute,, Berkeley, CA, 1995, Tech. Rep. TR [17] A. Bhattacharya ad P. K. Chattopadhyay, Bogeography-Based Optmzato for Dfferet Ecoomc Load Dspatch Problems, IEEE Trasactos o Power Systems, Vol. 5, No., pp , May 010. [18] X. S. Yag, Frefly algorthm, Levy flghts ad global optmzato, Research ad Developmet Itellget Systems XXVI, Sprger, Lodo UK, pp , 010. [19] X.-S. Yag, S. S. Sadat Hosse, ad A. H. Gadom, Frefly Algorthm for solvg ocovex ecoomc dspatch problems wth valve loadg effect, Appled Soft Computg, Vol. 1, No. 3, pp , 01. [0] X. S. Yag, Frefly algorthms for multmodal optmzato, Proceedgs of the Stochastc Algorthms: Foudatos ad Applcatos (SAGA 09), Vol. 579 of Lecture Notes Computg Sceces, pp , Sprger, Sapporo, Japa, Oct R. Subramaa receved the B.E. degree Electrcal ad Electrocs Egeerg from Combatore Isttute of Techology the year 005 ad M.E degree Power Systems Egeerg from Govermet College of Techology, Combatore the year 007. He s curretly dog PhD the area of Power system cotrol ad Subramaa et al: A Effcet Meta Heurstc Algorthm to Solve Ecoomc Load Dspatch Problems 51

7 operato uder Aa Uversty. Presetly he s workg as a Assocate professor the Departmet of Electrcal ad Electrocs Egeerg at Akshaya College of Egeerg ad Techology, Combatore. Hs research terests are power system aalyss, power system cotrol ad operato, mathematcal computatos, optmzato ad Soft Computg Techques. K. Thaushkod receved hs B.E. degree Electrcal ad Electrocs Egeerg from College of Egeerg, Gudy the year197. He has receved M.Sc. (Egg.) degree from PSG College of Techology, Combatore the year1974. He has receved PhD degree Power Electrocs from Bharathyar Uversty the year Presetly he s the Drector of Akshaya College of Egeerg ad Techology, Combatore ad he s a former Sydcate Member, Aa Uversty of Techology, Combatore. Hs research terests are Power Electrocs Drves, Electrcal Maches, Power Systems, ad Soft Computg techques, Computer Networks, Image Processg ad Vrtual Istrumetato. A. Prakash receved the B.E degree Electrcal ad Electrocs Egeerg from Sr Nadhaam College of Egeerg ad Techology the year 005 ad M.Tech degree Power Electrocs ad Drves from PRIST Uversty, Tajore the year 011. He s curretly workg as a Assstat Professor the Departmet of Electrcal ad Electrocs Egeerg at Akshaya College of Egeerg ad Techology, Combatore. Hs research terests are Power System Modellg ad Aalyss ad Power Electroc applcatos to Power Systems. 5 Iraa Joural of Electrcal & Electroc Egeerg, Vol. 9, No. 4, Dec. 013