Solution of Large Scale Economic Load Dispatch Problem using Quadratic Programming and GAMS: A Comparative Analysis

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1 ISSN Eglad UK Joural of Iformato ad Computg Scece Vol. 7 No pp Soluto of Large Scale Ecoomc Load Dspatch Problem usg Quadratc Programmg ad GAMS: A Comparatve Aalyss Devedra Bse 1 Har Moha Dubey 1 Majaree Padt 1 ad B. K. Pagrah 2 1 Departmet of Electrcal Egeerg Madhav Isttute of Techology ad Scece Gwalor (M.P) Ida 2 Departmet of Electrcal Egeerg Ida Isttute of Techology Delh Ida (Receved March accepted Jue ) Abstract. Ths paper presets a comparatve aalyss of effcet ad relable moder programmg approach usg quadratc programmg (QP) ad geeral algebrac modelg system (GAMS) to solve ecoomc load dspatch (ELD) problem. The proposed methodology easly takes care of dfferet equalty ad equalty costrats of the power dspatch problem to fd optmal soluto. To valdate the effectveess of algorthm smulatos have bee performed over four dfferet cases.e. Crete Islad system of 18 thermal geeratg uts twety geeratg uts system wth losses practcal Tawa Power Compay (TPC) data whch cossts of 40 geeratg ut system ad a very large system cosstg of 110 geeratg ut. Results obtaed wth the proposed method have bee compared wth other exstg relevat approaches avalable lteratures. Expermetal results support the clam of profcecy of the method over other exstg techques terms of robustess ad most mportatly ts optmal search behavor. Keywords: Ecoomc load dspatch quadratc programmg geeral algebrac modellg system quadratc cost fucto operatg lmt costrats.. 1. Itroducto The dea behd ecoomc dspatch problem a power system s to determe the optmal combato of power output for all geeratg uts whch wll mmze the total fuel cost whle satsfyg load ad operatoal costrats. The ecoomc dspatch problem s very complex to solve because of ts massve dmeso a o-lear objectve fucto ad large umber of costrats. Varous vestgatos o the ELD have bee udertake tll date. Sutable mprovemets the ut output schedulg ca cotrbute to sgfcat cost savgs [1 2]. Also formato about formg market clearg prces s provded by t. To mprove the qualty of soluto lot of researches have bee doe ad varous methods have bee evolved so far the feld of ecoomc load dspatch. Several classcal optmzato techques such as the lambda terato approach the gradet method the lear programmg method ad Newto s method were used to solve the ELD problem [3]. Lambda terato method s the most commo whch has bee appled to solve ELD problems. But for effectve mplemetato of ths method the formulato must be cotuous. Though fast ad relable the ma drawback of the lear programmg methods s that they are assocated wth the pecewse lear cost approxmato [4]. I order to get the qualtatve soluto for solvg the ELD problems Artfcal Neural Network (ANN) techques such as Hopfeld Neural Network (HNN) [4] have bee used. The objectve fucto of the Ecoomc Dspatch problem s trasformed to a Hopfeld eergy fucto ad umercal teratos are appled to mmze the eergy fucto. The Hopfeld model has bee employed to solve the ED problems for uts havg cotuous or pecewse quadratc fuel cost fuctos ad for uts havg prohbted zoe costrats. I the covetoal Hopfeld Neural Network the put-output relatoshp for ts euros ca be descrbed by sgmod fucto. Due to the use of the sgmod fucto to solve the ED problems the Hopfeld model takes more terato to provde the soluto ad ofte suffers from large computatoal tme. Correspodg author E-mal address: harmohadubey@redffmal.com. Publshed by World Academc Press World Academc Uo

2 Joural of Iformato ad Computg Scece Vol. 7 (2012) No.3 pp Recetly dfferet heurstc approaches have bee proved to be effectve wth promsg performace. These clude evolutoary programmg (EP) [6] geetc algorthm (GA) [7] dfferetal evoluto (DE) [8] partcle swarm optmzato (PSO) [9] etc. Improved fast Evolutoary programmg algorthm has bee successfully appled for solvg the ELD problem [1 5]. Other algorthms lke Bogeography-Based optmzato(bbo) [10] Chaotc partcle swarm optmzato (CPSO) [11] ew partcle swarm wth local radom search (NPSO-LRS) [12] Self-Orgazg Herarchcal PSO [13] Bacteral foragg optmzato [14] mproved coordato aggregated based PSO [15] quatum-spred PSO [16] mproved PSO [17] HHS algorthm [18] ad HIGA [19] are some of the those whch have bee successfully appled to solve the ELD problem. I ths paper a comparatve aalyss of Quadratc programmg (QP) ad Geeral Algebrac Modelg System (GAMS) approach has bee proposed to solve ecoomc load dspatch problems. Quadratc programmg s a effectve tool to fd global mma for optmzato problem havg Quadratc objectve fucto ad lear costrats. The objectve fucto for the etre test cosdered here s Quadratc ature for all cases but the costrats are ot lear. Costrats are lberalzed by trasformato of varable techque ad the Quadratc programmg s appled recursvely tll the covergece s acheved [20]. Geeral Algebrac Modelg System (GAMS) [21] s a hgh-level model developmet evromet that supports the aalyss ad soluto of lear o lear ad mxed teger optmzato problems. GAMS s especally useful for hadlg large dmeso ad complex problem easly ad accurately. I ths paper the effectveess of the proposed algorthm s demostrated usg four stadard test problems () Crete Islad system of 18 thermal geeratg uts () 20 geeratg uts system wth losses () practcal large scale Tawa Power Compay (TPC) system cosstg of 40 geeratg uts ad (v) a very large system cosstg of 110 geeratg uts. The paper s orgazed as follows: Secto 2 provdes a bref descrpto ad mathematcal formulato of ELD problems. The cocept of QP ad GAMS s dscussed Secto 3 ad Secto 4 respectvely. The performace of both proposed approaches ad the smulato studes are dscussed Secto 5. Fally Secto 6 presets the coclusos. 2. Problem formulato I a power system the ut commtmet problem has varous sub-problems varyg from lear programmg problems to complex o-lear problems. The cocered ELD problem s oe of the dfferet o-lear programmg sub-problems of ut commtmet. The ELD problem s about mmzg the fuel cost of geeratg uts for a specfc perod of operato so as to accomplsh optmal geerato dspatch amog operatg uts ad retur satsfyg the system load demad cosderg power system operatoal costrats. The objectve fucto correspodg to the producto cost ca be approxmated to be a quadratc fucto of the actve power outputs from the geeratg uts. Symbolcally t s represeted as N cos t Mmze F G t f (P ) (1) 1 Where 2 f ( P ) a P bp c N G (2) s the expresso for cost fucto correspodg to th geeratg ut ad a b ad c are ts cost coeffcets. P s the real power output (MW) of th geerator correspodg to tme perod t. N G s the umber of ole geeratg uts to be dspatched. The costrats are: 1) Power Balace Costrats: JIC emal for subscrpto: publshg@wau.org.uk

3 202 Devedra Bse et al: Soluto of Large Scale Ecoomc Load Dspatch Problem usg Quadratc Programmg ad GAMS: A Comparatve Aalyss Ths costrat s based o the prcple of equlbrum betwee total system geerato ( P ) ad total system loads (P D ) ad losses (P L ). That s N G P P P D L 1 Where the trasmsso loss P L s expressed usg B- coeffcets [3] gve by N N P L 1 j 1 G G N G P B j P j 1 B 0 P B The followg codtos for optmalty ca be obtaed after applyg lagraga multpler ad K.T. codto 2 a P + b = (1-2 j1 00 N G 1 (3) (4) B j ) = (5) 2) The Geerator Costrats: The power geerated by each geerator should be wth ts lower lmt P m ad upper lmt P max so that m max P P P (6) 3. Quadratc Programmg Algorthm Quadratc Programmg s a effectve optmzato method to fd the global soluto f the objectve fucto s quadratc ad the costrats are lear. It ca be appled to optmzato problems havg oquadratc objectve ad olear costrats by approxmatg the objectve to quadratc fucto ad the costrats as lear. For all the problems the objectve s quadratc but the costrats are also quadratc so the costrats are to be made lear [20]. The o lear equatos ad equaltes are solved by the followg steps. Step 1: To talze the procedure allocate lower lmt of each plat as geerato evaluate the trasmsso loss ad cremetal loss coeffcets ad update the demad. P = P m x = 1 - j1 B j P ad ew old PD = PD + P L Step 2: Substtute the cremetal cost coeffcets ad solve the set of lear equatos to determe the cremetal fuel cost λ as. λ P ew D b 0. 5 a b 0. 5 a Step 3: Determe the power allocato of each plat b ew a P a 2 x (9) If plat volates ts lmts t should be fxed to that lmt ad oly the remag plats oly should be cosdered for the ext terato. (7) (8) JIC emal for cotrbuto: edtor@jc.org.uk

4 Joural of Iformato ad Computg Scece Vol. 7 (2012) No.3 pp Step 4: Check for covergece s the tolerace value for power balace volato. Step 5: Carry out the steps 2-4 tll covergece s acheved. ew P PD P (10) For all the above four steps the objectve s quadratc but the costrats are also quadratc so the costrats are to be made lear. Mmze XH X T + f T X Subjected to KX R X m X X max X = x x x x f = f f f f R = R R R R T m H s a Hessa matrx of sze ad A s a m matrx represetg equaltes. For the ecoomc dspatch wth losses the quadratc programmg algorthm ca be effectvely mplemeted by defg the matrces H f K ad R. H= dag a 1 a2 a... x1 x2 x K = [1 1 1] 1 matrx ad f = b x 1 1 b x R = PD + P L old 2 2 L b... x 4. Geeral Algebrac Modelg System (GAMS) The Geeral Algebrac Modelg System (GAMS) s a hgh-level model specally desged for modelg lear olear ad mxed teger optmzato problems. GAMS ca easly hadle large ad complex problems. It s especally useful for hadlg large complex problems whch may requre much revso to establsh a accurate model. Coverso of lear to olear optmzato s also very smple. Models ca be developed solved ad documeted smultaeously matag the same GAMS model fle. The basc structure of a mathematcal model coded GAMS has the compoets: sets data varable equato model ad output [22] ad the soluto procedure are show below. Optmzato Model Formulato GAMS Model descrpto preprocessg solver GAMS soluto report Global ad Local search method for olear optmzato Optoal calls for other solvers ad exteral Programs Fg. 1: GAMS modelg ad soluto procedure JIC emal for subscrpto: publshg@wau.org.uk

5 204 Devedra Bse et al: Soluto of Large Scale Ecoomc Load Dspatch Problem usg Quadratc Programmg ad GAMS: A Comparatve Aalyss STEPS FOR PROBLEM FORMULATION WITH GAMS GAMS formulato follows the basc format as gve below: 1. SETS Declarato Assgmet of members 2. Data (PARAMETERS TABLES SCALARS) Declarato Assgmet of values 3. VARIABLES Declarato Assgmet of type Assgmet of bouds ad/or tal values (optoal) 4. EQUATIONS Declarato Defto 5. MODEL ad SOLVE statemets 6. DISPLAY statemets (optoal) 5. Result ad Dscusso The Quadratc programmg ad GAMS have bee appled o four dfferet stadard systems. Test case I cossts of Crete Islad system of 18 thermal geeratg uts Test case II cossts of 20 geeratg uts system wth losses Test case III cossts of practcal Tawa Power Compay (TPC) 40 geerators system ad Test case IV cossts of a large scale system cosstg of 110 geeratg uts.the programs were wrtte MATLAB 7.8 for quadratc programmg ad mplemetato o GAMS wth system cofgurato Core 2 Duo processor ad 3GB RAM Test case 1 Crete Islad system of 18 thermal geeratg uts havg quadratc (Covex) cost fucto: The parameters of all thermal uts are take from [23] ad the maxmum power demad of the system set at MD = MW. The results are compared wth λ-terato ad Bary GA [23] RGA [23] ad ABC [24] for ths system. The summarzed results of test case 1 for dfferet demads wthout loss for the QP ad GAMS algorthm are lsted Table: 1 ad the comparatve results are provded Table 2 whch shows that QP ad GAMS both provdes superor result the earler reported results; But GAMS provdes much better result the QP Test case 2 The system cossts of twety geeratg uts havg quadratc cost fucto wth geeratg ad trasmsso loss coeffcet ad power demad s set at 2500 MW. The parameters of all thermal uts ad loss coeffcet are take from [25].The results are compared wth λ-terato ad Hopfeld Model [25] BBO [10] ad SA [26] methods for ths system. The results obtaed by quadratc programmg approach ad GAMS are lsted Table: 3. It ca be clearly see from Table: 3 the proposed GAMS provdes better results as compared to other reported evolutoary algorthm techques lke λ-terato Hopfeld Model BBO ad SA Test case 3 A 40 ut practcal ED system of Tawa Power Compay (TPC) s employed as ths example uses quadratc (covex) ut cost fuctos. The put data of the etre system are gve [27]. I ths case there are two dfferet load demads 9000 MW ad MW wthout trasmsso losses are cosdered. The JIC emal for cotrbuto: edtor@jc.org.uk

6 Joural of Iformato ad Computg Scece Vol. 7 (2012) No.3 pp results are compared wth VSDE [27] ad SA [26] methods for ths system. The results obtaed by quadratc programmg approach ad GAMS are lsted Table: 4. TABLE 1: RESULT OF 18 UNIT SYSTEM (MD= MW) TABLE 2: COMPARISION OF RESULT OF 18 UNIT SYSTEM 5.4. Test case 4 A large scale system cosstg of 110 geeratg uts system s employed ths example uses quadratc (covex) ut cost fuctos wthout losses.the put data of the etre system s take from [28]. To vestgate the robustess of the large system here there are three dfferet low medum ad hgh power demad of MW MW ad MW are cosdered. The results are compared wth Aalytcal approach [29] SA [30] SAB [30] SAF [30] ad RQEF [31] methods for ths system. The results obtaed by quadratc programmg approach ad GAMS are lsted Table: 5 ad the comparatve results are provded Table: 6 whch shows that QP ad GAMS both provdes better results as compared to other reported evolutoary algorthm techques lke Aalytcal approach SA SAB SAF ad RQEA But GAMS provdes much better result the QP. JIC emal for subscrpto: publshg@wau.org.uk

7 206 Devedra Bse et al: Soluto of Large Scale Ecoomc Load Dspatch Problem usg Quadratc Programmg ad GAMS: A Comparatve Aalyss TABLE 3: COMPARISION OF RESULT OF 20 UNIT SYSTEM (PD=2500 MW) JIC emal for cotrbuto: edtor@jc.org.uk

8 Joural of Iformato ad Computg Scece Vol. 7 (2012) No.3 pp TABLE 4: BEST POWER OUTPUT FOR FORTY GENERATING UNITS SYSTEM JIC emal for subscrpto: publshg@wau.org.uk

9 208 Devedra Bse et al: Soluto of Large Scale Ecoomc Load Dspatch Problem usg Quadratc Programmg ad GAMS: A Comparatve Aalyss TABLE 5: BEST POWER OUTPUT FOR 110 GENERATING UNITS SYSTEM JIC emal for cotrbuto: edtor@jc.org.uk

10 Joural of Iformato ad Computg Scece Vol. 7 (2012) No.3 pp TABLE6: COMPARISION OF RESULTS FOR 110 UNITS SYSTEM (Cost ($/h)) 6. Cocluso I ths paper Quadratc Programmg (QP) ad Geeral Algebrac Modelg System (GAMS) for optmzato have bee used for solvg four practcal power dspatch problems. Case I cosstg Crete Islad system 18 geeratg uts wth quadratc cost characterstcs wthout trasmsso loss whch s vestgated by chage percetage of maxmum demad ad comparso s made wth λ-terato Bary GA RGA ad ABC. Based o the smulated results performace comparso amog four above lsted dfferet methods we ca say that QP ad GAMS provdes superor result tha prevously reported methods. I Case II the system cossts of twety geeratg uts havg quadratc cost fucto wth loss coeffcet ad obtaed result s compared wth λ-terato Hopfeld Model RGA ad ABC algorthms. JIC emal for subscrpto: publshg@wau.org.uk

11 210 Devedra Bse et al: Soluto of Large Scale Ecoomc Load Dspatch Problem usg Quadratc Programmg ad GAMS: A Comparatve Aalyss I Case III a 40 geeratg uts data of practcal ELD system of Tawa Power Compay (TPC) s employed as example uses quadratc (covex) ut cost fuctos whch s vestgated o two dfferet load demad ad comparso s made wth VSDE ad SA reported lterature. Ad fally Case IV a large scale system cossts of 110 ut geeratg uts s employed uses quadratc (covex) ut cost fuctos wthout losses to vestgate the robustess of algorthm. Ths vestgated o three dfferet hgh medum ad low power demads. Ad compared to those obtaed wth RGA ad SGA ad Hybrd GA reported lterature. The comparso shows that GAMS performs better the above metoed methods. The GAMS algorthm has superor features cludg qualty of soluto ad good computatoal effcecy. Therefore ths results shows that GAMS s a promsg techque for solvg complcated problems power system. Ackowledgemet The authors are thakful to Drector Madhav Isttute of Techology & Scece Gwalor (M.P) Ida for provdg support ad facltes to carry out ths research work. 7. Refereces [1] B. H. Choudhary ad S. Rahma A revew of recet advaces ecoomc dspatch IEEE Tras Power Sys. Vol. 5 No. 4 pp [2] H. H. Happ Optmal power dspatches a comprehesve survey IEEE Tras. Power Apparatus Syst. PAS-96 pp [3] A. J. Wood ad B. F. Wolleberg Power Geerato Operato ad Cotrol Wley. New York 2d ed [4] J. H. Park Y. S. Km I. K. Eom ad K. Y. Lee Ecoomc Load Dspatch for prcewse Quadratc Cost Fucto Usg Hopfeld Neural IEEE Tras. o Power Systems Vol. 8 No. 3 pp [5] K.Y. Lee Fuel cost mmzato for both real ad reactve power dspatches IEE Proc C Ge. Trasm. Dstrb. Vol. 131 No. 3 pp [6] H.T. Yag P.C. Yag ad C.L. Huag Evolutoary Programmg based ecoomc dspatch for uts wth osmooth fuel cost fuctos IEEE Tras. Power Syst. Vol. 11 No. 1 pp [7] D. C. Walter ad G. B. Sheble Geetc algorthm soluto of ecoomc dspatch wth valve-pot loadg IEEE Tras. Power Syst. Vol. 8 No. 3 pp [8] L. S. Coelho ad V. C. Mara Combg of chaotc dfferetal evoluto ad quadratc programmg for ecoomc dspatch optmzato wth valve-pot effect IEEE Tras. Power Syst. Vol. 21 No.2 pp [9] J. B. Park K. S. Lee J. R. Sh ad K.Y. Lee A partcle swarm optmzato for ecoomc dspatch wth osmooth cost fuctos IEEE Tras. Power Syst. Vol. 20 No. 1 pp [10] A. Bhattacharya ad P. K. Chattopadhyay Bogeography-Based optmzato for dfferet ecoomc load dspatch problems IEEE Tras. Power Syst. Vol. 25 No. 2 pp [11] C. Jej M. Xaoqa L. Lxag ad P. Hapeg Chaotc partcle swarm optmzato for ecoomc dspatch cosderg the geerator costrats Eergy Coverse Maage Vol. 48 pp [12] A. Immauel Selvakumar ad K. Thaushkod A ew partcle swarm optmzato soluto to o-covex ecoomc dspatch problem IEEE Tras Power Syst. Vol. 22 No. 1 pp [13] K. T. Chaturved M. Padt ad L. Srvastava Self-Orgazg Herarchcal Partcle Swarm Optmzato for No-Covex Ecoomc Dspatch IEEE Tras. Power Syst. Vol. 23 No. 3 pp [14] B. K. Pagrah ad V. R. Pad Bacteral foragg optmzato elder mead hybrd algorthm for ecoomc load dspatch IET Geer. Trasm. Dstrb. Vol. 2 No. 4 pp [15] G. Joh Vlachogas ad K. Y. Lee Ecoomc load dspatch a comparatve study o heurstc optmzato techques wth a mproved coordated aggregato-based PSO IEEE Tras. Power Syst. Vol. 24 No. 2 pp. JIC emal for cotrbuto: edtor@jc.org.uk

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