Analysis of Economic Load Dispatch Using Genetic Algorithm

Transcription

1 Internatonal Journal of Applcaton or Innovaton n Engneerng & Management (IJAIEM) Web Ste: Emal: edtor@aem.org Analyss of Economc Load Dspatch Usng Genetc Algorm Arunpreet Kaur 1, Harnder al Sngh 2 and Abhshek Bhardwa 3 1 Assstant rofessor, Electrcal Engneerng Department, BUEST, Badd , H.., Inda 2 Assstant rofessor, Electrcal Engneerng, SBSSTC, Ferozpur , unab, Inda 3 Assstant rofessor, Electrcal Engneerng Department, BUEST, Badd , H.., Inda ABSTRACT Genetc algorm s a versatle meod n e feld of optmzaton over e past few years. Ths paper presents an economc load dspatch problem of ermal generator usng Genetc Algorm (GA) meod. The proposed meod has been appled on 3 generator system and 6 generator systems. The system consdered here s e lossless system. The results obtaned show a sgnfcant mprovement n generator fuel cost whle at e same tme satsfyng varous equalty and nequalty constrants. Keywords: Economc Load Dspatch, Genetc Algorm. 1. INTRODUCTION Electrcal power systems are desgned and operated to meet e contnuous varaton of power demand. In power system mnmzng, e operaton cost s very mportant. Economc Load Dspatch (ELD) s a meod to schedule e power generator outputs w respect to e load demands, and to operate e power system most economcally, or n oer words, we can say at man obectve of economc load dspatch s to allocate e optmal power generaton from dfferent unts at e lowest cost possble whle meetng all system constrants Over e years, many efforts have been made to solve e ELD problem, ncorporatng dfferent knds of constrants or multple obectves rough varous maematcal programmng and optmzaton technques. The conventonal meods nclude Newton- Raphson meod, Lambda Iteraton meod, Base ont and artcpaton Factor meod, Gradent meod, etc [1]. However, ese classcal dspatch algorms requre e ncremental cost curves to be monotoncally ncreasng or pece-wse lnear [2]. The nput/output characterstcs of modern unts are nherently hghly nonlnear (w valve-pont effect, rate lmts etc) and havng multple local mnmum ponts n e cost functon. Ther characterstcs are approxmated to meet e requrements of classcal dspatch algorms leadng to suboptmal solutons and erefore, resultng n huge revenue loss over e tme. Consderaton of hghly nonlnear characterstcs of e unts requres hghly robust algorms to avod gettng stuck at local optma [3]. The classcal calculus based technques fal n solvng ese types of problems. In s respect, stochastc search algorms lke genetc algorm (GA) [4]-[9], evolutonary strategy (ES) [10]-[12], evolutonary programmng (E) [13], partcle swarm optmzaton (SO) [14] and smulated annealng (SA) may prove to be very effcent n solvng hghly nonlnear ELD problem wout any restrctons on e shape of e cost curves. Alough ese heurstc meods do not always guarantee e global optmal soluton, ey generally provde a fast and reasonable soluton (sub optmal or near global optmal). 2. ECONOMIC LOAD DISATCH 2.1 Economc load dspatch The Economc Dspatch can be defned as e process of allocatng generaton levels to e generatng unts, so at e system load s suppled entrely and most economcally. For an nterconnected system, t s necessary to mnmze e expenses. The economc load dspatch s used to defne e producton level of each plant, so at e total cost of generaton and transmsson s mnmum for a prescrbed schedule of load. The obectve of economc load dspatch s to mnmse e overall cost of generaton 2.2 Generator Operatng Cost The total cost of operaton ncludes e fuel cost, cost of labour, supples and mantenance. Generally, costs of labour, supples and mantenance are fxed percentages of ncomng fuel costs. The power output of fossl plants s ncreased sequentally by openng a set of valves to ts steam turbne at e nlet. The rottlng losses are large when a valve s ust opened and small when t s fully opened. Fgure 1 Smple model of a fossl plant Volume 3, Issue 3, March 2014 age 240

2 Internatonal Journal of Applcaton or Innovaton n Engneerng & Management (IJAIEM) Web Ste: Emal: edtor@aem.org Fgure 1.shows e smple model of a fossl plant dspatchng purposes. The cost s usually approxmated by one or more quadratc segments. The operatng cost of e plant has e form shown n Fgure 2. Output ower (MW) mn g g Fgure 2 Operatng costs of a fossl fred generator The fuel cost curve may have a number of dscontnutes. The dscontnutes occur when e output power s extended by usng addtonal bolers, steam condensers, or oer equpment. They may also appear f e cost represents e operaton of an entre power staton, and hence cost has dscontnutes on parallelng of generators. Wn e contnuty range e ncremental fuel cost may be expressed by a number of short lne segments or pece-wse lnearzaton. The mn g s e mnmum loadng lmt below whch, operatng e unt proves to be uneconomcal (or may be techncally nfeasble) and g s e mum output lmt. 3. FORMULATION OF ELD ROBLEM 3.1 Obectve Functon The obectve of e economc dspatch problem s to mnmze e total fuel cost of ermal power unts subected to e equalty and nequalty constrants of a power system. The smplfed cost functon of each generator can be represented as a quadratc functon as gven n (2). NG F( g ) F ( g ) 1 (1) 2 F ( ) a b c Rs hr (2) g g g / Where a, b, c are cost coeffcents for unt, F ) s e total cost of generaton, g s e generaton of ( g plant. 3.2 Equalty and Inequalty Constrants Actve power balance equaton For power balance, an equalty constrant should be satsfed. The total generated power should be e same as total load demand plus e total transmsson lne loss. NG 1 g D loss where D s e total load demand and s e total lne loss Mnmum and mum power lmts Generaton output of each generator should be le between mum and mnmum lmts. The correspondng nequalty constrants for each generator are Where generaton ( 1,2,...., NG ) (4) mn s e lower permssble lmt of real power generaton, g (3) g s e upper permssble lmt of real power Volume 3, Issue 3, March 2014 age 241

3 Internatonal Journal of Applcaton or Innovaton n Engneerng & Management (IJAIEM) Web Ste: Emal: edtor@aem.org 4. GENETIC ALGORITHM A Genetc Algorm (GA) s a search technque used n computng to fnd exact or approxmate solutons to optmzaton and search problems. Genetc algorms are categorzed as global search heurstcs. Genetc algorms are a partcular class of Evolutonary Algorms (EA) at use technques nspred by evolutonary bology such as nhertance, mutaton, selecton, and crossover. Genetc algorms are mplemented n a computer smulaton n whch a populaton of abstract representatons (called chromosomes or e genotype of e genome) of canddate solutons (called ndvduals, creatures, or phenotypes) to an optmzaton problem evolves toward better solutons. Tradtonally, solutons are represented n bnary as strngs of 0s and 1s, but oer encodngs are also possble. The evoluton usually starts from a populaton of randomly generated ndvduals and happens n generatons. In each generaton, e ftness of every ndvdual n e populaton s evaluated, multple ndvduals are stochastcally selected from e current populaton (based on er ftness), and modfed (recombned and possbly randomly mutated) to form a new populaton. The new populaton s en used n e next teraton of e algorm. Commonly, e algorm termnates when eer a mum number of generatons has been produced, or a satsfactory ftness level has been reached for e populaton. If e algorm has termnated due to a mum number of generatons, a satsfactory soluton may or may not have been reached. Once we have e genetc representaton and e ftness functon defned, GA proceeds to ntalze a populaton of solutons randomly, and en mprove t rough repettve applcaton of mutaton, crossover, nverson, and selecton operators. 4.1 Representaton Genetc Algorms are derved from a study of bologcal systems. In bologcal systems evoluton takes place on organc devces used to encode e structure of lvng bengs. These organc devces are known as chromosomes. A lvng beng s only a decoded structure of e chromosomes. Natural selecton s e lnk between chromosomes and e performance of er decoded structures. In GA, e desgn varables or features at characterze an ndvdual are represented n an ordered lst called a strng. Each desgn varable corresponds to a gene and e strng of genes corresponds to a chromosome. Chromosomes are made of dscrete unts called genes. 4.2 Encodng Normally, a chromosome corresponds to a unque soluton x n e soluton space. Ths requres a mappng mechansm between e soluton space and e chromosomes. Ths mappng s called an encodng. In fact, GA works on e encodng of a problem, not on e problem tself. The applcaton of a genetc algorm to a problem starts w e encodng. The encodng specfes a mappng at transforms a possble soluton to e problem nto a structure contanng a collecton of decson varables at are relevant to e problem. 4.3 Decodng Decodng s e process of converson of e bnary structure of e chromosomes nto decmal equvalents of e feature values. Usually s process s done after de-catenaton of e entre chromosome to ndvdual chromosomes. The decoded feature values are used to compute e problem characterstcs lke e obectve functon, ftness values, constrant volaton and system statstcal characterstcs lke varance, standard devaton and rate of convergence. The stages of selecton, crossover, mutaton etc are repeated tll some termnaton condton s reached. The equvalent decmal nteger of bnary strng s obtaned as Where y b s e l 1 2 b ( 1,2,..., L) 1 (5) bnary dgt of e strng, l s e leng of e strng, L s e number of strngs or populaton sze. The contnuous varable can be obtaned to represent a pont n e search space accordng to a fxed mappng rule,.e. mn mn y ( 1,2,..., L) (6) l 2 1 mn Where s e mnmum number of varable, s e mum value of varable, y s e bnary coded value of e strng 4.4 Intalzaton Intally many ndvdual solutons are randomly generated to form an ntal populaton. The populaton sze depends on e nature of e problem, but typcally contans several hundreds or ousands of possble solutons. Tradtonally, e populaton s generated randomly, coverng e entre range of possble solutons (e search space). Occasonally, e solutons may be "seeded" n areas where optmal solutons are lkely to be found. 4.5 Evaluaton Sutablty of e solutons s determned from e ntal set of soluton of e problem. For s sutablty determnaton, we use a functon called ftness functon. Ths functon s derved from e obectve functon and used n successve genetc operaton. The evaluaton functon s a procedure for establshng e ftness of each chromosome n e populaton and s very much applcaton orentated. Snce Genetc Algorms proceed n e drecton of evolvng e Volume 3, Issue 3, March 2014 age 242

4 Internatonal Journal of Applcaton or Innovaton n Engneerng & Management (IJAIEM) Web Ste: Emal: edtor@aem.org fttest chromosomes, and e performance s hghly senstve to e ftness values. In e case of optmzaton routnes, e ftness s e value of e obectve functon to be optmzed. enalty functons can also be ncorporated nto e obectve functon, n order to acheve a constraned problem. 4.6 Ftness Functon The Genetc algorm s based on Darwn s prncple at The canddates, whch can survve, wll lve, oers would de. Ths prncpal s used to fnd ftness value of e process for solvng mzaton problems. Mnmzaton problems are usually transferred nto mzaton problems usng some sutable transformatons. Ftness value f (x) s derved from e obectve functon and s used n successve genetc operatons. The ftness functon for mzaton problem can be used e same as obectve functon F (X ) The ftness functon for e mzaton problem s: f ( x) F ( X ) (7) For mnmzaton problems, e ftness functon s an equvalent mzaton problem chosen such at e optmum pont remans unchanged. The followng ftness functon s often used n mnmzaton problems: F( X ) 1/(1 f ( x)) (8) Here f (x) s ftness functon and F (X ) s obectve functon. 4.7 Selecton Durng each successve generaton, a proporton of e exstng populaton s selected to breed a new generaton. Indvdual solutons are selected rough a ftness-based process, where ftter solutons (as measured by a ftness functon) are typcally more lkely to be selected. Certan selecton meods rate e ftness of each soluton and preferentally select e best solutons. Oer meods rate only a random sample of e populaton, as s process may be very tmeconsumng. Most functons are stochastc and desgned so at a small proporton of less ft solutons are selected. Ths helps keep e dversty of e populaton large, preventng premature convergence on poor solutons. opular and well-studed selecton meods nclude roulette wheel selecton and tournament selecton. 4.8 Reproducton The next step s to generate a second generaton populaton of solutons from ose selected rough genetc operators: crossover (also called recombnaton), and/or mutaton. For each new soluton to be produced, a par of "parent" solutons s selected for breedng from e pool selected prevously. By producng a "chld" soluton usng e above meods of crossover and mutaton, a new soluton s created whch typcally shares many of e characterstcs of ts "parents". New parents are selected for each new chld, and e process contnues untl a new populaton of solutons of approprate sze s generated. Alough reproducton meods at are based on e use of two parents are more "bology nspred, some research suggests more an two "parents" are better to be used to reproduce a good qualty chromosome. These processes ultmately result n e next generaton populaton of chromosomes at s dfferent from e ntal generaton. Generally e average ftness wll have ncreased by s procedure for e populaton, snce only e best organsms from e frst generaton are selected for breedng, along w a small proporton of less ft solutons, for reasons already mentoned above. 4.9 Termnaton Ths generatonal process s repeated untl a termnaton condton has been reached. Common termnatng condtons are: 1.) A soluton s found at satsfes mnmum crtera 2.) Fxed number of generatons reached 3.) Allocated budget (computaton tme/money) reached 4.) The hghest rankng soluton's ftness s reachng or has reached a plateau such at successve teratons no longer produce better results 5.) Manual nspecton 6.) Combnatons of e above Flow Chart of Genetc Algorm Volume 3, Issue 3, March 2014 age 243

5 Internatonal Journal of Applcaton or Innovaton n Engneerng & Management (IJAIEM) Web Ste: Emal: edtor@aem.org Fgure 3 Flow Chart of Genetc Algorm 5. ROOSED ALGORITHM The step-wse procedure s outlned below: 1. Read data, namely cost coeffcents, a, b, c, no. of teratons, leng of strng, populaton sze, probablty of mn crossover and mutatons, power demand and and. 2. Create e ntal populaton randomly n e bnary form. 3. Decode e strng, or obtan e decmal nteger from e bnary strng usng Eq. (5) 4. Calculate e power n MW generated from e decoded populaton by usng Eq. (9) mn mn y l ( =1,2,...NG, =1,2,...L) (9) 2 1 Where L s e number of strngs or populaton sze, y s e bnary coded value of e substrng 5. Check If If, en set, en set mn ( f mn 6. Fnd ftness f f ) en set f f and If en set 7. Fnd populaton w mum ftness and average ftness of e populaton. 8. Select e parents for crossover usng stochastc remander roulette wheel selecton meod. 9. erform sngle pont crossover for e selected parents. 10. erform mutaton 11. If e number of teratons reaches e mum, en go to step 12. Oerwse, go to step The ftness at generates e mnmum total generaton cost s e soluton of e problem. 6. NUMERICAL RESULTS AND DISCUSSION The result of ELD after e mplementaton of proposed GA meod s dscussed. The programs are mplemented n MATLAB The performance s evaluated wout consderng losses usng 6 generator and 3 Generator test system. The coeffcents of fuel cost and mum and mnmum power lmts are gven n Table 1 [1], [15]and Table 3[16] respectvely. The power demand s consdered to be 450 (MW) and 850 (MW) respectvely. The descrpton of e results obtaned by utlzng GA wout consderng losses s detaled herew. The leng of strng n e populaton has been taken as 96 and 48 respectvely and e populaton sze has been assumed as 20 and 10 respectvely. The sngle strng represents ree substrngs, each of 16 bts. Therefore, populaton we have taken s 20 x 96 and 10 x 48 respectvely n bnary. The mutaton probablty s taken as 0.4. The crossover and mutaton ponts are chosen randomly. The optmal result of GA s shown below n Table 2 and Table 4. Volume 3, Issue 3, March 2014 age 244

6 Internatonal Journal of Applcaton or Innovaton n Engneerng & Management (IJAIEM) Web Ste: Emal: edtor@aem.org Table 1: Fuel Cost Coeffcents for 3 generator System Unt mn a b c no Table 2: Optmal Result of GA for 3 generators 1 (MW) (MW) (MW) Total ower(mw) Total Cost(Rs/MWh) Table 3: Fuel Cost Coeffcents for 6 generator System Unt mn a b c No Table 4: Optmal Result of GA for 6 generators (MW) (MW) (MW) (MW) (MW) (MW) Total ower(mw) Total Cost(Rs/MWh) CONCLUSION In s Study, Genetc Algorm s successfully mplemented to mnmze e fuel cost of e generators on two dfferent set of problems w two dfferent load demands. It has been observed at genetc algorm s capable of optmzng any knd of problems rrespectve of load demand. The results dscussed above are obtaned after sgnfcant reducton n Fuel Cost of Generators and satsfes each and every constrant. REFERENCES [1] A. J. Wood, B. F. Wollenberg, ower generaton, operaton and control, John Wley & Sons, New York, [2] Ndul Snha, Chakrabar R. and. K. Chattopadhyay, Evolutonary programmng technques for economc load dspatch, IEEE Transactons on evolutonary computaton, Vol. 7, pp.83-94, [3] Ndul Snha, Chakrabart,. K. Chatopadhyay, Improved fast evolutonary program for economc load dspatch w non-smoo cost curves, IE (I) Journal. EL Vol. 85, [4] Hroyuk Mor, Takuya Horguch, Genetc algorm based approach to economc load dspatchng, IEEE Transactons on power systems, Vol. 1, pp , Volume 3, Issue 3, March 2014 age 245

7 Internatonal Journal of Applcaton or Innovaton n Engneerng & Management (IJAIEM) Web Ste: Emal: edtor@aem.org [5] C. L. Chang, Genetc based algorm for power economc load dspatch, IET Gener. Transm. Dstrb, 1, (2), pp , [6] S. H. Lng, H. K. Lam, H. F. Leung and Y. S. Lee, Improved genetc algorm for economc load dspatch w valve- pont loadngs, IEEE Transactons on power systems, Vol. 1, pp , [7] R. Ouddr, M. Rahl and L. Abdelhakem Kord, Economc dspatch usng genetc algorm: applcaton to western algera s electrcal power network, Journal of Informaton Scence and Engneerng 21, pp , [8] Mmoun Younes, Mostafa Rahl, Economc power dspatch usng e combnaton of two genetc algorms, Journal of Electrcal and Electroncs Engneerng, Vol. 6, pp , [9] Chra Achayuakan, S.C.Srvastava, A genetc algorm based economc load dspatch soluton for eastern regon of egat system havng combned cycle and cogeneraton plants, IEEE Transactons on power systems, Vol. 1, pp , [10] Mmoun Younes, Mostefa Rahl, Lahour Abdelhakem Kordak, Economc power dspatch usng Evolutonary Algorm, Journal of Electrcal Engneerng Vol. 57, No. 4, pp , [11] June Ho ark, Sung Oh Yang, Hwa Seok Lee, Young Moon ark, Economc load dspatch usng evolutonary algorms, IEEE Transactons on power systems, Vol. 2, pp , [12] Grdhar Kumaran, V. S. R. K. Mouly, Usng evolutonary computaton to solve e economc load dspatch problem, IEEE Transactons on power systems, Vol. 3, pp , [13] J. H. ark, S. O. Yang, K. J. Mun, H. S. Lee and J. W. Jung, An applcaton of evolutonary computatons to economc load dspatch w pecewse quadratc cost functons, IEEE Transactons on power systems, Vol. 69, pp , [14] R. C. Eberhart and J. Kennedy, A new optmzer usng partcle swarm eory, roceedngs of e 6. Internatonal Symposum on Mcro Machne and Human Scence, Nagoya, Japan, pp , [15] Ah Kng R. T. F. and Rughoopu H. C. S., Eltst Multobectve Evolutonary Algorm for Envronmental/Economc Dspatch, IEEE Congress on Evolutonary Computaton, Canberra, Australa, vol. 2, pp , [16] Dhllon J. S., art S. C. and Koar D.., Mult obectve stochastc optmal ermal power dspatch, bd, pp , Volume 3, Issue 3, March 2014 age 246