ECONOMIC DISPATCH SOLUTION USING A REAL-CODED GENETIC ALGORITHM

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1 Acta Electrotechnca et Informatca o. 4, Vol. 5, 005 ECOOMIC DISATCH SOLUTIO USI A REAL-CODED GEETIC ALGORITHM * Hamd BOUZEBOUDJA, ** Abdelkader CHAKER, *** Ahmed ALLALI, **** Bakhta AAMA * Electrotechncal Department, Faculty of Electrcal Engneerng, USTO, B. 505 El M naouar, Oran, Algera, Tel.: , E-mal: hbouzeboudja@yahoo.fr ** Laboratory of Electrcal etwork, E..S.E.T, E..S.E.T, B. 74 El M'naouar, Oran, Algera, Tel.: , E-mal: chaker@ecole.enset-oran.dz *** Electrotechncal Depatment, Faculty of Electrcal Engneerng, USTO B. 505 El M naouar, Oran, Algera, Tel.: , E-mal: allal@unv-usto.dz **** Electrotechncal Department, Faculty of Electrcal Engneerng Unversty Djllal Labes, Unversty Djllal labes, B. 98 Sd-bel-abbes, Algera, Tel.: , E-mal: naamasabah@yahoo.fr SUMMARY In ths paper, an effcent and practcal real-coded etc algorthms (RCGAs) has been proposed for solvng the economc dspatch problem. The objectve s to mnmze the total eraton fuel cost and keep the power flows wthn the securty lmts. For each problem of mzaton n etc algorthms(gas) there are a large number of possble encodngs. Although bnary representaton s usually appled to power msaton problems, n ths letter we use a RCGAs, wtch s a modfed GAs employng real valued vectors for representaton of the chromosomes. The use of real valued representaton n the GAs has a number of advantages n numercal functon mzaton over bnary encodng []. The effcency of the GAs s ncreased as there s no need to convert chromosomes to the bnary type, less memory s requred, there s no loss n precson by dscretzaton to bnary or other values, and there s greater freedom to use dfferent etc operators. A non-unform arthmetc crossover operator was ntroduced nto the RCGAs [][]. We used a non-unform arthmetc crossover operator produces a complmentary par of lnear combnatons produced from random proportons of the parents. The non-unform mutaton operator s used to nject new etc materal nto the populaton and t s appled to each new structure ndvdually[]. A gven mutaton nvolves randomly alterng each e wth a small probablty. The proposed technque mproves the qualty of the soluton and speed of converce of the algorthm. The RCGAs developed s compared wth a bnary-coded etc algorthm (BCGAs) and classcal msaton technque of Quas- ewton (Broyden-Fletcher-Goldfarb-Shanno) [3]. The proposed approach has been tested on the IEEE 5-bus system [4]. Keywords: economc dspatch, real-coded etc algorthm, bnary-coded etc algorthm, non-unform arthmetc crossover. ITRODUCTIO The economc dspatch (ED) problem s one of the most mportant operatonal functons of the modern clay energy management system. The purpose of the ED s to fnd the mum eraton among the exstng unts, such that the total eraton cost s mnmzed whle smultaneously satsfyng the power balance equatons and varous other constrants n the system. The lterature of the ED problem and ts soluton methods are surveyed n [5] and [6]. However, t s realzed that the conventonal technques become very complcated when dealng wth ncreasngly complex dspatch problems, and are further lmted by ther lack of robustness and effcency n a number of practcal applcatons. Recently, a global mzaton technque known as GAs whch s a knd of the probablstc heurstc algorthm has been studed to solve the power mzaton problems. The GAs may fnd the several sub-mum solutons wthn a realstc computaton tme. The effcency and the robustness of the proposed RCGAs are demonstrated by test functons. Then the RCGAs wth smulated nonunform arthmetc crossover, eltsm and a nonunform mutaton are appled to ED problem. In order to nvestgate ts performance, the RCGAs developed s compared wth a BCGAs prevously employed and classcal msaton technque of Quas-ewton (Broyden-Fletcher-Goldfarb-Shanno) [3] The proposed approach has been tested on the IEEE 5-bus system [4].. ROBLEM FORMULATIO The ED problem may be expressed by mnmzng the fuel cost of erator unts under constrants. Dependng on load varatons, the output of erators has to be changed to meet the balance between loads and eraton of a power system. The power system model conssts of n eratng unts already connected to the system. The ED problem can be expressed as: Mn F ( ) () ( ) ( a + b c ) F = + () ISS Faculty of Electrcal Engneerng and Informatcs, Techncal Unversty of Košce, Slovak Republc

2 Economc Dspatch Soluton Usng a Real-Coded Genetc Algorthm Where a, b and c are the cost coeffcents of the -th erator and s the number of erators ncludng the slack bus. s the real power output of the -th erator. F ( ) s the operatng cost of unt ( $/h). Subjects to the followng constrants: mn Where D L = L max mn B max for,, (3) D L B = 0 (4) + j Gj j= B 0 + B : total demand : transmsson losses : maxmum eraton output of the -th erator : mnmum eraton output of the -th erator : coeffcents of transmsson losses 00 (5) GAs s a eral stochastc mzaton algorthm that was orgnally developed for solvng unconstraned problems. By applyng an exteror penalty functon we transform a constraned non-lnear ED problem nto an unconstraned problem. We can rewrte the problem shown n () as Fm ( G, rk ) = F ( ) + B. h (6) r Where the value of the penalty coeffcent r k s checked at each teraton. The constant B s defned as B > 0 B = 0 f f k h 0 h = 0 h s the equalty constraned defned as h = D L 3. GEETIC ALGORITHMS (7) (8) GAs are well-known stochastc methods of global mzaton based on the evoluton theory of Darwn [7]. They have successfully been appled n dfferent real-world applcatons. GAs was orgnally developed for solvng unconstraned problems. Recently, many varants of GAs have been developed for solvng constraned nonlnear programmng. Most of these methods were based on penalty formulatons that transform () nto an unconstraned functon F m ( G,r k ) (6), consstng of a sum of the objectve and the constrants weghted by penaltes, and use GAs to mnmze F m ( G,r k ). GAs, unlke strct mathematcal methods, have the apparent ablty to adapt to nonlneartes and dscontnutes commonly found n power systems [8]. The basc dea behnd GAs s to mathematcally mtate the evoluton process of nature. The algorthms are based on the evaluaton of a set of solutons, called populaton. The populaton s treated wth etc operatons. At the teraton the populaton X conssts of a number of ndvduals x j, that s, solutons, where s called a populaton sze. The populaton s ntalzed by randomly erated ndvduals. The ndvduals can be encoded usng ether bnary or real numbers. We use the latter because of ther popularty. Each ndvdual xj = (x,,x n ) s a vector of varables. Each varable s a real number. The sutablty of an ndvdual s determned by the value of the objectve functon, to be called a ftness functon. A new populaton s erated by the etc operatons selecton, crossover and mutaton. arents are chosen by selecton and new offsprngs are produced wth crossover and mutaton. All these operatons nclude randomness. The success of the mzaton process s mproved by eltsm where the best ndvduals of the old populaton are coped as such to the next populaton. 4. REAL-CODED GEETIC ALGORITHM For each problem of mzaton n GAs there are a large number of possble encodngs. Although bnary representaton s usually appled to power msaton problems, n ths letter we use a RCGAs, wtch s a modfed GAs employng real valued vectors for representaton of the chromosomes. The use of real valued representaton n the GAs has a number of advantages n numercal functon mzaton over bnary encodng [][]. The effcency of the RCGAs s ncreased as there s no need to convert chromosomes to the bnary type, less memory s requred, there s no loss n precson by dscretzaton to bnary or other values, and there s greater freedom to use dfferent etc operators. For the real valued representaton, the k-th chromosome C k can be defned as follows [][][9]: C k =[ k, k,, kn ] k=,,,popsze (9) where popsze means populaton sze and k s the eraton power of the -th unt at k-th chromosome. Reproducton nvolves the creaton of new offsprng from the matng of two selected parents or matng pars. It s thought that the crossover operator s manly responsble for the global search property of the GA. A non-unform ISS Faculty of Electrcal Engneerng and Informatcs, Techncal Unversty of Košce, Slovak Republc

3 Acta Electrotechnca et Informatca o. 4, Vol. 5, arthmetc crossover operator was ntroduced nto the RCGAs [][][9]. We used a non-unform arthmetc crossover operator produces a complmentary par of lnear combnatons produced from random proportons of the parents. The heurstc crossover operator produces a chld that s a lnear extrapolaton away from the better parent along the drecton of the vector jonng the two parents. Two chromosomes, selected randomly for crossover, C and C j, may produce two offsprng, C + and C j +, whch s a lnear combnaton of ther parents,.e., C + = a. C + ( -a). C j C j + = ( -a). C + a. C j where a s a random number n range of [0,]. (0) () The non-unform mutaton operator s used to nject new etc materal nto the populaton and t s appled to each new structure ndvdually[]. A gven mutaton nvolves randomly alterng each e wth a small probablty. Let us suppose C = (c,, c,, c n ) a chromosome and c Є [up, low ] a e to be mutated. ext, the e, c ', resultng from the applcaton of nonunform mutaton []. The e c ' can be defned as follows: c ' = C + mut_scale. d. randn () Where: d = up low mut_scale s normally set to 0. randn s ormally dstrbuted random numbers. An eltst GAs search s used and guarantees that the best soluton obtaned so far n the search s retaned and used n the followng eraton, and thereby ensurng no good soluton already found can be lost n the search process. The selecton s based on the cost of parent vectors F m ( ) wth the correspondng cost of offsprng vectors F m ( ) n ths populaton. The best / vector havng mnmum cost, whether parent vector p or offsprng vector / s selected for the new parent for the next eraton. An non-unform arthmetc crossover operator s used. After crossover s completed, non-unform mutaton s performed. In the mutaton step, a random real value makes a random change n the m- th element of the chromosome. After mutaton, all constrants are checked whether volated or not. If the soluton has at least one constrant volated, a new random real value s used for fndng a new value of the m-th element of the chromosome. Then, the best soluton so far obtaned n the search s retaned and used n the followng eraton. The RCGAs process repeats untl the specfed maxmum number of eratons s reached. The flowchart of RCGAs s shown n Fg.. Generate Intal populaton Generaton Maxmum eratons? o Evaluaton of Ftness Functon ut Into a Fle Yes Choose Good Combnaton Yes Check Constrans o 5. ECOOMIC DISATCH USI GEETIC ALGORITHM Reproducton Stop RCGAs s a probablstc search technque, whch erates the ntal parent vectors dstrbuted unformly n ntervals wthn the lmts and obtans global mum soluton over number of teratons. The mplementaton of RCGAs s gven below. The ntal populaton s erated after satsfyng the equaton (3). The elements of parent vectors ( ) are the real power outputs of eratng unts dstrbuted unformly between ther mnmum and maxmum lmts. The ftness functon s used to transform the cost functon value nto a measure of relatve ftness. The ftness functon s gven n equaton (6). on-unform Arthmetc on-unform Mutaton Generaton = Generaton + Fg. Flowchart of RCGAs ISS Faculty of Electrcal Engneerng and Informatcs, Techncal Unversty of Košce, Slovak Republc

4 4 Economc Dspatch Soluton Usng a Real-Coded Genetc Algorthm 6. SIMULATIO RESULTS The proposed approach s tested on the IEEE 5- bus system[4]. The cost functons n dollars per hour were as follows: F ( G ) = G +.8 G + 40 F ( G ) = G +.7 G + 60 F 3 ( G3 ) = 0.00 G3 +. G3 +00 F 4 ( G4 ) = G4 +.0 G4 + 5 F 5 ( G5 ) = G5 +.9 G5 + 0 ower eraton lmts: 00 G G G G G5 300 The system load D was 730 MW. Transmsson losses L are computed usng the B coeffcents. The proposed method was mplemented n Matlab 5.3 wth -III 73MHz system. The parameter settngs to execute RCGAs are: opulaton sze pop = 30 umber of teratons max = 300 robablty of crossover c = 0.8 robablty of mutaton m = 0.08 The mnmum cost and actve power eratons are presented n Tab.. G G G3 G4 G5 cost ($/h) Tme (sec) In Tab., the results of proposed method (RCGAs) are compared wth the results of BCGAs and BFGS. It s seen that there s a neglgble dfference n the mal values of cost between the RCGAs and BCGAs. The BFGS method produced a hgher operaton cost than other methods. The RCGAs demonstrated faster converce then BCGAs. The total computatonal tme of the RCGAs s far less than for the BCGAs. G G G3 G4 G5 RCGAs BCGAs BFGS cost ($/h) tme (sec) Tab. Results of RCGAs compared wth BCGAs and BFGS methods Tab. Results of RCGAs Fg. shows the eraton cost evoluton durng the teratve procedure. 7. COCLUSIO In ths paper a new RCGAs has been presented and compared wth a BCGAs and classcal msaton technque of BFGS, the proposed method has been appled to the economc dspatch problem. The proposed technque mproves the qualty of the soluton and reduce the computaton tme. REFERECES Fg. The eraton cost evoluton durng the teratve procedure [] Z. Mchalewcz: Genetc Algorthms + Data Structures = Evoluton rograms, nd ed, Berln, Sprnger Verlag, 994. [] Z. Mchalewcz: Genetc Algorthms + Data Structures = Evoluton rograms, ew York, Sprnger Verlag, 99. [4] R. B. Gungor,. F. Tsang, B. Webb: A technque for mzng real and reactve ISS Faculty of Electrcal Engneerng and Informatcs, Techncal Unversty of Košce, Slovak Republc

5 Acta Electrotechnca et Informatca o. 4, Vol. 5, power schedules, IEEE Trans on as 90, p , 97. [5] H.H. Happ: Optmal power dspatch A comprehensve survey, IEEE Trans. ower App. Syst., vol. AS-96, pp , 977. [6] B. H. Chowdhury and S. Rahman: A revew of recent advances n economc dspatch, IEEE Trans. on ower Systems, vol. 5, pp , ovember 990. [7] D. E. Goldberg : AG exploraton. Optmsaton et apprentssage automatque, Edton Addson Wesley France 99. [8] Benjamn W. Wah, Y-Xn Chen: Constraned Genetc Algorthms and ther Applcatons n on-lnear Constraned Optmzaton, roc. th Internatonal Conference on Tools Artfcal ntellce, ovember 000. [9] Francsco Herrera, Manuel Lozano: Gradual Dstrbuted Real-Coded Genetc Algorthms, IEEE Transactons on Evolutonary computaton, Vol. 4,., Aprl 000. BIOGRAHY Hamd Bouzeboudja was born on In 993 he graduated at the Electrotechncal Department of the Faculty of Electrcal Engneerng at Unversty (USTO) n Algera. He defended hs Magster n the feld of mal power flow problems n 996; hs thess ttle was "Optmal ower Flow". Hs scentfc research s focusng an practcal methods based on etc algorthms for solvng the economc dspatch problem of complex systems. Abdelkader Chaker s a rofessor n the Department of Electrcal Engneerng at the ESET, n Oran Algera. He receved a h.d. degree n Engneerng Systems from the Unversty of Sant-etersburg. Hs research actvtes nclude the control of large power systems, multmachne multconverter systems, the unfed power flow controller. Hs teachng ncludes neural process control and real tme smulaton of power systems. Ahmed Allal was born on In 987 he graduated at the Electrotechncal Department of the Faculty of Electrcal Engneerng at Unversty (USTO) n Algera. He defended hs Magster. n the feld of mal power flow problems n 990; hs thess ttle was "Optmal Dstrbuton of Actve owers Usng Lnear rogrammng wth Losses Cost Mnmzaton". Hs scentfc research s focusng an control and real tme smulaton of power systems, and study of the Dynamc stablty of the networks electrcal supply. Bakhta aama was born on In 00 she graduated at the Department of Electrotechncs of the Faculty of Electrcal Engneerng at Unversty Djllal Labes n Algera. She defended her Magster. n the feld of mal power flow problems n 004; hs thess ttle was "Optmal ower flow usng etc algorthms". Hs research actvtes focusng an practcal methods based on etc algorthms for large scale power system. ISS Faculty of Electrcal Engneerng and Informatcs, Techncal Unversty of Košce, Slovak Republc