Economic Dispatch using a Genetic Algorithm: Application to Western Algeria s Electrical Power Network

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1 JOURNAL OF INFORMATION SCIENCE AND ENGINEERING 2, (2005) Short Paper Ecoomic Dispatch usig a Geetic Algorithm: Applicatio to Wester Algeria s Electrical Power Network Power Systems Optimizatio Laboratory Faculty of Electrical Egieerig Uiversity of Sciece ad Techology of Ora El M aouer, Ora, 3000 Algeria ouiddir@hotmail.com {rahlim, hkoridak}@yahoo.fr A geetic algorithm is used to solve a ecoomic dispatch problem. The chromosome cotais oly the ecodig of a ormalized icremetal cost system. Therefore, the total umber of bits of a chromosome is etirely idepedet of the umber of uits. I the first case, the trasmissio lie losses are calculated usig the Newto-Raphso method ad kept costat. I the secod case, the trasmissio lie losses are cosidered as a liear fuctio of the real geerated power. The coefficiets are calculated usig the Gauss-Seidel method. This method has bee applied to the wester part of the Algeria power etwork, ad the results have bee foud to be satisfactory compared with other results obtaied usig classical methods. Keywords: power trasmissio losses, ecoomic dispatch, geetic algorithm, ormalized icremetal cost system, power systems, miimizatio, optimal load flow. INTRODUCTION I a electrical power system, a cotiuous balace must be maitaied betwee electrical geeratio ad varyig load demad, while the system frequecy, voltage levels, ad security also must be kept costat. Furthermore, it is desirable that the cost of such geeratio be miimal [, 2]. I additio, the divisio of load i the geeratig plat becomes a importat operatio as well as a ecoomic issue which could be solved at every load chage (%) or every 2-3 miutes. Research techiques have bee successfully used to solve optimal load flow problems by usig liear or o liear programmig, but these algorithms are geerally limited to covex regular fuctios. May fuctios are multi-modal, discotiuous ad ot differetiable. Stochastic samplig methods have bee used to optimize these fuctios. Whereas traditioal resolutio techiques use the characteristics of the problem to determie the ext samplig poit (e.g., gradiet, Hessias, liearity ad cotiuity), stochastic resolutio techiques make o such assumptios. Istead, the ext sampled poit is determied Received July 30, 2003; revised November 2, 2003 ad March 25, 2004; accepted August 9, Commuicated by Chi-Teg Li. 659

2 660 based o stochastic samplig or decisio rules rather tha o a set of determiistic decisio rules. Geetic algorithms have bee used to solve difficult problems with objective fuctios that do ot possess properties such as cotiuity, differetiability ad so forth [3-6]. These algorithms maitai ad maipulate a set of solutios ad implemet a survival of the fittest strategy i their search for a better solutio. I our case, a geetic algorithm is used to solve the ecoomic dispatch problem uder some equality ad iequality costraits. The equality costrait reflects a real power balace, ad the iequality costrait reflects the limit of real geeratio. The voltage levels ad security are assumed to be costat i both cases. The proposed approach has bee applied to the wester part of the Algeria power etwork, ad the results have bee judged satisfactory. 2. OBJECTIVE The ecoomic dispatch problem, which is used to miimize the cost of productio of real power, ca geerally be stated as follows: Mi Fi( Pi) () i= Subject to: Pi = D + PL (2) i= P i, mi P i P i, max, (3) where, geerally, Fi(Pi) is a quadratic curve: Here: Fi(Pi) = c i + b i Pi + a i Pi 2 (4) a i, b i ad c i are the kow coefficiets; : umber of geerators; Pi: real power geeratio; D: real power load; P L : real losses. 3. OVERVIEW OF THE GENETIC ALGORITHM Geetic algorithms are resolutio algorithms based o the mechaics of atural selectio ad atural geetics. They combie survival of the fittest amog strig structures

3 ECONOMIC DISPATCH USING A GENETIC ALGORITHM 66 with structured yet radomized iformatio exchage to form a resolutio algorithm with some of ma s capacity for survival. I every geeratio, a ew set of artificial creatures (strigs) is created by usig bits ad pieces from the fittest of the old; a occasioal ew part is used for good measure. While radomized, geetic algorithms are o simple radom walk, they efficietly exploit historical iformatio to speculate o ew research poits with expected improved performace [3, 5]. Geetic algorithms are essetially derived from a simple model of populatio geetics. The three prime operators associated with the geetic algorithm are reproductio, crossover, ad mutatio. Reproductio is a process by which idividual strigs are copied accordig to their fitess values. Copyig strigs accordig to their fitess values meas that strigs with higher values have a higher probability of cotributig oe or more offsprig i the ext geeratio. Crossover is a importat compoet of geetic algorithms, takig two idividuals ad producig two ew idividuals as show i Fig.. Paret A: 00 0 Paret B: 0 0 Child A: 00 0 Child B: 0 0 Fig.. Diagram of simple crossover. Although reproductio ad crossover search ad recombie existig chromosomes, they do ot create ay ew geetic material i the populatio. Mutatio is capable of overcomig this shortcomig. It ivolves the alteratio of oe idividual to produce a sigle ew solutio as show i Fig. 2. Child A: 000 New child A: 0000 Fig. 2. Biary mutatio. Fig. 3 shows the geetic algorithm flow chart used i this study.

4 662 Iitializatio Evaluatio Termiatio Selectio Recombiatio Mutatio Fig. 3. Geeral flow chart used i this study. 4. GENETIC ALGORITHM SOLUTION The ecodig ad decodig techiques, costraied geeratio output calculatio, ad the fitess fuctio are described i more detail below. 4. Ecodig ad Decodig I this paper, the proposed approach uses the λ equal system (equal icremetal cost system) criterio as its basis. λ m is the ormalized icremetal cost system, where 0 λ m. The advatage of usig the λ system is that the umber of bits of a chromosome will be etirely idepedet of the umber of uits. Te bits, however, represet λ m. Fig. 4 shows the ecodig diagram of λ m [, 6]. d d 2 d 3 d 4 d 5 d 6 d 7 d 8 d 9 d 0 x x x x x x x x x x where d i Є {0, }, i =, 2,, 0 Fig. 4. Ecodig diagram of λ m. The decodig of λ m ca be expressed as follows: λ m i = ( dx2 ), (5) where d i Є {0, }, i =, 2,, 0. i

5 ECONOMIC DISPATCH USING A GENETIC ALGORITHM 663 The relatioship betwee the icremetal cost value λ ad the ormalized icremetal cost system λ m is λ = λ mi + λ m (λ max λ mi ), (6) where λ mi ad λ max represet the iitially computed miimum ad maximum values: λ mi dfi( Pi, mi) = mi dpi ad (7) λ max dfi( Pi, max) = max dpi 4.2 Geeratio Output If the Lagrage fuctio methods ad the Kuh-Tucker [6] coditios are applied to the costraied optimizatio, the ecoomic dispatch problem ca be reformulated as follows: λ L i= i= (8) L( P, λ) = Fi( Pi) + ( D + P Pi), which, after some rearragemet of terms, becomes L( P, λ) = Fi( Pi) λ( Pi PL) + λ( D), i= i= (9) PF i (2a i P i + b i ) = λ for P i, mi P i P i, max PF i (2a i P i + b i ) λ for P i = P i, max (0) PF i (2a i P i + b i ) λ for P i = P i, mi where PF i is the pealty factor of uit i, give by PFi =. () P Pi L 4.3 Fitess Fuctio The fitess fuctio for the miimizatio problem is geerally give as the iverse of the objective fuctio. I this paper, the fitess fuctio is give by the relatio Fit =. (2) + Fi

6 Parameter Selectio The geetic algorithm has a umber of parameters that must be selected. These iclude populatio size, crossover, ad mutatio probability: populatio size = 0, crossover probability = 0.85, mutatio probability = TEST SYSTEM AND RESULTS The proposed method was applied to the electrical etwork i wester Algeria (Fig. 5) to assess the suitability of the algorithm. The fuel cost (i Nm 3 /hr) equatios for the two geerators are F (P ) = P P 2, F 2 (P 2 ) = P 2 +.7P 2 2, subject to 30 P 50 (MW), 0 P 2 70 (MW), D = 505 MW Fig. 5. Electrical etwork i wester Algeria. The total load was 505 MW, ad the trasmissio lie losses were 5.94 MW after calculatio usig the Newto-Raphso method [2, 7]. Two cases were cosidered. I the first case, the trasmissio lie losses were calculated ad kept costat, ad i the secod, the trasmissio lie losses were cosidered as a liear fuctio of real geerated power.

7 ECONOMIC DISPATCH USING A GENETIC ALGORITHM 665 Table. Trasmissio lie data i p.u. k - m Impedace Lie chargig j0.005 j j j j j j j j j j j j0.070 j j j j j j j j j j0.07 j j j j j0.096 Table 2. Bus data i p.u. N Bus type Real power Reactive power Referece Load Load Load Load Load Load Load Load Load Load Productio Table 3. Results for case. GA Fletcher-Reeves [8] Fletcher [8] Soelgaz* [8] P optimal (MW) P optimal 2 (MW) P L (MW) Fuel cost (Nm 3 /h) Computig time(s) / Geeratio umber / * Soelgaz: Algeria Electricity ad Gas Board.

8 Case The trasmissio lie losses were calculated ad kept costat (P L = 5.94 MW). The power balace equatio the became: P + P 2 = MW. The results for the real geerated optimal power, miimum fuel cost, ad computig time are give i Table Case 2 The trasmissio lie losses were cosidered as a liear fuctio of real geerated power. The coefficiets were calculated usig the Gauss-Seidel method [8, 9]: P L = 0.089P P 2. The power balace equatio was, therefore, 0.98P P 2 = 505 MW. The results for the real geerated optimal power, miimum fuel cost, ad computig time are give i Table 4. Table 4. Results for case 2. GA Fletcher-Reeves [9] Fletcher [9] Soelgaz* [9] P optimal (MW) P optimal 2 (MW) P L (MW) Fuel cost (Nm 3 /h) Computig time(s) / Geeratio umber / 6. INTERPRETATIONS I the first case, the losses as determied usig Newto-Raphso method are kept costat (5.94 MW) for the three methods, ad they are equal to the losses recorded by Soelgaz. A better cost has bee obtaied usig the geetic algorithm method as compared with the Fletcher ad Fletcher-Reeves methods. A gai of Nm 3 /year of gas has bee obtaied. If the Soelgaz costs were cosidered, this would be the equivalet to a 2.35% profit. I the secod case, the losses are liearly formulated, which makes it possible to reduce them by a sigificat degree. Although the Fletcher ad Fletcher-Reeves methods give losses that are lower tha those obtaied usig the geetic algorithm, the latter gives a better productio cost ad a profit evaluated at Nm 3 /year of gas, which would be the equivalet of 2.82% of the productio cost of Soelgaz.

9 ECONOMIC DISPATCH USING A GENETIC ALGORITHM CONCLUSIONS The determiatio of the steady-state operatig coditio of the optimal power system is a o-liear problem. A geetic algorithm solutio has bee developed i this paper, based o the Lagrage method. The umerical results i both cases idicate that the proposed method ca be used to determie the optimum cotrol for the geeratio of power with the miimum fuel cost ad lower trasmissio lie losses, ad with accurate results obtaied i a short eough period of time to be compatible with o-lie applicatios. REFERENCES. P. H. Che ad H. C. Chag, Large-scale ecoomic dispatch by geetic algorithm, IEEE Trasactios o Power Systems, Vol. 0, 995, pp M. Rahli ad P. Pirotte, Optimal load flow usig sequetial ucostraied method (SUMT) uder power trasmissio losses miimizatio, Electric Power Systems Research Joural, 999, pp I. Hoube, Méthodes du plus proche voisi appliquées à la stabilité trasitoire des réseaux électriques, Thèse de doctorat Es scieces Uiversité de Liège, C. Wag ad S. M. Shahidehpour, Effects of ramp-rate limits o uit commitmet ad ecoomic dispatch, IEEE Trasactios o Power Systems, Vol. 8, 993, pp D. E. Goldberg, Geetic Algorithm i Search Optimizatio ad Machie Learig, Addiso Wesley, A. Bakirrtzis, V. Petridis, ad S. Karzalis, Geetic algorithm solutio to ecoomic dispatch problem, i Proceedigs of IEE Geeratio, Trasmissio, ad Distributio, Vol. 4, 994, pp R. Ouiddir ad M. Rahli, Optimal load flow usig geetic algorithm: applicatio to the west Algeria power etwork, Simposio Iteracioal la Calidad sobre de la Eergia Electrica (SICEL 200), R. Ouiddir ad M. Rahli, Optimal load flow usig geetic algorithms uder power trasmissio losses miimizatio, Iteratioal Coferece o Modelig ad Simulatio i Techical ad Social Scieces (MS2002 Cogress), M. Rahli, Applicatio d ue ouvelle méthode de programmatio o liéaire à la répartitio écoomique des puissaces actives du réseau Ouest algérie, Coferece o Modellig ad Simulatio o Electric Systems (CMSES 94), 994, pp

10 668 Rabah Ouiddir was bor o Jue, 9, 96 i Ora, Algeria. He received his B.S. degree i Electrical Egieerig from the Electrical Egieerig Istitute of the Uiversity of Scieces ad Techology of Ora (USTO) i 988, the M.S. degree from the Electrical Egieerig Istitute of the Uiversity of Sidi Belabbes (Algeria) i 993. He is curretly Professor of Electrical Egieerig at The Uiversity of Sidi Belabbes (Algeria). His research iterests iclude operatios, plaig ad ecoomics of electric eergy systems, as well as optimisatio theory ad its applicatios (Evolutioary Algorithm). Mostefa Rahli was bor o October 24, 949 i Mocta-Douz, Mascara, Algeria. He received his B.S. degree i Electrical Egieerig from the Electrical Egieerig Istitute of the Uiversity of Scieces ad Techology of Ora (USTO) i 979, the M.S. degree from the Electrical Egieerig Istitute of the Uiversity of Scieces ad Techology of Ora (USTO) i 985, ad the Ph.D. degree from the Electrical Egieerig Istitute of the Uiversity of Scieces ad Techology of Ora (USTO) i 996. From 987 to 99, he was a visitig professor at the Uiversity of Liege (Motefiore s Electrical Istitute) Liege (Belgium) where he worked o Power Systems Aalysis uder Professors Pol Pirotte ad Jea Louis Lilie. He is curretly Professor of Electrical Egieerig at the Uiversity of Scieces ad Techology of Ora (USTO), Ora, Algeria. His research iterests iclude operatios, plaig ad ecoomics of electric eergy systems, as well as optimizatio theory ad its applicatios. Lahouari Abdelhakem-Koridak was bor o March 2, 966 i Ora, Algeria. He received his B.S. degree i Electrical Egieerig from the Electrical Egieerig Istitute of the Uiversity of Scieces ad Techology of Ora (USTO) i 993, the M.S. degree from the Electrical Egieerig Istitute of the Uiversity of Scieces ad Techology of Ora (USTO) i He is curretly Professor of Electrical Egieerig at the Uiversity of Scieces ad Techology of Ora (USTO). His research iterests iclude operatios, plaig ad ecoomics of electric eergy systems, as well as optimisatio theory ad its applicatios.