ECONOMIC POWER DISPATCH USING THE COMBINATION OF TWO GENETIC ALGORITHMS

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1 ISTANBUL UNIVERSITY JOURNAL OF ELECTRICAL & ELECTRONICS ENGINEERING YEAR VOLUME NUMBER : 2006 : 6 : 2 (75-8) ECONOMIC POWER DISPATCH USING THE COMBINATION OF TWO GENETIC ALGORITHMS Mmoun YOUNES Mostafa RAHLI Département de géne mécanque, Unversté de Sd Bel Abbés, BP 89 Sd DJllal, SBA, Algére E-mal: younesm@yahoo.fr ABSTRACT Lately, one notces an ncrease n the applcatons of the genetc algorthms (GA) through several felds, and one noted that all the utlsateurs(exploteurs) these algorthms wonder on the choce of the values of the genetc operators n order to ncrease the performances of the algorthm. The objectve of ths artcle tres to solve ths problem, by usng two genetc algorthms one to determne the values of the genetc operators and the other to optmze the functon cost. Numercal results on a test system consstng of 3 thermal unts show that the proposed approach has an ablty to fnd the better solutons than the conventonal genetc algorthm.. Keywords: Dspatchng, Genetc algorthm (GA).. INTRODUCTION The theory of natural evoluton mantans that adaptve changes to speces occur through natural selecton, whereby ndvduals better suted for survval are more lkely to reproduce and hence ncrease the representaton of ther genetc materal n the gene pool. The combnng of genetc materal from two such ndvduals can produce an offsprng even better suted to the envronment. Ths s how speces evolve,.e., sex s a mechansm for exchangng genetc materal and hence for speces to rapdly adapt to ther envronment. To address the shortcomngs of classcal determnstc optmzaton algorthms, a number of algorthms based on natural selecton and embedded randomness have been developed. One such class of algorthms are the populatonbased methods of Evolutonary Computaton (EC). EC s a class of search and optmzaton technques based on the Darwnan evolutonary model. The development of EC can be traced back to the 960s and the Evolutonary Strateges of Rechenberg and Schwefel n Germany and the Genetc Algorthms (GAs) of Holland [6] n the Unted States (see [8] for further hstorcal background). One of ther prmary features s that they search through a large populaton of solutons for the one whch optmzes a gven evaluaton or ftness functon. GA explore better the space of research to the total research of the optmum compared to other methods lke the tradtonal methods (détermnstes)[2]. Nevertheless these methods soufrent on the determnaton and the choce of the values of the genetc operators (operator of crossng, operator of change and sze of the populaton). Receved Date : Accepted Date:

2 76 Economc Power Dspatch Usng The Combnaton Of Two Genetc Algorthms In ths artcle one tres to fnd a soluton wth ths problem, by usng the combnaton of two genetc algorthms the frst to determne the genetc operators and the second for the optmzaton of the objectve functon (cost). The method was appled to a network real 62 bus. Ths artcle s organzed as follows. Secton 2 s developed to the Objectve and Mathematcal formulaton. In secton 3 wet expose the genetc algorthm to solve the problem of ED s developed. Secton 4 contans the obtaned results whch show the potental of the genetc algorthm. In concluson, n secton 5, several sutable remarks are gven to conclude ths paper. 2. OBJECTIVE AND MATHEMATICAL FORMULATION The probleme of the economc whch exst to mnmze the cost of producton of the real power can generally be stated as folows[,2,3,4]: Mn NG C P ( ) Under the followng constrants: mn P Q mn max () P P (2) max Q Q (3) NG ND P = PDj + j= NG = ND j= Dj P L Q Q + P L (4) (5) where generally C P = a + b P + c P (6) ( ) 2 and a,b,c are the known coeffcents and NG, s number of generator; ND, s number of loads; P, s real power generaton; Q, s reactve power generaton; P, s real power load; Q, Dj s reactve power load; PL and QL are respectvely real and reactve losses. The nonlnear programmng problem can be formally stat as: Dj Mnmze: f ( x) Subject to m lnear and /or nonlnear equalty constrants h x = 0 =,, ( ) m and ( m) p lnear and/or nonlnear nequalty constrants ( x) 0 = m +, p g, the nonlnear programmng problem s converted nto a sequence of unconstraned problems[4] by defnng the p functon as follows m p k k k 2 k k k P( x, r ) = f ( x ) + h ( x ) r ln g ( x ) (7) k r m+ Where r k > 0 s called the coeffcent of penalty. 3. GENETIC ALGORITHM IN ECONOMIC POWER DISPATCH 3.. GA Appled to economc power dspatch A smple Genetc Algorthm s an teratve procedure, whch mantans a constant sze populaton P of canddate solutons. Durng each teraton step (generaton) three genetc operators (reproducton, crossover, and mutaton) are performng to generate new populatons (offsprng), and the chromosomes of the new populatons are evaluated va the value of the ftness wtch s related to cost functon. Based on these genetc operators and the evaluatons, the better new populatons of canddate soluton are formed. Wth the above descrpton, a smple genetc algorthm s gven as follow [6,7]:. Generate randomly a populaton of bnary strng 2. Calculate the ftness for each strng n the populaton 3. Create offsprng strngs through reproducton, crossover and mutaton operaton. 4. Evaluate the new strngs and calculate the ftness for each strng (chromosome). 5. If the search goal s acheved, or an allowable generaton s attaned, return the best chromosome as the soluton; otherwse go to step 3. We now descrbe the detals n employng the smple GA to solve the economc power dspatch Chromosome codng and decodng GAs work wth a populaton of bnary strng, not the parameters themselves. For smplcty and convenence,bnary codng s used n ths paper [9,0]. Wth the bnary codng method, the actve

3 Economc Power Dspatch Usng The Combnaton Of Two Genetc Algorthms 77 generaton power set of 4-bus test system ( P G2 P G and ) would be coded as bnary strng of 0 and wth length L and L2 (may be dfferent), respectvely. Each parameter P has upper bound b and lower bound a.the choce of m and m2 for the parameters s concerned wth the resoluton specfed by the desgner n the search space. In the bnary codng method, the bt length m and the correspondng resoluton R s related by b a R = 2 m (8) As result, the Pg set can be transformed nto a bnary strng (chromosome) wth length m and then the search space s explored. Note that each chromosome present one possble soluton to the problem. For example, suppose the parameter doman of ( and P G 2 ) whch s presented n Table : Table. Parameters set of P Bus mn P max P (MW) If the resoluton (R, R2) s specfed as (0.235, 0.294). we have m=8 and m2=8.then the parameter set ( )can be coded accordng to the followng (Table 2): Table 2. Codng of a ($/hr) P b ($/MW.hr) parameter set P G(MW) code P G2(MW) code c ($/MW 2.hr ) MW) , , , , If the canddate parameters set s (94.455,72.940), then the chromosome s a bnary strng The decodng procedure s the reverse procedure. The frst step of any genetc algorthm s to generate the ntal populaton. A bnary strng of length L s assocated to each member (ndvdual) of the populaton. The strng s usually known as a chromosome and represents a soluton of the problem. A samplng of ths ntal populaton creates an ntermedate populaton Thus some operators (reproducton, crossover and mutaton) are appled to ths new ntermedate populaton n order to obtan a new one. The process, that starts from the present populaton and leads to the new populaton, s named a generaton when executng a genetc algorthm (Table 3). Table3. Frst generaton of GA process for 4 bus example Chr Intal populaton P G (MW) P G2 (MW) Fcost ($/h) Crossover Crossover s the prmary genetc operator, whch promotes the exploraton of new regons n the search space. For a par of parents selected from the populaton the recombnaton operaton dvdes two strngs of bts nto segments by settng a crossover pont at random,.e. Sngle Pont Crossover. The segments of bts from the parents behnd the crossover pont are exchanged wth each other to generate ther offsprng[,2,3]. The mxture s performed by choosng a pont of the strngs randomly. And swtchng ther segments to the left of ths pont. The new strngs belong to the next generaton of possble solutons. The strngs to be crossed are selected accordng to ther scores usng the roulette wheel [8]. Thus, the strngs wth larger scores have more chances to be mxed wth other strngs because all the copes n the roulette have the same probablty to be selected.

4 78 Economc Power Dspatch Usng The Combnaton Of Two Genetc Algorthms 3.4. Mutaton Mutaton s a secondary operator and prevents the premature stoppng of the algorthm n a local soluton. The mutaton operator s defned by a random bt value change n a chosen strng wth a low probablty of such change. The mutaton adds a random search character to the genetc algorthm, and t s necessary to avod that, after some generatons, all possble solutons were very smlar ones. All strngs and bts have the same probablty of mutaton. For example, n the strng , f the mutaton affects to tme bt number sx, the strng obtaned s and the value of P G2 change from MW to MW Reproducton Reproducton s based on the prncple of survval of the better ftness. It s an operator that obtans a fxed number of copes of solutons accordng to ther ftness value. If the score ncreases then the number of copes ncreases too. A score value s of assocated to a gven soluton accordng to ts dstance of the optmal soluton (closer dstances to the optmal soluton mean hgher scores) Descrpton of Genetc Algorthm The cost functon s defned n [] Our objectve s to search ( )n ther admssble lmts to acheve the optmsaton problem of EPD. The cost functon F(x) takes a chromosome (a possble( )) and returns a value. The value of the cost s then mapped nto a ftness value Ft( ) so as to ft n the genetc algorthm. To mnmse F(x) s equvalent to gettng a maxmum ftness value n the searchng process. A chromosome that has lower cost functon should be assgned a larger ftness value. The objectve of EPD has to be changed to the maxmsaton of ftness to be used n the smulated roulette wheel as follows: Fmax - F j(, 2) f Fmax F j(, 2) ftness= 0; otherwse 3.7. Other parameters The operators of the genetc algorthm are guded by a certan number of parameters fxed n advance, whose ther values nfluence the success or not algorthme genetc. parameters are: These Sze of the populaton N, and L s the length of the codng of each ndvdual, (n the case of the bnary codng): If N s too large the computng tme of the algorthm can prove very sgnfcant, and f N s too small, t can converge too quckly towards a bad chromosome. Ths mportance of the sze s prmarly due to the concept of parallelsm mplct whch mples that the number of ndvduals treated by the algorthm s at least proportonal to the cube of the number of ndvduals. In general, the value of the sze of the populaton[3] s between 30 and 50 ndvduals Probablty of crossng Pc: It depends on the form of the selectve functon. Its choce s n general heurstc (just lke for P m. The hgher t s, the more the populaton sudden of sgnfcant changes. The usual rate s selected between 60% and 00%. Probablty of changes P m Ths msses s general E dregs weak (between 0.% and 5%), snce one hgh msses s lkely to lead to has research too alé has tore. Rather than to reduce P m another way of preventng that the best ndvduals are faded s to uses the taken back explct one of the elte n some propo R ton. Thus, very often, best the 5%, for example, of the populaton are drectly reproduced wth dentcal, the operator of reproducton not playng whereas one the 95% remanders. That s called has éltste strategy. 4. SIMULATION 4..Smulaton condtons The GAGA has been developed by the use of Matlab verson 6.5. the tested wth Pentum 4, 500 MHz PC wth 28 MB RAM. The proposed algorthms n ths paper have been compared to the conventonal GA by applyng to the test system [4] whch conssts of 3 thermal generators. The coeffcents a, b, c, e and f of the heat-rate functons appearng n Eqs. (6) and operaton lmts of the generators used n the test system are tabulated n Table 4. Total load demand of the system s 2520 (MW), and 3 generators should satsfy ths load demand economcally. The parameter values used for GA are gven n Table 5.

5 Economc Power Dspatch Usng The Combnaton Of Two Genetc Algorthms 79 Table 4. Characterstc of 3 Generators of network 62 bus power penalty, equaton (7), s used. Owng to the randomness n the GA approach, The best Type mn max -2 P P a b c *0 actve power dspatch solutons together wth the (MW) (MW) ($/hr) ($/MW.hr) ($/MW 2.hr) assocated power found by the GAGA are tabulated n Table 2. For comparson purposes, the dspatch solutons obtaned by the GA,GA2,GA3 and GA4, are summarzed n Table Smulaton results: The GAGA, GA, GA2, GA3 and GA4 have been evaluated on the above power system. The adopted parameters n the algorthms are gven n Table 2. The objectve functon wth the actve The results n Tables 2 show that the dspatch solutons determned by the GAGA lead to lower actve power than that found by thega, GA2, GA3 and GA4, whch confrms that the GAGA s well capable of determnng the global or nearglobal optmum dspatch soluton. In addton, the results summarzed n Table 2 show that the proposed GAGA s about two tmes faster than GA2 n speed. The optmzaton search procedures by the GAGAand GA,GA2,GA3 and GA4 s shown n Fgure 3. It can be seen that, by usng the adaptve probabltes of crossover and mutaton, the teratons for convergence can be reduced greatly. Table 5. results of GAGA compared wth classcal GA for the Networks 62-bus system GAGA GA GA 2 GA3 GA4 Pc Pm N crossover sngle pont sngle pont sngle pont sngle pont sngle pont operator selecton tournament tournament tournament tournament tournament operator ,289.4, ,333.76, , , , Cost ($/h) Tme (S)

6 80 Economc Power Dspatch Usng The Combnaton Of Two Genetc Algorthms cost($/h) 2,20E+008 2,00E+008,80E+008,60E+008,40E+008,20E+008,00E+008 8,00E+007 6,00E+007 4,00E+007 2,00E+007 0,00E+000-2,00E generaton B C D E F Fg. 8. Convergence characterstc curves of the GAGA(B),GA(C),GA2(D),GA3(E) and GA4(F). 5. CONCLUSIONS A method composed by two genetc algorthms (GAGA) has been developed for determnaton of the global or near-global optmum soluton for optmal actve power dspatch of power systems. In the prncpal genetc algorthm, the probabltes of crossover (pc), mutaton ( pm) and sze of the populaton (N), were chosen by second genetc algorthme. The performance of the proposed method demonstrated through ts evaluaton on the 62-bus power system shows that the GAGA s able to undertake global search wth a fast convergence rate and a feature of robust computaton. From the smulaton study, t has been found that the GAGA converges to the global optmum. REFERENCES [] R. Fletcher, Practcal Methods of Optmsaton, John Wlley & Sons, 986. [2] M. Rahl, Appled Lnear and Nonlnear Programmng to Economc Dspatch, Ph.D. Thess, Electrcal Insttute, USTO, Oran, Algera, 996. [3] G.W. Stagg and A.H. E Abadh. Computer Methods n Power System Analyss McGraw Hll Book Company, New York (968). [4] D.M. Hmmelblau. In: Appled Non Lnear Programmng McGraw Hll, New York (972). [5] M. Younes, M. Rahl and A. kordak., Dspatchng Economque par l ntellgence Artfcelle, Proceedng ICEL 2005,U.S.T. Oran, Algera, Vol.02, pp.6-2,3-5 November [6] Holand, J. H, " Adaptaton n natural and Artfcal Systems. " The unversty of Mchgan press, Ann Arbor, USA, 975. [7] D. E. Goldberg Genetc Algorthms n Search, Optmzaton and Machne Learnng, Addson Wesley Publshng Company, Ind. USA, 989.

7 Economc Power Dspatch Usng The Combnaton Of Two Genetc Algorthms 8 [8] T. B ack, F. Ho_mester, and H.-P. Schwefel. A survey of Evoluton Strateges. In R. Belew and L. Booker, edtors, Proceedngs of the fourth nternatonal conference on genetc algorthms, pages 2-9. Morgan Kaufmann: San Mateo, CA, 99. [9] L. La, J. T. Ma, R. Yokoma, M. Zhao Improved genetc algorthms for optmal power flow under both normal and contngent operaton states Electrcal Power & Energy System, Vol. 9, N.5, pp , 997. [0] B. S. Chen, Y. M. Cheng, C. H. Lee, A Genetc Approach to Mxed H2/H00 Optmal PID Control, IEEE Control system, pp , October 995 []M.Younes, M.Rahl: La Répartton Optmale des pussances en utlsant l algorthme génétque,iceee 2004, Laghouat, Aprl, 24-26, 2004 [2]N. Snha, R. Chakrabart, and P. K. Chattopadhyay: Evolutonary programmng technques for economc load dspatch. IEEE Trans. Evol. Comput., vol. 7, no., pp , Feb [3]M.Younes, M.Rahl, H Kordak : Genetc/ Evolutonary Algorthms and applcaton to power systems, PCSE 05, O. E. Bouagh, May 9-, Unv, Algera, [4] Wong K.P., Wong Y.W., Genetc and genetc/smulated-annealng approaches to economc dspatch, IEE Proc. 4 (5) (994) 507_/53.