GENETIC ALGORITHM APPLICATION IN ECONOMIC LOAD DISTRIBUTION

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1 Internatonal Journal on Techncal and Physcal Problems of Engneerng (IJTPE) Publshed by Internatonal Organzaton of IOTPE ISS IJTPE Journal December 013 Issue 17 Volume 5 umber 4 Pages GEETIC ALGORITHM APPLICATIO I ECOOMIC LOAD DISTRIBUTIO.M. Tabatabae 1, A. Jafar 1.S. Boushehr 1, K. Dursun 3 1. Electrcal Engneerng Department, Seraj Hgher Educaton Insttute, Tabrz, Iran n.m.tabatabae@gmal.com, al.jafar.860@gmal.com, nargesboush@yahoo.com. Taba Elm Internatonal Insttute, Tabrz, Iran 3. Electrcal Engneerng Department, Ostfold Unversty College, Fredrkstad, orway, kaml.dursun@hof.no Abstract- In ths paper we try to present Genetc Algorthm and how to use t for solvng Economc Dspatch problem. In the frst secton, we present on what bass does t work, and why Genetc Algorthm (GA) s useful and what problems can be solved by t. In the second secton, we present theory of GA lke representaton of desgn varables, representaton of objectve functon and constrants, genetc operators, and n thrd secton we revew the GA steps for solvng problem, and n fourth secton we present classc Economc Dspatch by Genetc Algorthm and how to use t for solvng ED problems, lke present of constrants and varables. In ffth secton we use MATLAB programng tool for wrte a smple program for solvng a smple problem of ED that wrtten program presented n paper attachment. In sxth secton we compare results of GA and Lagrangan method for compare GA performance. To end we have concluson of ths paper. Keywords: Genetc Algorthm (GA), Economc Dspatch, Optmzaton Algorthms. I. ITRODUCTIO In some of optmzaton problems, number of varables n the problem s contnues and number of them s dscontnues. Also search space n ths problems sometmes s non-convex or dscontnuous. These two factors along wth other factors proposed n combned optmzaton and contnuous optmzaton causes use of standard methods of optmzaton go neffcent, and n terms of computatonal are very expensve. In other words, solvng above problems wth classcal optmzaton methods, causes local optmal soluton n the neghborhood of the startng pont. A possble way to solve such complex optmzaton problems usng a method called a Genetc Algorthm that t has the ablty to fnd a global optmal soluton wth a hgh probablty of success. In facts the Genetc Algorthm s a search technque for fndng approxmate solutons to optmzaton problems usng concepts such as bology nhertance and mutatons. Ths algorthm that t based on Darwn's theory of evoluton s bult, frst, the varables are coded wth the approprate bnary strngs then usng computer smulatons laws struggle to survve constantly run and more approprate dscplnes are n fact an optmal soluton can be obtaned. The Genetc Algorthm s a method based on probabltes, to ensure that gven results are best values run the algorthm several tmes and compare results. However, the probablty of fndng the global optmum response n the event of the use of the approprate values for the parameters of the algorthm s huge [1-4]. II. GEETIC ALGORITHM DEFIITIO A. Representaton of Desgn Varables In GAs, the desgn varables are represented as strngs of bnary numbers, 0 and 1. For example, f a desgn varable x s denoted by a strng of length four (or a fourbt strng) as ( ), ts nteger (decmal equvalent) value wll be =5. If each desgn varable x, =1, n s coded n a strng of length q, a desgn vector s represented usng a strng of total length nq. For example, f a strng of length 5 s used to represent each varable, a total strng of length 0 descrbes a desgn vector wth n=4. The followng strng of 0 bnary dgts denote the vector (x 1 =18, x =3, x 3 =1, x 4 =4): Fgure 1. Example of strng length In general, f a bnary number s gven by b q b q-1,, b b 1 b 0, where b k =0 or 1, k=1,, q then ts equvalent decmal number y (nteger) s gven by: q y b (1) k 0 k k Ths ndcates that a contnuous desgn varable x can only be represented by a set of dscrete values f bnary representaton s used. If a varable x (whose bounds are gven by x l and x u ) s represented by a strng of q bnary numbers, as shown n Equaton (1), ts decmal value can be computed as: 88

2 u l q l x x k q k 1 k0 x x b () Thus f a contnuous varable s to be represented wth hgh accuracy, we need to use a large value of q n ts bnary representaton. In fact, the number of bnary dgts needed (q) to represent a contnuous varable n steps (accuracy) of x can be computed from the relaton: u l q x x 1 (3) x For example, f a contnuous varable x wth bounds 1 and 5 s to be represented wth an accuracy of 0.01, we need to use a bnary representaton wth q dgts where q or q 9. Equaton () shows why GAs are naturally suted for solvng dscrete optmzaton problems [3, 5]. B. Representaton of Objectve Functon and Constrants Because Genetc Algorthms are based on the survval of the fttest prncple of nature, they try to maxmze a functon called the ftness functon. Thus GAs are naturally sutable for solvng unconstraned maxmzaton problems. The ftness functon, F(X), can be taken to be same as the objectve functon f(x) of an unconstraned maxmzaton problem so that F(X) = f(x). A mnmzaton problem can be transformed nto a maxmzaton problem before applyng the GAs. Usually the ftness functon s chosen to be nonnegatve. The commonly used transformaton to convert an unconstraned mnmzaton problem to a ftness functon s gven by: 1 F( X) (4) 1 f( X) It can be seen that Equaton (4) does not alter the locaton of the mnmum of f(x) but converts the mnmzaton problem nto an equvalent maxmzaton problem. A general constraned mnmzaton problem can be stated as: Mnmze f(x) subject to g (X) 0; =1,,, m and h j (X) 0; j=1,,, p. Ths problem can be converted nto an equvalent unconstraned mnmzaton problem by usng the concept of penalty functon as: m p j j (5) mn ( X ) f ( X ) r g ( X ) R h ( X ) 1 j1 where r and R j are the penalty parameters assocated wth the constrants g (X) and h j (X), whose values are usually kept constant throughout the soluton process. In Equaton (5), the functon g ( X ), called the bracket functon, s defned as: g( X ) f g( X ) 0 g ( X) (6) 0 f g ( X) 0 In most cases, the penalty parameters assocated wth all the nequalty and equalty constrants are assumed to be the same constants as: r =r; =1,,, m and R j =R; j=1,,, p, where r and R are constants. The ftness functon, F(X), to be maxmzed n the GAs can be obtaned, smlar to Equaton (4), as: 1 F( X) (7) 1 ( X ) The Equatons (5) and (6) show that the penalty wll be proportonal to the square of the amount of volaton of the nequalty and equalty constrants at the desgn vector X, whle there wll be no penalty added to f(x) f all the constrants are satsfed at desgn vector X [3-5]. C. Genetc Operators The soluton of an optmzaton problem by GAs starts wth a populaton of random strngs denotng several (populaton of) desgn vectors. The populaton sze n GAs (n) s usually fxed. Each strng (or desgn vector) s evaluated to fnd ts ftness value. The populaton (of desgns) s operated by three operators reproducton, crossover, and mutaton to produce a new populaton of ponts (desgns). The new populaton s further evaluated to fnd the ftness values and tested for the convergence of the process. One cycle of reproducton, crossover, and mutaton and the evaluaton of the ftness values s known as a generaton n GAs. If the convergence crteron s not satsfed, the populaton s teratvely operated by the three operators and the resultng new populaton s evaluated for the ftness values. The procedure s contnued through several generatons untl the convergence crteron s satsfed and the process s termnated. The detals of the three operatons of GAs are gven below [3-5]. D. Reproducton Reproducton s the frst operaton appled to the populaton to select good strngs (desgns) of the populaton to form a matng pool. The reproducton operator s also called the selecton operator because t selects good strngs of the populaton. The reproducton operator s used to pck above average strngs from the current populaton and nsert ther multple copes n the matng pool based on a probablstc procedure. In a commonly used reproducton operator, a strng s selected from the matng pool wth a probablty proportonal to ts ftness. Thus f F denotes the ftness of the strng n the populaton of sze n, the probablty for selectng the th strng for the matng pool (p ) s gven by: F p, 1,...,n (8) n j1 F j ote that Equaton (8) mples that the sum of the probabltes of the strngs of the populaton beng selected for the matng pool s one. The mplementaton of the selecton process gven by Equaton (8) can be understood by magnng a roulette wheel wth ts crcumference dvded nto segments, one for each strng of the populaton, wth the segment lengths proportonal to the ftness of the strngs as shown n Fgure 1. By spnnng the roulette wheel n tmes (n beng the populaton sze) and selectng, each tme, the strng chosen by the roulette-wheel ponter, we obtan a matng 89

3 pool of sze n. Snce the segments of the crcumference of the wheel are marked accordng to the ftness of the varous strngs of the orgnal populaton, the roulettewheel process s expected to select F/F copes of the th strng for the matng pool, where F denotes the average ftness of the populaton: 1 n Fj n j 1 F (9) Fgure. Roulette-Wheel selecton scheme In Fgure 1, the populaton sze s assumed to be 6 wth ftness values of the strngs 1,, 3, 4, 5, and 6 gven by 1, 4, 16, 8, 36, and 4, respectvely. Snce the ffth strng (ndvdual) has the hghest value, t s expected to be selected most of the tme (36% of the tme, probablstcally) when the roulette wheel s spun n tmes (n=6 n Fgure 1). The selecton scheme, based on the spnnng of the roulette wheel, can be mplemented numercally durng computatons as follows. The probabltes of selectng dfferent strngs based on ther ftness values are calculated usng Equaton (8). These probabltes are used to determne the cumulatve probablty of strng beng coped to the matng pool, p by addng the ndvdual probabltes of strngs 1 through as: P p (10) j1 j Thus the roulette-wheel selecton process can be mplemented by assocatng the cumulatve probablty range P -1 -P to the th strng. To generate the matng pool of sze n durng numercal computatons, n random numbers, each n the range of zero to one, are generated (or chosen). By treatng each random number as the cumulatve probablty of the strng to be coped to the matng pool, n strngs correspondng to the n random numbers are selected as members of the matng pool. By ths process, the strng wth a hgher (lower) ftness value wll be selected more (less) frequently to the matng pool because t has a larger (smaller) range of cumulatve probablty. Therefore, strngs wth hgh ftness values n the populaton, probablstcally, get more copes n the matng pool. It s to be noted that no new strngs are formed n the reproducton stage; only the exstng strngs n the populaton get coped to the matng pool. The reproducton stage ensures that hghly ft ndvduals (strngs) lve and reproduce, and less ft ndvduals (strngs) de. Thus the GAs smulate the prncple of survval-of-the-fttest of nature [1, 3-5]. E. Crossover After reproducton, the crossover operator s mplemented. The purpose of crossover s to create new strngs by exchangng nformaton among strngs of the matng pool. Many crossover operators have been used n the lterature of GAs. In most crossover operators, two ndvdual strngs (desgns) are pcked (or selected) at random from the matng pool generated by the reproducton operator and some portons of the strngs are exchanged between the strngs. In the commonly used process, known as a sngle-pont crossover operator, a crossover ste s selected at random along the strng length, and the bnary dgts (alleles) lyng on the rght sde of the crossover ste are swapped (exchanged) between the two strngs. The two strngs selected for partcpaton n the crossover operators are known as parent strngs and the strngs generated by the crossover operator are known as chld strngs. For example, f two desgn vectors (parents), each wth a strng length of 10, are gven by: (Parent1) X { } 1 (Parent ) X { } The result of crossover, when the crossover ste s 3, s gven by: ( Offsprng1) X { } 3 ( Offsprng) X { } 4 Snce the crossover operator combnes substrngs from parent strngs (whch have good ftness values), the resultng chld strngs created are expected to have better ftness values provded an approprate (sutable) crossover ste s selected. However, the sutable or approprate crossover ste s not known beforehand. Hence the crossover ste s usually chosen randomly. The chld strngs generated usng a random crossover ste may or may not be as good as or better than ther parent strngs n terms of ther ftness values. If they are good or better than ther parents, they wll contrbute to a faster mprovement of the average ftness value of the new populaton. On the other hand, f the chld strngs created are worse than ther parent strngs, t should not be of much concern to the success of the GAs because the bad chld strngs wll not survve very long as they are less lkely to be selected n the next reproducton stage (because of the survval-of-the-fttest strategy used). As ndcated above, the effect of crossover may be useful or detrmental. Hence t s desrable not to use all the strngs of the matng pool n crossover but to preserve some of the good strngs of the matng pool as part of the populaton n the next generaton. In practce, a crossover probablty, p c s used n selectng the parents for crossover. Thus only 100p c percent of the strngs n the matng pool wll be used n the crossover operator whle 100(1 p c ) percent of the strngs wll be retaned as they are n the new generaton (of populaton) [1, 3-5]. 90

4 F. Mutaton The crossover s the man operator by whch new strngs wth better ftness values are created for the new generatons. The mutaton operator s appled to the new strngs wth a specfc small mutaton probablty, p m. The mutaton operator changes the bnary dgt (allele s value) 1 to 0 and vce versa. Several methods can be used for mplementng the mutaton operator. In the sngle-pont mutaton, a mutaton ste s selected at random along the strng length and the bnary dgt at that ste s then changed from 1 to 0 or 0 to 1 wth a probablty of p m. In the bt-wse mutaton, each bt (bnary dgt) n the strng s consdered one at a tme n sequence, and the dgt s changed from 1 to 0 or 0 to 1 wth a probablty p m. numercally, the process can be mplemented as follows. A random number between 0 and 1 s generated/chosen. If the random number s smaller than p m, then the bnary dgt s changed. Otherwse, the bnary dgt s not changed. The purpose of mutaton s (1)- to generate a strng (desgn pont) n the neghborhood of the current strng, thereby accomplshng a local search around the current soluton, ()- to safeguard aganst a premature loss of mportant genetc materal at a partcular poston, and (3)- to mantan dversty n the populaton [1, 3-5]. As an example, consder the followng populaton of sze n = 5 wth a strng length 10: Here all the fve strngs have a 1 n the poston of the frst bt. The true optmum soluton of the problem requres a 0 as the frst bt. The requred 0 cannot be created by ether the reproducton or the crossover operators. However, when the mutaton operator s used, the bnary number wll be changed from 1 to 0 n the locaton of the frst bt wth a probablty of np m. ote that the three operator s reproducton, crossover, and mutaton are smple to mplement. The reproducton operator selects good strngs for the matng pool, the crossover operator recombnes the substrngs of good strngs of the matng pool to create strngs (next generaton of populaton), and the mutaton operator alters the strng locally. The use of these three operators successvely yelds new generatons wth mproved values of average ftness of the populaton. Although, the mprovement of the ftness of the strngs n successve generatons cannot be proved mathematcally, the process has been found to converge to the optmum ftness value of the objectve functon. ote that f any bad strngs are created at any stage n the process, they wll be elmnated by the reproducton operator n the next generaton. The GAs have been successfully used to solve a varety of optmzaton problems n [1, 3-5]. III. GEETIC ALGORITHM PROCEDURE The computatonal procedure nvolved n maxmzng the ftness functon F(x 1, x,, x n ) n the Genetc Algorthm can be descrbed by the followng steps: a. Choose a sutable strng length l=nq to represent the n desgn varables of the desgn vector X. Assume sutable values for the followng parameters: populaton sze m, crossover probablty pc, mutaton probablty p m, permssble value of standard devaton of ftness values of the populaton (S f ) max to use as a convergence crteron, and maxmum number of generatons ( max ) to be used an a second convergence crteron. b. Generate a random populaton of sze m, each consstng of a strng of length l=nq. Evaluate the ftness values F, =1,,, m of the m strngs. c. Carry out the reproducton process. d. Carry out the crossover operaton usng the crossover probablty pc. e. Carry out the mutaton operaton usng the mutaton probablty p m to fnd the new generaton of m strngs. f. Evaluate the ftness values F, =1,,, m of the m strngs of the new populaton. Fnd the standard devaton of the m ftness values. g. Test for the convergence of the algorthm or process. If S f (S f ) max, the convergence crteron s satsfed and hence process may be stopped. Otherwse, go to step h. h. Test for the generaton number. If max, the computatons have been performed for the maxmum permssble number of generatons and hence the process may be stopped. Otherwse, set the generaton number as =+1 and go to step (c) [1, 3-5]. A look at the steps mentoned s n Fgure 3. Fgure 3. Flowchart of bnary GA [3] IV. CLASSIC ECOOMIC DISPATCH BY GEETIC ALGORITHM Another type of method that s used to solve the classc Economc Dspatch problem s Genetc Algorthm. The theoretcal foundaton for GA was frst descrbed by Holland and was extended by Goldberg. GA provdes a soluton to a problem by workng wth a populaton of ndvduals each representng a possble soluton. Each possble soluton s termed a chromosome. ew ponts of the search space are generated through GA operatons, known as reproducton, crossover, and mutaton. These operatons consstently produce ftter offsprng through successve generatons, whch rapdly lead the search toward global optmal [7-9, 1, 13, 14]. 91

5 A. GA Based ED Soluton The classc economc dspatch problem can be stated: mnmze F F ( P ) (11) 1 G D loss 1 G P P P (1) PG PD Ploss 0 (13) 1 Addng penalty factor h 1 to the volaton of power outputs, we can combne Equatons (13) and (1): FA F ( PG ) h1 PG PD Ploss (14) 1 1 The value of the penalty factor should be large so that there s no volaton for unt output at the fnal soluton. Snce GA s desgned for the soluton of maxmzaton problems, the GA ftness functon s defned as the nverse of Equaton (14). F 1/ F (15) ftness A In the economc dspatch problem, the problem varables correspond to the power generatons of the unts. Each strng represents a possble soluton and s made of substrngs, each correspondng to a generatng unt. The length of each substrng s decded based on the maxmum/mnmum lmts on the power generaton of the correspondng unt and the soluton accuracy desred. The strng length, whch depends on the length of each substrng, s chosen based on a trade-off between soluton accuracy and soluton tme. Longer strngs may provde better accuracy but result n more soluton tme. Thus the step sze of the unt can be computed as follows: [, 3] PG max PG mn (16) n 1 V. SIMULATIO To show a smple example we assume a system that has two power generators whch fuel cost functons by $/h s as follow: F P P1 (17) F P P The powers are determned by MW and total load s 600MW whch the losses are neglected. Generators constrants (by MW) are as follow: 00 P 1 450, 00 P 450 (18) In ths paper we use Matlab software programng tool for wrte a program for solve ths smple problem. The results are as follow: P MW, P MW (19) and the optmum fuel cost s equal to 4991$/h [, 6-9]. VI. COMPARISO RESULTS OF GEETIC ALGORITHM AD LAGRAGIA METHODS For comparson we need lagrangan method results, the Lagrangan method results same as follow: P 370 MW, P 30 MW (0) 1 By comparson ths results wth results of Genetc Algorthm we understand the accuracy of GA s good f we choose parameters approprate. Also speed of GA for searchng n the bg area and more number of generators s better than other search methods. Another advantage s that GA don t need to dfferental of cost functon, and use cost functon to fnd better result. Ths property s very mportant when my cost functon curve sn t contnues or cost functon s pece lnes. Table 1. Results of three algorthms for solvng assumed ED [14] Parameters ACO GA Lagrangan P load (MW) P1 (MW) P (MW) Cost ($/h) Fgure 4. Mnmum of cost functon Fgure 5. Value of Power output of unt 1 Fgure 6. Value of power output of unt VII. COCLUSIOS In ths paper we have seen that genetc algorthms can be a powerful tool for solvng problems and for smulatng natural systems n a wde varety of scentfc felds. An optmzaton problem because of ncrease securty and decrease costs n power systems always was 9

6 mportant. So methods of optmzaton n power systems should have hgh accuracy and hgh speed. Some of ths methods have some constrant that can affect performance of networks, and we try to troubleshootng ths problems. Genetc Algorthm s method that try to troubleshootng ths problem, GA has hgh speed, global search, hgh accuracy and compatble wth restrctons. Ths advantages make GA as powerful optmzaton method. So we can learn GA and use t n any problem that need to optmzaton. REFERECES [1] S.R. Sngresu, Engneerng Optmzaton - Theory and Practce, 4th Edton, John Wley & Sons, 009. [] J. Zhu, Optmzaton of Power System Operaton, IEEE Press Seres on Power Engneerng, John Wley & Sons Inc., 008. [3] R.L. Haupt, S.E. Haupt, Practcal Genetc Algorthms, nd Edton, John Wley & Sons Inc., 004 [4] F. Merrkh Bayat, Optmzaton Algorthms Inspred by ature, Zanjan Unversty, Zanjan, Iran, 01. [5] M. Melane, An Introducton to Genetc Algorthms, MIT Press, [6].M. Tabatabae, F. Rajab Optmzaton of Power Generaton Dstrbuton wth Genetc Algorthm, 9th Baku Internatonal Congress on Energy, Ecology and Economy, pp , Internatonal Ecoenergy Academy, Baku, Azerbajan, June 7-9, 007. [7] Economc Dspatch, Concepts, Practces and Issues FERC, Staff Palm Sprngs, Calforna, ov. 13, 005. [8] H. Saadat, Power System Analyss, McGraw-Hll, [9] H.H. Happ, Optmal Power Dspatch, IEEE Trans. on Power Apparatus and Systems, PAS-93, pp , [10] A.J. Wood, B.F. Wollenberg, Power Generaton, Operaton and Control, Wley Interscence Publcaton, John Wley & Sons Inc., [11] A.G. Bakrtzs, P.. Bskas, C.E. Zoumas, V. Petrds, Optmal Power Flow by Enhanced Genetc Algorthm, IEEE Transactons on Power Systems, Vol. 17, o., pp.9-36, May 00,. [1] G.B. Sheble, K. Brttg, Refned Genetc Algorthm- Economc Dspatch Example, IEEE/PES Wnter Meetng, Paper 94 WM PWRS, [13] K.P. Wong, Y.W. Wong, Genetc and Genetc/Smulated-Annealng Approaches to Economc Dspatch, Inst. Elect. Eng., Gener. Transm. Dstrb., Vol. 141, o. 5, pp , Sep [14].M. Tabatabae, A. Jafar,.S. Boushehr, K. Dursun, Ant Colony Algorthm Applcaton n Economc Load Dstrbuton, Internatonal Journal on Techncal and Physcal Problems of Engneerng (IJTPE), Issue 16, Vol. 5, o. 6, pp , September 013. BIOGRAPHIES aser Mahdav Tabatabae was born n Tehran, Iran, He receved the B.Sc. and the M.Sc. degrees from Unversty of Tabrz (Tabrz, Iran) and the Ph.D. degree from Iran Unversty of Scence and Technology (Tehran, Iran), all n Power Electrcal Engneerng, n 1989, 199, and 1997, respectvely. Currently, he s a Professor n Internatonal Organzaton of IOTPE. He s also an academc member of Power Electrcal Engneerng at Seraj Hgher Educaton Insttute (Tabrz, Iran) and teaches power system analyss, power system operaton, and reactve power control. He s the General Secretary of Internatonal Conference of ICTPE, Edtor-n-Chef of Internatonal Journal of IJTPE and Charman of Internatonal Enterprse of IETPE all supported by IOTPE. He has authored and coauthored of sx books and book chapters n Electrcal Engneerng area n nternatonal publshers and more than 130 papers n nternatonal journals and conference proceedngs. Hs research nterests are n the area of power qualty, energy management systems, ICT n power engneerng and vrtual e-learnng educatonal systems. He s a member of the Iranan Assocaton of Electrcal and Electronc Engneers (IAEEE). Al Jafar was born n Zanjan, Iran n He receved the B.Sc. degree n Electrcal Engneerng from Abhar Branch, Islamc Azad Unversty, Abhar, Iran n 011. He s currently the M.Sc. student n Seraj Hgher Educaton Insttute, Tabrz, Iran. He s the Member of Scentfc and Executve Commttees of Internatonal Conference of ICTPE and also the Scentfc and Executve Secretary of Internatonal Journal of IJTPE supported by Internatonal Organzaton of IOTPE ( Hs research felds are power system analyss and operaton, and reactve power control. arges Sadat Boushehr was born n Iran. She receved her B.Sc. degree n Control Engneerng from Sharf Unversty of Technology (Tehran, Iran), and Electronc Engneerng from Central Tehran Branch, Islamc Azad Unversty, (Tehran, Iran), n 1991 and 1996, respectvely. She receved the M.Sc. degree n Electronc Engneerng from Internatonal Ecocenergy Academy (Baku, Azerbajan), n 009. She s the Member of Scentfc and Executve Commttees of Internatonal Conference of ICTPE and also the Scentfc and Executve Secretary of Internatonal Journal of IJTPE supported by Internatonal Organzaton of IOTPE ( Her research nterests are n the area of power system control and artfcal ntellgent algorthms. Kaml Dursun was born n Ankara, Turkey, He receved the M.Sc. and the Ph.D. degrees from The orwegan Techncal Unversty, all n Power Electrcal Engneerng, n 1978 and 1984, respectvely. He worked wth ABB n several countres. Currently, he s an Assocate Professor of Power Engneerng at Ostfold Unversty College (Fredrkstad, orway). He s the secretary of Internatonal Conference of ICTPE. Hs research nterests are n the area of effcent power dstrbuton, energy management systems and vrtual e- learnng educatonal systems. 93