Archive of SID. A Novel Pareto-Optimal Solution for Multi-Objective Economic Dispatch Problem.

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1 IRANIAN JOURNAL OF ELECTRICAL AND COMPUTER ENGINEERING, VOL. 6, NO., SUMMER-FALL 007 A Novel Pareto-Optmal Soluton for Mult-Obectve Economc Dspatch Problem S. Muraldharan, K. Srkrshna, and S. Subramanan Abstract The modern power system has become very complex n nature wth conflctng requrements. Heavy load demands and dstrbuted generaton compromse the cost of generaton. Also the polluton hazards and nherent losses contrbute towards an neffcent system. A scenaro of coordnaton between cost, emsson and loss wll echo a pareto-optmal generaton. The oft handled teratve approaches and optmzaton technques have been replaced by a Dynamc Programmng technque nvolvng novel recursve approach for realzng producton cost mnmzaton, wth an emsson constraned and loss reduced condton. A study of test system exhbts the computatonal effcency and accuracy of the soluton. Index Terms Economc Dspatch (ED), emsson constraned economc dspatch (ECED), emsson dspatch (EmD), self adaptve dynamc programmng (SADP). E I. INTRODUCTION CONOMIC dspatch (ED) s defned as the process of allocatng generaton levels to the generatng unts so that the system load s suppled entrely and most economcally []. Many economc dspatch approaches [], []-[7] have been proposed by varous researchers to formulate and solve ths problem. Reference [] provded a revew of the advances n ths feld. Equal ncremental cost condton has been the crteron for economc schedulng of power generaton among unts n a plant [8]-[0]. Wthn a plant, n the absence of transmsson loss, the ncremental producton cost of a plant exactly represents the ncremental cost condton among unts. In general, optmum plant cost equaton can be represented by a second order polynomal usng curve-fttng technques. Ths formulaton could also be done analytcally. The solutons are of mathematcal nterest when appled for cost mnmum condton. Increased power generaton and rse n polluton levels go hand n hand. Mnmum emsson condton provdes only theoretcal nsght to the soluton technques. Hence, the plant cost equaton must take nto account economc power generaton wth emsson restrctons. Ths leads to a small devaton n optmum producton cost as obtaned from economc generaton condton. When the emsson becomes controlled, the dspatch becomes Emsson Manuscrpt receved July, 006; revsed November 8, 006. S. Muraldharan s wth the Department of Electrcal & Electroncs Engneerng, MEPCO Schlenk Engneerng College, Svakas , Tamlnadu State, Inda (e-mal: yes_mural@ yahoo.com). K. Srkrshna s wth the Department of Electrcal & Electroncs Engneerng, K. L. N. College of Engneerng, Madura 630 6, Tamlnadu State, Inda (e-mal: drksr@yahoo.com). S. Subramanan s wth the Department of Electrcal Engneerng, Annamala Unversty, Annamala Nagar , Tamlnadu State, Inda (e-mal: sanaycdm@yahoo.co.n). Publsher Item Identfer S (07) /07$0 007 JD Constraned Economc Dspatch (ECED). Varous penalty costs of emsson have been attempted here for chastsng the emsson released. Any dspatch neglectng transmsson losses s a study by tself for solated plants alone. In a mult-area, mult-plant system, the transmsson loss n between plants s consderable. Ths forces a coordnaton to be adopted between cost, emsson, and loss. Papers over the past two decades have hghlghted very extensve teratve soluton procedures nvolvng. Formulaton of total system equaton. Dfferentaton of ths equaton 3. Incremental cost of receved power 4. Curve fttng formulae 5. Iteratve technques wthout approxmatons 6. Non-teratve soluton methods after makng approxmatons. The best generaton pursues a smultaneous realzaton of varous obectves lke cost, emsson, and loss each at ther mnmum. Achevng an absolute mnmum of all these three obectves concurrently n a real tme system s a near mpossblty. A best all-round soluton attanable n such a case becomes the optmal soluton for ths multobectve problem. A summary of several algorthms of ECED s dscussed n []. A combned handlng of economc and mnmum emsson dspatch by ntroducng a prce penalty factor s gven n [3]. In ths paper, the prce factor coordnates the emsson wth actual fuel costs and ths s approxmated by a curve analyss. Ths prce penalty factor and Back Propagaton Neural Network soluton s presented n [4]. To avod local mnmum and to acheve global mnmum, the authors employ momentum, and adaptve learnng rate. The Hopfeld neural network approach [5] for optmal economc/envronmental dspatchng suffers from the number of teratons nvolved and the dffculty wth the Hopfeld coeffcents. A systematc procedure for determnng the approprate coeffcents for the obectve functon has been solved usng Fuzzy Logc [6]. Reference [7] presented an analyss of the performance of the Hopfeld Neural Network methods for economc-emsson dspatch problem. Sequental quadratc programmng as a tool to solve the two-obectve problem, after transformng t nto a sngle obectve usng weghng factor s adopted n [8]. The choce of weghng factors reflectng the operator s preference for one of the obectves nfluences the overall optmzaton procedure. The mplementaton of bobectve generaton dspatch by Genetc algorthm [9] s realzed wthn the functon by blendng power balance equaton nto t. Mult-obectve decson makng problem has been attempted n [0], [] usng weghted mn-max

2 MURALIDHARAN et al.: A NOVEL PARETO-OPTIMAL SOLUTION FOR MULTI-OBJECTIVE ECONOMIC DISPATCH PROBLEM 3 method. To get the optmal effcent dspatch from the nonnferor set, fuzzy set theory has been appled. A new evolutonary algorthm for envronmental/economc power dspatch was proposed n [], targetng a true multobectve problem wth competng and non-commensurable obectves. The 60 s saw the ntroducton of Dynamc programmng as a tool for solvng unt commtment. In the earler methods [3], [4], the commtment of generatng unts was determned ndependently for every tme perod. A heurstc truncated wndow DP [5], employs a varable wndow sze accordng to the load demand ncrements. Fuzzy dynamc programmng (FDP) approach wth a fuzzy obectve functon characterzed by the fuzzy set related to total cost s used n [6]. An teratve dynamc programmng method for solvng economc dspatch ncludng transmsson lne losses [7] used a modfed cost functon to obtan the optmal dspatch. A conventonal Dynamc programmng procedure conssts of two parts:. An evaluaton of all possble confguratons from the begnnng to the end of the problem, and. Back-track operaton from the end to the begnnng over the optmal path. In contrast to ths, Self Adaptve Dynamc Programmng (SADP) technque provdes at once the generaton schedule of ndvdual generators n an analytcal form for a large number of varables wthout the tradtonal Lagrange form. Unlke lnear programmng, there s a dearth of standard mathematcal formulaton for the Dynamc Programmng. Therefore, problem solvng nvolves - developng the functonal equatons for the problem and solvng functonal equatons for determnng the optmal soluton. Increase n the number of states at each stage s the curse of dmensonalty n the lterature of Dynamc Programmng. The result s spectacular n computatonal savngs, f the state varables are three or less. Aganst ths background, t has been establshed that the format developed n ths paper can even be extended to hgher number of state varables wth well-defned mathematcal approach. Hence, t has been aptly called Self Adaptve Dynamc Programmng (SADP) approach. Ths paper presents SADP technque elmnatng common Lambda approach for loss ncluded emsson constraned economc dspatch problem. The regular quadratc cost equatons, a penalty factor for emsson and a prce factor for chargng the transmsson losses are ncorporated. A way of recordng the mpact of fossl fuel based electrcal energy producton on human health and the envronment s acheved by estmatng a monetary value for the generated emssons. The blendng of emsson wth actual fuel costs s facltated by the use of a prce penalty factor (F/E rato). The prce factor (g), whch monetzes the transmsson loss, has been defned as the rato of the cost of generaton to the power of generaton (F/P rato). Ths analytc concept gves encouragng results very closely followng the conventonal methods. In ths work, a plant havng sx generators has been consdered for study. Proposed algorthm can also be extended for a larger system. II. DYNAMIC PROGRAMMING Programmng (DP) s a mathematcal technque []- [] dealng wth the optmzaton of multstage decson process. The word programmng s used n the mathematcal sense of selectng an optmum allocaton of resources and t s dynamc as t s partcularly useful for problems where decsons are taken at several dstnct stages. Dscrete, contnuous, determnstc, as well as probablstc models can be solved by ths method. A system n ts ntal state, descrbed by a vector s N, fnally reaches the state s 0 as a result of certan decsons denoted by the vector d. The transformaton T N can be functonally explaned as s 0 = TN ( sn, d). Let a real valued functon ψ N( sn, d) called the obectve or the return functon be assocated wth the transformaton ( T N ) whch measures the effectveness of the decsons made and the output that results from these decsons. The obectve s to determne a gven nput s N to optmze (mnmze or maxmze) ψ N subect to the constrant s0 = TN( sn, d). Ths multstage problem s decomposed nto stages, where N, and s represents the nput at the th stage. Startng from the ntal state s N, the system s consdered to pass through successve states sn, sn, sn, sn 3,... s, s before reachng the fnal state s 0. Thus each state s s the functon of the nput state s and the decson vector d,.e. s = T ( s, d ). There results a stage return functon f ( s, d ). In addton, the return functon ψ N s a functon of stage returns,.e., ψ N = ψ N( fn, fn,... f, f). From the above dscusson, t would seem to suggest that f ψ N s of the form ψ N = fnο fn... fο f where ο represents a composton operator ndcatng ether addton or multplcaton, then ψ N = fnο ψ N, where ψ N = f N ο fn... fο f. It s possble to separate all ψ N, ψ N,... ψ successvely n ths order, and thus the recursve equaton may now be proposed as F ( s) = mn[ f ο F ( s )], N Wth F( s ) = mn f d d subect to s = T( s, d ), N. Ths type of approach consttutes the backward recurson. Ths backward recurson can be convenently used only when optmzaton wth respect to a specfc nput s N s needed, and the output s 0 s not consdered. To optmze the system wth respect to a prescrbed output s 0, t would therefore be natural to reverse the drecton,.e. treat s as the functon of s and d, and substtute s = T( s, d ), N. Also express stage returns as functons of stage output and then proceed from stage N to stage. Such a procedure, known as the forward recursve approach s adopted n ths paper. III. FORMULATION OF ECED PROBLEM WITH LIMITED LOSSES A fnal operatng condton wth mnmum producton cost at reduced emsson rate whle achevng an acceptable loss value leads to a mult-obectve problem whch s dealt successfully n ths work. In general, an emsson constraned economc dspatch problem starts wth a mathematcal cost equaton () [$/hr], whch s modeled to represent each ndvdual generator n terms of generaton and cost coeffcents and a mathematcal emsson equaton () [kg/hr], whch s

3 4 IRANIAN JOURNAL OF ELECTRICAL AND COMPUTER ENGINEERING, VOL. 6, NO., SUMMER-FALL 007 formulated to relate the emsson coeffcents wth the ndvdual generaton. F ( P) = a P bp c () E ( P ) = d P ep f () where, P s the ndvdual generaton from unt ; a, b and c are ts cost coeffcents and d, e and f are ts emsson coeffcents. Transmsson losses are gven by P l, where Pl = P BP. Here P and B are n the form of matrces representng power generaton and transmsson loss coeffcents respectvely. And P s the transpose of P. A load balance equaton wll mpose constrant upon generaton as n P Pl Pd = 0 (3) = where P d s the total system load demand and P l s the transmsson loss. A generaton lmt wll also act as a constrant over the operatng range of ndvdual generators.e. P P P (4) mn max An apprecable ncrease n the volume or weght of emsson s governed by the magntude of generaton, whch n turn governs the cost and hence the economcal operaton of the system. These costs are coordnated wth a penalty cost of emsson (h), gven by h = F E (5) where F, E are the cost and emsson respectvely correspondng to specfc condtons of generaton ncludng the lmts of generaton (mnmum and maxmum) and average costs of generaton as gven below and dscussed fully n [3] h = F E h = F E max max max mn mn mn n ave = ( max mn ) com = ( ave) = h h h h h n For a sample three generator system, the emsson constraned cost equaton [$/hr] s 3 t = = f (( ap bp c) h( dp ep f)) (6) The cost of transmsson losses n between the plants s accounted wth the actual fuel costs by a prce factor (g) whch s taken as g = F P (7) where, F & P are n turn the cost and generaton correspondng to th generator for specfc condtons of generaton ncludng the lmts of generaton and the average costs of generaton as gven n (8). gmax = Fmax Pmax gmn = Fmn Pmn n (8) gave = ( gmax gmn ) gcom = ( gave) n = For a sample three-generator system, modfed form of cost equaton [$/hr] now becomes 3 3 ( a hd ) P ( b he ) P ft = ( c h f ) g ( PB P ) = (9) = = 3 3 ( ap bp c g( PB P) = = where, a = ( a hd ), b = ( b he ) and c = ( c h f). Now the loss formula for the frst generator can be modfed as PL = PB P = PB P P B ( P P ) P B3( P3 P ) = PB ( α α α3) = PB (0) where, B s the modfed form of self-coeffcent. Usng ths, cost equaton [$/hr] for the frst generator can be rewrtten as f = ( ap bp c g( P B)) = (( a gb) P bp c) Substtutng a = a gb, we arrve at the cost equaton for the frst generator as f = ( ap bp c) () Smlar cost equatons can be derved for the second and the thrd generators respectvely. The whole formulaton as observed s purely analytc n nature wth hgh possblty for accurate solutons. A best choce s chosen for the penalty factor of emsson and the prce factor of loss. An optmum penalty factor for emsson arrved at a prevous paper [4] s made use of n ths work. IV. IMPLEMENTATION OF RECURSIVE APPROACH TO ECED PROBLEM Let s be the output from the th stage of ths multstage problem. Consderng the three-stage system, s 3 s the output from the thrd stage and t equals P P P3. Outputs from earler stages are respectvely, s = P P = s3 P3 (for the second stage) and s = P = s P (for the frst stage). Now f ( s ) = mn ( a P bp c ) () 0< P< s Snce c, c and c 3 are constants, they are removed from the respectve equatons and ther sum can be added to the cost equaton [$/hr] at the end. f ( s ) = mn ( a P b P f ( s )) 0< P< s mn ( ap bp 0< P< s f( s P)) ap bp a s P b s 0< P< s P = = mn ( ( ) ( )) (3) For the second generator, mnmum s attaned when the above equaton (3) s dfferentated wth respect to P and equated to 0. Ths gves the value of P n terms of s and constant,.e. P = As B (4) where A = a (a a) and B = ( b b) (a a) Smlarly for the thrd generator,

4 MURALIDHARAN et al.: A NOVEL PARETO-OPTIMAL SOLUTION FOR MULTI-OBJECTIVE ECONOMIC DISPATCH PROBLEM 5 TABLE I COST, EMISSION COEFFICIENTS AND POWER LIMITS FOR SIX GENERATOR SYSTEM Unt a b c d e f Mn load (MW) Max Load (MW) TABLE II LOSS COEFFICIENTS FOR SIX-GENERATOR SYSTEM ( 0-6 ) f ( s ) = mn ( a P b P f ( s )) $/hr 3 3 0< P3< s = a3 P3 b3 P3 f 0< P3< s3 s3 P3 mn ( ( )) $/hr Solvng further we arrve at the value of P as 3 " " " " " (a a A 4a A a A ) s (a A B " " ab b ba aab ba b3) P3 = " " " " (a a A 4a A a A a ) 3 3 (5) (6).e., P = As B Substtuton of cost coeffcents, emsson coeffcents and the total load on the system n the above equaton wll provde the optmum generaton for the thrd generator. Proceedng n ths fashon and expandng sequentally as per the equatons F( P) = ap bp c and s = s P where vares from to 6, we can arrve at the equatons for all the sx generators n the gven test system. Substtuton of cost, emsson and loss coeffcents and also the load n the equaton for P wll yeld the generaton of th generator under optmum condton. Ths procedure can also be extended to any n generator system and ths brngs out the effcacy of SADP. The analytcal nature of the method used here ensures hgh accuracy and the same s aptly demonstrated by a flowchart n Fg.. A penalty factor for emsson and a prce factor for transmsson loss have helped the suggested recursve technque to acheve ths smple analytcal form. A trangularzaton has been adopted for the loss coeffcent matrx, whch has made the dscussed dynamc programmng approach also sutable for loss ncluded condton. Whle attemptng to attan the obectve, a suboptmal pont usng the above technque s found for the emsson constraned economc dspatch condton, whch neglects loss. Then the same procedure s appled for emsson constraned economc dspatch condton wth loss, usng modfed form of B-coeffcent matrx [8]. In the subsequent approach, a few teratons are requred so that the same format suts the total generaton wth loss ncluson. Thus, an all round satsfactory performance forms the bass of system plannng. The best performance addresses to all the three obectves mentoned n ths paper, to be at ther best possble values. Snce TABLE III COST, EMISSION AND LOSS (FOR 800 MW) WITH VARIED PRICE FACTORS Cost ($/hr) Emsson (kg/hr) Loss (MW) g mn g max g ave g com TABLE IV COMPARISON OF RESULTS FOR SIX GENERATOR SYSTEM (FOR 700 MW) Conventonal Lambda teratve Method Proposed Method Method n [4] P (MW) P (MW) P3 (MW) P4 (MW) P5 (MW) P6 (MW) Total cost ($/hr) Total Emsson (kg/hr) Total Loss (MW) smultaneous realzaton of ther mnma s mpossble, the comparson chart helps to fnd a near optmal soluton satsfyng mult-obectve crteron, wth a small devaton from ther ndvdual mnma. As a subsequent process, varous values are consdered for prce factor of loss. Ths s done on a systematc bass wheren the cost comparson, emsson comparson and loss comparson are performed. The comparson chart has been proposed for an overall mult-obectve performance montorng. V. RESULTS AND DISCUSSION A study s made on IEEE sx-generator, 30-bus test system [], [6], [9] wth cost and emsson coeffcents and power lmts as gven n Table I and loss coeffcents taken from Table II. Cost, emsson and loss results, for a sample load of 800 MW obtaned by recursve method for varous prce factors are presented n Table III. A sngle prce factor (g) does not provde a soluton for the best confguraton, where all the three obectves are at ther mnmum. It necesstates a comprehensve study about varous prce factors (8). Ths table also helps to dentfy the prce factor wth whch compromse between the obectves are satsfed. Comparson of total cost, total emsson and total loss (for 900 MW) under ECED condton wth loss restrcton for varous prce factor combnatons are graphcally presented n Fg.. Ths pctoral representaton smplfes the task of dentfyng an all round soluton. Results obtaned for 700 MW usng recursve approach have been compared wth that arrved through conventonal Lambda teratve method and quck method [4], n Table IV. Accuracy of the proposed algorthm has been

5 6 IRANIAN JOURNAL OF ELECTRICAL AND COMPUTER ENGINEERING, VOL. 6, NO., SUMMER-FALL 007 Fg. 3. Cost comparson. Fg.. SADP Flowchart. Fg.. Comparson of total cost, total emsson and total loss (for 900 MW). endorsed wth ths table, where the results of the proposed algorthm match wth that of the conventonal and also score over the method n [4]. Comparson of teratons n Table V portrays the superorty n computatonal speed (n terms of the number of teratons requred to arrve at the soluton) of SADP wth that of conventonal Lambda teratve method. Also, Table VI presents the CPU tme requrement for the proposed algorthm when runnng on a computer wth Pentum-IV.5Ghz processor and 8Mb RAM. TABLE V NUMBER OF ITERATIONS REQUIRED FOR VARIOUS LOADS IN SIX GENERATOR SYSTEM Number of teratons Load Proposed Conventonal Lambda teratve method method 500 MW MW MW MW MW TABLE VI CPU TIME REQUIRED FOR SIX GENERATOR SYSTEM UNDER VARIOUS LOADS FOR THE PROPOSED ALGORITHM Load CPU tme n sec. 500 MW MW MW MW MW.0970 The cost and emsson under ECED condton are compared wth ther matchng parts under pure economc dspatch and pure emsson condtons n Fgs. 3 and 4. These graphs llustrate the fact that the proposed algorthm provdes a best compromse n cost and emsson, snce mnmum of both cannot be acheved smultaneously. Fg. 5 clearly proects the reducton n the number of teratons requred for the proposed approach when compared wth the conventonal teratve method. Fg. 6 vsually presents the CPU tme requrement for the SADP approach. VI. CONCLUSION A novel form of dynamc programmng technque s presented n ths paper. All the three factors of power dspatch vz. cost, emsson and loss are combned and the pareto-optmal economc dspatch for emsson constraned and loss-restrcted case s the obectve of ths paper. The twn obectves of cost and emsson are conflctng n nature and a compromse has to be reached to obtan an acceptable power dspatch strategy wthn the varous system constrants. Mnmum cost, mnmum emsson, and mnmum loss condtons are ndvdually of theoretcal fancy and are not a smultaneously realzable phenomenon. Therefore, any economc power dspatch problem must consder emsson propertes and loss ncluson. Emsson lmts, must be

6 MURALIDHARAN et al.: A NOVEL PARETO-OPTIMAL SOLUTION FOR MULTI-OBJECTIVE ECONOMIC DISPATCH PROBLEM 7 Fg. 4. Emsson comparson. Fg. 5. Comparson of number of teratons. Fg. 6. CPU tme requrement for varous loads. satsfed even though there s a slght devaton wth regard to economy. An attempt to reduce loss s mandatory. Whle achevng ths, the regular long and laborous conventonal Lambda approach nvolvng ncremental producton cost and ncremental transmsson losses s totally avoded. In spte of ths, the realzaton of total cost mnmzaton n producton cost, wth an emsson constraned and loss reduced condton has come about. From Table III, the prce factor g mn s found to be an acceptable soluton n realty. It s hoped that the varaton n the penalty cost of emsson for emsson restrcton and varaton n prce factor n chargng transmsson losses can provde better performance soluton. Ths requres an ndepth study for an overall comparson and thus refnement of economc generaton may be a realstc achevement. REFERENCES [] B. H. Chowdhury and S. Rahman, "A revew of recent advances n economc dspatch," IEEE Trans. Power Syst., vol. 5, no. 4, pp , Nov [] J. H. Talaq, F. El-Hawary, and M. E. El-Hawary, A summary of envronmental/ economc dspatch algorthms, IEEE Trans. on Power Syst., vol. 9, no. 3, pp , Aug [3] C. Palanchamy and K. Srkrshna, "Economc thermal power dspatch wth emsson constrant," Journal of Insttuton. of Engneers (Inda), vol. 7, pp. -8, Apr. 99. [4] P. S. Kulkarn, A. G. Kothar, and D. P. Kothar, "Combned economc and emsson dspatch usng mproved Backpropagaton neural network," Electrc Machnes and Power Systems, vol. 8, no., pp. 3-44, Jan [5] T. D. Kng, M. E. El-Hawary, and Feral El-Hawary, "Optmal envronmental dspatchng of electrc power systems va an mproved Hopfeld neural network model," IEEE Trans. on Power Syst., vol. 0, no. 3, pp , Aug [6] Y. H. Song, S. Wang, P. Y. Wang, and A. T. Johns, "Envronmental/Economc dspatch usng Fuzzy logc controlled Genetc algorthms," IEE Proc. Gen., Trans. & Dstrb., vol. 44, no. 4, pp , 997. [7] Y. Denurek and A. Demroren, "Economc and mnmum emsson dspatch," WSEAS Trans. on Systems, vol. 3, no., pp , Jun [8] P. K. Hota, R. Chakrabart, and P. K. Chattopadhyay, "Economc emsson load dspatch wth lne flow constrants usng sequental quadratc programmng technque," Journal of Insttuton. of Engneers (Inda), vol. 8, pp. -5, Jun [9] S. L. Surana, and P. S. Bhat, "Emsson controlled economc dspatch usng Genetc algorthm," Journal of Insttuton. of Engneers (Inda), vol. 8, pp , Mar. 00. [0] J. S. Dhllon, S. C. Part, and D. P. Kothar, "Mult-obectve decson makng n stochastc economc dspatch," Int. J. Electrc Machnes and Power Systems, vol. 3, no. 3, pp , 995. [] J. S. Dhllon, S. C. Part, and D. P. Kothar, "Stochastc economc emsson load dspatch," Electrc Power Systems Research, vol. 6, no. 3, pp , 993. [] M. A. Abdo, "A novel mult-obectve evolutonary algorthm for solvng envronment/economc dspatch problem," n Proc. 4th PSCC Conference, Sevlla, 00. [3] P. G. Lowery, "Generatng unt commtment by Dynamc programmng," IEEE Trans. on Power App. and Syst., vol. 85, no. 5, pp , May 966. [4] A. K. Ayoub and A. D. Patton, "Optmal thermal generatng unt commtment," IEEE Trans. on Power App. and Syst., vol. 90, no. 4, pp , Jun. 97. [5] Z. Ouyang and S. M. Shahdehpour, "An ntellgent dynamc programmng for unt commtment applcaton," IEEE Trans. on Power Syst., vol. 6, no. 3, pp , Aug. 99. [6] C. C. Su. and Y. Y. Hsu, "Fuzzy dynamc programmng: an applcaton to unt commtment," IEEE Trans. on Power Syst., vol. 6, no. 3, pp. 3-37, Aug. 99. [7] Z. -X. Lang and J. D. Glover, "A zoom feature for a dynamc programmng soluton to economc dspatch ncludng transmsson losses," IEEE Trans. on Power Syst., vol. 7, no., pp , May 99. [8] K. Krchmayer, Economc Operaton of Power Systems, New York: John Wley & sons, 958. [9] I. J. Nagrath and D. P. Kothar, Power System Engneerng, New Delh: TMH Publshers, 994. [0] A. J. Wood and B. F. Wollenberg, Power Generaton, Operaton and Control, New York: John Wley Inc., 984. [] R. Bellman, Dynamc programmng, Prnceton Unversty Press, 957. [] D. Bertsekas, Dynamc Programmng: Determnstc and Stochastc Models, Prentce Hall, 987. [3] S. Muraldharan, K. Srkrshna, and S. Subramanan, "A novel dynamc programmng approach to emsson constraned economc dspatch," n Proc. of Int. Conf. on Robotcs, Vson, Informaton and Sgnal Processng, pp. 3-36, Penang, Malaysa, Jul [4] D. P. Kothar and J. S. Dhllon, Power System Optmzaton, Prentce Hall, 004. S. Muraldharan obtaned B.E n Electrcal Engneerng from Madura Kamara Unversty n 994 and M.S n Software Systems from Brla Insttute of Technology & Scence, Plan n 997. He s servng as a

7 8 IRANIAN JOURNAL OF ELECTRICAL AND COMPUTER ENGINEERING, VOL. 6, NO., SUMMER-FALL 007 Selecton Grade Lecturer n the Department of Electrcal & Electroncs Engneerng at MEPCO Schlenk Engneerng College, Svakas, Inda. He s currently pursung hs doctoral research at SASTRA, Tanore. Hs felds of nterest nclude electrcal machnes, power systems, and artfcal ntellgence. K. Srkrshna obtaned B.E n Electrcal Engneerng from Annamala Unversty n 96 and M.E n Power Engneerng from Indan Insttute of Scence n 964. He had hs Doctoral degree n 98. He had hs career as Head of the Department n Thyagaraar College of Engneerng, Madura and later at K.L.N College of Engneerng, Madura tll 000. He served as Prncpal, Sr Sowdambka College of Engneerng n Arupukkota, Inda tll 003. He s presently Prof. & Head, Department of Electrcal and Electroncs Engneerng, K.L.N College of Engneerng, Madura, Inda. Hs felds of nterest nclude electrcal energy conservaton, nducton machnes, and power systems S. Subramanan obtaned B.E n Electrcal Engneerng from Madura Kamara Unversty n 989 and M.E. n Power Systems from Madura Kamara Unversty n 990. He had hs Doctoral degree from Annamala Unversty n 00. He s servng as a Professor n Electrcal Engneerng at Annamala Unversty, Annamala Nagar, Inda. Hs felds of nterest nclude power system economcs, electrcal machne desgn, and voltage stablty.