A Hybrd Model for Congeston Management wth Real and Reactve Power Transacton Narayana Prasad Padhy Yog Ra Sood Abdel Moamen. M. A. Maruth Kumar H. O. Gupta Abstract-- In ths paper, an effcent and practcal hybrd model has been proposed for congeston management analyss for both real and reactve power transacton under deregulated envronment of power system. The proposed hybrd model determnes the optmal blateral or multlateral transacton and ther correspondng load curtalment n two stages. In the frst stage classcal gradent descent optmal power flow algorthm has been used to determne the set of feasble curtalment strateges for dfferent amount of real and reactve power transactons. Whereas n the second stage fuzzy decson opnon matrx has been used to select the optmal transacton strategy consderng ncrease n prvate power transacton, reducton n percentage curtalment, and ts correspondng change n per unt cost and hence proft as fuzzy varables. The performance of the proposed algorthm has been tested usng modfed IEEE 30 bus test system. The solutons so obtaned are found to be qute encouragng and relable refer to both utlty and customers. Index Terms-- Optmal Power Flow, Gradent Descent Algorthm, Congeston Management And Fuzzy Decson Systems. I. INTRODUCTION HE completely unbundled electrc power markets are Tmanly consstng of companes (GENCOs), transmsson companes(transcos), dstrbuton companes (DISCOs), energy brokers and an ndependent system operator(iso). The operaton of transmsson system under TRANSCOs s expected to reman a regulated monopoly whch wll allow open, non-dscrmnatory and comparable access wthout congeston to all producers and consumers of the deregulated system[1]. The term congeston has come to power systems from economcs n conuncton wth deregulaton although congeston was present on power systems before deregulaton. Then t was dscussed n terms of steady state securty, and basc obectve was to control generator output so that the system remaned secured at the lowest cost. Conflcts between securty and economcs could be traded off Narayana Prasad Padhy s wth the Department of Electrcal Engneerng, Indan Insttute of Technology, Roorkee 247667, Inda(nppeefee@tr.ernet.n) H. O. Gupta, Y. R. Sood, Abdel Moamen, M. A. and Maruth Kumar are wth the Department of Electrcal Engneerng, Indan Insttute of Technology, Roorkee 247 667, Inda. wthn one decson-makng entty. Whle ths process sounds qute exact, the expected costs of less secure operaton could not be accurately quantfed and the lmts themselves could develop a great deal of flexblty when there was more to be saved by purchasng them. Here we are concerned wth transmsson congeston and whch s defned as consequence of network constrants characterzng a fnte network capacty that prevents the smultaneous delvery of power from an assocated set of power transactons or when the producers and consumers of electrc energy desre to produce and consume n amounts that would cause the transmsson system to operate at or beyond one or more transmsson lmts, then the system s sad to be congested. So fnally controllng both the and loads so that transfer lmts of the transmsson system are properly taken care s known as congeston management[2]. In the deregulated power system the challenge of congeston management for the transmsson system operator (ISO) s to create a set of rules that ensure suffcent control over producers and consumers (generators and loads) to mantan and acceptable level of power system securty and relablty n both the short term (real-tme operaton) and the long term whle maxmzng market effcency[4]. The rules must be robust, because there wll be many aggressve enttes seekng to explot congeston to create market power and ncreased profts for themselves at the expense of market effcency. The rules should also be far n how they affect partcpant, and they should be transparent, that s, t should be clear to all partcpants why a partcular outcome has occurred. If there s no congeston, there s one zone prce throughout the system, and the generators are pad the same prce for ther energy as the loads pay. When there s congeston, zone prces dffer, each generator pad dfferent prce. Deregulaton n power market leads to decentralzed dspatch of generated power and due to whch, power system operaton poses greatest challenge wthn the compettve market n handlng both blateral and multlateral transactons together or even separately. Transmsson lnes are stressed or congested proportonally wth ncreasng transactons. Whereas the above sad problem can be solved partally by ncreasng the transmsson lne capabltes usng FACTS devces[3] or do not encourage prvate 0-7803-7519-X/02/$17.00 2002 IEEE 1366
transactons. In ths paper t has been suggested to encourage the prvate transactons wth mnmum curtalment on the customers loads. So to counter ths problem, congeston has been managed based on curtalment on loads wthn a specfed operatonal lmt tll we obtan an optmal curtalment strateges for both blateral and multlateral transactons of real and reactve power. There exst three basc types of curtalment strateges mplemented by the ISO n collaboraton wth market partcpants n the optmal transmsson dspatch model. Pont to pont curtalment: The strategy concerns ndvdual contracts n whch the curtalment n load must be same as n a blateral transacton. Group curtalment: The concern of group curtalment s to make possble a group transfer wthout group curtalment. If the power supples wthn a group have to be curtaled then ths shortfall has to be spread across the loads. Separate Curtalment: The concern of ths strategy s to mnmze the change to every nected or extracted power blocks at the generator bus and the load bus of a group based on wllngness to pay factor. ar Curtalment: If requred we may have to curtal all the load ponts lnearly to make the system free from congeston. Specfc Curtalment: Ths concern wth the curtalment of loads normally near congested lnes n the system. In the proposed hybrd model a dfferent procedure has been adopted for chargng the customers under congeston compared to other exstng methods[5-6] such as Prce Area Congeston Management, Buyback Congeston Management and US transacton based soluton. Fnally In developng countres lke Inda the above sad proposals may not be vald due to certan practcal and poltcal reasons. So n the proposed model, along wth lnear curtalment, curtalment on the customers load based on ther mportance and wllngness to pay prce n dfferent buses has been adopted to encourage both blateral and multlateral transactons. If the transacton s possble only through the consumers load curtalment then the proft obtaned through transactons can be transferred to the customers to reduce ther per unt consumpton cost. Among the varous factors consdered to evaluate curtalment strategy for congeston management, the factors lke reducton n power transacton, percentage curtalment and varaton n per unt cost are consdered to be most fuzzy n nature[8-10]. So based on fuzzy opnon matrx approach the optmal transacton and curtalment strategy has been obtaned to help both customers and prvate transacton partes n ther decson makng[10-11]. II. STEP-BY-STEP ALGORITHM OF THE PROPOSED MODEL Step 1: Select the transacton(blateral or multlateral). Step 2: If blateral then enter the bus numbers takng part n the transacton and the load to be transacted. Otherwse enter the bus numbers assocated n the multlateral transacton and the group load that to be transferred. Step 3: Run the OPF for the IEEE 30 bus test system and calculate the per unt cost of the system wth and wthout transacton. 1. Prepare the database for the system ncludng lne data, bus data, generator data and tap settng of the transformers. data ncludes the nformaton of the lnes such as resstance, reactance and shunt admttance. Bus data ncludes the nformaton of the generators, loads connected at each and every bus. The generator data ncludes the cost coeffcent of the generators and real power lmts. 2. Formaton of Y bus usng lne resstance, reactance, shunt elements and tap changng rato. 3. Assume sutable values of voltage magntude at all the buses excludng swng bus and ts angle for all the buses, also set the error for calculated actve and reactve power. 4. Assume a set of control varables U, such as real power (It may be real power or voltage magntude or ts angle at voltage control bus). 5. Calculate real and reactve power usng formula for all buses. P V V Y cos θ + δ δ, =1,2,3...N (1) Q = N N = V V Y ( ) sn ( θ + δ δ ),=1,2,3...N (2) Where N s total no of buses. 6. Calculate error for real and reactve power between specfed and calculated for load buses and only real power for voltage control buses. If t s wthn tolerable lmt goto 11 else contnue the next steps. 7. Calculate Jacoban matrx usng formula P P δ V J 1 J 2 J =. = (3) Q Q J 3 J 4 δ V 8. Calculate voltage magntude and angle ncrement usng formula (except reference bus) θ = V P Q 1 [ J ] (4) 1367
9. Calculate new bus voltage magntude and ts angle on all buses (except refrence bus) V new =V old + V (5) θ new =θ old + θ (6) 10. Goto 5. 11. Calculate [λ] usng equaton 1 g f [ λ] =. x x (7) g where s Jacoban matrx (3); f s the cost functon x of the generators(cost functons are n terms of and voltage & angle n the non-reference bus and at reference bus respectvely). 12. Calculate f usng equaton T f g f= [ λ] u + u (8) 13. If f s close to zero then goto 16 else contnue. 14. Calculate new value of control varables. U new =U old + U (9) Where U = -C. f C s correcton factor 15. Goto 5. 16. Calculate cost usng the formula and stop. Cost= gnbus =1 a 2 p + bp + c $/hr (10) Assumng cost functon s quadratc. 17. Calculate the per unt cost. Step 4. Check for the transmsson lne lmts (MVA lmts) If any of the lne lmts are volated, curtal the remanng loads lnearly, curtal the load at one and both the buses across the congested lne tll congeston s elmnated. Step 5. Check for percentage of curtalment on remanng loads. If t s greater than a fxed percentage (we have consdered t to be 7) then these blateral contracts should not be encouraged and go to step6. Else calculate the correspondng blateral transacton load, percentage of curtalment and per unt cost. Step 6. If percentage of curtalment on remanng loads s more than 7, curtal the transacton power by steps (say 2 of the maxmum power ) tll the system s free from congeston. Step 7. Apply fuzzy opnon matrx approach, dscussed n detal n the secton 4 for optmal selecton of the transacton among feasble transactons. III. RESULTS AND DISCUSSIONS The performance of the proposed hybrd model has been tested wth the modfed IEEE 30 bus test case system. In whch both blateral and multlateral real and reactve power transactons have been ntroduced. In case 1 both real and reactve power blateral transacton between bus No 5 and 22 has been dscussed. Three dfferent curtalment strateges such as lnear, two pont and sngle pont have been adopted for the above case1. Table 1A, 1B and 1C shows the transacton load, percentage curtalment, per unt cost and the lne congested wth real power transfer for lnear, two pont and sngle pont curtalment respectvely, whereas Table 1D, 1E and 1F shows the transacton load, percentage curtalment, per unt cost and the lne congested wth reactve power transfer for lnear, two pont and sngle pont curtalment respectvely. TABLE 1A. BILATERAL TRANSACTION OF REAL POWER BETWEEN BUS NO. 5 AND 22 curtalment n System per unt 25.00 11.00 2.8731 27 24.50 9.50 2.8869 27 24.00 8.50 2.8960 27 23.50 7.00 2.9097 27 23.00 6.00 2.9188 27 22.50 4.50 2.9325 27 22.00 3.00 2.9461 27 21.50 2.00 2.9551 27 21.00 0.50 2.9686 27 20.50 0.00 2.9730 27 27 s n between bus 10 and 21 TABLE 1B. TWO POINT LINEAR CURTAILMENT AT THE BUS NO. 10 AND 21 FOR THE BILATERAL TRANSACTIONOF REAL POWER BETWEEN BUS NO. 5 AND 22 curtalment n 25.00 17.50 2.9603 27 24.50 15.50 2.9617 27 24.00 13.50 2.9631 27 23.50 11.50 2.9646 27 23.00 9.50 2.9661 27 22.50 7.00 2.9680 27 22.00 5.00 2.9694 27 21.50 3.00 2.9709 27 21.00 1.00 2.9723 27 20.50 0.00 2.9730 27 27 s n between bus 10 and 21 1368
TABLE 1C SINGLE POINT CURTAILMENT AT BUS NO. 10 FOR THE BILATERAL TRANSACTION OF REAL POWER BETWEEN BUS NO. 5 AND 22 curtalment n 25.00 17.00 2.9637 27 24.50 15.00 2.9648 27 24.00 13.00 2.9660 27 23.50 11.00 2.9671 27 23.00 9.00 2.9681 27 22.50 7.00 2.9692 27 22.00 5.00 2.9703 27 21.50 3.00 2.9714 27 21.00 1.00 2.9725 27 20.50 0.00 2.9730 27 lne no 27 s between bus no 10 and 21 TABLE 1D BILATERAL TRANSACTION OF REACTIVE POWER BETWEEN BUS NO. 5 AND 22 Load n MVAR curtalment n 30.00 6.50 2.9796 27 29.40 3.50 2.9798 27 28.80 0.50 2.9799 27 28.20 0.00 2.9797 27 27 s n between bus 10 and 21 TABLE 1E TWO POINT LINEAR CURTAILMENT AT THE BUS NO. 10 AND 21 FOR THE BILATERAL TRANSACTION OF REACTIVE POWER BETWEEN BUS NO. 5 AND 22 Load n MVAR curtalment n 30.00 9.50 2.9801 27 29.40 5.00 2.9800 27 28.80 0.50 2.9800 27 28.20 0.00 2.9797 27 27 s n between bus 10 and 21 TABLE 1F. SINGLE POINT CURTAILMENT AT BUS NO. 21 FOR THE BILATERAL TRANSACTION OF REACTIVE POWER BETWEEN BUS NO. 5 AND 22 Load n MVAR Reactve power curtalment n 30.00 6.50 2.9801 27 29.40 3.50 2.9800 27 28.80 0.50 2.9800 27 28.20 0.00 2.9797 27 27 s n between bus 10 and 21 In case 2 both real and reactve power blateral transacton between bus No 8 and 15 has been dscussed. Three dfferent curtalment strateges such as lnear, two pont and sngle pont have been adopted for the above case2. Table 2A, 2B and 2C shows the transacton load, percentage curtalment, per unt cost and the lne congested wth real power transfer for lnear, two pont and sngle pont curtalment respectvely, whereas Table 2D, 2E and 2F shows the transacton load, percentage curtalment, per unt cost and the lne congested wth reactve power transfer for lnear, two pont and sngle pont curtalment respectvely. TABLE 2A BILATERAL TRANSACTION OF REAL POWER BETWEEN BUS NO. 8 AND 15 Congeste d 20.00 3.00 2.9512 18 19.60 1.50 2.9649 18 19.20 0.50 2.9739 18 18.80 0.00 2.9782 18 18 s n between bus 12 and 15 TABLE 2B. TWO POINT LINEAR CURTAILMENT AT THE BUSNO. 12 AND 15 FOR THE BILATERAL TRANSACTIONOF REAL POWER BETWEEN BUS NO. 8 AND 15 curtalment n 20.00 16.50 2.9688 18 19.60 10.50 2.9723 18 19.20 5.00 2.9755 18 18.80 0.00 2.9784 18 18 s n between bus 12 and 15 TABLE 2C. SINGLE POINT CURTAILMENT AT BUS NO. 15 FOR THE BILATERAL TRANSACTION OF REAL POWER BETWEEN BUS NO. 8 AND 15 curtalment n 20.00 11.50 2.9756 18 19.60 7.00 2.9767 18 19.20 3.50 2.9776 18 18.80 0.00 2.9784 18 18 s n between bus 12 and 15 TABLE 2D. BILATERAL TRANSACTION OF REACTIVE POWER BETWEEN BUS NO. 8 AND 15 30.00 8.50 2.9802 18 29.40 4.50 2.9804 18 28.80 0.50 2.9806 18 28.20 0.00 2.9803 18 18 s n between bus 12 and 15 TABLE 2E TWO POINT LINEAR CURTAILMENT AT THE BUS NO. 12 AND 15 FOR THE BILATERAL TRANSACTION OF REACTIVE POWER BETWEEN BUS NO. 8 AND 15 30.00 95.00 2.9805 18 29.40 52.00 2.9802 18 28.00 4.00 2.9806 18 28.20 0.00 2.9803 18 18 s n between bus 12 and 15 1369
TABLE 2F. SINGLE POINT CURTAILMENT AT BUS NO. 15 FOR THE BILATERAL TRANSACTION OF REAL POWER BETWEEN BUS NO. 8 AND 15 30.00 50.00 2.9806 18 29.40 26.00 2.9806 18 28.00 2.00 2.9806 18 28.20 0.00 2.9803 18 18 s n between bus 12 and 15 In case 3 both real and reactve power multlateral transacton between bus No 4,15 and 24, 27 has been dscussed. Two dfferent curtalment strateges such as lnear and sngle pont have been adopted for the above case3. Table 3A and 3B shows the transacton load, percentage curtalment, per unt cost and the lne congested wth real power transfer for lnear and sngle pont curtalment respectvely, whereas Table 3C and 3D shows the transacton load, percentage curtalment, per unt cost and the lne congested wth reactve power transfer for lnear and sngle pont curtalment respectvely. TABLE 3A MULTILATERAL TRANSACTION OF REAL POWER BETWEEN BUS NO. 4, 15(13, 11 MW INITIAL) AND 24, 27(15, 9 MW INITIAL) 24.00 19.50 2.7990 31 23.52 17.50 2.8180 31 23.04 15.50 2.8370 31 22.56 13.50 2.8558 31 22.08 11.50 2.8746 31 21.60 9.50 2.8932 31 21.12 7.00 2.9165 31 20.64 5.00 2.9349 31 20.16 2.50 2.9578 31 19.68 0.50 2.9759 31 19.20 0.00 2.9803 31 31 s n between bus 24 and 22 TABLE 3B SINGLE POINT CURTAILMENT AT BUS NO. 24 FOR THE MULTILATERAL TRANSACTION OF REAL POWER BETWEEN BUS NO. 4, 15(13, 11 MW INITIAL) AND 24, 27(15, 9 MW INITIAL) 24.00 29.50 2.9733 31 23.52 26.00 2.9741 31 23.04 23.00 2.9748 31 22.56 20.00 2.9756 31 22.08 16.50 2.9765 31 21.60 13.5 2.9772 31 21.12 10.5 2.9779 31 20.64 7.50 2.9787 31 20.16 4.00 2.9795 31 19.68 1.00 2.9802 31 19.20 0.00 2.9803 31 31 s n between bus 24 and 22 TABLE 3C MULTILATERAL TRANSACTION OF REACTIVE POWER IN BETWEEN BUS NO. 4, 15(13, 11 MVAR INITIAL) AND 24, 27(15, 9 MVAR INITIAL) Load n MVAR LINE CONGESTED 24.00 37.50 2.9772 31 23.52 30.50 2.9775 31 23.04 23.00 2.9779 31 22.56 16.00 2.9782 31 22.08 9.00 2.9787 31 21.60 1.50 2.9792 31 21.12 0.00 2.9792 31 31 s n between bus 24 and 22 TABLE 3D SINGLE POINT CURTAILMENT AT BUS NO. 24 FOR THE MULTILATERAL TRANSACTION OF REACTIVE POWER IN BETWEEN BUS NO. 4, 15(13, 11 MVAR INITIAL) AND 24, 27(15, 9 MVAR INITIAL) Load n MVAR curtalment n 24.00 27.50 2.9795 31 23.52 22.50 2.9795 31 23.04 17.00 2.9795 31 22.56 12.00 2.9794 31 22.08 6.50 2.9794 31 21.60 1.50 2.9793 31 21.12 0.00 2.9792 31 31 s n between bus 24 and 22 IV. FUZZY OPINION MATRIX APPROACH Among the varous factors consdered to evaluate a transacton, the factors lke ncrease n prvate power transacton, reducton n customers load curtalment and proft based on the transacton are consdered to be most fuzzy n nature. In the ordnary set representaton, f any transacton leads to no curtalment then ts grade s 1 otherwse t wll be 0. The fuzzy values and the relatve grades for the above mentoned fuzzy varables are gven by Fuzzy Values Relatve grades Excellent 0.9 Very Good 0.8 Good 0.6 Average 0.5 Far 0.4 Bad 0.2 In general the opnon of the publc and utlty vares due to so many reasons and to avod such dscrmnatons the author has consdered two dfferent experts n decson makng one belongs to utlty and the other from publc. To test the performance of the proposed model the results obtaned for Case 1 and Table 1A has been consdered. Only seven transacton cases wth curtalment percentage less than 7 have been consdered for analyss. The experts opnon about the transactons based on the reducton n customers load curtalment s gven below n the form of a matrx. 1370
Reducton n load curtalment Utlty Expert(E1) Publc Expert(E2) T1 0.9 0.2 T2 0.8 0.4 T3 0.6 0.5 T4 0.5 0.6 T5 0.5 0.8 T6 0.4 0.9 T7 0.2 0.9 Increase n power transacton: Utlty Expert(E1) Publc Expert(E2) T1 0.8 0.6 T2 0.9 0.6 T3 0.8 0.5 T4 0.6 0.4 T5 0.5 0.4 T6 0.4 0.2 T7 0.2 0.2 Proft: Utlty Expert(E1) Publc Expert(E2) T1 0.6 0.8 T2 0.6 0.6 T3 0.6 0.4 T4 0.5 0.4 T5 0.4 0.4 T6 0.4 0.2 T7 0.2 0.2 The fuzzy decson set regardng reducton n customers load curtalment s obtaned by usng Hurwtz rule[10], HV = αh + ( 1 α) L Where α s the optmsm pessmsm ndex and taken as 0.8. Now for reducton n customers load: The hghest grade for all the transactons are obtaned as below: H = (0.9, 0.8, 0.6, 0.6, 0.8, 0.9, 0.9) Smlarly the lowest grades for all the canddates are obtaned as: L = (0.2, 0.4, 0.5, 0.5, 0.5, 0.4, 0.2) The hurwtz fuzzy decson set based on reducton n customers load curtalment s, F c =[(0.76, T1), (0.72, T2), (0.58, T3), (0.58, T4), (0.74, T5), (0.80, T6), (0.76, T7)] For ncrease n prvate power transacton: The hghest grade for all the transactons are obtaned as below: H = (0.8, 0.9, 0.8, 0.6, 0.5, 0.4, 0.2) Smlarly the lowest grades for all the canddates are obtaned as: L = (0.6, 0.6, 0.5, 0.4, 0.4, 0.2, 0.2) The Hurwtz set for ncrease n prvate power transacton s, F t =[(0.76, T1), (0.84, T2), (0.74, T3), (0.56, T4), (0.48, T5), (0.36, T6), (0.20, T7)] For overall proft: The hghest grade for all the transactons are obtaned as below: H = (0.8, 0.6, 0.6, 0.5, 0.4, 0.4, 0.2) Smlarly the lowest grades for all the canddates are obtaned as: L = (0.6, 0.6, 0.4, 0.4, 0.4, 0.2, 0.2) The Hurwtz set for ncrease n overall proft s, F p =[(0.76, T1), (0.60, T2), (0.60, T3), (0.48, T4), (0.40, T5), (0.36, T6), (0.20, T7)] The overall fuzzy decson set has been obtaned as follows: F = [F c,f t,f p ] = [(0.76, T1), (0.84, T2), (0.74, T3), (0.58, T4), (0.74, T5), (0.80, T6), (0.76, T7)] From the above decson set, transacton T2 has the hghest grade and hence t may be selected. V. CONCLUSIONS In the proposed hybrd model transactons those are makng the system free from congeston as well as helpng the utlty n ganng fnancal beneft have been encouraged. Based on the classcal programmng multple solutons are obtaned consderng a tolerable percentage of curtalment on loads. To overcome the above sad dffcultes and specfy the best transacton that to be encouraged has been evaluated usng a smple fuzzy opnon matrx. The proposed model performed effcently wth IEEE 30 bus test system and the same can be extended to any practcal network. The proposed model s manly free from complex mathematcal formulatons and provdes qute encouragng results. VI. REFERENCES [1] Perveen Kumar and S. C. Srvastava. Congeston Management n Deregulated Market A case study on an Indan Power System. NPSC, Bangalore, Inda, Dec.2000, pp. 191-196. [2] R. S. Fang and A. K. Davd. Transmsson Congeston Management n Electrcty Market. IEEE Transactons on Power System, Vol.14, No.3, August 1999, pp. 877-883. [3] Kankar Bhattacharya and Jn Zhong. Reactve Power as an Ancllary Servce. IEEE Transactons on Power System, Vol.16, No.2, May 2001, pp. 294-300. [4] Harry Sngh, Shangyou and Alex papalexopalos. Transmsson Congeston Management n Compettve Electrcty Market. IEEE Transactons on Power System,Vol.13, No.2, May 1998, pp. 672-680. [5] R. S. Fang and A. K. Davd. Optmal Dspatch Under Transmsson Contracts. IEEE Transactons on Power System, Vol. 14, No. 2, May 1999, pp. 732-737. [6] Rchard P. Chrste, F. Wollenberg and Ivar Wangensteen. Transmsson Management n the Deregulated Envronment. Proceedng of IEEE, Vol.88, No.2, Feb.2000, pp. 170-195. [7] Hermann W. Dommel and Wllam F. Tnney. Optmal Power Flow Solutons. IEEE Transactons on Power Apparatus and Systems, Vol. PAS-37, No.10, October 1968, pp. 1866-1876. [8] Deutsch, S. J. and Malmborg, C., A Fuzzy Set Approach to Data Set Evaluaton for Decson Support, IEEE Transacton on Systems, Man and Cybernetcs, Vol. SMC-15, No. 6, Nov/Dec 1985, pp. 777-783. [9] Hauffman, A. Theory of Fuzzy Sets, Volume 1, Academc Press, New York, 1975. [10] Ramachandran, V. and Sankaranarayana, V. Fuzzy Concepts Appled to Statstcally Decson Makng Methods, Proceedngs of 15 IFIP Internatonal Conference, 1991. [11] Rau, N. S., Transmsson Loss and Congeston Cost Allocaton An Approach Based on Responsblty VII. BIOGRAPHIES Narayana Prasad Padhy obtaned hs degree of engneerng and Master of Engneerng n 1990 and 1993, respectvely. In 1997, he obtaned hs Ph.D. degree from Anna Unversty, Chenna, Inda. He oned Brla 1371
Insttute of Technology & Scence as an Assstant Professor n 1997. He s presently workng as assstant professor wth the department of Electrcal Engneerng, Indan Insttute of Technology, Roorkee, Inda. He taught course n Basc Electrcal Engneerng, Power Systems and Artfcal Intellgence. Hs feld of nterest s FACTS, Restructrng and Wheelng studes of Deregulated Power System and Artfcal Intellgence Applcatons to Power System Optmzaton Problem, 1372