Equivalent Line Limit Calculation Using Available Thermal Transfer Capability

Transcription

1 Equvaent Lne Lmt Cacuaton Usng vaabe herma ransfer Capabty Wonhyeok Jang Department of Eectrca and Computer Engneerng Unversty of Inos at Urbana-Champagn Urbana, Champagn County homas J. Overbye Department of Eectrca and Computer Engneerng Unversty of Inos at Urbana-Champagn Urbana, Champagn County bstract Poer system equvaents have been used for faster smuatons, but not much nterest as on cacuatng equvaent ne mts and ths mts the use of equvaents n smuatons. uthors used tota transfer capabty n assgnng equvaent ne mts n prevous papers and ths paper utzes avaabe transfer capabty n dong so. detaed procedure of the proposed method s descrbed th a sma -bus case. n exact souton case and a non-exact souton case exampes are provded. For nonexact souton cases, soutons are bounded th upper and oer estmates. so, the proposed method apped to the IEEE 8-bus case and resuts are provded and compared th those th tota transfer capabty. Keyords Equvaent ne mts; avaabe transfer capabty; poer system equvaent, poer transfer dstrbuton factors. I. INRODUCION Poer system equvaents (PSEs) have been used for decades snce the frst equvaencng method named after the deveoper Ward as ntroduced n ate 90s []-[]. hs tradtona Ward method and ts varants are based on the Kron reducton hch performs Gaussan emnaton on a part of netork (caed externa area) to equvaent the components n that area and retans the area of nterest (caed nterna or study area) for study [5]. Boundary buses beteen the to areas are attached th ney created generators and oads to keep the ne fo from the externa area to the nterna area. s the externa buses are equvaenced th the Kron reducton, fcttous equvaent nes are created beteen ther frst neghbor buses and they do not have assocated therma mts. It s easy to make a sma equvaent case here t s not possbe to assgn mts on equvaent nes that resut n a the operatng ponts same as those n the fu system. In genera, commerca toos assgn zero to equvaent ne mts hch ndcates any amount of poer can fo on those nes. hs aspect restrcts the use of PSEs as ne mts pay a key roe n many poer system anayses. Other recent methods have been deveoped to create PSEs that capture dfferent attrbutes of the fu system such as poer transfer dstrbuton factors (PDFs), ocatona margna prces, and optma poer fos [6]-[8]. Even though other equvaencng methods have been deveoped for varous appcatons not much of nterest have been focused on mts of equvaent nes. Recenty, authors have produced a coupe of methods that cacuate equvaent ne mts usng tota transfer capabty [9]- []. hose methods use PDFs to keep the tota transfer capabty (C) beteen the boundary buses. hose methods ony use the netork tsef droppng a the bus njectons hence are operatng pont ndependent. When there s no exact souton, they offer upper, oer and/or best estmates hen equvaent ne mts do not have exact soutons. Usng unoaded systems s good n one aspect as they ony focus on netork and therefore may be more fexbe th varous fo patterns. Hoever, a probem th usng C for unoaded systems n cacuatng equvaent ne mts occurs hen there are movements of bus njectons hch frequenty occur n equvaencng. fter they assgn mts on equvaent nes and reoad the netork and t can be chaengng to sove poer fo especay hen many generators and oads n the externa area are moved to boundary buses or nto the study area n arge cases. Of course, t s more dffcut to sove optma poer fo thout consderng these movement of bus njectons. herefore, ths paper orks th oaded systems consderng a the bus njectons and ther movement n cacuatng equvaent ne mts. he proposed method n the paper uses avaabe transfer capabty (C). ccordng to the defnton of C n the NERC document, C consders exstng transmsson commtments (EC), capacty beneft margn (CBM) and transmsson reabty margn (RM) []. C C EC CBM RM () When cacuatng Cs and Cs, there are a fe system constrants typcay consdered; votage mts, anguar stabty mts, therma mts, and pre- and post-contngency condtons. Snce ony the therma mts are consdered n ths paper, avaabe therma transfer capabty (C) s defned as the C th ony consderng therma mts. so, the to margns are gnored for smpcty. II. PROPOSED MEHOD. Smuaton Setup he Max/Hungaran method presented n [9] s used n the paper n cacuatng equvaent ne mts th C. Hoever, assumed smuaton envronment s the same as that n the topdon approach presented n [] as t s better to be used for arge-scae systems n future appcatons. hs means that ony the fu case and ts reduced equvaent case are needed and no /7/$ IEEE Copyrght (c) 07 IEEE. Persona use of ths matera s permtted. Hoever, permsson to use ths matera for any other purposes must be obtaned from the IEEE by sendng a request to pubspermssons@eee.org. hs paper as presented at the exas Poer and Energy Conference (PEC), Coege Staton, X, February 07.

2 extra nformaton s requred to kno n assgnng mts. Informaton such as the order of bus emnaton or movement of bus njectons from the externa area to the study area are not needed n the smuaton envronment th the top-don approach. he mts of equvaent nes n the reduced case s the objectve to cacuate. the data needed n assgnng equvaent ne mts are n the gven to cases. hs assumpton be benefca for anybody ho ants to use ths proposed method as the ony nput s the fu case and the correspondng equvaent case. Of course, the poer fo of the fu case needs to be soved thout an error because ths method uses C hch requres the amount of ne fos. B. Crteron of Equvaent Lne Lmt Cacuaton he crteron for the successfu equvaent ne mt cacuaton s bus-to-bus C matchng beteen the fu and the equvaent cases. More specfcay, the method tres to match the Cs beteen pars of buses for equvaent nes n the equvaent case th those for the same bus pars n the fu case. Here, for equvaent ne mt cacuaton, ony the nes that are emnated aong th the externa buses denoted by L = { [, L]} are consdered n cacuatng Cs n the fu case. he retaned nes do not need to partcpate n the process of equvaencng and cacuatng equvaent ne mts as they are unchanged th ther mt n the reduced case. herefore, the Cs of ony emnated nes are used n cacuatng equvaent ne mts. he C for emnated ne for transacton, denoted as can be cacuated th the mt of ne, denoted as F, PDF of ne for the same transacton, denoted as φ, and the exstng commtment on ne, denoted as M. F,, F, When a PDF vaue s cose to zero, the transacton has tte mpact on the mtng ne. It shoud be noted that snce C s dependent of the drecton of transactons, the C vaues of a transacton from bus x to bus y s dfferent from those of a transacton from bus y to bus x. he C for transacton s the mnmum of a the Cs of the emnated nes. L () mn () hs ndcates ho much more poer can be transferred for transacton through the netork oaded th the aready exstng commtment before there s any voaton on ne mts n the system. PDF vaues used n cacuatng C are obtaned th the ossess dc approxmaton hch ony consders rea poer []. C. Probem Formuaton When a bus or a group of adjacent buses, k, are emnated n the fu case and there are equvaent nes n the reduced case created beteen the frst neghbor buses of k, as denoted by S = {s [, S]}. he set of equvaent nes are denoted by L = { τ τ [, L ]} and there are L = ( S ) number of mts to be determned. In the reduced case, the C vaues of transactons of bus pars for equvaent nes ony consderng poer fong on equvaent nes τ can aso be cacuated as F,, F, here tde ndcates vaues are from the reduced case and F τ are the unknon equvaent ne mts to be cacuated. he C for transacton n the reduced case s the mnmum of a the Cs of the equvaent nes τ. mn (5) L Snce the crteron of assgnng equvaent ne mts s to match Cs beteen bus pars for equvaent nes th those for the same bus pars n the fu case, exact souton cases occur hen () (6) s C vaues are drecton dependent, there shoud be a reference drecton for transactons so that the cacuaton of PDF vaues and C vaues have a consstent sgn conventon. hs can be done by sovng the reduced case and set the drecton of equvaent ne fos as the reference for the transactons of nterest. One may argue that the other drecton aso be consdered. Combnaton of both drectons of fo n each transacton can be computatonay very heavy n arge scae cases hence ths paper focuses on a snge drecton based on the soved poer fo. (6) can be soved for the ony unknon, equvaent ne mt dependng on the sgn of PDF vaue on the correspondng equvaent ne n the reduced case as, F, (7), When the PDF vaue of an equvaent ne for a transacton s zero, hch means there s on mpact of the transacton on the ne, the C vaue for the transacton on the ne becomes nfnty as shon n (). herefore, the mt on the equvaent ne shoud be nfnty to accommodate the C vaue. III. EXMPLE CSES -bus test case s used to ustrate the step-by-step procedure of the proposed method. Both an exact souton case

3 and a non-exact souton case are demonstrated th the test case. When there s no exact souton, upper and oer estmates of equvaent ne mts are cacuated usng the Max/Hungaran method presented n [9]. so, the resuts of ts appcaton to the IEEE 8-bus case s provded.. -bus est Case th Exact Souton Fg. depcts the fu -bus case th ne oadngs on the eft and PDF vaues for a transacton from bus to bus on the rght. Lnes have a mt n MV and a reactance n p.u. Bus s the externa area to be equvaenced makng a reduced -bus case n Fg.. sack 7% 5% 70 MV 0. pu 50 % 00 MV % 0% 60 MV 0. pu 50 Mvar sack PDF % PDF 70 MV 0. pu 50 PDF 00 MV PDF PDF 60 MV 0. pu Fg.. Fu -bus case th ne oadng (eft) and PDFs from bus to bus (rght) Fg. aso shos ne oadngs on the eft and PDF vaues for a transacton from bus to bus on the rght. 7% 50 sack 0 % 50 Mvar 50 PDF % sack Fg.. Reduced -bus case th ne oadng (eft) and PDFs from bus to bus (rght) he C vaue for a transacton from bus to bus consderng ony emnated nes can be cacuated usng () and () as (,) mn L 0.5 (,) (,), (,) (,), (,) (,) PDF 9% PDF % PDF mn,, L mn 9.6, 0.5, 559. L Lkese, the other to C vaues can be cacuated and a the C vaues beteen the boundary buses are shon abe. hese C vaues need to be matched th those n the reduced case for the same par of transactons as foos. mn L (9) PDF 50 Mvar 50 Mvar (8) BLE CS BEWEEN BOUNDRY BUSES IN -BUS CSE FOR EXC SOLUION Bndng ne () (, ) (, ) 0.5 (, ) (, ) 0. (, ) (, ) 8.6 he drecton of transactons are determned by the drecton of fos on the equvaent nes n the reduced case. (9) can be expanded for the -bus test case as (,) (,) (,) (,) mn (,), (,), (,) (,) (,) (,) (,) mn (,), (,), (,) (0) (,) (,) (,) (, mn,, ) (,) (,) (,) Dependng on the sgn of PDF vaues on the equvaent nes n the reduced case, one of the three equatons n () can be used to formuate equaty equatons n (0). he frst equaton for transacton (,) n (0) can be rertten usng (7) and (8) as F(,) (,) F(,) (,) F(,) (,) mn,, 0.5 (,) (,) (,) (,) (,) (,) F(,).0 F(,) 9.7 F(,). mn,, () he denomnators on the eft hand sde can be mutped for each equvaent ne and nequaty constrants can be made as foos. F 6.50, F 7.6, F.50 () (,) (,) (,) hs can be done for the other to equatons and a the nequaty constrants can be put n a matrx as () Each ro corresponds to each transactons and each coumn corresponds to each equvaent ne mt. so, a the entres n each coumn has to be smaer than or equa to the correspondng equvaent ne mt. One entry for each ro must have an equaty constrant so that the C for the transacton can be determned. Moreover, one entry for each coumn must have an equaty constrant so that the correspondng equvaent ne mt can be determned. Crteron for exact souton: f the maxmum entry n each coumn beongs to a dfferent ro, those maxmum vaues are the exact souton for the equvaent ne mts. hs s because they satsfy a the equaty and nequaty constrants. Often, just choosng the maxmum vaue n each coumn provdes an exact souton. Hence, ths -bus test case has an

4 exact souton as F (,) = 6.50, F (,) =.5, and F (,) =.7. From experment of other cases, entres n the Ps matrx can be a negatve vaue hch s not possbe hen usng Cs for unoaded systems. Negatve entres may be converted to zero as a ne mt shoud be a postve vaue from a practca pont of ve. he C vaues th the exact souton mts can be cacuated n the reduced case usng () and (5) to see f they reay yed the exact C vaues from the fu case. he C vaue for transacton from bus to bus s cacuated for vadaton as foos. (,) (,) (,) (,) mn,, 0.5 (,) (,) (,) mn,, () mn 0.5, 89., 86. B. -bus est Case th Non-Exact Souton Not every emnaton of a bus or a group of buses resuts n an exact souton case. hs s because equvaent ne mts are not cacuated the ay equvaent ne admttances are cacuated by Kron reducton durng the equvaencng process. Fg. shos a fu -bus test case th ne oadng on the eft and PDF vaues for a transacton from bus to bus on the rght. Bus s the externa bus to be emnated from the fu case to create a reduced -bus case n Fg.. sack 65% 60% 70 MV 0. pu 50 Mvar 00 MV 50 Mvar % % 60 MV 0. pu sack PDF % PDF 70 MV 0. pu 50 Mvar 00 MV PDF 50 Mvar PDF PDF 60 MV 0. pu Fg.. Fu -bus case th ne oadng (eft) and PDFs from bus to bus (rght) Fg. aso shos ne oadng on the eft and PDF vaues for the same transacton as the fu case on the rght. he dfference from the exact souton case above s that bus no has a generator attached to t njectng poer nto the system and bus does not have a generator any more. When a bus th bus njecton s beng emnated durng an equvaencng process, the Ward equvaent and ts varant spt the bus njecton and attach a porton of t to ts frst neghbor buses to mantan fo on retaned nes unaffected. nother ay s to just remove t from the externa area hen the bus njecton s far from the study area so that t does not have much mpact on fdety of smuaton resuts. so, some appcaton moves the bus njecton as a hoe to a bus n the study area and attach oads n other buses to keep the same ne fos []-[5]. No matter ho bus njectons from the externa area are redstrbuted nto the study area durng equvaencng, the proposed method does not need to kno. It just needs a fu case and a reduced case to assgn equvaent ne mts to keep the C vaues as cose as those n the fu case. sack 65% 50 Mvar 0 9 % sack 50 Mvar PDF % Fg.. Reduced -bus case th ne oadng (eft) and PDFs from bus to bus (rght) When a bus th bus njecton s beng emnated durng an equvaencng process, the Ward equvaent and ts varant spt the bus njecton and attach a porton of t to ts frst neghbor buses to mantan fo on retaned nes unaffected. nother ay s to just remove t from the externa area hen the bus njecton s far from the study area so that t does not have much mpact on fdety of smuaton resuts. so, some appcaton moves the bus njecton as a hoe to a bus n the study area and attach oads n other buses to keep the same ne fos []-[5]. No matter ho bus njectons from the externa area are redstrbuted nto the study area durng equvaencng, the proposed method does not need to kno. It just needs a fu case and a reduced case to assgn equvaent ne mts to keep the C vaues as cose as those n the fu case. he C vaues beteen the boundary buses of bus can be cacuated usng () and () as shon n abe. s ths s dfferent ne fos from the exact souton case, the C vaues are aso dfferent. PDF 9% PDF % BLE CS BEWEEN BOUNDRY BUSES IN -BUS CSE FOR NON-EXC SOLUION Bndng ne () (, ) (, ) 87.0 (, ) (, ) 76. (, ) (, ) 69.6 he Ps matrx for nequaty constrants can be obtaned th (0) as foos PDF PDF (5) No, choosng the maxmum vaue n each coumn does not meet the Crteron for exact souton as both the nd and rd

5 coumn has ther maxmum vaues at the nd ro. hs does not satsfy equaty constrants for transacton (,), hch s the rd ro. he poer fo for the transacton s overestmated as no entry n the ro s enforced. herefore, ths set of entres s the upper estmate of equvaent ne mts; F (,) = 0.7, F (,) =.09, and F (,) = 8.0. Loer estmates can be obtaned by enforcng the equaty constrant for a transactons. hs causes some of the nequaty constrants to be voated as ths souton underestmate the poer fo n some transactons. Gven the Ps matrx for the non-exact souton case, the Hungaran method s utzed for underestmate [6]. he Hungaran method aso knon as Kuhn-Munkres agorthm s a combnatora optmzaton agorthm to sove assgnment probems. he frst step s to create a matrx th mt voaton cost for each entry. he mt voaton cost s defned by the sum of voaton for a entres n each coumn. Let the Ps be an m by n matrx and then the procedure to obtan the matrx th mt voaton cost s as foos. a. Create a matrx th the same sze as the Ps matrx. b. Wth the st entry n each coumn n Ps, compare th a the other entres n ts coumn. c. Sum the dfferences beteen tsef and arger entres n ts coumn and put the vaue n the ne matrx. d. Repeat b and c for a the other entres n each coumn. he matrx th mt voaton cost for the -bus non-exact souton case s obtaned th the above procedure as (6) No, ths becomes a resource assgnment probem here one entry n each ro and each coumn has to be chosen to that the sum of a the chosen entres s mnmzed. ppyng the Hungaran method, the seected entres are zero n the st and the nd coumn and 5. for the rd coumn. hs combnaton mnmzes the sum of mt voaton cost. herefore, the oer estmates for equvaent ne mts are F (,) = 0.7, F (,) =.09, and F (,) =.6. he mt for equvaent ne (,) s the ony dfference from the overestmates. he C vaues th the upper estmates and oer estmates are cacuated n the reduced case usng () and (5) to see ho much Cs are overestmated and underestmated, respectvey, comparng th those n the fu case as shon n abe. BLE COMPRISON OF CS BEWEEN HE FULL ND HE REDUCED CSE FOR NON-EXC SOLUION Fu -bus case () / overestmate Reduced -bus case / underestmate (,) (,) (,) When the maxmum entry n each coumn n the Ps matrx s chosen for upper estmates, the C for transacton from bus to bus s overestmated about 9. he th the oer estmates determned th the Hungaran method, the C for transacton from bus to bus s underestmated about 8.%. C. IEEE 8-bus Case he proposed method s apped to the IEEE 8-bus case. he buses to be emnated are seected as the same buses used n [0] to compare the resuts. Whe [0] uses the unoaded IEEE 8-bus case ony consderng the netork tsef, ths paper uses the hoe netork. he Ward equvaencng s used to make a reduced 6-bus case and bus njectons n the externa area are redstrbuted nto the boundary buses n the reduced case. so, the same groupng of maxma adjacent externa buses s performed as [0] resutng n mutuay ndependent subgroups of buses to be emnated. Fg. 5 shos the resut of the emnatng subgroups from the IEEE 8-bus case usng C. he numbers of x-axs ndcate the bus numbers n each subgroup. Number of boundary buses of each group s shon th crces and normazed rms C vaues for each group n the reduced cases are shon th squares. Normazed rms C Exact mts Upper/Loer mts Number of boundary buses Number of boundary buses Fg. 5 Resut of emnatng subgroups of 56 buses n IEEE 8-bus case usng C

6 n nterestng pont n ths resut s that the groups th nonexact souton are dfferent from [0]. When usng C n [0], a subgroup th buses 9, 0, and, and the other th buses 8, 9, and are the non-exact souton cases he usng C n ths paper, a subgroup th buses 5,5, 5 and 58, and the other th buses 8,8,8,95,96 and 97 resut n upper and oer mts. Snce subgroup emnatons s used here, the number of f-ns does not need to be consdered hen equvaencng as t s n [0]. here are three subgroups th just one boundary bus. hese subgroups do not have any assocated equvaent nes, thus ther normazed rms C s zero. abe compares ne oadng beteen the fu case and the reduced case hen soved for poer fo. he reduced case th oer mts have one addtona ne overoaded compared to the fu case. hs ne s one of the equvaent nes hen the subgroup of buses 8, 8, 8, 95, 96 and 97 s emnated and ts boundary buses are 77, 80, 85 and 9. Hoever, ths ne s not overoaded hen the reduced case th assgn th upper ne mts. BLE COMPRISON BEWEEN FULL ND REDUCED CSE FOR LINE LODING From o CID Lmt (MV) Loadng (%) Fu case Eq. case / oer mts Eq. case / upper mts IV. COMPUIONL SPEC Let S be the number of boundary buses of a group of adjacent buses that are beng emnated. here are ( S ) number of equvaent nes created beteen the boundary buses. he number of transacton to consder s aso the same. he number of PDF cacuaton s ( S ) [(S ) + S] for both the fu case and the reduced case. he number of C cacuatons n the fu case s ( S ). For non-exact souton cases, the Hungaran method requres the computaton compexty n the order of ( S ) hch s not sma. For arger cases, ths coud be a probematc as the number of boundary buses ncreases. Choosng the group of externa buses th feer number of boundary buses oud be se for faster smuatons. V. CONCLUSIONS hs paper ntroduced a ay to assgn mts on equvaent nes by tryng to keep avaabe therma transfer capabty n the reduced case as cose as possbe to those n the fu case. Gven the fu case and the reduced case, the proposed method can cacuate exact equvaent ne mts f possbe. When exact souton does not exst, the method provdes upper and oer estmates. estng th arger cases shoud be the next step snce computaton oud be very expensve th the current method as the number of boundary buses ncrease. so, arger cases are harder to sove poer fo as more amount of bus njectons are moved to boundary buses. CKNOWLEDGMEN hs ork as supported by the U.S. Department of Energy through the Consortum for Eectrc Reabty echnoogy Soutons (CERS). REFERENCES [] J. B. Ward, Equvaent crcuts for poer-fo studes, EEE rans. Poer pparatus and Systems, vo. 68, pp. 7 8, 99. [] W. F. nney and J. M. Brght, daptve reductons for poer fo equvaents, IEEE rans. Poer Systems, vo. PWRS-, no., pp. 5 60, May 987. [] S. Deckmann,. C. Pzzoante,. J. Montce, B. Stott, O. sac, Studes on Poer System Load Fo Equvaencng, IEEE rans. Poer pparatus and Systems, vo. PS-99, no. 6, pp. 0-0, Nov./Dec [] F. C. schmonet, J. F. Verstege, n externa system equvaent for onne steady-state generator outage smuator, IEEE rans. Poer pparatus and Systems, vo. PS-98, no., pp , 979. [5] G. Kron, ensor nayss of Netorks, Wey, 99. [6] X. Cheng and.j. Overbye, "PDF-based Poer System Equvaents," IEEE rans. Poer Systems, vo. 0, no., pp , Nov [7] S. Hao and. Papaexopouos, "Externa netork modeng for optma poer fo appcatons," IEEE rans. Poer Systems, vo. 0, no., pp , 995. [8] D. Sh, D.L. Shahan, N.L, D.J. yavsky, J.. aber, R.D. Zmmerman and W.D. Schuze, Optma Generaton Investment Pannng: Pt. : Netork Equvaents, n Proc. 0 North mercan Poer Symposum (NPS), Champagn, IL, Sept. 0. [9] W. Jang, S. Mohapatra,. J. Overbye and H. Zhu, Lne Lmt Preservng Poer System Equvaent, n Proc. 0 IEEE Poer and Energy Conference at Inos (PECI), Champagn, IL, - Feb. 0. [0] S. Mohapatra, W. Jang and. J. Overbye, Equvaent Lne Lmt Cacuaton for Poer System Equvaent Netorks, IEEE rans. Poer Systems, Vo. 9, No. 5, pp. 8-6, Sep. 0. [] W. Jang, S. Mohapatra and. J. Overbye, oards a ransmsson Lne Lmt Preservng gorthm for Large-scae Poer System Equvaents, n Proc. Haa Internatona Conference on System Scences (HICSS), Kaua, Haa, 5-8, Jan. 05. [] Whte Paper on the MOD Standards, North mercan Eectrc Reabty Counc, Ju. 0. [] J. Wood and B. F. Woenberg, Poer Generaton, Operaton, and Contro. Ne York: Wey, 996, pp. 08. [] D. Sh, D.L. Shahan, N.L, D.J. yavsky, J.. aber, R.D. Zmmerman and W.D. Schuze, Optma Generaton Investment Pannng: Pt. : Netork Equvaents, n Proc. 0 North mercan Poer Symposum (NPS), Champagn, IL, Sept. 0. [5] J. aber, D. Shahan, R. Zmmerman, C. Marquet, M. Zhang, W. Schuze, R. Schuer, and S. Whtey, Mappng energy futures usng the superopf pannng too: n ntegrated engneerng, economc and envronmenta mode, n Proc. Haa Internatona Conference on System Scences (HICSS), Jan. 7 0, 0, pp [6] H.W. Kuhn, he Hungaran Method for the ssgnment Probem, Nava Research Logstcs Quartery, vo., no. -, pp. 8-97, Mar. 955.