Transmission line parameter identification using PMU measurements

Transcription

1 rnsmission line prmeter identifition using PM mesurements Di hi,*,, Dniel J. lvsk, Kristin M. Koellner, Nim Logi, Dvid E. Wheeler hool of Eletril Computer nd Energ Engineering, Arizon tte niversit, empe, A , A lt iver Projet, Phoeni, A 857-5, A MMA Aurte knowledge of trnsmission line (L impedne prmeters helps to improve ur in rel settings nd power flow modeling. o improve L prmeter estimtes, vrious lgorithms hve een proposed in the pst to identif L prmeters sed on mesurements from Phsor Mesurement nits (PM s. hese methods re sed on the positive sequene L models nd n generte urte positive sequene impedne prmeters for full-trnsposed L when mesurement noise is sent; however these methods m generte erroneous prmeters when the L s re not full-trnsposed or when mesurement noise is present. PM field-mesure dt re often orrupted with noise nd this noise is prolemti for ll prmeter identifition lgorithms, prtiulrl so when pplied to short trnsmission lines. his pper nlzes the limittions of the positive sequene L model when used for prmeter estimtion of L s tht re untrnsposed nd proposes novel method using liner estimtion theor to identif L prmeters more relil. his method n e used for the most generl se: short/long lines tht re full trnsposed or untrnsposed nd hve lned/unlne lods. esides the positive/negtive sequene impedne prmeters, the proposed method n lso e used to estimte the zero sequene prmeters nd the mutul impednes etween different sequenes. his pper lso emines the influene of noise in the PM dt on the lultion of L prmeters. everl se studies re onduted sed on simulted dt from AP to vlidte the effetiveness of the new method. hrough omprison of the results generted this novel method nd severl other methods, the effetiveness of the proposed pproh is demonstrted. KE WOD: PM, P-snhronized phsor mesurement; positive sequene trnsmission line model; trnsmission line impedne prmeters; liner estimtion theor. NODCON Aurte trnsmission line (L impedne prmeters re of gret importne in power sstem opertions for ll tpes of sstem simultions, suh s trnsient stilit, stte estimtion et., nd re used s the sis for protetive rel settings. L prmeters in the pst hve een estimted engineers sed on the tower geometries, ondutor dimensions, estimtes of tul line length, ondutor sg, nd other ftors []. hese lulted prmeters re sed on ssumptions nd pproimtions. With the development of the PM tehnolog, snhronized phsors offer the possiilit of llowing urte estimtion of trnsmission line prmeters. Aurte knowledge of L impedne prmeters helps to: mprove ur in rel settings. mprove post-event fult lotion nd thus led to quiker restortion of the sstems. mprove trnsmission-line modeling for sstem simultions, suh s stte estimtion lultions. Determine when the model for trnsmission line in the entrlized dtse hs not kept pe with modifitions to tht trnsmission line, suh s the insertion of series pitors, etension of the line, *Correspondene to: Di hi, Deprtment of Eletril Engineering, Arizon tte niversit, empe, A , A E-mil: Di.hi@su.edu

2 re-ondutoring of the line, et. everl methods hve een proposed in the pst to identif L prmeters using PM mesurements [3-6]. One two-port ACD prmeter sed method is proposed in [3]. his method utilizes two smples of snhronized mesurements from eh terminl of the L to identif the ACD prmeters; from these hin prmeters the impedne prmeters n e lulted. n this work, we refer to this method s the two mesurement method. Another simpler method proposed in [4] requires onl one smple from the two terminls of L to lulte the L impednes diretl; this method is heneforth referred to s the single mesurement method the uthors. oth methods in [3] nd [4] hve drwks. First the do not perform well when there is noise in the phsor mesurements. eond, these methods re sed on the positive sequene L model whih is suitle onl for full trnsposed L s; when the L s re untrnsposed or not full trnsposed, ppling these methods will led to onsiderle errors in the lulted prmeters. eferene [5] proposes method sed on the distriuted L model nd uses nonliner estimtion theor to generte n optiml estimtor of the fult lotion nd L prmeters. With redundnt smpling of mesurements, this method redues the effets of rndom noise to errors in the lulted prmeters, ut this method still hs limittion sine it is sed on the positive sequene L model onl nd neglets the unlne in the sstem. t is well known tht for full trnsposed L s, the three sequene networks re ompletel deoupled nd the positive sequene impedne prmeters re determined onl the positive sequene voltges nd urrents. However, for untrnsposed L s or L s tht re not full trnsposed, the three sequene networks will e mutull oupled; using onl the positive sequene mesurements in these ses to estimte the positive sequene prmeters will generte inurte prmeter estimtes. n this pper, the limittion of positive sequene L model for the purposes of prmeter estimtion is ddressed nd novel method is proposed. he new method n e used to identif L prmeters for the most generl se: short/long trnsposed/untrnsposed lines with lned/unlned lod onditions. he method n e used to lulte the prmeters for oth short L s with the nominl pi model nd long L s with n equivlent pi model. he new method is sed on the liner estimtion theor nd emplos multiple PM mesurements; it genertes stisftor results even when the mesurements re orrupted with noise. his pper is orgnized s follows: etion introdues the single nd doule mesurement methods. Derivtion of the new method is presented in setion 3 long with disussion of the limittions of the using the positive sequene L model for prmeter estimtion. n setion 4, severl se studies sed on simulted dt from AP re introdued nd the effetiveness of the new method is vlidted. he min onlusions of the work re summrized in setion 5.. HE NLE AND DOLE MEAEMEN MEHOD A generl three phse L model is shown elow in Figure, where ( ( ( ( [ ], ( ( ( ( [ ],,. one smple of mesurements we men the time-domin snhronized phsor mesurements (ontining the sme time stmp of voltge nd urrent tken from ll three phses t oth ends of trnsmission line.

3 Figure. 3-phse trnsmission line model From nodl nlsis, we n write the following two equtions ( nd ( for Figure : ( ( ( ( he voltge nd urrent vriles in the equtions ove re ll phse-frme-of-referene quntities. n order to trnsform these quntities to sequene-frme-of-referene quntities, we need to ppl the phse-to-sequene trnsformtion mtri, whih is defined s: A where j e he following reltionships then hold etween the phse quntities nd sequene quntities: ( ( A ( ( A A A A A multipling oth sides of ( nd ( the mtri A, these equtions n e rewritten s: ( A A ( A A ( (3 ( A A ( (. (4 For full trnsposed trnsmission lines, there re two independent omponents in the phse series impedne mtri, mutul nd, sine self nd self mutul. he phse-frme-of-referene series impedne mtri hs the following form: self mutul mutul mutul self mutul mutul (5 As result, the sequene series impedne mtri for full trnsposed line is well known to e digonl of the form: mutul self

4 where ( i j is the self impedne for eh sequene network ij A A (6 he sequene shunt suseptne mtri is lso found to e digonl following the sme derivtion. herefore, eh of (3 nd (4 n e roken up into three independent equtions, of whih the positive sequene equtions re:. ( (7 ( (8 his mens tht the sequene networks re full deoupled nd the positive sequene impedne prmeters re onl determined the positive sequene voltges nd urrents. Hene, s is well known, for full trnsposed L s nd perfet mesurements (without noise, we n get urte positive sequene impedne prmeters using onl the positive sequene phsor mesurements even if the urrents flowing through the trnsmission line re unlned. As mentioned erlier, two prmeter estimtion methods sed on this positive sequene model hve een proposed; the single mesurement nd doule mesurement methods. he single mesurement method [4] is derived solving (7 nd (8 for nd equtions:. he impedne prmeters n e otined following the two ( ( (9 ( ( Compred with the single mesurement method, the method proposed in [3] (referred to in this pper s the doule mesurement method utilizes two smple of mesurements nd lultes the impedne prmeters using two step proedure. First the ACD prmeters of the L re estimted using the following hin prmeter equtions, whih is sed on the two-port network model []: A ( C D ( A (3 where,, C D (4, Positive sequene phsor mesurements from smple #,,, Positive sequene phsor mesurements from smple #

5 olving the four omple equtions (~(4 with four unknowns sed on Crmer s ule gives: det A (5 det (6 det C (7 where V det V D (8 det V V One the hin prmeters re lulted, the impedne prmeters n e lulted diretl from the following reltionships: A.5 (9 ( C.5 ( ( D.5 ( 3. AN OPMAL PAAMEE EMAON MEHOD FO NANPOED LNE he limittion with the single nd doule mesurement methods (s we will show lter is tht the re sensitive to noise [4], nd re found lking when the trnsmission line is untrnsposed nd operted under unlned onditions. One ojetive of this pper is to rrive t n optiml prmeter identifition method tht not onl is less sensitive to noise, ut lso is pplile to the se where the L s re untrnsposed. Following derivtion similr to tht of the previous setion, when the trnsmission line is not full trnsposed, the sequene impedne mtri hs the following form: A A (3 where ij (i j is the mutul impedne etween different sequene networks. his result is well known: For L tht is untrnsposed or not full trnsposed, the sequene impedne mtri is not digonl nd there is mutul oupling etween the three sequene networks. As result, we will hve to tke into ount of the effets of negtive nd zero sequene omponents when we lulte the positive sequene impednes under lned/unlned loding onditions. 3. Desription of the proposed model n the 3-phse L nominl/equivlent pi model, the shunt dmittne mtri is omprised of two prts: the shunt ondutne (rel prt nd the shunt suseptne (imginr prt. Compred to shunt suseptne, shunt ondutne is negligile, nd thus, in the proposed method, it is negleted. Eqution ( nd ( n e written s: ( j (4

6 For trnsmission line, the impedne mtri, j ( (5, is lws smmetril nd n e written s: (6 ine the inverse mtri of is smmetril, denote s to otin: P P (7 For eqution (4, multipling oth sides genertes: P ewriting eqution (8 nd (5 into mtri formt: ( j (8 j (9 j (3 where (,, or (,, or Further epnding eqution (9 nd (3 ields 6 omple equtions: j ( (3 j ( (3 j ( (33 j ( ( ( (34 j ( ( ( (35 j ( ( ( (36

7 n equtions (3~(36, notiing (,,,,, or is omple numer, we define: j (,,,,, or For the purpose of otining n optiml estimte of the nd prmeters in (3~(36, we epnd these 6 omple equtions into rel equtions. Due to limited spe, these rel equtions re not listed here ut n e found in APPENDX A nd the detiled derivtions re presented in [7]. he disussion elow is sed on these rel equtions. n order to generte simple nd uniform epression for the prolem, the following definitions re mde: Define X to e the mesurement vetor, whih is known nd n e lulted from the PM mesurements. ht is, X [,,..., 4 ] [e(, m(, e(,..., e(, m(,..., e(, m(,..., e(, m(...] (37 where e(. nd m(. ield the rel nd imginr prt of the input rgument, respetivel. Define to e the unknown prmeter vetor, whih is omposed of the unknown impedne prmeters. ht is, [,,..., 4 ] [,,,,...,,,,...,,,,,,, ] (38 From the definition (37, we m further define mesurement vetor: [ 9,, 3, 9 3, 5, 3 7,,, 4, 4, 6, 8 4 ] (39 sed on the definitions ove, we further rrnge these equtions into mtri formt s: H (4 where H is mtri formulted from the equtions referened ove nd ontins mesurements. And the mesurement vetor,, ontins PM voltge nd urrent mesurements [7]. 3. Proposed optiml estimtor n order to estimte the prmeters for L, ll methods previousl disussed require 3-phse voltge nd urrent PM phsors mesurements from oth terminls of the line. We will refer the phsor mesurements tken t one time instnt s one smple. he reltionships mong the vlues in one smple re desried (4. ht is, rel equtions re needed to desrie the interdependenies of one smple. sing multiple smples of PM mesurements, the underdetermined set of equtions, (4, eomes n overdetermined set of equtions nd stte estimtion tehniques n e used to estimte the L s prmeters. Assuming N PM smples tken t N distint time instnts re ville, we n set up *N equtions, using, H, to hve the following dimensions respetivel: *N, *N 8, 8. And finll we rrive t tpil over-determined liner stte estimtion prolem. sing the unised lest squre estimtor [8], the est estimtion of the unknown vetor is found to e: One ( H H H (4 is known, the phse impedne prmeters n e retrieved esil sed on (38. After the phse impedne mtri is lulted, ppling the sequene trnsformtion mtri will ield the sequene impedne prmeters. he equtions used here re shown elow:

8 A A A ( P A (4 A A (43 For noisless PM mesurements, onl two PM smples re needed to lulte the impedne prmeters urtel. Otherwise, multiple PM smples re neessr to inrese the redundn so tht liner estimtion theor nd d dt detetion nd elimintion n e onduted. Clssil methods for d dt detetion n e found in [8-]. n the lssil methods, d dt re identified primril sed on the stud of the model sled residuls. First, ll the dt re utilized to ondut the prmeters estimtion. he sled residuls for eh smple of mesurements re heked. he smples ontining lrge unepeted errors will hve sled residuls muh lrger thn the verge vrine, whih tull n help to identif those d PM dt smples. After removing these d dt, the liner estimtion tehnique is utilized one gin to lulte the impedne. Cse studies showing how this method ompres with the single nd doule mesurement methods will e presented in the net mjor setion. n the net susetion we show how this method n e modified to ount for mutul oupled L s. 3.3 Applition of the model to mutull oupled L s Mn trnsmission lines re loted djent to eh other, or re under-uilt with lower voltge trnsmission/distriution lines. hese djent or under-uild lines indued voltges whih skew the PM mesurements. hese indued voltges use the estimted impedne prmeters to e erroneous when the methods utilizing onl the positive omponents re used. And these errors re onsiderle if the mutul oupling is signifint nd the L is not full trnsposed [7]. We show net tht the method proposed here m e modified to ompenste for the voltges indue the mutull oupled L s. he effet of mutul oupling is to indue voltge in series with the trnsmission line ondution voltge drop. o ompenste for this effet, this indued voltge must e sutrted from the voltge mesurements. his indued voltge n e lulted sed on the mutul indutne nd the urrent phsor mesurements of the trnsmission line induing the voltge on the line to e studied. One the indued voltges for ll three phses re estimted, these indued voltges n e sutrted from the voltge drop ross the trnsmission line mking smll hnge to the left side of eqution (8 s: P indued ( j (44 where indued is the estimted indued voltge phsors lulted from PM mesurements from other trnsmission lines. his ide ws tested nd stisftor results were otined in [7]. Due to the spe limittion, detiled informtion nd test results will not e presented in this pper. 4. CAE DE n this setion, 3 se studies re presented to demonstrte the proedures nd vlidte the effetiveness of the proposed method. he Alterntive rnsients Progrm (AP is emploed to uild 3-phse trnsmission line model nd to generte snhrophsor mesurements []. One 3 kv trnsmission line is simulted using tpil phsil prmeters (e.g., tower geometr, ondutor tpe otined from lt iver Projet (P

9 [APPENDX ]. he trnsmission lines re uilt in AP using the LCC ojets. he LCC suroutine utomtill lultes the impedne prmeters of the L s one the phsil prmeters re input. hese impedne prmeters lulted the LCC suroutine re tken s the true prmeters of the L s, nd re used s referene vlues to e ompred with the lulted prmeters. euse the method we propose requires redundnt mesurements otined from different loding onditions, time vring lod is modeled on the reeiving-end of the trnsmission lines. he lod urve is set to e sinusoidl with period of 4 hours. t vries etween one % to 8% of the mimum line pit. o mke sure tht eh smple represents different lod ondition, smples re tken ever five minutes. Another ojetive of this setion is to demonstrte the limittion of the positive sequene L model when pplied to prmeter estimtion of untrnsposed L s. n order to do tht, two lgorithms proposed in [3] nd [4] re emploed to lulte the impedne prmeters for n untrnsposed L with unlned loding onditions. he results otined from these two lgorithms re ompred with the results from the proposed method. 4. Cse for the full trnsposed L o vlidte the proposed prmeter estimtion method, the following eperiment ws onduted. A 4.5-km-long ompletel trnsposed L with the prmeters shown in Appendi A ws modeled in AP with diurnl sinusoidll vring lod s desried ove. Voltge nd urrent phsors for different loding onditions were smpled from the AP output nd no noise ws dded to these mesurements. hese mesurements served s input to the method proposed nd the prmeters tht hrterize the trnsmission line re estimted. ine the line is full trnsposed, there re si non-zero prmeters tht represent the L; these re: the positive, zero nd negtive sequene series impednes nd shunt suseptnes. Appling the proposed method to the simulted snhrophsor mesurements, the results shown in le re otined. As epeted, le shows tht urte estimtes re hieved the proposed method. he other mutul impednes (e. g.,, not shown re lulted the lgorithm ut re essentill zero s epeted. 4. Cse for the untrnsposed L n setion, this pper illustrtes the limittion of the positive sequene model nd the methods sed on this model. n this susetion, the methods proposed in referene [3] nd [4] re tested with snhrophsor dt otined from one untrnsposed L with unlned lod. his eperiment ws the sme s tht desried in etion 4. eept the line ws untrnsposed nd the lod ws unlned. n the simultion with 4% unlned lods were pplied to the L. he degree of unlne is defined in this pper s the rtio of negtive le. Optiml estimtor for full trnsposed line with proposed method Quntit eferene vlues Optiml estimtes Error(% in Error(% in X or j j6.99.%.6% ( j j6.99.%.6% ( j j.857.4%.44% ( ( j5.98e-5 j5.95e-5 -.6% ( j5.98e-5i j5.95e-5 -.6% ( j.7633e-5i j.765e-5 -.9%

10 sequene urrent to the positive sequene urrent ( %. he results for these two methods re shown in le nd le, whih show inurte impedne prmeters re lulted for untrnsposed line with unlned lod when the single nd doule mesurement methods re used. le. Clulted prmeters for untrnsposed line with unlned lod using the doule mesurement method in [3] Quntit eferene vlues Clulted vlues Error(% % ( X % ( 5.88e e % ( le. Clulted prmeters for untrnsposed line with unlned lod using single mesurement method in [4] Quntit eferene vlues Clulted vlues Error(% % ( X % ( 5.88e e % ( le V. Clulted prmeters for untrnsposed line with unlned lod using proposed method Quntit eferene vlues Optiml estimtes Error(% in Error(% in X or j j.87 %.6% ( ( *.8839+j j6.994 %.9%.3-j j.396 % % ( ( -.54-j j.36.49%.77% ( -.54-j j.36.49%.77% ( j j.54.3% % (.3-j % %.437+j j.5 % % ( (.763e-5.765e-5 -.8% ( 5.88e-5 5.9e-5 -.8% * (.585e-6.587e-6 -.3% (.585e-6.585e-6 - % * ( e e % *,,,, he performne of the proposed method is evluted for the sme L with the sme degree of 4% unlne. ine the L is untrnsposed, the impedne mtri lulted is full mtri. he results re shown in le V, where lose greement n e found etween the referene vlues nd the optiml estimtes. t

11 should e noted tht, with the proposed method, not onl the digonl elements ut lso the non-digonl elements in the impedne mtri n e lulted urtel. 4.3 Cse for untrnsposed L with noise in the mesurements PM dt re time tgged with ur of etter thn miroseond nd mgnitude ur tht is etter thn %. However, this potentil performne is not hieved in tul field instlltions due to unised rndom mesurement noise nd ised errors from instrumenttion hnnels [], []. Figure shows the mgnitude nd phse ngle of voltge phsor mesurements tken from PM. As shown in the two plots, there re spikes nd omple mesurement noise ehvior in the phsor mesurements. he spikes in the plots represent d dt, whih m e removed through d dt detetion tehniques [8-]. However, mn times, it is ver diffiult nd m even e impossile to filter out ll dt orrupted with noise. As will e shown, the single nd doule mesurement methods re sensitive to this noise, where s the optiml prmeter estimtion lgorithm is le to redue the effets of this noise on the prmeter estimtion. Another omplition rought on noise is tht its effet on prmeter estimtion is more pronouned on shorter trnsmission lines. o emine the effet of noise on prmeter estimtion in generl nd its effet on line length in prtiulr, we onduted eperiments to eplore the reltionship etween the ur of the estimted prmeters nd the length of trnsmission line under nois mesurement onditions. he noise onsidered ws uss noise with men nd % stndrd devition. For ertin length of trnsmission line, we rn the AP simultion of the untrnsposed line nd otined N smples of the snhrophsor mesurements. hese N smples of mesurements re referred to s one set of mesurements. hen we dded different noise vetors to the snhrophsor mesurements to form M different sets of mesurements. And then we pplied the proposed method nd the single nd doule mesurement methods to eh set of the nois mesurements to estimte the impednes of the L. ppling these methods to the M different sets of nois mesurements, we generted different estimtes of the impedne prmeters for the trnsmission line desried in APPENDX. Assuming tht the impedne prmeters estimted this w were normll distriuted, we lulted the 95% onfidene intervl of these prmeters, tht is, the rnge in whih our estimted prmeter fell 95% of the time. We repeted this proess for different lengths of trnsmission lines so tht we ould otin the onfidene intervls of the impedne prmeters for different lengths of L s. he stndrd devition of the error in the prmeter versus length of the trnsmission line is shown in Figure 3. imilr plots for X nd re shown in Figure 4 nd Figure 5. Figure eorded mgnitude nd phse ngle of voltge phsor from PM

12 tnd devition of errors in X(% tnd devition of errors in (% 6 5 single-mesurement method in [4] doule mesurement method in [3] proposed method Length of L (km Figure 3. tndrd devition of Errors in lulted for L s of different lengths single-mesurement method in [4] doule mesurement method in [3] proposed method Length of L (km Figure 4. tndrd devition of errors in X for L s of different lengths

13 tnd devition of errors in (% 8 6 single-mesurement method in [4] doule mesurement method in [3] proposed method Length of L (km Figure 5. tndrd devition of errors in for L s of different lengths Figure 3-Figure 5 show the stndrd devition (D of errors in lulted impedne prmeters s funtion of L lengths when the PM mesurements re orrupted uss noise (, %. Figure 4 shows tht for 4.5-km long trnsmission line (n in-servie line for whih we were sked lt iver Projet to estimte the prmeters, with uss noise (, % dded, the D s of errors in lulted X reh out 8%, 5%, nd 8% of the true vlue the single mesurement method, the doule mesurement method nd the proposed method, respetivel. As the length of the trnsmission line inreses, the D s of the errors derese nd the D s lulted the proposed method re lws muh smller thn the single or doule mesurement methods. f the trnsmission line is longer thn 5 kilometers, with the sme sets of noise dded, the devition in the error of the lulted series retne will e smller thn % for the proposed method while the D s the other two methods remin t out 4%. hese figures lerl show the dvntge of the proposed method over the other two methods. And the proposed method is prtiulrl superior to the other two methods for short trnsmission lines. 5. CONCLON his ojetive of this pper is to ompre vrious methods for estimting the prmeters for L s (under trnsposed nd untrnsposed onfigurtions nd under lned nd unlned onditions using snhrophsor mesurements. Prtiulrl of interest, is the pplition of these methods to short L s. n this pper, it is shown tht the positive sequene model of L hs limittions when used to onstrut the defining equtions for L prmeter estimtion. Prmeter estimtion methods sed on the positive sequene model lone re shown to perform prtiulrl poorl when pplied to short untrnsposed nd unlned trnsmission lines. his pper proposed novel optiml prmeter estimtion method to urtel estimte the impedne prmeters for generl trnsposed/untrnsposed L with lned or unlned loding. his method is sed on the liner

14 estimtion theor nd ssumes vililit of redundnt PM mesurements. As stted erlier, d dt detetion method n lso e utilized to identif nd remove d PM dt smples. everl se studies re onduted nd the performne of this optiml estimtor is ompred with other methods. he enourging results shown here re for tpil trnsmission line onfigurtion. 6. L OF AEVAON AND MOL 6. Arevitions AP P LCC PM D P L Alterntive rnsients Progrm lol Positioning stem line/le onstnt Phsor Mesurement nit stndrd devition lt iver Projet trnsmission line 6. mols A sequene trnsformtion mtri 3-phse shunt suseptne mtri (3 3 3-sequene shunt suseptne mtri (3 3 (,... self shunt suseptne of one phse or mutul shunt suseptne etween phses; element of (,,... shunt suseptne of one sequene or mutul shunt suseptne etween different sequenes; elements of det( determinnt of ( sending (reeiving end phse urrent vetor (3 ( (,, sending (reeiving end phse urrent ( sending (reeiving end sequene urrent vetor (3 ( (,, sending (reeiving end sequene urrent m( imginr prt of numer e( rel prt of numer ( sending (reeiving end phse voltge vetor (3 ( (,, sending (reeiving end phse voltge ( sending (reeiving end sequene voltge vetor (3 ( (,, sending (reeiving end sequene voltge

15 inverse of P (3 3 (,,,... elements in P 3-phse shunt dmittne mtri (3 3 (,,... self shunt dmittne of one phse or mutul shunt suseptne etween phses; element of 3-sequene shunt dmittne mtri (3 3 (,,,... self shunt suseptne of one sequene or mutul shunt suseptne etween different indued sequenes; elements of vetor (3 omprised of indued voltges (due to mutul oupling on 3-phse trnsmission line 3-phse series impedne mtri (3 3 (,,... self series impedne of one phse or mutul series impedne etween phses; element of 3-sequene series impedne mtri (3 3 (,,,... self series impedne of one sequene or mutul series impedne etween different sequenes; elements of inverse of 7. APPENDX A Comple equtions (3~(36 re epnded into rel equtions. defining j (,,,,, or, these equtions n e written s: e( e( m( m( e( m( m( m( e( m( (A. e( e( m( m( e( m( m( m( e( m( (A. e( e( m( m( e( m( m( m( e( m( (A. 3 m( m( m( e( (A. 4 m( m( m( e( (A. 5 m( m( m( e( (A. 6 m( m( e( e( m( e( e( e( m( e( (A. 7 m( m( e( e( m( e( e( e( m( e( (A. 8

16 m( m( e( e( m( e( e( e( m( e( (A. 9 e( V V e( V V e( V V m( (A. e( e( e( V m( (A. m( m( e( where e(. nd m(. ield the rel nd imginr prts of the input rgument, respetivel. m( (A. 8. APPENDX he phsil prmeters (e.g., tower geometr, ondutor tpe of the trnsmission line used in the simultions s desried in setion 4 re shown in le.. he soil resistivit is ssumed to e in the simultion. le... rnsmission line phsil prmeters used in simultions Phse. etne No (ohm/mile AC ohm m Outer rdius of Condutor Horizontl Vertil Vertil the ondutor resistne t Freq distne from undle height undle height (inh (with no skin effet (ohm/mile AC the enter of the tower (feet t tower (feet t mid-spn (feet *sstem frequen is 6 Hz; ll phse ondutors re single-undled ondutors ACKNOWLEDEMEN he uthors thnk lt iver Projet for the support in rring out this reserh work. EFEENCE. J. Dunn lover, Mulukutl. rm, homs Overe. Power sstem nlsis nd design. homson-engineering: A, 7.. North Amerin nhrophsor nititive Performne & tndrds sk em. nhrophsor Mesurement Aur Chrteriztion. Novemer E. Wilson,. A. evenergen, D. L. Mh. Clultion of trnsmission line prmeters from snhronized mesurements. Eletri Mhines nd Power stems 999; 7: Di hi, Dniel J. lvsk, Nim Logi, Kristin M. Koellner. dentifition of short trnsmission-line prmeters from snhrophsor mesurements. 4th North Amerin Power mposium (NAP 8; un Lio, Mlden Kezunovi. Optiml estimte of trnsmission line fult lotion onsidering mesurement errors. EEE rnstions on Power Deliver 7; (3:

17 6. A. P.. Meliopoulos,. Frdnesh. Phsor mesurement pplitions-rnsmission line loss mesurement nd prmeter identifition; Monitoring for trnsformer unlne opertion. Fult nd Disturne Anlsis & Preise Mesurements in Power stem Conferene, Arlington, VA, Di hi. tilizing snhrophsor tehnolog to determine trnsmission line impedne prmeters, M. thesis. Arizon tte niversit Dougls C. Montgomer, Elizeth A. Pek,. eoffre Vining. ntrodution to Liner egression Anlsis. New ork: John Wile & sons: A. Aur, A.. Eposito. Power sstem stte estimtion-heor nd implementtion. Mrel Dekker: New ork, 4.. J. ringer nd W. tevenson. Power sstem nlsis. Mrw-Hill: New ork, Dommel H.W. Eletromgneti rnsients Progrm eferene Mnul (EMP heor ook. PA: Potrlnd, Oregon, A. P.. Meliopoulos,. J. Cokkinides, F. lvn, nd. Frdnesh. P-nhronized Dt Aquisition: ehnolog Assessment nd eserh ssues. Proeedings of the 39th Annul Hwii nterntionl Conferene on stem iene, Hwii, Jnur 4-7, 6.