Univesity of Kentucky UKnowledge Univesity of Kentucky Doctoal Dissetations Gaduate School NEW ACCURATE FAULT LOCATON ALGORTHM FOR PARALLEL TRANSMSSON LNES Pamote Chaiwan Univesity of Kentucky, joechaiwan@yahoo.com Click hee to let us know how access to this document benefits you. Recommended Citation Chaiwan, Pamote, "NEW ACCURATE FAULT LOCATON ALGORTHM FOR PARALLEL TRANSMSSON LNES" (). Univesity of Kentucky Doctoal Dissetations. 83. htts://uknowledge.uky.edu/gadschool_diss/83 This Dissetation is bought to you fo fee and oen access by the Gaduate School at UKnowledge. t has been acceted fo inclusion in Univesity of Kentucky Doctoal Dissetations by an authoized administato of UKnowledge. Fo moe infomation, lease contact UKnowledge@lsv.uky.edu.
NEW ACCURATE FAULT LOCATON ALGORTHM FOR PARALLEL TRANSMSSON LNES DSSERTATON A dissetation submitted in atial fulfillment of the equiements fo the degee of Docto of Philosohy in the College of Engineeing at the Univesity of Kentucky By Pamote Chaiwan Lexington, Kentucky Diecto: D. uan Liao, Pofesso of Electical and Comute Engineeing Lexington, Kentucky Coyight Pamote Chaiwan
ABSTRACT OF DSSERTATON NEW ACCURATE FAULT LOCATON ALGORTHM FOR PARALLEL TRANSMSSON LNES Electic owe systems have been in existence fo ove a centuy. Electic owe tansmission line systems lay an imotant ole in caying electical owe to customes eveywhee. The numbe of tansmission lines in owe systems is inceasing as global demand fo owe has inceased. Paallel tansmission lines ae widely used in the moden tansmission system fo highe eliability. The aallel lines method has economic and envionmental advantages ove single cicuit. A fault that occus on a owe tansmission line will cause long outage time if the fault location is not located as quickly as ossible. The faste the fault location is found, the soone the system can be estoed and outage time can be educed. The main focus of this eseach is to develo a new accuate fault location algoithm fo aallel tansmission lines to identify the fault location fo long double-cicuit tansmission lines, taking into consideation mutual couling imedance, mutual couling admittance, and shunt caacitance of the line. n this eseach, the equivalent P cicuit based on a distibuted aamete line model fo ositive, negative, and zeo sequence netwoks have been constucted fo system analysis duing the fault. The new method uses only the voltage and cuent fom one end of aallel lines to calculate the fault distance. This eseach aoaches the oblem by deivation all equations fom ositive sequence, negative sequence, and zeo sequence netwok by using KL and KCL. Then, the fault location is obtained by solving these equations. EMTP has been utilized to geneate fault cases unde vaious fault conditions with diffeent fault locations, fault tyes and fault esistances. Then the algoithm is evaluated using the simulated data. The esults have shown that the develoed algoithm can achieve highly accuate estimates and is omising fo actical alications.
KEWORDS: Distibuted aamete line model, Paallel tansmission line, Equivalent P cicuit, Mutual couling imedance, Fault location Pamote Chaiwan Student s Signatue Setembe, Date
NEW ACCURATE FAULT LOCATON ALGORTHM FOR PARALLEL TRANSMSSON LNES By Pamote Chaiwan D.uan Liao Diecto of Dissetation D. Zhi David Chen Diecto of Gaduate Studies
DEDCATON This dissetation is dedicated to my aents M. Dee Chaiwan and Ms. Sumontha Chaiwan
ACKNOWLEDGMENTS am heatily thankful to my adviso, D. uan Liao, whose guidance, encouagement, and suot fom the initial to the final level enabled me to develo an undestanding of this eseach. This dissetation would not have been ossible without his hel. also would like to thank D. uming Zhang, D.Jimmie Cathey, and D. Alan Male fo thei suot in a numbe of ways to seve on the Dissetation Advisoy Committee. would like to thank D. Zhongwei Shen of the MAT Pogam to seve as the Outside Examine. t is my leasue to thank the faculty membes who made this dissetation ossible. also would like to thank my aents and family membes, who have been waiting to see my success and my fiends Anthony M. King and Sam tantasook fo thei suot. iii
TABLE OF CONTENTS ACKNOWLEDGMENTS.. iii LST OF TABLES.... vi LST OF FGURES.... viii CHAPTER ONE.... NTRODUCTON. BACKGROUND......4. Symmetical Comonent and Sequence Netwoks... 4. Positive Sequence Comonent.. 5. Negative Sequence Comonent......6.3 Zeo Sequence Comonent...7. Unsymmetical Faults...... Unsymmetical Faults Classification....... oltage and Cuent Netwok Equations in Sequence Comonent.3 Analysis of Unbalanced Faults...3. Single Line-to Gound Faults...3. Line-to Line Faults 5.3.3 Double Line to Gound Faults...8 CHAPTER TWO. REEW OF LTERATURES Review of Existing Fault Location Algoithm... CHAPTER THREE.8 iv
PROPOSED NEW FAULT LOCATON ALGORTHM FOR PARALLEL TRANSMSSON LNES......8. Model Used.....8. Poosed Equivalent P Cicuit Model fo New Fault Location Algoithm fo Paallel Tansmission Lines....3. Positive Sequence Netwok...3. Negative Sequence Netwok.....35.3 Zeo Sequence Netwok.....38.4 Poosed Distibuted Paamete Line Model Based Algoithm...4.5 Poosed New Method to Estimate Fault Distance and Fault Resistance.. 45.5. Poosed Algoithm.. 46.5. The Bounday Condition fo aious Faults.5 CHAPTER FOUR...5 EALUATON STUDES...5. Results of the Existing Algoithm fo Fault Location Estimation of aious Tyes of Faults and aious Fault Resistances.... 5. Results of the Poosed Algoithm with aious Tyes of Fault and aious Fault Resistances... 59 3. oltage and Cuent Wavefoms at Teminal P duing Fault with aious Tyes of Faults.......67 4. Estimated Fault Location and Fault Resistance...83 v
CHAPTER FE..9 CONCLUSON. 9 BBLOGRAPH.....93 TA....97 vi
LST OF TABLES Table 3., Paametes e km of zeo-sequence netwoks of a aallel line 3 Table 3., Paametes e km of ositive-sequence netwoks of a aallel lines 3 Table 3.3, Souce imedance at P and Q......3 Table 4., Fault location estimation fo vaious tyes of faults and vaious fault esistances at 5 of 3 km:(.67.u.) of existing algoithm.5 Table 4., Fault Resistances estimation fo vaious tyes of faults at 5 of 3km: (.67.u.) of existing algoithm... 5 Table 4.3, Fault location estimation fo vaious tyes of faults and vaious fault esistances at of 3 km:(.333.u.) of existing algoithm..53 Table 4.4, Fault Resistances estimation fo vaious tyes of faults at of 3 km: (.333.u.) of existing algoithm....54 Table 4.5, Fault location estimation fo vaious tyes of faults and vaious faults esistances at of 3 km: (.667.u.) of existing algoithm.55 Table 4.6, Fault Resistances estimation fo vaious tyes of faults at of 3 km: (.667.u.) of existing algoithm. 56 Table 4.7, Fault location estimation fo vaious tyes of faults and vaious fault esistances at 5 of 3 km: (.833.u.) of existing algoithm..57 Table 4.8, Fault Resistances estimation fo vaious tyes of faults at 5 of 3 km: (.833.u.) of existing algoithm.58 vii
Table 4.9, Fault location estimation fo vaious tyes of faults and vaious fault esistances at 5 of 3 km of oosed algoithm.. 59 Table 4., Fault Resistances estimation fo vaious tyes of faults at 5 of 3 km of oose algoithm.... 6 Table 4., Fault location estimation fo vaious tyes of faults and vaious fault esistances at of 3 km..... 6 Table 4., Fault Resistances estimation fo vaious tyes of faults at of 3 km.... 6 Table 4.3, Fault location estimation fo vaious tyes of faults and vaious fault esistances at of 3 km... 63 Table 4.4, Fault Resistances estimation fo vaious tyes of faults at of 3 km...64 Table 4.5, Fault location estimation fo vaious tyes of faults and vaious fault esistances at 5 of 3 km....65 Table 4.6, Fault Resistances estimation fo vaious tyes of faults at 5 of 3 km....66 Table 4.7, Estimated fault location and fault esistance......83 Table 4.8, % Eo Estimated fault location and fault esistance..88 viii
LST OF FGURES Figue., Positive sequence comonent...5 Figue., Negative sequence comonent.....6 Figue.3, Zeo sequence comonent....7 Figue.4, Comonent of hase a....7 Figue.5, Comonent of hase b....8 Figue.6, Comonent of hase c....8 Figue.7, Thee unbalanced hasos a, b, and c obtained fom thee set of balanced hasos.. 9 Figue.8, Single Line to gound fault on hase a... Figue.9, Single Line to gound fault on hase a with fault imedance..4 Figue., Line-to- Line fault. 5 Figue., Line-to- Line fault with fault imedance......7 Figue., Double Line-to gound fault...8 Figue 3., System diagam used in the develoment of the new algoithm...9 Figue 3., Equivalent P cicuit of ositive sequence netwok of the system duing the fault.... 3 Figue 3.3, Equivalent P cicuit of negative sequence netwok of the system duing the fault....35 Figue 3.4, Equivalent P cicuit of mutually couled zeo-sequence netwok of the system duing the fault...38 ix
Figue 4., oltage wavefoms of hase a to gound fault on line bus P.. 67 Figue 4., oltage wavefoms of hase a to gound fault on line bus P.. 68 Figue 4.3, Cuent wavefoms of hase a to gound fault on line bus P.. 69 Figue 4.4, Cuent wavefoms of hase a to gound fault on line bus P.. 7 Figue 4.5, oltage wavefoms of hase b to c fault on line bus P....7 Figue 4.6, oltage wavefoms of hase b to c fault on line bus P....7 Figue 4.7, Cuent wavefoms of hase b to c fault on line bus P....73 Figue 4.8, Cuent wavefoms of hase b to c fault on line bus P....74 Figue 4.9, oltage wavefoms of BCG fault on line bus P......75 Figue 4., oltage wavefoms of BCG fault on line bus P 76 Figue 4., Cuent wavefoms of BCG fault on line bus P 77 Figue 4., Cuent wavefoms of BCG fault on line bus P 78 Figue 4.3, oltage wavefoms of ABC fault on line bus P 79 Figue 4.4, oltage wavefoms of ABC fault on line bus P 8 Figue 4.5, Cuent wavefoms of ABC fault on line bus P 8 Figue 4.6, Cuent wavefoms of ABC fault on line bus P 8 x
CHAPTER ONE. NTRODUCTON Powe tansmission systems have been in existence fo ove a centuy. Powe tansmission systems lay an imotant ole in caying electical owe to customes eveywhee. The numbe of tansmission lines in owe systems is inceasing as global demand fo owe has exanded. Cuently, the bulk tansmission of electical owe is done by means of aallel lines which ae widely used in the moden tansmission systems. The aallel lines method has economic and envionmental advantages ove single cicuit. Unfotunately, a fault that occus in one at of the owe system, such as a geneato o owe tansmission line, can destoy the whole system if the fault location is not located as quickly as ossible. The faste the fault location is found, the soone the system can be estoed and outage time can be educed. The double-cicuit tansmission line is used moe often than the singlecicuit and the incile of distance elaying states that the imedance measued by a elay is ootional to the distance of that elay to the fault. Theefoe, by measuing the imedance it can be detemined whethe the line being otected is faulted o not. Unfotunately, thee ae seveal ways fo the following to be eos in accuately measuing a fault location, and they should be taken into full consideation.
. The self-imedance, mutual imedances and mutual admittance The ositive sequence mutual imedances and the negativesequence mutual imedances ae about 3-5% of its own self-imedances. The zeo-sequence mutual imedances ae about 5-55% of the zeosequence self-imedances. Thus, the eo occus if the calculation of the fault location consides only the self-imedances.. Shunt caacitance Fo long-length tansmission lines (moe than 5 miles o 4 km), the line is consideed to have a shunt caacitance instead of lumed aametes fo the calculation of exact fault location. f lumed aametes ae used, then eos will occu. 3. Fault esistance The fault esistance can only be detemined using the algoithm that will be oosed in this dissetation. Theefoe, it cannot be used as inut to detemine the fault location. 4. Souce imedance The changing of the souce imedance without changing the setting of the fault calculation equiment can cause an eo in the accuate calculation of fault location. 5. Caacitance voltage tansfome 6. The classification of the tansmission line Lack of knowledge of the classification of tansmission lines can lead to eo in the calculation of the accuate fault location. Thee ae
thee classes of tansmission lines: shot, medium, and long tansmission lines. n the shot-length line class (less than 5 miles o 8 km), the shunt caacitance is consideed so small that can be ignoed. We only conside the seies of esistance R and inductance L. n the medium-line class (5 to 5 miles o 8 to 4 km), the caacitance will be eesented as two caacitos each equal to half the line caacitance, which is known as the nominal-π model. n the long-length line class (moe than 5 miles o 4 km), the line is consideed to have distibuted aametes instead of lumed aametes. This will ovide accuate esults. t is efeed to as the equivalent-π model since it has lumed aametes which ae adjusted so that they ae equivalent to the exact distibuted aamete model. The uose of this eseach is to imove the fault distance estimation fo long aallel tansmission lines, taking into consideation mutual couling imedance and mutual couling admittance. The distibuted tansmission line aametes model will be emloyed. 3
. BACKGROUND n ode to estimate the fault distance, the following concets needs to be exlained:. Symmetical Comonent and Sequence Netwoks. Unsymmetical Faults. Symmetical Comonent and Sequence Netwoks The well-known theoy of symmetical comonent that was intoduced by Chales Legeyt Fotescue is vey useful to solve the oblems fo unbalanced condition on owe systems. Accoding to his theoy, unbalanced thee hase faults can be esolved into thee sets of balanced thee hase systems by using the method of symmetical comonents that consists of: 4
. Positive sequence comonent, which consists of thee hasos with equal magnitudes and aat fom each othe, and hase sequence ae the same as oiginal hasos. c a b Figue. Positive sequence comonent 5
. Negative sequence comonent, which consists of thee hasos with equal magnitudes and aat fom each othe, and hase sequence ae oosites of the oiginal hasos. b a c Figue. Negative sequence comonent 6
.3 Zeo sequence comonent, which consists of thee hasos with equal magnitudes and zeo hase dislacements fom each othe. a b c Figue.3 Zeo sequence comonent n the owe system that consists of thee hases such as a, b, and c a a + a + a a a a a Figue.4 Comonents of hase a 7
b b + b + b b b b b Figue.5 Comonents of Phase b c c + c + c c c c c Figue.6 Comonents of hase c 8
Then hase a, b, and c can be obtained as follow: a c b Figue.7 Thee unbalanced hasos a, b, and c that wee obtained fom thee set of balanced hasos Whee a.5 + j.866 (.) a 4.5 j.866 (.) a 3 36. + j (.3) b a (.4) b a a (.5) b a a (.6) c a (.7) c a a (.8) c a a (.9) Thus a a + a + a (.) 9
b a + a a + a a (.) c a + a a + a a (.) a a a b a a a A a (.3) c a a a a whee A a a (.4) a a Then A a a (.5) 3 a a a a a a a a b A b (.6) 3 a a a c c We have a 3 ( a + b + c ) (.7) a 3 ( a + a b + a c ) (.8) a ( 3 a + a b + a c ) (.9) Fo cuent we also have a a + a + a (.) b a + a a + a a (.) c a + a a + a a (.) a ( 3 a + b + c ) (.3) a 3 ( a + a b + a c ) (.4)
a 3 ( a + a b + a c ) (.5). Unsymmetical Faults. Unsymmetical faults can be classified into:.. Single line to gound fault: a-g, b-g, and c-g.. Line to line faults: ab, bc, and ca..3 Double line to gound fault: abg, bcg, and cag..4 Thee hase fault: abc As the unbalanced fault occus in the owe system duing the fault, the unbalanced cuent will go into the system. The method of symmetical comonent will be utilized to calculate the cuent on the system.. oltage and Cuent netwok equation in Sequence comonent The voltages in electic owe system ae assumed to be balanced until the fault occued. Only ositive sequence comonent of the e-fault voltage f is consideed. This can be witten in the matix fom as Z a (.6) a E f Z a (.7) a Z a (.8) a a a E f Z a Z a (.9) Z a
.3 Analysis of Unbalanced Faults [8].3. Single line-to gound faults Single line to gound faults occus when one of any thee lines falls is on the gound. Assume that the fault occued on hase a with zeo fault imedance as shown in figue.. Since the fault imedance is zeo and load cuent is neglected, then at the fault oint a b c (.3) a Fault a Sending b Teminal C Figue.8 Single Line to gound fault on hase a The fault condition can be conveted to symmetical comonent as We get a a + a + a (.3) a a a a a b (.3) 3 a a a c
a a a a 3 (.33) With the fault cuent f a (.34) a a a E f a Z a 3 a E f Z a 3 a Z a 3 Z Z Z a a 3 a a 3 a a 3 (.35) (.36) (.37) (.38) Thus a + a + a Z a + E 3 f Z a Z a 3 (.39) 3 f a 3E f Z +Z +Z (.4) We can assume that the sequence comonent must be connected in seies and shot cicuited because a + a + a and a a a (.4) 3
Assume that the fault occu on hase a though imedance Z f then to the gound. a a Z f Sending b Teminal C Figue.9 Single Line to gound fault on hase a with fault imedance At the fault oint we have a Z f a (.4) b c (.43) By using method of symmetical comonent a + a + a ( a + a + a )Z f (.44) a a a a a b (.45) 3 a a a c a a a a 3 (.46) a + a + a 3 a Z f (.47) O a + a + a a 3Z f (.48) 4
Equating these equations line to gound fa f Z +Z +Z +3Z f (.49) Then a a a is the cuent injecting to the fault fo the single.3. Line-to line faults Line to line fault occus when two lines come to contact to each othe. Assume that the fault is on hase b and c with no fault imedance. The fault conditions fo this tye of fault ae: a b c b c (.5) a Sending b Teminal Fault c Figue. Line-to- Line fault 5
By using method of symmetical comonent Then a a a a a 3 b (.5) a a a c b a (.5) a 3 (a a ) b (.53) a 3 (a a) b (.54) We can assume that a (.55) a (a 3 a ) b (.56) a 3 (a a) b (.57) We can assume that a a (.58) O a + a (.59) The symmetical comonent when b c a a a a a b (.6) 3 a a a c b We get a a (.6) 6
Then the bounday conditions ae a, a + a and a a (.6) Assume that the fault is on hase b and c with fault imedance. f Z f is in the ath between b and c the fault conditions fo this tye of fault ae: a b fc b c b Z f (.63) a Sending b Teminal fb Z f c fc Figue. Line-to- Line fault with fault imedance By using method of symmetical comonent Then fa fa a a fb (.64) 3 fa a a fb fa (.65) fa 3 (a a ) fb (.66) fa 3 (a a) fb (.67) 7
We can assume that fa fa (.68).3.3 Double Line-to gound faults Double line to gound faults occu when any two lines of thee lines comes in contact with the gound. Assume that the fault occus on hase b and hase c though imedance Z f to gound. The fault conditions fo this tye of fault ae fa fb fc fb + fc Z f (.69) a Sending b Teminal fb c fc fb + fc Z f Figue. Double Line-to gound fault fa fa a a fb (.7) 3 fa a a fc fa 3 fb + fc (.7) 8
Since b c fb + fc Z f (.7) Then b c 3Z f fa (.73) Use the method of symmetical comonent to find fb a a a a a b (.74) 3 a a a c a 3 ( a + a b + a c ) (.75) a ( 3 a + a b + a c ) (.76) Thus a a (.77) a ( 3 a + b + c ) (.78) Because b c 3 a a + b (.79) ( a + a + a ) + 3Z f fa (.8) a + a + 3Z f fa (.8) We obtain a 3Z f fa a (.8) a a 3Z f fa (.83) 9
The fault cuent can be obtained as Z fa fa (.84) Z +Z +3Z f fa f Z + Z Z+3Z f Z+Z+3Z f (.85) fa fa Z +3Z f Z +Z +3Z f (.86)
CHAPTER TWO REEW OF LTERATURES Review of Existing Fault Location Algoithm Many oosals fo imoving fault distance estimation fo aallel tansmission lines have been develoed and esented in the ast. n [], the mutual couling imedances between the aallel tansmission lines ae esented. Zeo-sequence mutual imedances ae about 5-55% of the zeo-sequence self-imedances and will lead to significant eo if the calculation of the fault location does not take it into account. n [], the authos oose a method fo aallel tansmission line fault location using one-end data. Data obtained fom the fault lines and sound line ae utilized to deive the sequence hase voltage and sequence hase cuent equations at the elay location to calculate the fault distance by eliminating the tems containing the sequence cuent fom the othe end. With the bounday condition, the fault distance estimation can be obtained. Howeve, while this method is indeendent of fault esistance, load cuents, souce imedance, and emote in-feed, yet shunt caacitance is neglected which might lead to eos in the calculation of fault distance. n [3], one teminal algoithm using local voltages and cuent nea end of the faulted line has been emloyed. The zeo-sequence cuent fom the nea end of the healthy line is used as the inut signals. The authos use the comensation techniques to comensate fo the eos that cause fom the fault esistance.
n [4], an adative otective elaying scheme fo aallel-line distance otection is oosed. A detailed algoithm is used to imove the distance otection efomance fo aallel lines affected by mutual couling effect. The algoithm takes into account the zeo-sequence cuent of the aallel cicuit to comensate fo the mutual effect. To imove efomance, the algoithm solves the oblem based on zeo sequence on the aallel line, the line oeating status, and the default zeo-sequence comensation facto, esectively. J. zykowski, E. Rosolowski, and M. Mohan Saha [5] oosed a fault location algoithm fo aallel tansmission lines by using the voltage and cuent hasos at one end. The comlete measuement of the thee-hase voltages and thee-hase cuents fom a faulted line and a healthy line ae measued by the fault locato. The fault cuent is calculated without the zeo sequence by setting the cuent to zeo to exclude the zeo sequence comonents. Accoding to the availability of comlete measuements at one end, the deived algoithm is a vey simle fist-ode fom because the fault location algoithm does not include any souce imedance, then the algoithm is not influenced by the vaying souce imedances o fault esistance n A. Wiszniewski [6], an algoithm fo locating fault on tansmission line has been oosed. The accuacy of the fault location is affected by the fault esistance since the fault cuent though the fault esistance shifts in hase with the cuent measued at the end of the line. The algoithm will comensate and accuately locate the fault.
The authos in [7] esent an algoithm that deals with non-eath faults on one of the cicuits of aallel tansmission line. Thee voltage equations fom one end to a faulty line to the fault oint wee established based on symmetical comonent. Then adding these thee equations togethe to fom the equation with fault cuent and fault esistance as unknowns. By alying Kichhoff voltage law(kl) the fault cuent can be exessed as a function of fault location. Then fault esistance and fault location can be obtained by solving those equations. This algoithm does not conside shunt caacitance which may cause eos fo the long tansmission lines. The authos of [8] oose a technique fo using the data fom two teminals of the tansmission line to estimate fault location. The lumed aamete line model is adoted and the shunt caacitance fo long tansmission line is comensated in an iteative calculation. This technique is indeendent fom the fault tye, fault esistance, load cuent, and souce imedance. Synchonization of data is not equied fo this technique. Real-time communication is not needed fo this analysis, only the off-line ost-fault analysis. A new digital elaying technique fo aallel tansmission lines is esented in [9]. This technique uses only one elay at each end of the two teminals. The technique ovides a simle otection technique without equiing any comlex mathematics while avoiding softwae and hadwae comlications. The autho in [] ooses a novel digital distance-elaying technique fo tansmission line otection. Two elays instead of fou ae used fo a aallel 3
tansmission line. One is at the beginning and the othe is at the end. Each elay eceives thee voltages and six cuent signals fom the aallel line. This technique comaes the measued imedance of the coesonding hase. t solves the comlexities of the tye of faults, high fault esistance, mutual effects, and cuent in-feed. n [], the otection of double-cicuit line using wavelet tansfom is oosed. The authos oose using the oweful analyzing and decomosing featues of wavelet tansfom to solve the oblems in a double tansmission line when otected by a distance elay. The technique uses thee-line voltages and six-line cuent of the aallel tansmission lines at each end. The algoithm is based on a comaison of the detailed coefficient of coesonding hases. The oosed method will eliminate oblems such as high fault esistance, cosscounty fault, mutual couling effect, cuent in-feed, and fault nea a emote bus. A high-esistance fault on two teminal aallel tansmission lines is esented in []. The ae discusses the oblems faced by a conventional non-ilot distance elay when otecting two teminal aallel tansmission lines. These oblems include gound fault esistance, efault system conditions, mutual effects of aallel lines, and shunt caacitance influences. The ae also esents a detailed analysis of imedance by the taking into account the elaying oint, mutual effects of aallel lines, shunt caacitance influences, and the system extenal to the otected line. The authos in [3] oose avoiding unde-eaching in twin cicuit lines without esidual cuent inut fom the aallel line. The mutual couling effect is 4
one oblem fo tansmission line otection fom single hase-to-eath faults on multile cicuit towes. The zeo sequence of the lines gets mutual couled causing an eo in the imedance seen by the elay. This causes the distance otection elay at one end of the faulty line to oveeach and the elay at the othe end to unde-each, which may lead to false ti of the healthy line. The authos oose the chaacteistic exession fo the effectiveness exeienced by a double cicuit with and without mutual couling and develo a non-iteative micoocesso-based eal-time algoithm fo comuting fault distance and zeosequence comensation in the distance elay scheme. Refeence [4] esents a method to locate the faults location in aallel tansmission lines without any measuements fom the healthy line cicuit. The ae discusses a new one-end fault location algoithm fo aallel tansmission lines. The method consides the flow of cuents fo the zeo sequence and utilizes the elation between the sequence comonents of a total fault cuent elevant fo single hase-to gound faults. This allows eflecting the mutual couling effect unde hase-to gound faults without using the zeo sequence cuent fom the healthy line cicuit. n [5] the tansmission line fault location methods have been esented. nstead of using both voltage and cuent, the method utilizes only the voltage as an inut and eliminates the use of cuent that caused eos because of the satuation of cuent tansfome. The fault location algoithms used unsynchonized voltage measued duing the fault. The algoithm also consides 5
shunt caacitance. The souce imedances ae assumed to be available at two teminals. Authos [6] esent the method fo deiving an otimal estimation of the fault location that can detect and identify the bad measuement to minimize the measuement eos fo imoving the fault location estimation. The deivation is based on the distibuted aamete line model and fully consides the effect of shunt caacitance. Autho [7] esents the deivation of the equivalent P cicuit fo the zeosequence netwoks of a double-cicuit line based on distibuted aamete model. The autho alies the symmetical comonent tansfomation that esult in ositive sequence, negative sequence, and zeo sequence. The mutual couling effect is taking into account fo zeo sequence analysis and the effects of shunt caacitance and a long line effect is consideed. Moe efeences can be found in [8]-[4] egading the studied subject. The algoithm based on lumed aamete model is esented in [5] to intoduce the eos fo long tansmission lines. The algoithm needs only the magnitude of the cuent fom diffeent teminal that is the diffeent cuent in diffeent cicuit measued at the same teminal. Because of this algoithm is develo in thee teminal aallel tansmission line, thus each teminal netwok should be conveted to an equivalent thee teminal netwok. The algoithm needs only the diffeences of the cuent, thus synchonization of the teminal is not equied. This algoithm is indeendent of the fault esistance and any souce imedances. 6
Refeence [6] esents a method to locate the faults location in aallel tansmission lines due to the mutual couling effects between cicuits of the lines by using the data fom only one end of the line. The algoithm is based on modifying the imedance method using modal tansfomation that tansfom the couled equations of the tansmission lines into decouled equations, then the elimination of the mutual effects esulting in an accuate estimation fo the fault location. 7
CHAPTER THREE PROPOSED NEW FAULT LOCATON ALGORTHM FOR PARALLEL TRANSMSSON LNES Each of the eseach oosals cited above fo detemining the tansmission line fault location has its own advantages and disadvantages, deending on the availability of the system measuement. n this eseach, will exloe new methods fo extacting a moe accuate estimation of fault location in long aallel tansmission lines by using the equivalent P cicuit based on a distibuted aamete line model. The new method, assuming the local voltage and cuent ae available, will fully conside the mutual couling imedance, the mutual couling admittance and shunt caacitance fo high ecision in fault distance estimation. This eseach builds uon and extends the wok of [] by accuately consideing the shunt caacitances of lines.. MODEL USED The new method uses only the voltage and cuent fom one end of aallel lines to calculate the fault distance []. This method is indeendent of the fault esistance, emote infeed, and souce imedance. This method is using shunt caacitance based on distibuted aamete line model and mutual couling between lines instead of lum aamete to imove the fault distance estimation fo aallel tansmission lines. 8
3 P 3 3 Q 4 4 4 f Figue3.. System diagam used in the develoment of the new algoithm To get the oosed algoithm to wok, the owe system model shown in fig. is used to develo the method fo imoving fault distance estimation fo aallel tansmission lines using only voltage and cuent fom only one end of the aallel tansmission lines. This owe system model consists of two geneatos, two aallel tansmission lines and fou buses:,, 3, and 4. We have assumed that one of the aallel lines is exeiencing a fault at bus 4. Afte we have finished with the model, ATP-EMTP (Altenative Tansient Pogam), secial softwae fo the simulation and analysis tansient in owe system, will be used fo the simulation and analysis. The model will be designed to study tansient state while fault occus in the owe system. ATP-EMTP has been utilized to geneate fault cases unde vaious fault conditions with diffeent fault 9
locations, fault tyes and fault esistances. ATP-EMTP will give the oututs in voltage, cuent, owe and enegy vesus times. All of the outut files fom ATP-EMTP simulation will be saved as an.at file and then conveted to a data file in.l4 fomat fo Matlab (Matix laboatoy) to use fo analysis. n the othe wods, EMTP will give an outut in time domain signals that is the simulation of the fault condition. Then we will use Matlab to convet time domain signals to fequency domain by using FFT to get hasos to use as inut fo the algoithm. n this eseach we assume that one of the aallel lines PQ that is 3 km long was selected to exeience the fault at F. The fault is km away fom bus P with ohms fault esistance. The system has the base voltage of 4 k and fequency is 5 Hz. The tansmission lines ae fully distibuted and the aametes of the tansmission lines ae obtained fom the table below: Table 3. Paametes e km of zeo-sequence netwoks of a aallel line Paamete Seies imedance(ohm/km) Mutual imedance(ohm/km) Shunt admittance(s/km) alue.68+j.37.3+j.638 j.78e-6 Mutual admittance y m (S/km) j.64e-6 3
Table 3. Paametes e km of ositive-sequence netwoks of a aallel lines Paamete alue Seies imedance(ohm/km).6+j.353 Shunt admittance(s/km) j4.66e-6 Table 3.3.Souce imedance at P and Q Paamete Teminal P Teminal Q Positive-sequence souce imedance(ohm).39+j9.7544.4745+j8.698 Zeo-sequence souce imedance(ohm).87+j8.4968.689+j3.967 oltages and cuents data in the system model at teminal P have been geneated unde vaious fault tyes and fault conditions. The data wee utilized in the algoithm in []. Liao, S. Elangovan, Digital Distance Relaying Algoithm fo Fist-Zone Potection fo Paallel Tansmission Lines, Poc.-Gene. Tansm. Distib. EE, 998, 45, (5),.53-536., to imlement and evaluate the simulated data fo fault distance and fault esistance The esults shown above will be used to comae with the esults of my oosed algoithm. 3
. Poosed Equivalent P Cicuit Model fo New Fault Location Algoithm fo Paallel Tansmission lines The symmetical comonent theoy will be used to design the model. Shunt caacitance, mutual admittance, and mutual imedance have to be consideed fo zeo sequence.. Positive Sequence Netwok The ositive sequence, the negative sequence, and zeo sequence netwoks of the aallel tansmission line ae deicted in Figue, Figue, and Figue 3 esectively. The aallel cicuits ae assumed to have the same aamete. Buses ae denoted by P and Q, while R is the fault location. Figue 3.. Equivalent P cicuit of ositive sequence netwok of the system duing the fault 3
n figue 4, the following notations ae adoted:, q ositive sequence voltage duing the fault at P and Q, ositive sequence voltage duing the fault at R at line and, q ositive sequence cuent duing the fault at P and Q at line, q ositive sequence cuent duing the fault at R at line, q Positive sequence cuent duing the fault at P and Q at line, q ositive sequence cuent duing the fault at R at line Z, q Z equivalent seies imedance of the line PR and QR at line Z, q Z equivalent seies imedance of the line PR and QR at line, q equivalent shunt admittance of the line PR and QR at line, q equivalent shunt admittance of the line PR and QR at line f ositive sequence fault cuent at R l fault distance fom P to R in mile o km The equivalent line aametes ae calculated based on the distibuted aamete line model as [7]: Z c z s y s (3.) γ s z s y s (3.) Z c z s y s (3.3) γ s z s y s (3.4) 33
Whee Z c chaacteistic imedance of the line γ s oagation constant of the line Z c chaacteistic imedance of the line γ s oagation constant of the line z s, s y ositive sequence seies imedance and shunt admittance of line e mile o km, esectively. z s, s y ositive sequence seies imedance and shunt admittance of line e mile o km, esectively. Z Z c sinh(γ s l ) (3.5) Z q Z c sinh[γ s (l l )] (3.6) Z Z c sinh(γ s l ) (3.7) Z q Z c sinh[γ s (l l )] (3.8) q q Z c tanh γ sl tanh γ s (ll ) Z c Z c tanh γ sl tanh γ s (ll ) Z c (3.9) (3.) (3.) (3.) 34
. Negative Sequence Netwok Figue 3.3.Equivalent P cicuit of negative sequence netwok of the system duing the fault n figue 5, the following notations ae adoted:, q Negative sequence voltage duing the fault at P and Q, Negative sequence voltage duing the fault at R at line and esectively, q Negative sequence cuent duing the fault at P and Q at line, q Negative sequence cuent duing the fault at R at line, q Negative sequence cuent duing the fault at P and Q at line 35
, q Negative sequence cuent duing the fault at R at line Z, q Z equivalent seies imedance of the line PR and QR at line Z, q Z equivalent seies imedance of the line PR and QR at line, q Equivalent shunt admittance of the line PR and QR at line, q Equivalent shunt admittance of the line PR and QR at line f Negative sequence fault cuent at R l Fault distance fom P to R in mile o km The equivalent line aametes ae calculated based on the distibuted aamete line model as[7]: Z c z s y s (3.3) γ s z s y s (3.4) Z c z s y s (3.5) γ s z s y s (3.6) Whee Z c chaacteistic imedance of the line γ s oagation constant of the line Z c chaacteistic imedance of the line γ s oagation constant of the line 36
z s y, s ositive sequence seies imedance and shunt admittance of line e mile o km, esectively. z s y, s ositive sequence seies imedance and shunt admittance of line e mile o km, esectively. Z Z c sinh(γ s l ) (3.7) Z q Z c sinh[γ s (l l )] (3.8) Z Z c sinh(γ s l ) (3.9) Z q Z c sinh[γ s (l l )] (3.) q q Z c tanh γ sl tanh γ s (ll ) Z c Z c tanh γ sl tanh γ s (ll ) Z c (3.) (3.) (3.3) (3.4) 37
.3 Zeo Sequence Netwok Figue 3.4.Equivalent P cicuit of mutually couled zeo-sequence netwok of the system duing the fault n figue 6, the following notations ae adoted: q, zeo sequence voltage duing the fault at P and Q, zeo sequence voltage duing the fault at R at line and, q zeo sequence cuent duing the fault at P and Q at line, q zeo sequence cuent duing the fault at R at line 38
, q zeo sequence cuent duing the fault at P and Q at line, q zeo sequence cuent duing the fault at R at line Z, Z q equivalent seies imedance of the line PR and QR at line Z, Z q equivalent seies imedance of the line PR and QR at line, q equivalent shunt admittance of the line PR and QR at line, q equivalent shunt admittance of the line PR and QR at line, m Z, y Z m total equivalent self and mutual shunt admittance total equivalent self and mutual seies imedance shelf shunt admittance of the line e unit length y m z mutual shunt admittance between line e unit length self-seies imedance between lines e unit length z m mutual seies imedance between lines e unit length f zeo sequence fault cuent at R l fault distance fom P to R in mile o km n the mode domain, define Z cm (z z m ) (y + y m ) (3.5) Z cm (z z m ) y (3.6) γ m (z z m )(y + y m ) (3.7) γ m (z + z m )y (3.8) 39
Z [Z cm sinh(γ m l ) + Z cm sinh(γ m l )] (3.9) Z [Z cm sinh(γ m l ) + Z cm sinh(γ m l )] (3.3) Z q Z cm sinh γ m (l l ) + Z cm sinh γ m (l l ) (3.3) Z q Z cm sinh γ m (l l ) + Z cm sinh γ m (l l ) (3.3) Z m [Z cm sinh(γ m l ) Z cm sinh(γ m l )] (3.33) Z mq Z cm sinh γ m (l l ) Z cm sinh γ m (l l ) (3.34) tanh(γ ml ) Z cm tanh(γ ml ) Z cm (3.35) (3.36) q tanh(γ m (ll ) ) (3.37) Z cm q tanh(γ m (ll ) ) (3.38) Z cm m tanh(γ ml ) Z cm mq tanh(γ m (ll ) ) Z cm tanh(γ ml ) (3.39) Z cm tanh(γ m (ll ) ) Z cm (3.4) 4
4.4 Poosed Distibuted Paamete Line Model Based Algoithm The distibuted aamete line model will be adoted fo the long tansmission lines. Based on the sequence netwoks, the following equations ae obtained: Positive Sequence: + Z + (3.4) + + (3.4) Z + + (3.43) + + (3.44) q q q Z + + (3.45) q q q q q + + (3.46) q q q q Z + + (3.47) q q q q q + + (3.48) ( ) se f q R + + (3.49) Z (3.5)
4 Negative Sequence: + Z + (3.5) + + (3.5) Z + + (3.53) + + (3.54) q q q Z + + (3.55) q q q q q + + (3.56) q q q q Z + + (3.57) q q q q q + + (3.58) ( ) se f q R + + (3.59) Z (3.6)
43 Zeo Sequence: We have: Z Z Z Z dx d m m (3.6) + + y y y y y y dx d m m m m (3.6) Whee,, z z self seies imedance e unit length of line and line esectively, y y self shunt admittance e unit length of line and line esectively Tansfomation matices and i T z z z z T m m v m m i z z T (3.63) + + m m v m m m m i y y T y y y y y y T (3.64) Then we can define a a a a T v (3.65) A A A A T v (3.66)
44 The following equations ae deived: m m m m o m Z Z + + + + (3.67) m m + (3.68) m m m m o m Z Z + + + + (3.69) m m + (3.7) mq mq q mq q q q q q mq oq mq q q q q q Z Z + + + + (3.7) mq q mq q q q q + (3.7) mq mq q mq q q q q q mq oq mq q q q q q Z Z + + + + (3.73) mq q mq q q q q + (3.74) ( ) se f q R + + (3.75) These equations fom the basis fo develoing the fault location algoithm fo diffeent tyes of faults as descibed in the next section.
45.5 Poosed New Method to Estimate Fault Distance and Fault Resistance The new method will aoach the oblem by deiving all equations fom ositive sequence, negative sequence, and zeo sequence netwok by using KL and KCL. Then, this eseach will emloy function in Matlab ogam called Fsolve fo iteative calculation. An a-g tye of fault will be consideed fist, and then the othe tyes of fault will be tackled late. The bounday condition fo an a-g fault is + + se se se (3.76) The tansfomation below will be adoted: a a a a c b a (3.77) Whee 3 j a o + (3.78) The zeo, ositive, and negative sequence of each hase can be deived as follow: c b a a a a a 3 (3.79) And the same fo cuent: c b a a a a a 3 (3.8)
.5. Poosed Algoithm This eseach aoaches the oblem by deiving all equations fom ositive sequence, negative sequence, and zeo sequence netwok by using KL and KCL. Then, the fault location is obtained by solving these equations. The Newton-Rahson aoach can be used to solve the unknowns as follows. Define the following function vecto: The Jacobian matix J(x) is calculated as: Whee Jij (x) the element in i th ow and j th column of J(x) f i, i, (3.8) f eal(f) (3.8) f imaj(f) (3.83) f(x) [f (x), f (x)] T (3.84) Jij (x) f i (x) x j, i,.,, j,, (3.85) The unknown can be obtained following an iteative ocedue. n the k th iteation, the unknowns ae udated using equation Whee x k+ x k x (3.86) x [J(x k )] f(x k ) (3.87) x k, x k+ the value of x befoe and afte k th iteation, esectively; x udate fo x and k th iteation; k iteation numbe stating fom 46
47 The iteation can be teminated when the udate x is smalle than the secified toleance. The unknown vaiables can be obtained by solving these equations and then bounday condition fo each tye of faults will be emloyed. Positive Sequence: _ Z left (3.88) _ left (3.89) Z (3.9) (3.9) q q q Z (3.9) _ q q q q q Z ight (3.93) _ q q q q q Z left + (3.94) _ q q q q q left (3.95) q f + (3.96) _ f f se R left (3.97)
48 Negative Sequence: _ Z left (3.98) _ left (3.99) Z (3.) (3.) q q q Z (3.) _ q q q q q Z ight (3.3) _ q q q q q Z left + (3.4) _ q q q q q left (3.5) q f + (3.6) _ f f se R left (3.7)
49 Zeo Sequence: m Z Z (3.8) _ m Z Z left (3.9) ( ) m left left (3.) ( ) _ m left (3.) ( ) _ mq q left A (3.) + mq q mq q qt Z Z A Z A Z left _ (3.3) ( ) q qt mq q left left (3.4) q f + (3.5) _ f f se R left (3.6)
.6 The bounday condition fo vaious faults: A-G fault se + se + se (3.) B-C fault se + a se + a se se + a se + a se > se se (3.8) Whee a B-C-G fault se + a se + a se se + a se + a se > se se (3.9) ABC fault se (3.) The fault location is obtained based on se, se and se and the bounday conditions. Let us take hase A to gound fault as an examle: Define: f se + se + se (3.) Then, we get a vecto of eal equations F [eal(f); imag(f)]; (3.) The unknown vaiables ae l andr f. Then the Newton-Rahson method can be used to find the unknown vaiables. An initial value of.5 fo l and zeo fo R f can be used. 5
CHAPTER FOUR EALUATON STUDES This chate comaes the esults between the Digital Distance Relaying Algoithm fo Fist-Zone Potection fo Paallel Tansmission Lines with the oosed algoithm.. Results of the existing algoithm fo Fault location estimation of vaious tyes of faults and vaious fault esistances. The fault location fo vaious tyes of faults and vaious fault esistances ae esented in Table 4.. The fault esistances, the estimated fault distance of each tye of faults ae given in column,,3,4, and 5 esectively. Table 4. Fault location estimation fo vaious tyes of faults and vaious fault esistances at 5 of 3 km: (.67.u.) of existing algoithm Fault Resistance Ω Fault Tyes a-g b-c b-c-g a-b-c.668.669.669.67.673.674.674.679.678.679.679.689 5
The estimated fault esistances fo vaious tyes of faults and vaious actual fault esistances ae esented in Table 4.. The actual fault esistance is in column ; the estimated fault esistances of each tye of faults ae given in column, 3, 4 and 5 esectively. Table 4. Fault Resistances estimation fo vaious tyes of faults at 5 of 3 km: (.67.u.) of existing algoithm Actual Fault Resistance Ω Fault Tyes a-g b-c b-c-g a-b-c 9.93 4.9668 4.9668 9.949 99.434 49.6793 49.6793 99.993 98.373 99.983 99.983 98.37 n Table 4.3, the estimated fault location fo vaious tyes of faults ae esented in column, 3, 4, and 5.The fault esistance fo each tye of fault ae given in the fist column. 5
Table 4.3 Fault location estimation fo vaious tyes of faults and vaious fault esistances at of 3 km: (.333.u.) of existing algoithm Fault Resistance Ω Fault Tyes a-g b-c b-c-g a-b-c.334.335.335.335.3347.3358.3358.3364.3353.3364.3364.3376 The fault esistance fo vaious tyes of faults and vaious fault location ae esented in Table 4.4. The actual fault esistances ae in the fist column, the estimated fault esistances of each tye of faults ae given in the second, thid, fouth, and fifth column esectively. 53
Table 4.4 Fault Resistances estimation fo vaious tyes of faults at of 3 km: (.333.u.) of existing algoithm Actual Fault Tyes Fault Resistance Ω a-g b-c b-c-g a-b-c 9.893 4.975 4.975 9.9363 98.9859 49.56 49.56 99.6 97.89 99.95 99.95 97.65 Table 4.5 esents the estimated fault location fo vaious fault tyes and vaious fault esistances, vaious fault esistances ae given in column. The estimated fault locations ae esented in column, 3, 4, and 5. 54
Table 4.5 Fault location estimation fo vaious tyes of faults and vaious fault esistances at of 3 km: (.667.u.) of existing algoithm Fault Resistance Ω Fault Tyes a-g b-c b-c-g a-b-c.6733.68.68.689.67.679.679.6774.6686.6774.6774.6739 The fault esistance fo vaious tyes of faults and vaious fault location ae esented in Table 4.6. The actual fault esistances ae in the fist column, the estimated fault esistances of each tye of faults ae given in the second, thid, fouth, and fifth column esectively. 55
Table 4.6 Fault Resistances estimation fo vaious tyes of faults at of 3 km: (.667.u.) of existing algoithm Actual Fault Tyes Fault Resistance Ω a-g b-c b-c-g a-b-c 9.787 4.9458 4.9458 9.7789 99.89 48.7538 48.7538 97.9493.64 97.9598 97.9598 97.9 Table 4.7 esents the estimated fault location fo vaious fault tyes and vaious fault esistances, vaious fault esistances ae given in column. The estimated fault locations ae esented in column, 3, 4, and 5. 56
Table 4.7 Fault location estimation fo vaious tyes of faults and vaious fault esistances at 5 of 3 km: (.833.u.) of existing algoithm Fault Resistance Ω Fault Tyes a-g b-c b-c-g a-b-c.8466.866.866.868.8379.8546.8546.8475.888.8474.8474.834 The fault esistance fo vaious tyes of faults and vaious fault location ae esented in Table 4.8. The actual fault esistances ae in the fist column, the estimated fault esistances of each tye of faults ae given in the second, thid, fouth, and fifth column esectively. 57
Table 4.8 Fault Resistances estimation fo vaious tyes of faults at 5 of 3 km: (.833.u.) of existing algoithm Actual Fault Tyes Fault Resistance Ω a-g b-c b-c-g a-b-c 9.3445 4.459 4.459 8.777 99.4489 44.985 44.985 94.475.46 94.77 94.77 4.463 We have noticed that the eo occus when the distance and Rf is inceasing. To imove the accuacy of fault distance estimation fo aallel tansmission lines, we use the algoithm that we oose with the same system model that we have ceated ealie. The method in estimating the fault location in long aallel tansmission lines by using the equivalent P cicuit is based on a distibuted aamete line model. The new method, howeve, assuming the local voltage and cuent ae available, fully consides the mutual couling imedance, the mutual couling admittance and shunt caacitance fo high ecision in fault distance estimation. The following shows the esults of the oosed algoithm. 58
. Results of the Poosed Algoithm with aious Tyes of Faults and aious Fault Resistances. The fault location fo vaious tyes of faults and vaious fault esistances ae esented in Table 4.9. The fault esistances, the estimated fault distance of each tye of faults ae given in column,,3,4, and 5 esectively. Table 4.9 Fault location estimation fo vaious tyes of faults and vaious fault esistances at 5 of 3 km of oosed algoithm. Fault Resistance Ω Fault Tyes a-g b-c b-c-g a-b-c 49.9879 49.9957 49.9957 49.9939 49.98 49.9797 49.9797 49.9554 49.9534 49.9576 49.9576 49.99 The estimated fault esistances fo vaious tyes of faults and vaious actual fault esistances ae esented in Table 4.. The actual fault esistance is in column ; the estimated fault esistances of each tye of faults ae given in column, 3, 4 and 5 esectively. 59
Table 4. Fault Resistances estimation fo vaious tyes of faults at 5 of 3 km of oose algoithm. Actual Fault Tyes Fault Resistance Ω a-g b-c b-c-g a-b-c.8 4.9935 4.9935.4.44 5.6 5.6..495.98.98.7 n Table 4., the estimated fault location fo vaious tyes of faults ae esented in column, 3, 4, and 5.The fault esistance fo each tye of fault ae given in the fist column. 6
Table 4. Fault location estimation fo vaious tyes of faults and vaious fault esistances at of 3 km: Fault Resistance Ω Fault Tyes a-g b-c b-c-g a-b-c 99.973 99.9945 99.9945 99.9876 99.9669 99.9743 99.9743 99.9463 99.9368 99.9474 99.9474 99.89 The fault esistance fo vaious tyes of faults and vaious fault location ae esented in Table 4.. The actual fault esistances ae in the fist column, the estimated fault esistances of each tye of faults ae given in the second, thid, fouth, and fifth column esectively. 6
Table 4. Fault Resistances estimation fo vaious tyes of faults at of 3 km: Actual Fault Tyes Fault Resistance Ω a-g b-c b-c-g a-b-c 9.993 4.997 4.997 9.9999.69 5.7 5.7.9.548.8.8.9 Table 4.3 esents the estimated fault location fo vaious fault tyes and vaious fault esistances, vaious fault esistances ae given in column. The estimated fault locations ae esented in column, 3, 4, and 5. 6
Table 4.3 Fault location estimation fo vaious tyes of faults and vaious fault esistances at of 3 km: Fault Resistance Ω Fault Tyes a-g b-c b-c-g a-b-c 99.976...53.5.737.737.68.58.59.59.34 The fault esistance fo vaious tyes of faults and vaious fault location ae esented in Table 4.4. The actual fault esistances ae in the fist column, the estimated fault esistances of each tye of faults ae given in the second, thid, fouth, and fifth column esectively. 63
Table 4.4 Fault Resistances estimation fo vaious tyes of faults at of 3 km: Actual Fault Tyes Fault Resistance Ω a-g b-c b-c-g a-b-c 9.9433 5. 5. 9.988 99.875 49.956 49.956 99.853 99.4956 99.858 99.858 99.935 Table 4.5 esents the estimated fault location fo vaious fault tyes and vaious fault esistances, vaious fault esistances ae given in column. The estimated fault locations ae esented in column, 3, 4, and 5. 64