A new accurate algorithms based on mathematical modeling of two parallel transmissions lines system (TPTLS) as influenced by the mutual effect to determine the fault location is discussed in this work. The distance relay measures the impedance to the fault location which is the positive-sequence. The principle of summation the positive-, negative-, and zero-sequence voltages which equal zero is used to determine the fault location on the TPTLS. Also, the impedance of the transmission line to the fault location is determined. These algorithms are applied to single-line-to-ground (SLG) and double-line-to-ground (DLG) faults. To detect the fault location along the transmission line, its impedance as seen by the distance relay is determined to indicate if the fault is within the relay’s reach area. TPTLS under study are fed from one- and both-ends. A schematic diagrams are obtained for the impedance relays to determine the fault location with high accuracy.
New accurate algorithmsmathematical modelingfault locationscheme diagramsparallel transmission linesmutual effectSLG and DLG faultsIntroduction
An accurate detections of fault location of the high voltage transmission lines (TL) is a critical subject of the researches and engineering for long time. This action explains that a fault point can be resolved extremely not long after a fault happens along a transmission line to save time which is expected to resume the activity of the line and advantage power utilities and customers [1]. An algorithm for fault location dependent on Fourier investigations of a faulted system was proposed [2,3]. The algorithm encapsulates a precise fault location by estimating just nearby end information. Its basic hypothesis was concentrated through computerized recreations on a model force framework. The fault locator has been created in [4,5], which compute the reactance of a flawed line, with a smaller scale processor, utilizing the one-terminal voltage and current information of the transmission line. Another fault location calculation applied to multi-terminal two equal transmission lines was proposed [6–8]. The calculation utilized similar AC contributions as defensive transfers dependent on the greatness of the differential flow at every terminal.
Another strategy for deficiency area estimation was proposed [9,10] utilizing information from the two closures of the TL with no requirement for information synchronization. Two new strategies were proposed for fault location in equal twofold circuit multi-terminal TL by utilizing stage voltages and flows data from CCVT’s and CT’s at every terminal [11–15]. The two strategies were applied to stage to-ground and stage-to-stage flaws. Another issue area strategy dependent on six-sequence fault segments was produced for equal lines dependent on the issue examination of a joint equal TL [16,17]. A calculation for precisely finding TL single line-to-ground flaws with insignificant data of voltage and current was created [18]. A microprocessor-based fault locator was portrayed [19–22], which utilized a novel remuneration strategy to improve precision. An exact fault location method that utilized post-flaw voltage and current determined at both line closes were proposed [23,24]. A fault location technique for one of two exclusively transposed equal transmission lines was depicted [25–30]. Parallel TL under issue were decoupled into the normal part net and differential segment net [31–35]. Ground fault area of TL dependent on time contrast of appearance of ground-mode and airborne mode voyaging waves was examined in [36–38].
In this research work, a new accurate algorithm based on mathematical modeling of two parallel transmissions lines system (TPTLS) as influenced by the mutual effect to determine the fault location is discussed. The distance relay measures the impedance to the fault location which is the positive-sequence. The principle of summation the positive-, negative-, and zero-sequence voltages which equal zero, of Ef1 +Ef2 +Ef0 = 0, is used to determine the fault location on the TPTLS. Also, the impedance of the transmission line to the fault location is determined. These algorithms are applied to single-line-to-ground (SLG) and double-line-to-ground (DLG) faults. To detect the fault location along the transmission line, its impedance as seen by the distance relay is determined which indicates if the fault within the reach of the relay. TPTLS under study are fed from one- and both-ends. A schematic diagrams are obtained for the impedance relays to determine the fault location with high accuracy.
Mathematical Modeling of TPTLS Fed on One-End
Fig. 1 shows a two parallel transmission lines system (TPTLS) TL1 and TL2, which are protected by two distance relays R at bus 1. Where; If: fault current; IB: line current from bus 1; EB1: bus voltage at bus 1; It: total current fed from the generator at bus 1.
Two parallel transmission lines system (TPTLS) fed on one-end
Determination of the Fault Location without Mutual Coupling between TPTLSSLG Fault
Fig. 2 shows the sequence impedance networks of SLG fault occurred at one-line of the TPTLS at percentage (fraction) n of the line length where, the sequence networks are connected in series in SLG fault to determine the fault currents as:
It=Ifand their componentsIt1=If1,It2=If2,andIt0=If0
If1=If2=If0=(1/3)If
where, IB0, IB1, and IB2: Zero-, positive-, and negative-sequence currents at bus 1; It0, It1, and It2: Zero-, positive-, and negative-sequence total current fed from the generator at bus-1.
where, n: percentage (fraction) of line length that is location of fault has happen. From Eq. (3) the bus sequence currents are equal, i.e.,
IB1=IB2=IB0
Because It1 = It2 = It0.
(A) Equivalent positive-, (B) Equivalent negative-, and (C) Equivalent zero-sequence networks of parallel transmission line systems fed from one-end without taking into account the mutual effect
At bus 1, the voltage sequence-components EB1 is:
EB1=Eg1-It1(xg1+xt)EB2=-It2(xg2+xt)EB1=-It0xt
The generator G voltage (emf) Eg1=1∠0 pu.
EB=EB1+EB2+EB0
The voltage at fault location Ef is:
Ef=Ef1+Ef2+Ef0=0
From Fig. 2:
EB1-Ef1IB1=nZ1
EB2-Ef2IB2=nZ2
EB0-Ef0IB0=nZ0
Adding Eqs. (8)–(10) and using Eqs. (4), (6) and (7) to obtain:
EB-0IB1=n(Z1+Z2+Z0)
From Eq. (11), n(cal): calculated percentage (fraction) of line length is:
n(cal)=(EB/IB1)Ztotal
where,
Ztotal=Z1+Z2+Z0
Therefore, the impedance of the transmission line to fault location ZR which measured by the distance relay is:
ZR=n(cal)Z1=(EB/IB1)ZtotalZ1
Fig. 3 shows a schematic diagram of the distance relay that determine the fault location on the faulted TL based on the TL impedance ZR which indicates if the fault is within the relay’s reach area.
The schematic diagram for the implementation of the impedance relay
DLG Fault
Fig. 2 shows the sequence impedance networks of DLG fault occurred at one-line of the TPTLS at percentage (fraction) n of the line length where, the sequence networks are connected in parallel in DLG fault to determine the fault currents as:
It=If,and their componentsIt1=If1,It2=If2andIt0=If0
Again; IB0, IB1, and IB2: Zero-, positive-, and negative-sequence currents at bus 1; It0, It1, and It2: Zero-, positive-, and negative-sequence total current fed from the generator at bus 1. Eq. (3) can be applied in this case and the voltage sequence-components EB1 at bus 1 is calculated as:
EB1=(a2Eg1)-It1(xg1+xt)EB2=-It2(xg2+xt)EB0=-It0xt
The generator G voltage (emf) Eg1=1∠0 pu.
EB=EB1+EB2+EB0
The voltage at fault location Ef is:
Ef=Ef1+Ef2+Ef0=0
In analogy with Eqs. (8)–(10), a three similar equations can be written:
EB1-Ef1IB1=nZ1
EB2-Ef2IB2=nZ2=(EB2-Ef2IB1)(-a)(Z2eq+Z0eqZ0eq)
EB0-Ef0IB0=nZ0=(EB0-Ef0IB1)(-a2)(Z2eq+Z0eqZ2eq)
where,
IB2=-(IB1a)(Z0eqZ2eq+Z0eq)IB0=-(IB1a2)(Z2eqZ2eq+Z0eq)
where; Z0eq and Z2eq: total (equivalent) impedances of the zero- and negative-sequence networks without the mutual effect, respectively.
(EB2-Ef2IB1)=nZ2(-1a)(Z0eqZ2eq+Z0eq)
(EB0-Ef0IB1)=nZ0(-1a2)(Z2eqZ2eq+Z0eq)
Adding Eqs. (19), (22) and (23), and using Eqs. (17) and (18), one can write:
Eq. (25) is a cubic complex coefficients equation:
An3+(B-D(EBIB1))n2+(C-E(EBIB1))n-(F(EBIB1))=0
Solving this equation, Eq. (26), using MATLAB software to determine the calculated percentage (fraction) n(cal), which expresses where the fault is located, (n(cal) is the real one of three solutions). Therefore, the impedance of the transmission line to fault location ZR which measured by the distance relay is:
ZR=n(cal)Z1
Fig. 4 shows a schematic diagram of the distance relay that determine the fault location on the faulted TL based on the TL impedance ZR which indicates if the fault is within the relay’s reach area.
The schematic diagram for the implementation of the impedance relay
Determination of Fault Location of TPTLS as Affected by Mutual CouplingSLG Fault
Fig. 5 shows the sequence impedance networks of SLG fault occurred at one-line of the TPTLS at percentage (fraction) n of the line length where with mutual effect between TPTLS.
Itm=IBm=Ifm
Their components Itm1 = Ifm1 = IBm1, Itm2 = Ifm2 = I and Itm0 = Ifm0 = IBm0:
Ifm1=Ifm2=Ifm0=(1/3)Ifm
(A) Equivalent positive-, (B) equivalent negative-, and (C) equivalent zero-sequence networks of TPTLS fed from one-end taking into account the mutual effect
From Eqs. (28) and (29) the bus sequence currents are equal, i.e.,
IBm1=IBm2=IBm0
The voltages at fault location Efm and at the bus location EBm are:
where, Z1m, Z2m, and Z0m: The positive-, negative- and zero-sequence mutual impedances between the parallel TL. Adding Eqs. (33)–(35), and using Eqs. (29)–(32) to obtain:
EBm-0IBm1=n22(Zm-Ztotal)+nZtotal
where, Zm = Z1m +Z2m +Z0m. The solution of the quadratic Eq. (36) determines the calculated percentage (fraction) nm(cal) as:
Therefore, the distance relay can measure the impedance ZRm which is expressed as:
ZRm=EBm1-Efm1Iam1
where, Iam1: the current of n(Z1 − Z1m) branch. Fig. 6 shows a schematic diagram of the distance relay that determine the fault location on the faulted TL based on the TL impedance ZRm which indicates if the fault is within the relay’s reach area.
The schematic diagram for the implementation of the impedance relay
DLG Fault
Fig. 5 shows the sequence impedance networks of DLG fault occurred at one-line of the TPTLS at percentage (fraction) n of the line length with mutual effect.
Itm=IBm=Ifm
Their components Itm1 = Ifm1 = IBm1, Itm2 = Ifm2 = IBm2 and Itm0 = Ifm0 = IBm0. The voltage sequence-components EBm at bus 1 is calculated as:
where,
IBm2=-(IBm1a)(Z0meqZ2meq+Z0meq)IBm0=-(IBm1a2)(Z2meqZ2meq+Z0meq)
where; Z0meq and Z2meq: total (equivalent) impedances of the zero- and negative-sequence networks with taking into account the mutual effect, respectively.
Solving this equation, Eq. (50), using MATLAB software to determine the calculated percentage (fraction) nm(cal), which expresses where the fault is located (nm(cal) is the real one of the four solutions). Therefore, the distance relay can measure the impedance ZRm which is expressed as:
ZRm=EBm1-Efm1Iam1
where Iam1 is the current that passes through n(Z1 − Z1m) branch. The schematic diagram for the implementation of the impedance relay is as shown in Fig. 7, which determine the fault location on the faulted TL based on the TL impedance ZRm which indicates if the fault is within the relay’s reach area.
The schematic diagram for the implementation of the impedance relay
Mathematical Modeling of TPTLS Fed on Both-Ends
Fig. 8 shows a two parallel transmission lines system (TPTLS) TL1 and TL2, which are protected by two distance relays R at bus-1. Where; If: fault current; IB: line current from bus-1; EB1, EB2: bus voltages at bus-1 and bus-2 respectively; It1, It2: total currents fed from the generators G1 at bus-1 and G2 at bus-2.
The two parallel transmission lines system (TPTLS) fed on both-ends
Determination of Fault Location Without Mutual Coupling Between TPTLS
Fig. 9 shows the sequence impedance networks of SLG fault occurred at one-line of the TPTLS at percentage (fraction) n of line length with taking into account the mutual effect. The delta connection of the parallel TL converted to star connection for simplifying the circuit. Fig. 10 shows the positive-sequence network after such conversion.
Za1=0.5n(1-n)Z1Zb1=0.5n(1-n)Z1
Zc1=0.5nZ1
Za0=0.5n(1-n)Z0Zb0=0.5n(1-n)Z0Zc0=0.5nZ0
(A) Equivalent positive-, (B) equivalent negative-, and (C) equivalent zero-sequence networks of parallel transmission line systems fed from both-ends without taking into account the mutual effect
Delta-to-star conversion of positive-sequence network
The total current It which supplied from generator G1 at bus-1 are:
where, the emf of generators G1 and G2 are (Eg1=Eg2=1∠0) pu.
EB=EB1+EB2+EB0
The fault voltage sequence-components Ef are:
Ef1=EB1-(nZ1IB1)Ef2=EB2-(nZ2IB2)Ef0=EB0-(nZ0IB0)
where their sum is equal to zero, i.e.,
Ef=Ef1+Ef2+Ef0=0
In analogy with Eqs. (8)–(10), a three similar equations can be written with aiding of Fig. 9. Adding these equations and using Eqs. (57) and (59), one can write:
where, CF: compensating factor which is expressed as:
CF=IB0IB1
So, the calculated percentage (fraction) n(cal) which is a define of fault location on the TL:
n(cal)=(EB/IB1)(Z1+Z2+Z0CF)
And, the measured impedance that seen by the relay ZR is:
ZR=n(cal)Z1=(EB/IB1)(Z1+Z2+Z0CF)Z1
Fig. 11 shows a schematic diagram of the distance relay that determine the fault location on the faulted TL based on the TL impedance ZR which indicates if the fault is within the relay’s reach area.
The schematic diagram for the implementation of the impedance relay
Determination of Fault Location with Mutual Coupling of TPTLS
Fig. 12 shows the total (equivalent) sequence impedance networks, positive- negative- and zero-sequence networks, of TPTLS at percentage (fraction) n of the line length with mutual effect. The delta connection of the parallel TL converted to star connection for simplifying the circuit. Fig. 13 shows the positive-sequence network after such conversion.
(A) Equivalent positive-, (B) equivalent negative-, and (C) equivalent zero-sequence networks of parallel transmission line systems fed from both-ends with mutual effect
Delta-to-star conversion of the positive-sequence network
The total current Itm components which supplied from generator G1 at bus-1 are:
In analogy with Eqs. (8)–(10), a three similar equations can be written with aiding of Fig. 13. Adding these equations and using Eqs. (69) and (71), one can write:
Solving this equation, Eq. (75), using MATLAB software to determine the calculated percentage (fraction) nm(cal), which expresses where the fault is located, (nm(cal) is the real one of the four solutions). Therefore, the distance relay can measure the impedance ZRm which is expressed as:
ZRm=EBm1-Efm1Iam1
where, Iam1 is the current of n(Z1 − Z1m) branch. The schematic diagram for the implementation of the impedance relay is as shown in Fig. 14 that determine the fault location on the faulted TL based on the TL impedance ZRm which indicates if the fault is within the relay’s reach area.
The schematic diagram for the implementation of the impedance relay
Results and Discussion
A MATLAB code is applied to the strategy which is explained before in Sections 2 and 3 to decide the percentage (fractions) n(cal) and nm(cal) characterizing the fault location of two parallel transmission lines system (TPTLS) with or without taking into account the mutual coupling between the parallel lines. The positive-sequence impedances ZR and ZRm which seen by the distance relay R are determined at various places of the fault. This makes it conceivable to decide the relay under-reach at proportion for various positions of the fault. SLG and DLG faults are examined when the system is fed from one- or both-ends.
For SLG faults, the proposed algorithms can determine exactly where the location of the fault whatever the system is fed from one- or both-ends and with or without mutual coupling between the TPTLS as listed in Tabs. 1–4. Also, the proposed algorithms are worked well in case of DLG faults fed from one- or both-ends and with or without mutual coupling between the TPTLS as listed in Tabs. 5 and 6.
Calculated fault location <italic>n</italic><sub>(<italic>cal</italic>)</sub> and <italic>n</italic><sub><italic>m</italic>(<italic>cal</italic>)</sub> seen by distance relay <italic>Z<sub>R</sub></italic> and <italic>Z<sub>Rm</sub></italic>, and reach–ratio for SLG faults with and without mutual effect of one-end systems
% along line length, ngiven
% of line length at fault, n(cal)
Line impedance as seen by the phase-distance relay, ZR
% of line length at fault, nm(cal)
Line impedance as seen by the phase-distance relay, ZRm
Reach-ratio ((ZRm/ZR) −1)%
10
10
0.0012+j 0.0034
10
0.0012+j 0.0034
−0.0725
20
20
0.0025+j 0.0069
20
0.0025+j 0.0069
−0.1528
30
30
0.0037+j 0.0103
30
0.0038+j 0.0103
−0.2421
40
40
0.0049+j 0.0138
40
0.0050+j 0.0137
−0.3420
50
50
0.0062+j 0.0172
50
0.0063+j 0.0170
−0.4546
60
60
0.0074+j 0.0206
60
0.0077+j 0.0204
−0.5823
70
70
0.0087+j 0.0241
70
0.0090+j 0.0237
−0.7286
80
80
0.0099+j 0.0275
80
0.0104+j 0.0270
−0.8976
90
90
0.0111+j 0.0309
90
0.0118+j 0.0303
−1.0952
100
100
0.0124+j 0.0344
100
0.0133+j 0.0335
−1.3291
Calculated fault location <italic>n</italic><sub>(<italic>cal</italic>)</sub>, <italic>n</italic><sub><italic>m</italic>(<italic>cal</italic>)</sub>, zero-sequence impedances <italic>Z</italic><sub>0<italic>R</italic></sub> and <inline-formula id="ieqn-12"><alternatives><inline-graphic xlink:href="ieqn-12.png"/><tex-math id="tex-ieqn-12"><![CDATA[$Z_{0\textit{Rm}}$]]></tex-math><mml:math id="mml-ieqn-12"><mml:msub><mml:mrow><mml:mi>Z</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn><mml:mstyle class="text"><mml:mtext class="textit" mathvariant="italic">Rm</mml:mtext></mml:mstyle></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>, and ratio (<inline-formula id="ieqn-13"><alternatives><inline-graphic xlink:href="ieqn-13.png"/><tex-math id="tex-ieqn-13"><![CDATA[$Z_{0\textit{Rm}}/Z_{0R}$]]></tex-math><mml:math id="mml-ieqn-13"><mml:msub><mml:mrow><mml:mi>Z</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn><mml:mstyle class="text"><mml:mtext class="textit" mathvariant="italic">Rm</mml:mtext></mml:mstyle></mml:mrow></mml:msub><mml:mo>/</mml:mo><mml:msub><mml:mrow><mml:mi>Z</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn><mml:mi>R</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>) for SLG faults with mutual effect of one-end systems
% Along line length, ngiven
Zero-sequence impedance up to fault location, Z0R
Zero-sequence impedance up to fault location, Z0Rm
Zero-sequence ratio (Z0Rm/Z0R)
10
0.0026+j 0.0131
0.0027+j 0.0135
1.0281
20
0.0052+j 0.0263
0.0055+j 0.0278
1.0593
30
0.0078+j 0.0394
0.0084+j 0.0431
1.0942
40
0.0104+j 0.0525
0.0115+j 0.0596
1.1334
50
0.0130+j 0.0656
0.0149+j 0.0774
1.1779
60
0.0156+j 0.0788
0.0185+j 0.0969
1.2288
70
0.0182+j 0.0919
0.0225+j 0.1185
1.2874
80
0.0208+j 0.1050
0.0269+j 0.1427
1.3559
90
0.0234+j 0.1181
0.0318+j 0.1701
1.4368
100
0.0260+j 0.1313
0.0374+j 0.2018
1.5338
Calculated fault location <italic>n</italic><sub>(<italic>cal</italic>)</sub> and <italic>n</italic><sub><italic>m</italic>(<italic>cal</italic>)</sub> seen by distance relay <italic>Z<sub>R</sub></italic> and <italic>Z<sub>Rm</sub></italic>, and reach-ratio for SLG faults with and without mutual effect of both-end systems
% along line length, ngiven
% of line length at fault, n(cal)
Line impedance as seen by the phase-distance relay, ZR
% of Line length at fault, nm(cal)
Line impedance as seen by the phase-distance relay, ZRm
Reach ratio ((ZRm/ZR) −1)%
10
10
0.0062+j 0.0172
10
0.0062+j 0.0173
0.4140
20
20
0.0124+j 0.0344
20
0.0124+j 0.0345
0.3370
30
30
0.0185+j 0.0516
30
0.0185+j 0.0517
0.2456
40
40
0.0247+j 0.0688
40
0.0247+j 0.0689
0.1354
50
50
0.0309+j 0.0859
50
0.0309+j 0.0859
0.0000
60
60
0.0371+j 0.1031
60
0.0371+j 0.1029
−0.1705
70
70
0.0433+j 0.1203
70
0.0433+j 0.1198
−0.3916
80
80
0.0494+j 0.1375
80
0.0495+j 0.1364
−0.6899
90
90
0.0556+j 0.1547
90
0.0557+j 0.1527
−1.1139
100
100
0.0618+j 0.1719
100
0.0618+j 0.1685
−1.7637
Calculated fault location <italic>n</italic><sub>(<italic>cal</italic>)</sub>, <italic>n</italic><sub><italic>m</italic>(<italic>cal</italic>)</sub>, zero-sequence impedances <italic>Z</italic><sub>0<italic>R</italic></sub> and <inline-formula id="ieqn-19"><alternatives><inline-graphic xlink:href="ieqn-19.png"/><tex-math id="tex-ieqn-19"><![CDATA[$Z_{0\textit{Rm}}$]]></tex-math><mml:math id="mml-ieqn-19"><mml:msub><mml:mrow><mml:mi>Z</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn><mml:mstyle class="text"><mml:mtext class="textit" mathvariant="italic">Rm</mml:mtext></mml:mstyle></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>, and ratio (<inline-formula id="ieqn-20"><alternatives><inline-graphic xlink:href="ieqn-20.png"/><tex-math id="tex-ieqn-20"><![CDATA[$Z_{0\textit{Rm}}/Z_{0R}$]]></tex-math><mml:math id="mml-ieqn-20"><mml:msub><mml:mrow><mml:mi>Z</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn><mml:mstyle class="text"><mml:mtext class="textit" mathvariant="italic">Rm</mml:mtext></mml:mstyle></mml:mrow></mml:msub><mml:mo>/</mml:mo><mml:msub><mml:mrow><mml:mi>Z</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn><mml:mi>R</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>) for SLG faults with and without mutual effect of both-end systems
% along line length, ngiven
Zero-sequence impedance up to fault location, Z0R
Zero-sequence impedance up to fault location, Z0Rm
Zero-sequence ratio (Z0Rm/Z0R)
10
0.0130+j 0.0656
0.0130+j 0.0635
0.9691
20
0.0260+j 0.1313
0.0259+j 0.1278
0.9741
30
0.0390+j 0.1969
0.0389+j 0.1929
0.9805
40
0.0520+j 0.2626
0.0519+j 0.2595
0.9888
50
0.0650+j 0.3282
0.0650+j 0.3282
1.0000
60
0.0780+j 0.3938
0.0782+j 0.4003
1.0160
70
0.0910+j 0.4595
0.0917+j 0.4787
1.0406
80
0.1040+j 0.5251
0.1060+j 0.5703
1.0836
90
0.1170+j 0.5907
0.1244+j 0.6978
1.1770
100
0.1300+j 0.6564
0.1869+j 1.0074
1.5313
Calculated fault location <italic>n</italic><sub>(<italic>cal</italic>)</sub> and <italic>n</italic><sub><italic>m</italic>(<italic>cal</italic>)</sub> seen by distance relay <italic>Z<sub>R</sub></italic> and <italic>Z<sub>Rm</sub></italic>, and reach-ratio for DLG faults with and without mutual effect of one-end systems
% along line length, ngiven
% of line length at fault, n(cal)
Line impedance as seen by the phase-distance relay, ZR
% of line length at fault, nm(cal)
Line impedance as seen by the phase-distance relay, ZRm
Reach ratio ((ZRm/ZR) −1)%
10
10
0.0012+j 0.0034
10
0.0012+j 0.0034
−0.0929
20
20
0.0025+j 0.0069
20
0.0025+j 0.0069
−0.1962
30
30
0.0037+j 0.0103
30
0.0037+j 0.0103
−0.3115
40
40
0.0049+j 0.0138
40
0.0049+j 0.0137
−0.4413
50
50
0.0062+j 0.0172
50
0.0062+j 0.0171
−0.5884
60
60
0.0074+j 0.0206
60
0.0074+j 0.0204
−0.7564
70
70
0.0087+j 0.0241
70
0.0087+j 0.0238
−0.9502
80
80
0.0099+j 0.0275
80
0.0099+j 0.0271
−1.1763
90
90
0.0111+j 0.0309
90
0.0111+j 0.0304
−1.4434
100
100
0.0124+j 0.0344
100
0.0124+j 0.0337
−1.7636
Calculated fault location <italic>n</italic><sub>(<italic>cal</italic>)</sub>, <italic>n</italic><sub><italic>m</italic>(<italic>cal</italic>)</sub>, zero-sequence impedances <italic>Z</italic><sub>0<italic>R</italic></sub> and <inline-formula id="ieqn-23"><alternatives><inline-graphic xlink:href="ieqn-23.png"/><tex-math id="tex-ieqn-23"><![CDATA[$Z_{0\textit{Rm}}$]]></tex-math><mml:math id="mml-ieqn-23"><mml:msub><mml:mrow><mml:mi>Z</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn><mml:mstyle class="text"><mml:mtext class="textit" mathvariant="italic">Rm</mml:mtext></mml:mstyle></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>, and ratio (<inline-formula id="ieqn-24"><alternatives><inline-graphic xlink:href="ieqn-24.png"/><tex-math id="tex-ieqn-24"><![CDATA[$Z_{0\textit{Rm}}/Z_{0R}$]]></tex-math><mml:math id="mml-ieqn-24"><mml:msub><mml:mrow><mml:mi>Z</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn><mml:mstyle class="text"><mml:mtext class="textit" mathvariant="italic">Rm</mml:mtext></mml:mstyle></mml:mrow></mml:msub><mml:mo>/</mml:mo><mml:msub><mml:mrow><mml:mi>Z</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn><mml:mi>R</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>) for DLG faults with and without mutual effect of one-end systems
% along line length, ngiven
Zero-sequence impedance up to fault location, ZR
Zero-sequence impedance up to fault location, ZRm
Zero-sequence ratio (Z0Rm/Z0R)
10
0.0026+j 0.0131
0.0027+j 0.0135
1.0280
20
0.0052+j 0.0263
0.0055+j 0.0278
1.0590
30
0.0078+j 0.0394
0.0084+j 0.0431
1.0937
40
0.0104+j 0.0525
0.0115+j 0.0595
1.1328
50
0.0130+j 0.0656
0.0149+j 0.0773
1.1771
60
0.0156+j 0.0788
0.0185+j 0.0968
1.2277
70
0.0182+j 0.0919
0.0225+j 0.1184
1.2860
80
0.0208+j 0.1050
0.0269+j 0.1425
1.3542
90
0.0234+j 0.1181
0.0318+j 0.1698
1.4347
100
0.0260+j 0.1313
0.0374+j 0.2015
1.5313
From Tabs. 1, 3 and 5, the calculated values of percentage (fraction) n(cal) and nm(cal) that obtained from the proposed algorithms are exactly equal to the ngiven which is defining the fault location on one-faulted TL of the TPTLS.
In one-end systems, the impedance relay is under-reach due the mutual coupling between the TPTLS, ZRm is less than ZR and the reach-ratio is given in negative values in Tabs. 1 and 3 while in systems fed from both-ends, the reach-ratio is positive close to bus-1 and negative values for far faults from bus-1, Tabs. 3 and 4.
Fig. 15A shows that the relay R can see currents flowing in the same direction of a-f1 of systems that feeds from one-end. Therefore, the line impedance that seen by the relay ZR is small. In Fig. 15B the faults accrue far away from bus-1, so the current passes through long section of TL, a-f2 and the increase of mutual coupling is more noticeable in the relay ZRm, over ZR. This is why reach-ratio values are small for faults close to bus-1 and increased while increasing the distance of faults far away from bus-1, Tabs. 1, 3 and 5. Not only the impedance is seen by the relay, but also the zero-sequence impedance Z0Rm exceeds Z0R, the impedance without mutual coupling as given in Tabs. 2, 4, and 6.
The flowing currents in the TPTLS fed from one-end for (A) near-bus fault, and (B) far-bus fault on
Fig. 16A shows the relay that can see currents in the opposite direction of a-f1 portion for faults close to bus-1 in systems fed from both-ends. ZRm exceeds ZR which mean the relay is overreach or positive-value as listed in Tab. 3 without considering the mutual coupling between the TPTLS. Fig. 16B shows that as fault moves far away from bus-1, the relay can see the current that flowing in the longer portion a-f2 of the faulted line. So, ZRm is less than ZR, which mean reach ratio is negative without taking into account the effect of mutual coupling as listed in Tab. 3. In the case of fault occurred at the mid-way of the faulted line length, the relay will not be in over- or under-reach in Tabs. 3 and 4, Fig. 16C.
The flowing currents in the TPTLS fed from both-end for (A) near-bus fault, (B) far-bus fault, and (C) mid-way fault on
Fig. 17 shows that the zero-sequence ratio (Z0Rm/Z0R) for SLG faults increases at the fault moves along the line length toward the far-bus. This is simply explained by the increasing effect of the mutual coupling between parallel lines. The zero-sequence ratio for DLG faults is almost the same as for SLG faults in Fig. 17.
Zero-sequence ratio for SLG and DLG faults against fault location <italic>n<sub>given</sub></italic>
Conclusions
Based on mathematical computation of the fault location of parallel transmission lines with taking into account the effect of mutual coupling between parallel lines, the conclusions can be listed as:
An accurate algorithms of determination of fault location on parallel transmission lines fed from one- or both-ends with or without considering the mutual coupling is considered in details of the research work. These algorithms are applied to SLG and DLG faults occurred on one of the paralleled transmission lines.
The impedances that the distance relay sees with or without mutual coupling in one- or both-ends feeding systems in SLG or DLG faults are calculated.
A schematic diagrams of the proposed algorithms are obtained based on the mathematical modeling of fault location which indicates if the fault is within the relay reach area.
Funding Statement: The authors received no specific funding for this study.
Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.
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Bus currents as a function of fault current and total current supplied to bus 1. Concerning Fig. A1, one can write: