Accurate Fault Location Modeling for Parallel Transmission Lines Considering Mutual Effect

: 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.


Introduction
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][7][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][12][13][14][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][20][21][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][26][27][28][29][30]. Parallel TL under issue were decoupled into the normal part net and differential segment net [31][32][33][34][35]. Ground fault area of TL dependent on time contrast of appearance of ground-mode and airborne mode voyaging waves was examined in [36][37][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 E f 1 + E f 2 + E f 0 = 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. Fig. 1 shows a two parallel transmission lines system (TPTLS) TL 1 and TL 2 , which are protected by two distance relays R at bus 1. Where; I f : fault current; I B : line current from bus 1; E B1 : bus voltage at bus 1; I t : total current fed from the generator at bus 1.  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:

SLG Fault
I t = I f and their components I t1 = I f 1 , I t2 = I f 2 , and I t0 = I f 0 (1) where, I B0 , I B1 , and I B2 : Zero-, positive-, and negative-sequence currents at bus 1; I t0 , I t1 , and I t2 : 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., Because I t1 = I t2 = I t0 .
The voltage at fault location E f is: From Fig. 2: From Eq. (11), n (cal) : calculated percentage (fraction) of line length is: where, Therefore, the impedance of the transmission line to fault location Z R which measured by the distance relay is: Again; I B0 , I B1 , and I B2 : Zero-, positive-, and negative-sequence currents at bus 1; I t0 , I t1 , and I t2 : 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 E B1 at bus 1 is calculated as: The generator G voltage (emf) E g1 = 1∠0 pu.
The voltage at fault location E f is: In analogy with Eqs. (8)-(10), a three similar equations can be written: where, where; Z 0eq and Z 2eq : total (equivalent) impedances of the zero-and negative-sequence networks without the mutual effect, respectively.
Adding Eqs. (19), (22) and (23), and using Eqs. (17) and (18), one can write: Eq. (24) can be written in another form: where, 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 Z R which measured by the distance relay is:

Determination of Fault Location of TPTLS as Affected by Mutual Coupling 2.2.1 SLG 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.
Their components I tm1 = I fm1 = I Bm1 , I tm2 = I fm2 = I and I tm0 = I fm0 = I Bm0 : From Eqs. (28) and (29) the bus sequence currents are equal, i.e., The voltages at fault location E fm and at the bus location E Bm are: With reference to Fig. 5, one can write: where, Z 1m , Z 2m , and Z 0m : The positive-, negative-and zero-sequence mutual impedances between the parallel TL. Adding Eqs. (33)- (35), and using Eqs. (29)-(32) to obtain: where, Z m = Z 1m + Z 2m + Z 0m . The solution of the quadratic Eq. (36) determines the calculated percentage (fraction) n m(cal) as: Therefore, the distance relay can measure the impedance Z Rm which is expressed as: where, I am1 : the current of n(Z 1 − Z 1m ) 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 Z Rm which indicates if the fault is within the relay's reach area.
Their components I tm1 = I fm1 = I Bm1 , I tm2 = I fm2 = I Bm2 and I tm0 = I fm0 = I Bm0 . The voltage sequence-components E Bm at bus 1 is calculated as: The generator G voltage (emf) E g1 = 1∠0 pu. The voltage at fault location E fm is: In analogy with Eqs. (8)-(10), a three similar equations can be written with aid of Fig. 5. where, where; Z 0meq and Z 2meq : total (equivalent) impedances of the zero-and negative-sequence networks with taking into account the mutual effect, respectively.
Adding Eqs. (43), (46) and (47), and using Eqs. (41) and (42), one can write: Eq. (48) can be written in another form: where, Eq. (49) represents a fourth-order equation with complex coefficients: where, Figure 7: The schematic diagram for the implementation of the impedance relay Solving this equation, Eq. (50), using MATLAB software to determine the calculated percentage (fraction) n m(cal) , which expresses where the fault is located (n m(cal) is the real one of the four solutions). Therefore, the distance relay can measure the impedance Z Rm which is expressed as: where I am1 is the current that passes through n(Z 1 −Z 1m ) 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 Z Rm which indicates if the fault is within the relay's reach area. Fig. 8 shows a two parallel transmission lines system (TPTLS) TL 1 and TL 2 , which are protected by two distance relays R at bus-1. Where; I f : fault current; I B : line current from bus-1; E B1 , E B2 : bus voltages at bus-1 and bus-2 respectively; I t1 , I t2 : total currents fed from the generators G 1 at bus-1 and G 2 at bus-2.  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. The total current I t which supplied from generator G 1 at bus-1 are:

Determination of Fault Location Without Mutual Coupling Between TPTLS
The current sequence-components I B which supplied from generator G 1 (Appendix A) are: The voltage sequence-components E B at bus-1 are: where, the emf of generators G 1 and G 2 are (E g1 = E g2 = 1∠0) pu.
The fault voltage sequence-components E f are: where their sum is equal to zero, i.e., 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: So, the calculated percentage (fraction) n (cal) which is a define of fault location on the TL: And, the measured impedance that seen by the relay Z R is: Fig. 11 shows a schematic diagram of the distance relay that determine the fault location on the faulted TL based on the TL impedance Z R which indicates if the fault is within the relay's reach area.  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.  The total current I tm components which supplied from generator G 1 at bus-1 are:

Determination of Fault Location with Mutual Coupling of TPTLS
The sequence components of the current in the branch n(Z 1 −Z 1m ), n(Z 2 −Z 2m ), and n(Z 0 − Z 0m ) in Fig. 12, respectively (see Appendix A) are: The voltage-sequence components E B at bus-1 are: where the emf of generators G 1 and G 2 are (E g1 = E g2 = 1∠0) pu. The sequence components of the voltage at the fault E f are: where their sum is zero, i.e.,: 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: where, CF m : compensating factor which is expressed as: Eq. (72) representing the total (sum) impedances between fault location and the bus-1 which is re-expressed as: where,

Eq. (74) represents a cubic complex coefficients equation:
where, Solving this equation, Eq. (75), using MATLAB software to determine the calculated percentage (fraction) n m(cal) , which expresses where the fault is located, (n m(cal) is the real one of the four solutions). Therefore, the distance relay can measure the impedance Z Rm which is expressed as: where, I am1 is the current of n(Z 1 −Z 1m ) branch. The schematic diagram for the implementation of the impedance relay is as shown in Fig. 14 Figure 14: 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 n m(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 Z R and Z Rm 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.
From Tabs. 1, 3 and 5, the calculated values of percentage (fraction) n (cal) and n m(cal) that obtained from the proposed algorithms are exactly equal to the n given 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, Z Rm is less than Z R 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-f 1 of systems that feeds from one-end. Therefore, the line impedance that seen by the relay Z R is small. In Fig. 15B the faults accrue far away from bus-1, so the current passes through long section of TL, a-f 2 and the increase of mutual coupling is more noticeable in the relay Z Rm , over Z R . 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 Z 0Rm exceeds Z 0R , the impedance without mutual coupling as given in Tabs. 2, 4, and 6.     . 16A shows the relay that can see currents in the opposite direction of a-f 1 portion for faults close to bus-1 in systems fed from both-ends. Z Rm exceeds Z R 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-f 2 of the faulted line. So, Z Rm is less than Z R , 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 overor under-reach in Tabs. 3 and 4, Fig. 16C.  Fig. 17 shows that the zero-sequence ratio (Z 0Rm /Z 0R ) 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.

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.