A Fault Location Algorithm for Multi-Section Combined Transmission Lines Considering Unsynchronized Sampling

: This paper introduces a fault location technique for multi-section combined transmission lines, utilizing unsynchronized fault record data. The approach determines the intersection points of the voltage magnitudes computed using the fault current and voltage records of each terminal and the constant line parameters of each section. Finally, the real fault location is identified based on the calculated inverse angle of equivalent impedance. To evaluate the performance of the proposed algorithm, a three-section combined transmission line is constructed using the PSCAD/EMTDC software version 5.0.0, and various fault scenarios are simulated and tested employing the proposed algorithm. The results indicate that the proposed algorithm is applicable to various types of faults in a multi-section combined transmission line; the estimations are accurate.


Introduction
An overhead transmission line is typically used to send electricity from power stations to substations near consumers.However, the use of underground cables has become imperative and is mandated by the environmental constraints imposed on construction sites in urban areas.This has triggered the development of hybrid transmission systems that combine both overhead and underground lines.Precise fault location minimizes the time and expense associated with deploying crews to search for faults.The enhancement of customer service and ensuring that consumers experience minimal interruptions improve the power system performance and pinpoint weak or vulnerable points [1,2].Moreover, accurate fault location enhances the availability and reliability of power systems [3].As the constant line parameters of overhead and underground transmission lines differ significantly, the accuracies of homogenous transmission line fault location methods [4][5][6][7] must be re-examined.A fault location approach that considers the non-homogeneity of combined transmission lines is required.Furthermore, it is crucial to take into account the effects of the unsynchronized sampling resulting from varying record starting times in different fault recorders when implementing a multi-terminal fault location algorithm.

Literature Review
This section highlights promising fault location approaches for combined transmission lines.The existing algorithms are based on neural networks, traveling wave evaluations, and impedance characteristics.
The use of artificial neural networks for fault detection, classification, and localization is discussed in [8,9].The model of [8] utilizes the phasors of the three-phase voltage and current as inputs.In contrast, the approach of [9] requires only instantaneous current data.Although both methods are very accurate, they are applicable to only homogeneous transmission lines.The reports in [10][11][12] used artificial neural networks to estimate the Energies 2024, 17, 703 3 of 17

Our Motivation and Contributions
The inhomogeneity of combined underground/overhead transmission lines renders fault location techniques based on distributed parameters difficult regardless of whether single-or dual-ended applications are employed.Apart from the effects of fault resistance, single-ended methods do not handle variations in the positive-sequence impedance per unit length well; this parameter is critical when seeking to determine the fault location.Thus, the precision of traditional algorithms fluctuates considerably as the fault distance varies.Although several approaches seek to locate faults on multi-section combined transmission lines, most require high sampling rates or synchronized sampling, which are expensive and difficult to achieve in real-world applications.The main objective of this paper is the development of a fault location algorithm that addresses the abovementioned challenges.The process is divided into two stages.First, the intersection points between the voltage magnitudes, computed using the fault currents and voltages recorded at each terminal, are determined employing an incremental increase approach.Next, a real fault identification algorithm is applied to the identified intersection point.If this is not a real fault location, that point is rejected, and the process continues until a real fault is located.The proposed algorithm offers several advantages:

•
Fault locations on combined transmission lines can be accurately estimated even when the fault record sampling data are not synchronized, • The performance of the proposed algorithm remains accurate under various types of faults, fault resistances, fault inception angles, and sources of impedance variation,

•
Faults are accurately located even if they lie very close to interconnection points, • The proposed algorithm can be modified to accommodate different numbers of sections according to real-world applications.
The remainder of this study is organized as follows.In Section 2, the details of the proposed fault location algorithm are described.In Section 3, the estimated fault location for different fault conditions depending on different fault resistance, fault locations, fault types, and phase errors are presented.Section 4 concludes the study and mentions planned future work.

The Fault Location Algorithm
This section presents the principles and approaches utilized for fault location estimation on a combined transmission line with unsynchronized sampling consideration.To facilitate understanding in the subsequent section, a list of relevant acronyms and their definitions is provided in Table 1.

Voltages and Currents along the Homogenous Transmission Line
The single-line diagram in Figure 1 shows a homogenous transmission line of total distance L. V S , I S , V R , and I R are the three-phase voltage and current measurements at terminals S and R, respectively.V S1 , I S1 , V R1 , and I R1 are the positive-sequence voltage and current components derived from V S , I S , V R , and I R by using symmetrical component transformation.The positive-sequence voltage and current along the transmission line at distance x from terminal S then can be computed based on the S and R terminal measurement data, as written in Equations ( 1) and (2) [34].

Voltages and Currents along the Three-Section Combined Transmission Line
In contrast to the homogeneous transmission line, where the voltage and current along the line can be calculated using the measurement data from each terminal, for an inhomogeneous line, these values can be computed solely based on the voltage and current at the nearest interconnection point.The voltage and current at the interconnection

Voltages and Currents along the Three-Section Combined Transmission Line
In contrast to the homogeneous transmission line, where the voltage and current along the line can be calculated using the measurement data from each terminal, for an inhomogeneous line, these values can be computed solely based on the voltage and current at the nearest interconnection point.The voltage and current at the interconnection point are determined based on the voltage and current up to the terminal.
The single-line diagram in Figure 2 shows a three-section combined transmission line of total distance L. The transmission line is segmented into the three components P1, P2, and P3, where L P1 and L P2 are the lengths from terminal S to the interconnection points M and N. V S , I S , V R , and I R are the three-phase voltage and current measurements at terminals S and R, respectively.V S1 , I S1 , V R1 , and I R1 are the positive-sequence voltage and current components derived from V S , I S , V R , and I R .The positive-sequence voltage and current along the transmission line at a specific point distance x from terminal S can then be computed based on the S and R terminal fault record data using Equations ( 3) and (4).
where V XS1 , I XS1 , V XR1 , and I XR1 are the positive-sequence voltages and currents at a point along the line at distance x from terminal S. Similarly, V MS1 , I MS1 , V NS1 , I NR1 , V MR1 , I MR1 , V NR1 , and I NR1 are the positive-sequence voltages and currents at the intersection points M and N, derived with the aid of Equations ( 5)-( 8), respectively: (5) where Z C1_Pi = Z 1_Pi /Y 1_Pi and γ 1_Pi = Y 1_Pi × Z 1_Pi are the positive-sequence characteristic impedance and positive-sequence propagation constant of section Pi (where i = 1, 2, 3) of the transmission line.Z 1_Pi and Y_Pi are the positive-sequence impedance and admittance, respectively, of section Pi of the transmission line.
Energies 2024, 17, x FOR PEER REVIEW 6 of 18 where ZC1_Pi = Z 1_Pi / Y 1_Pi and γ 1_Pi = Y 1_Pi × Z 1_Pi are the positive-sequence characteristic impedance and positive-sequence propagation constant of section Pi (where i = 1, 2, 3) of the transmission line.Z 1_Pi and Y_Pi are the positive-sequence impedance and admittance, respectively, of section Pi of the transmission line.

The Voltage Magnitude Intersection Points
As explained in the previous section, the voltage and current at a given point along the transmission line can be computed using the voltages and currents measured at the terminals with constant line parameters.Figure 3 indicates the magnitude of the computed voltage at various points along a three-section combined transmission line; these are derived from the data obtained at both terminals.At the fault locations, the magnitude of the computed voltage (VXS1, VXR1) is identical.This can be written as: A point F on the transmission line that satisfies ( 9) is identified as a fault point.The

The Voltage Magnitude Intersection Points
As explained in the previous section, the voltage and current at a given point along the transmission line can be computed using the voltages and currents measured at the terminals with constant line parameters.Figure 3 indicates the magnitude of the computed voltage at various points along a three-section combined transmission line; these are Energies 2024, 17, 703 6 of 17 derived from the data obtained at both terminals.At the fault locations, the magnitude of the computed voltage (V XS1 , V XR1 ) is identical.This can be written as: A point F on the transmission line that satisfies ( 9) is identified as a fault point.The algorithm commences by setting the initial location, x, to zero and subsequently determines the differences [the d(x) values] between the computed voltage magnitudes based on the measurements from both terminals.The calculation proceeds as follows in Equation (10): As combined transmission lines are inhomogeneous, fault point localization becomes challenging when employing the Newton-Raphson or modified secant method.We use an incremental increase method.Figure 3 indicates that the different value d(x) decreases as x moves closer to the intersection point.Therefore, the algorithm increases x as follows: In order to generalize the increasing rate α to be applicable to different systems, the algorithm is conducted based on the per unit quantity.After evaluation with different values, the increasing rate in Appendix A was selected based on a balance between the convergence speed and accuracy.
This process continues until d(x) converges to the selected perturbation value δ, indicating that Equation ( 9) is proximity-satisfied.However, if the algorithm cannot find the intersection point (x exceeds 1 or the iteration number attains the maximum threshold), the result at the previous sample will be used instead.

Real Fault Localization
If a fault occurs on a transmission line, all fault-crossing computations render either Equation ( 1) or (2) inaccurate, the exception being when the computation point is precisely aligned with the fault location F. This error can introduce a problem that leads to the satisfaction of Equation ( 9) at a point F' other than the real fault location F, as illustrated in Figure 4.The consequence is the potential misidentification of the real fault location; a more accurate fault location identification algorithm is imperative.

Real Fault Localization
If a fault occurs on a transmission line, all fault-crossing computations render either Equation ( 1) or (2) inaccurate, the exception being when the computation point is precisely aligned with the fault location F. This error can introduce a problem that leads to the satisfaction of Equation ( 9) at a point F' other than the real fault location F, as illustrated in Figure 4.The consequence is the potential misidentification of the real fault location; a more accurate fault location identification algorithm is imperative.
To address this issue, this paper introduces a novel fault location identification algorithm based on the inverse angle of the equivalent impedance.This leverages insights into the inverse angle that enhance the precision and reliability of the fault localization.By integrating this algorithm into the iterative algorithm discussed in the previous section, spurious fault locations are systematically rejected, and the value of x is incrementally adjusted until the true fault location F is identified.
If a fault occurs on a transmission line, all fault-crossing computations render either Equation ( 1) or (2) inaccurate, the exception being when the computation point is precisely aligned with the fault location F. This error can introduce a problem that leads to the satisfaction of Equation ( 9) at a point F' other than the real fault location F, as illustrated in Figure 4.The consequence is the potential misidentification of the real fault location; a more accurate fault location identification algorithm is imperative.To address this issue, this paper introduces a novel fault location identification algorithm based on the inverse angle of the equivalent impedance.This leverages insights into the inverse angle that enhance the precision and reliability of the fault localization.By integrating this algorithm into the iterative algorithm discussed in the previous section, spurious fault locations are systematically rejected, and the value of x is incrementally adjusted until the true fault location F is identified.
Figure 5a shows the interconnections between the sequence networks for singlephase-to-ground faults on the transmission line.ZS0, ZR0, ZS1, ZR1, ZS2, and ZR2 are the zeropositive-and zero-negative-sequence sending-end and receiving-end source impedances, respectively.Similarly, ZLS0, ZLR0, ZLS1, ZLR1, ZLS2, and ZLR2 are the zero-positive-and zeronegative-sequence transmission line impedances on the sending-end and receiving-end sides.VS and VR are the sending-end and receiving-end voltages and RF the fault resistance.Figure 5b-d show the interconnections of the sequence networks for phase-tophase, phase-to-phase-to-ground, and three-phase faults on the transmission line, respectively.From each network in Figure 5, the simplified network in Figure 6 can be derived.Figure 5a shows the interconnections between the sequence networks for single-phaseto-ground faults on the transmission line.Z S0 , Z R0 , Z S1 , Z R1 , Z S2 , and Z R2 are the zeropositive-and zero-negative-sequence sending-end and receiving-end source impedances, respectively.Similarly, Z LS0 , Z LR0 , Z LS1 , Z LR1 , Z LS2 , and Z LR2 are the zero-positive-and zeronegative-sequence transmission line impedances on the sending-end and receiving-end sides.V S and V R are the sending-end and receiving-end voltages and R F the fault resistance.Figure 5b-d show the interconnections of the sequence networks for phase-to-phase, phaseto-phase-to-ground, and three-phase faults on the transmission line, respectively.From each network in Figure 5, the simplified network in Figure 6 can be derived.Where Zeq, the equivalent impedance, varies by the specific type of fault.Despite the distinctive characteristics of the fault types, all Zeq values are inductive in nature.Using the equivalent network of Figure 6, the equivalent impedance Zeq can be calculated as:  Where Zeq, the equivalent impedance, varies by the specific type of fault.Despite the distinctive characteristics of the fault types, all Zeq values are inductive in nature.Using the equivalent network of Figure 6, the equivalent impedance Zeq can be calculated as: Where Z eq , the equivalent impedance, varies by the specific type of fault.Despite the distinctive characteristics of the fault types, all Z eq values are inductive in nature.Using the equivalent network of Figure 6, the equivalent impedance Z eq can be calculated as: where V F and I F are the positive-sequence voltage and current at the fault point.The phasor quantities can be computed using Equations ( 13) and ( 14), respectively: Finally, the inverse angle of the equivalent impedance φ IEZ can be derived as: where V ′ XS1 (x) is the complex conjugate of V XS1 (x) and ang() is the angle computation operator for the phasor quantity.
The inherently inductive characteristics of the equivalent impedance specifically position the inverse phase angle φ IEZ on the complex plane.When the distance x yielded by the iterative method of the above section is an actual fault location, the inverse phase angle φ IEZ falls within the fourth quadrant.Conversely, if x is not the location of a real fault, φ IEZ lies outside the fourth quadrant.This distinctive behavior distinguishes real from misidentified faults.By analyzing the locations of the inverse phase angles on the complex plane, the misidentified fault locations reported by the voltage magnitude intersection point algorithm are rejected until the actual fault is identified.
The flowchart of our new fault location algorithm for combined transmission lines wherein sampling is unsynchronized is shown in Figure 7.
Energies 2024, 17, x FOR PEER REVIEW 9 of 18 the iterative method of the above section is an actual fault location, the inverse phase angle φ IEZ falls within the fourth quadrant.Conversely, if x is not the location of a real fault, φ IEZ lies outside the fourth quadrant.This distinctive behavior distinguishes real from misidentified faults.By analyzing the locations of the inverse phase angles on the complex plane, the misidentified fault locations reported by the voltage magnitude intersection point algorithm are rejected until the actual fault is identified.The flowchart of our new fault location algorithm for combined transmission lines wherein sampling is unsynchronized is shown in Figure 7.

The Test Environment
To evaluate the performance of the presented technique, the 154 kV double-circuit, three-section combined transmission line shown in Figure 8 was simulated with the aid

The Test Environment
To evaluate the performance of the presented technique, the 154 kV double-circuit, three-section combined transmission line shown in Figure 8  The double circuit was considered to be two independent single circuits; the new fault location technique could thus be applied to each line individually.The algorithm was implemented employing the user-defined model of PSCAD/EMTDC.The sampling frequency was 4800 Hz or 80 samples/cycle; the fundamental frequency was 60 Hz.The method of [35] was used to remove the decaying DC offset and to derive the fundamental phasors of each measured signal.The performance of the algorithm was assessed based on the error percentage:

Influence of the Phasor Synchronization Error and Fault Types
Several simulations were performed using the new algorithm.We varied the fault types and resistance and the phasor synchronization errors.For example, Figure 9 shows the three-phase voltages and currents recorded at the S and R terminals assuming a singlephase-to-ground fault occurs at a 29.5 km distance from the S bus in the cable section, thus very close to the connection to the overhead transmission line (and the fault resistance is therefore 0 Ω).The phase error between the data recorded at both terminals is 90°.The double circuit was considered to be two independent single circuits; the new fault location technique could thus be applied to each line individually.The algorithm was implemented employing the user-defined model of PSCAD/EMTDC.The sampling frequency was 4800 Hz or 80 samples/cycle; the fundamental frequency was 60 Hz.The method of [35] was used to remove the decaying DC offset and to derive the fundamental phasors of each measured signal.The performance of the algorithm was assessed based on the error percentage:

Influence of the Phasor Synchronization Error and Fault Types
Several simulations were performed using the new algorithm.We varied the fault types and resistance and the phasor synchronization errors.For example, Figure 9 shows the three-phase voltages and currents recorded at the S and R terminals assuming a singlephase-to-ground fault occurs at a 29.5 km distance from the S bus in the cable section, thus very close to the connection to the overhead transmission line (and the fault resistance is therefore 0 Ω).The phase error between the data recorded at both terminals is 90 • .
By applying the presented approach with constrained parameters in Appendix A to the fault-recorded data in Figure 9, the estimated fault location result in Figure 10 is provided.It is evident in Figure 10 that the proposed algorithm is applicable for estimating the fault location on a multi-section combined transmission line with an acceptable convergence time, even though the recorded data from both terminals are not synchronized.The estimation error falls to below 1% within the first two cycles of the fault.After three cycles, the estimated error is 0.015%, corresponding to a common protective relay tripping time.
The full test results using single-phase-to-ground faults with different fault resistances and phasor synchronization angle errors are shown in Table 2.The fault resistance ranged from 0 to 50 Ω and the phasor synchronization error from 0 to 180 • .Three different faults at distances of 10, 50, and 80 km from terminal S were created on the combined transmission line.The estimated fault locations were captured three cycles after the fault initialization.The estimation errors listed in Table 2 show that the errors for the same fault resistances were very similar, although the phasor synchronization angle errors and Energies 2024, 17, 703 10 of 17 fault locations differed.This indicates that the algorithm is accurate; it is not affected by phasor synchronization errors.The smallest estimation error was less than 0.002% for a single-phase-to-ground fault with a fault resistance of 0 Ω 10 km from the S terminal.Although the estimation error increases as the fault resistance rises, the maximum error was only 0.623% for a single-phase-to-ground fault with a resistance of 50 Ω 10 km from the S terminal.Other fault locations exhibited the same lowest errors (0.004%) for 0 Ω fault resistances.The highest errors were 0.381 and 0.599% for fault resistances of 50 Ω 50 and 80 km from the S terminal, respectively.The fault resistance more significantly affects faults in underground cables than faults in overhead transmission lines, reflecting the effects of cross-bonding on fault location algorithms.
frequency was 4800 Hz or 80 samples/cycle; the fundamental frequency was 60 Hz.The method of [35] was used to remove the decaying DC offset and to derive the fundamental phasors of each measured signal.The performance of the algorithm was assessed based on the error percentage:

Influence of the Phasor Synchronization Error and Fault Types
Several simulations were performed using the new algorithm.We varied the fault types and resistance and the phasor synchronization errors.For example, Figure 9 shows the three-phase voltages and currents recorded at the S and R terminals assuming a singlephase-to-ground fault occurs at a 29.5 km distance from the S bus in the cable section, thus very close to the connection to the overhead transmission line (and the fault resistance is therefore 0 Ω).The phase error between the data recorded at both terminals is 90°.By applying the presented approach with constrained parameters in Appendix A to the fault-recorded data in Figure 9, the estimated fault location result in Figure 10 is provided.It is evident in Figure 10 that the proposed algorithm is applicable for estimating the fault location on a multi-section combined transmission line with an acceptable convergence time, even though the recorded data from both terminals are not synchronized.The estimation error falls to below 1% within the first two cycles of the fault.After three cycles, the estimated error is 0.015%, corresponding to a common protective relay tripping time.The full test results using single-phase-to-ground faults with different fault resistances and phasor synchronization angle errors are shown in Table 2.The fault resistance ranged from 0 to 50 Ω and the phasor synchronization error from 0 to 180°.Three different faults at distances of 10, 50, and 80 km from terminal S were created on the combined transmission line.The estimated fault locations were captured three cycles after the fault initialization.The estimation errors listed in Table 2 show that the errors for the same fault resistances were very similar, although the phasor synchronization angle errors and fault locations differed.This indicates that the algorithm is accurate; it is not affected by  Testing was next expanded to include phase-to-phase, phase-to-phase-to-ground, and three-phase faults; we comprehensively evaluated how the new algorithm performed under various fault scenarios.The conditions used during the single-phase-to-ground fault tests were maintained (to ensure consistency).
Table 3 shows the performance of the algorithm when phase-to-phase faults develop.The estimation errors at different phase synchronization angle errors but at the same fault resistance are similar; however, the difference increases at higher fault resistances.The error variations at the various fault locations are similar.Specifically, the minimum estimation errors for faults at distances of 10, 50, and 80 km from the S terminal were 0.005, 0.018, and 0.003% and the maximum errors 0.495, 0.458, and 0.757%, respectively.Table 4 shows that the new algorithm performs well when phase-to-phase-to-ground faults develop under the described conditions.The algorithm accurately locates the faults; the minimum and maximum errors are 0.001 and 0.477%.The response of the proposed algorithm to three-phase faults is shown in Table 5.For a fault resistance of 0 Ω, a noticeable error (0.542%) occurs at the 50 km point of the overhead line when the input phase error is 180.As the fault resistance increases, the error becomes insignificant compared to the previous cases.The proposed algorithm was also verified under various fault inception angles.In this case, single-phase-to-ground faults with different fault resistances and locations were simulated with fault inception angles varying from 0 to 180 • .The testing results in Table 6 indicate that the proposed algorithm remains effective and unaffected by variation in the fault inception angles.Moreover, the algorithm reliability was assessed by varying the source impedance.Table 7 demonstrates the test results for single-phase-to-ground faults at different locations along the transmission line with 0 Ω fault resistance.It can be seen that the proposed algorithm can estimate the fault location accurately with different source impedances.[31] A comparison between the proposed algorithm and the algorithm in [31] is shown in Table 8, which shows the estimation results and percentage error for different types of faults, a 10 Ω fault resistance, a 0 • synchronous angle (synchronous measurements), and different fault locations.As seen from Table 8, the proposed algorithm achieves a similar accuracy compared to the algorithm in [31].While the proposed algorithm may have a slower calculation speed, it is capable of handling unsynchronized sampling, a feature the algorithm presented in [31] lacks.

1 ⁄
Y 1 represents the positive-sequence characteristic impedance and γ 1 = √ Y 1 × Z 1 is the positive-sequence propagation constant.Z 1 and Y 1 are the positive-sequence series impedance and shunt admittance of the transmission line.Energies 2024, 17, x FOR PEER REVIEW 5 of 18 VXR1 IXR1 = ½ e γ (L x) + e γ (L x) −ZC1(e γ ( x) − e γ (L x) ) −1/ZC1(e γ ( x) − e γ (L x) ) e γ x + e γ (L x) represents the positive-sequence characteristic impedance and γ = √ Y 1 × Z 1 is the positive-sequence propagation constant.Z1 and Y1 are the positive-sequence series impedance and shunt admittance of the transmission line.

Figure 1 .
Figure 1.A single-line diagram of a homogenous transmission line.

Figure 1 .
Figure 1.A single-line diagram of a homogenous transmission line.

Figure 2 .
Figure 2. A single-line diagram of a three-section combined transmission line.

Figure 2 .
Figure 2. A single-line diagram of a three-section combined transmission line.

Figure 3 .
Figure 3. Computed voltage magnitudes along the transmission line based on the S and R terminal fault record data.

Figure 3 .
Figure 3. Computed voltage magnitudes along the transmission line based on the S and R terminal fault record data.

Figure 4 .
Figure 4. Computed voltage magnitudes along a transmission line based on S and R terminal fault record data when there are two intersection points.

Figure 4 .
Figure 4. Computed voltage magnitudes along a transmission line based on S and R terminal fault record data when there are two intersection points.

Figure 5 .
Figure 5. Interconnections among the sequence networks for (a) a single-phase-to-ground fault; (b) a phase-to-phase fault; (c) a phase-to-phase-to-ground fault; and (d) a three-phase fault.

Figure 6 .
Figure 6.Equivalent interconnection network for different types of faults.

Figure 5 .Figure 5 .
Figure 5. Interconnections among the sequence networks for (a) a single-phase-to-ground fault; (b) a phase-to-phase fault; (c) a phase-to-phase-to-ground fault; and (d) a three-phase fault.

Figure 6 .
Figure 6.Equivalent interconnection network for different types of faults.

Figure 6 .
Figure 6.Equivalent interconnection network for different types of faults.

Figure 7 .
Figure 7. Flowchart of the fault location algorithm.

Figure 7 .
Figure 7. Flowchart of the fault location algorithm.

Figure 9 . 18 Figure 9 .
Figure 9. Single-phase-to-ground fault records of (a) the voltage at the S terminal; (b) the current at the S terminal; (c) the voltage at the R terminal; (d) the current at the R terminal.

Figure 10 .
Figure 10.Localization of a single-phase-to-ground fault at 29.5 km distance from the S terminal.

Figure 10 .
Figure 10.Localization of a single-phase-to-ground fault at 29.5 km distance from the S terminal.

Table 1 .
The relevant acronyms and their definitions.

Table 2 .
Test data on single-phase-to-ground faults at different locations along the transmission line with variation in the fault resistance and phase errors.

Table 3 .
Test results for phase-to-phase faults at different locations along the transmission line with variation in the fault resistance and phase errors.

Table 4 .
Test results for phase-to-phase-to-ground faults at different locations along the transmission line with variation in the fault resistance and phase errors.

Table 5 .
Test results for three-phase faults at different locations along the transmission line with variation in the fault resistance and phase errors.

Table 6 .
Test results for single-phase-to-ground faults at different locations along the transmission line.The fault resistances and fault inception angles varied.

Table 7 .
Test results for single-phase-to-ground faults along the transmission line.The source impedance varied.Comparison of the Proposed Algorithm with the Algorithm in