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Fault Location Theory

Six single-ended impedance-based methods for estimating fault distance.

In Detego

See the Fault Location Guide for how to configure and use this tab in the app. This page covers the mathematical theory behind all six algorithms.

Overview

All single-ended impedance-based fault location methods start from the same circuit equation. Looking from the relay terminal into a faulted line of total positive-sequence impedance $Z_{1L}$ , the measured loop voltage is:

Faulted line voltage equation

V_S = m \cdot Z_{1L} \cdot I_S + R_F \cdot I_F

Where

$V_S$ Loop voltage phasor at the relay terminal
$m$ Per-unit fault distance (0 = relay, 1 = remote end)
$Z_{1L}$ Total positive-sequence line impedance (complex)
$I_S$ Loop current phasor at the relay terminal
$R_F$ Fault resistance (real, in ohms)
$I_F$ Total fault current through the fault resistance

The first term is the voltage drop across the line up to the fault point. The second term is the voltage drop across the fault resistance.

If the fault resistance $R_F$ were zero (a bolted fault), solving for $m$ would be trivial: $m = V_S / (Z_{1L} \cdot I_S)$ . In practice, fault resistance is never zero -- arc resistance, tower footing resistance, and vegetation contact resistance all contribute. The $R_F \cdot I_F$ term contaminates the measurement and causes errors in the distance estimate.

Each algorithm below uses a different strategy to eliminate or minimise the influence of fault resistance. They range from the simplest (Reactance) to the most sophisticated (Eriksson, Novosel), trading complexity for accuracy under increasingly difficult conditions.

Distance vs time — the reactance method produces a time-varying distance estimate. During the fault window, the stable portion is averaged to produce the final result.

Reactance Method

The reactance method is the simplest fault location technique. It exploits the fact that fault resistance is purely real: it contributes only to the resistive component of the measured impedance and has zero imaginary part. By extracting only the imaginary part (reactance) of the loop impedance ratio, the fault resistance term is eliminated -- provided the fault current $I_F$ is in phase with the relay current $I_S$ .

Reactance method (IEEE C37.114)

m = \frac{\text{Im}(V_S / I_S)}{\text{Im}(Z_{1L})}

Where

$m$ Per-unit fault distance
$V_S$ Loop voltage phasor
$I_S$ Loop current phasor (with k_0 compensation for ground loops)
$Z_{1L}$ Total positive-sequence line impedance
$\text{Im}(\cdot)$ Imaginary part operator

The numerator is the measured loop reactance. The denominator is the total line reactance. Their ratio gives the per-unit distance.

When It Works Well

The reactance method is accurate when the fault current

I_F

and the relay current

I_S

are approximately in phase (homogeneous system, no significant load flow). It is always available as a fallback because it requires no pre-fault data or source impedances.

Limitation: When $I_S$ and $I_F$ are not in phase -- due to remote infeed, load flow, or non-homogeneous source impedances -- the $R_F \cdot I_F / I_S$ term has a non-zero imaginary part that leaks into the reactance measurement, causing distance error proportional to fault resistance.

Takagi Method

The Takagi method improves on the reactance method by using superimposed current (also called incremental current) to cancel the fault resistance term. The superimposed current is the difference between the fault-period current and the pre-fault current. In a homogeneous system, the superimposed current is in phase with the total fault current $I_F$ , which makes it an effective proxy for eliminating the resistance contribution.

Takagi method

m = \frac{\text{Im}(V_S \cdot \Delta I^*)}{\text{Im}(Z_{1L} \cdot I_S \cdot \Delta I^*)}

Where

$V_S$ Loop voltage phasor
$I_S$ Loop current phasor (with k_0 compensation for ground loops)
$\Delta I$ Superimposed current = I_{S,\text{fault}} - I_{S,\text{pre}}
$\Delta I^*$ Complex conjugate of the superimposed current
$Z_{1L}$ Total positive-sequence line impedance

Why It Eliminates Fault Resistance

Substituting the voltage equation into the numerator gives two terms: the line impedance term and the fault resistance term. The fault resistance term becomes $\text{Im}(R_F \cdot I_F \cdot \Delta I^*)$ . Since $R_F$ is real and $I_F$ is approximately in phase with $\Delta I$ (in a homogeneous system), the product $I_F \cdot \Delta I^*$ is approximately real, and its imaginary part is approximately zero. The fault resistance contribution vanishes.

Requirements

The Takagi method requires pre-fault data to compute the superimposed current. The pre-fault phasors are extracted from a DFT window before fault inception. If no fault inception time is available (either auto-detected or user-specified), the method cannot be used. It works for all fault types (ground, phase-phase, and three-phase).

Modified Takagi Method

The Modified Takagi method replaces the superimposed current with zero-sequence current as the reference for cancelling fault resistance. The zero-sequence current is approximately in phase with the fault current for ground faults, serving the same purpose as the superimposed current in the standard Takagi method -- but without requiring any pre-fault data.

Modified Takagi method

m = \frac{\text{Im}(V_S \cdot I_0^*)}{\text{Im}(Z_{1L} \cdot I_S \cdot I_0^*)}

Where

$V_S$ Loop voltage phasor
$I_S$ Loop current phasor (with k_0 compensation for ground loops)
$I_0$ Zero-sequence current = (I_a + I_b + I_c) / 3
$I_0^*$ Complex conjugate of the zero-sequence current
$Z_{1L}$ Total positive-sequence line impedance

Using the residual current 3I_0 = I_a + I_b + I_c gives the same result -- the factor of 3 appears in both numerator and denominator and cancels.

Applicability: Ground faults only (SLG and DLG). For phase-to-phase faults without ground involvement, zero-sequence current is zero and the method is undefined. No pre-fault data is required, making it useful when fault inception timing is unavailable.

Angle Correction

The full Modified Takagi formulation includes an optional angle correction

e^{-jT}

to compensate for non-homogeneous systems, where

T

is the angle between the zero-sequence current and the fault current. When

T = 0

(the default), the correction term equals 1 and has no effect.

Eriksson Method

The Eriksson method uses knowledge of the source impedances at both terminals to formulate a quadratic equation in $m$ . By incorporating the system topology, it can correctly handle non-homogeneous systems where the Takagi method loses accuracy.

Eriksson quadratic

m = \frac{B \pm \sqrt{B^2 - 4C}}{2}

Where

$B$ = a - e \cdot b / f
$C$ = c - e \cdot d / f

Coefficient Definitions

The six coefficients are computed from the loop phasors and system impedances. Define the auxiliary complex quantities:

Eriksson coefficients

\begin{aligned} a &= \text{Re}\!\left(1 + \frac{Z_R}{Z_{1L}} + \frac{V_S}{Z_{1L} \cdot I_S}\right) \\[6pt] b &= \text{Im}\!\left(1 + \frac{Z_R}{Z_{1L}} + \frac{V_S}{Z_{1L} \cdot I_S}\right) \\[6pt] c &= \text{Re}\!\left(\frac{V_S}{Z_{1L} \cdot I_S} \cdot \left(1 + \frac{Z_R}{Z_{1L}}\right)\right) \\[6pt] d &= \text{Im}\!\left(\frac{V_S}{Z_{1L} \cdot I_S} \cdot \left(1 + \frac{Z_R}{Z_{1L}}\right)\right) \\[6pt] e &= \text{Re}\!\left(\frac{\Delta I}{Z_{1L} \cdot I_S} \cdot \left(1 + \frac{Z_R + Z_S}{Z_{1L}}\right)\right) \\[6pt] f &= \text{Im}\!\left(\frac{\Delta I}{Z_{1L} \cdot I_S} \cdot \left(1 + \frac{Z_R + Z_S}{Z_{1L}}\right)\right) \end{aligned}

Where

$Z_S$ Local source positive-sequence impedance
$Z_R$ Remote source positive-sequence impedance
$\Delta I$ Superimposed current (same as Takagi)
$V_S, I_S$ Loop voltage and current phasors

Root Selection

The quadratic yields two roots. The physically meaningful root is selected by checking which value falls in the range $0 \leq m \leq 1$ . If both roots are valid, the root closest to the Takagi estimate is preferred as a consistency check. If neither root is in range, the result is flagged as unreliable.

Requirements: Pre-fault data (for $\Delta I$ ) and source impedances $Z_S$ and $Z_R$ at both terminals. These are typically estimated from fault studies or relay settings.

Novosel Method

The Novosel method operates in the positive-sequence domain, which eliminates dependence on fault type and zero-sequence compensation. It uses an iterative approach with a distribution factor to estimate the total fault current, then explicitly solves for both fault distance and fault resistance.

Positive-sequence voltage equation

V_1 = m \cdot Z_{1L} \cdot I_1 + R_F \cdot I_{1,F}

Where

$V_1$ Positive-sequence voltage at the relay (from Fortescue transformation)
$I_1$ Positive-sequence current at the relay
$I_{1,F}$ Positive-sequence fault current (estimated via distribution factor)
$R_F$ Fault resistance

Distribution Factor

The distribution factor estimates what fraction of the total fault current flows from the local terminal, based on the network impedances:

Positive-sequence distribution factor

C_1(m) = \frac{Z_R + (1 - m) \cdot Z_{1L}}{Z_S + Z_{1L} + Z_R}

Where

$C_1(m)$ Distribution factor (depends on fault location m)
$Z_S$ Local source positive-sequence impedance
$Z_R$ Remote source positive-sequence impedance

The fault current is then estimated as $I_{1,F} = I_1 / C_1(m)$ . Since $C_1$ depends on $m$ , and $m$ depends on $I_{1,F}$ , the solution is iterative.

Iterative Solution

The algorithm proceeds as follows:

Initialise $m_0$ from the reactance method: $m_0 = \text{Im}(V_1/I_1) / \text{Im}(Z_{1L})$
Compute $C_1(m_0)$ and $I_{1,F} = I_1 / C_1(m_0)$
Solve for $m$ by separating real and imaginary parts of the voltage equation and eliminating $R_F$ :

Closed-form m update

m = \frac{\text{Re}(V_1) \cdot \text{Im}(I_{1,F}) - \text{Im}(V_1) \cdot \text{Re}(I_{1,F})}{\text{Re}(Z_{1L} \cdot I_1) \cdot \text{Im}(I_{1,F}) - \text{Im}(Z_{1L} \cdot I_1) \cdot \text{Re}(I_{1,F})}

If $|m_{\text{new}} - m_{\text{old}}| < 0.001$ , the solution has converged. Otherwise, update $m_0 = m_{\text{new}}$ and repeat from step 2.
Once converged, compute fault resistance from the imaginary part: $R_F = (\text{Im}(V_1) - m \cdot \text{Im}(Z_{1L} \cdot I_1)) \;/\; \text{Im}(I_{1,F})$

Convergence is typically achieved within 3--5 iterations for well-conditioned systems. The method explicitly produces both $m$ and $R_F$ , which is useful for validating the result.

Requirements: Source impedances $Z_S$ and $Z_R$ . Works for all fault types because it uses positive-sequence quantities exclusively.

Radial (Apparent Impedance) Method

The radial method is the simplest possible fault location technique. It computes the fault distance as the ratio of the measured loop impedance magnitude to the total line impedance magnitude — the apparent impedance approach.

Radial method

m = \frac{|Z_{\text{loop}}|}{|Z_{1L}|}

Where

$m$ Per-unit fault distance
$|Z_{\text{loop}}|$ Magnitude of the measured loop impedance
$|Z_{1L}|$ Magnitude of the total positive-sequence line impedance

Unlike the reactance method (which uses only the imaginary part), the radial method uses the full impedance magnitude. This means fault resistance is not separated — any fault resistance adds directly to the measured impedance and inflates the distance estimate. The method reports a fault resistance of zero because it cannot distinguish line impedance from fault resistance.

When to Use

The radial method is appropriate for simple radial feeders where the fault resistance is expected to be low (bolted or near-bolted faults). It provides a quick sanity check and is always available without pre-fault data or source impedances. On networked systems or with significant fault resistance, prefer the Reactance or Takagi methods.

Measurement Loops

All algorithms (except Novosel, which uses positive-sequence quantities) require a measurement loop that selects the appropriate voltage and current phasors based on fault type. The loop selection follows IEEE C37.114.

Fault Type	Loop Voltage $V_S$	Loop Current $I_S$
SLG-A	$V_a$	$I_a + k_0 \cdot (I_a + I_b + I_c)$
SLG-B	$V_b$	$I_b + k_0 \cdot (I_a + I_b + I_c)$
SLG-C	$V_c$	$I_c + k_0 \cdot (I_a + I_b + I_c)$
LL-AB	$V_a - V_b$	$I_a - I_b$
LL-BC	$V_b - V_c$	$I_b - I_c$
LL-CA	$V_c - V_a$	$I_c - I_a$
DLG-AB, DLG-BC, DLG-CA	Dominant faulted phase (e.g. $V_a$ )	$I_{ph} + k_0 \cdot (I_a + I_b + I_c)$
3-Phase	Any phase-phase (e.g. $V_a - V_b$ )	Corresponding current (e.g. $I_a - I_b$ )

Phase-phase loops inherently cancel the zero-sequence component and do not use $k_0$ compensation. For line-to-line faults without ground involvement, these loops are the natural choice.

For double line-to-ground (DLG) faults, the phase-to-ground loop of the dominant faulted phase (the phase carrying the highest fault current) with $k_0$ compensation provides the most accurate fault location. The phase-to-phase loop between the two faulted phases does not account for the zero-sequence ground return path, leading to significant overestimation. This approach is consistent with how numerical distance relays handle DLG faults internally (e.g., Schneider P43x configures PG loops for 2-phase-to-ground measurement).

Zero-sequence compensation factor

k_0 = \frac{Z_0 - Z_1}{3 \cdot Z_1}

Where

$Z_0$ Zero-sequence line impedance (complex)
$Z_1$ Positive-sequence line impedance (complex)

k_0 is a complex number. Typical values for overhead lines: magnitude 0.5--0.7. For cables: magnitude 0.3--0.5.

k₀ Compensation Formats

Different relay manufacturers use different input conventions for zero-sequence compensation. All are mathematically equivalent and convertible to the standard $k_0$ used internally.

Standard (Z₁ / Z₀ Input)

The user provides $Z_1$ and $Z_0$ as magnitude and angle. The compensation factor is computed as:

Standard k_0 from impedances

k_0 = \frac{Z_0 - Z_1}{3 \cdot Z_1}

Where

$Z_0$ Zero-sequence line impedance (mag + angle)
$Z_1$ Positive-sequence line impedance (mag + angle)

Used by SEL (auto mode), ABB (KN computation). This is the default input mode.

Direct (k₀ Magnitude + Angle)

The user enters $k_0$ directly as magnitude and angle. No conversion is needed -- the value is used as-is. This matches SEL manual mode (k0M1), ABB KN, Siemens K0, and Schneider kG (Method 1).

Siemens RE/RL + XE/XL

Siemens relays (e.g. 7SA612) express the compensation as two separate real/reactive ratios. The conversion to the standard complex $k_0$ is:

Siemens ratio conversion

k_0 = \frac{RE + j \cdot XE}{R_1 + j \cdot X_1}

Where

$RE$ = (RE/RL) \cdot R_1, where R_1 = |Z_1| \cos(\angle Z_1)
$XE$ = (XE/XL) \cdot X_1, where X_1 = |Z_1| \sin(\angle Z_1)
$RE/RL$ Ratio of earth resistance to line resistance (Siemens setting)
$XE/XL$ Ratio of earth reactance to line reactance (Siemens setting)

RE/RL and XE/XL are dimensionless ratios. Typical values: RE/RL = 1.0--3.0, XE/XL = 2.0--4.0.

Phase-Only (Schneider Method)

Some Schneider relays (e.g. P437) use a phase-only ground loop computation where the compensated current is computed differently:

Phase-only ground loop

I_{\text{comp}} = I_{\text{phase}} \cdot (1 + k_G)

Where

$I_{\text{comp}}$ Compensated loop current
$I_{\text{phase}}$ Faulted phase current only
$k_G$ Ground compensation factor (magnitude + angle)

In the phase-only method, k_G has a DIFFERENT numerical value than k_0 for the same line, because the computation formula is structurally different.

Compare with the standard method where the residual current $I_a + I_b + I_c$ is used: $I_{\text{comp}} = I_{\text{phase}} + k_0 \cdot (I_a + I_b + I_c)$ . The phase-only method only uses the faulted phase current, which avoids reliance on the other two phases but requires a different compensation value.

Algorithm Comparison

The six algorithms represent a spectrum of complexity and accuracy. The following diagram and table summarise their relative strengths.

Figure — Algorithm Performance Comparison

With fault resistance present, the simple reactance method overestimates the distance because the resistive component of V/I contaminates the measurement. Takagi and Modified Takagi use superimposed and zero-sequence currents respectively to cancel the fault resistance effect.

Method	R_F Handling	Data Required	Best For
Reactance	Assumes in-phase I_S/I_F	None (always available)	Bolted faults, quick estimate, fallback
Takagi	Eliminates via superimposed current	Pre-fault phasors	General-purpose, best single-source method
Modified Takagi	Eliminates via zero-sequence current	Ground fault (I₀ > 0)	Ground faults without pre-fault data
Eriksson	Solves quadratic using source Z	Pre-fault + source impedances	Non-homogeneous systems, strong infeed
Novosel	Iterative with distribution factor	Source impedances	All fault types, explicit R_F estimate
Radial	Includes R_F in distance (no separation)	None (always available)	Bolted faults on radial feeders, quick sanity check

Cross-Validation

When multiple algorithms are available, their results can be cross-validated. If all methods agree within a few percent, confidence in the result is high. When they disagree significantly, it suggests a problem: incorrect measurement loop, wrong line parameters, CT saturation, or an evolving fault. Investigate the disagreement before trusting any single result.

Statistical Summary

Each algorithm produces a time-varying distance estimate across the fault window. The per-sample estimates are summarised using robust statistical measures. The median distance is the headline result because it is resistant to outliers — a few extreme samples (from DFT transients or CT saturation) do not shift the result. The spread is quantified using the Median Absolute Deviation (MAD).

Median fault distance

\tilde{m} = \text{median}(m[1], m[2], \ldots, m[N])

Where

$\tilde{m}$ Median per-unit distance over the fault window
$N$ Number of samples in the fault window
$m[i]$ Distance estimate at sample i

Median Absolute Deviation (MAD)

\text{MAD} = \text{median}(|m[i] - \tilde{m}|)

Where

$\text{MAD}$ Median of the absolute deviations from the median

MAD is converted to a standard-deviation-equivalent (σ) by multiplying by 1.4826, which assumes an underlying normal distribution.

Robust sigma

\sigma_{\text{MAD}} = 1.4826 \times \text{MAD}

Where

$\sigma_{\text{MAD}}$ MAD-based sigma (comparable to standard deviation for normal data)
$1.4826$ Consistency factor for normal distribution

The confidence level is determined by comparing the MAD to the line length. This produces a percentage that scales naturally with the protected line — a 0.5 km spread on a 10 km line is significant, but the same spread on a 200 km line is excellent.

MAD / Line Length	Confidence	Interpretation
< 5%	High	Consistent measurements with low scatter. Distance estimate is reliable.
5% – 15%	Medium	Moderate scatter. Distance estimate is usable but should be interpreted with caution.
> 15%	Low	High scatter. Distance estimate has significant uncertainty. Check for CT saturation, incorrect line parameters, or evolving faults.

Fallback When Line Length Is Not Set

When the line length is zero or not configured, the confidence calculation falls back to a coefficient of variation (CV) approach using the MAD-based sigma divided by the absolute median distance. The same 5%/15% thresholds apply.

PreviousFault Classification

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Fault Location Theory

Six single-ended impedance-based methods for estimating fault distance.

In Detego

See the Fault Location Guide for how to configure and use this tab in the app. This page covers the mathematical theory behind all six algorithms.

Overview

Faulted line voltage equation

V_S = m \cdot Z_{1L} \cdot I_S + R_F \cdot I_F

Where

$V_S$ Loop voltage phasor at the relay terminal
$m$ Per-unit fault distance (0 = relay, 1 = remote end)
$Z_{1L}$ Total positive-sequence line impedance (complex)
$I_S$ Loop current phasor at the relay terminal
$R_F$ Fault resistance (real, in ohms)
$I_F$ Total fault current through the fault resistance

The first term is the voltage drop across the line up to the fault point. The second term is the voltage drop across the fault resistance.

Distance vs time — the reactance method produces a time-varying distance estimate. During the fault window, the stable portion is averaged to produce the final result.

Reactance Method

Reactance method (IEEE C37.114)

m = \frac{\text{Im}(V_S / I_S)}{\text{Im}(Z_{1L})}

Where

$m$ Per-unit fault distance
$V_S$ Loop voltage phasor
$I_S$ Loop current phasor (with k_0 compensation for ground loops)
$Z_{1L}$ Total positive-sequence line impedance
$\text{Im}(\cdot)$ Imaginary part operator

The numerator is the measured loop reactance. The denominator is the total line reactance. Their ratio gives the per-unit distance.

When It Works Well

The reactance method is accurate when the fault current

I_F

and the relay current

I_S

are approximately in phase (homogeneous system, no significant load flow). It is always available as a fallback because it requires no pre-fault data or source impedances.

Takagi Method

Takagi method

m = \frac{\text{Im}(V_S \cdot \Delta I^*)}{\text{Im}(Z_{1L} \cdot I_S \cdot \Delta I^*)}

Where

$V_S$ Loop voltage phasor
$I_S$ Loop current phasor (with k_0 compensation for ground loops)
$\Delta I$ Superimposed current = I_{S,\text{fault}} - I_{S,\text{pre}}
$\Delta I^*$ Complex conjugate of the superimposed current
$Z_{1L}$ Total positive-sequence line impedance

Why It Eliminates Fault Resistance

Requirements

Modified Takagi Method

Modified Takagi method

m = \frac{\text{Im}(V_S \cdot I_0^*)}{\text{Im}(Z_{1L} \cdot I_S \cdot I_0^*)}

Where

$V_S$ Loop voltage phasor
$I_S$ Loop current phasor (with k_0 compensation for ground loops)
$I_0$ Zero-sequence current = (I_a + I_b + I_c) / 3
$I_0^*$ Complex conjugate of the zero-sequence current
$Z_{1L}$ Total positive-sequence line impedance

Using the residual current 3I_0 = I_a + I_b + I_c gives the same result -- the factor of 3 appears in both numerator and denominator and cancels.

Angle Correction

The full Modified Takagi formulation includes an optional angle correction

e^{-jT}

to compensate for non-homogeneous systems, where

T

is the angle between the zero-sequence current and the fault current. When

T = 0

(the default), the correction term equals 1 and has no effect.

Eriksson Method

Eriksson quadratic

m = \frac{B \pm \sqrt{B^2 - 4C}}{2}

Where

$B$ = a - e \cdot b / f
$C$ = c - e \cdot d / f

Coefficient Definitions

The six coefficients are computed from the loop phasors and system impedances. Define the auxiliary complex quantities:

Eriksson coefficients

\begin{aligned} a &= \text{Re}\!\left(1 + \frac{Z_R}{Z_{1L}} + \frac{V_S}{Z_{1L} \cdot I_S}\right) \\[6pt] b &= \text{Im}\!\left(1 + \frac{Z_R}{Z_{1L}} + \frac{V_S}{Z_{1L} \cdot I_S}\right) \\[6pt] c &= \text{Re}\!\left(\frac{V_S}{Z_{1L} \cdot I_S} \cdot \left(1 + \frac{Z_R}{Z_{1L}}\right)\right) \\[6pt] d &= \text{Im}\!\left(\frac{V_S}{Z_{1L} \cdot I_S} \cdot \left(1 + \frac{Z_R}{Z_{1L}}\right)\right) \\[6pt] e &= \text{Re}\!\left(\frac{\Delta I}{Z_{1L} \cdot I_S} \cdot \left(1 + \frac{Z_R + Z_S}{Z_{1L}}\right)\right) \\[6pt] f &= \text{Im}\!\left(\frac{\Delta I}{Z_{1L} \cdot I_S} \cdot \left(1 + \frac{Z_R + Z_S}{Z_{1L}}\right)\right) \end{aligned}

Where

$Z_S$ Local source positive-sequence impedance
$Z_R$ Remote source positive-sequence impedance
$\Delta I$ Superimposed current (same as Takagi)
$V_S, I_S$ Loop voltage and current phasors

Root Selection

Requirements: Pre-fault data (for $\Delta I$ ) and source impedances $Z_S$ and $Z_R$ at both terminals. These are typically estimated from fault studies or relay settings.

Novosel Method

Positive-sequence voltage equation

V_1 = m \cdot Z_{1L} \cdot I_1 + R_F \cdot I_{1,F}

Where

$V_1$ Positive-sequence voltage at the relay (from Fortescue transformation)
$I_1$ Positive-sequence current at the relay
$I_{1,F}$ Positive-sequence fault current (estimated via distribution factor)
$R_F$ Fault resistance

Distribution Factor

The distribution factor estimates what fraction of the total fault current flows from the local terminal, based on the network impedances:

Positive-sequence distribution factor

C_1(m) = \frac{Z_R + (1 - m) \cdot Z_{1L}}{Z_S + Z_{1L} + Z_R}

Where

$C_1(m)$ Distribution factor (depends on fault location m)
$Z_S$ Local source positive-sequence impedance
$Z_R$ Remote source positive-sequence impedance

The fault current is then estimated as $I_{1,F} = I_1 / C_1(m)$ . Since $C_1$ depends on $m$ , and $m$ depends on $I_{1,F}$ , the solution is iterative.

Iterative Solution

The algorithm proceeds as follows:

Initialise $m_0$ from the reactance method: $m_0 = \text{Im}(V_1/I_1) / \text{Im}(Z_{1L})$
Compute $C_1(m_0)$ and $I_{1,F} = I_1 / C_1(m_0)$
Solve for $m$ by separating real and imaginary parts of the voltage equation and eliminating $R_F$ :

Closed-form m update

m = \frac{\text{Re}(V_1) \cdot \text{Im}(I_{1,F}) - \text{Im}(V_1) \cdot \text{Re}(I_{1,F})}{\text{Re}(Z_{1L} \cdot I_1) \cdot \text{Im}(I_{1,F}) - \text{Im}(Z_{1L} \cdot I_1) \cdot \text{Re}(I_{1,F})}

If $|m_{\text{new}} - m_{\text{old}}| < 0.001$ , the solution has converged. Otherwise, update $m_0 = m_{\text{new}}$ and repeat from step 2.
Once converged, compute fault resistance from the imaginary part: $R_F = (\text{Im}(V_1) - m \cdot \text{Im}(Z_{1L} \cdot I_1)) \;/\; \text{Im}(I_{1,F})$

Convergence is typically achieved within 3--5 iterations for well-conditioned systems. The method explicitly produces both $m$ and $R_F$ , which is useful for validating the result.

Requirements: Source impedances $Z_S$ and $Z_R$ . Works for all fault types because it uses positive-sequence quantities exclusively.

Radial (Apparent Impedance) Method

Radial method

m = \frac{|Z_{\text{loop}}|}{|Z_{1L}|}

Where

$m$ Per-unit fault distance
$|Z_{\text{loop}}|$ Magnitude of the measured loop impedance
$|Z_{1L}|$ Magnitude of the total positive-sequence line impedance

When to Use

Measurement Loops

Fault Type	Loop Voltage $V_S$	Loop Current $I_S$
SLG-A	$V_a$	$I_a + k_0 \cdot (I_a + I_b + I_c)$
SLG-B	$V_b$	$I_b + k_0 \cdot (I_a + I_b + I_c)$
SLG-C	$V_c$	$I_c + k_0 \cdot (I_a + I_b + I_c)$
LL-AB	$V_a - V_b$	$I_a - I_b$
LL-BC	$V_b - V_c$	$I_b - I_c$
LL-CA	$V_c - V_a$	$I_c - I_a$
DLG-AB, DLG-BC, DLG-CA	Dominant faulted phase (e.g. $V_a$ )	$I_{ph} + k_0 \cdot (I_a + I_b + I_c)$
3-Phase	Any phase-phase (e.g. $V_a - V_b$ )	Corresponding current (e.g. $I_a - I_b$ )

Phase-phase loops inherently cancel the zero-sequence component and do not use $k_0$ compensation. For line-to-line faults without ground involvement, these loops are the natural choice.

Zero-sequence compensation factor

k_0 = \frac{Z_0 - Z_1}{3 \cdot Z_1}

Where

$Z_0$ Zero-sequence line impedance (complex)
$Z_1$ Positive-sequence line impedance (complex)

k_0 is a complex number. Typical values for overhead lines: magnitude 0.5--0.7. For cables: magnitude 0.3--0.5.

k₀ Compensation Formats

Different relay manufacturers use different input conventions for zero-sequence compensation. All are mathematically equivalent and convertible to the standard $k_0$ used internally.

Standard (Z₁ / Z₀ Input)

The user provides $Z_1$ and $Z_0$ as magnitude and angle. The compensation factor is computed as:

Standard k_0 from impedances

k_0 = \frac{Z_0 - Z_1}{3 \cdot Z_1}

Where

$Z_0$ Zero-sequence line impedance (mag + angle)
$Z_1$ Positive-sequence line impedance (mag + angle)

Used by SEL (auto mode), ABB (KN computation). This is the default input mode.

Direct (k₀ Magnitude + Angle)

The user enters $k_0$ directly as magnitude and angle. No conversion is needed -- the value is used as-is. This matches SEL manual mode (k0M1), ABB KN, Siemens K0, and Schneider kG (Method 1).

Siemens RE/RL + XE/XL

Siemens relays (e.g. 7SA612) express the compensation as two separate real/reactive ratios. The conversion to the standard complex $k_0$ is:

Siemens ratio conversion

k_0 = \frac{RE + j \cdot XE}{R_1 + j \cdot X_1}

Where

$RE$ = (RE/RL) \cdot R_1, where R_1 = |Z_1| \cos(\angle Z_1)
$XE$ = (XE/XL) \cdot X_1, where X_1 = |Z_1| \sin(\angle Z_1)
$RE/RL$ Ratio of earth resistance to line resistance (Siemens setting)
$XE/XL$ Ratio of earth reactance to line reactance (Siemens setting)

RE/RL and XE/XL are dimensionless ratios. Typical values: RE/RL = 1.0--3.0, XE/XL = 2.0--4.0.

Phase-Only (Schneider Method)

Some Schneider relays (e.g. P437) use a phase-only ground loop computation where the compensated current is computed differently:

Phase-only ground loop

I_{\text{comp}} = I_{\text{phase}} \cdot (1 + k_G)

Where

$I_{\text{comp}}$ Compensated loop current
$I_{\text{phase}}$ Faulted phase current only
$k_G$ Ground compensation factor (magnitude + angle)

In the phase-only method, k_G has a DIFFERENT numerical value than k_0 for the same line, because the computation formula is structurally different.

Algorithm Comparison

The six algorithms represent a spectrum of complexity and accuracy. The following diagram and table summarise their relative strengths.

Figure — Algorithm Performance Comparison

Method	R_F Handling	Data Required	Best For
Reactance	Assumes in-phase I_S/I_F	None (always available)	Bolted faults, quick estimate, fallback
Takagi	Eliminates via superimposed current	Pre-fault phasors	General-purpose, best single-source method
Modified Takagi	Eliminates via zero-sequence current	Ground fault (I₀ > 0)	Ground faults without pre-fault data
Eriksson	Solves quadratic using source Z	Pre-fault + source impedances	Non-homogeneous systems, strong infeed
Novosel	Iterative with distribution factor	Source impedances	All fault types, explicit R_F estimate
Radial	Includes R_F in distance (no separation)	None (always available)	Bolted faults on radial feeders, quick sanity check

Cross-Validation

Statistical Summary

Median fault distance

\tilde{m} = \text{median}(m[1], m[2], \ldots, m[N])

Where

$\tilde{m}$ Median per-unit distance over the fault window
$N$ Number of samples in the fault window
$m[i]$ Distance estimate at sample i

Median Absolute Deviation (MAD)

\text{MAD} = \text{median}(|m[i] - \tilde{m}|)

Where

$\text{MAD}$ Median of the absolute deviations from the median

MAD is converted to a standard-deviation-equivalent (σ) by multiplying by 1.4826, which assumes an underlying normal distribution.

Robust sigma

\sigma_{\text{MAD}} = 1.4826 \times \text{MAD}

Where

$\sigma_{\text{MAD}}$ MAD-based sigma (comparable to standard deviation for normal data)
$1.4826$ Consistency factor for normal distribution

MAD / Line Length	Confidence	Interpretation
< 5%	High	Consistent measurements with low scatter. Distance estimate is reliable.
5% – 15%	Medium	Moderate scatter. Distance estimate is usable but should be interpreted with caution.
> 15%	Low	High scatter. Distance estimate has significant uncertainty. Check for CT saturation, incorrect line parameters, or evolving faults.

Fallback When Line Length Is Not Set

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