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Directional Assessment

Three-phase directional analysis with 90° quadrature connection, earth fault direction, and negative-sequence polarisation for determining fault direction.

In Detego

See the Directional Assessment Guide for how to use this tab and interpret the results. This page covers the mathematical theory.

90° Quadrature Connection

The 90-degree quadrature connection is the standard method for phase fault directional relaying. Each phase current is compared against the cross-phase (line) voltage — the voltage across the other two phases — which provides a polarizing reference that remains healthy during single-phase faults.

Standard Polarizing Pairs

Operating Current	Polarizing Voltage	Line Voltage Formula	Phase Element
$I_a$	$V_{bc}$	$V_b - V_c$	Phase A
$I_b$	$V_{ca}$	$V_c - V_a$	Phase B
$I_c$	$V_{ab}$	$V_a - V_b$	Phase C

For a balanced system, each line voltage has magnitude $\sqrt{3}$ times the phase voltage and leads the corresponding phase voltage by 30°. The quadrature connection ensures the polarizing voltage is derived from unfaulted phases, maintaining a reliable reference even when one phase voltage collapses during a close-up fault.

Directional Angle

The directional angle is the fundamental measurement for directional determination. It represents the angular displacement between the operating current and the polarizing voltage, adjusted by the relay characteristic angle (RCA).

Directional Angle

\theta_{\text{dir}} = \angle I_{\text{operate}} - \angle V_{\text{polarise}} - \text{RCA}

Where

$I_operate$ Phase angle of the operating current phasor
$V_polarise$ Phase angle of the quadrature polarizing voltage phasor
$RCA$ Relay Characteristic Angle (shifts the operate boundary)

The result is normalized to the range -180 to +180 degrees. If |θ_dir| ≤ 90° → FORWARD, else → REVERSE. If the margin is below the INDETERMINATE threshold → INDETERMINATE.

Directional Torque

The directional torque is the scalar product of the voltage and current magnitudes projected onto the maximum torque angle. It provides both the direction (sign) and the confidence (magnitude) of the directional decision.

Directional Torque

T = |V| \times |I| \times \cos(\theta_{\text{relative}})

Where

$|V|$ Magnitude of the polarizing voltage phasor
$|I|$ Magnitude of the operating current phasor
$\theta_{relative}$ Directional angle relative to the MTA

Positive torque → forward fault (operate zone). Negative torque → reverse fault (restrain zone).

The margin percentage indicates how deep into the operate or restrain zone the operating point sits: 100% at the MTA (maximum sensitivity), 0% at the ±90° boundary.

Directional element operating characteristic. The MTA (Maximum Torque Angle) defines peak sensitivity. Faults in the operate region produce positive torque (forward); faults in the restrain region produce negative torque (reverse).

In the Detego user interface, the directional torque is labeled P₀ (zero-sequence active power) and displayed in watts (W) or kilowatts (kW). The formula and interpretation are identical — positive values indicate forward, negative values indicate reverse.

Relay Characteristic Angle (RCA)

The RCA shifts the maximum sensitivity direction to align with the expected fault current angle. The correct RCA depends on the network configuration and earthing arrangement and the network earthing method.

Phase Fault RCA Presets

Preset	RCA	Application
90°-30°	$30°$	Plain feeders (overhead lines, cables) — zero-sequence source behind relay
90°-45°	$45°$	Transformer feeders — zero-sequence source in front of relay
Custom	User-defined	Free input for non-standard configurations

Earth Fault RCA Presets

The system earthing method fundamentally changes the angle relationship between zero-sequence voltage and current during a ground fault. The RCA must be set accordingly:

Earthing Type	RCA	Directional Method
Resistance earthed	$0°$	Fault current predominantly resistive, in phase with voltage
Solidly earthed (distribution)	$-45°$	Distribution networks with moderate X/R ratio (default)
Solidly earthed (transmission)	$-60°$	Transmission networks with high X/R ratio
Insulated (unearthed)	$90°$	Capacitive current directional
Petersen coil (resonant)	$0°$	Wattmetric (zero-sequence active power)
Custom	User-defined	Free input for non-standard earthing arrangements

Critical: Earth Fault RCA Polarity

Solidly earthed systems require a negative RCA (-45° for distribution, -60° for transmission) because the zero-sequence current lags the zero-sequence voltage reversal. A common error is using RCA = 0° for solidly earthed systems, which places the maximum torque angle 45-60° away from the actual fault current direction and can cause incorrect directional decisions on boundary faults.

Negative-Sequence Directional

The negative-sequence directional element provides an alternative polarization method for unbalanced faults. It is particularly useful for:

Close-up faults where all three phase voltages collapse and quadrature polarization fails
High-resistance earth faults where zero-sequence quantities are small
Cross-country faults producing complex zero-sequence patterns

Negative-Sequence Direction

\text{If } |\angle I_2 - \angle(-V_2)| \leq 90° \Rightarrow \text{FORWARD}

Where

$I_2$ Negative-sequence current phasor (from symmetrical components)
$V_2$ Negative-sequence voltage phasor (from symmetrical components)
$-V_2$ Reversed V2 phasor (angle + 180°)

Signal strength |V2| × |I2| indicates reliability. Low values mean the fault is balanced and neg-seq polarization is unreliable.

Residual Voltage (3V0)

The residual voltage is the vector sum of the three phase voltages. In a balanced three-phase system, the three voltage phasors cancel exactly, producing zero residual. During ground faults, the voltage imbalance produces a non-zero residual that represents three times the zero-sequence voltage.

Residual Voltage

3V_0 = V_a + V_b + V_c

Where

$V_a, V_b, V_c$ Phase voltage phasors from the three-phase VT
$3V_0$ Residual voltage (three times the zero-sequence voltage)

A broken-delta VT secondary winding produces this quantity directly. Detego computes it from individual phase voltage phasors.

Residual Current (3I0)

The residual current is the vector sum of the three phase currents, representing the current flowing through the ground return path. Like residual voltage, it is zero under balanced conditions and non-zero during ground faults.

Residual Current

3I_0 = I_a + I_b + I_c

Where

$I_a, I_b, I_c$ Phase current phasors from the three-phase CT
$3I_0$ Residual current (three times the zero-sequence current)

Serves as the operating quantity for earth fault directional elements.

Zero-Sequence Active Power

Zero-sequence active power provides a directional determination specifically designed for Petersen coil (resonant grounded) systems where the conventional overcurrent-based directional method fails due to the high neutral impedance limiting ground fault current.

Zero-Sequence Active Power

P_0 = |V_0| \times |I_0| \times \cos(\theta_{V_0} - \theta_{I_0})

Where

$V_0$ Zero-sequence voltage magnitude
$I_0$ Zero-sequence current magnitude
$\theta_{V_0}$ Zero-sequence voltage angle
$\theta_{I_0}$ Zero-sequence current angle

Positive P0 indicates forward fault direction; negative indicates reverse.

Wattmetric Sensitivity

The wattmetric method is sensitive to the resistive losses in the faulted feeder (cable dielectric losses, conductor resistance). Even when the total ground fault current is only a few amperes, the active power component reliably indicates direction because the reactive components (capacitive and inductive) cancel at the Petersen coil resonance point.

INDETERMINATE Zone

The directional decision has three outcomes: FORWARD, REVERSE, or INDETERMINATE.

When the operating point falls near the ±90° boundary between the operate and restrain zones, the directional angle provides very little confidence in the declared direction. Small errors in CT polarity, measurement noise, or transient harmonics can flip the result from one zone to the other.

Detego introduces an INDETERMINATE zone around the boundary. When the margin percentage falls below a configurable threshold, the result is declared INDETERMINATE rather than forcing a binary FORWARD/REVERSE decision. This is displayed as "LOW CONFIDENCE" in the user interface.

CT Polarity Verification

When direction is INDETERMINATE, verify the CT polarity convention. Current transformers installed in opposite orientation will shift the directional angle by 180°, potentially placing a clearly forward fault in the reverse zone. Check that the primary current direction convention (outward from bus = positive) matches the recording.

Margin Formula

The margin formula measures how deep into the operate or restrain zone the operating point sits. It provides a symmetric confidence metric that works identically for both FORWARD and REVERSE results.

Directional Margin

M = \frac{|\,|\theta_{\text{rel}}| - 90°\,|}{90°} \times 100\%

Where

$\theta_{rel}$ Relative angle (directional angle minus RCA)
$M$ Margin percentage (0% at boundary, 100% at MTA or anti-MTA)

This formula works symmetrically for both FORWARD and REVERSE results. At the MTA (θ_rel = 0°) the margin is 100%. At the anti-MTA (θ_rel = ±180°) the margin is also 100%. At the ±90° boundary, margin is 0%.

A previous common implementation error was using a formula that only measured the distance from the Maximum Torque Angle (MTA). Such a formula gives 0% margin for all REVERSE results regardless of confidence — a clearly reverse fault at the anti-MTA would incorrectly show 0% margin instead of 100%. The symmetric formula above correctly represents confidence for both directions.

Wattmetric Method for Petersen Coil Networks

On Petersen coil (resonant grounded / compensated) networks, the Petersen coil inductance is tuned to nearly cancel the total network capacitance to ground. During an earth fault, the inductive current from the coil and the capacitive current from the healthy feeders flow in opposition, leaving only a small residual current $I_0$ .

Wattmetric (P₀) directional zones — compensated network

On compensated (Petersen coil) networks, I₀ is nearly perpendicular to V₀ on both faulted and healthy feeders. The angular margin between I₀ and the 90° boundary is only ~2° (shown oscillating), making angular-based direction unreliable. The wattmetric method uses the sign of P₀ = |V₀||I₀|cos(θ) instead: positive P₀ reliably indicates the faulted feeder regardless of the narrow margin.

The residual current $I_0$ is almost entirely reactive, flowing at approximately 90° from $V_0$ . This means the angular margin between $I_0$ and the operate/restrain boundary is inherently thin — often just 2–5°. Cycle-by-cycle variations caused by transient harmonics, CT saturation, or measurement noise can cause the directional angle to oscillate across the boundary, flipping the declared direction between FORWARD and REVERSE.

The wattmetric method resolves this ambiguity by using the sign of the zero-sequence active power $P_0$ instead of the angular margin. While the reactive components cancel at the Petersen coil resonance point, the active power component — arising from feeder conductor resistance and cable dielectric losses — consistently indicates the faulted feeder:

$P_0 > 0$ — FORWARD: the fault is on this feeder
$P_0 < 0$ — REVERSE: the fault is on another feeder

The sign of the zero-sequence active power determines the fault direction.

Earthing Type Classification

The system earthing arrangement determines the zero-sequence current magnitude and angle during earth faults. Different earthing types produce characteristic patterns in zero-sequence measurements, and correct identification of the earthing type is essential for selecting the appropriate directional method and RCA.

Detego analyzes multiple indicators from the fault data to classify the earthing type:

$Z_0$ angle: inductive (Petersen coil), capacitive (insulated), mixed (solidly earthed), resistive (resistance earthed)
$V_0$ displacement: near 100% of phase voltage for compensated/insulated, small for solidly earthed
$I_0$ magnitude: very small for compensated, large for solidly earthed
$Z_0$ magnitude: very high for compensated/insulated, low for solidly earthed

Characteristic Indicators by Earthing Type

Earthing Type	$Z_0$ Angle	$V_0$ Displacement	$I_0$ Magnitude	$Z_0$ Magnitude
Solidly earthed	Mixed (30–70°)	Small	Large	Low
Resistance earthed	Small (<30°)	Small–moderate	Moderate	Moderate
Petersen coil	Inductive (~-90°)	Large (~100%)	Very small	Very high
Insulated	Capacitive (~+90°)	Large (~100%)	Small	Very high

Windowed Directional Averaging

A single-point directional measurement at one time instant can be affected by transient harmonics, CT saturation, or measurement noise. To improve reliability, windowed averaging computes the directional result at every cycle through the fault duration.

The final direction is determined by majority vote: the direction with the most cycles wins. The percentage of agreeing cycles indicates confidence. The mean directional angle and mean margin over the window provide additional context for interpreting the result.

Results are reported as, for example: "FORWARD — 85% of cycles (17 forward · 3 reverse · 0 indeterminate)".

Value for Petersen Coil Networks

Windowed averaging is particularly valuable for Petersen coil networks where the wattmetric

P_0

may oscillate near zero on boundary faults. Aggregating over multiple cycles smooths out cycle-by-cycle noise and provides a statistically meaningful directional assessment.

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Directional Assessment

Three-phase directional analysis with 90° quadrature connection, earth fault direction, and negative-sequence polarisation for determining fault direction.

In Detego

See the Directional Assessment Guide for how to use this tab and interpret the results. This page covers the mathematical theory.

90° Quadrature Connection

Standard Polarizing Pairs

Operating Current	Polarizing Voltage	Line Voltage Formula	Phase Element
$I_a$	$V_{bc}$	$V_b - V_c$	Phase A
$I_b$	$V_{ca}$	$V_c - V_a$	Phase B
$I_c$	$V_{ab}$	$V_a - V_b$	Phase C

Directional Angle

\theta_{\text{dir}} = \angle I_{\text{operate}} - \angle V_{\text{polarise}} - \text{RCA}

Where

$I_operate$ Phase angle of the operating current phasor
$V_polarise$ Phase angle of the quadrature polarizing voltage phasor
$RCA$ Relay Characteristic Angle (shifts the operate boundary)

The result is normalized to the range -180 to +180 degrees. If |θ_dir| ≤ 90° → FORWARD, else → REVERSE. If the margin is below the INDETERMINATE threshold → INDETERMINATE.

Directional Torque

T = |V| \times |I| \times \cos(\theta_{\text{relative}})

Where

$|V|$ Magnitude of the polarizing voltage phasor
$|I|$ Magnitude of the operating current phasor
$\theta_{relative}$ Directional angle relative to the MTA

Positive torque → forward fault (operate zone). Negative torque → reverse fault (restrain zone).

The margin percentage indicates how deep into the operate or restrain zone the operating point sits: 100% at the MTA (maximum sensitivity), 0% at the ±90° boundary.

Relay Characteristic Angle (RCA)

Phase Fault RCA Presets

Preset	RCA	Application
90°-30°	$30°$	Plain feeders (overhead lines, cables) — zero-sequence source behind relay
90°-45°	$45°$	Transformer feeders — zero-sequence source in front of relay
Custom	User-defined	Free input for non-standard configurations

Earth Fault RCA Presets

The system earthing method fundamentally changes the angle relationship between zero-sequence voltage and current during a ground fault. The RCA must be set accordingly:

Earthing Type	RCA	Directional Method
Resistance earthed	$0°$	Fault current predominantly resistive, in phase with voltage
Solidly earthed (distribution)	$-45°$	Distribution networks with moderate X/R ratio (default)
Solidly earthed (transmission)	$-60°$	Transmission networks with high X/R ratio
Insulated (unearthed)	$90°$	Capacitive current directional
Petersen coil (resonant)	$0°$	Wattmetric (zero-sequence active power)
Custom	User-defined	Free input for non-standard earthing arrangements

Critical: Earth Fault RCA Polarity

Negative-Sequence Directional

The negative-sequence directional element provides an alternative polarization method for unbalanced faults. It is particularly useful for:

Close-up faults where all three phase voltages collapse and quadrature polarization fails
High-resistance earth faults where zero-sequence quantities are small
Cross-country faults producing complex zero-sequence patterns

Negative-Sequence Direction

\text{If } |\angle I_2 - \angle(-V_2)| \leq 90° \Rightarrow \text{FORWARD}

Where

$I_2$ Negative-sequence current phasor (from symmetrical components)
$V_2$ Negative-sequence voltage phasor (from symmetrical components)
$-V_2$ Reversed V2 phasor (angle + 180°)

Signal strength |V2| × |I2| indicates reliability. Low values mean the fault is balanced and neg-seq polarization is unreliable.

Residual Voltage (3V0)

Residual Voltage

3V_0 = V_a + V_b + V_c

Where

$V_a, V_b, V_c$ Phase voltage phasors from the three-phase VT
$3V_0$ Residual voltage (three times the zero-sequence voltage)

A broken-delta VT secondary winding produces this quantity directly. Detego computes it from individual phase voltage phasors.

Residual Current (3I0)

Residual Current

3I_0 = I_a + I_b + I_c

Where

$I_a, I_b, I_c$ Phase current phasors from the three-phase CT
$3I_0$ Residual current (three times the zero-sequence current)

Serves as the operating quantity for earth fault directional elements.

Zero-Sequence Active Power

P_0 = |V_0| \times |I_0| \times \cos(\theta_{V_0} - \theta_{I_0})

Where

$V_0$ Zero-sequence voltage magnitude
$I_0$ Zero-sequence current magnitude
$\theta_{V_0}$ Zero-sequence voltage angle
$\theta_{I_0}$ Zero-sequence current angle

Positive P0 indicates forward fault direction; negative indicates reverse.

Wattmetric Sensitivity

INDETERMINATE Zone

The directional decision has three outcomes: FORWARD, REVERSE, or INDETERMINATE.

CT Polarity Verification

Margin Formula

The margin formula measures how deep into the operate or restrain zone the operating point sits. It provides a symmetric confidence metric that works identically for both FORWARD and REVERSE results.

Directional Margin

M = \frac{|\,|\theta_{\text{rel}}| - 90°\,|}{90°} \times 100\%

Where

$\theta_{rel}$ Relative angle (directional angle minus RCA)
$M$ Margin percentage (0% at boundary, 100% at MTA or anti-MTA)

Wattmetric Method for Petersen Coil Networks

Wattmetric (P₀) directional zones — compensated network

$P_0 > 0$ — FORWARD: the fault is on this feeder
$P_0 < 0$ — REVERSE: the fault is on another feeder

The sign of the zero-sequence active power determines the fault direction.

Earthing Type Classification

Detego analyzes multiple indicators from the fault data to classify the earthing type:

$Z_0$ angle: inductive (Petersen coil), capacitive (insulated), mixed (solidly earthed), resistive (resistance earthed)
$V_0$ displacement: near 100% of phase voltage for compensated/insulated, small for solidly earthed
$I_0$ magnitude: very small for compensated, large for solidly earthed
$Z_0$ magnitude: very high for compensated/insulated, low for solidly earthed

Characteristic Indicators by Earthing Type

Earthing Type	$Z_0$ Angle	$V_0$ Displacement	$I_0$ Magnitude	$Z_0$ Magnitude
Solidly earthed	Mixed (30–70°)	Small	Large	Low
Resistance earthed	Small (<30°)	Small–moderate	Moderate	Moderate
Petersen coil	Inductive (~-90°)	Large (~100%)	Very small	Very high
Insulated	Capacitive (~+90°)	Large (~100%)	Small	Very high

Windowed Directional Averaging

Results are reported as, for example: "FORWARD — 85% of cycles (17 forward · 3 reverse · 0 indeterminate)".

Value for Petersen Coil Networks

Windowed averaging is particularly valuable for Petersen coil networks where the wattmetric

P_0

may oscillate near zero on boundary faults. Aggregating over multiple cycles smooths out cycle-by-cycle noise and provides a statistically meaningful directional assessment.

PreviousOvercurrent Protection NextDifferential Protection