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Protection Quality

CT saturation detection, superimposed delta protection, DC offset measurement, and X/R ratio estimation.

In Detego

See the Compute Tab Guide for how to create this signal via the Compute Tab. This page covers the mathematical theory.

CT Saturation Index

Current transformer saturation occurs when the CT core flux exceeds its linear region, causing the secondary waveform to clip. This distorts all downstream measurements -- phasors, RMS, impedance, and differential quantities all become unreliable once CT saturation is present. The CT Saturation Index quantifies the degree of saturation using the ratio of the second harmonic to the fundamental.

CT Saturation Index

\text{CT Saturation Index (\%)} = \frac{|H_2|}{|H_1|} \times 100

Where

$H1$ Fundamental frequency (1st harmonic) magnitude from sliding DFT
$H2$ Second harmonic magnitude from sliding DFT

A Hanning window is applied to the H2 component to reduce spectral leakage from the sharp clipping edges.

The second harmonic is the dominant byproduct of the asymmetric clipping that CT saturation produces. A saturated CT waveform has its positive (or negative) peaks flattened while the opposite half-cycle remains intact, generating even-order harmonics with $H_2$ being the largest.

CT Saturation Threshold

An H2/H1 ratio exceeding 10% is widely used as an indicator of significant CT saturation. Above this level, protection measurements derived from the saturated current channel should be interpreted with caution.

CT Saturation Visualization

The diagram below compares the ideal CT secondary with a saturated one driven by a fully offset fault current. Note how the secondary collapses during core saturation and recovers as the DC offset decays, with the H2/H1 ratio exceeding the 10% threshold throughout the saturated interval.

CT secondary saturation from a fully offset fault current (X/R ≈ 15). The saturated secondary collapses as core flux exceeds the saturation level, producing sharp recovery spikes. The H2/H1 ratio rises well above the 10% threshold during active saturation and subsides as the DC offset decays.

Effects on Protection

Overcurrent elements: Saturated RMS values underestimate true fault current, potentially causing delayed or failed trip.
Distance elements: Distorted current phasors shift the apparent impedance, causing reach errors.
Differential elements: Unequal saturation on HV vs. LV CTs creates false differential current, risking misoperation.

Superimposed Delta Protection

Superimposed delta quantities isolate the change component of a signal by subtracting the value from exactly one power frequency cycle earlier. Under steady-state conditions, the delta signal is zero because the waveform repeats every cycle. At the instant of a fault, the change in current or voltage produces a sharp transient spike in the delta signal.

Superimposed Delta

\Delta I(t) = I(t) - I(t - T)

Where

$I(t)$ Current sample at time t
$T$ One power frequency cycle period (20 ms at 50 Hz, 16.67 ms at 60 Hz)

The delta signal spikes at fault onset and decays within one cycle as the new steady-state is established.

Modern protection relays use superimposed delta quantities as the basis for ultra-high-speed fault detection. Unlike fundamental-frequency-based methods (RMS, phasors) that require at least one full cycle to settle, superimposed quantities detect the fault from a single sample -- the very first sample where the current deviates from its pre-fault trajectory.

Applications

Superimposed delta is used in modern relay platforms for sub-cycle tripping (operating times under 5 ms), travelling-wave fault location, and as the primary fault inception detector in Detego's automatic fault detection algorithm.

DC Offset

When a fault occurs on an AC power system, the fault current is not purely sinusoidal. The circuit inductance prevents an instantaneous change in current, so a decaying DC component is superimposed on the AC fault current to satisfy the initial conditions at the instant of fault inception. The magnitude of this DC offset depends on the point-on-wave at which the fault occurs.

DC Offset (half-cycle extraction)

I_{\text{DC}}(t) = \frac{\max(x) + \min(x)}{2}

Where

$x$ Samples within a half-cycle window centered at time t
$max(x)$ Maximum sample value in the half-cycle window
$min(x)$ Minimum sample value in the half-cycle window

For a pure sinusoid, max + min = 0 (symmetric about zero). A non-zero result indicates DC offset.

The DC component is largest when the fault occurs at a voltage zero crossing (maximum asymmetry) and zero when the fault occurs at the voltage peak (fully symmetric fault). Maximum asymmetry produces a first-peak fault current that can be nearly twice the symmetric peak value, which is critical for equipment ratings and breaker duty assessment.

Asymmetrical fault current with DC offset decay. The DC component decays exponentially with time constant τ = X/(ωR). Higher X/R ratios produce slower decay and greater peak asymmetry.

Breaker Considerations

High DC offset at the time of current interruption increases the energy the circuit breaker must dissipate. Standards such as IEC 62271-100 define asymmetrical breaking current ratings that account for the DC component at contact parting time.

X/R Ratio

The X/R ratio of the fault circuit determines how quickly the DC offset decays. It is computed from the DC decay time constant $\tau$ using an exponential fit of the measured DC component.

X/R from DC Decay

\frac{X}{R} = 2\pi f \tau

Where

$f$ System frequency (50 or 60 Hz)
$\tau$ DC decay time constant from exponential fit (seconds)

Exponential Fitting Method

The time constant $\tau$ is determined by fitting the DC envelope to a decaying exponential. Taking the natural logarithm transforms the exponential into a linear regression problem:

Linearized DC Decay

\ln|I_{\text{DC}}(t)| = \ln(I_{\text{DC}_0}) - \frac{t}{\tau}

Where

$I_{DC_0}$ Initial DC magnitude at fault inception
$t$ Time from fault inception
$\tau$ Decay time constant (the negative reciprocal of the slope)

Linear regression of ln|I_DC| vs. t yields slope = -1/τ.

The fitting procedure applies several quality filters:

Minimum DC fraction: The DC component must exceed a configurable minimum fraction of the AC peak-to-peak amplitude. Below this level, the fault is nearly symmetric and the DC signal is too small to fit reliably.
Minimum fit points: A sufficient number of half-cycles of DC data are required for a statistically meaningful regression. Shorter recordings produce unreliable time constant estimates.
Result clamping: The computed X/R ratio is clamped to a physically reasonable range. Values outside this range indicate fitting errors or non-physical conditions.

Interpreting X/R Values

The X/R ratio has direct engineering significance:

Low X/R (< 5): Fast DC decay. Common in distribution systems and resistive faults. The current becomes symmetric within 1-2 cycles.
Moderate X/R (5-15): Typical for sub-transmission faults. DC decays within 3-5 cycles.
High X/R (> 15): Slow DC decay. Typical of transmission-level faults near large generators. The DC component persists for many cycles, potentially causing CT saturation and breaker interruption challenges.

Engineering Significance

A high X/R ratio (> 15) means the DC component may still be significant when the breaker attempts to interrupt the fault current. This increases the asymmetrical breaking duty and may affect breaker coordination. It also increases the likelihood of CT saturation on the faulted circuit, which can distort the measurements used by backup protection elements.

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Protection Quality

CT saturation detection, superimposed delta protection, DC offset measurement, and X/R ratio estimation.

In Detego

See the Compute Tab Guide for how to create this signal via the Compute Tab. This page covers the mathematical theory.

CT Saturation Index

\text{CT Saturation Index (\%)} = \frac{|H_2|}{|H_1|} \times 100

Where

$H1$ Fundamental frequency (1st harmonic) magnitude from sliding DFT
$H2$ Second harmonic magnitude from sliding DFT

A Hanning window is applied to the H2 component to reduce spectral leakage from the sharp clipping edges.

CT Saturation Threshold

CT Saturation Visualization

Effects on Protection

Overcurrent elements: Saturated RMS values underestimate true fault current, potentially causing delayed or failed trip.
Distance elements: Distorted current phasors shift the apparent impedance, causing reach errors.
Differential elements: Unequal saturation on HV vs. LV CTs creates false differential current, risking misoperation.

Superimposed Delta Protection

Superimposed Delta

\Delta I(t) = I(t) - I(t - T)

Where

$I(t)$ Current sample at time t
$T$ One power frequency cycle period (20 ms at 50 Hz, 16.67 ms at 60 Hz)

The delta signal spikes at fault onset and decays within one cycle as the new steady-state is established.

Applications

DC Offset

DC Offset (half-cycle extraction)

I_{\text{DC}}(t) = \frac{\max(x) + \min(x)}{2}

Where

$x$ Samples within a half-cycle window centered at time t
$max(x)$ Maximum sample value in the half-cycle window
$min(x)$ Minimum sample value in the half-cycle window

For a pure sinusoid, max + min = 0 (symmetric about zero). A non-zero result indicates DC offset.

Asymmetrical fault current with DC offset decay. The DC component decays exponentially with time constant τ = X/(ωR). Higher X/R ratios produce slower decay and greater peak asymmetry.

Breaker Considerations

X/R Ratio

The X/R ratio of the fault circuit determines how quickly the DC offset decays. It is computed from the DC decay time constant $\tau$ using an exponential fit of the measured DC component.

X/R from DC Decay

\frac{X}{R} = 2\pi f \tau

Where

$f$ System frequency (50 or 60 Hz)
$\tau$ DC decay time constant from exponential fit (seconds)

Exponential Fitting Method

The time constant $\tau$ is determined by fitting the DC envelope to a decaying exponential. Taking the natural logarithm transforms the exponential into a linear regression problem:

Linearized DC Decay

\ln|I_{\text{DC}}(t)| = \ln(I_{\text{DC}_0}) - \frac{t}{\tau}

Where

$I_{DC_0}$ Initial DC magnitude at fault inception
$t$ Time from fault inception
$\tau$ Decay time constant (the negative reciprocal of the slope)

Linear regression of ln|I_DC| vs. t yields slope = -1/τ.

The fitting procedure applies several quality filters:

Minimum DC fraction: The DC component must exceed a configurable minimum fraction of the AC peak-to-peak amplitude. Below this level, the fault is nearly symmetric and the DC signal is too small to fit reliably.
Minimum fit points: A sufficient number of half-cycles of DC data are required for a statistically meaningful regression. Shorter recordings produce unreliable time constant estimates.
Result clamping: The computed X/R ratio is clamped to a physically reasonable range. Values outside this range indicate fitting errors or non-physical conditions.

Interpreting X/R Values

The X/R ratio has direct engineering significance:

Low X/R (< 5): Fast DC decay. Common in distribution systems and resistive faults. The current becomes symmetric within 1-2 cycles.
Moderate X/R (5-15): Typical for sub-transmission faults. DC decays within 3-5 cycles.
High X/R (> 15): Slow DC decay. Typical of transmission-level faults near large generators. The DC component persists for many cycles, potentially causing CT saturation and breaker interruption challenges.

Engineering Significance

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