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Frequency Tracking

Per-cycle DFT frequency and rate of change of frequency (ROCOF) for generator protection and islanding detection.

In Detego

Create this signal via the Compute Tab. This page covers the mathematical theory behind the computation.

Overview

Frequency is one of the most critical parameters in power system operation. It reflects the real-time balance between generation and load: when generation exceeds load, frequency rises; when load exceeds generation, frequency falls. Detego provides two frequency-related computed signals, both DFT-based and restricted to voltage channels for measurement integrity.

System frequency during a generation-load imbalance event. The smooth blue line shows the true frequency; the green staircase shows the per-cycle DFT measurement (one value held per cycle). The dashed amber line shows ROCOF on the right axis. Red dotted lines indicate under/over-frequency relay thresholds.

Computed Signals

1. Frequency (Per-Cycle DFT)

The frequency signal is computed using a sliding one-cycle DFT that extracts the fundamental phasor angle at every sample. The DFT phase angle is unwrapped to produce a continuous, monotonically increasing angle that eliminates $\pm 180°$ discontinuities. The frequency is then computed as one averaged value per complete cycle, held constant across all samples in that cycle — producing a characteristic stepped (staircase) output.

Per-cycle frequency

f = f_{\text{nom}} + \frac{\theta_{\text{end}} - \theta_{\text{start}}}{2\pi \cdot T_{\text{cycle}}}

Where

$f$ Measured frequency (Hz)
$f_{nom}$ Nominal system frequency (50 or 60 Hz)
$\theta_{start}, \theta_{end}$ Unwrapped DFT phase at the start and end of the cycle
$T_{cycle}$ Cycle duration (seconds)

One frequency value is computed per cycle and held constant across all samples in that cycle.

Phase unwrapping

\theta[i] = \theta[i-1] + \text{unwrap}(\angle X[i] - \angle X[i-1])

Where

$X[i]$ Fundamental DFT phasor at sample i
$\text{unwrap}()$ Constrains the phase delta to [-\pi, \pi]

Unwrapping ensures the phase angle is continuous and monotonic, preventing discontinuities from corrupting the frequency calculation.

The stepped output produces one value per cycle rather than noisy per-sample estimates. This matches relay-grade cyclic frequency measurement — modern numerical frequency relays also compute frequency on a per-cycle basis to reject transient noise and provide stable readings for protection decisions.

Voltage channels only

Voltage channels are recommended for frequency measurement (per IEEE C37.118.1) because current waveforms may contain DC offset, harmonics, and CT saturation artifacts that corrupt the fundamental phasor angle.

2. ROCOF (Rate of Change of Frequency)

The Rate of Change of Frequency (ROCOF) is the time derivative of the cycle-averaged frequency. It is computed as a finite difference between consecutive cycle-averaged frequency values, then smoothed with a multi-cycle moving average to reduce noise.

Rate of change of frequency

\frac{df}{dt} = \frac{f[c] - f[c-1]}{\Delta t}

Where

$f[c]$ Cycle-averaged frequency at cycle c (Hz)
$\Delta t$ Time between cycle centres (seconds)

The raw finite difference is smoothed with a multi-cycle moving average before output.

Like the frequency signal, ROCOF uses a stepped output — one value held per cycle. This matches the per-cycle update rate of the underlying frequency measurement and avoids amplifying noise through differentiation of per-sample estimates.

ROCOF sensitivity

ROCOF is inherently noisy because it differentiates an already-estimated quantity. The multi-cycle smoothing window trades response speed for measurement stability. ROCOF is used for islanding detection (IEEE 1547) and under-frequency load shedding rate monitoring.

Engineering Context

Frequency Protection

Frequency relays are classified by their function code and operating direction:

Under-frequency (81U) — trips when frequency falls below a threshold. Used in under-frequency load shedding (UFLS) schemes to arrest frequency decline by disconnecting blocks of load.
Over-frequency (81O) — trips when frequency rises above a threshold. Used for generator over-speed protection.
Generator protection — frequency is monitored for reverse power, loss of field, and abnormal frequency conditions that can damage turbine blades.
Load shedding (UFLS) — multi-stage automatic scheme that progressively disconnects load at lower frequency thresholds to prevent cascade generator tripping.

Islanding Detection

Islanding occurs when a distributed generator continues to operate after being disconnected from the grid. ROCOF relays (ANSI device 81R) are one of the primary methods for detecting islanding — when the island forms, the generation-load imbalance causes a rapid frequency change that the ROCOF relay detects.

ROCOF-based anti-islanding is specified in IEEE 1547 for distributed generation interconnection. The ROCOF signal in Detego can be used to assess whether a ROCOF relay should have operated and to measure the actual $df/dt$ seen at the relay location.

Signal Comparison

Aspect	Frequency	ROCOF
Measures	System frequency (Hz)	Rate of change (Hz/s)
Method	Per-cycle DFT + phase unwrapping	Finite difference of frequency
Output	Stepped (one value per cycle)	Stepped (one value per cycle)
Channel	Voltage only	Voltage only
Use case	81U/81O protection, frequency trending	Islanding detection, rate-of-change relays

PreviousPhasor Extraction NextEnvelope Analysis

Loading documentation...

Frequency Tracking

Per-cycle DFT frequency and rate of change of frequency (ROCOF) for generator protection and islanding detection.

In Detego

Create this signal via the Compute Tab. This page covers the mathematical theory behind the computation.

Overview

Computed Signals

1. Frequency (Per-Cycle DFT)

Per-cycle frequency

f = f_{\text{nom}} + \frac{\theta_{\text{end}} - \theta_{\text{start}}}{2\pi \cdot T_{\text{cycle}}}

Where

$f$ Measured frequency (Hz)
$f_{nom}$ Nominal system frequency (50 or 60 Hz)
$\theta_{start}, \theta_{end}$ Unwrapped DFT phase at the start and end of the cycle
$T_{cycle}$ Cycle duration (seconds)

One frequency value is computed per cycle and held constant across all samples in that cycle.

Phase unwrapping

\theta[i] = \theta[i-1] + \text{unwrap}(\angle X[i] - \angle X[i-1])

Where

$X[i]$ Fundamental DFT phasor at sample i
$\text{unwrap}()$ Constrains the phase delta to [-\pi, \pi]

Unwrapping ensures the phase angle is continuous and monotonic, preventing discontinuities from corrupting the frequency calculation.

Voltage channels only

2. ROCOF (Rate of Change of Frequency)

Rate of change of frequency

\frac{df}{dt} = \frac{f[c] - f[c-1]}{\Delta t}

Where

$f[c]$ Cycle-averaged frequency at cycle c (Hz)
$\Delta t$ Time between cycle centres (seconds)

The raw finite difference is smoothed with a multi-cycle moving average before output.

ROCOF sensitivity

Engineering Context

Frequency Protection

Frequency relays are classified by their function code and operating direction:

Under-frequency (81U) — trips when frequency falls below a threshold. Used in under-frequency load shedding (UFLS) schemes to arrest frequency decline by disconnecting blocks of load.
Over-frequency (81O) — trips when frequency rises above a threshold. Used for generator over-speed protection.
Generator protection — frequency is monitored for reverse power, loss of field, and abnormal frequency conditions that can damage turbine blades.
Load shedding (UFLS) — multi-stage automatic scheme that progressively disconnects load at lower frequency thresholds to prevent cascade generator tripping.

Islanding Detection

Signal Comparison

Aspect	Frequency	ROCOF
Measures	System frequency (Hz)	Rate of change (Hz/s)
Method	Per-cycle DFT + phase unwrapping	Finite difference of frequency
Output	Stepped (one value per cycle)	Stepped (one value per cycle)
Channel	Voltage only	Voltage only
Use case	81U/81O protection, frequency trending	Islanding detection, rate-of-change relays

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