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Power Quantities

Active, reactive, and apparent power, power factor, and instantaneous vs. averaged output modes.

In Detego

View power measurements in the Power Quantities tab. This page covers the mathematical theory behind the computation.

Overview

Power quantities describe the rate of energy transfer between source and load. Detego computes four power signals from paired voltage and current channels: active power, reactive power, apparent power, and power factor.

The relationship between these quantities is captured by the power triangle, where apparent power is the hypotenuse, active power is the adjacent side, and reactive power is the opposite side, with the power factor angle between them.

The Power Triangle

Power triangle showing the relationship between active power (P), reactive power (Q), and apparent power (S). The power factor angle θ determines how much of the apparent power does useful work.

Power triangle relationship

S^2 = P^2 + Q^2

Where

$S$ Apparent power (VA) -- total power delivered by the source
$P$ Active power (W) -- real power doing useful work
$Q$ Reactive power (VAR) -- power oscillating between source and load

This Pythagorean relationship holds for sinusoidal (non-distorted) waveforms. For distorted waveforms, a distortion power component D appears: S^2 = P^2 + Q^2 + D^2.

Signal Types

1. Active Power (P)

Active power represents the real power doing useful work -- heating resistive loads, driving motors, producing light. It is the average value of the instantaneous power over one cycle. Active power always flows in one direction (source to load, or load to source in generation). In protection engineering, active power direction is used for reverse power protection (ANSI 32) on generators and for directional power flow monitoring.

The Active Power signal supports two output modes, selectable via a toggle in the signal builder:

Averaged mode (default)

DFT-based active power

P = V_{\text{RMS}} \times I_{\text{RMS}} \times \cos(\theta)

Where

$V_RMS$ RMS voltage magnitude from the sliding DFT phasor
$I_RMS$ RMS current magnitude from the sliding DFT phasor
$theta$ Angle difference between voltage and current phasors (power factor angle)

The averaged mode uses phasor-derived quantities to produce a smooth, one-cycle-averaged active power trace. This is the standard representation for steady-state analysis.

Instantaneous mode

Instantaneous power

p(t) = v(t) \times i(t)

Where

$v(t)$ Instantaneous voltage sample
$i(t)$ Instantaneous current sample

The instantaneous mode computes the sample-by-sample product of raw voltage and current. For a sinusoidal system, instantaneous power oscillates at twice the system frequency (100 Hz for a 50 Hz system, 120 Hz for 60 Hz). The oscillation is centered on the active power value, with the amplitude determined by the reactive power component. Positive values indicate energy flowing from source to load; negative values indicate energy flowing back from load to source, which occurs during portions of the cycle when reactive elements return stored energy.

2. Reactive Power (Q)

Reactive power

Q = V_{\text{RMS}} \times I_{\text{RMS}} \times \sin(\theta)

Where

$theta$ Angle difference between voltage and current phasors

Positive Q indicates inductive reactive power (current lagging voltage). Negative Q indicates capacitive reactive power (current leading voltage).

Reactive power represents energy oscillating between source and load without performing useful work. Inductive loads (motors, transformers) absorb reactive power to maintain their magnetic fields; capacitive loads (capacitor banks, cable charging) supply reactive power. While reactive power does no net work, it increases the total current flowing in the system, causing additional $I^2R$ losses and requiring larger conductor ratings.

3. Apparent Power (S)

Apparent power

S = V_{\text{RMS}} \times I_{\text{RMS}}

Where

$V_RMS$ RMS voltage magnitude
$I_RMS$ RMS current magnitude

Apparent power is the total power including both active and reactive components. It represents the full capacity demanded from the source and determines equipment sizing -- transformers, cables, and switchgear are rated in VA (or kVA, MVA) because they must carry the full apparent power current regardless of how much of it does useful work. Apparent power is always greater than or equal to active power; they are equal only when the power factor is unity (purely resistive load).

4. Power Factor (PF)

Power factor

\text{PF} = \cos(\angle V - \angle I) = \frac{P}{S}

Where

$angle V$ Voltage phasor angle
$angle I$ Current phasor angle
$P$ Active power
$S$ Apparent power

The power factor is the ratio of useful (active) power to total (apparent) power. It ranges from 0 to 1, where 1 means all power is doing useful work (purely resistive load) and 0 means all power is reactive (purely inductive or capacitive load). In practice, industrial loads typically operate at power factors between 0.8 and 0.95 lagging (inductive). Utilities penalize customers with low power factors because the excess reactive current increases system losses.

Lagging vs. leading

A lagging power factor means current lags voltage (inductive load -- motors, transformers). A leading power factor means current leads voltage (capacitive load -- capacitor banks, lightly loaded cables). The sign of Q indicates the type: positive Q is lagging (inductive), negative Q is leading (capacitive).

How Detego Computes Power

Power quantities in Detego are computed using the sliding DFT phasor method:

At each time step, the voltage and current phasors are extracted using the one-cycle DFT with DC offset removal (see Phasor Extraction).
The RMS magnitudes $V_{\text{RMS}}$ and $I_{\text{RMS}}$ are taken from the phasor magnitudes.
The power factor angle $\theta = \angle V - \angle I$ is computed from the phasor angles.
Active, reactive, and apparent power are computed from the formulas above.

When the Active Power signal is set to instantaneous mode, the phasor step is bypassed and the output is the sample-by-sample product of the raw voltage and current waveforms.

Line-to-Line Voltage Derivation

Many COMTRADE recordings from numerical relays only contain line-to-line voltages ( $V_{AB}$ , $V_{BC}$ , $V_{CA}$ ) rather than phase-to-neutral voltages ( $V_{AN}$ , $V_{BN}$ , $V_{CN}$ ). Since power calculations require phase-to-neutral voltages paired with their respective line currents, Detego derives the phase-to-neutral quantities from the available line-to-line measurements.

Fundamental Identity

Line-to-line voltages are defined as the difference between two phase voltages:

Line-to-line definitions

V_{AB} = V_A - V_B, \quad V_{BC} = V_B - V_C, \quad V_{CA} = V_C - V_A

Adding all three: V_AB + V_BC + V_CA = 0. This identity always holds regardless of system balance.

Because the three line-to-line voltages always sum to zero, only two independent measurements are needed. If one is missing, Detego derives it from the other two.

Deriving Phase-to-Neutral Voltages

Given at least two line-to-line phasors, the phase-to-neutral phasors are obtained by solving the system of equations above. The result is:

Phase-to-neutral from line-to-line

V_{AN} = \frac{V_{AB} - V_{CA}}{3}, \quad V_{BN} = \frac{V_{BC} - V_{AB}}{3}, \quad V_{CN} = \frac{V_{CA} - V_{BC}}{3}

Where

$V_{AN}, V_{BN}, V_{CN}$ Derived phase-to-neutral voltage phasors
$V_{AB}, V_{BC}, V_{CA}$ Measured (or derived) line-to-line voltage phasors

These formulas are derived under the assumption that the sum of the three phase-to-neutral voltages is zero ( $V_A + V_B + V_C = 0$ ). In symmetrical component terms, this means the zero-sequence voltage is zero. All arithmetic is performed in the complex (phasor) domain.

Why zero-sequence is lost

Line-to-line voltages inherently eliminate the zero-sequence component. Each line-to-line voltage is a difference between two phases, and because zero-sequence is identical in all three phases, it cancels in every subtraction:V_AB = (V_0 + V_1 + V_2) − (V_0 + a²V_1 + aV_2) = V_1(1 − a²) + V_2(1 − a)The derived phase-to-neutral voltages therefore contain only positive-sequence and negative-sequence components. For most power analysis this is sufficient, since zero-sequence power is negligible in normal operation and during phase-to-phase faults.

PreviousSequence Components NextImpedance Analysis

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Power Quantities

Active, reactive, and apparent power, power factor, and instantaneous vs. averaged output modes.

In Detego

View power measurements in the Power Quantities tab. This page covers the mathematical theory behind the computation.

Overview

The Power Triangle

Power triangle showing the relationship between active power (P), reactive power (Q), and apparent power (S). The power factor angle θ determines how much of the apparent power does useful work.

Power triangle relationship

S^2 = P^2 + Q^2

Where

$S$ Apparent power (VA) -- total power delivered by the source
$P$ Active power (W) -- real power doing useful work
$Q$ Reactive power (VAR) -- power oscillating between source and load

This Pythagorean relationship holds for sinusoidal (non-distorted) waveforms. For distorted waveforms, a distortion power component D appears: S^2 = P^2 + Q^2 + D^2.

Signal Types

1. Active Power (P)

The Active Power signal supports two output modes, selectable via a toggle in the signal builder:

Averaged mode (default)

DFT-based active power

P = V_{\text{RMS}} \times I_{\text{RMS}} \times \cos(\theta)

Where

$V_RMS$ RMS voltage magnitude from the sliding DFT phasor
$I_RMS$ RMS current magnitude from the sliding DFT phasor
$theta$ Angle difference between voltage and current phasors (power factor angle)

The averaged mode uses phasor-derived quantities to produce a smooth, one-cycle-averaged active power trace. This is the standard representation for steady-state analysis.

Instantaneous mode

Instantaneous power

p(t) = v(t) \times i(t)

Where

$v(t)$ Instantaneous voltage sample
$i(t)$ Instantaneous current sample

2. Reactive Power (Q)

Reactive power

Q = V_{\text{RMS}} \times I_{\text{RMS}} \times \sin(\theta)

Where

$theta$ Angle difference between voltage and current phasors

Positive Q indicates inductive reactive power (current lagging voltage). Negative Q indicates capacitive reactive power (current leading voltage).

3. Apparent Power (S)

Apparent power

S = V_{\text{RMS}} \times I_{\text{RMS}}

Where

$V_RMS$ RMS voltage magnitude
$I_RMS$ RMS current magnitude

4. Power Factor (PF)

Power factor

\text{PF} = \cos(\angle V - \angle I) = \frac{P}{S}

Where

$angle V$ Voltage phasor angle
$angle I$ Current phasor angle
$P$ Active power
$S$ Apparent power

Lagging vs. leading

How Detego Computes Power

Power quantities in Detego are computed using the sliding DFT phasor method:

At each time step, the voltage and current phasors are extracted using the one-cycle DFT with DC offset removal (see Phasor Extraction).
The RMS magnitudes $V_{\text{RMS}}$ and $I_{\text{RMS}}$ are taken from the phasor magnitudes.
The power factor angle $\theta = \angle V - \angle I$ is computed from the phasor angles.
Active, reactive, and apparent power are computed from the formulas above.

When the Active Power signal is set to instantaneous mode, the phasor step is bypassed and the output is the sample-by-sample product of the raw voltage and current waveforms.

Line-to-Line Voltage Derivation

Fundamental Identity

Line-to-line voltages are defined as the difference between two phase voltages:

Line-to-line definitions

V_{AB} = V_A - V_B, \quad V_{BC} = V_B - V_C, \quad V_{CA} = V_C - V_A

Adding all three: V_AB + V_BC + V_CA = 0. This identity always holds regardless of system balance.

Because the three line-to-line voltages always sum to zero, only two independent measurements are needed. If one is missing, Detego derives it from the other two.

Deriving Phase-to-Neutral Voltages

Given at least two line-to-line phasors, the phase-to-neutral phasors are obtained by solving the system of equations above. The result is:

Phase-to-neutral from line-to-line

V_{AN} = \frac{V_{AB} - V_{CA}}{3}, \quad V_{BN} = \frac{V_{BC} - V_{AB}}{3}, \quad V_{CN} = \frac{V_{CA} - V_{BC}}{3}

Where

$V_{AN}, V_{BN}, V_{CN}$ Derived phase-to-neutral voltage phasors
$V_{AB}, V_{BC}, V_{CA}$ Measured (or derived) line-to-line voltage phasors

Why zero-sequence is lost

PreviousSequence Components NextImpedance Analysis