Restrained Differential Math Basis

The formulas in this calculator are based on two core SEL restrained differential definitions.

SEL-487: Operate is the phasor sum of winding currents, and Restraint is the scalar sum of winding currents.

SEL-387: Operate is the phasor sum, but Restraint is the scalar average rather than the scalar sum.

How This Calculator Applies Them

SEL-487: Restraint is modeled as \( \mathsf{|W1| + |W2|} \). Operate is modeled as \( \mathsf{|W1 - W2|} \).

SEL-387: Restraint is modeled as \( \mathsf{(|W1| + |W2|) / 2} \). Operate is the same as the 487: \( \mathsf{|W1 - W2|} \). If testing more than two windings simultaneously, the averaging divisor must be adjusted to match the number of windings included. This calculator only supports a two winding test.

Both relays use \( \mathsf{W1 - W2} \) for the operate term rather than a phasor sum because secondary injection test currents are set up with opposing winding polarities, making the scalar difference equivalent to the phasor sum in this context.

Since OP and RES are both magnitudes, they are always non-negative. Therefore absolute values are required and the graph is restricted to the first quadrant — negative values of either axis have no physical meaning in this context.

Per-Unit Conversion

Before applying operate/restraint equations, winding currents must be converted to per unit by dividing each winding current by its corresponding tap value. The OP/RES equations in this tool assume per-unit inputs.

To convert results back to secondary amps (or back to per-unit forms), the same equations can be rearranged algebraically.

Characteristic Line Equations

The O87P and 87U settings define horizontal lines across the graph, independent of restraint:

\( \mathsf{OP = O87P} \)

\( \mathsf{OP = 87U} \)

SEL-487 — both slope lines originate from the origin:

Slope 1: \( \mathsf{OP = \frac{SLP1}{100} \times RES} \)

Slope 2: \( \mathsf{OP = \frac{SLP2}{100} \times RES} \)

Slope 2 has the same form as Slope 1 but a steeper setting, and is only applied while CON is asserted (high security mode).

SEL-387 — Slope 1 originates from the origin, but Slope 2 does not. Instead, Slope 2 diverges from Slope 1 at the IRS1 breakpoint, ensuring continuity at the knee:

Slope 1 (RES ≤ IRS1): \( \mathsf{OP = \frac{SLP1}{100} \times RES} \)

Slope 2 (RES > IRS1): \( \mathsf{OP = \frac{SLP1}{100} \times IRS1 + \frac{SLP2}{100} \times (RES - IRS1)} \)

Slope 2 is a line with slope SLP2/100 passing through the point where Slope 1 meets IRS1. It does not pass through the origin. The overall 387 trip boundary follows Slope 1 below IRS1 and Slope 2 above it, floored by O87P throughout.

For both relays, the 87R trip boundary at any given restraint is the greater of the O87P floor and the applicable slope line.

Testing the SEL-487E Adaptive Slope 2

Unlike the SEL-387, the 487E applies Slope 2 only while in high security mode. High security mode is activated when the relay detects an external (through) fault, indicated by assertion of the CON word bit. While CON is asserted, the operate current must exceed the Slope 2 threshold — rather than Slope 1 — for an 87R trip to be declared.

Entering High Security Mode

Simulate a through fault by injecting currents equivalent to any point below the Slope 1 trip characteristic and greater than the DIRTR setting (1.2 pu RES default). When the relay identifies this as an external fault, it asserts CON and enters high security mode. CON will remain asserted for a minimum of 3 cycles and up to one second, or until the relay determines the fault is internal to the differential zone.

Changing currents too abruptly can cause the relay to drop out of high security mode and revert to the Slope 1 characteristic.

Finding the Slope 2 Trip Value

Two approaches are valid and both can produce a result within the relay's ±5% margin of error:

Binary search: Pulse operate values just below and just above the Slope 2 line, monitoring for trip/no-trip. Each pulse must complete within the 3 cycle minimum CON assertion window. Bracket inward until the trip boundary is identified.

Linear ramp: Start the through fault at a point just below the Slope 2 boundary — close enough that a slow ramp can reach it within one second. Ramp one winding's current toward Slope 2 at the slowest rate that crosses the line before CON times out. The point where the relay trips is the measured Slope 2 value.

A successful linear ramp test shows the following SER sequence:

1. CON asserts — relay enters high security mode
2. 87R asserts — trip on Slope 2
3. CON deasserts — high security mode timer expires

If CON deasserts before 87R asserts, the relay exited high security mode before the trip threshold was reached. Any subsequent trip occurred on Slope 1, not Slope 2.

Click Here To Calculate Suggested Start Point

Harmonic Restraint — Quick Overview

When harmonic restraint is enabled, the relay adds a portion of the harmonic content measured in the differential signal back into the restraint calculation. This raises the effective trip boundary, making the relay less likely to trip on transformer inrush or over-excitation.

Base Restrained Trip Equation

Without harmonic restraint, the Slope 1 trip condition is:

\( \mathsf{OP \gt \frac{SLP}{100} \times RES} \)

Harmonic-Augmented Equation

With harmonic restraint enabled, an additive term raises the trip threshold:

\( \mathsf{OP \gt \frac{SLP}{100} \times RES + harm\_add} \)

Computing the Harmonic Addition

The harmonic currents are derived from the operating current and the user-entered harmonic percentage settings:

\( \mathsf{I_{h2} = \frac{H_{2nd\%}}{100} \times I_{OP}} \)

\( \mathsf{I_{h4} = \frac{H_{4th\%}}{100} \times I_{OP}} \)

Each harmonic component is scaled by a gain factor derived from the relay’s PCT2 (2nd harmonic) and PCT4 (4th harmonic) settings:

\( \mathsf{k_2 = \frac{100}{PCT2}} \)

\( \mathsf{k_4 = \frac{100}{PCT4}} \)

The total harmonic addition is the sum of the scaled harmonic currents:

\( \mathsf{harm\_add = k_2 \times I_{h2} + k_4 \times I_{h4}} \)

Expanded Form

Substituting the harmonic current definitions:

\( \mathsf{harm\_add = \frac{100}{PCT2} \times \frac{H_{2nd\%}}{100} \times I_{OP} \;+\; \frac{100}{PCT4} \times \frac{H_{4th\%}}{100} \times I_{OP}} \)

Which simplifies to:

\( \mathsf{harm\_add = \frac{H_{2nd\%}}{PCT2} \times I_{OP} \;+\; \frac{H_{4th\%}}{PCT4} \times I_{OP}} \)

Key Observations

• Lower PCT2/PCT4 settings → larger gain factors → more restraint (harder to trip).
• The O87P floor is independent of harmonic restraint.
• For the purposes of this calculator, harmonic restraint is applied only to Slope 1 for trip-status evaluation.
• On the graph, dashed lines show the augmented slope characteristic when harmonic restraint is active.