GE SR745 Differential Slope: Calculating Test Currents

Problem: Predicting the Trip Point Instead of Trusting It

Secondary injection testing of a percentage differential element on a transformer relay is only a valid test if you can calculate the expected trip current independently of the relay. Without that number you are recording whatever the relay does and calling it a pass. The worked example below uses a real setting and application set:

Step 1 - Base Currents and CT Secondary Values

Compute rated line current on each winding, then the corresponding CT secondary current. Three-phase: I = MVA x 1000 / (sqrt(3) x kV_LL).

The two secondary currents are not equal, so the relay must apply an internal magnitude (tap) correction derived from the MVA and voltage settings you entered. Balanced through-current therefore corresponds to a secondary injection ratio of:

k_mag = I2_sec / I1_sec = 3.701 / 2.789 = 1.327
I_W2_inject = 1.327 x I_W1_inject   (magnitude only, before phase compensation)

Step 2 - Resolve the "x CT" Base Before You Do Anything Else

A setting of 0.2 x CT can be referenced to two different bases, and the numbers differ by nearly a factor of two on winding 1. Verify which one the manual specifies; do not assume.

If the relay normalizes both windings to a common per-unit base (the usual reason a relay asks for MVA and kV), the second row is the one that produces an internally consistent characteristic and the one your spreadsheet should carry. Prove it during commissioning with the pickup test in Step 5.

Step 3 - Phase Compensation for Dyn1

A Dyn1 transformer introduces a 30 degree displacement between winding 1 and winding 2 currents plus zero-sequence isolation. A numerical differential relay removes this internally by applying a delta-type combination to one winding's currents. Two consequences for testing:

1. Three-phase balanced injection: phase-shift the winding 2 source set by the compensation angle so the differential currents read near zero. Confirm this by reading the relay's metered differential per phase - target is a few percent of tap, not exactly zero, because of CT ratio mismatch and source error.
2. Single-phase injection into the delta-compensated winding: the compensation algebra (for example (Ia - Ib) / sqrt(3)) splits one injected phase current into two differential elements at 1 / sqrt(3) = 0.577 of the injected magnitude. Your calculated trip current on that winding must be divided by 0.577 (multiplied by 1.732) to get the amps the test set must output.

Step 4 - Dual-Slope Characteristic Equations

Define Id = operate (differential) current and Ir = restraint current, both in the relay's per-unit base. Below the breakpoint the element operates when:

Id > max( Pickup , S1 x Ir )        for Ir <= Breakpoint
with Pickup = 0.2, S1 = 0.25, Breakpoint = 1.0

Note that 0.2 / 0.25 = 0.8 pu: below 0.8 pu restraint the pickup setting governs, above it the 25 % slope governs.

Above the breakpoint there are two common implementations. They diverge badly at 98 % slope, so identify yours:

Form B implies a vertical step of 0.73 pu at the breakpoint - a region where increasing fault current can move the operating point from trip to no-trip. If the relay's own characteristic plot shows no step, you have Form A. Test one point at Ir = 2.0 pu: 1.23 pu versus 1.96 pu operate current resolves it in a single shot.

Step 5 - Restraint Definition and the Spreadsheet

The restraint quantity is the third variable manufacturers do not standardize. For a two-winding relay with compensated per-unit currents I1 and I2:

Operate current is the phasor sum of the compensated winding currents: Id = |I1 + I2|, where the polarity convention is such that through-current cancels. Build the sheet in this order:

1. Inputs: W1 test amps, W1 angle, W2 test amps, W2 angle, MVA, kV1, kV2, both CT ratios, pickup, S1, BP, S2.
2. Normalize: I1_pu = I_W1 / 2.789, I2_pu = I_W2 / 3.701 (transformer-rated base) or divide by 5.0 (CT-rated base). Keep the base selectable with a flag cell.
3. Compensate: apply 0.577 and the phase rotation to whichever winding the relay treats as delta.
4. Compute: Id, Ir, then the threshold from the Step 4 equation with the correct branch.
5. Verdict: IF(Id > Threshold, "TRIP", "NO TRIP"), plus a margin cell (Id / Threshold - 1) x 100 in percent.

Step 6 - Test Procedure and Verification

1. Metering check first. Inject a known single-phase current into each winding separately and compare the relay's metered secondary amps to the test set. Any error above roughly 2 % means the CT ratio or base settings are wrong, and every later result is invalid.
2. Pickup test. Winding 2 open, ramp winding 1. With one end open, Id = Ir = I1_pu, so the slope line (Id = 0.25 Ir) can never be reached and the element trips at the pickup setting. Predicted W1 secondary amps = pickup_pu x 2.789 A, divided by 0.577 if W1 is the compensated winding. Record the ratio of measured to predicted - that ratio tells you conclusively which per-unit base of Step 2 is in use.
3. Slope 1 test. Inject opposing currents at a restraint point of about 0.5 x BP and again just below BP. Hold I2 fixed, ramp I1 until trip, and compare to the sheet. Expect the measured slope to land near 25 % with a few percent tolerance.
4. Breakpoint / slope 2 test. Run the Ir = 2.0 pu point to select Form A or Form B, then add a third point at Ir = 3.0 pu to confirm the 98 % gradient.
5. Harmonic inhibit. Second-harmonic inrush restraint and fifth-harmonic overexcitation restraint will block an otherwise correct trip. Confirm the test set output is a clean sinusoid, or disable the inhibit functions for the slope test and restore them afterward. A "failed" relay that trips only with a pure sine source is a test-set harmonic problem, not a relay problem.
6. Timing. Log the operate time at 2x the calculated threshold on each slope segment and trend it across the fleet.

Interpreting Repeat Failures

Before condemning a population of relays, eliminate these in order, since each one produces symptoms identical to a defective differential element:

Document the resolved answers to the three ambiguities - per-unit base, restraint definition, above-breakpoint form - once per relay model and firmware revision, and reuse them across the fleet. That single page turns the spreadsheet from an estimate into an acceptance criterion.