Kundur's Example 13.2, Against Dynawo

Every other check in this module is self-supporting: a closed form, an identity, a reduction, an oracle. Those catch a great deal, but there is one thing they cannot catch — a formulation that is internally consistent and physically wrong. For that you need a second implementation.

The case, and why this one

Kundur's Power System Stability and Control Example 13.2, as Dynawo ships it: a sixth-order machine with no regulators, an infinite bus, two parallel lines, a fixed-ratio transformer, a bolted fault at the line junction and a line trip on clearing.

It was chosen because every element is something gridoxide models exactly — no tap changers, no limits, no saturation, no fifth-order machine. Nothing has to be excused before the comparison starts, which is what makes a disagreement mean something.

Dynawo's repository ships its own solver's reference curves alongside the case, so the comparison needs no Dynawo install — the same arrangement tests/data/ucte/ and tests/data/iidm/ already use for pypowsybl.

What the reference file settles before a single step

Two things, checked first because a disagreement in either would make the rest meaningless:

• theta at t = 0 is 1.224004 rad. gridoxide's own initialization, from an entirely independent derivation, puts δ₀ within 6e-5 of it. That is the rotor-frame convention agreeing with a second implementation — much stronger evidence than any self-consistency check.
• PmPu is 0.903 while PGenPu is 0.900 on the machine base. The difference is exactly the 0.003 pu stator copper loss, which is the statement that Dynawo's mechanical power, like gridoxide's, is the air-gap power.

The operating point is checkable by hand against the book, too: P = 0.9 pu and E_t = 1.0∠28.34°, and gridoxide's power flow reproduces the file's stated terminal angle of 0.49445 rad to 1e-4.

The agreement

The disagreement, and what it turned out to be

After clearing the two separate steadily. That was chased rather than absorbed into a tolerance, and what it is not was established first, each by experiment:

• not step size — gridoxide's answer is converged to 1e-4 rad between h = 4 ms and h = 0.0625 ms;
• not the nominal frequency — at 60 Hz the machine loses synchronism outright;
• not the tripped branch — tripping the other line, or neither, gives a wholly different trajectory;
• not the sign of the q-axis damper coupling that the model's derivation left open — flipping it moves the answer by under 1e-3 rad, which also means this case does not settle that question either way.

Localizing it needed separating the power-angle relation from the accumulated angle. At matched rotor angle in the first tenth of a second after clearing, gridoxide's terminal power is 0.6% below Dynawo's, and the voltage error steps at the clearing instant rather than growing smoothly. So it is algebraic, not numerical.

Reading Dynawo's own Modelica settled it:

udPu = (Ra + RTfo)·idPu − omegaPu·lambdaqPu
uqPu = (Ra + RTfo)·iqPu + omegaPu·lambdadPu
2·H·der(omegaPu) = cmPu·PNomTurb/SNom − cePu − DPu·(omegaPu − omegaRefPu)
PePu = cePu·omegaPu

Dynawo keeps ω on the speed-voltage terms and writes the swing equation in torque. gridoxide makes the classical RMS approximation in both places: ω ≈ 1 in the stator, and the swing equation in power. The two differ by exactly a factor of ω, so they agree at synchronous speed and part in proportion to the speed deviation — which is the observed pattern exactly: nil at t = 0, 0.04% early in the fault where ω − 1 = 0.0015, and 0.6% after clearing where ω − 1 = 0.009.

Neither form is wrong. Kundur §13.3 states the approximation explicitly; Sauer & Pai keep the terms.

Acting on the attribution

Both forms are now available — "speed_voltages": true in a document, --speed-voltages on the command line, speed_voltages=True in Python — and turning the full one on drives the discrepancy to nothing:

That is the proof rather than the argument. Measuring a discrepancy and attributing it by reading the reference's source is one thing; acting on the attribution and watching the number collapse by a factor of twenty-four is another. Had the ω ≈ 1 assumption not been the cause, switching it off would have moved the offset somewhere arbitrary rather than to zero.

What remains — 0.026% — is genuinely unexplained, and is bounded by a gate so it stays visible.

The approximation is still the default, for two reasons: it is what makes the phasor formulation coherent in the first place, and every closed-form gate in this crate, the equal-area criterion above all, is derived from the power form. A run that wants Dynawo's answer asks for Dynawo's model.

This is what an external gate is for: a difference invisible to every self-consistency check, because both formulations are internally perfect.

What is not yet done

A second external reference. ANDES is installed and its power flow reproduces the case — once its lines are given matching Vn1/Vn2, since it defaults them to 110 kV and silently rescales every impedance — but its own initialization then fails on the hand-built case, with residuals of 0.16 in the bus-3 angle equation, and the machine loses synchronism where both other tools keep it. That is a setup problem rather than a finding, and it is recorded as outstanding rather than reported as a result.