Symmetrical Components and the Fault Equations

The problem page says gridoxide solves in the phase domain and reports in both. This page is the mathematics of that: the transform, the sequence networks IEC 60909 is written in, the boundary conditions each fault type imposes, and how those conditions are stamped into a phase-domain matrix.

The transform

Any three phase quantities decompose into three balanced sets. With \(a = e^{j2\pi/3} = 1\angle 120°\),

\[ \begin{pmatrix} X_a \\ X_b \\ X_c \end

\underbrace{\begin{pmatrix} 1 & 1 & 1 \\ 1 & a^2 & a \\ 1 & a & a^2 \end{pmatrix}}_{A} \begin{pmatrix} X_0 \\ X_1 \\ X_2 \end{pmatrix}, \qquad \begin{pmatrix} X_0 \\ X_1 \\ X_2 \end{pmatrix} = \frac{1}{3} \begin{pmatrix} 1 & 1 & 1 \\ 1 & a & a^2 \\ 1 & a^2 & a \end{pmatrix} \begin{pmatrix} X_a \\ X_b \\ X_c \end{pmatrix} \]

Three useful identities fall straight out of \(1 + a + a^2 = 0\):

\[ X_0 = \tfrac{1}{3}\bigl(X_a + X_b + X_c\bigr), \qquad X_a = X_0 + X_1 + X_2, \qquad \sum_{\text{phases}} X = 3X_0 \]

The first is the whole reason zero sequence matters: it is the average of the three phases, so it is non-zero exactly when the three do not sum to zero — which for currents means current is returning through the ground. No ground path, no zero sequence. That single fact explains most of the qualitative behaviour on this page.

The three sequences are named for what they do to a machine: positive rotates the normal way, negative rotates backwards, zero does not rotate at all.

shortcircuit::fortescue is the \(A^{-1}\) above, and it is the exact inverse of the \(A\) that network::fortescue_to_phase builds the phase-domain Y-bus with, so a result projected into sequences and pushed back returns where it started.

Why the sequences decouple

For a network whose three phases are symmetric, \(A^{-1} Y_{abc} A\) is block-diagonal:

\[ Y_{012} = \begin{pmatrix} Y_0 & & \\ & Y_1 & \\ & & Y_2 \end{pmatrix} \]

Three independent networks, each a third the size, none talking to the others. That is what makes the sequence domain the natural place to write a standard in, and it is why IEC 60909 states its formulae in \(Z_0, Z_1, Z_2\).

For a passive network \(Z_2 = Z_1\): a line does not care which way the phasors rotate. \(Z_0\) is different — usually two to four times \(Z_1\) for an overhead line, because the return path is the earth rather than a conductor. gridoxide takes it from the input directly: a source's z01_ratio is the multiplier \(Z_0 / Z_1\), and lines carry r0/x0 of their own.

The couplings are what the fault introduces. A fault is the one place the three phases are not treated alike, so it appears in the sequence domain as an interconnection between the three otherwise independent networks — and the pattern of that interconnection is the fault type.

The four fault types as boundary conditions

Reduce each sequence network to its Thévenin equivalent at the faulted bus — \(Z_1, Z_2, Z_0\) looking in, with the pre-fault voltage \(E = c,U_n/\sqrt{3}\) driving the positive-sequence one — and every fault type becomes a way of wiring the three together.

Three things in that table are worth stating out loud.

The \(3\) in the line-to-ground numerator and the \(3Z_f\) are the same \(3\). All of \(I_a\) flows through the fault impedance, but \(I_a = 3I_0 = 3I_1\) because the three sequence currents are equal, so referred to one sequence network the impedance appears as \(3Z_f\).

Two-phase has no zero-sequence term at all, because there is no ground return. This is the sharpest available check on a short-circuit result: if a two-phase fault reports a zero-sequence component, something is wrong.

Where \(Z_0 = Z_1 = Z_2\), a line-to-ground fault draws exactly the same current as a three-phase one — \(3E/3Z = E/Z\). That degeneracy is easy to hit in a test fixture (PGM's z01_ratio defaults to 1) and easy to mistake for a bug. Where \(Z_0 > Z_1\), which is the normal case, the line-to-ground current is smaller; near a solidly grounded transformer, where \(Z_0\) can be less than \(Z_1\), it is larger, and that is why the line-to-ground case sometimes sizes the breaker.

The phase domain, and why gridoxide uses it

The sequence formulae above require reducing the network to a Thévenin equivalent at the fault, per sequence, per fault location. The phase domain instead keeps the whole \(3N \times 3N\) nodal system

\[ I_N = Y_{bus}, U_N, \qquad U_N = \bigl[,U_{1a}, U_{1b}, U_{1c},\ U_{2a}, \ldots,\bigr]^{T} \]

and writes the fault's boundary conditions into it directly, by rewriting rows and columns at the faulted bus. Two reasons this is the better choice here:

• Unbalanced faults fall out without special-casing. A fault type is a different set of rows to rewrite, not a different network to assemble.
• It is what power-grid-model does, so its fifteen short-circuit fixtures cross-validate gridoxide's numerics directly.

\(Y_{bus}\) is built by transforming each element's sequence admittances into phase coordinates with \(A\), which for a symmetric element gives the familiar circulant

\[ Y_{abc} = A \begin{pmatrix} y_0 & & \\ & y_1 & \\ & & y_2 \end{pmatrix} A^{-1} = \begin{pmatrix} y_s & y_m & y_m \\ y_m & y_s & y_m \\ y_m & y_m & y_s \end{pmatrix}, \quad y_s = \tfrac{y_0 + 2y_1}{3},\ \ y_m = \tfrac{y_0 - y_1}{3} \]

with \(y_2 = y_1\). The mutual term \(y_m\) vanishes exactly when \(y_0 = y_1\), which is the statement that a network with identical sequence impedances has no phase coupling at all.

Stamping the fault

An impedance fault and a bolted fault are not the same formula evaluated at a limit, and the code does not treat them that way.

An impedance fault adds admittance. For a line-to-ground fault on phase \(p\) at bus \(k\), one entry:

\[ Y[3k{+}p,\ 3k{+}p] \mathrel{+}= y_f \]

A two-phase fault between \(p_1\) and \(p_2\) ties them to each other rather than to ground, so it stamps the usual four-entry branch:

\[ Y[p_1,p_1] \mathrel{+}= y_f,\quad Y[p_2,p_2] \mathrel{+}= y_f, \quad Y[p_1,p_2] \mathrel{-}= y_f,\quad Y[p_2,p_1] \mathrel{-}= y_f \]

A bolted fault has \(y_f = \infty\), and letting an infinity into the matrix produces NaN, not a large current. So the faulted bus's equations are replaced: its voltage is known to be zero, so the unknown in that position becomes the injected current instead. In the matrix that is a column erased and a \(-1\) written on the diagonal, with the corresponding right-hand side set to zero.

The two-phase case needs different surgery again, because its constraint is \(U_{p_1} = U_{p_2}\) rather than either being zero: the two columns are folded together rather than erased, and one row becomes the difference equation. Two-phase-to-ground needs both operations at once — the phases are tied to each other and the pair is tied to ground — which is why it takes the column surgery even with a finite impedance.

The practical consequence for input data: r_f = x_f = 0 means bolted, and that is the default when a fault omits them. A fault that draws no current is not something the component can express, because it would not be a fault.

The cost of the phase domain

The phase domain needs the network to have a path to ground. An ungrounded network has a singular zero-sequence system in the phase domain — it does not in the sequence domain, where the zero-sequence network is simply absent — and comes back as ShortCircuitError::Singular.

In practice a real network is grounded somewhere; the case that bites is a small test model with a delta winding on both sides of a node. gridoxide follows power-grid-model in adding a tiny artificial susceptance to ground such a winding, purely so the matrix stays non-singular. It is a numerical device, not physics, and its exact value is arbitrary — a point that matters when comparing against reference data, as tests/data/pgm/short_circuit/README.md records.

Reading a result

Every result carries the sequence view alongside the phase quantities, and it is the fastest way to tell whether an answer is sane:

A second useful check: a delta winding blocks zero-sequence current. If a node sits behind one and still shows zero-sequence voltage, the transformer's zero-sequence branch is wrong. Both properties are asserted directly in python/tests/test_short_circuit.py.

The worked example carries all of this through on numbers small enough to check by hand.