The Short-Circuit Problem

Motivation

Power flow and state estimation both describe a network that is working. Short-circuit analysis describes one that has just failed: a conductor has touched ground, or two phases have touched each other, and for the next few cycles an enormous current flows through whatever path the impedances allow.

The number that comes out — the initial symmetrical short-circuit current \(I_k''\) — is one of the most consequential in power engineering. A circuit breaker is chosen because it can interrupt it. A conductor is sized because it can survive it for the fraction of a second before protection clears. A protective relay is set because it can distinguish it from load. Get it wrong in one direction and equipment explodes; wrong in the other and faults go undetected.

What makes this not a power flow

Three things, and all three are deliberate simplifications that IEC 60909 makes so the answer does not depend on the operating point:

Loads and generation are ignored entirely. Not linearized, not approximated — dropped. A short-circuit current is a property of the network's impedances, and the standard isolates it from whatever the dispatch happened to be that afternoon. This is the assumption that makes a short-circuit study a property of the grid rather than of a moment. Shunt admittances are kept: a shunt is a passive element, not a dispatch decision.

Sources sit behind a fixed EMF, scaled by a voltage factor \(c\) rather than by their own setpoint — see below.

The problem is linear. With every load gone, what remains is a linear circuit: impedances, fixed voltage sources, and a fault. One sparse factorization, no initial guess, no iteration count, no convergence to fail — the same character as the constant-admittance linearization, and for the same reason. A short-circuit calculation either has a solvable network or it does not.

The voltage factor c

Rather than model the pre-fault voltage profile, IEC 60909 replaces every source EMF with \(c \cdot U_n / \sqrt{3}\), where \(c\) is a single tabulated factor:

The two columns are two different studies, not two guesses at one answer:

• \(c_{max}\) gives the largest credible fault current. This is what breaker ratings and thermal withstand are sized against.
• \(c_{min}\) gives the smallest. This is the protection-sensitivity study: if the relay can still see a fault at the end of the feeder under the least favourable conditions, it will always see it.

gridoxide exposes the choice as VoltageScaling::{Maximum, Minimum} and nothing else — see Deliberate omissions. The standard distinguishes a 6% and a 10% low-voltage tolerance for \(c_{min}\); like power-grid-model, gridoxide assumes 10%.

One detail worth stating because it is easy to get wrong: the factor replaces the source's u_ref rather than multiplying it. A source declared at 1.02 p.u. contributes \(c\), not \(1.02c\). That is precisely what isolates the result from the operating point, and it is invisible in every reference fixture because they all set u_ref = 1.0.

Formulation: the phase domain

Two formulations are in common use, and they are not approximations of each other — they are the same problem in different coordinates.

The sequence domain (0-1-2) is what IEC 60909 itself is written in, and what most textbooks teach. Build three decoupled networks — zero, positive, negative — reduce each to a Thévenin equivalent at the fault point, and connect them in the pattern the fault type dictates: positive alone for a three-phase fault, all three in series for single-phase-to-ground, and so on.

The phase domain (abc) keeps the full \(3N \times 3N\) nodal system and writes the fault's boundary conditions directly into it, rewriting the rows and columns at the faulted bus:

\[ I_N = Y_{bus} , U_N \]

gridoxide solves in the phase domain. Two reasons. First, unbalanced faults fall out without special-casing each one — a fault type is a different set of rows to rewrite, not a different network to assemble. Second, and decisively for this project, it is what power-grid-model does, so its fifteen short-circuit fixtures cross-validate gridoxide's numerics directly. That is the same standard every other numeric feature here is held to.

Results are then reported in both bases: phase quantities, and symmetrical components via the inverse Fortescue transform. The sequence view is where a fault type's signature is legible at a glance, which is most of why it is worth having — see Fault Types.

Both formulations are written out in Symmetrical Components and the Fault Equations, and A Fault Current, by Hand computes three of them on a one-node system where every step is a calculator away.

The cost of that choice

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.

This is inherited from the formulation, and power-grid-model documents the same limitation for itself. 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 a winding that has no zero-sequence path of its own, 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 in detail.

Islands

A bus with no path to any source cannot carry a fault current, so a fault declared there returns zero and the bus is reported de-energized and pinned at zero volts. The energized part of the network is solved on its own.

This is the same contract every other solver in gridoxide offers — see Multi-Island Power Flow — and it exists for the same reason: a sourceless component has no reference, and inventing one would be guessing.

Deliberate omissions

gridoxide implements the voltage factor \(c\) and nothing else of IEC 60909's wider parameterization. In particular it does not model:

• Study type (sub-transient / transient / steady-state) and the associated machine reactances. powsybl's short-circuit API models these; gridoxide does not read the machine data they need.
• Configurable initial voltage profiles — user-supplied voltage ranges, or seeding from a previously solved power flow.
• Peak, breaking and thermal-equivalent currents (\(i_p\), \(I_b\), \(I_{th}\)), which are derived from \(I_k''\) by further standard factors.

Each is a real capability elsewhere. None is implemented here because no fixture in this tree can validate it, and this project's rule is that a number it prints is a number something checked.

Using it

From the CLI:

gridoxide short-circuit network.json --scaling max

From Rust:

#![allow(unused)]
fn main() {
use gridoxide::pgm::shortcircuit::short_circuit_from_pgm;
use gridoxide::shortcircuit::{ShortCircuitOptions, VoltageScaling};

let opts = ShortCircuitOptions { scaling: VoltageScaling::Maximum };
let (net, report) = short_circuit_from_pgm(&input, 1e6, 50.0, opts)?;
for fault in &report.faults {
    println!("fault {}: {:.1} A on phase a", fault.id, fault.i_f[0]);
}
for branch in &report.branches {
    println!("branch {}: {:.1} A into its `to` end", branch.id, branch.i_to[0]);
}
}

From Python:

import gridoxide

result = gridoxide.short_circuit("network.json", scaling="max")
for fault in result.faults:
    print(fault.id, fault.i_f)
for branch in result.branches:
    print(branch.id, branch.i_from, branch.i_to)

The report covers five component kinds: nodes (voltages, and their symmetrical components), faults, sources, branches — lines, transformers and links alike, with per-phase currents at both terminals, each on its own node's base — and shunts. Branch currents are what a protection study grades against; a fault current alone says what the fault draws, not what any particular breaker sees. Fully open branches and inactive shunts have no entry, and a branch that no closed terminal connects to an energized node is reported de-energized, carrying zero.

The fault itself is declared in the input document, as power-grid-model's fault component: a node, a fault type, an optional phase, and an optional fault impedance. A fault with no r_f/x_f is a bolted fault — a dead short — not one that draws no current.