Short-Circuit Analysis

What short-circuit analysis is

A power-flow solution describes the normal operating state of a network. Short-circuit analysis answers a different question: what current flows when the network faults — and therefore what the switchgear must be able to interrupt, what busbars and conductors must withstand mechanically and thermally, and whether protection relays will detect the fault and trip selectively. Both extremes matter: the maximum fault current sizes the equipment; the minimum fault current is the one a protection scheme must still be able to see.

How it is commonly done: IEC 60909

The industry-standard method is IEC 60909. It is deliberately not a time-domain simulation: it is a quasi-static calculation with defined safety margins, so results are reproducible and conservative without detailed dynamic machine models.

Equivalent voltage source at the fault location. The entire network is reduced to the short-circuit impedance seen from the faulted bus, driven by a single equivalent source. All other sources are removed; loads and line charging are neglected. The initial symmetrical short-circuit current at nominal voltage Un follows from that impedance Zk:

\[I_k'' = \frac{c \, U_n}{\sqrt{3}\,\lvert Z_k \rvert}\]

The voltage factor c absorbs the difference between nominal and actual pre-fault voltage plus a safety margin. It is tabulated by voltage level and by objective: c_max (for example 1.10 in HV networks) for maximum currents, c_min (0.95 or 1.00) for minimum currents.

Derived quantities. Everything else follows from the initial current: the peak current (first crest, mechanical stress)

\[i_p = \kappa \,\sqrt{2}\; I_k'', \qquad \kappa \approx 1.02 + 0.98\, e^{-3R/X}\]

— which is why the resistance-to-reactance ratio at the fault matters, not just the impedance magnitude — plus the breaking current at contact separation and the thermal equivalent current for the fault duration.

Where the fault current comes from. A synchronous machine feeds the fault through its subtransient reactance (x''_d); a network feeder is represented by an equivalent impedance derived from its declared maximum short-circuit power and its R/X ratio; asynchronous motors contribute briefly and matter mostly for the peak current and for faults close to the motor — the standard defines when they may be neglected. Neglecting motor contributions makes the maximum current an underestimate, which is the non-conservative direction; a careful implementation must say so when it does it.

Unbalanced faults. Single line-to-earth — the most common real fault — line-to-line and two-phase-to-earth faults are computed with symmetrical components: the fault current follows from combinations of the positive-, negative- and zero-sequence networks. The zero-sequence network is where transformers dominate: their vector group and star-point earthing decide whether a zero-sequence current path exists at all.

How the network is solved: Z-bus, and Takahashi

For one fault location the needed quantity is a single diagonal element of the bus impedance matrix: with the short-circuit admittance matrix Y_sc (series impedances only, source admittances on the diagonal, no loads or shunts), the fault impedance at bus i is

\[Z_{ii} = \left( Y_{sc}^{-1} \right)_{ii}\]

Nobody inverts the full matrix for this. The practical method is the Z-bus column solve: factorize Y_sc once (sparse LU), then obtain column i of the inverse with one forward/backward substitution against the i-th unit vector. That column contains Z_ii for the fault current and the off-diagonal entries needed for the voltage sag and the branch contributions around the fault.

What the Takahashi method adds. For a sweep over all buses the column approach costs one triangular solve per bus. Takahashi, Fagan and Chen (1973) showed that all inverse elements matching the sparsity pattern of the LU factors — in particular the entire diagonal — can be computed in a single backward pass over the factors, at a cost comparable to the factorization itself. This sparse inverse (also known as the Takahashi equations) is the classic power-system technique for exactly this job: every Z_ii of a large network in one pass, without ever forming the dense inverse. A full-network fault-level study therefore scales like one factorization, not like n solves.

The equations. Write the (scaled, permuted) factorization as $A = L\,D\,\tilde U$ with $L$ unit lower triangular, $D$ the pivot diagonal, and $\tilde U$ unit upper triangular, and let $Z = A^{-1}$. Multiplying $Z A = I$ and $A Z = I$ out and rearranging gives the two Erisman-Tinney identities

\[Z = D^{-1}L^{-1} + (I - \tilde U)\,Z, \qquad Z = \tilde U^{-1}D^{-1} + Z\,(I - L).\]

Because $D^{-1}L^{-1}$ is lower triangular and $I - \tilde U$ is strictly upper triangular, the first identity expresses every diagonal and upper entry of $Z$ through entries of $Z$ with LARGER row index; the second does the same for the lower entries through larger column indices:

\[Z_{jj} = \frac{1}{d_j} - \sum_{k > j} \tilde U_{jk} Z_{kj}, \qquad Z_{ij} = -\sum_{k > i} \tilde U_{ik} Z_{kj} \;\; (i < j), \qquad Z_{ij} = -\sum_{k > j} Z_{ik} L_{kj} \;\; (i > j).\]

Processing columns from $n$ down to $1$ therefore needs no entry that has not been computed yet. The key structural result is that, restricted to the sparsity pattern of $(L + U)^{\mathsf T}$ (the FILLED factors), these sums only ever reference entries inside that same pattern - the recurrence is self-contained, and the whole selected inverse costs one backward pass over $\mathrm{nnz}(L) + \mathrm{nnz}(U)$ entries, comparable to the factorization itself. The diagonal of the original matrix stays inside that pattern as long as the row and column pivot orders coincide, which is what UMFPACK's symmetric strategy produces on the structurally symmetric $Y_{sc}$; the implementation checks exactly this and counts every out-of-pattern reference defensively.

Sparlectra implements both approaches: per-bus faults use the column solve, and all-bus sweeps can opt into the Takahashi sparse inverse with runShortCircuit!(net; sweep_method = :takahashi). One selected-inverse pass per island then replaces the per-bus solves (measured 34x to 264x over the serial sweep between 2000 and 16000 buses, growing with size). The results agree with the default sweep_method = :solves to machine precision (about 1e-15 relative) but not bitwise, which is why the solve-based sweep remains the default; islands where the method does not apply (an unsymmetric UMFPACK pivot ordering, a pattern-closure violation) fall back to column solves automatically. Threaded sweeps (runtime.parallel.*) and the Takahashi pass compose: the selected inverse removes the per-bus solve cost, the threads cover whatever solves remain.

How Sparlectra does it

Available today: the data, and its completeness. Every CGMES import harvests the short-circuit source data into CGMESImportResult.shortcircuit — read, never silently altered, values in CGMES units:

Whether a given delivery could support a calculation is measurable: shortCircuitCoverage(result.shortcircuit) reports, per class, the record count and per attribute the fill rate; printShortCircuitCoverage renders it, and every CGMES run writes the same view into cgmes.log.

The calculation. runShortCircuit! computes the balanced three-phase initial symmetrical current:

result = importCGMES(path = ["grid.zip", "boundary.zip"])
sc = runShortCircuit!(result; case = :max)          # or :min; buses = :all default
sc = runShortCircuit!(result.net, result.shortcircuit; buses = ["Bus_1_220"], case = :min)
printShortCircuitResult(sc)

Per fault bus you get Ik'' (kA), Sk'' (MVA), and κ/i_p from the R/X ratio at the fault location (IEC 60909-0 method b, capped at 1.8 below 1 kV / 2.0 above). The positive-sequence short-circuit matrix uses series branch impedances only — loads, line charging and shunt compensators are dropped per the standard. Four source types feed the fault:

• synchronous machines: x''_d on machine base, converted to network base, with the §6.6.3 fictitious resistance so the R/X ratio at the fault stays meaningful;
• network feeders (ExternalNetworkInjection): equivalent impedance from the declared initial short-circuit current and R/X ratio — at the connection point the declared current is reproduced exactly;
• boundary equivalents with a declared positive-sequence impedance;
• asynchronous motors: locked-rotor impedance Z_M = (1/(I_LR/I_rM)) · U_rM²/S_rM per §6.7, with S_rM taken from the rated apparent power or formed from mechanical power, efficiency and power factor. Motors raise only the maximum current; they never enter the minimum case.

Voltage factors follow IEC 60909-0 Table 1 by voltage level (c_max 1.05/1.10, c_min 0.95/1.00); short_circuit.c_factor (or the c_factor keyword) is a scalar expert override for verification runs. The per-bus path is a Z-bus column solve — one sparse LU factorization per island and case, one triangular solve per fault bus; the Takahashi sparse inverse remains the designated optimization for all-bus sweeps on large networks.

Safety flags — read them. A short-circuit result is safety-relevant, so every substituted default and every skipped contribution is flagged on the affected result rows themselves (contains_defaulted_data plus a reason list), not only logged. The documented substitutions: a machine without x''_d gets 0.2 pu on machine base; a feeder without an R/X ratio gets R = 0.1·X; a motor without a locked-rotor R/X ratio gets the §6.7.2 guidance value (0.10/0.15 for MV motors, 0.42 for LV). A motor or feeder whose impedance cannot be formed at all is skipped and its whole island is flagged: the maximum current is then a lower bound — the non-conservative direction for equipment rating. Buses in islands without any source report status = :no_source with NaN currents instead of fabricated numbers.

Current limitations. The transformer impedance correction K_T (IEC 60909-0 §6.3.3) and the generator correction K_G (§6.6.3) are not applied — the harvested data does not carry the needed rated power factors; their absence biases Ik'' slightly high on transformer-/generator-near buses. The LV c_max variant 1.10 (for +10 % voltage-tolerance bands) and a configurable per-voltage-level c table are not available. Unbalanced faults (single line-to-earth, line-to-line) and the derived breaking/thermal quantities are not supported yet; they require the zero-sequence model whose source data (transformer vector groups and earthing) is already harvested. The reference tests derive their expected values analytically from the IEC formulas (full derivations in test/test_short_circuit.jl).

Web UI. The PowerFlow form's Short circuit button runs both cases (no power-flow solve involved) and writes short_circuit_max.csv / short_circuit_min.csv plus a result-page summary; it is only selectable for CGMES deliveries that carry short-circuit source data — see Web UI.

References

• IEC 60909-0, Short-circuit currents in three-phase a.c. systems — Part 0: Calculation of currents; IEC 60909-4 for worked verification examples.
• K. Takahashi, J. Fagan, M.-S. Chen: Formation of a sparse bus impedance matrix and its application to short circuit study, PICA 1973.
• ENTSO-E CGMES profile documentation (EquipmentShortCircuit profile).
• CGMES Import — how the source data arrives and is validated.