Short-Circuit Analysis

Sparlectra computes balanced three-phase short-circuit currents (Ik'' maximum and minimum, peak current i_p) via runShortCircuit! and the Web UI's Short circuit button. Unbalanced faults (single line-to-earth, line-to-line) are not supported.

The maximum fault current sizes switchgear, busbars and conductors; the minimum is the one a protection scheme must still see.

How it is commonly done: IEC 60909

IEC 60909 is a quasi-static calculation with defined safety margins, reproducible and conservative without dynamic machine models.

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

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

The voltage factor c absorbs the difference between nominal and pre-fault voltage plus a safety margin, tabulated by voltage level and 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. The peak current (first crest, mechanical stress) is

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

so the R/X ratio at the fault matters, not only the impedance magnitude (the generic factor; Sparlectra reports the method-b form of the result section). Sparlectra computes $I_k''$, $i_p$ and $\kappa$ (method b); breaking and thermal equivalent currents are not computed. None of the three depends on the system frequency.

Where the fault current comes from. A synchronous machine feeds the fault through its subtransient reactance (x''_d), a network feeder through an equivalent impedance from its declared short-circuit power and R/X ratio; asynchronous motors contribute briefly, mostly to the peak current. A unit behind a full-size converter (wind, solar, battery) feeds a current its control limits; IEC 60909-0:2016 models it as a current source. Neglecting motors or converters underestimates the maximum current, the non-conservative direction, so an implementation must say when it does it.

Unbalanced faults. Single line-to-earth (the most common real fault), line-to-line and two-phase-to-earth faults use symmetrical components; transformer vector group and star-point earthing decide whether a zero-sequence path exists at all.

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

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

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

The Z-bus column solve factorizes Y_sc once (sparse LU) and obtains column i of the inverse with one triangular solve against the i-th unit vector; the column holds Z_ii and the off-diagonal entries for voltage sag and branch contributions.

What the Takahashi method adds. Takahashi, Fagan and Chen (1973) showed that all inverse elements inside the sparsity pattern of the LU factors, the entire diagonal among them, follow from one backward pass over the factors at a cost comparable to the factorization: an all-bus sweep 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, $\tilde U$ unit upper triangular, and let $Z = A^{-1}$. Multiplying $Z A = I$ and $A Z = I$ out gives the Erisman-Tinney identities

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

\[D^{-1}L^{-1}\]

is lower triangular and $I - \tilde U$ strictly upper triangular, so the first identity gives every diagonal and upper entry of $Z$ from entries with larger row index, the second the lower entries from larger column indices:

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

Columns processed from $n$ down to $1$ need no entry not yet computed. Restricted to the sparsity pattern of $(L + U)^{\mathsf T}$ (the filled factors), the sums reference only entries inside that pattern, so the selected inverse costs one backward pass over $\mathrm{nnz}(L) + \mathrm{nnz}(U)$ entries. The diagonal stays inside the pattern while row and column pivot orders coincide, which UMFPACK's symmetric strategy produces on the structurally symmetric $Y_{sc}$; the implementation checks this and counts every out-of-pattern reference.

Per-bus faults use the column solve, all-bus sweeps the Takahashi sparse inverse, one pass per island. sweep_method = :auto (default) applies the pass to islands with at least short_circuit.takahashi_min_buses buses (default 50); :takahashi and :solves force one method for every island. Results agree with :solves to about 1e-15 relative, not bitwise; islands where the method does not apply (unsymmetric UMFPACK pivot ordering, pattern-closure violation) fall back to column solves. The service and Web UI paths honor short_circuit.sweep_method. Threaded sweeps (runtime.parallel.*) compose with the pass.

How Sparlectra does it

The data. Every CGMES import harvests the short-circuit source data into CGMESImportResult.shortcircuit and onto the network (net.sc_sources), so an export or a run of the network alone has it (read, never altered, CGMES units). The source classes keep the records of the objects the import mapped onto the network; a skipped one (terminal disconnected, out of service) feeds no fault current:

shortCircuitCoverage(result.shortcircuit) reports the record count per class and the fill rate per attribute; 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: Ik'' (kA), Sk'' (MVA), and κ/i_p from the R/X ratio of $Z_k$ at the fault location (IEC 60909-0 method b: the reported value is $\kappa = \min\bigl(1.15\,(1.02 + 0.98\, e^{-3R/X}),\ \kappa_{\max}\bigr)$ with $\kappa_{\max} = 1.8$ up to and including 1 kV and $2.0$ above). Five source types feed the fault:

• synchronous machines: x''_d on machine base, converted to network base, with the §6.6.3 fictitious resistance;
• network feeders (ExternalNetworkInjection): equivalent impedance from the declared initial short-circuit current and R/X ratio, reproduced exactly at the connection point;
• 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 from the rated apparent power or from mechanical power, efficiency and power factor; motors enter only the maximum case;
• converter-connected units (CGMES 3.0 PowerElectronicsConnection: wind, solar and battery parks, mapped with cgmes_import.map_power_electronics): current sources (IEC 60909-0:2016 clause 6.9), maximum case only;
• synchronous machines on a wind or solar unit (CGMES SynchronousMachine on a WindGeneratingUnit or SolarGeneratingUnit, IIDM generator with energySource WIND or SOLAR): the same current sources, see below.

A branch built from a CGMES EquivalentBranch enters with its series impedance as delivered, also with a negative r: a network equivalent is no equipment. A leg of a three-winding transformer in its star equivalent does the same, since only the sum of two legs is physical. The report names the number of such branches with negative r in the islands of the fault buses, one line for the equivalent branches and one for the three-winding legs.

A converter feeds $I_{sk} = \mathrm{maxIFault} \cdot S_r / (\sqrt{3}\, U_r)$ from its ratedS and ratedU. Its share is superposed on the equivalent voltage source. With $Z$ the impedance matrix of the island in which the voltage sources are short-circuited (the matrix of the first term) and $j$ over the converter buses, the current at fault bus $i$ is

\[I_k'' = \frac{c \, U_n}{\sqrt{3}\,\lvert Z_{ii} \rvert} + \frac{1}{\lvert Z_{ii} \rvert} \sum_j \lvert Z_{ij} \rvert \, I_{sk,j}\]

The magnitudes are added, which bounds the phasor sum from above for any angle of the converter currents. κ applies to the first term only, $i_p = \kappa \sqrt{2}\, I''_{vs} + \sqrt{2}\, I''_{conv}$; the reported zk_ohm, R/X and κ are those of $Z_{ii}$.

An island fed by converters alone has no voltage source and no $Z_{ii}$. At a bolted fault IEC 60909 neglects loads and shunt branches, so the whole current of every converter of the island flows into the fault. Every bus of such an island reports status = :ok with the arithmetic sum $I_k'' = \sum_j I_{sk,j}$ over the converters of the island, referred to the voltage of the fault bus, and $i_p = \sqrt{2}\, I_k''$ without κ; zk_ohm, R/X and κ are NaN. The sum assumes all currents in phase and is therefore an upper bound: the row is flagged with the reason converter-only island, arithmetic sum, no voltage source, but not marked as a lower bound (a skipped unit in the island still marks it). With no converter feeding (factor off, no maxIFault) the island stays :no_source.

maxIFault comes from CIM17. No CGMES 3.0 profile (EQ, SC, SSH, DY, OP) defines it, so a conforming delivery has none and its converters feed nothing and flag their island (next paragraph). The current source applies where a delivery carries the attribute beyond the profile, or where the source data is built in code.

short_circuit.converter_current_factor (default 0.0, off) fills the gap: a converter without maxIFault then feeds that multiple of its rated current, $I_{sk} = k \cdot S_r / (\sqrt{3}\, U_r)$, and the rows of its island carry the substitute as a reason instead of the lower-bound flag. A stated maxIFault always wins over the factor; a stated value that is not positive feeds nothing and flags the island with that value. The factor is a case key: the configuration file, the case configuration file and an API override can set it. The keyword of runShortCircuit! is converter_current_factor.

The fault current of a converter is a manufacturer figure, not a standard value, so the factor has no numeric default and has to be set for the case. As orientation, converters typically feed about 1.0 to 1.5 times their rated current. Left off, every converter unit feeds nothing and its island is reported as a lower bound.

A synchronous machine on a wind or solar unit is a converter unit as well: a CGMES SynchronousMachine whose GeneratingUnit is a WindGeneratingUnit or SolarGeneratingUnit, an IIDM generator with energySource WIND or SOLAR. It takes the converter path, not the machine path: $I_{sk} = k \cdot S_r / (\sqrt{3}\, U_r)$ with $k$ = short_circuit.converter_current_factor, $S_r$ from ratedS, else from the maximum active power of the unit (CGMES GeneratingUnit.maxOperatingP, IIDM maxP, taken at power factor 1). A rated power of 9999 MVA and above and a maximum of 9999 MW and above are placeholders, not sizes. With the factor off or without a size the machine feeds nothing and flags its island as a lower bound, like a converter without maxIFault or ratedS. Its record carries the unit class in the field converter_unit; an SCF file keeps the field in its sc_source row, so the rule holds after an SCF round trip. The CGMES export writes the unit with its wind or solar class, so the rule also holds after a CGMES round trip (CGMES Export).

Voltage factors follow IEC 60909-0 Table 1 by voltage level (c_max 1.05/1.10, c_min 0.95/1.00).

Closed switches and links. Buses joined by closed links (busbar couplers, retained CGMES switches) are contracted into one node, the same groups the power flow contracts. Every bus of a fed group reports the Ik'' and the island of that node, also a bus that hangs on the network through a switch alone. Machines, feeders and motors on any bus of the group feed the node, and their reasons appear on every row of the group. A branch between two buses of one group is shorted by the link and left out. A group without any branch stays :isolated.

Safety flags. Every substituted default and every skipped contribution is flagged on the affected result rows (contains_defaulted_data plus a reason list). A machine without x''_d gets 0.2 pu on its size (rated power, else maximum active power, else the network base with a reason of its own; a maximum of 9999 MW and above and a rated power of 9999 MVA and above are placeholders, not sizes, and the reason names a placeholder rated power; a machine with x''_d and a placeholder rated power takes the network 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 is skipped and its island flagged: the maximum current is then a lower bound, the non-conservative direction. Every row of such an island carries lower_bound = true (CSV column lower_bound, count sc_lower_bound_rows in the run metadata); a row on a default alone is flagged but has lower_bound = false. A converter without maxIFault (with short_circuit.converter_current_factor off) or without ratedS (and a converter-connected unit of a network whose source data has no record for it, and a machine on a wind or solar unit with the factor off or without a size) feeds nothing and flags its island the same way in the maximum case; a missing ratedU is replaced by the bus nominal voltage and flagged. The minimum case needs no flag, because a missing source lowers the current. Buses in islands without any source report status = :no_source with NaN currents; an island with feeding converters alone gets their sum (above). A machine without data counts as a source on the default in every format, so a run is succeeded only when every source carries its data (next section).

Source data and run status

One rule holds for the source data of every format:

A succeeded run with flagged rows gives the reason short_circuit_flagged_lower_bound when at least one row is a lower bound (a skipped source), else short_circuit_flagged_defaults (defaults or a substituted converter current only).

The rule lives in the library: runShortCircuit! stores the status and its reason on the result (status, status_reason; failed with short_circuit_data_missing also when no bus could be evaluated), and the service reports them unchanged. The three levels and where they show:

printShortCircuitResult starts every table with the head line, for example run status: warning (short_circuit_partial_defaults), 4 of 4 rows flagged, 2 rows with a lower bound.

The quantity is x''_d of a synchronous machine, the short-circuit current of a feeder, the reactance of an equivalent injection, the locked-rotor ratio of a motor. Converters do not enter this count; one without maxIFault flags the rows of its island instead (with the factor set, as a substitute).

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 lacks the rated power factors), so Ik'' is biased slightly high near transformers and generators. The LV c_max variant 1.10 (+10 % voltage-tolerance bands) and a per-voltage-level c table are not available. Unbalanced faults need the zero-sequence model, whose source data (transformer vector groups and earthing) is already harvested. The breaking current $I_b$, the thermal equivalent current and $\kappa$ by method c are not computed.

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.