Slack Bus and External Grid Sources

A power flow needs one bus with a fixed voltage; a real grid connection is a source behind a finite impedance. IEC 60909-0 gives that impedance from the declared short-circuit data, and an auxiliary-bus formulation carries it into the power flow and the short-circuit calculation without changing the solver (addExternalGrid!, see Implementation in Sparlectra).

Why the load flow needs a slack

For $n$ buses with nodal admittance matrix $Y_\mathrm{bus}$ the injections follow from the voltages:

\[I = Y_\mathrm{bus} V, \qquad S_i(V) = V_i \, \overline{(Y_\mathrm{bus} V)_i}, \qquad i = 1, \dots, n.\]

The load flow asks for the voltages with $S_i(V) = S_{\mathrm{spec},i}$. In rectangular coordinates ($V_k = V_{r,k} + j V_{i,k}$) that is $2n$ real equations in $2n$ real unknowns: square, yet not solvable as posed, for two reasons.

1. The losses are unknown before the solution. Summing the bus balances gives

\[\sum_{i=1}^{n} S_i(V) = S_\mathrm{loss}(V),\]

where $S_\mathrm{loss}$ collects the series losses and the shunt and charging consumption of every branch and shunt. With the injection specified at every bus a solution exists only if $\sum_i S_{\mathrm{spec},i} = S_\mathrm{loss}(V^\ast)$ at the yet unknown solution. At least one bus must leave its injection free.

2. The equations fix no angle reference. The residuals of PQ and PV buses are invariant under a uniform rotation of every phasor:

\[S_i\!\left(e^{j\delta} V\right) = S_i(V), \qquad \bigl\lvert e^{j\delta} V_i \bigr\rvert = \lvert V_i \rvert .\]

Differentiating at $\delta = 0$ puts the rotation generator $jV$ in the kernel of the full Jacobian at every point:

\[J(V)\,(jV) = 0 ,\]

with $jV$ read as the real state vector $[-V_i;\, V_r]$. A PQ/PV-only system is structurally singular: a solution comes as a one-parameter family of rotated copies, and Newton has no isolated root. One angle must be fixed.

The slack bus repairs both defects. Its complex voltage is fixed (magnitude and angle) and its two balance equations are dropped: the fixed angle removes the rotation kernel, the dropped equations free the slack injection for losses and imbalance.

The reduced system has $2(n-1)$ equations in $2(n-1)$ unknowns and is generically regular. The slack injection is an output:

\[S_\mathrm{ref} = V_\mathrm{ref}\, \overline{(Y_\mathrm{bus} V)_\mathrm{ref}} .\]

The rectangular solver implements this reduction: the state is $x = [\,V_r(\text{non-slack});\; V_i(\text{non-slack})\,] \in \mathbb{R}^{2(n-1)}$, and the slack voltage enters only as data through the $Y_\mathrm{bus}$ coupling of its neighbors (Solver Guide).

Reference priority

A case file names its slack. Which unit takes over when there is none, or when the named one is gone, comes up in three places: an AC island without a reference, the outage of the slack unit in N-1, and auto_slack on a case without a slack. Every unit carries a reference priority for that (ProSumer.referencePriority, the referencePriority keyword of addProsumer! and addExternalGrid!) with the semantics of the CGMES attribute: 0 no preference, 1 the strongest candidate, a larger number weaker. It orders candidates; it does not make a unit the slack by itself. The order is the same in all three places:

1. the smallest positive priority wins;
2. between equal priorities, and among units without one, reference_candidate_rank decides: a network injection before a machine, a unit that regulates the voltage of its own bus before one that does not, then the larger unit;
3. equal keys go to the smallest bus index.

An island without a reference considers its voltage-controlled units first and its other generating units only when it has none; a static var compensator is never the first choice, it carries no active power.

The island report ac_islands.csv names the chosen bus and the reason (reference priority or strongest unit), the N-1 result names the bus that took over. examples/others/exp_island_reference_priority.jl shows it on the shipped two-island case data/scf/two_islands_prio.scf.json.

What the ideal slack idealizes away

A bus whose voltage is constant whatever current it supplies is an ideal voltage source with zero internal impedance:

\[V(I) = U_\mathrm{ref} \quad \text{for every } I \qquad\Longleftrightarrow\qquad Z_\mathrm{th} = 0 .\]

No real grid connection behaves like this. Three artifacts follow:

1. Infinite voltage stiffness. The bus voltage shows no reaction to loading; the profile around the reference bus comes out too good.
2. Concentrated balance. The whole loss-plus-imbalance term lands on one machine, although primary control spreads it over many. The distributed active-power slack in the Solver Guide addresses the balance and keeps the voltage stiffness ideal.
3. No short-circuit contribution. The initial symmetrical short-circuit current at a fault with impedance $Z_k$ to the source is $I_k'' = c\,U_n / (\sqrt{3}\,\lvert Z_k \rvert)$; for $Z_k \to 0$ it diverges. An ideal slack carries no finite short-circuit datum (Effect on the short-circuit calculation).

The external grid as an IEC 60909-0 network feeder

IEC 60909-0 models the connection to a superordinate network, the network feeder (German: Netzeinspeisung), as an ideal source behind an impedance derived from two declared quantities: the initial symmetrical short-circuit apparent power $S_{kQ}''$ (or current $I_{kQ}''$) at the connection point Q, and the ratio $R_Q/X_Q$:

\[I_{kQ}'' = \frac{S_{kQ}''}{\sqrt{3}\; U_{nQ}}, \qquad Z_Q = \frac{c\, U_{nQ}}{\sqrt{3}\; I_{kQ}''} = \frac{c\, U_{nQ}^2}{S_{kQ}''},\]

with $U_{nQ}$ the nominal voltage at the connection point and $c$ the voltage factor of the case. The impedance splits by the ratio:

\[X_Q = \frac{Z_Q}{\sqrt{1 + (R_Q/X_Q)^2}}, \qquad R_Q = \left(\frac{R_Q}{X_Q}\right) X_Q .\]

Without a known ratio, IEC 60909-0 permits $R_Q/X_Q = 0.1$ ($X_Q = 0.995\, Z_Q$) for high-voltage feeders; the short-circuit engine substitutes the same value and flags the affected result rows.

The finite $Z_Q$ gives the source a finite stiffness. Its terminal characteristic in normal operation is

\[V_t = U_\mathrm{ref} - Z_Q\, I_t ,\]

and for a load $P + jQ$ drawn through the feeder the longitudinal voltage drop is approximately

\[\Delta V \approx \frac{R_Q\, P + X_Q\, Q}{\lvert V_t \rvert} .\]

A weak grid (small $S_{kQ}''$, large $Z_Q$) shows large voltage swings under load, which the ideal slack suppresses. The usual grid-strength measure is $\mathrm{SCR} = S_{kQ}'' / S_\mathrm{load}$.

Voltage factor in the load flow. The feeder impedance in the load flow uses $c = 1$; $c_\mathrm{max}$ and $c_\mathrm{min}$ are short-circuit safety margins (nominal versus pre-fault voltage) and belong to the short-circuit cases.

Taking the source into the load-flow equations

Variant 0: ideal representation (the slack as implemented)

The connection bus is the slack: $V_t \equiv U_\mathrm{ref}$, internal impedance neglected. This is the default and correct whenever the feeder is strong relative to the studied network. The short-circuit data ($S_k''$, $R/X$) is carried alongside and changes no load-flow result.

Variant A: augmented equations (explicit source current)

Keep the connection bus $t$ as an ordinary bus and add the source current $I_s \in \mathbb{C}$ injected at $t$ as a new unknown, coupled by the terminal equation

\[U_\mathrm{ref} - Z_Q\, I_s - V_t = 0 .\]

The bus-$t$ residual gains a source term, bilinear in the new unknown,

\[r_t \;=\; V_t\,\overline{(Y_\mathrm{bus} V)_t} \;-\; S_{\mathrm{spec},t} \;-\; V_t\, \overline{I_s} \;=\; 0 ,\]

and the terminal equation, split into real and imaginary parts, adds two real equations for the two new real unknowns:

\[\begin{bmatrix} U_{\mathrm{ref},r} \\ U_{\mathrm{ref},i} \end{bmatrix} - \begin{bmatrix} R_Q & -X_Q \\ X_Q & R_Q \end{bmatrix} \begin{bmatrix} I_{s,r} \\ I_{s,i} \end{bmatrix} - \begin{bmatrix} V_{t,r} \\ V_{t,i} \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} .\]

No bus is eliminated: $2n + 2$ unknowns and $2n + 2$ equations. Unlike the all-PQ system it is regular: the constraint contains the fixed phasor $U_\mathrm{ref}$ and anchors the angle, and the free source current absorbs the loss balance. The price: two new Jacobian columns (derivatives of the bus-$t$ rows with respect to $I_{s,r}, I_{s,i}$) and two new rows ($-I_2$ with respect to $V_{t,r}, V_{t,i}$, the negative $2 \times 2$ impedance block with respect to the current), new residual and block types for state indexing, Jacobian assembly, step control and active-set bookkeeping.

Variant B: auxiliary internal bus (equivalent, reuses the machinery)

The same physics fits the unmodified formulation when the internal source node gets its own bus. Add one auxiliary bus $q$, fix $V_q = U_\mathrm{ref}$ (bus $q$ is the slack) and connect it to the terminal bus $t$ with a series branch of impedance $Z_Q$ and zero shunt admittance. The branch stamps into the admittance matrix as usual,

\[Y_{qq} \leftarrow Y_{qq} + \frac{1}{Z_Q}, \qquad Y_{tt} \leftarrow Y_{tt} + \frac{1}{Z_Q}, \qquad Y_{qt} = Y_{tq} \leftarrow -\frac{1}{Z_Q},\]

and the terminal bus becomes an ordinary solved bus (PQ, or PV if something else regulates it).

Equivalence to Variant A. The branch current

\[I_s = \frac{V_q - V_t}{Z_Q} = \frac{U_\mathrm{ref} - V_t}{Z_Q} ,\]

is Variant A's terminal equation solved for $I_s$. Substituted into the bus-$t$ balance it reproduces Variant A's residual; the constraint rows become the nodal equations of bus $q$, which the slack reduction removes. Variant B is Variant A with the source current eliminated analytically: same solution, two fewer unknowns, no new equation types.

(Variant B has $2((n{+}1)-1) = 2n$ unknowns: the new bus is the slack and drops out of the state.) With unchanged equation types, Y-bus assembly, Jacobian, Q-limit switching, island handling and step control apply unchanged.

Per-unit value of the branch. With system base $S_\mathrm{base}$ and the voltage base $U_{nQ}$ at the connection point,

\[z_\mathrm{pu} = \frac{Z_Q}{Z_\mathrm{base}} = \frac{c\, U_{nQ}^2 / S_{kQ}''}{U_{nQ}^2 / S_\mathrm{base}} = \frac{c\, S_\mathrm{base}}{S_{kQ}''} ,\]

with $c = 1$ for the load-flow branch. Example: $S_{kQ}'' = 3000\,\mathrm{MVA}$ on a $100\,\mathrm{MVA}$ base gives $z_\mathrm{pu} = 1/30 \approx 0.0333$, split by $R_Q/X_Q$. The impedance base is the nominal voltage of the connection bus itself ($Z_\mathrm{base} = U_{nQ}^2 / S_\mathrm{base}$), so no $(U_{nQ}/U_\mathrm{base})^2$ factor applies.

The stiff limit

For $S_{kQ}'' \to \infty$ the branch impedance vanishes, $z_\mathrm{pu} = c\,S_\mathrm{base}/S_{kQ}'' \to 0$, and the terminal voltage converges to the internal one:

\[\lvert V_t - U_\mathrm{ref} \rvert = \lvert Z_Q \rvert \, \lvert I_s \rvert \;\longrightarrow\; 0 ,\]

since $I_s$ stays bounded (it converges to the injection current of the ideal slack). Variant B degenerates continuously to Variant 0 (a regression test: a huge $S_k''$ must reproduce the ideal-slack solution). Numerically the limit is hostile: the stamped admittance $1/z_\mathrm{pu} \propto S_{kQ}''$ dominates the rows and columns of buses $q$ and $t$, and the Jacobian's conditioning degrades linearly in $S_k''$. Do not emulate an ideal feeder with an enormous $S_k''$; use Variant 0.

What changes in the results

• The terminal voltage and angle become load-dependent: the angle reference sits on the internal bus $q$, so $V_t$ has a nonzero angle and a magnitude below (or above, for reverse flow) $U_\mathrm{ref}$.
• The source's power depends on the measuring point: injection at the internal node and power arriving at the terminal differ by the branch loss $\Delta S = (R_Q + jX_Q)\,\lvert I_s \rvert^2$.
• The neighborhood sees a realistic voltage profile; the differences decay with electrical distance from the connection point.

Unchanged: one angle reference per island, the loss balance absorbed by one free injection, every other bus keeps its PQ/PV role.

Implementation in Sparlectra

Effect on the short-circuit calculation

The IEC 60909-0 method (runShortCircuit!) replaces the operating state by the equivalent voltage source at the fault location: sources enter only through their internal impedances to ground, loads and line charging are dropped, and the network reduces to the driving-point impedance $Z_{ff}$ seen from the fault bus $f$:

\[I_k''(f) = \frac{c\, U_n(f)}{\sqrt{3}\; \lvert Z_{ff} \rvert} .\]

The per-island short-circuit matrix is assembled from the series branch impedances; every source contributes a shunt admittance on the diagonal of its connection bus. The feeder's stamp is

\[Y_{tt} \leftarrow Y_{tt} + \frac{1}{R_Q + jX_Q} ,\]

with $R_Q, X_Q$ from the equations above at the $c$ of the case. Several feeders on one bus stack as parallel admittances.

Consistency. For a network of a single feeder, a fault at the connection point sees $Z_{ff} = Z_Q$ and recovers the declared current exactly:

\[I_k'' = \frac{c\, U_n}{\sqrt{3}} \cdot \frac{S_{kQ}''}{c\, U_n^2} = \frac{S_{kQ}''}{\sqrt{3}\, U_n} = I_{kQ}'' .\]

The voltage factor cancels by construction.

Maximum and minimum case. The maximum case combines $S_{k,\max}''$ with $c_\mathrm{max}$ (equipment rating), the minimum case $S_{k,\min}''$ with $c_\mathrm{min}$ (protection sensitivity). Minimum data is optional: a feeder without it is skipped in the minimum case and the affected rows are flagged.

Two consequences:

1. An ideal slack contributes nothing to a short circuit. $Z = 0$ has no finite admittance stamp; taken literally it would short the equivalent source. "This bus is the slack" is load-flow bookkeeping, not short-circuit data: an island whose only source is the load-flow slack reports status = :no_source, which is why an external-grid element must carry $S_k''$ and $R/X$.
2. The load-flow representation does not influence the short-circuit result. IEC 60909-0 ignores the load-flow state (the pre-fault voltage is replaced by the $c$-factor convention), so Variant 0 and Variant B give the same short-circuit result. In Variant B the auxiliary branch is inert: the feeder admittance is stamped at the physical connection bus, the internal bus has no path to ground of its own, and a dead-end branch carries no fault current. Two rules hold: the feeder record stays anchored at the physical connection bus, and the internal bus is excluded from fault sweeps. The differing voltage factors ($c = 1$ in the load-flow branch, $c_\mathrm{max}/c_\mathrm{min}$ in the stamp) never meet in one calculation.

Peak current $i_p$, the $\kappa$ factor, the Z-bus column solve and the flag semantics: Short-Circuit Analysis.

Summary

References

• IEC 60909-0:2016, Short-circuit currents in three-phase a.c. systems, Part 0: Calculation of currents (network feeder model, voltage factor $c$, minimum/maximum cases).
• Bergen & Vittal, Power Systems Analysis; Oeding & Oswald, Elektrische Kraftwerke und Netze (slack/PV/PQ split, Newton power flow).