Slack Bus and External Grid Sources

Why the load flow needs a slack

Consider a network of $n$ buses with nodal admittance matrix $Y_\mathrm{bus}$. The bus 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 problem asks for the complex voltages $V$ such that the computed injections match the specified ones, $S_i(V) = S_{\mathrm{spec},i}$. In rectangular coordinates ($V_i = V_{r,i} + j V_{i,i}$) that is $2n$ real equations in $2n$ real unknowns — formally square, yet not solvable as posed, for two structural reasons.

1. The losses are unknown before the solution. Summing all bus balances gives an exact conservation identity: the injections add up to what the network elements absorb,

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

where $S_\mathrm{loss}$ collects the series losses and the shunt/charging consumption of every branch and shunt. If the injection were specified at every bus, a solution could only exist if the specified values happened to satisfy $\sum_i S_{\mathrm{spec},i} = S_\mathrm{loss}(V^\ast)$ at the (yet unknown) solution — a measure-zero coincidence. At least one bus must therefore leave its injection free, to absorb the a-priori unknown loss-plus-imbalance term.

2. The equations fix no angle reference. All residual quantities used at PQ and PV buses are invariant under a uniform rotation of every voltage 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 $\delta = 0$ shows that the rotation generator $jV$ lies in the kernel of the full (unreduced) Jacobian at every point:

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

A PQ/PV-only system is thus structurally singular: if a solution exists at all, it comes as a one-parameter family of rotated copies, and Newton's method has no isolated root to converge to. One voltage angle must be fixed to select a representative.

The slack bus repairs both defects at once. One bus — the slack, also called the reference (REF) bus — has its complex voltage fixed entirely (magnitude and angle), and its two balance equations are dropped from the system. Fixing the angle removes the rotation kernel (an admissible variation must leave the slack entries untouched, which $jV$ does not); dropping the balance equations frees the slack injection to absorb losses and imbalance. The classical bus-type split is:

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

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

Sparlectra's rectangular solver implements exactly this reduction: the state vector is $x = [\,V_r(\text{non-slack});\; V_i(\text{non-slack})\,] \in \mathbb{R}^{2(n-1)}$, and the slack voltage is set once and never solved — it enters the equations only as data through the $Y_\mathrm{bus}$ coupling terms of its neighbors (see the Solver Guide).

What the ideal slack idealizes away

Because its voltage is held constant no matter what current it supplies, the slack bus is — read as a physical device — 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. The idealization produces three distinct artifacts:

1. Infinite voltage stiffness. The bus voltage shows no reaction to loading; the voltage profile in the electrical neighborhood of the reference bus comes out systematically too good.
2. Concentrated balance. The entire loss-plus-imbalance term lands on one machine, although in a real interconnection primary control spreads it over many units. This artifact is addressed — orthogonally to this page — by the distributed active-power slack described in the Solver Guide, which distributes the balance role but keeps the voltage stiffness ideal.
3. No usable 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 therefore carries no finite short-circuit datum at all — see the short-circuit section below.

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 internal impedance derived from two declared quantities: the initial symmetrical short-circuit apparent power $S_{kQ}''$ (or equivalently the current $I_{kQ}''$) at the connection point Q, and the resistance-to- reactance 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 considered case. The impedance splits by the declared ratio:

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

When no exact ratio is known, IEC 60909-0 permits assuming $R_Q/X_Q = 0.1$ (equivalently $X_Q = 0.995\, Z_Q$) for high-voltage feeders; Sparlectra's short-circuit engine applies the same substitution and flags the affected result rows rather than substituting silently.

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 component of the 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 — precisely the behavior the ideal slack suppresses. The ratio of the feeder's short-circuit power to the connected load, $\mathrm{SCR} = S_{kQ}'' / S_\mathrm{load}$, is the usual grid-strength measure.

Voltage factor in the load flow. The factor $c$ is a short-circuit safety concept (it absorbs the difference between nominal and actual pre-fault voltage plus a margin). When the feeder impedance is used as an operating-point model inside the load flow, $c = 1$ is the appropriate choice; $c_\mathrm{max}$/$c_\mathrm{min}$ belong exclusively to the short-circuit cases.

Taking the source into the load-flow equations

Variant 0 — ideal representation (the slack as implemented)

The external grid is represented by making its connection bus the slack: $V_t \equiv U_\mathrm{ref}$, internal impedance neglected. This is today's behavior, and it remains the correct default whenever the feeder is strong relative to the studied network. The short-circuit data ($S_k''$, $R/X$) is then carried alongside the network but changes no load-flow result.

Variant A — augmented equations (explicit source current)

The direct way to put the source into the equations keeps the connection bus $t$ as an ordinary bus and introduces the source current $I_s \in \mathbb{C}$ injected at $t$ as a new unknown, coupled by the source's terminal equation:

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

Two things change in the system:

• The bus-$t$ residual gains a source term. With the extra injection, the power balance at $t$ reads

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

  so the residual now depends on the new unknown through the bilinear term $V_t \overline{I_s}$.

• Two new real equations and two new real unknowns. Splitting the terminal equation into real and imaginary parts:

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

No bus is eliminated in this formulation: the system has $2n + 2$ real unknowns ($2n$ voltage components plus $I_{s,r}, I_{s,i}$) and $2n + 2$ equations ($2n$ power balances plus the two constraint rows). It is square and — unlike the all-PQ system — regular, because the constraint contains the fixed phasor $U_\mathrm{ref}$ and is therefore not rotation invariant: it anchors the angle. Likewise the free source current absorbs the loss balance. Both classical slack roles are taken over by the source equations.

The price is structural: the Jacobian gains two new columns (the derivatives of the bus-$t$ power rows with respect to $I_{s,r}, I_{s,i}$, from $\partial (V_t \overline{I_s})/\partial I_s$) and two new rows (the constraint derivatives, $-I_2$ with respect to $V_{t,r}, V_{t,i}$ and the negative of the $2 \times 2$ impedance block with respect to the current). These are new residual and block types: state indexing, Jacobian assembly, damping/line-search/trust-region trial states, and the active-set bookkeeping would all need to learn about the extra state. That is a core solver modification.

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

The same physics fits into the unmodified formulation by giving the internal source node an explicit bus. Add one auxiliary bus $q$, fix its voltage $V_q = U_\mathrm{ref}$ — bus $q$ is the slack — and connect it to the terminal bus $t$ with a series branch of impedance $Z_Q$ (zero shunt admittance). The branch stamps into the admittance matrix in the standard way,

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

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

Equivalence to Variant A. The branch current is

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

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

The bookkeeping comparison:

(Variant B has $2((n{+}1)-1) = 2n$ unknowns: one bus more, but the new bus is the slack and drops out of the state.) Because Variant B changes nothing in the equation types, the entire existing machinery — sparse Y-bus assembly, analytic rectangular Jacobian, Q-limit active-set switching, island handling, damping/merit/trust-region step control — applies unchanged. This is the decisive argument for Variant B in an existing Newton implementation.

Per-unit value of the branch. With system base $S_\mathrm{base}$ and the voltage base equal to $U_{nQ}$ 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 as argued above. Example: $S_{kQ}'' = 3000\,\mathrm{MVA}$ on a $100\,\mathrm{MVA}$ base gives $z_\mathrm{pu} = 1/30 \approx 0.0333$, split by $R_Q/X_Q$. If the bus nominal voltage differs from the voltage base, the general form $z_\mathrm{pu} = c\, (U_{nQ}/U_\mathrm{base})^2\, S_\mathrm{base}/S_{kQ}''$ 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 therefore degenerates continuously to Variant 0 — the ideal slack is the exact limit of the non-ideal source, which makes a sharp regression test: a huge $S_k''$ must reproduce the ideal-slack solution to tight tolerance.

Numerically the limit is hostile: the stamped admittance $1/z_\mathrm{pu} \propto S_{kQ}''$ grows without bound, the rows and columns of buses $q$ and $t$ become dominated by the $\pm 1/z$ entries, and the Jacobian's conditioning degrades linearly in $S_k''$. An "almost ideal" feeder via an artificially enormous $S_k''$ is therefore the wrong tool in production — use the ideal representation (Variant 0) directly when ideal behavior is wanted.

What changes in the results

Compared with the ideal slack, the non-ideal source changes:

• Terminal voltage and angle become load-dependent. The angle reference now 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. The injection at the internal node and the 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 stiffness artifact of the ideal slack disappears; differences decay with electrical distance from the connection point.

What does not change: the network still has exactly one angle reference per island, the loss balance is still absorbed by exactly one free injection, and every other bus keeps its PQ/PV role.

Required additional functions

Putting the above into a package needs a small, well-defined API surface (implemented in Sparlectra as issue #299):

1. A constructor, addExternalGrid!. One call that (a) creates the load-flow side — in the ideal variant exactly what a manually added slack-type injection produces today, via the existing prosumer path, no new solver concept; and (b) converts the declared short-circuit data at add time into the feeder record the short-circuit engine already consumes: $I_{kQ}''$ from $S_{kQ}''$ and the bus nominal voltage, plus the $R/X$ ratios, for the maximum and (optionally) minimum case. Validation (positive $S_k''$, $S_{k,\min}'' \le S_{k,\max}''$, non-negative ratios) happens here, not in the engine.
2. A native short-circuit data container on the network. The engine is deliberately duck-typed over the record contract of the CGMES short-circuit harvest; a field-identical native container (NativeShortCircuitData) lets programmatically built networks — and MATPOWER imports, which carry no short-circuit attributes at all — feed the same engine without touching it.
3. A convenience overload runShortCircuit!(net; buses, case, c_factor) that reads the native container, so a natively built network runs the identical calculation path as a CGMES delivery.
4. Copy-path safety. Every code path that reconstructs or copies a network must carry the container along; a regression test copies a net with an external grid and runs the short circuit on the copy.
5. Optionally, the non-ideal load-flow variant as a flag on the constructor (internal_impedance = true): create the auxiliary bus and the series branch through the existing bus/branch APIs, mark the auxiliary bus as the slack, and tag both as internal so reports can distinguish them.

Equally important is what is not needed: with Variant B there are no new residual types, no new Jacobian blocks, no changes to step control or active-set logic — no solver function is added or modified. The entire "how does the equation system change" question is answered inside the network model.

Effect on the short-circuit calculation

The short-circuit method (IEC 60909-0, as implemented by runShortCircuit!) replaces the operating state by the equivalent voltage source at the fault location: all sources are removed and represented only by 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 $f$:

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

In Sparlectra the per-island short-circuit matrix is assembled from the series branch impedances only, and 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, evaluated with the $c$ of the considered case. Several feeders on one bus stack as parallel admittances — physically parallel infeeds.

Consistency of the feeder model. For a network consisting of a single feeder, a fault at the connection point sees $Z_{ff} = Z_Q$, and the declared current is recovered 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 of the feeder equation — the model returns precisely the datum the grid operator declared.

Maximum and minimum case. The maximum case combines the declared $S_{k,\max}''$ with $c_\mathrm{max}$ (equipment rating); the minimum case combines $S_{k,\min}''$ with $c_\mathrm{min}$ (protection sensitivity). Minimum data is optional: a feeder without declared minimum values is skipped in the minimum case and the affected rows are flagged — the existing safety-flag contract of the engine, not a new mechanism.

The two central consequences for the slack-versus-source question:

1. An ideal slack contributes nothing to a short circuit. Its idealization $Z = 0$ has no finite admittance stamp; taken literally it would short the equivalent source and produce an unbounded current. The property "this bus is the slack" is load-flow bookkeeping, not short-circuit data. An island whose only "source" is the load-flow slack therefore reports status = :no_source — which is exactly why an external-grid element must carry $S_k''$ and $R/X$: without them, no fault current can be attributed to the grid connection at all.
2. The load-flow representation does not influence the short-circuit result. IEC 60909-0 deliberately ignores the load-flow state — the pre-fault voltage is replaced by the $c$-factor convention — so it is irrelevant whether the load flow modeled the external grid ideally (Variant 0) or non-ideally (Variant B). In Variant B the auxiliary branch is even inert in the short-circuit network: the feeder's 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. Only two rules must hold: the feeder record stays anchored at the physical connection bus, and the fictitious internal bus is excluded from fault sweeps (a "fault" there has no physical meaning). The differing voltage factors — $c = 1$ in the load-flow branch, $c_\mathrm{max}/c_\mathrm{min}$ in the short-circuit stamp — never conflict, because the two representations are never combined in one calculation.

Peak current $i_p$, the $\kappa$ factor, the Z-bus column solve, and the flag semantics are described in 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).
• Standard load-flow formulations of the slack/PV/PQ split and the Newton power flow: Bergen & Vittal, Power Systems Analysis; Oeding & Oswald, Elektrische Kraftwerke und Netze.