Ideal Switches and Zero-Impedance Branches

Motivation

Real networks contain connections that are, by design, not really "branches" in the sense the Powerflow page assumes: breakers, disconnectors, bus-bar couplers, jumpers — elements meant to either tie two points electrically together with (idealized) zero impedance, or fully separate them, with no impedance value in between and no partial state.

Modeling one as an ordinary branch runs straight into the admittance formulation the rest of that page relies on. A branch's contribution to the bus admittance matrix is built from \(Y = 1/Z\); with \(Z = 0\) that's an infinite entry, not a large-but-finite one. And this isn't just a numerical inconvenience to work around — a closed zero-impedance connection isn't naturally a "branch" at all. Its physical meaning is an equality constraint, \(V_i = V_j\), not a current flowing in proportion to a voltage difference. That's a fundamentally different kind of equation from every other term in \(f(x)\), which is exactly why real tools solve this with dedicated mechanisms rather than a variant of the ordinary branch stamp.

Three approaches

1. Topological reduction (merge before you formulate)

The most direct fix is to never let the zero-impedance connection reach the admittance formulation in the first place. Build a graph of nodes connected by closed zero-impedance edges, find its connected components, and treat each component as a single node in the actual power-flow model — an open switch is simply not an edge, so it doesn't merge anything.

This adds no new equations, no new unknowns, and no numerical stiffness. The costs are that it happens before the solver ever runs, as a separate graph pass over the input model, and that the two original terminals lose their distinct identity — nothing downstream can distinguish "flow through the left side of the coupler" from "flow through the right side" once they've become one node. Switching state also can't change between solves without redoing the reduction and rebuilding the model.

2. Large-admittance regularization

A second approach keeps the zero-impedance connection as an ordinary branch, but assigns it a series admittance many orders of magnitude larger than anything else in the network (say 10⁶ S in a 10 kV network) and zero shunt admittance. The same nodal equations that already exist for every other branch then force near-equality of the two bus voltages as a natural numerical consequence — no new equation type, no graph pass, just an extreme parameter value. Open/closed status needs no special handling either: it's the same connection-status flag every other branch already has.

The trade-off is conditioning: introducing one admittance value five or six orders of magnitude larger than the rest of the matrix widens its dynamic range substantially, which is exactly the kind of thing a direct sparse solver's pivoting has to work harder to stay accurate through as a network grows. That's a real cost, paid deliberately in exchange for needing no separate topological pre-processing stage at all — the ideal connection is just a branch, as far as the rest of the solver is concerned.

3. Equality-constrained augmented system

The third approach takes the constraint interpretation from the Motivation section literally: instead of merging the two nodes away or approximating the constraint with an extreme admittance, add the equation \(V_i = V_j\) (and, in polar form, \(\theta_i = \theta_j\)) directly into the Newton-Raphson system as its own row. Since adding an equation without adding an unknown would leave the system over-determined, a pair of new "dummy" variables is introduced alongside it — one at each of the two buses, equal and opposite, representing the (otherwise unmodeled) power flowing through the ideal connection to make the constraint hold. The system stays exactly square: one new equation, one new pair of unknowns.

A complication this approach has to handle and the other two don't: a loop of zero-impedance branches produces a linearly dependent constraint set (the last edge's constraint is implied by the others), which would make the augmented system singular. The standard fix is to compute a minimum spanning tree over each connected group of zero-impedance branches and constrain only the tree edges, leaving the redundant non-tree edges as inactive constraints that can be reactivated if the tree has to be re-routed after a later switching operation.

Compared to the other two, this is real bookkeeping — the spanning-tree maintenance in particular has no analogue in either simpler approach — but it's also the only one of the three that keeps both original terminals numerically distinct after the fact, which matters if anything downstream needs to report or reason about flow through each side of the connection separately.

Where this fits in gridoxide today

gridoxide's solver core has no node-breaker layer: every Bus reaching network::build_ybus is already a fully-resolved electrical node. Topology resolution is a precondition of the Bus/Line/Transformer model, so the choice above is made once, at import.

The rule: identity, not impedance

Merge only when the element has no identity in the output model. Otherwise it stays a branch.

The deciding evidence is what the fixtures assert. All four upstream power-flow cases containing a link publish that link's own current, and vision-validation-network publishes its full p/q/s at both ends. Two state-estimation fixtures publish per-node injections either side of a link. Merging deletes the branch those numbers describe, so it is unavailable wherever they are asserted — branch satisfies all eight, merge satisfies two.

Note this is decided by what the element is, not by what is attached to it. An earlier version of this rule asked whether the endpoints carried appliances, and vision-validation-network's link 74 refutes it: the link joins two nodes with no appliances at all — apparently the safest possible merge — and power-grid-model still reports 1.964 MW flowing through it.

The threshold row exists because a connection can be electrically zero-impedance without being declared as one. ill-conditioned-by-line-meshed carries a line at 7.07e-9 p.u.; only a value-based test catches that, which is powsybl's framing rather than power-grid-model's.

The numbers, and why they were measured

Both constants were chosen by sweeping, not derived, because the two calculation types pull in opposite directions:

• Power flow wants a link stiff. The drop across it is \(\Delta V = I/y\), and the fixtures check node voltages and the link current at 1e-5 relative.
• State estimation wants it soft. \(G = H^{T}WH\) squares the admittance, so power-grid-model's 1e8 becomes 1e16 in the gain matrix.

The window is about one order of magnitude wide. That narrowness is the argument for treating these as regularization parameters with measured values rather than physical constants: a network far outside these fixtures' power scale may need them re-measured, and if no value serves both, approach 3 is the exit — it imposes \(V_i = V_j\) exactly, with no large number anywhere.

topology::ZERO_IMPEDANCE_THRESHOLD is 1e-7 p.u. on the same basis: measured across all 86 branches in the committed PGM fixtures and every CGMES one, it sits above the single pathological line at 7.07e-9 and below the smallest legitimate branch (1.0e-6 in PGM, 2.92e-6 in CGMES), so it disturbs nothing currently modelled.

Detection and treatment are separate numbers. The threshold only decides whether a branch is an ideal connection; a branch it catches is clamped to IDEAL_CONNECTION_Y, the same admittance a declared link gets. One number cannot do both jobs: it has to sit below every legitimate branch, yet clamping merely to that level leaves \(|Y|\) as high as 1e7 — inside the range measured as singular for state estimation, and 35x stiffer than a link. Separating them satisfies both, and says the right thing besides: a line that short is an undeclared link, so it should be treated as one.

powsybl's own threshold is 1e-8, but its default treatment is the equality-constrained formulation rather than a clamp, so it never has to reconcile the two roles in a single value.

Consequences of the merge, where it is used

CGMES inherits approach 1's limitations directly: switching state cannot change between solves without re-importing, and no per-side flow is reportable across a merged switch. There is also an empirical reason that side merges rather than stamping branches — it was tried, and the AC Newton-Raphson solve diverged on FullGrid with 20-odd such branches active at once, which is exactly the conditioning cost approach 2 carries.

Tool reference

powsybl uses both approaches 1 and 3, at different layers and for different reasons: closed switches that its bus/branch view merges away never reach the solver at all, while retained switches that survive into a node/breaker solve get the augmented-system treatment.