When the general model collapses

Page status: controlled positive-sequence specialization with factor- and network-level executable witnesses; global decision equivalence and external review remain open.

The general multiconductor, multi-terminal model is a baseline for preserving meaning. It is not a claim that every study needs all of that structure. This chapter gives a positive derivation of the familiar balanced positive-sequence bus–branch model and lists the assumptions that make the collapse admissible.

The balanced invariant subspace

Let a three-phase two-terminal factor use the phase order $(a,b,c)$ and the Fortescue matrix $\mathbf F$

This is the classical symmetrical-coordinate construction introduced by Fortescue [12]. The transform itself is always invertible; sequence decoupling is not. Decoupling requires the phase-domain operator, including connected series/shunt factors and grounding, to leave the sequence subspaces invariant.

\[\mathbf v_{abc}=\mathbf F\mathbf v_{012}, \qquad \mathbf F=\begin{bmatrix}1&1&1\\1&a^2&a\\1&a&a^2\end{bmatrix}, \qquad a=e^{\mathrm j2\pi/3}.\]

The positive-sequence subspace is

\[\mathcal V_+=\{\mathbf F[0,V_1,0]^{\mathsf T}:V_1\in\mathbb C\}.\]

An exactly balanced operating point has phase voltages and currents in this subspace, with the corresponding phase shifts. Balance is a restriction on the admissible state and injections; it is not created by drawing one edge per bus pair.

Factor-level assumptions

Consider a linear series or nominal-$\pi$ factor with phase-domain relation $\mathbf i=\mathbf Y\mathbf v$. A sufficient exact-collapse contract is:

1. compatible phase sets: every retained terminal has the same ordered three-phase set, with no unresolved neutral or earth port;
2. cyclic symmetry: each series and shunt matrix commutes with the cyclic phase permutation, equivalently it is circulant in the declared phase coordinates;
3. balanced boundary data: sources, injections, and measurements are restricted to $\mathcal V_+$;
4. sequence-compatible grounding: zero- and negative-sequence return paths are either absent from the study or represented by fixed factors whose constraints are not queried;
5. two-terminal closure: each device is already a two-terminal factor, or a multi-terminal device has an explicitly verified positive-sequence compilation;
6. decision symmetry: controls are common to all phases, and limits, costs, and admissible states are invariant under the same phase symmetry;
7. observation restriction: the study does not ask for phase-specific, neutral-to-ground, negative-sequence, zero-sequence, or internal winding quantities.

Under these assumptions, $\mathbf F^{-1}\mathbf Y\mathbf F$ is diagonal in sequence coordinates, and the positive-sequence block is invariant:

\[\mathbf Y_{012}=\mathbf F^{-1}\mathbf Y_{abc}\mathbf F =\operatorname{diag}(Y_0,Y_1,Y_2), \qquad I_1=Y_1V_1.\]

Transposition alone is therefore not an exact positive-sequence reduction. It can support a balanced or approximately cyclic model, but exact restricted closure still requires the operator, grounding, devices, limits, controls, and observations to respect the declared sequence symmetry.

The federated applicability guard PSK-000009 makes this boundary executable through BMOPFTools contract positive_sequence_collapse_applicability. The compact check verifies circulant series/shunt factors, phase order, and the scalar positive-sequence relation, while requiring explicit declarations for balanced boundary data, sequence-compatible grounding, two-terminal closure, phase-symmetric decisions, and positive-sequence observations. Its pass is a restricted relation check; it does not certify an unbalanced feeder or any phase-specific, neutral/earth, protection, internal-device, objective, or solver observation.

For a nominal-$\pi$ factor, the same statement applies to the series and shunt blocks separately. The positive-sequence network is therefore a derived two-terminal scalar complex network with one voltage and current per bus/arc, provided the factor library and decision constraints close under the restriction.

The generated witness experiments/generated/positive-sequence-collapse-witness.json diagonalizes one circulant impedance matrix to numerical precision and records a non-circulant perturbation that mixes sequences. The companion experiments/generated/balanced-transmission-witness.json assembles a three-bus, two-arc nominal-$\pi$ network with the same cyclic series and shunt factors. It solves the reduced phase-domain equations and the scalar positive-sequence equations independently, embeds the scalar voltages and branch currents, and checks their residuals. The result is a network-level positive example for the declared balanced observation family, not a global decision-equivalence theorem.

The generated comparison experiments/generated/balanced-transmission-independent-reproduction.json repeats the phase-domain and scalar solves with a separate standard-library complex Gaussian-elimination implementation. It matches the Julia voltages, residuals, and branch-current checks row-for-row. This is independent numerical reproduction of the declared fixture, not independent review of the modelling assumptions or a standards-aligned transmission validation.

The geometry is a projection aid: the circulant factor leaves the positive-sequence axis invariant, while the perturbed factor produces an off-axis residual. The residual $\rho_+$ is a coordinate diagnostic until propagated through constraints and decisions.

Network-level derivation

Let $C_+$ restrict a general model to the positive-sequence subspace and let $E_+$ embed a positive-sequence state into phase coordinates. Under the factor, boundary, and decision assumptions above,

\[\mathcal F_{+}=C_+(\mathcal F_{abc}), \qquad E_+(\mathcal F_{+})\subseteq\mathcal F_{abc},\]

and the declared observations agree on the embedded balanced states as

\[h_{abc}\circ E_+=\widehat h_+.\]

This is a restricted factorization on $E_+(\mathcal F_+)$. It is not the stronger global identity $h_{abc}=\widehat h_+\circ C_+$ on arbitrary phase-domain states.

Thus the positive-sequence model is exact for the restricted observation family $H_+$. It is not exact for the unrestricted phase-domain feasible set unless every feasible phase-domain point is balanced, which is a much stronger statement.

For an approximately balanced model, define the residual of a phase-domain state $v$ by

\[\rho_+(\mathbf v)=\left\|\mathbf v-E_+C_+\mathbf v\right\|.\]

This is a coordinate residual, not a decision-error bound. A voltage residual must be propagated through the factor equations and constraints before it can support an engineering approximation claim.

What collapses, and what does not

The running fixture intentionally fails several guards: it has four-wire lines, an explicit grounded neutral, nonuniform terminal sets, a phase permutation, unbalanced loads, full coupled matrices, and a three-winding transformer. It is therefore a test of the general model, not a balanced positive-sequence witness. The transmission specialization must be a separately declared fixture or a parameterized subcase with its own residual and checks.

Decision consequence

The positive-sequence collapse is exact for a restricted decision problem when the feasible-set inclusion, recovery embedding, and observation factorization above hold. If a contingency opens one phase, a relay observes zero sequence, a transformer has phase-dependent taps, or a conductor limit becomes active, the decision domain has left $\mathcal F_+$. The general port–factor model or an explicitly guarded intermediate model is then required.