Non-proportional three-phase four-wire parallel case

Page status: guarded AC decision case with self-checked and independently reproduced numerical evidence; broader global claims remain open.

This case reuses the canonical member-versus-aggregate distinction from the first scalar counterexample; its new content is the non-proportional four-wire current map and joint quadratic containment certificate.

The shared certificate geometry and decision-gap plate are introduced in the multiconductor parallel case; this chapter reuses that visual contract while changing the member data and solving the non-proportional four-wire decision problem.

The preceding phase–neutral example deliberately used proportional member matrices. This case removes that simplification while retaining the book's general baseline: ordered $(a,b,c,n)$ conductors, full mutual coupling, an explicit neutral return, unbalanced constant-power loading, individual member limits, and a decision objective.

Reciprocal non-proportional members

Two series-only members $\ell_1ij$ and $\ell_2ij$ share buses $i$ and $j$. Their full impedance matrices are complex symmetric and have positive resistive and reactive diagonals. The second matrix is not a scalar multiple of the first: after fitting the best complex scalar $\rho$, the infinity-norm residual is

\[\|\mathbf Y_{\ell_2}-\rho\mathbf Y_{\ell_1}\|_\infty=0.3663, \qquad \frac{\|\mathbf Y_{\ell_2}-\rho\mathbf Y_{\ell_1}\|_\infty} {\|\mathbf Y_{\ell_2}\|_\infty}=0.1219.\]

Both members follow

\[\mathbf I_{\ell i j}=\mathbf Y_\ell(\mathbf U_i-\mathbf U_j),\]

with component limits $|I_{\ell i j,c}|\le0.72$ p.u. at both ends. A balanced four-wire slack supplies an unbalanced wye load with phase directions

\[(s_a,s_b,s_c)= (0.70+0.14\mathrm j,\ 0.55+0.12\mathrm j,\ 0.42+0.09\mathrm j),\]

all multiplied by the served-load decision $\alpha$. Each phase-to-neutral voltage magnitude lies in $[0.88,1.05]$ p.u.; neutral KCL is explicit.

Joint componentwise redundancy certificate

Because $\mathbf Y_{\ell_1}$ is nonsingular, the common voltage drop can be eliminated between the member laws:

\[\mathbf I_{\ell_2 i j}=\mathbf K\mathbf I_{\ell_1 i j},\qquad \mathbf K=\mathbf Y_{\ell_2}\mathbf Y_{\ell_1}^{-1}.\]

For retained component discs $|I_{\ell_1 i j,k}|\le I^{\max}_{\ell_1,k}$, the exact worst-case magnitude of candidate component $c$ is

\[\max |I_{\ell_2 i j,c}|= \sum_k |K_{ck}|I^{\max}_{\ell_1,k}.\]

The equality is constructive: the independent complex currents can choose phases that align every term in row $c$. Nonsingularity ensures that every retained current vector corresponds to a voltage drop. Thus the row-norm test is necessary and sufficient for each candidate component to be implied jointly by all retained component limits.

For $(a,b,c,n)$, the certified worst cases are respectively $(0.1773,0.1710,0.1647,0.1636)$ p.u., all well below $0.72$ p.u. Since a series-only reverse-end current changes only sign, the same proof covers $\ell ji$. The target can remove all four $\ell_2$ component constraints while retaining its line law, identity, recovered currents, and possible use by other observations.

This extends the one-constraint PSD result in TR-PAR-005. It does not yet cover singular retained maps, shunt currents, limits implied jointly by several different retained members, or topology- and decision-dependent parameters. The four-wire nominal-pi case next adds distinct from/to shunt currents and certifies the full stacked terminal map.

Decision results

The exact-pruned target removes four constraints and agrees with the source to $1.7\times10^{-14}$ in objective value. The naive target has the same model size but serves 60% more of the load direction by violating the binding $\ell_1$ phase-$a$ constraint. Again, size does not determine fidelity. The certified worst-case currents are absolute p.u. magnitudes; the loading columns are current divided by the 0.72 p.u. member rating.

The unbalanced solution has neutral voltage $0.02796$ p.u.; this is not a balanced transmission case with a cosmetic fourth coordinate. All four conductors participate in the coupled member recovery and redundancy proof.

Independent checks

Three checks use different seams:

1. JuMP and Ipopt solve the source, lifted, pruned, and naive nonlinear models.
2. A LinearAlgebra-only finite-difference Newton continuation and bisection reproduces the source boundary at $1.1274329171$, within $1.4\times10^{-8}$ of Ipopt, with power-flow residual below $5\times10^{-16}$.
3. BMOPFTools' public line_yprim reconstructs each ordered four-wire primitive from the stored impedance entries and matches the direct admittance blocks to $10^{-12}$.

Together these establish claim TR-PAR-006; they do not constitute a global optimality proof for an arbitrary nonconvex AC OPF.

Run:

julia --project=experiments experiments/run_four_wire_parallel_ac.jl
julia --project=experiments experiments/test/four_wire_parallel_ac.jl

The generated certificate is experiments/generated/four-wire-parallel-ac-certificate.json.