Four-wire nominal-pi parallel case

Page status: guarded nominal-$\pi$ decision case with executable certificates; singular shunted refusal and state/voltage-dependent recomputation are explicit, while global extensions remain open.

This is the nominal-$\pi$ extension of the canonical parallel-member failure in the first scalar counterexample. Only the additional shunt-current observations and both-end limit contract are new here.

The shared certificate geometry and decision-gap plate are introduced in the multiconductor parallel case; the nominal-$\pi$ chapter then adds the full two-end primitive and its refusal and recomputation guards.

The series-only four-wire case uses the same current magnitude at opposite member ends. A nominal-$\pi$ line removes that shortcut: shunt current depends on the local terminal voltage, and $\mathbf I_{\ell i j}$ is generally not $-\mathbf I_{\ell j i}$. Redundancy must therefore be certified on the full two-end primitive, as in the scalar setting of [1].

Full terminal-current map

For series admittance $\mathbf Y_\ell$ and end shunts $\mathbf Y^{\mathrm{sh}}_{\ell,i}$ and $\mathbf Y^{\mathrm{sh}}_{\ell,j}$, define

\[\begin{bmatrix} \mathbf I_{\ell i j}\\ \mathbf I_{\ell j i} \end{bmatrix} = \underbrace{ \begin{bmatrix} \mathbf Y_\ell+\mathbf Y^{\mathrm{sh}}_{\ell,i}&-\mathbf Y_\ell\\ -\mathbf Y_\ell&\mathbf Y_\ell+\mathbf Y^{\mathrm{sh}}_{\ell,j} \end{bmatrix}}_{\mathbf A_\ell} \begin{bmatrix} \mathbf U_i\\ \mathbf U_j \end{bmatrix}.\]

The experiment adds unequal diagonal charging blocks at the two ends of both reciprocal, non-proportional $(a,b,c,n)$ members from TR-PAR-006. The retained primitive $\mathbf A_{\ell_1}$ is nonsingular, with condition number $2922.9$. Hence

\[\begin{bmatrix} \mathbf I_{\ell_2 i j}\\ \mathbf I_{\ell_2 j i} \end{bmatrix} =\mathbf A_{\ell_2}\mathbf A_{\ell_1}^{-1} \begin{bmatrix} \mathbf I_{\ell_1 i j}\\ \mathbf I_{\ell_1 j i} \end{bmatrix}.\]

Applying the complex-polydisc row norm to this eight-dimensional recovery map gives an exact worst-case candidate magnitude no larger than $0.17793$ p.u. for any of the eight $ij$ or $ji$ components, below the $0.72$ p.u. rating. All member-2 terminal limits are therefore jointly implied by the full set of member-1 terminal limits.

The invertibility guard matters. The series-only full primitive is singular because a common endpoint-voltage shift produces no series current; that case uses the reduced voltage-drop coordinate of TR-PAR-006. Nominal-$\pi$ shunts make absolute terminal voltage observable, so silently applying the series recovery would omit charging current.

The certificate records the retained-map condition number, a normalized backward error for the recovery solve, and a relative margin for every deleted limit. It rejects a map that is too ill-conditioned or a limit whose margin is numerically ambiguous. The result is exact for the nominal matrices and centered complex discs; uncertainty in those matrices requires a separate robustness analysis.

For reproducibility, the implementation rejects a retained map when its 2-norm condition number exceeds $\kappa_{\max}=10^8$; it also rejects a candidate row when its relative margin is below $\varepsilon_{\mathrm{margin}}=10^{-8}$. These are numerical certification guards, not physical operating limits. The present fixture has $\kappa_2(\mathbf A_{\ell_1})=2922.9$, comfortably inside the declared conditioning bound.

Guarded extensions: singular, jointly retained, and state-dependent

The companion witness experiments/generated/guarded-parallel-reduction-witness.json makes three boundaries executable:

TR-PAR-JOINT-001 records the scope of the middle row: the support sum is an exact linear-disc implication when every retained limit is imposed together. The generated witness now uses three retained discs. It is not yet a full nonlinear AC result for several retained members.

Jointly retained constraints (TR-PAR-JOINT-001, TR-PAR-AC-JOINT-001)

A separate three-member AC probe crosses this linear certificate into the nonlinear network model. It sets $I_{\ell_3}=0.10I_{\ell_1}+0.10I_{\ell_2}$, assigns member 3 a $0.15$ p.u. component limit, and obtains a joint support bound of $0.144$ p.u. The source and exact-pruned formulations both solve at served fraction $1.2401762$ (gap below $7\times10^{-14}$). This is claim TR-PAR-AC-JOINT-001: a fixed-map, locally solved AC witness, not a global nonlinear optimality theorem.

The same certificate includes an independent finite-difference damped-Newton continuation and bisection check of the source boundary. It brackets the boundary within $6.0\times10^{-9}$ served-fraction units, with power-flow residual below $1.1\times10^{-15}$; its boundary differs from the Ipopt source solve by about $1.3\times10^{-8}$. This is an independent numerical reproduction of the declared branch, not a proof of global AC optimality.

The same three-member certificate now includes TR-PAR-STATE-001, a finite four-state admittance envelope. At the base, higher-admittance, lower-admittance, and phase-selective unbalanced states, the member maps, joint support certificate, and source and exact-pruned AC formulations are rebuilt. Pruning remains exact in all four local solves, while the optimal served value changes across states. The state rows also carry independent boundary checks. This is finite state-dependent evidence, not a global control-policy or nonlinear optimality guarantee. The companion three-member-state-envelope-independent-reproduction.json reconstructs all four state boundaries with a separate standard-library Newton/bisection implementation.

This does not certify a singular shunted primitive or arbitrary AC control policy. It records the guard and the correct next representation, rather than treating a pseudoinverse or a frozen nominal map as an exact rewrite.

Singular-map refusal (TR-PAR-SINGULAR-001)

The generated nominal-$\pi$ certificate now includes singular-map refusal probes. One constructs a rank-deficient neutral series map with zero endpoint shunts; a second retains a nonzero from-end neutral shunt while leaving the to-end neutral shunt absent. Both verify that the full two-end recovery matrix is singular after realification and that the redundancy evaluator refuses it. The fallback is an endpoint-voltage or factor-coordinate formulation; this is a refusal witness, not a pseudoinverse-based reduction.

For the series-only singular fixture, the certificate also demonstrates the valid reduced-coordinate fallback. Writing $\Delta\mathbf U=\mathbf U_i- \mathbf U_j$ gives $\mathbf I_{\ell_1}=\mathbf Y_{\ell_1}\Delta\mathbf U$ and $\mathbf I_{\ell_2}=\mathbf Y_{\ell_2}\Delta\mathbf U$; the declared neutral rows are zero and are retained as an explicit invariant. The reduced recovery map is therefore exact on the endpoint-voltage-drop coordinate even though the full two-end terminal map remains rank deficient. This is a scoped series-only result: it does not use a pseudoinverse and does not extend to singular shunted maps or a global nonlinear AC theorem. This scoped executable result is claim TR-PAR-SINGULAR-001.

State-conditioned maps (TR-PAR-STATE-001)

The same certificate includes scoped state-conditioned map probes: changing the endpoint shunts changes the nominal-π primitive, so the base-state map is rejected off-state and the shifted map is recomputed. A voltage-dependent shunt probe makes the stronger point that the map changes with terminal voltage, not merely with a separately declared state. These records provide the local guard needed for a control/state-dependent study; they do not provide a global robust AC bound or an optimizer-independent equivalence theorem.

The certificate now also solves the two declared shunt states with their own maps. The served-fraction boundary changes from $1.1286205497$ in the base state to $1.1285736287$ after the shunt shift. Re-running the exact-pruned formulation at the shifted state agrees with its shifted source formulation to $4.5\times10^{-14}$. Thus the probe demonstrates a state-conditioned full-AC decision calculation and exact pruning after recomputation; it does not justify reusing the base map or claim a global control policy theorem.

The generated evidence now evaluates a finite three-state envelope: the base state, the shifted state above, and a reverse shunt shift. Each state rebuilds its nominal-$\pi$ map, rechecks the joint limit certificate, and solves both the source and exact-pruned AC formulations. All three states are locally solved, with maximum source/pruned objective gap below $1.2\times10^{-13}$. This is finite declared-state evidence, not a bound over a continuous control or uncertainty set.

The finite envelope is summarized once here; the map and both local nonlinear solves are rebuilt for every row.

AC decision comparison

The load directions, explicit neutral KCL, phase-to-neutral voltage bounds, and served-load objective are retained from the preceding four-wire case. Receiving-bus power uses the negative $ji$ terminal current, including the receiving-end shunt.

The exact-pruned target deletes eight constraints and agrees with the source to $1.2\times10^{-13}$ in objective value. The same-size naive formulation serves substantially more load by violating the binding member-1 limit. The certified worst-case currents are absolute p.u. magnitudes; the loading columns are current divided by the corresponding 0.72 p.u. rating.

Independent checks and scope

BMOPFTools' line_yprim reconstructs both complete nominal-$\pi$ primitives from their series and from/to shunt entries and matches the direct matrices to $10^{-12}$. A separate finite-difference Newton continuation and bisection reproduces the source boundary at $1.1286205634$, within $1.4\times10^{-8}$ of Ipopt, with residual below $4\times10^{-15}$.

This establishes claim TR-PAR-007 for fixed nominal-$\pi$ members whose retained full primitive is nonsingular. The accompanying refusal and recomputation probes expose, but do not solve, singular shunted maps, voltage-dependent shunts, tap or switching states, or implication by constraints distributed across several retained members.

Run:

julia --project=experiments experiments/run_pi_four_wire_parallel_ac.jl
julia --project=experiments experiments/test/pi_four_wire_parallel_ac.jl

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