Parameterized transformer tap decisions

Page status: exact parameterized tap-factor construction with solver-backed evidence; broader control domains remain open.

A fixed transformer primitive is a snapshot, not a decision model. If a tap is continuous or discrete, substituting its start value before optimization removes feasible operating points, the decision identity, and any objective or constraint that depends on the tap.

This chapter compiles an adjustable winding transfer into an exact parameterized factor. It does not select a tap and does not claim that one fixed admittance represents all positions.

Typed tap domain

For transformer $x$ and adjustable winding $k$, let $t_{xk}$ be a stable decision identity. The compact contract declares one of

\[\mathcal D_{xk}^{\mathrm{cont}} =[\underline t_{xk},\overline t_{xk}], \qquad \mathcal D_{xk}^{\mathrm{disc}} =\{t_{xk}^{(1)},\ldots,t_{xk}^{(m)}\}.\]

It also records the start value, but the start is initialization data—not a replacement for the domain. Values must be finite and positive; discrete positions must be unique and ordered. Every adjustable winding has a unique decision identifier such as tap/x1/winding/2.

The first executable parameterization is a scalar ganged tap applied to all labelled coils on one winding:

\[a_{xkc}(t_{xk}) =t_{xk}a^0_{xkc}, \qquad c\in\mathcal C_{xk}.\]

The base coefficient $a^0_{xkc}$ may itself be complex and need not be equal across coils. The scalar tap parameterization is a declared model family, not an inference from a field named tap. Independent per-phase controls require distinct decision identities and a richer domain.

A family of exact fixed-linear factors

Stack all retained decisions as $\mathbf t_x$. The power-dual transfer from the previous chapter becomes

\[\mathbf B_x(\mathbf t_x) =\mathbf T_x(\mathbf t_x)\mathbf A_x.\]

At every admissible decision $\mathbf t_x\in\mathcal D_x$, the terminal factor is

\[\mathbf Y_x(\mathbf t_x) =\mathbf B_x(\mathbf t_x)^{\mathrm H} \mathbf Y_x^{\mathrm{coil}} \mathbf B_x(\mathbf t_x) +\mathbf S_x^{\mathsf T}\mathbf Y_x^0\mathbf S_x +\mathbf Y_x^{\mathrm{ground}}.\]

The corresponding winding-side leakage current is retained as

\[\mathbf I_x^{\mathrm{w,leak}}(\mathbf t_x) =\mathbf T_x(\mathbf t_x)^{\mathrm H} \mathbf Y_x^{\mathrm{coil}} \mathbf T_x(\mathbf t_x)\mathbf A_x\mathbf U_x.\]

The compiler stores this parameterized relation and the decision domain. Its evaluator accepts a complete admissible decision assignment, substitutes that same value into the transfer, and invokes the certified fixed-linear compilation TR-XFMR-004. Missing, additional, continuous out-of-range, or unlisted discrete values return structured rejections.

Exactness for decision problems

Let $z$ contain network states and other controls. For constraints and an objective that use the declared terminal and recovered component quantities, the source feasible set has the form

\[\mathcal F_x =\left\{ (z,\mathbf t_x): \mathbf t_x\in\mathcal D_x, \quad g_x(z,\mathbf t_x)=0, \quad h_x(z,\mathbf t_x)\leq0 \right\}.\]

TR-XFMR-005 maps the decision by identity,

\[\widehat{\mathbf t}_x=\mathbf t_x,\]

and uses the same pointwise component relations. Hence it preserves the declared feasible set and any objective $f(z,\mathbf t_x)$ expressed through those interfaces. This is an exact compilation because the tap remains a variable; it is not a claim that the resulting optimization problem is convex or easy.

By contrast, evaluating the start value $\mathbf t_x^0$ produces only the slice

\[\mathcal F_x^{\mathrm{frozen}} =\{(z,\mathbf t_x)\in\mathcal F_x: \mathbf t_x=\mathbf t_x^0\}.\]

Unless the domain is already a singleton or the decision is provably irrelevant to the requested result, this is an inner restriction rather than an exact compilation.

Discrete decision witness

The illustrative contract layers a discrete winding-2 tap $\{0.95,1.00,1.05\}$ on the running WYE/WYE/DELTA transformer. It does not modify the canonical fixture or claim these are nameplate settings. At a fixed balanced boundary-voltage witness, winding 2 is at $0.97$ of its nominal voltage. The decision problem minimizes the maximum winding-2 leakage current subject to the original $2200\ \mathrm A$ winding limit.

Both the direct source evaluation and parameterized target retain feasible positions $\{1.00,1.05\}$ and select $1.05$. Freezing the factor at its $1.00$ start value gives objective $1903.716\ \mathrm A$ rather than $1232.656\ \mathrm A$, a gap of $671.060\ \mathrm A$. It therefore loses the optimal decision even though the frozen point itself is feasible.

Across all three positions, the recorded source/target terminal-admittance difference is zero to machine precision. The largest component complex-power residual is $4.88\times10^{-9}\ \mathrm{VA}$ at the SI-scaled witness.

Continuous and coordinate tests

The same compiler accepts a continuous interval. The executable test uses $[0.94,1.06]$, evaluates an interior value $1.013$, and rejects $1.061$. Reordering a winding's labelled base coefficients changes only their stored coordinates: alignment produces the same terminal factor at the same retained tap value.

Malformed domains are rejected before a start snapshot is constructed. The guards cover missing or duplicate decision identities, unsupported coefficient parameterizations, nonpositive or unsorted positions, invalid continuous bounds, and start values outside the declared domain.

Model boundary

The current rule covers scalar ganged magnitude taps whose fixed excitation and internal-grounding factors do not depend on tap. Phase-angle controls, independent phase taps, mechanically coupled devices, tap-dependent leakage, deadbands, switching costs, operation-count limits, and automatic control logic need additional typed relations. A solver adapter must encode the retained continuous or discrete factor without changing its domain or current recovery semantics.

The machine-readable result is certificate TR-XFMR-005. The transformer tap AC decision case now couples this factor to network voltage, neutral-KCL, power-balance, and recovered-current constraints rather than holding the boundary-voltage witness fixed.

For a concrete BMOPF source/target pair, companion executable contract transformer_tap_domain_preservation checks the narrower question of whether a mapped continuous tap interval was retained. It can prove that dropping tap_min and tap_max replaced an adjustable interval by a fixed singleton, but a passing interval check is not a substitute for this chapter's pointwise factor certificate or the AC-network decision comparison. The stable link is PSK-000005; implementation details, Finding codes, and the minimized fixture remain owned by BMOPFTools.