Transformer primitive admittance
Every transformer and regulator model in this specification has an exact nodal admittance block (a primitive admittance, or Yprim): the complex matrix $\textcolor{brown}{\mathbf{Y}_x}$ over the element's terminal nodes such that
\[\textcolor{blue}{\mathbf{I}_x} = \textcolor{brown}{\mathbf{Y}_x}\,\textcolor{blue}{\mathbf{U}_x},\]
where $\textcolor{blue}{\mathbf{U}_x}$ stacks the node voltages (each terminal to ground) and $\textcolor{blue}{\mathbf{I}_x}$ the currents into the element at those terminals (out of the bus). This is the object OpenDSS exposes as a PD-element Yprim (Export Y / DumpYprim) and that PowerModelsDistribution assembles internally, so it is the natural artifact for cross-tool validation: an informed reader can compare our block against their own implementation term-by-term, before any topology or Kirchhoff law enters.
This page gives $\textcolor{brown}{\mathbf{Y}_x}$ in symbolic form for all six transformer/regulator models — the Transformers subtypes (single_phase, center_tap, wye_delta, delta_wye, n_winding) and the Regulators (single_phase_autotransformer, open_delta_regulator). The OPF constraints stamp the same relations as nodal current injections, so the two paths cannot drift apart (a test pins each solved setpoint against $\textcolor{brown}{\mathbf{Y}_x}\textcolor{blue}{\mathbf{U}_x}$). Symbols are defined in Notation and the loss elements ($\textcolor{brown}{Z^{\text{fr}}_x}$, $\textcolor{brown}{Z^{\text{to}}_x}$, $\textcolor{brown}{Y_0}$, $\textcolor{brown}{y_n}$) in Transformers §4.
1. Conventions and construction
| Item | Choice |
|---|---|
| Units | SI — siemens, volts, amperes. No per-unit base. |
| Node | a (bus, terminal) pair; $\textcolor{blue}{\mathbf{U}_x}$ stacks them in the order listed for each subtype. |
| Current sign | $\textcolor{blue}{\mathbf{I}_x}$ = current into the element (out of the bus), matching OpenDSS Yprim. |
| Turns ratio | $\textcolor{red}{N}=\textcolor{red}{N_0}\,\textcolor{red}{t}$ with nameplate $\textcolor{red}{N_0}=\textcolor{red}{v^{\text{nom}}_{\text{fr}}}/\textcolor{red}{v^{\text{nom}}_{\text{to}}}$ and fixed tap multiplier $\textcolor{red}{t}$ (tap, default 1). For center_tap, $\textcolor{red}{v^{\text{nom}}_{\text{to}}}$ is the per-leg rating. |
| Winding leakage | $\textcolor{brown}{Z^{\text{fr}}_x}=\textcolor{red}{R_1}+\textcolor{brown}{j}\textcolor{red}{X_1}$ (r/x_series_from), $\textcolor{brown}{Z^{\text{to}}_x}=\textcolor{red}{R_2}+\textcolor{brown}{j}\textcolor{red}{X_2}$ (r/x_series_to). |
| No-load shunt | $\textcolor{brown}{Y_0}=\textcolor{red}{G_0}+\textcolor{brown}{j}\textcolor{red}{B_0}$ (g/b_no_load), across winding 2 (the to-side coil) — the OpenDSS placement. Inductive: $\textcolor{red}{B_0}<0$. |
| Neutral grounding | $\textcolor{brown}{y_n}=1/(\textcolor{red}{R_n}+\textcolor{brown}{j}\textcolor{red}{X_n})$ (r/x_neutral_from, r/x_neutral_to): an internal branch from a shared neutral terminal to earth — a diagonal add at that node (OpenDSS's Yprim gains exactly $\textcolor{brown}{y_n}$ on the neutral diagonal). |
The construction. Every block is built in the textbook primitive-admittance × connection-matrix form
\[\textcolor{brown}{\mathbf{Y}_x} = \textcolor{brown}{\mathbf{C}}^{\text{T}}\,\textcolor{brown}{\mathbf{y}_{\text{prim}}}\,\textcolor{brown}{\mathbf{C}} \;+\; \text{(winding-2 shunt)}\;+\;\text{(neutral diagonal }\textcolor{brown}{y_n}),\]
where $\textcolor{brown}{\mathbf{C}}$ maps node voltages to per-core winding voltages and $\textcolor{brown}{\mathbf{y}_{\text{prim}}}$ is the (block-)diagonal primitive of the winding pairs. Because $\textcolor{brown}{\mathbf{y}_{\text{prim}}}$ is symmetric, so is the result:
A transformer built from linear impedances and ideal cores is a reciprocal network, so $\textcolor{brown}{\mathbf{Y}_x}=\textcolor{brown}{\mathbf{Y}_x}^{\text{T}}$ (a plain transpose — not Hermitian; the network is lossy, so $\textcolor{brown}{\mathbf{Y}_x}\neq\textcolor{brown}{\mathbf{Y}_x}^{\text{H}}$). Every block below satisfies $\lVert\textcolor{brown}{\mathbf{Y}_x}-\textcolor{brown}{\mathbf{Y}_x}^{\text{T}}\rVert_\infty<\epsilon$; any asymmetry is a sign/convention bug.
\[\textcolor{brown}{\mathbf{Y}_x}\]
is finite only for nonzero leakage. An ideal ($\textcolor{brown}{Z}\to 0$) winding has infinite primitive admittance and a singular block; OpenDSS always carries a small leakage, and so must any exported case. A zero-leakage unit that still has core loss exports its (singular) shunt-only block rather than zeros.
2. single_phase (wye–wye, Γ-model)
Per phase-pair $k$: HV node $p$, LV node $q$. The two coils lump into one leakage referred to HV (the Γ model). With $\textcolor{brown}{y}=1/\textcolor{brown}{Z}$ and the no-load shunt split over the $\textcolor{red}{n_c}$ pairs,
\[\textcolor{brown}{Z} = \textcolor{red}{t}^2\,\textcolor{brown}{Z^{\text{fr}}_x} + \textcolor{red}{N}^2\,\textcolor{brown}{Z^{\text{to}}_x}, \qquad \textcolor{brown}{Y_0'} = \textcolor{brown}{Y_0}/\textcolor{red}{n_c},\]
so the off-nominal tap scales the from-coil leakage by $\textcolor{red}{t}^2$ (equivalently $\textcolor{brown}{Z}=\textcolor{red}{N}^2(\textcolor{brown}{Z^{\text{to}}_x}+\textcolor{brown}{Z^{\text{fr}}_x}/\textcolor{red}{N_0}^2)$, the to-referred leakage held at nominal — matching OpenDSS's turns-scaled Yprim). The $2\times2$ block for pair $k$ (nodes $[p,q]$), with $\textcolor{brown}{\mathbf{C}}=[\,1\;\;-\textcolor{red}{N}\,]$ and the shunt on the to node, is
\[\begin{bmatrix}\textcolor{blue}{I_p}\\ \textcolor{blue}{I_q}\end{bmatrix} = \begin{bmatrix} \textcolor{brown}{y} & -\textcolor{red}{N}\textcolor{brown}{y}\\[2pt] -\textcolor{red}{N}\textcolor{brown}{y} & \textcolor{red}{N}^2\textcolor{brown}{y}+\textcolor{brown}{Y_0'} \end{bmatrix} \begin{bmatrix}\textcolor{blue}{U_p}\\ \textcolor{blue}{U_q}\end{bmatrix}.\]
The full block is block-diagonal over the $\textcolor{red}{n_c}$ phase-pairs. Each grounded neutral (from and/or to) that a phase-pair returns to adds $\textcolor{brown}{y_n}$ to that neutral node's diagonal.
3. center_tap (split-phase, 3-winding)
Five nodes $[\,p,\,m,\,a,\,g,\,c\,]$: HV-phase, HV-neutral, LV-leg-1, LV-centre-tap, LV-leg-2. This is a genuine 3-winding unit (1 HV + 2 series-aiding LV legs sharing $g$). Refer all arms to the LV base:
\[\textcolor{brown}{y_1}=\textcolor{red}{N}^2/(\textcolor{red}{t}^2\textcolor{brown}{Z^{\text{fr}}_x}),\qquad \textcolor{brown}{y_2}=1/\textcolor{brown}{Z^{\text{to}}_x},\qquad \textcolor{brown}{Y_\Sigma}=\textcolor{brown}{y_1}+2\,\textcolor{brown}{y_2}.\]
The stored primary arm is in nominal ohms; its physical impedance scales by tap², so the referred primary admittance is independent of the primary tap. This matches the non-unity-tap OpenDSS fixtures for #393.
Eliminating the internal star node gives the symmetric 3-port star admittance in the winding-terminal ordering $[\text{HV-ref},\,\text{leg-1},\,\text{leg-2}]$, with $\textcolor{brown}{\mathbf{y}}=(\textcolor{brown}{y_1},\textcolor{brown}{y_2},\textcolor{brown}{y_2})$:
\[[\textcolor{brown}{\mathbf{Y}_{\text{3port}}}]_{ij}= \begin{cases} \textcolor{brown}{y_i}(\textcolor{brown}{Y_\Sigma}-\textcolor{brown}{y_i})/\textcolor{brown}{Y_\Sigma} & i=j,\\[4pt] -\textcolor{brown}{y_i}\textcolor{brown}{y_j}/\textcolor{brown}{Y_\Sigma} & i\neq j. \end{cases}\]
The connection matrix maps node voltages $[p,m,a,g,c]$ to the three winding voltages — the HV winding sees $(\textcolor{blue}{U_p}-\textcolor{blue}{U_m})/\textcolor{red}{N}$, leg 1 sees $\textcolor{blue}{U_a}-\textcolor{blue}{U_g}$, and leg 2 sees $\textcolor{blue}{U_g}-\textcolor{blue}{U_c}$ (both legs the same polarity about the centre tap — series-aiding):
\[\textcolor{brown}{\mathbf{C}}= \begin{bmatrix} 1/\textcolor{red}{N} & -1/\textcolor{red}{N} & 0 & 0 & 0\\ 0 & 0 & 1 & -1 & 0\\ 0 & 0 & 0 & 1 & -1 \end{bmatrix}.\]
With the no-load shunt across winding 2 = LV leg 1 (the $a$–$g$ coil, $\textcolor{brown}{\mathbf{c}}_{ag}=\mathbf{e}_a-\mathbf{e}_g$) and the two optional neutral-grounding branches — $\textcolor{brown}{y_n^{\text{fr}}}$ at the HV neutral $m$ (node 2) and $\textcolor{brown}{y_n^{\text{to}}}$ at the centre tap $g$ (node 4):
\[\textcolor{brown}{\mathbf{Y}_x}= \textcolor{brown}{\mathbf{C}}^{\text{T}}\,\textcolor{brown}{\mathbf{Y}_{\text{3port}}}\,\textcolor{brown}{\mathbf{C}} +\textcolor{brown}{Y_0}\,\textcolor{brown}{\mathbf{c}}_{ag}\textcolor{brown}{\mathbf{c}}_{ag}^{\text{T}} +\textcolor{brown}{y_n^{\text{fr}}}\,\mathbf{e}_m\mathbf{e}_m^{\text{T}} +\textcolor{brown}{y_n^{\text{to}}}\,\mathbf{e}_g\mathbf{e}_g^{\text{T}}.\]
This reproduces OpenDSS's own transformer Yprim to machine precision. The star arms come from the symmetric short-circuit split $\textcolor{red}{X_{1,\star}}=(\textcolor{red}{X_{HL}}+\textcolor{red}{X_{HT}}-\textcolor{red}{X_{LT}})/2$, $\textcolor{red}{X_{2,\star}}=(\textcolor{red}{X_{HL}}+\textcolor{red}{X_{LT}}-\textcolor{red}{X_{HT}})/2$.
4. wye_delta (Yd) and delta_wye (Dy)
Take $\textcolor{red}{n_\phi}=3$. Node ordering $[\,w_1,w_2,w_3,w_n,\,d_1,d_2,d_3\,]$ (wye phases, wye neutral, delta nodes). Each core couples a wye phase-to-neutral winding voltage with a delta line-to-line winding voltage; the $6\times7$ connection matrix is
\[\textcolor{brown}{\mathbf{C}}= \begin{bmatrix}\mathbf{I}_3 & -\mathbf{1} & \mathbf{0}\\[2pt] \mathbf{0} & \mathbf{0} & \textcolor{red}{\mathbf{D}}\end{bmatrix}, \qquad \textcolor{red}{\mathbf{D}}=\begin{bmatrix}1&-1&0\\0&1&-1\\-1&0&1\end{bmatrix},\]
where $\textcolor{red}{\mathbf{D}}$ (forward delta) is used for Yd and $\textcolor{red}{\mathbf{D}}^{\text{T}}$ (backward delta) for Dy. The effective per-winding turns ratio and the wye-referred short-circuit admittance are
\[\textcolor{red}{n^{\text{eff}}}= \begin{cases}\sqrt3/\textcolor{red}{N} & \text{Yd},\\ \textcolor{red}{N}\sqrt3 & \text{Dy},\end{cases} \qquad \textcolor{brown}{y_t}=\frac{1}{\textcolor{brown}{Z_{\text{sc}}}},\quad \textcolor{brown}{Z_{\text{sc}}}=\textcolor{brown}{Z^w}+\frac{\textcolor{red}{n_\phi}}{(\textcolor{red}{n^{\text{eff}}_0})^2}\,\textcolor{brown}{Z^d},\]
with $\textcolor{brown}{Z^w},\textcolor{brown}{Z^d}$ the wye- and delta-side leakages (mapped by which side is from) and $\textcolor{red}{n^{\text{eff}}_0}$ the nominal (tap-1) ratio. A non-nominal tap scales the short-circuit impedance by $(\textcolor{red}{n^{\text{eff}}_0}/\textcolor{red}{n^{\text{eff}}})^2$ on the tapped (wye=from) side. The ideal coil relation is $\textcolor{blue}{U^d_k}=\textcolor{red}{n^{\text{eff}}}\,\textcolor{blue}{U^w_k}$, so with $\textcolor{red}{a}=1/\textcolor{red}{n^{\text{eff}}}$ the per-core primitive (rows $[w_k,d_k]$) is
\[\textcolor{brown}{\mathbf{y}^{(k)}_{\text{prim}}}=\textcolor{brown}{y_t} \begin{bmatrix}1 & -\textcolor{red}{a}\\ -\textcolor{red}{a} & \textcolor{red}{a}^2\end{bmatrix}.\]
Stamping $\textcolor{red}{n^{\text{eff}}}$ instead of $\textcolor{red}{a}$ builds the inverse transformer — its no-load point lands at $\textcolor{blue}{U^d}=\textcolor{blue}{U^w}/\textcolor{red}{n^{\text{eff}}}$ instead of $\textcolor{red}{n^{\text{eff}}}\textcolor{blue}{U^w}$ (an $\sim(\textcolor{red}{n^{\text{eff}}})^2$ error) — yet passes every symmetry and passivity check. Only the OpenDSS Yprim cross-check catches it.
The nodal block is $\textcolor{brown}{\mathbf{Y}_x}=\textcolor{brown}{\mathbf{C}}^{\text{T}}\textcolor{brown}{\mathbf{y}_{\text{prim}}}\textcolor{brown}{\mathbf{C}}+\textcolor{brown}{Y_0}\,(\text{winding-2 shunt})+\textcolor{brown}{y_n}\,(\text{wye-neutral diagonal})$, with the shunt a delta of $\textcolor{brown}{Y_0}/\textcolor{red}{n_\phi}$ branches on the LV delta (Yd) or phase-to-neutral on the LV wye (Dy). For the common lossless-delta case ($\textcolor{brown}{Z^d}=0$, $\textcolor{brown}{y_w}=1/\textcolor{brown}{Z^w}$, no shunt) it evaluates to the explicit symmetric block
\[\textcolor{brown}{\mathbf{Y}_x}=\textcolor{brown}{y_w} \begin{bmatrix} \mathbf{I}_3 & -\mathbf{1} & -\textcolor{red}{a}\textcolor{red}{\mathbf{D}}\\[4pt] -\mathbf{1}^{\text{T}} & 3 & \textcolor{red}{a}\,\mathbf{1}^{\text{T}}\textcolor{red}{\mathbf{D}}\\[4pt] -\textcolor{red}{a}\textcolor{red}{\mathbf{D}}^{\text{T}} & \textcolor{red}{a}\,\textcolor{red}{\mathbf{D}}^{\text{T}}\mathbf{1} & \textcolor{red}{a}^2\textcolor{red}{\mathbf{D}}^{\text{T}}\textcolor{red}{\mathbf{D}} \end{bmatrix}.\]
The wye↔delta cross blocks are transposes of each other; the $3$ on the wye neutral is the zero-sequence admittance path, so a zero from-side impedance makes that row singular.
5. n_winding (general, WYE and/or DELTA)
For $\textcolor{red}{n_W}\ge 2$ windings, start from the OpenDSS-style short-circuit matrix $\textcolor{brown}{\mathbf{Z}_B}$ (the $(\textcolor{red}{n_W}-1)\times(\textcolor{red}{n_W}-1)$ leakage referred to winding 1, built from the pairwise $\textcolor{red}{x_{\text{sc}}[i,j]}$ and per-winding resistances) and invert it, $\textcolor{brown}{\mathbf{Y}_B}=\textcolor{brown}{\mathbf{Z}_B}^{-1}$. Expanding with winding 1 as the reference node and de-referring by the turns ratios $\textcolor{red}{\mathbf{D}}=\operatorname{diag}(\textcolor{red}{N_k})$ gives the per-winding admittance
\[\textcolor{brown}{\mathbf{Y}_w}=\textcolor{red}{\mathbf{D}}^{-1}\,\textcolor{brown}{\mathbf{C}_{\text{ref}}}^{\text{T}}\,\textcolor{brown}{\mathbf{Y}_B}\,\textcolor{brown}{\mathbf{C}_{\text{ref}}}\,\textcolor{red}{\mathbf{D}}^{-1}, \qquad [\textcolor{brown}{\mathbf{C}_{\text{ref}}}]_{i,:}=\mathbf{e}_1-\mathbf{e}_{i+1}.\]
This $\textcolor{red}{n_W}\times\textcolor{red}{n_W}$ winding admittance is stamped onto the terminal nodes, once per phase $p$, through a connection-aware coil incidence $\textcolor{brown}{\mathbf{P}}$: a WYE coil maps to its phase-neutral pair, a DELTA coil to its phase-phase pair (selected by the winding's delta_roll, the vector-group orientation). Summing over phases and adding the winding-2 shunt,
\[\textcolor{brown}{\mathbf{Y}_x}=\sum_{p}\ \textcolor{brown}{\mathbf{P}}_p^{\text{T}}\,\textcolor{brown}{\mathbf{Y}_w}\,\textcolor{brown}{\mathbf{P}}_p \;+\;\textcolor{brown}{Y_0}\,\text{(winding-2 coil)}.\]
This is the exact multi-winding leakage star; the 2-winding subtypes above are its closed-form specialisations.
6. Regulators
Both regulator subtypes replace the nameplate ratio with the fixed-tap effective ratio $\textcolor{red}{n^{\text{eff}}}$ from tap_ratio $\textcolor{red}{a}$ and regulator_type: $\textcolor{red}{n^{\text{eff}}}=1/\textcolor{red}{a}$ (Type B, default) or $\textcolor{red}{a}$ (Type A). Because an autotransformer is not galvanically isolated, the leakage is the autotransformer sum $\textcolor{brown}{Z}=\textcolor{brown}{Z^{\text{fr}}_x}+(\textcolor{red}{n^{\text{eff}}})^2\textcolor{brown}{Z^{\text{to}}_x}$ and the no-load shunt sits on the from side.
single_phase_autotransformer. Nodes $[\,t^{\text{ph}}_{\text{fr}},\,t^{\text{ph}}_{\text{to}},\,t^{n}_{\text{fr}},\,t^{n}_{\text{to}}\,]$, one core spanning phase-to-neutral on each side. With $\textcolor{brown}{y_t}=1/\textcolor{brown}{Z}$, the primitive and connection are
\[\textcolor{brown}{\mathbf{y}_{\text{prim}}}=\textcolor{brown}{y_t} \begin{bmatrix}1 & -\textcolor{red}{n^{\text{eff}}}\\ -\textcolor{red}{n^{\text{eff}}} & (\textcolor{red}{n^{\text{eff}}})^2\end{bmatrix}, \qquad \textcolor{brown}{\mathbf{C}}= \begin{bmatrix}1 & 0 & -1 & 0\\ 0 & 1 & 0 & -1\end{bmatrix},\]
giving $\textcolor{brown}{\mathbf{Y}_x}=\textcolor{brown}{\mathbf{C}}^{\text{T}}\textcolor{brown}{\mathbf{y}_{\text{prim}}}\textcolor{brown}{\mathbf{C}}$ plus the from-winding shunt. Reducing to the two phase nodes (neutrals at 0) recovers the familiar $\bigl[\begin{smallmatrix}\textcolor{brown}{y_t} & -\textcolor{red}{n^{\text{eff}}}\textcolor{brown}{y_t}\\ -\textcolor{red}{n^{\text{eff}}}\textcolor{brown}{y_t} & (\textcolor{red}{n^{\text{eff}}})^2\textcolor{brown}{y_t}\end{smallmatrix}\bigr]$; every column of the full $4\times4$ sums to zero (current conservation through the shared neutral).
open_delta_regulator. Two line-to-line cores across the phase pairs implied by connection (ABBC/BCAC/CABA), each with its own tap $\textcolor{red}{n^{\text{eff}}_j}$ and primitive $\textcolor{brown}{y_{t,j}}\bigl[\begin{smallmatrix}1 & -\textcolor{red}{n^{\text{eff}}_j}\\ -\textcolor{red}{n^{\text{eff}}_j} & (\textcolor{red}{n^{\text{eff}}_j})^2\end{smallmatrix}\bigr]$ across the from/to pair $(p,q)$, summed with the from-side shunt:
\[\textcolor{brown}{\mathbf{Y}_x}=\sum_{j=1}^{2}\ \textcolor{brown}{\mathbf{C}}_j^{\text{T}}\,\textcolor{brown}{\mathbf{y}_{\text{prim},j}}\,\textcolor{brown}{\mathbf{C}}_j.\]
This is the device's natural line-to-line admittance — the "unspecified neutral" matrix of Yan et al. (2018), IEEE Trans. Smart Grid 9(3):2224–2234, doi:10.1109/TSG.2016.2609440, Eq. (11): the shared phase carries $2\textcolor{brown}{y_t}$ on its diagonal (both regulators) and the from↔to coupling scales as $\textcolor{red}{n^{\text{eff}}}$ and $(\textcolor{red}{n^{\text{eff}}})^2$ — the autotransformer factor, not an isolated-transformer ratio. The galvanic straight-through of the shared phase (the paper's common-neutral model, its Eq. 14) is a topological constraint imposed in the OPF, not folded into this primitive — folding it in would conflate the device admittance with one particular elimination of the shared node.
7. Validation and implementation
The blocks are built by transformer_yprim (nwinding_yprim for n_winding) and exported node-aligned via export_yprim / write_yprim, for direct comparison with OpenDSS Export Y. Four gates hold each block to the physics:
| Gate | Check |
|---|---|
| Symmetry | $\lVert\textcolor{brown}{\mathbf{Y}_x}-\textcolor{brown}{\mathbf{Y}_x}^{\text{T}}\rVert_\infty<\epsilon$ for every block. |
| Closed form | the single_phase $2\times2$ block matches §2 exactly (unit-test oracle). |
| Power balance | $\mathfrak{R}[\textcolor{blue}{\mathbf{U}}^{\text{H}}\textcolor{brown}{\mathbf{Y}_x}\textcolor{blue}{\mathbf{U}}]=\sum_k\lvert\textcolor{blue}{I_k}\rvert^2\textcolor{red}{R_k}$ (winding resistive loss) for arbitrary $\textcolor{blue}{\mathbf{U}}$. |
| OpenDSS cross-check | dump Yprim per fixture, permute to the same node order, assert $\lVert\textcolor{brown}{\mathbf{Y}_x}^{\text{bmopf}}-\textcolor{brown}{\mathbf{Y}_x}^{\text{dss}}\rVert_\infty<\text{tol}$ — catches turns-ratio direction, $\sqrt3$ scaling, delta phase-shift, and shunt placement in one shot. |
A fifth gate ties this export to the OPF constraints: at a solved power-flow setpoint, the nodal currents $\textcolor{brown}{\mathbf{Y}_x}\textcolor{blue}{\mathbf{U}_x}$ match the OPF's own winding-current variables node-by-node, including off-nominal fixed taps — so the admittance export and the optimisation model cannot diverge.
The Yd/Dy OPF winding-current variables are not the physical terminal currents used here: the OPF's $\textcolor{blue}{I^w_k}=\textcolor{red}{n^{\text{eff}}}\,\textcolor{blue}{I^{\text{phys}}_{w,k}}$ and $\textcolor{blue}{I^d_k}=\textcolor{blue}{I^{\text{phys}}_{d,k}}/\textcolor{red}{n^{\text{eff}}}$ carry the ratio scaling. The OPF is internally consistent (its voltage equations and current coupling use the same scaled convention throughout), so it produces correct voltages and power flows; this export uses the physical terminal currents via $\textcolor{brown}{\mathbf{C}}^{\text{T}}\textcolor{brown}{\mathbf{y}_{\text{prim}}}\textcolor{brown}{\mathbf{C}}$ and is therefore directly comparable to OpenDSS.