Lines

A line is a multi-conductor power cable or overhead line, modelled as a nominal $\Pi$ equivalent: a series impedance with a shunt admittance half-section at each end. Parts 1–5 state the foundational (physics) model; part 6 records how BMOPFTools realises it. Symbols are defined in Notation.

1. Data model

A line is an entry of the top-level line object, keyed by its string ID $\ell$. It carries exactly one impedance source: either a referenced linecode (per-metre matrices scaled by length) or inline absolute matrices.

The oneOf schema rule requires either linecode + length or at least R_series_1_1 + X_series_1_1 inline (and then no linecode).

Linecode

A linecode $c$ stores per-metre matrices shared across lines of the same type.

2. Input symbols

Parameter construction

The line's series impedance and shunt admittances come from one source:

\[\text{linecode: }\quad \textcolor{brown}{\mathbf{Z}^{\text{s}}_\ell} = \textcolor{brown}{\mathbf{Z}^{\text{s}}_c}\,\textcolor{red}{L_\ell}, \qquad \textcolor{brown}{\mathbf{Y}^{\text{sh}}_{\ell ij}} = \textcolor{brown}{\mathbf{Y}^{\text{sh}}_{\ell ji}} = \tfrac{1}{2}\,\textcolor{brown}{\mathbf{Y}^{\text{sh}}_c}\,\textcolor{red}{L_\ell};\]

or the inline absolute matrices are used directly (never scaled by length). Inline data permits independent from- and to-side shunts $\textcolor{brown}{\mathbf{Y}^{\text{sh}}_{\ell ij}}\neq\textcolor{brown}{\mathbf{Y}^{\text{sh}}_{\ell ji}}$.

3. Variables

A line has $n_\ell = |\textcolor{purple}{\mathbf{N}_{\ell i}}|$ conductors. The series current flowing from $i$ toward $j$ is the independent unknown:

\[\textcolor{blue}{\mathbf{I}^{\text{s}}_{\ell ij}} \in \mathbb{C}^{n_\ell}.\]

The terminal current $\textcolor{blue}{\mathbf{I}_{\ell ij}}$ (what enters the bus's KCL) and the shunt current $\textcolor{blue}{\mathbf{I}^{\text{sh}}_{\ell ij}}$ are the other line currents; the equalities of part 4 relate them to $\textcolor{blue}{\mathbf{I}^{\text{s}}_{\ell ij}}$ and the bus voltages.

4. Equality constraints

Ohm's law (series voltage drop)

Across the series impedance, for the forward orientation $\ell ij$:

\[\textcolor{blue}{\mathbf{U}_j}[\textcolor{purple}{\mathbf{N}_{\ell j}}] = \textcolor{blue}{\mathbf{U}_i}[\textcolor{purple}{\mathbf{N}_{\ell i}}] - \textcolor{brown}{\mathbf{Z}^{\text{s}}_\ell}\,\textcolor{blue}{\mathbf{I}^{\text{s}}_{\ell ij}}.\]

The impedance matrix is full: off-diagonal entries couple the voltage drop on one conductor to the current in another.

Shunt currents

Each $\Pi$ half-section draws a current set by its admittance and the local bus voltage:

\[\textcolor{blue}{\mathbf{I}^{\text{sh}}_{\ell ij}} = \textcolor{brown}{\mathbf{Y}^{\text{sh}}_{\ell ij}}\,\textcolor{blue}{\mathbf{U}_i}, \qquad \textcolor{blue}{\mathbf{I}^{\text{sh}}_{\ell ji}} = \textcolor{brown}{\mathbf{Y}^{\text{sh}}_{\ell ji}}\,\textcolor{blue}{\mathbf{U}_j}.\]

Terminal current and series-current conservation

The current entering the line at each bus is the series current plus that end's shunt current, and the two directional series currents are equal and opposite:

\[\textcolor{blue}{\mathbf{I}_{\ell ij}} = \textcolor{blue}{\mathbf{I}^{\text{s}}_{\ell ij}} + \textcolor{blue}{\mathbf{I}^{\text{sh}}_{\ell ij}}, \qquad \textcolor{blue}{\mathbf{I}_{\ell ji}} = \textcolor{blue}{\mathbf{I}^{\text{s}}_{\ell ji}} + \textcolor{blue}{\mathbf{I}^{\text{sh}}_{\ell ji}}, \qquad \textcolor{blue}{\mathbf{I}^{\text{s}}_{\ell ij}} + \textcolor{blue}{\mathbf{I}^{\text{s}}_{\ell ji}} = \mathbf{0}.\]

The terminal currents $\textcolor{blue}{\mathbf{I}_{\ell ij}},\textcolor{blue}{\mathbf{I}_{\ell ji}}$ are the contributions this line makes to KCL at buses $i$ and $j$ (see Buses §4).

5. Inequality constraints

Cartesian variable bounds

A box may be placed on the series-current variable $\textcolor{blue}{\mathbf{I}^{\text{s}}_{\ell ij}}$ to bound the search. Because the engineering limit constrains the total (terminal) current, the box on the series part is the limit inflated by the worst-case shunt contribution $\rho_k$ on that row:

\[|\mathfrak{R}(\textcolor{blue}{I^{\text{s}}_{\ell ij,k}})| \le \textcolor{red}{I^{\max}_{\ell ij,k}} + \rho_k, \qquad |\mathfrak{I}(\textcolor{blue}{I^{\text{s}}_{\ell ij,k}})| \le \textcolor{red}{I^{\max}_{\ell ij,k}} + \rho_k,\]

with $\rho_k \le \sum_{j'} |\textcolor{brown}{Y^{\text{sh}}_{\ell ij,kj'}}|\,\textcolor{red}{U^{\max}_{i,j'}}$ from the endpoint voltage caps. This is a solver-conditioning aid; it is implied by the engineering bound below.

Engineering bounds

Thermal current limit on the terminal current, at both ends, for all conductors including neutral:

\[\textcolor{blue}{\mathbf{I}_{\ell ij}}\circ(\textcolor{blue}{\mathbf{I}_{\ell ij}})^{*} \le \textcolor{red}{\mathbf{I}^{\max}_{\ell ij}}\!\circ\textcolor{red}{\mathbf{I}^{\max}_{\ell ij}}, \qquad \textcolor{blue}{\mathbf{I}_{\ell ji}}\circ(\textcolor{blue}{\mathbf{I}_{\ell ji}})^{*} \le \textcolor{red}{\mathbf{I}^{\max}_{\ell ij}}\!\circ\textcolor{red}{\mathbf{I}^{\max}_{\ell ij}}.\]

Apparent-power limit (optional, from s_max), with the terminal power $\textcolor{blue}{\mathbf{S}_{\ell ij}} = \textcolor{blue}{\mathbf{U}_i}\circ(\textcolor{blue}{\mathbf{I}_{\ell ij}})^{*}$:

\[\textcolor{blue}{\mathbf{S}_{\ell ij}}\circ(\textcolor{blue}{\mathbf{S}_{\ell ij}})^{*} \le \textcolor{red}{\mathbf{S}^{\max}_{\ell ij}}\!\circ\textcolor{red}{\mathbf{S}^{\max}_{\ell ij}}.\]

Per-line angle difference (optional va_diff_min/va_diff_max). For each conductor, with $\textcolor{blue}{z}=\textcolor{blue}{U_{i,\cdot}}\,(\textcolor{blue}{U_{j,\cdot}})^{*}=c+\textcolor{brown}{j}s$:

\[\tan(\textcolor{red}{\theta^{\Delta,\min}_\ell})\, c \ \le\ s \ \le\ \tan(\textcolor{red}{\theta^{\Delta,\max}_\ell})\, c.\]

6. Implementation in BMOPFTools

Realisation

• Rectangular Ohm's law. The complex KVL is stamped as its two real parts per conductor $k$ (branch.jl:_add_line_constraints!): $\mathfrak{R}(\textcolor{blue}{U_i})-\mathfrak{R}(\textcolor{blue}{U_j}) = \sum_{j'}(\mathbf{R}_{kj'}\mathfrak{R}(\textcolor{blue}{I^{\text{s}}})-\mathbf{X}_{kj'}\mathfrak{I}(\textcolor{blue}{I^{\text{s}}}))$ and $\mathfrak{I}(\textcolor{blue}{U_i})-\mathfrak{I}(\textcolor{blue}{U_j}) = \sum_{j'}(\mathbf{R}_{kj'}\mathfrak{I}(\textcolor{blue}{I^{\text{s}}})+\mathbf{X}_{kj'}\mathfrak{R}(\textcolor{blue}{I^{\text{s}}}))$.
• Shunt currents are not variables. Where part 4 writes $\textcolor{blue}{\mathbf{I}^{\text{sh}}}=\textcolor{brown}{\mathbf{Y}^{\text{sh}}}\textcolor{blue}{\mathbf{U}}$ as a current, the code introduces no variable for it. It builds the affine JuMP expression $\textcolor{brown}{\mathbf{Y}^{\text{sh}}}\textcolor{blue}{\mathbf{U}}$ (shunt.jl:_shunt_current!) and substitutes it in place wherever $\textcolor{blue}{\mathbf{I}^{\text{sh}}}$ appears — the KCL contribution and the terminal-current limit. The terminal current $\textcolor{blue}{\mathbf{I}_{\ell ij}}=\textcolor{blue}{\mathbf{I}^{\text{s}}_{\ell ij}}+\textcolor{brown}{\mathbf{Y}^{\text{sh}}_{\ell ij}}\textcolor{blue}{\mathbf{U}_i}$ is thus an expression, not a variable. This is exact — it eliminates the shunt-current variables and their defining equalities analytically.
• Series-current alias. Only the from-side series current is a variable (cr_fr/ci_fr, variables.jl:_add_line_variables!); the conservation law $\textcolor{blue}{\mathbf{I}^{\text{s}}_{\ell ji}}=-\textcolor{blue}{\mathbf{I}^{\text{s}}_{\ell ij}}$ is applied by making the to-side an AffExpr alias -cr_fr, so it is never a separate variable or equality.
• Conductor counts of the impedance matrix and both terminal maps must match exactly; a mismatch is refused (E.INT.LINE_DIM_MISMATCH), never truncated.
• Cartesian box is _limit_current_box! in branch.jl, added only when every voltage cap feeding the row is known. The thermal/apparent-power limits are the quadratics on the total-current and total-power expressions; the angle limit is branch.jl:_add_line_angle_constraints!.

Source map

Reconciliation notes (data model)