Shunts

A shunt is a fixed admittance connected between a set of bus terminals (and, implicitly, ground). It models capacitor banks, grounding impedances, and similar passive elements. Parts 1–5 state the foundational model; part 6 records how BMOPFTools realises it. Symbols are defined in Notation.

1. Data model

A shunt is an entry of the top-level shunt object, keyed by its string ID $h$.

2. Input symbols

The matrix is stored row-first: entry $(k,j)$ is field G_k_j / B_k_j. Ground is never indexed (its voltage is zero); zero-filled rows/columns should be removed by trimming the terminal map.

3. Variables

None. A shunt introduces no unknown — its current is determined entirely by the bus voltage and its fixed admittance.

4. Equality constraints

The current drawn into the shunt at its terminals is linear in the bus voltage:

\[\textcolor{blue}{\mathbf{I}_{h}} = \textcolor{brown}{\mathbf{Y}_{h}}\,\textcolor{blue}{\mathbf{U}_i}[\textcolor{purple}{\mathbf{N}_{h}}].\]

This current is the shunt's contribution to KCL at bus $i$ (leaving the bus toward the admittance/ground). Examples: a single-entry $\textcolor{brown}{Y_{h,nn}}$ on terminal $n$ grounds the neutral through an impedance; a delta pattern $\textcolor{brown}{Y_{\text{cap}}}\,\textcolor{red}{\mathbf{M}^{\Delta}}$ on $\{a,b,c\}$ models a delta capacitor bank.

5. Inequality constraints

None. With a fixed admittance the current follows the voltage; a current limit would risk trivial infeasibility and is not imposed.

6. Implementation in BMOPFTools

Realisation

• No variables — in-place substitution. Exactly as for the line $\Pi$-shunt, the code builds the affine expression $\textcolor{brown}{\mathbf{Y}_{h}}\textcolor{blue}{\mathbf{U}_i}$ as JuMP AffExpr terms (shunt.jl:_shunt_current!) and subtracts them directly from the KCL accumulator (shunt.jl:_add_shunt_constraints!) — no shunt-current variable or defining equality is created.
• Grounded terminals are absent from the voltage dictionary and contribute nothing (their voltage is zero), so grounding admittances collapse naturally.
• Per row/column $k,j$: $\mathfrak{R}(\textcolor{blue}{I_{h,k}}) = \sum_j(\mathbf{G}_{kj} v^r_j - \mathbf{B}_{kj} v^i_j)$, $\mathfrak{I}(\textcolor{blue}{I_{h,k}}) = \sum_j(\mathbf{G}_{kj} v^i_j + \mathbf{B}_{kj} v^r_j)$.

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