BranchLumpedConstant

Source: GridKit/Model/EMT/Component/Branch/BranchLumpedConstant/README.md

BranchLumpedConstant Model

BranchLumpedConstant represents a lumped-parameter EMT transmission line. The nominal \(\pi\)-model is obtained by spatially discretizing the telegrapher equations over a segment of length \(\Delta x\), with a half shunt placed at each port. Series current \(\mathbf{i}\) is directed from bus 1 to bus 2. Bus residual current injections are positive into buses. All electrical parameter matrices are \(3 \times 3\) and capture self and mutual coupling between phases.

Figure 1: Lumped constant EMT branch model

Model Parameters

Symbol	Units	Description	Note
\(\mathbf{R}'\)	[\(\Omega\)/m]	Series resistance matrix per unit length	\(\mathbb{R}^{3 \times 3}\)
\(\mathbf{L}'\)	[H/m]	Series inductance matrix per unit length	\(\mathbb{R}^{3 \times 3}\)
\(\mathbf{G}'\)	[S/m]	Shunt conductance matrix per unit length	\(\mathbb{R}^{3 \times 3}\)
\(\mathbf{C}'\)	[F/m]	Shunt capacitance matrix per unit length	\(\mathbb{R}^{3 \times 3}\)
\(\Delta x\)	[m]	Line segment length	\(\mathbb{R}\)

Model Derived Parameters

\[\begin{split}\begin{aligned} \mathbf{R} &= \mathbf{R}'\Delta x & \mathbf{G} &= \mathbf{G}'\Delta x \\ \mathbf{L} &= \mathbf{L}'\Delta x & \mathbf{C} &= \mathbf{C}'\Delta x \end{aligned}\end{split}\]

Model Variables

Internal Variables

Differential

Symbol	Units	Description	Note
\(\mathbf{i}\)	[A]	Series branch current, directed bus 1 to bus 2	\(\mathbf{i} = [i_a, i_b, i_c]^T \in \mathbb{R}^3\)

Algebraic

None.

External Variables

External variables enter component model equations but are owned by other components. The EMT bus at each port owns the voltage variable and provides the equation needed to have a balanced system of equations.

Differential

Symbol	Units	Description	Note
\(\mathbf{v}_1\)	[V]	Port voltage at bus 1, owned by bus 1	\(\mathbf{v}_1 = [v_{1,a}, v_{1,b}, v_{1,c}]^T \in \mathbb{R}^3\)
\(\mathbf{v}_2\)	[V]	Port voltage at bus 2, owned by bus 2	\(\mathbf{v}_2 = [v_{2,a}, v_{2,b}, v_{2,c}]^T \in \mathbb{R}^3\)

Algebraic

None.

Model Equations

Differential Equations

\[0 = \mathbf{R}\,\mathbf{i} + \mathbf{L}\dot{\mathbf{i}} + \mathbf{v}_2 - \mathbf{v}_1\]

Algebraic Equations

None.

Bus Residual Contributions

The lumped line contributes to the KCL residual at each port bus. Each expression is accumulated into the owning bus residual.

\[\mathbf{i}^\text{inj}_1 := - \dfrac{\mathbf{G}}{2}\,\mathbf{v}_1 - \dfrac{\mathbf{C}}{2}\,\dot{\mathbf{v}}_1 - \mathbf{i}\]

\[\mathbf{i}^\text{inj}_2 := - \dfrac{\mathbf{G}}{2}\,\mathbf{v}_2 - \dfrac{\mathbf{C}}{2}\,\dot{\mathbf{v}}_2 + \mathbf{i}\]

Initialization

The initialization assumes a balanced three-phase system. Given bus voltages \(\mathbf{v}_1(0)\), \(\mathbf{v}_2(0)\) and their time derivatives \(\dot{\mathbf{v}}_1(0)\), \(\dot{\mathbf{v}}_2(0)\) from the EMT bus, and the power flow phasor series current \(I = |I| \angle \theta\), the initial series current is:

\[\begin{split}\mathbf{i}(0) = \sqrt{2}\,|I| \begin{bmatrix} \cos(\theta) \\ \cos(\theta - \tfrac{2\pi}{3}) \\ \cos(\theta + \tfrac{2\pi}{3}) \end{bmatrix}\end{split}\]

The initial derivative is then given by the series branch equation for DAE consistency:

\[\dot{\mathbf{i}}(0) = \mathbf{L}^{-1}\left(\mathbf{v}_1(0) - \mathbf{v}_2(0) - \mathbf{R}\,\mathbf{i}(0)\right)\]

Model Outputs

Candidate monitorable outputs include the series branch current components \(i_a\), \(i_b\), and \(i_c\).

Port current injection expressions are documented above as \(\mathbf{i}^\text{inj}_1\) and \(\mathbf{i}^\text{inj}_2\).