LineLumped

LineLumped represents an \(N\)-phase, \(K\)-conductor lumped EMT line over length \(\Delta x\). Series current \(\mathbf{i}_{12}\) is directed from terminal 1 to terminal 2.

Block Diagram

Figure 1: LineLumped model

The conductor-to-phase mappings are shown in the equations and omitted from the diagram for clarity.

Model Parameters

Define the phase- and conductor-index sets

\[\mathcal{N} = \{1,\ldots,N\}, \qquad \mathcal{K} = \{1,\ldots,K\}\]

Symbol	Units	JSON	Description	Note
\(N\)	[-]	N	Number of phases	Required, positive integer
\(K\)	[-]	K	Number of conductors	Required, positive integer
\(\mathbf{c}\)	[-]	conductors	Conductor phase-index list	\(\mathbf{c} \in \mathcal{N}^K\)
\(\Delta x\)	[m]	dx	Line segment length	Required, positive

Parameter Validation

\[\begin{split}\begin{aligned} N &\in \mathbb{Z}_{>0} \\ K &\in \mathbb{Z}_{>0} \\ \mathbf{c} &\in \mathcal{N}^K \\ \{c_k \mid k \in \mathcal{K}\} &= \mathcal{N} \\ \Delta x &> 0 \end{aligned}\end{split}\]

Model Derived Parameters

\[\begin{split}P_{\phi,nk} = \begin{cases} 1, & n = c_k \\ 0, & n \ne c_k \end{cases}, \quad n \in \mathcal{N},\quad k \in \mathcal{K}\end{split}\]

Model Ports

Symbol	Port	Type	Units	Description	Note
\(\mathbf{v}_1\)	v1	Input	[V]	Terminal 1 bus voltage	\(\mathbf{v}_1 \in \mathbb{R}^N\)
\(\mathbf{v}_2\)	v2	Input	[V]	Terminal 2 bus voltage	\(\mathbf{v}_2 \in \mathbb{R}^N\)
\(\mathbf{i}_1\)	i1	Output	[A]	Current injection at terminal 1	\(\mathbf{i}_1 \in \mathbb{R}^N\)
\(\mathbf{i}_2\)	i2	Output	[A]	Current injection at terminal 2	\(\mathbf{i}_2 \in \mathbb{R}^N\)

Submodels

Symbol	Description	Type	Order	JSON	Inputs	Outputs
\(\mathbf{z}\)	Per-unit-length series impedance	VectorFit	\(KQ_{\mathbf{z}}\)	Zp	\(\mathbb{R}^K\)	\(\mathbb{R}^K\)
\(\mathbf{y}_1\)	Per-unit-length shunt admittance at terminal 1	VectorFit	\(KQ_{\mathbf{y}}\)	Yp	\(\mathbb{R}^K\)	\(\mathbb{R}^K\)
\(\mathbf{y}_2\)	Per-unit-length shunt admittance at terminal 2	VectorFit	\(KQ_{\mathbf{y}}\)	Yp	\(\mathbb{R}^K\)	\(\mathbb{R}^K\)

Yp provides one coefficient set; the two terminal instances maintain independent states.

Submodel Validation

\[\mathrm{rank}(\mathbf{E}^{\mathbf{z}}) = K\]

Submodel Wiring

None.

Model Variables

Internal Variables

Differential

Symbol	Units	Description	Note
\(\mathbf{i}_{12}\)	[A]	Series current from terminal 1 to terminal 2	\(\mathbf{i}_{12} \in \mathbb{R}^K\)

Algebraic

Symbol	Units	Description	Note
\(\mathbf{i}_1^\mathrm{sh}\)	[A]	Shunt current at terminal 1	\(\mathbf{i}_1^\mathrm{sh} \in \mathbb{R}^K\)
\(\mathbf{i}_2^\mathrm{sh}\)	[A]	Shunt current at terminal 2	\(\mathbf{i}_2^\mathrm{sh} \in \mathbb{R}^K\)

External Variables

Differential

Symbol	Units	Description	Note
\(\mathbf{v}_1\)	[V]	Terminal 1 voltage owned by EMT bus	\(\mathbf{v}_1 \in \mathbb{R}^N\)
\(\mathbf{v}_2\)	[V]	Terminal 2 voltage owned by EMT bus	\(\mathbf{v}_2 \in \mathbb{R}^N\)

Algebraic

None.

Model Equations

Internal Equations

Differential

\[0 = \Delta x\,\mathbf{z}[\mathbf{i}_{12}] + \mathbf{P}_\phi^\mathsf T(\mathbf{v}_2-\mathbf{v}_1)\]

Algebraic

\[\begin{split}\begin{aligned} 0 &= \Delta x\,\mathbf{y}_1[\mathbf{P}_\phi^\mathsf T\mathbf{v}_1] + 2\mathbf{i}_1^\mathrm{sh} \\ 0 &= \Delta x\,\mathbf{y}_2[\mathbf{P}_\phi^\mathsf T\mathbf{v}_2] + 2\mathbf{i}_2^\mathrm{sh} \end{aligned}\end{split}\]

External Equations

\[\begin{split}\begin{aligned} \mathbf{i}_1 &\leftarrow \mathbf{P}_\phi( \mathbf{i}_1^\mathrm{sh} - \mathbf{i}_{12} ) \\ \mathbf{i}_2 &\leftarrow \mathbf{P}_\phi( \mathbf{i}_2^\mathrm{sh} + \mathbf{i}_{12} ) \end{aligned}\end{split}\]

Initialization

None.

Monitors

Monitor	Units	Description	Note
i12	[A]	Series current from terminal 1 to terminal 2	\(\mathbf{i}_{12} \in \mathbb{R}^K\)
i_sh1	[A]	Shunt current at terminal 1	\(\mathbf{i}_1^\mathrm{sh} \in \mathbb{R}^K\)
i_sh2	[A]	Shunt current at terminal 2	\(\mathbf{i}_2^\mathrm{sh} \in \mathbb{R}^K\)