LineDistributed

LineDistributed represents an \(N\)-phase, \(K\)-conductor distributed EMT line.

Block Diagram

Figure 1: LineDistributed 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	\(\mathbf{c} \in \mathcal{N}^K\)

Parameter Validation

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

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
\(\mathbf{y}_1^\mathrm{c}\)	Characteristic admittance at terminal 1	VectorFit	\(KQ_{\mathbf{y}^\mathrm{c}}\)	Yc	\(\mathbb{R}^K\)	\(\mathbb{R}^K\)
\(\mathbf{y}_2^\mathrm{c}\)	Characteristic admittance at terminal 2	VectorFit	\(KQ_{\mathbf{y}^\mathrm{c}}\)	Yc	\(\mathbb{R}^K\)	\(\mathbb{R}^K\)
\(\mathbf{h}_{21}\)	Propagation from terminal 2 to terminal 1	Propagation	Composite	H	\(\mathbb{R}^K\)	\(\mathbb{R}^K\)
\(\mathbf{h}_{12}\)	Propagation from terminal 1 to terminal 2	Propagation	Composite	H	\(\mathbb{R}^K\)	\(\mathbb{R}^K\)

Yc and H each provide one coefficient set. The model assumes a reciprocal uniform line, so

\[\mathbf{Y}_1^{\mathrm{c}}(s)=\mathbf{Y}_2^{\mathrm{c}}(s), \qquad \mathbf{H}_{12}(s)=\mathbf{H}_{21}(s)\]

The two terminal-admittance instances maintain independent states, and the two directional propagation instances maintain independent states and histories.

Submodel Validation

The characteristic-admittance fits must be stable, proper, and positive real. Together with the propagation fits, they must produce a passive line model.

Submodel Wiring

\[\begin{split}\begin{aligned} \mathbf{i}_1^\mathrm{c} &\leftarrow \mathbf{y}_1^\mathrm{c}[\mathbf{P}_\phi^\mathsf T\mathbf{v}_1] \\ \mathbf{i}_2^\mathrm{c} &\leftarrow \mathbf{y}_2^\mathrm{c}[\mathbf{P}_\phi^\mathsf T\mathbf{v}_2] \\ \mathbf{i}_1^\mathrm{inc} &\leftarrow \mathbf{h}_{21}[\mathbf{i}_2^\mathrm{ref}] \\ \mathbf{i}_2^\mathrm{inc} &\leftarrow \mathbf{h}_{12}[\mathbf{i}_1^\mathrm{ref}] \end{aligned}\end{split}\]

Model Variables

Internal Variables

Differential

None.

Algebraic

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

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

None.

Algebraic

\[\begin{split}\begin{aligned} 0 &= -\mathbf{i}_1^\mathrm{ref} + 2\mathbf{i}_1^\mathrm{c} - \mathbf{i}_1^\mathrm{inc} \\ 0 &= -\mathbf{i}_2^\mathrm{ref} + 2\mathbf{i}_2^\mathrm{c} - \mathbf{i}_2^\mathrm{inc} \end{aligned}\end{split}\]

External Equations

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

Initialization

Initialize the characteristic-admittance and propagation submodels.

Monitors

Monitor	Units	Description	Note
i_c1	[A]	Characteristic-admittance current at terminal 1	\(\mathbf{i}_1^\mathrm{c} \in \mathbb{R}^K\)
i_c2	[A]	Characteristic-admittance current at terminal 2	\(\mathbf{i}_2^\mathrm{c} \in \mathbb{R}^K\)
i_inc1	[A]	Incident current at terminal 1	\(\mathbf{i}_1^\mathrm{inc} \in \mathbb{R}^K\)
i_inc2	[A]	Incident current at terminal 2	\(\mathbf{i}_2^\mathrm{inc} \in \mathbb{R}^K\)