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updated EMT docs partially
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‎GridKit/Model/EMT/Bus/README.md‎

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\end{aligned}
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```
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### Model Derived Parameters
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### Derived Parameters
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None.
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### Submodels
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## Submodels
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None.
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### Submodel Validation
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None.
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## Initialization
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The initial bus voltage is given by the parameter vector $\mathbf{v}_0$:
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### Input Initialization
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```math
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\mathbf{v}(0)=\mathbf{v}_0
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\begin{aligned}
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\mathbf{i}^{\mathrm{inj}}_m
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&\leftarrow \text{connected-device current starts},
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\quad m\in\{1,\ldots,M\}
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\end{aligned}
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```
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Only $\mathbf{v}(0)$ is initialized. The solver computes the initial
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bus-voltage derivative from the differential residual.
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### Internal Initialization
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Initialization assigns the parameterized bus-voltage start:
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```math
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\mathbf{v} \leftarrow \text{parameterized bus-voltage start}
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```
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The differential residual determines the initial bus-voltage derivative.
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### Output Initialization
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None.
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## Monitors
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‎GridKit/Model/EMT/Component/Line/LineDistributed/README.md‎

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# LineDistributed Model
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`LineDistributed` represents a distributed EMT line with characteristic
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admittance $\mathbf{y}_c$ and propagation model $\mathbf{h}$.
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`LineDistributed` represents an $N$-phase, $K$-conductor distributed EMT line.
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## Block Diagram
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## Model Parameters
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Symbol | Units | JSON | Description | Note
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----------------- | ----- | ------------ | ------------------------------------------------------ | ----
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Symbol | Units | JSON | Description | Note
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------ | ----- | ---- | ----------- | ----
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$N$ | [-] | `N` | Number of phases | Required, positive integer
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$K$ | [-] | `K` | Number of conductors | Required, positive integer
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$\mathbf{P}_\phi$ | [-] | `conductors` | Permutation matrix mapping each conductor to its phase | $\mathbf{P}_\phi \in \mathbb{R}^{N \times K}$
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### Parameter Validation
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```math
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\begin{aligned}
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N &> 0 \\
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K &> 0 \\
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\mathbf{P}_\phi &\in \mathbb{R}^{N \times K}
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\end{aligned}
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```
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### Derived Parameters
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None.
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### Model Derived Parameters
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## Submodels
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The number of phases $N$ and conductors $K$ are the row and column counts of
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$\mathbf{P}_\phi$, respectively.
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Submodel | Type | Order | Inputs | Outputs
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-------- | ---- | ----- | ------ | -------
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Characteristic admittance $f^{\mathbf{y}_c}$ | [`VectorFit`](../../../Operators/Rational/VectorFit/README.md) | $Q_{\mathbf{y}_c}$ | $\mathbf{v}_{1},\mathbf{v}_{2}\in\mathbb{R}^N$ | $\mathbf{i}^{\mathrm{sh}}_{1},\mathbf{i}^{\mathrm{sh}}_{2}\in\mathbb{R}^K$
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Propagation $f^{\mathbf{h}}$ | [`Propagation`](../../../Operators/Shift/Propagation/README.md) | $Q_{\mathbf{h}}$ | $\mathbf{u}_1,\mathbf{u}_2\in\mathbb{R}^K$ | $\mathbf{i}^{\mathrm{inc}}_{1},\mathbf{i}^{\mathrm{inc}}_{2}\in\mathbb{R}^K$
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### Submodels
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### Submodel Validation
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Submodel | Inputs | Parameters | Outputs
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-------- | ------ | ---------- | -------
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[`VectorFit`](../../../Operators/Rational/VectorFit/README.md) characteristic admittance $\mathbf{y}_c$ | $\mathbf{v}_{1}\in\mathbb{R}^N$ | `Yc` | $\mathbf{i}^{\mathrm{sh}}_{1}\in\mathbb{R}^K$
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[`VectorFit`](../../../Operators/Rational/VectorFit/README.md) characteristic admittance $\mathbf{y}_c$ | $\mathbf{v}_{2}\in\mathbb{R}^N$ | `Yc` | $\mathbf{i}^{\mathrm{sh}}_{2}\in\mathbb{R}^K$
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[`Propagation`](../../../Operators/Shift/Propagation/README.md) $\mathbf{h}$ | $2\mathbf{i}^{\mathrm{sh}}_{2}-\mathbf{i}^{\mathrm{inc}}_{2}$ | `H` | $\mathbf{i}^{\mathrm{inc}}_{1}\in\mathbb{R}^K$
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[`Propagation`](../../../Operators/Shift/Propagation/README.md) $\mathbf{h}$ | $2\mathbf{i}^{\mathrm{sh}}_{1}-\mathbf{i}^{\mathrm{inc}}_{1}$ | `H` | $\mathbf{i}^{\mathrm{inc}}_{2}\in\mathbb{R}^K$
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```math
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\begin{aligned}
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f^{\mathbf{y}_c} &: \mathbb{R}^N \rightarrow \mathbb{R}^K \\
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f^{\mathbf{h}} &: \mathbb{R}^K \rightarrow \mathbb{R}^K
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\end{aligned}
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```
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## Model Variables
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#### Algebraic
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None.
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Symbol | Units | Description | Note
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------ | ----- | ----------- | ----
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$\mathbf{i}^{\mathrm{sh}}_{1}$ | [A] | Shunt current at terminal `1` | $\mathbf{i}^{\mathrm{sh}}_{1}\in\mathbb{R}^K$
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$\mathbf{i}^{\mathrm{sh}}_{2}$ | [A] | Shunt current at terminal `2` | $\mathbf{i}^{\mathrm{sh}}_{2}\in\mathbb{R}^K$
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$\mathbf{i}^{\mathrm{inc}}_{1}$ | [A] | Incident current at terminal `1` | $\mathbf{i}^{\mathrm{inc}}_{1}\in\mathbb{R}^K$
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$\mathbf{i}^{\mathrm{inc}}_{2}$ | [A] | Incident current at terminal `2` | $\mathbf{i}^{\mathrm{inc}}_{2}\in\mathbb{R}^K$
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### External Variables
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External variables are owned by the EMT bus.
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#### Differential
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Symbol | Units | Description | Note
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-------------------------------------|-------|------------------------------------------|-----
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Symbol | Units | Description | Note
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------ | ----- | ----------- | ----
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$\mathbf{v}_{1}$ | [V] | Terminal `1` voltage owned by EMT bus | $\mathbf{v}_{1} \in \mathbb{R}^N$
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$\mathbf{v}_{2}$ | [V] | Terminal `2` voltage owned by EMT bus | $\mathbf{v}_{2} \in \mathbb{R}^N$
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```math
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\begin{aligned}
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\mathbf{i}^{\mathrm{sh}}_{1} &=
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\mathbf{y}_c*\mathbf{v}_{1} \\
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f^{\mathbf{y}_c}(\mathbf{v}_{1}) \\
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\mathbf{i}^{\mathrm{sh}}_{2} &=
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\mathbf{y}_c*\mathbf{v}_{2} \\
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f^{\mathbf{y}_c}(\mathbf{v}_{2}) \\
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\mathbf{i}^{\mathrm{inc}}_{1} &=
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\mathbf{h}*\left(
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f^{\mathbf{h}}\left(
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2\mathbf{i}^{\mathrm{sh}}_{2}
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\mathbf{i}^{\mathrm{inc}}_{2}
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\right) \\
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\mathbf{i}^{\mathrm{inc}}_{2} &=
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\mathbf{h}*\left(
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f^{\mathbf{h}}\left(
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2\mathbf{i}^{\mathrm{sh}}_{1}
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\mathbf{i}^{\mathrm{inc}}_{1}
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## Initialization
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Let subscript $0$ denote initial values. Initial values satisfy the wiring
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equations:
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### Input Initialization
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```math
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\begin{aligned}
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\mathbf{i}^{\mathrm{sh}}_{1,0} &= \mathbf{y}_c*\mathbf{v}_{1,0} \\
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\mathbf{i}^{\mathrm{sh}}_{2,0} &= \mathbf{y}_c*\mathbf{v}_{2,0} \\
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\mathbf{i}^{\mathrm{inc}}_{1,0} &= \mathbf{h}*\left(2\mathbf{i}^{\mathrm{sh}}_{2,0} - \mathbf{i}^{\mathrm{inc}}_{2,0}\right) \\
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\mathbf{i}^{\mathrm{inc}}_{2,0} &= \mathbf{h}*\left(2\mathbf{i}^{\mathrm{sh}}_{1,0} - \mathbf{i}^{\mathrm{inc}}_{1,0}\right)
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\mathbf{v}_1,\mathbf{v}_2
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&\leftarrow \text{terminal-bus voltage}
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\end{aligned}
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```
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### Internal Initialization
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Initialization assigns the internal line currents in dependency order:
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```math
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\begin{aligned}
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\mathbf{i}^{\mathrm{sh}}_{1}
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&\leftarrow f^{\mathbf{y}_c}(\mathbf{v}_{1}) \\
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\mathbf{i}^{\mathrm{sh}}_{2}
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&\leftarrow f^{\mathbf{y}_c}(\mathbf{v}_{2}) \\
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\mathbf{i}^{\mathrm{inc}}_{1}
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&\leftarrow
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f^{\mathbf{h}}\left(
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2\mathbf{i}^{\mathrm{sh}}_{2}
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-
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\mathbf{i}^{\mathrm{inc}}_{2}
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\right) \\
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\mathbf{i}^{\mathrm{inc}}_{2}
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&\leftarrow
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f^{\mathbf{h}}\left(
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2\mathbf{i}^{\mathrm{sh}}_{1}
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-
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\mathbf{i}^{\mathrm{inc}}_{1}
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\right)
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\end{aligned}
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```
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### Output Initialization
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Current-injection ports are assigned from initialized internal line currents.
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```math
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\begin{aligned}
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\mathbf{i}^{\mathrm{inj}}_{1}
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&\leftarrow
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\mathbf{P}_\phi\left(
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\mathbf{i}^{\mathrm{inc}}_{1}
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-
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\mathbf{i}^{\mathrm{sh}}_{1}
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\right) \\
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\mathbf{i}^{\mathrm{inj}}_{2}
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&\leftarrow
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\mathbf{P}_\phi\left(
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\mathbf{i}^{\mathrm{inc}}_{2}
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-
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\mathbf{i}^{\mathrm{sh}}_{2}
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\right)
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\end{aligned}
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```
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