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general correctness for lumped
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‎GridKit/Model/EMT/Component/Line/LineLumped/README.md‎

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# LineLumped Model
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`LineLumped` represents a lumped-parameter EMT transmission line.
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The nominal $\pi$-model is obtained by spatially discretizing the telegrapher equations over
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a segment of length $\Delta x$, with a half shunt placed at each port.
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Series current $\mathbf{i}$ is directed from bus 1 to bus 2. Bus residual
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current injections are positive into buses. All electrical parameter matrices
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are $3 \times 3$ and capture self and mutual coupling between phases.
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`LineLumped` represents a lumped-parameter EMT transmission line using a
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$\pi$-section equivalent over length $\Delta x$. Series current $\mathbf{i}$
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is directed from bus 1 to bus 2. Bus residual current injections are positive
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into buses.
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Notes:
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- For now, only constant `R`, `L`, `G`, and `C` parameters are supported. These
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should remain valid, but the more general interface should also accept
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per-unit-length $\mathbf{Z}'(s)$ and $\mathbf{Y}'(s)$ as `Rational` models.
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When only the $\mathbf{D}$ and $\mathbf{E}$ terms are used, this reduces to
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$\mathbf{Z}'(s)=\mathbf{R}'+s\mathbf{L}'$ and
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$\mathbf{Y}'(s)=\mathbf{G}'+s\mathbf{C}'$ without adding algebraic or
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differential variables.
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<div align="center">
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<img align="center" src="../../../../../../docs/Figures/EMT/LineLumped/diagram.svg">
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## Model Parameters
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For phase count $N$ and conductor count $K$:
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Symbol | Units | Description | Note
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-----------------|----------------|------------------------------------------|---------------------------------
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$\mathbf{R}'$ | [$\Omega$/m] | Series resistance matrix per unit length | $\mathbb{R}^{3 \times 3}$
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$\mathbf{L}'$ | [H/m] | Series inductance matrix per unit length | $\mathbb{R}^{3 \times 3}$
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$\mathbf{G}'$ | [S/m] | Shunt conductance matrix per unit length | $\mathbb{R}^{3 \times 3}$
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$\mathbf{C}'$ | [F/m] | Shunt capacitance matrix per unit length | $\mathbb{R}^{3 \times 3}$
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$\mathbf{P}_\phi$ | [-] | Permutation matrix mapping each conductor to its phase | $\mathbf{P}_\phi \in \mathbb{R}^{N \times K}$
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$\mathbf{R}'$ | [$\Omega$/m] | Series resistance matrix per unit length | $\mathbb{R}^{K \times K}$
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$\mathbf{L}'$ | [H/m] | Series inductance matrix per unit length | $\mathbb{R}^{K \times K}$
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$\mathbf{G}'$ | [S/m] | Shunt conductance matrix per unit length | $\mathbb{R}^{K \times K}$
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$\mathbf{C}'$ | [F/m] | Shunt capacitance matrix per unit length | $\mathbb{R}^{K \times K}$
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$\mathbf{Z}'$ | [$\Omega$/m] | Series impedance per unit length | $\mathbb{C}^{K \times K}$; future compatibility: supports passing a convolutional model
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$\mathbf{Y}'$ | [S/m] | Shunt admittance per unit length | $\mathbb{C}^{K \times K}$; future compatibility: supports passing a convolutional model
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$\Delta x$ | [m] | Line segment length | $\mathbb{R}$
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## Model Derived Parameters
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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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``` math
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\begin{aligned}
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\mathbf{R} &= \mathbf{R}'\Delta x & \mathbf{G} &= \mathbf{G}'\Delta x \\
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\mathbf{L} &= \mathbf{L}'\Delta x & \mathbf{C} &= \mathbf{C}'\Delta x
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\mathbf{R} &= \mathbf{R}'\Delta x & \mathbf{G} &= \dfrac{\mathbf{G}'\Delta x}{2} \\
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\mathbf{L} &= \mathbf{L}'\Delta x & \mathbf{C} &= \dfrac{\mathbf{C}'\Delta x}{2}
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\end{aligned}
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```
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Symbol | Units | Description | Note
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-----------------|--------|-----------------------|---------------------------------
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$\mathbf{i}$ | [A] | Series line current, directed bus 1 to bus 2 | $\mathbf{i} = [i_a, i_b, i_c]^T \in \mathbb{R}^3$
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$\mathbf{i}$ | [A] | Series conductor current, directed bus 1 to bus 2 | $\mathbf{i} \in \mathbb{R}^K$
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#### Algebraic
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None.
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### External Variables
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External variables enter component model equations but are owned by
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other components. The EMT bus at each port owns the voltage
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variable and provides the equation needed to have a balanced system
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of equations.
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#### Differential
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Symbol | Units | Description | Note
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-----------------|--------|--------------------------|------------------
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$\mathbf{v}_1$ | [V] | Port voltage at bus 1, owned by bus 1 | $\mathbf{v}_1 = [v_{1,a}, v_{1,b}, v_{1,c}]^T \in \mathbb{R}^3$
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$\mathbf{v}_2$ | [V] | Port voltage at bus 2, owned by bus 2 | $\mathbf{v}_2 = [v_{2,a}, v_{2,b}, v_{2,c}]^T \in \mathbb{R}^3$
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$\mathbf{v}_1$ | [V] | Port voltage at bus 1, owned by bus 1 | $\mathbf{v}_1 \in \mathbb{R}^N$
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$\mathbf{v}_2$ | [V] | Port voltage at bus 2, owned by bus 2 | $\mathbf{v}_2 \in \mathbb{R}^N$
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#### Algebraic
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### Differential Equations
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The series relation is written in convolution form as $0=\mathbf{z}*\mathbf{i}+ \mathbf{P}_\phi^T(\mathbf{v}_2 - \mathbf{v}_1)$. For constant parameters this can be expressed as:
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``` math
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0 = \mathbf{R}\,\mathbf{i} + \mathbf{L}\dot{\mathbf{i}} + \mathbf{v}_2 - \mathbf{v}_1
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0 = \mathbf{R}\,\mathbf{i} + \mathbf{L}\dot{\mathbf{i}} + \mathbf{P}_\phi^T(\mathbf{v}_2 - \mathbf{v}_1)
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```
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### Algebraic Equations
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The lumped line contributes to the KCL residual at each port bus.
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Each expression is accumulated into the owning bus residual.
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More generally, the shunt conductor current is
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$\mathbf{y}*\mathbf{P}_\phi^T\mathbf{v}$; for constant parameters this becomes:
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``` math
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\mathbf{i}^\text{inj}_1 := - \dfrac{\mathbf{G}}{2}\,\mathbf{v}_1 - \dfrac{\mathbf{C}}{2}\,\dot{\mathbf{v}}_1 - \mathbf{i}
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```
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``` math
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\mathbf{i}^\text{inj}_2 := - \dfrac{\mathbf{G}}{2}\,\mathbf{v}_2 - \dfrac{\mathbf{C}}{2}\,\dot{\mathbf{v}}_2 + \mathbf{i}
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\begin{aligned}
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\mathbf{i}^\text{inj}_1
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&:= \mathbf{P}_\phi\left(
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- \mathbf{G}\,\mathbf{P}_\phi^T\mathbf{v}_1
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- \mathbf{C}\,\mathbf{P}_\phi^T\dot{\mathbf{v}}_1
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- \mathbf{i}
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\right) \\
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\mathbf{i}^\text{inj}_2
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&:= \mathbf{P}_\phi\left(
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- \mathbf{G}\,\mathbf{P}_\phi^T\mathbf{v}_2
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- \mathbf{C}\,\mathbf{P}_\phi^T\dot{\mathbf{v}}_2
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+ \mathbf{i}
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\right)
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\end{aligned}
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```
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## Initialization
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The initialization assumes a balanced three-phase system. Given bus
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voltages $\mathbf{v}_1(0)$, $\mathbf{v}_2(0)$ and their time
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derivatives $\dot{\mathbf{v}}_1(0)$, $\dot{\mathbf{v}}_2(0)$ from
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the EMT bus, and the power flow phasor series current
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$I = |I| \angle \theta$, the initial series current is:
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The model initializes from an initial series current vector $\mathbf{i}(0)$
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consistent with the selected conductor basis. For a three-conductor balanced phasor
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current $I = |I| \angle \theta$, one common choice is:
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``` math
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\mathbf{i}(0) = \sqrt{2}\,|I|
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DAE consistency:
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``` math
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\dot{\mathbf{i}}(0) = \mathbf{L}^{-1}\left(\mathbf{v}_1(0) - \mathbf{v}_2(0) - \mathbf{R}\,\mathbf{i}(0)\right)
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\dot{\mathbf{i}}(0) = \mathbf{L}^{-1}\left(\mathbf{P}_\phi^T\left(\mathbf{v}_1(0) - \mathbf{v}_2(0)\right) - \mathbf{R}\,\mathbf{i}(0)\right)
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```
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## Model Outputs
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Candidate monitorable outputs include the series line current components
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$i_a$, $i_b$, and $i_c$.
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Candidate monitorable outputs include the components of the series line current
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$\mathbf{i}$.
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Port current injection expressions are documented above as
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$\mathbf{i}^\text{inj}_1$ and $\mathbf{i}^\text{inj}_2$.

‎GridKit/Model/EMT/Component/Line/README.md‎

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The snippets below show model composition only. Final `.case.json` syntax may
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change when the EMT line models are implemented.
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### Universal Line Model
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### Nominal Pi Model
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The ULM is the general case where a `Rational` model defines the
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characteristic admittance $\mathbf{Y}_c$ and a `Propagation` model defines the
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propagation function $\mathbf{H}$.
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The nominal pi model is the constant-parameter lumped case.
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```js
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{
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"class": "LineDistributed",
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"class": "LineLumped",
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"params": {
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"conductors": [...],
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"Yc": {
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"R": ...,
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"L": ...,
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"G": ...,
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"C": ...,
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"dx": ...
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}
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}
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```
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### Frequency-Dependent Pi Model
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The frequency-dependent pi model keeps the lumped topology and replaces the
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constant line matrices with rational parameter models.
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Note: using a `Rational` model with only $\mathbf{D}$ nonzero is an equivalent
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way to define the nominal pi model.
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```js
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{
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"class": "LineLumped",
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"params": {
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"Z": {
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"class": "Rational",
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"params": {
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"D": ...,
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"residues": [...]
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}
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},
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"H": {
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"class": "Propagation",
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"Y": {
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"class": "Rational",
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"params": {
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"tau": [...],
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"dt_min": ...,
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"D": ...,
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"E": ...,
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"poles": [...],
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"C": ...,
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"B": ...
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"residues": [...]
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}
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}
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},
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"dx": ...
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}
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}
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```
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}
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}
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```
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### Universal Line Model
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The ULM is the general case where a `Rational` model defines the
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characteristic admittance $\mathbf{Y}_c$ and a `Propagation` model defines the
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propagation function $\mathbf{H}$.
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```js
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{
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"class": "LineDistributed",
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"params": {
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"conductors": [...],
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"Yc": {
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"class": "Rational",
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"params": {
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"D": ...,
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"E": ...,
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"poles": [...],
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"residues": [...]
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}
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},
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"H": {
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"class": "Propagation",
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"params": {
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"tau": [...],
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"dt_min": ...,
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"poles": [...],
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"C": ...,
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"B": ...
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}
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}
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}
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}
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```

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