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Update modeling documentation structure (#128)
* Initial update to GENROU docs * Initial update to GENSAL docs * Initial update to load docs * Initial BusFault docs * Initial branch docs update * Correct load math markers --------- Co-authored-by: abirchfield <abirchfield@users.noreply.github.com> Co-authored-by: pelesh <peless@ornl.gov>
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‎src/Model/PhasorDynamics/Branch/README.md‎

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# Branch Model
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# Transmission Line Branch Model
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Transmission lines and different types of transformers (traditional, Load
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Tap-Changing transformers (LTC) and Phase Angle Regulators (PARs)) can be
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modeled with a common branch model.
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## Transmission Line Model
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The most common circuit that is used to represent the transmission line model
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is $`\pi`$ circuit as shown in Figure 1. The positive flow direction is into
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buses. Commonly used convention is to define positive direction to be from
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Figure 1: Transmission line $`\pi`$ equivalent circuit
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</div>
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Here
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``` math
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Z = R + jX
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```
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and
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## Model Parameters
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Symbol | Units | Description | Note
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------------|---------|---------------------------------| ------
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$R$ | [p.u.] | Branch series resistance |
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$X$ | [p.u.] | Branch series reactance |
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$G$ | [p.u.] | Branch shunt conductance |
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$B$ | [p.u.] | Branch shunt susceptance |
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### Model Derived Parameters
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Note the difference between little-g and big-G, little-b, big-B in these equations.
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``` math
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Y = G + jB,
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```
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where $`R`$ is line series resistance, $`X`$ is line series reactance, $`B`$ is
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line shunt charging, and $`G`$ is line shunt conductance. As can be seen from
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Figure 1 total $`B`$ and $`G`$ are separated between two buses. The current
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entering bus 1 can be obtained from Kirchhoff's current law as
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```math
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I_1 = y(V_2 - V_1) - \frac{Y}{2} V_1,
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```
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where $`V_1`$ and $`V_2`$ are respective bus voltages and
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```math
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y = \frac{1}{Z} = \frac{R}{R^2+X^2} + j\frac{-X}{R^2+X^2} = g + jb.
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```
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Similarly, current entering bus 2 is given as
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```math
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I_2 = y(V_1 - V_2) + \frac{Y}{2} V_2.
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```
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These equations can be written in a compact form as:
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```math
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\begin{bmatrix}
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I_{1}\\
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I_{2}
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\end{bmatrix}
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= \mathbf{Y}
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\begin{bmatrix}
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V_{1}\\
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V_{2}
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\end{bmatrix}
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```
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where:
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```math
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\mathbf{Y}_{TL}=\begin{bmatrix}
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-(g + jb) - \dfrac{G+jB}{2} & g + jb \\
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g + jb & -(g + jb) - \dfrac{G+jB}{2}
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\end{bmatrix}
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\begin{aligned}
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g &=\dfrac{R}{R^2 + X^2} \\
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b &= -\dfrac{X}{R^2 + X^2}\\
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\end{aligned}
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```
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### Branch contributions to residuals at adjacent buses
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After some algebra, one obtains expressions for real and imaginary components
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for the currents entering adjacent buses:
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```math
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I_{r1} = -\left(g + \frac{G}{2}\right) V_{r1} + \left(b + \frac{B}{2} \right) V_{i1}
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+ g V_{r2} - b V_{i2}
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```
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## Model Variables
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```math
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I_{i1} = -\left(b + \frac{B}{2} \right) V_{r1} - \left(g + \frac{G}{2}\right) V_{i1}
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+ b V_{r2} + g V_{i2}
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```
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### Internal Variables
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```math
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I_{r2} = g V_{r1} - b V_{i1}
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- \left(g + \frac{G}{2}\right) V_{r2} + \left(b + \frac{B}{2} \right) V_{i2}
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```
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#### Differential
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None.
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```math
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I_{i2} = b V_{r1} + g V_{i1}
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- \left(b + \frac{B}{2} \right) V_{r2} - \left(g + \frac{G}{2}\right) V_{i2}
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#### Algebraic
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Symbol | Units | Description | Note
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------------|---------|---------------------------------| ------
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$I_{r1}$ | [p.u.] | Terminal current, real component, bus 1 | Read by bus
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$I_{i1}$ | [p.u.] | Terminal current, imaginary component, bus 1 | Read by bus
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$I_{r2}$ | [p.u.] | Terminal current, real component, bus 2 | Read by bus
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$I_{i2}$ | [p.u.] | Terminal current, imaginary component, bus 2 | Read by bus
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### External Variables
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#### Differential
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None.
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#### Algebraic
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Symbol | Units | Description | Note
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------------|---------|---------------------------------| ------
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$V_{r1}$ | [p.u.] | Terminal voltage, real component, bus 1 | owned by bus object
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$V_{i1}$ | [p.u.] | Terminal voltage, imaginary component, bus 1 | owned by bus object
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$V_{r2}$ | [p.u.] | Terminal voltage, real component, bus 2 | owned by bus object
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$V_{i2}$ | [p.u.] | Terminal voltage, imaginary component, bus 2 | owned by bus object
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## Model Equations
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### Differential Equations
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None.
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### Algebraic Equations
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``` math
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\begin{aligned}
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0 &= - I_{r1} -\left(g + \dfrac{G}{2}\right) V_{r1} + \left(b + \dfrac{B}{2}\right) V_{i1} + g V_{r2} - b V_{i2}\\
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0 &= I_{i1} - \left(b + \dfrac{B}{2}\right) V_{r1} - \left(g + \dfrac{G}{2}\right) V_{i1} + b V_{r2} + g V_{i2}\\
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0 &= I_{r2} + g V_{r1} - b V_{i1} - \left(g + \dfrac{G}{2}\right) V_{r2} + \left(b + \dfrac{B}{2}\right) V_{i2}\\
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0 &= I_{i2} + b V_{r1} + g V_{i1} - \left(b + \dfrac{B}{2}\right) V_{r2} - \left(g + \dfrac{G}{2}\right) V_{i2}
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\end{aligned}
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```
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## Transformer Branch Model
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# Transformer Branch Model
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**Note: Transformer model not yet implemented**
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‎src/Model/PhasorDynamics/BusFault/README.md‎

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# Bus Fault
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Represents an impedance fault at a bus. This device can exist in two states, on or off, controlled by the user. Following a state change, generally the solver needs to be reset as this is a discrete event.
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## Model Parameters
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Symbol | Units | Description | Note
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------------|---------|---------------------------------| ------
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$R$ | [p.u.] | Fault resistance |
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$X$ | [p.u.] | Fault reactance |
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$U$ | [unitless] | Binary status $$\in \{0, 1\}$$ | Set by user to put fault on or off.
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### Model Derived Parameters
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``` math
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\begin{aligned}
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G &=\dfrac{R}{R^2+ X^2} \\
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B &= -\dfrac{X}{R^2 + X^2}\\
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\end{aligned}
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```
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## Model Variables
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### Internal Variables
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#### Differential
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None.
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#### Algebraic
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Symbol | Units | Description | Note
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------------|---------|---------------------------------| ------
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$I_r$ | [p.u.] | Terminal current, real component | Read by bus
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$I_i$ | [p.u.] | Terminal current, imaginary component | Read by bus
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### External Variables
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#### Differential
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None.
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#### Algebraic
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Symbol | Units | Description | Note
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------------|---------|---------------------------------| ------
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$V_r$ | [p.u.] | Terminal voltage, real component | owned by bus object
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$V_i$ | [p.u.] | Terminal voltage, imaginary component | owned by bus object
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## Model Equations
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### Differential Equations
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None.
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### Algebraic Equations
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``` math
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\begin{aligned}
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0 &= -I_{r} + U (-G V_{r} + B V_{i}) \\
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0 &= -I_{i} + U (-B V_{r} - G V_{i})
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\end{aligned}
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```

‎src/Model/PhasorDynamics/Load/README.md‎

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Load modeling is one of the more complex aspects of power system dynamics.
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The simplest model, which is used for this challenge problem, is to model
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the load as a complex shunt impedance with the impedance given by:
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the load as a complex shunt impedance $$ R + jX $$.
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## Model Parameters
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Symbol | Units | Description | Note
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------------|---------|---------------------------------| ------
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$R$ | [p.u.] | Load resistance |
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$X$ | [p.u.] | Load reactance |
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### Model Derived Parameters
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``` math
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Z = R + jX
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```
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where $`R`$ is the load resistance, $`X`$ is the load reactance. The current
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drawn by the load is then obtained as
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```math
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I_{\mathrm{load}} = \frac{V_{\mathrm{bus}}}{Z},
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\begin{aligned}
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G &=\dfrac{R}{R^2 + X^2} \\
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B &= -\dfrac{X}{R^2 + X^2}\\
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\end{aligned}
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```
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where $`V_{bus}`$ is the voltage on the bus to which the load is connected.
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After some algebra, one obtains expressions for real and imaginary components
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for the currents entering the bus:
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```math
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I_{r} = -g V_{r} + b V_{i}
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```
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```math
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I_{i} = - b V_{r} - g V_{i}
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```
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where
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```math
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g = \frac{R}{R^2+X^2} ~~~\mathrm{and}~~~ b = \frac{-X}{R^2+X^2}.
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## Model Variables
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### Internal Variables
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#### Differential
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None.
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#### Algebraic
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Symbol | Units | Description | Note
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------------|---------|---------------------------------| ------
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$I_r$ | [p.u.] | Terminal current, real component | Read by bus
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$I_i$ | [p.u.] | Terminal current, imaginary component | Read by bus
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### External Variables
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#### Differential
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None.
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#### Algebraic
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Symbol | Units | Description | Note
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------------|---------|---------------------------------| ------
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$V_r$ | [p.u.] | Terminal voltage, real component | owned by bus object
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$V_i$ | [p.u.] | Terminal voltage, imaginary component | owned by bus object
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## Model Equations
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### Differential Equations
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None.
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### Algebraic Equations
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``` math
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\begin{aligned}
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0 &= I_{r} +G V_{r} - B V_{i} \\
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0 &= I_{i} +B V_{r} + G V_{i}
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\end{aligned}
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```

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