# EXDC1

_Source: `GridKit/Model/PhasorDynamics/Exciter/EXDC1/README.md`_

> [!NOTE]  
> This documentation is not in the standard format and EXDC1 is not scheduled to be developed as of 06/26/2025.

   
```{image} ../../../../../Figures/EXDC1.JPG
:align: center
```

   
  Figure 1: Exciter EXDC1 model. Figure courtesy of [PoweWorld](https://www.powerworld.com/WebHelp/).

## Nomenclature

### Inputs
- $V_{REF}$ - voltage reference set point
- $E_{C}$ - output from the terminal voltage transducer
- $V_{S}$ -  power system stabilizer output signal (if present)
- $V_{UEL}$ and $V_{OEL}$ - limiters

### Differential Variables
- $V_{t}$ - terminal voltage (2 is sensed $V_{t}$)
- $V_{B}$ - input to a voltage regulator (3)
- $V_{R}$ - voltage regulator output also know as exciter field voltage (4)
- $V_{F}$ - stabilizing feedback signal (5)
### Parameters
- $T_{R}$ - filter time constant, sec (0)
- $K_{A}$ - voltage regulator gain (40)
- $T_{A}$ - time constant, sec (0.1)
- $T_{B}$ - lag time constant, sec (0)
- $T_{C}$ - lead time constant, sec (0)
- $V_{RMAX}$ - maximum control element output, pu (1)
- $V_{RMIN}$ - minimum control element output, pu (-1)
- $K_{E}$ - exciter field resistance line slope margine, pu (0.1)
- $T_{E}$ - exciter time constant, sec (0.5)
- $K_{F}$ - rate feedback gain, pu (0.05)
- $T_{F1}$ - rate feedback time constant, sec (0.7)
- $E1$ - field voltage value, 1 (2.8)
- $SE1$ - saturation factor at E1, (3.7)
- $E2$ - field voltage value, 2 (3.7)
- $SE2$ - saturation factor at E2, (0.33)

## Equations
First block
```{math}
\dfrac{dV_{t}}{dt}=\dfrac{1}{T_{R}}(E_{C}-V_{t})
```
Second block
```{math}
\dfrac{dx_{1}}{dt}=\dfrac{1}{T_{B}}((V_{REF}-V_{t}-V_{F}+V_{S}+V_{UEL}+V_{OEL})-V_{B})
```
```{math}
V_{B}=x_{1}+\dfrac{T_{C}}{T_{B}}(V_{REF}-V_{t}-V_{F}+V_{S}+V_{UEL}+V_{OEL})
```
Third block
```{math}
\dfrac{dV_{R}}{dt} = \begin{cases}
   \dfrac{1}{T_{A}}(K_{A}V_{B}-V_{R}) &\text{if } V_{RMIN}<=V_{R}<= V_{RMAX}\\
   0 &\text{if } V_{B}>0 \text{  and  } V_{R}>=V_{RMAX} &\text{ also then } V_{R}=V_{RMAX}\\
   0 &\text{if } V_{B}<0 \text{  and  } V_{R}<=V_{RMIN} &\text{ also then } V_{R}=V_{RMIN}\\
\end{cases}
```
Fourth block
```{math}
\dfrac{d\dfrac{E_{FD}}{\omega}}{dt}=\dfrac{1}{T_{E}}(V_{R}-\dfrac{(K_{E}+S_{E})E_{FD}}{\omega})
```
Feedback loop
```{math}
\dfrac{dx_{2}}{dt}=-\dfrac{V_{F}}{T_{F1}}
```
```{math}
V_{F}=x_{2}+\dfrac{K_{F}}{T_{F1}}\dfrac{E_{FD}}{\omega}
```
Saturation is modeled using an alternative quadratic function, with the value of Se specified at two points :
```{math}
Sat(x) = \begin{cases}
   \dfrac{B(x-A)^2}{x} &\text{if } x>A \\
   0 &\text{if } x<=A
\end{cases}
```
same as with the synchronous machines. There are two solutions, and one where $A<1$ should be chosen.
 
## Initialization
```{math}
V_{t}=V_{t_{0}}
```
```{math}
E_{C}=V_{t_{0}}
```
```{math}
(V_{REF}-V_{t}-V_{F}+V_{S}+V_{UEL}+V_{OEL})=V_{B}
```
```{math}
V_{R}=V{R_{0}}
```
```{math}
V_{B}=\dfrac{V{R_{0}}}{K_{A}}
```
```{math}
\dfrac{E_{FD}}{\omega}=\dfrac{E_{FD_{0}}}{\omega}
```
```{math}
V_{R}-\dfrac{(K_{E}+S_{E})E_{FD}}{\omega}=0
```
```{math}
V_{F}=0
```
```{math}
x_{2_{0}}=-\dfrac{K_{F}}{T_{F1}}\dfrac{E_{FD}}{\omega}