# SEXS-PTI

_Source: `GridKit/Model/PhasorDynamics/Exciter/SEXS-PTI/README.md`_

## **Simplified Excitation System Model (SEXS-PTI)**

### Block Diagram

Simplified excitation system model.

   
```{image} ../../../../../Figures/SEXS_PTI_DIAGRAM.png
:align: center
```


  Figure 1: Exciter SEXS-PTI model. Figure courtesy of [PowerWorld](https://www.powerworld.com/WebHelp/)

### Model Parameters

Symbol          | Units  | Description                                   | Typical Value | Note
----------------|--------|-----------------------------------------------|---------------|------
$T_A$           | [sec]  | Numerator time constant of lag-lead block     |               |
$T_B$           | [sec]  | Denominator time constant of lag-lead block   |               |
$T_E$           | [sec]  | Exciter field time constant                   |               |
$K$             | [p.u.] | Voltage regulator gain                        |               |
$E_{fd}^{\max}$ | [p.u.] | Maximum excitation output                     |               |
$E_{fd}^{\min}$ | [p.u.] | Minimum excitation output                     |               |

PowerWorld/PSS/E SEXS_PTI data often gives $T_A/T_B$ as a ratio. GridKit stores
$T_A$ and $T_B$ separately, so convert ratio-format data with
$T_A = (T_A/T_B)T_B$ before passing parameters to the model.

### Model Variables

#### Internal Variables

##### Differential

Symbol    | Units  | Description                       | Note
----------|--------|-----------------------------------|-----
$V_R$     | [p.u.] | Lag-lead block state              |
$E_{fd}$  | [p.u.] | Exciter field voltage output      |

##### Algebraic

Symbol    | Units  | Description                       | Note
----------|--------|-----------------------------------|-----
$V_{tr}$  | [p.u.] | Terminal voltage error signal     |

#### External Variables

##### Differential

None.

##### Algebraic

Symbol          | Units  | Description                                  | Note
----------------|--------|----------------------------------------------|-----
$E_C$           | [p.u.] | Compensated machine terminal voltage magnitude | Computed from bus voltage
$V_{ref}$       | [p.u.] | Reference voltage                            | Set during initialization
$V_S$           | [p.u.] | Stabilizer output                            | Optional, defaults to zero
$V_{OEL}$       | [p.u.] | Over-excitation limiter signal               | Constant zero until modeled
$V_{UEL}$       | [p.u.] | Under-excitation limiter signal              | Constant zero until modeled

### Model Equations

#### Differential Equations

The SEXS-PTI differential equations, as derived from the model diagram. Define the pre-limit derivative of $E_{fd}$

```{math}
f = \dfrac{1}{T_E}\left[-E_{fd} + \dfrac{K}{T_B}(-V_R + T_A V_{tr})\right]
```

so that $\dot E_{fd}$ can be written in piecewise form compactly.

```{math}
\begin{aligned}
  \dot V_R      &= -V_{tr} + \dfrac{1}{T_B}(-V_R + T_A V_{tr}) \\
  \dot E_{fd}   &=
  \begin{cases}
     f
        &  \text{if } (E_{fd}^{\min} < E_{fd} < E_{fd}^{\max}) & \lor \\
        &  \quad (E_{fd} \leq E_{fd}^{\min} \land f > 0)       & \lor \\
        &  \quad (E_{fd} \geq E_{fd}^{\max} \land f < 0)            \\
     0  &  \text{else}
  \end{cases}
\end{aligned}
```

In simulation the piecewise form above is replaced with a smooth approximation where $\phi$ is GridKit's smooth anti-windup indicator. See [CommonMath: Anti-Windup Indicator](../../../common-math.md#derived-functions) for its definition, behavior, and design rationale.

#### Algebraic Equations

```{math}
\begin{aligned}
0&=-V_{tr}-E_C+V_{ref}+V_S+V_{OEL}+V_{UEL}
\end{aligned}
```

### Initialization

The generator initializes the EFD signal first. SEXS-PTI then reads that value
as $E_{fd,0}$ and assumes steady state with $V_S=V_{OEL}=V_{UEL}=0$:

```{math}
\begin{aligned}
E_C &= \sqrt{V_r^2+V_i^2} \\
V_{tr,0} &= \dfrac{E_{fd,0}}{K} \\
V_{R,0} &= (T_A - T_B)V_{tr,0} \\
V_{ref} &= E_C + V_{tr,0}
\end{aligned}
```

All derivatives initialize to zero.