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Add EXPIC1 exciter documentation stub
lukelowry Jun 1, 2026
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docs: complete EXPIC1 README [skip ci]
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272 changes: 272 additions & 0 deletions GridKit/Model/PhasorDynamics/Exciter/EXPIC1/README.md
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# **Proportional/Integral Excitation System Model (EXPIC1)**

EXPIC1 is a proportional/integral excitation system with terminal-voltage
sensing, a PI regulator, cascaded regulator filters, stabilizing feedback,
potential/current-source scaling, rectifier loading, exciter limits, saturation,
and an exciter field-voltage state.

Notes:
- Internal voltage and current signals are on model base unless otherwise stated.
- The rectifier loading block $F_{\mathrm{ex}}=f(I_N)$ is the source AC-exciter
loading curve from Fig. 1; it is not a CommonMath helper.
- If $K_P=0$ and $K_I=0$, the diagram sets $V_B=1$.
- If $T_E=0$, the source diagram states $E_{\mathrm{fd}}=E_0$; the exciter
field state becomes algebraic.

## Block Diagram

Standard model of the EXPIC1 Exciter.

<div align="center">
<img align="center" src="../../../../../docs/Figures/PhasorDynamics/EXPIC1_diagram.png">

Figure 1: Exciter EXPIC1 model. Figure courtesy of [PowerWorld](https://www.powerworld.com/WebHelp/)
</div>

## Model Parameters

Symbol | Units | JSON | Description | Typical Value | Note
------------------------------------|----------|-----------|---------------------------------------------------------|---------------|------
$T_R$ | [sec] | `Tr` | Transducer time constant | 0.0 | Block name: `Tr`; if zero, $E_T$ is algebraic
$K_A$ | [p.u.] | `Ka` | PI regulator gain | 1.0 | Block name: `Ka`
$T_{A1}$ | [sec] | `Ta1` | PI regulator numerator time constant | 0.0 | Block name: `Ta1`
$V_{R1}^{\max}$ | [p.u.] | `Vr1` | PI regulator upper output limit | 1.0 | Source label: `VR1`
$V_{R2}^{\min}$ | [p.u.] | `Vr2` | PI regulator lower output limit | -1.0 | Source label: `VR2`
$T_{A2}$ | [sec] | `Ta2` | First denominator time constant in regulator filter | 0.0 | Block name: `Ta2`
$T_{A3}$ | [sec] | `Ta3` | Numerator time constant in regulator filter | 0.0 | Block name: `Ta3`
$T_{A4}$ | [sec] | `Ta4` | Second denominator time constant in regulator filter | 0.0 | Block name: `Ta4`
$V_R^{\max}$ | [p.u.] | `Vrmax` | Maximum regulator output before source multiplier | 1.0 | Block name: `Vrmax`
$V_R^{\min}$ | [p.u.] | `Vrmin` | Minimum regulator output before source multiplier | -1.0 | Block name: `Vrmin`
$K_F$ | [p.u.] | `Kf` | Stabilizing feedback gain | 0.0 | Block name: `Kf`
$T_{F1}$ | [sec] | `Tf1` | First feedback denominator time constant | 0.0 | Block name: `Tf1`
$T_{F2}$ | [sec] | `Tf2` | Second feedback denominator time constant | 0.0 | Block name: `Tf2`
$E_{\mathrm{fd}}^{\max}$ | [p.u.] | `Efdmax` | Maximum exciter input limit | 5.0 | Block name: `EFDMAX`
$E_{\mathrm{fd}}^{\min}$ | [p.u.] | `Efdmin` | Minimum exciter input limit | -5.0 | Block name: `EFDMIN`
$K_E$ | [p.u.] | `Ke` | Exciter field-resistance line-slope margin | 0.1 | Block name: `Ke`
$T_E$ | [sec] | `Te` | Exciter time constant | 0.5 | Block name: `Te`; if zero, $E_{\mathrm{fd}}=E_0$
$E_1$ | [p.u.] | `E1` | First saturation voltage point | 2.8 | Block name: `E1`
$S_E(E_1)$ | [p.u.] | `SE1` | Saturation value at $E_1$ | 0.08 | Block name: `Se1`
$E_2$ | [p.u.] | `E2` | Second saturation voltage point | 3.7 | Block name: `E2`
$S_E(E_2)$ | [p.u.] | `SE2` | Saturation value at $E_2$ | 0.33 | Block name: `Se2`
$K_P$ | [p.u.] | `Kp` | Potential-source voltage coefficient | 0.0 | Source label: `KP`; forms $V_E$
$K_I$ | [p.u.] | `Ki` | Potential-source current coefficient | 0.0 | Source label: `KI`; forms $V_E$
$K_C$ | [p.u.] | `Kc` | Rectifier loading current coefficient | 0.0 | Block name: `Kc`; forms $I_N$

### Parameter Validation

Invalid EXPIC1 parameter sets are rejected by the following checks.

```math
\begin{aligned}
&T_R \ge 0,\quad T_{A1}\ge 0,\quad T_{A2}\ge 0,\quad T_{A3}\ge 0,\quad T_{A4}\ge 0 \\
&T_{F1}\ge 0,\quad T_{F2}\ge 0,\quad T_E\ge 0 \\
&V_{R2}^{\min}\le V_{R1}^{\max},\quad V_R^{\min}\le V_R^{\max},\quad E_{\mathrm{fd}}^{\min}\le E_{\mathrm{fd}}^{\max}
\end{aligned}
```

The saturation points are either disabled together or define a valid positive
two-point quadratic fit.

### Model Derived Parameters

The saturation curve is fitted from the two supplied saturation points. If both
saturation factors are zero, use $S_A=0$ and $S_B=0$. Otherwise:

```math
\begin{aligned}
C &= \sqrt{\dfrac{S_E(E_2)}{S_E(E_1)}} \\
S_A &= \dfrac{C E_1 - E_2}{C - 1} \\
S_B &= \dfrac{S_E(E_1)}{(E_1 - S_A)^2}
\end{aligned}
```

The source calculation uses explicit real and imaginary terminal voltage/current
components:

```math
\begin{aligned}
V_{\mathrm{src}}^{\mathrm{r}} &= K_P V_{\mathrm{r}} - K_I I_{\mathrm{i}} \\
V_{\mathrm{src}}^{\mathrm{i}} &= K_P V_{\mathrm{i}} + K_I I_{\mathrm{r}}
\end{aligned}
```

## Model Variables

### Internal Variables

#### Differential

Symbol | Units | Description | Note
------------------------------------|--------|---------------------------------------------------------|------
$E_{\mathrm{fd}}$ | [p.u.] | Field-voltage output state | State 1 in Fig. 1; algebraic when $T_E=0$
$E_T$ | [p.u.] | Sensed terminal voltage | State 2 in Fig. 1; source label: `Sensed Vt`; algebraic when $T_R=0$
$V_A$ | [p.u.] | PI regulator output | State 3 in Fig. 1
$x_{R1}$ | [p.u.] | First regulator filter state | State 4 in Fig. 1; source label: `VR1`
$V_R$ | [p.u.] | Regulator output before source multiplier | State 5 in Fig. 1; source label: `VR`
$V_{F1}$ | [p.u.] | First feedback filter state | State 6 in Fig. 1; source label: `VF1`
$V_F$ | [p.u.] | Stabilizing feedback output | State 7 in Fig. 1; source label: `VF`

#### Algebraic

Symbol | Units | Description | Note
------------------------------------|--------|---------------------------------------------------------|------
$e_V$ | [p.u.] | Voltage-error signal after feedback | Summing junction after $E_T$
$V_{\mathrm{src}}^{\mathrm{r}}$ | [p.u.] | Real component of the source expression | From terminal voltage/current components
$V_{\mathrm{src}}^{\mathrm{i}}$ | [p.u.] | Imaginary component of the source expression | From terminal voltage/current components
$V_{\mathrm{src}}$ | [p.u.] | Potential/current source magnitude | Nonnegative source magnitude
$I_N$ | [p.u.] | Normalized exciter loading current | Source label: `IN`; satisfies $V_{\mathrm{src}}I_N=K_C I_{\mathrm{fd}}$ when source scaling is active
$F_{\mathrm{ex}}$ | [p.u.] | Rectifier loading factor | Source label: `FEX`; source curve $F_{\mathrm{ex}}=f(I_N)$
$V_B$ | [p.u.] | Source multiplier after rectifier loading | Product of $V_{\mathrm{src}}$ and $F_{\mathrm{ex}}$, or 1 when $K_P=K_I=0$
$E_0$ | [p.u.] | Limited exciter input | Limited by $E_{\mathrm{fd}}^{\min}$ and $E_{\mathrm{fd}}^{\max}$
$S_E$ | [p.u.] | Saturation coefficient evaluated at $E_{\mathrm{fd}}$ | Uses derived saturation curve

### External Variables

#### Differential

None.

#### Algebraic

Symbol | Units | Description | Note
------------------------------------|--------|---------------------------------------------------------|------
$E_C$ | [p.u.] | Compensated terminal voltage magnitude | Source label: `EC`
$V_{\mathrm{ref}}$ | [p.u.] | Voltage-control reference | Source label: `VREF`
$V_{\mathrm{uel}}$ | [p.u.] | Under-excitation limiter input | Source label: `VUEL`; optional, defaults to zero
$V_S$ | [p.u.] | Stabilizer input signal | Source label: `VS`; optional, defaults to zero
$V_{\mathrm{oel}}$ | [p.u.] | Over-excitation limiter input | Source label: `VOEL`; optional, defaults to zero
$V_{\mathrm{r}}$ | [p.u.] | Terminal-voltage real component | Source label: `VT`
$V_{\mathrm{i}}$ | [p.u.] | Terminal-voltage imaginary component | Source label: `VT`
$I_{\mathrm{r}}$ | [p.u.] | Terminal-current real component | Source label: `IT`
$I_{\mathrm{i}}$ | [p.u.] | Terminal-current imaginary component | Source label: `IT`
$I_{\mathrm{fd}}$ | [p.u.] | Machine field current | Source label: `IFD`

## Model Equations

### Differential Equations

```math
\begin{aligned}
0 &= -T_R\dot E_T - E_T + E_C \\
0 &=
-\dot V_A
+ \text{antiwindup}\!\left(
V_A,
K_A e_V,
V_{R2}^{\min},
V_{R1}^{\max}
\right) \\
0 &= -T_{A2}\dot x_{R1} - x_{R1} + V_A \\
0 &= -T_{A4}\dot V_R - V_R + x_{R1} + T_{A3}\dot x_{R1} \\
0 &= -T_{F1}\dot V_{F1} - V_{F1} + V_R \\
0 &= -T_{F2}\dot V_F - V_F + K_F\dot V_{F1} \\
0 &= -T_E\dot E_{\mathrm{fd}} + E_0 - (K_E + S_E)E_{\mathrm{fd}}
\end{aligned}
```

CommonMath defines the [Anti-Windup](../../../../CommonMath.md#anti-windup-indicator)
target and smooth approximation.

### Algebraic Equations

```math
\begin{aligned}
0 &= -e_V + V_{\mathrm{ref}} + V_{\mathrm{uel}} + V_S + V_{\mathrm{oel}} - E_T - V_F \\
0 &= -V_{\mathrm{src}}^{\mathrm{r}} + K_P V_{\mathrm{r}} - K_I I_{\mathrm{i}} \\
0 &= -V_{\mathrm{src}}^{\mathrm{i}} + K_P V_{\mathrm{i}} + K_I I_{\mathrm{r}} \\
0 &= -V_{\mathrm{src}}^2
+ \left(V_{\mathrm{src}}^{\mathrm{r}}\right)^2
+ \left(V_{\mathrm{src}}^{\mathrm{i}}\right)^2 \\
0 &=
\begin{cases}
-I_N & K_P=0\ \text{and}\ K_I=0 \\
-V_{\mathrm{src}}I_N + K_C I_{\mathrm{fd}} & \text{otherwise}
\end{cases} \\
0 &= -F_{\mathrm{ex}}
+ \begin{cases}
1 & K_P=0\ \text{and}\ K_I=0 \\
f(I_N) & \text{otherwise}
\end{cases} \\
0 &= -V_B
+ \begin{cases}
1 & K_P=0\ \text{and}\ K_I=0 \\
V_{\mathrm{src}}F_{\mathrm{ex}} & \text{otherwise}
\end{cases} \\
0 &= -E_0 + \text{clamp}(V_B V_R, E_{\mathrm{fd}}^{\min}, E_{\mathrm{fd}}^{\max}) \\
0 &= -S_E + S_B\,q(E_{\mathrm{fd}} - S_A)
\end{aligned}
```

CommonMath defines helper targets for [clamp](../../../../CommonMath.md#derived-functions)
and the primitive [quadratic ramp](../../../../CommonMath.md#primitives) $q$.
The rectifier loading function $f(I_N)$ is the source curve shown in Fig. 1.
The $V_{\mathrm{src}}$ residual uses the nonnegative branch of the squared
source-magnitude equation.

## Initialization

For a standard unsaturated start, the machine initializes
$E_{\mathrm{fd},0}$ and $I_{\mathrm{fd},0}$ first. EXPIC1 reads those values,
sets all internal derivatives to zero, and evaluates:

```math
\begin{aligned}
E_{T,0} &= E_{C,0} \\
V_{\mathrm{src},0}^{\mathrm{r}} &= K_P V_{\mathrm{r},0} - K_I I_{\mathrm{i},0} \\
V_{\mathrm{src},0}^{\mathrm{i}} &= K_P V_{\mathrm{i},0} + K_I I_{\mathrm{r},0} \\
V_{\mathrm{src},0}
&= \sqrt{
\left(V_{\mathrm{src},0}^{\mathrm{r}}\right)^2
+ \left(V_{\mathrm{src},0}^{\mathrm{i}}\right)^2
} \\
0 &=
\begin{cases}
-I_{N,0} & K_P=0\ \text{and}\ K_I=0 \\
-V_{\mathrm{src},0}I_{N,0} + K_C I_{\mathrm{fd},0} & \text{otherwise}
\end{cases} \\
F_{\mathrm{ex},0} &=
\begin{cases}
1 & K_P=0\ \text{and}\ K_I=0 \\
f(I_{N,0}) & \text{otherwise}
\end{cases} \\
V_{B,0} &=
\begin{cases}
1 & K_P=0\ \text{and}\ K_I=0 \\
V_{\mathrm{src},0}F_{\mathrm{ex},0} & \text{otherwise}
\end{cases} \\
S_{E,0} &= S_B\,q(E_{\mathrm{fd},0} - S_A) \\
E_{0,0} &= (K_E + S_{E,0})E_{\mathrm{fd},0} \\
V_{R,0} &= \dfrac{E_{0,0}}{V_{B,0}} \\
x_{R1,0} &= V_{R,0} \\
V_{A,0} &= x_{R1,0} \\
V_{F1,0} &= V_{R,0} \\
V_{F,0} &= 0 \\
e_{V,0} &= \dfrac{V_{A,0}}{K_A} \\
V_{\mathrm{ref},0}
&= e_{V,0} + E_{T,0} + V_{F,0}
- V_{\mathrm{uel},0} - V_{S,0} - V_{\mathrm{oel},0}
\end{aligned}
```

This closed-form start requires nonzero $K_A$ and $V_{B,0}$, inactive PI and
exciter limits, and residual consistency with the source curve. When
$K_P$ and $K_I$ are not both zero, it also requires $V_{\mathrm{src},0}\ne 0$.
If $T_E=0$, the final exciter residual is algebraic and requires
$E_{\mathrm{fd},0}=E_{0,0}$. Starts that bind the PI regulator, cascaded
regulator, or exciter limits are outside these closed-form equations.

## Model Outputs

Output | Units | Description | Note
----------------|--------|-------------------------------------|------
`efd` | [p.u.] | Field-voltage output | $E_{\mathrm{fd}}$
`et` | [p.u.] | Sensed terminal voltage | $E_T$
`va` | [p.u.] | PI regulator state | $V_A$
`vr1` | [p.u.] | First regulator filter state | $x_{R1}$
`vr` | [p.u.] | Regulator output | $V_R$
`vf1` | [p.u.] | First feedback filter state | $V_{F1}$
`vf` | [p.u.] | Stabilizing feedback output | $V_F$
`vb` | [p.u.] | Source multiplier | $V_B$
`in` | [p.u.] | Normalized exciter loading current | $I_N$
`fex` | [p.u.] | Rectifier loading factor | $F_{\mathrm{ex}}$
`se` | [p.u.] | Saturation coefficient | $S_E$
1 change: 1 addition & 0 deletions GridKit/Model/PhasorDynamics/Exciter/README.md
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Expand Up @@ -14,4 +14,5 @@ device internal voltage.
There are a few standard Exciter models
- IEEE Type 1 Excitation Model (See [IEEET1](IEEET1/README.md))
- IEEE DC1 Excitation Model (See [EXDC1](EXDC1/README.md))
- EXPIC1 Excitation Model (See [EXPIC1](EXPIC1/README.md))
- Simplified Excitation System Model (See [SEXS-PTI](SEXS-PTI/README.md))
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