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Add EXAC2 exciter documentation stub
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243 changes: 243 additions & 0 deletions GridKit/Model/PhasorDynamics/Exciter/EXAC2/README.md
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# **IEEE Type AC2 Excitation System Model (EXAC2)**

EXAC2 is an IEEE Type AC excitation system with a voltage transducer, lead-lag
input compensation, limited voltage-amplifier state, low-value gate, voltage
regulator limits, exciter alternator state, stabilizing feedback, rectifier
loading, saturation, and optional speed multiplier.

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.
- The source diagram labels the optional multiplier input as `Speed`; GridKit
uses machine speed deviation, so the enabled multiplier is $1+\omega$.

## Block Diagram

Standard model of the EXAC2 Exciter.

<div align="center">
<img align="center" src="../../../../../docs/Figures/PhasorDynamics/EXAC2_diagram.png">

Figure 1: Exciter EXAC2 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, $V_C$ is algebraic
$T_B$ | [sec] | `Tb` | Lag time constant for voltage-regulator input lead-lag | 0.0 | Block name: `Tb`
$T_C$ | [sec] | `Tc` | Lead time constant for voltage-regulator input lead-lag | 0.0 | Block name: `Tc`
$K_A$ | [p.u.] | `Ka` | Voltage-amplifier gain | 40.0 | Block name: `Ka`
$T_A$ | [sec] | `Ta` | Voltage-amplifier time constant | 0.1 | Block name: `Ta`
$V_A^{\max}$ | [p.u.] | `VaMax` | Maximum voltage-amplifier output | 1.0 | Block name: `VAMAX`
$V_A^{\min}$ | [p.u.] | `VaMin` | Minimum voltage-amplifier output | -1.0 | Block name: `VAMIN`
$K_B$ | [p.u.] | `Kb` | Regulator pre-limit gain after low-value gate | 1.0 | Block name: `KB`
$V_R^{\max}$ | [p.u.] | `Vrmax` | Maximum voltage-regulator output | 1.0 | Block name: `Vrmax`
$V_R^{\min}$ | [p.u.] | `Vrmin` | Minimum voltage-regulator output | -1.0 | Block name: `Vrmin`
$T_E$ | [sec] | `Te` | Exciter alternator time constant | 0.5 | Block name: `Te`
$K_L$ | [p.u.] | `Kl` | Field-current limiter feedback gain | 0.0 | Block name: `KL`
$K_H$ | [p.u.] | `Kh` | Regulator feedback path gain | 0.0 | Block name: `KH`
$K_F$ | [p.u.] | `Kf` | Stabilizing feedback gain | 0.05 | Block name: `Kf`
$T_F$ | [sec] | `Tf` | Stabilizing feedback time constant | 0.7 | Block name: `Tf`; if zero, $V_F$ is algebraic
$K_C$ | [p.u.] | `Kc` | Rectifier loading current coefficient | 0.0 | Block name: `Kc`; forms $I_N$
$K_D$ | [p.u.] | `Kd` | Demagnetizing factor feedback gain | 0.0 | Block name: `Kd`
$K_E$ | [p.u.] | `Ke` | Exciter field-resistance line-slope margin | 0.1 | Block name: `Ke`
$V_{\mathrm{lr}}$ | [p.u.] | `VLr` | Low-value gate lower reference | 0.0 | Source label: `VLR`
$E_1$ | [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`
$s_{\mathrm{spd}}$ | [binary] | `Spdmlt` | Speed multiplier flag | 0 | Block name: `Spdmlt`; 1 enables the speed multiplier

### Parameter Validation

Invalid EXAC2 parameter sets are rejected by the following checks.

```math
\begin{aligned}
&K_A > 0 \\
&T_R \ge 0,\quad T_A > 0,\quad T_B \ge 0,\quad T_C \ge 0,\quad T_E > 0,\quad T_F \ge 0 \\
&T_B > 0\quad\text{or}\quad(T_B = 0\ \text{and}\ T_C = 0) \\
&V_A^{\min} \le V_A^{\max},\quad V_R^{\min} \le V_R^{\max} \\
&s_{\mathrm{spd}} \in \{0,1\}
\end{aligned}
```

The saturation points follow the same two-point validation used by other
exciter READMEs: both saturation values are zero, or both points define a valid
positive 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}
```

## Model Variables

### Internal Variables

#### Differential

Symbol | Units | Description | Note
------------------------------------|--------|---------------------------------------------------------|------
$V_E$ | [p.u.] | Exciter alternator voltage state before output multipliers | State 1 in Fig. 1; source label: `VE`
$V_C$ | [p.u.] | Sensed compensated voltage | State 2 in Fig. 1; source label: `Sensed Vt`; algebraic when $T_R=0$
$V_A$ | [p.u.] | Limited voltage-amplifier output | State 3 in Fig. 1; source label: `VA`
$x_{\mathrm{ll}}$ | [p.u.] | Lead-lag block state | State 4 in Fig. 1; source label: `VLL`
$V_F$ | [p.u.] | Stabilizing feedback washout output | State 5 in Fig. 1; source label: `VF`; algebraic when $T_F=0$

#### Algebraic

Symbol | Units | Description | Note
------------------------------------|--------|---------------------------------------------------------|------
$e_V$ | [p.u.] | Voltage-regulator input error before lead-lag block | Summing junction after sensed voltage
$V_{\mathrm{ll}}$ | [p.u.] | Lead-lag output | Input to voltage amplifier
$V_H$ | [p.u.] | Regulator feedback path signal | Block name: `KH`
$V_L$ | [p.u.] | Field-current limiter low-value gate input | Block name: `KL`; lower reference $V_{\mathrm{lr}}$
$V_{\mathrm{lv}}$ | [p.u.] | Low-value gate output | Lesser of amplifier path and limiter path
$V_R$ | [p.u.] | Voltage-regulator output | Limited by $V_R^{\min}$ and $V_R^{\max}$
$S_E$ | [p.u.] | Saturation coefficient evaluated at $V_E$ | Uses derived saturation curve
$I_N$ | [p.u.] | Normalized exciter loading current | Source label: `IN`; satisfies $V_E I_N=K_C I_{\mathrm{fd}}$
$F_{\mathrm{ex}}$ | [p.u.] | Rectifier loading factor | Source label: `FEX`; source curve $F_{\mathrm{ex}}=f(I_N)$
$V_{\mathrm{fe}}$ | [p.u.] | Exciter feedback signal | Sum of saturation/resistance and $K_D I_{\mathrm{fd}}$ paths
$E_{\mathrm{fd}}$ | [p.u.] | Field-voltage output | Output after rectifier loading and optional speed multiplier

### 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`
$V_S$ | [p.u.] | Stabilizer input signal | Source label: `VS`; optional, defaults to zero
$V_{\mathrm{uel}}$ | [p.u.] | Under-excitation limiter input | Source label: `VUEL`; optional, defaults to zero
$V_{\mathrm{oel}}$ | [p.u.] | Over-excitation limiter input | Source label: `VOEL`; optional, defaults to zero
$I_{\mathrm{fd}}$ | [p.u.] | Machine field current | Source label: `IFD`
$\omega$ | [p.u.] | Machine speed deviation | Source label: `Speed`; optional when $s_{\mathrm{spd}}=0$

## Model Equations

### Differential Equations

```math
\begin{aligned}
0 &= -T_R\dot V_C - V_C + E_C \\
0 &= -T_B\dot x_{\mathrm{ll}} - x_{\mathrm{ll}} + e_V \\
0 &= -T_A\dot V_A
+ \text{antiwindup}\!\left(
V_A,
-V_A + K_A V_{\mathrm{ll}},
V_A^{\min},
V_A^{\max}
\right) \\
0 &= -T_E\dot V_E + V_R - V_{\mathrm{fe}} \\
0 &= -T_F\dot V_F - V_F + K_F\dot V_E
\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_S + V_{\mathrm{uel}} + V_{\mathrm{oel}} - V_C - V_F \\
0 &= -T_B(V_{\mathrm{ll}} - x_{\mathrm{ll}}) + T_C(e_V - x_{\mathrm{ll}}) \\
0 &= -V_H + K_H V_{\mathrm{fe}} \\
0 &= -V_L + V_{\mathrm{lr}} + K_L V_{\mathrm{fe}} \\
0 &= -V_{\mathrm{lv}} + \text{min}(V_A + V_H,\ V_L) \\
0 &= -V_R + \text{clamp}(K_B V_{\mathrm{lv}}, V_R^{\min}, V_R^{\max}) \\
0 &= -S_E + S_B\,q(V_E - S_A) \\
0 &= -V_E I_N + K_C I_{\mathrm{fd}} \\
0 &= -F_{\mathrm{ex}} + f(I_N) \\
0 &= -V_{\mathrm{fe}} + (K_E + S_E)V_E + K_D I_{\mathrm{fd}} \\
0 &= -E_{\mathrm{fd}}
+ \left(1+s_{\mathrm{spd}}\omega\right)F_{\mathrm{ex}}V_E
\end{aligned}
```

CommonMath defines helper targets for [min and 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.
When $T_B=T_C=0$, the lead-lag block is bypassed so $V_{\mathrm{ll}}=e_V$.

## Initialization

The machine initializes $E_{\mathrm{fd}}$ and $I_{\mathrm{fd}}$ first. For a
standard unsaturated start, EXAC2 reads those values, sets all internal
derivatives to zero, and first solves the coupled rectifier-loading equations:

```math
\begin{aligned}
0 &= -V_{E,0}I_{N,0} + K_C I_{\mathrm{fd},0} \\
0 &= -F_{\mathrm{ex},0} + f(I_{N,0}) \\
0 &= -E_{\mathrm{fd},0}
+ \left(1+s_{\mathrm{spd}}\omega_0\right)F_{\mathrm{ex},0}V_{E,0}
\end{aligned}
```

Then evaluate the feedback path:

```math
\begin{aligned}
S_{E,0} &= S_B\,q(V_{E,0} - S_A) \\
V_{\mathrm{fe},0} &= (K_E + S_{E,0})V_{E,0} + K_D I_{\mathrm{fd},0}
\end{aligned}
```

Then solve the low-value gate and voltage-regulator chain:

```math
\begin{aligned}
V_{R,0} &= V_{\mathrm{fe},0} \\
V_{H,0} &= K_H V_{\mathrm{fe},0} \\
V_{L,0} &= V_{\mathrm{lr}} + K_L V_{\mathrm{fe},0} \\
V_{\mathrm{lv},0} &= \dfrac{V_{R,0}}{K_B} \\
V_{A,0} &= V_{\mathrm{lv},0} - V_{H,0} \\
V_{\mathrm{ll},0} &= \dfrac{V_{A,0}}{K_A} \\
V_{C,0} &= E_{C,0} \\
V_{F,0} &= 0 \\
x_{\mathrm{ll},0} &= e_{V,0} = V_{\mathrm{ll},0} \\
V_{\mathrm{ref},0}
&= e_{V,0} + V_{C,0} + V_{F,0}
- V_{S,0} - V_{\mathrm{uel},0} - V_{\mathrm{oel},0}
\end{aligned}
```

This standard start requires $1+s_{\mathrm{spd}}\omega_0\ne 0$,
$V_{E,0}\ne 0$, $K_A\ne 0$, $K_B\ne 0$, inactive $V_A$ and $V_R$ limits, and
the low-value gate selecting the amplifier path. Starts with active low-value
gate limiting or saturated regulator states are outside these closed-form
equations.

## Model Outputs

Output | Units | Description | Note
----------------|--------|-------------------------------------|------
`efd` | [p.u.] | Field-voltage output | $E_{\mathrm{fd}}$
`ve` | [p.u.] | Exciter alternator voltage state | $V_E$
`vc` | [p.u.] | Sensed compensated voltage | $V_C$
`va` | [p.u.] | Voltage-amplifier state | $V_A$
`vr` | [p.u.] | Voltage-regulator output | $V_R$
`vll` | [p.u.] | Lead-lag output | $V_{\mathrm{ll}}$
`vf` | [p.u.] | Stabilizing feedback state | $V_F$
`vlv` | [p.u.] | Low-value gate output | $V_{\mathrm{lv}}$
`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))
- EXAC2 Excitation Model (See [EXAC2](EXAC2/README.md))
- Simplified Excitation System Model (See [SEXS-PTI](SEXS-PTI/README.md))
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