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Add ESAC6A exciter documentation stub
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237 changes: 237 additions & 0 deletions GridKit/Model/PhasorDynamics/Exciter/ESAC6A/README.md
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# **IEEE Type AC6A Excitation System Model (ESAC6A)**

ESAC6A is an IEEE Type AC excitation system with sensed terminal-voltage
feedback, cascaded regulator lead-lag blocks, voltage-regulator limits, an
exciter alternator state, field-current feedback limiting, 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 ESAC6A Exciter.

<div align="center">
<img align="center" src="../../../../../docs/Figures/PhasorDynamics/ESAC6A_diagram.png">

Figure 1: Exciter ESAC6A 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
$K_A$ | [p.u.] | `Ka` | Voltage-regulator gain | 40.0 | Block name: `Ka`
$T_A$ | [sec] | `Ta` | Regulator denominator time constant | 0.1 | Block name: `Ta`
$T_K$ | [sec] | `Tk` | Regulator numerator time constant | 0.0 | Block name: `Tk`
$T_B$ | [sec] | `Tb` | Lag time constant for second lead-lag block | 0.0 | Block name: `Tb`
$T_C$ | [sec] | `Tc` | Lead time constant for second lead-lag block | 0.0 | Block name: `Tc`
$V_A^{\max}$ | [p.u.] | `VaMax` | Maximum first regulator block output | 1.0 | Block name: `VAMAX`
$V_A^{\min}$ | [p.u.] | `VaMin` | Minimum first regulator block output | -1.0 | Block name: `VAMIN`
$V_R^{\max}$ | [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`
$V_{\mathrm{fe}}^{\mathrm{lim}}$ | [p.u.] | `Vfelim` | Feedback-limiter summing-junction reference | 0.0 | Source label: `VFELIM`
$K_H$ | [p.u.] | `Kh` | Feedback-limiter gain | 1.0 | Block name: `KH`
$V_H^{\max}$ | [p.u.] | `Vhmax` | Maximum feedback-limiter lead-lag output | 1.0 | Block name: `VHMAX`; lower limit is zero
$T_H$ | [sec] | `Th` | Feedback-limiter denominator time constant | 0.0 | Block name: `TH`
$T_J$ | [sec] | `Tj` | Feedback-limiter numerator time constant | 0.0 | Block name: `TJ`
$K_C$ | [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`
$E_1$ | [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 ESAC6A parameter sets are rejected by the following checks.

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

## 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 $T_R=0$
$x_A$ | [p.u.] | First regulator lead-lag denominator state | State 3 in Fig. 1; source label: `TA Block`
$x_{\mathrm{ll}}$ | [p.u.] | Second lead-lag denominator state | State 4 in Fig. 1; source label: `VLL`
$V_F$ | [p.u.] | Stabilizing feedback signal | State 5 in Fig. 1; source label: `VF`

#### Algebraic

Symbol | Units | Description | Note
------------------------------------|--------|---------------------------------------------------------|------
$e_V$ | [p.u.] | Voltage-regulator input error before first lead-lag | Summing junction after sensed voltage
$V_A$ | [p.u.] | Limited first regulator lead-lag output | Limited by $V_A^{\min}$ and $V_A^{\max}$
$V_{\mathrm{ll}}$ | [p.u.] | Second lead-lag output | Input to $V_R$ summing junction
$V_H$ | [p.u.] | Feedback-limiter lead-lag output before $K_H$ | Limited by 0 and $V_H^{\max}$
$V_H^{\mathrm{pre}}$ | [p.u.] | Feedback-limiter lead-lag output before limits | Bypasses to $V_F$ when $T_H=T_J=0$
$V_R$ | [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, $K_D I_{\mathrm{fd}}$, and feedback-limiter 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`
$V_{\mathrm{uel}}$ | [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
$I_{\mathrm{fd}}$ | [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_A\dot x_A - x_A + K_A e_V \\
0 &= -T_B\dot x_{\mathrm{ll}} - x_{\mathrm{ll}} + V_A \\
0 &= -T_E\dot V_E + V_R - V_{\mathrm{fe}} \\
0 &= -T_H\dot V_F - V_F + V_{\mathrm{fe}}
\end{aligned}
```

### Algebraic Equations

```math
\begin{aligned}
0 &= -e_V + V_{\mathrm{ref}} + V_{\mathrm{uel}} + V_S - V_C - V_F \\
0 &= -V_A
+ \text{clamp}\!\left(
x_A + \dfrac{T_K}{T_A}(K_A e_V - x_A),
V_A^{\min},
V_A^{\max}
\right) \\
0 &= -T_B(V_{\mathrm{ll}} - x_{\mathrm{ll}}) + T_C(V_A - x_{\mathrm{ll}}) \\
0 &= -V_H^{\mathrm{pre}}
+
\begin{cases}
V_F, & T_H = T_J = 0 \\
V_F + \dfrac{T_J}{T_H}(V_{\mathrm{fe}} - V_F), & T_H > 0
\end{cases} \\
0 &= -V_H + \text{clamp}\!\left(V_H^{\mathrm{pre}}, 0, V_H^{\max}\right) \\
0 &= -V_R + \text{clamp}(V_{\mathrm{ll}} - K_H V_H, 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}} + V_{\mathrm{fe}}^{\mathrm{lim}} \\
0 &= -E_{\mathrm{fd}}
+ \left(1+s_{\mathrm{spd}}\omega\right)F_{\mathrm{ex}}V_E
\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.
When $T_B=T_C=0$, the second lead-lag block is bypassed. When $T_H=T_J=0$, the
feedback-limiter lead-lag block is bypassed before the 0-to-$V_H^{\max}$ clamp.

## Initialization

The machine initializes $E_{\mathrm{fd}}$ and $I_{\mathrm{fd}}$ first. For a
standard unsaturated start, ESAC6A reads those values and sets all internal
derivatives to zero. First solve 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:

```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} + V_{\mathrm{fe}}^{\mathrm{lim}} \\
V_{F,0} &= V_{\mathrm{fe},0} \\
V_{H,0}^{\mathrm{pre}} &= V_{F,0} \\
V_{H,0} &= \text{clamp}(V_{H,0}^{\mathrm{pre}}, 0, V_H^{\max}) \\
V_{R,0} &= V_{\mathrm{fe},0} \\
V_{\mathrm{ll},0} &= V_{R,0} + K_H V_{H,0} \\
V_{A,0} &= V_{\mathrm{ll},0} \\
x_{A,0} &= V_{A,0} \\
x_{\mathrm{ll},0} &= V_{\mathrm{ll},0} \\
V_{C,0} &= E_{C,0} \\
e_{V,0} &= \dfrac{x_{A,0}}{K_A} \\
V_{\mathrm{ref},0} &= e_{V,0} + V_{C,0} + V_{F,0} - V_{\mathrm{uel},0} - V_{S,0}
\end{aligned}
```

This standard start requires $1+s_{\mathrm{spd}}\omega_0\ne 0$,
$V_{E,0}\ne 0$, inactive $V_A$, $V_R$, and $V_H$ limits, and nonsingular
regulator gains/time constants. Starts that bind those limits 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.] | First regulator output | $V_A$
`vll` | [p.u.] | Second lead-lag output | $V_{\mathrm{ll}}$
`vf` | [p.u.] | Feedback-limiter state | $V_F$
`vh` | [p.u.] | Feedback-limiter output | $V_H$
`vr` | [p.u.] | Voltage-regulator output | $V_R$
`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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## Types
There are a few standard Exciter models
- ESAC6A Excitation Model (See [ESAC6A](ESAC6A/README.md))
- IEEE Type 1 Excitation Model (See [IEEET1](IEEET1/README.md))
- IEEE DC1 Excitation Model (See [EXDC1](EXDC1/README.md))
- Simplified Excitation System Model (See [SEXS-PTI](SEXS-PTI/README.md))
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