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Add ESST4B exciter documentation stub
lukelowry May 31, 2026
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docs: complete ESST4B README [skip ci]
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243 changes: 243 additions & 0 deletions GridKit/Model/PhasorDynamics/Exciter/ESST4B/README.md
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# **IEEE Type ST4B Potential- or Compound-Source Controlled-Rectifier Exciter Model (ESST4B)**

ESST4B is a static excitation system with compensated-voltage sensing, an outer
proportional/integral voltage regulator, a lag block, an inner
proportional/integral regulator with exciter-output feedback, low-value
over-excitation limiter gating, and potential- or compound-source rectifier
scaling.

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
controlled-rectifier loading curve from Fig. 1; it is not a CommonMath helper.
- The potential-source calculation uses explicit real and imaginary terminal
voltage/current components; the diagram's complex expression is not used as
model-equation notation below.

## Block Diagram

Standard model of the ESST4B Exciter.

<div align="center">
<img align="center" src="../../../../../docs/Figures/PhasorDynamics/ESST4B_diagram.png">

Figure 1: Exciter ESST4B model. Figure courtesy of [PowerWorld](https://www.powerworld.com/WebHelp/)
</div>

## Model Parameters

Symbol | Units | JSON | Description | Typical Value | Note
------------------------------------|--------|-------------|---------------------------------------------------------|---------------|------
$T_R$ | [sec] | `Tr` | Compensated-voltage transducer time constant | 0.0 | Block name: `Tr`; if zero, sensed voltage is algebraic
$K_{\mathrm{pr}}$ | [p.u.] | `Kpr` | Outer regulator proportional gain | 1.0 | Block name: `KPR`
$K_{\mathrm{ir}}$ | [p.u./s] | `Kir` | Outer regulator integral gain | 0.0 | Block name: `KIR`
$V_R^{\max}$ | [p.u.] | `Vrmax` | Maximum outer regulator output | 1.0 | Block name: `VRMAX`
$V_R^{\min}$ | [p.u.] | `Vrmin` | Minimum outer regulator output | -1.0 | Block name: `VRMIN`
$T_A$ | [sec] | `Ta` | Regulator lag time constant | 0.0 | Block name: `Ta`; if zero, $V_A$ is algebraic
$K_{\mathrm{pm}}$ | [p.u.] | `Kpm` | Inner regulator proportional gain | 1.0 | Block name: `KPM`
$K_{\mathrm{im}}$ | [p.u./s] | `Kim` | Inner regulator integral gain | 0.0 | Block name: `KIM`
$V_M^{\max}$ | [p.u.] | `VmMax` | Maximum inner regulator output | 1.0 | Block name: `VMMAX`
$V_M^{\min}$ | [p.u.] | `VmMin` | Minimum inner regulator output | 0.0 | Block name: `VMMIN`
$K_G$ | [p.u.] | `Kg` | Exciter-output feedback gain into inner regulator | 0.0 | Block name: `KG`
$K_P$ | [p.u.] | `Kp` | Potential-source voltage coefficient magnitude | 0.0 | Source label: `KP`
$K_I$ | [p.u.] | `Ki` | Potential-source current coefficient | 0.0 | Source label: `KI`
$V_B^{\max}$ | [p.u.] | `VbMax` | Maximum rectifier source multiplier | 999.0 | Block name: `VBMAX`
$K_C$ | [p.u.] | `Kc` | Rectifier loading current coefficient | 0.0 | Block name: `Kc`; forms $I_N$
$X_L$ | [p.u.] | `Xl` | Source reactance term in potential-source calculation | 0.0 | Source label: `XL`
$\theta_P$ | [deg] | `ThetaPDeg` | Potential-source coefficient angle | 0.0 | Source label: `thetaP`; forms $K_P^{\mathrm{r}}$ and $K_P^{\mathrm{i}}$
$V_G^{\max}$ | [p.u.] | `VgMax` | Maximum exciter-output feedback signal | 999.0 | Block name: `VGMAX`; ceiling on $K_G E_{\mathrm{fd}}$

### Parameter Validation

Invalid ESST4B parameter sets are rejected by the following checks.

```math
\begin{aligned}
&T_R \ge 0,\quad T_A \ge 0 \\
&V_R^{\min} \le V_R^{\max},\quad V_M^{\min} \le V_M^{\max} \\
&V_B^{\max} > 0,\quad V_G^{\max} \ge 0
\end{aligned}
```

### Model Derived Parameters

The potential-source coefficient is resolved into real scalar components:

```math
\begin{aligned}
K_P^{\mathrm{r}} &= K_P\cos\theta_P \\
K_P^{\mathrm{i}} &= K_P\sin\theta_P
\end{aligned}
```

Here $\theta_P$ is converted from degrees before evaluating the trigonometric
functions.

## Model Variables

### Internal Variables

#### Differential

Symbol | Units | Description | Note
------------------------------------|--------|---------------------------------------------------------|------
$V_M$ | [p.u.] | Inner regulator output | State 1 in Fig. 1
$V_C$ | [p.u.] | Sensed compensated voltage | State 2 in Fig. 1; source label: `Sensed Vt`; algebraic when $T_R=0$
$V_A$ | [p.u.] | Lagged outer-regulator output | State 3 in Fig. 1; algebraic when $T_A=0$
$x_R$ | [p.u.] | Outer regulator integral state | State 4 in Fig. 1; source label: `VR`

#### Algebraic

Symbol | Units | Description | Note
------------------------------------|--------|---------------------------------------------------------|------
$e_V$ | [p.u.] | Voltage-error signal into outer regulator | Summing junction after sensed voltage
$V_R$ | [p.u.] | Limited outer regulator output | Limited by $V_R^{\min}$ and $V_R^{\max}$
$V_G$ | [p.u.] | Limited exciter-output feedback signal | $K_G E_{\mathrm{fd}}$ limited by $V_G^{\max}$
$e_M$ | [p.u.] | Inner regulator error | $V_A$ minus $V_G$
$V_{\mathrm{lv}}$ | [p.u.] | Low-value gate output | Lesser of $V_M$ and $V_{\mathrm{oel}}$
$V_{\mathrm{src}}^{\mathrm{r}}$ | [p.u.] | Real component of the potential-source expression | From terminal voltage/current components
$V_{\mathrm{src}}^{\mathrm{i}}$ | [p.u.] | Imaginary component of the potential-source expression | From terminal voltage/current components
$V_E$ | [p.u.] | Potential- or compound-source voltage magnitude | Nonnegative source magnitude
$I_N$ | [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 $F_{\mathrm{ex}}=f(I_N)$
$V_B$ | [p.u.] | Rectifier source multiplier | Limited by $V_B^{\max}$
$E_{\mathrm{fd}}$ | [p.u.] | Field-voltage output | Product of low-value gate and $V_B$

### External Variables

#### Differential

None.

#### Algebraic

Symbol | Units | Description | Note
------------------------------------|--------|---------------------------------------------------------|------
$V_{\mathrm{comp}}$ | [p.u.] | Compensated voltage input | Source label: `VCOMP`
$V_{\mathrm{ref}}$ | [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 a high value when omitted
$V_{\mathrm{r}}$ | [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 V_C - V_C + V_{\mathrm{comp}} \\
0 &=
-\dot x_R
+ \text{antiwindup}\!\left(
V_R,
K_{\mathrm{ir}}e_V,
V_R^{\min},
V_R^{\max}
\right) \\
0 &= -T_A\dot V_A - V_A + V_R \\
0 &=
-\dot V_M
+ \text{antiwindup}\!\left(
V_M,
K_{\mathrm{im}}e_M,
V_M^{\min},
V_M^{\max}
\right)
\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_C \\
0 &= -V_R + \text{clamp}(K_{\mathrm{pr}}e_V + x_R, V_R^{\min}, V_R^{\max}) \\
0 &= -V_G + \text{min}(K_G E_{\mathrm{fd}}, V_G^{\max}) \\
0 &= -e_M + V_A - V_G \\
0 &= -V_{\mathrm{lv}} + \text{min}(V_M, V_{\mathrm{oel}}) \\
0 &= -V_{\mathrm{src}}^{\mathrm{r}}
+ K_P V_{\mathrm{r}}
- X_L K_P^{\mathrm{i}} I_{\mathrm{r}}
- \left(K_I + X_L K_P^{\mathrm{r}}\right)I_{\mathrm{i}} \\
0 &= -V_{\mathrm{src}}^{\mathrm{i}}
+ K_P V_{\mathrm{i}}
+ \left(K_I + X_L K_P^{\mathrm{r}}\right)I_{\mathrm{r}}
- X_L K_P^{\mathrm{i}} I_{\mathrm{i}} \\
0 &= -V_E^2
+ \left(V_{\mathrm{src}}^{\mathrm{r}}\right)^2
+ \left(V_{\mathrm{src}}^{\mathrm{i}}\right)^2 \\
0 &= -V_E I_N + K_C I_{\mathrm{fd}} \\
0 &= -F_{\mathrm{ex}} + f(I_N) \\
0 &= -V_B + \text{min}(V_E F_{\mathrm{ex}}, V_B^{\max}) \\
0 &= -E_{\mathrm{fd}} + V_{\mathrm{lv}}V_B
\end{aligned}
```

CommonMath defines helper targets for [min and clamp](../../../../CommonMath.md#derived-functions).
The rectifier loading function $f(I_N)$ is the source curve shown in Fig. 1.

## Initialization

For a standard unsaturated start, the machine initializes
$E_{\mathrm{fd},0}$ and $I_{\mathrm{fd},0}$ first. ESST4B reads those values,
sets all internal derivatives to zero, and evaluates:

```math
\begin{aligned}
V_{C,0} &= V_{\mathrm{comp},0} \\
V_{\mathrm{src},0}^{\mathrm{r}}
&= K_P V_{\mathrm{r},0}
- X_L K_P^{\mathrm{i}} I_{\mathrm{r},0}
- \left(K_I + X_L K_P^{\mathrm{r}}\right)I_{\mathrm{i},0} \\
V_{\mathrm{src},0}^{\mathrm{i}}
&= K_P V_{\mathrm{i},0}
+ \left(K_I + X_L K_P^{\mathrm{r}}\right)I_{\mathrm{r},0}
- X_L K_P^{\mathrm{i}} I_{\mathrm{i},0} \\
V_{E,0} &=
\sqrt{
\left(V_{\mathrm{src},0}^{\mathrm{r}}\right)^2
+ \left(V_{\mathrm{src},0}^{\mathrm{i}}\right)^2
} \\
0 &= -V_{E,0}I_{N,0} + K_C I_{\mathrm{fd},0} \\
F_{\mathrm{ex},0} &= f(I_{N,0}) \\
V_{B,0} &= \text{min}(V_{E,0}F_{\mathrm{ex},0}, V_B^{\max}) \\
V_{\mathrm{lv},0} &= \dfrac{E_{\mathrm{fd},0}}{V_{B,0}} \\
V_{M,0} &= V_{\mathrm{lv},0} \\
V_{G,0} &= \text{min}(K_G E_{\mathrm{fd},0}, V_G^{\max}) \\
e_{M,0} &= 0 \\
V_{A,0} &= V_{G,0} \\
V_{R,0} &= V_{A,0} \\
x_{R,0} &= V_{R,0} \\
e_{V,0} &= 0 \\
V_{\mathrm{ref},0} &= V_{C,0} - V_{\mathrm{uel},0} - V_{S,0}
\end{aligned}
```

This closed-form start requires $V_{E,0}\ne 0$, $V_{B,0}\ne 0$, inactive
$V_R$, $V_M$, $V_G$, and $V_B$ limits, and the low-value gate selecting $V_M$.
Starts with active low-value gate limiting or saturated PI states are outside
these closed-form equations.

## Model Outputs

Output | Units | Description | Note
----------------|--------|-------------------------------------|------
`efd` | [p.u.] | Field-voltage output | $E_{\mathrm{fd}}$
`vm` | [p.u.] | Inner regulator output | $V_M$
`vc` | [p.u.] | Sensed compensated voltage | $V_C$
`va` | [p.u.] | Lagged outer-regulator output | $V_A$
`vr` | [p.u.] | Outer regulator output | $V_R$
`vg` | [p.u.] | Exciter-output feedback signal | $V_G$
`vlv` | [p.u.] | Low-value gate output | $V_{\mathrm{lv}}$
`ve` | [p.u.] | Potential-source voltage magnitude | $V_E$
`vb` | [p.u.] | Rectifier source multiplier | $V_B$
`in` | [p.u.] | Normalized exciter loading current | $I_N$
`fex` | [p.u.] | Rectifier loading factor | $F_{\mathrm{ex}}$
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))
- IEEE ST4B Excitation Model (See [ESST4B](ESST4B/README.md))
- Simplified Excitation System Model (See [SEXS-PTI](SEXS-PTI/README.md))
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