Automatic Voltage Regulators (AVR)

AVR are used to determine the voltage in the field winding $v_f$ (or $V_f$) in the model.

Fixed AVR `[AVRFixed]`

This is a simple model that set the field voltage to be equal to a desired constant value $v_f = v_{\text{fix}}$.

Simple AVR `[AVRSimple]`

This depicts the most basic AVR, on which the field voltage is an integrator over the difference of the measured voltage and a reference:

\[\begin{align} \dot{v}_f = K_v(v_{\text{ref}} - v_h) \tag{1a} \end{align}\]

AVR Type I `[AVRTypeI]`

This AVR is a simplified version of the IEEE DC1 AVR model:

\[\begin{align} \dot{v}_f &= -\frac{1}{T_e} \left[ V_f(K_e + S_e(v_f))-v_{r1} \right] \tag{2a} \\ \dot{v}_{r1} &= \frac{1}{T_a} \left[ K_a\left(v_{\text{ref}} - v_m - v_{r2} - \frac{K_f}{T_f}v_f\right) - v_{r1} \right] \tag{2b} \\ \dot{v}_{r2} &= -\frac{1}{T_f} \left[ \frac{K_f}{T_f}v_f + v_{r2} \right] \tag{2c} \\ \dot{v}_m &= \frac{1}{T_r} (v_h - v_m) \tag{2d} \end{align}\]

with the ceiling function:

\[\begin{align*} S_e(v_f) = A_e \exp(B_e|v_f|) \end{align*}\]

AVR Type II `[AVRTypeII]`

This model represents a static exciter with higher gains and faster response than the Type I:

\[\begin{align} \dot{v}_f &= -\frac{1}{T_e} \left[ V_f(1 + S_e(v_f))-v_{r} \right] \tag{3a} \\ \dot{v}_{r1} &= \frac{1}{T_1} \left[ K_0\left(1 - \frac{T_2}{T_1} \right)(v_{\text{ref}} - v_m) - v_{r1} \right] \tag{3b} \\ \dot{v}_{r2} &= \frac{1}{K_0 T_3} \left[ \left( 1 - \frac{T_4}{T_3} \right) \left( v_{r1} + K_0\frac{T_2}{T_1}(v_{\text{ref}} - v_m)\right) - K_0 v_{r2} \right] \tag{3c} \\ \dot{v}_m &= \frac{1}{T_r} (v_h - v_m) \tag{3d} \end{align}\]

with

\[\begin{align*} v_r &= K_0v_{r2} + \frac{T_4}{T_3} \left( v_{r1} + K_0\frac{T_2}{T_1}(v_{\text{ref}} - v_m)\right) \\ S_e(v_f) &= A_e \exp(B_e|v_f|) \end{align*}\]

Excitation System AC1A `[ESAC1A]`

The model represents the 5-states IEEE Type AC1A Excitation System Model:

\[\begin{align} \dot{V}_m &= \frac{1}{T_r} (V_{h} - V_m) \tag{4a} \\ \dot{V}_{r1} &= \frac{1}{T_b} \left(V_{in} \left(1 - \frac{T_c}{T_b}\right) - V_{r1}\right) \tag{4b} \\ \dot{V}_{r2} &= \frac{1}{T_a} (K_a V_{out} - V_{r2}) \tag{4c} \\ \dot{V}_e &= \frac{1}{T_e} (V_r - V_{FE}) \tag{4d} \\ \dot{V}_{r3} &= \frac{1}{T_f} \left( - \frac{K_f}{T_f}V_{FE} - V_{r3} \right) \tag{4e} \\ \end{align}\]

with

\[\begin{align*} I_N &= \frac{K_c}{V_e} X_{ad}I_{fd} \\ V_{FE} &= K_d X_{ad}I_{fd} + K_e V_e + S_e V_e \\ S_e &= B\frac{(V_e-A)^2}{V_e} \\ V_{F1} &= V_{r3} + \frac{K_f}{T_f} V_{FE} \\ V_{in} &= V_{ref} - V_m - V_{F1} \\ V_{out} &= V_{r1} + \frac{T_c}{T_b} V_{in} \\ V_f &= V_e f(I_N) \\ f(I_N) &= \left\{\begin{array}{cl} 1 & \text{ if }I_N \le 0 \\ 1 - 0.577 I_N & \text{ if } 0 < I_N \le 0.433 \\ \sqrt{0.75 - I_N^2} & \text{ if } 0.433 < I_N \le 0.75 \\ 1.732(1-I_N) & \text{ if } 0.75 < I_N \le 1 \\ 0 & \text{ if } I_N > 1 \end{array} \right. \end{align*}\]

on which $X_{ad}I_{fd}$ is the field current coming from the generator and $V_{h}$ is the terminal voltage, and $A,B$ are the saturation coefficients computed using the $E_1, E_2, S_e(E_1), S_e(E_2)$ data.

Simplified Excitation System `[SEXS]`

The model for the 2 states excitation system SEXS:

\[\begin{align} \dot{V}_f &= \frac{1}{T_e} (V_{LL} - V_f) \tag{5a} \\ \dot{V}_r &= \frac{1}{T_b} \left[\left(1 - \frac{T_a}{T_b}\right) V_{in} - V_r \right] \tag{5b} \end{align}\]

with

\[\begin{align*} V_{in} &= V_{ref} + V_s - V_h \\ V_{LL} &= V_r + \frac{T_a}{T_b}V_{in} \\ \end{align*}\]

on which $V_h$ is the terminal voltage and $V_s$ is the PSS output signal.

Excitation System ST1 `[EXST1]`

The model represents the 4-states IEEE Type ST1 Excitation System Model:

\[\begin{align} \dot{V}_m &= \frac{1}{T_r} (V_{h} - V_m) \tag{6a} \\ \dot{V}_{rll} &= \frac{1}{T_b} \left(V_{in} \left(1 - \frac{T_c}{T_b}\right) - V_{rll}\right) \tag{6b} \\ \dot{V}_{r} &= \frac{1}{T_a} (V_{LL} - V_{r}) \tag{6c} \\ \dot{V}_{fb} &= \frac{1}{T_f} \left( - \frac{K_f}{T_f}V_{r} - V_{fb} \right) \tag{6d} \\ \end{align}\]

with

\[\begin{align*} V_{in} &= V_{ref} - V_m - y_{hp} \\ V_{LL} &= V_{r} + \frac{T_c}{T_b} V_{in} \\ y_{hp} &= V_{fb} + \frac{K_f}{T_f} V_r \\ V_f &= V_r \\ \end{align*}\]

on which $V_h$ is the terminal voltage.

Excitation System EXAC1 `[EXAC1]`

The model represents the 5-states IEEE Type EXAC1 Excitation System Model:

\[\begin{align} \dot{V}_m &= \frac{1}{T_r} (V_{h} - V_m) \tag{7a} \\ \dot{V}_{r1} &= \frac{1}{T_b} \left(V_{in} \left(1 - \frac{T_c}{T_b}\right) - V_{r1}\right) \tag{7b} \\ \dot{V}_{r2} &= \frac{1}{T_a} (K_a V_{out} - V_{r2}) \tag{7c} \\ \dot{V}_e &= \frac{1}{T_e} (V_r - V_{FE}) \tag{7d} \\ \dot{V}_{r3} &= \frac{1}{T_f} \left( - \frac{K_f}{T_f}V_{FE} - V_{r3} \right) \tag{7e} \\ \end{align}\]

with

Excitation System ST8C `[ST8C]`

The model represents the 5-states IEEE Type ST8C Excitation System Model:

\[\begin{align} T_r \dot{V}_m &= V_{h} - V_m \tag{8a} \\ \dot{x}_{a1} &= \begin{cases} V_{\pi_{in}} & \text{if } V_{\pi_{min}} < K_{pr} V_{\pi_{in}} + K_{ir} x_{a1} < V_{\pi_{max}} \\ 0 & \text{otherwise} \end{cases} \tag{8b} \\ \dot{x}_{a2} &= \begin{cases} I_{fd_{diff}} & \text{if } V_{a_{min}} < K_{pa} I_{fd_{diff}} + K_{ia} x_{a2} < V_{a_{max}} \\ 0 & \text{otherwise} \end{cases} \tag{8c} \\ T_a \dot{x}_{a3} &= \begin{cases} K_a \pi_{out} - x_{a3} & \text{if } V_{r_{min}} < x_{a3} < V_{r_{max}} \\ 0 & \text{otherwise} \end{cases} \tag{8d} \\ T_f \dot{x}_{a4} &= K_f I_{fd} - x_{a4} \tag{8e} \\ \end{align}\]

with

\[\begin{align*} V_{\pi_{in}} &= V_{ref} + V_s - V_m \\ I_{fd_{ref}} &= \text{clamp}(K_{pr} V_{\pi_{in}} + K_{ir} x_{a1}, V_{\pi_{min}}, V_{\pi_{max}}) \\ I_{fd_{diff}} &= I_{fd_{ref}} - x_{a4} \\ \pi_{out} &= \text{clamp}(K_{pa} I_{fd_{diff}} + K_{ia} x_{a2}, V_{a_{min}}, V_{a_{max}}) \\ I_{N1} &= \frac{K_{c1} I_{fd}}{V_e} \\ F_{ex} &= f(I_{N1}) \\ V_{b1} &= \min(F_{ex} V_e, V_{B1_{max}}) \\ V_{b2} &= 0 \quad \text{(feedforward current not implemented)} \\ E_{fd} &= V_{b1} x_{a3} + V_{b2} \\ f(I_N) &= \left\{\begin{array}{cl} 1 & \text{ if }I_N \le 0 \\ 1 - 0.577 I_N & \text{ if } 0 < I_N \le 0.433 \\ \sqrt{0.75 - I_N^2} & \text{ if } 0.433 < I_N \le 0.75 \\ 1.732(1-I_N) & \text{ if } 0.75 < I_N \le 1 \\ 0 & \text{ if } I_N > 1 \end{array} \right. \end{align*}\]

on which $I_{fd}$ is the field current from the generator, $V_{h}$ is the terminal voltage, and $V_s$ is the PSS output signal.

Notes

The rectifier function $f(I_N)$ models the characteristic of a three-phase full-wave rectifier
Terminal current feedforward ($K_{i2}$) is not currently implemented and must be set to 0.0
Voltage compensation is not implemented ($V_e = K_p$)

Excitation System ST6B `[ST6B]`

The model represents the 4-states IEEE Type ST6B Excitation System Model:

\[\begin{align} T_r \dot{V}_m &= V_{h} - V_m \tag{9a} \\ \dot{x}_{i} &= \begin{cases} V_{\pi_{in}} & \text{if } V_{a_{min}} < K_{pa} V_{\pi_{in}} + K_{ia} x_{i} < V_{a_{max}} \\ 0 & \text{otherwise} \end{cases} \tag{9b} \\ T_{da} \dot{x}_{d} &= -K_{da} V_{\pi_{in}} - x_{d} \tag{9c} \\ \dot{V}_{g} &= \frac{1}{T_g} (K_g E_{fd} - V_g) \tag{9d} \\ \end{align}\]

with

\[\begin{align*} V_{\pi_{in}} &= V_{ref} + V_s - V_m \\ \text{PI output} &= \text{clamp}(K_{pa} V_{\pi_{in}} + K_{ia} x_{i}, V_{a_{min}}, V_{a_{max}}) \\ \text{PD output} &= x_{d} + \frac{K_{da}}{T_{da}} V_{\pi_{in}} \\ V_a &= \text{PI output} + \text{PD output} \\ \text{FF output} &= (V_a - V_g) K_m + K_{ff} V_a \\ V_{r1} &= \max((I_{lr} K_{ci} - X_{ad} I_{fd}) K_{lr}, V_{r_{min}}) \\ V_{r2} &= \text{clamp}(\text{FF output}, V_{r_{min}}, V_{r_{max}}) \\ V_r &= \min(V_{r1}, V_{r2}) \\ E_{fd} &= V_r V_m \\ \end{align*}\]

on which $X_{ad} I_{fd}$ is the field current from the generator, $V_{h}$ is the terminal voltage, and $V_s$ is the PSS output signal. Finally, the derivative block implements a high-pass filter:

\[\text{PD output} = x_d + \frac{K_{da}}{T_{da}} u, \quad T_{da} \dot{x}_d = -\frac{K_{da}}{T_{da}} u - x_d\]

while the current limiter ensures:

\[V_{r1} = \max((I_{lr} K_{ci} - X_{ad} I_{fd}) K_{lr}, V_{r_{min}})\]

Notes

The derivative term uses a high-pass filter implementation
Current limiting is active when field current exceeds the reference value $I_{lr}$
The exciter output is the minimum of voltage-limited and current-limited values
During initialization, $I_{lr}$ is typically adjusted to match operating conditions