Prime Movers and Turbine Governors (TG)

This section describes how mechanical power is modified to provide primary frequency control with synchronous generators. It is assumed that $\tau_{\text{ref}} = P_{\text{ref}}$ since they are decided at nominal frequency $ω = 1$.

Fixed TG `[TGFixed]`

This a simple model that set the mechanical torque to be equal to a proportion of the desired reference $\tau_m = \eta P_{\text{ref}}$. To set the mechanical torque to be equal to the desired power, the value of $\eta$ is set to 1.

TG Type I `[TGTypeI]`

This turbine governor is described by a droop controller and a low-pass filter to model the governor and two lead-lag blocks to model the servo and reheat of the turbine governor.

\[\begin{align} \dot{x}_{g1} &= \frac{1}{T_s}(p_{\text{in}} - x_{g1}) \tag{1a} \\ \dot{x}_{g2} &= \frac{1}{T_c} \left[ \left(1- \frac{t_3}{T_c}\right)x_{g1} - x_{g2} \right] \tag{1b} \\ \dot{x}_{g3} &= \frac{1}{T_5} \left[\left(1 - \frac{T_4}{T_5}\right)\left(x_{g2} + \frac{T_3}{T_c}x_{g1}\right) - x_{g3} \right] \tag{1c} \\ \tau_m &= x_{g3} + \frac{T_4}{T_5}\left(x_{g2} + \frac{T_3}{T_c}x_{g1}\right) \tag{1d} \end{align}\]

with

\[\begin{align*} p_{\text{in}} = P_{\text{ref}} + \frac{1}{R}(\omega_s - 1.0) \end{align*}\]

TG Type II `[TGTypeII]`

This turbine governor is a simplified model of the Type I.

\[\begin{align} \dot{x}_g &= \frac{1}{T_2}\left[\frac{1}{R}\left(1 - \frac{T_1}{T_2}\right) (\omega_s - \omega) - x_g\right] \tag{2a} \\ \tau_m &= P_{\text{ref}} + \frac{1}{R}\frac{T_1}{T_2}(\omega_s - \omega) \tag{2b} \end{align}\]

TG Simple `[TGSimple]`

This turbine governor represents a simple first-order model with droop control. It's a 1-state model suitable for basic frequency response studies.

\[\begin{align} T_m \dot{\tau}_m &= \left(P_{\text{ref}} + d_t (\omega_{\text{ref}} - \omega)\right) - \tau_m \tag{3a} \end{align}\]

The TGSimple uses a linear relationship between frequency deviation and power output and a first order lag response with time constant $T_m$. It is suitable for studies where detailed turbine dynamics are not critical since provides basic primary frequency response through droop control.

TGOV1 `[SteamTurbineGov1]`

This represents a classical Steam-Turbine Governor, known as TGOV1.

\[\begin{align} \dot{x}_{g1} &= \frac{1}{T_1} (\text{ref}_{in} - x_{g1}) \tag{4a} \\ \dot{x}_{g2} &= \frac{1}{T_3} \left(x_{g1}^\text{sat} \left(1 - \frac{T_2}{T_3}\right) - x_{g2}\right) \tag{4b} \end{align}\]

with

\[\begin{align} \text{ref}_{in} &= \frac{1}{R} (P_{ref} - (\omega - 1.0)) \tag{4c} \\ x_{g1}^\text{sat} &= \left\{ \begin{array}{cl} x_{g1} & \text{ if } V_{min} \le x_{g1} \le V_{max}\\ V_{max} & \text{ if } x_{g1} > V_{max} \\ V_{min} & \text{ if } x_{g1} < V_{min} \end{array} \right. \tag{4d} \\ \tau_m &= x_{g2} + \frac{T_2}{T_3} x_{g1} - D_T(\omega - 1.0) \tag{4e} \end{align}\]

GAST `[GasTG]`

This turbine governor represents the Gas Turbine representation, known as GAST.

\[\begin{align} \dot{x}_{g1} &= \frac{1}{T_1} (x_{in} - x_{g1}) \tag{5a} \\ \dot{x}_{g2} &= \frac{1}{T_2} \left(x_{g1}^\text{sat} - x_{g2}\right) \tag{5b} \\ \dot{x}_{g3} &= \frac{1}{T_3} (x_{g2} - x_{g3}) \tag{5c} \end{align}\]

with

\[\begin{align} x_{in} &= \min\left\{P_{ref} - \frac{1}{R}(\omega - 1.0), A_T + K_T (A_T - x_{g3}) \right\} \tag{5d} \\ x_{g1}^\text{sat} &= \left\{ \begin{array}{cl} x_{g1} & \text{ if } V_{min} \le x_{g1} \le V_{max}\\ V_{max} & \text{ if } x_{g1} > V_{max} \\ V_{min} & \text{ if } x_{g1} < V_{min} \end{array} \right. \tag{5e} \\ \tau_m &= x_{g2} - D_T(\omega - 1.0) \tag{5f} \end{align}\]

HYGOV `[HydroTurbineGov]`

This represents a classical hydro governor, known as HYGOV.

\[\begin{align} T_f\dot{x}_{g1} &= P_{in} - x_{g1} \tag{6a} \\ \dot{x}_{g2} &= x_{g1} \tag{6b}\\ T_g \dot{x}_{g3} &= c - x_{g3} \tag{6c}\\ \dot{x}_{g4} &= \frac{1 - h}{T_w} \tag{6d} \end{align}\]

with

\[\begin{align} P_{in} &= P_{ref} - \Delta \omega - R x_{g2} \tag{6e} \\ c &= \frac{x_{g1}}{r} + \frac{x_{g2}}{rT_r} \tag{6f} \\ h &= \left(\frac{x_{g4}}{x_{g3}}\right)^2 \tag{6g}\\ \tau_m &= h\cdot A_t(x_{g4} - q_{NL}) - D_{turb} \Delta\omega \cdot x_{g3} \tag{6h} \end{align}\]

DEGOV `[DEGOV]`

This turbine governor represents the IEEE Type DEGOV Diesel Engine Governor Model. It's a 5-state model commonly used for diesel generators.

\[\begin{align} T_1 \dot{x}_{ecb1} &= -\Delta\omega - T_1 x_{ecb1} - x_{ecb2} \tag{7a} \\ \dot{x}_{ecb2} &= x_{ecb1} \tag{7b} \\ (T_5 + T_6) \dot{x}_{a1} &= y_1 - (T_5 + T_6) x_{a1} - x_{a2} \tag{7c} \\ \dot{x}_{a2} &= x_{a1} \tag{7d} \\ \dot{x}_{a3} &= K y_2 \tag{7e} \end{align}\]

with

\[\begin{align*} \Delta\omega &= \omega - \omega_{sys} \\ y_1 &= \frac{T_4}{T_1 T_2} (-\Delta\omega) + \frac{T_3 - T_1 T_4 / (T_1 T_2)}{T_1 T_2} x_{ecb1} + \frac{1 - T_4 / (T_1 T_2)}{T_1 T_2} x_{ecb2} \\ y_2 &= \frac{0}{T_5 T_6} y_1 + \frac{T_4 - (T_5 + T_6) \cdot 0 / (T_5 T_6)}{T_5 T_6} x_{a1} + \frac{1 - 0 / (T_5 T_6)}{T_5 T_6} x_{a2} \\ y_{delay} = e^{-s T_d} x_{a3} P_m &= T_d y_{delay} \cdot \omega \\ \tau_m &= \frac{P_m}{\omega} \end{align*}\]

Note that the model includes a time delay $T_d$ applied to the final actuator output x_{a3}.

DEGOV1 `[DEGOV1]`

This is an enhanced version of the DEGOV model with optional droop control. It's a 5-state or 6-state model depending on the droop flag.

\[\begin{align} T_1 \dot{x}_{g1} &= \text{ll\_in} - T_1 x_{g1} - x_{g2} \tag{8a} \\ \dot{x}_{g2} &= x_{g1} \tag{8b} \\ (T_5 + T_6) \dot{x}_{g3} &= y_1 - (T_5 + T_6) x_{g3} - x_{g4} \tag{8c} \\ \dot{x}_{g4} &= x_{g3} \tag{8d} \\ \dot{x}_{g5} &= K y_2 \tag{8e} \end{align}\]

If droop flag = 1, an additional state is added:

\[\begin{align} T_e \dot{x}_{g6} &= \tau_e - x_{g6} \tag{8f} \end{align}\]

with

\[\begin{align*} \text{ll\_in} &= P_{ref} - (\omega - 1.0) - \text{feedback} \cdot R \\ \text{feedback} &= \begin{cases} x_{g5} & \text{if droop\_flag} = 0 \\ x_{g6} & \text{if droop\_flag} = 1 \end{cases} \\ y_1 &= \frac{0}{T_1 T_2} \text{ll\_in} + \frac{T_3}{T_1 T_2} x_{g1} + \frac{1}{T_1 T_2} x_{g2} \\ y_2 &= \frac{0}{T_5 T_6} y_1 + \frac{T_4}{T_5 T_6} x_{g3} + \frac{1}{T_5 T_6} x_{g4} \\ P_m &= x_{g5}(T_d) \cdot \omega \\ \tau_m &= \frac{P_m}{\omega} \end{align*}\]

Notes

When droop flag = 0: feedback comes from actuator output (speed droop)
When droop flag = 1: feedback comes from electrical power (load droop)

WPIDHY `[WPIDHY]`

This model represents a Woodward PID Hydro Turbine Governor, a 7-state model designed for hydro turbine control.

\[\begin{align} T_{reg} \dot{x}_{g1} &= \text{reg} \cdot x_{in} - x_{g1} \tag{9a} \\ \dot{x}_{g2} &= K_i' \cdot \text{pid\_input} + \frac{K_p'}{K_i'} x_{g2} \tag{9b} \\ T_a \dot{x}_{g3} &= \text{pid\_out} - x_{g3} \tag{9c} \\ T_a \dot{x}_{g4} &= -\frac{K_d'}{T_a} \text{pid\_input} - x_{g4} \tag{9d} \\ T_b \dot{x}_{g5} &= x_{g3} - x_{g5} \tag{9e} \\ \dot{x}_{g6} &= \text{clamp}(x_{g5}, V_{min}, V_{max}) \quad \text{with windup protection} \tag{9f} \\ \frac{T_w}{2} \dot{x}_{g7} &= \text{power\_at\_gate} + T_w x_{g7} - x_{g7} \tag{9g} \end{align}\]

with

\[\begin{align*} x_{in} &= \tau_e - P_{ref} \\ \text{pid\_input} &= x_{g1} - (\omega - \omega_{sys}) \\ K_p' &= (-T_a K_i) + K_p \\ K_d' &= (T_a^2 K_i) - (T_a K_p) + K_d \\ K_i' &= K_i \\ \text{pi\_out} &= \text{pid\_input} \cdot K_p' + x_{g2} \\ \text{pd\_out} &= x_{g4} + \frac{K_d'}{T_a} \text{pid\_input} \\ \text{pid\_out} &= \text{pi\_out} + \text{pd\_out} \\ \text{power\_at\_gate} &= f(\text{clamp}(x_{g6}, G_{min}, G_{max})) \\ P_m &= \text{clamp}(x_{g7}, P_{min}, P_{max}) - D(\omega - \omega_{sys}) \\ \tau_m &= \frac{P_m}{\omega} \end{align*}\]

Note that this model includes:

Includes nonlinear gate-to-power mapping function
Water inertia modeled with lead-lag compensation

PIDGOV `[PIDGOV]`

This model represents a PID Turbine Governor, typically used for hydro applications. It's a 7-state model with configurable feedback.

\[\begin{align} T_{reg} \dot{x}_{g1} &= R_{perm} \cdot x_{in} - x_{g1} \tag{10a} \\ \dot{x}_{g2} &= K_i \cdot \text{pid\_input} \tag{10b} \\ T_a \dot{x}_{g3} &= \text{pi\_out} - x_{g3} \tag{10c} \\ T_a \dot{x}_{g4} &= -\frac{K_d}{T_a} \text{pid\_input} - x_{g4} \tag{10d} \\ T_a \dot{x}_{g5} &= (x_{g3} + \text{pd\_out}) - x_{g5} \tag{10e} \\ \dot{x}_{g6} &= \text{clamp}\left(\frac{x_{g5} - x_{g6}}{T_b}, V_{min}, V_{max}\right) \quad \text{with windup protection} \tag{10f} \\ \frac{T_z}{2} \dot{x}_{g7} &= \text{power\_at\_gate} + T_z x_{g7} - x_{g7} \tag{10g} \end{align}\]

with

\[\begin{align*} x_{in} &= \begin{cases} P_{ref} - \tau_e & \text{if feedback\_flag} = 0 \\ P_{ref} - x_{g6} & \text{if feedback\_flag} = 1 \end{cases} \\ \text{pid\_input} &= x_{g1} - (\omega - \omega_{sys}) \\ \text{pi\_out} &= K_p \cdot \text{pid\_input} + x_{g2} \\ \text{pd\_out} &= x_{g4} + \frac{K_d}{T_a} \text{pid\_input} \\ T_z &= A_{tw} \cdot T_w \\ \text{power\_at\_gate} &= f(\text{clamp}(x_{g6}, G_{min}, G_{max})) \\ P_m &= x_{g7} - D_{turb}(\omega - \omega_{sys}) \\ \tau_m &= \frac{P_m}{\omega} \end{align*}\]

Note that this model includes:

Configurable feedback: electrical power or gate position
Modified integrator with rate limiting for gate servo
Water column dynamics with surge tank effects
Nonlinear gate-to-power characteristic curve