Tgov1

Source: GridKit/Model/PhasorDynamics/Governor/Tgov1/README.md

Steam Turbine-Governor Model (TGOV1)

Block Diagram

Standard model of the stream turbine

Figure 1: Governor TGOV1 model. Figure courtesy of PowerWorld

Model Parameters

Symbol	Units	Description	Typical Value
\(R\)	[p.u.]	Droop Constant	0.05
\(T_1\)	[sec]	Valve Time Delay	0.5
\(T_2\)	[sec]	Turbine Numerator Time Constant	2.5
\(T_3\)	[sec]	Turbine Delay	7.5
\(P_v^{\max}\)	[p.u.]	Max Valve Position	1
\(P_v^{\min}\)	[p.u.]	Min Valve Position	0
\(D_t\)	[p.u.]	Turbine Damping Coefficient	0

Model Variables

Internal Variables

Differential

Symbol	Units	Description	Note
\(P_{tx}\)	[p.u.]	Turbine Power (State 1 in Fig. 1)
\(P_v\)	[p.u.]	Valve Position (State 2 in Fig. 1)

Algebraic

Symbol	Units	Description	Note
\(P_m\)	[p.u.]	Mechnical Power to Generator	Read by a Machine Model

External Variables

Differential

Symbol	Units	Description	Note
\(\omega\)	[p.u.]	Machine Speed Deviation	Read from a Machine Model

Algebraic

Symbol	Units	Description	Note
\(P_{ref}\)	[p.u.]	Reference Power	Either a constant parameter or external variable

Model Equations

Differential Equations

The TGOV1 differential equations, as derived from the model diagram. Define the pre-limit derivative of \(P_v\)

\[f = \dfrac{1}{T_1}\left[-P_v + \dfrac{1}{R}(P_{ref} - \omega)\right]\]

so that \(\dot P_v\) can be written in piecewise form compactly.

\[\begin{split}\begin{aligned} \dot P_{tx} &= P_v - \dfrac{1}{T_3}(P_{tx}+T_2P_v) \\ \dot P_v &= \begin{cases} f & \text{if } (P_v^{\min} < P_v < P_v^{\max}) & \lor \\ & \quad (P_v \leq P_v^{\min} \land f>0) & \lor \\ & \quad(P_v \geq P_v^{\max} \land f<0) \\ 0 & \text{else} \end{cases} \end{aligned}\end{split}\]

Algebraic Equations

The algebraic equation dictating the mechnical power output.

\[\begin{split}\begin{aligned} P_m &= \dfrac{1}{T_3}(P_{tx}+T_2P_v) - D_t \omega \\ \end{aligned}\end{split}\]

In simulation the piecewise form above is replaced with a smooth approximation where \(\phi\) is GridKit’s smooth anti-windup indicator. See CommonMath: Anti-Windup Indicator for its definition, behavior, and design rationale.

Initialization

At steady state we assume that \(P_v\) is at or within its limits. This implies the initial conditions are a function of \(P_m\) which is equal to the electric torque.

\[\begin{split}\begin{aligned} P_{tx} &= (T_3-T_2) P_m\\ P_v &= P_m\\ \dot P_{tx} &=0\\ \dot P_v &=0\\ \end{aligned}\end{split}\]

And if the reference power is a constant parameter, we can determine the value by solving the steady state equations.

\[\begin{split}\begin{aligned} P_{ref} &= R P_m\\ \end{aligned}\end{split}\]