GASTPTI

GASTPTI is a gas turbine-governor model with speed-droop fuel control and an exhaust-temperature low-value selector.

Warning

GridKit does not yet apply the associated generator’s Governor Response Limits modes Down Only and Fixed to GASTPTI. Normal response is always used.

Notes

• PowerWorld caps its load reference at \(A_T\) during transient simulation; GridKit does not.

• The GASTD-only dbL/dbH speed deadband is not part of GASTPTI.

• Unlike PowerWorld, GridKit rejects rather than swaps reversed \(V^{\min}\) and \(V^{\max}\) values.

Block Diagram

Figure 1: GASTPTI governor model. Figure courtesy of the PowerWorld GAST_PTI model reference.

Model Parameters

Symbol	Units	JSON	Description	Typical Value	Note
\(R\)	[p.u.]	R	Permanent speed droop	0.05	Speed deviation per component-base power deviation
\(T_1\)	[s]	T1	Fuel-valve time constant	0.4
\(T_2\)	[s]	T2	Fuel-flow time constant	0.1
\(T_3\)	[s]	T3	Exhaust-temperature time constant	3.0
\(A_T\)	[p.u.]	At	Ambient-temperature load limit	1.0	Component base
\(K_T\)	[p.u.]	Kt	Exhaust-temperature feedback gain	2.0
\(V^{\max}\)	[p.u.]	Vmax	Upper valve response limit	1.0	Component base
\(V^{\min}\)	[p.u.]	Vmin	Lower valve response limit	0.0	Component base
\(D^\mathrm{turb}\)	[p.u.]	Dturb	Turbine damping coefficient	0.0	Component-base power per speed deviation
\(T^\mathrm{rate}\)	[MW]	Trate	Turbine rating	Required	Same-valued MVA component base; GridKit addition

Parameter Validation

A valid GASTPTI parameter set must satisfy the following conditions:

\[\begin{split}\begin{aligned} R &> 0 \\ T_1,T_2,T_3 &\ge 0 \\ A_T,K_T,D^\mathrm{turb} &\ge 0 \\ T^\mathrm{rate} &> 0 \\ V^{\min} &\le V^{\max} \end{aligned}\end{split}\]

Model Derived Parameters

Let \(\epsilon_T=10^{-3}\ \mathrm{s}\). Accepted time constants below \(\epsilon_T\) are raised to that floor in place:

\[\begin{split}\begin{aligned} T_x &\leftarrow \max\!(T_x,\epsilon_T), && x\in\{1,2,3\} \\ S^{\mathrm{base}} &\leftarrow 10^6 T^\mathrm{rate} \\ k_{\mathrm{base}} &= \dfrac{S^{\mathrm{sys}}}{S^{\mathrm{base}}} \end{aligned}\end{split}\]

Multiplication by \(k_{\mathrm{base}}\) converts system-base power to component base. \(S^{\mathrm{sys}}\) and \(S^{\mathrm{base}}\) are stored in VA.

Model Ports

Name	Port	Init	Description
speed	Input	Known	Optional machine speed deviation; defaults to zero
pref	Input	Unknown	Optional active-power/load reference; latches its initialized value when unattached
pmech	Output	Known	Required mechanical power output

Model Variables

Internal Variables

Differential

Symbol	Units	Description	Note
\(x_V\)	[p.u.]	Fuel-valve state	State 1 in Fig. 1; component base
\(x_F\)	[p.u.]	Fuel-flow state	State 2 in Fig. 1; component base
\(x_T\)	[p.u.]	Exhaust-temperature feedback state	State 3 in Fig. 1; component base

Algebraic

Symbol	Units	Description	Note
\(V_D\)	[p.u.]	Speed/load fuel demand	Component base
\(V_T\)	[p.u.]	Temperature-limit fuel demand	Component base
\(V\)	[p.u.]	Low-value selector output	Component base
\(P_\mathrm{m}\)	[p.u.]	Mechanical power output	System base

External Variables

Differential

None.

Algebraic

Symbol	Units	Description	Note
\(\omega\)	[p.u.]	Machine speed deviation	Optional speed; defaults to zero
\(P^\mathrm{ref}\)	[p.u.]	Active-power/load reference	Optional pref; system base

Model Equations

Smooth functions: antiwindup, min.

Internal Equations

Differential

The initialized constants \(V_{\mathrm{resp}}^{\min}\), \(V_{\mathrm{resp}}^{\max}\), and \(s^{\mathrm{valve}}\) are defined under Internal Initialization.

\[\begin{split}\begin{aligned} 0 &= -\dot{x}_V + \dfrac{s^{\mathrm{valve}}}{T_1} \text{antiwindup}\!( x_V,V-x_V; V_{\mathrm{resp}}^{\min},V_{\mathrm{resp}}^{\max} ) \\ 0 &= -\dot{x}_F + \dfrac{1}{T_2}(-x_F+x_V) \\ 0 &= -\dot{x}_T + \dfrac{1}{T_3}(-x_T+x_F) \end{aligned}\end{split}\]

Algebraic

\[\begin{split}\begin{aligned} 0 &= - \omega + R(k_{\mathrm{base}}P^\mathrm{ref}-V_D) \\ 0 &= -V_T + A_T+K_T(A_T-x_T) \\ 0 &= -V + \text{min}(V_D,V_T) \\ 0 &= -k_{\mathrm{base}}P_\mathrm{m} + x_F-D^\mathrm{turb}\omega \end{aligned}\end{split}\]

External Equations

None.

Initialization

Input Initialization

\[\begin{split}\begin{aligned} \omega &\leftarrow \text{machine speed deviation} \\ P_\mathrm{m} &\leftarrow \text{machine mechanical power} \end{aligned}\end{split}\]

Internal Initialization

\[\begin{split}\begin{aligned} x_F &\leftarrow k_{\mathrm{base}}P_\mathrm{m} +D^\mathrm{turb}\omega \\ x_V,x_T &\leftarrow x_F \\ V_T &\leftarrow A_T+K_T(A_T-x_F) \\ m_T &\leftarrow V_T-x_F \\ (V_{\mathrm{resp}}^{\min},V_{\mathrm{resp}}^{\max}) &\leftarrow (\min(V^{\min},x_F),\max(V^{\max},x_F)) \end{aligned}\end{split}\]

\[\begin{split}s^{\mathrm{valve}} \leftarrow \begin{cases} 1 & V_{\mathrm{resp}}^{\min}\lt V_{\mathrm{resp}}^{\max} \\ 0 & V_{\mathrm{resp}}^{\min}=V_{\mathrm{resp}}^{\max} \end{cases}\end{split}\]

\[\begin{split}(V_{D},V) \leftarrow \begin{cases} ( V_T-\text{iramp}\!(m_T), x_F ) & s^{\mathrm{valve}}=1 \\ ( x_F, \text{min}\!(x_F,V_T) ) & s^{\mathrm{valve}}=0 \end{cases}\end{split}\]

When \(s^{\mathrm{valve}}=1\), initialization requires a finite positive \(m_T\) so iramp is defined. All candidates and response bounds are validated before state, derivatives, or signals are changed; failed initialization is atomic.

A response limit that excludes the initialized fuel flow is widened to include it, and a warning is logged. This matches PowerWorld’s default Modify Limits and Run treatment of initial limit violations.

Output Initialization

\[P^\mathrm{ref} \leftarrow \dfrac{1}{k_{\mathrm{base}}} \left(V_D+\dfrac{\omega}{R}\right)\]

Initialization preserves the machine-seeded system-base \(P_\mathrm{m}\). An attached pref signal receives the initialized reference; an unattached port latches that value for subsequent residual evaluations.

Monitors

Monitor	Units	Description	Note
pmech	[p.u.]	Mechanical-power output	\(P_\mathrm{m}\); system base
xvalve	[p.u.]	Fuel-valve state	\(x_V\); component base
xflow	[p.u.]	Fuel-flow state	\(x_F\); component base
xtemp	[p.u.]	Exhaust-temperature feedback state	\(x_T\); component base
vload	[p.u.]	Speed/load fuel demand	\(V_D\); component base
vtemp	[p.u.]	Temperature-limit fuel demand	\(V_T\); component base

Appendix A: iramp

For a positive smooth-ramp output \(v>0\) and CommonMath smoothing parameter \(\mu\),

\[\text{iramp}(v) = v+\dfrac{1}{\mu}\log(1-\exp(-\mu v))\]

This is the positive-range inverse of GridKit’s smooth ramp.