IEEEG1

IEEEG1 is a steam turbine-governor model with speed deadband, a governor lead–lag, rate- and position-limited governor output, optional nonlinear governor gain, turbine bowl and reheat stages, and separate high-pressure and low-pressure mechanical-power outputs.

Notes

• Input and output powers are on the turbine-rating base when Trate > 0; otherwise the connected machine MVA base is used.

• The dashed dbL/dbH speed deadband block is only for IEEEG1D. IEEEG1 uses the Type 1 no-offset db1 block documented with CommonMath deadband1.

Block Diagram

Figure 1: Governor IEEEG1 model. Figure courtesy of PowerWorld

Model Parameters

Symbol	Units	JSON	Description	Typical Value	Note
\(K\)	[p.u.]	K	Governor speed-control gain	20.0
\(T_1\)	[s]	T1	Governor lead–lag denominator time constant	0.0
\(T_2\)	[s]	T2	Governor lead–lag numerator time constant	0.0
\(T_3\)	[s]	T3	Governor output servo time constant	0.1
\(U_\mathrm{o}\)	[p.u./s]	Uo	Maximum opening rate	0.1	Source label: UO
\(U_\mathrm{c}\)	[p.u./s]	Uc	Maximum closing rate	-0.1	Source label: UC
\(P^{\max}\)	[p.u.]	Pmax	Maximum governor output	1.0	Source label: PMAX
\(P^{\min}\)	[p.u.]	Pmin	Minimum governor output	0.0	Source label: PMIN
\(T_4\)	[s]	T4	Turbine bowl time constant	0.3	State 3 in Fig. 1
\(K_1\)	[p.u.]	K1	High-pressure fraction from turbine bowl	0.2	Top output branch
\(K_2\)	[p.u.]	K2	Low-pressure fraction from turbine bowl	0.0	Bottom output branch
\(T_5\)	[s]	T5	Reheater time constant	5.0	State 4 in Fig. 1
\(K_3\)	[p.u.]	K3	High-pressure fraction from reheater	0.3	Top output branch
\(K_4\)	[p.u.]	K4	Low-pressure fraction from reheater	0.0	Bottom output branch
\(T_6\)	[s]	T6	Crossover time constant	0.5	State 5 in Fig. 1
\(K_5\)	[p.u.]	K5	High-pressure fraction from crossover	0.5	Top output branch
\(K_6\)	[p.u.]	K6	Low-pressure fraction from crossover	0.0	Bottom output branch
\(T_7\)	[s]	T7	Double-reheat time constant	0.5	State 6 in Fig. 1
\(K_7\)	[p.u.]	K7	High-pressure fraction from double reheat	0.0	Top output branch
\(K_8\)	[p.u.]	K8	Low-pressure fraction from double reheat	0.0	Bottom output branch
\(D_{\omega}\)	[p.u.]	db1	Type 1 speed deadband threshold	0.0	uses CommonMath deadband1
\(\epsilon\)	[p.u.]	Eps	Nonlinear gain smoothing/curve tolerance	0.0	Source nonlinear gain setting
\(D_{\mathrm{gv}}\)	[p.u.]	db2	Governor-output backlash/deadband width	0.0	nonzero support must be explicit
\(P^{\mathrm{rate}}\)	[MW]	Trate	Optional turbine-rating power base	0.0	Trate > 0 defines the governor base

The optional nonlinear governor gain curve is represented by source points:

Symbol	Units	JSON	Description	Typical Value	Note
\(G_V^{(k)}\)	[p.u.]	Gv1-Gv6	Governor-output curve input point \(k\)	0.0	Source labels: Gv1 through Gv6
\(P_{\mathrm{GV}}^{(k)}\)	[p.u.]	Pgv1-Pgv6	Governor-output curve value point \(k\)	0.0	Source labels: Pgv1 through Pgv6

Parameter Validation

A valid IEEEG1 parameter set must satisfy the following conditions:

\[\begin{split}\begin{aligned} &T_1,T_2,T_3,T_4,T_5,T_6,T_7 \ge 0,\quad T_3>0 \\ &T_1 > 0\quad\text{or}\quad(T_1 = 0\ \text{and}\ T_2 = 0) \\ &U_\mathrm{c} < 0 < U_\mathrm{o},\quad P^{\min}\le P^{\max} \\ &D_{\omega}\ge 0,\quad D_{\mathrm{gv}}\ge 0,\quad P^{\mathrm{rate}}\ge 0 \\ &G_V^{(1)} < G_V^{(2)} < \cdots < G_V^{(6)} \\ &0 \le P_{\mathrm{GV}}^{(1)} \le P_{\mathrm{GV}}^{(2)} \le \cdots \le P_{\mathrm{GV}}^{(6)} \end{aligned}\end{split}\]

Model Derived Parameters

The governor component base and nonlinear governor-output curve are:

\[\begin{split}\begin{aligned} S_{\mathrm{gov}}^{\mathrm{base}} &= \begin{cases} P^{\mathrm{rate}} & P^{\mathrm{rate}} > 0 \\ S^{\mathrm{machine}} & \text{otherwise} \end{cases} \\ N_{\mathrm{GV}}(x) &= P_{\mathrm{GV}}^{(1)} + \sum_{k=1}^{5} \text{linseg}\!( x;\, G_V^{(k)},\, G_V^{(k+1)},\, P_{\mathrm{GV}}^{(k+1)} - P_{\mathrm{GV}}^{(k)} ) \end{aligned}\end{split}\]

Model Ports

Name	Port	Init	Description
speed	Input	TBD	Machine speed deviation
pref	Input	TBD	Governor reference
paux	Input	TBD	Auxiliary power input
pmech_hp	Output	TBD	High-pressure mechanical-power output
pmech_lp	Output	TBD	Low-pressure mechanical-power output

Model Variables

Internal Variables

Differential

Symbol	Units	Description	Note
\(P_{\mathrm{GV}}\)	[p.u.]	Governor output	State 1 in Fig. 1
\(x_{\mathrm{LL}}\)	[p.u.]	Governor lead–lag state	State 2 in Fig. 1
\(x_4\)	[p.u.]	Turbine bowl state	State 3 in Fig. 1; denominator \(T_4\)
\(x_5\)	[p.u.]	Reheater state	State 4 in Fig. 1; denominator \(T_5\)
\(x_6\)	[p.u.]	Crossover state	State 5 in Fig. 1; denominator \(T_6\)
\(x_7\)	[p.u.]	Double-reheat state	State 6 in Fig. 1; denominator \(T_7\)

Algebraic

Symbol	Units	Description	Note
\(\omega_{\mathrm{db}}\)	[p.u.]	Deadbanded speed deviation	Defined by CommonMath deadband1
\(y_{\omega}\)	[p.u.]	Lead–lag-conditioned speed signal	Output of \(K(1+sT_2)/(1+sT_1)\)
\(e_G\)	[p.u.]	Governor command error	Sum of references minus speed and output feedback
\(r_G\)	[p.u./s]	Rate-limited governor derivative target	Limited by \(U_\mathrm{c}\) and \(U_\mathrm{o}\)
\(P_{\mathrm{GV}}^{\mathrm{nl}}\)	[p.u.]	Nonlinear governor gain output	Output of db2/curve branch
\(P_\mathrm{m}^{\mathrm{HP}}\)	[p.u.]	High-pressure mechanical-power output	Source label: PMECH_HP
\(P_\mathrm{m}^{\mathrm{LP}}\)	[p.u.]	Low-pressure mechanical-power output	Source label: PMECH_LP

External Variables

Differential

None.

Algebraic

Symbol	Units	Description	Note
\(\omega\)	[p.u.]	Machine speed deviation	Source label: Speed
\(P_{\mathrm{ref}}\)	[p.u.]	Governor reference	Source label: Pref
\(P_{\mathrm{aux}}\)	[p.u.]	Auxiliary power input	Source label: Paux; optional, defaults to zero

Model Equations

Smooth functions: antiwindup, clamp, deadband1, deadband2, linseg.

Internal Equations

Differential

\[\begin{split}\begin{aligned} 0 &= -T_1\dot x_{\mathrm{LL}} - x_{\mathrm{LL}} + K\omega_{\mathrm{db}} \\ 0 &= -\dot P_{\mathrm{GV}} + \text{antiwindup}\!( P_{\mathrm{GV}}, r_G; P^{\min}, P^{\max} ) \\ 0 &= -T_4\dot x_4 - x_4 + P_{\mathrm{GV}}^{\mathrm{nl}} \\ 0 &= -T_5\dot x_5 - x_5 + x_4 \\ 0 &= -T_6\dot x_6 - x_6 + x_5 \\ 0 &= -T_7\dot x_7 - x_7 + x_6 \end{aligned}\end{split}\]

Algebraic

\[\begin{split}\begin{aligned} 0 &= -\omega_{\mathrm{db}} + \text{deadband1}(\omega;\ -D_{\omega},\ D_{\omega}) \\ 0 &= -y_{\omega} + \begin{cases} K\omega_{\mathrm{db}}, & T_1 = T_2 = 0 \\ x_{\mathrm{LL}} + \dfrac{T_2}{T_1}(K\omega_{\mathrm{db}}-x_{\mathrm{LL}}), & T_1 > 0 \end{cases} \\ 0 &= -e_G + P_{\mathrm{ref}} + P_{\mathrm{aux}} - y_{\omega} - P_{\mathrm{GV}} \\ 0 &= -r_G + \text{clamp}\!\left(\dfrac{e_G}{T_3}; U_\mathrm{c}, U_\mathrm{o}\right) \\ 0 &= -P_{\mathrm{GV}}^{\mathrm{nl}} + N_{\mathrm{GV}}\!(\text{deadband2}(P_{\mathrm{GV}}; -D_{\mathrm{gv}}, D_{\mathrm{gv}})) \\ 0 &= -P_\mathrm{m}^{\mathrm{HP}} + K_1x_4 + K_3x_5 + K_5x_6 + K_7x_7 \\ 0 &= -P_\mathrm{m}^{\mathrm{LP}} + K_2x_4 + K_4x_5 + K_6x_6 + K_8x_7 \end{aligned}\end{split}\]

When \(T_1=T_2=0\), the governor lead–lag block is bypassed so \(y_{\omega}=K\omega_{\mathrm{db}}\).

External Equations

None.

Initialization

For a power-flow start, require \(\omega=0\) and initialize all internal derivatives to zero:

\[\begin{split}\begin{aligned} P_{\mathrm{aux}} &\leftarrow 0,\quad \text{if omitted} \\ \omega_{\mathrm{db}} &\leftarrow \text{deadband1}(0;\ -D_{\omega},\ D_{\omega}) \\ x_{\mathrm{LL}} &\leftarrow K\omega_{\mathrm{db}} \\ y_{\omega} &\leftarrow x_{\mathrm{LL}} \end{aligned}\end{split}\]

Solve for \(P_{\mathrm{GV}}\) and \(P_{\mathrm{GV}}^{\mathrm{nl}}\) using the supplied high- and low-pressure mechanical powers:

\[\begin{split}\begin{aligned} 0 &= -P_{\mathrm{GV}}^{\mathrm{nl}} + N_{\mathrm{GV}}\!(\text{deadband2}(P_{\mathrm{GV}}; -D_{\mathrm{gv}}, D_{\mathrm{gv}})) \\ 0 &= -P_{\mathrm{m}}^{\mathrm{HP}} + (K_1+K_3+K_5+K_7)P_{\mathrm{GV}}^{\mathrm{nl}} \\ 0 &= -P_{\mathrm{m}}^{\mathrm{LP}} + (K_2+K_4+K_6+K_8)P_{\mathrm{GV}}^{\mathrm{nl}} \end{aligned}\end{split}\]

\[\begin{split}\begin{aligned} x_4,x_5,x_6,x_7 &\leftarrow P_{\mathrm{GV}}^{\mathrm{nl}} \\ P_{\mathrm{ref}} &\leftarrow P_{\mathrm{GV}} + y_{\omega} - P_{\mathrm{aux}} \end{aligned}\end{split}\]

Initialization requires the effective governor output to lie inside \(P^{\min}\) and \(P^{\max}\) and the opening/closing rate limits to be inactive.

Monitors

Monitor	Units	Description	Note
pmech_hp	[p.u.]	High-pressure mechanical-power output	\(P_\mathrm{m}^{\mathrm{HP}}\)
pmech_lp	[p.u.]	Low-pressure mechanical-power output	\(P_\mathrm{m}^{\mathrm{LP}}\)
pgv	[p.u.]	Governor output	State 1
leadlag	[p.u.]	Governor lead–lag state	State 2
bowl	[p.u.]	Turbine bowl state	State 3
reheater	[p.u.]	Reheater state	State 4
crossover	[p.u.]	Crossover state	State 5
double_reheat	[p.u.]	Double-reheat state	State 6