IEEEG1

Source: GridKit/Model/PhasorDynamics/Governor/IEEEG1/README.md

IEEE Type 1 Speed-Governor Model (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.

• Source governor-response settings may modify \(U_o\), \(U_c\), \(P^{\max}\), and \(P^{\min}\) before the equations are evaluated.

• PSSE IEEEG1 source data may omit db1, db2, nonlinear-gain points, and turbine rating; those omitted features must be documented as inactive rather than silently dropped.

Block Diagram

Standard model of the IEEEG1 Governor.

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	Block name: K
\(T_1\)	[sec]	T1	Governor lead-lag denominator time constant	0.0	Block name: T1
\(T_2\)	[sec]	T2	Governor lead-lag numerator time constant	0.0	Block name: T2
\(T_3\)	[sec]	T3	Governor output servo time constant	0.1	Block name: T3
\(U_o\)	[p.u./s]	Uo	Maximum opening rate	0.1	Source label: UO
\(U_c\)	[p.u./s]	Uc	Maximum closing rate	-0.1	Source label: UC
\(P^{\max}\)	[p.u.]	Pmax	Maximum governor output	1.0	Block name: PMAX
\(P^{\min}\)	[p.u.]	Pmin	Minimum governor output	0.0	Block name: PMIN
\(T_4\)	[sec]	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
\(T_5\)	[sec]	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
\(T_6\)	[sec]	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
\(T_7\)	[sec]	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	Block name: db1; 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	Block name: db2; 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

Invalid IEEEG1 parameter sets are rejected by the following checks. If source governor-response settings adjust limits, apply these checks to the effective values used by the equations.

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

Model Derived Parameters

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

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

CommonMath defines the linear segment helper used by \(N_{\mathrm{GV}}\).

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 \(U_c\) and \(U_o\)
\(P_{\mathrm{GV}}^{\mathrm{nl}}\)	[p.u.]	Nonlinear governor gain output	Output of db2/curve branch
\(P_m^{\mathrm{HP}}\)	[p.u.]	High-pressure mechanical-power output	Source label: PMECH_HP
\(P_m^{\mathrm{LP}}\)	[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

Differential Equations

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

CommonMath defines the Anti-Windup target and smooth approximation.

Algebraic Equations

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

CommonMath defines helper targets and smooth approximations for deadband1, deadband2, clamp, and linseg. When \(T_1=T_2=0\), the governor lead-lag block is bypassed so \(y_{\omega}=K\omega_{\mathrm{db}}\).

Initialization

Initialization is performed by evaluating the steady-state residuals in dependency order. Let subscript \(0\) denote initial values and set all internal derivatives to zero. For a standard power-flow start:

\[\begin{split}\begin{aligned} \omega_0 &= 0 \\ P_{\mathrm{aux},0} &= 0 \\ \omega_{\mathrm{db},0} &= \text{deadband1}(0,\ -D_{\omega},\ D_{\omega}) \\ x_{\mathrm{ll},0} &= K\omega_{\mathrm{db},0} \\ y_{\omega,0} &= x_{\mathrm{ll},0} \end{aligned}\end{split}\]

Given initialized high- and low-pressure mechanical powers, solve the turbine chain by choosing \(P_{\mathrm{GV},0}\) so that the turbine fractions reproduce the connected machine operating point:

\[\begin{split}\begin{aligned} P_{\mathrm{GV},0}^{\mathrm{nl}} &= N_{\mathrm{GV}}\!\left(\text{deadband2}(P_{\mathrm{GV},0}, -D_{\mathrm{gv}}, D_{\mathrm{gv}})\right) \\ x_{4,0}=x_{5,0}=x_{6,0}=x_{7,0} &= P_{\mathrm{GV},0}^{\mathrm{nl}} \\ P_{m,0}^{\mathrm{HP}} &= (K_1+K_3+K_5+K_7)P_{\mathrm{GV},0}^{\mathrm{nl}} \\ P_{m,0}^{\mathrm{LP}} &= (K_2+K_4+K_6+K_8)P_{\mathrm{GV},0}^{\mathrm{nl}} \\ P_{\mathrm{ref},0} &= P_{\mathrm{GV},0} + y_{\omega,0} - P_{\mathrm{aux},0} \end{aligned}\end{split}\]

This closed-form start requires the effective governor output to lie inside \(P^{\min}\) and \(P^{\max}\) and the opening/closing rate limits to be inactive. Starts where governor response limits fix the limits to the initial condition must document those effective limits before applying the residuals.

Model Outputs

Output	Units	Description	Note
pmech_hp	[p.u.]	High-pressure mechanical-power output	\(P_m^{\mathrm{HP}}\)
pmech_lp	[p.u.]	Low-pressure mechanical-power output	\(P_m^{\mathrm{LP}}\)
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