GENROU

Sixth-order round-rotor synchronous machine. See the shared conventions.

Notes

• \(X''_q=X''_d\) (round rotor assumptions)

• \(X''_{d}\) does not saturate

• Same relative amount of saturation occurs on both \(d\) and \(q\) axis

Block Diagram

Figure 1: GENROU. Figure courtesy of PowerWorld

Model Parameters

Symbol	Units	JSON	Description	Typical Value	Note
\(P_0\)	[p.u.]	p0	Initial active power injection	1.0
\(Q_0\)	[p.u.]	q0	Initial reactive power injection	0.0
\(H\)	[s]	H	rotor inertia	3
\(D\)	[p.u.]	D	damping coefficient	0
\(R_\mathrm{a}\)	[p.u.]	Ra	winding resistance	0
\(T'_{d0}\)	[s]	Tdop	Open circuit direct axis transient time const.	7
\(T''_{d0}\)	[s]	Tdopp	Open circuit direct axis sub-transient time const.	0.04
\(T'_{q0}\)	[s]	Tqop	Open circuit quadrature axis transient time const.	0.75
\(T''_{q0}\)	[s]	Tqopp	Open circuit quadrature axis sub-transient time const.	0.05
\(X_d\)	[p.u.]	Xd	Direct axis synchronous reactance	2.1
\(X'_{d}\)	[p.u.]	Xdp	Direct axis transient reactance	0.2
\(X''_{d}\)	[p.u.]	Xdpp	Direct axis sub-transient reactance	0.18
\(X_q\)	[p.u.]	Xq	Quadrature axis synchronous reactance	0.5
\(X'_{q}\)	[p.u.]	Xqp	Quadrature axis transient reactance	0.5
\(X''_{q}\)	[p.u.]	Xqpp	Quadrature axis sub-transient reactance	0.18
\(X_{\ell}\)	[p.u.]	Xl	Stator leakage reactance	0.15
\(S_{10}\)	[p.u.]	S10	Saturation factor at 1.0 pu flux	0
\(S_{12}\)	[p.u.]	S12	Saturation factor at 1.2 pu flux	0
\(S^\mathrm{base}\)	[MVA]	mva	Machine power base	100

Parameter Validation

None.

Model Derived Parameters

\[\begin{split}\begin{aligned} G &= \dfrac{R_\mathrm{a}}{R_\mathrm{a}^2+(X''_q)^2} & B &= -\dfrac{X''_q}{R_\mathrm{a}^2+(X''_q)^2}\\ S_A &= \min\left(\dfrac{1.2\sqrt{S_{10}/S_{12}} +1}{\sqrt{S_{10}/S_{12}} +1}, \dfrac{1.2\sqrt{S_{10}/S_{12}} -1}{\sqrt{S_{10}/S_{12}} -1}\right) & S_B &= \dfrac{S_{12}}{(S_A-1.2)^2} \\ X_{d1} &= X_d-X'_d & X_{q1} &= X_q-X'_q \\ X_{d2} &= X'_d-X_\ell & X_{q2} &= X'_q-X_\ell\\ X_{d3} &= (X'_d-X''_d)/X_{d2}^2 & X_{q3} &= (X'_q-X''_q)/X_{q2}^2 \\ X_{d4} &= (X'_d-X''_d)/X_{d2} & X_{q4} &= (X'_q-X''_q)/X_{q2} \\ X_{d5} &= (X''_d-X_\ell)/X_{d2} & X_{q5} &= (X''_q-X_\ell)/X_{q2}\\ X_{qd} &= (X_q-X_\ell)/(X_d-X_\ell) \\ f_\mathrm{base} &= f_\mathrm{sys} \end{aligned}\end{split}\]

When \(S_{12}=0\), \(S_A=S_B=0\).

System bases \(f_\mathrm{sys}\) [Hz] and \(S^\mathrm{sys}\) [MVA] are taken from the system at initialization.

Model Ports

Name	Port	Init	Description
bus	Bus	Known	Terminal bus voltage and current-balance residuals
pmech	Input	Unknown	Mechanical-power input; held constant when unconnected
efd	Input	Unknown	Field-voltage input; held constant when unconnected
speed	Output	Known	Machine speed-deviation output

Model Variables

Internal Variables

Differential

Symbol	Units	Description	Note
\(\delta\)	[rad]	Machine internal rotor angle
\(\omega\)	[p.u.]	Machine speed deviation	Optionally read by governor or stabilizer component
\(E'_q\)	[p.u.]	Quadrature axis transient flux
\(\psi'_d\)	[p.u.]	Direct axis subtransient flux
\(\psi'_q\)	[p.u.]	Quadrature axis subtransient flux
\(E'_d\)	[p.u.]	Direct axis transient flux

Algebraic

Symbol	Units	Description	Note
\(\psi''_q\)	[p.u.]	Total q-axis subtransient flux
\(\psi''_d\)	[p.u.]	Total d-axis subtransient flux
\(\psi''\)	[p.u.]	Machine total subtransient flux
\(k_{\mathrm{sat}}\)	[p.u.]	Saturation coefficient
\(V_d\)	[p.u.]	Machine internal voltage, d-axis
\(V_q\)	[p.u.]	Machine internal voltage, q-axis
\(T_\mathrm{e}\)	[p.u.]	Electrical torque
\(I_d\)	[p.u.]	Terminal current, d-axis
\(I_q\)	[p.u.]	Terminal current, q-axis
\(I_r\)	[p.u.]	Terminal current, real component on network reference frame	Machine base
\(I_i\)	[p.u.]	Terminal current, imaginary component on network reference frame	Machine base

External Variables

Differential

None.

Algebraic

Symbol	Units	Description	Note
\(V_r\)	[p.u.]	Terminal voltage, real component on network reference frame	owned by bus object
\(V_i\)	[p.u.]	Terminal voltage, imaginary component on network reference frame	owned by bus object
\(P_\mathrm{m}\)	[p.u.]	Mechanical power from the prime mover	System base
\(E_{\mathrm{fd}}\)	[p.u.]	Field winding voltage from the excitation system	Machine base

Model Equations

Smooth functions: \(q\).

Internal Equations

Differential

\[\begin{split}\begin{aligned} \dot\delta &= \omega \cdot 2\pi f_\mathrm{base} \\ \dot\omega &= \dfrac{1}{2H}\left(\dfrac{(S^\mathrm{sys}/S^\mathrm{base})P_\mathrm{m}-D\omega}{1+\omega} - T_\mathrm{e}\right)\\ \dot{E}'_{q} &= \dfrac{1}{T'_{d0}} ( E_{\mathrm{fd}}-E'_{q}-X_{d1} (I_d+X_{d3}(E'_{q}-\psi'_{d}-X_{d2}I_d)) -\psi''_{d}k_{\mathrm{sat}} )\\ \dot{\psi}'_{d} &= \dfrac{1}{T''_{d0}}(E'_{q}-\psi'_{d}-X_{d2}I_d)\\ \dot{\psi}'_{q} &= \dfrac{1}{T''_{q0}}(E'_{d}-\psi'_{q}+X_{q2}I_q)\\ \dot{E}'_{d} &= \dfrac{1}{T'_{q0}} ( -E'_{d}+X_{q1} (I_q-X_{q3}(E'_{d}-\psi'_{q}+X_{q2}I_q)) + X_{qd}\psi''_{q}k_{\mathrm{sat}} ) \\ \end{aligned}\end{split}\]

Algebraic

\[\begin{split}\begin{aligned} 0 &= -\psi''_{q} -E'_{d}X_{q5} - \psi'_{q}X_{q4} \\ 0 &= -\psi''_{d} +E'_{q}X_{d5} + \psi'_{d}X_{d4}\\ 0 &= -\psi'' +\sqrt{(\psi''_{d})^2+(\psi''_{q})^2} \\ 0 &= -k_{\mathrm{sat}} + S_B q(\psi''-S_A) \\ 0 &= -V_d -\psi''_{q}(1+\omega)\\ 0 &= -V_q +\psi''_{d}(1+\omega)\\ 0 &= -T_\mathrm{e} +(\psi''_{d} - I_dX''_d)I_q-(\psi''_{q} - I_qX''_d)I_d \\ 0 &= -I_d + I_r \sin(\delta) - I_i \cos(\delta) \\ 0 &= -I_q + I_r \cos(\delta) + I_i \sin(\delta) \\ 0 &= -I_r + G (V_d \sin(\delta) + V_q \cos(\delta) - V_r) - B (-V_d \cos(\delta) + V_q \sin(\delta) - V_i) \\ 0 &= -I_i + B (V_d \sin(\delta) + V_q \cos(\delta) - V_r) + G (-V_d \cos(\delta) + V_q \sin(\delta) - V_i) \end{aligned}\end{split}\]

External Equations

The terminal currents are added to the bus residuals on system base:

\[\begin{split}\begin{aligned} I_r^{\mathrm{bus}} &\leftarrow I_r^{\mathrm{bus}} + \dfrac{S^\mathrm{base}}{S^\mathrm{sys}} I_r \\ I_i^{\mathrm{bus}} &\leftarrow I_i^{\mathrm{bus}} + \dfrac{S^\mathrm{base}}{S^\mathrm{sys}} I_i \end{aligned}\end{split}\]

Initialization

The power-flow solution gives \(V_r\), \(V_i\), \(I_r\), and \(I_i\). At synchronous speed, the total subtransient-flux magnitude is available directly in the network frame and is independent of rotor angle:

\[\begin{split}\begin{aligned} \psi'' &\leftarrow \sqrt{ (V_r + R_\mathrm{a} I_r - X''_q I_i)^2 +(V_i + R_\mathrm{a} I_i + X''_q I_r)^2 } \\ k_{\mathrm{sat}} &\leftarrow S_B q(\psi''-S_A) \\ k'_{\mathrm{sat}} &\leftarrow 1 + X_{qd}k_{\mathrm{sat}} \\ X_{\mathrm{sat},\delta} &\leftarrow k'_{\mathrm{sat}}X''_d + X_q-X''_q \\ \delta &\leftarrow \operatorname{atan2}( (V_i+R_\mathrm{a}I_i)k'_{\mathrm{sat}}+X_{\mathrm{sat},\delta}I_r, (V_r+R_\mathrm{a}I_r)k'_{\mathrm{sat}}-X_{\mathrm{sat},\delta}I_i ) \end{aligned}\end{split}\]

With \(\delta\) known, the rotor-frame currents, voltages, flux states, field voltage, and mechanical power follow directly from the steady-state model equations above.

Monitors

Monitor	Units	Description	Note
ir	[p.u.]	Terminal current, real component \(\dfrac{S^\mathrm{base}}{S^\mathrm{sys}}I_r\) in the network frame	Oriented leaving the machine; system base
ii	[p.u.]	Terminal current, imaginary component \(\dfrac{S^\mathrm{base}}{S^\mathrm{sys}}I_i\) in the network frame	Oriented leaving the machine; system base
p	[p.u.]	Active power \(P=\dfrac{S^\mathrm{base}}{S^\mathrm{sys}}(V_rI_r+V_iI_i)\)	Oriented leaving the machine; system base
q	[p.u.]	Reactive power \(Q=\dfrac{S^\mathrm{base}}{S^\mathrm{sys}}(V_iI_r-V_rI_i)\)	Oriented leaving the machine; system base
delta	[rad]	Machine internal rotor angle \(\delta\)
omega	[p.u.]	Machine speed deviation \(\omega\)	\(\omega=0\) at synchronous speed
speed	[p.u.]	Per-unit machine speed	\(1+\omega\)