GenClassical

GENCLS is the 2nd order synchronous machine model: a constant transient EMF behind the transient reactance. See the General Synchronous Machine Model for general synchronous machine information.

Notes

• No field dynamics: the efd input is the EMF behind \(X'_d\) and is held constant when unconnected.

• No saliency and no saturation.

Model Parameters

Symbol	Units	JSON	Description	Typical Value	Note
\(P_0\)	[p.u.]	p0	Initial active power injection	1.0	System base; required initialization source
\(Q_0\)	[p.u.]	q0	Initial reactive power injection	0.0	System base; required initialization source
\(S^\mathrm{base}\)	[MVA]	mva	GENCLS component power base	100.0
\(H\)	[s]	H	Rotor inertia	3.0
\(D\)	[p.u.]	D	Damping coefficient	0.0
\(R_\mathrm{a}\)	[p.u.]	Ra	Armature resistance	0.0	Component base
\(X'_d\)	[p.u.]	Xdp	Direct-axis transient reactance	0.2	Component base

Parameter Validation

None.

Model Derived Parameters

\[\begin{split}\begin{aligned} G &= \dfrac{R_\mathrm{a}}{R_\mathrm{a}^2+(X'_d)^2} \\ B &= -\dfrac{X'_d}{R_\mathrm{a}^2+(X'_d)^2} \\ k_\mathrm{base} &= \dfrac{S^\mathrm{sys}}{S^\mathrm{base}} \end{aligned}\end{split}\]

Multiplying by \(k_\mathrm{base}\) converts system base to component base. \(S^\mathrm{sys}\) and \(S^\mathrm{base}\) are stored in VA; \(f^\mathrm{sys}\) is the system frequency base in Hz.

Model Ports

Name	Port	Init	Description
bus	Bus	Known	Terminal bus voltage
pmech	Input	Unknown	Mechanical-power input
efd	Input	Unknown	Field-voltage input
speed	Output	Known	Speed-deviation output

Model Variables

Internal Variables

Differential

Symbol	Units	Description	Note
\(\delta\)	[rad]	Rotor angle
\(\omega\)	[p.u.]	Speed deviation	Exported through speed when assigned

Algebraic

Symbol	Units	Description	Note
\(T_\mathrm{e}\)	[p.u.]	Electrical torque	Component base
\(I_\mathrm{r}\)	[p.u.]	Terminal current, real component	Component base
\(I_\mathrm{i}\)	[p.u.]	Terminal current, imaginary component	Component base

External Variables

Differential

None.

Algebraic

Symbol	Units	Init	Description	Note
\(V_\mathrm{r}\)	[p.u.]	Known	Terminal voltage, real component	Bus input
\(V_\mathrm{i}\)	[p.u.]	Known	Terminal voltage, imaginary component	Bus input
\(P_\mathrm{m}\)	[p.u.]	Unknown	Mechanical power	Optional signal port pmech; system base
\(E_\mathrm{fd}\)	[p.u.]	Unknown	Field voltage	Optional signal port efd; component base

Model Equations

Internal Equations

Differential

\[\begin{split}\begin{aligned} 0 &= -\dot\delta + 2\pi f^\mathrm{sys}\omega \\ 0 &= -\dot\omega + \dfrac{1}{2H}\left(\dfrac{k_\mathrm{base}P_\mathrm{m}-D\omega}{1+\omega}-T_\mathrm{e}\right) \end{aligned}\end{split}\]

Algebraic

\[\begin{split}\begin{aligned} 0 &= -T_\mathrm{e} + GE_\mathrm{fd}^2 - E_\mathrm{fd}[(GV_\mathrm{r}-BV_\mathrm{i})\cos(\delta)+(BV_\mathrm{r}+GV_\mathrm{i})\sin(\delta)] \\ 0 &= -I_\mathrm{r} + E_\mathrm{fd}(G\cos(\delta)-B\sin(\delta)) - GV_\mathrm{r} + BV_\mathrm{i} \\ 0 &= -I_\mathrm{i} + E_\mathrm{fd}(B\cos(\delta)+G\sin(\delta)) - BV_\mathrm{r} - GV_\mathrm{i} \end{aligned}\end{split}\]

External Equations

\[\begin{split}\begin{aligned} \Delta I^\mathrm{bus}_\mathrm{r} &\mathrel{+}= \dfrac{I_\mathrm{r}}{k_\mathrm{base}} \\ \Delta I^\mathrm{bus}_\mathrm{i} &\mathrel{+}= \dfrac{I_\mathrm{i}}{k_\mathrm{base}} \end{aligned}\end{split}\]

Initialization

Input Initialization

\[\begin{split}\begin{aligned} V_\mathrm{r}, V_\mathrm{i} &\leftarrow \text{terminal-bus voltage} \\ P_0, Q_0 &\leftarrow \text{power-flow injection on system base} \end{aligned}\end{split}\]

Internal Initialization

All internal derivatives are initialized to zero:

\[\begin{split}\begin{aligned} I_\mathrm{r} &\leftarrow k_\mathrm{base}\dfrac{V_\mathrm{r}P_0+V_\mathrm{i}Q_0}{V_\mathrm{r}^2+V_\mathrm{i}^2} \\ I_\mathrm{i} &\leftarrow k_\mathrm{base}\dfrac{V_\mathrm{i}P_0-V_\mathrm{r}Q_0}{V_\mathrm{r}^2+V_\mathrm{i}^2} \\ \delta &\leftarrow \text{arg}[V_\mathrm{r}+jV_\mathrm{i}+(R_\mathrm{a}+jX'_d)(I_\mathrm{r}+jI_\mathrm{i})] \\ \omega &\leftarrow 0 \\ T_\mathrm{e} &\leftarrow V_\mathrm{r}I_\mathrm{r}+V_\mathrm{i}I_\mathrm{i}+R_\mathrm{a}(I_\mathrm{r}^2+I_\mathrm{i}^2) \end{aligned}\end{split}\]

Output Initialization

\[\begin{split}\begin{aligned} P_\mathrm{m} &\leftarrow \dfrac{T_\mathrm{e}}{k_\mathrm{base}} \\ E_\mathrm{fd} &\leftarrow |V_\mathrm{r}+jV_\mathrm{i}+(R_\mathrm{a}+jX'_d)(I_\mathrm{r}+jI_\mathrm{i})| \end{aligned}\end{split}\]

Monitors

Monitor	Units	Description	Note
ir	[p.u.]	Terminal current, real component	System base; oriented leaving the machine
ii	[p.u.]	Terminal current, imaginary component	System base; oriented leaving the machine
p	[p.u.]	Active power	System base; \(V_\mathrm{r}I_\mathrm{r}+V_\mathrm{i}I_\mathrm{i}\)
q	[p.u.]	Reactive power	System base; \(V_\mathrm{i}I_\mathrm{r}-V_\mathrm{r}I_\mathrm{i}\)
delta	[rad]	Rotor angle
omega	[p.u.]	Speed deviation	\(\omega=0\) at synchronous speed
speed	[p.u.]	Per-unit machine speed	\(1+\omega\)