GenClassical

Source: GridKit/Model/PhasorDynamics/SynchronousMachine/GenClassical/README.md

Classical Generator

An electrical machine model with two differential variables (i.e. second-order model) is often called classical generator model. While its predictive ability is limited, it is useful for studies of grid network properties. Mathematically, it is equivalent to a driven damped pendulum model.

Model Parameters

Symbol	Units	Description	Note
\(P_0\)	[p.u.]	initial active power injection
\(Q_0\)	[p.u.]	initial reactive power injection
\(H\)	[s]	rotor inertia
\(D\)	[p.u.]	damping coefficient
\(R_a\)	[p.u.]	winding resistance
\(X_{dp}\)	[p.u.]	machine reactance parameter
\(S_\mathrm{mach}\)	[MVA]	machine power base

Model Derived Parameters

• \(G = \dfrac{R_a}{R_a^2 + X_{dp}^2} ~~~\) equivalent stator winding conductance

• \(B = \dfrac{-X_{dp}}{R_a^2 + X_{dp}^2} ~~~\) equivalent stator winding susceptance

• \(f_\mathrm{base} = f_\mathrm{sys} ~~~\) frequency base taken from the system at initialization

• \(S_\mathrm{mach,VA} = 10^6 S_\mathrm{mach} ~~~\) derived machine base used for machine-base/system-base conversions

Model Variables

Internal Variables

Differential

Symbol	Units	Description	Note
\(\delta\)	[rad]	machine power angle
\(\omega\)	[p.u]	machine speed deviation	Optionally read by a governor or a stabilizer component

Algebraic

Symbol	Units	Description	Note
\(T_{e}\)	[p.u.]	electrical torque
\(I_r\)	[p.u.]	machine real injection current	read by bus
\(I_i\)	[p.u.]	machine imaginary injection current	read by bus

Note: All three can be expressed as a function called by the model equations. We add these as variables as they are needed for outputs.

External Variables

External variables enter component model equations but are owned by other components. The other components also provide equations needed to have a balanced system of equations.

Differential

None.

Algebraic

Symbol	Units	Description	Note
\(V_r\)	[p.u.]	machine bus real voltage	owned by a bus object
\(V_i\)	[p.u.]	machine bus imaginary voltage	owned by a bus object
\(P_m\)	[p.u.]	mechanical power input	owned by governor, constant if no governor is connected to the machine
\(E_p\)	[p.u.]	field winding voltage	owned by exciter, constant if no exciter is connected to the machine

Model Equations

Differential Equations

\[\begin{split}\begin{aligned} \dot{\delta} &= \omega \cdot 2\pi f_\mathrm{base} \\ \dot{\omega} &= \frac{1}{2H}\left( \frac{P_{m} - D\omega}{1+\omega} - T_{e}\right) \end{aligned}\end{split}\]

Algebraic Equations

\[\begin{split}\begin{aligned} 0 &= T_{e} - \left( G E_p^2 - E_p \left[(G V_r - B V_i)\cos\delta + (B V_r + G V_i)\sin\delta \right]\right) \\ 0 &= I_r + G V_r - B V_i - E_p(G \cos\delta - B \sin\delta) \\ 0 &= I_i + B V_r + G V_i - E_p(B \cos\delta + G \sin\delta) \end{aligned}\end{split}\]

As noted earlier, all three algebraic equations can be expressed as functions and substituted directly in the component and bus equations, respectively. We use redundant variables for modeling convenience.

Initialization

To initialize the model, given bus voltages \(V_r\), \(V_i\), and initial generator injection active and reactive power, \(P\) and \(Q\), we take following steps to initialize the system:

Complex power is defined as

\[S=VI^{*}\]

or

\[P + jQ = (V_r + j V_i)(I_r - j I_i).\]

From here, we compute injection currents from the initial power injection and bus voltages as

\[\begin{split}\begin{aligned} I_r &= \frac{PV_r + QV_i}{V_r^2 + V_i^2} \\ I_i &= \frac{PV_i - QV_r}{V_r^2 + V_i^2} \end{aligned} \end{split}\]

We substitute the expressions above into equations for current injections and obtain

\[\begin{split}\begin{aligned} E_p \sin\delta &= \dfrac{-B I_r + G I_i}{G^2 + B^2} + V_i \\ E_p \cos\delta &= \dfrac{G I_r + B I_i}{G^2 + B^2} + V_r \end{aligned}\end{split}\]

By dividing these two equations, we get an expression for the machine angle at the steady state:

\[\delta = \arctan \dfrac{E_i}{E_r} \, ,\]

And by squaring and adding them, we get an expression for the field winding voltage at the steady state

\[E_p = \sqrt{E_r^2 + E_i^2} \, ,\]

where

\[\begin{split}\begin{aligned} E_r &= \dfrac{G I_r + B I_i}{G^2 + B^2} + V_r \, ,\\ E_i &= \dfrac{-B I_r + G I_i}{G^2 + B^2} + V_i \, . \end{aligned}\end{split}\]

Next, we set the machine speed deviation to zero:

\[\omega = 0\]

Now, we can compute the electrical torque and set the mechanical torque to be equal to the electrical:

\[\begin{split}\begin{aligned} T_{e} &= G E_p^2 - E_p \left[ (G V_r - B V_i ) \cos\delta + (B V_r + G V_i )\sin\delta \right] \\ P_{m} &= T_{e} \end{aligned} \end{split}\]

With this, we initialize the machine at a steady state.

Model Outputs

Symbol	Units	Description	Note
\(I_r\)	[p.u.]	Terminal current, real component on network reference frame	Oriented leaving the machine, system base
\(I_i\)	[p.u.]	Terminal current, imaginary component on network reference frame	Oriented leaving the machine, system base
\(P\)	[p.u.]	Active power, \(V_rI_r+V_iI_i\)	Oriented leaving the machine, system base
\(Q\)	[p.u.]	Reactive power, \(V_iI_r-V_rI_i\)	Oriented leaving the machine, system base
\(\delta\)	[rad]	Machine internal rotor angle
\(\omega\)	[p.u.]	Machine speed deviation	\(\omega=0\) at synchronous speed