modelSynchGen

Information

Synchronous Generator Model

Detailed synchronous generator model connected to the grid, based on d-q decomposition.

Voltage-Current Relation

$$ \left[\begin{matrix}R_a+R_e & x_q'+x_e\\ -x_d'-x_e & R_a+R_e\end{matrix}\right]\left[\begin{matrix}I_d \\ I_q\end{matrix}\right]= \left[\begin{matrix}E_d'+V_s\sin\delta_e \\ E_q'-V_s\cos\delta_e\end{matrix}\right] $$

where:

• \(R_a\) and \(R_e\) are phase winding and equivalent network resistances
• \(x_d\), \(x_q\), \(x_d'\), \(x_q'\) are d-/q-axis normal and transient reactances
• \(x_e\) is equivalent network reactance
• \(I_d\), \(I_q\) are d-/q-axis currents
• \(E_d'\), \(E_q'\) are d-/q-axis transient voltages
• \(V_s\) is network RMS voltage
• \(\delta_e\) is phase shift angle

Phase Shift Angle Dynamics

$$ \frac{\mathrm{d}\delta_e}{\mathrm{d}t} = (\omega - \omega_s)\frac{n_p}{2} $$

where \(n_p\) is number of poles, \(\omega\) and \(\omega_s\) are generator and grid angular velocities.

Swing Equation

$$ \frac{\mathrm{d}\omega}{\mathrm{d}t}=\frac{\dot{W}_s-P_e}{J\omega} $$

Transient Operation

$$ \begin{array}{c} T_{qo}'\frac{\mathrm{d}E_d'}{\mathrm{d}t} =-E_d' + (x_q' - x_q)I_q \\ T_{do}'\frac{\mathrm{d}E_q'}{\mathrm{d}t} = -E_q' + (x_d - x_d')I_d + E_f \end{array} $$

where \(T_{do}'\) and \(T_{qo}'\) are d-/q-axis transient open-circuit time constants.

Excitation System

Field voltage dynamics:

$$ \frac{\mathrm{d}E_f}{\mathrm{d}t} = \frac{-E_f + K_E\left(V_{tr}-V_t-V_{stab}\right)}{T_E} $$

where \(K_E\) is excitation system gain, \(T_E\) is excitation time constant, \(V_{tr}\) is voltage reference set point, and \(V_t = \sqrt{\left(E_d'-R_aI_d-x_q'I_q\right)^2+\left(E_q'-R_aI_q+x_d'I_d\right)^2}\) is terminal voltage.

Stabilization

$$ \frac{\mathrm{d}V_{stab}}{\mathrm{d}t} = \frac{-V_{stab} + K_F\frac{\mathrm{d}E_f}{\mathrm{d}t}}{T_{FE}} $$

where \(K_F\) is stabilizer gain and \(T_{FE}\) is stabilizer time constant.

Output Power

Active and reactive power:

$$ \begin{array}{c} P_e = 3\left(E_d'I_d+E_q'I_q\right)\\ Q_e = \sqrt{9V_t^2I_t^2-P_e^2} \end{array} $$

where \(I_t=\sqrt{I_d^2+I_q^2}\) is terminate current.

Connectors

• RealInput: turbine shaft power
• RealOutput: angular velocity and frequency

Parameters

User specifies: nominal active/reactive powers, phase winding resistance, number of poles, network parameters (equivalent resistance/reactance, RMS voltage, grid angular velocity), d-/q-axis reactances and time constants, field voltage limits, excitation/stabilizer gains and time constants, moment of inertia, friction factor, and initialization options.

Note: For more advanced modeling, consider using generator models from OpenIPSL.

More details in [Sharefi2011].

Parameters

Connectors

Components