Question

A synchronous generator of reactance 1.20 p.u. is connected to an infinite bus bar through transformers and a line of total reactance of 0.6 p.u. The generator no load voltage is 1.20 p.u. and its inertia constant is H = 4 MWs/MVA. The resistance and machine damping may be assumed negligible. The system frequency is 50 Hz. If the generator is loaded to 50% of the maximum power limit, then the frequency of natural oscillations is

• A. 4.76 Hz
• B. 0.764 Hz
• C. 3.52 Hz
• D. 0.467 Hz

Answer 1

The correct option is B 0.764 Hz

${\delta }_0={\sin }^{-1}\left(\frac{P_e}{P_m}\right)={\sin }^{-1}\left(0.5\right)=30{}^{\circ }$

$P_e=\frac{EV}{X}\sin {\delta }_0$

$\frac{d{P}_e}{d\delta }=\frac{EV}{X}\cos {\delta }_0=\frac{1.2\times 1}{1.8}\cos 30{}^{\circ }$

$=0.577MW(p.u.)/elec.rad$

$M=\frac{GH}{\pi f}=\frac{1\times 4}{\pi \times 50}=0.025s^2/elec.rad$

$\omega ={\left(\frac{\left(\frac{d{P}_e}{d{\delta }_0}\right)}{M}\right)}^{1/2}={\left(\frac{0.577}{0.025}\right)}^{1/2}=4.8rad/\sec$

Frequency of natural oscillation,

$f=\frac{\omega }{2\pi }=0.764Hz$