Question

A 50 Hz synchronous generator is connected to an infinite bus through a line. The per unit reactances of the generator and the line are j0.3 p.u. and j0.2 p.u. respectively. The generator no load voltage is 1.1 p.u. and that the infinite bus is 1.0 p.u. The inertia constant of the generator is 3 MW-sec/MVA. If the generator is loaded to its maximum power transfer capacity and a small perturbation is given, then the resulting natural frequency of oscillations is _____Hz.

• A. 1.53

Answer 1

The correct option is A 1.53
Natural frequency of oscillations,

$f_n=\sqrt{\frac{{\begin{array}{c}\frac{\left(\partial P_e\right)}{\partial \delta }\end{array}|}_{\delta 0}}{M}}$

$GH=\frac{1}{2}M\omega$

$M=\frac{GH}{\pi f}=\frac{1\times 3}{\pi \times 50}=0.019$

$P_e=\frac{|E||V|}{X}\sin {\delta }_0$

$0.6P_{max}=P_{max}sin{\delta }_0$

${\delta }_0=si{n}^{-1}\left(0.6\right)={36.87}^{\circ }$

${\begin{array}{c}\frac{\partial P_e}{\partial \delta }\end{array}|}_{{\delta }_0}=P_{e}cos{\delta }_0=\frac{1.1}{0.5}cos\left({36.87}^{\circ }\right)=1.76p.u.$

$f_n=\sqrt{\frac{1.76}{0.019}}=9.62rad/sec$

$f_n=\frac{9.62}{2\pi }=1.53Hz$