A synchronous generator is to be connected to an infinite bus through a transmission line of reactance X = 0.2 pu, as shown in figure the generator data is as follows:

X¹ = 0.1 pu, E¹ = 1.0 pu, H = 5 MJ/MVA, mechanical power P_m = 0.0 pu, $$\omega $$_B = 2 $$\pi \times $$50 rad/sec. All quantities are expressed on a common base.

The generator is initially running on open circuit with the frequency of the open circuit voltage slightly higher than that of the infinite bus. If at the instant of switch closure $$\delta = 0$$ and $$\omega = {{d\delta } \over {dt}} = {\omega _{init}},$$ compute the maximum value of $${\omega _{init}}$$ so that the generator pulls into synchronism.

$$\int {\left( {{{2H} \over {{\omega _B}}}} \right)\omega d\omega + {P_e}d\delta = 0} $$

A transmission line has a total series reactance of 0.2 pu. Reactive power compensation is applied at the midpoint of the line and it is controlled such that the midpoint voltage of the transmission line is always maintained at 0.98 pu. If voltage at both ends of the line are maintained at 1.0 pu, then the steady state power transfer limit of the transmission line is

A long lossless transmission line has a unity power factor (UPF) load at the receiving end and an ac voltage source at the sending end. The parameters of the transmission line are as follows:
Characteristic impedance $${Z_c} = 400\Omega ,\,\,$$, propagation constant $$\,\beta = 1.2 \times {10^{ - 3}}\,\,rad/km,\,\,$$ and length $$\,l = 100\,km.\,\,$$ The equation relating sending and receiving end questions is $${V_s} = {V_r}\,\cosh \,\,\left( {\beta l} \right) + j\,Z{}_c\,\,\sinh \left( {\beta l} \right){{\rm I}_R}$$ Complete the maximum power that can be transferred to the UPF load at the receiving end if $$\left| {{V_s}} \right| = 230\,\,kV.\,\,$$

Consider a long, two-wire line composed of solid round conductors. The radius of both conductors is $$0.25$$ cm and the distance between their centers is $$1$$m. If this distance is doubled, then the inductance per unit length.