The figure shows the single line diagram of a single machine infinite bus system.

The inertia constant of the synchronous generator $$𝐻 = 5$$ $$MW-s/MVA.$$ Frequency is $$50Hz.$$ Mechanical power is $$1$$ pu. The system is operating at the stable equilibrium point with rotor angle $$\delta $$ equal to $${30^ \circ }$$. A three phase short circuit fault occurs at a certain location on one of the circuits of the double circuit transmission line. During fault, electrical power in pu is $$\,{P_{\max }}\,\,\sin \delta .\,\,\,$$. If the values of $$\delta $$ and $$d$$$$\delta $$$$/dt$$ at the instant of fault clearing are $${45^ \circ }$$ and $$3.762$$ radian/s respectively, then $$\,{P_{\max }}$$ (in pu) is _________.

For a fully transposed transmission line

A 183-bus power system has 150PQ buses and 32 PV buses. In the general case, to obtain the load flow solution using Newton-Raphson method in polar coordinates, the minimum number of simultaneous equations to be solved is ___________.

A function f(t) is shown in the figure. The Fourier transform F($$\mathrm\omega$$) of f(t) is