$2.$ A round$-$rotor generator is connected to a large power $($infinite bus$)$ system through a transformer and a double circuit transmission line as shown in the figure. The per$-$unit equivalet circuit including the generator's reactance $\backslash ($ X$\_\{$d$\}\backslash )$ and $\backslash ($ X$\_\{$d$\}^\{\backslash$prime$\}\backslash ),$ transformer's reactance $\backslash ($ X$\_\{$T r f$\}\backslash ),$ reactance of the transmission line $\backslash ($ X$\_\{$L$\}\backslash )$ and the infinite bus system reactance $\backslash ($ X$\_\{$S y s$\}\backslash )$ based on generator's ratings is also given below.

$($a$)$ If the generator is loaded with $0\mathrm{.}8$ pu of real power and $0\mathrm{.}46$ pu of reactive power $($lagging$)$ and the generator's terminal volatge is $1\mathrm{.}11$ pu $,$ find the operating point. Calculate the steady$-$state synchronizing power coefficient and stability margin.

$($b$)$ For the unregulated generator operating at the point calculated in $($a$),$ the per$-$unit inertia constant $\backslash ($ H$=\backslash )\backslash (5\backslash$mathrm$\{$MWs$\}/\backslash$mathrm$\{$MVA$\}\backslash )$ and the damping coefficient $\backslash ($ D$=3\backslash$mathrm$\{$pu$\}\backslash ).$ Using the classical model for the generator, determine the small$-$signal angle dynamics after a small disturbance $($shift on the initial condition$)$ is applied. $(\backslash (\backslash$mathrm$\{$f$\}=50\backslash$mathrm$\{$~Hz$\}\backslash )).$ Calculate transient synchronizing power coefficient, damping ratio and time$-$response in angle and speed.

$($c$)$ Repeat $($a$)$ and $($b$)$ assuming that an AVR $($Automatic Voltage Regulator$)$ is in action.