Two three$-$phase, rated $30$ MW each, $\backslash (66-\backslash$mathrm$\{$kV$\}\backslash ),$ Y $-$connected synchronous generators A and B operating in parallel supply a load of $30$ MW at $66$ kV and $0\mathrm{.}8$ pf lagging. The synchronous reactance of the two generators $\backslash ($ A $\backslash )$ and $\backslash ($ B $\backslash )$ are $\backslash (\backslash$mathrm$\{$j$\}10\backslash$Omega $\backslash )$ and $\backslash (\backslash$mathrm$\{$j$\}12\backslash$Omega $\backslash ),$ respectively. At no$-$load condition, the frequencies of the two generators $\backslash ($ A $\backslash )$ and $\backslash ($ B $\backslash )$ are $52$ Hz and $51$ Hz $,$ respectively. At full load condition, the frequencies of both generators drop to $49$ Hz $.$

$($a$)$ Select $37\mathrm{.}5$ MVA and $66$ kV as base values and take the load bus as the reference bus $($i$.$e$.,$ set its voltage phase angle as $\backslash (0^\{\backslash$circ$\}\backslash )),$ draw the impedance diagram of this system $($in p$.$u$.$ values$).$

$(5$ Marks$)$

$($b$)$ What is the system frequency and how much real power is supplied by each generator? Show the answers with real values in Hz and MW$.$

$(6$ Marks$)$

$($c$)$ If the total load $($including both real and reactive power$)$ is equally shared by the two generators, calculate the respective power angle $\backslash (\backslash$delta $\backslash )$ of the two generators.

$(6$ Marks$)$

$($d$)$ Generator A$\text{'}$s power angle is now changed to $\backslash (3^\{\backslash$circ$\}\backslash ),$ while its internal emf is kept constant and the total load is also constant. What will be the real power output of the two generators A and B $?$ Show the answers with real values in MW$.$

$(8$ Marks$)$