Consider a $3-$phase, $11$kV$,35$MVA star$-$connected generator protected by differential $(87$G$)$ and earth$-$fault $(50$N$)$ relays. The CTs used are rated at $2000/5.$

For differential relay $(87$G$),$ the operating current of differential relay $($I$\_$set $)$ is $150$ mA $.$ The protection engineer noticed that the CTs exhibit a voltage difference where the CT$\_1$ has a voltage of $80\%$ of CT$\_2.$ The relay, with a resistive impedance $($R$\_$r $)$ of $90,$ is connected to the midpoint of the pilot wires. The distance between the two CTs is $200$ meters, and the pilot wire has a resistance of $0\mathrm{.}005/$ meter.

The direct through$-$fault $($ I$\_$f $)$ is calculated to be $25379$ A

a$)$ Determine the stabilizing resistance $($R$\_$stab $)$ needed to be connected in series with the relay to achieve a stability factor of $3.$ For the earth$-$fault relay $(50$N$)$;

b$)$ Determine the value of the earth resistor needed to allow only $15\%$ of the winding to be left unprotected. c$)$Considering the calculated resistor in $($b$)$ what would be the maximum earth$-$fault current?