Question $1$

A typical one$-$machine system connected to an infinite bus is shown in Figure Q$1.$ The system consists of one synchronous machine, one transformer, two parallel transmission lines and an infinite bus. The various data are:

Inertia constant of the synchronous machine, $H=5M\frac{J}{M}VA$

Transient reactance of the generator, $x_d^{\text{'}}=0\mathrm{.}3p.u.$

Transformer leakage reactance, $x_t=0\mathrm{.}2$ p$.$u$.$

Line reactance, $x_{L1}=x_{L2}=0\mathrm{.}3$ p$.$u$.$

Supply frequency, $f=50Hz$

Real power delivered to the infinite bus, $P_e=0\mathrm{.}8$ p$.$u$.$

The infinite bus voltage, $V=1\mathrm{.}0$ p$.$u$.$

Apply Equal Area Criterion $($EAC$)$ for the following cases:

a$)$ A temporary three$-$phase symmetrical fault occurs at point $P,$ in Figure Q$1$ for a short time and then the fault is cleared. Both lines are intact. Determine the system stability, the critical clearing time and critical clearing angle.

$2$EEE$40007$ S$2,2024$

b$)$ A three$-$phase symmetrical fault occurs at $Q,$ the middle point of one of the lines shown in Figure Q$1.$ The fault is cleared by isolating the faulted line by opening the circuit breakers at both ends. Determine the system stability, the critical clearing time and critical clearing angle.

C$)$ A three$-$phase symmetrical fault occurs at point $R$ shown in Figure Q$1.$ The fault is cleared by isolating the faulted line by opening the circuit breakers at both ends. Determine the system stability, the critical clearing time and critical clearing angle.