GATE 2023 EE Q.41: solution and answer

Official answer: DSource: Official answer key, GATE 2023 EE (IIT Kanpur)

1. In load-flow analysis each bus has four variables, $P$, $Q$, $|V|$ and $\theta$, and two of them are specified. The pair that is specified decides the bus type: $|V|$ and $\theta$ for the slack bus, $P$ and $|V|$ for a $P$-$|V|$ bus, and $P$ and $Q$ for a $P$-$Q$ bus.

2. Bus 3 is an infinite bus with $|V_3|=1.0\text{p.u.}$ and ${\theta }_3=-15^{\circ }$ both fixed. It is therefore the slack (reference) bus.

3. Bus 2 has the alternator output fixed at $P_2=200\text{MW}$ and $Q_2=40\text{MVAr}$. Both $P$ and $Q$ are specified, so bus 2 is a $P$-$Q$ bus.

4. Bus 1 has voltage $V_1=1.05\angle {\theta }_1\text{p.u.}$ and source current $I=|I|\angle ({\theta }_1\pm 90^{\circ })$. The complex power injected at bus 1 is $S_1=V_1{I}^{\ast }$. With $I^{\ast }=|I|\angle (-{\theta }_1\mp 90^{\circ })$, this gives $$S_1=1.05|I|\angle ({\theta }_1-{\theta }_1\mp 90^{\circ })=\mp j1.05|I|.$$

5. The real part of $S_1$ is zero for every value of $|I|$, so $P_1=0$ is known. $Q_1=\mp 1.05|I|$ changes with the source current, so it is not specified. $|V_1|=1.05\text{p.u.}$ is held fixed. With $P$ and $|V|$ specified, bus 1 is a $P$-$|V|$ (voltage-controlled) bus.

6. Result: Bus 1 is a $P$-$|V|$ bus, Bus 2 is a $P$-$Q$ bus and Bus 3 is the slack bus. This is option (D).