Text problem $6\mathrm{.}43.$ For a three bus power system assume bus $1$ is the swing with a per unit voltage of $,$ bus $2$ is a PQ bus with a per unit load of $2\mathrm{.}0+$j$0\mathrm{.}5,$ and bus $3$ is a PV bus with $1\mathrm{.}0$ per unit generation and $1\mathrm{.}0$ voltage setpoint.$$ The per unit line impedances are$$j$0\mathrm{.}1$ between buses $1$ and $2,$j$0\mathrm{.}4$ between buses $1$ and $3,$ and j$0\mathrm{.}2$between buses $2$ and $3.$ Using a flat start, use the Newton$-$Raphson approach to determine the first iteration phasor and voltages at buses $2$ and $3.$

To aid in checking your solution, Y$-$bus is:

$-$j$12\mathrm{.}5$ j$10$ j$2\mathrm{.}5$ j$10-$j$15$ j$5$ j$2\mathrm{.}5$ j$5-$j$7\mathrm{.}5$

Hints if you are having trouble getting the correct answer:$$ You only need to use the equations associated with P$2,$ Q$2,$ and P$3,$ to compute the unknown voltage at bus $3$ and unknown angles at buses $2$ and $3.$The initial mismatch is related to these active and reactive power injections. $$When computing the Jacobian, note the squared V$2$ term in the reactive power equation for Q$2.$