QUESTION $1$

a$.$ Consider Fig Q$1$ below, the magnitude of voltage at bus $1$ is adjusted to $1\mathrm{.}5$ p$.$u$,$ the scheduled loads at buses $2$ and $3$ are as marked on the diagram, line impedance are marked in per unit on a $100$MVA base and the line charging susceptance are neglected.

i Derive the admittance matrix for the network.

$(5$mks$)$

ii Using the Gauss$-$seidel method, in one iteration, determine the phasor values assuming the initial phasor values at each bus are as follow

$V_1^{(0)}=1\mathrm{.}05pu$

$V_2^{(0)}=0\mathrm{.}9800-j0\mathrm{.}0599$

$V_3^{(0)}=1\mathrm{.}0001-j0\mathrm{.}05$

Find the next node phasor voltages at bus $2$ and $3.(4$ mks $)$

iii Use the result obtained in $($ii$)$ above to find the slack bus power$-S_{1.}(3$ mks $)$