Problem $1.(60$ points$)$

The above figure shows the one line diagram of a three$-$phase power system whose bus impedance matrix is

Positive sequence and negative sequence: $\left[\begin{array}{ccc}2\mathrm{.}2& 1\mathrm{.}0& 1\mathrm{.}2\\ 1\mathrm{.}0& 6\mathrm{.}0& 3\\ 1\mathrm{.}2& 3& 5\mathrm{.}6\end{array}\right]$ohm;

Zero sequence: $\left[\begin{array}{ccc}1\mathrm{.}0& 0\mathrm{.}5& 3\\ 0\mathrm{.}5& 8\mathrm{.}0& 3\\ 3& 3& 6\mathrm{.}0\end{array}\right]$ ohm. The current source at node $2$ is $\frac{10}{{?}_{30}}A,\frac{10}{{?}_{-}}90$ A $,$ and $\frac{10}{{?}_{150}}$A for phase $A,B$ and C $,$ respectively. There are no other sources.

$(1)$ Find the phase voltage of each node during normal operations. $(10$ points$)$

$(2)$ Suppose there is a phase A to ground fault with fault resistance of $5$ ohm occurring at node $1,$

$(2\mathrm{.}1)$ draw the connected sequence networks for analyzing this fault

$(10$ points$)$

$(2\mathrm{.}2)$ find the sequence fault currents at the fault point $(10$ points$)$

$(2\mathrm{.}3)$ find the phase fault currents at the fault point $(10$ points$)$

$(2\mathrm{.}4)$ find the sequence voltages of each node during the fault $(10$ points$)$

$(2\mathrm{.}5)$ find the phase voltages of each node during the fault $(10$ points$)$