$($c$)$ A radial distribution system with source at one side of the feeders, each with maximum load based on MVA rating is shown in Figure Q$\mathrm{.}1($c$).$ Impedance of the transformer is referred to its own rating and impodance of each transmission line is $1\mathrm{.}2$ pu with base of $100$MVA,$13\mathrm{.}2kV.$ Assume that all impedances are pure reactance. The RSI of each relay is also indicated in the figure.

$($i$)$ Prove that for a $3-$phase fault at Bus $1,$ Bus $2$ and Bus $3,$ the fault currents are $1367$ A$,2187$ A and $1673$ A$,$ respectively.

$(4$ marks$)$

$($ii$)$ Based on the fault currents given in Q$\mathrm{.}1($c$)($i$),$ propose the Time Setting Multiplier $($TSM$)$ of all relays to achieve a well$-$coordinated system. The TSMs of relay R$1$ and R$4$ are all set to $0\mathrm{.}1.$ The discrimination time $($DT$)$ is $0\mathrm{.}4$ seconds. The relay characteristic operating time $($RCOT$)$ is given by:

RCOT$=\frac{0\mathrm{.}14}{PS{M}^{0\mathrm{.}02}-1}$

and the TSM ranges from $0\mathrm{.}05$ to $1.$

$(12$ marks$)$

Figure Q$\mathrm{.}1($c$)$