$4($a$)$ Consider the two$-$area power system shown in Figure $1.$ The marginal costs of systems $\backslash ($ A $\backslash )$ and $\backslash ($ B $\backslash )$ are given by the following expressions.

$($i$)$ Calculate the marginal costs of the two areas when they are operated in an isolated market and as a single market.

$($ii$)$ Calculate the marginal costs of the two areas when the capacity of the interconnection is $50$ MW $.$

$($iii$)$ Use a diagram to demonstrate the supply curves of the two areas, their generation output, demand, and the energy flow along the interconnection in $($ii$).$

$($b$)$ A three$-$node system is depicted in Figure $2.$ Table $4$ provides the capacity and marginal cost for all $3$ generators. Table $5$ provides the reactance and capacity of the three branches. Calculate the nodal marginal price for nodes $1,$ and $3.$

Table $4$

Table $5$