F$=$cP$^(2)+$bP$+$a Problem $1(20$ points$)$

The distribution network consists of two microgrids $($MGs$).$ Each of them has one gas turbine to supply their local load and the generation characteristics are presented in Table I. Also, they could exchange power via one power line with $90$ kW capacity. The load demands of the two MGs in this two$-$hour period are given in Table II$.$ Use the Lagrangian Relaxation method to determine the dispatch of the two gas turbines, power exchanges, and the incremental cost of the system for the two$-$hour period. $($Hint: The initial power exchange from MG$1$ to MG$2$ is set as $20$ kW $,$ and the iteration step size is set as $10.$ At most five iterations would be enough.$)$

Table I. Gas turbine operation characteristics $($cost function is denoted by $\backslash ($ F$=\backslash$mathrm$\{$c$\}$ P$^\{2\}+\backslash$mathrm$\{$b$\}$ P$+\backslash$mathrm$\{$a$\}\backslash ))$

Table II$.$ Forecasted load demand of the two MGs for two$-$hour period $($kW$)$