Question #$4$

Consider the $2$ bus $($slack$-$load$)$ power system shown in the figure below. An objective function to be minimized with respect to the control variables $P_2$ and $Q_2$ has the form: $f=(P_1+P_2-1{)}^2.$

The load flow solution $($start point in OPF $)$ with $P_2=-3\mathrm{.}5$ pu and $Q_2=-1pu$ is $V_2=\frac{0\mathrm{.}9084}{{?}_{-}}0\mathrm{.}4416$rad.

Considering the state variables ${}_2$ and $|V_2|,$ Calculate:

$4\mathrm{.}1)$ The optimal load flow Jacobian matrix $H_x,$ as well as the derivatives $H_u,f_u,$ and $f_x.$

$4\mathrm{.}2)$ The Lagrange multiplier $$ and total derivates $d\frac{f}{d}u$ at the starting point.

$4\mathrm{.}3)$ Suggested control variable change$-$step du toward optimal solution and find the new values of the control variables. $($Assume search parameter $=0\mathrm{.}7)$