A power system has two $(2)$ generating plants and the power is being dispatched economically with $P_1=150MW$ and $P_2=275MW.$ The loss coefficients are:

$B_{11}=0\mathrm{.}10{10}^{-12}M{W}^{-1}$

$B_{12}=0\mathrm{.}01{10}^{-2}M{W}^{-1}$

$B_{22}=0\mathrm{.}13{10}^{-12}M{W}^{-1}$

To raise the total load on the system by IMW will cost an additional $$200$ per hour. Find:

a$)$ The penalty factor for plant $(1).$

$(12$ marks$)$

b$)$ The additional cost per hour to increase the output of plant $(1)$ by IMW.

$(8$ marks$)$