Example $7\mathrm{.}7$

The fuel cost in $\frac{\text{\$}}{h}$ of three thermal plants of a power system are

$C_1=200+7\mathrm{.}0P_1+0\mathrm{.}008P_1^2$

$C_2=180+6\mathrm{.}3P_2+0\mathrm{.}009P_2^2$

$C_3=140+6\mathrm{.}8P_3+0\mathrm{.}007P_3^2$

where the powers are in MW$.$ The plant outputs are subject to the following limits

$10MW{P}_{1}85MW$

$10MW{P}_{2}80MW$

$10MW{P}_{3}70MW$

For this example, assume the power loss is given by the simplified expression

$P_{L(pu)}=0\mathrm{.}0218P_{1(pu)}^2+0\mathrm{.}0228P_{2(pu)}^2+0\mathrm{.}0179P_{3(pu)}^2$

where the loss coefficients are specified per unit on a $100-$MVA base. Determine the optimal dispatch of generation when the total system load is $150$ MW $.$

Problem One

Repeat Example $7\mathrm{.}7$ in the textbook $($page $48$ in the class notes$)$ for $P_D=100MW.$ Show detailed steps. Assume ${}^{(1)}=8.$