$($a$)$ Describe the mathematical formulation and Lagrange function for a take or pay fuel contract.

$($b$)$ The "Cut and Shred Paper Company" of northern Minnesota has two power plants. One burns coal and the other burns natural gas supplied by the Texas Gas Company from a pipeline. The paper company has ample supplies of coal from a mine in North Dakota, and it purchases gas as take$-$or$-$pay contracts for fixed periods of time. For the $8-$h time period shown later, the paper company must burn $15,050mcf$ of gas $($where $1\mathrm{.}0mcf=1000$ ft $3).$

The fuel costs to the paper company are:

Coal $0\mathrm{.}60$ $$/$MBtu

Gas $2\mathrm{.}0$$$mcf($where $1mcf=1000f{t}^3)$

The gas is rated at $1100Bt\frac{u}{f}{t}^3$

Input$-$output characteristics of generators:

Unit $1($coal unit$)$: $H_1(P_1)200+8\mathrm{.}5P_1+0\mathrm{.}002P_1^{2}MBt\frac{u}{h}$

$0\mathrm{.}8607\frac{\text{\$}}{?}$ Unit $2(gas$ unit$)$ : $H_2(P_2)300+6\mathrm{.}0P_2+0\mathrm{.}0025P_2^{2}MBt\frac{u}{h},50$

Load $(both$ load periods are $4h$ long$)$:

$\begin{array}{cc}\text{P}\text{e}\text{r}\text{i}\text{o}\text{d}& \text{L}\text{o}\text{a}\text{d}(MW)\\ 1& 400\\ 2& 650\end{array}$

Assume both units are $on-$line for the entire $8h.$

$i.$ Write down the economic dispatch equations for each period $j.$

$ii.$ Verify that a value for $(gamma),of0\mathrm{.}8607\frac{\text{\$}}{m}ef$ satisfies the total gas consumption constraint. Accordingly find the resulting unit schedules for each period$\mathrm{.}50$

Unit $2(gas$ unit$)$ : $H_2(P_2)300+6\mathrm{.}0P_2+0\mathrm{.}0025P_2^{2}MBt\frac{u}{h},50$

Load $(both$ load periods are $4h$ long$)$:

$\begin{array}{cc}\text{P}\text{e}\text{r}\text{i}\text{o}\text{d}& \text{L}\text{o}\text{a}\text{d}(MW)\\ 1& 400\\ 2& 650\end{array}$

Assume both units are $on-$line for the entire $8h.$

$i.$ Write down the economic dispatch equations for each period $j.$

$ii.$ Verify that a value for $(gamma),of0\mathrm{.}8607\frac{\text{\$}}{m}ef$ satisfies the total gas consumption constraint. Accordingly find the resulting unit schedules for each period.