The Katahdin Paper Company in Millinocket, Maine, operates a hydroelectric generating station on the Penobscot River. Water is piped from a dam to the power station. The rate at which the water flows through the pipe varies, depending on external conditions. The power station has three different hydroelectric turbines, each with a known (and unique) power function that gives the amount of electric power generated as a function of the water flow arriving at the turbine. The incoming water can be apportioned in different volumes to each turbine, so the goal is to determine how to distribute water among the turbines to give the maximum total energy production for any rate of flow.

The following quadratic models were determined for the power output of each turbine, along with the allowable flows of operation:

$KW_1 = (-18.89 + 0.1277 Q_1 - 4.08 \cdot 10^{-5}Q_1^2 )(170 - 1.6 \cdot 10^{-6} Q_T^2)$ $KW_1 = (-24.51 + 0.1358 Q_2 - 4.69 \cdot 10^{-5}Q_2^2 )(170 - 1.6 \cdot 10^{-6} Q_T^2)$ $KW_1 = (-27.02 + 0.1380 Q_3 - 3.84 \cdot 10^{-5}Q_3^2 )(170 - 1.6 \cdot 10^{-6} Q_T^2)$ $250 \leq Q_1 \leq 1110, \quad 250 \leq Q_2 \leq 1110, \quad 250 \leq Q_3 \leq 1225$

where \( Q_k \) is the flow through the \( k \)th turbine (in cubic feet per second), \( KW_k \) is the power generated by the \( k \)th turbine (in kilowatts), and \( Q_T = Q_1 + Q_2 + Q_3 \) is the total flow through the station.

If all turbines are being used, we we wish to determine the flow \( Q_k \) to each turbine that will give the maximum total energy production.

1. Write the statement of this problem as a program \( (P) \) with the notation from equation (22). Label the objective function, as well as the inequality constraints accordingly. Write a necessary condition for optimality and verify that is satisfied by the obvious solution \( Q_1=1110~\mathrm{ft}^3/\mathrm{s}\), \( Q_2=1110~\mathrm{ft}^3/\mathrm{s}\), \( Q_3=1225~\mathrm{ft}^3/\mathrm{s} \).

2. Use Frank-Wolfe method with initial guess \( Q_1 = Q_2 = Q_3 = 250~\mathrm{ft}^3/\mathrm{s} \). Perform at least one step.

3. We would like to impose the extra condition \( Q_1 + Q_2 +Q_3 = 2400~\mathrm{ft}^3/\mathrm{s} \). Perform any necessary modifications in \( (P) \) to transform it into a program \( (P’) \) with only equality constraints. (Hint: use slack and surplus variables). Use a projection method to solve this program, with initial guess \( Q_1 = Q_2 = Q_3 = 800~\mathrm{ft}^3/\mathrm{s} \). Perform at least one step.