Looking for full steps to solve this with answer The input-output curve of a gas

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Looking for full steps to solve this with answer

The input-output curve of a gas-fired generating unit is approximated by the following function: H(P) 1209.3 P 0.0025 P MJ/h This unit has a minimum stable generation of 200 MW and a maximum output of 500 MW. The cost of gas is 1.20SMJ. Over a 6-h period, the output of this unit is sold in a market for electrical energy at the prices shown in the table below. Period 3 6 Price (SMWh)12.510133.51511 Assuming that this unit is optimally dispatched, is initially on-line and cannot be shut down, calculate its operational profit or loss for this period.

Explanation / Answer

The above function is maximised when it is differentiated and output equates to zero.

I.e. dH/dP = 9.3 + 0.005P = 0

So P = 9.3/0.005 = 1860 MJ

Now cost of gas = $1.2 per MJ

Hence total cost of gas = 1.2*1860 = $2232

Now dividing this energy over a period of 6 hours.

Hence per hour energy distribution = 1860/6 = 310 MJ

Now total price of energy = (310*12.5) + (310*10) + (310*13) + (310*13.5) + (310*15) + (310*11)

= 3875+3100+4030+4185+4650+3410

= $23250

Hence total profit = Total price - Total cost

= 23250 - 2232

= $ 21,018.00

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