Star Power Company is a power company in the Midwest region of the United States. Star buys and sells energy on the spot market. Star can store power in a high-capacity battery that can store up to 60 kWh (kilowatt hours). During a particular period, Star can buy or sell electricity at the market price known as LMP (Locational Marginal Price). The maximum rate that power can be injected or withdrawn from the battery is 20 kWh per period. Star has forecasted the following LMP’s for the next 10 periods:

Period ...... LMP ($/kWh)

1 ........... $ 5

2 ...........$27

3 ...........$ 2

4........... $25

5 ...........$22

6 ...........$29

7 ...........$24

8 ...........$20

9 ...........$61

10 ...........$66

The battery is full at the beginning of period 1; that is, at the start of the planning horizon, the battery contains 60 kWh of electricity.

a. Develop a linear programming model Star Power can use to determine when to buy and sell electricity in order to maximize profit over these 10 weeks. What is the maximum achievable profit?

b. Your solution to part (a) should result in a battery level of 0 at the end of period 10. Why does this make sense? Modify your model with the requirement that the battery should be full (60 kWh) at the end of period 10. How does this impact the optimal profit?

c. To further investigate the impact of requirements on the battery level at the end of period 10, solve your model from part (b) with the constraint on the ending battery level varying from 0 kWh to 60 kWh in increments of 10 kWh. Develop a graph with profit on the vertical axis and required ending battery level on the horizontal axis. Given that Star has not forecasted LMPs for periods 11, 12, and so on, what ending battery level do you recommend that Star use in its optimization model?