Answer the following questions:

1. Cheapo Electrons is an electricity retailer. The table below shows the load that it forecast its consumers would use over a 6-h period. Cheapo Electrons purchased in the forward market and the power exchange exactly enough energy to cover this forecast. The table shows the average price that it paid for this energy for each hour. As one might expect, the actual consumption of its customers did not exactly match the load forecast and it had to purchase or sell the difference on the spot market at the prices indicated. Assuming that Cheapo Electrons sells energy to its customers at a flat rate of 24.00 $/MWh, calculate the profit or loss
that it made during this 6-h period. What would be the rate that it should have charged its customers to break even?

Period                            1       2      3    4    5    6
Load Forecast (MWh) 120 230 310 240 135 110
Average cost ($/MWh) 22.5 24.5 29.3 25.2 23.1 21.9
Actual load (MWh) 110 225 330 250 125 105
Spot price ($/MWh) 21.6 25.1 32 25.9 22.5 21.5

2. The input–output curve of a gas-fired generating unit is approximated by the following function: H(P) = 120 + 9.3 P + 0.0025 P² MJ/h. This unit has a minimum stable generation of 200 MW and a maximum output of 500 MW. The cost of gas is 1.20 $/MJ. Over a 6-h period, the output of this unit is sold in a market for electrical energy at the prices shown in the table below.
Period               1      2   3    4    5 6
Price ($/MWh) 12.5 10 13 13.5 15 11
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.

3. Repeat the calculation of Problem 2 assuming that the cost curve is replaced by a three-segment piecewise linear approximation whose values correspond with those given by the quadratic function for 200 MW, 300 MW, 400 MW and 500 MW.

4. Assume that the unit of Problem 2 has a start-up cost of $500 and that it is initially shut down. Given the same prices as in Problem 2, when should this unit be brought on-line and when should it be shut down to maximize its operational profit? Assume that dynamic constraints do not affect the optimal dispatch of this generating unit.

5. Borduria Generation owns three generating units that have the following cost functions:
Unit A: 15 + 1.4 PA + 0.04 P_A $/h
Unit B: 25 + 1.6 PB + 0.05 P²_B $/h
Unit C: 20 + 1.8 PC + 0.02 P²_C $/h
How should these units be dispatched if Borduria Generation must supply a loadof 350 MW at minimum cost?

6. How would the dispatch of Problem 5 change if Borduria Generation had the opportunity to buy some of this energy on the spot market at a price of 8.20 $/MWh?

* Demand response (DR) (Milligan/Kirby) questions:
MK1: Write a short paragraph that argues that demand response should be treated equivalent to generation, provided that it behaves in the same way as a generator.
MK2: Based on the WWSIS, why might one make the argument that DR might be an economic approach to mitigate the relatively few times that contingency reserves (or possibly load) may not be met?
MK3: An aluminum smelter plant has decided to bid as a demand response resource. It has decided to respond based on price signals - when electricity prices are high the smelter would reduce load. Of the 5 types of demand response we discussed, which type appears to be the most appropriate for the smelter? What impact, if activated, would this have on the load (net of the smelter) that the generation would need to provide?
For the MKx questions, please write a short, succinct answer. Hint: focus on the core of the question and develop an answer that directly addresses the issue with limited, if any, extraneous discussion. Focus on causality and directness.

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MK1: Any demand response is like a materializing factor post supply. So it behaves as a generation in the mechanism behind the equilibrium dynamics. But at the same time equilibrium cannot be introduced without the generator as well and that part of the game is provided by demand because it is through this interaction of demand and supply that we reach a situation of equilibrium.
MK2: In a situation of crisis any form of revenue acts as a source of respite for the firms. So it is no surprise that DR might be an economic respite from shortage in contingency reserves due to excess load.
MK3: Clearly the firm likes to reduce load with an increase in prices so it perceives a decline it demand due to an increase in prices. Clearly this is a situation where the demand curve is downwards sloping and the supply curve is elastic. If activated this will keep on reducing prices to the extent that the market eventually fails....
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