This assignment is due at 4 pm on Wednesday 19 October. You should also submit the spreadsheet that you use for your calculations.

The marks will be given entirely for the written document – the spreadsheet will be looked at in order to check that you carried out the calculations correctly (particularly if the numbers seem strange). So you need to ensure that the written document can be read on its own: make sure to include any figures or tables that you want to refer to. If you are using formulae to carry out calculations in the spreadsheet then you need to also give these in the written document. You should take care in your use of English in the submission: you will lose marks if the document is not written clearly. There is a page limit on the submission of 12 pages including a cover page and all tables and figures (but you will probably need no more than 6 pages)

The spreadsheet needs to be developed from scratch starting from the data that I provide. You must not

use any part of another group’s spreadsheet.

Samson Hydro operates a pumped hydro storage facility. When power costs are low water can be pumped up into a raised reservoir and when power costs are high the same water can be released to generate power. The total usable capacity of the reservoir is 9,000,000 litres (which we write as 9,000 kl or kilolitres)

There are only three states that the plant at Samson Hydro can be in: (a) pumping water up into the reservoir using power, (b) idle, (c) allowing water to flow out of the reservoir to generate power. When pumping water Samson Hydro requires 290 MW and during this phase a total of 14 kl per minute can be pumped. When water is flowing out of the dam, the rate of flow is 16 kl per minute and this will generate 260 MW of power.

There is a loss of efficiency in storing power in this way (it costs more power to pump than Samson Hydro generates through letting water out, even though the flows are greater when Samson Hydro is generating). This will mean that it makes sense for there to be significant periods when the plant is idle.

The spreadsheet gives 6 weeks of price data (42 days, or 1008 hours). These are electricity prices in € per MWh and are taken from the prices in Denmark for a period starting on 9 September 2021. (Note that prices can occasionally be slightly negative)

Part 1 (16 Marks)

Suppose that a single daily schedule of operation will be determined that will be used throughout the six-week period. It is desired to maximize average daily profit with the reservoir at a level of 4,000 kl at the beginning and end of every day (12 midnight) and allowing for the maximum capacity of the reservoir. Formulate this as a linear program.

Use the notation ������,��� for the price in hour ��� of day ���. You may wish to use, as variables for each of 24 hours, the proportion of the hour pumping, and the proportion of the hour generating, 48 numbers in total - since the proportion idle can then be deduced. You may also want to define a variable for the reservoir level at the end of hour ���. You need to carefully write down expressions (using Sigma notation) for the objective and each of the constraints, as well as specifying which variables are non-negative.

Then solve the optimization problem. Find the maximum average daily profit and the daily schedule of operation that achieves this.

Part 2 (28 marks)

Now consider a scenario in which a decision needs to be made without knowledge of future prices. Suppose that Samson Hydro needs to decide on what to do for the next hour (proportions of time pumping; or generating; or idle) on the basis of the current price and reservoir level. We consider a simple linear threshold policy of the following form:

(A) At the end of hour ��� calculate a number ������ = αyt + β������ where yt is the reservoir level at the end of hour ��� and ������ is the price in hour ���. Here ��� and ��� are parameters to be chosen.

(B) The policy is to pump in the following hour (hour ��� + 1) if ���t ≤ ���, generate in hour ��� + 1 if Wt ≥ ���, and do nothing if ������ is between these two values. Here ��� and ��� are parameters to be chosen.

We allow ���, ���, ��� and ��� to depend on the time of day. The version of this policy that you should use has two different values for these parameters: one for the morning (first 12 hours before midday) and one for the afternoon/evening (second 12 hours, after midday). For decisions on whether to pump for the hour from 12 noon to 1 pm you need to use the second set of parameters (applying these to the prices in the period from 11 to noon and the reservoir level at noon). Please be careful about the timing of this switch over. Because of the linearity we can normalise so that ��� = 1 in both cases. For the first hour of the day please use a fixed policy based on your answer to Part 1, rather than looking at the last hour of the previous day.

In fact, the policy cannot be applied in exactly this manner – so your first task is to determine a version of this policy that will ensure that the reservoir level stays positive and below 9,000 kl and returns to a 4,000 kl at the end of the day (12 midnight). Except where forced to do otherwise by these constraints the policy will, as before, pump as much as possible if Wt ≤ ���, generate as much as possible if ������ ≥ ���, and be idle if ������ is between these two values. Note that, for example, the constraint of needing to be at 5,000 kl at midnight may force Samson Hydro to pump, even if ������ > ���. Please describe your modified policy carefully.

(Hint: Consider adding a restriction that the reservoir level (in 1000s of litres) at the end of hour ��� lies in the region between the values min(9000, 4000 + (24 − ���)60 × 16) and max(0, 4000 −

(24 − ���)60 × 14) . )

Now find values for the parameters ���, ��� and ��� with both am and pm values for each (so 6 parameters in total) that optimize the average profit made over the 42 days of data. Note that this is a nonlinear optimization problem. Because there is more than one local minimum and the linear decision rule