Generally, domestic fuel bills are proceed in a linear manner, which in maths we usually write as y=mx+c. For a fuel bill the fixed part (c) is called a standing charge, often given as a charge per day, while the element m is the charged cost per unit of energy, probably expressed in kilowatt-hours, kWh.

My supplier is currently charging 42.24p per day and 28.02p/kWh for electricity and 27.22 per day and 7.34 p/kWh for gas.

1. A two-person family, the Adams, in a 75m² house has a typical annual consumption of 7.5 MWh gas, mostly heating, and 1.5MWh of electricity. Estimate their annual energy bill.

2. The Barracloughs live in a 115m² house and used 11.5Mwh of gas and 3.9MWh of electricity. Calculate their bill on the same price structure.

3. On the assumption that all the gas was used for space heating, express the two gas bills as a price per square metre.

4. The Barracloughs install solar panels and reduce their electrical consumption to 2.6MWh. We assume that actually they have the same consumption but the difference has been used directly off the roof supply. What was the apparent saving, in money?

5. in the first full year the panels generated 2500 kWh and the excess not used at the time was passed to the grid, exported. For this the supplier pays 3.5p per kWh, though that rate used to be as much as ten times higher. How much money is paid back to the Barracloughs?

6. If the solar panels cost £5000 and there are no changes to the prices, how many years would it take to return the capital cost, ignoring things like inflation or interest that the same money could have produced?

7. If electricity goes up in price relative to everything else and if the Barracloughs continue with the same behaviour, does this make the panels seem like a better idea? Suppose the cost per unit of electricity doubles (both in and out) two years after installation; how does this change the time it takes to recover the capital cost?

8. The Chapman family live in a 260m² house and consume 12.5MWh of gas and 3.7MWh of electricity, even with solar panels. Relative to the other families, comment on the Chapmans' use of gas for heating.

9. (Harder) At a national level, a third of electricity in the UK is generated from gas. (True in 2022). If gas were to double in price (from G to 2G), make an estimate of how the price of electricity, E, might change.

10. (Still hard) On the rates I have given you, electricity is between 3 and 4 times more expensive than gas per MWh. Wholesale gas prices jumped by a factor of 4, so from your answer to Q9 you can judge how electricity will change in price. From this, you can estimate how general household energy bills will change, once all the increases have been passed on to customers.

If you express the prices in terms of each other, E=𝞪G, then 3 <𝞪< 4, which is the first sentence of Q10 turned into maths notation. Then express the old bill B= Eₘ+Gₘ in terms of Gₘ (Gas money). Next, look at the three families, who use volumes of gas and electricity so that the energy units of Eᵥ and Gᵥ (volumes of E and G in MWh) can be expressed as Gᵥ=kEᵥ (since E is smaller, I say 3<k<5 and Eₘ = E.Eᵥ ). Now use these two relationships to show how B = Eₘ+Gₘ = E.Eᵥ + G.Gᵥ moves to a new value, which we might call B'. It's going to be at least 𝞪 times more expensive, while k>𝞪.

1. 1000kWh = 1MWh. 365 days of standing charge is added to the energy costs;

(365*0.4224 + 1500*0.2802 ) +(365*0.2722 +7500*0.0734) = 154.18 + 420.30 +99.35 + 550.50

so Gas cost is £ 649.85 , Electricity is £574.48 and the grand total is £1224.33

2. The standing charges are the same as Q1.

(154.18 + 3900*0.2802 ) +(99.35 +11500*0.0734) = (154.18 + 1092.78 ) +(99.35 +844.10)

= 1246.96 + 943.45 = £2190.41

3. A: 649.85/75 = £8.66(5) per sq. metre. B: 943.45/115 = £8.20(4) per sq. metre.

4. 3.9-2.5=1.3 1300kWh @ 0.2803 = £364.26.

5. 1200kWh excess, (2500-1300) suggests 1200*0.035 = £42. Not very much at all. The family would be very much better off if they changed their habits to use as much as possible when their panels are generating power. 28.02p/3.5p is a factor of eight; it is eight times as expensive to pay for importing energy as to use their own. This might even justify having a domestic, house-based battery.

6. The saving per year is 364.26+42 = £406.26 which suggests 12.3 years for the return.

7. Each year's gain is £406.26,and double the gain for the other years (T) suggests that 5000 = 406.26(2+2T) => 12.3=2+2T => T = 5.15 years. So seven years after installation the capital is recovered.

8. The Chapmans are using 48kWh per sq.m. where both Adams and the Barracloughs use 100 kWh per sq.m. So the Chapmans use the heat more effectively; suggestions include that (i) they have a very much better insulated house (ii) they keep the house at a markedly lower temperature (iii) they heat fewer rooms in the winter.

9. Let the gas price be G. let the electricity pricce be E. If gas moves from G to 2G and if this price applies to electricity generation, then a third of electricity suddenly costs twice as much, so the electricity price goes up by a third of what it was, to four thirds of its previous price. the new prices are 2G for gas and 4E/3 for electricity.

Electricty per unit is 27.22/7.34 = 3.71 times more expensive before any price changes.

10. G->4G means a third of electricity has risen in cost by three times as much so the increase is three thirds, literally doubling the electricity price. So where old bills were G+E they will move to 4G+2E.

The three example families show that, in a reflection of the price structure, more gas is used than electricity. Before there are adjustments to consumption (colder houses, fewer rooms heated, etc) each example uses between 3 and 5 times as much gas as they do electricity (Adams 7.5/1.5= 5; Barracloughs 11.5/3.9 = 2.95; Chapmans 12.5/3.7 = 3.38)

So, if the ratio is k the old total was E+kG and the new total will be 2E+4kG. If k=3, then 2E+12G, k=4 => 2E+16G, k=5 => 2E+20G. But, on the figures I have given, E = 3.71G, so this means that the 'old' bills were E+G = 4.7G and the new bills would be k=3 => 2E+12G = 19.4G, k=4 => 2E+16G = 23.4G, k=5 => 2E+20G = 27.4G. Divide each of these by the old total, 4.7G and we have k=3 =>19.4G/4.7G = 4.1 times the old price, k=4 => five time the old price and k=5 => 5.8 times the old price.

So when gas jumps by a factor of 4, the energy bills also jump by between 4 and 6. Not only do we have a leap in household costs but that will cause all sorts of costs to jump and so we have inflation (the same money buys less). So the people with very littel spare money are in difficulty. thebusinesses that use a lot of gas for their energy supply (a steel mill, for example) are is suddenly dire straits, as their costs will leap upwards. Gas suppliers have all sorts of issues with cash flow, too, as they have tyo pay the new wholesale price immediately and it takes time to pass that on to customers. This is the sort of issue we have been seeing in the first half of 2022.

DJS 20220515