Variability of heating demand throughout a year.

Friends, as I drift off to sleep at night, I often reflect on the subtle wonder of heat pumps, and last night I imagined this article in my head. However writing the article has proved more difficult than I had envisaged in my sleepy reverie.

There are lots of technical details, so in case you need to leave early, the question I wanted to answer was this:

• How many days a year does a heat pump (or indeed any other kind of heater) need to work at full power to keep a dwelling warm? And what fraction of the time does it need work at, say, half-power?

The answer for the region around Heathrow Airport is shown in the graph below.

Click on image for a larger version. Based on 3 years of daily data, the graph shows the typical number of days per year that the given fraction of full heating power is required. Full power is the power required to keep the dwelling at 20 °C on the coldest day in the last three years. Adding up the individual data points, one can see, for example, that for typically 69 days per year, the required heating power is between 40% and 60% of the maximum. Heating power between 80% and 100% is only required on 12 days per year.

This graph is useful because it allows one to estimate the costs of running a gas boiler or a heat pump throughout the whole year. And it also one to assess how a battery can be used with a heat pump to reduce running costs during the winter.

So let’s see how to calculate the graph above.

1. To begin

The first step is to estimate how the heating demand varies through the year. To do this I downloaded daily heating degree-day (HDD) data from degree days.net for my neighbourhood airport, Heathrow, for the last 3 years. I used the 16.5 °C HDD data because this corresponds roughly with the heating demand for a dwelling kept at 20 °C.

Click on image for a larger version. 3 years of daily heating demand data based on the meteorological station at Heathrow Airport UK, just 10 km from my home. You can see the two cold spells we had last winter (22/23) in December and January.

I then divided each data point by the maximum heating demand – which was 18.8 °C. I then expressed the heating demand as a fraction of this maximum.

Click on image for a larger version. 3 years of daily heating demand data based on the meteorological station at Heathrow Airport UK, just 10 km from my home. Data are expressed as a fraction of the maximum heating demand which occurred in January 2023.

In this graph the data are just the same as in the first graph, but the vertical axis is now labelled by the fraction of maximum heating demand. Looking at this graph we can see that there are:

• Lots days in which heating demand was less than 10% of maximum,
• Just a few days in which heating demand was greater than 90% of maximum
• Lots of days where heating was in the range between 10% and 90% of maximum.

This graph applies to any kind of heater, gas boiler or heat pump – it’s just describing the variation in heating demand through the year that must be met in order to keep a dwelling at the same temperature.

2. Let’s make a histogram

Fascinating as the above graph is, it does not tell us what we want to know! We want to be able to estimate the fraction of the time that the heat pump operates in the range between (say) 10% and 20%, or in the range between (say) 50% and 75%. To work this out we need to re-structure the data. Here are some graphs of the re-structured data with the heating power divided into twenty 5% bands: 0-5%, 5.1% to 10%, …..95.1% to 100%.

Click on image for a larger version. Histogram of 3 years of daily heating demand data based on the meteorological station at Heathrow Airport. See text for explanation.

Each point on the graph shows the percentage of the heat demand in each band. So for example the above graph tells us that:

• 22.4% of the time the heating demand is less than  5% of full power.

Or considering several bands together,

• The heating required is between 40% and 60% of full power for 5.1% + 5.9% + 4.2% + 3.8% = 19.0% percent of the year
• The heating required is above 80% of full power for 0.9% + 0.9% + 1.0% + 0.5% = 3.3% percent of the year.

We can also usefully re-draw the graph expressing the frequency of each level of heat demand as a likely number of days per year on which that heating demand will be required.

Click on image for a larger version. The graph shows the typical number of days per year that the given fraction of full heating power is required. Adding up the individual data points, one can see, for example, that for typically 69 days per year, the required heating power is between 40% and 60% of the maximum. Heating power between 80% and 100% is only required on 12 days per year.

Grouping the data into 20% bands we see that there are around 160 days – generally known as “summer” – with very low heating demand. There are a further (roughly) 160 days with medium heating demand (between 20% and 60%) and finally (roughly) 45 days with high heating demand (between 60% and 100%).

3. Let’s think about a dwelling with a particular heating demand

I find the above analysis fascinating, but it becomes even more interesting if one considers a specific dwelling with (say) 5 kW of maximum heating demand. We can now re-draw the above graph in a number of ways.

Click on image for a larger version. For a dwelling with a maximum heating demand of 5 kW, the graph shows the typical number of days per year that a particular average heating power per day was required to keep the dwelling at 20 °C. For example, for typically 69 days per year, the required heating power is between 2 kW and 3 kW.

The graph above now has the heat pump power in kilowatts (of heat) showing that – for example – the average daily heating power is between 3 and 4 kilowatts (of heat) for typically 36 days per year.

If a heater delivers on average 3 kW of heating power for a day it will deliver 3 kW × 24 hours = 72 kWh/day of heat energy into the dwelling. So we can replot the graph again but this time labelling the horizontal axis with the heat energy delivered per day (kWh/day).

Click on image for a larger version. For a dwelling near Heathrow with a maximum heating demand of 5 kW, the graph shows the typical number of days per year that a particular number of kWh of heating required to keep the dwelling at 20 °C. For example, for typically 12 days per year, the required heating energy exceeded 96 kWh/day.

And now we can begin to see something useful. The graph tells us that – on average – for 23.8 days a year, the dwelling requires ~48 kWh/day of heating. So if we multiply 23.8  days × 48 kWh/day we get 1,142 kWh. So we can now work how much heat is delivered in each operating band. Remember that although there are not many days in the high-power band, lots of heat is delivered on each of those cold days. And likewise, although there are lots of low-power days, not much heat is delivered on those days.

Click on image for a larger version. For a dwelling near Heathrow with a maximum heating demand of 5 kW, the graph shows the typical amount of heat (kWh) delivered at each power level. For example, Of the 13,900 kWh required for the whole year, 1,300 kWh were delivered at between 80% and 100% power, and 4,300 kWh were delivered at between 40% and 60% power.

This graph yields lots of information:

• Adding up all the data we see that throughout the year the heat delivered amounts to 13,900 kWh.
• If this had been delivered by a 90% efficient gas boiler then the boiler would have consumed 15,400 kWh of gas per year. And the Rule of Thumb would have suggested that “right size” of heat pump would be 5.3 kW – quite close to the actual ‘perfect’ heat pump size.
• We see that the coldest 12 days of the year require 1,300 kWh of heating – around 9% of the annual heat load – and that the bulk of the heating (10,700 kW or 77%) is delivered on milder days when the average heating power is between 1 kW and 4 kW.

4. Cost

The discussions so-far has just centred on the heat delivered to the dwelling. Nothing so far has been about costs, or the specific advantages of using a heat pump. The above discussion could apply to electrical heaters or a gas boiler. The graph below shows the distribution of costs across the different power levels expressed as cost per day, assuming heating at 8p/kWh. As I write this typical gas tariff.

Click on image for a larger version. For a dwelling near Heathrow with a maximum heating demand of 5 kW, the graph shows the typical distribution of cost per day (£) if each unit of heating delivered costs £0.08 £/kWh. For example, 9% of the annual bill would be incurred on the coldest days, each day costing more than £8/day.

The wonder of heat pumps is that they can deliver more than one unit of heat energy for each unit of electrical energy consumed. The ratio of the two is called the Coefficient of Performance or COP, and it changes with the operating parameters of the heat pump. In particular, at lower external temperatures when the heat pump is working hardest, the COP is lowest. The graph below shows a guess for how the COP of a heat pump might change with heating demand – a proxy for outside temperature.

Click on image for a larger version. The graph below shows a guess for how the COP of a heat pump might change with heating demand – a proxy for outside temperature. At 100% heating demand the outside temperature was around – 2 °C.

Assuming this variation we can now work out how much electrical power – which we need to pay for – was required by the heat pump to deliver the necessary thermal power.

Click on image for a larger version. For a dwelling near Heathrow with a maximum heating demand of 5 kW, the graph shows the typical amount of heat (kWh) delivered at each power level, and the amount of electricity (kWh) required to deliver that heat. For example, on the coldest days when the heat pump was operating above 80% capacity, 1,300 kWh of heat were delivered using only 484 kWh of electricity.

This analysis tells us that over the entire year 3,800 kWh of electricity would be required to pump the 13,900 kWh of heat – a seasonally-averaged COP of 3.6.

• If one paid 28 p/kWh for electricity this would cost £1,064 to deliver the heat.
• If one paid 32 p/kWh for electricity this would cost £1,216 to deliver the heat.
• If we were to use gas a 90% efficient gas boiler to deliver the same 13,900 kWh of heat at 7p/kWh this would cost £1,081.

To the accuracy of this calculation, the costs of using gas at £0.07/kWh or a heat pump at £0.28p/kWh are similar. Of course the carbon costs are dramatically different. Using a gas boiler would emit 3,500 kg of CO2 compared with emissions of only 874 kg when using a heat pump: a 75% reduction.

5. Making the heat pump option cheaper

Friends. we now reach the point where I explain why I have taken you on this long journey. I didn’t want to emphasise it at the start of the article because it seemed so complicated – but if you have made it this far, I think will now get the point!

Wouldn’t it be great if the heat pump could not only supply low carbon electricity, but could also do it at significantly lower cost – significant enough to justify the capital expenditure of a heat pump?

To achieve lower cost, one needs to spend even more money and use a heat pump with a battery to allow off-peak electricity purchases: But what size battery should one buy? And how much will it save? Now we are getting to the serious questions!

So first we need to re-jig the graphs above to show how electricity use is distributed across the different amounts of heat delivered per day.

Click on image for a larger version. For a dwelling near Heathrow with a maximum heating demand of 5 kW, the graph shows the typical number of days in which a particular amount of electricity (kWh) is  to deliver heating to maintain 20 °C.

So now I will consider my current tariff with Octopus: they keep changing its name so I won’t confuse you, but it means I can buy electricity at 7.5 p/kWh for 6 hours at night: the rest of the time electricity costs 32p/kWh. I use this to run the house and charge up the 13.5 kWh Powerwall battery.  The battery then has to run the house for 18 hours until the next cheap period. If the capacity is not enough then I have to buy some full price electricity!

Considering only space heating – not hot water or other appliances – and assuming 100% battery efficiency, we see that for the days where the heating demand in that 18 hours amounts to less than 13.5 kWh, we could run the house entirely on cheap electricity. So when daily demand is less than 18 kWh of electricity, the heating demand could be met entirely with cheap-rate electricity.

If we had more batteries, then we could store more cheap electricity. The graph below shows an estimate for the impact that different sizes of batteries would have on the annual heating bill.

Click on image for a larger version. For a dwelling near Heathrow with a maximum heating demand of 5 kW, the graph shows the typical annual cost (excluding standing charges) of using a heat pump  to deliver heating to maintain 20 °C.

Please note: this calculation covers just the cost of the consumed units of electricity: it ignores many relevant factors such as:

• Standing charges.
• The use of electricity to do other essential activities.
• Inefficiencies charging and discharging the battery.
• The capital costs of the heat pump and the battery.
• Other more complex tariff structures.
• Use of the batteries to export electricity.
• Any use of solar panels.

Reflections

Friends, I conceived of this article in a reverie, and I am having difficulty finishing it – it’s like a dream from which I cannot wake up! But here’s the bottom line:

• Using a battery with a heat pump, but without Solar PV is an unusual combination. However I thought I would investigate to see if it might possibly be a useful combination.
• Using a heat pump with a battery can reduce annual running costs – but even for a modest 5 kW of peak heat loss (i.e. 72 kWh of heat/day), one needs a large battery – around 10 kWh to make a significant dent in the annual bill.

If you have read this far, I would just like to personally say “Thank you, and Congratulations” – and I hope you don’t feel short-changed at the conclusion.