Assessing Powerwall battery degradation: End of Winter of 22/23

Friends, three months ago in December 2022, I wrote about my technique for estimating the degradation of the capacity of my Tesla Powerwall 2 battery (Link).

The idea was to consider data from days in which the battery is charged to 100% capacity at night, and then discharges fully to 0% during the day. This only happens in winter when household demand is high and solar PV generation is low. After correcting for any solar generation one can make a reasonable estimate for the practical working capacity of the battery.

Click on diagram for a larger version. Illustration of day in which the Powerwall fully discharged.

We have now had another 60 days during which the battery discharged fully and using the same technique I described previously, I have re-evaluated the degradation of battery capacity. The results are shown below in two graphs: it’s the same data in both graphs they are just plotted on different times scales.

Taking all the data into account, the trend line suggests that battery capacity is degrading at roughly 2.6% per year. Since the battery has a nominal capacity of 13.5 kWh, this corresponds to a loss of capacity of 0.35 kWh/year.

Click on image for a larger version. Full discharge capacity plotted versus date. The blue dots show all the data and the dots surrounded with a pink circle show data with solar contribution during the day. The trend line suggests capacity is being lost at 2.6%/year.

The Powerwall was installed in March 2021 and if the degradation were to continue at 2.6%/year in future years then the battery would lose 10% of its capacity by the winter of 2024, and 20% of its capacity by the winter of 2028.

Click on image for a larger version. Full discharge capacity plotted versus date. The blue dots show all the data and the dots surrounded with a pink circle show data with solar contribution during the day. The trend line suggests capacity is being lost at 2.6%/year.

What to do?

Nobody wants their £10,000 battery to be losing capacity. But there is very little I can do about it! It’s inherent in the nature of the batteries and of the charge/discharge cycles they experience.

One option would be to prevent the battery from fully discharging by forcing it to retain, say, 1 kWh in reserve. However, while this might reduce the rate of degradation, it would deprive me of the battery capacity that I was hoping to preserve!

So my plan is to do nothing. If the degradation continues at this rate I will still have a 10 kWh battery in 2031 – and that’s still a very useful size of battery.

If I feel the need for more storage, my thought is that it would probably eventually make sense to upgrade one of my solar PV inverters to be a hybrid inverter, and then add extra battery capacity in the loft. These batteries will have the Lithium Iron Phosphate chemistry that is supposed have very low degradation. This option would likely cost much less than buying a replacement – or additional – Powerwall.

Of course, by the time this becomes important Powerwalls may have fallen in price and be readily available (!). Or the use of batteries in vehicles for domestic storage may have become commonplace.

In short, this is tomorrow’s problem.

 

Tariff Calculation Spreadsheet

Friends, as you may or may not know, Saturday night is the night of week I like spend documenting large spreadsheets. Lucky me!

In this article I will be describing how to use the spreadsheet I have developed over the last couple of weeks and used in the last couple of articles. The spreadsheet allows one to estimate the likely costs of using particular electricity tariffs – the Octopus GO, FLUX and COSY tariffs – if your dwelling has a domestic battery and solar PV panels. My hope is that it will help people make rational choices about which option is best for them.

Download

The Excel spreadsheet can be downloaded from this link. If you are downloading this macro-enabled file on a Windows computer, then the macros will probably be blocked by default. To change this you may need to first close the file in Excel, then right-click on the file’s icon and select the ‘Properties’ pane. Here you should see a tick box labelled “unblock”. If you unblock the file then it should work correctly.

Please note, the spreadsheet comes with no guarantee of anything at all, and it can at times be very slow to re-calculate.

Structure

The spreadsheet consists of two parts.

• At the top are several boxes that set the parameters of the simulation, and which show the results. I think of this as a ‘dashboard’
• Below this are 8,760 rows, one for each hour of the year. The spreadsheet proceeds through the year hour-by-hour simulating the flows of electricity from solar PV panels and the grid, into and out of batteries and your dwelling.

Throughout the spreadsheet, cells which are ‘inputs’ i.e. cells that you might reasonably want to change, have a yellow background with red text. Cells which are ‘outputs’ i.e. which contain the results of calculations have a red background with white or yellow text.

Click on image for a larger version. The visual appearance of ‘inputs’ and ‘outputs’.

The Dashboard: Overview

Click on image for a larger version.

The dashboard has controls in four regions.

• Regions 1 & 2 are about the tariff being simulated and the prices of the tariffs in each of their cheap, medium and high periods.
• Region 3 sets the parameters of the battery and the charging strategy, the amount of solar PV generation, and the likely household load.
• Region 4 contains the results of the simulation.

The Dashboard: Tariff Section

Click on image for a larger version.

After selecting the tariff to be investigated in the drop down menu, the selected tariff will appear in the box on the far right.

The prices of the different tariff stages can be changed if you desire.

The timings of the tariffs cannot be changed on the spreadsheet. They are set in a Visual Basic macro with the code below. If you understand this sort of thing then you can see how you could adapt the spreadsheet to a different set of tariff timings.

Click on image for a larger version.

The Dashboard: Main Settings

Click on image for a larger version.

The settings for the simulation have four main parts.

The Battery Settings are as follows

• Battery capacity describes the amount of electrical energy the battery can store.
• Round trip efficiency accounts for the energy lost as electricity is stored in the battery and then later drawn from the battery.
• Charge Rate is the rate at which battery charges. This limits how much electricity can be stored in the battery over a given period.
• The initial state of charge is the assumed state of charge at midnight on the 1st January.

Click on image for a larger version.

The cells referring to summer and winter should really be in the neighbouring box about charging strategy [Edit: they have now been moved]. Why are they there? For some systems, it can be sensible to switch between different charging strategies through the year, depending on the amount of solar energy available. For my own system:

• In winter I charge the battery at night using cheap rate electricity and the battery then runs down during the day.
• In summer, there is no need to charge at night because the battery is charged for free during the day by solar PV electricity.

The two boxes allow one to choose the days of the year on which to switch strategy.

In the ‘Grid Charging Strategy‘ box one can choose between charging when electricity is cheap, and/or charging in the hour before electricity becomes expensive. Either option can be chosen to be active in summer, winter or both.

In the solar factor box, one can set the expected amount of solar generation expected. This is generated by scaling the hour-by-hour solar data by the factor shown in the box.

Note that selecting a target amount will give an accurate result if the average solar generation from 2005 to 2016 is selected in the drop down menu at the bottom of the box. If one selects solar data from a particular year then the amount of generation will vary in line with that years variable output.

Click on image for a larger version.

The final box simulates how electricity demand from the household changes through the year.

Click on Image for a larger version.

Demand consists of two components: a steady consumption every day, and a component which peaks in mid-winter simulating the use of a heat pump.

The length of winter depends on a setting from 1 to 5 as shown in the figure below.

Click on Image for a larger version.

How the simulation works.

Each row of the spreadsheet from row 48 downwards calculates the state of charge of the battery, the household demand, and the solar PV according to the settings on the dashboard.

The simulation works row-by-row proceeding hour-by-hour through the year. The columns are as follows

• A: Index
• B: The day of the year expressed as a decimal day.
• C: The hour of the day, used for calculating the appropriate tariff
• D: The season of the year WINTER or SUMMER according to dashboard settings
• E: The Tariff Rate (cheap, medium, or high) determined by the Visual Basic code described above.
• F: G: & H: The hourly background & variable consumption – and their total.
• I: The daily average of the demand
• J: The time of day expressed as a fraction of a whole day
• K: Blank
• L: The 3-day running average of solar generation (used for plotting)
• M: The hour-by-hour solar data selected in the drop down menu in the SOLAR box. This is looked up from the data table in columns AE to AQ
• N: Modified demand: this is the difference between current demand and the amount of solar PV currently being generated.
• O: The amount of energy delivered to the battery (if positive) or drawn from the battery (if negative). This is calculated according to the current state of charge of the battery.
• P: The amount of energy drawn from the grid (if positive) or sent to the grid (if negative). This is calculated according to the current state of charge of the battery.
• Q, R, S, & T : Imports: If column P indicates that electricity has been imported, columns R, S and T list the amount of it in each tariff charging rate: these columns are then totalised at the top to calculate the annual amount to be paid.
• U, V, W, & X : Exports: If column P indicates that electricity has been exported, columns U, W and X list the amount of it in each tariff charging rate: these columns are then totalised at the top to calculate the annual amount of revenue accrued.
• Y: Hourly cost of grid imports
• Z: Hourly amount due from grid exports
• AA: Depending on the charging strategy this is when the battery charges from the grid. Household consumption is also met by the grid during these periods.
• AB: The amount of electrical energy sent to the battery.
• AC: The current state of charge of the battery.
• AD: Blank
• AE to AQ. Hourly solar data for the years 2005 to 2012 downloaded for the EU Sunshine database for my 4,000 kWh/year solar system in Teddington.

Graphs

Click on Image for a larger version.

Scrolling down  the spreadsheet will show four graphs which can be helpful in understanding what is happening.

The first graph shows three quantities charted across each day of the year:

• The 3-day average of the selected solar data
• The daily household demand
• The capacity of the battery

The second graph shows the state of charge of the battery charted across each day of the year.

The third and fourth graphs show the state of charge of the battery during a typical summer and a typical winter day.

More Tariff Calculations: GO, FLUX, COSY and more

Friends, in the previous article I estimated the annual cost of running a household with various “time-of-use” (TOU) tariffs i.e. bargains with the electricity company in which the price of a unit of electricity varies through the day.

The article seemed to strike a chord with many people asking several “But what if..?” questions. This article is an attempt to answer some of those questions.

The reason that I think this is an important issue as TOU tariffs are likely to become more popular.

Why? Because getting people to avoid consuming during peak hours is cheaper and greener than building a new power station to meet that demand. It’s also cheaper and greener than using gas in an existing power station to meet that peak demand.

And as domestic batteries and solar PV installations become ever more common, avoiding consumption at peak times has become more achievable for ‘ordinary’ people.

But it has also become all but impossible to work out which TOU tariff will be cheaper (or greener). And this puts us in danger of perpetuating the confusopoly that currently exists in energy tariffs.

The only way I know to work out which tariff is cheaper is to simulate an entire year hour-by-hour with realistic consumption, solar PV and battery storage. And that is exactly what I did: please read the previous article if you want to familiarise yourself with that.

Click on image for a larger version. The concept of a Confusopoly was invented by Scott Adams.

When I began to write this article, I planned to look at

• Different Tariffs (Octopus’s GO, COSY, and FLUX tariffs – alongside the standard variable tariff)
• The effect of solar generation
• The effect of battery storage capacity
• The effect of a small (10 kWh/day) or a large (20 kWh/day) domestic consumption
• Seasonal Strategies

However as I began this program of work I realised I had bitten off more than I could chew.

So below I will look at some of these issues, but people’s requirements are so specific that I thought the best way to be helpful was to make the spreadsheet easier to use and that is what I have tried to do.

The Excel spreadsheet can be downloaded from this link. If you are downloading this macro-enabled file on a Windows computer, then the macros will probably be blocked by default. To change this you may need to right-click on the file and select the ‘Properties’ pane. Here you should see a tick box labelled “unblock”. If you unblock the file then it should work correctly. The next article will describe how the spreadsheet is structured.

One thing I won’t be looking at is the effect of charging electric vehicles. Sorry: I don’t have experience of this.

Tariffs

In this article I will investigate four tariffs: Octopus’s so-called COSY, GO, and FLUX tariffs, and a standard variable tariff (SVT) – currently (February 2023) set at 34p/kWh. The assumed costs and times of operation are shown in the figure below.

Click on Image for a larger version. Illustration of the variation in price through the day for electricity imports (left) and exports (right) on the Octopus Go FLUX and COSY tariffs. Also shown as a dotted line is current standard variable tariff (SVT).

The daily-average import prices of the FLUX and COSY tariffs are about 34p/kWh, the same as the SVT, but the average GO tariff is 37p/kWh.

The daily-average export prices of the FLUX and COSY tariffs are 22p/kWh and 15p/kWh, the same as the SVT, but the average GO export price is just 4p/kWh.

Solar PV and Demand

The standard demand that I used last time was based on my own home. It consisted of a baseload of 10 kWh/day and then a seasonally variable heating demand peaking at 15 kWh/day. The solar PV that I used previously was based on my own home generation of just under 4,000 kWh/year. But of course these are only relevant to me – and there are many combinations relevant to other people.

For example, the figure below shows four combinations of high and low demand and high and low solar generation. It’s clear that these are very different situations and that the likely cost savings will be very different in each scenario. And this does not include changing the size of the battery.

Click on Image for a larger version. Four scenarios showing the variation of daily demand and daily solar PV generation through the year. The upper graphs show high demand and the lower graphs show low demand. Solar PV generation varies from just under 2,000 kWh to just under 8,000 kWh. Also shown as a dotted line is 13.5 kWh of battery storage available in a Tesla Powerwall.

Pre-discussion

Before looking at any results it is important to understand the ways in which it is potentially possible to generate savings compared to the standard variable tariff (SVT).

• Using the battery alone it is possible to generate savings by avoiding peak tariffs.
• Using solar alone it is possible to generate savings by using solar electricity instead of drawing electricity from the grid. But solar PV doesn’t match demand minute-by-minute, sometimes oversupplying and sometimes undersupplying.
• Using battery and solar, can generate savings in several ways:
  • By avoiding peak and standard tariffs
  • By improved self-utilisation of solar electricity
  • By exporting electricity.

Bearing these factors in mind, let’s look at some results.

Charging Strategies with a 13.5 kWh Battery: No Solar PV

I compared GO, COSY and FLUX to the SVT without considering any solar PV generation, but instead I used four different charging strategies:

1. No strategy – i.e. the battery is not used at all.
2. Pre-charge the battery for an hour before the peak rate.
3. Charge the battery as much as possible in the cheap rates
4. Pre-Charge the battery before peak rate AND charge the battery during cheap rate.

The SVT for this scenario came to £1,849/year and the four strategies above differed in cost as shown below:

Click on Image for a larger version. Annual savings or extra costs compared to the £1,849 SVT when using different tariffs. This chart shows the impact of using a battery alone with no solar PV. See text for details.

• Not using the battery with COSY and GO costs several hundred more pounds per year. So don’t use one of these tariff’s if you don’t have a battery or solar!
• Pre-charging for one hour before the peak rate resulted in modest savings (~7%) compared with the SVT.
• Charging during cheap rate resulted in big savings on the GO tariff (-39%), modest savings on the FLUX tariff(-20%), but a significant increase on the COSY tariff (+17%).
• Pre-charging and Cheap Rate charging together saves a lot of money with GO, only a little (12%) with FLUX and is more expensive with COSY.

At first I thought the simulation must be in error to for COSY to cost more than SVT. But looking at the charging details I saw that in summer (when demand was lower) using the ‘charge when cheap’ rule led to overcharging on COSY because it has two cheap rates just 3 hours apart. This could be avoided with smarter programming.

My conclusion from this is that the only way to significantly save money when using a battery alone is with a tariff such as GO which offers very low prices (12p/kWh) and a large battery able to store almost a whole day’s consumption.

Charging Strategies with modest solar PV and two sizes of batteries

Next I considered a situation with a modest solar PV installation (~2000 kWh/year corresponding to ~ 6 south-facing panels) and either a small (5 kWh) or a large (13.5 kWh) battery. I then compared the different charging strategies I used in the previous section with each of the GO, COSY and FLUX tariffs to the SVT with no solar PV.

Click on Image for a larger version. Annual savings or extra costs compared to the £1,849 SVT when using different tariffs. This chart shows the impact of using a small battery (left) or large battery (right) with a small solar PV installation. See text for details.

Comparing these results with the previous ‘No Solar’ results, it is immediately obvious that – with or without a battery – even a modest solar PV installation saves a serious amount of money. The average saving from the ‘No Battery’ scenario is 33% compared to the SVT.

Pre-charging from the grid for one hour before peak rate results in extra savings on all tariffs averaging 41% independent of battery size.

Moving to a larger battery (13.5 kWh vs 5 kWh) is only really of significant benefit on the GO tariff (56% vs. 33%). For the FLUX tariff the extra battery capacity increases savings only from 33% to 42%.

Annual Variability

Next I considered the effect of annual variability. I estimated the annual cost for a scenario similar to that in my home using the GO and FLUX tariffs and solar data for each year from 2005 to 2016.

Click on Image for a larger version. Top: Annual variability in solar PV generation (kWh/year). Middle: Annual variability in estimated year cost using the GO tariff. Bottom: Annual variability in estimated year cost using the FLUX tariff. See text for details.

The year-to-year variation in solar PV yield was roughly ±10% over the years 2005 to 2016. Unsurprisingly, good solar years led to lower overall costs on both GO and FLUX and vice versa.

But there is a curious feature in the calculations. Using the FLUX tariff the average cost over the individual years 2005 to 2016 was similar to the cost calculated using the average solar data. But for the GO tariff this was not the case. Instead, the average cost over the individual years 2005 to 2016 was much higher than the cost calculated using the average solar data.

I am not sure why this is, but it may be a feature of the fact that the averaged solar data has less variability than data for any individual year.

Other scenarios 

At this point I realised that I was becoming overwhelmed. 

I realised that there was no way to systematically summarise the results of using different tariffs across all the possibilities of demand, solar PV, tariff and charging strategy. So rather than trying to calculate everything myself, I resolved to tidy up the spreadsheet and make it reasonably suitable for other people to use. The dashboard of the revised Version 3 is shown below. In the next article I will describe how to fill out the spreadsheet.

The Excel spreadsheet can be downloaded from this link. If you are downloading this macro-enabled file on a Windows computer, then the macros will probably be blocked by default. To change this you may need to right-click on the file and select the ‘Properties’ pane. Here you should see a tick box labelled “unblock”. If you unblock the file then it should work correctly.

Click on Image for a larger version. Image of the dashboard from the revised spreadsheet for evaluating the costs of different tariffs with different demand, generation, and storage scenarios.

Summary

I compared GO, COSY and FLUX tariffs to the SVT in a couple of scenarios. My conclusions are that:

• Using a battery alone is only really valuable if one uses a big battery to download very cheap electricity.
• However, even modest amounts (e.g. 2,000 kWh) of Solar PV can be very beneficial, and a battery increases the savings possible still further.
• Because the average solar data has less variability than the solar data from any individual year, some tariff’s may not generate a typical cost when using the averaged solar data.

Beyond this, I’m afraid you will need to do these calculations for yourself. More details in the next article…

 

Tariff Calculations: Octopus GO versus Octopus FLUX

Friends, I have struggled to write this article. As you may have noticed, it has taken weeks.

I started writing after I was asked on Twitter about a new electricity tariff called ‘FLUX’ offered by Octopus Energy. Would it be cheaper or more expensive than their ‘GO’ tariff?

It’s a simple question and one that is worth asking. But it is very hard to answer because it involves both hourly details, but also seasonal changes.

I could see how to get at an answer but I have struggled with my waning technical skills. Imagine if you will, an old boxer going in for one fight too many. Finding themselves on the ropes, they face the unavoidable and inevitable reality of their own decline. But bravely they struggle and finish the fight bruised and defeated, but with their dignity in tact. Similarly I have found my prowess with Excel and Visual Basic to be much diminished, but I have somehow battled through.

GO and FLUX

The Octopus GO tariff which I currently use offers 4 hours of electricity for 7.5p/kWh between 00:30 and 04:30 each day. The rest of the time the cost is 40.75p/kWh. Exports of electricity are paid for at 4.1 p/kWh.

During winter, I buy electricity cheaply at night, and then use it during the day. For most of December and January, the battery could not supply the house for the whole day and I had to purchase electricity at full price for a few hours on those days. Overall, the average price I paid was around 12p/kWh in those two months.

Click on Image for a larger version. Illustration of the variation in price through the day for electricity imports (left) and exports (right) on the Octopus Go and Octopus FLUX tariffs.

The Octopus FLUX tariff is more complicated. It has a cheap rate in the night, but only for 3 hours 02:00 to 05:00 and not so very cheap (20.4 p/kWh). But it also has a more expensive rate (47.5 p/kWh) during peak demand hours from 16:00 to 19:00 each day. The rest of the time the cost is 34 p/kWh.

Initially FLUX looks much worse than GO, but the twist is that FLUX offers much higher rates for exporting electricity: 9.4 p/kWh, 22 p/kWh, 36.5 p/kWh for the cheap medium and high rates respectively. These figures should be compared with the miserly 4.1 p/kWh on the GO tariff.

There are so many variable quantities that I really had no idea which tariff would be cheaper. The results of my calculations appear obvious in retrospect, but that didn’t make the calculations any easier! My conclusions are that:

• For small solar PV installations (<~4,000 kWh/year), the big savings from using the night time electricity on GO outweigh the gains from exporting electricity at a good price.
• For large solar PV installations (>~6,000 kWh/year), this situation is reversed: The savings from using the night time electricity on GO are outweighed by the gains from exporting electricity on the FLUX tariff.
• For medium-sized solar PV installations, the two tariffs have similar costs.

Click on Image for a larger version. Estimates of the annual cost of electricity on the Octopus Go and Octopus FLUX tariffs as a function of the amount of solar generation. This applies to my household – see text for details – and assumes a 13.5 kWh storage battery. 

It turns out that, if you have the capability to export lots of solar PV, then the FLUX tariff could result in very low – and even negative – electricity bills. In retrospect, this is sort of obvious, but it was not obvious at all to me when I began.

But the spreadsheet I developed for the calculation allowed me to do the calculations for different sizes of battery and different amounts of solar PV generation, so I’ve investigated the matter a little more deeply below.

Sadly, because the spreadsheet is Macro-enabled, for security reasons I can’t link to it from this blog and many users wouldn’t be able to download it anyway. But if you really want a copy, please ask for a copy in the comments and I will send it to you somehow. But be warned that the spreadsheet is complicated and slooooow. On my computer it takes around 1 minute to evaluate the yearly calculation.

[Update: I think you can download the spreadsheet from this Dropbox Link]

Let me explain how I made the calculation and then I’ll discuss a few more details.

How to work out which tariff is cheaper

To answer this question I wrote a spreadsheet which modelled the electricity use in a household hour-by-hour for an entire year i.e. the spreadsheet has 365 x 24 = 8,760 rows.

For each hour of the year I estimated:

• The household demand: I modelled this as being the sum of a fixed amount each day (10 kWh/day) plus an amount used for heating that peaked in winter at 25 kWh/day.
• Solar PV: Using the EU sunshine database, I downloaded hour-by-hour sunshine data for my house location from 2005 to 2016, and then averaged this to give a typical solar generation year.
• I then worked out how to supply the household demand.
  • If Solar Power exceeded demand, then the excess was used to charge the battery, and if the battery was full, the excess was exported.
  • If Solar Power was less than demand, then the solar power offset the imported electricity.
  • During winter, the battery was fully charged during the cheap hours.
  • I estimated the battery to have a round-trip storage efficiency of 90%.

The spreadsheet and associated VB Macros took days to debug, but here are the results.

Household Demand

The modelled daily demand is shown below along with the EU sunshine database estimate of PV generation amounting to ~ 4,000 kWh/year. Basic electricity demand is ~ 10 kWh/day but peaks at 25 kWh/day in mid-winter due the heat pump, and amounts to ~ 5,000 kWh/year.

Click on Image for a larger version. Graph showing the modelled daily household demand throughout the year, and the 3-day average of solar generation. The solar data is the average of the years 2005 to 2012 estimated for my location and array size in Teddington.

The relationship between Solar PV supply and household demand is such that one needs to use two different strategies depending on the time of the year.

• In the Winter: the battery is charged using cheap rate electricity and discharges during the day – sometimes running out at night.
• In the Summer: there is no night time charging and the battery charges during the day and discharges during the night.

These two modes are illustrated in the graphs below.

The first graph shows a week in winter under the two different tariffs. The four hours of cheap electricity under the GO tariff allows the battery to charge to full, but the FLUX tariff only has three hours of cheap electricity so the battery only charges to around 10 kWh. The battery then discharges to run the household, and is partially supported by the weak solar generation, but typically runs out well before the end of the day.

Click on Image for a larger version. The state of charge of the battery through 7 days in winter. The upper graph shows the Octopus GO tariff which allows the battery to be fully re-charged each night. The lower graph shows the Octopus FLUX tariff which only has enough cheap hours to enable partial filling of the battery. The solar generation is also shown in yellow.

The second graph shows a week in summer. At this time of year, solar generation is enough to run the household and charge the battery during the day, with enough left over for export.

Click on Image for a larger version. The state of charge of the battery through 7 days in summer. Also shown is the solar generation is also in yellow and electricity exports in grey.

The switch between the summer and winter strategies is made on day 90 and day 270 – an arbitrary choice but one which corresponds roughly to the point where the 3-day average of solar exceeds the average household demand. The graph below shows the state of charge of the battery throughout the entire year on both tariffs.

Click on Image for a larger version. The state of charge of the battery through the entire year. The top graph shows the estimate for the GO tariff and the lower graph shows the estimate for the FLUX tariff.

Costs

The simulation runs hour-by-hour through the entire simulated year. For each hour, I estimated how much electricity was imported and exported, and then applied the appropriate tariff rate. This allowed me to summarise the situation for my home as below.

Click on Image for a larger version. Results of calculations of cost of running my household on (top) the GO tariff and (bottom) the FLUX tariff.

Both tariffs offer the possibility of running a home very cheaply: with annualised energy bills in the range £30/month to £50/month. However the GO tariff appears to be cheaper in this simulation £34/month compared with £50/month for FLUX.

The analysis shows why: being able to fill up with electricity at 7.5 p/kWh reduces the cost the electricity dramatically – £298/year versus £648/year. The improved rates for export on the FLUX tariff (£209/year versus £38/year) aren’t enough to make up for that.

Discussion#1: The effect of extra solar generation 

My conclusion is that for me, with my existing 3,800 kWh/year PV installation, the GO tariff is more economical.

But having recently had extra panels installed, this conclusion may not hold. The difference in annual cost between the two tariffs is ~£193 and the typical difference between the FLUX and GO export tariffs is ~ £0.18. So if the new system could export ~1,000 kWh more in summer, then the balance could easily shift.

And indeed, that is what the simulations show. Notice that for 8,000 kWh of generation the annual cost of electricity would be negative i.e. the house would be a bona fide power station!

Click on Image for a larger version. Lower graph: estimates of the annual cost of electricity on the Octopus GO and Octopus FLUX tariffs as a function of the amount of solar generation. Upper graphs: Details of how the the import costs and export and rewards vary on the FLUX tariff (left) and GO tariff (right). [NOTE: The original graph had an erroneous curve plotted. This was updated at 23:27 on 23/2/2023]

If the newly installed panels generate as much as I hope, then the annual generation may approach 6,000 kWh and in this case, the FLUX tariff would be marginally cheaper.

Discussion#2: The effect of battery size 

Whilst I was making these calculations, I thought it would also be interesting to look at the effect of battery size. For my home – with solar PV generation of ~3,800 kWh/year – the simulations suggest that bigger batteries are better – no news there – but that above roughly 10 kWh the additional savings are minimal.

Click on Image for a larger version. Lower graph: estimates of the annual cost of electricity on the Octopus GO and Octopus FLUX tariffs as a function of battery size with ~ 3,800 kWh of solar generation. Upper graph: Details of how the the import costs and export and rewards vary on the FLUX tariff (left) and GO tariff (right).

This is a relief to me. It means that as the batteries degrade, the system itself is likely to continue to perform well for many years.

Discussion#3: Strategy 

Friends, life is complicated enough without having to consider battery management strategy. Nonetheless, this is where we are!

Observant readers may have noticed that I made no specific efforts to avoid consuming energy at peak hours because it doesn’t happen very often. But if it could be done, then on the FLUX tariff, there would be a reduction in both costs and carbon emissions during these dirtiest hours of the day.

The problem for me is that I am not sure whether the occult Tesla logic which controls my battery, is smart enough to avoid using electricity at peak times. If it could achieve this, then on days when the battery might be expected to run out early, the system might preemptively charge in the middle of the day (at 34p/kWh) and so avoid consuming grid electricity during the peak hours when the equivalent electricity would cost 47.5p/kWh.

For a load of around 1 kW for 3 hours the potential saving would be around 45p/day which over 60 days of winter might amount to ~£27/year.

Errors and Mistakes

Friends, writing this article has been very difficult, and I must warn you although I have carried out many checks, I might easily have made some errors. Sorry. Please feel free to point them out to me when you spot them. The results appear to be about right for my own situation and so I have modest confidence that the errors are not too major.

But overall, despite the fact there are errors and mistakes, I think this spreadsheet offers a tool for evaluating the complex interaction of solar generation, battery storage, and time-of-use tariffs. I hope it helps.

Carbon Dioxide Accounting: Why I hate it.

Friends, the turn of the year is the time when Carbonistas such as myself look at their carbon dioxide accounts. Like all accounting it is tedious but sort of important.

However carbon dioxide accounting depresses me more than regular accounting because I can hardly believe any of the numbers!

Allow me to explain…

The Big Picture

Click on image for a larger version. The red line on the graph shows estimated emissions from my household if I had not undertaken any refurbishment. The data are calculated month by month out to 2040. The green line shows actual estimated emissions from my household. The black dotted-line shows the additional effect of paying Climeworks to remove CO2 from the atmosphere on my behalf.

The aim of my activities and expenditure over the last three years has been to reduce ongoing carbon dioxide emissions from all aspects of my life, but targeting especially my home.

The graph above shows how I expect household emissions to accumulate based on various assumptions. Notice the scales: the horizontal scale extends out to 2040, my targeted date of death, and the vertical scale is in tonnes of carbon dioxide. Tonnes!

• The red line shows how I would expect emissions to accumulate if I had made no alterations to the house.
• The green line shows how I expect emissions to accumulate based on the current plan. This is based on the amount of electricity I draw from the grid.
• The dotted black line accounts for the activities of Climeworks who have promised to permanently remove 50 kg of CO2/month in my name. This line is dotted because I don’t personally have any evidence that Climeworks are actually removing CO2 from the atmosphere.

The net effect of my efforts will hopefully by 2040 amount to around 78 tonnes of CO2 emissions which do not take place. But in honesty, I am not very sure about these numbers.

Assumptions, Assumptions, Assumptions

Working out the data for this graph involves estimating the amount of electricity and gas that the household has consumed (not so hard) – and will consume in future (a bit harder, but still not crazily difficult).

However it also involves associating an amount of carbon dioxide with each unit of gas or electricity used – the so-called carbon intensity (CI) measured in kilograms of CO2 per kilowatt hour (kgCO2/kWh) of gas or electricity. And I genuinely do not know what numbers to use for these CI’s.

Allow me to explain my difficulty.

Assumptions for gas

For gas, a hypothetical 100% efficient boiler would produce around 0.18 kgCO2/kWh.

But it also takes energy – and thus emissions – to extract and deliver the gas to my boiler, and these emissions should also be associated with my consumption.

However, allocating these ‘up stream‘ emissions is not straightforward. It will differ depending on the source of gas e.g. from the North Sea (~+0.013 kgCO2/kWh) or liquified natural gas shipped from (say) the US (~+0.035 kgCO2/kWh). And also it will vary with the distance gas is pumped through the gas distribution network.

And then there is the giant smelly elephant in the room: leaks.

The gas network leaks. At every point from gas platforms to our homes, leaks are very significant. Probably around 1% of the gas we consume leaks, and some of the burned gas escapes without combustion.

When methane leaks it enters the atmosphere, staying for around 10 years before reacting to form CO2 and H2O . And during that 10 years or so, it warms the atmosphere much more intensively than CO2. Averaged over 20 years – methane is around 80 times more powerful greenhouse gas than CO2.

So a leak of 1% anywhere from the gas well to our homes increase the carbon intensity associated with methane by approximately  1% x 80 x 0.18 = 0.144 gCO2/kWh. Combined with upstream emissions this practically doubles the carbon intensity of burning methane compared with the value used by most web sites.

The only way to really know the amount CO2 emitted associated with gas use, is to use no gas at all: anything multiplied by zero is zero.

Assumptions for electricity

As difficult as it is to truly know the  appropriate carbon intensity (CI) to associate with gas consumption, it is much more difficult to know the appropriate CI to associate with electricity consumption. This is because electricity is generated from several different sources, each with its own characteristic CI.

For example, as I type this, this web site tells me that the carbon intensity of the electricity I am using is 0.065 kgCO2/kWh, but this web site tells me that the carbon intensity of the electricity I am using is 0.101 kgCO2/kWh. Which should I believe? I just don’t know.

Both figures will change depending on the composition of generating technologies, but they have (I suppose) made different assumptions about how to account for some emissions. I have previously written to the web sites to ask but received no reply.

Click on image for larger version. Data from MyGridGB and National Grid on carbon intensity. the two sites give answer which differ by 0.036 kgCO2/kWh – amounting to ~30% discrepancy.

But what if I want to draw some extra electricity? If I switch on a tumble dryer, this extra demand must be met by a source which can be switched on to meet that demand, and in practice, this is always gas-fired generation, which is nominally assigned a CI of 0.45 kgCO2/kWh.

So I have to choose whether to allocate an average CI (0.101 or 0.065 kgCO2/kWh) or a marginal CI (0.45 kgCO2/kWh) to my consumption. How do I decide what is my average consumption and what is marginal? I genuinely do not know.

And additionally, the same elephant (methane leaks) that was in the room for gas consumption, is still in the room for electricity derived from gas-fired power stations. Accounting for leaks, the contribution to the average CI of gas-fired generation could practically double from 0.45 kgCO2/kWh to 0.81 kgCO2/kWh which is almost as bad as coal-fired electricity generation.

And there are similar problems accounting for electricity exported from – say – solar panels. In principle, each extra kWh exported displaces a kWh that would have been generated by gas-fired generation. And so exports of solar electricity are avoiding emissions of CO2 at the marginal rate for gas-fired emissions (0.45 kgCO2/kWh). But should this also include the effect of methane leaks avoided?

And since the CI of grid electricity is changing all the time, should I do my accounting in (say) half-hour periods? Or should I use day or night averages? Or weekly, monthly or yearly averages?

Click on image for a larger version. graph showing the variation in CI with time of day: using electricity at night is generally a bit greener because the fraction of electricity generated by wind and nuclear power is greater. Data from the Carbon Intensity web site.

And some argue that the CI of grid electricity varies from region to region! They argue that in regions where there is lots of renewable generation the ‘local’ CI is low. But this ignores the fact that it is essentially a single grid, and that if these regions were isolated, the grid would not be able to function.

Click on image for a larger version. Map showing regional sub grids together with an indication (by colour) of the ‘local’ carbon intensity. Data from the Carbon Intensity web site.

So what do I do?

Friends, this is why I hate carbon accounting: just changing the accounting basis can apparently change emissions associated with electricity or gas consumption depending on where they take place, and how many leaks are associated with the consumption. And part of this is real, and part is conventional practice, which ignores critical issues like methane leaks.

So one can find oneself making spreadsheets of enormous complexity in search of an accounting accuracy that is ultimately unattainable.

So in the face of all this complexity and ambiguity I assign the same carbon intensity to gas and electricity (imports and exports) of 0.230 kgCO2/kWh.

• For gas this is a bit higher than higher than estimates that add upstream emissions but much lower than estimates that account for methane leaks.
• For electricity this is roughly the average CI for the years 2019 to 2022 as specified on the MyGridGB web site. If this figure changes significantly in 2023 I will update it.

Click on image for a larger version. Map showing carbon intensity averaged over one year showing the systematic reduction in CI. Data from the MygridGB web site.

The graph at the head of the article shows progress so far and how I anticipate things unfolding over the years. In calculating that graph I disregarded…

• Exports of solar electricity which could be considered to be avoiding emissions by displacing gas-fired generation.
• The share of a wind farm that I bought and which should start generating from November 2023. Again, this could be considered to be avoiding emissions by displacing gas-fired generation.

If add these in to my projections, (CI = 0.23 kgCO2/kWh) then the outlook looks better. However, the uncertainties in all the numbers here are so great that I just don’t know if any of it is correct. That’s why all the lines are dotted.

Overall, I know that household gas consumption is zero and therefore so are emissions, no matter what the CI. And this year I expect that we will be more or less off-grid i.e. taking no electricity from the grid – for roughly 6 months. And so I know emissions during that period will be zero. In short, just minimising grid consumption is probably the best way to ensure that associated carbon dioxide emissions are low.

Click on image for a larger version. The red line on the graph shows estimated emissions from my household if I had not undertaken any refurbishment. The data are calculated month by month out to 2040. The green dotted line shows estimated emissions from my household accounting for electricity exported in the summer as ‘negative emissions’ i.e. I have avoided someone else emitting CO2. The black dotted-line shows the additional effect of paying Climeworks to remove CO2 from the atmosphere on my behalf. The blue dotted-line shows the ‘negative emissions’ effect of shares in a wind farm due to begin generating in November 2023.

The Great Carbon Dioxide Accountant in the Sky

Friends, on the sacred slopes of Mauna Loa in Hawaii, there is a carbon dioxide accountant far greater than I.

Click on image for larger version. Mauna Loa CO2 observatory: the  location of the great Carbon Dioxide Accountant in the sky.

Patiently this accountant has been monitoring the concentration of carbon dioxide in the atmosphere since 1959, the year of my birth.

This accountant:

• Does not care about which value of carbon intensity I use in my calculations.
• Does not care about whether I used the correct estimate for embodied carbon in my solar panels or triple-glazing
• Cannot be sweet-talked with promises of future emissions reductions.

They just measure the concentration of carbon dioxide in the Earth’s atmosphere.

When the volcano is not erupting, this accountant publishes their results daily. And this global accountant shows that whatever we are doing is just not enough.

Even if this curve stabilised at its current value of around 420 ppm, the Earth would not cool. But this curve is not stabilising – it is still rising – and it is our actions that are causing this – and only our actions can stop it.

Click on image for larger version. Black Curve: Monthly average atmospheric carbon dioxide concentration versus time at Mauna Loa Observatory, Hawaii (20 °N, 156°W). Red Curve: Fossil fuel trend of a fixed fraction (57%) of the cumulative industrial emissions of CO₂ from fossil fuel combustion and cement production. This fraction was calculated from a least squares fit of the fossil fuel trend to the observation record. Data from Scripps CO₂ Program.

 

 

Setback? Should you lower heating overnight?

Friends, it has become a fact of my conversational life that people ask me questions about heating their homes. And one of the most common questions I am asked is whether or not people should turn down their heating overnight? This is referred to by heating engineers as overnight ‘setback’.

Up until today I have only had a generic and unsatisfactory answer:

“Well, it depends…”.

But last night I finally saw how the question could be answered and then I wrote a spreadsheet to test out my idea. And now I can I answer more fulsomely. My answer will now be

“Well, it depends, but it’s marginal, either way”.

Before I get into the gory details, let me just outline that this article is about working out which strategy uses less energy. In contrast to this dull, but understandable, utilitarian perspective, because any gains are marginal, I would urge you to keep doing whatever makes your home a place of joy.

This is a long and technical article, and follows on from a previous dull post in which I estimated the heat capacity of house. Sorry. If you are looking for something lighter, please allow me to recommend this article about candles! But if you really want to know the details, read on.

The Question

The question people are really asking is this:

• If I set back the temperature from (say) 20 °C during the day to (say) 16 °C at night I will save energy.
• But then if I want the house to be (say) 20 °C when I get up, I need to apply additional or boost heating for (say) a couple of hours before I get up and this requires extra energy.
• On balance, will applying this ‘setback’ save energy? Use extra energy? Or will it make no difference?

The Answer

The answer is as follows,

• If the heating is 100% efficient, then a setback period will always save energy. It’s generally not a big saving, but it is always a saving.
• If the efficiency of heating varies with power, then a setback period may save energy, or may not.

Specifically

• Direct electrical heating with fan heaters, storage heaters or infrared heaters.
  • A setback period will always save energy.
  • The longer the setback period and the lower the temperature, the greater the saving.
• Gas heating with a gas boiler
  • The efficiency of a modern condensing gas boiler is generally in the range 85% ± 5%.
  • If the efficiency falls with increased power – which can happen – then the small saving in energy can be offset by the decreased efficiency of the boiler at high power.
• Heat Pump
  • The efficiency of an Air Source Heat Pump (ASHP) is generally around 300%.
  • If the efficiency falls with increased power – which can happen – then the small saving in energy can be offset by the decreased efficiency of the ASHP at high power.

Spreadsheet Calculation

To calculate the energy savings I wrote a spreadsheet that simulates the way energy flows into and out of a house over a 24 hour period.

I modelled 4 separate ‘modes’ of heating the house.

• The steady state: is where the temperature is stable and the input heating power immediately flows out of the house.
• Off: is the condition on entering the setback period where there is initially no heating and the temperature falls as the house loses heat ‘naturally’. The rate at which the house cools depends upon the heat transfer coefficient and the heat capacity of the house.
• Setback: is where the temperature is stable at a lower temperature than in the steady state and the input heating power immediately flows out of the house.
• Boost: is where the heating power is increased to rapidly raise the temperature of the house.  The rapidity with which the house cools depends upon the heat transfer coefficient and the heat capacity of the house.

Click on image for larger version. Illustration of the temperature variation during a setback period. The dotted orange line shows the data for the situation where the house is maintained at the same temperature 24/7 and the red line shows data for the modelled ‘setback’.

I then divided the day into 0.1 hour periods and calculated (a) the heating energy and (b) the fuel consumed, in each of these periods. I then summed up the heating power and fuel consumed taking account of the possibility that the efficiency might be different in each phase.

Click on image for larger version. Illustration of the cumulative use of heat energy through a setback period. The dotted orange line shows the data for the situation where the house is maintained at the same temperature 24/7 and the red line shows data for the modelled ‘setback’.

Unfortunately there are a large number of variables that can be – well – varied: specifically

• Internal Steady State Temperature
• External Temperature
• Setback Temperature
• Length of setback period
• Length of boost period/Boost power
• Heat Transfer Coefficient/Thermal Resistance
• House Heat Capacity/Time constant
• Heating Efficiency in each stage

If you want to play with the spreadsheet you can download the Excel™  spreadsheet here:

• Setback Modelling – for sharing,

I must warn you it is an experimental spreadsheet and I can give no guarantees that it is error-free. In fact I can almost guarantee it is error-strewn!

It’s all about the boost!

The reason that the balance of benefits in a ‘setback’ is nuanced is because of the use of ‘boost’ power.

Without boost power, or with only a low power boost, the internal temperature will only return to its set temperature slowly.

For example, in the illustration below, the boost power is 4 kW while the steady state power is 3 kW. Even with this extra 1 kW, the boost takes several hours to bring the temperature back to the set point.

Click on image for larger version. When the boost power is low, it takes a long time to return the house to its set temperature. The model settings are shown at the top. The upper graph shows temperature versus time and the lower graph shows the cumulative energy used during the day. On each of the graphs the dotted orange line shows the data for the situation where the house is maintained at the same temperature 24/7 and the red line shows data for the modelled ‘setback’.

Obviously this is isn’t satisfactory because the internal temperature remains well below the set point for hours after it should have stabilised. However the savings (12%) compared with maintaining the steady state continuously are substantial!

If we increase the boost power to 6 kW, then the internal temperature returns relatively rapidly to the set point, and the savings are still a reasonable 9% compared with maintaining the steady state.

Click on image for larger version. Increasing the boost power reduces the time to return the house to its set temperature. The model settings are shown at the top. The upper graph shows temperature versus time and the lower graph shows the cumulative energy used during the day. On each of the graphs the dotted orange line shows the data for the situation where the house is maintained at the same temperature 24/7 and the red line shows data for the modelled ‘setback’.

However, if the heating in the boost phase is less efficient than the heating in the steady state, then these energy savings can easily disappear. For example, in the situation below, the boost heating efficiency is reduced from 90% to 80%. This has exactly the same thermal behaviour as the example above, but would now use 3% more fuel.

Click on image for larger version. If the boost heating is 10% less efficient than the steady state heating, then the energy savings can be wiped out. The model settings are shown at the top. The upper graph shows temperature versus time and the lower graph shows the cumulative energy used during the day. On each of the graphs the dotted orange line shows the data for the situation where the house is maintained at the same temperature 24/7 and the red line shows data for the modelled ‘setback’.

The examples above might be appropriate to the behaviour of gas boilers which sometimes condense water vapour in their exhaust fumes less effectively when operating at higher power.

But exactly the same principle could also apply with a heat pump. To pump higher thermal power it might be necessary to raise the temperature of the water flowing in the radiators.

The example below has the same power settings as the examples above, but now assumes an efficiency of 300% (i.e. COP = 3) for all heating phases. You can see that overall energy consumed is 3 times lower than in the examples above.

Click on image for larger version. If the heating efficiency is the same for all heating phases, then the energy savings are the same whether that efficiency is 90% or 300%. The model settings are shown at the top. The upper graph shows temperature versus time and the lower graph shows the cumulative energy used during the day. On each of the graphs the dotted orange line shows the data for the situation where the house is maintained at the same temperature 24/7 and the red line shows data for the modelled ‘setback’.

But if the efficiency in the boost phase falls to 250% from 300%, then once again, the savings are reversed and the setback strategy actually costs more than the steady state strategy.

Click on image for larger version. If the heating efficiency in the boost phase is 250% rather than 300%, then the setback strategy costs more energy. The model settings are shown at the top. The upper graph shows temperature versus time and the lower graph shows the cumulative energy used during the day. On each of the graphs the dotted orange line shows the data for the situation where the house is maintained at the same temperature 24/7 and the red line shows data for the modelled ‘setback’.

Conclusions

Developing this simulation has helped me understand some of the basic physics behind the use of setback strategies. And so my advice has developed from “Well, it depends…” to “Well, it depends, but it’s marginal, either way”.

The critical factor is the relative efficiency of heating at higher power (boost) compared with heating at lower power (steady state). If the boost heating is even marginally less efficient than the steady state heating then any energy savings are reduced, and may even be reversed.

This begs the question of whether there is any way to know what is the situation in a particular household. And my first thought is, “No“. Without detailed measurements, there is no way to tell!

 

Estimating the heat capacity of my house

Friends, the spell of cold weather at the start of December 2022 has led to me breathlessly examining data on the thermal performance of the heat pump and the house.

During this period, outside temperatures fell as low as -5 °C and average daily temperatures were below 0 °C. In order to try to keep the internal temperature constant, I studied measurements of internal temperature taken every 2 minutes. The data were pretty stable, only rarely falling outside the bound of 19.5 °C ± 0.5 °C.

But looking in detail, I noticed a curious pattern.

Click on image for a larger version. Two graphs from the period 6th to 18th December 2022. The upper graph shows the air temperature in the middle of the house. At around 01:30 each night the temperature fell sharply. The lower graph shows the rate of change of the air temperature versus time (°C/hour). From this graph it is clear that the rate at which the temperature fell was approximately -0.95 °C/hour.

The upper graph shows sharp falls in temperature at 01:30 each night. These were caused by the heat pump switching to its hot water heating cycle. Prior to this, the heat flowing into the house from the heat pump was more-or-less balanced by the heat flowing out. But when the heat pump switches to heating the domestic hot water, there was no heating from the heat pump and the internal temperature fell.

The lower graph shows the rate of change of the air temperature (°C/hour) versus time over the same period. From this graph it is clear that the rate at which the air temperature fell during the domestic hot water cycles was approximately 0.95 °C/hour.

With a little mathematical analysis (which you can read here if you care) this cooling rate can be combined with knowledge of the heat transfer coefficient (which I estimated a couple of weeks ago) to give estimates of (a) the time constant for the house to cool and (b) the effective heat capacity of the house.

Analysis: Time Constant 

The time constant for the house, is the time for the temperature difference between the inside and outside of the house to fall to ~37% of its initial value after the heating is removed.

The time constant is estimated as (the initial temperature difference) divided by (the initial cooling rate). In this case  the initial temperature difference was typically ~20 °C and the initial cooling rate was 0.95 °C/hour, so the time constant of the house is roughly 21 hours. Sometimes it’s useful to express this in seconds: i.e. 21 x 3,600 = 75,600 seconds.

This suggests that if we switched off all the heating when the house was at 20 °C and the external temperature was 0 °C, the house would cool to roughly 7.4 °C after 21 hours. Intuitively this seems right, but for obvious reasons, I don’t want to actually do this experiment!

Note that this time constant is a characteristic of the house and does not vary with internal or external temperature.

Analysis: Thermal Resistance  

A couple of weeks ago I posted an analysis of the heating power required to heat our house as the ‘temperature demand’  increased as the external temperature fell. The summary graph is shown below.

Click on graph for a larger version. Graph of average heating power (in kW) versus temperature demand (°C) for the first 10 days of December 2022.

From this I concluded that Heat Transfer Coefficient (HTC) for the house was around 165 W/°C.

The inverse of the HTC is known as the thermal resistance that connects the inside of the house to the external environment. So the thermal resistance for the house is ~ 1/165 = 0.00606 °C/W.

Analysis: Heat Capacity  

A general feature of simple thermal analyses is that the time constant, thermal resistance and heat capacity are connected by the formula:

Time constant = Thermal resistance x Heat Capacity

Since we have estimates for the time constant (75,600 s) and the thermal resistance (0.00606 °C/W) we can this estimate the heat capacity of the house as 12,474,000 joules per °C.

This extremely large number is difficult to comprehend, but if we change to units more appropriate for building physics we can express the heat capacity as 3.5 kWh/°C. In other words, if the house were perfectly insulated, it would take 3.5 kWh of heat to raise its temperature by 1 °C.

We can check whether the number makes sense by noticing that the main mass of the house is the bricks from which it is built. A single brick weighs ~3 kg and has a heat capacity of ~2,400 J/°C. So thermally it looks like my house consists of 12,474,000/2,400 ~ 5,200 bricks.

However this estimate is too small. Even considering just the 133 square metres of external walls, if these have the equivalent of 120 bricks per square metre that would come to ~16,000 bricks.

So I think this heat capacity estimate just applies the heat capacity of the first internal parts of the house to cool. This refers to all the surfaces in contact with the air. So I think this is the effective heat capacity for cooling just a degree or two below ambient.

Why did I bother with this? The ‘Setback’ Problem

Friends, sometimes I go upstairs and forget why I went. And sometimes I start analysing things and can’t remember why I started! Fortunately, in this case, I had a really good reason for wanting to know the effective heat capacity of my house.

When I analysed the heat flows previously, I have had to assume that the temperature of the house was stable i.e. that there was a balance between the heat flowing in and the heat flowing out. As long as the temperature of the fabric of the house is stable, then it is neither storing or releasing heat.

However this isn’t enough if we want to understand some very common problems in the thermal physics of houses, such as “the setback problem”: this is the question of whether it’s smart to reduce the temperature of a dwelling (say) overnight and then to re-heat it once again in the morning. To answer this question we need to know about the rate at which a house cools down (it’s time constant) which is equivalent to knowing its heat capacity.

And that is why I have done this prolonged and tedious analysis. The next article will be an analysis of ‘The Setback Problem’. And it will be much more exciting!

Assessing Powerwall battery degradation

Click on image for a larger version. Three screenshots from my phone showing the performance of the battery on 9th and 17th January 2022, and 11th December 2022. The key data concerns the total amount of energy discharged from the Powerwall. See text for details.

Friends, the Tesla Powerwall2 battery that we installed in March 2021 has transformed the way we use electricity and allowed us to go off-grid for prolonged periods each year. I have no regrets.

But lurking at the back of my mind, is the question of battery degradation.

This phenomena arises due to parasitic chemical reactions that occur as the battery approaches either full charge or full discharge. These reactions ‘capture’ some lithium and remove its ability to be used to store charge. Hence one expects the capacity of a battery to decline with extended use, particular near the extremes of battery capacity.

This particularly affects batteries used for domestic applications as they are often charged fully and then discharged fully – particularly in the winter.

The extent of the degradation depends on the specific chemistry of the battery. More modern battery chemistries labelled as ‘LiFePO4: Lithium Iron Phosphate” perform better than the previous best in class so-called “NMC: Nickel Manganese Cobalt”. Unfortunately, the Powerwall2 uses NMC batteries. This article has a comparison of the properties of different lithium-ion battery chemistries.

Battery degradation is a real phenomenon, but unsurprisingly, battery manufacturers do not make it straightforward to spot. I first looked at this about a year ago, but I don’t think my analysis was very sensible.

I now think I have a better method to spot degradation, and 20 months after installation, initial degradation is apparent.

Method. 

The new method looks at data from winter days during which the battery is discharged from full to empty, with little or no solar ‘top up’.

In winter our strategy is to charge up the battery with cheap electricity (currently 7.5p/kWh) between 00:30 and 04:30 and to run the house from this until the battery is empty. When it’s cold and the heat pump is working hard we can use up to 30 kWh/day and so the nominal 13.5 kWh of stored electricity is not enough to run through the day. So we run out of battery typically in the early evening and then run off full-price electricity until we can top up again.

The run-time can be extended by a top-up from the solar PV system, which can be anything from 0 kWh in overcast conditions, up to around 7 kWh in full December sun.

My idea is to measure the Powerwall’s total discharge and to compensate for any solar top up. By restricting measurements to days when the battery goes from full to empty, I don’t have to rely on estimates of battery remaining capacity. These days mainly occur in December and January.

For example, today (12 December 2022), the battery was charged to 100% at 04:30 and discharged 12.8 kWh to give 0% just after midday. I the estimate battery capacity as 12.8 kWh.

But on 7 December 2022, the battery was charged to 100% at 04:30 and discharged 15.5 kWh to give 0% just 22:00. This was a sunny day and the battery was topped up by 3.0 kWh of solar. I thus estimate battery capacity as 15.5-3.0 = 12.5 kWh.

In this latter case the way to compensate for the 3.0 kWh of charging is not clear. Why? Because the 3 kW of solar is used to charge the battery and so this may be done with say 95% efficiency (say) in which case only 2.85 kWh of solar energy would be stored. So there is some ambiguity in data which is solar compensated, but for this analysis I am ignoring this difficulty.

The data are shown below:

Click on image for a larger version. Graph showing total Powerwall discharge after compensating for any solar top-up. See text for details.

Discussion. 

The nominal capacity of the Powerwall2 is 13.5 kWh. This is – presumably – the stored electrical energy of the batteries when they are fully charged. To be useful, this energy must be discharged and converted to AC power, and this cannot be done with 100% efficiency.

Considering the data from the winter of 2021/22, the average Full-to-Empty discharge was 13.1 kWh, and so it looks like the discharge losses were around 3%. I think this is probably a fair estimate for the performance of a new battery.

The data show a considerable amount of scatter: the standard deviation is around 0.2 kWh. I am not sure why this is. Last winter, the battery would sometimes only charge to 99% rather than 100% and I corrected for this. That is why the capacity data do not lie entirely on exact tenths of a kWh.

Considering the data from the winter of 2022/23, the average Full-to-Empty discharge is currently 12.8 kWh. This represents a reduction in capacity of 2.3% (0.3 kWh) compared with last winter. However, there is a whole winter ahead with another 50 or so full discharges before spring and that average could well fall.

If the trend continued then battery capacity would fall to 10 kWh in around 2030. That would still be a useful size battery, and by that time hopefully a newer (and cheaper!) model will be available.

I’ll be keeping an eye on this and will write an update at the end of the winter season. But I thought it was worth publishing this now in case fellow battery owners wanted to monitor their own batteries in a similar way.

 

Cold Weather Measurements of Heat Transfer Coefficient

Friends, it’s winter and the weather is reassuringly cold: average daily temperatures in Teddington are around 0 °C. And as I wrote the other week, that offers the opportunity to make measurements of the Heat Transfer Coefficient of a dwelling.

People with Gas Boilers

This is especially valuable for people with gas boilers who are thinking about getting a heat pump.

When the outside temperature is around 0 °C, the average heating power required to heat the majority of UK homes is typically in the range 5 kW to 10 kW.

Most gas boilers have a full power of 20 kW to 30 kW and so can heat a home easily. To keep the temperature just right, the boilers cycle on and off to reduce their average output to the required level. For most houses there is no possibility that a boiler will be undersized.

Heat pumps operate differently. They are typically less powerful than boilers and the maximum heat pump output must be chosen to match the maximum heat requirement of the house.

By measuring the daily use of gas (kWh) by a boiler on a cold day one can estimate the size of heat pump required to heat the dwelling to an equivalent temperature.

I wrote about this at great length here, but at its simplest one just takes the amount of gas used on a very cold day (say 150 kilowatt hours) and divides by 24 (hours) to give the required heat pump power in kilowatts (150/24 = 6.3 kW).

People with Heat Pumps

But the cold weather is not just for people with gas boilers: Heat pump custodians and people heating their house electrically can gain insights when it’s cold.

Starting on 1st December I looked up:

• the average daily temperature;
• the daily heat output from the heat pump (in kWh);
• the daily electricity consumption of the heat pump (in kWh);

The internal temperature was a pretty stable 20 °C throughout this period. So I first worked out the so-called temperature demand: that’s the difference between the desired internal temperature and the actual external temperature.

I then plotted the daily heat output from the heat pump (in kWh) versus the average daily temperature demand (°C). The data fell on a plausible straight line as one might expect. Why? Because the colder it is outside, the faster heat flows out through the fabric of the dwelling, and the greater the rate at which one must supply heat to keep the temperature constant.

In the graph below I have re-plotted this  but instead of using the average daily heat output from the heat pump (in kWh) I have divided this by 24 to give the average daily heat pump power in kilowatts.

Click on graph for a larger version. Graph of average heating power (in kW) versus temperature demand (°C) for the first 10 days of December 2022. Notice that the line of best fit does not go through the origin.

The maximum heat output from the 5 kW Vaillant Arotherm plus heat pump varies with the external temperature, but for flow temperatures of up to 45 °C, it exceeds 5.6 kW.

The maximum daily average power for the first 10 days of December is just over 3 kW, so I think the heat pump will cope well in even colder weather. Indeed I could probably have got away the next model down. But it does seem to be a general rule of heat pumps that one ends up with the model one size above the size one actually needs.

Click on image for a larger version. Specifications for the Vaillant Arotherm plus heat pump. For the 5 kW model at an external temperature of -5 °C and heating water for radiators to between 40 °C and 45 °C, the maximum output is between 5.6 kW and 6 kW.

The slope and intercept of the graph

The slope of the graph is approximately 0.166 kW/°C or 166 W/°C. This figure is known as the Heat Transfer Coefficient for a dwelling. It is the figure which characterises the so-called fabric efficiency of a dwelling.

However, as I noted many years ago when I looked at this problem using gas boiler measurements, the straight line does not go through the origin. The best-fit line suggests zero power output when the external temperature is 2.8 °C below the internal temperature.

This would imply that the heat flow through the fabric of the building was not proportional to the difference between the inside and outside temperature.

The reason for this is that there are other sources of heating in the house, and not all the heat pump output goes into the house. Specifically:

• People: each person heats the house with around 100 W, about 2.4 kWh/day.
• Electrical Items: All the electricity consumed by items in the house ends up as heat. For my home this amounts to around 10 kWh/day.
• Hot Water: Heat pump output that heats domestic hot water is mostly lost when the hot water is used. My guess for this house is that this amounts to around 3 kWh.day.

So to estimate the actual amount of heat dissipated in the house I should really take the heat pump output and:

• Add 2.4 kWh/day for each person in the house
• Add 10 kWh/day for all the electrical items
• Subtract 3 kWh/day for the hot water lost.

Together this amounts to adding 9.4 kWh/day to each heating estimate. Pleasingly, plotting the same graph with these corrections, the graph now intercepts within 0.5 °C of the origin. To me this indicates that I am now accounting for all the significant sources of heat within the house reasonably well.

Click on graph for a larger version. Graph of average heating power (in kW) versus temperature demand (°C) for the first 10 days of December 2022.

I haven’t included any solar gain in these estimates because at this time of year solar gain is generally very low unless a home has large south facing windows. Previously I have noted that solar gain seemed to be much more important in spring and autumn with longer and generally sunnier days.

COP

I also took the opportunity to evaluate how the daily averaged coefficient of performance (COP) varies with external temperature. This is based on readings from a heat meter and an electricity meter which monitors the heat pump.

The COP values below 3 are a little bit lower than I would like, but still acceptable. On subsequent cold days I will be seeing if there are adjustments I can make to the heat pump operation which can improve this.

Click on graph for a larger version. Graph of daily average COP versus daily average outside temperature (°C) for the first 10 days of December 2022. Extrapolating the trend suggests that the COP would reach unity at a temperature of – 12 °C.

Out of curiosity, I also evaluated the heat output using the Vaillant SensoApp. The figures were massively in error. For example, on 10th December the app suggested the total heat delivered to the house by the heat pump was 58 kWh. In fact the correct answer was 73.9 kWh.

Conclusion

Cold weather offers an opportunity to assess the so-called Fabric Efficiency of a dwelling by direct measurements of its Heat Transfer Coefficient.

The cold weather will be with us for a few more days so there’s still a chance to make measurements in your dwelling.

Annual variation in location of sunrise and sunset

Friends, we are just 16 days away from the winter solstice, and those of us who live in the UK are dismally aware that at this time of year, the Sun only rises above the horizon for about 8 hours.

And not only does the Sun rise above the horizon for a short time, it also fails to rise very high in the sky.

Together these factors make for poor yields from solar panels.

This article is about a third factor which I feel is sometimes overlooked: the location of sunrise and sunset.

The location of sunrise#1

I asked OpenAI Chatbot about this:

How does the location of sunrise vary through the year in the UK?

The location of sunrise in the UK varies significantly throughout the year. In the winter months, sunrise occurs in the southeast, while in the summer months, it appears in the northeast. The amount of daylight hours in the UK also increases as the year progresses, with the summer months having the longest days and the winter months having the shortest.

This is an excellent answer. Accurate and concise. I followed up:

How far north of east does the sun rise at the summer solstice?

At the summer solstice (the longest day of the year), the sun rises as far north as 70° north latitude in the UK, which is around halfway between East Anglia and the Shetland Islands.

In contrast, this answer is utter nonsense! So I guess I will have to write this article myself!

The location of sunrise#2

I was interested in the location of sunrise because of the new panels I am installing will face about 22° north of east – not a very favourable location.

I looked up data for each week of the year from The Time and Date website: the data below are relevant to London, but you can look up data for many other locations worldwide if you are interested.

Click on image for a larger version. This is an extract from tables at the Time and Date web site. It has both the time of sunrise and sunset and the angle of sunrise and sunset measured clockwise from due North.

I then collated the results and plotted them through the weeks of the year.

Click on image for a larger version. This graph shows how the location of sunrise and sunset vary through the year. Angles of sunrise and sunset measured clockwise from due North.

The graph above shows that the phrase: “the Sun rises in the East and sets in the West” is only approximately true. For 6 months of the year, the Sun rises north of East and sets north of West.

My New Solar Panels
This is probably not news to anyone, but I found it interesting, because I am putting solar panels on my home that face north of East.

Click on image for a larger version. Google Maps view of my house showing existing solar panels in blue and the new panels in Yellow. For 6 months of the year between spring and autumn equinoxes, the panels should produce a useful solar yield in the morning.

After plotting these lines on the map, and noting which houses the lines intercepted, I was able to translate them onto a photograph to show the expected location of sunrise through the year.

Click on image for a larger version. Photograph showing the ‘panels-eye’ view of the street-scene at the back of the house For 6 months of the year between spring and autumn equinoxes, the panels should produce a useful solar yield in the morning.

Considering all the panels on the house,- including the 12 installed in November 2020 – in summer the system should generate from early dawn – only just after 4 a.m. in mid-summer, to almost 8:00 p.m. So despite the poor orientation, the Easy-PV calculator suggests the 5 panels will generate 1,338 kWh per year (268 kWh/panel) compared with 3,860 kWh year from the original 12 panels (322 kWh/panel).

Click on image for a larger version. Charts showing the angular extent of daytime through the year. The orientation of the three sets of panels on the different roofs is shown as red arrowed lines.

Along with the 5 panels on the roof, I have installed three panels on the flat roof which are only at 12° to the horizontal. The Easy-PV calculator suggests these 3 panels will generate 919 kWh per year (306 kWh/panel), although I am not sure I properly accounted for shading.

Click on image for a larger version. The same photograph as above but now showing the panels on the flat roof.

Summary
Sadly, although the panels have been installed for more than month, no inverter has been installed and they have not been connected to the grid. Apparently, this will happen “tomorrow”.

But if the output is as I anticipate, then next year the system will generate around 6 MWh. The amount we draw from the grid should be slightly reduced as I hope we will be off-grid for 6 months rather than 4.5 months year. So considered over a year, cumulative generation should be roughly twice as much as we draw from the grid.

Consequently – considered over a year – we should export almost as much as we import, which is getting close to one definition of carbon neutrality. This is my dream!

Click on image for a larger version. Cumulative PV generation for 2022 is just under 4 MWh, in line with the MCS guidance when the system was installed. Cumulative Grid Consumption is expected to be just over 3 MWh this year. The dotted purple line shows anticipated generation next year.