Saturday, November 7, 2015

Home heat pump efficacy

Our house is electrically heated, which is a bad idea in the long run. Trying to save on electricity, we got a heat pump installed in September 2014. Now that it's been running for a year, I decided to take a look at some numbers.

Our electric utility company (Helen) provides a service from which I can download my usage data for each hour. I downloaded all the data that was available and started playing around with it in Octave.

Figure 1. Electricity usage over the observed period.

Figure 2. Correlation between outside temperature and electricity usage.

My plan was to first create a model of our electricity usage pre heat pump, using the outside temperature as the explaining variable. I would then look at how large the discrepancy between our recent electricity usage is compared to that model.

Looking at the correlation between outside temperature and electricity usage in Figure 2, it seems that there is a reasonably linear relationship whenever the temperature is below 10 degrees Celsius. As I am only really interested in using the heat pump as a heating device during the winter, I will hence fit the model to cold weather data only.

Figure 3. Our electricity usage averaged over days.

Our average daily electricity usage is shown in Figure 3. Our usage increases near midnight, but there is no sudden increase even though the water heater turns on every night. This is because the heater turns on at different times on different days of the week, as controlled by the utility company. Also, since our daily routines follow a weekly pattern, it perhaps makes more sense to look at average weekly usage.

Figure 4. Our electricity usage averaged over weeks.
In Figure 4, the spikes of the water heater turning on are clearly visible. It seems that the heater turns on at around 23:00 most days, but on Wednesday and Saturday at 21:00. Also our daytime electricity usage seems to be a bit higher on weekends than weekdays. Night time usage is about the same. Makes sense.

Figure 5. A second look at the electricity usage versus temperature. This is the electricity usage in the training data smoothed using the weekly average profile.

When the weekly variation is removed from the data, we get a smoothed version of electricity usage. Figure 5 shows the smoothed electricity usage versus temperature in the training data, i.e. the cold weather data from before the heat pump was installed. It seems to have a much reduced variance compared to the original data in Figure 2 and thus seems to allow a better first order polynomial approximation.

Figure 6. Electricity usage, model fit and model prediction. Model fit is how the model reproduces the training data, while prediction is what the model predicts the usage would be.

Figure 7. Model fit error and model prediction error in kW.

Figures 6 and 7 show the first order polynomial model fit and its prediction. Figure 7 in particular shows the difference between the model output and actual data (i.e. positive values means the model estimates greater electricity usage). The model prediction error in magenta seems to be ever so slightly positive, which would translate to us using just a bit less power than before (about 6% less). I was expecting a huge difference, so this comes as quite a disappointment.

There are, however, some caveats. The weather was quite warm last winter, so there wasn't that much need for heating. Also, a big thing is that we have gone from a 2 person household to a 4 person household within the data period. This probably means increased water consumption, which in turn increases electricity use.

Lets approach the problem from a different direction. Instead of looking at how much less electricity we use now compared to a model of pre heat pump usage, we can model both cases and compare these models.

Figure 8. Pre heat pump installation and post heat pump installation electricity usage (smoothed with weekly average profile).

Figure 8 shows the smoothed electricity usage versus the outside temperature. The blue data is the same that is shown in Figure 5, but the red data is measurements from post heat pump installation. It's not very clearly visible as last winter wasn't very cold, but it seems that the post heat pump installation data has a slightly shallower slope, which would indicate that heating the house uses less power than before.

Figure 9. Model fit to pre and post heat pump installation data.

Figure 9 shows the result of fitting first order polynomials in the data. The models are
  •  Pre heat pump installation
    • P = 2.94 kW - 0.121 kW/C * T
  • Post heat pump installation
    • P = 2.70 kW - 0.092 kW/C * T,
where P is the used electric power in kW and T is the outside temperature in Celsius. That is, the electric power required for heating per degree Celsius has reduced by about 24%. This makes me feel slightly less disappointed, but it's still nowhere near what I was expecting. However, the post heat pump installation data isn't very reliable as there is still so little of it available. Hopefully I'll have better data by spring time.

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