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.|
|Figure 4. Our electricity usage averaged over weeks.|
|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,