A few people I've talked to have suggested that running the heating continuously in a house might actually reduce the amount of oil burned. I was a bit sceptical so decided to do a little science.

Thinking simply, the longer a house is warm the longer it'll be losing heat and therefore the more heat will need to be replaced so obviously not letting the house cool down, say overnight or while you're out, will result in a higher heating bill.

But, I can think of a few reasons why this might not be completely true and how it might be possible for varying temperatures to result in higher consumption:

- If you have a modulating boiler, heating continuously might result in the boiler spending less time running at full pelt which might be more efficient.
- Heating continuously might result in lower return temperatures from the radiators so, if you have a condensing boiler, the boiler working in condensing mode more of the time.
- If the house is not up to temperature when that's wanted (first thing in the morning or on getting in) there's a temptation to turn the thermostat up and then, maybe, not turn it down soon enough resulting in the house being run at a higher temperature than is really needed for comfort.
- Perceived temperature is a function of both the air temperature and the radiant temperature of the surroundings (walls, floor, ceiling, …) - each counts about equally - so if the building fabric is cool the air needs to be about an equal amount warmer to compensate. In a well ventilated house without heat-recovery ventilation this will result in significantly higher ventilation heat losses. Continuous heating is likely to leave the building fabric a bit warmer.
- Allowing the house to cool down is likely to result in a certain amount of condensation in and on the building fabric. When it's next warmed up some of the energy goes towards evaporating this again. Thinking through the significance of this is left as an exercise for the reader or the future.

So, having established that my measurement of boiler run time is reasonable, I decided to try running the heating continuously for, roughly, alternate weeks in November and December. Actually I ran it continuously for the following inclusive runs of days:

2018-11-11 Sun | / | 2018-11-17 Sat |

2018-11-26 Mon | / | 2018-12-01 Sat |

2018-12-09 Sun | / | 2018-12-15 Sat |

Normal operation is for heating to be on from 07:00 to 23:00 with it being manually switched off when I go out, in which case it comes on again at 15:00 or 21:00 depending on whether I'm out in the day or the evening.

Switching from timed to continuous was done during the evening of the day before so the heating then ran for the last hour of that day and overnight. Switching from continuous to timed was done during the last day indicated so the last hour of the “continuous” days weren't actually heated.

Just comparing the boiler run times in different types of weeks is likely to be wildly misleading because of changes in the weather so I needed a figure to represent the heating requirement caused by the weather. In this leaky house wind speed is almost as important as outside temperature in determining that.

Using a tiny bit of physical intuition I plucked the following equation out of thin air to model the heating requirement:

heatreq = (b - t)·(1 + k·v^{p})

where:

heatreq is an arbitrarily scaled number proportional to the amount of heating required.

b is a base outside temperature below which heating is indicated.

t is the actual outside temperature (°C).

k is a constant indicating the relative importance of the wind speed.

v is the wind speed (m/s).

p is a power the wind speed is raised to indicate the ventilation air change rate.

Temperatures and wind speeds came from METARs reported for Wick airport 30 km to the NE and at a similar elevation and distance from the sea.

For some houses the amount of solar input would likely also be a significant factor. However, this is the more northerly half of a semi facing south east with relatively small windows so I assume this is less relevant here.

Determining b, k and p is a matter of least-squares regression. Given that I only ever had a tenuous notion of how the closed-form method of doing a least-squares linear regression works there didn't seem to be a lot of point even thinking about one for this equation. Instead, I did a successive approximation on the data for the whole of November and December resulting in:

b = | 21.3839 |

k = | 0.016284 |

p = | 1.693317 |

These numbers are not terribly stable, that is, they can continue to drift around a bit with successive iterations of the approximations. Or, to put it another way, the least-squares sum of the residuals is not overly sensitive to the exact values chosen.

I classified days as “continuous” or “timed”. Further, the “timed” days were classified whether I was in for most of the day or out for at least an hour or so (i.e., more than just a quick trip to my site to pick up post or whatever). For continuous heating days I didn't bother if I was in or out assuming it'd make no more than a tiny difference to the heating requirement.

Doing a scatter plot of the proportion of each day that the boiler was running against the heating requirement gave:

The day with the most extreme heating requirement, 43.98 thingummies, the purple cross just below the brown “continuous” regression line towards the top right with a boiler run proportion of 0.474 (i.e., the boiler running 47.4% of the day) was 2018-12-15 Saturday, the last day of continuous running, when it wasn't amazingly cold (showing as 5° or 6°C for the whole 24 hours) but was very windy, building from 12 to 21 m/s (26 to 47 mph) with gusting to half as much again.

Doing simple linear regressions of the boiler run proportion against the modelled heating requirement for each of the continuous and timed day types gives the brown and yellow lines respectively.

The brown line is above the yellow line so continuous heating does, as I really expected, use more oil.

But, noting that the y axis starts at 0.1, not zero, it's nothing like as much more as you might expect for the proportional extra number of hours of heating (24 instead of 16 so half as much again).