Good old physics tells us that it is impossible to not have to put extra energy in to get the temperature up without putting extra energy in. I havent seen anything in your numbers to indicate that the cop is significantly better during the recovery period, so I don’t see any gain there. Especially as the outside temperature from 22-4am is about the same as in the time period after that (and often lower). 

As you say that your house is badly insulated, so it must be the thermal mass that leads to the slow loss of inside temperature and hence a low change in the inside outside delta. 

if we knew your heat loss we can calculate what the one or two degree heat loss represents in terms of energy and calculate what that means. For example, assuming your heat loss is 7kw at deltat 22K:

7kW/22K=0.318 kW/K heat loss per degree.

so in the extreme case of iat dropping by 2 degrees over the six hours, so the average temperature will be 1 degree lower and have 1/22 less energy over that time that you would have if the heating had been on. But now you have to put that energy back in and by definition it’s the same amount of energy as you had lost. It looks like the time to recover varies somewhat, but even when the air temperature has recovered that building mass will still be recovering, so the iat does not represent the full picture of the recovery. Let’s say it takes 4 hours. 
if oat was 5C during that time your heat loss is (19-5)/22*7=4.455 and would have been (20-5)/22*7=4.773 kw if the heating had been on. Your saving during that period has hence been (4.773-4.455)*6=1.908kwh. At cop 2 that is 1.908/3=0.636

so you didn’t heat the house for 6 hours to 1K below 20, so you didn’t put in 4.455kw*6=26.73 kW and will have to put that back in to heat the house up. if there was a difference of cop during these periods you could hence save additionally (let’s say cop 3 vs cop 4):

26.73kwh/3-26.73kwh/4=2.228 kWh 

so your maximum electricity saving is less than 3kwh out of 24h*4.455kw/3.7=28.897 kWh. 
the problem is that you could only get the improved cop if you switched it back on much later when the oat has improved, I..e. Middayish. 

I suspect your curve fitting is to the data that includes the cutback and recovery and hence expects the usage during recovery. 

Posted by: @cathoderay

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But the odd things is, there is no obvious extra energy being put back in during the recovery period. Yet the recovery certainly happens, and in good time. There is no visible extra use during the recovery period to be offset against the undoubted savings during the setback, so in effect i get the full benefit of the setback.

@jamespa — thoughts? Or anyone else?

I apologise that I have been silent on this for a while, my mind has been elsewhere, not least fighting Vodafone (long story, in essence they are obstructing my desire (and right) for landline number portability and giving me the run around like there was no tomorrow).

I haven't yet caught up with all the above but from a quick read there seems to be some further progress.  However my general feeling about this topic remains as it was and can be reasonably be summed up in:

• Unless the house cools significantly, the energy lost from the house cant reduce significantly during a setback relative to the no setback case.  All energy lost from the house must be replaced to restore the house to its starting condition. 

• If a house cools by only 2-3 degrees for a few hours, the percentage reduction in energy lost (and thus the reduction in energy we need to supply to the house) is very small.  For example if it cools by a maximum of 3 degrees for 8 hours in 24, then the saving is very roughly 0.5*3/20*8/24, = 2.5%

Posted by: @adrian

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• Good old physics tells us that it is impossible to not have to put extra energy in to get the temperature up without putting extra energy in.

• Until we can explain observations, however accurately recorded, or have sufficient robust statistical evidence, we cannot infer anything at all about how observations might apply to circumstances other than the specific ones in which they were made

I will try to get back to this once my telephone problems are solved (or when I have some time to spare where I am not in the place to solve them)!

There are some rather weird page layout things happening at the moment which I tried to fix by checking the underlying html (hence the notes saying I edited them attached to the above two comments, I didn't change any of the wording, just looked at the html). Putting that aside, here are my thoughts.

@robs — thanks for your thoughts. Solar gain (in daylight hours, not during the setback period and initial recovery) and wind chill may have had an effect on IAT, and defrosts certainly did (and on COP in the latter case), they can be seen in the chart (LWT below RWT) but what we are primarily looking at here is energy use, observed when it is known, and expected (predicted) when it is not known, eg what it would have been had I not had a setback. Observed is observed, and unless we are going to say my monitoring system is up the spout, which I don't think it is, then it is what it is. Yes. we should look very hard at whether the measurements are wrong, but I don't think they are, or if they are, they are only out by a small and not significant amount. The reason I think they are OK is that I have independent ways of checking the two key measurements, OAT and energy, a separate temperature data logger for the former, and an independent kWh meter for the latter. The OAT is very much a local OAT, because it is affected by the exhaust air coming from the heat pump, and so is more accurately the air intake temperature (AIT, the sensor is inside outdoor unit, on the air intake side), and it is typically a degree or two below the wider OAT, but we know that, and much more importantly the AIT is the primary, and primary by a long way, determinant of the set LWT temperature, and so ultimately the energy in. Never mind the R and R squared values, we can see that in this chart ( a repost added here for reference), there is a very tight correlation between the OAT/AIT and the energy in:

The scatter about the line is random variation which is to be expected, indeed I would be rather worried if it wasn't there. @adrian "I suspect your curve fitting is to the data that includes the cutback and recovery and hence expects the usage during recovery." No, this plot is only for data points when the heat pump was running, it includes recovery periods but not the setbacks. It is also space heating only, no DHW heating. 

What that chart tells me is that I can make valid predictions about the energy in, ie expected values, once I know the OAT/AIT. Indeed, when I do that (make predictions), the predicted (expected values) fall where they should. There is of course an air of self fulfilling prophesy about this, since I am in effect just reversing the mathematics, but that just underlines how robust this method is. It just works (the observed values on the X axis are setback hour values, zero energy in):

The predicted (expected) values are exactly where they should be. This model (because that is what it is, even if I don't use that term) works. Given an OAT/AIT, we can very accurately predict the energy in. Yes in the real world there will be some random variation, but when we come to sum the hourly values over a longer period (24 hours), the randomness means the variations will cancel themselves out.  

The essential point here is that the equation (shown in the upper chart above) is the best predictor we have of expected energy in values, given a known OAT/AIT. Behind the scenes, it takes into account all of the physics, both that which we know about, and that which we don't know about, and gives us a summary in that equation.   

"Also, you've assumed that once the IAT has recovered the house fabric has also returned to its pre-setback state, I'd suggest that this is not a safe assumption and the fabric will continue to re-absorb energy. As a result your calculated expected values include additional energy (thus inflated) as a result."

I am not sure about this. The IAT is what matters to the inhabitants. If the fabric is still re-absorbing energy despite a recovered IAT, and I agree it may be, the heat pump and it's control logic doesn't know this (it doesn't know what the IAT is, let alone the energy flux in the fabric), it only knows and only uses the OAT/AIT to set the set LWT and so in due course the energy in. Yes, it does also know the RWT, but, subject to being corrected, because this is a grey area that may be relevant, I don't think that is used in any way to set the LWT, which is merely derived from the WCC.

Furthermore, I think your logic may be the wrong way round. If the house is in reality somehow sucking in more energy during the recovery period, then my predicted/expected should be too low (ie deflated, rather than inflated) as a result. But in fact they are neither: the predictions match the observations.

@adrian — "Good old physics" - next you will be invoking "old Father time"! My contentions is that empirical results, once you have ruled out measurement error, always trump theoretical results. I don't doubt the laws of physics for one second, but I can and sometimes do doubt their application. If a theory (model) comes up with a different answer to the observed data, then there is something wrong with the model, and it need to be revised. That is the scientific method. Hypotheses, test, modify. My very simple model works, the predicted (expected) values are very close to the observed values when we have both. If we have another hypothetical model that incorporates other factors and somehow produces a different answer that doesn't fit the facts (observed values), then we need to look again at that hypothesis.   

"I havent seen anything in your numbers to indicate that the cop is significantly better during the recovery period, so I don’t see any gain there." It is visibly worse, unsurprisingly as the OAT is lower, and there are defrost cycles.

"if we knew your heat loss we can calculate what the one or two degree heat loss represents in terms of energy and calculate what that means." We do know what my heat loss is, a monitored heat pump makes this easy to measure. Given a steady IAT, the heat loss = the energy delivered to the building, ie energy out from the heat pump. So far this season, I have been running without a setback, and the IAT has been pretty steady for most of the time, with some milder weather followed by some colder nights:

I can plot hourly mean OAT against hourly energy out ie delivered to the house:

which suggests a heat loss of 9.5kWh per hour (9.5kW). The outliers? Not enough of them to matter.

"At cop 2 that is 1.908/3=0.636" — no doubt an innocent typo, but that should be "At cop 2 that is 1.908/2=0.954". But the real problem I see with your method is that it relies on energy out, and then assumptions about COP to derive energy in. That is two places where error can creep in, the assumptions about energy out, and the assumptions (let's face it they are guesstimates) about COP. My method on the other hand (a) relies on energy in which is more robust and is directly the number we are interested in and (b) makes no assumptions about COP, because it doesn't need to. Put prosaically, my method points a finger at the face of the moon, whereas your points at the dark side of the moon, and then comes to some conclusions about the light side of the moon.

That said, I think it might be possible to do a more robust calculation using your method. We do have all the data we need, IAT, OAT, the heat loss equation (above, which could be redone with inside/outside delta T on the X axis), to determine the heat loss, and we could estimate the COP at various OATs. We could then do an hour by hour calculation, and finally sum them over 24 hours. But do we need to do that? I'm not sure we do, but then again it might be interesting to do it. 

I also have some concern about an excessive focus on the energy balance in the fabric. What matters to the inhabitants is not that energy balance, but the IAT. Is it possible that while the IAT is OK, the house is still out of balance, and so in fact the conservation of energy (which I don't dispute) doesn't apply in the short term, but only in the (much) longer term? Yes, the house loses X amount of energy during the setback, and X ultimately needs to be put back in, but what about the short term. Maybe the fabric is forever playing catch up, while the IAT is OK for the inhabitants? My earlier question about the duration of setback is relevant here: longer setbacks (days or weeks) do mean real savings. At what point, and why, do the savings start to become more questionable?

Finally at the end of a rather long post, there is this chart:

from this post, which gives more detail and a statistical analysis, but in brief this uses a quasi-epidemiological method to attempt to answer the question: are setback and non-setback days both part of the same general population (ie no difference between them) or are they each part of their own independent population (ie they are fundamentally different). This plot (done in R) includes the 95% confidence intervals (grey shaded areas), and visually it strongly suggests two different populations ie they are fundamentally different. The statistical analysis (which may be open to challenge) concluded "The setback running as recorded in this data set appears to use about 20-23% less energy compared to no setback running."    

I must repeat again a truth universally acknowledged, that this is a single analysis in want of a singular answer. It only applies to my house in the conditions that applied during the time period under analysis.

Posted by: @jamespa

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not least fighting Vodafone (long story, in essence they are obstructing my desire (and right) for landline number portability and giving me the run around like there was no tomorrow).

If it's any comfort (of the you are not alone variety) I have a very similar long running problem with Plusnet. It's like arguing with a brick wall.  

BTB the page layout problem with this forum page persists at least for me, but I can now at least select text and have it placed in a quote in the comment I am writing. For my last post that didn't happen, so I had to manually "put quotes in quotes".