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It is to decide between two possibilities for heating of rooms during the absence in the building:

  • heating constantly to 20 °C (68 °F) or
  • heating 1 hour before coming home with 25 °C (77 °F) with the same effect that the room temperature is 20 °C (68 °F) at 5 p.m.

The following conditions has the example:
No people are at home 9 hours in the time between 8 a.m. and 5 p.m.
The outside temperature is 0 °C (32 °F).
The normal inside temperature shall be 22 °C (72 °F).
The building has a medium insulation, so that the room temperature decrease from 22 °C (72 °F) to 18 °C (65 °F) after 4 hours.

Which method of heating is the best to achieve low energy costs?

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  • What type of heating system? Natural gas forced air? Electric forced air? Electric baseboard? Boiler/radiators? Not sure where you get "with 25 C (77 F)" - most heating systems producemuch hotter temperatures (whether forced air or baseboard or radiator) - generally producing the same temperature output no matter what the desired ambient temperature. – manassehkatz-Moving 2 Codidact Jan 1 '20 at 15:09
  • In the specific case are installed hot water radiators supplied by gas boiler. 25 C (77 F) is the set example maximum temperature, which is not reached of 1 hour - setting of any other high temperature has the same effect. – gotwo Jan 1 '20 at 16:01
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The engineering theory is that a greater differential in temperature causes a faster loss of energy, so maintaining a constant temperature is more costly.

This theory was proven by tests of Twin Homes by The Canadian Centre for Housing Technology. Several papers have been written about studies at these homes, here is a rather thorough one https://www.healthyheating.com/downloads/Thermostats/nrcc48361.pdf

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  • It is a interesting practice study. The interesting main result for that is an energy saving of 13% for set back temperature from 22°C (72°F) to 16°C (61°F) in nighttime but also daytime, in total over 14 hours (23:00-6:00, 9:00-16:00). Of course some special conditions of the building, location and so on are to be considered. – gotwo Jan 5 '20 at 12:26
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Keeping an unoccupied building heated or cooled is always wasteful, unless you're up against some sort of environmental boogeyman like pipe freeze or condensation. The idea that "it costs more to cool it down/reboot/etc." is pure bunk.

  • It does not take a half gallon of fuel to start an engine.
  • PC's do not surge 5000 amps during the 30 seconds of startup.
  • Nor do fluorescent lights, though there's a nugget of truth here: every startup ages the tube slightly (The $2 tube. You can greatly improve this by spending $4 premium on a programmed-start ballast, or just go LED.)
  • A/C runs continuously for awhile when you start it back up, but the savings from when it was not running is huge.
  • Ditto heat.

There is nothing magical about "1 hour". The house should reach target temperature the moment you walk in the door, and not one minute sooner. If, based on conditions, it will take 40 minutes to do that, then it should start 0:40 before. If it'll take 1:30, start 1:30 before. Smart 'stats like the Nest are good at this; they base it on learned experience and indoor/outdoor ambient temperature differences. "But wait, I don't remember fitting and cabling up an exterior thermostat on my Nest!" No, the Nest got the outside temp by saying "Hey Google, I'm at zip 60061. What's my outdoor temp?" That's why a Nest needs the Internet.

On a thermostat, never overshoot the temperature and come back down. What you're trying to do is kick the furnace into overload to make it get hot faster. Furnaces don't work like that; they only have "on" or "off", 0% or 100% (a few have intermediate settings, but all furnaces understand when the temp is -5 of target, go 100%.

The only thing you could possibly accomplish is, on a heat pump system, get the system to run the primary and emergency heat at the same time, and that is uneconomical. The primary heat will get there in a reasonable time.

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    Thank you. As is often the case, your answer included some of what I was trying to get at in comments (but which I didn't make an answer as I really wasn't sure how to state it clearly). I hadn't thought about the Nest knowing outside temp that way - but it makes perfect sense - no different from the temp on my cell phone. – manassehkatz-Moving 2 Codidact Jan 1 '20 at 18:16
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I've been involved with many efficiency studies about this during my years with the power company. All of them came up with the same results, more efficient to start heating/cooling one hour before coming home. In all cases, the units ran more when set to heat/cool all day. It was even more efficient if the first one home turned on the unit. This wasn't the results the power company wanted because it caused peak generating demands in the evenings.

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  • Can you explain what you mean with "it was even more efficient if the first one home turned on the unit."? Why is it for a (my) house efficient turned on the unit e.g. 12 a.m. at first of all other houses? – gotwo Jan 5 '20 at 12:28
  • @gotwo I was basing it on normal work schedules. Heating/air conditioning empty space is inefficient. Setting thermostat to go on one hour before anyone gets home is more efficient than having unit run all day but less efficient than just turning on the unit when the first person gets home. – JACK Jan 5 '20 at 14:16
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The amount of time it takes to raise the temperature depends not only on the inside and outside temepratures, but also the wind, rain or snow, even humidity makes a difference.

I would not overcomplicate this. I would just set the programmable thermostat to say 60* while the house is empty, and raise it to 72* an hour before you get home. The heat will run full bore for an hour which should be enough to get it reasonably comfortable. Experiment from there.

Setting the thermostat higher does not make sense. When the thermostat calls for heat, the heat runs full bore until the desired temperature is reached, then it stops. (It can be a little more complicated with two stage etc. but that's the basic idea.)

So if it's not up to 72 (or close) when you get home, and you're uncomfortable, switch over a little earlier.

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