This is effectively what a thermostat does.

The problem is that the controller won’t know how well insulated each room is, how cold it is outside (including wind speed), which doors and windows are open and when, what people or devices are doing in each room.

The way thermostats solve this is by creating a closed loop where they react to how the room reacts to their actions.

Depending on how your heaters work you’ll likely need some dynamic component to react to these unforeseen changes unless you can live with the temperature being very unstable.

To get a rough idea of how long the heaters will have to run you can look at each room in for the last n days and see if the heater’s runtime was long enough to (on average) hold your target temperature. Dividing the average temperature with the target temperature will give you an idea whether they were on for too long or too short. (If the heaters have thermostats you’ll likely need to subtract a small amount from that value so that it will settle at the minimum required heating time)

If that value is close to 1.0 you know that on those days the heating time was just about perfect.

Once that is the case you can take the previous days heating time and divide it up over the cheapest hours. The smaller of a value n you choose the more reactive the system will be but it will also get a little more unstable. Depending on your house and climate this system described here might simply be unsuitable for you because it takes too long to react to changes.

There are many other ways to approach this very interesting problem. You could for example try to create a more accurate model incorporating weather and other data with machine learning. That way it could even do rudimentary forecasting.

I have had collegues that did entire PHD’s on this topic with giant simulations and machine learning… It’s just to say that this is a very difficult problem to solve exactly, so go easy on yourself and find a simple approximation or something good enough to get you the edge on the prices you pay for heating. I see some good ideas in the comments already, but keep it simple is my best advice.

Maybe you can check how versatile made it: https://github.com/jmcollin78/versatile_thermostat Works like a charme here

My thermostats compute this for me. This is all the ones I have on the ground floor (other ones aren’t in HA yet). It’s been a bit cold last night, around -14°C. Oh, and the chart is for yesterday.

“Cuisine” is a pretty large room of about 250-300 sq ft, and it’s in kind of an open space with “Salon”, hence why the latter doesn’t run very often. “Bureau” is my office, about 150 sq ft, and “SDB RDC” is a bathroom / laundry room, a bit less than 100 sq ft.

If you’re measuring the temperature in the room currently, you could try trending it yourself. Start the heater, and see how quickly the temperature rises (e.g., degrees per hour). Call this Rate 1.

Then turn off the heat and see how quickly the temperature drops. Call this Rate 2. For the formula below, make it a positive number.

Assuming the weather conditions are similar and the room temperature doesn’t change too much during data collection:

Rate of heat loss = Heater power * Rate 2 / (Rate 1 + Rate 2)

This number could be impacted by the weather: temperature, wind and insolation (affected by time of day, time of year, latitude, and cloud cover). It’s also impacted by room conditions (temperature, slade position, how many times the door is opened), so you’d need to do a few trials to get a sense for thr impact of different variables.

You’ve probably already thought of this, but your strategy is going to result in noticeable swings in temperature in the room, because ypure going to do a lot of heating at once when prices are cheap, then turn off the heating and let the room cool. Compare that to a thermostat that tries to maintain a constant temperature.

Sounds like a fun project - good luck! I’d love to hear updates here as you go.

I might be able to help you get an approximation.
Though quite honestly, you’d probably save time and effort by just using a thermostat in each room, and setting HA not to power the heat on when pricing is >75% day average, since you know ahead of time, and it’ll probably save you about as much.

Overall, you’ll need to calculate the heat loss per delta T per room.
You’ll likely have to do this by best estimate, then perhaps a bit of real-life experimentation with a 1KW heater, or similar.

For each room, make a list of the external walls, their composition, and the depth of each layer of material depth.*
Look up the rough WMK (watts of heat lost per square meter (at 1m thick), per degree difference from the outside to the inside) for each material.
Take those values, and use them to make an estimate of the room loss.
So for example, a room with 1 external wall of 10m2, made of 215mm brick (0.7WMK), and 50mm of fibreglass insulation ( 0.038WMK).
1÷(((1÷0.7)×0.215)+(1÷0.038)×0.05) = 0.61 WMK.
Multiplied by 10m2, 6.1Watts per degree difference.
So if it’s 0 outside, and 20 inside, 120W lost by that wall.

There are a lot of other factors that you can bring in (air tightness, rsi, thermal gain). But this will give you a low ballpark number you can start working off for each room.
So long as you’re supplying more than the room is losing, it will gradually heat.
If you want to work out how much it will take to warm all the objects/walls, that’s a whole other calculation. But if you’re maintaining temps, it doesn’t matter so much.

Oh, and a thin sheet of glass will lose about 5W/m2/k. Scary!

* I’m ignoring internal walls, as I’m assuming you’d plan to keep the house roughly the same temperature. And it would get complicated.

https://loadcalc.net/

That’s for sizing equipment, but in the top left you can change the design conditions to get two points and interpolate between them.

Why not just use standard thermostat functionality: set the target temp a bit higher when rates are low and a bit lower when rates are high. It won’t be perfect but it’ll stay more comfortable and you don’t have to over complicate things.

One thing you don’t mention is whether you have any way to store heat, to even out the times when your heat is off. Some of this is thermal mass in the room and maybe that’s all you can do. My parents house when I was a kid had thermal storage radiators that worked really well. The heating elements were on a timer, so only came on overnight when rates were lower, but only heated the bricks or oil or whatever storage medium was inside. The radiators were essentially an insulated box so were cool to the touch. Then, during the day, the thermostat simply controlled a fan circulating air through the radiator to pick up heat as needed.

I’ve used https://www.bestheating.ie/btu-calculator to decide the power of my new boiler, so far it is working well. But as other said, this is likely a very rough approximation.