Thermostat Logic

Suppose you live in an all-electric house in a town lime Bonavista; You will find yourself fighting on two opposing fronts:-

(1) Keep your thermostats at a constant value throughout winter; that is; maintain your house's rooms at a constant temperature day and night. In my case that would mean keeping each room at 18 degrees Celsius.

(2) Keep your thermostats at a fixed values throughout winter; that is; maintain your house's rooms at a constant temperature of 18c during the day and at 15c during the night.

I maintain different day and night temperatures for four of my rooms, and fixed (24 hours per day) for two rooms. My electricity bill remains at 10,000 Kwh per year, year after year.

Prove that option 1 is more efficient than option 2. Or if you prefer, prove that option 2 is better than option 1.

You could run option1 this winter, then option 2 next winter, but then you would have to factor in hour-by-hour temperatures for both years, and (all houses are leaky) hour-by-hour wind speeds for both years. Best of luck with those factors!

We need a logical solution to our proof.

Suppose we start with option 1 and set all six rooms to 18c. Our annual bill will come to an unpredictable value. We cannot know that value until the year is finished, because we cannot know the temperature and wind profiles for that year until that meter has stopped spinning for the year.

Now suppose that for just one hour in the year we lower the thermostat in room 1 to 6c for just one hour. That may or that may not affect the total, and we will never be able to compare two values here.

Now suppose that for just one week in the year we lower the thermostat in room 1 to 6c for that one week. That may or that may not affect the total, and we will never be able to compare two values here.

Now suppose that for the year we lower the thermostat in room 1 to 6c. For the entire year. That must affect the total, and we can be certain that we will have consumed less energy than had we left that room set to 18c all year round.

There must be a spectrum along which our energy consumption varies in near-zero increments depending on the number of hours that we enforce that lower setting.

Does this happen? Yes! I keep my laundry at 6c year-round so that my water pipes do not freeze.

As well I keep my guest room at 6c year round, because I do not want my jars of bottled fruit to freeze.

If you are still convinced, think about my energy consumption if I move my preserves from the guest room into the laundry and leave that guest room unheated. I could disconnect the wires from the baseboard heater.

With programmable thermostats, with at least two settings ("wake up in the morning" and "go to bed at nighttime") we can practice Option 1 or Option 2 as we see fit. There will be some spectrum between option 1 and our settings for option 2.

Since Option 1 introduces some spectrum, then there must remain a continuous spectrum that runs to a lower value for option 2.

We still will not know where along that line option 2 is guaranteed to be better than option 1.

We know only that somewhere along that line option 2 is guaranteed to be better than option 1.

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