Optimal HVAC control

2010 August 10

My wife and I sometimes argue about how the A/C should be run. She generally wants it turned off when I am still in favor of having it run. Her argument goes something like: “I don’t feel hot, so why should we be wasting the electricity”. Our house has a fair amount of thermal mass since it has old fashioned plaster walls, so in order to get it to not run in the evening you have to crank the temp up to 80 or 82 degrees F or so, then you suffer this high heat until 2 AM because there’s no airflow through the house, even though the outside temp might be 65 or so. My wife generally says this when the internal air temp is around 76 F, and the A/C is running about a 30% duty cycle, and often when she is sitting directly under an A/C vent.

Sure, the ideal thing would be to install a whole house fan, but that has its own problems (where to put it, and breaking out the ceiling plaster, framing in the fan, installing electrical, as well as upgrading the attic ventilation and soforth).

The question I have is what is the optimal set point control so that we accomplish the following: the internal temperature of the house equilibriates to 75 degrees F in the least amount of time, and the A/C fan shuts off for the rest of the night, while still never having the internal air temperature drop below 70 F.

Let’s put it a little more precisely, we want some kind of “badness” function which increases away from 75F and goes to infinity at either 70F or 82F (these are approximately the limits of our comfort zone). Then we want to minimize the functional $$\int {\rm badness}(T(t)) + a \times {\rm isrunning}(t) dt$$ where $$T(t)$$ is the temperature at time t, and $$a$$ is the tradeoff parameter that sets the relative scale of badness from having the fan run, vs. badness for having the temperature wrong.

In my next article (when the baby gives me a chance) I will propose a badness function, and a dead simple model of the thermodynamics of a house.

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