Energy loss through an insulator is exactly linear to temperature.

The problem is that people who would have otherwise tolerated 50F temperatures without heating by putting on a sweater are pulling their old oil column heater out of the storage.

One other problem is that one of the ways people have been making their heating more efficient is by using heat pumps, and their efficiency gain drops off with lower temperatures until they become no more efficient than the resistive heaters they replaced during the kind of unusually cold weather Texas is apparently seeing. This makes the peak to average power usage problem even worse.

More modern heat pumps can be efficient at up to around -10 degF[1]. Higher efficiency federal/local rebates are not bad across the US, so I wouldn't be surprised if these were installed in some TX homes. It's still a capacity problem if average use is 10hr/day instead of 2hr/day and your house has no gas furnace heating. I would guess most TX (Dallas/Austin) homes probably do have a gas furnace, but maybe not in coastal/south TX (Houston).

[1] https://daikincomfort.com/go/aurora/

Are their heat pumps linked to outside if their efficiency is so affected? The models I see generally pipe underground, where the temperature is mostly stable and isn't affected by a week of unusually cold weather.

The overwhelming majority of heat pumps across the US are air-source (rather than ground-source), so "yes, overwhelmingly".

Oh you're right.

Time-dependent solutions to temperature change across a gradient are exponential decay functions that solve diff eqs based on Newton's law of cooling. It depends mostly on the insulation and the temperature difference.

While it is 10 F outside, the central heating system for my apartment in a 6-story building cannot maintain temperature. It is 66 F and gradually falling. I would guess it would be 2.5kW running 24/7.

25KW more likely

Probably closer to this than 2.5kw. 2.5kw on a decently modern heat pump should be providing 25k btu. About enough to heat a 2000sqft house ~50f over ambient.

I have a 2.4kW heater in my bedroom...

Just in case anyone is about to miss this detail:

That heater probably doesn't run continously.

For a well designed shared system you can get away with less than n x what an individual unit needs.

(That said, I too guess 2.5kW is too small for what GP mentioned.)

Energy usage is superlinear below a certain point. As I recall, about 25% of homes in Texas use heat pumps. but below 30-ish degrees, heat pumps don't work and they have to switch to resistance-based heating So, demand will start rising a whole lot faster.

There are newer heat pumps that work below 30 degrees, but they're more expensive and not typically used in Texas for obvious reasons.