You're not left with a flat battery at the end of the night. Many vehicles are combined in intelligent systems which work together to ensure that the vehicles have the energy they need (which is easy to set in all the systems I've seen) but provide enough grid support to make this work.

Remember that even my little town car (Renault Zoe) has a 52kWh battery.... which would run my house for five days. So the energy stored in these systems can be considerable.

The people doing these things have thought a lot about it. Take a look at this video - it's a bit 'puff piece' but shows what one way of doing it looks like:

https://www.youtube.com/watch?v=fKItLGPdN0k