[RedDevil]

The power needed to electrify all cars is way less than total domestic electricity usage. Not that that could happen in less than a decade; total EV production rises fast, but not to the order of being able to flip the market that fast. The transition would be smooth by necessity.
EV owners frequently report their energy bills hardly rise despite frequent home charging.

So no, EV's will not dramatically impact grid stability. Rather, they will improve it as most charging is done at night when the load is relatively low.

EVs can be used to help stabilize the grid with smart charging or even resupplying the net when needed (or the house when the grid does go down after all). Or have a Powerwall to do just that, always.
The technology already exists, all we need is the will to implement it.