Forget about the plants. It’s the grid that counts.
And remember that there is input and output and that they must be balanced at all times.
These days most plants are automated - but not all of them. And most homes and businesses are automated as well, in the sense that you don’t have to hold a button down for the electricity to run - but not all of them.
Assume that all humans instantly disappear. Everything that happens next depends on the particulars, but here’s what would generally happen.
The grid is now like a pencil balanced on its point. Some plants would go off right away. The grid would react automatically to some extent isolating these plants from the system and shifting resources. But the other end is shifting as well because the normal flows have been disrupted. All the portions of the grid don’t talk to one another automatically, as we’ve just seen again. Some areas would go black. Without human operators to shift the loads, the blackout would spread.
Most major power plants would shut off automatically as soon as this happens. All the nuclear power plants in the northeast did during the blackout. No meltdowns.
How long for this to occur? Probably no more than minutes.
The longer the system went without human intervention, the wider the blackout would become.
Worldwide, you would see this pattern recur with regional variations.
Some areas, some buildings, some places would retain power for a while. Many businesses these days - ironically, a much higher percentage in the third world where they are used to blackouts - have backup generators that would automatically go on when the grid failed. These would stay on until whatever powered them - gasoline or propane or whatever - ran out.
In industrialized countries, water and sewage systems run off the grid. They will also stop very soon. And unless you had a Day the Earth Stood Still situation in which the alien kept all the planes flying and stopped cars without accidents and whatever, there would be a million crashes, fires, and catastrophes that would damage a fair amount of society.
Beyond that, the scenarios are just too numerous to make sense of them. Certainly some survivors could live on: VernWinterbottom already mentioned the Amish and there are millions of people elsewhere in the world who don’t depend on electricity.
But there would be no Internet. And no SDMB. So why bother living?