I'm assuming there would still be some electrical load because appliances and machines would be left running, but that load would drop over time. It would drop sharply at first as heavy load machines stopped running, then the next major drops would probably be caused by grid failures over weeks or months.
Will power plants continue to produce and deliver electricity in the houses?
How long the plants keep running depends heavily on how much management they need, of course. It's possible to automate a lot of stuff even without looking at things like AI management.
A coal plant would run out of fuel fast (as mentioned in another answer) but a nuclear power plant typically refuels every 18-24 months and hydro/wind/solar stuff doesn't have the fuel problem exactly.
However, I think the thing that would kill the grid first is automatic safeguards that trip when the power generation and load get uneven. You can't just generate all the power you want regardless of load, you have to manage generation to match the load. That requires careful, ongoing control in the US power grid. If people stopped doing that, the power grid might start going down in hours. I don't know how automated that is at this point in time, but given what I've read from people who work in the power industry, it seems to require a lot of human effort to keep things running smoothly even day to day.
Regardless of how the grid is managed, it's likely that any active generation power plant like nuclear or coal would simply shut down automatically if people stopped showing up.
How soon will each power station break down?
Coal / Nuclear
Coal plants would run out of fuel or shut down quickly without workers. Nuclear plants should automatically shut down too, if the engineers who designed it were smart. Neither plant would break down per se, but I'd expect both coal and nuclear to stop generating power inside 24 hours or so.
Wind turbines need significant maintenance and management too, as it happens. Electrical components fail, gearboxes need fluid changes and sometimes need to be replaced, and you can't just let a wind turbine do its own thing no matter what the wind is doing. If the wind is too high, it can destroy the tower. Who controls which way the turbine is facing? Is that automated? It could be, and it's an important question.
If the turbines just stopped getting maintenance and didn't get killed by high winds, and were able to stay pointed into the wind, from some maintenance numbers I found online I'd expect the wind farm to be down to <30% capacity in one year and completely dead by ~18 months.
Hydro plants are pretty low maintenance, but again, require a lot of management. This is a simpler problem than the wind farm problem, since it's just deciding where to route water through the dam. But something needs to make those decisions. If it's computer controlled, and the computer also manages the reservoir, the plant could conceivably stay functional for years just producing enough electricity to keep itself powered on. Eventually corrosion would take its toll and turbines would start to fail, but at minimal load that could take a decade or more.
Hydro dams would probably get shut down by clogged intakes before the hardware failed, assuming computer management. So, that could mean months to years depending on what kind of debris problems they have.
If there is no automatic management of the system, though, it's a big fat "it depends". In what state was the hydro plant when everyone vanished? Were the turbines running full bore to match the peak power load for the day? Was it flood season, and they were letting a lot of water through to keep the reservoir level manageable? Neither situation would be good for the plant. If the plant was in the wrong state at the time of the great vanishing, it could be damaged in short order, or drain the reservoir in the dry season. So, days to months and the plant would be down.
Solar panels are pretty durable. I believe a typical solar panel is expected to last ~25 years. They do degrade with time, and naturally the output drops if they are dusty. Snow would stop them working at all, of course.
The weak point in a solar system is the battery bank. Lead acid batteries require maintenance. Connections corrode and electrolyte evaporates off. Left unattended, getting hit with a charge/discharge cycle every day, the battery bank in a house might only last 3 months, and certainly not a lot longer. Some of the batteries would be killed from the lack of maintenance, while others would only require new distilled water to work fairly well again.
There are other battery technologies that work fine with solar and require less maintenance, but they're substantially more expensive than typical lead acid batteries. So the system could be designed to be more robust, but probably wouldn't have been.
Again, there's still the grid maintenance to worry about too. Transformers fail, switches fail, power lines fall, etc. Once that stuff fails, some of it is effectively irreplaceable. Even fixing a power line is a nontrivial task. I think the power grid would start failing inside 6 months, which would start cutting off parts of the city.
How long it will be possible to use the electrical energy in the abandoned houses of this city?
I take it you want each district to be powered by its own independent grid tied to one of the types of generators?
Assuming automatic control of the power grid so it didn't just shut down, ballpark estimates:
- Nuclear - <24 hours
- Coal - <24 hours
- Wind - Grid will probably fail first from lack of maintenance
- Hydro - Until the grid failed
- Solar - Until the grid failed, but it depends, with a bit of elbow grease
years are possible
Solar is the best bet for long term power, because even if the houses don't have their own panels you can simply relocate and rewire panels from the solar farm to feed the houses you want to feed. That assumes the person doing it has a clue and some basic tools, including a multimeter, of course.
Does disaster happen (like the explosion of a nuclear reactor)?
Probably not. If the grid is entirely dependent on human control with no automation or safeguards, grid failures would probably start fires.
Which stations will last longer?
Quite a lot of this comes down to "Well, it depends how they designed the systems." In most cases, automation isn't done simply because it's cheaper to pay a load of people $20-30 an hour to watch dials and push buttons (or aim windmills) than it is to design, build, and retrofit an automatic system. That sort of automation typically becomes standard over decades, as new systems are built to replace old, worn out ones. The exception is when human costs rise to the point where it's cheaper to just retrofit the automatic system. That's why fast food restaurants are starting to replace workers with robots.
So if what you want in your world is for power to be available even after the people all vanish, it's certainly possible with current technology if, for example, the society placed a high value on the perceived safety of removing human error from the equation.
There are also other ways to get power. There would probably be emergency generators around, in places like hospitals. Some people would own them for camping or as backups (I have a <30lb 1000W Yamaha generator that's been amazingly useful). You can get power from welder/generator machines. RVs have generators, and so do some refrigerated trailers. Gasoline is usable for a year or more, and diesel can last quite a while longer if properly treated. It's also possible to modify internal combustion engines to run on alcohol. So, there are lots of ways to get power in a city even if the grid is down, if you're not worried about people stopping you looting the place.