Assume the drone is launched from from outside the property of the plant and does not fly over the property boundary of the plant. Location is the United States. (If laws by state differ then focus on the mid Atlantic PA, NY, NJ). Assume all "point source" power plants, coal, gas, nuclear etc.
Unlike Germany, in the United States, ownership of the ground doesn't give control of the airspace. Once you're off the ground, you're under the exclusive jurisdiction of the Federal Aviation Administration.
Many of the larger power plants, such as Hoover Dam, are covered by flight restrictions that prohibit you from flying close to them. The current NOTAM providing those restrictions is here, and incorporates by reference the map of restrictions here. The local natural-gas peaking plant, on the other hand, has no such restriction, and I could freely fly over it if I were willing to deal with a tangle of power lines.
There's also no general restriction on photographing power plants. The closest open airspace to Hoover Dam is only about 1200 feet away, so if you can get your drone there, you can freely take photographs.
You are not allowed to cross onto someone else's property with a drone unless you have permission, which for a power station you don't. Trespass with a drone is always treated as knowingly and intentional, even if it is a navigational error. Only when one loses control of the drone, it wouldn't be a trespass, but a violation of the regulations on drone flight.
Those regulations say also that one is only allowed to use a drone within eyesight and when one has clear control over it. To get out of the eyesight and into the "fly remotely with camera control" field you have a special license to use a drone this way, and that license doesn't allow you to cross property lines either but mandates you to acquire clearing from air traffic control.
And above that, you have to respect restricted airspace and no-fly zones which extend for example around airports. One such rule on permanently restricted air zones requires any airborne vehicle to stay above 1500 feet (or more) in an area around nuclear facilities - which is above the area where you may operate drones legally. Examples are ED-R 8, 24, and 41, all spanning 0.8 nautical miles around the center of the nuclear facility and barring most air traffic 24/7 unless one has clearing from ATC.
If you can manage to do all the aforementioned parts, you can do photos all you want, but because of the distances you might need to launch your drone and keep it, the photos might be worth nothing - especially for nuclear facilities, due to any drone counting as air traffic and thus having to abide by the no-fly zone around it. Which means they are grounded.