In the UK our mains standard is 230 V AC at 50 hertz. So I understand that the generators at the power stations need to spin at exactly 3000 RPM . This is true no matter what the source is (nuclear, coal, gas, hydroelectric etc). Let's use steam as an example here. How do they use the steam to spin the generator at a constant 50 hertz? It doesn't seem plausible to control the speed with the amount of steam passed though to that precise speed. So how do they do it?
Electronic – How do power stations maintain 50 hertz
generatormains
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Your question is very general, and so is this answer.
When a power plant creates power like the Hoover Dam, it can provide 2.07 GW of electrical power. My question is what does this mean? I assume from Faraday’s law that the induced voltage across the generator coil produces an current, and this combination (P = VI) is the actual power but I'm sure that my thinking is naïve. Can someone roughly sketch out how the electrical power of a power plant is computed?
From a mechanical perspective
"2.07 GW" means that the peak output of the power plant is 2.07 GW. This is most likely a series of smaller units, say 20 × 100 MW units = 2.0 GW.
The generator is a converter of mechanical energy into electrical energy. So to generate 2.07 GW of electrical energy, an equivalent amount of mechanical energy has to be provided. In the case of the Hoover Dam, the mechanical energy is provided by water falling to a lower elevation, giving up its gravitational potential energy in the process.
From this perspective, you can think of the maximum electrical output power of a power plant as the maximum rate at which it can convert mechanical energy into electrical energy, factoring in the efficiency of the conversion process.
The rate at which mechanical power is generated is a mechanical engineer's problem. For a hydroelectric station, the mechanical power would depend on the water pressure, turbine size, and various design parameters. For a wind turbine, the mechanical power would be set by the radius of the blades. And so on.
From an electrical perspective
Yes, the electrical energy produced follows Faraday's Law and Ohm's Law, though for an AC system, V and I are sinusoids which may not be in phase, and P ≠ VI. Rather, apparent power (volt-amperes) S = VI, and real power (watts) P = VI cos ɸ.
Other complications include electrical losses (per Ohm's law) and magnetic losses (eddy currents induced in metallic parts).
If possible, what kind of voltages and currents are power plants producing before the Step Up transformers? Of course, this varies from one power plant to another.
Regarding typical voltages
In my experience, small generators (i.e. diesel gen-sets) generate directly at the utilisation voltage, say 415 V here in Australia.
Larger power station units generate at a medium voltage like 11kV before stepping up to transmission voltage, i.e. 132 kV.
I imagine a medium voltage like 11kV is preferred vs. a high voltage like 33kV, because less insulation is required on the windings and the rotating parts may be physically lighter.
Regarding typical currents
An aeroderivative gas turbine, i.e. the General Electric LM6000, is typically rated about 45 MW and might have a 60 MVA alternator attached to it. Calculation of the three-phase line current at 11kV is left as an exercise to the reader. Don't forget your √3.
A coal power station unit might be rated 400 MVA at 22kV. See "Tarong Power Station" in QLD, Australia, which consists of four large units like this. Again, calculation of the line current is left as an exercise.
Note: I am at home and hence don't have access to my reference material at work. The above numbers are indicative, so treat them with a grain of salt.
If you are curious as to the exact operating principles and theory of an AC generator, I would encourage you to look up a textbook on electric machinery. My personal favourite is Mulukutla Sarma's Electric Machines. Check your university library for a copy.
In a coal or nuclear plant, the thermal power changes very slowly, perhaps 10-20% per hour. To have power available for spinning reserve, the steam turbines are run at a lower power than the boilers, the main throttle is set so that there is some steam available but not used. The excess steam bypasses the turbine and its energy is wasted. If more power is required, the steam valve can be carefully opened and the power delivered increases. For a big steam turbine this might still take 30 seconds.
So to a first approximation the slow thermal plants consume fuel for the full total of actual power + spinning reserve.
You have a good question about the timescales. At the shortest timescales, fractions of a second, the frequency is passively stabilised by the inertia of all the generators (and rotating loads). At longer timescales it's entirely up to the control systems adjusting the power of each generator, and depends on the transient power response of the generator.
Some time back I found a very good presentation by John Undrill, called "Power Plant / System Dynamics and Control" presented at a NREL / EPRI workshop, May 2013. I can't find a copy of the document to link to now, see if you can find a cached copy somewhere.
Best Answer
From UK's National Grid:
Figure 1. The graph shows Frequency data to a 15 second resolution over the hour up to 2016-05-21 13:31 BST.
To answer your questions:
Not constantly. Over a day it averages to exactly 50 Hz. This used to be a requirement to keep all mains powered clocks in synch. I have one on my electrical supply day/night meter.
Yes, they all run in synch.
It is, in fact, possible, provided there is enough steam being generated to supply the peak load. With a single steam generator supplying an island, for example, the scheduler would plan ahead using weather information, TV schedules (the famous 10 million kettles going on in the big-game half-time), large load user schedules, etc., to have the thermal plant generating enough steam in anticipation. Meanwhile, while demand is low, the plant has to vent the excess steam to atmosphere. Yes, this is wasteful.
In practice, most grids have a mix of base-load thermal plant with other fast response generators such as gas turbine and hydro which can be switched in and out quickly.
The UK also has capability to import power from France and Ireland by underwater DC interconnects. These allow connection of multiple national grids without synchronising problems.