Does a power station directly control the supplied power or voltage? The electric grid's "job" is to provide to the end user a steady sinusoidal A/C voltage with an RMS value of 120 V and a frequency of 60 Hz. (That's in the United States - other countries have other standards for voltage and frequency. Also, the voltage coming directly out of the power stations is much higher than this, but is stepped down at a series of transformers between the power station and the end user.)
The details of the electric grid are extremely complicated, but at a 10,000-foot level, we can massively abstract the entire grid to an equivalent circuit consisting of an A/C voltage source (the power stations) and a resistor (a load that consists of the aggregation of all of the end users' appliances). Assuing that the appliances are ohmic, the relevant quantities are the electrical power $P$, voltage $V$, current $I$, and resistance $R$, related by the usual formulas $P = IV$ and $V = IR$.
In physics it's often a bit tricky to distinguish between the independent and dependent variables, but my understanding is that the most natural choice of independent quantities are the power $P$ and the resistance $R$, which in turn determine the voltage and current via $V = \sqrt{PR}$ and $I = \sqrt{P/R}$. Here's my reasoning, which may well be way off base:

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*The end users directly and independently control which appliances are on or off. In a (highly) idealized scenario where literally everything is turned off and there is no current leakage, these appliances would collectively have an infinite equivalent resistance. Every time an end user turns on an appliance, they close a switch that slightly decreases the equivalent resistance of the load, and every time they turn off an appliance, they slightly increase the equivalent resistance. The equivalent resistance is therefore an independent quantity $R(t)$ whose time dependence is physically undetermined and unpredictable (although, of course, in practice subject to large-scale patterns determined by the time of day and year).

*Almost all modern power stations (except for photovoltaic solar stations) generate electricity by physically spinning a turbine and then converting its rotational kinetic energy into A/C electricity via an alternator. The station operators directly control the amount of mechanical power that gets converted into electricity. (I'm not entirely sure how they do this - each turbine must spin at a constant angular speed $\omega$ in order to maintain the correct A/C frequency, so I guess the operators somehow modulate the torque $\tau_\parallel$ that the steam applies to the turbines?) The supplied power $P(t)$ is therefore the other independent quantity.1
Assuming that this is roughly correct, the reason that the voltage stays roughly constant around $V_\text{target}$ as the equivalent resistance changes over time is that the power plant is continuously monitoring the effective resistance $R(t)$ and (directly) modulating the supplied power to be $P(t) \approx V_\text{target}^2/R(t)$. When done correctly, this targeting yields $V(t) = \sqrt{P(t) R(t)} \approx V_\text{target}$ and $I(t) = \sqrt{P(t) / R(t)} \approx V_\text{target} / R(t)$. 2
(Meanwhile, there's a whole other feedback loop that keeps the frequency stable. This mechanism needs to be much more precise, because superposing mis-synchronized A/C voltage sources leads to beats and/or destructive interference that uniformly decreases the RMS voltage. My non-expert understanding is that for real-world A/C electricity, the frequency is extremely stable, but the voltage amplitude randomly fluctuates quite a bit, on the order of 10%. This seems very much compatible with the story above.)
I'm obviously sweeping a huge amount of complex detail under the rug, but is this story roughly correct? The electricity generation mechanism directly controls the supplied power, which is "manually" pegged to a target voltage? In particular, if you were to turn off all of the feedback mechanisms in the power station and replace it with a simple driven generator, while letting the load continue to fluctuate over time, am I correct that the delivered power would be much closer to constant than the delivered voltage? (This would, of course, be completely impractical for a real-world electric system, because then every time someone turned on an appliance, that would decrease the voltage delivered to every other appliance.)

1 Things are different when the power is supplied by a battery rather than by a generator. My understanding is that if you power a time-varying load with a battery, then the voltage will remain much closer to constant than the power (whereas with a simple externally driven generator, it's the power that stays closer to constant).
2 There's clearly some philosophical ambiguity as to whether it's the voltage determining the power or the power determining the voltage, depending on whether you treat the power station's "motivation" as endogenous or exogenous. But I hope that my question makes clear that I'm talking about the direct physical causation.
 A: I've done a little bit of digging into how the power grid regulates the supplied voltage and frequency, and I think I have a conceptually accurate answer to this question.
Most of what I thought was correct: the end users control the effective load resistance $R(t)$ and demanded power $P(t)$ exogenously, so the power plant has no direct control over that.
But I missed something very important: for a given AC generator, the generated voltage $V$ and and frequency $\omega$ are not independent, but are directly proportional. We can already see this with a toy model of an AC generator consisting of a rigid loop of wire rotating in a uniform magnetic field with angular frequency $\omega$: the voltage around the wire has amplitude $V = \Phi \omega$, where $\Phi$ is the magnetic flux through the loop when its normal vector is parallel to the magnetic field.
A turbine generator being turned by torque $\tau$ applied by steam or liquid water at angular frequency $\omega$ converts the mechanical power $P = \tau \omega$ to electrical power $P = I V = V^2/R$. When a new load comes on unexpectedly, the electromagnetic resistive torque on the turbine increases. The applied mechanical torque (set by the amount of steam hitting the turbine blades) doesn't react right away, so the torques unbalance and the turbine's rotation briefly begins to slow down. The operators' control loop detects this and very quickly increases the amount of mechanical torque that the steam delivers to the turbine, pushing the angular velocity back up to the target. And vice versa when the load decreases. The operators aren't directly regulating the voltage at all; instead, they're regulating the frequency, and since $V \propto \omega$, this automatically keeps the supplied voltage approximately constant as well.
So the chain of causation is:

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*Initially, $\tau \omega = P_\text{load}$.

*Customer changes the demanded load $P_\text{load}$

*Change in electrical power draw changes the electromagnetic torque on the turbine

*Now-unbalanced torque causes the turbine to speed up or slow down toward an angular velocity $P_\text{load}/\tau$.

*Generator operators quickly change the mechanical torque delivered to the turbine in order bring its rotation frequency back to the nominal $\omega$. This also brings the supplied voltage back up to the nominal value, while changing the mechanical power delivered to the turbine to match the new demanded load.

In the abstract, the operators are trying to keep the supplied mechanical torque equal to $P_\text{load}/\omega_\text{nominal}$. But in practice, they're more directly trying to just keep the rotational frequency fixed to the target, and in doing so they automatically supply the right amount of power.
There's a very important detail that this explanation is sweeping under the rug, which is that there are many synchronized generators in a typical grid, not just one. This makes the control loop much less demanding, because the networked generators collectively create a completely automatic electromagnetic feedback loop that keeps them all synchronized. So any new loads are (roughly speaking) automatically distributed equally across all of the generators, as long as they aren't too large. So larger grids are more stable, and small changes in loads in a large grid have only a modest effect on the frequency of each individual generator.
