One factor overlooked by many is decay heat. Once you shut down, the reactor continues producing substantial heat for some time. This is a serious concern in a reactor that has lost coolant.
Here is post I wrote a few years ago on the topic:
When a nuclear power plant is running, the fissions generate a whole boatload of different isotopes. Back in high school, I remember having the distinct impression that a nuclear reactor followed a distinct set of deterministic fissions, producing a neat chain of fission products, mostly due to pictures like this.
It just ain’t so. Everything is smashing around so much that you are likely to find many different isotopes in the mess that is produced, though the products do follow a sort of nonrandom distribution, an interesting curve that some have called the Dolly Parton Curve.
Anyway, among all of that stuff that is generated are many unstable isotopes with vanishingly-small half lives and some with half lives on the order of billions of years.
The heat that runs the power plant comes mostly from the kinetic energy of the fission products being tossed about, with a small percentage (~7%) of full power coming from the decay of the fission products.
See the chart on page 6 of this document (Warning: PDF) for a little insight into the way decay heat works.
Once the reactor is shut down, meaning the neutron population goes below self-sustaining, the reactor still retains that steady-state decay heat of ~7-8%
Immediately, the fast-decaying isotopes start to peter out, resulting in the decay heat dropping, as shown in the curve in the PDF.
After a day, the power is < 1%. After a week, around 0.1%. After a year, around 0.05%. After ten years, .0025%
This means that if you were operating at 1000MW, immediately after shutdown you have to deal with 70MW of energy in the reactor. After a week, you still have 1MW of heat being generated. After a year, half a megawatt. This is why spent cores still need babysitting.
The decay heat is, in fact, being used during full power operations: it is always present and always around 7-8% of your total power output.
There are systems for preventing loss of coolant, typically referred to as the “reactor fill” system, but that brings its own problems: a reactor that is designed to automatically increase power when cooler water is introduced (meaning more steam is being drawn from the steam generator) is a good thing in most cases, but when you suddenly introduce a slug of cold water from the reactor fill system, this might just add enough reactivity to overcome the negative reactivity of the control rods and therefore cause bad things to happen.
And about those rods. They are indeed designed to absorb neutrons. They do not perform the role of moderator. In a typical pressurized water reactor (the only kind of plant I ever worked in), the rods are raised at startup and lowered at shutdown, with little movement during normal operations. The rods are not used to throttle power.
In these reactors (PWR), the water is the moderator. The water slows down the fast neutrons that come from fission so that they have a better chance of interacting with other nuclei. It turns out that slow “thermal” neutrons are better at causing fission than fast ones, so a moderator is used to bring the speeds down.