How to power "Power Functions" devices from an electric wall plug? In this video showing an example of Great Ball Contraption, we can see (at 1:35) that some trains powers are used to power Power Functions motors:


  


That must be more convenient than having to change batteries after several hours.



What parts are involved, from the electric wall plug to the final motor?

I guess that the following parts are used:


Power Adaptor (70931) plugged on wall plug, that provide current through a wire connected to… 
Train Speed Regulator 9V (2868b) that outputs current through electric studs
a Power Functions Extension Wire connected on electric studs that provide a Power Functions-specific plug (8886)
a Power Functions motor, e.g L motor:


Are my guesses correct?

If I remember correctly, I have already seen in some videos that the Technic Control Center II (2840c02) was used in place of Train Speed Regulator. Does it work too?

The Train Speed Regulator 9V are Technic Control Center II are incredibly cheap, are they the cheaper way to power devices from an electric wall plug? The Power Adaptor is relatively expensive, but it's probably possible to find a cheaper and compatible adaptor.
What you describe is indeed the most common way to GBC modules, but it's worth noting that one adaptor+regulator can typically power more modules (it depends on the modules), sometimes only one end of the extension cable is used, when placed on the regulator, it effectively provides a power functions "plug", and if you want to power more modules, the wiring can get a little easier to handle, when every module is essentially connected to a regulator.

Another option is an NXT with the official rechargable battery mounted and an adaptor plugged in, and then of course NXT cables to an NXT motor (I don't remember if NXT->PF cables exists, but NXT->9V cables does and you have mentioned that the PF extension cables can attach to 9V). I have also seen modules with an RCX, I think that has a plug for an adaptor directly, but I'm not sure, I don't think I've seen GBC modules powered by EV3's yet, but there's probably a solution. - Probably more expensive, but having an NXT in a module allows other possibilities, especially if you also put in some sensors.

As Mr. Shiny and New 安宇 mentions the rechargable battery box is a third solution.

All of these solutions have the advantage that they only use official unmodified LEGO parts.