Anything on the power grid would need to be isolated before it would be even remotely usable/repairable. Otherwise any power would just fly off into nothingness. Even booting up a full size power plant would be worthless if you were hooked up to the national grid and thereby attempting to use it to power the entire continental US. And considering power lines run underground as well, I would say that is entirely out of the question.
Generators and such would likewise need to be disconnected from the grid, both for that reason and because who the hell knows what's flowing through those wires. For all we know, the massive solar flares going on (because, with Cataclysm's lore the way it is, any disaster than can happen probably is; so let's face it, there's probably massive solar flares) could send a giant jolt of electricity down the line, making everything explode.
What I think would be best is something akin to a more dangerous version of Minecraft's Industrialcraft mod. A variety of machines of varying usefulness which draw power from a generator system through some sort of cable setup. You know how you're told not to daisy-chain extension cords and power strips? Yeah. That could be your method of setting up wiring. Draw too much current, and suddenly you've got a fire on your hands, right in the middle of your base. This power system could be entirely separate from the mobile battery-drive system, though possibly interconnected through some form of rechargable batteries (less uses per volume & weight). Overall, we would want much less emphasis on stored power (the MFSUs and such from IC are just silly), with no more than a temporary buffer for avoiding instantaneous fluctuations (or not, if you don't have the skills to put one together) and regulating a constant current.
some possible machines:
Fume hood: removes any smoke or other gaseous substance from its tile when turned on (useful for any sort of advanced crafting generating toxic gasses with other machines, or just putting near a fire pit)
CNC machine: metal tools & parts; possibly including some which are unattainable through normal crafting (it would actually make sense to have a lot of vehicle parts craftable only with this) [computer skill, mechanic skill]
Computer: craft software components for advanced robotic equipment [computer skill]; stored on USB drives... play a video game about the pre-apocolypse world for happiness (like reading a non-skill book)
Stove: (already sorta exists in a defunct state); power it for most of the things you can do with an open fire; smokeless
The general idea being to add in more high end crafting items, particularly pulling in the holy skill trinity which is electronics, mechanics, and computers. They're just begging for more high-end robotics stuff; especially since we already see many robots in game. Skitterbots, ect, are found in labs; making a craftable friendly one like the manhacks and turrets would add more endgame stuff (robot diplomancer gooo!) without much actual work. So yeah, there's a lot that can be done in that general direction.