While standing in front of them, I have theorized that it would be possible to design a unit that needed neither mains power nor a battery that would have to be replaced periodically. You could use the water flow to charge either a capacitor or a rechargeable battery. Just put a little turbine in the flow, connected to a mini generator that would recharge the power source. The system could be designed such that if usage was low, it would automatically flush to recharge the battery/capacitor before it ran down. (Yes, this would waste water, but in many areas this is not a major issue, and would be preferable to having to replace batteries.)
I posted my theory in a similar thread here on the SDMB, and another poster provided a link to a system that worked exactly that way. I was pretty proud of myself for figuring it out.