Disk drives with battery backed write caches are a requirement for any system where data corruption is considered a system failure. We use these at my job for things other than just our data bases. It doesn't need to be just SSD's either; traditional platter-based hard drives have this feature as well.
Normally you use an HDD without any write cache and if you do use a write cache along the way (on the HDD or on the server) you make sure you are battery backed.
SSD is a bit different since even if you do not use write caching on the SSD there is a non-trivial amount of meta data that is kept in RAM and needs to be written safely to the media. You need quite a bit of juice to do all this work which is entails keeping most of the hardware working.
Even the HDD has enough capacitor power to park the head back and lock it safely. But I believe that you don't really need all hardware operating at full capacity, only enough spin to rotate and to pull the head back and the spin continues even if you don't power it for the parking time.
Both HDD and SSD do not guarantee much about an IO that was started writing but an acknowledgement wasn't sent about it. SCSI standard from which all disks derive requirements doesn't require anything in such a case and leaves it as undefined.
