Partition labels and UUIDs are used in Linux to get around the fact that device names in general are not stable. My recommendation is to uniquely label each USB drive. Depending on the filesystem, the commands
e2label /dev/sdb1 mylabel
dosfslabel /dev/sdb1 mylabel
make a label for the partition. After this you can mount the partition using the command
mount LABEL=mylabel /mntpoint
and never again have to refer to device names (which are constantly changing). In your case you may also want to put a suitable line in /etc/fstab to specify the mount point.
More importantly, however, would be to solve the power supply issues before your data gets corrupted or hardware damaged. Note that the low-level formatting in a USB stick or SD card can get damaged by low power to the point where the storage device doesn't work anymore.