batteries made specifically for outdoor strobe usage consist of a deep cycle battery and an inverter (pure sine wave, I believe)
different manufacturers make battery packs for their brand of lights
there are a few to choose from, there's a cheap one from Jinbei, there's higher end models from Elinchrom and Profoto and Hensel
there are power supplies from Dynalite (XP1100 I believe)
if you want to use a strobe outdoors with a battery pack that's non-propriety you can't just use any power supply, it has to be able to take a surge of twice the wattage stated on the power supply at least, and the power supply has to provides a pure sine wave.
I've tried it before, it's very difficult. The only one that seems to work ok is Alienbees Vagabond unit