Up to 50, I think it's clear the WoL is using their anger to fuel their powers. Post 50, it becomes their 'love' for Eorzea, or at least to protect those that can't protect themselves. Sid, in the other hand, is using anger all the way to at least 58... IMO, using Rielle as a justification for his revenge based on what Ishgard did to his family, his mentor, and Fray. It isn't until the penultimate quests that he is able to put that aside and become more focused on protecting others, IMO.

As for the fuel source.... Emotions, positive or negative, drawing upon personal aether. However, unlike WMGs, the DRKs refuel their own aether by stealing their foes aether. This prevents the DRK from burning themselves out.

Just speculation based on the mechanics of the class.