Alliance energy refills 1 flag every four hours for a maximum of 3 flags.
It’s rate never changes (faster or slower); everyone has the same system.
Refill flasks are available for purchase every now and then, and it is possible to get a free flask from Mystic Vision or a Rare Wanted chest, etc.
ALSO, when you level up, all of your energy fills to maximum: world energy (red flag), Raid energy (blue), and alliance (purple). Some folks are careful NOT to level up until they have used all flags, and/or wait until they have a Titan to expend six flags on it at once (their original 3, plus the level up 3).