Certainly, energy in some form (glucose) is necessary for brain to work. This part is obvious. The question is, how much energy do different brain tasks require. By the way, the brain is doing some work even when I am not thinking, even when I am not using willpower. So the question is, does the brain need more glucose for tasks that require willpower, as opposed to tasks that do not require willpower?

Only if the willpower-related brain tasks need more glucose than willpower-unrelated tasks, only then we can treat the willpower as a limited resource. But if it would happen to be the opposite, if willpower-related and willpower-unrelated tasks need the same energy, then we could take the willpower as unlimited resource (and the glucose as simply the fixed cost of living).