I'm addicted to black tea. If I don't have two cups, daily, I get bad headaches.

It's not caffeine, because coffee doesn't substitute, neither does green tea.

So it must be some stimulant unique to black tea.

Maybe theobromine, theophylline, or L-Theanine? (see http://www.healthline.com/nutrition/stimulants-in-tea - maybe a questionable source...)