Hobby servo can draw as much as 1A when stall. You definitely need high mAh rating battery, i.e. Lipo 3000mAh and above, which should last you 15-20mins. Common AA battery is out of the question. The current draw is too high for this type of battery, and cannot maintain decent voltage. Don't be surprise if your robot is jumpy, and reset frequently. Lipo may rated at 7.4v, but at 100% charge capacity (direct from charger), it is rated at 8.4v. Once it begin to discharge (the first 10% discharge), the voltage will drop to 7.4v and stabilize.
Hobby servos are rated from 4.5-6.0v, so you need to have good voltage regulator circuit to take care the 8.4v. Typical linear voltage regulator are rated up to 1A only, do not use that to regulate 20 servos, you will blow your regulator in no time. Either you use a higher rating VR (costly), passing transistors method, or simply get 1 regulator for each servo.
Heat dissipation is another area which you want to look into. If power is not handle properly, your controller may reset frequently.
If you want to skip that, try 5x NiMH rechargable battery, 2500mAh. Fully charge 5x NiMH still deliver slightly more than 7.0v. A single silicon diode on each servo will do the trick.