Can I run my MacBook Pro on just AC power?
For some operations certain models require more power than the Power Supply alone can deliver at a given time. For this incident it relies on additional power from the battery. It is to be assumed that that is your machines issue. You'll probably need to get that battery replaced. The price is usually between 120-150 at an Apple Certified Repair Center in Austria. 

*My information is based on statements from AppleCare and their internal chat service for Certified Technicians.