Your issue is not normal obviously but this could be a sign that something is wrong with the hardware.
The best adviseadvice is to go to an Apple store or an Apple Authorized Service Provider and let them look at it.
The issue could be your battery, it could be the MagSafe board or it could be the logic board.
I wish I could help you more but without seeing the unit with my own eyes and taking it apart I can't say for certain what it is. I doubt it is software and therefore encourage you to reach out and find someone who is certified to open Macs. https://locate.apple.com/countryreach out and find someone who is certified to open Macs.