Why unplugging adapter makes backlight turn off? Closing/opening screen turns backlight on. (Latitude D531)

We have an issue where unplugging battery charger (adapter), the screen backlight turns off. If we look carefully, we can still see that there IS something on the screen. Closing and re-opening the lid helps the backlight to turn on. If i don't re-plug the adapter, the backlight still works after re-opening the lid.

Here at workplace we could live with that, but when taking computer at home, the screen backlight turns automatically at some minute interval.

I'm trying to figure out what could be the problem (tried swapping some parts, because we have some as our business only buys DELL Latitude notebooks): 

• This is not adapters fault, because I tried a different one and it didn't solve the issue.
• This is not a battery fault, because i tried a different one and it didn't solve the issue.
• Could it be that current varies at that point and computer switches to battery power that this issue is more painful at home?
• Could it be the screen inverter? I could understand that the backlight would not work at all, but this issue is interesting as it happens only when switching to battery power.

Has anyone had the same issues and has any suggestions prior to testing inverter?