display goes black if on battery

Hello,

my EliteBook 8560w was working as expected. But since 3 days, the screen goes black after startup when I reach the login screen and I am on battery. If I have the power cable connected, everything is working.

Also, after booting on battery - if I connect the power cable, the screen comes back. If I am on battery and connect a second screen via VGA, I have a picture too.

What can be the problem, that I have no picture, when I boot on battery or disconnect the power cable?

Thanks, and best regards,
Thomas