laptop powers off when unplugged

I have an Acer Aspire V5-571P. It has been upgraded to Windows 10. After upgrading, it would shut down whenever I removed the power cord. It looked like the battery was not charging, and it showed the level was 0%. So, I purchased a new battery. It has worked fine for a number of months. (not getting much use, but worked every time) Recently it has started acting up again. Whenever I pull out the power cord, it shuts down. It will not start up unless it is plugged in. The battery level says it is 100%. What is going on with this computer? Can it be fixed? Is it actually a bad battery again? Was it before?
Thanks for any help you can offer.