battery - What could be wrong with my laptop, which shuts down unexpectedly?

I've searched quite a few forums and questions on this website, but felt like none matched my exact problem.

Simply said, my laptop shuts down without any notice. By 'shut down' in this question I always mean: turns off immediately, without following the correct procedure for shutting down. This instant shutdown also seems to imply that the laptop takes a much longer time to boot the first time after, but I don't think permanent damage is caused after such a 'crash'. At least I hope so.

This problem only occurs when the charger is not connected to the laptop, thus, pretty much ruling out the possibility that there is a heating problem or something similar. Even when the battery is removed and the charger is plugged in, no problems occur. I tested this extensively, but the problems just never occur when the charger is plugged in, not even when playing a computer game, for example.

Since my battery was over two and a half years old and had been used a lot, I decided to buy a new one, thinking that I could use the extra capacity even if battery turned out not to be the culprit. With the new battery, the problem persists. The shutdowns might have become slightly less frequent, but I haven't tested this so thoroughly.

Just to clarify: the problem is not that the battery charge indicator is broken and the laptop shuts down because the battery is dead. After such a shutdown, I can start my computer without connecting the charger and continue working. Today, I worked for three hours on battery power, without recharging. In these three hours, four shutdowns occured, spread more or less equally in time.

I have already reinstalled my operating system (Windows 10), but to no avail. In fact, the problem (still with my old battery) existed before updating from Windows 8 to 10. I have an anti-virus programme which is up to date. I have not yet completely reset the computer and wiped the harddrive; I would like to keep this as a last resort, for obvious reasons.

As a non-expert, the only thing I can think of is a hardware problem, some kind of faulty communication between the battery and the system. I'd rather not have my laptop tinkered with unnecessarily, so I'd like to hear your opinion before bringing my laptop to the manufacturer (and paying a high price, as the warranty has already expired).

Looking forward to hearing from you