Without connecting on a charger, I cannot turn on my laptop.

My laptop has a condition that puzzled me a lot, which is that I sometimes (yes it's randomly and I couldn't find out the regulation) cannot turn it on without connecting on a charger even there's 100% power in the battery and the battery is operating really well. After turning it on, it works as a normal one even I pull off the charger.
My laptop is Lenovo Carbon X1 and it's under the warranty. It has had this condition since I bought it. Last week I asked a service man from Lenovo to fix it. He changed the battery for me, and it could be turned on without charger at that time, but still after several hours I turned it off, it cannot be turned on without the charger again.
I also found that every time after shutdown the laptop, I can turn it on without the charger. But sometimes after cooldown for several hours, it can only be activated by the charger even it has 100% power in the battery.
I don't want to take my charger and try to find a charging plug every time when I want to use my laptop in the public place. How can I solve this problem? And how can I describe this kind of problem to the repair personnel of the Lenovo?