I'm not sure what a WiFi spot has to do with anything.
Can you be more specific? You are saying the small light on your power adapter is green right? When it is green, what does the charge indicator say? Is your battery fully charged? 97-100%?
More importantly, if you let the battery run down some, like to 60% and then you plug your adapter back in, does the light stay green? Or does it turn red?