Where is the smoke coming from? Given the multiple symptoms, I'd start by feeling the alternator to see if it's uncommonly hot. If it is, it has a shorted diode that hasn't burned open yet. Once it does it won't get hot but it will only develop one third of its maximum capacity which will run the battery down. Check the serpentine belt too to see if it's slipping and overheating. Suspect the spring-loaded tensioner if it is.