Honda FR-V 2.2 Diesel 2007 – Battery – alternator

Hello, as a family we bought a Honda FR-V in November 2012 with about 102000 km. I had problems with the battery in December, so that I had changed it after charging etc. in January for a new battery. The second battery is now empty again. I have tested the battery voltage several times, with the following results: Cold engine with interior ventilation – 11.7 volts Warm engine with interior ventilation and light 14.04 volts War also 14.3 volts I have measured 15,2 volts with a warm engine (but only once) I was already at the workshop, this one has presented me with an estimate in which I should spend 830 € for a new alternator. How can I measure if my alternator is faulty? Ampere/Volt? Is it normal that I get less than 12 volts with a cold engine? I would be happy about your answers. Greeting Rudi