But how it supposed to go from 12V to 15.7V for the Peltier

The 15.7V rating is the maximum voltage for the module, not it's recommended operating voltage. You don't want to get near it. To approach that voltage would require a very effective heat sink to dissipate all the heat being produced. It's also possible that heat is not the only issue and that voltage alone might damage the internal junctions of the device - I'm not sure.

 

A MOSFET could be used to drive the fan -- simple and inexpensive. I have done it many times. (Also include a flyback diode.)

See this diagram -- replace motor with fan: http://www.nerdkits.com/videos/motors_and_microcontrollers_101/

 

Given that your controller is 'smart', you don't need an electronic PID controller. In your case, PID is an algorithm for sensing temperature changes and responding with bursts of current calculated to approach as quickly as possible while avoiding too much overshoot and oscillation. The magnitude of these current changes will be computed in the RPi according to the PID algorithm.

 

Thank you, guys. I'm studying you solutions so far.