I have wondered whether using a small windmill (on the roof) to power the compressor directly could work. This to prevent the energy loss of converting mechanic

I have wondered whether using a small windmill (on the roof) to power the compressor directly could work. This to prevent the energy loss of converting mechanical energy to electrical and electrical to mechanical again. Combining this with a heat/cold storage which is filled there is a lot of wind to be used when there is less, seems to be good idea as well.