12 Coil Delta Configuration for 6 Phase

Assuming this is a wind turbine alternator design, the power output is limited by the power available at the input shaft and the acceptable temperature rise of the copper conductors. More phases would give you lower rectified ripple (smoother DC output), but not more power.

Depending on your turbine design, you may get better low speed performance with more coils. Hard to be specific with so little information about the design.

I agree with the view that power generated would not go up by increasing the number of coils and /or increasing the number of phases. A generator is rather a converter of mechanical power to electrical power. It does not produce power it converts. So it would convert only that power which is made available to it by the prime mover for conversion.

as far increasing the number phases from 3 to 6 and coils from 9 to 12, coils per phase would go down from 3 coils per phase (in case of 3 phase) to 2 coils per phase (in case of 6 phase 12 coils). This would result in substantial reduction in phase voltage, off course other parameters remaining constant like flux density , speed etc. So same power at reduced voltage would mean increased current. There would by many other complications like harmonics etc.

You cannot generate more power as you are hoping.

The improvement is in a smoother DC since you will be rectifying 6 phases.

The voltage can be adjusted by controlling the excitation as usual on a synchronous generator (before rectification...)

Extra cost will be in the rectifier needing more diodes...

If you are using an existing stator and modifying the coils configuration, then the copper qty will be almost the same (weight) since you are limited with the size of the grooves...

definitely no increase without increase of the input power from prime mover.(if your are talking about the generator becoming more capable to produce higher output, then no dice unless drastic change in overall size to accomodate more copper ways or sizes...