The problem when i proved \$3S_Y=S_\Delta\$ by using \$S=I^2 Z\$ in three phase circuit
The phase current in the star configuration (with Z/3) converted from delta (with Z) is wrong-



The phase current in the star configuration (with Z/3) converted from delta (with Z)will be 

$$I'_{p} = \frac{V_{L}\sqrt{3}}{Z}$$

$$I'_{p} = 3I_{p}$$

which gives you 
$$S^{\Delta} = 9I^{2}_{p}Z$$

and 
$$S^{Y} = 3I^{2}_{p}Z$$

hence

$$3S^{Y} = S^{\Delta}$$