The conduction current is same as displacement current when source is:

$\begin{array}{r}i=\frac{dQ}{dt}\\ =C\frac{dV}{dt}\end{array}$

If voltage across the capacitor is constant (In case of DC),  $\frac{dV}{dt} = 0$

then i = 0.

In case of DC, the displacement current is zero.

Hence, the conduction current is same as displacement current when source is AC only.