Let $\Gamma$ be a principal connection on a principal bundle $\pi:P\to M$ and let $\Lambda$ be a linear connection on $M$. We describe all possible natural prolongations of $\Gam$, with respect to $\Lam$, to principal connections on the principal gauge prolongation $W^rP$ of $P$. For $r=1,2$ we give the full classification, for $r\ge 3$ we give a base of natural operators which generates all possible natural principal connections on $W^rP$.