  We obtain expansions of ergodic integrals for $\Z^d$-covers of compact
self-similar translation flows, and as a consequence we obtain a form of weak
rational ergodicity with optimal rates. As examples, we consider the so-called
self-similar $(s,1)$-staircase flows ($\Z$-extensions of self-similar
translations flows of genus-$2$ surfaces), and a variation of the Ehrenfest
wind-tree model.
