We give a sufficient condition for a countable model $M$ of PA to be expandable to an $\omega$-model of AST with absolute $\Omega$-orderings. The condition is in terms of saturation schemes or, equivalently, in terms of the ability of the model to code sequences which have some kind of definition in $(M, \omega)$. We also show that a weaker scheme of saturation leads to the existence of wellorderings of the model with nice properties. Finally, we answer affirmatively the question of whether the intersection of all $\beta$-expansions of a $\beta$-expandable model $M$ is the set $\mathrm{RA}(M, \omega)$--the ramified analytical hierarchy over $(M, \omega)$. The results are based on forcing constructions.