To get a set theory without the power set axiom, I could just take an existing set theory like ZF or ZFC, and remove the power set axiom. However, perhaps I would have to be careful how to formulate the other axioms then, or have to add some sentences that were provable before in the presence of the power set axiom as additional axioms.
So if I need a set theory without the power set axiom, it seems wiser to use a theory already investigated in sufficient detail by somebody else. Of course, the theory should be sufficiently "well behaved" so that it is still used at least occasionally. (If ZF or ZFC without the power set axiom should turn out to be such theories, then of course they also qualify as an answer.)