Obstructions to deformations of DG modules

Let k be a field and n greater than or equal to 1. There exist a differential graded k-module (V,d) and various approximations to a differential d + td(1) + t(2)d(2) + ... + t(n)d(n) on V[[t]]; one of which gives a non-trivial deformation, another is obstructed, and another is unobstructed at order n. The analogous problem in the category of k-algebras in characteristic zero remains a long-standing open question.