On possible sums from multiset of powers of $d$

Let $A$ be a finite multiset of powers of a positive integer $d>1$. We describe the structure of the set $\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}(A)$ of all sums of submultisets of $A$, and in particular give a criterion of $\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}(A)=\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}(B)$ for two multisets $A,B$.