Principal subspaces for the affine Lie algebras in types $D$, $E$ and $F$

We consider the principal subspaces of certain level $k⩾1$ integrable highest weight modules and generalized Verma modules for the untwisted affine Lie algebras in types $D$, $E$ and $F$. Generalizing the approach of G. Georgiev we construct their quasi-particle bases. We use the bases to derive presentations of the principal subspaces, calculate their character formulae and find some new combinatorial identities.