  We construct a family of vertex algebras associated to the current algebra of
finite-dimensional abelian Lie algebras along with their modules and
logarithmic modules. We show this family of vertex algebras and their modules
are quasi-conformal and strongly $\N$-graded and verify convergence and
extension property needed in the logarithmic tensor category theory for
strongly graded logarithmic modules developed by Huang, Lepowsky and Zhang.
