  In this article we derive the expression of \textit{renormalized energies}
for unit-valued harmonic maps defined on a smooth bounded domain in
\(\mathbb{R}^2\) whose boundary has several connected components. The notion of
renormalized energies was introduced by Bethuel-Brezis-H\'elein in order to
describe the position of limiting Ginzburg-Landau vortices in simply connected
domains. We show here, how a non-trivial topology of the domain modifies the
expression of the renormalized energies. We treat the case of Dirichlet
boundary conditions and Neumann boundary conditions as well.
