  In this paper we study the problem of model reduction of linear network
systems. We aim at computing a reduced order stable approximation of the
network with the same topology and optimal w.r.t. H2 norm error approximation.
Our approach is based on time-domain moment matching framework, where we
optimize over families of parameterized reduced order models matching a set of
moments at arbitrary interpolation points. The parameterization of the low
order models is in terms of the free parameters and of the interpolation
points. For this family of parameterized models we formulate an
optimization-based model reduction problem with the H2 norm of error
approximation as objective function while the preservation of some structural
and physical properties yields the constraints. This problem is nonconvex and
we write it in terms of the Gramians of a minimal realization of the error
system. We propose two solutions for this problem. The first solution assumes
that the error system admits a block diagonal observability Gramian, allowing
for a simple convex reformulation as semidefinite programming, but at the cost
of some performance loss. We also derive sufficient conditions to guarantee
block diagonalization of the Gramian. The second solution employs a gradient
projection method for a smooth reformulation yielding (locally) optimal
interpolation points and free parameters. The potential of the methods is
illustrated on a power network.
