  In this paper we consider a geothermal energy storage in which the
spatio-temporal temperature distribution is modeled by a heat equation with a
convection term. Such storages often are embedded in residential heating
systems and control and management require the knowledge of some aggregated
characteristics of that temperature distribution in the storage. They describe
the input-output behaviour of the storage and the associated energy flows and
their response to charging and discharging processes. We aim to derive an
efficient approximative description of these characteristics by a
low-dimensional system of ODEs. This leads to a model order reduction problem
for a large scale linear system of ODEs arising from the semi-discretization of
the heat equation combined with a linear algebraic output equation. In a first
step we approximate the non time-invariant system of ODEs by a linear
time-invariant system. Then we apply Lyapunov balanced truncation model order
reduction to approximate the output by a reduced-order system with only a few
state equations but almost the same input-output behavior. The paper presents
results of extensive numerical experiments showing the efficiency of the
applied model order reduction methods. It turns out that only a few suitable
chosen ODEs are sufficient to produce good approximations of the input-output
behaviour of the storage.
