  This paper examines model parameter estimation in dynamic power systems whose
governing electro-mechanical equations are ill-conditioned or singular. This
ill-conditioning is because of converter-interfaced power systems generators'
zero or small inertia contribution. Consequently, the overall system inertia
decreases, resulting in low-inertia power systems. We show that the standard
state-space model based on least squares or subspace estimators fails to exist
for these models. We overcome this challenge by considering a least-squares
estimator directly on the coupled swing-equation model but not on its
transformed first-order state-space form. We specifically focus on estimating
inertia (mechanical and virtual) and damping constants, although our method is
general enough for estimating other parameters. Our theoretical analysis
highlights the role of network topology on the parameter estimates of an
individual generator. For generators with greater connectivity, estimation of
the associated parameters is more susceptible to variations in other generator
states. Furthermore, we numerically show that estimating the parameters by
ignoring their ill-conditioning aspects yields highly unreliable results.
