  We consider the problem of stability analysis for distribution grids with
droop-controlled inverters and dynamic distribution power lines. The inverters
are modeled as voltage sources with controllable frequency and amplitude. This
problem is very challenging for large networks as numerical simulations and
detailed eigenvalue analysis are impactical. Motivated by the above
limitations, we present in this paper a systematic and computationally
efficient framework for stability analysis of inverter-based distribution
grids. To design our framework, we use tools from singular perturbation and
Lyapunov theories. Interestingly, we show that stability of the fast dynamics
of the power grid depends only on the voltage droop gains of the inverters
while, stability of the slow dynamics, depends on both voltage and frequency
droop gains. Finally, by leveraging these timescale separation properties, we
derive sufficient conditions on the frequency and voltage droop gains of the
inverters that warrant stability of the full system. We illustrate our
theoretical results through a numerical example on the IEEE 13-bus distribution
grid.
