  The energy mix of future power systems will include high shares of wind power
and solar PV. These generation facilities are generally connected via
power-electronic inverters. While conventional generation responds dynamically
to the state of the electric power system, inverters are power electronic
hardware and need to be programmed to react to the state of the system.
Choosing an appropriate control scheme and the corresponding parameters is
necessary to guarantee that the system operates safely. A prominent control
scheme for inverters is droop control, which mimics the response of
conventional generation. In this work, we investigate the stability of coupled
systems of droop-controlled inverters in arbitrary network topologies.
Employing linear stability analysis, we derive effective local stability
criteria that consider both the overall network topology as well as its
interplay with the inverters' intrinsic parameters. First, we explore the
stability of an inverter coupled to an infinite grid in an analytic fashion and
uncover stability and instability regions. Secondly, we extend the analysis to
a generic topology of inverters and provide mathematical criteria for stability
and instability of the system. Last, we showcase the usefulness of the criteria
by examining two model systems using numerical simulations. The developed
criteria show which parameters might lead to an unstable operating state.
