Small-signal stability of power systems with voltage droop

The stability of inverter-dominated power grids remains an active area of research. This paper presents novel sufficient conditions for ensuring small-signal stability in lossless and constant $R/X$ grids with highly heterogeneous mixes of grid-forming inverters that implement an adapted $V$-$q$ droop control. The proposed conditions can be evaluated in the neighborhood of each bus without information on the rest of the grid. Apart from the presence of $V$-$q$ droop, no additional assumptions are made regarding the inverter control strategies, nor is dynamical homogeneity across the system assumed. The analysis is enabled by recasting the node dynamics in terms of complex frequency and power, resulting in transfer functions that directly capture the small-signal frequency and amplitude responses to active and reactive power imbalances. These transfer functions are directly aligned with typical design considerations in grid-forming control. Building on an adapted small-phase theorem and viewing the system as a closed feedback loop between nodes and lines, the derived stability conditions also yield new insights when applied to established inverter control designs. We demonstrate in simulations that our conditions are not overly conservative and can identify individual inverters that are misconfigured and cause instability.