Small-signal Stability of Grid-tied Inverter Networks Saber Jafarpour a , Victor Purba b , Sairaj V. Dhople b , Brian Johnson c , Francesco Bullo a a Department of Mechanical Engineering and the Center of Control, Dynamical Systems and Computation, University of California Santa Barbara, CA 93106, USA b Department of Electrical and Computer Engineering at the University of Minnesota, Minneapolis, MN 55414 USA c Department of Electrical Engineering at University of Washington, Seattle, Washington, WA 98195 USA Abstract This paper considers the small-signal stability of electrical networks composed dominantly of three-phase grid-following inverters. We identify a suitable time-scale decomposition for the inverter dynamics and, using singular perturbation theory, we obtain an analytic sufficient condition for the small-signal stability of the network. In contrast to the alternative of performing an eigenvalue analysis of the full-order network dynamics, our analytic sufficient condition has the benefit of reducing computational complexity and yielding insights on the role of network topology and constitution as well as inverter filter and control parameters on small-signal stability. Our numerical analysis for an inverter network with radial topology validates the approach and illustrates that, in a wide parametric regime, our analytic condition coincides with the exact stability threshold. Key words: networks of inverters, dynamical system analysis, stability analysis 1 Introduction Problem description and motivation The ongo- ing shift from fossil-fuel-driven synchronous generators to power-electronics-interfaced renewable energy is lead- ing to changes in how power grids are modeled, an- alyzed, and controlled. While synchronous generators are generally rated at several hundreds of MVA and in- stalled on the transmission backbone, power electron- ics inverters are distributed across both transmission and distribution subsystems and are generally much smaller in capacity. Moreover, synchronous generators have large rotating masses that absorb supply-demand ⋆ This paper was not presented at any IFAC meeting. This work was supported in part by the U.S. Department of En- ergy (DOE) Solar Energy Technologies Office under Con- tract No. DE-EE0000-1583 and the National Science Foun- dation under award ECCS-1453921. Corresponding author S. Jafarpour. Tel. +1 805-893-5169 Email addresses: saber.jafarpour@engineering.ucsb.edu (Saber Jafarpour), purba002@umn.edu (Victor Purba), sdhople@umn.edu (Sairaj V. Dhople), brianbj@uw.edu (Brian Johnson), bullo@engineering.ucsb.edu (Francesco Bullo). fluctuations and limit frequency excursions during tran- sients, whereas inverters typically have very different dynamics—attributable dominantly to their digital con- trollers [39]—and they possess no moving parts. In sum- mary, future grids will have a highly distributed archi- tecture as inverters assume a more prominent role, and this will concomitantly necessitate the development of compatible models and analysis approaches to ensure stability and reliability. Broadly speaking, there are two main modes of control for the inverters in power networks. The first is grid- forming mode, in which terminal voltages of the invert- ers are regulated with the understanding that the invert- ers dictate the voltage magnitude and frequency in the network. Prior works in the literature on grid-forming controls include design of distributed PI controllers for secondary frequency control and power sharing in induc- tive networks with radial topology [33,34] and meshed topology [31], stability analysis using the virtual oscil- lator model [8,35], model-order reduction using singular perturbations [21], and designing controllers for a paral- lel network of identical grid-forming inverters using pas- sivity [36]. The second mode of control is grid-following, in which inverters inject currents while synchronized to Preprint submitted to Automatica 8 February 2019 arXiv:1902.02478v1 [math.OC] 7 Feb 2019