Small-signal stability of power systems with voltage droop

The paper’s Proposition 1 states decentralized small‑signal stability conditions for lossless grids with V–q droop: two frequency-domain inequalities on each 2×2 nodal transfer matrix and a node‑wise droop bound αn ≥ 2∑m Ỹnm V°m−1 cos(φ°n−φ°m); see (3)–(5) in the paper’s statement and Appendix D for the proof outline . The network linearization is Tnet(s) = (1/2) Ũ† (1/s) Jnet Ũ with Jnet Hermitian, and the authors show an edge-wise decomposition under the α-bound that renders Tnet semi-stable and frequency-wise semi-sectorial . Lemma 4 proves the nodal inequalities (3)–(4) are equivalent to positive definiteness of the Hermitian part of Tn(jω) (‘strict accretivity’) . The candidate solution argues via passivity: the nodal blocks −Tn are strictly negative real by the same inequalities, the network block is lossless with an energy W, and a summed storage gives Ṡ ≤ −ε∥u∥², implying internal small-signal stability. This aligns with the paper’s structure (nodes stable and sectorial; network semi-stable and semi-sectorial), albeit using passivity/KYP instead of small-phase theory. Minor issues in the candidate solution include claiming the φ–φ block is positive definite (it is Laplacian-like and has a uniform-angle zero mode; cf. the explicit zero mode noted for Jnet) and that the storage is radially unbounded; these do not overturn the main conclusion but need tightening .