algorithm Theorem: optimal droop [FD, Simpson-Porco, & Bullo ’13, Zhao, Mallada, & FD ’14] The following statements are equivalent: (i) the economic dispatch with cost coefficients αi is strictly feasible with global minimizer (θ , u ). (ii) ∃ droop coefficients Di such that the power system possesses a unique & locally exp. stable sync’d solution θ. If (i) & (ii) are true, then θi ∼θ i , ui =−Di (ωsync −ω∗), & Di αi = Dj αj . similar results for non-quadratic (strictly convex) cost & constraints similar results in transmission ntwks with DC flow [E. Mallada & S. Low, ’13] & [N. Li, L. Chen, C. Zhao, & S. Low ’13] & [X. Zhang & A. Papachristodoulou, ’13] & [M. Andreasson, D. V. Dimarogonas, K. H. Johansson, & H. Sandberg, ’13] & . . . 14 / 32