Spontaneous synchrony in power-grid networks

S1. Power-grid data. The data required for power flow calculations were obtained as follows. For the 10-generator system, known as the New England test system, the parameters were taken from ref. 47. For the 3and 50-generator systems, the parameters were taken from refs. 30 and 48, respectively. The data for the Guatemala and Northern Italy systems were provided by F. Milano (University of Castilla – La Mancha), and the data for the Poland system were provided as part of the MATPOWER software package49. The dynamic data required to calculate the synchronous state, determine its stability, and simulate equation (2) in the main text, are not all available for the real power grids, and were obtained as follows. For all systems, we assumed that before any optimization the damping coefficient and droop parameter satisfy Di + 1/Ri = 50 per unit for all generators. The parameters xd,i and Hi are available for the three test systems from the respective references mentioned above. For the Guatemala, Northern Italy, and Poland systems, we estimated xd,i and Hi using the strong correlation observed in the test systems between each of these parameters and the power Pi injected by generator i into the network, as shown in Supplementary Fig. S1. The estimated values are xd,i ≈ 92.8P−1.3 i and Hi ≈ 0.04Pi, where Pi is in megawatts, xd,i is in per unit, and Hi is in seconds.