Nonlinear stability measures of synchronised states in a power-grid model

Nonlinear stability of desired dynamics in multi-stable systems (systems with more than one attracting state) depends on the shape and size of its basins of attraction. ‘Basin stability’ estimates the volume of a state’s basin of attraction and estimates the probability that a random initial condition evolves towards the state. If properties of the random initial conditions used are analysed, then basin stability can also provide estimates of the shape of the basin of attraction, but which are coarse-grained and lack details of small-scale features of its boundary. The closest approach of the basin boundary to the state can be computed via an optimisation procedure, providing minimum perturbation amplitudes to leave the desired region. Minimal disturbances are missed by basin stability estimations (by two orders of magnitude in perturbation energy for transition to fluid turbulence) and so offer a complimentary nonlinear stability measure to basin stability.

Minimal disturbances for desynchronisation are computed in the ‘swing equation’, a network of secondorder Kuramoto oscillators which acts as a simple model for power-grid dynamics, in small fournode power-grids and a complex model UK power-grid. The amplitudes of minimal disturbances vary non-monotonically with the number of connections in the grid, depending on the details of the dynamical evolution of the perturbation across the grid. A comparison between the amplitude of minimal disturbances and basin stability for a range of nodal powers and dissipation rates shows that these nonlinear stability measures evolve independently, emphasising the need for both measures to be used in the design of nonlinear systems. The desynchronised dynamics of large power-grids are investigated in detail; an asymptotic expansion is developed to explain the grid wide dynamics of single-node ‘dead-end’ desynchronisation events.