Rate of change of frequency under line contingencies in high voltage electric power networks with uncertainties

In modern electric power networks with fast evolving operational conditions, assessing the impact of contingencies is becoming more and more crucial. Contingencies of interest can be roughly classified into nodal power disturbances and line faults. Despite their higher relevance, line contingencies have been significantly less investigated analytically than nodal disturbances. The main reason for this is that nodal power disturbances are additive perturbations, while line contingencies are multiplicative perturbations, which modify the interaction graph of the network. They are, therefore, significantly more challenging to tackle analytically. Here, we assess the direct impact of a line loss by means of the maximal Rate of Change of Frequency (RoCoF) incurred by the system. We show that the RoCoF depends on the initial power flow on the removed line and on the inertia of the bus where it is measured. We further derive analytical expressions for the expectation and variance of the maximal RoCoF, in terms of the expectations and variances of the power profile in the case of power systems with power uncertainties. This gives analytical tools to identify the most critical lines in an electric power grid.