Distribution of the Sizes of theBblackouts in Power Grids, Synthetic Models, and the Motter and Lai Model under different Dynamical Rules and Criteria of Overload

Studies of the sizes of the blackouts in real grids and computer simulation models of them using the direct current approximation suggest that the resulting blackout sizes are distributed as a power law when using the standard criterion of resilience, the so-called N-1 condition: The grid must safely operate in the event of a failure of any single line. At any stage of the cascade, one of the lines whose load exceeds the maximum values imposed by the N-1 condition fails and immediately all the currents in the grids are redistributed adjusting to the new topology. On the contrary, when the grid is modeled with a uniform tolerance proportional to its initial current for all the nodes and one removes all the overloaded lines simultaneously at each stage of the cascade, the distribution of the sizes of the blackouts is bimodal as in a first the order phase transition, resulting in either a very small blackout or a very large blackout. Here we reconcile both approaches by looking at how the blackout distribution changes with model parameters and different dynamic rules of failure of the overloaded lines. We also study the Motter and Lai model of betweenness overload and find similar results, suggesting that the physical laws of flow are not the determinant factor in the problem.