Supplementary File: Revealing Cascading Failure Vulnerability in Power Grids using Risk-Graph

In current literatures, many simulation methods have been proposed to investigate the vulnerability of power grids [1]. Generally speaking, high-level statistical models, e.g. the CASCADE model [2], provide some statistical and probabilistic methods to study cascading failures; historical data methods try to find failure patterns from the historical blackout records [3]; deterministic models are the most important and prevalent models in studying the vulnerability of power grids, within which N − x contingency analysis [4] is the biggest family; network models mainly employ the topological properties of power grid networks, e.g. “betweenness” and “degree”, to mimic the power distribution in power grids. In addition, network theory models are widely adopted to study the malicious attack strategies against power grids. There are two prevailing network models, the recoverable model [5] and the non-recoverable model [6]. The major difference between them is the assumption of load definition and load redistribution after occurrences of failures or overloading of nodes/links [7]. Under the recoverable model, the load of each node/link is defined as betweenness [5]. To calculate the betweenness of nodes/links, we need to find the shortest path between each pair of generator and distribution substations. For example, if there are NG generator and ND distribution substations, there are NG × ND generator-distribution pairs. The betweenness of a node/link is then defined as the number of such shortest paths going through this node/link. When one or more nodes/links are knocked down, the shortest paths that originally pass through them need to be detoured, and then the load of surviving nodes/links is redistributed by recalculating the betweenness. Compared to the recoverable model, the load of a node/link under the non-recoverable model is defined as the multiplication of its degree with the