## Robustness and Vulnerability of Power Grids
### Natural Failures and Robustness  
We use term \(N(0)\) to denote the number of nodes in \(\Omega(0)\) at \(t=0\) and incorporate a failure at a node \(i\) by removing it from the network. As a result, the load of node \(i\) is then distributed to its neighbouring nodes, which are shown in Equations (1)-(6). If the resultant load at any neighbouring node exceeds its capacity, this node fails. The load redistribution is repeated until the loads of all remaining nodes fall below their respective capacities, i.e., the system reaches another stable condition. The number of failed nodes during the process is denoted by \(f_{i}\). Simulating this cascading process \(n\) times yields \(f_{1}\), \(f_{2}\),...,\(f_{n}\), i.e., starting from the set of nodes \(\Omega(0)\), randomly selecting one node as failed and distributing its load in the network until the system stabilizes. The expected value of the discrete distribution thus obtained is called the normalized avalanche index, which indicates the probability of the occurrence of cascading failure in a power system, and is computed as follows:  
\[a_{r}=\frac{\sum f_{i}}{n(N(0)-1)}\,. \tag{7}\]  
We studied the behaviour of the avalanche index \(a_{r}\) after natural failures upon the tunable parameter \(\alpha\in[0,2]\) and the tolerance parameters \(\beta\in[0,1]\) and \(\gamma\in[0,1]\). These parameters affect each other and together determine the avalanche factor \(a_{r}\). Simulations were carried out for each triple \((\alpha,\beta,\gamma)\) with \(\alpha\) ranging from 0.1-2.0 with a step size of 0.1 and both \(\beta\) and \(\gamma\) ranging from 0-1.0 with a step size 0.01. The simulation will produce in each case a discrete distribution of the ratios between the number of failed nodes and the total number of nodes in \(\Omega(0)\). In order to quantify the extent of cascading failures in different types of networks, Figure 1a-c illustrates the ratios of \(a_{r}\) falling into different subranges of the tolerance parameters' ranges \(0\leq\beta,\gamma\leq 1\) for the 20 values of \(\alpha\) mentioned above.  
Remarks:  
* For scale-free power grids, Figure 1a shows the fractions of \(a_{r}\) being zero or falling into one of the ten intervals \(\frac{l}{10}<a_{r}\leq\frac{l+1}{10}(l=0,1,\cdots,9)\), respectively. As \(\alpha\) increases, the probability of a large cascading failure (\(a_{r}>0.9\), marked in red) becomes lower, and the probability of no cascading failure (\(a_{r}=0\), marked in blue) becomes higher. In particular, it is found that for \(\alpha=2\), the highest probability (77%) of not encountering a cascading failure is achieved. Scale-free power grid networks are considered robust under natural failure conditions, and they become more robust when \(\alpha\) increases, which means that, in natural failure mode, heavy-loading networks are more robust than light-loading ones.  
* For small-world power grids, Figure 1b shows that the fractions of \(a_{r}\) marked in red and blue do not depend predominantly on \(\alpha\), which means that parameter \(\alpha\) has less influence on \(a_{r}\) in the natural failure mode. Small-world power grid networks are still robust under natural failure conditions, but less robust than scale-free ones. The highest probability of not encountering a cascading failure is only 39%.
* For ER power grids, Figure 1c reveals that there are no obvious variations as \(\alpha\) increases, which indicates that parameter \(\alpha\) has no influence on \(a_{r}\) under natural failure conditions. The probability of not encountering a cascading failure is constantly 95%. Hence, Erdos-Renyi power grids are the most robust ones among the three typical network types under natural failure conditions.  
Figure 1: Probabilities of cascading failures under natural failure conditions for three network types. ER, Erdős-Renyi.