## 3 Results
### Cluster Risk to Transmission Owner (TO)
#### 3.2.2 Company Risk Assessment  
The cluster statistics presented in the previous sections can be used to evaluate a company risk caused by clusters of overlapping outages. The impact I of a cluster can be defined, for example, as its size or, in more sophisticated way, as the sum of equivalent MVA values of transmission elements involved in this cluster. The likelihood of a cluster can be estimated as follows. The expected number \(n_{k}(7)\) of clusters of size k over 7 years for a company A is estimated by:  
\[n_{k}(7)=N_{k}(7)^{*}Inv(A)/Inv(TADS) \tag{1}\]  
where \(N_{k}(7)\) is the number of clusters of size k in the TADS data from 2013 to 2019, listed in the last column of Table 2, Inv(A) is the company A inventory of transmission elements of the 200 kV and above reportable in TADS, and Inv(TADS) = 8910 is the average annual TADS inventory of the 200 kV+ from 2013 to 2019 (for \(k\leq Inv(A)\)). Finally, the number \(n_{k}(1)\) of clusters of size k for a given year can be estimated by \(n_{k}(7)/7\) (again under the assumption of stationarity of the number of clusters). Finally, the estimates \(n_{k}(1)\) are plugged into formula (2) for the company cluster risk R(1 year):  
\[R(1\;year)=\Sigma_{k}\;I_{k}\;^{*}n_{k}(1), \tag{2}\]  
where k is the size of a cluster and \(I_{k}\) is the impact of a cluster of size k. Estimates \(n_{k}(1)\) can be further used to calculate likelihood of a cluster of a given size for an hour, a day etc. The company risk, as defined by (2), is proportional to a company inventory and the time period for which the risk is estimated. For example, assuming that a cluster impact is defined as the cluster size (\(I_{k}=k\)), for a company X with the transmission inventory of 62 elements with voltages above 200 kV, the 1-year cluster risk R(1 year) = 6.9 is calculated from 1-year estimates of number of clusters listed in Table 4.  
For two actual TOs that report to TADS, anonymized companies A and B with similar inventory of about 62 elements a year, the numbers of observed clusters from 2013 to 2019 were as follows: company A experienced three clusters of size 2; company B had 13 clusters of size 2, six clusters of size 3, and two clusters of size 4 (the total of 20 clusters). Figure 5: Distribution of number of clusters by Transmission Owner (TO) for 7 years (2013–2019). These empirical data show that for company B the suggested methodology provides good estimates of the number of clusters based on the combined inventory alone; however, it is not the case for company A. More general evaluation of the cluster risk estimates is illustrated in Figure 6. Figure 6A shows a histogram of the expected 7-year cluster risk for all TOs that reported in TADS from 2013-2019 and had at least one transmission element above 200 kV, with the cluster risk for each company calculated by formula (2) adjusted for 7 years. Figure 6B shows a histogram of actual cluster impact for the same set of TOs; for these calculations a likelihood of a cluster of a given size is replaced with the number of the clusters the TO had in 2013-2019. The histograms (A) and (B) are reasonably close in the middle parts of the corresponding distributions. The predictable differences between them are as follows. The distribution (A) starts with a small positive value, as even a TO with one reportable element has a positive expected cluster risk defined by formula (2), while the distribution (B) starts with value 0 since some TOs with few elements had no clusters over the years 2013-2019. A more prominent difference between (A) and (B) is observable in their right-hand tails. Distribution (B) has as outliers the companies that experienced multiple and/or large clusters caused, for example, by wildfires and hurricanes during the years 2013-2019 while distribution (A) of the expected cluster risk by definition consists of the sample mean values for the companies of the same size and outliers tend to be \"absorbed\". If a time frame of the analysis increases, the histograms (A) and (B) are expected to become closer and closer.