# WAMs Based Eigenvalue Space Model for High Impedance Fault Detection
## 3 HIF Detection in Eigenvalue Space
### Framework for Zones of Protection in Eigenvalue Space  
The output of the Subspace estimator is used to build eigenvalue clusters. These clusters are created using \(k\)-means clustering techniques. The clusters are dynamic, and can be updated at user specified intervals or whenever new data becomes available. In this work, simulated PMU readings collected over a 24-h period were used to train the clustering algorithm. The load profile, illustrated in Figure 6, was modeled after Houston, TX with data provided by ERCOT [29].  
In order to facilitate the creation of the zones of protection and improve their reliability, statistical metrics are used to evaluate the suitability of the clusters. As part of this work a metric referred to as the Eigenvalue Drift Coefficient (EDC) was developed. After the clustering algorithm has converged, vectors containing the Euclidean distance between pairs of eigenvalues in each cluster are created. From this set of values, the mean distance  
Figure 5: Estimated eigenvalues in eigenvalue space.  
Figure 6: Houston load 24-h profile in PU [29].  
and standard deviation within each cluster is computed. The EDC of each cluster is calculated as follows:  
\[EDC=\frac{\sigma}{\mu} \tag{19}\]  
where \(\sigma\) is the standard deviation, and \(\mu\) is the mean of the distances.  
Clusters with a high EDC tend to have a relatively uniform distribution compared to clusters with a low EDC. Clusters with high EDC yield zones of protection which are more secure. The clustering step is repeated until all clusters produce acceptable EDC values. Polynomial curve fitting is then used to define the zones of protection. Low and high EDC clusters are illustrated in Figures 7 and 8, respectively. High EDC values allow for the use of lower degree polynomials for curve fitting.  
The number of clusters used by the algorithm is determined iteratively, starting with 6 clusters, and decreasing the number of clusters until EDCs higher than 1.7 are achieved for all clusters. At EDCs higher than 1.7, polynomials of degree 3 and less can be used to produce the zones of protection. For clusters with an EDC at or less 1.7, polynomials of degrees 3 and less might not provide a suitable fit. In addition to having to utilize higher degree polynomials for curve fitting, a low EDC also leads to protection zones where most of the eigenvalues are not at the center of the zone. Off-centered eigenvalues ultimately lead to zones with inconsistent safety margins, which could lead to false positives if eigenvalues  
Figure 8: High EDC cluster.  
Figure 7: Low EDC cluster.  
near the edge of the zone drift outside during normal operation, or delayed reaction during an abnormal condition for eigenvalues located in a particularly wide area of the zone.  
Safety margins are created using the mean \(\mu\) of the cluster, and a multiplier corresponding to the magnitude of \(\mu\). For higher values of \(\mu\), the algorithm uses smaller multipliers, while higher multipliers are used for smaller values of \(\mu\). Two zones of protection are used for each cluster. An inner zone where the eigenvalues reside during normal operation, and an outer zone that generates an immediate reaction. If an eigenvalue drifts to the area in between the two zones, a timer is started. If the eigenvalue returns to the inner zone before the timer expires, no action is taken. However, if the eigenvalue doesn't return to the inner zone before the time expires or continues to drift beyond the second zone, action is taken. For this work, a time period equivalent to the cluster update time was used.