# Reliability of Supply and the Impact of Weather Exposure and Protection System Failures +
## 5 Conclusions  
In this paper we have shown a compact and generalized method of including protection system failures in power system reliability analysis, based on a graph-theoretical approach. The use of adjacency matrices in the approach is novel. This incorporates the topology of the grid and possible paths of failure propagation due to protection system misoperation into the reliability analysis as a set of matrix operations. The method was extended further to account for time-series of failure probability in the analysis, allowing for inclusion of both protection system misoperation and time-varying failure probabilities throughout the year due to weather exposure.  
Our investigations confirm the hypothesis that the combination of failure bunching effects and protection system failures adversely impact power system reliability. A case study was presented to show the effect on reliability of supply when weather exposure and protection system misoperation were implemented into the analysis. The case study shows that taking protection system reliability and the adjacency of transmission lines into account can have a large impact on the contribution to annual ENS from certain cut-sets due to the propagation of protection system failures. Since protection system misoperation follows an initial failure, propagating failures cluster around periods of high failure probability from other causes, and further increase risks associated with failure bunching effects. Thus, taking a time-series approach to capture time-varying failure rates including protection system failures can more accurately quantify the reliability of supply. This is especially important for more frequently occurring second order cut-sets. Taking the time-series approach would also be especially relevant when considering multiple operating states throughout the year. The results of the analysis can support decision makers in risk-based prioritization of grid hardening, inspection or maintenance efforts.  
The compact and generalized method presented can be implemented with ease on large power systems. The results can be used to prioritize preventive and corrective measures aiming to reduce risks associated with unwanted events in the power system. The use of adjacency matrices to incorporate dependencies between component outages in the reliability evaluation was exemplified using an approximate method and an analytical time-series method. The use of the probability matrices could, however, also be implemented into Monte Carlo based tools to the same effect.  
Conceptualization, E.S.K.; methodology, E.S.K.; original draft preparation, E.S.K.; review and editing, G.H.K.; visualization, E.S.K.; supervision, G.H.K. All authors have read and agreed to the published version of the manuscript.  
The research leading to these results has received funding through the project \"Analysis of extraordinary events in power systems\" (HILP) (Grant No. 255226), co-funded by the Research Council of Norway, Stantett and Fingrid.  
Not applicable.  
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.