# Reliability of Supply and the Impact of Weather Exposure and Protection System Failures +
## 4 Case Study  
A test case is constructed to exemplify the combined effect of failure bunching due to weather and protection system failures on power systems reliability. The test case is based on the topology and annual failure rates from the 24-bus IEEE Reliability Test System (RTS) [36] with added protection systems. Only Permanent Outage Rates (POR) for the branches are considered. A representation of the system with numbered branches can be seen in Figure 4. A contingency enumeration approach is employed to evaluate the reliability of the system. A consequence analysis is performed at peak load, using an AC OPF algorithm with load shedding, as described in [37] yielding interrupted power at load points for the different cut-sets. The calculated interrupted power is used together with the unavailability of the cut-sets to calculate their respective contributions to annual Expected Energy Not Supplied (ENS) of the system. Time series of hourly probability of failure due to wind conditions is calculated according to the method outlined in [19]. The time series covers 30 years of hourly estimated failure probabilities due to wind for actual lines in the Norwegian transmission system based on historical weather. This incorporates correlated weather exposure between lines in the data material. For the time-series, 75 percent of the failure probability is assumed to be constant, while 25 percent is scaled wind dependent failure probability varying at an hourly interval. Transformers and cables are assumed unaffected by wind, and are given a constant failure rate throughout the year. Annual failure rates due to permanent outages for the branches and the associated outage durations can be found in [36]. All protection systems are assumed to have an annual failure rate \(\lambda_{i}^{s}=0.025\). The probability of missing operation of the protection system is assumed to be \(P_{s}^{s,m}=0.0205\). The probability of unwanted non-selective tripping of the protection systems is assumed to be \(P_{l}^{s,u}=0.007\). Repair of protection system units, relevant to FT2, is assumed to be 2 h. FT3 and FT4 are associated with a 0.5 h switching time. These parameter values are in line with the case study presented in [17]. Unavailability of the different cut-sets are calculated in four ways: (1) A base case using an approximate method of reliability evaluation not including protection system failures [34]. (2) An approximate method including protection systems failures, as outlined in Section 3.1. (3) A time-series method including wind-dependent failure rates, as outlined in [10]. (4) A time-series method including wind-dependent failure rates adjusted for protection system faults, as outlined in Section 3.2. A simplification is done compared  
Figure 4: Twenty-four-bus Reliability Test System (RTS) [36] with component labels. Overhead transmission line unless otherwise noted. to the original approach in [10] when calculating time-series of unavailability, where restoration times are assumed to be constant values, rather than distributions. Adjacency matrices are created and weighted according to dependent probabilities and failure rates following the procedure in Section 3 to incorporate protection system reliability into the analysis. The results can be illustrated for the single line, branch 35, in Figure 5, where vertices are transmission lines and the directed edges show the adjacency between the lines. The line-style of the edges indicates the source line protection system side (A-side is dashed, B-side is dotted). To accumulate results, all edges leading into a vertex are summed up to see the contribution of FT3 and FT4 from the adjacent lines. Note the double connection between lines 34 and 35 appearing, causing line 34 to contribute to FT3 and FT4 at line 35 through two separate paths. Figure 6 shows a 40 hour sample of the probability of failure for the individual components in the cut-set \(\{26,27\}\), as well as the calculated unavailability of the cut-set, to illustrate the method.