\section{Case Study}\label{sec:Case Study}

A case study was conducted using a dataset of 83 TCC measurements from 38 HVCBs from the Icelandic TSO. The health index model was employed to analyse the 83 TCC measurements, feature points from the TCC measurements were identified, and diagnostic data was computed. The relevant diagnostic data was then used to quantify a score for the amount of degradation of each failure mechanism using the algorithm in Table \ref{tab:EJS_alg}. After evaluating the scores for all failure mechanisms and combining them with the corresponding RPN values in Figures \ref{tab:RPN_Ls} and \ref{tab:RPN_Pm}, the health indices were calculated using the algorithm in Table \ref{tab:HI_alg}. 

Kernel density estimation (KDE) \cite{wkeglarczyk2018kernel} with a Gaussian kernel is used to estimate and plot the probability density function (pdf) for the resulting health indices of the HVCB data set, as shown in \ref{fig:HI_density_plot}. Compared with histograms, KDE produces a smoother pdf since it is not dependent on the subjective choice of the number of bins. This is especially useful when comparing two or more distributions since they may have different bin sizes.

\begin{figure}[]
\centering
\includegraphics[width=\linewidth]{IMG/HI_density_plot.pdf}
\caption{Distribution of health indices for the HVCB data set.}
\label{fig:HI_density_plot}
\end{figure}

The boundary cases of the HVCBs in the data set with highest/lowest $HI\_\lambda^s$ and $HI\_P^m$ values are presented in Table \ref{tab:EJS} with the corresponding diagnostic data and scores for the failure mechanisms assessed using TCC measurements. The TCC measurements with the lowest and highest health indices for $HI\_\lambda^s$ are shown in Figures \ref{fig:TCC_Ls_low} and \ref{fig:TCC_Ls_high} while the TCC measurements with the lowest and highest health indices for $HI\_P^m$ are shown in Figures \ref{fig:TCC_Pm_low} and \ref{fig:TCC_Pm_high}.

\begin{table*}[]
\centering
\caption{Diagnostic data and scores for the boundary cases. Units for t\_latch and t\_open are ms; units for max\_r are ohms; units for avg\_I are amps; units for noise are amps per ms.}
\label{tab:EJS}
\begin{tabular}{cccccccccccccc}
\toprule
 & \multicolumn{6}{c}{\textbf{Diagnostic data}} & \multicolumn{7}{c}{\textbf{Scores}} \\ \cline{2-14} 
\textbf{Case} & \textbf{t\_latch} & \textbf{noise} & \textbf{avg\_I} & \textbf{t\_open} & \textbf{max\_r} & \textbf{ab\_anom} & \textbf{FM01} & \textbf{FM02} & \textbf{FM03} & \textbf{FM04} & \textbf{FM06} & \textbf{FM07} & \textbf{FM12} \\ \midrule
Lowest $HI\_\lambda^s$ & 0.50 & 36.85 & 2.08 & 21.50 & 15222 & 0.01 & 1.68 & 1.00 & 1.00 & 4.00 & 4.00 & 4.00 & 4.00 \\
Highest $HI\_\lambda^s$ & 11.00 & 0.33 & 2.66 & 49.05 & 15222 & 0.11 & 4.00 & 4.00 & 4.00 & 4.00 & 4.00 & 3.66 & 3.66 \\
Lowest $HI\_P^m$ & 10.40 & 0.01 & 0.44 & 58.85 & 15222 & 0.44 & 4.00 & 4.00 & 4.00 & 1.00 & 4.00 & 1.04 & 1.04 \\
Highest $HI\_P^m$ & 1.60 & 0.01 & 0.49 & 29.28 & 15222 & 0.04 & 3.18 & 3.99 & 3.99 & 4.00 & 4.00 & 4.00 & 4.00 \\
 \bottomrule
\end{tabular}
\end{table*}

\begin{figure}[]
\centering
\includegraphics[width=\linewidth]{IMG/Test 35.pdf}
\caption{TCC measurements with the lowest $HI\_\lambda^s$.}
\label{fig:TCC_Ls_low}
\end{figure}

\begin{figure}[]
\centering
\includegraphics[width=\linewidth]{IMG/Test 5.pdf}
\caption{TCC measurements with the highest $HI\_\lambda^s$.}
\label{fig:TCC_Ls_high}
\end{figure}

\begin{figure}[]
\centering
\includegraphics[width=\linewidth]{IMG/Test 15.pdf}
\caption{TCC measurements with the lowest $HI\_P^m$.}
\label{fig:TCC_Pm_low}
\end{figure}

\begin{figure}[]
\centering
\includegraphics[width=\linewidth]{IMG/Test 2.pdf}
\caption{TCC measurements with the highest $HI\_P^m$.}
\label{fig:TCC_Pm_high}
\end{figure}

FM01 was assessed based on the latch time (t\_latch). The average latch time for the entire fleet was determined to be 2.2 ms. Any latch time below this threshold was considered a potential unsecured latch and thus given a downgraded degradation score. FM02 and FM03 were assessed based on the level of noise detected in the TCC measurement. The noise level was determined for the entire fleet by calculating the derivative of the coil current past point D, which should remain constant. Higher values were considered a potential indicator of FM02 and FM03. FM04 was evaluated based on the average current during the test (avg\_I) and HVCB tripping time (t\_open). The average current of the fleet during testing was determined to be 2.2 amps, and the acceptable tripping time was determined to be 35 ms from specification sheets. The maximum resistance (max\_r) was extracted from the condition data and used for scoring FM06. For 72 out of 83 HVCBs, the normal maximum resistance when the HVCB had tripped increased to 15,221 ohms. The assessment of FM07 and FM12 was based on the anomalies found on the TCC while the latch was being pressed (ab\_anom). Therefore, the magnitude of the anomalies between points A and B was quantified. The resulting health indices and condition parameters for the boundary cases of $HI\_\lambda^s$ and $HI\_P^m$ are presented in Table \ref{tab:Cond}.

\begin{table}[]
\centering
\caption{Health indices and condition parameters for the boundary cases. }
\label{tab:Cond}
\begin{tabular}{ccccc}
\toprule
 & \multicolumn{2}{c}{\textbf{Health indices}} & \multicolumn{2}{c}{\textbf{Condition parameters}} \\ \cline{2-5}
\textbf{Case} & \textbf{$HI\_\lambda^s$} & \textbf{$HI\_P^m$} & \textbf{$\lambda^s$} & \textbf{$P^m$} \\ \midrule
Lowest $HI\_\lambda^s$ & 0.569 & 1.000 & 0.044 & 0.021 \\
Highest $HI\_\lambda^s$ & 1.000 & 0.980 & 0.028 & 0.021 \\
Lowest $HI\_P^m$ & 0.797 & 0.824 & 0.035 & 0.024 \\
Highest $HI\_P^m$ & 0.933 & 1.000 & 0.030 & 0.021 \\ \bottomrule
\end{tabular}
\end{table}

The coefficients of the Cox PHM were determined for an outage database of the life histories of 464 HVCBs from the Icelandic transmission grid and was used to calculate the baseline failure rate $\lambda_0$. The outage database contained information on HVCB uptime, whether a failure was observed in its lifetime, and covariates such as voltage level, manufacturer, age and substation. For covariates to be used in the Cox PHM, they must satisfy the Cox PHM assumption, which states that the ratio of the hazards for different covariates must be constant over time. The voltage and manufacturer covariates passed the Cox PHM assumption, but the age and substation covariates failed the assumption. Since the manufacturer covariate would reveal sensitive information regarding relative failure rates between the manufacturers, the case study was completed using the voltage covariate only.

The condition parameter $\lambda^s$ was obtained using Equation \ref{eq:Ls}, and the model results were multiplied by the calibration coefficient in Equation \ref{eq:CC}. The historically observed failure rate from the outage database was 0.0334 f/yr. The effect of calibration is seen in Figure \ref{fig:calibration}. The condition parameter $P^m$ was obtained using Equation \ref{eq:Pm}. However, in contrast to the calculation of $\lambda^s$, for the calibration of $P^m$ no database existed with statistics about HVCBs not tripping when called for, since they are rare events and not commonly logged. Therefore, the baseline value for $P_0$ used was 0.0205 as taken from literature \cite{vadlamudi2013dependability} \cite{vadlamudi2014impact}.

\begin{figure}[]
\centering
\includegraphics[width=\linewidth]{IMG/Calibration.pdf}
\caption{Distribution of failure rates for the HVCB data set before and after calibration to the historical average.}
\label{fig:calibration}
\end{figure}

The 4-bus test system \cite{Kjolle-Gjerde_TR_A7175_OPAL} illustrated in Figure \ref{fig:OPAL_net} was selected to demonstrate in a transparent manner how HVCB condition impacts the power system reliability indices. This also facilitated the comparison between the reliability indices when considering HVCB condition and the results obtained in \cite{vadlamudi2013dependability} \cite{vadlamudi2014impact} for the same test system without considering HVCB condition.

\begin{figure}[]
\centering
\includegraphics[width=2.5in]{IMG/OPAL Network.pdf}
\caption{The 4-bus test system single-line diagram.}
\label{fig:OPAL_net}
\end{figure}

The transmission line equivalent failure rates were calculated using Equations \ref{eq:PT3} to \ref{eq:lam_eq}, where additional reliability data for the test system's components can be found in Table \ref{tab:Reliability_data}.

\begin{table}[]
\centering
\caption{4-bus test system reliability data \cite{Kjolle-Gjerde_TR_A7175_OPAL}}
\label{tab:Reliability_data}
\begin{tabular}{ccc}
\toprule
\textbf{Component} & \textbf{$\lambda$ (f/yr)} & \textbf{r (hours)} \\ \midrule
Line 1 & 2 & 20 \\
Line 2 & 3 & 15 \\
Line 3 & 4 & 12 \\
Line 4 & 5 & 10 \\
Breakers & 0.025 & 2 \\ \bottomrule
\end{tabular}
\end{table}

The contingency analysis results of the 4-bus test system using the steps described in Sec. \ref{sec:cont_anal} are shown in Table \ref{tab:lookup}. A line status of 0 means that the line is up or in service, while a status of 1 indicates that the line is down or out of service. The load demand ($P$) was kept constant, and a lossless transmission system was assumed. The maximum loading capacity for all lines was assumed to be 135 MW.

\begin{table*}[]
\centering
\caption{Contingency analysis for the 4-bus test system. The units of \textbf{P}, \textbf{P\_{int}} and \textbf{SAC} are MW.}
\label{tab:lookup}
\begin{tabular}{ccccccccccc}
\toprule
\multirow{2}{*}{\textbf{State}} & \multicolumn{4}{c}{\textbf{Line status}} & \multicolumn{3}{c}{\textbf{Load 1}} & \multicolumn{3}{c}{\textbf{Load 2}} \\ \cline{2-11} 
 & \textbf{1} & \textbf{2} & \textbf{3} & \textbf{4} & \textbf{P} & \textbf{SAC} & \textbf{P\_{int}} & \textbf{P} & \textbf{SAC} & \textbf{P\_{int}} \\ \midrule
0 & 0 & 0 & 0 & 0 & 100 & 100 & 0 & 75 & 75 & 0 \\
1 & 0 & 0 & 0 & 1 & 100 & 100 & 0 & 75 & 75 & 0 \\
2 & 0 & 0 & 1 & 0 & 100 & 77.14 & 22.86 & 75 & 57.86 & 17.14 \\
3 & 0 & 1 & 0 & 0 & 100 & 77.14 & 22.86 & 75 & 57.86 & 17.14 \\
4 & 1 & 0 & 0 & 0 & 100 & 100 & 0 & 75 & 75 & 0 \\
5 & 0 & 0 & 1 & 1 & 100 & 100 & 0 & 75 & 0 & 75 \\
6 & 0 & 1 & 0 & 1 & 100 & 0 & 100 & 75 & 75 & 0 \\
7 & 0 & 1 & 1 & 0 & 100 & 0 & 100 & 75 & 0 & 75 \\
8 & 1 & 0 & 0 & 1 & 100 & 100 & 0 & 75 & 75 & 0 \\
9 & 1 & 0 & 1 & 0 & 100 & 57.14 & 42.86 & 75 & 42.86 & 32.14 \\
10 & 1 & 1 & 0 & 0 & 100 & 42.86 & 57.14 & 75 & 32.14 & 42.86 \\
11 & 0 & 1 & 1 & 1 & 100 & 0 & 100 & 75 & 0 & 75 \\
12 & 1 & 0 & 1 & 1 & 100 & 100 & 0 & 75 & 0 & 75 \\
13 & 1 & 1 & 0 & 1 & 100 & 0 & 100 & 75 & 75 & 0 \\
14 & 1 & 1 & 1 & 0 & 100 & 0 & 100 & 75 & 0 & 75 \\
15 & 1 & 1 & 1 & 1 & 100 & 0 & 100 & 75 & 0 & 75 \\ \bottomrule
\end{tabular}
\end{table*}

Approximate methods were employed to assess the reliability of the 4-bus test system. Initially, default parameters from the literature \cite{vadlamudi2013dependability} \cite{vadlamudi2014impact} were used without considering HVCB condition with $\lambda^s$ = 0.025, $P^m$ = 0.0205 and $P^u$ = 0.007. Then, for each of the 83 HVCBs in the condition data set, resulting values of the condition parameters $\lambda^s$ and $P^m$ were assumed in the test system.

The adjacency matrices for the test system were specified as per Equations \ref{eq:A_matrix} and \ref{eq:B_matrix}.

\begin{equation}
A = \begin{pmatrix}
0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0
\end{pmatrix}
\label{eq:A_matrix}
\end{equation}

\begin{equation}
    B = \begin{pmatrix}
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0
\end{pmatrix}
\label{eq:B_matrix}
\end{equation}

The cut sets, as determined by the results of the contingency analysis for delivery point 1, are states 2, 3, 6, 7, 9 and 10, and the cut sets for delivery point 2 are states 2, 3, 5, 7, 9 and 10. These states are cut sets in this system because they result in a partial or complete disconnection between the generation facilities and the delivery points under up to second-order contingencies. If the transmission lines within a cut set were not adjacent, as determined by the adjacency matrices, Equations \ref{eq:par_lam} to \ref{eq:par_U} were used to compute the equivalent basic reliability indices of the two non-neighbouring lines, which represent a parallel system. If the lines within a cut set were adjacent, as determined by the adjacency matrices, Equations \ref{eq:step1} to \ref{eq:dep_r} were used to compute the equivalent basic reliability indices of the two neighbouring lines, which account for common-mode failures. The reliability indices were quantified using Equations \ref{eq:Pint} to \ref{eq:ENS}. This resulted in one value of EENS (as well as other reliability indices) for each of the HVCBs in the condition data set. This EENS value is a quantitative measure of how the HVCB's technical condition affects the reliability of the test system.

A sensitivity analysis \cite{saltelli2008global} was performed on the condition parameters $P^m$ and $\lambda^s$. The EENS was observed as the condition parameters varied within their expected range. A linear regression on the form $EENS = \beta_0 + \beta_1 x$, for $x$ being either $P^m$ or $\lambda^s$, was applied to the results. $\beta_1$ specifies the magnitude and direction of the EENS response to the condition parameters and was used as an indicator of sensitivity. The inputs used in the sensitivity analysis are presented in Table \ref{tab:sen_inputs}. The description of the input values used for the condition parameters is as follows:

\begin{itemize}
    \item Default - The default values used in the literature
    \item Scenario 1 - The maximum possible values that could occur when scoring failure mechanisms with TCC measurements
    \item Scenario 2 - The maximum possible values that could occur when scoring all failure mechanisms.
\end{itemize}
Scenarios 1 and 2 are used to illustrate the \emph{potential} impact of technical condition on component-level and system-level reliability, which will be higher than the impact quantified for the specific HVCBs in the condition data set.

\begin{table*}[]
\centering
\caption{Inputs to the sensitivity analysis}
\label{tab:sen_inputs}
\begin{tabular}{cccccc}
\toprule
\multirow{2}{*}{\textbf{Parameter}} & \multicolumn{3}{c}{\textbf{Input values}} & \multicolumn{2}{c}{\textbf{Variation from Default}} \\ \cline{2-6}
 & \textbf{Default} & \textbf{Scenario 1} & \textbf{Scenario 2} & \textbf{Scenario 1} & \textbf{Scenario 2} \\
 \midrule 
$\lambda^s$ & 0.0250 & 0.0710 & 0.0770 & 184.07 \% & 208.06 \% \\
$P^m$ & 0.0205 & 0.0302 & 0.0557 & 47.13 \% & 171.83 \% \\ \bottomrule
\end{tabular}
\end{table*}

The sensitivity analysis results for $\lambda^s$ are displayed in Figures \ref{fig:LP1_ls} and \ref{fig:LP2_ls} and summarised in Table \ref{tab:sen_Ls}.
The sensitivity analysis results for $P^m$ are displayed in Figures \ref{fig:LP1_Pm} and \ref{fig:LP2_Pm} and summarised in Table \ref{tab:sen_Pm}.
In addition to the three scenario cases, the figures show the data points for each of the HVCBs in the condition data set being inserted in the test system. 
The EENS increases both when \( \lambda^s \) is increased and when \( P^m \) is increased. This shows how deteriorated component condition negatively impacts the reliability indices.

\begin{figure}[]
\centering
\includegraphics[width=\linewidth]{IMG/LP1_Ls_MW.pdf}
\caption{Sensitivity of the system EENS to \(\lambda^s\) for load point 1 in the test system.}
\label{fig:LP1_ls}
\end{figure}

\begin{figure}[]
\centering
\includegraphics[width=\linewidth]{IMG/LP2_Ls_MW.pdf}
\caption{Sensitivity of the system EENS to \(\lambda^s\) for load point 2 in the test system.}
\label{fig:LP2_ls}
\end{figure}

\begin{table*}[]
\centering
\caption{Outputs from the sensitivity analysis of $\lambda^s$}
\label{tab:sen_Ls}
\begin{tabular}{ccccccc}
\toprule
\multirow{2}{*}{\textbf{Load point}} & \multicolumn{3}{c}{\textbf{EENS (MWh)}} & \multicolumn{2}{c}{\textbf{Variation from Default}} & \multirow{2}{*}{\textbf{$\beta_1$}} \\ \cline{2-6}
 & \textbf{Default} & \textbf{Scenario 1} & \textbf{Scenario 2} & \textbf{Scenario 1} & \textbf{Scenario 2} &  \\ \midrule
LP1 & 2230.74 & 2239.73 & 2240.90 & 0.40 \% & 0.46 \% & 195.5 \\
LP2 & 1675.61 & 1682.37 & 1683.25 & 0.40 \% & 0.46 \% & 146.7 \\ \bottomrule
\end{tabular}
\end{table*}



\begin{figure}[]
\centering
\includegraphics[width=\linewidth]{IMG/LP1_Pm_MW.pdf}
\caption{Sensitivity of the system EENS to \(P^m\) for load point 1 in the test system.}
\label{fig:LP1_Pm}
\end{figure}

\begin{figure}[]
\centering
\includegraphics[width=\linewidth]{IMG/LP2_Pm_MW.pdf}
\caption{Sensitivity of the system EENS to \(P^m\) for load point 2 in the test system.}
\label{fig:LP2_Pm}
\end{figure}

\begin{table*}[]
\centering
\caption{Outputs from the sensitivity analysis of $P^m$}
\label{tab:sen_Pm}
\begin{tabular}{ccccccc}
\toprule
\multirow{2}{*}{\textbf{Load point}} & \multicolumn{3}{c}{\textbf{EENS (MWh)}} & \multicolumn{2}{c}{\textbf{Variation from Default}} & \multirow{2}{*}{\textbf{$\beta_1$}} \\ \cline{2-6}
 & \textbf{Default} & \textbf{Scenario 1} & \textbf{Scenario 2} & \textbf{Scenario 1} & \textbf{Scenario 2} &  \\ \midrule 
LP1 & 2230.74 & 2238.73 & 2259.87 & 0.36 \% & 1.31 \% & 827.2 \\
LP2 & 1675.61 & 1681.96 & 1698.76 & 0.38 \% & 1.38 \% & 657.2 \\ \bottomrule
\end{tabular}
\end{table*}