\section{Introduction}\label{sec:Introduction}

Modelling the reliability of the electrical power system is important for effective system planning and development. Power system reliability analysis utilises analytical or simulation methods to quantify the reliability indices \cite{singh2018electric}. The failure rates of components are an important input for the reliability analysis. Traditionally, components of the same class are assigned the same failure rate, which is assumed to be time-invariant. However, the failure rates of components can vary depending on the amount of wear-out (or ageing), which can be determined by their condition data. Condition monitoring data is a physical measure of some aspects of a component's health. When utilized correctly, it can lead to improved failure rate estimation for individual components and, consequently, more accurate system reliability evaluation. Thus, suitable methodologies are needed to estimate failure rates based on the component's technical condition.

\subsection{Related work}

Li \cite{li2002incorporating} proposed a method to account for ageing failures in power system reliability evaluation by defining an unavailability due to ageing failures. This approach enabled the analysis of the effects of aged cables on system reliability. The results showed that age had a significant impact on the reliability indices. However, the limitation of this study was that failure rates do not always increase due to age alone but rather due to the condition of a component as a result of usage \cite{he2009hybrid}. Therefore, methodologies were needed to integrate condition data into power system reliability analysis. Awadallah \textit{et al.} \cite{awadallah2014influence} demonstrated the use of an Arrhenius-Weibull probability distribution to model the effects of thermal stress on transformer end-of-life failures. This approach and others \cite{he2007transformer} \cite{he2009hybrid} made it possible to include the effect of the transformer hot spot temperature on power system reliability. To include numerous condition measurements, Jahromi \textit{et al.} \cite{jahromi2009approach} analysed transformer condition data and aggregated the measurements into a single component health index. The health index was used to assist in decision-making regarding end-of-life asset replacement.

However, health indices are not direct inputs to system-level reliability analysis, but rather, failure rates are required. Therefore, Toftaker \textit{et al.} \cite{toftaker2023accounting} presented a methodology capable of integrating transformer condition data into power system reliability analysis by creating a condition-dependent failure rate from the component's health index. The methodology used information from a database that contained condition data from scrapped components to account for preventive retirements. To the best of our knowledge, only studies on the ageing and condition of cables and transformers have been reported in the literature for power system reliability.

 High-voltage circuit breakers (HVCBs) are critical system components that should be accounted for in system-level reliability analysis. HVCBs introduce new failure modes due to their distinct role in the power system as part of protection systems for transmission lines. To capture this, four protection system fault types (FTs) were described by Vadlamudi \textit{et al.} \cite{vadlamudi2013dependability} \cite{vadlamudi2014impact} \cite{gjerde2016enhanced} to calculate an equivalent line failure rate: FT1 was related to the line's own failure rate, FT2 was due to the HVCB tripping without a command, FT3 was due to the HVCB failing to trip on command, and FT4 was due to relay selectivity issues. Mathematical models of the HVCB fault types were formulated and incorporated into a methodology \cite{Kjolle-Gjerde_TR_A7175_OPAL} for reliability studies based on approximate methods. Kiel \textit{et al.} \cite{kiel2020reliability} generalised the four mathematical models to incorporate the topological dependencies in the system through the use of an adjacency matrix. 

\subsection{Outline and contributions}

This paper's main contribution is the proposal of a methodology for integrating HVCB condition data into power system reliability analysis. It also examines the sensitivities between the system-level reliability indices and the HVCB condition parameters.

A health index model where HVCB condition data was used to quantify a health index using a framework based on failure modes, effects and criticality analysis (FMECA) was conceptualised in \cite{grant2023health}. The present work extends the health index model by segmenting the HVCB failure mechanisms to create a health index for each HVCB failure mode: 1) tripping without a command and 2) failing to trip on command. Algorithms are also presented to assess the onset of failure mechanisms from condition monitoring data and scoring the degradation associated with these failure mechanisms. A condition parameters model that translated health indices into condition-dependent failure rates using the Cox Proportional Hazards Model was developed in \cite{Grant_RAMS}. The present work extends this condition parameters model to quantify condition-dependent HVCB reliability parameters for each of the two HVCB failure modes. The last step of the methodology integrates the above component-level models with existing methods \cite{vadlamudi2013dependability} \cite{vadlamudi2014impact} \cite{gjerde2016enhanced} \cite{Kjolle-Gjerde_TR_A7175_OPAL} \cite{kiel2020reliability} related to protection system failures and approximate methods for power system reliability analysis.

Enabled by the methodological contributions described above, this paper aims to answer the following research questions:
\begin{enumerate}
    \item How can HVCB condition data be integrated into power system reliability analysis?
    \item How does the condition of an HVCB influence the likelihood of the HVCB tripping without a command or failing to trip on command?
    \item What are the effects of unwanted and missing HVCB operation on the system-level reliability indices?
    \item What are the sensitivities between the reliability indices and the HVCB condition parameters?
\end{enumerate}

The rest of this paper is structured as follows: Section \ref{sec:Methodology} presents the proposed methodology for incorporating HVCB condition data into power system reliability analysis. Section \ref{sec:Case Study} presents a case study with real HVCB condition data from the Icelandic transmission system and investigates the system-level effects on a 4-bus test system. A discussion is provided in Section \ref{sec:Discussion} before conclusions are drawn in Section \ref{sec:Conclusion}.