## 4 Case Study  
The case study presented in this paper models the reliability of a steam turbine. It extends the analysis presented by [PERSON] and [PERSON] [25] who illustrated the benefits of condition monitoring. Specifically, [PERSON] and [PERSON] proposed a model that can enable a conditioning monitoring process by adding some sensors for checking the status of the steam turbine subsystems and avoids undesired failures. This modelling can be implemented by the use of a 2-input PAND gate that takes as first input the condition monitoring sensor and a second input the system to monitor. In this way, the PAND triggers only if the condition monitoring sensors fail before the system is monitored. Figure 6(a) shows the Dynamic Fault Tree of the steam turbine, while Table 8 shows the main parameters of the system components [25]. As can be seen, the simplified model of the steam turbine, based on a FMEA, is characterized by the relevant failure modes that affect the functioning of the Diaphrags and the Rotor Blade. The condition monitoring depends on the sensors S1, S2, S3 and S4 which, respectively, monitor the status condition of the Diaphrags, of the Vibration faults, of the Steam Temperature and, lastly, of the Turbine. What has been missed in the original formulation [25] so far described, is that condition monitoring can actually enable preventive maintenance with the benefit of extending the remaining useful life of the system. In order to do that, it is necessary to code a model that must account for the restoration of those basic events which can be characterized by a repair rate. As shown in Table 8, in the model thereby proposed, the basic events which can benefit from a restoration are the Crack Formation and Debris Penetration (as highlight with asterisk). Moreover, for these two events, it is assumed that they follow a Weibull distribution in order to model the deterioration effect. This hypothesis suits better than the random fault behavior proposed in [46], that makes use of the exponential distribution. Therefore, the formula of Equation (5) can be used again with \(\beta=1.2\) and \(\gamma=1/\lambda_{\rm i}\). As for the time of restoration, the repair rate for the Crack Formation or the Debris Penetration has been set to \(1/168\) h, which corresponds to a restoration mean time of a week. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**ID** & **Description** & **System** & **Dist.** & **Params** & **Disyt** & **Params** \\ \hline BE1 & \begin{tabular}{c} Steam \\ Temperature \\ Steam \\ Humidity \\ Debris \\ Penetration * \\ \end{tabular} & Diaphragms & Exp. & \(\lambda=2.857\times 10^{-5}\) & - & - \\ BE3 & \begin{tabular}{c} Steam \\ Humidity \\ Debris \\ Penetration * \\ \end{tabular} & Exp. & \(\lambda=1.176\times 10^{-4}\) & - & - \\ \hline BE4 & \begin{tabular}{c} Vibration \\ Crack \\ Formation * \\ Steam \\ Temperature \\ \end{tabular} & Rotor Blade & Exp. & \(\gamma=7003\) & Exp. & \(1/168\) \\ \hline S1 & \begin{tabular}{c} Wireless \\ Accelerometer \\ \end{tabular} & Diaphrags & Exp. & \(\lambda=8\times 10^{-5}\) & - & - \\ S2 & \begin{tabular}{c} Accelerometer \\ \end{tabular} & Rotor Blade & Exp. & \(\lambda=1.74\times 10^{-4}\) & - & - \\ S3 & \
\begin{tabular}{c} Thermometer \\ \end{tabular} & Rotor Blade & Exp. & \(\lambda=1\times 10^{-6}\) & - & - \\ S4 &
\begin{tabular}{c} Tachometer \\ \end{tabular} & Turbine & Exp. & \(\lambda=80\times 10^{-6}\) & - & - \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Parameters of the Dynamic Fault Tree Basic Events. The other basic events, designated to model the occurrence of an undesired process condition (temperature, humidity and vibration) not related to a fault of a steam turbine component, cannot be characterized by a repair rate, therefore they are considered as nonrepairable events. As for the sensors, it is assumed that the failure event refers to a measuring error due to a wrong calibration or loss of precision, rather than a breakage. Therefore, also for these two basic events, a repair rate is not applicable.