## 4 Reliability Analysis  
Once the proposed propulsive system was outlined, a reliability analysis was carried out, as flight safety enhancement was the main objective of the present research. To this end, a Markov model has been used, as suggested in [22]. According to this approach, the reliability of a system is calculated in terms of availability as the probability of existing in a state of usability, which is based on the transition rates from a pre-existent \"healthy state\" and the transition rates from a previously failed to a \"healthy\" state in each time interval. In its simplest form, the availability of the system in Figure 11 is given by the probability associated with the state \"healthy\", while the transition from one state to the other is characterized by the parameters \(\lambda\) and \(\mu\), which are failure and repair rates, respectively, defined as:  
\[\lambda= \frac{1}{\text{MTBF}} \tag{2}\] \[\mu= \frac{1}{\text{MTTR}} \tag{3}\]  
where MTBF and MTTR are the mean time between failure of and the mean time to repair a given component. In complex systems such as that being examined, a reliability assessment is related to the particular configuration of the system itself, so that all possible states can be enumerated and consequently their probability can be calculated. In the present work, the reliability of the developed system is compared to the reference conventional system, which is the system before hybridization. The conventional system (only thermal) is made of three distinct components, namely the engine, the clutch, and the gearbox. It follows that the system can assume eight different states, summarized in Table 5, where the working state of a component or system is denoted by 1 and the failed state by 0 and whose transitions are represented in Figure 12.  
Thus, the availability of the system coincides with the probability of state 1, calculated as:  
\[\text{P}_{1}=\frac{\mu_{\text{E}}\cdot\mu_{\text{C}}\cdot\mu_{\text{GB}}}{( \mu_{\text{E}}+\lambda_{\text{E}})\cdot(\mu_{\text{C}}+\lambda_{\text{C}}) \cdot(\mu_{\text{GB}}+\lambda_{\text{GB}})} \tag{4}\]  
where the subscripts E, C, and GB of failure rate, \(\lambda\), and repair rate, \(\mu\), refer to the engine, clutch, and gearbox, respectively. From Equation (4), an availability of 0.9910 is obtained. Figure 11: Markov chain of a two-state system. As the hybrid system is made of six components (battery, EM2, EM1, engine, clutch, and gearbox), the number of states is now 64. The states and the corresponding Markov chain for the proposed hybrid system are given in Appendix A. It is assumed that gearbox failure always leads to system failure, while clutch failure is not critical, as long as the system can work in electric mode. The failure and repair rates employed in the calculation of system availability are given in Table 6[22, 36, 37]. The availability of the hybrid system, calculated as the sum of the probabilities of the working system states, is equal to 0.9999. As expected, the redundancy and versatility of the hybrid system allow for a higher availability with respect to the conventional system, demonstrating the efficacy of hybridization in increasing safety. \begin{table}
\begin{tabular}{c c c} \hline \hline  & \(\lambda\) **(1/y)** & \(\mu\) **(1/y)** \\ \hline Engine & 2.52802 & 361.6846 \\ Generator (EM1) & 0.01111 & 4151.659 \\ Battery & 0.0006 & 848.0155 \\ Motor (EM2) & 0.01825 & 2591.716 \\ Clutch & 0.032259 & 16.10294 \\ Gearbox & 0.2545 & 3981.818 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Failure and repair rates of involved components. Figure 12: Markov chain of the conventional system.