## 6 Conclusions  
Many studies and statistics show that relying on hybrid electric vehicles will increase in the coming years due to many reasons such as the depletion of conventional sources of energy and environmental concerns. The causes of uneasiness regarding HEVs are their reliability and availability. To overcome this worry, several researchers stated the importance of detecting minor defects through prognosis and calculating the RUL of the system.  
RUL is one of the most important parameters to be calculated in the case of defects in a system because it gives information on the time needed before this defect becomes a total failure and affects the operation of the studied system. Additionally, it helps in generating a decision on action needed after the occurrence of the defect. Answers to the following questions will be elaborated: Should we stop the system immediately? Is it cost effective to stop the system? How much time can this system still run before there is a total failure where corrective action needs to be taken?  
In our studied case, the defect is a minor crack in one magnet in a surface permanent magnet machine used in hybrid electric vehicles. The health state of the magnet is very important in this application. The magnets are one of the essential elements in the permanent magnet machine that support the air gap magnetic field; subsequently, they support the generated torque and power.  
The suggested RUL calculation study will calculate the time needed before the crack deepens and reach a graver state. This is performed after the detection of its presence using HMM model where torque, vibration, and temperature are used as model inputs.  
The important factors for detecting the presence of preliminary cracks in the magnet and calculating their RUL before a total fracture occurs are: first, to know how long this machine can remain working, especially since a PMM with a crack in one magnet can still operate but with a lower efficiency; second, the possibility of taking a corrective action, which maybe repairing or replacing the cracked magnet.  
The proposed strategy was able, after detecting the presence of preliminary cracks in one magnet by HMM, to calculate the remaining useful life of the defective piece of magnet.  
In article [37], an RUL calculation for the same type of machine for the case of turn-to-turn short-circuit faults is presented. In this article, an RUL calculation strategy is suggested for the case of demagnetization faults. For the future, the work in this field to elaborate a global RUL calculation model encountering different types of faults that may occur in the electrical machine and in the whole electrical propulsion system of the vehicle is continuous.  
Figure 13: Magnet number of cycles versus crack depth.  
**Author Contributions:** Writing--review & editing, R.G., G.H. and N.M. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research received no external funding.  
**Data Availability Statement:** Data available on request, from the corresponding author, due to privacy restrictions.  
**Conflicts of Interest:** The authors declare no conflict of interest.