## 1 Introduction  
Prognosis and RUL calculations are imperative perceptions to assure the healthy operation of systems and predict its total failure long before it occurs [1, 2, 3]. Hence, applying prognosis in critical applications such as HEV becomes mandatory [2]. There are three categories of prognostic methods: heuristic, model-based, and data-based methods [4, 5, 6, 7, 8, 9]. HMM is a useful data-based prognostic method to predict the sequence of state changes in a system based on the sequence of observations. Hence, it is recommended for systems having a finite internal states that generate a set of external observations. The internal states of the system are invisible to an outward observer. The current state is directly dependent on the immediate previous state; this sequence is called the Markov process [10, 11]. Many researchers started applying HMM for diagnosis and prognosis of electrical and mechanical faults in electrical machines [12]; the considered mechanical fault is mainly a crack in the bearing. However, none applied prognosis when faults such as demagnetization, being the most severe fault in machines such as permanent magnet machines [13], are integrated. In paper [14], the author highlights the importance of detecting imminent faults in electric machines used in critical applications such as hybrid electric vehicles and avionics. It proves that hidden Markov model, used in diagnostics, is useful to be used as a prognostic technique. It presents two methods based on the hidden Markov models for the prediction of coming faults. They are based on pattern recognition, which is a data-driven approach commonly used in the field of faults recognition and diagnostic. Paper [15] applies prognosis techniques to detect electrical faults using a hidden Markov model and estimates the remaining useful life of relevant equipment using an estimation approach based on the probability of state failure. The observables of the model are time and frequency features extracted from the machine's torque measurement. To train the HMM, experimental observation probability densities are used due to limited available data. The investigated fault is a turn-to-turn short circuit; the research aims to detect the presence of turn-to-turn short circuits and estimate their severity from the extracted torque features. The author in [11] declares that HMM is an advantageous method in diagnosis, prognosis, and condition monitoring fields. Currently, the wide availability of different types of sensors (vibration, temperature, torque) encourage researchers to apply HMM. It has been noted that HMMs can still be useful even if little training data and little prior knowledge about the system exists. In this paper, the observed data are the current of the electric motor. This measured parameter is used to detect the presence of faults, define the current state of the machine, and predict its future state. The remaining useful life calculation is a new important concept that is beneficial to be applied in all electrical fields [16]. The RUL calculation and estimation for different elements constituting the electrical system of electric and hybrid electric vehicles is relatively new; however, it is becoming mandatory to assure health and proper operation [17; 18; 19; 20]. In [21], the author highlights the importance of prognosis and RUL calculation specially for particular systems such as electrical machines in industrial applications. A data-driven prognostics method is used where a health index (HI) is proposed to predict the RUL of the selected electrical machine. Results show the advantage of this method over the traditional direct RUL prediction. In hybrid electric vehicle applications, most researchers that discuss the RUL calculation apply it to the battery of the system. In [22], the RUL calculation is performed for a lithium battery used in a hybrid electric vehicle. The RUL estimation evaluate consecutive probability distributions of degrading battery states where a particle filter algorithm is built.