# Remaining Useful Life Prediction of Broken Rotor Bar Based on Data-Driven and Degradation Model
## 6 Prognosis Results  
In this paper, an exponential degradation model is used to estimate the RUL of the SCIM with a broken rotor bar. The inputs of the degradation model are the constructed HIs from the torque, stator current, and speed signals. The profile of the extracted HIs is shown in Figure 8. Among these HIs curves, the HI-torque and HI-current signals curves have the most apparent degradation trend with a monotonic increase in the whole life following the resistance variation. On the other hand, the HI-speed trend does not show any variation until a few days before failure. Therefore, the HI-torque and HI-current are chosen as the inputs to the degradation model. The process of estimating a broken rotor's RUL is triggered once the degradation is detected based on HI-torque and HI-current. The degradation model is used to detect the time evolution trends of HI-torque and HI-current, to update its parameters by looking at newly available observations, and to estimate when the FT is crossed. The parameters of this model are determined using the HI values. At the beginning, the parameters are initialized as \(\theta=1\), \(\beta=1\), \(var(\theta,\beta)=10^{6}\), and \(\phi=-1\), so that the degradation model is mostly relies on the observed HI data. The exponential model can evaluate the significance of the exponential slope by looking at the generated trends of the HIs. Once a significant slope of HIs is detected, the model deletes the previous observations and restarts its estimation based on the original priors. The prediction results of the proposed approach are given in Figure 9. It can be observed that, once the degradation threshold is detected (day 11), the degradation model starts to estimate the RUL by predicting new values of the HI-torque and HI-current. Using the updated window of the HIs values, the parameters of the model are calculated and the slope of the model is updated to reflect the evolving changes in each HI. The process is repeated until the predicted value reaches the FT. In our work, the EOL or FT is considered a threshold assumed to be the last value of HI-torque and HI-current before machine destruction. It can be seen from Figure 9a,b that, from the very beginning when the algorithm starts estimating the RUL, i.e., after the detection of degradation, the estimates of the RUL are more or less close to their calculated values. Looking at the predicted trajectory based on the HI-current illustrated in Figure 9b, it can be observed that the degradation model did not follow the false alarm. As a significant part of the proposed prognostic methodology, the RUL prediction contributes to better understanding the false alarms that the machine gives and to anticipating  
Figure 8: The different health indicators computed from the torque, current, and speed signals. the time of failure. In terms of predicting the time of EOL, the estimated RUL of the SCIM with a broken rotor bar is 5 and 8 days based on the HI-torque and HI-current, respectively. To evaluate the prediction accuracy of the proposed approach, several performance metrics are calculated, including \(\alpha-\lambda\), root mean square error (\(RMSE\)), normalized root mean square error (\(NRMSE\)), mean absolute percentage error (\(MAPE\)), and fitness degree \(R^{2}\)[23]. The \(\alpha-\lambda\) performance metric evaluates the error bounds specified by \(\alpha\) of the estimated RUL and the relative distance specified by \(\lambda\), on time, of a given point from a broken rotor's EOL [46]. The \(RMSE\) affords an estimation of the error by measuring the mean distance between the predicted RUL and its ground truth. It is expressed as  
\[RMSE=\sqrt{\frac{1}{N}\Sigma_{i=1}^{N}(Act-Pre)^{2}} \tag{13}\]  
where \(Act\) and \(Pre\) are the actual value and predictive value, respectively. \(N\) is the length of data sample.