# Energy-Based Prognosis of the Remaining Useful Life of the Coating Segments in Hot Rolling Mill
## 3 Approach
### Method for RUL Prediction in Hot Rolling Mills
\[E\bigg{(}\frac{dW}{dN}\bigg{)}=\frac{d\overline{W}}{dN}\approx\frac{\Delta W}{ \Delta N}=\frac{\Sigma W}{N_{t}} \tag{2}\]  
In Equation (2), \(N_{t}\) is the number of completely processed products, E is the mean value operator whereas the upper hyphen also stands for mean value, \(\Delta W\) stands for the increase of the embodied energy in the system, \(\Delta N\) is the increase of the number of fully processed products, and \(\Sigma W\) the total excitation energy stored (the sum of the energy contributions \(W\) of the products produced so far). The mean rate of wear progress is continuously updated as the production procedure goes on. Given the embodied energy capacity (\(W_{p}\)) of the tool (or a group of tools depending on the case study structure), which has been offline estimated in the baseline phase, the remaining life is estimated as the number of pieces that are predicted to be processed until segment exchange. The RUL prediction \(N\) is calculated via the following equation:  
\[N=\left| \begin{array}{c}W_{p}-W_{N}\\ \bigg{(}\frac{d\overline{W}}{dN}\bigg{)}\bigg{|}_{N=N_{t}}\end{array} \right| \tag{3}\]  
where \(W_{N}\) is (equally to \(\Sigma W\) mentioned in Equation (2) the total energy added from the previously processed products, and \(N_{t}\) is the number of fully processed products. \([x]\) symbolizes the \"floor\" under x, i.e., the greater integer less or equal to x. This is because the RUL value must be an integer. To have a more flexible prediction, a constant prediction tolerance of \(\pi\Delta N\) pieces is applied to the discrete RUL prediction \(N\). The prediction is then transformed into a closed interval, defined shown as below:  
\[\left\{ \begin{array}{lcl}[N-\Delta N,N+\Delta N]&,&N-\Delta N>0\\ [0,N+\Delta N]&,&N-\Delta N\leq 0\end{array} \right. \tag{4}\]  
The procedure described up to Equation (4) is performed to each cylinder separately. The machine's final RUL forecast interval is the lower RUL interval among these two. The interval prediction described in the mathematical format (4) is valid if and only if the mean rate of wear progress lays inside the open interval (\(W1\), \(W2\)):  
\[W1=\frac{W_{p}-W_{N}}{N+\Delta N+1}\;\;\;,\;\;\;W2=\left\{ \begin{array}{lcl} \frac{W_{p}-W_{N}}{N-\Delta N-1}&,&N>\Delta N+1\\ \infty&,&N\leq\Delta N+1\end{array} \right. \tag{5}\]  
The probability of the RUL interval prediction to be verified is computed by assuming a normal distribution for the mean rate of wear progress, integrating the distribution from \(W1\) to \(W2\). \[C=\int\limits_{W2}^{W1}f\left(\frac{d\overline{W}}{dN}\middle|\mu,\;\sigma^{2 }\right)d\left(\frac{d\overline{W}}{dN}\right) \tag{6}\]  
The integral in Equation (6) is the confidence of the prediction for f (x\(|\mu\), \(\sigma^{2}\)) being the normal probability density function. The distribution parameters are defined with the Maximum Likelihood Estimation (MLE) method [46]. The designed prediction system saves the updates of the mean rate of wear progress, each time a product leaves the machine, and makes use of an updates vector to re-apply the MLE. The applied RUL estimation software is designed to provide displayed comments in a User Interface (UI) for each prediction's confidence. A confidence assessment scale is adopted, accompanied by the assigned confidence labels. The procedure described up to here is repeated continuously, until it is time for replacing the segments. The assigned confidence labels are presented in Table 1.