# Energy-Based Prognosis of the Remaining Useful Life of the Coating Segments in Hot Rolling Mill
## 5 Industrial Case Study and Approach Validation
### Prediction Method Testing and Validation  
The data values provided by the industrial company were grouped based on their timestamps, according to the format presented in Table 2. There, the first column is the data timestamp, the second and the third refer to segment surface temperatures of the machine's cylinders (top is for cylinder A and bottom is for cylinder B), and the last two refer to the hydraulic forces. Every two consecutive timestamps correspond to the processing of a single product. Table 3 presents the different datasets that contain historical data captured during production and were used for validating the method. These provided data correspond to periods where no change of the coating segments had occurred, but do not correspond on the full duration of segment usage (from the time they were first mounted to the time they were replaced). \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Timestamp**} & \multicolumn{2}{c}{**Segment Top Surface**} & \multicolumn{2}{c}{**Segment Bottom**} & \multirow{2}{*}{**Cylinder Hydraulic A**} & \multirow{2}{*}{**Cylinder Hydraulic B**} \\  & **Temperature (Celsius Degrees)** & & & \\  & & **(Celsius Degrees)** & & \\ \hline
2020-01-22T09-26:09 & [189, 189, 102] & [548, 549,..., 90] & [-29, -16,..., -67] & [15, 3,..., -40] \\
2020-01-22T09-26:37 & [262,  
Each dataset corresponds to a different period, during which a specific set of six segments was used. Two different types and a different total number of trailing arms were processed during the periods captured by the available datasets. Moreover, a different number of trailing arms was processed in each captured period. To evaluate our approach, nine Monte Carlo prediction experiments were conducted, with each one using a single dataset as its input, to evaluate the prediction accuracy (output) of the discussed approach. In each prediction, the total energy that was absorbed from each cylinder is calculated and the higher of these two energies is considered as the dataset's embodied energy capacity \(W_{p}\). This capacity is assumed as the selected wear level for changing the segments for each distinct dataset. The objective of this validation strategy is to prove mathematically that the method's prediction accuracy is independent of which wear level is selected as the replacing point of the segments so that the findings and the conclusions of the research are trustable. For performing the prediction experiments, the development of a specialized code in a software environment is required. In the proposed approach, data PSD functions were estimated with Welch's averaged periodogram, and numerical integration was performed via the trapezoidal method. The tolerance for constructing the RUL estimation interval was set to \(\Delta N=\mp 40\) products. For the evaluation process, the prediction update horizon that is referred to in Section 4 was set to 1, to generate more predictions during the Monte Carlo experiments. Indicative outcomes of prediction experiments are shown in Figures 7 and 8. In Figure 7, the resulted segment wear from each product is indicated, in terms of added energy. The \(x\)-axis indicates the number of pieces that have been produced, and it is used as a discretization of time, and the \(y\)-axis shows the energy W that is added to each cylinder. Cylinder A refers to the upper cylinder of the rolling mill and cylinder B to the lower one. As it is shown in Figure 7, each processed trailing arm has a different energy contribution to segment wear, confirming the non-stationarity mentioned in Section 2. The statistical parameters of the added energy (mean, variance) change during the production timeline. Non-stationarity in wear exists because, in practice, it is impossible to attain the same processing conditions for each product.