# 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  
The flow of the proposed approach is presented next, in Figure 2:  
Figure 2 shows that, in real-time and while the manufacturing process is underway, RUL (remaining products to proceed) is calculated based on the following:  
1. The remaining embodied energy capacity of the segments. 2. The mean rate of wear progress. In this case, RUL calculates the remaining products that can be produced before the coating of the segment reaches its useful life. Under the assumption that someone a priori had decided to change the segments when they reach a certain wear stage, this energy level is considered as the segments' embodied energy capacity. Tool breakage could also be selected as the aforementioned wear stage. The embodied energy capacity is calculated from past sensor data (baseline/historic data), measuring key parameters that affect wear and converting them into energy concentration. For each baseline session, the concentrated amount of energy from the beginning of usage of the segments until their replacement is calculated. Due to data disturbances (e.g., floor vibration from nearby machinery) [15], the baseline energies will probably have a small variance. The mean of the baseline energy values is defined as the tools' embodied energy capacity. The number of baseline sessions is defined such that the variance of the baseline energies is low enough. Figure 2: Flowchart of the proposed approach. A feature representing the excitation energy from each processed product is constructed in real-time from the normalized force (F) and temperature (T) measurements. Normalization is performed by removing the data mean from the data values and then dividing with their standard deviation. After performing an energy analysis [36] on these two time-series for each machine's cylinder, we found that each cylinder's excitation energy is as follows:  
\[W=\int\limits_{f_{a}}^{f_{b}}G_{FF}(f)df+\int\limits_{f_{a}}^{f_{b}}G_{TT}(f) df-\int\limits_{f_{a}}^{f_{b}}|G_{TT}(f)|df \tag{1}\]  
where \(f\) is the frequency in Hz; \(G_{FF}(f)\) and \(G_{TT}(f)\) are the Power Spectral Densities (PSDs) of the normalized signals F and T, respectively; and \(G_{TF}(f)\) is the cross-PSD between F and T. In Equation (1), the term that contains the cross-PSD of F and T is removed to remove the part of T that is due to (\"linearly\" explained by) F. It should be noted that W is a mathematical quantity that is proportional to the energy that the processing of each product adds to the system. The norm in the last term is used because the cross-PSD part is a complex function of frequency. Integration boundaries \(f_{a}\) and \(f_{b}\) are defined by the acquired signals' bandwidth. For a sampling frequency \(f_{s}\), according to Shannon theorem, the maximum frequency in which signal information can be accessed is \(f_{b}=f_{s}/2\). The lower integration frequency, \(f_{a}\), is determined from the sensor's specifications. It is the lowest band that the sensor can record data information. Energy W is calculated after the end of the processing of each product, and it is added to the already stored energy of the measuring cylinder, reducing the remaining capacity of its segments. The next step is to estimate the mean rate of wear progress. The mean rate of wear progress is the mean added energy from each product that would result in the same product size as in the present period. This is calculated in Equation (2). The values of each product's energy contribution W are collected, forming a sample that is real-time updated after the machine finishes each product's processing.