## 3 Predictor of the Harvestable Power from Vibrations (PHPV) Design
### Proposed Predictor  
Since the industrial process operates seven days a week and 24 h a day, the data stored in the database was initially divided into days. Each time slot corresponds to one minute, as shown in Figure 8, where it is also highlighted the history of the data used to predict the harvestable power. The predictor proposed in this work takes advantage of the results obtained in previous predictors. Since it has been shown in previous works that the EWMA method fails at the appearance of power peaks [7; 44], to improve the precision in the estimation of the power, the gap between the estimated and actual power during the previous time slots was taken into consideration. \[\hat{p}_{d}\left(t,d\right)=\sum_{i=1}^{N_{D}}\alpha_{i}\cdot p_{d-i}(t)+\sum_{j=0 }^{N_{D}}\beta_{j}\cdot p_{d-j}(t-1)+\sum_{u=1}^{T}\gamma_{u}(p_{d}(t-u)-\hat{p}_ {d}(t-u)) \tag{8}\]  
where \(\hat{p}_{d}\left(t,d\right)\) represents the estimated power at time \(t\) of day \(d\). \(N_{D}\) is the number of previous days taken into account in the estimate, and \(\alpha_{i}\), \(\beta_{j}\), and \(\gamma_{u}\) are, respectively, the weights of the filters allowing consideration of the powers measured in the last days at the same time, that of the previous days at the earlier time slots and finally, and the gap between the powers measured and estimated during the previous time slots. Optimum weights were determined to minimize RMSE. \(T\) represents the number of slots, making it possible to optimize the prediction. Using the algorithm defined by Equation (8), the optimal weight \(\left(\alpha_{\text{opt}},\quad\beta_{\text{opt}},\quad\gamma_{\text{opt}}\right)\) for each estimate was determined then the error was calculated to find the optimal value of \(T\). The evolution of the RMSE for the vibration on the gearbox at the output side, according to \(T\) and \(N_{D}\), is represented in Figure 9. The figure shows that the RMSE decreases with the number of days considered in the database. For all the curves, the optimal value of \(T\) is around 25; this value of \(T\) was used for the rest of the analyses. Figure 8: Database to PHPV design. Figure 9: RMSE according to the number of the previous times slots \(T\). In the following, the performance achieved with the PHPV is compared with previous predictors, such as the EWMA defined by the algorithm in Equation (4) and the PHEV determined by the algorithm in Equation (7). The performance criteria considered in this paper are the maximum absolute error \(E_{a_{\max_{i}}}\), and the RMSE \((RMSE_{i})\), respectively defined as below:  
\[\left\{ \begin{aligned} & E_{a_{\max_{i}}}=\max(abs\ (p_{i}-\hat{p}_{i}))\\ & RMSE_{i}=\sqrt{\mathbb{E}(p_{i}-\hat{p}_{i})^{2}}\end{aligned} \right. \tag{9}\]  
where \(\mathbb{E}\) is the expectation and \(p_{i}\) denotes the real power and \(\hat{p}_{i}\), the predicted power. The PHPV predictor's performance compared to those achieved with the EWMA and PHEV predictor is represented in Figure 10. For the twelve measurement points, the improvements obtained with the PHPV predictor compared to the PHEV predictor, about the relative error and the RMSE are reported in Table 2. For each measurement point shown in Figure 10, the representation is made around the most pronounced peaks, and the main observations are as follows:  
* Compared to the EWMA (curve in black dotted line), we observed shifts in the appearance of the power peaks with the real power represented in solid black lines. This can give rise to an overestimation of the performance of a WS at a given instant and an underestimation of its performance at the next time slot. * The signals predicted by the PHEV algorithm (in red) are more in phase with the real power in most cases. This can be explained by the fact that the algorithm, as defined in Equation (7), considers the time slots after the current time slots. However, there are considerable differences between the actual and predicted power.