## 3 Predictor of the Harvestable Power from Vibrations (PHPV) Design
### Brief State of the Art on Linear Power Prediction  
Many studies have been proposed in recent years to predict the amount of power harvestable from ambient energy sources [7; 43; 44; 45; 46]. Most of the proposed methods relate to solar energy, given the ease of access to the database of meteorological stations in different regions. The proposed methods are centered on the EWMA predictor [47]. The EWMA method defines a data sampling period [45]. More specifically, in [45], it considered data of the sun over seven days, and the observations were made every 30 min, corresponding to 48 time slots per day. The prediction method is based on the fact that on a typical day, the harvestable energy is similar to that harvested during the same time the previous day. Thus, the prediction of the harvestable power at the time \(t\) of day \(d\), \(\hat{p}_{d}(d,t)\) is defined in [45] as follows:  
\[\hat{p}_{d}(d,t)=\alpha\hat{p}_{d}(t-1)+(1-\alpha)p_{d-1}(t) \tag{3}\]  
where \(\hat{p}_{d}(t-1)\) is the estimated power for the previous period, and \(p_{d-1}(t)\) is the power harvested during the same period in the previous day; \(\alpha\) is a weighting factor. For different values of \(\alpha\), the prediction error is evaluated, and the optimal weight is obtained as being that which minimizes this error. In [45], an optimal weight of 0.5 was obtained corresponding to an error in the output current of a solar panel of 2.3 mA. As we can see, in this method, the evaluation of the power is based on an estimate of the power in the previous time slot, which contributes to an increased prediction error. However, an advantage of the method, as proposed in [45], lies in the fact that it does not require a high storage space. To improve the performance of the predictor, as defined in [45], the power at time \(t\) of day \(d\), \(\hat{p}_{d}(d,t)\) was estimated in [7; 43; 44] as follows:  
\[\hat{p}_{d}(d,t)=\alpha p_{d}(t-1)+(1-\alpha)p_{d-1}(t) \tag{4}\]  
with \(p_{d}(t-1)\), which represents the real power harvested during the previous time slot for the current day; the other terms are the same used in Equation (3). In Equation (4), the \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value and Unit** \\ \hline Stack area & 2.353 in\({}^{2}\) \\ Stack length & 1.81 in \\ Test voltage & 20 V \\ No-load displacement at test voltage & 0.04 mm \\ Blocking force at test voltage & 500 N \\ Capacitance & 125 nF \\ Resonant frequency at constant field & Values labeled in Figure 5 \\ Mechanical quality factor & 80 \\ Load resistance & 2.5 k\(\Omega\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Values of the model parameters for estimating the harvestable power. estimated power is determined using actual power data only, contrary to the algorithm proposed initially in Equation (3). This modification then contributes to minimizing the prediction error. While comparing the performance achieved with this algorithm defined by Equation (4) with that of other predictors described below, a prediction error of 33.26% was achieved for solar data recorded over 10 days in [43]. In [45], a prediction error of 28% was reached for solar data recorded over a year. Finally, using the vibration data recorded on the same process considered here, a relative error of 25.25% was achieved in [7]. To improve the performance of the predictor defined by Equation (4), the model in [43] was proposed and known as the Weather-Conditioned Moving Average (WCMA). The model considers solar data from previous days and estimates solar power \(\hat{p}_{d}(d,t)\) according to the algorithm below:  
\[\hat{p}_{d}(d,t)=\alpha\cdot p_{d}(d,t-1)+(1-\alpha)GAP_{k}\cdot M_{D}(d,t) \tag{5}\]  
where \(\alpha\) is the weight of the filter defined in the same way as in [45]; \(M_{D}(d,t)\) is the average of the power over the last \(D\) days at slot \(t\); finally, the factor \(GAP_{k}\) measures the deviation between current solar conditions and previous days. The proposed method yields an average relative error of only 10%.