# Piezoelectric Energy Harvesting Prediction and Efficient Management for Industrial Wireless Sensor
## 2 Case Study and Measured Vibrations
### Mechanical-Electric Conversion  
To estimate the harvestable energy of the measured vibrations, a Simscape model was used. Recall that the fluctuations that occur in the harvestable power were inevitable, regardless of the used harvesting method. Thus, the study of the conversion model was beyond the objectives of this work. It is proposed here to analyze the fluctuations of the vibrations due to the different operating speeds of the engines. The conversion system was built in the Simscape/Simulink Matlab environment using the piezo stack component. The structure and dynamics of this block corresponded to the constitutive equations of piezoelectricity developed in [44] and defined as follows:  
\[\left\{ \begin{array}{l}S=s^{E}T+dE\\ D=dT+\varepsilon^{T}E\end{array} \right. \tag{1}\]  
where \(S\), represents the strain sensor, \(T\) the stress tensor, and \(D\) and \(E\) the electric displacement vector and the electric field vector, respectively. \(\varepsilon^{T}\) is the dielectric permittivity, which is measured at constant stress. The piezo stack block was considered in [45], and the impact of the number of piezoelectric layers on the conversion efficiency was analyzed. It emerged from this work that the increase in the number of piezoelectric layers leads to a rise in mechanical losses. The Simscape model with the piezo stack composite has also been used in [46] to evaluate the harvestable vibrations on vehicle engines to extend the wireless sensor's lifetime necessary for the Vanet network in smart transport systems. The model's parameterization also makes it possible to use the characteristics extracted from the marketed components' datasheet. Most of the simulation parameters proposed here (Table 3) come from Mide technology's QP20W composite datasheet [47]. Figure 5: Frequency distribution of the measured vibrations. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Stack area \(\left(\text{mm}^{2}\right)\) & 45.974 \(\times\) 33.02 \\ Blocking Force \(\left(\text{N}\right)\) & 88.96 \\ Test voltage \(\left(\text{V}\right)\) & 40 \\ No-load displacement at test voltage volts \(\left(\text{\mu m}\right)\) & 0.016 \\ Electric capacitance \(\left(\text{nF}\right)\) & 145 \\ Resonant Frequency \(\left(\text{Hz}\right)\) & 1 Hz \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameter values for the piezo stack component in Simscape Simulink. The simulation diagram is shown in Figure 6 below. The schematic includes an excitation source for the transducer (the vibration data measured over the 31 days was considered). A Simulink to a physical signal (PS) converter was used to transform the data into a physical signal. The signal output is a force that then feeds the piezo stack composite. The voltage across the load (voltage) was converted into Simulink's physical signals and saved for further processing. Using vibration data recorded the first day, the optimal load resistance obtained for the parameter values reported in Table 3 is shown in Figure 7. The analyses were proposed for different values of the transducer's resonance frequencies. The results show that the best performance was achieved when the resonance frequency of the composite was set at 1 Hz. This result was consistent with that of Figure 5, where we can see that the acceleration peak on the first day is closer to 1 Hz. The value of optimal load resistance was 1.5 M\(\Omega\). The analysis leading to the results of Figure 7 was based on the average harvestable power. The fluctuations in power that occurred during the first four days in which the load resistance was set to 1.5 M\(\Omega\), are shown in Figure 8. Similar amounts of the harvestable power over the first four days, with an average power of around 1.26 mW is observed in Figure 8. The evolution of the harvestable power under a month's data is shown in Figure 9. The average harvestable power is 1.28 mW,  
Figure 6: Simulation diagram in Simscape. Figure 7: Dependence between the harvested power and load resistance.