# Power Control Optimization of an Underwater Piezoelectric Energy Harvester
## 6 Results  
After applying the optimization algorithm to the control system, we have obtained the following results. Figure 9 illustrates the curve of the extracted power from the energy harvester versus the proportional gain of the control law, \(\mathrm{K_{P}}\), for each Reynolds number. The optimal value of the spring constant, \(\mathrm{K_{spring}}\), for each case is also denoted. It is clearly visible in each plot that a maximum power is achieved in the range of examined \(\mathrm{K_{P}}\). The maximum extracted power at the lowest Reynolds number, \(\mathrm{Re}\) = 3000, is 3.95 \(\mathrm{\SIUnitSymbolMicro W}\) which corresponds to a water velocity of \(\mathrm{V_{water}}\) = 0.34 m/s. At the maximum Reynolds number, \(\mathrm{Re}\) = 12,000 with a \(\mathrm{V_{water}}\) = 1.36 m/s, the power is much larger reaching the value of 996.25 \(\mathrm{\SIUnitSymbolMicro W}\). For the intermediate water velocities of \(\mathrm{V_{water}}\) = 0.68 m/s and \(\mathrm{V_{water}}\) = 1.02 m/s corresponding to the \(\mathrm{Re}\) = 6000 and \(\mathrm{Re}\) = 9000, respectively, the mean power extracted is also considerable. The largest leap in the power is achieved in the range from \(\mathrm{Re}\) = 3000 to \(\mathrm{Re}\) = 9000 where the power is multiplied by almost 10 times. Table 4 represents the values of the extracted power for all water velocity cases and their corresponding optimal values of the spring constant \(\mathrm{K_{spring}}\) and proportional gain of the control law \(\mathrm{K_{P}}\).  
Figure 9: Power results along the proportional gain of the control law \(\mathrm{K_{P}}\) for different Reynolds numbers. The optimal \(\mathrm{K_{spring}}\) is also dented for each \(\mathrm{Re}\) case.  
The variation of the output voltage at different Reynolds numbers is shown in Figure 10. Note that Figure 10 represents the transient state from the initial state t = 0 s and only up to 5 s for better illustration of the wave form progress. It could be found that the output voltage of the piezoelectric vibrator increases as the Reynolds number increases. As expected, the maximum value of voltage is achieved by the case with the largest fluid velocity which corresponds to the case Re = 12,000.  
Figure 11 represents the power output evolution of the energy harvester and the optimal proportional gain of the control law, \(\mathrm{K_{p}}\), for each Reynolds number. Note that the parameter of the proportional gain keeps almost constant while the power seems to increase following an exponential law. The results of the current work show that thanks to the implementation of the new control law and optimization scheme, the power output of the energy harvester has significantly increased in comparison with previous studies of [PERSON] et al. (2016), [PERSON] et al. (2016), and [PERSON] et al. (2015). Furthermore, in a similar work carried out by [PERSON] et al. (2015) equipped with two piezoelectric beams and two cylinders, the power output (calculated by the sum of the upstream and downstream beams) obtained in the case of the 1 cm was around 1 \(\upmu\)W at the water velocity of 0.34 m/s. In the present study, the power output achieved at that velocity is four times larger with a more simple mechanical system.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Re** & \(\mathbf{V_{water}}\) **(m/s)** & **Power \(\upmu\)W** & \(\mathbf{K_{p}}\) **(Optimal)** & \(\mathbf{K_{spring}}\) **(Optimal)** \\ \hline
3000 & 0.34 & 3.95 & 4.412 & 7.2671 \(\times\) 10\({}^{-9}\) \\
6000 & 0.68 & 108.15 & 4.419 & 3.4594 \(\times\) 10\({}^{-8}\) \\
9000 & 1.02 & 395.12 & 4.459 & 6.5404 \(\times\) 10\({}^{-8}\) \\
12,000 & 1.36 & 996.25 & 4.452 & 1.1627 \(\times\) 10\({}^{-7}\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Energy harvester power output for all water velocity cases for all cases and their corresponding optimal values of the spring constant \(\mathrm{K_{spring}}\) and proportional gain of the control law \(\mathrm{K_{p}}\)  
Figure 10: Output voltage for each case at different Reynolds numbers.