# Power Control Optimization of an Underwater Piezoelectric Energy Harvester
## 5 Harvester Control Modeling
### Piezoelectric Power Control  
The proposed control law for the power control of the piezoelectric energy harvester is defined by the following equations. Equation (11) determines the piezoelectric current with the proportional gain of the control law, \(K_{p}\), and Equation (12) is formed by matching Equations (5) and (10). \[\mathrm{i_{1}}=\ K_{p}V_{1} \tag{11}\]  
\[K_{p}V_{1}\ =\ \alpha\frac{du_{1}}{dt}\ -\ C\frac{dV_{1}}{dt} \tag{12}\]  
We supposed that the electrical time constant is lower than the mechanical time constant. \[\mathrm{K_{p}V_{1}}\;+\;\mathrm{C}\frac{\mathrm{d}V_{1}}{\mathrm{d}t}\;=\;\alpha \frac{\mathrm{d}u_{1}}{\mathrm{d}t} \tag{13}\]  
Therefore, in piezoelectric steady response we have:  
\[\mathrm{K_{p}V_{1}}\;=\;\alpha\frac{\dot{r_{1}}}{\mathrm{K_{trans}}} \tag{14}\]  
Equation (14) is an approximation of the Equation (13) given that the term \(\mathrm{C}\frac{\mathrm{d}V_{1}}{\mathrm{d}t}\) could be negligible in a steady state of the piezoelectric. Since the cylinder angle \(\theta\) is very small, we can also assume that \(\theta\) is very similar to \(\sin(\theta)\), as stabilized in Equation (15). \[\mathrm{r_{1}}\;=\;\mathrm{a}\cdot\sin\theta\;=\;\mathrm{a}\cdot\theta \tag{15}\]  
The previous system can be reduced to the Equations (16)-(18). Equation (16) is a combination between Equations (8) and (10). \[\mathrm{J_{wt}}\frac{\mathrm{d}^{2}\theta}{\mathrm{d}t^{2}}\;=\;\mathrm{T_{ hydro}}-\mathrm{K_{spring}}\theta\;-\;\mathrm{f}\frac{\mathrm{d}\theta}{\mathrm{d}t}\;-\; \mathrm{a}\cdot\mathrm{F_{m}} \tag{16}\]  
\[\mathrm{K_{p}V_{1}}\;=\;\alpha\frac{\mathrm{a}}{\mathrm{K_{trans}}}\frac{ \mathrm{d}\theta}{\mathrm{d}t}\Rightarrow\mathrm{V_{1}}\;=\;\alpha\frac{ \mathrm{a}}{\mathrm{K_{p}\cdot K_{trans}}}\frac{\mathrm{d}\theta}{\mathrm{d}t} \tag{17}\]  
\[\mathrm{L_{1}\cdot\rho_{piezo}\cdot A}\frac{\mathrm{d}^{2}r_{1}}{\mathrm{d}t^{2 }}\;=\;-\mathrm{k}\frac{r_{1}}{\mathrm{K_{trans}}}\;-\;\frac{\alpha^{2}\cdot \mathrm{a}}{\mathrm{K_{p}\cdot K_{trans}}}\frac{\mathrm{d}\theta}{\mathrm{d}t} +\mathrm{F_{m}} \tag{18}\]  
The parameter \(\theta\) is introduced in Equation (18) as a variable state and the parameter \(\mathrm{r_{1}}\) is replaced by the formulation proposed in Equation (15). In this way, Equation (19) is obtained. Now the variable \(\mathrm{F_{m}}\) has been isolated to one side of the Equation (20) and everything else to the other side. \[\mathrm{L_{1}\cdot\rho_{piezo}Aa}\frac{\mathrm{d}^{2}\theta}{\mathrm{d}t^{2}} \;=\;-\mathrm{k}\cdot\mathrm{a}\frac{\theta}{\mathrm{K_{trans}}}\;-\;\frac{ \alpha^{2}\cdot\mathrm{a}}{\mathrm{K_{p}\cdot K_{trans}}}\frac{\mathrm{d} \theta}{\mathrm{d}t}+\mathrm{F_{m}} \tag{19}\]  
\[\mathrm{F_{m}}=\;\mathrm{L_{1}\cdot\rho_{piezo}\;Aa}\frac{\mathrm{d}^{2}\theta} {\mathrm{d}t^{2}}\;+\;\mathrm{k}\cdot\mathrm{a}\frac{\theta}{\mathrm{K_{trans} }}\;+\;\frac{\alpha^{2}\cdot\mathrm{a}}{\mathrm{K_{p}\cdot K_{trans}}}\frac{ \mathrm{d}\theta}{\mathrm{d}t} \tag{20}\]  
Now we proceed to introduce the variable \(\mathrm{F_{m}}\) defined by Equation (20) in the Equation (16). Therefore, a new equation is obtained, Equation (21). Note that some terms have been grouped to make it clearer. \[K_{p}V_{1}\ =\ \alpha\frac{du_{1}}{dt}\ -\ C\frac{dV_{1}}{dt} \tag{12}\]0  
After that, the terms of Equation (21) are ordered to achieve the Equation (22) which describes the energy harvester performance:  
\[K_{p}V_{1}\ =\ \alpha\frac{du_{1}}{dt}\ -\ C\frac{dV_{1}}{dt} \tag{12}\]1  
The hydro-dynamical performance of the harvester is defined by Equation (23):  
\[K_{p}V_{1}\ =\ \alpha\frac{du_{1}}{dt}\ -\ C\frac{dV_{1}}{dt} \tag{12}\]2  
The equivalent dynamic of the whole system can be considered as a second order linear system, see Equation (24). The chosen output is the rotational speed of the cylinder, because the extracted power depends on it. The input variable is the lift coefficient \(\rm C_{L}\) because this is the only term on Equation (24) that changes with the time. The lift coefficient has a sinusoidal time waveform as it is shown in [PERSON] et al. [50].