## 6 Results
### Hourglass and Spherical Buoy Comparison  
The free-response results of Section 3 indicated that the hourglass buoy possessed synergistic behavior between its buoyancy and dynamic FK forces. In this section, we compare the closed-loop response of the hourglass buoy to a spherical one using the control law of Equation (23).  
Two cases are considered: (1) a diameter of 5 m compared to the hourglass buoy whose total height was also \(2h=5\) m, and (2) a diameter of 7.5 m so that the power generated for both buoys was approximately equal. The properties of the two spherical buoys are given in Table 2 where the added mass and damping were estimated from [42].  
Figure 7 shows the energy extracted, control force, velocity, and position error, \(\zeta_{r}-\zeta\). The energy and control force plots show the results of both buoys. Since the tracking response was nearly identical for both buoys, the velocity and position error plots are shown for only the hourglass buoy.  
While the hourglass-shaped buoy shows about 160% increased energy extraction over the sphere, it should not be concluded that the hourglass shape is better than the spherical one for all conditions. An examination of the control force and velocity plots shows that the hourglass force is better aligned with the buoy velocity peaks. Thus, more energy is extracted. This could be due to differences in the mass properties of the two buoys and the effect this has on their responses resulting from their different nonlinear Froude-Krylov  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Feature** & **Symbol** & **Case 1** & **Case 2** & **Units** \\ \hline Radius & \(R\) & 2.5 & 3.75 & m \\ Draft & \(h_{d}\) & 2.5 & 3.75 & m \\ Mass & \(m\) & 32,725 & 110,446 & kg \\ Density & \(\rho_{b}\) & 500 & 500 & kg/m\({}^{3}\) \\ Added Mass & \(m_{a}\) & 14,019 & 47,492 & kg \\ Damping & \(b\) & 11,208 & 18,665 & N/(m/s) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Spherical buoy parameters.  
Figure 7: Closed–loop response and performance comparing the hourglass and spherical buoys. The hourglass buoy height and the spherical buoy diameter are both 5 m.  
force expressions. Furthermore, the hourglass buoy control amplitude is near twice that of the sphere.  
To explore this further, the radius of the spherical buoy was increased to 3.75 m to yield similar energy extraction as the hourglass shown in Figure 8. This resulted in larger control forces, but the phasing between the control force and velocity was not as efficient as with the hourglass buoy for these conditions.  
A comparison of the power used in the Figures above gives a partial indicator of the WEC's performance. As a performance metric, extracted power ignores the requirements placed on the PTO in terms of force and stroke. Equation (29) was used to capture this and allow comparison across different geometries. This power, force, amplitude (PFA) metric reduces commensurately with the PTO size in terms of maximum force and stroke.  
\[PFA=\frac{A_{P}}{A_{F}A} \tag{29}\]  
where \(A_{P}\) is the amplitude of the real power, \(A_{F}\) is the amplitude of the control force and \(A\) is the amplitude of the buoy position.  
The PFA for the hourglass is 0.99 compared to 0.88 for the 3.75-m diameter sphere, about a 13% improvement for the hourglass buoy. Since, in both cases, the tracking control was nearly perfect, the buoy position had no effect.