## 4 Simulation Results under Irregular Waves in the Time Domain
### Result of the CAV Controller  
In order to evaluate the performance of the CAV controller, we set the \(\Omega_{d}\) as a constant to run the same simulation previously performed under the equivalent irregular waves, having a PM spectrum with \(T_{p}=10\) s and \(H_{s}=2\) m.  
There are three factors that affect the CAV controller's performance: \(\Omega_{d}\), \(k_{p}\) and the inertia of the turbine and generator rotors \(J\). Note that when \(b_{g}\) is constant, the variation of \(\Omega\) decreases as \(J\) increases. We first sought to observe the influence of \(k_{p}\) in the time domain. Figures 7 and 8 depict the simulation results of the OWC-turbine combined system with the CAV controller in the time domain where \(k_{p}=0.05\) and \(1\), respectively, and \(\Omega_{d}=250\) rad/s. Through comparing the two figures, it is apparent that a \(k_{p}\) value of one provides a faster convergence speed with a smaller error bound of \(\Omega\). When \(k_{p}\) is at one, \(\Omega\) converges to the \(\Omega_{d}\). However, in both cases, we observe that the time interval of the maximum value of \(\eta_{t}\) is drastically reduced compared to that of the i-MET controller, as seen in Figure 5. Furthermore, as seen in Figures 7 and 8, the peaks of \(P_{t}\) and \(\eta_{t}\) are unmatched, causing the turbine's performance to have a negative effect on the total output. Figure 7: Simulation result of the first 100 s with the constant angular velocity (CAV) controller when \(\Omega_{d}=250\) rad/s and \(k_{p}=0.05\): (**a**) wave \(\lambda\) and water column positions \(x\); (**b**) airflow speed at the turbine duct \(V_{x}\) and pressure drop across the turbine \(\Delta p_{t}\); (**c**) turbine rotor angular velocity \(\Omega\) and its desired value \(\Omega_{d}\); (**d**) generator damping coefficient (control input) \(b_{g}\) and power extraction from the turbine \(P_{t}\); (**e**) turbine efficiency \(\eta_{t}\) and its maximum value \(\eta_{t,max}\). In order to more accurately observe the power extraction and the convergence of the angular velocity, we measured and compared their average values over a period of 5000 s. Figures 9-11 show the \(RMSE^{*}(\Omega)\), average turbine power extraction \(\bar{P}_{t}\) and the average turbine efficiency \(\bar{\eta}_{t}\), with respect to \(k_{p}\), when \(\Omega_{d}\) is at 250, 400 and 550 rad/s. As seen in Figures 9-11, \(RMSE^{*}(\Omega)\) converges to zero as \(k_{p}\) increases, as intended for control purposes. It should, however, be emphasized that the curve of \(RMSE^{*}(\Omega)\) with \(J=1\) is lower than that with \(J=5\) regardless of \(k_{p}\) in all cases. The CAV controller functions better when the inertia is smaller. This runs contrary to the results given when not using a controller. Hence, when not using a controller, a larger inertia results in smaller angular speed variation. Unlike \(RMSE^{*}(\Omega)\), which is the inverse of \(k_{p}\) independent of \(\Omega_{d}\), the average power extraction \(\bar{P}_{t}\) and average turbine efficiency \(\bar{\eta}_{t}\) behave differently depending on the reference angular velocity \(\Omega_{d}\). The optimal average power extraction and average turbine efficiency is achieved when \(\Omega_{d}=400\) rad/s (Figure 10). It should be noted that under these condition, when \(k_{p}\) increases, \(\bar{P}_{t}\), and \(\bar{\eta}_{t}\) also increase. However, when \(\Omega_{d}=550\) rad/s, the increase in \(k_{p}\) does not result in an overall increase in \(\bar{P}_{t}\) and \(\bar{\eta}_{t}\) (Figure 11).