## 4 Simulation Results under Irregular Waves in the Time Domain
### Result of the i-MET Controller  
We carry out the simulation of an OWC-turbine combined system with an i-MET controller given as (20) with (18) under the PM wave condition in the time domain using a fourth order Runge-Kutta method. Between all i-MET controller's gains \(k_{p}\), \(\gamma\) and \(\epsilon\), \(k_{p}\) had the greatest effect on the controlled result. Therefore, with fixed gains \(\gamma\) and \(\epsilon\) set at 100 and 0.01, respectively, we observe the difference in performance with varying \(k_{p}\) values. Figures 4 and 5 show the results of the first 30 s of the simulation, when \(k_{p}\) is set at 0.5 and 10, respectively. It can be observed in both graphs that \(\Omega_{d}\) changes consistently to follow \(\phi_{o}=1.03\) by (18) with the fluctuation of \(V_{x}\). Through comparisons between both graphs, however, it can be seen that a result of \(k_{p}=10\) produces a better performance than that of \(k_{p}=0.5\) in terms of control purposes, since \(\Omega\) tracks \(\Omega_{d}\) closer and more quickly. This yields that the longer duration of the instant efficiency \(\eta_{t}\) is able to reach the maximum value with \(k_{p}=10\). To improve the accuracy of our analysis, we obtained values for average power extraction from the turbine \(\bar{P}_{t}\), average turbine efficiency \(\bar{\eta}_{t}\) and the relative root mean squared error of \(\Omega\) (\(RMSE^{*}(\Omega)\)) by changing \(k_{p}\) to a value between 0.1 and 10 for 5000-s intervals (\(T_{m}\)). \(\bar{\eta}_{t}\) is expressed as the ratio between the time integral value of the pneumatic power and the power extraction from the turbine. \(RMSE^{*}(\Omega)\) is the relative root mean squared error of the \(\Omega\) with respect to average desired angular velocity \(\bar{\Omega}_{d}\) for the times when the controller is functioning. These can be expressed as follows:  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** & **Unit** \\ \hline \(a_{c}\) & 28.26 & m\({}^{2}\) \\ \(a_{t}\) & 0.0254 & m\({}^{2}\) \\ \(b_{t}\) & 0.03 & m \\ \(I_{0}\) & 5 & m \\ \(k_{h}\) & 283.87 & N/m \\ \(l_{t}\) & 0.03 & m \\ \(n_{t}\) & 10 & \\ \(r_{t}\) & 0.135  
\[\bar{P}_{t}=\frac{1}{T_{m}}\int_{0}^{T_{m}}P_{t}dt=\frac{1}{T_{m}}\int_{0}^{T_{m}} b_{g}\Omega^{2}dt, \tag{28}\] \[\bar{\eta}_{t}=\frac{\int_{0}^{T_{m}}b_{g}\Omega^{2}dt}{\int_{0}^{T _{m}}\Delta p_{t}\cdot Qdt},\] (29) \[RMSE^{*}(\Omega)=\frac{1}{\Omega_{d}}\sqrt{\frac{\sum\limits_{n} ^{t_{n,f}}\left\{\int_{t_{n,i}}^{t_{n,f}}\Omega^{2}dt\right\}}{\sum\limits_{n }\left\{t_{n,f}-t_{n,i}\right\}}}. \tag{30}\]  
Here, \(t_{n,i}\) and \(t_{n,f}\) represent the beginning and the end times, respectively, of when the controller is working at the \(n\)-th order, which is equivalent to \(b_{g}>0\). Figure 6 reveals that \(\bar{P}_{t}\) and \(\bar{\eta}_{t}\) increase as \(k_{p}\) increases. Beyond a \(k_{p}\) value of five, \(\bar{P}_{t}\) and \(\bar{\eta}_{t}\) reach a plateau, and no improvement of them can be observed. Conversely, as \(k_{p}\) increases, \(RMSE^{*}(\Omega)\) decreases drastically. Therefore, it appears that the optimal performance of the i-MET controller is reached as \(k_{p}\) increases. However, as seen in the Figure 5, when \(k_{p}=10\), \(b_{g}\) tends to generate abnormal peaks as \(\Omega_{d}\) converges zero. Figure 4: Simulation result of the first 30 s with the instantaneous-maximum efficiency tracking (i-MET) controller when \(k_{p}=0.5\): (**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 \(\eta_{t,max}\).