## 1 Introduction
[34] were the first to apply a phase controller to the OWC-WEC by making the damping coefficient proportional to the device's velocity. These studies all aimed at developing strategies to maximize the conversion efficiencies in the chamber stage (first stage). However, since OWC's overall efficiency is also influenced by the turbine stage (second stage), control strategies to ensure the turbine's efficiency would also be of benefit to the system. The power of air flow induced by the water column oscillation is transferred to the turbine, and the resultant turbine rotation in turn affects the motion of the water column. Thus, control strategies that consider the entire system including the turbine stage were introduced in [6; 35; 36; 37; 38; 39]. In [6; 35; 36], the frequency domain analysis is introduced to calculate a large number of simulations that consider various ocean states. It is assumed, however, that the turbine rotational speed remains constant, thereby resulting in a large rotational inertia. However, under these conditions, the turbine cannot respond in a low level of sea energy state. In addition, since this analysis is done in the frequency domain, the moment controller strategies in the time domain were not allowed. In [37; 38; 39], control strategies to avoid stalling were introduced, since when using the Wells turbine, stalling drastically reduces the efficiency. These analyses were based on the Newtonian mechanics, so performed in the time domain, and analyzing the effect of the control strategy impact on the turbine stage was carried out. Because the effect of turbine stage on the chamber stage is not taken into account, the effect of control strategy impact on the overall system efficiency was not analyzed. In summary on the previous studies on the optimization and control theory of OWC-WECs including turbine aerodynamics, it can be mainly categorized as the frequency domain analysis and time domain analysis; they have their own pros and cons. The analysis of the frequency domain takes into account the compressibility of air and provides the direct substitution of the wave spectrum into the analysis, but the turbine aerodynamics based on the Newtonian mechanics is impossible,which brings the assumption of the turbine's constant angular speed [6; 35; 36]. Otherwise, the time domain analysis is able to address the angular acceleration of the turbine rotor since it is based on the Newtonian mechanics [27; 28; 37; 38; 39]. Therefore, it is not necessary to assume the constant angular speed of the turbine rotor. We choose the time domain analysis in order to develop a real-time control law for OWEC-WECs considering turbine aerodynamics. We investigate a system that combines the hydrodynamics of the chamber, the aerodynamics of the impulse turbine and the modeling of an ideal DC generator, as shown in Figure 1. Unlike the analysis in the frequency domain, analysis in the time domain is based on the Newtonian mechanics, and it considers the angular acceleration of the turbine rotor. In the turbine aerodynamics, the pressure on the turbine, induced torque from the electric generator and turbine rotor inertia are taken into account and determine the angular acceleration of the turbine rotor in real time. Accordingly, it requires no assumption of constant rotor angular speed for the frequency domain, and rotor inertia acts as an important variable. We put forth two strategies for the controller. First is a strategy that maximizes the turbine's instant efficiency. The second strategy attempts to maintain the rotational speed at a constant rate. These two methods are differentiated by the definition of the reference angular velocity and can be induced by the Lyapunov stability theory [40; 41; 42; 43]. The aforementioned controllers can be implemented by a load control that is attached to the ideal DC generator. We investigate the performance of the controllers through simulation in the time domain under irregular waves.