# Modeling and Simulation of a Wave Energy Converter _Nwave_
## 3 Modeling of the Device's Dynamics
### Power Take Off Module Modeling  
A buoy undergoes complex motions in the heave, surge and pitch directions in the two dimensions. While heave and pitch receive restoring force due to buoyancy, the restoring force in the surge direction is solely dependent on the tension of the ropes. The further away it gets from the origin and in the surge (\(x\)) direction, the difference between angles \(q_{1}\) and \(q_{2}\) grows, leading it to return back to the origin. Rope drums (or rope pulleys) connected to the ropes repeat the winding and unwinding with the rope's back and forth movements. In order to deliver the torques from pulleys to the one-way electrical generator, bi-directional winding motions should be converted into a uni-directional rotation using a ratchet gear as seen in Figure 4. The use of the ratchet mechanism divides the power transmission processes as follows: when the buoy pulls on the rope, the rope pulls on the counterweight, making it rise. Simultaneously, it provides the driving torque to the generator. Conversely, when the counterweight falls, the driving torque is not transmitted to the generator, and it only provides the restoring force to the buoy through the rope tension. The reverse setting that driving torque is connected when the counterweight is falling is also possible. However, this mechanism is risky because unless the counterweight is sufficient, the load of the generator will result in the rope losing tension. Then, the rope can deviate from the pulley or drum, which can result in breaking of the rope. For this reason, the reverse setting requires careful consideration of the relationship between the generator load and the mass of the counterweight. In this paper, we deal only with the former ratchet mechanism, in which the driving torque is transmitted when the counterweight rises. Figure 4 shows the overall configuration of the PTO module. The angular velocities of the pulleys that Ropes 1 and 2 are pulled over are \(\omega_{1}\left(=l_{1}/r_{1}\right)\) and \(\omega_{2}\left(=l_{2}/r_{2}\right)\), respectively, where \(r_{1}\) and \(r_{2}\) are the radii of the pulleys. The angular velocities of the generator and the pulley connected to it are \(\omega_{g}\) and \(\omega_{p}\), respectively, which have the relation of \(\omega_{g}=\kappa\omega_{p}\) where \(\kappa\) is the radius ratio between the pulley and generator. For numerical simplicity, it is assumed that the inertia of the pulleys and shafts can be ignored. Then, the dynamics of the generator can be expressed as follows:  
\[\kappa^{2}J_{g}\dot{\omega}_{p}=\tau_{1}+\tau_{2}-\kappa\tau_{g}, \tag{12}\]  
where \(J_{g}\) is the inertia of the generator rotor. \(\tau_{1}\), \(\tau_{2}\) and \(\tau_{g}\) are the induced torques from Pulleys 1, 2 and the generator, respectively. \(\tau_{1}\) and \(\tau_{2}\) have the following relation with \(f_{1}\) and \(f_{2}\) in (2) expressed as:  
\[\tau_{i}=r_{i}f_{i},\quad\text{for}\;\;i=1,2.\]  
We assume that induced torque from the generator \(\tau_{g}\) is proportional to the angular velocity \(\omega_{g}\) and is given as follows:  
\[\tau_{g}=b_{g}\omega_{g}, \tag{13}\]  
where \(b_{g}\) is the generator damping coefficient. Then, the instantaneous power output from the generator is:  
\[P_{g}=\tau_{g}\omega_{g}=b_{g}\omega_{g}^{2}=\kappa^{2}b_{g}\omega_{p}^{2}. \tag{14}\]  
Furthermore, the time-averaged output power is denoted by \(\bar{P}\left(=\int Pdt\right)\). Since the LWT is premised [11], the output power is proportional to the square of the wave amplitude. Hence,  
Figure 4: Power take off (PTO) module consisting of counterweights, pulleys, including ratchet gears, and the generator. the output power per unit amplitude can be defined as \(P^{*}=P/\left(wave\,amplitude\right)^{2}\), and accordingly, time averaged power per unit amplitude is denoted as \(\bar{P}_{g}^{*}\)[12].