As shown in the following details, the DEIM point absorber is composed by two coaxial cylindrical buoys. The central buoy is fixed to the seabed, while the external is able to move vertically. Neglecting the fixed buoy, the sea wave energy converter can be modeled as a single body having one degree of freedom along the vertical axis
[
23]. The WEC behavior is simulated in the time domain, because of the non-linearity of the power take off (PTO) system. At this step, regular waves are assumed, in order to simplify the equations [
24]. This is a limitation for the model, because time domain approaches with irregular wave inputs would fit better in the general context [
25]. Nevertheless, the differences between the two approaches are limited, especially for short waves, like in the Mediterranean Sea [
26]. Furthermore, at this step our goal is to assess the interactions between sea waves and the energy converter device, evaluating the electrical energy production. In this way, it is possible to proceed with the design and development of the conversion. According to this goal, we believe that the simplified model of regular waves is adequate and acceptable. The vertical motion of the floating buoy is determined by solving the following equation, which considers the hydrodynamic forces
and the resistance forces
due to the power take off (PTO) system:
where
m is the total mass of the system and
represents its vertical acceleration. The hydrodynamic forces on the heaving buoy is evaluated by:
where
is the vertical coordinate at time
t, measured from the initial equilibrium position. The five terms on the right side of the equation represent the different forces acting on the buoy: (1) considers the added inertial force, due to the motion of fluid (having the mass
) caused by the buoy; (2) radiation damping force, due to the waves generated by buoy oscillations, where
is the radiation damping coefficient; (3) viscous damping force accounting for relative turbulent flow, where
is sea water density,
is the waterplane area of the body at rest,
is the viscous damping coefficient, set equal to one [
27], and
is the vertical velocity of the free water surface; (4) hydrostatic restoring force, where
g is gravity; and (5) vertical component of the excitation force, due to the incident waves on the assumedly fixed body, where
is force amplitude,
and
are, respectively, wave height and frequency and
α is the phase angle between the wave and the wave-induced heaving force. In turn, the excitation force can be decomposed into two contributions: the diffraction force considering the wave deformation, generated by the structure and the Froude-Krylov force, due to the undisturbed wave field. If the floating buoy is very small compared with the wavelength, diffraction force can be neglected, so the excitation force is can be assumed equal to only the Froude-Krylov [
28]. In the present analysis, the full expression of the excitation force is considered. The resistance force,
, generated by the PTO system, is modeled as:
where
is the electromagnetic braking force, caused by the electrical power production inside the linear generator, and
is the elastic force of the spring system, connected to the translator, which is calculated by:
where
is the elastic stiffness constant of the spring. The electromagnetic force can be evaluated applying the Faraday’s law and the Maxwell equations to the electro-magnetic structure of sea wave energy converter. A simplified analytical model, presented by Thorburn and Leijon [
29], is used to calculate the voltage generated in the stator,
:
where
is the magnetic field evaluated in stator tooth,
is the width of a stator tooth,
d is the width of the stator stack,
p is the total number of poles,
is the number of slots per pole and phase,
c is the number of coils in a slot,
is the pole width and
δ is the load angle. Considering the energy supply to a purely resistive load and using the equivalent electric circuit, the voltage,
, and the current,
, measured at the terminals of each phase are, respectively, evaluated by:
where
is internal resistance of windings,
is circuit inductance and
is load resistance. Finally, the output power, (
t), and the electromagnetic force,
, are calculated, respectively, by:
where
is the phase index and
is the generator efficiency. Several simulations were run, changing the main control parameters of the sea wave energy converter and the state, taking in account the conditions that maximize the electrical energy production. In particular, this condition occurs if the system is in resonance with the dominant wave frequency. A resonant point absorber system has a significantly higher power absorption, thanks its enhanced amplitude and speed in vertical motion [
30]. However, for small devices, such as the DEIM point absorber presented here, the resonant frequency of the device tends to values higher than the typical sea state frequency, and so, the resonant condition is practically impossible to achieve. One possible solution consists in increasing the natural period of oscillation of a point absorber, adding a totally submerged mass with neutral buoyancy connected to the floating buoy [
31,
32]. The additional inertia allows the decreasing of the natural frequency of the device, according with the following equation:
where
is the total mass of the WEC and
is the total added mass at the frequency of the incident wave. In order to archive the resonance condition of the energy converter to the typical range of sea wave frequency, a spherical body was added as the submerged object, changing the natural frequency of the system. The simulations showed that the maximum power output is achieved when the system resonates with waves having a peak period of 5.5 s, while the optimum natural period of oscillation ranges between 6 and 7 s. For the buoy, the frequency-dependent coefficients of added mass, radiation damping and excitation force were pre-calculated with the commercial software, ANSYS AQWA [
33]. The code is based on the boundary element method and on the linear potential wave theory, which is a suitable approximation for the modeling of point absorbers [
34].