*4.1. RAO*

The first result presents the Response Amplitude Operator (RAO) measured in the absence of applied load (free oscillation). The RAO is defined as the ratio between the total extension of the oscillation and the incident wave height. Note that although the generated waves are regular, the signal is not "exactly periodic" in time. The incident wave component is separated from the reflected one by an array of wave gauges, through the Zelt–Skielbreya [25] method. The incident wave height used to evaluate the RAO is assumed to be the *H*rms characteristic value of the incident component. In analogy, the oscillation amplitude used for the RAO is the rms value (among the possible averages, the "rms" is chosen since this characteristic value preserves the total energy). The generated period is found to be equal to the target.

Figures 5–7 present the RAO for the different water depths, function of the period T and the target wave height H.

**Figure 6.** Measured RAO. Free oscillations, water depth = 0.40 cm.

In all cases, the RAO grows almost proportionally with the tested wave periods (as a consequence of the horizontal stretching of the wave orbits with T ranging from 1–4 s).

For T = 1 and 2 s, the RAO behavior and, consequently, the oscillation angle, is similar for the three water depths (d = 0.35, 0.40, 0.45 m). In particular, for T = 1 s (i.e., 3 s at prototype scale) the device oscillations is approximately 2◦/cm, independent from H and d. For instance, for H = 8 cm, the oscillations are 2◦/cm × 8 cm = 16◦, and there is one oscillation every second. For T= 2 s, the RAO is approximately 6◦/cm.

For T = 4 s, the RAO depends significantly both on water depth and on wave height: (a) for H=2 and 4 cm, the oscillation increases with increasing water depths; (b) for H = 6 cm the oscillation is constant for the three tested water depths; (c) for H = 8 cm, the oscillation decreases with water depth.

To explain this behavior, a number of factors should be considered. The torque applied by the wave depends on the wave frequency, the wave orbital shape, the oscillation range (i.e., the obliquities spanned by the device during its motion), the nonlinear shape of the wave (higher crests and longer troughs) and, the abrupt non-linear effect due to possible submergence of the device during its movement, so that high and small waves induce qualitatively different responses. The device oscillation range depends on water depth since the initial equilibrium position of the floater changes for the three cases. For d = 0.35 m, 0.40 m and 0.45 m, the initial rest angle (with respect to the horizontal) is approximately 35◦, 50◦ and 60◦.

When the device is restrained by the PTO, the device movements are significantly reduced and the RAO is obviously much smaller. Figure 8 shows the RAO for the case with L2 = 4.1 Nm, that was found to give a slightly larger power for all wave periods. Figure 8 shows that the RAO measured in correspondence to H = 8 cm was consistently larger than the case H = 4 cm.

**Figure 8.** Measured RAO. Optimal load applied, water depth = 0.40 cm.

#### *4.2. Measured Efficiency*

The efficiency is obtained as the ratio between the converted energy *E*<sup>c</sup> and the incident wave energy, *E*<sup>i</sup> = ρ*gH*<sup>2</sup> i/8.

The incident wave energy flux is proportional to the group celerity and therefore, quantitatively, the highest power production is achieved for longer waves. However, in relative terms, the device efficiency decreases with the wave period, having an opposite behavior of the RAO achieved without load. In fact, larger RAOs do not correspond to a more effective transfer mechanism from wave to mechanical energy, and in fact the largest oscillations are achieved in stationary conditions, when the wave energy transfer mechanism is low. The RAO is rather affected by the wave kinematic.

Figure 9 shows the efficiency measured in the wave flume. The higher efficiency is approximately 35%, measured for T = 1 s. The case H = 8 cm is more relevant, being associated to a larger power output.

**Figure 9.** Measured Efficiency. Optimal load applied, water depth = 40 cm.

### *4.3. Wave Attenuation*

The wave reflection and transmission are certainly affected by the device movements and hence, as seen in the previous chapters, by the PTO load. Figures 10 and 11 show, for the case with optimal PTO load, the reflection coefficients defined as the ratio between reflected and incident wave height, and the transmission coefficient defined as the ratio between transmitted and incident wave height. As expected, the transmission coefficient increases for long periods. It is in fact easy to understand that the device cannot limit the transmission of slow oscillations (long period).

**Figure 11.** Transmission coefficient. Optimal load applied, water depth = 40 cm.

#### *4.4. Energy Balance*

The obtained results (*k*R, *k*<sup>T</sup> and *η*) are checked by means of an energy balance condition. Energy flux conservation requires that, once the stationary conditions are reached, the incident energy flux is given by the sum of reflected, transmitted, dissipated and converted ones. For horizontal bed, the energy flows with constant group celerity, and since the channel width is also constant, the equation can be written as balance of energies:

$$E\_\mathbf{i} = E\_\mathbf{r} + E\_\mathbf{t} + E\_\mathbf{c} + E\_\mathbf{d} \tag{2}$$

The wave energy is proportional to the square of the wave height. If such proportionality factor is ρg/8, Equation (2) becomes:

$$
\rho \text{g} / 8 \,\text{H}^2\text{i} = \rho \text{g} / 8 \,\text{H}^2\text{ r} + \rho \text{g} / 8 \,\text{H}^2\text{ l} + E\_\text{c} + E\_\text{d} \tag{3}
$$

Let's define the efficiency *η* as the ratio between converted and incident wave energy, and *ε* as the sum of the measurement errors and the ratio between the energy dissipated by the device movements and the incident wave energy. Dividing all terms in Equation (3) by the incident wave energy we find:

$$1 = k^2 \, \_\text{r} + k^2 \, \_\text{t} + \eta \, + \, \varepsilon \tag{4}$$

Figure 12 shows this balance, accounting for the first 3 terms in the RHS of Equation (4). It shows how the incident energy is distributed into reflected, transmitted and harvested energy, among the different tested wave periods.

**Figure 12.** Distribution of incident wave energy. Optimal load applied, H = 8 cm, d = 40 cm.

The sum of the three terms is expected to be always lower than 1 (i.e., 100%), since the dissipations in water are usually not small (in absence of moment errors, *ε* > 0). However, even assuming *ε* = 0, for T > 2 s, the sum is larger than 100%, and this can only be ascribed to measurement errors. However, the error is not significant (slightly above 5%) and the measured quantities may be considered sufficiently accurate.

#### **5. Conclusions**

The dynamics and the efficiency of a new WEC named EP4 were tested in the wave flume of Padova University. Tests were performed in scale 1:10 with respect to a possible application in front of a low-lying sandy coast, at 4 m water depths, where the installation could benefit by its double purpose of harvesting the wave energy and protecting the shore.

The EP4 efficiency was obtained with a novel control system characterized by a PTO load delayed with respect to the device movements.

The device appears to resonate at a very high period (19 s at prototype scale), and it is suggested that the design is modified to reduce the added mass associated to its movements, so that the natural periods are close to the main wave periods, to utilize the motion resonance effect and therefore to achieve even better performance under short waves.

The peculiar PTO used to restrain the device motion resulted very efficient and easy to build. The resistive force, simulating the generator, was designed to be zero when the flap changes direction, in order to allow the WEC to gain some kinetic energy. Rotations were accurately measured by a HD video camera, and automatically post-processed to obtain the rotational amplitude in time. Incident and reflected waves were measured by an array of 4 wave gauges, and transmission by a 5th gauge.

The device efficiency was found to be 35% for periods of 1 s (3 s at full scale) and a little less for longer periods, characterized by a higher energy content. Since only a single value of PTO delay was tested, an optimized control strategy may lead to larger efficiency also for long waves.

Irregular wave conditions, naturally associated with a lower performance, will probably be tested through additional experiments. In this case, the PTO will be equipped with a flywheel to smooth the irregularities of the wave field. To force the rotation of the shaft along the same direction, a gear and clutch system can be used: during the motion inversion, before the flywheel is engaged, the application of the resistive load would be delayed in similitude to the tested conditions.

The performance in terms of coastal protection followed the expectations: The measured wave transmission coefficient was respectively 75% and 90% for waves with period of 1 and 2 s (3 to 6 s at full scale).

**Author Contributions:** Conceptualization, L.M., M.V., P.R., C.F.; Methodology, L.M., M.V., P.R., C.F.; Investigation, L.M., M.V., P.R., C.F.; Writing and Editing, L.M., M.V., P.R., C.F.; Supervision, L.M., M.V., P.R., C.F.

**Funding:** This research received no external funding.

**Acknowledgments:** We thank Dario Bernardi, inventor of the EP4, for sharing the information on his device. This paper is a deep revision of the note presented at SDEWES 2018 Conference, code SDEWES2018.00422, with title "EXPERIMENTAL INVESTIGATION ON THE DYNAMICS OF A FLOATING WEC WITH PTO PHASE CONTR" and selected for resubmission at Energies.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


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*Article*
