3.3.4. Free Decay Tests

Free decay tests are executed to determine the decay response. The heave displacement z allows us to calculate the logarithmic decrement *Λ*, the corresponding damping ratio *ξ<sup>d</sup>* and the natural period *Tn*. This *T<sup>n</sup>* should correspond with the one obtained from the radiation tests discussed in Section 3.3.2. Conceiving the WEC as a mass-spring-damper system, *ξ<sup>d</sup>* follows from *Λ* [31]:

$$\Lambda = \frac{1}{n-1} \ln \left( \frac{\chi\_1}{\chi\_n} \right) = \frac{2 \pi \mathfrak{f}\_d}{\sqrt{1 - \mathfrak{f}\_d^2}} \tag{14}$$

in which *x*<sup>1</sup> and *x<sup>n</sup>* are the values of the first and the n-th peak of the free decay motion z. For un underdamped system (0 < *ξ<sup>d</sup>* < 1), the oscillations of the WEC buoy fade exponentially over time and tend toward zero, yielding the envelope:

$$z\_{\varepsilon}(t) = z\_{A} \cdot \exp(-\mathfrak{F}\_{d}\omega\_{n}t) \tag{15}$$

in which *z<sup>A</sup>* is the amplitude of the free WEC response [31]. The angular natural frequency *ω<sup>n</sup>* of the WEC buoy can be calculated from the damped natural frequency *ω<sup>d</sup>* :

$$
\omega\_n = \frac{2\pi}{T\_n} = \frac{\omega\_d}{\sqrt{1 - \xi\_d^2}} \tag{16}
$$

Table 7 gives an overview of the executed free decay tests, with indication of the start position of the WEC buoy *zstart*, *CC*, *n* used Equation (14), *ξ<sup>d</sup>* and *Tn*. A positive *zstart* corresponds to a submerged start position and a negative value corresponds to an elevated start position. The WEC buoy is submerged or elevated with a torque command. Once the torque is set equal to zero, the decay motion is initiated. Test\_089 and Test\_129 yield a *T<sup>n</sup>* equal to 1.19 s, corresponding to the value obtained by the bode plots of the radiation tests displayed in Figure 11. When the motor and gearbox with pinion are removed in Test\_170, a *T<sup>n</sup>* equal to 1.00 s is obtained. This lower *T<sup>n</sup>* can be attributed to the removed inertia within the PTO drivetrain since removing inertia results in a lower *Tn*. Test\_089 and Test\_129 yield a *ξ<sup>d</sup>* of 0.13 and 0.12, respectively, corresponding to an underdamped system. Given this low *ξ<sup>d</sup>* , *ω<sup>n</sup>* will closely approximate *ω<sup>d</sup>* .

**Table 7.** Overview of the performed free decay tests.


The limited draft in combination with the flat bottom will introduce important bottom slamming effects for tests where the WEC buoy re-enters the water after being lifted out [32]. These bottom slamming forces are assessed in Test\_161 and Test\_162, for which the start position of the WEC buoy is completely lifted out of the water. In addition, nonlinear viscous drag effects occur. The semi-empirical Morison equation describes the viscous force *Fvis* in function of a viscous drag coefficient *C<sup>D</sup>* [33]:

$$F\_{\rm vis} = -0.5 \,\rho \, S \, \mathcal{C}\_D \, \dot{X} |\dot{X}| \tag{17}$$

This *C<sup>D</sup>* can be identified from experiments or from fully viscous modeling methods based on the Navier–Stokes equations [34–36]. The experimental quantification of bottom slamming effects and nonlinear viscous drag effects is not addressed in the presented article.

Test\_170 considers multiple free decay tests for the WEC buoy with the motor and gearbox with pinion removed. While z is obtained by both the laser and encoder for the other tests, only laser measurements are available when the motor is removed. The WEC buoy is manually pushed down or lifted from the measurement bridge.

Figure 16a displays the WEC buoy displacing a water volume, equivalent to the volume of the WEC buoy, after being dropped from −0.202 m in Test\_161. The concentric circular radiated waves can be observed. Bottom slamming forces are obtained from *Floadcells*. Figure 16b shows the free decay z time series and corresponding exponential decay envelopes according to Equation (15) for Test\_089, Test\_129 and Test\_170.

Test\_089 with an elevated start position of -0.089 m results in a quasi-equivalent decay response as Test\_078 with a submerged start position of 0.078 m. In order to allow a comparison of the symmetry of the WEC system, Test\_089 is plotted with inversed sign.

#### *3.4. Power Absorption*

Figure 17 displays the four quadrants where the WEC PTO system acts as a motor or generator, according to the adopted sign convention [37].

When *FPTO* and the resultant *z*˙ have an opposite sign, energy will be extracted from the waves and the PTO acts as a generator, occurring in quadrant II and IV. When *FPTO* and the resultant *z*˙ have an identical sign, energy will be consumed and the PTO acts as a motor, occurring in quadrant I and III. The defined convention in the Simulink model is that a net positive absorbed power value *Pabs* corresponds to mechanical power absorption

1

and a net negative *Pabs* corresponds to adding power to the system. Therefore, a minus sign is added to the multiplication of *FPTO* with *z*˙ in order to obtain *Pabs*:

$$P\_{\rm abs} = -F\_{\rm PTO} \cdot \dot{z} \tag{18}$$

Note that this calculation results in instantaneous values. For regular waves, averaging over a number n of wave periods T results in the averaged absorbed power value *Pabs*:

$$\overline{P}\_{\rm abs} = \frac{1}{nT} \int\_0^{nT} P\_{\rm abs}(t)dt \tag{19}$$

#### 3.4.1. Impedance Matching

Impedance matching can be applied to enable maximum power transfer between two oscillatory systems, the ocean waves and the WEC PTO system. Since impedance is defined as a complex value, the PTO generally has a resistance component (real part) and a reactance component (imaginary part). The maximum power transfer theorem says that the maximum possible power is delivered to the PTO when the PTO impedance *ZPTO* (load impedance or input impedance) is equal to the complex conjugate (represented by \*) of the impedance of the source *Z<sup>i</sup>* (intrinsic impedance or output impedance). For two impedances to be complex conjugates their resistances must be equal and their reactances must be equal in magnitude and opposite in sign. With the WEC impedance model given by Equation (13), an optimal PTO is obtained by *ZPTO*(*ω*) [12,38]:

$$Z\_{\rm PTO}(\omega) = Z\_i^\*(\omega) = \mathcal{B}(\omega) - i \left( \omega (M + A(\omega)) - \frac{K}{\omega} \right) \tag{20}$$

The frequency response of the Proportional Integral Derivative (PID) controller is given by [12]:

$$FRF\_{PID}(\omega) = \frac{(i\omega)^2 K\_D + i\omega K\_P + K\_I}{i\omega} \tag{21}$$

*KP*, *K<sup>I</sup>* and *K<sup>D</sup>* are the proportional, integral and derivative gains of the controller, respectively. The rearrangement of Equation (21) results in the PID controller impedance:

$$\mathbf{Z\_{PID}}(\omega) = i\omega \mathbf{K\_D} + \mathbf{K\_P} - i\frac{\mathbf{K\_I}}{\omega} = \mathbf{K\_P} + i(\omega \mathbf{K\_D} - \frac{\mathbf{K\_I}}{\omega}) \tag{22}$$

Given Equations (20) and (22), the proportional gain for the Proportional (P) controller, equivalent to the damping coefficient *CPTO*,*<sup>P</sup>* for the resistive control strategy, is given by:

$$\mathcal{C}\_{\text{PTO},P}(\omega) = |Z\_i^s(\omega)| = \sqrt{\mathcal{B}^2(\omega) + \left(\omega(M + A(\omega)) - \frac{K}{\omega}\right)^2} \tag{23}$$

Given Equations (20) and (22), the proportional and integral gain for the Proportional Integral (PI) controller, equivalent to *CPTO*,*PI* and the spring coefficient *KPTO*,*PI* for the reactive control strategy, are given by:

$$\mathcal{C}\_{\text{PTO},\text{PI}}(\omega) = \text{Re}\{Z\_i^\*(\omega)\} = \mathcal{B}(\omega) \tag{24}$$

$$\mathcal{K}\_{\text{PTO},\text{PI}}(\omega) = \omega \cdot \text{Im}\{\mathcal{Z}\_i^\*(\omega)\} = \omega \cdot \left(-\left(\omega(M + A(\omega)) - \frac{K}{\omega}\right)\right) = -\omega^2 (M + A(\omega)) + K \tag{25}$$

The presented study considers only P and PI control. Gu et al. present the implementation of a PID controller in a frequency domain model for a heaving point absorber WEC [39]. In this case the derivative controller acts on the acceleration term, corresponding to mass control.

#### 3.4.2. Causal Impedance Matching P and PI Controller

For any causal system, the complex conjugate of the impedance will produce an acausal system. Since a WEC is a causal system, an acausal system is obtained for *Z* ∗ *i* (*ω*). Therefore, to implement Equation (23) or Equations (24) and (25) perfectly across all wave frequencies, the future velocity of the WEC has to be known, resulting in an acausal controller implementation [12]. However, the controller does not need to operate perfectly at all wave frequencies simultaneously and can be designed to work well in a restricted range of frequencies. The energy in a typical sea state ranges over at most a single decade of frequencies, with the sea state changing appreciably only over the course of hours. This band-limited and slowly varying nature of ocean waves allows to utilize a causal realization of the impedance matching approach. Approximating *Z* ∗ *i* (*ω*) in the peak frequency *ω<sup>p</sup>* of the design sea state is straightforward to implement in the Simulink model. The impedance of the causal impedance matching P controller is equal to *CPTO* defined in Equation (24) [12,38]:

$$Z\_P(\omega) = \mathbb{C}\_{\text{PTO}}(\omega\_p) \tag{26}$$

The impedance of the causal impedance matching PI controller is an interpolation of the impedance given by Equation (22) in *ω<sup>p</sup>* [12,38]:

$$Z\_{\rm PI}(\omega) = \mathbb{C}\_{\rm PTO}(\omega\_p) - i \cdot \frac{K\_{\rm PTO}(\omega\_p)}{\omega} \tag{27}$$

Based on the experimentally determined *Z<sup>i</sup>* in Test\_112, the impedance of a causal impedance matching P controller and PI controller can be determined, according to Equations (26) and (27), respectively. Figure 18a shows the gain and phase of these controllers, for an interpolation point of T = 1.50 s. Since in the power absorption tests various wave conditions are considered, Figure 18b plots *CPTO*,*P*(*ω*) (Equation (23)), *CPTO*,*PI*(*ω*) (Equation (24)) and *KPTO*,*PI*(*ω*) (Equation (25)) in function of the wave frequency. The resulting control parameters for T = 1.50 s and T = 2.00 s are indicated. As expected, *KPTO*,*PI*(*ω*) = 0 kg/s<sup>2</sup> for *T<sup>n</sup>* = 1.19 s.

(**a**) (**b**)

**Figure 18.** Radiation Test\_112: (**a**) Bode plot intrinsic impedance and causal impedance matching P and PI controller; (**b**) Coefficients causal impedance matching P and PI controller.

3.4.3. Resistive Control

The baseline control strategy is resistive control, equivalent to causal impedance matching P control as defined in Equation (26). *FPTO*,*<sup>P</sup>* is equal to *z*˙ multiplied with a positive *CPTO*:

$$F\_{\rm PTO,P} = -\mathbb{C}\_{\rm PTO} \cdot \sharp \tag{28}$$

A minus sign is added, since *FPTO*,*<sup>P</sup>* should oppose *z*˙. Tables 8 and 9 provide an overview of the performed resistive control tests, for regular and irregular waves, respectively. The applied *C<sup>C</sup>* is given. For the regular waves, characterized by T and H, a range of *CPTO* is tested during a single test. For the irregular waves, characterized by a JONSWAP wave spectrum with *γ* = 3.3, *H<sup>s</sup>* and *Tp*, a fixed *CPTO* is applied.


**Table 8.** Overview of the resistive control strategy tests for regular waves.

**Table 9.** Overview of the resistive control strategy tests for irregular waves.


Figure 19a shows a snapshot of the z and *z*˙ time series for Test\_100. Figure 19b shows the corresponding *CPTO* input, with a stepwise increase from 400 kg/s to 450 kg/s, and the *Pabs* output.

(**a**) (**b**) **Figure 19.** Resistive control test (Test\_100): (**a**) z and *z*˙ time series; (**b**) *Pabs* and *CPTO* time series.

#### 3.4.4. Reactive Control

Reactive control, equivalent to causal impedance matching PI control as defined in Equation (27), aims to bring the WEC into resonance by the addition of a spring. *FPTO*,*PI* is composed of *z*˙ multiplied with a positive *CPTO* and z multiplied with negative *KPTO*.

$$F\_{\rm PTO,PI} = -\mathcal{C}\_{\rm PTO} \cdot \dot{z} - K\_{\rm PTO} \cdot z \tag{29}$$

Again, a minus sign is added, since *FPTO*,*PI* should oppose *z*˙. Table 10 gives an overview of the performed reactive control tests for regular waves, characterized by H and T. *C<sup>C</sup>* is fixed at 0.6 and a range of *CPTO* and *KPTO* is tested during each test.


**Table 10.** Overview of the reactive control strategy tests for regular waves.

Figure 20a shows a snapshot of the z and *z*˙ time series for Test\_138. The Simulink control model did not impose constraints on *z*, *z*˙, *z*¨, allowing these high values for z and *z*˙. The stroke is mechanically limited to 0.25 m by the micro switches, displayed in Figure 3. Figure 20b shows the corresponding *CPTO* input, with a stepwise increase from 60 kg/s to 72 kg/s, and the *Pabs* output. The net *Pabs* is calculated as the period averaged difference between the integrated generated power (positive *Pabs*) and the integrated added power (negative *Pabs*), according to Equation (19) and the sign convention as defined in Figure 17. *<sup>K</sup>PTO* is equal to <sup>−</sup>1520 kg/s<sup>2</sup> for the displayed time window.

**Figure 20.** Reactive control test (Test\_138): (**a**) z and *z*˙ time series; (**b**) *Pabs* and *CPTO* time series.

3.4.5. Power Absorption Comparison between the Resistive and Reactive Controller

Two regular wave conditions, characterized by H = 0.09 m, T = 2.00 s and by H = 0.09 m, T = 1.50 s, respectively, are considered to compare the presented resistive and reactive control strategy. *Pabs* is calculated with n = 5 in Equation (19). For resistive control, *CPTO*,*<sup>P</sup>* is obtained according to Equation (23) and for reactive control, *CPTO*,*PI* and *KPTO*,*PI* are obtained according to Equations (24) and (25), respectively. Figure 18b displays the obtained control parameters for T = 2.00 s and T = 1.50 s. For *CPTO*,*PI*, lower values than experimentally indentified are adopted. For the regular wave with H = 0.09 m and T = 2.00 s, resistive control (Test\_100) with *CPTO* = 450 kg/s yields *Pabs* = 1.89 W and reactive control (Test\_138) with *CPTO* = 72 kg/s and *KPTO* = −1520 kg/s yields *Pabs* = 10.26 W. For the regular wave with H = 0.09 m and T = 1.50 s, resistive control (Test\_103) with *CPTO*

= 200 kg/s yields *Pabs* = 3.09 W and reactive control (Test\_147) with *CPTO* = 70 kg/s and *KPTO* = −1120 kg/s yields *Pabs* = 11.75 W. Figure 21 visualizes these results.

**Figure 21.** *Pabs* for Test\_100, Test\_103, Test\_138 and Test\_147.

For the reactive controller, a significant increase in power absorption compared to the resistive controller is observed, which is according to the literature [40]. Figure 20a shows that resonance occurs as *zmax* = 0.165 m is significantly higher than H/2 = 0.045 m. To properly assess the power performance of the reactive controller compared to the resistive controller, irregular waves should be considered, as discussed in Section 3.4.2.

The amplification on z requires the motor to add energy, which has a certain efficiency. Apart from larger peaks in *FPTO* and *Pabs*, a major drawback of reactive control is the energy loss by dissipative processes inherent to the back-and-forth energy exchange between the PTO and the WEC buoy, especially when the magnitude of the exchanged energy is comparable to, or even significantly larger than, the net absorbed energy [41]. The presented research does not consider the PTO efficiency, nor the damping force, nor the reactive force. Strager et al. present a method to determine the optimal reactive control parameters for a given combination of non-ideal PTO efficiency and monochromatic wave frequency [42].

#### **4. Discussion and Conclusions**

Within the 'WECfarm' project, two test campaigns are performed at the AAU wave basin: (a) a testing of the first WEC in April 2021 and (b) a testing of a two-WEC array in February 2022, in preparation of five-WEC array tests. The main objective of the 'WECfarm' project is to cover the scientific gap on experimental data necessary for the validation of recently developed (non-linear) numerical models. The presented article discusses experimental testing of an isolated WEC, being the first test campaign within the 'WECfarm' project. The primary objective of evaluating the hydrodynamics, electromechanics, control platform, DAQ and structural performance of the WEC is to allow extending the setup to a five-WEC array.

The WEC buoy is a truncated cylinder with a high diameter to draft ratio to increase radiation and WEC–WEC interactions. The PTO system of the WEC is a PMSM connected to a gearbox powering a rack and pinion system. A configuration of three air bushings guarantees a permanent layer of air between the guide shafts and the bushings, resulting in zero-friction linear guiding. The WEC control and data acquisition are realized with a Speedgoat Performance real-time target machine, offering the possibility to implement advanced WEC array control strategies in the MATLAB-Simulink model. This unique PTO system in combination with the real-time target machine makes active and accurate PTO control possible.

The drivetrain (motor, gearbox, rack and pinion) friction is assessed by manually moving the WEC buoy up and down, from below the loadcells and under a zero torque command. Based on the empirical relationship between *Floadcells* and *z*˙, a simplified model based on Coulomb and viscous friction is determined. The resulting friction compensation model *FComp* is implemented in Simulink as a torque augmentation on the torque command.

A SID approach is adopted, considering the WEC system to be composed of two SISO models, the radiation and the excitation model. The radiation tests in calm water, with *FPTO* chirp-up and chirp-down noise signals as input and *z*˙ as output, yield the

intrinsic impedance *Z<sup>i</sup>* . The excitation tests with the WEC buoy fixed, with as input various JONSWAP spectra and as output *Floadcells*, yield the wave excitation force coefficients *He*(*ω*). The assumption that WG measurements at sufficient distance from the WEC buoy may be used to characterize *η* on the location of the WEC is confirmed. Free decay tests characterize the WEC buoy decay response, confirming that *T<sup>n</sup>* = 1.19 s, obtained by the radiation tests.

Power absorption tests are executed with the resistive and reactive control strategy. The tests with regular waves are executed with 'real-time' tuning of the control parameters in the Simulink model. Adopting an impedance matching approach, the optimal *CPTO*,*<sup>P</sup>* for resistive control and the optimal *CPTO*,*PI* and *KPTO*,*PI* for reactive control are calculated from *Z<sup>i</sup>* . For the two selected regular wave conditions, characterized by H = 0.09 m, T = 2.00 s and by H = 0.09 m, T = 1.50 s, the reactive controller results in a significant higher averaged absorbed power *Pabs* compared to the resistive controller. However, the presented research does not consider PTO efficiency, which is detrimental for reactive control.

The implementation of the friction compensation model *FComp* proved to be a good methodology to partly compensate Coulomb and viscous friction attributed to the drivetrain. The experimentally determined radiation and excitation model yields a simple, though accurate, model of the WEC system. The intrinsic impedance, resulting from the radiation test, is used to design a causal impedance matching P and PI controller, equivalent to a resistive and reactive controller. Although this approach was only tested limitedly, mainly for regular waves, the successful extension to a multiple-WEC array, considering irregular waves, is confirmed. The testing of the isolated 'WECfarm' WEC proved to be successful and the extension of the setup to a five-WEC array will allow it to comply with the future research objectives.

**Author Contributions:** Conceptualization, methodology and analysis, T.V.; execution of the experiments, T.V., F.F., L.D.B., H.C. and B.D.W.; data curation, T.V.; writing—original draft preparation, T.V.; writing—review and editing, V.S., F.F., M.V. and P.T.; funding acquisition for the experimental setup, V.S. and P.T. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work is supported by the the FWO (Fonds Wetenschappelijk Onderzoek-Research Foundation Flanders), Belgium, through the following funding: (1) Timothy Vervaet is Ph.D. fellow (fellowship 11A6919N); (2) Vasiliki Stratigaki is a FWO postdoctoral researcher (fellowship 1267321N) and has been granted the 'FWO Research Grant' for constructing the WEC experimental set-up (FWO-KAN-DPA376). The travel expenses of Timothy Vervaet, Louis De Beule, Hendrik Claerbout and Bono De Witte for conducting the experimental campaign at the AAU wave basin were funded by four WECANet COST Action CA17105 Short Term Scientific Missions (STSMs). COST (European Cooperation in Science and Technology) is supported by the EU Framework Programme Horizon 2020. COST is a funding agency for research and innovation networks.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The datasets resulting from the WECfarm project will be made available in due time on https://www.awww.ugent.be, accessed on 23 August 2022.

**Acknowledgments:** Michiel Herpelinck and Brecht De Backer were involved in the design of the WECfarm WEC in the framework of a master dissertation supervised by Prof. Kurt Stockman from the Department of Electromechanical, Systems and Metal Engineering of Ghent University. Brecht De Backer initiated the Autodesk Inventor drawings of the WEC. OAV (https://www.oavco.com/, accessed on 23 August 2022) is acknowledged for providing educational discount on the OAV 40 mm air bushings used in the experimental setup as well as for providing technical support [14]. Beckhoff, Wittenstein and Speedgoat are acknowledged for providing technical support. Aalborg University, Denmark is acknowledged for making the wave basin available from 12–18 April 2021 for the presented test campaign.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

### **Abbreviations**

The following abbreviations are used in this manuscript:


### **References**

