**1. Introduction**

A point absorber wave energy converter (WEC) consists of a floating or submerged body to capture energy from different wave directions. The point absorber diameter should preferably be in the range of 5–10% of the prevailing wavelength [1]. Due to its ability to absorb energy from different directions, this WEC type is particularly suitable to put in arrays. In a WEC array, hydrodynamic interactions between the WECs occur through radiation and diffraction of waves. Both constructive and destructive interactions will occur between individual WECs within a WEC array called near-field interactions.

**Citation:** Vervaet, T.; Stratigaki, V.; Ferri, F.; De Beule, L.; Claerbout, H.; De Witte, B.; Vantorre, M.; Troch, P. Experimental Modelling of an Isolated WECfarm Real-Time Controllable Heaving Point Absorber Wave Energy Converter. *J. Mar. Sci. Eng.* **2022**, *10*, 1480. https:// doi.org/10.3390/jmse10101480

Academic Editor: Constantine Michailides

Received: 9 September 2022 Accepted: 7 October 2022 Published: 11 October 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

Göteman et al. define directions for future research necessary for a better WEC array optimization approach [2]. There is a need for available real life data for the validation of WEC array modeling and optimization. This is a research gap for the full wave energy sector, not only in the optimization of WEC arrays. In computational fluid dynamics-based numerical wave tanks (CNWT), the Power Take-Off (PTO) system is mostly modeled as a linear spring–damper system, not representing realistic PTO dynamics and inefficiencies and undermining the overall model fidelity [3]. To validate CNWT considering WECs, it is desirable to incorporate a realistic, nonlinear PTO model. As the computational power capabilities increase yearly, so do the numerical model's capabilities, stressing the need for experimental data to validate the model. However, publicly available databases from WEC array experiments are scarce. Vervaet et al. identified 17 experimental campaigns on point absorber WEC arrays, carried out during the last decades [4]. This limited number of experimental campaigns is due to the high cost of constructing and testing in wave basin facilities, as well as due to the complexity of the experiments and related instrumentation [5]. Therefore, the 'WECfarm' project aims to deliver a dataset to cover the research gap on the need for publicly available real life and reliable data to validate these new advanced numerical models. Vervaet et al. discuss the state of the art in physical modeling of point absorber WEC arrays and the identification of research gaps, resulting in design specifications of the 'WECfarm' experimental setup [4].

The 'WECfarm' experimental setup consists of an array of five heaving point-absorber WECs, designed as a unique test bench for future innovative WEC array research, able to address the current requirements and research gaps on physical WEC array testing. Given the limited number of five WECs, the WECfarm WEC array is not classified as a large WEC array, as the Manchester Bobber 25-WEC array [6], the PerAWaT project 24-WEC array [7] and the WECwakes project 25-WEC array [8]. Vervaet et al. discuss the features of the experimental setup, for which the most important ones are summarized below [4]. The WEC buoy is designed to be generic, being a truncated cylinder with a draft of 0.16 m and a radius of 0.30 m. The high diameter-to-draft ratio of 3.75 yields a flat Response Amplitude Operator (RAO) response and high resonance bandwidth, enhancing WEC–WEC interactions. The WECs are equipped with a Permanent Magnet Synchronous Motor (PMSM), addressing the need for WEC array tests with an accurate and actively controllable PTO. The air bushings linear guiding system excludes guiding friction in the power absorption measurements. The WEC array control and data acquisition are realized with a Speedgoat (Speedgoat, Köniz, Switzerland) Performance real-time target machine, offering the possibility to implement advanced WEC array control strategies in the MATLAB-Simulink environment. Wave basin testing with 'WECfarm' WEC arrays targets to include long- and short-crested waves and extreme wave conditions, representing real sea conditions. Within the 'WECfarm' project, two experimental campaigns have been performed at the Aalborg University (AAU) wave basin: (a) testing of the first WEC in April 2021, addressed in the presented article; (b) testing of a two WEC array in February 2022. An experimental campaign with a five WEC array, in the new wave basin; the Coastal and Ocean Basin (COB) in Ostend (Belgium) [9], is scheduled in 2023.

Friction characterization tests are performed to quantify the drivetrain (motor, gearbox, rack and pinion) introduced friction, whereafter a Coulomb and viscous based friction model for partial compensation of the drivetrain friction is implemented in the MATLAB-Simulink control model. Beatty et al. used a Proportional Integral (PI) force control to minimize the error between the target and measured forces for the physical model of a Wavestar WEC [10]. However, Bacelli et al. stress that closing the loop around a force sensor may induce negative consequences for the design of higher level control loops [11]. The presented friction compensation methodology provides an alternative for closing the PTO force feedback loop around the force sensor.

Coe et al. present a WEC control design based on the principle of impedance matching [12]. The control parameters yield from the complex conjugate of the intrinsic impedance, determined by radiation system identification (SID) tests, where the WEC is excited by a

torque noise signal in calm water. Bacelli et al. performed SID tests for an isolated heaving point absorber WEC [13]. This article discusses the application of SID tests on an isolated 'WECfarm' WEC. The SID methodology will be extended to arrays with two to five WECs. Application of the impedance matching methodology for WEC arrays will yield valuable data and insights on WEC-WEC interactions and WEC array control optimization. Within the presented test campaign, the resistive and reactive controller are tested for a selection of regular wave conditions. This testing campaign allows a full evaluation of the WEC prior to extending the setup to five WECs.

Section 2 provides a detailed overview of the experimental setup, with a focus on the 'WECfarm' WEC, instrumentation and wave basin setup. The experimental results of the drivetrain friction model characterization tests, the SID tests and the power absorption tests are discussed in Section 3. A summary of the findings and conclusions are presented in Section 4.

#### **2. Experimental Setup**

#### *2.1. WECfarm WEC and Instrumentation*

For a detailed discussion on the design of the 'WECfarm' five-WEC array, the reader is referred to [4]. In this article, we consider a single, isolated 'WECfarm' WEC. Figure 1 shows a 3D rendering of the final design of the device. The used right-handed coordinate system has its origin at the intersection of the still water level (SWL) with the vertical axis through the center of the WEC buoy. This allows us to express displacements of the WEC buoy relative to the SWL. The x-axis corresponds with the positive wave propagation direction. The y-axis follows from the motor sign convention: a positive torque results in a downward motion of the WEC buoy. Therefore, the z-axis is pointed downwards to define positive forces, displacements, velocities and accelerations.

**Figure 1.** Rendering of the 'WECfarm' WEC, made with Autodesk Inventor (Autodesk, San Rafael, CA, USA).

To exclude friction in the linear guiding, 40 mm OAV (OAV Air Bearings, Princeton, NJ, USA) air bushings are used. The air bushings are characterized by a load versus pressure curve, where one air bushing can cope with a maximum radial load of 720 N, for a nominal pressure of 5.5 bar [14]. A configuration of three OAV 40 mm air bushings guarantees a permanent layer of air between the guide shafts and the bushings for the most extreme wave conditions, resulting in zero-friction linear guiding on the condition

of proper alignment. The PTO system of the WEC is designed as a PMSM connected to a gearbox powering a rack and pinion system. The pinion pitch circle radius *Rpinion* is equal to 0.0212205 m. A Wittenstein (Wittenstein, Igersheim, Germany) single-stage gearbox 'NPR 025S-MF1-4 -2E1-1S' with ratio i = 4 is connected to a Beckhoff (Beckhoff Automation, Verl, Germany) PMSM 'AM8542-2E11-0000' with an inertia of 6.17 kg cm<sup>2</sup> , a rated torque of 3.97 Nm and a rated speed of 1200 RPM, for 230 V AC power supply [15,16]. The velocity on the pinion will be four times less than on the motor shaft, while the torque on the pinion will be four times more than on the motor shaft. The Beckhoff PMSM is powered and controlled by a Beckhoff motor drive type 'AX5103-0000-0212'. The hydrodynamic part of the WEC consists of an Acrylonitril-Butadieen-Styreen (ABS) thermofolded truncated cylindrical buoy, covered with a Polymethylmethacrylate (PMMA) plate. Figure 2a shows a 2D rendering of the WEC buoy with its dimensions. The WEC buoy is 0.32 m high and designed with a draft of 0.16 m. This draft corresponds with a submerged volume of 0.03683 m<sup>3</sup> . Therefore, the mass of the WEC buoy and hydrodynamically activated parts on top of it is 36.83 kg.

(**a**) (**b**) **Figure 2.** 'WECfarm' WEC: (**a**) 2D rendering of the WEC buoy, dimensions in m; (**b**) sensors and their respective location on the WEC.

Figure 3 shows a scheme of the data acquisition and control flow for the 'WECfarm' setup with the isolated WEC, with a legend indicating the signal type. The MATLAB-Simulink real-time control model is built on the host PC and loaded on the Speedgoat Performance real-time target machine by Ethernet communication. This target machine runs the Simulink model and processes the input/output (I/O) at a sample frequency of 1000 Hz. In this context, the high sample frequency corresponds with the defined 'real-time' terminology. For each test, the various time series of each logged Simulink signal are saved within a single MATLAB structure.

A scheme on the bottom of Figure 3 shows the sensor input for the Speedgoat IO133 terminal board. The accelerometer ADXL335 (Analog Devices, Norwood, MA, USA) is used to measure the acceleration of the WEC buoy in the heave direction and is attached on top of the rack, the furthest position on the WEC from the water. The accelerometer has a linearity of ±0.3 % of the Full Scale Output (FSO) [17]. Three Tedea Huntleigh (Vishay Precision Group, Malvern, PA, USA) 50 kg load cells are placed between the hydrodynamic part (=the buoy) and the electromechanical part (=the motor) to measure the actual applied forces. The load cells with accuracy class C3 have a total error (per OIML R60) of 0.020% of the rated output [18]. A configuration of at least three load cells is required to avoid torsion and bending influencing the measurements. The mass of the three load cells together is equal to 0.682 kg, the mass above the load cells *mtop* is equal to 27.610 kg and the mass below the load cells is equal to 8.534 kg, which results in a total hydrodynamically activated mass m of 36.83 kg. A TLE analog weight transmitter (Laumas Elettronica, Montechiarugolo, Italy) is used to amplify these three analog signals and to sum them to one analog signal. In case the WEC is locked, the wave heave excitation force *F<sup>e</sup>* can be measured. In case the motor is active, the load cells measure the PTO force *FPTO*. The upper micro switch and the

lower micro switch are used as safety limit switches. It is necessary to limit the amplitude of the WEC buoy displacement to prevent the guiding system damaging the structure. The laser sensor Micro-Epsilon optoNCDT 1420-500 (Micro-Epsilon, Ortenburg, Germany) is installed as a backup for the motor encoder to measure the displacement of the WEC buoy relative to the SWL. The laser sensor has a linearity of ±500 µm, equivalent to ±0.1% of the FSO [19]. Moreover, the laser sensor can be used for displacement measurements for tests without the motor. Three pneumatic indicators, one for each air bushing, are used as a visual safety indicator in the pneumatic circuit. As long as the air bushing is provided with a certain air pressure, the red balloon in the indicator stays inflated, confirming the air bushings are pressurized. Figure 2b shows a picture of the WEC as installed at the AAU wave basin with indication of the location of the sensors.

**Figure 3.** General data acquisition and control flow for the isolated 'WECfarm' WEC.

The motor drive, the Speedgoat IO133 terminal board, DC power supply and loss current switches are centralized in the control cabinet. Figure 3 shows a picture of the inside of the control cabinet. During the experimental campaign, the VTI (VTI Instruments Corporation, Irvine, CA, USA) wave gauge sensor system was put inside the control cabinet to establish the connection of the VTI wave gauge sensor analog output with the Speedgoat IO133 terminal board analog input. The VTI wave paddle trigger is used for synchronization by providing a constant voltage signal from the moment the wave paddles are activited. The seismic accelerometer (PCB 393B04) is placed on top of the steel frame to quantify possible vibrations of this frame. Vibrations of the frame and resonance in particular are to be avoided, since these affect the measurement quality of the other sensors.

1

The torque request in the Simulink model is sent by the EtherCAT (Ethernet for Control Automation Technology) communication protocol to the Beckhoff motor drive as a Master Data Telegramm (MDT) process parameter. On the other hand, the Speedgoat target machine can receive by EtherCAT communication Amplifier Telegramm (AT) process parameters from the Beckhoff motor drive. The Beckhoff PMSM input and output signals are sent from and to the Beckhoff motor drive by the One Cable Technology (OCT), which allows to power the motor and process feedback. The motor drive receives the absolute position within one revolution at an 18 bit resolution from the single-turn absolute encoder. This encoder allows real-time determination of the state (position and velocity) of the WEC buoy. The drive provides the motor with a certain current, corresponding to a torque by multiplication with the torque constant of 1.91 Nm/A [16].

The uncertainty in the measured values with the given instrumentation is minimized by the selection of sensors with high resolution, high accuracy and low linearity error. It is important that the sensors are correctly calibrated, with a zero offset for the equilibrium position with draft 0.16 m, represented in Figure 2a. Besides the uncertainty related to sensor measurements, it is important to quantify the uncertainty of the physical testing results. Lamont-Kane et al. identified five distinct sources of uncertainty for physical testing of WEC-arrays [20]: Spatial variation of the wave-field within the wave basin; temporal variation of the wave-field from one repeat to another; the repeatability of model response for any single individual WEC; the reproducibility of model response between various nominally identical WECs (not applicable for the test campaign with a single, isolated WEC in the presented article) and the variation in the time-series of an incident irregular wave train. The quantification of these sources of uncertainty for the performed experimental campaign is not addressed in the presented article.

#### *2.2. Wave Basin Setup*

The experimental campaign took place at the AAU wave basin of the Ocean and Coastal Engineering Laboratory of the Department of the Built Environment [21]. The wave basin measures 14.60 m × 19.30 m × 1.50 m (length × width × depth) with an active test area of 8.00 m × 13.00 m (length × width). The wave generation system is 13 × 1.5 m (width × height) with 30 individually controlled wave paddles (snake type configuration). The system allows accurate generation of 3D waves due to narrow vertically hinged paddles (0.43 m segment width) with maximum wave height up to 0.45 m (at 3.0 s period) and typical maximum significant wave height *H<sup>s</sup>* in the range of 0.25–0.30 m. The wave basin is equipped with passive wave absorber elements. The AwaSys wave generation software is able to generate regular, irregular, solitary waves, execute 2-D and 3-D active wave absorption (reflection compensation) and generate 2nd order irregular unidirectional and multidirectional waves [22].

Seven resistive wave gauges (WGs) are installed in the wave basin to measure incident, diffracted, radiated and reflected waves during the tests. Figure 4a shows the planview layout of the wave basin with the central location of the WEC buoy, WGs, wave generation system and passive absorption. The interdistance between the different WGs and the WEC buoy are indicated in m and waves are generated from the bottom of Figure 4a. These seven WGs are also displayed in Figure 5a. The numbering is based on the used analog input ports of the Speedgoat IO133 terminal board.

Figure 4b shows a 'bird's-eye' perspective of the experimental setup at the AAU wave basin. The WEC is attached to the bridge over the wave basin, complying with the spirit level requirements. Waves are generated from the top right corner in Figure 4b. The host PC for the Simulink control and the PC with the AwaSys software are located in line with the bridge, on the bottom right corner in Figure 4b. The water level is set equal to 1.010 m, which results in an equal positive and negative heave stroke with a magnitude of 0.25 m. This water level should be kept constant during the experimental campaign, since changing water level yields an offset for the position measurements from the laser sensor and motor encoder. A water density *ρ* of 1000 kg/m<sup>3</sup> is taken into account.

Figure 5a shows a picture of the overview of the setup as schematized in Figure 3. Figure 5b shows the coordinate system and sign convention on a picture of the experimental setup, as adopted in Figures 1 and 2a. During this test campaign, waves are generated in the x-direction, corresponding to the direction of the greatest stiffness of the WEC.

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

**Figure 4.** Experimental setup of the isolated 'WECfarm' WEC at the AAU wave basin: (**a**) wave basin planview layout with dimensions in meters; (**b**) bird's-eye perspective picture towards the wave paddles.

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

**Figure 5.** Experimental setup of the isolated 'WECfarm' WEC at the AAU wave basin: (**a**) picture with indication of the subsystems; (**b**) picture with indication of the coordinate system and sign convention.

## **3. Results**

#### *3.1. Test Matrix*

Table 1 gives an overview of the different types of tests that are performed. Based on their underlying purpose, they are subdivided in three categories. The next sections discuss the different test categories.


**Table 1.** Overview of the types of performed tests.

#### *3.2. Friction Model*

The air bushings coefficient of friction is a function of air shear from motion, not from surface contact. Therefore, a friction coefficient of 0.00008 in log-scale due to the contribution from air molecules and gravitation can be taken into account [23]. To obtain the air gap between the shaft and the air bushing, a compressor with a two stage air filter supplies these bushings with clean and dry air under a nominal pressure of 5.5 bar. The gearbox break-away torque *T*<sup>01</sup> is expected to be around 0.20–0.40 Nm, with convergence towards lower values for a longer operational lifetime [24]. The motor static friction *M<sup>R</sup>* is reported to be equal to 0.02 Nm [15]. As a result, the total drivetrain friction experienced by the WEC will be mainly determined by the Coulomb friction attributed to the gearbox. Moreover, additional Coulomb and viscous friction attributed to the rack and pinion will occur. Based on the empirical characterization of the actual friction, a static friction model is constructed, discussed in Section 3.2.1 [25].

#### 3.2.1. Friction Characterization Tests

The friction of the WEC with the motor and gearbox installed is characterized by a zero torque command on the motor. The WEC buoy is manually moved up and down from below the load cells with following elements in the test sequence: slowly at a targeted constant velocity close to 0.0 m/s and an accelerated motion with amplitude up to 0.40 m/s. Figure 6a shows a picture of a friction characterization test in the empty AAU wave basin. The tests are repeated in the filled AAU wave basin to benefit from buoyancy forces.

(**a**) (**b**) **Figure 6.** Friction characterization tests (**a**) in the empty AAU wave basin; (**b**) with application of lateral loading in the-x-direction.

During the execution of the lifting procedure, the force acting on the loadcells *Floadcells*, the position z, the velocity *z*˙ and the acceleration *z*¨ are measured. The first element of the test sequence allows us to determine the Coulomb damping coefficient *CCou*, while the second element allows us to determine the viscous damping coefficient *CVis*.

The friction force *FFriction* to compensate for equals *Floadcells* reduced by the acceleration force *Facc* caused by *mtop* and by the rotational inertia of the motor and the gearbox. To take this rotational inertia of the motor and the gearbox into account in *Facc*, it is expressed as a mass in the heave direction, i.e., the mass attributed to the motor and gearbox inertia *mMGI*:

$$m\_{MGI} = \frac{I\_{motor} \cdot i^2 + I\_{garkbox}}{R\_{pinion}^2} = \frac{6.17 \text{ kg cm}^2 \cdot 4^2 + 0.71 \text{ kg cm}^2}{(2.12205 \text{ cm})^2} = 22.08 \text{ kg} \tag{1}$$

with *Jmotor* the motor inertia equal to 6.17 kg cm<sup>2</sup> [16] and *Jgearbox* the gearbox inertia equal to 0.71 kg cm<sup>2</sup> [26]. It is anticipated that the actual inertia will be higher, as the inertia of the rack and pinion is not taken into account in Equation (1). The resulting *FFriction* is:

$$F\_{\text{Friction}} = F\_{\text{loadcells}} - F\_{\text{acc}} = F\_{\text{loadcells}} - \left(-\mathbb{E} \cdot (m\_{\text{top}} + m\_{\text{MGI}})\right) \tag{2}$$

In case the motor and the gearbox are not installed and the air bushings work properly, *Facc* should equal *Floadcells*, resulting in *FFriction* = 0 N. Figure 7a shows the *z* and *z*˙ time series for the friction characterization Test\_054. Figure 7b shows the time series of *Floadcells*, *Facc*, *FFriction* and the fitting of *CVis* and *CCou*. *Floadcells* exceeds *Facc* due to the addition of the drivetrain (motor, gearbox and rack and pinion) friction.

**Figure 7.** Friction characterization WEC with gearbox and motor (**a**) *z* and *z*˙ time series (Test\_054); (**b**) Force time series (Test\_054).

Fitting a model to *FFriction* results in the friction compensation model *FComp*, given by Equation (3).

$$F\_{\text{Comp}} = \begin{cases} 0 & \text{for } -\sharp\_{\text{Bou}} \le \sharp \le \sharp\_{\text{Bou}} \\ -(-\mathsf{C}\_{\text{Vis}} \cdot \sharp - \mathsf{C}\_{\text{Cou}} \cdot \text{sign}(\sharp))\mathsf{C}\_{\text{C}} & \text{for } \quad \sharp < -\sharp\_{\text{Bou}} \, ||\, \sharp > \sharp\_{\text{Bou}} \end{cases} \tag{3}$$

Note that an additional minus sign is necessary to have *FComp* in the same direction as *z*˙. The model is composed of a Coulomb part proportional to the sign of the velocity and a viscous part proportional to the velocity. A velocity boundary *z*˙*Bou* is imposed to avoid rapid sign switches for the Coulomb friction term at low velocities. Consequently, no friction compensation is taken into account between −*z*˙*Bou* and *z*˙*Bou*. These boundaries correspond approximately with the noise range on the velocity feedback from the motor encoder. Moreover, it is preferred to partly compensate for the friction to preserve a realistic PTO, expressed by the compensation factor *CC*. Figure 8a shows the *Floadcells* to *z*˙ mapping for the friction characterization Test\_054, which allows us to fit the parameters of *FComp*. Table 2 provides the obtained values for *CVis*, *CCou*, *CC*, and *z*˙*Bou*.

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

**Figure 8.** Force to velocity mapping from the WEC friction characterization tests: (**a**) No lateral loading (Test\_054); (**b**) No lateral loading (Test\_054) versus −20 kg lateral loading in the x-direction (Test\_057).

**Table 2.** *CVis*, *CCou*, *C<sup>C</sup>* and *z*˙*Bou* for the resulting *FComp*.


Figure 7b shows the time series of *FComp*. The proper functioning of the velocity boundaries can be observed as less sign switches occuring for *FComp* (yellow curve) compared to the fitting of *CVis* and *CCou* (magenta curve). The difference between *FFriction* (green curve) and *FComp* (yellow curve) is the remaining PTO friction.

The friction characterization tests are repeated with different lateral loading conditions, as an approximation for the surge or sway wave excitation force. In the wave propagation direction (x-direction) −20 kg is applied with a tension spring and rope around the WEC buoy, shown in Figure 6b (Test\_057). This loading of 196 N corresponds approximately with a surge wave excitation force for a wave with period T = 1.0 s and wave height H = 0.40 m, according to linear potential flow simulations with the open-source software package openWEC [27], with the integration of the Boundary Element Method (BEM) code Nemoh. The *Floadcells* to *z*˙ mapping in Figure 8b shows no increased *Floadcells* for Test\_057 compared to Test\_054, demonstrating the proper functioning of the air bushings. In the Simulink model, the uncompensated input motor torque *τmotor*,*uncomp* augmented with *FComp*(*Rpinion*/*i*) yields the compensated input motor torque *τmotor*,*comp*:

$$
\tau\_{\text{motor},comp} = \tau\_{\text{motor},uncomp} + F\_{\text{Comp}} \frac{R\_{ppinion}}{i} \tag{4}
$$

Note that the friction compensation in Equation (4) is implemented as a feedback control structure, since the velocity output of the WEC is used. Commonly, a friction compensation feedforward control structure is adopted when tracking a reference is the objective, which is not the case here [28]. Table 3 provides an overview of all the executed friction characterization tests, with indication of the filling of the AAU wave basin, lateral loading force, lateral loading direction and applied air pressure. The Test\_ID is the identification number of each performed test.


**Table 3.** Overview of the friction characterization tests.

#### *3.3. System Identification*

#### 3.3.1. Linear Decomposed Wave–WEC Interaction Model

To perform a SID of the WEC, a linear decomposition model formulation for the wave–WEC interaction is adopted. The problem is separated into radiation and excitation components. Distinct tests are performed for both the radiation and the excitation components to determine the WEC system response, whereafter radiation and excitation frequency response functions (FRFs) can be constructed from the discrete frequency components. At a high level of abstraction, the WEC can be considered as a system with two inputs: the surface elevation *η* and *FPTO*, as displayed in Figure 9. When the WEC system, consisting of the PTO and the WEC buoy, is assumed to be linear for small motions and small waves, superposition can be applied. The WEC system is considered to be composed of two single input single ouput (SISO) models, the radiation and the excitation model, as displayed in Figure 9 [12,13,29]:

**Figure 9.** WEC block diagram based on the dual single input single ouput (SISO) radiation/diffraction model.

The radiation model is obtained by computing the ratio of the FRF of the output *z*˙ to the FRF of the input *FPTO*, resulting in the admittance *G*(*ω*):

$$G(\omega) = \frac{\check{X}(\omega)}{\widehat{F}\_{\text{PTO}}(\omega)}\tag{5}$$

The quantities are expressed as a function of the angular frequency *ω*. The upper case indicates that these variables are all in the frequency-domain, while the hat symbol ^ denotes that these variables are complex quantities. The control parameters for the impedance matching controller are determined from the intrinsic impedance *Zi*(*ω*), defined as the inverse of *G*(*ω*):

$$Z\_i(\omega) = \left(G(\omega)\right)^{-1} = \frac{\hat{F}\_{\rm PTO}(\omega)}{\hat{X}(\omega)}\tag{6}$$

Mechanical impedance is a measure of the opposition to motion from a source when a potential is applied, defined as the ratio of force (potential) to velocity (flow) in Equation (6). Note that *X* b˙ is used instead of *Z* b˙ for the FRF of *z*˙, to avoid confusion with the defined *Z<sup>i</sup>* . The excitation model is obtained by computing the ratio of the FRF of the output excitation force *F<sup>e</sup>* , measured as *Floadcells*, to the FRF of the input *η* at the location of the WEC, resulting in the excitation force coefficients *He*(*ω*):

$$H\_{\mathfrak{E}}(\omega) = \frac{\widehat{F}\_{\mathfrak{E}}(\omega)}{\widehat{\eta}(\omega)}\tag{7}$$

In the first instance, the above-discussed SID tests are carried out in "open loop", which means no output feedback is considered, represented in Figure 9 by no feedback arrow. A "closed loop" is obtained at any time the WEC is controlled and *FPTO* is calculated based on an output measurement. In Figure 9, *FPTO* depends on *z*˙, as *FPTO* = *Cz*˙, with C representing the control system dynamics. In case of resistive damping, C is a negative constant in this formulation. As a result, the frequency-domain WEC equation of motion, as displayed in Figure 9 as the dual SISO model, is given by:

$$
\widehat{F}\_{\rm PTO} + H\_\varepsilon(\omega)\widehat{\eta} = \widehat{F}\_{\rm PTO} + \widehat{F}\_\varepsilon = Z\_i(\omega)\dot{X} \tag{8}
$$

By applying the superposition principle, the decomposition in the radiation and excitation model is given by:

$$\hat{X} = \frac{1}{Z\_i(\omega)}(\hat{F}\_{\rm PTO} + H\_\varepsilon(\omega)\hat{\eta}) = \frac{1}{Z\_i}\hat{F}\_{\rm PTO} + \frac{H\_\varepsilon(\omega)}{Z\_i(\omega)}\hat{\eta} \tag{9}$$

3.3.2. Impedance Formulation and Radiation Tests

When the Fourier transform F differentiation property is applied to write *z*¨ in terms of *z*˙:

$$\mathcal{F}[\vec{z}(t)] = i\omega \dot{X}(\omega) \tag{10}$$

the point absorber WEC equation of motion can be written in terms of *X* b˙ (*ω*) [12,30]:

$$m\operatorname{i}\omega\,\widehat{X}(\omega) = -(\mathcal{B}(\omega) + \operatorname{i}\omega\,A(\omega))\widehat{X}(\omega) + \frac{K}{\operatorname{i}\omega}\widehat{X}(\omega) + \widehat{F}\_{\text{PTO}} + \widehat{F}\_{\text{\textquotedblleft}} \tag{11}$$

where *m* = 36.83 kg, *A*(*ω*) is the added mass coefficient and *B*(*ω*) is the hydrodynamic damping coefficient. The hydrostatic stiffness coefficient *K* is given by:

$$\mathbf{K} = \rho \mathbf{g} \mathbf{S} \tag{12}$$

where *S* is the cross-sectional area of the WEC buoy at the SWL equal to 0.283 m<sup>2</sup> , *ρ* = 1000 kg/m<sup>3</sup> and g the gravitational acceleration equal to 9.81 N/kg. Rearrangement of Equation (11) based on Equation (9), results in *Zi*(*ω*):

$$Z\_i(\omega) = \frac{\hat{\mathbf{F}}\_{\text{PTO}} + \hat{\mathbf{F}}\_{\varepsilon}}{\hat{X}} = \mathcal{B}(\omega) + i \left(\omega(m + A(\omega)) - \frac{K}{\omega}\right) \tag{13}$$

*Zi*(*ω*) is experimentally determined by executing a forced oscillation test in the AAU wave basin, without waves generated by the wave paddles. In Equation (13), *F<sup>e</sup>* equals zero and *FPTO* is a chirp signal with a frequency spectrum covering the bandwidth of interest [13]. Since it is experimentally more convenient to use *FPTO* as an input and measure the output *z*˙, *Z<sup>i</sup>* is obtained as the inverse of *G*, defined by Equations (5) and (6).

Table 4 shows an overview of the performed radiation tests. The chirp-up PTO input torque is defined with an initial frequency of 0.0 Hz, a target time of 220 s and a frequency at target time of 4.0 Hz. The chirp-down PTO input torque is defined with an initial frequency

of 4.0 Hz, a target time of 220 s and a frequency at target time of 0.0 Hz. The noise signal is defined with an amplitude of 1.0 Nm, multiplied by the defined PTO gain to control the absolute maximum heave amplitude of the WEC buoy *zmax*. Most of the test were executed with *C<sup>C</sup>* = 0.0, without the friction compensation model discussed in Section 3.2 included. Test\_112 and Test\_113 consider *C<sup>C</sup>* = 0.6, which is the WEC system to build the resistive and reactive controller on.


Figure 10a displays a radiation test for the 'WECfarm' WEC, where the circular radiated waves can be observed. Figure 10b shows the *FPTO* and *z*˙ time series, for which *z*˙ displays a resonance. According to Equation (4), a *τmotor*,*uncomp* of 1.0 Nm corresponds with a *FPTO* of 188.5 N.

**Figure 10.** Radiation test: (**a**) Picture of the setup in the AAU wave basin; (**b**) *FPTO* and *z*˙ time series (Test\_059).

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

The calculated *Z<sup>i</sup>* can be displayed as a bode plot with gain and phase, given in Figure 11 for Test\_021, Test\_059, Test\_066, Test\_112 and Test\_113 [12]. Comparing Test\_021 with Test\_059 confirms that the chirp-up and chirp-down signal yields the same *Zi*(*ω*). Test\_066 with a gain of 0.5 and *zmax* = 0.041 m results in a higher identified *Z<sup>i</sup>* compared to Test\_059 with a gain of 1.0 and *zmax* = 0.116 m, stressing the importance of covering motion amplitudes representative for the WEC during operation. Test\_112 and Test\_113 demonstrate how the implementation of *FComp* alters the WEC system dynamics. As expected, *z*˙ is more amplified when the friction is compensated. The WEC resonance frequency *f<sup>n</sup>* is equal at 0.84 Hz, corresponding to a natural period *T<sup>n</sup>* = 1.19 s.

**Figure 11.** Bode plot of the experimentally identified *Z<sup>i</sup>* for Test\_021 , Test\_059, Test\_066, Test\_112 and Test\_113.

#### 3.3.3. Excitation Tests

*He*(*ω*) is experimentally determined by locking the WEC in equilibrium position, as defined in Figure 2a, imposing a frequency rich *η* signal and measuring *Floadcells*. The WEC is fixed by deactivating the motor drive, resulting in an active holding brake. Application of Equation (7) results in *He*(*ω*).

Table 5 shows an overview of the performed excitation tests. The wave input generated by the wave generation system is a JONSWAP wave spectrum, defined by a peak enhancement factor *γ* of 3.3 and the mentioned significant wave height *H<sup>s</sup>* and peak period *Tp*. In the case of Test\_133 with regular waves, H and T are mentioned. Using a start signal from the wave paddles and recording the wave paddle motion, allow for a deterministic comparison of the same sea state over different control inputs and types.

**Table 5.** Overview of the performed excitation tests.


Figure 12a displays an excitation test for the WECfarm WEC, where the incoming and diffracted waves can be observed. Figure 12b shows the *Floadcells* and *ηWG*<sup>11</sup> time series for Test\_074. When a wave crest (negative *η*) passes the WEC buoy, the load cells are compressed, resulting in a positive *Floadcells*. When a wave trough (positive *η*) passes the WEC buoy, the load cells are under tension, resulting in a negative *Floadcells*.

Figure 13a shows the resulting *F*b *loadcells*, equivalent to *F*b*<sup>e</sup>* , and *<sup>η</sup>*<sup>b</sup> for Test\_074 and Test\_077. The calculated *H<sup>e</sup>* can be displayed with a gain and phase, given in Figure 13b for Test\_074 and Test\_077. Frequency smoothing has been performed on *F*b *loadcells*, *<sup>η</sup>*<sup>b</sup> and the gain and phase of *H<sup>e</sup>* . The Gaussian-weighted moving average over a window of 30 frequency intervals of 0.0042 Hz has been taken. Figure 13b confirms the higher *H<sup>e</sup>* for lower wave frequencies, as noticed in Figure 13a.

**Figure 12.** Excitation test: (**a**) Picture of the setup in the AAU wave basin; (**b**) *Floadcells* and *ηWG*<sup>11</sup> time series (Test\_074).

**Figure 13.** Wave excitation Test\_074 and Test\_077: (**a**) *F*b *loadcells* and *<sup>η</sup>*b, frequency smoothened over a window of 30 frequencies; (**b**) *He*, frequency smoothened over a window of 30 frequencies.

To calculate *He*(*ω*) based on the most straightforward conceptual definition, a WG is placed on the location of the WEC buoy when the WEC buoy is not present and then record the *η* time series. The WEC buoy is then put in place and the same wave time series are run again, this time measuring the force on the WEC buoy caused by the waves. For control purposes, this procedure is of little use, since it is clearly not possible to measure *η* at the point where the WEC is located, once the WEC is in place. Therefore, *η* obtained with WG 11 is used as an approximation for *η* on the location of the WEC buoy. It is shown by Bacelli et al. that if the distance between the WEC buoy and the WG is increased, the term describing the diffracted waves at the WG becomes small enough and the original model given by Equation (9) can be adopted [13].

To verify the assumption that the data of WG 11 of these excitation tests could be used for this purpose, at the end of the test campaign some tests with wave spectra are executed with the WEC buoy removed and WG11 moved to the position of the WEC buoy, equilinear with the other WGs. For these tests, the interdistance WG 6 to WG 11 is 1.31 m and the interdistance WG 11 to WG 8 is 0.98 m. Figure 14 shows a picture of this layout of the WGs.

**Figure 14.** Picture of WGs layout for *η* measurements with WG 11 on the location of the WEC buoy.

Three JONSWAP spectra and one regular wave are tested, defined in Table 6.


**Table 6.** Overview of *η* measurement tests with WG11 on the location of the WEC buoy.

The top Figure 15a shows the time series comparison between *η* measured by WG11 in excitation Test\_074 and *η* measured by WG11 in Test\_172. The bottom Figure 15a shows this comparison for Test\_050 and Test\_173.

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

**Figure 15.** Wave excitation tests (Test\_074 and Test\_050) and wave field accuracy tests (Test\_172 and Test\_173): (**a**) *<sup>η</sup>WG*<sup>11</sup> time series; (**b**) *<sup>η</sup>*b, frequency smoothened over a window of 30 frequencies.

For both wave conditions, a minor difference between the two time series can be observed. Figure 15b shows *<sup>η</sup>*<sup>b</sup> for Test\_074 compared to Test\_172 and *<sup>η</sup>*<sup>b</sup> for Test\_050 compared to Test\_173. Figure 15a,b confirm that *η* obtained with WG 11, according to the layout of the WGs in Figure 4a, can be used as an approximation for *η* on the location of the WEC buoy.
