Wave Energy Converter’s Slack and Stiff Connection: Study of Absorbed Power in Irregular Waves

Abstract

Two different concepts of wave energy converter coupled to the novel C-GEN linear generator have been studied numerically, including the evaluation of different buoy sizes. The first concept has a slack connection between the buoy and the generator on the seabed. Another concept is based on a stiff connection between the buoy and the generator placed on an offshore platform. Three different approaches to calculate the damping force have been utilized within this study: the optimal damping coefficient, R-load, and $RC$-load. R-load is a model for the load applied to a grid-connected generator with passive rectification. $RC$-load is a model for a phase angle compensation applied to a system with active rectification. The radiation forces originating from the oscillatory motion of the buoy have been approximated using the transfer function in the frequency domain and the vector fitting algorithm. A comparison of the approximation methods is presented, and their accuracy has been evaluated. The advantage of the vector fitting method has been shown, especially for higher approximation orders which fit the transfer function with high accuracy. The study’s final results are shown in terms of the absorbed power for the sea states of March 2018 at Wave Hub, UK.

1. Introduction

The abundant potential of wave energy [1] for the green energy market is captivating for engineers and scientists across the world. Many mechanisms contribute to the motions of the ocean surface: atmospheric pressure gradients, wind, moon gravitation, and even seismic activity. The waves essential in wave energy conversion are the surface waves created by wind. Six degrees of freedom characterize the free motion of a floating body. A wide range of motion causes a variety of concepts to capture the kinetic and potential energy of the wave. A large number of wave energy converters (WECs) have been designed to absorb wave energy. However, very few of them have been deployed offshore, and there is currently no leading technology [2,3]. An economically feasible technology still has to be found because of a combination of large forces, low speeds, and a harsh offshore environment.

The conventional classifications of WECs are as follows: (1) attenuators, terminators, and point absorbers [4,5]; (2) oscillating water columns (OWC), overtopping devices, and oscillating bodies [6,7]. Other technologies outside of these classifications have been proposed [2]. For example, submerged pressure differential devices [3].

An overview of WECs’ modeling can be found in [8], such as frequency models, wave-to-wire (time-domain approach) and computational fluid dynamics models. The time-domain models describe a WEC’s dynamics. For example, wave-to-wire modeling of an OWC using a rigid piston model can be found in [9].

The point absorber is one of the most commonly used WECs, characterized as a WEC independent from wave direction, which utilizes a floater sized to capture energy from a limited bandwidth of waves. Point absorbers are specifically designed to harvest energy from waves with wavelengths much larger than the dimensions of the absorber. The most efficient energy capture can be achieved if the device is matched to the wave climate for maximum energy extraction. A mooring or gravitational foundation keeps the WEC in place and acts as the oscillation reference frame. An example of point absorber technology is the WEC concept developed by the wave energy research group from Uppsala University (UU’s WEC) [10].

This paper aims to assess the practicality of two point absorber topologies by estimating the absorption of power. In simulations, real wave data collected by a wave buoy during March 2018 at Wave Hub, UK, were used. The first concept is similar to UU’s WEC, where the generator is fixed on the seabed. The connection between the buoy and the translator is slack (Figure 1a). In the second concept, the generator is placed on a fixed four-legged platform; the connection is stiff (Figure 1b). In the simulations, the novel C-GEN direct drive linear generator [11,12] is considered as the electrical power take-off (PTO). The C-GEN (Figure 1c) is a multi-stage air-cored permanent magnet generator technology [11]. It consists of a modular stator and a translator section, shown in Figure 2. The stator coils are encased in an epoxy material to protect them against water ingress and corrosion [12]. The generator design differs from conventional PTO designs by adopting a flooded airgap allowing operation without seals or complex bearing arrangements.

The slack concept model for UU’s WEC was validated with experimental data [13]. During the experiment, various resistive loads were tested for different sea states. Due to the absence of full-scale experimental data for C-Gen it is still a problem to carry out error analysis or validation of the simulations.

Offshore platforms are widely used in the oil, gas and offshore wind energy sector. A review on the integration of WECs and large floating platforms can be found in [14]. Nguyen et al. [14] highlighted the variety of floating platforms moored by tethers, mooring lines, dolphin-frame guide systems, and pier/quay wall systems. Incorporating WECs into offshore platforms can allow for easier installation and access for operation and maintenance compared to submerged alternatives. The future of offshore renewable energy may include co-located generators, such as wind and wave installed jointly. For example, a study on the interaction between wave-wind hybrid concepts can be found in [15]. Lee et al. [15] proposed a large moored floating platform, able to carry four wind turbines and 24 WECs. Our study suggests utilizing a four-legged platform configuration, adapted from [16]. To the best of our knowledge, the integration of WECs on a four-legged platform has not yet been investigated.

As the first step of the WEC concept assessment, mathematical modeling and numerical simulations can be used to investigate the system behavior. Calculations on the possible outcomes of different WEC concepts bypass the high financial burden of experimental trials [17]. Modeling of WEC complex dynamics can be achieved by different methods. The potential flow theory is often used to find hydrodynamic properties of the buoys [18,19]. This method has been widely in use since 1974, due to its relative simplicity and time efficiency [20]. The buoy motion can be found as the solution in the time domain to the Cummins equation [21] based on Newton’s second law and the linear representation of the hydrodynamic forces as radiation force, hydrostatic stiffness force and excitation force. The advantages and limitations of different simulation methods based on the state-space model are reviewed in [22].

In this paper, time-domain simulations are performed using state-space modeling. It is computationally demanding to solve the convolution which is present in the Cummins equation. By including several bodies within simulations, the system response may contain multiple resonance peaks. One way is to approximate the radiation term by means of a transfer function that can be found using vector fitting. It was proposed for engineering problems from high-voltage power systems to microwave systems and high-speed electronics in [23], but is also used in hydrodynamic applications. In [24], the vector fitting was adapted to calculate dynamics of a hinged five-body WEC, consisting of a cylinder linked to four spheres, as well as for an array of 17 bodies. In another study [25], the calibration of a vessel model was performed: the response amplitude operators (RAOs) were calculated using vector fitting and then compared to experimental data. Our study compares two different methods to approximate the radiation force: one method is based on transfer function approximation in the frequency domain as described in [26], the other method performs the rational approximation by vector fitting [23]. The advantages of using vector fitting are discussed.

The absorbed power is usually determined by a damping coefficient, i.e., the proportionality coefficient between the PTO force and the translator velocity [27]. The PTO force is the electromagnetic force allowing for the transformation of the kinetic loads into electrical energy. The power output for different damping coefficients is calculated, and the optimal damping coefficient for each sea state is found to maximize the power output. Similar evaluations for optimal damping are presented in [28]. Various approaches were proposed to increase the absorbed power of WECs. For example, in [29], Falnes reviewed methods to control the oscillations to approach the optimum interaction between the wave and the WEC. Another review on power improvement for direct-drive WECs via electric control is given in [30]. Hong et al. concluded that direct-drive point absorbers are more concerned with the electrical control of the WEC than other designs. In [31], Wang et al. used a parallel capacitor circuit to estimate the power output with and without cable losses. A resistive load is discussed for the calculation of absorbed power in [32], and it is a model of a grid-connected generator with passive rectification. The $RC$-load is a model for phase angle compensation applied to a system with active rectification so that the current is in phase with the emf. Our study only deals with electrical phase compensation and rectification via $RC$-load. The model does not contain explicit rectifier solutions. Instead, a simplified approach using the lumped generator circuit with a connected load is considered to reduce the computational burden.

The paper is organized as follows. First, the models of slack and stiff WEC concepts are described, and the simplifications are highlighted. Then, the approximation of the radiation force using state-space modeling is presented. The details on the damping force calculation via a constant damping coefficient, R-load and $RC$-load are given. It is followed by a comparison of the two state-space methods: transfer function fitting in the frequency domain (TFFD) and rational approximation by vector fitting (RAVF). The best performing method is used to estimate the average power using March 2018 wave data from Wave Hub, UK [33]. Furthermore, the results of the calculations in terms of the average and peak powers are presented for each concept. Finally, the results are discussed, and conclusions on the buoy size are drawn.

2. Method

2.1. WEC Model

In this paper, models of two WEC concepts are studied numerically. The first concept describes a slack connection between the buoy and a generator affixed to the seabed (Figure 1a). The second concept details a stiff connection between the buoy and the translator (Figure 1b), where the generator is on a four-legged platform with the buoy placed underneath. We consider in this publication the C-GEN direct drive linear generator, developed by the Edinburgh University research group. The specific characteristics of the C-GEN generator can be found in

Table 1. Parameters of the C-GEN generator.

Parameters	Values
Mass of Translator	5600 kg
Number of stages	4
Translator length	2 m
Stroke length	3 m
Pole width	0.083 m
Number of coils	18
Number of turns	258
Number of poles	24
Peak velocity	1 m/s
Phase Voltage, rms	240 V
Windings phase resistance	13.35 $\mathsf{\Omega }$
Windings phase inductance	0.21 H
Flux density of airgap	0.45 T

. Since the primary source of movement for the point absorber is the heave movement, only one degree of freedom is assessed.

2.1.1. Slack Connection

In the slack connection concept, the generator is placed on the seabed, and the heave movement of the buoy lifts the translator while gravitational force pulls the translator back down. When the connection line becomes slack, the buoy displacement may be greater than the translator displacement, as the buoy is not fixed vertically above the generator. According to Newton’s second law of motion, different forces are acting on the buoy and translator, which are illustrated accordingly in Figure 3. Following this classification, two separate equations can be formulated, for the buoy and the translator correspondingly:

$$m_b{\ddot{z}}_b\left(t\right)=f_{gb}+f_{wb}\left(t\right)+f_b+f_e\left(t\right)+f_r\left(t\right)+f_{hs}\left(t\right),$$

(1)

$$m_t{\ddot{z}}_t\left(t\right)=f_{gt}+f_{pto}\left(t\right)+f_{wt}\left(t\right)+f_{es}\left(t\right),$$

(2)

where $m_b,m_t$ are the mass of the buoy and translator; ${\ddot{z}}_b\left(t\right),{\ddot{z}}_t\left(t\right)$ are the accelerations of the buoy and translator correspondingly; $f_{gb}$, $f_{gt}$ are the gravitational forces on the buoy and translator; $f_b$ is the buoyancy force; $f_e\left(t\right)$ is the excitation force; $f_r\left(t\right)$ is the radiation force; $f_{hs}\left(t\right)$ is the hydrostatic restoring force; $f_{pto}\left(t\right)$ is the damping force of the power take-off; $f_{es}\left(t\right)$ is the end-stop force; $f_{wb}\left(t\right)$ and $f_{wt}\left(t\right)$ are the wire force for the buoy and the translator respectively, calculated as:

$$f_{wb}=-f_{wt}=\left\{\begin{array}{cc}{k}_w(z_b-z_t),& for{z}_b>z_t;\\ 0,& \mathrm{otherwise}.\end{array}\right.$$

(3)

where $k_w$ is the wire stiffness coefficient.

The excitation force is a sum of incident and scattered waves. However, if the body’s dimensions are less than the wavelength $\lambda$, which is held for the point absorber, scattering may be neglected [34]. A detailed description of the calculation of the forces can be found in [35].

2.1.2. Stiff Connection

A linear generator placed on a fixed platform above the heaving buoy is a novel concept considered in this work. Such an arrangement is interesting for hybrid solutions such as a combination of wind, wave, or solar and for isolated offshore platforms with different purposes. The offshore platform integration may give more accessible access for maintenance of the generator and the possibility of installing replaceable batteries and other energy storage on the platform [15].

The buoy and the translator displacements are assumed to be equal, resulting in the same displacement of the buoy and translator $z=z_b=z_t$. Therefore the equation of motion has the following form:

$$(m_b+m_t)\ddot{z}\left(t\right)=f_e\left(t\right)+f_r\left(t\right)+f_{hs}\left(t\right)+f_{pto}\left(t\right)+f_{es}\left(t\right),$$

(4)

where the forces are defined as for Equations (1) and (2).

2.2. Hydrodynamic Properties of Buoys

A balance on the buoy choice is essential: the buoy should be able to lift the heavy translator and provide high enough mechanical power. Smaller buoys provide low absorbed power and may not give enough buoyancy force for the translator. Therefore, buoys smaller than 1 m radius are not considered.

The buoys geometry is created using the software Multisurf, and the hydrodynamic properties are calculated using boundary element method based software WAMIT. The dimensions and the mass are given in

Table 2. Buoy dimensions used for the two concepts.

Radius, m	Draft, m	Height, m	Mass, kg
1	2.4	3.5	2200
2	0.7	2	3000
3	0.3	1	4000
4	0.2	1	6000

.

A critical component to assess the accuracy of the numerical simulation is RAO. It is normally defined as the ratio between the linear motion response and the amplitude of the incident wave. RAO for 1 m waves is found as:

$$RAO\left(\omega \right)=\frac{{\widehat{F}}_e\left(\omega \right)}{-{\omega }^2{A}_{33}\left(\omega \right)+j\omega B_{33}\left(\omega \right)+C_{33}}$$

(5)

where ${\widehat{F}}_e\left(\omega \right)$ is the excitation force complex amplitude; $A_{33}\left(\omega \right)$ is the added mass; $B_{33}\left(\omega \right)$ is the radiation damping; $C_{33}$ is the hydrostatic stiffness matrix.

The hydrodynamic properties of the buoys for the two concepts differ due to the presence of a four-legged platform. The dimensions of the platform are 12 × 12 m, and the diameter of each supportive leg is 1 m. Increasing the distance between the buoy and platform legs will decrease the hydrodynamic interaction.

Different wave heading angles are tested for the buoy with a radius of 4 m, placed beneath the four-legged platform. Figure 4 shows RAO (Figure 4a) and directions relative to the buoy (Figure 4b) for 0${}^{\circ }$ and 45${}^{\circ }$. The RAOs show a minor difference for the different wave heading angles, and the case of 0${}^{\circ }$ wave heading angle was chosen for the analysis.

Figure 5 shows the added mass $A_{33}$ and the radiation damping $B_{33}$ in heave for the cylindrical buoy of 4 m radius. For the slack WEC concept, the curves are smooth with a single peak for each radiation term. In the stiff connection case, there are several resonance peaks along the frequency axis. The approximation of these hydrodynamic parameters requires an accurate fitting method.

2.3. Damping Force

The optimum values of damping force are calculated for each sea state in March 2018.

The power absorbed by a WEC is calculated as:

$$P=f_{damp}v,$$

(6)

where v is the translator velocity; $f_{damp}$ is the damping force. The damping force is calculated using three different approaches.

First, the optimal damping coefficient was used to calculate the average absorbed power in given sea conditions. The damping force, in this case is as follows:

$$f_{damp1}=c_{damp}v,$$

(7)

where $c_{damp}$ is the constant damping coefficient.

In the second approach, the optimal resistive load $R_l$ is a model of a grid-connected generator with a passive rectification. This passive rectification represents the damping and the load of the generator and could be equivalent to the diode rectification. The single-phase connection circuit is shown in Figure 6a.

The electrical frequency of the linear generator is calculated as:

$${\omega }_e=\frac{\pi }{{\tau }_p}v,$$

(8)

where ${\tau }_p$ is the pole width.

For the resistive load, the damping force can be calculated such as:

$$f_{damp2}=\frac{3E_{rms}^2(R_g+R_l)}{{(R_g+R_l)}^2+{\left({\omega }_e{L}_g\right)}^2}\frac{1}{v},$$

(9)

where $R_g$ is the resistance of generator coils; $L_g$ is the inductance of generator coils; $\omega$ is the electrical frequency; $E_{rms}$ is the root mean square induced phase voltage, calculated as in [32]:

$$E_{rms}=\frac{N{\mathsf{\Phi }}_0{\omega }_e}{\sqrt{2}},$$

(10)

where N is the number of the turns in the coils; ${\mathsf{\Phi }}_0$ is the magnetic flux amplitude of the time varying magnetic flux $\mathsf{\Phi }$ [32].

In the third approach (Figure 6b), the $RC$-load is a model for a phase angle compensation applied to a grid-connected generator with active rectification. Active rectification does not affect the passivity constraint on the system, and it works as reactive power compensation to obtain higher damping for the passive system. The $RC$-load damping force is calculated as follows:

$$f_{damp3}=\frac{3E_{rms}^2(R_g+R_l)}{{(R_g+R_l)}^2+{(-\frac{1}{{\omega }_e{C}_l}+{\omega }_e{L}_g)}^2}\frac{1}{v},$$

(11)

where $C_l$ is the capacitance of the load and all other variables are the same as in Equation (9).

2.4. Radiation Force

The radiation force in Equations (1) and (4) is necessary to estimate the buoy dynamics. Only one degree of freedom, heave, is considered in this paper. The radiation force arises from the change in momentum of the fluid, due to the buoy motion and it can be written as follows [36]:

$$f_r=A_{\infty }\ddot{z}+{\int }_0^{t}k(t-\tau )\dot{z}\left(\tau \right)d\tau .$$

(12)

where $A_{\infty }$ is the constant infinite frequency added mass; $k\left(t\right)$ is the radiation impulse-response function.

Equation (12) contains the convolution integral, representing the fluid memory effect. Time-domain models with convolution terms are computationally demanding and difficult for implementation [37]. The computational time depends on the time step, simulation length, and degrees of freedom. State-space models can be used to approximate the convolution in linear systems [38].

The radiation term results in the following linear approximation [39]:

$$\dot{x}\left(t\right)=Ax\left(t\right)+B,$$

(13)

$$y\left(t\right)=Cx\left(t\right),$$

(14)

where A is the state matrix; B is the control matrix; C is the observation matrix; $y\left(t\right)$ is the actual approximation of the convolution term; $x\left(t\right)$ is the vector of state variables.

Our study compares TFFD and RAVF approximation methods and applies them to proceed further with the state-space model. The first approximation method uses transfer function fitting in the frequency domain as described by Taghipour and Perez in [26]. The second one uses rational approximation by vector fitting, developed by Gustavsen [23,40,41].

In the frequency domain, the impulse response function $k\left(t\right)$ can be written in a complex form as a transfer function (TF) [38]:

$$K\left(\omega \right)=B_{33}\left(\omega \right)+j\omega (A_{33}\left(\omega \right)-A_{\infty }),$$

(15)

The properties of $K\left(\omega \right)$ must satisfy conditions at the zero and infinite angular frequency $\omega$, such as in [26]:

• Damping $B\left(\omega \right)$ tends to zero when $\omega$ tends to zero. The difference $A\left(0\right)-A\left(\omega \right)$ becomes finite, resulting in ${lim}_{\omega \to 0}K\left(j\omega \right)=0$.
• Damping $B\left(\omega \right)$ tends to zero when $\omega$ tends to infinity. Therefore, ${lim}_{\omega \to \infty }K\left(j\omega \right)=0$.
• Passivity of the system. It ensures that there is no own energy generated by the system, the energy is only stored or dissipated, supplied by the excitation force of the wave. The influence of passivity in linear and nonlinear control systems is shown in [42].

2.4.1. Transfer Function in the Frequency Domain

The TFFD approach is based on the approximation of the TF. Taghipour et al. [26] describe the frequency domain identification problem. The TFFD can be divided into the following steps:

• Defining the initial weights for the fitting of TF by the rational function;
• Calculation of the hydrodynamic coefficients using the least square method;
• Improving the fit by choosing an appropriate weight vector, corresponding to a minimal chosen error;
• A passivity check; roots with a positive real part are identified, and passivity reinforcement is performed: the sign of negative real parts for the unstable roots is flipped according to the convolution properties described above.

A parametric model, such as TF in equation (15) can be fitted with a rational function, using an appropriate approximation order. The rational fitting function can be written as follows:

$$G\left(s\right)=\frac{P\left(s\right)}{Q\left(s\right)}=\frac{p_m{s}^m+p_{m-1}{s}^{m-1}+...+p_0}{s^n+q_{n-1}{s}^{n-1}+...+q_0},$$

(16)

where $s=j\omega$ is the vector holding frequency samples; m and n are related by $n=m+1$ [38].

The best fitting result is identified by an equality error method [43] based on complex curve fitting by Levi [44] as:

$$arg\underset{b,a}{min}\sum _{k=1}^{n}W\left(k\right){|G\left(j\omega \left(k\right)\right)A\left(\omega \left(k\right)\right)-B\left(\omega \left(k\right)\right)|}^2,$$

(17)

where W is the weighting function; $A\left(\omega \right(k\left)\right)$ and $B\left(\omega \right(k\left)\right)$ are the added mass and the radiation damping at the frequency $\omega \left(k\right)$ correspondingly; n is the number of frequencies.

The TFFD results in a poor fitting if the resonant poles are close to the imaginary axis and at low frequencies. Deschrijver et al. [45] noted that unbalanced weighting of Equation (17) depends on the selected basis functions, and it may reduce the quality of the fitting for high frequencies.

2.4.2. Rational Approximation by Vector Fitting

The RAVF method, developed by Gustavsen [23,40,41] targets the problems of modeling power system transients, such as the effects arising from eddy currents or from relaxation phenomena in dielectrics. Vector fitting is a reformulation of the Sanathanan–Koerner iteration [40], which can be written as [45]:

$$arg\underset{b_l,a_l}{min}\sum _{k=0}^n{\left|\frac{A^l\left(\omega \left(k\right)\right)-B^l\left(\omega \left(k\right)\right)}{W^{l-1}\left(k\right)}-\frac{W^l\left(k\right))}{W^{l-1}\left(k\right)}G\left(j\omega \left(k\right)\right)\right|}^2,$$

(18)

where $l=1,...,L$ is the iteration step.

The RAVF approach requires the use of complex starting poles distributed over the considered frequency interval. Regarding the properties of the transfer function $K\left(\omega \right)$ for the radiation force calculations, the rational function for approximation has the following form:

$$f\left(s\right)=\sum _{n=1}^N\frac{c_n}{s-a_n},$$

(19)

where $c_n$ are the complex residues; $a_n$ are the complex poles.

Considering the number of fitted polynomials is less than the number of frequencies, the problem can be solved using the following steps:

• Calculation of poles, using the initial given set: a default starting set of poles can be provided as starting poles, in an iterative process these poles are improved;
• Calculation of residues, made by the least square method;
• A passivity check, where the new poles are ensured to be stable and passivity is reinforced; if needed, the sign of the unstable poles’ real parts are inverted.

3. Accuracy of the Model

Comparison of the Two Approximations

A comparison between TTFD and RAVF on buoys with a 2 m and 4 m radius is completed to choose the best performing method for the power estimation. The calculation time and the relative error are estimated, and the approximation order is varied to find the best fit. The analysis has been carried out on a stationary computer with four physical cores running on 3.50 GHz with 16 GB memory.

The goodness of fit is assessed by the correlation coefficient $k_c$ and root relative squared error ${ϵ}_r$. The best fit corresponds to $k_c$ being close to 1 and ${ϵ}_r$ being close to zero. The correlation coefficient is calculated as follows:

$$k_c=\frac{{\sum }_{i=1}^N(x_i-x_{av})(y_i-y_{av})}{\sqrt{{\sum }_{i=1}^N{(x_i-x_{av})}^2{\sum }_{i=1}^N{(y_i-y_{av})}^2}},$$

(20)

where $x_i$ is the predicted value by the model of the record i; $x_{av}$ is the average of the modeled data set of N values; $y_i$ is the target value for the comparison; $y_{av}$ is the average of the target values.

The error is calculated using ${ϵ}_r$:

$${ϵ}_r=\sqrt{\frac{{\sum }_{i=1}^N{(y_i-x_i)}^2}{{\sum }_{i=1}^N{(y_i-y_{av})}^2}}.$$

(21)

Table 3. Comparison of the two approaches for the buoy with a radius of 2 m for different approximation orders N.

N	${\mathit{k}}_{\mathit{r}}$		${\mathit{ϵ}}_{\mathit{r}}$		Time, s
	TFFD	RAVF	TFFD	RAVF	TFFD	RAVF
3	0.9959	0.7908	0.7273	7.5642	0.0935	2.1992
4	0.9997	0.9989	0.0446	0.0848	0.0211	0.7260
5	0.9996	0.7925	0.0005	26.9476	0.0153	0.8855
6	0.9999	0.9999	0.0268	0.0828	0.1129	0.6632
7	0.9999	0.6079	0.0221	51.5098	0.0771	0.7957
8	0.9999	0.9999	0.0159	0.0607	182.18	0.6523
9	−	0.9999	−	0.0686	−	0.6288
10	−	0.9878	−	1.5843	−	1.7884
11	−	0.7368	−	7.5926	−	0.8021
12	−	0.8741	−	2.6359	−	0.7282
13	−	0.9995	−	0.4739	−	0.6668
14	−	0.9017	−	4.5169	−	0.7067
15	−	0.7720	−	0.5195	−	0.7279
16	−	0.9999	−	0.0532	−	3.5091
17	−	0.9976	−	0.6541	−	1.4792
18	−	0.9999	−	0.0362	−	0.7475
19	−	0.4588	−	20.67	−	0.7801
20	−	0.9999	−	0.0321	−	1.9464
21	−	0.9992	−	0.4968	−	0.8284
22	−	0.9999	−	0.0402	−	0.7147
23	−	0.9999	−	0.0349	−	0.6947
24	−	0.9999	−	0.0487	−	0.7139
25	−	0.9999	−	0.0315	−	0.7282

presents errors, correlation coefficients and the computational time for the approximation of the radiation term using TFFD and RAVF. Different approximation orders were tested. TFFD shows good results for small fitting orders, with the best order equal to eight. The values are not provided for the cases of higher approximation order when the convergence has not been achieved (marked with a dashed line). The corresponding best fitting result for the RAVF is the 25th order. Regarding the calculation time, the TFFD approach was quick for the lower orders of fitting, taking 0.01 s. The calculation time for RAVF stays in the range between 0.6 s and 2 s.

Figure 7 shows the transfer functions, calculated with the approximation order of four and eight. From Table 3 it can be noted that a higher approximation order provides a better approximation to the original peaky transfer function. Figure 8 shows one of the best curve fittings, achieved using an approximation order of 16 by RAVF.

Figure 9 shows RAOs, obtained by the approximation order of four (Figure 9a) and eight (Figure 9b). Although order eight shows poor fitting results for TF, the RAOs fit well.

Table 4. Comparison of the two approaches for the buoy with a radius of 4 m for the different order of approximation N.

N	${\mathit{k}}_{\mathit{r}}$		${\mathit{ϵ}}_{\mathit{r}}$		Time, s
	TFFD	RAVF	TFFD	RAVF	TFFD	RAVF
3	0.9992	0.9982	0.0708	1.1462	0.0865	2.3621
4	0.1964	0.9987	7$·{10}^{64}$	1.2971	0.0194	0.8734
5	0.1964	0.9991	4$·{10}^{65}$	0.8850	0.0114	0.8990
6	0.1964	0.9991	10${}^{54}$	1.2095	0.0276	0.7253
7	0.1964	0.9991	2.5$·{10}^{42}$	1.1208	0.0674	0.7466
8	−	0.9998	−	0.9605	−	0.6875
9	−	0.9997	−	1.1580	−	0.7518
10	−	0.9999	−	0.7258	−	0.7378
11	−	0.9998	−	0.8217	−	0.7028
12	−	0.9998	−	0.8386	−	0.7286
13	−	0.9999	−	0.7882	−	0.7305
14	−	0.9997	−	1.0562	−	0.7556
15	−	0.9991	−	0.8054	−	0.8148

shows the results calculated for the largest buoy with a radius of 4 m. TFFD assessment shows that the only good fit is achieved for the fitting order of three. The approximation orders 4–7 provide very poor accuracy, and after the seventh order, the method fails to converge. The RAVF shows great fitting results for all considered orders, with the best equal to 10 ($k_c\approx$ 1, ${ϵ}_r$ = 0.7).

It is worth noting that TFFD may fail to achieve a satisfactory TF fit for large floaters with small drafts, resulting in erroneous numerical calculations in the time domain. On the contrary, the RAVF achieves satisfactory results, especially for higher orders of approximation.

In this paper, the TFFD method is used for the slack connected buoys and RAVF for the stiff concept.

4. Results

4.1. Irregular Waves

The absorbed power of WEC is subject to seasonal variations. Therefore, the values of power vary strongly during the year due to the low energy resource in summer and high energy resource in winter. Further investigations on the seasonal power absorption are needed to assess the possibilities of future installations offshore. In this paper, only one month is considered to investigate the performance of the generator in irregular waves. In detail, the seasonal variation of an OWC in the Mediterranean sea is shown in [46].

In our study, the wave data collected by the wave buoy from Wave Hub, Cornwall UK, was used to simulate an irregular wave input for the WEC concepts. The wave data are open access and provided in [33]. Wave elevations are recorded during March of 2018 and measured by a Waverider buoy. In [47] a comparison between monthly, seasonal and annual variations of wave power at Wave Hub is presented. The annual average wave power density is estimated to be 20 kW/m—the highest power density of 40 kW/m in winter and the lowest wave resource of 10 kW/m in summer. The chosen spring month (March 2018) can be considered as a close representation of the yearly average of the wave climate, due to the spring and autumn seasons providing average wave power, as shown in [47].

The original sampling frequency of the data was 1.28 Hz. The water surface elevation was recorded into 1487 raw files of 30 min sampling periods. The data have been interpolated linearly to increase the sampling frequency up to 2.56 Hz. Missing data points are flagged as “9999”, as a part of quality control identifying unrealistic data which might have been caused by, e.g., a power failure [48]. All flagged data points have been removed before proceeding with the wave data.

The wave power density equation is valid for deep water, where the water depth is larger than half the wavelength [49] and can be found as follows:

$$J=\frac{\rho g^2}{64\pi }H{s}^{2}Te,$$

(22)

where $\rho$ is the water density; g is the gravitational constant; $Hs$ is the significant wave height; $Te$ is the wave energy period.

Figure 10 shows J dependent on the wave height and wave energy period. The mean value for March 2018 corresponds to 20.2 kW/m.

4.2. Absorbed Power

The presented models assess the operation of WEC in irregular sea states of Wave Hub, UK. The choice of the concepts is related to the currently absent hydrodynamic concept for C-GEN, and hydrodynamic absorption has not yet been studied for this generator. Therefore, the first concept is taken from the existing concept (Uppsala University WEC). The second concept is novel, utilizing an offshore platform for a possible co-located renewable energy power plant.

The present study is based on the following assumptions. The generator is fixed on the platform for the stiff connection and the sea bed in the slack concept. The sea depth is constant and incoming waves are unidirectional. In the following constraints, the only motion inside the linear generator is heave, and the buoy motion is limited to heave. The potential wave theory implies the small amplitude motion condition. Therefore, higher values of $Hs$ are considered to be qualitative.

The PTO damping force is taken into account when estimating the power absorption of the linear generator. Figure 11 and Figure 12 present the absorbed power for the slack and stiff connection concept, respectively, calculated for each different buoy radius: 1 m, 2 m, 3 m and 4 m. The averaged values of active power are found for each sea state and the results are shown for the damping forces $f_{damp1}$, $f_{damp2}$ and $f_{damp3}$, calculated according to (7), (9) and (11) respectively.

Capture width ratio is calculated to estimate the hydrodynamic efficiency of the buoys [50], found as:

$$C_w=\frac{P_{avg}}{J{r}_b}$$

(23)

where $r_b$ is the buoy radius.

Table 5. Capture width ratio and average values of power for each case.

Concept	${\mathit{r}}_{\mathit{b}}$, m	${\mathit{P}}_{\mathit{m}\mathit{a}\mathit{x}}^{{\mathit{C}}_{\mathit{d}\mathit{a}\mathit{m}\mathit{p}}}$, kW	${\mathit{P}}_{\mathit{m}\mathit{a}\mathit{x}}^{\mathit{R}}$, kW	${\mathit{P}}_{\mathit{m}\mathit{a}\mathit{x}}^{\mathit{R}\mathit{C}}$, kW	${\mathit{C}}_{\mathit{w}}^{{\mathit{C}}_{\mathit{d}\mathit{a}\mathit{m}\mathit{p}}}$, %	${\mathit{C}}_{\mathit{w}}^{\mathit{R}}$, %	${\mathit{C}}_{\mathit{w}}^{\mathit{R}\mathit{C}}$, %
	1	2.12	1.92	1.95	10.49	9.49	9.62
Slack	2	7.57	3.61	3.80	18.71	8.93	9.39
connection	3	10.91	3.83	4.08	18.98	6.32	6.72
	4	12.89	3.96	4.23	15.94	4.90	5.24
	1	2.22	1.97	2.00	10.96	9.75	9.89
Stiff	2	8.22	3.66	3.87	20.33	9.06	9.58
connection	3	20.63	3.86	4.13	34.01	6.36	6.80
	4	32.04	3.76	4.03	39.62	4.65	4.98

summarizes the average absorbed power $P_{avg}$ and the capture width ratio $C_w$, calculated for the slack and stiff concepts, varied buoy sizes, and damping force approaches.

The hydrodynamic capture width is up to 40% for the large buoys in both concepts by the optimal damping coefficient approach. On the contrary, the results by the equivalent generator circuit indicate the smallest buoy as being the most efficient.

The primary difference between the considered WEC concepts is the wire force, absent in the stiff connection, and the influence of the platform legs on the buoy hydrodynamics. The simulation shows the potential with the stiff concept being up to 32 kW with the largest buoy. However, in the model, the scattering was neglected; therefore, the results for the stiff connection may be overestimated.

The values of average power for the smallest buoy are about 2 kW in all considered cases. It shows that the R- and $RC$-loads achieved comparable results with the well-known constant damping approach [13]. Figure 13 shows absorbed power for both WEC concepts as being very similar. The difference is mainly by the optimal damping coefficient approach against the equivalent circuit.

Larger buoys provide a more distinct difference between the concepts. The positive influence of the wire force and slacking of the wire can be seen in the numerical results. The limiting condition on the absorbed power can be noted for the load models, such as R- and $RC$-loads, and it will be further discussed below.

The optimization of the resistive load is partially achieved only for the smallest considered buoy. For larger buoys, the optimal value of the resistive load is the initial value for the optimization. Such a small resistive load is physically nearly a zero load; therefore, the generator is short-circuited and provides no output power. The phase shift compensation model increases the absorbed power by 7%.

Table 6. Peak values of active power for each case.

Concept	${\mathit{r}}_{\mathit{b}}$, m	${\mathit{P}}_{\mathit{m}\mathit{a}\mathit{x}}^{{\mathit{C}}_{\mathit{d}\mathit{a}\mathit{m}\mathit{p}}}$, kW	${\mathit{P}}_{\mathit{m}\mathit{a}\mathit{x}}^{\mathit{R}}$, kW	${\mathit{P}}_{\mathit{m}\mathit{a}\mathit{x}}^{\mathit{R}\mathit{C}}$, kW
	1	72.58	45.08	47.49
Slack	2	336.81	73.50	76.85
connection	3	588.60	84.95	88.04
	4	1032.91	84.27	85.80
	1	78.45	46.92	46.81
Stiff	2	310.11	65.93	68.96
connection	3	1058.40	64.83	69.68
	4	1373.41	77.74	80.90

presents peak powers $P_{max}$ for the WEC concepts, different buoy sizes and damping forces. A direct-drive linear generator in the wave environment provides high power peaks due to the intermittent nature of the waves. The peaks introduce a challenge to the WEC design: the return of the investments based on the output. The large capacity of the system is needed to survive large power peaks, while the actual absorbed power is significantly lower. It is especially critical for the larger buoys. Therefore, an economic assessment should be carried out in further research to find an efficient and inexpensive solution.

Figure 14 shows the translator dynamics relative to the wave elevation and velocity for the stiff concept. The buoy of 1 m radius was used in the calculations.

Small volume and size buoys generally provide less mechanical energy than larger and heavier buoys. However, the generator design is essential to achieve power absorption. Large buoys for small generators may not lead to extensive energy absorption. Therefore, the buoy size optimization should be carried out together with PTO-size optimization as in [51]. Moreover, from an economic view, the material costs for the large buoys may not be worth the investment. Further research on the costs is needed to assess the optimal buoy size.

5. Discussion

The damping coefficient $c_{damp}$ is found to maximize the average power absorption by the WEC in a given sea state. The damping force is linear, and this approach is independent of the type of generator.

In the case of the permanent magnet linear generator, the generator characteristics should be taken into account, such as the magnetic flux amplitude, the total number of winding turns, pole width, synchronous inductance and phase resistance. As a result, the damping coefficient is not constant but depends on the translator velocity [32]. Moreover, the maximum achievable damping coefficient for a sea state may be less than with a constant damping coefficient, calculated with the first approach.

Including the $RC$-load is aimed to replicate phase shift compensation, such that the current is in phase with emf, i.e., zero d-axis current. The results have shown that the absorbed power is significantly higher for larger buoys for the $RC$-load approach. However, the optimization of R-load is not achieved for buoys of 2, 3 and 4 m radius, perhaps, due to the features of the C-GEN linear generator: high phase resistance of about 13 $\mathsf{\Omega }$. However, the absorbed power for a buoy of 1 m radius is partially optimized.

As shown in [32], the damping coefficient $c_{damp2}$ reaches the maximum when $v\to 0$. Namely,

$$c_{damp2}^{max}=\underset{v\to 0}{lim}\frac{f_{damp2}}{v}.$$

(24)

From Equation (9),

$$c_{damp2}^{max}=\underset{v\to 0}{lim}\frac{3E_{rms}^2(R_g+R_l)}{{(R_g+R_l)}^2+{\left(\omega L_g\right)}^2}\frac{1}{v^2}=\frac{3E_{rms}^2}{(R_g+R_l)}\frac{1}{v^2}.$$

(25)

For a given generator, the load resistance $R_l$ is the only varying parameter in (25). Then the damping coefficient $c_{damp2}^{max}$ is maximum if and only if $R_l\to 0$. Hence, the bounding value of the damping coefficient $c_{damp2}^{\ast }$ is given by the equation:

$$c_{damp2}^{\ast }=\frac{3E_{rms}^2}{R_g}\frac{1}{v^2}.$$

(26)

For example, in the calculations for the 1 m buoy the maximum achieved $c_{damp}^{max}$ = 26,474 Ns/m with the average $c_{damp}^{avg}$ = 12,805 Ns/m, while for R-load, this value could only reach the maximum of 8922 Ns/m. Figure 15 shows damping coefficients, calculated for the damping approaches for the 1 m buoy. The maximum damping coefficient for the largest buoy of 4 m radius is 95,547 Ns/m, while the R-load cannot overcome the limit as shown above. Therefore, the difference between the results on damping force approaches occurs (see Figure 13 and Figure 14). It is minor for a 1 m buoy, compared to larger buoys, as shown in Table 5.

The generator dimensions need to be chosen carefully for a WEC with such varying power levels. If the generator phase resistance could be reduced by one-fifth, higher output power values could be achieved. The results clearly show that by choosing an oversized buoy (2 m and larger) for this particular generator setup, most of the available energy would not be absorbed in the system, as shown by R- and $RC$-loads. Since the generator is actually underdamped compared to the optimum damping (see Figure 14), not all available and possible energy is absorbed. Moreover, a large part of it is lost due to the generator inner resistance. Regarding the difference between the two WEC topologies, the absence of wire force positively influences the results in the stiff concept. If the limit on $c_{damp2}^{max}$ allowed for higher values, the results by R- and $RC$-loads would give equivalent values to the results seen from the damping coefficient approach for larger buoys (see Table 5).

6. Conclusions

Two point absorber WEC concepts coupled to the novel liner permanent magnet generator C-GEN have been investigated. Different buoy sizes have been tested in the model: 1 m, 2 m, 3 m and 4 m radius buoys. A strong frequency-dependent hydrodynamic response characterizes large buoys with small drafts in the stiff connection concept. It requires efficient approximation methods to calculate the radiation force. The conventional TFFD has been compared with the RAVF method widely used in power systems. It has been shown that RAVF achieves excellent results for large buoys with small drafts, especially for higher orders of approximation.

In the present study, real wave data from Wave Hub, Cornwall, UK, have been utilized to calculate the absorbed power. The hydrodynamic capture width ratio has been shown to be about 10% for the smallest considered buoy. The corresponding average absorbed power for all cases is about 2 kW. The model of phase shift compensation by $RC$-load has shown up to a 7% increase in average absorbed power.

Damping coefficient calculations have shown a strong dependency on the power absorption due to the buoy size and the potential to obtain up to 32 kW with the largest buoy via the stiff connection. The absence of the wire force positively influenced the results. However, the maximum damping coefficient for R- and $RC$-loads for a sea state may be less than the constant damping coefficient. The maximum damping coefficient of the largest buoy of 4 m radius is 95,547 Ns/m, while the R-load cannot overcome the limit. The results have shown that the generator dimensions need to be chosen carefully for a WEC. The choice of oversized buoys, such as of 2 m radius and larger, would result in the loss of a large portion of energy available in the system. A power level closer to the power levels calculated for larger buoys by the damping coefficient approach can be achieved for R- and $RC$-loads if the internal generator resistance is reduced.