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
-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
-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
-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.
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:
where
is the water density;
g is the gravitational constant;
is the significant wave height;
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 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
,
and
, calculated according to (
7), (
9) and (
11) respectively.
Capture width ratio is calculated to estimate the hydrodynamic efficiency of the buoys [
50], found as:
where
is the buoy radius.
Table 5 summarizes the average absorbed power
and the capture width ratio
, 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
-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 -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 presents peak powers
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 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 -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 -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 . However, the absorbed power for a buoy of 1 m radius is partially optimized.
As shown in [
32], the damping coefficient
reaches the maximum when
. Namely,
For a given generator, the load resistance
is the only varying parameter in (
25). Then the damping coefficient
is maximum if and only if
. Hence, the bounding value of the damping coefficient
is given by the equation:
For example, in the calculations for the 1 m buoy the maximum achieved
= 26,474 Ns/m with the average
= 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
-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
allowed for higher values, the results by
R- and
-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 -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 -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 -loads if the internal generator resistance is reduced.