Based on an earlier conceptual model of a two body system point absorbing wave energy converter tuned to resonance in Swedish west coast sea states, an extended coupled hydrodynamic, mechanic, and electromagnetic model has been developed. The hydrodynamic characteristics of the two body system are studied in the frequency and time domain, while its response to real Swedish west coast sea states are studied in the time domain, by using a wave energy converter model with two independently moving bodies connected to a direct driven linear generator with non-linear damping. The two body system wave energy converter gives nearly 80% power capture ratio in irregular waves. The resonant behaviour is shown to be sensitive to the shape of the spectrum, and the distance between the two bodies is shown to have a large effect on the power absorption.

Since the start of wave energy research’s second heyday in the 21st century, many concepts are now taking the step towards full scale sea trials.1 Some showing that it might be a feasible way to meet the increasing energy demands, though many issues remain to be solved or improved: grid connection, durability, and increased energy absorption to mention a few. The last issue needs a full system approach with coupled hydrodynamic, mechanic, and electromagnetic equations that include the whole chain from primary energy conversion from ocean waves to grid connection. Many concepts for wave energy conversion have been tested, and the point absorber system has been one of the main concepts that have been investigated since the research begun,2–4 which is also the focus in this paper.

The increase of energy absorption has since the start of wave energy research been focused on tuning the system to oscillate in resonance with the incoming wave. It is well known that a point absorber system in resonance with the incident wave will achieve increased amplitude and speed, and thereby transfers more energy than a system working off resonance.4,5 One method to force the system into resonance via active control was proposed by Budal and Salter independently in the mid 70s. Budal later proposed the method to latch the point absorber at fixed positions, and thereby achieve approximate phase control.6,7 Besides the need for a very powerful mechanical or electromagnetic breaking mechanism, this approach requires non-causal information of the excitation force.8 Some algorithms are being implemented on a theoretical level at present.9 

In order to avoid expensive and vulnerable control system and breaking mechanisms, another strategy would be to use a passive system to engineer the frequency response of the point absorber so as to be in resonance with the dominant sea states at the chosen location. In addition, it is also important to keep the power capture ratio at low levels in extreme sea states, to reduce the maximum loads on the system. This would also have the secondary effect of power smoothing. One way of shifting the frequency response of the system is by increasing the inertia of the moving parts. This have led to the idea of a point absorber system with two bodies, one acting as a surface buoy, extracting energy, while the lower is passive and adds the desired inertia.10–13 It is important to have the two bodies a sufficient distance apart so as not to get destructive interference. All tests on the two body system (TBS) thus far have shown promising results on a theoretical level with a 60% absorption in irregular waves and a significant decrease in optimal load damping.12 Laboratory experiments with adjustable inertia have shown to give a 60% power capture ratio in irregular waves.2 

The wave energy converter (WEC) concept named the Lysekil Project that is being developed at Uppsala University is a point absorber system with a direct-driven linear generator connected to a semi-submerged buoy.14 The first full-scale experimental prototype was installed at the Swedish west coast in mars 2006. Since then seven more WECs have been deployed at the test site, in the spring of 2009, three WECs were connected to an underwater substation.3,15,16 Now as the simulations have been confirmed with full-scale experiments, steps can be taken to improve the parts of the system. One of the first steps was to develop a theoretical model for a two body system tuned to resonance for Swedish west coast sea states.12 Following this, the next step will be to extend the model to a fully coupled hydrodynamic, mechanic, and electromagnetic model that simulates the two bodies connected to a direct-driven linear generator with a non-linear load.

A TBS WEC connected to a non-linear DC-voltage, sub-sea cable, and resistive load is described in this paper. The hydrodynamic characteristics of the two bodies are obtained from impulse response functions calculated using the BEM (Boundary Element Method) code wamit® and implemented in Simulink® that model the coupled hydrodynamic, mechanic, and electromagnetic system. The model is forced by time series of wave elevation gathered by a wave measurement buoy at the research test site.

The research test site is located approximately 2 km offshore just south of the town Lysekil, situated between a northern marker (58° 11′850′′N 11° 22′460′′E) and a southern marker (58° 11′630′′N 11° 22′460′′E). It is sheltered by small islands to the north, and to the south is the small islet of Klammerskäret where a surveillance tower is deployed. The seabed is fairly level with an average depth of 25 m. A 2.9 km long sea cable connects the WECs to a measurement station on a nearby island. The measurement station is equipped with resistive dump loads with different load values and a diode rectifier that allow the WEC to be loaded at different DC-levels.18 Wave elevation data are collected approximately 50 m from the WEC. Variations in the water level due to tides and air pressure variations are very small at this site and have been neglected in this study. For the measurements, a commercial system was used: the Datawell Waverider buoy. The data were obtained at a sampling rate of 2.56 Hz. At the test site, 44% of the annual energy flux occurs for sea states with an energy period Te in the interval 4–7 s and a significant wave height Hs in the interval 1–3 m.19 For more information on the Uppsala University research test site, see Leijon et al.20 

Ocean waves with an amplitude that is much smaller than the wave length can with good accuracy be described by potential linear wave theory.17 This assumes an ideal fluid, i.e., incompressible, irrotational, and non-viscous, which implies that the velocity potential φ satisfies the Laplace equation 2φ=0. Since the waves are assumed to have small amplitude compared to wave length, the dynamic free surface boundary condition can be linearized. A rigid body has 6° of freedom; in these simulations, we restrict the movement of both bodies to heave only since this is the dominating motion. Due to linearization, the hydrodynamic forces acting on the buoy can be split into two components; the excitation force Fe and the reaction force Fr caused by radiation.5 The excitation force is given by a convolution of the wave elevation η(t) and the non-causal impulse response function fe(t),

Fe(t)=fe(t)*η(t).
(1)

Since the reaction force is a direct consequence of the buoy’s motion, it is causal and the Kramers-Kronig relations imply that the radiation matrix R(ω) and added mass matrix ma(ω) are independent. The radiation impedance in the time domain h(t) is given by the inverse Fourier transform of the radiation matrix,

h(t)=2π0R(ω)cos(ωt)dω,
(2)

The reaction force can be written as a convolution product with the buoy’s vertical velocity y·(t) according to

Fr(t)=-h(t)*y·(t).
(3)

The hydrodynamic parameters fe, R, and ma for the TBS have been calculated using the package wamit®. The hydrostatic force Fh is dependent on the surface buoy’s vertical displacement from equilibrium y(t) according to

Fh=-ρgAy(t),
(4)

where ρ is the density of sea water, g is the acceleration of gravity, and A is the water plane area of the surface buoy.

An undamped oscillator’s natural frequency ωn is set by the cylindrical buoy’s water plane area, the systems mass, and added mass according to

ωn=ρgAm+ma,
(5)

where m is the systems total mass and ma its total added mass at the frequency of the incident wave. It has previously been shown that, by using an additional submerged body, its own mass and added mass can be used to shift the systems natural frequency to coincide with the frequency of the waves at the site.12 If the lower submerged body is placed deep enough not to interfere destructively with the surface buoy and having neutral buoyancy, it will act as a passive energy storage mechanism that adds desired inertia to the system. The TBSs resonant behavior can be visualized by looking at the systems’ response amplitude operator, H, in the frequency domain,

H(ω)=fe-ω2(ma+m)+iωR+ρgA,
(6)

where the caret ⁁ denotes the Fourier transform. In the frequency domain analysis of Eq. (6), a simpler approach is taken, with the TBS modeled as being rigidly coupled. Thus, the hydrodynamic parameters fe, R, and ma are calculated for the whole system, not individually for each body as in the time domain analysis later on. The system is in resonance when ω=ωn, and by selecting a suitable volume for the lower submerged body, the TBS may be tuned to have a resonance frequency that coincides with the sea state at the location. This feature is shown in Fig. 4.

FIG. 4.

The un-damped response amplitude operator H for TBS40 (solid line) and for SBS (dashed line).

FIG. 4.

The un-damped response amplitude operator H for TBS40 (solid line) and for SBS (dashed line).

Close modal

The TBS WEC consists of a semi-submerged vertical cylinder connected by a line to the lower submerged body, and the latter has neutral buoyancy, see Fig. 1. A sphere has been chosen as the geometry for the submerged body in this study to minimize the drag force. The drag force for a sphere is approximately 60% lower than for a cylinder moving along its axis. The drawback with this solution is the loss in added mass, since the sphere has lower added mass for the same volume, and thus, a larger sphere has to be chosen to get the desired inertia. The difference in excitation force and radiation is negligible when comparing the two geometries.11 The lower body is connected via a line and a guiding system to the translator of the direct-driven linear generator. The surface and lower body are allowed to move in heave as constrained by the connecting flexible line, and their positions are labeled y1 and y2, respectively. The connection between them and between the lower body and the translator is modeled as a very stiff spring kl. The generator is a three phase direct-driven linear generator with Nd-Fe-B type permanent magnets on the octagonal translator whose vertical position is labeled yt. The stroke length of the translator Lst is set to 4 m. The generator is also equipped with upper and lower end stop springs ke to protect it from mechanical shock loads in waves higher than the design condition. The generator is connected to a diode rectifier with a DC link capacitor connected via a 2.9 km sub-sea cable to a resistive load on shore.

FIG. 1.

Mechanical components of the TBS wave energy converter.

FIG. 1.

Mechanical components of the TBS wave energy converter.

Close modal

The hydrodynamic and mechanic parameters are modeled as being linear while the electrical power take-off is non-linear. The non-linear viscous or drag force experienced by the submerged body can be approximated by FD=0.5ρA2y·22CD with the drag coefficient CD = 0.42 for a sphere. For a velocity of 1 m/s, this gives a drag force of around 4 kN that corresponds to about 4% of the excitation force. This approximated drag force is an over estimate since for a proper value of the drag force, the velocity y·2 should be the relative velocity between the submerged body and the water particles. This is also supported by a study made on the losses due to viscous forces on a submerged pontoon in Hals et al.21 They found that the losses were 4% of the absorbed power for a CD of 1. Thus, with a CD of 0.42, the losses should be small, and viscous forces are therefore excluded in our model.

We set up a set of second order differential equations that describe the forces that create the TBSs dynamics,

(ma,1+m1)y··1(t)=fe,1(t)*η(t)-h1(t)*y·1(t)-ρgA1y1(t)-Fl,1,
(7)
(ma,2+m2)y··2(t)=fe,2(t)*η(t)-h2(t)*y·2(t)-Fl,2,
(8)
(mt)y··t(t)=Fl,2-Fem-Fe,u+Fe,l.
(9)

Here, the index 1,2 represents the upper and lower body, respectively. The hydrodynamic parameters are calculated individually for each of the two bodies, and for the submerged body, they are extracted for two depths 40 and 10 m below the surface. The connection between the surface buoy y1 and the lower body y2 as well as between the lower body and the translator yt is modeled as a very stiff spring with the line force Fl,1,2 as follows:

Fl,1={kl(y1-y2)ify1>y20,ify1y2,
(10)
Fl,2={kl(y2-yt)ify2>yt0,ify1y2,
(11)

where kl is the line spring constant. The upper Fe,u and lower Fe,l end stop spring forces are modeled by

Fe,u={-keΔytifyt>1.890,ifyt1.89,
(12)
Fe,l={keΔytifyt<1.890,ifyt1.89,
(13)

where ke is the end stop spring constant. The conversion from mechanical to electrical energy is expressed by the electromagnetic damping force Fem. The electrical real power PR is taken for calculating the reaction force,

Fem(t)=PR(t)y·t(t),
(14)
PR=3EgIgcos(φ),
(15)

where, Eg is the voltage drop over generator, Ig is the current in the generator, and φ is the phase lag. When the translator moves in and out of the stator, its contribution to the power production changes and that is given by the active area of the stator Aact. This position dependent relation is given as

Aact={0,if|yt|12(lt+ls)1,if|yt|12(lt-ls)1ls(12(lt+ls)-|yt|),else,
(16)

where lt and ls are the length of the translator and stator, respectively. This relation, Eq. (16), is used in calculating the generated voltage.

The three-phase electric circuit, see Fig. 2, consists of an uncontrolled rectifier with a DC link capacitor, sub-sea cable, and a resistive load. The rectifier is modelled using generic diode models available in SimPowerSystems block set from Simulink®. The subsea cable is modeled by its equivalent π model.

FIG. 2.

The simplified three-phase electric circuit consisting of the linear generator, diode rectifier, sub-sea cable, and resistive load.

FIG. 2.

The simplified three-phase electric circuit consisting of the linear generator, diode rectifier, sub-sea cable, and resistive load.

Close modal

The system’s ability to extract power is strongly dependent on the damping from generator and load. The damping sets the bandwidth of the resonance peak, and it is necessary to have a resonance peak that covers as much as possible of the wave spectrum. For this configuration, an optimum load resistance Rload of 2.5 Ω has been selected.

The translator and the electric circuit are modelled including the frictional losses, copper losses, and iron losses in the generator and the losses in the sub-sea cable and load. However, in this study, we intend to compare the behavior of the TBS to the original system without the additional submerged body, here named single body system (SBS). We have, therefore, excluded losses in the sea cable and load, and the presented absorption values in Fig. 5 are on the generator terminal.

FIG. 5.

(Color) Power capture ratio Prat vs. energy period Te for TBS40 (black curve), TBS10 (red curve), and for SBS (blue curve). The result for wave records 15 (green cover) and 17 (yellow cover) that are presented by their corresponding wave power spectrum in Figs. 11(a) and 11(b) is highlighted for TBS40. The lines are intended as guides for the eye.

FIG. 5.

(Color) Power capture ratio Prat vs. energy period Te for TBS40 (black curve), TBS10 (red curve), and for SBS (blue curve). The result for wave records 15 (green cover) and 17 (yellow cover) that are presented by their corresponding wave power spectrum in Figs. 11(a) and 11(b) is highlighted for TBS40. The lines are intended as guides for the eye.

Close modal
FIG. 11.

(Color) Wave power spectrum for (a) wave record 15 and (b) wave record 17. The sea state for wave record 15 is Te = 4.85 s, Hs = 0.76 m, and for wave record 17, it is Te = 5.01 and Hs = 0.95 m. The corresponding surface buoy position for TBS40 is inserted in the figures with surface buoy position y1 (red curve) and wave amplitude η (blue curve).

FIG. 11.

(Color) Wave power spectrum for (a) wave record 15 and (b) wave record 17. The sea state for wave record 15 is Te = 4.85 s, Hs = 0.76 m, and for wave record 17, it is Te = 5.01 and Hs = 0.95 m. The corresponding surface buoy position for TBS40 is inserted in the figures with surface buoy position y1 (red curve) and wave amplitude η (blue curve).

Close modal

The TBSs ability to extract power can be expressed by the power capture ratio, Prat, i.e., how much of the available power it converts to electricity,

Prat=PRD1kTeHs2100.
(17)

Here, kTeHs2 is the total time averaged wave power per meter wave front. The energy period Te and significant wave height Hs are obtained from the wave spectrum as Te=m-1/m0 and Hs=4m0 with m being the spectral moments. Using the deep water approximation, the constant k is given by k=ρg2/64π. The simulations are forced by 50 measured wave records with sea states ranging from an energy period of 4.1–8.3 s and significant wave heights from 0.4–4.1 m. The wave records are 50 s long re-sampled to 100 Hz. Two wave records with significant difference in spectral shape but similar sea state are given special attention in order to compare the TBS sensitivity to the spectral shape. The two wave records are labeled (15) with Te = 4.85, Hs = 0.76 m and (17) with Te = 5.01 s, Hs = 0.95 m. All other main parameters are given in Table I.

Table I.

Parameters for the TBS wave energy converter.

D1  Diameter for surface buoy 6.0 m 
D2  Diameter for lower body 4.9 m 
d1  Draft for surface buoy 0.72 m 
L1  Length between bodies 40 m, 10 m 
Ls  Stroke length 4 m 
lt  Length of translator 2 m 
ls  Length of stator 2 m 
m1  Mass of surface buoy 6930 kg 
m2  Mass of lower body 61601 kg 
mt  Mass of translator 14 ton 
kl  Spring constant for line 450 kN/m 
ke  Spring constant for end stop 215 kN/m 
Rg  Generator resistance 0.8 Ω/phase 
Rload  Load resistance 2.5 Ω 
Rsc  Sub-sea cable resistance 0.5 Ω/phase 
D1  Diameter for surface buoy 6.0 m 
D2  Diameter for lower body 4.9 m 
d1  Draft for surface buoy 0.72 m 
L1  Length between bodies 40 m, 10 m 
Ls  Stroke length 4 m 
lt  Length of translator 2 m 
ls  Length of stator 2 m 
m1  Mass of surface buoy 6930 kg 
m2  Mass of lower body 61601 kg 
mt  Mass of translator 14 ton 
kl  Spring constant for line 450 kN/m 
ke  Spring constant for end stop 215 kN/m 
Rg  Generator resistance 0.8 Ω/phase 
Rload  Load resistance 2.5 Ω 
Rsc  Sub-sea cable resistance 0.5 Ω/phase 

The hydrodynamic parameters are calculated using wamit® and implemented in a Simulink® model where the modeling can be divided into the following four blocks:

  • Surface buoy model

  • Submerged body model

  • Translator model

  • Generator model

The block diagram representation of the full model is given in Fig. 3.

FIG. 3.

Block diagram representation of the coupled hydrodynamic/mechanic and electromagnetic model that simulates the TBS WEC.

FIG. 3.

Block diagram representation of the coupled hydrodynamic/mechanic and electromagnetic model that simulates the TBS WEC.

Close modal

The power capture ratio Prat on the generator terminal for the TBS with the lower body submerged to 40 m (TBS40), 10 m depth (TBS10) and for the SBS with the same model settings for 50 wave records is shown in Fig. 5.

For a point absorber to be in resonance in frequencies corresponding to a regular ocean sea state, the draft of the buoy has to be large, thus its radiating capabilities decrease, leading to low absorption in all frequencies. While the amplitude response can be tuned by the generator damping, the frequency response have in this case been tuned by adding inertia, see Fig. 4. The size of the lower submerged body is chosen so as to give the TBS a resonant frequency that coincides with the dominating sea state at the Swedish west coast. It has an energy period in the range of 4–7 s or a corresponding frequency range of 0.9–1.6 rad/s.

A first indication of the difference in amplitude and frequency response comparing the TBS and SBS is given in Fig. 4. The TBS40 shows a clearly resonant behavior within the range for the sea state at the Swedish west coast, with the peak at 1.2 rad/s, i.e., a wave period of 5.2 s. The SBS shows resonant behavior at a higher frequency, around 1.8 rad/s, which is outside the desired frequency range. Thus, by increasing the inertia, the TBS can be tuned to have a natural period of oscillation that coincides with the sea state. However, when the system is subjected to the velocity dependent damping, the amplitude and frequency response will be modified, as will be clear from the time domain analysis.

TBS40 has a maximum absorption of nearly 80% of the incoming wave for wave record 15 where the sea state has an energy period of 4.85 s, see Fig. 5. For higher wave periods, the power capture ratio decreases quite rapidly for TBS40 down to about 10% for the largest waves. A curve fit is made to get a hint of the average which shows a peak between 4.5 and 4.8 s with about 58% absorption for the TBS40. Note that the wave period for highest absorption is shifted from 5.2 s in the harmonic un-damped analysis, Fig. 4, to around 4.7 s for the TBS40 subjected to a velocity dependent damping, Fig. 5. Thus, when the system is subjected to high damping, the frequency response is shifted towards higher frequencies. For TBS10, the depth effect is clearly visible with the peak for the curve fit reaching a power capture ratio of 37%, and wave record 15 yields a power capture ratio of nearly 50%. The SBS shows a steady decrease from 20% to 5% power capture ratio. For the SBS, no resonant behavior is visible since the resonant peak is outside the wave period range. The resonant behavior for the TBS compared to the SBS is viewed more in detail in Figs. 810, where the vertical position of the surface buoy, lower body, and translator is plotted, and in Figs. 12(a) and 12(b), where the translator velocity is presented.

FIG. 8.

(Color) TBS40 surface buoy position y1 (red curve), lower body position y2 (black curve), and translator position yt (not visible) vs. the wave amplitude η (blue curve) for wave record 15. The translator follows the surface buoy, and these curves can not be distinguished.

FIG. 8.

(Color) TBS40 surface buoy position y1 (red curve), lower body position y2 (black curve), and translator position yt (not visible) vs. the wave amplitude η (blue curve) for wave record 15. The translator follows the surface buoy, and these curves can not be distinguished.

Close modal
FIG. 12.

(Color) Translator velocity for TBS40 (red curve), TBS10 (black curve), and SBS (green curve) for (a) wave record 15 and (b) wave record 17.

FIG. 12.

(Color) Translator velocity for TBS40 (red curve), TBS10 (black curve), and SBS (green curve) for (a) wave record 15 and (b) wave record 17.

Close modal
FIG. 10.

(Color) SBS surface buoy position y1 (red curve) and translator position yt (green curve) vs. the wave amplitude η (blue curve) for wave record 15.

FIG. 10.

(Color) SBS surface buoy position y1 (red curve) and translator position yt (green curve) vs. the wave amplitude η (blue curve) for wave record 15.

Close modal

In order to achieve good power absorption for a two body system, the bodies have to be placed a sufficient distance apart, which is obvious when comparing the difference in power capture ratio between TBS40 and TBS10 in Fig. 5. Physically speaking, the resonance behavior has to be decoupled from the radiation properties of the surface buoy. This was already mentioned in Alves et al.11 This feature can be seen by studying the radiated wave from the surface buoy. The radiated wave from the surface buoy is plotted in Fig. 6 for TBS40, TBS10, and SBS, where we have also included TBS5 with the lower body at 5 m depth.

FIG. 6.

Amplitude of radiated wave from a harmonically oscillating buoy with a period T = 4.7 s, TBS40 (solid line), TBS10 (long dashed line), TBS5 (short dashed line), and SBS (not visible). The radiated waves for the SBS and TBS40 coincide and can not be distinguished. The radiated wave is plotted from the buoy’s outer rim 3 m from the center.

FIG. 6.

Amplitude of radiated wave from a harmonically oscillating buoy with a period T = 4.7 s, TBS40 (solid line), TBS10 (long dashed line), TBS5 (short dashed line), and SBS (not visible). The radiated waves for the SBS and TBS40 coincide and can not be distinguished. The radiated wave is plotted from the buoy’s outer rim 3 m from the center.

Close modal

The radiated waves for the SBS and TBS40 coincide and cannot be distinguished, thus, for the TBS40, the radiated waves from the surface buoy do not feel the presence of the lower body. When the lower submerged body is placed closer to the surface buoy, the radiated wave starts to feel the presence of the lower body, resulting in decreased amplitude. From Fig. 6, we can clearly see that the radiating capabilities of the surface buoy are affected in a negative way by the lower body if it is placed too close.

Furthermore, if the lower body is placed too close to the surface, it will radiate waves and also be exerted by excitation force from the wave that is out of phase with the excitation force on the surface buoy, and both forces will interfere destructively. If the lower body is placed deep enough, both the excitation forces on, and radiation from, the lower body are very low as can be seen in Figs. 7(a) and 7(b).

FIG. 7.

Excitation force fe (a) and radiation impedance h (b) impulse response functions for the surface buoy (solid line) and the submerged body at 40 m depth (short dashed line) and 10 m depth (long dashed line).

FIG. 7.

Excitation force fe (a) and radiation impedance h (b) impulse response functions for the surface buoy (solid line) and the submerged body at 40 m depth (short dashed line) and 10 m depth (long dashed line).

Close modal

The excitation force on the lower body submerged to 10 m shows a value that is about 7%–8% of the excitation force on the surface buoy and in an opposite direction, see Fig. 7(a). The lower body will then counteract the surface buoy’s movement. The radiation from the lower submerged body is very low even at 10 m depth as can be seen in Fig. 7(b).

Since the lower body has neutral buoyancy and high inertia, the translator mass has to be sufficiently high to maintain a stiff coupling between the bodies and the translator. Otherwise the motion of the two bodies and translator will be out of phase for long periods of time, inevitably leading to high snap loads. The translator mass must, on the other hand, not be too large since the translator mass is directly proportional to the draft of the surface buoy. The buoy’s ability to extract energy decreases with increasing draft as it loses its radiating capabilities. An evaluation of the translator mass needed to reach a stiff coupling has been made, and with a translator mass of 14 ton, the desired behavior has been reached as can be seen in Fig. 8. A translator mass of 14 ton gives a draft of 0.72 m for the surface buoy. For higher translator mass, the power absorption decreases.

With a translator mass of 14 ton, a more or less stiff coupling between the three bodies can be achieved where the surface buoy, lower body, and translator following each other as can be seen in Fig. 8, except for a small deviation at the turning points where the lower body deviates slightly from the surface buoy and translator, and this is due to the flexibility in the line.

For a point absorber in resonance, the buoy velocity is in phase with the excitation force, and this implies that the buoy is phase shifted −90° relative to the wave elevation. Such resonant behavior can be seen for the TBS40 in Fig. 8. For smaller waves, there are no resonant behavior, and it does not respond to the smallest waves as can be seen between 20–35 s; the latter can be explained by the high inertia in the oscillation that over powers the contribution from the smallest waves and the system’s oscillation decay until the surface buoy encounters a larger wave again. The draft of the buoy is 0.72 m and the height 2 m, so the surface buoy will never leave the surface or get completely submerged due to its large oscillation.

The systems behavior is similar for the TBS10, see Fig. 9, except for a lower amplitude response, visualizing the depth effect that yields a rather large difference in overall power absorption between TBS40 and TBS10, as can be seen in Fig. 5. For the SBS case, there is no resonant behavior at all as can be seen in Fig. 10. Instead, the surface buoy follows the wave, and the clear phase difference between the wave and surface buoy as can be seen in Figs. 8 and 9, which correspond to a resonant behavior are no longer present. It can also be seen in Fig. 10 that the SBS follows even the smallest waves.

FIG. 9.

(Color) TBS10 surface buoy position y1 (red curve), lower body position y2 (black curve), and translator position yt (not visible) vs. the wave amplitude η (blue curve) for wave record 15. The translator follows the surface buoy, and these curves can not be distinguished.

FIG. 9.

(Color) TBS10 surface buoy position y1 (red curve), lower body position y2 (black curve), and translator position yt (not visible) vs. the wave amplitude η (blue curve) for wave record 15. The translator follows the surface buoy, and these curves can not be distinguished.

Close modal

By comparing the difference in divergence among the data points in Fig. 5 for TBS and SBS, one can see the resonant TBSs sensitivity to the shape of the wave power spectrum. This is especially evident when comparing the result from wave records 15 and 17 for TBS40 in Fig. 5 where the difference in power capture ratio is about 34 percentage units, although the difference in energy period between wave records 15 and 17 is only 0.2 s. The wave power spectra for wave records 15 and 17 are presented in Figs. 11(a) and 11(b) with corresponding surface buoy positions for TBS40 inserted.

Although the wave power spectrum for the wave record in data point 17, Fig. 11(b), is well within the resonance bandwidth of the TBS, it yields a considerably lower power capture ratio than for data point 15, Fig. 11(a). This is due to that wave power spectrum 15 being more concentrated around the resonance frequency while wave power spectrum 17 is broader, as can be seen in Figs. 11(a) and 11(b). A spectral width parameter can be estimated from the spectral moments as ɛ=[(m2m0-m12)/m12]1/2.22 For wave power spectrum 15, ɛ = 0.39, while for wave power spectrum 17, ɛ = 0.46. By comparing the surface buoy positions for TBS40 for wave records 15 and 17, which are inserted in Figs. 11(a) and 11(b), we can see that there is pronounced resonant behavior for wave record 15, but not for wave record 17.

The resonant behavior for TBS40 for wave record 15 is even more evident when viewing the translator velocity in Fig. 12(a), where the velocity oscillates smoothly with decreasing amplitude from 0 to 30 s compared to the more stochastic distribution for the translator velocity in the SBS case that more or less follows the wave. By comparing the translator velocity for TBS40 for wave records 15 and 17, Figs. 12(a) and 12(b), we see that even though wave record 17 has higher wave amplitude, Figs. 11(a) and 11(b), the translator velocity for wave record 15 is higher and thus produce more power.

The significant decrease in power capture ratio with the lower body at 10 m instead of 40 m seems too large to be explained solely by the disturbance in the radiating capabilities for the surface buoy. It seems more likely that this depth effect is a result of the combination of the disturbance in the radiating capabilities, in Fig. 6, together with the 180° phase shifted excitation force on the lower body, in Fig. 7(a).

The losses due to viscous forces on the lower submerged body are excluded from the model since an approximation of its magnitude shows of a very small value; this is also supported by the work in Hals et al.21 

The large decrease in power capture ratio for the TBS for larger wave periods is a desired phenomenon since it will act both as a filter for extreme loads as well as a power smoother.

In this model study, no consideration has been taken to the electrical or mechanical survivability of the system. The high damping needed to achieve the desired power output creates very high currents that will put high strain on sensitive electrical components. Although a more or less stiff coupling have been reached with a sufficiently high translator mass, the increased velocity for the TBS will subject the line between the bodies and translator as well as the generator end stops to higher forces then for the SBS. It is also important to mention that what we present is the power at the generator terminal and not the load power, since any losses in the transmission to shore are neglected in this study.

A coupled hydrodynamic, mechanic, and electromagnetic model for a two body system wave energy converter connected to a non-linear load has been described. A more or less stiff coupling has been achieved between the three bodies when using a translator mass of 14 ton. With the lower submerged body at 40 m depth, a resonant behaviour has been achieved with almost 80% power capture ratio, and with the lower body at 10 m depth, the resonance behaviour remains but with a lower amplitude response that decreases the power capture ratio to 50%, showing the importance of the depth effect. This depth effect seems to arise from the combination of the counteracting excitation force on the lower body and the disturbance of the radiating capabilities of the surface buoy. We finally show that the TBSs resonant behaviour is sensitive to the shape of the wave power spectrum where a narrow spectrum gives a much higher power capture ratio.

This project was supported by the Swedish Energy Agency, Vattenfall AB, Statkraft AS, Fortum, ÅF Group, The Swedish Association of Graduate Engineers, Seabased AB, Vinnova, Uppsala University, the Gothenburg Energy Research Foundation, Draka Cable AB, the Göran Gustavsson Research Foundation, Vargöns Research Foundation, Falkenberg Energy AB, the Foundation for the Memory of J. Gust Richter, the Wallenius Foundation, and Swedish Research Council Grant No. 621-2009-3417.

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