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This thesis is based on the following papers, which are referred to in the text by their Roman numerals.
I Engström, J., Waters, R., Stålberg, M., Strömstedt, E.,
Eriksson, M., Isberg, J., Henfridsson, U., Bergman, K., Asmussen, J., and Leijon, M., "Offshore experiments on a direct-driven Wave Energy Converter", Proc. of the 7th European Wave and Tidal Energy Conference, Porto, Por-tugal, (2007)
II Waters, R., Engström, J., Isberg, J., and Leijon, M., "Wave climate off the Swedish west coast", Renewable Energy 34(6), 1600-1606, (2009)
III Tyrberg, S., Stålberg, M., Boström, C., Waters, R., Svens-son, O., Strömstedt, E., Savin, A., Engström, J., Langhamer, O., Gravråkmo, H., Haikonen, K., Tedelid, T., Sundberg, J., and Leijon, M., "The Lysekil Wave Power Project: Status update", Proceedings of the 10th World Re-newable Energy Conference, WREC, Glasgow, UK, (2008)
IV Langhamer, O., Wilhelmsson, D., and Engström, J., "Arti-ficial reef effect and fouling impacts on offshore wave power foundations and buoys - a pilot study", Estuarine, Coastal and Shelf Science 82(3), 426-432, (2009)
V Leijon, M., Waters, R., Rahm, M., Svensson, O., Boström, C., Strömstedt, E., Engström, J., Tyrberg, S., Savin, A., Gravråkmo, H., Bernhoff, H., Sundberg, J., Isberg, J., Ågren, O., Danielsson, O., Eriksson, M., Lejerskog, E., Bolund, B., Gustafsson, S., and Thorburn, K., "Catch the Wave to Electricity: The Conversion of Wave Motion to Electricity Using a Grid-Oriented Approach", IEEE Power & Energy Magazine 7(1), 50-54, (2009)
VI Engström, J., Eriksson, M., Isberg, J., and Leijon, M., "Wave Energy Converter with enhanced amplitude re-sponse at frequencies coinciding with Swedish west coast sea states by use of a supplementary submerged body", Journal of Applied Physics 106(6), 064512-5, (2009)
VII Isberg, J., Engström, J., and Leijon, M., "Depth variation of energy transport in fluid gravity waves" Journal of Re-newable and Sustainable Energy 2(2), 023104, (2010)
VIII Tyrberg. S., Svensson, O., Kurupath, V., Engström, J., Strömstedt, E., and Leijon, M., ”Wave Buoy and Translator Motions - On-Site Measurements and Simulations, IEEE Journal of Oceanic Engineering, 36(3), 377-385, (2011)
IX Li, W., Engström, J., Hai, L., Bontemps, S., Waters, R., Isberg, J., and Leijon, M., "Optimization of the Dimensions of a Gravity-based Wave Energy Converter Foundation Based on Heave and Surge Forces", Proc. of the 9th Euro-pean Wave and Tidal Energy Conference, Southampton, UK, (2011)
X Lejerskog, E., Gravråkmo, H., Savin, A., Strömstedt, E., Tyrberg, S., Haikonen, K., Krishna, R., Boström, C., Rahm, M., Ekström, R., Svensson, O., Engström, J., Ekergård, B., Baudoin, A., Kurupath, V., Hai, L., Li, W., Sundberg, J., Waters, R., and Leijon, M., "Lysekil Research Site, Swe-den: A Status Update" Proc. of the 9th European Wave and Tidal Energy Conference, Southampton, UK, (2011)
XI Engström, J., Isberg, J., Eriksson, M., and Leijon, M., "To-tal instantaneous energy transport in polychromatic fluid gravity waves at finite depth", Submitted to Journal of Re-newable and Sustainable Energy (2011)
XII Engström, J., Isberg, J., Eriksson, M., and Leijon, M., "Properties of the energy transport for plane-parallel poly-chromatic surface gravity waves in waters of arbitrary depth", Submitted to IEEE Journal of Oceanic Engineer-ing, (2011)
XIII Kurupath, V., Engström, J., Ekergård, B., Isberg, J., and Leijon, M., "Modelling and Simulation of Linear Wave En-ergy Converter", Submitted to IET Renewable Power Gen-eration, (2011)
XIV Engström, J., Kurupath, V., Isberg, J., and Leijon, M., "A resonant Two Body System for a point absorbing Wave En-ergy Converter with direct-driven linear generator", Ac-cepted for publication in Journal of Applied Physics, (02 November 2011)
Other contributions by the author that is not included in the thesis. XV Langhamer, O., Wilhelmsson, D., and Engström, J., “De-
velopment of invertebrate assemblages and fish on offshore wave power”, Proc. of the ASME 28th International Confer-
ence on Ocean, Offshore and Arctic Engineering, OMAE 2009, OMAE2009-79239, Honolulu, USA
XVI Boström. C., Svensson. O., Rahm. M., Lejerskog. E., Savin. A., Strömstedt, E., Engström, J., Gravråkmo, H., Haiko-nen, K., Waters, R., Björklöf, D., Johansson, T., Sundberg, J., and Leijon, M., "Design proposal of electrical system for linear generator wave power plants" Presented at the 35th Annual Conference of the IEEE-Industrial-Electronics-Society (IECON 2009), Porto, Portugal, (2009)
XVII Gravråkmo, H., Leijon, M., Strömstedt, E., Engström, J., Tyrberg, S., Savin, A., Svensson, O., and Waters, R., "De-scription of a torus shaped buoy for wave energy point ab-sorber", Presented at the conference Renewable Energy 2010, Yokohama, Japan, (2011)
Reprints were made with permission from the respective publishers.
Contents
Introduction...................................................................................................17 1.1. Wave energy conversion ...................................................................17 1.2. Wave energy......................................................................................19
1.2.1 Wave energy resource.................................................................19 1.2.2. Wave energy transport properties ..............................................20
1.3. Wave energy conversion devices ......................................................21 1.3.1. Point absorber ............................................................................22 1.3.2. Two-body point absorber...........................................................23
1.4 Aim of the thesis ................................................................................23
2. The Lysekil wave energy project ..............................................................25 2.1 The Lysekil test site............................................................................25 2.2 The technology...................................................................................27
3. Theory of energy in ocean waves and energy absorption for a point absorber.........................................................................................................30
3.1 Potential linear wave theory ...............................................................30 3.2 Ocean wave spectrum.........................................................................32 3.3 Energy transport of ocean waves........................................................34 3.4 Energy absorption for a point absorber ..............................................36
3.4.1 Energy absorption for a two-body point absorber ......................39
4. Methods to calculate the energy of ocean waves and modelling the point absorber.........................................................................................................40
4.1. Wave climate and sea states ..............................................................40 4.1.1 Wave data measured at the Lysekil test site ...............................40 4.1.2 Hindcast model wave data ..........................................................41 4.1.3 Empirical spectra ........................................................................41
4.2. Energy transport of ocean waves at finite depth ...............................41 4.2.1. Depth variation of the energy transport .....................................42 4.2.2. Total instantaneous energy transport .........................................42
4.3 Point absorber model..........................................................................45 4.3.1 Defining the geometry ................................................................45 4.3.2 Calculation of hydrodynamic coefficients ..................................46 4.3.3 Point absorber model with linear damping .................................47 4.3.4 TBS point absorber model with non-linear damping..................49
5. Results and discussion ..............................................................................55 5.1. Wave climate at the Swedish west coast ...........................................55 5.3. Energy transport of ocean waves at finite depth ...............................58
5.3.1. Depth variation of energy transport ...........................................58 5.3.2. Total instantaneous energy transport .........................................59
5.4. TBS point absorber model.................................................................63 5.4.1. Linear damping..........................................................................63 5.4.2. Non-linear damping ...................................................................65 5.4.2.1 Grid connection .......................................................................70
6. Conclusions...............................................................................................75
7. Suggestions for future work......................................................................76
8. Summary of Papers ...................................................................................77
9. Svensk sammanfattning ............................................................................83
10. Acknowledgements.................................................................................86
Bibliography .................................................................................................88
Nomenclature
Chapter 3 – Theory
Chapter 3.1 – Potential linear wave theory
Symbol SI Unit Quantity kg/m3 Density t s Time x, y, z m Cartesian coordinates g m2/s Acceleration of gravity
m2/s Fluid velocity potential a m Wave amplitude m Free surface elevation
h m Water depth k m-1 Wave number
rad/s Angular frequency m Wave length
Chapter 3.2 – Ocean wave spectrum
Symbol SI Unit Quantity m Sea surface displacement zn m Fourier coefficients n - Harmonics S m2s/rad Power spectrum f Hz Frequency mk m2/sk Spectral moment Hs or Hm0 m Significant wave height Te or Tm0-1 s Energy period SBS m2s/rad Bretschneider spectrum
m rad/s Modal angular frequency SPM m2s/rad Pierson-Moskowitz spectrum U19.5 m/s Wind speed at 19.5 m above sea level
Chapter 3.3 – Energy transport of ocean waves
Symbol SI Unit Quantity j W/m2 Energy transport j W/m2 Time averaged energy transport
totj W/m Total time averaged energy transp. jtot W/m Total instantaneous energy transp. vg m/s Group velocity p N/m2 Dynamic pressure vx, vy, vz m/s Fluid velocity K, L, M, I - Integration kernels
Chapter 3.4 – Energy absorption for a point absorber
Symbol SI Unit Quantity P W Power, absorbed Pmax W Power, maximum absorbed Pub W Power, upper bound Pc W Incident power J W/m Energy transport for harmonic
wave T s Harmonic wave period H m Harmonic wave height V m3 Buoy volume L m Characteristic length of body
r m2/s Velocity potential, radiation
e m2/s Velocity potential, excitation
d m2/s Velocity potential, diffraction
0 m2/s Velocity potential, incident wave
n - Normal vector S m2 Wetted surface area Fe N Excitation force Fr N Radiation force Fh N Hydrostatic force
n rad/s Natural frequency A m2 Area m kg Mass ma kg Added mass
ef N/m Transfer func. excitation force
R kg/s Transfer func. radiation resistance H ms/rad Response amplitude operator
Chapter 4 – Method
Chapter 4.2.2 – Total instantaneous energy transport
Symbol SI Unit Quantity U - Integration kernel - Hermite-Weber function H - Hermite polynomial - Kronecker delta
Chapter 4.3 – Point absorber model
Symbol SI Unit Quantity U, V - B-spline basis function M - Number of panels m a
kg m
Mass of buoy and translator Buoy radius
Fmech N Mechanical force Fgen N Generator force Fhyd N Hydrodynamic force ks N/m Spring constant Nm/s Generator damping coefficient Pratio % Power capture ratio y m Body position 1,2 - Surface(1) and submerged(2) body yt m Translator position
am kg Added mass at infinite frequency
h kg/s Radiation impedance m1 kg Mass of surface buoy m2 kg Mass of submerged body mt kg Mass of translator Fl N Line force Fe N End stop spring force Fem N Electromagnetic damping force PR W Electrical real power eg V Voltage in the generator ig A Current in the generator VDC link V Voltage on the DC link Aact m2 Active area of the stator ls m Length of stator lt m Length of translator D m Diameter d m Draft for surface buoy L1 m Distance between bodies Ls m Stroke length
kl N/m Spring constant for line ke N/m Spring constant for end stop Rg Generator resistance Rload Load resistance Rsc Sub-sea cable resistance
Abbreviations
WEC Wave Energy Converter L1 - L9 Names of the Lysekil project WEC’s AC Alternating Current DC Direct Current OWC Oscillating Water Column BEM Boundary Element Method WAMIT Wave Analysis Massachusetts Institute of Technology PTO Power Take Off TBS(40,10) Two-body system SBS Single Body System WAM WAve prediction Model SWAN Simulating WAves Nearshore DFSBC Dynamic Free Surface Boundary Condition BBC Bottom Boundary Condition KFSBC Kinematic Free Surface Boundary Condition PWM Pulse Width Modulation IGBT Insulated-Gate Bipolar Transistor VSI Voltage Source Inverter
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Introduction
The uneven distribution of sunlight over the earth creates wind patterns that blow from high pressure zones to areas of low pressure. Friction between the wind and sea surface creates waves. This is an ascending chain in energy conversion where the energy density increases from sunlight to waves, giv-ing the world’s oceans a renewable source of free energy on the order of terawatts that has to this date not been utilized commercially. When the wind stops to blow, the wave energy will continue to travel with low losses as swells. Thus, wave energy can be expected to have smaller power fluctua-tions over longer periods of time than other renewable energy sources such as wind and solar power.
1.1. Wave energy conversion
While the wind energy industry has seen a continuously exponential growth during the last decades1, the pattern has not been the same for wave power since the research begun in the 70’s [1]. In 2002 there existed 340 patents regarding wave energy conversion, with the first known patent dating back to 1799 [2]. But it was not until the oil crisis of the 70’s that the research started on a larger scale. At that time, several research programs started with both private and public support mainly by the European countries UK, Por-tugal, Sweden, Denmark, Ireland, and Norway [2]. The last years have seen a large increase in activity. Several wave power concepts, in Europe as well as internationally, are taking steps toward full-scale offshore testing. Many concepts are also advancing into the first commercial plants or large demon-stration pilot plants2, 3, 4[1-6].
1 www.gwec.net, Wind power drives global renewable energy growth, Global Wind Energy Council, retrieved 2011 08 09 2 www.seabased.com, retrieved 2011 08 09 3 www.wavebob.com, retrieved 2011 08 09 4 www.wavegen.co.uk/news_press release 8 july 2011 mutriku opening.htm, retrieved 2011 08 09
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Figure 1.1: Typical total instantaneous energy transport from a sea state measured at the Lysekil wave energy research test site.
Ocean waves vary greatly in amplitude and frequency which gives large short-term energy fluctuations, see Figure 1.1. The dramatic nature of ocean waves is clearly visible as the power changes from 0 kW/m to 120 kW/m in a few seconds. This places special demands on the design of the electrical system since the output voltage needs to be prepared for the grid that relies on an even power distribution, see Paper V. To avoid expensive buffering systems such as large capacitor banks, the need for even power distribution is thought to be solved by interconnected wave energy converters in large wave power parks [7, 8, 9], see also Papers V and XIII. Another of the major challenges is the extreme forces leading to maximum power levels up to 45 times higher than the average, see Paper II. Certain electrical components are sensitive to the high currents related to extreme forces. Associated problems are expensive over-dimensioning, difficulties to make mooring last and me-chanical overloads. One of the major issues for wave energy technologies should be to keep the maintenance work to a minimum since offshore work is associated with high costs. Due to the high maintenance and installation cost together with the harsh environment for wave energy conversion, it is necessary to utilize its advantage when compared to other renewable sources as wind and solar energy; namely the high energy density. It is therefore of the utmost importance to have a high power capture ratio in order to be eco-nomically feasible. For larger wave power parks one also needs to consider
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the demands made by society and possible conflicts with other interests. Thus, in order for wave power to be a profitable power technology in the future, one has to consider the entire system from the wave to the grid.
1.2. Wave energy
It is important to have a detailed knowledge of the sea state and wave cli-mate while planning and designing a wave energy converter system for vari-ous reasons. The total time averaged energy transport sets the energy poten-tial and sea state. The energy potential gives a hint of how much power can be produced, and the sea state has to be known in order to design and opti-mize the technology. The depth distribution of the energy transport sets the absorbing body’s absorbing capabilities. The total instantaneous transport is of importance for an active control mechanism of the converter damping that relies on real time information of the wave. Finally, it is also important to know the extreme waves in order to design the technology for mechanical and electrical overload.
1.2.1 Wave energy resource
An estimate of the global wave power resource accounts for about 2 TW [10]. Due to the general wind pattern, the oceans’ eastern shores are where the most energetic wave climates are encountered, see Figure 1.2.
6050
4030
1520
304050
70
100
30
20
15
10
20
3050
50
304050
70100 50
40
20
1020
40
5040
20
1540
60
40
70 50
100
20
15
40
100 15 15
100
20
Figure 1.2: The worlds wave energy resource given in kW/m [10].
The most energetic wave climates at any given location can be found south of South America due to the circumpolar storms and the unlimited fetch. The
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Atlantic coast in Europe is estimated to have an average resource of 290 GW out of which 32 - 48 GW is estimated to be technically achievable [11]. The most energetic sites in Europe are found outside Ireland and Scotland with average energy transports of 75 kW/m [11]. It is also worth noting that the average winter power level along the European Atlantic coast is twice as high as the average annual, coinciding with the energy demands of the Euro-pean society [11]. It is well known that the wave climate off the coast of Sweden is relatively calm in comparison to the coasts facing the oceans of the world. The Norwegian coast, only 300 km to the west, has an average wave climate with an energy transport up to an order of magnitude higher [12]. The mean energy transport in the Skagerrak region off the Swedish west coast has been estimated to be about 6 kW/m [13]. The documented average significant wave height during the time period from 1 April 1980 to 13 January 1981 at Väderöarna was 2.46 m and with a mean wave period of 5.42 s. The highest expected wave for a 100 year period at the same location was calculated to 18.3 m [14].
1.2.2. Wave energy transport properties
It has long been known that the energy of surface gravity waves travel with the group velocity. Reynolds [15] was first to show the connection between group velocity and energy transport, and Rayleigh [16] expanded the theory to cover waters of finite depth. Tulin [17] treated the case of energy transport of monochromatic surface gravity waves at arbitrary depth more extensively, where he examined each of the transport components, kinetic, gravity poten-tial and surface tension. An expression for the total instantaneous energy transport for waters of infinite depth was developed by Isberg et al. [18]. Besides the work done by Reynolds, Rayleigh, Tulin and more recently the work by Isberg, the case of energy transport in surface gravity waves has remained relatively untouched. The focus has instead been towards the non-linear interactions in the energy transfer mechanism [19, 20]. In the finite water regime, the description of energy transport in ocean waves is limited to its total time averaged transport and the spectral distribution and given by phase-averaged models such as SWAN (Simulating WAves Nearshore) [21] or by energy spectra such as the TMA spectrum [22] or the Pierson Mosko-witz spectrum [23], both methods giving a more general view. With the phase-averaged models, the sea surface elevation can be reconstructed in space and time while accounting for shallow water effects such as refraction, diffraction, triad and quadruplet wave-wave interactions. See e.g. ref. [24] for a review on the phase-averaged models.
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1.3. Wave energy conversion devices
Several concepts have been proposed with the goal to convert the kinetic and potential energy of ocean waves for human purposes [1]. The different solu-tions on how to absorb the wave energy can be classified into three major categories:
• Oscillating Water Column (OWC) • Overtopping devices • Wave activated bodies
The first category refers to an oscillating water column inside a chamber which, by means of the fluctuating air pressure in the chamber, drives a tur-bine, e.g. the Limpet OWC5, see Figure 1.3(a). The second category uses a structure where the waves are guided into an ascending ramp and spill over at a certain threshold. The increased potential energy can then be used, e.g. the Wave Dragon [3], see Figure 1.3(b). The last category includes all sys-tems that are based on wave induced motions of solid bodies. For example several bodies moving relative to each other such as the Pelamis [5], or one body moving relative to an internal reference point or fixed to the ground, e.g. the PowerBuoy6, see Figure 1.3(c). The focus in this thesis falls into the third category which includes the system of a point absorber moving relative to a fixed reference at the seabed.
Figure 1.3: (a) shows the concept of an Oscillating water column, (b) the concept of an Overtopping device and in (c) the concept of a Point absorber. The picture is adapted from [25].
5 www.wavegen.co.uk, retrieved 2011-08-09 6 www.oceanpowertechnologies.com, retrieved 2011-08-09
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1.3.1. Point absorber
The concept of a point absorber converting the heaving and surging motions of waves to electric energy has been one of the main concepts that have been investigated since wave power research began [26-30]. The idea is simple; the wave gives rise to motions in the point absorbing buoy which is directly transmitted to a Power Take Off (PTO). A point absorbing buoy is defined as a floating body with a characteristic length that is much smaller than the typical wave length of ocean waves [28]. The PTO might be hydraulic [30] or electric [4, 31]. Models of point absorber systems with non-linear PTO have been verified with good accuracy to full-scale experiments, see e.g. ref. [27] and Paper V. Full-scale experimental results from the Lysekil project Wave Energy Converter (WEC) show a power capture ratio on the generator terminal of 24 % excluding iron and mechanical losses, at wave periods of 4.5 s [25].
The shape of the buoy has been shown to have no significant impact re-garding the energy absorption since a point absorbing buoy with a reason-able size has a natural period of oscillation that is higher than the frequency for ocean waves [32]. The larger the buoy, the more energy can be absorbed is the fundamental idea. However, if the buoy is too large, the diffraction forces will dominate and the buoy will no longer work as a point absorber. Further, it is not the amount of power that is absorbed that is interesting, it is how much of the available power the device converts that is interesting. Thus, one needs to find a point absorber configuration that absorbs as much power as possible from the incoming wave and this includes the size of the buoy, the inertia of the system as well as the damping from the PTO. An optimization algorithm has been evaluated and it was found that it is more advantageous to use several small units rather than a few big ones for a given area [33]. Maniar and Newman [34] showed that array interactions may increase the power absorption of each device if the wave length to spac-ing ratio is ideal. The latter has led to wave energy converter devices with closely spaced arrays of buoys such as Wavestar7.
It is well understood that a point absorber having a natural period of oscil-lation that coincides with the incident wave will achieve increased amplitude and speed and thereby transfer more energy from the wave to the PTO [28, 35, 36]. Methods to physically control the oscillation of a WEC to coincide with the dominant period of the ocean waves were proposed by Budal [37] and Salter [38] independently in the mid 70’s. Two different control strate-gies have been suggested, continuous and discrete. The first uses a continu-ous control mechanism where information to the controller needs to contain both measurements as well as predictions of the near future. Input to the controller can be the measured and predicted displacement of the body, 7 wavestarenergy.com, retrived 2011-09-28
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which can be transferred linearly by convolution integrals to the optimum load for the WEC system [36]. With discrete control the oscillation is con-trolled at a finite number of times during a wave period. In the case of an oscillating point absorber, the method of latching was proposed in the late 1970’s independently by Budal [39], Guenther [40] and French [41]. Latch-ing is obtained by blocking the motion of the body when the velocity is zero, i.e. at the turning points, and then releasing the body when the velocity of the body is in phase with the predicted excitation force on the body from the incident wave. Besides the need for a very powerful mechanical or electro-magnetic breaking mechanism, this approach requires non-causal informa-tion of the excitation force [42]. Simulation on a heaving point absorber system using latching indicated a threefold increase in yearly power produc-tion compared to the same system without phase control. It also gave a steadier output power [43].
1.3.2. Two-body point absorber
In order to avoid expensive and vulnerable control systems and breaking mechanisms, one way of shifting the frequency response of the system is by increasing the inertia of the moving parts. This has 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 tuning the point absorber to resonance [44, 29, 45], see also Paper VI. All tests on the two-body system thus far have shown promising results on a theoretical level with a 60 % absorption in irregular waves and a significant decrease in opti-mal load damping, see Paper VI. An experimental setup with a point ab-sorber working against a pulley with supplementary mass has shown to give a 60 % power capture ratio in irregular waves [26].
1.4 Aim of the thesis
A research team of 23 scientists at the division of Electricity within Uppsala University are currently developing and optimizing a technology to convert the energy of ocean waves to electricity. This is a highly multidisciplinary research area, in which the work presented in this thesis forms a part. The main focus is on: (i) the sea state that sets the design criteria for the wave energy converter, (ii) different wave energy transport properties for ocean waves traveling in waters of finite depth and (iii) increasing the conversion efficiency of energy from the waves to the generator and load. The latter includes coupled hydrodynamic, mechanic and electromagnetic state equa-tions to describe the dynamics of the wave energy converter.
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The work is presented by first giving a brief introduction to the wave en-ergy research project carried out at Uppsala University, the so called Lysekil project, see Chapter 2. In that chapter the technology and the test site is be described together with a short description of what has been done so far.
This is followed in Chapter 3 by an explanation of the fundamental theory that has been used to carry out the calculations in Chapter 4. Chapter 3 starts with a description of potential linear wave theory that forms the basis for the work done in the papers. Then three different ways to present the wave cli-mate and sea state (measured time series of wave elevation, hindcast model wave data and empirical spectra) that are used in this thesis are presented. This is followed by a presentation of expressions that describe different wave energy transport properties that will be used to compare with derived solutions for waves traveling in waters of finite depth. The fundamental the-ory for energy absorption of a point absorbing wave energy converter is pre-sented with a special emphasize on resonance and methods to achieve reso-nance (latching and two-body systems).
Chapter 4 presents the methods that have been used to derive the results in Chapter 5. It starts with a brief survey of how the three different methods to describe the wave climate and sea states have been used. This is followed by a description on how different properties for the energy transport of ocean waves in waters of finite depth are derived. Finally Chapter 4 describes the method that has been used to model point absorbing wave energy converters, both the conventional as well as the conceptual resonant Two-Body System, including both linear and non-linear load characteristics.
The most significant results regarding the wave climate at the Swedish west coast, energy transport in waters of finite depth and from the Two-Body System point absorber model for three different load conditions are pre-sented and discussed in Chapter 5.
The results are condensed in a list of conclusions in Chapter 6. Sugges-tions for future work are given in Chapter 7 and a brief summary of the in-cluded papers is given in Chapter 8. Finally, a short summary in Swedish can be found in Chapter 9.
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2. The Lysekil wave energy project
Since 2002, the division of Electricity within the Swedish Centre for Renew-able Electric Energy Conversion at Uppsala University has carried out wave power research with the goal to develop a technology to convert the energy of ocean waves to electricity. It is a technology that is both potentially prof-itable and environmentally friendly. A system for wave energy conversion called the Lysekil project has now been subjected to initial tests. The Lysekil project is a multi-disciplinary undertaking including a wide variety of areas such as power systems, electronics, hydrodynamics, mechanical construction and environmental studies.
2.1 The Lysekil test site
While the research is carried out at Uppsala University, the offshore experi-ments are carried out at the test site outside Lysekil at the Swedish west coast, see Figure 2.1. The Lysekil test site is situated between a northern marker (58º 11’ 850’’N 11º 22’ 460’’E) and a southern marker (58º 11’ 630’’N 11º 22’’ 460’’E) and covers about 40 000 m2. 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. A measurement station is located about 3 km to the south on the island Hermanö. The seabed is fairly level with an average depth of 25 m consisting mostly of sandy silt.
Figure 2.1: The test site outside Lysekil.
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Wave measurements started at the test site in April 2004 when a Datawell®8 Waverider buoy was deployed, see Figure 2.2(b) [46]. The first WEC, named L1, was deployed at the test site on March 13th, 2006 and was con-nected via a 2.9 km long sea cable to the measurement station on Hermanö, see Figure 2.2(a, f).
In the summer of 2008, a 12 m high lattice tower was deployed on the small islet of Klammerskäret, see Figure 2.2(h). The lattice tower is used for surveillance and measurement. A wind turbine together with solar panels and batteries are charging a network camera placed on the lattice tower with the purpose to study the position and movement of the buoys, see Paper VIII. In 2007 a total of 21 environmental buoys had been installed at the test site. The environmental buoys are used for studies of the environmental impact of the WEC and how biofouling will affect the buoy’s ability to extract energy, see Paper IV and Figure 2.2(c).
In February 2009, L2 and L3 where deployed and later that spring an un-derwater substation was deployed. The three WEC’s L1, L2 and L3 inter-connected to the underwater substation transferred power to shore, giving the first experimental data of a wave power park [7]. See Figure 2.2(d, g) for pictures of the underwater substation and L2, L3. In October 2009, L9 was deployed. L9 had an octagonally shaped stator to increase the electromag-netic damping, increase the voltage and also increased translator weight that is used as the retracting force instead of springs. In November 2010 four more WEC’s were deployed, L4, L5, L7 and L8. Experimental results from L1, L2, L3 and L9 are found in Papers I, V and X.
Figure 2.2 shows a collection of photos of the technology and facility that are installed at the research test site. More thorough information on the Lysekil test site is found in Papers I, III, V and X.
8 www.datawell.nl, retrieved 2011-08-09
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a) b)
d) f)
c)
g) h) i)
e)
Figure 2.2: The top (guiding system) of L1 and the line are shown in (a). The Datawell® Waverider wave measurement buoy can be seen in (b). The environ-mental buoys for environmental studies are shown in (c). The underwater substation under construction can be seen in (d). First voltage pulses from L1 after the deploy-ment in March 2006 in (e). A sea anemone on L1 can be seen in picture (f). L2 and L3 the day before deployment in the Lysekil harbor can be seen in (g). The lattice tower in (h) and finally the buoy that was first used on L1 is shown in picture (i).
2.2 The technology
The Lysekil project uses a point absorber system with a direct-driven linear generator placed on a gravity foundation on the seabed with a buoy at the sea surface that absorbs energy from the waves, see Figure 2.3. Inside the gen-erator, a translator is connected to the buoy via a line and a guiding system, see Figure 2.2(a). The translator has Neodymium-Iron-Boron magnets mounted between aluminum spacers. In WEC’s L1, L2 and L3 springs are attached to the translator to retain tension in the line. In WEC’s L4, L5, L7, L8 and L9 increased translator weight is used as the retracting force. End
28
stop springs are mounted at the top to handle the mechanical overload in waves higher than the design conditions. The translator moves inside a stator with a quadratic cross section (L9 has an octagonal cross section) that con-sists of a conventional three phase winding (L7 and L8 use a one phase winding). The problem with the varying output power from each device is thought to be solved by interconnecting several devices in large clusters, see refs. [7, 9] and Papers V and XIII. The linear generator L1 was designed for a nominal power of 10 kW and to withstand high overload during periods of strong sea states. The main characteristics of L1 are presented in Table 2.1. For a more thorough description of the WEC technology and a detailed study of the magnetic circuit see refs. [25, 47-51].
Table 2.1: Some of the main characteristics of the WEC (L1).
Main features of the WEC, L1 Buoy type Cylinder Buoy radius 1.5 m Buoy height 0.8 m Buoy draft 0.4 m Buoy mass 1000 kg Generator nominal power 10 kW Nominal voltage 200 V Nominal speed 0.67 m/s Winding resistance 0.44 Winding inductance 11.7 mH Air gap 3 mm Number of stator sides 4 Pole width 50 mm Vertical stator length 1264 mm Vertical translator length 1867 mm Translator weight 1000 kg Spring constant 6.20 kN/m
Although the WEC consist of a complex multi-physical chain of energy con-version, the principal operation is quite simple, see Figure 2.3. When the buoy encounters a wave crest, the wave makes the buoy move upwards and thereby pulls the translator in the generator. Some of the kinetic energy in the translator is induced as voltage in the stator winding and the remaining energy is stored as mechanical energy in the springs. The stored energy in the spring is induced in the stator winding when the springs pull down the buoy in the following wave trough.
29
Figure 2.3: A conceptual sketch of the Lysekil project Wave energy converter.
In the initial tests of L1, the output voltage from the WEC was measured over the resistive load at the measurement station, see Paper I and X. The next step was to load L1 with a non-linear load, using a diode rectifier and a capacitive filter. This was done in order to test the capacitive filters power smoothing effect and as a more realistic load condition [50]. With the launch of the underwater substation, L1, L2 and L3 could be interconnect to a non-linear load configuration where inverters are used to actively control the DC level, see Figure 2.4 and ref. [7]. Experimental results from the intercon-nected L2 and L3 are found in Paper X. Finally, a resonance circuit was in-stalled in September 2010 with the aim to increase the damping of the gen-erators and thereby increase the output power. The circuit is a combination of a rectifier bridge and a resonance circuit [51]. Experimental results of L9 connected to the resonance circuit are found in Paper X.
Figure 2.4: A schematic sketch of the power transmission system.
30
3. Theory of energy in ocean waves and energy absorption for a point absorber
Most waves in the oceans are created by the wind. Friction between the wind and sea surface creates ripples or capillary waves that grows depending on the time they are exposed to the wind and on how much the waves are dis-turbed by the seabed. Gravity is the main restoring force which affects the movement of the waves. The waves continue to grow until the phase speed of the wave crest equals the speed of the wind. At this stage; a fully devel-oped sea has been created. Surface gravity waves transport energy with low losses until they reach the coast where they loose their energy due to break-ing waves and friction against the seabed. The theory in this thesis only ap-plies to surface gravity waves, which are the waves of interest for wave en-ergy conversion.
For an object to be a good wave energy absorber it has to be a good wave maker; this is the classical statement describing the nature of a point ab-sorber. A point absorber is defined as a body having a characteristic length that is much smaller then the wave length [28]. The fundamental idea with wave energy conversion using a point absorber is to have a damped oscillat-ing motion with a body that converts the energy of the wave to a load.
3.1 Potential linear wave theory Fluid gravity waves with a small amplitude a compared to the wavelength , can be described by linear potential wave theory [52, 53, 54]. Assuming an incompressible, non-viscous and irrotational fluid, the fluid motion can be expressed by a fluid velocity potential satisfying the Laplace equa-tion 02 . Further, for small amplitudes, the boundary conditions can be linearized and the free surface wave elevation is replaced with the undis-turbed fluid surface (z = 0) where the velocity potential satisfies the two boundary conditions:
The linearized Dynamic Free Surface Boundary Condition (DFSBC):
31
.0zt
g (3.1)
and the linearized Kinematic Free Surface Boundary Condition (KFSBC):
.0 tz z
(3.2)
where g is the acceleration of gravity. Finally, the Bottom Boundary Condi-tion (BBC) specifies that the fluid velocity must be parallel to the sea bed, see Figure 3.1.
Figure 3.1: Definitions for plane-parallel surface gravity wave.
For a plane-parallel wave propagating in the positive x-direction in water of depth h, the velocity potential can be written as an integral over its harmonic components as:
.cosh
coshˆ2
1,, de
hk
hzktzx tkxi (3.3)
where the angular frequency must have the same sign as the wave number k. The Fourier transform of the potential ˆ is defined as:
.,0,0ˆ tdet ti (3.4)
and is related to k by the dispersion relation:
.tanh2 khgk (3.5)
32
With the fluid boundary conditions linearized and with the velocity potential defined, solutions to the properties of the energy of ocean waves and the hydrodynamic forces acting on the buoy can be found. Thus, potential linear wave theory forms the basis for the theoretical work done in this thesis.
3.2 Ocean wave spectrum
In order to describe the nature of real ocean waves they are written as a combination of harmonic components, i.e. a Fourier series. If the sea surface displacement, now defined as (t)= (x0,y0,t), is assumed to be a continuous function measured over the time T, then it can be decomposed into a infinite series of cosine and sine functions over the time interval T/2 t -T/2, ac-cording to:
.2
n
nftinezt (3.6)
Here f is the fundamental frequency and nf are harmonics of the fundamental frequency:
.2sin2cos2 tnfinfte nfti (3.7)
and
....2,1,01 2
2
2 ndtetT
z
T
T
nftin
(3.8)
where zn is the Fourier transform of (t). Now a spectral density function or power spectrum can be defined by:
.2 *nn zzfnfS (3.9)
where *nz is the complex conjugate of nz .
Different moments can be calculated from the wave power spectrum that contains useful statistical information. The kth spectral moment are defined as:
.dffSfm kk (3.10)
33
Two often used parameters are the significant wave height Hs and energy period Te, defined as:
.4 00mHH ms
(3.11)
and
.0
110 m
mTT me
(3.12)
There are three general ways to create a spectrum describing ocean waves. The first is to use time series of measured wave elevation. Secondly, one can use a hindcast model, i.e. a simulated sea state derived from wind data and calibrated by wave measuring buoys and satellite altimeter data. The last is to create a spectrum using empirical methods. There exist several empirical methods for different purposes. One describing a broad-banded spectrum of ocean waves is the Bretschneider spectrum [55] defined as:
.4
25.1 425.125
4meHS s
mBS
(3.13)
The Bretschneider spectrum is a two parameter spectrum where Te defines the modal angular frequency m. Another often used spectrum is the Pier-son-Moskowitz spectrum [23]; it describes a fully developed sea where the wind is in equilibrium with the waves according to:
.4
5.19
5
2Ug
PM eg
S (3.14)
Here = 8.1·10-3, = 0.74 and U19.5 is the wind speed at 19.5 m above sea surface.
In this thesis all three methods are used; time series of wave data col-lected by the wave measurement buoy at the Lysekil test site are used throughout the whole thesis, hindcast models made by Fugro OCEANOR9 in Paper II and finally the Bretschneider and Pierson-Moskowitz spectra are used in Paper VII.
9 www.oceanor.no, retrieved 2011 08 10
34
3.3 Energy transport of ocean waves
The energy content of a surface gravity wave is concentrated to the surface and decreases exponentially with depth, for a = 60 m wave as much as 50 % of the energy is transported in the upper 5 m of the wave. Further, the energy transport is dependent on the group velocity of the wave which has a large influence on the transport properties as we shall see later on. To find solutions for a surface gravity wave the water depth can be assumed to have infinite depth, i.e. using the deep water approximation. As a rule of thumb, the deep water approximation is said to be valid when h > /2. In this thesis, existing solutions of wave energy transport properties for the deep water domain are expanded to cover waters of finite depth and the theories are compared, see Paper VII, XI and XII.
The total time averaged energy transport per unit width of a plane-parallel wave totj can be found for any wave spectrum S by using the well known expression:
0
0
.h
gtot dSvgdzzjj (3.15)
where the bar denotes the time average, the density of water and vg is the group velocity that is defined for intermediate water depth as:
.tanh2sinh
21
2
1kh
g
kh
khvg
(3.16)
and when using the deep water approximation it is defined as:
.2,
gv dwg
(3.17)
Further, the wave spectrum’s distribution of energy over depth can be found by using the deep water approximation according to:
0
22
.2
dkSeg
zj zkdw
(3.18)
By integrating (3.18) over z, using the integration limit z - we get the total time averaged energy transport for waters of infinite depth according to:
35
02
2
0
2
, .64
1
2 sedwdwtot HTg
dSg
dzzjj (3.19)
which is the well known expression 2seHkT for the energy transport in the
deep water limit. The energy transport vector per unit time and area for surface waves
equals the dynamic pressure p times the fluid velocity v and can thus be de-scribed by the velocity potential as:
.,,,,,,zyxt
pvpvpvjjjj zyxzyx (3.20)
which is the Eulerian description of the fluid motion. For a plane-parallel wave propagating in waters of infinite depth in the
positive x-direction, the instantaneous energy transport vector can be found as a convolution integral of the wave elevation according to:
tdtttgzLtdtttgzKgtzj
tdtttgzMtdtttgzKgtzj
z
x
,,,
,,,. (3.21)
The work done by the volume of water to the left of the origin (x < 0) on the volume to the right of the origin (x > 0) equals the total instantaneous energy transport through the surface (x = 0) and is expressed for deep water waves as:
22
2
,
,
,2
tt
ttttIwith
tdtdttttttIg
tj dwtot
. (3.22)
where K, L, M and I are integration kernels that are explained in more detail in ref. [18].
36
3.4 Energy absorption for a point absorber
It can be shown that a heaving axisymmetric body interfering with harmonic waves can at most absorb Pmax = J /2 where J is the incoming energy trans-port J = g2TH2/32 [28, 58, 59]. The volume V of an immersed body also puts an upper bound Pub, based on the wave period T and the wave height H, of how much power it can absorb when in optimum destructive interference Pub = V g H/4T [35]. The theoretical upper bound Pub and the incident power Pc = LJ, where L is the characteristic length of the body, are universal for any axisymmetric system and set the upper limits for the power absorp-tion. On the other hand, the actual absorbed power P is unique for the system and dependent on the damping from the PTO, the geometry of the buoy and the inertia of the moving parts. To achieve optimum destructive interference, i.e. resonance, the point absorber system needs to more or less obey both optimum amplitude and optimum phase response. For an electrical applica-tion this is often referred to as Impedance matching. For a given buoy ge-ometry, the optimum amplitude response can be reached by choosing the correct damping by the load. However, optimum amplitude response does not necessarily mean maximum overall power production since a narrow and high resonance peak only cover a small fraction of the sea state. Thus, for maximum overall power production it might be better to have a lower and broader resonance peak. A reasonably sized buoy has an optimum phase response that lies outside the frequency of regular ocean waves, i.e. < 1.4 rad/s, due to its low inertia. Increasing the draft gives the desired increase in inertia that is necessary to achieve the desired phase response. However, an increased draft also leads to a decrease in amplitude response since the en-ergy density of the wave decreases exponentially with depth. The gain in phase response leads to a loss in amplitude response. In order to elude this dilemma and to avoid using too large buoys, the systems phase response can be tuned by some control mechanism and this will be discussed further in chapter 3.4.1. Strictly speaking, this can only be done when the source and load are purely linear, but a useful approximate matching can still be done with a non-linear load as will be shown in chapter 5.4.2. An extensive dis-cussion in this area can be found in [56].
There are three major force regimes regarding wave/body interaction; dif-fraction, inertia and viscous forces, see Figure 3.2. In order for a body sub-jected to a wave to act as a point absorber it has to be sized correctly in com-parison to the characteristics of the wave. For a large body, when /L < 5, diffraction forces are significant and a smaller amount of inertia is converted from the wave to the body. When the body is much smaller than the wave length and H/L > 10, flow separation occurs and viscous or “drag” forces are significant. In this case one can expect energy losses due to viscous dissipa-tion, which also sets a limit for the use of linear wave theory.
37
Figure 3.2: Area of dominance for the dominating force regimes regarding wave/body interaction. Here L is a characteristic length of the body subjected to a wave with wave length and wave height H. The black area represents a cylinder of 6m diameter subjected to Swedish west coast sea states. The picture is adapted from [57].
Taking the characteristic length for the vertical cylinder used in the Lysekil project, subjected to Swedish west coast sea states, see Figure 5.2, the stud-ied case will occupy the black area in Figure 3.2. And as can be seen, the studied case stays within the area for acting like a point absorbing buoy ex-cept for the smallest waves when diffraction starts to be significant. The latter sets an upper bound for a point absorber in Swedish west coast sea states.
A body subjected to a wave has six degrees of freedom, surge, sway, heave, pitch, roll, and yaw. In this thesis the motion is mostly restricted to heave since it is the motion of interest for absorption and the dominating motion as can be seen in Paper IX. The hydrodynamic forces acting on the point absorbing buoy are calculated by solving the fluid velocity potential via the pressure it exerts on the buoy. Due to linearity, the velocity compo-nent can be split up into two parts:
.re (3.23)
Here e is the velocity potential for the excitation force and r the velocity potential associated with the radiated wave. The velocity potential is found by solving the Laplace’s equation 02 on the wetted surface S of the buoy, where both e and r satisfy the boundary conditions DFSBC, KFSBC and BBC.
This is done in a two step process. The excitation problem calculates the scattering of an incident wave on a body fixed in space:
S
de ndSiF .0 (3.24)
38
where n is the normal vector representing heave mode. If the body is suffi-ciently small compared to the wave length, (3.24) can be simplified further by neglecting the diffraction term d, the resulting force is called the Froude-Krylov force. The radiation problem is calculated for an oscillating body in calm water:
S
rr ndSiF . (3.25)
The velocity potential for the excitation force e contains information of both the undisturbed incident wave 0 as well as the diffracted wave d, see Fig-ure 3.3.
Figure 3.3: The excitation problem (top) and radiation problem (lower). The veloc-ity potential for e = excitation force, 0 = incident wave, d = diffraction and r = radiation.
Subsequently, the hydrodynamic forces can be divided into three force com-ponents according to:
.)( hre FtFtFtF (3.26)
where the excitation force Fe is the force that will transfer motion to the body while the radiation force Fr is associated with radiating waves and thus dampens the motion. The hydrostatic force Fh is calculated for a static buoy in calm water. The hydrodynamic coefficients can be obtained by a BEM-code such as WAMIT® or as closed form expressions see e.g. ref. [60]. For a more thorough description of the wave/device interaction see e.g. refs. [57, 61].
39
3.4.1 Energy absorption for a two-body point absorber
For a point absorber in resonance in harmonic waves, the buoy’s velocity is in phase with the excitation force and this implies that the buoy is phase shifted -90 degrees relative to the wave elevation. As mentioned in previous chapter, to achieve this, the phase response of the point absorber has to be engineered by some control mechanism to coincide with the frequency of ocean waves. To accomplish this, the method of latching has been proposed; the buoy is released when it is in phase with the excitation force of the wave. By using a two-body system, the mass and added mass of the lower sub-merged body is used to tune the system to an optimum phase response. The increased inertia in the system delays the motion so it coincides with the phase of the excitation force. This is apparent when looking at an un-damped oscillator’s natural frequency n, which is set by the cylindrical buoy’s water plane area A, the system’s total mass m and total added mass ma according to:
.a
n mm
gA (3.27)
From (3.27) we can see that by using an additional submerged body, its own mass and added mass can be used to shift the system’s natural frequency to coincide with the frequency of the waves at the site, see Paper VI and XIV. If the lower submerged body is placed deep enough not to interfere destruc-tively 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 resonant behavior can be further visualized by looking at the system’s response amplitude operator H , in the frequency domain:
.ˆ
ˆˆ
2 gARimm
fH
a
e (3.28)
where ef and R are transfer functions for the excitation force and radiation resistance. The response amplitude operator H (3.28) is plotted for the two-body system tuned to resonance in Swedish west coast sea state and for a conventional point absorber in Figure 5.8.
40
4. Methods to calculate the energy of ocean waves and modelling the point absorber
Time series of wave elevation collected by the wave measurement buoy located at the Lysekil test site are used to describe the sea state at the test site in terms of energy scatter diagram in Paper I. The time series are further used to force the point absorber model in Paper IV, VI, XIII and XIV. Hind-cast wave data are used to describe the wave energy potential and the charac-teristic sea states of the Swedish west coast in Paper II. Expressions for wave energy transport properties for polychromatic surface gravity waves travel-ling in waters of finite depth are developed and compared to existing solu-tions that use the deep water approximation; see Paper VII, XI and XII. A point absorber model based on the Lysekil project WEC has been developed, see Papers VI and XIII, and is further extended to a Two-Body System point absorber (TBS); see Paper VI and XIV. The point absorber model is also used to estimate the biofouling effect on the WEC’s energy absorption in Paper IV. Hydrodynamic coefficients are provided in Paper VIII.
4.1. Wave climate and sea states
The wave climate and sea states are described in three different ways in this thesis.
4.1.1 Wave data measured at the Lysekil test site
A Datawell® Waverider wave measurement buoy located 50 m from the WECs’ continuously measures the sea state at the test site. Half hour long time series of the wave elevation , sampled with 2.56 Hz are collected. For each time series a power spectrum S according to (3.9) is calculated. The time series are used to force the WEC model and the power spectra are used to calculate different properties of the waves that describe the sea state.
41
4.1.2 Hindcast model wave data
The wave data used to describe the wave climate at the Swedish west coast in Paper II were purchased from and prepared by Fugro OCEANOR. For more detailed information of the wave data generation, see ref. [62]. The wave data consist of half hour time series for every sixth hour without inter-ruption during the period 1997 to 2004. The data are created by the com-bined use of wave measurement buoys, satellite altimeter measurements and wind driven hindcast models. The hindcast model used in this case is the WAM-model (WAve prediction Model) developed by the European Centre for Medium-range Weather Forecast [63]. Data from measurement buoys and satellites have been used to calibrate the WAM-model. For the five near-shore sites, where the depth is between 15–30 m, the SWAN model has been used which is prepared with information of the coastline and bathymetry [21]. The SWAN model can thereby account for the multiple effects when the wave feels the presence of the seabed. The location of site 9, see Figure 5.2, is chosen so as to coincide with the wave measurement buoy at the Lysekil test site. Comparison has then been made in Paper II between the SWAN-model data and the data from the Lysekil test site wave measurement buoy.
4.1.3 Empirical spectra
The empirical Bretschneider and Pierson-Moskowitz spectra are used in Paper VII to compare the energy transport’s depth distribution to a measured ocean wave spectrum.
4.2. Energy transport of ocean waves at finite depth
Based on existing theory for deep water waves, see Chapter 3.3, expressions have been derived for the energy transport as a function of depth and the total instantaneous transport’s development over time for polychromatic waves in waters of finite depth. The theory is limited to plane-parallel waves in an ideal fluid with linearized boundary conditions, thus deriving solutions to the Laplace equation as convolution integrals of the wave elevation in an Eulerian reference frame. Since we look at plane-parallel waves propagating in the x-direction and z being negative in the direction from the surface to the sea bed, the y-component of the transport vector j vanishes. The focus is on the difference between existing solutions for waves in the deep water regime and derived solutions for waves travelling over waters of finite depth. This is done in order to look at the magnitude of error made when using the deep
42
water solutions. With finite depth we consider waves travelling in waters of intermediate depths, due to the linear approach in this work. For truly shal-low water waves (h<0.05 ) the non-linear effects are too large for these solu-tions to be valid. See, e.g., Willebrand [20], where the contributions of the non-linear interactions were found to be small for ocean waves, but signifi-cant for shallow water waves.
4.2.1. Depth variation of the energy transport
From the energy transport vector (3.20), the time averaged energy transport zj through a vertical surface element can be defined as:
2/
2/
1 T
T
TwithdtxtT
zj . (4.1)
where T is the total time averaged over. Equation (4.1) is solved by differen-tiating the velocity potential (3.3) with respect to t and x, using the DFSBC (3.1) and the Fourier representation of the velocity potential ˆ (3.4). Finally, by shifting the integration variable to the wave number k and after some manipulation, the discrete depth distribution of the energy transport is written as:
0
2
2sinh
2sinh2
2cosh1
2cosh1
2dkS
hk
khkh
hk
hzkgzj . (4.2)
The measured time series of wave elevation from the Lysekil test site enters (4.2) via the linearized DFSBC and the spectrum in (4.2) is defined as
ˆˆTS where the asterix (*) denotes the complex conjugate. Equation (4.2) is compared to the deep water solution (3.18) for two meas-ured sea states in Paper VII. In the same paper, the solution’s (4.2) sensitiv-ity to the shape of the spectra is investigated by comparing results for meas-ured spectra to empirical Bretschneider (3.13) and Pierson-Moskowitz spec-tra (3.14).
4.2.2. Total instantaneous energy transport
To find an expression for the total instantaneous energy transport in waters of finite depth jtot(t) the approach is similar to the derivation of the energy transport’s depth variation above, starting by differentiating the velocity potential with respect to x and t. To get the total, the energy transport has to be vertically integrated to the sea bed z = -h. The latter is a bit more cumber-
43
some, where the explicit dependence of the hyperbolic functions is removed by means of trigonometric identities, and non-dimensional integration vari-ables are introduced by Froude scaling the wave number and angular fre-quency. The solution reads:
.,2
2
tdtdh
gt
h
gt
h
gUtttt
gtjtot
(4.3)
The vertically integrated energy transport as a function of time (4.3) is then given as a convolution integral, quadratic in the wave elevation where the integration kernel U reads:
.2
1, 212
22
1
22
21
2
2
1
121
2211 ddekk
kkU ii (4.4)
The integration kernel U is computed numerically by the fast Fourier trans-form and shown in Figure (4.1).
Fig. 4.1: A 3D plot of the integration kernel U( 1, 2) in (a) and a contour plot of log(|U( 1, 2)|) in (b). U is oscillatory in character with minimum value min(U) = -0.018 and has a (integrable) singularity at (0,0)
Equation (4.3) is used to investigate the magnitude of error made when using the integration limit z - i.e. the deep water solution (3.22), in Paper XI and XII. In Paper XI, (4.3) is used to study the solutions response to different vertical integration limits, i.e. water depths, for 10 sea states from the Lyse-kil test site.
Intuitively the transport of energy in a wave train is always positive or zero in the direction of the wave when looking at the wave from a fixed ref-erence point at the sea bed. This was also proven in ref. [18] for the deep
44
water solution (3.22). Whether or not this is the case for the finite depth so-lution (4.3) is investigated by numerical methods. This is done by finding the minimum value of (4.3) at t = 0 s by using (g = h = 1), integrating over t´, t´´ and by using the Hermite spectral method with Hermite-Weber functions:
)(!2
1)( 2/2
xHen
x nx
nn
n=0,1,2,3,... (4.5)
where Hn(x) is the n-th Hermite polynomial. The Hermite–Weber functions, which are orthonormal, i.e.:
mnnm dxxx )()( . (4.6)
form an orthonormal basis of L2(R). In addition, they are simultaneously eigenfunctions of the one dimensional Schrödinger harmonic oscillator op-erator and eigenfunctions of the unitary Fourier transform [64]. The wave elevation ˆ is approximated by expanding it in the subspace of L2(R) spanned by Nnn ,..,1,0, , where N is an integer, as:
N
nnnu
0
)(ˆ . (4.7)
Then the total energy transport at t = 0 s can be written:
.)()(
)2(0
21222
21
22
21
2
2
1
11
0 02
ddkk
kkAwith
Auuj
nmmn
N
mmnn
N
nmtot
(4.8)
The matrix Amn is calculated by numerical integration and the eigenvalues and eigenfunctions of this matrix are computed. If there exists at least one negative eigenvalue, the corresponding eigenfunction will make the total instantaneous energy transport negative at t = 0 s and this is investigated in Paper XII.
45
4.3 Point absorber model
The WEC’s ability to extract energy from the ocean wave depends on the geometry of the buoy, the mass of the moving parts together with the damp-ing from the generator and load. The work to model the WEC is carried out in a three step process: The geometry of the axisymmetric point absorbing buoy is defined and
meshed in the 3D CAD tool MultiSurf®10. The hydrodynamic coefficients for the point absorbing buoy are calcu-
lated for harmonic waves using the BEM-code WAMIT®11. The hydrodynamic coefficients are implemented in the point absorber
WEC model and the system’s response to measured time series of wave elevation are calculated and presented in the time domain.
4.3.1 Defining the geometry
The axisymmetric buoy is defined in the 3D CAD tool MultiSurf®, see Fig-ure 4.1. The created geometry file is prepared for direct implementation in WAMIT®. The geometry can be meshed or defined continuously by B-spline basis functions. In this thesis the geometries are described by B-spline basis functions. The order of the B-spline basis functions is set by MultiSurf® and in this thesis a third degree polynomial has been used. B-spline basis func-tions are described further in the following chapter.
Figure 4.1: A cylindrical buoy extracted from MultiSurf®.
10 www.aerohydro.com, retrieved 2011-08-15 11 www.wamit.com, retrieved 2011-08-15
46
4.3.2 Calculation of hydrodynamic coefficients
WAMIT® is a radiation/diffraction 3D panel program for linear analysis of the interaction between surface gravity waves and offshore structures. It has previously been used in wave power applications, where verification to both large-scale and small-scale experiments show good agreement [27, 65]. The linearized boundary value problem is solved using Green’s theorem by inte-grating the diffraction and radiation velocity potentials in closed surfaces at the body boundary, see chapter 3.4. Throughout this thesis the “higher-order method” has been used (chapter 6 in WAMIT® user manual, see Footnote 11). In the higher-order method the velocity potential around the body is described by a tensor product of B-spline basis functions giving continuous distribution of the potential. The body is defined by patches where the patches are divided into panels by the intersections (in B-spline terminology often referred to as knots) between the B-spline functions. The B-spline basis function is defined as:
.,11
uM
ijiij
M
j
VuUu (4.9)
In the higher-order method the accuracy depends on the order of the B-spline basis functions Ui(u) and Vj( ) and by their number of panels Mu and Mv, see Equation (4.9). The order is set when designing the body in MultiSurf®. Mu and Mv are the number of panels each patch will be divided into. Since this thesis uses simple axisymmetric geometries, a default value for panel subdi-vision provided by WAMIT® has been used. The default setting maximizes each panel size so it is proportionally sized to the body dimension and the wave length. The hydrodynamic coefficients extracted from WAMIT® in this thesis are the excitation force, added mass, radiation resistance and the radi-ated wave in the frequency domain and impulse response functions for exci-tation force and radiation impedance in the time domain. The water depth is set to infinite and the linearized surface boundary condition implies that no consideration has to be taken regarding the top of the geometry; the latter could otherwise lead to problems such as waves overtopping the geometry. The geometry can thereby be seen as being infinitely high. Convergence tests are carried out by the WAMIT® code and are further validated in ref. [65].
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4.3.3 Point absorber model with linear damping
A point absorber system is modeled as an oscillating mass, spring, and damper system. This is the lowest order system where the whole system is linear and rigidly coupled, see Figure 4.2.
Figure 4.2: Conceptual sketch of the point absorber system.
As a point absorber in this case we consider a partially submerged vertical cylinder with radius a. The dotted sphere in Figure 4.2 represents the sub-merged sphere that is used to model the TBS in Paper VI. The connection between the buoy and the linear generator is modeled as a stiff rod. Thus, we restrict the buoy motion from six degrees of freedom to heave only. The generator is modeled as a linear damper, i.e. the generator force Fgen is di-rectly proportional to the velocity z of the system with the constant of pro-portionality being the damping coefficient . This configuration also models the damping as being 100 % active during the whole stroke length meaning that the translator/stator is active during the whole stroke length. This linear mechanical damper is used to simulate a purely resistive load. A spring with spring constant ks acts as a restoring mechanical force Fmech proportional to the system’s vertical position z. The Lysekil project WEC has previously been modeled with a linear damping using the same approach, see [66].
The forces that make the cylinder move from its equilibrium position z = 0 m can be described by the state equation:
.hregenmech FFFFFzm (4.10)
48
here m is the total mass of the moving parts and z is the vertical acceleration of the rigid system. The excitation force in the frequency domain is de-scribed by:
.ˆˆˆee fF (4.11)
where the caret ( ˆ ) denotes a Fourier transform, ˆ is the amplitude of the undisturbed incident wave. ef is a transfer function describing the relation between the excitation force and the incident wave. The force associated with radiation is proportional to the vertical speed z of the rigid system and is given in the frequency domain as:
.ˆˆˆˆ zmiRF ar (4.12)
where R is the radiation resistance and am is the added mass. The transfer function for the excitation force ef together with the radiation resistance R and added mass am are calculated using WAMIT®. A transfer function H describing the point absorbers amplitude response z to the incident wave ˆ can be formulated as:
.ˆ
ˆ22
sa
e
kagRimm
fH (4.13)
The first term in the denominator in (4.13) contains information of the inertia of the system; both the static mass m and the dynamic added mass ma. For the TBS in Paper VI these terms include the mass and added mass from the submerged sphere. The second term in the denominator contains information about the damping on the system from the generator and from the point absorbing buoy’s wave radiation R, where a linear damping coefficient is used in Paper IV and VI, representing a direct-driven linear generator con-nected to a resistive load. The two last terms are the hydrostatic and spring coefficient respectively, which will be in equilibrium in calm sea to keep a pre-tension in the system.
The power consumed by the linear damper, which models the generator, is dependent on the damping from the generator and the vertical speed squared according to:
.1
0
2T
dtzT
P (4.14)
49
Here P is the average power over the time interval T. Optimal damping, or optimal amplitude response, for power production is studied in Paper VI. The system’s vertical position z in the time domain is found by taking the convolution of the impulse response function H(t) with the amplitude of the incident wave (t), where the impulse response function H(t) is the inverse Fourier transform of (4.13). A power capture ratio Pratio can be defined as the ratio of the absorbed power by the damper P and the energy transport dwtotj , subjected to the characteristic length of the buoy as:
.2 ,dwtot
ratio ja
PP (4.15)
where dwtotj , is the energy transport of the incident wave as given in (3.19). Results for the TBS with linear resistive load are presented in Paper VI and the excitation force in both heave and surge has been used to optimize the WEC’s gravity-based foundation in Paper IX. The response amplitude op-erator (4.13) is also used to estimate the biofouling effect on the energy ab-sorption in Paper IV.
4.3.4 TBS point absorber model with non-linear damping
Based on the conceptual point absorber system with two bodies connected to a direct-driven linear generator in Paper VI, the model is extended by treat-ing the three objects, surface buoy, submerged body and translator inde-pendently as constrained by a flexible line. The linear damping coefficient in Paper VI is extended to an electromagnetic non-linear electric circuit. Since the damping is velocity dependent, the simulation is carried out in the time domain.
The TBS WEC consists of a semi-submerged vertical cylinder connected by a flexible line to the lower submerged body with neutral buoyancy, see Figure 4.3. A sphere has been chosen as the geometry for the submerged body in this study to minimize the drag force. The lower body is connected via a flexible 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 there the positions are labeled y1 respectively y2. The connection between them and between the lower body and the translator is modeled as a very stiff spring kl. The genera-tor is a three-phase direct-driven linear generator with Nd-Fe-B type perma-nent magnets on the octagonal translator whose vertical position is labeled yt. The stroke length of the translator Lst is increased from the conventional 1.7 m to 4 m to prepare for the increased amplitude response. The generator is also equipped with upper and lower end stop springs ke to protect it from
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mechanical shock loads in waves higher than the design condition. The gen-erator is connected to a diode rectifier with a DC-link capacitor and a 2.9 km sub-sea cable transmits power to a resistive load on shore.
Figure 4.3: Mechanical components of the conceptual TBS WEC.
The hydrodynamic parameters are calculated using WAMIT® and imple-mented in a Simulink® model where the modelling can be divided into the following four blocks: Surface buoy model Submerged body model Translator model
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Generator/load model
The Simulink® model is described in more detail in Paper XIII. This coupled hydrodynamic, mechanic and electromagnetic TBS WEC model is described by a system of coupled non-linear second order differential equations:
.1,11111,111, lea FtygAtythttftymm (4.16)
.2,222,222, lea Ftythttftymm (4.17)
.,,2, leueemltt FFtFFtym (4.18)
here the index (1, 2) represents the upper and lower body respectively and (·,··) denotes velocity and acceleration. The mechanical forces Fl,(1,2) and Fe,(u,l) are the upper and lower line spring force and end stop spring force, see Figure 4.3. The excitation force is given by a convolution of the wave eleva-tion (t) and the non-causal impulse response function fe(t):
ttftF ee . (4.19)
Since the hydrodynamic reaction force is a direct consequence of the buoys motion it is causal and the Kramers-Kronig relation implies that the radiation matrix R( ) and added mass matrix ma( ) are mutually related. The radia-tion impedance h(t) in the time domain is given by the inverse Fourier trans-form of the radiation matrix:
0
.cos2
dtRth (4.20)
and the added mass matrix can be treated explicitly as the constant
aa mm . The hydrodynamic reaction force can be written as a convolu-tion product with the buoy’s vertical velocity ty according to:
.tythtFr (4.21)
The hydrodynamic parameters are calculated individually for each of the two bodies and for the submerged body they are extracted for two depths, 40 m (TBS40) and 10 m (TBS10) below the surface. The conversion from me-chanical to electrical energy is expressed by the electromagnetic damping force Fem. The electrical real power PR is taken to calculate the reaction force:
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.ty
tPtF
t
Rem
(4.22)
.1
0
T
ggR dttiteT
tP (4.23)
Here eg is the voltage at the generator terminal and ig the current in the gen-erator. From (4.23) a power capture ratio is defined as in (4.15). When the translator moves in and out of the stator its contribution to the power produc-tion changes and that is given by the active area of the stator Aact. This posi-tion dependent relation is given as:
elseylll
llyif
llyif
A
tsts
stt
stt
act
,2
112
1,1
2
1,0
. (4.24)
where lt and ls are the length of the translator and stator respectively. This relation (4.24) is used in calculating the generated voltage. The three-phase electric circuit, see Figure 4.4, consists of an uncontrolled rectifier with a DC link capacitor, a sub-sea cable and a resistive load. Experimental results from offshore experiment with the Lysekil WEC using a similar damping configu-ration can be found in ref. [50].
Figure 4.4: The simplified three-phase electric circuit consisting of the linear gen-erator, diode rectifier, sub-sea cable and resistive load.
The damping from the electrical system is controlled by the resistive values of the load that sets the translator velocity in Paper XIV. The optimal damp-ing that gives the highest power PR is found by empirical methods. 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, this study is intended to compare the behavior of
53
the TBS to the original system without the additional submerged body. Thus, losses in the sea cable and load are excluded and the presented absorption values in Paper XIV are on the generator terminal.
The electric circuit is further extended with the resistive load exchanged by a grid scheme, see Figure 4.5.
Figure 4.5: Circuit diagram of the grid scheme.
The grid scheme uses a three-phase diode rectifier and a DC link capacitor. The diode rectifier is rated for a maximum voltage rating of 1800 V, a cur-rent rating of 600 A and the DC link capacitor value is 12 F. The values are chosen so as to be consistent with the actual values at the Lysekil test site. The inverter and the step up transformer are modeled as a mathematical equivalent model. It is implemented in this manner to avoid long simulation times due to the high frequency PWM switching. Inverter and transformer losses are neglected in this model, as are any additional filter requirements for the grid connection. The presented power values in Figure 5.25 are on the DC link and the damping is set by a fixed DC voltage VDC link = 60 V.
The main parameters of the TBS presented in Paper XIV are given in Ta-ble 4.1.
Table 4.1: Main parameters for the TBS. 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 14000 kg kl Spring constant for line 450 kN/m ke Spring constant for end stop 215 kN/m Rg Generator resistance 0.8 /phase Rload Load resistance 2.5 (Paper XIV) Rsc Sub-sea cable resistance 0.5 /phase VDC link Voltage on the DC link 60 V (Chapter 5.4.2.1)
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The power capture ratio (4.15) on the generator terminal for 50 measured sea states is presented in Paper XIV. The same paper also presents the buoy, submerged body and translator position together with the translator velocity. Chapter 5.4.2.1 presents results of the TBS40 and SBS connected to the grid, including power capture ratio (4.15) based on the power (4.23) on the DC-link for 50 measured sea states, force on the translator and generator line current. Hydrodynamic coefficients in the time domain are provided in Pa-pers VIII and XIII.
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5. Results and discussion
5.1. Wave climate at the Swedish west coast
In Paper II it is found that the average energy transport in offshore Skagerrak is 5.2 kW/m and that at the nearshore sites it has decreased down to 2.8 kW/m. The dominating directions for the waves in Skagerrak are south-westerly from the North Sea. This explains why the energy transport in Kat-tegat is 2.4 kW/m; the most energetic waves created on the North Sea are blocked by Denmark and do not reach Kattegat. The dominating wind direc-tion for the two southernmost sites in Skagerrak is westerly. The discrepancy from the other Skagerrak sites may also be explained by the presence of Denmark; the longer waves feel the presence of the shallow bottom and get refracted towards Denmark, i.e. towards the west, see Figure 5.1.
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Figure 5.1: Locations of the 13 studied sites. The wave roses show the distributions in time of the direction for the waves. Site 9 corresponds to the test site for the Lysekil project, site 5 is presented in Figure 5.2 and site 6 are used to calculate the annual production for TBS40 and SBS in Table 5.1.
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The wave climate gives the design parameters for the power level that the WEC should be designed for. It is shown in Paper I that during the first three months of operation, L1 had to handle power levels 28 times higher than the average. Over the 8 years of measurement in Paper II, the most powerful energy transport reaches a half-hour average of 250 kW/m, which is 45 times higher than the average. Calculations in Paper II estimate that the statistical 100 year wave in offshore Skagerrak can measure up to 12.6 m, which is significantly lower than the estimation of 18.3 m made in ref. [14]. A com-bined energy and scatter diagram such as the one in Figure 5.2 can give a hint of the sea states the WEC should be designed for.
Figure 5.2: Combined scatter and energy diagram for site 5. Colors show annual energy transport per meter wave front [kWh/(m*year)]. Isolines present the energy transport in accordance with Equation (3.19). Results are based on 6 hour averages covering the years 1997-2004. The region indicated by the solid line encloses 44 % of the total annual energy.
It is shown in Figure 5.2 that in offshore Skagerrak, 44 % of the annual en-ergy transport is bounded by a distinct region between 4 - 7 s in energy pe-riod and 1 - 3 m in significant wave height. The mentioned parameters point out one of the design challenges; a wave energy converter cannot be de-signed to have an installed power corresponding to the 100-year wave, but must nevertheless be constructed to withstand these power levels.
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5.3. Energy transport of ocean waves at finite depth
The validity of using the deep water approximation in waters of intermediate depth (10-50 m) is investigated with the use of derived expressions for the depth variation and total instantaneous energy transport. This is done by using a typical Swedish west coast sea state collected by the wave measure-ment buoy at the Lysekil test site. The modal parameters Hs = 2.43 m and Te = 6.25 s extracted from the power spectrum describe the time series of wave elevation.
5.3.1. Depth variation of energy transport
Around 50 % of the energy transport is transported in the upper 5 m of the water column as can be seen in Figure 5.3. Furthermore, the deep water solu-tion gives a too low value for the energy transport compared to the finite depth theory. There is discrepancy of about 6 % in this case, see Figure 5.3. This discrepancy is present in the whole water column and increases with depth.
Figure 5.3: The dashed line show the time averaged energy transport as a function of depth for finite depth zj and the solid line presents the results for deep water the-ory zjdw
. The water depth h for the calculations is 25 m and the total transport for finite depth theory is 18.2 kW/m and 17.1 kW/m for deep water theory.
The same comparison is made in Paper VII but for a weaker sea state (Hs = 1.13 m, Te = 4.7 s). As one might suspect, there is no major discrepancy be-tween the finite and deep water theory in that case, since most of the waves in that wave record have wavelengths that obey the deep water criteria. The
59
energy depth profiles sensitivity to the shape of the power spectrum is inves-tigated in Paper VII where solutions to the measured spectrum are compared to the Bretschneider and Pierson-Moskowitz spectrum with significant dif-ference in the spectral shape. The solutions agreed and showed no sensitivity to the spectral shape.
The feature of higher energy transport for finite depth theory can be ex-plained by a higher group velocity, since the transport is proportional to an increase in the group velocity, see (3.15). This is well illustrated by compar-ing the group velocity for finite depth theory and deep water theory; see Figure 5.4(a).
Figure 5.4: The group velocity vg for finite depth theory at 25 m water depth (dashed) vs. the group velocity for deep water theory vg,dw (solid) in (a), and the discrete power spectrum that is used for the calculations in (b). The power spectrum S in (b) has a significant wave height Hs of 2.43 m and an energy period Te of 6.25 s.
When using a water depth of 25 m, the finite depth theory gives a higher group velocity for waves within the range of 0.4 < < ~1.2 rad/s. It thus covers considerable amounts of the used wave spectrum, see Figure 5.4(b), and would cover more of a more powerful spectrum, such as could be ex-pected at a more exposed coastal area than the Swedish west coast.
5.3.2. Total instantaneous energy transport
The discrepancy between the finite and the deep water theory is further in-vestigated by looking at the total instantaneous energy transport for waves in waters of finite depth. This is done in Figure 5.5(b), where (3.22) and (4.3) have been used to show the total instantaneous energy transport for the time series of wave elevation used in Figure 5.3.
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Figure 5.5: An excerpt of the time series of the wave elevation used for the calcu-lations in (a) and the corresponding total instantaneous energy transport for the finite depth theory jtot (red) versus the deep water theory jtot,dw (black), in (b). The water depth h for the finite depth theory is set to 25 m.
Figure 5.5(b) shows the dramatic behavior of wave energy transport. Follow-ing the wave profile, the energy transport increases from zero to over 100 kW/m in just a few seconds. The feature of higher transports for the finite depth theory already visible in Figure 5.3 is clearly seen for the total instan-taneous transport as well. This is most evident for the highest peaks, where for example at t = 662 s the finite depth theory gives a transport of 80 kW/m compared to the deep water theory’s 70 kW/m, a difference of about 12 %. For the smaller peaks, the feature of higher transports for the finite depth solution disappears. This is expected since the wavelength to depth ratio decreases and therefore these waves obey the deep water criteria better.
In Paper XI the total instantaneous energy transport for different water depths and infinite depth are compared for a sea state with Hs = 1.15 m, Te = 5.6 s.
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Figure 5.6: Total instantaneous energy transport jtot for different water depths h. For comparison; the black line shows the energy transport calculated using the infinite depth expression jtot,dw.
A water depth of 12 m clearly gives the highest transport while for deeper depths the discrepancy decreases and the result for 60 m and infinite depth coincides. A wave with a period of 5.6 s has a wave length of approximately 50 m, thus when propagating at a water depth of 60 m it obeys the deep wa-ter criteria. The highest energy transport in Figure 5.6 is 25 kW/m while the corresponding transport for the deep water regime is 21 kW/m, a difference of about 16 %. Another interesting feature from Figure 5.6 is that both maxima and minima, approximately, coincide for finite and deep water the-ory.
The minimum value of the double integral in (4.3) at t = 0 s is studied by using the Hermite spectral method; see (4.8) and Paper XII. This is done in order to see if the total instantaneous energy transport always is positive in the direction of the propagating wave. The smallest eigenvalue is min = -7.0·10-4, i.e. there is at least one negative eigenvalue and consequently the corresponding eigenfunction makes the total energy transport negative at t = 0 s. This leads to the surprising conclusion that the direction of the total in-stantaneous energy transport can be opposite to the direction that the waves travel. The normalized waveform that gives the highest negative energy transport at t = 0 s is shown in Figure 5.7(a) together with the corresponding total instantaneous energy transport in Figures 5.7(b, c).
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Figure 5.7: The normalized wave form in (a) that gives the highest negative total instantaneous energy transport jtot in (b), with an enlargement of the negative part of the energy transport around t = 0 s in (c). In this figure the water depth h is set to 25 m.
In Figure 5.7(c) we can actually see that the total instantaneous energy trans-port jtot has a very small negative value, -11 W, at t = 0 s. Thus, for short durations, the total instantaneous energy transport can have a direction that is opposite the direction of the propagating waves in a frame fixed to the sea-bed.
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5.4. TBS point absorber model
By adding a submerged body, the resonance frequency is shifted towards lower frequencies as can be seen in Figure 5.8. This is the fundamental idea with the TBS since the device can be tuned to the actual sea state at the site.
Figure 5.8: The un-damped response amplitude operator H for TBS40 (solid line) and for a single 6 m diameter cylinder (dashed line). This TBS uses a 6 m diameter cylindrical buoy and a 4.9 m diameter submerged sphere.
By using a 6 m cylindrical buoy and a 4.9 m submerged sphere, the un-damped system is tuned to have an optimum amplitude response at frequen-cies that coincide with the Swedish west coast wave climate, as can be seen in Figure 5.8. With this solution, the criteria for a high amplitude response are met since the surface buoy has the bottom surface close to the sea surface and thus use as much as possible of the excitation force while the phase re-sponse criteria are met by the submerged body that provide the desired iner-tia in terms of free added mass. The phase response will be shifted somewhat towards higher frequencies when applying damping as we shall see later on.
5.4.1. Linear damping
It is shown in Paper VI that when using a 3 m radius cylinder the power cap-ture ratio is about 32 % at optimal damping; while for the TBS with a 3 m
64
radius sphere added to the system, the power capture ratio is increased to 60 %, see Figure 5.9.
Figure 5.9: Power capture ratio Pratio vs. energy period Te for 34 different sea states at optimal damping. Time series of wave elevation measured at the Lysekil test site are used as sea states. The lines are intended as a guide for the eye.
After the peak in absorption for the TBS with a 3 m radius sphere at 4.5 s, the power capture ratio decreases with the increasing energy period. This is a desired pattern since it will even out the power fluctuation and also leads to a relative decrease in the load on the generator end stop. With a 4 m radius sphere the TBS has an optimal amplitude response at ~6 s. The peak is also much higher and narrower than for the 3 m sphere; see Figure 2 in Paper VI. From Figure 5.9 we can see that when increasing the radius of the sphere it affects the power capture ratio. The results for the 4 m radius sphere are much more scattered since a more peaked resonance bandwidth makes the system more sensitive to the shape of the wave spectrum, see Figure 5.9. This feature is studied in more detail in the next chapter.
According to (4.14), the absorbed power is dependent on the damping co-efficient times the translator speed squared. Hence, an increased translator speed gives a decrease in optimal damping for the same power. This can be seen in Figure 5.10. Since the TBS gets increased translator speed due to its enhanced amplitude response; the damping for the TBS is lower than for the single cylinder for the same power capture ratio.
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Figure 5.10: Power capture ratio Pratio as a function of the damping coefficient . The sea state is a measured time series of wave elevation with Hs =0.8 m and Te =4.5 s.
The single cylinder has an optimal power capture ratio at a damping coeffi-cient of about 160 kNs/m. For the TBS the optimal damping coefficient is decreased to 60 kNs/m, as can be seen in Figure 5.10. This behavior opens up the possibility to construct smaller units with the TBS while keeping the same nominal power.
5.4.2. Non-linear damping
In Paper XIV, two configurations for the TBS are presented, TBS40 with the submerged body at 40 m water depth and TBS10 with the submerged body at 10 m water depth. The submerged body is placed at different water depths in order to investigate the depth effect as will be discussed further on. The results for the TBS are presented for 50 measured time series of wave eleva-tion that are collected at the Lysekil test site. The TBS40 and TBS10 are compared to the conventional system with one body, here called the Single Body System (SBS).
The 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 Figure 5.11. 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 better view of the distribution of power capture ratio over energy period in Figure 5.11. The curve fit for the TBS40 shows a peak be-tween 4.5 and 4.8 s with about 58 % power capture ratio.
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Figure 5.11: Power capture ratio Prat vs. energy period Te for TBS40 (black), TBS10 (red) and for SBS (blue). The result for wave record 15 (green cover) and 17 (yellow cover) that are presented by their corresponding wave power spectrum in Figures 5.12(a) and 5.12(b) are highlighted for TBS40. The lines are intended as guides for the eye.
Note that the wave period for highest absorption is shifted from 5.2 s in the harmonic un-damped analysis, Figure 5.8, to around 4.7 s for the TBS40 subjected to a velocity dependent damping, Figure 5.11. Thus, when the system is subjected to high damping, the peak frequency response is shifted towards higher frequencies. For TBS10 the depth effect is clearly visible. The peak for the curve fit reach a power capture ratio of 37 %, see the red line in Figure 5.11, and wave record 15 yields a power capture ratio of 50 %. The SBS shows a steady decrease from 20 % to a 5 % power capture ratio. For the SBS, no resonant behavior is visible since the resonance peak is out-side the wave period range. The resonant behavior for the TBS compared to the SBS is viewed more in detail in Figures 5.12 and 5.13, where the vertical position of the surface buoy, submerged body and translator are plotted.
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Figure 5.12: Wave power spectrum for (a) wave record 15, (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 s and Hs = 0.95 m. The corresponding surface buoy position for TBS40 is inserted in the figures with surface buoy position y1 (red) and wave amplitude (blue).
For a point absorber in resonance, the buoy’s vertical velocity is in phase with the excitation force and this implies that the buoy is phase-shifted -90 degrees relative to the wave elevation. Such resonant behavior can be seen for the TBS40 in Figure 5.12(a). The system does not respond to smaller waves since the high inertia in the oscillation overpowers the contribution from the smallest waves and the system’s oscillation decays 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 com-pletely submerged due to its large oscillation. By comparing the difference in divergence among the data points in Figure 5.11 for TBS and SBS, one
68
can see the resonant TBS’s sensitivity to the shape of the wave power spec-trum. This is especially evident when comparing the result from wave record 15 and 17 for TBS40, where the difference in power capture ratio is about 34 percentage units, although the difference in energy period between wave record 15 and 17 is only 0.2 s. Figure 5.12(a) and 5.12(b) show that wave power spectrum 15 is more concentrated around the resonance frequency while wave power spectrum 17 is broader. Thus, wave record 15 contains more waves with the desired frequency leading to the clear resonant behav-ior with amplified buoy amplitude as can be seen in Figure 5.12(a), but not in Figure 5.12(b). This points out the reason for not using an even larger submerged body as presented in Figure 5.9, since it will only lead to higher divergence among the data points, leading to a decrease in overall power production.
With a translator mass of 14000 kg, a more or less stiff coupling between the three bodies can be achieved where the surface buoy, lower body and translator follow each other as can be seen in Figure 5.13, except for a small deviation at the turning points where the lower body deviates slightly from the surface buoy and the translator. This is due to the flexibility in the line.
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Figure 5.13: TBS40 in (a), TBS10 in (b) and SBS in (c). The figure shows the sur-face buoy position y1 (red), lower body position y2 (black) and translator position yt (green) vs. the wave amplitude (blue) for wave record 15. The translator follows the surface buoy in (a) and (b) and these curves can not be distinguished.
The system’s behavior is similar for the TBS40 and TBS10 as can be seen in Figures 5.13(a) and 5.13(b), except for a lower amplitude response for TBS10, visualizing the depth effect that yields a rather large difference in overall power absorption between TBS40 and TBS10, as can be seen in Fig-ure 5.11. For the SBS case there is no resonant behavior at all as can be seen in Figure 5.13(c). Instead, the surface buoy follows the wave and the clear
70
phase difference between the wave and surface buoy that can be seen in Fig-ures 5.13(a) and 5.13(b), which corresponds to a resonant behavior are no longer present. It can also be seen that the SBS follows even the smallest wave.
The depth effect that arises from the difference in distance between the submerged body and buoy is clearly visible as a difference in amplitude re-sponse in Figures 5.13(a) and 5.13(b) and a significant decrease in converted power in Figure 5.11. This means that in order to achieve good power ab-sorption for the TBS, the two bodies have to be placed a sufficient distance apart. This effect is investigated in Paper XIV in terms of the radiated wave from the buoy and the excitation and radiation force on the two bodies. It was stated in ref. [29] that the resonant behavior has to be decoupled from the radiation properties of the surface buoy. In Figure 6 in Paper XIV the radiated wave from the surface buoy is plotted in for TBS40, TBS10 and SBS, where we have also included TBS5 with the lower body at 5 m depth. The radiated wave for SBS and TBS40 coincide while the TBS10 and TBS5 radiated a wave with lower amplitude. If the submerged body is placed too close to the buoy it will be affected by the excitation force from the wave that is out of phase with the excitation force on the surface buoy and these forces will interfere destructively. The excitation force on the submerged body at 10 m depth are about 7-8 % of the excitation force on the surface buoy and in an opposite direction, see Figure 7(a) in Paper XIV. With the submerged body placed at 40 m depth, the excitation force exerted is very small. The significant decrease in power capture ratio with the submerged 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, together with the 180 degree phase-shifted exci-tation force on the submerged body.
5.4.2.1 Grid connection
With a grid scheme included in the TBS point absorber model, a more realis-tic damping characteristic is applied, see figure 4.5. The average force on the translator per absorbed power on the DC-link gives an indication of the wear of the components in the WEC-system. The average force on the translator per absorbed power for TBS40 and SBS is plotted for 50 sea states in figure 5.14.
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Figure 5.14: Average force on the translator per average absorbed power PR vs. total time averaged wave energy transport
dwtotj ,. TBS40 (black) and SBS (red). The lines
are intended as a guide for the eye. (Unpublished results)
The TBS40 has the highest power capture ratio at an energy period of around 4.8 s, see figure 5.11, and this corresponds roughly to a sea state of around 5 kW/m. In that area the force per absorbed power on the TBS40 is lower than for the SBS as we can see in Figure 5.14. For more energetic sea states, out-side the resonance bandwidth, the force on the TBS40 increases since the power capture ratio decreases. The overall picture is that the forces increase for energetic sea states when using the TBS, but the forces on the TBS per absorbed power are lower than on the SBS for the design sea states, that is, the sea state that the device most often is subjected to. The high forces at the most energetic sea states are nevertheless a challenge that will set higher demands on the mechanical construction. But during normal operation the wear on the equipment are less when using the TBS, since the device has to be designed to withstand the maximum forces.
The power capture ratio calculated from the absorbed power on the DC-link for TBS40 and SBS is plotted for 50 sea states in Figure 5.15.
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Figure 5.15: Power capture ratio Pratio vs. total time averaged wave energy transport
dwtotj ,. TBS40 (black) and SBS (red). The lines are intended as a guide for the eye.
(Unpublished results)
With TBS40 connected to the grid, the absorbed power on the DC-link is still substantially higher than for the SBS as can be seen in Figure 5.15. For the TBS40, the power capture ratio decreases from a maximum value of 80 % for 4 kW/m energy transport down to 20 % for 60 kW/m. For the most energetic sea states the difference between TBS and SBS is small. This de-sired phenomenon indicates a smoother output power which means that the need for a large buffering capacity can be reduced. Since the voltage and current levels follow the produced power, a better utility of the electrical components can be expected and this is shown in Figure 5.16, where the generator line current ig is plotted for TBS40 and SBS.
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Figure 5.16: Absolute value of the generator line current ig for TBS40 (black) and SBS (red). Results from a sea state with an energy period Te = 6.46 s and a signifi-cant wave height Hs = 2.52 m. (Unpublished results)
The current levels in the generator for the TBS40 are higher than for the SBS as can be seen in Figure 5.16. However, the number of samples with a cur-rent above 70 A, which is the maximum current for the SBS, are relatively few. Whereas the current levels are more evenly distributed for the TBS40, indicating that the electrical components are better utilized since they have to be dimensioned to handle the maximum current levels.
The non-linear viscous or drag force experienced by the submerged body on the TBS can be approximated by DD CyAF 2
225.0 with the drag coeffi-cient CD = 0.42 for a sphere. For a typical 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 overestimate since for a proper value of the drag force, the velocity 2y 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 ref. [67]. There they found that the losses were 4 % of the absorbed power for a CD of 1. Thus, in this model with a CD of 0.42 the losses should be small and the viscous forces are therefore excluded from the model.
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By using the power curve in Figure 5.11 and the wave energy scatter dia-gram for site 6, see Figure 4 in Paper II, an estimate can be made of the dif-ference in annual power production for the TBS40 and SBS, see Table 5.1.
Table 5.1: Calculated annual power production for TBS40 and SBS. The power production is based on the power capture ratio Pratio in Figure 5.11, and the sea state Hs, Te and their number of entries are based on the wave energy scatter diagram from site 6 in Paper II. (Unpublished results)
Te [s] Pratio TBS40 [%] Pratio SBS [%] Hs [m] 0-2 2-3 3-4 4-5
4-5 55 20 2178 0 0 0
5-6 40 15 1348 213 0 0
6-7 28 10 376 360 69 0
7-8 18 8 157 83 94 22
Annual power TBS40 292 MWh
Annual power SBS 111 MWh
Table 5.1 shows that by using the TBS40, the annual power production can be almost threefold compared to the SBS. The actual numbers should be used carefully since it is a rather rough estimate but as a comparison it gives a hint of what could be expected. It is also interesting to note that this three-fold increase coincides well with the estimated threefold increase in annual power production using latching in ref. [43].
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6. Conclusions
The average energy transport in offshore Skagerrak is found to be 5.2 kW/m, and has decreased to 2.8 kW/m in nearshore Skagerrak. In off-shore Skagerrak, 44 % of the annual energy transport is found in a distinct area between 4-7 s in energy period and 1-3 m in significant wave height.
Solutions for the wave energy transport’s depth distribution and its total
instantaneous energy transport have been developed for polychromatic waves travelling in waters of finite depth. Comparisons with solutions us-ing the deep water approximation show a higher transport for the finite depth solution in the whole water column and as much as 16 % higher in the total instantaneous transport. For weaker sea states the solutions agree asymptotically. It is shown that the energy depth profiles exhibit little sensitivity to the shape of the spectrum. The energy transport is also shown to be able to have a direction that is opposite to the direction of the wave propagation.
A TBS WEC model tuned to be in resonance in Swedish west coast sea
states has been developed based on the Lysekil WEC. Forced by meas-ured sea states from the Lysekil test site, the TBS model gives a signifi-cant increase in power production for both linear and non-linear load compared to a conventional point absorber system. Since the TBS results in increased translator speed due to resonance; the required damping for the TBS is lower than for the SBS for the same power capture ratio. Due to the TBS’s resonant nature, it is shown to be sensitive to the shape of the wave spectrum it is exposed to. Forces per absorbed power at design sea states are lower for the TBS than the SBS, while the picture is the op-posite for more energetic sea states. Power capture ratio for the TBS de-creases from the top values at design sea states to almost coincide with the SBS at the most energetic sea states. It can also be seen that the cur-rent in the generator are more evenly distributed for the TBS. These fea-tures indicate a better utilization of the equipment and also help with smoothing the output power and thus decreasing the need for a large buff-ering capacity.
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7. Suggestions for future work
The work in this thesis only account for the operation of a single WEC. This should be extended to simulations of coupled multiple WEC’s, in-cluding both hydrodynamic as well as electric coupling between different units. This is required in order to estimate the production from a future farm of WEC’s and to study the power smoothing effects using several interconnected devices. This should also include array studies of the TBS, which might lead to even better power smoothing effects.
Now when the TBS has been simulated with a realistic grid connection,
next step is to carry out full scale experimental test to verify this model, where insights can be drawn on e.g. the viscous losses on the submerged body and the higher extreme forces that arise when the translator hits the end stop of the generator.
Extend the derived expressions for the total instantaneous energy trans-
port over finite depth to include a sloping sea bed, i.e. h(x).
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8. Summary of Papers
Paper I Offshore experiments on a direct-driven Wave Energy Converter
This paper mainly presents experimental results based on data obtained from the first three months of the installed WEC (L1) at the research test site out-side Lysekil. The results are in terms of output voltage and absorbed energy. Results are obtained for three load values and it is found that a resistive load of 4.9 gives the highest energy absorption, independent of the sea state. It also gives a description of the wave climate at the test site at the time of op-eration. A short description of the experimental set-up is included.
The author was responsible for writing the paper. The author presented the peer-reviewed paper at the 7th European Wave and Tidal Energy Conference, Porto, Portugal, (2007) Paper II Wave climate off the Swedish west coast
This paper summarizes eight years (1997-2004) of wave data to thoroughly describe and discuss the wave climate of the Swedish west coast. The data consist of 13 sites offshore and nearshore, from the southern part of Kattegat to the northernmost part of Skagerrak. The data are derived from the hind-cast models WAM and SWAN, and verified at one site by Uppsala Universi-ties’ wave measurement buoy at the research test site outside Lysekil. Large losses in energy transport are observed between offshore and nearshore sites. The average energy transport at offshore Skagerrak is found to be 5.2 kW/m, whilst for the nearshore site the transport is 2.8 kW/m. The average energy transport in Kattegat is measured to 2.4 kW/m. The statistical hundred year wave is calculated to be up to 12.6 m in offshore Skagerrak.
The author is responsible for the literature review for the paper which is included in the introduction. The author also participated in writing the dis-cussion. Renewable Energy 34(6), (2009)
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Paper III The Lysekil Wave Power Project: Status Update
The paper gives a status update of the Lysekil project together with some previously published results. The activity within most of the research areas is described briefly and a hint of the upcoming activity is given. Absorbed power as a function of sea states is presented both for the first operation with resistive dump load and the latter operation period when the electrical sys-tem was prepared for voltage rectification.
The author is included in the paper as having participated in many parts of the Lysekil project. The paper was presented by Simon Tyrberg at the 10th World Renewable Energy Conference in Glasgow, UK, (2008) Paper IV Artificial reef effect and fouling impacts on offshore wave power foun-dations and buoys – a pilot study
This paper describes a study of the environmental risks with offshore wave energy installations and the potentially positive effects on the marine envi-ronment. The study area is the research test site outside Lysekil. The study investigates the colonization of foundations by invertebrates and fish as well as fouling assemblages on buoys. The influence of complexity (holes) and surface orientation of the foundations on colonization and species composi-tion during three years of submergence are examined. The biofouling effects on the energy absorption of the buoy are examined by means of a hydrody-namic model, based on experimental results of fouling assemblages on mark-ing buoys. The results show that it is possible to develop and design wave power foundations, to cater for crabs and lobsters, and possibly enhance potentially positive effects on biomass for these species. Simulations indi-cate that biofouling has no significant effect on the energy absorption of the buoy.
The author is responsible for the hydrodynamic simulations and the part in the paper describing this. Estuarine, Coastal and Shelf Science 82(3), (2009) Paper V Catch the Wave to Electricity: The Conversion of Wave Motions to Electricity Using a Grid-Oriented Approach
This paper gives an overview of the wave energy research that is carried out by the Swedish Centre for Renewable Electric Energy Conversion at Upp-sala University, from the first fundamental ideas to the planned wave energy park. The paper emphasizes the grid-oriented approach of the conversion
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from wave energy to electricity. The general motto for the Lysekil project, that mechanical complexity should be kept to a minimum is pointed out. The three-step experimental verification of the theoretical simulation is presented together with the first results from the offshore operation. The results show excellent agreement between the predicted and the measured power absorp-tion for a 4.9 resistive dump load.
The author participated in the writing of the paper and participated in many parts of the Lysekil project. IEEE Power & Energy Magazine 7(1), (2009)
Paper VI Wave Energy Converter with enhanced amplitude response at frequen-cies coinciding with Swedish west coast sea states by use of supplemen-tary submerged body
This paper presents a conceptual idea of a WEC with adjustable inertia in the form of a supplementary object submerged below the motions of the waves. The two-body system is based on and developed with the Lysekil project in mind and thus tuned for the sea state at the Lysekil test site. The theoretical model is based on potential linear wave theory and the power capture ratio are studied for real ocean wave data collected at the test site. The two-body system are found to get enhanced amplitude response at frequencies coincid-ing with the sea state at the test site, resulting in an increase in energy con-version of a factor of two. The two-body system also has a decreased opti-mal damping for energy absorption of about 60 %.
The author performed most of the work on the paper. Journal of Applied Physics 106(6), (2009) Paper VII Depth variation of energy transport in fluid gravity waves
In this paper expressions for the time averaged energy transport at finite depth as a function of distance from the surface are derived. The results are presented as the energy transport for two examples of sea states measured near Lysekil off the west coast of Sweden. They are compared to results using the simpler deep water theory. In addition the measured sea states are fitted to the Pierson–Moskowitz and Bretschneider spectra, and the energy distribution is calculated using these spectra. The latter is done in order to investigate to what degree the energy transport is sensitive or insensitive to details in the sea state.
The author participated to a large degree in the analysis and writing of the paper. Journal of Renewable and Sustainable Energy 2(2), (2010)
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Paper VIII Wave Buoy and Translator Motions - On-Site Measurements and Simu-lations
This paper presents real sea measurements of buoy motion and translator motion for a point absorber type wave energy converter with a linear genera-tor. Buoy motion has been measured using two different systems: a land-based optical system and a buoy-based accelerometer system. The data have been compared to simulations from a Simulink model for the entire system. The purpose of the paper is to make comparisons of different systems for measuring buoy motion, to compare the measured motions of buoy and translator, and qualitatively describe their relation.
The author is responsible for the hydrodynamic modelling in the paper. IEEE Journal of Oceanic Engineering 36(3), (2011) Paper IX Optimization of the Dimensions of a Gravity-based Wave Energy Con-verter Foundation Based on Heave and Surge Forces
This paper presents optimization studies on the dimensions of a gravity-based wave energy converter foundation. The optimization is focused on the undesirable motion of the gravity-based foundation, particularly a study of the tip and lift phenomena regarding the heave and surge forces. The optimi-zation is intended to estimate the appropriate dimensions for the gravity-based foundation with respect to the specific wave climates at the test site. The aim of the study is to avoid the unnecessary construction costs and pro-mote the competitiveness of the whole wave energy converter.
The author participated in setting up the hydrodynamic modelling. The paper was presented by Wei Li at the 9th European Wave and Tidal Energy Conference, Southampton, UK, (2011) Paper X Lysekil Research Site, Sweden: A Status Update
The paper gives a status update of the Lysekil project together with some previously published results. Experimental results include power absorption and power production from the generators connected to different buoys un-der different load configurations. Upcoming work at the test station are pre-sented briefly e.g. the preparation to connect the system to the grid and the fact that a second underwater substation is being built for underwater con-nection of at least seven wave energy converters.
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The author is included in the paper as having participated in many parts of the Lysekil project. The paper was presented by Erik Lejerskog at the 9th European Wave and Tidal Energy Conference, Southampton, UK, (2011) Paper XI Total instantaneous energy transport in polychromatic fluid gravity waves at finite depth
In this paper an expression for the total instantaneous energy transport for polychromatic fluid gravity waves at finite depth based on potential theory with linearized free surface boundary conditions is derived. With finite depth, the validity of the theory is limited to intermediate depths, since the non-linear effects are too large for pure shallow water waves. The results are obtained from time series of wave elevation collected by the wave measure-ment buoy located at the Lysekil test site at the Swedish west coast. The results are presented for different water depths and compared to results based on the deep water approximation.
The author performed the analysis, numerical calculations and wrote the paper. Submitted to Journal of Renewable and Sustainable Energy, (2011) Paper XII Properties of the energy transport for plane-parallel polychromatic sur-face gravity waves in waters of arbitrary depth
This paper compare existing solutions for the energy transport of ocean waves based on the deep water approximation to derived solutions for the wave energy transport as a function of depth and time for waves in waters of finite depth. One measured sea state from the Lysekil test site is used throughout the paper to estimate the error made when using the deep water approximation for different properties of the energy transport. The direction of the energy transport is also investigated using Hermite-Weber functions.
The author performed the analysis, numerical calculations and wrote the paper. Submitted to IEEE Journal of Oceanic Engineering, (2011) Paper XIII Modeling and Simulation of Linear Wave Energy Converter
This paper presents a comprehensive model of the Lysekil project linear synchronous wave energy converter encompassing the buoy, electric genera-tor and the grid connection converters. The modelling was done using Mat-lab/Simulink. The objective of the simulations presented here is the study of
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the wave generator behavior in relation to its grid connection strategy. The generator model includes end spring effects, translator end effects, reaction force and winding and cable impedances. The rectifier, DC-link energy stor-age and inverters are also included for a complete analysis. Wave data are taken from the measurement buoy at the Lysekil test site. The model is com-putationally fast and provides the opportunity to develop farm models. The farm model is simulated with three wave generators and the results are pre-sented.
The author participated in the writing of the paper and is responsible for the implementation of the hydrodynamic part. Submitted to IET Renewable Power Generation, (2011) Paper XIV A resonant Two Body System for a point absorbing Wave Energy Con-verter with direct-driven linear generator
A Two-Body System point absorbing Wave energy converter model tuned to resonance in Swedish west coast sea states are presented in this paper. The conceptual model is based on the Lysekil project wave energy converter. 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 resonant behavior’s sensitiv-ity to the shape of the spectrum and the distance between the two bodies effect on the power absorption are studied for a set of sea states.
The author is responsible for the conceptual design and wrote the paper, while the modelling was done in collaboration with Venugopalan Kurupath. Accepted for publication in Journal of Applied Physics, (02 November 2011)
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9. Svensk sammanfattning
De havsvågor som är intressanta för vågkraft är ytvågor som är skapade av friktionen mellan vinden och vattenytan. Den ojämna fördelningen av solin-strålning över jorden skapar vindströmmar från hög till lågtryck, storskaliga vindmönster som ständigt skapar vågor på haven. Energitätheten ökar från sol-vind till våg och när det slutat blåsa så fortsätter vågorna att färdas med små energiförluster som dyningar. Detta ger att världens oceaner har en för-nybar källa av effekt i storleksordningen terrawatt, som uppskattas kunna ha en betydande roll för framtidens ökande energibehov. De senaste åren har forskning för utveckling av vågenergiomvandling tagit fart, mycket tack vare den ökande miljödebatten samt insikten att de dominerande energikällorna är ändliga. Flera projekt, både privata och universitetsbaserade tar nu de sista stegen i utvecklingen mot fullskaliga vågkraftsparker och kommersialise-ring.
Omvandlingen från vågenergi till elektrisk energi är en tvärvetenskaplig uppgift, innehållandes allt från hållfasthetsberäkningar och generatordesign till hydromekanik och miljöstudier. Omvandlingen sker via flera energiom-vandlingsprocesser som var för sig måste förstås samt hur de är kopplade som en helhet. Teknologin står inför många utmaningar där en av de största är att aggregaten måste vara designade för att tåla effekttoppar som är upp till 45 ggr större än den installerade effekten.
Forskningen och utvecklingen av ett vågkraftskoncept startades av avdel-ningen för elektricitetslära inom Swedish Centre for Renewable Electric Energy Conversion vid Uppsala universitet år 2002. Konceptet som kallas Lysekilsprojektet är baserat på en direkt-driven linjär generator kopplad via en lina till en boj vid ytan. Havsvågornas irreguljära rörelser överförs från bojen via en lina till generatorn där en i både amplitud och frekvens kraftigt varierande spänning induceras i statorn. Denna spänning måste sedan om-vandlas till att ha en frekvens som är anpassad till det elektriska nätet. 2006 installerades det första aggregatet (L1) vid testanläggningen strax utanför Lysekil. Därmed har de teoretiska modellerna som beskriver hydrodynami-ken och generatorn nu kunnat valideras av fullskaliga experimentella resul-tat. Sedan dess har ytterligare åtta aggregat (L2-L9) sjösatts samt ett under-vattensställverk. Tre aggregat har varit uppkopplade mot ställverket samti-digt och därmed utgjort en småskalig vågkraftspark. Inom kort kommer ett nytt ställverk sjösättas som gör det möjligt att koppla upp aggregaten på nätet.
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Denna avhandling fokuserar på energitransporten i havsvågorna samt den hydrodynamiska interaktionen mellan vågorna och bojen. Det senare innefat-tar kopplade ekvationer för både våg/boj-interaktionen samt elektromagne-tiska ekvationer som beskriver generatorn och den elektriska kretsen.
Uttryck för energitransporten för vågor i vatten med ändligt djup har ta-gits fram för att uppskatta felmarginalen för motsvarande uttryck som an-vänder den enklare djupvattenteorin. Detta har gjorts både för hur energin är fördelad över djup samt hur den förändras över tid. Resultaten är baserade på uppmätta tidsserier av vågamplitud från testanläggningen i Lysekil. Resulta-ten visar att användandet av djupvattenteorin ger för låga energitransporter med upp till 16 procent i den totala momentana effekten. Detta antas bero på att djupvattenteorin inte tar hänsyn till den ökande grupphastigheten då vå-gor når grundare vatten. Förutom för de svagaste vågklimaten som uppfyller djupvatten kraven så är resultaten entydiga för samtliga testade vågklimat och skillnaden ökar med ökande vågperiod. Vidare visas att energitranspor-ten i vatten med ändligt djup kan ha en riktning som är motsatt vågriktning-en.
Ett av huvudmålen med avhandlingen har varit att öka energiabsorptionen i systemet. För att veta för vilket vågklimat aggregatet ska vara anpassat så är en studie gjord av vågklimaten i Skagerrak och Kattegatt. Baserat på det, så har en konceptuell tvåkropps punkt-absorbator modell tagits fram som är baserad på Lysekils aggregatet. Ett koncept där det mellan bojen vid ytan och generatorn vid botten finns en sfär med neutral flytkraft placerad. Syftet med den extra kroppen är att dess egenvikt samt s.k. added mass ska öka trögheten i systemet och därigenom påverka egensvängningsperioden. Mo-dellen använder linjär vågteori för överföringen av energi från våg till boj medan generatorn och den elektriska kretsen är simulerad på tre olika sätt; som en linjär resistiv last, som en olinjär resistiv last med likriktare samt i form av en nätuppkoppling. Vågdata för att driva modellen är tidsserier av vågamplitud, vilka insamlas av projektets vågmätningsboj vid testanlägg-ningen.
Det visar sig att genom att ändra diametern på sfären så kan systemets egensvängningsperiod anpassas till typiska perioder för svenska vågklimat. Simuleringar av tvåkroppsmodellen visar att systemet kommer i resonans och ger en kraftig ökning i energiomvandlingseffektivitet för både linjär och olinjär last. Simulering ger upp till 80 procents energiomvandling med tvåk-roppsystemet uppkopplat på nätet. Systemet visar sig vara känsligt för fre-kvensfördelningen i vågspektrat men trots det så visar resultat på en trefaldig ökning av den årliga energiproduktionen baserat på åtta års vågdata från västkusten. Den förhöjda hastigheten i translatorn ger även att dämpnings-faktorn kan sänkas med 60 procent, vilket ger möjligheten att konstruera mindre aggregat med bibehållen nominell effekt. Resultat visar även att kraf-ten per absorberad effekt på translatorn samt strömmen i generatorn för två-kroppsmodellen är jämnare fördelade. Detta indikerar ett bättre utnyttjande
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av utrustningen i systemet vilket är viktigt ur kostnadssynpunkt. Två-kroppsmodellen absorberar mest vid de mest förekommande vågklimaten och betydligt mindre i kraftigare vågklimat. Detta indikerar på en jämnare uteffekt samt ett minskat lagringsbehov. En osäkerhet med resultaten är att viskositeten ignorerats. Detta antas kunna vara mer tydligt i tvåkomponent systemet med dess ökade våta yta samt högre hastigheter vilket kan leda till ökad turbulens och därmed vissa energiförluster. Jämförelse med liknande studier visar dock att de troligtvis är små, men endast experimentella studier kan ge ett fullgott svar. Detta kommer författaren att utföra under nästa år.
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10. Acknowledgements
Jan Isberg, min huvudhandledare. Du är en stor inspirationskälla, och tack för att du alltid tar dig tid att svara på mina frågor. Tack för att du låter mig arbeta så fritt. Mats Leijon, min biträdande handledare samt projektledare. Tack för att du ger mig möjligheten att få arbeta så fritt med det här intressanta projektet. Din drivkraft är inspirerande. Jag skulle villja tacka alla finansiärer till projektet igen (nämnda i artiklar-na). Min doktorandtjänst har varit finansierad av Sveriges Centrum för För-nybar Energiomvandling (CFE) där VINNOVA och Energimyndigheten står som huvudfinansiärer. J. Gust. Richters Minnesfond har bidragit med anslag för fullskaligt experiment av tvåkroppssystemet. ÅF, Wallenbergstiftelsen, C F Liljewalchs resestipendiefond och Stiftelsen Futura har bidragit med rese-stipendier till konferenser jag deltagit i.
Mikael Eriksson and Venugopalan Kurupath. Thanks for all the help, great ideas, interesting collaboration and for sharing your great knowledge. And for showing such a great interests in my work. Tack Cecilia Boström, Rafael Waters, Halvar Gravråkmo, Mikael Bergkvist, Anders Goude, Mårten Grabbe, Simon Tyrberg och Magnus Rahm för er värdefulla korrekturläsning av avhandlingen.
Kalle Haikonen. För mycket roligt har hänt under de här åren för att jag ska kunna skriva ner det här. Vågkraftsgruppen, tack för all hjälp och ser fram emot ett fortsatt gott sam-arbete. Thanks, everyone at the Division of Electricity for the help with a lot of things and for sharing this inspiring place with you. Gunnel Ivarsson, Ulf Ring och Thomas Götschl, Tack för all hjälp och för att ni håller ordning på den här avdelningen.
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Min familj och alla mina vänner, stort tack för allt stöd och för att ni finns där. Sofia Johansson. Älskar dig mest av allt.
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