Modelling and Experimental Verification of Direct Drive Wave Energy Conversion: Buoy-Generator Dynamics

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(1)Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 287. Modelling and Experimental Verification of Direct Drive Wave Energy Conversion Buoy-Generator Dynamics MIKAEL ERIKSSON. ACTA UNIVERSITATIS UPSALIENSIS UPPSALA 2007. ISSN 1651-6214 ISBN 978-91-554-6850-7 urn:nbn:se:uu:diva-7785.

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(133) List of Papers. This thesis is based on the following papers, which are referred to in the text by their Roman numerals. I. II. III. IV. V. VI. VII VIII. IX. Danielsson, O., Eriksson, M., Leijon, M., "Study of a Longitudinal Flux Permanent Magnet Linear Generator for Wave Energy Converters" International Journal of Energy, 2006, 30:1130-1145 Leijon, M., Danielsson, O., Eriksson, M., Thorburn, K., Bernhoff, H., Isberg J., Sundberg, J., Ivanova, I., Sjöstedt, E., Ågren, O., Karlsson, K.E., Wolfbrandt, A., "An electrical approach to wave energy conversion" Renewable energy 31, 2006, pp. 13091319 Thorburn, K., Karlsson, K-E.,Wolfbrandt, A., Eriksson, M., Leijon, M., "Time stepping finite element analysis of variable speed synchronous generator with rectifier" Applied Energy 83, 2006, pp. 371-386 Eriksson, M., Thorburn, K., Bernhoff, H., Leijon, M. "Dynamics of a Linear Generator for Wave Energy Conversion" OMAE 2004, 23:th International Conferance on Mechanics and Arctic Engineering, 2004, Vancouver, British Columbia, Canada, June 20-25 Eriksson, M., Isberg, J., Leijon, M., "Hydrodynamic Modelling of a Direct Drive Wave Energy Converter" International Journal of Engineering Science 43, 2005, pp. 1377-1387 Eriksson, M., Isberg, J., Leijon, M., "Theory and Experiment of an Elastically Moored Cylindical Buoy", IEEE Journal of Oceanic Engineering, vol 31, NO. 4, October 2006, pp. 959-963 Isberg, J., Eriksson, M., Leijon, M., "Instantaneous Energy Flux in Fluid Gravity Waves" Submitted to Phys. Rev. E, Dec 2006 Waters, R., Stålberg, M., Danielsson, O., Svensson, O., Gustafsson, S., Strömstedt, E., Eriksson, M., Sundberg, J., Leijon, M., "Experimental results from sea trials of an offshore wave energy system" Applied Physics letter 90, 034105 2007 Eriksson, M., Waters, R., Svensson, O., Isberg, R., Leijon, M., "Wave power absorption: Experiment and Simulation" Submitted to Journal of Applied Physics, March 2007. 5.

(134) X. XI. Wolfbrandt, A., Eriksson, M., Isberg, J., Leijon, M., "Simulation of a Linear Generator for Wave Power Absorption -PartI: Modeling" Submitted to IEEE Transaction on Energy Conversion, March 2007 Eriksson, M., Wolfbrandt, Waters, R., Svensson, O., Isberg, J., Leijon, M., "Simulation of a Linear Generator for Wave Power Absorption -Part II: Verification", Submitted to IEEE Transaction on Energy Conversion, March 2007. The author has also contributed with inputs to the following paper which not is relevant for this thesis or the results are present in some of the included papers. A. B. C. D. E. Stålberg, M., Waters, R., Eriksson, M., Danielsson, O., Thorburn, K., Bernhoff, H., Leijon, M. "Full-Scale Testing of PM Linear Generator for Point Absorber WEC" Sixth European Wave Energy Conferance, August 29th-September 2nd, 2006, Glascow, Scotland Danielsson, O., Leijon, M., Thorburn, K., Eriksson, M., Bernhoff, H., "A Direct Drive Wave Energy Converter - Simulations and Experiment" OMAE 2005, "24:th International Conference on Mechanics and Arctic Engineering, Halkidiki, Greece, 12-17 June 2005 Danielsson, O., Thorburn, K., Eriksson, M., Leijon, M., "Permanent Magnet Fixation Concepts for Linear Generator" Fifth European Wave Energy Conferance, 17-20 September, Cork, Ireland Bolund, B., Segergren, E., Solum, A., Lundström, L., Lindblom, A., Thorburn, K., Eriksson,M., Nilsson, K., Ivanova, I.,Danielsson, O., Eriksson, S., Bengtsson, H., Sjöstedt, E., Isberg, J., Sundberg, J., Bernhoff, H., Karlsson, K-E., Wolfbrandt, A., Ågren, O., Leijon, M. "‘Rotating and Linear Synchronous Generators for Renewable Electric Conversion - an Update of the Ongoing Research Projects at Uppsala University" Procedings from NORPIE 2004 conference 14-16 June 2004, Trondheim, Norway Bolund, B., Thorburn, K., Sjöstedt, E., Eriksson, M., Segergren, E., Leijon, M., "Upgrading Generators with new Tools and High Voltage Technology" International Journal on Hydropower and Dams Vol. 11, issue 3, May 2004, pp. 104-108. Reprints were made with permission from the publishers.. 6.

(135) Glossary of symbols Symbol. SI unit. Quantity. σ γ φ φ0 φd λ μ0 μr η ρ ν ω A B D E H I Jf Jm M v a A f ac d E f Fr Fe Fem H. A/V m. Conductivity. Nm/s. Generator damping factor. m2 /s. Velocity potential. m2 /s. Velocity potential for incident wave. m2 /s. Velocity potential for diffracted wave. m. Wave length. V s/Am. Permeability of vacuum Relative permeability. m. Wave elevation. Kg/m3. Density. m2 /s. Cinematic viscosity. rad/s. Angular frequency. Tm. Magnetic vector potential. T. Magnetic flux density. C/m2. Displacement field. V/m. Electric field. A/m. Magnetic field. A. Current. A/m2. Free current density. A/m2. Magnetization current density. A/m. Magnetization. m/s. Velocity. m. Buoy radius Active stator length. m. Buoy draft. J. Energy. Hz. Frequency. N. Radiation force. N. Excitation force. N. Electromagnetic force. m. Wave height. 7.

(136) Hs. m. Significant wave height. h. m. Water depth. J. W/m. Energy flux. Jpm. A/m2. PM-current density. Js. A/m2. Source current density. k. 1/m. Wave number. ku. N/m. Spring constant upper endstop. kl. N/m. Spring constant lower endstop. kw. N/m. Spring constant wire. ks. N/m. Spring constant. L. H. Inductance. ls. m. Stator length. lp. m. Piston length. ma. kg. Added mass. m∞. kg. Added in infinity frequency limit. m. kg. Mass. P. W. Power. patm. Pa. Atmospheric pressure. p. Pa. Pressure. R. Ω. Resistance. Rr. kg/s. Radiation resistance. S. m2 /Hz. Wave power spectrum. Tp. s. Energy period. Wc. J. Co-energy. Z. kg/s. Radiation impedance. Bold symbols denotes a vector quantity and hat denotes a Fourier transformed quantity.. 8.

(137) Contents. 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Wave energy conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Overview of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Energy absorption with point absorbers . . . . . . . . . . . . . . . . . . . . . 2.1 Overview of the proposed concept . . . . . . . . . . . . . . . . . . . . . . 2.2 Linear generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Energy absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Wave Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Potential wave theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Buoy water interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Time series and power spectrum . . . . . . . . . . . . . . . . . . . . . . . 3.4 Wave energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Validity of potential wave theory . . . . . . . . . . . . . . . . . . . . . . . 4 Generator Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Modelling approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Field based modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Circuit theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Calculation of Impedances . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Mechanical damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Buoy geometry and damping factor . . . . . . . . . . . . . . . . . . . . . 5.2 Performance of simulation approaches . . . . . . . . . . . . . . . . . . . 5.3 Energy absorption with rectified armature voltage . . . . . . . . . . 6 Experiment and verification of models . . . . . . . . . . . . . . . . . . . . . . 6.1 Experimental generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Offshore experiment I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Offshore experiment II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Energy absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Experiment and modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Suggestions for future work . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Summary of papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Svensk sammanfattning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 11 11 13 14 17 17 19 21 25 25 27 29 30 33 35 35 36 38 40 40 43 43 45 47 49 49 50 52 57 57 58 59 61 63 67 69.

(138) Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 10. 71.

(139) 1. Introduction. Today’s huge energy consumption is largely based on fossil fuels, like coal and oil. These sources have a limited lifetime and are not environmentally friendly. Every year, carbon dioxide is released into the air, which contributes to the greenhouse effect. It is necessary to find alternative energy sources if we want to have the same welfare in the future. One of the major problems so far in extracting energy from renewable sources is to reach energy prices that can compete with the low cost of fossil fuels. The unfriendly environment that a wave power device is exposed to makes it harder to find a device with good economical potential compared for example to wind power, even if the energy density is higher. On the other hand wave power is an unused renewable energy source with huge potential. For Sweden it has the benefit to follow the season variation of energy demands. In wave power there are a great variety of different concepts, some better than others. With research and experience these many concepts will converge to a few with better potential than the others. This is a process that takes time but as the time proceeds we get closer to a concept which is economically competitive.. 1.1 Wave energy conversion What makes wave energy interesting is that it has the highest energy density among all renewable energy sources [1]. Further, it is available up to 90% of the time at a given site, this can be compared with solar and wind power which tend to be available just 20-30% of the time [2]. The higher degree of utilization for wave power can be explained by considering waves as energy a storage created by winds. Even when it has stopped blowing waves remain for a long time, and waves can travel very long distances with very small energy losses. Another advantage for countries like Sweden, where autumns and winters are very energy demanding periods, the natural seasonal variability of wave energy follows the electrical demands. For solar energy is it the opposite. If wave energy is so interesting and oceans contain so much environmental friendly energy why are there no large commercial plants? From a technical point there are many challenges to overcome in the attempt to extract energy from waves and to reach economic feasibility. Some of the concepts have to struggle with low velocities, typically 0.1Hz. A generator requires 500 times higher frequency, typically [1]. This puts high demands on the generators and gearboxes. 11.

(140) Figure 1.1: Illustration of different principles for wave energy absorption [3]. 1. Vertical oscillating buoy moving relative sea bottom, 2. Vertical oscillating buoy moving relative a damping plate, 3. Heaving and pitching buoy moving relative a damping plate, 4. Two buoys moving relative each other, 5. Floating device with a oscillatory rotation motion around a fixed axle, 6. A flap following the water motion back and forth, 7. Vertical oscillating buoy moving relative a fixed chamber, 8. The wave elevation causes pressure difference which drives a turbine, 9. The same principle as 8, for offshore use. 10. The wave causes pressure difference which drives a turbine 11. The waves are led into a narrowing channel; it makes an increase in wave amplitude. Some of the water is spilling in to a reservoir where a turbine is place at the outlet.. Another factor, the irregular behaviour of ocean waves makes it difficult to obtain maximum efficiency of a device over the entire range of excitation frequency. Furthermore, the irregularities cause a power fluctuation that puts 12.

(141) higher demands on generators; they have to handle electrical overloads. The peak power can be ten times larger than the average power. Power fluctuations also cause problems for grid companies. But the most challenging is the survivability all concepts have to face in the extreme conditions that the oceans offer. The wave power devices are exposed to salt water, biological growth and under some periods very rough wave conditions. The structural loading in the event of extreme weather conditions, such as hurricanes, may be as high as 100 times the average loading [1]. In the design of a power device there are two extremes; an underestimation or an over estimation of the design loads for a device. In the first case total or partial destruction of the device can be expected. In the latter case, very high construction costs thus making the technology non competitive. Extracting energy from the ocean waves are not something new. The first technique was patented more than 200 years ago. The first patent is from France in 1799 by Girard and son. From 1855 to 1973 there are 340 patents documented in Great Britain [1], which have carefully been documented by Leishman and Scobie [4]. In figure 1.1 different types of conversion principles are illustrated schematically, number one illustrates the principle that is studied in this thesis. It is during the last 30 years that major’s steps have been taken, it begun in the mid 1970: s during the oil crisis. Norway was one on the first countries and one of the leading in wave power research. Research has also been conducted in Sweden. The two best known concepts are the IPS buoy [5] and the hose pump [6]. So far most researchers have concentrated on the hydrodynamic aspects of different converters and not so much on how to transform the motion created by a wave to electric energy. Independently on the chosen technology it is crucial to have an overall solution from wave absorber to grid connection. As this thesis will show the absorption is very much dependent on the extraction of electric power.. 1.2 Potential The available energy in the world’s oceans is estimated to be between 200 and 5000GW, mostly found in offshore locations [7]. This can be compared to the world’s technical potential of hydro power which is estimated to 2200GW of which only 25% is exploited [7]. Most energy is found at the coastlines of countries facing the major oceans, see figure 1.2. The wave climate around the Swedish coast is different with smaller waves and less energy compared to the major oceans, which put higher demand on the extraction technology. In an article from 1979 [8] the total annual energy flux on the Swedish coast was estimated to 40-60TWh, of that 3-15TWh was estimated to be utilizable. In a article from 2002 the technically available resource is approximately 5-10TWh [1]. The amount utilizable energy is very much dependent on the 13.

(142) 60. 50 50 40. 30 20. 30 100. 50. 15 20 30 40 50 70. 40. 30. 40. 40 100 50. 10. 15. 70. 50. 15. 20 15 20. 15 15. 10 20 40. 30. 100. 50. 20 40. 40. 40 60 70 100. 20. 20. 50 100. Figure 1.2: Approximate yearly average global distribution of wave power levels in kW per meter of wave front [9].. technology. Some devices delivers power only if the waves are large enough, i.e. smaller waves will not contribute. Studies of the technically potential for Swedish climates has been made [10, 11]. The total electricity consumption is approximately 150ThW/year, where about half of the energy comes from hydropower and the other half from nuclear power, and approximately 1% comes from wind power.. 1.3 Overview of the thesis This work is a part of a larger project carried out at Uppsala University, which aims to investigate and develop the wave energy concept described in chapter 2. Within the project two theses have been written, one focused on the linear generator written by Oskar Danielsson [12]. He has designed and verified calculations of our first prototype of the linear generator [13, 14, 15]. Further, he has studied longitudinal end effects [16]. The second thesis is about system aspects and is written by Karin Thorburn [17]. She has studied connections of many generators in a farm [18, 19, 20, 21, 22]. The focus on this thesis lies in modelling the converter, from incident waves to output power. Different modelling approaches have been tested due to accuracy and simulation time. The ability to absorb energy, the dynamic behaviour and damping has been studied to understand the limitations and possibility with this type of concept. The hydrodynamic models used have the limitation to work for a converter in operation condition, not for extreme waves and survivability. The second chapter gives the idea behind and an explanation of the concept which this thesis is based on. Further, an introduction to linear generators and different known methods for increasing the power absorption for a point absorber are explained and discussed. 14.

(143) The third chapter introduces the reader to potential wave theory, which has been used trough out the thesis. Potential wave theory is a well known theory used to simulate wave-body interaction and to calculate the energy content in waves. Moreover, equations for calculating the instantaneous power flux in a wave are presented. These equations are derived from potential wave theory for infinite depth; the derivation is presented in paper [23]. The fourth chapter presents the different model approaches that are used for modelling the generator. Further, an explanation of the approaches and the kind of simulation they are applicable to are discussed. The following chapters summarize and discuss the results presented in the papers. The different modelling approaches are presented and some of the more important results are highlighted. They are followed by a discussion. Finally, a summary of the most important conclusions of the thesis is given, followed by a summary of the papers and the contribution of the author are given. The goal with the thesis is to clarify the dynamic behaviour and absorption of the proposed concept for wave energy conversion and to investigate important parameters for an effective system.. 15.

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(145) 2. Energy absorption with point absorbers. The concept studied in the thesis is a point absorber [24] connected to a direct driven linear generator located at the sea floor. This section presents an overview of the proposed concept describing the different parts from buoy to grid connection. Moreover, a short explanation is presented of how energy is absorbed from the ocean waves as different techniques used to increase the energy absorption.. 2.1 Overview of the proposed concept The proposed concept studied in this thesis is of type 1 in figure 1.1, a direct driven linear generator connected to a heaving buoy (point absorber) [25, 26]. Figure 2.1 illustrates a farm of converters where one converter unit is opened and the different parts are visible.. Figure 2.1: Illustration of a farm of converters where one converter unit is opened and the different parts are visible.. 17.

(146) Each converter consists of a floating buoy, from the buoy a rope is connected to the piston of a linear direct driven generator located at the seabed. A spring is attached from the lower piston end to a concrete foundation on the sea bottom. In wave crests the buoy and piston moves upwards and energy is stored as potential and mechanical energy in the piston and spring respectively. In a wave trough the stored energy accelerates the piston downwards. The piston is covered with rows of permanent magnets of alternating polarity. When it moves relative to the stator an alternating magnetic field in the stator coils induces a voltage giving rise to currents in the conductors. The current in turn affects the piston with a force opposite to the direction of motion. Controlling the power output from the generator makes it possible to affect the dynamics of the whole system. A concept using direct drive generators results in a armature voltage that varies in both amplitude and phase, due to the irregular ocean waves and the oscillating motion. Figure 2.2 shows experimental data from the test site in Lysekil, which clearly illustrates the irregular armature voltage and the output power. Variations are on both the short time scale, a few seconds, and the long timescale, hours and days.. Phase voltage (V). 75 50 25 0 −25 −50 −75 110. 115. 120 Time (s). 125. 130. (a) 4. Power (kW). 3. 2. 1. 0 110. 115. 120 Time (s). 125. 130. (b). Figure 2.2: Example of experimental data from Lysekil, measured by Olle Svensson. a) Phase voltage from a generator in real waves. b) Generated power for voltage in a). The figure shows short time scale variations in power as variations in both amplitude and frequency for the voltage.. 18.

(147) Short time scale variations correspond to irregularities from one wave to the next wave, whereas a long time scale variation depends on changes in weather. The voltage has to be refined before connection to grid. The refinement consists of converting the irregular voltages to voltages with constant frequency and amplitude corresponding to the grid. At the first stage we intend to rectify the voltage from the unit to a constant DC voltage with passive diode rectifiers. The constant DC voltage is then inverted and transformed to the main grid frequency and amplitude [25]. The proposed concept consists of many small wave power converters connected into a cluster of many hundred or even thousand units [20]. Each unit is connected at the same DC level and the current from each unit is added before inverting to grid frequency. Connecting a large number of units will smooth out the power fluctuations on the short time scale. Further, a failure to one or a few units will not affect the power production considerably, as would be the case for a few larger units. The wave energy converters are designed to have their eigenfrequencies different from the frequency spectrum of incident waves; i.e. buoy is intended to follow the wave elevation. In future designs it is possible to include different control system such as latching or some kind of power control system with wave prediction algorithms to increase the energy absorption, see section 2.3.. 2.2 Linear generator In many applications that involve linear/straight motions it can be advantageous to use linear generators or actuators to reduce the complexity of the mechanical interface. Today there exist a number of different linear machines [27, 28]. Wave power devices are traditionally using conventional high speed rotating generators for energy conversion. This requires a system which converts the slow linear/rotation motion of the wave energy absorber to a high speed rotating motion. Hydraulic systems and gearboxes are used for that purpose [29]. An alternative would be direct driven energy converters, such approach requires low speed generators and in some cases they have to be linear [30, 31, 32, 33, 34, 35]. The first electric linear motor was developed more than 100 years ago in the USA. However, linear generators are fairly new for wave energy applications. The idea with direct drive generators is to reduce the complexity of the mechanical interface and there by reduce the number of movable parts and to minimize the mechanical losses. In this way the need for maintenance will hopefully be kept to a minimum. The mechanical interface is in this way replaced with an electrical interface. The electrical interface can be expected to have a longer life time and require less maintenance. Within this project a special type of generator has been developed to fit the demands and required properties for an energy converter that is properly 19.

(148) (a). (b). Figure 2.3: Illustration of the main parts in the linear generator, a. The four sided piston ant the stators, b. Cross section of piston and stator.. working and adapted to the wave motion. The generator has to handle electric overloads and have a high damping factor for low speeds. A high damping factors means that the generator develops a high counteracting electromagnetic force that will damp the piston motion. 20.

(149) The generator developed, is a three phase linear synchronous generator with surface mounted permanent magnets [13, 14, 36, 37, 38]. The piston (or rotor for a traditional rotating generator) is covered with rows of permanent magnets of alternating polarity. Moreover, the magnets used are Nd-Fe-B magnets that have very high remanence, about four times higher than ferrite magnets. The magnet rows are separated with aluminum spacers, see figure 2.3. The stator is made of laminated electrical non-oriented steel sheets and isolated copper conductors. The conductors are wound in slots (holes) in the stator steel and forms closed loops or coils. A low speed generator is larger in size than a high speed generator for the same power and voltage rating. An increase in size implies more magnets and stronger attractive forces between piston and stator. As the attractive forces increase rapidly with the decreased air-gap this puts very high demands on precision and a robust supporting structure.. 2.3 Energy absorption Absorbing wave energy for conversion means that energy has to be removed from the waves. This is the same as producing a wave that interferes destructively with the incoming waves; it can be argued that a good wave absorber is closely related to its ability to generate waves [39]. Figure 2.4 illustrates the principle how energy is absorbed from an incident wave in this 2D example. The oscillating absorber is of infinite length and the incident waves are plane parallel, see figure 2.4a. If the buoy oscillates in heave a symmetric radiated wave is created, figure 2.4b, if it is in resonance with the incident wave and the amplitude is optimal it has been shown that it is possible to absorb 50% of the energy in the incident wave [40, 41, 42]. For a buoy oscillating in pitch motion, an unsymmetrical radiated wave is created, see figure 2.4c. In this case it is possible to absorb more than 50% of the incident wave energy [41, 42]. With a combination of heave and pitch it is possible to absorb all incident energy, se figure 2.4d. Experiment with Salter’s duck has shown that absorption of more than 80% of the energy in a monochromatic incident wave is possible [43]. The parameters that are connected to the ability to absorb energy are excitation force, radiation impedance and damping. The first two parameters are dependent on the buoy geometry. The last parameter, the damping, has to do with the generator characteristics and how energy is extracted from the generator. By using linear theory for buoy-water interaction and assuming that the rope never slacks, 100% active piston/stator area, and that the electromagnetic damping force is proportional to the piston velocity [41], gives that absorbed energy can be written as,. 21.

(150) (a). (b). (c). (d). Figure 2.4: a. Incident wave b. Symmetric wave radiation by heaving ocsillation c. Antisymmetric wave generation by pitching oscillation d. Superposition of the generated waves on the incident wave results in complete absorption of the incident wave. [39, 42]. 2 γ Fê 1 Pa = 2 (γ + Rr )2 + (ω m + ω ma ˘ks /ω − ρ gπ a2 )2. (2.1). Where γ is the damping factor, Rˆ r the radiation resistance, m piston and buoy mass, ma the added mass, a the buoy radius, ks the spring constant, and Fê the excitation force (see chapter 3 for further explanation the hydrodynamic parameters). By letting,. ω m + ω ma ˘ks /ω − ρ gπ a2 = 0,. (2.2). maximizes the absorbed power Pa . From the relation 2.2 the natural frequency of the converter is expressed as, . ω=. 22. k + ρ gπ a2 m + ma. (2.3).

(151) Figure 2.5: Wave elevation and vertical displacement of buoy position as function of time [39]. a. Wave elevation, b.Vertical displacement of latched buoy (artificial resonance), c. Vertical displacement of oscillating buoy in resonance.. The natural frequency of the converter has only to do with the buoy/piston mass and the added mass of the buoy (the added mass depends on the buoy shape). When the wave frequency coincides with the natural frequency of the converter we have resonance in the system. A vertical axis-symmetric point absorber (a heaving body which has a much smaller diameter than the wave length) absorbs at most the energy transported by the incident wave front of width equal to the wave length divided by 2π [41]. This width is named the absorption width. By tuning the natural frequency of the system to coincide with the wave frequency, the system will be in resonance. This is called phase control. In resonance there is a phase difference of 90 degrees between wave and buoy motion and the excitation force is in phase with the buoy velocity. The natural frequency can be tuned by varying mass and spring constant, as discussed above. For a real wave with a broad wave spectrum it will be difficult to use the resonance optimally. Instead an artificial resonance can be obtained by something called latching. In latching a lock and release mechanism forces the motion to have a phase difference of 90 degree to incident wave. A comparison of buoy motion for a resonant and a latched system is given in figure 2.5. The second parameter to optimize is the oscillator’s amplitude; this is adjusted by the damping factor. A larger damping will decrease the oscillator amplitude and the velocity. In the proposed concept the damping is dependent on the electric load. By changing load and in turn the power outtake it will be possible to control the absorption. Latching could be implemented by a heavy non linear damping, i.e. to have a very large damping that almost makes the piston to stand still for low velocities. When the accelerating forces becomes too high and the buoy motion is out of phase the damping decreases to a normal operation condition. Such non-linear behaviour could be achieved using power electronics. There are a number of control strategies how to control the phase and amplitude to optimize the power extraction from the waves [44]. The most advanced strategies 23.

(152) predict the incident wave for some seconds in advance to calculate optimal motion for maximal energy absorption. Simulations of wave energy converters similar to the proposed concept show an increase in power absorption for a system with motion control. A simulation of a heaving buoy moving relative a fixed reference made by H. Eidsmoen [45], showed that the year average power production was three times higher for a system with a phase controlled motion than without. Furthermore, a controlled system gave a more steady power output. In an article by K. Nielsen [46], simulation and experiment of a point absorber with hydraulic power take off, showed that latching increased the absorbed power 50% in regular waves but with only 4-8% for irregular waves.. 24.

(153) 3. Wave Theory. Modelling water waves is a very complex task. Simple problems quickly become very computational demanding. Depending on required accuracy and purpose of the simulations different approximations can be made to simplify the theories. Trough out this thesis small amplitude linear potential wave theory has been used. For a wave energy converter in normal operation condition this theory valid, see section 3.5. To model survivability of a converter in rough conditions other theories have to be adopted.. 3.1 Potential wave theory Potential wave theory is frequently used to describe water waves with small amplitude compared with its wave length [47, 48, 41, 49]. The theory can be derived from the conservation of mass and momentum equations;. ∂ρ + ∇ · (ρ v) = 0 ∂t. (3.1). ∂v 1 1 + (v · ∇)v = − ∇Ptot + ν ∇2 v + f, ∂t ρ ρ. (3.2). Where v is the velocity vector of the flowing fluid element, ρ the density, Ptot the pressure of the fluid, ν the cinematic viscosity constant, and f external forces per unit volume. In this case, with a gravity wave, f = (0, 0, ρ g). Moreover, the following assumptions are made, • Ideal fluid An ideal fluid has no viscosity, i.e. ν = 0. This will result in no friction losses in the fluid. • Incompressibility For an incompressible fluid the continuity equation 3.1 gives ∇ · v = 0 • Irrotational It is also assumed that the fluid is rotational free, i.e. ∇ × v = 0.. For an irrotational motion the velocity can be written as the gradient of a velocity potential, v = ∇φ , inserting this into the incompressible continuity equation results in the Laplace equation for the potential, ∇2 φ = 0.. (3.3) 25.

(154) The boundary conditions are the Cinematic Free Surface Boundary Condition (CFSBC), the Dynamic Free Surface Boundary Condition (DFSBC), Bottom Boundary Condition (BBC) and Periodic Lateral Boundary Condition (PLBC). An illustration of the geometry and definition of parameters of the boundary value problem is found in figure 3.1. Moreover, surface tension is neglected. The DFSBC is derived from Bernoulli’s equation. Letting the pressure at the surface, p = 0, which follows from the transformation, p → p − patm , gives the DFSBC,. ∂φ 1 (3.4) ∇φ · ∇φ + gη = 0 at y = 0 ∂t 2 The CFSBC is derived by noting that a fluid particle at the surface remains there. Mathematically this is written, D ∂ {y − η (x,t)} = {y − η (x,t)} + v · {y − η (x,t)} = 0 Dt ∂t Which gives,. (3.5). ∂φ ∂η ∂φ ∂η = + (3.6) ∂y ∂t ∂x ∂x At the seabed, which is a rigid surface, there can be no normal velocity, hence, ∂φ =0 (3.7) ∂z By assuming small amplitude compared to wavelength and water depth, boundary conditions 3.4 and 3.6 can be linearized, ∂φ + gη = 0 at y = 0 ∂t. (3.8). ∂φ ∂η = at y = 0 (3.9) ∂y ∂t An analytical solution to the linearized boundary value problem is derived by using separation of variables and applying PLBC, φ (x,t) = φ (x + λ ,t) = φ (x,t + T ). The solution for a particular frequency ω is given by, φω =. Hg cosh(k(h + z)) sin(kx − ω t) 2ω cosh(kh). (3.10). Where ω is the wave angular frequency, k is the wave number, z is the depth, H is twice the amplitude, and h is the distance from surface to bottom. The surface wave elevation is,. ηω (x,t) =. 26. H sin(kx − ω t) 2. (3.11).

(155) Figure 3.1: Definition of parameters for the boundary value problem.. 3.2 Buoy water interaction The buoy-wave interaction can be divided into two separate problems, where each problem gives rise to a force. These two forces are excitation force and radiation damping force. The excitation force is calculated with the buoy kept fixed with a constant draft. An incident harmonic plane parallel wave impinge on the buoy, see figure 3.2a. The velocity potentials is solved for the boundary value problem, where the total potential is the sum of the potential corresponding from the incident wave and the diffracted potential created by the fixed buoy. The excitation forces for a harmonic wave with frequency ω is calculated by integrating the pressure over the wet surface S of the buoy. The expression for excitation force in the frequency domain for oscillation mode j is given by, Fê, j (ω ) = iωρ. S. (φˆ0 + φˆd )n j dS = fê, j (ω )ηˆ (ω ). (3.12). Where ω is the angular velocity of the incident wave, n j is the normal for oscillating mode j, φˆ0 is the incident velocity potential, φˆd is the diffracted wave potential, fê, j (ω ) is the excitation force factor, and ηˆ (ω ) is the wave elevation. To determine the radiation force it is assumed that the buoy oscillates with the angular velocity ω in the absence of incident waves, see figure 3.2b. The solution to the boundary value problem gives the radiated velocity potential φr . As in the excitation problem the radiation force is calculated by integrating the pressure over the wet surface S of the buoy. The alternative representation is by multiply the radiation impedance, Z j j , with the buoy velocity, vˆ j . The expression for radiation force in the frequency domain for oscillation mode j is given by, Fˆr, j = iωρ. S. (φˆr n j )dS = −Z j j vˆ j. (3.13) 27.

(156) The buoy water interaction is determined by the excitation force and radiation force, the excitation force will drive the buoy while the radiation force will damp the buoy motion. To give an intuitive understanding of the radiation impedance we divide it into a real and imaginary part, Z j j = R j j + iX j j = R j j + iω m j j. (3.14). Where the real part is radiation resistance and the imaginary part is the added mass multiplied with the angular frequency. The radiation resistance gives rise to a counteracting force on the buoy which is in phase with the buoy velocity and is associated with active power losses. This force will damp the buoy motion; the energy loss in kinetic energy is transformed into a radiated outgoing wave. As an example, consider a free oscillating buoy. This buoy will create an outgoing wave; the energy content in this wave comes from loss of the buoys kinetic energy. After some time all buoy energy has been transformed into waves and the buoy stands still. The so called added mass ma has the dimension of mass and corresponds to a volume of water that is moved when the buoy is moving. The added mass is a function of angular velocity and it is also dependent of the buoy shape. A large added mass give the buoy more inertia, the eigenfrequency of the buoy is close related to added mass. For simpler geometries such as cylinders and spheres it is possible to give closed expression for the excitation force factor and radiation impedance [50], the hydrodynamical parameters can also be calculated using software such as WAMIT [51]. The buoy-water interaction presented is a linear system, where the excitation force factor and radiation impedance is transfer functions relating the acting forces to incident wave elevation and buoy velocity. Until now the theories is for the frequency domain. In the time domain multiplications and transfer functions becomes convolutions, denoted ∗, and impulse response functions respectively. The excitation force in the time domain is written as, Fe,t (t) = ft (t) ∗ vt (t). (3.15). And the radiation force is given as, Fr,t (t) = −Zt (t) ∗ vt (t). (3.16). In general the added mass does not vanish in the infinite frequency limit. This causes problems when one wishes to apply Fourier transforms. To remedy this problem, the added mass in the infinite limit frequency, m∞ , is subtracted from the added mass and is treated separately [41]. The added mass and radiation resistance are closely related to each other, from the radiation resistance it is possible to calculate the added mass and vice versa. They are related by Kramers-Kronig relations [41]. This gives the possibility to express the radiation force as a function of buoy velocity or buoy acceleration, Fr,t (t) = −k(t) ∗ vt (t) − m∞ v˙t (t) = −h(t) ∗ v˙t (t) − m∞ v˙t (t) 28. (3.17).

(157) (a). (b). Figure 3.2: (a) Excitation problem, (b) Radiation problem.. Where k(t) =. 2 π. h(t) =. 2 π. ∞ 0. ∞ 0. 2 Rˆ r (ω ) cos(ω t)d ω = − π. ∞ 0. ω [ma (ω )˘m∞ ] sin(ω t)d ω (3.18). (ma (ω ) − m∞ ) cos(ω t)d ω = −. 2 π. ∞ ˆ Rr (ω ) 0. ω. sin(ω t)d ω (3.19). For a more thorough reading about body wave interaction see for example [52] or [53].. 3.3 Time series and power spectrum An ocean wave can be represented in different ways depending on the field of application. In simulations it is in most cases preferable to use raw wave data, i.e. time series of wave elevation. The other way of representing a wave is by its wave power spectrum. However this neglects any phase information. The power spectrum can be calculated from the wave elevation data. The wave power spectrum S( f ) is defined as, 29.

(158) S( f ) =. 2 | T0. T0. η (t)e2π f t dt|2 ,. 0. (3.20). where T0 is the total measured time. In this form the power spectrum gives no information of the direction the waves is propagating. This information can also be included, then this is called the directional power spectrum [49]. From the power spectrum different moments can be calculated, the n:th moment of a spectrum mn is defined as, mn =. ∞ 0. f n S( f )d f. (3.21). Where f is the wave frequency. The moments give statistical information of the wave. Important quantities are the significant wave height, Hs , and the energy period, Tp , defined as, √ Hs = Hm0 = 4 m0 Tp = Tm0−1 = (. m−1 ) m0. (3.22) (3.23). An older definition of significant wave height is the average height of the highest third of the waves, H1/3 .. 3.4 Wave energies The energy in an ocean wave can be divided into two parts, kinetic and potential. The kinetic energy is due to the water fluid motion, and potential energy is due to the wave elevation. The total wave energy content in a plane harmonic wave on deep water is half kinetic and half potential energy. The total energy can be calculated from the wave power spectrum, as E = ρg. ∞ 0. S( f )d f. (3.24). The wave energy transport denoted J , is the energy passing a fictious vertical wall of unit width is defined as, J = ρg. 0 −∞. pvx dz. (3.25). Where vx is the horizontal particle velocity and p is the pressure. The average energy transport is calculated from significant wave height and the energy period, by, J = cTp Hs2 30. (3.26).

(159) Where the constant c = ρ g/32π . The energy can be considered to be transported with the group velocity. The relation between energy transport and wave energy is given by, J = vg E. (3.27). In paper [23] we have derived an expression for the instantaneous energy flux, J(t), from time series of the wave elevation for waves of arbitrary shape. Furthermore have we derived an expression for the energy flux vector j = ( jx , jy , jz ). The energy flux vector corresponds to the Poynting vector in electrodynamics. The expression is derived using linear wave theory for infinite depth. The expression for the energy flux is given by. The energy transport through a vertical plane perpendicular to the wave is given as a double convolution, . . ρ g2 ∞ ∞ |t | + |t | η (t − t)η (t − t)dt dt (3.28) 2π −∞ −∞ t 2 + t 2 In figure 3.4 is a plot of the energy flux for a harmonic wave. As can be seen the energy transport seems always to be positive, which as well is proven in paper [23]. The expression for the energy flux vector for a wave travelling in the positive x-direction is given by, J(t) =. jx = ρ g. ∞ −∞. . . . K(−z/g,t − t)η (t )dt ·. ∞ −∞. M(−z/g,t − t)η (t )dt . jy = 0. jz = ρ g. ∞ −∞. K(−z/g,t − t)η (t )dt ·. (3.29) (3.30). ∞ −∞. L(−z/g,t − t)η (t )dt . (3.31). Where the integration kernels K, L, and M are given by, 2 1 −τ K(ζ , τ ) = e 4ζ 2 πζ. (3.32). 2 τ −τ L(ζ , τ ) = √ 3/2 e 4ζ 4 πζ. (3.33). . |τ |. |τ | − 4τζ2 2√ζ x2 1 − e e dx (3.34) 2πζ 2πζ 3/2 0 The last two factors of the M kernel is called Dawson’s integral, for which it exist efficient numerical computation algorithms [54]. The modulus of power flux vector for a harmonic wave is shown in figure 3.3. Moreover the flux vector for an enlargement of figure 3.3 is shown in figure 3.4. M(ζ , τ ) =. 31.

(160) Figure 3.3: (a) Wave elevation, (b) Integrated energy flux, (c) Horizontal enery flux.. Figure 3.4: Enlargement of the energy flux in figure 3.3. 32.

(161) 3.5 Validity of potential wave theory Linear potential wave theory assumes that the wave height is much smaller than the wave length. The fluid motion generated by any moving object must also be correspondingly small. Figure 3.5 roughly indicates in what region the theory is valid. In the figure H is the wave height, a is the typical diameter of the object, and λ is the wave length. When H/a is large, so that the particle paths are long compared to the object diameter, flow separation occurs and this changes substantially the flow predicted by the inviscid theory. This area is marked ’viscous’ in the figure. When λ /a is large, so that the wavelength is much larger than the typical diameter of the object, the wave field is little modified by the object and wave diffraction is relatively unimportant. This are is marked ’diffraction’ in the figure. The square in figure 3.5 illustrates the domain of interests for simulation this thesis is based on. Typical wave periods are 4-8 seconds, wave heights is 0-4m, and buoy diameters 2-5m. As can be seen the potential wave theory is well suited for modeling the buoy wave interaction.. Figure 3.5: The figure illustrates domains where viscous and diffraction forces are of importance. The square indicates the domain of intrests for simulation in this thesis [52].. 33.

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(163) 4. Generator Theory. In the thesis three approaches of generator modelling have been used. The different models are of varying complexity and are suitable for different kinds of simulations. The simple model is very fast and rough while the most complex model is computationally demanding and very accurate. The models are: Field based model, Equivalent circuit model, and Mechanical model.. 4.1 Modelling approaches • Field based modelling: In the Field based model the time dependent electromagnetic field solved by a FE method. External electric circuit for load and motion equation is coupled to the field calculations and these are solved simultaneously in time. From the fields are losses, electromagnetic force etc calculated. This modelling gives a near complete picture of the generator and its properties. Moreover, from the fields and current distributions a temperature analysis can be made. This approach of modelling the generator is very computationally heavy and is useful mainly to design new generators. • Circuit theory: In the equivalent circuit model the generator is represented by a system of four coupled windings. Three of them represent the armature phase winding and the fourth is fictious representing the field winding or in this case the permanent magnets. Inductances in the coils are pre-calculated by stationary FE solutions of the field based models. Moreover, the inductances are piston position and electric load dependent. This approach of modelling the generator gives the same accuracy as the field based models but is much faster. This approach opens up for system analysis of complex system, such as studies of different load conditions and farms etc. • Mechanical Damping: In this model the electric system, generator and load, is translated into a mechanical damping function. With this approach only the mechanical system can be studied, such as energy absorption, prediction of energy production and design of the mechanical structure as springs and end-stops. This way of modeling is very fast and large amount of data can be produced that statistical investigations of energy absorption etc can be based on.. 35.

(164) 4.2 Field based modeling The fundamental and governing equations in electromagnetics is Maxwell’s equations, ∇×E = −. ∂B ∂t. ∇×H = Jf +. ∂D ∂t. (4.1). (4.2). ∇·B = 0. (4.3). ∇·D = ρf. (4.4). Where E is the electric field, D is the electric flux density, H is the magnetic field, and B is the magnetic flux density. J and ρ are current and charge density respectively. In addition to Maxwell’s equation material properties is needed, these can be formulated by the following constitutive equations, B = μ H,. (4.5). J = σE. (4.6). Where μ is the non-linear permeability, represented by a B-H curve that can be found in data sheets from manufactures of electrical steel. The conductivity is represented by σ , which is a scalar constant here. For numerical computational reasons the magnetic flux density is written as a magnetic vector potential, B = ∇ × A. Restricting the field problem to two dimensions an expression for the field can be derived [55, 56], 1 ∂ Az ∇· ∇Az + σ (4.7) = Js + Jpm μr μ0 ∂t Where Az is the z-component of the magnetic vector potential. The current densities in stator windings are denoted Js , whereas Jpm is a fictous current density that models the permanent magnets. The field calculations are coupled to external algebraic equations describing the electric load connected to the generator, the external circuit includes factors which compensates for end effects which are not included in the two dimensional field formulations. The external equations are, Ia + Ib + Ic = 0 Ua + Rs Ia + Lsend 36. ∂ La ∂ Ib −Ub − Rs Ib − Lsend = Vab ∂t ∂t. (4.8). (4.9).

(165) ∂ Lc ∂ Ib −Ub − Rs Ib − Lsend = Vcb , (4.10) ∂t ∂t where Ua , Ub , and Uc are the phase voltages, Ia , Ib , and Ic are the phase currents, Vab and Icb are the line voltages, Rs is the phase resistance, and is the coil end inductance. The field problem is solved for a rotating generator. To speed up the simulation time a moving boundary in the air gap is adopted, this condition is used to simulate that the rotor rotates inside the stator. By using this method we do not have to re-mesh for every time step. It has the limitation that the air gap is constant resulting in that no excentrities can be studied. Moreover, cyclic boundary conditions are used that makes it possible to only simulate one pole of the generator. Figure 4.1 shows the mesh and the resulting field for one calculation pole/cell/domain. The induced counteracting electromagnetic force which is present when power is delivered by the generator is calculated from the change in magnetic field energy in the air gap, the force is expressed as, Uc + Rs Ic + Lsend. dWc (4.11) dx Where Wc is the co-energy in the airgap and x denotes the piston position. Fem =. (a). (b). Figure 4.1: Illustration of one calculation domain, (a) The mesh, (b) The resulting field. 37.

(166) 4.3 Circuit theory In the equivalent circuit theory the generator is represented by inductances, resistances and voltages sources. The circuit representation a generator can be seen as a system of four coupled windings. The first three windings represent the stator coils, the fourth winding which is a fictitious winding represent the permanent magnets. Figure 4.2 shows the equivalent circuit for a Y-connected three phase generator; in this particular case Y-connected impedance, consisting of inductance in series with a resistance, acts as a load. The terminal voltage, U, for the circuit can be written under any load condition as, dL dI +ω I+E (4.12) dt dθ Where R is the resistance matrix, I is the current vector, L is the inductance matrix, and E is the induced emf. These parameters are defined as, U = RI + L. ⎡. Laa Lab Lac. ⎤. ⎢ ⎥ L = ⎣ Lba Lbb Lbc ⎦ Lca Lcb Lcc ⎡. Ea. ⎤. ⎡. La f. (4.13) ⎤. d ⎢ ⎥ ⎥ ⎢ E = ⎣ Eb ⎦ = τω ⎣ Lb f ⎦ I f dθ Ec Lc f. (4.14). I = [Ia , Ib , Ic ]T. (4.15). U = [Ua ,Ub ,Uc ]T. (4.16). I = diag {Ra , Rb , Rc }. (4.17). The machine inductances are represented by a matrix L with nine elements, where the three diagonal element are the three phases self inductance and the off diagonal element representing different combinations of the mutual inductances. The inductance matrix is pre-calculated by stationary FEM solutions the field problem for a particular generator design. The inductance matrix is electric load and piston position dependent and is stored as a database. In paper [57] shows how the inductances can look like for a generator with a certain stator winding configuration. In section 4.3.1 follow a brief explanation of how the impedances are calculated. The permanent magnets are modelled by fictitious windings supplied from a current source with a constant current, I f . The current is linearly proportional to coercivity of the magnet times the magnet height in the direction of magnetization. 38.

(167) This representation is for a conventional rotating generator where rotor angular velocity ω gives the rotor angle as, θ = ω t + θ0 . The transformation and approximation of the rotating to a linear generator is here made with a scale factor τ (θ ), it is defined as active stator length divided with total stator length. The scale factor reduces the induced emf when the piston leaves the stator. Additional equations are needed to give a closed electric circuit, these equations represents the load and approximates the end-effects, which is not incorporated in the inductance model. For example for a Y-connected load the additional equations are given by, Ia + Ib + Ic = 0. (4.18). ΔUab + (Rs + Rload )ΔIab + (Lsend + Lload ). ∂ ΔIab = 0 ∂t. (4.19). ΔUbc + (Rs + Rload )ΔIbc + (Lsend + Lload ). ∂ ΔIbc = 0 ∂t. (4.20). Where ΔUab ≡ Ua −Ub , ΔUbc ≡ Ub −Uc , ΔIab ≡ Ia − Ib , and ΔIbc ≡ Ib − Ic . The end effects are included as inductances as described in the previous section. The electromagnetic force developed to counteract the motion is calculated as, Fem =. Tdev , r. (4.21). where r is the inner radius of the stator and Tdev is the developed torque. The electromagnetic torque is calculated by a method based on energy conservation principle. Tdev is expressed by, 1 dW Tdev = (U j I j − Rs I 2j ) − (4.22) ∑ ωr a,b,c dt Where ωr is mechanical angular speed of the rotor, ω = ω /p, p is the pole number, and W is stored magnetic energy.. Figure 4.2: Definition of impedances in circuit representation of a three phase generator connected to a load.. 39.

(168) 4.3.1 Calculation of Impedances The inductance coefficients describe the relation between flux linkage and currents. The method used for calculating the inductance matrix in the simulation used in papers is the Enhanced Incremental Energy Method, EIEM [58, 59, 60]. The method requires stationary finite element solution under current perturbation for each machine winding and provides detailed inductance variations as function of rotor position. After the operating solution has been computed the differential reluctivity is evaluated. This computation is carried out in each step of the non-linear FE-analysis as a part of a Newton iteration process. The linear generator is seen as a rotating machine with a very large radius. The inductance is considered as a function of rotor position angle, θ , which is expressed as a fourier serie. Hence, the derivative of the inductances with respect to θ can easily be obtained. For a more detailed explanation see paper [57].. 4.4 Mechanical damping In contradistinction to the very accurate generator models described above a simple generator model has been developed, this model is very rough and acts as a non-linear damper in the mechanical system. The damping is a function of velocity, load and piston position. The damping force is derived from the relation between force, power and piston velocity, P = F x˙. (4.23). F = γ x˙. (4.24). Where g is the damping constant or function for the linear or the non-linear case respectively. For a constant velocity the damping is proportional to active power in the circuit and it is expressed as,. γ = 3∑. 1 ui 2 Ri x˙. (4.25). where γ is the damping, Ri is the resistance, ui the voltage, and x˙ is the piston velocity. The effects on the force when the piston leaves the stator in taken into account by a factor which is the quote between active piston length and stator length, this expressed as, ⎧ ⎪ ⎨ 0 A f ac = 1 ⎪ ⎩ 1 1 ls. 40. if. |x| ≥ 12 (l p + ls ). |x| ≤ 12 (l p − ls ) if else 2 (l p + ls ) − |x|. (4.26).

(169) Where ls and l p is the stator and piston length respectively. The damping force is then expressed as, Fem = A f ac γ x˙. (4.27). The total absorbed power is given by, P = γ A f ac x˙2. (4.28). 41.

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(171) 5. Modelling. Development and improvement of the next generation wave energy converters is very much dependent on computer simulations. This chapter presents in brief the developed models of the converter used in the thesis. Furthermore, some results based on simulations are presented.. 5.1 Buoy geometry and damping factor To achieve high energy absorption it is necessary to adopt the buoy motion in an optimal way to fit present wave climate. The dynamics is totally determined by the buoy shape, mass of moving parts and the damping. A simplified linear model of the system is presented in paper [61]. The model includes heave motion only and the buoy and piston is connected by a stiff rod. Further, the generator represents by a constant damping factor. The buoy amplitude in the time domain, z, is given as, z = H ∗ η,. (5.1). where H is a transfer function describing the wave energy converter dynamics, η is wave elevation, the ∗ denotes convolution. The transfer function in the frequency domain is given by, Hˆ =. fê , −ω (ma + m) + iω (γ + Rˆ r )ρ gπ a2 ks. (5.2). where ma is added mass, m is the total mass of piston and buoy, γ is damping factor, a is buoy radius, ks is spring constant, Rˆ r is the radiation resistance, and fê is the excitation force. The hydrodynamic parameters ma and Fe are pre-calculated using commercial software WAMIT [51]. For a converter of the size studied in this thesis it is very difficult to reach resonance by an increase in mass. Another possibility is to use a buoy with a smaller diameter, see figure 5.1a. Moreover, the buoy shape has very little impact on the power absorption as long as the system is off resonance, at least for the studied shapes that were a cylinder with flat-, conical- and spherical bottom. This has to be studied more carefully before conclusions can be drawn. The different studied buoy geometries have only significant impact for frequencies near the resonance frequency. The highest resonance and narrowest peak was obtained for a 43.

(172) 25 Flat−shaped bottom Cone−shaped bottom Spherical−shaped bottom. 22.5. Power Capture Ratio (%). 20 17.5 15 12.5 10 7.5 5 2.5 0 0. 10. 20. 30. (a). 50 60 γ (kNs/m). 70. 80. 90. 100. 70. 80. 90. 100. (b). 2.5. 30. R=0.2m, d=0.7m R=0.5m, d=0.7m R=1.0m, d=0.7m R=1.5m, d=0.7m R=1.5m, d=0.4m R=2.0m, d=0.4m R=2.0m, d=0.3m. 2 1.75. 27.5 25 22.5 Power Capture Ratio (%). 2.25. abs(H) (m/Hz). 40. 1.5 1.25 1 0.75. 20. R=0.2m, d=0.7m R=0.5m, d=0.7m R=1.0m, d=0.7m R=1.5m, d=0.7m R=1.5m, d=0.4m R=2.0m, d=0.4m R=2.0m, d=0.3m. 17.5 15 12.5 10 7.5. 0.5. 5. 0.25 0 0. 2.5. 1. 2. 3 ω (rad/s). (c). 4. 5. 6. 0 0. 10. 20. 30. 40. 50 60 γ (kNs/m). (d). Figure 5.1: (a) Transfer function H for cylindrical buoy with flat-, conic- and spherical- bottom., (b) to (a) corresponding power capture ratio as function of damping factor for ω = 1. , (c) Transfer function H for cylindrical buoy with different radius., (d) to (c) corresponding power capture ratio as function of damping factor for ω = 1. Unpublished work.. cylinder with spherical bottom, lowest and widest peak for a cylinder with flat bottom, see figure 5.1a. To reach a high absorption it is necessary to damp the buoy motion as effectively as possible, the damping is proportional to the generated power. The damping factor has to be increases with buoy radius for obtain optimal energy absorption as long as the converter is off resonance 5.1c,d. Simulations show that the resonance frequency of the energy converter and the frequency of characteristic wave spectrum coincide when the buoy diameter is small, for a flat bottom buoy these coincide for a radius of approximately 0.2m, this for an incident wave with Te ≈ 6s. For larger buoy radius the resonance frequency increases. 44.

(173) 5.2 Performance of simulation approaches Within this thesis a number of different simulation approaches has been used to simulate the wave energy converter in operation. Common to all approaches is that potential wave theory is used to model the buoy wave interaction. Hydrodynamic parameters such as radiation resistance and added mass for the studied buoy geometry are pre-calculated using WAMIT [51] or from analytical expressions derived by Bhatta [50]. It should be mentioned again that all models used for a converter are in normal operation, for survivability more advanced models for the buoy water interaction have to be used. The difference in the models lies in the generator/load and the coupling between these different parts. The generator models are presented in detail chapter 4.1. Different approaches are suited for different type of simulation and in different stage of the research. The most computational heavy model is also the one giving the most information. The different models are illustrated in figure 5.2 and explained below. • Approach I (Field based model) This model is based on two equation of motion, one for the buoy and for the piston. The buoy is restricted to move in heave only. The buoy-wave interaction is given by the excitation force coefficient and the radiation impedance for the given buoy geometry, these parameters are given in the frequency domain. When the distance between buoy and piston is shorter than the rope, the motion equation decouples. The generator is modelled by the field based model, see chapter 4. To the generator an electric circuit is connected, which is a π -link and a resistive load. This model is illustrated in figure 5.2. This result in an expanded stiffness matrix, where the external equations are added rows. External equations include the electric load, the coupled motion equation for buoy and piston. The hydrodynamic forces acting on the buoy are included, the excitation force is calculated in advance since it depends only on the incident wave elevation, but the damping force is a convolution of the impulse response function and the buoy velocity history. For the implicit formulated system semi-implicit RungeKutta time integration methods of different orders are used. It can handle events, i.e. the simulations can be interrupted at arbitrary times; equations can be replaced by other equations and parameters can be set to new values. This gives a possibility to use different equations in different phases of the simulation. For example the motion equation can be decoupled when there is a slack on the wire or it can simulate a diode when it turns on and off. • Approach II (Equivalent Circuit model) This model is basically the same as the previous. The difference lies in the generator model. In this case the generator is represented by an inductance matrix, which is pre-calculated from stationary FE solutions to the field problem, see chapter 4. The inductances are load and velocity dependent. • Approach III (Non-linear damping) 45.

(174) Figure 5.2: Illustration of the different simulation approaches.. This model implemented in the commercial software SIMULINK. The buoy is restricted to move in heave only; its equation of motion is coupled to the equation of motion for the piston, when the distance is smaller than the rope length they decouple. The generator is modelled as a nonlinear damper, see chapter 4. This time dependent problem solves by using SIMULINK’s numerical solvers. In this case the hydrodynamic parameters have to be given as an impulse response function, see paper [62] for details. • Approach IV (Linear damping) 46.

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