Wave energy conversion based on multi-mode line absorbing systems

Transcription

1 Wave energy conversion based on multi-mode line absorbing systems A thesis submitted to The University of Manchester for the degree of Doctor of Philosophy in the Faculty of Engineering and Physical Sciences 215 Efrain Carpintero Moreno School of Mechanical, Aerospace and Civil Engineering

2 Contents Contents... 2 List of figures... 6 List of tables... 1 Nomenclature Abstract Declaration Copyright Statement Dedication Acknowledgement The Author Chapter I Introduction Global status of renewable energy Renewable energy in UK Wave energy resource in UK seas Renewable energy in Mexico Wave energy resource in Mexican seas Research objectives Specific Objectives Summary Chapter II Literature review Current devices classified by converter type Modelling techniques Linear wave diffraction theory CFD models Spectral wave modelling Performance evaluation of WECs experimental modelling Response in extreme conditions Ocean energy research at the University of Manchester

3 The Manchester Bobber Summary Chapter III Linear modelling of multi-body wave energy converter Linear wave theory Modelling of a single wave energy converter Application to a wave energy converter Numerical modelling in heave for a two-float WEC Two-body converter configuration work principle Discretization of the heave equations for the converter floats coupled system Obtaining vertical floats velocities for the couple floats system Obtaining body forces with WAMIT Converter optimization for the two-body configuration Optimization criteria Optimization for two-float array with a 1/4 th geometrical scale... 6 Hydrodynamic and excitation forces Case 4b Prediction of the power output Case 4b Results for different float configurations Optimization of the mechanical force applied to the system Summary Chapter IV Experimental method Description of the experiments Wave definition... 7 Irregular waves - JONSWAP spectrum... 7 Wave power formulation Froude scaling law Facilities Manchester facilities Plymouth facilities Wave system models Operation system for the three-float converter Experimental procedure

4 Wave definition for testing at the University of Manchester Wave definition for testing at Plymouth University Wave field evaluation Experimental procedure for testing with the converter in place Data post processing to obtain the power output Obtaining the rod velocity for 1/4 scale model Obtaining the rod velocity for 1/8 scale model Summary Chapter V Experimental investigation of multi-body WEC M4 wave energy converter the converter evolution Development of the two-float system Two-float system parameter investigation Development of three-float system Summary Chapter VI Evaluation of a three-float wave energy system at two scales Scaled laboratory models /4 th scale models /8 th scale models Experiments with a 1/4 th laboratory scale model Experimental results in regular waves Experiment results in irregular waves Experiments with a 1/8 th laboratory scale model Regular waves Irregular waves Results comparison across scales Summary Chapter VII Power projection for full-scale prototypes Power projection method Wave energy resource - Scatter plots Defining the action range of a tuned device at each site Power projection for deployment sites

5 Results for a converter with λs = Results for a converter with λs = Predicted power variation with the length scale factor Summary Chapter VIII Discussion and Conclusions Overview Discussion Systematic methodology for converter design Power projection finding Conclusions Contribution of this research Future work Mooring system design Survivability study Feasibility study at the specific selected place Investigation of the political and environmental regulations of suitable sites for positioning a full-scale prototype Appendix A Appendix B References Word count 47,574 5

6 List of figures Figure 1.1: Global electricity energy production on 214, [1] Figure 1.2: Global energy resources, wave power per meter of wave front [kw/m], [2] Figure 1.3: Annual average wave height [m] and power [kw/m] for UK seas [12] Figure 1.4: Wave period occurrence at UK seas Figure 1.5: Annual average wave energy density in Mexican seas [kw/m], ( 26 Figure 1.6: Annual average wave height, Hmean, in Mexican seas [m], ( 26 Figure 1.7: Annual average zero-crossing wave period, Tz, in Mexico seas [s], ( 26 Figure 2.1: Point absorbers, current technology Figure 2.2: Attenuators, current technology Figure 2.3: Terminators, current technology Figure 2.4: Oscillating water columns, current technology Figure 2.5: Overtopping devices, current technology Figure2.6: Schematic approach at still position of Manchester Bobber (courtesy of Prof Stansby) Figure 3.1 Boundary values for linear waves, adapted from Dean & Dalrymple [93] Figure 3.2: Wave energy differential mass approach. [68] Figure 3.3: Schematic representation of the work principle for the two-body converter configuration Figure 3.4: Freedom degrees for a simple floating body [94] Figure 3.5: Sensitivity analysis of the different methodology for body representation in WAMIT runs Figure 3.6: a) WAMIT floats array, b) Body representation by an a finite number of panels Figure 3.7: Hydrostatic coefficients, a) added mass, b) radiation damping Figure 3.8: Excitation force on converter floats due to heave mode of motion Figure 3.9: Power output for the two-float array (Case 4b); without (solid line) and with (dotted line) drag effects Figure 3.1: Power output at different float arrays configurations; case 1a (black line), case 2a (green line), case 3a (red line), case 4b (blue line), and case 5a (purple line) Figure 3.11: Power output (left), vertical float response (centre and right for float 1 and 2, respectively) at different mechanical applied forces for a wave amplitude of.1 m, B99, max = [(N-s)/m] Figure 3.12: Power contours as function of mechanical force, Rmech, and angular wave frequencies, w. 67 6

7 Figure 4.1: Manchester wave flume Figure 4.2: Wave basin at the Marine Institute at Plymouth University Figure 4.3: Working principle for the three-float array model Figure 4.4: Wave gauges array at the University of Manchester Figure 4.5: Wave gauges array at Plymouth University Figure 4.6: Calibration curve for the three wave gauges used at the University of Manchester flume s Figure 4.7: Strain gauge calibration, schematic configuration (left side), calibration curve (right side) Figure 4.8: Wave energy converter location at the University of Manchester flume Figure 4.9: PTO system for 1/4 scale model Figure 4.1: PTO system for 1/8 scale model Figure 4.11: Geometrical analysis to obtain Md Figure 5.1: Two-float configurations for experimental testing Figure 5.2: Power variation with wave period for Cfg1 (Table 5.2) for five different wave heights Figure 5.3: Power variation with wave period for Cfg2 (Table 5.2) for five different wave heights Figure 5.4:Comparison of the average power output with wave period for Cfg3 and Cfg4 at H =. 2 m.. 96 Figure 5.5: Comparison of the average power output with wave period for Cfg4 and Cfg5, H =. 2 m Figure 5.6: Comparison of the average power output with wave period for Cfg4 and Cfg6, H =. 2 m Figure 5.7: Schematic wave energy converter diagram with the three-float array Figure 5.8: Average power with wave period for the two-float and three-float array, and H=.2 m Figure 5.9: Average power output with wave period for different floats spacing, S1, and H=.2 m Figure 5.1: Average power output with wave period for five different float 3 drafts, d3, and H =. 2 m Figure 5.11: Average power output with wave period for different floats spacing, S2, and H =. 2 m.. 13 Figure 5.12: Average power output with wave period for different pivot point, h, heights and H =. 2 m Figure 5.13: Average power output with wave period for configurations with different d1 and H =. 2 m Figure 5.14: Average power output with wave period for different wave height Figure 6.1: Manchester device - Sketch of configuration with flat bases, dimensions [m]

8 Figure 6.2: Manchester device - Sketch of configuration with rounded bases, dimensions [m] Figure 6.3: Plymouth device - Sketch of configuration for large scale, dimensions [m] Figure 6.4: Power output comparison with wave period for converter with flat and rounded base floats Figure 6.5: Power output comparison with wave period for different wave hiehgts Figure 6.6: Variation of water surface elevation, η, moment, Md, angle, θ, and power output with time for regular waves of H. 23 m and T 1. 5 s. Converter with rounded base floats Figure 6.7: Variation of capture width ratio (based on Te) with ratio of peak period to Tp to resonant heave period for float 3, Tr3 (from WAMIT model) for uni-directional irregular waves and γ = Figure 6.8: Variation of water surface elevation, η, moment, Md, angle, θ, and power output with time for uni-directional irregular waves of Hs. 43 m, Tp s and γ = Figure 6.9: Variation of capture width ratio (based on Te) with ratio of peak period, Tp, to resonant heave period for float 3, Tr3 (from WAMIT model) for uni-directional irregular waves with γ = 1, Figure 6.1: Variation of capture width ratio (based on Te) with ratio of peak period, Tp, to resonant heave period for float 3, Tr3 (from WAMIT model) with γ = 3. 3 and s =, 3, 5, rounded-base floats Figure 6.11: Variation of capture width ratio (based on Te) with ratio of peak period to Tp to resonant heave period for float 3, Tr3 (from WAMIT model) with γ = 1. and s =, 3, 5, rounded-base floats Figure 6.12: Variation of average power output with real wave height for different wave periods Figure 6.13: Variation of average power output with wave height square for different wave periods Figure 6.14: Variation of water surface elevation, η, moment, Md, angle, θ, and power output with time for regular waves of H. 19 m and T 2. 5 s. Large scale model Figure 6.15: Variation of average power output with real wave height for different peak wave periods Figure 6.16: Variation of average power output with real wave height for different peak wave periods Figure 6.17: Variation of water surface elevation, η, moment, Md, angle, θ, and power output with time for uni-directional regular waves of Hs. 238 m and Tp 2. 6 s. Large scale model Figure 6.18: Converter performance for different ranges of mechanical damping Figure 6.19: Variation of capture width ratio (based on Te) with ratio of peak period to Tp to resonant heave period for float 3, Tr3 (from WAMIT model) irregular waves with γ = 3. 3 and s =, 3 and Figure 6.2: Variation of capture width ratio (based on Te) with ratio of peak period to Tp to resonant heave period for float 3, Tr3 (from WAMIT model) irregular waves with γ = 3. 3 and γ = Figure 6.21: Variation of power output with ratio of peak period to Tp to resonant heave period for float 3, Tr3 (from WAMIT model) irregular waves at significant wave height target of Hs =. 2 m

9 Figure 6.22: Comparison of the Capture width ratio between different laboratory scale models for unidirectional irregular waves with γ = 3. 3 and different significant wave heights Figure 6.23: Capture width ratio comparison between different laboratory scale models for spreading (s = 3) irregular waves with γ = 3. 3 and different significant wave heights Figure 6.24: Capture width ratio comparison between different laboratory scale models for spreading (s = 5) irregular waves with γ = 3. 3 and different significant wave heights Figure 7.1: Specific locations for power capture projections with M4 wave energy converter Figure 7.2: Wave energy occurrence and capture width for the selected places Figure 7.3: Variation of the average power output (left hand side) and power/mass ratio (right hand side) with the device length/predominant wave length ratio for the economics analysis

10 List of tables Table 1.1: Electricity trend [1] [MToe] Table 1.2: Carbon budgets for the period [8, 9] Table 1.3: Trend of total energy production coming from green energies [1] Table 1.4: Trend of total electricity production coming from green energies [1] Table 1.5: Renewables energies contribution to electricity production in UK [11] Table 1.6: Trend of total electricity production coming from green energies [1] Table 1.7: Renewables energies contribution to electricity production in Mexico [11] Table 2.1: Key points of the point absorbers devices Table 2.2: Key points of the point absorbers devices Table 2.3: The Salter s Duck characteristics Table 2.4: Oscillating water columns technical aspects Table 2.5: Overtopping devices Table 3.1: Different sizing floats configuration Table 4.1: Scaling factors [68] Table 4.2: Wave parameters at full-scale prototype Table 5.1: Versions of the two float 1/4th scale model for experimental testing Table 5.2: Floats and wave Parameters for experimental phase Table 5.3: Floats and wave Parameters for experimental phase Table 5.4: Geometrical parameters for some configuration using a the three-float array Table 5.5: Distance variation of spacing between float 2 and float Table 5.6: Optimal converter configuration by the experimental phase Table 6.1: Mass distribution and inertias for flat base configuration with origin at hinge point Table 6.2: Heave natural periods for flat base floats based on WAMIT Table 6.3: Mass distribution and inertias for rounded base configuration with origin at hinge Table 6.4: Heave natural periods for rounded base floats based on WAMIT Table 6.5: Mass distribution and inertias the for 1/8th scale with origin at hinge Table 6.6: Heave natural periods for rounded base floats based on WAMIT

11 Table 6.7: Parameters for comparison of the capture width ratio- s = Table 6.8: Parameters for comparison of the capture width ratio - s = Table 6.9: Mechanical damping applied in both scale models at each wave period tested with directional spreading swell waves with s = Table 6.1: Parameters for comparison of the capture width ratio - s = Table 7.1: Sites locations [121, 122] Table 7.2: Scatter plot for the Wave Hub, UK Occurrence (%) [121], note shaded areas do not contribute to power generation (56.7%), λs = Table 7.3: Scatter plot for the Wave Hub, UK Occurrence (%) [121], note shaded areas do not contribute to power generation (7.26%), λs = Table 7.4: Wave data for power capture projections with a full-scale prototype with λs = 4, and converter system with a length scale factor tuned at each specific place Table 7.5: Capture width ratio at each bin of zero-crossing periods for a device with λs = Table 7.6: Power matrix for a converter system with λs = 4 at the Wave Hub [kw] Table 7.7: Energy yield for a converter system with λs = 4 at the Wave Hub [kwh/y] Table 7.8: Capture width ratio at each bin of zero-crossing periods for a device with λs = Table 7.9: Power matrix for a converter system with λs = 81 at the Wave Hub [kw] Table 7.1: Energy yield for a converter system with λs = 81 at the Wave Hub [kwh/y] Table 7.11: Power projection results summary for all investigated sites Table 7.12: Power projections for the converter system with different scales and the wave data obtained for the Wave Hub site

12 Nomenclature A Added mass coefficient a I Incident wave amplitude [m] a R Reflected wave amplitude [m] a wz Added mass [kg] A wp Projected area onto the water plane [m 2 ] B Radiation damping coefficient B d Mechanical damping from experiments [Nms] b Width of the wave front [m] b r Radiation damping [Ns m] b v Viscous damping [Ns m] b p Mechanical damping [Ns m] C Spring constant [N/m] C g Group wave velocity [m/s] C pto Multiplication damping factor CW Capture width CW r Capture width ratio d Water depth [m] E p Potential energy [J] E k Kinetic energy [J] F a Added mass force [N] F e Excitation force [N] F e Complex amplitude of the excitation force [N] F ext External force [N] F mech Mechanical force [N] F PTO Damping force [N] F r Radiation force [N] F z Excitation force on z direction [N] f Wave frequency [Hz] f p Peak wave frequency [Hz] f n Natural frequency [Hz] g Gravitational acceleration [m s 2 ] H Wave height [m] H s Significant wave height [m] k Wave number [1/m] K R Reflection coefficient k s Mooring line stiffness [N/m] l Actuator rod displacement [m] l Actuator rod velocity [m s] l mid Distance from the hinge normal to the actuator axial line [m] l m Characteristic laboratory model length [m] l p Characteristic prototype length [m] M Mass [kg] M d Moment around the hinged joint [N-m] N Number of mooring lines P mech Mechanical power [W] Power output [W] P out 12

13 P w Regular wave power [W] P w,irreg Irregular wave power [W] p Pressure [N/m 2 ] R Mechanical damping for numerical model [Ns m] S Stiffness [kg/s 2 ] s Spreading factor T Wave period [s] T e Energy wave period [s] T p Peak wave period [s] T z Zero-crossing wave period [s] t Time [s] U Velocity [m/s] U Complex velocity [m/s] u Water particle velocity in x direction [m/s] w Water particle velocity in z direction [m/s] z Vertical displacement [m] z Vertical velocity [m/s] z Vertical acceleration [m s 2 ] x Horizontal displacement [m] x Horizontal velocity [m/s] x Horizontal acceleration [m s 2 ] γ Peak enhancement factor η Water surface elevation [m] λ Wave length [m] λ s Length scale factor [m] ρ Water density [kg/m 3 ] ω Angular wave frequency [rad/s] ω n Natural angular wave frequency [rad/s] Γ Gamma function 13

14 Abstract The University of Manchester Efrain Carpintero Moreno Doctor of Philosophy Wave energy conversion based on multi-mode line absorbing systems March 215 Wave energy conversion remains a promising technology with substantial renewable resources to be exploited in many parts of the world. However to be commercially attractive more effective conversion is desirable. There is scope for increasing power capture by use of several bodies responding with several modes, some or all of which may undergo resonance for frequencies within a wave climate. This theme is explored here with a floating moored line absorber system where the relative motion generates power by incorporation of a damper to represent the power take off. To be most effective the bodies should be responding in anti-phase requiring spacing between adjacent bodies of half a wavelength. First a converter design including two bodies is investigated experimentally and numerically responding solely in heave. The bodies have drafts to provide resonant frequencies within a wave spectrum, the stern diameter is as large as possible within the inertia regime and the bow diameter is optimised to provide maximum power. Experiments showed this system to be limited since the desirable anti phase heave modes were contaminated with other modes for off resonance response considerably reducing power generation. To stabilise motion in the desired modes another small float was introduced as the bow float rigidly connected by a beam to the mid float with the added benefit of adding forcing due to surge and pitch to some degree (following Prof Peter Stansby s design). The sizes of the three floats increase from bow to stern, causing the line absorber to align with the wave direction. This system was optimised through experiments varying float spacing, diameter, draft and the hinge point above the mid float about which relative angular motion occurs. These experiments were undertaken at small scale in the wide Manchester University flume at about 1/4 th scale. Regular and random (JONSWAP) waves were investigated including directionality and different spectral peakedness factor. Corresponding experiments were undertaken at five time larger scale (about 1/8 th ) in the wave basin at the COAST laboratory of Plymouth University. These tests were for a flatbased floats; the mechanical damping coefficient for larger scale was within the range for the smaller scale tests after appropriate (Froude) scaling. Tests at Manchester showed that the more rounded base floats (the mid float being hemi spherical) provided improved power capture. Device effectiveness is defined in terms of capture width ratio; that is the average power divided by the wave power per metre divided by the wavelength, defined by the energy period in the case of irregular waves. The experiments showed that capture width ratios were greater than 25% in regular waves and greater than 2% in irregular waves across a broad range of wave periods. With rounded base floats capture width ratios over 2% were achieved for a broad range of wave frequencies up to a maximum greater than 35%. Limited experiments at larger scale showed that increasing the damping coefficient could increase power capture by about 5%. Characterisation by capture width ratio is convenient for determining annual energy yield from scatter diagrams. This was undertaken for six sites of interest for wave energy conversion. It was assumed that the greatest power to weight ratio determines the most economic device; it was found that large devices could produce very large average power, for example average power of 2 MW, but the optimum power/weight ratio occurred at smaller scale, with average power typically.3 MW. 14

15 Declaration No portion of the work referred to in the thesis has been submitted in support of an applicant for another degree or qualification of this or any other university or other institute of learning. 15

16 Copyright Statement i. The author of this thesis (including any appendices and/or schedules to this thesis) owns certain copyright or related rights in it (the Copyright ) and s/he has given The University of Manchester certain rights to use such Copyright, including for administrative purposes. ii. Copies of this thesis, either in full or in extracts and whether in hard or electronic copy, may be made only in accordance with the Copyright, Designs and Patents Act 1988 (as amended) and regulations issued under it or, where appropriate, in accordance with licensing agreements which the University has from time to time. This page must form part of any such copies made. iii. The ownership of certain Copyright, patents, designs, trademarks and other intellectual property (the Intellectual Property ) and any reproductions of copyright works in the thesis, for example graphs and tables ( Reproductions ), which may be described in this thesis, may not be owned by the author and may be owned by third parties. Such Intellectual Property and Reproductions cannot and must not be made available for use without the prior written permission of the owner(s) of the relevant Intellectual Property and/or Reproductions. iv. Further information on the conditions under which disclosure, publication and commercialisation of this thesis, the Copyright and any Intellectual Property and/or reproductions described in it may take place is available in the University IP Policy (see in any relevant Thesis restriction declarations deposited in the University Library, The University Library s regulations (see /regulations) and in The University s policy on Presentation of Theses. 16

17 Dedication To my lovely daughter, wife and family Acknowledgement I would like to thank my supervisors Prof Peter Stansby and Dr Tim Stallard and express all my gratitude for their support, encouragement and guidance since the first day of this wonderful journey called PhD. With their help the understanding of the concepts cover in this research was achieved. There have past four years with limited contact with my family, but their infinite love and support are always with me. I have not a single word to describe the huge thank to you. I also would like to express my sincere gratitude to all my colleagues and friends for all their support during these years. With colleagues and friends like you guys, the time in the office is really enjoyable. I want to express all my thankfulness to Prof Teresa Alonso-Ragado and Dr Sergio Areas (The University of Guanajuato, Mexico) for their advice and coaching to obtain the position as PhD student in the University of Manchester. And last but not less important, I like to thank so much to CONACYT for all the financial support and recommendations in order to the completion of this degree. 17

18 The Author The author has previously completed a Master in Mechanical Engineering at the University of Guanajuato, Mexico. During the PhD, the author has been involved in experiments and analysis of the WECwakes project funded by EU FP7 HYDROLAB IV programme (conference papers have been published with collaboration of the author as results of this investigation), SuperGen UKCMER doctoral training programme workshop Mooring and Reliability. Two journals and two conference papers have been published (one journal and one conference paper with a submitted status) related with the subject developed during the PhD with the collaboration of the author. 18

19 Chapter I Introduction World energy demand is mainly supplied by fossil fuels. Although fossil fuels could cover the current needs there are uncertainties about how long fossil fuels presently in reservoirs can cover the future world requirements. With the constant increase in energy demand and concerns over climate change, the renewable energies such as wind, solar, biofuels, hydro, and ocean energy (tidal and wave energy) take an important role in reducing dependence on fossil fuels and the environmental risk due to global warming. In recent years, there has been increasing interest in electricity generation from ocean waves. Many concepts for wave energy capture have been proposed, but no single design is yet commercially viable. Many of these devices generate electricity from wave-induced oscillations of a floating body. The power output of such systems is maximized when the system response is in resonance conditions. It is not straightforward to attain resonance over the wide range of wave frequencies that occur at an offshore site, since resonance response is over a narrow frequency bandwidth for wave energy conversion from a single device. A possibility for increasing power output per wave energy converter is to consider a system with several response modes, for example an array of several bodies each with a different natural period within the typical range of wave periods for a given site. Wave energy conversion has two major challenges; 1) generation of electricity at a competitive rate, and 2) survival of extreme loads on the wave energy converter in storm conditions. This research investigates the response, power capture, and device optimization for a novel wave energy converter with multiple-modes of motion Global status of renewable energy The global electricity demand has shown a trend in annual increase of around 3% between 2 and 213 [1], see Table 1.1. This annual increase together with the global warming risk emphasises the importance of the power generation method, which so far is highly dependent on fossil fuel combustion where a high level of carbon emissions are produced as greenhouse gases. Since the Kyoto protocol in 1997, member countries have implemented many policies for decarbonising in order to address global warming. UK and 19

20 Mexican actions to address these concerns have been considered in this research because the author has Mexican nationality and the host University is in the UK. In order to have a power supply free of carbon emissions renewable energy plays a crucial role. Wind, solar and hydro power generation are established technologies. However, other technologies are also being investigated, such as ocean energy (wave and tidal power generation). In 213, 28% of the total world electricity generation was covered by renewable energy, including hydroelectric power generation [1]. Nevertheless, the wave energy contribution to the total electricity production is negligible as the wave energy industry is still in its infancy. Figure 1.1 presents a general overview of the green or renewable power generation by most countries, where it is possible to see that UK and Mexico are delivering an electricity production from green energies below 2% of the total production, 16.1 and 13.7% respectively. There are just a few countries with more than 5%: Norway (97.9%), Colombia (73.7%), Brazil (77.1%), Venezuela (68.8%), Canada (62.7%), Portugal (62.5%) and Sweden (53.2%). Table 1.1: Electricity trend [1] [MToe] World UK Mexico Table 1.1: Continuation [MToe] (% increase/year) (% increase/year)

21 Figure 1.1: Global electricity energy production on 214, [1]. Nevertheless, the wave energy resources around the world are quite abundant. A general wave energy resource overview is given by Figure 1.2 [2]. These figures are in good agreement with the theoretical offshore wave energy resource assessment performed by Gunnar [3]. The numerical model used by Gunnar was calibrated against satellite altimeter data (Topex, Jason-1) for a 1-year period between 1997 to 26 with intervals of 6 hours on a.5º latitude and longitude grid, and the model outputs were validated against measured buoy data (The World Waves database). Many other wave energy assessments have been presented by [4-6], giving similar results to those figures given at Figure 1.2. Figure 1.2: Global energy resources, wave power per meter of wave front [kw/m], [2]. 21

22 1.2. Renewable energy in UK In order to reduce the environmental impact due to high levels of greenhouse gas emission, the UK government has been set an ambitious target for 25, which would reduce the UK emissions by 8% with respect to the UK emissions baseline in 199 [7]. This reduction was planned to be gradually achieved with specific targets at time intervals of five years as is shown in Table 1.2 [8, 9]. At the same time, the energy consumption provided by renewable energy has been set to 15% by 22 and between 3-45% by 23 [1]. Table 1.3 presents an increasing trend in the total energy production coming from renewable energies in UK [1]. However to reach the proposed targets, strong actions must be taken since the 22 target is high in relation to current trends. In the last 4 years, the electricity production in the UK by green technologies has experienced a significant growth, doubling in four years; from 8.% (21) to 16.1% (213), Table 1.4 [1]. Nonetheless, these figures could be considerably improved taking advantage of the ocean energy potential around the UK, which so far contributes less than 1%. The 22 target percentage on Table 1.4 was estimated using the figures for 213, where the 5.8% of the total power includes 16.1% of the total electricity generated by renewable energy. Table 1.5 shows the contribution to electricity generation from each green energy source. Table 1.2: Carbon budgets for the period [8, 9]. 1 st carbon budget (28-212) 2 nd carbon budget ( ) 3 th carbon budget ( ) 4 th carbon budget ( ) Carbon budget level [MtCO 2e] Reduction respect to the baseline [%] Table 1.3: Trend of total energy production coming from green energies [1]. Year Target 22 % Table 1.4: Trend of total electricity production coming from green energies [1]. Year Target 22 %

23 Table 1.5: Renewables energies contribution to electricity production in UK [11]. UK Hydro Ocean Wind Solar Bioenergy Production [GWh] Contribution [%] 2.3 Less than Less than Wave energy resource in UK seas The UK is surrounded by seas with enormous wave energy transmission to the coast; specially at north west of Scotland (North Atlantic), where the annual average of wave power reaches up to 6 kw per metre of wave front [12]. Figure 1.3 shows the annual average of wave height [m] and power [kw/m], in left and right pictures respectively. It is possible to see that the annual average of offshore wave height for the UK seas is up to 3.25 meters. These factors together with the wave period and ocean bathymetry are key parameters for input to the design of an appropriate wave energy converter for a specific site. The wave period determines the resonance condition for the converter, the ocean bathymetry defines water depth which determines the choice of a fixed or floating device, since at large depth the economics of a fixed structure is prohibitive. The wave power density together with the converter capture width (power from a length of wave crest which is converted into mechanical power for conversion to electricity) give a realistic estimation of the power capture. Figure 1.3: Annual average wave height [m] and power [kw/m] for UK seas [12]. Reproduced from Crown Copyright. 23

24 The wave period occurrence for UK seas is presented in Figure 1.4 [13], where the UK seas are divided into eight geographic regions as depicted on the left side: Northern North Sea (NNS), Central North Sea (CNS), Southern North Sea (SNS), English Channel (EC), Celtic Sea (CS), Irish Sea (IS), Hebrides (H), and Western Shetland Shelf (WSS). On the right side, it is possible to identify that the wave period in the North Sea decreases from the north to the south. The CS, EC, H, and WSS have a wide range of spectral peak periods (covering about 1 s) where the swell conditions provide a great contribution to the wave climate. The SNS and IS have narrow ranges due to the wave generation dominated by wind; with spectral peak periods between 4-9 s and 3-8 s, respectively. a) b) Figure 1.4: Wave period occurrence at UK seas. a) geographic division, b) wave period occurrence for NNS (, dark blue), CNS (, pink), SNS (, green), EC (x, cyan), CS (*, dark purple), IS (, brown), H (+, orange), WSS (, blue), [13] Renewable energy in Mexico Mexico is a nation with its economy based on oil production (9 th in the world for oil production [14]), and is thus highly dependent on fossil fuel combustion. However, Mexico has a great potential in green energy, which has been exploited very little so far. Mexico is facing the climate change problems by setting some regulation for electrical energy generation. This is a two stage process; a) 35% of the total electrical energy generated must come from renewable sources by 224, and b) at least 5% of total 24

25 electricity production by 25 [15]. Table 1.6 presents an irregular trend in the total green energy production in Mexico [1], where it is possible to see that the actual green energy production is less than a half of the target for 224. This means considerable improvement is necessary in order to achieve the desired target. Hence, additional ways to supply green energy should be found. Table 1.7 shows the different green energy contributions in Mexico, and it can be seen that the ocean energy is not contributing although Mexico is surrounded by the Pacific Ocean on the west and the Gulf of Mexico on the east. For this reason, in the next section an overview of the wave energy resources in Mexico is presented. Table 1.6: Trend of total electricity production coming from green energies [1]. Target Target Year % Table 1.7: Renewables energies contribution to electricity production in Mexico [11]. Mexico Geothermic Hydro Ocean Wind Solar Bioenergy Production [GWh] Contribution [%] Less than 1 Around 1 Wave energy resource in Mexican seas Recently, wave energy conversion has experienced increasing interest in Mexico, and some work has been developed at UNAM and CICESE [16-19] funded by the government. The assessed wave resources are given in Figures 1.5 to Figure 1.7. The power density in Mexican seas reaches up to 45 kw per metre of wave front, particularly at Baja California Sea (see Figure 1.5), and the wave height average for this zone is around 2 m (see Figure 1.6). The wave period around Mexican seas varies between 4-12 s with the shorter periods on the Gulf of Mexico (4-6 s), and from 9 to 12 s in the Pacific Ocean decreasing from the northern part towards the Equator (see Figure 1.7). 25

26 Figure 1.5: Annual average wave energy density in Mexican seas [kw/m], based in Hs and Te ( Figure 1.6: Annual average wave height, H mean, in Mexican seas [m], ( Figure 1.7: Annual average zero-crossing wave period, T z, in Mexico seas [s], ( 26

27 1.4. Research objectives The objective of this research is to optimize the power capture of a wave energy converter designed to have good capture width performance and ease of deployment, and to quantify the annual energy yield due to sea state conditions at chosen deployment sites. In order to improve the wave energy converter performance, it is necessary to understand the converter behaviour, so specific objectives are defined below. Specific Objectives Conduct an experimental programme to quantify converter performance based on a multi-body line absorber with multi-mode response including resonance with a geometrically-scaled model for a representative range of wave frequencies and wave amplitudes. Develop a numerical model to predict the energy converter response and power output at different wave conditions. Quantify the annual energy yield considering realistic wave characteristics for particular locations using the experimental work which provides the capture width ratios Summary Ambitious targets to reduce carbon emission or greenhouse gases have been agreed around the world. The UK government has proposed a structured plan to reduce carbon emissions by 8% by 25, relative to the carbon emission baseline for UK in 199. The Mexican government has presented a plan to obtain a least 5% of its electricity from renewable energy by 25. Both plans are divided into several phases in order to monitor the target achieved at the end of each phase; these phases correspond to time periods of five and ten years for UK and Mexico, respectively. These actions encourage assessment of marine energy as a promising energy source which has a high level of energy concentration for UK offshore sites, notably for North West of Scotland. Despite to the lower level of wave energy in Mexican seas, they still have a significant potential due to the long coastline. 27

28 In order to take advantage of the wave energy around UK and Mexican seas an investigation of new more effective forms of wave energy system is undertaken here for offshore conditions where wave energy is higher than for nearshore sites. The systems have to be moored and floating for larger water depths, typically greater than 2 m. Multibody line absorbers with multi-mode response are considered with variable resonance responding to wave conditions. Device response, power capture and survivability are investigated. First two bodies will be considered. 28

29 Chapter II Literature review Many concepts of wave energy conversion have been proposed and investigated in recent years, which transform energy transported by ocean waves into electrical energy. These converters can be grouped in five categories: point absorbers, attenuators, terminators, oscillating water columns and overtopping devices. The wave energy converter design ideally involves experimental and numerical modelling in order to get a suitable prediction of the converter performance. Different numerical model techniques have been utilized in order to predict the power performance and the wave-structure interaction such as linear wave diffraction theory in time and frequency domains for regular and irregular waves and computational fluid dynamics (CFD) models. The present section gives a general description of the current state of the industry and a general description of the modelling techniques Current devices classified by converter type Point absorbers are devices moving in heave, pitch, surge, or some combination of them. The main characteristic of these devices is that they are relatively small compared with the wavelength of the incident wave. The power absorption by point absorbers is maximized at a resonance condition, where the capture width (ratio between power captured and the incident wave power per metre) is given by 1/2π, 1/π and 1/π wavelengths for heave, pitch and surge, respectively [2]. These capture widths for a single point absorber are for a narrow band of frequencies. Examples of wave energy converters using a point absorber concept comprising a single unit are: Wavebob [21, 22], Archimedes Wave Swing (AWS) [23-25], CETO [22, 26], PS Frog [27, 28], Oyster [29, 3]. Other converter designs using an array of point absorbers in order to increase the power generation are: WaveStar [31, 32], Manchester bobber [33-35], Langlee [36]. Most of the point absorbers mentioned above are devices working in a single mode of motion with the exception of the PS Frog, Figure 2.1d, which generates power output by surge and pitch motion. Wavebob comprises two axisymmetric floats (Figure 2.1a), one floating at the free-surface and the other totally submerged, the system is connected to seabed by a mooring line; the power by this converter is due to heave motion of the floating buoy at the free surface. 29

30 AWS is a completely submerged device fixed at the sea bed (see Figure 2.1b), this converter comprises a stationary structure and a heaving part (cap); a volume of air is trapped between the stationary structure and the cap. Therefore, due to pressure of the sea water above, the air is pressurised and depressurised driving the cap which is attached to a hydraulic actuator to produce electrical energy. CETO and Oyster are wave energy converters which pump high pressure water to a power plant station onshore, the power conversion by these devices is carried out by hydraulic turbines in the power plant. The main difference between these converters is that CETO is a heaving point absorber which is totally submerged, see Figure 2.1c, while Oyster is a surging point absorber fixed at sea bed, Figure 2.1e. Manchester Bobber and Wave Star have a fixed support structure with arrays of heaving floats, Figure 2.1g and 2.1f, respectively. But power generation is totally different; on the Manchester Bobber the power is taken by a permanent-magnet motor driven by an inertia system as explained in section 2.5, while the power in Wave Star is captured due to hydraulic actuators as described by Hansen et al. [12] and Ferri et al. [13]. Langlee is a floating structure containing surging water wings where the power is captured by hydraulic actuators [17], Figure 2.1h. Table 2.1 shows a summary of the key points in the current technology for point absorbers. a) Wavebob [37] b) AWS [25] 3

31 c) CETO courtesy of d) PS Frog [27] e) Oyster [3] f) Wave Star [31] g) Manchester Bobber [34] h) Langlee [22] Figure 2.1: Point absorbers, current technology. WEC Peak Power [kw] Table 2.1: Key points of the point absorbers devices. WEC Position Water depth [m] Wavebob 5 Floating Over 5 Main characteristics -It comprises two axisymmetric circular floats; one completely submerged and a floating. -Linear generator. -Heaving device. No. of units Status 1 Closed Table 2.1 to be continued in the next page 31

32 AWS 25 Floating Over 1 CETO 24 Submerged 25-5 PS Frog 2 Floating Over 7 Oyster 8 WaveStar 2 Manchester Bobber 5 Langlee 132 Fixed at seabed Fixed platform Fixed platform Semisubmerged Around 12 - ~2 m 4-1 -Comprises 12 floating axisymmetric circular cells of 16 m of diameter. -Capacity factor of 25% -Heaving device -Multipurpose device; freshwater and power production, axisymmetric circular float of d=11 m. -Heaving device -A flat floating paddle roughly of 25 m and 3 m, for depth and width respectively. - Sliding of a powertake-off mass -Pitching and surging device. -Flat paddle of 18 m wide and 11 m height. -Power generation on shore -Surging device -2 hemispherical floats array. -Average power around 4 kw -Heaving device -Array of 1 hemispherical-ended cylindrical floats. -Heaving device -3x5 m structure -Up to 2% of CW -Surging device 12 Active 1 Active 1-1 Closed 2 Active 1 Closed 1 Active Attenuators (also called line absorbers) are wave energy converters with a predominant dimension aligned with the incident wave direction. Pelamis Wave Power [38, 39], and Anaconda [4] are common examples of this kind. Pelamis is a floating converter which comprises five cylindrical segments which produce power by the relative pitch motion, Figure 2.2a. A single unit of Pelamis has a rated capacity of 75 kw. Pelamis Wave Power is the most investigated project so far and it was the first design to generate electricity for a national grid. After a testing phase between 24 and 27 some modifications were made giving a new design called Pelamis 2. At the time of writing, the Pelamis team have some projects which aim to reach up to 1 MW in a farm comprising 1 devices. Anaconda wave energy converter is a totally submerged 32

33 closed rubber tube filled with water, Figure 2.2b, where bulge waves are generated due to the external surface waves. These bulge waves are used to produce power at one end of the tube (stern) by activating a hydraulic actuator. Table 2.2 presents a summary of the main characteristics about the attenuators presented in this section. a) Pelamis [41] b) Anaconda [4] Figure 2.2: Attenuators, current technology. WEC Table 2.2: Key points of the point absorbers devices. Rated Water WEC Power depth Main characteristics position [kw] [m] Pelamis 75 Floating - Anaconda 1 Submerged 2-18 m long and 4 m diameter. -Up to 2 m of capture width (CW of 11%). -Pitching device. - Long distensible rubber tube. -15m long and 7 m diameter. -Capture width above 3 m. No. of units Status 1 Closed 1 Active Terminators, these converters have the principal axis perpendicular to the incident wave direction, Figure 2.3. One example of this kind is Salter s Duck [42, 43] developed by Professor Stephen H. Salter at the University of Edinburgh. The Duck is comprised by a finite number of asymmetric rotary segments which face the incoming waves to produce power output. A complete summary of the work done with this converter can be found in the Joao Cruz book [38]. Table 2.3 gives some information about this converter, a detailed analysis is given by the provided references. 33

34 a) The Salter s Duck [43] Figure 2.3: Terminators, current technology. WEC Rated Power [kw] Table 2.3: The Salter s Duck characteristics. WEC Water depth position [m] Salter s Duck 375 Floating 3 Main characteristics -Maximum performance peaks around 7 s and 11.5 s. -Pitching device. No. of units Status - Closed An oscillating water column (OWC) is composed of a partially-submerged structure which forms a chamber with an opening so that pressures from ocean waves cause enclosed water to oscillate which in turn causes air in the chamber above the water to oscillate through an orifice containing a Wells turbine and a generator. This turbine rotates in the same direction regardless of the direction of the air flow. OWCs were widely investigated in the very early stage of ocean wave energy due to easy accessibility on a shoreline. Some examples of this technology are LIMPET [44, 45], Figure 2.4a, which was developed at Queen s University of Belfast by Professor Trevor Whittaker with a rated capacity of 25 kw, the Mighty Whale [46, 47], Figure 2.4b, which is an array of OWC with a maximum efficiency of 15% (the percentage of the captured power upon the wave power incident over the device width), and the Ocean Energy buoy [48], Figure 2.4c. Some technical aspects about these converters are given in Table

35 a) LIMPET [44] b) Mighty Whale [47] c) Ocean Energy Buoy (www. oceanenergy.ie) Figure 2.4: Oscillating water columns, current technology. WEC Table 2.4: Oscillating water columns technical aspects. WEC Water depth Main characteristics position [m] Rated Power [kw] LIMPET 25 Onshore - Mighty Whale 11 Floating Over 4 -Deployment on shore to easy electricity delivery kw of average power. -Operational power plant. -Three oscillating water columns kw of average power. -Maximum power output around 6-7 s of wave period. -Operated at full scale between 1998 and 22. No. of units Status 1 Active 1 Closed OE buoy - Floating Active Overtopping devices have a curved ramp facing the incoming waves, the waves propagate over the ramp and then drop vertically into a reservoir, storing the water as potential energy which is drained by ducts located over the bottom of the reservoir; these contain hydraulic turbines to transform the kinetic energy into electrical energy. Different designs of wave energy converter using the overtopping principle have been proposed; some of 35

36 which are depicted in Figure 2.5. Wave Dragon [49-51] (Figure 2.5c), the tapered channel wave power device (Tapchan, Figure 2.5b) [52] and Sea Wave Slot cone Generator (SSG, Figure 2.5b) [53, 54] are examples of this kind of wave energy converter. Some technical design aspects for these devices are given by Table 2.5. a) SSG [53] b) Tapchan [55] c) Wave Dragon [51] Figure 2.5: Overtopping devices, current technology. WEC Rated Power [W] WEC position SSG 163 On shore - Tapchan 35 On shore - Wave Dragon 4 Floating 2-5 Table 2.5: Overtopping devices. Water depth [m] Main characteristics -Four hydraulic turbines (Kaplan turbines) connected by a vertical shaft. -Overall turbine efficiency between 1-26% -Estimated an annual production of 32 MWh/y. - 4 m for the channel entrance and 17 m long. -A consistent power delivery -A prototype operation stated on 1985 in a Norway island. -It comprises wave reflectors which increase the wave height by about 5 to 75%. -A full scale prototype for 2 kw/m of wave resource will comprise a 5 m^3 reservoir No. of units Status 1 Active 1-1 Closed 36

37 2.2. Modelling techniques Several techniques to analyse wave-structure interaction and wave field perturbation by the converter presence have been developed. However a single one cannot solve all aspects for any system. Each one has a different set of characteristics which make it more or less suitable depending on the problem to be solved and the available computational resources. In principle the most general approach is through Computational Fluid Dynamics (CFD) models but this can be very expensive in terms of computational resources. A summary of the modelling techniques is presented in this section. Linear wave diffraction theory The linear wave theory is based on potential flow theory defined by a velocity potential satisfying the continuity equation. Potential flow assumes incompressible, inviscid, and irrotational flow. There are two other assumptions for a linearised solution which are: the wave height must be much smaller than the wave length and much smaller than the water depth; if the body is responding, the body motion must also be small. An early application of potential flow theory was undertaken by Havelock in 194 [56]. Since then, many studies of offshore structure loading and wave-structure interaction analysis have been based on linear wave theory. Commercial software such as WAMIT, AQWA and ANSYS have been developed for an efficient solution of the diffraction problems for single bodies or/and array of bodies. Recently at the University of Manchester, Bellew [33] used the linear theory to calculate the forces on an array of floating hemisphere bodies due to interaction with a train of incident waves and compared with experimental results demonstrating that linear wave theory is applicable to closelyspaced arrays in many situations except at high frequencies. Generally, linear wave diffraction theory is applicable to wave-structure interaction in small to moderate sea states. Numerical predictions are in good agreement with experimental measurements and CFD simulations [57, 58]. However, the limitations in the linear wave theory become apparent with large sea states due to the violation of the small wave height criterion. Techniques using the linear wave theory have been developed such as frequency-domain, semi-analytical approaches (point absorber method [59]), the plane wave theory [6], multiple scattering [61], direct matrix method [62]), and time-domain formulation. 37

38 CFD models CFD Models have had success in engineering areas such as turbo-machinery, combustion and aeronautics. A CFD model solves the Navier Stokes equations which are derived from the mass and momentum conservation. CFD models are based on finite element method (FEM), finite volume method (FVM), finite difference method (FDM) or spectral elements method (SEM) [63]. Recently, CFD has become an alternative to simulate the wavestructure interactions. In contrast with linear diffraction wave theory, CFD models may include the full nonlinear problems with viscous effects and two-phase (water-air) flow which makes these models an attractive option to simulate extreme wave loading as reported by Westphalen [64]. In addition to the advantages described above, CFD offers an option for analysing the power output and response for different converter configurations in order to find the optimal configuration. With CFD it is possible to change the converter geometry or configuration with minimal cost. There is not the necessity to build different physical models, but the accuracy of the results is highly dependent in the assumptions made in the CFD model, particularly in relation to meshing and turbulence modelling. Some examples where these models have been applied are given by Omidvar et al. [65] using SPH, Wood [66], and Agamloh et al. [65-67]. Spectral wave modelling Wind waves are generally represented as a spectrum with JONSWAP commonly used. Spectral wave propagation inshore is modelled through the conservation of wave action which includes source terms due to wind generation [68]. These models are thus classified as phase-averaged predicting the spectral evolution with bathymetry due to shoaling, refraction, wind forcing, bottom friction and wave breaking dissipation. Since the wave spectral models solve the conservation equation, any structure (or wave energy converter) may be represented as an energy absorber and the radiated energy may be prescribed. There are two open source codes available: SWAN model developed at The University of Delft [69], and the TOMAWAC model created in the Électricité de France [7]. Two different methods to represent arrays of wave energy converters in a spectral wave model have been developed: supra-grid and sub-grid [71], where the array of converters is represented with a single coefficient of energy transmission or each device may be treated 38

39 as a source/sink of wave energy (a local grid point). Some examples of the spectral modelling of wave energy converters are given by Folley and Whittaker [72], Millar et al. [73], and Smith et al. [72-74] Performance evaluation of WECs experimental modelling The wave energy converters presented above require the problem of economic viability to be addressed. To be credible with governmental and private institutions (investors), the converters must show a good performance for a range of sea states at the target location for deployment. However, no design has proved economically viable to date. Many studies on theoretical converter performance assessment have been based on linear wave theory and small-scale and compare their results against experiments with smallscale models in flumes or basin facilities, since the investigation reported by Evans et al. [75], where theoretical and experimental performance results have an agreement of 98.2% just for small and regular waves. Therefore, regular waves are mainly used to understand the principle of energy conversion rather than the converter performance assessment. Capture width is a useful measure for evaluating the converter performance and it is given by the ratio between the power absorbed by the converter and the wave power per metre width of wave crest. The optimum performance is achieved when the natural frequency of converter oscillation equals the wave frequency and the mechanical damping produced by the Power take off (PTO) system is equal to the radiation damping [76, 77]. The mechanical damping applied by the PTO system may be adjusted to optimize the converter performance since this damping controls the converter response. Several techniques to control the converter performance have been proposed; one of the most investigated is latching control [77], which attempts to minimise the difference between the phase of the converter oscillation velocity and the excitation force induced by the incoming waves. De Becker et al. [78] studied an array of 12 heaving point absorbers and found that the power capture is more uniform through the whole array by applying a constraint technique (response constrain for example) rather than unconstrained arrays. De Becker applied different constraint methods such as: optimum parameters (OP) for single body, diagonal optimization (DO) through the array and an individual optimization (ID). It was seen that the OP method was not an efficient way to improve the array performance due to not all 39

40 converter units reach the optimum working conditions simultaneously, while the ID method increases the array performance by 14% compared with the DO and gives a more uniform response to all converter units which could be beneficial for an effective simple maintenance programme Several theoretical studies on single or arrays of wave energy converters have been published following Budal et al. [59]. However, in contrast with the theoretical researches, just a few experimental studies have been published. One of the first experimental studies was reported by Budal et al. [79] who reported the response and power absorption for a couple of hemispherical-ended heaving floats finding good agreement with point absorber theory for the interaction factor, q, defined by the ratio of the average power of an array of floats divided by the sum of average powers from the same number of floats in isolation. An experimental study was conducted by Stallard et al. [35] for different arrays of closelyspaced heaving point absorbers. A positive interaction was seen where the mechanical power observed for some floats produced almost twice that for the same system in isolation. In general, it was seen that the power output can be improved for closely-spaced arrays in comparison with isolated devices for wave periods greater than the natural period for the device and may be reduced for periods smaller than the natural period. Stratigaki et al. [8], report an experimental study of an array of 25 axisymmetric heaving floats subject to several sea state conditions including spread seas with a spreading parameter (s = 1). Stratigaki et al. found that the wave height was reduced up to 18.1% downwave by the presence of the array of converters which affects the other wave energy converters and the environment itself Response in extreme conditions One of the most significant problems for wave energy converters is that the design should be robust enough to survive extreme wave conditions. Theoretical and experimental methodologies have been developed for modelling wave energy converters under extreme conditions, a review of the existing methodology is presented by Coe and Neary [81]. In order to avoid damage to the converter system, some developers lock the PTO system when the converter is subject to extreme conditions, e.g. Beatty et al. [82]. However, this action causes a different wave loading over the converter structure which should be 4

41 analysed. Stallard et al. [83, 84] selected the geometry of a heaving point absorber such that the immersed geometry change due to the small change of mass reduced the float response to within an acceptable range during storm conditions without the PTO system engaged. In other words, the intermittently submerged converter geometry is used to provide different natural periods which are determined by the water plane area and draft. Some studies of the extreme waves around UK seas have been carried out. One example is the wave mapping in UK waters [85], where extreme waves up to 18 m in wave height have been projected using the available data with a return period of 1 years. This report suggests that any design of a marine structure should survive extreme loads caused by waves up to 18 m in wave height (for specific locations see the extreme wave distribution map for further references). In other words, any fixed or floating wave energy converter should be able to manage the extreme conditions for the particular deployment site of interest. A computational model for extreme conditions has to be nonlinear requiring a general CFD method such as Volume of Fluid (VOF) or Smoothed Particle Hydrodynamics (SPH) Ocean energy research at the University of Manchester Ocean wave energy research at the University of Manchester dates back 1 years, primarily concerned with the design of a wave energy converter known as Manchester Bobber which consists of a closely-spaced array of heaving floats, Stansby et al. [86]. Experimental studies with 1/1 scale-model of Manchester Bobber were carried out at University of Manchester facilities in order to quantify the converter response, and a single 1/1 float scale model was tested at NaREC. The Manchester Bobber An illustration of a 1 float Manchester Bobber system is shown in Figure 2.6. The system consists of an array of floats each connected to a counterweight by means of a cable which is supported by a couple of pulleys. This counterweight is used to maintain tension on the cable and it is always above water, any modification in the counterweight mass alters the immersed volume of the float hence modifies the natural frequency of the oscillating float and connected counterweight. The nearest pulley to the float is on a shaft with a freewheel 41

42 clutch. On the other side of the clutch, a flywheel is attached which provides inertia force to the rotation system. The rotary system drives a permanent-magnet generator that transforms the mechanical energy into electrical energy. Figure2.6: Schematic approach at still position of Manchester Bobber (courtesy of Prof Stansby). Lok [87] developed the control strategies for Manchester Bobber identifying the drive-train parameters to optimize the annual energy. Weller [33, 34] conducted an experimental study of the Manchester Bobber in order to obtain the device response and power output for a single float and different array arrangements in both regular and irregular waves. The variation of response and power output for the float position within the array was quantified. In addition, the non-linear response of a single float under the influence of extreme waves was analysed for a later comparison to numerical models. Bellew [33] conducted a numerical investigation into the heave response and power output quantification for an arrangement of floats. Alexandre [88] analysed the modification of irregular wave conditions due to energy extraction by a long string of Bobber-type devices and investigated the resultant change of nearshore processes using the spectral wave model (SWAN). Omidvar [89] used the SPH method to simulate the wave-structure interactions, under extreme conditions. Good agreement of the SPH model with the experimental data obtained by Stallard et al. [35] and Weller et al. [9] was found. 42

43 2.6. Summary In this chapter, general information about wave energy converters and some examples of the existing technologies were presented. It is clear that no single best solution has been identified. All wave energy converters presented in this section are prototypes under research investigation in order to find the key parameters to allow improvements to be implemented; to date there are no wave energy devices in commercial operation. Unfortunately, wave energy conversion is still suffering strong criticism which has produced the closure of some wave energy conversion programmes such as Wavebob, Manchester Bobber, LIMPET, PS Frog and now Pelamis. Numerical predictions using appropriate techniques and experimental work must be applied in order to improve the understanding of the wave-converter interactions, so that the wave energy converter capture width can be improved. Thus, the electricity cost will be reduced, making the wave energy conversion an economical and realistic option to support electrical energy production. From the current technology review, the necessity to improve the capture width by wave energy converters is clear. At present, the capture width varies between 1-2% of a wavelength. This compares with power coefficients for wind turbines (25-35%, [91]), and efficiencies of power plants with combined cycle (4-6% [92]). However these figures are not equivalent and the cost per kwh of produced energy should be compared instead (cost-benefit analysis). Wave energy converters with around 2-4% of capture width ratio are desirable. 43

44 Chapter III converter Linear modelling of multi-body wave energy Idealized models to simulate the wave field, wave energy converter response and power performance have been used as the starting point for designing a new wave energy conversion concept. In this research, linear wave theory and the harmonic oscillator system have been employed in order to optimize the performance for a new design of wave energy conversion concept proposed by Professor Peter K. Stansby at the University of Manchester. This section presents the linear wave theory and the converter modelling as a harmonic oscillator system where the excitation force is due to the incoming waves. Force coefficients are determined from the potential flow solver WAMIT Linear wave theory Ocean waves propagate in a slightly viscous fluid and over a non-uniform bed surface. However, the viscous effects for the main body of the fluid motion are concentrated in thin layers at the bottom and free-surface; the motion of the fluid can be considered as irrotational. Also the water can be considered incompressible. Therefore, with these assumptions the potential flow exists which satisfies the equation of continuity such that φ = Eq. 3.1 where φ is the velocity potential in the vertical plane (x,z) where x is the coordinate in the direction of the wave propagation and z the vertical coordinate. In other words, the twodimensional approach for the velocity potential equation is used in the x-z plane. Linear waves are a small sinusoidal surface disturbance with the free-surface elevation given by η = H 2 cos(kx ωt) Eq. 3.2 where H is the wave height, k is the wave number, t is time, and ω is the angular wave frequency such that k = 2π λ Eq. 3.3 ω = 2π T Eq. 3.4 where λ is the wavelength and T is the wave period. 44

45 The unsteady Bernoulli equation for an irrotational flow is given by 1 2 (u2 + w 2 ) + gz + p ρ + φ = Eq. 3.5 t where u = φ x and w = φ z. For small perturbation of the wave height; p, φ, u, w. u 2 and w 2 may be ignored, therefore, the problem becomes linear. Assuming φ is cyclic such that φ = Z sin(kx ωt), where Z = f(z) Eq. 3.6 Substituting into Laplace s equation for irrotational and incompressible condition 2 φ x φ z 2 = Eq. 3.7 and taking derivatives of the potential velocity and substituting in Eq. 3.7 gives 2 Z z 2 sin(kx ωt) + k2 sin(kx ωt) Z = Eq. 3.8 The general solution to Eq. 3.8 is given by Z = Ae kz + Be kz Eq. 3.9 φ = (Ae kz + Be kz ) sin(kx ωt) Eq. 3.1 Applying the boundary value problem approach as depicted in Figure 3.1 in order to determine the constants in Eq. 3.1 where the boundary conditions are given by: Bottom boundary condition; this condition implies that the water particles adjacent to the seabed cannot cross the boundary (the seabed is assumed impermeable and the normal velocity is zero) and it is described by φ z =, on z = d Eq where d is the water depth, and 45

46 Substituting condition to Eq. 3.1 and rearranging this equation gives φ = 2Be kd cosh[k(d + z)] sin(kx ωt) Eq Dynamic free-surface boundary condition; this condition states that the pressure at the free-surface should be zero (gauge) at any position at any time. Therefore, the unsteady Bernoulli equation is applied over the free-surface such as z = η = 1 φ g t = 2ωB g ekd cosh(kd) cos(kx ωt) Eq At initial conditions of x = and t =, the water surface elevation is given by η = H 2 Eq Hence, 2Be kd = Hg 2ω cosh(kd) Eq Finally, the potential velocity is given by φ = Hg 2ω cosh[k(z + d)] sin(kx ωt) Eq cosh(kd) Kinematic free-surface boundary condition; this boundary condition determines that the free-surface is free to deform in the absence of an external force and it is given by η = φ t z, on z = Eq This boundary condition requires that the velocity of any particle in the free-surface must be equal to the velocity of the free-surface itself. 46

47 Taking derivatives in Eq results an important relationship between the wave number and the angular wave velocity known as the linear dispersion relationship given by ω 2 = gk tanh(kd) Eq Figure 3.1 Boundary values for linear waves, adapted from Dean & Dalrymple [93]. The energy content in ocean waves is a combination of kinetic and potential energy. The energy conservation analysis for ocean waves begins by considering an elemental mass of water as depicted in Figure 3.2, where the element mass is given by δm = ρη(δx)b Eq where b is the width of the wave crest, δx is small, ρ is the water density and η is the water surface elevation which is given by Eq Therefore, the potential and kinetic energy content in the elemental mass is given by δe P = 1 2 ρgη2 (δx)b Eq. 3.2 δe k = 1 2 ρ(u2 + w 2 )bδxδz Eq where u and w are the velocities. 47

48 The final expression for the total potential energy content in a wavelength, λ, is E P = ρgh2 b 8 λ cos 2 (kx ωt)dx = ρgh2 λb 16 Eq The kinetic energy content in a wavelength, λ, is given by E k = 1 2 ρd d λ (u2 + w 2 )dxdz = ρgh2 λb 16 Eq Finally, the total energy becomes E = E p + E k = ρgh2 λb 8 Eq Figure 3.2: Wave energy differential mass approach. [68]. The energy flux propagated in the wave direction is usually known as wave power and it is given by P = 1 T T d pdy udt = ρgh2 b 8 c g Eq

49 where c g is the group velocity given by. c g = c 2 [1 + 2kd ] Eq sinh(2kd) 3.2. Modelling of a single wave energy converter Due to the nature of the ocean waves, a simple floating wave energy converter can be modelled as a driven harmonic oscillator system, where the excitation forces due to the incoming waves are applied to the converter floats. Damping forces are due to the radiated waves resulting from the float velocity, mechanical damping which is due to the damped actuator system representing the power take off system and to friction losses by moving parts (e.g. bearings). Applying Newton s second law and the driven harmonic oscillator formulation to the motion of a body, the resulting equation which describes the movement of the body is given by Mx = F ext Rx Cx Eq where M F ext R C x is the body mass is the external force is the mechanical damping coefficient is the spring constant is the instantaneous position of the body Application to a wave energy converter A wave energy converter is subject to the excitation force, F e, associated with a stationary body due to the incident and scattered waves, the radiation force, F r, as result of radiated waves produced by the body oscillations, the added mass force, F a, due to the body acceleration, the hydrostatic restoring force, F res, due to its position in relation to the horizontal surface and the mechanical damping force, F damp, caused by the damper 49

50 representing the PTO system and friction at moving parts. Therefore, the equation of motion (Eq. 3.27) for a wave energy converter is described by Mx = F e F r F a F damp F res Eq It is assumed that the mechanical damping force and the hydrostatic restoring force vary linearly with the velocity and the displacement, respectively. Thus, the equation of motion (Eq. 3.28) becomes Mx = F e F r F a Rx Sx Eq where, R is the mechanical damping coefficient and S is the hydrostatic stiffness. The forces and motions are considered periodic in sinusoidal waves and so the complex amplitude can be used to simplify the calculation. Using the complex amplitude the differentiation and the integration of the complex amplitude can be obtained multiplying the complex velocity with iω and 1, respectively. Therefore, Eq then becomes iω MiωU e iωt = F e F r F a RU e iωt + i ω SU e iωt Eq. 3.3 here x = 1 iω U e iωt, x = U e iωt, and x = iωu e iωt where means the complex amplitude, U is the body velocity, and ω is the angular wave frequency. The radiation damping force and the added mass force proportional to body motion are given by F r F a = Bx Ax Eq where B and A are known as the radiation damping coefficient and the added mass coefficient, respectively. In terms of the complex amplitude this hydrodynamic forcing is given by F r F a = (B + iaω)u e iωt Eq

51 Substituting these forces into the equation of motion (Eq. 3.3), this equation becomes MiωU e iωt = F e + ( (B + R) iωa + i ω S) U e iωt Eq Solving Eq for velocity, finally, the velocity is given by U e iωt F e = (B + R) + iω (M + A S Eq ω 2) Assuming that the mechanical power of a moving device is given by the device velocity, U, times a damping force applied, F PTO, by a power take off system, the mechanical power which the converter can attain is such that P mech (t) = F PTO (t)u(t) = F PTO (t)u e iωt Eq Numerical modelling in heave for a two-float WEC The numerical modelling provides a prediction of the converter response and an estimation of the power capture. In order to evaluate the numerical model accuracy, generally, experimental work using geometrically-scaled models may be carried out, and results compared with the numerical model predictions. If the agreement between the numerical and experimental results is satisfactory, the numerical model can be used to evaluate the energy converter feasibility at prototype scale. The numerical model accuracy is dependent on a number of simplifications made to the actual problem under investigation. The wave energy converter under investigation in this work comprises two or three floats depending on the device configuration. However, a numerical model in the frequency domain considering just the two-float array (in this thesis) was carried out in order to aid optimization of the floats sizing and spacing between floats. Two-body converter configuration work principle The design of the wave energy converter commenced with a two-float array Figure 3.3a, where the power was taken off by the actuator which simulates the PTO system. It was placed between the floats linkage mechanism and a vertical bar fixed centrally on float 1. 51

52 The relative motion between floats drives the PTO system, therefore, an antiphase motion between floats was desired to produce the largest relative displacement between floats. In other words, the optimum distance between float centres is given by half the wave length (or predominant wave length for an irregular wave field). For the first converter design the response system was limited to heave relative motion between floats by means of the twobeam linkage system, Figure 3.3b. The captured power by the system is calculated by the relative vertical velocity of the floats multiplied by the total mechanical damping force, F mech, which is obtained from the sum of the mechanical damping force applied by the actuator and friction losses at moving parts, such that P out = F mech (U 2 U 1 ) Eq where U 1 and U 2 are the vertical velocities for floats 1 and 2, respectively. a) b) Figure 3.3: Schematic representation of the work principle for the two-body converter configuration. For the numerical modelling in heave motion, each float of the system is considered as a free floating body in order to obtain the excitation, radiation damping and added mass forces. Then, a linear system of equations for the coupled heave motion for floats 1 and 2 is solved in order to obtain the vertical velocities using the modelling for a wave energy converter explained in section 3.2, where the mechanical damping force (damping caused by the PTO system) is an applied force proportional to the relative vertical velocity of the floats. The constant of proportionality is defined in terms of a multiplication factor of the radiation damping coefficient for float 2. The numerical modelling for the coupled heave motion is described in the following. A free-floating body can oscillate in six different modes of motion, three rectilinear motions (surge, sway and heave), and three angular motions (roll, pitch and yaw), see 52

53 Figure 3.4 [94]. The heave motion is the particular motion of interest for the numerical model described here. Figure 3.4: Freedom degrees for a simple floating body [94]. Newton s second law applied to the motion in the vertical direction (z direction) for a single body results in the general equation for heave [94] which is given as mz = a wz z b r z b v z z b p z ρga wp z Nk s z + F z cos(ωt + α z ) Eq where a wz A wp b r b v b p F z g k s m N t z z z α ρ ω Added mass Projected area onto the water plane Radiation damping coefficient Viscous damping coefficient Mechanical damping coefficient Excitation force on z direction Gravitational acceleration Mooring line stiffness Float mass Number of mooring lines Time Float vertical position Float vertical velocity Float vertical acceleration Angular phase Water density Angular wave frequency 53

54 Discretization of the heave equations for the converter floats coupled system The two axisymmetric circular floats of the energy converter are shown in Figure 3.3a, and the equations of motion for the system in heave (Eq.3.37) are developed considering their interaction assuming: Viscous force formulation as suggested in the Morison equation. Mooring lines effects are neglected Using the linear wave theory to obtain the excitation, radiation and added mass forces (from WAMIT) Therefore, the equation of heave motion for float 1 is given by (m 1 + a wz33 )z 33 + (a wz39 )z 99 = b r33 z 33 b r39 z ρa wp1c d1 z 33 z 33 ρga wp1 z 33 Eq F z33 and for float 2 the equation becomes (m 2 + a wz99 )z 99 + (a wz93 )z 33 = b r99 z 99 b r93 z ρa wp2c d2 z 99 z 99 ρga wp2 z 99 Eq F z99 where C d is the drag coefficient, which can be obtained from the experimental values calculated by Hoerner [95], and the excitation force for each float can be calculated analytically in some simple cases, or numerically using software such as WAMIT ( used in this case). The drag coefficient in all numerical simulations performed in this section is defined as C d = 1.1, which is a conservative value within the range of drag coefficients values found in [96] for a smooth circular cylinder in an oscillatory flow. Subscripts in Eq and 3.39 refer to the common terminology used for the identification of the mode of motion for a system with several devices. Thus, 33 means pure heave motion of float 1, 99 is pure heave motion for float 2, while 39 and 93 are the cross terms produced by the bodies interaction. 39 refers to the influence in heave motion of float 1 caused by heave motion of float 2, and vice versa for 93. Finally, subscripts 1 and 2 mean the corresponding physical property for float 1 and float 2, respectively. 54

55 Obtaining vertical floats velocities for the couple floats system Solving the linear system of Eqs and 3.39, and applying the damped harmonic oscillator approach as described in section 3.2, the vertical floats velocities are given by F U = [B + R + ω (M + A S ω 2) i] Eq. 3.4 where capital and bold letters mean matrices or vectors which define the hydrostatic and hydrodynamic components for both floats. These matrices are given by Added mass matrix A = [ A 33 A 39 A 93 A 99 ] Radiation damping matrix B = [ B 33 B 39 B 93 B 99 ] Body mass matrix Stiffness matrix M = [ M 1 M 2 ] S = [ S 1 S 2 ] Force matrix F = [ F 3 F 9 ] The mechanical damping matrix, R, is the external force due to the damper (PTO system), which is due to relative vertical velocity between floats and it is defined in terms of the 55

56 maximum radiation damping, B max, for the bigger float over the whole analysed frequency range. Therefore, the mechanical damping matrix becomes R = C pto B max [ ] where, C pto, is a multiplication factor which gives the possibility to find the optimum value for the mechanical damper to produce power output as explained in section 3.5. The stiffness coefficient is given by S = ρga wp Eq Finally with vertical velocities for both floats, it is possible to find the power output for the wave energy converter, which is given by Eq. 3.36, where the mechanical force, F mech, for the numerical model is defined as F mech = R(U 2 U 1 ) Eq Then, substituting the mechanical force into the power output formulation (Eq. 3.36), power output is that P out = R(U 2 U 1 ) 2 Eq Obtaining body forces with WAMIT Body forces due to wave excitation (incident waves and diffraction on a fixed body), radiation damping (due to body velocity) and added mass (due to body acceleration) are generally defined in terms of coefficients determined from potential flow analysis based on linear wave theory. The classical linear solution method is to represent the body as wave source terms and solve as a boundary integral or element problem. The body is represented as source panels. There are well established commercial codes which undertake this analysis, the most notable being WAMIT [97]. This code also allows second-order effects but this requires additional source terms to represent the water surface and is not considered here. This potential flow analysis does of course neglect viscous effects which need to be represented as separate terms as Eq and

57 The body or structure can be a fixed or a floating body. WAMIT makes use of two main subprograms; POTEN and FORCE which solve for the velocity potential and the hydrodynamic forces, respectively. A single body or multiple bodies can be simulated using a finite or infinitely large water depth. The WAMIT manual gives further information to simulate a specific problem under investigation. Two different methods in WAMIT can be used to obtain these forces, these are: 1. The high-order method; where the body geometry is represented by B-spline, MultiSurf models, or an explicit analytical formulation. The potential velocity is represented by B-splines in a continuous manner and the fluid velocity on the body surface is evaluated by analytical differentiation, and 2. The low-order method; where the body geometry is represented by an ensemble of panels, and the problem solution is approximated by the average values on each panel [97, 98]. In most applications the high-order method provides a more accurate solution, faster convergence and requires lower computational resources. A sensitivity test for a circular flat-based cylinder (which can represent one of the converter floats) was performed with high-order and low-order methods in order to investigate any discrepancy between them, and very small discrepancies were found (see Figure 3.5). Figure 3.5 shows the vertical excitation forces produced by the wave-float interaction for a case where the body representation used an analytical formulation (high-order method, solid black line) and for the panel method (low-order method, dashed red line, 836 panels). Figure 3.5a shows the added mass coefficient, which has a negligible difference between the high and the loworder method. The radiation damping coefficient and the excitation force show differences of the same order as the added mass differences, Figure 3.4b and Figure 3.4c, respectively. The difference found between the high-order and low-order method are much less than 1%. The low-order method may be generally used to obtain the hydrodynamics forces for any converter configuration (two-body or three-body device). This is considered accurate given the very small difference between high and low-order methods for specific simple cases. 57

58 F 3 [N] B 3 [(N-s)/m] A 3 [kg] T [s] a) T [s] b) T [s] c) Figure 3.5: Sensitivity analysis of the different methodology for body representation in WAMIT runs. a) added mass on float due to heave, b) radiation damping coefficient on float due to heave, c) excitation force on float due to heave. 58

59 3.4. Converter optimization for the two-body configuration This section determines the values for the geometrical parameters; diameter and draft (mass), adjusted on float 1 and 2 in order to optimize the power capture of the converter using the numerical model described above. Optimization criteria First step in the converter optimization is identifying the range of wave frequencies to which the converter shall be subject (the most common range of frequencies for a specific site at full scale). The aim of the optimization is to attain maximum power conversion. Therefore, the predominant wave frequency (period) defines the geometrical parameters for float 2 (basically the float draft), such that the float 2 reaches the resonance condition at the predominant wave frequency. The diameter for this float should be as big as possible in order to maximize the power capture. However, this diameter should not be bigger than a fifth of the predominant wave length to be in the inertia regime [99]. Geometrical parameters for float 1 (bow, Figure 3.4) are defined in order to have the resonance condition at a different wave frequency than float 2 (which will provide a wider wave frequency range for power generation), generally, around 2 to 25% larger wave frequency for resonance than for float 2. The diameter for this float is defined in order to produce maximum power. However, it should be as small as possible to avoid strong wave diffraction which will produce motion reduction on float 2 (reducing the power output), to facilitate natural alignment with the incoming waves and to save material cost The distance between float centres is given by half of the predominant wave length for anti-phase motions of the floats. Therefore, the diameters are limited by this spacing in order to both avoid physical interference and reduce float interaction. Many theoretical and some experimental studies have been published on the effect of inter-float spacing and configuration on the response and power output of arrays of wave energy converters, such as [59, 1]. In the remainder of this section, the converter optimization with the numerical model is discussed considering a 1/4 scale. Several experiments with the same aim were carried out with a laboratory scale model (1/4) which are presented in Chapter five of this thesis. 59

60 Optimization for two-float array with a 1/4 th geometrical scale For a defined wave frequency range of.6 f 1.2 Hz; the f =.9 Hz is defined as natural frequency for float 2 of the converter system ( f r2 =.9 Hz) following the optimization criteria described above. This wave frequency produces a wave length around 1.36 m. Therefore, suitable float spacing should be between.6 s.8 m which will produce nearly anti-phase motions of the floats for wave frequencies near the middle of this frequency range. Then, the wave frequency for resonance conditions by float 1 should be given as f r1 = 1.1 Hz. Angular resonant frequency for a heaving device [68] is given by ω n = ρga wp a w + m Eq The wave frequency for resonance conditions is f r = ω n /2π Eq The selection of the diameter of each float as explained above is determined considering the float 2 diameter up to 2% of the predominant wave length, to be in the inertia regime capturing the phase of a wave and allowing diffraction to be neglected for deployment of an array. For a selected diameter, the required resonance conditions for both floats are mainly dependent on the float draft and so this dimension is determined by Eq for resonance. With these considerations, several numerical and experimental converter configurations have been tested following the criteria described and the results indicate that the best configuration for power generation is the one given for case 4 in Table 3.1. A summary of some different geometrical configurations analysed with the numerical model is presented in Table 3.1. An explanation of the numerical model analysis is given in the next section for a single case, 4b. 6

61 Table 3.1: Different sizing floats configuration. Float 1 Float 2 Energy s Diameter Draft f converter r ω n Diameter Draft f r ω n [m] [m] [m] [Hz] [rad/s] [m] [m] [Hz] [rad/s] a Case 1 b c d Case 2 a Case 3 a ,.6 a b c Case 4 d ,.6 e f g Case 5 a Hydrodynamic and excitation forces Case 4b The WAMIT simulation is briefly described here. Figure 3.6a shows a diagram of the twofloat array for WAMIT simulation, where each float is represented by an ensemble of a finite number of panels. Figure 3.6b shows the panel ensemble used to represent the circular cylinder with the rounded corner at the float base; this figure gives just a quarter of the body float since the symmetry in x and y axis was used. From Figure 3.6b, it is possible to see a non-uniform panel distribution having a finer mesh at the rounded edge for a more accurate solution. A mesh analysis for three different body meshes was carried out in order to assess mesh dependence. Mesh 1 comprised 18, mesh 2 comprised 855 and mesh 3 contained 29 panels per quadrant. Results of the added mass, radiation damping and excitation force coefficients for the WAMIT simulations were found to be within 3% a) b) Figure 3.6: a) WAMIT floats array, b) Body representation by an a finite number of panels. 61

62 In order to obtain the hydrodynamic forces a range of wave frequencies was set between.75 f 1.25 Hz. Figure 3.7 presents the hydrodynamic coefficients; a) added mass and b) radiation damping, where the top left and bottom right are the added mass (Figure 3.7a) and radiation damping (Figure 3.7b) coefficients due to motion of float 1 and float 2, respectively. While top right represents the coefficients on float 2 due to motion of float 1 and vice versa for the bottom left figure as described on 3.2. From Figure 3.7a, it is possible to see that the added mass for float 1 is between 5.5 a kg for the given range of wave frequencies, which means 66-78% of the displaced mass, whereas for float 2, the added mass is bounded by a kg (55-6% in terms of the displaced mass). From Figure 3.7b, it is possible to see that the radiation damping coefficient for float 1 has a maximum value around 12 [(N-s)/m] for a wave frequency around.85 [Hz], whereas the radiation damping coefficient for float 2 does not have a maximum value for this range of wave frequencies, but the bigger values are at low wave frequencies; the highest value is around 24.6 [(N-s)/m]. This value defines the mechanical damping force (PTO damping) with a multiplication factor, see Eq Figure 3.8 shows the excitation force for float 1 and float 2, Figure 3.8a and 3.8b, respectively, and in Figure 3.8c, the float responses are shown. From Figure 3.8c, it is evident that the presence of the float 2 has a significant influence on the float 1 response. When float 2 is oscillating near to the resonance condition (f r2.9 Hz), the radiated waves produce an extra motion on float 1 compared with the float 1 response in isolation (dashed black line) and a motion reduction when wave frequency is bigger than 1 Hz approximatively. In others words, a positive interaction for the wave frequencies around the resonance for float 2 and a negative interaction for wave frequencies greater than 1 Hz. Whereas, the float 1 does not produce the equivalent effect on float 2, since the radiated wave of float 1 are not as significant. The same effect is seen in the vertical excitation force for float 1 (Figure 3.8a), where the force has a minimum value for an wave frequency around 1.25 Hz (corresponding to the trough in the float 1 response between the resonance condition for each float). The vertical excitation force and response for float 2 do not present apparent modifications by the presence of float 1 as shown in Figure 3.8b and Figure 3.8c. 62

63 F 3 [N] F 9 [N] d z [cm] B 9,3 [(N-s)/m] B 9,9 [(N-s)/m] B 3,3 [(N-s)/m] B 3,9 [(N-s)/m] A 9,3 [kg] A 9,9 [kg] A 3,3 [kg] A 3,9 [kg] f [Hz] a) f [Hz] f [Hz] f [Hz] b) Figure 3.7: Hydrostatic coefficients, a) added mass, b) radiation damping f [Hz] f [Hz] f [Hz] a) b) c) Figure 3.8: Excitation force on converter floats due to heave mode of motion. a) excitation force on float 1, b) excitation force on float 2. H =.2 m (solid blue line), H =.25 m (dotted red line), H =.3 m (dash-dot green line), H =.35 m (dashed black line), and c) free response for float 1 and 2, red and blue line, respectively and the dashed black lines correspond to the float responses at isolation condition. 63

64 P out [W] Prediction of the power output Case 4b Once the hydrodynamic forces have been obtained, which are taken as an input into the velocity equation for the coupled system (Eq. 3.34), it is possible to obtain power output from Eq. 3.36, which is given by the relative velocity between the floats multiplied by the mechanical damping applied by the PTO system. This equation is given by P out = C pto R(U 2 U 1 ) 2 Eq Evaluating the power output when C pto = 1, the resulting power curve is given by Figure 3.9 where the solid line is the power output neglecting the drag effects, and dotted line is the power output considering drag as explained above (Eq and Eq with C d = 1.1). From this figure, it is possible to see that the influence of drag is more significant when float 2 is close to resonance conditions with power reductions up to 16% f [Hz] Figure 3.9: Power output for the two-float array (Case 4b); without (solid line) and with (dotted line) drag effects. 64

65 P out [W] Results for different float configurations This section presents the results from the numerical model for five different device configurations presented in Table 3.1 in order to identify the converter configuration that maximises average power output. Figure 3.1 shows the power output for the different cases listed in Table 3.1, where it is possible to see that the converter performance is period and converter configuration dependent. The converter configuration that maximises power for a range of wave frequencies between Hz is by case 3a (purple line), whereas for frequencies smaller than 1 Hz; case 4b (blue line) gives the converter configuration that maximises power capture. Configurations given by cases 3a and 4b produce highest power capture at two different ranges, but taking into account the whole range of wave periods; it was found that case 4b produce 1% more average power than case 3a and up to 8% greater power output for wave frequencies smaller than 1 Hz. In Fig.3.1 power without the drag effects is shown by solid lines, while with drag is shown by the dashed lines (drag effects as specified in 3.38 and 3.39 equations with a drag coefficient of 1.1). The drag effect reduces average power by up to 5%, 16%, 24.7%, 1% and 12.5% for cases 1-5, respectively f [Hz] gure 3.1: Power output at different float arrays configurations; case 1a (black line), case 2a (green line case 3a (red line), case 4b (blue line), and case 5a (purple line). 65

66 P out [W] d z [m] d z [m] Finally, it is possible to conclude that from the different analysed configurations the converter configuration which gives better results in terms of power outputs is that one given by case 4b. With this numerical model in heave, the two-float configuration has been optimized, however, future converter optimizations for the three-float array will be considered Optimization of the mechanical force applied to the system. The mechanical force applied by the PTO system damps the float motion. Accordingly, this force determines the performance for the wave energy converter. Therefore, the ideal range of forces is investigated at this stage. A range of.1 C pto 2 with increments of.1 in Eq is analysed in order to find the ideal range of the mechanical force applied. Figure 3.11 presents the power capture (left) and float responses (centre and right for float 1 and 2, respectively) for different mechanical forces applied to system using the numerical model in heave presented above for the converter configuration defined by case 4b. From this figure, it is possible to see that the converter performance is improved at low damping forces, and the best results are given with mechanical damping less than or equal to the maximum radiation damping..4 Power.16 Float 1.18 Float f [Hz] f [Hz] f [Hz] R mech =.5B 99,max R mech =1.B 99,max R mech =2.B 99,max R mech =3.B 99,max R mech =4.B 99,max R mech =5.B 99,max R mech =6.B 99,max Figure 3.11: Power output (left), vertical float response (centre and right for float 1 and 2, respectively) at different mechanical applied forces for a wave amplitude of.1 m, B 99,max = 24.6 [(N-s)/m]. 66

67 f [Hz] Figure 3.12 presents the power contours as a function of the mechanical damping and wave frequency. From this figure, it is clear that the wave energy converter performance is highly dependent on the applied mechanical damping, which has an optimal range for best results around 1 to 5 [Ns/m]. The maximum power output for the two-float array given by case 4b is around.38 [W] for a mechanical damping of 17 [Ns/m]. The ideal mechanical force range resulting from the numerical model is quite small. In others words, it is difficult to achieve at 1/4 th scale physical model due to available options for pneumatic or hydraulic actuators. However, this scale model is mainly used for understanding of the converter response and future system improvements R mech [(N-s)/m] Figure 3.12: Power contours as function of mechanical force, R mech, and angular wave frequencies, w Summary A numerical model in heave for a two-float array is used in order to investigate the influence of float geometry, drag and mechanical damping on the response amplitude and average power output. Results for five different converter configurations were presented 67

68 and it was found that the maximum power output is given by a converter with float diameters of.3 m and.4 m and float drafts of.12 m and.19 m for float 1 and float 2, respectively. Drag has a considerable influence on the converter performance reducing average power up to 24.7%. Therefore, the numerical model without the consideration of drag forces will generate unreliable results. However, it is important to take into account that this power prediction was calculated considering the idealised oscillation in heave motion for both floats. The inclusion in the numerical model of other modes of motion will be useful to have more accurate power predictions. Modes of motion that substantially influence response can be defined by means of experimental observations (Chapter V). This would inform development of the numerical model with the inclusion of different modes of motion. In Chapter IV, the experimental methodology for the wave energy converter response and power output analysis is presented. Chapter V presents the experimental work carried out with a two-float array for converter development and system optimization. The latest version of the converter configuration includes a three-float array which has been experimentally tested in 1/4 th and 1/8 th scales this experimental work is presented in Chapter VI. Finally, Chapter VII presents some power projections with the numerical and experimental findings. The numerical investigation presented in this chapter illustrated the power capture potential of a wave device comprising two heaving floats of differing dimensions. Wave energy concepts and studies limiting float motion to heave only have relied upon a supporting strut or guide [9, 11-14] affixed to a fixed, or much larger floating, structure. This is partly because, in practice, floating bodies intended for relative motion in heave only would also respond in pitch. Depending on the linkage between floats (e.g. parallelogram linkage) and system parameters, motion in such modes will alter response and power output relative to the heave-only predictions. An experimental approach is used for further evaluation of free-floating wave energy systems including both two- and threefloat systems. 68

69 Chapter IV Experimental method The experimental phase in wave flumes or wave basins for the development of a new concept of wave energy conversion is a key stage, since it allows identification of the main variables which influence the converter performance in an accessible and controlled environment. This in turn allows improvements or redesign of the wave energy converter. This chapter presents the experimental procedure and facility description for the wave energy converter under investigation. 4.1 Description of the experiments The experimental phase conducted with the wave energy converter model using two different geometrically-scaled models was divided into two different stages, or two different kinds of experiments, which are classified as: 1. Analysis of the converter behaviour: experiments with regular waves were conducted in order to identify the main mode of motion of the converter and for the system optimization, and 2. Converter performance assessment: the converter performance was assessed in both regular and irregular waves, which allows identification of the converter capability to transform the wave power into mechanical power. For both experimental stages, combinations of a representative range of the most common frequencies and wave amplitudes around the UK seas were defined. The laboratory sea states were created using wavemakers from Edinburgh Designs at two different wave flume facilities; a wide wave flume in the hydraulics laboratory at the University of Manchester and the wave basin at the Marine Institute of Plymouth University. It is well known that wavemakers generally have differences between the target wave heights and output values, therefore, for each combination of frequency and amplitude the real wave properties (amplitude and frequency) were measured without the device in position using an arrangement of wave gauges at the wave energy converter position. Wave simulation, facilities, model, and test procedure description are given in following sections. 69

70 4.2. Wave definition Ocean waves are caused by natural phenomena such as: motion of celestial bodies causing tides and motion of the tectonic plates (submarine earthquakes) causing tsunamis, and (of concern here) waves generated by wind blowing over the ocean surface (with typical periods between 4 to 8 s). Wind directly generates waves which may be extreme and when generated in the ocean, these waves may travel large distances, e.g. across oceans, resulting in waves of relatively long period called swell waves (waves with periods larger than 9 s). Wind waves are defined by their height and frequency spectrum which is determined by the wind speed, duration and fetch. Small regular waves may be defined by linear wave theory and irregular or random waves are considered to be the sum of a number of sinusoidal linear waves representing a spectrum. In extreme conditions linear wave theory is inadequate, waves may break, and nonlinear theory must be used such as stream function theory if non breaking and a more general CFD approach if breaking, such as Volume of Fluid (VOF) [15] or Smoothed Particle Hydrodynamics (SPH) [16]. The wave energy converter under investigation is designed to work at intermediate-deep water conditions (2-5 m), where the ocean waves can be considered as swell (long waves with periods of 9s and above) or wind waves (waves with a typical range of periods between 4 and 9s) which depend on the location of interest. Swell and wind waves may be considered as regular and irregular waves, respectively. This section presents the wave power formulation and the representation of an irregular wave field using the JONSWAP spectral approach. Irregular waves - JONSWAP spectrum Ocean waves are random in nature. If uni-directional they are defined by a spectrum, and this can be modified by a spreading function to represent varying directionality. The spectrum is defined by extreme value probability functions, typically significant wave height and peak period/frequency at the spectral peak. Extreme values for wave height are typically estimated using functions such as Rayleigh, Gumbel or Weibull. Different empirical formulations for the wave energy spectrum have been found depending on wind speed, duration and fetch. The most common forms are Bretschneider [17], 7

71 Pierson-Moskowitz [18], and JONSWAP [19]. The JONSWAP (the Joint North Sea Wave Project) spectrum is the most general form and is used here, given by S(f) = αg2 (2π) 4 f 5 e( 4 5 (f p f ) 4 ) γ e 1 2 (f f p 1 ) σ Eq where α is a constant with value of.81 for the northern North Sea. The JONSWAP spectrum relates the wave energy density to the wind speed and the fetch length, f is the wave frequency, f p is the peak wave frequency, γ is the peak enhancement factor and σ defines the width of the spectrum, with values of σ = { σ a =.7 for f > f p σ b =.9 for f < f p Eq The energy of a uni-directional irregular wave is distributed over a range of frequencies as described above. Wave spectral representations which include both angular directional and frequency distribution are known as directional spectrum, these spectra can be mathematically represented as S(f, θ) = S(f) D(f, θ) Eq where S(f, θ) is the directional spectrum function, S(f) is the uni-directional energy spectrum, D(f, θ) is the angular spreading function, f is the wave frequency and θ is the wave direction relative to the principal direction. The total energy content in a directional spectrum must be equal to the energy content in the corresponding uni-directional spectrum such as π S(f) D(f, θ)dθdf = S(f) π Eq Several idealized models have been proposed for the spreading function in [11, 111]. The cos s (θ) spreading function was used in this research work in order to generate directional spectra, this function is defined by 71

72 D(f, θ) = ( 2(2s 1) π ) (Γ2 (s + 1) Γ(s + 1) ) coss (θ θ ) Eq. 4.5 where Γ is the gamma function, θ the main wave direction, and s is the constant spreading factor which is a function of wave frequency and wind e.g. Mitsuyo et al. [11], and Hasselmann et al. [118]. Wave power formulation The formulation for wave power per metre for regular waves based on linear theory is given by Eq. 3.25, and wave power for irregular waves in deep water is defined as a half of the wave power for regular waves given by P w,irreg = 1 2 (1 8 ρgh s 2 C g ) Eq. 4.6 where H s is significant wave height (the mean height of the highest one third of waves) and k for group velocity is based on the energy period, T e. T e =.78T p for wind waves (γ = 1 in Eq. 4.1) and T e =.84T p for swell waves ( γ = 3.3 in Eq. 4.1). γ is the spectral peakedness factor in the JONSWAP spectrum used for all these tests. Although the expression by Eq. 4.6 is for deep water it is accurate to within 12% for the intermediate waves considered here and it provides a convenient form of non-dimensionalisation Froude scaling law Scale models are used to replicate behaviour in a controlled environment. Experiments at reduced scale are often of value for identification of parameters and response modes that substantially influence performance. By means of a combination of the scaled model characteristics and working conditions at full scale, an indication can be obtained of the overall device performance at full scale. However, all physical quantities need to be properly scaled. Scaling methods based on Froude or Reynolds numbers are appropriate depending on the flow conditions. Froude number represents the ratio between inertia force and gravity force, while Reynolds numbers gives the ratio of inertia force and viscous force. 72

73 Ocean surface waves are gravity-driven, therefore, Froude scaling is most appropriate in this case. In other words, the equality in Froude number for the geometrically-scale model and full scale prototype will ensure that gravity forces (wave forces) are correctly scaled. The Reynolds number is used to characterise the nature of the flow, i.e. whether it is laminar or turbulent. The scaling analysis begins by defining the length ratio between the laboratory geometrically-scale model and the full-scale prototype, length scale factor, which is given by λ s = l p l m Eq. 4.7 where l p and l m are the characteristic length for prototype and scaled model, respectively. Applying the Froude scaling law it is possible to obtain the scaling factors for different properties, which are presented on Table 4.1 [68]. Table 4.1: Scaling factors [68]. Quantity Multiplication factor Length [m] λ s Time [s] λ s Velocity [m/s] λ s Mass [kg] 3 λ s Volume [m 3 ] λ s 3 Force [N] 3 λ s Pressure [Pa] λ s Power [W] 3.5 λ s Damping [Nms] λ s 4.5 Inertia [kgm 2 ] λ s 5 This does mean that Reynolds number and Weber number are not scaled correctly. As surface tension effects are small Weber number is not significant. However Reynolds number (UL/ν) where U is the velocity of the object relative to the fluid, L, is the characteristic length of the object and ν is kinematic viscosity. This scales the ratio of viscous and inertia forces effects and determines the magnitude of drag coefficient notably. 1.5 At small scale, the Reynolds number will be smaller than at full scale by a factor of λ s and this determines turbulence transition which can affect the drag coefficient. 73

74 4.4. Facilities Manchester facilities The hydraulics laboratory at the University of Manchester has a wave flume with test section 5 m wide, 15 m long, and typical water depth of.45 m. The wave flume comprises eight wave paddles of sector-carrier piston type from Edinburgh Design located at one end of the flume, at the other end (18.3m approximatively) an energy absorbing beach is located. This kind of wavemaker produces stable waves at a distance of three times the water depth in front of the wave paddles [112]. Regular waves, any irregular spectra such as the Pierson-Moskowitz, JONSWAP, and Gaussian spectra, focussed waves, directional short-crested waves with spreading function cos s (θ) and cos 2s ( θ ) waves can be generated. Waves are generated using the OCEAN 2 software [112]. For the experimental work at Manchester facilities, the wave flume is filled with fresh water (ρ = 1 kg/m 3 ) with.45 m of water depth. Therefore, any distances further than 1.35 m from the wave paddles are suitable to place the device. Figure 4.1 shows an experimental schematic set up used for the testing phase. The device is placed around 3 m away from the wave maker and anchored to the wave flume structure by means of a horizontal string, the encoder and load cell are connected to a National Instrument computer running a LabVIEW program for data collection. Figure 4.1: Manchester wave flume. 74

75 Plymouth facilities The Marine Institute of the Plymouth University has an ocean wave basin 35 m long and 15.5 m wide, operable at different water depths (up to 3 m with a raisable floor). The basin comprises 24 bottom-hinged flap wave makers, which can generate regular waves up to.9 m in wave height at.4 Hz of wave frequency and regular waves with heights above.2 m for a range of wave frequencies,.166 < f < 1 Hz of. The flap paddles are ideal for simulating deep water conditions with sinusoidal or random form. A grid of cleats at 1 m x.5 m centres arranged across the central floor section allowing attachment of a mooring line. Regular and irregular waves are created as in the Manchester University wave flume using the OCEAN software. A motion video capture system may be used in order to record the wave energy converter at any time during the experiments, this system comprises a set of 6 Qualysis cameras which record the position of the device with 6 degree of freedom. Figure 4.2 shows a picture of the wave basin at the Plymouth University, see also A water depth of 2.9 m was used for the experiments. Figure 4.2: Wave basin at the Marine Institute at Plymouth University. For both facilities, a wave gauge system of the capacitance sensor type is used. This kind of wave gauge measures the immersed distance of a wire with a dielectric sheath by means of the capacitance between the wire and the water, which is used as ground, [113]. 75