Critical Assessment, Developments and Applicability for Economical Optimisation Revised Version PhD ThesisDefended in public at Aalborg University19 September 2014 FrancescoFerri Department of Civil Engineering, The Faculty of Engineering and Science, Aalborg University, Aalborg, Denmark River Publishers 9788793237247 The research presented in this thesis was financially supported by the Danish Council for Strategic Research under the Programme Commission on Sustainable Energy and Environment (Contract 09-067257, Structural Design of Wave Energy Devices). Most of all, I would like to thank Jens Peter and Aurélien. Jens Peter and his wise point of view, guided me during the whole PhD period, smoothing my rough edges off and helping me whenever it was needed. Aurélien is an example to me in both pedagogical and technical aspects. I believe that sharing one’s wisdom is the best, but most difficult, thing to do, and Aurélien can do it so naturally. I would like to acknowledge all the colleagues at the Department of Civil Engineering; a special thanks goes to Peter Frigaard, for his way to be a manager, to Vivi Søndergaard, for her valuable corrections, and to Helle Schrøder Hansen, Pernille Bisgaard Pedersen, for their help with any bureaucratic matters, which are still obscure to me. I would like to thank Niels Drustrup and Nikolaj Holk for their support in the laboratory work. The Wave Energy Research Group, me included, would be in panic without them. I would like to broadly thank every single person I have been working with: both lucky and unlucky collaborations do exist, and they always teach you something new. A special thank goes to Vittoria University Institute for Integrated Energy System, to Fraunhofer Institute for Wind Energy and Energy System Technology (IWES), to Bologna University, to Chalmers University and to École Central de Nantes. Thank you to my old and new family. The wild development of the world from the 1769, years in which James Watt patented his steam engine, brought utterly to the actual, real or assumed, economical, political, environmental and energetic crisis. The answer to the question “how to solve these problems?" is a tangled unsolved discussion, but talking about renewable energy partially ravels the problem out. Wave energy is a large, mostly untapped, renewable energy resource. It has the potential to contribute significantly to the future energy mix, but the sector has not yet rolled off into the market in consequence of a number of technical and non-technical issues. These can be efficiently summarised in the cost of the energy produced by the various wave energy converters: If compared with other renewable energy technologies the cost of energy from the ocean waves is still significantly higher. Holding the comparison it also important to notice that there is not a clear front runner in the wave energy sector, which fades effort and funding over a too broad frame. In order to assist efficient development and analysis of wave energy converters and therefore to accelerate the sector progression towards commercialisation, a generally applicable, efficient and reliable wave-to-wire model tool is needed. A wave-to-wire model identifies the relation from the source of energy of a particular location to the expected device productivity. The latter being expressed in terms of electricity fed into the grid. The model needs to output a coarse picture of the actual status of the different devices and their power productivity, which is used afterwards to sieve promising concepts out. In a macro-scale the work can be divided into two main contributions First, highlight the complementarity between numerical simulation and laboratory experience, in what can be efficiently summarised as: reliable model. Numerical models are per se meaningless, but easy to manipulate, fast and relatively cheap. Physical models are complex, slow and expensive but realistic if adequately implemented. Since there is no real need of “new", but more of “how can we get the best of what we have", the numerical model used is entirely based on well established methods. The experimental data is used as a check point to verify the direction of the numerical path. Second, shed light on what should be the objective of the sector: minimisation of the cost of energy. Two different techniques to reduce the cost of energy are compared: the former maximises the system revenue (income) by acting on the control logic, while the second extends the first methods adding a penalty term due to the effect of the control logic on the structural design. Once more, both methods are based on well established or standard techniques. Med den industrielle udvikling efter 1769, tidspunktet hvor James Watt tog patent på dampmaskinen, er der opstå et økonomisk, politisk, miljømæssig og energimæssig krise – hvad enten man anser denne for antagelig eller reel. Svaret på spørgsmålet ”hvordan løser man disse problemer?” er en kompliceret og ufuldendt diskussion, men med introduktionen af vedvarende energiproduktion, er der givet et bud på, hvordan man til dels kan løse disse problemer. Bølgeenergi er en uudnyttet vedvarende energikilde, som har potentiale til at bidrage betydeligt til fremtidens blandede energiforsyning. Dog har bølgeenergisektoren endnu ikke udbudt denne mulighed på markedet grundet en række tekniske og ikke-tekniske årsager. Disse kan mest effektivt opsummeres ved at på pege produktionsudgifterne som opstår ved brug af de forskellige energitransformatorer: Sammenlignet med andre vedvarende energiteknologier, er produktionsudgifterne stadigt betydeligt højere, når man udnytter energi fra havets bølger. Hvis man bliver ved denne sammenligning, er det også vigtigt at være opmærksom på, at der ikke er en klar frontløber i bølgeenergisektoren, hvilket gør at indsats og finansering spredes over et bredt felt. For at kunne bidrage til en effektiv udvikling og analyse af bølgeenergikoncepter og dermed bidrage til en acceleration af hele sektorens fremgang mod kommercialisering, er det nødvendigt med et effektivt og på lideligt wave-to-wire værktøj, som er generelt anvendeligt. En wave-to-wire model identificerer relationen mellem en energikilde på et bestemt sted og enhedens forventede produktion. Sidstnævnte er udtrykt i form af elektricitet, der leveres til elnettet. Modellen skal kunne give et groft billede af den faktiske status af sektoren og dens potentiale, og senere bruges til at finde frem til de mest lovende koncepter. På et makroniveau kan arbejdet fordeles på to hovedindsatser. Først skal komplementaritet mellem numerisk simulation og eksperimentel erfaring fremhæves, hvilket mest effektivt kan opsummeres som: Pålidelig model. Numeriske modeller er i sig selv meningsløse men lette at manipulere, hurtige og relativt billige, hvor fysiske modeller er komplekse, langsomme og dyre, men realistiske hvis implementeret i tilstrækkelig grad. Siden der ikke er et faktisk behov for ”noget nyt” men snarere ”at få det bedste ud af hvad vi har”, er den anvendte numeriske model udelukkende baseret på veletablerede metoder. Eksperimentel data bliver anvendt som verifikation til at styre retningen af den numeriske udvikling. Dernæst, kaste lys på hvad der bør være målet for sektoren: Minimering af de samlede energiomkostninger. To forskellige teknikker til at reducere omkostningerne er sammenlignet: Den første maksimerer systemomsætning (indkomst) ved at ændre på kontrolstrategien, mens den anden udvider de første metoder ved at tilføje et straf baseret på kontrolstrategiens indflydelse på det strukturelle design. Begge modeller er igen baseret på veletablerede eller standardiserede teknikker. The thesis is presented as a collection of the following eight papers, given in Appx. A. Paper 1: Ferri, F; Sichani, M T; Frigaard, P B. A case study of short-term wave forecasting based on FIR filter:optimisation of the power production for the Wavestar device. Proceeding of the 22nd International Ocean and Polar Engineering Conference, Rhodos, Greece, 2012. Paper 2: Ferri, F; Kramer, M M; Pecher, A. Validation of a wave-body interaction model by experimental tests. Proceeding of the 23rd International Ocean and Polar Engineering Conference, Anchorage, Alaska, 2013. Paper 3: Wehmeyer, C; Ferri, F; Skourup, J; Frigaard, P B. Experimental Study of an Offshore Wind Turbine TLP in ULS Conditions. Proceeding of the 23rd International Ocean and Polar Engineering Conference, Anchorage, Alaska, USA, 2013. Paper 4: Zurkinden, A S; Ferri, F; Beatty, S; Kofoed, J P; Kramer, M M. Non-Linear Numerical Modeling and Experimental Testing of a Point Absorber Wave Energy Converter. Ocean Engineering, pp. 11-21, vol. 78, 2014. Paper 5: Ferri, F; Ambühl, S; Fischer, B; Kofoed, J P. Balancing power output and structural fatigue of wave energy converters by means of control strategies. Energies, March 2013. Paper 6: Wehmeyer, C; Ferri, F; Andersen, M T; Pedersen, R R. Validated state space model of a TLP including a flexible topside in non-linear regular waves. Submitted to Energies, April 2014. Paper 7: Angelelli, E; Zanuttigh, B; Ferri, F; Kofoed, J P. Experimental assessment of the mooring influence on the power output of a Wave Activated Body floating WEC. Proceeding of the 10th European Wave and Tidal Energy Conference, Aalborg, Denmark, 2013. Paper 8: Ferri, F; Andreoni, G; Perisic, N; Lavelle, J; Kofoed, J P. Cable-free Floating Object Tracking Using an Image Processing Approach. Proceeding of the 10th European Wave and Tidal Energy Conference, Aalborg, Denmark, 2013. This thesis has been submitted for assessment in partial fulfillment of the PhD degree. The thesis is based on the submitted or published scientific papers which are listed above. Parts of the papers are used directly or indirectly in the extended summary of the thesis. As part of the assessment, co-author statements have been made available to the assessment committee and are also available at the Faculty. The thesis is not in its present form acceptable for open publication but only in limited and closed circulation as copyright may not be ensured. Other relevant publications from the author, not appended to this document, are listed hereafter. List of submitted publications: Markus, D; Ferri, F; Wuchner, R; Frigaard, P B; Bletzinger, K U. Complementary numerical-experimental benchmarking for shape optimization and validation in waves and currents. Submitted to Computer & Fluid. 2014. List of published publications: Ferri, F; Kracht, P. Implementation of a Hydraulic Power Take-Off for wave energy application: deliverable D4.7. Department of Civil Engineering, Aalborg University, Aalborg. DCE Technical Reports, nr. 157, 2013. Ferri, F; Kramer, M. Laboratory experiments on Wavestar device: Definition and comparison of hydrodynamic coefficients: deliverable D4.1-3. Department of Civil Engineering, Aalborg University, Aalborg. DCE Technical Reports, nr. 158, 2013. Angelelli, E; Zanuttigh, B; Martinelli, L; Ferri, F. Physical and numerical modelling of mooring forces and displacements of a wave Activated Body Energy converter. Proceeding of the 33rd International Conference on Ocean, Offshore and Arctic Engineering, San Francisco, California, USA, 2014.

Symbols and Acronyms	Description
ACRONYMS
A-WEC	Attenuator WEC
AEP	Annual Energy Production
BEM	Boundary Element Method
BIEM	Boundary Integral Equation Method
CAPEX	Capital Expenditure
CCC	Complex Conjugate Control
CF	Cost Factor
CFD	Computational Fluid Dynamic
CIES	Control, Instrumentation and Electrical System
CoE	Cost of Energy
CoG	Centre of Gravity
CW	Capture Width
DNS	Direct Numerical Simulation
DoF	Degree of Freedom
E	Turnover or Revenue
EoM	Equation of Motion
FFT	Fast Fourier Transform
FIR	Finite Impulse Response
FLS	Fatigue Limit State
FOWT	Floating Offshore Wind Turbine
FRF	Frequency Response Function
fWEC	floating WEC
GHG	Green House Gasses
HS	Hydrodynamic Sub-system
HSS	Hosting Structure System
IFFT	Inverse Fast Fourier Transform
IMU	Inertial Motion Unit
IRF	Impulse Response Function
KC	Keulegan-Carpenter Number
LES	Large Edge Simulation
LUCF	Land use change and forestry
MPC	Model Predictive Control
MS	Main Structure
O&M	Operation and Maintenance
OFS	Onshore Facility System
OPEX	Operational Expenditure
OT	Over Topping
OWC	Oscillating Water Column
P	Proportional Control
PA-WEC	Point Absorber WEC
PAC	Phase and Amplitude Control
PaM	Panel Method
PCS	Power Connection System
PD	Proportional Derivative Control
PI	Proportional Integral Control
PID	Proportional Integral Derivative Control
PIDc	Proportional Integral Derivative Control with compensation of the radiation term
PSD	Power Spectral Density
PTO	Power Take-Off
RANSE	Reynold-averaged Navier-Stokes Equation
RS	Reaction System
SDWED	Structural Design of Wave Energy Devices
SKS	Station Keeping System
SN	Stress Number of Cycles Curve
SPH	Smoothed Particle Hydrodynamics
T-WEC	Terminator WEC
TLP	Tension Leg Platform
ULS	Ultimate Limit State
WEC	Wave Energy Converter
WAB	Wave Activated Body
SYMBOLS
A	Wave Amplitude
A(i)	Cross-sectional area of the critical structural detail for the i-th control strategies
Ap	Body Projected Area
C1(i)	Fraction of the Total Investment Cost over the WEC LifetimeDependent on the Control Strategy
C2	Fraction of the Total Investment Cost over the WEC LifetimeIndependent on the Control Strategy
CA	Radiation Damping Coefficient Matrix
Cc	Damping Control Coefficient
CD	Viscous Drag Coefficient
	Linearised Viscous Drag Coefficient
CM	Added Mass Coefficient Matrix
CM∞	Limit of the Added Mass Coefficient Matrix for ω ⇝ ∞
CO2	Carbon Dioxide
Ctot	Total investment cost over the WEC lifetime
D	Characteristic Length of the Body
DPTO	Constant PTO Force Level
dt	Time Discretisation Step
f	Wave Frequency
fD	Viscous Drag Force, Time Domain
FD	Viscous Drag Force, Frequency Domain
	Wave Excitation Force Vector, Time Domain
	Wave Excitation Force Vector, Frequency Domain
	Wave Excitation Force Frequency Coefficient Vector
	External Force Vector acting on the i-th Body, Time Domain
	Mooring Force Vector, Frequency Domain
fmax	Maximum Control Load
fp	Wave Peak Frequency
fPTO	PTO Force Acting on the Body, Time Domain
FPTO	PTO Force Acting on the Body, Frequency Domain
	Constrained PTO Force, Time Domain
	Theoretical Commanded PTO Force by the Main LogicController, Frequency Domain
	Radiation Force Vector, Time Domain
	Radiation Force Vector, Frequency Domain
	Resultant Force Vector, Frequency Domain
fu	Control Load, Time Domain - Equivalent to fPTO
G(s)	Generic Linear Time Invariant Transfer Function
h	Time Discretisation Step
H	Wave Height
	Transformation Matrix associated to the vector
He	Hessian Matrix of the MPC
hex	Impulse Response Function from the Surface Elevation to the Excitation Force
Hex	Transfer Function from the Surface Elevation to the Excitation Force
	Impulse Response Function from the Surface Elevation to the Reference Velocity
	Transfer Function from the Surface Elevation to the Reference Velocity
Hm0	Spectral Estimation of the Significant Wave Height
hopt	Impulse Response Function from the Excitation Force to the Reference Velocity
Hopt	Transfer Function from the Excitation Force to the Reference Velocity
hRAD	Impulse Response Function of the Radiation Force
Ic	Inertia Matrix of the Rigid Body around CoG
j	Current Time Instant
J	Cost Function
k	Wave Number
Kc	Stiffness Control Coefficient
ke	Rate of Discount
Km	Linearised Mooring Stiffness Matrix
kpto	Stiffness of the PTO system
L	Lagrangian
m	Mass of the Rigid Body
M	Mass Matrix of the Body
Mc	Mass Term of the Feed-Back Controller
Mpto	Mass of the PTO system
n	Number of Generalised DoFs
NIRF	Order of the IRF
Np	Number of Panels
Np	Number of Steps in the Prediction Horizon
ODE	Ordinary Differential Equations
p	Percentage of the Total Investment Costs Affected by the Control Strategy
P	Matrix to Map the MPC State Vector Increment in the Output Space
Pi	Potential Energy of i-th Body
pr	Probability of Occurrence
Q	Weight Matrix in the MPC formulation
	Body Displacement Vector in the Generalised DoFs Time Domain
	Body Velocity Vector in the Generalised DoFs Time Domain
	Body Acceleration Vector in the Generalised DoFs TimeDomain
qi	i-th Generalised DoF
	Skew Symmetric Matrix of the Vector
Sj(ω)	Wave Spectrum
T	Wave Period
Tf	Upper Bound of the Time Integration
Td	Input Matrix for the Viscous Drag Force
Ti	Kinetic Energy of i-th Body
Tp	Wave Peak Period
TPTO	Input Matrix for the PTO Force
u[j…j + Np -1]	Optimal MPC Trajectory
	Body Velocity Vector Time Domain
	Body Velocity Vector Frequency Domain
	Body Acceleration Vector Time Domain
	Body Acceleration Vector Frequency Domain
vA and vB	state of the PTO interconnection points
x	x-axis direction or heading direction
x	State Vector in the MPC Formulation
y	y-axis direction or sideway direction
z	z-axis direction or vertical direction
SUBSCIPTS
ref	The subscript identify the reference control strategy
GREEK LETTERS
α	Spectral Intensity Factor
Υ	Peak Enhancement Factor
Δυ	Increment of the Wave Excitation Force
Δu	Increment of the MPC Control Variable
ζpto	Critical Damping Ratio of the PTO system
η	Surface Elevation
λ	Wave Length
	Body Displacement Vector Time Domain
	Body Displacement Vector Frequency Domain
ρ	Water Density
ς	Fluid Particle Velocity
τυ	Matrix to Map the MPC Disturbance Increment in theOutput Space
τu	Matrix to Map the MPC Control Increment in the OutputSpace
τ	Time Constant of the PTO system
ω	Angular Frequency
ωpto	Natural Frequency of the PTO system

1.1 Green House Gas (GHG) and Carbon Dioxide (CO2) pro capita emission, (World Resource Institute, 2010). The data excluded the contribution from the land use change and forestry (LUCF) 1.2 Worldwide distribution of the global energy consumption, (Mcginn et al., 2013). 1.3 Worldwide distribution of the electrical energy consumption, (Mcginn et al., 2013). The class “Other renewable” includes wind, solar, geothermal and biofuels used to generate electricity 1.4 Map of the wave energy resource variability from season to season (Cruz, 2007) 1.5 Pre-commercial WECs listed in (Gadonneix et al., 2010), from left to right and from top to bottom: McCable Wave Pump, WavePlane, OceanLinx, Oyster, WaveGen Limpet, OPT Ocean Power Technology, Wavestar, IPS OWEC, SEABASED, Pelamis, SEEWEC, AWS Ocean Energy, Poseidon, OE Ocean Energy, Pico OWC. Only WECs developed in physical models are shown 1.6 Performance and readiness index matrix (Weber, 2012). The blue dots represent some of the patented WECs, and the yellow line represents the actual prospective of the sector. The green line represents a viable solution to reduce the cost of energy (CoE) of WECs 1.7 From left to right: Wavestar WEC, Dexa wave WEC and Weptos WEC. The photograph of the Wavestar shows the large-scale prototype tested in Hanstholm, DK, while the other two are graphical representations 1.8 Thesis outline diagram 1.9 Objective of the thesis and interconnection between sub-modules. The colour legend is located at the left-bottom of the figure. Greyed texts and boxes represent alternative solution and methodologies not used in this thesis. The figure is inspired by (Taghipour, 2008) 2.1 Wave-to-wire model definition and WEC system breakdown 2.2 Top: general PTO scheme for WEC of the WAB type. Green and cyan colours are used to distinguish between hydraulic and direct drive systems respectively. Bottom: sketch of the interconnection for a PTO system with the Wavestar WEC. 3.1 Types of error and framework of validation and veriication of numerical models. Figure inspired by (ASME, 2009) 3.2 Different wave excitation force regimes for a matrix of wave states, in agreement with Chakrabarti (1987). H: wave height, D: characteristic length of the object, λ: wave length. Each red dots represents a different wave period (T) and H duo. The numbering is rising in T irst and later in H 3.3 Mesh convergency study for a cylinder (diameter=10 m and draft=10 m). Np is the number of panels and nd is the smaller panel dimension divided by the cylinder diameter. Top left: Wave excitation force amplitude coefficients per unit of wave amplitude in function of the wave frequency. Top right: Evolution of the normalised error of the wave excitation problem in function of Np, with respect to the case Np = 2136. Bottom left: Radiation damping (CA) and added mass (CM) coefficients per unit of body velocity and acceleration respectively in function of the body motion frequency. Bottom right: Evolution of the normalised error of the radiation problem in function of Np, with respect to the case Np = 2136 3.4 Code-to-code verification of PTO models. Left side: Direct drive permanent magnet generator (PMG) with torque control response (blue line) vs second order transfer function response (red line); responses for unitary step and sinusoidal excitation. Right side: Constant pressure hydraulic PTO response (black line) vs approximated Coulomb damper response (red line); velocity-force curve and responses for irregular wave excitation 3.5 Assessment of the AEP from the location SD and WEC power matrix 3.6 Scale PTO force/velocity characterisation compared with an ideal approximated Coulomb damper model. On the right side the physical PTO model is shown in the inal coniguration 3.7 Measured displacement time series. Colour map: red - reference signal, blue -optical signal and green - accelerometer integrated signal 3.8 Validation of PTO models. Left side: Time domain comparison between measured (black), first-order transfer function (blue) and second-order transfer function (red) responses to a step in the commanded force, normalised with the end force. On the bottom side the discrepancy time series between models and measurement is plotter. Right side: Power spectral density of the measured and modelled data 3.9 Validation of mooring models. Time domain comparison between measured (red), quasi-static model (black) and dynamic model (blue) responses to a sinusoidal motion of the fairlead. Left side: Period of oscillation 1.9. Right side: Period of oscillation 1.3. The periods are normalised by the natural period of the floater in heave 3.10 Time series comparison between simulated and measured force acting on the Wavestar WEC single floater for two control configurations: P (upper plot) and PI (lower plot). Colour map: measured data (green line) and simulated data (blue line) 4.1 General layout of the time domain controller implemented with the PAC scheme 4.2 Optimal control prediction stage. From top to bottom and from left to right: Time series comparison between measured (green) and estimated (blue) wave excitation force. Time series of the calculated (black) and estimated (magenta) optimal velocity trajectory. Optimal feasible (blue) and unfeasible (red) position and velocity trajectories. The set of feasible position is delimited by black dotted lines 4.3 MPC with receding horizon control principle. The blue dots represent the first sample of the optimal control trajectory used at each time sample. Based on (Li et al., 2012) 4.4 Comparison of the AEP for the Wavestar WEC for the Hanstholm SD with different control scheme and cases. Cases: 1 - Unconstrained linear model; 2 - Unconstrained weakly non-linear model; 3 - Constrained weakly non-linear model 4.5 Flow diagram of the combined control and fatigue analysis methodology. Model inputs: WEC specification and SD. Model output: minimal cost function. 4.6 Comparison of CF for different p values and different control strategies. Colour map: p=0% (blue), p=10% (green), p=20% (magenta) B.1 (A) - Large-scale prototype installed in 2009 near Hanstholm (DK). The machine fed electricity into the grid until it was moved to the harbour for reconfiguration in September 2013. (B) - Schematic representation of a single floater of the Wavestar WEC. θ represent the rotational DoF and A identifies the pivoting point of the loater. The power is extracted by means of an hydraulic PTO system, represented on the igure by its actuator only B.2 Small-scale (1:20) physical model of the Wavestar WEC single floater. Points A, B and C correspond to the ones sketched in Fig. B.1 B.3 Communication flow diagram between host computer, target computer and WEC B.4 Absorbed power in function of the damping coefficients (Cc) for two different sea states. IRB1: Hm0 = 0.051 m and Tp = 1 s. IRB1: Hm0 = 0.08 m and Tp = 1.25 s. In both cases the wave steepness is near 3.5 %. A JONSWAP spectrum with γ = 1 is used for the wave generation. The sample time of each test is ive minutes B.5 Angular velocity versus PTO moment, measured (dots) and requested (full line) C.1 Old (top) and newest (bottom) conigurations of the Weptos WEC. The A-shaped platform is the grey (metal) coloured body and the rotors are the orange and yellow coloured bodies. Images' source: (Weptos, 2014) C.2 Meshed Weptos WEC. The black arrows define the inertial coordinate system, while the local coordinate systems are represented by the x-axis (red arrow) and y-axis (green arrow) C.3 Capture width ratio in function of Tp for the Hanstholm SD at the scale 1:20. The blue line represents the solution of the model depicted in Fig. C.2, and the red line represents the same results with the application of a correction factor to account for the gap between rotors 3.1 Relative occurrence of different wave states (pr) from six years, buoy measurements located at 6332100N, 474700E, water depth: 17 m ([Hm0] = m, [Tp] = s). Both parameters defines the mean value over an interval. The adopted discretisation is 1 s in Tp and 0.5 m in Hm0 3.2 Operational sea states parameters for the Danish North Sea As presented in the title, the thesis analyses technical matters related to one important and untapped renewable energy source. More specifically, the implementation and analysis of wave energy conversion system models and their utilisation are the primary points. The word “Wave-to-wire” defines inputs and outputs of the sought model, the former being the source of energy (ocean waves) and the latter being the usable absorbed energy feed into the electrical grid (wire). The reason for needing to talk about renewable energy is a matter of finding the right motivations. In his book, MacKay (2008) found three objective catalysts to lead the debate (paraphrasing the text): 1 - Even though the scenario is rather uncertain, measurements and indications say that using fossil fuels is changing the climate (Cubasch et al., 2013). Climate change is correlated to the greenhouse effect, which in turn is driven by the modified rate of production/absorption of the so called greenhouse gases. This unbalance or transient scenario is blamed on several human activities, which often regress to fossil fuels combustion. 2 - Fossil fuels as oil, gas and coal are a nonrenewable resource, thus it seems reasonable to imagine that their cheap supply will run out within a “short” timeframe. Production of such resources must inevitably reach a maximum, or “peak”, but the discussion about “how to locate the Oil Peak” is scattered and subjective. According to IEA (2013) based on the actual knowledge, the oil production will peak around 2020, but as argued in (BP, 2013), this information is misleading due to the technical evolution. It does not really matter when, the point is the presence of a production limit. Alternative energy sources are thus needed. Furthermore, instead of setting a valuable raw material (fossil fuels) on fire, it could be wise to use it to produce high quality goods. 3 - The stability and security of the energy supply is not of secondary importance. Assuming that fossil fuels are available, they are oddly distributed in few countries of the world, and ruled by even fewer companies. In this scenario, there is a real risk of making the economy of a whole state vulnerable to the whims of that restricted group of people. Thus the independency on those energy sources is a valuable option. On top of these motivations, two others, but non only, can be added to the discussion: 4 - The environmental aspects should never be forgotten. An undesired by-product of the fossil fuel activities is the environmental pollution. The consequence of accidental pollution of the water, soil and air has been made evident once more with the Deepwater Horizon accident (BP, 2010), while the estimation of the operational pollution rate of fossil fuels is still somewhat unclear. The environment pollution matters greatly because it has direct and indirect consequences on the human wellness, especially in future generations. Fig. 1.1. Green House Gas (GHG) and Carbon Dioxide (CO2) pro capita emission, (World Resource Institute, 2010). The data excluded the contribution from the land use change and forestry (LUCF). 5 - The growth of a new sector brings new job opportunities in both research and industrial corporations. In 2012, 5.7 million people have been estimated to be employed directly and indirectly in the renewable energy sector (Ferroukhi et al., 2013), mostly in solar and biofuels. But estimation says that the number can triple in accordance with the actual development planes. Each of the above motivations get even more important when looking at the prospected trends of development of different countries. As shown in Fig. , the pro capital pollution rate of emerging countries like China and India are well below the actual rate of developed countries like USA, Australia, Canada and Europe. The pro capita pollution rate is expressed in ton of carbon dioxide (CO2) and green house gasses (GHG) emission per person, excluding the contribution from the land use change and forestry (LUCF). If the living standard of these countries are exported to or imported by the emerging ones, there is a real risk of a sharp increase in the climate change rate, a short-term depletion of fossil fuels and a general shortage of electricity supply. Regardless of the chosen motivation, the need for alternative sources of energy is evident.

Even if the renewable energy sources are considered to be unlimited, it is possible to generally define a resource as “renewable” when its production/consumption rate is balanced in a human timeframe. In addition, in order to be sustainable its utilisation should result in an environmental benefit. The latter point excludes nuclear, coal and Oil & Gas resources from the list because their exhaustion time frame is still somehow unclear de facto. Since renewable energies are often less concentrated, widely distributed, unstable, etc., their exploitation needs to be a summation of small contributions. In a global frame, renewable energy accounts for 19 % of the total consumed energy (Mcginn et al., 2013). As shown in Fig. , this fraction is equally shared between traditional biomass – mainly wood -used to cook or heat, and modern renewables. In this subset, we account for hydropower, biomass, solar PV and thermal, geothermal, wind and biofuels. Fig. 1.2. Worldwide distribution of the global energy consumption, (Mcginn et al., 2013). The worldwide electrical energy consumption is around 13 % of the global energy consumption and almost 22 % is provided by renewable sources, Fig. . Among them, the hydropower has the larger share, around 77 % of the electrical energy from renewables.

The wave energy sector is currently not a noticeable contributor but it has a high potential to play a significant part to the world energy mix in the future. The practically exploitable wave power potential has been assessed to be up to 3-3.7 TW, (Mørk et al., 2010), which is about a fourth of the global demand and roughly the double of the global electrical consumption. As shown in Fig. , if an overall efficiency of 10 % could be achieved, the energy produced from ocean waves may cover 20-30 % of the global electrical consumption. The wave energy resource is well distributed in temperate and subtropical zones of both hemispheres. Further, the wave energy resource eases the energy supply process, being close to the highly dense populated zones (coastlines). Compared with oil and gas offshore structures, wave energy converters are associate with a smaller environmental risk, while sharing a low visual impact that makes the resource well accepted by the community, (Chozas, 2013). Waves occur in all sizes and forms owing to the magnitude of the force acting on the water. Small objects impacting the water surface will generate short waves, while the gravitational attraction of the moon and sun will generate long waves (tides). The wave energy converter (WEC) mainly interacts with wind generated waves that are the consequence of the odd dynamic pressure distribution of the air phase (wind) exerted on the water surface. As a result of the distance between generation point and closest shoreline in the downstream direction (fetch), as well as the strength of the blowing wind and water depth, the waves will reshape toward the least dissipative form, i.e. gravitational waves. Gravitational waves can travel long distance with negligible energy dissipation, as long as the associate dynamic pressure field does not interact with the seabed. Thus, the energy content is reduced in the near-shore zone either in friction or breaking wave phenomenas. Since the waves are wind driven, and the wind is a result of the uneven distribution of the irradiated energy by the sun over the earth surface, it follows that waves are just a second derivative of the solar energy. Fig. 1.3. Worldwide distribution of the electrical energy consumption, (Mcginn et al., 2013). The class “Other renewable” includes wind, solar, geothermal and biofuels used to generate electricity. The advantages of the wave energy resource is given by its predictability (Chozas, 2013) and the higher energy density if compared with wind and solar. On the contrary, higher density means higher ratio between extreme and operational energy, which entails higher structural costs. Thus the source variability over seasons becomes a critical point. Unlucky, as shown in Fig. , the lower variability can be found in low density inhabited areas of the southern hemisphere, away from the energy request. Another interesting point is the complementarity between wind and wave source, (Chozas, 2013), which well matches with the concepts of renewable energy mix. Even if rooted back at the beginning of the twentieth century, the modern wave energy sector first rose in the seventies together with the oil crises when Masuda (1971) created the first navigation buoy driven by an oscillating water column WEC. From those years in which the basement of the sector had been developed, according to Nielsen (2012) and Gadonneix et al. (2010), hundreds of patents have been released even if the sector went through different phases of interest. Nowadays, sevral converters (Fig. ) have reached the pre-commercial stage; this shows that the embedded energy potential has been considered enough to attract both stakeholders and developers. Fig. 1.4. Map of the wave energy resource variability from season to season (Cruz, 2007) Fig. 1.5. Pre-commercial WECs listed in (Gadonneix et al., 2010), from left to right and from top to bottom: McCable Wave Pump, WavePlane, OceanLinx, Oyster, WaveGen Limpet, OPT Ocean Power Technology, Wavestar, IPS OWEC, SEABASED, Pelamis, SEEWEC, AWS Ocean Energy, Poseidon, OE Ocean Energy, Pico OWC. Only WECs developed in physical models are shown. All the proposed concepts conceived so far can be grouped in three macro-systems, classified by the working principle of the primary energy capture: Oscillating Water Column (OWC) Overtopping (OT) Wave Activated Body (WAB) The proposed classification is based on (Drew et al., 2009; Falcão, 2010). Others possible subdivisions are based on, for example, the distance from shore, the power take off chain mechanism, the orientation to the wave front, etc. For further details about classification and working principle see (Clement et al., 2002; Drew et al., 2009; Brooke, 2003; Falcão, 2010; Nielsen, 2012). The WAB class can be subdivided in three group based on the dimension and orientation of the body: point absorbers (PA-WEC) -small body compared with the wave length, attenuator (A-WEC) aligned with the wave propagation direction and terminator (T-WEC) perpendicular to the wave propagation direction. Until the last decade, in absence of a standardised procedure, the development of different concepts went in random directions leading to various issues. The most relevant ones are the credibility gap between stakeholder, community and developers as well as the investment losses. In order to contain and manage the growing sector, standard protocols have been defined in the EquiMar project, (Ingram et al., 2011). Its aim is to give a generalised step by step method to guide developers from the paper sketch to the commercial device, while encouraging the use of small-scale physical tests, together with numerical simulations. Another important contribution to the sector was recently introduced by Weber (2012), with the concept of performance and readiness indexes matrix, Fig. . The methodology presents the optimal path with balanced indexes, aiming for the minimal cost of energy (CoE), while minimising the overall investment cost in the development stage. In order to achieve this goal, the variation of the performance index should be steep in the starting phase with small scale while flattening when a satisfactory performance is conceived; the other index should follow the complementary path -green line in Fig. . According to the author, the analysis of the best pre-commercial WECs shows an overly high readiness index for the corresponding performance index yellow line and blue dots in Fig. -leading to a prospected CoE too high if compared with other renewable resources. Various limiting parameters do exist for the sector to be able to become a viable solution in a short time frame. The predominant one is the (levelised) CoE: too high if compared with other sources of energy. For WEC, but more generally for off-shore structures, the overall lifetime cost is driven by structural costs, which are ascribable to extreme and fatigue loads (design loads, (Carbon Trust and DNV, 2005)). As presented by Fitzgerald (2009) and Previsic and Shoele (2013), the WEC cost breakdown reveals that bottle-necks are located mainly in structural costs and later in the mooring and PTO costs. But in a case with a large number of small units, i.e. for a PA-WECs farm the operation and maintenance (O & M) procedures will have an important share too. Another important parameter is the number of different WEC concepts, too large to make possible to focus the research effort on a few promising devices. This large distribution of “feasible” solutions is tailored by the uncertainties of the performance capability, evaluated in model tests. Fig. 1.6. Performance and readiness index matrix (Weber, 2012). The blue dots represent some of the patented WECs, and the yellow line represents the actual prospective of the sector. The green line represents a viable solution to reduce the cost of energy (CoE) of WECs. Both, numerical and physical models have been used to analyse and compare different concepts, and together with the progress in computer science, the focus moved from laboratory based analysis towards a numerical/physical integrated analysis. Nowadays, numerical wave-to-wire models of WECs are a well-known methodology of analysis of WECs, and they cover a large part of the available literature of the sector. Most of the works presented are based on the linearised Diffraction/Radiation theory inherited from the Oil & Gas and naval sectors, (Hansen et al., 2012; Abraham and Kerrigan, 2013; Babarit and Clement, 2006; Fusco and Ringwood, 2013; Beatty et al., 2008; Costello et al., 2011) etc., though in the last years more and more focus has been given to the extension of the linear problem with non-linear contributions, (Sclavounos, 2012; Guerinel et al., 2013; Bhinder et al., 2011). The need of non-linear terms in the numerical model formulation is tailored by the violation of the assumptions of the underlined linearised theory. The latter is based on small waves and body motion where the body wetted surface does not change significantly in time and viscous effects are negligible due to the small velocities. In contrast, a WEC is requested to move as much as achievable in specific degree of freedoms (DoFs), which bring to the afore mentioned violations. So far, the most comprehensive analysis of WECs based on a weakly non-linear wave-to-wire model is presented in (Babarit et al., 2012). Different WEC systems at different locations are compared in terms of power performance, while highlighting the level of uncertainty of the results. In addition to the large utilisation and development of wave-to-wire models, the com-putational power growth brought at the top the WECs optimisation matter. Optimisation or efficiency maximisation of WEC is a hot topic of research which can be easily and cheaply achieved with numerical models. So far, mainly two techniques have been used: hydrodynamic optimisation based on the shape optimisation (Alves et al., 2007; Gilloteaux and Ringwood, 2010) and efficiency maximisation through control schemes. The energy maximisation of wave energy converter by mean of control strategy is a vast topic of research. The close form solution of the maximisation problem is one of the earliest finding of the sector, and in the last decades the interest moved toward real-time implementation of the control strategy, (Hals, 2010; Fusco and Ringwood, 2013; Babarit et al., 2009; Cretel et al., 2011a; Hals et al., 2011a; Price, 2009; Fusco and Ringwood, 2010a; Nielsen et al., 2013). However, the recent works presented by Borgarino et al. (2012) and by Zurkinden et al. (2013) highlighted the importance of the structural design in the global optimisation of WECs. In contrast to this large number of works in the numerical environment, the literature regarding the validation of numerical results based on physical model tests is somehow loose, (Durand et al., 2007; Lopes et al., 2009; Paredes et al., 2013), and mostly based on simple geometries.

In order to cope with the need for renewable and complementary energy plans, the wave energy sector offers an important research field due to the high embedded potential. But the cost of the produced energy causes the resource to be largely untapped. With the aim of assisting efficient development and analysing WECs and thereby accelerating the sector progression towards commercialisation, a generally applicable, efficient and reliable wave-to-wire model tool is needed. As a consequence of the large number of different WEC concepts, the model is seldomly generally applicable, and focus has been given to the WECs of the wave activated body (WAB) type. The efficiency and reliability of the model are defined once the overall objective is stated. Within the framework of the Structural Design of Wave Energy Devices (SDWED), which granted this work to a great extent, the word “efficient” means low computational cost and “reliable” means an uncertainty level as small as attainable. Three different WECs defines the base of analysis, namely Wavestar (2014), Dexawave (2014) and Weptos (2014). They are listed in increasing order of complexity from a model point of view, and presented in Fig. . Among the other before-mentioned issues, four points are considered critical in the implementation of a wave-to-wire model and they define the objective of the thesis. Identification of the critical sub-systems for a general WAB WEC. Albeit similar to a large extent, WECs of the OT and OWC types are not considered as a part of this work. The selection of the key sub-systems is led by the analysis of predominant loads acting on the WEC. Analysis of the available model for the selected sub-systems and identification of the best cost-effective solutions. The definition of “best” is defined as a trade-off between computational time, accuracy and precision, being the last one related to the actual stage of the wave energy sector. Entailed by the large number of concepts, what is needed is mostly a rough sieving of the available devices rather than a detailed analysis of them. Therefore, the best model will have a rather coarse precision keeping the computational time to the lower achievable level and the accuracy at the highest possible level. Fig. 1.7. From left to right: Wavestar WEC, Dexa wave WEC and Weptos WEC. The photograph of the Wavestar shows the large-scale prototype tested in Hanstholm, DK, while the other two are graphical representations. Validation of the numerical models by means of physical tests. In order to quan-tify/reduce the uncertainty of the numerical results. These steps are indispensable to create a fair comparison without exception, or reducing the gap, between numerical and physical results. Economical optimisation of WECs. The first step will be the increment of the WEC absorbed energy by means of advanced control strategies, which will be then balanced by structural fatigue analysis. The main attempt is dual: implementation of widely known control strategies into a WEC physical model in order to understand the real applicability of those techniques and to apply well established fatigue analysis procedures in order to introduce a structural variable into the economical optimisation of the WEC. A corollary objective of the thesis, tailored by a specific need found in the physical tests, is the implementation of a non-contact position and orientation tracking system.

The thesis is organised in a collection of papers including a thorough introduction to the work. The general and specific problems are stated in the introduction, while the detailed analysis of the issues is given in Appx. A which collects all the papers. Two other appendixes are included, Appx. B and Appx. C, which give details about the Wavestar WEC physical model and the Weptos WEC numerical model. The latter describes the first set of interim results obtained for the Weptos WEC, but due to large range of uncertainty expected in function of the approximation used the model has not been incorporated in the thesis as such. The introduction is divided in five chapters as indicated in Fig. . In Ch. 1, the general leading motivation is given together with the previous and current status of the wave energy sector. The chapter attempts to shed light on the current issues faced by the sector which are limiting the exploitation of the wave energy resource, seeding the objectives of the work in those elements which have been considered key points. In Ch. 2, the system breakdown and qualitative analysis of a generic WEC of the WAB type are given. The result of the quantitative analysis is the definition of critical sub-components from a model point of view. For each of the subsystems, the actual status of development is analysed to identify the key points, which are then addressed in detail in Chaps. 3 and 4. Also, the relevant literature is listed. Fig. 1.8. Thesis outline diagram. Ch. 3 is divided in two main sections. Sec. 3.1 lists the available numerical models for each of the selected sub-systems, whilst the ones considered relevant are further detailed. As already introduced in the context of this work, “relevant”, previously “best”, is selected in agreement with the relatively low level of precision required in the analysis. Therefore, low computational cost methods with the highest -achievable -precision will be preferred. The second part of the chapter, Sec. 3.2, describes the physical models used and the validation of the numerical results, giving a critical discussion over the limitation of the proposed methodologies. The resultant model is subsequently used as a base for the optimisation of the WECs discussed in Ch. 4. Similar to the previous chapter, 4 is divided in two main sections. Sec. 4.1 briefly describes five well know control techniques. Also, their implementation is described, highlighting the main difficulties and deviations from the reading theory. In the second part of the chapter, Sec. 4.2, the control laws introduced are used together with standard structural fatigue methods to find a balanced optimal economical design. In the last section the focus will stay put on a simplified version of the Wavestar WEC, reducing the system to a single degree of freedom to ease the analysis of the system. Ch. 5 summarises the main outcome of the work presented in the thesis, together with the prospected future works. Fig. 1.9. Objective of the thesis and interconnection between sub-modules. The colour legend is located at the left-bottom of the figure. Greyed texts and boxes represent alternative solution and methodologies not used in this thesis. The figure is inspired by (Taghipour, 2008)

A wave energy converter (WEC) can be defined as a dynamic system with one or more degree of freedoms (DoFs), used to transform the wave energy content into “useful” — typically electrical — energy. The definition infers the wave as the input, and the “useful” energy as the output of the system. The term useful has been highlighted because along with the electrical power production, a WEC can have other uses, for example potable water production (Nolan and Ringwood, 2006). In the following, as already introduced previously, only WECs of the WAB type are considered. On a macro scale, the device breakdown shows six main subsystems, listed below and exemplified in Fig. with relative interconnection: Main Structure (MS) -Hydrodynamic System (HS) plus Hosting Structure System (HSS) Power Take Off (PTO) System Reaction System (RS) or Station Keeping System (SKS) Control, Instrumentation and Electrical System (CIES) Power Connection System (PCS) Onshore Facility System (OFS) Even though each of these elements is indispensable for the system to work, only the first four element of the list will be further discussed hereafter. The choice can be justified by the objective of the work presented in the thesis: the definition of global and steady state parameters, rather than transient and specific ones. Further, the PTO and the CIES systems will be discussed together since they are tightly linked. The nomenclature has been chosen in agreement with (Hamedni et al., 2014), developed during the SDWED project. Fig. 2.1. Wave-to-wire model definition and WEC system breakdown.

The main structure of a WEC is a polymorph element that changes from device to device; Fig. gives some examples of devices. It is unrealistic to define a general shape, but from a mechanical point of view the system can be seen, to a great extent, either as a single rigid body or a multi body system. Deviations from this assumption may be given by the Anaconda WEC or similar types, (Chaplin et al., 2007), although they can be considered as a multi-body system once the single body size is properly defined based on structural eigenmode analysis. The main structure either holds, or is, the hydrodynamic subsystem (HS) of the WEC. The HS is defined as the element through which the device interacts with the energy source; thus it is the place where the energy conversion starts. For WECs with a large emerged, or non interacting with waves, volume, the main structure will be composed by the HS and the Hosting Structure System (HSS), the latter being the house of the CIES. Besides converting the wave energy content into mechanical energy at the HS, the MS is required to last extreme loads, prevent the contact between water moisture and electrical part, withstand corrosion, as well as ease and reduce the cost of operational and maintenance (O&M) procedures. The MS placement is as such a key point. As introduced in Ch. 1, the CoE is the limiting factor for most WECs conceived so far. The power absorbed in operational sea states and the related turnover (income associated with the energy production) is too small if compared with the total cost of the system over its lifetime. The cost of the system is made by capital expenditure (CAPEX) and operational expenditure (OPEX) in first approximation. In many cases, the total cost is driven by structural capital costs. It is important to highlight that for point absorber WECs installed in large numbers the O&M could become as important as the structural cost. But the matter is not further described in the thesis. When the structural cost is the bottleneck, both the ultimate limit state (ULS) and the fatigue limit state (FLS) design parameters should be considered in the first stage of analysis (Veritas, 2013; DNV, 2010). The ULS corresponds to the maximum load-carrying resistance caused by extreme events. The latter is defined for WECs as sea states associated with high energy density and small likelihood, i.e. one event over ten, fifty, hundred years. As a consequence to most of the wave energy content being concentrated near the free surface, a body placed closer to the water surface will access a larger pool of energy. While this is convenient in operational sea states, it turns harmful in extreme conditions. The power per unit width of wave front associated with operational and extreme sea-states can be 5-50 and >2000 kW/m respectively. The WEC needs a survivability strategy in order to reduce the magnitude of the design loads associated to extreme sea states and thereby reduce the capital cost. Different survivability scenarios have been proposed like submergences, shape-variation, control, etc. (Folley and Chaplin, 1998). On the other hand, the device can be placed somewhere away from the high energetic zone, which reduces the extreme/operational loads ratio and the convertible energy too. From another perspective, the choice of the design standard is also a fundamental element. In contrast to the Oil & Gas industry where stringent rules or guidelines need to be applied as a consequence of the high magnitude associated with the system breakdown, for the wave energy sector the risk associated with the system failure can be lowered to the same level as in the offshore wind sector. Briefly, the magnitude of the failure is mainly the cost of the device without any major implication for human life risk, environmental risk, etc. Although specific standards have been proposed in recent years (Carbon Trust and DNV, 2005), so far the wave energy sector needs to refer to the wind sector, whose standardisation is more mature (Veritas, 2013). The other structural design parameter cited is the FLS. In spite of the large number of load cycles that the WEC will withstand through its expected lifetime, fatigue is generally considered to have a smaller impact on the design of the system. This assumption can be shared to a great extent for WECs with a resistive controller, but when an active control strategy is used the conclusion is not so trivial; for the definition of the controller types see Ch. 4. In fact, when an active control strategy is adopted, the amplitude of the PTO duty cycle will undergo a large increase if compared with a resistive controller. The load amplification causes a greater stress accumulation on the structure and a subsequent growth of capital cost. Besides the CoE, as already introduced in the previous chapter, the large number of WEC concepts is also partially liable of the slow development rate of the sector. Among the patented concepts — more than hundred (Gadonneix et al., 2010) — only few of them have reached the pre-commercial stage, whilst the majority is still in conceptual/small-scale test. This large variability is imputable directly to the MS, since both PTO and reaction systems rely on more mature and standardised technologies. A vast case history — mostly based on numerical methods — describes the functionality of the different concepts (Chaplin et al., 2007; Hansen et al., 2012; Pecher et al., 2012b; Silva et al., 2013; Mackay et al., 2012; Seidel et al., 2012; Beatty et al., 2008; Babarit et al., 2012) in operational conditions. Despite their importance, extreme events are rarely considered from a numerical point of view (Palm et al., 2013a) and more present in the literature concerned with physical modelling (Zanuttigh et al., 2013). On the contrary, the dissemination of results of large scale prototypes is loose, (Marquis et al., 2010; Pecher et al., 2012a). To a great extent the numerical analysis is based on the linearised hydrodynamic problem solution (see Ch. 3), inherited from Oil & Gas and the naval industry. But if a prediction of the economical effect of extreme loads needs to be endorsed, then non-linear waves are the base for the design (Carbon Trust and DNV, 2005). Regrettably, only few examples of utilisation of fully non-linear solvers are available (Yu and Li, 2011; Palm et al., 2013a; Westphalen et al., 2009) due to their considerable computational cost. Similar to the lack of research in extreme sea-states, the literature about the influence of the control strategy into the fatigue design of the WEC is also limited (Borgarino et al., 2012; Zurkinden et al., 2013).

From what has been previously introduced, two issues are considered for further analysis in the next chapters. First, the definition of a fast numerical methodology able to output reliable results in different control scenarios in operational sea states. The term “results” accounts for the calculation of absorbed power and loads. Although the wave structure interaction can be evaluated with different levels of accuracy, only the methods which infer a small computational time can be used to compare a large number of different systems. The validation procedure is needed to estimate the level of uncertainties expected in the results. The topic is further covered in Ch. 3, and it is based on the implementation of standard numerical methods and their analysis in physical models. In addition, as discussed in Ch. 4, the numerical model is used as a base of optimisation of the given WEC, not only from an energy perspective but also from an economical one, pushing toward a minimisation of the actual CoE. At this stage, FLS will be used in the model too. Second, the predictability of ULS or other extreme loads using a simple and low computational cost model. The topic is further covered in Appx. A.3 and Appx. A.6, and it is based on the implementation of standard numerical methods and their analysis in physical models. Although the model describes a floating offshore wind turbine (FOWT), similarities with large WECs could allow a partial extension of the methods to the wave energy sector. Due to the large variability of WEC concepts, exceptions from what stated above are likely to exist. For example, whenever the HS is also hosting the CIES, insulation and maintenance may become important elements too, meaning that their influence should not be disregarded even in the first stage of analysis.

A PTO system is defined as the chain of processes which transforms the available energy at the HS, i.e. activated body, reservoir, chamber, etc., into electricity to be delivered to the grid. The definition infers the encompassment of a control law (CIES system) into the PTO system. The PTO system is the key element of a WEC, first of all because it is the distinguishing element between a floating body and an energy converter. The dynamic response of a WEC is highly correlated to both the PTO type and the control law. The PTO exerts time varying loads on the HS, and widely on the MS, which affect the absorption capability of the system. The absorption capability of a WEC are commonly quantified in term of the capture width (CW), defined as the ratio between absorbed energy and incoming energy per meter of wave front. Although each WEC developer/company has the tendency to develop its own PTO system, few main different classes of the PTO type are widely used into the wave energy sector. In particular for WECs of the WAB type — the focus of this thesis — two main system are adopted: Fig. 2.2. Top: general PTO scheme for WEC of the WAB type. Green and cyan colours are used to distinguish between hydraulic and direct drive systems respectively. Bottom: sketch of the interconnection for a PTO system with the Wavestar WEC. Mechanical direct driven system Hydraulic system For the sake of simplicity in this thesis, the direct drive class also includes those PTO systems where the connection between the HS and the generator is obtained by means of mechanical elements but excluding any fluid. For example, PTO with gear boxes is considered a direct driven system. Further, along with the hydraulic systems, the air compressed systems are also a possible solution, but their utilisation is bounded by the limited overall efficiency (Lemofouet and Rufer, 2005). The PTO system receives high harmonic forces (up to MN) with long oscillation periods (from 1-20 s), and it needs to fulfil the required grid specification. Furthermore, it is important to bear in mind the aggressive environmental conditions where the device is situated (i.e. high salinity, distance from land, high load in storm condition, water spray, slamming load, etc.). It is possible to draft a common scheme for the two PTO classes, see Fig. , where the available energy, either kinetic or potential, is transferred from the HS into the generator, in parallel with a control scheme, and a storage system (if needed); the transformation can be either single or multistage.

This is the simpler PTO type among the two, which makes it the first studied solution in the early research years in the sector: a reduced number of components is a basic request especially in offshore matters. The generator, either linear or rotative, and the HS are directly linked via mechanical connectors, such as mooring line, gearbox, pulley or belt. The main cons were inefficiency, high cost and weight of the overall system, but in recent years, technical improvements led by the wind sector have made this type of PTO a viable solution for a small size WEC. The design airgap velocity have been reduced from 60 m/s to 0.5 m/s fitting with PA-WEC (Mueller, 2002; Mueller and Baker, 2002; Shek et al., 2008; Binh et al., 2012; Ivanova et al., 2005; Polinder et al., 2005; Polinder et al., 2007; Eriksson et al., 2005). In addition, enhancements in power electronics (Brooking et al., 2002), together with advances in the sector of battery and ultra capacitor, have also meant an improvement on the quality of the energy fed into the grid, which means a possible match between the produced electricity quality and the grid code. Common examples of PA-WEC using linear mechanical direct driver are Archimede Wave Swing, Uppsala Point Absorber and the “L10” Buoy. Even though linear generators are commonly used in direct drive WEC, rotating generators can also be adopted. For example, the Weptos WEC, (Pecher et al., 2012b), adopts a rotating generator driven by a cluster of activated bodies mounted in a single large structure. The main disadvantage of rotating generator is the minimum angular velocity required by the generator to work properly, which forces the utilisation of a gearbox with related challenges.

In this class of PTO, a fluid flux generated by a hydraulic piston (actuator) connected to the HS impels the motion of a motor that is coupled to a rotating generator. The fluid can be both water and oil, even though the utilisation of the latter allows a better control on performance and durability of the system. Although hydraulic PTO systems are robust and designed to work with large forces and small velocities, which coincide with the characteristic framework of WECs, the complexity of the system is a limiting factor in off-shore applications. They do normally adopt accumulators even though the utilisation of a power electronic unit can replace this need, i.e. Wavestar WEC (2014). According to Kamizuru et al. (2012) and Costello et al. (2011), the hydraulic circuit can either use or not use a bridge rectifier at the output of the actuator. The rectified solution is simpler but less efficient and controllable, but in both cases the system is far more complex compared to the direct drive option. Results of both numerical, (Costello et al., 2011; Hansen et al., 2012; Falcão, 2007; Zhang et al., 2012) and physical, (Marquis et al., 2010; Henderson, 2006; Lasa et al., 2012) WEC models with hydraulic PTO systems have been extensively reported in recent years. A diverse model has been recently presented by Hansen et al. (2013), where a multi-chamber hydraulic PTO system has been used as a simulation base and built in full scale too. Common examples of WECs with hydraulic PTO system are: Wavestar, CPT, AquaBUOY, OPT, Wavebob for the PA-WEC type, Pelamis, Dexadevice for the A-WEC type and Oyster for the T-WEC type.

The control logic is normally included in the definition of the PTO system due to their strong bound. The control system defines how the PTO system works, and in turn affects the operation of the WEC. The modification of the control logic is the most straightforward way to modify the efficiency of the WEC and therefore to modify its revenue. A robust literature about the implementation and optimisation of the control logic within the wave sector is available. Some examples are:Hansen et al. (2013); Fusco and Ringwood (2013); Cretel et al. (2011a); Babarit and Clement (2006); Binh et al. (2012); Falcão (2007); Fossen (2011); Falnes (2002); Hals (2010); Abraham and Kerrigan (2013), but more details are given in Ch. 4. But most of the example available are based on numerical simulation and only few example of implementation of advance control strategies in physical models are available.

From the previously introduced issues, two are considered for further analysis in the next chapters. First, due to the need of a fast numerical method, it is of primary interest to solve the implementation of a PTO model problem by simplified transfer function and lumped models. In those models, only the eigenfrequency of the PTO within the wave/WEC frequency range is then needed. The topic is further covered in Ch. 3 where examples of simplified PTO models are given for both numerical and physical cases. Second, a critical point in the PTO/control system is the implementation of advanced control strategies, and the evaluation of their true capability, in physical models. Due to the bad scalability of PTO systems, the implementation of advanced control schemes in laboratory tests is often unfeasible. The topic is further covered in Ch. 3 where two examples oh the implementation of PTO in physical models is given.

Besides from the first order motion induced by linear waves, offshore structures are subject to forces, such as wave, current and wind, that are causing a drift in the mean position. Reaction or station keeping systems are adopted in every WEC to achieve the following tasks: Maintain the floating structure on station, plus/minus a certain tolerance, under operational and extreme condition. Reduce/remove load on the electrical connection. Ensure alignment of directional WEC. Avoid impact with other structures (ships, WECs, etc.). Nevertheless, reaction systems should not wrongly affect the power production of WECs, have low cost, high reliability level, low environmental impact and require little inspection and maintenance. As shown in Fitzgerald (2009), the RS has a large share in the overall lifetime cost of the converter, thus its cost optimisation goes along with the economical optimisation of the energy produced by a WEC. Similar to the main structure, the station keeping system shall be designed to last extreme events, whose loads can exceed operation loads by a decade or more. Further, the variability of WEC design does not allow an a priori best configuration, which instead needs to match the WEC and location characteristics. But based on simple considerations, it is possible to identify two different classes: Mooring system for large WEC with self-referencing body, i.e. FO3, Weptos, Wave Dragon. Mooring system for activated body WEC without referencing body or small size self-referencing WEC. On the one hand, in class 1 the reaction system does not influence the dynamic behaviour of the HS to a great extent, i.e. only the long period motion of the referencing body is modified. On the other hand, in class 2, while the reaction system needs to be still in compliance with the stationkeeping requirement, the WEC will undergo a modification of the overall body response in both static and dynamic ranges. For this class, disregarding the mooring model— either the full description or a linearised version— will increase the uncertainties of the predicted power performance of the device heavily. A first example of the direct implementation of the mooring system into the power absorption maximisation of a PA-WEC is presented in (Richter et al., 2013). The design of reaction systems for a WEC is a difficult task, mainly because the technical design needs to be balanced by fundamental non-technical matters. First in place comes the risk based design for off-shore renewable converters, (IEC/TC, 2013). In order to reduce cost of energy of WEC systems, the mooring system should be designed with a lower risk level than an Oil & Gas offshore platform. The utilisation of a recent standard from the offshore wind sector (Veritas, 2013) will lead to a cost reduction of the mooring system in consequence of the reduction of the amount of material employed. In light of the weight of the reaction system in the overall lifecycle cost of the WEC — ca 20 % (Fitzgerald, 2009), its cost reduction is an important matter. But a large number of failures of WECs at real sea deployments — directly related to a failure of the stationkeeping system — has caused a lack of faith in the public opinion toward the wave energy sector. Paraphrasing Gadonneix et al. (2013), the sector underestimates the embedded technical difficulties, increasing the discontent of observers, stakeholders and community. So, the mooring system needs to increase the associated level of reliability to be accepted, which is the diametrically opposite of what has been shortly stated. Specific design rules have been inherited from the offshore industry, i.e. naval and Oil & Gas , but a WEC differs from those applications by the dynamic response. Ships and platforms are meant to reject the wave disturbance while WECs are tuned to be in resonance with it. The velocity associated with the stationkeeping of a WEC is larger than the one of ships and platforms, therefore non linear processes as internal viscous damping need to be accounted with a dynamic cable model, (Johanning et al., 2007). Several works addressed this issue with both experimental and numerical analysis, i.e. (Dubuque, 2011; Ruiz-Minguela et al., 2008; Rhinefrank et al., 2010; Zanuttigh et al., 2013; Parmeggiani et al., 2013; Palm et al., 2013a; Paredes et al., 2013). The above mentioned numerical examples are all based on a fully dynamic model of the mooring cable. The main drawback of these methods is the high computational time, which mostly limits their utilisation in optimisation or fatigue assessment routines. These types of analysis are commonly conducted with a quasi-static approach, (Vicente et al., 2011; Richter et al., 2013). While waiting for the computation time of dynamic model to drastically drop, an alternative approach has been proposed by Fitzgerald and Bergdahl (2008) where the fully dynamic model is used to tune the frequency response of the mooring system including both stiffness and damping terms.

Based on the needs of the analysis presented in this thesis, a linear approach with constant parameter is used. However, dynamic simulation or physical model test of the station keeping system are then used to calibrate the linear parameter of one of those models. The mooring analysis is restricted to the only case of spread mooring systems, where each of the mooring cable is constituted by a simple suspended chain, without intermediate bodies. The influence of the full reaction system into the power capability of a floating WEC is further analysed in Appx. A.7 with experimental tests.

As suggested by the EquiMar protocol, the performance index method and others, the feasibility of the WEC concepts and its potential performance should be evaluated and optimised as early as possible in the development process. The utilisation of models, numerical or physical, is a necessary step therefore. In the modelling phase, different sources of errors are expected to deviate the model output from the unknown truth. A general overview of the problem is described in Fig. which is based on the analysis presented in (ASME, 2009). The methodology (validation and verification) estimates the accuracy of a numerical simulation in two steps. The verification step establishes the accuracy of the code based on a benchmark with available analytical solutions. The validation step estimates the simulation modelling error range based on a comparison with appropriate experimental results. The errors which lie between the truth and model output are measurement (physical), algorithm (numerical) and structural (physical and numerical). A brief description of the errors is given in the following. Measurement errors are the easiest to identify and quantify. They are related to the instrumentation accuracy and precision, to the error introduced by the physical environment i.e. wave basin reflection, and to the operation of the experiments i.e. wave generation technique, consistency on the procedures, etc. Algorithm errors are defined by the solver error i.e. truncation errors, discretisation, etc., and by the parametric errors i.e. model inputs and sensitivity of the model to their variation. Structural errors are related to model inadequacy and simplification. For a physical model, one possible source of structural error is the scaling law used. For example, the underestimation of the scale effect can lead to a modification of the ratio between predominant forces. For a numerical model, one possible source of structural error is the model approximation and its underlying theory, for example using a potential solver for a viscous dominated flow. The following sections provide further details.

Simulation are mainly helpful in earning insight to the dynamic behaviours and interactions that are often not promptly evident from theory. Even though simulations are often chosen to study transient dynamics, compare conceptual designs, risk assessment, etc. They are merely a representation of a theory that is abstracted from observation of the physical process. Numerical models need to be realistic, simple and easy to manipulate; these are conflicting requirements. Realistic models are seldom simple, and simple models are seldom realistic. Thus, the definition of what is considered relevant infers features and behaviours that are pertinent and those can not be neglected in the model. From the key elements listed in the previous chapter, steady state and global parameters are considered pertinent features in the model selection. In point of fact, the different methodologies applied to calculated power performances and structural fatigue damages of WECs (Ch. 2) are based on long simulations, thus the computational cost of the models needs to be considered. This requirement is in accordance with the aims of the SDWED project, which is partially funding the work presented in this document. The main project objective is the definition of a generally applicable, efficient and reliable wave-to-wire model tool, used to assist an efficient development and analysis of WECs. Fig. 3.1. Types of error and framework of validation and verification of numerical models. Figure inspired by (ASME, 2009). For each of the elements discussed in Ch. 2, the available models are listed and the ones considered relevant in the framework of this thesis will be further discussed.

The combination of hydrostatic and hydrodynamic pressure fields exerts variable loads on any floating structure exposed to the wave structure. There are several models with different levels of accuracy to solve the time varying loads on the structure, where the common ones are: Morison’s Equation - Semi empirical non-linear model PaM - Panel Method (also known as Boundary Element Model (BEM), Boundary Integral Equation Method (BIEM) and Diffraction/Radiation problem) CFD/(u)RANSE -(unsteady) Reynold-averaged Navier-Stokes Equation SPH - Smoothed Particle Hydrodynamics CFD/LES - Large Edge Simulation CFD/DNS - Direct Numerical Simulation The different model types are listed in ascending order of required computational time/cost and model complexity. The DNS method is considered not usable for any practical application for the time being due to the excessive required computational time. Simulations of WECs based on CFD methods are possible (Yu and Li, 2011; Palm et al., 2013a), but their utilisation is limited to short-time simulations in consequence of their large computational cost, i.e. extreme condition assessment. Mostly the (u)RANSE turbulence method is used, while LES is still too heavy from a computational point of view to be widely used in practical applications. So far, the Morison and PaM models are the common adopted methodology when a fast analysis needs to be carried out. An example is the assessment of time average parameters such as annual energy production (AEP) and capture width (CW) of the WECs. The relevant bibliography includes a broad spectrum of case studies, which was efficiently summarised in the work presented by Babarit et al. (2011); Babarit et al. (2012). The studies show, based on a hybrid model, the power performances of the predominant WECs at different European locations on the Atlantic coast. In this context, the term “hybrid“, or weakly non-linear model (3.18), infers a combination of the linear Diffraction/Radiation (PaM) solution and simplified non-linear contributions, such as the viscous drag term from the Morison’s equation or non-linear hydrostatic models. The verification of the PaM solution is presented in (Sarpkaya, 2010; Hulme, 1982) for two geometries: a cylinder and a hemisphere respectively. The verification of the Morison’s equation is presented in (Morison et al., 1950) for a cylinder. The choice of a hybrid model is inherited from the analysis of predominant forces acting on the structure, which can be conducted using non-dimensional parameters. Using a non-dimensional analysis brings a reduction of the structural error on the numerical model Fig. , increasing the adequacy of the model for the specific case. For bluff bodies (Faltinsen, 1993) subject to oscillatory flows, one can refer to the Keulegan-Carpenter (KC) number in first approximation. The KC number describes the relative importance of viscous over inertial forces in oscillatory flows, and it is defined as: Fig. 3.2. Different wave excitation force regimes for a matrix of wave states, in agreement with Chakrabarti (1987). H: wave height, D: characteristic length of the object, λ: wave length. Each red dots represents a different wave period (T) and H duo. The numbering is rising in T first and later in H. Vmax is the maximum velocity, T is the period of oscillation and D is the characteristic length of the object. Three flow regime regions can be drafted: KC<2 -inviscid, irrotational flow: inertia dominated model (Radiation/Diffraction model) 10>KC>2 -transient region: drag and inertia model (Hybrid model) KC>10 -turbulent flow: drag dominated model (Morison eq.) PA-WECs are assumably nested in the intermediate set, while the A-WEC types are likely to be inertia dominated. For the case of the T-WECs type, the model depends on the size of the system. A comprehensive discussion over the topic can be found in (Faltinsen, 1993; Chakrabarti, 1987). The non-dimensional analysis of a single floater of the Wavestar WEC is summarised in Fig. . The plot shows the different excitation force regimes as presented in (Chakrabarti, 1987) in function of the KC=H/D number (deep water approximation) and the diffraction parameter πD/λ for different wave states (red dots). Here, H is the wave height and λ is the wave length. The concepts of wave excitation force will be discussed later on. As seen in Fig. the viscous drag loads will have a negligible weight in the excitation force exerted by the wave on the Wavestar WEC floater. This approach has been used as a base study for the work developed in both Appx. A.4 and Appx. A.6, respectively for the Wavestar WEC and a FOWT of the tension leg platform (TLP) type. Even though FOWTs are not specifically included in title of the thesis, their analysis has been part of the work due to their partial similarity with large size floating WECs with self reference structure, e.g. Weptos WEC, FO3 WEC.

The method solves the Laplace equation for an inviscid, incompressible, irrotational fluid, also called ideal fluid. The 3D problem is mapped into a surface problem (panel methods) and the velocity potential in the fluid domain is solved from the given boundary condition. The first level of analysis is coupled with a linearised kinematic and dynamic free-surface boundary condition that is valid for small waves (H/λ≪ 1), i.e. the limit of linear wave in which the wavy surface can be approximated by the water shape at rest, thus applying first order Taylor expansion. This assumption greatly simplifies the problem because the free surface is known before-hand and is not part of the problem to be solved. In the limit of small body motion and small waves, the loads acting on the structure can be approximated by a linear function of the surface elevation. Further, if steady state conditions are sought, the loads have the same frequency as the waves that are causing them. In this framework, the hydrodynamic problem can be separated in two problems, called the radiation and excitation problems. The excitation problem defines the loads exerted by the passing waves on the structure held at the equilibrium position. Due to the linearisation of the free surface boundary condition, the wetted surface is constant. The excitation problem is further divided in two contributions, called diffraction and Froude-Krylov. The diffraction problem is valid when D ≥ λ, thus the wave field near the body is affected by the stationary body, so that there is no flux on the body surface. The Froude-Krylov problem is valid when D« λ, thus the wave field is not affected by the presence of the body, and the velocity potential on the wetted surface equals the incident (wave) potential. When the first order approximation does not hold the modification of the Froude-Krylov contribution with the instantaneous wetted surface integration is the most straightforward alternative to the linearised problem (Guerinel et al., 2013). The excitation force vector is calculated as a summation of both diffraction and Froude-Krylov contributions. The excitation loads are proportional to the incident waves, for each of the DoFs of the body. A single rigid body can be described by six DoFs, these are commonly defined in the offshore sector as surge (translation in the x axis), sway (translation in the y axis), heave (translation in the z axis, vertical direction), roll (rotation about the x axis), pitch (rotation about the y axis) and yaw (rotation about the z axis). Therefore, a general force vector is composed by three forces and three moments. The relation between the excitation force and the incident wave is a function of the wave frequency, and it can be defined in the frequency domain as: is the complex excitation force coefficients vector, A(ω,kx)is the wave amplitude function of the wave frequency (ω), wave number (k) and spatial coordinate (x). The radiation problem defines the loads exerted by the body motion induced waves on the body itself in otherwise still water. In the absence of constraints, a single rigid body freely moves in six modes, which have been defined above. Each elements of the radiation force vector has a term proportional to the body acceleration (Added Mass) and a term proportional to the body velocity (Radiation Damping). The coefficients are function of the body’s motion frequency. The radiation force vector is defined in the frequency domain as: Here, CM(ω)and CA(ω)are the added mass and radiation damping matrix respectively, and are the body acceleration, body velocity and body displacement vectors defined in the frequency domain, and iis the complex operator. Both CM(ω)and CA(ω)are square matrixes of dimension nDoF × nDoF, being nDoF the number of DoF of the structure. Softwares like WAMIT (WAMIT, 2014), ANSYS AQWA (ANSYS, 2014) and Nemoh (open-source) (LHEEA-ECN, 2014) solve the diffraction/radiation problem by using the panel method in frequency domain. The results of the frequency domain representation is influenced by the quality of the surface discretisation. In order to reduce/quantify this source of error, a sensitivity study of the result variation in function of the number of discretisation points is needed. Fig. shows the results of the sensitivity analysis for a cylindrical floater with diameter of 10 m and draft of 10 m. The number of panel (Np) is varied from 56 to 2136, the latter being the reference solution.

The Morison equation (Morison et al., 1950) is a semi-empirical equation which encompasses both inertia and viscous loads. In the context of this work, the focus will stay on the viscous contribution only. The importance of this contribution for a WEC ought to be deducted from its working principle. Contrary to Oil & Gas and FOWT structures, which are designed to avoid resonance phenomena, WECs are especially tuned to have at least one eigenfrequency in the wave frequency range. A direct consequence of this condition is the non-negligible influence of viscous drag loads on the motion of the structure caused by the amplified body velocity. The resonance condition is not only obtained in the design phase, but also with application of advance control strategies as explained in Sec. 4.1. In such conditions, the assumption of irrotationality of the flow is broken and flow separation may occur, resulting in a reduced motion amplitude. The viscous drag contribution (fD) is implemented as a quadratic term in function of the relative velocity between body (v) and fluid particles (ς). fD can be defined in the time domain as: Here, ρ is the water density, Ap is the body surface projection in the plane perpendicular to the axis of application of the force and CD is the drag coefficients. Due to the quadratic term, fD cannot be defined in the frequency domain as such. The frequency domain force (FD) can be obtained from a linearisation of the above equation, as: Fig. 3.3. Mesh convergency study for a cylinder (diameter=10 m and draft=10 m). Np is the number of panels and nd is the smaller panel dimension divided by the cylinder diameter. Top left: Wave excitation force amplitude coefficients per unit of wave amplitude in function of the wave frequency. Top right: Evolution of the normalised error of the wave excitation problem in function of Np, with respect to the case Np = 2136. Bottom left: Radiation damping (CA) and added mass (CM) coefficients per unit of body velocity and acceleration respectively in function of the body motion frequency. Bottom right: Evolution of the normalised error of the radiation problem in function of Np, with respect to the case Np = 2136. where is the linearised viscous drag coefficients. Here, V(ω)and ς(ω)are the frequency domain pairs of the respective time domain variable presented in (3.4). A commonly adopted linearisation technique is based on the energy conservation approach as described in (Sarpkaya, 2010). Through the thesis, the frequency/time domain pairs of a quantity are represented with upper and lower case letters respectively. Appx. A.4, Appx. A.5 and Appx. A.6 address the influence of the drag contribution for the Wavestar WEC floater and for a FOWT. In agreement with the non-dimensional analysis shown in Fig. , the viscous drag contributions are negligible for a passive controlled WEC — not in resonance — while in case of an active/advanced control strategy the viscous drag term can have an impact in the overall force summation. For example, for the Wavestar the viscous drag contribution is as high as 20 % as shown in Appx. A.5. In the case of the FOWT, the viscous drag contribution is high for two reasons: a relatively small characteristic length of the body close to the water surface and a flat bottom with sharp edges. One of the critical points in the present viscous drag formulation is the definition of the point where the force is evaluated. As suggested by Faltinsen (1993), the right position needs to be empirically estimated, though 25 % of the wave height from the still water level in the downward direction could be a first guess. The other critical point is the definition of the CD coefficients in waves. As presented in Appx. A.4, an experimental approach can be used, but due to the small contribution of the viscous drag loads, the calculated coefficients could be affected by a high level of uncertainty. An alternative and promising approach presented by Bhinder et al. (2011) is an iterative fitting procedure based on CFD results which has been used to evaluate the CD coefficient for a cylinder in harmonic waves. In this way, the experimental noise issues are solved, but the definition of a proper turbulent model becomes a key parameter though. Another approach can be seen inAppx. A.6 where the structure has been decomposed in simple subelements, and the contribution from each of them included in the formulation of the full structure. The advantage is the ease in the definition of the drag coefficient for each sub elements as described in (Faltinsen, 1993).

Besides the hydrodynamic loads, the other important contribution to the wave structure interaction is given by the hydrostatic force. The hydrostatic force vector is the results of the combined action of loads — gravitational and buoyancy — acting on the body in still water. In the limit of the small body motion the hydrostatic force vector is proportional to the displacement of the body, and it can be defined in the frequency domain as: Khy is the linear hydrostatic stiffness matrix. When the small body approximation is not valid, the integration of the hydrostatic pres-sure over the instantaneous wetted surface could be used to account for the linearisation error. This non-linear contribution need to be coupled with the non-linear Froude-Krylov contribution, which is also calculated on the instantaneous wetted surface. Even though these non-linear contributions can be of high relevance for the numerical models of the WECs they have been considered out of the scope of the thesis. The hydrostatic stiffness matrix is obtained from the PaM softwares cited above.

The PTO system with embedded controller is of paramount importance for the dynamic behaviour of the WEC. Fig. shows the schematics of the two PTO systems mostly used with a WEC of the activated body type: hydraulic and direct drive. As input the PTO model receives the state (position, velocity, acceleration) of the interconnection points (vA and vB), and it outputs the force (FPTO) acting on the MS and RS plus the energy absorbed by the system. Based on the controllability of the PTO force, two main model sets can be identified for the systems shown in Fig. . For the case of good controllability, i.e. direct drive or hydraulic PTO with variable pressure (4-quadrant motor branch), the PTO loads will be governed by the chosen control law. Therefore, the loads commanded by the controller will be modified by the response of the PTO. This approximation is based on the following assumption. In order to ensure controllability of the PTO, the characteristic time scale of the low level controller -i.e. magnetic flux and pressure signals -is designed to be much smaller in magnitude than the time scale of the controlled load. Therefore the PTO response can be approximated with only a linear time invariant transfer function (G(s)). For a less controllable system, i.e. constant pressure hydraulic PTO, the description of the full model can be simplified with a piecewise linear function, such as a Coulomb damper. It is important to notice that the discontinuity of the Coulomb damper is often replaced by a linear transition region around the zero velocity point, in order to avoid stiffness and instability in the numerical model (Babarit et al., 2012). This equals the adoption of a check valve dynamic in the full description of the hydraulic PTO. As shown in Appx. A.7 the Coulomb model also fits well in a friction dominated PTO system, often applied in small-scale laboratory systems. The code-to-code verification of the approximated systems, for the case of direct drive PTO (controllable) and constant pressure hydraulic PTO (less controllable) is given in Fig. . In both cases, the benchmark is the full numerical model of the PTO system. On top of the former motivation leading to a model simplification, it is also necessary to point out the need of consistency in the wave-to-wire model. Indeed, the overall accuracy of the wave-to-wire model will be determined by the part of the system with lower accuracy, in this case the hydrodynamic model. Therefore, an over-complex PTO model will not increase the overall accuracy of the wave-to-wire model. In general, the force exerted by the PTO system can be described by one of the following simplified models: The transfer function (G(s)) and the Coulomb damper model can be represented as: Mpto, kpto, ωnpto, τpto and ζpto are respectively the characteristic mass, stiffness, natural frequency, time constant and critical damping ratio of the system, and DPTO is the constant PTO force, v is the relative boundaries velocity and vL is the limit velocity of the linear range. For a pure Coulomb damper, the latter variable becomes zero. So far, the domain identity of FPTO has not been explicitly specified, but in order to avoid inconsistency in the definition of the equation of motion, the proper domain needs to be tagged. A linear commanded load can be expressed in both the time (fPTO(t)) and the frequency (FPTO(ω)) domains by applying forward or inverse Fourier transform. On the other hand, for non-linear models the force exerted by the PTO can be defined in time domain only. The time domain formulation is also used when fPTO(t)is constrained. The application of constraint is often needed in order to bound the results of the linear model when excessively large motion or loads are obtained. For example, the application of an advance control strategy to tune the WEC into resonance, see Appx. A.5, will lead to a displacement response operator per unit of incident wave height tending to infinity, which infers infinite velocity and power too. These conditions are physically unrealistic and either a displacement or a velocity or a PTO load constraint is required. The constrained PTO force can be obtained by: where g(vA, vB, f(PTO))is a general non-linear function of the boundary state and PTO type.

The loads commanded by the controller has a major influence on the power performances and dynamic response of the WEC. It is a common technique to define a control law linear to the WEC velocity (V) and displacement (Ξ) in the first stage of analysis — proportional-integral (PI) control. Fig. 3.4. Code-to-code verification of PTO models. Left side: Direct drive permanent magnet generator (PMG) with torque control response (blue line) vs second order transfer function response (red line); responses for unitary step and sinusoidal excitation. Right side: Constant pressure hydraulic PTO response (black line) vs approximated Coulomb damper response (red line); velocity-force curve and responses for irregular wave excitation. Cc and Kc are the controller damping and stiffness coefficients. For Kc = 0 the controller reverts to a simple resistive or passive controller -proportional (P) control. The bandwidth of the controller is relatively narrow, and the energy extraction is only significant in the close neighbourhood of the resonance frequency of the oscillator (WEC). For Kc ≠ 0 the controller is called active PI controller. The compensation of the intrinsic stiffness of the WEC causes the system to be in resonance with the wave frequency over a broader range, inferring a larger absorbed power. Further details about advanced control laws are given in Ch. 4.

Besides the heavy economical weight of reaction systems in the overall cost of the WEC, their influence on the dynamic behaviour of the WEC is in many cases strongly correlated to the system type. For example, bottom fixed reaction systems, e.g. Wavestar support structure, are assumed to be infinitely rigid and they will not be considered in the numerical model. For mooring systems associable to class 1 (Ch. 2) in operational sea states, the numerical model can be reverted to a simple stiffness coefficient, obtained from the force/displacement curve of the complete reaction system set-up. The force displacement curve can be obtained either from quasi-static or dynamic numerical models or from experimental tests. For this type of system, the expected motion is relatively small in consequence of the large ratio between structure dimension and wave length in operational sea states. Therefore, both velocity and acceleration of the fairleads tend to be at zero, which means convergency between dynamic and quasi-static model of the reaction system. For mooring system associable to class 2, the numerical modelling needs to describe both conservative and non conservative loads. The term conservative is used as a synonym for no energy dissipation. In this case, the utilisation of linear stiffness coefficients as well as a quasi-static description will only partially describe the reaction system loads. In order to solve this problem, a dynamic model of the mooring cable needs to be used. In the framework of the SDWED project, the implementation of the dynamic reaction system models in the equation of motion of WECs has been pursued. The dynamic model of the reaction system (MOODY (Palm et al., 2013b)) developed by Chalmer University uses the finite element method with high-order polynomial basis function and discontinuous elements. The spatial discretisation is based on the local discontinuous Galerkin method (Fitzgerald and Bergdahl, 2008). Although the dynamical model of the reaction system needs to be applied to fully define the impact of the mooring system in the overall body motion, the actual computational cost of the integrated model is currently too high to simulate a full set of sea states. This high computational cost is partially related to the fact that the software is implemented in Matlab, thus a sharp drop of the computational cost is expected if the software will be implemented in other languages. For the time being, one of the available solutions found is the one proposed by Fitzgerald and Bergdahl (2008). However, this method has not been tested in the timeframe of the work described in this thesis. The general mooring force vector is expressed in the frequency domain as: where Km is the linearised mooring stiffness matrix.

For all the cases presented in this thesis except for the FOWT work, the waves are mod-elled using the linear potential theory. The model is once more based on the assumption of unviscid, irrotational, incompressible fluid, and small waves (H/λ). Two types of waves are used through the work: regular (monochromatic) and irregular (multichromatic). Further, only 2D long-crested waves are adopted. Even though most of the real sea waves are irregular, the utilisation of regular waves is often used as a first stage of analysis due to implicit simplification of the problem. For a pure harmonic wave excitation force, the WEC model simplifies to a mass-spring-damper system with constant coefficient. Moreover, the usage of regular wave quickly induces a steady state behaviour of the WEC response, which entails a reduction of the time needed to reach steadiness of parameters like absorbed power, maximum load, etc. Few wave periods are enough to quantify the problem. Of course regular waves do not reproduce the natural variability of the energy source and cannot be used to estimate the behaviour of the WEC in a real location. So, irregular sea states are used, which can be modelled by a simple superposition of independent regular waves under the assumption of linear theory. It is important to notice that in case of irregular sea states the simulation time needed to be long enough to reach steadiness of the sough parameters. As suggested in the Equimar project, after 500-1000 waves the steady state is reached (Ingram et al., 2011, pg. 197). The wave model is described in frequency domain by a standard spectrum (Sj(ω)), which defines the distribution of the wave power spectral density (PSD) across the frequency range. For a regular wave the spectrum reverts to a simple impulse, while for irregular sea states one of the commonly used model is the parametrised JONSWAP spectrum. Different types of parametrisation are possible and hereafter the one described in (Frigaard and Andersen, 2010) is presented: where f is the wave frequency, fp is the peak wave frequency defined as 1/Tp where Tp is the wave peak period, α is the spectral intensity and γ is the peak enhancement factor. The sea state is fully defined by three parameters only, Hm0, Tp and γ. Large values of γ(>3.3) identify a narrower distribution of the energy about the peak wave period, while when γ become one, the JONSWAP spectrum coincides with the Pierson-Moskowitz spectrum, which describes a fully-developed sea state. The generation of a time domain signal from the PSD has been realised using two different generation methods in this work: random phase and white noise filtering. The random phase is a deterministic method, which maps exactly the discretised target PSD function into a finite time series by the means of inverse Fourier transform (IFFT). The single harmonic components are combined with random phases. The white noise filtering is a probabilistic method, where the discretised target spectrum is firstly converted in a time domain filter, and then applied to a white noise time series (input stochastic process). The resultant stochastic process is coloured as defined in the target PSD distribution. The method is probabilistic because the PSD of the output matches the target PSD only in a probabilistic sense: only if the length of the time series go to infinity do the two PSDs coincide. The time domain filter is obtained with the same methodology described in Appx. A.1, once the target PSD is formulated accordingly. Different assumptions are used for the FOWT case where the analysis is focused on ULS conditions. According to (Veritas, 2013), the ULS are described by non-linear wave models, the non-linearity order being function of the water depth. For the tested conditions, the stream function wave theory developed by Dean and Dalrymple (1991) minimised the error with the measured surface elevation in experimental tests, and it has been used as input for the model thus. It is important to bear in mind that using a non-linear wave model involves a consistent growth of complexity in case of irregular sea-states, which often re-sults in a reduction of the order of the non-linear model toward a second-order Stoke wave.

The main loads acting on the WEC, and described so far, are assembled in the equation of motion of the WEC using the Newton-Euler formulation. The nomenclature used from now onwards is consistent with the one adopted in the Nemoh PaM software, being the source of the hydrodynamic model. For a single body WEC, the equation of motion (EoM) about its centre of gravity (CoG) is defined in the frequency domain as: is the total force vector acting on the body and Mis the mass matrix of the body -including both mass (diagonal) and inertia matrix. Using the forces defined above (3.14) becomes: Here, TPTO and TD are input matrixes used to map the PTO and linearised viscous drag forces into the body DoFs. The dependency of the variables to ω is not represented for compactness. The EoM can be described at any point, once the kinematic of the system is defined. Kinematic describes the motion of objects without considering the cause for the motion. In particular, the EoM around a generic point CO can be defined as: is the transformation matrix associated to the position vector from CoG to CO defined as: where I and 0 are identity and zero matrix of appropriate dimension and is the skew symmetric matrix associated with the cross product between the position vector and the angular acceleration. For a comprehensive discussion over the topic see (Fossen, 2011) When FD and FPTO become non-linear the frequency domain formulation of the EoM cannot be used. By taking the inverse Fourier transform, (3.15) is mapped into the time-domain, leading to a system of integro-differential equations; Cummins’ equation (Cummins, 1962): Here, j identifies each of the Ndofequations, being Ndof the number of DoFs of the body, CMj,∞ represents the limit of the added mass coefficient for is the impulse response function (IRF) of the radiation force, and are the body acceleration, body velocity and body displacement vectors defined in the time domain, and the lower case f is used for the time domain pair of the respective frequency domain force, i.e. fex is the wave excitation force in the time domain and Fex is the wave excitation force in the frequency domain. The IRF can be obtained using a frequency sampling method from the frequency response function (FRF) (Parks and Burrus, 1987). hRAD vanish for t < 0and it is called causal IRF (not anticipative system). The integral term in (3.18) is a convolution integral. fex can be obtained using either a time or a frequency domain method. The frequency domain method is similar to the random noise method (Sec. 3.1.4). The wave spectrum is multiplied by the wave excitation FRF, and the phase of the resultant FRF is summed to a random phase and mapped in the time domain by taking the inverse Fourier transform. The time domain method is based on the convolution of the wave excitation IRF (hex) with the time series of the surface elevation (η). hex is non-causal (anticipative system) and the outputs of the system depends on past, current and future inputs. The result of the convolution between hex and η is delayed in time by a factor NIRF/2* dtwhere NIRFis the order of the IRF and dtis the time distance from one sample to the other. From a simulation view point the non-causality of the wave excitation force is not of particular interest since η is know from the beginning to the end, but this issue matters greatly for the controllability of the WEC in real-time applications. Indeed, in order to evaluate fex at the current time a short-time prediction of the surface elevation is needed, causing a growth of the model uncertainty. The topic will be further discussed in Ch. 4. For an array of Nb bodies, the equation of motion does not change in shape, if the bodies are mechanically uncoupled, i.e. an array of PA-WECs, or connected to a common fixed reference frame, i.e. the Wavestar WEC (multi-body system with fixed reference frame). In this last case, the motion of each floater is mechanically decoupled from each other under the assumption of infinitely rigid reference frame. Whilst for a multi-body system with moving reference, the EoM needs to be adapted to the specific case. On the one hand, the formulation of the problem in the multi-body frame eases its definition, but on the other hands if the number of bodies is large, the dimension of the problem can result in a large demand in terms of computational power. A good alternative to solve the issue is the usage of generalised DoF. In this framework, the standard six DoFs of a rigid body are extended with extra DoFs, which describe alternative modes of motion of the structure. The Weptos WEC is an example of a multi-body WEC. The system described by Pecher et al. (2012b) is composed by 40 floaters, therefore the number of DoF of the whole system is 246. If a generalised approach is used instead, the system dimension reduces to 46. The definition of generalised DoF is a built-in functionality in WAMIT and Nemoh. While WAMIT outputs also the hydrostatic and mass matrix for the generalised problem, Nemoh does not. Since part of the work conducted in this thesis is based on Nemoh results, once the hydrodynamic problem is defined in terms of generalised DOFs, all the others contributions need to be defined accordingly. The formulation of the mass matrix for the system can become a non-trivial task if both translation and rotation of coordinate systems need to be endorsed in the Newton-Euler method. In order to reduce the risk of error in the mass matrix formulation, an energy based approach (Euler-Lagrange) is used in the following. The Euler-Lagrange formulation is commonly used in multi-body solvers from robotic application, and it seems to be a growing topic of research in the wave energy sector too, (Ó’Catháin et al., 2008; Mccomb et al., 2013). In general for a multi-body system with n generalised DoFs (qi with i ranging form 1 to n), the Euler-Lagrange equations can be written as: where is the external force vector acting on the i-th body and L is the Lagrangian, defined from the kinetic energy (Ti) and the potential energy (Pi) of each element of the system as: where M is the mass matrix of the system in terms of generalised DoF, obtained from the application of constraints in the multi-body system, see Appx. C for further details. and are the vectors of the body displacement and velocity in the generalised DOFs. In the first stage of analysis, the mass matrix can be linearised, therefore the EoM can be formulated using the Newton-Euler as: where ) and Mare the mass matrix and its linearised version respectively, Tris the summation matrix used to sum up the different contributions for each body acting on the same DoF and the vector of forces and moment is obtained from the force summation. is the vectors of the body acceleration in the generalised DOFs.

The objective of the simulations is the definition of time averaged parameters as the annual energy production (AEP) or the capture width (CW) and the estimation of loads time series of a given WEC in a given location. The AEP is the two dimensions’ sum of the matrix obtained by multiplying the scatter diagram (SD) and the power matrix of the specific WEC at the given location. The SD defines the wave climate at the specific location using a 2D histogram where the probability of occurrence (pr) of each sea states is given in function of Hm0 and Tp. The base wave model is the JONSWAP, and the parameter (γ) is assumed fixed for each location, although it is a function of the . Two different SDs have been used in the simulations, both related to Danish locations and summarised in Tab. and Tab. . For the Danish section of the North Sea, the γ parameter is set to 3.3. The power matrix is defined from the mean power associated at each sea state of the SD. The mean power is estimated using the wave-to-wire model of the given WEC. The procedure to calculate the AEP is exemplified in Fig. . The system of ordinary differential equations (ODE) was solved in Matlab and Simulink using the built-in ode-solvers (Murphy, 2011; MATLAB, 2013). Due to the simplification used in the sub-models, the system of ODE is often non-stiff, and to a large extent an explicit Runge-Kutta scheme of order 2 or 4 guarantees a truncation error small enough for the accuracy of the model. A system of ODEs is defined non-stiff when every eigenfrequency of the system lays in a similar frequency range. The truncation error of a solver is defined as the difference between the true solution and the approximated one. The truncation error is defined from the order of the solver. For the Runge-Kutta scheme of order 4, the local truncation error is O(h5)while the global truncation error is O(h4). Local refers to the instantaneous error and global to the accumulated error. his the time discretisation.

A physical model reproduces a real life process on a different scale. The main benefit in physical modelling is the incorporation of all contributions, and if a proper physical set-up is used, a physical model gives trustable results. But the main deficiencies of physical modelling are the relatively large cost and amount of time required compared with some numerical tools, and the presence of unavoidable errors, see Fig. . Physical models should be seen mostly as a design and model validation tool, more than a design tool itself, especially in light of the large growth of computational power in recent years. Fig. 3.5. Assessment of the AEP from the location SD and WEC power matrix. Tab. 3.1. Relative occurrence of different wave states (pr) from six years, buoy measurements located at 6332100N, 474700E, water depth: 17 m ([Hm0] = m, [TP] = s). Both parameters defines the mean value over an interval. The adopted discretisation is 1 s in Tp and 0.5 m in Hm0.

Hm0/TP	0.5	1.5	2.5	3.5	4.5	5.5	6.5	7.5
0.25	-	-	-	0.04	0.04	0.02	0.01	-
0.75	-	-	-	0.07	0.17	0.11	0.05	0.01
1.25	-	-	-	-	0.06	0.11	0.05	0.01
1.75	-	-	-	-	-	0.06	0.05	0.02
2.25	-	-	-	-	-	0.01	0.05	0.02
2.75	-	-	-	-	-	-	0.01	0.02
3.25	-	-	-	-	-	-	-	0.01

Tab. 3.2. Operational sea states parameters for the Danish North Sea

Wave State #	1	2	3	4	5
Hm0	[m]	1.1	2.0	3.0	4.0	5.0
Tp	[s]	6.1	7.4	8.7	9.8	10.7
pr	[-]	0.448	0.225	0.108	0.051	0.024

As discussed in the chapter preamble, different sources of uncertainty are ascribable to the experimental results, mainly grouped in structural and measurement. The structural errors are mainly linked to the scaling law used to define the physical set-up. Given that it would be optimal to maintain the mutual interactions between dominating loads unaltered, it is not possible to simultaneously scale gravitational, inertial and viscous forces, unless modification of the so-called constant parameter is endorsed, i.e. gravitational acceleration, medium viscosity, etc. In marine applications, the commonly used scaling laws are Froude and Reynolds. Generally, the physical analysis of steady state parameters of large bodies is carried out using Froude scaling law (bulk properties), while both fast transient phenomena localised at the boundary of the volume and small body dynamics are ascribable to the Reynolds scaling law. As shown in Fig. , the scaling law should be decided based on the results of non-dimensional analysis of the flow regimes. For the specific case of the Wavestar presented in the figure, the complete dynamic similarity using Froude scaling law will not be achieved, because viscous drag loads will not be scaled correctly. However, as presented in Appx. A.4 the relative weight of viscous loads is less than 5 % for a passive controller set-up, thus the viscous scaling error is considered of small impact in the overall summation. The quality of the results is directly related to the scaling factor: the larger it is, the smaller the scale induced errors are. A commonly adopted scale frame for experimental wave energy converter models is between 1:50 and 1:10, being first a proof of concepts and second a design and feasibility study. For extreme cases, e.g. FOWT in ULS, a scale ratio as low as 1:100 is still considered in the correct range, (Ingram et al., 2011). Larger scales are considered to be beyond the scope of this work. Another important source of structural error is related to the implementation of a PTO system in the physical model. This is most probably the critical component in the physical model definition. PTO systems generally do not scale with Froude scaling law, besides there are manufacture limitations lead by electrical component efficiency, material strength, which retain the scaling of full-size PTOs. The most important consideration to bear in mind when reproducing a scaled PTO system is to keep the reaction force at the interface between the PTO system and moving body similar to the target system. An example of PTO model scalability can be found in Appx. A.7. Although it could be possible to scale a linear electrical actuator or a hydraulic PTO to match the required load range in the experiments, the weight of the system would restrain its applicability. The choice has been directed towards a linear air piston, whose force/velocity response resembles to a great extent an approximated Coulomb damper model, Fig. . The variation of the constant loads, used for example to optimise the mean absorbed power for the given scatter diagram, was achieved by modification of the PTO lever arm. This makes the pair with the modification of the constant pressure level for a hydraulic PTO. It should be noted that although the air piston is quite small, its weight is already at the edge of the weight reserved for the PTO system. Besides structural errors, measurement errors need to be identified, quantified and mitigated if possible. The instrumentation errors are often considered of less importance (≤ 1%) once the instrument range and measured range are consistently chosen. The main environmental error for a wave energy application is the index of wave reflection characteristic of the wave basin. This source of uncertainties can be quantified with a 3D analysis of the wave field, with and without the structure in water. A standard index of reflection for mild slope beach is below 10 % in terms of wave amplitude. The operation errors are often subjective and more difficult to estimate. One source of operational error is the modification of the system response induced by a measurement system. Of particular interest is the case of the slack moored floating WEC (fWEC). The assessment of the position and orientation in the 3D space is fundamental to quantify power performance as well as the response operator of the structure. The measurement system should not interfere with the system dynamics, meaning no additional weight or loads should be ascribable to its presence. An interesting approach to solve the problem is shown in Appx. A.8, where a video based methodology inherited from the augmented reality sector has been presented. Fig. summarises the comparison of experimental results between the presented methodology and other two measuring devices: a potentiometer based system and an inertial measurement unit (IMU) system. The potentiometer based system is the reference signal (red line), but its utilisation in a fWEC is limited due to the presence of a spring to tightly wind up the connecting thread. IMU (green line) is accurate and precise with respect to the structure orientation, but since they measure accelerations they suffer of error accumulation in the double integration which cause the signal to drift. In order to avoid this drift, the raw signal needs to be filtered with a high pass filter ending in a poor accuracy of the system in case of slow motion of the fWEC. The video based (blue line) system shows a good agreement in all the tested frequency ranges. It is important to highlight the main limitation of the video based system: the presence of a singularity in the tracking algorithm that induces discontinuities in the results. Fig. 3.6. Scale PTO force/velocity characterisation compared with an ideal approximated Coulomb damper model. On the right side the physical PTO model is shown in the final configuration. Fig. 3.7. Measured displacement time series. Colour map: red - reference signal, blue - optical signal and green - accelerometer integrated signal. For a general description of the laboratory testing procedures and issues, see (Ingram et al., 2011).

Due to the strong assumptions adopted, e.g. ideal fluid, in the numerical model definition, the validation step is a crucial point. In this section, the comparison between experiment and numerical data is briefly summarised, the detailed description is presented in Appxs. A.2, A.4, A.7, A.3, A.6. The comparison between experiments and numerical results traces the structure of Sec. 3.1.

For the Wavestar single floater described in Appx. B, the wave body interaction is decomposed in three terms: hydrostatic, radiation and wave excitation force. All three sets of test are run in a condition where the other forces are zero or known: the hydrostatic tests are run in the limit of zero velocity and acceleration without waves, the radiation tests are run from the knowledge of the hydrostatic force and without waves, the wave excitation tests are run with standstill floater and waves acting on it. In order to simplify the analysis, only one harmonic per test is used, and in order to check the model linearity the response to different input amplitudes is analysed. The frequency domain coefficients are obtained by minimisation of the error between models and measured loads in the least square sense. As presented in Appx. A.2, the agreement between numerical and experimental results is sometimes counterintuitive at first glance. Especially for the small amplitude waves and motions, where the higher accuracy of the reading theory is expected, the measurement uncertainties become as important as the measured value itself leading to erroneous results. The data presented in Appx. A.2 for the wave excitation force was biased by an offset in the calibration function. The correct frequency domain response is re-presented in Appx. A.4.

Approximated models similar to the ones presented in Fig. can be used and compared against experimental test data. The results for the linear electrical motor used in the Wavestar set-up are presented in Fig. . Two simplified transfer function models are used with order one (blue dots) and two (red dots). As highlighted in the frequency domain plot (right side in Fig. ), the second-order model represents the main dynamics in the frequency range of interest, the latter being 0.5− 2[Hz]. Moreover, the first-order model does not fit the “truth” in the high frequency range, underestimating the response of the actuator.

As introduced above, the simplified description of the mooring force (3.12) of mooring system of class 2 is somehow underestimating the energy dissipation, which occurs in the mooring cable. As presented in Fig. , for a long period of oscillation (left side) the quasi-static, measured and dynamic mooring tension responses have similar magnitude and a small phase shift. But for shorter waves, the quasi-static approximation fails to describe the energy dissipation process, highlighted in the phase shift with respect to the measurement data and dynamical solution too. Fig. 3.8. Validation of PTO models. Left side: Time domain comparison between measured (black), firstorder transfer function (blue) and second-order transfer function (red) responses to a step in the commanded force, normalised with the end force. On the bottom side the discrepancy time series between models and measurement is plotter. Right side: Power spectral density of the measured and modelled data. Fig. 3.9. Validation of mooring models. Time domain comparison between measured (red), quasi-static model (black) and dynamic model (blue) responses to a sinusoidal motion of the fairlead. Left side: Period of oscillation 1.9. Right side: Period of oscillation 1.3. The periods are normalised by the natural period of the floater in heave.

The comparison between the EoM (3.15,3.18) and the measured data is the last step in the model validation. The Wavestar single floater is used once more as example due to its intrinsic simplicity. In addition for this case study it is possible to estimate the variation of the model error in function of control strategy. This point is particularly important because by applying an advanced control strategy the linear potential theory is likely to be violated in consequence of the amplitude amplification induced by the resonance condition. The estimation of the uncertainties in function of the control strategy is fundamental to balance the results of the control optimisation presented in Ch. 4. Fig. shows the time series of measured (green) and simulated (blue) force acting on the PTO connection points for the two control strategy presented in (3.11). The normalised mean square error for the P control case (upper plot) is 92 % while for the PI control case (lower plot) is 81 % as expected. Fig. 3.10. Time series comparison between simulated and measured force acting on the Wavestar WEC single floater for two control configurations: P (upper plot) and PI (lower plot). Colour map: measured data (green line) and simulated data (blue line).

As stated in the introduction of the thesis, Sec. 1.1.1, the current CoE of WECs is, in general, still significantly higher than other more mature renewable energy technologies, thus the sector has not yet reached a competitive economical level to roll off into the market. The WECs tested and studied so far have a relative small turnover compared to the high capital cost, which to a large extent is driven by the structural loads in extreme conditions. The turnover is defined as the income associated with the energy production. On the one hand, the simplest approach to increase the economic viability of WECs is to increase their turnover by means of improved control strategies. The first part of the chapter deals with the brief introduction of the different control approaches used to maximise the energy absorbed by the WEC and related issues. On the other hand, whether this method defines a CoE minimiser for WECs is not obvious. An advance controller infers larger turnover but also larger design loads, the latter being related to the structural cost of the WEC. In Sec. 4.2 the focus will stay put on assembling a simple methodology toward a more balanced optimisation of WECs from an economic prospective. The results are extrapolated from simulation of the numerical model, whose validation was presented in Ch. 3.

A controller identifies a set of processes which monitor, and alter if needed, the state of a dynamic system by manipulating its inputs. In WAB WECs, one of the primary objectives of the controller is to modify the absorbed power — either mechanical or electrical — by manipulating the force exerted by the PTO on the HS. A high absorbed power level entails correct and higher velocity, high machinery force and correct phase (timing). The concept of maximum energy absorption has been independently realised by Evans (1981) and Falnes (2002), back in the 1970s. Named complex conjugate control (CCC) and phase and amplitude control (PAC) by the authors, they provide an upper bound for the WAB WEC capability. In recent years, though, more and more studies have been presented on the topic, mostly dealing with approximation of the afore mentioned “best“ solution. Besides the control law proposed in (3.11), other alternatives are possible, commonly classified in two groups: passive and active. In this context, the term “passive“ infers a purely resistive control strategy characterised by an unidirectional energy flux, i.e. the linear passive damper control (P) presented in (3.11) for Kc = 0. The term “active“ refers to those control strategies where in order to enlarge the absorbed power some energy is fed back into the HS during a part of the wave cycle using a bidirectional energy flux. In the “passive“ class, besides the P controller, the latching and declutching strategies are worth mentioning. Latching and declutching controllers were first studied by Budal et al. (1982), and later applied to complex system. Several authors focused on the implementation of latching and declutching controller for a floating WEC, in both physical and numerical models, i.e. (Babarit and Clement, 2006; Durand et al., 2007; Lopes et al., 2009; Ringwood and Butler, 2004; Babarit et al., 2009). However, their implementation remain marginal due to the embedded limitations, such as non-causality of the control scheme, performance losses in multi DoFs systems and abrupt loads variation; as recently pointed out by Cretel et al. (2011b). On the contrary, the P controller is often identified as a lower power performance bound for WECs due to the simple applicability. An example of its implementation on physical and numerical models can be found in (Hansen et al., 2012; Marquis et al., 2010). The “active“ group includes a broad range of controllers, appealing for the substantial theoretical increment of the power performance if compared with the P controller. Two main controller sub-classes exist: non-causal and causal. The former relies on the prediction of either the body velocity or the excitation loads to define the maximum absorption trajectory, while the latter approximates the non-causal model with a causal one in order to remove the uncertainties related with the prediction, Perdigão and Sarmento (1989). Examples of non-causal controllers are both CCC and PAC, while by far the most known causalised controller are the feed-back ones (Hansen et al., 2012; Nielsen et al., 2013). The PI controller presented in (3.11) identifies one possible configuration. It should be noted that every causalised controller is intrinsically sub-optimal in multi-frequency sea-states. Recently, the optimal control theory within the framework of model predictive control (MPC) has been introduced, addressing the problem of constraint control. The most relevant examples of MPC implementation are given in (Hals et al., 2011b; Richter et al., 2013; Brekken, 2011; Cretel et al., 2011a; Abraham and Kerrigan, 2013). MPC defines a stand-alone class, which spans in both active and passive controller sets, in function of the defined constrained problem (Cretel et al., 2011a). Further, the MPC includes both causal and non-causal controllers, given that the model disturbance — the wave induced load — can be either defined from causal and non-causal excitation force model. For a comprehensive description of viable control solutions see, (Price, 2009; Hals et al., 2011a).

The following discussion is valid for a single degree of freedom WAB WEC, but similar conclusions can be sketched for multi-DoF systems whenever the presence of cross-coupling terms is included in the formulation. As described in Falnes, 2002, Ch.6, for a single DoF WEC of the WAB type the maximum energy absorption happens when the incident wave frequency matches the body natural frequency, i.e. resonance condition. CCC and PAC define two alternative closed form solutions for the maximum energy absorption control trajectory. On the one hand, as analysed in (Korde, 2000; Hals et al., 2011a) the complex conjugate approach is anti-causal and its implementation requires forecast of the body velocity. Since the system is highly damped the auto-correlation length of the body velocity is relatively short and its prediction, based on past observation, becomes uncertain (Nielsen, 2012); since the prediction of the velocity is not reliable, the controller is not practically feasible. On the other hand, the phase and amplitude approach relies on the prediction of the surface elevation. For a relative narrow-band wave spectrum the prediction of the surface elevation is considered more reliable than the body velocity as shown in (Fischer and Kracht, 2012; Fusco and Ringwood, 2010b) and Appx. A.1. The PAC is further discussed and its block diagram exemplified in Fig. . The general scheme used for the controller is a velocity tracking, whose reference signal is the optimal body velocity at the actual instant of time t = k(v[k]). The reference signal is calculated either from (4.1) or (4.2), based on the information of the excitation force, which is in turn calculated from the surface elevation (η). Equations (4.1) and (4.2) are time/frequency domain Fourier transform (FFT) pairs, where capital letters are used for the frequency domain variables. Hopt in the transfer function from the excitation force to the reference velocity, Hex is the transfer function from the surface elevation to the excitation force and Hηv is the transfer function from the surface elevation to the reference velocity. If η(t)is known a priori, the velocity can be obtained from (4.2) by forward and inverse FFT application. The real time implementation of the controller is somewhat non-trivial. The whole process can be divided in three steps. Short-term prediction of the surface elevation in order to complete the non-causal part of Hex; Evaluation of the optimal velocity at the actual instance of time from the actual and past value of fex[k,k − 1, …]; Minimisation of the error between reference and true signals. The first step in the controller definition is addressed in Appx. A.1. The wave excitation force is predicted in the short-time using a deterministic method, which is implemented by finite impulse response (FIR) filters. The FIR filter technique has been selected for its simple implementation and phase linearity, but other approaches are possible, i.e. infinite impulse response filter and neural network (Schoen et al., 2008; Schoen et al., 2011; Fischer and Kracht, 2012; Fusco and Ringwood, 2010a; Fusco and Ringwood, 2008). The FIR filter is implemented by using the frequency sampling design methods (Parks and Burrus, 1987). The reference velocity is obtained from the predicted excitation force by using a state observer, commonly in the form of Hilbert transform or Kalman filters (Fagley, 2012; Fusco and Ringwood, 2010a). The observer defines the frequency and amplitude of the excitation force. The former output is used to assess the actual transfer function Hopt, which is then multiplied by the excitation force amplitude to give the reference velocity. As an alternative, the first two steps can be joined using a FIR mapping from the surface elevation — measured some distance in the up-wave direction — directly into the optimal velocity. The filter is designed as per the excitation force prediction, but the fitted transfer function is now rather than Hex. Fig. shows the comparison between measured and estimated wave excitation force and optimal velocity trajectory, together with the position/velocity diagram. Fig. 4.1. General layout of the time domain controller implemented with the PAC scheme. The error between reference and actual velocity (e[k]) is the input for velocity controller. Different formulations of the latter have been tested, i.e. resonant controller (Zmood and Holmes, 2003), internal model controller (Francis and Wonham, 1976), lead-lag compensator (Bakshi, 2009), but the best tracking quality (phase and amplitude) and disturbance rejection was obtained with a simple PI controller. The signal presented in Fig. (right hand side) for the Wavestar WEC single floater already highlights one of the major limitation of the method. The large amplitude of the commanded optimal trajectory can easily become un-realistic. For the presented case in the optimal trajectory the floater is required to be completely submerged (lower condition) and out of the water (upper condition) for a large part of the time series, being the upper and lower conditions sketched in dotted black lines. This large non-linearity already happens for relative small waves, i.e. in Fig. the used sea state has Hm0=0.02m and Tp=1.2s, model scale 1:20 and floater diameter 0.25 m. The sketched boundaries are a function of the floater displacement as well as the available emerged volume due to the non-symmetry in the horizontal plane. The other important limitation of the PAC is the inability to handle constraints. Both position and velocity constraints can be handled by defining feasible and not-feasible set for the optimal solution (Hals et al., 2011a; Fusco and Ringwood, 2013), although the controller will present abrupt variation of velocity and position which could undermine its robustness. On the contrary, the application of PTO constraint is not easily implementable. The control law required the utilisation of large amount of reactive power, which often overcomes the PTO limits, and as shown in Appx. A.5 the saturation of the PTO load invalidate the performance of the controller to a great extend. Implementation of the PAC has been attempted in a physical model of the Wavestar WEC, for further information see Appx. B. The study has seen physical tests issues like wave directionality, presence of an interaction between the body motion and the measured surface elevation up-wave, performance degradation induced by noise and PTO delay in the velocity tracker and brought the physical implementation of the controller to an end. Although each of these issues could ideally be solved (Fusco and Ringwood, 2010a; Frigaard and Brorsen, 1995), the growing implementation complexity is not balanced by the growth of power performance numerically estimated if compared with a simple PI controller in realistic (constraint) cases, see Appx. A.5. Fig. 4.2. Optimal control prediction stage. From top to bottom and from left to right: Time series comparison between measured (green) and estimated (blue) wave excitation force. Time series of the calculated (black) and estimated (magenta) optimal velocity trajectory. Optimal feasible (blue) and unfeasible (red) position and velocity trajectories. The set of feasible position is delimited by black dotted lines.

Practical alternatives to the PAC are feed-back and MPC. Generally, latching and de-clutching schemes present theoretical and practical implementing limitations which have reduced their importance. Among the others, the one important limitation is related to the effect of the abrupt control loads variation commanded by the controllers. The application of non-smooth PTO loads will induce vibration on the structural and mechanical compo-nents, which can revert in larger need of O&M procedures. The effect cannot be grasped in a fully rigid numerical model of WEC, though a hydro-elastic model should be used in the analysis. The implementation of latching and declutching is not further considered, due to the excessive model complexity. The full feed-back controller law (Nielsen et al., 2013) can be formulated in time domain as: where fu is the commanded loads, Cc, Kc and Mc are the damping, stiffness and mass control coefficients, t is the actual instant of time and hRAD is the IRF of the frequency radiation function. The last term of the right hand side of (4.3) defines the causalised compensation of the radiated energy. The controller can handle both velocity/position and force constraints. Velocity and position constraints can be implemented by adding a load that penalises the constraint violation. The PTO constraints can be solved using an anti-windup procedure, which resets the error accumulation in the integral term of the controller (Åström and Hägglund, 2005). The results from the simulations presented in Appx. A.5, and reported in Fig. , summarise the theoretical difference between the three of the of the five different configurations derived from (4.3) listed below: where the acronyms P, I, D and c stand for proportional, integral, derivative and convolution compensation respectively. As presented in (Hansen et al., 2012) the bandwidth of the PI, the PD and the PID controllers does not differ significantly, therefore only the PI controller was used in the analysis. All of them can tune the natural frequency of the system to the one of the incoming wave. On the contrary, the P controller cannot match the incoming wave frequency, therefore the performance will diminish as the wave frequency gets away from the natural frequency of the oscillator. A direct aftereffect is a larger step of efficiency between P and PI/PD controllers, while the efficiency variation between the other cases is rather flat. It is important to bear in mind that besides the growth of the power performance, the utilisation of advance control strategy will also induce a growth of the model uncertainties, as introduced in Fig. for the P and PI controllers. This behaviour further reduces the true difference between the different controllers, especially in the active controller cases. Kc, Mc and Cc coefficients are obtained by matching the impedance of the specific WEC; the last coefficient can be obtained from a stochastic analysis of the input process (Nielsen et al., 2013) also. The capability of tuning the controller coefficients is a great advantage for its implementation in physical models. In fact, even if the calculated coefficients from the solution of the linear PaM problem are accurate, they will be different from the real system ones for the summation of structural model errors in both numerical and physical models, Fig. . This fact together with an efficiency similar to the one of PI scheme bounds the applicability of the PID and PIDc controller in physical models. The larger number of parameters and induced complexity is not compensated by the increased power performance. The PI and PD have similar characteristics but the latter is affected negatively by the higher noise level of the acceleration measurement than the position measurement. MPC defines together with the PAC the upper limit for the power capability of the WEC in unconstrained cases. Indeed, the two controllers revert to the same “optimal” solution when the PTO is ideal. On the contrary, when the PTO presents limitations, the constraints (position, velocity, PTO loads and PTO loads variation) are directly included in the cost function formulation (J) for the MPC, leading to a theoretical “optimal” constraint solution. The term “optimal” means maximisation of the mean absorbed power. The cost function J is obtained from the discretisation of the average absorbed power over a time period T. where denotes the current time instant, fu is the control load, Np is the number of time steps of the prediction horizon, Q is the weight matrix, Δu is the increment of the control variable and Δν is the increment vector of the wave excitation force. The matrices P, τu and τν map the state vector as well as the control and the disturbance increment sequences into the output space. The product urepresents the Hessian matrix (He). The problem is convex with unique solution when . The overall optimisation problem with the constraint on the control force (fmax) can be written as: The working principle of the MPC is shown in Fig. . At each time step (j) the optimal trajectory (u[j...j+Np - 1]) which maximises J — the absorbed power — is calculated for the given feasible sets. Although the optimal control trajectory is defined in the whole prediction horizon, applying the receding horizon principle (Kwon et al., 2005), only the first sample (u[k]) of the sequence is used, allowing the controller to react to any unpredicted future disturbance. Fig. 4.3. MPC with receding horizon control principle. The blue dots represent the first sample of the optimal control trajectory used at each time sample. Based on (Li et al., 2012). The performance of the MPC leans to the ones of the phase and amplitude controller when Np approaches infinity. But practically, Np is a trade-off between computational cost and accuracy. The computational time of the quadratic problem is cubic in Np. A prediction horizon equal to the period of one full wave is a good first guess (Soltani and Sichani, 2013; Cretel et al., 2011b). The performance of the MPC is also affected by the uncertainty of the system disturbance prediction (wave excitation force) over the control horizon. The results presented in Appx. A.5 do not account for this source of error, so they define an upper limit for the MPC formulation. For the Wavestar WEC single floater applied to the Hanstholm scatter diagram in presence of active constraints, and in case of a perfect prediction of the excitation force, the MPC is capable to extract roughly 25 % more energy per year if compared with the active feed-back scheme. It is important to bear in mind that due to the absence of the disturbance prediction error, the results from the MPC should be regarded only as a upper limit of the controller capability. If the prediction error is included, the gap between the MPC results and the feed-back control results is reduced, but the quantification of this effect is not publicly available yet. Practical implementations of MPC are not available in literature. Among the others, two of the major issues are the computational cost of the method when constraints are active and the reduced capacity of the controller to be tuned to the true physical environment. Fig. summarises the application of the above mentioned control strategies for the Wavestar WEC single floater for the Hanstholm scatter diagram Tab. . Fig. 4.4. Comparison of the AEP for the Wavestar WEC for the Hanstholm SD with different control scheme and cases. Cases: 1 - Unconstrained linear model; 2 - Unconstrained weakly non-linear model; 3 - Constrained weakly non-linear model.

As presented so far, minimising the CoE by means of control can generally be defined as a 1D optimisation problem. Indeed only the AEP was considered. On the one hand, this methodology will reduce the CoE by varying the turnover of the WEC, but on the other hand, the larger loads compared to a P control exerted by the PTO on the structure will induce a different load scenario on the structural elements. Following, different load scenarios will (probably) modify the design specification leading to different CAPEX and OPEX values for the structure. Since the structural-related costs cover a large share of the overall lifetime cost of WECs, then the CoE will be affected too. Due to the fact that the control law will be suppressed — or modified — in extreme sea states, it is possible to foresee only a modification of the cyclical loads acting on the structure, and accordingly its fatigue design. Standards for oil and gas (see e.g. (ISO, 2007)) as well as offshore wind turbines (IEC, 2005; Veritas, 2013; GmbH, 2005) and (offshore) steel structures (DNV, 2010; CEN, 2005) recommend the use of SN curves for fatigue analyses of structural designs. The SN curve characterises the material performance concerning fatigue and shows the relation between the number of load cycles at a given stress amplitude leading to fatigue failure. Although fatigue response and control law are interdependent, the implementation of a fully coupled system is restrained by the complexity of the methodology used to quantify the fatigue behaviour. Nevertheless, at the first stage of analysis one could focus on the implementation of a sequential approach instead. The sequential methodology presented in Appx. A.5 combines power performances and fatigue analysis results of a WEC into a cost factor (CF), which eases the selection of an economically best control law. Fig. sketches the flow diagram of the proposed method, going from the WEC and location definitions, down to the identification of the minimum CF. The definition of the AEP assessment is given in Fig. while literature and definitions related to the fatigue model are given in Appx. A.5. Indeed, the fatigue analysis is not part of the work conducted in this thesis, but results of a collaborative task, and specifications on the fatigue model are not given hereafter. Fig. 4.5. Flow diagram of the combined control and fatigue analysis methodology. Model inputs: WEC specification and SD. Model output: minimal cost function. The derivation of the cost factor model is described in the following. Excluding the dependency on the discount rate, the CoE for a specific WEC is calculated as the ratio between total investment cost (Ctot) and turnover (E) through the project lifetime (Davey et al., 2009). Ctot is defined as the summation of capital expenditures (CAPEX) and operational expenditures (OPEX) over the design lifetime, while Ecan be considered in first approximation proportional to the AEP of the device, by the constant parameter ke(rate of discount). Because both investment cost and turnover are affected by the control strategy adopted, a dependency on the i-th control strategy (Sec. 4.1) is introduced using the term (i). Moreover Ctot can be divided in two contributions: one proportional to the control strategy (C1(i)) and another independent of it (C2). Here, p represents the percentage of the total investment costs affected by the control strategy. For example, in the case of the Wavestar WEC single floater, C1(i) includes the CAPEX of the arm structure as well as the CAPEX of the PTO system mainly, but also any expected variation of the cost of O&M cost ascribable to the control law. Similarly, C2 includes the CAPEX of the full structure, PTO and arm excluded, as well as commissioning/decommissioning costs, electricity connection and O&M costs not related to the control strategy. Assuming a linear relation between cost and structural dimension (cross-sectional area) and defining a certain controller, i.e. the P controller, as a base of comparison, the association between the CoE and the above mentioned CF can be defined as: A(i) and Aref are respectively the cross-sectional area of the critical structural detail for the i-th and the reference control strategies, determined in the fatigue model, and Ctot,ref is the total investment cost over the device lifetime for the reference control strategy. The linear relation between cost and structural dimension (cross-sectional area) assumes a fixed length of the detail in the analysis. Under this assumption the cost is directly proportional with the cross-sectional area of the structural detail. A different control strategy entails a modification of the AEP, but also a variation of the design cross-sectional area of the selected structural detail, reflected in the CF. The model relies on the definition of the parameter p, which is WEC specific and in general rather uncertain. In order to study the influence of p on the overall results a sensitivity analysis needs to be carried out. In Appx. A.5, the whole algorithm is applied to the Wavestar WEC single floater case, subject to the Hanstholm SD, with the P controller as reference case. The main result is reported in Fig. where the CF is graphed in function of the adopted control strategy and in function of the p parameter. It should be noted that for the case p = 0 the optimisation problem reverts to a simple control optimisation problem. The p parameter is varied within the range 0− 20%, chosen in agreement with the wind turbine sector due to the absence of specific data. In the offshore wind sector the capital expenditure varies between 50-70 % of the overall lifecycle costs of an offshore wind turbine (Berkhout et al., 2012). The costs of the turbine itself make about one third of it (Blanco, 2009). That is, the value of p is very likely to lie well below 20 %. For the specific test case, two main conclusion can be drawn. First, the derivative of the output variation in the parameter space is small, so the propagation of the uncertainty on the p parameter in the overall model is rather limited. Second, the MPC is the CF minimiser, but due to the assumption of perfect knowledge of the future in the actual implementation, the comparison results are positively biased. As cited before, the prediction error, which is the base of the MPC, is not implemented in the present formulation and the quantification of its effect is still unclear. Based on the actual stage of development and if the judgment is balanced by implementation simplicity, tunability, etc., the cost-effective solution is the PI controller. In any case, it seems clear that the implementation of an active control strategy reduces the CF by a factor two in the best scenario if compared with a standard passive controller technique. Fig. 4.6. Comparison of CF for different p values and different control strategies. Colour map: p=0% (blue), p=10% (green), p=20% (magenta). Even though the main assumption adopted — linear relation between WECs cost and structural dimensions — is rather stringent, the CF still gives a valuable tool to gain insight into the economic potential of a WEC in function of the applied control strategy.

The risk of a future energy crisis and environmental deterioration is real if political and economical plans are not considered by now. In order to keep the growth of average world temperature below 2 °C (Birol et al., 2013, [450 scenario]) a sharp and utter change of the overall pollution rate is needed and the change needs to happen in this period of time. Wave energy is a large — mostly untapped — energy resource. It can contribute to the future energy mix if a number of initially underestimated issues are solved within a short period of time. As introduced in Ch. 1, the most important limitation of the sector to roll-off into the market is the actual or estimated CoE, which is affected by technical and non-technical matters. To further complicate the problem, the large number of patented devices hinders the concentration of the development effort over the few promising concepts, if any. In agreement with the objective of the SDWED project, which partially funded this work, the definition of a general applicable and reliable numerical model of WECs (wave-to-wire model) is paramount for the sector. The model should be used to compare different WECs technologies and output a reduced range of candidates to be further developed and/or optimised. Time average parameters are used as the base of comparison, the most important being the AEP of WECs at specific locations. The assessment of the AEP relies on relatively long time simulations that together with the large number of WECs and suitable locations shed light on the need for a computationally fast wave-to-wire model. Due to the large number of WEC types only WAB WECs are considered in the analysis. Numerical wave-to-wire modelling of WECs is not a novel topic: in literature hundreds of example can be found for several WEC types. But two main problems of the proposed solution are found and addressed in this work. On the one hand, the numerical model needs to be compared with experimental data in order to validate the reading theory and the embedded approximations. The comparison between numerical and physical models is presented in Ch. 3. Most of the examples are related to the Wavestar WEC, due to the concept’s simplicity, which eases the analysis. On the other hand, the main limitation of the analysis based on numerical simulation presented so far is the obstinate maximisation of the energy produced by the WEC, which only addresses part of the CoE minimisation problem. The classical energy maximisation approach and the proposed CoE minimisation approach are both presented in Ch. 4. The base of the CoE minimisation approach is the following: the maximisation of the energy absorbed by the WEC is obtained via modification of the control logic, which entails a modification of the loads exerted by the PTO on the structure. Therefore, the control strategy will have a (quantifiable) influence in both capital and operational costs related to the structural and lately in the CoE of the WEC. The overall approach can be classified in three sequential steps. 1 -In Ch. 2, a general WAB WEC system breakdown is presented, highlighting the predominant and representative components. The breakdown tool is the first step of analysis where the definition of what is considered relevant defines features that will be, or will not be, included in the model in a macro scale. As concluded in Ch. 2, three main subcomponents are considered fundamental for the wave-to-wire model: hydrodynamic subsystem, power take off subsystem with annex control logic and reaction subsystem. For each of those, a set of issues based of the following study is listed. Their importance lies ether in the active contribution on the main dynamic of the WEC or in the economical share or most likely on both. 2 -In Ch. 3, the available numerical and physical modelling techniques are presented and skimmed on the basis of the set of problems highlighted in the breakdown procedure. This part represents the stage two of the analysis: the numerical model is further refined based on the required objective, and all subcomponents and overall system need to be compared with experimental data sets. The need of a fast wave-to-wire model excludes linear and non-linear models based on the volume discretisation, such as CFD and SPH methods. Waiting for the computational power of computers to drastically increase, they can be used to assess fast transient phenomenas more than time average parameters. Solution of the Diffraction/Radiation problems based on panel methods and Morison’s equation are two viable solutions. The free software Nemoh is a promising solution (free of charge and solution in line with the other competitors) but the computational time appears to be a bottleneck. For the case of the Weptos WEC Appx. C a convergency study has not been run for time issues. But the Nemoh code is expected to be further developed at the end of the code competition proposed by NREL (Open-WARP). Due to the high velocity expected for the converter, if compared with Oil & Gas or FOWT structures, a hybrid or weakly non-linear model is the correct solution based on the non-dimensional analysis. The comparison between numerical hydrodynamic model and experimental data can be summarised in two points: first, the uncertainties of the experimental data need to be quantified in order not to bias the comparison, and second the larger the motion of the system the large the numerical error is, due to uncontemplated contributions, i.e. slamming, overtopping, etc. PTO systems and their control logic differentiate a WEC from a traditional floating structure. The two common types of PTO deployed in WAB WECs are direct drive and hydraulic. Although their full numerical models are still affordable from a computational cost view point, they can be efficiently substituted by a simplified model reducing the computational time by a factor 2-10. This change is mainly ascribable to the removal of the higher harmonics, which stiffen the system of ODE. The model can be found by an interpolation of the characteristic PTO curves, similar to a principal component analysis of the numerical system. The comparison with experimental data shows a good general agreement for different cases. The reaction systems are fundamental for the position compliance of floating WAB WECs. Although their implementation in the wave-to-wire model is based on a simple stiffness matrix, results shows that, in operational conditions, a large dissipation of energy happens around the cable. The general solution would require the application of dynamic solvers based on multi-body or FE methods, but their computational cost is still far too high. The implementation of a linear stiffness-damping reaction system model based on experimental or dynamic solvers data appears the cost-effect solution, but its implementation is reserved for future analysis. One of the most important points achieved in this work is presented in the effect of the control strategy in the overall error of the wave-to-wire model. The application of an advanced control strategy does have an influence in the model prediction uncertainty, which needs to be used carefully when different control strategies are compared to each other. 3 -In Ch. 4, the two different optimisation methodologies are presented: the first is a 1-D optimisation based on the modification of the control strategy, and the second is a 2-D optimisation based on the control strategy and its effect on the structural design. In the context of this thesis, “optimisation” is considered a synonym for CoE minimisation. The application of advanced control strategies is the simplest way to modify the eco-nomical viability of a WEC. Several standard techniques are presented in literature and five of them have been compared. The energy absorbed, response in a constraint scenario, tunability and implementation simplicity are the criteria used to select among the different control logics. The PI controller seems to be superior to all the others as a trade off between selection criterion. Although MPC shows better results, the judgment is weighted by its ideal implementation: the results are a function of the prediction of the future forces, and perfect knowledge of the future has been assumed in the implementation. Since the load exerted by the PTO on the structure is a function of the control strategy, the latter is likely to induce a modification of the fatigue behaviour of the structure. The quantification of these effects is the based of the 2-D optimisation of the WEC. Although the best solution should be a fully coupled numerical model, the complexity of the fatigue analysis restrained its implementation. Therefore, a sequential approach has been used. The CF is introduced to analyse the effect of the control strategy on the COE with respect to a reference scenario, in this case P control. The results for the Wavestar WEC are in line with the 1-D optimisation. The PI controller can reduce the CF by a factor 1.5-2 in realistic constraint scenarios, which is second only to the MPC reduction. Same as before, the MPC cannot really be considered in the selection due to its idealised implementation.

The definition of a general and reliable wave-to-wire model and the implementation of a 2-D optimisation methodology are, among others, two important steps for the quantification of the true potential of the wave energy sector. In continuation of the work presented in this thesis, the most straightforward future step is the validation of the wave-to-wire model for different test cases, and possibly the extension of the model for the other two types of WEC. Although the first set of interim results for the Weptos WEC are promising the number of approximations used does not allow to draw any major conclusion on the model validity. In order to obtain a general wave-to-wire model, the formulation of the equation of motion needs to be automatised and the Lagrangian approach seems to be one of the possible candidates. Once the extended validation of the wave-to-wire model is achieved, the procedure presented in the 2-D CoE minimisation needs to be applied too. The 2-D CoE analysis presented in this thesis is only a “toward” steps. Its importance lies in shedding light on the true problem of the CoE minimisation more than the brutal energy maximisation. But its implementation is only in an embryonic form. First of all, the result of the model should be validated by FE structural analysis, and secondly the fatigue model should be implemented in the control algorithm ending in a fully coupled system. The natural framework of this global optimisation problem is the MPC scheme. In this frame of research, the appraisal of the unbiased capability of the MPC including the uncertainty of the disturbance prediction is a relevant topic and it is expected to be analysed in the upcoming future. In addition the implementation of the MPC in physical models and the quantification of the uncertainties of the numerical model when MPC is considered are two other topics for future development of the MPC framework. Personally I think that in order to facilitate the development of a generally applicable and reliable wave-to-wire model, we need to create a free flux of information between research centres in terms of numerical knowledge and experimental data sets.

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ABSTRACT: Short-term wave forecasting plays a crucial role for the control of a wave energy converter (WEC), in order to increase the energy harvest from the waves, as well as to increase its life time. In the paper it is shown how the surface elevation of the waves and the force acting on the WEC can be predicted using FIR filter. The predictors have been validated in laboratory with unidirectional regular and irregular waves. Here a single point absorber, (1:20) scale of the Wavestar device, is used. The results show that it is possible to predict wave and forces acting on the device using a properly designed FIR filter KEY WORDS: FIR FILTER; NUMERICAL MODEL; ACAUSAL FUNCTION; WEC. Ferri F.^a, Kramer M.M.^a and Pecher A.F.^{a a} Department of Civil Engineering, Aalborg University, Aalborg, Denmark. In Proceeding of the 23rd International Offshore and Polar Engineering Conference (ISOPE), Anchorage, Alaska, USA, 2013. ABSTRACT: Within the wave energy field, numerical simulation has recently acquired a worldwide consent as being a useful tool, besides physical model testing. The main goal of this work is the validation of a numerical model by experimental results. The numerical model is based on a linear wave-body interaction theory, applied for a point absorber wave energy converter. The results show that the ratio floater size/wave amplitude is a key parameter for the validity of the applied theory. KEY WORDS: WAVE ENERGY CONVERTERS; PHYSICAL MODEL TESTING; NUMERICAL MODEL VALIDATION; HYDRODYNAMIC COEFFICIENTS. Wehmeyer C.^a, Ferri F.^b, Skourup J.^c and Frigaard P.B.^{b a} Rambøll Offshore Wind, Esbjerg, Denmark; ^b Department of Civil Engineering, Aalborg University, Aalborg, Denmark; ^c Rambøll Port and Geostructures, Esbjerg, Denmark. In Proceeding of the 23rd International Offshore and Polar Engineering Conference (ISOPE), Anchorage, Alaska, USA, 2013. ABSTRACT: An extensive model test program has been carried out in order to assess the behavior of a tension leg moored substructure as support of a Floating Offshore Wind Turbine (FOWT). The floater was inspired by an industrial design. The tests focused on the ultimate limit state (ULS) behavior, therefore no aerodynamic or gyroscopic effects were included, i.e. the turbine hub was represented by a lumped mass, and focus given to wave forces and dynamic behavior. The model tests have been conducted in the 3D deep water basin of the Hydraulics and Coastal Engineering Laboratory at the University of Aalborg at a scale of 1:80. The model tests were made with a range of monochromatic, bichromatic and irregular waves. All waves are modeled long crested and were run with and without sub and super harmonics. Three different structure layouts were tested, i.e. the tests were run with substructure only, with a rigid tower representation and with a flexible tower representation. Three submerged load cells measured the response of the tendons, and two accelerometers measured the response the total structure, being located at the substructure – tower interface and in the nacelle. The paper describes the setup of the test and a first set of interim results. KEY WORDS: FLOATING OFFSHORE WIND TURBINE; PHYSICAL MODEL TEST; TENSION LEG PLATFORM. Zurkinden A.S.^a, Ferri F.^a, Beatty S.^b, Kofoed J.P.^a and Kramer M.M.^{a a} Department of Civil Engineering, Aalborg University, Aalborg, Denmark; ^b Department of Mechanical Engineering, University of Victoria, Victoria, Canada. In Ocean Engineering, vol. 78, pp. 11-21, 2014. ABSTRACT: A time domain model is applied to a three-dimensional point absorber wave energy converter. The dynamical properties of a semi-submerged hemisphere oscillating around a pivot point where the vertical height of this point is above the mean water level are investigated. The numerical model includes the calculation of the non-linear hydrostatic restoring moment by a cubic polynomial function fit to laboratory test results. Moreover, moments due to viscous drag are evaluated on the oscillating hemisphere considering the horizontal and vertical drag force components. The influence on the motions of this non-linear effect is investigated by a simplified formulation proportional to the quadratic velocity. Results from experiments are shown in order to validate the numerical calculations. All the experimental results are in good agreement with the linear potential theory as long as the waves are sufficiently mild i.e. H/λ ≤ 0.02. For steep waves, H/λ ≥ 0.02 however, the relative velocities between the body and the waves increase thus requiring inclusion of the non-linear hydrostatic restoring moment to effectively predict the dynamics of the wave energy converter. For operation of the device with a passively damping power take-off the moment due to viscous drag is found to be negligible. KEY WORDS: WAVE ENERGY CONVERTER; POINT ABSORBER; NON-LINEAR HYDROSTATIC MOMENT; NON-LINEAR DRAG MOMENT; LINEAR POTENTIAL THEORY. Ferri F.^a, Ambühl S.^a, Fischer B.^b and Kofoed J.P.^{a a} Department of Civil Engineering, Aalborg University, Aalborg, Denmark; ^b Division Control Engineering and Energy Storages, Fraunhofer Institute for Wind Energy and Energy System Technology, Kassel, Germany. In Energies, vol. 7, pp. 2246-2273, 2014. ABSTRACT: In order to reduce the cost of electricity produced by wave energy converters (WECs), the benefit of selling electricity as well as the investment costs of the structure has to be considered. This paper presents a methodology for assessing the control strategy for a WEC with respect to both energy output and structural fatigue loads. Different active and passive control strategies are implemented (proportional (P) controller, proportional-integral (PI) controller, proportional-integral-derivative with memory compensation (PID) controller, model predictive control (MPC) and maximum energy controller (MEC)), and load time-series resulting from numerical simulations are used to design structural parts based on fatigue analysis using rain-flow counting, Stress-Number (SN) curves and Miner’s rule. The objective of the methodology is to obtain a cost-effective WEC with a more comprehensive analysis of a WEC based on a combination of well known control strategies and standardised fatigue methods. The presented method is then applied to a particular case study, the Wavestar WEC, for a specific location in the North Sea. Results, which are based on numerical simulations, show the importance of balancing the gained power against structural fatigue. Based on a simple cost model, the PI controller is shown as a viable solution. KEY WORDS: PASSIVE CONTROL; ACTIVE CONTROL; MODEL PREDICTIVE CONTROL (MPC); FATIGUE ANALYSIS; WAVE ENERGY CONVERTER (WEC); WAVESTAR. Wehmeyer C.^a, Ferri F.^b, Andersen M.T.^b and Pedersen R.R.^{a a} Rambøll Offshore Wind, Esbjerg, Denmark; ^b Department of Civil Engineering, Aalborg University, Aalborg, Denmark. In Submitted to Energies. Date of submission May 2014. ABSTRACT: The rising demand for renewable energy solutions is forcing the established industries to expand and continue evolving. For the wind energy sector the vast resources in deep sea locations have encouraged the research towards the installation of turbines in deeper waters. One of the most promising technologies able to solve this challenge is the floating wind turbine foundation. For the ultimate limit state where higher order wave loads have a significant influence, a design tool that couples non-linear excitations with structural dynamics is required. To properly describe the behaviour of such a structure a numerical model is proposed and validated by physical test results. The model is applied to a case study of a tension leg platform with a flexible topside mimicking the tower and a lumped mass mimicking the rotor-nacelle assembly. The model is additionally compared to current commercial software, where the need for the coupled higher order dynamics proposed in this paper becomes evident. KEY WORDS: FLOATING WIND TURBINE; TLP; NON-LINEAR WAVE; PHYSICAL MODEL TEST; ULTIMATE LIMIT STATE WAVE LOAD. Angelelli E.^a, Zanuttigh B.^a,FerriF.^b and Kofoed J.P.^{b a} DICAM, University of Bologna, Bologna, Italy; ^b Department of Civil Engineering, Aalborg University, Aalborg, Denmark. In Proceeding of the 10th European Wave and Tidal Energy Conference (EWTEC), Aalborg, Denmark, 2013. ABSTRACT: The paper presents the preliminary results of new physical tests carried out in the directional wave basin of Aalborg University (DK). The devices under exams are two floating 7 Degrees of Freedom Wave Activated Bodies moored with a spread system composed by 4 steel chains. The devices were subject to ordinary North Sea wave climate conditions and deployed in 1:60 scale. The main purpose of this paper is to analyse the performance of a Wave Energy Converter considering the interdependencies among energy production, loads on real moorings and device movements. The mooring effects on power production and on device movements are specifically investigated by varying the chain pre-tension level. Results suggest that the power production optimization is achieved with a slack mooring system providing a quasi-static response to the ordinary wave attacks. KEY WORDS: FLOATING WAVE ENERGY CONVERTERS; WAVE ACTIVATED BODY; EXPERIMENTS; SPREAD MOORING; CHAIN PRE-TENSION; POWER PRODUCTION. Ferri F.^a, Andreoni G..^b, Persic N.^a, Levelle J.^a and Kofoed J.P.^{a a} Department of Civil Engineering, Aalborg University, Aalborg, Denmark; ^b Laboratory of Automation and Robotics of DEIS, University of Bologna, Bologna, Italy. In Proceeding of the 10th European Wave and Tidal Energy Conference (EWTEC), Aalborg, Denmark, 2013. ABSTRACT: In this paper, an optical non-contact low budget method for tracking the position of multi-object is presented for marine/off-shore laboratory applications. Particular focus is given at the wave energy field and the analysis of floating wave energy converters dynamics. The measurement of the position and orientation is often a key point in this context, achieved nowadays with different standard technologies. Typically a base requirement for a measurement system is to be accurate and precise, without affecting the dynamical and statical behaviour of the system itself. Video based systems satisfy these points, but they are typically prohibitively expensive. Through the article a low budget one-camera video based system is presented and compared with other two standard methods: direct physical measurement and inertial motion unit (IMU). The system was tested both, in a dry environment and in waves. Results show an average error below 4and precision in the low frequency components but also increased high frequency noise components. Further investigation will be addressed to the implementation of a hybrid system, where the corrupted high frequency information can be replaced by the IMU signal. KEY WORDS: IMAGE PROCESSING; PHYSICAL MODEL; FLOATING OBJECT; AUG-MENTED REALITY; OPEN SOURCE. The objective of this appendix is to give details for the physical model of the Wavestar WEC single floater developed at Aalborg University, Civil Engineering Department by Morten Kramer. The Wavestar WEC (Wavestar, 2014) is a multi point absorber that extracts the wave energy potential from the motion of floaters induced by the passing waves. The floaters are attached to a fixed platform by arms. The hinge between the floater arm and the platform restrains the motion of the floater except for the rotation around the hinge axis, resulting in one rotational DoF for each floater. The platform is held stationary by four bottom fixed piles. Fig. B.1 shows the large-scale prototype installed near Hanstholm (DK). The machine is composed by four bottom fixed piles and two floaters, each of them 5 m in diameter, and represents a section of the prospected complete machine with 20 floaters. The length of the floater arms is ∼10 m. Fig. B.1 (A) - Large-scale prototype installed in 2009 near Hanstholm (DK). The machine fed electricity into the grid until it was moved to the harbour for reconfiguration in September 2013. (B) - Schematic representation of a single floater of the Wavestar WEC. θ represent the rotational DoF and A identifies the pivoting point of the floater. The power is extracted by means of an hydraulic PTO system, represented on the figure by its actuator only. Fig. B.1 (B) shows a schematic representation of a single floater of the Wavestar WEC. The pivoting point is represented by the point A. The rotational degree of freedom and its relative positive direction are represented with the greek letter θ. Fig. B.2 Small-scale (1:20) physical model of the Wavestar WEC single floater. Points A, B and C correspond to the ones sketched in Fig. B.1. Fig. B.2 shows the small-scale physical model of the Wavestar WEC single floater. The selected scale is 1:20 with a resulting floater diameter of 0.25 m. The floater is equipped with a laser, to measure the motor stroke and therefore the floater rotation, with a load cell, to measure the load acting on the PTO arm, and with a linear servo electrical motor acting like and active PTO system. The connection point C is moved on the rear of point A with respect to the large scale prototype. The floater is made hollow accounting for the usage of ballast material. The electrical motor is fully controlled by a xPC connection. The connection is built in the Matlab/Simulink environment, see Fig. B.3. Briefly, the requested operations, such as the control law, the data calibration, etc., are defined in a Simulink block diagram in the host computer. The information is compiled in c-code and sent to the target computer where a real time operating system (xPC) applies the custom operations. The xPC communicates with a I/O board where the data from the load cell and laser is collected. The raw data is treated and used to defined the set-point, either the position of, or the force acting on the floater. The set point is sent to the motor controller (low level controller) which tracks the set-point using a feed-back scheme (PID controller). Fig. B.3 Communication flow diagram between host computer, target computer and WEC. An example of the capability of the physical model is given in Fig. B.4. The figure shows the comparison between the numerical calculated (full line) and the measured (dots) mean absorbed power for two different sea states, IRB1 and IRB2, in function of the damping coefficient (Cc). The sea states are defined as: IRB1: Hm0 = 0.051 m and Tp = 1 s IRB2: Hm0 = 0.08 m and Tp = 1.25 s. The JONSWAP wave spectrum with γ = 1 and the random noise technique were used for the wave generation. Fig. B.4 Absorbed power in function of the damping coefficients (Cc) for two different sea states. IRB1: Hm0 = 0.051 m and Tp = 1 s. IRB1: Hm0 = 0.08 m and Tp = 1.25 s. In both cases the wave steepness is near 3.5 %. A JONSWAP spectrum with γ = 1 is used for the wave generation. The sample time of each test is five minutes. The adopted controller is a simple passive scheme proportional to the velocity of the floater. Fig. B.5 shows the comparison between the measured and requested linear relation between floater angular velocity and exerted PTO moment. The signal variance around the red line is ascribable to both measurement noise, inner control latency, and friction in the hinge (bearing). Fig. B.5 Angular velocity versus PTO moment, measured (dots) and requested (full line) The objective of this appendix is to describe the implementation of a wave-to-wire model for the Weptos WEC, in extension of the methodology given in Ch. 3. The Weptos WEC (Weptos, 2014) is a multi-body system that extracts the wave energy potential from the motion of floaters induced by the passing waves. The floaters are connected to a main floating platform. The platform is composed by two legs arranged in an A-shaped configuration with variable opening angle. The A-apex defines the bow of the system. The floaters are evenly distributed along two axes, one for each sides of the platform (port and starboard). These axes are parallel to the axes defined by the platform legs. Each floater (rotor) is hinged to the platform, and the connection leaves only one rotational DoF between platform and rotor. The rotor shape is based on the well-known Salter’s duck (Salter, 1974), where the tip of the duck is pointing outwards. Fig. C.1 Old (top) and newest (bottom) configurations of the Weptos WEC. The A-shaped platform is the grey (metal) coloured body and the rotors are the orange and yellow coloured bodies. Images’ source: (Weptos, 2014) Fig. C.1 shows two different configurations: the top one is the old configuration, which was tested in the CCOB facility in Santander, Spain (Pecher et al., 2012b), while the bottom one is the newest configuration under development. They only differ in the design of the platform, and in the number of rotors for each axis, e.g 20 vs 10. The device is moored and the mooring system allows the orientation of the WEC in 360°.

The proposed numerical model of the Weptos WEC is based on the old configuration, due to the presence of experimental data for the validation procedure, but the methodology is generally applicable to multi-body WECs. Due to the complexity of the system, the first stage of analysis uses a fully linear numerical model based on the solution of the Diffraction/Radiation problems, i.e. gyroscopic, viscous drag and non-linear hydrostatic contributions are considered negligible. The last two terms are expected to have an important share in the force summation for each rotor, and their implementation is part of the future works list. Due to the presence of a floating platform, the motion of each element of the WEC affects the motion of the rest of the system, and for this reason the system needs to be modelled as a multi-body system. The constrained motion between platform and rotor can be modelled either explicitly or implicitly. The former solution makes use of additional springs or algebraic equations to bound the motion of the constrained DoFs: the system is firstly formulated as a loose cluster of bodies and the constraints are applied afterwards. The utilisation of additional springs with high stiffness coefficient should be discouraged, because the resulting system of equations is stiff and slow to solve thus. In contrast, the implicit method re-maps the total number of DoFs of the loose system into the so called generalised DoFs, discarding all the unavailable DoFs. For the case of the Weptos with 40 rotors (old configuration) the explicit approach leads to a system with 246 (6*40+6) DoFs, while the implicit approach leads to 46 (40+6) generalised DoFs. It is important to bear in mind that, the position of a rigid body in the 3D space is completely determined by 6 DoFs. Due to the important reduction of the system’s order, the implicit approach is considered hereafter. It is important to highlight that similar methods have been proposed in recent years (Taghipour, 2008; Ruehl et al., 2014; Ó’Catháin et al., 2008), and the overall methodology is not new as such, but its application in wave energy conversion systems is an important topic of research. Fig. C.2 shows the meshed geometry relative to the Weptos WEC and the coordinate systems used. The black arrows define the inertial coordinate system, while the local coordinate systems are represented by the x-axis (red arrow) and y-axis (green arrow). The vertical axis is pointing upwards and it is not represented for graphical reasons. The main differences with the model shown in Fig. C.1 are: Simplified platform, the legs are made of two cylinders, and the transversal beam is absent. The space between the rotors is set to zero, in order to remove the error of the linear solution for those small gaps. The platform can only move in three DoFs, x-direction, z-direction and rotation around y-axis. This approximation is possible because only 2D long crested waves travelling in the direction parallel to the x-axis of the WEC are used. Fig. C.2 Meshed Weptos WEC. The black arrows define the inertial coordinate system, while the local coordinate systems are represented by the x-axis (red arrow) and y-axis (green arrow). The EoM of the multi-body WEC with generalised DoFs does not differ from the EoM of a single rigid body WEC (3.15,3.18). Both WAMIT and Nemoh softwares can calculated the linearised solution of the Diffraction/Radiation problems for a system with generalised DoFs. In contrast, ANSYS Aqwa does not use generalised DoFs and the EoM is solved with the explicit formulation of the constraint. The software Nemoh has been used for the purpose. Three main steps are needed to formulated the EoM of the Weptos WEC, given the geometry description and the Nemoh solutions. Define the mass matrix in the generalised base; Map the Nemoh output into the generalised base; Map the hydrostatic stiffness matrixes into the generalised base.

The mass matrix is formulated using an energy based approach as partially described in (3.19, 3.20). Briefly, the Lagrangian (L) is defined as the summation of the kinetic (Ti) and potential (Pi) energy for each body of the system. Ti is defined as: Here, mi and Ii are the mass and the inertia matrix, and and are the linear velocity vector and the angular velocity vector of the CoG. All the variable are related to the i-th body. It is possible to introduce the the geometric Jacobian matrix (J), which relates the generalised velocity vector to the velocity — linear and angular — of the point of interest (pi), The Jacobian matrix is decomposed into a linear velocity term (Jv) and a angular velocity term (Jw). Substituting (C.2) into (C.1) brings to: The summation of the i-th contributions over the number of bodies brings to: can be linearised by assuming small rotations — cos(θ)=1 and sin(θ)=θ—, and deleting all the second or higher order terms. The Jacobian matrix can be formulated using the constraint (joint) information. Ji is further decomposed as: where is a column vector of dimension 6x1 describing the connectivity between the bodies and j identifies the j-th generalised degree of freedom. is defined as: Here, is the vector defining the generalised DoF in the inertial coordinate system, i.e. for the vertical displacement , and is the position vector between the CoG of the i-th body and pi, defined in the inertial coordinate system.

The need of a matrix to map from the Nemoh output to the generalised base is linked to the way how Nemoh generates the output file. The mapping matrix is merely a summation matrix. The hydrodynamic coefficients of the platform are defined at its own CoG. Each of the rotors will have the same DoF of the platform, in this case surge, heave and pitch, plus the additional rotation about its own axis. The summation matrix needs to sum the rotors surge, heave and pitch to the ones of the platform. For the case study the summation matrix Tr is shaped as: The matrix is exemplified for a simplified system with only two rotors (Ndof=5) for graphical reasons, where Ndof is the number of generalised DoFs. The red and orange boxes represent the rotors rotational DoFs, while the first three rows of the matrix define the summation of each body in the platform DoFs.

The hydrostatic stiffness matrixes are evaluated for the platform and one rotor at their CoG in the local coordinate system. The stiffness matrix is a 6x6 matrix defined for the canonical 6 DoFs of a rigid body. The mapping rule from the generalised DoFs to the generalised hydrostatic force vector is defined as: Here, is the matrix ([Ndof x (6*Nb)]) to map from the generalised DoFs to the canonical 6 DoFs for each independent body, expressed in the body local coordinate system, and is a matrix ([(6*Nb)x(6*Nb)]) obtained from the concatenation of the hydrostatic stiffness matrix for each body, e.g where the rotor matrix is repeated Nb-1 times. Nb is the number of bodies of the WEC, in this case 41, 40 rotors plus 1 platform. is obtained from the linearised transformation matrixes between the different point of the structure, expressed in the inertial coordinate system. The hydrostatic stiffness matrix for the platform (Kp) and for the rotor (Kr) are calculated in Nemoh.

In order to define the Weptos numerical model, the information about mooring and PTO models are needed. First, the mooring system is modelled as a linear spring, acting on the surge DoF only. Therefore, the stiffness matrix is a zeros matrix with the entry (1,1) equal to the stiffness coefficient. The latter is obtained from the interpolation of the force/displacement plot as presented in (Pecher et al., 2012b). The PTO model is based on an ideal actuator model, i.e. unitary transfer function, and on a P-control law, resistive controller where the PTO force is proportional to the body velocity. The damping coefficient (Cc) of the controller was selected based on an energy maximisation principle, using the simplex method, (Lagarias et al., 1998). For the system specification see (Pecher et al., 2012b). The results reported below are given for the Hanstholm SD. The SD is scaled using the Froude scaling law with a scale ration 1:20. Fig. C.3 shows the capture with ratio (CWr) in function of Tp. CWr is defined as the ratio between the mean absorbed power at the WEC and the wave power per unit of wave front multiplied by the absorption length of the WEC. The blue line represents the solution of the system presented in Fig. C.2, and the red line is the same curve but scaled by a factor 0.24/0.29. The latter represents the ratio between the real rotor width and the approximated rotor width used to remove the gap between rotors. Fig. C.3 Capture width ratio in function of Tp for the Hanstholm SD at the scale 1:20. The blue line represents the solution of the model depicted in Fig. C.2, and the red line represents the same results with the application of a correction factor to account for the gap between rotors. The results presented in Fig. C.3 are in line with the results published in (Pecher et al., 2012b). The trend of CWr in function of Tp has a similar shape, even though the numerical model overestimate the maximum CWr of about 40 %. This error comes from a summation of different approximations, the most important being listed hereafter: PTO model: The PTO model of the physical WEC is based on the overrunning clutch mechanism, which entails energy extraction in one direction only (upstroke) and torque transfer only if the rotor velocity is higher then the PTO axle velocity. The numerical model of the PTO considers the mono-directional power absorption by dividing the absorbed power by a factor two, but does not consider any other non-linearity or frictions. The torque is constantly transferred from rotor to ideal generator. Hydrodynamic model: The hydrodynamic model does not include any viscous drag dissipation, neither quadratic nor linearised, but the KC number of the rotors goes from 5 to >10. Therefore, the viscous drag contribution should have a significant impact in the force summation Due to the large number of DoFs it has not been possible to run a mesh convergency study.