HYDRODYNAMIC BEHAVIOUR OF FLOATING WIND TURBINES Difference between linear versus non-linear wave effects on wave induced fatigue loads in a spar-type floating wind turbine

(1)

HYDRODYNAMIC BEHAVIOUR OF FLOATING WIND

TURBINES

Difference between linear versus non-linear wave effects on wave induced

fatigue loads in a spar-type floating wind turbine

Jorrit Roy Bergsma

November 2015

(2)

(3)

Hydrodynamic behaviour of floating wind turbines

Difference between linear versus non-linear wave effects on wave induced fatigue loads in a

spar-type floating wind turbine

Master of Science Thesis

Author: Jorrit Roy Bergsma

Committee: Prof. dr. ir. R.H.M. Huijsmans

TU Delft Chairman

Dr.ir. A. Romeijn

TU Delft

Dr.ir. S.A. Miedema

TU Delft

Dr.ir. P.L.C. van der Valk

Siemens

Ir. M. Tilanus

Siemens

(4)

(5)

RELEVANCE OF FLOATING WIND TURBINE

v

S

UMMARY

Huge developments in bottom founded turbines have been made since the start of offshore wind energy, mainly in Europe. In many countries outside of Europe, steep coasts permit no bottom-fixed foundations. The solution for this problem is floating wind turbines (FWT). Up to now, several concepts of FWTs have been designed and a couple of prototypes have been installed. Research on the different concepts of FWTs has advanced over the past ten years, and several aero-elastic tools have been developed. Non-linear wave effects have often been ignored in modelling the hydrodynamic behaviour of FWTs, under the assumption that they are significantly smaller than linear wave effects. In addition, only a few studies have been conducted on fatigue loads in floating wind turbines.

Therefore, the objective of this thesis is to assess the non-linear hydrodynamic effects on wave induced fatigue loads in a spar-type FWT. Non-linear hydrodynamic effects, or second-order wave effects, cause extra loads on the turbine at the sum and difference frequencies of the incident waves. The sum and difference frequencies cover a frequency range that can overlap with structural eigenfrequencies of the FWT, which could lead to resonant excitations that cause significant fatigue damage to the structure. Therefore, it is important to investigate how second order wave loads affect the fatigue response of a FWT. The analysis is focused on the fatigue loads at the tower bottom, since its design is fatigue driven. The main contributor for the fatigue at the tower bottom is the oscillations of the bending moment. The overall goal of this thesis is to answer the question whether and when second-order wave effects should be included in computer simulations of FWTs.

In Matlab, a hydro-elastic model of a spar-type FWT is developed based on the Hywind Demo concept. The model is a multibody description that captures the first structural eigenmode. The first eigenmode is of particular interest, as it overlaps with the wave’s non-linear sum-frequencies. The floater is exposed to first- and second-order wave loads, which are determined using the Morison equation that takes the motion of the floater into consideration. The mooring lines are included as linear springs and account for the horizontal stability. The model simulates the hydrodynamic excitation forces and dynamic response in the time-domain. The wave kinematics are modelled using the Siemens Wave Generator tool, which accounts for linear and non-linear wave kinematics, including Wheeler stretching and a diffraction correction. The loads and responses caused by second-order hydrodynamics are analyzed and compared with the first-order hydrodynamic responses. The load cycles of the response are translated to fatigue loads via rain flow counting of half-cycles.

Two studies have been conducted in this thesis. The first study contains a statistical analysis of the fatigue loads, for different simulation durations and different number of realizations per sea state. The results of this study show that the equivalent fatigue load varies more for shorter simulation duration and the root mean square error is higher in these cases. The mean of the estimated fatigue loads appear independent of simulation duration.

The second study focuses on the difference in fatigue loads caused by linear versus non-linear waves. The results of 1-hour simulations of multiple sea states with increasing severity are compared. In

(6)

INTRODUCTION

vi

calm sea states, the fatigue loads at tower bottom are 5% to 20% higher for non-linear waves than for linear waves. The main reason is the extra energy at frequencies around twice the peak frequency, related to the second-order sum frequencies. These frequencies overlap with the tower bending eigenfrequency and are strongly amplified in the response.

In contrast to calm sea states, the rougher sea states cause 3% to 10% higher fatigue loads for linear waves comparing with non-linear waves. The linear waves contain more energy around the wave peak frequency and around the pitch eigenfrequency. Significantly more energy is present around these frequencies in the bending moment response, leading to higher fatigue loads.

Finally, it is concluded that the overall occurrence probability of sea states that lead to higher fatigue loads when non-linear effects are included, lies between 18% and 63%. For these sea states, the estimated fatigue loads are 5% to 20% higher for non-linear waves than for linear waves.

(7)

vii

A

CKNOWLEDGEMENTS

A year ago I started my thesis with great ambitions and little experience. For (at least) once in my life I wanted to develop a numerical model in some fancy programming language. I was interested in floating structures and their interaction with waves and I was fascinated by wind turbines. So why not combine these ingredients and make a hydro-servo-elastic model of a floating wind turbine? Let’s take some numerical computing software, add some wind, a pinch of mooring accompanied with a touch of structural flexibility and spice it up with some non-linear waves. Stir this all until it has a solid 3D texture, and voila... The five course dinner served in Matlab is ready.

A great idea when you are an experienced chef. But for me, this bar was too high. This resulted in a tough graduation process, with many disappointments. In my mind I climbed over many walls, carrying a dead horse with me. And often I was so preoccupied with everything that still had to be done, that I forgot what road I had already left behind. That is the downside of last year. However, the upside is that I did learn programming in a fancy language. I did develop a numerical model of a floating wind turbine. I did analyze wave-structure interaction. I did gain the experience I believe a good engineer should have. And above all, I learned dealing with the disappointments, I faced my limits and I got acquainted with hidden traps that will not surprise me again. And not to forget, I once again realized what challenges you can overcome when having good people around you. Therefore, I would like to thank you all.

To begin with Siemens Wind Power, I would like to express my gratitude for providing me the opportunity to do this project. In particular my supervisors Mauk and Paul, who have taught me a lot about the topic and remembered me every now and then that I was walking a long road that I had to finish by taking small steps. I also want to thank Sven, Bas and Michiel for their help with multi-bodies, VDIs and struggles with SWAG.

Furthermore I would like to thank the other graduate students at Siemens, for the enjoyable office hours, the coffees, the beers, the bowling, the Toko Bali, and of course – the sauna times. Also, thanks to all other colleagues from the wind department, who made my time as pleasant as it has been. Who knows, we will meet as colleagues again.

From the TU Delft I would like to thank professor Huijsmans and Arie Romeijn for their constructive feedback and guidance through the learning process of this master thesis.

In the after-office hours I could always count on my house mates. Witek, Remco, Bauke and Henk: thanks for the beers and the nice ‘WAPs’. We are a great team, and I hope we will even have greater times now the last one of us is an official ‘yup’ as well!

Heit and mem, Yorick and TNB, thank you for all the support this year. The first MSc. Bergsma is ready to explore the world! Klaas, Rita and Jasper, thank you for always being interested and supportive.

Finally, and above all, Anne, thank you for your unconditional support, your listening and for making this entire year a little lighter. I am looking forward to exploring the future together.

(8)

INTRODUCTION

(9)

ix

L

IST OF

T

ABLES

TABLE 1.1-MAIN PROPERTIES OF HYWIND DEMO [2] 3

TABLE 1.2–NATURAL FREQUENCIES AND DAMPING RATIO OF HYWIND DEMO [2] 3 TABLE 2.1-STRUCTURAL PROPERTIES OF INDIVIDUAL BODIES OF HWDEMO 15

TABLE 2.2–STRUCTURAL DAMPING 20

TABLE 2.3–STRUCTURAL SPRING STIFFNESS 21

TABLE 2.4-PROPERTIES OF MOORING SYSTEM 22

TABLE 2.5–OVERVIEW OF COMPARED TOLERANCE SETTINGS 34

TABLE 3.1-COMPARISON OF EIGENPERIODS AND DAMPING LOGARITHMIC DECREMENT OF PLATFORM MOTIONS 40

(14)

INTRODUCTION

xiv

L

IST OF

F

IGURES

FIGURE 1.1-THREE FLOATING CONCEPTS 2

FIGURE 1.2-DIMENSIONS OF HYWIND-OC3 TURBINE 4

FIGURE 1.3-HYWIND CONCEPT WITH 'CRAW-FOOT' MOORING CONNECTION 8 FIGURE 2.1-ORIENTATION OF GLOBAL AND LOCAL AXIS-SYSTEMS OF MDOF MODEL 14 FIGURE 2.2-SET-UP OF MDOF MODEL WITH REFERENCE TO PARAGRAPHS WHERE PARTS ARE EXPLAINED 14 FIGURE 2.3–LOCATION OF CENTRE OF MASS (B#CM) AND NODES (B#N#) OF BODIES 15 FIGURE 2.4-SCHEMATIC OVERVIEW OF 2 RIGID BODIES INTER-CONNECTED WITH 6DOF SPRINGS 15

FIGURE 2.5-STABILIZING EFFECT OF BUOYANCY FORCE - 24

FIGURE 2.6-DEFINITION OF A HARMONIC REGULAR WAVE [34] 24

FIGURE 2.7-FIRST- AND SECOND-ORDER IRREGULAR WAVE SIMULATION.

HS=15,5 M,TP=17,8 S, Γ=1,7,N=21600,ΔT=0,5 S [35] 29

FIGURE 2.8-FLOWCHART OF WAVE KINEMATICS MODULE SWAG 29

FIGURE 2.9-WHEELER STRETCHING METHOD [35] 30

FIGURE 2.10-DIFFERENT WAVE FORCE REGIMES.HERE Λ IS WAVE LENGTH [5] 30 FIGURE 2.11–MACCAMY-FUCHS DIFFRACTION CORRECTION FOR THE INERTIA COEFF. (A)AND PHASE

LAG (B) AS A FUNCTION OF THE DIFFRACTION PARAMETER [34] 31 FIGURE 2.12-LOCATION OF GRID POINTS WHERE WAVE KINEMATICS ARE CALCULATED.IN VERTICAL

DIRECTION MORE DENSELY PACK AROUND MWL 32

FIGURE 2.13-COMPARISON OF DIFFERENT TOLERANCE SETTINGS,RT(RELATIVE TOLERANCE)

AND AT(ABSOLUTE TOLERANCE) 35

FIGURE 2.14-DETERMINE TOTAL FATIGUE DAMAGE WITH SN CURVE ([36]) 37

FIGURE 3.1-STATIC EQUILIBRIUM TEST 40

FIGURE 3.2-COMPARISON OF FREE DECAY TESTS [2] 41

FIGURE 3.3-COMPARE MOTION RESPONSE TO HARMONIC WAVES WITH T10 S AND H6 M 42 FIGURE 3.4-COMPARE INTERNAL FORCES TO HARMONIC WAVES WITH T10 S AND H6 M 43

FIGURE 3.5-NATURAL FREQUENCY OF TOWER FOR-AFT BENDING 44

FIGURE 4.1-SCATTER DIAGRAM FAROE ISLANDS [19] 49

FIGURE 4.2–EFL FOR 40 DIFFERENT SEEDS OF SEA STATE HS=5,49M AND TP=11,3S 50

FIGURE 4.3–ESTIMATED RMS ERROR FOR EFL AS FUNCTION OF NUMBER OF SEEDS 51 FIGURE 4.4–MEDIAN AND VARIANCE OF EFL FOR DIFFERENT SEA STATES AND SIMULATION TIMES.

RESP.20,20,10 SEEDS, FOR 10MIN,1HR,3HR. 52

FIGURE 4.5–BOXPLOTS OF EFL FOR DIFFERENT “SNAP-SHOTS” IN 60-MIN SIMULATION.20 SEEDS 52 FIGURE 4.6–MEDIAN AND VARIANCE OF RELATIVE DIFFERENCE IN EFL BETWEEN LINEAR AND

NON-LINEAR WAVES, FOR SIMULATION TIMES 53

FIGURE 4.7-WAVE SPECTRUM,HS:6S AND TP:10M.LINEAR (INPUT) AND NON-LINEAR (OUTPUT) SPECTRUM 53

FIGURE 4.8-PSD GRAPHS OF WAVE ELEVATION OF SEA STATE HS:5,49 M,TP11,30 S 55

(15)

xv FIGURE 4.10–PSD OF WAVE ELEVATION – RELATIVE DIFFERENCE 56 FIGURE 4.11-PSD OF HORIZONTAL HYDRODYNAMIC FORCE - NORMALIZED ABSOLUTE DIFFERENCE 57 FIGURE 4.12-PSD OF HORIZONTAL HYDRODYNAMIC FORCE – RELATIVE DIFFERENCE 58 FIGURE 4.13-PSD OF TOWER BENDING MOMENT - NORMALIZED ABSOLUTE DIFFERENCE 59 FIGURE 4.14-PSD OF TOWER BENDING MOMENT – RELATIVE DIFFERENCE 59 FIGURE 4.15–FREQUENCY-AMPLIFICATION RELATION BETWEEN HORIZONTAL EXTERNAL FORCE AT -20 M

BELOW MWL AND BENDING MOMENT AT TOWER BOTTOM, FOR DIFFERENT DAMPING RATIO’S Ζ 60 FIGURE 4.16–DIFFERENCE IN EFL OF BENDING MOMENT AT TOWER BOTTOM FOR MULTIPLE

REALIZATIONS OF FIVE DIFFERENT SEA STATES 61

FIGURE 4.17–EFL OF FIVE DIFFERENT SEA STATES, LINEAR AND NON-LINEAR WAVES 61 FIGURE 4.18-EFL OF BENDING MOMENT AT TOWER BOTTOM FOR SEA STATES WITH DIFFERENT HS 62 FIGURE 4.19-SCATTER DIAGRAM OF HYWIND SCOTLAND SITE AT BUCHAN DEEP FOR PERIOD 1958-2010 63 FIGURE 7.1-ROTATING RIGID BODY WITH POINT P IN GLOBAL (OXYZ) AND IN LOCAL (OXYZ) COORDINATES 71 FIGURE 7.2-A ROTATING RIGID BODY B(OXYZ) WITH A FIXED POINT O IN A REFERENCE FRAME G(OXYZ).

LEFT: VELOCITY VECTOR.RIGHT: ACCELERATION VECTOR 74

FIGURE 7.3-SECOND-ORDER WAVE FREQUENCY MATRIX.LEFT: SUM-FREQ.RIGHT: DIFF-FREQ 81 FIGURE 7.4-WAVE SPECTRA WITH DIFFERENT HS,TP, DEPTH AND CUT-OFF FREQUENCY 83

FIGURE 7.5-HYDRODYNAMIC ADDED MASS AND DAMPING FOR OC3-HYWIND SPAR, SIMILAR BEHAVIOR

AS HYWIND DEMO [13] 86

FIGURE 7.6-FORCE-DISPLACEMENT RELATIONSHIPS OF THE OC3-HYWIND MOORING SYSTEM IN 1D,

SIMILAR BEHAVIOR AS HYWIND DEMO [13] 87

FIGURE 7.7-PSD OF ABSOLUTE DIFFERENCE BETWEEN LINEAR AND NON-LINEAR WAVE ELEVATION

(NL-L)- DISPLACEMENT OF ENERGY OVER FREQUENCIES IS MARKED WITH BURGUNDY DASHED LINE 91 FIGURE 7.8-PSD OF WAVE ELEVATION - DIFFERENCE BETWEEN LINEAR AND NON-LINEAR WAVES

(NL MINUS L) NORMALIZED WITH POWER AT PEAK FREQUENCY 92

FIGURE 7.9-PSDWAVE ELEVATION - NORMALIZED ABSOLUTE DIFFERENCE 92 FIGURE 7.10-PSD OF WAVE ELEVATION – ABSOLUTE DIFFERENCE BETWEEN LINEAR AND NON-LINEAR

WAVES (NL MINUS L) 92

FIGURE 7.11-PSD OF WAVE ELEVATION – ABSOLUTE DIFFERENCE BETWEEN LINEAR AND NON-LINEAR

WAVES (NL MINUS L) 93

FIGURE 7.12-PSD OF HORIZONTAL HYDRODYNAMIC FORCE - DIFFERENCE BETWEEN LINEAR AND NON-LINEAR WAVES (NL MINUS L) NORMALIZED TO POWER AT PEAK FREQUENCY –NB: INCORRECT Y-LABELS 93 FIGURE 7.13-PSD OF HORIZONTAL HYDRODYNAMIC FORCE - DIFFERENCE BETWEEN LINEAR AND NON-LINEAR

WAVES (NL MINUS L) NORMALIZED TO POWER AT PEAK FREQUENCY–NB: INCORRECT Y-LABELS 93 FIGURE 7.14-PSD OF HORIZONTAL HYDRODYNAMIC FORCE – ABSOLUTE DIFFERENCE BETWEEN LINEAR

AND NON-LINEAR WAVES (NL MINUS L) 94

FIGURE 7.15-PSD OF HORIZONTAL HYDRODYNAMIC FORCE – ABSOLUTE DIFFERENCE BETWEEN LINEAR

(16)

INTRODUCTION

xvi

FIGURE 7.16-PSD OF BENDING MOMENT TOWER BOTTOM - DIFFERENCE BETWEEN LINEAR AND NON-LINEAR WAVES (NL MINUS L) RELATIVE TO POWER AT TOWER EIGENFREQUENCY–NB: INCORRECT Y-LABELS 94 FIGURE 7.17-PSD OF BENDING MOMENT TOWER BOTTOM - DIFFERENCE BETWEEN LINEAR AND NON-LINEAR

WAVES (NL MINUS L) RELATIVE TO POWER AT TOWER EIGENFREQUENCY 95 FIGURE 7.18-PSD OF BENDING MOMENT TOWER BOTTOM – ABSOLUTE DIFFERENCE BETWEEN LINEAR AND

NON-LINEAR WAVES (NL MINUS L) 95

FIGURE 7.19-PSD OF BENDING MOMENT AT TOWER BOTTOM – ABSOLUTE DIFFERENCE BETWEEN LINEAR

AND NON-LINEAR WAVES (NL MINUS L) 95

FIGURE 7.20-PSD GRAPH OVERVIEW OF TOWER BENDING MOMENT FOR SMOOTH SEA STATE 96 FIGURE 7.21-PSD GRAPH OVERVIEW OF TOWER BENDING MOMENT FOR SEVERE SEA STATE 96

(17)

xvii

L

IST OF

A

BBREVIATIONS

BEM Blade element momentum

BFWT Bottom founded wind turbine

BHawC Bonus horizontal axes wind turbine code CFD Computational fluid dynamics

CoB Center of buoyancy

CoG Center of gravity

DLC Design load case

DOF Degree of freedom

EFL Equivalent fatigue load

EoM Equation of motion

FEM Finite element model

FTA Floater tower assembly

FWT Floating wind turbine

HAWC2 Horizontal axes wind turbine code 2 HAWC2 Horizontal axis wind turbine code FFT Fast fourier transformation

JONSWAP Joint Offshore North Sea WAve Project

L Linear

MBD Multi body dynamics

MDOF Multi degrees of freedom

MW Megawatt

MWL Mean water level

NL Non linear

NREL National renewable energy laboratory (US) OC3 Offshore Code Comparison Collaboration

PM Pierson-Moskowitz

RMS Root mean square

STD Standard deviation

SWAG Siemens Wave Generator

TLP Tension leg platform

TMT TMT method (multi body dynamics)

(18)

(19)

1 I

NTRODUCTION

1.1 Relevance of floating wind turbine

For the past decade wind energy technology is rapidly growing, and by many it is considered to be one of the most promising sources of renewable energy. Accelerated by international regulations, a growing number of governments change their policy and focus on the development of wind farms. Wind turbines were initially designed for on land installation, and for some decades an increase of both the number of installed turbines and the maximum wind power capacity is to be seen [9]. However, some obstacles appeared with the increase of onshore wind turbine installation. One being the environmental impact, such as visual pollution and noise, on people living close to wind turbines, and another being the limitation of suitable areas for large high capacity wind farms. These restrictions turned the focus more and more on offshore wind energy. Offshore wind farms have as key advantages higher and more constant wind speeds, available area for large wind farms and lack of visual impact. A downside of offshore wind farms is the costs being higher than onshore wind farms.Equation Chapter (Next) Section 2

Since the start of offshore wind energy huge developments in bottom founded turbines have been made, leading to a constant growing annually amount of installed power, mainly in Europe. In many countries outside of Europe, steep coasts do not permit for bottom-fixed foundations. Here floating wind turbines are a solution (from now on often abbreviated as FWT: floating wind turbine). Another advantage of floating offshore wind turbines is that deeper waters are to be found further offshore where higher and more constant wind speeds are present.

Hitherto several concepts of FWT have been designed, and prototypes of a couple of concepts have yet been installed. One of these yet-installed concepts is SWAY®, a single moored, spar-type floater, of which a 1:6 scale prototype is built. Another promising concept is WindFloat, whose floating foundation is a semi-submersible concept. One of the best known concepts is Hywind Demo, a spar construction with catenary mooring, built in 2009. It has proven its success and yet the follow-up is in its design phase, comprising the development of a small farm of five 6MW turbines at the Scottish coast, called Hywind Scotland. Hywind Demo is chosen to be the reference concept in this thesis. The main reasons for this

(20)

INTRODUCTION

2

choice are the availability of a great amount of (verification) data and the in-house knowledge within Siemens. .

The FWT is exposed to ever varying wind, wave and current loads. These loads induce a lot of stress variations, which on their turn cause fatigue loads. In the design of a FWT fatigue loads are one of the main design drivers. Therefore it is of great importance to understand which physical phenomena have an effect on these fatigue loads. One of the underexposed phenomena is the influence of second order wave effects on fatigue loads. This leads us to the topic of this thesis: investigating the difference between

first order wave effects and second order wave effects on the wave induced fatigue loads of a spar-type float offshore wind turbine.

1.2 Description of FWT concept

Different concepts exist for floating wind support structures. A summary of various floating concepts is formulated by Bossler [3]. The majority of the total of 80 different concepts from this summary can be divided in three fundamental concepts, categorized on the ability to derive stability in the floating condition (see Figure 1.1).

 Ballast stabilized (low centre of gravity), e.g. spar

 Mooring stabilized, e.g. tension leg platform (TLP)

 Buoyancy or water plane stabilized, e.g. semi-submersible Figure 1.1 - Three floating concepts

1.2.1 Hywind concept –a spar-type FWT

1.2.1.1 Spar floater

Within this thesis a spar-type FWT is evaluated, based on the Hywind concept. A spar-type FWT is a ballast stabilized system, where a long hollow cylindrical support structure accounts for the buoyancy force and a large mass in the lower end of the spar allows for stability. The horizontal position is maintained by the tension of (catenary) mooring lines. The water plane area is relatively small, in order to minimize wave excitation of the system.

The FWT has six platform degrees of freedom (DOFs), three translations and three rotations. The two translations in the horizontal plane are sway (left-right) and surge (for-aft) and the translation in the vertical direction is heave. The two rotations around the horizontal axis are roll (left-right, around the x-axis) and pitch (for-aft, around the y-x-axis) and the rotation around the vertical axis is called yaw.

(21)

DESCRIPTION OF FWT CONCEPT

3

1.2.1.2 Hywind Demo properties

A catenary moored spar-type FWT inspired by the Hywind Demo concept is considered in this study. Hywind Demo, from Statoil, is world’s first full scale spar-type FWT and is installed at the Norwegian coast. Hywind Demo is composed of three parts: a spar-buoy substructure, a tower and a Siemens 2.3 MW wind turbine generator. Based on Hywind Demo other designs called OC3-Hywind and Hywind Scotland have been developed.

Figure 1.1 presents as basic image of a spar-buoy FWT. The hull and tower are made of steel and are ballasted with gravel and water at the lower end. The substructure, also referred to as floater, or spar(-buoy), is designed to support the tower-turbine assembly and provides for a desirable sea performance. The floater is a long cylinder, with a reduced diameter at the water surface to mitigate hydrodynamic loads in the wave zone. At the lower end the spar-buoy is ballasted with gravel and water to lower the centre of gravity which accounts for stability. Three catenary mooring lines are attached to the floater, slightly underneath the centre of buoyancy with angles of 120 degrees in between. The cables are anchored in the seabed. The spar supports the tower, and the tower supports the wind turbine generator.

In Table 1.1 the main properties of Hywind Demo are presented, a more extensive description is provided in [2]. In Table 1.2 the natural frequencies and damping ratios are set-out.

Table 1.1 - Main properties of Hywind Demo [2]

Item Dimension

Turbine Siemens 2.3-MW

Water depth (m) 200

Draft (m) 100

Hub height above MWL (m) 65

Displacement (m) 5036

Static centre of buoyancy (m) under MWL -51,03

Diameter at MWL (m) 6,0

Diameter at bottom (m) 8,3

Mass total FWT (ton) 5086

Centre of gravity of total FWT (m) -67,48

Mass moment of inertia Ixx and Iyy floater (ton*m2) 29,24·108

Mass moment of inertia Izz (ton*m2) 9,17·104

Height of mooring connection (m) -53,0

Table 1.2– Natural frequencies and damping ratio of Hywind Demo [2]

Item _{Frequency [Hz]}Natural _{Damping ratio δ}

Surge/ Sway 0,009 0,25

Heave 0,036 0,22

Roll / Pitch 0,040 0,13

Yaw 0,055 0,10

1st tower bending moment 0,46 0,01

2nd tower bending moment 1,667 --

(22)

INTRODUCTION

4

Figure 1.2 - Dimensions of Hywind-OC3 turbine

1.3 Previous research on FWTs

Research on different concepts of FWTs has advanced over the past ten years. Relevant research has been conducted on both FWT in general and spar-supported FWT specifically. An overview of the relevant previous research is presented in the following paragraphs, ordered per subject. The outcome of previous studies forms the starting point of this thesis and gives guidance to the relevant modelling methods and possible simplifications.

First the start of FWT modelling is addressed, and its relation with the modelling of bottom founded wind turbines. Then the specific research on spar-type FWTs is set out, followed by the modelling of hydrodynamics. Subsequently studies on modelling mooring lines are presented and thereafter the aerodynamics. Finally, the research on fatigue loads and applied methods of structural modelling are explained.

At the end of each paragraph a conclusion is stated which summarizes the most relevant information for this thesis based on the outcome of previous research.

1.3.1 Start of modelling floating wind turbines

Theories related to bottom-founded wind turbines (BFWT) have been used as a starting point of research on FWT. For fixed offshore wind turbines uncoupled modelling could be sufficient under certain conditions [11]. However, this is not valid for floating wind turbines, due to the important dynamic coupling of the tower-nacelle and the floater [18], since aerodynamic and hydrodynamic damping and excitation forces are strongly affected by one another through the relative motions. The nonlinear hydro-elastic response of the floater and mooring system, together with nonlinear wave loading, makes the system modelling even more complex.

Where the modelling of bottom founded structures could be done in the frequency domain, which is computationally faster and especially suitable for fast computations that do not require maximum

(23)

PREVIOUS RESEARCH ON FWTS

5 accuracy (preliminary design), the modelling of FWTs should be done in the time domain. Due to transient behaviour and nonlinear coupled dynamics of the floater, tower and rotor, the equations of motions need to be solved at each time step.

Modelling is done in the time domain, since non-linearity cannot be included in a frequency domain analysis.

1.3.2 Platform motions

An important operational issue for a spar is the pitch motion. Large pitch motions can introduce a varying angle of thrust which could affect gyroscopic moments and stability. Pitch motions also change the angle between the plane of the rotor and the ambient wind direction, worsening the power generation. An analysis of Karimirad and Moan [17] showed that the occurrence of large pitch motions is possible due to resonant excitation.

Pitch and roll stability are mainly maintained by the structures' pitch and roll stiffness. The stiffness increases with increasing metacentric height [20] (i.e. the vertical distance between the centre of mass and the metacentre), and increasing relative distance between the centres of gravity and buoyancy [15]. The stability is also influenced by the tension on the mooring lines and the position of the connection point of the mooring lines [10]. In addition, the control system can have a significant influence on the platform's pitch motions [30].

Karimirad and Moan [16] concluded that the motions of the platform strongly influences the power production through surge motions of the nacelle, initiated by both platform surge and pitch motions. Furthermore Sultania and Manuel [32] found that a combination of short and long term sea states may result in a unique condition where the highest bending moment on the tower and the platform occur. For determining these bending moments correctly, the often used assumption that the floater is infinitely stiff is not applicable.

Pitch motions are of specific interest for the simulations. A flexible floater will be modelled to be able to determine internal bending moments more accurately.

1.3.3 Modelling of hydrodynamics

Hydrodynamics are incorporated in existing simulation tools as a combination of incident-wave kinematics and hydrodynamic loading models [12]. The hydrodynamic loads result from the integration of the dynamic pressure over the wetted surface of the structure, i.e. Froude-Krylov force. The loads include contributions from inertia forces (added mass), linear and nonlinear viscous drag (radiation), incident-wave scattering (diffraction), current and restoring buoyancy force.

1.3.3.1 Hydrodynamic theories

One of the most frequenly used wave theories in offshore engineering is Linear Airy wave theory. Linear Airy wave theory is not applicable for steep-sided and wave breaking waves, found in shallow waters. Since the Hywind concept is present in deeper water, linear wave theory is generally applied. Linear Airy

(24)

INTRODUCTION

6

wave theory yields wave kinematics, i.e. a description of the velocity and acceleration of water particles in space and time.

Besides linear hydrodynamic loads, also non-linear second-order hydrodynamic loads do exist. Second-order hydrodynamic loads are proportional to the square of the wave amplitude and have frequencies equal to both the sum and the difference of the multiple-incident wave frequencies of an irregular sea state. Although the natural frequencies of a floating structure are designed to be outside the wave energy spectrum, the second-order forces can excite these frequencies. Despite the forces normally being small in magnitude, the resonant effect can be important. These second order effects are the focus of this thesis and are further explained in paragraphs 1.3.3.2 and 2.4.

Another widely used theory is potential flow. In this theory a velocity field of the water is described as the gradient of a scalar function: the velocity potential. For a potential flow, irrotationality is assumed, which makes it valid for several applications. However, potential flow is not applicable for structures with large translational displacements, and lies therefore outside of this thesis’ scope.

Also a few nonlinear hydrodynamic theories are formulated, including a Rankine-Airy based panel method and computational fluid dynamics (CFD) [27]. These methods are effective for the simulation of the behaviour of a FWT in extreme waves. The downside of these theories is that they are rather complex and computational expensive.

1.3.3.2 First and second order wave effects

Often second order wave effects are left out of scope, since their influence is expected to be rather small in comparison with first order loads. However, in cases where the frequency ranges of second order waves overlap with structural eigenfrequencies the second order waves could lead to significant loads due to resonance effects. The effect of second order waves on fatigue loads on the turbine (floater and tower combination) has not been investigated yet, but some studies have been conducted on other effects of first and second order waves.

A study performed by Lucas [25], on the comparison of first- and second-order hydrodynamics confirms the importance of including the second-order wave effects when modelling a FWT. He has found that for the OC3-Hywind the second-order excitation force in stochastic waves is important in surge and pitch modes, although it still is smaller than the first-order excitation force.

Another study on the effect of second-order forces was performed by Roald [29]. Wave forces on a spar were investigated using diffraction software WAMIT. Second-order forces appeared to be small for the OC3-Hywind spar. But where the forces excite platforms eigenfrequencies, some second-order motion response was seen. In this study a single rigid body formulation was used, so a conclusion on the effect of sum-frequencies on structural responses and loads could not be drawn. Therefore the question on the influence of second order wave effects on fatigue loads remains unanswered.

Karimirad [14] found that the heave motion is affected by the drift and second order forces at the heave resonant frequency. The different frequency forces are found to not have a significant effect on the tension of the mooring lines and other motions. It is important to note that Karimirad used linear and no nonlinear wave kinematics for calculating the second order loads.

(25)

7 The studies of Roald and Karimirad do not give an analogous explanation of the influence of second-order wave forces. Several motions and rotations seem to be significantly influenced, but no answer can be given to the question when second order wave effects are relevant to take into account considering fatigue loads. Some limitations to the proposed method have been identified in the research of Roald [29]. First, the tower flexibility cannot be taken into account in WAMIT, leading to inaccuracies in the simulation of second-order quantities for structural loads. The second drawback is that viscous effects (drag), which would likely damp some of the second-order motion response, are not accounted for when solving the equations of motion in WAMIT.

1.3.3.3 Morison equation

A study on the modelling aspects of FWT by Karimirad [14] investigated different hydrodynamic modelling strategies based on the Morison equation, pressure integration and panel methods. The results of the study show that either the Morison equation considering the instantaneous position of the floater or first order hydrodynamic forces based on the panel method together with considering the quadratic viscous forces can provide accurate results.

In the original Morison equation the heave excitation force is not taken into account. Heave excitation forces can be included by accounting for hydrodynamic pressure (pressure integration method) in heave direction.

Second order wave effects appear to have some influence on heave motions and second order motions response within the platforms natural frequency range is seen. The effect of difference frequency forces on mooring line tension and other motions is negligible. The effect on fatigue loads and the relation to the excitation of structural eigenfrequencies is not investigated yet, thus is the focus of this thesis.

The Morison equation where the instantaneous position of the floater is accounted for will be used to calculate wave loads. Besides that, the pressure integration method is used to account for excitation forces in the heave direction.

1.3.4 Mooring line dynamics

The mooring lines mainly contribute to the stability in the horizontal direction and allow the floater to surge smoothly. Rigid body motions in surge and sway direction have a very low natural frequency and they are governed by the stiffness of the catenary mooring lines. The yaw motions are also mooring stabilized, due to a 'craw-foot' configuration at the connection point as is shown in Figure 1.3. The platform's pitch and roll motions are mainly ballast-stabilized.

In a study on mooring line model fidelity of different FWT platform types [8] the quasi-static model agrees very well with a dynamic FEM model. Here it was concluded that quasi-static mooring models are accurate for analyzing the response of OC3-Hywind. Caution should be taken in extreme sea conditions, because the results will become less accurate.

In the Hywind design the fairleads are located very near the platform’s pitch and roll axes. Besides that, the platform’s low natural frequencies keep its velocities relatively low, resulting in small motions of

(26)

INTRODUCTION

8

the mooring lines and little contribution to damping in these DOFs. The contribution to the damping ratio in the yaw direction is more significant because the spar-buoy’s inertia and hydrodynamic damping in the yaw DOF are small. According to the inviscid linear hydrodynamics approach, the platform would not have any hydrodynamic yaw damping contribution. However, yaw damping is introduced as an additional linear damping term, to match empirical results from Statoil. For the tensions in the mooring lines the quasi-static model causes significant under-prediction of the inner mooring loads, however this is not relevant for this study. In extreme conditions results will become less accurate.

From [14] it was concluded that second order wave forces do not significantly affect the motion and tension responses of the mooring system. Furthermore it was found that mooring damping and inertia forces seem important for tension responses of individual mooring lines. However, mooring-line damping had no significant effect on the platform motion responses. This was also found in a study of Kvittem et al. [22]. So it could be concluded that quasi-static force-displacement relationships are sufficient for dynamic analyses of a FWT.

The mooring lines will be modelled by a quasi-static force-displacement relationship in surge, sway and yaw direction. The influence of the mooring system on the heave motions is neglected. Furthermore mooring line inertia and damping effects are neglected.

Figure 1.3 - Hywind concept with 'craw-foot' mooring connection

1.3.5 Fatigue loads

In comparison with the subjects already discussed, relatively little research has been conducted on fatigue loads in a FWT. The fatigue assessment of a semi-submersible wind turbine [22] showed that the blade passing frequency resonance in the tower and pitch motion of the platform were the most significant contributors to fatigue damage in the tower and a semi-submersible pontoon. Here only first order waves were simulated because the simulation tool FAST was not able to model second order waves.

(27)

9 Hereby resonance effects due to overlapping frequency ranges of second order waves and elastic modes have not been investigated.

Another study [7] analyzed the influence of simulation time on computed loads and fatigue loads. It was shown that the dependence of simulation length of loads due to hydrodynamics and floating platform motions did not affect ultimate loads for the FWT OC3-Hywind. A larger number of 10-minute simulations led to the same loads as longer simulations, on the condition that the total simulation time was the same. For fatigue loads it appeared that there is greater sensitivity in the loads to the method of counting unclosed cycles, as compared to simulation length [7]. Hence is decided to count half-cycles during rain flow counting for fatigue loads.

Fatigue loads will be determined at relevant positions along the floater-tower assembly for many different load cases. Furthermore unclosed-cycle counting of half cycles will be applied in the fatigue loads analysis.

1.3.6 Wind loads

The modelling of aerodynamics is out of the scope of this thesis. Not including wind simplifies the analysis, and makes the influence of (non-) linear waves on fatigue loads more clear. The downside is that some significant influences of wind on the total fatigue loads are not accounted for.

From several studies (i.a. [11], [18]) can be concluded that the resultant shear forces and bending moments due to wind forces and due to wave forces individually are from the same order of magnitude. Under the assumption that these (maximum) loads are representative for fatigue loads as well, one could say that excluding wind loads means that roughly half of the fatigue loads are not included. On the other hand, this means that wave forces account for a significant part of the fatigue loads, so it is relevant to look at the influence of second order wave effects. This is a coarse conclusion, that needs to be confirmed in a future research.

Wind loads will be excluded in this study. The exclusive inclusion of waves is regarded as justifiable, because the wave loads are of the same order of magnitude as the wind loads.

1.3.7 Structural modelling

In most studies the dynamic response of the spar structure to wind and wave loads is evaluated in terms of rigid body degrees of freedom of the floating substructure [10]. Since the floater is a steel cylinder with large diameter the structural stiffness is high and often assumed to be rigid. However, some studies show that the hydro-elasticity of the spar could be of significant influence.

In a comparison study [15] it was found that there are significant differences in the spar-type floater motions between a rigid model and an elastic model. In certain degrees of freedom the rigid body experienced greater motions than the elastic model due to the lack of structural damping. Hence flexible modelling of the spar and tower might be of significant importance for the determination of fatigue loads. For analyzing the general motions and structural responses at an acceptable accuracy, a rigid body formulation of the spar-buoy is accurate enough, instead of multibody elastic modelling. The spar platform is almost rigid compared with the tower [17]. Besides, the natural frequencies of rigid body

(28)

INTRODUCTION

10

motions are very low compared with the natural frequencies of the elastic modes. The rigid body motions (or platform motions) have such a low natural frequency that these are in the second order wave difference frequency range, and could be sensitive for second order wave drift force.

However, the main reason to waive a rigid body formulation is the fact that fatigue loads cannot accurately be determined at different points along the structure. Furthermore, as already explained, the fact is that the first eigenmode of the tower is in the frequency range second order sum frequencies, and therefore of particular interest for this study. So it is twofold to choose to model the spar-buoy and tower as flexible structures

Multibody formulation is a proper method for modelling a flexible structure. It can account for large-amplitude motions and it is relatively fast from a computational point of view.

The floater and tower will be modelled as a flexible structure using a multibody dynamic formulation. This way the first two elastic eigenmodes are included. Furthermore the fatigue loads can be accurately determined at different points along the structure..

1.4 Project formulation

1.4.1 Research questions

In the previous paragraphs it is already explained that the focus of this study will be on the influence of first order and second order wave effects on fatigue loads in the spar-type FWT. To capture the essence of this thesis the most relevant research questions are set out here.

The main research question is formulated as:

What is the difference between the first order wave effects versus second order wave effects on the wave induced fatigue loads of a spar-type floating offshore wind turbine?

Related sub-questions are:

 When do second order wave effects have a significant influence on the wave induced fatigue loads in comparison to first order wave effects? Cq. When do second order wave effects need to be included in the hydrodynamic analysis?

 What simulation duration is necessary to capture the important effects of second order waves?

 What is the influence of the number of seeds per sea state on the variance of the estimated fatigue loads?

1.5 Thesis outline

In chapter 2 a profound description of the self-made MDOF model is provided. This includes the applied theories for the structural modelling, the wave kinematics, the related hydrodynamic forces, the time integration method of solving the equations of motion, the computation method of the fatigue loads and the underlying assumptions of the applied methods.

(29)

THESIS OUTLINE

11 In chapter 3 the verification of the MDOF model is described. The verified characteristics of the model are the static equilibrium position, the natural frequencies and damping values of the separate platform motions, the natural frequency of tower bending and the response to waves in terms of motion and internal forces.

Chapter 4 provides an overview of the studies that have been conducted and the related results. First the results of the statistical analysis are set out, in which it is analyzed what the influence of simulation time and number of random seeds is on the estimated fatigue loads. Second the analysis of the difference in fatigue loads due to linear and non-linear waves for multiple load cases is described.

In chapter 5 the conclusions and recommendations are treated.

(30)

(31)

2 M

ODEL SET

-

UP

For studying the difference in fatigue loads due to linear and non-linear waves, a hydro-elastic model has been developed. The set-up of this model is explained in this chapter. This includes the applied theories for structural modelling, the wave kinematics and related hydrodynamic forces are described, just like the time integration method for solving the equations of motion, the computation method of the fatigue loads and the underlying assumptions of the applied methods.

2.1 Overview of model

The hydro-elastic model describes the hydrodynamic and structural response of a spar-type FWT, as explained in paragraph 1.2.1, induced by hydrodynamic forces. The FWT (Figure 1.2) is modelled as a semi-flexible structure. The model consist of two coupled bodies, interconnected with springs, based on multibody theory. The model is a 2D description, hence it is not able to account for directionality of loads in the horizontal plane, but it does allow for the modelling of the forces in the governing directions. Each body has three degrees of freedom: X, Z and Θ (two translations and one rotation respectively). The motion (position, velocity and acceleration) of the centre of gravity of each body is calculated in global coordinates. Consequently the motion of the FWT, as a response to external forces, is described in space and time. The output of the model also includes internal structural forces and bending moments, inertia forces and external hydrodynamic, hydrostatic, gravity and mooring forces.

Figure 2.2 presents the structure of the hydro-elastic model. The mooring module calculates the mooring loads on the platform Fmooring and the hydrostatic & hydrodynamic module calculates the

hydrostatic and hydrodynamic loads on the structure Fhydro. All loads are combined and included in the

equations of motion, which are subsequently solved at every time step. The result of this process is the state vectorXS, which contains the position, velocity and acceleration of m elements in 2D, resulting in

m



3 DOFs. Since the FWT is build-up of m number of bodies, with each 3 DOFs, the model is referred to as MDOF model. Equation Chapter (Next) Section 2

(32)

MODEL SET-UP

14

Figure 2.1 - Orientation of global and local axis-systems of MDOF model

Figure 2.2 - Set-up of MDOF model with reference to paragraphs where parts are explained

2.2 Structural module

2.2.1 Physical representation

The MDOF model is a 2D model with m×3 degrees of freedom: X, Z, and Θ. In the final model of this study the structure is divided in 2 connected bodies, one for the spar and one for the tower. The bodies can individually rotate and translate, and their motions are described in global and local axis system as defined in Figure 2.1. The equations of motion of the multibody assembly are derived using multibody theory, which is explained in paragraph 2.2.2.

Structural properties

Mass,dimensions,stiffness… §2.2

Equation of Motion

ODE45: time integration solver §2.5

Hydrostatic forces

Buoyancy and gravity §2.4.1

Hydrodynamic forces

Morison equation §2.4.6

Wave kinematics

SWAG tool – 1st_{and 2}nd_order

§2.4.2 - §2.4.5

Mooring forces

Spring stiffness §2.3

Output

, , , _hydro, _mooring, _internal

X X X F F F

§2.2.2

Fatigue loads

Rainflow counting §2.6

(33)

STRUCTURAL MODULE

15 The spar and tower are both represented by one rigid body. This is the minimal set of bodies to represent the first bending mode. When more bodies are included higher modal shapes could be taken into account. This is at the expense of computation time, since more degrees of freedom are included. Higher modes are out of scope in this study, since these modes have higher frequencies that are out of range of the wave frequencies included in this study.

The two rigid bodies together represent the floater tower assembly (FTA) which varies along the length of the structure in diameter, wall thickness, mass and external forces. In Figure 2.3 the positions of the bodies, their centre of mass and related nodes are shown. The main structural properties of the individual bodies are presented in Table 2.1. The bodies are connected with springs with 3 DOF, so they act in 1 rotational and 2 lateral directions (see Figure 2.4). The stiffness of the springs determines the bending behaviour of the structure. The stiffness possesses a linear relationship with deflection and is depending on structural properties of the bodies. The stiffness properties of each DOF are presented in Table 2.3. The stiffness of the springs are chosen such that the significant bending modes are included. The stiffness of the springs is determined based on structural properties. This is further explained in paragraph 2.2.4. Also structural damping is included, its determination and values are explained in paragraph 2.2.3.

Table 2.1 - Structural properties of individual bodies of HWDemo

Property Body 1 Body 2

Length (m) 111 54

Vertical position of lower node wrt MWL (m) -100 11

Diameter (m) 8,3-6,5 6,5-2,3

Mass / meter (kg∙106_/m) _{See appendix C} _{See appendix C}

Mass (kg∙106₎ _4,71 _0,38

Figure 2.3 – Location of centre of mass

(34)

MODEL SET-UP

16

2.2.2 Multibody dynamics applied method

Multibody theory is used to model the spar as a flexible structure, that allows for large rotations and displacements. This paragraph describes the applied multibody dynamics (MBD) method, the set up of the EoM, the inclusion of structural damping and stiffness, and finally the calculation of motions and loads.

The MBD method that is applied to derive the equations of motions at every time step is the TMT-method, which is proposed in the lecture notes ’Multibody Dynamics B’ [33]. The derivation of the equations of motion is explained in Appendix D. Solely the end results are presented here.

The equations of motion are derived for two rigid bodies with each 3 DOF, that are connected by springs with stiffness in 3 DOF and dampers in 3 DOF. The equations of motion of the whole system, including the springs and dampers, in vector-matrix notation is:

{ T } T{ T T } spring d damp      T MT q T F Mg D σ D σ (2.1) with T transformation vector

M full mass matrix

TT_MT _{reduced mass matrix of all bodies and springs}

q second order time derivative of the state vector in generalized coordinates

F external forces

g convective acceleration vector

T

D DD xv,i( )i J D Q( , )v ,first order difference matrix of spring element σ

σspring vector of spring forces and torques

T

dD dDD xv,i( )i J D x Q( ( ), )v i first order difference matrix of damper element σ

σdamper vector of damper forces and torques

The first term, _{T MT , where the original mass matrix is transformed to the reduced mass matrix has}T

been the inspiration for the name of this TMT-method.

In summary the TMT-method is a convenient way to build up the EoMs in a structured manner. The end result is a set of equations that describe the motion of the whole system in independent generalized coordinates. Via a single transformation the motion is found for all other relevant parts (in normal coordinates) of the system. For the FWT the independent generalized coordinates are the CoGs of the bodies. This will be further explained in the next paragraph.

2.2.2.1 State vectors of the FWT system

A state vector of independent generalized coordinates qj is defined. The set of generalized coordinates

contains the minimal number of coordinates, c.q. DOF, that is needed to describe the motions of the whole system. These are the 3 DOF of each body. Once the position (2 DOF) and the orientation (1 DOF) of the centre of mass of each rigid body is known the position of other points at the bodies can be determined.

The set of generalized coordinates of the FWT is:

j[X Z  X Z  ]T

(35)

17 The set of three coordinates describes the position and orientation of the centre of mass of body 1 in global coordinates. When BCM₁is defined as a vector with all 3 DOF of body 1, and continuing this for the other bodies in the same way , the results are as follow:.

i[X Zi ii]T BCM (2.3) j        BCM q BCM₂ 1 _(2.4)

Next, a state vector x_iis defined that contains all coordinates of the centre of mass of the bodies and the coordinates of other selected points at the structure. The other selected points, referred to as dependent coordinates, are the positions of the nodes of each body, which means the upper- and lower end of each body. This is shown in Figure 2.3, which represents the floater bottom, floater top, tower bottom and tower top. Furthermore the mooring attachment point, CoB, and the point at MWL are included in the state vector. The positions of the rigid bodies bottom and top nodes are defined as:

i 1 i 1 i i i B N B N i B N B N [ ] [ ] T T X Z X Z   B N B N ₂ ₂ 1 2 (2.5) M k M                      BCM BCM B N B N x 1 1 1 2 (2.6)

A transformation vectorT_k is defined, such that state vectorx_kcan be determined by a kinematic transformation ofq_i, so

k k( )qj

x = T (2.7)

The transformation vector T_k contains the relations between the coordinates of the state vector x_k and the generalized coordinates qj, i.e. the relations between the locations of the CoG of the bodies and the

locations of other selected points, among which the body nodes.

beam,1 k beam,1 M M M beam,M M M M beam M 1 M M M , ] ] ] ] ] ] ] ] ] ] [ [ [ [ [ [ [ [ [ [ T T T T T T T T T T X Z X Z X Z X Z X Z R L R L R X Z L R L                    _ _ _{ }         _ _ _              T 1 1 1 2 1 1 1 1 2 1 1 1 1 2 1 2 0 0 0 0 (2.8)

(36)

MODEL SET-UP

18

2.2.2.2 Mass characteristics

The turbines mass matrix consists of mM₃₃ matrices, which are the systems mass and its mass moment of inertia, represented by diagonal terms in the mass matrix. Since the CoG of each body is located in the origin of the local coordinate system, no off-diagonal mass terms are present. Body 2, representing the tower, has additional mass and inertia, and a vertically shifted CoG, due to the RNA. The mass properties are based on data from [13].

, jj , yy yy M M I M M I                      M 1 1 1 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (2.9)

The mass properties of the bodies are determined based on dimensions of the tower and floater as described in Table 2.1. Basic mechanics is used to determine the mass distribution, mass moments of inertia and CoG (Steiner’s rule) of the individual bodies. A discretization of the FWT’s mass and moment of inertia is given in Appendix C.

2.2.2.2.1 Added mass

Accelerations of the floater in water induces forces that are related to added mass. The added mass terms are included via an extra mass term in the mass matrix in the EoM. This implies a linearization of the added mass its frequency-dependence, which is regarded as acceptable based on [13] and Figure 7.5 in Appendix C.

The added mass term is related to the weight of the volume of water that is displaced when the floater accelerates through the water. Due to coupling between surge and pitch, off-diagonal terms are present as well. The definition of added mass as it is included in the model is presented below. The added mass in surge and pitch direction are obtained by using strip theory [5]. For the heave motion the submerged disk method is applied. a12 and a21 are zero, since the spar hull is a vertical and symmetric

body. subm water a a₁₁V  g C (2.10) water bot a   r 3 22 1 8 2 3 (2.11) ( ) ( ) ( ) CoB CoG

CoB CoG CoB CoG

a a z z a a z z a z z                A 11 11 33 22 2 11 11 0 0 0 0 (2.12)    _{ } _   A AM M 33 33 66 66 33 33 0 0 0 (2.13)

(37)

19

2.2.2.3 External forces

In equation (2.1) the vector F is the term that includes all external forces. In the equation of motion these are transformed to forces and bending moments around the CoGs of the bodies, as shown in equation (2.15). All external forces together are:

, ,

ext  hydro static hydro dynamic  mooring gravity

F F F F F (2.14)

The influence of the mooring stiffness and hydrostatic stiffness, and the influence of the hydrodynamic damping, are included as external forces in the model. Where damping and stiffness are related to platform velocity and displacement respectively. The influence of gravity is also included as an external force. The force vectors include a description of the forces and moments around the CoG of the bodies, therefore the force vector has a dimension 1×3M and is defined as:

, , , , ( ) x z x z F F T t F F T                          F 1 1 2 2 (2.15)

 The hydrostatic forces Fhydro,static are represented by the hydrostatic restoring forces due to

buoyancy, as explained in paragraph 2.4.1.

 The hydrodynamic forces Fhydro,dynamic are forces due to the movement of water particles (wave

kinematics) and the movement of the structure through the water. The wave kinematics are

explained in paragraphs 2.4.2-2.4.5. The hydrodynamic forces are explained in detail in paragraph 2.4.6.

 The mooring forces Fmooring are incorporated as a linear stiffness. This is explained in paragraph

2.3.

 The gravity forces Fgravity are obviously downwards forces which are related to the mass times the

gravitational acceleration.

,

gravity i i

F M g (2.16)

 Where i, is the number of the body. Fgravity acts downwards in the CoG of the bodies. Together

with the buoyancy force the sum of the gravity forces of the bodies does account for a restoring moment around the CoG of the whole platform. This restoring moment effect is included via the hydrostatic force, as explained in paragraph 2.4.1.

2.2.3 Structural damping properties

2.2.3.1 Determination of structural damping

The structural damping is accounted for as dashpots between the bodies of the MDOF-model. The damping of the dashpots as provided in Table 2.2 is determined by taking a percentage of the critical damping, called damping ratio. The damping ratio of ζ = 1 % is chosen based on [13], which is the value for welded steel in working stress level. The damping vector is defined as :

(38)

MODEL SET-UP

20



   

B 2 K M (2.17)

2.2.3.2 Implementation of structural damping

The damping of the dashpots that represent the structural damping of the FTA is represented in equation (2.1) by the term T .

damp

dD σ This term is incorporated via the virtual power balance, as explained in paragraph 2.2.2. The relative velocity of the dashpot element is described in terms of the independent generalized coordinates as.

v v( )i v v,i( )i i

v D x v D x x (2.18)

The term _dDT_{in equation (2.1) is the first order velocity difference matrix}

v,i( )i

D x in generalized coordinates, and is obtained by taking the Jacobian of D x_v( )_i , which describes the dashpot constraints.

v( )i X X D x Z Z         _{  }      1 2 1 2 1 2 (2.19) v,i( )i ( ( ), )v i dDD x J D x Q (2.20)

The damping vector B as presented in Table 2.2 composes the damping force vector σdamp, via:

( ) damp D xv i σ B (2.21) X Z B B B            B (2.22)

The forces in the dashpots between the bodies are known once T damp

dD σ is known. Together with the structural springs, which will be explained in the subsequent paragraph, the dashpot forces form the total internal forces at the node.

Table 2.2 – Structural damping

Damping DOF Damping value

BX [N/m/s] 1,22·107

BZ [N/m/s] 3,18·107

BΘ [N/rad/s] 4,71·107

2.2.4 Structural stiffness properties

2.2.4.1 Determination of structural stiffness

The structural stiffness is accounted for as springs between the bodies of the MDOF-model. The stiffness of the springs as provided in Table 2.3 is determined using mechanics on the stiffness of cylinders. An enhanced explanation of the basic mechanics is to be found in [28].

The spring acts in three degrees of freedom. At the inter-connection the spring is incorporated as a constraint in the external force term in the EoM. A single spring is defined as a vector with three spring constants for all 3 DOF:

T

x z

(39)

21 The shear stiffness corresponds with a bar loaded by a force in a direction perpendicular on its longitudinal axis, i.e. a force required to produce a unit deflection.

x GA K

L

 (2.24)

The spring stiffness in vertical direction corresponds with the elongation stiffness of the cylinder. This is defined as: z EA K L  (2.25)

The bending stiffness KΘ is defined as:

xx EI K

L

 (2.26)

The final applied bending stiffness is a factor 2,17 lower than calculated with equation (2.26), in order to get the right tower bending eigenfrequency. This factor is determined via tuning the rotational stiffness to agree with eigenfrequency of 0,44 Hz. Equation (2.26) overestimates the stiffness, because it is based on the bending of a cylinder with constant diameter and wall thickness, while the tower diameter and wall thickness decrease over height. It is important to correctly include the bending stiffness, since this determines the eigenfrequency of for-aft bending of the tower. The correct implementation of the tower stiffness is verified in paragraph 3.3.1.

Table 2.3 – Structural spring stiffness

Spring DOF Stiffness value

KX [N/m] 2,45·108

KZ [N/m] 6,36·108

KΘ [N/rad] 4,71·1010

2.2.4.2 Implementation of structural stiffness

The stiffness of the springs that represents the structural stiffness of the FTA is represented in equation (2.1) by the term _{D σ}T _{. This term is incorporated via the virtual power balance, as explained in paragraph}

2.2.2. The relative displacement of the spring element is described in terms of the independent generalized coordinates as.

v D xv( )i v D x xv,i( )i i

     (2.27)

The term _{D in equation (2.1) is the first order difference matrix}T

v,i( )i

D x in generalized coordinates, and is obtained by taking the Jacobean of D x_v( )_i , which describes the spring constraints.

( ) v i X X D x Z Z      _  _       1 2 1 2 1 2 (2.28) v,i( )i ( , )v D x J D Q   D (2.29)

The spring stiffness vector K as presented in Table 2.2 compose the spring force vector , via:

v

D

 

σ Κ (2.30)

HYDRODYNAMIC BEHAVIOUR OF FLOATING WIND TURBINES Difference between linear versus non-linear wave effects on wave induced fatigue loads in a spar-type floating wind turbine

HYDRODYNAMIC BEHAVIOUR OF FLOATING WIND

TURBINES

Difference between linear versus non-linear wave effects on wave induced

fatigue loads in a spar-type floating wind turbine

Jorrit Roy Bergsma

November 2015

Hydrodynamic behaviour of floating wind turbines

Difference between linear versus non-linear wave effects on wave induced fatigue loads in a

spar-type floating wind turbine

Master of Science Thesis

Author: Jorrit Roy Bergsma

Committee: Prof. dr. ir. R.H.M. Huijsmans

TU Delft Chairman

Dr.ir. A. Romeijn

TU Delft

Dr.ir. S.A. Miedema

TU Delft

Dr.ir. P.L.C. van der Valk

Siemens

Ir. M. Tilanus

Siemens

S

UMMARY

A

CKNOWLEDGEMENTS

TABLE OF CONTENTS

L

IST OF

T

ABLES

L

IST OF

F

IGURES

L

IST OF

A

BBREVIATIONS

1 I

NTRODUCTION

1.1 Relevance of floating wind turbine

1.2 Description of FWT concept

1.2.1 Hywind concept –a spar-type FWT

1.2.1.1 Spar floater

1.2.1.2 Hywind Demo properties

1.3 Previous research on FWTs

1.3.1 Start of modelling floating wind turbines

1.3.2 Platform motions

1.3.3 Modelling of hydrodynamics

1.3.3.1 Hydrodynamic theories

1.3.3.2 First and second order wave effects

1.3.3.3 Morison equation

1.3.4 Mooring line dynamics

1.3.5 Fatigue loads

1.3.6 Wind loads

1.3.7 Structural modelling

1.4 Project formulation

1.4.1 Research questions

1.5 Thesis outline

2 M

ODEL SET

-

UP

2.1 Overview of model

2.2 Structural module

2.2.1 Physical representation

2.2.2 Multibody dynamics applied method

2.2.2.1 State vectors of the FWT system

2.2.2.2 Mass characteristics

2.2.2.2.1 Added mass

2.2.2.3 External forces

2.2.3 Structural damping properties

2.2.3.1 Determination of structural damping

2.2.3.2 Implementation of structural damping

2.2.4 Structural stiffness properties

2.2.4.1 Determination of structural stiffness

2.2.4.2 Implementation of structural stiffness