Juni 2021
Modelling the Resilience of
Offshore Renewable Energy System Using Non-constant Failure Rates
Mussie Abraham Beyene
Masterprogram i förnybar elgenerering
Teknisk- naturvetenskaplig fakultet UTH-enheten
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Ångströmlaboratoriet Lägerhyddsvägen 1 Hus 4, Plan 0
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Box 536 751 21 Uppsala
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018 – 471 30 03
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http://www.teknat.uu.se/student
Modelling the Resilience of Offshore Renewable Energy System Using Non-constant Failure Rates
Mussie Abraham Beyene
Offshore renewable energy systems, such as Wave Energy Converters or an Offshore Wind Turbine, must be designed to withstand extremes of the weather environment.
For this, it is crucial both to have a good understanding of the wave and wind climate at the intended offshore site, and of the system reaction and possible failures to different weather scenarios.
Based on these considerations, the first objective of this thesis was to model and identify the extreme wind speed and significant wave height at an offshore site, based on measured wave and wind data. The extreme wind speeds and wave heights were characterized as return values after 10, 25, 50, and 100 years, using the Generalized Extreme Value method.
Based on a literature review, fragility curves for wave and wind energy systems were identified as function of significant wave height and wind speed. For a wave energy system, a varying failure rate as function of the wave height was obtained from the fragility curves, and used to model the resilience of a wave energy farm as a function of the wave climate. The cases of non-constant and constant failure rates were compared, and it was found that the non-constant failure rate had a high impact on the wave energy farm's resilience. When a non-constant failure rate as a function of wave height was applied to the energy wave farm, the number of Wave Energy Converters available in the farm and the absorbed energy from the farm are nearly zero. The cases for non-constant and an averaged constant failure of the
instantaneous non-constant failure rate as a function of wave height were also compared, and it was discovered that investigating the resilience of the wave energy farm using the averaged constant failure rate of the non-constant failure rate results in better resilience. So, based on the findings of this thesis, it is recommended that identifying and characterizing offshore extreme weather climates, having a high repair rate, and having a high threshold limit repair vessel to withstand the harsh offshore weather environment.
Tryckt av: Uppsala ELEKTRO-MFE 21001 Examinator: Irina Temiz Ämnesgranskare: Jens Engström Handledare: Malin Göteman
Producing power from renewable energy is crucial due to the climate change caused by Greenhouse Gases (GHG) emissions as well as the depletion of the fossil fuels.
The world’s energy source has shifted to renewable energy sources as a consequence of the 1970s oil crisis, the depletion of fossil fuels, and, most importantly, climate change. Because of the energy available in offshore winds and waves, offshore re- newables have a great potential to contribute to the global energy demands. Ocean waves are untapped source of energy, with considerable potential for energy extrac- tion. Compared to more mature renewable energy technologies like wind and solar energy, wave energy technologies have not yet reached a commercial scale, despite the fact that waves often carry more energy and have a lower intermittency. Wind farms have been commercialized for nearly 30 years, and they are now thriving, particularly in Europe.
Offshore renewable energy can be harvested by Offshore Wind Turbine (OWT) and Wave Energy Converter (WEC), but in addition to absorbing the energy effi- ciently, they must also be designed to withstand high wind speeds and high ocean waves.
So, in order to tap this massive amount of offshore renewable energy, weather conditions like extreme wind speed and significant wave height should be modelled and characterized at the location where the farms are installed. The main goal of this paper was to contribute to the development of this untapped offshore renewable energy resource by modeling and characterizing the aforementioned weather climate using extreme statistical modeling, as well as examine how the wave energy farm performed if this severe weather environment causes various failures.
First, my sincere gratitude goes to the Almighty God for sustaining me this far in pursuit of my life.
No words could adequately express my gratitude to the Global Electricity Part- nership organization (GSEP), headquartered in Canada, for funding my studies here in Sweden. I would also like to thank Uppsala University for supporting me in my education from the beginning to the end of this master’s degree.
Furthermore, I would like to thank Dr.Malin Göteman, my supervisor, for her close mentoring and assistance. I believe that finishing this thesis would be incred- ibly challenging without her close assistance. I would also like to thank Dr.Jens Engström, my subject reader, for his cooperation.
I would also like to express my gratitude to my program coordinator, Dr. Irina Temiz, for her moral and intellectual guidance. In addition, I would like to express my thankful to my study counsellor, Dr. Juan de Santiago, for his support and advice during my graduate studies.
Finally, I would like to express my heartfelt gratitude to my family and friends, who have always been there for me.
WECs Wave Energy Converters WEC Wave Energy Converter
WT Wind Turbine
WTs Wind Turbines
OWT Offshore Wind Turbine
PTO Power Take-Off
OWC Oscillating Water Columns
PV Photovoltaic
OTS Overtopping Systems
PA Point Absorbers
OB Oscillating Bodies
MWh megawatt-hour
MW megawatt
UU WEC Uppsala University Wave Energy Converter RD Research and Development
BBDB Backward Bent-Duct Buoy
CWR Capture Width Ratio
VAWT Vertical Axis Wind Turbine HAWT Horizontal Axis Wind Turbine
GHG Greenhouse Gases
CDF Cumulative Distribution Function
EVT Extreme Value Theory
POT Peak Over Threshold
BM Block Maxima
GEV Generalized Extreme Value
iid independent and identically distributed GEVD Generalized Extreme Value Distribution NDBC National Data Buoy Centre
IGBT Insulated Gate Bipolar Transistor NREL National Renewable Energy Laboratory FEA Finite Element Analysis
FAST Fatigue Aerodynamics Structure Turbulence CAE Computer Aided Engineering
pdf Probability Density Function
ICDF Inverse Cumulative Distribution Function OREDA Offshore Reliability Data
SWH significant wave height
ISO International Organization for Standardization
1 INTRODUCTION 2
1.1 Problem Statement . . . 3
1.2 Objectives . . . 3
1.3 Thesis Outline . . . 3
2 THEORY 4 2.1 Wave Energy Technology and Farms . . . 4
2.1.1 Wave Energy Technology . . . 4
2.1.1.1 Oscillating Water Columns (OWC) . . . 4
2.1.1.2 Oscillating Bodies (OB) . . . 5
2.1.1.3 Overtopping Systems (OTS) . . . 6
2.1.2 Wave Energy Farms . . . 6
2.2 Wind Energy Technology and Farms . . . 7
2.2.1 Wind Turbine Technology . . . 7
2.2.2 Wind Energy Farms . . . 9
2.3 Offshore Survivability and Risk Analysis . . . 9
2.4 Threat characterization . . . 10
2.4.1 Generalized Extreme Value (GEV) Method . . . 10
2.4.2 Block Maxima (BM) Method . . . 11
2.5 Vulnerability Assessment of the System . . . 12
2.5.1 Vulnerability of Wind Energy Systems . . . 12
2.5.2 Vulnerability of Wave Energy Systems . . . 16
3 METHODOLOGY 17 3.1 Threat Characterization . . . 17
3.2 Wave Farm Availability . . . 17
3.2.1 Absorbed Power from the Wave Energy Farm . . . 18
3.2.2 Numbers of Active Wave Energy Converters (WECs) with Constant Failure Rate . . . 19
3.3 Failure rates as function of wave height . . . 20
4 RESULT AND DISCUSSION 25 4.1 Threat Characterization . . . 25
4.1.1 Fitting of Maxima Dataset to Weibull Distribution . . . 25
4.1.2 Data Analysis with Cumulative Distribution Function . . . . 26
4.1.3 Return Levels of the Hazards at Different Return Periods . . . 27
4.1.4 Quantitle-Quantile Model Fitness Check . . . 29
4.2 Wave farm availability . . . 29
4.2.1 Number of Active WECs for Constant Failure Rate . . . 30
4.2.2 Absorbed Power for Constant Failure Rate . . . 33 4.3 Failure rates as function of wave height . . . 36
4.3.1 Resilience of Wave Energy Farm Using Varying Failure Rate with Constant Random Failure Rates . . . 36 4.3.2 Comparison the Resilience of the Wave Energy Farm Using
Varying Failure Rate and its Average Constant Failure Rate . 38
5 CONCLUSION AND FUTURE WORK 43
5.1 Conclusion and Limitations . . . 43 5.2 Future work . . . 43
References 45
Appendices 48
A Appendices 49
1.1 Steps for WEC design[4] . . . 2
2.1 Oscillating Water Columns (OWC) WEC[1] . . . 5
2.2 Uppsala University Wave Energy Converter (UU WEC)[1] . . . 5
2.3 Yearly increase of turbine size [19] . . . 8
2.4 Power curve of wind turbine [18] . . . 8
2.5 Wind speeds versus average failure rate[33] . . . 14
2.6 Offshore generator failure(Solid) versus onshore generator failure(grey)[33] 14 2.7 Different approaches of fragility curve versus wave height [34] . . . 15
2.8 3D fragility curve surface of foreaft overturning bending moment in response to both wind speed and wave height [34] . . . 15
3.1 Rectangular wave energy farm representation. The total number of WECs is calculated by multiplying the number of WECs in each row by the number of rows. . . 19
3.2 Bathtub curve [40] . . . 21
3.3 Analogous function used to fit fragility curve of WECs . . . 22
3.4 Relationship between time steps in one year and significant wave height 23 3.5 Instant failure rate as a function of significant wave height . . . 24
4.1 Block Maxima of extreme wind speed and significant wave height . . 26
4.2 Probability Density Function (PDF)of wind speed and significant wave height . . . 26
4.3 Cumulative Probability of maxima significant wave height . . . 27
4.4 Cumulative probability of maxima wind speed . . . 27
4.5 Return level of signifcant wave height . . . 28
4.6 Return level of extreme wind speed . . . 28
4.7 Return values of both hazards at different return periods . . . 29
4.8 Quantile-Quantile Plot of Maxima Wind Speeds . . . 29
4.9 Number of WECs when failure rate is λ = 0.22/year and a repair rate of µ = 365/year . . . 30
4.10 Number of WECs when failure rate is λ = 11/year and a repair rate of µ = 2190/year . . . 31
4.11 Number of WECs when failure rate is λ = 2.5/year and a repair rate of µ = 4.3/year . . . 32
4.12 Number of WECs when failure rate is λ = 11/year and a repair rate of µ = 4.3/year . . . 33
4.13 Absorbed power when failure rate is λ = 0.22/year and a repair rate of µ = 365/year, comparably to Figure 4.9. The annual absorbed
energy is 20.18 GWh. . . 34
4.14 Absorbed power when failure rate is λ = 2.5/year and a repair rate of µ = 4.3/year, related to Figure 4.11. The annual absorbed energy is 13.86 GWh. . . 35
4.15 Absorbed power when failure rate is λ = 11/year and a repair rate of µ = 4.3/year. The annual absorbed energy is 6.46 GWh. . . 35
4.16 Resilience of wave energy farm in relation to varying failure rate as a function of the wave height and a repair rate of µ = 365/year . . . 37
4.17 Resilience of wave energy farm in relation to varying failure rate as a function of the wave height and a repair rate of µ = 4.3/year . . . 37
4.18 Resilience of wave energy farm in relation to varying failure rate as a function of the wave height and a repair rate of µ = 2190/year . . . . 38
4.19 Relationship between significant wave height (SWH),varying failure rate and its average constant failure rate . . . 39
4.20 Comparison of operational number of WECs using a varying failure rate as a function of the wave height and using its average constant failure rate of λ = 0.0678/year with a repair rate of µ = 2190/year . . 40
4.21 Comparison of absorbed energy using a varying failure rate as a func- tion of the wave height and using its average constant failure rate of λ = 0.0678/year with a repair rate of µ = 2190/year . . . 40
4.22 Comparison of operational number of WECs using a varying failure rate as a function of the wave height and using its average constant failure rate of λ = 0.0678/year with a repair rate of µ = 4.3/year . . . 41
4.23 Comparison of absorbed energy using a varying failure rate as a func- tion of the wave height and using its average constant failure rate of λ = 0.0678/year with a repair rate of µ = 4.3/year . . . 41
A.1 Backward Bent-Duct Buoy (BBDB) prototype at Galway Bay,Ireland[6] 49 A.2 100 second time window of load voltage,speed of the translator,position of the translator, and three-phase power output[8] . . . 50
A.3 Monopile Wind Turbine (WT),gravity base WT,Jacket type wind tur- bine WT[1] . . . 51
A.4 Detailed subsystem of WECs [2] . . . 51
A.5 Maximum thresholds of significant wave height and wind speed for 30days . . . 52
A.6 Absorbed power when very high failure rate of λ = 11/year and a high repair rate of µ = 2190/year were applied. The annual absorbed energy is 18.99 GWh. . . 53
A.7 Wave Energy Converter (WEC) Technologies[5] . . . 54
A.8 Backward Bent-Duct Buoy (BBDB) WEC[6] . . . 54
A.9 Uppsala University WEC at Swedish coast,Lysekil[10] . . . 55
A.10 Wave dragon Overtopping Systems (OTS)[11] . . . 55
A.11 a)Vertical Axis Wind Turbine (VAWT) b) Horizontal Axis Wind Tur- bine (HAWT) [1] . . . 56
A.12 Main components of Wind Turbine (WT) system [16] . . . 56
A.13 Greater details of Horizontal Axis Wind Turbine (HAWT) compo- nents [17] . . . 57 A.14 5 MW offshore wind turbine model studied at National Renewable
Energy Laboratory[34] . . . 57 A.15 Foreaft towertop deflection fragility surface with respect to wind speed
and wave height[35] . . . 58 A.16 Foreaft bending moment fragility surface with respect to wind speed
and wave height[35] . . . 58
4.1 Resilience of wave energy farm using constant failure rate and repair rate . . . 36 4.2 Resilience of wave energy farm using non-constant failure rate as a
function of wave height and a constant repair rate . . . 38 4.3 Resilience of wave energy farm using non-constant failure rate as a
function of wave height and its average constant failure rate . . . 41
INTRODUCTION
Coal, petroleum, and natural gas continue to be the leading energy sources [1].
However, the concern of fossil fuel depletion in the future, along with concerns over climate change and over national energy protection when fossil fuel concentrations are limited to certain areas, has compelled the quest for renewable energy resources [1]. The European Commission aims to reduce emissions of polluting greenhouse gas by 85% to 95% by 2050 compared to 1990 levels of greenhouse gas emissions [2].
Renewable energy is generated from natural cycles that replenish themselves on a regular basis, such as wind, waves, solar, geothermal, biomass [1]. Wind energy has been identified as one of the viable alternatives to the global climate change and energy crises, as well as a potential sustainable, safe, and reliable energy source. In addition, the world’s oceans hold enormous amounts of wave energy that, if properly used, could continue in solving the world’s energy crisis [1]. Offshore wind, wave, and tidal energy in the US is estimated at 8.75 billion megawatt-hour (MWh)(800 million households) per year [3]. As of the moment, the majority of offshore wind farms have already been installed in Europe, and the associated energy prices are getting lower, but they remain above that of Photovoltaic (PV) systems. The degree of maturity of wave and tidal technologies is less, with fewer than 1000 megawatt (MW) of installed capacity. For the design of wave and wind energy devices, long-term distributions of environmental parameters such as wave height, wave period, and wind speed are needed. Long-term knowledge is required for assessing risk of failure due to the environmental loads. Wave and tidal energy are not commercially deployed due to their high initial costs [3]. In figure 1.1, the WEC design procedure has been addressed.
Figure 1.1: Steps for WEC design[4]
1.1 Problem Statement
Because of the high weather climates, such as high wind speeds and significant wave heights, offshore sites have a high source of energy. In order to ensure the resilience of offshore energy farms, it is necessary to model and characterize the weather cli- mate when considering harvesting this untapped offshore renewable energy resource.
As a result, this study provided at least a preliminary basis for the extraction of this powerful green renewable energy by identifying weather climate extremes and modelling the resilience of wave energy farms. This problem was solved by using the concept of extreme statistical methods to model and characterize the extremes of wind speed and significant wave height, as well as using the constant and non- constant failure rates as a function of wave height to model the resilience of wave energy farms in terms of the number of WECs available in the farm and its absorbed power.
1.2 Objectives
The thesis aims broadly to model and characterize the resilience of offshore renewable energy systems. It aims to accomplish three main objectives. The first objective is to model and characterize extreme wind speeds and significant wave heights threats at an offshore site, based on measured wind and wave data. The second objective is to identify both wave power and wind power plants’ vulnerability in terms of fragility curves, based on a literature study, and to translate this vulnerability to a non-constant failure rate that is a function of the wave height. The third objective is to model the resilience of a wave farm, in terms of the number of available WECs, as a function of the wave climate and the varying failure rate.
1.3 Thesis Outline
In Chapter 1, the study started with a brief introduction. Chapter 2 discussed a theoretical description of wave energy technology and farms, wind energy tech- nology and farms, offshore survivability and risk analysis. In Chapter 3, many of the methodologies that were used to accomplish the goals were presented. The ob- jective’s results and discussion were presented in Chapter 4 . Finally, Chapter 5 concluded the thesis findings.
THEORY
2.1 Wave Energy Technology and Farms
This section discusses WEC technology and a wave energy farm. A theory of how WEC technology works is discussed, as well as its classification and the general theory of the wave energy farm are discussed.
2.1.1 Wave Energy Technology
Wave Energy Converter (WEC) technologies can be categorized based on their oper- ating principles, location (shoreline, near-shore, or offshore), and size (point absorber versus large absorber) [5]. Wave energy technologies can be classified into three types based on their working principles [1]: Oscillating Water Columns (OWC), Overtop- ping Systems (OTS), and Oscillating Bodies (OB).
Similarly, as shown in the figure of Appendix A.7, the paper [5] outlined the classification of wave energy technology based on its working principle.
2.1.1.1 Oscillating Water Columns (OWC)
The Oscillating Water Columns (OWC) as shown in the figure of Appendix A.7 can be either fixed or floating. The fixed form of OWC has a concrete or steel frame immersed in water open to the sea. This is referred to as a collector [1]. As incident waves enter and exit the water column, an air trapper column above the water column directs air into a turbine, which drives a coupled electrical generator.
This concept is depicted in Figure 2.1, and it is the first technology known as first generation wave energy technologies [1].The fixed OWC is installed on or near the shoreline, and the wave energy absorbed decreases as the wave climate decreases as it approaches the shoreline. The wave-powered navigation buoy and the big kaimei barrage are examples of floating OWCs built for deep water and developed by Yoshio Masuda in Japan [5]. Later, the geometry of the floating OWC was modified to the Backward Bent-Duct Buoy (BBDB), which consists of an L-shaped duct, a buoyancy chamber, an air chamber, and a Power Take-Off (PTO) (air-turbine with generator), as shown in the figure of Appendix A.8 [6].
The BBDB has a high Capture Width Ratio (CWR). Aside from that, the mooring cost and required mooring force are low due to the slow forward movement against the wave direction. The usable amount of wave energy that has been fully
Figure 2.1: Oscillating Water Columns (OWC) WEC[1]
absorbed and captured by WECs is referred to as CWR. It is represented mathe- matically as a ratio of absorbed power (P in kW) to wave resource (J in kW/m) [7]. As a result, the unit is meter (m). The BBDB prototype, which was tested for three years at sea in Galway Bay, Ireland, is included in Appendix A.
2.1.1.2 Oscillating Bodies (OB)
The vast majority of offshore WEC technologies are Oscillating Bodies (OB). The wave energy potential at deep water offshore location is large [5], but drawbacks include expensive maintenance and requirement of long underwater electrical cables for transmission. One kind of OB is the Point Absorbers (PA) WEC. The point absorber’s buoy has a horizontal size that is much smaller than the incident wave length [1]. Uppsala University Wave Energy Converter (UU WEC) is a single-body PA that uses heave motion (up and down) between a buoy on the ocean floor and a fixed linear generator on the seabed. As shown in 2.2, they are connected by a tight line that allows movement between them.
Figure 2.2: Uppsala University Wave Energy Converter (UU WEC)[1]
The translator’s springs assist in restoring force during the trough cycle of the wave and smoothing power generation by storing energy during the crest cycle of the wave [8]. The translator goes up as the buoy is pulled by the incoming wave. Ac- cording to Faraday’s law (because there are permanent magnets on the translator), this interaction produces pulsating output voltage at the linear generator voltage’s stator, and some of the energy is retained as mechanical energy in the spring. The accumulated energy is then induced in the stator during the wave trough as the spring pulls the translator down [9]. Appendix A.2 contains a 100-seconds time window of voltage at the load, total three-phase power output, speed of the transla- tor ,and position of the translator [8]. According to reference [10], which is depicted in the figure of Appendix A.9, researchers from Uppsala University deployed various buoys with their linear generator at Lysekil on Sweden’s west coast.
2.1.1.3 Overtopping Systems (OTS)
Overtopping Systems (OTS) are submerged WEC with a reservoir to hold ocean waves, and the kinetic energy of the wave is converted to potential energy. When the reservoir is complete, the water returns to the ocean and strikes the water turbine as it moves, producing electricity [11]. The reservoir can assist in smoothing out power generation [1]. The figure in Appendix A.10 depicts floating Overtopping Systems (OTS).
In general, as overall failure rate of a WEC is the sum of the failure rates of its individual subsystems. The WEC subsystem, which is comprised of common components to all types of WECs such as mooring, structure, PTO, and power transmission, as a concise four part sub system to aid comprehension of Section 3.2.2 [12]. The subsystem is presented in Appendix A.4, as noted in reference [2].
2.1.2 Wave Energy Farms
Following the end of the oil crisis in the 1980s, the pace at which harvesting energy from ocean waves slowed. However, it grew rapidly after the European Commission began funding the wave energy industry in their Research and Development (RD) policy in the 1990s [1]. The change in pressure caused by the sun’s heating causes winds to blow over a long stretch, causing ocean waves to be created [1]. Wave energy can be conceived of intuitively as a concentrated and dense form of solar energy [13]. To capture energy from a wave, a structure that can respond appro- priately to the force exerted by the waves must explicitly interfere with the wave [14]. The incoming wave moves the buoy, but the body fixed to the sea bed does not. Controlling such reaction forces increases the amount of energy extracted from ocean waves. According to reference[1], the wave power produced by ocean waves is usually expressed in terms of power per unit length, and it ranges between 20 and 70 kW/m2 depending on the latitude of the offshore site. A wave energy farm consists of a large number of WECs, and the hydrodynamic interactions between them tend to balance out the fluctuating power production [15].
2.2 Wind Energy Technology and Farms
The theoretical aspects of wind turbine technology and wind energy farms are dis- cussed in this section.
2.2.1 Wind Turbine Technology
A Wind Turbine (WT) captures kinetic energy of the wind with its air foiled shaped blades and convert it into rotational mechanical energy of the wind turbine ro- tor, which is then converted into electrical power by a synchronous or induction generator located on top of the WT tower [16]. The wind turbine rotor powers a low-speed shaft, which is stepped by a gear box and the generator is then powered by a high-speed shaft due to the increased rotational speed by gearbox [17]. The shaft, generator, and gearbox are all housed within a nacelle.Wind Turbine (WT) technologies are divided into two types based on their operating rotor principle:
Vertical Axis Wind Turbine (VAWT) and Horizontal Axis Wind Turbine (HAWT).
The VAWT has a vertical rotor shaft, and the advantage of this technology is that the wind direction has little impact on the electricity production. As a consequence, when the wind direction is highly variable, the VAWT is recommended [1].
The HAWT’s rotor blade shaft and electric generator are situated at the top of the wind turbine tower. The most common number of propeller blades used is three, and they should be oriented in the same direction as the wind. When the wind blows, the blade propeller lifts and rotates at a slower speed. Besides, the rotor blade is tapered to optimize kinetic energy[1]. HAWT is more widely used on the market because it has a high efficiency turbine, a high power density, a low cut-in wind speed and a low cost per unit power output [18]. Both technologies are depicted in the figure of Appendix A.11 [1].
Wind turbine blades, a gearbox, a generator, a power electronics converter, a power transformer, and an electric grid substation are the main components of a wind energy system, as illustrated in in the figure of Appendix A.12 [16].
The tower, nacelle, rotor hub, wind speed sensors, pitch drives, and pitch yaws are among the other mechanical components mentioned in reference [19]. Wind speed sensors provide information on wind speed and direction, and the tower, nacelle, and rotor hubs mechanically support the rotor blades. The figure in the Appendix A.13 also depicts the HAWT in greater depth [16].
As shown in equation 2.1 the power output of the turbine is directly proportional to the square of the rotor diameter and the cubic of the wind speed [17].
Pwind = 1
2ρAv3 (2.1)
where ρ is the density of the air, A is the swept area of the rotor, and v is the instantaneous wind speed.
Installation of a large wind turbine diameter is preferable to installation of a small wind turbine diameter due to the simplicity of its economy and technology [19]. As a result, the scale of the wind turbine is increased on an annual basis, as illustrated in figure 2.3 [19].According to reference [20], turbines are as large as football stadiums. Appendix A.3 contains details on various wind turbine support
structures, such as monopile structures, gravity-based structures, and jacket support structures [1].
Figure 2.3: Yearly increase of turbine size [19]
Using a power curve, it is often simpler to visualize the output power versus wind speed. Figure 2 depicts the wind turbine power curve, which is a graph of power output versus mean wind speed [18]. The output power begins to be achieved when the minimum threshold level, known as the cut-in speed, is reached. And, as the speed increases from the cut-in speed to the rated speed, the output power increases (the maximum speed that it can produce power). Then, if the mean wind speed approaches the rated speed, even though the wind speed is increased, no output power is produced due to the power control being enabled. Finally, when the wind speed reaches the cut-out speed, the wind turbine comes to a halt and is shielded from damage [18].
Figure 2.4: Power curve of wind turbine [18]
2.2.2 Wind Energy Farms
Knowledge of wind characteristics aids in wind turbine design, the creation of wind measurement techniques, and, finally, the selection of a suitable wind farm location [18]. Wind turbines are clustered together to form a wind park, and the construction of a wind farm offshore is expensive due to the need for a strong foundation and submarine cable to carry power to the shoreline electric substation [19]. Though the wind turbine concept dated back to the 18th century, the concept of developing a wind farm offshore was first suggested in 1930 and first introduced in 1972 by Dr.
William E Heronemus, a professor at the University of Massachusetts [1]. The wind speed is stronger in offshore, and the speed between the cut-in speed (the minimum speed at which electricity can be produced) and the rated power (the secure max- imum limit speed) increases wind power output [21]. Offshore deployment, on the other hand, is often more costly, with some studies saying that the operation and maintenance of offshore wind turbines accounts for 30% of total energy costs [21].
Reference [19] estimates that the capacity of offshore wind farms could be 40 GW by 2020, while the cumulative wind capacity is 760 GW by 2020.
2.3 Offshore Survivability and Risk Analysis
Failure rate is defined as the number of times the component or system fails per unit of time [2]. The failure rate is primarily measured in failures per year or in failures per million hours due to the small numbers they generally exhibit. If, for example, the component failure rate is 0.05 per year, that component is expected to fail once every twenty years [2]. Reliability data is either not available due to limited field knowledge or is maintained confidential, ensuring competitive advantages and intellectual property in the context of different project developments [12]. Different failure rates for Wave energy components are given in the reference [2], and there are different failure rates for different components in the handbook of oil company handbook reference [22].
Risk is always connected with what may take place in the future [23]. According to reference [23], the response to the following three questions defines risk analysis:
“what can go wrong?”, “what is the probability of that happening?”, “what are the consequences?”. The extreme high wave heights and high wind speed, respectively, can be what can go wrong for wave power parks and wind farm parks.
Natural hazards or threats are devastating natural calamities that endanger lives, environment and infrastructure. These natural hazards are defined by their location, degree of severity and frequency of occurrence [24]. Examples of severe weather may be severe hurricanes, strong winds, exceptionally large waves, or extremely high temperatures, etc.
Resilience is the capacity to prepare and adapt to changing circumstances and to withstand and rapidly recover from disturbance [24].
Some authors have merged threat analysis and system topology into one coherent study and is called system fragility analysis. This system fragility analysis allows us to understand the post-disaster capability of the system after a catastrophic incident by determining which components will always fail and which will continue to function [24].
Research groups have merged threat study, system topology, and system oper-
ation into a single study. This can be referred as serviceability evaluation. This permits one to estimate the ability of the system to meet power supply demand over time from the wind power farm or wave power farms [24].
2.4 Threat characterization
Long-term extreme wind speed or extreme high wave height are estimated using the Generalized Extreme Value (GEV) method. This branch of statistics deals with the upper and lower tails of the probability distribution. It is divided into two methods, Peak Over Threshold (POT) and Block Maxima (BM) methods. The definition and discussion of BM will be presented in Section 2.4.2. The POT technique fits a distribution over a threshold to the peaks of grouped excesses, the excesses being the data minus the threshold in a group, and measuring return values considering the speed of cluster occurrence [25]. The greatest drawback of the POT method is that it is sensitive to select a threshold value [26]. The BM method, on the other hand, contains ambiguity by missing a lot of useful data in each block of the chosen range [26].
The International Organization for Standardization (ISO) 19901–1 sets out guide- lines for the estimation of return periods not more than 4 times longer than the length of the data set when determining return values for the design of offshore structures [27]. It means that the time series data taken from the National Data Buoy Centre (NDBC) should have at least a quarter-long record of data when com- pared to the n-year return period [26]. The buoy measurement or hindcast model can be used to obtain a time series of long-term weather conditions. In order to use the extreme statistical method to predict the weather condition of the n-year return period, the designers must find a high quality data source with appropriate time series [26]. The risk of extreme weather conditions should be taken into account when designing coastal structures, engineering marine energy conversion systems, scheduling vessel transportation routes and maximizing public safety in the coastal zone [27]. The construction of ship, offshore and coastal structures requires a strong knowledge of the most extreme wind and wave conditions that they will have to experience throughout their lifetime [28].
2.4.1 Generalized Extreme Value (GEV) Method
GEV Method is used to predict extreme weather conditions, but the uncertainty of the estimates increases as the duration of the return period year gets longer [27].
The most difficult work for marine and offshore researchers is to describe the severe weather conditions during the n-year return period. As the n-year return duration is well beyond the historical data reported, the researcher uses the Extreme Value Theory (EVT) to extrapolate the n-year return weather situation, which will, of course, have some uncertainties [26]. The Cumulative Distribution Function (CDF) of Generalized Extreme Value Distribution (GEVD) is shown in Equation 2.2 [29].
G(z) = (
expn
−1 + z−µσ −1 o
: 6= 0 exp −exp z−µσ
: = 0 (2.2)
where µ is the location parameter, σ is the scale parameter, is the shape parameter, z is the maxima wave heights or wind speeds (in this thesis)
When the shape parameter, , is zero, positive, and negative, the Generalized Extreme Value Distribution (GEVD) is known as Gumbel (Type I), Frechet (Type II), and Weibull (Type III) distributions, respectively.
2.4.2 Block Maxima (BM) Method
One common approach to extreme value analysis is to fit a parametric model, such as the GEV model, using the BM approach [30]. The Block Maxima (BM) method is a classification of the theory of extreme value, which divides the observations into blocks of equal size and non-overlapping, and from each divided block takes a maximum observation [31]. Block sizes of one year are common, but the findings can be sensitive to block size. In the paper [30], various block sizes are being tried to investigate this. In addition, the BM method is implicitly believed to be independent and identically distributed (iid). In the sense of climate change, this method would presume that extremes can be considered stationary during each time frame, i.e.
that extremes are stationary during the 30-year reference period and the 30-year prediction period. If the influence of any long-term trend is small relative to the other variability, this may not be a very unrealistic estimate [30]. The GEVD fits every block maxima dataset in the BM method [32].
The following are some of the reasons why only the BM extreme method can be used [31]:
• Maximum observation may only be available, such as one-year maximum ob- servation only.
• Where the observations are not entirely iid, the block maximum approach might be preferable.For instance, annual periodicity may exist in the case of yearly maxima, or short-range dependency may exist within blocks but not between blocks.
• The BM approach is easier to enforce because block duration values, such as annual data, naturally exist in several instances.
In order to use the BM extreme method to estimate a 50 year return period of extreme significant wave height, a minimum of 20 year recorded data from the buoy station should be used in accordance with reference [26]. The Equation for BM is represented by Equation 2.3
Ln = M ax (L1, L2, L3, , , , , Li) (2.3) where L1, L2, . . . .Li is a long series data of n observations. The Li are typically used to represent values of a process that are measured on a regular basis, such as hourly over a course of a year.
Maxima values, Ln,can be linearly renormalized as equation 2.4 in order to avoid the incorrect fitting distribution of Ln[25]. The bnand an are the scale and location parameter of the GEVD method, respectively.
Ll = Lm− bn an
(2.4)
Both the sequence constants an and bn should be positive. If and only if we make the right choices of an and bn, the location and scale of the renormalized value, Ll, will become more stable as the number of observations, n, increases [25].
After blocking the series of independent series data L1, L2. . . .Li into annually maxima blocks Lm1, Lm2. . . .Lmi , the fitting quantile distribution of the maxima block data set can be obtained by inverting equation 2.2 as shown in equation 2.5 [25].
Zp = µ − σ 1 − {− log (1 − p)}−
: for 6= 0
µ − σ log {− log (1 − p)} : for = 0 (2.5) where G (Zp) is 1 − p, Zp is the return level of the hazards. Zp is the return level associated with the 1p return period [25]. The Zp level is expected to be exceeded on average once every 1p year [25]. More precisely, Zp is exceeded by the annual maximum in any given year with the probability, p [25]. Equation 2.5 is written in terms of yp = − log (1 − p)and shown in Equation 2.6 to draw logarithmic scale plot of zp with − log (1 − p)) [25].This makes the tail of the distribution compressed, allowing for the display of return level estimates for long return periods.
Zp = µ − σ 1 − yp−
: for 6= 0
µ − σ log yp : for = 0 (2.6)
When zp is logarithmically plotted against yp, or similarly if zp is plotted against logyp, then the plot is linear in the case of µ = 0. If µ < 0, the plot is convex, with an asymptotic limit as p approaches zero at µ − σ ; if > 0, the plot is concave, with no finite bound [25]. This is called return level plot [25]. Return level plots are particularly useful for both model presentation and validation because of their ease of interpretation and the fact that the scale chosen compresses the tail of the distribution, highlighting the extrapolation effect [25].
2.5 Vulnerability Assessment of the System
This section discusses the damage states of each system’s vulnerable component by reviewing different fragility curves’ literature reviews and evaluating their failure probabilities in relation to wind speed and wave height.
2.5.1 Vulnerability of Wind Energy Systems
The more fault there is in offshore wind energy farm, the more material and la- bor needed to repair it, so wind farm designers tend to use wind turbines with a low failure rate and limited maintenance [33]. A failure is defined as a visit to a turbine place beyond a regular allotted service time and leading to material con- sumption. The word ’material’ covers everything used in the turbines, including the consumables (such as carbon brushes) to new parts in the turbine (such as complete Insulated Gate Bipolar Transistor (IGBT) units and full generators) [33].
λ = PI
i=1
PK
1 ni,k/Ni PI
i=1Ti/8760 (2.7)
where λ is failure rate per turbine in one year, Ti is the total time period in hours, K is number of subassemblies, Ni is the number of turbines, I is the number of intervals for the data are collected. The cumulative number of failures per year for each turbine is seen in Equation 2.7 [33]. The numerator is the cumulative number of the failures of the turbine across all periods. The overall number of all periods in hours is the denominator, separated by hours in one year.
The failure rate and repair rate can also be expressed as seen in equation 2.8 and 2.9.
λ = 1
MTTF (2.8)
where MTTF is the Mean Time to Failure.
R = 1
MTTR (2.9)
where MTTR is the Mean Time to Repair (downtime during maintenance).
The failure rate is sometimes misunderstood as a probability of failure, but it should be understood as the severity or frequency of failure [2].
The most common causes of failure for offshore wind turbines are pitch and hy- draulic systems. About a 13% of all system failures are due to pitch and hydraulic systems. Other components is the second most common cause of faults, accounting for almost 12.2% of all faults. The ‘Other Components’ group includes failures of auxiliary parts such as lifts, ladders, hatches, door seals, and nacelle seals. The generator, gearbox, and blades are the third, fourth, and fifth most common causes of offshore failure, with 12.1%, 7.6%, and 6.2%, respectively [33]. The generator is the largest contributor to major turbine repairs category. Oil and valve faults account for about 30% of all pitch/hydraulic failures, with actuator, sludge, and pump fixes and replacements accounting for the remaining 20%. Failures such as spills, unscheduled oil switches, and unscheduled oil top-ups are also examples of oil problems. Sensor failures and leakage are common sludge problems. The majority of valve, accumulator, and pump problems are overcome by replacing the valve, ac- cumulator, and pump [33]. With roughly 25%of failures, door hatch and skylight problems are the most common contributors to the ‘other components’ failure cat- egory. Covers, bolts, lights, and lift replacements include the top five problems, each accounting for around 5% of the total failure rate in this category [33]. With about 31% of all failures in the generator failure category, slip ring problems are the most common cause of generator failures [33]. Bearing concerns, problems with the generator grease pipes, rotor problems, and fan replacements are the remaining four items in the top five.
Offshore wind turbines that are subjected to strong wind speeds, according to reference [33], have a higher failure risk and is illustrated in Figure 2.5. To analyze the difference between the offshore and onshore, the considered offshore wind speed is 8.2 m/s and the considered onshore wind speed is around 6.3 m/s.
The probability of an offshore generator failing is almost eight times that of an onshore generator failing and is depicted in Figure 2.6, according to the paper [33]. According to reference [33], the severe high wind speeds at offshore locations, the difficulties of accessing offshore locations, the higher rating capacity of offshore turbines, and the harsh climate will all contribute to an offshore risk failure being greater than an onshore failure. Outside nacelle, towers and blades are exposed to
Figure 2.5: Wind speeds versus average failure rate[33]
the rough offshore climate. Despite the fact that the power electronics converters and generator are well-sealed within the nacelle, they can also be subjected to extreme conditions when maintenance is performed [33]. According to reference [33], by the third operating year of a wind farm, the average failure rate for an offshore wind turbine rises to about 10 failures per turbine per year. Minor fixes account for 80%
of the total, major repairs for 17.5%, and major replacements for 2.5%. The cost of
Figure 2.6: Offshore generator failure(Solid) versus onshore generator fail- ure(grey)[33]
operating and maintaining offshore wind farms is currently 6 million EUR/year, or 25% to 28% of total costs.
The figure in Appendix A.14 shows a 5 MW offshore wind turbine with a monopile foundation built on a model from the National Renewable Energy Labo- ratory (NREL), with dispersed wind and wave loading, and a vulnerability analysis in terms of fragility behavior was performed for this turbine in reference [34]. For the turbine’s design load case, a 50 year return period of wind and wave loading was also investigated [34].
Figure 2.7 depicts various methods for estimating the fragility curve of an off- shore wind turbine as wave height increases. The methods used in reference [34]
were Finite Element Analysis (FEA) using SAP2000 software, Fatigue Aerodynam- ics Structure Turbulence (FAST) using Computer Aided Engineering (CAE) and Morison’s equations. Figure 2.7 shows that the failure probability for FEA, FAST,
Figure 2.7: Different approaches of fragility curve versus wave height [34]
and Morison’s equation at a wave height of 25 m is almost 10%, 50%, and 70%, respectively. As a result, compared to the other approaches, the FEA approach has less bending moment (failure probability) above the mudline of the monopile foundation (failure probability). The reason that FEA has a lower failure probabil- ity is not because it is the best method, but because each method’s offshore wind turbine modeling and other specifications are different. According to reference [34], the FAST method was nominated as the best approach for offshore wind turbine vulnerability analysis for extreme wind speed and extreme significant wave height for the reason that the FAST approach data was validated with experimental data and had a fragility curves among the other approaches.
Figure 2.8: 3D fragility curve surface of foreaft overturning bending moment in response to both wind speed and wave height [34]
Figure 2.8 illustrates the overturning bending moment above the mudline base in relation to wind speed and wave height using the FAST method, according to reference [34]. When the speed is 30 m/s and the wave height is 15 m/s, the fail- ure probability is approximately 68%. The extreme wave height at the target sit (Nantucket Sound, USA, Massachusetts) is 3.5 m, and it has a nearly 50% failure probability when the wind speed reaches 28 m/s, as shown by the dotted line.
According to reference [35], the blade dip deflection has a failure probability of 99% at a extreme wind speed of 75 m/s and an extreme wave height of 20 m, whereas the overturning bending (flexural) moment from the mudline has a failure probability of 98%.
The figure in Appendix A.15 shows a three-dimensional analysis of the fore-aft tower tip deflection fragility versus the independent variables of wind speed and wave height. When the wind speed is kept constant at 5 m/s and the wave height is increased from 1 m to 20 m, the probability failure rises from 0% to 15%, as shown in the figure in Appendix A.15. In contrast, the failure probability rises from 3%
to 56% when the wave height keeps at 2 m and the wind speed rises from 3 m/s to 75 m/s. This implies that the wind speed has a much greater effect on top tower deflection failure probability than the wave height [35]. Furthermore, when extreme wind speeds, 75 m/s, and extreme significant wave heights, 20 m, are taken into account, the failure probability is around 93%.
2.5.2 Vulnerability of Wave Energy Systems
The failure rate of a WEC is difficult to assess, as there are so many different technologies and so few notable commercially deployed WECs [36]. According to reference [37], one of the options for applying the failure rate for wave energy farms is to use the established failure rate tailored to the target farm location. Based on this, the paper [12] used a failure rate of λ = 0.555/year for a slack mooring sys- tem, λ = 0.47/year for power lines, λ = 2.42/year for hydraulic power take off, and λ = 1.91/year for steel hull on the Pelamis type WEC. Using the fault tree analysis, the cumulative WEC failure rate is estimated to be 1.386/year [2]. According to reference [38], the failure rate of the linear generator is estimated to be 1.9526/year.
The failure rate in reference [36] varied from zero to ten times a year, with a major applied to a failure rate of 1.752/year.
Profitability has been previously being imperiled by WEC failures, which have also shut down private companies [4]. Motivations to better understand WEC design responses and loads have grown as a result of the need to ensure device survival and avoid unnecessary structural costs [4]. Several WEC design methods have been developed at the same time as a result of RD on the development of WEC.
METHODOLOGY
3.1 Threat Characterization
This section discusses the GEV method for modeling and characterizing wind speed and significant wave height.
The extreme value methods presented in this thesis were applied to data mea- sured at the 46005 in-situ buoy station in the west Washington coast, USA [39].
This research project began with data collection from the NDBC. The collected data were available hourly on the reference [39] for 41 years, from 1976 to 2019, except for 2005, 2009, 2013. The aim of the thesis’s first objective is to characterize extreme wind speed and extreme wave height. The BM EVT was applied to these raw data in accordance with equation 2.3 and modelled into the annual block size.
The EVT concept relies on the statistical behavior of equation 2.3, and it is represented by the intuition of all the hourly dataset taken from the NDBC in this research project. The Li typically represent values of a process measured regularly – in this thesis, for example, the yearly measurement of wind speed and significant wave height is taken into consideration so that the maximum measured value is, Ln, over n observation units.
Then, as shown in Equation 2.2, an GEVD is used to fit the modelled data from the buoy measurements to the Gumbel, Frechet, and Weibull distributions.
As a result of equation 2.2, the fitting distributions can be easily identified as Gumbel (type I), Frechet (type II), and Weibull (type III), and the fitted estimated parameters can then be used to characterize the n return values of the extreme wind speed and extreme significant wave height. By inferring the shape parameter() only, the maxima datasets at hand indicates which type of distributions fitted them well.
Finally, the return levels of wind speed and significant wave height after some return periods are characterized by Equation 2.5.
3.2 Wave Farm Availability
The purpose of this section is to investigate the resilience of the wave energy farm using constant failure rates.
3.2.1 Absorbed Power from the Wave Energy Farm
This section discussed and analyzed methods of absorbing untapped ocean power.
The energy flux in real irregular ocean waves can be calculated as shown in equation 3.1.
J = ρg2T H2
64π (3.1)
where J is the energy flux per unit length of a real irregular ocean waves at deep water in W/m, ρ is the mass density of the sea water, H is the significant wave height, T is the time period of the wave, and g is the gravitational acceleration (9.81m/s2).
Once the energy flux passes through the WEC is known as in equation 3.1, the absorbed power by the WEC is then related by equation 3.2.
CW = Pabsorbed
J (3.2)
where CW is the capture width of the buoy, (width of the incoming ocean wave absorbed by WEC in m), and Pabsorbed is the absorbed power by a WEC.
Equation 3.2 can be written in terms of capture width ratio (also known as power capture ratio) in Equation 3.3 if the diameter of the WEC is considered to be D.
CWR = Pabsorbed
J D (3.3)
where CWR is the capture width ratio (unitless) of the buoy, and D is the diameter of the WEC.
The wave energy farm is composed of N WECs that are spaced apart by a distance d and organized in a grid lattice structure with L rows perpendicular to the incoming ocean waves and n WECs within each row, as illustrated in Figure 3.1.The power or energy flux available (J) from the incoming ocean waves in the first row of the wave energy plant is the same with Equation 3.1. Thus, according to Equation 3.2,the absorbed power by the n WECs in the first row will be Prow1= n ·CW · J. As a result, the available energy flux per meter(J2) reaching row 2 is equivalent to the incoming wave energy flux to row 1(J) minus the power absorbed by row 1(Prow1),as illustrated in equation 3.4
J2 = J
1 − CW d
2
(3.4) Similarly, the available energy flux at row 3 will be equal to the incoming energy flux from row 2 minus the absorbed power by row 2, as shown in Equation 3.5.
J3 = J
1 − CW d
3
(3.5) And therefore, the energy flux available in the last nth row is equivalent to what is seen in equation 3.6.
Jn= J
1 − CW d
n
(3.6)
Figure 3.1: Rectangular wave energy farm representation. The total number of WECs is calculated by multiplying the number of WECs in each row by the number of rows.
Finally, as seen Equation 3.7,the wave energy park’s absorbed power is the summa- tion of all the power absorbed by each row in the park.
PT otabs =
L
X
i
CW.n.J
1 −CW d
i−1
(3.7)
3.2.2 Numbers of Active Wave Energy Converters (WECs) with Constant Failure Rate
This section of the methodology addressed how many WECs are in the wave energy parks, as well as how many WECs will remain if the wave energy farm is subjected to a constant failure rate.
In respect of the failure rate and repair rate, Wave Energy Converter (WEC) numbers in the wave energy farm for a whole year can be expressed in Equation 3.8[36].
W (t) = W (t0)
1 − λ
λ + µ(1 − e−(λ+µ)t
(3.8) where W (t) is the number of WECs at each hour, W (t0)is the number of WECs during the beginning of the wave park, t is the time in each hour, λ is the failure rate, and µ is the repair rate.
The total number of WECs at each hour(tn) can be calculated by subtracting the total number of failed WECs from the total number of WECs at the previous hour((tn−1)) and then adding the repaired WECs as shown in Equation 3.9 [36].
W (tn) = W (tn−1)(1 − λ) + [(W (t0) − W (tn−1)] µ (3.9)
where W (tn) is the total number of WECs at each hour(tn), W (tn−1) is the total number of WECs at the previous hour (tn−1), and W (tn−1) is the total number of failed WECs.
As shown in equation 3.9, the failure rate and repair rate of WECs can have an impact on the number of WECs active in a wave energy farm.
It should be noted that the failure and repair rates used in this thesis are ed- ucated assumptions due to lack of data as there no much commercially deployed WECs. It was estimated using Offshore Reliability Data (OREDA) [22] prepared by SINTEF Industrial Management, which contains failure rates for various sub- systems in the offshore oil and gas sector, as well as other research studies such as [12] that take into account adjustment factors for environmental and other fac- tors. Since the WECs design subsystem consists of mooring, structure, PTO, and transmission as discussed, the majority of the components may be estimated using offshore oil and gas components, though certain adjustments are necessary. So, the overall WEC failure rate assumed in this thesis are from λ = 0 /year to λ = 15 /year. The mean time to repair considered here was from 24 hours to 365 days. So, the corresponding repair rate is the inverse of each of these mean time to repair in a year, yielding 365/year and 12.16/year, respectively.
Furthermore, the initial number of point absorber type WECs was assumed 250, each with a 50 kW rated power. The buoy has a diameter of 4 m and a capture width of 0.8 m. So, the width of the incoming ocean wave captured by the buoy (CWR) is calculated 20% of the diameter of the buoy (because CWR = CWD 100%).
In the wave energy farm, the separation distance between each WEC in a grid layout structure is assumed 30 m.
3.3 Failure rates as function of wave height
This section develops a method for determining the active number of WECs in the wave energy farm with varying failure rates.
As with the bathtub curve, the failure rate can vary with time. Bath tub curves show that the failure rate is high during the initial installation of the farm due to the immaturity phase of the implementation; however, the failure rate gradually reduces and becomes steady after a few years and the generated power is high; then the failure rate rises due to wear and tear over time. The bathtub curve is shown in Figure 3.2[40]. Rather than considering time, as is the case with the bathtub curve, this thesis investigated a changing failure rate function as a function of significant wave height.
Failure rate can be represented as a ratio of the Probability Density Func- tion (pdf) to a reliability, as shown in equation 3.16. To begin, as shown in equation 3.11, pdf was expressed as the derivative of the failure probability (CDF) of the de- veloped fragility curve. The fragility curve used for this thesis is shown in Figure 2.7.
However, it is difficult to approximate the failure rate for this fragility curve to non- zero values for the first ranges of significant wave height, so an analogous function fitting the behavior of this fragility curve was opted, as shown in Equation 3.15. The developed method was then finalized based on this and equation 3.9.The following parts deepen this method and include its necessary conceptualization theory.
Figure 3.2: Bathtub curve [40]
The CDF is also known as the failure probability function or the unreliability function. It indicates the failure probability of a component prior to a given parame- ter in the horizontal axis of the fragility curves, as seen in this example. Wind speed and significant wave height are the parameters defined in this research project. The CDF can also be represented as the total area to the left from the point (extreme wind speed or extreme significant wave height) on the horizontal axis as shown in Equation 3.10.
F (H) = Z H
0
f (H) dH (3.10)
where F (H) is the Cumulative failure probability function (CDF), f(H) is the failure probability density function (pdf), and H is the significant wave height.
The probability density function is written in Equation 3.11 which is the inverse of Equation 3.10 .
f (H) = dF (H)
dH (3.11)
The reliability function indicates the probability that the component will survive above the prescribed wind speed or significant wave height. It can be expressed as in equation 3.12.
R(H) = 1 − F (H) (3.12)
where R(H) is the reliability function.
The failure rate function can be written as a ratio of the probability density function and the reliability function as shown in Equation 3.13.
λ(H) = f (H)
R (H) (3.13)
where λ(H) is the failure rate function.
Due to the fact that the slope of fragility curves is not constant, the changing failure rate was quantified using the derivative concept and the slope of the fragility curve’s tangent line. To approximate the changing failure rate of the fragility curve
of the Figure 2.7,the horizontal axis was first split into smaller infinitesimals intervals of the significant wave height (∆H).
dF (H)
dH = lim
∆H→0
(FH0 + ∆H) − (FH0)
∆H (3.14)
where ∆H is the distance from the point of tangency, (FH0 + ∆H) is the y-value failure probability not far from the point of tangency, and (FH0)is the y-value failure probability at the point of tangency.
The slope of the tangent line of the sample fragility curve was then represented the limit description of the derivative, as seen in equation 3.14.This means that the slope of this tangent line was expressed mathematically in equation 3.14 as the distance from the point of tangency become infinitely smaller.
In this thesis, the derivative in Equation 3.11 was computed numerically by discretizing the wave height H and computing the slope at each step.
However, since it is difficult to approximate the failure probability until 15 m of the significant wave height to a non-zero magnitude, this sample fragility curve is fitted with an analogous function as seen in Equation 3.15.
F (H) = e−
1+k(H−ba )−1k
(3.15) where k is 0.3, a is 1, and b = ak.
Figure 3.3 depicts a figure illustrating the assumed analogous fragility function in respect to significant wave height.
Figure 3.3: Analogous function used to fit fragility curve of WECs
Then, the relationship between time in hours and instant significant wave height was analyzed for a year. In the same way, the relationship between instant significant wave height and instant failure rate was finally analyzed.
Figure 3.4: Relationship between time steps in one year and significant wave height Figures 3.4 and 3.5 show this relationships between time and instant signifi- cant wave height, as well as instant significant wave height and instant failure rate, respectively.
In addition, an average constant failure rate is computed in MATLAB using vectors of the varying instant failure rate. The resilience of the wave energy farm using the average constant failure rate was then compared to the resilience of the wave energy farm using the varying failure rate.
The failure rate function can then be written as shown in Equation 3.16 by substituting Equation 3.12 into equation 3.13.
λ(H) = f (H)
1 − F (H) (3.16)
Figure 3.5: Instant failure rate as a function of significant wave height
RESULT AND DISCUSSION
4.1 Threat Characterization
After modelling the raw data for each year, the maximum wind speed and maximum significant wave height are resulted as in Figure 4.1 . With some exceptions due to other factors when the data was measured at the NDBC, almost the maximum of both hazards rises and falls at the same time each year.
4.1.1 Fitting of Maxima Dataset to Weibull Distribution
The maxima data were fitted by using the GEVD as shown in equation 2.3. The best fit distribution to the block maxima data from the type I (Gumbel), type II (Frechet), and Type III (Weibull) Generalized Extreme Value probability distribution is the type III GEV probability distribution. By inferring the shape parameter (), the best fitting from the families of GEVD can be identified. The best fitting probability distribution for the BM data is the type III Generalized Extreme Value probability distribution as the fitdist method in MATLAB fitted the shape parameter, , to be negative. The scale parameter, location parameter and shape parameter were estimated by type III GEV probability distribution. From the type III (Weibull) families of Generalized Extreme Value Distribution, the maxima block dataset can be fitted with probability density function. Figure 4.2 illustrates the pdf of extreme wind speed and extreme significant wave height using the type III (Weibull) GEV distribution.
Figure 4.1: Block Maxima of extreme wind speed and significant wave height
Figure 4.2: Probability Density Function (PDF)of wind speed and significant wave height
As expected, the shape of the weibull has no tail or is short tailed, with no extremes after certain points, as illustrated in Figure 4.2. Furthermore, the height of the ocean wave’s pdf over a specific point on the horizontal axis is greater than the pdf of the wind speed. Because maximum wind speed data ranges from 15 to 30 m/s, which is greater than wave height, it covers a larger area than pdf of wave height.
4.1.2 Data Analysis with Cumulative Distribution Function
The CDF of a maxima of wind speeds and significant wave heights is shown in Figures 4.4 and 4.3, respectively. In Figure 4.3 ,for example, shows that with an accumulated probability of 60 % that the significant wave height will be less than or equal to 11 m. Similarly, with an accumulated probability of 50 % that the wind speed will be less than or equal to 21.9 m in Figure 4.4.
Figure 4.3: Cumulative Probability of maxima significant wave height
Figure 4.4: Cumulative probability of maxima wind speed
4.1.3 Return Levels of the Hazards at Different Return Pe- riods
Figures 4.5 and 4.6 show the extreme significant wave height and extreme wind speed return levels at 10, 25, 50 and 100 years. As shown in Figure 4.5, the ex- pected return level of the extreme significant wave height at 10 and 100 return years is 12.64 m and 13.67 m, respectively. This plot is done using the Inverse Cumulative Distribution Function (ICDF) in MATLAB using the equation 2.6. Thus, at least once every 10 years, the level of significant wave height exceeded 12.64 m. This can
also be seen in Figure 4.1, which shows that the extreme significant wave height took place in 1988 and then again around in 1998.
Similarly, as shown in Figure 4.6, the return level of extreme wind speed for return years of 10 and 100 years is 25.25 m/s and 27.49 m/s, respectively. The extreme wind speed is thus expected to exceed 25.25 m/s at least once in the 10th return year. According to this, Wind Turbines (WTs) and Wave Energy Converters (WECs) in offshore farms should be designed to meet this type of extreme hazard.
Figure 4.5: Return level of signifcant wave height
Figure 4.6: Return level of extreme wind speed
As Figure 2.8 is applied to this thesis, the extreme wind speed that was char- acterized is 26.98 m/s and a significant wave height of 13.49 m for 50 year return period. This characterized return level of extreme wind speed and extreme signifi- cant wave height can be seen in Figure 2.8 with almost 60% of failure probability.
Figure 4.7: Return values of both hazards at different return periods
As a result, the risk of failure is too high, and if this location had been chosen for an offshore wind farm site, the wind turbine designer would have been notified and would have considered the characterized wind speed in advance.
4.1.4 Quantitle-Quantile Model Fitness Check
Figure 4.8 shows the quantile-quantile plot of the sample maxima wind speed datasets using Type III GEVD (Weibull distribution) model. As shown there, the line is al- most linear, so the GEVD type III model is a good fit for the maxima wind speed datasets.
Figure 4.8: Quantile-Quantile Plot of Maxima Wind Speeds
4.2 Wave farm availability
This section presents and discusses the resilience of the wave energy farm by inves- tigating how the wave power farm performed under conditions of constant failure
