CRANFIELD UNIVERSITY Micheil Gordon Integrated Fatigue Analysis of an Offshore Wind Turbine and Monopile Foundation School of Energy, Environment and Agrifood Advanced Mechanical Engineering MSc Academic Year: 2014 - 2015 Supervisor: Professor Feargal Brennan September 2015
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CRANFIELD UNIVERSITY
Micheil Gordon
Integrated Fatigue Analysis of an Offshore Wind Turbine and
Monopile Foundation
School of Energy, Environment and Agrifood
Advanced Mechanical Engineering
MSc
Academic Year: 2014 - 2015
Supervisor: Professor Feargal Brennan
September 2015
CRANFIELD UNIVERSITY
School of Energy, Environment and Agrifood
Advanced Mechanical Engineering
MSc
Academic Year 2014 - 2015
Micheil Gordon
Integrated Fatigue Analysis of an Offshore Wind Turbine and
Monopile Foundation
Supervisor: Feargal Brennan
September 2015
This thesis is submitted in partial fulfilment of the requirements for
Appendix C Integration of Airy Linear Wave Theory over Depth ................... 83
Appendix D Integration of Stoke’s 2nd Order Wave Theory over Depth ........ 87
Appendix E Actuator Disk Theory ................................................................. 89
Appendix F Wind Turbulence Intensity Factor .............................................. 91
Appendix G Wind Speed Distribution ............................................................ 92
Appendix H Wind Turbulence ....................................................................... 97
Appendix I Finding the Tower Top Stiffness ............................................... 104
Appendix J Finding the MWL Stiffness ....................................................... 105
Appendix K Mudline Wind Bending Stress Transfer Function ..................... 108
Appendix L Turbulent Thrust Force PSD .................................................... 110
Appendix M List of S-N Curves ................................................................... 112
vii
LIST OF FIGURES
Figure 1: Global wind power (Gsanger and Pitteloud, 2013) .............................. 1
Figure 2: S-N curves for steel structures in seawater with cathodic protection (Det Norske Veritas, 2012) .................................................................................. 5
Figure 3: Time based fatigue determination of fatigue damage from wave loading (Passon, 2015) ............................................................................................ 7
Figure 7: Free surface elevation time series from JONSWAP spectrum .......... 22
Figure 8: Ranges of validity for a variety of wave theories (Det Norske Veritas, 2014) ......................................................................................................... 23
Figure 9: Water particle motion (Veldkamp and Van Der Tempel, 2005) ......... 24
Figure 10: Actuator disk model (Manwell et al., 2009) ...................................... 30
Figure 11: Number of occurrences of 10min wind speed intervals in one year with wind speed bins 1m/s wide ........................................................................ 33
Figure 12: Kaimal spectrum for mean wind speed from 3.5m/s to 24.5m/s and with a turbulence intensity of 12% ............................................................. 35
Figure 13: Offshore wind system modelled as a 1 degree of freedom mass-on-pole system (Van Der Tempel, 2006) ........................................................ 36
Figure 14: Transfer function of tower top displacement for the NREL reference turbine with its respective foundation properties (peak=0.6330Hz) ........... 40
Figure 15: Transfer function of MWL displacement for the NREL reference turbine with its respective foundation properties (peak=8.1652Hz) ....................... 42
Figure 16: Transfer function for mudline bending stress from wind loading ..... 44
Figure 17: PSDs of the turbulent thrust force on the rotor at each operational mean wind speed with a 12% turbulence intensity .................................... 46
Figure 18: Process to find the mudline bending stress spectrum from wind loading .................................................................................................................. 48
Figure 25: Mudline bending stress from wave loading ..................................... 58
Figure 26: Comparing results using Airy and Stokes 2nd order wave theories .. 66
Figure 27: Significance of the Drag term in the Morison Equation (no marine growth) ....................................................................................................... 67
Figure 28: Significance of the Drag term in the Morison Equation (with marine growth) ....................................................................................................... 68
Figure 29: In-phase versus out of phase superposition (Kühn, 2001) .............. 72
Figure 30: Significant wave height around the UK (ABP mer, 2008) ................ 81
Figure 31: Airy and Stoke’s 2nd order wave theories (Det Norske Veritas, 2010) .................................................................................................................. 82
Figure 32: Turbulence intensity as a function of mean wind speed (Van Der Tempel, 2006) ........................................................................................... 91
Figure 33: Turbulence intensity as a function of mean wind speed (Burton et al., 2011) ......................................................................................................... 91
Figure 34: Weibull probability density function when 𝑼 = 6m/s (Manwell et al., 2009) ......................................................................................................... 93
Figure 35: Yearly average wind speed at 100m elevation in European waters (Van Der Tempel, 2006) ............................................................................ 94
Figure 36: Weibull distribution for mean annual wind speed = 10m/s and SD=4.8 .................................................................................................................. 95
Figure 37: Example of the wind spped probability for a 1m/s wind speed bin (Lynn, 2011) ......................................................................................................... 95
Figure 38: Ten minute mean wind speed from 0.5m/s to 9.5m/s with turbulence superimposed on top ................................................................................. 98
Figure 39: Ten minute mean wind speed from 10.5m/s to 19.5m/s with turbulence superimposed on top ................................................................................. 99
Figure 40: Ten minute mean wind speed from 20.5m/s to 29.5m/s with turbulence superimposed on top ............................................................................... 100
Figure 41: One month of wind speed data ..................................................... 101
Figure 42: Wind loading over a one month period .......................................... 103
Figure 43: Finding the tower top displacement transfer function .................... 104
ix
LIST OF TABLES
Table 1: Site reference parameters .................................................................. 14
Table 2: Turbine and foundation reference parameters ................................... 16
Table 3: Wave parameters (Det Norske Veritas, 2010, Van Der Tempel, 2006) .................................................................................................................. 18
Table 4: Wave number determination using two methods ................................ 26
Table 5: Finding the tower top stiffness ............................................................ 39
Table 10: S-N curves for most frequently used structural details – Reproduced from DNV-J101 (Det Norske Veritas, 2014) ............................................. 112
Equation 26: Frequency response function for displacement ........................... 38
Equation 27: Transfer function for the tower top displacement......................... 40
Equation 28: Transfer function for the MWL displacement ............................... 42
xi
Equation 29: Transfer function for mudline bending stress from wind loading.. 45
Equation 30: Definition of PSD (1) .................................................................... 46
Equation 31: Definition of PSD (2) .................................................................... 47
Equation 32: Response spectrum of the mudline bending stress from wind loading ....................................................................................................... 47
Equation 33: Phasor form of a complex number .............................................. 50
Equation 34: Transfer function for mudline bending stress from wave loading 52
Equation 35: PSD wave loading integrated over depth .................................... 54
Equation 36: Response spectrum of the mudline bending stress from wind loading ....................................................................................................... 55
Equation 37: Damage equivalent stress range using Kühn’s unweighted equivalent method ..................................................................................... 62
Equation 38: Damage equivalent stress range direct superposition ................. 63
Equation 39: SN curve ..................................................................................... 63
Equation 40: Power coefficient (1) .................................................................... 89
Equation 41: Rotor power ................................................................................. 89
Equation 42: Power coefficient (2) .................................................................... 89
Equation 43: Weibull probability distribution ..................................................... 92
as well as the momentary loading as a result of wind gusts, vibrations from
turbine start and stop and any resonance induced loading (Manwell et al., 2009).
As previously mentioned, to accommodate all the possible loading regimes
requires significantly detailed information regarding all aspects of the system and
deployment site. Such an analysis would need to be carried out in the time
domain to include the non-linear loading interactions (Van Der Tempel, 2006).
Consequently the analysis would require significant computational and time
resources. This kind of study is far beyond the scope of this investigation and it
has been shown that for design purposes, linearized approaches are acceptable
(Det Norske Veritas, 2014, Van Der Tempel, 2006, Yeter et al., 2014, Seidel,
2014). To some extent, the non-conservative approach of neglecting such non-
linearities will be mitigated by adopting a conservative collinear wind and wave
direction assumption. This assumption is acceptable for locations where the
prevailing winds are onshore, but becomes less acceptable for locations
frequently experiencing offshore winds (Arany et al., 2014). That said, it should
be noted, if a 90° misalignment occurs between the wind and wave direction,
minimal damping prevails and such conditions will govern the fatigue design
(Passon and Branner, 2014). Again, more detailed information regarding the
specific installation site is required to take these additional factors into
consideration.
4.4 Results
The accuracy of the results from this investigation are invariably proportional to
the legitimacy of the assumptions and the limitations inherent in the employed
methodology. Some of the major assumptions have already been discussed in
the preceding sections, however further discussion on a number of issues will be
presented here with respect to the final results.
70
4.4.1 Simulation Length
As previously discussed, the standard simulation length is 10 minutes (600
seconds) which is the length employed throughout this investigation. However,
according to a consensus group of experts in the field, this length should be
increased to between 1200 seconds to 3600 seconds (Van Der Tempel, 2006).
Furthermore, due to the IFFT procedure used, the frequency vector resolution
was chosen to correspond to the simulation length required (i.e. 600 points). This
investigation choose not to increase the simulation length, due primarily, to the
additional computational resources which were found to increase exponentially
with simulation length. In addition, the wave stress spectrum was determined only
once for a 600 second time period and repeated ‘N’ – number of times as required
by the Weibull distribution analysis. It is suggested that with additional time and
computational resources, this study would benefit from a more detailed and
comprehensive sensitivity analysis that considers the effects of increasing the
simulation time length and varying the frequency vector resolution.
Despite the limitations in the methodologies, this investigation was careful to
follow standard practise which deems 600 seconds sufficient (Van Der Tempel,
2006) and that the length of the frequency vector should be adequate to maintain
the details contained within the spectral shape (Barltrop and Adams, 1991). To
verify the latter has been upheld, all spectra used have been plotted and provided
throughout this study or in the relevant appendices.
4.4.2 Spectral Multiplication
In section 2.8.4, the mudline bending stress spectrum from wave loading was
described. The methodology chose to ignore all frequencies beyond 2Hz as a
result of the propensity of the JONSWAP spectrum to tend to zero as the
frequency increases. As a result, the system excitation from wave loading is
relatively small. The system’s displacement transfer function from wave loading
was found to have a peak frequency of 8.186Hz, which is at a significant distance
from the peak frequencies of the JONSWAP spectrum which occurs, for the
reference parameters, at around 0.15Hz. Consequently, a significant proportion
of the response spectrum is cancelled out. If sea state parameters decrease the
71
JONSWAP peak frequency will increase and less cancellation will occur,
however, significant decreases are unlikely for the North Sea, which suggests
that system resonance from waves is unlikely.
4.4.3 System Response
Another potential source of error, with respect to wind and wave loading, are the
methods used to establish system response. As previously mentioned, data
regarding the peak frequency of the tower top displacement transfer function, was
taken directly from the literature, applied to this study, and used to establish the
system’s stiffness. Making the necessary assumptions, this was in turn used to
find the system stiffness at the mean water level and the subsequent response
from wave loading. Although the peak frequency of the transfer function found in
the literature was quoted for the NREL 5MW reference turbine, the methods used
to establish this transfer function and the detailed parameters and assumptions
on which it was based, were not explored further. Herein lies an area of great
uncertainty, and without significant time and investment to verify this assumption,
the data presented from the wind and wave loading analysis must be approached
with caution. To improve confidence in the results, a complete and detailed Finite
Element Analysis is suggested, which can be used to establish a significantly
more accurate estimation of the system’s stiffness and subsequently the
response transfer functions.
4.4.4 Final Results Analysis
The first two methods of superposition used in this study were the out-of-phase
spectral superposition, and Kühn’s unweighted equivalent method. These two
methods were selected on the basis of their proven accuracy and validity (Van
Der Tempel, 2006, Kühn, 2001). Interestingly, the results from these two
approaches were found to return exactly the same result, when rounded to one
decimal place. According to Kühn (2001), this is expected due to Pythagoras’s
law which can be considered as an out-of-phase superposition of the damage
equivalent stress range values. For a further discussion of Kühn’s method and
its derivation refer to Kühn (2001). As expected the final results from these two
methods are the least conservative, and are thought to provide the most realistic
72
estimation of the fatigue life for the reference turbine and selected environmental
parameters.
The in-phase spectral superposition is more conservative than the previous two
methods, yet less conservative than the final direct superposition approach. With
reference to the final figures, the in-phase methods yielded a 14% more
conservative result than the out-of-phase and unweighted equivalent methods.
The reason for this increase in conservativism can be explained once again by
the phase. When the wind and wave phases are aligned, the result is an
amplification in magnitudes relative to the magnitudes of each spectra, and no
cancellation takes place. Figure 29, reproduced from Kühn (2001), illustrates the
difference between the in-phase and out-of-phase superposition.
Figure 29: In-phase versus out of phase superposition (Kühn, 2001)
Unfortunately, as suggested by Kühn (2001) the in-phase approach is far too
conservative and if possible, alternative superposition methods should be used.
It was hypothesized that the final method of superposition would yield overly
conservative results and was included for comparison purposes. The model’s
results confirmed this assertion, which returned a 45% more conservative fatigue
73
life estimation than the results from the out-of-phase and unweighted equivalent
approaches.
Before carrying out the various superposition approaches, a prediction of the
outcomes was established from data available in the literature. Using methods
that are known to be less accurate than others may appear counterproductive,
however, this was undertaken to verify, to some extent, that the model is capable
of returning values that are in line with what is expected for a given methodology.
This final point becomes even more significant when direct result comparison
from similar investigations was, regrettably, not possible. This is due to the very
specific nature of every fatigue analyses performed for the offshore wind industry
and the confidential nature of the findings. Consequently, no publicly available
comparative data could be found, to the Author’s knowledge. Subsequently a
complete model validation was not possible, and is suggested as an area for
future investigations.
Although the results appear to conform to the methodological expectations from
the superposition processes, the model has also returned higher than expected
fatigue live predictions for the offshore wind turbine and monopile foundation
structure. The precise reason for this, is regrettably beyond the scope of this
investigation, however it is likely that an accumulation of assumptions made
throughout the methodological process has a significant effect on the final fatigue
figures. However, these assumptions were imperative in order to proceed through
the analysis and to establish a working fatigue model. As previously stated, it is
speculated that the un-verified assumption pertaining to the system stiffness
presents a very large area of uncertainty. With additional time, sensitivity
analyses could contribute to the verification or rejection of such unverified
assumptions and help to establish how variable the end fatigue life is to
fluctuations in each assumption made.
Finally, it is also worth noting, that similar investigations of this nature, rarely use
IFFT to return to the time domain for the determination of the damage equivalent
stress range, as there are accepted methods to obtain this directly from the
superimposed stress spectrum (Van Der Tempel, 2006). It would therefore be of
74
great interest, to future studies, to compare frequency domain damage equivalent
stress range calculations with IFFT and RFC derived damage equivalent stress
range, to identify any discrepancies and explore their possible reasons.
Despite the limitations discussed in this chapter, every care has been taken to
verify, where reasonably possible, that the Matlab scripts used to generate and
process data, are returning reasonable results for that specified methodology,
theory and known input parameters. These verifications have been presented
and discussed throughout the entirety of this investigation.
4.5 Areas for Future Investigations
Throughout the execution of this research project, areas deemed worthy of further
attention have been noted and are summarised in this section below:
The effects of neglecting a transition piece on the fatigue life
How many iterations are required to observe a convergence in the damage
stress equivalent values to more than one decimal place
Given the importance of frequency domain analysis for the offshore industry,
this investigation suggests that future studies that explore the limits of validity
for this fundamental assumption would be beneficial
FEA analysis to establish system stiffness and subsequent model validation.
Explore the limits of validity of neglecting wave drag in the frequency domain
calculation of fatigue life
Run the model for a known case with known and detailed parameters to
establish validity
Identify the extent of discrepancies in the damage equivalent stress range
values from using conventional frequency domain methods and the less
conventional IFFT with RFC method used in this investigation
Further sensitivity analyses with the purpose of establishing the sensitivity of
the fatigue life to the effects of each un-verified assumption made
75
5 CONCLUSIONS
The aim of this study was to present a numerical model for the fatigue analysis
of offshore wind turbines with monopile foundations. This investigation has
achieved the outlined aim, addressed all stated objectives and presented and
discussed the methodologies and findings. The outcomes from the study agree
with data presented in the literature and indicate that there is no difference
between using the out-of-phase spectral superposition or Kühn’s unweighted
equivalent method when rounded to one decimal place. This agreement in the
results provides confidence for the methodological processes used however, the
unrealistically augmented final fatigue life has raised questions regarding the
validity of a number of the assumptions. To validate the model and to increase
confidence in the assumptions it is suggested that significantly more detailed
information is required and that further sensitivity analyses should be performed.
In conclusion, the model presented in this investigation is the result of extensive
and detailed research that draws on a wide variety of technical resources to arrive
at a working offshore wind turbine fatigue model. Although the model has its
limitations it is now in a strong position to be taken forward as the basis for future
investigations looking into the fatigue lives of offshore wind turbines. With the
number of offshore wind farms increasing, and with deeper water and harsher
conditions expected, it is imperative that fatigue life investigations continue to
ensure system safety and economic viability is maintained for the foreseeable
future.
77
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APPENDICES
Appendix A UK Significant Wave Heights
Figure 30: Significant wave height around the UK (ABP mer, 2008)
82
Appendix B Equations for Airy Linear and Stoke’s 2nd
Order Wave Theories
Figure 31: Airy and Stoke’s 2nd order wave theories (Det Norske Veritas, 2010)
83
Appendix C Integration of Airy Linear Wave Theory
over Depth
In order to establish the wave loading on an offshore monopile foundation, it is
necessary to integrate the water particle modulus of the velocity multiplied by the
velocity (|𝑢|𝑢) as well as the water particle acceleration �̇� over the depth.
C.1 Integration of |u|*u with respect to z
Referring back to horizontal water particle velocity equation (Equation 10):
𝑢(𝜙, 𝑧; 𝑡) = 𝑎𝑖2𝜋𝑓𝑖 ×cosh [𝑘(𝑧 + 𝑑)]
sinh (𝑘𝑑)× cos (𝜙 − 2𝜋𝑓𝑖𝑡)
Rearranging to isolate the term to be integrated and expanding the cosh bracket
gives:
𝑢(𝜙, 𝑧; 𝑡) = 𝑎𝑖2𝜋𝑓𝑖 ×cos(𝜙 − 2𝜋𝑓𝑖𝑡)
sinh(𝑘𝑑)× cosh (𝑘𝑧 + 𝑘𝑑)
It should be noted that the purpose of multiplying |u| by u rather than including u2
is to avoid a sign change which complicates the integration further. However it
can be seen that the only term to be integrated is the cosh(𝑘𝑧 + 𝑘𝑑) term, and
since cosh is always positive, this term can be treated as cosh2 (𝑘𝑧 + 𝑘𝑑) which
can then be multiplied by the modulus of the remaining terms.
𝐿𝑒𝑡 𝐴 = 𝑎𝑖2𝜋𝑓𝑖 ×cos (𝜙 − 2𝜋𝑓𝑖𝑡)
sinh (𝑘𝑑)
Thus:
|𝑢|𝑢 = |𝐴|𝐴 × cosh2 (𝑘𝑧 + 𝑘𝑑)
The integration can now be carried out.
∫ |𝑢|𝑢0
−𝑑
𝑑𝑧 = |𝐴|𝐴 ∫ cosh2 (𝑘𝑧 + 𝑘𝑑) 0
−𝑑
𝑑𝑧
Integrate by making a substitution:
𝐿𝑒𝑡 (𝑘𝑧 + 𝑘𝑑) = 𝑣
84
Cannot directly integrate cosh2(𝑣) refer to trigonometric identities. Note:
cosh(𝑣) =𝑒𝑣 + 𝑒−𝑣
2
And that:
cosh2(𝑣) = (𝑒𝑣 + 𝑒−𝑣
2)
2
cosh2(𝑣) =1
4× (𝑒𝑣 + 𝑒−𝑣)2
Expanding the brackets gives:
cosh2(𝑣) =1
4(𝑒2𝑣 + 2 + 𝑒−2𝑣)
Substituting back into the integral:
∫ |𝑢|𝑢0
−𝑑
𝑑𝑧 = |𝐴|𝐴 ×1
4∫ (𝑒2𝑣 + 2 + 𝑒−2𝑣)
0
−𝑑
𝑑𝑧
Need to different (𝑘𝑧 + 𝑘𝑑) to make the 𝑣 the subject of the integration:
𝑑𝑣
𝑑𝑧= 𝑘
𝑑𝑧 =𝑑𝑣
𝑘
Thus:
∫ |𝑢|𝑢0
−𝑑
𝑑𝑧 = |𝐴|𝐴 ×1
4𝑘∫ (𝑒2𝑣 + 2 + 𝑒−2𝑣)
0
−𝑑
𝑑𝑣
Integration can now take place:
∫ |𝑢|𝑢0
−𝑑
𝑑𝑧 = |𝐴|𝐴 ×1
4𝑘× [
1
2𝑒2𝑣 + 2𝑣 −
1
2𝑒−2𝑣]
−𝑑
0
Simplifying slightly gives:
85
∫ |𝑢|𝑢0
−𝑑
𝑑𝑧 = |𝐴|𝐴 ×1
8𝑘× [𝑒2𝑣 + 4𝑣 − 𝑒−2𝑣]−𝑑
0
Before substitute back in for 𝑣 convert back to a trigonometric term if possible:
sinh(𝑣) =𝑒𝑣 − 𝑒−𝑣
2
2sinh(2𝑣) = 𝑒2𝑣 − 𝑒−2𝑣
Thus:
∫ |𝑢|𝑢0
−𝑑
𝑑𝑧 = |𝐴|𝐴 ×1
8𝑘× [2sinh(2𝑣) + 4𝑣]−𝑑
0
Substituting back in for 𝑣:
∫ |𝑢|𝑢0
−𝑑
𝑑𝑧 = |𝐴|𝐴 ×1
8𝑘× [2sinh[2(𝑘𝑧 + 𝑘𝑑)] + 4(𝑘𝑧 + 𝑘𝑑)]−𝑑
0
Finally:
∫ |𝑢|𝑢0
−𝑑
𝑑𝑧 = |𝐴|𝐴 ×1
8𝑘× [2sinh(2𝑘𝑧 + 2𝑘𝑑) + (4𝑘𝑧 + 4𝑘𝑑)]−𝑑
0
Apply the limits gives:
∫ |𝑢|𝑢0
−𝑑
𝑑𝑧 = |𝐴|𝐴 ×1
8𝑘× [2sinh(2𝑘𝑑) + 4𝑘𝑑]
This is the final equation used for |𝑢|𝑢 in the Morison Equation.
C.2 Integration of water particle acceleration with respect to z
The same procedure was employed for the water particle acceleration given that
exactly the same term in the equation is integrated, (cosh [𝑘(𝑧 + 𝑑)]).
Thus:
∫ �̇�0
−𝑑
𝑑𝑧 = 𝑎𝑖(2𝜋𝑓𝑖)2 ×sin (𝜙 − 2𝜋𝑓𝑖𝑡)
sinh (𝑘𝑑)× ∫ cosh [𝑘(𝑧 + 𝑑)]
𝑑
−𝑑
𝑑𝑧
86
Which can be performed by substituting (𝑘𝑧 + 𝑘𝑑) = 𝑣.
∫ �̇�0
−𝑑
𝑑𝑧 = 𝑎𝑖(2𝜋𝑓𝑖)2 ×sin (𝜙 − 2𝜋𝑓𝑖𝑡)
sinh (𝑘𝑑)× ∫ cosh(𝑣) 𝑑𝑧
𝑑
−𝑑
∫ �̇�0
−𝑑
𝑑𝑧 = 𝑎𝑖(2𝜋𝑓𝑖)2 ×
sin (𝜙 − 2𝜋𝑓𝑖𝑡)
sinh (𝑘𝑑)×
1
𝑘∫ cosh(𝑣) 𝑑𝑣
𝑑
−𝑑
∫ �̇�0
−𝑑
𝑑𝑧 = 𝑎𝑖(2𝜋𝑓𝑖)2 ×sin (𝜙 − 2𝜋𝑓𝑖𝑡)
sinh (𝑘𝑑)×
1
𝑘[sinh(𝑣)]−𝑑
0
Applying the limits and substituting back in for 𝑣:
∫ �̇�0
−𝑑
𝑑𝑧 = 𝑎𝑖(2𝜋𝑓𝑖)2 ×
sin (𝜙 − 2𝜋𝑓𝑖𝑡)
sinh (𝑘𝑑)×
1
𝑘[sinh(𝑘𝑧 + 𝑘𝑑)]−𝑑
0
∫ �̇�0
−𝑑
𝑑𝑧 = 𝑎𝑖(2𝜋𝑓𝑖)2 ×sin (𝜙 − 2𝜋𝑓𝑖𝑡)
sinh (𝑘𝑑)×
1
𝑘[sinh(𝑘𝑑)]
Cancelling the sinh terms gives:
∫ �̇�0
−𝑑
𝑑𝑧 = 𝑎𝑖(2𝜋𝑓𝑖)2 ×sin (𝜙 − 2𝜋𝑓𝑖𝑡)
𝑘
87
Appendix D Integration of Stoke’s 2nd Order Wave
Theory over Depth
D.1 Integration of |u|*u according to Stoke’s 2nd Order Wave
Theory over Depth
Once Airy theory has been integrated over the depth Stokes theory can be easily
integrated by modifying the same equations used to integrate Airy. This is
possible due to only very minor differences in the integrand. The final integration
of stokes is presented below.
Stoke’s 2nd order wave water particle velocity is given as:
𝑢𝑆𝑡𝑜𝑘𝑒′𝑠 = 𝑢𝐴𝑖𝑟𝑦 +3𝑎𝑖
2𝜋2𝑓𝑖
𝜆×
cos[2(𝜙 − 2𝜋𝑓𝑖𝑡)]
sinh4(𝑘𝑑)× cosh [2(𝑘𝑧 + 𝑘𝑑)]
To integrate let:
3𝑎𝑖2𝜋2𝑓𝑖
𝜆×
cos[2(𝜙 − 2𝜋𝑓𝑖𝑡)]
sinh4(𝑘𝑑)= 𝐵
Therefore:
∫ |𝑢|𝑢𝑆𝑡𝑜𝑘𝑒′𝑠
0
−𝑑
𝑑𝑧
= ∫ |𝑢|𝑢𝐴𝑖𝑟𝑦
0
−𝑑
𝑑𝑧 + |𝐵|𝐵 ×1
16𝑘
× [2sinh(4𝑘𝑧 + 4𝑘𝑑) + (8𝑘𝑧 + 8𝑘𝑑)]−𝑑0
Giving:
∫ |𝑢|𝑢𝑆𝑡𝑜𝑘𝑒′𝑠
0
−𝑑
𝑑𝑧 = ∫ |𝑢|𝑢𝐴𝑖𝑟𝑦
0
−𝑑
𝑑𝑧 + |𝐵|𝐵 ×1
16𝑘× [2sinh(4𝑘𝑑) + (8𝑘𝑑)]
D.2 Integration of water particle acceleration according to
Stoke’s 2nd Order Wave Theory over Depth
Following the same process as described above the water particle acceleration
can be integrated over the water depth relatively easily.
88
Stoke’s 2nd order wave water acceleration is given as:
�̇�𝑆𝑡𝑜𝑘𝑒′𝑠 = �̇�𝐴𝑖𝑟𝑦 +12𝑎𝑖
2𝜋3𝑓𝑖2
𝜆×
sin[2(𝜙 − 2𝜋𝑓𝑖𝑡)]
sinh4(𝑘𝑑)× cosh [2(𝑘𝑧 + 𝑘𝑑)]
Integration becomes:
∫ �̇�𝑆𝑡𝑜𝑘𝑒′𝑠
0
−𝑑
𝑑𝑧
= ∫ �̇�𝐴𝑖𝑟𝑦
0
−𝑑
𝑑𝑧 +12𝑎𝑖
2𝜋3𝑓𝑖2
𝜆×
sin[2(𝜙 − 2𝜋𝑓𝑖𝑡)]
sinh4(𝑘𝑑)
× ∫ cosh[2(𝑘𝑧 + 𝑘𝑑)] 𝑑𝑧0
−𝑑
Integrating and applying the limits yields:
∫ �̇�𝑆𝑡𝑜𝑘𝑒′𝑠
0
−𝑑
𝑑𝑧 = ∫ �̇�𝐴𝑖𝑟𝑦
0
−𝑑
𝑑𝑧 +12𝑎𝑖
2𝜋3𝑓𝑖2
𝜆×
sin[2(𝜙 − 2𝜋𝑓𝑖𝑡)]
sinh4(𝑘𝑑)×
1
2𝑘sinh(2𝑘𝑑)
These equations are then used in the Morison Equation to determine the total
wave force on the monopile.
89
Appendix E Actuator Disk Theory
It is known that the power coefficient (Cp) is represented by Equation 40, where
𝑃 is the power of the rotor.
𝐶𝑃 =𝑃
1
2𝜌𝑎𝐴𝑟𝑜𝑡𝑜𝑟𝑈3
Equation 40: Power coefficient (1)
And that the power of the rotor is described by:
𝑃 =1
2𝜌𝑎𝐴𝑟𝑜𝑡𝑜𝑟𝑈3[4𝑎(1 − 𝑎)2] Equation 41: Rotor power
Thus equating Equation 40 and Equation 41 gives:
𝐶𝑃 = 4𝑎(1 − 𝑎)2 Equation 42: Power coefficient (2)
If it was possible to extract 100% of the wind energy Cp would equal 0.59,
however this is highly unlikely with the maximum efficiency achieved by large
scale wind turbines currently around 50% (Cp = 0.5) (Lynn, 2011). Therefore, if
an assumption is made regarding the turbine power coefficient, the value of the
axial induction factor can be found iteratively. Taking NREL turbine power
coefficient equal 0.482 (Jonkman et al., 2009), the axial induction factor was
found iteratively as 0.1785973 (see Table 8 below).
Table 8: Finding the axial induction factor for a turbine with a Cp=0.482
Cp a Desired Cp
0.5625 0.25 0.482
0.554496 0.24
0.545468 0.23
0.535392 0.22
0.524244 0.21
0.512 0.2
0.498636 0.19
0.484128 0.18
0.482613356 0.179
90
Cp a Desired Cp
0.481087008 0.178
0.482461248 0.1789
0.482309023 0.1788
0.482156682 0.1787
0.482004223 0.1786
0.482000105 0.1785973
With reference to Equation 42, and taking the initial value of the axial induction
factor as 0.25, the value of 𝑎 was found by adjusting 𝑎 until the desired value of
Cp was found. Table 8 presents the process used to find 𝑎 and which was found
to be 0.1785973.
91
Appendix F Wind Turbulence Intensity Factor
Figure 32: Turbulence intensity as a function of mean wind speed (Van Der
Tempel, 2006)
Figure 33: Turbulence intensity as a function of mean wind speed (Burton et al.,
2011)
92
Appendix G Wind Speed Distribution
The Weibull probability distribution can be obtained using the following function
(Equation 43) and for further information see Manwell et al. (2009).
𝑝(𝑈) = (𝑘
𝑐) × (
𝑘
𝑐)
𝑘−1
exp [− (𝑈
𝑐)
𝑘
] Equation 43: Weibull probability
distribution
Where:
𝑝(𝑈) = Probability density function
𝑘 = Shape factor
𝑐 = Scale factor
𝑈 = Freestream velocity
The ‘k’ and ‘c’ parameters can be determined using a variety of methods, with
some significantly more complex than others. The simplified methods are used in
this investigation as the focus is on the ability to simulate a North Sea offshore
environment rather than to generate very location specific environmental
conditions. If this model is used in a detailed site specific study further
investigation is suggested in establishing the ‘k’ and ‘c’ parameters, refer to
Manwell et al. (2009) for alternative determination approaches.
According to Justus (1978) cited in Manwell et al. (2009) a good approximation
for k when 1 ≤ k < 10, is given by Equation 44.
𝑘 = (𝜎�̅�
�̅�)
−1.086
Equation 44: Shape factor ‘k’
Once k is established using the above formula, c can be established using Lysen
(1983)’s approximation given in Equation 45.
𝑐 = �̅� × (0.568 +0.433
𝑘)
−1𝑘
Equation 45: Scale factor ‘c’
Where:
93
�̅� = Long term mean wind speed
The higher the value of k, the sharper the peak becomes, indicating less variation
in the overall wind speed at the site of interest (see Figure 34). Figure 34
reproduced from Manwell et al. (2009) demonstrates a number of Weibull
probability density functions for different values of k.
Figure 34: Weibull probability density function when �̅� = 6m/s (Manwell et al.,
2009)
Using the Weibull distribution the probability of occurrences of a wind speed
range can be established. The annual average wind speed encountered in the
North Sea is around 10m/s from Figure 35, and a typical value of the shape factor
for offshore conditions is around 2.2 (Van Der Tempel, 2006). According to Van
Der Tempel (2006) the shape factor reduces in value when moving from offshore
to onshore as a result of the latent cooling and heating which increases the
onshore wind speed variability when compared to offshore.
94
Figure 35: Yearly average wind speed at 100m elevation in European waters (Van
Der Tempel, 2006)
The hub height of the NREL reference turbine used in this investigation is at a
height of 90m above the mean water level (MWL) which rests on top of a monopile
foundation 20m in length and assumed equal to the depth of 20m. It has been
assumed that the annual average offshore wind speed at the hub height of 90m
can be represented with an annual average wind speed of 10m/s and a k value
of 2.2. Thus, taking k equal to 2.2 results in a wind speed standard deviation (SD)
of 4.8 (from Equation 44). These values have been assumed to represent general
conditions experienced in the North Sea.
The next procedure is to use the Weibull pdf to find the number of occurrences
for a given wind range using the following procedure:
1. Establish the Weibull pfd from zero to 30m/s using the procedure
described above (see Figure 36)
95
Figure 36: Weibull distribution for mean annual wind speed = 10m/s and SD=4.8
2. Assuming constant wind speed over ten minutes therefore there will be
52,560 ten minute stationary periods in one year
3. Divide the distribution up into bins 1m/s wide and find the probability at
centre of each bin which will occur at 0.5, 1.5, 2.5…(see Figure 37)
Figure 37: Example of the wind spped probability for a 1m/s wind speed bin
(Lynn, 2011)
96
4. Multiply the probability of each bin centre by the number of ten minute
stationary periods in one year to yield the number of times that the
specific wind speed range is experienced in the course of one year (see
Figure 11).
5. Employ the actuator disk theory to find the thrust of the wind turbine at
each mean bin wind speed
97
Appendix H Wind Turbulence
The process used to generate the Kaimal spectrum is given here, as
recommended in the relevant standard DNV J101 (Det Norske Veritas, 2014).
𝑆�̅�10(𝑓) = 𝜎2
4𝐿𝑘
�̅�10
(1 +6𝑓𝐿𝑘
�̅�10)
5/3 Equation 46: Kaimal spectrum
𝐿𝑘 = 5.67𝑧 𝑓𝑜𝑟 𝑧 < 60𝑚
and
𝐿𝑘 = 340.2𝑚 𝑓𝑜𝑟 𝑧 ≥ 60𝑚
Equation 47: Integral scale
parameter
Where:
𝑆�̅�10(𝑓) = Kaimal PSD at a given mean wind speed
𝑓 = Frequency
𝐿𝑘 = Integral scale parameter
In order to plot the spectrum the standard deviation for a given ten minute wind
stationary period must be determined.
With reference to the turbulence intensity equation (Equation 20), this
investigation has established the standard deviation over the full range of ten
minute mean wind speeds taking the turbulence intensity factor at a constant
value of 12%. Assuming a turbine hub height greater than 60m and a constant
turbulence intensity of 12% Figure 12 demonstrates the Kiamal PSD for mean
wind speeds from 3.5m/s (3m/s = cut in speed) to 24.5m/s (25m/s = cut out
speed).
An example of the wind speed turbulence for mean wind speeds from 0.5m/s to
29.5m/s with 1m/s interval bins is given in the following three figures.
98
Figure 38: Ten minute mean wind speed from 0.5m/s to 9.5m/s with turbulence
superimposed on top
99
Figure 39: Ten minute mean wind speed from 10.5m/s to 19.5m/s with turbulence
superimposed on top
100
Figure 40: Ten minute mean wind speed from 20.5m/s to 29.5m/s with turbulence
superimposed on top
Now a wind turbulence model has been established the total wind speed time
series over the course of one year can be created by repeating the process the
required number of times as established by the Weibull probability distribution.
Referring Figure 11 and to the number of ten minute periods with a mean wind
speed of 0.5m/s is 231, therefore the process is repeated 231 times at a mean
wind speed of 0.5m/s before proceeding to determine the time series at 1.5m/s.
If this is repeated for the entire Weibull distribution the wind speed time series for
the entire year can be generated. However the computational resources required
is extensive, therefore a simplified approach has been used in this investigation.
By calculating the distribution over the course of one month a much more
manageable dataset is obtained. The results from this analysis are provided in
Figure 41.
101
Figure 41: One month of wind speed data
102
H.1 Wind Loading From Wind Turbulent Time Series
The offshore wind turbine wind loading regime is significantly dominated by the
thrust of the turbine and as such the wind loading from tower and nacelle has
been neglected.
To establish the turbine thrust this investigation has employed the actuator disk
theory as described in section 2.4.3.1 and the turbulent one month wind speed
data from the previous section (2.4.3.3). The remaining required parameters are
found from the NREL reference turbine. See Jonkman et al. (2009)). Table 9
outlines the parameters used for the loading calculations.
Table 9: Turbine thrust calculation parameters
Parameter Units Value Comments
Air density (ρ) [kg/m3] 1.225 Taken at standard conditions (sea level
at 15°C) (Manwell et al., 2009)
Rotor diameter [m] 128 (Jonkman et al., 2009)
Rotor area [m2] 12868 (Jonkman et al., 2009)
Axial induction factor
(a) [] 0.1785973 See section 2.4.3.1
Using the data presented in Figure 41, the parameters given in Table 9 and
referring to Equation 18 and Equation 19 the wind loading can be established at
each wind speed during the course of one month. The results are presented
below.
103
Figure 42: Wind loading over a one month period
104
Appendix I Finding the Tower Top Stiffness
Figure 43, describes the iterative process used to find the tower top stiffness
assuming a peak frequency of 0.633Hz can be applied.
Figure 43: Finding the tower top displacement transfer function
105
Appendix J Finding the MWL Stiffness
The deflection of the structure can be established using the Double-Integration
method (Pytel and Kiusalaas, 2011) which yields a deflection equation at any
height, which in this case is the height anywhere along the turbine tower (see
Pytel and Kiusalaas (2011) for the full derivation and method explanation).
Equation 48 demonstrates the equation of the elastic curve used to establish the
deflection at the mean water level.
𝑑2𝛿
𝑑𝑧2=
𝑀
𝐸𝐼
Equation 48: Differential equation of the
elastic curve
𝑀 = 𝐹 × 𝑧 Equation 49: Moment
Where:
𝛿 = Deflection
𝑧 = Vertical distance up the turbine tower (z=0 at the mudline)
𝑀 = Moment
𝐹 = Force
𝐸 = Young’s modulus (210GPa from Jonkman et al. (2009))
𝐼 = 2nd moment of inertia
Given that the stiffness is now known at the top of the turbine where z=length,
the stiffness at the point where z=20 (MWL) can be established using the
following procedure.
𝐸𝐼𝛿 = ∬ 𝑀 𝑑𝑧 𝑑𝑧 + 𝐶1 + 𝐶2
𝐸𝐼𝛿 = ∬ 𝐹(ℎ − 𝑧) 𝑑𝑧 𝑑𝑧 + 𝐶1 + 𝐶2
Equation 50: Equation of the elastic
curve
Where:
106
ℎ = Height of the turbine (90m above the MWL and 20m below = 110m)
Preform the integration:
𝐸𝐼𝑑𝛿
𝑑𝑧= ∫ 𝐹(ℎ − 𝑧)𝑑𝑧
𝑧
0
𝐸𝐼𝑑𝛿
𝑑𝑧= ∫ (𝐹ℎ − 𝐹𝑧) 𝑑𝑧
𝑧
0
𝐸𝐼𝑑𝛿
𝑑𝑧= 𝐹ℎ𝑧 − 𝐹
𝑧2
2
𝐸𝐼𝛿 = ∫ 𝐹ℎ𝑧 − 𝐹𝑧2
2𝑑𝑧
𝑧
0
𝐸𝐼𝛿 = 𝐹ℎ𝑧2
2− 𝐹
𝑧3
6
Equation 51: Differential equation of the
elastic curve
Now take the situation where z=h:
𝐸𝐼𝛿 =𝐹ℎ3
2−
𝐹ℎ3
6
= 𝐹ℎ3 (1
2−
1
6)
𝐸𝐼𝛿 =𝐹ℎ3
3
Equation 52: Simplified equation of the
elastic curve where z=h
By equating Equation 52 with the equation for stiffness (force over deflection) a
new equation for stiffness is obtained as a function of height up the structure:
𝑘 =𝐹
𝛿=
3𝐸𝐼
ℎ3
Equation 53: Stiffness
Where:
𝑘 = Stiffness
107
𝛿 = Deflection
Now two equations can be presented, one for the known case where the stiffness
(k1) is equal to 143200N/m (see Table 5) and one for (k2) at the MWL where the
stiffness is unknown.
𝑘1 =3𝐸𝐼
ℎ13 = 143200N/m
Equation 54: Stiffness at hub height
𝑘2 =3𝐸𝐼
ℎ23
Equation 55: Stiffness at the MWL (1)
Where:
ℎ1 = Hub height from the seabed (110m)
ℎ2 = MWL (20m from seabed)
Assuming a constant value of I across the tower the following equation can be
used to find the stiffness at the MWL:
𝑘2 = 𝑘1 ×ℎ1
3
ℎ23
= 143200 ×1103
203
= 𝟐𝟑𝟖𝟐𝟒𝟗𝟎𝟎𝑵/𝒎
Equation 56: Stiffness at the MWL (2)
Finally the transfer function can be established for the wave loading using the
stiffness estimated at the MWL.
108
Appendix K Mudline Wind Bending Stress Transfer
Function
The following procedure is carried out using the flexure formula (Equation 57) to
find translate the displacement transfer function into a stress transfer function:
𝜎 =𝑀𝑦
𝐼
Equation 57: Flexure Formula (1) (Gere and
Goodno, 2009)
Which can be re-written in terms of the loading:
𝜎 =𝐹 × 𝐿 × 𝑦
𝐼 Equation 58: Flexure Formula (2)
And from Equation 53 it was found that:
𝛿 =𝐹ℎ3
3𝐸𝐼
Equation 59: Deflection as a function of
height
Thus by equating Equation 58 and Equation 59 the following relationship is
established:
𝜎 = 𝛿 × [3𝐸𝑦
𝐿2]
Equation 60: Bending stress in terms of
displacement
Where:
𝜎 = Bending stress [Pa]
𝐼 = 2nd Moment of inertia of the cross-sectional area [m4] (assumed constant
across entire structure)
𝑀 = Bending moment [Nm]
𝐸 = Young’s modulus (210GPa from Jonkman et al. (2009))
𝐹 = Force [N]
𝐿 = Length [m] (110m from hub to mudline)
109
𝛿 = Deflection [m]
𝑦 = Distance from the neutral axis [m] (in this case at tower top = D/2=1.935m)
With reference to Equation 60 a linear relationship between bending stress at the
mudline and the tower top displacement can be observed. By multiplying
|𝐻(𝑠)𝛿𝑡𝑜𝑝| (Equation 27) by 3𝐸𝑦
𝐿2 from Equation 60 the wind loading mudline
bending stress transfer function can be found (see proceeding equations).
|𝑋(𝑠)𝑡𝑜𝑝
𝐹(𝑠)𝑤𝑖𝑛𝑑| × (
3𝐸𝑦
𝐿2) = |𝐻(𝑠)𝜎𝑤𝑖𝑛𝑑|
|𝑋(𝑠)𝑡𝑜𝑝
𝐹(𝑠)𝑤𝑖𝑛𝑑| × (
3 × 210 × 109 × 1.935
1102)
|𝑋(𝑠)𝑡𝑜𝑝
𝐹(𝑠)𝑤𝑖𝑛𝑑| × (100747934)
= |𝐻(𝑠)𝜎𝑤𝑖𝑛𝑑|
|𝐻(𝑠)𝜎𝑤𝑖𝑛𝑑| = |𝜎(𝑠)𝑤𝑖𝑛𝑑
𝐹(𝑠)𝑤𝑖𝑛𝑑|
Equation 61: Transfer function for
mudline bending stress from wind
loading
Where:
|𝐻(𝑠)𝜎𝑤𝑖𝑛𝑑| = Wind loading mudline bending stress transfer function
𝜎(𝑠)𝑤𝑖𝑛𝑑 = Mudline bending stress from wind loading
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Appendix L Turbulent Thrust Force PSD
The required spectrum was established for all operational mean wind speeds
analysed using the following equations as described by Arany et al. (2014).
𝑆𝐹𝑤𝑖𝑛𝑑(𝑓) = 𝜌𝑎
2𝐴𝑟𝑜𝑡𝑜𝑟2𝐶𝑇
2�̅�2𝜎2�̃��̅�10(𝑓) Equation 62: PSD of the turbulent thrust
force on the rotor
�̃��̅�10(𝑓) =
𝑆�̅�10(𝑓)
𝜎2
Equation 63: Normalised Kaimal
spectrum
𝜎 = 𝐼𝑡 × �̅�10
Equation 64: Ten minute wind speed
standard deviation (from Equation
20)
Where:
𝑆𝐹𝑤𝑖𝑛𝑑(𝑓) = PSD of the turbulent thrust force on the rotor
�̃��̅�10(𝑓) = Normalised Kaimal spectrum
𝜌𝑎 = Density of air
𝐴𝑟𝑜𝑡𝑜𝑟 = Rotor area
𝐶𝑇 = Thrust coefficient
𝐼𝑡 = Turbulence intensity
�̅�10 = Ten minute mean wind speed
𝜎 = Ten minute wind speed standard deviation
As previously mentioned the thrust coefficient can be found using BEM theory
which is complex and time consuming, alternatively it can estimated using
Frohboese et al. (2010)’s thrust coefficient method which has been shown to yield
relatively accurate, conservative results for the majority of offshore wind turbines
within their operating wind speeds (Frohboese et al., 2010). The trust coefficient
used for this investigation was estimated using Frohboese et al. (2010)’s method
which is provided by the following equation:
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𝐶𝑇 =3.5(2�̅�10 − 3.5)
�̅�102
Equation 65: Thrust coefficient
estimation (Frohboese et al., 2010)
Referring back to Equation 62, it should be noted that all parameters except for
the Normalised Kaimal spectrum, originate from the actuator disk theory and have
been squared.
To determine the PSD of the turbulent thrust force on the rotor of the NREL
reference turbine the following data was used. The NREL turbine has a cut in
wind speed of 3m/s and a cut out wind speed of 25m/s. Assuming the turbine
thrust outside of the turbine’s operating conditions can be neglected, which can
be justified based on the low occurrence of mean wind speeds above 25m/s and
below 3m/s (see Weibull distribution in section 2.4.3.2), the PSD of the turbulent
thrust force on the rotor at each operational mean wind speed can be established.
Figure 17 presents 22 PSD’s of the turbulent thrust force on the rotor for each
mean wind speeds ranging from 3.5m/s up to 24.5m/s with increments of 1m/s to
cover all the operational wind speeds.
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Appendix M List of S-N Curves
Table 10: S-N curves for most frequently used structural details – Reproduced