Stochastic Dynamic Analysis of an Offshore Wind Turbine Structure by the Path Integration Method
"> Figure 1
<p>Prototype and equivalent dynamic model of the offshore wind turbine (OWT). (<b>a</b>) The OWT supported by bucket foundation; (<b>b</b>) the equivalent dynamic model of the OWT.</p> "> Figure 2
<p>Transfer function (TRF) from wind speed to tower-top load [<a href="#B41-energies-12-03051" class="html-bibr">41</a>].</p> "> Figure 3
<p>The finite element model (FEM) of OWT supported by bucket foundation.</p> "> Figure 4
<p>The first two mode shapes obtained from the modal analysis.</p> "> Figure 5
<p>Implementation procedure of fast Fourier transform (FFT) convolution for Probability density function (PDF).</p> "> Figure 6
<p>Turbulence standard deviation for the normal turbulence model (NTM) [<a href="#B12-energies-12-03051" class="html-bibr">12</a>].</p> "> Figure 7
<p>Wind excitation spectrum for the wind condition of Case 3 and transfer function (TRF) for the NREL 3 MW wind turbine.</p> "> Figure 8
<p>Relative wind excitation spectrum (Sff) and the second-order filtered spectrum for the wind condition of Case 3.</p> "> Figure 9
<p>Joint probability density functions (PDFs) of (<span class="html-italic">x</span><sub>1</sub>, <span class="html-italic">x</span><sub>2</sub>) extracted from the regular path integration (PI) method and from the Monte Carlo simulation (MCS) for the wind condition of Case 3: (<b>a</b>) Joint PDFs from the regular PI method; (<b>b</b>) joint PDFs from the MCS.</p> "> Figure 10
<p>Marginal PDFs of (displacement, velocity): (<b>a</b>) PDFs of displacement; (<b>b</b>) PDFs of velocity.</p> "> Figure 11
<p>Relative wind excitation spectra and PDF of displacement calculated by FFT-PI for different cases: (<b>a</b>) Relative wind excitation spectral density for Case 1, 3, and 5; (<b>b</b>) PDF of displacement calculated by FFT-PI for Case 1, 3, and 5; (<b>c</b>) Relative wind excitation spectral density for Case 2, 3, and 4; (<b>d</b>) PDF of displacement calculated by FFT-PI for Case 2, 3, and 4.</p> ">
Abstract
:1. Introduction
2. Materials and Methods
2.1. Equivalent Dynamic Model of the OWT
2.2. Linear Filter Method for Stochastic Wind
2.3. Finite Element Modeling
2.4. The Path Integration Method
3. Results and Discussion
3.1. Stochastic Dynamic Response of the OWT
3.2. Parameter Influences of Wind Conditions
3.3. Reliability Assessment
4. Conclusions
- The stochastic dynamic analysis of the OWT under horizontal stochastic wind excitation was numerically investigated by the PI method. The probability density function (PDF) of the joint response obtained by the 4D regular PI and FFT-based PI methods coincided very well with that of the Monte Carlo simulation (MCS), which demonstrated that these two PI methods provide reliable and reasonable results for such a dynamic system describing the OWT.
- The FFT-based PI method held advantages over the MSC and regular PI methods considering computation efficiency and accuracy. Meanwhile, the reliability based on the marginal PDF of displacement and the total probability law could be calculated when the OWT was subjected to stochastic wind excitation.
- The influences of the horizontal mean wind speed and turbulence standard deviation on the relative wind excitation spectrum and the joint PDFs were also investigated. The turbulence standard deviation had a comparatively larger impact than that of horizontal mean wind speed in this study. Therefore, when one assesses the safety of an OWT under horizontal stochastic excitation, turbulence standard deviation is one of the most important aspects that should be taken into account.
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Nomenclature
OWT | Offshore wind turbine |
PI | Path integration |
SDOF | Single-degree-of-freedom |
PDF(s) | Probability density function(s) |
FFT | Fast Fourier transform |
MCS | Monte Carlo simulation |
FP | Fokker–Planck |
2D/4D/6D | Two-/four-/six-dimensional dynamic systems |
GPU/CPU | Graphics/center processing unit |
TRF | Transfer function |
FEM | Finite element model |
SDE | Stochastic differential equation |
RKM | Runge–Kutta–Maruyama |
TPD | Transition probability density |
m, c, k | Mass, damping, stiffness |
u, , | Displacement, velocity, acceleration |
, , | Natural frequency, critical damping, damping ratio |
Swind, SFF | turbulence wind spectrum, wind excitation spectrum |
Displacement, velocity, relative wind excitation, state variable in filter equations | |
Parameters in Ito process X | |
, | Transition PDF, previous PDF |
Jacobi determinant, Dirac delta function symbol | |
Vhub, σk | Horizontal mean wind speed, turbulence standard deviation |
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Parameters | Values |
---|---|
Blade number | 3 |
Hub height above sea level (m) | 90 |
Tower diameter base, top (m) | 4.3, 3.2 (linear variation) |
Tower thickness base, top (mm) | 50, 30 (linear variation) |
Rotor-nacelle mass (t) | 180 |
Foundation mass (t) | 2700 |
Material | Young’s Modulus (GPa) | Poisson’s Ratio | Density (kg/m3) |
---|---|---|---|
Steel | 200.0 | 0.30 | 7850 |
Concrete | 36.0 | 0.20 | 2500 |
Depth | Young’s Modulus (MPa) | Poisson’s Ratio | Cohesion (kPa) | Friction Angle (°) |
---|---|---|---|---|
12.0 | 15.0 | 0.30 | 9.0 | 31.0 |
12.0 | 31.5 | 0.35 | 4.4 | 33.5 |
12.2 | 20.0 | 0.30 | 4.7 | 33.2 |
13.3 | 33.0 | 0.23 | 10.8 | 29.1 |
14.1 | 39.0 | 0.25 | 5.2 | 33.0 |
Mode Order | Eigenvalue rad/s | Model |
---|---|---|
1st | 1.98 | For-aft |
2nd | 15.52 | For-aft |
Cases | Mean Wind Speed Vhub (m/s) | Turbulence Standard Deviation σk (m/s) | Turbulence Intensity I | α | β | γ |
---|---|---|---|---|---|---|
Case 1 | 16 | 2.884 | 0.180 | 0.009 | 0.134 | 0.534 |
Case 2 | 20 | 2.472 | 0.124 | 0.010 | 0.140 | 0.484 |
Case 3 | 20 | 2.884 | 0.144 | 0.009 | 0.141 | 0.565 |
Case 4 | 20 | 3.296 | 0.165 | 0.010 | 0.139 | 0.638 |
Case 5 | 24 | 2.884 | 0.120 | 0.009 | 0.145 | 0.590 |
Cases | Case 1 | Case 2 | Case 3 | Case 4 | Case 5 |
---|---|---|---|---|---|
Reliability (%) | 99.84 | 99.90 | 99.75 | 99.41 | 99.71 |
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Zhao, Y.; Lian, J.; Lian, C.; Dong, X.; Wang, H.; Liu, C.; Jiang, Q.; Wang, P. Stochastic Dynamic Analysis of an Offshore Wind Turbine Structure by the Path Integration Method. Energies 2019, 12, 3051. https://doi.org/10.3390/en12163051
Zhao Y, Lian J, Lian C, Dong X, Wang H, Liu C, Jiang Q, Wang P. Stochastic Dynamic Analysis of an Offshore Wind Turbine Structure by the Path Integration Method. Energies. 2019; 12(16):3051. https://doi.org/10.3390/en12163051
Chicago/Turabian StyleZhao, Yue, Jijian Lian, Chong Lian, Xiaofeng Dong, Haijun Wang, Chunxi Liu, Qi Jiang, and Pengwen Wang. 2019. "Stochastic Dynamic Analysis of an Offshore Wind Turbine Structure by the Path Integration Method" Energies 12, no. 16: 3051. https://doi.org/10.3390/en12163051