Identification of Wind Load Exerted on the Jacket Wind Turbines from Optimally Placed Strain Gauges Using C-Optimal Design and Mathematical Model Reduction
Abstract
:1. Introduction
2. Theory of Wind Load Identification for Offshore Wind Turbines
2.1. Frequency-Domain Dynamic Load Identification Theory
2.2. Ill Posedness of Systems of Linear Equations
2.3. Methods for Linear Systems
- (1)
- Moore–Penrose pseudo-inverse method;
- (2)
- TRM + L-curve: solving a system of linear equations using the TRM, where the regularisation parameter is derived by the L-curve criterion;
- (3)
- TRM + GCV: solving a system of linear equations using the TRM, where the regularisation parameter is derived using the GCV criterion;
- (4)
- TSVD + L-curve: solving a system of linear equations using the TSVD method, where the regularisation parameter is derived using the L-curve criterion;
- (5)
- TSVD + GCV: solving a system of linear equations using the TSVD method, where the regularisation parameter is derived using the GCV criterion.
3. Mathematical Model for Dynamic Load Identification in the Frequency Domain
3.1. Determination of Sensor Orientation
3.2. Analysis of the Effects of Wave Loading on Wind Turbine Structures
3.3. Effects of Wind Loads on Wind Turbine Tower Structures
3.4. Determination of Initial Sensor Position
3.5. Mathematical Modelling and Analysis of the Effects of Load Identification
4. Sensor Arrangement Optimisation Methods
- (1)
- For the variation in the condition number of in the initial mathematical model, as shown in Figure 13, the frequency response function matrix corresponding to the maximum condition number is selected. For this frequency response function matrix , compute all its remaining matrices to obtain , ,..., and .
- (2)
- Calculate all remaining condition numbers of the matrix , that is, the condition numbers of , ,..., and .
- (3)
- Select the residual matrix corresponding to the smallest number of residual conditions; that is, the matrix becomes , completing the first reduction.
- (4)
- Calculate all remaining condition numbers of the matrix , that is, the condition numbers of , , …, and .
- (5)
- Select the residual matrix corresponding to the smallest residual condition number; that is, the matrix becomes , completing the second reduction.
- (6)
- This continues until the matrix is reduced to . The change in the frequency response function matrix condition number as the number of reductions increases is shown in Figure 14. As shown in this figure, the condition number decreased from more than 500 to approximately 15, showing an obvious change.
- (7)
- Compare matrix with matrix to determine the final measurement point location. The final locations of the measurement points are shown in Figure 15b.
- (1)
- Select the frequency response function matrix corresponding to the peak of the condition number in Figure 13 and randomly select m rows in this frequency response function matrix to obtain the initial matrix and the remaining matrix .
- (2)
- Select a row from the remaining matrix and add it to to obtain the expanded matrix . There are (n–m) expanded matrices in total.
- (3)
- Calculate the condition numbers of all the expanded matrices and obtain the expanded matrix corresponding to the minimum condition number.
- (4)
- Delete one row from the expanded matrix to obtain the original dimension matrix . There are (m + 1) original dimension matrices in total.
- (5)
- Calculate the condition numbers of all the original dimension matrices to obtain the original dimension matrix corresponding to the minimum condition number.
- (6)
- Steps (2)–(4) are the exchanges of measurement point positions. Repeat steps (2)–(4) until the number of conditions is no longer reduced, or the number of sequential exchanges is defined artificially, and the loop is jumped out when the number of runs is achieved. The entire process is shown in Figure 18.
- (7)
- Compare the final matrix with the total matrix to determine the location of the final measurement point. The final location of the measurement point is shown in Figure 15a.
5. Frequency-Domain Fatigue Analysis
- (1)
- The narrowband approximation, which employs Equation (25) for damage calculation, represents a more conservative methodology [46].
- (2)
- The bandwidth correction method involves the application of a correction to Equation (27), as expressed in Equation (26)
- (3)
- Fatigue damage can be computed using an approximate probability density function formula, widely acknowledged as the Dirlik formula [50].
6. Algorithm Validation
6.1. Identification of Wind Loads at a Wind Speed of 10 m/s
6.2. Identification of Wind Loads at a Wind Speed of 12 m/s
6.3. Identification of Wind Loads at a Wind Speed of 24 m/s
7. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Nomenclature
The force in the X-direction | |
The force in the Y-direction | |
The force in the Z-direction | |
The moment in the X-direction | |
The moment in the Y-direction | |
The moment in the Z-direction | |
Frequency response function matrix | |
The superscript denotes the conjugate transpose | |
Excitation PSD matrix | |
Response PSD matrix | |
Eigenvalue | |
Eigenvector | |
Pseudo-response vector | |
Pseudo-excitation vector | |
Unit harmonic excitation | |
The superscript denotes generalised inversion | |
Error in the response data | |
Left singular vector matrix | |
Right singular vector matrix | |
Identity matrix | |
Singular value matrix | |
Matrix norm | |
Determinant of a matrix | |
The superscript denotes transpose | |
The regularisation parameter | |
Stress in the X-direction | |
Stress in the Y-direction | |
Shear stress | |
First principal stress | |
Third principal stress | |
Angle between the direction of the principal stress and the x-axis | |
Mean wind speed | |
Friction velocity | |
Karman constant | |
Fatigue damage value | |
Zeroth-order spectral moment | |
Parameters of the S-N curve | |
Parameters of the S-N curve | |
Bandwidth correction factor | |
Average number of cycles per unit time | |
latitude | |
horizontal wind speed power spectrum | |
horizontal pulsating wind speed root variance | |
ground roughness length | |
gradient height | |
Coriolis constant |
Appendix A
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Node Number | SNR (dB) | SNR (dB) | SNR (dB) |
---|---|---|---|
Node 1 | 39.0920 | 38.8284 | 39.9995 |
Node 2 | 52.2057 | 37.0709 | 32.1821 |
Node 3 | 52.7516 | 37.7086 | 33.5127 |
Node 4 | 50.9226 | 36.7145 | 32.8440 |
Node 5 | 50.8579 | 38.9891 | 35.6809 |
Node 6 | 50.7789 | 40.1269 | 37.3964 |
Node 7 | 52.4072 | 41.6726 | 50.6944 |
Node 8 | 52.7076 | 33.0891 | 44.4267 |
Node 9 | 49.7327 | 27.8918 | 40.9444 |
Node 10 | 47.3923 | 27.9521 | 40.2653 |
Node 11 | 44.9178 | 26.0324 | 40.7441 |
Node 12 | 43.7101 | 24.7581 | 40.8920 |
Node 13 | 47.6613 | 38.2679 | 43.3208 |
Node 14 | 55.9286 | 36.2682 | 40.8216 |
Node 15 | 56.9565 | 39.3064 | 42.9643 |
Node 16 | 56.9636 | 39.5892 | 41.6206 |
Node 17 | 56.3502 | 42.3164 | 44.4427 |
Node 18 | 55.9141 | 43.9866 | 46.2169 |
Node 19 | 43.6565 | 39.7544 | 43.4904 |
Node 20 | 37.4465 | 34.2639 | 36.1277 |
Node 21 | 29.7839 | 32.0071 | 31.5327 |
Node 22 | 33.4227 | 34.3829 | 31.2402 |
Node 23 | 33.9734 | 35.4146 | 30.3160 |
Node 24 | 33.9923 | 36.1781 | 29.2770 |
Distance from Root (m) | Chord (m) | Twist (deg) | Thickness (%) | Pitch Axis (%) | Foil Section Number |
---|---|---|---|---|---|
0 | 2.90 | 20 | 99.99 | 50.00 | 1 |
10 | 3.82 | 12.28 | 65.85 | 50.85 | 1 |
20 | 4.29 | 6.91 | 36.28 | 47.75 | 2 |
30 | 3.41 | 3.57 | 32.76 | 43.21 | 2 |
40 | 2.63 | 1.20 | 31.19 | 39.87 | 2 |
50 | 2.06 | −0.49 | 29.67 | 37.28 | 3 |
60 | 1.61 | −1.43 | 26.83 | 36.99 | 3 |
70 | 1.17 | −0.28 | 22.27 | 37.76 | 4 |
78.9 | 0.05 | 2.54 | 21.00 | 49.20 | 4 |
Distance from Root (m) | Centre of Mass (x’) (%) | Centre of Mass (y’) (%) | Mass Axis Orientation (deg) | Radius of Gyration Ratio |
---|---|---|---|---|
0 | 0.00 | 50.01 | 0.00 | 1.00 |
10 | −0.09 | 49.74 | 19.35 | 0.83 |
20 | −0.28 | 47.65 | 10.28 | 0.56 |
30 | −0.75 | 44.75 | 7.44 | 0.54 |
40 | −0.79 | 42.26 | 4.59 | 0.50 |
50 | −0.88 | 40.20 | 2.34 | 0.49 |
60 | 0.09 | 40.12 | 0.61 | 0.45 |
70 | 2.86 | 40.08 | 1.29 | 0.38 |
78.9 | 2.53 | 49.53 | 3.04 | 0.20 |
Parameter | Value |
---|---|
Latitude (deg) | 34 |
Surface roughness | 0.01 |
Average wind speed (m/s) | 8 |
Angular velocity of earth’s rotation (rad/s) | |
Height of hub centre of wind turbine (m) | 103 |
Component | TRM + L-Curve | TRM + GCV | TSVD+ L-Curve | TSVD + GCV | Moore–Penrose |
---|---|---|---|---|---|
() | 6.60 × 108 | 1.23 × 108 | 6.24 × 108 | 2.12 × 108 | 2.68 × 107 |
() | 4.91 × 107 | 2.65 × 107 | 8.58 × 107 | 2.24 × 107 | 1.42 × 107 |
() | 2.09 × 108 | 6.33 × 107 | 9.02 × 107 | 6.46 × 107 | 8.08 × 107 |
() | 8.44 × 109 | 5.01 × 109 | 6.27 × 109 | 2.89 × 109 | 8.28 × 108 |
()) | 3.44 × 1011 | 2.16 × 1011 | 2.88 × 1011 | 2.68 × 1011 | 9.74 × 109 |
()) | 4.46 × 1011 | 1.17 × 1011 | 2.16 × 1011 | 1.14 × 1011 | 6.61 × 109 |
Component | MAE (COD) | MAE (DOD) | MAE (Initial) |
---|---|---|---|
() | 1.01 × 108 | 1.33 × 108 | 2.68 × 107 |
() | 2.69 × 107 | 7.20 × 107 | 1.42 × 107 |
() | 9.84 × 107 | 2.91 × 108 | 8.08 × 107 |
() | 1.60 × 109 | 1.62 × 1010 | 8.28 × 108 |
()) | 6.15 × 109 | 3.27 × 1010 | 9.74 × 109 |
()) | 6.60 × 109 | 1.40 × 1010 | 6.61 × 109 |
Node Number | Fatigue Damage Values Calculated by Simulation | Fatigue Damage Values Calculated from Identified Load Spectra | Percentage of Relative Error |
---|---|---|---|
Node A | 6.37 × 10−4 | 6.43 × 10−4 | 0.9% |
Node B | 7.17 × 10−4 | 7.09 × 10−4 | 1.2% |
Node C | 1.87 × 10−5 | 1.86 × 10−5 | 0.2% |
Node D | 8.26 × 10−4 | 8.26 × 10−4 | 0.1% |
Node E | 8.63 × 10−4 | 8.65 × 10−4 | 0.2% |
Node F | 1.00 × 10−3 | 9.95 × 10−4 | 0.5% |
Component | MAE |
---|---|
() | 1.34 × 108 |
() | 2.99 × 108 |
() | 1.17 × 108 |
() | 2.27 × 109 |
()) | 7.31 × 109 |
()) | 8.03 × 109 |
Node Number | Fatigue Damage Values Calculated by Simulation | Fatigue Damage Values Calculated from Identified Load Spectra | Percentage of Relative Error |
---|---|---|---|
Node A | 2.43 × 10−3 | 2.51 × 10−3 | 3.3% |
Node B | 1.17 × 10−3 | 1.15 × 10−3 | 1.8% |
Node C | 1.17 × 10−5 | 1.17 × 10−5 | 0.0% |
Node D | 1.51 × 10−3 | 1.51 × 10−3 | 0.4% |
Node E | 1.14 × 10−3 | 1.14 × 10−3 | 0.1% |
Node F | 1.60 × 10−3 | 1.58 × 10−3 | 1.3% |
Component | MAE |
---|---|
() | 1.41 × 108 |
() | 3.41 × 108 |
() | 1.56 × 108 |
() | 2.29 × 109 |
()) | 1.17 × 1010 |
()) | 1.10 × 1010 |
Node Number | Fatigue Damage Values Calculated by Simulation | Fatigue Damage Values Calculated from Identified Load Spectra | Percentage of Relative Error |
---|---|---|---|
Node A | 3.36 × 10−4 | 3.39 × 10−4 | 0.9% |
Node B | 1.86 × 10−3 | 1.83 × 10−3 | 1.8% |
Node C | 2.66 × 10−5 | 2.66 × 10−5 | 0.1% |
Node D | 2.10 × 10−3 | 2.10 × 10−3 | 0.0% |
Node E | 7.48 × 10−4 | 7.48 × 10−4 | 0.0% |
Node F | 1.61 × 10−3 | 1.59 × 10−3 | 1.7% |
Component | MAE |
---|---|
() | 3.40 × 108 |
() | 1.03 × 109 |
() | 5.98 × 108 |
() | 6.64 × 109 |
()) | 3.01 × 1010 |
()) | 3.45 × 1010 |
Node Number | Fatigue Damage Values Calculated by Simulation | Fatigue Damage Values Calculated from Identified Load Spectra | Percentage of Relative Error |
---|---|---|---|
Node A | 6.23 × 10−5 | 6.59 × 10−5 | 5.7% |
Node B | 2.14 × 10−4 | 2.26 × 10−4 | 4.2% |
Node C | 7.31 × 10−5 | 7.68 × 10−5 | 5.1% |
Node D | 5.68 × 10−4 | 5.74 × 10−4 | 1.1% |
Node E | 4.13 × 10−3 | 4.29 × 10−3 | 3.9% |
Node F | 6.35 × 10−5 | 6.68 × 10−5 | 5.2% |
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Zhu, F.; Zhang, M.; Ma, F.; Li, Z.; Qu, X. Identification of Wind Load Exerted on the Jacket Wind Turbines from Optimally Placed Strain Gauges Using C-Optimal Design and Mathematical Model Reduction. J. Mar. Sci. Eng. 2024, 12, 563. https://doi.org/10.3390/jmse12040563
Zhu F, Zhang M, Ma F, Li Z, Qu X. Identification of Wind Load Exerted on the Jacket Wind Turbines from Optimally Placed Strain Gauges Using C-Optimal Design and Mathematical Model Reduction. Journal of Marine Science and Engineering. 2024; 12(4):563. https://doi.org/10.3390/jmse12040563
Chicago/Turabian StyleZhu, Fan, Meng Zhang, Fuxuan Ma, Zhihua Li, and Xianqiang Qu. 2024. "Identification of Wind Load Exerted on the Jacket Wind Turbines from Optimally Placed Strain Gauges Using C-Optimal Design and Mathematical Model Reduction" Journal of Marine Science and Engineering 12, no. 4: 563. https://doi.org/10.3390/jmse12040563