# Sensitivity Analysis of Modal Parameters of a Jacket Offshore Wind Turbine to Operational Conditions

^{1}

^{2}

^{3}

^{*}

*JMSE*– Recent Advances and Future Perspectives)

## Abstract

**:**

## 1. Introduction

_{0}), which is often the same for the FA and SS directions, should therefore lie within three possible regions: soft–soft, soft–stiff (the common one), and stiff–stiff, as shown in Figure 1. Three design types are: (a) soft–soft design where f

_{0}≤ f

_{1P,min}in which the structure is very flexible, (b) soft–stiff design, meaning the f

_{0}lies between 1P and 3P with safety margins (f

_{1P,max}≤ f

_{0}≤ f

_{3P,min}), and (c) stiff–stiff design where f

_{0}> f

_{3P,max}with a very stiff support structure. The stiff–stiff design is considered a conservative design and requires a very stiff and massive tower and foundation with a higher cost of construction and installation than a soft–stiff support structure design. So, the soft–stiff design is the economical solution for avoiding the resonance of the wind turbine system. The optimal design of a wind turbine system requires the consideration of any change in the system frequency due to the environmental/operational conditions, which is discussed in this paper. The OWT’s dynamic response to wind and wave loads also depends on the natural frequency of the wind turbine system due to the dynamic nature of the loads and the relatively low first frequency of the system. A fatigue assessment can be performed for the OWT using environmental conditions during its service lifetime by continuous monitoring during the operation of the OWT [16,17].

## 2. Materials and Methods

#### 2.1. B2 OWT at the BIWF

#### 2.1.1. B2 OWT Structural Information

^{2}. The jacket and tower are connected with a rigid transition piece (TP) in the model. The TP mass is 42.8 tons and the mass of the deck legs attached to it is 38.0 tons, leading to an assumed total mass of 80.8 tons for the TP. The RNA mass is specified as 430 tons with the center of mass 4.2 m upwind of the yaw axis and 2.1 m above the tower top. The mass moment of inertia matrix [I] of the RNA at the tower top was computed from the OpenFAST model as given in Equation (1). The values are not shown, as they are protected under a nondisclosure agreement.

_{xx}, I

_{yy}, and I

_{zz}are moments of inertia, and I

_{xy}, I

_{yx}, I

_{xz}, I

_{zx}, I

_{yz}, and I

_{zy}are products of inertia; x, y, and z are the nacelle frame coordinates, as shown later in the paper in Figure 7.

#### 2.1.2. Instrumentation of Monitoring System and Identified Modal Parameters

#### 2.1.3. System Identification Results

#### 2.2. Modeling of B2 OWT

#### 2.2.1. OpenSees and SAP2000 Models

#### 2.2.2. OpenFAST Model

_{g}follows the nominal wind direction (positive downwind) and the z-axis is vertical. The FA and SS directions are assumed to be in the x

_{n}and y

_{n}directions, respectively, and are perpendicular to each other.

**Figure 7.**OpenFAST main coordinate system [57].

#### 2.2.3. Model–Data Correlation

_{1},φ

_{2}are two mode shapes that are compared, and by design 0 ≤ MAC ≤ 1. A MAC value of 1 means that the mode shapes are identical, and a MAC value of 0 means perpendicular mode shapes. Here, MAC values are in the range of 0.92–0.97 which indicates a good match between the OpenFAST model and the identified modes. The OpenSees and SAP2000 models result in identical natural frequencies and have MAC = 1 for their mode shapes. Therefore, only the OpenSees model is included in Table 1.

## 3. Results

#### 3.1. Numerical Sensitivity Analysis of Modal Parameters

#### 3.1.1. Considered Environmental/Operational Conditions

#### 3.1.2. Sensitivity of Modal Parameters to Considered Conditions in OpenFAST

#### 3.1.3. Sensitivity of Modal Parameters with Respect to Individual Operational Parameters

#### Effect of Wind Speed

_{T}) for different wind speeds are given in Table 3. The maximum C

_{T}is 0.61 (wind speed of 11 m/s) and the lowest is 0.16 (wind speed of 31 m/s). Therefore, the thrust force and aerodynamic stiffening effect decrease in Region 3 with the increase in wind speed, resulting in a decrease in the natural frequencies of the FA mode. Reduction in the TSR also results in less aerodynamic damping.

#### Effect of Pitch Angle of Blades

#### Effect of Rotor Speed

#### Effect of Nacelle Yaw Angle

#### Effect of Mean Sea Level

#### Effect of Soil Stiffness

^{9}Pa (initial soil spring value) to 2.36 × 10

^{9}× 20 Pa with increments of 2.36 × 10

^{9}× 0.5 is considered in the OpenSees FE model of the B2 OWT, where the soil spring magnifying coefficient increases from 1 to 20. As shown in Figure 14, the vertical soil spring has small effects (less than 1% stiffening effect) on both the FA and SS natural frequencies. A similar study was performed for effects of lateral and rotational soil springs, and it was concluded that the effects of lateral and rotational soil springs are much smaller than vertical springs and thus negligible. However, soil springs are reported to affect the monopiles more significantly [58].

#### 3.1.4. Analytical Sensitivity

#### Aerodynamic Damping—Analytical Expressions

_{c}is the critical damping equal to 2mω

_{n}, c

_{adp}is the aerodynamic damping term equal to $\frac{B}{2}\rho {c}_{l\alpha}\mathsf{\Omega}{s}_{1b}$, S

_{1}

_{b}(m

^{3}) is the first moment of area of one blade, B is the number of blades, Ω is the rotational speed, $\rho $ is the atmospheric density, C

_{lα}is an assumed lift curve slope (assumed to be constant along the blade span, i.e., before stall), m is the RNA mass, and ω

_{n}is the first natural frequency of the wind turbine. The parameters used for the B2 OWT are S

_{1}

_{b}= 7080 m

^{3}, B = 3, Ω = 1.204 rad/s (11.5 rpm), $\rho $ = 1.225 kg/m

^{3}, C

_{lα}= 2$\pi $ rad

^{−1}, m = 4.3 × 10

^{5}kg, and ω

_{n}= 1.71 rad/s (f

_{n}= 0.272 Hz). This leads to an aerodynamic damping ratio of ζ = 6.7%. On the other hand, the numerical model’s total damping ratio is 8.5% at the rated rotor speed and rated wind speed, shown in Figure 9. The simulation damping ratio is computed as 1% for a parked turbine, which is the structural Rayleigh damping. The damping ratio at the rated wind speed includes both structural and aerodynamic damping. So, the total analytical damping ratio is 7.7%, which is the summation of the structural damping of 1% and the aerodynamic damping of 6.7%. The total analytical damping of 7.7% is in reasonable agreement with the numerical damping ratio of 8.5% from OpenFAST, as shown in Figure 10. The identified damping ratios from measured data, shown in Figure 4, are between 3 and 15% for wind speed greater than or equal to the rated wind speed, and the analytical damping ratio is within this range.

#### Rotational Effects on Stiffness

#### 3.2. Sensitivity Analysis Using Experimental Data

_{1}to X

_{6}) are selected as the inputs to multivariable linear regression models with modal parameters (natural frequencies and damping ratios of the FA mode) as the outputs. The input (or predictor) variables, X

_{1}to X

_{6}, are the normalized wind speed, rotor speed, power, yaw angle, wind misalignment, and collective pitch angle of the blades. All the input values are available in the SCADA data of the turbine and are averaged over 10-min windows. The output or response, Y, is either the first FA natural frequency or the first FA damping ratio, estimated using the acceleration measurements through SSI-DATA [70]. All variables are scaled to be in the range [0, 1]. Three models are considered in this sensitivity study: (1) the standard multiple linear regression model, (2) the multiple linear regression model with interaction terms, and (3) the polynomial degree 2 regression model with interaction terms. The model forms 1–3 are represented in Equations (5)–(7), respectively. The regression coefficients are estimated through a least squares problem.

- $\widehat{y}$: prediction of Y based on X
_{i}= x_{i}for any i = 1, …, 6

^{2}, which is used to measure the goodness-of-fit. It provides the measure of fit in a form of the proportion of variances, and it takes a value between 0 and 1. R

^{2}= 1 means all the variability in the output (i.e., frequency and damping) can be explained by the inputs (i.e., perfect fit), and R

^{2}close to zero indicates that the regression model cannot explain any of the variability in the output (i.e., worst fit).

^{2}statistic for the FA frequency of all models is relatively low, and it increases slightly with additional model complexity. This means that these simple regression models can account for some variabilities in the first FA frequency and damping ratio. For example, R

^{2}is 0.48 for Model 3, indicating that 48% of the variability in the output (first FA frequency) has been explained by the regression model. The R

^{2}for the first FA damping ratio is larger than those for the FA frequency, 0.64 for Model 3; therefore, the variability of the first FA damping ratio is better explained by the regression models. The R

^{2}of the first FA damping ratio also increases slightly when adding model complexity.

## 4. Discussion

## 5. Summary and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Normalized power spectral density for typical wind and wave loads, and the range of rotor-induced excitation ranges (1P and 3P ranges) to avoid resonance of the Block Island OWT.

**Figure 2.**Wind turbines B1–B3 (from

**left**to

**right**) standing in the Block Island Wind Farm, RI, USA (Photo: Michael Dwyer).

**Figure 3.**Instrumentation layout of sensors along with the BIWF-B2 turbine, reprinted from [43], with permission from Elsevier, 2023.

**Figure 4.**Identified first FA/SS natural frequencies and damping ratios of the B2 turbine using experimental data of one week of operation (21 April 2021 to 27 April 2021) and one day of idling (14 June 2021). Red stars and blue triangles are related to the idling and operating turbine, respectively.

**Figure 9.**Campbell diagram of the B2 OWT turbine for the first FA and SS bending modes according to the OpenFAST model, in the cases of flexible blades (1st Tower FA and 1st Tower SS) and rigid blades (1st Tower FA-Rigid and 1st Tower SS-Rigid), and the analytical damping ratio for the first FA mode.

**Figure 10.**Campbell diagrams for the first FA and SS tower bending modes of the OWT in the parked condition (rotor rpm = 0).

**Figure 11.**Campbell diagrams for the first FA and SS tower bending modes of the B2 OWT operating in Region 3*.

**Figure 12.**Campbell diagrams for the first FA and SS tower bending modes of the B2 OWT operating at a constant wind speed of 7 m/s and varying rotor speed.

**Figure 13.**Yaw angle effect on the first FA and SS tower bending modes of the B2 OWT, operating at the rated wind speed of 11 m/s and rotor speed of 11.5 rpm.

**Figure 14.**Effect of vertical soil spring on the first FA and SS tower bending modes of the B2 OWT in a parked condition.

Mode | OpenFAST Freq. [Hz] Flexible | OpenFAST Freq. [Hz] Rigid | OpenSees Freq. [Hz] | Mean (std) Identified Freq. [Hz] | Mean (std) MAC of OpenFAST vs. Identified | Mean (std) MAC of Opensees vs. Identified |
---|---|---|---|---|---|---|

First SS | 0.275 | 0.277 | 0.288 | 0.292 (0.001) ^{1} | 0.97 (0.04) ^{1} | 0.97 (0.04) ^{1} |

First FA | 0.272 | 0.279 | 0.291 | 0.313 (0.007) | 0.92 (0.06) | 0.92 (0.06) |

Second SS | 1.66 | 1.60 | 1.54 | 1.98 (0.016) | 0.95 (0.03) | 0.97 (0.03) |

Second FA | 2.41 | 1.87 | 1.77 | 2.24 (0.076) | 0.94 (0.07) | 0.93 (0.07) |

^{1}Mean (standard deviation) is computed over 144 data sets collected on 22 April 2021.

**Table 2.**Selected wind speeds, rotor speeds, and pitch angle of blades for OpenFAST simulations of the B2 OWT.

Region | Wind Speed (m/s) | Rotor Speed (rpm) | Blade Pitch (°) |
---|---|---|---|

Region 1 | 0.0 | 0.0 | 0 |

Region 2 | 3.0 | 3.9 | 0 |

7.0 | 8.9 | 0 | |

11.0 | 11.5 | 0 | |

Region 3 * | 15 | 11.5 | 9 |

19 | 11.5 | 14 | |

23 | 11.5 | 19 | |

27 * | 11.5 * | 24 * | |

31 * | 11.5 * | 29 * |

**Table 3.**Mean of the tip speed ratio and rotor aerodynamic thrust coefficient over time, obtained from the OpenFAST simulations of the turbine operating at the rated rotor speed, varying wind speed, and zero pitch angle of blades in Region 3*.

Wind Speed (m/s) | Rotor Aero C_{T} (-) | TSR (-) |
---|---|---|

11 | 0.61 | 8.19 |

15 | 0.45 | 5.98 |

19 | 0.33 | 4.70 |

23 | 0.20 | 3.92 |

27 | 0.18 | 3.33 |

31 | 0.16 | 2.82 |

Model | FA Frequency | FA Damping |
---|---|---|

1 | 0.38 | 0.59 |

2 | 0.46 | 0.63 |

3 | 0.48 | 0.64 |

Regression Coefficient | Related Variable Name | FA Frequency | FA Damping |
---|---|---|---|

Intercept (${\widehat{\beta}}_{0}$) | Constant | 0.30 | 1.80 |

${\widehat{\beta}}_{1}$ | Wind speed | −0.01 | −1.48 |

${\widehat{\beta}}_{2}$ | Rotor speed | 0.06 | 3.90 |

${\widehat{\beta}}_{3}$ | Power | −0.03 | 3.21 |

${\widehat{\beta}}_{4}$ | Yaw | −0.01 | −0.81 |

${\widehat{\beta}}_{5}$ | Misalignment | −0.01 | −0.95 |

${\widehat{\beta}}_{6}$ | Pitch | −0.01 | −1.57 |

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## Share and Cite

**MDPI and ACS Style**

Partovi-Mehr, N.; Branlard, E.; Song, M.; Moaveni, B.; Hines, E.M.; Robertson, A.
Sensitivity Analysis of Modal Parameters of a Jacket Offshore Wind Turbine to Operational Conditions. *J. Mar. Sci. Eng.* **2023**, *11*, 1524.
https://doi.org/10.3390/jmse11081524

**AMA Style**

Partovi-Mehr N, Branlard E, Song M, Moaveni B, Hines EM, Robertson A.
Sensitivity Analysis of Modal Parameters of a Jacket Offshore Wind Turbine to Operational Conditions. *Journal of Marine Science and Engineering*. 2023; 11(8):1524.
https://doi.org/10.3390/jmse11081524

**Chicago/Turabian Style**

Partovi-Mehr, Nasim, Emmanuel Branlard, Mingming Song, Babak Moaveni, Eric M. Hines, and Amy Robertson.
2023. "Sensitivity Analysis of Modal Parameters of a Jacket Offshore Wind Turbine to Operational Conditions" *Journal of Marine Science and Engineering* 11, no. 8: 1524.
https://doi.org/10.3390/jmse11081524