Research on Parameter Influence of Offshore Wind Turbines Based on Measured Data Analysis

Abstract

Offshore wind turbines are prone to structural damage over time due to environmental factors, which increases operational costs and the risk of accidents. Early detection of structural damage through monitoring systems can help reduce maintenance costs. However, under complex external conditions and varying structural parameters, existing methods struggle to accurately and quickly detect damage. Understanding the factors that influence structural health is critical for effective long-term monitoring, as these factors directly affect the accuracy and timeliness of damage identification. This study comprehensively analyzed 5 MW offshore wind turbine measurement data, including constructing a digital twin model, establishing a surrogate model, and performing a sensitivity analysis. For monopile-based turbines, sensors in x and y directions were installed at four heights on the pile foundation and tower. Via Bayesian optimization, the finite element model’s structural parameters were updated to align its modal parameters with sensor data analysis results. The update efficiencies of different objective functions and the impacts of neural network hyperparameters on the surrogate model were examined. The sensitivity of the turbine’s structural parameters to modal parameters was studied. The results showed that the modal flexibility matrix is more effective in iteration. A 128-neuron, double-hidden-layer neural network balanced computational efficiency and accuracy well in the surrogate model for modal analysis. Flange damage and soil degradation near the pile mainly impacted the turbine’s health.

1. Introduction

Over the past two decades, the installed capacity of offshore wind turbines has experienced a remarkable expansion. In 2023 alone, a cumulative 117 GW of wind power capacity was integrated into the global grid. China is the nation with the largest renewable energy power generation system globally. China’s renewable energy installed capacity reached 1450 GW by the end of 2023, surpassing its thermal power generation capacity for the first time [1]. Despite the booming wind turbine market worldwide, the lifespan and stability of the wind turbines already in service present challenges that the wind turbine market must address for its sustainable development [2,3]. Structural health monitoring (SHM) is a technology commonly utilized in lifespan assessment and stability monitoring [4,5]. This paper combines SHM technology with structural inversion technology based on Bayesian optimization to monitor the health status of a specific offshore wind turbine pile foundation. The study explores the implications of potential damage conditions of the wind turbine on structural stability and lifespan.

SHM is a specialized field focused on the continuous monitoring of civil structures and plays a crucial role in validating structural design and ensuring the safety inspection of construction [3,6,7]. A typical SHM system obtains information, such as structural loads, structural responses, and environmental factors, through a sensing system installed on the structure and then collects and analyzes this information via a transmission system and a processing system. The number of monitored high-rise buildings is increasing. The modal information (natural frequencies, damping ratios, modal shapes, etc.) of high-rise buildings is sensitive to structural changes, and SHM based on vibration information has been widely investigated. Reynders et al. proposed a comprehensive set of automated operational modal analysis (AOMA), which achieved good results and inspired the subsequent work of other scholars [8]. However, the work of Reynders et al. neglected the influence of structural uncertainties. When monitoring operational structures such as wind turbines that are located in a relatively harsh environment, the traditional SHM process based on vibration information may experience a reduction in accuracy during the system analysis due to effects such as the harmonic effect [9] and the gyroscopic effect [10,11]. To alleviate the harmonic effect, Civera [12] et al. incorporated a low-pass filter in AOMA to cut off high-frequency signals (2023). Nevertheless, the harmonic frequencies generated by the offshore wind turbine rotor are low and close to the main frequency, making a low-pass filter unsuitable for offshore wind turbines.

Devriendt [13] et al. adopted a band-pass filter in the AOMA of offshore wind turbines, specifying a frequency range for interception (2013). However, the rotational speed of the wind turbine hub under operating conditions is positively correlated with the environmental wind speed, which also leads to the instability of the frequency of the harmonics received by the wind turbine [14]. Artificially specifying a frequency range for interception makes it difficult to ensure that SHM can accurately monitor structural state changes in a dynamic environment. In view of the above characteristics, this paper introduces the Kalman filter (KF) to suppress the components with low stability in the signal and reduce the impact of harmonic excitation.

After obtaining the modal parameters of the structure through AOMA, a digital twin (DT) is developed. A DT is a virtual model that accurately replicates the physical information of the wind turbine. It can calculate the remaining lifespan of the structure, simulate the stress condition of the structure with preset external loads, and contribute to the exploration of the mechanical characteristics of the structure [15,16,17]. This model mainly stems from minimizing the difference between the virtual output and the measured output, where the output usually refers to the structural modal parameters obtained through AOMA. For wind turbines, the use of a DT that automatically updates the virtual model according to the measured modal parameters can effectively enhance the monitoring efficiency of changes in structural reliability. The model update methods can be divided into two categories: deterministic optimization [18] and Bayesian inference [19]. Both of these methods have their own advantages and disadvantages in engineering and have been widely adopted. In wind turbines, due to the severe external environment, the measured modal parameters inevitably contain errors caused by factors such as wind force and mechanical operation, which subsequently affect accuracy during the DT update.

The main work of this research is to conduct health monitoring of offshore wind turbines using DT, which includes two major steps: using the stochastic subspace identification (SSI) method to conduct a systematic analysis of the measured data of offshore wind turbines to determine the modal parameters of the turbines, and using the measured modal parameters and the modal parameters of the DT model to construct an objective function for the deterministic optimization of the model.

The measured data of the offshore wind turbines are obtained from the data collected during the two-month operation of a supervisory control and data acquisition (SCADA) system of an offshore wind turbine in the East China Sea. After preprocessing these data with KF, a systematic analysis is carried out, using SSI to obtain the natural frequencies and modal shapes of the offshore wind turbines. Subsequently, the superstructure of the wind turbine is simplified, and an initial finite element model of the offshore wind turbine is constructed. The modal parameters obtained from the systematic analysis are used to optimize the model and update the material parameters of the parts of the wind turbine that are more prone to damage. In this process, different dynamic fingerprints are used to construct objective functions, and their performances are compared. Finally, a sensitivity analysis of the potential damage conditions of the wind turbine is conducted using DT.

In Section 1, the basic parameters and wind data processing of the monitored offshore wind turbine structure are introduced. In Section 2, the design and theory for data preprocessing, analysis, and digital twin model establishment are presented. In Section 3, the digital twin model establishment for an East China Sea wind turbine is demonstrated and the parameter impact on structural health is discussed via sensitivity analysis. Finally, in Section 4, the significance and deficiencies of this work are discussed. This study is distinctive in considering the influence of harmonic interference of offshore wind turbines. A systematic analysis of the potential damage conditions of the wind turbine is carried out using the measured data in combination with Bayesian optimization. These analyses are beneficial for providing guidance in the design and maintenance process of the wind turbine pile foundation.

2. Dataset

2.1. Structure Introduction

This study used data collected from a 5 MW, monopile-based, offshore wind turbine located in the East China Sea. The turbine has a height of 107.96 m, with a pile leg having an outer diameter ranging from 7 m to 8 m and a wall thickness of 0.075 m, and a tower frame with an outer diameter ranging from 4.02 m to 7 m and a wall thickness of 0.036 m to 0.045 m.

The turbine was instrumented with a series of measurement points. Eight measurement points were set up on its tower, which were distributed at four horizontal heights (as shown in Figure 1). These points were used for measuring the accelerations in the x and y directions, respectively.

Signal acquisition was conducted from 26 May to 30 July 2019, spanning 90 days. The sampling frequency was set at 20 Hz with continuous 24 h operation, where the collected data were segmented into 10 min intervals, each containing 12,000 data points (equivalent to 10 min duration). The acquired signals included horizontal acceleration of the pile foundation, ambient wind speed, and rotor speed. Throughout the monitoring period, no known structural damage or sensor malfunctions occurred, and variations in modal parameters were primarily driven by environmental factors.

2.2. Rotation Data Processing

Accelerometers were installed to monitor the vibration of the turbine in response to operational and environmental forces. The acceleration sensors were installed on the tower wall of the wind turbine, forming a coordinate system in the x and y directions, as shown in Figure 2.

The nacelle of the offshore wind turbine adjusts its yaw angle following the wind direction for maximum wind energy utilization. As the coordinate system does not change with the yaw angle, the collected acceleration data need processing based on the nacelle’s yaw angle, converting x and y accelerations to fore–aft (FA) and side–side (SS) ones as per Equations (1) and (2):

(1)$a_{FA}=a_x\cos ({150}^{\circ }-\theta )+a_y\sin ({150}^{\circ }-\theta )$

(2)$a_{SS}=a_x\sin ({150}^{\circ }-\theta )-a_y\cos ({150}^{\circ }-\theta )$

Among them, $a_x$ and $a_x$ are the acceleration data collected by the acceleration sensors in the x and y directions, respectively; $a_{FA}$ and $a_{SS}$ are the accelerations of the wind turbine in the along-wind direction and cross-wind direction, respectively; $\theta$ represents the yaw angle of the wind turbine; and $150°$ is the deviation angle between the coordinate system formed by the sensors and the north direction.

3. Methods

3.1. Kalman Filter

To validate the Gaussian distribution assumption of system process noise in Kalman filtering, we perform statistical characterization of sensor-collected noise signals. The key metrics are calculated as follows:

(3)$Skew(X)=E[({\displaystyle \frac{X-\mu }{\sigma }}{)}^3]={\displaystyle \frac{k_3}{{\sigma }^3}}={\displaystyle \frac{k_3}{k_2^{3/2}}}$

(4)$Kurt(X)=E[({\displaystyle \frac{X-\mu }{\sigma }}{)}^4]={\displaystyle \frac{E[{(X-\mu )}^4]}{{(E[{(X-\mu )}^2])}^2}}$

The above formulas represent Skewness and Kurtosis, respectively. When the absolute value of Skewness is less than 0.5 and the absolute value of Kurtosis is less than 1.2, the signal can be considered to follow a normal distribution. The calculation results of the signals from the eight channels in this study are presented in Table 1:

Next, the signal stability is quantitatively evaluated by calculating the Sample Entropy of the signals before and after preprocessing, using the following equation:

(5)$SampEn=-\ln {\displaystyle \frac{A}{B}}$

The average Sample Entropy of the denoised data across the eight channels is calculated to be 0.31, and the variance fluctuation range is 17%, indicating that the signal stability is satisfactory.

The periodic operation of the rotor of an offshore wind turbine generates harmonics corresponding to the rotational frequency, and the energy of the forced vibration caused by these harmonics may be close to that of the structural free vibration [20]. When the harmonic frequency is close to the modal frequency, the harmonic mode overwhelms the structural mode, which brings interference to the AOMA and reduces the analysis accuracy of AOMA [21]. However, the variance of the distribution of harmonic modes in the frequency domain is relatively high. Therefore, it is feasible to distinguish between harmonic modes and structural modes from a statistical perspective. In this paper, a data-driven method is first adopted to predict the structural acceleration data. Under the assumption of normal distribution for both the predicted data and the measured data, the two are combined to obtain the output with the highest probability, so as to suppress the components with poor stability in the signal. The specific process is as follows:

Construct the state Equation (6) and the observation Equation (7) of the system as follows:

(6)$x_k=A{x}_{k-1}+B{u}_{k-1}+{\omega }_k$

(7)$z_k=H{x}_k+{\nu }_k$

In the equations, $x_k$ is the state vector, $u_k$ is the system control vector, ${\omega }_k$ is the system noise vector, $z_k$ is the observation vector, and $v_k$ is the observation noise vector. A, B, and H are the state transition matrix, the control matrix, and the observation matrix, respectively.

It is assumed that the system noise vector and the observation noise vector are positive definite, symmetric, uncorrelated, and conform to the normal distribution with an expectation of 0:

(8)$\mathrm{E}(\omega )=0,\mathrm{cov}(\omega )=\mathrm{E}(\omega {\omega }^{\mathrm{T}})=Q$

(9)$\mathrm{E}(v)=0,\mathrm{cov}(v)=\mathrm{E}(v{v}^{\mathrm{T}})=R,\mathrm{E}(\omega v^{\mathrm{T}})=0$

Then, the recursive calculation ${\widehat{x}}_k^{-}\in R^n$ at the design time step is the state vector derived from combining prior knowledge with Equation (6) at the $k-1$ moment:

(10)${\widehat{x}}_k^{-}=A{\widehat{x}}_{k-1}+B{u}_{k-1}$

Taking into account the influence of harmonics and other noises, the prior predicted value of the state vector ${\widehat{x}}_k^{-}$, the innovation ${\omega }_k$, and the observed value $z_k$ are combined to establish ${\widehat{x}}_k$ as the prediction for the next moment. First, the deviations between the prior predicted value of the algorithm and the final predicted value of the algorithm and the measured data should be established.

(11)$e_k^{-}=x_k-{\widehat{x}}_k^{-}$

(12)$e_k=x_k-{\widehat{x}}_k$

The covariance of ${\widehat{x}}_k^{-}$ and ${\widehat{x}}_k$ should be solved.

(13)$P_k^{-}=\mathrm{E}[e_k^{-}{e}_k^{-\mathrm{T}}]$

(14)$P_k=E[e_k{e}_k^T]$

To reduce the proportion of harmonics in the signal, the Kalman gain matrix should be solved as the weight by using the covariance of the difference between the algorithm output and the measured data. The predicted data and the observed data are then linearly combined with weights:

(15)$P_k^{-}=A{P}_{k-1}{A}^{\mathrm{T}}+Q$

(16)$K_k=P_k^{-}{H}^{\mathrm{T}}{(H{P}_k^{-}{H}^{\mathrm{T}}+P)}^{-1}$

(17)${\widehat{x}}_k={\widehat{x}}_k^{-}+K_k(z_k-H{\widehat{x}}_k^{-})$

(18)$P_k=(I-K_{k}H)P_k^{-}$

In the equations, $K_k$ and $P_k$ are the Kalman gain matrix and the filtering offset matrix, respectively. Equations (17) and (18) are the update equation for the final output of the algorithm and the update equation for the filtering offset matrix, respectively.

3.2. Stochastic Subspace Identification

The SSI covariance (COV) is regarded as one of the best and most mature operational modal analysis methods [21,22]. This paper briefly introduces the main steps in the implementation of this method in the program.

First, a Hankel matrix for the response signals of the structure is constructed:

(19)$H=\left[\begin{array}{c}{Y}_{\mathrm{p}}\\ Y_{\mathrm{f}}\end{array}\right]={\displaystyle \frac{1}{\sqrt{p}}}\left[\begin{array}{ccccc}{y}_0& y_1& y_2& \cdots & y_{b-1}\\ y_1& y_2& y_3& \cdots & y_b\\ y_2& y_3& y_4& \cdots & y_{b-1}\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ y_{a-1}& y_a& y_{a+1}& \cdots & y_{a+b-1}\\ y_a& y_{a+1}& y_{a+2}& \cdots & y_{a+b}\\ y_{a+1}& y_{a+2}& y_{a+3}& \cdots & y_{a+b+1}\\ y_{a+2}& y_{a+3}& y_{a+4}& \cdots & y_{a+b+2}\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ y_{2a-1}& y_{2a}& y_{2a+1}& \cdots & y_{2a+b-1}\end{array}\right]$

In Equation (19), a generally takes a value that is half of the sampling frequency to meet the Nyquist frequency requirement, and b represents the number of columns of the Hankel matrix. The value of b is deduced inversely according to the total amount of data M in the time domain, that is:

(20)$b=M-2a+1$

where $y_n\in R^{l\times 1}$ is the n-th group of data in the time domain, and l is the number of channels of the signal. $Y_P\in R^{la\times b}$ and $Y_f\in R^{la\times b}$ in Equation (19) represent the past matrix and the future matrix, respectively.

A Toeplitz matrix is constructed by assembling the past matrix and the future matrix, which is the key to the COV in this method.

(21)$T_{\parallel a}=Y_f{Y}_p^{\mathrm{T}}$

Next, a singular value decomposition on Equation (21) is performed to obtain:

(22)$T_{\parallel a}=\left[\begin{array}{cc}{U}_1& U_2\end{array}\right]\left[\begin{array}{cc}{S}_1& 0\\ 0& 0\end{array}\right]\left[\begin{array}{c}{V}_1^{\mathrm{T}}\\ V_2^{\mathrm{T}}\end{array}\right]=U_1{S}_1{V}_1^{\mathrm{T}}$

In Equation (22), $U_1\in R^{la\times la}$ and $V_1\in R^{la\times la}$ are unitary matrices composed of the left and right singular vectors corresponding to nonzero singular values, respectively, and $S_1$ is a diagonal matrix of nonzero singular values arranged in descending order:

(23)$O_a=U_1{S}_1^{1/2}$

(24)$A=O_a{(1:l,:)}^{†}\ast O_a(l+1:2l,:)$

The observation matrix $O_a\in R^{la\times la}$ is solved through Equation (23), and the system matrix A is obtained by multiplying the upper and lower partition matrices of the observation matrix as shown in Equation (24). Next, to solve the modal parameters, an eigenvalue decomposition of A is performed, as shown in Equation (25):

(25)$A=\mathrm{\Psi }\mathrm{\Lambda }{\mathrm{\Psi }}^{-1}$

In Equation (25), $\mathsf{\Lambda }=diag\left({\mu }_i\right)$ is a diagonal matrix of eigenvalues of A, and $\Psi$ is a complex eigenvector matrix. The eigenvalue solution of the continuous system is as follows:

(26)${\lambda }_i={\displaystyle \frac{\ln ({\mu }_i)}{\mathrm{\Delta }t}}$

In a continuous system, the eigenvalues of the system exist in conjugate form, so the modal parameters can be expressed by Equation (27):

(27)${\lambda }_s^c,{\lambda }_s^{c^{\ast }}=-{\xi }_s{\omega }_s\pm j{\omega }_s\sqrt{1-{\xi }_s^2}$

Finally, the modal information is obtained as follows, where $f_i$, ${\xi }_i$, and ${\mathrm{\Phi }}_i$ correspond to the i-th order (natural frequency, damping ratio, and mode shape).

(28)$f_i={\displaystyle \frac{\left|{\lambda }_i\right|}{2\pi }}$

(29)${\xi }_i={\displaystyle \frac{{\lambda }_i+\lambda {\ast }_i}{2\sqrt{{\lambda }_i\lambda {\ast }_i}}}$

(30)${\mathrm{\Phi }}_i=C\ast {\Psi }_i$

3.3. Finite Element Model Establishment

Modeling was carried out in accordance with the wind turbine structural parameters designed by Goldwind Science and Technology Co., Ltd. (Urumqi, Xinjiang, China). The superstructure was modeled using a simplified design approach. (The specific simplified parameters are presented in Table 2). By simplifying the superstructure into equivalent mass points, the computational cost was effectively reduced. Additionally, the soil surrounding the pile was modeled as a solid to accurately simulate the pile–soil interaction. The normal behavior was configured as a ‘hard’ contact with penalty function constraints, and the tangential behavior’s friction formula was also set as a penalty function.

The wind load that was distributed along the pile exhibited a relatively minor influence on the stress of the pile. Consequently, our primary attention was directed toward the loads that act on the hub. In accordance with Bernoulli’s principle, within a system, the sum of the kinetic energy, gravitational potential energy, and pressure potential energy remains constant. Hence, when the wind turbine is in operation and a sudden reduction in pressure occurs, it gives rise to loads on the turbine. An equivalent concentrated load can be derived from the Bernoulli Equation [11]:

(31)$F_{areo}=0.5{\rho }_a\pi R_T^2{V}_S^2(1+2{\xi }_S/V_S)C_T$

(32)$C_T=4a_0(1-a_0)$

In this context, ${\rho }_a$ denotes the air density, which is set at 1.225 kg/m³. $R_T$ represents the hub radius, with the unit being meters. $V_S$ and ${\xi }_S$ stand for the average wind speed and the fluctuating wind speed, respectively. $C_T$ is the thrust coefficient, and $a_0$, which is the induction factor, is typically assigned a value of 0.5.

In accordance with the specifications put forward by the European Committee for Standardization, it is advisable to utilize the Kaimal spectrum [23] as the wind speed spectrum for the purpose of simulating the wind field:

(33)${\mathrm{S}}_f(n)={\displaystyle \frac{4x}{6.667n{(2+x^2)}^{5/3}}}$

(34)$x={\displaystyle \frac{n\mathrm{L}(z)}{{\overline{v}}_z}}$

(35)$\mathrm{L}(z)=300{[{\displaystyle \frac{z}{200}}]}^{0.67+0.05\mathrm{m}(z_0)}$

n represents frequency, and x is defined as $x={\displaystyle \frac{1200\mathrm{m}}{{\overline{v}}_z}}$, where ${\overline{v}}_z$ represents the average wind speed at height z above the pile’s sea level. $z_0$ designates the height of the point on the pile where the wind speed is zero relative to the sea level.

The wind speed spectrum having been identified, the linear wave superposition method was employed for this analysis as it is deemed the most appropriate approach to superimpose the cosine functions generated by the Kaimal wind speed spectrum. The time-domain graph illustrating the generated wind speed is presented in Figure 3.

The Morison equation was adopted for the simulation of irregular wave loads:

(36)$\mathrm{d}F=\mathrm{d}{F}_d+\mathrm{d}{F}_1=C_D\rho {\displaystyle \frac{D}{2}}u\left|u\right|\mathrm{d}z+C_M\rho {\displaystyle \frac{\mathsf{\pi }{d}^2}{\partial t}}\mathrm{d}z$

The parameters in Equation (36) are presented as follows: $C_D$ stands for the drag coefficient (also known as the resistance coefficient), $C_M$ is the inertia force coefficient, $D$ is the diameter of the pile, $\rho$ is the density of seawater (taken as 1025 kg/m³), $u$ represents the flow velocity, and ${\displaystyle \frac{\partial u}{\partial t}}$ represents the acceleration of the water flow.

For the simulation of irregular waves, we chose to continue using the linear wave superposition method. According to the linear wave theory, the flow velocity and acceleration in Equation (36) can be calculated using the following formulas:

(37)$\mathrm{u}(z,t)=\omega {\displaystyle \frac{\cosh (kz)}{\sinh (kd)}}\eta (t)$

(38)${\displaystyle \frac{\partial u(zt)}{\partial t}}={\omega }^2{\displaystyle \frac{\cosh (kz)}{\sinh (kd)}}\eta (t)$

In Equations (37) and (38), $\eta \left(t\right),k,\omega ,d,z$ are the wave surface elevation, wave number, circular frequency of the wave, water depth, height of the load-acting point, respectively, where $z$ takes 0 when below the sea surface and the distance from the point to the sea surface when above it.

Regarding the wave surface elevation function in this context, we chose to use the Pierson–Moskowitz (P-M) two-parameter spectrum revised by the International Towing Tank Conference for the simulation:

(39)$\eta (t)={\displaystyle \sum _{i=1}^m\sqrt{2S_{\eta \eta }({\hat{\omega }}_1)\mathrm{\Delta }{\omega }_i}\cos ({\tilde{\omega }}_{1}t+{\epsilon }_i)}$

The parameters in Equation (10) are defined as follows. $S_{\eta \eta }\left(\omega \right)$, which plays a crucial role in characterizing the wave energy distribution, is the P-M wave energy spectral density function ($S_{\eta \eta }\left(\omega \right)={\displaystyle \frac{173H_s^2}{T^4{\omega }^5}}{e}^{{\displaystyle \frac{-691}{{\omega }^4{T}^4}}}$). $m$, whose value is determined based on specific requirements and usually ranges from 50 to 100, represents the number of superimposed cosine waves. $\mathsf{\Delta }{\omega }_i$ refers to the frequency difference between the superimposed cosine functions, while ${\tilde{\omega }}_i$ stands for the representative frequency of the $i$-th cosine function. $H_S$ is the selected average wave height that influences the overall wave characteristics, and $T$ is the wave period. Additionally, ${\epsilon }_i$ is the initial phase value of the corresponding wave generated by random numbers, which adds variability to the wave representation.

Based on these defined parameters, by simultaneously solving Equations (9)–(12), an equation containing $dF$ and $dz$ could be obtained. Integrating this equation enabled us to derive the concentrated load that is equivalent to the wave force. Calculating the moment of this load allowed us to obtain the equivalent moment.

For instance, under Wave Condition 1 in this paper, we selected an average wave height of 6 m, a wavelength of 25 m, along with setting $C_D=1$ and $C_M=2$. The sampling frequency was configured to be within the range of 0.1 Hz to 5.1 Hz, and the time domain was set to 600 s. With a step size of 0.1 s, the simulation was carried out six times. The time-domain diagram of wave height is shown in Figure 4.

3.4. Design of the Model Updating the Objective Function

In the process of model updating, the modal parameters of the structure were regarded as the eigenvalues of the structure. Selecting an appropriate objective function to measure the differences among these structural modal parameters was of great significance for the accurate updating of the model. In this research, several objective functions were chosen, namely the modal assurance criterion (MAC), the modal shape curvature (MSC), and the modal flexibility matrix (MFM). These parameters have commonly been utilized as damage parameters in many studies [24,25,26]. However, there has been a scarcity of research focusing on the selection of these damage parameters within the context of offshore wind turbines. Hence, this paper explores how different objective functions influence both the convergence speed and the quality of the iterative population during the model updating process of offshore wind turbines.

MAC is a statistical criterion. Its main advantage is that it can clearly reflect significant differences among modal parameters. Nevertheless, it is not highly sensitive to subtle differences. The MAC does not rely on the frequency response matrix or the system matrix. Instead, it can be analyzed solely based on the modal shapes of the structure. Therefore, it is typically used to conduct cluster analysis by expanding the shapes obtained from system analysis or to compare the analytically acquired shapes with the measured ones. The specific formula for the MAC value is as follows:

(40)$\mathrm{MAC}(r,q)={\displaystyle \frac{{\left|{\left\{{\phi }_A\right\}}^H\left\{{\phi }_X\right\}\right|}^2}{({\left\{{\phi }_A\right\}}^H\left\{{\phi }_A\right\})({\left\{{\phi }_X\right\}}^H\left\{{\phi }_X\right\})}}$

In this formula, $\left\{{\phi }_A\right\}$ and $\left\{{\phi }_X\right\}$ represent the vectors of the analytical mode shape and the measured mode shape, respectively. The value of the MAC falls within a range from 0 to 1. Specifically, the closer the MAC value is to 1, the more closely the analytical mode shape resembles the measured mode shape.

The MSC is an index that is derived through the application of the central difference approximation. Its notable advantage is that, due to its sensitivity to multiple damages, it can effectively identify the alterations in the structural modal information when structures are subjected to combined damages. The specific formula for the MSC value is presented as follows (since the analyzed data are discretely sampled, the finite difference method was adopted to approximate the second-order spatial derivatives for computational tractability):

(41)$k_{ij}={\displaystyle \frac{{\mathrm{\Phi }}_{i(j-1)}-2{\mathrm{\Phi }}_{ij}+{\mathrm{\Phi }}_{i(j+1)}}{{(\mathrm{\Delta }x)}^2}}$

Here, $k_{ij}$ represents the MSC corresponding to the i-th modal shape of the j-th measuring point of the structure, ${\mathsf{\Phi }}_{ij}$ represents the modal shape of the j-th measuring point corresponding to the i-th modal shape of the structure, and $\mathsf{\Delta }x$ represents the distance between each measuring point in the formula.

Based on this, the MSC of the j-th measuring point in each modal shape is calculated according to $k_{ij}$. Then, its mean value is set as the objective function, as presented below:

(42)$D{F}_{\mathrm{MSC}}^j={\displaystyle \frac{1}{m}}{\displaystyle \sum _{i=1}^m\left|k_{ij}^m-k_{ij}^a\right|}$

In this context, m stands for the modal order that is of interest in this study. Specifically, $k_{ij}^m$ and $k_{ij}^a$ denote the MSC corresponding to the j-th measuring point within the i-th modal shape, which are based on the measured data and the analysis results, respectively. Meanwhile, ${DF}_{MSC}^j$ represents the MSC damage coefficient of the j-th measuring point, and this is what we refer to as the objective function.

The MFM can be derived solely based on the modal parameters of the structure. Its sensitivity to changes in structural characteristics is also used as an indicator for structural damage:

(43)$\left[F\right]={\displaystyle \sum _{i=1}^m\frac{1}{{\omega }_i^2}\left\{{\varphi }_i\right\}}{\left\{{\varphi }_i\right\}}^T$

In this context, m denotes the number of modal orders under consideration. $\left\{{\mathsf{\Phi }}_i\right\}$ comprises the i-th modal shape and the natural frequency of the structure. Specifically, ${\omega }_i$ represents the i-th natural frequency, and $\left[F\right]$ represents the MFM of the structure. Then, the difference between the measured modal parameters and the MFM obtained from the analysis results can be expressed as follows:

(44)$\left[\mathrm{\Delta }F\right]=\left[F^m\right]-\left[F^a\right]$

$\left[F^m\right]$ and $\left[F^a\right]$, respectively, represent the MFM of the measured modal parameters and that of the analysis results. To be specific, the j-th column of $\left[\mathsf{\Delta }F\right]$ represents the MFM difference of the j-th measuring point of the structure, denoted as $\left\{{\mathsf{\Delta }F}_j\right\}$. Additionally, the difference feature of the i-th point of the structure, which is the maximum value of this vector, is represented by $\underset{i}{\max }\left|{\mathsf{\Delta }F}_{ij}\right|$.

With the above definitions in place, the curvature type of the MFM damage factor for the j-th measuring point of the structure can be solved in the following manner:

(45)$D{F}_{\mathrm{MFM}}^j={\displaystyle \frac{\underset{i}{\max }\left|\mathrm{\Delta }{F}_{i(j-1)}\right|-2\underset{i}{\max }\left|\mathrm{\Delta }{F}_{ij}\right|+\underset{i}{\max }\left|\mathrm{\Delta }{F}_{i(j+1)}\right|}{{(\mathrm{\Delta }x)}^2}}$

Among them, ${DF}_{\mathrm{MFM}}^j$ represents the structural MFM damage parameter, with $\mathsf{\Delta }x$ denoting the distance between adjacent measuring points.

3.5. Uncertain Parameters

For offshore wind turbine structures, the mechanical properties of the structure itself as well as the contact between the pile and the soil are capable of reflecting the various degrees of damage inflicted on the offshore wind turbine by the real environment. In this research, uncertain parameters were set for updating with respect to four components: the tower, flange, pile foundation, and the soil surrounding the pile. This enabled the finite element model to correspond to the common causes of wind turbine damage (as cited) and reflect the damage to the measured structure more accurately.

The uncertain samples were the elastic moduli of the tower, flange, pile foundation, and the soil around the pile, denoted as $E_t,E_f,E_p,$ and $E_s$, respectively. To achieve this, Latin hypercube sampling was initially employed to acquire samples of these uncertain parameters. Subsequently, these samples were utilized to train the surrogate model, enhancing the update speed of the finite element model. The value ranges of the samples were specified as follows: $210 \mathrm{M}\mathrm{p}\mathrm{a}>E_t>310 \mathrm{M}\mathrm{p}\mathrm{a}$, $110 \mathrm{M}\mathrm{p}\mathrm{a}>E_f>210 \mathrm{M}\mathrm{p}\mathrm{a}$, $210 \mathrm{M}\mathrm{p}\mathrm{a}>E_p>310 \mathrm{M}\mathrm{p}\mathrm{a}$, and $90 \mathrm{M}\mathrm{p}\mathrm{a}>E_s>200 \mathrm{M}\mathrm{p}\mathrm{a}$.

3.6. Digital Twin

The digital DT technology proposed in this study establishes a real-time bidirectional mapping between the physical wind turbine and its virtual model through three key innovations:

(1). Dynamic Data-Driven Updating: Unlike conventional FEM updating that relies on periodic experimental data (e.g., modal tests), the DT integrates real-time sensor data from the SCADA system (e.g., accelerations, wind speeds, rotor speeds) and updates structural parameters via Bayesian optimization (Section 3.5). This enables the DT to reflect the structural state under actual operational conditions (Section 4.2).

(2). Closed-Loop Integration: The DT combines KF sensor signals, SSI-based operational modal analysis, and neural network surrogate modeling (Section 3.3, Section 3.4 and Section 3.5) to achieve automated parameter calibration. In contrast, traditional FEM updating requires manual intervention for data processing and model recalibration.

(3). Context-Sensitive Adaptability: The DT prioritizes parameters critical to structural health (e.g., flange stiffness and soil–pile interaction) identified through sensitivity analysis (Section 4.3), whereas conventional FEM updates often focus on global parameter adjustments.

3.7. Technical Roadmap

The principal task of this paper was to carry out modal analysis of offshore wind turbine structures based on measured data. This information was then employed to reflect the parameters of the actual structure in the finite element model. A database was constructed via this finite element model, and sensitivity analysis was utilized to investigate the influence of offshore wind turbine structural parameters on structural health. The specific workflow is detailed in Figure 5.

4. Discussion

4.1. Signal Preprocessing of Wind Turbines

Before starting system analysis, preprocessing of signals from mechanical structures is of paramount importance and serves as a fundamental prerequisite. In the realm of offshore wind turbines, the signals obtained by sensors comprise a wide variety of components. Notably, there are inherent modal responses that are induced by broadband environmental excitations. Harmonic excitations originating from the wind turbine hub, along with wind-induced vibrations resulting from the rotation of blades, are also present.

If the effects caused by these excitations are not effectively alleviated, errors in data interpretation are likely to occur during the system analysis. Such errors, if they transpire, unavoidably lead to deviations and misguidance in subsequent research and decision-making processes. As a result, the precise assessment and efficient management and control of the entire offshore wind turbine system can be adversely impacted.

In this research, Kalman filtering was utilized to perform preprocessing on the signals gathered by eight sensors on the offshore wind turbine, as depicted in Figure 6.

After processing, the spiky phenomenon in the data was significantly diminished. The cause of this phenomenon is that the high-frequency components within the signal have been effectively suppressed. The high-frequency components stem from environmental excitations, whereas the modal information of the structure is typically concentrated in the low-frequency range. Thus, it can be concluded that Kalman filtering has effectively achieved the noise reduction work for the offshore wind turbine signals.

Finally, to verify whether the important information of the signal is preserved before and after filtering, the Frequency Energy Retention Ratio (FERR) is introduced for signal analysis, with the formula as follows:

(46)$\mathrm{FERR}(f_i)={\displaystyle \frac{{\displaystyle {\int }_{f_i-\mathrm{\Delta }f}^{f_i+\mathrm{\Delta }f}{\left|X_{KF}(f)\right|}^{2}df}}{{\displaystyle {\int }_{f_i-\mathrm{\Delta }f}^{f_i+\mathrm{\Delta }f}{\left|X_{orig}(f)\right|}^{2}df}}}\times 100\%$

where $\mathrm{\Delta }f$ represents the frequency bandwidth, and $X_{KF}$ and $X_{orig}$ denote the Fourier spectra of the corrected and original signals, respectively.

According to the analysis results, the energy retention rate in the 0–5 Hz range is approximately 67.8%, while in the 5–10 Hz range, it is about 95.7%. This distribution ratio aligns well with the noise characteristics.

4.2. System Analysis

Offshore wind turbines are operational structures that exhibit distinct mechanical properties in the along-wind and cross-wind directions while in operation. Hence, it becomes imperative to conduct analyses on their characteristics from diverse directions. In this section, the signals of an offshore wind turbine located in the East China Sea and operating over a one-week period were chosen and then grouped into sets, with each set covering a time interval of 10 min, for the purpose of conducting separate system analyses.

In the research content related to Figure 7, a rating was carried out for the stability of structural frequency points. The rating basis encompassed two aspects, namely hard indicators and soft indicators. Those frequency points that exhibited stability in the three aspects of frequency, damping ratio, and mode shape were assigned a marked value of 4.0. If a frequency point failed the test of damping ratio stability, it was correspondingly marked as 3.0. If it did not pass the assessment of mode shape stability, it was marked as 2.0. The frequency points that were only stable in the frequency dimension yet did not meet the stability requirements regarding damping ratio and mode shape were marked as 1.0. The frequency points that did not pass the stability tests in all three dimensions of frequency, damping ratio, and mode shape were marked as 0.0.

Upon further in-depth analysis, it can be observed that in the mechanical environment of the offshore wind turbine during its operation, the reason why the mechanical properties in the FA direction and the SS direction display significant differences lies in the influence of wind loads. Under the same wind speed conditions, when specific analyses were separately carried out for the FA direction and the SS direction, the obtained stability charts demonstrated a high level of similarity in the overall trend, and the modal response laws of the two were consistent.

To accurately quantify the dynamic characteristics of the offshore wind turbine in the actual operating environment, the dynamic response data monitored by the wind turbine within a one-week period were systematically calculated. With the assistance of the environmental condition information synchronously collected by the SCADA system, the modal parameters of the offshore wind turbine corresponding to each sample were strictly sorted according to the average wind speed value. A wind speed-natural frequency relationship diagram was drawn up, as shown in Figure 8. A comparison of the change trends of the first-order natural frequencies in the FA direction and the SS direction showed that the former was more sensitive to the change in wind speed and had a more intense response. However, from the overall perspective, both directions were positively correlated with the wind speed. When the wind speed increased, the natural frequency also increased.

The internal mechanism behind the increase in the natural frequency in the FA direction, mainly originated from the torque induced by the gyroscopic effect during the rotation of the wind turbine. This torque coupled with the inherent mode of the wind turbine itself disrupted the original modal balance, altered the structural stiffness characteristics, and promoted the continuous increase in the natural frequency. The natural frequencies in both the FA direction and the SS direction increased to a certain extent, which can be partly attributed to the limitations and misjudgments of the SSI method in the process of modal identification. As the wind speed increases, the rotational speed of the wind turbine hub also accelerates accordingly, resulting in excitation signals that highly coincide with the rotational frequency or multiple frequencies of the wind turbine. The frequencies of these excitation signals are very close to the first-order natural frequency of the wind turbine and increase synchronously with the continuous increase in the rotational speed, interfering with the accurate judgment of the SSI method and causing deviations in the modal identification results, which are misjudged as a substantial increase in the natural frequency.

Figure 9 depicts the modal shapes of the offshore wind turbine acquired through the SSI approach and compares them with those of the finite element model. The computed results demonstrated that the modal shapes of the finite element model were largely in accordance with those of the measured structure, confirming the validity of the finite element model.

4.3. Establishing a Surrogate Model

Deep-learning technology is efficient in learning the correlations among parameters. It can map the intricate relationships between the structural parameters and modal parameters of offshore wind turbines into a neural network structure, augmenting the efficiency of model optimization. In this paper, a fully connected neural network was constructed using PyTorch (1.4.0) and trained, based on a database comprising the model parameters and those of the finite element model. The neural network’s training set consists of input data comprising four-dimensional material parameters, while the output parameters are formed by combining four-dimensional modal shape samples with one-dimensional first-order natural frequency samples. The dataset configuration includes a validation set of 100 samples and a training set of 300 samples. For the training strategy, the ReLU activation function was employed to mitigate gradient vanishing issues, MSE (Mean Squared Error) was adopted as the loss function, and input/output data underwent normalization and standardization processes prior to training. Concurrently, the selection strategy of neural network structure parameters for offshore wind turbines was thoroughly analyzed.

In this research, the design of the neural network architecture was systematically optimized through Bayesian optimization. To objectively determine the optimal network configuration, we implemented a Bayesian optimization framework that automatically explores the hyperparameter space of hidden layer dimensions. This data-driven strategy effectively balances model complexity and computational efficiency while avoiding subjective empirical selection. The optimization process evaluated 50 neural architectures with hidden layer dimensions ranging from 32 to 256 neurons, guided by the Tree-structured Parzen Estimator (TPE) algorithm to efficiently navigate the high-dimensional parameter space. These settings allowed a systematic analysis of the impact of the model capacity on the learning effect and enabled an endeavor to identify the optimal network configuration.

To guarantee the robustness and reliability of the evaluation outcomes, the division of the experimental dataset introduced the k-fold cross-validation method, with k = 4 for dataset partitioning. This approach can cyclically divide the dataset into training, validation, and test sets, allowing each piece of data to have the opportunity to engage in the evaluation process in different roles, effectively avoiding the problems of uneven data distribution and overfitting, resulting from a single random division. This process comprehensively validated the performance of the model on various data subsets.

Through integrating Bayesian optimization with k-fold cross-validation, we established a dual validation mechanism for both architectural selection and performance evaluation. The Bayesian optimization process yielded four distinct architectural configurations, characterized by different dimensions in their various hidden layers: [135, 180], [153, 60], [213, 65], and [229, 38]. These architectural configurations exhibit distinct topological characteristics in terms of model complexity, while simultaneously demonstrating their capability to achieve superior predictive performance across the dataset.

Finally, a detailed comparative analysis of the performance of surrogate models with different hidden-layer-dimension configurations was conducted, and the results are presented in Figure 10. As shown, all optimized network architectures achieved satisfactory prediction accuracy.

To enhance computational efficiency in practical applications while balancing accuracy and efficiency, this study further calculated and compared the time required for each group of surrogate models to generate 1000 outputs. The results are depicted in Figure 11. The computation times for the four structural schemes remained comparable, ranging from 0.125 s to 0.18 s. However, when the hidden layer dimensions were set to 135 and 180, the prediction accuracy was significantly higher than that of the other three groups. Consequently, surrogate models with hidden-layer dimensions of 135 and 180 were selected in this research to perform structural parameter optimization for the finite element model of the offshore wind turbine.

Finally, the accuracy of the neural networks with hidden-layer dimensions of 135 and 180 was validated through K-fold cross-validation (K = 4) using data generated by the finite element model. The K-fold cross-validation results are shown in Figure 12. The figure substantiates that the neural network architectures with hidden-layer dimensions of 135 and 180 demonstrate validated accuracy through cross-validation procedures.

4.4. Model Parameter Optimization and Sensitivity Analysis

Upon successfully obtaining the modal parameters of the measured structure of the offshore wind turbine, the structural parameter optimization of the finite element structure was accomplished by constructing an objective function based on the modal parameters of the finite element model. In this paper, the chosen objective functions were MAC, MSC, and MFM. All three functions were employed to measure the differences among modal parameters. There has been a scarcity of research on the impacts of these different objective functions on offshore wind turbines.

This study employed the aforementioned three objective functions for Bayesian optimization and evaluated their iterative performance based on convergence speed during the optimization process, as shown in Figure 13. The results demonstrate that although all three objective functions achieved convergence after multiple iterations, MFM exhibited the most superior convergence performance while also achieving the highest final accuracy. This enhanced performance likely stems from its unique capacity to holistically incorporate both mode shape similarity and frequency consistency within the objective function framework. Notably, MFM systematically integrates modal frequency information through its inherent mathematical formulation, whereas conventional indices such as MSC and MAC focus solely on spatial correlation of vibrational patterns. This dual consideration mechanism enables MFM to capture essential system-level dynamic characteristics. Therefore, this study selected MFM as the objective function for Bayesian optimization.

Upon the completion of the structural optimization, a high-precision finite element model was acquired. Based on this model, this study conducted an analysis of the sensitivity of the structural parameters of the offshore wind turbine. The sensitivity factors encompassed the elastic moduli of the tower, flange, pile foundation, and the soil surrounding the pile of the offshore wind turbine. The evaluation indicators comprised the modal shapes and natural frequencies of the offshore wind turbine.

The analysis method employed was the Sobol index method, which was used to quantify the global sensitivity of the impact of model input variables on output variables and decompose the variance of output values to determine the contribution of each input variable and variable group to the output uncertainty. The analysis results are presented in Figure 14, where ${Ms}_1$, ${Ms}_2$, ${Ms}_3$, and ${Ms}_4$ denote the modal shapes at four measuring points on the pile foundation and tower of the wind turbine, respectively, and $f_n$ represents the first-order natural frequency of the wind turbine.

Based on Figure 14, the following conclusions can be derived:

The values of the mode shapes at each measurement point and the value of the first-order natural frequency exhibit a positive correlation with the elastic moduli of all parts of the structure.

Where the influence of a single parameter on the modal parameters of the wind turbine is taken into account, the elastic modulus of the soil surrounding the pile has a negligible impact on the modal shapes of the structure. Instead, it is predominantly influenced by the elastic modulus at the flange. This aligns with the reality that, in practical projects, the majority of wind turbine accidents stem from damage occurring at the flange.

When solely considering the impact of a single parameter on the modal parameters of the wind turbine, regarding the first-order natural frequency of the wind turbine, apart from the flange, the elastic modulus of the other parts of the pile structure still demonstrates a relatively low correlation. Its value is primarily influenced by the soil around the pile.

Taking account of the influence of the interaction among multiple variables, the impact of the elastic modulus of the wind turbine structure itself (along with that of the soil around the pile) on the modal shapes of the structure has witnessed a significant increase. Among these factors, the elastic modulus at the structure’s flange still retains the highest level of sensitivity.

With regard to the first-order natural frequency of the wind turbine, there is not a notable difference between the first-order sensitivity index and the total-order sensitivity index. Mathematically, this is due to the antagonistic effect among the four structural parameters in this model being rather weak or the synergy being relatively low. From a practical standpoint, however, this might be because the first-order natural frequency is excessively influenced by the elastic modulus of the soil around the pile, which consequently causes the influence of other parameters on the first-order natural frequency to be overshadowed.

5. Conclusions

This paper conducted a study on the correlation between structural parameters and modal parameters of a 5 MW offshore wind turbine in the East China Sea. Sensors installed at the pile foundation and tower section were used to collect dynamic responses. The harmonic components within these responses were preprocessed via the KF, followed by the application of the SSI method to conduct operational modal analysis to obtain the modal parameters of the real structure.

The construction of the objective function involved several damage indices of the modal parameters from both the offshore wind turbine’s finite element model and those in the FA direction of the actual structure. Through a comparison of multiple common damage indices, the MFM was determined to be the most efficient objective function. Subsequently, Bayesian optimization was employed to optimize the structural parameters of the finite element model, resulting in a high-precision model that aligned more closely with the measured structure. Leveraging this optimized model, a database encompassing the structural parameters and modal parameters of the offshore wind turbine was established. A surrogate model of structural parameters and modal parameters, based on the fully connected neural network, was trained using this database. Based on this surrogate model, a sensitivity analysis of each structural parameter of the wind turbine was carried out.

For a systematic analysis of the measured data, this study integrated the on-site measured environmental data with the analysis outcomes of the dynamic response data of the structure over a week. The analysis results obtained in both the FA and SS directions were largely consistent with the theoretical results and the previous work of others, validating the credibility and accuracy of the systematic analysis part of this study.

The verification of the rationality of the initial finite element model was based on a comparative analysis with the modal shapes of the measured structure. The comparison results demonstrated good similarity in the FA direction between the two, and the characteristics of the modal shapes were basically in alignment, confirming the reliability and validity of the finite element model in this direction.

During the training process of the surrogate model, experiments were conducted on the selection of the model’s hyperparameters to strike a balance between computational cost and analysis accuracy. The final results indicated that when dealing with the modal parameter problem of this offshore wind turbine, the double-hidden-layer neural network with a dimension of 128 exhibited both good accuracy and low computational cost.

Finally, in the sensitivity analysis of the structural parameters of the offshore wind turbine, the sensitivities of the elastic moduli of the tower, flange, pile foundation, and soil around the pile to the modal parameters were analyzed, considering both single variables and variable groups. The analysis findings revealed that when only a single variable was taken into account, the elastic modulus of the flange played a decisive role in determining the modal shapes of the wind turbine. Even in the case of variable groups, it remained the factor with the greatest influence, which was consistent with the engineering fact that most wind turbine accidents are caused by flange damage. On the other hand, the first-order natural frequency of the wind turbine was primarily influenced by the soil around the pile, and the distribution of sensitivity was not significantly affected under both preconditions. This was attributed to the strong linear relationship between the structural parameters and the first-order natural frequency. It was also possible that the excessive sensitivity of the soil around the pile overshadowed the correlations among the influencing factors.

In summary, this study’s analysis of structural parameter sensitivity enhances offshore SHM practicality through mechanistic insights and applicable strategies. Mechanistically, flange stiffness governs localized damage (e.g., modal distortion), while pile–soil properties dominate global dynamics (e.g., frequency shifts), clarifying priority parameters for SHM: real-time tracking of flange integrity and pile–soil degradation via frequency trends. Practically, these findings guide sensor optimization (e.g., strain monitoring at flanges) and multi-level alarms using digital twins to map sensitivity parameters. Lightweight algorithms enabled by surrogate models support lifecycle assessment without additional hardware costs, bridging data-driven monitoring to mechanism-informed maintenance for proactive offshore structural management.

Deploying offshore SHM systems faces critical challenges: (1) sensor durability and environmental resilience: high-precision sensors must withstand harsh marine conditions (e.g., corrosion, biofouling) while maintaining long-term stability; (2) multi-source data fusion and real-time constraints: balancing computational efficiency (e.g., edge processing of SSI-based modal analysis) with communication bandwidth limitations for high-frequency data streams; (3) model generalization and parameter drift: adaptive updating mechanisms are required to address aging-induced structural changes (e.g., pile–soil interaction evolution) and ensure surrogate model fidelity.

Future advancements should integrate hybrid mechanistic-data-driven approaches, embedded self-diagnostic sensors, and federated learning for distributed model refinement. Digital twin frameworks could map sensitivity parameters (e.g., flange strain, frequency shifts) to prioritized maintenance actions, bridging theoretical insights to proactive offshore structural management.

Author Contributions

Conceptualization, R.K. and T.Z.; methodology, R.K.; software, R.K.; writing—original draft, R.K. and T.Z.; writing—review and editing, J.Z.; visualization, C.L.; supervision, T.Z.; funding acquisition, J.Z. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Data are contained within the article.

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Tables

Figure 1 Sensor positions of offshore wind turbines.

Figure 2 Installation coordinates of sensors on offshore wind turbines.

Figure 3 Time-history diagram of simulated wind loads.

Figure 4 Time-history diagram of simulated wave height.

Figure 5 Flowchart of the model correction method based on Bayesian optimization.

Figure 6 Utilizing Kalman filtering to perform preprocessing on the data obtained from eight sensors on the wind turbine, with the aim of reducing the components that display relatively low stability. (a) Sensor 1; (b) Sensor 2; (c) Sensor 3; (d) Sensor 4; (e) Sensor 5; (f) Sensor 6; (g) Sensor 7; (h) Sensor 8.

Figure 7 Stability charts of the offshore wind turbine. (a) Fore–aft direction; (b) side–side direction.

Figure 8 Correlation between the first-order natural frequencies of the offshore wind turbine with respect to the wind speed. (a) Fore–aft direction; (b) side–side direction.

Figure 9 Comparison of modal shapes between the finite element model and the measured structure (1st–4th order): (a) corresponds to the 1st modal shape, (b) corresponds to the 2nd modal shape, (c) corresponds to the 3rd modal shape, and (d) corresponds to the 4th modal shape.

Figure 10 Comparison of predictions with different hyperparameters of the surrogate model.

Figure 11 Comparison of accuracy and time cost of the surrogate model with different hyperparameters.

Figure 12 Training and validation loss convergence (Mean ± Std).

Figure 13 Demonstration of the iteration diagrams of the three objective functions.

Figure 14 Diagram of sensitivity analysis of wind turbine structural parameters to modal parameters: (a) first-order sensitivity index; (b) total-order sensitivity index.

Table 1

Verification of normal distribution for raw data.

Channel	Skewness	Kurtosis
1	−0.011	0.031
2	0.153	0.205
3	0	0.018
4	0.093	0.217
5	−0.046	0.012
6	−0.018	−0.07
7	0	−0.023
8	0.016	−0.031

Table 2

Nacelle and blade parameters.

Component	Weight/kg	Center of Gravity/m			Moment of Inertia/kg∙m2
		X	Y	Z	Ixx	Iyy	Izz
RNA	3.5 × 105	−4.30	2.16 × 10−3	2.64	4.37 × 107	2.353 × 107	2.542 × 107

References

1. Global Wind Energy Council. 2023; Available online: https://gwec.net/ (accessed on 21 December 2023).

2. Ju, S.H.; Huang, Y.C. Analyses of offshore wind turbine structures with soil-structure interaction under earthquakes. Ocean Eng.; 2019; 187, 106190. [DOI: https://dx.doi.org/10.1016/j.oceaneng.2019.106190]

3. Yang, Y.; Li, Q.S.; Yan, B.W. Specifications and applications of the technical code for monitoring of building and bridge structures in China. Adv. Mech. Eng.; 2016; 9, 1687814016684272. [DOI: https://dx.doi.org/10.1177/1687814016684272]

4. Martinez-Luengo, M.; Shafiee, M. Guidelines and cost-benefit analysis of the structural health monitoring implementation in offshore wind turbine support structures. Energies; 2019; 12, 1176. [DOI: https://dx.doi.org/10.3390/en12061176]

5. Nielsen, J.S.; Tcherniak, D.; Ulriksen, M.D. A case study on risk-based maintenance of wind turbine blades with structural health monitoring. Struct. Infrastruct. Eng.; 2021; 17, pp. 302-318. [DOI: https://dx.doi.org/10.1080/15732479.2020.1743326]

6. Annamdas, V.G.M.; Bhalla, S.; Soh, C.K. Applications of structural health monitoring technology in Asia. Struct. Health Monit.; 2017; 16, pp. 324-346. [DOI: https://dx.doi.org/10.1177/1475921716653278]

7. Klikowicz, P.; Salamak, M.; Poprawa, G. Structural health monitoring of urban structures. Procedia Eng.; 2016; 161, pp. 958-962. [DOI: https://dx.doi.org/10.1016/j.proeng.2016.08.833]

8. Reynders, E.; Pintelon, R.; De Roeck, G. Uncertainty bounds on modal parameters obtained from stochastic subspace identification. Mech. Syst. Signal Process.; 2008; 22, pp. 948-969. [DOI: https://dx.doi.org/10.1016/j.ymssp.2007.10.009]

9. Badrzadeh, B.; Gupta, M. Practical experiences and mitigation methods of harmonics in wind power plants. IEEE Trans. Ind. Appl.; 2013; 49, pp. 2279-2289. [DOI: https://dx.doi.org/10.1109/TIA.2013.2260314]

10. Moynihan, B.; Mehrjoo, A.; Moaveni, B.; McAdam, R.; Rüdinger, F.; Hines, E. System identification and finite element model updating of a 6 MW offshore wind turbine using vibrational response measurements. Renew. Energy; 2023; 219, 119430. [DOI: https://dx.doi.org/10.1016/j.renene.2023.119430]

11. Pezeshki, H.; Pavlou, D.; Adeli, H.; Siriwardane, S. Gyroscopic effects of the spinning rotor-blades assembly on dynamic response of offshore wind turbines. J. Wind Eng. Ind. Aerodyn.; 2024; 247, 105698. [DOI: https://dx.doi.org/10.1016/j.jweia.2024.105698]

12. Civera, M.; Sibille, L.; Fragonara, L.Z.; Ceravolo, R. A DBSCAN-based automated operational modal analysis algorithm for bridge monitoring. Measurement; 2023; 208, 112451. [DOI: https://dx.doi.org/10.1016/j.measurement.2023.112451]

13. Devriendt, C.; Jordaens, P.J.; De Sitter, G.; Guillaume, P. Damping estimation of an offshore wind turbine on a monopile foundation. IET Renew. Power Gener.; 2013; 7, pp. 401-412.

14. Wei, S.; Han, Q.; Peng, Z.; Chu, F. Dynamic analysis of parametrically excited systems under uncertainties and multi-frequency excitations. Mech. Syst. Signal Process.; 2016; 72, pp. 762-784.

15. Kim, C.; Dinh, M.C.; Sung, H.J.; Kim, K.H.; Choi, J.H.; Graber, L.; Yu, I.-K.; Park, M. Design, implementation, and evaluation of an output prediction model of the 10 MW floating offshore wind turbine for a digital twin. Energies; 2022; 15, 6329. [DOI: https://dx.doi.org/10.3390/en15176329]

16. Tao, F.; Xiao, B.; Qi, Q.; Cheng, J.; Ji, P. Digital twin modeling. J. Manuf. Syst.; 2022; 64, pp. 372-389. [DOI: https://dx.doi.org/10.1016/j.jmsy.2022.06.015]

17. Zhang, R.; Wang, F.; Cai, J.; Wang, Y.; Guo, H.; Zheng, J. Digital twin and its applications: A survey. Int. J. Adv. Manuf. Technol.; 2022; 123, pp. 4123-4136.

18. Bartilson, D.T.; Jang, J.; Smyth, A.W. Sensitivity-based singular value decomposition parametrization and optimal regularization in finite element model updating. Struct. Control Health Monit.; 2020; 27, e2539.

19. Kamariotis, A.; Chatzi, E.; Straub, D. A framework for quantifying the value of vibration-based structural health monitoring. Mech. Syst. Signal Process.; 2023; 184, 109708.

20. Weijtjens, W.; Lataire, J.; Devriendt, C.; Guillaume, P. Dealing with periodical loads and harmonics in operational modal analysis using time-varying transmissibility functions. Mech. Syst. Signal Process.; 2014; 49, pp. 154-164.

21. Dai, K.; Wang, Y.; Lu, W.; Ren, X.; Huang, Z. Investigation of the stochastic subspace identification method for on-line wind turbine tower monitoring. Nondestructive Characterization and Monitoring of Advanced Materials, Aerospace, and Civil Infrastructure; SPIE: Bellingham, WA, USA, 2017; Volume 10169, pp. 602-608.

22. Kaimal, J.C.; Wyngaard, J.C.; Isumi, Y.; Coté, O.R. Spectral characteristics of surface-layer turbulence. Q. J. R. Meteorol. Soc.; 1972; 98, pp. 563-589.

23. Calidori, A.; Bernagozzi, G.; Castellaro, S.; Landi, L.; Diotallevi, P.P. An FDD-based modal parameter-less proportional flexibility-resembling matrix for response-only damage detection. J. Civ. Struct. Health Monit.; 2024; 14, pp. 401-429.

24. Ciambella, J.; Vestroni, F. The use of modal curvatures for damage localization in beam-type structures. J. Sound Vib.; 2015; 340, pp. 126-137.

25. Lei, S.; Mao, K.; Li, L.; Xiao, W.; Li, B. Direct method for second-order sensitivity analysis of modal assurance criterion. Mech. Syst. Signal Process.; 2016; 76, pp. 441-454.

26. Greś, S.; Döhler, M.; Andersen, P.; Mevel, L. Kalman filter-based subspace identification for operational modal analysis under unmeasured periodic excitation. Mech. Syst. Signal Process.; 2021; 146, 106996.