LI Ying, WANG Bin , LIU Qiang, GAO Shan and LU Sujie

1)Chinese-German Institute of Engineering, Zhejiang University of Science and Technology, Hangzhou 310023, China

2)Key Laboratory of Far-Shore Wind Power Technology of Zhejiang Province, Hangzhou 311122, China

3)PowerChina Huadong Engineering Corporation Limited, Hangzhou 311122, China

4)Zhejiang Huadong Mapping and Engineering Safety Technology Co., Ltd., Hangzhou 310014, China

Abstract In the actual measurement of offshore wind turbines (OWTs), the measured accelerations usually contain a large amount of noise due to the complex and harsh marine environment, which is not conducive to the identification of structural modal parameters. For OWTs with remarkably low structural modal frequencies, displacements can effectively suppress the high-frequency vibration noise and amplify the low-frequency vibration of the structure. However, finding a reference point to measure structural displacements at sea is difficult. Therefore, only a few studies on the use of dynamic displacements to identify the modal parameters of OWTs with high-pile foundations are available. Hence, this paper develops a displacement conversion strategy to study the modal parameter identification of OWTs with high-pile foundations. The developed strategy can be divided into the following three parts: zero-order correction of measured acceleration, high-pass filtering by the Butterworth polynomial, and modal parameter identification using the calculated displacement. The superiority of the proposed strategy is verified by analyzing a numerical OWT with a high-pile foundation and the measured accelerations from an OWT with a high-pile foundation. The results show that for OWTs with high-pile foundations dominated by low frequencies, the developed strategy of converting accelerations into displacements and then performing modal parameter identification is advantageous to the identification of modal parameters, and the results have high accuracy.

Key words offshore wind turbine; high-pile foundation; modal parameter identification; baseline drift; low-frequency noise

1 Introduction

Wind energy is characterized by enormous reserves and renewable, clean, and pollution-free power (Baoet al., 2020).Research on wind energy has become an area with excellent development potential in the aspect of new energy and has received increasing attention from countries worldwide (Gaoet al., 2020). Compared with onshore wind power, offshore wind power has the advantages of high wind speed, low wind shear, low turbulence, high output, and a long service life (Zhaoet al., 2020). Consequently, offshore wind power has also received extensive attention due to its vast energy reserves, continuous resource stability, and low environmental impact, and offshore wind turbines (OWTs)have been rapidly developed in recent years (Liuet al., 2019).

However, with the development of OWTs, issues with their safety have also been the subject of increasing concern(Zhanget al., 2020). Over the past 10 years, safety accidents affiliated with OWTs have frequently occurred, with 1000 accidents transpiring in 2014 alone (Gaoet al., 2020). In addition, the price of offshore wind energy is considerably higher than that of onshore wind energy due to the high initial installation and maintenance costs (Dinget al., 2020).Therefore, preventing accidental damage to OWTs has become a critical issue (Huanget al., 2020). Modal parameter identification has been successfully used for OWTs to evaluate the safety characteristics of the structure; however, stimulating the real structural modes is impossible due to the large rigidity of the structural foundation and the weak wave energy (Lianet al., 2020). Moreover, the signal-to-noise ratio of the measured accelerations is low due to the influence of complex environmental noise, which increases the difficulty of modal parameter identification(Liuet al., 2020a). Liuet al. (2020b)introduced ship- based excitation to excite the real structural modes to overcome the aforementioned problem. However, this method inevitably causes damage and destruction to the structure.

Studies have shown that for structures dominated by lowfrequency vibration, such as bridges, using dynamic displacements for modal parameter identification is better than acceleration-based modal parameter identification (Zhenget al., 2019). This finding is due to the sensitivity of the acceleration to high-frequency noise and the dominance of low-frequency vibrations in the displacement (Honget al.,2010). For OWTs dominated by low frequencies, the use of dynamic displacements for modal parameter identification suppresses the high-frequency noise components, which is advantageous for the identification of the true structural modes using measured accelerations (Gaoet al., 2021a).However, the use of dynamic displacements to identify the modal parameters of OWTs remains rare because dynamic displacement is a relative concept and the measurement requires fixed reference points, which is difficult to achieve in a real offshore measurement environment (Kim and Sohn,2017). Therefore, the current modal parameter identification method for OWTs is based on accelerations (Donget al.,2018).

Theoretically, acceleration and displacement have a strict mathematical calculation relationship; that is, the corresponding displacement can be calculated by integrating the acceleration twice. However, in the calculation process,each integration step will inevitably introduce the influence of the initial terms, corresponding to the initial velocity and the initial displacement (Arias-Lara and De-la-Colina, 2018).Especially in the second integration step, in addition to the baseline drift caused by the initial displacement, a drift term caused by the integration of the initial velocity generated by the first integration will also emerge (Davidet al.,2002). Moreover, the initial velocity and displacement of the structure for the measured data are usually unknown;thus, the dynamic displacement calculated by directly integrating the measured acceleration twice does not match the real displacement, thereby seriously inducing possible drift problems in the integrated results (Chiu, 1997). Hence,integration cannot be directly used to obtain dynamic displacements for related research (Liuet al., 2021).

Based on the nature of the drift terms caused by integration, this paper develops an improved modal parameter identification strategy for OWTs with high-pile foundations by removing the drift terms from the displacements calculated by integrating the measured accelerations. This strategy first fits the measured accelerations by using the least-square method, removes the fitted component from the accelerations, and then performs high-pass filtering on the accelerations to eliminate low-frequency errors. Afterward, the corrected accelerations are integrated to obtain the corresponding velocities and the average value of the velocity is removed, thereby eliminating the linear drift caused by the initial velocity in the integrated displacement. Finally, the corrected dynamic displacement is used as the input, and the stochastic subspace identification (SSI)method (Prioriet al., 2018)is used for modal parameter identification.

The remainder of this paper is organized as follows. First,a detailed theoretical derivation of the proposed strategy,including the conversion of measured accelerations into the corresponding displacements and identification of the modal parameters of the structure from the calculated displacements, is conducted. A numerical OWT is established in ANSYS in the third section, and the dynamic response of the structure under wave excitation, including acceleration and displacement, is calculated. Then, the calculated acceleration and displacement characteristics are compared, and the accuracy of using the developed strategy to convert the acceleration into displacement is verified. Finally, the modal parameter identification results using the calculated accelerations and displacements are compared.In the fourth section, measurements are performed on an OWT with a high-pile foundation, and the acceleration of the structure under the excitation of waves is recorded.Furthermore, ship collision loads are introduced to excite the structure to obtain reference results of the structural modes, and the eigensystem realization algorithm (ERA)method is used to identify the structural modal parameters under the collision. The measured accelerations and the calculated displacements of the structures under environmental excitation are then used for modal parameter identification, and the results are compared and discussed.

2 Preliminary Review

2.1 Trifunac and Lee (1990)

The method by Trifunac and Lee is based on the adjustment of the baseline. In each step of integration, the acceleration, velocity, and displacement are all subjected to highpass filtering. At present, this method has been incorporated into the latest signal processing technology and has been widely used for the processing of seismic signals. The flowchart of the method is shown in Fig.1.

Fig.1 Flowchart of the method by Trifunac and Lee (1990).

2.2 Darragh et al. (2004)

This method aims to correct the baseline error contained in the record by maintaining a static permanent displacement. To achieve this purpose, the velocity adjustment can be realized by one of the three following functional forms:linear fitting, bilinear piecewise continuous fitting, and quadratic fitting. The best-fitting function is then differentiated and removed from the original acceleration. The flowchart of the method is shown in Fig.2.

Fig.2 Flowchart of the method by Darragh et al. (2004).

2.3 Chiu (1997)

This method aims to fix the main errors of the acceleration baseline. The main baseline errors found in these data include the acceleration drift constant, low-frequency instrument noise, low-frequency background noise, small initial velocity and displacement values, and operating errors,including errors due to data processing. The method also includes the following three main steps: least-square fitting of the velocity record, high-pass filtering of the acceleration, and subtraction of the initial velocity value:

Step 1: Performing least-square fitting on the velocity before filtering can effectively eliminate the baseline drift of the acceleration.

Step 2: Performing the filtering step removes the linear trend and other low-frequency errors from the acceleration.

Step 3: Subtracting the initial velocity eliminates the linear trend of the displacement.

This method also reduces the necessary steps to adjust the baseline error, minimizing the data manipulation side effects.

3 Modal Parameter Identification Based on Modified Integrated Accelerations

3.1 Zero-Order Correction of Measured Accelerations

In the vibration data of OWTs in a marine environment,in addition to structural vibration information, structural vibrations caused by environmental noise will inevitably emerge and can be expressed as (Wanget al., 2019)

whereas(t)andan(t)represent the structural and noise information, respectively. In addition to the two aforementioned components, when the acceleration sensor is used for testing, the baseline drift of the data will inevitably be induced by the sensor error as follows:

wherebis the baseline error due to the used sensor.

After acquiring the acceleration of the structure, the corresponding velocity can be obtained through integration,which can be expressed as

where(t)represents the acceleration after subtracting the mean value.

To estimate the structure displacement correctly, the least-square method is first used to fit the calculated velocityv(t), and the fitted curve can be expressed as (Chistyakova and Chistyakov, 2020)

wherepandqrepresent the intercept and the slope of the fitting curve, respectively, which can be expressed as

where the overbar represents the average of the corresponding symbol. The fitted curve can then be subtracted from the acceleration(t)as

wherel'(t)represents the derivative ofl(t)considering timet.

3.2 Displacement Estimation by Using Butterworth High-Pass Filtering

The data drift induced by the baseline drift of the acceleration sensor and the initial velocity has been removed by using the aforementioned steps. To remove the inaccuracy caused by low-frequency noise, a Butterworth filter is introduced as (Pintoet al., 2019)

wherenrepresents the order of the filter,ωcrepresents the cutoff frequency,ωprepresents the edge frequency of the passband, andis the value at the edge of the passband.

A corrected acceleration can be required as(t)after implementing the high-pass Butterworth filter. The corresponding velocity can then be calculated by integration as

and the corresponding displacement can be calculated as

3.3 Modal Parameter Identification Based on Reconstructed Displacements

After calculating the displacement of the structure, the estimated datacan be arranged into a Hankel matrixH0|2i-1with 2iblock rows andjblock columns, wherej=s- 2i+ 2, and the Hankel matrix can be written as (Shokraviet al., 2020)

wherepandjrepresent the past and future, respectively.

The line space output of the future is then projected to the line space output of the past as

The following can be obtained by performing singular value decomposition on matrixiP(Zhouet al., 2019):

In Eq. (11), the projection matrixiPis equal to the product of the observability matrixΓiand the Kalman filter state sequenceKias

where the extended observable matrix can be expressed as

whereCandAare the system matrixes andKican be expressed as

The matrix [A,C] is transformed into modal coordinates after calculating the system matrixesAandC, and the matrix [Λ,CΨ] can be obtained. The diagonal matrixΛcontains the modal frequency and damping information of the structure, and the matrixCΨindicates the information at the nodes.

From the relationshipΛc= lnΛ/Δt, the eigenvalues in the continuous time domain are calculated from those in the discrete time domain, and then the modal frequency and damping are calculated from the real and imaginary parts,respectively. The characteristic rootΛcalways appears in the form of a pair of complex conjugate numbers as (Peter and Bart, 1996)

whereωqandξqare the undamped modal frequency and damping factor, respectively, for theqth mode as

3.4 Procedure of the Developed Algorithm

The procedure of the proposed approach can be described by the following three steps (the flowchart is shown in Fig.3.

Fig.3 Flowchart of the proposed method.

Step 1: Zero-order correction of measured accelerations:first, integrate the measured acceleration to calculate its corresponding velocity as Eq. (3). Then, perform least-square fitting on the calculated velocity as Eqs. (4)and (5). Finally, subtract the first derivative of the fitting term from the measured acceleration as Eq. (6).

Step 2: Low-frequency noise removal and acceleration integration: introduce the Butterworth filter as Eq. (7)to filter out the low-frequency noise in the acceleration and integrate the corrected acceleration to calculate its corresponding displacement as Eq. (9).

Step 3: Modal parameter identification using the corrected displacement: use the estimated displacement to construct a Hankel matrix as Eq. (10)and solve the Hankel matrix to calculate the modal frequency and damping as Eqs. (17)and (18), respectively.

4 Numerical Study

A numerical OWT with a high-pile foundation is first used to verify the accuracy and superiority of the proposed strategy. The purpose of using this numerical example is threefold. First, the properties of acceleration and displacement at the same position of the structure are compared theoretically. Then, the accuracy of the developed acceleration-based strategy for the conversion of the corresponding displacement is verified. Finally, modal parameter identification of the calculated acceleration and converted displacement is performed to test the superiority of the modal parameter identification method based on the converted displacement.

4.1 Model Introduction and Dynamic Response Analysis

The prototype of the OWT used in this case has a highpile foundation, including a total of eight pile legs. The height of the tower is 84 m, and the diameter is 6 m. The height and diameter of the high-pile cap are 5 and 15 m,respectively. The length and diameter of each pile leg are 94 and 1.5 m, respectively, with a slope of 6:1. The generating power of the OWT is 4 MW, and the impeller diameter is 130 m. First, finite element software ANSYS is used to model the OWT. The model comprises three parts,including the tower, the cap, and the eight legs. The tower section is constructed by beam elements, which are divided into 31 sections of different lengths, diameters, and thicknesses with elevations of 8 – 89 m. The pile cap is located under the tower and is constructed by a solid element with a bottom elevation of 2 m. The eight pile legs are distributed in a circle with a diameter of 6 m and are constructed by pipe elements divided into five parts according to the different environments around the legs: the interior of the pile cap is filled with air, concrete in air, water, and soil. The final model comprises a total of 30794 elements and 29954 nodes. The finite element model is shown in Fig.4. The modal parameters of the structure can be calculated by the software after modeling the structure, and the first two modal frequencies of the structure are 0.3512 and 1.7003 Hz.

Fig.4 Finite element model of the offshore wind turbine with high-pile foundation.

The wave loads should first be simulated when calculating the dynamic response. Under extreme conditions,the wave height and wave period are set to 5 m and 9 s in the calculation, respectively. The JONSWAP spectrum is used to simulate random waves, and the Morrison formula is used to calculate the wave force acting on the turbine. In addition, considerable noise will inevitably emerge in real marine environments; therefore, Gaussian white noise with a power of 100 dBW is added to the wave force. Fig.5 presents the simulated wave force. Upon obtaining the wave force, the simulated wave force is applied as input to the connection between the tower and the high-pile foundation of the OWT. The dynamic response of the structure is then calculated.

Fig.5 Simulated wave force in the time domain.

Figs.6(a)and (c)show the calculated acceleration and displacement in the time domain, respectively, while Figs.6(b)and (d)illustrate the results in the frequency domain.Comparing the two results, the calculated acceleration is significantly affected by noise, especially the frequency part.However, the displacement is dominated by low- frequency vibration, which suppresses the high-frequency noise in the motion. Consequently, dynamic displacement is beneficial for modal parameter identification for OWTs with low modal frequencies.

Fig.6 Calculated acceleration in the (a)time and (b)frequency domains; the calculated displacement in the (c)time and (d)frequency domains.

4.2 Displacement Reconstruction Based on Calculated Accelerations

The following first verifies the accuracy of the developed strategy of transforming accelerations into displacements. A period of acceleration is intercepted from the calculated dynamic acceleration for analysis to consider the environmental noise and the initial conditions (including the initial velocity and displacement)of the structure simultaneously. Herein, the acceleration from 150 s to 250 s is selected for the displacement transform, and the selected acceleration is demonstrated in Fig.7(a). Fig.7(b)shows the displacement calculated by directly integrating the acceleration twice. The displacement calculated after integration has extremely drifted due to the influence of the initial conditions.

Fig.7 (a)Used acceleration in the time domain; (b)Displacement calculated by directly integrating the acceleration.

The acceleration is first integrated when using the developed strategy for displacement transformation to obtain its corresponding velocity, as indicated by the blue line in Fig.8. The acceleration first needs a zero-order correction before performing the second integration. The least-square method is introduced to fit the velocity obtained by the integration, and the fitting result is shown as the red line in Fig.8. The zero-order correction of the acceleration can be achieved by deriving the fitted curve and subtracting the derivation result from the acceleration.

Fig.8 Calculated velocity and the fitting curve.

The zero-order correction aims to eliminate the baseline drift caused by the acceleration sensor. In addition, the lowfrequency noise error in the acceleration affects the displacement transformation. Therefore, the Butterworth polynomial is introduced to filter out the low-frequency error.The integrated displacement after integrating the modified acceleration twice is shown in Fig.9. The figure reveals that the displacement obtained by the developed strategy is in good agreement with the calculated displacement, which also confirms the accuracy of the developed accelerationbased strategy for displacement transformation.

Fig.9 Calculated and transformed displacements obtained by the developed strategy.

4.3 Comparison of the Modal Parameter Identification Results

The modal parameter identification stability diagram is introduced to compare the identification results of the calculated acceleration and transformed displacement. Particularly, for the same mode, if the modal frequency error of the two identification results is within 1% with the increase in the modal order used, then the error of the identification result of the damping ratio is within 5%; that is,

The result of the identification is then marked as stable;otherwise, the result is considered unstable. Fig.10 shows the following: if the identification result is stable, then it is represented by a red ‘×’; if the result is unstable, then it is signified by a blue ‘°’. Figs.10 and 11 show the stability diagram results using the calculated acceleration and transformed displacement for modal parameter identification.In the analysis, the modal order is set to increase gradually from 0 to 50. The use of transformed displacement for modal parameter identification is advantageous in identifying the low-frequency modes of the structure. When the modal order is 8, the converted displacement can be used to identify the stable poles at the first structural mode. The stable poles can be identified for the second- order structural mode when the modal order reaches 16. When the modal order reaches 16, the stable poles of the first-order structural mode can be identified by using the calculated acceleration. The modal order for the second- order mode must reach 32 to identify stable poles. Therefore, when using transformed displacements for modal parameter identification, small modal order can be used for OWTs dominated by low frequencies to identify stable results.

Fig.10 Stability diagram of modal parameter identification using the calculated acceleration.

Fig.11 Stability diagram of modal parameter identification using the transformed displacement.

Table 1 lists the structural modal frequencies outputted by the software and the results using the calculated acceleration and transformed displacement. The two-order modal frequencies of the structure are all identified, but the modal frequencies determined by the transformed displacement are close to the calculation results of the software. The difference between the first-order modal frequency identified by the acceleration and the calculation result of the software is 2.59%, and the error of the identification result using the transformed displacement is only 0.32%. For the second-order modal frequency, the identification accuracy using the transformed displacement is increased by 7.33%.

Table 1 Identified results of the first two-order modal frequencies

5 Field Test Study

Actual measurements of an OWT with a high-pile foundation were conducted to verify the effectiveness of the proposed strategy when applied to measured structural vibration data. The actual measurements are divided into two parts. The first part introduces ship collision loads to excite the structure, and the ERA method is used to identify the modal parameters of the structure. The result is then utilized as a relatively accurate reference result. The second part measures the vibration acceleration of the structure under wave excitation and then uses the measured acceleration and the displacement reconstructed by the proposed strategy to identify the modal parameters. The results are then compared and discussed. The selected OWT is located near Rudong County in Zhejiang Province, China. The geographical location of the wind farm where the tested OWT was located is shown in Fig.12.

5.1 Measurement Process

A total of five three-way acceleration sensors (model MEAS 4803A-0002C000488)produced by the American company MEAS were used in the measurement to obtain the vibration acceleration of the structure. The sensors were installed on the inner wall from the bottom to the top of the OWT. The height of each sensor is 0, 17.2, 41.8, 60.2, and 75.1 m. The vibration acceleration is collected by the acquisition instrument, which was produced by Germany IMC Integrated Measurement and Control Co., Ltd. (model CRONOSPL16-DCB8). The data acquisition instrument contains 64 channels, and the collected data are converted into an ASCII file for the analysis program to call through IMC FAMOS software. In addition, a ship is used to impact the structure to simulate the pulse excitation on the structure to obtain a relatively accurate modal result. The tested OWT and the main equipment used in the measurement are shown in Fig.13.

5.2 Modal Parameter Identification Under Hammer Impact

The modal frequency and damping of the OWT under operating conditions are usually unknown in a real marine environment. Therefore, a relatively accurate answer is necessary to compare the modal parameter identification results using the measured acceleration and the transformed displacement. Herein, the ERA method was used to identify the modal parameters of the OWT, and the result was utilized as a reference answer for subsequent comparisons.The basic idea of the ERA method is to construct the Hankel matrix according to the impulse response of the system and calculate the modal parameters through the Hankel matrix.Thus, the ship collision is used as an impulse excitation to act on the turbine, and the ERA method is used to obtain the true modal parameters of the structure. With a sampling frequency of 200 Hz, the recorded acceleration of the top acceleration sensor in they-direction is illustrated in Fig.14. This figure reveals that the amplitude of the measured acceleration of approximately 800 s has fluctuations, which also indicates a collision occurrence at this moment.

Fig.12 Geographical location of the test OWT. SCS, the South China Sea.

Fig.13 Tested OWT and the equipment used in the measurement.

Fig.14 Measured (a)accelerations and (b)displacements.

In the calculation, the modal order is set as 500, and the time from 779 s to 784 s is selected to identify the modes in the measured acceleration. To determine the accuracy of the calculated modal information, the seabed position of the OWT is set to the default as a fixed zero point, and the measured accelerations of the five arranged sensors are identified. The vibration shape diagram of the corresponding mode of the structure can then be drawn by combining the positions of the five sensors. The first three modes of the turbine can be determined by selecting the modal parameters that conform to the first three modes of the vibration law of the structure from the identification results.Fig.15 confirms that the first three-order modal frequencies of the structure identified by the ERA method are 0.3599,1.6921, and 2.9813 Hz. The first three orders of modal damping ratio are 0.0176, 0.0152, and 0.0261.

Fig.15 First three-order formations of the test OWT.

5.3 Reconstruction of Displacements from Measured Accelerations

In addition to the excitation of a ship collision, measurements of the OWT under wave loads were also conducted. By using a sampling frequency of 200 Hz and a recording time of 1800 s, the recorded accelerations measured by the sensor on the top of the structure in they- direction are shown in Fig.16(a). Herein, the acceleration from 600 s to 700 s shown in Fig.16(b)is selected for analysis to demonstrate the proposed displacement conversion strategy and compare the modal parameter identification using the measured acceleration and converted displacement.

Fig.16 Measured (a)accelerations and (b)displacements.

The measured acceleration must be zero-order corrected first according to the proposed strategy to reconstruct the displacement corresponding to the measured acceleration.The velocity is calculated by directly integrating the acceleration, and the integrated velocity is displayed in Fig.17 as the blue line. The integrated velocity displays severe drift due to the numerous noise components in the measured acceleration. Moreover, the baseline drift error is inevitably caused by the limitation of the sensor during the measurement. Therefore, the least-square method is first used to fit the integrated velocity, and the fitting curve is illustrated in Fig.17 as the red line. The red curve fitted by the least-square method sufficiently characterizes the drift term in the integrated velocity.

Fig.17 Calculated velocity by integrating the acceleration and the fitting curve.

The derivative of the fitting term can be calculated after obtaining the fitting drift term of the directly integrated velocity, and then the obtained derivative term can be removed from the measured acceleration, that is, the zeroorder correction of the measured acceleration. In addition to the error caused by the baseline drift of the acceleration sensor in the measured acceleration, the low-frequency noise caused by the marine environment will also produce errors in the integration displacement. Therefore, the Butterworth polynomial is introduced to filter the low- frequency noise in the measured acceleration, and the filtered result is shown in Fig.18. A comparison with the measured acceleration shows that the corrected acceleration is smoother than the directly measured acceleration,and the sudden change caused by noise is reduced.

Fig.18 Measured acceleration and the modified acceleration.

The corresponding velocity can be calculated by integrating the acceleration after the modified acceleration is obtained. Notably, an unknown initial velocity is introduced again after integration, and the integration produces an unwanted drift term. Therefore, the average value of the integrated velocity should be removed first, and then the modified velocity should be integrated to obtain the corresponding displacement. The calculated displacement is shown in Fig.19. Compared with the acceleration in Fig.18,the transformed displacement is smooth and the noise interference is less. Thus, the anti-noise performance of the displacement is improved, and the displacement is conducive to identifying the modal parameters of the structure.

Fig.19 Transformed displacement by using the developed strategy.

5.4 Comparison of the Modal Parameter Identification Results

Similarly, the stability diagram is introduced, and the modal parameter identification is performed on the measured acceleration and the converted displacement. In the calculation, the dimension of the projection matrix is set to 500, and the stability value is set to 1%. With an increase in the modal order from 5 to 120, the obtained stability diagrams of using the acceleration and displacement are shown in Figs.20 and 21, respectively. The two figures reveal that using acceleration can identify only a small portion of the stable poles at the first-order mode of the structure, while the second- and third-order modes of the structure can no longer be identified. However, in the result using the transformed displacement, in addition to the first-order mode of the structure, the second- and third-order modes can also be identified. This finding is due to the suppression of highfrequency noise and vibration of the structure in the integration process and the amplification of low-frequency vibration, thereby facilitating the identification of low-frequency structural modal parameters.

Tables 2 and 3 also respectively list the modal frequency and modal damping ratio identification results using the measured acceleration, the transformed displacement, and the ERA method. Compared with using the measured acceleration to identify only the first-order mode of the structure, using the transformed displacement can help determine the first three-order modes of the structure. The identification accuracy has also been markedly improved for the determination of the first-order mode. The identification accuracy of the first-order modal frequency is improved by 5%, and the accuracy of the first-order modal damping ratio is improved by 84.09%. Overall, for OWTs dominated by low-frequency vibration, the displacement transformed by the proposed strategy is conducive to identifying modal parameters.

Fig.20 Stability diagram of the modal parameter identification using the calculated acceleration.

Fig.21 Stability diagram of the modal parameter identification using the transformed displacement.

Table 2 Identified results of the first three-order modal frequencies

Table 3 Identified results of the first three-order modal damping ratio

6 Conclusions

A strategy based on zero-order correction and Butterworth polynomial for estimating the displacement of OWTs with high-pile foundations is developed, and the estimated displacement is applied to the identification of structural modal parameters. The developed strategy removes the baseline drift problem caused by the acceleration sensors and filters out the low-frequency noise in the measured accelerations, thereby realizing the transformation of the dynamic displacement of OWTs. Through a comparison with the acceleration-based modal parameter identification strategy, the displacement-based modal parameter identification strategy is advantageous to OWTs. For example,Fig.6 compares the acceleration and displacement of the same structure at the same position in the frequency domain.The results show that the high-frequency components can be suppressed and the low-frequency components will be amplified by integrating the acceleration. Therefore, the use of displacement for modal parameter identification is advantageous for low-frequency-dominated structures. In addition, the introduction of the stability diagram demonstrates stable pole results identified using the transformed displacement, as shown in Figs.20 and 21. Finally, by comparing the calculated modal parameter identification results, the developed displacement-based modal parameter identification strategy not only can identify high-order structural modes but also has high accuracy, as shown in Tables 2 and 3, respectively. Notably, focusing on the accuracy of the reconstruction of displacements and the selection of the filtering parameters is necessary when using the proposed method. If the parameters are selected improperly, then the removal accuracy of drift items and the accuracy of identified modal parameters will be seriously affected. Future research will focus on the integral relationship between acceleration, velocity, and displacement to establish a displacement reconstruction method for nonlinear motion response reconstruction.

Acknowledgements

The authors acknowledge the financial support of the National Natural Science Foundation of China (Nos. 5207 1301, 51909238 and 52101333), the Zhejiang Provincial Natural Science Foundation of China (No. LHY21E0900 01), the Zhejiang Provincial Natural Science Foundation of China (No. LQ21E090009), and the support of the research on structural modeling analysis for the offshore wind turbines subjected to the multi-source coupled factors with the ship collision safety evaluation funded by Powerchina Huadong Engineering Corporation Limited.