Under the action of wind load, the foundation ring of the fan will generate stress concentration and alternating stress, leading to fatigue failure. Based on the average wind speed obtained from the supervisory control and data acquisition (SCADA) system, the orthogonal expansion method of random pulsating wind field and the number theory point selection method are used to calculate and simulate the corresponding pulsating wind speed time series, and then calculate the wind speed time series and wind load time series. Based on the ABAQUS finite element software, a model of a 5 MW wind turbine is established. The modal analysis is used to obtain the modal vibration pattern and the intrinsic frequency to verify the reasonableness of the structural modeling. Subsequently, the wind load time series is applied to the wind turbine model to obtain the stress time series using instantaneous modal dynamic analysis (Modal dynamics). The stress time series at the location of maximum stress concentration in the foundation ring is extracted, and the stress time series with the same stress amplitude is obtained through conversion, which has Wiener characteristics. After obtaining the stress amplitude using the rainflow counting method, the fatigue reliability is calculated based on the structural fatigue reliability analysis method considering the threshold crossing duration of the random process. This research holds reference significance for the calculation of fatigue reliability under approximate working conditions.
Most collapse accidents of wind turbine structures are caused by fatigue damage to the concrete foundation or connecting bolts of the wind turbine under wind load, leading to noticeable sliding displacement and structural collapse. Currently, although a large amount of analysis has been conducted on the reliability of wind turbine structures, the various uncertainties present in the natural environment and the operation process make it difficult to accurately assess the fatigue reliability of wind turbine structures using constant reliability statistical indicators. Therefore, research on the fatigue reliability of wind turbines is relatively limited. Therefore, it is of great practical significance and economic value to use realtime monitoring data from supervisory control and data acquisition (SCADA) systems to establish a structural model of wind turbines and conduct fatigue reliability analysis. Condition monitoring is an important means to improve the operational safety of wind turbines and is of great significance for the safe and efficient utilization of the turbine structure^{[19]}. Monitoring techniques for wind turbine structures mainly include ultrasonic testing, thermal imaging, process parameter monitoring, performance parameter monitoring, and Xray imaging techniques. Currently, the most widely used realtime monitoring system for wind turbine structures is the SCADA system, which can monitor the operation status of all components or subsystems of the turbine structure in realtime. It integrates information collection, condition monitoring, and parameter adjustment, providing technical support for the longterm safe operation of wind turbine structures.
The SCADA system of wind turbines has the capability of realtime monitoring and data acquisition, and many scholars have utilized this system to solve problems that are more in line with actual operating conditions. Kusiak
In recent years, there have been frequent incidents of wind turbine foundation ring detachment from the surrounding concrete or damage to the connection between the foundation and the upper tower structure. These incidents have resulted in excessive tower oscillation, turbine alarms, shutdowns, and even collapses, causing significant economic losses that are difficult to estimate. The main cause of these problems is excessive fatigue loads. Therefore, the fatigue reliability analysis of wind turbine structures has attracted widespread attention from researchers. For example, Velarde
This article focuses on the fatigue reliability of the foundation ring of wind turbines. The reliability analysis of the structure under longterm wind loads belongs to the field of structural dynamic reliability analysis, which involves two failure mechanisms: exceedance failure mechanism^{[2830]} and fatigue failure mechanism^{[31,32]}. Structural fatigue failure is caused by the accumulation of damage due to the structure exceeding the safety threshold multiple times during the response process. Therefore, the probability analysis of multiple threshold crossings in stochastic processes should be considered in fatigue reliability assessment.
Based on recent advancements in the analysis of stochastic processes and multiple threshold crossings^{[33,34]}, as well as the availability of realtime monitoring data through SCADA systems to determine average wind speeds, this study employs the orthogonal expansion method^{[22]} and the number theory method^{[35]} to simulate the pulsating wind speedtime series. By generating the stochastic wind speedtime series, the wind load acting on the wind turbine can be calculated. The wind turbine structure is modeled using ABAQUS finite element software, and the obtained wind loadtime series is applied to the model to obtain the stresstime series at the stress concentration of the wind turbine foundation ring. Considering the significant impact of the duration of the stochastic response process beyond the damage threshold (referred to as crossingthreshold duration) on the fatigue reliability of the wind turbine structure, it is important to use a structural fatigue reliability analysis method that considers the stochastic process of crossingthreshold holding time. This approach is more representative of the actual fatigue reliability analysis of the wind turbine foundation ring, and it has practical implications for engineering applications.
ABAQUS is utilized to conduct finite element modeling and analysis of a 5 MW wind turbine generator. The main parameters of the wind turbine are as follows. The height of the generator rotor is 80 m, the diameter of the rotor is 115 m, the rated wind speed is 11.9 m/s, and the design life is 20 years. The nacelle of the turbine weighs 131.142 t. The rotor consists of three blades and a hub, with each blade’ length being 55 m. The total weight of the rotor is 68.283 t, and the rated rotational speed is 14.3 rpm. The wind turbine tower is composed of three sections of variable crosssection steel tower, with geometric parameters shown in
The geometric parameters of each tower of the wind turbine





Lower section  14.86  4.5  4.5  0.05 
middle section  28.315  4.5  3.9  0.05 
upper section  32.102  3.9  3.07  0.02 
In order to obtain realtime operational condition factors of wind turbines, most wind farms are equipped with SCADA systems, which can monitor and collect data in real time, with a data collection interval of seconds. There are limitations in obtaining data through SCADA systems, such as incomplete and inaccurate data,
The wind speed time series at a certain height
where
In order to simplify the calculation, the tower of a 75 m high wind turbine tower is divided into 0~15, 15~45 and 45~75 m, and the wind speed at the center of the three sections is taken as the average wind speed of the tower in the section, i.e., the average wind speed at 7.5, 30 and 60 m. The conversion for the average wind speed at each height is expressed as follows.
where
In the case where the power spectral density (PSD) function of the pulsating wind speedtime series
where
According to the Literature^{[22]}, it can be obtained that the stochastic process
where
For the correlation matrix
where the deterministic function
Construct the corresponding autocorrelation function of the virtual pulsating wind displacement random process
where
Because the wind speed at the center point of each tower section and at the impeller is mainly considered, for Eq. (3), there is^{[22]}
where {
The reaction pulsating wind characteristic stochastic process
where
If the stochastic process of pulsating wind speed and its virtual wind displacement can be assumed to be the Gaussian process, then the sets of random variables {
0~15 m wind speedtime series.
15~45 m wind speedtime series.
45~75 m wind speedtime series.
Wind speedtime series at the impeller under normal operating conditions of the wind turbine.
The standard value of wind loads perpendicular to the surface of the wind turbine tower according to “Standard for design of highrising structure”^{[36]} is calculated according to
where
Given the wind speed time series acting on the tower body, the corresponding wind load acting on the wind turbine tower body is calculated according to
where
According to the Bernoulli's theory the relationship between wind speed and wind pressure can be obtained as
where
For the wind load on the tower blades, it can be divided into two specific cases: normal operating conditions and extreme load conditions. Due to the obtained data in this paper being wind speeds obtained during normal operation of the wind turbine, the effects of extreme loads are not considered for now. Under normal operating conditions, the thrust force of the blade due to the wind load can be calculated according to the following equation, based on the momentumblade element theory.
where
Thrust coefficients of the wind turbine at different wind speeds.
The wind speedtime series of the wind turbine tower at 0~15, 15~45, 45~75 m is brought into Eq. (15) respectively, and the total wind loadtime series acting on each section of the tower can be calculated out, as given in
Total wind load acting on 0~15 m tower bodytime series.
Total wind load acting on 15~45 m tower bodytime series.
Total wind load acting on 45~75 m tower bodytime series.
The wind speedtime series at the impeller of the wind turbine under normal operating conditions is brought into Eq. (17) to obtain its corresponding total wind loadtime series, as shown in
Total wind load acting on the impeller of the fan under normal operating conditionstime series.
The focus of this study is on the foundation ring, so the model has been simplified for ease of modeling, mainly divided into three major parts: (1) upper structure; (2) tower body; and (3) foundation. The upper structure consists of the hub, blades, and nacelle, while the foundation includes the concrete foundation and the foundation ring. The tower body is modeled based on the actual geometric parameters in
Material properties






C40 Concrete  2500  32.5  0.2  2.39  26.8 
Q345 Steel  7850  200  0.3  345  345 
Steel Q345 hardening curve
414  465  543  606  671  708  773  
0  0.0428  0.903  0.1384  0.1782  0.2175  0.2663 
Since the focus of this study is on the fatigue reliability and fatigue life of the foundation ring, the modeling of the steel reinforcement in the concrete foundation and the modeling below the foundation are abandoned. Tie connections are used between the hub and nacelle, nacelle and tower body, and tower body and foundation ring. The concrete foundation and the foundation ring are in facetoface contact, with Coulomb friction contact in the tangential direction and a friction coefficient of 0.35. The normal contact is treated as hard contact. Regarding the boundary conditions, the concrete foundation ground is assumed to have a fully fixed boundary condition since the modeling of the foundation is not considered. The mesh division for each component is as follows: the blades and hub use solid elements C3D10, the nacelle, concrete foundation, and tower body use solid elements C3D8R, and the foundation ring uses shell elements S4R. The overall mesh division is shown in
Overall meshing.
Foundation ring meshing.
Since the wind speed collected in real time by the SCADA system in this study is the wind speed on the windward side, the wind load is only applied to the windward side of the turbine rotor and tower body. The total wind load received at heights of 015, 1545, 4575 m, and at the rotor is established as “amplitude” based on the corresponding relationship between time and load values. In the ABAQUS finite element software, the 75 m high tower body is divided into three sections, and reference points RP1 to RP3 are established at the center of each section. Each reference point is coupled with the corresponding surface of the tower body, and the loads acting on each section of the tower body are applied to the reference points. As for the total wind loadtime series acting on the turbine rotor, since the main focus of the study is on the foundation ring, a concentrated load method is used to apply the load to the center of the rotor hub.
Based on the ABAQUS model established in Section "Establishment of wind turbine model in ABAQUS", in order to ensure that the wind turbine does not collapse due to resonance between loworder natural frequencies and rotor rotation frequencies (The difference between the higher order intrinsic frequency of the wind turbine and the rotational frequency of the impeller is large, and due to the role of structural damping, the higher order part of the dynamic response attenuates very quickly), modal analysis is conducted before performing transient modal dynamic analysis in ABAQUS finite element software. Therefore, the wind turbine model is first subjected to modal analysis to obtain its loworder natural frequencies.
Modal analysis is performed in ABAQUS, without considering the influence of loads, only setting boundary conditions. This study only aims to obtain the first four natural frequencies of the wind turbine, so the subspace iteration method is used for faster calculation results. The calculated natural frequencies of the wind turbine are shown in
Wind turbine first 4th order mode shapes; (A) Firstorder mode shapes; (B) Secondorder mode shapes; (C) Thirdorder mode shapes; (D) Fourthorder mode shapes.
Wind turbine first 4th order natural frequency
1  2  3  4  
0.37680  0.37759  0.6420  0.80267 
The rated speed of the 5 MW wind turbine rotor is known to be 14.3 r/min. From calculations, the rotational frequency of the rotor can be determined as
Relative difference between the first 4 orders of the wind turbine
1  2  3  4  
58.12  58.45  169.41  236.83  
47.29  47.18  10.20  12.28 
In the transient modal dynamic analysis, Rayleigh damping is considered. According to
where
By substituting the first and second natural frequencies of the wind turbine from
The wind load calculated in Section "Calculation of wind loadtime series" is applied to the wind turbine model. Then, the Rayleigh damping coefficients α and β are input into the modal dynamics analysis step of the ABAQUS finite element software to perform transient modal dynamic response analysis, thereby obtaining the stresstime series. The stress contour map of the foundation ring, which is the main focus of the study, is shown in
Stress cloud of wind turbine foundation ring.
Stresstime series in the stressconcentrated region of the wind turbine foundation ring.
The model used above is a 5 MW wind turbine. Because of its high power generation rate, it is more commonly used in the field of wind power generation, especially in some areas rich in wind resources (including the seaside). The dimensions and material properties of the wind turbine are obtained from the manufacturer. In performing the ABAQUS finite element modeling and analysis, the structure is modeled in a simplified manner, such as the superstructure is modeled in an equivalent manner for ease of computation and generalizability, which can be used for the study of this type of 5 MW wind turbine. This paper focuses on the fatigue reliability at the maximum stress at the edge of the flange on the base ring of a wind turbine. The fatigue reliability of the flange edge on the base ring is analyzed by the fatigue reliability calculation method considering the crossingthreshold duration, and a comparative analysis is made by the traditional method. The method is generalized and can be used for other types and sizes of wind turbine structures, and is also applicable to the calculation of stressfatigue reliability of structures in different environments.
According to the “Standard for design of steel structures”^{[39]}, the stress amplitude of lower than the fatigue cutoff limit [Δσ_{L}], that is, Δσ = σ_{max}  σ_{min} < [Δσ_{L}], will not lead to fatigue damage. Therefore, it can be concluded that fatigue damage is mainly related to the stress amplitude, and not directly related to the maximum and minimum stress values. Therefore, the stresstime series of the maximum stress region of the foundation ring is obtained by subtracting its mean value, as shown in
Stresstime series in the stressconcentration region of the modified wind turbine foundation ring.
The stresstime series of a structure under load can be considered as a stochastic process. And some stresstime histories of structures can be approximated as Wiener processes, which have the characteristics of a Wiener process.
The sample curves for the Wiener process.
Based on the progress of the analysis of multiple threshold crossings of stochastic processes in recent years^{[33]}, the structural fatigue reliability analysis method considering the crossingthreshold duration of stochastic processes is used to analyze the fatigue reliability of structures. The calculation formula is as follows^{[40]}.
where
The connection between the foundation ring and the tower is made using a flange connection. The maximum stress generated at the upper flange of the foundation ring can significantly reduce the fatigue strength at the upper flange and bolt locations. The research object is selected as the location of the maximum stress at the upper flange of the foundation ring, and its stresstime series is shown in
Stresstime series.
Statistics of stress amplitude
19.70  21.05  22.40  23.74  25.09  26.44  27.79  29.14  
103.45  321.84  804.60  1,011.49  965.52  413.79  114.94  183.91  
0.02069  0.06437  0.16092  0.2023  0.1931  0.08276  0.02299  0.03678  
30.48  31.83  33.18  34.53  35.87  37.22  38.57  39.92  
149.43  333.33  160.92  195.40  149.43  34.48  22.99  34.48  
0.02989  0.06667  0.03218  0.03908  0.02989  0.0069  0.0046  0.0069 
According to the data in
Statistics on the number of threshold crossings (



[0, 50]  6,472  64.72% 
(50, 100]  2,351  23.51% 
(100, 150]  890  8.9% 
(150, 200]  234  2.34% 
(200, 250]  45  0.45% 
(250, 300]  6  0.06% 
(300, 350]  2  0.02% 
The next step is to calculate the value on the right side of Eq. (20), i.e., calculating
Values of U (
0  1  2  3  4  5  6  
  303  151  101  76  61  50  
7  8  9  10  11  12  13  
43  38  34  30  28  25  23  
14  15  16  17  18  19  20  
22  20  19  18  17  16  15 
The probability of statistically obtaining the number of crossings
The fatigue reliability curves for flange stress concentration areas on the base ring (
The fatigue reliability of flange stress concentration areas on the base ring (
0  1  2  3  4  5  6  
100  99.98  97.24  88.45  79.24  71.37  64.72  
7  8  9  10  11  12  13  
59.80  56.45  53.32  50.08  48.46  46.29  44.52  
14  15  16  17  18  19  20  
43.79  42.05  41.22  40.27  39.39  38.46  37.59 
The fatigue reliability of flange stress concentration areas on the base ring (
0  1  2  3  4  5  6  
100  99.62  87.98  72.75  61.98  54.80  48.83  
7  8  9  10  11  12  13  
44.82  41.76  39.49  36.86  35.71  33.77  32.50  
14  15  16  17  18  19  20  
31.82  30.60  29.88  29.17  28.50  27.77  26.98 
From
Based on realtime monitoring data from the SCADA system, this paper combines the ABAQUS finite element software for modeling and a new structural fatigue reliability analysis method that considers the stochastic process of exceeding the threshold duration. The fatigue reliability of the maximum stress at the upper flange of the foundation ring of the wind turbine structure under normal operating conditions was calculated. The main research work and conclusions are as follows.
(1) Under normal operating conditions, the wind speed time series of each tower section is calculated by wind speed exponential model, orthogonal expansion method of random pulsating wind field and number theory point selection method in number theory, and then the wind load time series of each tower section and blade is calculated using the wind load calculation formula of standard for design of highrising structures and the momentumleaf vein theory. The wind loads are determined based on the realtime wind speed data collected by the SCADA system, so it should be a more reliable result.
(2) Establish a simplified model in ABAQUS finite element software in line with the engineering profile. The reasonableness of the structural model of the wind turbine was first verified by modal analysis. Subsequently, the wind load is applied to the model for instantaneous modal dynamics analysis to obtain the structural stress time course. The results show that the maximum stress region of the foundation ring is located at the edge of the upper flange, which is consistent with the engineering reality.
(3) Fatigue reliability in the region of maximum stress in the base ring is calculated using a structural fatigue reliability analysis method that considers random processes’ crossingthreshold duration. The results are compared with those of the traditional method to demonstrate the superiority and reasonableness of the method that takes into account the calculation of crossingthreshold durations.
Conceptualization, methodology, supervision: Zhang Z
Formal analysis, writingoriginal draft preparation: Liu Y
Investigation, writingreview and editing: Li W
The data are available from the corresponding author upon reasonable request.
This work was jointly supported by the Talent Recruitment Project of Hunan Province, China (grant no. 2023TJZ17), National Natural Science Foundation of China (grant no. 52478130), Regional Scientific and Technological Cooperation and Exchange Project of Hunan Association for Science and Technology (2024SKXKJ09).
All authors declared that there are no conflicts of interest.
Not applicable.
Not applicable.
© The Author(s) 2024.