Fatigue reliability analysis of wind turbine tower under random wind load
Introduction
In the past 20 years, wind power has continuously developed at a high growth rate. By the end of 2017, the cumulative installed capacity of wind power in the world has reached 539.581 GW; the newly installed capacity is 52.573 GW at 2017.
The capacity of the wind turbine is growing larger and larger, indicating the importance of understanding the vibration response and fatigue characteristics of the wind turbine structure. For a highly flexible structure such as an onshore wind turbine, the wind load is its control load, which acts extremely frequently, causing cyclic stress and deformation of the structure. Many researchers have studied the wind-induced vibration of the wind turbine structure. Bazeos et al. [1] used the refined and simplified finite element model to study the vibration characteristics of the wind turbine tower, emphasizing the necessity of refined finite element analysis in predicting stress at a specific position. Quilligan et al. [2] developed the Lagrangian energy conservation method for establishing the motion equation in the downwind direction of the wind turbine, and these equations are solved in the frequency domain. In addition, methods for converting wind spectral density into wind speed time series based on Fourier inverse transform are also frequently used [3], [4].
Given that the wind turbine is subjected to long-term dynamic load, the fatigue problem is prominent. It is thus necessary to accurately predict the fatigue damage of the wind turbine structure, especially in the place where the dynamic stress at the tower bottom is high. This kind of fatigue failure can be observed in other slim structures subjected to wind loads, such as transmission towers and traffic poles [5], [6], [7].
The structural fatigue damage can be affected by many factors, among which load is the most important one. Wind load has essentially non-negligible randomness. Therefore, it is a natural choice to analyze structural fatigue damage by using the reliability analysis method.
Structural reliability has been studied for more than 60 years, while the history of studying structural fatigue reliability is much shorter. This early research began with Wirsching [8]. Byers [9] reported the research status and progress of the theory and application of structural fatigue reliability assessment. Freudenthal [10] proposed a fatigue-reliability design method using a stress-strength interference model. Murty et al. [11], Jing and Jow [12] studied the distribution of fatigue strength and its application in reliability calculation.
Wind-induced vibration fatigue of high-rise structures is essentially a random variable amplitude. Previous research in this field is mainly based on the theory of fatigue cumulative damage, and the fatigue life is calculated by using the S-N curve. Peil [13] used the rain flow method to calculate the wind-induced fatigue of the mast; Huang and Hancock [14], Chaudhury and Dover [15], Chow and Li [16] and many other researchers proposed several equivalent stress amplitude methods. Wirsching [17] proposed an equivalent narrow-band method which is applied to the marine structure. Based on the Wirsching’s equivalent narrow-band method, Holmes [18] proposed a simplified method to estimate the upper and lower bounds of wind-induced fatigue damage of high-rise structures. With such a perspective, Repetto and Solari [19] have developed an advanced model to estimate an upper bound, a lower bound and an accurate value of the actual broad band stress process based on the bi-modal assumption. Ding and Chen [20] re-evaluated the TB (Benasciutti and Tovo [21]) spectral method for the fatigue analysis of broad-band Gaussian and non-Gaussian wind load effects. A refined formulation is also proposed by them to account for the influence of spectral shape on fatigue damage. Moreover, Gong et al. [22] estimated the long-term extreme response of operational and parked wind turbine and then proposed a new method to perform the estimation of structural response using annual maximum wind speed distribution, which takes into account the independent number of wind speed process in terms of extremal index.
The methods for solving reliability include Monte Carlo method [23], First Order Second Moment method [23], and Response Surface method [24]. The main goal of the reliability analysis based on the First Order Second Moment method is to find the second-order moment statistics of the random structural response. The basic idea of the Response Surface method is to present the input and output variables of the function as standard normal distribution variables of the polynomials, and the coefficients of each variable are determined by collocation points. The failure probability and reliability index are then obtained through the functional equation. Although the Monte Carlo method has good applicability, it is at the expense of a large amount of calculation. In recent years, Chen and Li [35], [25], [26], [27] have developed a class of probability density evolution methods for random structural responses. With these methods, the cluster of evolutionary probability density curves of structural responses can be accurately and quantitatively obtained. Therefore, it is convenient to directly calculate the reliability of the structure under random loads according to the specified displacement. For wind turbine structures, the fatigue damage caused by the random wind is prominent, but the results of the fatigue reliability analysis of the wind turbine structure are still rare.
Based on the idea of probability density evolution, this paper constructs a virtual stochastic process, so that the fatigue damage in the time domain is the sectioned random variable of the virtual stochastic process. Furthermore, the probability density evolution equation [27] is established, and the probability distribution of the fatigue damage is obtained. The fatigue reliability of the wind turbine structure is given by integrating the probability of fatigue damage in the safe domain.
Section snippets
Random fluctuating wind field
In order to calculate the probability density of fatigue damage, the structural dynamic equation under the wind load needs to be solved, and the stress time histories at the tower bottom under wind speeds, corresponding to a certain matrix Θ, can be obtained. The matrix Θ reflects the randomness of the excitation. Then, the rain flow method is applied to count the number of stress cycles. The number of stress cycles here is substituted into the Miner criterion, where the S-N curve and the
Probability density evolution method for fatigue reliability analysis
After orthogonally expanding the wind load, the expansion result can be substituted into the dynamic response governing equation, and the governing equation can be solved to obtain the dynamic response for various elements of Θ. By substituting the dynamic response into the probability density evolution equation and integrating the solution for Θ, the probability density of the dynamic response can be obtained. The solution is as follows.
Fatigue cumulative damage theory
The algorithm for fatigue damage D in Section 3 is described in this section. Damage refers to the degree of damage to the material under cyclic loading. It is generally represented by a dimensionless parameter D. When D = 0, the material is intact. When D > 1, it indicates that the material has reached its fatigue life.
The fatigue cumulative damage theory is used to analyze the structural fatigue damage under the random load. The commonly accepted theory is the Palmgren-Miner linear fatigue
Calculation parameters
The wind turbine used in this case study is located in a coastal wind farm in Shandong province, China. The rated power of the wind turbine is 1.5 MW, and the height of the wind turbine hub is 70 m. According to the measured wind speed and wind direction results of the wind farm, the annual average wind speed at the wind turbine hub height (70 m) is 6.62 m/s. The frequency of occurrences of wind speed between 3 m/s and 25 m/s accounts for 93.0%. The annual average wind power density is 331.8 W/m
Conclusions
This paper mainly studies the fatigue reliability analysis of the tower of onshore wind turbines. The conclusions are as follows:
- (1)
By using the “orthogonal expansion method” and the “Number-Theoretical method” of random dynamic action, the random fluctuating wind field is expanded into a set of scattered points to calculate the wind load. The numerical calculation of the dynamic response of the wind turbine is verified with the measured data, showing the rationality of the dynamic analysis
Acknowledgments
The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China (Nos. 51779224 and 51579221), and Zhejiang Basic Public Welfare Research Program (No. LHZ19E090002).
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