Method for Evaluating the Reliability and Competitive Failure of Wind Turbine Gearbox Bearings Considering the Fault Incubation Point

Abstract

Aiming to resolve the problem where the reliability of gearbox bearings of wind turbines is easily affected by random impact, this paper puts forward a reliability evaluation method for the competitive failure of gearbox bearings of wind turbines considering the fault incubation point. Firstly, we use Weibull distribution to simulate the fault latent process of wind turbine gearbox bearings. Secondly, a natural degradation model of gearbox bearing is established based on the Wiener process. Then, we model the random impact arrival frequency and impact intensity through a homogeneous Poisson process and normal distribution, respectively. Finally, based on considering the fault incubation point, the natural degradation of the bearing, the instantaneous degradation caused by impact, and the decline in the impact resistance of the bearing, a reliability evaluation model of gearbox bearings of wind turbines is established. A high-speed bearing of a gearbox from a wind farm in northern China is selected for simulation analysis. The results show that the proposed method can better describe the reliability decline process of the gearbox bearings of wind turbines, which has a specific guiding significance for the maintenance of wind turbines.

1. Introduction

As a clean and pollution-free renewable energy, wind energy has developed rapidly in recent years. According to the statistics released by the Global Wind Energy Council in 2016, the new capacity of the global wind power market exceeded 54.60 GW, and the total global installed capacity reached 486.74 GW. By 2050, the installed capacity of wind turbines in China will reach 2.40 TW, accounting for 33.80% of the power energy structure [1,2,3]. Wind turbines which run in a harsh natural environment for a long time are affected by extreme weather and loads. The performance of their components inevitably declines gradually with the change in running time, eventually leading to failures [4]. Wind turbines are complex systems composed of multiple components, including generators, hubs, blades, gearboxes, and other structures. Among them, the gearbox contains a large number of bearing gear structures which easily accelerate aging during operation and then lead to the failure of the wind turbines. According to some investigations, the damage rate of the gearbox of onshore wind turbines is about 9.8%, in which the mechanical failure is caused by bearing failure in more than 80% of cases. The downtime caused by gearbox failure accounts for about 19.4% of wind turbine downtime, even as high as 50% in offshore wind turbines. Although the gearbox is not the most frequent component of wind turbines, gearbox failure is the largest cause of downtime, and the maintenance cost is very high [5,6,7,8]. Therefore, it is of great significance to accurately evaluate the reliability of gearbox bearings of wind turbines to guide the operation and maintenance of wind turbines.

Due to the development of reliability theory, various methods exist to evaluate the reliability of mechanical structures, such as gearbox bearings. For example, Li et al. [9,10] proposed a physical modeling method based on finite element analysis, and Li et al. [11] proposed a statistical analysis method based on data. Data acquisition has become relatively easy with the development of Internet of Things (IoT) perception and advanced sensing technology. Statistical analysis methods based on big data, especially the stochastic process method, are widely used.

Performance degradation modeling based on the stochastic process is an important method for studying the reliability of products with small samples, high reliability, and long life. The Wiener process is often used to describe the variation law of mechanical structure degradation to evaluate the remaining life. Li et al. [12] considered the remaining life of AC contactors based on the Wiener process; Jin et al. [13,14] put forward a degradation modeling method based on the binary Wiener process and evaluated the life of bearings and gear pumps through two degradation quantities whose changing laws are in line with Wiener process. Deng et al. [15] established a residual life prediction model based on threshold distribution by constructing a single-performance degenerate Wiener process model and fusing a multi-performance degenerate Wiener process model to evaluate the residual life of computer numerical control (CNC) machine tools. However, some examples in the literature [12,13,14,15] used the method of natural degradation to model reliability, ignoring the impact of random shocks on the system, and the evaluation results may be higher than the real reliability. During the operation of wind turbines, due to the sudden change of extreme load, the gearbox is intermittently impacted, which leads to sudden failure; the Poisson process [16,17], normal distribution [18], exponential distribution [19], and so on are commonly used to describe the impact.

The failure of a mechanical structure is often affected by multiple factors, and the typical failure forms of the gearbox are bearing failure, gear fatigue, wear, fracture, insufficient lubrication, and so on [20]. Therefore, a competitive failure model considering both degenerate and sudden failure has apparent advantages in reliability evaluation.

Competitive failure has always been a hot topic in reliability research. Zhu et al. [21,22] evaluated a system’s reliability by studying the competitive process of sudden failure caused by random impact and degradation failure caused by the degradation process, considering the impact on degradation trajectory and degradation rate. Zhao et al. [23] described the degradation process of a low-voltage switchgear through the Wiener process, simulated sudden failure through exponential distribution, and described different failure situations through the ratio of median life of degradation failure to accidental failure to evaluate the reliability of a low-voltage switchgear. Hao et al. [24] considered various fatigue failure modes of wind turbine gearbox wear and put forward a competitive failure model which comprehensively considers interface characteristics, mechanical properties, and the residual stress gradient and evaluates the reliability of wind turbine gearboxes. In reference [25], a degradation index of wind turbines gearbox was constructed by supervisory control and data acquisition (SCADA) monitoring data, the degradation failure of gearbox bearing was described by the Wiener process, the sudden failure was simulated by Weibull distribution, and the reliability of the bearing was evaluated by considering the correlation of two failure modes. However, for the gearbox bearings of wind turbines, the failure rate function of the bearings in the degradation process meets the “bathtub curve” [26,27], and the existing reliability models mostly model and analyze from the loss failure period but seldom consider the accidental failure period of gearbox bearings of wind turbines, so the evaluation results may be lower than the real reliability.

Therefore, this paper proposes a reliability evaluation method for the competitive failure of gearbox bearings of wind turbines considering the fault incubation point. Firstly, the Weibull distribution is used to simulate the fault incubation process of gearbox bearings from the accidental failure period to the wear failure period. Secondly, the Wiener process is used to model the natural degradation process of gearbox bearings during the wear failure period. Then, random shock arrival frequency and impact intensity are simulated using a homogeneous Poisson process and normal distribution. Finally, the reliability of gearbox bearings is evaluated based on the fault incubation process of the gearbox bearings, the acceleration of the degradation process during the wear failure period, and the impact resistance of the degradation process.

The main contributions of this paper are as follows:

(1) Compared with the method of reliability modeling considering natural degradation, the proposed method considers the influence of natural degradation and random impact on bearing reliability at the same time and puts forward a competitive failure model of gearbox bearings of wind turbines which describes the real decline process of gearbox bearing reliability more comprehensively and avoids the possible “overestimation” phenomenon;

(2) Compared with the competitive failure reliability modeling method that only considers the loss failure period, this paper considers the unexpected failure period of the bearing on the basis of the competitive failure model and models the bearing failure latent process through Weibull distribution and puts forward the competitive failure model of gear bearings of wind turbines considering the failure latent process, thus avoiding the possible “underestimation” phenomenon.

The other parts of this paper are arranged as follows. Section 2 introduces the causes of gearbox bearing failure in wind turbines. Section 3 models the failure and random impact of wind turbine gearbox bearings. In Section 4, the reliability model of competitive failure of wind turbines gearbox bearings considering fault incubation point is constructed. Section 5 is the process of model parameter estimation. Section 6 is the experimental verification part. Finally, Section 7 ends the paper by summarizing the main conclusions.

2. Introduction to Bearing Failure of Gearbox of Wind Turbine

A wind turbine is a device that converts mechanical energy into electrical energy, and the function of the gearbox is to transfer the low-speed kinetic energy generated by the wind wheel to the generator and obtain a higher speed. The three-stage gearbox with two-stage planetary gears and a one-stage parallel shaft has been widely used because of its compact structure and high transmission efficiency. Its basic structure is shown in Figure 1.

The bearings of the gearbox shown in Figure 1 can be divided into bearings of the low-speed stage, bearings of the intermediate stage, and bearings of the high-speed stage. Excepting the bearings of the high-speed stage, which are deep-groove ball bearings, all the other bearings are tapered roller bearings. The failure rate of the high-speed bearings is higher than that of the other bearings because their rotating speed is faster, and their working conditions are worse [28]. When the wind turbine runs with load for a long time, it may be affected by factors including sudden changes of wind direction and extreme weather. If it is accompanied by insufficient lubrication, it may cause the transmission bearing to wear and break. The fault condition of a gearbox bearing is shown in Figure 2.

When the bearing fails, it is usually accompanied by abnormal vibration or temperature rise. By analyzing the vibration data and temperature data of the gearbox bearing, the decline process of bearing reliability can be analyzed. However, the vibration data are easily disturbed by noise, which affects the final bearing reliability evaluation results. Therefore, this paper uses more easily available and stable temperature data to evaluate the degradation and reliability decline process of the high-speed bearing of a gearbox, in which the temperature data are measured by a temperature sensor installed on the outer ring of the bearing.

3. Failure Modeling of Gearbox Bearing of Wind Turbines

3.1. Model Hypothesis

With the performance degradation of wind turbines, gearbox bearings may have a loose structure and insufficient lubrication, and then the performance degradation of bearings is reflected in the rise of the temperature index. Therefore, this paper describes the performance degradation process of bearings based on the temperature index of gearbox bearings. To simplify the analysis, the following assumptions are made in combination with the actual situation and existing research results when evaluating the reliability of gearbox bearings of wind turbines:

(1) As shown in Figure 3, the performance degradation of gearbox bearings of wind turbines consists of two stages. The first stage is the accidental failure period, during which it is difficult to intuitively find the abnormal phenomenon of bearings from the monitoring data. However, once the equipment starts to run, the fault has already begun to hatch. The second stage is the wear-and-tear failure period. During this process, the degradation trend of gearbox performance due to the long-term operation of the wind turbine becomes more apparent. The fault incubation point is the dividing point between the accidental and wear failure periods.

In the accidental failure stage, because the performance degradation trend is relatively small, and the operation is stable, Weibull distribution is often used to simulate the fault incubation point [29]. For the loss failure stage, the Wiener process is often used to describe the performance degradation process [30,31]. When the degradation process reaches the threshold, the gearbox bearing of wind turbines degrades and fails;

(2) As shown in Figure 4, during the operation of wind turbines, there are two failure modes of the gearbox bearings: degradation failure and sudden failure. When the bearing is impacted by extreme load, it accelerates the degradation process of the wear and failure stage; with the degradation of the gearbox bearing, its impact resistance also decreases continuously. When the degradation reaches the degradation failure threshold $H$, the bearing degenerates and fails; when the intensity of random impact exceeds the impact resistance $D\left(t\right)$ of the bearing, that is, the random impact amplitude $V$ exceeds the bearing’s sudden failure threshold $D\left(t\right)$, the bearing suddenly fails. The two failure modes compete with each other and influence each other. When any failure mode occurs, the gearbox bearing fails.

3.2. Fault Incubation Point of Gearbox Bearing Based on Weibull Distribution

When studying the reliability and life of industrial equipment, the Weibull distribution is especially suitable for equipment with cumulative failure distribution, so it is widely used in the life or reliability evaluation of various pieces of equipment or components. From the historical fault data of wind turbines, it can be verified that the fault distribution of wind turbine gearboxes conforms to Weibull distribution law [32], that is, the failure rate of a gearbox is high in the initial stage of operation, and its failure form is often related to human factors such as system configuration and maintenance, the failure rate of equipment remains at a low level in the middle stage of operation, and it has typical loss characteristics at the end of the operation, such as bearing wear failure caused by insufficient lubrication. So, this paper uses Weibull distribution to simulate the fault incubation point $I$ of a wind turbine gearbox bearing, that is, $I\sim Weibull\left(\delta ,\eta \right)$, and the probability density function (PDF) is:

$$f_I(I)=\frac{\delta }{\eta }{\left(\frac{I}{\eta }\right)}^{\delta -1}\exp \left[-{\left(\frac{I}{\eta }\right)}^{\delta }\right]$$

(1)

where $\delta$ is the shape parameter of the Weibull distribution, and $\eta$ is the size parameter of the Weibull distribution. Through the change of shape parameter $\delta$ and scale parameter $\eta$, the Weibull distribution can approximately describe the distribution law of various failure rates and has good generalization.

3.3. Modeling of Gearbox Bearing Degradation Based on the Wiener Process

The Wiener process is often used to analyze the variation law of performance degradation when equipment is affected by internal and external factors. In the stage of wear and tear, the bearing temperature fluctuates and rises with the deterioration of gearbox bearings. Therefore, this paper uses a one-dimensional Wiener process with drift to model the degradation process of gearbox bearings and selects gearbox bearing temperature as the characteristic quantity of gearbox bearing performance degradation. The expression of bearing temperature degradation $g\left(t\right)$ at time $t$ is [12]:

$$g\left(t\right)=g_0+\alpha t+\beta B\left(t\right)$$

(2)

where $g_0$ is the initial value of bearing temperature degradation, indicating the initial value of the temperature degradation at the beginning of the bearing wear failure process in Figure 4a; and $\alpha$ is the drift parameter, which indicates the average change rate of the degradation amount per unit time. When the degradation amount shows an increasing trend, $\alpha >0$; otherwise, $\alpha <0$. $\beta$ is the diffusion parameter, which indicates the random fluctuation or variability of the degradation amount per unit of time; and $B\left(t\right)$ is standard Brownian motion, which is used to represent the random fluctuation of the model. Then, the probability density function of $g\left(t\right)$ is:

$$f_g(g\left(t\right))=\frac{1}{\sqrt{2\pi {\beta }^{2}t}}\exp \left[-\frac{{(g\left(t\right)-g_0-\alpha t)}^2}{2{\beta }^{2}t}\right]$$

(3)

3.4. Random Impact Modeling

During the operation of wind turbines, due to extreme loads such as extreme wind load or low temperature, the gearbox bearings are impacted. Then, with the increase in lubricating oil viscosity, the lubricating ability decreases, which may lead to bearing fracture and sudden failure. In the existing research, the homogeneous Poisson process is widely used to model the frequency of random shocks [33]. Assuming that the number of random shocks is $N\left(t\right)$ at cut-off time $t$, the probability density function of $N\left(t\right)=m$ is:

$$P\left(N\left(t\right)=m\right)=\frac{{\left(\lambda t\right)}^m}{m!}\exp \left(-\lambda t\right)$$

(4)

where $\lambda$ is the arrival rate of the homogeneous Poisson process. To describe random impact strength $V_i$, the normal distribution is used [34], that is, $V_i\sim N\left(\mu ,{\sigma }^2\right)$, and its probability density function is:

$$f_V^N\left(V_i\right)=\frac{1}{\sqrt{2\pi }\sigma }\exp \left[-\frac{{\left(V_i-\mu \right)}^2}{2{\sigma }^2}\right]$$

(5)

where $\mu$ is the mean value of random impact strength, and ${\sigma }^2$ is the variance of random impact strength.

4. Reliability Model of Competitive Failure of Gearbox Bearings Considering Fault Incubation Point

As shown in Figure 2, due to the difference in the degradation performance between the accidental failure period and wear failure period of gearbox bearings of wind turbines, their reliability can be calculated in two parts with the fault incubation point as the boundary:

$$R\left(t\right)={\displaystyle {\int }_0^{t}R\left(t|t\le I\right)·f_I\left(I\right)dI}+{\displaystyle {\int }_t^{+\infty }R\left(t|t>I\right)·f_I\left(I\right)dI}$$

(6)

4.1. Reliability When $t\le I$

Assuming that the number of random impacts reaches $m$ at time $t\le I$, because the gearbox bearing is in the accidental failure period at this time, its degradation process is not obvious, so its reliability is as follows:

$$R\left(t|t\le I\right)=R_{th}\left(t|t\le I\right)·R_{tf}\left(t|t\le I,N\left(t\right)=m\right)$$

(7)

where $R_{th}\left(t|t\le I\right)$ represents the soft degradation failure reliability of the bearing at $t\le I$. Because the degradation amount of the bearing in the accidental failure period is almost unchanged, $R_{th}\left(t|t\le I\right)=1$; $R_{tf}\left(t|t\le I,N\left(t\right)=m\right)$ represents the reliability of the bearing’s sudden hard failure under $m$ impacts, assuming that random impact strength $V_i$ has the following relationship with bearing’s impact resistance $D_0$:

$$D_s=V_i-D_0$$

(8)

According to Equation (5), the probability density function of $D_s$ can be deduced as follows:

$$f_V^N\left(D_s\right)=\frac{1}{\sqrt{2\pi }\sigma }\exp \left[-\frac{{\left(D_s-\mu +D_0\right)}^2}{2{\sigma }^2}\right]$$

(9)

Then, $R_{tf}\left(t|t\le I,N\left(t\right)=m\right)$ is:

$$R_{tf}\left(t|t\le I,N\left(t\right)=m\right)={\displaystyle \sum _{m=0}^{\infty }P\left(N\left(t\right)=m\right)·{\left({\displaystyle {\int }_{-\infty }^0{f}_V^N\left(D_s\right)d{D}_s}\right)}^m}$$

(10)

4.2. Reliability When $t>I$

When $t>I$, the reliability evaluation of gearbox bearings needs to consider three parts: the degradation failure process of bearings during wear failure, the acceleration of random impact in the degradation process, and the influence of the degradation process on impact resistance.

4.2.1. Degenerate Soft Failure Reliability at $t>I$

According to the engineering experience, the instantaneous degradation increment caused by a single random impact is positively correlated with the impact amplitude, so this paper assumes that the relationship between the instantaneous degradation increment $g_{\sum 1}\left(t\right)$, and the impact amplitude $V_i$ is:

$$g_{\sum 1}\left(t\right)=\phi {\displaystyle \sum _{i=1}^m{V}_i}$$

(11)

where $\phi >0$ is the random impact coefficient, and $V_i$ is the amplitude of the $i\mathrm{th}$ impact. The sum of $m$ independent identically distributed normal distributions still obeys the normal distribution, and $\phi$ is a constant, so there is $g_{\sum 1}\left(t\right)\sim N\left(m\phi \mu ,m{\phi }^2{\sigma }^2\right)$, and the probability density function of the instantaneous degradation increment $g_{\sum 1}\left(t\right)$ generated by random impact is:

$$f_{g_1}\left(g_{\sum 1}\left(t\right)|N\left(t\right)=m\right)=\frac{1}{\sqrt{2\pi m}\phi \sigma }\exp \left(-\frac{{\left(g_{\sum 1}\left(t\right)-m\phi \mu \right)}^2}{2m{\phi }^2{\sigma }^2}\right)$$

(12)

According to Equations (3) and (12), the probability density function of the total degradation of gearbox bearings can be derived as follows:

$$f_{g_{\sum }}\left(g_{\sum }\left(t\right)|N\left(t\right)=m\right)={\displaystyle {\int }_0^{+\infty }{f}_g\left(g\left(t\right)\right)}·f_{g_1}\left(g_{{}_{\sum }}\left(t\right)-g\left(t\right)\right)dg$$

(13)

Then, when $t>I$, the degraded soft failure reliability of gearbox bearing is:

$$R_{th}\left(t-I|N\left(t\right)=m-m_1\right)={\displaystyle {\int }_0^{H}f\left(g_{\sum }\left(t-I\right)|N\left(t\right)=m-m_1\right)}d{g}_{\sum }$$

(14)

where $H$ is the threshold of bearing degradation failure.

4.2.2. Reliability of Sudden Hard Failure When $t>I$

With the increase of gearbox bearing degradation, its ability to withstand fatal impact decreases, as shown in Figure 2. The relationship between the bearing hard failure threshold and degradation is as follows:

$$D\left(t\right)=D_0-\xi g_{\sum }\left(t\right)$$

(15)

where $\xi$ is the degradation coefficient of the hard failure threshold, and the difference between random impact amplitude $V_i$ and hard failure threshold $D\left(t\right)$ is $D_S\left(t\right)$. Then,

$$D_S\left(t\right)=V_i-D_0+\xi g_{\sum }\left(t\right)$$

(16)

If the gearbox bearing is impacted $m$ times in $\left(I,t\right)$, the probability density function of $D_S\left(t\right)$ is:

$$f_{D_s}\left(D_s\left(t\right)\right)=\frac{{\displaystyle {\int }_0^{+\infty }{f}_V^N\left(V_i\right)\ast f_g\left(\frac{D_s\left(t\right)-V_i-D_0}{\xi }|N\left(t\right)=m\right)d{V}_i}}{\xi }$$

(17)

When $t>I$, the reliability of sudden hard failure of the gearbox bearing is:

$$R_{tf1}\left(t-I|t>I,N\left(t\right)=m\right)={\left[{\displaystyle {\int }_{-\infty }^0{f}_{D_s}\left(D_s\left(t\right)\right)d{D}_s}\right]}^m$$

(18)

4.2.3. Total Reliability When $t>I$

Assuming that the number of random impact arrivals is $m$, in which they reach $m_1$ before fault incubation point $I$, the reliability expression of the bearing is:

$$R\left(t|t>I\right)={\displaystyle \sum _{m=0}^{+\infty }\begin{array}{l}P\left(N\left(t\right)=m\right)·\\ \left[{\displaystyle \sum _{m_1=0}^{m}P\left(N\left(I\right)=m_1\right)R_{tf}\left(I|N\left(I\right)=m_1\right)R_{th}\left(t-I|N\left(t\right)=m-m_1\right)}\right]·\\ \left[{\displaystyle \sum _{m_1=0}^{m}P\left(N\left(I\right)=m_1\right)R_{tf}\left(I|N\left(I\right)=m_1\right)R_{tf1}\left(t-I|N\left(t\right)=m-m_1\right)}\right]\end{array}}$$

(19)

5. Model Parameter Estimation

5.1. Parameter Estimation of Wiener Process

The degradation model based on the Wiener process includes the drift coefficient $\mu$ and the diffusion coefficient $\sigma$. In this paper, the performance degradation of gearbox bearings is calculated by the maximum likelihood estimation method. The total sample size is $P$, and it is assumed that $N$ samples are degraded and will fail. According to the nature of the Wiener process, the performance degradation increment $\Delta X^i$ obeys the normal distribution related to the time scale, that is:

$$\Delta X^i=X\left(t_i+\Delta t_i\right)-X\left(t_i\right)\sim N\left(\mu \Delta t_i,{\sigma }^2\Delta t_i\right)\hspace{1em}i=1,2,\cdots ,n$$

(20)

The probability density function of $\Delta X^i$ is:

$$f\left(\Delta X^i|\mu ,\sigma \right)=\frac{1}{\sqrt{2\pi {\sigma }^2\Delta t_i}}\exp \left(-\frac{{\left(\Delta X^i-\mu \Delta t_i\right)}^2}{2{\sigma }^2\Delta t_i}\right)$$

(21)

The likelihood function is:

$$L\left(\mu ,\sigma \right)={\displaystyle \prod _{j=1}^N{\displaystyle \prod _{i=1}^{n}f\left(\Delta X^i\right)}}$$

(22)

The likelihood function takes partial derivatives of $\mu$ and $\sigma$, respectively, and makes the partial derivatives zero to obtain the maximum likelihood estimate of $\mu$ and $\sigma$ as follows:

$$\widehat{\mu }=\frac{1}{N}{\displaystyle \sum _{j=1}^N\left(\frac{1}{n}{\displaystyle \sum _{i=1}^n\frac{\Delta X^i}{\Delta t_i}}\right)}$$

(23)

$$\widehat{\sigma }=\frac{1}{N}{\displaystyle \sum _{j-1}^N\sqrt{\frac{1}{n}{\displaystyle \sum _{i=1}^n\frac{{\left(\Delta X^i-\widehat{\mu }\Delta t_i\right)}^2}{\Delta t_i}}}}$$

(24)

5.2. Parameter Estimation of Weibull Process

The failure rate function of two-parameter Weibull distribution is:

$$h_0(t)=\frac{\delta }{\eta }{\left(\frac{t}{\eta }\right)}^{\delta -1}$$

(25)

It is observed that the failure rate function of the Weibull distribution only depends on the running time, so the estimation of parameters $\delta$ and $\eta$ only needs to count the failure time $t_j$. Assuming that $M$ samples in the total sample will fail, the likelihood function is constructed according to the probability density function of $h_0(t)$ as follows:

$$L(\beta ,\eta )=\underset{}{\overset{}{{\displaystyle \prod _{j=1}^M\frac{\delta }{\eta }{\left(\frac{t_j}{\eta }\right)}^{\delta -1}\exp \left[-{\left(\frac{t_j}{\eta }\right)}^{\delta }\right]}}}$$

(26)

After the logarithm of the likelihood function, the partial derivatives of $\delta$ and $\eta$ are obtained respectively, and Newton’s method is used for the iterative solution so that only the maximum likelihood estimates of the Weibull distribution parameters related to time can be obtained.

6. Results

6.1. Input Data

To verify the validity of the proposed reliability evaluation method, this paper selects the high-speed bearing of the gearbox of a particular wind turbine in a wind farm in northern China for simulation analysis. The initial temperature of the front bearing of the gearbox of this type is 25.4 °C, the degradation failure temperature is 95 °C, and the maximum speed of the wind turbine system is 18 r/min. Some samples of the gearbox bearing fault data are shown in

Table 1. Partial samples of bearing fault data.

Bearing number	1	2	3	4	5	6
Life/months	90	98	109	115	131	133
Bearing number	7	8	9	10	11	12
Life/months	135	144	153	157	170	183

, and these fault samples come from many wind turbines of the same type in this wind farm:

6.2. Model Parameters

According to Table 1, the fault incubation point $I$ is estimated to obtain $\widehat{\delta }=2.1246$, $\widehat{\eta }=634.392$, and the probability density function of $I$ is:

$$f_I(I)=\frac{2.1246}{634.392}{\left(\frac{I}{634.392}\right)}^{1.1246}\exp \left(-{\left(\frac{I}{634.392}\right)}^{2.1246}\right)$$

(27)

In order to better reflect the degradation process of the gearbox bearings, the gearbox temperature is arranged in time sequence to obtain $\left(t_0,c_0\right),\left(t_1,c_1\right),\cdots ,\left(t_n,c_n\right)$. The existing research shows that the gearbox temperature front axle is positively correlated with the speed of the wind wheel [35], so the speed data $\left(t_0,r_0\right),\left(t_1,r_1\right),\cdots ,\left(t_n,r_n\right)$ are extracted. The temperature data are processed by the formula $e_i=c_i/r_i$, and the performance degradation of each sampling point $\left(t_0,e_0\right),\left(t_1,e_1\right),\cdots ,\left(t_n,e_n\right)$ is obtained, where $t_0<t_1<\cdots <t_n$ and $e_i$ are the degradation of gearbox performance. Because there may be abnormal data and random fluctuation data in the monitoring data, the sliding window method with a window width of $k$ is used to process the data from $t_0$ to $t_n$, and the performance degradation $x_m$ of the gearbox at time $t_m$ is:

$$x_m=\frac{1}{k}{\displaystyle \sum _{i=m-k+1}^m{e}_i}$$

(28)

According to the data in Section 6.1, the degradation failure threshold of the front bearing of this type of gearbox is $H=5.28 °\mathrm{C}·\min /\mathrm{r}$. Based on the degradation of the temperature performance of the gearbox’s front bearing, the Wiener process’s parameters are determined by the maximum likelihood estimation method, and other model parameters are determined by reference [36]. The specific model parameters are shown in

Table 2. Model parameters.

Model Parameter Description	Distribution Form	Parameter Estimation Value	Parameter Source
Natural degradation of gearbox bearing temperature	Wiener	$\begin{array}{l}\alpha =0.301\\ \beta =5.304\end{array}$	Maximum likelihood estimate
Random impact frequency	Possion	$\lambda =0.2$	[36]
Random impact strength	Normal	$\begin{array}{l}\mu =6.13\\ \sigma =1.77\end{array}$	[36]
Random impact coefficient	Constant	$\phi =0.2$	[36]
Degenerate failure threshold	Constant	$H=5.28 °\mathsf{C}·\min /\mathrm{r}$	/
Bearing impact resistance	Constant	$D_0=8.5$	[36]
Degradation coefficient of impact resistance of bearings	Constant	$\xi =0.1$	[36]

.

6.3. Assessment Result

This paper proposes a reliability evaluation method for the competitive failure of gearbox bearings of wind turbines considering the fault incubation point. To verify the actual performance of the models, three reliability models considering different factors are constructed for comparison, and the model descriptions are shown in

Table 3. Model description.

Model 1	Reliability model considering only natural degradation
Model 2	Competitive failure reliability model considering fault incubation point
Model 3	Reliability evaluation model considering only competitive failure

.

According to the parameters in Table 2, the reliability curves of the three models are calculated and are shown in Figure 5.

As shown in Figure 5, the evaluation results of the reliability evaluation model only considering the natural degradation process show that the bearing reliability declines slowly, which is quite different from the life of the gearbox bearing of the wind turbines in Table 1. By the 50th month, the reliability of bearing is only 0.177, which is “underestimated” to some extent compared with the data in Table 1. The evaluation results of the competitive failure reliability model considering the fault incubation point are as follows: the gearbox bearing of wind turbines has high reliability in the first 50 months, and the reliability of the bearing decreases seriously in 50–100 months, and the reliability is at a low level after the 125th month, which is consistent with the bearing life data of the bearing in Table 1.

The performance of each reliability evaluation model is quantitatively compared by calculating the mean time to failure (MTTF). The calculation method of MTTF is the integration of reliability curves.

Table 4. MTTF comparison of model evaluation.

Models	MTTF/Months
Model 1	140.6
Model 2	93.8
Model 3	52.3

shows the MTTF comparison of the evaluation results of the three models.

Based on the AGMA6006 standard and the basic rated life of bearings as the judgment basis [37], the average minimum rated life of gearbox bearings of wind turbines is 70,000 h (about 97.2 months). Compared with the minimum rated life, Model 1, Model 2, and Model 3 errors are 44.7%, 3.5%, and 46.2%, respectively. Therefore, the evaluation error of the proposed method is reduced by 41.2% and 42.7% compared with Model 1 only considering natural degradation and Model 3 only considering competitive failure.

6.4. Parameter Sensitivity Analysis

The reliability evaluation method proposed in this paper considers the fault incubation point of the gearbox bearing of wind turbines. Additionally, it considers the natural degradation of the bearing, the acceleration of random impact on the degradation process, and the influence of the degradation process on the impact resistance of the bearing. To verify the reliability evaluation method’s sensitivity, we adjust the analysis model’s key parameters, and the results are shown in Figure 6, Figure 7, Figure 8 and Figure 9.

As shown in Figure 6, Figure 7, Figure 8 and Figure 9, when the $\alpha ,\lambda ,\xi$ value of the model increases or the $\alpha ,\lambda ,\xi$ value decreases, the decline speed of the reliability of the gearbox bearing of the wind turbines is accelerated. When the value of $\alpha$ increases, the natural degradation speed of the bearings accelerates; when the value of $\lambda$ increases, the frequency of random impacts on the bearings increases; when the value of $\eta$ increases, the impact resistance of the bearing decreases faster; when the value of $\delta$ decreases, the fault incubation point of the bearing is advanced, so the reliability of the bearing decreases even more. In addition, by observing Figure 7, it can be seen that the change in $\lambda$ value has an obvious influence on the reliability of gearbox bearings, so it can be inferred that the frequency of random impact is the main factor affecting the reliability of bearings, so if you want to prolong the service life of gearbox bearings, you should try to reduce the impact frequency $\lambda$.

7. Conclusions

Aiming to resolve the problem where existing competitive failure reliability modeling methods of wind turbine gearbox bearings only considering the wear failure period lead to large evaluation errors, in this paper, a reliability evaluation method of the competitive failure of gearbox bearings of wind turbines considering the usual latent points is proposed. The high-speed bearing of a gearbox from a wind farm in northern China is selected for simulation analysis, and the following conclusions are drawn:

(1)

A reliability model for the competitive failure of gearbox bearings of wind turbines considering the fault incubation point is proposed, and the accidental failure period and wear failure period of bearing degradation are considered at the same time, which can better describe the reliability decline process of gearbox bearings of wind turbines compared with a model only considering wear failure period. The simulation results show that the MTTF error of the proposed method is 3.5%, which is 41.2% and 42.7% lower than that of the existing methods that only consider natural degradation and competitive failure, respectively;

(2)

Sensitivity analysis of key parameters $\alpha ,\lambda ,\xi ,\delta$ in the model is carried out. The results show that with the increase in value $\alpha ,\lambda ,\xi$ or the decrease in value $\delta$, the decline speed of wind turbine gearbox bearing reliability is accelerated. Among them, the change in $\lambda$ value has an obvious influence on the reliability of bearings, so if it is necessary to extend the service life of gearbox bearings, we must try to reduce the impact frequency $\lambda$;

(3)

The limitation of the evaluation method proposed in this paper is that the evaluation process requires higher data quality. Higher data quality will make the results of the parameter estimation more consistent with the real distribution, and if the data quality is lower, the evaluation error may increase.