AN Zong-wen(安宗文), XU Jie(許 潔), ZHANG Xiao-ling(張小玲)

1 School of Mechatronics Engineering, Lanzhou University of Technology, Lanzhou 730050, China 2 School of Mechatronics Engineering, University of Electronic Science and Technology of China, Chengdu 610054, China

Reliability Analysis of Wind Turbine Gearbox Based on the Optimal Confidence Limit Method

AN Zong-wen(安宗文)1*, XU Jie(許 潔)1, ZHANG Xiao-ling(張小玲)2

1SchoolofMechatronicsEngineering,LanzhouUniversityofTechnology,Lanzhou730050,China2SchoolofMechatronicsEngineering,UniversityofElectronicScienceandTechnologyofChina,Chengdu610054,China

Based on the zero-failure data of 30 Chinese 1.5 MW wind turbine gearboxes (WTGs), the optimal confidence limit method was developed to predict the reliability and reliability lifetime of WTG. Firstly, Bayesian method and classical probability estimation method were introduced to estimate the value interval of shape parameter considering the engineering practice. Secondly, taking this value interval into the optimal confidence limit method, the reliability and reliability lifetime of WTG could be obtained under different confidence levels. Finally, the results of optimal confidence limit method and Bayesian method were compared. And the comparison results show that the rationality of this estimated range. Meantime, the rule of confidence level selection in the optimal confidence limit method is provided, and the reliability and reliability lifetime prediction of WTG can be acquired.

windturbinegearbox(WTG);theoptimalconfidencelimitmethod;confidencelevel;zero-failuredata;reliability

Introduction

As the core part of energy transmission, wind turbine gearboxes (WTGs) are often working under high-speed, heavy duty and high strength circumstances, so WTGs are vulnerable to injuries[1]. Statistics show that gearbox failures are still the main reason for the failure of wind turbines, approximately 20% of the total number of faults[2]. Therefore, it is necessary to analyze and predict the reliability of WTGs. However, the study on Chinese wind turbines starts late, the operating time of WTGs is not very long. The reliability data collected from a wind farm of WTGs are performed as zero-failure data, that is, during the operation time, WTGs haven’t occur failure.

The research about zero-failure data is a new field in recent years, but it is demanded urgently in practical projects[3]. So, the study on zero-failure data has attracted more and more attention, and some achievements have been obtained already. Chenetal. discussed the reliability confidence limit of zero-failure data when the distribution type of product was known with censored data, and the optimal confidence limit method was defined and calculated in their researches, which was very suitable for application analysis[4]. Therefore, the optimal confidence limit method has become a representative achievement for reliability assessment under zero-failure data. Hanetal. also had done a lot of researches about the optimal confidence limit method and its application to reliability analysis[5-7]. The optimal confidence limit method and its application have been widely studied and develop, which provides a practical analysis method for reliability assessment under zero-failure data[8].

Based on the above background, 30 Chinese 1.5 MW WTGs are chosen as the analysis objects. And based on their zero-failure data, the optimal confidence limit method is used to predict the reliability of WTGs, and the confidence level selection for reliability prediction is discussed.

1 Collection of Zero-Failure Data

Assume thatktimes censored tests are performed on a group of products, the end time aret1,t2, …,tk(t1

Based on the operating records of 30 WTGs in 52 months, all of them turn out to be working normally. The zero-failure data of WTGs are represented as follows (unit of time: d):

ti=[ 824, 877, 889, 897, 904, 908, 1006, 1020, 1021,
1040, 1051, 1063, 1072, 1096, 1100,
1111, 1129, 1163, 1216, 1228, 1233,
1251, 1269, 1276, 1295, 1300, 1302,
1325, 1330, 1330], (i=1, 2, …, 30).

Here, the samples’ numbers aren1=n2=…=n30=1, andn1+n2+…+n30=30.

2 Optimal Confidence Limit Method

Chenetal.[10]proposed the optimal confidence limit method, and provided a specific calculation formula for reliability assessment and reliability lifetime prediction under a certain distribution[4-5]. Because of the two-parameter Weibull distribution is a flexible distribution, with strong adaptability, assume that the lifetime distribution of WTG obeys two-parameter Weibull distribution, and the lifetime distribution function is

F(t)=P(T≤t)=1-e-(t/η)m，t>0.

(1)

Reliability model is

R(t)=e-(t/η)m，t>0.

(2)

wheremis the shape parameter, assume 00 is the scale parameter.

2.1 The 1-αlower confidence limit forR(t)

When shape parameter and scale parameter of Weibull distribution are unknown, if an interval of shape parameter can be acquired. For example, assumem1≤m*≤m2, the lower confidence limit of reliabilityRL(t) can be calculated as:

(3)

Put this model into Weibull distribution, then

(4)

(5)

(6)

The 1-αis defined as the confidence level[11]. The difference between confidence level and reliability is: confidence level reflects the credibility in test results, while the reliability reflects the quality level of the components or system themselves. Theαis called the significant level, and its value is given by people according to the actual working condition .

2.2 The 1-αlower confidence limit fort(R)

Assumingη>0andm1≤m≤m2, the lower confidence limit of reliability lifetimetL(R) can be calculated as:

tL(R)=

(7)

(8)

3 Estimation of an Interval for Shape Parameter

Due to the research of Chinese WTG starting late, the shortage of experienced information is a thorny issue. Therefore, in this paper, Bayesian method and conservative classical probability estimation method are introduced to analyze the zero-failure data of WTGs, and relevant shape parameters can be obtained. But the basic idea of the two methods is the curve-fitting method.

The steps using curve-fitting method to estimate reliability are discussed as follows[12].

(1) Get the estimation of the probabilitypj=P{T<τj},j=1, 2, …,κat timeτj.

(3) Calculate the reliability using the fitted distribution curve. The steps (2) and (3) are not difficult to be done. The crucial point is the first step—how to estimate the probabilitypj=P{T<τj}.

Firstly, according to the group forming criterion from Ref. [13], the zero-failure data are divided into 5 groups, which are listed in Table 1.

Table 1 Zero-failure data of WTGs after grouped

Let the minimum time of jthtestasitscensoredtime,denotedbyτj, and the corresponding number njareshowninTable1,sj=nj+nj +1+…+nκ(κ=5,j=1, 2, …, 5) represents the number of gearboxes still working normally when test time reached or exceed.

The process to estimate shape parameters of both Bayesian method and classic probability estimation method are given as follows.

3.1 Bayesian method

By using of Bayesian hypothesis, the uniform distribution in [0,λκ] is chosen as the prior distribution of the probabilitypκ, its prior distribution denoted asπ(pκ) is defined as

(9)

Here the parameterλκis finally decided by experts’ experience.

Then, Bayesian estimation ofpjis

(10)

whererj=sj+τκ/τj-1,j=1, 2, …,κ.

According to the actual operation of Chinese WTGs, andλκ’s selection principle, takeλκ=0.5.

Therefore the relevant data for Bayesian method are displayed in Table 2.

Table 2 The relevant data of Bayesian method

3.2 Classic probability estimation method

Based on classic probability estimation method, the probabilitypj=p(T<τj) is calculated as

(11)

Table 3 The relevant data of classical probability estimation method

From what have been discussed above, the shape parameters of Weibull distribution, estimated by Bayesian method and classic probability estimation method, are 3.833 and 4.222, respectively. Taking the two values as reference, an interval for shape parameter is estimated, and the practice of engineering is taken into consideration as well, putm1=3 andm2=4.5, that is 3≤m≤4.5. Then bring this range into Eqs.(6) and (7) the 1-αlower confidence limit for reliability and reliability lifetime of WTGs can be obtained.

4 Reliability Analysis of WTGs Based on the Optimal Confidence Limit Method

4.1 The reliability prediction of WTGs

Takingm=3.833,η=2267.756 into Eq. (2) the WTG’ reliability model based on Bayesian method can be obtained

R(t)=e-(t/2 267.756)3.833.

(12)

According to statistics, the wind turbines’ effective work time is 5620 h (about 234 d) a year at a wind farm. Then the reliability of WTGs operating 10 years (about 2340 d) can be calculated.

R(2340)=32.38%.

Taking 3≤m≤4.5 into Eq.(6), the 1-αlower confidence limit of WTGs’ reliability are estimated under different confidence levels and working times. And the comparison between Bayesian method and the optimal confidence limit method of reliability prediction can be seen in Table 4.

Table 4 The comparison between Bayesian method and the optimal confidence limit method of reliability prediction

From Table 4, we can conclude as follows.

(1) When 1-α=0.8, the results of the two methods are basically consistent. From this, the reasonable of the range estimated by Bayesian method and classic probability estimation method can be verified.

(2) When for the short-term reliability forecasting, increasing 1-αappropriately, can make the reliability predictions closer to the actual operation situation; while for the long-term reliability prediction, 1-αshould not be too high, otherwise, the predictions are too small, and there are no practical significance.

According to the rule of 1-αobtained in the above analysis, the optimal confidence limit method is applied to

predicting the reliability of WTGs under the situation of running 9 and 10 years. Since 9 and 10 years are long time predictions, 1-α=0.8 has been taken in this paper. Then under the current operating conditions, the reliability of WTGs operating 9 years is about 44.54%; running for 10 years is about 27.27%.

4.2 The reliability lifetime prediction of WTGs

Reliability lifetime is a lifetime corresponding to a given reliability. Taking 3≤m≤4.5 into Eq. (7), the 1-αlower confidence limit for WTGs’ reliability lifetime is estimated under different reliability. And the comparison between Bayesian method and the optimal confidence limit method for reliability lifetime prediction can be seen in Table 5.

Table 5 The comparison between Bayesian method and the optimal confidence limit method of reliability lifetime prediction (unit: d)

From Table 5, we can find: (1) the results of the two methods are basically consistent with each other when 1-α=0.975; (2) when doing the prediction of T,inthesituationthatrequirehighR, 1-αshould be reduced appropriately to avoid the “conditions are too harsh”; while the lower R, 1-αshould be taken higher accordingly, in order to avoid the estimation results “blind optimism”.

Based on the conclusion above, three types of reliability lifetime, characteristic lifetime (the lifetime corresponding to R=0.37),mediumlifetime(thelifetimecorrespondingtoR=0.5),andaveragelifetime(0.693timesofaveragelifetimeisequivalenttomediumlifetime),arechosentobepredictedbyusingtheoptimalconfidencelimitmethod.Duetothecorrespondingreliabilityofthethreekindsoflifetimeisnothigh,therefore,choosing1-α=0.975. Then, the characteristic lifetime of WTGs is about 9.76 years, the medium lifetime is about 8.65 years, and average lifetime is about 12.49 years.

5 Conclusions

In this paper, Bayesian method combined with the classic probability estimation method are developed to analyze the reliability of WTGs based on the zero-failure data of WTGs, and the intervals of shape parameter is estimated considering the engineering practice.

Based on the value interval shape parameter, the optimal confidence limit method is applied to predicting the reliability and reliability lifetime of WTGs. The results compared between the optimal confidence limit method and Bayesian method can verify that the value interval of shape parameter is reasonable. Meanwhile, the rule of confidence level selection can be obtained when using the optimal confidence limit method to predict the reliability and reliability lifetime.

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Foundation item: National Natural Science Foundation of China(No.51265025)

1672-5220(2014)06-0839-04

Received date: 2014-08-08

* Correspondence should be addressed to AN Zong-wen, E-mail: anzongwen@163.com

CLC number: TB114.3; TH114 Document code: A