Research Article Wind Turbine Gearbox Fault Diagnosis...

Research ArticleWind Turbine Gearbox Fault Diagnosis Method Based onRiemannian Manifold

Shoubin Wang, Xiaogang Sun, and Chengwei Li

School of Electrical Engineering and Automation, Harbin Institute of Technology, Harbin 150001, China

Correspondence should be addressed to Shoubin Wang; [email protected]

Received 27 February 2014; Accepted 30 March 2014; Published 16 April 2014

Academic Editor: Weichao Sun

Copyright © 2014 Shoubin Wang et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Asmultivariate time series problemswidely exist in social production and life, fault diagnosismethod has provided people with a lotof valuable information in the finance, hydrology, meteorology, earthquake, video surveillance, medical science, and other fields. Inorder to find faults in time sequence quickly and efficiently, this paper presents a multivariate time series processing method basedon Riemannian manifold. This method is based on the sliding window and uses the covariance matrix as a descriptor of the timesequence. Riemannian distance is used as the similarity measure and the statistical process control diagram is applied to detect theabnormity of multivariate time series. And the visualization of the covariance matrix distribution is used to detect the abnormity ofmechanical equipment, leading to realize the fault diagnosis. With wind turbine gearbox faults as the experiment object, the faultdiagnosis method is verified and the results show that the method is reasonable and effective.

1. Introduction

At present, the environmental pollution and the energyshortage become increasingly prominent, while wind poweras a representative of the new clean energy has made a leapforward in recent years. Wind power has advantages such aslow cost, being clean and pollution-free, short constructionperiod, and small occupation space, which will be more andmore obvious in energy issues like curbing carbon dioxideemissions, stabilizing oil price fluctuations, and so on. Sowind resources have gotten the more and more attention ofgovernments around the world [1].

With the increase of large wind generator capacity, thesystem structure is increasingly complex, but at the sametime, frequent accidents of wind power, from the commonfaults such as bearing wear of wind turbine itself, the geartooth broke, and shaft misalignment to the burning of windturbine itself, blade drop, and the collapsing of windmachine,not only can cause the blackouts but also can produce serioussafety accidents and huge economic losses [2, 3]. In viewof the above, the fault diagnosis and the safe operation ofwind turbines have gradually become an important researchdirection in the development of wind power.

Sweden, Finland, and Germany’s wind power failurestatistics data show that failures will occur in thewind turbineblade, hydraulic, electrical, and machine driven systems, inwhich themost common one is the electrical system fault andthe major downtime is caused by gear faults. Statistics alsopoint out that the larger the wind turbine size, the higher thefrequency of the fault [4, 5].

Gearbox uses the mechanical transmission parts.Whether the operation of the gear box is normal has a greatimpact on the whole machine or unit. Improper design,manufacture, maintenance, and operation are the maincauses of the gear box faults [6]. Therefore to improve thereliability of gear box operationmeans to improve the level ofoperation maintenance, to monitor the operation conditionof gear box, and to troubleshoot the problems. According tothe statistics, in the gear box, gear failure is the main one,about 60% of the total number of all kinds of faults. With thedevelopment of related disciplines, new methods of signalanalysis and fault diagnosis technology will be applied togearbox fault diagnosis [7].

Researching the more accurate wind turbine fault diag-nosis technology, improving the operation’s reliability ofwind power turbines, and realizing the conversion from

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014, Article ID 153656, 10 pageshttp://dx.doi.org/10.1155/2014/153656

2 Mathematical Problems in Engineering

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the preventive maintenance to the state detection make theturbine’s early failure be found in time and prevent downtimeaccidents. Early failure is made to be found in time anddowntime accidents are prevented by researching the moreaccurate wind turbine fault diagnosis technology, improv-ing the operation’s reliability of wind power turbines, andrealizing the conversion from the preventive maintenance tothe state detection. On the other hand, that can also solveproblems such as the lack of equipment regular maintenance,the maintenance of excess, and the improper maintenance,which has profound significance to the development of windturbines.

Based on the wind turbine operation and the analysisof the structural characteristics, the goal of the researchersbecomes how to improve the efficiency of fault diagnosis andvisualization. It is necessary to use the new fault diagnosismethod. In this paper, by analyzing test data, wind turbine

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gearbox fault diagnosis methods based on Riemannian man-ifold are proposed. And the method realized wind turbinegearbox fault diagnosis correctly.

2. Research Status and Method

2.1. Research Status. Fault monitoring and diagnosis technol-ogy research originated in the US, which focused on faultmechanism, detection, diagnosis, and prediction in aerospacesystem at first and then on the important equipment in otherindustries. Domestic fault diagnosis technology was devel-oped on the basis of the introduction of overseas advancedtechnology. The research was initially focused on the faultmechanism, diagnosis methods of all kinds of mechanicalequipment, and simplemonitoring anddiagnosis instrument,and then the online monitoring and fault diagnosis deviceswere gradually developed domestically.

Mathematical Problems in Engineering 3

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In 1961, the mechanical failure prevention group estab-lished by United States Naval Research Laboratory activelyengaged in the development of diagnostic technology. In thelate 60s and the early 70s, the machine care center headedby R. A. Collacott started the research of fault diagnosistechnology in UK. In China, Baoshan Iron & Steel GroupCompany and Taiyuan Iron & Steel Company became thepilot units to carry out diagnosis technology research in1983 and vibration, infrared, and iron spectrum laboratorieswere established firstly, which achieved some results in thefault detection, diagnosis, and prediction. Chinese Societyof Vibration engineering established the society of faultdiagnosis to develop fault diagnosis technology based on thetheory and algorithm in May 1987. In recent years, someChinese colleges and universities, such as Xi’an Jiaotong Uni-versity, Northeastern University, or Shenyang University ofTechnology, have established the engineering centers of fault

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diagnosis, contributing positively to the development andpopularization of fault diagnosis technology. Fault diagnosistechnology absorbs the new achievements of modern scienceand develops rapidly from theory to practice, which involveslots of fields, such as computer, sensor and testing technology,signal analysis andprocessing, forecasting, automatic control,system identification, artificial intelligence, machine dynam-ics and vibration engineering, and other fields, and has thecharacteristics of strong engineering applicability and closerelation to high technology and so forth [8, 9].

Fault diagnosis process generally includes diagnosisobjects, sensors, signal processing system, state recognition,diagnosis decision-making, and maintenance. Signal pro-cessing technology plays an important role in the correctdiagnosis fault, which is the core of diagnosis. So the tradi-tional signal processing methods is introduced briefly in thenext section.


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2.2. Signal Processing Methods. Signal processing techniquesinclude fault signal detection and processing, where vibra-tion, noise, temperature, pressure, flow, current, and voltageare usually detected. Signal processing theory is based onmathematical statistics and stochastic process. The basis ofthe correct mechanical fault diagnosis includes the appropri-ate signal processing method, the accurate processing, andthe obvious expression.

Vibration monitoring of rotating machinery and otherlarge machinery is commonly used in practice. It needs theinstallation of the sensors to measure the vibration of themain parts of wind turbine, such as displacement, velocity,and acceleration, and then to process all kinds of measureddata, in order to judge the fault type and location. Thismachine fault diagnosis method is relatively accurate and canlocate the fault more directly, so it is one of the commonlyused methods in the field of fault diagnosis. The theoreticalbasis of vibrationmonitoring is as follows: during the processof manufacture, installation, and operation, the failure like

error, crack, and spalling of the gear, shaft, bearings, and othermechanical parts will accumulate gradually and become theexciting source of vibration expressed in vibration signal withthe period of the part’s rotating cycle.Themain signal analysismethods are time domain analysis, and frequency-domainanalysis.

Traditional signal processing methods include Fouriertransform, wavelet analysis, and the empirical mode decom-position (EMD). Fourier transform establishes the bridgebetween the signals in time domain and frequency domainand provides time domain and frequency-domain analysis.The time domain analysis method is to use time domainsignal for wavelet analysis. The time domain analysis methodis to use time domain signal to analyze and give the results. Ina project, directly measured signal is usually the time domainsignal, and the time domain analysis is more effective in thesignal containing obvious harmonic, periodic, or instanta-neous pulse component. As a result, it is the most simple andmost direct monitoring method. Because the generation and


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development of failure often cause the dynamic changes ofthe signal frequency, it often needs to obtain the frequency-domain information of the signal for better understanding ofthe dynamic characteristics of diagnosed equipment.

Fourier transform can decompose a complex timedomain signal into the finite or infinite frequency harmoniccomponent for the profounder level information. Spectrumanalysis has become the main content of the mechanicalequipment failure vibration diagnosis. Wavelet transformconverts the signal in the time-scale plane to observe it indifferent scales and decomposes it in different frequencybands, leading to the obvious outline and details of thesignal. So it has the multiresolution ability. Empirical modedecomposition is to stabilize a signal (or its reciprocal) and todecompose the signal in different scale of fluctuation or trendleading to producing a series of data having the different scalecharacteristics.

3. Multivariate Time Series Processing MethodBased on Riemannian Manifold

Multivariate time series processing method based on Rie-mannian manifold is based on sliding window and usesthe covariance matrix as a descriptor of the time sequence,in which Riemannian distance is used as the similaritymeasure and the statistical process control (statistical processcontrol, SPC) diagram as evaluation to detect the abnormityof multivariate time series. And the visualization of thecovariancematrix distribution is used to detect the abnormityof mechanical equipment, so as to realize the fault diagnosis.

Riemannian manifold 𝑀 is a local European topolog-ical space and a differential manifold having a continuousRiemannian metric. In the manifold 𝑀 each point has asmall neighborhood which is a diffeomorphism with a smallneighborhood of the European space. Geometry in the space


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should be based on the distance between the two infinite nearpoints, (𝑥

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in the view of the positive definite quadratic form identifiedby the square of differential arc length that is the positivedefinite symmetric matrices composed of function, which isthe Riemannian metric.

3.1. The Standardized Time Series. Because the unstandard-ized sequence comparison does not make any sense for datamining, before any data processing the first thing to do shouldbe the data standardization, which makes it form a timeseries with a mean of 0 and a standard deviation of 1. Thestandardized sequence considers both the dependence on theoriginal observation data in time series and the interferenceof random fluctuations. What is important is that it can


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reduce the computational complexity of algorithmmaintain-ing the information of the original sequence effectively at thesame time:

𝑋 =(𝑋𝑖−mean (𝑋))std (𝑋)

, (1)

where 𝑋 is a time serie, mean(𝑋) is mean value, and std(𝑋)is standard deviation.

3.2. Sliding Window Partition and Covariance Matrix Calcu-lation. Consider an 𝑁-channel time sequence. If the lengthof the sliding window is 𝑇, that means the original sequenceis divided into a number of the subsequence with the lengthof𝑇, each subsequence of themultivariate time series forms amatrix𝑋

𝑖= [𝑥𝑡+𝑇𝑖, . . . , 𝑥

𝑡+𝑇𝑖+𝑇−1], and𝑋

𝑖∈ R𝑁×𝑇. The space

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covariance matrix of subsequence is estimated by the samplecovariance matrix:

𝑃𝑖=1

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where 𝑃𝑖is the sample covariance matrix and 𝑇 is the length

of the sliding window.Because covariance matrix is the symmetric and positive

semidefinite structure belonging to positive definite symmet-ric matrix of Riemannian manifold, a series of operationsprovided by the Riemannian geometry can be used to dealwith the covariance matrix. For spatial information of thetime series contained in space covariance matrix, all ofthe algorithms based on distance can be represented usingRiemannian distance.

3.3. The Reference Covariance Matrix and the RiemannianDistance Calculation. After the estimation of the slidingwindow sample covariance matrix with (1), the referencematrix 𝑃 is estimated by self-adaption. The formula is asfollows:

𝑃𝑡+1= (𝑃𝑡)1/2

[(𝑃𝑡)−1/2

𝑃(𝑃𝑡)−1/2

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(𝑃𝑡)1/2

, (3)

where 𝑃𝑡is the reference matrix of the previous iterations,

𝑃 is the current covariance matrix, and 𝛼 is a coefficient ofself-adaptive speed. The Riemannian distance between eachcovariance matrix 𝑃

𝑖and reference covariance matrix 𝑃 is

defined as

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∑

𝑛=1

log2 (𝜆𝑛), (4)

where 𝑑𝑅is the Riemannian distance and 𝜆

𝑛is the eigenvalue

of 𝑃−1/2𝑃𝑃−1/2.


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3.4. The Distance Threshold Calculation according to the3𝜎 Principle of Statistical Process Control. Statistical processcontrol diagram is composed of three parts:

UCL: up control line, the values of 𝜇 + 3𝜎;CL: center line, the values of 𝜇;LCL: low control line, the values of 𝜇 − 3𝜎,

where two important parameters are the following.𝜇 is position parameter and average, the center of the

distribution, and expectations. It reflects the overall compre-hensive ability.𝜎 is scale parameter, degree, and standard deviation of

the distribution. It reflects the degree to which the real valuedeviates from the expected value. The higher the value, themore scattered the data.

Control chart principle belongs to the small probabilityevent principle: if any point is out of the bound, the abnormal

judgment is obtained. If the small probability events donot occur actually and the above abnormal phenomenon isfound, the control chart is the graphicalmethod of hypothesistest.

According to the 3𝜎 principle, the boundary value th(th = 𝜇 + 3𝜎) can be estimated by using the mean 𝜇and standard deviation 𝜎 of the Riemannian distance of thecovariance matrix and the reference matrix. But accordingto the experience, when whether the abnormal phenomenonoccurs is detecting, the boundary value th usually is definedas

th = 𝜇 + 2.5𝜎, (5)

where th is boundary value, 𝜇 ismean value, and𝜎 is standarddeviation.

If 𝑑𝑅> th, it proves that time series is abnormal.


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4. The Experimental Results and Analysis

The equipments used in the test are two low-frequencyvibration acceleration sensors and two normal accelerationsensors.

Consider the experiment mainly aimed at the gear box;four sensors were located to detect equipment failure morefully, in which the low-frequency vibration accelerationsensors were placed on the input shaft and the annular gearand the ordinary vibration acceleration sensors were placedon the intermediate shaft and high speed shaft. And thesampling frequency of 500Hz, long time sampling, and thetotal sampling time more than 10min were used for the twochannels of the input shaft and the annular gear of gear box;the sampling frequency of 2000Hz and the total samplingtime close to 10min were used for the two channels of theintermediate shaft and the high speed shaft.

In order to verify the fault diagnosis methods, this paperchooses the gearbox data in normal and fault state foranalysis.

Because of the large amount of data, they are divided into7 groups and the analysis and fault diagnosis aremainly basedon number 2 group and number 3 group given below.

The fault diagnosis steps are as follows:(1) the parts working condition analysis according to the

time domain signal diagram,(2) the parts working condition determination based on

the spectrum diagram,(3) the parts working condition judgment based on the

process control diagram of Riemann manifolds,(4) the system working condition determination con-

sidering the time domain analysis, the frequencyspectrum analysis, and the process control chart ofRiemann manifolds.

4.1. Analysis of Data 2

4.1.1. Time Domain Analysis of Data 2. The time domainwaveform graphs of the four time series belonging to Data


2 are shown as in Figures 1, 2, 3, and 4. It can be seen thatthe vibration signal is flat in the time domain diagram andthe impact vibration signal does not occur. It leads to thepreliminary judgment that no failure occurs.

4.1.2. Spectrum Analysis of Data 2. The responding spectro-grams are shown as in Figures 5, 6, 7, and 8. Accordingto them, the trouble-free characteristic information can bedetermined and the working condition of system is normal.

4.1.3. Process Control Charts Analysis of Data 2 Based onRiemannManifolds. At first, the phase space is reconstructedfor each measured data. For comparison, let the embeddingdimension m be 3, and the delay time tau is determined bythe autocorrelation method.

The process control diagrams based on Riemannianmanifolds are shown in Figures 9 and 10. The diagrams showthat the Riemannian distance 𝑑

𝑅is less than a boundary value

th (𝑑𝑅< th), leading to the trouble-free judgment of system.

According to the time domain diagrams, spectrum dia-grams, and Riemann control charts, there is not any failureor malfunction detected in Data 2.

4.2. Analysis of Data 3

4.2.1. Time Domain Analysis of Data 3. The time domainwaveform graphs of the four time series belonging to Data 3are shown as in Figures 11, 12, 13, and 14. The time domaindiagrams show that vibration signal leads an impact phe-nomenon, but the amplitude is small and the phenomenonis not obvious. The system working condition is hardlydetermined.

4.2.2. Spectrum Analysis of Data 3. The responding spectro-grams are shown as in Figures 15, 16, 17, and 18. It can be seenthat different simple harmonic waves appear on themeasuredpoints on the input shaft and the high speed shaft from thespectrum diagram. It is particularly serious on the measuredpoint of the input shaft and the fault can be determined.

4.2.3. Process Control Charts Analysis of Data 3 Based onRiemann Manifolds. Control chart based on Riemannianmanifold is shown in Figures 19 and 20. In Riemann controlcharts, the abnormal points appear on the intermediate shaftand the high speed shaft.

Considering the analysis of the time domain, frequencyspectrum, and the Riemann control charts, Data 3 is judgedin the fault state. The method can diagnose faults effectivelyand accurately, and the judgment is consistent with theexperimental results.

5. Conclusion

Based on the sliding window, this paper uses the covariancematrix as a descriptor of the time sequence to realize thespatial filtering of the multidimensional time series andRiemannian distance as the similarity measure to realizethe anomaly detection of multivariate time series by the

assistance of the statistical process control charts in projectsand the visualization of the distribution of covariancematrix.By the analysis of the failure data of wind turbine gearboxvibration testing, it is proved that the method based onRiemannian manifold can diagnose faults effectively andquickly.

Riemannian manifold method has a certain advantage inthe treatment of multivariate time series, but the researcheson fault diagnosis of mechanical system is less. The proposedmethod is the new method for fault diagnosis and expandsthe research scope. What is more, it also can be applied to thefault diagnosis of other rotating machinery.

The future research should combine this method withthe advanced control algorithm (such as neural network,genetic algorithm, wavelet analysis, and Fourier transform),to eliminate, analyze, and process the data, leading to themore intuitive and effective results. In addition, the windturbines are the complex mechanical transmission system.So the research on the feature information extraction andmechanism of the failure should be strengthened.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

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