Fault Diagnosis Method Based on Fractal Theory and Its Application in Wind Power Systems

ZHAO Ling(赵玲)，HUANG Da-rong，2(黄大荣)，SONG Jun(宋军)

(1.College of Information Science and Engineering，Chongqing Jiaotong University，Chongqing 400074，China;2.National Key Laboratory Incubation Base of Bridge Structural Engineering，Chongqing Jiaotong University，Chongqing 400074，China)

Introduction

Usually，due to the influence of various factors，the dynamics behavior of mechanical component in failure is non-linear and the vibration signal is non-stationary and non-periodic signal with transient characteristics.In rotating machinery，the failure of rolling bearing can result in the deterioration of machine running condition.Effective detection and diagnosis of the incipient fault of rolling element can assure the running reliability of machine.Generally，extracting the fault feature from the vibration signal to detecting the fault can effectively reduce the possibility of catastrophic damage and the downtime.The random fractal is an important and active branch of current non-linear analysis，and it is particularly suitable for the research of various nonlinear phenomena［1］.The fractal dimension can quantitatively describe the non-linear behavior of vibration signal，and is broadly applied to the field of fault diagnosis.

The time domain and frequency domain analyses are always relatively independent in classical signal analysis methods，and it is impossible to express the characteristic in both time and frequency domains simultaneously.Wavelet packet transform is also a timedomain analysis method，and its unique characteristic of multi-scale makes it a good solution to analyze nonstationary signal［2－3］.For the non-stationary and timevarying characteristics of vibration signal extracted from running machinery，neural network presents a new procedure to solve complex identify and diagnose problems，due to its self-learning，adaptive，robustness and extensive capacity.

This paper presents a new fault diagnosis procedure using the wavelet transform，fractal technology and neural network for machinery systems.

1 Noise Reduction of Signal

The wind turbine vibration signal is always dis-turbed by the background noise and the fault components，such as bearing and gear in the rotation part，and shows periodic non-stationary characteristic.Meanwhile，the alternate load generated by the nonstationary wind on the transmission system makes the vibration signal show Gaussian noise and non-linearity characteristic.Analyzed the key components in the gear-box，the fault frequency can be calculated.

For the pretreatment of wind turbine vibration signal，a de-noising method based on wavelet is put forward in this paper.Considered the wind turbine structures，the signals are non-stationary，and its parameters are time-varying.But for early fault signals，the fault feature is not strong enough and drowned in the strong noise，and it is difficult to be extracted.In this case，the traditional filter methods can not separate the noise and useful signal.The wavelet de-nosing method has better analysis effect，while has some difficulties in the selection of wavelet base and decomposition level.Aimed at adverse working conditions and strong noise，‘sym8’wavelet can be taken as wavelet base to carry out the decomposition and the decomposition level can be set as 5，also‘sqtwolog’rule can be chosen as the soft threshold.The experiment results reveal that the method can considerably improve the capability of feature extraction and incipient fault diagnosis under strong noise background，as shown in Fig.1.

Fig.1 Vibration signal waveform before and after de-nosing

2 Fault Characteristics Exaction for WindPowerSystem Basedon Wavelet Packet Transform

The wavelet packet transform is an extension of wavelet transform，and it can solve the“low resolution in high frequency”problem of binary wavelet transform.To carry out the wavelet packet decomposition of vibration signalx(t)，the following recurrence formulas can be used.

The substance of wavelet packet decomposition is making the signal get through the high-pass and lowpass combination filterhkandgk，and the signal is decomposed into high-and low-frequency parts.After decomposition，data reduces by 50%，and the amount of data is compressed.The band width Δf，decomposing layerjand the sampling frequencyfsof wavelet packet decomposition meet following relation.

After the wavelet packet decomposition，the amplitude，energy，mean value，variance and kurtosis，etc.of the vibration signal in each frequency band can be selected as the characteristic parameters.In this paper，the energy in special band is extracted as the signal characteristics.The steps［3］of characteristics extraction of wavelet are as follows.Firstly，the sampled signals are decomposed by using wavelet packet.Experientially，the number of decomposition layers can be set as 5，then according to the results of initial decomposition，the perfect number of decomposition layers can be chosen，and the decomposition is carried again.LetSrepresent the original signal，and(i，j)represent thej-th node ofi-th layer in the wavelet packet decomposition tree，i=0，1，2，…，N;j=0，1，2，…，2N－1，whereNis the number of decomposition layers.The wavelet in each frequency band is restructured.Denote the signals in all frequency bands asS1，S2，…，Si，…，Sn，and the energy of each frequency band can be calculated by using

wherexik(i=1，2，…，16;k=0，1，…，m)represents the amplitude of discrete point of restructured signalSi，mis the number of sample point in the duty cycle.Comprehensively analyze all the sampled vibration signals，choose the most concentrated frequency band of fault signal change，and normalize the selectedmfrequency bands as

The vibration wave，for misaligned rotor in rotating speed of 1 500 r/min，is shown in Fig.1(a).The signal sampling frequency is 2 000 Hz，after decomposing the sampled signal，in 5th decomposition layer，the bands 1，2，4，5，7 and 9 with band width Δf=31.2 Hz are chosen as the characteristic band where centralizing the fault signal change;the energy statistics，and the wavelet decomposition are shown in Fig.2(a)and(b)，respectively.

Fig.2 Waveform and Statistics of sampled signal

The energies in 6 frequency bands form a six-dimensional vector，which is normalized as ［0.36，0.40，0.10，0.02，0.03，0.02］.

3 Fractal Characteristics Analysis for Vibration Signal

As well known，the non-linear dynamic and chaos theories can be used to describe the irregular broadband signals in non-linear dynamical systems，and some interesting physical information and useful features can be extracted from such signals.The fractal is a group of objects or systems with self-similarity in a somewhat technical sense or in all scales.Such an object needs not to exhibit exactly the same structure in all scales，but the similar type of structures must appear.According to the point of view of fractal geometry，though the complex objects are chaotic，they have no scales or self-similarity.Scale-free or self-similarity is often regarded as a criterion to determine if an object has the fractal characteristics，and if it does，the fractal methods can be used to analyze it.

Figuer 3 shows the logarithmic curve of the sampled signal and its least squares fitting，and the slope of fitting line is the fractal dimension.Then，the fractal scale-free range can be determined.The determination of the scale-free interval means the sampled signal can be analyzed with the fractal methods.

Fig.3 Logarithmic curve of sampled signal

The fault vibration signal ofwind power system is time-varying and irregular，and within a certain scale，it has fractal characteristic，so the structure characteristic，i.e.fractal dimension，can be extracted from it and used as the eigenvector for fault quantitative identification［4－5］.

4 Grid Fractal and Fractal Dimension Calculation

The fractal theory developed from non-linear dy-namic and chaos theories is a promising new tool to interpret physical systems with irregular time-domain analysis scale.For given discrete information，there are several kinds of fractal modes，and the grid fractal is a fractal dimension easier to be implemented.It divides the Euclidean spaceRnintoΔgrids as small as possible.When formally isometric separating，i.e.，it uses the cube with side lengthΔandndimension to separate the setXinto digital points set.The number of points in setXin the discrete space can be denoted asNΔ.Then，kcount points in the different grid widths can be obtained.Thus，Δgrid is magnified asKΔgrid，andNKΔrepresents the count point of setXin the discrete space(the distance isKΔ).

Setxk=log(k)，yk=log(NkΔ)，k=1，2，…，K，then the slope of straight line structured byMk(xk，yk)is the grid dimension.The actual signal sampling is calculated by using the grid points dimension as the points set of discrete space.

The grid dimension of vibration signal can be calculated as follows.According to the calculation principle of discrete information grid fractal，the real sampling time isT，the sampling interval is Δt，and the vibration signal isx1，x2，…，xn，wherenis the number of sample points，n=T/Δt，and Δtdepends on signal state.Define the fractal dimension of signal as

It reflects the signal characteristic.For normal signal and various unusual signals of machinery，the values ofdjshould be different，and it is used as a characteristic variable to judge different signal status.

For the six typical status of rotor，i.e.normal，imbalance，misaligned，oil film eddy，oil vibration，surging and rotating stall，the sampling is carried out in the same cycleTi(T1=0.05，T2=0.1，T3=0.2，T4=0.4，T5=0.8，T6=0.16 ms).By using formula(4)，the dimensiondican be obtained，andTiis used as the characteristic fractal dimension of sampling correspondingly，as shown in Tab.1.

5 Wavelet Neural Network

The nonlinear mapping function of wavelet neural network can be used for classifying the faults，even for the concurrent and multiple faults［8－12］，and its convergence is obviously faster than the BP neural network.Its structure is similar to the three-layer BP network，including input layer，hidden layer and output layer.The wavelet function or scaling function is used as its activation function，as shown in Fig.4.

Fig.4 Structure of wavelet neural network

To improve the performance of wavelet neural network，an accelerating algorithm can be used to increase the convergence speed and avoid the convergence at a local minimum.In this paper，the improved adaptive BP algorithm is used for training the wavelet neural network［8］，as shown in Fig.5.

The whole process of network can be divided into two stages.In the first stage，the calculation is carried out from the input layer.Calculate the output of each layer according to the input sample.Find the output of output layer.This is a forward propagation.In the second stage，the weight has to be corrected.Calculate and revise each layer from the output layer.This is a backward propagation，i.e.the error backward propa-gation algorithm.These two processes run iteratively，till convergence.

Fig.5 Algorithm flowchart

The dynamic adjustment method for each adjustable parameters of the learning rate is

For showing the superiority of wavelet neural network，based on BP algorithm，we structure a traditional neural network，which has the same input，output and hidden nodes as the wavelet network.Select 17 samples to train the wavelet network and BP network.The numbers of input and output nodes are 5，the number of hidden nodes is 10，and system error is 0.01.After the network learning，the time costs of these two networks training is shown in Tab.2.It can be seen that，for the same training error，the training time of wavelet neural network is much shorter than that of the traditional BP network and it has stronger tolerance ability.

Tab.2 Training time costs of two networks

6 Fault Diagnosis Method Based on Fractal，Wavelet and Neural Network

The model of fault diagnosis is shown in Fig.6.The real status of equipment is online monitored by sing sensors and the data of rotor is collected.After data pretreatment，the data is analyzed by using wavelet.If there is something wrong with the equipment，it will have a greater impact on the energy of the signal in each frequency band.Therefore，the signal energy in each frequency band can be taken to structure the characteristic vector to extract the fault characteristic effectively.Firstly，wavelet decomposing the sampled signals are decomposed by using the wavelet packet;reconstruct the wavelet is restructured in each frequency band to obtain the energyEiof signal in each frequency band;choose the most concentrated frequency bands of signal change caused by faults are chosen，and then the selected frequency bands are normalized and compose the original characteristic vector with the calculated fractal dimensions of sampled signals in different sampling cycles，d=［E1，E2，…，Em，D1，D2，…，Dn］T.

Fig.6 Flowchart of fault Diagnosis

The improved wavelet neural network can be trained by data;and considered the actual fault output state pattern，the weight is adjusted to establish the wavelet neural network for fault online monitoring.Set the number of input and output nodes as the fault characteristic numberk.The initial number of hidden nodes can be set by using experience formula，and adjusted in the training.During the network training，set the learning step lengthηas 10，the inertia factorαas 0.01，and system error as 0.001.Because the Db6 wavelet of the Doubechies wavelets has the orthogonality，compactness and good property，it can be selected as the wavelet neural network neuron.

For standard thep-th sample ［dp1，dp2，…，dpm］［0，0，…1，…，0］，when the sample output matrix is identity matrix，the corresponding status or fault is 1，and the non-corresponding status is 0.The sample output is decimal fraction.When recognizing，the distance function is used for judgment.

7 RotorFaultDiagnosisinWind Power System

In the online monitoring and fault diagnosis for wind power system，the waveform of vibration signal is colleted by using sensors.When the system is in failure，the vibration waveform shows the different characteristics depending upon the fault type.The proportions of harmonic energy in different frequency bands are also different.Moreover，the severity of system fault determines the complexity of vibration wave.Because the vibration signal shows non-equilibrium and fractal property，the fractal dimension of vibration waveform can be used to represent the characteristic of system fault.There exists some correspondence between the fractal dimension and fault type.Therefore，the calculated fractal dimension can be used to judge the fault type.

Rotor is a main part of the rotating machinery，and its vibration signal reflects the mechanical fault information in the amplitude，frequency and time domains.In this paper，six typical states，shuch as normal，unbalance，misalignment，oil film eddy，oil film whip，surge and rotating stall，are selected as the input samples of network diagnosis model，and the corresponding fault outputs are defined as the target sample.The energy characteristic vectorT=［E1，E2，E3，E4，E5，E6］can be obtained，as described in Section 1.For all the vibration signal，the fractal dimension can be found out，as stated in Section 4，and the dimension characteristic vectorD=［D1，D2，D3，D4，D5，D6］can be obtained.The dimension characteristic vectorD=［D1，D2，D3，D4，D5，D6］is used as the input vector of wavelet neural network.

The number of input nodes is set as 6，the number of output nodes equals to the number of input nodes，i.e.6，and the number of hidden nodes can be determined by using the experience formula，initially set as 10 and adjusted in the training.Let learning step length beη=10，inertia factorα=0.01，and select Db6 wavelet as the wavelet neural network neuron.When the system error is required as 0.001，the network can stop learning in 49 steps，as shown in Fig.7.

Fig.7 Error curve of wavelet neural network of wind power system

Then，the trained network is used to diagnose the vibration signal in Fig.1.The energy characteristic vector and dimension characteristic vector are input into network，and a vector［0.002，0.978，0.001，0.023，0.006，0.002］outputs.According to the distance function，the fault diagnosis result is misaligned，and it is consistent with the actual situation and shows that this method is valuable.

8 Conclusions

There always exists noise in sampled signal;therefore the noise reduction is indispensable.With the de-noising，some information in signal will be removed，thus，it is necessary to choose the correct soft threshold.In the each stage of signal processing，choosing suitable method to minimize the error can reduce the error accumulation.

The wavelet analysis is especially suitable for processing the non-stationary signal.By transforming the sampled vibration signals，the fault characteristic can be extracted according to the energy distribution in each frequency band after decomposition.By using the fractal dimension as the fault characteristic parameter，the difficulties of traditional methods on fault characteristic extraction and analysis are overcome effectively.

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