Abstract—Shaft vibration of steam turbine usually has characteristics of strong harmonics and typical sub-Gaussian distribution. In the process of startup and shutdown of units, there are always some changes of harmonic components with nonlinear coupling reciprocity each other. That makes the technology of power spectral analysis based on second-order statistics restricted in application. This paper made a deep research on the measured vibration signals of a large steam turbine in its slow-down process. Used traditional methods and higher order spectral analysis (HOSA) methods separately to analyze the shaft vibration signals and bearing vibration signals. And it clarified the characteristic of nonlinear harmonic coupling between the fundamental frequency of shaft rotation and its harmonics. The result shows that bispectrum and 1(1/2) dimensional spectrum based on third-order cumulant can clearly reveal the large changes of higher order spectral characteristics when the unit is under the varying operating conditions. And they can also reveal the changes of non-Gaussian and nonlinear characteristics for the vibration signals. HOSA has a high sensitivity and can effectively restrain the Gaussian noise of the measured vibration signals. So it builds a good new foundation for feature extraction and fault diagnosis. Keywords—steam turbine; slow-down; bispectrum; 1(1/2)

dimensional spectrum; quadratic phase coupling

I. INTRODUCTION T present China’s power structure mainly depends on

coal-fired power generation. In order to improve the economy of coal-fired plants, reduce the cost of

construction and operation, meet the requirement of energy saving and environmental protection, the equipments of thermal power plants have been developed in aspects of large scale, complexity, high parameter and high automation. In recent years, most of China’s new turbine generator units are large supercritical (ultra-supercritical) units with the capacity of 600 MW and above. Because the structure of large unit is relatively complex and the practical operation experience is quite few, there are some widespread concerns of large-scale

Zhou Yanbing, a doctoral candidate comes from Key Laboratory of Condition Monitoring and Control for Power Plant Equipment, North China Electric Power University, Beijing 102206, China. (e-mail: zhouyanbing@ncepu.edu.cn)

Liu Yibing, a professor comes from Key Laboratory of Condition Monitoring and Control for Power Plant Equipment, North China Electric Power University, Beijing 102206, China. (e-mail: lyb@ncepu.edu.cn)

An Hongwen, a doctoral candidate comes from Key Laboratory of Condition Monitoring and Control for Power Plant Equipment, North China Electric Power University, Beijing 102206, China. (e-mail: anhongwen@ncepu.edu.cn)

Che Liming, a Master degree candidate comes from North China Electric Power University, Beijing 102206, China. (e-mail: cheliming@ncepu.edu.cn)

units, for example, enhance the safety and reliability in the process of operation, avoid serious accidents and improve economic efficiency.

Shaft vibration of steam turbine can reflect the operation state of the unit, and it’s easy to be measured. Whether in the process of normal running or startup and shutdown, unit’s abnormal states or faults can immediately react to the abnormal changes of the shafting amplitude. So the shafting vibration monitoring and the signal analysis methods are widely applied in the condition monitoring and fault diagnosis for turbines. The shafting vibration usually has a characteristic of strong harmonics, and the energy mainly concentrates in the low-order harmonic components. The shafting faults or the change of states show great changes of some harmonic components. So technologies of spectral analysis are widely used in fault feature extraction. However, large unit has a complex structure, intricate operation condition, and nonlinear rotor-bearing system. So the measured vibration signals are usually unstable, and often contain non-Gaussian nonlinear components as well as all kinds of interference noises. Especially in process of startup and shutdown of units, the rotational speed of shaft needs to pass some critical speeds, so the shaft vibration signals and bearing seat vibration signals are protean daedal, and non-Gaussian nonlinear information is very prominent and complex, which bring difficulty in extracting vibration features and evaluating running states. Spectral analysis methods based on Fourier transform are restricted, because they can’t explain the quadratic phase coupling phenomenon caused by the nonlinear characteristics of mechanical equipments, in addition, they can’t process non-Gaussian random signals existent in the engineering practical signals.

Vibration monitoring and fault diagnosis technologies for steam turbine have been researched dozens of years, and abundant achievements and practical experience have been accumulated. Along with the developments of signal analysis and fault diagnosis, new methods have been constantly applied in vibration signal analysis and fault feature extraction of steam turbine in recent ten years. HOSA is the expansion of correlation analysis and spectral analysis, can excavate deeper statistics information of signals, and has special properties in dealing with non-Gaussian signals and nonlinear signals. So HOSA has been widely used in fault diagnosis field of various types of equipments and systems such as compressor [1], bearing [2], [3], gear [4]-[6], control system [7] and rotating machinery[8], [9]. HOSA has gradually become an effective technology for machinery vibration signal analysis and fault feature extraction. However it is seldom applied in the actual vibration signal analysis of large steam turbine generator [10].

Higher Order Spectral Analysis for Vibration Signals of the Large Steam Turbine in Slow-Down Process Zhou Yanbing, Liu Yibing, An Hongwen and Che Liming

A

This paper took actual measured vibration signals of a large steam turbine in slow-down process as an example. With the purpose of studying the higher order statistical properties of turbine vibration signals, traditional methods and HOSA methods have been separately used and compared. The paper revealed the non-Gaussian nonlinear quadratic phase coupling information hidden in the vibration signals whose predominant components are harmonics. It built a good new foundation for feature extraction and fault diagnosis.

II. THEORETICAL BASIS

A. The Second-order Statistics and Their Limitations Second-order statistics such as autocorrelation function

rx(τ) and power spectral density function Px(ω) are usually used as the statistical characteristics of random signal x(t). rx(τ) and Px(ω) are defined as:

( ) { ( ) ( )}xr E x t x tτ τ= + (1) j( ) ( )e dx xP r ωτω τ τ∞ −

−∞= � (2) Where τ and ω are time delay variable and frequency

variable, E{⋅} is the expectation calculation symbol. From the above definition, it’s available that:

2 ( )d ( )dxx t t P ω ω∞ ∞−∞ −∞=� � (3)

The equation shows the physical significance of power spectral density function. It reflects power distribution of the signal in the frequency axis (in other words, the contribution of some frequency component ω to the signal). Power spectrum can also be expressed as:

2( ) ( ) ( ) | ( ) |xP X X Xω ω ω ω∗= = (4) Where X(ω) is Fourier transform of the signal (spectrum),

and it’s a complex function. X*(ω) is the conjugate function of X(ω), and |X(ω)| is the amplitude function (amplitude spectrum) of X(ω). It’s clear that power spectrum contains only the power amplitude information of every frequency components, while the phase information is lost. Quadratic phase coupling is a harmonic distortion phenomenon between harmonic components in weakly nonlinear systems. When rotating mechanical equipments are malfunctioning, quadratic phase coupling phenomena of vibration signals usually appear, and second-order statistics are helplessness about these.

B. Third-order Cumulant, Bispectrum and 1(1/2) Dimetional Spectrum Third-order cumulant is the most common higher order

statistic, bispectrum and 1(1/2) dimensional spectrum are based on third-order cumulant, and they are the most common higher order spectra. If x(t) is a random signal with zero mean, its third-order cumulant is defined as:

3 1 2 1 2( , ) { ( ) ( ) ( )}xc E x t x t x tτ τ τ τ= + + (5) It can be proved that third-order cumulants of random

signals with symmetrical distribution (such as Gaussian distribution) equal to zero. Thus third-order cumulant has a role of restraining Gaussian noise. This is an important property, which is also the advantage of bispectrum and 1(1/2) dimensional spectrum based on third-order cumulant.

Bispectrum is defined as 2 dimensional Fourier transform of third-order cumulant:

1 1 2 2( )1 2 3 1 2 1 2( , ) ( , ) d dj

x xB c e ω τ ω τω ω τ τ τ τ∞ ∞ − +−∞ −∞= � � (6)

It can be seen that bispectrum is the function of 2 frequency independent variables (ω1 and ω2). Bispectrum reflects the interaction among ω1, ω2 and ω1+ω2.

Similar to power spectrum, bispectrum satisfies the following relationship:

1 2 1 2 1 2( , ) ( ) ( ) ( )xB X X Xω ω ω ω ω ω∗= + (7) Third-order cumulant is a 2 dimentional function. If τ1=τ2

in C3x(τ1, τ2), the diagonal slice of third-order cumulant will be a 1 dimentional function:

3 ( ) { ( ) ( ) ( )}xC E x t x t x tτ τ τ= + + (8) 1(1/2) dimensional spectrum is defined as 1 dimensional

Fourier transform of diagonal slice of third-order cumulant: 2�j

3( ) ( )e dfxS f C ττ τ∞ −

−∞= � (9) 1(1/2) dimensional spectrum can suppress Gaussian noise,

preserve the quadratic phase coupling information of signals, and reduce the amount of calculation. So it’s conducive to fault feature extraction.

III. AN EXAMPLE FOR VIBRATION ANALYSIS OF STEAM TURBINE IN SLOW-DOWN PROCESS

A. The Description of Unit and Its Vibration Measurement Take a large steam turbine with the capacity of 850MW of

some power plant as a research object. The main steam temperature is 535 , and the main steam pressure is 18.2MPa. The working rotational speed of the unit is 3000 rpm. The steam turbine has a structure of a single shaft, four cylinders and four exhausts. It consists of a high pressure cylinder, an intermediate pressure cylinder and two low pressure cylinders. The steam turbine, the generator and the exciter compose the unit. From the steam turbine to the exciter, the unit’s rotor is supported by eight sliding bearings, which are respectively called the first to the eighth bearing. The shaft system structure of the unit is shown in Fig. 1.

Fig. 1. The shaft system structure of large turbine unit.

By using the actual monitoring system, make a continuous measurement for the unit in its slow-down process with about 25 minutes long, which changes from working rotational speed to turning rotational speed (about 200 rpm). Make the measurement for every bearings in order to survey 2 channels of shaft vibration signals with X-Y vertical direction and 1 channel of bearing vibration signal, which can been obtained from the turbine supervisory instrumentation (TSI). Then convert the actual signals into digital signals, and input them into a computer for subsequent analysis and processing. Sampling frequency is 800 Hz. This paper takes the X-direction shaft vibration signal and the bearing vibration

signal of the first bearing near the high pressure cylinder as examples.

B. The Time-domain and Frequency-domain Analysis of Vibration Signals There are the time series of shaft vibration signal and

bearing vibration signal of the first bearing from top to bottom in Fig. 2. From the amplitude change of shaft vibration signal, we learn that the unit’s rotational speed declines from 3000 rpm to 200 rpm within 800 seconds. The slow-down process has at least three obvious amplitude changes, which are corresponding to the three critical speeds of shaft system in the 1st bearing. However, the variation of the bearing vibration signal is relatively complex. There are also some resonances occurred in slow-down process, which are corresponding to several critical speeds. But the bearing vibration is not synchronous with the shaft vibration when they pass their own critical speeds. In addition, the random components in the bearing vibration signal are more obvious by comparing to the shaft vibration.

Fig. 2. Measured signals of shaft vibration and bearing vibration.

Divide the above 1500s long vibration signal into 100 sections, and make use of power spectral analysis to process first 4096 spots of each section. According to the time sequence, arrange every power spectra to form a waterfall map show in Fig. 3, whose upper part is the waterfall map of shaft vibration signal and lower part is that of shaft vibration signal. It has clearly reflected the change information of vibration signals’ fundamental frequency and its lower order harmonics along with the change of rotational speed. The shaft vibration signal has obvious peaks in the fundamental frequency and its second-order harmonic, while the bearing vibration signal has more peaks in first 6 harmonics. The resonance peaks of the shaft vibration signal’s fundamental frequency probably appear in 45Hz, 37Hz and 22Hz. However with regard to the bearing vibration signal, its resonance peaks probably appear in 46Hz, 38Hz, 32Hz, 27Hz and 19Hz. Both of their resonance frequencies are different.

These differences between shaft vibration and bearing vibration are mainly related to the dynamic characteristics of their respective systems. The relative shaft vibration of unit may be considered as the response of the rotor-bearing supporting system. So the shaft vibration is mainly influenced by the intrinsic properties of rotor-bearing system and rotor’s excited force. When rotational speed of the rotor passes its critical speed, the natural frequency of system is consistent with the rotational frequency, and at that moment there will be resonance appeared. The resonance peak’s shape depends on the damping and the rigid characteristic of the bearing’s oil film. However the bearing vibration reflects the absolute vibration of the supporting bearing seat. The system composed by the bearing seat and the foundation has own intrinsic properties. The excitation sources of the bearing vibration include the vibration of rotor, foundation, cylinders and pipelines, etc. Since there are a variety of excitation sources, which not only come from the periodic excited force of rotor vibration, but also have various random excited forces caused by the fluid in pipeline, so the bearing vibration characteristic is more complicated, contains more resonance frequencies, and is not necessarily corresponding to the rotor’s resonance frequencies.

Fig. 3. Power spectral waterfall map of shaft vibration and bearing vibration.

Excluding the above analysis results, the signal’s time-domain waveform and the frequency-domain spectral analysis can no longer provide more information; can not

reveal the change of non-Gauss characteristic and the nonlinear phase coupling information of the shaft rotational frequency in slow-down process. This is also the limitation of the classical analysis methods.

C. Bispectral Analysis In order to reveal the quadratic nonlinear phase coupling

characteristics of unit’s vibration signals in slow-down process, we make use of bispectrum to analyze the above 100 sections of sample vibration signals. The calculation of bispectral estimation adopts the direct method. Each small piece of a sample has 128 points, and FFT length is 512 points.

By means of 2 dimensional contour maps, Fig. 4 shows the bispectral analysis results of the shaft vibration signals. The contour line scale has been unified. Take the bispectral analysis of shaft vibration as an example. It separately shows the bispectral analysis results of the second, the eighth, the forty-fourth and the hundredth section of samples from top to bottom in Fig. 4, which are respectively corresponding to the status of before slow-down, the first time passing critical speed, the third time passing critical speed and turning. It is easy to find that the amplitude of bisepctrum has obvious peak values at the place of fundamental frequency and its harmonics when the rotational speed approaches the critical speeds, which explained that non-Gaussian characteristic of vibration signals caused by resonance is quite strong and the phenomenon of quadratic phase coupling is prominent. However, the amplitude values of bispectrum are very small under the turning state, which explained that non-Gaussian characteristic of vibration signals is too weak and the phenomenon of quadratic phase coupling is not obvious. This variation trend of nonlinear feature is nearly consistent with that of the energy of frequency-domain spectral analysis. Likewise, bispectral analysis of bearing vibration signals has the similar result too. However in the bispectral diagram, the change information of quadratic phase coupling is not very obvious, and it’s a little difficult to extract this nonlinear feature.

D. 1(1/2) Dimensional Spectral Analysis We make use of 1(1/2) dimensional spectral analysis for

the shaft vibration sample signals used in Fig. 4, the obtained results separately correspond to the status of before slow-down, the first time passing critical speed, the third time passing critical speed and turning. Analysis results are shown in Fig. 5. Especially when the rotor’s rotational speed approaches the critical speed at the first time, the amplitude value of 1(1/2) dimensional spectrum is the biggest. It explained that the quadratic nonlinear information of shaft vibration signals in these conditions are mainly concentrated in the places of fundamental frequency and its second-order harmonic frequency, other frequency components don’t participate in phase coupling.

Fig. 4. Bispectra of shaft vibration signals in different conditions.

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Fig. 5. 1(1/2) dimensional spectra of shaft vibration signals

in different conditions.

Fig. 6. 1(1/2) dimensional spectral waterfall maps of

shaft vibration and bearing vibration Although the power spectral analysis reflects the energy

information, it loses the phase information. 1(1/2) dimensional spectrum can make up this defect well. In order to conveniently compare the difference between energy

change of vibration signals and phase coupling change in slow-down process, 1(1/2) dimensional spectral waterfall maps of shaft vibration signals and bearing vibration signals are shown in Fig. 6. Compared with the waterfall maps of power spectrum in Fig. 3, it can be found that the unit passed many critical rotational speeds in slow-down process, not only the energy of amplitude increase, quadratic phase coupling also deepen. What’s more, the energy and quadratic nonlinearity of shaft vibration are mainly concentrated on the fundamental frequency and its second-order harmonic frequency, while those of bearing vibration are existed at each lower order harmonic frequencies. The reasons of this difference between shaft vibration and bearing vibration are the different vibration sources and different vibration mechanisms. In addition, the frequencies corresponded to the peaks of power spectrum are consistent with those of 1(1/2) dimensional spectrum under the critical speed, but their amplitude relationship is different. The amplitude value of power spectrum at fundamental frequency is far greater than that at other harmonic frequencies, however they have little difference in 1(1/2) dimensional spectrum. It indicated that no matter the shaft vibration signals or the bearing vibration signals, even if the energy of harmonic frequencies is weak, it is still existed strong quadratic phase coupling and obvious nonlinearity.

IV. CONCLUSIONS In this paper, methods of higher order statistics analysis

were applied to analyze vibration signals and extract features of large steam turbine in slow-down process, and we could obtain the following conclusions: 1) Made use of time-domain methods and

frequency-domain methods to analyze the vibration signals of steam turbine in slow-down process. The differences of vibration sources and mechanism between shaft vibration and bearing vibration have been expounded. From the standpoints of amplitude and energy, the changes of fundamental frequency and its harmonic frequencies have been analyzed

2) Because bispectral analysis and 1(1/2) dimensional spectral analysis which were both based on third-order cumulant had advantages of suppressing Gaussian noise, highlighting the harmonic components, they were very suitable for the research of higher order statistic characteristics and nonlinear feature extraction. Especially the method of 1(1/2) dimensional spectrum, which contained the quadratic nonlinear information of vibration signals and revealed the relationship of phase coupling, could not only clearly reflect the change of unit’s vibration states, but also make up the defect of the classical spectral analysis, and it was an excellent kind of characteristic index.

ACKNOWLEDGMENT This work was supported by the National 863 High

Technology Research and Development Program Funding (2008AA04Z410).

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