**6. Statistical symptoms**

330 Mechanical Engineering

not change substantially with time. In fact this is compatible with the main mechanisms of

also valid. Such case is illustrated by an example shown in Fig.7a. On the other hand, for a considerable lifetime consumption, the slope will initially increase with time (Fig.7b) to a

(a) (b)

(c)

Although quantitative assessment results should not be generalized over different turbine types, it has been estimated that, for components from the blade frequency range, vibration velocity vs. time plots with linear slope values below about (10 20) 10-6 (mm/s)/day are

For the harmonic frequency range, the value of 10-4 (mm/s)/day may be accepted as a very rough estimate. Accelerated damage may result in a value higher by an order of magnitude

substantially lower than

*b*.

Fig. 7. (a) Linear fit slope vs. time: 230 MW unit, front low-pressure turbine bearing, horizontal direction, 6.3 kHz band; (b) the same, 200 MW unit, low-pressure turbine casing rear/ right side, horizontal direction, 3.15 kHz band; (c) vibration velocity vs. time: 200 MW unit, high pressure/intermediate pressure bearing, vertical direction, 8 kHz band. Data smoothing: peak trimming at *c* = 1.5 followed by three-point averaging. Red line in (c)

typical for normal lifetime consumption (natural damage) with

 << *<sup>b</sup>* are

) resembles an exponen-

lifetime consumption, i.e. fatigue and creep, for which linear approximations for

*<sup>b</sup>* even exponential fit fails (Fig.7c).

point wherein linear approximation is no longer acceptable and *S*(

tial curve. For

close to

represents exponential fit.

### **6.1 Dispersion measures**

Up to this point, the deterministic approach has been employed. It may be argued that this is not compliant with the statistical nature of vibration generation mechanisms. What is more important, however, is the fact that statistical approach allows for eliminating problems resulting from the influences of control and interference. The basic idea may be summed up as 'if we cannot get rid of it, then try to make use of it'.

The main assumption in the statistical approach is that a symptom is a random variable rather than a deterministic function of machine condition parameters. Parameters of this random variable also depend on object condition and thus may be themselves accepted as symptoms (sometimes they are referred to as meta-symptoms, in order to stress that they are not directly measurable physical quantities). The idea of determining such symptoms is shown schematically in Fig.8.

Fig. 8. The idea of statistical symptom determination: parameters pertaining to measured symptom value distribution are determined within a time window .

Obviously elements of the control and interference vectors are also random variables. Moreover, for a given turbine at some fixed location, it is reasonable to assume that statistical parameters of these random variables do not change with time, so each of them is characterized by a time-invariant probability distribution. Now, let us assume that we determine probability distribution of a vibration-based symptom *S* (say, vibration velocity level in a given frequency band, measured in a given point) in a manner shown in Fig.8. If it can be shown that

Vibration-Based Diagnostics of Steam Turbines 333

Comparison of these five dispersion measures is shown in the example presented in Fig.9, which refers to a natural damage (last measurement was performed shortly before rotor

Fig. 9. K-200 unit, front high-pressure turbine bearing, vertical direction, 8 kHz band.

(), (c) *m*(), (d) *q*(), (e) (), (f) ( ) represents symptom trend estimated for the entire period

*<sup>i</sup>*) are consecutive

); data window containing

where *i* is the number of data points contained in the time window, *S*(

symptom value readings and *St*(

(a) symptom time history, (b)

20 measurements.

under consideration.

$$
\hat{\varepsilon}S/\hat{\varepsilon}\mathbf{R}\_i = f(\mathbf{X}) \text{ and/or } \hat{\varepsilon}\mathbf{S}/\hat{\varepsilon}\mathbf{Z}\_i = f(\mathbf{X})\text{.}\tag{25}
$$

then the distribution of *S* will obviously change with condition parameters. Intuitively we may expect that with deteriorating technical condition both *S*/*Ri* and *S*/*Zi* will increase as *<sup>b</sup>*, i.e. *S* shall be more and more sensitive to control and interference parameters. It can be shown on the basis of a suitably modified Energy Processor model that this is exactly the case (Gałka and Tabaszewski, 2011), providing that measurement errors are excluded. Thus, a measure of vibration level dispersion may be accepted as a diagnostic symptom.

Standard deviation is the most commonly used dispersion measure. From the point of view of this application, it has two main deficiencies. First, normal distribution is tacitly assumed, which in general is not the case. Second, standard deviation is very sensitive to outliers, and obviously no data smoothing can be employed in estimating dispersion. Although standard deviation has yielded basically encouraging results (Gałka, 2008b), robust measures are far superior. Mean absolute difference between two consecutive measurement results was first proposed (Gałka, 2008b) and () time histories have been found much more regular and easier to interpret than those of (). Other possibilities include median absolute deviation about the median *m*, defined as

$$m \equiv \text{Med}[S-\text{Med}(S)] \, , \tag{26}$$

which in fact consists in centering the data around median rather than mean value, and interquartile range given by:

$$q = Q\_{\mathbb{S}}(S) - Q\_{\mathbb{I}}(S). \tag{27}$$

where *Qi* is the *i*th quartile:

$$Q\_1 = F^{1}(0.25), \ Q\_2 = F^{1}(0.5), \ Q\_3 = F^{1}(0.75) \ . \tag{28}$$

*F* being the cumulative distribution function. Obviously, for a symmetrical distribution these two approaches are equivalent, but with a heavy-tailed distribution this is not the case. For the normal distribution, both *m* and *q* are constant multiples of .

Time window width is obviously a compromise. Larger yields better estimation of dispersion but it has to be kept in mind that the approach schematically shown in Fig.8 is in fact based on the assumption that

$$\bigwedge\_{i}^{X\_{i}} (\theta + \delta \theta) \approx X\_{i}(\theta) \,. \tag{29}$$

If this condition is not fulfilled, centering data around any value 'averaged' over the entire time window becomes groundless. This is certainly the case when is close to *<sup>b</sup>*. In analyzing time series one should speak in terms of deviations from the trend rather than from some mean value corresponding to the entire time window. This in fact explains why () yields better results, as differences between consecutive measurement results better represent such deviations. Another symptom may be thus proposed, in the form of

$$\varepsilon = \frac{\sum\_{i=1}^{n} \left[ \left| S(\theta\_i) - S\_t(\theta\_i) \right| \right]}{n} \,\,\,\,\tag{30}$$

then the distribution of *S* will obviously change with condition parameters. Intuitively we may expect that with deteriorating technical condition both *S*/*Ri* and *S*/*Zi* will increase

can be shown on the basis of a suitably modified Energy Processor model that this is exactly the case (Gałka and Tabaszewski, 2011), providing that measurement errors are excluded. Thus, a measure of vibration level dispersion may be accepted as a diagnostic symptom.

view of this application, it has two main deficiencies. First, normal distribution is tacitly assumed, which in general is not the case. Second, standard deviation is very sensitive to outliers, and obviously no data smoothing can be employed in estimating dispersion. Although standard deviation has yielded basically encouraging results (Gałka, 2008b), robust measures are far superior. Mean absolute difference between two consecutive

 *m* = Med[*S* – Med(*S*)] , (26) which in fact consists in centering the data around median rather than mean value, and

 *q* = *Q*3(*S*) – *Q*1(*S*), (27)

 *Q*1 = *F*-1(0.25), *Q*2 = *F*-1(0.5), *Q*3 = *F*-1(0.75) , (28) *F* being the cumulative distribution function. Obviously, for a symmetrical distribution these two approaches are equivalent, but with a heavy-tailed distribution this is not the case.

is obviously a compromise. Larger

dispersion but it has to be kept in mind that the approach schematically shown in Fig.8 is in

*i i*

If this condition is not fulfilled, centering data around any value 'averaged' over the entire

analyzing time series one should speak in terms of deviations from the trend rather than from some mean value corresponding to the entire time window. This in fact explains why

represent such deviations. Another symptom may be thus proposed, in the form of

*n*

*i*

 

1

) yields better results, as differences between consecutive measurement results better

*i ti*

 

() () 

*S S*

*n*

 

*X X* ( ) () 

was first proposed (Gałka, 2008b) and

found much more regular and easier to interpret than those of

include median absolute deviation about the median *m*, defined as

For the normal distribution, both *m* and *q* are constant multiples of

*i*

time window becomes groundless. This is certainly the case when

*<sup>b</sup>*, i.e. *S* shall be more and more sensitive to control and interference parameters. It

as 

Standard deviation

measurement results

interquartile range given by:

where *Qi* is the *i*th quartile:

Time window width

(

fact based on the assumption that

*S*/*Ri* = *f*(**X**) and/or *S*/*Zi* = *f*(**X**) , (25)

is the most commonly used dispersion measure. From the point of

(

> (

.

) time histories have been

yields better estimation of

is close to

*<sup>b</sup>*. In

. (29)

, (30)

). Other possibilities

where *i* is the number of data points contained in the time window, *S*(*<sup>i</sup>*) are consecutive symptom value readings and *St*() represents symptom trend estimated for the entire period under consideration.

Comparison of these five dispersion measures is shown in the example presented in Fig.9, which refers to a natural damage (last measurement was performed shortly before rotor

Fig. 9. K-200 unit, front high-pressure turbine bearing, vertical direction, 8 kHz band. (a) symptom time history, (b) (), (c) *m*(), (d) *q*(), (e) (), (f) (); data window containing 20 measurements.

Vibration-Based Diagnostics of Steam Turbines 335

there is virtually no correlation with the blade components, *r* being about –0.1. The root

Fig. 10. Time histories of vibration amplitudes: K-200 unit, rear intermediate-pressure turbine bearing, axial direction, 4*f*0 (a) and 1000 Hz (b) components. Vertical lines indicate repair.

In addition to being sensitive to outliers, the Pearson correlation coefficient is deficient in that a linear relation is assumed. Both above-mentioned disadvantages may be eliminated or at least alleviated to some extent by using non-linear and more robust measures of

> *d N N* <sup>2</sup> <sup>1</sup>

> > , given by

*N N*

( 1)

*i d*

( 1)

2 2 6

, (33)

, given by

. (34)

correlation. These include the Kendall rank correlation coefficient

and Spearman rank correlation coefficient

3 Detailed case study may be found in (Gałka, 2008a)

1

cause was thus different.3

replacement). It is easily seen that they all increase with , almost monotonically, but () is obviously influenced by outliers and hence is of 'step-like' form. Both *m*() and *q*() are more regular, but certainly () and () are superior; in particular, the latter is most regular and almost perfectly monotonic. The 'dynamics' of these symptoms is also noteworthy: during the period covered by observation they increase roughly by one order of magnitude. They may be thus considered highly sensitive to lifetime consumption. It may also be noted that a marked increasing tendency starts well before rotor replacement (about four years). Symptoms of this type may thus provide an 'early warning', with a lead long enough e.g. to re-schedule maintenance or purchase replacement parts.

#### **6.2 Correlation measures**

The use of a correlation measure in vibration-based condition assessment is twofold. First, we may check the very existence of correlation or, more precisely, determine if it is 'weak' or 'strong'. This may be very useful, because – as already noted – in steam turbines several possible malfunctions sometimes produce similar changes of vibration characteristics. Such approach is thus applicable in qualitative diagnostics. Second, we may study how a correlation measure changes with time and utilize the results for a quantitative diagnosis.

The most commonly used measure of correlation is the Pearson product-moment correlation coefficient *r*, given by the normalized covariance

$$r = \frac{E\{(S\_1 - \eta\_1)(S\_2 - \eta\_2)\}}{\sqrt{E\{(S\_1 - \eta\_1)^2\}E\{(S\_2 - \eta\_2)^2\}}},\tag{31}$$

where *E* denotes expected value and

$$
\eta\_1 = E(S\_1), \ \eta\_2 = E(S\_2) \tag{32}
$$

This measure is very sensitive to outliers (see e.g. Maronna, Martin and Yohai, 2006), but is often sufficient for a qualitative diagnosis. The basic idea stems from the fact that if two symptoms can be shown to be correlated, we may infer that they are dependent, i.e. that their changes have been caused by the same condition parameter. The reverse is not true: if two random variables are not correlated, this does not imply that they are independent.

Fig.10 shows two vibration time histories recorded with a 200 MW turbine that suffered an intermediate-pressure rotor failure and secondary fracture of steam guiding fences. Manifestation of this failure in vibration patterns was quite complex. It may be noted, however, that before repair both these components tended to increase simultaneously. Several other components from the blade frequency range – up to about 2 kHz – behaved in a similar manner. We may thus suspect that comparatively high level of the 4 *f*0 component was a result of the fluid-flow system failure. This is corroborated by correlation analysis; for 23% CPB spectra bands from 800 Hz to 2 kHz coefficients of correlation with the 4 *f*0 component ranged from *r* = 0.689 to *r* = 0.912, while for two other turbines of the same type |*r*| was below 0.2 (in several cases negative). Shortly after the repair the 4 *f*0 component increased again, eventually reaching even substantially higher level; this time, however,

almost perfectly monotonic. The 'dynamics' of these symptoms is also noteworthy: during the period covered by observation they increase roughly by one order of magnitude. They may be thus considered highly sensitive to lifetime consumption. It may also be noted that a marked increasing tendency starts well before rotor replacement (about four years). Symptoms of this type may thus provide an 'early warning', with a lead long enough e.g. to

The use of a correlation measure in vibration-based condition assessment is twofold. First, we may check the very existence of correlation or, more precisely, determine if it is 'weak' or 'strong'. This may be very useful, because – as already noted – in steam turbines several possible malfunctions sometimes produce similar changes of vibration characteristics. Such approach is thus applicable in qualitative diagnostics. Second, we may study how a correla-

The most commonly used measure of correlation is the Pearson product-moment correlation

1 12 2 2 2 11 22

 

> 

, (31)

2 = *E*(*S*2) . (32)

{( )( )} {( ) } {( ) }

This measure is very sensitive to outliers (see e.g. Maronna, Martin and Yohai, 2006), but is often sufficient for a qualitative diagnosis. The basic idea stems from the fact that if two symptoms can be shown to be correlated, we may infer that they are dependent, i.e. that their changes have been caused by the same condition parameter. The reverse is not true: if two random variables are not correlated, this does not imply that they are

Fig.10 shows two vibration time histories recorded with a 200 MW turbine that suffered an intermediate-pressure rotor failure and secondary fracture of steam guiding fences. Manifestation of this failure in vibration patterns was quite complex. It may be noted, however, that before repair both these components tended to increase simultaneously. Several other components from the blade frequency range – up to about 2 kHz – behaved in a similar manner. We may thus suspect that comparatively high level of the 4 *f*0 component was a result of the fluid-flow system failure. This is corroborated by correlation analysis; for 23% CPB spectra bands from 800 Hz to 2 kHz coefficients of correlation with the 4 *f*0 component ranged from *r* = 0.689 to *r* = 0.912, while for two other turbines of the same type |*r*| was below 0.2 (in several cases negative). Shortly after the repair the 4 *f*0 component increased again, eventually reaching even substantially higher level; this time, however,

*ES S*

1 = *E*(*S*1),

*ES ES*

tion measure changes with time and utilize the results for a quantitative diagnosis.

) are superior; in particular, the latter is most regular and

, almost monotonically, but

) and *q*( () is

) are more

replacement). It is easily seen that they all increase with

re-schedule maintenance or purchase replacement parts.

coefficient *r*, given by the normalized covariance

where *E* denotes expected value and

independent.

*r*

() and (

regular, but certainly

**6.2 Correlation measures** 

obviously influenced by outliers and hence is of 'step-like' form. Both *m*(

there is virtually no correlation with the blade components, *r* being about –0.1. The root cause was thus different.3

Fig. 10. Time histories of vibration amplitudes: K-200 unit, rear intermediate-pressure turbine bearing, axial direction, 4*f*0 (a) and 1000 Hz (b) components. Vertical lines indicate repair.

In addition to being sensitive to outliers, the Pearson correlation coefficient is deficient in that a linear relation is assumed. Both above-mentioned disadvantages may be eliminated or at least alleviated to some extent by using non-linear and more robust measures of correlation. These include the Kendall rank correlation coefficient , given by

$$\tau = 1 - \frac{2d\_{\Lambda}}{N(N-1)} \, ^{\prime} \tag{33}$$

and Spearman rank correlation coefficient , given by

$$\rho = 1 - \frac{6\sum d\_i^2}{N(N^2 - 1)}\,. \tag{34}$$

<sup>3</sup> Detailed case study may be found in (Gałka, 2008a)

Vibration-Based Diagnostics of Steam Turbines 337

Fig. 11. Plots of Pearson (a), modified Pearson (b), Kendall (c) and Spearman (d) correlation coefficients: 200 MW unit, front high-pressure turbine bearing, vertical direction, 5 kHz and

of diagnostic information. Such selection may employ the Singular Value Decomposition method, known from linear algebra (see e.g. Cempel, 2003). This approach has been applied to a number of steam turbines and results have been found very encouraging (Gałka, 2011a). In general, components generated by rotor stages are more informative in this respect than those generated by bladed diaphragms. Best results have been obtained with high-pressure turbines; this is not particularly surprising, due to high temperature and pressure, which

Steam turbines, which are of vital importance for any economy, have always been at the leading edge of technical diagnostics development. A variety of vibration monitoring systems is available on a commercial scale, usually tailored to individual needs. Some of them are referred to as 'diagnostic systems', which is not always strictly true, as many

In general, qualitative diagnostics is currently based on well-established procedures and rules, especially for harmonic components. Quantitative condition assessment seems to be at

6.3 kHz bands. Data window containing 25 measurements.

contribute to accelerated lifetime consumption.

merely provide data for diagnostic reasoning.

**7. Conclusion** 

In these formulae, *N* is the number of scores (elements) in two data samples, *d* denotes symmetric difference distance and *di* are differences between individual ranks. It should be noted here that both and are calculated on the basis of ranks rather than standard deviations and therefore more suitable for analyzing time series (Salkind, 2007). They are thus more appropriate when dealing with correlation as a function of .

As mentioned earlier, for a rapidly developing damage one should speak in terms of a deviation from the trend than of some mean or expected value. We may therefore, as suggested in (Gałka, 2011a), introduce yet another correlation measure, tentatively termed 'modified Parsons coefficient' *r*. Using the notation from Eqs.(30) and (31), *r* is given by

$$r' = \frac{E\left\{ \left( S\_1 - S\_{t1} \right) \left( S\_2 - S\_{t2} \right) \right\}}{\sqrt{E\left\{ \left( S\_1 - S\_{t1} \right)^2 \left( S\_2 - S\_{t2} \right)^2 \right\}}}\,\,\,\tag{35}$$

Let us assume that lifetime consumption degree *D* = /*<sup>b</sup>* is the only condition parameter that is taken into account. Then Eq.(9) for a given symptom *S* may be rewritten as

$$S = f(D\_1, R\_1, R\_2, \dots, R\_{\delta}, Z\_1, Z\_2, \dots, Z\_m) \,. \tag{36}$$

Within the framework of the Energy Processor model, the influence of *D* on *S* is purely deterministic and *S*(*D*) is a monotonically increasing function. As *D* approaches unity, both *S* and *dS*/*dD* tend to infinity (cf. Eq(14)), so equal increments of *D* will result in increasing increments of *S*:

$$D \to \mathbf{1} \Rightarrow \Delta S = S(D + \Delta D) - S(D) \to \infty \quad \text{(\$\Delta D = const.)} \,, \tag{37}$$

and this will hold for all symptoms. Correlation is thus expected to increase with *D*, as for any two symptoms *Sj* and *Sk* both will, to a growing extent, be dominated by *D* rather than other factors and thus become more deterministic with respect to *D*. Speaking in a descriptive manner, Eq.(35) may be viewed as revealing a competition between the random (represented by *Ri* and *Zi*) and the deterministic (represented by *D*). The above argumentation suggests that for *D* 1 the latter should prevail and consequently a measure of correlation should increase in value. This phenomenon has been termed the 'Old Man Syndrome'.4

Fig.11 shows comparison of the above four correlation measures plotted against time for the same unit as in Fig.9 (albeit for different frequency bands). All plots exhibit a more or less pronounced 'saddle', which was found to have resulted from an overhaul which 'decorrelated' the symptoms to a certain extent. The terminal increasing section is, however, evident.

Due to a large number of vibration-based symptoms generated by a typical multi-stage steam turbine, the number of pairs to be analyzed in terms of correlation is large, of the order of a few dozen or more. It is, however, possible to select those with the highest content

<sup>4</sup> To the author's best knowledge, this term has been first used in the context of technical diagnostics by Cempel (see Cempel, 1991).

In these formulae, *N* is the number of scores (elements) in two data samples, *d* denotes symmetric difference distance and *di* are differences between individual ranks. It should be

deviations and therefore more suitable for analyzing time series (Salkind, 2007). They are

As mentioned earlier, for a rapidly developing damage one should speak in terms of a deviation from the trend than of some mean or expected value. We may therefore, as suggested in (Gałka, 2011a), introduce yet another correlation measure, tentatively termed 'modified Parsons coefficient' *r*. Using the notation from Eqs.(30) and (31), *r* is given by

112 2

*ES S S S*

that is taken into account. Then Eq.(9) for a given symptom *S* may be rewritten as

*ES S S S*

 *t t*

 *S* = *f*(*D*, *R*1, *R*2, ..., *Rk*, *Z*1, *Z*2, ..., *Zm*) . (36) Within the framework of the Energy Processor model, the influence of *D* on *S* is purely deterministic and *S*(*D*) is a monotonically increasing function. As *D* approaches unity, both *S* and *dS*/*dD* tend to infinity (cf. Eq(14)), so equal increments of *D* will result in increasing

 *D* 1 *S* = *S*(*D* + *D*) – *S*(*D*) (*D* = const.) , (37) and this will hold for all symptoms. Correlation is thus expected to increase with *D*, as for any two symptoms *Sj* and *Sk* both will, to a growing extent, be dominated by *D* rather than other factors and thus become more deterministic with respect to *D*. Speaking in a descriptive manner, Eq.(35) may be viewed as revealing a competition between the random (represented by *Ri* and *Zi*) and the deterministic (represented by *D*). The above argumentation suggests that for *D* 1 the latter should prevail and consequently a measure of correlation should increase in value. This phenomenon has been termed the 'Old Man

Fig.11 shows comparison of the above four correlation measures plotted against time for the same unit as in Fig.9 (albeit for different frequency bands). All plots exhibit a more or less pronounced 'saddle', which was found to have resulted from an overhaul which 'decorrelated' the symptoms to a certain extent. The terminal increasing section is, however,

Due to a large number of vibration-based symptoms generated by a typical multi-stage steam turbine, the number of pairs to be analyzed in terms of correlation is large, of the order of a few dozen or more. It is, however, possible to select those with the highest content

4 To the author's best knowledge, this term has been first used in the context of technical diagnostics by

11 22

*t t*

2 2

/

thus more appropriate when dealing with correlation as a function of

*r*

Let us assume that lifetime consumption degree *D* =

are calculated on the basis of ranks rather than standard

.

. (35)

*<sup>b</sup>* is the only condition parameter

noted here that both

increments of *S*:

Syndrome'.4

evident.

Cempel (see Cempel, 1991).

 and 

Fig. 11. Plots of Pearson (a), modified Pearson (b), Kendall (c) and Spearman (d) correlation coefficients: 200 MW unit, front high-pressure turbine bearing, vertical direction, 5 kHz and 6.3 kHz bands. Data window containing 25 measurements.

of diagnostic information. Such selection may employ the Singular Value Decomposition method, known from linear algebra (see e.g. Cempel, 2003). This approach has been applied to a number of steam turbines and results have been found very encouraging (Gałka, 2011a). In general, components generated by rotor stages are more informative in this respect than those generated by bladed diaphragms. Best results have been obtained with high-pressure turbines; this is not particularly surprising, due to high temperature and pressure, which contribute to accelerated lifetime consumption.
