Results of calculations for six symptoms;
For complex objects, condition assessment is usually based on indirect symptoms related to residual processes such as vibration, noise, heat generation, etc. The number of available symptoms is often large, and it is necessary to select those which are most representative (i.e., sensitive to condition parameters). Such selection may be based on singular value decomposition (SVD). An alternative approach is proposed that employs information content measures. In order to obtain a reliable condition assessment and prognosis of its evolution (in particular, remaining useful life estimation), certain preprocessing of experimental data is necessary. This involves, among others, issues such as life cycle normalization or identification and removal of outliers. Suitable procedures are proposed and discussed. Example is presented for vibration-based symptoms of steam turbine technical condition.
- diagnostic symptom
- technical condition
- information content
Measurement (acquisition of data that contain information on object condition)
Qualitative diagnosis (or recognition-identification and localization of failures and malfunctions)
Quantitative diagnosis (estimation of damage advancement)
Prognosis (forecast for object operation in future)
In structural health monitoring and condition-based maintenance, the third and fourth steps are of particular importance. Quantitative diagnosis is in fact an estimation of the current object condition. Once this has been accomplished, a prognosis may follow, which basically means remaining useful life (RUL) estimation on the basis of certain criteria. This is extremely important for proper and safe operation and cost-effective maintenance of complex and critical machinery.
Evolution of object condition may be described in terms of the
For many objects it is impracticable or inconvenient to describe condition evolution in terms of the hazard function (or failure density). An alternative approach is based on the analysis of energy transformation and dissipation mechanisms, which leads to the energy processor model [1, 7]. This model implies that object condition is estimated in an indirect manner, from measurable physical quantities referred to as
for the former and
for the latter; in both cases,
Large and complex objects usually generate many diagnostic symptoms, and their number in fact has no upper limit. It has to be kept in mind that values of these symptoms depend not only on condition parameters. If all symptoms
This chapter is devoted mainly to symptom evaluation and selection methods based on the analysis of information content measures. Some attention shall, however, also be paid to the method employing the singular value decomposition, the first that has been used for this purpose.
Suitability of symptom evaluation methods has been verified for a number of vibration-based symptoms generated by steam turbines operated at utility power plants. Details on symptom generation mechanisms may be found, e.g., in [1, 10, 11]. Absolute vibration velocity was recorded in the form of 23% constant percentage bandwidth (CPB) spectra, at points located at bearings and low-pressure turbine casings. Piezoelectric accelerometers were used with magnetic mountings, which allows for a frequency range well above 10 kHz. This implies that both “harmonic” (i.e., resulting directly from rotational motion) and “blade” (i.e., generated by the fluid flow system) components are recorded. Vibration amplitudes in frequency bands determined from turbine vibrodiagnostic models [1, 11] are the diagnostic symptoms to be evaluated. It has to be stressed here that presented methods are valid for a broad class of various diagnostic symptoms, irrespective of their physical origin.
2. Singular value decomposition
Singular value decomposition (SVD) is well known from linear algebra; concise description can be found, e.g., in . To the author’s best knowledge, the idea to employ this method in technical diagnostics goes back to the late 1990s . Application for vibration-based symptoms has shown this method to give consistent results .
The first step is to represent symptom value database in the form of an
According to  and following notation used herein, the
This means that this fault can be expressed in terms of left-singular or right-singular vectors, which are generally interpreted as “input” and “output” [13, 15]. In the case of system condition evolution, “input” represents condition parameters and “output” represents symptoms. Obviously, the second discriminant, given by Eq. (10), is of practical use here, as condition parameters are typically nonmeasurable.
SVD analysis may be performed using one of available software packages. In practical applications the first step is to analyze individual singular values. For a comparatively new object, the descent of consecutive singular values is rather slow; this means that dominant failure mode has not yet appeared. On the other hand, with considerable lifetime consumption degree, the first singular value dominates. Examples are shown in Figure 2. They refer to vibration-based symptoms generated by steam turbine fluid flow systems. In both cases illustrated in Figure 2, there are six such symptoms. For a turbine with a few dozen thousand hours logged (Figure 2a), contributions of the first three singular values into generalized damage are 36, 29, and 17%, respectively. For the second turbine (Figure 2b), which has logged well over 200,000 hours, corresponding values are 48, 24.5, and 10%—the difference is clearly seen. The second step is to calculate contributions of individual symptoms into several (e.g., three) first singular values. Corresponding graphs are shown in Figure 3. For the first turbine, dominant symptoms cannot be identified, although we may infer that symptom numbers 1 and 5 can be skipped. For the second turbine, however, dominance of symptom numbers 5 and 6 is clearly seen, and they may be judged most sensitive to the fluid flow system lifetime consumption.
3. Information content measures
3.1. The idea
The abovementioned energy processor model is, by its very nature, deterministic. From Eq. (5), however, it is clearly seen that symptom values depend not only on deterministic condition parameters
For a given object operated at a given location, it is reasonable to assume that
Investigations of information content and its measures were pioneered by Claude E. Shannon. In his fundamental work , he introduced an information content measure
Shannon entropy was originally introduced for verbal communication; hence, a discrete random variable is involved. A diagnostic symptom in the sense of the energy processor model is in general continuous, so a derivative of
It may be added here that several other entropy types have been proposed, e.g., by Hartley , Rényi , or Tsallis . Their use, however, has been limited. Hartley entropy is a specific case of the Shannon entropy, while Rényi entropy may be viewed by its generalization. Both Rényi and Tsallis entropies involve certain adjustable parameters of rather unclear physical meanings, which are generally difficult to estimate.
For the purpose of condition symptom evaluation, the time window procedure may be employed. A window containing sufficient number of
3.2.1. Distribution type
Obviously, in order to employ the abovementioned procedure, symptom distribution type has to be determined. In general, distributions of diagnostic symptom values are of the right-hand tailed type . Weibull and gamma distributions are commonly used, with the probability density functions given by
Diagnostic symptom time histories often exhibit a considerable number of outliers. According to , “an outlying observation, or outlier, is one that appears to deviate markedly from other members of the sample in which it occurs”; there is no generally accepted precise definition. From the point of view of information theory, outliers are equivalent to noise. As with the definition, there is no universal method for removing outliers. The “three-sigma rule,” which is often used for this purpose, is not applicable to distributions with long right-hand tails . Three-point averaging  merely flattens outliers instead of removing them. The author has suggested a procedure referred to as “peak trimming” , based on comparison of a data point with two adjacent points. If for the
Fitting continuous distributions to experimental symptom value histograms within the time window limits require at least weak stationarity. This implies that for every symptom
Trend may be determined by fitting a suitable function to experimental symptom time history. Weibull and Fréchet functions may be used for this purpose; for low values of
3.2.4. Abrupt changes
Complex and costly machines like, for example, power-generating units are usually designed for long service life. During the period between commissioning and final withdrawal from use, they are usually subject to various processes of maintenance, repair, and overhaul. Each of them introduces changes of object properties, which influence both diagnostic symptom generation mechanisms and their propagation from origin to measurement points. So far, it has been assumed (tacitly) that each
Figure 7 clearly shows that, if fitting continuous function to experimental data is expected to yield consistent results, abrupt changes should be eliminated. In principle this is relatively simple. Each life cycle and hence each symptom life curve are characterized by the so-called logistic vector , which describes its “quality.” This vector may be replaced by its scalar measure
This idea may seem simple, but precise determination of the moment of transition from a life cycle to the next one may be problematic. Sufficient operational documentation is not always available, and transitions are often masked by random fluctuations. A method for their detection is thus necessary. Such method may be based on techniques originally developed for statistical process control.
In the 1920s Walter A. Shewhart developed a tool for determining whether a process (e.g., manufacturing) is under control, known as the process control chart. If that was the case, no modifications of process or control were needed; otherwise, an intervention was necessary, in order to restore stable and controlled operation . In 1954 E.S. Page proposed a more sensitive process control chart, employing cumulative sum and consequently named CUSUM . His approach consisted in introducing a quantity originally referred to as a “quality number,” developing an algorithm to estimate its changes and establishing a quantitative criterion. In general this quality number is a statistical parameter. If this procedure is employed for mean value, it can be used for detecting abrupt changes .
Let us assume that a variable
defines the figure of merit. Cumulative sum
3.2.5. Representativeness factor
It may be said, in a descriptive manner, that ICM is a measure of the degree of process organization around a monotonically increasing trend. However, the rate of this increase should also be taken into account in symptom evaluation. Organization may take place around a weakly increasing curve; such symptom is only weakly sensitive to object condition evolution and as such is of little use, despite marked ICM decrease. A measure is thus required that would combine
representativeness factor is then defined as
Measurement data for the first example were obtained with the intermediate-pressure turbine of a 260 MW power-generating unit; the first measurement was performed shortly after commissioning, and available data cover the period of almost 10 years. Vibration velocity was recorded at the front and rear bearings, in three mutually perpendicular directions. Components generated by turbine fluid flow system are contained in four 23% CPB bands, which give 24 available symptoms. Of these, as many as 13 symptoms have revealed no increasing trend; this may be attributed to comparatively short period of operation, as evolution of the fluid flow system condition is usually rather slow. For the remaining 11 symptoms, measured values were normalized, and peak trimming was performed (Eqs. (21) and (22), with
Continuous entropy time histories are in some cases rather irregular, but nonetheless six of them exhibit a decreasing trend; an example is shown in Figure 11. For these six cases, representativeness factor was calculated in accordance with Eq. (29). Results are shown in Table 1. It is easily seen that the values of
Figure 12 shows contributions of all 11 symptoms that exhibit an increasing trend into the first three singular values. It may be noted that results are basically consistent with those shown in Table 1. The main differences are:
Comparatively high contributions of symptom number 5, which has a low representativeness factor
Better result for symptom number 18
Comparatively high contributions of symptom number 9, which is absent in Table 1 (lack of entropy decreasing trend)
|Symptom number||Symptom description (kHz)||Value of ||Entropy decrease rate||Representativeness factor|
|1||FB-V 3.15||11.24||0.960||85.44 × 10−3|
|2||FB-V 4||10.64||0.905||85.07 × 10−3|
|5||FB-H 3.15||500.0||0.010||0.02 × 10−3|
|16||RB-V 6.3||52.63||0.775||14.73 × 10−3|
|18||RB-H 4||52.63||0.497||9.44 × 10−3|
|24||RB-A 6.3||55.56||0.637||11.47 × 10−3|
Before commenting on these findings, a second example will follow, this time for a comparatively old 200 MW unit with over 200,000 hours logged; available database covers over 16 years. Fluid flow system of the high-pressure turbine generates vibration components that are contained in ten 23% CPB frequency bands. Given two bearings and three directions, this means that as many as 60 symptoms have to be analyzed. In order to simplify the picture, a two-stage procedure was employed . First, for every measurement point and direction, two dominant symptoms were selected using the SVD approach. Twelve symptoms selected in this manner were then analyzed with both SVD and ICM methods. Results are shown in Figure 13 and Table 2.
|Symptom number||Symptom description (KHz)||Representativeness factor|
|1||FB-V 6.3||−0.63 × 10−3|
|2||FB-V 8||−15.30 × 10−3|
|3||FB-H 5||6.97 × 10−3|
|4||FB-H 6.3||5.57 × 10−3|
|5||FB-A 6.3||8.84 × 10−3|
|6||FB-A 8||3.77 × 10−3|
|7||RB-V 5||−1.93 × 10−3|
|8||RB-V 8||1.17 × 10−3|
|9||RB-H 6.3||0.20 × 10−3|
|10||RB-H 8||4.14 × 10−3|
|11||RB-A 2||−2.52 × 10−3|
|12||RB-A 8||5.69 × 10−3|
In Table 2, cases with
In order to comment on these two examples, it has first to be noted that neither SVD nor ICM approach can be considered a reference one. It seems that discrepancies between the results obtained with both may be attributed to at least two possible causes. First, preprocessing of measurement data is based on relatively simple procedures, and their inherent deficiencies—such as inadequate robustness—may influence the final result. Second, the SVD method does not disqualify cases with entropy increase, which are rejected within the ICM approach. This question requires further study. As pointed out in , it seems justified to state that symptoms selected on the basis of
In this chapter, a relatively straightforward and simple method is presented for evaluation of diagnostic symptoms from the point of view of their suitability for assessment and prognosis of technical condition evolution. For this purpose, the proper choice of symptoms is of prime importance. This is particularly important for complex objects that generate a large numbers of various symptoms. In most cases it is very difficult, or even impossible, to make such choice in a direct manner, even with extensive knowledge on object layout and operation. The proposed method is based on an analysis of an information content measure as a function of time, and the basic assumption is that the greater is general damage advancement, the more deterministic, and hence predictable the symptom becomes. It turns out, however, that in order to obtain reliable results certain preprocessing of measurement data is mandatory. Results of this method have been compared with those obtained from singular value analysis, which had been earlier proposed and tested. This approach has been applied for vibration-based symptoms of steam turbines operated by power plants and shown to give consistent results. In general it can be applied to any symptom, irrespective of its physical origin, as well as for other machines or structures. In the author’s view, possible further development should be concentrated on the preprocessing of measurement data and improvement of the representativeness factor. Other information content measures might also be worth considering; however, the best results have so far been obtained with continuous entropy.
Conflict of interest
The author of this text has no conflict of interest to declare.