=Paper=
{{Paper
|id=Vol-2667/paper11
|storemode=property
|title=Analysis of hydraulic unit operation stability according to its vibration monitoring results
|pdfUrl=https://ceur-ws.org/Vol-2667/paper11.pdf
|volume=Vol-2667
|authors=Anastasiya Alekseeva,Irina Karpunina,Vladimir Klyachkin
}}
==Analysis of hydraulic unit operation stability according to its vibration monitoring results ==
https://ceur-ws.org/Vol-2667/paper11.pdf
Analysis of hydraulic unit operation stability
according to its vibration monitoring results
Anastasiya Alekseeva Irina Karpunina Vladimir Klyachkin
Standardization department Department of general professional Department of applied mathematics
Ulyanovsk Design Bureau of disciplines and Informatics
Instrumentation Ulyanovsk Civil Aviation Institute Ulyanovsk State Technical University
Ulyanovsk, Russia Ulyanovsk, Russia Ulyanovsk, Russia
age-89@mail.ru karpunina53@yandex.ru v_kl@mail.ru
Abstract—During hydraulic unit steady - state operation it case, when Shewhart control charts can be applied. It is not
is necessary to support its functioning stability. The analysis always, that usual Shewhart control charts are quick enough
of vibration results for correlated values is carried out with in revealing the stability violation. The various ways of their
multivariate statistical control methods: the process average efficiency improvement are used. Such as: special form
control is done based on Hotelling’s algorithm, when structures searching on the chart, warning limit
multivariate dispersion control is done through the generalized introduction, process monitoring with memory charts
variance algorithm. The article investigates the efficiency of application (cumulative sum and exponentially weighed
generalized variance algorithm: how fast the generalized moving average control charts) , etc. The efficiency of this
variance test chart reacts to a hydraulic unit vibration stability
or that statistical tool application depends on the type of the
prone breakdown. The investigation revealed that the
hydraulic unit operation stability versus multivariate
most hazardous for the current process kind of breakdown.
dispersion is not always appropriately assessed through a It might be a rapid rise of average or process dispersion, its
standard generalized variance algorithm. To improve the trend, etc.
monitoring sensitivity to a prone breakdown, it is reasonable to For the correlated values multivariate statistical
modify this algorithm with a search of non-random structures monitoring the control methods are used: the monitoring of
on the corresponding chart, with a warning limit and a process average is done based on Hotelling’s algorithm,
exponentially weighed moving average (EWMA) on a
when multivariate dispersion control is done through the
generalized variance.
generalized variance algorithm. After certain time intervals
Keywords—statistical process control, multivariate scattering, the samples are taken, and for each sample there is an
generalized dispersion, control chart estimated Hotelling’s value and generalized variance, i.e.
controllable values covariance matrix determinant; the
I. INTRODUCTION alternation of this parameter characterizes the scattering
process stability [7-11]. This approach is applied in different
During hydraulic unit steady-state operation it is
domains [12-15].
necessary to provide for its functioning stability. In fact,
vibration instability might lead to emergencies and The hydraulic unit vibration monitoring data were
extraordinary cases with dramatic consequences. The analyzed: there were 10 values to assess: the vibration of
example of Sayano-Shushenskaya hydraulic electro power lower Х1 and upper Х3 generator set bearing, upstream and
station hydraulic unit destruction with multiple fatalities in on the RH coast Х2, Х4, hydraulic turbine shaft vibration
result is the most vivid illustration hereto. downstream Х5 and on the RH coast Х6, hydraulic generator
shaft vibration Х7, Х8, and also hydraulic turbine cover
The vibration value data during motion monitoring, in
vibration Х9,Х10.
online mode, are applied to hydraulic unit control stand, and
if necessary, when the vibration data processing system Figure 1 shows multivariate charts, plotted within
predicts its significant increase, the load is reduced. The Statistica [16] system by two correlated values Х6-Х8 (the
analysis of the data applied can be carried out in different significant correlation is available between these two values,
ways [1-4]. the significant correlation by Student criteria at significance
level equal to 0.05; sample correlation coefficient equal to
One of the approaches, widely used in the technical
r = 0.61). Both charts testify to vibration stability:
process stability monitoring, is the statistical control
Hotelling’s value does not exceed the limit (13.756),
method. The data monitoring is performed, the process non-
generalized dispersion is also within the limits (limit is
random deviation is revealed in result: the monitored data
14.514).
are to be located within the limits of the corresponding
confidence intervals. By deviation we mean the graphical It is worth saying that the limits mentioned above are
location of one of the points on the chart beyond the limit. determined by means of statistics methods and are not the
At the same time the physically monitored data are still limits for vibration; these are the limits of the existing
within the limits, however, the statistics reveals the process confidential interval (CI). Their violation means stability
instability [5-6]. Shewhart control charts are applied to breakdown, though the limit values remain within the limits
monitor the independent values: both mean level and yet. Timely reaction to such breakdown incidents excludes
process dispersion are monitored simultaneously. The the emergency situation.
standards assume the application of average values and
range charts or standard deviation, as well as individual However, not only beyond-the-limit controlled statistics
observation and moving range charts. In the vibration testifies to the process failure, but different special form
monitoring of a hydraulic unit some readings of vibration structures on the chart do. Along with it, the mentioned
pick-ups are not correlated with the others and this is the above methods do not always react effectively to the process
Copyright © 2020 for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0)
Data Science
prone breakdowns. Hotelling’s algorithm controlling 1
p
(4)
multivariate level of average is well enough studied in this b1
p
( n j );
respect [5-9], which is not the case with multivariate ( n 1) j 1
dispersion control.
p p p
1 , (5)
b2
2p
( n j )[ ( n k 2 ) ( n k ) ]
( n 1) j 1 k 1 k 1
and the assessment of target generalized dispersion |0| is
determined as per the learning sample. Provided the low
control limit LCL as per the formula (3) turns out to be
negative, zero value is taken.
Vibration stability break down is testified by the location
of at least one point on the chart of the generalized
dispersion beyond one of the limits, that means that the
process is steady if the following inequality is true:
LCL < |St| < UCL,
where t means the number of the controlled sample.
For the quality rating of algorithm sensitivity to the
Fig. 1. Multivariate charts.
process, prone breakdown average sample run length is
The aim of the investigation is to increase the efficiency applied, i.e. the number of observations done within the
of hydraulic unit vibration monitoring in its operation values period of time between the moment of the initial breakdown
multivariate dispersion criteria through the assessment of its occurrence and the moment of the breakdown finding.
generalized variance algorithm sensitivity: how fast the For the experiment purpose a set of samples, similar to
generalized variance test chart reacts to a hydraulic unit real ones in motion, were simulated. The bench-mark data
vibration stability prone breakdown. are vector of mean values and correlated values covariance
II. GENERALIZED VARIANCE ALGORITHM SENSITIVITY matrix. The algorithm of simulating multinomial random
ASSESSMENT variables is used.
Generalized variance algorithm is in fact the check for For the simulated samples different failures of process
the hypothesis of covariance matrix equality of the vibration scattering are introduced, and the number of samples from
process to the set value 0. For each moment of time t a the moment of the introduced failure till the moment of the
sample covariance matrix St, is formed, the elements of process running beyond the warning limit on the plotted
which are as following: charts of the generalized variance is determined. Averaging
these data for all the samples we will get an average run
1
s jkt ( x ijt x j )( x ikt x k ) , (1) length.
n 1
Figure 2 shows the results of the carried out experiments
xijt is the result of observation i as per index j in sample t with multivariate dispersion of two correlated values. There
(i = 1,…, n, n is the sample size, j, k = 1, …, p, p is the was simulated a dispersion abrupt increase by 1.25 times
quantity of the monitored values, t = 1, …, m, m is the
(sample value of the determinant of covariance matrix was
number of samples taken for the vibration analysis). The
multiplied by d = 1.25), by 1.5, by 1.75, by 2 times. The
determinant |St| of matrix (1) is the generalized dispersion of
instantaneous sampling t. corresponding values of d are plotted on the diagram on its
horizontal axis. The vertical axis shows the values of
The estimated covariance average is also calculated as average run length L(d), estimated by 1000 simulated
per the whole sample population : samples.
1 m , (2)
s jk s jkt
m t 1
which forms the covariance matrix S; its determinant |S| is
used as the assessment of target generalized dispersion |0|.
While plotting the control chart, sample values of
generalized dispersion |St| for each sample t are taken.
The generalized dispersion chart limits are determined as
per the following formula:
UCL
|0| (b1 u1-/2 b 2 ), (3) Fig. 2. Average run length in result of the experiments.
LCL
The results of the experiments (experimental results)
where u1-/2 is the quintile of normal distribution policy 1 were approximated by the regression parabola relation, built
– /2, is the significance (probability of false alarm); the in the environment of Excel spread sheets (trend line):
coefficients are calculated as per the following formulae:
VI International Conference on "Information Technology and Nanotechnology" (ITNT-2020) 47
Data Science
L(d) = -5.36d 2 + 13.50d – 3.028 warning limit UWL and lower warning limit LWL) is done
through Markovian chain similar to average charts limit
Determination factor R2 = 0.993 indicates the high estimation [19-21].
quality of the plotted model. Using this relation and
The calculation results can be presented as follows:
knowing which scattering increase value is jeopardizing (or
critical) for the tested item, we can assess the quality of a UWL
generalized variance algorithm and make corrections in the 0 ( b1 B b2 ) , (6)
LWL
process of multivariate dispersion control.
Similar results were achieved for other sets of correlated В coefficient is determined from the tables [21] as per the
values. number of points between the warning and control limits. It
is reasonable to check all three variants in practice: 2, 3 or 4
Let us assume that for two vibration values monitoring points are between the limits.
the abrupt increase in dispersion by 1.6 times is hazardous.
Then the mentioned formula means that the generalized
variance chart will find this breakdown after L(1.6) = 4.8
samples. Sometimes this value is inadmissible: within this
period of time the vibration will cause unintended
consequences. In this case it is necessary to change the
control procedure in order to improve its sensitivity.
III. GENERALIZED VARIANCE ALGORITHM SENSITIVITY
IMPROVEMENT METHODS
Fig. 4. Three consecutive points in a raw between the warning and control
To improve the control efficiency one may use several limits on the generalized variance chart.
different approaches: to analyse the non-random structures One more approach, providing dispersion monitoring
on the chart of generalized variance, to introduce an efficiency increase under certain conditions, is the use of
additional warning limit, to apply exponentially weighed exponentially weighed moving average on a generalized
moving average (EWMA) on a generalized variance. variance (figure 5). The tests revealed that this chart senses
Analysing the non-random structures on the generalized the abrupt increase of the dispersion faster than the usual
variance we proceed from the assumption that generalized chart of generalized variance.
variance algorithm is based on the use of normal distribution The values of exponentially weighed moving average
(ND) (three-sigma rule), so to reveal the defect the same EWMA, plotted on the chart, is calculated as per the
types of structures could be used as for Shewhart control following formula:
charts [17-18]. The space between the central line and upper
limit is divided into three; the width of each one is equal to E t (1 ) E t 1 t , (7)
one standard deviation. The non-random structures, whose
probability is commensurable with the probability of a false where λ means the parameter of exponential smoothening
warning, are (figure 3): (0 < λ < 1).
а) at least one point runs beyond the limit, The position of the control limits of the exponentially
b) at least two out of three consecutive points above weighed moving average control chart for the generalized
the central line run beyond two sigma limit, dispersion is determined as per the following formula:
c) at least four out of five consecutive points above the U СС
0 H Et ,
(8)
central line run beyond one sigma limit, L СС
d) six increasing or decreasing points in a raw (trend), where Н means the parameter, specifying the position of the
etc. limits (as a rule it is assumed that Н = 3); the standard
deviation of exponentially weighed moving average can be
found as per the formula:
1 (1 ) ,
2
2 2t (9)
Et
n 2
where means the assessment of the generalized
dispersion standard deviation.
Fig. 3. Chart of non-random structures on the generalized variance.
The introduction of a warning limit increases the
sensitivity of the generalized variance control chart (Fig. 4).
The position of such a limit line is assessed according to the
number of points between the warning and control limit
lines, considered to be an abnormality (usually two, three, or
Fig. 5. Chart of exponentially weighed moving average on a generalized
four). The estimation of the warning limit position (upper variance.
VI International Conference on "Information Technology and Nanotechnology" (ITNT-2020) 48
Data Science
IV. CONCLUSION [9] S. Bersimis, S. Psarakis and J. Panaretos, “Multivariate Statistical
Process Control Charts: An Overview,” Quality and reliability
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functioning stability monitored as per vibration monitoring [10] Y.C. Tan, M.H. Lee and W.W. Winnie, “An improved switching
multivariate dispersion criteria is not always appropriately rule in variable sampling interval Hotelling's Т2 control chart,”
assessed through the generalized variance standard Institute of Electrical and Electronics Engineers, pp. 1412-1416,
2015.
algorithm . The dispersion increase is often found too late,
when vibration may cause harmful circumstances. To [11] M.H. Lee, “Variable sampling rate Hotelling’s Т2 control chart with
runs rules,” South African Journal of Industrial Engineering, vol. 23,
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ACKNOWLEDGMENT Kuvayskova, “Selection of aggregated classifiers for the prediction
of the state of technical objects,” CEUR Workshop Proc., vol. 2614,
The investigation is carried out supported by the joint pp. 361-367, 2019.
research grant from Russian Foundation for Basic Research [14] V.N. Klyachkin and I.N. Karpunina, “The Analysis of technical
and Ulyanovsk region government, project 18-48-730001. object functioning stability as per the criterion of monitored
parameters multivariate dispersion,” CEUR Workshop Proc., vol.
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