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 The conducted experiment revealed that hydraulic unit Engineering International, vol. 23, рр. 517-543, 2007. 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, increase the sensitivity of monitoring to prone breakdowns it no. 1, pp. 122-129, 2012. is reasonable to modify this algorithm by the search of non- [12] V.R. Krasheninnikov, V.N. Klyachkin and Yu.E. Kuvayskova, random structures on the corresponding chart, by “Models updating for technical objects state forecasting,” introducing a warning limit, or by the use of exponentially Proceedings of the 3rd Russian-Pacific Conference on Computer weighed moving average on a generalized variance. Technology and Applications, IEEE, pp. 1-4, 2018. [13] D.A. Zhukov, V.N. Klyachkin, V.R. Krasheninnikov and Yu.E. 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. REFERENCES 1903, pp. 28-31, 2017. [1] Corporate standard of NPO “Innovations in power engineering”. [15] V.N. Klyachkin, K.S. Shirkunova and A.D. Bart, “Analysis of the Hydro-Electrical Power Plant. Methods to assess technical condition Stability of the chemical composition of wastewater in the of the main equipment, Company standard, 70238424.27.140.001- production of printed circuit boards,” Ecology and Industry of 2011. Russia, vol. 23, no. 5, pp. 47-51, 2019. [2] Corporate standard of JSC RUSHYDRO. “Vertical Hydraulic [16] V. Borovikov, “Statistica: Art of Data Analysis,” Advanced, SPb: Units”. Methodological instructive regulations to check and rectify Petersburg, 2001. the alignment defect. RUSHYDRO company standard 02.01.91- [17] J. Carlos García-Díaz, “The ‘effective variance’ control chart for 2013. monitoring the dispersion process with missing data,” Industrial [3] L.S. Kuravsky and S.N. Baranov, “Technical diagnostics and Engineering, vol. 1, no. 1, pp. 40-45, 2007. monitoring based on capabilities of wavelet transforms and [18] D. Wheeller and D. Chambers, “Statistical Process Control. relaxation neural network,” Insight-Non-Destructive Testing and Business Optimization with Shewhart Control Charts,” Мoscow: Condition Monitoring, vol. 50, no. 3, pр. 127-132, 2008. Alpina Business books, 2009. [4] P.V. Repp, “The system of technical diagnostics of the industrial [19] Yu.A. Kropotov, A.Yu. Proskuryakov and A.A. Belov, “Method for safety information network,” Journal of Physics: Conference Series, forecasting changes in time series parameters in digital information vol. 803, no. 1, 012127, 2017. management systems,” Computer Optics, vol. 42, no. 6, pp. 1093- [5] D.C. Montgomery, “Introduction to Statistical Quality Control,” 1100, 2018. DOI: 10.18287/2412-6179-2018-42-6-1093-1100. New York: John Wiley and Sons, 2009. [20] A.I. Maksimov and M.V. Gashnikov, “Adaptive interpolation of [6] T.P. Ryan, “Statistical Methods for Quality Improvement,” New multidimensional signals in differential compression,” Computer York: John Wiley and Sons, 2011. Optics, vol. 42, no. 4, pp. 679-687, 2018. DOI: 10.18287/2412-6179- 2018-42-4-679-687. [7] C. Fuchs and R.S. Kennet, “Multivariate quality control: Theory and Applications,” New York: Marcel Dekker, 1998. [21] Government standards GOST R 50779.41-96 (ISO 7873-93). “Statistical methods. Control charts for arithmetic average with [8] C. Lowry and D.C. Montgomery, “A review of multivariate control warning limits,” Standards publishing house, 1996. charts,” Technometrics, vol. 27, p. 800-810, 1995. VI International Conference on "Information Technology and Nanotechnology" (ITNT-2020) 49