Process Control and Instrumentation 2000 (PCI2000), 26-28 July 2000, Glasgow
DETECTION AND DIAGNOSIS OF UNIT-WIDE OSCILLATIONS *
N.F. Thornhill , S.L. Shah+ and B. Huang+ * Department of Electronic and Electrical Engineering, University College London, Torrington Place, London WC1E 7JE, UK +
Corresponding author. E-mail:
[email protected]
Department of Chemical and Materials Engineering, University of Alberta, Edmonton, Canada T6G 2G6
ABSTRACT This article addresses the detection and diagnosis of oscillations in process measurements. It uses the power spectra of the measurements because the spectra are insensitive to the phase lags caused by time delays and process dynamics. Principal components analysis (PCA) can be applied to the power spectra for detection of groups of oscillatory process measurements. In cases where a well defined periodic oscillation is detected then diagnosis of the root cause is possible using a measure of harmonic distortion. Examples using this measure are discussed and its suitability for diagnosis assessed. INTRODUCTION It is important to detect and to diagnose the causes of oscillations in process operation because a plant running close to a product quality limit is more profitable than a plant that has to back away because of variations in the product (Martin et al, 1992). This paper addresses the detection and diagnosis of oscillation in cases where the oscillation appears in several process variables. The work reported here fits into a broader context of disturbance detection and feature diagnosis (Love and Simaan, 1988; Bakshi and Stephanopoulos, 1994; Davis et.al., 1996). Oscillations are one special class of disturbance. Hägglund (1995) described a method for the on-line detection of oscillation within a control loop and other authors have also considered the problem (Rengaswamy and Venkatasubramanian, 1995; Thornhill and Hägglund, 1997). Harris et.al (1996) reported plant-wide control loop assessment in which spectral analysis was useful. An issue is to determine the root cause of a plant-wide oscillation. It is not easy to determine cause and effect especially when recycles are present or when physical influences propagate in the opposite direction to process flows, for instance when oscillations in a downstream flow control loop cause variations in level of a feed tank. Our work uses spectral signatures. It exploits similarities between the spectra of signals in the plant to determine which process variables have similar oscillations. It also exploits differences between the spectra, specifically differences in the harmonics, to judge which is the root cause of the plant-wide oscillation. Three case studies are presented. One was from a refinery unit where the cause of oscillation was known. Another was from pilot plant to which a deliberate periodic disturbance was applied. There were also known non-linearities in the valve characteristics for this plant. In both cases the distortion factor pointed to the correct root cause. The third was from a SE Asian refinery involving thirty seven measurements. In that case clusters were observed in a spectral principal components analysis which highlighted groups of variables having similar oscillatory behaviour.
METHODS Detection of unit-wide oscillations: When a unit-wide oscillations is suspected (using Hägglund, 1995, for example) then the spectra of the measurements involved may be compared to one another. A visual comparison is adequate for small numbers of process measurements, but for larger data sets the method of spectral principal components analysis (Tabe et.al., 1998) may be applied. If the oscillations are related then one principal component would describe most of the variability in the spectra. In spectral PCA the columns of the data matrix, X, are the power spectra P( f ) of the signals over a range of frequencies up to one half the sample frequency. Smoothed single-sided power spectra may be calculated using the averaged periodogram method (Welch, 1967), or the fast Fourier transform may be used, for instance using MATLAB (The Mathworks, Natick, MA) m process variables → P1 ( f1 ) ... Pm ( f1 ) N frequency X = .. .. .. channels P ( f ) ... P ( f ) ↓ m N 1 N A full PCA decomposition reconstructs the X matrix as a sum over m orthonormal basis functions t 1 to t m which are spectrum-like functions having N frequency channels arranged as a column vector: X = t 1 ( w1,1 ... w1,m ) + t 2 ( w2,1 ... w2,m ) + ... + t m ( wm,1 ... wm,m ) A description of the majority of the variation in X can often be achieved by truncating the PCA description. If all the variables had similar spectral features then one term would describe most or all of the variability in the spectra. In other cases more terms may be needed. The following is a three-term PCA model in which the variation of X that is not captured by the three terms appears in an error matrix E: X = t 1 ( w1,1 ... w1,m ) + t 2 ( w2,1 ... w2,m ) + t3 ( w3,1 ... w3,m ) + E Each spectrum in X may be represented graphically. When three t-vectors are in use the i’th spectrum maps to a point having the co-ordinates w1,i , w 2,i and w 3,i in a threedimensional space called a weightings plot. Similar spectra have similar w-coordinates. Therefore such groups form clusters. Diagnosis of the root cause: There are several possible causes of oscillation. One common cause is the presence of non-linearity such as a valve dead-band or a faulty instrument which sets up a limit cycle in a control loop. The oscillation caused by the faulty valve is liable to act as a disturbance elsewhere in the unit. Limit cycles are often non-sinusoidal and therefore have harmonics at multiples of the fundamental frequency. If unit-wide oscillation is due to a limit cycle then a good candidate for the root cause is the process variable with the richest set of harmonics. The reason is that the dynamic behaviour of the rest of the unit generally has low-pass characteristics and
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smooths the signal, thus removing harmonics. This work used a calculation of a distortion factor to aid diagnosis of oscillations cause by limit cycles. The distortion factor, D, is: D=
Ptot − Pfund Ptot
where Ptot is the total power in the spectrum and Pfund is the power in frequency channels k to k + A occupied by the fundamental harmonic: N
Ptot = ∑ P( f i )
and
i =1
Pfund =
k +A
∑ P ( fi )
i =k
The calculation of D requires an inspection of the spectrum to determine a suitable range k to k + A . In this work the spectra were assessed by hand. The task is not onerous in cases where a cluster of process variables having a common oscillation has been detected because the values of k and k + A need be determined only once and can then be used for all the spectra in that cluster. D cannot be determined in cases with no welldefined oscillation and no spectral peak. The process variable having the highest distortion factor has relatively more power in the harmonics. It is thus the main candidate for the root cause and is the one to which maintenance effort should be directed first. CASE STUDIES Refinery diagnosis example: time trends
TC FC steam
temperature
AC
analyser
air
spectra
steam flow
off gas
0
2000 time/s
4000
0.001
0.01 frequency/Hz
Figure 1. Refinery unit with oscillatory process measurements
Figure 1 shows schematic for a refinery unit with oscillatory process measurements (courtesy BP Amoco Oil). The time trends and spectra of the oscillating steam flow, temperature and analyser variables are also shown. It was known that the steam flow control loop had a faulty instrument at the time when the data were collected. The three spectra had similar peaks at the frequency of oscillation and one principal component explained 94% of the spectral variation thus confirming the visual observation that the process variables had a common oscillatory feature. The feature to the right of the main peak in the steam flow spectrum is a harmonic because it appears at double the frequency of the main peak. Distortion factors were
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determined for the three variables as follows:- Steam flow: D = 0.281 ; Temperature: D = 0.011 ; Analyser: D = 0.007 . The distortion factor was highest for the steam flow variable. Thus the distortion factor suggests the steam flow control loop was the root cause, which matches with what the refinery control engineer knew about this problem. The analyser is further from the root cause than the temperature and it had the smallest distortion factor. Therefore it is concluded that the dynamics of the unit cleaned the oscillation as it moved further from its source. Pilot plant example: time trend
log of power spectrum HW Valve
FT hot water
TC
level
FC
temperature
FT
steam
cold water
CW flow
flow sp
LC
CW flow set point LT
CW valve
TT
steam valve 0
Figure 2. Pilot plant schematic
200 time/s
400
0 0.2 0.4 frequency/Hz
Figure 3. Time trends and spectra
Figure 2 shows a schematic of pilot plant in the Computer Process Control group of the Department of Chemical and Materials Engineering, University of Alberta. The plant was run with a disturbance to the hot water flow, which was switched periodically on and off. Figure 3 shows the time trends and spectra of the measured variables, while Figure 4 shows a signal flow for the plant annotated with the distortion factors. The spectra in figure 3 show that several of the measurements had harmonics, but that the root cause disturbance (the hot water flow) has the most pronounced harmonic content. Its distortion factor was also largest at 0.43. Figure 4 shows that the distortion factor generally reduced as the influence of the disturbance propagated through the plant. An exception, however, was the cold water flow which had a higher D value than the cold water valve that precedes it in the signal flow graph. The explanation for this finding is that there was a second source of nonlinearity in the plant because the characteristic of the cold water valve in flow control loop was not linear. This result therefore illustrates an additional benefit of the Dmeasure; it can pinpoint sources of non-linearity in a plant. level HW flow
CW set pt 0.094
0.43 temperature
0.088
CW valve
CW flow
0.067
0.074
0.220
0.228
steam valve
steam flow (not measured)
Figure 4. Signal flow graph for pilot plant, with distortion factors
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Plant-wide oscillation detection: The final example shows the results of PCA analysis of 37 process measurements from a SE Asian refinery. The schematic is not available for reasons of confidentiality, therefore this example shows the ability of PCA in detection of clusters of oscillating variables but without any diagnosis. Figure 5 shows the three dimensional weightings plot for spectral PCA analysis of the measurements. Three principal components explained 90% of the variability of the spectra, and some clusters were observed. For instance, the white squares were characterised by positive values of w1 and hence were described by the t 1 vector that had a sharp spectral peak at 0.06 min −1 (an oscillation period of 16.7 min ). By contrast, a three term time-domain PCA explained only 40% of the variability in the data set. The reason for the enhanced performance of spectral PCA is that it is insensitive to the time delays and phase lags caused by process dynamics that are well known to cause problems with time-domain PCA (e.g. Ku et.al., 1995). Therefore it is concluded that spectral PCA offers advantages for detection of plant wide oscillations.
0.25 t 1 t 2 t
w
3
3
-0.25 1.0
0.001 w
w
2
0
0
1.0
0.01
0.1
frequency/min
1.0 -1
1
Figure 5. results of spectral PCA on refinery data CONCLUSIONS This paper has outlined approaches to the detection and diagnosis of unit-wide oscillations. The use of spectral PCA has proved to be a reliable and robust means of detecting groups of process variables with similar oscillatory features. In cases of oscillations caused by limit cycling the distortion factor was a useful statistic. The distortion factor was also sensitive to a non-linear valve characteristic. It is noted, however, that the calculation of distortion factor required manual intervention. First it was necessary to determine that a distinct oscillation was present since D is not a reliable statistic when the signal to noise ratio is low. Also the range of frequency channels that defined the fundamental frequency had to be selected manually. Therefore the statistic shows promise but needs refinement. These results motivate further work towards the automation of high level diagnosis for cause and effect reasoning, for example using signed digraphs as proposed by Vedam and Venkatasubramanian (1999). ACKNOWLEDGEMENTS Nina Thornhill gratefully acknowledges financial support from NSERC and support from members of the CPC group of the University of Alberta for this collaborative study. The authors would like to thank M.A. Shoukat Choudhury and R. Bushan Gopaluni for help with conducting the experimental runs for the study.
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