2.2 Case studies
2.2.2 The Tennessee Eastmen process
The Tennessee Eastmen problem is widely applied in the chemical and process engineering practice to test the industrial applicability of the developed process monitoring techniques([8, 29, 30, 32]). This benchmark problem has all the characteristics that an operating chemical process does, so it is suitable to evaluate the performance of the developed time series segmentation methodology.
The Tennessee Eastmen process consist of five major unit operations: a reactor, a product condenser, a vapor-liquid separator, a recycle compressor and a product gas stripper. Two products are produced by two simultaneous gas-liquid exothermic
reactions and a byproduct is generated by two additional exothermic reactions.
The process has 12 manipulated variables, 22 continuous measurements and 19 composition measurements. The process is sampled with the sample time of 0.1h. The simulator of the process was developed by Downs and Vogel in [58].
The control system used for dynamic simulations is decentralized control strategy created by Ricker [59]. The simulator includes a set of programmed disturbances listed in Table A.1 in Appendices. To be able to utilize the dynamic PCA the process variables listed in Table A.2.
To check the performance of the proposed time-series segmentation methodologies (in Algorithm 2.1 and Algorithm 2.2) the following operation scenario is considered with disturbances included: step in A/C feed ratio at 40th hour, random variation in C feed temperature at 60th hour, slow drift in reaction kinetics at80thhour, sticking of condenser cooling water valve at100thhour and an unknown type disturbance at 120th hour (1st,10th,13th,15th and 19th disturbance in Table A.1 in Appendices). This way 6 different segments are expected if the disturbances change the correlation structure of the input-output variables.
Data matrix is constructed as [yTk yTk−1uTk uTk−1] to build the dPCA model . Throughout our segmentation process the first 31 principal components were applied which explain 97% of process variance. 15000 samples (150 hours long) from normal operation data are applied for the analysis and the first 3000 samples are utilized to compute the initial covariance matrix.
Results of the time-series segmentation
Similarly to the previous case, conventional process monitoring indicators - the Hotelling T2 and Q reconstruction error - are applied to detect the disturbance introduced above. Since the first 3000 (30 hours long time scale) samples are applied to initialize the covariance matrix, the value of these indicators is 0 in these sample times as depicted in Figure 2.6.
The confidence limits for the process are determined by using Eq (2.13) and Eq (2.14). Changes in the correlation structure are detectable either in the Hotelling T2 metric or the Q reconstruction error. These metrics cannot be utilized apart from each other: e.g. the random variation in C feed temperature is not detected in Hotelling T2 plot but in the Q reconstruction error plot and e.g. slow drift in reaction kinetics cannot be detected just by observing Q reconstruction error plot since it can be detected in HotellingT2 plot. To follow the adaptation of the dPCA
0 20 40 60 80 100 120 Slow drif in reaction kinetics
Step in A/C feed ratio Random variation in C feed temp.
Figure 2.6: HotellingT2, Q metrics and value of forgetting factor in the considered time scale of TE process
model to the changes in the correlation structure the value of forgetting factor is examined, depicted in Figure 2.6.
As the forgetting factor shows having the most significant necessity for adaptation in the 40th and 60th hour when the step change in A/C feed ratio and random variation in C feed temperature occur. In the rest of the considered scenario the value of the forgetting factor does not change relevantly (it is close to 1). It indicates the correlation structure does not change as significantly as in the previously mentioned cases.
At first the off-line segmentation is evaluated.In the considered scenario the number of expected segments is six assuming that every disturbances change the correlation structure. This expectation should be modified after the examination of Figure 2.6, which hint less than six different operation regimes. The most important question is to determine the number of desired segments. Two cases were examined:
in the first case the number of desired segments is 10, in the second one it is 20.
The results of the bottom-up scenarios are summarized in Figure 2.7. It confirms the expectations based on the examination of Figure 2.6, which means that the most significant change in the correlation structure occurs in the 40th hour. Effect of
0 20 40 60 80 100 120 140 0
5 10
Time (h)
Bottom−up segmentation, desired number of segments=20
Any Process Data
Bottom−up segmentation, desired number of segments=10
Any Process Data
Figure 2.7: Results of different segmentation scenarios of TE process
rest of disturbances is significantly lower, however they can be detected even with using low number assumed segments. The only exception is the unknown type disturbance, occurs in the120thhour, which can not be detected in case of 10 desired number of segments. It is in accordance with small changes of the forgetting factor in that particular time scale, depicted in Figure 2.6. The number of false segment border detection is quite low despite of defining high number of desired segments.
As second the on-line segmentation methodology is applied (Algorithm 2.2).
The result of the time-series segmentation scenario is shown in Figure 2.7. It is quite similar to the bottom-up scenario with 10 desired segments, since just the unknown type disturbance is not detected and every other disturbances are indicated. The closely adjacent segment borders (like around 40th hour) indicates that in that particular time scale the considered system is in a transient state and adapts to the occurred disturbance.
Both of the off-line and on-line methodologies are capable to detect changes in correlation structure and rate their effects. The change of step in A/C feed ratio at 40th hour and random variation in C feed temperature at 60th hour can be easily detected as depicted in Figure 2.6 in the plot of the forgetting
factor. These disturbances cause significant changes in correlation structure. The rest of the disturbances cannot cause such change in the correlation structure (since the forgetting factor does not decrease as significantly as before), but e.g.
detailed investigation of the classical of HotellingT2metric we can highlight these disturbances. The off-line time-series segmentation algorithm is convergent, which means that the most significant disturbances are detected first and the less significant in the end. It is shown in Figure 2.7, in the first and second plot. The segment borders converged to the most significant disturbance as first (step in A/C feed ratio at 40th) and then detect the other disturbances. This convergence is the reason of multiple ("thicker") segment borders. The unknown type disturbance at120thhour has the less significant effect on the correlation structure, hence it is not detected in case when the number of desired segments was 10.
The same statements can be taken by considering the on-line version of the proposed methodology. Besides detecting the changes in the correlation structure, this algorithm is also capable to rank the effect of the disturbances, however it detects the effect of the slow drift in reaction kinetics (occurred in80th hour) later than the off-line version, depicted in Figure 2.7, in the third plot. Similarly to the off-line version it neglects the effect of the unknown type disturbance at120th hour. These capabilities make the on-line version applicable for real time detection and ranking the effect of occurred disturbances, which can support the effective operation of the chemical technologies.