4.3 Fuzzy Cluster based Fuzzy Segmentation
5.1.2 Application example
In this section the proposed algorithms will be applied to the data- and model-based product quality monitoring and control of a polyethylene plant at Tiszai Vegyi Kombin´at (TVK) Ltd., which is the largest Hungarian polyolefine production company. The monitoring of a medium and high-density polyethylene (MDPE, HDPE) plant is considered. For more details see also Example 4.2. The main properties of products of the HDPE (Melt Index (MI) and density) are controlled by the reactor temperature, monomer, comonomer and chain-transfer agent concentrations.
The polymerization unit is controlled by a Honeywell Distributed Control System (DCS), and the relevant process variables are collected and stored by the Honeywell Process History Data-module. The proposed process monitoring tool has been implemented independently from the DCS; the database of the historical process data is stored by a MySQL SQL-server. Most of the measurements are available in every 15 seconds on process variables which consist of input and output variables: the comonomer hexene, the monomer ethylene, the solvent isobutene and the chain transfer agent hydrogen inlet flowrates and temperatures (u1,...,4 = FCin6,C2,C4,H2 and u5,...,8 = TCin6,C2,C4,H2), the flowrate of the catalyst (u9 = Fcatin), and the flowrate, the inlet and the outlet temperatures of the cooling water (u10,...,12=Fwin, Twin, Twout).
The prototype of the proposed process monitoring tool has been implemented in MATLAB with the use of the Database and Kalman filter Toolboxes.
The Model of the Process
The model used in the state-estimation algorithm contains the mass, components and energy bal-ance equations to estimate the mass of the fluid and the formulated polymer in the reactor, the concentrations of the main components (ethylene, hexene, hydrogen and catalyst) and the reactor temperature. Hence, the state-variables of this detailed first-principles model are the mass of the fluid and the polymer in the reactor (x1 = GF and x2 =GP E), the chain transfer agent concentration (x3=cH2), monomer, comonomer and catalyst concentration in the loop reactor (x4=cC2,x5=cC6 and x6 =ccat), and reactor temperature (x7 = TR). Since there are some unknown parameters re-lated to the reaction rates of the different catalysts applied to produce the different products, there are additional state-variables: the reaction rate coefficients x8=kC2,x9=kC6,x10=kH2.
With the use of these state variables the main model equations are formulated as follows:
dGF F(.)means mass rate, c(.)p means the specific heat of the (.) component, and ∆Hi represents the heat of theith reaction.
For the feedback to the filter, measurements are available on the chain transfer agent, monomer and comonomer concentration (y1,2,3=x3,4,5), reactor temperature (y4=x7) and the density of the slurry in the reactor (y5=ρslurry, which is related tox1 andx2). The concentration measurements are available only in every 8 minutes.
The dimensionless state variables are obtained by the normalizing of the variables, xn= x−xmin
xint ,
where xmin is a minimal value and xint is the interval of the variable (based on a priori knowledge, e.g. the operators’ experiences if available). The values of the input and state variables have not been depicted in the figures presented in the next sections because they are secret so not publishable.
Parameters of the Segmentation Algorithms
The results studied in the next sections have been obtained by setting the initial process noise covari-ance matrix toQ=diag(10−4), the measurement noise covariance matrix toR=diag(10−8), and the initial state covariance matrix toP0=diag(10−8). The values of these parameters heavily depends on
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Figure 5.1: Screeplot for determining the proper number of principal components in case of datasets presented in (a) Example 5.1 and (b) Example 5.2, respectively.
the analyzed dataset. That is why the proper normalization method has an influence on the results.
However, the parameters above can be used to estimate the state variables not only the datasets presented in the next sections, but also other datasets that contain data from production of other products in different operation conditions but in the same reactor and produced by the same type of catalyst. In these cases the state estimation algorithm was robust enough related to the parameters above, they can be varied in the range of two orders of magnitude around the values above.
For the segmentation algorithm some parameters have to be chosen in advance, one of them isthe number of principal components. This can be done by the analysis of the eigenvalues of the covariance matrices of some initial segments. This method was used in Chapter 4 in Section 4.3.4. The datasets shown in Figure 5.3 and in Figure 5.4 were initially partitioned into ten segments. As Figure 5.1 illustrates, the cumulative rate of the sum of the eigenvalues shows that five PCs are sufficient to approximate the distribution of the data with 97% accuracy in both cases.
Another important parameter isthe number of segments. Unlike the segmentation method pre-sented in Chapter 4, the number of segments should be defined before the segmentation because the hierarchical clustering applied in this section is not able to determine this value. One of the applicable methods is presented by Vasko et al in [169]. This method is based on permutation test so as to determine whether the increase of the model accuracy with the increase of the number of segments is due to the underlying structure of the data or due to the noise. In this section the simplified version of this method has been used. It is based on the relative reduction of the modelling error (see (4.1) and (5.1)):
RR(c|T) = cost(STc−1)−cost(STc)
cost(STc−1) (5.14)
whereRR(c|T) is the relative reduction of error whencsegments are used instead of c−1 segments.
As it can be seen in Figure 5.2, significant reductions are not achieved by using more than 5 or 6 segments in case of both datasets. Similar figures can be obtained by Algorithm 2.
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Figure 5.2: Determining the number of segments by Algorithm 3in case of datasets presented in (a) Example 5.1 and (b) Example 5.2, respectively.
Example 5.1 (Monitoring of process transitions). In this study a set of historical process data covered 100 hours period of operation has been analyzed. These datasets include at least three segments because of a product transition around the45th hour (see Figure 5.3). Based on the relative reduction of error in Figure 5.2 (a), the algorithm searched for five segments (c= 5).
The results depicted in Figure 5.3 show that the most reasonable segmentation has been obtained based on the covariance matrices of state estimation algorithm (Algorithm 3). The segmentation obtained based on the estimated state variables is similar: the boundaries of the segment that contains the transition around the 45th hour are nearly the same, and the other segments contain parts of the analyzed dataset with similar properties. Contrary to these nice results, when only the measured input-output data were used for the segmentation the algorithm was not able to detect even the process transition.
It has to be noted that Algorithm 3 can be found more reasonable than Algorithm 2, because one additional parameter has to be chosen in the last case: the forgetting factor, α in the recursive estimation of the covariance matrices in (5.2). The result obtained byAlgorithm 2is very sensitive to its choice. The α= 0.95seemed to be a good trade-off between robustness and flexibility.
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Figure 5.3: a., b.: Segmentation basedAlgorithm 1; c., d.: Segmentation based onAlgorithm 2,; e., f.: Segmentation based onAlgorithm 3; a., c., e.: Input variables: Fin, Fin, Fin, Fin, Fin, Tin, Tout;
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Example 5.2 (Detection of changes in the catalyst productivity). Beside the analysis of the process transitions, the time-series of ”stable” operations have also been segmented to detect interesting patterns of relatively homogeneous data. For this purposeAlgorithm 3was chosen from the methods presented above, because it gives good results in case of product changes. One of these results can be seen in Figure 5.4, which shows a 120-hour long production period without any product changes.
Based on the relative reduction of error in Figure 5.2 (b), the number of segments was chosen to be equal to six (c= 6).
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Time (hour)
Ethylene and isobutane
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Time (hour)
Hexene
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Time (hour)
Hydrogen and catalyst
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Time (hour)
Temperature of cooling water
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Time (hour)
Reactor temperature
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Time (hour)
Concentrations
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Time (hour)
Density of slurry
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Time (hour)
Reaction rates
Figure 5.4: Segmentation based on the error covariance matrices.
The homogeneity of a historical process data set can be characterized by the similarity of the segments that can be illustrated as a dendrogram (see Figure 5.5).
1 5 2 4 6 3
0 1 2 3 4
x 10−4
Level
Figure 5.5: Similarity of the found segments.
This dendrogram and the border of the segments give a chance to analyze and to understand the hidden processes of complex systems. E.g. in this example these results confirm that the quality of the catalyst has an important influence in productivity. During the 20, 47, 75, 90th hours of the
presented period of operation changes between the catalyst feeder bins happened. The segmentation algorithm based on the estimated state variables was able to detect these changes that had an effect to the catalysis productivity, but when only the input-output variables were used segments without any useful information were detected.
It has to be noted that the borders of the segments given byAlgorithm 2 andAlgorithm 3are similar also in this case, but the dendrograms are different. This is because that the segments without product transition are much more similar to each other than in case of the time-series which contains a product transition. So it is a more difficult problem to differentiate segments of operations related to the minor changes of the technology, like the changes of the catalyst productivity. This phenomena can also be seen in the dendrogram: the values that belong to the axis of ordinates are smaller with one or two order(s) of magnitude in case of a time-series without product transition. In case of product transition not only the borders of the segments are similar but also the shape of the dendrograms are nearly the same. This shows that both algorithms are applicable for similar purposes.
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5.1.3 Conclusions
This section presented the synergistic combination of state-estimation and advanced statistical tools for the analysis of multivariate historical process data. The key idea of the presented segmentation algorithm is to detect changes in the correlation among the state-variables based on theira posteriori covariance matrices estimated by a state-estimation algorithm. PCA similarity factor can be used to analyze these covariance matrices. Although the developed algorithm can be applied to any state-estimation algorithms, the performance of the filter has huge effect on the segmentation. The applied DD2 filter has been proven to be accurate, and it was straightforward to include a varying number of parameters in the state vector for simultaneous state and parameter estimation, which was really useful for the analysis of the reaction kinetic parameters during process transitions. The application example showed the benefits of the incorporation of state estimation tools into segmentation algorithms.