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Distributed model projection based transition processes recognition and quality-related fault detection
Author’s Accepted Manuscript Distributed Model Projection based Transition Processes Recognition and Quality-related Fault Detection Yuchen He, Le Zhou, Zhiqiang Ge, Zhihuan Song www.elsevier.com
PII: DOI: Reference:
S0169-7439(16)30356-2 http://dx.doi.org/10.1016/j.chemolab.2016.10.001 CHEMOM3323
To appear in: Chemometrics and Intelligent Laboratory Systems Received date: 4 December 2015 Revised date: 11 September 2016 Accepted date: 4 October 2016 Cite this article as: Yuchen He, Le Zhou, Zhiqiang Ge and Zhihuan Song, Distributed Model Projection based Transition Processes Recognition and Quality-related Fault Detection, Chemometrics and Intelligent Laboratory Systems, http://dx.doi.org/10.1016/j.chemolab.2016.10.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Distributed Model Projection based Transition Processes Recognition and Quality-related Fault Detection Yuchen He1, Le Zhou2, Zhiqiang Ge1, Zhihuan Song1 1
State Key Laboratory of Industrial Control Technology, College of Control Science and Engineering, Zhejiang University,
Hangzhou 310027, Zhejiang, China 2
School of Automation and Electrical Engineering, Zhejiang University of Science and Technology,
Hangzhou 310023, Zhejiang, China
Abstract In this paper, a novel transition process identification algorithm based on distributed model projection (DMP) is proposed for clustering nonlinear transition data and monitoring the variations in the transition process. Compared to several alternative identification methods, the DMP algorithm considers both the correlations between variables and correlations between samples. Also, a framework is proposed to combine DMP algorithm and hierarchical clustering to derive an optimal clustering results through a large amount of individual trials of the DMP algorithm. Based on the offline classification results, a transition process is divided into several sub-segments and each of them can be characterized by a stable model. Then the online identification and monitoring methods are carried out based on the sub-models established in those segments. Finally, the Tennessee Eastman (TE) benchmark process is utilized to demonstrate the performance of the proposed process identification and monitoring strategy. Compared to previous works, the proposed algorithm is shown to be superior both in identification and monitoring.
Keywords: Transition process; Distributed model projection; Hierarchical clustering; Transition identification; Fault detection
To whom all correspondence should be addressed. Email: [email protected] Tel: +86-571-87951442
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1. Introduction Recently, data-based multivariate calibration methods have been widely used in modern industries for the purpose of process monitoring and diagnosis [1, 2]. Among them, the classical multivariate statistical process control (MSPC) algorithms have achieved great successes in the last few decades, such as principal component analysis (PCA), partial least squares (PLS), principle component regression (PCR), kernel partial least squares (KPLS) and their advanced variants [3-7]. However, most of these methods are assumed to run in a single and stable operating condition. In most cases, the real industrial processes are operated in different conditions due to fluctuations in raw materials, set-point changes, aging of equipment and seasoning effects. In order to solve this problem, many methods have been proposed for the purpose of multimode modeling and process monitoring [1, 8-13]. Particularly, Zhu et al. [14] proposed a clustering method for multimode recognition in which an ensemble clustering algorithm is employed. However, the performance of ensemble clustering depends on the proper choice of the initial value of the clustering method. Zhang et al. [12]proposed a subspace separation method where a common subspace and a specific space are used to describe the multimode characteristics. Practically, it is difficult to construct such two subspaces explicitly due to the complexity of multimode processes. It is also noticed that a stable operation status cannot be immediately switched to another one, indicating that there are transitions between two adjacent stable modes. These unstable processes should also be considered for multimode modeling and monitoring. It is always a challenging and sophisticated issue using traditional methods. For the purposes of transition identification and monitoring, Zhao et al.[15, 16], Ge et al.[17], Wang et al.[18], Tan et al.[19] and Yao et al.[20] proposed different algorithms for stable and transient process identification as well as multimode monitoring in recent years. In the above methods, transitions should be identified from multimode process in the offline step so that calibration models can be -2-
established using proper training transient data. In Zhao’s method[15], the duration of a transient process was defined from the center of a stable mode to the center of its neighboring mode so that the transition process is able to be described using adequate observations. The transition process recognition method was set manually instead of being identified based on the process data. It is quite unreasonable that stable modes would disappear and the whole process only consists of several transitions according to such definition. Process information was taken into account in Wang’s method where a complete transition process recognition and modeling method was proposed. In Wang’s methods[18], a transient process was separated into several sub-segments by the k-means method and each of them was characterized using linear models. Unfortunately, the Euclidean distance between different samples cannot fully reveal the variations and trends of complicated chemical process [21, 22]. In the past few decades, the MSPC methods have proved itself in chemical process modeling. Hence, Yao et al. [20]used a series of different PCA models to describe the behavior of a chemical process. A single PCA model was created for each sub-segment and the PCA similarity index was adopted to analyze the difference between each sub-segments. It indicates that the unstable nonlinear transition process is divided into several approximately stable linear processes where the classical MSPC methods can be employed. However, the assumption that variable relationships in each sub-model are linear correlated is still invalid in practice. In order to solve the above problems, we propose a novel algorithm called distributed model projection (DMP) in this paper. In the DMP algorithm, the transition process is recognized by an iterative method. Since the beginning part of a transition is very different from the ending part, it is still assumed that a transition can be divided into several sub-models, in which it can be treated as a stable process. The amount of clusters can be properly determined by the subtractive clustering algorithm (SCM). Each of these clusters is supposed to be a stable mode or a transition. Also, nonlinear relationship between process -3-
data and quality data is an important issue in chemical processes. For this purpose, the DMP algorithm is proposed by using the KPLS model to describe the behavior of a process. The algorithm starts with several initial models which are established by the randomly initialized data in each cluster. In order to capture the characteristics of a process accurately and avoid the noises caused by the single observation, the classical moving window strategy is adopted to create several overlapping windows across the whole process. According to the theory of principle-components-similarity[23], two data blocks with similar principle components subspaces may have similar statistical characteristics to each other. In other words, two data blocks with small prediction errors are considered to be similar to each other when these two blocks are projected to the same model. Hence, each moving window in a multimode process is projected to the initial models to calculate corresponding prediction errors. A window having the smallest error with respect to a model is assigned to the corresponding cluster. In such way, a transition process is initially recognized and divided into several segments (sub-models). On the basis of the current models, the windows in each cluster are statistically similar to each other. However, it is noticed that the initial models are established based on the randomly initialized data. Hence, these models should then be updated using all available samples within each cluster. After the new models are derived, all the windows are projected to these models to determine the new assignment of each window. And the iteration will continue until no further variation is found in the clustering results. In each single cluster, statistically similar windows are grouped together and dissimilar windows are assigned to different clusters. The clustering results of the DMP algorithm indicate the similarity between different moving windows. Besides, the DMP algorithm is considered as a nonhierarchical clustering method which may converge to a local optimal solution [21, 22]. It means that the clustering result in one trial by using DMP algorithm -4-
may be different from the result in another one. Compared to nonhierarchical clustering methods, global optimal solutions can be derived using hierarchical clustering methods. In order to obtain global optimal clustering results, a hierarchical clustering method is adopted by combining the clustering results of a large amount of DMP trials. The similarity information derived by the DMP algorithm can be the input of any common hierarchical method. After the transition process is recognized, the corresponding fault detection methods can be derived. For most multimode process monitoring methods, it only involves the process data where the abnormal situations of process can be monitored while the variations of the final product quality cannot be detected. When the quality data can be measured, the relationships between the process variables and the quality variables should also be included to enhance the ability of process and quality monitoring. In chemometric region, some supervised methods have been proposed for single and multimode process modeling [11, 24]. However, little attention has been paid on the quality-related transition process monitoring. Hence, a quality-related transition fault detection method is also studied in this paper. The rest of the paper is organized as follows: a brief description of transition process is given in Section 2. In Section 3, offline transition recognition by using the DMP algorithm and hierarchical clustering explanation are introduced explicitly. Then an online identification and quality-related process monitoring method is introduced in Section 4. A TE benchmark process is demonstrated in Section 5 to evaluate the performance of the proposed method. Finally, some conclusions are made.
2. Transition process description and analysis Differing from stable modes, a transition process is a trend from one stable mode to another. Hence, the statistical characteristics of a transition cannot be well depicted using the traditional MSPC method -5-
because of intricate relationships between variables. At the beginning of each transition, the statistical characteristic is similar to the previous stable mode. At the end, it is accordingly similar to the following stable mode. Therefore, it is difficult to handle transition modeling and monitoring issues using classical stable mode methods. Furthermore, considering the existence of control loops in an industrial process, ‘irregularity’ sometimes occurs in a transition [15, 16, 25]. It means that the present sample may contain similar characteristics with previous samples several sampling intervals before. Considering the statistical characteristics of transitions, several recognition and modeling steps are used in this paper. Firstly, the observations are divided into several clusters utilizing the DMP algorithm, where a stable process can be demonstrated using a single cluster while the transition process contains several smaller sub-models, so that the traditional MSPC methods can be employed in each cluster. After that, the corresponding process monitoring scheme can be established in each stable and transient mode. A new sample needs to be classified into a certain cluster once it is measured. Then the sample can be monitored using the corresponding fault detection algorithm. In the next two sections, the offline recognition and online identification algorithm will be introduced in detail.
3. Offline multimode identification and transition recognition A multimode process usually consists of several stable processes and transitions. Commonly, different stable processes and transitions should be identified before online monitoring. The conventional clustering methods were mainly carried out based on the Euclidean distance. Hence, the relationships between different samples were emphasized and two samples with small Euclidean distance between them were considered to belong to one cluster [18, 22]. It might work under the assumption that the relationships between different variables are simple. Considering the complexity in the process control, such assumption -6-
is often invalid in continuous chemical process. Usually, chemical processes contain variations, especially peaks and valleys, and it is therefore difficult to illustrate the behavior of a process merely by traditional Euclidean-distance-based clustering methods. For the purpose of process control and fault detection, a growing number of successes have achieved by the MSPC methods, such as PCA and PLS. It is more appropriate to introduce MSPC methods into process analysis and multimode identification. Inspired by the principle-components-similarity analysis[23], two data blocks are considered to be similar to each other when both residuals are very small with respect to the same model. In fact, a single mode can be interpreted as a group of samples with similar characteristics which can be well reveal by one single MSPC model. By calculating the corresponding model residuals, a process is divided into several segments each of which can be represented through a MSPC model. In the DMP algorithm, the similarity between windows is interpreted as a kind of model fitness rather than Euclidean distance. Besides, the DMP algorithm, like other nonhierarchical methods, will probably converge to a local optimal solution. Therefore, a hierarchical clustering manner using the results of DMP algorithm will be adopted in sub-Section 3.3 to give a global optimal clustering result. Since nonlinearity is a kind of common properties for most industrial processes, the classical KPLS model is employed for DMP algorithm considering the nonlinear cross-correlations between variables while it can also be readily used for quality-related fault detection. Besides, a classical moving window strategy is used for model projection to capture a time period of observation. This will help understand and analyze the behavior of a process through time segments rather than individual samples. The determination of window length L and window moving step M is a significant issue and has been discussed explicitly by previous works [14]. For more details about the selection of moving window parameter, please refer to Palazoglu’s work [14, 26]. Firstly, the KPLS model is introduced firstly in the next sub-section to give a better understanding about the DMP algorithm. -7-
3.1 Preliminaries 3.1.1 KPLS modeling In PLS, the relationships between the process and quality variables are extracted. Since the correlations between them are nonlinear in most chemicals, a kernel PLS model is employed in this paper, which is given as:
TPT E Y TQT F where x1 x2 T
mapping : xi
M
T
T
N S
( 1)
is the kernel form of process data X ,
indicates a nonlinear
Transformation (xi ) F . The quality prediction on testing data can be calculated
as follows:
ˆ = K U(TT KU)1 TT Y Y t t
( 2)
where K t and K are the kernel matrices of testing and training data, respectively. T=[t1 t 2 ...t A ] and
U = [u1 u2 ...u A ] are score matrices for process and quality data in the feature space of training data, respectively. A represents the number of principle components. Y is the training quality data. The prediction error of testing quality data Yt ( Nt q) with N t samples and q variables can be calculated as follows:
ˆ et Yt Y t
( 3)
For more details for KPLS modeling, please refer to the previous works [27-29]. Suppose that the sum of squares of residuals corresponding to the prediction error is Et
Nt
p
e i 1 j 1
2 ij ,t
where eij ,t represents an
element in the residual matrix et . From eq.(, it can be seen that the testing data can be well described by a certain KPLS model if its prediction residuals are small enough.
3.2 DMP algorithm -8-
In this sub-section, the DMP algorithm is introduced in detail. The DMP algorithm is implemented in an iterative way. In each iterative run, the overlapping moving windows in the whole process are projected to a series of KPLS models. The windows are assigned to the right cluster according to the corresponding residuals. In the next run, new KPLS models are established by all samples in each cluster. The windows are then reassigned to each cluster by calculating the residuals with respect to the new models. When no further variation appears in the clustering result, the iteration procedure of this trial stops. Commonly, it is significant to choose the initial models for the DMP algorithm, which typically starts with a pre-defined number of clusters. However, the amount of clusters is always unknown in industrial processes. In this paper, a SCM is introduced to estimate the amount of clusters [30]. Since the beginning part of a transition is very different from the ending part, a continuous transient process is approximately divided into several small sub-segments each of which is randomly initialized to create the initial KPLS models. After the parameter initialization is accomplished, the initial KPLS models are employed to classify moving windows into different clusters. It can be easily understood that a model will fit data well when the prediction errors of these data for the particular model are small enough. In this paper, the moving windows in the process are projected into those initial KPLS models to determine which cluster they are assigned to. Hence, the proposed algorithm is named as distributed model projection. After the projection is accomplished in this single run, these windows are classified into different clusters. In each cluster, windows are similar to each other. In order to prepare the next window projection, the KPLS models should also be updated based on the current samples in each cluster. Therefore, the new KPLS models are derived and the projection will continue until no further variation exists in the clustering result. For summary, the schematic of the DMP algorithm is shown in Figure 1 while the clustering steps using DMP algorithm are -9-
listed in Table 1. [Figure 1 about here] [Table 1 about here]
3.3 Hierarchical aggregation for the DMP algorithm As a nonhierarchical clustering method, the iteration using the DMP algorithm will converge rapidly to a local optimal solution [21, 22]. A hierarchical manner is therefore adopted by aggregating the clustering results derived by a large amount of individual DMP trials. The clustering result using the DMP algorithm in the bth trial is denoted as Gb (b=1,…,B). Gb is an Nb×C binary matrix where Nb and C are the number of windows and the number of clusters, respectively. in which
g b represents an element at ( nw, c ) of Gb,
g b is set to 1 if the nw th window is assigned to the c th cluster in the final clustering result of
the bth DMP trial. Each element of Gb can be expressed as follows:
1 gb (nw, c) 0
if window nw is assigned to cluster c in trial b otherwise
( 4)
A hierarchical aggregation is implemented by concatenating the clustering results of all B individual DMP trials:
Gall [G1 , G2 ,..., GB ]
( 5)
Then, an N×N similarity matrix can be obtained as follows
R 1
1 T Gall Gall B
( 6)
where 1 is a matrix consisted of ones. Rwz, which is an element of R, represents the dissimilarity between the window w and the window z. It is noticed that the dissimilarity matrix can be the input to any common hierarchical explanation to give a graphic dendrogram, which indicates the global optimal clustering results. -10-
For a single sample, it may belong to different clusters due to the utilization of the moving window. Hence, the samples at the edge of each cluster should be classified more clearly. The edges between different clusters are determined based on the concept of probability. It is assumed that ni ,l is the number of the times that the lth sample is assigned to the ith(i=1,…,C) cluster. The probability can be estimated as follows:
pi ,l
ni ,l
( 7)
ni ,l ni 1,l
If pi ,l is bigger than pi 1,l , it can be thought that the lth sample belongs to the ith cluster.
3.4 Multimode identification and transition recognition After the global optimal clustering results are obtained, it still needs to be clear whether each cluster should belong to a stable process or a part of transition process. So the statistical characteristics of transitions are analyzed. Different from stable modes, the features of transitions can be summarized as follows: 1) Durations of transitions are much smaller compared to stable modes; 2) A transition only occurs between two different stable modes; 3) ‘Irregularity’ happens within transitions sometimes; 4) The states of transitions may remain stable among small neighboring sample points. For these reasons, some parameters are discussed as follows for distinguishing stable processes and transitions. Since the durations of a transition is much shorter than that in stable modes,
is defined as
the minimal length of one stable mode[18]. A segment is considered as a stable mode if its length is larger than . Otherwise, it may belong to a transition and cannot well describe the characteristics of a stable mode. If a sub-model is identified between two same stable models, it is considered that such mode is caused -11-
by system noises instead of a transient mode. Besides, the situation that two separate segments in one transition share the same model is considered normal. The latter segment is treated as a kind of ‘irregularity’ [25, 31]. If the length of a transient segment is smaller than
which is defined as the least duration of the
transient segment[18], then the segment is treated as noise and should be deleted. Note that the duration of a transient sub-segment is much different from the duration of a stable mode or noise. Therefore, it is very convenient to select parameters
and . For better understanding about the offline transition
identification, a simple example is presented in Figure 2. [Figure 2 about here]
In Figure 2, a process containing two stable processes and a transition is divided into six clusters with ten segments. The segment whose length is larger than
is considered as a stable process. Therefore,
Segment 1, 8 and 10 are clustered as stable processes. The other segments may belong to the transition process. Meanwhile, segment 4 whose length is smaller than
is considered as noise which should be
deleted. Segment 5 and segment 7 share the same model. The ‘irregularity’ can be clearly seen in segment 7. Segment 8 and segment 10 belong to the same cluster. Therefore, segment 9 is not a transient segment and should be deleted. After the analysis of offline transition identification, the identification results are modified and shown in Figure 3. That is, segment i and segment vii belong to the stable modes. Meanwhile, segments ii-vi belong to a transient process. [Figure 3 about here]
In the above case, the whole process can be divided into 6 different clusters. Each of these clusters has its corresponding KPLS model, which reveals a different kind of data correlation. The flowchart of -12-
multimode offline identification is shown in Figure 4. [Figure 4 about here]
4. Online identification and fault detection for multimode processes After transitions are identified using offline DMP algorithm, the quality-related fault detection algorithm can also be made. Firstly, an online sample identification is discussed. Usually, the system state stays in the stable modes and a transition starts after a stable mode. The current online sample is denoted by the hth sample. Firstly, it should be clear that which stable mode the hth sample starts with [18]. It is assumed that there are S all different stable modes, and each of them is depicted by a KPLS model. In order to monitor the online samples, Hotelling’s T in this paper. The statistics T
2
2
statistics and SPE statistics are employed
can be defined as follows:
T 2 tt 1tt where t t is the corresponding score vector of test data and
( 8)
1 T T T represents the covariance of N
score matrix T in the feature space of training data. The statistics SPE for each sample can be defined as follows[32] :
SPE et etT
The confidence limits for T
2
( 9)
and SPE can be further defined as:
A( N 2 1) T ~ FA, N A, N ( N A)
( 10)
SPE ~ 2
( 11)
2
where is the significance level. and are the magnitude and the freedom degree of SPE statistic, respectively. These two parameters can be approximately represented by -13-
b / 2a and
2a 2 / b in which a and b are the estimated mean and variance, respectively [32]. The main steps of multimode processes online identification can be summarized as follows: 1) Determine which stable mode the h-1th sample belongs to using the control limits of all S all stable 2
modes. If T and SPE are less than the corresponding control limits of stable model i while exceed the control limits of stable model j ( j i ) , the h-1th online sample is considered as a 2
normal one and it belongs to S i . The procedure will go to step 2. If T and SPE go beyond all corresponding control limits of possible model, it is considered as a faulty sample and an alarm occurs; 2) Suppose the h-1th sample belongs to cluster i , monitor the hth sample using the control limits of S i . If T
2
and SPE are under the control limits of model i , the hth sample is still considered as a
sample of Si and the next online sample will be monitored using the same model. Otherwise, the hth sample may belong to a transition or a kind of fault and it goes to step 3; 3) Monitor the hth sample using the successive transient sub-model. According to practical conditions of industrial production, the process switching trajectory is always fixed to produce standardized materials. If a process operation state changes, it must switch to the beginning of a specific transient process instead of other sub-models. Hence, the hth sample can be considered as a faulty one if its statistics for subsequent sub-model still exceed the corresponding control limits. Otherwise, the hth sample is thought to belong to this transition and the rest online sample will be identified in the same manner. The flowchart of online identification and monitoring for transition sample is shown in Figure 5. [Figure 5 about here]
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5. Case study 5.1 TE process description and simulation design The Tennessee Eastman (TE) Industrial Process is widely used to demonstrate various monitoring methods, which mainly consists of five work units. In a TE process shown in Figure 6, 41 measurements and 12 manipulated variables are collected. For more detailed settings of TE benchmark, please refer to Chiang’s book[33]. In order to demonstrate the superiority of the proposed algorithm for transition offline identification, a case with six different stable modes is designed. Hence, there are five transitions among these stable modes. 60000 samples are collected in this case for offline transition identification. In the previous paper, 33 or 16 variables are often chosen for process modeling and monitoring method validation. The main purpose of these selections is to emphasize the data inner-relationship while ignoring the process structures. In this case, all quality variables in the stream 6 are selected in order to show the variations on the output variables in a single stream. The process variables with small variations are not selected in order to reduce the computational complexity both in online and offline steps. Hence, 28 selected process variables are employed. The dimension of the system is 60000×34 with 28 process variables and 6 quality variables which are shown in Table 2 and Table 3, respectively. The detailed trajectories of process variables and quality variables are shown in Figure 7. [Figure 6 about here] [Table 2 about here] [Table 3 about here] [Figure 7 about here]
5.2 Offline transition identification -15-
The proposed DMP algorithm is applied to offline identification in this sub-section. In order to give a better understanding about the edge determination, moving window strategy is employed and the window parameters should be derived. In this case study, the window parameters L=19, M=7 are chosen according to Zhu’s work [14, 26]. There are 8569 overlapping window across the whole process. Figure 8 shows the dendrogram of clustering results. Different branches correspond to different clusters of windows: 1A (7239-8569 & 5721), 2A(5821-7147 & 3091-3092), 2B(7149-7238), 3A(1-1445 & 4290), 3B(1446-1696), 4A(1697-2856), 4B(2857-3090 & 3093-3140), 5A(3141-4289), 5B(4291-4381), 6A(4382-5720 & 7148), 6B(5722-5820). It should be noted that there are 4 small branches (2857-2864 | 2865-2884 | 2885-2975 & 3093-3140 | 2976-3090) in part 4B. While 3 small branches are found in part 2B (7149-7155 | 7156-7167 & 7190-7236 | 7168-7189), 3B (1446-1490 & 1514-1568 | 1491-1513 | 1569-1694), 5B (4291-4300 | 4301-4326 | 4327-4381) and 6B (5722-5729 | 5730-5755 & 5775-5820 | 5756-5774), respectively. According to the probability in eq.(, the window clustering results are transferred to sample clustering results. There are 30 different segments classified into 22 different clusters in the process. The durations of the ten largest segments are 10108, 9373, 9343, 9289, 8134, 8043, 882, 805, 637 and 385 sampling intervals, respectively. It can be seen that there is a big difference between 8043 and 882 sampling intervals. The segment with duration of 8043 is most likely to be a stable mode. And the segment with duration of 882 is probably a sub-segment in a transition. According to the rules in sub-Section 3.4, the duration of the shortest stable mode. Hence,
is defined as
is set to 8043 [18]. The durations of the shortest ten
segments are 7, 7, 7, 14, 49, 52, 56, 70, 84 and 133 sampling intervals, respectively. According to Wang’s method[18],
is set to 49, since it is hard to characterize the segments less than 49 samples. Therefore,
segments whose durations are less than 49 are considered as noise and these segments should be eliminated. The final offline identification results are shown in Figure 9(a). -16-
[Figure 8 about here] To evaluate the performance of the algorithm, the ICA-PCA method in Wang’s work [18] is employed for comparison. The results of offline identification from the two approaches are both shown in Figure 9. Six stable modes (S1-S6) can be clearly identified using the proposed method in Figure 9(a). Meanwhile, the statistical characteristics of transitions can be described by several smaller sub-models and there are several kinds of ‘irregularity’ in these transitions. The phenomenon of ‘irregularity’ is found at 10597~10981, 21650~21985, 40424~40745 and 50329~50657, respectively. In these transitions, the data statistical characteristics are similar to the samples that are several sampling intervals before. It means that there are obvious fluctuations within these transitions. These samples are considered as a part of normal transitions instead of noise sub-segments. In Figure 9(b), six stable modes can also be identified using Wang’s method. However, mismatches are found around the edges of some transitions such as T23S1 and T56S1, since the method mainly is determined by the Euclidean distance. In the clustering results using Wang’s method[18], samples in the same cluster may not have similar variable correlations. Also, the ‘irregularity’ is considered as noise and should be eliminated. Therefore, there are a few absences in some transitions. For more comprehensive descriptions, the identification results of 17000th ~ 25000th samples using the DMP algorithm and Wang’s method are shown in Figure 10 and Figure 11, respectively. In Figure 10, the ‘irregularity’ occurs at 21650~21985. The transition is divided into five segments, which can be described using four sub-models (T23S1, T23S2, T23S3 and T23S4). The fifth and the third segment share the model T23S3. [Figure 9-11 about here] In Figure 11, the transition is divided into four parts with individual models. Meanwhile, an ‘irregularity’ is found from 21374 to 21635. Unfortunately, this irregularity is considered as a noise segment and should be -17-
eliminated in Wang’s method[18], so that the neighboring segments of ‘irregularity’ are chosen for online identification and monitoring. For instance, an online sample, which should be clustered to the irregularity, may be recognized to another segment in the transition. It can raise the risks of the Type I and Type II errors. The trajectories of the 34 variables in the samples from 17000 to 25000 are shown in Figure 12. It can be seen that the result of offline identification using the proposed algorithm is similar to the real situation. At the same time, mismatches are found both in stable modes and transitions identification using Wang’s method. [Figure 12 about here]
5.3 Online monitoring for transitions In this case, another 5000 online samples including two stable modes and one transition are used for the validation of online monitoring performance. The two stable modes belong to S2 and S3, respectively. The transition belongs to T23. The system condition changes at the 1500th sample. Three faults are introduced at the 1700th online sample including a step fault, a noise fault and a ramp fault in the separator level set-point. In order to validate the superiority of the proposed monitoring algorithm, Wang’s method is also adopted for comparison [18]. [Figure 13 about here] The monitoring results for faulty data and normal data using the proposed algorithm are shown in Figure 13. In Figure 13(a) and Figure 13(b), the false alarm rates for normal data are both very low except for the edge of each cluster. These models fit well on most samples in the process. In Figure 13(c) and Figure 13(d), the monitoring model is switched as soon as the system status changes and the false alarm rates are also very low in the samples from 1500 to 1700. Then a step fault is introduced at the 1700th sample and can be detected immediately by both statistics. Alarms occur at the 1737th sample and 1755th -18-
sample using the T 2 and SPE statistics, respectively. The fault can also be detected using the subsequent models. Figure 13(c) ~ (h) are the monitoring results for ramp fault and noise fault, respectively. It can be seen that both faults are detected very quickly. In addition, the false alarm rates are kept at a very low level in both statistics. [Figure 14 about here] Figure 14 shows the monitoring results for normal and faulty data using Wang’s method. PCA is used in the stable mode monitoring, while both ICA and PCA are employed for the transition monitoring. ICA is utilized to monitor the non-Gaussian components in the transition, and PCA is applied for the residual part monitoring. For most samples, the three faults cannot be well detected using ICA. Although the monitoring results of PCA are good compared to the performance of ICA, the time delay is still very big. Furthermore, the monitoring result deteriorates when the noise fault occurs. Almost half of the faulty samples are not detected. It is mainly because that there are offline misclassification and model mismatches in the online step. In Wang’s method, ‘irregularity’ is considered as noise and should be deleted. Therefore, there is no corresponding training samples. Besides, the first part of T23 is also misclassified in the offline identification. As a consequence, online samples which should be clustered as the ‘irregularity’ are misclassified into other segments, which may raise the missing alarm rates. Moreover, compared to Wang’s ICA-PCA monitoring method, the correlation between process data and quality data is taken into consideration in our proposed algorithm. KPLS models are employed to monitor the transitions and stable modes. The monitoring results using KPLS has a superior performance compared to ICA-PCA method both in transitions and stable modes. The fault detection rates and missing alarm rates using two methods are shown in Table 4. [Table 4 about here] -19-
The above simulation shows the superiority of the proposed method in two main aspects. Firstly, the proposed offline identification algorithm can well identify transitions and stable modes. The offline identification results are almost the same as the real situations. Furthermore, the ‘irregularity’ in transitions is also emphasized in the offline steps. Secondly, faults in the process are detected rapidly and accurately using the proposed quality-related online fault detection method. All the faults can be detected immediately once they occur and the false alarms are kept at a very low level simultaneously.
6. Conclusions A novel algorithm for quality-related multimode identification and process monitoring is developed in this paper. Firstly, a DMP based offline multimode identification method is proposed. Differing from stable modes, transitions are divided into several small segments, each of which is demonstrated by a single KPLS model. Hierarchical explanations are then implemented using the clustering results of all individual DMP trials. The ‘irregularity’ in transitions is also analyzed and verified both theoretically and experimentally. Online transition identification and the process monitoring are carried out based on the established model after the offline identification is finished. Finally, the Tennessee Eastman Process is employed to evaluate the effectiveness of the proposed algorithm. Compared to the traditional monitoring methods, it can be seen that the proposed method has superior performance both in offline recognition and online monitoring.
Acknowledgement This work is supported by the National Natural Science Foundation of China (61573308) and the National Science Council, R.O.C.
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Wang, F., et al., Process monitoring based on mode identification for multi-mode process with transitions. Chemometrics and Intelligent Laboratory Systems, 110 (2012) 144-155.
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Tan, S., et al., Multimode Process Monitoring Based on Mode Identification. Industrial & Engineering Chemistry Research, 51 (2011) 374-388.
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Yao, Y. and F. Gao, Phase and transition based batch process modeling and online monitoring. Journal of Process Control, 19 (2009) 816-826.
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Zhu, Z., Z. Song, and A. Palazoglu, Transition Process Modeling and Monitoring Based on Dynamic Ensemble Clustering and Multiclass Support Vector Data Description. Industrial & Engineering Chemistry Research, 50 (2011) 13969-13983.
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Figure captions Figure 1. The schematic of the DMP algorithm. Figure 2. The schematic of clustering results using the proposed method. Figure 3. Offline identification results. Figure 4. The flowchart of multimode offline identification. Figure 5. Transition online identification and monitoring. Figure 6. The Tennessee Eastman chemical process. Figure 7. Variables for offline identification. Figure 8. Hierarchical explanation by aggregating the results of the DMP algorithm. Figure 9. Results of offline identification on the TE process using (a) the proposed method, (b) Wang’s method. Figure 10. Result of offline identification from the 17000th to the 25000th sample using the proposed method. Figure 11. Result of offline identification from the 17000th to the 25000th sample using Wang’s method. Figure 12. The 17000th~25000th samples for all variables. Figure 13. Monitoring results for normal data using the proposed online monitoring algorithm (a)(b) T2 and SPE statistics for normal data, (c)(d) T2 and SPE statistics for step fault, (e)(f) T2 and SPE statistics for ramp fault, (g)(h) T2 and SPE statistics for noise fault. Figure 14. Monitoring results using Wang’s method for (a) Normal data, (b) Step fault, (c) Ramp fault, (d) Noise fault.
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Table captions Table 1. Clustering steps using the DMP algorithm Table 2. Process variables in the TE process Table 3. Quality variables in the TE process Table 4. Comparison of fault detection rates and missing alarm rates.
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Moving windows
... Model 1
... iS
Model
Model C
EiS ,k
E1,k
EC ,k
Minimum
... Cluster 1
... Cluster
iS
Cluster C
Results unchanged
Iteration convergence
Figure 1. The schematic of the DMP algorithm.
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Models update for the next iterative run
Figure 2. The schematic of clustering results using the proposed method.
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Figure 3. Offline identification results.
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Random initialization
Random initialization
DMP algorithm
DMP algorithm
Clustering results
Clustering results
Random initialization
...
DMP algorithm Clustering results
Hierarchical aggregation
Edge determination
Check “irregularity”& delete noise segments
Final identification results
Figure 4. The flowchart of multimode offline identification.
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Online sample
Next sample
SPE SPElimit T2 Tlimit2
The next model/segment
Yes
Current sample
current segment
No
No
Two consecutive models unsatisfied
Yes
Fault
Figure 5. Transition online identification and monitoring.
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Figure 6. The Tennessee Eastman chemical process.
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Figure 7. Variables for offline identification.
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Figure 8. Hierarchical explanation by aggregating the results of the DMP algorithm.
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Figure 9. Results of offline identification on the TE process using (a) the proposed method, (b) Wang’s method.
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Figure 10. Result of offline identification from the 17000th to the 25000th sample using the proposed method.
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Figure 11. Result of offline identification from the 17000th to the 25000th sample using Wang’s method.
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Figure 12. The 17000th~25000th samples for all variables.
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Figure 13. Monitoring results for normal data using the proposed online monitoring algorithm (a)(b) T2 and SPE statistics for normal data, (c)(d) T2 and SPE statistics for step fault, (e)(f) T2 and SPE statistics for ramp fault, (g)(h) T2 and SPE statistics for noise fault.
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Figure 14. Monitoring results using Wang’s method for (a) Normal data, (b) Step fault, (c) Ramp fault, (d) Noise fault.
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Table 1. Clustering steps using the DMP algorithm 1.
Normalize the process data and quality data.
2.
Cluster number is obtained using the subtractive clustering method.
3.
Initial KPLS models are established using the randomly initialized data.
4.
All windows in the process are projected into the KPLS models to obtain the clustering results using the criterion of minimal residual values.
5.
New KPLS models are established according to the clustering results in the last step.
6.
Go back to step 4 until the iteration converges.
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Table 2. Process variables in the TE process No.
Process variables
No.
Process variables
1
A feed(stream 1)
15
Stripper level
2
D feed(stream 2)
16
Stripper pressure
3
E feed(stream 3)
17
Stripper underflow(stream 11)
4
A & C feed(stream 4)
18
Stripper temperature
5
Recycle flow(stream 8)
19
Stripper steam flow
6
Reactor feed
20
Reactor cooling water outlet
rate(stream 6) 7
Reactor pressure
temperature 21
Separator cooling water outlet temperature
8
Reactor level
22
D feed flow(stream 2)
9
Reactor temperature
23
E feed flow(stream 3)
10
Purge rate(stream 9)
24
A feed flow(stream 1)
11
Product separator
25
A & C feed flow(stream 4)
26
Separator pot liquid
temperature 12
Product separator level
flow(stream 10) 13
Product separator
27
Stripper liquid product
pressure 14
Product separator
flow(stream 11) 28
Reactor cooling water flow
underflow(stream 10)
Table 3. Quality variables in the TE process No.
Quality variables
No.
Quality variables
1
A constituent(stream 6)
4
D constituent(stream 6)
2
B constituent(stream 6)
5
E constituent(stream 6)
3
C constituent(stream 6)
6
F constituent(stream 6)
Table 4. Comparison of fault detection rates and missing alarm rates Faults | fault detection rates (%)
The proposed method
Wang’s method
Step fault
98.91
99.09
Ramp fault
96.42
93.27
Noise fault
99.15
47.63
Normal data | false alarms (%)
0.16
3.32
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Highlights
> A strategy considering transition identification and fault detection is proposed for transition process. > A distributed model projection algorithm based on the moving window strategy is employed to identify transitions and stable modes in a process > A hierarchical aggregation based on the results of DMP algorithm in each individual trial is also implemented > A novel offline/online identification is developed for transitions and stable modes > Case study of transitions is designed to illustrate the superiority of the developed method >
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PII: DOI: Reference:
S0169-7439(16)30356-2 http://dx.doi.org/10.1016/j.chemolab.2016.10.001 CHEMOM3323
To appear in: Chemometrics and Intelligent Laboratory Systems Received date: 4 December 2015 Revised date: 11 September 2016 Accepted date: 4 October 2016 Cite this article as: Yuchen He, Le Zhou, Zhiqiang Ge and Zhihuan Song, Distributed Model Projection based Transition Processes Recognition and Quality-related Fault Detection, Chemometrics and Intelligent Laboratory Systems, http://dx.doi.org/10.1016/j.chemolab.2016.10.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Distributed Model Projection based Transition Processes Recognition and Quality-related Fault Detection Yuchen He1, Le Zhou2, Zhiqiang Ge1, Zhihuan Song1 1
State Key Laboratory of Industrial Control Technology, College of Control Science and Engineering, Zhejiang University,
Hangzhou 310027, Zhejiang, China 2
School of Automation and Electrical Engineering, Zhejiang University of Science and Technology,
Hangzhou 310023, Zhejiang, China
Abstract In this paper, a novel transition process identification algorithm based on distributed model projection (DMP) is proposed for clustering nonlinear transition data and monitoring the variations in the transition process. Compared to several alternative identification methods, the DMP algorithm considers both the correlations between variables and correlations between samples. Also, a framework is proposed to combine DMP algorithm and hierarchical clustering to derive an optimal clustering results through a large amount of individual trials of the DMP algorithm. Based on the offline classification results, a transition process is divided into several sub-segments and each of them can be characterized by a stable model. Then the online identification and monitoring methods are carried out based on the sub-models established in those segments. Finally, the Tennessee Eastman (TE) benchmark process is utilized to demonstrate the performance of the proposed process identification and monitoring strategy. Compared to previous works, the proposed algorithm is shown to be superior both in identification and monitoring.
Keywords: Transition process; Distributed model projection; Hierarchical clustering; Transition identification; Fault detection
To whom all correspondence should be addressed. Email: [email protected] Tel: +86-571-87951442
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1. Introduction Recently, data-based multivariate calibration methods have been widely used in modern industries for the purpose of process monitoring and diagnosis [1, 2]. Among them, the classical multivariate statistical process control (MSPC) algorithms have achieved great successes in the last few decades, such as principal component analysis (PCA), partial least squares (PLS), principle component regression (PCR), kernel partial least squares (KPLS) and their advanced variants [3-7]. However, most of these methods are assumed to run in a single and stable operating condition. In most cases, the real industrial processes are operated in different conditions due to fluctuations in raw materials, set-point changes, aging of equipment and seasoning effects. In order to solve this problem, many methods have been proposed for the purpose of multimode modeling and process monitoring [1, 8-13]. Particularly, Zhu et al. [14] proposed a clustering method for multimode recognition in which an ensemble clustering algorithm is employed. However, the performance of ensemble clustering depends on the proper choice of the initial value of the clustering method. Zhang et al. [12]proposed a subspace separation method where a common subspace and a specific space are used to describe the multimode characteristics. Practically, it is difficult to construct such two subspaces explicitly due to the complexity of multimode processes. It is also noticed that a stable operation status cannot be immediately switched to another one, indicating that there are transitions between two adjacent stable modes. These unstable processes should also be considered for multimode modeling and monitoring. It is always a challenging and sophisticated issue using traditional methods. For the purposes of transition identification and monitoring, Zhao et al.[15, 16], Ge et al.[17], Wang et al.[18], Tan et al.[19] and Yao et al.[20] proposed different algorithms for stable and transient process identification as well as multimode monitoring in recent years. In the above methods, transitions should be identified from multimode process in the offline step so that calibration models can be -2-
established using proper training transient data. In Zhao’s method[15], the duration of a transient process was defined from the center of a stable mode to the center of its neighboring mode so that the transition process is able to be described using adequate observations. The transition process recognition method was set manually instead of being identified based on the process data. It is quite unreasonable that stable modes would disappear and the whole process only consists of several transitions according to such definition. Process information was taken into account in Wang’s method where a complete transition process recognition and modeling method was proposed. In Wang’s methods[18], a transient process was separated into several sub-segments by the k-means method and each of them was characterized using linear models. Unfortunately, the Euclidean distance between different samples cannot fully reveal the variations and trends of complicated chemical process [21, 22]. In the past few decades, the MSPC methods have proved itself in chemical process modeling. Hence, Yao et al. [20]used a series of different PCA models to describe the behavior of a chemical process. A single PCA model was created for each sub-segment and the PCA similarity index was adopted to analyze the difference between each sub-segments. It indicates that the unstable nonlinear transition process is divided into several approximately stable linear processes where the classical MSPC methods can be employed. However, the assumption that variable relationships in each sub-model are linear correlated is still invalid in practice. In order to solve the above problems, we propose a novel algorithm called distributed model projection (DMP) in this paper. In the DMP algorithm, the transition process is recognized by an iterative method. Since the beginning part of a transition is very different from the ending part, it is still assumed that a transition can be divided into several sub-models, in which it can be treated as a stable process. The amount of clusters can be properly determined by the subtractive clustering algorithm (SCM). Each of these clusters is supposed to be a stable mode or a transition. Also, nonlinear relationship between process -3-
data and quality data is an important issue in chemical processes. For this purpose, the DMP algorithm is proposed by using the KPLS model to describe the behavior of a process. The algorithm starts with several initial models which are established by the randomly initialized data in each cluster. In order to capture the characteristics of a process accurately and avoid the noises caused by the single observation, the classical moving window strategy is adopted to create several overlapping windows across the whole process. According to the theory of principle-components-similarity[23], two data blocks with similar principle components subspaces may have similar statistical characteristics to each other. In other words, two data blocks with small prediction errors are considered to be similar to each other when these two blocks are projected to the same model. Hence, each moving window in a multimode process is projected to the initial models to calculate corresponding prediction errors. A window having the smallest error with respect to a model is assigned to the corresponding cluster. In such way, a transition process is initially recognized and divided into several segments (sub-models). On the basis of the current models, the windows in each cluster are statistically similar to each other. However, it is noticed that the initial models are established based on the randomly initialized data. Hence, these models should then be updated using all available samples within each cluster. After the new models are derived, all the windows are projected to these models to determine the new assignment of each window. And the iteration will continue until no further variation is found in the clustering results. In each single cluster, statistically similar windows are grouped together and dissimilar windows are assigned to different clusters. The clustering results of the DMP algorithm indicate the similarity between different moving windows. Besides, the DMP algorithm is considered as a nonhierarchical clustering method which may converge to a local optimal solution [21, 22]. It means that the clustering result in one trial by using DMP algorithm -4-
may be different from the result in another one. Compared to nonhierarchical clustering methods, global optimal solutions can be derived using hierarchical clustering methods. In order to obtain global optimal clustering results, a hierarchical clustering method is adopted by combining the clustering results of a large amount of DMP trials. The similarity information derived by the DMP algorithm can be the input of any common hierarchical method. After the transition process is recognized, the corresponding fault detection methods can be derived. For most multimode process monitoring methods, it only involves the process data where the abnormal situations of process can be monitored while the variations of the final product quality cannot be detected. When the quality data can be measured, the relationships between the process variables and the quality variables should also be included to enhance the ability of process and quality monitoring. In chemometric region, some supervised methods have been proposed for single and multimode process modeling [11, 24]. However, little attention has been paid on the quality-related transition process monitoring. Hence, a quality-related transition fault detection method is also studied in this paper. The rest of the paper is organized as follows: a brief description of transition process is given in Section 2. In Section 3, offline transition recognition by using the DMP algorithm and hierarchical clustering explanation are introduced explicitly. Then an online identification and quality-related process monitoring method is introduced in Section 4. A TE benchmark process is demonstrated in Section 5 to evaluate the performance of the proposed method. Finally, some conclusions are made.
2. Transition process description and analysis Differing from stable modes, a transition process is a trend from one stable mode to another. Hence, the statistical characteristics of a transition cannot be well depicted using the traditional MSPC method -5-
because of intricate relationships between variables. At the beginning of each transition, the statistical characteristic is similar to the previous stable mode. At the end, it is accordingly similar to the following stable mode. Therefore, it is difficult to handle transition modeling and monitoring issues using classical stable mode methods. Furthermore, considering the existence of control loops in an industrial process, ‘irregularity’ sometimes occurs in a transition [15, 16, 25]. It means that the present sample may contain similar characteristics with previous samples several sampling intervals before. Considering the statistical characteristics of transitions, several recognition and modeling steps are used in this paper. Firstly, the observations are divided into several clusters utilizing the DMP algorithm, where a stable process can be demonstrated using a single cluster while the transition process contains several smaller sub-models, so that the traditional MSPC methods can be employed in each cluster. After that, the corresponding process monitoring scheme can be established in each stable and transient mode. A new sample needs to be classified into a certain cluster once it is measured. Then the sample can be monitored using the corresponding fault detection algorithm. In the next two sections, the offline recognition and online identification algorithm will be introduced in detail.
3. Offline multimode identification and transition recognition A multimode process usually consists of several stable processes and transitions. Commonly, different stable processes and transitions should be identified before online monitoring. The conventional clustering methods were mainly carried out based on the Euclidean distance. Hence, the relationships between different samples were emphasized and two samples with small Euclidean distance between them were considered to belong to one cluster [18, 22]. It might work under the assumption that the relationships between different variables are simple. Considering the complexity in the process control, such assumption -6-
is often invalid in continuous chemical process. Usually, chemical processes contain variations, especially peaks and valleys, and it is therefore difficult to illustrate the behavior of a process merely by traditional Euclidean-distance-based clustering methods. For the purpose of process control and fault detection, a growing number of successes have achieved by the MSPC methods, such as PCA and PLS. It is more appropriate to introduce MSPC methods into process analysis and multimode identification. Inspired by the principle-components-similarity analysis[23], two data blocks are considered to be similar to each other when both residuals are very small with respect to the same model. In fact, a single mode can be interpreted as a group of samples with similar characteristics which can be well reveal by one single MSPC model. By calculating the corresponding model residuals, a process is divided into several segments each of which can be represented through a MSPC model. In the DMP algorithm, the similarity between windows is interpreted as a kind of model fitness rather than Euclidean distance. Besides, the DMP algorithm, like other nonhierarchical methods, will probably converge to a local optimal solution. Therefore, a hierarchical clustering manner using the results of DMP algorithm will be adopted in sub-Section 3.3 to give a global optimal clustering result. Since nonlinearity is a kind of common properties for most industrial processes, the classical KPLS model is employed for DMP algorithm considering the nonlinear cross-correlations between variables while it can also be readily used for quality-related fault detection. Besides, a classical moving window strategy is used for model projection to capture a time period of observation. This will help understand and analyze the behavior of a process through time segments rather than individual samples. The determination of window length L and window moving step M is a significant issue and has been discussed explicitly by previous works [14]. For more details about the selection of moving window parameter, please refer to Palazoglu’s work [14, 26]. Firstly, the KPLS model is introduced firstly in the next sub-section to give a better understanding about the DMP algorithm. -7-
3.1 Preliminaries 3.1.1 KPLS modeling In PLS, the relationships between the process and quality variables are extracted. Since the correlations between them are nonlinear in most chemicals, a kernel PLS model is employed in this paper, which is given as:
TPT E Y TQT F where x1 x2 T
mapping : xi
M
T
T
N S
( 1)
is the kernel form of process data X ,
indicates a nonlinear
Transformation (xi ) F . The quality prediction on testing data can be calculated
as follows:
ˆ = K U(TT KU)1 TT Y Y t t
( 2)
where K t and K are the kernel matrices of testing and training data, respectively. T=[t1 t 2 ...t A ] and
U = [u1 u2 ...u A ] are score matrices for process and quality data in the feature space of training data, respectively. A represents the number of principle components. Y is the training quality data. The prediction error of testing quality data Yt ( Nt q) with N t samples and q variables can be calculated as follows:
ˆ et Yt Y t
( 3)
For more details for KPLS modeling, please refer to the previous works [27-29]. Suppose that the sum of squares of residuals corresponding to the prediction error is Et
Nt
p
e i 1 j 1
2 ij ,t
where eij ,t represents an
element in the residual matrix et . From eq.(, it can be seen that the testing data can be well described by a certain KPLS model if its prediction residuals are small enough.
3.2 DMP algorithm -8-
In this sub-section, the DMP algorithm is introduced in detail. The DMP algorithm is implemented in an iterative way. In each iterative run, the overlapping moving windows in the whole process are projected to a series of KPLS models. The windows are assigned to the right cluster according to the corresponding residuals. In the next run, new KPLS models are established by all samples in each cluster. The windows are then reassigned to each cluster by calculating the residuals with respect to the new models. When no further variation appears in the clustering result, the iteration procedure of this trial stops. Commonly, it is significant to choose the initial models for the DMP algorithm, which typically starts with a pre-defined number of clusters. However, the amount of clusters is always unknown in industrial processes. In this paper, a SCM is introduced to estimate the amount of clusters [30]. Since the beginning part of a transition is very different from the ending part, a continuous transient process is approximately divided into several small sub-segments each of which is randomly initialized to create the initial KPLS models. After the parameter initialization is accomplished, the initial KPLS models are employed to classify moving windows into different clusters. It can be easily understood that a model will fit data well when the prediction errors of these data for the particular model are small enough. In this paper, the moving windows in the process are projected into those initial KPLS models to determine which cluster they are assigned to. Hence, the proposed algorithm is named as distributed model projection. After the projection is accomplished in this single run, these windows are classified into different clusters. In each cluster, windows are similar to each other. In order to prepare the next window projection, the KPLS models should also be updated based on the current samples in each cluster. Therefore, the new KPLS models are derived and the projection will continue until no further variation exists in the clustering result. For summary, the schematic of the DMP algorithm is shown in Figure 1 while the clustering steps using DMP algorithm are -9-
listed in Table 1. [Figure 1 about here] [Table 1 about here]
3.3 Hierarchical aggregation for the DMP algorithm As a nonhierarchical clustering method, the iteration using the DMP algorithm will converge rapidly to a local optimal solution [21, 22]. A hierarchical manner is therefore adopted by aggregating the clustering results derived by a large amount of individual DMP trials. The clustering result using the DMP algorithm in the bth trial is denoted as Gb (b=1,…,B). Gb is an Nb×C binary matrix where Nb and C are the number of windows and the number of clusters, respectively. in which
g b represents an element at ( nw, c ) of Gb,
g b is set to 1 if the nw th window is assigned to the c th cluster in the final clustering result of
the bth DMP trial. Each element of Gb can be expressed as follows:
1 gb (nw, c) 0
if window nw is assigned to cluster c in trial b otherwise
( 4)
A hierarchical aggregation is implemented by concatenating the clustering results of all B individual DMP trials:
Gall [G1 , G2 ,..., GB ]
( 5)
Then, an N×N similarity matrix can be obtained as follows
R 1
1 T Gall Gall B
( 6)
where 1 is a matrix consisted of ones. Rwz, which is an element of R, represents the dissimilarity between the window w and the window z. It is noticed that the dissimilarity matrix can be the input to any common hierarchical explanation to give a graphic dendrogram, which indicates the global optimal clustering results. -10-
For a single sample, it may belong to different clusters due to the utilization of the moving window. Hence, the samples at the edge of each cluster should be classified more clearly. The edges between different clusters are determined based on the concept of probability. It is assumed that ni ,l is the number of the times that the lth sample is assigned to the ith(i=1,…,C) cluster. The probability can be estimated as follows:
pi ,l
ni ,l
( 7)
ni ,l ni 1,l
If pi ,l is bigger than pi 1,l , it can be thought that the lth sample belongs to the ith cluster.
3.4 Multimode identification and transition recognition After the global optimal clustering results are obtained, it still needs to be clear whether each cluster should belong to a stable process or a part of transition process. So the statistical characteristics of transitions are analyzed. Different from stable modes, the features of transitions can be summarized as follows: 1) Durations of transitions are much smaller compared to stable modes; 2) A transition only occurs between two different stable modes; 3) ‘Irregularity’ happens within transitions sometimes; 4) The states of transitions may remain stable among small neighboring sample points. For these reasons, some parameters are discussed as follows for distinguishing stable processes and transitions. Since the durations of a transition is much shorter than that in stable modes,
is defined as
the minimal length of one stable mode[18]. A segment is considered as a stable mode if its length is larger than . Otherwise, it may belong to a transition and cannot well describe the characteristics of a stable mode. If a sub-model is identified between two same stable models, it is considered that such mode is caused -11-
by system noises instead of a transient mode. Besides, the situation that two separate segments in one transition share the same model is considered normal. The latter segment is treated as a kind of ‘irregularity’ [25, 31]. If the length of a transient segment is smaller than
which is defined as the least duration of the
transient segment[18], then the segment is treated as noise and should be deleted. Note that the duration of a transient sub-segment is much different from the duration of a stable mode or noise. Therefore, it is very convenient to select parameters
and . For better understanding about the offline transition
identification, a simple example is presented in Figure 2. [Figure 2 about here]
In Figure 2, a process containing two stable processes and a transition is divided into six clusters with ten segments. The segment whose length is larger than
is considered as a stable process. Therefore,
Segment 1, 8 and 10 are clustered as stable processes. The other segments may belong to the transition process. Meanwhile, segment 4 whose length is smaller than
is considered as noise which should be
deleted. Segment 5 and segment 7 share the same model. The ‘irregularity’ can be clearly seen in segment 7. Segment 8 and segment 10 belong to the same cluster. Therefore, segment 9 is not a transient segment and should be deleted. After the analysis of offline transition identification, the identification results are modified and shown in Figure 3. That is, segment i and segment vii belong to the stable modes. Meanwhile, segments ii-vi belong to a transient process. [Figure 3 about here]
In the above case, the whole process can be divided into 6 different clusters. Each of these clusters has its corresponding KPLS model, which reveals a different kind of data correlation. The flowchart of -12-
multimode offline identification is shown in Figure 4. [Figure 4 about here]
4. Online identification and fault detection for multimode processes After transitions are identified using offline DMP algorithm, the quality-related fault detection algorithm can also be made. Firstly, an online sample identification is discussed. Usually, the system state stays in the stable modes and a transition starts after a stable mode. The current online sample is denoted by the hth sample. Firstly, it should be clear that which stable mode the hth sample starts with [18]. It is assumed that there are S all different stable modes, and each of them is depicted by a KPLS model. In order to monitor the online samples, Hotelling’s T in this paper. The statistics T
2
2
statistics and SPE statistics are employed
can be defined as follows:
T 2 tt 1tt where t t is the corresponding score vector of test data and
( 8)
1 T T T represents the covariance of N
score matrix T in the feature space of training data. The statistics SPE for each sample can be defined as follows[32] :
SPE et etT
The confidence limits for T
2
( 9)
and SPE can be further defined as:
A( N 2 1) T ~ FA, N A, N ( N A)
( 10)
SPE ~ 2
( 11)
2
where is the significance level. and are the magnitude and the freedom degree of SPE statistic, respectively. These two parameters can be approximately represented by -13-
b / 2a and
2a 2 / b in which a and b are the estimated mean and variance, respectively [32]. The main steps of multimode processes online identification can be summarized as follows: 1) Determine which stable mode the h-1th sample belongs to using the control limits of all S all stable 2
modes. If T and SPE are less than the corresponding control limits of stable model i while exceed the control limits of stable model j ( j i ) , the h-1th online sample is considered as a 2
normal one and it belongs to S i . The procedure will go to step 2. If T and SPE go beyond all corresponding control limits of possible model, it is considered as a faulty sample and an alarm occurs; 2) Suppose the h-1th sample belongs to cluster i , monitor the hth sample using the control limits of S i . If T
2
and SPE are under the control limits of model i , the hth sample is still considered as a
sample of Si and the next online sample will be monitored using the same model. Otherwise, the hth sample may belong to a transition or a kind of fault and it goes to step 3; 3) Monitor the hth sample using the successive transient sub-model. According to practical conditions of industrial production, the process switching trajectory is always fixed to produce standardized materials. If a process operation state changes, it must switch to the beginning of a specific transient process instead of other sub-models. Hence, the hth sample can be considered as a faulty one if its statistics for subsequent sub-model still exceed the corresponding control limits. Otherwise, the hth sample is thought to belong to this transition and the rest online sample will be identified in the same manner. The flowchart of online identification and monitoring for transition sample is shown in Figure 5. [Figure 5 about here]
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5. Case study 5.1 TE process description and simulation design The Tennessee Eastman (TE) Industrial Process is widely used to demonstrate various monitoring methods, which mainly consists of five work units. In a TE process shown in Figure 6, 41 measurements and 12 manipulated variables are collected. For more detailed settings of TE benchmark, please refer to Chiang’s book[33]. In order to demonstrate the superiority of the proposed algorithm for transition offline identification, a case with six different stable modes is designed. Hence, there are five transitions among these stable modes. 60000 samples are collected in this case for offline transition identification. In the previous paper, 33 or 16 variables are often chosen for process modeling and monitoring method validation. The main purpose of these selections is to emphasize the data inner-relationship while ignoring the process structures. In this case, all quality variables in the stream 6 are selected in order to show the variations on the output variables in a single stream. The process variables with small variations are not selected in order to reduce the computational complexity both in online and offline steps. Hence, 28 selected process variables are employed. The dimension of the system is 60000×34 with 28 process variables and 6 quality variables which are shown in Table 2 and Table 3, respectively. The detailed trajectories of process variables and quality variables are shown in Figure 7. [Figure 6 about here] [Table 2 about here] [Table 3 about here] [Figure 7 about here]
5.2 Offline transition identification -15-
The proposed DMP algorithm is applied to offline identification in this sub-section. In order to give a better understanding about the edge determination, moving window strategy is employed and the window parameters should be derived. In this case study, the window parameters L=19, M=7 are chosen according to Zhu’s work [14, 26]. There are 8569 overlapping window across the whole process. Figure 8 shows the dendrogram of clustering results. Different branches correspond to different clusters of windows: 1A (7239-8569 & 5721), 2A(5821-7147 & 3091-3092), 2B(7149-7238), 3A(1-1445 & 4290), 3B(1446-1696), 4A(1697-2856), 4B(2857-3090 & 3093-3140), 5A(3141-4289), 5B(4291-4381), 6A(4382-5720 & 7148), 6B(5722-5820). It should be noted that there are 4 small branches (2857-2864 | 2865-2884 | 2885-2975 & 3093-3140 | 2976-3090) in part 4B. While 3 small branches are found in part 2B (7149-7155 | 7156-7167 & 7190-7236 | 7168-7189), 3B (1446-1490 & 1514-1568 | 1491-1513 | 1569-1694), 5B (4291-4300 | 4301-4326 | 4327-4381) and 6B (5722-5729 | 5730-5755 & 5775-5820 | 5756-5774), respectively. According to the probability in eq.(, the window clustering results are transferred to sample clustering results. There are 30 different segments classified into 22 different clusters in the process. The durations of the ten largest segments are 10108, 9373, 9343, 9289, 8134, 8043, 882, 805, 637 and 385 sampling intervals, respectively. It can be seen that there is a big difference between 8043 and 882 sampling intervals. The segment with duration of 8043 is most likely to be a stable mode. And the segment with duration of 882 is probably a sub-segment in a transition. According to the rules in sub-Section 3.4, the duration of the shortest stable mode. Hence,
is defined as
is set to 8043 [18]. The durations of the shortest ten
segments are 7, 7, 7, 14, 49, 52, 56, 70, 84 and 133 sampling intervals, respectively. According to Wang’s method[18],
is set to 49, since it is hard to characterize the segments less than 49 samples. Therefore,
segments whose durations are less than 49 are considered as noise and these segments should be eliminated. The final offline identification results are shown in Figure 9(a). -16-
[Figure 8 about here] To evaluate the performance of the algorithm, the ICA-PCA method in Wang’s work [18] is employed for comparison. The results of offline identification from the two approaches are both shown in Figure 9. Six stable modes (S1-S6) can be clearly identified using the proposed method in Figure 9(a). Meanwhile, the statistical characteristics of transitions can be described by several smaller sub-models and there are several kinds of ‘irregularity’ in these transitions. The phenomenon of ‘irregularity’ is found at 10597~10981, 21650~21985, 40424~40745 and 50329~50657, respectively. In these transitions, the data statistical characteristics are similar to the samples that are several sampling intervals before. It means that there are obvious fluctuations within these transitions. These samples are considered as a part of normal transitions instead of noise sub-segments. In Figure 9(b), six stable modes can also be identified using Wang’s method. However, mismatches are found around the edges of some transitions such as T23S1 and T56S1, since the method mainly is determined by the Euclidean distance. In the clustering results using Wang’s method[18], samples in the same cluster may not have similar variable correlations. Also, the ‘irregularity’ is considered as noise and should be eliminated. Therefore, there are a few absences in some transitions. For more comprehensive descriptions, the identification results of 17000th ~ 25000th samples using the DMP algorithm and Wang’s method are shown in Figure 10 and Figure 11, respectively. In Figure 10, the ‘irregularity’ occurs at 21650~21985. The transition is divided into five segments, which can be described using four sub-models (T23S1, T23S2, T23S3 and T23S4). The fifth and the third segment share the model T23S3. [Figure 9-11 about here] In Figure 11, the transition is divided into four parts with individual models. Meanwhile, an ‘irregularity’ is found from 21374 to 21635. Unfortunately, this irregularity is considered as a noise segment and should be -17-
eliminated in Wang’s method[18], so that the neighboring segments of ‘irregularity’ are chosen for online identification and monitoring. For instance, an online sample, which should be clustered to the irregularity, may be recognized to another segment in the transition. It can raise the risks of the Type I and Type II errors. The trajectories of the 34 variables in the samples from 17000 to 25000 are shown in Figure 12. It can be seen that the result of offline identification using the proposed algorithm is similar to the real situation. At the same time, mismatches are found both in stable modes and transitions identification using Wang’s method. [Figure 12 about here]
5.3 Online monitoring for transitions In this case, another 5000 online samples including two stable modes and one transition are used for the validation of online monitoring performance. The two stable modes belong to S2 and S3, respectively. The transition belongs to T23. The system condition changes at the 1500th sample. Three faults are introduced at the 1700th online sample including a step fault, a noise fault and a ramp fault in the separator level set-point. In order to validate the superiority of the proposed monitoring algorithm, Wang’s method is also adopted for comparison [18]. [Figure 13 about here] The monitoring results for faulty data and normal data using the proposed algorithm are shown in Figure 13. In Figure 13(a) and Figure 13(b), the false alarm rates for normal data are both very low except for the edge of each cluster. These models fit well on most samples in the process. In Figure 13(c) and Figure 13(d), the monitoring model is switched as soon as the system status changes and the false alarm rates are also very low in the samples from 1500 to 1700. Then a step fault is introduced at the 1700th sample and can be detected immediately by both statistics. Alarms occur at the 1737th sample and 1755th -18-
sample using the T 2 and SPE statistics, respectively. The fault can also be detected using the subsequent models. Figure 13(c) ~ (h) are the monitoring results for ramp fault and noise fault, respectively. It can be seen that both faults are detected very quickly. In addition, the false alarm rates are kept at a very low level in both statistics. [Figure 14 about here] Figure 14 shows the monitoring results for normal and faulty data using Wang’s method. PCA is used in the stable mode monitoring, while both ICA and PCA are employed for the transition monitoring. ICA is utilized to monitor the non-Gaussian components in the transition, and PCA is applied for the residual part monitoring. For most samples, the three faults cannot be well detected using ICA. Although the monitoring results of PCA are good compared to the performance of ICA, the time delay is still very big. Furthermore, the monitoring result deteriorates when the noise fault occurs. Almost half of the faulty samples are not detected. It is mainly because that there are offline misclassification and model mismatches in the online step. In Wang’s method, ‘irregularity’ is considered as noise and should be deleted. Therefore, there is no corresponding training samples. Besides, the first part of T23 is also misclassified in the offline identification. As a consequence, online samples which should be clustered as the ‘irregularity’ are misclassified into other segments, which may raise the missing alarm rates. Moreover, compared to Wang’s ICA-PCA monitoring method, the correlation between process data and quality data is taken into consideration in our proposed algorithm. KPLS models are employed to monitor the transitions and stable modes. The monitoring results using KPLS has a superior performance compared to ICA-PCA method both in transitions and stable modes. The fault detection rates and missing alarm rates using two methods are shown in Table 4. [Table 4 about here] -19-
The above simulation shows the superiority of the proposed method in two main aspects. Firstly, the proposed offline identification algorithm can well identify transitions and stable modes. The offline identification results are almost the same as the real situations. Furthermore, the ‘irregularity’ in transitions is also emphasized in the offline steps. Secondly, faults in the process are detected rapidly and accurately using the proposed quality-related online fault detection method. All the faults can be detected immediately once they occur and the false alarms are kept at a very low level simultaneously.
6. Conclusions A novel algorithm for quality-related multimode identification and process monitoring is developed in this paper. Firstly, a DMP based offline multimode identification method is proposed. Differing from stable modes, transitions are divided into several small segments, each of which is demonstrated by a single KPLS model. Hierarchical explanations are then implemented using the clustering results of all individual DMP trials. The ‘irregularity’ in transitions is also analyzed and verified both theoretically and experimentally. Online transition identification and the process monitoring are carried out based on the established model after the offline identification is finished. Finally, the Tennessee Eastman Process is employed to evaluate the effectiveness of the proposed algorithm. Compared to the traditional monitoring methods, it can be seen that the proposed method has superior performance both in offline recognition and online monitoring.
Acknowledgement This work is supported by the National Natural Science Foundation of China (61573308) and the National Science Council, R.O.C.
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Figure captions Figure 1. The schematic of the DMP algorithm. Figure 2. The schematic of clustering results using the proposed method. Figure 3. Offline identification results. Figure 4. The flowchart of multimode offline identification. Figure 5. Transition online identification and monitoring. Figure 6. The Tennessee Eastman chemical process. Figure 7. Variables for offline identification. Figure 8. Hierarchical explanation by aggregating the results of the DMP algorithm. Figure 9. Results of offline identification on the TE process using (a) the proposed method, (b) Wang’s method. Figure 10. Result of offline identification from the 17000th to the 25000th sample using the proposed method. Figure 11. Result of offline identification from the 17000th to the 25000th sample using Wang’s method. Figure 12. The 17000th~25000th samples for all variables. Figure 13. Monitoring results for normal data using the proposed online monitoring algorithm (a)(b) T2 and SPE statistics for normal data, (c)(d) T2 and SPE statistics for step fault, (e)(f) T2 and SPE statistics for ramp fault, (g)(h) T2 and SPE statistics for noise fault. Figure 14. Monitoring results using Wang’s method for (a) Normal data, (b) Step fault, (c) Ramp fault, (d) Noise fault.
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Table captions Table 1. Clustering steps using the DMP algorithm Table 2. Process variables in the TE process Table 3. Quality variables in the TE process Table 4. Comparison of fault detection rates and missing alarm rates.
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Moving windows
... Model 1
... iS
Model
Model C
EiS ,k
E1,k
EC ,k
Minimum
... Cluster 1
... Cluster
iS
Cluster C
Results unchanged
Iteration convergence
Figure 1. The schematic of the DMP algorithm.
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Models update for the next iterative run
Figure 2. The schematic of clustering results using the proposed method.
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Figure 3. Offline identification results.
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Random initialization
Random initialization
DMP algorithm
DMP algorithm
Clustering results
Clustering results
Random initialization
...
DMP algorithm Clustering results
Hierarchical aggregation
Edge determination
Check “irregularity”& delete noise segments
Final identification results
Figure 4. The flowchart of multimode offline identification.
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Online sample
Next sample
SPE SPElimit T2 Tlimit2
The next model/segment
Yes
Current sample
current segment
No
No
Two consecutive models unsatisfied
Yes
Fault
Figure 5. Transition online identification and monitoring.
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Figure 6. The Tennessee Eastman chemical process.
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Figure 7. Variables for offline identification.
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Figure 8. Hierarchical explanation by aggregating the results of the DMP algorithm.
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Figure 9. Results of offline identification on the TE process using (a) the proposed method, (b) Wang’s method.
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Figure 10. Result of offline identification from the 17000th to the 25000th sample using the proposed method.
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Figure 11. Result of offline identification from the 17000th to the 25000th sample using Wang’s method.
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Figure 12. The 17000th~25000th samples for all variables.
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Figure 13. Monitoring results for normal data using the proposed online monitoring algorithm (a)(b) T2 and SPE statistics for normal data, (c)(d) T2 and SPE statistics for step fault, (e)(f) T2 and SPE statistics for ramp fault, (g)(h) T2 and SPE statistics for noise fault.
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Figure 14. Monitoring results using Wang’s method for (a) Normal data, (b) Step fault, (c) Ramp fault, (d) Noise fault.
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Table 1. Clustering steps using the DMP algorithm 1.
Normalize the process data and quality data.
2.
Cluster number is obtained using the subtractive clustering method.
3.
Initial KPLS models are established using the randomly initialized data.
4.
All windows in the process are projected into the KPLS models to obtain the clustering results using the criterion of minimal residual values.
5.
New KPLS models are established according to the clustering results in the last step.
6.
Go back to step 4 until the iteration converges.
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Table 2. Process variables in the TE process No.
Process variables
No.
Process variables
1
A feed(stream 1)
15
Stripper level
2
D feed(stream 2)
16
Stripper pressure
3
E feed(stream 3)
17
Stripper underflow(stream 11)
4
A & C feed(stream 4)
18
Stripper temperature
5
Recycle flow(stream 8)
19
Stripper steam flow
6
Reactor feed
20
Reactor cooling water outlet
rate(stream 6) 7
Reactor pressure
temperature 21
Separator cooling water outlet temperature
8
Reactor level
22
D feed flow(stream 2)
9
Reactor temperature
23
E feed flow(stream 3)
10
Purge rate(stream 9)
24
A feed flow(stream 1)
11
Product separator
25
A & C feed flow(stream 4)
26
Separator pot liquid
temperature 12
Product separator level
flow(stream 10) 13
Product separator
27
Stripper liquid product
pressure 14
Product separator
flow(stream 11) 28
Reactor cooling water flow
underflow(stream 10)
Table 3. Quality variables in the TE process No.
Quality variables
No.
Quality variables
1
A constituent(stream 6)
4
D constituent(stream 6)
2
B constituent(stream 6)
5
E constituent(stream 6)
3
C constituent(stream 6)
6
F constituent(stream 6)
Table 4. Comparison of fault detection rates and missing alarm rates Faults | fault detection rates (%)
The proposed method
Wang’s method
Step fault
98.91
99.09
Ramp fault
96.42
93.27
Noise fault
99.15
47.63
Normal data | false alarms (%)
0.16
3.32
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Highlights
> A strategy considering transition identification and fault detection is proposed for transition process. > A distributed model projection algorithm based on the moving window strategy is employed to identify transitions and stable modes in a process > A hierarchical aggregation based on the results of DMP algorithm in each individual trial is also implemented > A novel offline/online identification is developed for transitions and stable modes > Case study of transitions is designed to illustrate the superiority of the developed method >
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