Modeling and Monitoring for Transitions Based on Local Kernel

Article pubs.acs.org/IECR

Modeling and Monitoring for Transitions Based on Local Kernel Density Estimation and Process Pattern Construction Marcos Quiñones-Grueiro, Alberto Prieto-Moreno, and Orestes Llanes-Santiago* Automation and Computing Department, CUJAE, Havana, Cuba CP 19390 ABSTRACT: Usually, industrial processes have multiple operational modes due to different production strategies, external environmental variability, or changes in product specifications. Monitoring of multimode processes constitutes a challenging problem considering multiple steady-state operational regions and dynamic transitions. This paper proposes a novel method for the offline identification of stable modes and transitions based on a local kernel density estimation algorithm. The online monitoring scheme is based on mode identification and transition tracking. The Tennessee Eastman (TE) benchmark process is used as a case study to evaluate the performance of the proposal. As a result, stable modes are successfully isolated from transitions, even when these involve complex changes in the production mode. The results also demonstrate that the proposed scheme is capable of tracking mode changes, and finally, results monitored during transitions confirm the validity and efficacy of the new approach compared with previous works. trajectories are not exactly the same.21 However, assuming that variables from industrial processes operate under a control system, then transitions can be considered as repetitive processes6 and the above-mentioned issues are usually ignored. Then, the difference among the possible transitions between two stable modes are not considered as significant so patterns can be established allowing the creation of models for transition periods. Even ideally, optimal transition policies can be formulated using fundamental models and constrained optimization.22 The initial approaches to address transition modeling and monitoring can be found in papers published by Bhagwat et al.23 and Srinivasan et al.18 The first one is a model-based approach, and the second one is too complex to be applied in real industrial processes. The soft-transition multiple PCA method was proposed by Zhao et al.24 and Yao25 for batch processes based on user-specified parameters. The transitional mode between two phases is described as a model made up by two neighboring stable models, though such a consideration might have some difficulties when describing the characteristics of transitional mode perfectly. More recent methods were presented by Tan et al.21 and Zhu et al.19,26 In the first one, transition periods are isolated and a series of local PCA models are proposed for process monitoring. When this approach is used, transitions are modeled and monitored as continuous stable states which differ in covariance structure. Most of the time, during the transition the assumption of multivariate normal distribution, on which PCA monitoring model is based, is not complied with. Then, this monitoring model is not appropriate for the transition mode. Then, this monitoring model is not appropriate for the transition mode.

1. INTRODUCTION In the past decade the monitoring of modern industrial processes has received increasing attention due to higher requirements associated with production safety and quality. Since the employment of measurement systems generates a great amount of data, multivariate statistical process monitoring (MSPM) methods are widely used in both academic research and industrial applications.1−5 Some industrial processes change their operation mode due to different manufacturing strategies, product specifications, and economic considerations.6 Conventional MSPM approaches, such as the principal component analysis (PCA) and partial least-squares (PLS), may have unsatisfactory performance when directly used for multimode processes if a single operating region is assumed.7 Three main modeling frameworks have been proposed to address the multimodal feature. First, one single global model is developed considering all operation modes.8−10 Second, adaptative models demanding the frequent updating of models when changes in operation modes are required,11−13 and finally, multiple models can be built to fit each individual operating mode.14−16 Most of the proposed methods only focuses on process monitoring in stable modes, when process variables are running in one steady state, which means that the characteristics of variables, both mean and covariance are constant.17 Transition periods between themwith different statistical properties to each modehave been ignored. Some common transitions include startups, shutdowns, and grade changes.18 In practice, there is a lack of a sound knowledge of the process to label the process transitions using historical data19 thus, more efforts are still needed in modeling and monitoring the transition period.7,10,13,16 Generally, transitions between two stable modes are not unique, since they often do not have exactly the same duration and, even in complex scenarios, variables can follow different trajectories.20 In addition, variable responses are very sensitive to noise. Therefore, the characteristic patterns of the © XXXX American Chemical Society

Received: October 16, 2015 Revised: December 15, 2015 Accepted: December 29, 2015

A

DOI: 10.1021/acs.iecr.5b03902 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research Zhu et al.19,26 proposed three visualization tools for transition labeling based on dynamic ensemble clustering and multiple overlapping models for transition monitoring. However, the final isolation process of the transition is relatively subjective and the transition periods are not explicitly modeled. Hence, when a fault occurs during the transition, the situation cannot be well monitored when using these models.19 The main contributions of this paper include the following: First, a new method for offline identification of stable modes and transitions without the need of setting the number of modes. The method is based on a local kernel density estimation (KDE) clustering algorithm and the duration of stable modes. Second, the development of a novel fault detection scheme for multimode processes with transitions. Here, a new distance-based method is presented to improve the monitoring results during transitions. In order to test its performance, different types of faults are tested. The article is structured as follows. The preliminary knowledge of the KDE clustering algorithm is exposed in section 2. In section 3, the proposed method for offline modeling and the online monitoring scheme are described. The effectiveness of the proposal is shown by using the Tennessee Eastman (TE) benchmark process in section 4. Finally, the conclusions are drawn.

2.2. Clustering Based on KDE. Without assuming in advance a number of groups or a particular distribution for each group, an unsupervised density based clustering algorithm is proposed by Hinneburg et al.29,30 DENCLUE (DENsity-based CLUstEring) is presented in two main steps. First, samples are labeled on the basis of the local maximum of the multivariate probability distribution of the data set in which they converge. Second, some of the initial groups are joined based on a connectivity threshold parameter evaluated in their neighborhoods. Common density based methods define density as the number of sample points in a radius defined space. When using these approaches, the resulting groups may diverge depending on the radius selection.31 In order to address this issue, DENCLUE resorts the notion of kernel and density function. First Fd is denoted as the d-dimensional feature space. The neighborhood of a data object is given by an appropriate metric distance: F d × F d → 9 in the space Fd. The smoothness parameter h determines how much the influence of a data point depends on the distance to neighboring points. Kernel and Density Function. A kernel f unction is a f unction K: 9 d → 9 , K(x) ≥ 0 which has29

∫9

2. PRELIMINARIES 2.1. Clustering Approaches for Data Labeling in Multimode Processes. In multimode processes each stable state has its specific dynamics and duration time.27,28 The monitoring results depends, to a great extent, on the correctness of data labeling of time series. Furthermore, when no expert knowledge of the process is available, unsupervised learning methods are usually adopted. For this purpose, conventional clustering algorithms have been adapted, namely, k-means or fuzzy c-means.2 Stable modes are established according to the desired productivity or product quality and they have stable characteristics for a long time. Consequently, each one can be described by using one statistical model.21 On the other hand, a transitional mode is the transient state between two stable modes. During a transient stage, the process variables are constantly changing and exhibit nonstationary and nonGaussian characteristics. Then, the covariance structure is also varying and a single model is not suitable to characterize the process. If the transitions form part of the data set, window-based approaches of conventional clustering methods have been used taking time correlation of variables into account.6,19,26 When using these approaches, complex ensemble solutions or consensus criteria must be established in order to determine the number of stable modes. In the first case, rules for combining different algorithms must be set forth. In the second case, multiple runs of the same algorithm with different parameters are necessary, thus increasing the computational effort for the final parameter selection. In this article an offline modeling method for transition and stable modes isolation is proposed. It is a window approach of a clustering algorithm based on local KDE. It automatically determines the maximums of local density distributions thus requiring a rule for merging continuous distributions. No consensus criteria or ensemble solutions are required and there is not need to set the number of modes in advance. First, clustering based on KDE is presented.

d

K (x ) d x = 1

(1)

The density f unction is def ined as the sum of the kernels of all data points. Given N data objects described by a set of d-dimensional feature vectors D = {x1, ..., xN} ⊂ Fd, the density f unction is defined as f D (x ) =

1 Nhd

N

⎛ x − xi ⎞ ⎟ h ⎠

∑ K ⎜⎝ i=1

(2)

In principle, it is desirable for the kernel function to be a symmetric, continuous, and differentiable function. For simplicity, the following Gaussian kernel function is used for all data points K (x ) =

⎛ 1 ⎞ 1 exp⎜ − xT x⎟ d /2 ⎝ ⎠ 2 (2π )

(3)

The local maxima of the density function for each data point is referred to as its density attractor x*. Initial groups based on common density attractors are created. In order to deal with noise, the threshold parameter ξ is pre-established so as to only consider the density attractors complying with f(x*) ≥ ξ are considered. Next, neighboring groups fulfilling this threshold condition in their respective nearest points join together. The search for density attractors was initially accomplished by using a hill climbing procedure based on the gradient of the local density function. A modification was introduced by Hinneburg et al.30 which accelerates and improves the convergence process. The complete algorithm pseudocode can be found in ref 32. If the DENCLUE algorithm is directly applied on multimode process data, stable and transitional modes could hardly be isolated. The reason is that the parameter selection ξ is not trivial. Specifically, this is true when the distribution of the groups differs in shape due to the presence of transitions. Then, in order to address this issue, a modification of the second step of the algorithm is introduced. The proposal is based on the assumption that the stable modes are stationary ergodic processes since the industrial B


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Industrial & Engineering Chemistry Research processes are under control actions. This consideration is not valid during transition states because the process variables are changing to reach a new stable state. Then, the data set is continuously split using a constant window length. For each one, the first step of DENCLUE algorithm is applied and density attractors are identified. Finally, a new merging rule is established by using a local distance approach.

3. NEW OFFLINE MODELING METHOD AND ONLINE MONITORING SCHEME PROPOSALS 3.1. Offline Modeling Method for Mode Identification. Before applying clustering algorithm for the mode identification, the offline data set X ∈ 9 n × d , with n samples in a ddimensional space, is divided in a series of continuous data segments with a length of T samples. The selection of this parameter should be conditioned by the dynamic features of the process considering the compromise between computational load and accuracy in the isolation of stable modes. While the window length grows, the dynamic behavior of each variable is lost, the identification of a transition startup is delayed and, consequently, the detection of mode changes. On the contrary, if a small window length is selected, the influence of noise could lead to the division of a stable mode including false transition states. Without a sound knowledge of process characteristics, the window length can be selected as 2−3 times above the number of process variables, according to the modeling experience of multivariate statistical regression methods.33 Additionally, at the end of this section a method to set the window length and analyze the influence of this parameter in the identification of stable modes is proposed and presented. The main interest in the isolation of the stable modes lies in the possibility of applying classic MSPM methods, like those based on PCA, to monitor each one. The monitoring models based on PCA rely on the assumption of a good estimation of the covariance matrix. This matrix will better represent the relationship among variables while the sample size of a single mode is increased. Based on this, the minimum length of clustered samples representing a stable mode is defined as twice the length of the window T, S > 2T. Of course, the fact that samples are grouped based on the convergence with the same density attractor or neighboring density attractors should be taken into consideration. The offline mode identification algorithm is described in Figure 1. After time series are segmented, density attractors are searched for each window. Smoothing parameter selection has been suggested on the basis of several methods.34 When clusters differ in size and shape (e.g., one is tight and other one is sparse), the selection of a single bandwidth may fail to give an accurate estimation of data distribution. The nearest neighbor method, also known as local scaling, has been presented in refs 34 and 35 to overcome this issue. When using this technique a dynamic bandwidth hx is assigned for each data point x, then the proper bandwidth is selected in accordance with the local statistics of the surrounding points. The modified density function is34 1 fĥ (x) = x nhx

⎛ x − xi ⎞ ⎟ ∑ K⎜ hx ⎠ i=1 ⎝

Figure 1. Offline mode identification steps.

hx = dist(x , xknn)

(5)

and xknn is the k-nearest neighbor of x using Euclidean distance. The selection of k, and consequently hx, will affect the shape of the estimated probability density function. If k is too small, the distribution in a window may be wrongly estimated as two distributions. On the other hand, if it is too big, two different distributions would be estimated as the same one. There is a relationship between the window lengths T and k. Then, for each T, the number of neighbors should be estimated in order to obtain the best separability among the obtained stable modes. In order to evaluate the separability, several indexes can be used, such as the Davies−Bouldin,36 the Calinski−Harabasz37 or the gap statistic.38 However, this paper does not attempt to analyze the best separability measure. In order to set the evaluation range for k, the minimum must be established so as to avoid the influence of outliers and the maximum will be assumed as T/2 to prevent the oversmoothing of the estimated distributions. Sample points converging to the same density attractor in each window are associated with a common group. Following this, the distance neighborhood of each density attractor xatt is defined as dist n(xatt) = dist(xatt , xk th)

(6)

where xkth is its kth-nearest neighbor. Then, continuous density attractors groups X(t × d) are clustered based on the nearest neighbor criterion as follows dist(xatti , xattj) ≤ dist n(xatti) + dist n(xattj)

(7)

The evaluation of this rule depends on the value of kth. Hence, it will influence the merging of the windows. If it is too small, a stable mode could be mistakenly identified as a transition. Otherwise, if it is too big, the initial and last part of a transition could be wrongly appended to their closest stable

n

(4)

where hx is calculated as C


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Industrial & Engineering Chemistry Research

Figure 2. Univariate example of the offline mode identification method.

modes. To set kth equals to k is considered as reasonable to avoid the above-mentioned problems. Thus, this is the consideration applied to this paper. Finally, clustered data segments complying with the minimum stable mode requirement are isolated and the remaining continuous windows are grouped as transitions Xt. Each one, is divided into equal length windows Xst. Here, a good quality data set is assumed. This means that outliers, missing data and data alignment issues are addressed in a preprocessing step. The following steps summarize the procedure for the selection of parameters: 1. Set a minimum value for the window length T. It can be selected between 2 and 3 times the number of process variables. 2. Establish a criterion to determine whether resulting windows represent a stable mode or not. In this paper the condition S > 2T is used. 3. Select the amount in which T will be increased. Therefore, the increase will be almost equal to the number of variables. 4. Set the range to determine the appropriate value of k for each window length, in order to get the best separability among clusters representing stable modes. The minimum value should be selected so as to have few or no influence of outliers. A value of k = 10 was used in this paper. The maximum value selected was T/2 to prevent the oversmoothing of estimated distributions. 5. Set the value of kth to determine the neighborhood of each density attractor. Select the value of kth equal to k. 6. Establish a criterion to evaluate the separability among clusters representing stable modes. 7. The number of stable modes can be selected based on the following rules: • If the number of stable modes converge in the same value during n increments of the window’s length T, it takes this value. There are no rules to

select the value of n. We suggest value 4 as the minimum. • If there is not convergence in a specific number of stable modes, select the number of stable modes for which the best separability among the clusters can be reached. 8. Finally, set the window’s length T equal to the smallest value for which the number of stable modes is obtained in order to identify the beginning of each transition as soon as possible. Figure 2 shows an example of the univariate case. The variable time series is included in Figure 2a. For illustrative purposes a total of 450 samples are collected and a window length of 50 is selected. For each one, the kernel density distribution is shown in Figure 2b. In this figure, the local density distributions of the windows, which are part of a transition, are represented by a dashed line. In the first step of the proposed algorithm, the density attractors are identified for each window. The smoothing parameter is selected based on the twelfth nearest neighbor (k = 12). Similarly the neighborhood for each density attractor is calculated by selecting the twelfth nearest neighbor (kth = 12). In the first window, a single density attractor is located at 0.2 and in the second one, at 0.24. The respective distance neighborhoods are 0.04 and 0.03. The distance between the attractors (0.04) is less than the sum of their distance neighborhoods and both windows are merged. In contrast, for windows 3, 4, and 5, more than one density attractor is identified in each one. This is due to the fact that during the transition, the local density distributions could hardly be approximated by a normal distribution. Analogous to the first two windows, the last four also merged. Finally, two stable modes, Xs1 formed by 1−2 window and Xs2 formed by windows 6−9, were isolated, respectively. The three remaining continuous windows define a transition between stable modes. A distance based approach is proposed for their monitoring, instead of using local PCA models. D


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Figure 3. Online monitoring scheme.

The z selection represents a compromise between robustness and sensitivity when detecting abnormal operation modes. For instance, if it is selected as 1, many false alarms could be generated due to the presence of outliers or noise in the measurements. Besides, if it is too big, there would be an increase in the detection delay of faults. Figure 3 shows the flowchart for process monitoring. The mode determination for the first sample in based on the scheme presented in Figure 4. Assumingly, when the process starts, it will be operating normally with at least the first few samples, either on a stable mode or at the beginning of a transition. If the statistics of any of the stable modes are under control limits, this will be selected as the current mode. In case that more than one stable mode can be used, the one with the minimum squared prediction error (SPE) value should be selected. If the sample statistics do not fit any stable mode, the transition with the smallest distance to its first window will be established. Once a transition is taking place, if the statistic of the current window exceeds its control limit, the next one will be evaluated. In order to determine if a stable state has been reached, the statistics of every stable states are evaluated before confirming the presence of a fault. A stable state can be reached when a transition finishes or if it is due to a fast transition. Next, the

Following, a detailed description of the complete monitoring scheme. 3.2. Proposed Online Monitoring Scheme. The online monitoring scheme is based on mode identification. The identification of the operation mode is made implicitly in the proposed method evaluating the monitoring statistics of each stable mode and the beginning of each transition. Assumingly, when an operation mode changes, the process dynamic varies and, consequently, the current control limits will be violated. Following the identification of a stable mode or a transition startup, when the next sample arrives, it will be first evaluated to determine if it belongs to the identified state. If the statistical values are below the current control limits, then it will be assumed that the current mode is still the same. Otherwise, if at least one of the limits is violated for z continuous samples, there will be three possible situations: 1. If the process was in a stable mode, either a transition starts or a different stable mode has been reached due to a fast transition. 2. If the process was in a transitional mode, a stable mode has been reached. 3. It is either an abnormal operation mode or a new unlabeled mode. E


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Industrial & Engineering Chemistry Research Dl =

dist i ≤1 d limst

(11)

When a transient starts, consecutive transition windows are evaluated.

4. CASE STUDY In this section the proposed approach is tested on the Tennessee Eastman (TE) benchmark chemical process. 4.1. Process Description and Simulation Design. The TE process is a simulation benchmark created by the Eastman Chemical Company based on a real industrial plant39 and widely used for testing the performance of multimode monitoring approaches.6,13,16,19,21 It consist of five major unit operations: a reactor, a product condenser, a recycle compressor, a vapor−liquid separator, and a product stripper. A detailed description can be found in ref 1. The decentralized strategy proposed by Ricker40,41 is applied since it leads to less variability in product rate and quality and maintains long periods of on-spec operation without feedback from composition measurements. There are 41 process variables and 12 manipulated variables measured. In the present work 31 variables are used for process monitoring. These include 22 continuous measured variables at higher sampling rate and 9 manipulated variables. The recycle valve, steam valve, and agitator are excluded due to their almost steady state through the simulation. A subset of these features is not selected as in others works19,21,26 due to the fact that different faults do not affect the same variables. As mentioned in ref 39 all process measurements include Gaussian noise with standard deviation typical of the measurement type. The sampling interval is 0.01 h. The modes 1, 2, and 3 defined in ref 39 are used for testing in this paper. It must be remarked that in comparison with other works that take into account transitions for monitoring purposes,19,21,26 the ones simulated in this article involve greater changes in the dynamic of the process. This is because the changes of mode are given by the variation in the final product characteristics and the production rate. Consequently, an accurate isolation of the stable states for the characterization of modes and transitions is harder to achieve. Even more, the monitoring of the process is more complex due to the fact that the variables exhibit more significant nonlinearities during the transitions. Table 1 includes the simulation case of the normal process behavior including transitions. Three dynamic transitions are

Figure 4. Scheme to determine the mode of the first sample.

monitoring model for stable-state and transitional mode are presented. 3.3. Stable-State Monitoring Model. Fault detection for stable-state models is performed on the basis of principal component analysis (PCA)1,2 monitoring statistics T2, and SPE. For each reference normalized mode X, the PCA algorithm is performed to obtain the corresponding covariance matrix. Σ=

XTX n−1

(8)

Singular value decomposition (SVD) Σ = VΛVT is computed. In that, Λ ∈ Rd×d is a diagonal matrix which contains the eigenvalues of Σ, arranged in descending order of variance, and V contains the corresponding eigenvectors. Selecting the first a eigenvectors from V in P for each monitoring model, the respective statistics Ti2 and SPEi are calculated for each sampling instant i in the traditional way.1 The control limits for each stable-state mode are calculated by assuming that the exact covariance matrix of the population is unknown. The number of retained principal components (PCs) is determined by calculating the smallest number of loading vectors needed to explain a specific minimum percentage of the total variance.1 3.4. Transition Monitoring Model. Transition pattern tracking is achieved based on a distance evaluation approach. Continuous subtransition states are stored as a transitional mode between two stable modes. When control limits are exceeded for the stable-state model Xs, the current sampling should be evaluated to determine if it fits the initial subtransition state of any of the transient patterns. For each subtransition model Xst a limit based on the Mahalanobis distance is defined as d limst (Xst ) = max{dist(xj , Xst ):

∀ xj ∈ Xst }

Table 1. Simulation Case Study of Normal Process Behavior desired G/H mass ratio

desired production rate (kg/h)

samples

3 1 2 3

90/10 50/50 10/90 90/10

11111 14076 14076 11111

1−1000 1001−7000 7001−12000 12001−20000

(9)

present in the data set. Process runs initially in mode 3 for about 10 h, and switch to mode 1. After 60 h, the process changes to mode 2 for other 50 h and switch again to mode 3, for the last 80 h. The last transition is the longest in time due to a greater change in the characteristics of the final product. The objective of this paper is not to improve the monitoring results during stable modes, therefore all faults are tested during

and dist(xj , Xst ) = (xj − μst )Σst −1(xj − μst )T

mode

(10)

where μst is the mean vector and Σst the covariance matrix of Xst. For current sampling i, disti(xi, Xst) is calculated and considered under control if F


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Industrial & Engineering Chemistry Research transitions. In this paper, in contrast with previous works,19,21,26 where few types of faults are simulated, a total of eight different kinds of faults are evaluated. Detailed fault information is presented in Table 2. As can be observed, various types of faults are simulated during different transitions in order to test the detectability of the proposed monitoring method in these states.

Table 3. Simulation Case Study of Normal Process Behavior

Table 2. Simulation Cases of Faults in Different Transitional Modes case

no.

1

1

2

10

3 4

17 2

5

11

6 7

13 5

8

2+8

fault A/C feed ratio, B composition constant (stream 4) C feed temperature (stream 4) unknown B composition constant, A/C ratio constant (stream 4) reactor cooling water inlet temperature reaction kinetics condenser cooling water inlet temperature multistep step and random variation faults (fault 1 and 8)

type

transitional mode

step

3−1

random variation unknown step

3−1

random variation slow drift step step + random variation

a + 2ad + a 2 2d

mode stable mode A transition AB stable mode B transition BC stable mode C transition CD stable mode D

before, the simulated modes involve changes in product characteristics and production rate, and therefore, the isolated transient patterns have a longer duration. Four stable modes are identified and transition startups are clearly detected. Although there are really three production modes mixed in the data set, the fourth isolated stable mode differs from the first because a small part of its previous transitional mode merges. However, as will be checked, this issue is not relevant for process monitoring. The monitoring results of the simulation case with a different data set can be observed in Figure 5a and b using a logarithmic scale for the y axis. In Figure 5a the SPE statistic is used, but for visualization purposes; instead of using a separate chart for the Dl statistic when a transition is identified, its value is presented alternatively with SPE. The control limit of 1 previously established, allows to visually distinguish when a transition is in course. This is because only in special cases the control limit for a stable mode is equal to 1. Similarly, the results of the T2 statistic are presented in Figure 5b. Again, the Dl statistic is alternated in this chart. As it is shown, the monitoring scheme is capable of adapting the control limits in accordance with mode changes having a very low false alarm rate (FAR) of only 0.75%. This proves that if a good isolation of the stable modes is achieved with the proposed method for offline mode identification, each mode and transition will be correctly modeled. Compared with the paper presented in ref 19, our proposal shows an improvement in the FAR indicating that it is more appropriate for mode identification and transition tracking. For all cases, the simulation begin with 500 samples in stable mode before transition starts. Faults are introduced after 1000 samples from the beginning of the transition. After that, the simulation will be running for another 1000 samples. Figures 6 and 7 show the monitoring results for cases 1 and 2. For both cases the FAR is 0. In the first one, there is a detection delay of only 4.2 min and a fault detection rate (FDR) of 99.40%. Ignoring the fact that different types of mode are simulated in the articles presented by Tan et al. and Zhu et al.,19,21 if results are compared for this fault, a similar delay in the detection is accomplished and a better FDR and FAR is achieved. It must be noted that in these papers this is the single fault tested. In the case of the other work by Zhu et al.,26 other faults are simulated but no fault is tested during transition. For case 2, there is a delay of 34.2 min and a FDR of 94.41%. The simulated fault here is of random variation type and can be easily confused with the current transition pattern, so detection is difficult. However, as the duration of the transition is too long, the delay in detection is not considerable and the FDR is good. In case 6 (Figure 8), the FDR is 88.51%. The fault simulated here, is a slow drift type and there is a transition in course. Then, a considerable effect on the dynamic of the process takes

3−1 1−2 1−2 1−2 2−3 2−3

4.2. Model Parameters. The first parameter to set is the minimum value to evaluate T. In the experiments, the minimum T is selected between 2 and 3 times the number of variables, hence Tmin = 70 samples. The increase of T is set approximately in conformity to the number of variables. The selection of other parameters, namely, S, k, and kth has been established as previously proposed so that each parameter depends on T. The separability measure used to estimate k was the Davies−Bouldin criterion, since it has been found to be one of the most efficient.42 The number of stable modes obtained for T = 70 is 5. In this case, stable mode 2 is split in two continuous stable modes. When T was increased to 100, 130, 160, and 190, the number of isolated stable modes converged to 4. Then, the smallest window length T = 100 is selected so as to accurately identify the beginning of a transition and improve the sensitivity of the fault detection method during transitions. Consequently, the minimum stable mode duration is set as S = 300. The final smoothing parameter resulting from the experiment is selected on the basis of k = 27 nearest neighbor and consequently, kth = 27. The confidence level for T2 and SPE statistics is set as 99%. The required percent of variance explained for each stable state model was selected as 90. The number of retained components can change from mode to mode. The required number of continuous samples z to declare the fault detected is selected as 5. The rule to build a stable PCA model was the following, to be always checked to guarantee a minimum threshold for the sample size1 Tth =

samples 1−1000 1001−3400 3401−7000 7001−9100 9101−12000 12001−16300 16301−20000

(12)

where a is the number of PCs and d the number of monitoring variables. 4.3. Results and Discussion. After the offline mode identification algorithm was applied on the simulation case, the results obtained are summarized in Table 3. As it was stated G


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Figure 5. Monitoring results for the simulation case study of the normal process behavior.

Figure 6. Monitoring results for simulation case 1.

The main advantages of the proposed method for offline mode identification compared with others are as follows: (1) resulting stable modes and transitions are obtained in only one run of the algorithm; (2) no assumptions are made concerning the probability distribution of process variables; and (3) there is no need to establish the number of modes with anticipation. However, there are some limitations, the main one is that for certain processes, setting the parameter S can be relatively subjective, which determines the minimum duration of a stable mode. Furthermore, the selection of the window length T could be difficult for processes with very complex dynamics and this could affect the accurate isolation of the stable modes. The proposed monitoring scheme presents the following benefits: (1) it is capable of tracking complex changes of mode

too much time to be perceived. Consequently, the delay in the detection time is 69.6 min, being the greater among all simulated faults. For case 7 (Figure 9), the alarm time is of 2.4 min and it must be highlighted that, even when the control system masks the fault effect, there is an FDR of 99.70%. Table 4 summarizes the FAR, FDR, and delay in detection for each case. As shown, all FDR are above 80% thus confirming that there is a good detection capability, and for all faults, the FAR is 0. Some of the faults present a large delay in detection, mainly for the random variation and slow drift type. The results obtained indicate that the proposed monitoring scheme is capable of following complex dynamic transitions between modes and detect efficiently different types of faults. H


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duration of the stable modes. Then, a novel monitoring scheme for fault detection is proposed. In this one, a new monitoring method for transitions is included to achieve a good performance during these states. The Tennessee Eastman process is used as a case study, and three modes are simulated. In contrast with previous works, these modes involve changes of the final product characteristics and production rate. Even more, different types of faults are tested during transitions. The results prove the effectiveness of the proposal. Stable modes are successfully isolated from transitions. The monitoring scheme demonstrates that it is able to adapt the control limits according to the changes of mode. In addition, a good detection of different types of faults during transitions is

with very low FAR; (2) faults can be detected during the transitions with good FDR and relatively low delay in detection; (3) a simple visualization tool is provided for the operator to determine whether the process is in a stable mode or in transition; and (4) practical implementation can easily be accomplished since only mean vectors, covariance matrixes and control limits need to be stored for each stable mode and subtransition state. However, it must be noticed that good results depend on the correct isolation of stable modes from transitions.

5. CONCLUSIONS In this article, a new mode identification method is presented for offline partition of multimode process data in stable modes and transitions. The method is based on local KDE and the I


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Figure 9. Monitoring results for simulation case 7. (3) Macgregor, J.; Cinar, A. Monitoring, fault diagnosis, fault tolerant control and optimization: Data driven methods. Comput. Chem. Eng. 2012, 47, 111−120. (4) Ge, Z.; Song, Z.; Gao, F. Review of Recent Research on DataBased Process Monitoring. Ind. Eng. Chem. Res. 2013, 52, 3543−3562. (5) Kruger, U.; Xie, L. Statistical Monitoring of Complex Multivariate Processes; John Wiley & Sons, Ltd.: West Sussex, U.K., 2012. (6) Wang, F.; Tan, S.; Peng, J.; Chang, Y. Process monitoring based on mode identification for multi-mode process with transitions. Chemom. Intell. Lab. Syst. 2012, 110, 144−155. (7) Ma, H.; Hu, Y.; Shi, H. A novel local neighborhood standardization strategy and its application in fault detection of multimode processes. Chemom. Intell. Lab. Syst. 2012, 118, 287−300. (8) Hwang, D.-h.; Han, C. Real-time monitoring for a process with multiple operating modes. Control Engineering Practice 1999, 7, 891− 902. (9) Lane, S.; Martin, E. B.; Kooijmans, R.; Morris, A. J. Performance monitoring of a multi-product semi-batch process. J. Process Control 2001, 11, 1−11. (10) Ma, Y.; Shi, H. Multimode Process Monitoring Based on Aligned Mixture Factor Analysis. Ind. Eng. Chem. Res. 2013, 53, 786− 799. (11) Xie, X.; Shi, H. Dynamic Multimode Process Modeling and Monitoring Using Adaptive Gaussian Mixture Models. Ind. Eng. Chem. Res. 2012, 51, 5497−5505. (12) Ge, Z.; Song, Z. Online monitoring of nonlinear multiple mode processes based on adaptive local model approach. Control Engineering Practice 2008, 16, 1427−1437. (13) Tong, C.; Palazoglu, A.; Yan, X. An adaptive multimode process monitoring strategy based on mode clustering and mode unfolding. J. Process Control 2013, 23, 1497−1507. (14) Zhao, S. J.; Zhang, J.; Xu, Y. M. Monitoring of Processes with Multiple Operating Modes through Multiple Principle Component Analysis Models. Ind. Eng. Chem. Res. 2004, 43, 7025−7035. (15) Chen, J.; Liu, J. Mixture Principal Component Analysis Models for Process Monitoring. Ind. Eng. Chem. Res. 1999, 38, 1478−1488. (16) Ning, C.; Chen, M.; Zhou, D. Hidden Markov Model-Based Statistics Pattern Analysis for Multimode Process Monitoring: An Index-Switching Scheme. Ind. Eng. Chem. Res. 2014, 53, 11084−11095. (17) Zhang, S.; Wang, F.; Tan, S.; Wang, S.; Chang, Y. Novel Monitoring Strategy Combining the Advantages of the Multiple

Table 4. Monitoring Results of Simulation Cases case

FAR (%)

FDR (%)

delay in detection (min)

1 2 3 4 5 6 7 8

0 0 0 0 0 0 0 0

99.40 94.41 97.50 99.10 98.90 88.51 99.70 99.00

4.2 34.2 15.6 6 7.2 69.6 2.4 6.6

achieved. A high FDR, low FAR, and relatively short delays in detection are obtained depending on the type of fault. Further research could be made to improve fault detection in the case of incipient faults during transitions. Moreover the development of fault diagnosis methods for multimode processes with transitions could be a future research work.

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AUTHOR INFORMATION

Corresponding Author

*Phone: +53 7 2663280 E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors would like to thank the anonymous reviewers because their thought-provoking and insightful comments and corrections have been very useful for improving the quality of the paper.

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REFERENCES

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Modeling and Monitoring for Transitions Based on Local Kernel

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