Industrial Process Monitoring Based on KnowledgeâData Integrated

Article Cite This: Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

pubs.acs.org/IECR

Industrial Process Monitoring Based on Knowledge−Data Integrated Sparse Model and Two-Level Deviation Magnitude Plots Lijia Luo,* Shiyi Bao, Jianfeng Mao, and Zhenyu Ding Institute of Process Equipment and Control Engineering, Zhejiang University of Technology, Hangzhou 310014, China

ABSTRACT: Data-driven process monitoring methods use only process data to build monitoring models, while useful process knowledge is completely ignored. Consequently, data-driven monitoring models usually have poor interpretability, which may reduce fault detection and diagnosis capabilities. To overcome such drawbacks, a novel knowledge−data integrated sparse monitoring (KDISM) model and two-level deviation magnitude plots are proposed for industrial process monitoring. The basic idea of KDISM is to build a sparse and interpretable monitoring model by integrating process data with process knowledge. The procedure for building the KDISM model consists of four steps: (1) use basic process knowledge to analyze meaningful connections among process variables; (2) divide process variables into different groups according to variable connections; (3) construct a knowledge-based sparse projection (KBSP) matrix on the basis of the variable grouping results; (4) apply the KBSP matrix to process data to build the KDISM model. The KDISM model has good interpretability because the KBSP matrix reveals meaningful variable connections. The KDISM model is also able to eliminate redundant interference between variables owing to the sparsity of the KBSP matrix. These two advantages make the KDISM model well-suited for fault detection and diagnosis. Two fault detection indices are defined based on the KDISM model for detecting the occurrence of faults. The variable deviation magnitude (VDM) is defined to quantify deviations of variables from the normal values. Based on the VDM, two-level deviation magnitude plots are proposed for fault diagnosis, with the first-level groupwise VDM plot used to identify faulty variable groups while the second-level square VDM plot is used to identify faulty variables. The effectiveness and advantages of the proposed methods are illustrated by an industrial case study. techniques, such as principal component analysis (PCA),1 partial least-squares (PLS),6 locality preserving projections (LPP),7 global−local preserving projections (GLPP),8 and independent component analysis (ICA).9 The MSA techniques are used to project high dimensional process data into a low dimensional space defined by a set of latent variables, so that the important information contained in process data is captured by latent variables. Each latent variable is a combination of process variables, and it represents the underlying correlations among process variables. Based on the latent variables, a monitoring model is then built for fault detection and diagnosis. The process monitoring performance depends on whether the latent variables reveal useful

1. INTRODUCTION To meet increasingly stringent requirements on process safety and high product quality, process monitoring is becoming more and more important to industrial processes. Two major tasks of process monitoring are fault detection and fault diagnosis. Fault detection is to detect process faults caused by improper operation, equipment malfunction, disturbances, and so on. Fault diagnosis is to identify faulty process variables and diagnose the root causes of faults. Effective process monitoring can guarantee production safety, prevent equipment damage, and provide guidance for fault elimination. Over the past few decades, data-driven process monitoring methods have been attracting increasing attention.1−5 Because data-driven monitoring methods need only process data rather than engineering knowledge about industrial processes, they are easy to implement in industrial practice. Most existing data-driven monitoring methods were proposed based on multivariate statistical analysis (MSA) © XXXX American Chemical Society

Received: May 25, 2017 Revised: November 15, 2017 Accepted: December 14, 2017

A

DOI: 10.1021/acs.iecr.7b02150 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research

Figure 1. Illustration of variable connections in industrial processes.

sparse latent variables from process data using sparse MSA methods. The key issue is how to utilize the physical/chemical connections between process variables to build an interpretable monitoring model. In order to eliminate process faults, fault diagnosis should be carried out immediately after faults are detected. Two commonly used fault diagnosis methods are fault reconstruction15,16 and contribution plots.17 In the fault reconstruction method, several reconstructed samples are computed from the faulty sample by correcting the effect of faults along a group of possible fault directions {Ξi} that correspond to known fault modes.15 The most likely fault direction, i.e., fault mode, is then selected from {Ξi} according to whether the reconstructed sample corresponding to this fault direction is closest to a normal sample.15 Although the fault reconstruction method has higher fault diagnosis accuracy, it is not easy to implement in actual industrial processes because a group of known fault directions are required. Contribution plots show contributions of process variables to the fault detection indices (e.g., T2 and SPE statistics),17 and the variable with the largest contribution is identified as a faulty variable. Contribution plots are easy to calculate; however, they suffer from the smearing effect caused by the interference between variables.17 Because of the smearing effect, some normal variables may have larger contributions than actual faulty variables. This may lead to unreliable fault diagnosis results. In addition, both fault reconstruction and contribution plot methods require a process monitoring model. Their fault diagnosis capabilities therefore depend on the process monitoring models. Using an inappropriate monitoring model may reduce the fault diagnosis accuracy. Therefore, it is necessary to develop more effective fault diagnosis methods that not only have high fault diagnosis accuracy but also are easy to implement in industrial processes. In this paper, a novel knowledge-data-integrated sparse monitoring (KDISM) model and two-level deviation magnitude plots are proposed for fault detection and diagnosis in industrial processes. The KDISM model is built by integrating process data with process knowledge. The KDISM model has much better interpretability than the data-driven models built by MSA techniques, because it not only reveals meaningful

connections among process variables. However, traditional MSA techniques can only produce dense latent variables, with each dense latent variable consisting of all process variables. Such dense latent variables lead to two drawbacks when applied to fault detection and diagnosis. First, the dense latent variable cannot fully eliminate interference among process variables. Because of variable interference, the influences of faulty process variables on latent variables can be diminished by the fault-free process variables. Consequently, the dense latent variables may be insensitive to the occurrence of faults, which reduces the fault detection performance. Second, the dense latent variables do not explicitly reveal meaningful connections (e.g., control connections or other physical/ chemical connections) between process variables. Therefore, dense latent variables usually have poor interpretability, and they are unsuitable for fault analysis and diagnosis. To overcome these drawbacks, some sparse MSA techniques, e.g., sparse global−local preserving projections (SGLPP),10 sparse principal component analysis (SPCA)11−13 and variablecorrelation-based sparse modeling,14 have recently been proposed to produce sparse latent variables. Unlike the dense latent variable, a sparse latent variable consists of only a few process variables. Sparse latent variables have the potential advantage of being easier to interpret, and thus, they are suitable for fault analysis and diagnosis. A main difficulty in using sparse MSA techniques is the selection of the appropriate level of sparsity (i.e., the number of nonzero elements) for each sparse latent variable. The sparsity of a sparse latent variable is often determined by some penalty parameters.10−13 However, it is difficult to choose appropriate penalty parameters that can produce sparse latent variables with the optimal sparsity. In addition, there is no guarantee that the sparse latent variables can reveal meaningful variable connections. In fact, for a given industrial process, the physical/chemical connections between process variables are explicit, and they can be easily obtained from the process flow diagram (PFD) and the piping and instrumentation diagram (P&ID) of the process. Therefore, it is possible to build an interpretable monitoring model according to the physical/ chemical connections between variables, instead of extracting B


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the reactor in Figure 1a, the flow rate (v6) and the separator level (v4) in Figure 1b, the flow rate (v7) and the separator pressure (v3) in Figure 1b, or the flow rate (v6) and the valve position (v5) in Figure 1c. 2.2. Variable Grouping. According to the above four categories of variable connections, process variables in an industrial process can be divided into different groups. A possible variable grouping procedure is as follows: Step 1: Record all online measurable process variables to form the variable list for grouping. Step 2: Assign process variables with the control connection into the same group, so that each group represents a control loop in the industrial process. For example, variables v1 and v2 in Figure 1a are assigned into the same group as (v1, v2). Step 3: Assign process variables which have both reaction and location connections into the same group, so that each group represents a biochemical reaction in the industrial process. Step 4: Assign process variables which have both location and type connections into the same group, so that each group represents a physical link in the industrial process. For example, variables v1 and v2 in Figure 1b are assigned into the same group as (v1, v2). For two variables with both location and type connections, if one of them has been assigned into a group corresponding to a control loop or a biochemical reaction, then the other variable can be assigned into the same group. For example, variables v5, v6, and v7 in Figure 1c are assigned into the same group as (v5, v6, v7), because v5 and v7 have the control connection and v6 has both location and type connections with v5. Step 5: Remove all grouped variables from the variable list. Step 6: Divide n remaining process variables into n different groups, so that each variable represents a group. The above variable grouping procedure leads to the variable grouping result with two characteristics: (1) all process variables are included in variable groups; (2) there is little overlap between different variable groups. The first characteristic enables the knowledge−data integrated sparse monitoring model, which is built on the basis of the variable grouping result, to detect faults associated with any process variables. The second characteristic is helpful to eliminate the redundant interactions between variables and to simplify the monitoring model. After dividing all process variables into different groups, a subsequent step is to determine that variables in the same group are positively or negatively correlated. For two variables with the control connection, if the manipulated variable should be decreased in response to a reduction in the set point of the controlled variable, the correlation between the manipulated variable and the controlled variable is positive; otherwise, their correlation is negative. For two variables with reaction, location, or type connections, if the increase in one variable leads to the increase in the other variable, the correlation between two variables is positive; otherwise, their correlation is negative. 2.3. Knowledge-Based Sparse Projection Matrix. Based on the variable grouping result, a knowledge-based sparse projection (KBSP) matrix is constructed for building the process monitoring model. If m process variables have been divided into l variable groups, then a projection matrix P = [p1, ..., pl] ∈ 9 m × l is constructed, with each column pj (j = 1, ..., l) corresponding to a variable group. The elements in

variable connections but also eliminates the redundant interference between process variables. This advantage makes the KDISM model well-suited for fault detection and diagnosis. Based on the KDISM model, the T2 and SPE statistics are defined and used for fault detection. The variable deviation magnitude (VDM) is defined to quantify the deviations of variables from the normal values. Based on the VDM, two-level deviation magnitude plots are proposed for fault diagnosis. The first-level groupwise VDM plot is used to identify faulty variable groups. The second-level square VDM plot is used to identify faulty variables. The proposed methods are applied to the Tennessee Eastman process to illustrate their effectiveness and advantages. The rest of the paper is organized as follows. Section 2 introduces the procedure for building the KDISM model. The process monitoring method, consisting of two fault detection indices defined based on the KDISM model and the two-level deviation magnitude plots for fault diagnosis, is presented in section 3. Section 4 demonstrates the effectiveness and advantages of the proposed methods by an industrial case study. The conclusions are given in section 5.

2. KNOWLEDGE−DATA INTEGRATED SPARSE MODELING PROCEDURE 2.1. Variable Connection Analysis. The PFD and P&ID shows the process stream, the piping of the process, the interconnection of process equipment, and the installed instrumentation used to control the process. Therefore, the PFD and P&ID provide enough process knowledge for analyzing the physical/chemical connections among process variables. In most industrial processes, the physical/chemical connections among variables can be classified into four categories: (1) Control connection: If process variables constitute a control loop, there is the control connection between them, for example, variables (v1, v2), (v3, v5), or (v6, v7) in Figure 1a. (2) Reaction connection: If the chemical substance A is converted to the chemical substance B by biochemical reactions, there is the reaction connection between them. Thus, the flow rate and concentration of the chemical substance A have the reaction connections with those of the chemical substance B. (3) Location connection: If process variables locate in the same process equipment (including pipes) or in two directly connected process equipment, there is the location connection between these variables, for example, variables (v1, v2) or (v3, v4, v5) in Figure 1b. (4) Type connection: The type connection can be further subdivided into two classes: strict type connection and generalized type connection. If process variables belong to the same variable type (e.g., temperature, pressure, flow rate, or concentration of the same chemical substance), there is the strict type connection between these variables, for example, two pressures (v1, v2), two temperatures (v3, v4), or three flow rates (v8, v9, v10) in Figure 1c. If process variables belong to the similar variable type (e.g., temperature versus heat, flow rate versus valve position, liquid flow rate versus liquid level, or gas flow rate versus pressure), there is the generalized type connection between these variables, for example, the reactor temperature (v3) and the generated heat in C


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Industrial & Engineering Chemistry Research each column vector pj of the matrix P are determined by three rules: (I) If the ith (i = 1, ..., m) variable is in the jth group, the ith element pij in the jth vector pj satisfies pij ≠ 0; otherwise, pij ≠ 0. Besides, any two nonzero elements pij and pkj in the jth vector pj satisfy |pij| = |pkj|. (II) If the correlation between the ith and kth process variables in the jth variable group is positive, two nonzero elements pij and pkj in the jth vector pj satisfy pij = pkj; otherwise pij = −pkj. (III) Each vector pj is scaled to unit length, i.e., pTj pj = 1. The matrix P is named as the knowledge-based sparse projection (KBSP) matrix. This KBSP matrix reveals the control, reaction, location, and type connections between process variables. 2.4. Knowledge−Data Integrated Sparse Monitoring Model. A knowledge−data integrated sparse monitoring (KDISM) model is built by applying the KBSP matrix to process data. Let X = [x1, ..., x n]T ∈ 9 n × m denote the normalized training data set consisting of n samples and m process variables. A KDISM model is built as

section 2.2, process variables are grouped in the following order: (1) control connection, (2) reaction and location connections, (3) location and type connections. In other words, the variables with control connections are grouped first, followed by the variables with both reaction and location connections, and finally the variables with both location and type connections. Process variables can be grouped in a different order if necessary. For example, if the reaction connection is considered to be more important than the control connection in some industrial processes, variables with reaction connections can be grouped first, and then variables with control connections are grouped. For a given industrial process, the control, location, type, and reaction connections (as defined in section 2.1) among process variables are fixed, and they are explicitly described by the PFD and P&ID of the industrial process. Thus, different persons will obtain the same/similar variable groups and also the same/similar KBSP matrix, if they follow the same variable grouping procedure. This reduces the effects of human factors on the monitoring performance when the proposed KDISM method is applied to an actual industrial process.

3. PROCESS MONITORING METHOD 3.1. Fault Detection Indices. Two fault detection indices, i.e., the T2 and SPE (square prediction error) statistics,18 are used for fault detection. Based on the KDISM model, the T2 and SPE statistics of a sample x are computed by

T = XP X = TP+ + E E = X − TP+

(1)

m×l

T 2 = (t − t ̅ )T S−1(t − t ̅ )

where P ∈ 9 is the KBSP matrix, P+ denotes the Moore− Penrose pseudoinverse of P, T ∈ 9 n × l is a score matrix, and E ∈ 9 n × m is a residual matrix. A new sample x new ∈ 9 m is projected on the KDISM model by

2

SPE = || e ||2 = || x − P+Tt ||

(4)

where t and e are score and residual vectors of the sample x, t ̅ denotes the mean of all row vectors of the score matrix T in eq 1, and S is the covariance matrix of the score matrix T. Confidence limits of the T2 and SPE statistics are obtained by kernel density estimation (KDE). The probability density function of T2 statistics of all training samples is estimated by KDE19

t new = PTx new x new = P+Tt new + enew enew = x new − P+Tt new

(3)

(2)

Note that the KBSP matrix P not only reveals meaningful variable connections but also eliminates the redundant interference among process variables. These advantages make the KDISM model well suited for fault detection and diagnosis. 2.5. Discussion. The essential difference between the KDISM model and traditional data-driven monitoring models built by multivariate statistical analysis (MSA) methods (e.g., PCA) lies in the KBSP matrix. Unlike the projection matrix (e.g., the loading matrix of PCA) extracted by MSA methods from process data, the KBSP matrix is constructed according to physical/chemical connections among process variables. This KBSP matrix has two important features: interpretability and sparsity, because each column of the KBSP matrix corresponds to a variable group consisting of only a few variables with explicitly physical/chemical connections. To make the KBSP matrix easier to construct in industrial practice and to achieve the sparsity of the KBSP matrix, only the basic and direct physical/chemical connections (i.e., the control, reaction, location and type connections) among process variables are considered. Such physical/chemical variable connections can be easily obtained from the PFD and P&ID of the industrial process, without relying too heavily on the detailed engineering knowledge. Variable grouping is an important step in constructing the KBSP matrix. According to the variable grouping procedure in

f ̂ (y ) =

1 nθ

i y − Ti2 yz zz zz θ k {

∑ K jjjjj n

i=1

(5) 2

where n is the total number of training samples, Ti is the T2 statistic of the ith training sample xi, θ is a bandwidth parameter, and K(·) denotes a kernel function. The confidence limit, Tc, of the T2 statistic at significance level α (α = 0.01 or 0.05) is calculated by Tc

∫−∞ f ̂ (y) dy = 1 − α

(6)

The confidence limit, SPEc, of the SPE statistic can be obtained in a similar way. A process fault is detected when either the T2 or SPE statistic of a new sample exceeds the confidence limit Tc or SPEc. 3.2. Fault Diagnosis Using the Two-Level Deviation Magnitude Plots. Fault diagnosis aims to identify faulty variables responsible for the fault and infer the root cause of the fault. The key issue of fault diagnosis is to determine which process variables deviate significantly from the normal operating range. Luo et al.20 proposed to use the variable deviation magnitude (VDM) as a metric for quantifying the deviations of variables from normal values. A sample x can be expressed as20 D


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Figure 2. Offline modeling and online monitoring procedures.

x = x̅ + Λd

(7)

reactions in the industrial process. The second-level deviation magnitude plot shows the square VDM given by

where x̅ = [x1̅ , ..., xm̅ ]T is the mean of all training samples, Λ = diag (σ1, ..., σm) is a diagonal matrix consisting of standard deviations σi (i = 1, ..., m) of m variables in training samples, and d = [d1, ..., dm]T is the VDM from the mean x.̅ The normal range [d̲ , d̅ ] of the VDM is determined by the training data set. In a faulty sample, there may be at least one variable vi for which di < d̲ i or di > di̅ . To improve the fault diagnosis capability by using the meaningful variable connections in each variable group, twolevel deviation magnitude plots are proposed for online fault diagnosis. The first-level deviation magnitude plot shows the group-wise VDM given by

Dî = di 2

where di is the deviation magnitude of the ith variable. If D̂ i 2 exceeds the confidence limit, max( d̲ j 2 , dj̅ ), the ith process variable is considered faulty. The fault in question is most likely caused by the variable with the largest D̂ i. The two-level deviation magnitude plots can be used to identify faulty variable groups and faulty variables simultaneously. 3.3. Modeling and Monitoring Procedures. The process modeling and monitoring procedures are shown in Figure 2 and summarized as follows: Stage I: Of f line modeling Step I-1: Analyze the control, location, type, and reaction connections between process variables using the PFD and P&ID of the industrial process. Step I-2: Divide process variables into different groups according to variable connections. Step I-3: Construct the knowledge-based sparse projection matrix. Step I-4: Normalize the training data set X to zero mean and unit variance. Step I-5: Build the knowledge−data integrated sparse monitoring (KDISM) model. Step I-6: Determine the confidence limits of T2 and SPE statistics. Stage II: Online monitoring

m

Dj =

∑ pij 2 di 2 i=1

(9)

(8)

where j = 1, ..., l is the index of variable groups, pij is the ith element in the jth column pj of the KBSP matrix P, and di is m the deviation magnitude of ith variable. Note that ∑i = 1 pij 2 = 1 because each column pj of the matrix P satisfies p Tj pj = 1. Thus, the groupwise VDM represents the weighted average square deviation magnitude of variables in the same group. The variable group with the largest Dj most likely contains faulty variables. Due to the meaningful variable connections in each variable group, the groupwise VDM is able to associate the faults with control loops, physical links, or chemical E


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Figure 3. Simplified P&ID of the TE process.

4.2. Variable Grouping Results. According to variable connections shown in the P&ID of the TE process, 33 process variables are divided into 18 groups by following the variable grouping procedure in section 2.2. The variable grouping result is shown in Table 3. Variables in groups 1−13 have control connections, corresponding to control loops in the TE process (see Figure 3). The variable v14 (product separator underflow) is assigned into the group 9 because it has both location and generalized type connections with the variable v29 (separator pot liquid flow valve). Variables v11 (product separator temperature) and v22 (condenser cooling water outlet temperature) constitute the group 14 as they have both location and type connections. The remaining variables, v6 (reactor feed rate), v7 (reactor pressure), v16 (stripper pressure), and v20 (compress work), are separately divided into groups 15−18, because they do not have the control connection or the location and type connections with other variables. According to the three rules in section 2.3, a knowledge-based sparse projection matrix P with the size of 33 × 18 is constructed on the basis of the variable grouping result. Each column pj (j = 1, ..., 18) of P corresponds to a variable group. 4.3. Fault Detection Results. The knowledge-based sparse projection matrix was applied to the training data set to build the KDISM model. For comparison, a PCA monitoring model was also built by implementing PCA on the training data set. To build the PCA model, the knowledgebased sparse projection matrix P in eq 1 was replaced by a PCA loading matrix consisting of 18 loading vectors. The

Step II-1: Normalize a new sample xnew using the mean and variance of training data. Step II-2: Project xnew onto the KDISM model. Step II-3: Compute the T2 and SPE statistics of xnew. Step II-4: Check whether the computed T2 or SPE statistic exceeds the confidence limit? If not, skip to step II-6; otherwise, report a fault and go to step II-5. Step II-5: Diagnose the cause of the fault using the two-level deviation magnitude plots. Step II-6: Return to step II-1 and monitor the next sample.

4. CASE STUDY ON THE TENNESSEE EASTMAN PROCESS 4.1. Process Description. The Tennessee Eastman (TE) process is a well-known benchmark for testing the performance of various fault detection and diagnosis methods.21 Figure 3 shows a simplified P&ID of the TE process.22 There are five unit operations in this process: a reactor, a condenser, a vapor−liquid separator, a recycle compressor, and a product stripper. As shown in Table 1, 22 continuous measurements (v 1 −v 22 ) and 11 manipulated variables (v 23 −v 33 ) are monitored. Process disturbances in Table 2 can be used to simulate the abnormal operating status. A training data set and 21 fault data sets were generated under the normal condition and 21 abnormal operating conditions, respectively. Each data set contains 960 samples, and the disturbance is introduced after the 160th sample. F


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false alarm rate (FAR).23 A higher FDR, a shorter FDD, and a lower FAR represents better fault detection performance. Table 4 shows fault detection results of KDISM and PCA for 21 faults. A total of 18 faults, with the exception of faults 3, 9, and 15, are clearly detected by two methods. FDRs for faults 3, 9, and 15 are very low because these faults have less effect on process variables. The KDISM and PCA methods show comparable detection performance for faults 1, 2, 4, 6−8, 12− 14, 17, 18, and 21. However, for faults 5, 10, 11, 16, 19, and 20, FDRs of the KDISM T2 statistic are much higher than those of the PCA T2 and SPE statistics. In addition, KDISM has shorter FDDs than PCA for faults 10, 16, 19, and 20. To further verify the fault detection performance of the KDISM method, Table 5 compares the mean FDR (MFDR) and mean FAR (MFAR) of five methods for 18 detectable faults (with the exception of faults 3, 9, and 15). The MFDRs and MFARs of GLPP, SPCA, and GLSA were computed using the data reported in refs 8, 12, and24. As shown in Table 5, the KDISM T2 statistic has the largest MFDR, which is slightly higher than those of GLPP and SPCA, but much higher than those of GLSA and PCA. In particular, the MFDR of the KDISM T2 statistic increases by 33.4% and 22.7% as compared to the MFDRs of PCA T2 and SPE statistics. The results in Table 5 validate that the KDISM method has better fault detection performance than other four methods. Figures 4 and 5 show monitoring charts of KDISM and PCA methods for faults 10 and 16, respectively. As shown in Figure 4a, the KDISM T2 statistic detects the occurrence of fault 10 at the 180th sample, which is earlier than the fault detection time (at the 194th sample in Figure 4b) of the PCA method by 14 samples. Moreover, most of fault samples are detected by the KDISM T2 statistic, while more than half of fault samples are not detected by the PCA T2 and SPE statistics. Consequently, the KDISM has a higher FDR and a shorter FDD than PCA for fault 10, as shown in Table 4. Figure 5a shows that fault 16 is detected by the KDISM T2 statistic at the 167th sample, with a short FDD of 6 samples. In Figure 5b, the PCA method detects fault 16 at the 177th sample, with a longer FDD of 16 samples. More than 86% of fault samples are detected by the KDISM T2 statistic, but the PCA T2 and SPE statistics detect only a small part (less than 44%) of fault samples. Again, the KDISM outperforms PCA by virtue of the higher FDR and the shorter FDD. These two examples illustrate that how the KDISM method outperforms the PCA method by detecting faults more quickly and more accurately. 4.4. Fault Diagnosis Results. To identify the variables responsible for the faults, the two-level deviation magnitude plots are used for fault diagnosis. It should be noted that there are two difficulties in diagnosing the root causes of 21 faults in the TE process. First, with the exception of faults 6, 14, 15, and 21, the actual faulty variables responsible for other 17 faults in Table 2 are not monitored. It is thus impossible to identify these actual faulty variables directly. Second, faults may propagate among variables along the control loops or physical links between variables. If the fault detection time lags seriously behind the fault occurrence time, the fault in question may have propagated throughout the process when it was detected. This may mislead the fault diagnosis result. Nevertheless, the two-level deviation magnitude plots produce reliable fault diagnosis results for most of 21 faults. Figure 6a and b show the two-level deviation magnitude plots at the 161st sample (i.e., fault detection time) for fault 4. Contribution plots (see Appendix A) are shown in Figure 6c

Table 1. Monitoring Variables in the TE Process no.

variable name

units

no.

variable name

units

v1 v2 v3 v4

A feed (stream 1) D feed (stream 2) E feed (stream 3) A and C feed (stream 4) recycle flow (stream 8)

kscmh kg h−1 kg h−1 kscmh

v18 v19 v20 v21

deg C kg h−1 kW deg C

kscmh

v22

kscmh

v23

v7

reactor feed rate (stream 6) reactor pressure

v24

v8

reactor level

kPa gauge %

v9

reactor temperature

deg C

v26

v10

purge rate (stream 9) product separator temperature product separator level

kscmh

v27

deg C

v28

%

v29

v13

product separator pressure

kPa gauge

v30

v14

m3 h−1

v31

v15

product separator underflow (stream 10) stripper level

stripper temperature stripper steam flow compress work reactor cooling water outlet temperature condenser cooling water outlet temperature D feed flow valve (stream 2) E feed flow valve (stream 3) A feed flow valve (stream 1) A and C feed flow valve (stream 4) compressor recycle valve purge valve (stream 9) separator pot liquid flow valve (stream 10) stripper liquid product flow valve (stream 11) stripper steam valve

%

v32

%

v16

stripper pressure

v33

v17

stripper underflow (stream 11)

kPa gauge m3 h−1

reactor cooling water flow valve condenser cooling water flow valve

v5 v6

v11 v12

v25

deg C

% % % % % % %

%

%

%

Table 2. Disturbances in the TE Process no.

process variable

type

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

A/C feed ratio, B composition constant (stream 4), B composition, A/C feed ratio constant (stream 4) D feed temperature (stream 2) reactor cooling water inlet temperature condenser cooling water inlet temperature A feed loss (stream 1) C header pressure loss-reduced availability (stream 4) A, B, C feed composition (stream 4) D feed temperature (stream 2) C feed temperature (stream 4) reactor cooling water inlet temperature condenser cooling water inlet temperature reaction kinetics reactor cooling water valve condenser cooling water valve unknown unknown unknown unknown unknown valve for stream 4 was fixed at the steady state position

step step step step step step step random random random random random slow drift sticking sticking unknown unknown unknown unknown unknown constant

performance of KDISM and PCA models is tested by 21 fault data sets. Confidence limits of T2 and SPE statistics are set as 99%, corresponding to the significance level α of 0.01 in eq 6. The fault detection performance is quantified by three indices: fault detection rate (FDR), fault detection delay (FDD), and G


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Industrial & Engineering Chemistry Research Table 3. Group Division of 33 Process Variables in the TE Processa group no.

variable no.

variable connection

nonzero elements in the matrix P

group no.

1 2 3 4 5 6 7 8 9

(v1, v25) (v2, v23) (v3, v24) (v4, v8, v26) (v5, v27) (v21, v32) (v9, v32) (v10, v13, v28) (v12, v14, v29)

I I I I I I I I I

1/√2, 1/√2, 1/√2, 1/√2 1/√2, 1/√2 1/√3, 1/√3, 1/√3 1/√2, −1/√2 −1/√2, 1/√2 1/√2, 1/√2 1/√3, −1/√3, 1/√3 1/√3, −1/√3, −1/√3

10 11 12 13 14 15 16 17 18

variable no. (v15, (v17, (v18, (v19, (v11, v6 v7 v16 v20

v30) v33) v31) v31) v22)

variable connection

nonzero elements in the matrix P

I I I I II, III

1/√2, −1/√2 1/√2, 1/√2 −1/√2, 1/√2 1/√2, 1/√2 1/√2, 1/√2 1 1 1 1

“I”, “II”, and “III” denote the control, location, and type connections between variables.

a

faulty variable of fault 4. The contribution plot of the SPE statistic in Figure 6d does not clearly show which variable is faulty. The contribution plots therefore give a misleading diagnosis result for fault 4. Figure 7 shows two-level deviation magnitude plots and contribution plots at the 163rd sample (i.e., fault detection time) for fault 12. The two-level deviation magnitude plots indicate that the 14th variable group is the faulty group (see Figure 7a) and variables v11 (product separator temperature) and v22 (condenser cooling water outlet temperature) are faulty variables (see Figure 7b). Therefore, it is easy to infer that the fault 12 is caused by the disturbance to the cooling water system of the condenser. This diagnosis result is very close to the actual cause of fault 12random changes in the condenser cooling water inlet temperature (Table 2). The contribution plot of the T2 statistic in Figure 7c show that variables v17 (stripper underflow) and v30 (stripper liquid product flow valve) may be faulty. However, this diagnosis result is incorrect because variables v17 and v30 are not even

and d for comparison. As shown in Figure 6a, the sixth and seventh variable groups are identified as faulty groups, because their groupwise variable deviation magnitudes far exceed the corresponding confidence limits. Since the sixth and seventh variable groups represents two control loops associated with the cooling water system of the reactor, it is reasonable to infer that the fault 4 may occur in the cooling water system of the reactor. The variable deviation magnitude plot in Figure 6b shows that variables v9 (reactor temperature) and v32 (reactor cooling water flow valve) are faulty variables, because they have high deviation magnitudes which exceed the confidence limits. Therefore, according to the two-level deviation magnitude plots, it is easy to infer that the fault 4 is caused by the disturbance to the cooling water system of the reactor. This diagnosis result is consistent with the actual cause of fault 4a step change in the reactor cooling water inlet temperature (Table 2). However, the variable v15 (stripper level) is identified as a faulty variable in the contribution plot of the T2 statistic (see Figure 6c), which deviates far from the actual

Table 4. Fault Detection Results of KDISM and PCA for 21 Faultsa KDISM 2

T

PCA T

SPE

2

SPE

fault no.

FDR (%)

FAR (%)

FDD (ns)

FDR (%)

FAR (%)

FDD (ns)

FDR (%)

FAR (%)

FDD (ns)

FDR (%)

FAR (%)

FDD (ns)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

99.9 98.5 2.4 96.3 100.0 99.6 100.0 98.0 5.6 87.6 74.0 99.9 95.1 100.0 7.8 86.4 97.1 89.8 60.5 82.6 58.0

0.0 0.6 4.4 0.0 0.0 0.0 0.0 0.0 1.9 0.0 0.6 0.6 0.0 0.0 0.6 3.8 1.25 0.6 0.6 0.0 1.9

2 12 46 0 0 2 0 15 0 19 5 2 38 0 93 6 19 78 10 72 215

83.9 97.9 0.6 55.0 20.8 100.0 100.0 88.5 0.5 13.3 19.0 89.0 84.3 62.6 0.4 4.3 75.3 89.6 3.3 32.9 27.9

0.0 0.0 0.6 0.0 0.0 1.3 0.6 0.6 0.0 0.0 0.0 0.6 0.0 0.0 0.6 1.9 0.6 1.3 0.0 0.0 0.0

7 17 275 0 4 0 0 28 6 52 59 3 48 2 422 145 24 80 85 78 484

99.1 98.3 1.8 54.8 24.3 99.1 100.0 97.0 1.8 28.6 53.3 98.5 93.6 99.9 1.6 12.8 79.6 89.4 11.8 36.6 41.1

0.0 0.0 0.0 0.6 0.6 0.6 0.0 0.6 0.6 0.0 0.0 0.0 0.0 0.6 0.6 2.5 0.6 1.3 0.0 0.6 0.6

7 14 330 0 0 8 0 25 0 53 5 2 48 0 580 312 26 87 10 84 250

99.8 94.9 2.4 99.9 25.6 100.0 69.1 83.8 2.0 42.9 63.3 88.8 95.5 94.0 2.4 43.4 96.9 90.4 25.0 54.6 56.3

1.3 0.6 1.3 1.3 1.3 0.6 0.6 1.3 3.1 0.6 2.5 0.0 0.6 0.0 0.6 3.1 0.6 0.6 1.3 1.3 5.6

2 20 105 1 0 0 0 17 2 33 6 3 36 1 106 16 19 77 10 82 243

“ns” denotes the number of samples. Bold values highlight the better results of KDISM than PCA.

a

H


Article

Industrial & Engineering Chemistry Research Table 5. MFDRs and MFARs of 18 Detectable Faults method

KDISM 2

GLSA24

PCA 2

2

index

T

SPE

T

SPE

T

MFDR (%) MFAR (%)

90.2 0.5

58.2 0.4

67.6 0.5

73.5 1.3

74.4

GLPP8 2

SPCA12 2

SPE

T

SPE

T

SPE

72.3

88.3 0.8

69.9 0.6

87.3 0.8

66.4 0.2

Figure 4. Monitoring charts of (a) KDISM and (b) PCA for fault 10.

Figure 5. Monitoring charts of (a) KDISM and (b) PCA for fault 16.

remotely connected to the actual faulty variable of fault 12. In Figure 7d, the variables v11 and v22 make slightly higher contributions to the SPE statistic than other variables, but it is hard to determine whether these two variables are faulty. The above two examples illustrate how the two-level deviation magnitude plots produce more reliable fault diagnosis results than the contribution plots. Table 6 shows the faulty variable groups and faulty variables identified by the two-level deviation magnitude plots for 18 detectable faults. Because the root causes of faults 16−20 are unknown (see Table 2), it is difficult to determine whether or not the identified faulty variables for these faults are correct. Because the actual faulty variables of faults 6 and 14 are monitored, they are correctly identified by the two-level deviation magnitude plots (see Table 6). For faults 4, 5, 7, 11, and 12, the identified faulty variable groups and faulty variables (see Table 6) are closely related to the actual faulty variables (see Table 2). It is thus possible to correctly infer the root causes of these faults. For faults 1, 2, 8, 10, 13, and 21, the identified faulty variables in Table 6 deviate from the actual faulty variables in Table 2. This is due to two reasons

mentioned earlier. First, the actual faulty variables of faults 1, 2, 8, 10, and 13 are not monitored. Second, faults 2, 8, 10, 13, and 21 are detected with longer delays; thus, they have propagated from the actual faulty variables to other variables (e.g., faults 2, 10, 13, and 21) or have spread over many process variables (e.g., fault 8) when they were detected.

5. CONCLUSION A novel process monitoring method is proposed on the basis of the KDISM model and two-level deviation magnitude plots. The KDISM model is built by integrating process data with process knowledge (i.e., the actual physical/chemical connections among process variables). Compared with the datadriven models built by multivariate statistical analysis methods (e.g., PCA), an outstanding advantage of the KDISM model is that it not only reveal meaningful variable connections but also eliminates the redundant interference between variables. This advantage makes the KDISM model well-suited for fault detection and diagnosis. The VDM is used to quantify the deviations of variables from their normal values. Based on the VDM, two-level deviation magnitude plots are proposed for I


Article


Figure 6. Two-level deviation magnitude plots and contribution plots at 161st sample for fault 4. (a) Groupwise variable deviation magnitude, (b) square variable deviation magnitude, (c) contribution to the T2 statistic, and (d) contribution to the SPE statistic.

fault diagnosis, with the first-level groupwise VDM plot used to

T 2 = (t − t ̅ )T S−1(t − t ̅ )

identify faulty variable groups while the second-level square

= (x − x̅ )T PS−1PT(x − x̅ ) ÉÑ ÄÅ m ÉÑ ÄÅÅ m ÑÑ ÅÅ ÑÑ ÅÅ Ñ Å Ñ = ÅÅÅ∑ (xiT − xi̅ T)pî ÑÑÑS−1ÅÅÅÅ∑ p̂ Tj (xj − xj̅ )ÑÑÑÑ ÅÅ ÑÑ ÅÅ ÑÑÑÑ ÑÖ ÅÅÇ j = 1 ÅÇ i = 1 Ö

VDM plot is used to identify faulty variables. The two-level deviation magnitude plots have two advantages over the traditional contribution plots: (1) they do not suffer from the smearing effect caused by the interference among variables; (2)

m

=

the first-level groupwise VDM plot associates faults with

̂ −1p̂ Tj (xj − xj̅ ) ∑ (xiT − xi̅ T)pS i

where xi and xj (i, j = 1, ..., m) are ith and jth variables in a sample x, x̅ denotes the mean of all training samples, and p̂i and p̂j are ith and jth row vectors in the sparse projection matrix P. The contribution of the ith variable xi to the T2 statistic is defined as

industrial process by using the meaningful variable connections. These two advantages significantly improve the fault diagnosis ability and accuracy of the two-level deviation magnitude plots. The effectiveness and advantages of the

m

proposed methods are illustrated by an industrial case study.

ci = (xi − xi̅ )T pS ̂ −1 ∑ p̂ Tj (xj − xj̅ ) i

The results indicate that (1) the KDISM model outperforms

j=1

the PCA model in terms of higher fault detection rate and

= (xi − xi̅ )T pS ̂ −1p̂ iT(xi − xi̅ )T i

shorter fault detection delay and (2) the two-level deviation

m

+ (xi − xi̅ )T pS ̂ −1 i

magnitude plots have better fault diagnosis capability than the

∑

p̂ Tj (xj − xj̅ )T

j = 1, j ≠ i

contribution plots.

■

(A1)

i,j=1

control loops, physical links, or chemical reactions in the

= cii + cij

APPENDIX A: CONTRIBUTION PLOTS

(A2)

Equation A2 shows that the contribution of xi to the T2 statistic consists of the direct contribution (cii ≥ 0) and the

The T2 statistic in eq 3 can be rewritten as J


Article


Figure 7. Two-level deviation magnitude plots and contribution plots at 163rd sample for Fault 12. (a) Groupwise variable deviation magnitude, (b) square variable deviation magnitude, (c) contribution to the T2 statistic, and (d) contribution to the SPE statistic.

Table 6. Fault Diagnosis Results for 18 Detectable Faultsa fault FDTb 1 2 3 4 5 6 7 8 9 10 11

faulty groups

faulty variables

163 173

17 8

v16 v10, v28

161 161 163 161 176

6, 7 14 1 4 8, 14

v9, v32 v22 v1, v25 v4 v10, v11, v22, v28

180 166

1 6, 7

v1, v25 v9, v32

fault

FDT

faulty groups

faulty variables

12 13 14 15 16 17 18 19

163 199 161

14 2 6, 7

v11, v22 v2, v23 v9, v32

183 180 246 171

6 6 14 5, 6, 7

v21 v21 v11, v22 v5, v9, v32

20 21

233 215

5 5

v27 v5

where p̃i is the ith column of the matrix P+. The contribution of the variable xi to the SPE statistic is defined as cĩ = (xi − p̃ iTt)2

Note that residuals ei are smeared out over different process variables due to the embedded error.17 Thus, process variables which are in the normal range can also give high residuals due to the mismatch of the model in fault conditions. Therefore, the contribution plot of the SPE statistic also suffers from the smearing effect. In the contribution plots, the variable with the largest contribution is regarded as a faulty variable.

■

Bold values highlight the group or variable with the largest deviation magnitude. bFault detection time (the serial number of sample).

*Tel.: +86 (0571) 88320349. E-mail address: lijialuo@zjut. edu.cn. ORCID

cross-contributions with other variables (cij, i ≠ j). Note that the cross-contribution cij may be negative and cause the smearing effect in the contribution plot.17 Due to the smearing effect, normal variables may have larger contributions than a faulty variable. This leads to the incorrect fault diagnosis result. The SPE statistic in eq 4 is rewritten as m

Lijia Luo: 0000-0002-6040-6147 Shiyi Bao: 0000-0001-9700-577X Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This study was supported by the National Natural Science Foundation of China (no. 61304116).

m

∑ ei 2 = ∑ (xi − p̃ iTt)2 i=1

AUTHOR INFORMATION

Corresponding Author

a

SPE = e Te =

(A4)

i=1

(A3) K


Article


■

REFERENCES

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