Large-scale Supervised Process Monitoring Based on

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Process Systems Engineering

Large-scale Supervised Process Monitoring Based on Distributed Modified Principal Component Regression Mengyu Rong, Hongbo Shi, and Shuai Tan Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.9b02163 • Publication Date (Web): 06 Sep 2019 Downloaded from pubs.acs.org on September 6, 2019

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Industrial & Engineering Chemistry Research

Large-scale Supervised Process Monitoring Based on Distributed Modified Principal Component Regression Mengyu Rong, Hongbo Shi,* Shuai Tan Key Laboratory of Advanced Control and Optimization for Chemical Processes of Ministry of Education, East China University of Science and Technology, Shanghai 200237, P.R. China

ABSTRACT: Traditional process monitoring methods construct a single monitoring model to detect if a fault happened. But the development of industrial technology has made industrial processes increasingly complex and huge. When considering local or quality-related faults, especially tiny faults in complex large-scale industrial processes, which is difficult to accurately detect these faults using a single monitoring model. Therefore, a novel distributed process monitoring framework for quality-related fault detection is proposed in this paper. To availably deal with a large number of process variables in large-scale processes, this paper introduces the idea of community partitioning of complex networks to carry out subblock division. Here, the Fast unfolding algorithm is used for multi-block division of process variables. Then, modified principal component regression (MPCR) model is constructed in each subblock to detect quality-related and unrelated faults. To get an intuitive monitoring result, Bayesian fusion based on probability weighting is applied to combine the detection results of all subblocks. Afterwards, the cumulative contribution plot based on multi-block Fast-MPCR is used for fault diagnosis. The benefits of distributed MPCR models are illustrated through a numerical experiment and the Tennessee Eastman (TE) process, the results indicate the superiority compared with other monitoring methods. Keywords: Distributed process monitoring, Modified principal component regression, Fast unfolding algorithm, Cumulative contribution plot

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1. INTRODUCTION With the improvement of modern industrial production level and the rapid development of automatic control technology, the scale of industrial processes is expanding year by year, and the equipment complexity is getting higher. In the meanwhile, the possibility of occurring faults in industrial processes increases.1-3 In order to ensure the production safety and economic benefits of the company, process monitoring has received widespread attention.4-5 Since traditional process monitoring methods adopt a single model usually, some local fault information may be submerged, which may result in unsuccessful fault detection for large-scale processes. To reduce the complexity of the system and improve the accuracy of fault detection, the distributed process monitoring schemes have received extensive attention in academic circles in recent years.6-9 The results of the literature show that these distributed monitoring solutions represent better detection results in complex large-scale industrial processes than a single monitoring model. In general, process decomposition is a critical step for distributed process monitoring scheme. Traditional process decomposition is based on known prior knowledge or process mechanism.6,8 Zhu et al. proposed distributed parallel PCA according to operating units for large-scale plant-wide processes monitoring.6 When traditional principal model-based methods have been employed for monitoring industrial processes in the past few years, getting those models becomes very difficult and time-consuming in plant-wide processes because of the large equipment structure and complex operation conditions.10,11 Recently, data-driven multi-block methods that can automatically divide large-scale processes into multiple subblocks are of significant interests. Typically, Jiang and Yan used mutual information-based multi-block PCA for plant-wide process monitoring.12 Besides, Ge and Song proposed a distributed PCA monitoring model for the plant-wide process.13 Tong et al. 2


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used distributed statistical monitoring based on four-subspace construction and Bayesian inference.14 In order to select related variables. Jiang et al. presented fault-related variables selection for performance-driven distributed monitoring.2 These methods for dividing subblocks can decompose the system into low-dimensional sub-systems to reduce computing and communication resources. As we know, product quality plays a pivotal role in industrial processes, which directly determines the benefits of the production process. Once product quality in a large-scale industrial process changes, it will have a huge impact on the entire production process.15 Operators must locate faults timely and take effective measures to prevent continued production of substandard products. And eliminating the fault effects to other parts of the process. Therefore, quality-related fault detection is really necessary for industrial processes. Recently, with the rapid development of sensor and computer technology, many research results have been achieved in the field of data-driven process monitoring in which quality-related multivariate statistical process monitoring (MSPM) methods have been studied widely.16,17 Partial least squares (PLS) is the most common qualityrelated monitoring method in MSPM field, which combines the process variables matrix and quality variables matrix of industrial processes to extract the main components and builds the regression model.18 In order to improve the performance of process monitoring and accurately detect qualityrelated faults. Many modified PLS methods have been proposed in recent years. Representative methods include TPLS and CPLS.19,20 Then, these methods are extended to dynamic and nonlinear fields, which show good monitoring results. In order to decompose process variables into two orthogonal subspaces, such that, one of the subspaces is fully responsible for predicting quality while the other has no contribution any more. The modified PLS (MPLS) method is proposed.21 3



Based on this, in order to ensure the completeness of residual subspace decomposition, Peng et al., based on the MPLS model, generated two orthogonal subspaces for residual space to construct the EPLS model.22 Besides, PCR is another widely studied quality-related monitoring method. Ding applied PCR to fault diagnosis in an industrial hot strip mill.23 Wang proposed the linear and nonlinear PCR that solved the fault detection of linear and nonlinear processes.24 Although the quality-related fault is particularly important. The quality-unrelated fault may affect the stability of operations and generate huge energy consumption, so quality-related and unrelated faults need to be considered simultaneously. Given that some quality-supervised large-scale industrial processes monitoring, Liu et al. proposed multi-block CPLS for decentralized monitoring of cold rolling continuous annealing processes.25 In order to improve the accuracy of fault diagnosis, Choi and Lee presented multi-block PLS-based localized process diagnosis.26 While considering the large-scale industrial processes under the supervision of quality. How to effectively decompose the process to locate the local fault and accurately judge whether the fault is quality-related or not, that is one thing that needs to be researched and solved in depth.27, 28 In order to effectively solve these problems in complex large-scale processes, this paper presents a novel distributed process monitoring scheme to detect and diagnose quality-related and qualityunrelated faults. To availably divide a large number of process variables in the large-scale system, this paper utilizes Fast unfolding algorithm that applies the idea of community partitioning of complex networks. This technique divides process variables into multiple subblocks automatically. Firstly, the fast unfolding algorithm is suitable for massive network node problems, so this method can be used to process big data and a large number of process variables in large-scale industrial processes. Secondly, this method is an unsupervised method with simple steps and easy 4


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implementation. Thirdly, the accuracy of subblock division using this method is higher than that of the ordinary clustering method. Fourthly, the fast unfolding algorithm divides multiple subblocks without any prior knowledge. Therefore, Fast unfolding algorithm is adopted for distributed process monitoring in this paper. First of all, the correlation relationships between process variables form a complex network. Then Fast unfolding algorithm divides the complex network into multiple subblocks. The concept of modularity is introduced to get an optimal divided result. As for obtained multiple subblocks, MPCR model is used for quality-related and unrelated fault detection. The statistics of principal component subspace in MPCR monitor quality-related faults, and the statistics of residual subspace detect quality-unrelated faults. Finally, Bayesian inference combines the detection results of all subblocks, which give an intuitive monitoring result for industrial processes. When the fault is successfully detected, the corresponding fault identification and positioning need to be performed. Traditional contribution plot method is a commonly used fault diagnosis method.29-30 However, this method still has some shortcomings. When industrial processes are complex and there are a large number of variables included in these processes. The contributing plot method cannot accurately locate fault variables because useful fault information may be covered by irrelevant information. In order to solve this problem, many modified contribution plot methods have been proposed. Yue and Qin developed a reconstruction-based contribution plot (RBC) using a combined index for fault reconstruction and identification.31 Furthermore, considering the smearing effect of contribution plot leads to ambiguous diagnosis results, smearing-out in contribution plot based fault isolation approach was proposed, which effectively guarantees correct fault isolation of multiple sensor faults.32 In order to solve the vague problem of fault diagnosability using a single contribution plot for large-scale industrial processes. A novel fault diagnosis scheme 5



based on the multi-block cumulative contribution plot is proposed in this paper. According to the proposed method, only those subblocks whose monitoring statistics detect faults construct corresponding cumulative contribution plot for fault diagnosis while others are discarded. The advantage of implementing this method is that it can remove irrelevant information and enhance the ability to diagnose faults. The main contributions and novelty of this paper are summarized as follows: (1) A novel quality-related distributed fault detection and diagnosis strategy is proposed; (2) The modularitybased Fast unfolding algorithm divides process variables into multiple subblocks automatically; (3) Modified principal component regression (MPCR) is proposed to detect quality-related faults and quality-unrelated faults; (4) The idea of multi-block cumulative contribution plot is used for improving the ability of fault diagnosis. The rest parts of this paper are organized as follows: in section 2, the basic PCR model is briefly described and the statistics in principal component subspace and residual subspace are constructed to detect faults. Section 3 explains the distributed process monitoring scheme including Fast unfolding algorithm for multi-block partitioning, modified PCR technique, Bayesian fusion strategy and fault diagnosis method. Section 4 describes the process monitoring flow including offline and on-line monitoring process. A numerical experiment and the TE benchmark process is applied to demonstrate the effectiveness of the monitoring scheme in section 5. Conclusions are given in section 6.

2. PRINCIPAL COMPONENT REGRESSION-BASED FAULT DETECTION The process variables and quality variables are collected from an industrial process to form input and output matrices. PCR establishes the regression model for process monitoring.33 An input 6


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matrix X

n m

, which denotes n samples with m variables. Y

n l

is an output matrix, which

is consisted of n samples with l variables. X and Y are normalized to zero mean and unit variance. The main steps of PCR algorithm are as follows. In this approach, PCA is first performed on X. Singular value decomposition on the covariance matrix of X is to extract the main variability information, as follows:

1 N 1

XT X

P PT

Ppc

pc

Pres

0

0 res

Ppc Pres

(1)

After the loading vector P is obtained, the latent score vector T can be calculated as

T

XP

Tpc Tres

(2)

The input matrix X is divided into the principal component subspace and the residual subspace.

PpcTpc

X

PresTres

(3)

Then, solving the regression coefficient of PCR. It is easy to get the following relationship YT X N 1

XT X N 1

(4)

Substituting Equation (1) into Equation (4), we can easily get the expression of the regression coefficient as follow:

Y T XPpc

1 pc

PpcT

N 1 = = where Y.

YT

1/ 2 pc

1/ 2 pc

PpcT X T

T

N 1 PpcT

1/ 2 pc

PpcT

(5)

is the regression coefficient matrix between the input matrix X and the output matrix is the new regression coefficient matrix between a normalized projection x and predicted

output yˆ . Therefore, the PCR-based regression model is

7



yˆ

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x =

1/ 2 pc

YT

PpcT X T

T 1/ 2 pc

N 1 1/ 2 T P pc pc x

=

PpcT x

(6)

= x where 1/ 2 pc

x

PpcT x

(7)

x is the standardized process variable which removes redundancy and noise. Then, the singular

value decomposition is performed on

, which is

YT

1/ 2 pc

PpcT X T

T

N 1 =U re S 1/ 2

0

VreT VunT

(8)

=U re S 1/ 2VreT and y VreT x 1/ 2 pc

=VreT

(9)

PpcT x

where y , a key variable vector for the detection, is the projection of x onto y. Therefore, the prediction of y based on x is

yˆ

x = x

(10)

=U re S

1/ 2

y

Next, building the corresponding statistics for process monitoring. As for the quality-related fault detection in principal component subspace, the statistic is

Tx2

xPpc

1/ 2 pc

VreVreT

1/ 2 pc

PpcT x T

(11)

Similarly, for quality-unrelated fault detection in residual subspace, the resulting statistic is

Tr2

xPpc

1/ 2 pc un

V VunT

1/ 2 pc

PpcT x T

The thresholds of Tx2 and Tr2 statistics that obey F distribution are calculated as follows: 8


(12)

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J th ,T 2 x

J th ,T 2 r

l n2 1 n n l

A l n2 1 n n A l

F l, n l

F

A l, n A l

(13)

(14)

where l is the number of quality variables, this paper studies a single quality variable (l=1) as an output. n is the number of samples. A is the number of selected principal components. F l , n l is an F distribution with l and n-l degree of freedom with significance level .. F A l , n A l is also an F distribution with A-l and n-A-l degree of freedom with significance level ..

3. DISTRIBUTED MPCR BASED FAULT DETECTION AND DIAGNOSIS 3.1. Fast Unfolding Algorithm for Multi-block Division Fast unfolding algorithm, which was proposed by Blondel et al. in 2008, is a heuristic community partition method.34 The purpose of this algorithm is to find greater modularity to complete the optimal community partition. Before introducing Fast unfolding algorithm, the concept of modularity is explained.35 In order to evaluate the result of the community partition, Newman et al. proposed the concept of modularity and used modularity to measure the quality of community partition.36 In simple terms, Fast unfolding algorithm divides more densely connected nodes into the same community, so that getting a larger modularity value. In the end, the largest modularity value represents the optimal community partition. The goal of community partition is to make the connections within the divided community more compact, and the connections among the communities are sparse. Therefore, the greater modularity value means the closer the connection within the community that is the better division result of multiple subblocks. The modularity resulting from the community partition of the network is a scalar value between -1 and 1. The formula of the modularity value is as follows:

9



1 2h

Q 1 2

where h

Ai , j

ki k j

ci , c j

2h

i, j

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(15)

Ai , j is the weight of all edges in the network. Ai , j represents the weight between i, j

node i and j. If nodes i and j are directly connected, which means that Ai , j is the weight value connected between two points, otherwise Ai , j =0. ki

Ai , j

indicates the weight of the edges

j

connected to node i. ci represents the community to which the node i is assigned. The function ci , c j

is used to determine if ci and c j belong to the same community, and if so, returns 1;

otherwise, returns 0. In order to facilitate the understanding of the modularity, which can be explained as a difference value between the number of connected edges in a community and a random expectation in a community network structure. When the number of actually connected edges is higher than the random expectation, the nodes of this community have the tendency to disperse in several communities. For this understanding, the following formula is used to describe modularity Q. 2

Q c

where

in

in

tot

2h

2h

represents the inner weights in community c,

(16)

tot

is the weights of all the edges

connected to the nodes inside community c, which include inner sides and outer sides of the community. Fast unfolding algorithm is an iterative algorithm based on the modularity value of community. The main goal is to continuously divide the network so that the modularity value of the entire network continuously increases after the division. Fast unfolding algorithm mainly includes two phases. The first phase is called modularity optimization. It is mainly to divide each node into the community where its neighboring nodes are located, so the modularity value becomes larger. The 10


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second phase is called community aggregation, which is mainly to aggregate the community that is divided in the first step into a point. That is, reconstructing the network according to the community structure generated in the previous step. Repeat the above process until the structure in the network no longer changes. Process variables dataset X

x1 , x2 ,..., xm

n m

is collected under normal condition.

According to the correlation of process variables, a correlation network is established. That is based on K nearest neighbor (KNN) algorithm to select k most relevant variables for each variable. Crossvalidation is used to select the optimal k value. The criterion of cross-validation for selecting the optimal k value in this paper depends on the correlation coefficient. If the k value is too small, the network connection is simple, and two correlated variables may not be divided into the same subblock. On the contrary, if the k value is too large, the network connection is complicated, and eventually the two variables with weak correlation are divided into the same subblock. Then a correlation network is constructed. According to Fast unfolding algorithm, the network is divided into different communities/subblocks. The steps of the multi-block division are as follow: Step 1: Get a normal process variables dataset and standardize it X

x1 , x2 ,..., xm

n m

,

then calculate the correlation coefficient matrix of the dataset. Step 2: According to KNN algorithm, k variables are selected with the largest correlation coefficient for each variable, form a whole correlation network. Step 3: Assign each node of the network to different communities. There are as many communities as the number of nodes in the initial partition. Step 4: Each node attempts to divide into the community where its neighboring node is located, the modularity value is calculated at this time. The difference value of modularity HQ between 11



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partition before and after is judged. The largest change value HQ corresponds to the best partition result. If HQ is not positive, then giving up the division.

Q

in

ki ,in 2h

tot

ki

2

2h

2 in

tot

2h

2h

ki 2h

2

(17)

where ki ,in represents the weights of the edges connected to node i in community c. Step 5: Repeat the above process until the modularity value no longer increases. Step 6: Construct a new network graph. Each divided community as a new node in the new network graph. Repeat steps 3 and 4 until the community structure no longer changes. Finally, we get the result of the process decomposition that process variables are divided into multiple communities/subblocks. 3.2. MPCR for Fault Detection Based on the above process decomposition, assuming that all of the process variables X are divided into B subblocks X

X1 , X 2 ,..., X B . Among which X b

n mb

b 1, 2,..., B

represents

b-th subblock. n is the number of samples, and mb is the number of variables in the b-th subblock. Although PCR decomposes the coefficient matrix between process and quality variables, this method still cannot completely separate the quality-related/principal component subspace and quality-unrelated/residual subspace, which results in the problem with a high false alarm rate. Therefore, PCR method can be improved. According to the MPLS method performs an orthogonal decomposition on X.37 Applying the idea, the modified PCR (MPCR) is proposed. Compared with the MPCR method and the MPLS method, the MPCR method proposed in this paper has lower computational complexity and can reduce the false alarm rate. In MPCR, the original input variables are replaced by their principal components for least-squares regression algorithm. This method uses

12


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PCA as a preprocessing step to effectively extract the variation information of process variables. Therefore, for the two orthogonal subspaces generated by the MPCR method, not only the principal component subspace has full predictive ability for quality variables, but also the variation in the residual subspace is guaranteed to have little effect on quality variables. This paper uses MPCR for fault detection in each subblock. Firstly, PCA decomposition is performed on process variables. X b = Xˆ b + X b = Tb PbT + X b

(18)

where the number of principal components is determined by cumulative percent variance or crossvalidation methods.38 Next calculating the loading matrix of Y based on the least squares regression between T and Y.

SbT = TbT Tb

-1

TbT Y

(19)

Then the regression coefficient matrix between the process variables X b and the quality variables Y is calculated as follows.

Pb SbT

Kb

(20)

The QR decomposition is performed on the regression coefficient matrix K b .

Qˆ b , Qb

Kb where Qˆ b

mb l

, Qb

mb

mb l

, and Rb

l l

Constructing orthogonal projection matrices

Rb 0

(21)

.

and

Qˆ b Qˆ bT

Qb QbT

. mb mb

mb mb

(22) (23)

And Tre and Tun denote the score matrices of Xˆ b and X b , respectively.

Tre

X b Qˆ b

(24)

Tun

X b Qb

(25)

13



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Therefore, the process variable X b is projected into two orthogonal subspaces Xˆ b and X b as follows: Xˆ b

X b Qˆ b Qˆ bT

(26)

Xb

X b Qb QbT

(27)

where Xˆ b is the part related to Y, and X b is the part that has no contribution to Y. mb 1

As for the observed new online sample xnew

, the corresponding T 2 statistics of Xˆ b

and X b are applied to monitor the principal component subspace and the residual subspace. T xˆ new xˆ new

T xnew xnew

(28)

T x T Qb QbT Qb QbT x =tun tun

(29)

1

T TT ( x T Qˆ b )T re re N 1

Tx2

Tr2

x T Qˆ b Qˆ bT Qˆ b Qˆ bT x =t reT t re

( x T Qb )T

TunT Tun N 1

x T Qˆ b

Tx2lim

(30)

1

x T Qb

Tr2lim

(31)

The thresholds Tx2lim and Tr2lim of statistics are calculated in each subblock. If statistics Tx2 or Tr2 in any subblocks exceed the corresponding thresholds, which means the quality-related or quality-unrelated faults happened.

3.3. Statistics Fusion Based on Bayesian Inference After the results of fault detection of all subblocks have been obtained, it is inconvenient to monitor the statistics of all subblocks simultaneously in an industrial process. Especially for largescale industrial processes, this situation is easy to occur that one unit of the process is alarmed, but the rest parts are normal. Therefore, in order to provide an intuitive detection result of process state, the statistics of all subblocks need to be effectively combined.39,40 In this paper, Bayesian inference is applied to fuse the monitoring results of all subblocks. 14


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In the fusion strategy, N and F represent the normal condition and the fault condition, respectively. The fault probability of T 2 statistic in each subblock is calculated as: PT 2 xb F PT 2 F

PT 2 F xb

where PT

2

xb

is expressed as follows: PT 2 xb

where PT 2 xb N

PT 2 xb N PT 2 N

and PT 2 xb F

PT 2 xb F

where PT N and PT F Tb2 xb

2

(33)

PT 2 xb F PT 2 F

can be expressed as PT 2 xb N

2

(32)

PT 2 xb

are set as 1

e

Tb2 xb / Tb2,lim

T2

e b ,lim

and

(34)

/ Tb2 xb

(35)

, which are determined by confidence level.

represents the T 2 statistic of the current sample in the b-th subblock, and Tb2,lim is the

corresponding threshold. After the fault probabilities of the new data sample are obtained in all subblocks, the next step is to integrate these individual fault probabilities into a unified monitoring decision. Here, different weights are set for those fault probabilities in different blocks. Based on the fault conditional probability, the weighted fault probabilities form the final fused probabilistic statistics BICT 2 of all subblocks as follows:

B

BICT 2 b

! "" PT 2 xb F PT 2 F xb "" # $ B 1" " PT 2 xb F "& b 1 %"

where the weight for each sub-block is

PT 2 xb F B

(36)

. The control limit of BICT 2 is

. Usually,

PT 2 xb F b 1

the value of

is 0.01. If the statistics BICTˆ 2 or BICT 2 exceeds

unrelated faults are considered to happen. 15


, quality-related or quality-


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3.4. Cumulative Contribution Plot for Fault Diagnosis When the abnormal state of the process is detected, it is necessary to accurately locate the fault variables. Then the operator takes the appropriate measures to deal with the fault. The contribution plot represents the contribution rate of each variable while a fault happened. Due to the large number of process variables in large-scale industrial processes, the distributed multi-block Fast-MPCR method was proposed to detect faults. For this distributed monitoring scheme, this paper proposes a multi-block cumulative contribution plot to identify the location of the fault. First, we need to identify faulty subblocks and non-faulty subblocks. The cumulative contribution plot only need to be constructed for subblocks whose statistics exceed the control limit and are identified as faults. Suppose X bf represents a faulty subblock, the dataset can be expressed as

X bf

x1f , x 2f ,..., x cf

(37)

where c indicates the number of variables contained in the faulty subblock. Considering the detected quality-related faults and quality-unrelated faults in a subblock, the different calculation formulas of variables are used for the two fault types. To diagnose the quality-related fault and isolate fault variables, the contribution of the p-th variable x fp ( p 1, 2,..., c) in q-th fault sample x fp, q is calculated as follows:

CON ( Xˆ ) pf , q

T TT ( x fp, q Qˆ np )T re re N 1

1

x fp, q Qˆ np

(38)

where Qˆ np is p-th eigenvector of the regression coefficient between normal process variables and quality variables. Tre denotes the score matrices of Xˆ b . Similarly, in order to identify fault variables of the quality-unrelated fault, the contribution of the p-th variable x fp ( p 1, 2,L , c) in

16


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q-th monitored sample x fp, q can be expressed as follows:

CON ( X )

p f ,q

p f ,q

p T n

(x Q )

TunT Tun N 1

1

x fp, q Qnp

(39)

where Qnp is p-th eigenvector of the regression coefficient, which is the unrelated part between normal process variables and quality variables. Detailed derivation process of Qnp can refer to Equation (21). Tun is the score matrices of X b . Since a single sample cannot reasonably represent the entire faulty process. The cumulative contribution plot is adopted by samples based on moving window to express the contribution of each variable. Thus, the cumulative contribution of variable

x fp is given in the following equation: C ( Xˆ ) pf

d

CON ( Xˆ ) pf , q

(40)

CON ( X ) pf , q

(41)

q 1

d

C ( X ) pf q 1

where d is the number of fault samples, which is 10 in this paper. C ( Xˆ ) pf is the cumulative contribution of variable x fp while a quality-related fault happened in the faulty subblock. And

C ( X ) pf means the cumulative contribution of variable

x fp

while a quality-unrelated fault

happened. In order to reasonably determine the fault variable, the decision strategy for the control limit of cumulative variable contribution is proposed. For the cumulative contribution plot used in this paper, randomly select d samples from normal training dataset t times. The selected data is denoted as xnp ( p 1, 2,..., c) , where c is the number of variables in the faulty subblock. The parameter d is the same as the number of fault samples used in the cumulative contribution plot, that is d=10. In order to reasonably represent the characteristics of normal samples, here t is set as 1000. The

17



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cumulative contribution of variable xnp is given in the following equation: C ( Xˆ ) np

d

CON ( Xˆ ) np, q

(42)

q 1

where

CON ( Xˆ ) np, q

T TT ( x Qˆ np )T re re N 1 p n,q

1

xnp, q Qˆ np

(43)

Therefore, a total of t cumulative contribution will be obtained for the p-th variable, which can be represented as CT ( Xˆ ) np

C ( Xˆ ) np,1 C ( Xˆ ) np,2 L C ( Xˆ ) np,t

(44)

Then the control limit of the cumulative contribution for the p-th variable is calculated by kernel density estimation (KDE). Because KDE is an effective approach for nonparametric density estimation which can maximize the approximation of the sample data. It is worth noting that the control limits Th( Xˆ ) of cumulative contribution for each variable are different. If the cumulative contribution C ( Xˆ ) pf of the p-th variable of the test sample exceeds its control limit, then the p-th variable is identified as a fault variable.

4. PROCESS MONITORING FLOW BASED ON MULTI-BLOCK FASTMPCR 4.1. Summary of Off-line and Online Fault Detection Off-line modeling Step 1: The training sample dataset X is collected under normal conditions and standardizes it. Step 2: Divide X into multiple subblocks using Fast unfolding algorithm. Step 3: Apply MPCR technique to construct a fault detection model in each subblock. Step 4: Calculate the statistics Tx2 , Tr2 and their control limits in each subblock. 18


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Online monitoring Step 1: According to Fast unfolding algorithm, the new sample

xnew

is divided into

corresponding subblocks. Step 2: The statistics of MPCR models for online samples are calculated in each subblock. Step 3: The statistics of all subblocks are combined by Bayesian inference. A quality-related fault is detected if BICTˆ 2 '

. When BICTˆ 2 (

and BICT 2 >

, a quality-unrelated fault

r

happened.

4.2. Multi-block Cumulative Contribution Plot-based Fault Diagnosis Step 1: Divide all process variables of a fault sample into multiple subblocks. Step 2: select the subblocks that can detect the current fault, then calculate the quality-related score matrix t re and quality-unrelated score matrix tun of the fault sample. Step 3: Calculate the contribution of each variable in selected faulty subblocks by using Equation (38) if a quality-related fault is detected. If it is a quality-related fault, applying Equation (39). Step 4: Calculate the cumulative contribution of each variable for different fault types through Equation (40) and (41). Step 5: Identify the root cause of the fault according to the cumulative contribution plot. In order to better understand the proposed distributed monitoring strategy, Figure 1 gives the flowchart to explain how to detect and diagnose quality-related and unrelated faults for industrial processes.

19




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FDR

No. of effective alarms 100% Total fault samples

(46)

This section takes a numerical experiment and the TE benchmark process to demonstrate the availability of the proposed distributed scheme for process monitoring. In order to show the superiority of the scheme, MPLS, MPCR and PCR techniques are used for comparison. In this paper, the significant level

is 0.01.

5.1 The Numerical Experiment The numerical example is built to illustrate the availability of the monitoring solution. The numerical example is presented as follows.

X

x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 y

where si ~ U 0,1 [0, 1].

0.5768 0.3982 0.8291 0 0 0.1224 0.3564 0.5564 0.5402 0 1.3473 0 0.6456

0.3766 0.3566 0.4009 0.3578 0 0.5332 0.5489 0 0.6642 0.8946 0 0.5874 0

0 0 0.2435 1.7678 1.3936 0.6235 0 1.3255 0 0.6598 0 9.4320 0

0 0 0 0.8519 0.8045 0 0.7742 0 0 0.5632 1.6253 0 1.3428

s1 s2 s3 s4

5 4 1 1 0 0 6 2 3 0 4 4 5 X

i 1, 2,3, 4

e j ~ N 0,0.052

e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 e12 e13

(47)

v

(48)

are data sources which obey uniform distribution in the interval

j 1,...,13

and

v ~ N 0,0.012

represent zero-mean Gaussian

random noises. The numerical example consists of 13 process variables xi i 1, 2,...,13

and 1 quality

variable y . To perform the distributed monitoring scheme, firstly, executing process decomposition for 13 process variables. According to Fast unfolding algorithm (k=3), all process variables are 21



divided into 4 subblocks. Here, we adopt the idea of KNN to select the k most relevant variables for each variable, and the cross-validation method is used to determine the k value. Figure 2(a) is the original network graph of the process variables X, and Figure 2(b) shows the result of the process decomposition using Fast unfolding algorithm. Obviously, when the modularity takes the maximum value Q=0.59079, the process variables are decomposed into 4 parts. They are subblock 1 X1

) x1 , x2 , x3 , x9 * ,

X2

) x7 , x11 , x13 * . 10

subblock 2 X 2

) x4 , x5 , x8 , x12 * ,

4

5

width of the network

2 7 1 8

-4

13

-6

2

1

0 -2

8 3

-4

11

-6

-4

-2

0

11 2

4

6

10

-8

12 10 -8

5

4

-6

9

-8 -10 -10

4 2

6

0 -2

12

9

8

4 2

subblock 4

modularity=0.59079

6

6

) x6 , x10 * ,

subblock 3 X 3

10

3

8


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 22 of 46

8

-10 -10

10

7 6 -8

-6

-4

length of the network

-2

0

13 2

4

6

8

10


(a)

(b)

Figure 2. Fast unfolding algorithm for multi-block division of process variables. (a) Original network graph of the

process variables. (b) Subblock division results of the process variables (The same color represents one subblock)

In this numerical example, we can clearly see that if a fault occurs in x3 , it will influence y directly. It will not exert an effect on output y while the fault is added in x5 . Fault 1: x3

x3*

N

200 x f , x f is a quality-related fault.

Fault 2: x5

x5*

N

200 x f , x f is a quality-unrelated fault.

In this experiment, a total of training samples N=400. The number of test samples is also 400. For faults 1 and 2, the first 200 samples are normal data, and the last 200 samples are fault data. (1) Quality-related fault 1: x3

x3*

N

200 x f . A step change of x3 by fault magnitude f =

0.8 is introduced from 201st sample to the end. 22


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Figure 3 shows the results of fault detection using MPCR model in 4 subblocks. It is obvious that only the principal component subspace of subblock 1 detects the fault. Observing fault 1, only

x3 adds the fault x f , and x3 is divided into the first subblock. The detection results of Figure 3 are consistent with the actual fault condition, which indicates that MPCR is a useful method for fault detection. Observing the detection results of principal component and residual subspaces of subblock 1 in Figure 3, it can be clearly seen that MPCR can accurately decompose quality-related part. Figure 4 is the monitoring results of fault 1 using the four methods of (a)Multi-block FastMPCR, (b)MPLS, (c)MPCR, (d)PCR. Obviously, only the Txˆ2 statistics in Figure 4(a) detect the quality-related fault successfully. For fault 1, only the process variable x3 happens the fault and the fault magnitude f =0.8 is small. The other three comparative methods in Figure 4(b), 4(c), 4(d) cannot detect the tiny quality-related fault. Although neither of MPCR and PCR detects the qualityrelated fault well. Obviously, the alarm rate of the Tr2 statistic of the proposed MPCR method increases, the detection result means a fault occurs. But the detection rate of the ordinary PCR method in both subspaces is very low, and the detection result cannot prove that a fault occurs. Therefore, the results of Figures 4(c) and 4(d) indicate that the proposed MPCR method has better detection results compared to ordinary PCR. Comparing all results, the distributed Fast-MPCR monitoring scheme has better detection performance for those faults which are local occurring with small magnitude.

23



Page 24 of 46

Figure 3. Detection results of fault 1 using MPCR in each subblock.

(a)

(b)

(c)

(d) 24


Page 25 of 46

Figure 4. Detection results of Fault 1 in the numerical experiment from (a) multi-block Fast-MPCR, (b) MPLS, (c) MPCR, (d) PCR

Figure 5 is the cumulative contribution plot of fault 1. It can be seen from the detection results in Figure 3 and Figure 4(a) that fault 1 is a quality-related fault, and the fault only occurs in subblock 1. Hence, we only need to construct the cumulative contribution plot for subblock 1 which contains process variables x1 , x2 , x3 and x9 . Obviously, process variable x3 is identified as the fault variable, which is consistent with the setting. Therefore, the cumulative contribution plot based on multi-block Fast-MPCR proposed in this paper can accurately identify fault variables. Variable contribution

30

25

20

Contribution

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


15

10

5

0 1

2

3

4

Variable number

Figure 5. Cumulative contribution plot of subblock1 for Fault 1 in the numerical experiment

(2) Quality-unrelated fault 2: x5

x5*

N

200 x f . A step change of x5 by fault magnitude f =

1 is introduced from 201st sample to the end. The detection results of fault 2 in all subblocks are presented in Figure 6. When the process variable x5 adds the fault x f , it only occurs in subblock 2 for this quality-unrelated fault. Figure 6 shows that only the statistics of residual subspace in subblock 2 detect the occurrence of the fault, which is consistent with the actual setting. Figure 7 describes the monitoring results of multi-block Fast-MPCR and three comparative methods MPLS, MPCR and PCR for fault 2. Figure 7(a), 7(b), 25



Page 26 of 46

7(c) can accurately detect the quality-unrelated fault. However, the statistics of PCR cannot detect the occurrence of this fault in Figure 7(d). Obviously, these techniques MPCR, MPLS and multiblock Fast-MPCR that process variables are projected into two orthogonal subspaces have more accurate detection results for the quality-unrelated fault.

Figure 6. Detection results of fault 2 using MPCR in each subblock

(a)

(b)

26


Page 27 of 46

(c)

(d)

Figure 7. Detection results of Fault 2 in the numerical experiment from (a) multi-block Fast-MPCR, (b) MPLS, (c) MPCR, (d) PCR Variable contribution

140

120

100

Contribution

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


80

60

40

20

0 1

2

3

4

Variable number

Figure 8. Cumulative contribution plot of subblock 2 for Fault 2 in the numerical experiment

Figure 8 shows the diagnosis result of fault 2 using the cumulative contribution plot. From the detection results of Figure 6 and Figure 7(a), which can be seen that fault 2 is a quality-unrelated fault. And only subblock 2 which includes process variables

) x4 , x5 , x8 , x12 *

detects the fault.

Therefore, the cumulative contribution plot is constructed for subblock 2 to isolate the fault variable. Figure 8 represents that the second variable x5 is the fault variable. The diagnosis result is consistent with the setting. Table 1 illustrates the FDRs of fault 1 and fault 2 based on 4 different techniques. It can be observed that multi-block Fast-MPCR generates more accurate detection results compared to 27



Page 28 of 46

MPLS, MPCR and PCR for the quality-related fault. In Table 1, it is clearly seen that the three comparative methods cannot availably detect the locally tiny fault 1. As for the quality-unrelated fault 2, multi-block Fast-MPCR, MPLS and MPCR can effectively detect this fault. But, the statistic Tr2 in PCR almost never detects the quality-unrelated fault. Therefore, the monitoring performance

of PCR is not satisfactory for the quality-unrelated and quality-related faults. In short, multi-block Fast-MPCR generates the best monitoring performance in this numerical example. Table 1. FDRs (%) of fault 1 and fault 2 using 4 methods Fast-MPCR Faults

MPLS

MPCR

PCR

BIC Tx2

BIC Tr2

Tx2

Tr2

Tx2

Tr2

Tx2

Tr2

1 (quality-related)

96

96.5

0

100

0.5

100

1.5

7

2 (quality-unrelated)

5

100

0.5

100

1.5

100

4.5

1

5.2 Tennessee Eastman Process The TE process, developed by Downs and Vogel, has been widely used to test the performance of process monitoring schemes.41 The process consists of 5 operating units: a reactor, a condenser, a compressor, a separator and a stripper. There are four reactants A, C, D and E in the process reactor. After a series of chemical reactions, products G and H are produced finally.42 Component F is assumed to be a by-product. TE process includes a normal dataset and 21 faults dataset, the faults 1-15 are known. Because faults 3, 9 and 15 have a small amplitude, this paper selects the other 12 known faults to monitor. The training and testing data sets include 480 and 960 samples, respectively. All faults are introduced at the 161st sample. The entire process includes 53 process variables, in which 41 measurement variables and 12 manipulated variables.43 A total of 33 variables are selected as input variables for process monitoring, including 22 continuous measured variables (XMEAS(1)-(22)) and 11 manipulated variables (XMV(1)-(11)). The final product component G (XMEAS(35)) is taken as the quality 28


Page 29 of 46

variable.44 However, it can be noted that the quality is not available in real time. Because the component cost is delayed by 6 to 15 min generally. Thus, due to the importance of the quality variable, it is necessary to design a quality-related process monitoring scheme for the industrial process. According to Fast unfolding algorithm, the 33 process variables are divided into 8 subblocks. The termination condition of Fast unfolding algorithm is the maximum modularity value Q= 0.5588. Figure 9 depicts the network graph of process variables using Fast unfolding technique (k=3). Figure 9(a) is the original network graph of 33 process variables. In Figure 9(b), 8 different colors represent 8 subblocks, the nodes existed in the same color are the variables contained in the same subblock. The division results of 33 process variables are shown in Table 2. 10

1

26

27

20

25

8

4 24

14

16

28

6


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


13 10

4

11 29

2

7 12 3

0 15

5

-2 30

17

-4

31 33

-6

6

-10 -10

23

18

-8

22

8 19 -8

-6

-4

-2

32 0

21

2

9 2

4

6

8

10


(a)

(b)

Figure 9. Fast unfolding algorithm for division of process variables. (a) Original network graph of the process variables. (b) Subblock division results of the process variables (The same color represents one subblock). Table 2. The division results of process variables in the TE process. Block number

Process variables

1

x1 , x4 , x14 , x20 , x25 , x26 , x27

2

x2 , x9 , x21 , x22 , x23 , x32

3

x3 , x7 , x11 , x13 , x16 , x24

4

x6 , x8 , x18 , x19 , x31

5

x5 , x17 , x33

6

x10 , x28

7

x12 , x29

8

x15 , x30

In order to understand the meaning of the block division in Table 2, the analysis of each 29



subblock in Table 2 is as follows. Subblock 1 includes measured variables XMEAS(1) (A feed), XMEAS(4) (A and C feed), XMEA(14) (Separator underflow), XMEAS(20) (Compressor work) and manipulated variables XMV(3) (A feed flow), XMV(4) (A and C feed flow), XMV(5) (Compressor recycle valve). The process variables contained in subblock 1 indicates the variable correlation in stream 1, stream 4 and the compressor contained variables is relatively large. Similarly, Subblock 2 includes measured variables XMEAS(2) (D feed), XMEAS(9) (Reactor temperature), XMEA(21) (Reactor water temperature), XMEAS(22) (Separator water temperature), and manipulated variables XMV(1) (D feed flow), XMV(10) (Reactor cooling water flow). These variables show that the reactor-related variables are correlated with the stream 2 related variables. Subblock 3 has measured variables XMEAS(3) (E feed), XMEAS(7) (Reactor pressure), XMEAS(11) (Separator temperature), XMEAS(13) (Separator pressure), XMEAS(16) (Stripper pressure), and manipulated variables XMV(2) (E feed flow). These variables show that pressurerelated variables are highly correlated with stream 3 related variables. Subblock 4 has measured variables XMEAS(6) (Reactor feed rate), XMEAS(8) (Reactor level), XMEAS(18) (Stripper temperature), XMEAS(19) (Stripper steam flow), and manipulated variables XMV(9) (Stripper steam valve). These variables show that the reactor is related to the stripper for some variables. Subblock 5 has measured variables XMEAS(5) (Recycle flow), XMEAS(17) (Stripper underflow), and manipulated variables XMV(11) (Condenser cooling water flow). These variables show that there is a strong correlation among the flows of these devices. Subblock 6 has measured variables XMEAS(10) (Purge rate), and manipulated variables XMV(6) (Purge valve). These variables show purge rate and purge valve are highly correlated. Subblock 7 has measured variables XMEAS(12) (Separator level), and manipulated variables XMV(7) (Separator pot liquid flow). These variables 30


Page 30 of 46

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show separator level and separator pot liquid flow are correlated. Subblock 8 includes measured variables XMEAS(15) (Stripper level), and manipulated variables XMV(8) (Stripper liquid product flow). They show stripper level and stripper liquid product flow are highly correlated. In summary, considering the results of variable division of 8 subblocks, different subblocks are divided according to different variable attributes, different operating units or different streams. It is worth noting that not all variables in an operating unit are strongly correlated, and not all variables of the same attribute are strongly correlated. Therefore, the final result shows these subblocks divide the highly correlated variables into the same subblock according to different characteristics. As for the 8 subblocks after the process variables are divided, MPCR model is used to detect the faults in each subblock, and the statistics of principal component subspaces in all subblocks are fused by Bayesian inference. Similarly, combining all the statistics of residual subspaces in all subblocks. To show the benefits of the proposed distributed detection scheme, MPLS, MPCR and PCR are simulated and compared. For fairness, the number of latent variables in both MPCR and PCR is the same as that in PCA. In these 4 comparative methods, (BIC) Tx2 is constructed as statistics in the principal component subspace, and (BIC) Tr2 is established as statistics in the residual subspace. Before introducing the known 12 faults, it should be judged that these faults are quality-related or not. According to the judgment rule, there are three types of faults in the TE process.44,45 The first fault type [IDV (2, 6, 8, 10, 12, 13)] is considered to be the quality-related fault. The second fault type [IDV (1, 5, 7)] represents the quality-recoverable fault. The third type consists of the quality-unrelated fault, including [IDV (4, 11, 14)]. In the following, several typical fault cases will be discussed in detail. 31



(1) Quality-related fault Fault 2 is a step change in inert ingredients B, which happens in stream 4 of the TE process. Once this fault occurs, product component G in stream 9 deviates from normal value all the time. The red dotted line in Figure 10 shows the variation of the quality variable when fault 2 happens. From this figure, it can be seen that the quality variable fluctuates much in the event of fault 2 occurring from the 161st sample. The proposed multi-block Fast-MPCR technique compares with MPLS, MPCR and PCR techniques, detection results are shown in Figure 11. The Tx2 statistics of the principal component subspace in multi-block Fast-MPCR can quickly detect the quality-related fault and maintain a continuous alarm as Figure 11(a) shows. In Figure 11(b), the Tx2 statistics of MPLS detect the fault at the 200th sample and return to the normal state at the 300th sample. Observing Figure 11(c), the Tx2 statistics of MPCR detect the fault at about the 200th sample, but after the 300th sample, the statistics just exceed the threshold, which is prone to missing alarm. Figure 11(d) plots the Tx2 statistics returning to a normal state when the fault has occurred for a while. By comparing the results of the four techniques, it can be concluded that the proposed multiblock Fast-MPCR has a better detection performance for the quality-related fault.

32


Page 32 of 46

Page 33 of 46

Figure 10. The trend of quality variables in normal condition and three different fault conditions

(a)

(b)

(c)

(d)

Figure 11. Detection results of Fault 2 in the TE process from (a) multi-block Fast-MPCR, (b) MPLS, (c) MPCR, (d) PCR Subblock 2:Variable contribution

Subblock 4:Variable contribution

4

Contribution

Contribution

3 2 1 0

3 2 1 0

1

2

3

4

5

6

1

Variable number Subblock 6:Variable contribution

2

3

4

5

Variable number

30

Contribution

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


20 10 0 1

2

Variable number

Figure 12. Multi-block cumulative contribution plot of Fault 2 in the TE process

33



Figure 13. The real value of the identified fault variables in fault 2

Figure 12 shows the cumulative contribution plots of three subblocks 2, 4 and 6 which detect the fault. Since this is a quality-related fault, and only the statistics in quality-related subspace of multi-block Fast-MPCR exceed the control limits of in subblocks 2, 4 and 6. (Due to space limitations, detailed fault detection results of 8 subblocks are not given.) The diagnosis result of Figure 12 indicates the fourth and fifth variables in subblock 2, the third variable in subblock 4 and the two variables in subblock 6 are identified as fault variables. According to the division result of the subblocks in Table 2. The five fault variables are process variables x10 , x18 , x22 , x23 , x28 . Figure 13 is the real values of the identified fault variables. It can be clearly seen from the figure that these variables change significantly after the fault happened. Although the variable x23 only changes lightly, the multi-block cumulative contribution plot still recognizes the fault variable. The identified result indicates the accuracy of the fault diagnosis method. (2) Quality-recoverable fault Fault 5 is a step change which happens in condenser cooling water inlet temperature. Once the fault occurs, the product component G in stream 9 will change right away. After a period of time, due to the adjustment of the controller, the product component G returns to the normal state. The 34


Page 34 of 46

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rose-red dotted line (o-) in Figure 10 depicts the trend of the quality variable when the fault occurs. Obviously, this fault is recoverable to product quality. The monitoring results of the proposed multiblock Fast-MPCR, MPLS, MPCR and PCR techniques are shown in Figure 14. All the Tx2 statistics in Figure 14(a), 14(c) and 14(d) detect the quality-recoverable fault. For this kind of fault, the operator can perform fast processing or ignore the change, due to the recoverability of the fault. The detection result of Tx2 statistics in Figure 14(b) shows MPLS cannot detect the qualityrecoverable fault which is misjudged as a quality-related fault. Since the fault is quality-recoverable, the detailed diagnosis results are not given here.

(a)

(b)

(c)

(d)

Figure 14. Detection results of Fault 5 in the TE process from (a) multi-block Fast-MPCR, (b) MPLS, (c) MPCR, (d) PCR

(3) Quality-unrelated fault Fault 14 refers to the reactor cooling water valve sticking. When this fault occurs, the product 35



component G in stream 9 never be influenced. The yellow dotted line in Figure 10 represents the trajectory of the quality variable when the fault occurs, which shows that product quality remains stable state to the fault. Figure 15 gives the monitoring results of proposed multi-block Fast-MPCR, MPLS, MPCR and PCR techniques. The Tr2 statistics of residual subspace in Figure 15(b) detect the fault, but the Tx2 statistics reach the FDR of 6%. Both Tr2 and Tx2 statistics of MPCR detect the fault in Figure 15(c). Therefore, this method cannot define the fault is a quality-unrelated fault. In Figure 15(d), the FDR of the Tr2 statistics is 58%, and part of the Tx2 statistics exceeds the threshold. In contrast, only the Tr2 statistics detect the fault in Figure 15(a). Therefore, the proposed multi-block Fast-MPCR expresses the best detection performance for the qualityunrelated fault.

(a)

(b)

(c)

(d)

Figure 15. Detection results of Fault 14 in the TE process from (a) multi-block Fast-MPCR, (b) MPLS, (c) 36


Page 36 of 46

Page 37 of 46

MPCR, (d) PCR

Figure 16 is the multi-block cumulative contribution plot of the quality-unrelated fault 14. The detection results of the multi-block Fast-MPCR indicate that only subblocks 2 and 3 detect the fault. Referring to Table 2. Process variables x7 , x9 , x13 , x32 in subblock 2 and subblock 3 are identified as fault variables. The real values of these identified fault variables are showed in Figure 17. It can be clearly seen from the figure that the multi-block cumulative contribution plot not only can identify fault variables x9 and x32 with large fluctuations, but also can well recognize fault variables x7 and x13 with increasing fluctuations. Therefore, the cumulative contribution plot based on multiblock Fast-MPCR is an effective method for fault diagnosis. Subblock 2:Variable contribution

Contribution

60 40 20 0 1

2

3

4

5

6

Variable number Subblock 3:Variable contribution

30

Contribution

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


20 10 0 1

2

3

4

5

6

Variable number

Figure 16. Multi-block cumulative contribution plot of Fault 14 in the TE process

37



Page 38 of 46

Figure 17. The real value of the identified fault variables in Fault 14

Table 3 lists the FDRs of six quality-related faults using four different methods in the TE process. The best monitoring result for each fault has been bolded. Due to the compensation of the controller, when faults 1, 5 and 7 occur, the quality variable will return to normal quickly. It is meaningless to compare the FDRs for these three faults. So they are not listed in Table 3. As for these quality-related faults [IDV (2, 6, 8, 10, 12, 13)], it can be seen that the proposed multi-block Fast-MPCR technique has the highest FDRs compared to MPLS, MPCR and PCR in Table 3. Especially for faults 8, 10, 12 and 13, the FDRs have been significantly improved. Therefore, the multi-block Fast-MPCR achieves the best detection result for these quality-related faults. Table 3. FDRs of quality-related faults in the TE process MPLS

MPCR

PCR

Fast-MPCR

Txˆ2

Txˆ2

Txˆ2

BIC Txˆ2

2

91.75

95.25

32.37

98.50

6

99.12

97.62

97.62

99.5

Faults

8

62.5

75.12

72.62

96.75

10

20.00

19.37

15.12

46.25

12

79.75

72.62

69.37

95

13

78.38

75.25

74.12

94.63

Table 4 lists the detection results of quality-unrelated faults. It can be clearly seen from Table

38


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4, the BIC Txˆ2 statistics of multi-block Fast-MPCR have the highest FDRs for the three qualityunrelated faults. In short, the proposed multi-block Fast-MPCR strategy has the best monitoring performance for these faults. Obviously, PCR shows poor detection results for these faults. MPLS and MPCR can detect these faults well. But for fault 11, the proposed method demonstrates the higher FDR. Therefore, this proposed method is more advantageous for quality-unrelated fault detection than comparative methods. Table 4. FDRs of quality-unrelated faults in the TE process Faults

MPLS

MPCR

Txˆ2

Tx%2

Txˆ2

4

2.38

100.00

11

2.25

14

3.12

PCR T

Txˆ2

0.5

100.00

78.62

4.63

100.00

6

Fast-MPCR T

BIC Txˆ2

BIC Tx%2

1

1.88

3

100.00

80.00

1.63

4.25

4.25

84.87

100.00

6.5

57.5

2.37

100.00

2 x%

2 x%

Table 5 shows the FARs of all quality-related faults and quality-unrelated faults in the TE process. Comparing the FARs of the four methods for all faults, it is obvious that all methods show lower FARs. Although the proposed multi-block Fast-MPCR method has a FAR of 3.75 for fault 2, which is slightly higher than other comparative methods. The multi-block Fast-MPCR strategy has lower FARs for other faults, such as fault 10 and fault 12. Moreover, although the proposed monitoring strategy has a certain FARs when detecting faults, these FARs are less than 4%. Therefore, the proposed monitoring strategy still has a good performance for these faults detection. Table 5. FARs of quality-related faults and quality-unrelated faults in the TE process Faults

MPLS

MPCR

PCR

Fast-MPCR

Txˆ2

Tx%2

Txˆ2

Tx%2

Txˆ2

Tx%2

BIC Txˆ2

BIC Tx%2

2

0

0

0

0

0

1.875

3.75

1.875

6

0.625

0

0.85

0

2.5

0.625

2.125

0.625

8

0

0.625

0

2.35

0

0

2.75

1.782

10

0

0

0

0

3.68

0

0

0

12

2.5

1.25

1.75

1.25

0

0

1.16

1.875

13

0

0

0

0

0.35

0

0

0

4

0

0.625

0

0.625

0

0.625

1.5

2.5

11

1.25

0.625

0.625

0.625

1.86

0

1.75

0.625

14

0

0

0

0

0

0.625

2.75

1.25

39



6.

CONCLUSIONS In this paper, a novel distributed process monitoring scheme based on multi-block Fast-MPCR

technique is proposed for quality-related fault detection in large-scale industrial processes. Fast unfolding algorithm of complex networks is used to divide the process variables into multiple subblocks. Then, MPCR regression model is constructed in each subblock to carry out fault detection, and the detection results of all subblocks are fused by Bayesian inference. The syncretic statistics in different subspaces judge the fault type which is a quality-related fault, qualityrecoverable fault or quality-unrelated fault. The distributed monitoring scheme has clearer diagnostic logic to locate faults especially for the process with many variables. Therefore, the cumulative contribution plot based on multi-block Fast-MPCR is applied to identify fault variables. The proposed distributed multi-block Fast-MPCR scheme is applied in a numerical experiment and the TE process, which demonstrates the efficiency of multi-block Fast-MPCR technique compared with MPLS, MPCR and PCR, especially for quality-related faults. The diagnosis results of the multiblock cumulative contribution plot also show the effectiveness of the distributed monitoring scheme. But the increase of computational load is its drawback, there are rooms for improvement. The application of this monitoring scheme to dynamic and nonlinear large-scale processes can be the future research work.

AUTHOR INFORMATION Corresponding author: Hongbo Shi; E-mail: [email protected]; Tel: +86-021-64252189; Fax: +86-021-64252349

ACKNOWLEDGMENTS This work was supported in part by the National Natural Science Foundation of China under 40


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Grant 61703161 and Grant 61673173, Fundamental Research Funds for the Central Universities under Grant 222201714031.

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Table of Contents (TOC) Graphic:

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Large-scale Supervised Process Monitoring Based on

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