Review and Perspectives of Data-Driven Distributed Monitoring for

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

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Review and Perspectives of Data-Driven Distributed Monitoring for Industrial Plant-Wide Processes Qingchao Jiang,† Xuefeng Yan,*,† and Biao Huang‡ †

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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 ‡ Department of Chemical and Materials Engineering, University of Alberta, Edmonton, Alberta T6G2V4, Canada ABSTRACT: Process monitoring is crucial for maintaining favorable operating conditions and has received considerable attention in previous decades. Currently, a plant-wide process generally consists of multiple operational units and a large number of measured variables. The correlation among the variables and units is complex and results in the imperative but challenging monitoring of such plant-wide processes. With the rapid advancement of industrial sensing techniques, process data with meaningful process information are collected. Data-driven multivariate statistical plant-wide process monitoring (DMSPPM) has become popular. The key idea of DMSPPM is first decomposing a plantwide process into multiple subprocesses and then establishing a data-driven model for monitoring the process, in which process variable decomposition is important for guaranteeing the monitoring performance. In the current review, we first introduce the basics of multivariate statistical process monitoring and highlight the necessity of designing a distributed monitoring scheme. Then state-of-the-art DMSPPM methods are revisited. Finally, opportunities of and challenges to the DMSPPM methods are discussed.

1. INTRODUCTION To increase the production volume and improve the product quality, a modern plant-wide process generally consists of several subprocesses or operational units.1−3 The number of measured variables is large, the interactions among units are complex, and the process characteristics are diverse. These characteristics and complexities cause difficulties in modeling such plant-wide processes. Moreover, the process and environmental safety are to be considered. Process monitoring, as an efficient approach to ensure favorable operating conditions, is gaining an increasing amount of attention.4−6 Over the past decades, a large number of monitoring methods, which are generally classified as model-based,7,8 knowledge-based,9 and data-driven methods,10,11 have been available.12,13 The model-based methods usually require an accurate process model,7,8 whereas the knowledge-based methods rely mostly on expert experience.9 However, given the complexity of a large-scale plant-wide process, accurate models or expert knowledge are generally difficult to obtain. Moreover, given the rapid progress in industrial sensing and controlling techniques, a large amount of process data is available. These process data may contain meaningful process information; hence, data-driven monitoring methods are becoming a focus of research.14−16 Multivariate statistical process monitoring (MSPM) methods are among the most popular data-driven monitoring methods.17−19 The key idea of MSPM methods is to initially © XXXX American Chemical Society

extract features through a certain multivariate analysis method, project the high-dimension data into low-dimension spaces, and then establish monitoring statistics in the low-dimension spaces. For linear Gaussian processes, the most widely used MSPM methods include principal component analysis (PCA), partial least-squares (PLS), and canonical correlation analysis (CCA). Numerous extensions have been developed to handle more complex process characteristics, such as non-Gaussian, nonlinear, and dynamic characteristics. These classical MSPM methods can handle a general process effectively, but they possibly have limited efficiency in a large-scale plant-wide process. A plant-wide process usually contains several units and a large number of variables.1,6,20 Establishing a centralized MSPM model for these plant-wide processes may ignore the local process behavior and degrade the monitoring performance. Process variables are divided into several blocks, and an MSPM model in each block is established to monitor these large-scale processes. These variable division-based monitoring methods are generally called multiblock or distributed methods. The necessity and feasibility of using multiblock or distributed methods were analyzed by Jiang and Huang.1 The Received: Revised: Accepted: Published: A

April 30, 2019 July 5, 2019 July 8, 2019 July 8, 2019 DOI: 10.1021/acs.iecr.9b02391 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Review

Industrial & Engineering Chemistry Research

Figure 1. Data-driven distributed monitoring framework for plant-wide processes.

Figure 2. Structure of this article.

centralized scheme, thereby reducing the monitoring redundancy in the detection of a local fault. A design framework of data-driven distributed process monitoring, which includes process variable decomposition, local monitor design, and decision-making system design,1 is presented in Figure 1. Process variable decomposition divides process variables into several blocks according to certain rules and is related to variable clustering and variable selection. The

efficiency of process decomposition was evaluated from the perspectives of practical application and theoretical justification.1 From a practical application aspect, appropriate process data decomposition increases the fault-tolerance ability, monitoring reliability, and economic efficiency. From the theoretical justification aspect, the distributed monitoring scheme explores more local process behaviors than a B

DOI: 10.1021/acs.iecr.9b02391 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Review

Industrial & Engineering Chemistry Research Tz,2 α = χα2 (m)

local monitoring design establishes appropriate local monitors according to local process characteristics. Generally, multivariate statistical analysis methods are employed for designing the local monitors. The decision-making system design establishes a decision-making system to achieve a comprehensive evaluation of the entire process status. It should be noted that the term “data-driven distributed process monitoring” here means that after the process variable decomposition, local monitors and decision-making system are designed by data-driven methods. Therefore, although the process decomposition can be achieved based on process model, process knowledge, or process data analysis, so long as the local monitors and the decision making system are designed based on process data, the method is regarded as a data-driven method. Various data-driven multivariate statistical plant-wide process monitoring (DMSPPM) methods, that is, the local monitors are established based on the multivariate statistical analysis methods, have been developed within the framework. The remaining part of this review is organized as follows. First, we provide a brief introduction of the basic MSPM methods. The motivations for establishing distributed monitoring schemes are presented. Then, the state-of-the-art DMSPPM methods are given particular consideration. The process variable decomposition methods and the decisionmaking system design methods are revisited. Finally, the opportunities and challenges for DMSPPM methods are discussed. An illustration of the structure of the paper is illustrated in Figure 2.

where m = rank(Σz) represents the degrees of freedom. For a zf with Ξf, the T2 is expressed as follows: T 2f = zTf Σ−z 1zf ∼ χ 2 (m , ν 2) 2

NDR = prob(T 2f < Tth2 |f ≠ 0) = F χ 2(Tth2 ; m , ν 2)

(5)

Substituting eq 4 into eq 5, we can obtain the following: NDR = F χ 2(χα2 (m); m , ν 2)

(6)

where m, ν ) is the cumulative distribution function of χ2(m, ν ). The NDR is determined by the two parameters m and ν2 given a deterministic α. The plot of NDR that changes with m and ν2 is presented in Figure 3. Jiang et al.1,27 clarified Fχ (χ2α(m); 2 2

2

Figure 3. NDR that changes with parameters.

that for a given m, the NDR monotonically decreases with v(v ∈ (0, + ∞)). The NDR monotonically increases with m given a fixed noncentrality parameter v2. In a plant-wide process, an early stage fault generally affects only a local process section. Then, when all variables in the monitoring are involved, the control limits are loosened and thereby the fault detection is limited. Reducing the degrees of freedom by selecting the relevant fault features or increasing the noncentrality parameter by highlighting the fault information is important. 2.3. PCA-Based Process Monitoring. PCA decomposes a data matrix Z ∈ 9 N × m (N samples) into dominant and residual part components, as follows:28,29

(1)

where z* represents a within-control sample and Ξf represents a possible fault with Ξ determining the direction and f determining the magnitude. The following hypothesis testing is formulated to examine the status of the sample: H0, the null hypothesis: f = 0, fault-free; H1, the alternative hypothesis: f ≠ 0, faulty. When the null hypothesis is rejected, a fault is detected. Otherwise, the process is identified as normal. A statistic J, with Jth denoting the threshold, is established to examine whether the null hypothesis should be rejected. The false alarm rate (FAR, or significance level α) refers to the probability prob{J > Jth | f = 0}. The nondetection rate (NDR) refers to the probability prob{J < Jth | f ≠ 0}. 2.2. T2 test. Under the multivariate Gaussian assumption, the Hotelling T2 provides the lowest NDR for a given FAR24,25 without fault information. Therefore, T2 is a basic statistic for multivariate statistical fault detection21,23,26 and is constructed as follows: Tz2 = zT Σ−z 1z

(4)

where χ (m, ν ) is the noncentral Chi-squared distribution, m represents the degrees of freedom, and ν2 = (Ξf)TΣ−1 z (Ξf) is the noncentrality parameter. The NDR is expressed as follows:1 2

2. MSPM BASICS 2.1. Multivariate Statistical Fault Detection. Fault detection determines whether a fault exists in a process using the statistical framework of hypothesis testing.1,21−23 A measurement vector zf ∈ 9 m (of m sensors) with possible faults is expressed as follows: zf = z* + Ξf

(3)

T T Z = TpcPpc + TresPres

(7)

PCA searches for the loading vector P, such that P z preserves the largest variance of the original data given z ∈ 9 m , as follows: T

P = arg max(PT ΣzP)

(8)

P

Then corresponding monitoring statistics are developed as follows:21

(2)

2 T TPCA = zT Ppc Λ−pc1Ppc z

where Σz is the covariance matrix. The follows a Chisquared distribution, and the threshold is determined by the following:21 T2z

T Q PCA = (I − PpcPpc )z

C

(9) 2

T = zT (I − PpcPpc )z

(10)


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Industrial & Engineering Chemistry Research Table 1. PCA, CCA, and PLS Extensions Basics/ Extensions

PCA

CCA

PLS

Kernel PCA37Serial PCA,38Parallel PCA-KPCA,39 Deep PCA,40Locally weighted PCA41 Dynamic PCA,45Dynamic inner PCA46

Kernel CCA,67Mixture CCA,42Locally weighted CCA68 Dynamic CCA,33 Two-dimensional dynamic CCA47 Just-in-time PCA,51 Sensitive PCA,52 Probabilistic PCA,53−55 Fault- Probabilistic CCA,55 Modified CVA,57 relevant PCA.56 Generalized CCA58

Nonlinearity Dynamics Other Modifications

Kernel PLS,43Locally weighted PLS44 Dynamic PLS,48,49Dynamic inner PLS50 Total PLS (TPLS)59Concurrent PLS60

Table 2. Alterative MSPM Methods Methods

Characteristic

Specialties and usage

Independent component analysis (ICA).61,62 Fisher discriminant analysis63,64 Slow feature analysis (SFA)65−68

Maximize independence between components;

For monitoring non-Gaussian processes

Maximize the distance between different data sets; Extract slowly varying features from dynamic processes

Local linear embedding (LLE)69

Maintain local linear characteristics of samples

Neighborhood preserving embedding (NPE)70,71 Global-local structure analysis (GLSA)72−77 Support vector data description (SVDD)75,78−80

Preserve the local neighborhood structure on the data manifold Preserve global and local process data structures

When faulty data are available Simultaneously monitor process disturbance and dynamic anomalies To explore manifold data structure and local data behavior To explore manifold data structure and local data behavior To explore both global and local data behavior

Find the minimal hyperelliptic curve that contained almost all samples.

where I is the identity matrix. 2.4. CCA-Based Process Monitoring. Given x ∈ 9 p (of p sensors) and y ∈ 9 q (of q sensors), CCA projects two sets of data into the most correlated spaces. The canonical correlation vectors (CCVs) P and Q maximize the correlation between the projected data as follows:26,30

2.5. PLS-Based Process Monitoring. The PLS method projects X and Y into the low-dimensional latent variable data T given the predictor data X ∈ 9 N × p and the quality data Y ∈ 9 N × q, as follows:11,34 l X = TPT + E o o X o m o T o o Y = TQ + E Y n

(P , Q ) = arg max ρ(PTx)(QTy) (P , Q )

= arg max (P , Q )

(11)

(p , q) = arg max pT XT Yq

Equation 11 can be solved by applying SVD to matrix K, as follows:31

K = Σ−x 1/2ΣxyΣ−y 1/2 = U ⁄ VT (12) ÄÅ ÉÑ Å Ñ where ⁄ = ÅÅÅÅ diag(σ1 , μ , σk) 0 ÑÑÑÑ ∈ 9 p × q and k = rank (Σxy). 0 0 ÅÇ ÑÖ For fault detection, two residuals are generated as follows:

rx = PTx − ΣQ T y

(13)

ry = QTy − ΣT PTx

(14)

2

Ty2

(18)

p,q

The latent variables ti are sequentially computed to maximize the covariance between the deflated input data, namely, Xi = Xi−1 − ti−1pTi−1, X1 = X, and the output data. The latent variables are obtained through a mapping matrix as T = XG, where G = W(PTW)−1 and W is the weighting matrix consisting of vectors wi with ti = Xiwi.11,34 For a new predictor sample x, the reconstructed predictor sample x̂ and the predicted quality sample ŷ are calculated as follows: x ̂ = PGT x

(19)

y ̂ = QGT x

(20)

32,33

Then, T is constructed as follows: Tx2

(17)

where P and Q are the projection matrices and EX and EY are the residual matrices. The projection vectors are obtained by sequentially maximizing the covariance as follows:

PT ΣxyQ (PT ΣxP)1/2 (QT ΣyQ )1/2

For nonlinear and non-Gaussian process monitoring

=

rxT Σ−rx1rx

(15)

=

r yT Σ−ry1ry

(16)

Then, two monitoring statistics are constructed as follows: ij TTT yz T 2 z G x TPLS = xT Gjjj j N − 1 zzz k { −1

where Σrx = Ip − ∑∑T and Σry = Iq − ∑T∑ are the covariances. T2x examines whether a fault exists in input x, in consideration of the correlation information from y. Ty2 examines whether a fault exists in the output y in consideration of the correlation information from x. The threshold of the statistics can be obtained through a Chi-square distribution. The CCA method is more efficient than monitoring with PCA in detecting a fault that affects only the input or output.

Q PLS = x − x ̂

2

(21)

= (I − PPT )x

2

(22)

PLS is generally used for quality-related or key performance indicator (KPI)-related monitoring. 2.6. Extensions and Alternative. PCA, CCA, and PLS have corresponding special features and application situations. D


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According to the decomposition mechanism, the process variable decomposition methods can be classified as knowledge-based methods and total data-driven methods. Knowledge-based variable decomposition methods divide variables according to a mathematic model or process knowledge. Physical constraints or process topology are generally considered in the knowledge-based decomposition. Datadriven variable decomposition methods divide variables according to the variable characteristics reflected by process data. It is worth mentioning that even a large-scale process is decomposed through the knowledge-based methods; the local monitors and decision-making system can be constructed using data-driven methods such as PCA, PLS, and CCA, etc. Since variable relation and fault information are two key considerations in the process variable decomposition, the total data-driven methods can be further divided into variable relation-based methods and fault information-aided methods. (i) The variable relation-based methods first examine the variable relationship through certain criterion and then divide variables according to the relationship; the most correlated variables are divided into the same block, while variables in different blocks are as independent as possible. The key advantage of the variable relation-based methods is that they rely on only the normal operating data but not the difficult-toobtain fault information or fault data. However, as analyzed in ref 35, without considering fault information, the monitoring scheme cannot be guaranteed to be optimal for a certain fault. (ii) Fault information-aided methods divide variables according to available fault information or constructed fault data. Once fault information is available, it can be used to improve the monitoring performance for certain known faults. Optimization algorithms are generally involved to optimize the monitoring performance for the available faults. 3.1. DMSPPM with Knowledge-Based Variable Decomposition. The knowledge-based process variable decomposition methods generally divide the process variables according to prior process knowledge, thereby explaining the results and identifying the fault root cause. Based on a knowledge-based predetermined block division, data-driven multiblock PCA, multiblock PLS, or multiset CCA algorithms are performed and monitoring methods are developed. Several forms of the multiblock PCA (MBPCA) method have been developed for plant-wide processes. The key idea of MBPCA methods is to initially decompose a process into blocks and then simultaneously obtain local and global information. Westerhuis et al.81 provided comprehensive analysis on several MBPCA methods. The superscores of the consensus PCA method are identical to the scores of the regular PCA method that uses all variables. Moreover, the hierarchical PCA method suffers from convergence problems. Qin et al.82 provided unified analysis on MBPCA and PLS methods. The loading levels and scores of the consensus PCA methods are calculated directly from the regular PCA method. In addition, although the MBPCA and regular PCA methods are basically equivalent, decomposing the process benefits the fault root cause localization. The decentralized monitoring method was successfully applied to a polyester film process. Subsequently, Cherry and Qin83 proposed an MBPCA variable contribution method for fault root cause identification. The method was applied to metrology data as well as to a plasma stripper. Liu et al.84 developed a multilevel PCA model to obtain meaningful monitoring of the underlying process. The multilevel PCA model was applied to the continuous annealing

Disadvantages of the basic PCA, CCA, and PLS methods can be discussed from two aspects. First, basic PCA, CCA, and PLS generally focus on the linear Gaussian static processes but may not be suitable for monitoring a process with more complex characteristics such as nonlinear, non-Gaussian, and dynamic. Second, basic PCA, CCA, and PLS may not be suitable for monitoring large-scale plant-wide processes. The key concern is a global monitoring model may ignore local process behavior and involve remarkable redundancy in detecting a local fault. Detailed discussions are provided in refs 27, 35, and 36. To deal with complex process characteristics, several improvements and extensions of PCA, CCA, and PLS have been proposed. Some representatives are summarized as Table 1. Numerous other MSPM methods have also been proposed to address various process or data characteristics, and some alternative and extensions are summarized as Table 2. These MSPM methods can generally be employed to design the local monitors in a distributed monitoring scheme. The application status and trends of the basic PCA, CCA, and PLS methods and extensions are presented in Figure 4. The data are indexed from the Web of Science by searching the related keywords in the theme.

Figure 4. Application status and trends of basic MSPM methods and extensions.

3. DMSPPM PROCESS VARIABLE DECOMPOSITION AND LOCAL MONITORING DESIGN A design framework of data-driven distributed process monitoring, which includes process variable decomposition, local monitor design, and decision-making system design, has been proposed by Jiang and Huang.1 Among the three levels, process variable decomposition plays a key role in data-driven distributed monitoring and has a remarkable impact on the monitoring performance. The impact of variable decomposition on the monitoring performance is analyzed in detail from both geometric meaning and statistical hypothesis testing aspects. It was pointed out that both variable relationship and fault information should be considered to preserve the fault detectability in the variable decomposition. Besides, the process decomposition and local monitor design are generally considered simultaneously, because the process decomposition and the local monitor design should cooperate to achieve efficient monitoring. Consequently, the current work discusses the variable decomposition and local monitoring design in an integration manner. E


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Figure 5. Joint-individual monitoring scheme for multiunit processes.

process (CAP), through which the efficiency in fault cause interpretation was shown. A distributed parallel PCA modeling and monitoring scheme with big data was developed by Zhu et al.16 Multiblock PLS (MBPLS) methods have been proposed for quality-related or KPI -related monitoring. MacGregor et al.85 developed monitoring charts for subblocks as well as the entire process to enhance the fault detection and diagnosis ability. Lopes et al.86 used the MBPLS method in an industrial pharmaceutical process. The advantage of the MBPLS over the standard PLS method in this process was demonstrated. Choi and Lee87 further analyzed the T2 and Q statistics of subblocks and of the entire process and derived the block or variable contributions. The method was applied to a wastewater treatment process, and its efficiency was demonstrated. A recursive MBPLS monitoring scheme was proposed by Wang et al. to address the nonstationary monitoring condition of a plant-wide process.88 Kohonen et al.89 developed an MBPLS method for modeling a NIR data set. Liu et al.90 proposed a multiblock concurrent projection to the latent structure (CPLS) method for decentralized monitoring of the CAP. The fault detection and diagnosis information from predictor data and quality data were considered. The efficiency was demonstrated. Zhao and Sun91 proposed a multispace T-PLS (MsT-PLS) method to more clearly reveal local information. The central idea of the MsT-PLS method is to decompose the process variables into multiple variable spaces and then study the relationship among the multiple variable spaces from a cross-space perspective. The proposed algorithm was applied to the Tennessee Eastman process (TEP), and the efficiency was demonstrated. The multiset CCA (MCCA) method extends the CCA method to the multiple set forms.92 B sets of variables with xi ∈ 9 mi , i = 1, 2, ..., B are assumed to be available. The MCCA method takes multiple stages to find canonical variables pTi xi, such that the correlation between them is maximized. The optimization problem can be expressed as follows:

Details on the solution are provided in ref 92. For a plantwide process with several operational units, some joint features that are shared by all units usually exist. A local unit may contain joint features that are related to other units and individual features that are not related to the others. Thus, Wang et al.93 proposed an MCCA-based joint-individual monitoring method to consider the joint and individual features of multiple operational units. An illustration of the MCCA-based joint-individual monitoring method is shown in Figure 5. First, the process data are collected from different operational units, and the multiset process data are obtained. Second, based on the MCCA algorithm, the joint features of the entire plant and individual features of each subset were extracted. Then, the monitoring statistics were established to examine the entire process and the local units. The advantages of the proposed joint-individual monitoring scheme were theoretically analyzed. The efficiency was also illustrated through applications on the TEP. The joint-individual monitoring method was extended to parallel-running batch processes given its efficiency.94 Aside from the multiblock PCA, multiblock PLS, and multiset CCA methods, several other monitoring approaches were proposed to address various process characteristics. Ge and Song95 proposed a multiblock ICA-PCA-based fault detection approach for the global monitoring of non-Gaussian processes. The two-level monitoring scheme was further improved; it used PCA and SVDD methods to describe within-unit information and cross-information among processes.96 To address process nonlinearity, nonlinear modeling and monitoring extensions have been developed. Zhang et al.97 developed a multiblock KPLS (MBKPLS) method for the decentralized monitoring of large-scale nonlinear processes and applied it to the CAP. The MBKPLS method performed better than the MBPLS method because nonlinearity was considered. Moreover, the MBKPLS process possessed better monitoring performance than the KPLS method because more local information was explored. Liu et al. proposed a multiblock dynamic concurrent kernel CCA (MBDCKCCA) method to handle process nonlinearity and dynamics simultaneously in the CAP.98 Strip thickness-specific variables, process-specific variables, and thickness-process covariations were monitored, and the multiblock extension was designed to increase the fault isolation ability. More recently, Zhu et al. proposed a distributed Bayesian network approach.99 Process knowledge is utilized to decompose a process, and then the Bayesian

B

(p1 , p2 , ..., pB ) = arg max ρ = p1 , p2 ,..., pB

s. t .

1 B

∑ piT xixjTpj i≠j

B

∑ piT xixiTpi = 1 i=1

(23) F


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Figure 6. Illustration of the variable relation-based block division methods.

model is constructed to achieve distributed monitoring. These multiblock methods generally performed process decomposition using process knowledge, which is not always available. 3.2. DMSPPM with Variable Relation-Based Variable Decomposition. Several data-driven process variable decomposition methods have been developed. The key idea of the total data-driven methods is to initially decompose the process data according to certain characteristics exhibited by the process data and then establish local monitoring models. Grbovic et al. proposed a data-driven process variable decomposition method based on sparse PCA.100 Each component obtained from the SPCA defines a block, and included variables contribute to the block. Local monitors are then constructed, and decision fusion is carried out. By automatically dividing variables into several subfeature blocks, the distributed monitoring explores more local behaviors of a plant-wide process. Later on, Ge and Song101 developed a process decomposition method based on variable contributions to the principal components of the PCA method. Variables that have similar contributions to the same principal component directions are automatically divided into the same blocks. This decomposition method considered the linear correlation information and emplaced the most closely correlated variables into the same block. The PCA-based process decomposition method was also extended to multilevel PCR for quality prediction and analysis.102 Recently, Tong et al. developed a PCA-based variable decomposition method that divided variables according to their relevance to the dominant and residual subspaces.103 Through this decomposition, variables that contribute similarly to the same subspace are divided into the same block, thereby reducing the model complexity and improving the performance of detecting a local fault. These PCA-based variable decomposition methods consider only the linear correlation relationship among variables but ignore the high-order statistics information.

To consider higher-order statistical information among variables, Jiang and Yan104 proposed a mutual information (MI)-MBPCA monitoring scheme, which divided the variables according to the MI. The major advantage of the MI-based variable relationship evaluation is that it considers higher-order statistical information rather than linear correlation-based methods. A joint probability approach was used to identify the operation model, and the MI-based multiblock method was extended to multimode monitoring given that the most closely related variables were in the same block but most of the independent variables were in different blocks.105 Subsequently, Jiang and Yan106 developed a spectral clustering-based process decomposition method, which unified the variable relation-based process decomposition into the variable spectral clustering framework. An illustration of the spectral clusteringbased process decomposition is presented in Figure 6, and the main procedures are as follows: Step 1: Variable Relationship Evaluation. The correlation coefficient and MI are most widely used for evaluating the relationships between two variables, xi and xj. The correlation coefficient ρij considers the second-order statistical information and evaluates the linear relationship between xi and xj; the coefficient is calculated as follows: ρij =

cov(xi , xj) D(xi) D(xj)

(24)

The MI considers higher-order statistical information and evaluates linear and nonlinear relationships among variables, which can be calculated as follows: Iij =

ij p(x , x ) yz

∬x x p(xi , xj) logjjjjj p(x )i p(xj ) zzzzzdxi dxj i j

k

i

j

{

(25)

Step 2: Relationship Matrix Construction. Once the relationship between every pair of variables is evaluated, a G


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Industrial & Engineering Chemistry Research relationship matrix R can be constructed for all process variables, as follows: ÅÄÅ r11 r12 ... r1m ÑÉÑ ÑÑ ÅÅ Ñ ÅÅ ÅÅ r21 r22 ... r2m ÑÑÑ ÑÑ Å Å R = ÅÅ ÑÑ ÑÑ ÅÅ ∂ ∂ ∏ ∂ ÑÑ ÅÅ Ñ ÅÅ r r ÅÇ m1 m2 ... rmm ÑÑÖ (26)

over, determining which variables from the other units should be involved to improve the local unit monitoring performance is important. One approach is to evaluate the correlation with the other units through the CCA method. Jiang et al. proposed a regularized CCA-based distributed monitoring method to monitor an individual unit considering the information from the neighboring units.27 First, the CCA method is performed between a local unit and the neighboring units. Second, considering that the CCA monitoring incorporates only correlated variables, GA-based regularization is performed to eliminate unrelated variables. Then, an optimal local fault detection residual is generated, and a monitoring statistic is constructed. This CCA-based distributed monitoring scheme is suitable for a continuous process. For batch processes, a key unit monitoring scheme that incorporates the time-slice CCA method was proposed. The monitoring scheme was applied to an industrial injection molding process.19 For quality-related or KPI-related monitoring, Ge proposed a distributed predictive modeling (DPM) approach.113 The PCA method was used in each operational unit to extract the dominant information and reduce the data dimension. Then, a Gaussian process regression model was established based on the obtained principal component scores, through which the nonlinear relationship between the scores and the KPI variables were considered. The efficiency of the DPM method was illustrated through the application on the TEP. When a large number of measured variables exists, the presence of quality-irrelevant variables may degrade the quality-modeling performance. Therefore, quality-relevant plant-wide process monitoring is related to variable selection. A summary of the most widely used variable selection methods in PLS-based modeling has been presented in a previous study.114 More recently, sparse modeling methods have also been proposed for simultaneous dimension reduction and variable selection.115,116 These variable selection methods for PLS-based modeling have potential use in plant-wide process quality monitoring. Although the efficiency was presented, these variable relation-based process variable decomposition methods considered only the normal operating data but ignored the fault information. Jiang et al.35 noted that without considering the fault information, the performance could not be guaranteed to be optimal. 3.3. DMSPPM with Fault Information-Aided Variable Decomposition. In practice, the process data of constant or periodical faults could be available. Ghosh et al.117 discussed the impact of variable selection on the fault detection performance. A genetic algorithm (GA)-based variable selection approach was developed. Jiang et al.35 further discussed the impact of process decomposition on fault detectability from the geometric and probability aspects.1,35 The detectability of a fault relies on the variable relationship and the fault characteristics. When fault data are available, considering the fault information can benefit the block division and improve the monitoring performance. Certain performance-driven distributed PCA monitoring methods have been proposed. These performance-driven distributed monitoring methods incorporate the fault information and provide the best possible performance for data-available faults. When fault data are not available, certain randomized algorithm (RA)-based validation data generation approaches are proposed. Jiang and Yan proposed a parallel PCA-KPCA (P-PCA-KPCA) method that incorporated RA and GA for nonlinear process monitoring.39

where rij = ρij or rij = Iij. The relation matrix R is a real symmetric matrix. Step 3: Spectral Decomposition of the Relationship Matrix. Then, the real symmetric matrix R is spectrally decomposed as follows:

R = PΛs PT

(27)

where P contains the orthonormal vectors and Λ is a diagonal matrix containing the eigenvalues λs1 ⩾ λs2 ⩾... ⩾ λsm ⩾ 0. The elements pji in the projection matrix P indicate the importance of xj to the i-th latent factor. Then, the division of the variables is transformed into the division of the coefficient matrix P. The variables with effects that are similar to those of the latent factor are grouped into the same block. Step 4: Conventional Variable Clustering Methods of the Loading Matrix P. Assume that the number of blocks is determined as B. Then, certain classic clustering methods, such as K-means and fuzzy C means clustering algorithms, are used to cluster the variables. Details are provided in ref 106. Following the MI-based variable decomposition, several extensions and improvements are developed. Xu et al.107 proposed a minimal redundancy maximal relevance-based MBPCA monitoring method in which the relevance between the variables and the redundancy were considered in the block division. A relevant and independent multiblock quality monitoring method was proposed by Huang and Yan.108 Subsequently, Huang and Yan109 proposed an MI-based quality two-block monitoring approach that was relevant and independent. All variables were divided and grouped into a quality-relevant or quality-independent block according to the MI based on the quality variable. Then, KPCA monitors were established for monitoring the quality-relevant and qualityindependent blocks. To deal with process dynamics, Tong and Shi proposed a dynamic feature selection-based distributed monitoring that involves time series dynamic information.110 A correlation coefficient is employed to evaluate the variable relations and divide the sub-blocks. Li et al. proposed a sparse SFA-based process variable decomposition method that considers changes of both static and dynamic information.111 Variables that have both static and dynamic correlations are divided into the same block. More recently, Zhao and Sun proposed a cointegration analysis (CA)-based process variable decomposition method for distributed process monitoring of nonstationary processes subject to frequently varying conditions.112 Nonstationary variables are first selected by the augmented Dickey−Fuller test and then divided into sub-blocks based on a sparse CAbased iterative block decomposition algorithm. With this process variable decomposition, the dynamic process behavior is examined from both local and global viewpoints. In practice, certain key operational units are fault-sensitive, and these key operational units should be monitored emphatically. Monitoring the variables of a local unit only ignores the correlation information, whereas including all available variables introduces monitoring redundancy. Mores

H


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Figure 7. Illustration of the fault information-aided decomposition and distributed monitoring scheme.

fitness value can be calculated directly, as shown in eq 28. When fault data are unavailable, RA-based fault construction can be employed to generate fault validation data. Step 4: Gene Operation and Chromosome Evolution. The chromosome is subjected to gene operations, such as selection, mutation, and crossover, until the optimal solution is obtained, or a predefined maximum number of generations is reached. Recently, Jiang et al.120 proposed an RA fault construction integrated with an optimization approach to determine the optimal variable transmission structure. Local monitors were established by involving the related variables from the other units for performance improvement. The objective of the correlation-based variable transmission determination is to preserve the highest level of correlation with the least number of variables retained. Then, the fitness function is constructed as follows:

The key idea of the P-PCA-KPCA was to explore the complex relationships among variables through RA-based fault construction. The variables involved in the PCA or the KPCA model were determined by optimizing the performance of the validation data. The P-PCA-KPCA monitoring method was applied to the CSTR process. These performance-driven or RA-based fault construction methods could be regarded as fault information-aided methods, and extensions to the nonlinear and dynamic processes were also developed.118,119 An illustration of the fault information-aided process decomposition is presented in Figure 7. The key steps of the fault information-aided process decomposition include incorporating optimization methods to achieve variable selection, summarized as follows: Step 1: Chromosome Design. The first step in the optimization is designing the chromosome. Generally, the variables are encoded by genes in the chromosome, and the value of a gene indicates the presence or absence of a corresponding variable. Step 2: Fitness Function Construction. The second step in the optimization is designing the fitness function. The objective is to achieve the largest NDR value given a determined FAR. Then, the fitness function is generally determined as follows: min NDR = w

s . t . FAR =

min(−tr(Σk ) + λ|w|) w

s . t . tr(Σk ) ⩾ ηtr(Σk 0)

where w denotes the gene values in the chromosome. By optimizing the transmitted variables, the fault detection performance of the local unit is improved.

Number of samples (J ⩽ Jth |f ≠ 0)

4. DMSPPM DECISION-MAKING SYSTEM DESIGN Once the distributed local monitors are established, a decisionmaking system becomes necessary to provide comprehensive evaluation of process status. Various decision fusion methods, such as the voting method,16 the “or” rule,109 the support vector data description method,104 Dempster−Shafer fusion methods,121,122 and any of the Bayesian methods, have been proposed.35,123 A comprehensive review of decision fusion methods for fault detection and diagnosis is provided by Ghosh et al.124 For establishing the decision-making system in data-driven distributed monitoring, Bayesian fusion methods

Total number of faulty samples

Number of samples (J > Jth |f = 0) Total number of normal samples

(29)

⩽ FAR th (28)

where FARth is the threshold of FAR. Step 3: Fitness Evaluation. Validation samples are necessary to evaluate the fitness given the temporarily established local monitors. When fault data are available, the I


Review


networked distributed monitoring was developed, thereby improving the practical application feasibility.128

are the most widely used. Here we focus only on the Bayesian methods, including the Bayesian inference-based fault detection and Bayesian method-based fault diagnosis. 4.1. Bayesian Inference Statistic Fusion. The Bayesian inference comprehensive (BIC) statistic is a weighted form of the local monitor results. The fault probability given xb (variables from the b-th block) is defined as follows:35,123 P(Fault |xb) =

P(xb|Fault )P(Fault ) P(xb)

5. CONCLUSIONS AND PERSPECTIVES This paper provided a detailed review on the data-driven distributed monitoring methods for industrial plant-wide processes. First, the basic MSPM methods were revisited, and motivations for establishing a distributed monitoring model were clarified. Then special attention was given to the DMSPPM method for large-scale and multiunit processes. Key techniques in establishing a distributed monitoring model, including process variable decomposition, local monitor design, and decision making system design, were introduced. Special attention was paid to the DMSPPM process variable decomposition, which can generally be classified as the knowledge-based methods, the variable relation-based methods, and fault information-aided methods. The knowledge-based process variable decomposition methods divide the process variables according to prior process knowledge, thereby having advantages in explaining the results and identifying the fault root cause. However, accurate knowledge for the decomposition is not always available. The variable relation-based methods and fault information-aided methods belong to the data-driven decomposition methods and divide the process variables based on process data, whose key advantage is that the requirement on process knowledge is relaxed. If no fault information is available, the key consideration in the process decomposition is the variable relations, and the variable relation-based methods are generally used. Once fault information is available, the fault information can be considered to improve the monitoring performance for known faults. Although several efficient methods have been developed, the DMSPPM method is still in its infancy. Several issues should be addressed in the future. The challenges and perspectives based on experiences of the authors are as follows: 5.1. Complex Process Characteristics. Currently, most existing works focus on the linear static processes, whereas a practical industrial process can be characterized by complex process behaviors, such as nonlinearity, dynamic, and multiple operation mode. Even though efforts have been made to address these complex process characteristics and some methods have been developed, theoretical basics of distributed monitoring for these complex processes are still under investigation. Data-driven distributed plant-wide process monitoring with more complex process characteristics considered requires more discussion. 5.2. Large but Imbalanced Data. In the process operation history, a large amount of process data under normal operating conditions are available. These normal operating condition data contain meaningful process information, but they are generally unlabeled. The labeled data, such as the quality data or the fault data, are difficult to obtain because of technique or economic limitations. It is common that a large volume of normal operating data exists but with a smaller amount of labeled data, thereby causing two main challenges in plant-wide process modeling and monitoring. The first limitation is extracting meaningful information from a large volume of data. The second limitation is determining the meaningful representations from the imbalanced data. The issue of large but imbalanced data is under discussion. 5.3. Multisource and Heterogeneous Signal Fusion. Except for conventional process variables measured by classical

(30)

P(xb) = P(xb|Normal)P(Normal) + P(xb|Fault )P(Fault ) (31)

where P(Normal) is defined at the confidence level β, whereas P(Fault) is defined as 1 − β. The terms P(xb | Normal) and P(xb | Fault) with respect to statistic Jb (statistic of the b-th local monitor) are defined as follows: new

P(xb|Normal) = e−Jb

/ Jb , th

new

, P(xb|Fault ) = e−Jb,th / Jb

(32)

Jnew b

where is the Jb statistic of a new sample; and Jb, th is the corresponding threshold. The overall Bayesian inference comprehensive statistic is calculated as follows: | l o o o P(xb|Fault )P(Fault |xb) o o } B o o ∑ | P ( x Fault ) o j b=1 o j = 1 n ~

∑ omoo B

BIC =

(33)

Numerous distributed monitoring methods employ this Bayesian inference method to fuse fault detection results from local monitors. The effectiveness of this Bayesian inferencebased decision making system has been illustrated in literature reports.1,35,99,101,103,112,125 4.2. Bayesian Fault Diagnosis System. When a fault is detected, assigning the current fault status to the most closely related historical fault sets F = {f1, f 2, ..., f r} becomes necessary. Then, the fault diagnosis aims to determine the fault status F using the current evidence ec and the historical data set D. The Bayesian solution to the posterior p(F | ec, D) is stated as follows:126 p(F | e c , D ) =

p(e c | F , D )p(F | D ) p(e c | D )

(34)

where p(e | F, D) is the likelihood, p(F | D) is the prior of F, and p(ec | D) = ∑Fp(ec | F, D)p(F | D) is a scaling factor. Pernestal127 provided the data-driven solution to p(ec | F, D) as follows: c

p(e = ei|F = f j , D) =

ni | f + ai | f j

Nf + A f j

j

j

(35)

where ni | f j accounts for the historical samples with e = ei and F = f j; ai | f j accounts for the prior samples assigned to ei under f j; Nf j = ∑ini | f j, and Af j = ∑iai | f j. The historical samples and prior samples contribute to the likelihood, which allows adding prior knowledge to the diagnosis.127 This scheme is the basic Bayesian fault diagnosis framework, and various extensions have been proposed. Jiang et al.126 first applied the Bayesian fault diagnosis framework to the PCAbased fault diagnosis and discussed the evidence generation and selection issue.1 Subsequently, a Bayesian diagnosis method that handled asynchronous measurements for J


Review


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industrial sensors, certain new sensing signal types, such as images, audio signals, or vibrations, are employed in modern processes. Extracting meaningful features from these multisource and heterogeneous signals and fusing these signals to achieve efficient monitoring for plant-wide processes are challenges that need to be surmounted. The multisource and heterogeneous signal fusion-based plant-wide process monitoring methods are being studied. 5.4. Cause−Effect and Propagation Path Analysis. In a large-scale, plant-wide process, an early stage fault generally affects only a local part or a local unit of the process. However, because of the increasing fault magnitude and the interaction effect among units, the fault propagates to the other units and spreads throughout the entire process. Then, the cause−effect and the propagation path analysis benefit the fault isolation and process recovery. 5.5. Monitoring Performance Assessment. Currently, most monitoring algorithms are limited in terms of their theoretical analysis or are tested only through certain simulation processes. For example, the TEP has been widely used for monitoring performance assessment. However, even the most widely used TEP is simulated after considering the material balance but ignoring the energy balance. The characteristics of an actual process may be more complex. More reliable performance assessment techniques or testing equipment are desired to provide a benchmark testing platform for monitoring performance assessment.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Mailing address: East China University of Science and Technology, P.O. Box 293, MeiLong Road no. 130, Shanghai 200237, P. R. China. ORCID

Qingchao Jiang: 0000-0002-3402-9018 Xuefeng Yan: 0000-0001-5622-8686 Biao Huang: 0000-0001-9082-2216 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors gratefully acknowledge the support from National Natural Science Foundation of China (61603138 and 21878081), the Programme of Introducing Talents of Discipline to Universities (the 111 Project, B17017), and the Natural Science and Engineering Research Council of Canada.

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