Estimation of Disturbance Propagation Path Using Principal

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Estimation of disturbance propagation path using principal component analysis (PCA) and multivariate granger causality (MVGC) techniques Usama Ahmed, Daegeun Ha, Seolin Shin, Nadeem Shaukat, Umer Zahid, and Chonghun Han Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.6b02763 • Publication Date (Web): 02 Jun 2017 Downloaded from http://pubs.acs.org on June 8, 2017

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Industrial & Engineering Chemistry Research

Manuscript for Industrial & Engineering Chemistry Research

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Estimation of disturbance propagation path using principal

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component analysis (PCA) and multivariate granger causality

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(MVGC) techniques

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Usama Ahmed a, Daegeun Ha a, Seolin Shin a, Nadeem Shaukat b, Umer Zahid c, Chonghun

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Han +a

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a

School of Chemical and Biological Engineering, Seoul National University, Seoul 151-744, Republic of Korea

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Department of Nuclear Engineering, Seoul National University, Seoul 151-744, Republic of Korea

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Chemical Engineering Department, King Fahd University of Petroleum & Minerals, Dhahran, Saudi Arabia.

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Correspondance : Chonghun Han ([email protected])

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+

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[email protected]

To whom correspondence should be addressed. Phone: +82-02-880-1887. Fax: +82-02-880-1887. Email:

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Abstract:

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Process monitoring and fault diagnosis using the multivariate statistical methodologies has been

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extensively used in the process and product development industries for the last many decades.

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The fault in one process variable readily affects all the other associated variables which not only

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makes the fault detection process more difficult but also time consuming. In this study, PCA

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based fault amplification algorithm is developed to detect both the root cause of fault and the

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fault propagation path in the system. The developed algorithm project the samples on the

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residual subspace (RS) to determine the disturbance propagation path. Usually, the RS of the

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fault data is superimposed with the normal process variations which should be minimized to

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amplify the fault magnitude. The RS containing amplified fault is then converted into the co-

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variance matrix followed by singular value decomposition (SVD) analysis which in turn

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generates the fault direction matrix corresponding to the largest eigenvalue. The fault variables

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are then re-arranged according to their magnitude of contribution towards a fault which in turn

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represents the fault propagation path using an absolute descending order functions. Moreover,

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the multivariate granger causality (MVGC) algorithm is used to analyze the causal relationship

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among the variables obtained from the developed algorithm. Both the methodologies are tested

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on the LNG fractionation process train and distillation column operation where some fault case

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scenarios are assumed to estimate the fault directions. It is observed that the hierarchy of

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variables obtained from fault propagation path algorithm are in good agreement with the MVGC

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algorithm. Therefore, fault amplification methodology can be used in industrial systems for

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identifying the root cause of fault as well as the fault propagation path.

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Keywords: Principal Component Analysis (PCA), Multi variate granger causality (MVGC),

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Singular Value Decomposition (SVD), Fault detection, Fault propagation path estimation 2 ACS Paragon Plus Environment

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1. Introduction

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With the developments in the automation and control systems in the process and product

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industries, it is possible to collect enormous amount of data. However, analysis and interpretation

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of data is a key issue. Using various statistical tools, it is possible to quickly analyze the data to

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enhance both the process performance and reduce industrial waste, thereby, improving process

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economics. Online and offline process monitoring techniques are widely used in the industries to

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statistically analyze the process behavior. To ensure the smooth and safe operation of chemical

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plants, large number of sensors are usually used to record and analyze the data. However, with

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an increase in number of sensors, the chances for sensor faults in addition to the process faults

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have also been increased. Moreover, the occurrence of any fault in a system affects all the

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associated variables and disturbs their normal correlations which makes it difficult to detect the

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actual root cause of fault. Therefore, instant fault detection through analyzing the root cause and

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estimating fault propagation path in the system has always remained a key issue in process

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monitoring.

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Data-driven1 models have been widely used in semiconductor manufacturing2, chemical3 and

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steel industries4 exhibiting multi-level control hierarchy during the last few decades. Several

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univariate and multivariate statistical tools have been developed that can be efficiently used for

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both the fault detection and diagnosis. However, multivariate statistical methodologies are more

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preferred over the univariate techniques for analyzing the complex industrial system where

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process variables exhibits a strong correlations. Principal component analysis (PCA) is one of the

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effective multivariate statistical technique that finds its applications for process monitoring and



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control5, 6, fault detection and diagnosis7, 8, and sensor validation9, 10 in various process industries.

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The PCA converts the higher dimensional correlated data into lower dimensional un-correlated

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data while retaining most of the original information. It can linearly reduce the dimensionality

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while ignoring the nonlinearities in the process data. PCA models the process behavior in terms

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of process variables during normal operation and compares the variation in those variables

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during fault situation. Hotelling T2 statistics and Q-statistic (squared prediction error (SPE)) are

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the two fault detection indices commonly used for analyzing the variation in process variables.

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The T2 statistics represents the systematic part of process variation in principal component

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subspace (PCS), whereas, SPE shows the residual part of the process variation in the residual

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subspace (RS). The T2 statistics and SPE can be calculated for each sample and compared with

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the confidence limits to monitor the process faults. The occurrence of any fault in the process

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either affects the T2 statistic or SPE of the samples or even changes both the statistics in some

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cases. Usually both the indices exceed their critical values during fault situation, whereas, the

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process is considered to be normal if both the indices remain under the control limits. T2

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statistics is usually used for overall process monitoring as it measures the variation among

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samples and indicate their distance from the center of the model. As multiple sensors are

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simultaneously affected by a certain fault, the contribution plot of samples shows the

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involvement of many variables towards a fault that makes it difficult to identify the actual fault

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variable. On the contrary, SPE measures the difference/residuals between the sample and its

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projection on the model. It measures the sum of variations in the RS which is not explained by

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the principle components (PC). SPE is highly sensitive to even minor process variations due to

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its smaller value compared to the T2 statistic11. Qin etl al.12 also performed the detailed


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comparison for both the indices and showed that SPE can detect the fault more readily compare

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to the T2 statistics.

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The PCA models combined with the fault detection indices finds it applications in various

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industries12-17 to monitor the process variations and diagnose process faults. For an instance,

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Pehna et al.18 applied the PCA model to the nuclear power plant and used the fault detection

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indices to monitor the temperature variation in the nuclear reactors. Similarly, Ferrer19 used the

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T2 statistics to analyze the process shift in the automobile manufacturing industries. Landells et

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al.20 used the PCA methodology along with the statistical indices for early fault detection in

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refineries and other chemical production industries.

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Several studies have relied on PCA models and used fault detection indices for early fault

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detection in the industrial systems. However, little attention has been paid towards identifying

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the fault propagation path in addition to the fault detection. As the fault in one variable affects

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the other associated variables, it is important to estimate the fault propagation path direction

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moving across the variables. Recently, Hong et al. 21-23 developed the fault propagation path

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estimation algorithms based on the progressive PCA models. The algorithm uses the SPE

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contribution plots to detect the variable having high contribution and identifies it as the fault

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variable. The detected variable is then eliminated and new PCA model is developed to identify

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the next variable. The detection of fault variable from each model in turns represent the order of

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variables affected by a certain fault. This methodology can be applied to the small systems with

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limited number of variables, however, it can become more complex and time consuming with an

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increase in number of process variables or unit processes.

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Granger causality (GC) algorithms24 based on time series hypothesis is another tool that can

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evaluate the cause and effect relationship among variables. Developing the logical interpretation 5 ACS Paragon Plus Environment


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of causal analysis along with the process knowledge can help in detecting the root cause of the

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faults. GC algorithms not only find its application in process and energy industries25-27 but also in

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economics28 due to its ease of implementation and reliable interpretation of results. It uses a

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statistical hypothesis to predict whether one time series can affect the other time series or not.

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Additionally, the PCA models can be combined with the GC algorithms for efficient process

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monitoring. For instance, Li et al.29 developed the framework of locating the root cause of fault

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using both PCA and GC algorithms. Similarly, Ladman et al.27 integrates the process causality

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and topology information to develop a causal model that determine the disturbance propagation

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path. Moreover, Yuan et al.26 used the multilevel GC framework with clustering techniques for

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determining the causal relationship among plant-wide oscillatory variables. Ladman et al.30 also

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compared various statistical techniques for different industrial process and suggested that only

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one specific method is not powerful enough to predict the actual causal and effect relationship.

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Therefore, several techniques should be used in addition with the process knowledge to obtain

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successful results.

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The PCA models usually require a training data set for developing the model and to determine

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confidence limits for the fault detection indices. Then the real plant data or fault data can be used

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for detecting and diagnosing the fault. On the other hand, GC uses only the time series

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information of variables irrespective of the normal or fault data to evaluate the causal

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relationship among variables. The aim of this study is to develop a robust algorithm that can

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predict the fault propagation path in a system while verifying the causal relationship among those

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fault variables. Therefore, the fault amplification methodology is first applied to the conventional

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PCA model to amplify the fault magnitude to identify the disturbance propagation path in terms

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of process variables. Then, the standalone multivariate GC (MVGC) methodology is used to 6 ACS Paragon Plus Environment

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identify the cause and effect relationship among the fault variables. The results from both the

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models are then compared to ensure the reliability of the fault propagation path obtained and

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verifying the causal relationship among variables at the same time. The developed models are

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applied to the LNG fractionation process and distillation columns operation where some of the

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common fault case scenarios are assumed to estimate the fault directions.

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The paper is divided into four major sections. First section briefly explains the PCA model

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development process and the methodology for estimating the fault propagation path. The

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following section explains the time series multi-variate granger causality algorithm to determine

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the causal analysis among variables. The third section explains the LNG fractionation process

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taken as an exemplary case and discusses the results for the simulated fault case scenarios to

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determine the fault propagation path. Finally, the last section embodies the conclusion of the

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paper.

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2. Fault detection based on Principal Component Analysis (PCA) methodology

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2.1. Concept of the PCA approach

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PCA model is developed in this study using Matlab® which is a commercial tool that contains

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large number of build-in mathematical functions to handle large dimensional matrices. PCA

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model uses the redundant information in the data to linearly reduce the set of correlated variables

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into fewer principal components (PC’s) that are not correlated. PCA model requires a data

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matrix X ε ℜ ×

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where n and m represent the number of samples and corresponding variables, respectively.

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Usually, a large number of samples are included in the training data which are scaled to zero

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mean and unit variance to capture the normal process variations. The scaled data matrix X given



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in equation 1 is used to develop the PCA model where represents the sample of m variables.

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= [ … . . ] ε ℜ ×

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Two types of alternative techniques can be used to generate the principal components. First one

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is the singular value decomposition (SVD) method that directly decomposes the data matrix into

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(1)

score matrix T ε ℜ × , loading matrix P ε ℜ × and residual matrix E as given in equation 2, where l represents the number of PC’s13. The principle component scores (T) and loadings (P) in

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turn represents the samples and variables relationship, respectively.

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= + = +

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matrices represents the variation and noise in the system18, respectively. Moreover, the residual

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(2)

T holds the linear combination of matrix X with the vector P such that T = X P where and E and , matrix can be further decomposed into residual scores and loadings represented as respectively. Usually, the number of PC’s are equal to the number of variables (m) included in

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the training data set. Some of the common methodologies available to shortlist the number of

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PC’s includes scree test, parallel analysis, percent variance test and residual sum of square

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statistics. As no fix criteria is available to use any specific technique13, we have used the percent

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variance technique to select only those PC’s that could explain the cumulative variance (90-99%)

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of data. The alternate method is to find the covariance of the data matrix X followed by the

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application of eigenvalue decomposition as given in equation 3.

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=

(3)


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The eigenvalue decomposition generates the P (principal components) and the diagonal matrix

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enclosing eigenvalues arranged in the descending order corresponding to the l leading

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eigenvectors as represented in equation 4.

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Λ = diag {λ! , λ# , λ$ , . . . . , λ }

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(4)

The sample vector x ε ℜ can be projected on the principal component subspace (PCS) and residual subspace (RS) as shown in equation 5 and equation 6, respectively. % = &≡ ∴ & = ' = − % ≡ (I - )

(5) (6)

%+ ' =

(7)

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2.2 Fault detection algorithm

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2.2.1

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The most important part of the process monitoring is the fault detection. Hotelling’s T2 and SPE

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indices measures the variability in the PCS and RS, respectively. T2 statistic measures the

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variation of each sample in the PCA model, whereas, SPE measures the distance between sample

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and its projection on to the model. The statistical limits are developed for both the indices and

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are compared with each sample. The Hotelling’s T2 statistic can be measured for each sample x

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as given in equation 8, where ∑* represents the eigenvalues corresponding to the P loading

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vectors.

Measure of variability using fault detection indices

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= +∑, )

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The control limit for the T2 statistic exhibiting multi-variate normal distribution can be calculated

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using frequency-distribution as shown in equation 9.

(8)



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# -*,, α =

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Where a, n and α represents the number of principal components, number of samples and level

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of significance. On the other hand, the SPE of each sample can be calculated by using equation

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10 where I represents the identity matrix.

*+!) *

.*,*,α

(9)

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' ∥ ≡ ∥ +3 − ) ∥2 SPE =∥

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Jackseon et al.31 developed the methodology for determining the control limits of SPE as given

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in equations 11-14, where α and δα # denotes the level of significance and upper control limit,

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respectively.

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456 ≤ δα #

(11)

ℎ: ;α

Estimation of Disturbance Propagation Path Using Principal

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