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https://www.revistaespacios.com Pag. 30

Vol. 43 (07) 2022 • Art. 3

Recibido/Received: 03/05/2022 • Aprobado/Approved: 01/07/2022 • Publicado/Published: 15/07/2022

DOI: 10.48082/espacios-a22v43n07p03

Granger causality procedeture to diagnosis and failture in

industrial systems

Procedimiento de causalidad de Granger para diagnóstico y localización de fallas en

sistemas industriales

BECERRA-ANGARITA, Oscar F.

ALVAREZ-PIZARRO, Yuli A.

Abstract

Industrial process supervision is an important subject nowdays due to the increased requirement for

safer processes for operators and effective for companies. Control loops affected by disturbs, are

grouped with PCA, based on their increased variability and the causal relationships between them are

detected via Granger causality. A graph drawing algorithm allows indicating the source of the

disturbance. The procedure is applied to data from a simulated chemical process CSTR. The proposed

procedeture correctly indicated the sources of disturbances.

Key words: fault diagnosis, Granger causality, system identification

Resumen

La supervisión de procesos industriales es un tema importante en la actualidad debido a la creciente

necesidad de procesos más seguros para los operadores y efectivos para las empresas. Los lazos de

control afectados por perturbaciones se agrupan con PCA, en función de su mayor variabilidad y las

relaciones causales entre ellos se detectan mediante la causalidad de Granger. Un algoritmo de dibujo

de gráficos permite indicar la fuente de la perturbación. El procedimiento se aplica a datos de un

proceso químico simulado CSTR. El procedimiento propuesto indicaba correctamente las fuentes de

perturbaciones.

Palabras clave: diagnóstico de fallas, causalidad de Granger, identificación del sistemas

1. Introduction

Technical improvements in recent years have improved quality and productivity in the industry, but they have

also resulted in the creation of increasingly complex systems to study and keep until their final stage of life. Just

managing new equipment and performing process maintenance does not guarantee a safe environment for

operators. Process plant management remains predominantly a manual activity, equally is the detection of

process abnormalities and the diagnosis of their probable causes. Knowledge about the relationships between

Docente investigador. Ingenieria electrónica . Universidad de Investigación y Desarrollo - UDI. Colombia. obecerra2@udi.edu.co

Docente. Ingenieria de telecomunicaciones. Universidad Santo Tomás. Colombia. yuli.alvarez01@ustabuca.edu.co

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BECERRA-ANGARITA O.F. & ALVAREZ-PIZARRO Y.A. «Granger causality procedeture to diagnosis and failture

in industrial systems»

Pag. 31

process variables is important to support decisions that take you back to your normal and safe operating state.

According to industry statistics, 70 % of accidents are caused by human error (Nor, N. et al., 2020). Recent events

at large plants, such as the detonation of the Kuwait Mina Al-Ahmed petrochemical refinery, had estimated

damage of $ 400 million. Some other case is the explosion of Petrobras’s offshore platform, Brazil, which resulted

in losses of $ 5 USD billion (Nor, N. et al., 2020). Although these types of large-scale accidents are not usual,

minor accidents are common, generating both economic and human losses. This indicates that there is still a long

way to go to improve supervisory and fault diagnosis performance for industrial processes.

Fault detection and diagnosis is a major problem in control systems (Alauddin, M. et al., 2018). Chemical

processes, power plants, factories and others are cases where an undetected failure can contribute to disastrous

economic, environmental and social effects. Investigations have been conducted to diagnose faults and monitor

equipment degradation. Artificial intelligence systems such as neural networks and fuzzy systems have been

applied to alleviate these difficulties and enhance the process monitoring system. (Kirilova, E. G. et al., 2022)

provide an extensive review of the various applications of neural networks for chemical engineering purposes,

and comparisons to existing conventional methods are also shown, both in simulation and online

implementation. Isolation of faults that propagate their effects on the plant are investigated in (Yong, G. et al.,

2015). Causality detection methods are used, and a new method has been proposed in (Marques, V. et al., 2015).

In a similar way to the parameter estimation failure detection methods in (Lindner, B. et al., 2019) seeks to detect

failures through changes in the causal relationships between variables. In addition, allowing the detection of

changes in the causal relationship, the data generated in each iteration can be used to test the relevance of

statistical causality, which is very important to increase the reliability in this type of research. This character test

was performed for Granger causality using surrogate series in (Sysoev, I. et al., 2015).

Faults or disturbances are detrimental in moving process variables away from their references. Its detection and

indication of the source is more problematic because its effect disappears over time (He, Z. et al., 2018). The

usual procedure is to search the databases for the variables affected by the disturbance and investigate their

cause from the knowledge of the process. This paper proposes a methodology for automatic indication of the

source of disturbances. Indicating an incipient fault detection approach via detrending and denoising is proposed.

The article is organized as follows: The next section offers a method for detecting disturbances and grouping

control loops that have been affected using PCA, and discusses the steps required to calculate Granger causality

and proposes a methodology based on it. The methodology for diagnosing system failures will be outlined step

by step in subsection 2.10 as the application of the proposed procedeture. Section 3 application to data from a

simulated industrial system CSTR. The conclusions are presented in Section 4.

2. Fault Detection in industrial systems

An industrial process has numerous variables, some of which are explicit in its importance to the process and

some are difficult to know if the process truly depends on them, but the relationship between them must be

considered. The propose of fault detection is to determine whether a potential fault has occurred in the sensor

network. A model is generally constructed through the fault-free health-monitoring in structural health

monitoring (SHM) field data to describe the normality, after that a sensor fault detection index can be defined.

Due to the difference of model building, the fault detection stage can be divides into unilabiate control chart-

based , multivariate statistical analysis –base and residual based methods. The fault detection index is then

computed for the currently measured sensor data and compared to a decision threshold. The potential sensor

fault is determinate to occur after the fault detection index exceeds its corresponding threshold. (Yi, T. et al.,

2018)

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BECERRA-ANGARITA O.F. & ALVAREZ-PIZARRO Y.A. «Granger causality procedeture to diagnosis and failture

in industrial systems»

Pag. 32

A SHM system is usually equipped with different types of sensors , which constitute a sensor network, to collect

structural responses. Therefore, it is necessary to use multivariate statistics that allows the joint monitoring of

variables. Typically the and statistics are used to calculate thresholds for monitoring non-normal situations, for

that reason in SHM field, these are grouped using the PCA method (Huang, H. . et al., 2017) ,which in addition to

reducing the number of data worked, is one of the most frequently used due to simplicity of its application, high

diffusion in the literature and the possibility of statistical analysis of the process through fault detection index

that present a superior performance to univariate control diagrams.

2.1. Principal component analysis

Consider as a vector of N samples of m sensors, so each line represents a sampling of each sensor.

The matrix X must be normalized. It has zero mean and unit variance. The matrix X can be decomposed according

to the singular value decomposition algorithm (SVD) (Kruger, U. et al., 2012).

(1)

Covariance matrix

(2)

Where

𝑈" ∈ " 𝑅!"!

and

𝑉" ∈"𝑅#"#

are unitary matrices, and

Σ"𝜖"𝑅!"#

contains the non –negative real singular

values of decreasing magnitude along its main diagonal and zero off diagonal elements. Solving the equation (1)

is equivalent to solving an eigenvalue decomposition of the sample covariance matrix S. The eigenvalues of the

matrix S are placed from largest to smallest representing the variance of each component, being the first

eigenvalue representing the greatest variability of the set of variables (Kruger, U. et al., 2012).

(3)

Where the matrix P is related to the highest eigenvalues of the matrix S and the columns of the orthogonal matrix

𝑉

. To find the projections of the matrix X it is necessary to calculate the projection score matrix

𝑇

(Kruger, U. et

al. 2012).

(1)

The projection of the T is the estimated

𝑋

*=𝑇𝑃$

(2)

Residue values and estimated values

𝑋

can be obtained (Kruger, U. et al. 2012).

𝑟 = 𝑋 − 𝑋

*= (𝐼 − 𝑃𝑃$)𝑥

(3)

2.2. Hoteling’s Statistic

From the equation (2), assuming invertible S, it's possible to define equation (9):

𝑧 = Λ%&'(𝑉$𝑥

(4)

Nx m

XÂÎ

VUX

-1

SXXVV

»=L

{ , ,... }

diag

ll l

L=S S=

T XP=

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BECERRA-ANGARITA O.F. & ALVAREZ-PIZARRO Y.A. «Granger causality procedeture to diagnosis and failture

in industrial systems»

Pag. 33

Hoteling’s statistic is given by (Chiang, L. et al., 2001):

(5)

The

𝑇(

statistic is the square 2-norm of a normalized observation vector x. From the equation (7) a threshold can

be found to characterize the variability of the data in all observed dimensions. Turned over a significance level,

appropriate values for the

𝑇(

statistic threshold can be automatically determined by using the F distribution.

Given a α significance level, the threshold can be calculated (Chiang, L. et al., 2001):

(6)

2.3. Statistic Q

The Q statistic is the 2-norm square standard that measures the deviation of the residue generated by the

difference between the estimate and the observations made, since r is the residue, the statistic is given by

(Chiang, L. et al., 2001):

(7)

The Q distribution threshold was approached by Jackson and Mudholkar (Chiang, L. et al., 2001).

(8)

Where

𝜃)="

∑

𝜎

(), ℎ+= 1 − (,!,"

-,#

*%./& "

and

𝑐0

is the normal deviation corresponding to the percentage

(1 −

𝛼)

where α is the significance level.

2.4. Statistical contributions Hoteling’s Statistic and Q

Once a fault is detected, it is necessary to determine which variables have left the control zone, which can be

extremely complex for systems with many variables. One way to distinguish affected variables is to calculate the

contributions of each variable to the statistics at the time of failure and to obtain the set of variables that

contributed to the violation. For the

𝑇(

statistic, the contributions at failure times are calculated using the T

score matrix. Negative contribution values are taken to zero.

(9)

where

𝜆&

is the eigenvalue corresponding to the score matrix column and

𝑋)1*

is the normalized sample. To

calculate the contributions of the residue, the squared residue itself is used (Chiang, L. et al., 2001).

(10)

Tzz=

2(1)(1)(, )

()

mn n

TFmnm

nn m

Qrr=

02 20 0

2(1)

hc hh

qqq

éù

=++

êú

ëû

2,,

ji i j

CONT P X

()

CONT r=

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BECERRA-ANGARITA O.F. & ALVAREZ-PIZARRO Y.A. «Granger causality procedeture to diagnosis and failture

in industrial systems»

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2.5. Granger causality

In diverse studies of science, causal relationships are inferred by using temporal signals of interest collected from

a given physical process. Thus, methodologies that may infer from these data such relationships with a certain

property have been created and discussed. In 1956, the mathematician Wiener intuitively formulated the idea

of causality in the prediction of time series (Wiener, N. et al. 1956), which was formalized in the field of

Econometrics using linear regression models by economist Granger (Granger, C. W. et al. 1969). According to

Granger causality, given two variables x and y, if the inclusion of past observations of x helps reduce the

prediction error of y then x causes y. This method was used in a financial market application to investigate the

uncertainty in predicting output growth in emerging markets (Balcilar, M. et al. 2022) and used to determine cause

and effect relationships in the social sciences studying life expectancy versus air pollution trajectories in Nigeria.

(Nwani, S.E. et al. 2022), neuroscience (Barnett et al. 2014), (Seth, A. et al. 2015), in the field of industrial process

engineering for root cause detection of disturbance or oscillation (Lindner, B. et al. 2019), (Lucke, M et al. 2022),

and recent Granger causality review and advances (Shojaie, A. et al. 2022)

In most of the causal investigations we try to discuss single causes in deterministic situations and two conditions

are important for causal determinations suppose that an event x is a cause for event y:

Granger causality assumes that the future can cause neither the present nor the past. In the case of the variables

x and y, you can have the following situations:

1. y causes x : (y → x)

2. x causes y : (x → y)

3. Feedback occurs between the two variables: (x ↔ y)

4. There is no causal relationship.

Thus, what matters is whether there is a statistically significant cause and effect relationship between the

variables x and y, which only occurs if there is a correlation and temporal precedence relationship between them

(Granger, C. W. et al. 1969).

Some procedures must be performed before applying the Granger method to a set of variables of interest. It is

first necessary to check if there is a requirement to perform some kind of preprocessing to the signals and to

verify their seasonality. The definition of the order of the estimated models should be chosen based on some

selection criteria (AIC or BIC, for example), just as the models should be validated using some specific techniques.

One should also choose the statistical method to infer causality and the correct method for multiple comparisons

performed (Aguirre, L. A. et al. 2004).

2.6. Mathematical modeling

A model is the simplified representation of a real system whose purpose is to identify its most relevant aspects

without worrying about all the details. A system can be modeled using black box modeling. This modeling, also

known as identification, assumes that little or no prior knowledge of the process is required, so having only

available inputs and outputs, it is possible to obtain a mathematical model of the system under study (Ljung, L.

et al. 1998), (Aguirre, L. A. et al. 2004).

Among these forms of representation, we can highlight the representation by transfer function, by state space

and the discrete time representation. Regarding this last representation, consider the following model:

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BECERRA-ANGARITA O.F. & ALVAREZ-PIZARRO Y.A. «Granger causality procedeture to diagnosis and failture

in industrial systems»

Pag. 35

(11)

Where

𝑦(𝑡)

and

𝑢(𝑡)

are the system output and input respectively, q is the delay operator,

𝜀(𝑡)

is the white noise

residue and all variables are measured in time where

𝑡 = 1,2,3 … 𝑇𝐴(𝑞)

, and

𝐵(𝑞)

are arbitrary polynomials

defined by

𝐴

(

𝑞

)

= 1 − 𝑎&𝑞%& − 𝑎(𝑞%( … − 𝑎!$𝑞%!$

and

𝐵

(

𝑞

)

= 1 − 𝑏&𝑞%& − 𝑏(𝑞%( … − 𝑏!%𝑞%!%

the

parameters

𝑛2

and

𝑛3

refer to the number of parameters or the order of the chosen model. For this work it will

be considered that

𝑛2="𝑛2= 𝑝

in all cases analyzed, that is the number of poles is the same number of zeros.

The model presented in Equation (14) is called the autoregressive model with exogenous inputs (ARX), where

the acronym AR refers to the autoregressive part of the model given by

𝐴

(

𝑞

)

𝑢

(

𝑡

) the letter X refers to the entry

𝐵

(

𝑞

)

𝑢

(

𝑡

) is called exogenous variable x.

In this work, system identification techniques use models whose representation is the ARX. The estimation of its

parameters made using the least squares method, to properly adjust the input and output data and verify if there

is a cause and effect between these data. Significantly, for causality analysis, it is not known who is entering or

leaving the model. For this reason, we seek the causal relationship between the variables by testing them all as

input and output and verifying if the variables included in obtaining the models help to predict the variable tested

as output.

2.7. Model order selection criteria

To estimate a model using the ARX structure in Granger analysis, is necessary to choose the order of the model.

One way to make this choice is to use information criteria that minimize a residual function that is penalized by

the number of regressors used, thus looking for the most parsimonious model. In this direction, we look for the

model that regards the minimum possible parameters to estimate and explain the behavior of the study variable

(small error) (Aguirre, L. A. et al. 2004). In this work the following criteria are used: Akaike Information Criterion

(AIC), Bayesian Information Criterion (BIC) (Ljung, L. et al. 1998).

(12)

(13)

Where

𝑙𝑛

| is the napierian logarithm that determines the residual covariance matrix of unrestricted models

and measures the suitability of the model. Increasing the number of parameters allowed increases the number

of degrees of freedom, generating less prediction error variation or allowing for more accurate data adjustment.

Reducing error variability by increasing the number of parameters is balanced by a penalty imposed by the

reporting criterion. Therefore, how the beginning parts of the equations (15) and (16) measure the reduction in

residual variation, while the second parts penalize the inclusion of each condition. If the penalty is less than the

reduction in residual variability, the regressor should be integrated into the model. Differently, the regression

will bring more cost than benefit and should be excluded from the model, allowing selection of the order that

minimizes the criterion applied.

2.8. Granger Conditional Causality

When the number of variables are greater than

2(𝑛 > 2)

, the bivariate Granger analysis can be extended to a

multivariable case (Seth, A. K. et al. 2010). Suppose, for example, that exist three variables

𝑥&, 𝑥(, 𝑥-

and is

necessary to know if the

𝑥-

variable causes the

𝑥&

variable. So if excluding the

𝑥-

variable significantly increases

()() ()() ()Aq yt Bqut t

ln | |

AIC T

=S+

ln | |

Tpn

BIC T

=S+

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BECERRA-ANGARITA O.F. & ALVAREZ-PIZARRO Y.A. «Granger causality procedeture to diagnosis and failture

in industrial systems»

Pag. 36

the model prediction error with the

𝑥&

variable output as compared to the

𝑥&

model prediction error including

all variables, then it can be said that

𝑥-→ 𝑥&

. To demonstrate the mathematical formulation of this analysis,

consider n variables. An

𝑦4

vector of dimension

𝑛"!

will be used to represent the observed values of all variables

at the time instant t. Thus, the autoregressive vector of order p can be expressed by:

(14)

Where

𝐴5

is the square array of parameters of the VAR model. Systems can be represented in matrix form or by

the equation (15).

(15)

The condition is

𝑚 > 𝑛𝑝

necessary to avoid singularities in the matrix product

𝑥$𝑥

, so that the B coefficients of

the B matrix can be estimated using the least squares method by the equation (19).

𝐵

*=(𝑋$𝑋)%&𝑋$𝑌

(16)

To detect the causal relationship of the variable

𝑥-

in the variable

𝑥&

, first estimate the quadratic sum of the

residuals with the variable

𝑥&

as output and all others as input (unrestricted model):

𝑅𝑆𝑆&= S𝑦" − "𝑋𝛽

UV$(𝑦 − 𝑋𝛽

(17)

Where y and β correspond respectively to y column and

𝐵

* Then, we estimate the quadratic sum of the

constrained model residues in the same way using the equation (13), having the

𝑥&

variable as output and

excluding only the

𝑥-

variable as input

𝑅𝑆𝑆-&

.Then the F distribution test statistic applies

𝐹0(𝑝, 𝑚 − 𝑛𝑝)

under

the null hypothesis that the variable

𝑥-

does not cause the variable

𝑥&

(18)

The null hypothesis will be rejected if

𝑓

""6"!> 𝐹0

or the calculated p-value is less than the significance level α

used in the test. Thus, it is said that

𝑥-→ 𝑥&

. To apply the Granger conditional analysis, the prerequisite that

𝑚 ≥ 𝑛𝑝

must be attended. Knowing that

𝑚 = 𝑇 − 𝑝

the condition for applying the conditional Granger analysis

is:

(19)

2.9. Granger multiple variable Causality

Consider that it is desired to infer causality between the variables by considering the following models found in

(Seth, A. K. et al. 2010).

(20)

tltt

yAX

() ( )( ) ()m n m np np n m n

YXB

´´´´

=+S

31 1

RSS RSS

fRSS

mnp

( )

1Tpn³+

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( )

0.8 1 0.5 2 0.4 1 0.2 2

0.9 1 0.8 2

0.5 1 0.2 2 0.5 1

xk xk xk zk yk

yk yk yk

zk zk zk yk

=---+-+-+

=---+

=---+-+

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BECERRA-ANGARITA O.F. & ALVAREZ-PIZARRO Y.A. «Granger causality procedeture to diagnosis and failture

in industrial systems»

Pag. 37

This model contains 3 different variables which have an autoregressive structure, and for this case, the variable

x depends on itself and a regressor of z and one of y; The y variable does not depend on another variable, it just

depends on itself, and finally the z variable depends on itself and y. All variable models are simulated, and white

noise has been introduced.

𝜀&, "𝜀(

and

𝜀-

.The first step is to determine if the data is stationary, because as a

requirement, the statistics applied to infer causality assume the stationary of the variables. In the Figure 1, 10

windows with 100 samples each showing the mean and standard deviation of the variables are presented. The

graphical result indicates that the data is stationary, as expected, as it was generated to satisfy this requirement.

If the data had not met this condition, the data should be differentiated until the prerequisite is met. (Ljung, L.

et al. 1998).

Figure 1

Variability stationary test

Source: Authors

The following step is established on the AIC or BIC criteria to estimate a minimum order to model the information

(Aguirre, L. A. et al. 2004). In the Figure 2 The AIC and BIC criteria for the signal set are presented. The criteria

indicate a minimum order of 2 to model all signals, which is coherent if we compare the equation (23).

Figure 2

AIC and BIC criteria for x,y and z

Source: Authors

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BECERRA-ANGARITA O.F. & ALVAREZ-PIZARRO Y.A. «Granger causality procedeture to diagnosis and failture

in industrial systems»

Pag. 38

Determining the minimum order according to the criteria, the following step is to analyze the autocorrelation of

residuals that are within the confidence interval for all regressors. In Figure 3, it is possible to verify that the

second order models adequately represent the signals, so the residue is white noise.

In order to infer causality between the set of three variables x, y and z, to solve this case it is necessary to perform

6 independent hypothesis tests or

𝑀 = 𝑛

(

𝑛 − 1

) where n is the number of variables. Only will be considered

significant casualties, when the p-value is less than or equal to

𝑓𝐹0

for each direction tested.

In the Figure 4 The graph found by applying the Granger causality method is shown. Comparing the result

obtained with the original equation models in equation (23), all causal relationships between x, y and x is caused

by the other.

Figure 3

AIC and BIC criteria Autocorrelation of residues for x, y and z

Source: Authors

Two variables, just as y is not caused by any other variable, so no arrow is pointing at her. Being verified the

method with this simple example; the following will be presented the methodology to solve the problem of fault

diagnosis in a complex system.

Figure 4

Granger causality graph for x, y and z

Source: Authors

2.10. Proposed procedeture to fault diagnosis

To properly diagnose the failure the following steps were defined:

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Step 1 Define and select data for system operation point, this amount of time series must be sure that does not

have any disturbance, so the system is working properly. Then extract information such as mean and standard

deviation of the variables, to normalize the data. With this normal dataset, the PCA model include a number of

components representing as initial test recommendation at least 70% of the variance of the variables, this

percentage is chosen arbitrarily or by the knowledge about the system, it may be too low or too high for a

particular application. When the number of observation variables is large and the amount of data available is

relative small, the

𝑇(

statistic tends to be inaccurate representation of the in-control process behavior, especially

in the loading vector corresponding to the smaller singular values. Additionally, the smaller singular values are

prone to errors because these values contain small signal noise. Therefore, in this case the loading vectors

associated only with the larger singular values should be retained in calculating the

𝑇("

statistic (Chiang, L. . et

al., 2001).

Step 2. Using the normalized system operation point data, following the equations (9) and (11) to calculate

statistical thresholds

𝑇(

and Q.

Step 3. For each new sample compute the statistical thresholds

𝑇(

and Q, if the thresholds are violated. After a

failure is indicated, calculate with the equations (12) and (13), the contributions of each variable to the failure

are calculated for a time window before and after the failure to raise a sign of the failure with the contributions

of the two statistics. Select the variables with the highest contribution.

Step 4. Using the variables separated by the calculation of the contributions, use the Granger causality method,

to infer causality between the affected variables to find the root cause or the relationship map between the

variables, but before of that follow the next considerations:

- To calculate causality relations between variables use the same data used for the normalized system

operation point data in step 1, but only use the suggested variables indicated by the contribution in the

step 3.

- Determine if the data is stationary, because as a requirement, the statistics applied to infer causality

assume the stationary of the variables. If the data do not have this condition, the data should be

differentiated until the prerequisite is met (Ljung, L. et al. 1998).

- To estimate a model using the ARX structure in Granger analysis is necessary to choose the order of the

model using AIC and BIC criteria.

Step 5. Identify the source or the sources that originated the fault through a search in the directed graph

produced by the Granger causality method.

The main objective in fault diagnosis area is to determine the cause of the fault. Even if the cause is not detected

using this steps, the information about the control loop where the fault took place is very close to the cause of

the fault. This information is very useful for process knowledge and may help to complete the diagnosis.

3. Experimental development, case study

Continuous stirred tank reactor (CSTR), this system is a simulator modeled in the Fortran language of a chemical

reactor that takes an exothermic reaction work where the original Fortran code is available in (Finch, F. E. et

al.1989). The MIT-CSTR software consists of a single file that can be easily compiled by a standard Fortran

compiler using MATLAB. The system has 18 variables and allows you to enter 22 different faults. The process

diagram is presented in Figure 5. A substance “A” is inserted into the tank to bring out an exothermic chemical

reaction to obtain two different products B and C. This system accepts level and temperature control circuits and

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BECERRA-ANGARITA O.F. & ALVAREZ-PIZARRO Y.A. «Granger causality procedeture to diagnosis and failture

in industrial systems»

Pag. 40

valve actuators and a fluid pump. In Table 1 the names of the system variables and Table 2 list the 22 faults that

can be introduced. In this work we will use fault number 19, which is the fault that affects the level controller

reference

𝑆𝑃&

Figure 5

CSTR system diagram

Source: Finch,F.E .1989

Table 1

CSTR variable list

Number

Variable

Acronym

Feed concentration

Feed flow rate

Feed temperature

Reactor level

Product A Concentration

Product B concentration

Reactor temperature

Coolant flow rate

Product flow rate

Coolant inlet temperature

Coolant inlet pressure

PCW

Level controller output

Coolant controller output

Coolant set point

Inventory

Mol balance

Cooling water pressure drop

Effluent pressure drop

Source: Finch,F.E. 1989

CNT

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BECERRA-ANGARITA O.F. & ALVAREZ-PIZARRO Y.A. «Granger causality procedeture to diagnosis and failture

in industrial systems»

Pag. 41

Table 2

CSTR fault list

Number

Fault

Acronym

No fault

Blockage at tank outlet

Blockage in jacket

Jacket leak to environment

Jacket leak to tank

Leak from pump

Loss of pump pressure

Jacket exchange surface fouling

External heat source (sink)

Primary reaction activation energy

Secondary reaction activation energy

Abnormal feed flowrate

Abnormal feed temperature

Abnormal feed concentration

Abnormal cooling water temperature

Abnormal cooling water pressure

Abnormal jacket effluent pressure

Abnormal reactor effluent pressure

REP

Abnormal level controller setpoint

Abnormal temperature controller setpoint

Control valve 1 stuck

Control valve 2 stuck

Source: Finch,F.E. 1989

3.1. Simulation

The system was simulated for 200 minutes where the first 100 minutes the system was in its normal operating

mode, then at minute 100 fault number 19 from Table 2 is inserted, Note that this failure affects the

𝑆𝑃&

level

controller reference. In the ¡Error! No se encuentra el origen de la referencia. is shown in the time domain the

18 simulation variables.

ext

PCW

JEP

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BECERRA-ANGARITA O.F. & ALVAREZ-PIZARRO Y.A. «Granger causality procedeture to diagnosis and failture

in industrial systems»

Pag. 42

Figure 6

CSTR simulation with fault number 19 at 100 min

Source: Authors

It is possible to observe how many of the variables left their normal operating point, some of them did not return

to the previous operating point as others that, having a controller, it compensated the failure through its

procedure and returned to normal system operation.

3.2. Fault detection

The ¡Error! No se encuentra el origen de la referencia. shows the fault indication in the system, the PCA is trained

with the normal operation data (first 100 minutes), and was reduced to 8 components, representing the 70% of

the variability of the data set, the algorithm correctly indicated the time when the system left its normal

operation using the statistics

𝑇(

and Q.

Figure 7

Failure indication with Statistics

𝑇!

and Q at minute 200

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BECERRA-ANGARITA O.F. & ALVAREZ-PIZARRO Y.A. «Granger causality procedeture to diagnosis and failture

in industrial systems»

Pag. 43

Source: Authors

In the ¡Error! No se encuentra el origen de la referencia. the contributions of the statistics for the failure interval

presented, that is, the scores of each variable. This interval contains the set of samples of the 18 variables from

minutes 195 to 210 or 5 minutes before failure and 10 minutes after detection. The purpose of this graph is to

analyze which variables contributed at the time of failure so that the statistics exceeded their thresholds.

Figure 8

Contributions of variables to fault 19 in the 195 to 210 minute interval

Source: Authors

Based on the contributions of fault 19, it is possible to find a pattern or signature of the fault by filtering out

those moments that are most significant. For this, the sum of all contributions in the time window was performed

to calculate obtaining 100% of the contribution and from this value a threshold of 1% was used to indicate how

significant a variable was determined to exceed the threshold. The result of this procedure is shown in ¡Error!

No se encuentra el origen de la referencia.. This figure shows how from the 200th minute that fault 19 was

inserted in the system, the variables affected in the

𝑇(

statistic were 4, 8, 9, 12 and 13; for the Q statistic,

variables 4, 5, 7, 9, 12, 13 and 14. As expected, there are several variables in common between the two

signatures; these significant variables are used to infer causality between them in order to check the causal map.

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BECERRA-ANGARITA O.F. & ALVAREZ-PIZARRO Y.A. «Granger causality procedeture to diagnosis and failture

in industrial systems»

Pag. 44

Figure 9

Fault 19 in 2D signature, in the interval 195 to 210 minutes

Source: Authors

3.3. Result analysis, Granger’s causality inference on the affected variables

Succeeding, the next step is to assess whether the data set is stationary in order to apply the Granger method.

The set of variables chosen with the help of the failure signatures was 4, 5, 7, 8, 9, 12, 13 and 14 the test result

to check the variables seasonality was negative so it was necessary to differentiate once the data (Aguirre, L. A.

et al. 2004). After differentiating the data, the test was positive. The next step is to apply the AIC or BIC criteria

to establish a minimum order for the set of signals.

Figure 10

AIC and BIC criteria for selected CSTR variables

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BECERRA-ANGARITA O.F. & ALVAREZ-PIZARRO Y.A. «Granger causality procedeture to diagnosis and failture

in industrial systems»

Pag. 45

Source: Authors

The ¡Error! No se encuentra el origen de la referencia. shows the result of the AIC and BIC criteria. These criteria

indicate the minimum order to estimate the set of signals, for the AIC the order is close to 10 and in the BIC it is

in 4, these orders are an approximation or starting point since it must be verified if the autocorrelation of the

residue is in mostly within the confidence interval. Fulfilling this condition, we can say that the model adequately

represents the data set since the residue does not contain any more useful information for the model. In this

case, order 15 was found. The ¡Error! No se encuentra el origen de la referencia. corroborates that the residues

can be called white noise.

Figure 11

Autocorrelation of the residue with model of order 15 for selected variables CSTR

Source: Authors

The result of Granger's causality method can be seen in ¡Error! No se encuentra el origen de la referencia. the

graph shows the cause and effect interactions between the variables that were selected with the fault

signature. The gain obtained with the subscription managed to reduce the search area by selecting the

variables that most contributed to the failure. In the graph, the arrow indicates that the pointed variable is

caused by the variable where the arrow starts. By the graph, fault 19, which affects the 𝑆𝑃& controller

reference, directly affects three variables and generates a reciprocal effect, between the temperature

variables. Therefore, even without knowing the source of the failure, it would still be possible to identify it.

The concentration of product A was not indicated with relation to cause or effect of any other variable, which

does not indicate that it does not have, only that the method was unable to infer.

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in industrial systems»

Pag. 46

Source: Authors

4. Conclusions

The proposed procedure to fault diagnosis with Granger's causality method is reviewed under identification

framework which can be useful in detecting the root cause of a failure in an industrial system. As stated, the

order of the model must be done with caution, because, if it is poorly dimensioned, the causal relations between

the variables may appear or disappear. However, even when not chosen in an appropriate way, they provide

guidance for investigating the root of a failure. When system identification techniques are used to fit models to

paired combination of variables, and if the models are deemed to be inadequate as per the correlation tests,

then one can quickly infer that the variables do not interact. It is shown that, for a given order, the models are

fitted in both directions, then the cross-correlation between the input and residues allow one to infer causality

between the two variables. Since the method only requires that the model capture the information contained in

the input, the method is naturally extended to the multivariate case, when one variable can be affected by

others.

The proposed procedure is applied to simulated data from a benchmark problem; indicate the source of these

disturbances was proposed, only needing the indication of the normal operation period. The affected data by

the disturbance are grouped checking those whose error signal variance increased significantly. To diagnose the

source, Granger causality method was used, since it allows to build a directed graph that relates the grouped

variables. The method can be applied automatically each time that a disturbance is detected by the operation

the results are then validated and compared with Granger’s method. Measuring the similarity between the

current operating conditions and historical operating conditions, which can identify abnormal behavior. Thus it

would be possible to automatically detect the presence of disturbances and immediately indicate its source.

The Granger method assumes linearity on data. This assumption is fulfilled in this approach since during training

the variations of process variables keep close to their steady state values. An extension of the method for plants

with multiple operation points is possible using the concept of data clustering, selecting multiple PCA training

sets for different operation points. A test to calculate the distance from each new sample to each cluster is

performed to select the PCA model to be used.

References

Aguirre, L. A. (2004). Introdução à identificação de sistemas–Técnicas lineares e não-lineares aplicadas a

sistemas reais. Editora UFMG. https://doi.org/10.13140/RG.2.1.1616.7925

Figure 12

Granger causality graph for selected variables CSTR

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BECERRA-ANGARITA O.F. & ALVAREZ-PIZARRO Y.A. «Granger causality procedeture to diagnosis and failture

in industrial systems»

Pag. 47

Alauddin, M., Khan, F., Imtiaz, S., & Ahmed, S. (2018). A bibliometric review and analysis of data-driven fault

detection and diagnosis methods for process systems. Industrial & Engineering Chemistry

Research, 57(32), 10719-10735. https://doi.org/10.1021/acs.iecr.8b00936

Balcilar, M., Ike, G., & Gupta, R. (2022). The role of economic policy uncertainty in predicting output growth in

emerging markets: a mixed-frequency granger causality approach. Emerging Markets Finance and

Trade, 58(4), 1008-1026.

Barnett, L., & Seth, A. K. (2014). The MVGC multivariate Granger causality toolbox: a new approach to Granger-

causal inference. Journal of neuroscience methods, 223, 50-68.

https://doi.org/10.1016/j.jneumeth.2013.10.018

Chiang, L. H., Russell, E. L., & Braatz, R. D. (2001). Fault detection and diagnosis in industrial systems.

https://doi.org/10.1088/0957-0233/12/10/706

Finch, F. E. (1989). Automated fault diagnosis of chemical process plants using model-based

reasoning (Doctoral dissertation, Massachusetts Institute of Technology). Available:

http://hdl.handle.net/1721.1/14194

Granger, C. W. (1969). Investigating causal relations by econometric models and cross-spectral

methods. Econometrica: journal of the Econometric Society, 424-438. https://doi.org/10.2307/1912791

He, Z., Shardt, Y. A., Wang, D., Hou, B., Zhou, H., & Wang, J. (2018). An incipient fault detection approach via

detrending and denoising. Control Engineering Practice, 74, 1-12.

https://doi.org/10.1016/j.conengprac.2018.02.005

Huang, H. B., Yi, T. H., & Li, H. N. (2017). Sensor fault diagnosis for structural health monitoring based on

statistical hypothesis test and missing variable approach. Journal of Aerospace Engineering, 30(2),

B4015003. https://doi.org/10.1061/(ASCE)AS.1943-5525.0000572

Joe Qin, S. (2003). Statistical process monitoring: basics and beyond. Journal of Chemometrics: A Journal of the

Chemometrics Society, 17(8-9), 480-502. https://doi.org/10.1002/cem.800

Kirilova, E. G. (2022). Artificial Neural Networks: Applications in Chemical Engineering. In Modeling and

Simulation in Chemical Engineering (pp. 127-146). Springer, Cham. https://doi.org/10.1007/978-3-030-

87660-9_6

Kruger, U., & Xie, L. (2012). Statistical monitoring of complex multivatiate processes: with applications in

industrial process control. John Wiley & Sons. https://www.wiley.com/en-

sg/Statistical+Monitoring+of+Complex+Multivatiate+Processes:+With+Applications+in+Industrial+Process

+Control-p-9780470517246

Lindner, B., Auret, L., Bauer, M., & Groenewald, J. W. (2019). Comparative analysis of Granger causality and

transfer entropy to present a decision flow for the application of oscillation diagnosis. Journal of Process

Control, 79, 72-84. https://doi.org/10.1016/j.jprocont.2019.04.005

Lindner, B., Auret, L., Bauer, M., & Groenewald, J. W. (2019). Comparative analysis of Granger causality and

transfer entropy to present a decision flow for the application of oscillation diagnosis. Journal of Process

Control, 79, 72-84. https://doi.org/10.1016/j.jprocont.2019.04.005

Ljung, L. (1998). System identification. Wiley encyclopedia of electrical and electronics engineering, 1-

19. https://doi.org/10.1002/047134608X.W1046.pub2

ISSN-L: 0798-1015 • eISSN: 2739-0071 (En línea) - Revista Espacios – Vol. 43, Nº 07, Año 2022

BECERRA-ANGARITA O.F. & ALVAREZ-PIZARRO Y.A. «Granger causality procedeture to diagnosis and failture

in industrial systems»

Pag. 48

Lucke, M., Chioua, M., & Thornhill, N. F. (2022). From oscillatory to non-oscillatory disturbances: A comparative

review of root cause analysis methods. Journal of Process Control, 113, 42-67.

https://doi.org/10.1016/j.jprocont.2022.03.004

Marques, V. M., Munaro, C. J., & Shah, S. L. (2015). Detection of causal relationships based on residual

analysis. IEEE Transactions on Automation Science and Engineering, 12(4), 1525-1534.

https://doi.org/10.1109/TASE.2015.2435897

Nor, N. M., Hassan, C. R. C., & Hussain, M. A. (2020). A review of data-driven fault detection and diagnosis

methods: Applications in chemical process systems. Reviews in Chemical Engineering, 36(4), 513-553.

https://doi.org/10.1515/revce-2017-0069

Nwani, S. E. (2022). Air pollution trajectories and life expectancy in Nigeria. International Journal of Social

Economics. https://doi.org/10.1108/IJSE-06-2021-0351

Seth, A. K. (2010). A MATLAB toolbox for Granger causal connectivity analysis. Journal of neuroscience

methods, 186(2), 262-273. https://doi.org/10.1016/j.jneumeth.2009.11.020

Seth, A. K., Barrett, A. B., & Barnett, L. (2015). Granger causality analysis in neuroscience and

neuroimaging. Journal of Neuroscience, 35(8), 3293-3297. https://doi.org/10.1523/JNEUROSCI.4399-

14.2015

Shojaie, A., & Fox, E. B. (2022). Granger causality: A review and recent advances. Annual Review of Statistics

and Its Application, 9, 289-319. https://doi.org/10.1146/annurev-statistics-040120-010930

Wiener, N., & Masani, P. (1957). The prediction theory of multivariate stochastic processes. Acta

Mathematica, 98(1-4), 111-150. https://doi.org/10.1007/BF02404472

Yi, T. H., Huang, H. B., & Li, H. N. (2017). Development of sensor validation methodologies for structural health

monitoring: A comprehensive review. Measurement, 109, 200-214.

https://doi.org/10.1016/j.measurement.2017.05.064

Yong, G., Xin, W., & Zhenlei, W. (2015). Fault detection for a class of industrial processes based on recursive

multiple models. Neurocomputing, 169, 430-438. https://doi.org/10.1016/j.neucom.2014.08.107

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