Performance Monitoring Based on Characteristic Subspace

Copyright © IFAC Advanced Control of Chemical Processes Hong Kong, China, 2003

ELSEVIER

IFAC PUBLICATIONS www.elsevier.comllocale/ifac

PERFORMANCE MONITORING BASED ON CHARACTERISTIC SUBSPACE

Ming Guo

Shuqing Wang

Nationa! Kev Laboratorv of Industria! Control Techno!ogv Zhejiang Un iversitv. Hangzhou, 310027, PR.China

Abstract:

In the operation and control of chemical process, automatic data logging

systems produce large volumes of data. It is important for supervising daily operation that how to exploit the valuable information about nomlal and abnormal operation, significant disturbance and changes in operational and control strategies. In this paper, principal component analysis (PCA) is clarified its essence from the view of space, and every different subspace represents different operational mode and process performance. Based on that, distance between two subspaces is calculated to evaluate the difference between them. The method is illustrated by a case study of a fluid catalytic cracking unit (FCCU) reactor-regenerator system. Copvright © 2003 IFAC

Keywords:

Principal component analysis, statistical process control, space distance,

process performance monitoring, FCCU.

Zhang, et aI., 1996).

I. INTRODUCTION Advances in computer technology and application of

As far as process fault diagnosis is concerned,

advanced control theory have resulted in routine

statistical process monitoring via PCA involves the

collection and storage of large volumes of data in

use of Hotelling T C and Q (also known as Square

chemical plant. Massive amounts of stored data can

Prediction Error or SPE) charts. Fault is identified

be used for analysis of the process operation and

with contributions of process variables to SPE. It is

previous oecun'ences of abnormal situation. Principal

only valid for simple fault situation, and difficult to

component analysis (PCA) can extract valuable

identify the root causes. Zhang and Martin (1996)

information from large historical database. Notable

proposed fault direction to identify different fault. Fault diagnosis

applications of PCA in chemical engmeenng have

is

achieved by comparing the

and

direction of the current on-line measurements with

MacGregor, 1995; Kresta, et aI., 1991), disturbance

those of a database of known trajectories of identified

detection

fault

faults. This method based on angle measurement

diagnosis (Wang and Song, 2002) and process fault

does not make full use of principal component

diagnosis (Kano, et al.,2001; Dunia and Qin,1998;

information of faults, and only the first loading

been

m

process

(Ku

and

monitoring

StoreI',

(Nomikos

1995),

sensor

649

vector is used. Dunia and Qin (1998) analyze the

Similarly, the residual

faults using subspace approach. But they assume that

where

the fault effect is not propagated into the other

proposed

a

novel

statistical

C

subspace

variables, which restricts its application. Kano, et al.

(200 I)

x satisfies

x::= (/ - C)x == ex E S c

detectability, identifiability and reconstructability of

~Hm,

(3)

is projection operator on the residual

5 ,with dimeS) == m - k. From the view

of space, PCA divides the measurement space Srn (dim(Sm)=m) into two orthogonal subspaces, a principal component subspace and a residual subspace. That is,

process

monitoring method based on changes in the subspace which is spanned by several principal components. The method makes use of principal component

(4)

information sufficiently, and has better monitoring perfonnance

than

conventional

PCA

based

on

Principal

Hotelling T' and Q charts. In essence, the method

component

subspace

primarily

characterizes the measurement subspace. When a

proposed in this paper is similar to the one proposed change in variable correlation occurs, that

by Kano. et al.. Their work is not dealt with fault

IS,

space

identification, while our approach goes beyond the

Srn has a change, the bases of principal component

fault detection task. Once a fault is detected. we have

subspace also produce corresponding changes. Wc

proposed a method based on subspace distance to

call principal component subspace S as characteristic

identity the type of fault.

subspace.

The paper is structured as follows: the second section

For a certain chemical process, we can define fault

gives a more strict procedure of deduction for PCA

set

based on subspace distance. and proposes a method

{F,} ;'=1 according to the data recorded in

historical database and tec1'inologic information. We

of fault identification according to historical database.

denote Si, Si as the characteristic subspace of fault Fi,

The third section presents an application of the

F,

approach to FCCU reactor-regenerator system. The

corresponding loading vectors, respectively. that is,

respectively.

final sectIon summarizes the approach.

arc

spanned

by

the

5, == span(u l , 11 2 ,L ,", )

5i where

PCA BASED SUBSPACE

2.

They

::=

span( vi' v"L , vJ

dim(Si)==r,

generality,

(5)

suppose

dim(S,)=s.

s <:: r

(6)

Without The

loss

of

dimensions

of

subspace Si, S, can be detenllined by the percent of

2.1 Spacial Signification of PCA

contribution to the accumulative variances. The difference between F i and F, can be reflected by the

PCA decomposes a nomlalized sample vector into

difference of bases of their characteristic subspace. In

two portions.

x==x+x, where

x

E ~H

order to identify different fault, the distance between

(I)

two subspaces is used to measure the difference. Let

'" is the sample vector normalized to

zero mean and unit variance. The vector

x is

matrix

the

projection on the principal component subspacc S: .\- == ppT X == Cx

UT U == I , V TV == I .The

(2)

projection operator

from subspace SJ onto subspace Si can be represented

where P E ~H""k is the PCA loading matrix. and k?

as

1 is the number of PCs retained in the PCA model. (7)

The matrix C == pp T is projection operator on the principal component subspace S,

x

E

5 <,;;; 91'" , with

For any unit vector YES, ,that is

dim(S)=k. The columns of the loading matrix Pare the eigenvectors of the correlation matrix associated

projection on subspace Si is written as

.llyll,

::=

I, its

.v::= Cy.

with the k largest eigenvalues. the distance between two subspaces is defined as

650

Now,

d i .;

= maxllY - it .1'

(8)

(ii)

-

Si cS i

'

that is, the subspace

spanned by the model for the fault FJ

(9)

subject to

If the subspace

contains the

subspace spanned by the model for fault F i

,

we can

also get d=O. It means that the fault Fi is masked by

Since y is a unit vector in 5" it can be represented as

fault F, ' and they can not be distinguished from each

y t

where

is

Vt

=

the

( 10)

coordinate

correspond to bases

coefficients

VI, V2, "',

V,

with

,

other. In fact, they are mistaken for the same fault.

vector

IItl12

(iii) If the subspace S, = S;l. that

= I .

orthogonal,

According to Lagrange's method, we have

aLl at =0

and

aL I aA = 0,

they are

VTV = O. Thus. we can get

Amin (A) = 0 , that is , d= I. It means that the fault F"

L( t, A ) = Ily - 5'11 ~ + A (11yll ~ - I) Let

then

IS,

(11)

FI can be distinguished from each other to the most extent.

with substitution

of Eq.1 0 in Eq.ll, we get the following expression,

At = A t

2.2 Fault Diagnosis Based on Distance

( 12)

From where

A = VTUUTV.

The coordinate coefficients

A is

vector t is an eigenvector of thc matrix A, and

above

description,

the

distance

between

subspaces can be used to identify the different faults. We define a match function as follows,

the corresponding eigenvalue. The distance between

Pi)

two subspaces is obtained by the substitution of

= (1- d i ,) X 100'10

Eq.12 into Eq.8,

=I-

JI - A min (A,

I )

100%

x

( 14)

( 13) When a fault occurs. the loading vectors of the fault data are calculated through

dE [0,1].

Now, we prove the distance

PCA and used to

represent the bases of characteristic subspace of the Proof. .: A = VTUUTV = (UTV)T UTV,

:. A is nonnegative definite, that is,

fault. On the basis of that, the library of characteristic subspace of faults can be formed and represented as

A ;::: 0,

follows,

U = [iir+\ 'u'+2'" "UI/I]'

Let E

= [V, D], then E

Where Si is the characteristic subspace corresponding is

to the fault F i , and SF the subs pace set, and

the bascs of the measurement space Srn , with ET E = Ee ~ I. ... EE T = VU T + UU T = I, and ...

UU)

? 0, ...

UU T

::;

A=VTUUTV::;/.thatis, Therefore, 0 ::; d ::; I,

I .

( 15)

SF =[SI' 52. ''',5,,]

Suppose the bases of residual subs pace of fault F i is

11

the

number of faults. The currently monitored process measurements can

Thus,

then be analyzed using PCA. The calculated loading

Amin(A)::;1

vectors form the bases of the subspace corresponding to the current observations. Denoting the current

End.

characteristic subspace by Sen,. the matching degree between Seur and the every subs pace in SF can be

Thus. we have the following three special cases: (i)

measured by Eq.I4 respectively after a fault is

if the subspace Si = SJ ' that is , the two subspace

detected. If the matching degree between Seur and

are identical, then U=VQ, where Q is nonsingular orthogonal

matrix.

With

Eq.I3,

we

can

some subspace (for example, Si ) is very close to

get

100%, then the current abnormal occurrence may be

A mm (A) = 1, that is, d=O. In the case of this, the

probably ascribed to fault Fi . On the contrary, if the matching degree is close to zero, it may be least

fault F i , F, can be considered as the same fault.

ascribed to fault F,. Thus, fault identification can be performed

651

by

calculating

the

matching degree

between the characteristic subspace of the current

oil, distillate oil and assembled feed, flow rate of air

data and the library of subspace of known faults. As

of the first regenerator and the second regenerator,

already discussed, some faults may be masked, so

outlet temperature of riser, the heat of exchanger,

domain knowledge is further needed in that case to

carbon dioxide content in the flue gas from the first

analyze the results and determine which fault has

regenerator, oxygen content in the flue gas from the

occurred on earth.

second regenerator, the temperature in the high

In practical application, a diagnostic threshold is

density bed of the first and the second regenerator.

required to be defined in advance. The maximum of

During the simulation, random noise was added to

matching degree between the current data and the

the measurement and controller outputs.

faults in the library should be larger than the

ten

diagnostic threshold. Otherwise, if the maximum of

summarized in Table I. Sample time is 4 minutes.

matching degree is less than the diagnostic threshold.

Each instance is simulated for 1000 minutes, and the

that is. the current data subspace is not well matched

database is a 250X 14X I0 matrix.

data

instances

have

been

Altogether

generated

and

with any fault characteristic subspace in the library, then it is likely that a novel fault has occurred. Once the occun'ence of a novel fault is confirmed, the bascs of the current data subs pace can be stored in the

library.

Through

this

method,

diagnostic

knowledge about novel faults is progressively learnt and the library updated.

3.

CASE STUDY

3.1 Process Description

Fluid catalytic cracking unit (FCCU) is considered as

I: deaerator

2: 2nd regenerator

3: I st regenerator

one of the most important unit in the refinery. A

4:

5: heat exchangcr

6: riser

simplified flow diagram is shown in Fig.1. Briefly,

7: 2nd regenerator flue gas

the fresh feed and recycle sludge oil are preheated.

gas

mixed, and then enter into the riser reactor where

settler

9: product

10: feed

8: 1st regenerator flue 11: air

Fig. I FCCU Reactor-regenerator Flow Sheet

they contact regenerated catalyst and start the cracking reactions. The spent catalyst passes to thc

3.2 Data Analysis and Fault Identification

stcam stripping section and enters the regenerator where the coke on the catalyst is burnt off with air.

When the measurement data are obtained, data

The heat released by the combustion of coke is

reconciliation is pcrformed to validate the sensor data.

supplied to the endothermic cracking reactions. The

Then they are normalized to zero mean and unit

extra heat than what is required by cracking reactions

variance before the data of each instance is analyzed

is taken away by the heat exchanger outside the

by PCA. The distance between two corresponding

regenerator. The FCCU reactor-regenerator model

subspaces is calculated with Eq.13.

which is used in this paper can be referred to the

calculated results. From Table 2, it can be seen that

work done by Yang, et al. (1997).

the distance between subspaces of case 2, 3, 4 is

Table 2 is the

relatively small compared with other distance, which To generate an instance, the simulation is running at

indicates that their difference between them is

normal mode. When all parameters become stable, a

relatively small. This is because they are all the flow

disturbance or fault is introduced and at the same

disturbance of fresh feed, only different

time, data recording is started. Fourteen variables are

magnitude, and they have similar effect on the

chosen to be recorded, including temperature of feed

correlation of data. The distance between case 2,3,4

preheated, flow rate of recycle oil, sludge oil, slurry

and case 5 is relatively large compared with the one

652

111

the

between case 2,3,4. That is because case 5 has an adverse disturbance direction. The distance between

!OD'X,

other cases is large. So we can use the distance to

.," .,~

identify different fault or disturbance.

'"

HD'X,

on'l,

OD

For online monitoring, the data matrix representing

"

the current operating conditions is updated by

U ;0

1D'"

E

:W'"

moving the time-window step by step as proposed by

"

D'.;,

Kano er al (2001). PCA is applied to the data matrix,

H

and the distance between the subspace of current data

ID

l)

Cases

and the one of normal operation data is calculated with Eq.13 at each step. If the distance goes beyond

Fig 3

the given control limit, the process is judged to be

Matching degree of the current case with the cases in the historical database

out of normal oper
4.

respectively, and the match degrees between them are also obtained with matching

degree

is

Eq.14.

If the maximum of

larger than

the

CONCLUSIONS

The diagnosis of abnormal operation can be greatly

diagnostic

facilitated if similar system performance has been

threshold, then the fault in the library corresponding

recorded

to the maximum has probably occurred.

III

the

historical

database.

Principal

component analysis is among the most popular

In order to verify the method for fault identification,

methods

for

extracting

information

from

data.

the preheated temperature of fresh feed decreased 6K

Through PCA, features associated with different

at 100 minute in the simulation. The monitoring

faults can be identified and used in fault diagnosis.

results are shown in Fig.2, and the 95% warning limit

The features are the characteristic subspace spanned

which can be detennined by statistical method is also

by several loading vectors. Fault diagnosis can then

shown.

be performed by calculating the distance between the subspace of current data matrix and the one of known

ll.n

faults in the library. It can also deal with novel faults

u. .;

"

~

and learn diagnosis knowledge about novel faults.

u. I

This method is applied to monitoring the FCCU

u.

z

-

u.

reactor-regenerator system. The results have shown

-

that the method can successfully identify different

u I

faults, because it makes full use of information about several principal components. It is important to note lflllill

that although the approach is well founded, there are Fig 2

Monitoring results for FCCU reactor-

problems to be solved in real industrial application. It

regenerator system

is advisable to combine domain knowledge with data mining method to diagnosis fault.

When the distance is out of the control limit. the matching degrees between the current data and the faults in the historical database are calculated. and

5.

ACKNOWLEDGEMENTS

the results are shown in Fig.3. The diagnostic threshold is predefined as 0.80. It can be seen that the

The support from the National High Technology (863)

current case matches well with case 8 at 86.32%

Project

matching degree which is above the diagnostic

acknowledged.

threshold, and the fault is successfully identified.

653

(No:

2001 AA41311 0)

is

gratefully

Intellectual Laboratory System, 30, \79-\96.

6. REFERRENCES

Nomikos,P. and MacGregor, J.F.( \995). Multivariate Dunia, Rand Qin, S.J( \998). Subspace approach to multidimensional

fault

identification

SPC charts for monitoring batch processes.

Technometrics, 37( 1),41-59.

and

reconstruction. AIChE.1. ,44(8), 1813-\831.

Wang,

statistical

process

PCA with optimized sensor locations. 1. Pmc.

monitoring

Cont., 6, 735-744.

method using principal component analysis.

Yang, M. Y, Rong, G. and Wang, S.Q.( 1997). FCCU

Comp.Chem.Eng.,25,1\03-1113. Kresta, J., MacGregor, J. F. and Marlin,T.E.( \99\ ).

reactor-regenerator

Zhang, J., Martin, E.B. and Morris, A.J(l996). Fault

69( \ ),35-47.

w.,

Ku,

Storer. R.H.. and Georgakis, c.( 1995).

detection

and

Disturbance detection and isolation by dynamic

statistical

techniques.

component

principal

contro\.

o{Chemical Processes, Banff, Canada, 653-657.

Can.J. Chem.Eng.,

performance.

advanced

Inlernational Symposium on Advanced Control

Multivariate statistical monitoring of process operating

H.(2002).

Statistical process monitoring using improved

Kano, M., Hasebe,S. and Hashimoto, 1.(2001). A new multivariate

H.Q., Song, Z.H. and Wang,

using

multivariate

Chemical Engineering

Research l/nd Design, 74( 1),89-96.

Chemical


diagnosis

Table I Summary of the 10 simul
Description of cases

I

normal operation

2

a step increClse of 10% in fresh feed flow rate

3

a step increase of 20% in fresh feed flow rate

4


5

a step decrease of 30% in fresh feed flow rate

6

a step increase of 10°1., in air flow rate of 1st regenerator

7

a step increase of 10% in heat of heat removal system

8

decrease of 3K in

preheated temperature of fresh feed

9

increase of 3K in outlet temperature of riser reactor

10

increase of 3K in high density bed temperature of I st regenerator

Table 2 The distance between ten different cases

"

case I

case 7

case X

case :\

C:.tse 4

case5

case (.

0.3896

1J.57'1'1

1J.661')

0.5'153

0.3406

0.4191

0.7596

0.5754

0.5355

09454

0.9'198

0.'1881

1J.9227

0.9527

0.955:'

case 1

<.:"se 2

0

case 2

0.38%

0

01J(,78

0.2054

06251

0.'1915

0.'1427

case 3

0.5799

o.O(,n

0

01300

0.653';

0.'1'191

0.%2}

case

<)

c:'lse 10

4

0.661'1

02054

o UOO

0

0.69'10

0.9998

09653

0.946'1

0.9.160

0.9609

C
0.5953

0.(,251

0.6535

0.69'11J

0

0.9769

0.'1'158

0.935'1

0.9566

0.9510

case 6

0.341J6

0.9'115

0.99'11

0.'1'198

0.97(,9

0

08441

0.'1960

0.8684

0.<)92(,

ca~e

7

0.41'11

0.9427

0.9(,22

0.9053

0.9958

0.8441

0

0.'1991

0.9618

0')<)98

case X

0.75%

0.9454

0.9227

0.94(,'1

0')359

0.9%0

0.'199\

0

0.9747

0.6478

0.'156(,

0.8684

0.9618

0.'1747

0

0.9230

0.9510

0.'1926

0.9998

O.M7X

0.9230

0

l:
case 9

0.5754

0.'1'198

0.'1527

0.93(,0

case 10

0.5355

0.'1881

0.9555

0.960'1

654