FAULT DETECTION AND DIAGNOSIS VIA IMPROVED MULTIVARIATE STATISTICAL PROCESS CONTROL

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FAULT DETECTION AND DIAGNOSIS VIA
IMPROVED MULTIVARIATE STATISTICAL PROCESS CONTROL
NOORLISA BINTI HARUN
UNIVERSITI TEKNOLOGI MALAYSIA
FAULT DETECTION AND DIAGNOSIS VIA
IMPROVED MULTIVARIATE STATISTICAL PROCESS CONTROL
NOORLISA BINTI HARUN
A thesis submitted in fulfilment of the
requirements for the award of the degree of
Master of Engineering (Chemical)
Faculty of Chemical and Natural Resources Engineering
Universiti Teknologi Malaysia
JULY 2005
ii
iii
In the Name of Allah, Most Gracious, Most Merciful.
I humbly dedicate to…
my beloved family members
my best friend
those who has shaped my life and
those who has influenced my life on the right path.
iv
ACKNOWLEDGEMENT
Firstly, I would like to take this opportunity to thank my respectful
supervisor, Assoc. Prof. Dr. Kamarul Asri Ibrahim who has relentlessly given
guidance, constructive ideas and invaluable advice, enabling me to achieve the
objective of this dissertation. In addition I am very grateful to my family members,
no amount of gratitude could repay their kindness of being there as well as the
patience to put up with my whim.
I also would like to thank my fellow course mates and friends for the
assistances, advices and ideas in producing a resourceful, fruitful and practicable
research. May God bless all of you.
v
ABSTRACT
Multivariate Statistical Process Control (MSPC) technique has been widely
used for fault detection and diagnosis (FDD). Currently, contribution plots are used
as basic tools for fault diagnosis in MSPC approaches. This plot does not exactly
diagnose the fault, it just provides greater insight into possible causes and thereby
narrow down the search. Hence, the cause of the faults cannot be found in a
straightforward manner. Therefore, this study is conducted to introduce a new
approach for detecting and diagnosing fault via correlation technique. The correlation
coefficient is determined using multivariate analysis techniques, namely Principal
Component Analysis (PCA) and Partial Correlation Analysis (PCorrA). An industrial
precut multicomponent distillation column is used as a unit operation in this research.
The column model is developed using Matlab 6.1. Individual charting technique such
as Shewhart, Exponential Weight Moving Average (EWMA) and Moving Average
and Moving Range (MAMR) charts are used to facilitate the FDD. Based on the
results obtained from this study, the efficiency of Shewhart chart in detecting faults
for both quality variables (Oleic acid, xc8 and linoleic acid, xc9) are 100%, which is
better than EWMA (75% for xc8 and 77.5% for xc9) and MAMR (63.8% for xc8 and
70% for xc9). The percentage of exact faults diagnoses using PCorrA technique in
developing the control limits for Shewhart chart is 100% while using PCA is 87.5%.
It shows that the implementation of PCorrA technique is better than PCA technique.
Therefore, the usage of PCorrA technique in Shewhart chart for fault detection and
diagnosis gives the best for it has the highest fault detection and diagnosis efficiency.
vi
ABSTRAK
Proses Kawalan Statistik Multipembolehubah (MSPC) digunakan secara
meluas untuk mengesan dan mengenalpasti punca kesilapan. Pada masa kini, carta
penyumbang digunakan untuk mengenalpasti punca kesilapan dalam MSPC. Carta
ini tidak dapat mengenalpasti punca kesilapan dengan tepat yang mana ia sekadar
menunjukkan kemungkinan besar punca kesilapan dan membantu memudahkan
pencarian punca kesilapan. Oleh itu, punca kesilapan tidak dapat ditentukan secara
langsung. Justeru, kajian ini telah dijalankan bagi memperkenalkan satu pendekatan
baru untuk mengesan dan mengenalpasti punca kesilapan melalui teknik korelasi.
Pekali korelasi ditentukan melalui kaedah Analisis Statistik Multipembolehubah iaitu
Analisis Komponen Utama (PCA) dan Analisis Korelasi Separa (PCorrA). Turus
penyulingan multikomponen industri digunakan sebagai operasi unit. Model turus ini
dibangunkan menggunakan perisian Matlab 6.1. Teknik carta individu yang terdiri
daripada Shewhart, Exponential Weight Moving Average (EWMA), dan Moving
Average dan Moving Range (MAMR) digunakan bagi mengesan dan mengenalpasti
punca kesilapan. Berdasarkan kepada keputusan yang diperolehi, kecekapan carta
Shewhart dalam mengesan kesilapan untuk kedua-dua pembolehubah kualiti (Asid
oliek, xc8 dan asid linoleik, xc9) adalah 100% yang mana lebih baik berbanding
dengan carta EWMA (75% bagi xc8 and 77.5% bagi xc9)dan carta MAMR (63.8% bagi
xc8 and 70% bagi xc9). Peratusan mengenalpasti punca kesilapan secara tepat
menggunakan teknik PCorrA bagi carta Shewhart adalah 100% manakala PCA
adalah 87.5%. Ini menunjukkan bahawa penggunaan teknik PCorrA adalah lebih
baik berbanding teknik PCA. Oleh itu, penggunaan teknik PCorrA dalam carta
Shewhart bagi tujuan mengesan dan mengenalpasti punca kesilapan adalah yang
terbaik dengan peratusan kecekapan yang paling tinggi.
vii
TABLE OF CONTENTS
CHAPTER
1
TITLE
PAGE
TITLE PAGE
i
DECLARATION
ii
DEDICATION
iii
ACKNOWLEDGEMENT
iv
ABSTRACT
v
ABSTRAK
vi
TABLE OF CONTENTS
vii
LIST OF TABLES
xii
LIST OF FIGURES
xiv
LIST OF SYMBOLS
xvi
LIST OF ABBREVIATIONS
xx
LIST OF APPENDICES
xxi
INTRODUCTION
1.1 Introduction
1
1.2 Research Background
2
1.3 Objectives of the Research
4
1.4 Scopes of the Research
4
1.5 Contributions of the research
5
1.6 Chapter Summary
5
viii
2
LITERATURE REVIEW
2.1 Introduction
7
2.2 Definitions of Fault, Fault Detection and Fault Diagnosis
8
2.3 Multivariate Statistical Analysis, MSA
9
2.4 Review on Principal Component Analysis, PCA
10
2.4.1
Extension of Principal Component Analysis, PCA
method
2.4.1.1
Multi-Way PCA
2.4.1.2 Multi-Block PCA
15
2.4.1.3 Moving PCA
16
2.4.1.4
17
Dissimilarity, DISSIM
2.4.1.5 Multi-Scale PCA
18
2.4.2 PCA Adopted in Industries
19
2.4.3
20
PCA as Controller
2.4.4 Theory of PCA
2.5 Review on Partial Correlation Analysis, PCorrA
2.5.1 Theory of PCorrA
2.6 Chapter Summary
3
14
21
23
26
27
IMPROVED STATISTICAL PROCESS CONTROL CHART
3.1 Statistical Process Control, SPC
29
3.2 Traditional Statistical Process Control, SPC chart
30
3.2.1
Shewhart Chart
31
3.2.2
Exponential Weight Moving Average, EWMA Chart
35
3.2.3
Moving Average and Moving Range, MAMR Chart
39
3.3 Improve Statistical Process Control Chart
43
3.4 Chapter Summary
45
ix
4
DYNAMIC SIMULATED DISTILLATION COLUMN
4.1 Introduction
46
4.2 Process Description of Precut Column
48
4.3 The Information of Pre-cut column
49
4.4 Degree of Freedom (DoF) Analysis
51
4.5 Formulations of the Distillation Column Models
56
4.5.1
4.5.2
Mass Balance Equations
56
4.5.1.1
Bottom column model
58
4.5.1.2
Trays model
59
4.5.1.3
Reflux drum model
61
4.5.1.4
Pumparound drum model
62
Equilibrium Equations
63
4.5.2.1
66
Bubble Point Calculations
4.5.3
Summation Equation
68
4.5.4
Heat Balance Equation
68
4.5.5
Hydraulic Equations
69
4.6 Dynamic Simulation of the Study Column
70
4.7 Controller Tuning
74
4.7.1
Bottom Liquid Level Controller
75
4.7.2
Bottom Temperature Controller
76
4.7.3
Top Column Pressure Controller
78
4.7.4
Sidedraw Tray Liquid Level Controller
79
4.7.5
Pumparound Temperature Controller
80
4.7.6
Pumparound Flow Rate Controller
81
4.7.7
Reflux Flow Rate Controller
82
4.7.8
Controller Sampling Time, TAPC
82
4.8 Dynamic simulation Program Performance Evaluation
83
4.9 Chapter Summary
84
x
5
METHODOLOGY
5.1 Introduction
86
5.2 Variable Selection for Process Monitoring
87
5.3 Statistical Process Control Sampling Time, TSPC
90
5.4 Normal Operating Condition (NOC) Data Selection and 92
Preparation
5.4.1
Data pretreatment
94
5.4.2
Correlation Coefficient via Multivariate Analysis 94
Techniques
5.4.3
5.4.2.1
Principal Component Analysis, PCA
95
5.4.2.2
Partial Correlation Analysis, PCorrA
101
Control limits of Improved Statistical Process 103
Control, ISPC chart
5.5 Generated Out of Control, OC data
105
5.6 Fault Coding Monitoring Framework
109
5.7 Performance Evaluation of the FDD Method
111
5.7.1
Fault Detection Efficiency
111
5.7.2
Fault Diagnosis Efficiency
112
5.8 Chapter Summary
6
113
RESULTS AND DISCUSSIONS
6.1 Statistical Process Control Sampling Time, TSPC
116
6.2 Normal Operating Condition, NOC Data
117
6.2.1 Correlation coefficient via Principal Component 119
Analysis, PCA
6.2.2
Correlation
coefficient
via
Partial
Correlation 123
Analysis, PCorrA
6.3 Parameters to Design Improved Statistical Process Control 124
chart
6.4 Fault Detection and Diagnosis Method Performance
6.4.1
False alarm Analysis
127
127
xi
6.4.2
Fault Detection Efficiency Results and Results 129
Discussions
6.4.3 Fault Diagnosis Efficiency Results and Results 132
Discussions
6.5 Chapter Summary
7
136
CONCLUSIONS AND RECOMMENDATIONS
7.1 Introduction
138
7.2 Conclusions of Improved SPC framework
139
7.3 Recommendations to Future Research
141
LIST OF REFERENCES
142
APPENDICES
154
xii
LIST OF TABLES
TABLE NO.
TITLE
PAGE
4.1
Specification of the pre-cut column
49
4.2
Components composition and conditions of the pre-cut column
50
4.3
Number of variables
52
4.4
Number of equations
53
4.5
Control loops properties
55
4.6
Cohen-Coon controller settings
75
4.7
The step change results for bottom liquid level controller
76
4.8
Properties of bottom liquid level P controller
76
4.9
The step change results for bottom temperature controller
77
4.10
Properties of bottom temperature PID controller
77
4.11
The step change results for top column pressure controller
78
4.12
Properties of top column pressure PI controller
78
4.13
The step change results for sidedraw tray liquid level controller
79
4.14
Properties of sidedraw tray liquid level P controller
79
4.15
The step change results for pumparound temperature controller
80
4.16
Properties of pumparound temperature PID controller
81
4.17
Properties of pumparound flow rate P controller
81
4.18
Properties of reflux flow rate P controller
82
5.1
APC variables for each control loops
87
5.2
Normal correlation coefficient for process variables
88
5.3
SPC variables
88
xiii
5.4
The control limits of Improved SPC chart
104
5.5
Parameters to design EWMA
105
5.6
Description of sensor fault, valve fault and controller fault
107
5.7
Improved SPC chart statistic for FDD
108
5.8
Coding system for fault detection
109
5.9
Coding system for fault diagnosis
109
5.10
Fault coding system for Shewhart Chart
110
5.11
Fault coding system for EWMA Chart
110
5.12
Fault coding system for MAMR Chart
110
6.1
Time constant for each control loops
116
6.2
The skewness and kurtosis value of NOC data
118
6.3
Summary of the NOC data
118
6.4
Eigenvectors Matrix
119
6.5
Eigenvalues, variation explained by each PC and cumulative
119
variation
6.6
The correlation coefficient using different cumulative variation
121
6.7
The percentage difference for two PCs retained and three PCs
122
retained compared to six PCs
6.8
Simple correlation matrix
123
6.9
The correlation coefficient result using PCorrA
124
6.10
Constant to design Shewhart range chart
124
6.11
FDD result using different L and β to design EWMA chart
125
6.12
FDD result using PCA with different window size to design 126
MAMR chart
6.13
FDD result using PCorrA with different window size to design 126
MAMR chart
6.14
False alarm result corresponding to oleic acid composition
128
6.15
False alarm result corresponding to linoleic acid composition
128
6.16
The known fault causes
129
6.17
The known out of control, OC conditions
130
xiv
LIST OF FIGURES
FIGURE NO.
TITLE
PAGE
2.1
Multi-block PCA monitoring framework
15
3.1
Shewhart range and Shewhart individual chart during normal
34
condition
3.2
Shewhart range and Shewhart individual chart during faulty
34
condition
3.3
EWMA control chart under normal condition
38
3.4
EWMA control chart under faulty condition
38
3.5
Moving average and range control chart under normal condition
42
3.6
Moving average and range control chart under faulty condition
42
3.7
The implementation of correlation coefficient, Cjk in SPC chart
44
4.1
Pre cut column
48
4.2
Distillation column control loop
54
4.3
Distillation column schematic diagram
57
4.4
Bubble point calculation procedure
67
5.1
The procedure to calculate and implement TSPC
92
5.2
Example Scree plot
97
5.3
The procedures to determine correlation coefficient, Cjk via PCA 100
method
5.4
The procedures to determine correlation coefficient, Cjk via 103
PCorrA metho
5.5
The outlined of FDD algorithm
115
xv
6.1
The autocorrelation plot
117
6.2
Histogram plot of NOC data
118
6.3
Scree plot
120
6.4
Fault detection efficiency using different Improved SPC chart
131
6.5
Percentage of faults occurs in different region detected by each 132
Improved SPC chart
6.6
Fault diagnosis situation using PCA for oleic acid composition
133
6.7
Fault diagnosis situation using PCorrA for oleic acid
134
composition
6.8
Fault diagnosis situation using PCA for linoleic acid
134
composition
6.9
Fault diagnosis situation using PCorrA for linoleic acid
composition
135
xvi
LIST OF SYMBOLS
aij
-
Wilson constant
n
-
Number of retained principal component
A
-
Square matrix
Ab
-
Bottom column area
At
-
tray active area
Ai,Bi,Ci
-
Antoine constant for component i
A*,B*,C*,D*
-
Constant for liquid heat capacity
A2, D.'001 , D.'999
-
Constant for MA and MR control limit
β
-
Weighting factor
B
-
Bottom flow rate
Cj,k
-
Correlation coefficient between variable j and variable k
δ
-
Shifted value in standard deviation unit
D
-
Distillate flowrate
FDetect
-
Fault Detection
FDiagnose
-
Fault Diagnosis
∆H vap
-
Heat of vaporization
∆H vap, n
-
Heat of vaporization at normal boiling point
hN
-
Liquid enthalpy leaving each tray
hb
-
Enthalpy for bottom stream
how
-
Over weir height
H, h
-
CuSum control limits
HN
-
Vapor enthalpy leaving each tray
xvii
I
-
Identity matrix
Kc
-
Controller Gain
Kp
-
Static Gain,
L28
-
Liquid flowrate at tray 28
LH, P
-
Pumparound drum height
Lf
-
Liquid feed flowrate
LH, Re
-
Reflux drum liquid level
LH, B
-
Bottom liquid level
L
-
Width of the control limit
M
-
Number of observations
MB
-
Bottom molar hold-up
MW
-
Molecular weight
MRe
-
Reflux drum molar hold-up
MP
-
Pumparound drum molar hold-up
MN
-
Molar hold up at tray N
N
-
Number of stages
p
-
Number of observations
P
-
Pumparound flowrate
Pci
-
Critical pressure of component i
Pi sat
-
Vapor pressure for component i
Ptot
-
Total pressure of the system
ρ
-
Liquid density
QR
-
Reboiler heat duty
Q
-
Number of variables for response variables
R
-
Correlation matrix
rj,k
-
Correlation value between variable j and variable k
Re
-
Reflux flowrate
R
-
Universal gas constant
rk
-
Autocorrelation coefficient
xviii
S
-
Sidedraw flowrate
S
-
Variance-Covariance matrix
sj,k
-
Covariance value of variable j and variable k
s
-
NOC standard deviation
T
-
Temperature of the system
Tci
-
Critical temperature of component i
Tri
-
Reduced temperature of component i
Tb,i
-
Boiling point for component i
TAPC
-
APC sampling time
TSPC
-
SPC sampling time
t, u
-
Latent vectors
τ
-
Time Constant
τD
-
Derivative Time Constant
τI
-
Integral Time Constant
θ
-
Dead time
V
-
Eigenvector matrix
v
-
Eigenvector or loading vectors
V
-
Vapor flowrate
Vci
-
Critical volume for component i
Vi
-
Liquid molar volume of component i
WL
-
Weir length
ω
-
Critical accentric factor of component i
w
-
Un-normalized eigenvectors
x
-
NOC mean
x1
-
Shifted mean
xi
-
ith observation in the process
xij
-
Value of the i-th row and j-th column matrix
xijs
-
Standardized variable
x
-
Liquid phase mole fraction
xix
xS
-
Sidedraw mole fraction
xRe
-
Reflux mole fraction
xP
-
pumparound liquid mole fraction
xD
-
Distillate mole fraction
γi
-
Liquid phase activity coefficient for component i
y
-
Vapor phase mole fraction
zi
-
The ith EWMA statistic
Zci
-
Critical compressibility factor for component i
φi
-
Vapor fugacity coefficient for component i
Λij
-
Wilson binary interaction parameter for component i and j
xx
LIST OF ABBREVIATIONS
APC
-
Automatic Process Control
DoF
-
Degree of Freedom
EWMA
-
Exponential Weight Moving Average
FDD
-
Fault Detection and Diagnosis
ISPC
-
Improved Statistical Process Control
MA
-
Moving Average
MESH
-
Mass, Equilibrium, Summation, Heat
MR
-
Moving Range
MSA
-
Multivariate Statistical Analysis
MSPC
-
Multivariate Statistical Process Control
NIPALS
-
Non-iterative Partial Least Squares
NOC
-
Normal Operating Condition
OC
-
Out of Control
PCA
-
Principal Component Analysis
PCorrA
-
Partial Correlation Analysis
PV
-
Process Variable
QV
-
Quality Variable
SPC
-
Statistical Process Control
USPC
-
Univariate Statistical Process Control
xxi
LIST OF APPENDICES
APPENDIX
TITLE
PAGE
A
Column profile comparison
155
B
Parameter selection to design EWMA chart
158
C
Parameter selection to design MAMR chart
163
D
Number of fault causes
170
CHAPTER 1
INTRODUCTION
1.1
Introduction
A major technological challenge facing the processing industries is the need
to produce consistently high quality product. This is particularly challenging in high
demanding situations where processes are subject to varying raw material properties,
changing market needs and fluctuating operating conditions due to equipment or
process degradation. The need to provide industry with techniques, which enhance
process performance, will require new methodologies to be adopted that are capable
of being used across a spectrum of industrial processing operations and on a wide
range of products (Simoglou et al., 2000).
Modern industrial processes typically have a large number of operating
variables under closed loop control. These loops used to compensate for many types
of disturbances and to counteract the effect of set point change. This is necessary to
achieve high product quality and to meet production standards. Although these
controllers can handle many types of disturbances, there are faults in the process
such as line blockage, line leakage, sensor fault, valve fault and controller fault that
cannot be handled adequately. In order to ensure that process is operating at normal
operating condition as required, faults must be detected, diagnosed and removed.
These activities, and their management, are called as Statistical Process Control, SPC
(Miletic et al., 2004).
2
The combination of Statistical Process Control (SPC) charts and multivariate
analysis approach is used for Fault Detection and Diagnosis (FDD) in the chemical
process. Principal Component Analysis (PCA) and Partial Correlation Analysis
(PCorrA) are the techniques used in this study. A precut multicomponent distillation
column that has been installed with controllers is used as the study unit operation.
Improved Statistical Process Control method is implemented to detect and diagnose
various kinds of faults, which occur in the process.
1.2
Research Background
Statistical methods are used to monitor the performance of the process over
time in order to detect any process shifts from the target. Most of Statistical Process
Control (SPC) techniques involve operations on single response variables such as
weight, pH, temperature, specific gravity, concentration and pressure. This
traditional SPC is used to monitor and verify that the process remains in statistical
control based on small number of variables. The state of statistical control means that
the process or product variables remain close to the target. Normally the fault in the
process is seek through the usage of SPC chart, i.e., the final product quality
variables. Measuring quality variables alone are not enough to describe the process
performance (Kourti et al., 1996). In this study, both quality variables and process
variables are monitored. This can be done by developing the correlation coefficient
between quality variables and process variables using multivariate analysis
technique. In traditional SPC, once the quality variables detected out of statistical
control signal, it is then left up to process operators and engineers to try to diagnose
the cause of out of control using their process knowledge (MacGregor and Kourti,
1995).
Chemical processes are becoming better instrumented with the advances of
process sensors and data acquisition systems. There are massive amount of data
being collected continuously and these variables are often correlated. Multivariate
statistical analysis has been developed in order to extract useful information from
process data and utilize it for process monitoring (Kresta et al., 1991). Normally,
3
multivariate statistical analysis technique is combined with SPC chart using
Hotelling’ s T2 and Q statistics to plot the graph for fault detection and diagnosis.
Even though this conventional method provides good results for fault detection, there
is difficulty in applying this method to diagnose faults because it does not provide a
causal description of the process that can be used as the basis for fault diagnosis.
Currently, contribution plots are used as the basic tools for fault diagnosis in the
multivariate statistical approaches. However, this plot cannot exactly diagnose the
cause but it just provides greater insight into possible causes and thereby narrows the
search. The cause of the faults cannot be found in a straightforward manner.
To overcome the limitation of contribution plot using multivariate analysis
technique to diagnose fault, Improved SPC chart is introduced in this study.
Multivariate analysis approach is applied in the SPC realm procedure to detect and
diagnose the faulty condition in different approach. The multivariate analysis method
is used to develop the control limits of SPC charts. The correlation coefficient
calculated from multivariate analysis technique is applied to improve the SPC chart.
The quality variables data is incorporated in the control chart during the faulty
condition for fault detection purpose, while the process variables which has been
correlated with the quality variables is used for fault diagnosis. Therefore, the
Improve SPC charts applied is not only for quality variables but also for process
variables. By monitoring the process variables, the cause of faulty condition, which
affects the quality variables, could be identified directly. This will help the operator
to take proper action immediately after faults exist in the process.
4
1.3
Objectives of the Research
1. To determine the performance of the Improved Statistical Process Control
charts; which are Shewhart individual and Shewhart range, Exponential
Weight Moving Average (EWMA) and Moving Average and Moving
Range (MAMR) chart for fault detection.
2. To utilize multivariate analysis technique, Principal Component Analysis
(PCA), and Partial Correlation Analysis (PCorrA), for developing the
relationship between the process variable(s) and the quality variable(s) for
fault diagnosis.
1.4
Scopes of the Research
The scopes of the research are:
1.
A simulated precut multicomponent distillation column model from
the literature is used as a case study. The model consists of
controllers with various kinds of disturbances and operating
conditions.
2.
A set of Normal Operating Condition (NOC) data and a set of Out
of Control (OC) data are generated using this simulated distillation
column.
3.
Normal operating statistical model is developed using the
multivariate techniques, PCA and PCorrA via correlation
coefficient approach between the quality variable(s) and the process
variable(s).
4.
The Shewhart individual and Shewhart range, Exponential Weight
Moving Average (EWMA), and Moving Average and Moving
Range (MAMR) charts are developed.
5.
The faulty condition is incorporated into the process model in order
to see the performance of the control charts to detect the fault(s)
and to find the cause of the fault(s).
5
6.
The results of fault detection and diagnosis are then discussed
further.
1.5
Contributions of the Research
In this research, a new approach known as Improved Statistical Process
Control for fault detection and diagnosis is introduced. Multivariate analysis
technique, Principal Component Analysis, PCA and Partial Correlation Analysis,
PCorrA are used to develop the correlation coefficient between quality variables and
process variables. This technique is incorporated in the Improved SPC chart as a
fault diagnosis tools to find the cause of the faulty condition. Improved SPC charts
are applied for both quality variables and process variables which have been
correlated with quality variables of interest.
1.6
Chapter Summary
This thesis contains seven chapters. The first chapter comprises of the
introduction of the research, research background, objectives of the research, scopes
of the research and contributions of this research. Chapter 2 reviews the multivariate
analysis techniques and the concept of multivariate analysis technique used in this
study, which are Principal Component Analysis (PCA) and Partial Correlation
Analysis (PCorrA).
Chapter 3 consists of the concept of Statistical Process Control, SPC chart,
detail explanation on SPC charts; Shewhart, EWMA and MAMR and the way to
construct these charts. This chapter also presents the implementation of multivariate
analysis technique in SPC chart to develop Improve SPC chart.
Chapter 4 expresses the reasons for choosing distillation column as the case
study. This chapter also presents the dynamic modeling of the columns, formulation
6
of dynamic simulating algorithm, establishment of dynamic simulation program,
controller tuning and explanation on Automatic Process Control, APC time
sampling. The performances of the written simulation program are compared with
HYSYS simulator.
Chapter 5 mainly contains the procedure for fault detection and diagnosis.
This chapter contains variables selection to perform process fault detection and
diagnosis, determination of SPC sampling time, generating of normal operating
condition data, out of control data and the development of correlation coefficient
using PCA and PCorrA to relate the quality variables with process variables.
Chapter 6 presents the result and discussion of the research. The results are
systematically presented. Discussions are made on the performance of each
Improved SPC chart to detect known fault in the process and the efficiency of the
proposed method to conduct fault diagnosis with application of multivariate
statistical analysis. Chapter 7 contains the conclusions of the research and
suggestions to the future SPC research activities.
CHAPTER 2
LITERATURE REVIEW
2.1
Introduction
Malfunctions of plant equipment, instrumentation and degradation in process
operation increase the operating costs of any process industries. More serious
consequences are gross accident such as explosion. Although major catastrophes and
disasters from chemical plant failures may be infrequent, minor accidents are very
common, occurring on a day to day basis, resulting in many occupational injuries,
illnesses, and costing the society billions of dollars. The petrochemical industry
losses approximately $20 billion annually due to poor management in abnormal
detection events (Venkatasubramanian et al., 2003a). Chen et al. (2004) also
highlighted that the US-based petrochemical industry could save up to $10 billion
annually if abnormal process behavior could be detected, diagnosed and
appropriately dealt with. Effective monitoring strategy for early fault detection and
diagnosis is very important not only from a safety and cost viewpoints, but also for
the maintenance of yield and the product quality in a process. This area has also
received considerable attention from industry and academia alike and many
researches have been conducted and many industrial applications exist.
8
2.2
Definitions of Fault, Fault Detection and Fault Diagnosis
In general, fault is deviations from the normal operating behavior in the plant
that are not due to disturbance change or set point change in the process, which may
cause performance deteriorations, malfunctions or breakdowns in the monitored
plant or in its instrumentation. Himmelblau (1978) defines a fault as a process
abnormality or symptom, such as high temperature in a reactor or low product
quality. Faults can be categorized into the following categories (Gertler, 1998):
1. Additive process faults
Unknown inputs acting on the plant, which are normally zero. They cause a
change in the plant outputs independent of the known input. Such fault can be
best described as plant leaks and load.
2. Multiplicative process faults
These are gradual or abrupt changes in some plant parameters. They cause
changes in the plant outputs, which also depend on the magnitude of the
known inputs. Such faults can be best described as the deterioration of plant
equipment, such as surface contamination, clogging, or the partial or total
loss of power.
3. Sensor faults
These are discrepancies between the measured and actual values of individual
plant variables. These faults are usually considered additive (independent of
the measured magnitude), though some sensor faults (such as sticking or
complete failure) may be better characterized as multiplicative.
4. Actuator faults
These are discrepancies between the input command of an actuator and its
actual output. Actuator faults are usually handled as additive though, some
kind (such as sticking or complete failure) may be described as
multiplicative.
While in many cases the classification of a particular fault as additive or
multiplicative follows naturally from its nature, sometimes it may also be arbitrary.
9
Fault detection is a monitoring process to determine the occurrence of an
abnormal event in a process, whereas fault diagnosis is to identify its reason or
sources. The detection performance is characterized by a number of important and
quantifiable benchmarks namely:
1. Fault sensitivity: the ability of the technique to detect faults of reasonably
small size.
2. Reaction speed: the ability of the technique to detect faults with reasonably
small delay after their occurence.
3. Robustness: the ability of the technique to operate in the presence of noise,
disturbances and modeling errors, with few false alarms.
2.3
Multivariate Statistical Analysis, MSA
Multivariate Statistical Analysis (MSA) is based on development of
multivariate regression model in which multiple input variables are related to one or
several outputs. Multivariate analysis technique is combined with statistical process
control, SPC charts to form Multivariate Statistical Process Control (MSPC) (Wise
and Gallagher, 1996; Ralston et al., 2001; Yoon et al., 2003). MSA is applied to
extract useful information from process data and utilize it for process monitoring
using control chart (Kano et al., 2000a). This method takes the interdependence and
large volume of process characteristics into consideration. MSA may be seen as a
response to the complexity and quantity of information available in a modern
computerized installation for FDD on plant (Lennox et al., 1999).
Chemical processes and associated control systems have become very
complex due to product quality demand, economics, safety, and environmental
constraints. As a result, these processes are heavily instrumented and more process
data is readily available. Therefore, the application of MSA couple up with SPC
chart for monitoring, fault detection and diagnosis is appropriate since this method is
based on operating data that is readily available. In chemical processes, data based
approaches rather than model-based approaches have been widely used for process
10
monitoring, because it is often difficult to develop detailed physical models (Kano et
al., 2000a; Yoon and MacGregor, 2000). MSA only requires a good database of
normal historical data, and the models are quickly and easily built from this.
MSA has traditionally employed static linear analysis with normally
distributed variables. Recent research showed that the application of MSA is not
limited for linear process but also applicable for nonlinear and process transition
(Duchesne et al., 2002). The application of this method in various fields has been
widely published and many industrial applications exist. This chapter reviews the
literature on the theory and application of Principal Component Analysis (PCA) and
Partial Correlation Analysis (PCorrA). Both PCA and PCorrA are multivariate
analysis techniques used in this study.
2.4
Review on Principal Component Analysis, PCA
In this section, the overview of the multivariate statistical technique used,
namely Principal Component Analysis (PCA) is presented. Principal Component
Analysis is also known as projection to latent structures (Jackson, 1991;
Daszykowski et al., 2003). Most of the important information from historical data
collected during normal operating process was projected into low dimensional model
that have the most variance. As new data comes available, they are compared with
this normal low dimensional model to detect any abnormal behavior.
PCA is best suited for analysis of steady-state data containing linear
relationships between the variables. This method has since been used extensively in
analytical chemistry, social and economic science, and more recently in engineering
systems. The applications of this approach in various industrial processes have been
reported in literature by several authors and becoming more common in published
research.
11
PCA for process monitoring and fault detection and diagnosis has been
widely studied for continuous, batch and semibatch processes. Ralston et al. (2001)
applied PCA on the iron chloride waste stream to produce high purity sodium
chloride. Q residual was used to detect abnormal variance of a test data set from the
PCA model while the diagnosis was done using the Q residual and Q contribution
plots. Instead of using overall residual to compensate masking errors, Ralston et al.
(2001) used confidence limits on single residual. However, this method has a
drawback in which it required great deal of data to sort out for each variable to
calculate the confidence limits. PCA method is also applied for other continuous
processes such as solvent recovery plant (Lennox et al., 1999) and reactors in a
chloro-carbon production plant (Goulding et al., 2000). Both used Hotelling’s T2
statistic and Q statistic for fault detection and contribution plots for faults diagnosis.
Yoon and MacGregor (2001) introduced fault signature to diagnose fault causes
applied for Continuous Stirred Tank Reactor, CSTR system with feedback control.
For semibatch processes, Simoglou et al. (2000) has applied PCA in
industrial fluidised-bed catalytic reactor for process monitoring for fault detection
and diagnosis. Other researchers used extension of PCA method that is multi-way
PCA for semibatch and bath process. This method is discussed in details in Section
2.4.1.
PCA method has been actively used in recent years for research on sensor
fault detection and diagnosis. A data driven system model using PCA method was
adopted for detecting and diagnosing sensor faults in nuclear power plants
(Upadhyaya et al., 2003). The FDD approach has been applied to a steam generator
system. PCA model is used for fault isolation. The first principal component of this
model is used as a fault signature. For a given test case, the residual vector is
computed using the data-driven model. This vector then projected on to the selected
PC direction and the corresponding cosine of the angle is determined. If this measure
is close to unity (> 0.9), then the fault is isolated. This procedure is repeated for all
the fault directions. The drawback of this technique is that, it requires sufficient fault
data for each case as a fault signature for fault diagnosis.
12
Jiji et al. (2003) used PCA to monitor networked Early Warning Fire
Detection (EWFD) systems in Navy ship for both event detection and determination
of source location and growth of fire rate. Typical fire detection systems monitor the
individual fire detectors and display their responses. Since smoke detectors could not
discriminate between actual fires and the smoke byproducts, Hotelling’s T2 statistics
and the Q-statistics were employed for event detection. Subsequently, contribution
plots were used to determine the source location, rate of growth and to discriminate
between actual fires and their byproducts in adjacent departments. A key benefit of
using MSPC method is its faster detection of smoldering fires while maintaining
rapid detection of flaming fires. Reduction in detection times as much as 5 minute
was observed for the five smoldering fires tested.
Principal Component Analysis (PCA) method was developed to detect and
diagnose sensor faults in Air Handling Units (AHU) (Wang and Xiao, 2004). Sensor
faults were detected using the Q statistic (Squared Prediction Error, SPE). They were
isolated using the Q statistic and Q contribution plot supplemented by simple expert
rules. The PCA method in detecting and diagnosing sensor faults of the AHU were
tested and evaluated by simulation test. Tests show that the PCA method is a
valuable tool in AHU process monitoring, fault detection and isolation.
PCA method can detect the faulty condition in various applications and can
detect any change of relationship between variables. However, PCA method cannot
directly indicate which variables are responsible since it does not provide a causal
description of the process that can be used as the basis for fault diagnosis. Due to
this, the PCA method is not very intuitive and does not always lend operators and
engineers to easy interpretation (Singh and Gilbreath, 2002). Currently, contribution
plots are used as the basic tools for fault diagnosis in the multivariate statistical
approaches (Kourti and MacGregor, 1996; Kourti et al., 1996). These plots are
typically very powerful in diagnosing simple faults, but much less successful at
providing a clear isolation of complex faults. The variable making a large
contribution to the Q statistic is indicated to be the potential fault source (Wang and
Xiao, 2004). The contribution plot helps to reduce the possible fault sources, and
thus focuses on a smaller range of measurements among all variables and thereby
greatly narrows the research (MacGregor and Kourti, 1995; Yoon and MacGregor,
13
2000). Narrowing the possibilities of which variable is highly correlated with the
fault using contribution plot does not equivocally diagnose the cause of the faults
(Chiang et al., 2000).
Several studies have been conducted to overcome the limitation using
contribution plot for fault diagnosis. Yoon and MacGregor (2001) proposed fault
signature developed from prior faults to diagnose faults. A fault signature consists of
directions of movement of the process in both the PCA model space and in the
orthogonal residual space during the period immediately following its detection of
the fault. The fault signature bank or catalog would simply be the collection of these
fault signatures that are available to date. Once a new fault occurs its signature is
compared with those in the fault bank in order to identify the most likely cause.
However, this method requires sufficient fault data to load in fault signature bank to
cover all possible faults in a process. Wang et al. (2002) proposed an improved PCA
that uses two new statistics of Principal-Component-Related Variable Residual, PVR
and Common Variable Residual, CVR to replace Q statistics in conventional PCA.
The process data vector in residual subspace is projected to another two subspace and
the information is represented by PVR and CVR statistics. However, the drawback of
this method is calculating control limits for the new statistic is similar with the Q
statistic in conventional PCA. This result in the fault causes cannot be identified in a
straightforward manner.
PCA was also applied in data analysis to find the cause of the problem
occurred in the process. Santen et al. (1997) used score and loading plot via PCA
method to access the problem of hot spot in reactors of Shell petrochemical process.
The outcome showed that maldistribution in catalyst bed the most important factor
cause high temperature in some of adiabatic Shell reactors. Solving hot spot
problems has improved the performance of the process and generate extra savings
and benefits in the order of 2 millions dollars per annum.
14
2.4.1
Extension of Principal Component Analysis, PCA method
2.4.1.1 Multi-Way PCA
Conventional PCA is best for analyzing a two-dimensional matrix of data
collected from a steady state process, containing linear relationships between the
variables. Since these conditions are often not satisfied in practice, several extensions
of PCA have been developed. Nomikos and MacGregor (1994) proposed Multi-Way
PCA, which allows the analysis of a multi-dimensional matrix. Multi way method
organized the data into time ordered block which each represent a single sample or
process run. Three dimensional array data (I, batch samples x J, process variables x
K, time) is decomposed to two dimensional array (I x JK) data for easier analysis
(Wise and Gallagher, 1996). Projection of these three dimensions data into two
dimensions makes this method suitable and widely applied for batch processes.
Nomikos and MacGregor (1994) used simulated data obtained from a semibatch
reactor to monitor the process. Multi-Way PCA was applied to industrial batch
polymerization reactor using Hotelling’s T2 chart for fault detection and contribution
plots for fault diagnosis (MacGregor and Kourti, 1995; Nomikos, 1996; Kourti et al.,
1996). Multi-Way PCA was also applied using process data collected from an
industrial fed-batch fermentation process (Lennox et al., 2001). Lopes and Monezes
(1998) applied this method for an industrial antibiotic production processes to detect
faulty batches. Martin et al. (1999) used batch polymerization reactor to illustrate the
implementation of Multi-Way PCA. Instead of using T2 statistic, M2 statistics was
used to determine the confidence bound for data not normally distributed. Martin and
Morris (1996) proposed M2 statistics that an empirical density based approach which
the bounds calculated is based on density estimation.
Under the Multi-Way PCA monitoring framework, Q statistic is used as the
fault detection tool to detect abnormal batch variation whereas contribution plots are
used as the fault diagnosis tool to isolate the faulty process variables that are
responsible for the out-of-control situation. The disadvantage of contribution plot is
that, they cannot isolate the causes automatically without the presence of confidence
limit. The plant personnel need to decide whether the out-of-control situation is due
to single or multiple faulty process variables.
15
2.4.1.2 Multi-Block PCA
Extensions of basic PCA to handle very large processes via Multi-Block PCA
were made by MacGregor et al. (1994). This method permits easier modeling and
interpretation of a large matrix by decomposing it into smaller matrices or blocks.
The Multi-Block PCA enables plant wide monitoring. The monitoring framework for
fault detection and diagnosis can be viewed in Figure 2.1.
Process Variables
Unit
Operation 1
Unit
Operation 2
Block 1
Unit
Operation 3
Block 2
Block T2
Block 3
• • •
Unit
Operation N
• • •
Block N
Block SPE
Out of control
Contribution Plots
Figure 2.1: Multi-block PCA monitoring framework
The process variables are divided into several blocks with respective to
specific unit operation. Block Hotellings’ T2 control chart and Q statistic control
chart are used to detect out-of-control situation while contribution plots are used to
isolate faulty process variables that cause the out-of-control situation. This approach
relies on contribution plots for fault diagnosis; the shortcomings of this method still
exist and could not offer complete fault isolation.
16
2.4.1.3 Moving PCA
The concept of Moving Principal Components Analysis is based on the idea
that a change of correlation in between process variables can be detected by
monitoring the directions of principal components (Kano et al., 2000a; Kano et al.,
2001b). In order to evaluate the change of direction of each principal component, an
index based of the inner product between two principal components is defined.
The index proposed in MPCA contained information on the current PCs
directions with reference PCs directions to detect any non-conformance situation to
the reference models. Previous known fault data sets are used to construct PCA
models corresponded to various kinds of fault situation to diagnose the fault cause.
The drawbacks of this method are as follow:
a) There are a lot of PCA models, which representing various past fault
situations are needed.
b) Sufficient data have to be collected in order to construct PC models for each
fault situation.
17
2.4.1.4 Dissimilarity, DISSIM
Kano et al. (2000d) proposed monitoring method based on process data
distribution known as DISSIM. DISSIM method is based on the idea that a change of
operating condition can be detected by monitoring a distribution of time-series data,
which reflects the corresponding operating condition. The degree of dissimilarity
between data sets is determined in DISSIM method (Kano et al., 2000a; Kano et al.,
2000b; Kano et al., 2001a). Dissimilarity index was defined to evaluate the
difference between two data quantitatively. Dissimilarity index control chart is used
for fault detection. This index contains the information of the current data
distribution with reference data distribution. For fault diagnosis, each historical
known fault data set is used to construct the known fault PCA models, which
represents specific known fault data distribution. Besides, a similarity index is
introduced to compare the current fault data distribution to each previous known
fault PCA models. The proposed method is limited to PCs, which have similar
variances because the index cannot function well if the PCs are changed. Other
drawbacks of this method are sufficient data is required to construct every known
fault PCA models and non-ability to isolate new fault.
For fault diagnosis purpose, a contribution of each process variable to the
dissimilarity index is introduced for identifying the variables that contribute
significantly to an out-of-control value of the index (Kano et al., 2000c). However,
there is difficulty to identify exact fault causes for the process, which has many
feedback control loops and the process variables are complicatedly related to each
other.
18
2.4.1.5Multi-Scale PCA
Bakshi (1998) developed Multi-Scale Principal Components Analysis, MSPCA by combining PCA and wavelet analysis. PCA has the ability to extract the
relationship between the process variables and de-correlate cross correlation while
wavelet analysis has the ability to extract events at different scales, compress
deterministic features in a small number of relatively large coefficients, and
approximately decorrelate a variety of stochastic processes (Bakshi, 1999). MS-PCA
methodology determines separate PCA models at each scale to identify the scales
where significant events occur. MS-PCA method has been applied for fault detection
in industrial Fluidized Catalytic Cracker Unit, FCCU. Results showed that MS-PCA
detects the faulty condition faster than conventional PCA using Hotelling’s T2
statistic and Q statistic but the weakness of this method is that he didn’t propose fault
diagnosis method.
Kano et al. (2000a) applied MS-PCA to monitor problems of a simple two
dimension matrix array data obtained from Tennessee Eastman Challenge process.
Other researchers, Misra et al. (2002) proposed the combination of PCA and wavelet
analysis. In essence, the MS-PCA approach is the same as proposed by Bakshi
(1998). However, some differences have been introduced in their study such as
multi-scale fault identification technique to identify the type of fault and sensor
validation approach to serve as an early warning in case a fault of large magnitude is
present. An industrial gas phase tubular reactor system used in this work for process
fault diagnosis and sensor fault detection. The outcomes showed that the proposed
method was able to detect and identify faults and abnormal events earlier than the
conventional PCA approach. The disadvantage of this method is that, it requires
basic understanding of the physical and chemical principles governing the process
operation to help in clustering the highly correlated variables together before
constructing the PCA model. Multi-scale fault identification does not provide the
limits for contribution plot.
19
2.4.2 PCA Adopted in Industries
The multivariate statistical analysis approach has been in widely accepted and
utilized by industry.
Yoon et al. (2003) discussed the application of MSPC to a continuous process
at Honeywell, Geismar, LA. They illustrated the approach and its benefits, that is
improved process understanding, improved product quality, and improved bottom
line. MSPC was used for detecting of caking of HF furnaces and predicting fuel gas
flow rate to HF furnace. Based on normal operation data, PC (Principal Component)
model for furnace caking was developed. Furnace caking detected using Distance to
the Model in the X Plane (DModX) and Hotelling’s T2 statistics and contribution
plots used to diagnose the detected abnormal event. The caking model has been
good tool used by operators and engineers as a quick monitoring method for each
furnaces operation. The economic benefits were substantial. Return On Investment
(ROI) was estimated to be less than two months in the first year.
Miletic et al. (2004) discussed another application of PCA method in industry
involving continuous and batch process. Multivariate statistical analysis is proven to
be a useful technique for Dofasco and Tembec in many applications. Dofasco Inc. is
a fully integrated steel producer, serving customers throughout North America with
high quality of rolled and tubular steels and laser welded blanks from its facilities in
Canada, the United States and Mexico. Tembec Inc. is a leading integrated Canadian
forest products company principally involved in the production of wood products,
market pulp and papers. PCA method has been applied for a continuous slab caster
(Dofasco) for fault detection. The deployment of the PCA-based multivariate SPC
monitoring system has had a significant impact on operating performance. Dofasco
has enjoyed its highest productivity levels with an on-average 50% reduction in
breakouts of its first caster, over the last six years. The used of PCA in batch cooks
monitoring in sulfite pulp digester (Tembec) for fault detection has allowed operators
to compensate quickly for small variations in the process. The Multivariate statistical
applications developed at Dofasco and Tembec have delivered valuable business
results that can be related to company profits. The Multivariate statistical
applications is proven to be robust to many practical matters such as missing data,
20
and is proven to be amenable to various maintenance and updating strategies.
Multivariate statistical analysis provides models and control chart schemes that are
readily understood by production personnel once training are provided.
2.4.3
PCA as Controller
Chen et al. (1997) designed multivariate statistical controller to decrease the
variation in product quality without real time quality measurements. The cross
correlation and autocorrelation among the variables are taken into consideration to
model the dynamic PCA. Principal Component, PC scores are calculated at a new
sampling time and the difference between previous data and current data is
calculated to determine Squared Prediction Error, SPE. SPE control limit is
determined using reference data. If the SPE is above its control limits, the statistical
controller should be turned off because the structure of the process variables has
changed. The multivariate statistical controller utilizes set point in a conventional
control system to improve control performance. This method is applied in a
continuous process, binary distillation column and Tennessee Eastman Challenge
process. A multivariate statistical controller is built to replace the composition
controller in binary distillation column case study. Results showed that statistical
controller could detect and compensate disturbance faster than PI controller. A large
variation due to the disturbances is shrunk using multivariate statistical controller and
steady state offset is eliminated using the combination of statistical controller and
feedback controller.
Souza et al. (2004), proposed the joint use of two methodologies, Statistical
Process Control, SPC and Engineering Process Control, EPC for controlling the
quality of products and services. Statistical Process Control, SPC perform process
monitoring using measurements obtained in the process without prescribing a control
action. On the other hand, Engineering Process Control, EPC allows for the
prescription of changes in variables involved in the process, in order to make them
the closest possible to the desired target. Multivariate proportional feedback
controller is proposed for ceramic materials company that produces several types of
21
tiles. The error that was made in forecasting the disturbance is modeled according to
Exponentially Weighted Moving Average (EWMA) statistics. The value of the
smooth constant (lambda) offers the smallest forecast error of the series of
disturbance errors adjusted by EWMA statistics. The proposed method allows for the
adjustment of the productive process to be made while the parts or products are still
on the production line.
2.4.4 Theory of PCA
Principal Components Analysis is a multivariate technique, which is used to
explain the variance-covariance structure through the linear combinations of the
original variables. The advantages of this method rely on data reduction and
information extraction. Because of this reason, PCA is commonly classified as data
reduction technique. Mathematically, PCA relies on eigenvector decomposition of
the covariance or correlation matrix of the process variables. Let p is the number of
variables with m measurements, which are collected and arranged in matrix form.
The multivariate data set can be presented by m x p matrix where the rows represent
observations and the columns represent variables. For notation of brevity, the bold
capital letter signifies as matrix, and bold small letter is a vector and italic letter
means scalar values. The following matrix data, X consist of m observations and p
variables,
⎡ x11
⎢
x21
Xm,p = ⎢
⎢M
⎢
⎢⎣ xm1
x12 K x1 p ⎤
⎥
x22 K x2 p ⎥
⎥
M
M
⎥
xm 2 L xmp ⎥⎦
2.1
A set of original variables x1, x2, …, xp is transformed to a set of new variables PC1,
PC2, …, PCp. The newly formed variables are referred to as Principal Components
(PCs). The mathematical representations that describe the transformation are shown
as follow,
22
PC1 = x1v1,1 + x2 v1, 2 + K + x p v1, p
PC2 = x1v2,1 + x2 v2, 2 + K + x p v 2, p
2.2
M
PC p = x1v p ,1 + x2 v p , 2 + K + x p v p , p
Equation 2.2 is simplified to:
PC p , m = X m , p V p , p
2.3
V is the eigenvector matrix, which consists of eigenvector v1, v2, …, vp. Each
eigenvector vi is a column vector which contains the arrangement of elements as vi =
[vi,1 vi,2 … vi,p]. The eigenvectors matrix can be presented as following,
Eigenvectors matrix, V =[v1 v2
⎡ v1,1
⎢v
⎢ 2,1
⎢ .
… vp] = ⎢
.
⎢
⎢ .
⎢v1, p
⎣
v1, 2
v2, 2
.
.
.
v2 , p
... v1, p ⎤
... v2, p ⎥⎥
...
. ⎥
...
. ⎥
⎥
...
. ⎥
... v p , p ⎥⎦
2.4
PC is the principal components scores matrix. A particular principal
component score, pci is obtained via the product of data matrix, X (m,p) to the
particular eigenvector vi, which is shown in the following equation:
pc i ( mx1) = X ( mxp ) v i ( px1)
2.5
Equation 2.5 is used to determine the principal component score matrix PC,
which contains m scores for each principal component:
PC ( m , p ) = X ( m , p ) V( p , p )
2.6
The numbers of PCs are equal to number of variables in the original data set.
The first PC accounts for the greatest variation in the data while the second PC
accounts for maximum variation that is not captured by the first PC, and so on. The
23
original variables can also be represented in terms of newly formed variables as
follows
X ( m, p ) = PC ( m , p ) V(Tp , p )
2.7
In order to maintain the uniqueness of this technique for data reduction, only
several Principal Components will be used to represent most of the original data
variation. If A Principal Components are decided to retain with A < p, then Equation
2.7 can be written as:
X ( m , A) = PC ( m , A) V(TA, A) + E ( m , p − A)
2.5
2.8
Review on Partial Correlation Analysis, PCorrA
The purpose of PCorrA method is to determine the correlation between two
variables while keeping the other variable constant. PCorrA has been applied in
various research fields such as statistical analysis, food science, environmental
issues, horticulture, cardiology and psychiatry.
Ibrahim (1997) has introduced Partial Correlation Analysis (PCorrA) to be
applied in chemical process data by developing Multivariate Statistical Process
Control (MSPC) scheme known as Active SPC methodology. The method is similar
to those used in the feedforward Automatic Process Control (APC). Active SPC was
applied to nonlinear simulated Continuous Stirred Tank Reactor (CSTR).
Foucart (2000) proposed a decision rule using PCorrA regression method to
discard a principal component from the set of predictors variables (independent
variables). The selection of Principal Components to retain depends on the small
mean of the squared residuals.
24
Tsumoto et al. (2002) used PCorrA to analyze acid amino sequences. Partial
correlation used to examine the contribution of the mutated sites (35 types of mutated
amino-acid) on acid amino sequence i.e VH and VL chains of partially mutated antilysozyme antibody HyHEL-10 to the change of thermodynamic measurements
(measured values: temperature, pH, combining coefficients Ka, free energy change
DG, enthalpy change DH, entropy change TDS, and change of specific heat DCp).
The relations between change of selected thermodynamic measurements and
locations of mutated sites were determined by keeping some of the thermodynamic
measurements constant. The results demonstrated that two or three sites on each of
the VH and VL chains have high correlation to the change of thermodynamic
condition.
Shirley et al. (2003) investigated the sensitivity of the population dynamics
model to the input parameters by analyzing the response of total population size and
total number of infected individuals in simulated badger populations over 20 years to
variations in the model inputs. An individual based spatially-explicit model was
designed to investigate the dynamics of badger population and TB epidemiology in
real landscape. The relationships between input parameters and model output were
analyzed using partial correlation. Parameters with high partial correlation
coefficients give large effect on the model output and that parameter is identified as
key driving parameters at relatively low numbers of simulations.
Quemerais et al. (1998) investigated the factors that had the most influence
on the mercury concentrations in water using partial correlation. When the effects of
dissolved iron and manganese removed, dissolved mercury shows no relation to
dissolved organic carbon. On the contrary, when the effect of dissolved organic
carbon, DOC removed, dissolved mercury concentrations are related to dissolve iron.
The observed mercury partition in the waters of the St. Lawrence River enhanced by
the partial correlation between the distributions coefficients. Partial correlation
analyses indicate that mercury strongly related to iron and manganese, while organic
carbon being indirectly related.
25
Wissemeier and Zuhlke (2002) conducted a study on the relationship between
tipburn with growth data and climatic variables. From the regression coefficient
results, tipburn severity highly correlated with the sum of irradiation, the head fresh
weight, the sum of temperature, the sum of maximum daily irradiation and the day
light length 23 days prior to harvest were obtained. However, sum of irradiation had
the highest correlation coefficient with tipburn. Moreover, the sum of irradiation
remained a highly significant predictor for tipburn if other variables were statistically
eliminated by partial regression analysis. The intercorrelation between variables
statistically eliminated using partial regression analysis and this result the
significance of every independent variable predicting tipburn lost if the impact of the
sum of irradiation is eliminated statistically.
Hashimoto et al. (2003) study the effect of basic Fibroblast Growth Factor,
bFGF on the pathophysiology of schizophrenia. bFGF is a multifunctional growth
factor that has been implicated in a variety of neurodevelopmental processes.
Analysis of partial correlation coefficients showed that the increased bFGF levels
might not be attributable to antipsychotic medication for schizophrenic patients.
Onat et al. (1999) investigated the relationship of waist circumference (WC)
and waist-to-hip ratio (WHR) with a number of established risk factors and their
relevance to cardiovascular morbidity. The study was conducted on random sample
of Turkish general adult population. Partial correlation coefficients on univariate
analysis was done to determine the correlation between four variables i.e blood
pressure and plasma lipids and either WC or WHR, controlled for age. The results
showed that the correlation was highly significant but moderately weak in genders,
men and women.
26
2.5.1 Theory of PCorrA
Partial correlation is the correlation of two variables while controlling for a
third or more other variables. The partial correlation presents the incremental
predictive effect of one process variable from the collective effect of all others and is
used to identify the process variables that have the greatest incremental predictive
power when a set of process variable is ready in the regression variate (DeVor et al.,
1992).
If the two variables of interest are y and x and the control variables are z1, z2
... zn, then the corresponding partial correlation coefficient is ryx z1, z 2..., zn . The order of
partial correlation depends on the number of variables that are being controlled for.
The first order partials equation after removing the effect of variable z1 is
r yx z1 =
ryx − r yz1 rxz1
1− r
2
yz1
1− r
2.9
2
xz1
where,
ryx z1 = Partial coefficient of y and x after controlled z1
ryx = Correlation coefficient between y and x
ryz1 = Correlation coefficient between y and z1
rxz1 = Correlation coefficient between x and z1
The second order partial after controlled the effect of z1 and z2 is
ryx z1z 2 =
ryx z1 − ryz 2 z1rxz 2 z1
1 − ryz2 2 z1 1 − rxz2 2 z1
=
ryx z 2 − ryz1 z 2 rxz1 z 2
1 − ryz2 1 z 2 1 − rxz2 1 z 2
2.10
Reapply the Equation 2.10 to compute the next order partial equation. The highest
order possible for partial is
n=N–1
2.11
27
where,
n = The highest partial order
N = Numbers of process variables
and the general equation for higher order partial is
ryx z1, z 2, z 3,K, zn =
ryx z 2, z 3,K, zn − ryz1 z 2, z 3,K, zn rxz1 z 2, z 3,K, zn
1 − ryz2 1 z 2, z 3,K, zn 1 − rxz2 1 z 2, z 3,K, zn
2.12
As shown mathematically in Equation 2.12, this correlation is calculated by
separating the group of process variables into subgroup in which one or more
variables are held constant before determining the correlation among the other
variables.
2.6
Chapter Summary
Multivariate statistical analysis, MSA is based on development of
multivariate regression model in which multiple input variables are related to one or
several outputs. Multivariate analysis technique is combined with statistical process
control, SPC charts to form Multivariate Statistical Process Control, MSPC.
Multivariate analysis techniques like Principal Components Analysis, PCA and
Partial Correlation Analysis, PCorrA are used in this study.
Review on PCA method shows that this method has been widely used in
various applications such as process monitoring, fault detection, fault diagnosis,
process controllers and data analysis. PCA for process monitoring and fault detection
and diagnosis has been widely studied for continuous processes and batch and
semibatch processes. The advantage of this method relies on data reduction and
information extraction from historical database. A set of original variables is
transform to a set of new variables called as Principal Component, PC. The number
28
of PCs used is less than number of original variables. This multivariate statistical
analysis approach has been in widely accepted and utilized by industry. Several
companies have started utilizing this method. The extensions of PCA such as MultiWay PCA, Multi-Block PCA, Moving PCA, Dissimilarity and Multi-Scale PCA also
have been discussed.
PCorrA is another multivariate analysis technique. The purpose of PCorrA
method is to determine the correlation between two variables while keeping constant
the other variables. This correlation is done by separating the group of process
variables into subgroup in which one or more variables are held constant before
determining the correlation among the other variables. PCorrA has been applied in
various research fields such as statistical analysis, food science, environmental
issues, horticulture, cardiology and psychiatry.
CHAPTER 3
IMPROVED STATISTICAL PROCESS CONTROL CHART
3.1
Statistical Process Control, SPC
Statistical Process Control (SPC) monitors the performance of a process over
time in order to verify the process whether it is in a state of statistical control or in a
state out of control. A state of control exists when the process variables close to its
desired values and only common cause variation exist. Meanwhile a state out of
control is exists if process variables fall beyond the control limits and it shows that
assignable causes of variation exist in a process. If the process is in control, it is
assumed that the products produced from these processes will also be in control
(Doty, 1996). Common causes are small random changes in the process that cannot
be avoided. These small differences are due to the inherent variation present in all
processes. They consistently affect the process and its performance. Variation of this
type is only removable by making a change in the existing process. Assignable
causes, on the other hand, are large variation in the process that can be identified as
having specified cause. Assignable causes are causes that are not part of the process
on a regular basis. This type of variation arises because of specific circumstances.
Sources of variation can be found in the process itself, the material used, the
operator’s actions, or the environment (Summer, 2000).
30
SPC is an excellent tool for process monitoring which can be applied to any
process. There are many statistical tools that can be used and the most widely
applicable are the “Seven Basic Tools”. They are histogram, check sheet, pareto
chart, cause and effect diagram, defect concentration diagram, scatter diagram and
control chart.
3.2
Traditional Statistical Process Control, SPC chart
The function of Statistical Process Control (SPC) chart is to compare the
current state of the process against Normal Operating Condition (NOC). The NOC
condition exists when the process or product variables remain close to their desired
values or in statistical control. In contrast, the Out of Control (OC) occurs when fault
appears in the process. The fault or malfunction is designated when the process
departs from an acceptable range of observed variables. This happens when the
process is having degradation in performance and experiencing malfunction of the
equipment due to a failure. Malfunction on a piece of equipment is usually relatively
easy to detect compared to process degradation. Thus, early detection of faults either
failure or degradation is needed to reduce the occurrence of sudden failure on
equipment, to reduce personnel accidents and to assist in the operation of
maintenance program.
Control charting, the featured tool of SPC is used to distinguish between
common cause variation and assignable cause variation in order to prevent
overreaction and underreaction to the process. Control charts approach is based on
the assumption that a process subjected to a common cause variation will remain in a
state of statistical control under which process remain close to target. Therefore, by
monitoring the performance of a process over time, OC events can be detected as
soon as they occur. If the causes for such events can be diagnosed and the problem
can be corrected, the process is driven back to its normal operation
(Venkatasubramanian et al., 2003).
31
Individual Shewhart and Shewhart range, Exponential Weight Moving
Average, EWMA, and Moving Average and Moving Range, MAMR are examples of
SPC charts. They are used for individual data. The following section discussed the
theory of each SPC chart and the method to construct these charts.
3.2.1
Shewhart Chart
Shewhart chart represents the current observation data excluding the previous
information of the process data. It is usually necessary to monitor both mean value of
the variables and its variability. In this study, individual data for quality variables and
process variables is used in designing Shewhart chart. Current data is plotted to
monitor the process over the time. Both average and standard deviation were
determined using historical data, which represent the desired condition for the
process operation. Let x1, x2, …, xm denote individual data for variable with m
observations, then the average of the variable x is,
x=
x1 + x2 + K + xm
m
3.1
where,
x
= Individual data
x
= Arithmetic average
m
= Number of observation
The center line and control limits for individual Shewhart chart was determined
using Equation 3.2,
32
UCL = 3s
CL = x
3.2
LCL = −3s
where,
UCL
= Upper control limit
LCL
= Lower control limit
CL
= Centre line
s
= Standard deviation
Shewhart charts used in this study have control limits ± 3-standard deviation. Three
out of 1000 data will fall out of the limits or 99.73% of data fall within these limits.
The probability of data to fall beyond this control limits is very small to occur.
Process variability is monitored by plotting values of the sample range, R on
a control chart. The sample range, R is the difference between the maximum value
and the minimum value of the sample. In this study, sample range, R is calculated for
every two observations. Let R2, R3,…, Rm be the ranges for m observation. Then, the
average of sample range, R is
R=
R2 + R3 + K + Rm
m
where,
R
= Sample range
R
= Average of samples range
m
= Number of observations
3.3
33
The center line and control limits on the Shewhart range chart are as follows:
UCL = D.'001 R
CL = R
3.4
LCL = D
'
.999
R
'
D.'001 and D.999 are the constants that can be found in (Oakland, 1996).
where
UCL
= Upper control limit
LCL
= Lower control limit
CL
= Centre line
R
= Average of samples range
'
D.'001 , D.999
= Constant
Figure 3.1 and Figure 3.2 show the Shewhart individual and Shewhart range chart
during NOC and OC respectively.
34
Figure 3.1: Shewhart range and Shewhart individual chart during normal condition
Out of control
Out of control
Figure 3.2: Shewhart range and Shewhart individual chart during faulty condition
35
3.2.2 Exponential Weight Moving Average, EWMA Chart
The Exponential Weight Moving Average (EWMA) is a statistic tool for
monitoring the process that averages the data in a way that gives less and less weight
to data as they further removed in time. The decision regarding the state of control of
the process at any time depends on the EWMA statistic, which is exponential
weighted average of all prior data, including the most recent data.
This type of chart has some very attractive properties such as (Bower, 2000):
1.
All of the data collected over time may be used to determine the
control status of a process.
2.
EWMA schemes may be applied for monitoring standard
deviations in addition to the process mean.
3.
There exists the ability to use EWMA schemes to forecast values of
a process mean.
4.
The
EWMA
methodology
is
not
sensitive
to
normality
assumptions.
The EWMA statistic, zi for each variables with m observations, i=1,2, … , m,
is defined as
zi = βxi + (1 − β ) zi −1
3.5
For i = 1, the EWMA statistics is
z1 = βx1 + (1 − β ) z0
3.6
The values of zo can be the target values or average of the variable during normal
operating condition.
36
where,
zi
= EWMA statistics
zo
= Average of normal data or target value
x
= Individual data
β
= Weighting factor
m
= Number of observations
From Equation 3.5, it shows that the information from the past observation is
combined with the current observation. The result make EWMA has the ability to
detect a small shift in the process. The EWMA statistic, zi is a weighted of all
previous data, substituted in the right–hand side of the equation above to give
zi = βxi + (1 − β )[βxi −1 + (1 − β ) zi −2 ]
= βxi + β (1 − β ) xi −1 + (1 − β ) 2 z i −2
3.7
The EWMA control procedure can be made sensitive to a small or gradual
drift in the process by selecting appropriate value of weighting factor, β . In general,
the values of β in the interval 0.05 ≤ β < 0.25 work well in practice, with 0.05, 0.1,
and 0.2 being popular choices (Montgomery, 1996). A good rule of thumbs is to use
smaller values of β to detect smaller shifts. It is found that the width of control limit,
L=3 (the usual 3 standard deviation limit) works reasonably well, particularly with
the larger value of β . Hunter (1989) suggested using β = 0.4 which the weight
given to the current and previous observation data match closely to the weight given
to observations by Shewhart chart. The advantage of using small value of β is it will
reduce the width of the limits in order to give good performance of the control chart
(Montgomery, 1996).
37
The variance of the EWMA statistics, s z2i is given by
⎛ β
s z2i = s 2 ⎜⎜
⎝2−β
⎞
⎟⎟[1 − (1 − β ) 2i ]
⎠
3.8
The center line and control limits for the EWMA control chart as follows:
UCL = x + Ls
β
2−β
[1 − (1 − β ) 2i ]
CL = x
3.9
LCL = x − Ls
β
2−β
[1 − (1 − β ) 2i ]
where
UCL
= Upper control limit
LCL
= Lower control limit
CL
= Centre line
x
= Arithmetic average
L
= Width of the control limit
s
= Standard deviation
Importantly, the EWMA structure is insensitive to normality. This makes the EWMA
chart an attractive candidate in general when addressing small changes in a process.
Figure 3.3 and Figure 3.4 show the pattern of EWMA chart under NOC and OC
respectively.
38
Figure 3.3: EWMA control chart under normal condition
Out of control
Figure 3.4: EWMA control chart with faulty condition
39
3.2.3 Moving Average and Moving Range, MAMR Chart
The Moving Average and Moving Range (MAMR) control charts are used in
handling data, which are difficult, or time consuming to obtain. In the chemical
industry, the nature of certain production processes and analytical method entails
long time interval between consecutive results. The alternative of moving average
and moving range control charts uses the data differently and is generally preferred
for the following reasons (Oakland, 1996):
By grouping the data together, reacting to individual results and over
1.
control is less likely.
It is more meaningful because it uses the latest piece of data.
2.
The EWMA chart uses weighted averages as the chart statistic. But Moving
Average, MA is another type of time-weighted control chart based on simple,
unweighted moving average and assumes that the past and present data are equally
important. Therefore, the MA for i = 1, 2, … m observation in a window of size w is
defined as
MA =
xi + xi −1 + K + xi − w+1
w
3.10
where
MA
= Moving average statistics
x
= Individual data
w
= Moving window size
m
= Number of observations
The Moving Range, MR chart is used in order to control process spread. This
control chart is likely to be very sensitive to assumption of normality, independence
and to the presence of autocorrelation (Wetherill and Brown, 1991). MR statistic is
similar to MA statistic.
40
A range of observation data in every time window w is taken over instead of the
average.
MRi=max[xi-w+1]–min[xi-w+1]
3.11
At time period i= w+1, the oldest data in the moving average and moving
range set is dropped and the newest one added to the set. The subgroup for every
time moving is the same. In general, a large window size, w will result in a large
detection delay, while a small w will result in a less sensitive fault detection statistic.
Therefore, appropriate value of w must be chosen to suit with the normal operating
condition, NOC data.
If x denotes the target value or the arithmetic average used as the center line
of the control chart, then the 3 standard deviation control limits for MA is,
UCL = x + A2 R
LCL = x − A2 R
3.12
and for the MR chart the control limits is defined as,
UCL = D.'001 R
LCL = D.'999 R
3.13
41
where
UCL
= Upper control limit
LCL
= Lower control limit
CL
= Centre line
x
= Arithmetic average
R
= Average of samples range
A2
= Constant for moving average
'
D.'001 , D.999
= Constant
The average of sample range, R is determined using normal operating data. The
sample range, R for moving chart is the difference between the maximum value and
the minimum value of the sample in moving window. The way to calculate the
sample range, R is the same with MR statistics. For every moving window, the
sample range, R is calculated. A2, D.'001 and D.'999 are the constant depends on
window size, w that can be found in (Oakland, 1996). In general, the magnitude of
the shift of interest and w are inversely related. Smaller shifts would be guarded
against more effectively by longer span moving averages, at the expense of quick
response to large shift. Figure 3.5 and Figure 3.6 show the pattern of MAMR chart
under NOC and OC respectively.
42
Figure 3.5: Moving average and range control chart under normal condition
Out of control
Out of control
Figure 3.6: Moving average and range control chart with faulty condition
43
3.3
Improved Statistical Process Control Chart
Traditional SPC chart is often applied by constructing separate control charts
for each of the selected variables. These charts give signal out of statistical control if
an assignable cause of variation exists in the process. SPC is implemented by
monitoring either quality characteristics or process characteristics. Traditional SPC
approach is deficient for at least three reasons. First, the fault in the process is seeked
through the usage of SPC chart, i.e., the final product quality variables. Measuring
quality variables alone is not enough to describe the process performance (Kourti et
al., 1996). Second, observations of multiple process or product characteristics are
assumed to be unrelated. Almost any measure of product quality is dependent on a
sequence of preceding processes. Hence, measures of process characteristics are
often correlated (Singh and Gilbreath, 2002). Third, in traditional SPC, once the
quality variables showed out of statistical control signal, it is then left up to process
operators and engineers to try to diagnose the cause of out of control using their
process knowledge and one at a time inspection of process variables (MacGregor and
Kourti, 1995).
In order to improve the traditional SPC chart to detect and diagnose
assignable cause occur in the process involving massive data, multivariate analysis
technique such as Principal Component Analysis, PCA and Partial Correlation
Analysis, PCorrA are used to incorporate all these data to be analyzed and included
in constructing the SPC chart. PCA is modified and used to determine the cross
correlation between quality variables and process variables while PCorrA is used to
determine the correlation between a quality variable and selected process variable
after removing the effect of other process variables. The correlation coefficient, Cjk is
calculated using both multivariate techniques in order to improve traditional SPC
charts. Let xj be the quality variables and xk be the process variables.
The relationship between x j and x k can be written as,
x j = Cjk x k
3.14
44
The control limit for quality variable, x j in general is
LCL< x j <UCL
3.15
where UCL and LCL is upper control limit and lower control limit respectively.
Substituting Equation 3.14 into Equation 3.15 and rearrange it gives,
UCL
LCL
< xk <
C jk
C jk
3.16
Figure 3.7 shows the use of correlation coefficient corresponding to process
variables. Once fault has been detected using quality variables, the correlation
coefficient is used to construct the Improved SPC chart using process variables to
identify the variables that cause the fault.
Process
variable
Quality
variable
Process
Process
variable
Quality
variable
Fault diagnose
UCL/Cjk
LCL/Cjk
Fault detect
UCL
LCL
Cjk
Figure 3.7: The implementation of correlation coefficient, Cjk in SPC chart
45
The quality variables data is incorporated in the control chart during the
faulty condition for fault detection purpose, while the process variables which have
been correlated with the quality variables is used for fault diagnosis. For example,
when statistical data for oleic acid, xc8 fall beyond the control limits, it shows that
fault is detected. Then, the next task is to determine the cause of the fault. Process
variables charts are monitored for this task. If statistical data for feed flowrate fall
beyond the limits, then it is concluded that this variable cause the fault. The Improve
SPC charts are applied not only for quality variables but also for process variables.
Both quality variables and process variables are monitored continuously to detect and
diagnose faults. By monitoring the process variables, which have been correlated
with quality variables, the cause of faulty condition, which affects the quality
variables, could be identified directly. This will help the operator to take proper
action immediately after the faults exist in the process.
3.4
Chapter Summary
Statistical Process Control (SPC) is an excellent tool for process monitoring
which can be applied to any process. SPC only requires a good database of normal
historical data, and the models are quickly and easily built from this. New data is
then compared with the normal operation model to detect a change in system. The
control chart is used as a Fault Detection and Diagnosis (FDD) tools in SPC. The
function of SPC chart is to compare the current state of the process against Normal
Operating Condition (NOC). The control chart will give signal out of statistical
control if assignable cause variation or fault exists in the process. Shewhart
individual, Shewhart range, Exponential Weight Moving Average (EWMA), and
Moving Average and Moving Range (MAMR) are SPC charts to be used to detect
and diagnose fault in the process.
46
Traditional SPC chart has some drawbacks in detecting and diagnosing fault
in data rich environment because it ignores the correlation among the variables.
Therefore, multivariate analysis techniques such as Principal Component Analysis
(PCA) and Partial Correlation Analysis (PCorrA) are applied to build linear
relationship between quality variables and process variables to improve traditional
SPC chart. PCA is used to determine the cross correlation between quality variables
and process variables while PCorrA is used to determine the correlation between a
quality variable and selected process variable after removing the effect of other
process variables. The correlation coefficient using these multivariate analysis
techniques is applied on control charts for fault diagnosis purpose.
Both quality variables and process variables are monitored in this study. The
cause of faulty condition, which affects the quality variables, could be identified in a
straightforward manner and help the operator to take proper action immediately after
the faults occur in the process.
CHAPTER 4
DYNAMIC SIMULATED DISTILLATION COLUMN
4.1
Introduction
The proposed FDD algorithm can be applied to any kind of unit operations in
chemical industry. However, distillation column is chosen in this study because of a
few reasons. Distillation accounts for approximately 95% of the separation systems
used by the refining and chemical industry (Hurowitz et al., 2003). It has a major
impact upon the product quality, energy consumption, and plant throughout of these
industries. Maintaining the product quality within specification is the usual policy in
industry and it is very difficult to control the process at the desired target. The
sensitivity of product quality to disturbances decreases as impurities are reduced.
According to Shinskey (1984), 40 percent of the energy consumed in a plant
allocated to distillation.
Distillation control is a challenging endeavor due to (Hurowitz et al., 2003):
1. The inherent nonlinearity of distillation.
2. Severe coupling present for dual composition control.
3. Non-stationary behavior.
4. The severity of disturbances.
48
Since distillation column involves multivariable and has non-stationary
behavior in nature, multivariate analysis is an appropriate technique for FDD in the
process to avoid any degradation process and to maintain the product quality. A precut column from the Palm Oil Fractionation Plant is choosen as a study column. A
simulated model is developed based on the information from this pre-cut column
using Matlab.
4.2
Process Description of Precut Column
The fatty acid mixtures, caproic acid (C6), caprylic acid (C8), capric acid
(C10), lauric acid (C12), myristic acid (C14), palmitic acid (C16), stearic acid (C18),
oleic acid (C18), and linoleic acid (C18) enter the precut column in liquid form at
tray 14. The light components (C6-C10) are condensed by direct contact in the
pumparound section and recovered as distillate. Then, the distillate product is
pumped to storage. The heavy components (C12-C18) recovered at the bottom of pre
cut column. These components will be passed to other system in the fractionation
unit for further fractionation process. Figure 4.1 shows the schematic diagram of pre
cut column.
Figure 4.1: Pre cut column
49
4.3
The Information of Pre-cut column
The summaries of the specification of the precut column, components
composition and conditions are presented in Tables 4.1 and 4.2 respectively.
Table 4.1: Specification of the pre-cut column
Variable
Specification
Number of actual tray
28
Actual feed position
Tray 14 from the top
Plate spacing
50 cm
Plate diameter
120 cm
Plate type
Sieve tray
Type of reboiler
Total
Reflux rate
10 kmol/hr
Pumparound rate
49.6 kmol/hr
Sidedraw rate
63.6 kmol/hr
Reboiler duty
2.158 x 106 kJ/hr
Weir length
0.912 m
Weir height
0.076 m
50
Table 4.2: Components composition and conditions of the pre-cut column
Component
Composition (kmol/kmol)
Feed stream
Distillate stream
Bottom stream
0.002
0.023
0
0.051
0.528
0
0.043
0.438
0.001
0.518
0.006
0.574
0.156
0
0.173
0.068
0
0.075
Stearic acid, C-18
0.014
0
0.013
Oleic acid, C-18-1
0.121
0
0.134
Linoleic acid, C-18-2
0.020
0
0.023
Temperature
483.15 K
442.55 K
509.55 K
Pressure
2.5 bar
0.11 bar
0.13 bar
Hexanoic acid, C-6
(Caproic acid)
Octanoic acid, C-8
(Caprylic acid)
Decanoic acid, C-10
(Capric acid)
Dodecanoic acid, C-12
(Lauric acid)
Tetradecanoic acid, C-14
(Myristic acid)
Hexadecanoic acid, C-16
(Palmitic acid)
51
4.4
Degree of Freedom (DoF) Analysis
There is a set of equation called as MESH equations involved for the dynamic
simulation, which consist of mass balance equation, M, equilibrium equation, E,
summary equation, S and heat balance equation, H. Another equation is hydraulic
equation to describe the behavior of the column bottom, normal tray, feed tray, reflux
drum, pumparound drum, sidedraw tray and the reflux tray. The number of
independent equation must be equal to the number of independent variables in order
to get a unique solutions (Stephanopoulos, 1984). This rule must be followed to
avoid either redundant equations or variable conditions because these conditions
cause infinite number of solutions or no solution at all. The equation for DoF is
DoF = Number of independent variables – Number of independent equations 4.1
For multicomponents distillation column, the number of variables and equation is
determined to perform DoF. It is presented in Table 4.3 and Table 4.4 as in Luyben
(1963).
52
Table 4.3: Number of variables
Variables
General Form
No. of Variables
2MN
2 x 9 x 29
Tray liquid flow, L
N
29
Tray vapor flow, V
N
29
Tray hold up, Molar
N
29
Reflux flow rate
1
1
Distillate flow rate
1
1
Sidedraw flow rate
1
1
Pumparound flow rate
1
1
Liquid stream flow rate
1
1
Pumparound drum hold up
1
1
Reflux drum hold up
1
1
2M
2x9
Bottom flow rate
1
1
Bottom hold up
1
1
Vapor flow from reboiler
1
1
2M
2x9
2(N+1)
2 x 29
2
2
Reflux drum temperature and pressure
2
2
Pumparound stream temperature
1
1
Reboiler duty
1
1
Cooler duty
1
1
Tray composition (vapor and liquid)
Side draw composition
Bottom composition
Trays pressure and temperature
Pumparound drum temperature and
pressure
Total
720
Notes: Number of trays, N = 28 +1 (total reboiler) = 29
Number of components, M = 9
53
Table 4.4: Number of equations
Equations
Total mass balance (tray,
General Form
No. of Equations
N+3
31
(N + 3) x (M – 1)
31 x 8
2(N + 3)
2 x 31
N +3
31
M(N + 3)
9 x 31
N+3
31
reflux drum, pumparound
drum, bottom)
Component balance, liquid
composition
Summation equations for
vapor and liquid
composition
Hydraulic (liquid flowrate)
Equilibrium equation
(vapor composition)
Heat balance
Total
682
Degrees of Freedom
720 – 682 = 38
There are seven control loops installed in the simulated column model to
ensure the stable operation of the column. Figure 4.2 shows the schematic diagram of
the study column with the control loops. The control variables, manipulated variables
and measured disturbances for each control loops are listed in Table 4.5.
54
To vent
CL 6
CL 3
Cooling
Utility In
Di
TIC
CL 5
Pumparound
Point
FIC
PIC
P
LIC
FIC
HX-2
Reflux
Point
Sd
Re
CL 4
F
Pump-2
CL 2
Hot
Utility
In
HX-1
Hot
Utility
Out TIC
LIC
Pre-Cut Column
B
CL 1
Pump-1
Controller
TIC: Temperature Indicator Controller
FIC: Flow Indicator Controller
LIC: Level Indicator Controller
PIC: Pressure Indicator Controller
Equipment
HX: Heat Exchanger
Stream
B: Bottom Flow Rate
Re: Reflux Flow Rate
Sd: Sidedraw Flow Rate
P: Pumparound Flow Rate
F: Feed Flow Rate
Di: Distillate Flow Rate
Symbol
CL: Control Loop
Figure 4.2: Distillation column control loop
CL 7
Cooling Utility
Out
55
Table 4.5: Control loops properties
Control
Control
Manipulated
Measured
loop
variable
variable
disturbance
1
Bottom column
Bottom
Liquid flowrate at
liquid level
flowrate
tray 28
Bottom
Hot utility
Liquid flowrate at
temperature
flowrate
tray 28
Top column
Vapor flowrate
Vapor flowrate at
pressure
to vent
tray 2
Reflux
Reflux
Sidedraw
flowrate
flowrate
flowrate
Pumparound
Distillate
Pumparound
flowrate
flowrate
point
Side draw tray
Distillate
Liquid flowrate at
liquid level
flowrate
tray 1
Pumparound
Cold utility
Pumparound
temperature
flowrate
point
2
3
4
5
6
7
Seven control loops is equivalent to seven independent equations. Therefore, DoF is
equal to 31 (38 – 7 control loops). In order to define the system completely, 31
independent variables are needed to be specified. The defined variables are:
1. 28 trays pressure. Pressure varies linearly from top tray to bottom column
2. Reflux flowrate
3. Pumparound flowrate
4. Reboiler duty
56
4.5
Formulations of the Distillation Column Models
Mathematical models for distillation columns can be classified into two
categories (Hurowitz et al., 2003). The first category is ‘rigorous’ models which
consider the details of composition, temperature and flow on each plate. The second
category is based on some kind of interpolation which considers an overall
description of the column using fewer variables. In this study, rigorous column
model is chosen.
The mathematical models used in dynamic simulation should be as general as
possible so that it can handle a wide range of problems and processes. The model
starts from basic mass and energy balances as well as the equilibrium relationship
between phases. Various forms of MESH equations have appeared in the literature.
The arrangement of the sets of equations depended on the methods adopted in
solving the problems. In addition, the equations can be written in steady state form.
Other terms like tray efficiency, pump-around system, chemical reaction and thermal
efficiency can also be introduced in these MESH equations. Besides these MESH
equations (Wang and Henke, 1966), correlations are also needed to determine
equilibrium ratio, K’s, liquid enthalpy, h’s, vapor enthalpy, H’s, Francis weir
equations and hydraulic equations. The following section systematically presents the
distillation column dynamic modeling.
4.5.1
Mass Balance Equations
The mass balance equations involve total flow rate balance and components
flow rate balance for each tray. The equations are written in derivative form in order
to reveal the dynamic behaviors of the column. In order to get a better picture of
mass balance for reflux drum, pumparound drum, normal tray, sidedraw tray, reflux
tray, feed tray and bottom column, a distillation column schematic diagram is
presented in Figure 4.3.
57
To vent
Tray 1
V2, y2,i
Tray 2
V3, y3,i
Tray 3
Pumparound
P, xP,i
Sidedraw
S, xs,i
Liquid to
pumparound
L2, x2,i L, xL,i
Reflux Re, xRe,i
Distillate, D
Cooler
Tray 13
Tray 14
Liquid feed, Lf, xf, i
Inlet trays schematic
Tray 15
Tray N-1
VN, yN, i
LN-1, xN-1, i
Tray N
VN+1, yN+1, i
LN, xN, i
Tray N+1
Normal tray schematic
Tray 28
L29
V29
Reboiler
LH, B
Bottom Column
(Tray 29)
Bottom flow
rate, B
Bottom molar
hold up, MB
Bottom tray schematic
Figure 4.3: Distillation column schematic diagram
58
4.5.1.1 Bottom column model
The bottom column models consist of the following equations:
i)
Bottom flow rate has a very close relationship with the bottom liquid
level.
Bottom flow rate, B =
ii)
16.778ρL H , B
0.5
4.2
MW
The bottom molar hold-up relates to the bottom liquid level.
Bottom molar hold-up, MB =
L H , B AB ρ
4.3
MW
9
Liquid density, ρ = ∑ x i ρ i
4.4
i =1
iii)
Bottom height derivative expression
dL H , B
dt
iv)
=
L N −1 − V N − B
Aρ
( b )
MW
4.5
Components mass balance in term of mole fraction derivative expression
dx N ,i
dt
=
L N −1 x N −1,i − V N y N ,i − Bxi
Aρ
L H ,b ( b )
MW
4.6
where:
ρ
= liquid density
N
= number of trays
MW
= molecular weight
yi
= component vapor mole fraction
LH, B
= bottom liquid level
MB
= bottom molar hold up
xi
= component liquid mole
AB
= Bottom column cross sectional
fraction
VN
= vapor flow rate at tray-N
area
B
= bottom flow rate
59
4.5.1.2 Trays model
The pumparound tray (N=1) model involves the following equations:
i)
Total mass balance
dM 1
= P + V 2 − L1 − V1
dt
ii)
4.7
Component mass balance
dxi Px P ,i + V2 y 2,i − L1 x1,i − V1 x1,i
=
dt
MN
4.8
The sidedraw tray (N = 2) models involve the following equations:
i)
Total mass balance for sidedraw
dM 2
= L1 + V3 − S − L2 − V 2
dt
ii)
4.9
Component mass balance for sidedraw
dx i L1 x1,i + V3 y 3,i − Sx s − V 2 y 2,i − L2 x 2,i
=
dt
MN
4.10
The reflux tray (N = 3) models involve the following equations:
i)
Total mass balance for reflux
dM 3
= L2 + V 4 + Re − L3 − V3
dt
ii)
4.11
Component mass balance for reflux
dx i L2 x 2,i + V 4 y 4,i + Re x Re,i − L3 x 3,i − V3 y 3,i
=
dt
MN
4.12
60
The normal trays (N = 4 to 13 and 15 to 28) models involve the following
equations:
i)
Total mass balance
dM N
= LN −1 + VN +1 − LN − VN
dt
ii)
4.13
Component mass balance
dxi LN −1 xN −1,i + VN +1 y N +1,i − VN y N ,i − LN xN ,i
=
dt
MN
4.14
The feed trays (N = 14) models involve the following equations:
i)
Total mass balance for liquid feed
dM N
= LN −1 + VN +1 + L f − LN − VN
dt
ii)
4.15
Component mass balance for liquid feed tray
dxN ,i
dt
=
LN −1 xN −1,i + VN +1 y N +1,i + L f x f ,i − VN y N − LN xN ,i
MN
4.16
where:
Re
= reflux flowrate
P
= pumparound liquid flowrate
xRe
= reflux mole fraction
xP
= pumparound liquid mole fraction
MN
= molar hold up at tray N
LN
= liquid flow rate at tray-N
S
= sidedraw flowrate
xi
= component liquid mole fraction
xS
= sidedraw mole fraction
VN
= vapor flow rate at tray-N
Lf
= liquid feed flow rate
yi
= component vapor mole fraction
xf
= liquid feed mole fraction
N
= number of trays
61
4.5.1.3 Reflux drum model
The reflux drum models are similar to the bottom column model. The reflux
drum models consist of the following equations:
i)
The liquid stream flow rate has a very close relationship with the reflux
drum liquid level.
Liquid flowrate, L =
ii)
15.02 ρL H ,Re
0.5
4.17
MW
The reflux drum molar hold-up relates to the reflux drum liquid level.
Reflux drum molar hold-up, MRe =
L H , Re ARe ρ
4.18
MW
9
Liquid density, ρ = ∑ x i ρ i
4.19
i =1
iii)
Reflux drum height derivative expression
dL H , Re
dt
iv)
=
S − Re− L
A ρ
( Re )
MW
4.20
Components mass balance in term of mole fraction derivative expression
dx N ,i
dt
=
Sx s ,i − Re x Re,i − Lx L ,i
A ρ
L H , Re ( Re )
MW
4.21
where:
ρ
= liquid density
Re
= reflux flowrate
MW = molecular weight
xRe
= reflux mole fraction
S
= sidedraw flowrate
L
= liquid flow rate
xS
= sidedraw mole fraction
LH, Re
= Reflux drum liquid level
ARe
= Reflux drum cross sectional area
xi
= component liquid mole fraction
MRe
= reflux drum molar hold up
62
4.5.1.4 Pumparound drum model
The pumparound drum model consists of the following equations:
ii)
The distillate flow rate has a very close relationship with the pumparound
drum liquid level.
Distillate flowrate, L =
ii)
1.12 ρL H , P
0.5
4.22
MW
The pumparound drum molar hold-up relates to the pumparound drum
liquid level.
Pumparound drum molar hold-up, MP =
L H , P AP ρ
MW
4.23
9
Liquid density, ρ = ∑ x i ρ i
4.24
i =1
v)
Pumparound drum height derivative expression
dL H , P
dt
vi)
=
L−P−D
A ρ
( P )
MW
4.25
Components mass balance in term of mole fraction derivative expression
dx N ,i
dt
=
Lx L ,i − Px P ,i − Dx D ,i
A ρ
LH , P ( P )
MW
4.26
where:
ρ
= liquid density
P
= pumparound liquid flowrate
MW
= molecular weight
xP
= pumparound liquid mole fraction
LH, P
= pumparound drum liquid level
D
= distillate flow rate
AP
= pumparound cross sectional area
xD
= distillate mole fraction
MP
= pumparound drum hold up
63
4.5.2
Equilibrium Equations
Equilibrium equations involve using appropriate thermodynamic equations to
determine the phase equilibrium between vapor and liquid phases. Indirectly, the Eequations are used to determine the liquid phase compositions, vapor phase
compositions and temperature or bubble point of the tray.
The thermodynamic equations used for describing phase equilibrium and
bubble point calculations are based on the Wilson correlation (Walas, 1985) for the
liquid phase and the Virial correlation (Smith et al., 1996) for the vapor phase. The
Virial correlation is used to calculate the fugacity coefficient of the vapor phase
because the system pressure is less than atmospheric pressure. The correlation gives
a good approximation in this operating region (Smith et al., 1996). The Wilson
correlation is used to determine the activity coefficient of the liquid phase since the
system exhibits non-ideality behavior (Reid et al., 1987).
The Virial correlation is determined using the following equations.
ln φ k =
⎤
Ptot ⎡
1 n n
⎢ B kk + ∑ ∑ y i y j (2δ ik − δ ij )⎥
RT ⎣
2 i =1 j =1
⎦
4.27
δ ik = 2Bik − Bii − Bkk
4.28
δ ij = 2 Bij − Bii − B jj
4.29
where δii = 0, δkk = 0, δik = δki
Bij =
RTcij
Pcij
( Bijo + ω ij Bij1 )
4.30
where,
Bijo = 0.083 −
0.422
Trij1.6
4.31
Bij1 = 0.139 −
0.172
Trij4.2
4.32
64
Trij =
ω ij =
T
Tcij
4.33
ωi + ω j
4.34
2
Tcij = Tci Tcj
Pcij =
Z cij =
Vcij
4.35
Z cij RTcij
4.36
Vcij
Z ci + Z cj
4.37
2
⎛ 3 Vci + 3 Vcj ⎞
⎟
=⎜
⎜
⎟
2
⎝
⎠
3
4.38
Ptot
= total pressure of the system
Vci
= critical vol. for component i
T
= temperature of the system
Tri
= reduced temperature of
component i
R
= universal gas constant
n
= number of components
Tci
= critical temperature of component i
yi
= mole fraction of component i
Pci
= critical pressure of component i
φi
= vapor fugacity coefficient for
component i
Zci
= critical compressibility factor for
component i
ω
= critical accentric factor of
component i
65
The Wilson correlation is determined using the following equations:
n
xk Λ ki
n
ln γ i = 1 − ln ∑ x j Λ ij − ∑
j =1
k =1
∑x Λ
j =1
Λ ij =
⎛ − aij
exp⎜⎜
Vi
⎝ RT
Vj
4.39
n
j
kj
⎞
⎟⎟
⎠
4.40
where,
Λij = 1 for i=j
Λij ≠ Λji
γi
= liquid phase activity coefficient for
Λij
component i
xi
= liquid mole fraction for component i
aij
= Wilson constant
= Wilson binary interaction
parameter for component i and j
Vi
= liquid molar volume of
component i
The relationship between the liquid mole fraction and vapor mole fraction for
component i is shown below
yiφi Ptot = xi γ i Pi sat
4.41
where,
Pi sat = vapor pressure for component i
Equation 4.41 is the fundamental vapor liquid equilibrium that will be used in
this study. The bubble point calculation is based on this equation.
66
4.5.2.1 Bubble Point Calculations
The tray temperature depends on the tray pressure, liquid compositions and
vapor compositions. The bubble point calculations by using thermodynamic methods
are known as gamma-phi approach because the liquid and vapor phase coefficient are
determined using two different equations. The purpose of the bubble point
calculation is to determine the vapor phase compositions and temperature for a given
pressure and liquid phase compositions. Bubble point calculations require initial
guess of temperature because the calculations involve iterations. Good guess will
cause fast convergence to the solution. The procedure of bubble point calculations is
presented in Figure 4.4.
67
Fix: Pressure, P and liquid composition xi.Guess: Temperature
Set: φ i = 1 γ i = 1 Identify: Component j
Determine Psat for each component. Antoine Equation :
Pi sat (mmHg ) = Ai −
Bi
T ( K ) − Ci
4.42
Calculate γ i using equation 4.39 and 4.40
Calculate :
Pjsat =
Ptot
4.43
⎛ x i γ i ⎞⎛⎜ Pi
⎟⎟ sat
⎜
i =1 ⎝ φ i ⎠⎝ P j
sat
n
∑ ⎜⎜
T=
Bj
A j − ln Pjsat
⎞
⎟
⎟
⎠
−Cj
4.44
Recalculate: Pi sat (equa. 4.42) and T (equa.4.44)
Calculate: yi (equa. 4.41), φi (equa. 4.27 – 4.38), γ i (equa. 4.39– 4.40)
Recalculate: Pjsat (equa. 4.43) and Tnew
(equa.4.44)
Check: Tnew ~ T
Yes
Solution reach
Figure 4.4: Bubble point calculation procedure
No
68
4.5.3 Summation Equation
Summation Equations are used to determine the vapor phase and liquid phase
compositions.
8
x9 = 1 − ∑ xi
4.45
i =1
8
y9 = 1 − ∑ yi
4.46
i =1
4.5.4 Heat Balance Equation
The heat balance equations are used to determine the vapor flow rate leaving
each tray. The reference temperature is at 273K. The liquid enthalpy and vapor
enthalpy are calculated using the following equations:
Liquid enthalpy for a component at tray-N,
T
hi, N =
∫A
*
i
+ Bi*T + Ci*T 2 + Di*T 3 dT
273
T
⎡ *
Bi*T 2 C i*T 3 Di*T 4 ⎤
= ⎢ Ai T +
+
+
⎥
2
3
4 ⎦ 273
⎣
4.47
Vapor enthalpy for a component at tray-N,
T
Hi, N =
∫A
*
i
+ Bi*T + Ci*T 2 + Di*T 3 dT + ∆H vap ,i
4.48
273
Heat of Vaporization, ∆H vap,i
⎡ T −T ⎤
= ∆H vap , n , i ⎢ c , i
⎥
⎢⎣ Tc , i − Tb , i ⎥⎦
0.38
4.49
9
Liquid enthalpy leaving each tray, hN =
∑x h
i =1
i
i, N
4.50
69
9
Vapor enthalpy leaving each tray, HN =
∑y H
i =1
Vapor stream leaving each tray, VN =
i, N
LN −1hN −1 + VN +1H N +1 − LN hN
HN
Vapor stream leaving the reboiler, V29 =
Where:
i
Q R + ( L28 − B)h B
H 29
4.51
4.52
4.53
Ai* , Bi* , C i* , Di*
= liquid heat capacity constant of component i
∆H vap
= heat of vaporization
∆H vap, n
= heat of vaporization at normal boiling point
Tb,i
= boiling point
L28
= liquid flowrate at tray 28
hb
= enthalpy for bottom stream
QR
= reboiler duty
4.5.5 Hydraulic Equations
Francis Weir hydraulic equations are used to determine the liquid flow rate
leaving each tray and tray molar hold-up. Liquid flow rate leaving each tray depends
on the over weir height, how. The involved hydraulic equations are shown as follows:
M N ( MW )
ρ ( At )
Liquid level, LH, N
=
Overwier height, how
= LH − WL
⎛h ⎞
Tray liquid flow rate, LN = ⎜ ow ⎟
⎝ 750 ⎠
3/ 2
4.54
4.55
⎛ ρ (3600) ⎞
⎜⎜
⎟⎟
⎝ WL ( MW ) ⎠
4.56
70
where:
how
= over weir height
WL
= weir length
LH, N
= liquid level at tray-N
At
= tray active area
4.6 Dynamic Simulation of the Study Column
The development of dynamic simulated column involves three different
stages, which are pre-simulation, algorithm formulation and program formulation.
Pre-simulation activities consist of the following tasks:
a. Understanding the assumptions that are used in simulating the column. The
assumptions adopted in simulating the column are,
1. Liquid on the tray is perfectly mixed and incompressible. Compositions
of the liquid stream leaving the tray will be the same as the tray molar
hold up compositions.
2. Tray vapor hold ups are negligible because the column operating
pressures are less than 10 bars.
3. Vapor and liquid are in thermal equilibrium. The temperature for both
liquid vapor streams leaving the tray is similar.
4. Vapor and liquid are in the equilibrium phase.
5. Pressure is constant in each tray but varies linearly up the column from
bottom pressure to the top column pressure or in other words, pressure
drop for each tray is the same.
6. Coolant and steam dynamics are negligible.
7. Dynamic changes in internal energy on the trays are negligible compared
with the latent heat effects. So, energy balance on each tray is just
algebraic equations.
71
b. The next task is to acquire the inlet stream conditions. It is based on the data
from Palm Oil Fractionation Plant. The data in Table 4.2 was sufficient
enough for the inlet stream.
c. The study column is simulated using Design II simulator to obtain a set of
steady state results that comprises of the profile of the following column
parameters:
1.
Column temperature profile
2.
Column liquid and vapor flow rate profile
3.
Trays liquid and vapor composition profile
4.
Liquid and vapor enthalpy profile
5.
Bottom and distillate stream flow rate and compositions.
6.
Sidedraw, pumparound and reflux stream flowrate and compositions
Column temperature and liquid compositions profiles are used as the initial
guess for the dynamic simulation.
d. There are 31 DoF which means that 31 variables need to be fixed in order to
achieve the desired bottom and distillate flowrate which is 37.5kmole/hr and
4kmole/hr respectively. The fix variables are:
1.
28 tray pressure with each tray encounters 0.00095 bar pressure drop.
2.
Reflux flow rate (10 kmole/hr)
3.
Pumparound flowrate (49.6 kmole/hr)
4.
Reboiler duty (2.1576 x 106 kJ/h)
72
Algorithm formulation is to establish a proper solution sequence, which can
reveal the column behaviors. MESH equations and hydraulic equations are used to
model the column that describes the column behavior. The systematic procedures in
formulating the simulated column are elaborated as follows,
ƒ
First: The algorithm for the bottom column to the 4th tray
1. Initialize the following derivatives variables.
i. Bottom liquid level and compositions : 9 derivative variables
ii. Each tray molar hold-up and compositions : 225 derivative variables
There are 234 derivatives variables that have to be initialized since there are 234
derivative equations.
2. Calculate the bubble point from the bottom tray till 4th tray.
The tray pressure is fixed but the tray temperature and liquid compositions
are guessed. The bottom pressure is fixed at 0.133 bar and 0.00095 pressure
drop at each tray. The guessing values are referred from the Design II steadystate simulation results. The bubble point calculations are used to determine
the tray temperature and the vapor phase compositions.
3. Calculate the liquid and vapor enthalpy from the bottom tray till 4th tray.
Based on the TN, xi, N and yi, N as calculated in the previous step, liquid and
vapor enthalpy for each tray are determined via Equation 4.47 to 4.49.
4. Calculate the liquid flow rate using Francis Weir equations.
From the initial bottom liquid level, trays molar hold-up and reflux drum
liquid level, the following variables are determined from bottom stage till 4th
stage:
i)
Bottom flow rate by Equation 4.2
ii)
Liquid leaving each tray by Equation 4.54 – 4.56.
73
5. Calculate the vapor flow rate leaving each tray by energy balance.
Based on the calculated liquid and vapor enthalpy, the vapor flow rate leaving
each tray is determined by Equation 4.53 for reboiler stage and Equation 4.52
for other trays.
6. Evaluate the molar hold up and components composition derivatives
variables. For every time increment, h = 0.01 hour, the evaluation for each
derivative variable is started from the bottom stage till 4th stage using RungeKutta fourth order method.
7. Increase time step.
After all derivative variables at time = t are evaluated from the first algorithm
and the second algorithm is to continue the calculation steps until time = tend
Second: The algorithm for the 3rd tray to reflux drum, pumparound and the top
tray
1. Start at the reflux drum.
Initialize the derivative variables as in step 1 in first algorithm. Perform
bubble point calculation. Calculate the liquid and vapor enthalpy for all the
streams. Determine the liquid stream flowrate via Equation 4.17.
2. Pumparound drum.
The procedure as step 1 with the distillate flowrate calculated using Equation
4.22.
3. The step for 3rd tray to 1st tray is similar to first algorithm.
74
After the algorithm formulation, then the written program is performed and
has the following features:
1. Process variables like feed stream flow rate, feed stream temperature,
feed stream pressure, distillate pressure, bottom column pressure, reflux
flow rate, pumparound flowrate and reboiler duty are having randomly
generated values. In the commercial simulators, these variables can only
accept fix value.
2. These process variables have autocorrelation effect, which is one of the
industrial data characteristics.
3. The developed program has main program and sub-programs. The main
program merely contains sub-programs executing sequences. The
following calculations are written in sub-programs:
ƒ
Bubble point calculations.
ƒ
Thermodynamic model for vapor and liquid equilibrium
calculations
4.7
ƒ
Liquid and vapor enthalpy calculations.
ƒ
Runge Kutta fourth order calculations.
Controller Tuning
There are seven control loops in the study column. In order to be able to use
these controllers, it must first be tuned to the system. This tuning synchronizes the
controller with controlled variable, thus allowing the process to be kept at the desired
operating condition. The empirical tuning method, which is Process reaction curve
method developed by Cohen and Coon is used to tune the controllers. This method
used the open loop in which the control loop is broken between the system
measurement and the controller. A step change is introduced in the input variable at
steady state during normal operating condition. The output is recorded and the
parameters static gain, Kp, time constant, τ, and dead time, θ is determined and
recorded. Dead time describes how long it takes for the process to start responding
while the time constant is the time required to reach 63.2% of the full response to a
75
step change. These parameters are used to determine the controller settings using the
equations in Table 4.6,
Table 4.6: Cohen-Coon controller settings
Type of controller
Proportional, P
Proportional-Integral, PI
Proportional-IntegralDerivative, PID
Controller Gain,
Integral Time
Derivative Time
Kc
Constant, τ I
Constant, τ D
1 ⎛ τ ⎞⎛ θ ⎞
⎜ ⎟⎜1+ ⎟
K p ⎝ θ ⎠⎝ 3τ ⎠
θ ⎞
1 ⎛ τ ⎞⎛
⎜ ⎟⎜ 0.9 +
⎟
K p ⎝ θ ⎠⎝
12τ ⎠
θ⎜
1 ⎛ τ ⎞⎛ 4 θ ⎞
⎜ ⎟⎜ + ⎟
K p ⎝ θ ⎠⎝ 3 4τ ⎠
θ⎜
⎛ 30 + 3θ / τ ⎞
⎟
⎝ 9 + 20θ / τ ⎠
⎛ 32 + 6θ / τ ⎞
⎟
⎝ 13 + 8θ / τ ⎠
4
⎛
⎞
⎟
⎝ 11 + 2θ / τ ⎠
θ⎜
Sections 4.7.1 to 4.7.7 show the controller properties of each control loops.
The controller sampling time, TAPC also known as Automatic Process Control (APC)
time sampling is determined using the guideline proposed by Ogunnaike and Ray
(1994). The details is discussed in Section 4.7.8.
4.7.1
Bottom Liquid Level Controller
The control variable for the bottom liquid level control loop is the bottom
liquid level while the proposed manipulated variable is the bottom flow rate. The
value of the bottom flow rate is varied (various step changes) and the parameters
static gain, Kp, time constant, τ, and dead time, θ for each change of the value of the
bottom flow rate is recorded and shown in Table 4.7. However, only small changes
of the value of the bottom flow rate are carried out as the developed Matlab program
will fail to converge for large changes in the value of the bottom flow rate.
76
Table 4.7: The step change results for bottom liquid level controller
Change in flow
Static Gain, Kp,
Time Constant, τ,
Dead Time, θ ,
rate of bottom
m/(kmole/hr)
hour
hour
+0.5% of SS value
-1.048
0.1
0.01
+1.0% of SS value
-1.047
0.1
0.01
-0.5% of SS value
-1.048
0.1
0.01
-1.0% of SS value
-1.048
0.1
0.01
Average Value
-1.048
0.1
0.01
stream
The controller used for controlling the bottom liquid level is Proportional (P)
controller. Based on the average value of Kp, τ, and θ from Table 4.7, the controller
parameter for this loop Controller Gain, Kc is determined. The calculated parameter
and information of the bottom liquid level P controller are shown in Table 4.8.
Table 4.8: Properties of bottom liquid level P controller
Controller
Control
Manipulated
Kc,
Type
Variable
Variable
(kmole/hr)/m
P
Bottom
Bottom
-9.86
liquid level,
flow rate, B
LH, B
4.7.2
Bottom Temperature Controller
The reboiler of the study column is assumed to be a total reboiler with hot oil
as the heating fluid for the reboiler system. The control variable in the bottom
temperature control loop is the bottom temperature and the proposed manipulated
variable is the flow rate of the hot oil in the reboiler system. The hot oil is assumed
to have a heat capacity, Cp = 7.5 kJ/kg.K (Geankoplis, 1993) and a change in
temperature, ∆T = 40K. These two parameters are assumed to be constant. The
steady-state (SS) value of the hot oil flow rate is 7192 kg/hr. The flow rate of the hot
77
oil is varied (various step changes) and the static gain, Kp, time constant, τ, and dead
time, θ , for each change of the value of the flow rate of the hot oil is recorded and
shown in Table 4.9.
Table 4.9: The step change results for bottom temperature controller
Change in flow
Static Gain, Kp,
Time Constant, τ,
Dead Time, θ ,
rate of hot oil
K/(kg/hr)
hour
hour
+5% of SS value
0.038
0.3
0.06
+10% of SS value
0.038
0.3
0.06
-5% of SS value
0.038
0.2
0.06
-10% of SS value
0.038
0.3
0.06
Average Value
0.038
0.3
0.06
The controller used for controlling the bottom temperature is ProportionalIntegral-Derivative (PID) controller. Using the average value of Kp, τ, and θ from
Table 4.9, the controller parameters such as Controller Gain, Kc, Integral Time
Constant, τI, and Derivative Time Constant, τD, are calculated.
The
calculated
parameters and information of the bottom temperature PID controller are shown in
Table 4.10.
Table 4.10: Properties of bottom temperature PID controller
Controller
Control
Manipulated
Kc,
Type
Variable
Variable
(kg/hr)/K
PID
Bottom
Hot oil flow
179.87
temperature,
rate, Fhot
Tbot
τI, hour
τD, hour
0.13
0.02
78
4.7.3
Top Column Pressure Controller
The control variable for the top column pressure controller loop is the top
column pressure while the proposed manipulated variable is the top vapor flow rate.
The value of the top vapor flow rate is varied (various step changes) and Kp, τ, and θ
for each change of the value of the top vapor flow rate is recorded and shown in
Table 4.11.
Table 4.11: The step change results for top column pressure controller
Change in flow
Static Gain, Kp,
Time Constant, τ,
Dead Time, θ ,
rate of top vapor
bar/(kmole/hr)
hour
hour
+5% of SS value
-9.6
1.0
0.1
+10% of SS value
-9.6
1.0
0.1
-5% of SS value
-9.6
1.0
0.1
-10% of SS value
-9.6
1.0
0.1
Average Value
-9.6
1.0
0.1
stream
The controller used for controlling the top column pressure is ProportionalIntegral (PI) controller. Using the average value of Kp, τ, and td from Table 4.11, the
controller parameters such as Kc and τI are calculated. The calculated parameters and
information of the top pressure PI controller are shown in Table 4.12.
Table 4.12: Properties of top column pressure PI controller
Controller
Control
Manipulated
Kc,
Type
Variable
Variable
(kmole/hr)/Bar
PI
Top column
Top vapor
-0.95
pressure, PTop
flow rate, Vtop
τI, hour
0.28
79
4.7.4
Sidedraw Tray Liquid Level Controller
The control variable for the sidedraw tray liquid level control loop is the
sidedraw tray liquid level while the proposed manipulated variable is the distillate
flow rate. The value of the distillate flow rate is varied (various step changes) and
Kp, τ, and θ for each change of the value of the distillate flow rate is recorded and
shown in Table 4.13.
Table 4.13: The step change results for sidedraw tray liquid level controller
Change in flow
Static Gain, Kp,
Time Constant, τ,
Dead Time, θ ,
rate of bottom
m/(kmole/hr)
hour
hour
+5% of SS value
-0.023
0.2
0.05
+10% of SS value
-0.023
0.2
0.05
-5% of SS value
-0.021
0.2
0.05
-10% of SS value
-0.022
0.2
0.05
Average Value
-0.022
0.2
0.05
stream
The controller used for controlling the sidedraw liquid level is P controller.
Using the average value of Kp, τ, and θ from Table 4.13, the Kc of this controller is
calculated. The calculated parameter and information of the sidedraw tray liquid
level P controller are shown in Table 4.14.
Table 4.14: Properties of sidedraw tray liquid level P controller
Controller
Control
Manipulated
Kc,
Type
Variable
Variable
(kmole/hr)/m
P
Sidedraw
Distillate
-193.89
tray liquid
flow rate, D
level, LH, Sd
80
4.7.5 Pumparound Temperature Controller
The control variable for the pumparound temperature controller is the
temperature of the pumparound stream while the proposed manipulated variable is
the flow rate of the cooling water in the pumparound cooler system. The water has a
heat capacity Cp = 4.184 kJ/kg.K (Geankoplis, 1993) and a change in temperature,
∆T = 10K. These two parameters are assumed to be constant and the steady-state
(SS) value of the cooling water flow rate is 411.57 kmole/hr. The value of the
cooling water flow rate is varied (various step changes) and Kp, τ, and θ for each
change of the value of the cooling water flow rate is recorded and shown in Table
4.15.
Table 4.15: The step change results for pumparound temperature controller
Change in flow
Static Gain, Kp,
Time Constant, τ,
Dead Time, θ ,
rate of cooling
K/(kmole/hr)
hour
hour
+5% of SS value
-0.303
1.1
0.1
+10% of SS value
-0.303
1.1
0.1
-5% of SS value
-0.303
1.1
0.1
-10% of SS value
-0.303
1.1
0.1
Average Value
-0.303
1.1
0.1
water
The controller used for controlling the pumparound temperature is PID
controller. Using the average value of Kp, τ, and θ from Table 4.15, the controller
parameters such as Kc, τI and τD are calculated. The calculated parameters and
information of the pumparound temperature PID controller are shown in Table 4.16.
81
Table 4.16: Properties of pumparound temperature PID controller
Controller
Control
Manipulated
Kc,
Type
Variable
Variable
(kmole/hr)/K
PID
Pumparound
Cooling
22.71
temperature,
water flow
TP
rate, Fcw
4.7.6
τI, hour
τD, hour
0.46
0.07
Pumparound Flow Rate Controller
The control variable for the pumparound flow rate control loop is the
pumparound flow rate while the proposed manipulated variable is the liquid flow rate
prior to the pumparound stream.
Since this liquid stream directly affects the
pumparound flow rate and is in the same streamline with the pumparound flow rate,
no step changes were being carried out to determine the value of Kp, τ, and θ of this
control loop. A simple P controller with a Kc = 1.0 is used for this control loop. The
parameter and information of pumparound flow rate controller are shown in Table
4.17.
Table 4.17: Properties of pumparound flow rate P controller
Controller
Control
Manipulated
Type
Variable
Variable
P
Pumparound
Liquid flow rate
flow rate, P
prior to
pumparound stream
Kc
1.0
82
4.7.7
Reflux Flow Rate Controller
The control variable for the reflux flow rate control loop is the reflux flow
rate while the proposed manipulated variable is the liquid flow rate prior to the reflux
stream. Since this liquid stream directly affects the reflux flow rate and is in the
same streamline with the reflux flow rate, no step changes were being carried out to
determine the value of Kp, τ, and θ of this control loop. A simple P controller with a
Kc = 1.0 is used for this control loop. The parameter and information of reflux flow
rate controller are shown in Table 4.18.
Table 4.18: Properties of reflux flow rate P controller
Controller
Control
Manipulated
Type
Variable
Variable
P
Reflux flow
Liquid flow rate
rate, Re
prior to reflux
Kc
1.0
stream
4.7.8
Controller Sampling Time, TAPC
The controller sampling time, TAPC, of the study column is determined using
the guideline proposed by Ogunnaike and Ray (1994). They proposed using one
tenth to one twentieth of smallest time constant of a process for its controller
sampling time. From the controller tuning results obtained from Section 4.7.1 to
Section 4.7.7, the bottom liquid level-bottom flow rate process section of the study
column process has the smallest time constant, τ = 0.1 hour. Therefore, the controller
sampling time of the study column is set at TAPC = 0.01 hour (one tenth of the
smallest time constant of the study column process).
83
4.8
Dynamic Simulation Program Performance Evaluation
The performances of the written algorithm and program are evaluated based
on the ability of the developed algorithm and program to achieve steady state results
that close to the steady state results from the HYSYS simulator. For each aspect
comparison, the HYSYS simulator results are used as the basis. Percentage
difference is calculated as follow:
% Difference =[(HYSYS result – Matlab result)/ (HYSYS result)] x 100%
The steady state results comparisons for the temperature, pressure, liquid
flowrate, vapor flowrate, distillate flowrate and bottom flowrate are presented in
Appendix A. The result obtained from the comparison of bottom and distillate
composition, the maximum percent different was about 4%. It shows that the written
program is able to achieve bottom flow rate and distillate flow rate that are close to
the commercial simulator. On the other hand, the maximum percentage different for
each temperature and pressure column profile is less than 10%.
Therefore, it can be concluded that the developed dynamic simulation
program is able to produce results that close to the commercial steady state
simulators. Thus, the developed dynamic simulation program is ready to be used to
generate a set of nominal operation condition (NOC) data and a set of disturbance
data.
84
4.9
Chapter Summary
Distillation column is chosen as unit operation for the proposed FDD method.
Process description of the pre cut column from the Palm Oil Fractionation Plant and
the information of this column is included. The column contains nine components.
Prior to performing the dynamic modeling, DoF analysis was performed. There are
31 DoF for the study column. The fix variables are 28 tray pressures (pressure varies
linearly from top tray to bottom column), reflux flowrate, pumparound flowrate and
reboiler duty.
Formulations of the distillation column model require MESH equation and
hydraulic equations. The column model can be divided into eight sections i.e. reflux
drum, pumparound drum, normal tray, sidedraw tray, reflux tray, feed tray and
bottom column. The equilibrium equation is based on Wilson Correlation and Virial
Correlation to determine activity coefficient and fugacity coefficient of the liquid and
vapor respectively. The heat balance equations are used to determine the vapor flow
rate leaving each tray. Francis Weir hydraulic equations are used to determine the
liquid flow rate leaving each tray and tray molar hold-up.
The development of dynamic simulated column involved three stages, which
are pre-simulation, algorithm formulation and program formulation. Pre-simulation
activities consist of understanding the assumptions that are used in simulating the
column, acquiring the inlet stream conditions, simulating the study column using
simulator Design II and specifying the desired separation flowrates and fixing the
degree of freedom. Algorithm formulation is to establish a proper solution sequence,
which can reveal the column behaviors. After completing the algorithm formulation,
the written program performs using MATLAB software.
The empirical tuning method, which is Process reaction curve method
developed by Cohen and Coon is used to tune the controllers. The controller settings
for various controller types for each control loop have been calculated. Controller
85
sampling time, TAPC = 0.01 hour was determined using one tenth of the smallest time
constant of the study column process.
CHAPTER 5
METHODOLOGY
5.1
Introduction
The meaning of monitoring in this work refers to the condition where
measurable variables are checked with the limits, and alarms are generated when out
of control condition occur. The objective of monitoring in the study column is to
produce the bottom composition of oleic acid and linoleic acid between 0.134
kmol/kmol to 0.135 kmol/kmol and 0.024 kmol/kmol to 0.025 kmol/kmol. Faulty
conditions were detected when these variables exceeded the control limits.
Once fault has been detected, the next step is to determine the cause of the
out of control situation. Therefore, focusing the plant operators and engineers on the
subsystem(s) would most likely know where the fault occurred. This assistance
provided by the fault diagnosis scheme in locating the fault can effectively
incorporate the operators and engineers in the monitoring process scheme and
significantly reduce the time to recover in-control operations. The task of diagnosing
the fault can be rather challenging when the number of process variables is large, and
the process is highly integrated. Proper and immediate actions to variables which
cause the fault can prevent the effect of the fault from propagating to other variables
(Chiang and Braatz, 2003).
In this research, a precut multi component distillation is used as the study unit
operation. The monitoring proposed method also can be applied to other unit
operations as well. This simulated column is developed using MATLAB.
87
5.2
Variable Selection for Process Monitoring
There are seven control loops installed in the study column. These control
loops controlled the process by counteracts the effects of disturbances change and set
point change to maintain the process at the target. This concept is known as
Automatic Process Control (APC), which uses continuous adjustment on
manipulated variables to keep the process on target. APC variables are divided into
three groups i.e controlled variable, manipulated variable, and measured disturbance.
Table 5.1 lists APC variables for each control loop of the study column.
Table 5.1: APC variables for each control loops
Control variable
Manipulated variable
Measured disturbance
Bottom column liquid level
Bottom flowrate
Liquid flowrate at tray 28
Bottom temperature
Hot utility flowrate
Liquid flowrate at tray 28
Top column pressure
Vapor flowrate to vent
Vapor flowrate at tray 2
Side draw tray liquid level
Distillate flowrate
Liquid flowrate at tray 1
Pumparound temperature
Cold utility flowrate
Pumparound point
Pumparound flowrate
Distillate flowrate
Pumparound point
Reflux flowrate
Reflux flowrate
Sidedraw flowrate
In contrast, SPC does not control the process but rather performs a monitoring
function and gives signals when maintenance is needed in the form of identification
and removal the root causes. SPC seeks the deviation in the process by detecting the
changes in monitoring variables when the assignable causes occur in the process.
SPC variables can be categorized into two i.e quality variables and process variables.
Quality variables consist of oleic acid composition, xc8 and linoleic acid composition,
xc9 at bottom stream. A few variables were listed for process variables selection.
These variables are feed flowrate, feed temperature, reflux flowrate, pumparound
flowrate, pumparound temperature, reboiler heat duty, sidedraw flowrate and
distillate flowrate. Normal correlation coefficient is calculated to analyze these
variables quantitatively. Variables which have correlation coefficient greater than
0.05 were selected. Correlation coefficient which is less than 0.05 is considered
small. Changes in process variables which have small correlation coefficient do not
88
affect quality variables significantly. Using small correlation coefficient can cause
the control limits for process variables become wider. Based on the results obtained
as shown in Table 5.2, five process variables have correlation coefficient greater than
0.5.
Table 5.2: Normal correlation coefficient for process variables
Process variables
Correlation coefficient
Feed flowrate
0.610
Feed temperature
0.340
Reflux flowrate
0.522
Pumparound flowrate
0.490
Reboiler heat duty
0.050
Pumparound temperature
0.030
Sidedraw flowrate
0.005
Distillate flowrate
0.001
These process variables were selected from five locations i.e feed stream,
pumparound stream, reflux stream and bottom stream. Feed flowrate and feed
temperature of the study column are the overall disturbances. Even though the study
precut column has been installed with controllers, the probability of faults to occur is
possible due to line blockage, line leakage and sensor fault. Pumparound and reflux
flowrates are the controlled variables in APC system. These variables are maintained
at set point values. The controllers will take action if any changes occur due to
measured disturbances and set points. Both variables are set on target values as long
as the controllers are functioning well or there is no fault exists. Even though these
two variables have been controlled, the possibility of faults to occur is possible due
to line blockage, line leakage, sensor faults, valve sticking and controller fault.
Therefore, these variables are included in monitoring process to identify the causes
of faults.
89
Reboiler heat duty of the study column is not measured directly and the value is
calculated using Equation 5.1,
Qr = FhotCp∆T
5.1
where,
Fhot = Hot oil flowrate
Cp
= Specific heat capacity
= 7.5 kJ/kg.K
∆T
= Temperature difference
= 40 K
Hot oil flowrate, Fhot is the manipulated variable in APC for maintaining the bottom
temperature at the desired value. This manipulated variable will change if the
controller takes action to compensate the effect of disturbance (liquid flowrate at tray
28) and set point (bottom temperature) change. Otherwise it could be due to faults
such as line blockage, line leakage, sensor faults, valve sticking and controller fault.
Table 5.3 lists the selected process variables to be monitored in this study.
Table 5.3: SPC variables
Location
Feed stream
Process variable
Feed flowrate, F
Feed temperature, Tf
Pumparound stream
Pumparound flowrate, P
Reflux stream
Reflux flowrate, Re
Bottom stream
Reboiler heat duty, Qr
During the monitoring process to detect and diagnose faulty conditions,
quality variables acted as indicator variables to show that the process is in the state of
statistical control or in the state of out of control. If any of the statistics value
calculated from these variables falls out of control limits, it shows that the fault
situation has taken place. On the other hand, process variables are used to find the
causes of the fault situation.
90
Process monitoring for FDD involves both APC and SPC variables. If the
disturbance or set point or both have changed, it is not considered as a fault. The
controller in the process will take action by counteracts the effect of disturbance and
set point change to keep the process on target. In order to detect the Out of Control
(OC) condition, the Improved SPC control limits are determined and the Improved
SPC charts are constructed. Process monitoring by the operator will be based on
these charts. Any fault detected in the process will lead the operator to find cause(s)
using the process variables data.
5.3
Statistical Process Control Sampling Time, TSPC
In practice, the dynamics of a typical chemical or manufacturing process
could cause the measurements to be autocorrelated (Bakshi, 1998). Therefore, the
right data collected at appropriate rate with sufficient information content must be
selected. Data for SPC analysis must be random with no autocorrelation. Statistical
monitoring might not function well for autocorrelated data (Kano et al., 2001).
Moreover, autocorrelation often reflects increased variability, which may cause
difficulty to distinguish between common cause variation and assignable cause
variation. Therefore, the data is collected at certain time interval called SPC
sampling time, TSPC to ensure the data is random and without autocorrelation. TSPC is
always greater than APC sampling time, TAPC. The measurement of the process
variables is less likely to be correlated when the sampling time for SPC is greater
than natural response time of the process, TAPC (Ogunnaike and Ray, 1994). The first
step to determine TSPC is to select a variable with maximum time constant, τ . Then,
autocorrelation coefficient is calculated using the observed data of this variable.
Autocorrelation is a correlation coefficient between two values of the same variable
at times ti and ti+k instead of correlation between two different variables. Given
vector of univariate data consist of m observations, x1, x2, … xm, the lag k
autocorrelation function, rk is defined as
91
∑
=
m− k
rk
i =1
( xi − x )( xi + k − x )
∑i=1 ( xi − x ) 2
m
5.2
where
rk
= Autocorrelation coefficient
x
= Individual data
x
= Arithmetic Average
m
= Number of observations
k
= Number of lags
The autocorrelation coefficient at any lag k is zero for the truly random
process. In practice, a computed autocorrelation coefficient rarely equals zero even
the observations are truly independent since sampling variation will generally result
in a small amount of correlation between observations (Alwan, 2000). The effect of
autocorrelation on the control limits of control chart for individual data will not be
significant until the lag one autocorrelation coefficient is 0.7 or higher (Woodall,
2000). The autocorrelation plot can be used for the following two purposes:
1. To detect non-randomness in data
2. To identify an appropriate time series model if the data are not random
In order to compute TSPC , the autocorrelation coefficient of a univariate data
is plotted with confidence bound, ± 2 / k (Wetherill and Brown, 1991). Any value
greater than confidence bound shows the presence of autocorrelation.
TSPC is
determined right after the autocorrelation coefficient plot reach the confidence
bound. Figure 5.1 summarizes the procedure to determine TSPC.
92
Select a variable with maximum
time constant, τ
Plot autocorrelation function and the
confidence bound
rk < confidence
No
bound
Yes
Calculate SPC time sampling,
TSPC
Use to take sample of all process
variables using this TSPC value
Figure 5.1: The procedure to calculate and implement TSPC
5.4
Normal Operating Condition (NOC) Data Selection and Preparation
SPC methods rely on formation of a statistical model based on historical
process data to establish normal operating behavior. The historical data is also known
as NOC, where data is collected when the process is in statistical control or quality
variables remain close to their desired values. This is the most important step in SPC
methodology since it is used to predict the future behavior of the process.
A set of quality variables and a set of process variables are collected and
arranged in the matrix form, X = [F Tf Re P Qr xc8 xc9] where the measurements of a
variable are arranged in the form of column vector. Some noises with zero mean
were imbedded into the MATLAB simulation program. The values are considered as
small random change in process variables such as feed flowrate, F, feed temperature,
93
Tf, reflux flowrate, Re, pumparound flowrate, P, and reboiler heat duty, Qr. The
data was collected at SPC time sampling interval to ensure no autocorrelation among
the measurements or to decorrelate the autocorrelation. This small random change is
also known as common cause variation that creates random data for quality variables
i.e oleic acid composition, xc8 and linoleic acid composition, xc9. To ensure a normal
distribution of a set of NOC data, the number of samples used must be large. Based
on the central limit theorem, if the sample size is large the theoretical sampling
distribution of the mean can be approximated closely with normal distribution. The
observation vectors are assumed to be independent and normally distributed. The
normal distribution plays an important role in the analysis of random signals for a
number of reasons (Gertler, 1998):
1. Many random physical processes can be rather accurately characterized by
the normal distribution.
2. The behavior of variables representing the combined effect of a large number
of phenomena approaches the normal distribution.
3. The normal distribution is relatively simple to handle mathematically.
The visual appearance of the normal distribution is a symmetric or bell
shaped curve. A histogram plot is used to show the distribution of data values. Other
parameters such as skewness and kurtosis were determined to measure the symmetry
of the data around the sample mean and the outlier-prone of a distribution
respectively. If skewness is negative, the data are spread out more to the left of the
mean than to the right and vice versa. The skewness of the normal distribution given
data (or any perfectly symmetric distribution) is zero. The kurtosis of the normal
distribution is 3 for distribution having 99.73% of its data fall between the limits
defined by mean plus and minus three standard deviation. Distributions that are more
outlier-prone than the normal distribution have kurtosis greater than 3 while
distributions that are less outlier-prone have kurtosis less than 3.
Once the NOC data set has been established, the data is treated before the
correlation coefficient between quality variables and process variables and the
confidence limits can be established for SPC charts.
94
5.4.1
Data Pretreatment
The NOC data is standardized before further analysis is carried out since the
variables have different units and wide range of data measurements. Each variable is
adjusted to zero average by subtracting off the original average of each column and
adjusted unit variance by dividing each column by its standard deviation. The
standardized equation is
xs =
x−x
s
5.3
where
xs
= Standardized variable
x
= Average
s
= Standard deviation
After the standardization, each variable have equal weights with zero average and
standard deviation equal to one (N (0, 1)).
5.4.2
Correlation Coefficient via Multivariate Analysis Techniques
The standardized NOC data is used to build the linear relationship between
quality variables i.e. oleic acid composition, xc8 and linoleic acid composition, xc9
and process variables i.e feed flowrate, F, feed temperature, Tf, reflux flowrate, Re,
pumparound flowrate, P, and reboiler heat duty, Qr using multivariate analysis
techniques, PCA and PCorrA. Let xj be the quality variables and xk be the process
variables. The relationship between x j and x k can be written as,
x j = Cjk x k
5.4
95
It has been argued that nonlinear model may be necessary to model
continuous processes. Actually, the model depends on the application. If the model is
used for monitoring, linear model is sufficient to describe process fluctuation around
an operating point (MacGregor and Kourti, 1995). The following section discusses
the detailed procedure to calculate the correlation coefficient using multivariate
analysis method.
5.4.2.1 Principal Component Analysis, PCA
The advantage of using PCA method is the developed NOC models can
represent the major process variations in a lower dimension way. The first step in
PCA methodology is to calculate the eigenvector matrix, V and the eigenvalue
matrix, L1/2 of standardized data using Singular Value Decomposition (SVD)
technique. Several authors have discussed this technique to determine the
eigenvector matrix and eigenvalue matrix (Chiang et al., 2000; Chen et al., 2001;
Ralston et al., 2001). This technique can be used to analyze the equation or matrices
that are either singular or numerically very close to singular. Standardized data
matrix, X (m,p), which has m variables with each variable has p measurements are
arranged in the form of m x p, where the measurements of a variable are organized in
the form of column vector. A standardized data matrix, X is decomposed into a
product of the column-orthogonal matrix U (m,p), orthogonal matrix V(p,p) and a
diagonal matrix L1/2. Equation 5.5 shows the fundamental identity of SVD. Bold
capital letter alphabet is matrix while small letter alphabet is vector and italic is
scalar.
X(m,p) = U(m,p)L1/2 (p,p)VT (p,p)
= u1 λ 1 v1T + u2 λ 2 v2T + … + up λ p vpT
5.5
96
where,
U
= Eigenvector matrix of XXT
V
= Eigenvector matrix of XTX
L1/2
= Singular value
= Diagonal matrix of positive square roots
of the eigenvalue of XTX
=
λ
λ
= Eigenvalue
The eigenvectors matrix, V(p,p), with square matrix contains eigenvectors v1, v2, …,
vp. Each eigenvector is a column vector which contains the arrangement of elements
[v1,1 v2,1 … vp,1], [v1,2 v2,2 … vp,2] … [v1,p v2,p … vp,p]. The eigenvectors matrix can be
written as,
⎡ v1,1
⎢v
⎢ 2,1
⎢ .
Eigenvectors matrix, V = [v1 v2 … vp] = ⎢
.
⎢
⎢ .
⎢v p ,1
⎣
v1, 2
v 2, 2
.
.
.
v p,2
... v1, p ⎤
... v2, p ⎥⎥
...
. ⎥
...
. ⎥
⎥
...
. ⎥
... v p , p ⎥⎦
5.6
The diagonal matrix L1/2 (p,p) with positive elements, λ 1, λ 2,…, λ p are called
eigenvalues of X. The eigenvalues explained the percentage of variations
corresponding to the eigenvectors. The eigenvalues are arranged in descending order
λ 1> λ 2 > ,…, > λ p > 0. This matrix is shown as,
⎡ λ1
⎢0
⎢
Eigenvalue matrix, L1/2 = ⎢ 0
⎢
⎢0
⎢0
⎣
0
λ2
0
0
0
0
0
0
0
.
0
0
0
.
0
0⎤
0 ⎥⎥
0⎥
⎥
0⎥
λ p ⎥⎦
5.7
97
Matrices of U, V and L1/2 have the following properties (Nash and Lefkovitch,
1976):
UTU = UUT =I
VTV = VVT = I
(L1/2)(L1/2) T = (L1/2) T (L1/2) =L
5.8
The next step is to determine the number of PCs to be retained. Reducing the
number of PCs means to reduce the dimension of the variability in the data. Three
rules of thumb are suggested to verify the number of PCs to be retained in the model
(Ralston et al., 2001). The rules of thumb involved:
1. Determining the ‘knee’ in the eigenvalue plot. The plot is referred to scree
plot, which is shown in Figure 5.2.
2. Determining the sudden jump in the eigenvalues.
3. Evaluating the percentage of variance to be retained.
Figure 5.2: Example Scree plot
98
The percentage of the variation to be explained by the eigenvalues can be fixed by
the user. Equation 5.9 shows the percent of variation that captured by ith eigenvalues.
λi (%) =
λi
λ1 + λ2 + ... + λ p
×100%
5.9
In order to find the number of eigenvalues to be retained, the percentage of
cumulative variation explained by j first eigenvalues is determined using equation
5.10.
λ1 + λ2 + ... + λ j
× 100%
λ1 + λ2 + ... + λ j + ... + λ p
5.10
Let assume that n out of p eigenvalues are kept or retained. Based on theoretical
study of PCA, the data matrix, X is related to PC after data reduction using Equation
5.11.
X = PC n VnT + E
5.11
The retained principal components [PC1, PC2, …, PCn], which form the Pn VnT term,
are associated with systematic variation in data while the residual principal
components [PCn+1, PCn+2, …, PCp], which form the residual matrix E are
considered of containing measurement errors (Seborg et al., 1996).
The eigenvectors and eigenvalues according to number of PCs retained are
used to determine the correlation coefficient between quality variables and process
variables. The equation is derived from the basic theory of PCA method. Equation
5.11 is simplified for both quality variable vector, xj and process variable vector, xk
and becomes,
xj = PCv Tj
5.12
xk = PCv Tk
5.13
99
Since both variables in standardized form with mean zero and standard deviation is
equal to one, the correlation coefficient, Cjk between xj and xk can be determined
using covariance equation,
Cjk = corr (xj, xk) = cov (xj, xk) = xjTxk
5.14
Substitute both equation 5.12 and equation 5.13 into equation 5.14. Therefore, the
correlation coefficient can be determined using eigenvector via the equation 5.15
Cjk = cov (xj, xk)
= xjTxk
= (PCv Tj )T (PCv Tk )
= v j PCTPC v Tk
5.15
The next step is to combine the PCA equation in Equation 2.3 with the equation
derived from SVD equation in Equation 5.5,
PC = XV = UL1/2 VT V = UL1/2
5.16
Equation 5.16 is used to replace PC in equation 5.15 produce the following equation,
Cjk = v j (UL1/2)T(UL1/2 )v Tk
= v j UTU (L1/2)T(L1/2 )v Tk
= v j Lv Tk
5.17
100
If n eigenvectors are involved, then the equation 5.17 can be rewritten as follows,
n
C jk = ∑ v jn vkn λn
5.18
i =1
where
n
= Number of retained eigenvectors
vk
= Eigenvectors of process variable
vj
= Eigenvectors of quality variable
The PCA procedures to determine the correlation coefficient is summarized in figure
5.3.
Standardized the NOC data
Determine the eigenvector matrix
via SVD technique
Arrange the eigenvalues respective to
eigenvector from the biggest to the smallest
Determined the number of eigenvectors to
be retained
Calculate the correlation coefficient, Cjk
Figure 5.3: The procedure to determine correlation coefficient, Cjk via PCA method
101
5.4.2.2 Partial Correlation Analysis, PCorrA
Partial correlation coefficient is defined as a correlation of selected quality
variable, xj and selected process variable, xk when the effects of other process
variable(s) have been removed from xj and xk. The initial step in PCorrA
methodology is to standardize NOC data. Then, the simple correlation coefficient
between two variables is determined. The simple correlation coefficient, rj,k is a
measure of linear association between the two variables. The simple correlation
coefficient and correlation matrix are determined from equations 5.21 and 5.22
accordingly. Variance is a measure of a variable’s dispersion or variation while
covariance is a measure of co-variation between two variables. The variance and
covariance of each variable are determined respectively from equations 5.19 and
5.20. Variance-covariance matrix is carried out in the analysis if the involved data
are in the same unit, same digit, and same magnitude order or the data are in
percentage. But, if the data or measurements are in different units like pressure,
temperature, flow rate, energy and so on, or the data are in different magnitude order,
then the analysis shall perform on the correlation matrix (Marriott, 1974).
∑ (x
m
Variance, s jk =
i
− x j )∑ (xi ,k − xk )
m
i, j
k
5.19
(m − 1)(m − 1)
⎡ s1,1
⎢s
⎢ 2,1
⎢ .
Covariance matrix, S = ⎢
.
⎢
⎢ .
⎢ s p ,1
⎣
s1, 2
s 2, 2
.
.
.
s p,2
Simple correlation coefficient, r jk =
s1, p ⎤
... s 2, p ⎥⎥
.
. ⎥
.
. ⎥
⎥
.
. ⎥
... s p , p ⎥⎦
...
s j ,k
s j , j s k ,k
5.20
5.21
102
⎡ r1,1
⎢r
⎢ 2,1
⎢ .
Normal Correlation matrix, R = ⎢
.
⎢
⎢ .
⎢r p ,1
⎣
where:
r1, 2
r2, 2
.
.
.
rp,2
... r1, p ⎤
... r2, p ⎥⎥
.
. ⎥
.
. ⎥
⎥
.
. ⎥
... r p , p ⎥⎦
i
: ith measurement
s 2j , j
: Variance of variable xj
xj
: Average value for variable xj
s j,k
: Covariance of variable xj and xk
5.22
Since five process variables involved in this study, therefore the order of partial
correlation is four. Correlation coefficient between selected quality variable, xj1 and
selected process variable, xk1 after removing the effect of other process variables, xk2,
xk3, xk4 and xk5 via PCorrA method is calculated using the Equation 5.23. The PCorrA
procedures is summarized in Figure 5.4.
C jk = C x
j 1 , xk 1
xk 2 , xk 3 , xk 4 , xk 5
=
rx
j 1 , xk 1
xk 3 , xk 4 , xk 5
1 − rx2 , x
j1
k2
− rx
j 1 , xk 2
xk 3 , xk 4 , xk 5
r
xk 3 , xk 4 , xk 5 xk 1 , xk 2 xk 3 , xk 4 , xk 5
1 − rx2k 1 , xk 2 xk 3 , xk 4 , xk 5
5.23
103
Standardized NOC data
Determine the simple correlation
coefficient via equation 5.21
Determine the order of partial
correlation via equation 2.11
Calculate the correlation coefficient, Cjk via
equation 5.23
Figure 5.4: The procedure to determine correlation coefficient, Cjk via PCorrA
method
5.4.3 Control limits of Improved Statistical Process Control chart
The control limits of Improved SPC chart is determined based on the
correlation between quality variables and process variables. Each Improved SPC
chart consists of two control limits, which are upper and lower limits. The upper line
will be the limit for the point greater than target value and the lower line for the point
less than target value. These limits are calculated based on the standardized NOC
data, which represent the desired process operation. Standardized data results of all
variables have zero mean and standard deviation of equal to one. The control limit is
determined so that the number of samples outside the control limits is 0.27% of the
entire samples. Many authors in literature used Shewhart chart with 3 standard
deviation limits (Woodall, 2000; Kahn et al., 1996; Montgomery, 1996; Oakland,
1996). Experience shows 3 standard deviation limits to be the most effective scheme
and seems to be an acceptable economic value (Woodall, 1996). The choice of size 3
standard deviation control limits gives a balance between statistical theory and
practical experience (Kahn et al., 1996). Most of statistical process control books
104
provide tables of the constant used such as D.'001 , D.'999 and A2 . Table 5.4 shows the
control limits of Improved SPC charts used in this study based on standardized data.
Table 5.4: The control limits of Improved SPC chart
Improved SPC
Quality variable
Process variable
chart
Control limit
Control limit
Shewhart individual UCL = 3
UCL = 3/Cjk ,
LCL = -3
LCL = -3/Cjk
UCL = D.'001 R
UCL = D.'001 R / C jk
LCL = D.'999 R
LCL = D.'999 R / C jk
Shewhart range
EWMA
UCL = + L
LCL = − L
MA
MR
β
[1 − (1 − β ) ]
2−β
β
2m
[1 − (1 − β ) ]
2−β
2m
UCL = + L
LCL = − L
β
2−β
β
2−β
[1 − (1 − β ) ]/Cjk
2m
[1 − (1 − β ) ]/Cjk
UCL = + A2 R
UCL = + A2 R / C jk
LCL = − A2 R
LCL = − A2 R / C jk
UCL = D.'001 R
UCL = D.'001 R / C jk
LCL = D.'999 R
LCL = D.'999 R / C jk
2m
There are two important parameters to design EWMA control charts i.e the
width of the control limit, L and the weighting factor, β . These two parameters
affect the efficiency of EWMA charts for FDD. The selection of these parameters in
this study is based on the performance of the chart to detect and diagnose small shift
in the process. Theoretically, smaller values of β can detect smaller shift in the
process. Several studies have been done and provide a range of value of L and
β (Montgomery, 1996). Table 5.5 shows several values of EWMA parameters to be
tested for sensitivity in this study.
105
Table 5.5: Parameters to design EWMA
5.5
Weighting factor, β
Width of the control limit, L
0.05
2.615
0.10
2.814
0.20
2.962
0.25
2.998
0.40
3.054
Generated Out of Control, OC data
In order to see the performance of Improved Statistical Process Control charts
some faults which were detected and diagnosed had been imbedded into simulated
distillation column. The model was run at steady state condition. The faulty
conditions were introduced into the process by increasing or decreasing the process
variables (F, Tf, P, Re, Qr) from the target, which caused the desired quality variables
(xc8, xc9) statistics fell beyond the control limits. The Out of Control, OC data were
collected during this condition and the data were arranged in matrix form and
standardized in the same way described for NOC data.
Fault causes were divided into three types of faults i.e sensor fault, valve fault
and controller fault. The effect of these three types of fault in closed loop is different
from those in the open loop operation. The differences are caused by the controller
actions to remove the deviation on the controlled variable. For open loop variable,
the fault will be of sensor fault or valve fault while for closed loop variable the fault
can be of sensor fault, valve fault or controller fault. In the case of simple sensor
fault, open loop and closed loops process give the same result on the measure
variables and the fault diagnosis is simple and relatively easy. This type of fault
occurs in a specific fault source and its effect is not propagated into other variables.
For the second type of fault that is valve fault, open loop and closed loop processes
give different result. Manipulated variables for closed loop will change and deviate
from its steady state values for a period of time when the fault occurs. The control
106
variables deviate from its set point until the controller brings the variables back to its
set points. But for the case of controller faults, both manipulated variables and
control variables will continuously deviate from the steady state value and set point
values respectively. Table 5.6 summarizes the effect of sensor fault, valve fault and
controller fault on the disturbance, manipulated and control variables.
107
Table 5.6: Description of sensor fault, valve fault and controller fault
Sensor Fault
•
•
D
For
open
•
loop
For
Controller Fault
open
•
loop
closed
loop
variables,
the
variables, the value of
of the variable changes
value of the variable
manipulated variable
abnormally. For closed
changes
(MV) AND control
loop variables, only the
For
value of the disturbance
variables,
(D) OR the manipulated
manipulated
variable (MV) OR the
(MV)
control variable (CV)
variable (CV) changes
changes abnormally.
abnormally together.
Control
limit
Fault
Control
limit
Steady
state
value
Steady
state
value
Control
D limit
Control
MV limit
Fault
only
For
variables, only the value
Control
limit
Fault
MV
CV
Valve Fault
abnormally.
closed
loop
variable
(CV)
both
changes
abnormally
variable
AND
together.
control
D
Steady
state
value
Control
MV limit
Fault
Steady
state
value
CV
Fault
Set
Point
CV
Control
limit
Fault
Control
limit
Set
Point
Steady
state
value
Steady
state
value
Control
limit
Fault
Set
Point
108
Single and multiple faults were considered in this study. The OC data that
were greater than ± 3standard deviation were divided into three regions. Region 1,
region 2, and region 3 refer to mean ± 4standard deviation, mean ± 5standard
deviation and over mean ± 5standard deviation respectively. The classification of
these OC data can alert the operators to the level of failure occurred in the process.
Improved SPC charts using quality variables were used to detect the faulty
condition in the process. Samples are taken over time and values of statistics are
plotted. The statistic for each Improved SPC charts is calculated using equation in
Table 5.7. The process is considered to be in control if the plotted statistic of the
quality variables falls within the control limits and out of control otherwise. On the
other hand, control charts based on process variables data are used to diagnose the
origin of the faults.
Table 5.7: Improved SPC chart statistic for FDD
Improved SPC chart
Chart statistics
Shewhart Individual
x = xi
Shewhart Range
Ri = max [xi-1+1] – min [xi-1+1]
EWMA
zi= β xi + (1- β )zi-1
MA
MAi =(xi + xi-1 + … + xi-w+1)/w
MR
MRi = max [xi-w+1] – min [xi-w+1]
109
5.6
Fault Coding Monitoring Framework
Fault coding system is proposed to perform the process fault detection and
diagnosis framework using Improved SPC chart. Using Improved SPC chart alone
may cause difficulties in observing the process by looking through all the Improved
SPC chart. The procedure for coding the control chart information are described as
follows:
i. Signal ‘1’ shows that:
ƒ
Statistical data for quality variables over the control limits
ƒ
Statistical data for process variables over the control limits
ii. Signal ‘0’ shows that:
ƒ
Statistical data for quality variables within the control limits
ƒ
Statistical data for process variables within the control limits
The faulty condition is detected when the quality variables chart give signal out of
statistical control. Once fault has been detected, process variables charts are used to
identify which variables caused the faulty condition. Table 5.8 and Table 5.9
summarize the application of coding system for fault detection and diagnosis.
Table 5.8: Coding system for fault detection
Observation
1
2
M
m
Oleic acid composition, xc8
1
1
M
0
Conclusion
Out of Control
Out of Control
M
In Control
Table 5.9: Coding system for fault diagnosis
Observation
Process variable
Re
P
0
0
Conclusion
1
F
0
Tf
1
Qr
0
2
1
0
0
0
0
M
m
M
0
M
0
M
0
M
0
M
0
Feed temperature cause
the fault
Feed flowrate cause the
fault
In Control
110
Fault coding systems for each Improved SPC chart used in this study are
shown in Table 5.10 to Table 5.12. Faulty condition is identified either Shewhart
Individual or Shewhart Range or both Shewhart Individual and Shewhart Range give
signal out of control situation. Moving Average and Moving Range chart used the
same interpretation as Shewhart Individual and Shewhart Range.
Table 5.10: Fault coding system for Shewhart Chart
Observation
Shewhart Individual
Shewhart Range
Conclusion
1
1
0
Out of control
2
0
1
Out of control
3
0
0
In control
M
M
M
M
m
1
1
Out of control
Table 5.11: Fault coding system for EWMA Chart
Observation
EWMA
Conclusion
1
1
Out of control
2
0
In control
M
M
M
m
1
Out of control
Table 5.12: Fault coding system for MAMR Chart
Observation
Moving Range
Moving Average
Conclusion
1
1
0
Out of control
2
0
1
Out of control
3
0
0
In control
M
M
M
M
m
1
1
Out of control
111
5.7
Performance Evaluation of the FDD Method
The performance of the FDD method using different Improved SPC chart is
evaluated from two different perspectives. First, the number of false alarm generated
during normal operating condition. Second, the successful of FDD method to detect
the fault and identify the correct process variable as fault cause for each fault
situation. The efficiency of the chart to detect and to diagnose the fault is discussed
and the equation used is mentioned in section 5.7.1 and 5.7.2 respectively.
5.7.1
Fault Detection Efficiency
The fault detection efficiency, ηFDetect. is determined using equation 5.24,
η FDetect =
Number of faults detected
Number of faults generated in the process
5.24
112
5.7.2 Fault Diagnosis Efficiency
The performance of a system in fault diagnosis is based on the ability of the
system to identify the correct process variable as fault cause for each fault situation.
The efficiency of the fault diagnosis system is evaluated based on the following
situations:
1. Successfully diagnosed fault
For every fault situation, the fault diagnosis tool is able to identify the
correct process variables as fault causes.
2. Unsuccessfully diagnosed fault
Unsuccessfully diagnosed faults can be categorized into three situations.
i. Partially diagnosed fault cause situation
Several of the known fault causes are identified by the fault diagnosis
tool. However, there are some fault causes are not identified.
ii. Over diagnosed fault cause situation
Over-diagnosed fault cause refers to the situation where the system
over-identifies other process variables that are not suppose to be the
fault causes.
iii. Undiagnosed fault cause situation
This situation shows that the known fault causes are totally
unidentified.
Therefore, the fault diagnosis efficiency, ηFDiag is determined using Equation
5.25
η FDiag =
Number of exact fault diagnosed
Total diagnosed fault situation
5.25
113
5.8
Chapter Summary
The methodology in this study is started with the variables selection for
process monitoring. There are two types of variables involved i.e Automatic Process
Contol (APC) variables and Statistical Process Control (SPC) variables. The APC
variables is categorized as control variables, manipulated variables and measured
disturbances while SPC variables is categorized as quality variables and process
variables.
Data for SPC analysis must be random with no existence of autocorrelation.
In order to ensure the data generated is random with no autocorrelation exist among
the variables, it must be collected at appropriate time. Therefore, SPC sampling time
is determined using autocorrelation plot with confidence bound at ± 2 / k . Then, a
set of historical data known as Normal Operating Condition (NOC) data is collected
when the process is in statistical control or quality variables remained close to their
desired values. The NOC data is subjected to small variation and normally
distributed in region ± 3 standard deviation. Before further analysis, the NOC data is
standardized to mean zero and unit variance since the variables have different units
and wide range of data measurements. By doing so, it gives equal weight to all
variables. The NOC data is collected for two purposes. The first one is to determine
the correlation coefficient, which relates the quality variable with process variables
via multivariate analysis techniques i.e Principal Component Analysis (PCA) and
Partial Correlation Analysis (PCorrA). The second purpose is to calculate the control
limits of Improved SPC charts for quality variables and process variables.
The faulty condition is introduced into the process by increasing or
decreasing the process variables from the target, which will cause the desired quality
variables statistics fall beyond the control limits. This known fault causes are divided
into three types of faults i.e sensor fault, valve fault and controller fault. Improved
SPC charts using quality variables are used to detect the faulty condition in the
process. On the other hand, control charts based on process variables data are used to
diagnose the origin of the faults. The Improved SPC chart used to compare the
current state of the process against NOC. If any points fall beyond the control limits,
114
the Improved SPC will give signal and it shows the Out of Control (OC) condition
has taken place. Fault coding system is proposed for easier interpretation of the
Improved SPC charts. Signal ‘1’ shows that the process is out of statistical control
while signal ‘0’ shows that the process is in control. The efficiency of the Improved
SPC chart is evaluated based on two aspects. First, the number of false alarm
generated during normal operating condition. Second, the successful of FDD method
to detect the fault and identify the correct process variable as fault cause for each
fault situation.
It is important to note that the implementation of the Improved SPC method
is to detect and diagnose fault in the process when the set point and disturbance
unchanged. This means that the controller in APC system are ready to counteracts
the effect of disturbance change and set point change as long as the controller
functions well. The methodology for developing the FDD systems for the simulated
precut distillation column is outlined in Figure 5.5.
115
Identify APC variables and SPC
variables for FDD purpose
Calculate SPC sampling time, TSPC
Generate data for both quality variables
and process variables at TSPC interval
Standardize data for each variable
Determine the correlation coefficient
between quality variables and process
variables using PCA and PCorrA
Calculate the control limits for each
control charts
99.73% of data fall
within control limits
No
Yes
NOC data is representable
Generate the OC data
Construct the Improved SPC
Performance evaluation of the
charts
developed FDD algorithm
Figure 5.5: The outlined of FDD algorithm
CHAPTER 6
RESULTS AND DISCUSSIONS
6.1
Statistical Process Control Sampling Time, TSPC
Table 6.1 summarizes the time constant of the control variables for each
control loops. Pumparound temperature has the highest time constant, so the
autocorrelation plot is based on pumparound cooler model.
Table 6.1: Time constant for each control loops
Control variable
Time constant, θ hr
Bottom column liquid level
0.10
Bottom temperature
0.28
Top column pressure
1.00
Side draw tray liquid level
0.20
Pumparound temperature
1.45
Figure 6.1 displays the autocorrelation coefficient of pumparound temperature versus
lag. As shown in this figure, the autocorrelation plot touched the upper bound
(+2standard deviation) during 216th observations. The autocorrelation plot is based
on data collected every 0.01 hr (APC sampling time, TAPC). Therefore, SPC sampling
time is equal to 2.16 hrs (216 x 0.01hr).
117
Figure 6.1: The autocorrelation plot
6.2
Normal Operating Condition, NOC Data
A set of historical data for both quality variables and process variables was
generated from the simulated dynamic distillation column. Each variable was
composed of 2000 observations during normal operating condition. The data was
sampled every 2.16 hr. 30 data were identified to have values greater than
± 3standard deviations. MATLAB programming was used to remove any outlier that
was greater than ± 3standard deviations. Therefore, after removing the outliers each
variable consisted of 1970 data observations. The histogram plot as in figure 6.2
shows the distribution of the NOC data. Figure 6.2 shows the distribution of NOC
data approximately normal with average at zeros and the data varies between
± 3standard deviations. The skewness and kurtosis in Table 6.2 show the value
closed to zero and three respectively. Based on the histogram and the kurtosis and
skewness values, it was decided that data were distributed normally and acceptable
for further analysis to construct the control charts. Table 6.3 shows the summary of
118
the NOC data. The NOC data is standardized using Equation 5.3 before proceed to
the next step in FDD methodology.
Figure 6.2: Histogram plot of NOC data
Table 6.2: The skewness and kurtosis value of NOC data
F
Tf
Re
P
Qr
xC8
xC9
Skewness
0.044
-0.020
0.021
-0.0057
-0.011
-0.0123
0.0192
Kurtosis
2.748
2.892
2.839
2.836
2.904
2.734
2.734
Table 6.3: Summary of the NOC data
F
Mean
Standard
Tf
41.558 483.15
Re
P
Qr
xC8
xC9
10
49.6
2.1576x106
0.13422
0.0246
0.03
0.02
283.93
2.965x10-5
7.77x10-4
0.05
0.093
41.71
483.42 10.087
49.659
2.1584 x106
0.13431
0.02509
41.41
482.88
49.542
2.1568 x106
0.13414
0.0242
deviation
Maximum
value
Minimum
value
9.912
119
6.2.1
Correlation coefficient via Principal Component Analysis, PCA
PCA is applied to the combined set of quality variables and process variables
together in data matrix, X. In principle there is a significant relations between the
two set of variables. Analysis of these variables together provides better detection of
fault than looking at them separately (Lennox et al., 1999). Singular Value
Decomposition, SVD was applied to determine the eigenvalues and eigenvectors of
NOC data. Table 6.4 shows the eigenvectors, V obtained. The eigenvalue versus PC
plot is shown in Figure 6.3 and a table of eigenvalues is shown in Table 6.5.
Table 6.4: Eigenvectors Matrix
P1
P2
P3
P4
P5
P6
P7
F
-0.3471
-0.6298
0.0122
-0.4557
-0.1321
0.4638
0.2061
Tf
0.2044
-0.4614
0.0240
0.4035
-0.7180
-0.2356
-0.1047
-0.2998
0.4138
-0.5878
0.2325
-0.3943
0.3918
0.1741
-0.2844
-0.0053
0.5771
0.6358
0.1566
0.3625
0.1610
Qr
0.0466
-0.4674
-0.5658
0.4134
0.5355
-0.0090
-0.0384
xc8
-0.5763
-0.0278
-0.0244
0.0174
0.0099
-0.6611
0.4768
xc9
0.5767
0.0070
-0.0008
0.0010
0.0139
0.0852
0.8123
Re
P
Table 6.5: Eigenvalues, variation explained by each PC and cumulative variation
Principal
Eigenvalue
Component
Difference
Variation
Cumulative
(∆= λ i+1- λ i)
Explained
Variation
(%)
(%)
42.9371
42.937
PC1
3.0056
PC2
1.0512
1.9544
15.0171
57.954
PC3
1.0097
0.0415
14.4242
72.378
PC4
0.9829
0.0268
14.0414
86.42
PC5
0.9506
0.0323
13.58
100
PC6
1.0314 x 10-7
0.9506
1.47 x 10-6
100
PC7
1.4219 x 10-9
0
2.03 x 10-8
100
120
After the eigenvectors and eigenvalues of the data matrix are obtained, the
next step is to determine the number of PCs to be retained. According to the first rule
of thumb, eigenvalues versus PC is plotted. From the scree plot in Figure 6.3, a
“knee” is clearly occurs at PC number two and PC number six. This indicates that
either one or six PCs should be retained in the model. Two PCs were preferred
because if more than half number of total Principal Components are required to
account for a reasonable variation, then the data cannot be summarized very well
with a few components and shall stick to the original data itself (Cliff, 1987).
Moreover, if six PCs are retained, this will violate the uniqueness of the PCA method
for data reduction.
Figure 6.3: Scree plot
For the second rule of thumb, the value of eigenvalues is used as an indicator
for the number of PC to be retained in the model. The value of eigenvalues was
shown in Table 6.5. The difference between PCs is calculated and the biggest
changes show the sudden jump in the eigenvalues. The average change in eigenvalue
between PCs for this data set is 0.5. The largest changes occur between PCs number
one and two (∆=1.9544) while the changes between PCs number five and six is
121
(∆=0.9506). Therefore, the best choice in this case is to select 2 PCs. In this data set,
second PC accounts for as much variance as 1.0512 of original variables.
For the third rule of thumb, the cumulative variation for the first ith PCs is
calculated. As shown in Table 6.5, the first two PCs explained 58% of the total
variance, which is about half of the total variance. Important information of original
data might be lost if small cumulative variation is used. Moreover, greater
eigenvalues usually describe the systematic variation and should be kept in the model
and Sharma (1996) suggested retaining eigenvalues greater than one. This is referred
to eigenvalues-greater-than-one rule. If two PCs retained were selected, important
information from third PCs which has eigenvalues greater than one might lost.
Therefore, three PCs retained are preferred which explained 72% of the total
variance and has eigenvalues greater than one. In order to select the number of PCs
to be retained, either two PCs or three PCs, the correlation coefficients using
eigenvectors and eigenvalues captured by two PCs, three PCs and six PCs were
determined. Six PCs, which explained 100% of the total variance is used as
reference. Table 6.6 lists the result of correlation coefficient calculated using
different cumulative variation.
Table 6.6: The correlation coefficient using different cumulative variation
Two PCs
Three PCs
Six PCs
Variables
xc8
xc9
xc8
xc9
xc8
xc9
F
0.6197
-0.6064
0.6194
-0.6064
0.6104
-0.6086
Tf
-0.3406
0.3510
-0.3412
0.3510
-0.3411
0.3419
Re
0.5073
-0.5167
0.5217
-0.5162
0.5220
-0.5212
P
0.4927
-0.4930
0.4785
-0.4935
0.4909
-0.4908
Qr
-0.0671
0.0774
-0.0532
0.0779
-0.0410
0.0853
The percentage difference for two PCs and three PCs retained compared to
six PCs is calculated and the result is recorded in Table 6.7. The overall result
obtained indicates that three PCs retained, which capture 72% of the total variance,
give the correlation coefficient closer to 100% variation than two PCs. The
percentage different for the correlation between quality variables and process
122
variables are small (less than 10%) except for correlation coefficient between
reboiler duty and oleic acid compositions. The highest different was about 64% using
two PCs while the percentage different was about 30% for three PCs retained.
Table 6.7: The percentage difference for two PCs and three PCs retained compared
to six PCs.
Two PCs
Three PCs
⎡ Two PCs - Six PCs ⎤
% ∆= ⎢
⎥⎦ x100%
Six PCs
⎣
⎡ Three PCs - Six PCs ⎤
% ∆= ⎢
⎥⎦ x100%
Six PCs
⎣
Variables
xc8
xc9
xc8
xc9
F
1.5236
-0.3615
1.4744
-0.3615
Tf
-0.1466
2.6616
0.0293
2.6616
Re
-2.8161
-0.8634
-0.0575
-0.9593
P
0.3667
0.4482
-2.5260
0.5501
Qr
63.6585
-9.2614
29.7561
-8.6753
Number of PCs retained will affect the fault detection and diagnosis results.
The correlation coefficient calculated based on the number of PCs retained must
close to the normal correlation using 100% variation. Three PCs retained has yielded
a result close to 100% variation. Therefore, the correlation coefficient calculated
using 3 PCs is used to develop the control charts for FDD.
123
6.2.2
Correlation coefficient via Partial Correlation Analysis, PCorrA
The simple correlation coefficients, which show the associations between
quality variables and process variables and among process variables, are presented in
Table 6.8.
Table 6.8: Simple correlation matrix
F
Tf
Re
P
Qr
xc8
xc9
F
1.0000
0.0018
-0.023
0.0029
0.0013
0.6104
-0.6086
Tf
0.0018
1.0000
-0.0379
-0.0129
0.0400
-0.3411
0.3419
Re
-0.023
-0.0379
1.0000
-0.0019
-0.0157
0.5220
-0.5212
P
0.0029
-0.0129
-0.0019
1.0000
-0.0288
0.4909
-0.4908
Qr
0.0013
0.0400
-0.0157
-0.0288
1.0000
-0.0410
0.0853
xc8
0.6104
-0.3411
0.5220
0.4909
-0.0510
1.0000
-0.9990
xc9
-0.6086
0.3419
-0.5212
-0.4908
0.0853
-0.9990
1.0000
The correlation coefficient results show that the process variables have
opposite correlation with oleic acid composition, xc8 and linoleic acid composition,
xc9. Feed flowrate, reflux flowrate, and pumparound flowrate are positively
correlated with oleic acid composition but negatively correlated with linoleic acid
composition. Feed flowrate is highly correlated with quality variables while reboiler
duty shows less correlation with quality variables.
Partial correlation is calculated between one of the process variables with
each quality variables while controlling for other process variables. The order of the
partial correlation depends on the number of process variables controlled for. Fourth
order partial correlation is calculated and the results are shown in Table 6.9. The
result obtained as shown in Table 6.9 shows that process variables are highly
correlated with the quality variables after the effect of other process variables has
been removed.
124
Table 6.9: The correlation coefficient result using PCorrA
6.3
F
Tf
Re
P
Qr
xc8
0.999
-0.999
0.999
0.999
-0.998
xc9
-0.999
0.999
-0.999
-0.999
0.999
Parameters to Design Improved Statistical Process Control chart
As has been mentioned in Chapter 5, three standard deviation limits is used to
construct the control limits for Shewhart individual for a few reason. In order to
calculate Shewhart range, the size of the window used is two. The constant used to
develop the control limits for Shewhart range using window size two is shown in
Table 6.10.
Table 6.10: Constant to design Shewhart range chart
Constant
Value
D.'001 ,
4.12
D.'999
0
A2
1.02
There are two important parameters to be selected to design EWMA chart i.e.
width of the control limit, L and the weighting factor, β . For this reason, one fault
case was introduced into the process. Pumparound flowrate was increased from 49.6
kmol/hr to 49.9 kmol/hr, which cause +0.6% deviation from target value. In order to
maintain the sensitivity of EWMA chart to detect small shift, the weighting factor
used starting from the smallest which is 0.5. This value was increased until the
EWMA chart can detect and diagnose the known fault. The EWMA statistics was
calculated and the chart was plotted for each parameters. Table 6.11 summarize the
result obtained using various EWMA parameters. Appendix B provides the EWMA
chart for each parameter. As shown in Table 6.11, EWMA chart using PCorrA
technique can identify the fault causes using weighting factor 0.05, 0.10, 0.20, 0.25,
and 0.4 but the chart can only detect the fault using weighting factor 0.4. On the
125
other hand, EWMA chart using PCA can only diagnose and detect fault using
weighting factor 0.4. Therefore, width of control limit 3.054 and weighting factor 0.4
are selected which give the best outcome in fault detection and diagnosis using both
techniques i.e PCorrA and PCA.
Table 6.11: FDD result using different L and β to design EWMA chart
Weighting
Width of the
Fault detection and diagnosis result
factor,
control limit,
β
L
FDetect
FDiagnose
FDetect
FDiagnose
0.05
2.615
Undetected
Diagnosed
Undetected
Undiagnosed
0.10
2.814
Undetected
Diagnosed
Undetected
Undiagnosed
0.20
2.962
Undetected
Diagnosed
Undetected
Undiagnosed
0.25
2.998
Undetected
Diagnosed
Undetected
Undiagnosed
0.40
3.054
Detected
Diagnosed
Detected
Diagnosed
PCorrA
PCA
To design the Moving Average and Moving Range, MAMR chart, the size of
moving window need to be decided. For this reason, one fault case was introduced
into the process. Feed flowrate was increased to 51 kmol/hr (+2.8% from the target
value). MAMR chart using three different window sizes i.e 3, 4 and 5 was used to
detect and diagnose this fault case. Larger window size may cause delay in detecting
and diagnosing the fault in the process. For each successive group, earlier result is
discarded and replaced by the latest. Tables 6.12 and 6.13 show the outcome of FDD
using both multivariate techniques, PCA and PCorrA with different window size.
MAMR charts for three different window sizes are shown in Appendix C.
Fault situation is detected or diagnosed when MA or MR statistic greater than the
control limits. It is also considered fault when both MA and MR statistic greater than
control limits. Results shown in Tables 6.12 and 6.13 indicate that Moving Average
chart have delay in detecting fault using all window sizes. Moving Average chart for
quality variables exceed the control limits one points later using window size three,
two points later using window size four and three points later using window size five
after the fault occurs. However, both MA and MR chart with window size three can
126
detect and diagnose known fault using PCA and PCorrA. Therefore, window size
three is selected to design MAMR chart for fault detection and diagnosis.
Table 6.12: FDD result using PCA with different window size to design MAMR
chart
Window
Fault detection and diagnosis result
size
3
MR
MA
FDetect
FDiagnose
Remarks
FDetect
FDiagnose
Remarks
Detected
Diagnosed
-
Detected
Diagnosed
One
point
delay
4
Detected
Diagnosed
-
Detected Undiagnosed
Two
points
delay
5
Detected Undiagnosed
-
Detected Undiagnosed
Three
points
delay
Table 6.13: FDD result using PCorrA with different window size to design MAMR
chart
Window
Fault detection and diagnosis result
size
MR
FDetect
3
FDiagnose
Detected Diagnosed
MA
Remarks
FDetect
FDiagnose
Remarks
-
Detected
Diagnosed
One
point
delay
4
Detected Diagnosed
-
Detected
Diagnosed
Two
points
delay
5
Detected Diagnosed
-
Detected
Diagnosed
Three
points
delay
127
6.4
Fault Detection and Diagnosis Method Performance
The performance of the FDD method was analyzed from two different
perspectives. First, the performance was analyzed with respect to the number of false
alarm generated during normal operating condition using different type of Improved
SPC chart. Second, the efficiency of the FDD method with respect to fault detection
and diagnosis during faulty condition was analyzed. Ideally, the monitoring data
should only affect by the fault during fault detection and diagnosis, FDD process.
However, the presence of disturbances and noises may cause an error and thus
interfering the monitoring results. Therefore, the generating data from the process for
monitoring must be maximally unaffected by these nuisance inputs so that it is robust
in the face of disturbances and noises. 100 data were generated without any known
fault inserted into the process for false alarm analysis.
6.4.1
False Alarm Analysis
False alarm occurred when Improved SPC chart for quality variables give
signal out of statistical control whereas the process variables are in statistical control.
Tables 6.14 and 6.15 show number of signal given by the Improved SPC charts
during normal operating condition for oleic acid composition, xc8 and linoleic acid
composition, xc9 respectively. One false alarm was detected using MAMR. Quality
variables charts give signal out of statistical control during observation 27th. There is
no false alarm found using Shewhart and EWMA.
128
Table 6.14: False alarm result corresponding to oleic acid composition
Improved
Number of false alarm
SPC
Process variables
Total
Quality
chart
variables
F
Tf
Re
P
Qr
xc8
Shewhart
0
0
0
0
0
0
0
EWMA
0
0
0
0
0
0
0
MAMR
0
0
0
0
0
1
1
Total
0
0
0
0
0
1
Table 6.15: False alarm result corresponding to linoleic acid composition
ISPC
Number of false alarm
chart
Process variables
Total
Quality
variables
F
Tf
Re
P
Qr
xc9
Shewhart
0
0
0
0
0
0
0
EWMA
0
0
0
0
0
0
0
MAMR
0
0
0
0
0
1
1
Total
0
0
0
0
0
1
The occurrence of false alarm may cause wrong action taken to the process
which operates at normal condition.
129
6.4.2 Fault Detection Efficiency Results and Results Discussions
Various kinds of faults were introduced to the simulated precut column at 80
locations. Number of single and multiple faults involving five selected process
variables i.e feed flowrate, F, feed temperature, Tf, reflux flowrate Re, pumparound
flowrate, P and reboiler heat duty, Qr are listed in Table 6.16.
Table 6.16: The known fault causes
Fault
Single
cause
fault
Multiple faults
Total
As part
As part
fault
of 2
of 3
causes
causes
causes
F
10
7
9
26
Tf
10
7
9
26
Re
13
8
6
27
P
11
8
6
25
Qr
6
0
15
21
Total
50
30
45
125
The quality variables i.e oleic acid composition, xc8 and linoleic acid composition, xc9
were monitored to identify the Out of Control (OC) condition. Number of OC
condition due to single faults and multiple faults detected by each Improved SPC
chart are shown in Table 6.17.
130
Table 6.17: The known out of control, OC conditions
Improved
Quality
Due to
Due to multiple
SPC chart
variable
single
faults
Total
OC
fault
2 causes
3 causes
50
15
15
80
50
15
15
80
33
13
14
60
34
15
15
62
24
13
14
51
28
13
15
56
condition
Shewhart Oleic acid,
xc8
Linoleic acid, xc9
EWMA
Oleic acid,
xc8
Linoleic acid, xc9
MAMR
Oleic acid,
xc8
Linoleic acid, xc9
Based on the result obtained in Table 6.17, the efficiency of each Improved
SPC chart was calculated to determine the performance of each chart to detect faults
in the process. The computed values of fault detection efficiency are shown in Figure
6.4. The results obtained shows that all known faults for both quality variables were
detected by Shewhart chart. EWMA chart can detect 75% of the known faults occurs
on oleic acid composition and 77.5% of the known faults occurs on linoleic acid
composition. MAMR chart has the lowest percentage, which are 63.8% and 70%
fault detected on oleic acid composition and linoleic acid composition respectively.
This shows that Shewhart chart performance in detecting faults is better than EWMA
and MAMR. Shewhart chart was plotted using 100% current data but MAMR
statistic was calculated using window size three consist of 33% current data and 67%
previous data. This caused detection delay using MAMR. EWMA statistic was
calculated using weighting factor, β = 0.4. This gives the EWMA statistic consists of
40% current data 60% previous data. Based on the percentage of current data and
previous data included during calculating the chart statistics, it shows that fault
detection efficiency decreases as the percentage of previous data involved in
calculating the statistics value for each charts increases.
131
FDetect efficiency
(%)
100
80
60
40
20
0
Shewhart
EWMA
MAMR
xc8
100
75
63.8
xc9
100
77.5
70
ISPC chart
xc8
xc9
Figure 6.4: Fault detection efficiency using different Improved SPC chart
Out of 80 faults inserted into the process, 17.5% fault cases identified fall in
region one (mean ± 4standard deviation), 25% fault cases fall in region two (mean
± 5standard deviation) and 57.5% fault cases fall in region three (beyond mean
± 5standard deviation). Figure 6.5 shows the result obtained for percentage of faults
detected in three different regions. Shewhart chart was able to detect all known faults
in all regions. EWMA chart was able to detect 2.5% fault cases in region one, 16.3%
fault cases in region two and 56.3% fault cases in region three while MAMR can
only detect fault in region two and region three. This indicates that Shewhart chart
was sensitive with small shift in the process while EWMA chart was sensitive with
moderate change in the process. MAMR chart started give signal OC condition for
large deviation occurs in the process.
FDetect efficiency
(%)
132
80.0
60.0
40.0
20.0
0.0
Shewhart
EWMA
MAMR
Region 1
17.5
2.5
0.0
Region 2
25.0
16.3
8.8
Region 3
57.5
56.3
55.0
OC region
Region 1
Region 2
Region 3
Figure 6.5: Percentage of faults occurs in different region detected by each Improved
SPC chart
6.4.3 Fault Diagnosis Efficiency Results and Results Discussions
The efficiency of Improved SPC chart for fault diagnosis depends on four
situations:
1. Exact diagnosed fault cause situation
2. Partially diagnosed fault cause situation
3. Over diagnosed fault cause situation
4. Undiagnosed fault cause situation
Improved SPC chart successfully diagnose fault if it can identify exactly fault cause
situations. If Improved SPC charts partially diagnosed, over diagnosed and
undiagnosed fault cause situation, then Improved SPC chart is considered failed to
diagnose fault causes. Improved SPC chart was constructed using process variables
data to diagnose faults. Discussions on the efficiency of Improved SPC charts to
diagnose faults are related with multivariate analysis technique used to calculate the
control limits for process variables. Figure 6.6 and Figure 6.7 provide fault diagnosis
results for oleic acid composition using PCA and PCorrA techniques respectively
while Figure 6.8 and Figure 6.9 for linoleic acid composition using PCA and PCorrA
133
techniques respectively. Number of exact fault diagnosed, partially fault diagnosed,
over fault diagnosed and undiagnosed fault for single and multiple faults are listed
detailed in Appendix D.
From the results obtained in Figures 6.6 and 6.8, show that Shewhart chart
using PCA technique was not able to diagnose all known faults in the process. Out of
80 fault causes insert into the process, Shewhart chart was able to identify only
87.5% fault causes which contribute out of statistical control to oleic acid
composition and linoleic acid composition. However, the number of exact fault
diagnosed by Shewhart is higher than EWMA and MAMR. Fault diagnosis using
PCorrA technique as shown in Figures 6.7 and 6.9 results 100% in diagnosing exact
fault causes using Shewhart chart. As shown in Figures 6.6 to 6.9, number of exact
fault diagnosed by EWMA chart and MAMR chart using PCorrA technique was
FDiagnose efficiency
(%)
higher than PCA technique.
100
50
0
Shewhart
EWMA
MAMR
Exact diagnosed
87.5
55
35
Partial diagnosed
6.25
11.25
18.75
Over diagnosed
0
0
0
6.25
33.75
46.25
Undiagnosed
ISPC chart
Exact diagnosed
Partial diagnosed
Over diagnosed
Undiagnosed
Figure 6.6: Fault diagnosis situation using PCA for oleic acid composition
134
FDiagnose efficiency
(%)
150
100
50
0
Shewhart
EWMA
MAMR
Exact diagnosed
100
75
63.75
Partial diagnosed
0
0
0
Over diagnosed
0
0
0
Undiagnosed
0
25
36.25
ISPC chart
Exact diagnosed
Partial diagnosed
Over diagnosed
Undiagnosed
FDiagnose efficiency
(%)
Figure 6.7: Fault diagnosis situation using PCorrA for oleic acid composition
100
50
0
Shewhart
EWMA
MAMR
Exact diagnosed
87.5
53.75
36.25
Partial diagnosed
6.25
13.75
21.25
Over diagnosed
0
0
0
6.25
32.5
42.5
Undiagnosed
ISPC chart
Exact diagnosed
Partial diagnosed
Over diagnosed
Undiagnosed
Figure 6.8: Fault diagnosis situation using PCA for linoleic acid composition
FDiagnose efficiency
(%)
135
150
100
50
0
Shewhart
EWMA
MAMR
Exact diagnosed
100
77.5
70
Partial diagnosed
0
0
0
Over diagnosed
0
0
0
Undiagnosed
0
22.5
30
ISPC chart
Exact diagnosed
Partial diagnosed
Over diagnosed
Undiagnosed
Figure 6.9: Fault diagnosis situation using PCorrA for linoleic acid composition
The performance of each Improved SPC chart to diagnose fault was affected
by the multivariate technique used to determine the control limit for process
variables. The calculated control limits for Improved SPC charts using PCorrA
technique performed better than PCA technique because the correlation coefficients
developed by this technique are closer to the actual value of the correlation
coefficients, representing the correlation between the selected process variables with
the quality variables of interest. Some of the process variables were kept constant
during determining the correlation between selected quality variable with one of the
process variables. By doing so, the correlation coefficient developed using PCorrA
was not affected by the variance of other process variables, which influence the
actual correlation between variables. The correlation coefficients obtained using
PCorrA for each process variables are close to one. Each Improved SPC charts either
exact diagnosed or undiagnosed known fault in the process. As shown in Figures 6.7
and 6.9, there is no partially diagnosed fault situation was identified.
The PCA method calculates the cross-correlation between variables
(interaction between variables) when determining the correlation coefficients
between the process variables and quality variables. The correlation coefficient
obtained using PCA is smaller than PCorrA. Small correlation coefficient, widen the
control limits of Improved SPC chart. As can be seen in Figure 6.6 to 6.9, number of
partially diagnosed and undiagnosed faults using PCA is higher than PCorrA. For
136
example as in Figures 6.6 and 6.7, EWMA chart failed to diagnose 45% fault causes
using PCA while 25% fault causes failed to diagnose using PCorrA. The correlation
coefficient calculated using PCA varies among the process variables. This caused
some of the process variables gave signal OC condition while others did not when
multiple faults involved. The results using PCA technique caused some multiple
faults cases are partially diagnosed. However, PCA technique which used 72%
variation of the original data is better than normal correlation, which used 100%
variation of original data.
6.5
Chapter Summary
A set of NOC data was generated from the simulated dynamic distillation
column to observe two quality variables of interest and five selected process
variables, each one composed of 1970 observations. NOC data was collected during
normal operating condition with quality variables closed to target value. In this
study, a sampling frequency at SPC sampling time that was 2.16 hrs enabled such an
assumption of zero autocorrelation to be satisfied.
The NOC data then was used to develop the correlation coefficient using
PCA and PCorrA. Three rules of thumbs are used to determine number of PCs to be
retained. Three Principal Components, PCs retained used in this study, which
captured 72% variation of the original data when developing the correlation
coefficient using PCA. Correlation coefficient calculated using three PCs retained
give result close to normal correlation which used 100% variation of original data.
Fourth order partial correlation is used to calculate the correlation coefficients using
PCorrA and the result obtained shows that selected process variable are highly
correlated with the quality variable after the effect of other process variables has
been removed.
In order to design Improved SPC chart, some parameters need to be selected
to determine the control limits using NOC data. Based on theoretical study from
literature, three standard deviation limits is used to construct the control limits for
137
Shewhart individual while Shewhart range is calculated using window size of two.
Sensitivity test was conducted to select parameters for EWMA chart MAMR chart.
The outcome from the test conducted give result; width of control limit equal to
3.054 and weighting factor equal to 0.4 are selected which give the best outcome in
fault detection and diagnosis using both techniques i.e PCorrA and PCA for EWMA
chart. The sensitivity test conducted for MAMR shows that MAMR was able to
detect and diagnose fault using window size three for both techniques i.e PCorrA and
PCA.
False alarm analysis was performed before faulty condition introduced into
the process to ensure the result obtained during faulty condition was not affected by
false alarm error. One false alarm case was identified for both quality variables using
MAMR chart. The performance of Improved SPC chart to detect and diagnose faults
during faulty condition was evaluated. For this purpose, 80 faults location has been
identified to insert various kinds of faults to the simulated precut column. Five
selected process variables with single and multiple faults were introduced into the
process. The results obtained show that Shewhart chart perform better than EWMA
and MAMR in detecting and diagnosing faults. Both multivariate analysis technique
i.e PCA and PCorrA give significant effect to the efficiency of each Improved SPC
chart to diagnose fault causes in the process. The correlation coefficients developed
using the PCorrA was better than PCA in representing the correlation between the
selected process variables and the quality variables of interest which provided high
efficiency in fault diagnosis.
CHAPTER 7
CONCLUSIONS AND RECOMMENDATIONS
7.1
Introduction
Process monitoring, early fault detection and diagnosis are important in
chemical industries for safety, maintaining product quality and reducing cost. A
procedure for Fault Detection and Diagnosis, FDD has been described. A new
approach known as Improved SPC charts is introduced to detect faults in the process
and identify variables, which cause the faults. The proposed method is demonstrated
on a simulated precut multicomponent distillation column.
Multivariate analysis technique i.e Principal Component Analysis, PCA and
Partial Correlation Analysis, PCorrA are utilized to determine the correlation
coefficient between quality variables and process variables. The correlation
coefficients are used to translate the control limits of SPC charts from quality
variables into process variables. This correlation is adopted in conventional SPC
charts to construct the control limits for process variables. The quality variables and
some of the process variables, which are correlated between the former and the later
variables are monitored. The quality variables data is incorporated in the Improved
SPC chart during the faulty condition for fault detection purpose, while the process
variables, which has been correlated with the quality variables is used for fault
diagnosis. The Improve SPC charts applied is not only for quality variables but also
for process variables.
139
7.2
Conclusions of Improved SPC framework
The proposed Improved SPC monitoring framework consist of the following
elements to perform false alarm analysis, fault detection and fault diagnosis:
1) False alarm analysis was conducted using normal operating condition data
to identify outliers.
2) Fault detection tool, which monitors the quality variables was used to
detect faulty condition in the process.
3) Fault diagnosis tools, which based on setting up the control limits for
process variables using multivariate analysis technique was used to
identify the causes of faulty condition.
4) Fault coding system, which uses ‘1’ to denote out of control limits
situations while uses ‘0’ to denote in control situations, was used to avoid
of using too many ISPC charts.
The methodology of the proposed method started with variables selections for
process monitoring. The variables involved are divided into two categories i.e APC
variables and SPC variables. Some of the selected SPC variables are part of APC
variables. Even though the simulated precut column has been installed with
controllers to keep the process on target, there is a probability of faults to occur due
to line blockage, line leakage, sensor faults, valve sticking and controller fault. It is
very important to highlight that the ISPC is implemented for the process change that
is not due to set point change and disturbance change since the controllers installed
are ready to counteract the effect of these changes.
Two sets of data i.e NOC data and OC data are generated from the simulated
precut column at TSPC interval. NOC data was subject to common cause variation,
which creates random behavior and used to construct the control limits for Improved
SPC chart and to develop the correlation coefficient via multivariate analysis
technique. On the other hand, OC data was subject to assignable cause variation due
to faulty condition occur in the process. The performance of the Improved SPC chart
is evaluated based on these two sets of data. Both NOC data and OC data are
140
standardized before further analysis since the variables have different units and wide
range of data measurements.
The proposed Improved SPC framework is analyzed in two different
perspectives. First, the number of false alarm generated during normal operating
condition. Second, the successful of the proposed method to detect the fault and
identify the correct process variable as fault cause for each fault situation. Based on
the result obtained, there is no false alarm given by Shewhart and EWMA charts
while MAMR chart has one false alarm case. Small number of false alarm produced
will result less risk of taking wrong action to the normal process.
Result on fault detection efficiency shows that Shewhart chart was able to
detect all known faults in the process. EWMA chart can detect 75% of the known
faults occurs on oleic acid composition, xc8 and 77.5% of the known faults occurs on
linoleic acid composition, xc9. MAMR chart has the lowest percentage of fault
detection efficiency, that is 63.8% and 70% fault detected on oleic acid composition
and linoleic acid composition respectively.
Multivariate analysis technique used to construct the control limits for
process variables gives significant effect to the efficiency of Improved SPC chart to
identify exact fault causes. Implementation of PCorrA provides 100% fault diagnosis
efficiency using Shewhart chart while PCA provides 87.5 % fault diagnosis
efficiency. EWMA and MAMR chart using PCorrA also give better performance
compared to PCA in terms of percentage of exact fault diagnosed. PCorrA technique
performed better than PCA technique because the correlation coefficients developed
by this technique are closer to the actual value of the correlation coefficients,
representing the correlation between the selected process variables with the quality
variables of interest.
All Improved SPC charts i.e Shewhart, EWMA and MAMR charts are able to
detect faults in the process. However, Shewhart chart gives the best performance in
detecting known faults in the process. While for fault diagnosis, implementation of
PCorrA in Improved SPC chart performed better than PCA. Among these charts,
Shewhart charts give the highest efficiency in exactly identifying fault causes using
141
PCorrA. The simplicity of the presentation of the Improved SPC charts based on
coding system used makes these charts easier to analyze and interpret. As a
conclusion, the research objective on studying the performance of Improved SPC
charts to detect faults and utilizing the multivariate analysis techniques to develop
the correlation between quality variables and process variables for fault diagnosis
have been achieved.
7.3
Recommendations to Future Research
In order to enhance the applicability of the proposed Improved SPC method,
the following suggestions are proposed:
i.
Future research activities can focus to apply the proposed approach
plant-widely, either in a simulation plant, pilot plant or real plant.
i.
The developed FDD tools for incoming research activity can be
applied on a multiple unit operations case study.
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APPENDIX A
COLUMN PROFILE COMPARISON
A.1
Comparisons on temperature
Stage
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
HYSYS
174.4598
176.0961
189.1816
216.3445
216.3445
216.3445
216.3445
216.3445
216.3445
216.3445
216.3445
218.5873
225.0159
229.3237
231.0301
231.0301
231.0301
231.0301
231.0301
231.0301
231.0301
231.0301
231.0301
232.416
235.0202
235.0202
235.0202
238.3387
238.3387
MATLAB
182.6361
186.668
193.2194
197.806
202.2502
207.1006
211.4466
214.4575
216.2516
217.3226
218.0954
218.9426
220.5438
225.9756
227.3566
228.0729
228.5256
228.8687
229.1604
229.4247
229.6729
229.905
230.1275
230.3494
230.5759
230.8451
231.2717
232.4103
234.9
% Difference
4.686
6.003
2.134
-8.568
-6.514
-4.272
-2.263
-0.872
-0.042
0.452
0.809
0.162
-1.987
-1.459
-1.590
-1.279
-1.084
-0.935
-0.809
-0.694
-0.587
-0.486
-0.390
-0.889
-1.891
-1.776
-1.594
-2.487
-1.442
155
A.2
Comparisons on pressure
Stage
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
HYSYS
0.1067
0.1076
0.1086
0.112717
0.11276
0.112802
0.112888
0.112974
0.11306
0.113403
0.114089
0.116833
0.119578
0.122322
0.122351
0.122379
0.122408
0.122494
0.12258
0.122665
0.123008
0.123694
0.125067
0.127811
0.1333
0.1333
0.1333
0.1333
0.1333
MATLAB
0.1088
0.1098
0.1107
0.1117
0.1127
0.1136
0.1146
0.1156
0.1166
0.1175
0.1185
0.1195
0.1204
0.1214
0.1224
0.1233
0.1243
0.1253
0.1262
0.1272
0.1282
0.1291
0.1301
0.1311
0.1321
0.133
0.134
0.135
0.1359
% Difference
1.968
2.044
1.933
-0.901
-0.052
0.707
1.516
2.324
3.131
3.612
3.866
2.282
0.687
-0.753
0.040
0.752
1.545
2.290
2.953
3.696
4.220
4.370
4.024
2.573
-0.900
-0.225
0.525
1.275
1.950
156
A.3
Comparisons on distillate composition
Distillate Stream
n-hexanoic acid
n-octanoic acid
n-decanoic
n-dodecanoic
n-tetradecanoic
n-hexadecanoic
stearic acid
oleic acid
linoleic acid
A.4
HYSYS
0.02148
0.51648
0.39755
0.01448
1.97E-06
6.72E-10
1.43E-14
2.53E-13
2.98E-13
MATLAB
0.02164
0.52339
0.41340
0.01394
0
0
0
0
0
% Difference
0.780
1.337
3.986
-3.742
0
0
0
0
0
MATLAB
0
0
0.00359
0.57694
0.17371
0.07521
0.01611
0.13497
0.02269
% Difference
0.00
0.00
0.389
0.902
-0.220
-0.170
-0.504
-0.212
-0.208
Comparisons on bottom composition
Bottom Stream
n-hexanoic acid
n-octanoic acid
n-decanoic
n-dodecanoic
n-tetradecanoic
n-hexadecanoic
stearic acid
oleic acid
linoleic acid
HYSYS
3.32E-14
2.18E-07
0.0035
0.5717
0.1740
0.0753
0.0162
0.1352
0.0227
157
APPENDIX B
PARAMETER SELECTION TO DESIGN EWMA CHART
B.1
Parameter Selection for EWMA Chart via PCA
The out of control data was generated during observation 10th. The pumparound
flowrate was increased from 49.6 kmol/hr to 49.9 kmol/hr. Figures B1 to B5 show the
EWMA charts using different width of the control limit, L and the weighting factor, β .
Figure B1: EWMA chart for xc8, xc9 and P using L=3.054 and β =0.4
158
Figure B2: EWMA chart for xc8, xc9 and P using L=2.998 and β =0.25
Figure B3: EWMA chart for xc8, xc9 and P using L =2.962 and β =0.20
Figure B4: EWMA chart for xc8, xc9 and P using L =2.814 and β =0.10
159
Figure B5: EWMA chart for xc8, xc9 and P using L =2.615 and β =0.05
B.2
Parameter Selection for EWMA Chart via PCorrA
The out of control data was generated during observation 10th. The pumparound
flowrate was increased from 49.6 kmol/hr to 49.9 kmol/hr. Figure B6 to B10 show the
EWMA charts using different width of the control limit, L and the weighting factor, β .
Figure B6: EWMA chart for xc8, xc9 and P using L =3.054 and β =0.4
160
Figure B7: EWMA chart for xc8, xc9 and P using L =2.998 and β =0.25
Figure B8: EWMA chart for xc8, xc9 and P using L =2.962 and β =0.20
Figure B9: EWMA chart for xc8, xc9 and P using L =2.814 and β =0.10
161
Figure B10: EWMA chart for xc8, xc9 and P using L =2.615 and β =0.05
162
APPENDIX C
PARAMETER SELECTION TO DESIGN MAMR CHART
C.1
Parameter Selection for MAMR Chart via PCA
The out of control data was generated during observation 10th. The feed flowrate
was increased from 41.557 kmol/hr to 51 kmol/hr. The control limits is calculated using
PCA method. Figures C1 to C6 show the MAMR charts using different window size.
xC8 mr
10
5
0
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
Number of observation
70
80
90
100
F mr
10
5
0
xC8 ma
2
0
-2
F ma
5
0
-5
Figure C1: MAMR chart for xc8 and F using window size of three
163
xC9 mr
10
5
0
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
60
50
40
Number of observation
70
80
90
100
F mr
10
5
0
xC9 ma
2
0
-2
F ma
5
0
-5
Figure C2: MAMR chart for xc9 and F using window size of three
xC8 mr
10
5
0
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
60
50
40
Number of observation
70
80
90
100
F mr
10
5
0
xC8 ma
2
0
-2
F ma
5
0
-5
Figure C3: MAMR chart for xc8 and F using window size of four
164
xC9 mr
10
5
0
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
60
50
40
Number of observation
70
80
90
100
F mr
10
5
0
xC9 ma
2
0
-2
F ma
5
0
-5
Figure C4: MAMR chart for xc9 and F using window size of four
xC8 mr
10
5
0
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
60
50
40
Number of observation
70
80
90
100
F mr
10
5
0
xC8 ma
2
0
-2
F ma
5
0
-5
Figure C5: MAMR chart for xc8 and F using window size of five
165
xC9 mr
10
5
0
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
60
50
40
Number of observation
70
80
90
100
F mr
10
5
0
xC9 ma
2
0
-2
F ma
5
0
-5
Figure C6: MAMR chart for xc9 and F using window size of five
C.2
Parameter Selection for MAMR Chart via PCorrA
The out of control data was generated during observation 10th. The feed flowrate
was increased from 41.557 kmol/hr to 51 kmol/hr. The control limits is calculated using
PCorrA method. Figures C7 to C12 show the MAMR charts using different window size.
166
xC8 mr
10
5
0
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
Number of observation
70
80
90
100
F mr
10
5
0
xC8 ma
2
0
-2
F ma
4
2
0
-2
Figure C7: MAMR chart for xc8 and F using window size of three
xC9 mr
10
5
0
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
60
50
40
Number of observation
70
80
90
100
F mr
10
5
0
xC9 ma
2
0
-2
F ma
4
2
0
-2
Figure C8: MAMR chart for xc9 and F using window size of three
167
xC8 mr
10
5
0
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
Number of observation
70
80
90
100
F mr
10
5
0
xC8 ma
2
0
-2
F ma
2
0
-2
Figure C9: MAMR chart for xc8 and F using window size of four
B
B
xC9 mr
10
5
0
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
Number of observation
70
80
90
100
F mr
10
5
0
xC9 ma
2
0
-2
F ma
2
0
-2
Figure C10: MAMR chart for xc9 and F using window size of four
B
B
168
xC8 mr
10
5
0
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
Number of observation
70
80
90
100
F mr
10
5
0
xC8 ma
2
0
-2
F ma
2
0
-2
Figure C11: MAMR chart for xc8 and F using window size of five
B
B
xC9 mr
10
5
0
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
Number of observation
70
80
90
100
F mr
10
5
0
xC9 ma
2
0
-2
F ma
2
0
-2
Figure C12: MAMR chart for xc9 and F using window size of five
B
B
169
APPENDIX D
NUMBER OF FAULT CAUSES
D.1
Number of fault causes for single fault and multiple faults
Table D1 to Table D4 listed the number of known fault, which exactly diagnosed,
partially diagnosed, over diagnosed and undiagnosed for single and multiple faults. The
results are arranged according to different Improved SPC chart used corresponding with
multivariate analysis technique applied to construct the control limits.
170
Table D1: Fault diagnosis result using PCA technique for oleic acid composition
Fault Diagnosis Results
OC Condition
Shewhart
EWMA
MAMR
1. Exact fault cause diagnosis
•
Known 1 fault cause situation
49
31
21
•
Known 2 fault cause situation
11
7
4
•
Known 3 fault cause situation
10
6
3
70
44
28
Total
2. Partially fault cause diagnosis
•
Known 1 fault cause situation
0
0
0
•
Known 2 fault cause situation
0
1
3
•
Known 3 fault cause situation
5
8
12
5
9
15
Total
3. Over diagnosed fault cause
•
Known 1 fault cause situation
0
0
0
•
Known 2 fault cause situation
0
0
0
•
Known 3 fault cause situation
0
0
0
0
0
0
Total
4. Undiagnosed fault cause
•
Known 1 fault cause situation
1
19
29
•
Known 2 fault cause situation
4
7
8
•
Known 3 fault cause situation
0
1
0
5
27
37
Total
171
Table D2: Fault diagnosis result using PCorrA technique for oleic acid composition
OC Condition
Fault Diagnosis Results
Shewhart
EWMA
MAMR
1. Exact fault cause diagnosis
•
Known 1 fault cause situation
50
33
24
•
Known 2 fault cause situation
15
13
13
•
Known 3 fault cause situation
15
14
14
80
60
51
Total
2. Partially fault cause diagnosis
•
Known 1 fault cause situation
0
0
0
•
Known 2 fault cause situation
0
0
0
•
Known 3 fault cause situation
0
0
0
0
0
0
Total
3. Over diagnosed fault cause
•
Known 1 fault cause situation
0
0
0
•
Known 2 fault cause situation
0
0
0
•
Known 3 fault cause situation
0
0
0
0
0
0
Total
4. Undiagnosed fault cause
•
Known 1 fault cause situation
0
17
26
•
Known 2 fault cause situation
0
2
2
•
Known 3 fault cause situation
0
1
1
0
20
29
Total
172
Table D3: Fault diagnosis result using PCA technique for linoleic acid composition
Fault Diagnosis Results
OC Condition
Shewhart
EWMA
MAMR
1. Exact fault cause diagnosis
•
Known 1 fault cause situation
49
31
24
•
Known 2 fault cause situation
11
6
3
•
Known 3 fault cause situation
10
6
2
70
43
29
Total
2. Partially fault cause diagnosis
•
Known 1 fault cause situation
0
0
0
•
Known 2 fault cause situation
0
2
4
•
Known 3 fault cause situation
5
9
13
5
11
17
Total
3. Over diagnosed fault cause
•
Known 1 fault cause situation
0
0
0
•
Known 2 fault cause situation
0
0
0
•
Known 3 fault cause situation
0
0
0
0
0
0
Total
4. Undiagnosed fault cause
•
Known 1 fault cause situation
1
19
26
•
Known 2 fault cause situation
4
7
8
•
Known 3 fault cause situation
0
0
0
5
26
34
Total
173
Table D4: Fault diagnosis result using PCorrA technique for linoleic acid composition
OC Condition
Fault Diagnosis Results
Shewhart
EWMA
MAMR
1. Exact fault cause diagnosis
•
Known 1 fault cause situation
50
34
28
•
Known 2 fault cause situation
15
13
13
•
Known 3 fault cause situation
15
15
15
80
62
56
Total
2. Partially fault cause diagnosis
•
Known 1 fault cause situation
0
0
0
•
Known 2 fault cause situation
0
0
0
•
Known 3 fault cause situation
0
0
0
0
0
0
Total
3. Over diagnosed fault cause
•
Known 1 fault cause situation
0
0
0
•
Known 2 fault cause situation
0
0
0
•
Known 3 fault cause situation
0
0
0
0
0
0
Total
4. Undiagnosed fault cause
•
Known 1 fault cause situation
0
16
22
•
Known 2 fault cause situation
0
2
2
•
Known 3 fault cause situation
0
0
0
0
18
24
Total
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