... Falsafah dan Sarjana secara penyelidikan, atau disertasi bagi pengajian secara kerja kursus dan penyelidikan, atau. Laporan Projek Sarjana Muda (PSM).
PSZ 19:16 (Pind. 1/97)
UNIVERSITI TEKNOLOGI MALAYSIA
BORANG PENGESAHAN STATUS TESIS+ JUDUL:
FlJZZY STATE SPACE MODELING FOR SOLVING INVERSE PROBLEMS IN MULTIVARIABLE DYNAMICS SYSTEMS
SESI PENGAJIAN: 2004/2005
RAZIDAH BINTI ISMAIL
Saya
(HURUF BESAR) mengaku membenarkan tesis (PSM/Sarjana/Doktor Falsafah)* ini disimpan di Perpustakaan Universiti Teknologi Malaysia dengan syarat-syarat kegunaan seperti berikut: 1. 2. 3. 4.
Tesis adalah hakmilik Universiti Teknologi Malaysia. PerpustaK~an Universiti TeknologiMalaysia dibenarkan membuat salinan untuk tujuan pengajian §~haja. Perpustakaan dibenarkan membuat salinan tesis ini sebagai bahan pertukaran antara institusi pengaj ian tinggi. **Sila tandakan (./ )
D
(Mengandungi maklumat yang berdarjah keselamatan atau kepentingan Malaysia seperti yang termaktub di dalam AKTA RAHSIA RASMI 1972)
SULIT
D
(Mengandungi maklumat TERHAD yang telah ditentukan oleh organisasi/badan di mana penyelidikan dijalankan)
TERHAD
[ : ] TIDAK TERHAD Disahkan oleh
(TANDATANGAN PENYELIA)
(TANDATANGAN PENULIS)
Alamat Tetap:
NO. 10, JALA~ LIMAU PAGAR 4/4F, PROF. MADYA DR. TAHIR BIN AHMAD
40000 SHAH ALAM,
Nama Penyelia
SELANGOR DARUL EHSAN Tarikh:
CATATAN:
~I-O~-o~
Tarikh:
* Potong yang tidak berkenaan. ** Jika tesis ini SULIT •
~l-OS--O~
atau TERHAD, sUa Iampirkan surat daripada pihak berkuasaJorganisasi berkenaan dengan menyatakan sekali sebab dan tempoh tesis ini periu dikelaskan sebagai SULIT atau TERHAD. Tesis dimaksudkan sebagai tesis bagi Ijazah Doktor Falsafah dan Sarjana secara penyelidikan, atau disertasi bagi pengajian secara kerja kursus dan penyelidikan, atau Laporan Projek Sarjana Muda (PSM).
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"We hereby declare that we have read this thesis and in our opinion this thesis is sufficient in terms of scope and quality for the award of the degree of Doctor of Philosophy"
Signature Name of Supervisor I
Assoc. Prof. Dr. Tahir Ahmad
Date
Signature Name of Supervisor II
Assoc. Prof. Dr. Shamsuddin Ahmad ~l - oS'" - oS"
Date
Signature Name of Supervisor III Date
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BAHAGIAN A - Pengesahan Kerjasama* Adalah disahkan bahawa projek penyelidikan tesis ini telah dilaksanakan melalui kerjasama antara
dengan
_
Disahkan oleh: Tarikh :
Tandatangan Nalna Jawatan (Cop raslni)
* Jika penyediaan tesis/projek Inelibatkan keljasclIna. BAHAGIAN B - Untuk Kegunaan Pejabat Sekolah Pengajian Siswazah Tesis ini telah diperiksa dan diakui oleh: Nama dan Alalnat Pen1eriksa Luar
Nalna dan Alalnat Penleriksa DalalTI I
Prof. Dr. Abu Osman Bin Md Tap Jabatan Matematik Fakulti Sains & Teknologi Kolej Universiti Sains & Teknologi Malaysia Mengabang Telipot 21030 Kuala Terengganu Terengganu Darul Iman
Prof. Madya Dr. Mohd Ismail Bin Abd Aziz Fakulti Sains UTM, Skudai
Pelneriksa Dalalll II
Nama Penyelia lain Uika ada) Disahkan oleh Penolong Pendaftar di Sekolah Pengaj ian Siswazah: Tandatangan
Tarikh:
Nalna
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FUZZY STATE SPACE MODELING FOR SOLVING INVERSE PROBLEMS IN MULTIVARIABLE DYNAMIC SYSTEMS
RAZIDAH BINTI ISMAIL
A thesis submitted in fulfilment of the requirements for the award of the degree of Doctor of Philosophy
Faculty of Science Universiti Teknologi Malaysia
MAY 2005
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ii
I declare that this thesis entitled "Fuzzy State Space Modeling for Solving Inverse
Problems in Multivariable Dynamic Systems " is the result of my own research except as cited in references. The thesis has not been accepted for any degree and is not concurrently submitted in candidature of any other degree.
~.u.~_
Signature
••.. __ .. _•.
Name
_____ RAZlRAH_~JNItJ.SMAIL
Date
.. _...•...•...
31-or-o~
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.. _
_
iii
To my husband Mohd Aminudin bin Md Amin. To my children Mohd Hani"t Mohd Zharif Nur Hanis/ Mohd 'A fif Nur Husna, Nurul Huda. In memory of my parents Hajah Rokiah binti Abdullah (deceased 10 August 2001) Haji Ismail bin Zainal (deceased 4 January 2002) AL-FATIHAH
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IV
ACKNOWLEDGEMENTS
In the Name of ALLAH s.w.t, The Most Beneficient, The Most Merciful. All praise is due only to ALLAH s.w.t, the Lord of the universe. Ultimately, only ALLAH s.w.t has given us the strength and courage to proceed with life in its entirety. His works are truly splendid and wholesome. His knowledge is truly complete with due perfection. I am highly indebted to many people for their assistance in the preparation of this thesis. I wish to express my sincere gratitude to my supervisor, Assoc. Prof. Dr. Tahir Ahmad for 11is excellent supervision, invaluable guidance, helpful discussions and continuous encouragement. His enthusiasm and thoughtful considerations, and his valuable suggestion throughout the preparation of this thesis are greatly appreciated. I would also like to extend my sincere gratitude an.d appreciation to my co-supervisors, Assoc. Prof. Dr Shamsuddin Ahmad and Assoc. Prof. Dr. Rashdi Shah Ahmad, for their valuable suggestions and guidance. Many thanks are also due to Assoc. Prof. Othman Ayob, from the Faculty of Mechanical Engineering, UTM and Prof. Dr. Mohd Nasir Taib from the Faculty of Electrical Engineering, UiTM for their valuable assistances and suggestions. I would also like to acknowledge with much appreciation the crucial role of the editors and referees or reviewers for all my publications. Their valuable comments and suggestions throughout the preparation of various publications have helped me to improve my work and are highly appreciated. My special thanks to Universiti Teknologi MARA (UiTM) and Jabatan Perkhidmatan Awam (JPA) for the financial support throughout the course of my study. I would also like to express my appreciation to my family: my dearest husband, who has always provided the encouragement and support that I truly need; my lovely children, who have always been the source of inspiration and pleasure; my parents, brothers and sisters for their continuous prayers for my success. Last but not least, my sincere thanks to all my colleagues and friends who kindly provided valuable and helpful comments in the preparation of the thesis, and to those involved directly or indirectly in the preparation of this thesis, whom I have not mentioned above.
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v
ABSTRACT
The maIn objective of this study is to develop a novel inverse modeling technique, known as Fuzzy State Space Model (FSSM).
This model is used for
optimization of input parameters in multivariable dynamic systems. In this approach, the flexibility of fuzzy modeling is incorporated with the crisp state space models proposed in the modern control theory.
The vagueness and uncertainty of the
parameters are represented in the model construction, as a way of increasing the available information in order to achieve a more precise model of reality. Subsequently, the inverse Fuzzy State Space algorithm is formulated for a multipleinput single-output system, which leads to the derivation of Modified Optimized Defuzzified Value Theorem. This algorithm is enhanced to address the optimization of parameters for a multiple-input multiple-output system, which leads to the derivation of an Extension of Optimized Defuzzified Value Theorem. The proofs of these theorems are presented. To facilitate the implementation of these algorithms, a semi-automated computational tool using Matlab® programming facilities is developed. The effectiveness of this modeling approach is illustrated by implementing it to the state space model of a furnace system of a combined cycle power plant. The results obtained in this application demonstrate that the proposed new modeling approach is reasonable and provides an innovative tool for decision-makers.
In
addition, the investigations on the properties and characteristics of FSSM have resulted in the derivation of some lemma and theorems related to convexity and normality of the induced solution of the model, and bounded stability of the Fuzzy State Space system. Finally, the properties of the induced solution of a single FSSM are generalized to the multi-connected systems of FSSM. related to the systems ofFSSM are also discussed.
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Some algebraic views
VI
ABSTRAK
Objektif utama kajian ini adalah untuk membangunkan satu teknik pemodelan songsang yang baru, dikenali sebagai Model Keadaan Ruang Kabur (MKRK). Model ini di gunakan untuk pengoptimuman parameter masukan dalam sistem dinamik ITIultipembolehubah. Dalam pendekatan ini, pemodelan kabur yang fleksibel digabungkan dengan model keadaan ruang rangup dari teori kawalan moden.
Kesamaran dan
ketidakpastian bagi parameter di wakili dalam pembentukan model, sebagai satu cara menambah maklumat supaya menghasilkan model yang lebih tepat.
Seterusnya,
algoritma Keadaan Ruang Kabur dibentuk untuk sistem pelbagai-masukan keluarantunggal, yang mana Teorem Nilai Penyahkaburan Optimum Ubahsuaian diterbitkan. Algoritma ini ditambahbaik untuk pengoptimuman parameter masukan bagi sistem pelbagai-masukan pelbagai-keluaran, yang mana Teorem Nilai Penyahkaburan Optimum Lanjutan diterbitkan. Bukti teorem-teorem tersebut ditunjukkan. Untuk memudahkan perlaksanaan algoritma ini, alat pengkomputeran separa-automotik menggunakan kemudahan pengaturcaraan Matlab® disediakan.
Keberkesanan
pendekatan pemodelan ini ditunjukkan dengan perlaksanaan keatas model keadaan ruang sistem relau bagi sebuah loji janakuasa kitar padu. Keputusan yan.g dihasilkan menunjukkan bahawa pendekatan pemodelan barn yang dicadangkan adalah berpatutan dan boleh menjadi satu alat yang innovatif kepada pembuat kataputus. Selain daripada itu, penyelidikan mengenai sifat dan ciri-ciri MKRK telah menghasilkan beberapa lemma dan teorem yang berkaitan dengan kecembungan dan normal bagi penyelesaian teraruh model , dan kestabilan terbatas bagi sistem Keadaan Ruang Kabur.
Akhir sekali, sifat-sifat penyelesaian teraruh bagi sebuah MKRK
dilanjutkan kepada beberapa sistem MKRK yang berhubung. Beberapa pandangan dari sudut algebra berkaitan dengan sistem MKRK juga dibincangkan.
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VII
TABLE OF CONTENTS
CHAPTER
TITLE
PAGE
THESIS STATUS DECLARATION SUPERVISOR'S DECLARATION TITLE PAGE DECLARATION DEDICATION
111
ACKNOWLEDGEMENTS
lV
ABSTRACT
V
ABSTRAK
VI
TABLE OF CONTENTS
VII
LIST OF TABLES
XII
LIST OF FIGURES
XIV
LIST OF SYMBOLS LIST OF APPENDICES
1
11
INTRODUCTION 1.1
Background and Rationale
XVII XXI
1 1
1.1.1
Approaches in Model Construction
4
1.1.2
Related Studies
8
1.2
Objectives of the Study
11
1.3
Implications of the Study
11
1.4
Scope of the Study
11
1.5
Outline of the Thesis
]3
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VIII
2
SYSTEM MODELING IN STATE SPACE REPRESENTATION
16
2.1
Introduction
16
2.2
Systems and System Models
17
2.3
Mathematical Modeling of a Control System
19
2.4
The State Space Modeling
23
2.5
Solutions of State Equations
28
2.5.1
28
2.6
3
Analytical Solution of State Equations
2.5.2 Computer Simulation of State Equations
29
Structural Properties of a Dynamic System
33
2.6.1
Stability
33
2.6.1.1 Lyapunov Stability
33
2.6.1.2 Input-Output Stability
34
2.7
Controllability and Observability
36
2.8
Inverse Problems in System Modeling
38
2.9
Uncertainties in System Modeling
40
2.10
Summary
41
FUZZY NUMBER AND FUZZY ARITHMETIC
43
3.1
Introduction
43
3.2
Membership Functions
44
3.3
Fuzzy Sets: Basic Concepts
46
3.3.1
48
Operations on Fuzzy Sets
3.3.2 Relation among Fuzzy Sets
50
3.4
a-cuts of Fuzzy Sets
51
3.5
The Extension Principle
52
3.6
Fuzzy Numbers
53
3.7
Arithmetic Operations on Fuzzy Numbers
55
3.7.1 The a-cut Method
55
3.7.2 The Extension Principle Method
56
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IX
3.8
4
Fuzzy Systems
56
3.8.1
Fuzzification
58
3.8.2
Fuzzy Environment
58
3.8.3
Defuzzification
58
3.9
Fuzzy Modeling
59
3.10
Fuzzy Algorithms
60
3.11
Related Studies on Fuzzy Arithmetic
61
3.12
Summary
62
FUZZY STATE SPACE MODELING
64
4.1
Introduction
64
4.2
Terminology
65
4.3
Representation of Uncertainty
66
4.4
Development of the Fuzzy State Space Model
68
4.5
Fuzzy State Space Algorithm for a MISO system
72
4.6
Convexity and Normality of the Induced Solution ofFSSM
75
4.7
Modified Optimized Defuzzified Value Theorem
78
4.8
Fuzzy State Space Algorithm for a MIMO system
81
4.9
Extension of the Optimized Defuzzified Value Theorem
85
4.10
A Computational Tool
86
4.10.1 Programs for a MISO system
88
4.10.2 Programs for a MIMO system
89
4.11
Stability of a Fuzzy State Space System
90
4.12
Summary
93
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x 5
IMPLEMENTATION OF FUZZY STATE SPACE MODEL ON THE FURNACE SYSTEM 94 5.1
Introduction
5.2
Modeling the Furnace System of a Combined Cycle Power Plant
95
5.3
The Furnace System
96
5.4
Model Assumptions
98
5.5
Mathematical Modeling of the Furnace System
99
5.6
State Space Modeling of the Furnace System
101
5.6.1
Development of the State Equation
102
5.6.2
Development of the Output Equations
103
5.7
5.8
The State Space Analysis of the Furnace System
105
5.7.1
Stability Analysis
107
5.7.2
Controllability and Observability
107
Implementation of the Fuzzy State Space Algorithm in the Furnace System (MISO)
5.9
5.10
109
5.8.1
Algorithm applied to Qir
109
5.8.2
Algorithm applied to Qis, Qes, PG and Qrs
114
5.8.3
Optimized Input Parameters
119
Implementation of the Fuzzy State Space Algorithm in the Furnace System (MIMO)
6
94
122
5.9.1
Algorithm applied to Qir and QiS
122
5.9.2
Algorithm applied to Qes and Po
126
5.9.3
Algorithm applied to Qrs
129
5.9.4
Optimized Input Parameters
131
Summary
134
SYSTEMS OF FUZZY STATE SPACE MODEL
136
6.1
Introduction
136
6.2
FSSM in Multi-connected Systems
137
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Xl
6.3
6.4
6.5
7
Properties of Induced Solution of FSSM in Multi-connected Systems
140
6.3.1
Convexity and Normality
141
6.3.2 Optimized Input Value Parameter
142
6.3.3 Bounded-Input Bounded-Output Stability
143
Algebraic View of Systems of FSSM
144
6.4.1
145
Common Feeder
6.4.2 Relations
147
Summary
148
CONCLUSION AND RECOMMENDATIONS
150
7.1
Summary of the Research
150
7.2
Significant Findings and Contributions
152
7.3
Recommendations for Future Research
154
156
REFERENCES
164 - 181
Appendices A to E
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XII
LIST OF TABLES
TABLE NO.
TITLE
PAGE
5.1
Input parameters specification
110
5.2
a-cuts of input parameters
112
5.3
Output parameters specification (Qir)
112
5.4
Endpoints of interval for input parameters (Qir)
114
5.5
Output parameters specification (Qis, Qes, PG and Qrs)
115
5.6
Endpoints of interval for input parameters (Qis)
118
5.7
Endpoints of interval for input parameters (Qes)
118
5.8
Endpoints of interval for input parameters (PG )
118
5.9
Endpoints of interval for input parameters (Qrs)
118
5.10
Calculated fuzzy value for the Furnace System (MISO)
119
5.11
Calculated norm for the Furnace System (MISO)
119
5.12
Optimized input parameters (MISO)
120
5.13
Calculated output parameters for Furnace System (MISO)
120
5.14
Comparison of optimized input parameters (MISO)
121
5.15
Output parameters specification (Qir and Qis)
124
5.16
Endpoints of interval for input parameters (Qir and QiS)
126
5.17
Output parameters specification (Qes and PG )
127
5.18
Endpoints of interval for input parameters (Qes and pG )
129
5.19
Endpoints of interval for input parameters (Qrs)
131
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XIII
5.20
Calculated fuzzy value and norm for the Furnace System (MIMO)
132
5.21
Optimized input parameters (MIMO)
132
5.22
Calculated output parameters for the Furnace System (MIMO)
132
Comparison of optimized input parameters (MIMO)
133
5.23
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XIV
LIST OF FIGURES
FIGURE NO.
TITLE
PAGE
1.1
Construction of the Fuzzy State Space Model
5
1.2
Linearization of a nonlinear dynamic system
6
1.3
Organization of the Thesis
15
2.1
Principle of Superposition
19
2.2
Transfer function or Input-Output representation of a system
21
2.3
State space representation of a system
22
2.4
State space representation of a linear multivariable system
26
2.5
State space representation from governing equations
27
2.6
Plots of the state and output responses
32
2.7
Plots of responses due to step functions
32
2.8
Mathematical Modeling: (a) Forward Problems (b) Inverse Problems
39
3.1
Crisp representation of "middle-aged"
45
3.2
Fuzzy representation of "middle-aged"
46
3.3
Features of the membership function
46
3.4
(a) convex fuzzy set
3.5
Complement of fuzzy set A
49
3.6
Union of fuzzy sets A and B
49
3.7
Intersection of fuzzy sets A and B
49
. (b) non-convex fuzzy set
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48
xv 3.8.
The intersection of two convex fuzzy sets
50
3.9
Examples of Fuzzy Numbers
54
3.10
Block diagram of a Fuzzy System
58
4.1
A triangular fuzzy number
68
4.2
Phases in developing a fuzzy algorithm
70
4.3
Flowchart for a Fuzzy State Space Algorithm
71
4.4
The intersection of the fuzzy desired parameter (a) on the maximum side of the induced parameter (b) on the minimum side of the induced parameter
80
The intersection of the desired parameter on the maximum side of the fuzzy induced parameter
87
The intersection of the desired parameter on the minimum side of the fuzzy induced parameter
87
5.1
Schematic diagram of a Boiler
97
5.2
Simplified block diagram of the Furnace System
101
5.3
The state response of the Furnace System
106
5.4
The output responses of the Furnace System
106
5.5
The Furnace System subjected to step function
108
5.6
The Furnace System subjected to impulse function
108
5.7
Fuzzification of input parameters (a) fuel flow (b) air flow (c) exhaust gas flow
111
5.8
Fuzzy value for the Furnace System (Qir)
113
5.9
Fuzzy value for the Furnace System (Qis)
116
5.10
Fuzzy value for the Furnace System (Qes)
116
5.11
Fuzzy value for the Furnace System (PG)
117
5.12
Fuzzy value for the Furnace System (Qrs)
117
5.13
Surface and contour plots for the Furnace System (Qir and QiS)
125
4.5
4.6
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XVI
5.14
Fuzzy value for the Furnace system (Qir and Qis)
125
5.15
Surface and contour plots for the Furnace system (Qes andpo)
128
5.16
Fuzzy value for the Furnace system (Qes and Po )
128
6.1
An example of a simple multi-connected system
136
6.2
Multi-connected systems of type A
140
6.3
Multi-connected systems of type B
140
6.4
SgFl
is a feeder of SgF2
146
6.5
SgF
is a common feeder of SgFl and SgF2
146
6.6
SFFis
the greatest common feeder of SgFI and SgF2
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147
XVII
LIST OF SYMBOLS
General symbols {X], X2, X3, ••• }
Set of elements Xl,
{xlp(x)}
Set determined by property P
(Xl, X2, X3, .•• , X n )
n-tuple
[Xij]
Matrix
[Xl, X2, X3, ••• , X n ]
Vector
11···11
Euclidean norm
X2, X3, ...
min
[Xl, X2, X3, •• • ,Xn ]
-
Minimum of Xl,
X2, X3, ••• , X n
max
[Xl, X2, X3, •• •,Xn ] -
Maximum of X],
X2, X3, ••. , X n
i,J', k
Arbitrary identifier (indices)
[a, b]
Closed interval of real numbers betweell a and b
[a, b)
Interval of real numbers closed in a and open in b
[a, 00)
Sets of real numbers greater than or equal to a
(a, b)
Open interval of real numbers between a and b
(a, b]
Interval of real numbers open in a and close in b
9t
Set of real numbers
9-\+
Set of positive real numbers Set of n-tuple of real numbers
~
Set of positive integers (natural numbers)
UorX
Universal set
A,B, C, A==B
...
Arbitrary sets (crisp or fuzzy) Set equality
Af-B AcB
Proper set inclusion
AcB
Set inclusion
A nB
Set intersection
Set inequality
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XViii
A uB
Set union
AxE
Cartesian product of sets A and B
o
Empty set
f
X~Y
Function ofjfromXinto Y
[I: X~ Y
Inverse function ofjfrom Y to X
R (X Y)
Relation on X x Y Implies If and only if Element of
E
Not an element of Such that There exist (at least one) For all Membership grade of x in fuzzy set A a-cut of fuzzy set A Strong a-cut of fuzzy set A sup A
Supremum of A
suppA
Support of fuzzy set A
h(A)
Height of fuzzy set A
core(A)
Core of fuzzy set A
alb
a divides b
gcd(a, b)
Greatest common divisor of a and b
gcf(a, b)
Greatest common feeder of a and b
SgF
Fuzzy State Space Model
[Uj, U2,
,Un]T
Input vector
Lv], Y2, ,Ym]T
Output vector
F li (x)
Degree of desirability of using a particular value x Fuzzy induced performance parameter Desired performance parameter A collection of interconnected FSSM systems Common feeder Greatest common feeder
COPYRIGHT © UiTM
XIX
Furnace System
CF
Fuel calorific value
J/kg
cpg
Specific heat of exhaust gases at constant pressure
Js/kgOK
cgs
Combustion gas specific heat capacity
Js/kgOK
WF
Fuel flow
kg/s
WA
Air flow
kg/s
WEG
Gas mass flow through the boiler
kg/s
WG
Exhaust gas (from the gas turbine) flow
kg/s
Win
Inlet mass flow
kg/s
W ou
Outlet mass flow
kg/s
wg
Flow of substances entering combustion
kg/s
hG
Exhaust gas (from the gas turbine) specific enthalpy
J/kg
hA
Air specific enthalpy
J/kg
hEG
Gas specific enthalpy
J/kg
hrej
Reference exhaust gases enthalpy
J/kg
hin
Inlet specific enthalpy
J/kg
hou
Outlet specific enthalpy
J/kg
Qir
Heat transferred by radiation to risers
J/s
Qis
Heat transferred by radiation to the superheater
J/s
Qrs
Heat transferred to reheater
J/s
Qgs
Heat transferred to superheater
J/s
Qes
Heat transferred to the economiser
J/s
Qin
Incoming heat flow
kg/s
Q
Heat flow
J/s
Tg
Gas temperature
T gs
Gas temperature at the superheater
Tst
Superheater metal tube temperature
T gr
Combustion gas temperature at the reheater
Trh
Reheater metal tube temperature
T ge
Combustion gas temperature at the economiser
Tel
Economiser metal tube temperature
T g1
Boiler exhaust gas temperature
Trej
Reference exhaust gases temperature condition
OK OK OK OK OK OK OK OK OK
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xx
Tm
~etal
tennperature
oK
Ts
Steann tennperature
oK
Rs
Stoichionnetric air/fuel volunne ratio
(-)
REG
Ideal gas constant for exhaust gases
y
The percentage excess air level
%
V
Volunne
m3
VF
Combustion channber volunne
nn3
(J'
Stefan-Boltznnan constant
f)
Burner tilt angle 0< ()