simulation of boiler drum process dynamics and...
TRANSCRIPT
1
1
Simulation of Boiler Drum Process Dynamics and Control
Jian Zhao
B. En,.
Department of Mechanical Engineering
McGill University
Montréal, Canada
A thesis submitted to the Faculty of Graduate Studies and Research
in partial fulfillment of the requirements f >r the aegree of
Master of Engineering
January 1992
@ Jian Zhao
1
1
Abstract
This thesis presents a mathematical process model for the dynamic analysis
of a vertical reheat boiler and the application of this model to the optimal design of a
drum water level controller. A numerical finite difference technique is used to formulate
this model.
The control system contains two loops, a feedforward loop using the steam
flow and drum pressure signais as the input and a feedback loop using the deviation of
the measured drum water level from its set point as an input. The feedback loop is an
incremental PlO controller with an adjustable proportion a 1 gain. The feedforward loop is
designed to directly actuolte the control devices before the "swell" and "shrinkage" in the
boiler water level occur. The feedforward controller output signal is summed along with
the output of the PlO controller to establish the set point for the c.ontrol actuator ThiS
scheme is effective because steam flow changes are immediately fed forward to change
the final feedwater set point for the control actuator. In this way, feedwater flow tracks
steam flow and any disturbances in the feedwater system will be arrested quickly
It is shawn that an incremental PlO controller plus adapt feedforward com
pensator can be successfully employed for the control of water level in such a plant.
Il
1 1
1
Résumeé
La thè~e présente un modèle mathématique de processus qUi peut servir dans
l'analyse dynamique d'une chaudière verticale à postcombustioh, et dans la conception
optimale d'un contrôleur de niveau d'eau du tambour. Pour l'élaboration de ce modèle,
on a utilisé UfI" tc-rttnique numérique différentielle.
le système de commande comporte deux boucles' une à commande directe
utilisant l'échappement de la vapeur et les indicateurs de pression du tambour, comme
donnée d'entrée, et l'autre, rétroactive, utilisant la déviation dans la mesure du niveau
d'eau du tambour, à partir de son ~oint de réglage. la boucle de commande rétroactive est
un régulateur proportionnel, intégral et différelltiel (PID) avec gam proportionnel ajustable
la boucle de commande directe tst conçue afin que l'organe de commande réagisse directe-
ment avant que ne surviennent le "gonflement" et la "diminution" du niveau d'eau dans la
chaudière. Les indications données par le contrôleur du niveau d'eau ajoutées aux résultats
obtenus par le régulateur PlO permettent d'établir le point de réglage du mécanisme de
commande. Ce procédé est efficace parce que les variations dans l'échappement de la
vapeur sont immédiatement et directement transmises et modifient le point de réglage
final du mécanisme d'alimentation d'eau. De cette façon le mécanisme d'alimentation
d'eau s'active dès qu'il y a échappement de vapeur le tambour se trouvant au niveau
du mécanisme d'alimentation d'eau arrêtera immédiatement toute pE'fturbatlon dans le
système d'alimentation d'eau.
On démontre ainsi qu'un régulateur PlO, utilisé avec un compensateur à cam-
J mande directe adaptable, peut-être employé avec succès comme contrôleur de niveau
d'eau dans les usines visées.
III
1 Acknowledgements
The author wishes to express his deep gratitude to Prof. louis J. Vroomen
and Dr. Paul J. Zsombor-Murray for their supervision of the research project and their
guidance, encouragement and patience during the course of this study.
Special acknowledgements are due to fellow members of the simulation and
modeling group: David Vaitekunas. Chen Huan-Wei and Eric VanWalsum.
A note of thanks is also extended to ail McRClM technicians and system
management persons who helped in the study of this project.
IV
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J
Contents
List of Figures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. '"
List of Tables . ........................................ .
Nomenclature . .................................. .
Chapter 1 Introduction ..................... .
1.1 Description of the Boiler .................. .
1.2 The Control of Modeled Boiler System by Simulation
1 2.1 Model Development Strategy ......... .
1 2.2 Control Simulator Development Strategy
Chapter 2 Process Model of the Drum Water level
System ............................ .
2.1 The Structure of the Drum Water level System ...
2.2 The Method to Develop the Model .. .... . ..
2.3 Process Model of the Feedwater Valve ....... .
2.4 Process Model of Recirculation loop
24.1 Downcomer Model ...
2.4.2 Riser Model . .
2.5 Process Model of the Feedwater line .. . .....
2.6 Process Model of the Drum ............... .
2.7 Steam Flow line Model . . . . . . . . . .. ..... . ....
VIII
XII
XIII
1
2
5
5
9
12
12
15
18
19
19
20
21
2?
25
v
1 Chapter 3 Oynamic Analysis of the Process Model . . . . . .. . 30
3.1 Solution of the Process Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 30
J.l.1 Runge-Kutta method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 31
3 1.2 The Application of Runge-K\Jtta Methods on the Model . . . . . . . . 33
3.2 Simulation Methods ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .. 35
3.2.1 Variable Step-Size Multistep Aigorithm .. . . . . . . . . . . . . . . . . .. .. 35
3.2.2 Simulation Program Structurf' ... . . . . . . . . . . . . . . . . . . . . . . . . . .. 44
3.3 Oynamic Behavior of the Process Model .. . . . . . . . . . . . . . . . . . . . . . . . .. 46
3.3.1 Time-Domain Dynamics of the Process Model . . . . . . . . . . . . . . . .. 47
3 32 Time-to-Frequency-Oomain Transformation Method . . . . . . . . . . 52
3.3.3 Frequency-Domain Oynamics tIf thp. Process Model. . . . . 55
Chapter 4 Control System Structure ............. . 65
4.1 The Outline of the Controlled Plant. . . . . . . . . .. ...... ............ 65
4 2 Drum W~ter Levet Control System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
Chapter 5 <Control Strategy . . . . . . . . . . . . . . . . . . . . .. ......... .... 71
5.1 Control Aigorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. ..... 71
5.1.1 Digital PlO Control Technique. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 71
5.1 2 Feedforward Control Aigorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 74
5.1.3 Combined PID-feed1orward Control Algorithm ................. 77
5.2 Refinements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .. 78
5.2.1 Data Sampling and Filtering . . . . . . . . . . . . . . . . . .. ............ 79
5 22 Improving the Differentiai Character of Output Response . . .. . . 80
~I
r 5.2.3 Antireset Windup ........................... .
5.3 The Control Program Structure. . . . . . . . . . . . .. . .. .
Chapter 6 Selection of Design Param<C!ters .
6.1 Ziegler-Nichols Method. . . . . . . . . . . . . . . .. . ...... .
6.2 The Feedback Controller Parameters Sett,"g ..... .
6.3 Influence of the Sampling Time At . ................ .
Chapter 7 Presentation of the Results ..
7.1 Closed-Ioop Behavior of the Process . . .. ..
7.1.1 Specification of Closed·loop Response ... .
7.1.2 Closed-Ioop 8ehavior Us,"g Feedback Control ..
7.1.3 Feedback-Feedforward Control 8ehavior ..
7.2 The Simulation Results ....
Chapter 8 Conclusions ......... .
8 1 Drum Water level System Model .. ..
8 2 Simulation Methods ............... .
8.3 Control Aigorithm .............. .
8.4 Future Work
References. . . .. ............. .. ....... . .. .
Appendix A. Sample Values of Frequency w ....
81
82
85
85
86
89
98
98
", 99
100
102
103
113
113
114
115
117
120
123
VII
1
List of Figures
1.1 Typical industrial boiler unit. . . . . . . . . . . . . . . . . . . . . . . . .. ........ .. 3
1.2 Boiler plant component models . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . .. 6
2.1 Schematic diagram of boiler drum system . . . . . . . . . . . . . . . . . . . . . . . . .. 13
2.2 The drum model - block diagram ................................. 15
2.3
2.4
31
3.2
Simplified recirculation loop diagram
Srmplified drum diagram.. ................... .
Flow chart of the programming procedure ....... .
The structure of the simulation program ., ............. .
19
23
36
45
3.3 Change in drum water level for 10 percent step increase ..... . . . . . . . .. 48
in feedwater flow n.' !
3.4 Change in steam discharge for 10 percent step increase .. . 48
in feedwater flC'w rate
3.5 Change in drum water level for 10 percent step increase ..... . . .. .... 49
in fuel flow rate
3.6 Change in stearn discharge for 10 percent step increase . . . . . . .. . .... 49
in fuel flow rate
3.7 Change in drum water level for a 50-psi step decrease ....... .... ... 50
in system pressure
3.8 Chang~ in steam discharge for a 50-psi step decrease . . . . . . . . . . . . . 50
in system pressure
3.9 Change in drum water level for 10 percent increase . . . . . . . . . . . . . . .. . 51
in steam flow rate
\/111
1 3.10 Frequency response of drum water level for 10 percent 59
step increase in feedwaler flow rate
3.11 Frequency response of steam discharge for 10 percent 60
step increase in feedwater flow rate
3.12 Frequency response of drum water level for 10 percent 61
step increase in fuel flow rate
3.13 Frequency response of steam dischetrge for 10 percent 62
step increase in fuel flow rate
3.14 Frequency response of drum water level for a 50-psI ..... 63
~tep decrease in system pressure
3.15 Frequency response of steam discharge for a 50-psI 64
step decrease in system pressure
4.1 Feedback control loop for drum water level system 67
42 General drum water level control system .. 69
5.1 PlO control with derivative on Pl :cess measurement 72
5.2 The feedforward control alg0f1thm 75
5.3 The adjustment of the gain Kc ........ . 82
5.4 The control program structure .......... . 83
6.1 The closed-Ioop responses from the initi'il and final controller
parameters .... .................. . ... . 90
6.2 Trall~ient response of the process for 10% increase ln steam flow
wlth sample interval ~t=l second ..... . 94
6.3 Transient response of the process for 10% increase ln steam flow
with sample interval ~t=2 second 95
J 6.4 Transient response of the process for 10% increase ln steam flow
wlth sample interval ~t=4 second. '" ... .. 96
1,1(
1 6.5 Transient response ct the process for 10% increase in steam flow
with sam pie interval At==8 second. . . . . . . . . . . . . . . . . .. .. ..
7.1 Step response of water level change for a 10 percent step in<.rease
97
in turbine governor valve area for difTerent values of Kc .. . . . . . . . . . . .. 105
7.2 Step response of water level change for a 10 percent step i nCiease
in turbine governor valve area for difTerent values of J(c
(Cont'd) ....................................... .
7.3 Step response of water level change for a 10 percent step i ncrease
in turbine governor valve area for difTerent values of TI
7.4 Step response of water level change for a 10 percent step Increase
in turbine governor valve Jrea for difTerent values of Tl
!Cont'd) ....................... " ............. .
7.5 Step response of water level change for a 10 percent step Increase
106
.. 106
107
in turbine govt:rnor valve area for difTerent values of TD 108
7.6 Step response of water level change for a 10 percent step Increase
in turbine governor valve area for dlfTerellt values of TD
(Cont'd) 109
7.7 Response of water lev el for a step change of 5 inches in set
point.. .. ............. " .......... .. . . ......... .. .. 109
7.8 Closed·loop response of drum water level change to a 10 per cent
step increase in governor valve area (feedback control) 110
7.9 Closed-Ioop response of drum water level change to a 10 per cent
step increase in governor valve area (feedforward-feedback
control) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .. ........ . ... 110
7.10 Step response of boiler's four most important variables for a 10
percent step increase in turbine governor valve area for optimum
controller settings ........................ . 111
x
1 7.11 Step response of boiler's four most important variables for a 10
percent step increase in turbine governor valve area for o~timum
controller settings (Cont'd) . . . . . . . . . . .. ....... .... .... 112
XI
..
1 List of Tables
2.1 Feedwater valve positioner characteristics ........................... 18
2.2 Values of constants Cl to C7 ...................................... 29
3.1 Sorne physical dimensions of the sample boiler. and the values
of its variables at steady-state 44 per cent full load .................. 34
:1
XII
Nomenclature
Symboll
A : flow cross-sectional area [ft2]
a: reset windup boundary
Cl to C7: tonstants
D : hydraulic diameter (ft)
e: error
f: friction factor
G: transfer function
g: acceleration of gravit y (ftjsec2]
Hl : pump head-flow characteristic
H2 : feedwater valve loss-coefficient lift characteristic
H3: specifie-volume-pressure relationship for saturdted vapor
H4: specifie-volume-pressure relationship for critical flow
T i
h: fluid enthalpy [Btu/lb]
)'111
1 i : number of simulat:on steps
./: 778 (ft-Ib/Btu)
/(: feedwater valve positioner gain
/(c : feedback controller gain
/(,: feedforward controller gain
L: feedwater valve lift (ft)
1 : pipe equivalent length [ft)
P: Controller output
PB: proportional band
p: steam generator pressure [psi]
Pl : feedwater pump discharge pressure (psi]
Q: heat transfer rate (Btu/ft3 - sec]
s : Laplace transform variable
T: steam temperature[ OF)
'l'OL: simulation tolerance
t : time (sec]
l v: steam drum volume [ft3]
XIV
~----------- ~-
1 v: specifie volume [ft3/lb)
W: fluid mass flow rate [lb/sec)
>[ . .J • quality of steam-water mixture leaving riser
Y: sampling value
y: water level [ft]
.,. . .. . z transform variable
f3: filtering coefficient
p: fluid density [lb/ft3)
(: feedwater valve positioner damping factor
f : coefficient of simulation algorithm
TD: differential period [sec]
TI : integral period [sec)
w: feedwater valve positioner natural frequency [sec- 1)
h.t : time increment [sec)
Subscripts
o : initial condition
cr : critical flow
xv
1 d: steam flow li ne
D: circulatin, flow li ne
dw: steam-water drum
e: feedwater line
f: saturated liquid
9: saturated vapor
m,: governor valve openin,
n' governor valve inlet
0: steady-state value
v: feedwater valve
w: steam-water mixture leaving the riser
Superscript
average
XVI
1
Chapter 1 Introduction
Although boiler controls have progressed to a certain extent, the role of the
boiler room operator has essentially remained the same durtng the last twenty years Even
today, boiler operators still physically check water levels at gauge glasses and vlsually
verify the pressure and temperature controls [6].
Thus safe and efficient operation of an industrial process depends on the
human operator as weil as the real-time process control system 80th elements must
perform weil together. The operator, in order to make decisions, depends upon the
availabillty, the timeliness, and the quality of the information presented to hlm by the
process control system. If the control system does not perfoim weil, the operator, no
matter how capable, will experience great difficulty ln ma king the nght decision On
the other hand, even if the control system does perform weil, the operator may not be
psychologically or technically prepared to make the nght deCISlon
Due to the infrequent occurrence of upset conditions ln the process, the abliity
of the operator and the control system to cope with these situations rernains untested
It is therefore quite clear that in order to efficiently train the operator and program the
control system, it is necessary to use simulation of both normal and abnorrnal operatlng
conditions, in real-time situations.
1 Introduction
Early studies of boiler process and control simulation tended towards the de~
velopment of complete transfer functions for the process and the controller. Then the
parameters of the controller were determined with convention al control techniques Even
when the process models are quite simple, this approach leads to controller transfer func-
tions which are either physically unfeasible or, at best, extremely complicated [2] With
the availability of modern high speed computers, it is now feasible to use direct digital
control (DOC) simulation of an entire plant.
This thesis discusses a method of using direct digital control and modern
computer techniques to combine the conventional PlO and feedforward control algorithms
Into a control simulator for a boiler drum system. In this project, a dynamic model of
a boiles drum syst~m will be presented first. Then the application of this model to
the optimal des;gn of a water level digital control simulator will be demonstrated The
performance of this control slmulator applied to this model will also be discussed It is part
of ongoing research at the McGill Research Centre for Intelligent Machines (McRClM) to
develop a boiler plant control simulator.
1.1 Description of the Boiler
The purpose of a boiler is to generate process steam, whether it be part of
a pulp and paper mill, a power station or on a marine vesse!. The system under study
is limited to a particular class of boilers: large-scale natural circulation fossil-fuel boilers
which opera te on any combination of oil, coal, natural gas, or wood. A dlagram of the
bOlier unit considered in this project is shown in Figure 1.1.
Fuel and air are delivered into the mill at a controlled rate. The mil! exhaust
fan blows the fuel-air mixture to burners in the combustion chamber or furnace. At the
2
t .~.
L
1
J
1 Introduction
Figure 1.1 Typical industrial builer unit
burners, the iir required for efficient combustion is supplied as secondary air, and the
fuel-air mixture is burned in suspension in the furnace. The combustion gases are wlth-
drawn from the furnace zone and pass, in succession, through the furnace eXit tubes, the
secondary-superheater tube bank, the pflmary-superheater tube bank and the economlzer
The flue gases finally pass through a heat exchanger, before belng reJected to the stack
at a temperature just in excess of the dew point of the sulphurous gas present
3
1 1 Introduction
Air is drawn from the top of the boilerhouse, where it is warm. It is then
passed through the heat exchanger, where it absorbs heat from the flue gas. By means
of ducts and dampers, a proportion of this air is used as primary air, the remainder is
distributed to the burner as secondary air.
Feedwater from the br;:er feedpump enters the boiler via the economizer,
where it absorbs sufficient heat to raise it to almost the saturation temperature corre
sponding to the boiler pressure. From the economizer, the feedwater flows through the
downcomers to horizontal headers at the bottom of the furnclce waterwalls. Water from
these headers is discharged into the waterwall tubes lining the furnace; heat is absorbed,
and partial evaporation takes place. The mixture of steam and water from the waterwalls
enter into the steam drum. In the drum, the saturated steam is separated from the water,
whlCh is recirculated via the downcomer-waterwall loop either by the difTerence between
flUld density, and/or a pump. The steam passes, in sequence, through the prlmary and
secondary superheater-tube banks and then to the plant, e.g., a turbine
From the control point of view, a boiler can be functionally dlvided into three
main loops comprising:
(a) the combustion loop
(b) the drum-feedwater loop
(c) the steam-temperature loop.
This thesis is limited to a study of the drum-feedwater loop.
To better understand the role of the individual elements of a boiler, consider
the component model diagram, Figure l.2. The inputs to the plant are air, fuel and
4
1
J
Introduction
water which ire converted using various energy processes to generate steam, the pJlnClpal
output, and combustion giS, ~ waste by-product. The diagram only illustrates the physlCal
processes which must be interfaced with the plant control system via control signal Inputs
Four elements of the boiler system are involved with the control of the drum water level
ln this project, the dynamic mathematical model of these elements was establsshed flrst,
then the direct digital control schemes were applled to these models
1.2 The Control of Modeled Boiler System by Sinll.lation
1.2.1 Model Development StratelY
Earlier water-Ievel control studies used "Iumped- parameter" 'frlresentation of
flow behavior in various system flow segments. The mathematical model was then slmu
lated on a real-time computer for the optlmization of the water-Ievel controller parameters
[1). In a lumped·parameter model, restrictive assumptlons pertalning to the spatial dlstJl
bution of the dependent variables within each segment must be made, thereby reduclng
the accuracy of the solution. Rigorous water-Ievel control studies must Include the slmul
taneous solution of the space-dependent and time-dependent conservation equatlons of
mass, momentum, and energy for the coolant flow through the steam plant. Dynamlc
models of this type will be discussed in this thesis.
For the purpose of digital-computer simulation, the boiler was subdlvided into
a number of sections, comprising:
1) feedwater valve/boiler t'eed pumps
2) recirculation loop
•
! • LI ..
1
1"" " "------..1; :i i e :. III • ....... Q
!~---~ i
DOWJiICOM8 MODIL
Figure 1.2 Boiler plant component models
1. Introduction
PUlLHIA'I'D MODIL
6
Introduction
1 3) drum
4) primary superheater
5) superheater spray
6) second.uy superheater
7) turbine governing stage
8) turbine
9) reheater spray
10) reheater
11) mills
12) combustion
13) superheater furnace
14) reheater furnace
Only those processes involved in item 1), 2), 3) and 7) will be dlscussed ln
this thesis.
By use of appropria te laws governing fluid dynamlcs, the empmcal. steady-
state heat-transfer relationships and the equations of state (nonllnear dlfferentlal and
algebraic) were written to describe the dynamic behavior of the above-listed four sections
of the boiler. These equations were then Ilnearized by consldenng only srnall variations
1 Introduction
• about a steady-state operating level. The resulting equations were used as the process
",odel by the simulated controller on the computer.
The major difficulty in the boiler's dynamic behavior analysis is the fact that
the whole system is very complex and contains n!Jmerous variables which are extremely
unwieldy to manipulate. The solution of the set of equations is as difticult to obtaln as the
synthesis of the actual mathematical description. large number of variables, nonlinearities,
and uncertainties in many phenomena ail contribute to the complexity of the problem.
One is then faced with the necessity of ma king some simpllfying assumptions ln order
to facilitate a solution. Oversimplification results in solutions which do not deswbe the
character of dynamic behavior properly, but too little or no simplification might Involve
unreasonable time and expense in obtaining a solution. Necessar, slmphfying assumptlons
may be classified under two types depending on the nature of simplification'
(a) certain physical phenomena are too complex to admit an exact mathematical
description, and some empirlcal or approximate formulation is required.
(h) the need to obtain a solution within a reasonable time for the set of equations
derived from analysis.
The simplifications may be made in the nature of flow or in the equations
themselves by eliminating nonlinearities, partial differentiation through appropriate tech-
niques (numerical analysis methods were used in this thesis). In the present study these
simplificatIons are combined in such a way that a set of linear ordinary differential equa-
tlons describes the dynamic behavior of the boiler drum system. Even after simplIficatIon,
the set of equations obtained in the following chapter is not tractable for hand so!ntion
and It is necessary to use machine computation.
8
1
. ...,
1 Introduction
Both the process model and the control simulatcr were written in the C
language on a SUN SPARCTM work station.
1.2.2 Control Simulator Development Strate~y
The control of the water level in the boiler drum and the feedwater flow is
extremely important to the operation of conventional steam generatll1g plants large
level variations can affert system power and hydrodynamic stabi!ity. Translent changes
in feedwater flow can also affect the plant power output and even create thermal shock
problems. In order to minimize these adverse efTects, it is deSifable even dunng large
fluctuations in steam demand to maintarn the water level close to a predetermmed set
point. The feedwater flow rate, should be adjusted in re~l-tlme to compensa te for the
level changes [21].
Boiler controls were essentially designed to control actlvity w,th,n a boder and
ensure its safe operation. Each boiler cons:sts of a pressure vessel, furnace a'ld burner,
working a~ an integral unit within a specific capacity If operated improperly. the bolier
experiences serious problems. Controls were developed to minlmlze these nsks
Separated by function, these devices fall into two areas of control, tlreslde
and waterside. Fireside devices monitor the burner and its operation They sense the
existence and strength of the f1ame using, in most cases, electronic, ultraviolet or mfrared
sensors. Fireside controls also check fuel pressure and temperature Waterside devices
control the water temperature, pressure and level in order to ersure safe operation The
control simulator developed in this thesis is only used to manlpulate the waterslde devlces
Pneumatic, analog, PlO control developed in the 1930'5 15 consldered to be
the first gener.:-tion of process control. Electronic, analog, PlO controllers, wh,ch appeared
9
-
1 1 Introduction
in the 1950'5, may be regarded u the second generation. In the 1970'5, direct digital
control (DOC) wu developed. But single loop PlO controllers are still in use where DOC
cannot be justified because of insufficient computing needs [20]. With the rapid growth in
high speed computers, particularly microcomputers, it has now become feasible to develop
a digital control simulator for an entire plant.
This project discusses a digital PlO combined with a feedforward scheme as
a control algorithm for a single loop digital control simulator. This digital control sim
ulator will be applied to a boiler drum. An attractive aspect of digital control is its
flexibility in implementing various algorithms. This has be reenforced by rapid growth
in software technology. Theoretically, because of time sampling and signal quantization,
digital PlO control cannot attain the same quatity as analog PlO control. But practically,
with powerful computers and modern software tools, it is possible to ma~e digital PlO
control perform as weil as or even better than its analog counterpart. The digital control
simulator includes five distinct blocks:
(a) feedwater valve
(b) recirculation loop
(c) feedwater line
(d) steam drum
(e) steam flow line
The first step to be taken in the control-system analysis is obvious from these
considerations. It consists of acquiring a complete understanding of the process to be con
trolled. The understanding begins with the formulation of a phenomenological description
10
IntroductIon
of the process. i.e., by following the physical ind chemical changes that occur along the
process path, and t:nds when transfer functions, that is, time-dependent functional re
lations between input (controlling) and output (controlled) variables of the process, are
obtained. This first step, which may be ca"ed "dynamic analysis", leads to the mathemati
cal description of the system so that an intelligent choice of the basic design configurations
for the cOlltroller becomes possible. The second step is to develop a real-time controller
based upon the results of the mathematical description of the boiler's dynam;c response
to various disturbances.
From this point of view, ,his digital water-Ievel controller study consists of
two parts. The first is a digital computer program simulating the dynamlC behavior of
the steam drum plant with length along the flow path and time as system independent
variables. This dynamic boiler drum plant model will be established in Chapter 2 Chapter
3 will present the dynamic behavior of this model both in the time-domain and frequency
domain. The time-to-freqllency domain transformation technique will also be shown
Then the drum model will be simulated using this technique. The second part 15 a digital
simulator for controller parameter optimization studies. The structure of the control
system will be described in detail in Chapter 4. In Chapter 5, the digital PID combined
with ;a feedforward control algorithm will be presented A filter which is used to Irnprove
the sampling quality will also be discussed. Chapter 6 will analyze the stabillty of the
control system. The controller parameter tuning technique will also be dlscu5sed The
performance of the model and the control algorithm used to simulate It Will be analyzed
in Chapter 7. Finally in Chapter 8. a summary of the flOdlOgs will provlde a baSIS for
further research into other aspects of process control.
11
1
Chapter 2 Process Model of the Drum Water Level System
The primary objective of this ch~pter is ta generate a drum water level system
model th~t beh~ves with reasonable accur~cy in representing the dynamic relationships
which exist between the gtnerated outputs, including: steam flow and water level, and
the manipulated input variable: feedwater v~lve position.
To be useful for analysis, the model is kept simple, but is still consistent with
the above requirements. Furthermore, the model must reflect the specifie values of ail
plant parameters ~t any ,iven operating level. This model can then be used to subject any
proposed control scheme ta a sensitivity an~lysis with respect ta certain variable boiler
pilrameters.
2.1 The Structure of the Drum Water Level System
The system under study is a typical industrial steam generator, and its drum
subsystem was modeled as the five sections shown in Figure 2.1.
1) Feedwater Valve - The feedwater valve regulates the feedwater flow from the
pumps. It consists of the feedwater valve itself and an electric-to-pneumatic
converter.
2. Protess Model of the Drum Waler level System
1
Figure 2.1 Schematic dialram of boiler drum system
13
1
1
2. Process Model of the Drum Water level System
2) Recirculation loop - The recirculation lûop includes a heating zone, riser, steam
separator and downcomer.
3) Feedwater Line - The cold water from the feed pumps go through the feedwater
line into the downcomer of the recirculation loop.
4) The Drum - The steam drum coUects the steam when it leaves the recirculation
loop.
5) Steam Flow Line - This is the steam outlet line from the drum. A governor
valve controls the mass f1ow.
Boilers operating below the critical point, except for once-through types, are
customarily provided with a steam drum in which saturated steam is separated from the
steam-water mixture discharged by the boiler tubes. Saturated liquid leaves the drum and
enters the downcomer while a saturated liquid vapor mixture enters the drum from the
risers. Saturated steam leaves the drum and enters the primary superheater. The drum
may also serve as a vessel for chemical boiler water treatment. However, the primary
functions of this drum are to provide a free, controllable, surface for the separation of
saturated steam from water and a housing for any mechanical separating devices.
Mass balance equations for water, steam and a liquid/vapor mixture have been
established for the drum water level system, from feedwater valve to primary superheater.
The controlled plant is shown in Figure 2.2.
The following basic assumptions have been adopted:
• 80th steam and liquid in the drum are at saturation temperature.
14
2. Proeess Model of the Drum Waler level System
1
Plll!DWA1D DaUM DOWNCOMEl IUSlll
VALVII
Filure 2.2 The drum model - block diagram
• This temperature is a function of drum pressure.
• There is negligible heat transfer along the length of the downcomer.
• There is no mass or energy storage in the downcomer and rlSer.
• The total circulating flow rate is constant.
2.2 The Method to Develop the Model
The model of the drum water level system was developed based upon the
mass, energy balance and the state equation of the system:
(a) Mass Balance: These are the well-known Navier-Strokes-type equations for
one-dimensional nonturbulent flow. Viscous friction is neglected 50 that the
15
1 2. Process Model of the Drum Water level System
velocity profile icross the flow is constant. However, a frictional-Ioss propor
tion~1 to the square of the lelocity is included in the momentum equation.
Continuity. momentum, and energy equations are applied with certain sim pli-
fyin, assumptions, mentioned later.
(h) Energy Balance: Empirical equations are used to determine the rate of heat
trinsfer from hot gas to superheated steam and to boiling liquid.
(c) State Equations: These are approximated from steam tables for saturated and
superheated steam about the steady-state operating conditions. The relations
are iSsumed to be linear for a given range of values of the variables.
It was mentioned in Chapter 1 that the first two types of equations involve
partial diff'erentiition as weil as nonlinearities. In order to facilitate a solution of these
complicated equations, it is necessary to reduce these equations to an ordinary linear-
equation form by applying small perturbations and diff'erence equation techniques.
Suppose an equation of the form:
(2 -1)
is to be reduced to linear ordinary differential equation form. In Equation (2 - 1) ft indicates time derivative and lm is the derivative with respect to the spa ce variable m.
ft is assumed that for small space intervals AI. the variables x, y, z, ... may be written as
linear functions of the variable m such that:
X2 - Xl ay Y2 - YI az z2 - z1 -ôm-= M 'am= M 'am= M ôx , ...
16
1 2. Process Model of the Drum Water Level System
where X2. !l2. '2 •... and Zl. YI. Zl. '" denote the value of the variables x. y. z . ...
Olt the end and at the beginning. respectively. of the space interval AI. Even though Tt,
YI. zb ... and X2. Y2, Z2 • ... are no longer functions of m, they are still functions of
time. x. y. z, ... ~. ~. ~ •... are now assumed to be thfl value of the variables at
the beginning of the space interval }.·f; hence Equation (2 - 1) can be written as
(2 - 2)
The Equation (2 - 2) is then perturbed about its steady-state operating con-
dition in order to eliminate the nonlinearities.
Hence it cOIn be written as:
(2 - 3)
where il dâ,t, ... cOIn be replaced by 1,(ll.xll . ... for sma" perturbations.
It is seen from Equation (2 - 3) that time derivative terms d;i, c!p" (~l, .. are treated as independent variables. and second or higher-order terms in perturbed vari
ables are neglected. The partial difTerentials i~ ./k. 3ft .... that form the coefficients
of the perturbed variables are evaluated at the initial steady-state operating condition
about which the dynamic behavior of the drum water level system is to be analyzed. As
a result of these simplifications, Equation (2 - 3) becomes a linear, first-order, ordinary
difTerential equation with constant coefficients in perturbed variables llXt, ll.T2. llYl.
17
1 2. Process Mode! of the Orum Water level System
2.3 Process Model of the Feedwater Valve
Condens~te is delivered to the inlet of the boiler feed pumps from the feedwater
heater. The feedwater flow from the pumps is regulated by the feedwater valve. The valve
positioner has built-in position feedb"k to eliminate drift in valve position. After analysis
and consultation with control equipment manufacturers, il simple second-order transfer
function was selected to ~pproximate the positioner's behavior [8).
L K
P = (w'n)2 + 2({w'n) + 1 (2 - 4)
The characteristics of the selected feedwater valve positioner are shown ln
Table 2.2.
Time for Full Valve Stroke Natural Frequency ColIn Dampins Coefficient' Seconds Radians/seconds
2 2.0 10 10 0.9 224 20 0.63 3.18 40 0.45 446
Table 2.1 Feedwater valve positioner characteristics
The electric-to-pneumatic converter can also be represented by a second-order
system having a natural frequency of 20 radians/second and an approximate damping
coefficient of 1.0. Consequently, the dyna mic characteristics of the converter are neglected
and the gain factor is incorporated in that of the valve positioner. A nominal valve speed
of 10% of full range per second was assumed.
18
2. Pro cess Model of the Drum Water Level System
2.4 Process Model of Recirculation Loop
A mathematical model of the recirculation loop was developed based upon tha t
of MacDonald (19). using several of his assumptions. For example. the system pressure,
feedwater f10w rate and the heat transfer from the combustion system are assumed con
stant during one control step. Figure 2.3 shows the simplified diagram of the recirculation
loop with the variables indicated along the path.
RIS ER DOWNCOMER.
Figure 2.3 Simplified recirculation loop diagram
2.4.1 Downcomer Model
As mentioned before. the downcamer is assumed ta have simple fluid-flow
19
1
(
2. Process Model of the Drum Waler level System
characteristics with no heat input or temperature difference between the top and the
bottom.
Usina acceptance test data (19) variations of downcomer out/et enthalpy he
with changes in feedwater flow, are approximated by the following quadratic:
he = -499.7251 We 2 + 655.7251 We + 365.0398
(-1162.3606We2 + 1525.2166We + 849.0826)
Btu/lb
kJ/kg (2 - 5)
An energy relationship between feedwater flow and circulating flow based on
adiabatic mixin, of two streams, is given by:
(2 - 6)
2 ••• 2 Riser Model
Feedwater discharging from the circulating pumps enters distribution headers
beneath the furnace sections. The water rises in the furnace riser tubes where boiling
takes place and the water-steam mixture enters the drum from the riser.
The enthalpy of the steam-water mixture leaving the risers is:
(2 -7)
The energy balance equation is given as:
Qw = Qws + Qwr = WD(h w - hD) (2 - 8)
20
1
J 2 Process Model of the Drum Ww~tr level System
2.5 Process Model of the Feedwater li ne
The feedwater line model is derived by application of the momentum equation
to the feedwater flow throulh the feedpump, feedwater valve and interconnectlng pIpes
The momentum equation for the feedwater flow is liven by,
(2 - 9)
ln the above equation. the contribution of momentum change, wh,ch 15 smalt,
i5 neglected. The pump di5charge pressure Pl is a glven function of mass flow rate and IS
obtained from the pump head-f1ow characteristic:
(2 - 10)
The feedwater valve 1055 coefficient Iv is a function of feedwater valve lift and
i5 obtained from the valve spec.:ification:
(2 - 11)
Introducing small perturbations around steady state for We. 1), ]JI. Iv and l"
expanding Equations (2 - 10) and (2 - 11) usÎng Taylor-Series about the steady state
point and neglectinl second and higher order terms, Equation (2 - 9) can be linearized.
It can be easily shown that this linearization pro cess gives:
21
2 Proeess Model of the Orum Water Level System
1 llWe(( dPI) _ 2(PI - plo) ~ IIp + AL (df tJ ) (6p)tJo
dWe 0 Weo dL 0 fvo (2 - 12)
Reuranging Equation (2 - 12) and deflning constants Cl and C2 as:
results in:
(2 - 13)
2.6 Process Model of the Drum
Figure 2.4 is the diagram of the drum showing various input and output vari-
ables. Two types of equations, mass-balance and energy-balance determine the behavior
of the drum sinee it is basically a eapacitance. In this study the efTect of the mass-transport
phenomena between the two phases is taken into consideration, but the water-Ievel varia-
tion due to bubble formation is neglected. In other words, it is assumed that evaporation
and condensation take place at the surface of the liquid phase. Henee the mathematieal
model of the drum dynamics takes the form of:
Mass balance:
22
1
f
2 Process Model of the Drum Water level System
RISBR:
Energy balance:
WDh w - (WD - We)hdw - Wdhd =
d dt«Vd - Vdw)Pdhd + VdwPdwhdw)
V APOR PHASE:
TO SUPERHEATERS:
ST'EAM FLOW RA'm TEMPERATURE DENSITY
MASS FLOW RATE _-..,.
PRESSURE TEMPERA1UIlE
DENSrrY VOLUME
(2 - 14)
(2 - 15)
QUALn'Y TEMPERATURE DENSrrY / ---. UQUlDPHASE: ~
,------ MASS fVV'v
-------r UQlllDLEVEL
DOWNCOMER: MASS PLOW RATE TEMPERATUR.E DENSITY
~ VOLUME 1'\.1'\./". FEEDWATER' DENSrrY --+--
TEMPERAruRE MASS FLOW RATE ~ TEMPERATURE
Figure 2.4 Simplified drum diagram
1
1
'l'·
2. Process Model of the Drum Water Level System
The following linear approximation of thermodynamic state relations are sat-
isfactory for conditions in the drum:
Pdw = -2.0132Pd + 49.7224 Ib/ Jt'; (-32.2483Pd + 796.4759 kg/m3)
Td = 20.5097Pd + 525.9288 oF (-6.3835Pd + 214.4049 OC)
Pd = 273.5046Pd + 539.5794 psi (1885.8142Pd + 3720.4000 kPa)
hdw = 32.5179Pd + 49&.3699 Btu/lb (75.6366pd + 1l59.2084 kJ/kg)
hd = -18.9583Pd + 1238.9003 Btu/lb (-44.0970Pd + 2881.6821 kJ / kg) (2 - 16)
They were calculated for saturated steam temperatures between 600 and
6500 F using data from Keenan and Keyes steam tables[5J.
It is convenient to define the funetion:
(2 -17)
where:
o.hd = (~hd) = _18.9583Btu I..!!!-. (_2.7530
kJ / kg)
8Pd sat lb ft 3 kg m3 (2 - 18)
o.hdw:= (8hdw ) = 32.5179Btu /~ (4.7220
kJ / kg)
8Pd sat lb Jt 3 kg ml (2 - 19)
24
1 2. Process Model of the Drum Water Level System
(ÔPdW) apdw = -ô = -2.0132
Pd .at (2 - 20)
The mass and energy balance equations (2 - 14) and (2 - 15) can now be
transformed to obtain:
(2 - 21)
1+ X d .!.tlU'e - -LWd - -WD Il' Pdw Pdw Pdw
-"dw = dt 1 + g(l-..eL)
Pdw
(2 - 22)
2.7 Steam Flow Line Model
The steam-f1ow li ne model consists of a steam f1o~,1 model and a governor
valve model. In the development of the steam f10w model, the steam flow rate IS lumped
with the steam generator drum volume. The steam f10w model is derived by application
of the momentum equation to the steam f10w from steam generator discharge nozzle to
governor valve.
(2 - 23)
ln the development of the above equations, the steam discharge through the
steam flow line is assumed to be 100% dry, therefore the contribution of momentum
change may be neglected. Moreover, the steam f10w volume is lumped with the steam
25
1 2 Process Model of the Drum Water level System
generator drum volume. Consequently, the time-dependent term in the momentum equa
tion can be ignored. The siturited vapor specifie volume Vg is a function of main pressure.
Since the steim flow pressure is viriable, it is issumed that Vg is calculated from steam
tables on the basis of average main pressure,
where
_ P+Pn P= 2
(2 - 24)
(2 - 26)
8y introducing small perturbations around steady state for p, pn, Vg and
Wd' expanding Equation (2 - 24) using Taylor-Series about the steady state point and
neglecting second and higher order terms, Equation (2 - 23) can be linearized as follows:
~ _ A = f.l.W:o ((dvg) ~- 2vgo At-I' ) P Pn 2 D A2 .r.= p + u! d 9 •• up 0 t'Ji 80
(2 - 26)
Rearringing Equation (2 - 26) and defining constants C3 and 04 as:
results in:
26
r 2. Process Model of the Drum Water Level Sy$tem
1 (2 - 27)
The steam velocity at the valve opening is related to stagnation conditions by
the steady state energy equation:
Uer = J2gJ(hg - hcr } (2 - 28)
The enthalpy hcr is found by maximizing. under isentropic condition. the mass
velocity given by the following equation:
(2 - 29)
Where p and h are function of pressure at the valve opening. The steam flow
rate through the governor valve is then given by:
(2 - 30)
where:
(2 - 31)
l and:
27
2. Process Model of the Drum Water level System
Per = CSPn (2 - 32)
Introducin, 5mall perturbition5 iround steady stite for Wd. Am and Per,
expandin, Equation (2 - 31) usin, Tiylor-Series about the steady state and neglecting
second and higher order ter ms. one obtiin:
Definin, constants C6 and C7 u:
results in:
rt _ W,o (dper) '-'6 - - -
Pero dpn 0
(2 - 33)
(2 - 34)
28
1 1
2. Process Model of the Drum Water level System
(onstants Values
Cl 4.103 C2 -0.5215 C3 1.038 C4 -0.1109 Cs 0.58 C6 0.11 C1 452.2
Table 2.2 Values of constants Cl to C7
.'.
29
1
1
Chapter 3 Oynamic Analysis of the Process Model
Once i dynamic process model has been developed, methods to solve this
model should be selected ind i dynamic analysis of this model should be done in order
to be able to design the suitable controller for this mode!. By solve, we mean thctt the
transient responses of the dependent variables can be found to sorne degree of accuracy by
numerically integriting the difTerential equations, given that appropriate initial values for
the dependent variables have been specified and that the inputs also have been specified
as functions of time. Oynamic analysis of the model will be based on time ana Irequency
response of the model.
ln this chapter, the method to solve the process model will be discussed first
then the simulation methods and simulating software will be described, finally the dynamic
behavior of the pro cess model will be presented.
3.1 Solution of the Process Model
Over the years applied mathematicians have developed a large number of nu
merical integration techniques from the simple (e.g., the Euler method) to the complicated
(e.g., the Runge-Kutta method) (13). Ali of these techniques represent sorne compromise
1 3. Oynamic Analysis of the Process Model
between computational effort (computing time) and accuracy. While a dynamic model
can always be 5OIved, there may be difficulties in obtaining useful numerical solutions
in some cases. A repertoire of modern computer programs for integrating ordinary dif·
ferential equations is available on most computer systems. Since most programs were
designed for general simulation, obtaining dynamic model solutions to systems composed
of a number of simulation equations may not be simply a straight forward process of using
standard integration routines. There will be a much larger expenditure of computer time
and cost. Secondly the cost of selecting, integrating and perfecting an extensive numerical
procedure is also high.
After a careful comparison and study of the available numencal techniques.
standard simulating programs and the entire process model, the Runge-Kutta method was
selected and it was decided to develop customized simulation software.
3,1.1 Runge-Kutta method
ln analyzing engineering systems, it is frequently necessary to solve sets of
simultaneous first-order differential equations. Such systems of equations are often used
to solve higher order difTerential equations which can be validly transformed into a set
of first-order difTerential equations. The process model in this project can be formulated
as a set of simultaneous first-order differential equations. Runge-Kutta methods are weil
suited to their solution. The application of these methods will be discussed.
Runge-Kutta formulas can be used to solve a pair of simultaneous first·order
differential equations of the form:
31
1 3. Dyn.mic Analysis of the Proeess Model
dy dz = J[z,y(x), u(z»)
du dz = F[z,y(x),u(z»)
(3 - 1)
where the initial values (y = YO, u = uo, when x = Zo) are known. Using the classical
fourth-order method described in (13). the following sets of equations are required:
where
where
kl = h[J(Zi' Yi, uïl)
k2 = h [J ( zi + ~, Yi + ~, ui + Y) ] k3 = h [J( xi + g, Yi +~, ui +~)]
k4 = h[J(Zi + h, Yi + k3, ui + q3»)
ql = h[F(Zi, Yi, Ui»)
q2 = ... [ F (Xi + g, Yi + ~, ui + y) ] q3 = h [F(Xi + g,Ui +~, ui +~)]
q4 = h[F(Xi + h, Yi + k:}, ui + q3»)
(3 - 2)
(3 -3)
(3 -4)
(3 - 5)
32
1 3. Dynamic Analysis of the Process Model
The function I(z, y, u) is used for calculating k values, sinee it is equal to
~,and k's are used in the recurrenceformula for 11. The funetion F(Z,lI,z' is used to
caleulate q values, since it is equal to ~, and q' 8 are used in the reeurrenee formula for
u values.
Runge-Kutta methods could be called single-step methods, sinee they require
only that Yi to be known in order to determine Yi+l' Thus th(~e methods are selfstarting.
Runge-Kutta methods were among the earliest methods employed in the numerical solution
of difTerential equations, and they are still widely used. As with any methods, they
possess certain advantages and disadvantages which must be weighed in considering thelr
suitability for a particular application. The principal advantage of Runge-Kutta methods
is their selfstarting feature and consequent ease of programming. One disadvantage IS the
req uirement that the function I( z, 11) must be evaluated for sever al sllghtly dlfTerent values
of x and 11 in every step of the solution (in every inerement of x by h) This repeated
determination of l(z,lI) may result in a less efficient method with respect to computing
time than other methods. But with today's high speed computers, this problem can be
easily solved and real-time simulation still can be obtained using the approach developed
in this thesis.
3.1.2 The Application of Runae-Kutta Methods on the Model
To be able to simulate the proeess model, sorne physical dimensions and
parameters of the sample boiler were adapted from Chien (4). These values are listed in
Table 3.1.
The solution of Equation (3 - 1) begins by substitut,"g the initial values of y
33
1
3. Oynamic Analysis of the Pro cess Model
Orum pressure Superh.at.r out let pressure Steam fIow rat. Feedwater rat, Quality of mixture leavin, riser Circulation rat,
Orum diameter Orum len,th
Variabl. values
Physical dimensions
1218 psi (8398.11 kPa) 1200 psi (8214 kPa)
32.6 lb/sec (14.1815 ki/sec) 32.6 lb/sec (14.1814 ki/sec)
4.3 per cent dry 990 lb/sec (449.064 ki/sec)
42ft (1 2802 m) 17 ft (5 1816 m)
Table 3.1 Some physical dimensions of the sample boiler, and the values of its variables at steady-state 44 per cent full load
and u, which must be known, into the given difTerential equations to obtain initial values
of the funetions 1 and F. Values of k1 and q1 are next obtained by multiplying the initial
values of 1 and F by h as indicated in Equtltion (3 - 3) and (3 - 5) and where h = tJ.x,
the step increment. When values of k1 and q1 are known, k2 and q2 are evaluated next,
then k3 and q), and finally k4 and q4. Then the recurrence formulas (Equation (3 - 2)
and (3 - 4)) are used to obtain values of y and u at x = Xi + h (Yi+1 and ui+d. These
new values of y and u are then used as starting values in the next iteration, to obtain
values of Yi+2 and Ui+2 at Xi + 2h, and so on, until the desired range of integration has
been reached.
ln this projeet, !(x, y, u) = Pd and F(x, y, u) = Vdw' By substituting the
initial values of We and Wd into Equation (2 - 21) and (2 - 22), the values of Pd and
Vdw ' required to determine k1 and ql, expressed in Equation (3 - 3) and (3 - 5), are
obtained. The other 3 k and q values are then determined. Ali four k and q are used
in the recurrence formulas, (3 - 2) and (3 - 4), to obtain values of Wei+l and Wdi+l'
34
J 3. Dynamic Analysis of the Process Model
These are then used in Equation (3-3) and (3-5) to recalculate k and q for substitution
into Equation (3 - 2) and (3 - 4) to obtain values of Wei+2 and Wdi+2' and 50 on An
outline of the procedure is shown in the flow chart of Figure 3.1. The initial values used
are:
t - 0 {We = 32.61b/sec (14.7874kg/sec) - Wd = 32.61b/sec (14.7874kg/sec)
3.2 Simulation Methods
The main problem in simulation is the solution of the modeling equations. If
these equations are complex or nonlinear or contain transcendental funetions, analytical
solutions are impossible. Therefore an iterative trial-and-error procedure must be devised
Based on the given initial conditions, a next step value is estimated by employing numerical
methods (Runge-Kutta in this thesis). Then this approximation is cheeked by predefined
convergent conditions to see if it satisfies them. If not, a new estimate 15 made and the
whole process is repeated until the iteration converges withlO the required "mits ln thls
trial-and-error procedure, the tolerance must be specified first and the step-slze must be
determined at each iteration to optimize the speed and aeeuraey of the simulation. The
details of the algorithm designed to meet the speed and aeeuraey requirements will be
described in this section. This is followed by a description of the simulation software in
which this algorithm is applied.
3.2.1 Variable Slep-Size Mullislep Aigorithm
ln the simulation of a difTerential equation:
35
•
f
DEFINI! STATEMBNT FUNCI10N
ItEADDATA
1. INmAL VALUE OPTIMB 2. TIMB INCRBMENT 3. PROO' INCRBMENT
.1NI11AL FBED WA TER FLO 5. lNJ11AL S"ŒAM FLOW
6. MAXIMUM 11MB
lNI'J1AlJZB PRINT 11MB
CALaJLAm"l AND RHO
CALCULATE tl,t2,k3 AND t4
3. Oynamic Analysis of the Pro cess Model
INCREMENT 11MB
N
INCllEMENT PRlNT11MB
Figure 3.1 Flow chart of the prolramminl procedure
36
1
T
3. Oynamic Analysis of the Process Model
y' = J(t,y) (3 - 6)
whenever a tolerance TOL > 0 is given, a minimum number of steps should be used to
ensure that the Ilobai error:
(3 - 7)
where:
yeti) - the exact value of the solution of the difTerentiai equation
Wi - the approximation obtained at the i-th step.
Generally speaking, integration step size must be varied during the procedure
in order to minimize the number of steps while controlling the global error. In this section
we examine a variable step-size multistep algorithmic technique that controls the error
efficiently through appropriate choice of step size. This algorithm is designed to avoid
waiting time with very smalt step sizes in regions of derivatives, and to avoid large step
sizes in analytic regions with higher order derivatives of great magnitude.
ln the study of the variable step-size multistep algorithm, the four-step Adams
Bashforth method was employed as predictor and the three-step Adams-Moulton method
as corrector in the error control procedure[13]. These methods are multistep methods
which use the approximation at more than one previous step to d~termine the approxi-
mation at the next step. Because of the 'multistep', these methods will generate a more
accu rate result than the 'single-step' method. The algorithm can be described as follows:
• Definitions:
37
1 3 Oynamic Analysis of the Process Modet
WPi: the predicted variable at i-th step.
WCi: the corrected variable at i-th step.
hi: the interval variable at i-th step.
q: the current step coefficient of hi
• Aigorithm:
h WPi = Wi-l + 24 (55f(ti-l! wi-d - 59f{ti-2! Wi-2)
+37/(ti-3! Wi-3) - 9f(I.-4, Wi-4»
h WCi = Wi-l + 24 (9f(ti! WPi) + 19f{ti-b Wi-l)
-5f{ti-2! Wi-2) + f(ti-3, Wi-3»
191WCi - wPil t: = 270hi
(3 - 8)
A difTerence equation method is said to be convergent with respect to the
differential equation it approximates if:
38
1 3 Dynamic Analysis of the Procm Model
(3 - 9)
Otherwise it is divergent. To better understand the local truncation error, we define that
the estimation at the next step is convergent when the error between the exact value and
the approximation is within the defined to/erance. Otherwise, we say the estimation is
divergent at this step. Under most conditions, the results will either converge or diverge
at the current step. In case of convergence, the local truncation error can be tolerated,
but perhaps at the cost of computation time. In this case, since the multistep method is
used in the algorithm and the new equally spaced starting values must be computed, a
change in step size for this method is much more costly in terms of functional evaluations
than single-step method. As a consequence, it is common practice to ignore the step-size
change whenever the local truncation error is between TOL/lO and TOL. The deflnltion
of TOL/lO rather than 0 is to avoid large computing cost. On the other hand, large
divergence cannot be permitted. In su ch a case, the algorithm follows a back-up procedure
as described as follows:
if: ERROR> TOL
then:
hi = qh j
tj = ti-l + qhi
If the local error exceeds the tolerance. the previous caleulalion 15 repeated
but with an estimated step size as a funetion of this error.
8ased on the algorithm discussed above, thl! following iterative procedure was
developed for the simulation program.
39
1 3. Oyn.mie Analysis o( the Protus Model
• Definitions:
Input:
a: start point
b: end point
Q: initial condition, y(o) = Q
hmox: maximum step size
hmin: minimum step size
Output:
Wi: the approximation of lI(tï) at i-th step
h: the step size at ;-th step
• Procedure:
(1) Set up a subalgorithm for the Runge-Kutu fourth-order method to be ca lied
RK4 which ICcepts as input a step size h and starting values Vo = y(XO,UO)
and returns (Zj, vj)lj = 1,2,3 defined by the following:
for j = 1,2,3
set:
40
1 3. Dynamic Analysis of the Process Model
kl = (h)f(Xj,Yi,"i)
k2 = (h)f( xi +~,Yi + ~,Ui + '1) k3 = (h)f(Xi +~,Yj + ~,Ui + 9f) k4 = (h)f(Xi + h,Yi + k3,Ui + 93)
91 = (h)F(Xi, Yi, ",)
92 = (h)F( xi +~,Yi + ~,Ui +~)
93 =(h)F(Xi+~,Yi +~,Ui+9f)
94 = (h)F(Xi + h, Yi + k3. ui + q3) 1
Yi+l = Yi + 6(k1 + 2k2 + 2k3 + k4)
1 ui+l = "i + 6(ql + 2q2 + 2q3 + q4)
(2) Input the initial conditions, by setting:
to = a;
wo = 0;
h = hmax;
Output (to, Wo)
[3] BiSed on the initial conditions, calculate the first, second and third estima-
tions. Cali RK4;
Set
N FLAG = 1; (Indicates computation (rom RK4)
M FLAG = 1; (MFLAG=O indicates acceptable computation)
i = 4;
t = t3 + h;
41
1 3 Oynlmic Analysis of the Process Model
[4] 8ased on the input functions, iterate ail the solutions until the final condition
is reached. While (ti-l ~ b or MFLAG = 1) do (5) to (9).
(5) 8ased on the startins values, calculate the values of predictor and corrector.
Set:
w P = Wi-l + 2~ (55J(ti-lt Wi-t> - 59J(ti-2, Wi-2)
+37/(ti-3, Wi-3) - 9/(ti-4, Wi-4»)
wc = Wi-l + ;. (9f(ti, WPi) + 19J(ti-b wi-d
-5/(ti-2,Wi-2) + !(ti-3,Wi-3»)
191WC- WPI t = 270hi
(6] Test the new estimation for convergence:
a) If ( ~ TOL, the solution has converged:
Set Wi = WC; (Result accepted)
MFLAG= 0;
and proceed to step (7).
b) Else, continue executing step (5] using the following transfer of
variables:
42
1 3 Oynamie Analysis of the Process Model
[7J The solution has now converled at the ,iven time step. The next estimation
should be set up:
i = i + 1;
NFLAG = O.
18J Before executing the next iter~tion, the local truncation error needs to be
further analyzed in order to optimize the step size.
a) If f ~ O.1TOL, the step size h should be increased.
Set HOLD = hi;
q = (T~L) 1/4
If q > 4 (given upper bound)
then set hi = 4hi
If hi > hmax then set hl = hmax.
If ti-l + 3h ~ b then set h = HOLD (Avoiâ terminating with
changes in step size)
b) If h f:. BOLD, the step size has been changed. The new starting
values at the current point need to be recalculated
Set NFLAG = 1;
Cali RJ(4; (recalcula te the new starting vdlues)
43
..
1
1
3. Dynamic Analysis of the Pro cess Model
Set i = i + 3; and proceed with step (9).
c) Else the result is rejected ind set up a new step size:
Set: MFLAG = 1;
( )1/4
Set: q = TgL
If q < 0.1 then set h. = O.lh.;
If hi < hm in then set hi = hmin.
(9) Set li = ti-l + hi return to step (5).
It should be emphasized that since the multistep methods require equal step
sizes for the stirting values, any change in step size necessitates recalculating new starting
values at that point. This is done by ca iii ni a Runge·Kutta subalgorithm (see Section
3.1.1). In addition, the coefficient q is generally given an upper bound to ensure that
a single unusually accurate approximation does not result in too large a step size. The
algorithm presented incorporates this safeguud with an upper bound of 4.
3.2.2 Simulation Prolram Structure
The simulation program was designed to carry out a simulation of the boiler
drum water level system as an initial dynamic analysis of its process model. Furthermore,
this simulation program will be employed as a control plant for the drum water level
44
1 3 Dynamic Analysis of the Process Model
soumON ALOORl'l1lM
INPUT PUNCI10N PltOCBSS OENDATOI MODEL
slMULAnON MONlTOa
PIt oœss U'I'PU1' 0
~--------------------------------------------------------.------
Figure 3.2 The structure of the simulation program
controller. In the design of the program, various general purpose numerical simulation
methods were adopted and developed.
Tite structure of the simulation program is shown in Figure 3 2. The program
consists of four parts: the plant model, the input function generator, the simulation
monitor and the solution algorithm. The plant model consists of the physical process
models configuring the drum water level system. The function generator, supplied by the
45
...
• 3. Dynamie Analysis of the Process Model
user. is used to deliver plant input 50 that the transient response of the plant can be
generated correspondine to the input. The simulation monitor carries out the following
funetions:
a) Steady-state initialization (t = to): a steady-state solution of the plant process
models is determined based on the function inputs at t = tO'
b) Transient simulation (to < t < t,,): the plant is simulated over the period
tt, - to based on the function generated inputs.
c) 1/0 discretization: the step size for each finite time solution is determined by
usine the variable step-size multistep algorithm described in Section 3.2.1.
Then. bued on the step size and input functions. the solution algorithm exe
cutes each model once per iteration until a convergent solution is obtained.
The simulation program is only the simulation of the plant model and is used
to analyze the dynamic behavior of the plant. The simulation program should be combined
with the plant controller to become a drum water level simulator. In the simulator. the
plant inputs will come from the operator through the controller and steady-state will be
reached after a transient process.
3.3 Dynamic 8ehavior of the Process Model
Employing the simulation program described in the last section. the dynamic
behavior of the plant model will be presented both in the time-domain and the frequency
domain in this section. By time-domain. we mean obtaining the dependence of the system
46
1 3. Oynamic Analysis of the Process Model
variables on time by $Olvin, the dift'erential equations aescribinc the system. These dy
namic funetions tell us what is happenin, in the real world (here in simulation) as time
increases. By frequency-domain, we are permitted to look at the dynamic relationships be
tween input variables and output variables. Through these relationships, the controller for
the desired plant can be designed. A time-to-frequency domain transformation technique
was al50 developed.
3.3.1 Time-Domain Dynamies of the Pro cess Model
The purpose of the time-domain dynamic analysis of the pro cess model is
essentially to present the behavior of the plant 50 that a better understanding of the drum
water level system can be reached. Secondly, the transient response of the process model
can be used as a reference for the control system design. In the time-domaln analysls, the
types of systems and types of disturbances should be determined first ln this proJect, the
system is a set of diff'erential equations and the disturbances are step changes in feedwater
f10w rate, fuel flow rate, system pressure and steam f10w rate.
Starting from a set of steady-state values for the main dependent variables,
the simulation program is employed to determine the non-linear time behavior of the water
level and the steam dischar,e due to step chanles in fuel fJow rate, feedwater flow rate
and system pressure. The tabulated time data (plotted in Figures 3.3 to 38) are then
used to approximate the transfer funetions which describe the uncontrolled behavior of
the recirculation loop mode!.
Figures 3.3, 3.5 and 3.7 show the transient response of drum water level y
when there are s~ p changes in feedwater f10w rate, fuel flow rate, or system pressure
respectively. Figures 3.4, 3.6 and 3.8 shows the response of steam dlscharge WB for a
41
1 3. Oynamic Analysis o( the Process Model
1
0.9
O.,
c: 0.1
l 0.6
1 0.5
•• f G.4
0.3
J 1 1 • 1
0.2
0.1
00 5 10 15 3D
Time, NCIlIIdI
Filure 3.3 Chan,. in drum water level for 10 percent step incruse in f •• dwat., flow ,at.
0
·1
·2
·3
..
., -6
·1
"0 5 10 15 1) 2S
"., .... Filure 3.4 Chan,. in stum discharae (or 10 percent step increase
in (eedwater f10w rate
48
r 1
3 Oynamic Analysis o( the Process Model
1 001
0
-0.1
c -0.2
l -cu
1 -00" J
1 -00'
-006
-G07
-001
-0090 10 15 3D li)
n-.~
Filure 3.5 Chan,e in drum water level (or 10 percent step increase in fuel f10w rate
102
-J 001
1 006
1 • o. ..
1 0.2
00 , 10 15 3D li)
nme. MaIDIII
" Figure 3.6 Chan,e in steam discharge for 10 percent step increase in fuel f10w rate
49
1
c
l 1 JI
1
J 1 1 • 1
3. Oynamic Analysis o( the Process Model
0.9
O.,
0.7
06
0.5
0.4
0.3
0.2
0.1
0 0 S 10 IS 20
Tune. eecoadI
Filure l.7 Chanle in drum water level (or a 50-psi step decrease in system pressure
40
30
Il
10
°0 S 10 lS 20 2S
11me, leCOIIdI
Figure 3.8 Chan,e in stum dischar,e (or a 50-psi step decrease in system pressure
30
50
1
cf
r 1 .. J
3. Oynamic Analysis of the Process Model
0.5
0
.0-'
-1
-1.5
-2 • 0 5 10 15 20 ~ 30 35 40 ~5
TIme, leCœdI
Figure 3.9 Change in drum water leI/el for 10 percent Increase in steam flow rate
st
step-change in feedwater flow rate, fuel flow rate and system pressure respectlvely The
response curves indicate that the y/We, y/W" y/p, Ws/We, Ws/HI, and W.~h/ transfer
functions can each be approximated with reasonable accuracy by a single tlme constant
ln Figure 3.9, the response of drum water level wlth 10 percent step Increase ln steam
flow rate is presented. ft is interesting to note that far sudden ch<lnge in thE' steam flow
rate, the water level initially goes up. and then starts decreasing at a more or less u niform
rate. This is similar to the swell observed in bailers when there is a sudden Increase in
load. The swell observed here is the result of sudden evaporatlan in the rlser because
of the drum-pressure drop. Hence the set of equations developed will predlct swell and
shrink, at least qualitatively.
ln a similar manner, any of the boiler variables used ln the derivation can be
51
1
3 Oynamic Analysis of the Process Model
found as a function of time and afore-mentioned input variables. The effect of load or
fuel changes, for example, on the circulation rate, quality of mixture leaving riser, wall
temperatures, and 50 forth, can be studied and some qualitative understanding of the
system dynamics can be obtained.
3.3.2 Time-to-Frequency-Domain Tran.formation Method
Theoretical/y, the frequency-domain dynamics can be obtained by substituti"g
lW for s in the system transfer function. Practically, however, the system transfer function
is not easily available. In most engineering applications the system is too complicated to
al/ow the formulation of a realistic transfer function. In this case, the frequency-domain
dynamics have to be obtained from the time dependent experimental test data (from
simulation data in this project). In this section, two types of time-to-frequency-domain
transformation techniques will be discussed and a comparison of these two methods will
be presented.
3.3.2.1 Step Testina
A plant operator makes changes from time to time in various input variables
such as feed rate and steam rate from one operating level to a new level. These step
response data are often easily obtained by merely recording the variables of interest for a
few hours or days of plant operation. These data can also be converted into frequency
response curve (17), basical/y by difFerentiating both curves in the frequency domain.
Consider a process with an input Q(t) and an output y(t). By definition, the
transfer function of the process is:
52
r 1
3 Oynamic AnalySIS of the PrOCt5S Model
G( ) = }'(s)
s Q(s)
Employing Laplace tr.nsformation and substituting .Ii = 1""', it gives
Employing Euler identity, Equation (3 - 11) becomes:
where:
A - iB G{rw) = C _ iD
[Ty A = Jo y(t)cos(wt)dt
[TJI B = Jo y(t)sin(wt)dt
[TQ C = Jo Q(t)co.<,(wt)dt
fTQ D = Jo Q(t)sm(CN,t)dt
Finally, rationalization of Equation (3 - 12) gives:
R G(' ) _ AG + J3 D
e zw - C2 + D2
AD-BG ImG(rw) = C2 + D2
(3 - 10)
(3 - 11)
(3 - 12)
(3 - 13)
(3 - 14)
(3 - 15)
where ReG(zw) and ImG(iw) denote the real and Imaglnary parts of r"(lW),
respectively. The amplitude ratio and phase angle of G( l....J) may be obtalned from
• 3 Oynamic Analysis of the Process Model
AR = IG(iw)1 = J ReG2(iw) + ImG2(iw) (3 - 16)
,p = LG(,w) = tan- 1(ImG(iw}/ReG(iw)
Thus, the task of coalculating frequency respons! data from step test data
reduces to bein, able to evaluate the integrals A. B. C. and D. expressed in Equation
(3 - 13), for known funetions y(t) and Q(t). The integrations are with respect to lime
between the definite limits of zero and the end of the period of time of interest, TJI for
y(t) and TQ for Q(t).
ln this project, y(t) is drum water level or drum pressure obtained from the
simulation monitor (see Section 32.2) and Q(t) is the step change in feedwater flow rate,
fuel flow rate 01 steam flow rate.
3.3.2.2 Digital Evaluation
Equation (3-11) is the fundamental time-to-frequency domain transformation
technique. It is simple and easy to program. But the main disadvantage of this method
is the oscillatory behavior of the sire and cosine terms olt high values of frequency (For
details see Section 3.3.3). This problem Will cause an inaeeuracy of the frequeney response
of the process.
To improve the transformation technique. Fourier transformation WolS em
ployed to evaluate Equation (3 - ll) directly. The Fourier integral transform (FIT) is
defined as:
(3 - 17)
54
1 3 Oynamlc Analysis of the Process Model
We c,n break up the total interval (0 to TJI) into a number of sublOtervals of
length llti. Then the FIT can be written, with no loss of ngor, as a sum of IOtervals:
FIT = t (f~1 y(t)e-awtdt) i=1 j,.-1
(3 - 18)
After several mathematical steps, finally, the approximated Fourier transfor-
mation of y( t) becomes:
fo X) -iwt ~ -;wt [(e- iWl1t; - 1 e- a..;l1t a ) (f-,,,'A/ I - 1 1 )]
y(t)e dt ~ ~ e 1-1 YI 2 - - !Ia-l --2-- - -o ;=1 w Ilt; r.,.) ",,' Ill, 1 .... •
(3 - 19)
As mentioned before, the input function of the process is a step change of
height h and duration D, thus its Fourier transformation is given by
(3 - 20)
3.3.3 Frequency-Oomain Oynamics of the Process Model
There are two ways to obtain the frequency response of the process Mathe-
matical methods, based on the equatlons that describe the system, are used to obtétln the
frequency response directly from the system transfer functlon. Expeflmer ",1 methods,
discussed ln Section 3 3 2, are used when a mathemattcal model of the system IS not
avallable or the model is too complex to express as a transfer functlon ln thls proJect,
expenmental methods were used to obtaln the frequency response of the drum water level
1
:1
3 Oynamic Analysis of the Process Model
system from its simulating time domain response (shown in Figures 3.3 to 3 8). In order
to develop the transformation program, the following rules were established:
• The time-domain response values of water level and drum pressure must be
obtained first from the simulation monitor at each sample time step.
• In the step testing method. substitute the value into Equation (3 -12) as y( t);
ln the digital evaluation method. substitute the value into Equation (3 - 19)
as Yi.
• Pick up a specifie numerical value of frequency w Equations (3 - 12) and
(3-19) arethen Integrated, giving one point on the frequency-response curve.
Then the frequency is changed and the integrations repeated, using the same
experlmental time functions y(t) and Q(t). Repeating thls for frequencies
over the range of interest gives the complete G(rw) The sample selection of
frequency w is listed in Appendix A.
Based on these rules, a transformation program was developed to generate the
frequency response (Bode plots in this project), presented in Figures 3 10 to 3.15 The
transformation program reads input and output data, caleulates the integration of Equa·
tlon (3 -12) or the Fourier transformation of Equation (3 -19), gets the transfer function
by dividing the Input and output data, anrJ prints out log modulus and phase angle at
d,fTerent values of frequency. From the Bode plot, a closed·form transfer function is then
estimated. Using a digital optimization program, the coefficients of the estimated transfer
function are varied systematiully until the closest possible fit betwee'1 the tabulated fre·
quency data and the approximating function is obtained. To check the correctness of the
transfer function, an appropriate program is employed to perform the inverse transforma·
56
r
1 1
3 Dynamlc AnalySIS of the Procus Model
tion 50 that the original time response can be reproduced and the accuracy of the tr ansfer
function verified. The transfer functions for the drum water level system are presented as
Z transforms due to the digital simulation of the pro cess They .He descnbed as
1) Drum pressure transfer function caleulated from tlme and frequency responses
due to a la percent step incruse in feedwater mass flow rate (shawn ln
Figure 3.3 and 3.10),
G _ 1.1730 - 2.4791z-1 + 4.1805z-2 - 2.3440z- 3 + 0 5674:- 4
1 - 57.6376 _ 225.9200z-1 + 291.6848z-2 - 230.7271:- 3 + 130.8541:-4 - 23.5294:- ~ (3 - 21)
2) Drum pressure transfer function calculated from tlme and frequency responses
due to a la percent step increase ln fuel flow rate (shown en FIgure 34 and
311).
G _ -2.4596 - 2.9372z- 1 - 1.2671z-2 - 1.1361z-3
2 - 31.7017 - 71.8710z-1 + 61.3656:-2 - 24.1952:-3 + 4.2391z-4 - 0 2402:-5
(3 - 22)
3) Drum pressure transfer function calclliated from time and frequency responses
due to a 10 percent step increase in steam flow rate (shown ln Figure 3 5 and
3.12),
0.1701 - 0.3454z- 1 + 0.2430z- 2 - 0.07l3:- 3
Cl = 2.1314 _ 6.5700z-1 + 7.9134z-2 - 4.4748:- 3 + :-4 (3 - 23)
51
• 3 Oynamic Analysis of the Process Model
4) Drum witer level trinsfer function calculated from tlme and frequency re
sponses due to , 10 percent step increase in feedwater mass flow rate (shown
in Figure 3.6 ,nd 3.13).
54.28 0 4 = 2.59 - 1.59z- 1 (3 - 24)
5) Drum witer level transfer function calcufated {rom time and frequency re-
sponses due to a 10 percent step increase in fuel flow rate (shown in Figure 3.7
and 3.14).
G _ 0.5969 - 0.6610z-1 + O.0875z-2
5 - 1.9870 _ 4.2774:-1 + 3.3172:-2 _ z-3 (3 - 25)
6) Drum water level transfer function calculated from time and frequency re-
spons!s due to a 10 percent step increase in steam mass flow rate (shown in
Figure 3.8 and 3.15).
G _ -151.13 + 151.13z-1
6 - 2.59 _ 1.59z-1 (3 - 26)
These six transfer funetions were estimated based on the particular boiler
model discussed in this thesis. The physical dimensions and variabl'! values of the boiler
are listed in Table 3.1. The coefficients of the estimated transfer funetions may vary
for different boilers. But the techniques developed in this thesis can be applled on the
estimation of other boilers.
58
• 't
1
, J
1 j
.'
3 Dynamlc Analysis of the Procus Model
o.,
0
~.5
-1
-u
-2
-2.S
-)
-B
... 100
_....J.. ____ .l.. ___ ... ____ ~_.J.~ _______ .... ~ ___ .J. __ • ...l--.Jr. ..1_ &. ~
10' 10'
FreqUftlCY. radi.1nIIIecond
100
ID
60
«)
20
0
-20
.«)
.(rO
-10
-100 101 10' 102
",......,.~
Figure 3.10 Frequency response of drl.m water level for 10 percent step incruse in feedwater flow ,ate
59
1
J J
J l 1
J
3 Oynamic Analysis of the Process Model
100
10
flO
40
3)
0
.3)
.«)
~
..JO
·100 101 10· 102
Pr.-c7.,..... ........
Figure 3.11 Frequency response ofsteam discharge (or 10 percent step incruse in feedwater f10w rate
60
1
T
3 Oyn.mic An.lysls of the P,oc~ss Model
,...-------.................................................. -------------------------------05 9 ,
0
-05
1 -1
-1.5
J -2
-2.5
-3
-3.5 1()O
t ft f • ft ft
10'
100
10
CIO
~
J 20
i 0
1 -20
~
~
-10
·100 lOI
, 1 •• ft !
10'
Fiaure 3.12 Frequency response of drum water level for 10 percent step incruse ln fuel flow r.te
lOI
102
61
1
3 Oynamic Analysis o( the Process Model
1.'
J JO.,
o
~, if
·50
·SS
J .(j()
~
f ·70
1 ·75
...,
. ., .!JO
101 10' 10a
JIreqINDcy. l''d'_'Iecœd
Filure 3.13 Frequeney response of stum dise"ar,e (Ot 10 percent step inerease in fuel f10w rate
62
1
3 Dynamlc AnollySIS of the Process Model
o.,
0
~., ---
J -1
.1.5
J ·2
.2,j
.)
.)-, UJI 10' 101
PrequaM:y. ndianahecoDd
100
10
lIO
40
J 20
l 0
j ·20
~
..a)
·10
·100 IéJl 10' lOI
1'reqIMIacy.~
L-. __________________________________ ----
Figure 3.14 Frequency response of drum water level for a ~O- psi step decrease ln system pressure
63
~
1
1
1
1
•
1
1 J
1 " l J
l Dynam,c Analysis o( the Process Model
0, ~--..-
1 -10
-20
-lO
~
-~
.«J
-70
-10
-90 101
Figure 3.15 Frequency response o( steam discharge (or a 50-psi step decrease in system pressure
101
64
1
Chapter 4 Control System Structure
ln the two proceeding chapters we discussed the uncontrolled dynamlc behavlor
of the drum water level system, from the model development to Its dynamlc analysis
Based on the results achieved, we may now design a controller to make the plant run
more automatically, efficiently and safely
ln this Chapter, the 'hardware' part of the control system will be presented
It consists of the outline of the controlled plant and the structure of the controller.
4.1 The Outline of the Controlled Plant
Instrumentation hardware has been revolutionized in the last several decades
Mechanical and pneumatic components have been replaced by microprocessors that serve
several control loops simultaneously. Despite ail these changes ln hardware, the basic
concepts of the control system structure and control algorithms remaln essentlally the
same as they were thirty years ago [17]. With powerful computers, It IS now easler to
Impleme'1t "ontrol structures by just rewriting the program But the obJect of controller
design is the same. achieve a control system that will give good, stable, robust control
1 4 Control System Structure
As mentioned in Chapter 1, a boiler can be functionally divided into three
main loops comprising:
1) the combustion loop
2) the drum-feedwater loop
3) the stum-temperature loop
ln this project only the drum-feedwater loop is Involved and the other two
loops are assumed to operate satisfactorily. Thus in the control of the drum water system
only the following variables are of interest:
• Manipulated variable' The feedwater flow can be changed in order to control
the drum water level.
• Controlled variables: The drum water level and drum pressure are controlled
to remain as constant as possible.
• load disturbance' A change of steam fJow will cause the drum water level and
drum pressure to depart from their setpoint and the control system must be
able to keep the plant under control despite the efTects of the disturbance.
When a fet:dback controlstrategy is implemented digitally, the controller input
'nd output must be digital (sampled) signais rather than continuous-time signais. Thus,
the continuous-time signal from the transmitter is sampled and converted periodically to
a digital signal by an analog-to-digital converter (AOC). A digital control algorithm is
then used to caleulate the controller output, also a digital signal. Before the controller
output is sent to a f'nal control element (feedwater valve ln thls project), this digital
66
1 4 Control System ~truclure
CONl'lOL VA1.VI SBNSOI
----e.t{> Njo.-.----.4 PkOC'ESS
IIP
l'ŒlD
...... _AIR SUPPLY PLOW TRANSMJTll!lt
-----t------ ------------------------------------------ ----------CONTROL
ROOM
1 COt-rTROLl.J!R AND
MANUAlJAI!rOMATIC S~Q ~ __________ S~~
Figure 4.1 Feedback control loop for drum water level system
signal is converted to a corresponding continut:: .. s-time signal by a d'gital-to-analog con·
verter (DAC). Alternatively. the digital signal can be converted to a sequence of pulses
representing the change in controller output. The continuous-time signal or pulse train is
then sent directly to a final control element that utilizes one or the other form of input
to change its position.
Figure 4.1 shows the control loop ;or the drum water level system wh.ch
consists of a sensor to detect the process variable, a transm,tter to convert the sensor
signal into a digital signal which the controller can understand, a controller that compares
61
4 Control Syst~m Structur~
thls pro cess sign~1 with ~ desired setpoint value and produces an appropriate controller
output signil ind i control valve thit changes the manipulated variable (feedwater flow
rate)
Ali elements shown in Figure 4.1 were simulated by mathematlCal models and
the controller part will be discussed further in the following sections.
4.2 Orum Water Level Control System
1.. water tube boilers, the drum level control system must ensure that the water
levells always malnlalned above the top of the risers/downcomers to prevent overheating
of the tubes. The drum level also must ")ot rise such that water from the drum is carried
over into the steam system. During start*up or under abnormal conditions, the plant
operator may want to set the position of the control valve hlmself instead of having the
(ontroller position it. Thus a manual/automatic override switch must be provided in the
controller. Certain desirable controller readout features include:
1) Indication of the value of the controlled variable: the signal from the trans
mitter.
2) Indication of the value of the signal being sent to the feedwater valve: the
controller output.
3) Indication of the setpoint.
Drum level is difficult to control because of the inverse response of the level
to changes in steam demand.
68
~ 1
4 Control S~st~m Structurt
r-----------------------.-----------------..
WATER LSVBL MEASURBD SETPOINT WATBJt LEVEL
PlD CONTROlLER
+ +
SETPOINT
CONTROL
ACTUATOR
DRUM PRESSURE
S1"BAM PLOW
FEEDPORW ARC
CONTROLLER
L-_______________ . __________________________ _
Figure 4.2 General drum water level control system
The schematic representation of the general control system is shown in Fig-
ure 4.2. The system contains two control loops, a feedback loop using the deviation of
the measured drum water level from its set point as an input and a feedforward loop using
the stearn flow signal as the input. The feedback loop is an increlT ~ntal PID controller
with an adjustable proportional gain.
As mentioned before, due to the drum pressure change caused by the steam
69
1 4 Control System Structure
load change, 'swell' or 'shrinkage' will occur. To improve the control quallty, a feedforward
loop was designed to use steam flow and drum pressure signais directly to move the control
devlce before the 'swel/' and 'shrinkage' in the boiler water level occur. A predesigned
function can be defined based on the response of the process during simulation. In this
project immediatel)' following the drum pressure and the steam flow signais, a step sIgnai is
generated by the feedforward control/er for fast control action. Steam flow is summed with
the output of the PlO controller to establish the set point for the control actuator. This
scheme is effective because steam f10w changes ar/.! immediately fed forward to change
the final feedwater set point for the control actuator. In this way feedwater flow tracks
steam flow. The drum level to feedwater cascade will quickly arre5t any disturbances in
the feedwater system.
The controlled device evaluated in this project is the feedwater 41,,\VII valve.
This valve is mathematically simulated by a second-order transfer function (for detail see
Chapter 2). Through the control of the valve, the feed water flow rate 15 manipulated
and the drum water level is control/ed following the changes in several variables.
70
Chapter 5 Control Strategy
ln previous chapters the dynamic behavlor of the drum water level process was
considered and sorne of the mathematical tools required to analyze the process dynamics
were developed. Furthermore the drum water level control system elements were outlmcd
Now the 'core' of the control system, the control algorlthm, may be consldered
ln this chapter, the incrementai PlO combined with feedforward control algo-
rithm will be introduced first, then an improvement of the algonthm will be discussed
5.1 Control Aigorithm
Ouring the past two decades, there has been widespread application of digital
control systems due to their f1exibility, computational power, and cost effectiveness [291
ln this section, the digital drum water level control technique, considered to be a digital
version of PlO and f(~edforward control, will be introduced. The final combined digital
PIO-feedforward control algorithm will also be developed.
5.1.1 Digital PlO Control Technique
ln most process PID control applications, the derivatlve action can be applied
'( .
, S Control Stratfgy
to either the error signal or just the.process measurement (PM). If it is applied to the
errar signal. step changes in set point will produce I.arge bumps in the control element
Therefore in this projett, the derivative action is applied only to the process measuremen\
signal as il enters the controller. The proportion al and integral action is then a pplied to
the d.fTerence between the setpoint and the output signal from the derivative unit (see
Figure 5.1)
MANtPUl.ATBD VAAWLB
---4If
COm'ROL O\1t'P\11'
PaOCllSS
FllI!I>IACk COI'n'lOLU!ll
PlACT10N
CONTltou..eo VAJUABLB
SETPOINT ..... ---
Figure 5.1 PlO control with derivalive on protess measurement
The analog PlO controller equation can be expressed as:
72
1
l
5 Control Stratt&y
- (Ilot de) P(t) = P + Kc e(t) + - e(t)dt + TD-, TI 0 ( t
(5 - 1)
A striightforward way of deriving a digital version of the ideal PID control
law in Equation (5 - 1) is to replace the integral and derivative terms by thelr dlscrete
equivalents. Thus, approximating the integral by a summation and the derivatlve by a
first-order backwird difference gives:
(5 - 2)
Equation (5- 2) is referred to as the position (ofm of the PlO control algorlthrn
since the actual controller output IS calculated.
An alternative approach is to use the velocity (ofm of the algorithm ln whlch
the change in controller output is calculated. It can be described as
where:
[(; = I\c llt
TI
J' /\cTD \d= -
Ilt
(5 - 3)
73
1
f
tages:
5 Control Str~tegy
Comparing Equition (5 - .2) with (5 - 3), the velocity form has three advan-
1) It inherently contiins some provision for antireset windup since the summation
of errors is not explicitly calculated.
2) The output âPn is in i form directly usable by final control elements that
require in input specifying change in position.
3) Wh en putting the contraller in automatic mode, that is, sWltching It from
manual operation, it is not necessiry to initiallze the output (P ln Equation
(5 - 2». The valve has been placed in the appropriate position during the
shrt-up procedure.
After further anilysis, the velocity form PlO algorithm was adopted and com
bined with feedforward control to be the final control algorithm in this project.
5.1.2 Feedforward Control Aigorithm
PlO control is i type of feedback control. In feedback control, an error must
be detected in a controlled variable before the feedback controller can take action to
change the manipulated variable. Input disturbances must upset the system before the
feedback controller can do anything.
It seems very reasonable that if we could detect a disturbance entering a pro
cess, we might begin to correct for it before it upsets the process. This is the basic idea of
feedforward control. If we can measure the disturbance, we can send this signal through
74
5 (ontrol StrattSy
a feedforward control algorlthm that makes appropriate changes ln the manlpulated varI
able so as to keep the controlled variable near its desired value ln the drurn water level
system, the disturbances (feedwater flow rate, steam f10w rate and fuel flow rate) can be
easily monitored from the simulated dynamic response of the process 50 the feedforward
control is weil suited to drum water level control.
r-----------------------------------.--------.-------------
DRUM PRBSSUJtB
STBAMFLOW
TRANSMl'ITBR
FEEDR>RWARD CONTROU..I!R
...... LEAD-_"""I""'LA_O~'D~
..... _RA....,n.--o __ ' ~
FEEOBACK SIGNAL ACUATOR
----------4 .. ~ ~ FEPJ)WATER PLOW
Figure 5.2 The feedforward control algorithm
PROCBSS
Wh en designing parameter-optimized feedforward control, one assumes a fixed
75
1
5 Control Strategy
(realizable) structure, i.e. the structure ind order of the feedforward algorithm are given
and the free pu~meters ~re ~djusted by pirimeter optimization. Herein, feed{orward
control structures of the form
(5 - 4)
are assumed.
Now the response of P,( k) to a step change of the disturbance variable
c/(k) = l(k) 15 considered. In thls project, a first order lag is used in the feedfor-
ward controller 50 that the change 10 the manipulated variable is not instantaneous. Thus
{or / = l. Equation (5 _. 4) becomes
(5 - 5)
For e ,(k) = tek) we have:
P,(O) = hO
P,(l) = (1 - fdPj(O) + hl
Pj (2) = - hP/el) + (1 - h)p/(O) + hl + h2
(5 - 6)
76
5 Control Strat~gy
The initial manipulated variable P,(O) equals hO or hoc ,(0) Therefore "0 can
be fixed sim ply by a suitable choice of P,(O), 50 that a definite manlpulatlng range can
be easily considered. 8y means of the given P,(O) the number of optllnlzed parallleters
for 1 = 1 is reduced to one parameter. For 1 = 1 the equations, together wlth Equation
(5 - 6), become:
ho = P,(O)
-P,(I) - K, h=-~--.o.-
P,(O) - 1\, hl = P,{l) - P,(O)(1 - Id
l' ho + Il} \,= 1+11
(5 - 7)
(5 - 8)
(5 - 9)
(5 - 10)
Hence in this project (1 = 1), the deSign of the feedforward element wlth a
prescribed initial manipulated variable PICO) leads to the optimizatlon of a single parame
ter P,(l). The computational effort for parameter optlmizatlon in thls case IS partlcularly
small.
As shown in Figure 5.2, the steam flow IS measured and this flow signai 15
multiplied bya dynamic (first order) and a constant (desired ratio K,) element The
output of the multiplier is then combined with the output of the PlO controller to become
a control signal applied to the manipulated varia~'"
5.1.3 Combined PID-feedforward Control Aigorithm
ln most engineering applications, feedforward control systems are Installed as
part of combined feedforward-feedback systems. The feedforward controller takes care ot
77
1 5 Control Strategy
the large ird frequent measurible disturbances. The fet:oDack controller takes care of any
errors thit come through the process becausf! of inaccuracies in the feedforward ,ontroller
or other unmeasured disturb.1nces. As shawn in Figure 4.2 the manlpulated variable
IS changed by bath the feedforward controller and the PlO controller. The algorithm
equation is formed by'
(5 - 11)
The final output equation is:
Pn = Pn-l + âP~ (5 - 12)
The parameters in equations (5 - 3) ta (5 -12) will be calculated and adjusted
by certain parameter seUing methods based on the results of the simulation. Details will
be presented in Chapter 6.
5.2 Refinements
As indicated in Chipter 4. when a digital computer is used for control. con·
tinuous measurements are converted into digital form by an analog ta digital converter
(AOC). First the signal must be sampled at discrete points in time and then the samples
must be dlgitized. The time interval between successive samples is referred to as the
sampling period ât. In order ta get reasonably accu rate measurements. an appropriate
sampling rate should be selected and il filter may be installed if the measurements are
nOlsy. Process-induced noise can arise from variations due ta mixing. turbulence, and
78
5 Control Str ategy
nonuniform multiphue flows. The effect of both process noise and measurement noise
can be reduced by signal filteri",. In this section, the data sampling and flltering will be
discussed first. Then the improvement of the controller output Will be presented
5.2.1 Data Samplinl and Filte,inl
Sampling and filtering were used to improve the input signal character with
real data measurement or simulation generated noise. The sample period can be adjusted
based on the process. To eliminate or reduce uror due to nOise and converSion, the
variables are sampled several times dUflng one sample period, and the average value 15
used as sample data value. It is called average value flltering The average value i5
calculated by
(5 - 13)
Where:
Yi = the ith time sampling value
Yi = the average filtered output value
N = number of samples
To obtain a satisfactory sample data value, the number of sample5, IV, has
to be chosen carefully. If N is too large, it will take \.00 long to obtain an average input
signal value. If N is tao small, the effect of filtering will be reduced.
79
1 5 Control StratelY
5.2.2 Improvinl the Differentiai Character of Output Response
Due to the large time del~y in the controlled plant, it is nece!isary ta modify
the theoretic~1 PlO control a1lorithm before it c~n be used in the actual control system.
ln this cOl'trol system, a first vrder filter hu been added to improve the output due ta
the difTerential component. This filter has the same function as the one on the rnput
side of the controller. But this case the measured signal is the output signal from the
combined feedback-feedforward controller and the output signal is the final control signal
to the drum water level system. The finite difference equation of this filter 15 as follows
P~ = i1Pn + (1 - Ct)P~_l (5 -- 14)
where:
Equation (5 - 14) indicates that the filtered measurement is a weighted sum
of the current measurement Pn and the filtered value at the previous sampling instant
P~-l' limitinl eues for /3 are:
/3 = 1: No filtering (the filter output is the raw measurement Pn).
/3 ~ 0: The current measurement is ignored.
ln electrical engineering parlance, the term 'filter' is synonymous wlth 'transfer
function,' sinee a filter transforms input signais to yield output signais ln EquatIon (5-2).
80
5 Control Strategy
the filter tr.nsforms the output signal from the combined feedback-feedforward controller
(Pn) to the final control signal (P~) to manipulate the feed water valve. Through this
filter, the final control signal is improved so that the change in the manipulated variable
of drum water level Îs not instantaneous.
5.2.3 Antirelet Windup
An inherent disadvantage of integral control is a phenomenon known as reset
windup. Recall that the integral mode causes the controller output to change as long as
the error e( t) :f O. When a sustained error occurs, the integral term becomes quite large
and the controller output eventually saturates. Further buildup of the integral term while
the controller is saturated is referred to as reset windup.
Reset windup occurs, for example, during the start-up ,of a batch proeess or
after a large set-point change. It can also occur as a consequence of a large sustaioed load
disturbance that is beyond the range of the manipulated variable. 50 this reset windup
must be recognized and corrected.
This is accomplished in a number or different ways, depending on the con
troller hardware and software used. In computer control syst~m, the reset windup can be
reduced by temporarily halting the integral control action whenever the contreller output
saturates. The integral action resumes when the output is no longer saturated. In the
actual operation of PlO control, the integral " :tion may be turned off and on many times
based on the reset windup situation. Thus, the gain, Kef should h adjustùle sinee the
conirol charaeter is changed while the integral action is turned off. As shown in Figure 5.3,
I\c is increased when the integral action turned off to produce a faster control action and
reset to the original value wt.en the integral iction is resumed.
81
1 INPUT Yo.lII
SBTPD
PlDACf
N
y
y
SBTPID
PDAcr
y
N
RBSIn'Kc
CLl!AR.PID
Filure 5.3 The .djustment o( the Iain Kr
5.3 The Control Program Structure
5 Control Strategy
ADJUSTKc
a.EA.RPD
As mentioned in Chapter 3, one of the main purposes of the simulation pro-
gram is to provide a virtual control plant for the drum water level controller Further,
the simulation program must be combined with the controller to become a drum water
82
1 5 Contr ,1 Strategy
CONnOL ,.... MODBL saumON ALOOJt.I1HM
AJ..OOart1IM
1
1
1 PIt
fNPur PUNCI10N PROCESS OI!NElA TOIt NODBL
i
SIMlJI..AnON MONn'OI
Filure 5.4 The control prol,.m structure
level simulator We are now al the position to develop a control program (controller) to
control the entire plant (simulation program).
Figure 5.4 shows the structure of the control program. The program consÎ!;.ts
of four parts: the 1;0 devices, the manual/automatic override switch, the control monitor
and the control algorithm. The control monitor is used to indicate the controlled variable.
the controfler output signal and the setpoint value. The manualjautomatic override switch
83
...
j
5 Control Strategy
is provided for the plant operator to control the valve by himself instead of havlng the
controller position it. The 1/0 device consists of two filters, one is employed on the
input side of the controller and the other is u5ed to improve the output behavlor of the
controfler. The 1/0 device carries out the following functions'
a) Replace the input function generator in the simulation .. ;ogram (see Section
3.2.2) and generate the control signal for the plant.
b) Measure the output variables of the plant and transmit these variables to the
controller.
c) Read the operating commands supplied by the user and transmit the com
mand~ to the controller.
8ased on the input commands and the measured vartables, the control algo
rithm accesses and calculates these input variables and sends out the manipulated signai
to the plant. The control algorithm has an antireset windup function. Whenever the
controller output saturates, the integral action Will be turned off untll the output IS no
longer saturated .
84
f
Chapter 6 Selection of Design Parameters
An important consequence of feedback control is that it can cause oSCIllatory
response. If the oscillation has a sm ail amplitude and damps out quickly, then the control
system performance is generally considered to be satlsfactory. However, under certain
Clrcumstances the oscillations may be underdamped and the amplitude IOcreases with
tlme until a physical limit is reached ln thi~ case, the control system is said to be
unstable. The most important dynamic aspect of any control system is its stability.
Theoretically, the stability of a control system can be analyzed by the roots of
Its characteristic equation, transfer function or on the Z plane. Practically, as mentioned
several times before, it is hard or even impossible to derive the transfer function of a real
system. Thus, in this chapter, the stability analysis of the water level control system will
make use of the transient response of ils closed-Ioop.
6.1 Ziegler-Nichais Method
ln order to obtain approximately optimal settings of parameters for continuo:Js
lime controllers with PIO-behavio" tuning rules are often applied. These rules were mostly
evolved for slow processel, and are based on experiments with a P-controller at the stability
1 6 Selt"ctlon cf Design P .. ulmetm
limit, or on time constants of the process. A survey of these rulE's 15 given in e.g. Isermann
[12}. Well-known rule~ al~ those by Ziegler and Nichols.
The Ziegler-Nichols (ZN) method consists of first finding the ultimate gain
Ku, at which value the loop is at the limit of stability with a proportional-only feedback
controller. The period ofthe resulting oscillation iscalled the ultimate period. Pu (minutes
per cyde). The ZN setting are then calculated from Ku and Pli for a PID controller by
1.' _ Au l\e --1.7
Pu TI= -
2
Pu TD=-
8
(6 - 1)
The ZN controller 5etting5 are pseudostandards in the control field They are
easy to find and to use and give reasonable performance 01'1 sorne Joops. There are many
loops where the ZN settings are not very good. They tend to be too underdamped for
most process control applications. But the ZN settings are uSf:lul as a place to start
6.2 The Feedback Controller Parameters Setting
After further study of the available parameter tuning methods. Ziegler- Nichols
was adopted as the basic parameter setting method in this project. Based or. the ZN
method, the following on-line trial and error procedure was developed to obtain the final
controller parameters:
• Definitions:
86
1
J
o Selection of Des"n Parameters
Proportional Band (PB): This is an alternative term instead of controller gain.
It is defined u:
V'/here Ke is the control/er gain. The term proportional band refers to the
range over which the error must change t(' drive the controller output over its
full range .
• Procedure.
[1] Takin, ail the integral and derivative action out of the controller, i.e., set Tl
at maximum (minutb per repeat) and TD at minimum (minutes).
[2] Set the PB at a high value, 200 in this project.
[3] Make a smailload change (10 per cent steam flow rate change in this project)
and observe the response of the controlled variable. The gain is low so the
response will be slugish.
[4J Reduce the PB by a factor of 2 (double the gain) and make another small
change in load.
(5) Keep reducin, PB, by repeatin, step [4], until the loop becomes highly un-
derdamped and oscilla tory. The gain at which this occurs is the ultimate gain
[6) Calculate Ke, TI and TD by employing the ZN method.
87
t
J
6 Selection of Design Paramrtprs
(7) Now st,rt bringin, in integr~1 "tion by reducing Tl by factols of 2, maklng
sm,1I disturbinces it eich vilue of TI ta see the effect
(8) Find the value of TI thit makes the loop hlghly underdamped and set TI at
twice this value.
(9) load changes should be used ta disturb the system and the derlvatlve should
be ,cting on the process measurement sign .. !. Flnd the value of T D that glves
the tightest control without arnplifymg the nOise 10 the process measulement
signal.
[101 Then reduce the PB by steps of 10 p'2r cent, untt! the desired specifICation on
damping coeffiCient or overshoot is satisfied
There ire many other tun;ng methods. One of the most Simple user; the step
response cf the process ta determine steady-state gain, tlme constant, and deadtlme
Then controller tunlng constants can be calculated from these values The ultlmate
gain and ultimate frequency approach were adopted ln thls proJect because of the very
significant problem of nonhnearity that exists in thls proJect and most other engineering
thermai·f\uid process systems Step testing drives the process away from ItS initiai steady
state and is. therefore, much more sensitive ta nonhnearlty than IS the c1osed-loop ultimate
gain method in which the process is held in a region near the initiai steady-state
Based on the above procedure, an optimlzatlon program was developed to <.al
culate the coefficients of the controller. Usina this program, the controller gains Kr, 1\ J'
TI and rD were adjusted, in practice, to obtain the 'best' closed-Ioop tranSlent responses
The 'best' closed·loop responses were those Judged to provlde an acceptable tranSlent
88
1 6 Selection of Design Parameters
behavior of the process, due to the input disturbances, and that were not expected to
improve with continued tri~1 arld error.
The initial and final controller p~rilmeters ilre shown ln Table 6.1. The initiai
parameter villues were obtiined by using the ZN metnod. To arrive at the final parameter
values shown, twelve step responses were calculated. with readJustment of tne controller
parameters by applying the trial and error procedure after the evaluation of each response
The transient responses ulculilted from the initial and final controller parameters are
shown in Figure 6.1. A comparison of the values is indicative of the extent to which the
trial and error adjustments of the controller parameters may be consldered as 'informed'
Conlroller Plrlmeters
Kc K 1 TI TD
Initlll Values Final Values
10 25 05 200 225 100 020 025
Table 6.1 Controller parlmele' lettin,s by usina ZN combined tnal and error method
The addition of the feedforward controller has no effect on the closed·loop
stabllity of the system for il lineilrized process. Thus, in thls proJect. the feedforward
controller is ildded dter the feedback controller is tuned. Details of the feedforward
controller tun,"g will be discussed in Chapter 7.
6.3 Influence of the Sampling Time ~t
As is weil known. sampled data controllers have generally inferior performance
than continuous control systems. This is sometimes explained by the fact that sampled
89
6 Seltet/on of DtSlgn P,,'amtltr5
1 1 1 • J
1 l 1 • 1
Il 1
:[ (\1
2
o
·2
1 1
(~----_---f
~L ____ ~I __ ~I ____ ~' ____ ~I __ ~I ___ ~~-4-__ ~ __ -J
o 20 40 60 10 100 120 1.0 160 110 D)
n-.~
a) The c10ged-loop response based on ZN paJ'1lDeaer Benin,.
:~'--~~'--~'--~t--~'--~'--~'-~
3
o
.1~--~--~--~~--~--·~--~--~----~--~--~ o 210 40 el 10 100 120 140 160 110 .,
na.,1ealIIdI
b) The doeed.loop leIpODIe bMcd ouille 1riII-1Dd-ew panmelU teUÙIIJ
Figure 6.1 The closed-Joop respor.$es (rom the initial and final controller pa
rametefl
90
1 6 SelectIon of DesIgn Parameters
signais contain less information than continuous signais. However, not only the InformatIon
but also the use of thls information IS of interest. As the class and the frequency spectrulI1
of the disturbince signais also plays an important mie, general remarks on the control
performance of sampled data systems ire difficult to make. However, for parameter-
optimized controllers one can issume in general that the control perform~nce deteriorates
with increasing sample time. Therefore, the sample time dlould be as small as possible
considering only the control performance.
ln selecting a sampling period, two questions must be considered'
1) How many mealjurement points does the computer monitor?
2) What is the best sampling period from a process point of view?
Control loops in general have to be designed such that the medium frequency
range cornes wlthin that range of the disturbanc~ signal spectrum where the magnitude
of the spectrum is low. In addition, disturbances with high and medium frequency corn-
ponenls must be flltered in order to avoid unnecessary variations in the the manipulated
variable. If disturbances up to the frequency Wmaz = Wl have to be contro:led approx-
im~tely as in .a continuous loop, the sampl~ time has to be chosen in accordance with
Shannon's sampling theorem [12):
1r At<--
- Wmaz (6 - 3)
This sampling theorem can be applied to select sampling times for an entire
control process. If a digital control system is connected to a single measurement point,
then that measurement can be sampled as ohen as desired, or as rapidly as the computer
91
1
1
6 Selection of Design Parameters
can sample. However, rlpid sampling of 1 large number of measurement points may
unnecessarily load th~ computer Ind restnct Its ability to perform other tasks Sampllng
too slowly can also reduce the efTectlventss of the feedback control system. especla"y
its abllr:t to cope with disturbances. $0 It is dlfflcult to make any genel d Ilzatlons 011
the selection of sampling period. A number of gUldeline and rules for selectrng t',e sam·
pllng period have beoen reported (29). However. since the optlm um samplrng peflod IS
application-specifie, it is still believed that the best way to select the samplrng pcriod IS
to evaluate the entire process.
The efTect of the ~ampling penod on control system performance for the water
level protess has been evaluated Figure 6 2 to 6.5 show the dlscrete values of the control
and the manipulated variables for the drum water level process after a step change of the
steam load variable for th~ sample times L'lt=l. 2, 4, and 8 seconds For the relatlv.:>ly
small sample tlme ~t =1 second one obtalns a close approXimatIon to the cOlltrol behavlor
of a continuous PID-controller. For ~t=2 second .. the contlnuous signai of the control
variable can still be estlmated falrly weil for the process However, th,s IS no longer valrd
for L'lt=4 and 8 second. It can be seen from Figure 6 2 to 65 that a sample tlme L'lt = 2
sec. compared with l:!.t = 1 sec whlch is gaod approXimation of the '_ontlouous case, ledds
only to a small deterioration in control performance. If only the control performance /5 of
interest the sample time can usually be greater than that required to dosely approxlmate
the continuous control loop.
It should be noticed that the process condition is another factor that affects
the selection of the sampling period. If proc.ess conditions change significantly, then it may
be necessary to change the sample period. For example. In thls project, If the feedwater
flow rate is significantly increased, the residence time and hence the tlme constant for tl.e
92
6 Selection of Desi,n Parameters
drum water level system il reduced. Consequently, it may be necessuy to use a smaller
samplin, period in order to achieve satisfactory control. A simpler, more conservative
approach would be to select a samplin, period that corresponds to the worst possible
" condition, that is, the smallest samplin, period. But in this project, the controller design
is based on an identified process model, and parameter estimation methods are used
for the identification, 50 the sample time should not be too small in order to avoid the
numerical difficulties which result from the approximate linear dependence in the system
equations for small samp,le time.
This discussion shows that the sample time has to be chosen according to
many requirements which are partially contradictory. Therefore suitable compromises
must be found in each case
93
1
l'
6. Selection of Design Parameters
6
5 . .
J .. 3
1 1 2 . • 1
1
0
·1
·2 0 20 40 60 10 100 120 140 160 110 200
Tune. seconda
lA
1.2 . . ,
1
t G.I
j G.6
DA
G.l
0
-4.2 0 20 40 60 10 100 120 140 !fiO 110 200
Tune. leCOIIda
Filure 6.2 Transient response of the process for 10% incruse in steam flow with sample interval At=l second
94
1
1
6 Selection of Design Parameters
, r
~
1 ]
l 2
1 • 1 olJ \f-
·1 1 -- •
·2 0 10 20 30 40 50 60 70 10 90 100
nme, ......
1.2,---r----,--...,..----.----.---,-------.---.----..-----.
o .•
t J
0.6
o..
~ "1 o· 0 10 » JO 40 » 60 70 10 90 ICI)
11mI. .....
Fieure 6.3 Transient response o( the process (or 10% increase in steam flow with sample interval ~t=2 second
95
1 6 Selection of Design Parameters
6
5
J • 3
l 1 2
• 1
0
·1
·2 0 5 10 15 210 2.S lO 35 40 ., !O
nme. lec:oadI
0.9
1 O.,
1 G.1
0.6
o.s
G.4 1 0.3
0 , 10 15 JO 25 30 " 40 4S ~
nn.. ....
Figure 6.4 Transient response of the proceu for 10% incruse in steam flow with sample interval At=4 second
96
1 6 Selection of Design P3rameters
I~r-------~------~------~------~------'
1000
1 l ~ 1 .. 0
J
.10001.--___ ....... ____ ~ ___ __' _________ __J
OSlO 15
1.05.-------.-----....----....------T'"------,
o."
1 1 0.9
0.1,
o.. V
0.7', , 10 ., 210 2S
'1\me,'-'
Filur. 6.5 Trlnsient response of the proc:ess (or 10% inclease in steam flow with Simple intervil L'.t==8 second
97
1
Chapter 7 Presentation of the Results
ln this chapter the simulation results of the drum water level control system
will be evaluated. First, the closed loop response of the process wlth and wlthout the
feedforward controller will be presented. Then, the simulation results of feedback control
and combined feedback-feedforward control Will be compared Flnally, the simulation
results of the process closed loop response with the optimlzed controller par a meter settmgs
will be presented. Ali tests were performed on the Sun SPARC ™ workstatlon
7.1 Closed-Ioop Behavior of the Process
There are a number of criteria by which the desired performance of a closed
loop system can be specified in the time domain. For example, we could speclfy that the
closed-Ioop system be critically damped so that there is no overshoot or OSCillation We
must then select the type of controller and set its tuning constants so that Il will glve,
when coupled with the process. the desired closed-Ioop response Naturally the control
specification must be physically attainable. In the analysis of the closed·loop behavlor, not
only the stability of the control system is of concern but the quality of control performance
should also be considered. The analysis of the control quality is very much based upon
the specifications of the closed-Ioop response.
1 7 Presentation of the Results
7.1.1 Specification of Cloled-Loop Relponse
There .re • number of time-domain specifi('ations for closed-Ioop response.
But the specification to be selected must be physically attainable for the entire process.
ln this project, the dynamic specifications can be listed belt)w'
1) Closed-Ioop damping coefficient
2) Overshoot: the magnitude by which the controlled variable swings past the
setpoint
3) Rise t.me (speed of response): the time it takes the process to come up to
the set point
4) Decay ratio: the ratio of maximum amplitudes of successive oscillations
5) Settlin, time: the time it takes for the amplitude of the oscillations to decay
to some fraction of the change in setpoint
6) Integr.1 of the squared error:
Notice that the first five of these assume an underdamped c1osed-loop system,
i.e., one that has sorne oscillatory nature (the ,hum water level system is underdamped)
ln the design of the controller. what we want is a reasonable compromise
between performance (tight control; small closed-Ioop time constants) and robustness
99
1 7 Presentation of the Results
(not too sensitive to ch~nges in process p~r~meters). This compromise IS ach,eved by
using ~ reason~ble clc~ed-Ioop d~mpinl coefficient to keep the real parts of the roots of
the closed-Ioop characteristic equ~tlon a reasonable distance from the IInaginary aXIS, the
point where the system becomes unstable.
The steady-st~te error is another specification. It 15 not a dynamic speCiflCa
-ion, but it is ~n import~nt performance criterion. In many loops ~ steady-state error of
zero is desired, i.e., the value of the controlled variable should eventually level out at the
setpoint.
7.1.2 Closed-Ioop 8ehavior Usina Feedback Control
The water level control system was evaluated by applying step changt's ln
the turbine governor valve area and observing the following tlme response water level,
steam pressure, steam mass flow rate and the feedwater flow rate The most Important
responses are the water level and feedwater flow rate The first one glves an indication
of the effectiveness of the control system ln performlOg Its main functlon The second
response shows the stability of the feedwater loop, wh,ch is the 'fastest'Ioop ln the system
and, as such, the most prone to exhibit unstable or underdamped behavlor
The effect of feedwater valve speed was first determlned Computer program
runs were conducted for different valve speeds, ranging from 2 to 40 second full stroke
time, without changing the controller parameter settings. Results of thls study Indicated
that the effect of the valve speed on controller performance was InslgniflCant A feedwater
valve with 20 second full stroke time was then selected since thls is an acceptable value
from the cost viewpoint.
100
l
-
7 Presentation of the Results
The effect of controller pu~meter seUings Kc, TI and TD' were then studied.
Computer progr~m runs, \Vith special ~ttention on system stability, were conducted. Re
sults of this study showed th~t the system performance is strongly affected by the settings
of A'c, TI and TD' By varyin, these parameters in a systematic way, a combination of
p.uameters was found for which the system response to a step change ln turbine gover
nor valve area was an optimum. The definition of 'optimal response' was based on the
inspection of the time response behavior (i.e., overshoot ar.d response time) rather than
a mathematical criterion.
Based on operating experience, the flow response of the fee·dwater loop (with
out water level and steam flow rate signais) was considered acceptable when the overshoot
was leu than 25 percent and the peak time not more than 2 second for a step change in
feedwater flow rate set point. In contrast, and in view of the lack of sufficient expenence
with large steam generators of the type studied here, it is more dif~cult to estabhsh a
performance criteria for the water level control system. However, it appears reasonable
to require that the water level return to the set point value within two minutes following
a step change in turbine governor valve area.
Studies on the controller parameter Kc indicate that the water level and feed
water flow response requirements impose lower and upper limits respectively, for this
parameter. However, there is a large range over which j(c can be varied without any
noticeable effect on system response. The upper and lower limits for the controller pa
rameter Kc were found to be equal to 0.5 and 3.0 respectively, which provides reasonable
freedom of final adjustment in the field.
Computer program runs on the controller parameter Tl show that, similar to
the case of j\c, the water-Ievel and feedwater response requirements respectively, impose
101
T
7 Presentltion of the Results
lower ind upper limits of this paumeter. Here, only i lower limlt of Tl = 0.5 sec
was established. The maximum possible value is best determlned under actual operatlng
conditions. However, an optimum response is achieved by choosing Tl = 10 sec Control
components for this optimum value are readlly avallable
The parlmeter TD' the differential time constant of the controller, also exerts
a large influence on system response. Computer program runs indicated that the optimal
value for TD is 0.25 sec, with upper li mit of 0 4 sec.
Typical water level responses for a vanety of controller parameter seUlngs are
shawn in Figures 7.2, 7 4, 7.6. These characteristics are determlned for a 10 percent
step increase in turbine governor valve area. In Figure 7.7, the water level behavlor is
shown for i step change in level set point
7.1.3 Feedbuk·Feedforward Control 8ehavior
The baSIC notion of feedforward control is to detect disturbances as they enter
the process and make adjustment in manipulated variables 50 that output variables are
held constant. We do not wait until the disturbance his worked .ts way through the
process and has disturbed everything to produce an error signal.
ln practice, many feedforward control systems are implemented by uSlng steady
state gain element (ratio control). In this project, in order to avoià any instantaneous
changes in the manipulated variables, a first-order lag is used in the feedforward controller.
Both dynamic element (first-order lag) and steady-state element feedforward control were
evaluated here.
102
1 7 Presentation of the Results
Figure 7.8 ind 7.9 show the closed-Ioop response of drum water level change
to a 10 per cent step increase in lovernor valve irea for PlO and feedforward-feedback
control. A comparl50n of Figure 7.8 with 7.9 indlcates that feedforward controller design
using a dynamic element can result in sigllificantly improved load responses even though
the ideil feedforwud controller is approximited by a lead-Iag unit (see Figure 5 2 for lead
lag unit). In particular, the combined feedback-feedforwird controller is very effec.tive. A
compari50n of the controller output signils in the bottom portions of Figure 7.8 and 7.9
shows that the improved performance does not require excessive control action since the
required controller output responses in Figure 7.9 are comparable ln magnitude to those
10 Figure 7.8.
The parameter tuning for the feedforward controller was discussed in Chapter
5. Based on Equation (5 - 7) to (5 - 10), we can see the only independent variable in
parameter optimintion is P,(l) Using a trial-and-error procedure, it is very easy to fine
tune the parameter P,(l).
7.2 The Simulation Results
The time variation of the four most important steam generator variables is
shown in Figure 7.11 for a 10 percent step increase in turbine governor valvf area and
optimum controller seuings.
The control system wu tested for susceptibility to drift and noise. The results
have shown that the control system is not greatly effected by drift in feedwater valve
position or controller Input signais. The level variation approaches zero after a step change
in turbine governor valve area and remains at zero. The feeriwater f10w rate approaches
103
1 7 Present,tlon of the Results
a new steady-state value and then holds constant. Further runs of the digital program,
wlth noise injected lOto the system by a noise generator, have Indlcated that the control
system is not partlCularly affected by the noise ln feedwater valve position or lellel and
flow measuremtnt signaIs.
104
1 7 Presentation of the Results
1 i .1
l
J l • 1
• 6
..
l
0
·2
... 0
Tune. leCOIlIb
K_cwl 0; ,_d-().23. U-5.0
3) 40 eo 10 100 13) 140 160 1. 200
......... IICXIIIIII
Figure 7.1 Step 'esponse of water level chan,e for a 10 percent step increase in turbine lovernor valve area for difTerent values of Kc
105
., " , f
1 7 Presentation of the Results
) 1 1 i .. J
2
0
·1
·2 0
V 3) 40 CIO 80 100 120 140 160 110 200
nme. secmdI
---------
Figure 7.2 Step response of water level change for a 10 percent step Increase in turbine governor valve aru for different values of ne (Cont'd)
.-------------------------------------"--
j l a
k
) ,----,---r---.,.--"T---.....---....----,---,---"T---
2
o
·1
·2
·3~ __ ~ __ ~ __ ~ ____ ~ __ -L __ ~ ____ L-__ ~ __ _L ___ ~
o 3) 40 60 80 100 120 140 lM 180 200
Tune. seconds
Figure 7.3 Step response of water level change for a 10 percent step increase in turbine lovernor vllve area for different values of Tl
1
106
1 7 Presentation of the Results
4 A
] U.S 0, K_cz2 0, ,_~.25
1 l
l a
1 0
V--1
-2 0 Il 40 60 10 100 120 140 160 110 lQO
TIme. leCOIIdI
J ] '_1=100: K_e-2.0,,_~.2J
l 2 1
1 0
·II....--~--~--~--~--~~--~--~--~--~---~ o Il 40 60 10 100 120 14C lC58 110 lQO
Filur. 7.4 Step response of water level chan,e (or a 10 percent step increue in turbine lovernor y.lve area for diff'erent values of TI (Cont'd)
107
1 7 Presenttltlon of the Results
j 1 o.
k
1 1 .1
!
3
2
1
o
-1
4
3
2
0
"
-2 0 :1)
60 10 100 120 140 160 110 200
Tune. seoonds
~---,--
t_cIaO 2.5, K_CZl2 O. U-=5
40 60 10 100 120 140 160 110 200
Tame. seconda
Figure 7.5 Step response o( water level change for a 10 percent step increase in lurbine lovernor valve area (or difTerent values of TD
108
•
f
7 Presentation of the Results
4 -T
] l_d~.3'; K_os2 O. '_18 '
1 2
l • 1 0
-1
·2 0 20 40 !JO 10 100 120 l.a 160 110 200
11me.1eCODdI
Filur. 7.6 Step response o( water level chan,e (or a 10 percent step increase in turbine ,overnor valve arel (or difTerent values o( rD (Cont'd)
1 l •• f
4
3
2
10 20 3D 40 70 10 90 100
Figure 7.7 Response of water level (or a step change of 5 inches in set point
109
1 1
1 7 Presentation of the Rtsults
1 l 1 .. 1
" 3
2
o
·1
·2~--~--~--~----~--~---4--~----~--~---J o :m .eo fj() 10 100 120 140 l(il) 110 200
Figure 7.8 Closed-Ioop response of drum water level change to a 10 per cent step incruse in lovernor valve area (feedback control)
3
1 2
l J .. 1 0
V -1
-2 0 :m 40 60 10 100 120 140 III 110 2100
1'InII, lIiCIOIIdI
Filure 7.9 Closed-Ioop response of drum water level change to al 10 per cent step increase in governor valve area (feedforward-feedback control)
110
•
(
7 Presentation of the Results
-------- -------_.-------------St
1 • 1
1 1
~f.1 ' , 1
j )
t ~ 2
• 1 .
0
·IL-__ ·~_~ __ ~ __ ~ ____ ~ __ ~ __ ~ __ ~ __ ~ __ ~ o ~ 4() 60 10 100 120 140 160 110 2100
4
2
1 f 0
J 1 -2
] J ~
1 -6 V
"0 20 ., CIO 10 100 UID 140 160 110 2100
TIme,"""
Filure 7.10 Step response of boile'. four most important variables for a 10 percent step increase in turbine lovernor valve aru for optimum controller set
tinls
111
1
l
L ..
1 PresentatIon o( the Reluits
4 ~ ].5 ~
1 ]
f 2.,
J 2
1 .. 1.'
1 O.,
0 0 ~) 40 60 10 100 120 140 160 110 200
nme, leCOIIdI
0 ~
-2
-4
1 ~
[ ...
J -10 .. -12
1 -14
-16
V -II
-20 0 20 40 60 10 100 120 140 160 110 200
11me, !eooadI
Filure 7.11 Slep response of boiler's four most important variables (or a 10 percent step increase in turbine governor valve area (or optimum controller settings
(Cont'd)
112
1
Chapter 8 Conclusions
The primary object of this thesis has been to develop a dynamic mathematical
model for the drum water level system and design a control system to apply on this model
as a controller. The model should be capable of simulating the transient responses in
both tlme and frequency domain. In the process , general simulation methods have been
developed. 8ased on the simulating transients of the process, an entire control algorithm
has been developed. The application of this model to the water level controller was tested
for stability and efficiency.
8.1 Drum Water level System Model
The model of the drum water level system has been derived based on the
conservati':m of mass, energy balance and the state equations of the system. Therefore,
the model is valid for liquid and low velocity steam f1ows, which ue the main variables
under study in this project.
The finite difFerence equation technique was chosen to develop the model for
two reasons. First by using this method, the complicated mass and energy equations can
be linearized so that the designed linear contro"er can be applied on this mode!. Second,
t 8 Conclusions
because the method is bued on relatively simple difTerence equations. a large system
simulator can be easily assembled from individual process models and therefore the method
is flexible. The linearized-model design method .1150 provides an analytlcal technique for
intergrating the controls of a strongly interacting process. The methods may .1150 be
particularly desirable under conditions of rapid load changes. and will have application for
improving the dynamic performance and traii;ïient stability of steam generating systems
using on li ne digital control.
It is irnperative to recognize that the water level swell is Inltiated from bOlier
pressure drop which causes rapid expansion of bubble volume in the water phase of both the
steam drum and the downcomer-riser loop. On the other hand. retardation of circulation
velocity reduces the bubble velocity in the water phase and therefore tends to increase the
water level swell. The simulated transient response of the model has proved that the model
can predict swell and shrink of the water level. The results .1150 support the feaslbllity of
using simplified linearized models for the online digital control of this nonlinear process.
8.2 Simulation Methods
The most important part of any process simulator is the solution algorithm. In
this thesis, the iteration method, a simple and fast optimum load allocation algorithm is
introduced. It is specifically adapted to the boiler drum water level system. The Iteration
m~thod is computationally compact. The great majority of instructions are nothing more
than elementary arithmeti..: or simple logical instructions. It can be practically and easlly
implemented as a computer program. The Ruga-KuUa method has been uscd as a
stad :'lg point to solve the linearized equations. The iteration method has in(orporated
a fine-tuning procedure which will maintain, but more usually improve upon, the cost
114
1
-
B Conclusions
correspondin, to this startin, point.
A larce portion of this work involved the development of variable step-size
multistep ilgorithms to optimize the speed and accuracy of the simulation by varying the
solution time step. In the development of the simulation algorithm. the four-step Adams
Bashforth method wu employed as predictor and the three-step Adams-Moulton method
as corrector in the error control procedure. These methods are multistep methods which
use approximation at more thin one previous step to determine the approximation at the
nex! step. Because of the 'multistep', these methods will generate a more accurate result
thin the 'one-step' method.
Employing the simulation method and time-to-frequency domain transforma
tion technique developed in this thesis, the dynamic behavior of the plant model was
evaluated both in the time-domain and the frequency-domain. In the time-domain, the
dependence of the system variables on time was obtained by solving the differential equa
tions describins the system. These dynamic functions tell us what is happening in the
real world (here in simulation) as time increases. In the frequency-domain, the dynamic
rel.ltionships between ih.,&ll variables and output variables were evaluated. From the time
and frequency domain response of the process, a set of closed-form transfer function was
estimated to describe the drum system.
8.3 Cor~t,ol Aigorithm
This thesis presented the development of a methodology for the design and
.n.llysis of drum water level system control and its application to the control of con ven
lional, drum-type. single reheat boilers. The design methodology is based on conventional
115
1 8 ConclusIons
control theory, incorpo"tes feedforw.rd, provides , method of state reconstruction, re-
t.ins the steady state accuracy adv'ntage of classical PlO controllers and is not dependent
upon unreason,ble model precision.
The application to the design .nd .nalysis of digital combined feedback-
feedforwifd control has been successful and the question as to whether or not the appli-
cation of modern digital control by simulation can improve the control of the conventional
boiler has been ,nswered. The improvement in dynamic response which can be obtdined
is shown to be quite significant. The simulation results were obtained using the hnearized
model from the nonlinear equations and the control system performed weil
The digital feedback·feedforward control scheme offers the posslbllity of achlev-
ing a closed·loop system performance which is comparable with the analogue scheme when
the sampling period is sufficiently short. In common with conventlonal analogue scheme
of control, digit,1 control systems may not require the development of the detalled process
dynamic model which 15 mandatory for convention al control schemes. The digital control
system was found to be more sensitive to the choice of sampling period than conventional
control schemes.
ln the study of digital feedback-feedforward control scheme, the selectIon and
adjustment of the control system parameters, to obtain acceptable transient responses,
presented very liule difficulty. Ali the systems were found to provide rapid seUling, with
zero steady·state offset in the controlled process variables, after input disturbances.
The study has iIIustrated the feasibility of applying optImal digital control to
a boiler model, based on minimization of a performance index related to state variables
and system inputs. A realistic stable system has been produced using the concepts of
116
, 8. Conclusions
dynJmic prl..orammIOI. Stability can be ensured by usinl the Ziegler-Nichols method.
The ZN me,"od has incorporated a controller parameter tuning procedure to improve the
system's s/,:~bility. Bued on the tuninl procedure, plus consldering the specifications of
the syste",'·, closed-Ioop behJvior, an optimal control performance has been obtained.
8.4 Future Work
The scope of development of mathematical model and control systems for
engineering thermal-fluid processes is virtually endless. Just for the boiler simulator alone,
many models and control s,'stems ha~e yet to be developed. From a modeling point of
view, any of the following could be possible subjects of future research:
• primary superheater
• superheater spray
• secondary superheater
• turbine
• reheater spray
• reheater
• mills
• combustion
• superheater furnace
117
1
t
8 Conclusions
• reheiter furnice
• fan ind pump
ln th~ modeling and simulation of any of the above elements, the methodology
introduced in this thesis could be applied. Idealy, for instance, in modelmg and simulatlng
the combustion loop of the boiler. the variable step-size multistep algorithm could be used
with the boiler efficiency curve to maintain the boiler operating at maximum efficiency.
From i control point of view, the following two loops could be subJects of
future study in the development of a boiler simulator.
• the combustion loop
• the steam-temperature loop.
The concept of two level control could be adapted ln a future development of
the whole boiler control simulator. The control of each individual process (e g combustion
loop) cou!d be considered as the low level controls. The control methods developed in
this thesis could be used in the control for each indlvldual loop at the lower level The
system level control would supervise the low level controls, calculate and process the input
variables. and transfer variables between the two levels and the low level controls. In the
development of the two level control for boiler simulation, parallel-processing could be
adopted since the two level concept would be well-sulted to parallelization. The system
level solution could be put on one processor to monitor solution convergence, calculate
time steps and trinsfer ~tream variables. The individual process models. distrlbuted in
space over the remaining parallel processors, would compute the single Iteration of the
solution without any externat requirements other than their input/output stream variables.
118
1 8 Conclusions
Further investigation mly reveal the true optimal system concerned with bal-
ancing boiler efficiency a,ainst loss of steam generation, and with steam-fJow-control at
the turbine governor valves under sliding pressure and temperature conditions The same
type of analysis may alsa be extended to the optimiution of the overall steam-generation
process, usine an inte,rated control system including effects of turbines and generators,
with energy input controlled from steam demand, and with the turbine valve regulated
for constant steam pressure.
119
1
,-..
References
References
(1] Amir, N. N., Abram, B., "Steam Generator Water-Level Control". rrans ASME, Journal of Basic Engineering. vol 88. 1966. pp 343-353.
(2] Anderson, J. H., "Dynamic Control of a Power Boiler". PROC. IEE. vol 116.
No. 7,1969.
(3] Auslander, D. M., Takihashi. Y., Tomizuka, M., "The Next Generation of Single loop Controllers: Hardware and Aigorithm for the Discrete/Decimal Process Controller", Trans ASM E, Journal of Oynamic Systems. Measurement, and Control, vol. 97, 1975, pp. 280.
[4] Chien, K. l., Ergin, E. J , Ling. C , Lee, A , "Dynamic Analysis of a BOlier".
T,ans ASME, vol. 80, 1958, pp. 1809-1819
[5] Chien, K. l.. Ergin. E. 1.. ling. c., "The Non-Interactmg Controller for a Steam Generating System", Control Engineering, vol 5. 1958, pp 95-101
[6] Connor, S., "Boiler Controls: Where They've Been, Where They're GOlng" ,
InTech Industry, April, 1989, pp. 36-38.
(7] Daniels, J. H., Enns, M., l-!ottenstine. RD., "Dynamic Representation of a Large Boiler-Turbine Unit", ASM E, Paper 61-5A-69, June. 1961.
[8] Ergin, E. 1., Ling. C, "Development of a Non-Interacting Controller for BOliers", Proe. lst InternationallFAC Conference. Moscow, vol. 4, 1960. pp 347
[9] Goldstein, P., uA Research Study on Internai Corrosion of Hign-Presure BOli
ers", Trans ASME, Journal of Engineering for Power, vol 90. 1968. pp. 21-37.
(10] Hougen, J. O., Hagber,. C. G., FrICke. l. H., Martin, O. R., "Process Identi
fication and Design", Chemical Engineering Progress. vol. 60, no 8, 1964.
[11] Hughes, F. M., Mallouppa, A., "Frequency Response Methods for Nuclear Station Boiler Control", Automatica, vol. 12. pp. 201-210, 1976
[12] Isermann, R., "Digital Control System" , Springer-Verlag. 1981.
(13] James, M. l.. Smith, G M., Wolford, J. C, "Applied Numerieal Methods
for Digital Computation", Harper & Row, Publishers, 2ed edltlon, New York.
1977.
[141 Jury, E. 1., "Sampled-Data Control Systems", John Wlley &. Sons. Ine .. New
York,1958.
120
1 References
(15) Kwan, J. P., Anderson, H. G., liA Mathematical Model of a 200MW Boiler", International Journal of Control, vol. 12, No. 6, 1970, op. 977-998.
(16) Lansin" E. G., 'Variable Pressure Peakin, BoHer, Operation, Testing. and Control". Trans ASME. Journal of Engineerint, for Power, vol. 97, 1975. pr 435-440.
(17J Luyben, W. L., "Process Modelin" Simulation. and Control for Chemical Engineers", McGraw-Hi". Inc., New York. 1990.
(18) MacDonald. J. P., Kwatny, K. G., Spare, J. H., "Non-linear Madel for Reheat Boiler-Turbine-Generator Systems Part 1 - General Description and Evaluation". Proc. 12th Joint Automatic Control Conference. Washington University. St. Louis, Missouri. August 11-13. 1971, pp. 219-226.
119) MacDonald, J. P., Kwatny, K. G .• Spare, J. H .• "Non-linear Madel for Reheat Boiler-Turbine-Generator Systems Part U - Development", Proc. 12th Joint Automatic Control Conference. Washington University, St. Louis, Missouri, August ll-13, 1971, pp. 227-236.
(20) MacDonald, J. P .• Kwatny, H. G., "Design and Analysis of Boiler-TurbineGenerator Con trois Using Optimallinear Regulator Theory". Proe. 12th Joint Automatic Contrel Conference, Washington University, St. Louis. Missouri, Au,ust ll-13, 1971.
(21) Morse, R. H., Brey, R. N., "An Electronic Feedwater Control System for the Experimental Boiling Water Reador", Nuclear Engineeril"î ~nd Science Conference, Session XVII, March 17-21, 1958.
(22) Nahavandi, A. N., yon Ho"en, R. F., "A Space-Dependent Cynalllic Analysis of Boilin, Water Reutor Systems", Journal of the American NucJear Society, Nuclear Science and Engineerin" vol. 20, 1964, pp. 392-413.
(23) Nicholson, H., "Integrated Control of a Nonlinear Boiler Madel", PROC. IEE, vol. ll4, no. 10, 1967, pp. 1569-1576.
(24) Nicholson, H., "Dynamic Optimization of a Boiler" PROC. IEE. vol. Ill, No. 8, 1964. pp. 1479-1500,.
(25J Poon, K. L., "EfTect of Rapid Steam Take-OfT on Natural Circulation and Water Level in Boilers", Trans ASME, Journal of Engineering for Power, vol. 97, 1975, pp. 645-654.
(26) Poon, K. L.. Chiu, P. c., "Changes in Water Level due ta Swell in a PoolBoilin, Plant", Journal of Mechanical Engineering Science, vol. 15, 1973, pp. 329-338.
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1 1 L
1 References
(27) Robinson. R. W., H~nson, R. A., "Obtaining Baseline Data for Boiler Optimization", ISA, InTech, pp. 61-64, 1984.
(28) Roffel, B., Rijnsdorp, J. E., "Process Dynamics, Control and Protection", Ann Arbor Science Publishers, Michigan, 1982.
(29) Seborg, D. E., Ed,u, T. F., Mellichamp, D. A., "Process Dynamics and Control", John Wihey Il Sons, Inc., New York, 1989.
(30) Takahashi, V., Tomizuka, M., Auslander, D. M., "Simple Discrete Control of Industrial Processes (Finite Time Setting Control Aigorithm for Single-Loop Digital Controller)", Trans ASME, Journal of Dynamic Systems, Measurement, and Control, vol. 97, 1975, pp. 354-361.
(31) Thompson, F. T., "A dynamic Model of a Drum-Type Boiler System", IEEE Transactions on Power Apparatus and Systems, vol. PAS-86, No. 5, 1967,
pp. 625-635.
[32] Unbehauen, H., Schmid, c., Bouiger, F., "Comparison and Application of DOC Aigorithms for a Heat Exchanger", Automatica, vol. 12. pp 393-402.
1976.
(33) Vaitekunas, D. A., "A Generic Dynamic Model for Crossflow Heat Exchangers With One Fluid Mixed" Master Th es;s , McGili University, 1990.
(34) Wu, S. Z., Wormley, D. N., Rowell, O., Griffith, P., "Boiler Implosion Control in Fuel Power Plants" Trans ASME, Journal of Oynamic Systems. Measurement. and Control, vol. 107, 1985, pp. 267-269.
[35] Zhao, J., Vroomen, L. J., Zsombor-Murray, P. J., "Design of Boile. Drum Level Control by Simulation", Proe. Computer Applications in Design, Simulation and Analysis, ISMM, New Orleans, March 5-7, 1990, pp. 253-256.
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Appendix A Sample Values of Frequency w
Appendix A. Sample Values of Frequency w
""1 = 0.010 ""18 = 0.501 ""35 = 5.012
""2 = 0.013 W19 = 0.631 ""36 = 5.623
""3 = 0.016 W20 = 0.794 ""31 = 6.310
W4 = 0.020 W21 = 1.000 ""38 = 7.079
Ws = 0.025 W22 = 1.112 ""39 = 7.943
W6 = 0.032 W23 = 1.259 W40 = 8.913
W7 = 0.040 W24 = 1.413 W41 = 10.00
W8 = 0.050 W2S = 1.585 W42 = 12.59
""9 = 0.063 W26 = 1.778 W43 = 15.85
""10 = 0.079 W21 = 1.995 "'-'44 = 19.95
Wu = 0.100 W28 = 2.239 W45 = 25.12
W12 = 0.126 W29 = 2.512 W46 = 39.81
""13 = 0.158 WlO = 2.818 W41 = 50.12
""14 = 0.200 Wl1 = 3.162 W48 = 63.10
""15 = 0.251 Wl2 = 3.548 W49 = 79.43
W16 = 0.316 Wl3 = 3.981 W50 = 100.0
Wl7 = 0.398 W34 = 4.467
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