Drum Boiler

Automatica 36 (2000) 363}378

Drum-boiler dynamicsq

K.J. As stroK m!,*, R.D. Bell"!Department of Automatic Control, Lund Institute of Technology, Box 118, S-221 00 Lund, Sweden

"Department of Computing, School of Mathematics, Physics, Computing and Electronics, Macquarie University, New South Wales 2109, Australia

Received 2 October 1998; revised 7 March 1999; received in "nal form 8 June 1999

Abstract

A nonlinear dynamic model for natural circulation drum-boilers is presented. The model describes the complicated dynamics of thedrum, downcomer, and riser components. It is derived from "rst principles, and is characterized by a few physical parameters.A strong e!ort has been made to strike a balance between "delity and simplicity. Results from validation of the model against uniqueplant data are presented. The model describes the behavior of the system over a wide operating range. ( 2000 Elsevier Science Ltd.All rights reserved.

1. Introduction

There are dramatic changes in the power industrybecause of deregulation. One consequence of this is thatthe demands for rapid changes in power generation isincreasing. This leads to more stringent requirements onthe control systems for the processes. It is required tokeep the processes operating well for large changes in theoperating conditions. One way to achieve this is to incor-porate more process knowledge into the systems. Therehas also been a signi"cant development of methods formodel-based control, see Garcia, Prett and Morari(1989), Qin and Badgwell (1997) and Mayne, Rawlingsand Rao (1999). Lack of good nonlinear process modelsis a bottleneck for using model-based controllers. Formany industrial processes there are good static modelsused for process design and steady-state operation. Byusing system identi"cation techniques it is possible toobtain black box models of reasonable complexity thatdescribe the system well in speci"c operating conditions.Neither static models nor black box models are suitablefor model-based control. Static design models are quite

qThis paper was presented at IFAC 13th World Congress, SanFrancisco, CA, 1996. This paper was recommended for publication inrevised form by Associate Editor T.A. Johansen under the direction ofEditor S. Skogestad.

*Corresponding author. Tel.: 00-46-46-222-8781; fax: 00-46-46-138-8118.

E-mail address: [email protected] (K. J. As stroK m)

complex and they do not capture dynamics. Black boxmodels are only valid for speci"c operating conditions.

This paper presents a nonlinear model for steam gen-eration systems which are a crucial part of most powerplants. The goal is to develop moderately complex non-linear models that capture the key dynamical propertiesover a wide operating range. The models are based onphysical principles and have a small number of para-meters; most of which are determined from constructiondata. Particular attention has been devoted to modeldrum level dynamics well. Drum level control is an im-portant problem for nuclear as well as conventionalplants, see Kwatny and Berg (1993) and Ambos, Duc andFalinower (1996). In Parry, Petetrot and Vivien (1995) itis stated that about 30% of the emergency shutdowns inFrench PWR plants are caused by poor level control ofthe steam water level. One reason is that the controlproblem is di$cult because of the complicated shrinkand swell dynamics. This creates a nonminimum phasebehavior which changes signi"cantly with the operatingconditions.

Since boilers are so common there are many modelinge!orts. There are complicated models in the form of largesimulation codes which are based on "nite element ap-proximations to partial di!erential equations. Althoughsuch models are important for plant design, simulators,and commissioning, they are of little interest for controldesign because of their complexity. Among the earlywork on models suitable for control we can mentionProfos (1955, 1962), Chien, Ergin, Ling and Lee (1958),de Mello (1963), Nicholson (1964), Thompson (1964),

0005-1098/00/$ - see front matter ( 2000 Elsevier Science Ltd. All rights reserved.PII: S 0 0 0 5 - 1 0 9 8 ( 9 9 ) 0 0 1 7 1 - 5

Fig. 1. Schematic picture of the boiler.

Quazza (1968, 1970), Caseau and Godin (1969), Kwanand Andersson (1970), McDonald and Kwatny (1970),Speedy, Bell and Goodwin (1970), Dolezal and Varcop(1970), McDonald, Kwatny and Spare (1971), Eklund(1971), As stroK m and Eklund (1972), As stroK m (1972), Bell(1973), Borsi (1974), Lindahl (1976), Tyss+, Brembo andLind (1976), Bell, Rees and Lee (1977) and Morton andPrice (1977). Boiler modeling is still of substantial inter-est. Among more recent publications we can mentionMa!ezzoni (1988, 1992, 1996), Klefenz (1986), Jarkovsky,Fessl and Medulova (1988), Unbehauen and Kocaarslan(1990), HoK ld (1990), Na and No (1992), Kwatny and Berg(1993), Na (1995).

The work presented in this paper is part of an ongoinglong-range research project that started with Eklund(1971) and Bell (1973). The work has been a mixture ofphysical modeling, system identi"cation and model sim-pli"cation. It has been guided by plant experiments inSweden and Australia. The unique measurements re-ported in Eklund (1971) have been particularly useful.A sequence of experiments with much excitation wereperformed on a boiler over a wide range of operatingconditions. Because of the excitation used, thesemeasurements reveal much of the dynamics of interest forcontrol. Results of system identi"cation experiments in-dicated that the essential dynamics could in fact be cap-tured by simple models, see As stroK m and Eklund (1972).However, it has not been easy to "nd "rst principlesmodels of the appropriate complexity. Many di!erentapproaches have been used. We have searched for thephysical phenomena that yield models of the appropriatecomplexity. Over the years the models have changed incomplexity both increasing and decreasing; empirical co-e$cients have been replaced by physical parameters asour understanding of the system has increased. The pa-pers As stroK m and Eklund (1972, 1975), As stroK m and Bell(1988, 1993) and Bell and As stroK m (1996) describe how themodels have evolved. The models have also been used forcontrol design, see Miller, Bentsman, Drake, Fahkfahk,Jolly, Pellegrinetti and Tse (1990), Pellegrinetti, Be-ntsman and Polla (1991), and Cheng and Rees (1997).Models based on a similar structure have been used forsimulation and control of deaerators, see Lu, Bell andRees (1997), and nuclear reactors, see Yeung and Chan(1990), HoK ld (1990), Irving, Miossec and Tassart (1980),Parry et al. (1995), Menon and Parlos (1992), Thomas,Harrison and Hollywell (1985), Schneider and Boyd(1985) and Kothare, Mettler, Morari, Bendotti andFalinower (1999).

2. Global mass and energy balances

A schematic picture of a boiler system is shown inFig. 1. The heat, Q, supplied to the risers causes boiling.Gravity forces the saturated steam to rise causing a

circulation in the riser-drum-downcomer loop. Feed-water, q

&, is supplied to the drum and saturated steam, q

4,

is taken from the drum to the superheaters and theturbine. The presence of steam below the liquid level inthe drum causes the shrink-and-swell phenomenonwhich makes level control di$cult. In reality the systemis much more complicated than shown in the "gure. Thesystem has a complicated geometry and there are manydowncomer and riser tubes. The out#ow from the riserspasses through a separator to separate the steam fromthe water. In spite of the complexity of the system it turnsout that its gross behavior is well captured by globalmass and energy balances.

A key property of boilers is that there is a very e$cientheat transfer due to boiling and condensation. All partsof the system which are in contact with the saturatedliquid}vapor mixture will be in thermal equilibrium. En-ergy stored in steam and water is released or absorbedvery rapidly when the pressure changes. This mechanismis the key for understanding boiler dynamics. The rapidrelease of energy ensures that di!erent parts of the boilerchange their temperature in the same way. For thisreason the dynamics can be captured by models of loworder. Drum pressure and power dynamics can, in fact,be represented very well with "rst-order dynamics asshown in As stroK m and Eklund (1972). At "rst it is surpris-ing that the distributed e!ects can be neglected for a sys-tem with so large physical dimensions.

Typical values of stored energy for two di!erent boilersare given in Table 1. The P16-G16 plant is a 160 MWunit in Sweden and the Eraring plant is a 660 MW unit inAustralia. The ratio of the energy stored in the metal tothat stored in the water is approximately 1 for P16-G16and 4 for the Eraring unit.

The numbers in Table 1 also give a measure of thetime it takes to deplete the stored energy at the gener-ated rate. Although the total normalized stored energyis approximately the same for both plants the fractionof the energy stored in water is much smaller for thelarger plant. This results in larger variations inwater level for the larger plant under proportionally

364 K.J. A sstro(m, R.D. Bell / Automatica 36 (2000) 363}378

Table 1Energy stored in metal, water, and steam for two boilers operating at rated pressure and temperaturebut at di!erent power generation conditions. The values are normalized with the power at theoperating conditions. The unit is J/W"s, the entries can thus be interpreted as time constants forthe di!erent storage mechanisms

Boiler Metal Water Steam Total

P16-G16 80 MW 641 739 64 1444P16-G16 160 MW 320 333 37 690Eraring 330 MW 1174 303 60 1537Eraring 660 MW 587 137 35 759

similar operating condition changes. This implies thatthe level control problem is more di$cult for largeboilers.

2.1. Balance equations

Much of the behavior of the system is captured byglobal mass and energy balances. Let the inputs to thesystem be the heat #ow rate to the risers, Q, the feedwatermass #ow rate, q

&, and the steam mass #ow rate, q

4.

Furthermore, let the outputs of the system be drumpressure, p, and drum water level, l. This way of charac-terizing the system is convenient for modeling. For simu-lation and control it is necessary to account for the factthat mass #ow rate q

4depends on the pressure by

modeling the turbine and the superheaters.To write the equations, let< denote volume, . denotes

speci"c density, u speci"c internal energy, h speci"c en-thalpy, t temperature and q mass #ow rate. Furthermore,let subscripts s, w, f and m refer to steam, water, feed-water, and metal, respectively. Sometimes, for clari"ca-tion, we need a notation for the system components. Forthis purpose we will use double subscripts where t de-notes total system, d drum and r risers. The total mass ofthe metal tubes and the drum is m

5and the speci"c heat

of the metal is Cp.

The global mass balance is

d

dt[.

4<

45#.

8<

85]"q

&!q

4(1)

and the global energy balance is

d

dt[.

4u4<

45#.

8u8<

85#m

5C

pt.]"Q#q

&h&!q

4h4.

(2)

Since the internal energy is u"h!p/. , the globalenergy balance can be written as

d

dt[.

4h4<

45#.

8h8<

85!p<

5#m

5C

pt.]

"Q#q&h&!q

4h4, (3)

where <45

and <85

represent the total steam and watervolumes, respectively. The total volume of the drum,downcomer, and risers, <

5is

<5"<

45#<

85. (4)

The metal temperature t.

can be expressed as a func-tion of pressure by assuming that changes in t

.are

strongly correlated to changes in the saturation temper-ature of steam t

4and thus also to changes in p. Simula-

tions with models having a detailed representation ofthe temperature distribution in the metal show thatthe steady-state metal temperature is close to the satura-tion temperature and that the temperature di!erencesalso are small dynamically. The right-hand side of Eq. (3)represents the energy #ow to the system from fuel andfeedwater and the energy #ow from the system via thesteam.

2.1.1. A second-order modelEqs. (1), (3), and (4) combined with saturated

steam tables yields a simple boiler model. Mathe-matically, the model is a di!erential algebraic system.Such systems can be entered directly in modelinglanguages such as Omola and Modelica and it can besimulated directly using Omsim, see Mattsson, Anders-son and As stroK m (1993) or Dymola. In this way we avoidmaking manual operations which are time consumingand error prone.

We will, however, make manipulations of the model toobtain a state model. This gives insight into the keyphysical mechanisms that a!ect the dynamic behavior ofthe system. There are many possible choices of statevariables. Since all parts are in thermal equilibrium it isnatural to choose drum pressure p as one state variable.This variable is also easy to measure. Using saturatedsteam tables, the variables .

4, .

8, h

4, and h

8can then be

expressed as functions of steam pressure. The secondstate variable can be chosen as the total volume of waterin the system, i.e. <

85. Using Eq. (4) and noting that <

5is

constant,<45

can then be eliminated from Eqs. (1) and (3)

K.J. A sstro(m, R.D. Bell / Automatica 36 (2000) 363}378 365

Table 2Numerical values of the terms of the coe$cient e

1at normal operating pressure

Boiler h#<

45/. 4

/p.

4<

45/h4/p

.8<

85/h8/p

m5C

p/t4/p

<5

P16-G16 80 MW 360 !40 2080 1410 85P16-G16 160 MW 420 !40 1870 1410 85Eraring 330 MW 700 !270 2240 4620 169Eraring 660 MW 810 !270 2020 4620 169

to give the following state equations:

e11

d<85

dt#e

12

dp

dt"q

&!q

4

e21

d<85

dt#e

22

dp

dt"Q#q

&h&!q

4h4, (5)

where

e11

".8!.

4

e12

"<45

L.4

Lp#<

85

L.8

Lp

e21

".8h8!.

4h4

e22

"<45Ah4

L.4

Lp#.

4

Lh4

Lp B#<

85Ah8L.

8Lp

#.8

Lh8

Lp B!<

5#m

5C

p

Lt4

Lp. (6)

This model captures the gross behavior of the boilerquite well. In particular it describes the response of drumpressure to changes in input power, feedwater #ow rate,and steam #ow rate very well. The model does, however,have one serious de"ciency. Although it describes thetotal water in the system it does not capture the behaviorof the drum level because it does not describe the distri-bution of steam and water in the system.

2.2. Further simplixcations

Additional simpli"cations can be made if we are onlyinterested in the drum pressure. Multiplying (1) byh8

and subtracting the result from (3) gives

h#

d

dt(.

4<

45)#.

4<

45

dh4

dt#.

8<

85

dh8

dt!<

5

dp

dt

#m5C

p

dt4

dt"Q!q

&(h

8!h

&)!q

4h#,

where h#"h

4!h

8is the condensation enthalpy.

If the drum level is controlled well the variations in thesteam volume are small. Neglecting these variations we

get the following approximate model:

e1

dp

dt"Q!q

&(h

8!h

&)!q

4h#, (7)

where

e1"h

#<

45

L.4

Lp#.

4<

45

Lh4

Lp#.

8<

85

Lh8

Lp

#m5C

p

Lt4

Lp!<

5.

The term <5

in e1

comes from the relation betweeninternal energy and enthalpy. This term is often neglectedin modeling, see Denn (1987). The relative magnitudes ofthe terms of e

1for two boilers are given in Table 2. The

terms containing Lh8/Lp and Lt

4/Lp are the dominating

terms in the expression for e1. This implies that the

changes in energy content of the water and metal massesare the physical phenomena that dominate the dynamicsof drum pressure. A good approximation of e

1is

e1+.

8<

85

Lh8

Lp#mC

p

Lt4

Lp.

Table 2 gives good insight into the physical mechanismsthat govern the behavior of the system. Consider forexample the situation when the pressure changes. Thechange in stored energy for this pressure change will beproportional to the numbers in the last two columns ofthe table. The column for the steam (Lh

4/Lp) indicates

that energy changes in the steam are two orders ofmagnitude smaller than the energy changes in water andmetal. The balance of the change in energy is used in theboiling or condensation of steam. The condensation #owrate is

q#5"

h8!h

&h#

q&#

1

h#A. 4<

45

dh4

dt#.

8<

85

dh8

dt

!<5

dp

dt#m

5C

p

dt4

dt B. (8)

Model (7) captures the responses in drum pressure tochanges in heat #ow rate, feedwater #ow rate, feedwatertemperature and steam #ow rate very well. An attractivefeature is that all parameters are given by steam tablesand construction data. The equation gives good insightinto the nonlinear characteristics of the pressure


response, since both e1

and the enthalpies on theright-hand side of the equation depend on the operatingpressure. To obtain a complete model for simulatingthe drum pressure a model for the steam valve has to besupplied.

The pressure model given by Eq. (7) is similar to themodels in de Mello (1963), Quazza (1968), As stroK m andEklund (1972), and Ma!ezzoni (1988). Models similar to(7) are included in most boiler models. Since model (7) isbased on global mass and energy balances it cannotcapture phenomena that are related to the distribution ofsteam and water in the boiler. Therefore it cannot modelthe drum level.

3. Distribution of steam in risers and drum

To obtain a model which can describe the behavior ofthe drum level we must account for the distributionof steam and water in the system. The redistribution ofsteam and water in the system causes the shrink-and-swell e!ect which causes the nonminimum-phase behav-ior of level dynamics, see Kwatny and Berg (1993). Onemanifestation is that the level will increase when thesteam valve is opened because the drum pressure willdrop, causing a swelling of the steam bubbles below thedrum level.

The behavior of two phase #ow is very complicatedand is typically modeled by partial di!erential equations,see Kutateladze (1959) and Heusser (1996). A key contri-bution of this paper is that it is possible to derive relative-ly simple lumped parameter models that agree well withexperimental data.

3.1. Saturated mixture quality in a heated tube

We will start by discussing the dynamics of water andsteam in a heated tube. Consider a vertical tube withuniform heating. Let . be the density of the steam}watermixture. Furthermore let q be the mass #ow rate, A thearea of the cross section of the tube, < the volume, h thespeci"c enthalpy, and Q the heat supplied to the tube. Allquantities are distributed in time, t, and space, z. Assumefor simplicity that all quantities are the same in a crosssection of the tube. The spatial distribution can then becaptured by one coordinate z and all variables are thenfunctions of z and time t.

The mass and energy balances for a heated section ofthe tube are

AL.Lt

#

Lq

Lz"0,

L. h

Lt#

1

A

Lqh

Lz"

Q

<.

Let a.

denote the mass fraction of steam in the #ow, i.e.the quality of the mixture, and let h

4and h

8denote the

speci"c enthalpies of saturated steam and water. The spe-ci"c internal energy of the mixture of steam and water is

h"a.h4#(1!a

.)h

8"h

8#a

.h#. (9)

In steady state we get

Lq

Lz"0,

Lqh

Lz"qh

#

La.

Lz"

QA

<

and it then follows from Eq. (9) that

a."

QA

qh#<

z.

Let m be a normalized length coordinate along the risersand let a

3be the steam quality at the riser outlet. The

steam fraction along the tube is

a.(m)"a

3m, 04m41. (10)

A slight re"nement of the model is to assume thatboiling starts at a distance x

0from the bottom of the

risers. In this case the steam distribution will be charac-terized by two variables a

3and x

0instead of just a

3. For

the experimental data in this paper it adds very little tothe prediction power of the model. For this reason we usethe simpler model although the modi"cation may beimportant for other boilers. There is actually a slip be-tween water and steam in the risers. To take this intoaccount requires much more complicated models. Thejusti"cation for neglecting this is that it does not havea major in#uence on the "t to experimental data.

The volume and mass fractions of steam are relatedthrough a

7"f (a

.), where

f (a.)"

.8a.

.4#(.

8!.

4)a

.

. (11)

It has been veri"ed that the simple model which usesa linear steam-mass fraction given by Eq. (10) anda steam-volume fraction given by Eq. (11) describes quitewell what happens in a typical riser tube. This is illus-trated in Fig. 2 which compares the steam distribution ina tube computed from Eqs. (10) and (11) with computa-tions from a detailed computational #uid dynamics codefor a riser tube in a nuclear reactor. The complex codealso takes into account that there is a slip between the#ow of steam and water. It is interesting to see that thesimple model captures the steam distribution quite well.

3.2. Average steam volume ratio

To model drum level it is essential to describe the totalamount of steam in the risers. This is governed by the


Fig. 2. Comparison of steady-state steam volume distribution cal-culated from Eqs. (10) and (11) (full lines) with results of numericalsolutions of detailed partial di!erential equation models (circles).

average volume fraction in the risers. Assume that themass fraction is linear along the riser as expressed byEq. (10) we "nd that the average volume fraction a6

7is

given by

a67"P

1

0

a7(m)"

1

a3P

a3

0

f (m) dm".

8.

8!.

4

]A1!.

4(.

8!.

4)a

3

lnA1#.

8!.

4.

4

a3BB. (12)

3.3. A lumped parameter model

Since we do not want to use partial di!erential equa-tions they will be approximated using the Galerkinmethod. To do this it will be assumed that the steam-mass quality distribution is linear, i.e. Eq. (10) holds, alsounder dynamic conditions.

The transfer of mass and energy between steam andwater by condensation and evaporation is a key elementin the modeling. When modeling the phases separatelythe transfer must be accounted for explicitly. This can beavoided by writing joint balance equations for water andsteam. The global mass balance for the riser section is

d

dt(.

4a67<

3#.

8(1!a6

7)<

3)"q

$#!q

3, (13)

where q3is the total mass #ow rate out of the risers and

q$#

is the total mass #ow rate into the risers. The globalenergy balance for the riser section is

d

dt(.

4h4a67<

3#.

8h8(1!a6

7)<

3!p<

3#m

3C

pt4)

"Q#q$#

h8!(a

3h##h

8)q

3. (14)

3.4. Circulation yow

For a forced circulation boiler downcomer #ow rate,q$#

is a control variable. For a natural circulation boiler

the #ow rate is instead driven by the density gradients inthe risers and the downcomers. The momentum balancefor the downcomer riser loop is

(¸3#¸

$#)dq

$#dt

"(.8!.

4)a6

7<

3g!

k

2

q2$#

.8A

$#

,

where k is a dimensionless friction coe$cient, ¸3

and¸$#

are lengths and A$#

is the area. This is a "rst-ordersystem with the time constant

¹"

(¸3#¸

$#)A

$#.

8kq

$#

.

With typical numerical values we "nd that the timeconstant is about a second. This is short in comparisonwith the sampling period of our experimental data whichis 10 s and we will therefore use the steady-state relation

12kq2

$#".

8A

$#(.

8!.

4)ga6

7<

3. (15)

3.5. Distribution of steam in the drum

The physical phenomena in the drum are complicated.Steam enters from many riser tubes, feedwater entersthrough a complex arrangement, water leaves throughthe downcomer tubes and steam through the steam valve.Geometry and #ow patterns are complex. The basicmechanisms are separation of water and steam and con-densation.

Let <4$

and <8$

be the volume of steam and waterunder the liquid level and let the steam #ow rate throughthe liquid surface in the drum be q

4$. Recall that q

3is the

#ow rate out of the risers, q&the feedwater #ow rate and

q$#

the downcomer #ow rate. The mass balance for thesteam under the liquid level is

d

dt(.

4<

4$)"a

3q3!q

4$!q

#$, (16)

where q#$

is the condensation #ow which is given by

q#$"

h8!h

&h#

q&#

1

h#A. 4<

4$

dh4

dt#.

8<

8$

dh8

dt

!(<4$#<

8$)dp

dt#m

$C

p

dt4

dt B. (17)

The #ow q4$

is driven by the density di!erences of waterand steam, and the momentum of the #ow entering thedrum. Several models of di!erent complexity have beenattempted. Good "t to the experimental data have beenobtained with the following empirical model:

q4$"

.4

¹$

(<4$!<0

4$)#a

3q$##a

3b(q

$#!q

3). (18)

Here<04$

denotes the volume of steam in the drum in thehypothetical situation when there is no condensation ofsteam in the drum and ¹

$is the residence time of the

steam in the drum.


3.6. Drum level

Having accounted for the distribution of the steambelow the drum level, we can now model the drum level.The volume of water in the drum is

<8$

"<85!<

$#!(1!a6

7)<

3. (19)

The drum has a complicated geometry. The linearizedbehavior can be described by the wet surface A

$at the

operating level. The deviation of the drum level l mea-sured from its normal operating level is

l"<

8$#<

4$A

$

"l8#l

4. (20)

The term l8

represents level variations caused bychanges of the amount of water in the drum and theterm l

4represents variations caused by the steam in the

drum.

4. The model

Combining the results of Sections 2 and 3 we can nowobtain a model that gives a good description of the boilerincluding the drum level. The model is given by thedi!erential equations (1), (3), (13), (14), and (16). In addi-tion there are a number of algebraic equations. Thecirculation #ow rate q

$#is given by the static momentum

balance (15), the steam #ow rate through the liquidsurface of the drum q

4$by (18), and the drum level l by

Eq. (20). The volumes are related through Eqs. (4) and(19). The model is a di!erential algebraic system, seeHairer, Lubich and Roche (1989). Since most availablesimulation software requires state equations we will alsoderive a state model.

4.1. Selection of state variables

State variables can be chosen in many di!erent ways. Itis convenient to choose states as variables with goodphysical interpretation that describe storage of mass,energy, and momentum. The accumulation of water isrepresented by the total water volume <

85. The total

energy is represented by the drum pressure p and thedistribution of steam and water is captured by thesteam-mass fraction in the risers a

3and the steam volume

in the drum <4$

.

4.2. Pressure and water dynamics

State equations for pressure p and the total amount ofwater <

85in the systems were obtained from the global

mass and energy balances, Eqs. (1) and (3). These equa-tions can be written as (5).

4.3. Riser dynamics

The mass and energy balances for the risers are givenby Eqs. (13) and (14). Eliminating the #ow rate out of therisers, q

3, by multiplying Eq. (13) by !(h

8#a

3h#) and

adding to Eq. (14) gives,

d

dt(.

4h4a67<

3)!(h

8#a

3h#)d

dt(.

4a67<

3)

#

d

dt(.

8h8(1!a6

7)<

3)

!(h8#a

3h#)d

dt(.

8(1!a6

7)<

3)

!<3

dp

dt#m

3C

p

dt4

dt"Q!a

3h#q$#

.

This can be simpli"ed to

h#(1!a

3)d

dt(.

4a67<

3)#.

8(1!a6

7)<

3

dh8

dt

!a3h#

d

dt(.

8(1!a6

7)<

3)#.

4a67<

3

dh4

dt

!<3

dp

dt#m

3C

p

dt4

dt"Q!a

3h#q$#

. (21)

If the state variables p and a3

are known the riser#ow rate q

3can be computed from Eq. (13). This

gives

q3"q

$#!

d

dt(.

4a67<

3)!

d

dt(.

8(1!a6

7)<

3)

"q$#!<

3

d

dt((1!a6

7).

8#a6

7.

4)

"q$#!<

3

d

dt(.

8!a6

7(.

8!.

4))

"q$#!<

3

LLp

((1!a67).

8#a6

7.

4)dp

dt

#<3(.

8!.

4)La6

7La

3

da3

dt. (22)

4.4. Drum dynamics

The dynamics for the steam in the drum is obtainedfrom the mass balance (16). Introducing expression (22)for q

3, expression (17) for q

#$and expression (18) for


q4$

into this equation we "nd

.4

d<4$

dt#<

4$

d.4

dt#

1

h#A. 4<

4$

dh4

dt#.

8<

8$

dh8

dt

!(<4$#<

8$)dp

dt#m

$C

p

dt4

dt B#a

3(1#b)<

3

d

dt((1!a6

7).

8#a6

7.

4)

"

.4

¹$

(<04$!<

4$)#

h&!h

8h#

q&. (23)

Many of the complex phenomena in the drum are cap-tured by this equation.

4.5. Summary

The state variables are: drum pressure p, total watervolume <

85, steam quality at the riser outlet a

3, and vol-

ume of steam under the liquid level in the drum <4$

. Thetime derivatives of these variables are given by Eqs. (5),(21), and (23). Straightforward but tedious calculationsshow that these equations can be written as

e11

d<85

dt#e

12

dp

dt"q

&!q

4,

e21

d<85

dt#e

22

dp

dt"Q#q

&h&!q

4h4,

e32

dp

dt#e

33

da3

dt"Q!a

3h#q$#

, (24)

e42

dp

dt#e

43

da3

dt#e

44

d<4$

dt

"

.4

¹$

(<04$!<

4$)#

h&!h

8h#

q&,

where h#"h

4!h

8and

e11

".8!.

4,

e12

"<85

L.8

Lp#<

45

L.4

Lp,

e21

".8h8!.

4h4,

e22

"<85Ah8

L.8

Lp#.

8

Lh8

Lp B#<

45Ah4L.

4Lp

#.4

Lh4

Lp B!<5#m

5C

p

Lt4

Lp,

e32

"A. 8

Lh8

Lp!a

3h#

L.8

Lp B(1!a67)<

3

#A(1!a3)h

#

L.4

Lp#.

4

Lh4

Lp Ba6 7<3

#(.4#(.

8!.

4)a

3) h

#<

3

La67

Lp

!<3#m

3C

p

Lt4

Lp, (25)

e33

"((1!a3).

4#a

3.

8)h

#<

3

La67

La3

e42

"<4$

L.4

Lp#

1

h#A. 4<

4$

Lh4

Lp#.

8<

8$

Lh8

Lp!<

4$

!<8$

#m$C

p

Lt4

Lp B#a3(1#b)<

3

Aa6 7L.

4Lp

#(1!a67)L.

8Lp

#(.4!.

8)La6

7Lp B,

e43

"a3(1#b)(o

4!o

8)<

3

La67

La3

,

e44

".4.

In addition steam tables are required to evaluateh4, h

8,o

4, o

8, Lo

4/Lp, Lo

8/Lp, Lh

4/Lp, Lh

8/Lp, t

4, and Lt

4/Lp

at the saturation pressure p. The results in Sections 5 and6 are based on approximations of steam tables withquadratic functions. More elaborated approximationswith table look-up and interpolation have been tried butthe di!erences in the dynamic responses are not signi"-cant.

The steam volume fraction a67

is given by Eq. (12), thevolume of water in drum <

8$by Eq. (19), the drum level

l by Eq. (20), and the downcomer mass #ow rate q$#

byEq. (15).

The partial derivatives of the steam volume fractionwith respect to pressure and mass fraction are obtainedby di!erentiating Eq. (12). We get

La67

Lp"

1

(o8!o

4)2Ao8

Lo4

Lp!o

4

Lo8

Lp B

A1#o8

o4

1

1#g!

o4#o

8go

4

ln(1#g)B (26)

La67

La3

"

.8

.4gA

1

gln(1#g)!

1

1#gB,where g"a

3(.

8!.

4)/.

4.

It is also of interest to know the total condensation#ow rate q

#5and the #ow rate out of the risers q

3. These

#ows are given by Eqs. (8) and (22), hence

a67"

.8

.8!.

4A1!

.4

(.8!.

4)a

3

lnA1#.

8!.

4.

4

a3BB,


<8$

"<85!<

$#!(1!a6

7)<

3,

l"<

8$#<

4$A

$

,

¹$"

o4<0

4$q4$

,

q2$#"

2.8A

$#(.

8!.

4)ga6

7<

3k

q#5"

h8!h

&h#

q&#

1

h#A. 4<

st

Lh4

Lp#.

8<

85

Lh8

Lp

!<5#m

5C

p

Lt4

Lp Bdp

dt,

q3"q

$#!<

3Aa6 7L.

4Lp

#(1!a67)L.

8Lp

#(o8!o

4)La6

7Lp B

dp

dt#(o

8!o

4)<

3

La67

La3

da3

dt.

4.6. Structure of the equations

Note that Eq. (24) has an interesting lower triangularstructure where the state variables can be grouped as(((<

85, p),a

3),<

4$) where the variables inside each paren-

thesis can be computed independently. The model canthus be regarded as a nesting of a second-, a third-, anda fourth-order model. The second-order model describesdrum pressure and total volume of water in the system.The equations are global mass and energy balances.There is a very weak coupling between these equationswhich is caused by the condensation #ow. The third-order model captures the steam dynamics in the risersand the fourth-order model also describes the dynamicsof steam below the water surface in the drum. The thirdequation is a combination of mass and energy balancesfor the riser, and the fourth equation is a mass balance forsteam under the water level in the drum.

Linearizing Eq. (24) shows that the system hasa double pole at the origin and poles at !h

#q$#

/e33

and!1/¹

$. One pole at the origin is associated with water

dynamics and the other with pressure dynamics. The poleassociated with pressure dynamics is at the origin be-cause the steam #ow is chosen as a control variable. Thepole moves into the left half-plane when the drum isconnected to the turbines. The poles !h

#q$#

/e33

and!1/¹

$are associated with dynamics of steam in the

risers and the drum.The nested structure re#ects how the model was de-

veloped. The third-order model is an improved version ofthe models in As stroK m and Bell (1988). In Bell and As stroK m(1996) we added drum dynamics as a fourth-ordermodel. The model in this paper is a re"ned version of thatmodel. Di!erent models of higher order have also beendeveloped.

4.7. Parameters

An interesting feature of the model is that it requiresonly nine parameters:

f drum volume <$,

f riser volume <3,

f downcomer volume <$#

,f drum area A

$at normal operating level,

f total metal mass m5,

f total riser mass m3,

f friction coe$cient in downcomer-riser loop k,f residence time ¹

$of steam in drum,

f parameter b in the empirical equation (18).

A convenient way to "nd the parameter k is to compute itfrom the circulation #ow rate. Perturbation studies haveshown that the behavior of the system is not very sensi-tive to the parameters.

The parameters used in this paper were based onconstruction data. Some of them were quite crude.Gray-box identi"cation, Bohlin (1991) was used ina comprehensive investigation in Eborn and S+rlie (1997)and S+rlie and Eborn (1999). Parameters were estimatedand hypothesis testing was used to compare severalmodel structures. The results showed that pressure dy-namics can be improved signi"cantly by increasing themetal masses. Signi"cant improvements can also be ob-tained by adjusting the coe$cients in the calibrationformula for the sensors.

In S+rlie and Eborn (1999) it was shown that thefriction coe$cient is not identi"able from the data. Thereis in fact a relation between the initial steam quality andfriction. A consequence of this is that it is highly desirablefor accurate modeling to measure the circulation #ow.This could also be an important signal to use in a levelcontrol system.

In Eborn and S+rlie (1997) hypothesis testing wasapplied to the models As stroK m and Bell (1988, 1993) (thirdorder) and Bell and As stroK m (1996) (fourth order). Thesestudies showed conclusively that the improvements ob-tained with the fourth-order model are signi"cant. Com-putations on "fth-order models with a more detailedrepresentation of the drum showed that the increasedcomplexity could not be justi"ed.

4.8. Equilibrium values

The steady-state solution of Eq. (24) is given by

q&"q

4,

Q"q4h4!q

&h&,

Q"q$#

a3h#,

<4$"<0

4$!

¹$(h

8!h

&)

.4h#

q&,


Fig. 3. Steady-state relation between steam volume ratio a67

and inputpower Q. The dashed curve shows the steam mass ratio a

3.

where q$#

is given by Eq. (15), i.e.

q$#"S

2.8A

$#(.

8!.

4)ga6

7<

3k

.

A convenient way to "nd the initial values is to "rstspecify steam #ow rate q

4and steam pressure p. The

feedwater #ow rate q&

and the input power Q are thengiven by the "rst two equations and the steam volume inthe drum is given by the last equation. The steam qualitya3

is obtained by solving the nonlinear equations

Q"a3h#S

2.8A

$#(.

8!.

4)ga6

7<

3k

a67"

.8

.8!.

4A1!

.4

(.8!.

4)a

3

lnA1#.

8!.

4.

4

a3BB.

(27)

The steam volume in the drum can then be computeddirectly.

Eq. (27) de"nes the steam volume ratio a67as a function

of the input power Q. This function which is shown inFig. 3 gives important insight into the shrink and swellphenomena. The curve shows that a given change ininput power gives a larger variation in average steamvolume ratio at low power. This explains why the shrinkand swell e!ects are larger at low power than at highpower.

4.9. Impact of modeling languages

Development of physical models is a tedious iterativeprocess. Di!erent physical assumptions are made,a model is developed and compared with experiments bysimulation, parameters may be "tted. Detailed investiga-tion of the results gives ideas for improvements andmodi"cations. It is a signi"cant e!ort to transform theequations to state space form because of the algebrainvolved. This is re#ected in the manipulations resultingin Eq. (24). Many intermediate steps have actually beenomitted in the paper. The modeling e!ort can be reducedsubstantially by using modeling languages such as

Dymola, Elmqvist (1978), Omola, Mattsson et al. (1993),or Modelica, Elmqvist, Mattsson and Otter (1998). Inthese languages the model is described in its most basicform in terms of di!erential algebraic equations. In ourcase this means that the basic mass and energy balances,such as (1) and (2), are entered into the system togetherwith the algebraic equations, such as (4). The softwarethen makes algebraic manipulations symbolically to sim-plify the equations for e$cient simulation.

5. Step responses

To illustrate the dynamic behavior of the model wewill simulate responses to step changes in the inputs.Since there are many inputs and many interesting vari-ables we will focus on a few selected responses. One inputwas changed and the others were kept constant. Themagnitudes of the changes were about 10% of the nom-inal values of the signals. To compare responses at di!er-ent load conditions the same amplitudes were used athigh and medium load.

5.1. Plant parameters

The parameters used were those from the Swedishpower plant. The values are <

$"40 m3, <

3"37 m3,

<$#"11 m3, A

$"20 m2, m

5"300 ,000 kg , m

3"

160,000 kg, k"25,b"0.3, and ¹$"12 s. The steam

tables were approximated by quadratic functions.

5.2. Fuel yow changes at medium load

Fig. 4 shows the responses of the state variables, thecirculation #ow rate q

$#, the riser #ow rate q

3, and the

total condensation #ow rate q#5

to a step increase in fuel#ow rate equivalent to 10 MW. Pressure increases atapproximately constant rate. The reason for this is thatsteam #ow out of the drum is constant. Total watervolume <

85increases due to the condensation that oc-

curs due to the increasing pressure. Steam quality at theriser outlet a

3"rst increases rapidly and then more grad-

ually. The volume of steam in the drum "rst increasesa little and it then decreases. The rapid initial increase insteam volume is due to the fast increase in steam from therisers. The decrease is due to the increased pressure whichcauses condensation of the steam. At the onset of the stepthere is a rapid increase in the outlet #ow rate from therisers. The #ow then decreases to match the downcomer#ow rate. The #ow rates are equal after about 30 s. Thecondensation #ow changes in a step-like manner.

5.3. Steam yow changes at medium load

Fig. 5 shows the responses to a step increase of 10 kg/sin steam #ow rate at medium load. Pressure decreases


Fig. 4. Responses to a step in fuel #ow rate of 10 MW at medium load.

Fig. 5. Responses to a step in steam #ow rate of 10 kg/s at mediumload.

Fig. 6. Responses to a step in fuel #ow rate of 10 MW at medium (solid)and high (dashed) loads.

Fig. 7. Responses to a step in steam #ow rate of 10 kg/s at medium(solid) and high (dashed) loads.

practically linearly because of the increased steam #ow.Total water volume also decreases because of increasedevaporation due to the decreasing pressure. Steam qual-ity at the riser outlet "rst increases rapidly due to thepressure decrease and it then decreases due to the in-creased circulation #ow rate. The volume of the steam inthe drum increases due to the decreased pressure. Thereis a very rapid increase of #ow out of the riser due to thepressure drop. After this initial transient the riser #owrate then decreases to match the downcomer #ow rate.There is a steady increase in both due to the decreasedpressure. The condensation #ow drops in an almoststep-like fashion because the pressure decreases atconstant rate.

5.4. Drum level responses

Since the behavior of the drum level is of particularinterest we will show step responses that give good in-sight into drum level dynamics for di!erent operatingconditions. It follows from Eq. (20) that drum level is thesum of l

8"<

8$/A

$and l

4"<

4$/A

$, which depend on

the volumes of water and steam in the drum. Drumpressure, steam mass, and volume fractions will also beshown. Responses for medium and high load will be givento illustrate the dependence on operating conditions.

Fig. 6 shows the response to a step in fuel #ow corre-sponding to 10 MW. The response in drum level is com-plicated and depends on a combination of the dynamics


Fig. 8. Comparison of model (solid line) and plant data (dots) forperturbations in fuel #ow rate at medium load.

of water and steam in the drum. The initial part of theswell is due to the rapid initial response of steam that wasalso seen in Fig. 4. The response in level is a combinationof two competing mechanisms. The water volume in thedrum increases due to increased condensation caused bythe increasing pressure. The volume of the steam in thedrum "rst increases a little and it then decreases becauseof the increasing pressure. Note that there are signi"cantchanges in steam quality and steam volume ratio for thedi!erent operating conditions. Compare with Fig. 3.

Fig. 7 shows the response to a step increase in steam#ow rate of 10 kg/s. There is a strong shrink and swelle!ect in this case too. The contributions from the vol-umes of steam and water have the same sign initially. Thewater volume will, however, decrease because of thesteam #ow.

6. Comparisons with plant data

Much of the model development was based on plantexperiments performed with the P16-G16 unit at OG re-sundsverket in MalmoK , Sweden in collaboration withSydkraft AB. The experiments are described in Eklund(1971) and As stroK m and Eklund (1972, 1975). They werecarried out in open loop with the normal regulatorsremoved. The signals were "ltered and sampled at a rateof 0.1 Hz. To ensure a good excitation of the processPRBS-like perturbations were introduced in fuel #owrate, feedwater #ow rate, and steam #ow rate. The inten-tion was to change one input signal in each experiment.To ensure that critical variables such as drum water leveldid not go outside safe limit we made manual correctionoccasionally. This means that several inputs werechanged in each experiment. The steam #ow ratechanged in response to pressure changes in all experi-ments because we were unable to control it tightly.

A large number of signals were logged during theexperiment. This proved to be very valuable because ithas been possible to use the data for a very large numberof investigations. The experiments were performed bothat high and medium load. In this paper we have used datawhere three variables, fuel #ow rate, feedwater #ow rate,and steam #ow rate were changed. This gives a total ofsix experiments which can be used to validate the model.

There were problems with the calibration of the #owrate measurement transducers and also some uncertaintyin the e!ective energy content of the oil. The approachused was to start with the nominal calibration values andthen make small corrections so that the long-term track-ing between the plant data and the model was as close aspossible. The change in all cases was well within thetransducers accuracy limits. Apart from that, no "ddlingwith the coe$cients was made. When showing the resultsin the following, we present the primary input variableand the responses in drum pressure and drum level.

6.1. Experiments at medium load

The results of experiments at medium load will bedescribed "rst. This is the operating condition where theshrink and swell phenomenon is most pronounced.

6.1.1. Fuel yow rate changesFig. 8 shows responses in drum pressure and drum level

for perturbations in fuel #ow rate. There is very goodagreement between the model and the experimental datafor drum pressure and drum water level. Note in particu-lar that there is an overshoot in the drum level responsefor step changes in fuel #ow. This is caused by theinteraction between the two state variables that describethe dynamics of steam under the liquid level in the drum.

6.1.2. Changes in feedwater yow rateFig. 9 shows the responses to changes in the feedwater#ow rate. The general character of the responses agreeswell. There are some deviations in the pressure responsesand some of the "ner details of the drum level are exag-gerated. The pressure does deviate in the 500}1500 sregion, but since the changes in pressure are small theresults are considered adequate.

6.1.3. Changes in the steam valveFig. 10 shows the responses to changes in the steam

valve. There is very good agreement between the modeland the experimental data. Note that there is a signi"cantshrink and swell e!ect which is captured very well by themodel.

6.2. Experiments at high load

6.2.1. Changes in fuel yow rateFig. 11 shows responses in drum pressure and drum

level for perturbations in fuel #ow rate. There is good


Fig. 9. Comparison of model (solid line) and plant data (dots) forperturbations in feedwater #ow rate at medium load.

Fig. 10. Comparison of model (solid line) and plant data (dots) forperturbations in steam #ow rate at medium load.

Fig. 11. Comparison of model (solid line) and plant data (dots) forperturbations in fuel #ow rate at high load.

Fig. 12. Comparison of model (solid line) and plant data (dots) forperturbations in feedwater #ow rate at high load.

agreement between the model and the experimental data.A comparison with Fig. 8 shows that the shrink and swelle!ect is much smaller at high load. It is interesting to seethat the model captures this.

6.2.2. Changes in feedwater yow rateFig. 12 shows the responses to changes in feedwater#ow rate. There are discrepancies in pressure from time400 to 1400. The total changes in pressure are small andminor variations in feedwater conditions can easily causevariations of this magnitude. The model exaggerates thelevel changes for rapid variations. Compare for exampledata in the interval 1500}2000.

6.2.3. Changes in the steam valveFig. 13 shows the responses to changes in the steam#ow rate. There is very good agreement between themodel and the experimental data. Note that the modelcaptures the drum level variations, particularly the swelland shrink e!ect very well. A comparison with Fig. 10shows that the shrink and swell e!ect is smaller at highloads. This is well captured by the model.

6.3. Comparison of behavior at high and medium load

The experiments have indicated that there are signi"-cant di!erences in behavior at high and medium loadsthat are well predicted by the model. We will now lookcloser at these di!erences.


Fig. 13. Comparison of model (solid line) and plant data (dots) forperturbations in steam #ow rate at high load.

Fig. 14. Comparison of behavior of drum water level at medium andhigh loads for perturbations in steam #ow rate. The model response isshown in solid lines and plant data is indicated by dots.

Fig. 15. Comparison of behavior of drum water level at medium andhigh loads for perturbations in fuel #ow rate. The model response isshown in solid lines and plant data is indicated by dots.

Fig. 16. Comparison of behavior of drum water level at medium andhigh loads for perturbations in fuel #ow rate. The model response isshown in solid lines and plant data is indicated by dots.

Fig. 14 compares the responses in drum level to steam#ow changes at high and medium loads. The steam valvechanges were almost the same in both experiments (as isshown in Figs. 10 and 13) but they were not identical.Because of the integrators in the model there are naturaldi!erences in the levels of the signals. Apart from thislevel shift the model matches the experiments very well.Fig. 14 also shows that the shrink and swell e!ect islarger at medium load. It is even larger at low loads.

Fig. 15 compares responses to changes in fuel #ow.Note the good agreement for dynamic responses betweenthe model and experiments. In this case there is a pro-nounced di!erence between the behaviors for mediumand high load. Fig. 16 compares responses to changes infuel #ow. We have taken a section of the data where there

are substantial rapid variations. Again we note the excel-lent agreement between model and experiments and wealso note the signi"cant di!erence between the behaviorsat medium and high loads.

7. Conclusions

A nonlinear physical model with a complexity that issuitable for model-based control has been presented. Themodel is based on physical parameters for the plant andcan be easily scaled to represent any drum power station.The model has four states; two account for storage oftotal energy and total mass, one characterizes steamdistribution in the risers and another the steam distribu-


tion in the drum. The model can be characterized bysteam tables and a few physical parameters.

The model is nonlinear and agrees well with experi-mental data. In particular, the complex shrink and swellphenomena associated with the drum water level are wellcaptured by the model. The model has a triangular struc-ture that can be described as (((<

85, p),a

3),<

4$), where the

states in each bracket can be determined sequentially.The linearized model has two poles at the origin and tworeal stable poles.

The model has been validated against plant data withvery rich excitation that covers a wide operating range.These experiments have given much insight into the be-havior of the system and they have guided the modelinge!ort. We believe that the approach used in this work canbe applied to other con"gurations of steam generators.

Model (24) can be simpli"ed by keeping only the domi-nant terms in expressions (25) for the coe$cients e

ij. This

could be useful for applications to model-based control.Preliminary investigations indicate that several terms canbe neglected without sacri"cing the "t to experimentaldata. A comprehensive study of this is outside the scopeof this paper. The model can also be re"ned in severalways. This will, however, require new measurements withfaster sampling rates.

Acknowledgements

The research has been supported by the Sydkraft Re-search Foundation and the Swedish National Board forIndustrial and Technical Development under contract97-04573. This support is gratefully acknowledged. Wewould also like to express our sincere gratitude to Syd-kraft AB for their willingness to perform experiments onplants to increase our understanding of their behavior.Useful comments on several versions of the manuscripthave been given by our colleagues J. Eborn, H. Tummes-cheit, and A. Glattfelder. We would also like to expressour gratitude to the reviewers for constructive criticism.

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Karl J. As stroK m is Professor and Head ofthe Department of Automatic Control atLund University since 1965. He has broadinterests in automatic control including,stochastic control, system identi"cation,adaptive control, computer control andcomputer-aided control engineering. Hehas supervised 44 Ph.D. students, writtensix books and more than 100 papers inarchival journals. He is a member of theRoyal Swedish Academy of EngineeringSciences (IVA) and the Royal Swedish

Academy of Sciences (KVA) and a foreign member of the US NationalAcademy of Engineering and the Russian Academy of Sciences. As stroK mhas received many honors including three honorary doctorates, theCallender Silver Medal, the Quazza Medal from IFAC, the RufusOldenburger Medal from ASME, the IEEE Control Systems ScienceAward and the IEEE Medal of Honor.

Rod Bell is an Honorary Senior ResearchFellow, a position he has held since Febru-ary 1998. His previous position was Headof the Computing Department which heheld since the beginning of 1994. He ob-tained his Ph.D. (Application of OptimalControl Theory to Industrial Processes) in1972 from the University of New SouthWales. He joined the academic sta! atMacquarie University in 1972. He becameinterested in computers (hardware andsoftware) in 1958 when he worked on one

of the "rst digital computers in Australia, UTECOM at the Universityof New South Wales. His current research interests lie mainly in theApplication of Control Theory to areas such as Industrial Processes(Power Stations in particular), National Economies, Management andRobotics. He is a member of IEEE. Among his personal interests arehorse riding, growing waratahs, bush walking and restoring Jaguarcars.


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Drum Boiler