Drum-boiler dynamics

Transcription

1 Automatica 36 (2000) 363}378 Drum-boiler ynamics K.J. As strok m* R.D. Bell Department of Automatic Control Lun Institute of Technology Box 118 S Lun Sween Department of Computing School of Mathematics Physics Computing an Electronics Macquarie University New South Wales 2109 Australia Receive 2 October 1998; revise 7 March 1999; receive in "nal form 8 June 1999 Abstract A nonlinear ynamic moel for natural circulation rum-boilers is presente. The moel escribes the complicate ynamics of the rum owncomer an riser components. It is erive from "rst principles an is characterize by a few physical parameters. A strong e!ort has been mae to strike a balance between "elity an simplicity. Results from valiation of the moel against unique plant ata are presente. The moel escribes the behavior of the system over a wie operating range Elsevier Science Lt. All rights reserve. 1. Introuction There are ramatic changes in the power inustry because of eregulation. One consequence of this is that the emans for rapi changes in power generation is increasing. This leas to more stringent requirements on the control systems for the processes. It is require to keep the processes operating well for large changes in the operating conitions. One way to achieve this is to incorporate more process knowlege into the systems. There has also been a signi"cant evelopment of methos for moel-base control see Garcia Prett an Morari (1989) Qin an Bagwell (1997) an Mayne Rawlings an Rao (1999). Lack of goo nonlinear process moels is a bottleneck for using moel-base controllers. For many inustrial processes there are goo static moels use for process esign an steay-state operation. By using system ienti"cation techniques it is possible to obtain black box moels of reasonable complexity that escribe the system well in speci"c operating conitions. Neither static moels nor black box moels are suitable for moel-base control. Static esign moels are quite This paper was presente at IFAC 13th Worl Congress San Francisco CA This paper was recommene for publication in revise form by Associate Eitor T.A. Johansen uner the irection of Eitor S. Skogesta. * Corresponing author. Tel.: ; fax: aress: [email protected] (K. J. As strok m) complex an they o not capture ynamics. Black box moels are only vali for speci"c operating conitions. This paper presents a nonlinear moel for steam generation systems which are a crucial part of most power plants. The goal is to evelop moerately complex nonlinear moels that capture the key ynamical properties over a wie operating range. The moels are base on physical principles an have a small number of parameters; most of which are etermine from construction ata. Particular attention has been evote to moel rum level ynamics well. Drum level control is an important problem for nuclear as well as conventional plants see Kwatny an Berg (1993) an Ambos Duc an Falinower (1996). In Parry Petetrot an Vivien (1995) it is state that about 30% of the emergency shutowns in French PWR plants are cause by poor level control of the steam water level. One reason is that the control problem is i$cult because of the complicate shrink an swell ynamics. This creates a nonminimum phase behavior which changes signi"cantly with the operating conitions. Since boilers are so common there are many moeling e!orts. There are complicate moels in the form of large simulation coes which are base on "nite element approximations to partial i!erential equations. Although such moels are important for plant esign simulators an commissioning they are of little interest for control esign because of their complexity. Among the early work on moels suitable for control we can mention Profos ( ) Chien Ergin Ling an Lee (1958) e Mello (1963) Nicholson (1964) Thompson (1964) /00/$- see front matter 2000 Elsevier Science Lt. All rights reserve. PII: S ( 9 9 )

2 364 K.J. Asstro( m R.D. Bell / Automatica 36 (2000) 363}378 Quazza ( ) Caseau an Goin (1969) Kwan an Anersson (1970) McDonal an Kwatny (1970) Speey Bell an Goowin (1970) Dolezal an Varcop (1970) McDonal Kwatny an Spare (1971) Eklun (1971) As strok m an Eklun (1972) As strok m (1972) Bell (1973) Borsi (1974) Linahl (1976) Tyss+ Brembo an Lin (1976) Bell Rees an Lee (1977) an Morton an Price (1977). Boiler moeling is still of substantial interest. Among more recent publications we can mention Ma!ezzoni ( ) Klefenz (1986) Jarkovsky Fessl an Meulova (1988) Unbehauen an Kocaarslan (1990) HoK l (1990) Na an No (1992) Kwatny an Berg (1993) Na (1995). The work presente in this paper is part of an ongoing long-range research project that starte with Eklun (1971) an Bell (1973). The work has been a mixture of physical moeling system ienti"cation an moel simpli"cation. It has been guie by plant experiments in Sween an Australia. The unique measurements reporte in Eklun (1971) have been particularly useful. A sequence of experiments with much excitation were performe on a boiler over a wie range of operating conitions. Because of the excitation use these measurements reveal much of the ynamics of interest for control. Results of system ienti"cation experiments inicate that the essential ynamics coul in fact be capture by simple moels see As strok m an Eklun (1972). However it has not been easy to "n "rst principles moels of the appropriate complexity. Many i!erent approaches have been use. We have searche for the physical phenomena that yiel moels of the appropriate complexity. Over the years the moels have change in complexity both increasing an ecreasing; empirical coe$cients have been replace by physical parameters as our unerstaning of the system has increase. The papers As strok m an Eklun ( ) As strok m an Bell ( ) an Bell an As strok m (1996) escribe how the moels have evolve. The moels have also been use for control esign see Miller Bentsman Drake Fahkfahk Jolly Pellegrinetti an Tse (1990) Pellegrinetti Bentsman an Polla (1991) an Cheng an Rees (1997). Moels base on a similar structure have been use for simulation an control of eaerators see Lu Bell an Rees (1997) an nuclear reactors see Yeung an Chan (1990) HoK l (1990) Irving Miossec an Tassart (1980) Parry et al. (1995) Menon an Parlos (1992) Thomas Harrison an Hollywell (1985) Schneier an Boy (1985) an Kothare Mettler Morari Benotti an Falinower (1999). 2. Global mass an energy balances A schematic picture of a boiler system is shown in Fig. 1. The heat Q supplie to the risers causes boiling. Gravity forces the saturate steam to rise causing a Fig. 1. Schematic picture of the boiler. circulation in the riser-rum-owncomer loop. Feewater q is supplie to the rum an saturate steam q is taken from the rum to the superheaters an the turbine. The presence of steam below the liqui level in the rum causes the shrink-an-swell phenomenon which makes level control i$cult. In reality the system is much more complicate than shown in the "gure. The system has a complicate geometry an there are many owncomer an riser tubes. The out#ow from the risers passes through a separator to separate the steam from the water. In spite of the complexity of the system it turns out that its gross behavior is well capture by global mass an energy balances. A key property of boilers is that there is a very e$cient heat transfer ue to boiling an conensation. All parts of the system which are in contact with the saturate liqui}vapor mixture will be in thermal equilibrium. Energy store in steam an water is release or absorbe very rapily when the pressure changes. This mechanism is the key for unerstaning boiler ynamics. The rapi release of energy ensures that i!erent parts of the boiler change their temperature in the same way. For this reason the ynamics can be capture by moels of low orer. Drum pressure an power ynamics can in fact be represente very well with "rst-orer ynamics as shown in As strok m an Eklun (1972). At "rst it is surprising that the istribute e!ects can be neglecte for a system with so large physical imensions. Typical values of store energy for two i!erent boilers are given in Table 1. The P16-G16 plant is a 160 MW unit in Sween an the Eraring plant is a 660 MW unit in Australia. The ratio of the energy store in the metal to that store in the water is approximately 1 for P16-G16 an 4 for the Eraring unit. The numbers in Table 1 also give a measure of the time it takes to eplete the store energy at the generate rate. Although the total normalize store energy is approximately the same for both plants the fraction of the energy store in water is much smaller for the larger plant. This results in larger variations in water level for the larger plant uner proportionally

3 K.J. Asstro( m R.D. Bell / Automatica 36 (2000) 363} Table 1 Energy store in metal water an steam for two boilers operating at rate pressure an temperature but at i!erent power generation conitions. The values are normalize with the power at the operating conitions. The unit is J/W"s the entries can thus be interprete as time constants for the i!erent storage mechanisms Boiler Metal Water Steam Total P16-G16 80 MW P16-G MW Eraring 330 MW Eraring 660 MW similar operating conition changes. This implies that the level control problem is more i$cult for large boilers Balance equations Much of the behavior of the system is capture by global mass an energy balances. Let the inputs to the system be the heat #ow rate to the risers Q the feewater mass #ow rate q an the steam mass #ow rate q. Furthermore let the outputs of the system be rum pressure p an rum water level l. This way of characterizing the system is convenient for moeling. For simulation an control it is necessary to account for the fact that mass #ow rate q epens on the pressure by moeling the turbine an the superheaters. To write the equations let < enote volume ϱ enotes speci"c ensity u speci"c internal energy h speci"c enthalpy t temperature an q mass #ow rate. Furthermore let subscripts s w f an m refer to steam water feewater an metal respectively. Sometimes for clari"cation we nee a notation for the system components. For this purpose we will use ouble subscripts where t enotes total system rum an r risers. The total mass of the metal tubes an the rum is m an the speci"c heat of the metal is C. The global mass balance is t [ϱ < #ϱ < ]"q!q (1) an the global energy balance is t [ϱ u < #ϱ u < #m C t ]"Q#q h!q h. (2) Since the internal energy is u"h!p/ϱ the global energy balance can be written as t [ϱ h < #ϱ h <!p< #m C t ] "Q#q h!q h (3) where < an < represent the total steam an water volumes respectively. The total volume of the rum owncomer an risers < is < "< #<. (4) The metal temperature t can be expresse as a function of pressure by assuming that changes in t are strongly correlate to changes in the saturation temperature of steam t an thus also to changes in p. Simulations with moels having a etaile representation of the temperature istribution in the metal show that the steay-state metal temperature is close to the saturation temperature an that the temperature i!erences also are small ynamically. The right-han sie of Eq. (3) represents the energy #ow to the system from fuel an feewater an the energy #ow from the system via the steam A secon-orer moel Eqs. (1) (3) an (4) combine with saturate steam tables yiels a simple boiler moel. Mathematically the moel is a i!erential algebraic system. Such systems can be entere irectly in moeling languages such as Omola an Moelica an it can be simulate irectly using Omsim see Mattsson Anersson an As strok m (1993) or Dymola. In this way we avoi making manual operations which are time consuming an error prone. We will however make manipulations of the moel to obtain a state moel. This gives insight into the key physical mechanisms that a!ect the ynamic behavior of the system. There are many possible choices of state variables. Since all parts are in thermal equilibrium it is natural to choose rum pressure p as one state variable. This variable is also easy to measure. Using saturate steam tables the variables ϱ ϱ h an h can then be expresse as functions of steam pressure. The secon state variable can be chosen as the total volume of water in the system i.e. <. Using Eq. (4) an noting that < is constant < can then be eliminate from Eqs. (1) an (3)

4 366 K.J. Asstro( m R.D. Bell / Automatica 36 (2000) 363}378 to give the following state equations: < p e #e t t "q!q < p e #e t t "Q#q h!q h (5) where e "ϱ!ϱ ϱ e "< p #< ϱ p e "ϱ h!ϱ h e "< h ϱ p #ϱ h p # < h ϱ p #ϱ h p t! < #m C p. (6) This moel captures the gross behavior of the boiler quite well. In particular it escribes the response of rum pressure to changes in input power feewater #ow rate an steam #ow rate very well. The moel oes however have one serious e"ciency. Although it escribes the total water in the system it oes not capture the behavior of the rum level because it oes not escribe the istribution of steam an water in the system Further simplixcations Aitional simpli"cations can be mae if we are only intereste in the rum pressure. Multiplying (1) by h an subtracting the result from (3) gives h t (ϱ < )#ϱ < h t #ϱ < h t!< p t t # m C t "Q!q (h!h )!q h where h "h!h is the conensation enthalpy. If the rum level is controlle well the variations in the steam volume are small. Neglecting these variations we get the following approximate moel: p e t "Q!q (h!h )!q h (7) where ϱ e "h < p #ϱ < h p #ϱ < h p t #m C p!<. The term < in e comes from the relation between internal energy an enthalpy. This term is often neglecte in moeling see Denn (1987). The relative magnitues of the terms of e for two boilers are given in Table 2. The terms containing h /p an t /p are the ominating terms in the expression for e. This implies that the changes in energy content of the water an metal masses are the physical phenomena that ominate the ynamics of rum pressure. A goo approximation of e is h e +ϱ < p #mc t p. Table 2 gives goo insight into the physical mechanisms that govern the behavior of the system. Consier for example the situation when the pressure changes. The change in store energy for this pressure change will be proportional to the numbers in the last two columns of the table. The column for the steam (h /p) inicates that energy changes in the steam are two orers of magnitue smaller than the energy changes in water an metal. The balance of the change in energy is use in the boiling or conensation of steam. The conensation #ow rate is q " h!h q # 1 h h ϱ < h t #ϱ < h t p!< t #m C t. (8) Moel (7) captures the responses in rum pressure to changes in heat #ow rate feewater #ow rate feewater temperature an steam #ow rate very well. An attractive feature is that all parameters are given by steam tables an construction ata. The equation gives goo insight into the nonlinear characteristics of the pressure Table 2 Numerical values of the terms of the coe$cient e at normal operating pressure Boiler h < / ϱ / ϱ < / / ϱ < / / m C / / P16-G16 80 MW 360! P16-G MW 420! Eraring 330 MW 700! Eraring 660 MW 810! <

5 K.J. Asstro( m R.D. Bell / Automatica 36 (2000) 363} response since both e an the enthalpies on the right-han sie of the equation epen on the operating pressure. To obtain a complete moel for simulating the rum pressure a moel for the steam valve has to be supplie. The pressure moel given by Eq. (7) is similar to the moels in e Mello (1963) Quazza (1968) As strok m an Eklun (1972) an Ma!ezzoni (1988). Moels similar to (7) are inclue in most boiler moels. Since moel (7) is base on global mass an energy balances it cannot capture phenomena that are relate to the istribution of steam an water in the boiler. Therefore it cannot moel the rum level. 3. Distribution of steam in risers an rum To obtain a moel which can escribe the behavior of the rum level we must account for the istribution of steam an water in the system. The reistribution of steam an water in the system causes the shrink-answell e!ect which causes the nonminimum-phase behavior of level ynamics see Kwatny an Berg (1993). One manifestation is that the level will increase when the steam valve is opene because the rum pressure will rop causing a swelling of the steam bubbles below the rum level. The behavior of two phase #ow is very complicate an is typically moele by partial i!erential equations see Kutatelaze (1959) an Heusser (1996). A key contribution of this paper is that it is possible to erive relatively simple lumpe parameter moels that agree well with experimental ata Saturate mixture quality in a heate tube We will start by iscussing the ynamics of water an steam in a heate tube. Consier a vertical tube with uniform heating. Let ϱ be the ensity of the steam}water mixture. Furthermore let q be the mass #ow rate A the area of the cross section of the tube < the volume h the speci"c enthalpy an Q the heat supplie to the tube. All quantities are istribute in time t an space z. Assume for simplicity that all quantities are the same in a cross section of the tube. The spatial istribution can then be capture by one coorinate z an all variables are then functions of z an time t. The mass an energy balances for a heate section of the tube are A ϱ t #q z "0 ϱ h t #1 qh A z "Q <. Let α enote the mass fraction of steam in the #ow i.e. the quality of the mixture an let h an h enote the speci"c enthalpies of saturate steam an water. The speci"c internal energy of the mixture of steam an water is h"α h #(1!α )h "h #α h. (9) In steay state we get q z "0 qh z "qh α z "QA < an it then follows from Eq. (9) that α " QA qh < z. Let ξ be a normalize length coorinate along the risers an let α be the steam quality at the riser outlet. The steam fraction along the tube is α (ξ)"α ξ 04ξ41. (10) A slight re"nement of the moel is to assume that boiling starts at a istance x from the bottom of the risers. In this case the steam istribution will be characterize by two variables α an x instea of just α. For the experimental ata in this paper it as very little to the preiction power of the moel. For this reason we use the simpler moel although the moi"cation may be important for other boilers. There is actually a slip between water an steam in the risers. To take this into account requires much more complicate moels. The justi"cation for neglecting this is that it oes not have a major in#uence on the "t to experimental ata. The volume an mass fractions of steam are relate through α "f (α ) where f (α )" ϱ α. (11) ϱ #(ϱ!ϱ )α It has been veri"e that the simple moel which uses a linear steam-mass fraction given by Eq. (10) an a steam-volume fraction given by Eq. (11) escribes quite well what happens in a typical riser tube. This is illustrate in Fig. 2 which compares the steam istribution in a tube compute from Eqs. (10) an (11) with computations from a etaile computational #ui ynamics coe for a riser tube in a nuclear reactor. The complex coe also takes into account that there is a slip between the #ow of steam an water. It is interesting to see that the simple moel captures the steam istribution quite well Average steam volume ratio To moel rum level it is essential to escribe the total amount of steam in the risers. This is governe by the

6 368 K.J. Asstro( m R.D. Bell / Automatica 36 (2000) 363}378 the #ow rate is instea riven by the ensity graients in the risers an the owncomers. The momentum balance for the owncomer riser loop is ) ( # q t "(ϱ!ϱ )α < g!k q 2ϱ A where k is a imensionless friction coe$cient an are lengths an A is the area. This is a "rst-orer system with the time constant Fig. 2. Comparison of steay-state steam volume istribution calculate from Eqs. (10) an (11) (full lines) with results of numerical solutions of etaile partial i!erential equation moels (circles). average volume fraction in the risers. Assume that the mass fraction is linear along the riser as expresse by Eq. (10) we "n that the average volume fraction α is given by α " α (ξ)" 1 α f (ξ)ξ" ϱ ϱ!ϱ ϱ 1! ln (ϱ!ϱ )α 1#ϱ!ϱ α ϱ. (12) 3.3. A lumpe parameter moel Since we o not want to use partial i!erential equations they will be approximate using the Galerkin metho. To o this it will be assume that the steammass quality istribution is linear i.e. Eq. (10) hols also uner ynamic conitions. The transfer of mass an energy between steam an water by conensation an evaporation is a key element in the moeling. When moeling the phases separately the transfer must be accounte for explicitly. This can be avoie by writing joint balance equations for water an steam. The global mass balance for the riser section is t (ϱ α < #ϱ (1!α )< )"q!q (13) where q is the total mass #ow rate out of the risers an q is the total mass #ow rate into the risers. The global energy balance for the riser section is t (ϱ h α < #ϱ h (1!α )<!p< #m C t ) " Q#q h!(α h #h )q. (14) 3.4. Circulation yow For a force circulation boiler owncomer #ow rate q is a control variable. For a natural circulation boiler ¹" ( # )A ϱ. kq With typical numerical values we "n that the time constant is about a secon. This is short in comparison with the sampling perio of our experimental ata which is 10 s an we will therefore use the steay-state relation kq "ϱ A (ϱ!ϱ )gα <. (15) 3.5. Distribution of steam in the rum The physical phenomena in the rum are complicate. Steam enters from many riser tubes feewater enters through a complex arrangement water leaves through the owncomer tubes an steam through the steam valve. Geometry an #ow patterns are complex. The basic mechanisms are separation of water an steam an conensation. Let < an < be the volume of steam an water uner the liqui level an let the steam #ow rate through the liqui surface in the rum be q. Recall that q is the #ow rate out of the risers q the feewater #ow rate an q the owncomer #ow rate. The mass balance for the steam uner the liqui level is t (ϱ < )"α q!q!q (16) where q is the conensation #ow which is given by q " h!h q # 1 h h ϱ < h t #ϱ < h t!(< #< ) p t #m C t. (17) The #ow q is riven by the ensity i!erences of water an steam an the momentum of the #ow entering the rum. Several moels of i!erent complexity have been attempte. Goo "t to the experimental ata have been obtaine with the following empirical moel: q " ϱ (<!< )#α q #α β(q!q ). (18) ¹ Here < enotes the volume of steam in the rum in the hypothetical situation when there is no conensation of steam in the rum an ¹ is the resience time of the steam in the rum.

7 K.J. Asstro( m R.D. Bell / Automatica 36 (2000) 363} Drum level Having accounte for the istribution of the steam below the rum level we can now moel the rum level. The volume of water in the rum is < "<!<!(1!α )<. (19) The rum has a complicate geometry. The linearize behavior can be escribe by the wet surface A at the operating level. The eviation of the rum level l measure from its normal operating level is l" < #< A "l #l. (20) The term l represents level variations cause by changes of the amount of water in the rum an the term l represents variations cause by the steam in the rum. 4. The moel Combining the results of Sections 2 an 3 we can now obtain a moel that gives a goo escription of the boiler incluing the rum level. The moel is given by the i!erential equations (1) (3) (13) (14) an (16). In aition there are a number of algebraic equations. The circulation #ow rate q is given by the static momentum balance (15) the steam #ow rate through the liqui surface of the rum q by (18) an the rum level l by Eq. (20). The volumes are relate through Eqs. (4) an (19). The moel is a i!erential algebraic system see Hairer Lubich an Roche (1989). Since most available simulation software requires state equations we will also erive a state moel Selection of state variables State variables can be chosen in many i!erent ways. It is convenient to choose states as variables with goo physical interpretation that escribe storage of mass energy an momentum. The accumulation of water is represente by the total water volume <. The total energy is represente by the rum pressure p an the istribution of steam an water is capture by the steam-mass fraction in the risers α an the steam volume in the rum < Pressure an water ynamics State equations for pressure p an the total amount of water < in the systems were obtaine from the global mass an energy balances Eqs. (1) an (3). These equations can be written as (5) Riser ynamics The mass an energy balances for the risers are given by Eqs. (13) an (14). Eliminating the #ow rate out of the risers q by multiplying Eq. (13) by!(h #α h ) an aing to Eq. (14) gives t (ϱ h α < )!(h #α h ) t (ϱ α < ) # t (ϱ h (1!α )< )!(h #α h ) t (ϱ (1!α )< ) p!< t #m C t t "Q!α h q. This can be simpli"e to h (1!α ) t (ϱ α < )#ϱ (1!α )< h t!α h t (ϱ (1!α )< )#ϱ α < h t p!< t #m C t t "Q!α h q. (21) If the state variables p an α are known the riser #ow rate q can be compute from Eq. (13). This gives q "q! t (ϱ α < )! t (ϱ (1!α )< ) "q!< t ((1!α )ϱ #α ϱ ) "q!< t (ϱ!α (ϱ!ϱ )) "q!< p ((1!α )ϱ #α ϱ ) p t #< (ϱ!ϱ ) α α α t. (22) 4.4. Drum ynamics The ynamics for the steam in the rum is obtaine from the mass balance (16). Introucing expression (22) for q expression (17) for q an expression (18) for

8 370 K.J. Asstro( m R.D. Bell / Automatica 36 (2000) 363}378 q into this equation we "n < ϱ ϱ #< t t # 1 h ϱ < h t #ϱ < h t!(< #< ) p t #m C t #α (1#β)< t ((1!α )ϱ #α ϱ ) " ϱ ¹ (<!< )# h!h h q. (23) Many of the complex phenomena in the rum are capture by this equation Summary The state variables are: rum pressure p total water volume < steam quality at the riser outlet α an volume of steam uner the liqui level in the rum <. The time erivatives of these variables are given by Eqs. (5) (21) an (23). Straightforwar but teious calculations show that these equations can be written as < p e #e t t "q!q < p e #e t t "Q#q h!q h p e t #e α t "Q!α h q (24) p e t #e α t #e < t " ϱ ¹ (<!< )# h!h h q where h "h!h an e "ϱ!ϱ ϱ e "< p #< ϱ p e "ϱ h!ϱ h e "< h ϱ p #ϱ h p #< h ϱ p #ϱ h p!< #m C t p e " h ϱ p!α h ϱ p (1!α )< # (1!α )h ϱ p #ϱ h p α < α #(ϱ #(ϱ!ϱ )α ) h < p!< #m C t p (25) α e "((1!α )ϱ #α ϱ )h < α ϱ e "< p # 1 h ϱ < h p #ϱ < h p!< t!< #m C p #α (1#β)< α ϱ p #(1!α )ϱ p #(ϱ!ϱ ) α p α e "α (1#β)(ρ!ρ )< α e "ϱ. In aition steam tables are require to evaluate h h ρ ρ ρ /p ρ /p h /p h /p t an t /p at the saturation pressure p. The results in Sections 5 an 6 are base on approximations of steam tables with quaratic functions. More elaborate approximations with table look-up an interpolation have been trie but the i!erences in the ynamic responses are not signi"- cant. The steam volume fraction α is given by Eq. (12) the volume of water in rum < by Eq. (19) the rum level l by Eq. (20) an the owncomer mass #ow rate q by Eq. (15). The partial erivatives of the steam volume fraction with respect to pressure an mass fraction are obtaine by i!erentiating Eq. (12). We get α p " 1 (ρ!ρ ) ρ ρ p!ρ ρ p α " ϱ α 1#ρ ρ 1 1#η!ρ #ρ ηρ ln(1#η) (26) ϱ η 1 η ln(1#η)! 1 1#η where η"α (ϱ!ϱ )/ϱ. It is also of interest to know the total conensation #ow rate q an the #ow rate out of the risers q. These #ows are given by Eqs. (8) an (22) hence α " ϱ ϱ!ϱ 1! ϱ ln (ϱ!ϱ )α 1#ϱ!ϱ ϱ α

9 K.J. Asstro( m R.D. Bell / Automatica 36 (2000) 363} < "<!<!(1!α )< l" < #< A ¹ " ρ < q q " 2ϱ A (ϱ!ϱ )gα < k q " h!h q # 1 h h ϱ < h p #ϱ < h p t!< #m C p p t q "q!< α ϱ p #(1!α ) ϱ p #(ρ!ρ ) α p p t #(ρ!ρ )< α α α t Structure of the equations Note that Eq. (24) has an interesting lower triangular structure where the state variables can be groupe as (((< p)α ) < ) where the variables insie each parenthesis can be compute inepenently. The moel can thus be regare as a nesting of a secon- a thir- an a fourth-orer moel. The secon-orer moel escribes rum pressure an total volume of water in the system. The equations are global mass an energy balances. There is a very weak coupling between these equations which is cause by the conensation #ow. The thirorer moel captures the steam ynamics in the risers an the fourth-orer moel also escribes the ynamics of steam below the water surface in the rum. The thir equation is a combination of mass an energy balances for the riser an the fourth equation is a mass balance for steam uner the water level in the rum. Linearizing Eq. (24) shows that the system has a ouble pole at the origin an poles at!h q /e an!1/¹. One pole at the origin is associate with water ynamics an the other with pressure ynamics. The pole associate with pressure ynamics is at the origin because the steam #ow is chosen as a control variable. The pole moves into the left half-plane when the rum is connecte to the turbines. The poles!h q /e an!1/¹ are associate with ynamics of steam in the risers an the rum. The neste structure re#ects how the moel was evelope. The thir-orer moel is an improve version of the moels in As strok m an Bell (1988). In Bell an As strok m (1996) we ae rum ynamics as a fourth-orer moel. The moel in this paper is a re"ne version of that moel. Di!erent moels of higher orer have also been evelope Parameters An interesting feature of the moel is that it requires only nine parameters: rum volume < riser volume < owncomer volume < rum area A at normal operating level total metal mass m total riser mass m friction coe$cient in owncomer-riser loop k resience time ¹ of steam in rum parameter β in the empirical equation (18). A convenient way to "n the parameter k is to compute it from the circulation #ow rate. Perturbation stuies have shown that the behavior of the system is not very sensitive to the parameters. The parameters use in this paper were base on construction ata. Some of them were quite crue. Gray-box ienti"cation Bohlin (1991) was use in a comprehensive investigation in Eborn an S+rlie (1997) an S+rlie an Eborn (1999). Parameters were estimate an hypothesis testing was use to compare several moel structures. The results showe that pressure ynamics can be improve signi"cantly by increasing the metal masses. Signi"cant improvements can also be obtaine by ajusting the coe$cients in the calibration formula for the sensors. In S+rlie an Eborn (1999) it was shown that the friction coe$cient is not ienti"able from the ata. There is in fact a relation between the initial steam quality an friction. A consequence of this is that it is highly esirable for accurate moeling to measure the circulation #ow. This coul also be an important signal to use in a level control system. In Eborn an S+rlie (1997) hypothesis testing was applie to the moels As strok m an Bell ( ) (thir orer) an Bell an As strok m (1996) (fourth orer). These stuies showe conclusively that the improvements obtaine with the fourth-orer moel are signi"cant. Computations on "fth-orer moels with a more etaile representation of the rum showe that the increase complexity coul not be justi"e Equilibrium values The steay-state solution of Eq. (24) is given by q "q Q"q h!q h Q"q α h < "<! ¹ (h!h ) q ϱ h

10 372 K.J. Asstro( m R.D. Bell / Automatica 36 (2000) 363}378 where q is given by Eq. (15) i.e. q " 2ϱ A (ϱ!ϱ )gα <. k A convenient way to "n the initial values is to "rst specify steam #ow rate q an steam pressure p. The feewater #ow rate q an the input power Q are then given by the "rst two equations an the steam volume in the rum is given by the last equation. The steam quality α is obtaine by solving the nonlinear equations Q"α h 2ϱ A (ϱ!ϱ )gα < k α " ϱ ϱ!ϱ 1! ϱ ln (ϱ!ϱ )α 1#ϱ!ϱ ϱ α. (27) The steam volume in the rum can then be compute irectly. Eq. (27) e"nes the steam volume ratio α as a function of the input power Q. This function which is shown in Fig. 3 gives important insight into the shrink an swell phenomena. The curve shows that a given change in input power gives a larger variation in average steam volume ratio at low power. This explains why the shrink an swell e!ects are larger at low power than at high power Impact of moeling languages Development of physical moels is a teious iterative process. Di!erent physical assumptions are mae a moel is evelope an compare with experiments by simulation parameters may be "tte. Detaile investigation of the results gives ieas for improvements an moi"cations. It is a signi"cant e!ort to transform the equations to state space form because of the algebra involve. This is re#ecte in the manipulations resulting in Eq. (24). Many intermeiate steps have actually been omitte in the paper. The moeling e!ort can be reuce substantially by using moeling languages such as Dymola Elmqvist (1978) Omola Mattsson et al. (1993) or Moelica Elmqvist Mattsson an Otter (1998). In these languages the moel is escribe in its most basic form in terms of i!erential algebraic equations. In our case this means that the basic mass an energy balances such as (1) an (2) are entere into the system together with the algebraic equations such as (4). The software then makes algebraic manipulations symbolically to simplify the equations for e$cient simulation. 5. Step responses To illustrate the ynamic behavior of the moel we will simulate responses to step changes in the inputs. Since there are many inputs an many interesting variables we will focus on a few selecte responses. One input was change an the others were kept constant. The magnitues of the changes were about 10% of the nominal values of the signals. To compare responses at i!erent loa conitions the same amplitues were use at high an meium loa Plant parameters The parameters use were those from the Sweish power plant. The values are < "40 m < "37 m < "11 m A "20 m m " kg m " kg k"25 β"0.3 an ¹ "12 s. The steam tables were approximate by quaratic functions Fuel yow changes at meium loa Fig. 4 shows the responses of the state variables the circulation #ow rate q the riser #ow rate q an the total conensation #ow rate q to a step increase in fuel #ow rate equivalent to 10 MW. Pressure increases at approximately constant rate. The reason for this is that steam #ow out of the rum is constant. Total water volume < increases ue to the conensation that occurs ue to the increasing pressure. Steam quality at the riser outlet α "rst increases rapily an then more graually. The volume of steam in the rum "rst increases a little an it then ecreases. The rapi initial increase in steam volume is ue to the fast increase in steam from the risers. The ecrease is ue to the increase pressure which causes conensation of the steam. At the onset of the step there is a rapi increase in the outlet #ow rate from the risers. The #ow then ecreases to match the owncomer #ow rate. The #ow rates are equal after about 30 s. The conensation #ow changes in a step-like manner Steam yow changes at meium loa Fig. 3. Steay-state relation between steam volume ratio α an input power Q. The ashe curve shows the steam mass ratio α. Fig. 5 shows the responses to a step increase of 10 kg/s in steam #ow rate at meium loa. Pressure ecreases

11 K.J. Asstro( m R.D. Bell / Automatica 36 (2000) 363} Fig. 4. Responses to a step in fuel #ow rate of 10 MW at meium loa. Fig. 6. Responses to a step in fuel #ow rate of 10 MW at meium (soli) an high (ashe) loas. Fig. 5. Responses to a step in steam #ow rate of 10 kg/s at meium loa. Fig. 7. Responses to a step in steam #ow rate of 10 kg/s at meium (soli) an high (ashe) loas. practically linearly because of the increase steam #ow. Total water volume also ecreases because of increase evaporation ue to the ecreasing pressure. Steam quality at the riser outlet "rst increases rapily ue to the pressure ecrease an it then ecreases ue to the increase circulation #ow rate. The volume of the steam in the rum increases ue to the ecrease pressure. There is a very rapi increase of #ow out of the riser ue to the pressure rop. After this initial transient the riser #ow rate then ecreases to match the owncomer #ow rate. There is a steay increase in both ue to the ecrease pressure. The conensation #ow rops in an almost step-like fashion because the pressure ecreases at constant rate Drum level responses Since the behavior of the rum level is of particular interest we will show step responses that give goo insight into rum level ynamics for i!erent operating conitions. It follows from Eq. (20) that rum level is the sum of l "< /A an l "< /A which epen on the volumes of water an steam in the rum. Drum pressure steam mass an volume fractions will also be shown. Responses for meium an high loa will be given to illustrate the epenence on operating conitions. Fig. 6 shows the response to a step in fuel #ow corresponing to 10 MW. The response in rum level is complicate an epens on a combination of the ynamics

12 374 K.J. Asstro( m R.D. Bell / Automatica 36 (2000) 363}378 of water an steam in the rum. The initial part of the swell is ue to the rapi initial response of steam that was also seen in Fig. 4. The response in level is a combination of two competing mechanisms. The water volume in the rum increases ue to increase conensation cause by the increasing pressure. The volume of the steam in the rum "rst increases a little an it then ecreases because of the increasing pressure. Note that there are signi"cant changes in steam quality an steam volume ratio for the i!erent operating conitions. 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 an swell e!ect in this case too. The contributions from the volumes of steam an water have the same sign initially. The water volume will however ecrease because of the steam #ow. 6. Comparisons with plant ata Much of the moel evelopment was base on plant experiments performe with the P16-G16 unit at OG resunsverket in MalmoK Sween in collaboration with Sykraft AB. The experiments are escribe in Eklun (1971) an As strok m an Eklun ( ). They were carrie out in open loop with the normal regulators remove. The signals were "ltere an sample at a rate of 0.1 Hz. To ensure a goo excitation of the process PRBS-like perturbations were introuce in fuel #ow rate feewater #ow rate an steam #ow rate. The intention was to change one input signal in each experiment. To ensure that critical variables such as rum water level i not go outsie safe limit we mae manual correction occasionally. This means that several inputs were change in each experiment. The steam #ow rate change in response to pressure changes in all experiments because we were unable to control it tightly. A large number of signals were logge uring the experiment. This prove to be very valuable because it has been possible to use the ata for a very large number of investigations. The experiments were performe both at high an meium loa. In this paper we have use ata where three variables fuel #ow rate feewater #ow rate an steam #ow rate were change. This gives a total of six experiments which can be use to valiate the moel. There were problems with the calibration of the #ow rate measurement transucers an also some uncertainty in the e!ective energy content of the oil. The approach use was to start with the nominal calibration values an then make small corrections so that the long-term tracking between the plant ata an the moel was as close as possible. The change in all cases was well within the transucers accuracy limits. Apart from that no "ling with the coe$cients was mae. When showing the results in the following we present the primary input variable an the responses in rum pressure an rum level Experiments at meium loa The results of experiments at meium loa will be escribe "rst. This is the operating conition where the shrink an swell phenomenon is most pronounce Fuel yow rate changes Fig. 8 shows responses in rum pressure an rum level for perturbations in fuel #ow rate. There is very goo agreement between the moel an the experimental ata for rum pressure an rum water level. Note in particular that there is an overshoot in the rum level response for step changes in fuel #ow. This is cause by the interaction between the two state variables that escribe the ynamics of steam uner the liqui level in the rum Changes in feewater yow rate Fig. 9 shows the responses to changes in the feewater #ow rate. The general character of the responses agrees well. There are some eviations in the pressure responses an some of the "ner etails of the rum level are exaggerate. The pressure oes eviate in the 500}1500 s region but since the changes in pressure are small the results are consiere aequate Changes in the steam valve Fig. 10 shows the responses to changes in the steam valve. There is very goo agreement between the moel an the experimental ata. Note that there is a signi"cant shrink an swell e!ect which is capture very well by the moel Experiments at high loa Changes in fuel yow rate Fig. 11 shows responses in rum pressure an rum level for perturbations in fuel #ow rate. There is goo Fig. 8. Comparison of moel (soli line) an plant ata (ots) for perturbations in fuel #ow rate at meium loa.

13 K.J. Asstro( m R.D. Bell / Automatica 36 (2000) 363} Fig. 9. Comparison of moel (soli line) an plant ata (ots) for perturbations in feewater #ow rate at meium loa. Fig. 11. Comparison of moel (soli line) an plant ata (ots) for perturbations in fuel #ow rate at high loa. Fig. 10. Comparison of moel (soli line) an plant ata (ots) for perturbations in steam #ow rate at meium loa. agreement between the moel an the experimental ata. A comparison with Fig. 8 shows that the shrink an swell e!ect is much smaller at high loa. It is interesting to see that the moel captures this Changes in feewater yow rate Fig. 12 shows the responses to changes in feewater #ow rate. There are iscrepancies in pressure from time 400 to The total changes in pressure are small an minor variations in feewater conitions can easily cause variations of this magnitue. The moel exaggerates the level changes for rapi variations. Compare for example ata in the interval 1500}2000. Fig. 12. Comparison of moel (soli line) an plant ata (ots) for perturbations in feewater #ow rate at high loa Changes in the steam valve Fig. 13 shows the responses to changes in the steam #ow rate. There is very goo agreement between the moel an the experimental ata. Note that the moel captures the rum level variations particularly the swell an shrink e!ect very well. A comparison with Fig. 10 shows that the shrink an swell e!ect is smaller at high loas. This is well capture by the moel Comparison of behavior at high an meium loa The experiments have inicate that there are signi"- cant i!erences in behavior at high an meium loas that are well preicte by the moel. We will now look closer at these i!erences.

14 376 K.J. Asstro( m R.D. Bell / Automatica 36 (2000) 363}378 Fig. 13. Comparison of moel (soli line) an plant ata (ots) for perturbations in steam #ow rate at high loa. Fig. 15. Comparison of behavior of rum water level at meium an high loas for perturbations in fuel #ow rate. The moel response is shown in soli lines an plant ata is inicate by ots. Fig. 14. Comparison of behavior of rum water level at meium an high loas for perturbations in steam #ow rate. The moel response is shown in soli lines an plant ata is inicate by ots. Fig. 16. Comparison of behavior of rum water level at meium an high loas for perturbations in fuel #ow rate. The moel response is shown in soli lines an plant ata is inicate by ots. Fig. 14 compares the responses in rum level to steam #ow changes at high an meium loas. The steam valve changes were almost the same in both experiments (as is shown in Figs. 10 an 13) but they were not ientical. Because of the integrators in the moel there are natural i!erences in the levels of the signals. Apart from this level shift the moel matches the experiments very well. Fig. 14 also shows that the shrink an swell e!ect is larger at meium loa. It is even larger at low loas. Fig. 15 compares responses to changes in fuel #ow. Note the goo agreement for ynamic responses between the moel an experiments. In this case there is a pronounce i!erence between the behaviors for meium an high loa. Fig. 16 compares responses to changes in fuel #ow. We have taken a section of the ata where there are substantial rapi variations. Again we note the excellent agreement between moel an experiments an we also note the signi"cant i!erence between the behaviors at meium an high loas. 7. Conclusions A nonlinear physical moel with a complexity that is suitable for moel-base control has been presente. The moel is base on physical parameters for the plant an can be easily scale to represent any rum power station. The moel has four states; two account for storage of total energy an total mass one characterizes steam istribution in the risers an another the steam istribu-

15 K.J. Asstro( m R.D. Bell / Automatica 36 (2000) 363} tion in the rum. The moel can be characterize by steam tables an a few physical parameters. The moel is nonlinear an agrees well with experimental ata. In particular the complex shrink an swell phenomena associate with the rum water level are well capture by the moel. The moel has a triangular structure that can be escribe as (((< p)α ) < ) where the states in each bracket can be etermine sequentially. The linearize moel has two poles at the origin an two real stable poles. The moel has been valiate against plant ata with very rich excitation that covers a wie operating range. These experiments have given much insight into the behavior of the system an they have guie the moeling e!ort. We believe that the approach use in this work can be applie to other con"gurations of steam generators. Moel (24) can be simpli"e by keeping only the ominant terms in expressions (25) for the coe$cients e. This coul be useful for applications to moel-base control. Preliminary investigations inicate that several terms can be neglecte without sacri"cing the "t to experimental ata. A comprehensive stuy of this is outsie the scope of this paper. The moel can also be re"ne in several ways. This will however require new measurements with faster sampling rates. Acknowlegements The research has been supporte by the Sykraft Research Founation an the Sweish National Boar for Inustrial an Technical Development uner contract This support is gratefully acknowlege. We woul also like to express our sincere gratitue to Sykraft AB for their willingness to perform experiments on plants to increase our unerstaning of their behavior. Useful comments on several versions of the manuscript have been given by our colleagues J. Eborn H. Tummescheit an A. Glattfeler. We woul also like to express our gratitue to the reviewers for constructive criticism. References Ambos P. Duc G. & Falinower C.-M. (1996). 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Unbehauen H. & Kocaarslan I. (1990). Experimental moelling an aaptive power control of a 750 MV once-through boiler. In Preprints IFAC 11th worl congress on automatic control vol. 11 Tallinn Estonia (pp. 32}37). Yeung M. R. & Chan P. L. (1990). Development an valiation of a steam generator simulation moel. Nuclear Technology }314. Karl J. As strok m is Professor an Hea of the Department of Automatic Control at Lun University since He has broa interests in automatic control incluing stochastic control system ienti"cation aaptive control computer control an computer-aie control engineering. He has supervise 44 Ph.D. stuents written six books an more than 100 papers in archival journals. He is a member of the Royal Sweish Acaemy of Engineering Sciences (IVA) an the Royal Sweish Acaemy of Sciences (KVA) an a foreign member of the US National Acaemy of Engineering an the Russian Acaemy of Sciences. As strok m has receive many honors incluing three honorary octorates the Callener Silver Meal the Quazza Meal from IFAC the Rufus Olenburger Meal from ASME the IEEE Control Systems Science Awar an the IEEE Meal of Honor. Ro Bell is an Honorary Senior Research Fellow a position he has hel since February His previous position was Hea of the Computing Department which he hel since the beginning of He obtaine his Ph.D. (Application of Optimal Control Theory to Inustrial Processes) in 1972 from the University of New South Wales. He joine the acaemic sta! at Macquarie University in He became intereste in computers (harware an software) in 1958 when he worke on one of the "rst igital computers in Australia UTECOM at the University of New South Wales. His current research interests lie mainly in the Application of Control Theory to areas such as Inustrial Processes (Power Stations in particular) National Economies Management an Robotics. He is a member of IEEE. Among his personal interests are horse riing growing waratahs bush walking an restoring Jaguar cars.