Tuning of a dynamic boiler model using a nonlinear multivariable optimisation method

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1 Preprnts of the 9th World Congress The Internatonal Federaton of Automatc Control Tunng of a dynamc boler model usng a nonlnear multvarable optmsaton method Tmo Yl-Foss Tampere Unversty of Technology, Department of Automaton Scence and Engneerng, P.O. Box 69, FI-330, Tampere, Fnland, (emal: tmo.yl-foss@tut.f) Abstract: Wde and complex process models set challenges to the modellng work. Especally the determnaton and tunng of the parameters of complex models are often laborous and tme-consumng. The strong cross-nterconnecton of modelled varables also makes tunng work more dffcult. Effcent tunng tools can be used to accelerate tunng work. In ths paper t s proposed how a nonlnear multvarable optmsaton method can be adapted for the tunng of the parameters of a dynamc crculatng fludzed bed (CFB) drum boler model, whch based on the frst prncple laws of mass and energy. In the fne-tunng example, the tuneable varables are the gas sde heat transfer coeffcents and the water sde local resstance coeffcents of the heat exchanger model blocks nto the flue gas duct. Keywords: Dynamc models, Modellng, Optmsaton, Crculatng Fludzed Bed Boler. INTRODUCTION Software envronments for modellng nclude usually ready model block lbrares or the opportunty for the dong of them. Model block lbrares can contan ndvdual process components models for example from tanks, valves, pumps and heat exchangers. The buldng of wde models accelerates by usng model block lbrares. It s easy to connect the ready submodel blocks nto wder process models. However, the values of the fxed parameters of model blocks have to be determned separately to correspond to the process to be modelled. On wde and complex process models the number of the fxed parameters can be very large n whch case the defnton work and tunng work may become laborous and may delay modellng work sgnfcantly. Interconnectons between modelled varables are also complcated by the tunng work (Halmevaara et al., 007, Yl-Foss, 03). The most fxed parameters of boler models can usually be determned straghtforward such as dmensons and natural constants. In ths context, the term fxed parameter determnes a constant s passed to a model before a smulaton run. The typcal fxed parameters of a process model are nformaton whch s related to the dmensonng of processes and can be for example lengths and dameters of ppes and heat transfer areas of heat exchanger. Usually the propertes of the structures are supposed to be constant. The determnaton of the ntal values of the state varables may be also laborous. The sutable values can be calculated or the model can be run to a desred state. The opportunty for the savng and loadng of state values facltates smulaton work. To mprove the modellng accuracy, some of the parameters may have to be trmmed based on the measured process data or other nformaton. To facltate model parameter tunng work dfferent tunng methods have been appled to dynamc power plant models. A method, called teratve Tunng (IRT), has been used to tune heat transfer coeffcents of a reheater model of a nuclear power plant (Halmevaara et al., 007). A nonlnear optmsaton algorthm that mnmzes the quadratc cost functon s also used n an other nuclear power plant model by Fazekas and Varga, 008. A crculatng fludzed bed boler model has been analysed and tuned usng a partcle flterng method by Ikonen et al, 0. The algorthm of the ntegral controller has also been appled nto the fne-tunng of the gas sde heat transfer coeffcents of a boler model (Yl- Foss et al., 0). The most sgnfcant challenges of the tunng of dynamc power plant models are the strong nterconnectons between process varables, nonlnearty and tme varant characterstcs. Power plant models usually descrbe water and flue gas flows and heat transfer between water and flue gas. The temperature dfferences and flow rates affect the transton of the heat. The varables depend strongly on each other. The nonlneartes are caused by the behavour of heat transfer coeffcents. The water sde heat transfer coeffcent depends on the propertes of water and the flow rate strongly. In the same way, the propertes of flue gas and flow rate affect also heat transfer from flue gas to heat transfer surfaces. Furthermore, the tme varant behavour can be caused by the varaton of the propertes of the fuel. (Yl-Foss, 03) In ths work t s presented how the model can be tuned wth an optmsaton method whch consders to the crossnteractons and nonlnearty of the model.. DYNAMIC PROCESS MODEL A dynamc process model of a crculatng fludzed (CFB) drum boler has been developed. The model represents the real boler process from the preheaters to the turbne. The Copyrght 04 IFAC 46

2 presented process model s based on the frst prncple laws of mass, energy, and momentum balances and expermental correlatons about reacton knetcs and heat transfer. The model was bult usng Smulnk and Matlab by The MathWorks. The man goal s that the dynamc model can be used for several purposes such as control desgn and process development. Fg.. A smplfed dagram of a natural crculaton drum boler process. The flue gas duct ncludes several heat exchanger elements such as economsers, superheaters, and ar preheaters. In the modelled process, the water from the feed water tank to the drum s preheated by the economsers (ECO, ECO and ECO3). The prmary (PSH and PSH) and secondary (SSH and SSH) superheaters ncrease steam temperature before the turbne. Superheated steam temperature s controlled usng spray water. Ar preheaters preheated combuston ar to the furnace. The wall tubes of the flue gas duct vaporzed water to the drum as well as the furnace walls. Thus, heat s transferred from flue gas to heat exchangers and the walls of the flue gas duct. The flue gas temperature drops along the duct. Heat flows from the flue gas to water, steam and ar are nteractng wth each other. The presented power plant model conssts of several blocks whch have been dvded nto blocks, water, gas and ar. The water sde of the process model s modelled usng several dentcal water sde model blocks n whch the values of parameters can be dfferent. The used water sde block able to handle the thermodynamcs of water, saturated water-steam mxture, and superheated steam under sub- and supercrtcal pressures. The developed water sde model block ncludes also the walls or the tubes of heat exchangers. The block can be used represent the water sdes of dfferent process components such the parts of an evaporator, superheaters and economsers (Yl-Foss et al., 0). The flue gas and ar sdes were also modelled usng model blocks. The separate model blocks have been developed for a furnace, cyclones, a loop seal, and a flue gas duct and ar preheaters. (Yl-Foss et al., 0). The developed model represents a natural crculaton drum boler. Fgure presents a smplfed dagram of a drum boler boler process. Drum bolers typcally consst of feed water and combuston ar preheaters, evaporator, drum, super- and reheaters. Boler water s partly vaporzed n vertcally mounted evaporator tubes of the furnace walls. After the evaporator, saturated water-steam mxture s separated nto saturated water and steam n the drum. The generated steam s replaced wth feed water. Ths boler water s crculated back to the nlet of the evaporator. Natural crculaton s caused by the densty dfference between the water flled down comer ppes and water-steam mxture flled rser tubes. After the drum superheaters convert saturated steam nto superheated steam. (Splethoff, 00) The process model ncludes also the dynamcs of the combuston and the fludzed bed materal. The dfferent types of fuels can be modelled usng the combuston model. In the furnace, the propagaton of combuston process and heat transfer depends on the amount of crculatng mass and suspenson densty. Heat flow from the flue gases and sold matter to the furnace walls s transferred by convectons and radaton. Sold matter and gases are separated from each other n cyclones. Sold matter flows back nto the furnace through the loop seal. In the loop seal heat transfer also takes place to the walls and other heat exchanger elements. From the cyclone flue gases flow to the flue gas duct secton of the boler. (Majanne and Köykkä, 009) Fg.. A smplfed example dagram of sub model blocks and some sgnals of two heat exchanger models. Fgure llustrates a smplfed example dagram of two heat exchangers of water or steam nto a flue gas duct. Each heat exchanger, such economaser or superheater, has been modelled on two model blocks: gas and water model blocks. These blocks have been connected to each other usng sgnals. The heat exchanger has been connected to the heat exchanger. Only some of the varables that contan the sgnal vectors are marked n Fgure. For example, Q g, 47

3 means the heat power [J/s] from the gas sde to the walls of the heat exchanger. The heat power depends several varables such flue gas temperature, wall temperature and gas sde heat transfer coeffcent, whch means heat transfer coeffcent between flue gas or suspenson matter and the wall of a heat exchanger. The heat power sgnal n turn has effect back to the surface temperature T s, [K] and the gas temperature T g, [K]. The water enthalpy h w, after the heat exchanger s also dependent on the heat power Q g,. The consecutve model blocks are to the seres connected. As ts consequence, that varatons of pressure p w,, enthalpy h w, and mass flow are conveyed forward n water model blocks. However, some of the physcal parameters may have to be tuned. For example t s dffcult to determne gas sde heat transfer coeffcents and coeffcents of local resstance to water and steam flows exactly wth physcal equatons. Ths reason, these coeffcents were chosen the tuneable parameters of the presented model to be tuned n ths work. Furthermore, the sgnfcance of the coeffcents affected a choce. Gas sde heat transfer coeffcents are small and flue gas dynamc s complcated (Hubka, 0). Because of ths the coeffcents tuned n the model. The coeffcent affects the heat transfer heat transfer Q g, from the flue gas to the surface of the tubes of the heat exchanger n the flue gas duct. The varable s the ndex of the heat exchanger. The heat transfer heat transfer can be calculated as (Incropera and DeWtt, 00) T - T - T - T Q& ( g,n, s, ) ( out, s, ) = A g, () g, a g, ht, æ T - ö g,n, Ts, lnç è Tg, out, - Ts, ø where α g, [W/(Km )] s the gas sde heat transfer coeffcent and A ht, [m ] s the heat transfer area. T s, [K] s the temperature of heat transfer surface as the tube walls of the heat exchanger. The T g,n, [K] s the temperature of the nlet flow of the flue gas to the modelled secton. T g,out, [K] s the temperature of the outlet flow. The gas sde heat transfer coeffcent α g, s can be determned as a g, = atunng, ( a g,conv, + a g, rad, ) () where α g,conv, [W/(Km )] s convectve heat transfer coeffcent and α g,rad, [W/(Km )] radant heat transfer coeffcent. In the model both coeffcents are calculated on equatons whch are found n the lterature. A more exact descrpton of the calculaton s found of an earler publcaton (Yl-Foss et al., 0). a tunng, [-] s the addtonal fne-tunng coeffcent of the gas sde heat transfer coeffcent. The default value of the coeffcent s. The coeffcents a tunng, of each heat exchanger models were specfed as parameters to be tuned. The coeffcents of the local resstance of the ppe system pressure drop are also dffcultly determned f the system s complex. The total pressure drop Δp w, [Pa] for forced convecton water, steam or saturated steam-water mxture flow can be determned as (Poro et al., 004) Dp = Dp + Dp + Dp + Dp (3) w, w,ff, w,l, w,ac, w,g, where Δp w,ff, [Pa] s the pressure drop due to frctonal resstance, Δp w,l, [Pa] s the pressure drop due to local flow obstructon, Δp w,ac, [Pa] s the pressure drop due to acceleraton of flow and Δp w,g, [Pa] s the pressure drop due to gravty. The pressure drop Δp w,l, due to local sngle-phase flow obstructon of water and steam can be calculated as (Poro et al., 004) w, w, G D pw,l, = x (4) w,l, r where ξ w,l, [-] s the local resstance coeffcent G w, s mass flux [kg/(m s)] and ρ w, [kg/m 3 ] s the densty of water or steam. The pressure drop Δp w,l, due to local two-phase flow obstructon of saturated steam-water mxture can be defned as (Poro et al., 004) { r ( - e ) + r } w,l, w,r,average, w, G D pl, = x (5) l, e w,s, w,r,average, where ρ w,l,, [kg/m 3 ] densty of (lqud) water and ρ w,s,, [kg/m 3 ] densty of steam. ε w,r,average, [-] s the average vod fracton. The local resstance coeffcent ξ w,l, can be estmated from approprate correlaton for dfferent flow obstructons. The local resstance coeffcents affect local pressure drops, whch are caused by a change n flow geometry and flow drecton. Examples branches, valves and bends drop pressure locally. Reference values and equatons for local resstance coeffcents are found n the lterature. However, the ablty to estmate to local pressure drops Δp w,l, of the complex ppe system s very dffcult than for other pressure drops Δp w,ff,, Δp w,ac, and Δp w,g,. Therefore, the coeffcents ξ l, of heat exchangers are consdered as parameters whch are expermentally tuned n ths work. The coeffcents are estmated before the tunng. The coeffcents are possble to tune usng the equaton x = (6) l, btunng,xl, where b tunng, [-] s the addtonal fne-tunng coeffcent of the local resstance coeffcent. The default value of the coeffcent s. The values of coeffcents affect the local pressure losses Δp w,l, and total pressure drops Δp w, of each heat exchangers. 3. NONLINEAR MULTIVARIABLE OPTIMASATION The model tunng problem can be usually formulated the optmsaton problem. Because the boler models are usually nonlnear and the models contan multvarable nteractons, nonlnear multvarable optmsaton s an obvous alternatve. In ths work, the optmsaton problem s solved usng the Matlab functon fmnsearch, whch fnd the mnmum of the cost functon. The functon s based on the multdmensonal 48

4 unconstraned nonlnear Nelder-Mead smplex method. Ths effcent dervatve-free drect search method optmzes the cost functon merely by comparng functon values. The basc algorthm of the method s qute smple to understand and easy to program and use. It s also popular n many felds of scence and technology. (Lagaras et al., 998, Chang, 0) Manly for these reasons the Nelder-Mead smplex method was chosen for ths work. Also some other optmsaton methods were tested wth the model. 4. MODEL FINE-TUNING In the fne-tunng example, the gas sde heat transfer coeffcent α g, and the local resstance coeffcent ξ l, of the heat exchangers ECO, ECO, ECO3, PSH, PSH, SSH and SSH of the presented drum boler model are tuned usng transfer fne-tunng coeffcents a tunng, and b tunng,. The heat exchanger model blocks are ndexed to the same order as above: the ndex of ECO s, the ndex ECO s, etc. The last heat exchanger SSH s the ndex s 7. The tuneable parameters are the manpulated nput varables of the optmsaton problem. Temperatures T g,out, or enthalpes h w, or heat powers Q g, and pressures p w, and ther combnatons can be chosen as the controlled varables. The choce s affected by the fact whether the values of the controlled varables to be reached for are known as n the ones calculated or to be measured from the process to be modelled. The selected target values of the controlled varables are ndvdual averages from a certan perod of the tme n the steady state stuaton. In ths tunng case, the values of the man part of temperatures T g,out, and T w,out, have been measured n a certan steady state stuaton of the boler process. T w,out, [K] means the water or steam outlet temperature from the heat exchanger. T w,out, can be calculated from pressures p w,out, and enthalpes h w,out, n the model. The values of the pressures p w, are less measured. Target values or setponts are not been measured drect are based on the calculated values usng a desgn calculaton. The calculated values can also be determned ndrectly usng measured values. Thus, the target values of the optmsaton case are measured values and calculated desgn values. When generally speakng choosng the target values, one must ensure to the relablty of the reference values. The cost functon J to be mnmsed s defned as J = e + e (7) T Dp 7 = ( pˆ - p ) e = k (9) p p, where k p, s the vector of the weght coeffcents and p w, s the vector of the target values of the total water or steam pressure drop of the heat exchanger model blocks. The process model s tuned to correspond to a certan steady state stuaton. The model and the optmsaton functon nteract wth each other durng the tunng smulaton, so the mnmsaton of the cost functon s more dffcult n the strong transent stuaton than the steady state stuaton. The subject of the presented tunng work s not the dynamc behavour of the model. If necessary, the model can be tested n dfferent operaton ponts f the comparson data s avalable. The dynamc valdaton must be made separately. To suppose also ths s data avalable from transent stuatons. The optmsaton functon s performed n parallel wth the process model durng the tunng procedure. The optmsaton functon changes the manpulated varables, a tunng, and ξ l,, wth the ntervals of 80 smulated seconds. The changes are made at long ntervals because the model s dynamc. The optmsaton functon must perceve the drectons of the controlled varables caused by the changes of the manpulated varables. Before the tunng smulaton, the process model s smulated aganst to the steady state stuaton lke values that have been measured from the process. Some controlles and varables set manual and constant durng the tunng smulaton run. Otherwse for example the steam temperature controllers dsturb optmsaton. The value of each weght coeffcents k p, s and the value of each weght coeffcents k p, s 0.3. Thus, the values of the temperature were set as the most mportant tunng targets. The unts that have been used n the cost functon are [ºC] and [bar]. Fgure 3 shows the value of the cost functon durng the tunng smulaton. The tunng smulaton begns at tme 0. The value of the functon becomes smaller consderably durng the smulaton w, w, where e T s a squared error of the water or steam outlet temperatures of the heat exchanger model blocks. The error e T can be defned as 7 = ( Tˆ - T ) e = k (8) T T, w,out, w,out, Cost functon where k T, s the vector of the weght coeffcents and T w,out, s the vector of the target values of the water or steam outlet temperatures. e p s a squared error of the outlet pressure of the heat exchanger model blocks. The error e p s wrtten as Smulated tme [s] Fg. 3. The cost functon durng the tunng smulaton. 49

5 Manpulated varables ECO ECO 4 PSH 5 PSH 6 SSH 7 SSH Relatve model error of outlet pressure [%] ECO ECO 4 PSH 5 PSH 6 SSH 7 SSH Smulated tme [s] Fg. 4. The manpulated varables (the gas sde heat transfer fne tunng coeffcents) durng the tunng smulaton. Manpulated varables ECO ECO 4 PSH 5 PSH 6 SSH 7 SSH Smulated tme [s] Fg. 5. The manpulated varables (the local resstance fne tunng coeffcents) durng the tunng smulaton. Relatve model error of outlet temperature [%] ECO ECO 4 PSH 5 PSH 6 SSH 7 SSH Smulated tme [s] Fg. 6. Relatve model errors of the water or steam outlet temperature of the heat exchanger model blocks durng the tunng smulaton The ndex of the heat exhanger Fg. 7. Relatve model errors of the water or steam outlet pressure of the heat exchanger model blocks durng the tunng smulaton. Fgure 4 presents how the gas sde heat transfer fne tunng coeffcents changed durng the tunng smulaton. At the begnnng of the smulaton, the optmsaton functon wll test all the manpulated varables one by one wth small changes. After the testng more varables are changed the same tme. Correspondngly Fgure 5 llustrates the changes n the local resstance fne tunng coeffcents. Fgure 6 shows relatve model errors of the model blocks of water or steam outer temperature of each heat exchanger durng the tunng smulaton. The error becomes smaller n almost all the model blocks. However, the errors of the heat exchanger model blocks ECO and ECO3 were not reduced Fgure 7 presents relatve model errors of the water or steam outer pressure durng the tunng smulaton. The pressure changes were mnor durng the smulaton. Because of ths, the pressure coeffcents were tuned once more separately based on Equaton 9. The value of each weght coeffcents k p, s. The gas sde heat transfer fne tunng coeffcents reman constant durng the second smulaton. Manpulated varables ECO ECO 4 PSH 5 PSH 6 SSH 7 SSH Smulated tme [s] Fg. 8. The manpulated varables (the local resstance fne tunng coeffcents) durng the tunng smulaton. 430

6 Relatve model error of outlet pressure [%] ECO ECO 4 PSH 5 PSH 6 SSH 7 SSH The ndex of the heat exhanger Fg. 9. Relatve model errors of the water or steam outlet pressure of the heat exchanger model blocks durng the tunng smulaton. Fgure 8 presents the local resstance fne tunng coeffcents durng the second tunng smulaton. The varaton s stronger than n Fgure 5. Fgure 9 shows that some of the blocks of the relatve model error s reduced by more than n Fgure 7. The result can be possbly be mproved by changng the cost functon, the weghtng coeffcents and the optmsaton tme step nterval. The results are satsfactory, especally for the outlet temperatures. In ths work, t was ntended to mnmse a certan process wholeness and not ndvdual varables. 5. CONCLUSIONS A dynamc process model of a crculatng fludzed (CFB) drum boler was presented. The gas sde heat transfer coeffcents and the local resstance coeffcents of water sde and of the heat exchanger model blocks of the flue gas duct are tuned usng nonlnear multvarable optmsaton method. The presented model contans multvarable nteractons and nonlnearty. In the fne-tunng example demonstrated that the chosen method was reduced model error n ths case. The method can be adapted to dfferent models. However, the use of the presented fne tunng method requres that the tunable parameters of the target s possble to change durng the tunng smulaton run. Halmevaara, K. and Hyötynem, H. (007). Dynaamsten smulontmallen parametren vrttämnen datapohjaslla tlastollslla menetelmllä, Proceedngs of Automaato 07, Fnnsh Socety of Automaton, Helsnk. Hubka, L. (0). Temperature Dynamc of Once-through Boler Based on Flue Gases Heat Transports, Preprnts of the 8th IFAC World Congress, , Mlano, Italy. Ikonen E., Kovacs, J., Aaltonen, H., Rtvanen, J., Selek, I. and Kettunen A. (0). Analyss and tunng of a CFB model usng partcle flterng, Proceedngs of the 8th IFAC Power Plant and Power System Control Symposum, Toulouse, France, -5 September 0. Incropera, F.P. and DeWtt, D.P. (00). Fundamentals of Heat and Mass Transfer. 5.p. USA, John Wley & Sons. Lagaras, J.C., Reeds, J. A., Wrght, M. H. and Wrght, P. E. (998). Convergence Propertes of the Nelder-Mead Smplex Method n Low Dmensons, SIAM Journal of Optmzaton, Vol. 9 Number, -47. Majanne, Y. and Köykkä, P. (009). Dynamc model of the crculatng fludzed bed boler, Preprnts of IFAC Symposum on Power Plants, Tampere, Fnland. Poro, I.L., Duffey, R.B. and Dumouchel, T.J. (004). Hydraulc resstance of fluds flowng n channels at supercrtcal pressures (survey), Nuclear Engneer and Desgn, Vol. 3, Splethoff, H. (00). Power Generaton from Sold Fuels,Sprnger-Verlag, Berln Hedelberg. Ylätalo, J. (009), Calculaton models for two-phase flow n steam boler tubes, Bachelor of Scence Thess, Lappeenranta Unversty of Technology, Fnland. Yl-Foss, T., Köykkä, P. and Majanne, Y. (0). A Generalzed Dynamc Water Sde Model for a Once- Through Benson Boler, Preprnts of the 8th IFAC World Congress, , Mlano, Italy. Yl-Foss, T., Köykkä P. and Majanne, Y. (0). A Tunng Tool for Gas Sde Heat Transfer Coeffcents of a Boler Model, Proceedngs of the 8th IFAC Power Plant and Power System Control Symposum, Toulouse, France, -5 September Yl-Foss, T. (03). Dynaamsten vomalatosmallen parametren määrttämnen ja vrttämnen, Proceedngs of AutomaatoXX Semnar, Fnnsh Socety of Automaton, May 03, Helsnk, Fnland. REFERENCES Chang, K-H. (0). Stochastc Nelder Mead smplex method A new globally convergent drect search method for smulaton optmzaton, European Journal of Operatonal Research, Volume 0, Issue 3, August 0, Fazekas, C. and Varga, I. (008). Modelng and Parameter Estmaton of a Nuclear Power Plant, 9th Internatonal PhD Workshop on Systems and Control: Young Generaton Vewpont, October 008, Izola, Slovena. 43