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1 Aalborg Univeritet Model-baed Control of a Bottom Fired Marine Boiler Solberg, Brian Willum; Kartenen, Clau M. S.; Anderen, Palle; Pederen, Tom Søndergård; Hvitendahl, Poul U. Publication date: 2005 Document Verion Publiher' PDF, alo known a Verion of record Link to publication from Aalborg Univerity Citation for publihed verion (APA): Solberg, B., Kartenen, C. M. S., Anderen, P., Pederen, T. S., & Hvitendahl, P. U. (2005). Model-baed Control of a Bottom Fired Marine Boiler.. General right Copyright and moral right for the publication made acceible in the public portal are retained by the author and/or other copyright owner and it i a condition of acceing publication that uer recognie and abide by the legal requirement aociated with thee right.? Uer may download and print one copy of any publication from the public portal for the purpoe of private tudy or reearch.? You may not further ditribute the material or ue it for any profit-making activity or commercial gain? You may freely ditribute the URL identifying the publication in the public portal? Take down policy If you believe that thi document breache copyright pleae contact u at vbn@aub.aau.dk providing detail, and we will remove acce to the work immediately and invetigate your claim. Downloaded from vbn.aau.dk on: April 25, 2017
2 MODEL-BASED CONTROL OF A BOTTOM FIRED MARINE BOILER Brian Solberg Clau M. S. Kartenen Palle Anderen Tom S. Pederen Poul U. Hvitendahl Aalborg Indutrie A/S, Gaværkvej 24, 9100 Aalborg, Denmark Dept. of Control Engineering, Aalborg Univerity, Aalborg, Frederik Bajervej 7C, 9220 Aalborg Øt, Denmark F.L.Smih A/S, Vigerlev All 77, 2500 Valby, Copenhagen, Denmark Abtract: Thi paper focue on applying model baed MIMO control to minimize variation in water level for a pecific boiler type. A firt principle model i put up. The model i linearized and an LQG controller i deigned. Furthermore the benefit of uing a team flow meaurement i compared to a trategy relying on etimate of the diturbance. Preliminary tet at the boiler ytem how that the deigned controller i able to control the boiler proce. Furthermore it can be concluded that relying on etimate of the team flow in the control trategy doe not decreae the controller performance remarkable. Keyword: Boiler, Dynamic modelling, LQG control, Etimator, Tet 1. INTRODUCTION The control of marine boiler mainly focue on minimizing the variation of team preure and water level in the boiler, keeping both variable around ome given et point. Up till now thi tak ha been achieved uing claical SISO controller. One uing the fuel flow to control the team preure and one uing the feed water flow to control the water level. A more efficient control can allow maller water and team volume in the boiler implying lower production cot and a more attractive product. The pecific boiler concerned in the preent work i a Miion TM OB boiler from AI product range. The boiler i a bottom fired one pa moke tube boiler hown in a blueprint in figure 1. The boiler conit of a furnace urrounded by water. In the top left ide of the boiler the team i led out and in the top right ide feed water i injected. Thi boiler differ from other boiler deign in two way: it i bottom fired and the flue ga pae traight through. Fig. 1. Blueprint of the Miion TM OB marine boiler which i produced by Aalborg Indutrie A/S. The challenge in thi work i to minimize the variation of water level to allow maller boiler geometry without compromiing preure performance. The control problem i complicated by the hrink-andwell phenomenon which introduce non-minimum
3 phae characteritic in the ytem. Thi phenomenon i een when the team flow or the feed water flow i abruptly increaed or decreaed. The purpoe of thi paper i to verify if MIMO control i uitable for bottom fired one pa moke tube boiler. Furthermore the benefit of uing the expenive and uncertain team flow meaurement compared to an etimate of thi diturbance in the control trategy i invetigated. The team flow influence the hrink-and-well phenomenon which make knowledge of thi quantity crucial in the control problem. Tet are performed at a full cale Miion TM OB boiler. ṁfu Tfu ṁa Ta ṁfp,o Tfp,o Flue Ga Pipe ṁfn,o Tfn,o qr,fn Furnace qc,fp qc,fn Metal qm qm w ṁ P T Steam Water Lw 2. BOILER MODEL The boiler model i put up uing firt principle a wa done in Åtröm and Bell [2000] for a drum boiler (for detailed information about the model derivation refer to Hvitendahl and Solberg [2004]). Throughout the paper ymbol and abbreviation are ued in equation and drawing and for explanation of thee refer to table 1 and 2 on page 7. The model conit of two part, a boiler model and a model of the actuator. Only the boiler model i derived in thi paper. Fig. 2. Block diagram of the four ub-ytem in the boiler model. Input and output are hown in the figure a well a flow and variable connecting the ub-ytem. Flue ga pip q c,f4 q c,f3 q o q i T o T i ṁfw Tfw Control Volume #4 Control Volume #3 2.1 Sytem Decompoition q c,f2 Control Volume #2 The boiler conit of two logically eparated part. One containing the heating part and one containing the water and team part. The heating part conit of the furnace and the flue ga pipe. The water and team part conit of all water and team in the boiler. The two part are interconnected by the metal eparating them i.e. the furnace jacket and the flue ga pipe jacket. Sub-ytem Model The boiler i divided into four ub-ytem for the purpoe of modelling. A block diagram of the boiler uing thee ubytem i hown in The Model Derivation The derivation i divided into ubection correponding to the four ub-ytem. The heating part ha been divided into four control volume two in the furnace (one radiation and one convection part) and two in the flue ga pipe (both convection part). Thi i done to get a more accurate etimate of the temperature ditribution throughout the heating part and to be able to better decribe the heat tranfer from the flue ga to the metal. Thi i illutrated in figure 3. The mean temperature in a control volume i et equal to the outlet temperature. The reaon for thi i that uing for intance a bilinear place dicretizing method introduce unwanted right half plane zero in a linear model. Furthermore the ma flow in a control volume i et equal to the q r,f1 Furnace Control Volume #1 Fig. 3. The heating part i divided into four control volume. input ma flow. Variation in preure and heat capacity c p,f of the flue ga in the heating part are diregarded wherea the denity variation are conidered a thee are much larger than variation in preure and heat capacity in the boiler operating range. The model of the control volume in the heating part can be found from the ma and energy balance of each control volume given a: and dρ f V = ṁ i ṁ o (1) d (ρ fv c p,f T o ) = ṁ i c p,f T i ṁ o c p,f T o Q (2) where Q i the heat delivered to the metal jacket. Combining the two balance equation give the following equation for the change in output temperature: dt o = ṁic p,f (T i T o ) Q ρ f V c p,f (3) Before finding the output ma flow ṁ o the change in denity ρ f of the flue ga mut be found.
4 The denity can be decribed uing the ideal ga equation and i given a: For the modelling purpoe a model of the water and team part of the boiler a illutrated in figure 4 i ued. ρ f = PM f R(T o K) (4) ṁ q where the M f i the molar ma of the flue ga and R i the ga contant, ee e.g. Serway and Beichner [2000]. Thi give the following equation for the change in denity: ṁ fw P q b ṁ b V dρ f = d PM f R(T o K) = ρ f T o K dt o (5) q fw q w b V b ṁ w b Q m w which together with (1) and (3) give the ma flow: V w ṁ o = 1 (T o K)c p,f (ṁ i c p,f (T i K) Q) (6) Furnace and Flue ga pipe The model of the four control volume in the heating part are almot identical and can be decribed by two equation for each control volume. One for expreing the change in temperature (3) and one for expreing the outlet ma flow being input to the next control volume (6). For each of the control volume the heat flow Q of equation 3 i either radiation or convection heat marked q r and q c in figure 3 repectively. Radiation heat q r,f1 i calculated a: q r,f1 = A f1 α r,f1 ((T f1 K) 4 (T m K) 4 ) (7) Convection heat q c,f2 i calculated a: q c,f2 = A f2 ṁ 0.8 f1 α c,f2(t f2 T m ) (8) Metal The dynamic of accumulated energy in the metal jacket eparating flue ga and water/team can be captured by mean of the energy balance. The metal i aumed to have the ame temperature in the entire volume a dynamic of thermal conduction for metal are fat. Thi give the following model of the metal part: dt m = Q f m Q m w ρ m V m,fj c p,m h 5 (9) where Q f m = q r,f1 q c,f2 q c,f3 q c,f4 i the energy flow upplied from the flue ga to the metal and Q m w = A mw (L w )α mw (T m T ) i the energy upplied to the water from the metal. Energy upplied to the water team part above the water urface i conidered negligible. Water/Steam Thi model ha the purpoe of decribing the team preure in the boiler P and the water level L w. The modelling i complicated by the hrink-and-well phenomenon which i caued by the ditribution of team bubble under the water urface (thi volume i abbreviated V b ). Fig. 4. Model for decribing the water and team part. For explanation of the abbreviation in the figure the reader i recommended to look in table 2. The total volume of water and team in the boiler i given a: V t = V w V V b, where V w i the water volume, V i the volume of the team pace above the water urface and V b i the volume of the team bubble below the water urface. To capture the dynamic of the water/team part the total ma and energy balance for the water/team part are conidered. The total ma balance for the water/team part i given a: d (ρ (V t V w ) ρ w V w ) = ṁ fw ṁ (10) from which the following expreion i found: ( (V t V w ) dρ ) dρ w V w dp dp f 66 dv w (ρ w ρ ) f 67 dp = ṁ fw ṁ h 6 (11) The total energy balance for the water/team part i given a: d (ρwvw(hw Pνw) ρ(vt Vw)(h Pν) ρ mv m,bj c p,mt ) = Q m w q fw q (12) which lead to the following differential equation: dh w dρ w ρwvw h wv w ρ (V t V w) dh dp dp dp h (V t V w) dρ dt V t ρ mv m,bj c p,m dp dp }{{ } f 76 dp dv w (h wρ w h ρ ) = q m w h fw ṁ fw h ṁ (13) f 77 h 7 It hould be noticed that the energy in the boiler metal jacket i included in the balance for the water/team part.
5 The two equation above only expre the preure and the water volume in the boiler. A the water level of interet in the control problem i given a: L w = (V w V b V o )/A w, another equation i needed for decribing the volume of team bubble V b in the water (the water level i meaured from the furnace top and V o i the volume urrounding the furnace and A w i the water urface area). To do thi the ma balance for the team bubble and the water are put up a: and d(ρ V b ) d(ρ w V w ) = ṁ w b ṁ b (14) = ṁ fw ṁ w b (15) repectively. The two flow ṁ b and ṁ w b are undetermined. Therefore an empirical equation i introduced. It expree the amount of team ecaping the water urface a: ṁ b = γ V b V w βṁ w b (16) By combining equation 14, 15 and 16 the final differential equation decribing the water/team part can be written a: ( ) dρ w dρ (1 β)v w V b dp dp }{{ } f 86 dp dv b ρ = (1 β)ṁ fw γ V b V w f 88 h 8 (1 β)ρ w f 87 dv w (17) Thi equation introduce V b in the model and hereby the hrink-and-well phenomenon. The Nonlinear Model The reulting overall nonlinear model of the boiler can be preented a below T f1 h T f2 h T f3 h T f4 h T ṁ = h 5 (18) f 66 f P h f 76 f V w h f 86 f 87 f 88 0 V b h T f4 h 9 F(x) ẋ h(x,u,d) where the firt order enor dynamic of the funnel temperature meaurement T f4 are included. The different matrix and vector entrie can be found in the equation derived earlier in thi ection, that i equation 3, 9, 11, 13 and 17. In practice the team flow i governed by everal valve combined with pipe reitance. Therefore a variable k(t) expreing pipe reitance and valve troke i introduced. ṁ i then given a: ṁ (t) = k(t) P (t) P atm (19) where P atm i the atmopheric preure and P (t) P atm i the differential preure over the valve. 2.3 Verification A parameter etimation ha been made to find etimate of the critical parameter in the model uch that it reflect the phyical boiler a well a poible. To verify the model a plot i made of the imulated preure and water level veru the meaured preure and water level, ee figure 5. P [bar] Steam preure, P Time [] L w [m] Water level, L w Time [] Fig. 5. Verification of boiler model. The olid line repreent the meaurement and the dahed line the imulation output. On the left ide the team preure i hown and on the right the water level. The input to the imulation i the meaured input to the phyical boiler ytem. The figure how good agreement between model and reality. 3.1 Strategy 3. CONTROLLER DESIGN Scheme The control trategy conit of two eparate control problem. One main controller, concerned in thi paper, in a cacade configuration with two actuator flow controller for fuel and feed water flow repectively. Compenator The control trategy i baed upon an LQG deign. The choice of an LQG deign wa inpired from a future goal of attempting to implement an MPC (Model baed predictive control) trategy capable of handling limitation in control ignal and tate. The LQ trategy i comparable to an MPC trategy without contraint. The deign i carried out in dicrete time. Part of the goal in thi control trategy i to compare the benefit of uing a team flow meaurement with that of a control trategy relying on etimate of the diturbance. Thi mean that the team flow diturbance (which i equivalent to the valve troke k introduced in the model) ha to be etimated along with proce tate. The valve troke k i the variable determining the load ituation of the boiler. A tep in k ha great influence on the ytem preure and water level due to hrink-andwell effect. A feed-forward in the controller from the valve troke i preumed to decreae the effect originated from the hrink-and-well phenomenon. To recontruct the effect of thi feed-forward a good etimate of the valve troke i needed.
6 3.2 Model The model decribing the boiler ytem (18) ha the form: F(x)ẋ = h(x,u,d) (20) where x = [T f1, T f2, T f3, T f4, T m, P, V w, V w, T f4 ]T i the tate vector, u = [ṁ fu, ṁ fw ] T i the input vector and d = [k, T fu, T fw ] T i the diturbance vector. The reaon why the air flow ṁ a i not included a an input i that the boiler i contructed with a fixed fuel/air ratio. Preceding the controller deign the model i linearized and the model order i reduced from nine to three leaving the tate vector: x = [P, V w, V w ] T. Thi new model wa found to decribe the ytem ufficiently preciely. The dicrete equivalent of the linear model i found and augmented by a model of the actuator controller dynamic reulting in the equation ytem: x (k 1) = Φx (k) Γ u(k) G d(k) (21) [ ] [ ] y(k) Hy 0 y (k) = = y a (k) 0 H x (k) (22) a = H x (k) where y = [P, L w ] and y a (k) correpond to output from the actuator model. 3.3 Control Law The goal of the controller i to minimize the variation in the water level L w and the team preure P from given et-point. The et-point are contant in normal operation of the boiler hence the purpoe i to reject the influence of the diturbance on the two parameter. The deign of the control law follow the principle outlined in Sørenen [1995]. The goal i to include diturbance in the controller to reject epecially the influence of change in the team flow valve poition k. Furthermore integral action i required to give offet free tracking of the reference. A both diturbance, reference and integral action are included in the performance index, the method require definition of a diturbance model, a reference model and an integral model. Augmented Sytem Model The original ytem tate vector i now augmented a x(k) = [x T (k),xt d (k),xt r (k),xt i (k)]t giving the model: Φ G H d 0 0 Γ 0 Φ x(k 1) = d Φ r 0 x(k) 0 u(k) H y 0 H r I 0 = Φx(k) Γu(k) (23) y(k) = [ H y ] x(k) = Hx(k) (24) e(k) = [ H y 0 H r 0 ] x(k) = H ex(k) (25) x i (k) = [ I ] x(k) = H i x(k) (26) A performance index with the purpoe of minimizing the error between reference and output, the integral tate and the control ignal can be et up a follow: I = ( e T (k)q 1e e(k) x T i (k)q 1i x i (k) (27) k=0 u T (k)q 2 u(k) ) State Feedback Minimizing the performance index reult in the well known control law: u(k) = [ L L d L r L i ]x(k) = Lx(k) (28) 3.4 Etimator The etimator mut recontruct tate not meaurable and give a current etimate ˆx of the tate vector x. Thi tate etimate will be input to the control law, which become u(k) = Lˆx(k). In the etimator deign the two firt component (x (k), x d (k)) of the augmented tate vector from equation 23 and 24 are of interet. In the real boiler ytem both proce and enor noie are preent. Including thee noie term a tochatic tate pace model for thee tate can be preented a: [ ] [ ] [ ] x (k 1) Φ = G H d x (k) x d (k 1) 0 Φ d x d (k) [ ] [ ] Γ w (k) 0 u(k) w d (k) and the output equation, [ ] y (k) y d (k) = [ ] [ H 0 0 H dy x (k) x d (k) ] [ v (k) v d (k) (29) ] (30) where w (k) and v (k) are proce noie and meaurement noie repectively. Both proce and meaurement noie are aumed to be uncorrelated zero-mean Gauian ditributed white noie equence. H dy i a matrix only electing the temperature diturbance a the team flow and hence the valve troke meaurement i not available (thee temperature are included in the etimator only to achieve meaurement filtering). 3.5 Etimator Gain Deign For derivation of the optimal etimator gain K ee e.g. Franklin et al. [1998]. Here jut note that the problem of finding the optimal etimator require knowledge of the proce and enor noie covariance matrice, Q and R repectively. Here the ytem decribed by equation 29 and 30 i conidered. Auming knowledge of Q and R the etimator gain can be found.
7 Covariance Matrice A dicued in Franklin et al. [1998] knowledge of R i often given from previou meaurement and enor accuracy wherea Q i a term accounting for unknown diturbance. The aumption of the proce noie being white i often ued becaue it implifie the reulting optimization problem. Phyically the proce noie can have any characteritic. In the preent work meaurement are available for determining the enor noie and the covariance matrix R i deigned diagonal containing the variance from the different meaurement on the diagonal. R = diag([σz 2 (1),..., σ2 z (p)]) (31) where [σz 2(1),..., σ2 z (p)] i a vector containing the pecific enor noie variance, where p i the dimenion of the meaurement vector. The proce noie in the boiler ytem i regarded a the diturbance, k the team flow valve troke, T fu fuel temperature and T fw the feed water temperature. But alo unknown diturbance might be preent and have to be conidered in the deign. w d i treated a known proce noie which i change in the diturbance known to occur. That leave w regarded a unknown diturbance on the ytem tate. Thi eem like a reaonable aumption a change in the diturbance input enter the ytem through the diturbance tate. Of coure the variance of w d can only be etimated and the problem of the noie being regarded a white till exit. The problem i that change in the team flow correponding to tep from middle load to maximum load are known to occur but thee change can for obviou reaon not be modelled a white noie. The proce noie covariance matrix Q can now be contructed diagonal with unknown proce noie element correponding to the ytem tate and reaonable variance expreing diturbance change correponding to the diturbance tate. Becaue of the under determined covariance matrix Q thi i ued a a deign parameter to achieve the bet etimator performance. The matrix i formed a: Q = diag([σud 2 (1),...,σ2 ud (n), σ2 d (1),...,σ2 d (l)]) (32) where [σud 2 (1),..., σ2 ud (n), σ2 d (1),..., σ2 d (l)] i a vector containing variance of the unknown diturbance and the known diturbance repectively. n i the ytem tate dimenion and l i the dimenion of the diturbance tate vector. 3.6 Cloed Loop Structure The cloed loop tructure of the LQG controller in the form ued here i preented in figure 6. Apart from the model matrice the figure contain the etimator gain matrix K and the feedback gain matrice L. L d = [L,L d ] and I y i a matrix electing the output P and L w. The tructure of the controller can be found in e.g. Sørenen [1995]. In thi cloed loop tructure the integral action in the compenator i included in u(k) Γ Γ x(k 1) x(k) y(k) z 1 H x(k 1) Ld Φ z 1 Φ Li Lr x(k) ˆx(k) H K 1 z 1 ȳ(k) ỹ(k) Iy - r(k) Sytem Etimator Controller Fig. 6. Cloed loop tructure of LQG controller. the controller directly on the difference between reference and output ignal. Another approach to incorporate the integral action through the etimator i dicued in e.g. Hvitendahl and Solberg [2004]. Including the meaurement of the team flow in the controller deign i aumed a practicable tak and i not illutrated here. 3.7 Stability It i well known that an oberver reduce the good tability margin exhibited by the LQ controller. For that reaon the tability of the deigned controller (with meaured team flow and etimated team flow) i invetigated to inure robutne of the cloe-loop ytem and find out if it i neceary to apply LTR (loop tranfer recovery). In figure 7 Nyquit plot of the open-loop ytem for both controller are hown. From the plot it can be een that both controller exhibit good tability margin even with the oberver introduced. Imag. axi Nyquit: full tate information Imag. axi Nyquit: etimator baed tate information Real Axi Real Axi Fig. 7. Nyquit plot of open-loop controlled ytem baed on full tate information (left) and etimated tate information (right). The olid line repreent the controller with meaured team flow and the dahed line the controller with etimated team flow. Both controller howed through imulation to behave and control the ytem well RESULTS Two tet were performed on AI Miion TM OB boiler - one for each deign. The tet conit of making tep change in the team valve troke correponding to a certain team flow auming a preure of 8 bar in the boiler. The change
8 are applied with three minute interval tarting at 1700 h. The equence i: h. The tet reult are hown in figure 8. L w [m] P [bar] Water level, L w Steam preure, P Time [] Water level, L w Steam preure, P Time [] Fig. 8. Verification of controller and evaluation of etimate of valve troke k. The plot on the left repreent the compenator with etimated k and the plot on the right the compenator with meaured k. The top plot how the water level and the bottom plot the team preure. From the plot it can be een that both controller are able to keep the water level and preure around the et point. Furthermore it can be een that there i no remarkable decreae in variation of water level when meauring the team flow. Wherea performance regarding preure variation i decreaed. 5. DISCUSSION It ha been verified that model baed MIMO control i uitable for control of one pecific cla of marine boiler (the bottom fired one pa moke tube boiler). When relying on etimate of the team flow it wa noted that there wa no remarkable difference regarding level variation wherea regarding preure the diturbance i eliminated more lowly. The meaurement ignal are contaminated by lot of meaurement noie corrupting etimate. It i expected that introduction of additional meaurement filtering and generation of a better etimate of the diturbance will reduce the influence of the diturbance on the preure. Future Work Much work till remain in the field of control of marine boiler. The reult preented in thi paper can be een a preliminary reult. The final goal i to minimize the team pace and water volume in the boiler. To achieve thi the final reult i expected to ue an MPC control trategy a thi can handle limitation on tate and control ignal. Table 1. Nomenclature. Symbol Unit Decription c p pecific heat capacity at C contant preure h pecific enthalpy k valve characteritic (valve Pa troke) m ma ṁ ma flow q energy flow A m 2 area K K =273 K (kelvin) L m level M molar ma mol P Pa preure Q added energy flow R ga contant Kmol T C temperature V m 2 volume α c m 2 heat tranfer contant for K convection heat α r m 2 K 4 heat tranfer contant for radiation heat β weight factor γ weight factor ρ denity ν Subcript a atm b c f fn fp fu fw i j m o r t w w m 3 m 3 pecific volume Table 2. Subcript. Decription air atmopheric bubble of team in water convection flue ga furnace flue ga pipe fuel feed water input jacket metal output radiation team above the water urface total water water urface Gene F. Franklin,. David Powell, and Michael L. Workman. Digital Control of Dynamic Sytem. Addion Weley Longman, Poul U. Hvitendahl and Brian Solberg. Modelling and multi variable control of a marine boiler. Mater thei, Aalborg Univeritet, Intitute of Electronic Sytem, Aalborg, Denmark, Raymond A. Serway and Robert. Beichner. Phyic for Scientit and Engineer with Modern Phyic. Saunder College Publihing, O. Sørenen. Optimal regulering. Aalborg Univeritet, Intitute of Electronic Sytem, Aalborg, Denmark, REFERENCES K.. Åtröm and R. D. Bell. Drum boiler dynamic. Automatica, 36: , 2000.
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