Predictive Control of a Boiler-turbine System

Transcription

1 Recent Researches n Crcuts and Systems Predctve Control of a Boler-turbne System JAKUB NOVAK, PR CHALUPA aculty of Aled Informatcs omas Bata Unversty n Zln Nam.G.Masaryka 5555, Zln CZCH RPUBLIC jnovak@fa.utb.cz Abstract: - In ths work, multle model control of a boler-turbne system s studed. Multle lnear local models are obtaned through ecewse lnearzaton. A bank of Kalman flters s constructed usng these local models. States and model arameters used for future redctons wthn the redctve control are udated onlne through comutaton of robablty densty functons. Smulatons show that the boler-turbne coordnated system can be successfully controlled by ths methodology. Key-Words: Power systems, Kalman flterng, Predctve control, Multle models, Boler-turbne, Pecewse lnearzaton Introducton he boler-turbne system s an essental art of a ower lant. he boler-turbne system exhbts nonlnear, tme-varyng, coulng behavor. Hence, the control of such system s comlex and challengng and a lnear model cannot cature the nonlnear dynamcs suffcently. he major control objectve of a boler-turbne system s to kee the outut of mechancal energy n balance wth the electrcal load demand whle mantanng the nternal varables such as drum steam ressure, temerature and drum water level wthn the desred ranges. Due to the varable demand for electrcty n a grd the ower lants are forced to change load frequently n a large magntude. As a result, the ower lants have to oerate n multle oeratng regmes and the nonlnear behavor becomes more sgnfcant durng these transtons between oeratng onts. It s essental that the develoed dynamc model can cature the dynamcs of the system n dfferent oeratng whle keeng the model relatvely smle, sutable for the desgn of feedback controllers. Numerous modelng and control methodologes have been aled for boler-turbne system. In [] multle model redctve control methodology where the system s modeled by ecewse lnear models s aled for control of a boler turbne unt. Gan schedulng aroach allowng the ossblty of large changes n oeratng condtons has been resented n []. uzzy schedulng model redctve controller where local models are equdstantly dstrbuted n the oeratng sace and the local models arameters are obtaned through lnearzaton s dscussed n []. Neuro-fuzzy network wth Controlled Auto-Regressve Integrated Movng Average CARIMA models and nterolaton based on the B-slnes functons has been successfully aled n [4]. Moon and Lee [5] resented a fuzzy controller that can udate the fuzzy rules adatvely by a smle set-ont errorcheckng rocess. he onlne learnng of Radal Bass uncton (RB) neural network has been tested n [6]. In [7] the dynamc fuzzy model, where local models are obtaned usng aylor seres around the nomnal onts, s resented. A two level herarchcal control scheme for boler-turbne system s mlemented n [8]. In ths aer, the controlled lant under consderaton s a 6 MW boler-turbne system that was reorted n [9]. he state s estmated usng the bank of Kalman flters each usng the arameters from dfferent oeratng ont. Multle Model Adatve stmaton (MMA) algorthm he multle model adatve estmaton (MMA) method s based on a bank of arallel Kalman flters (K), each tuned to descrbe the lant at a dfferent oeratng ont (g. ). he model s assumed to be lnear affne and of the form: x( k ) Ax( k) Bu( k) () y( k) Cx( k) Du( k) ISBN:

2 Recent Researches n Crcuts and Systems Outut of each K s then weghted by ts corresondng robablty based on the measurement hstory. g. shows a functonal block dagram of the MMA algorthm. Its rmary feature s a bank of Kalman flters oeratng n arallel, usng vectors of measurements y and control commands u as ther nut. ach Kalman flter has the same structure based on the lnearzed descrton of the rocess. At every samlng erod, each of these Kalman flters s roducng estmate of the state ( ) x k and resdual r( k ). he dea s that the model wth well-behaved resduals contans the arameters that best matches true arameters of the system. estng the hyothess whch model s the correct one s evaluated n the hyothess testng block. he ntal robablty of each hyothess beng correct s dstrbuted evenly: () / M () he outut redcton s gven by the mxture of condtonal robablty densty functons: where M ( y( k) u( k)) ( y k u( k )) () are the robabltes of each model beng the correct one and normalzed to. M m( t), (4) he redctve condtonal robablty densty functons are gven by the state-sace model as: y k u k y( k) x( k) x( k) u( k ) (5) where x( k) u( k) s the state-estmate rovded by the -th Kalman flter. One ste of the Kalman flter can be wrtten as: K ( k) A P( k) C I C P( k) C P( k ) A P( k) A Q K C P( k) C I K x( k ) A x( k) B u( k) K ( y C x( k) D u( k) ) (6) he condtonal robablty densty functon for known measurement nose has normal dstrbuton[] and can be comuted usng: ( y( k) x( k), ) N( y,( C PC ) ) y k ( C PC ) e ( ) ( ( ) x( k), e ) ex m/ r y( k) y( k) e e r ( k) r ( k) (7) he estmate of varance can be udated wth exonental forgettng wth factor as: S ( k ) e ( k ) ( k ) (8) g. Multle Model Adatve stmaton scheme ISBN:

3 Recent Researches n Crcuts and Systems where varables each ste: S ( k ) S ( k) ( k ) ( ( k) ) Boler-urbne System S and ( k ) are udated at rr C P( k ) C (9) he boler-turbne model used n ths aer was frst develoed by Bell and Astrom and has been oularly adoted n valdatng varous controllers for the boler-turbne system n smulaton. he arameters were estmated from the data collected from the Synvendska Kraft AB Plant n Malmo, Sweden. he rate ower of the lant s 6 MW. he model s a three nut, three outut, thrd order nonlnear system (g. ). he nuts are the ostons of the valve actuators that control the mass flow rates of fuel ( u n u), steam to the turbne ( u n u), and water to the drum ( u n u). he three major oututs are the electrcal ower ( y n MW), drum steam ressure ( y n kg/cm ), and drum water level ( y n m). he three state varables are the electrc ower ( x n MW), drum steam ressure ( x n kg/cm ), and the flud (steamwater) densty ( x n kg/m ). he model state equatons are gven as: 9/8 x.8u x.9u.5u x.7u.6 x.x 9/8 x 4 u (.u.9) x / 85 he oututs of the lant are gven: y x y x () ().7x acs qe / y where acs and qe are steam qualty and evaoraton rate (kg/s), resectvely. hey are gven by.58x.8x 5.6 acs x.94.4 x q.854u.47 x 45.59u.54u.96 e () Due to actuator lmtatons, all the control nuts u, u, u are subject to the followng constrants: u,,.7 u.7. u..5 u.5 () 4 Multle Lnearzaton of the Process In ths work the global model of the rocess s obtaned through ecewse lnearzaton. he nonlnear system: x( t) f ( x( t), u( t)) (4) y( t) f ( x( t), u( t)) n q where x( t) R, u( t) R, y( t) R reresent the states, nut value and outut value, resectvely. he system can be lnearzed around the oeratng ont usng the aylor s seres aroxmaton. hs results n a seres of M local lnear models of the form: x( t) A x( t) B u( t) (5) y ( t) C x( t) D u( t) where f ( x, u) f ( x, u) A x, u, B x u x, u g( x, u) g( x, u) C x, u, D x u x, u x x o x, u u o u, y y o y o o o o o o o o (6) he above equatons can be dscretzed to obtan a set of lnear systems n the form: x ( ) ( ) ( ) k Ax k Bu k (7) y( k) C x( k) D u( k) g. Boler-turbne model ISBN:

4 Recent Researches n Crcuts and Systems able. Oeratng onts of the boler-turbne dynamcs # # # #4 #5 x x x u u u he nonlnear system was lnearzed at 5 oeratng onts and able shows the values of nuts and states for these oeratng onts. 5 Predctve Control he state-sace model based redctve control s based on the tme-nvarant model: x( k ) Ax( k) Bu( k) (8) y( k) Cx( k) Du( k) he model of the rocess s obtaned at every samlng nterval and ts arameters are used for the entre redcton horzon H. he dscrete model (7) contans also the affne art that results from lnearzaton around non-zero steady-state: x( k ) Ax( k) Bu( k) he y( k) Cx( k) Du( k) M M M A = A, B = B, = M M M C = C, D = D, = (9) H -ste ahead outut redcton can be deduced: Y x( k) U Y yx yu y y( k ) u( k) y( k ) u( k ), U y( k H ) u( k H ) () where the matrces are gven as: CB D yu y CAB CB D H H H CA B CA B CB D C CA CA C CA, CA... C CA yx H () he comutaton of a control law of MPC s based on mnmzaton of the followng crteron J Y W Q Y W UR U () MPC where y( k j k ) s a j stes ahead redcton of the system, w( k j ) s a future reference trajectory and QR, are ostve defnte weghtng matrces. he mnmzaton of the crteron can be transformed nto a quadratc rogrammng roblem: JMPC u Hu fu () where matrx H and vector f are derved from model arameters gven by (). he quadratc roblem s usually solved numercally. As formulated, the nonlnear model redctve controller wll exhbt steady state offset n the resence of lant/model msmatch due to a lack of ntegral acton. o ntroduce an ntegral acton to remove steady-state error an ntegrator state must be added to the system: v( k) v( k ) ( w( k) y( k)) ( k) xk ( ) vk ( ) System matrces (9) are udated as follows: ( k ) A ( k ) Bu y( k) C ( k ) Du A B A, B, C I D W ( k) C C, D D, (4) (5) In order to mnmze the augmented state vk ( ) the cost crteron for MPC () s transferred to: where JMPC Y W Q Y W U R U X SX (6) X x( k) U (7) xx xu x ISBN:

5 Recent Researches n Crcuts and Systems xx xu x A A A H B AB B H H A B A B B H A A... (8) S () he smulaton reresents a frequent load demand change when the ower unt s n Automatc Generaton Control (AGC) mode []. he oeratng ont changes from # to #5 and then to #. g. resents the good erformance of multle-model redctve control and g.4 shows the outut of the redctve controller. he robablty that each model descrbes the lant at current samlng ont s dected on g Imlementaton In ths art, multle-model redctve control s aled to the Bell-Åström boler-turbne system. Multle lnear models were obtaned through lnearzaton n steady-state oeratng onts gven n able. he samlng erod was set to s due to the dynamcs of the rocess. he Kalman flter bank wth these local model was constructed wth ntal condton x [8 66,65 48]. he ntal estmate covarance matrx P was chosen to be: 4 4 P (9) 4 Saturaton constrants n the manulated varables are mosed to take nto account the mnmum/maxmum aerture of the valve regulatng the flow rates. he redcton horzon was set to samles as a result of usng dfferent values and comarng control erformances. he weghtng matrces Q,R,S assocated wth the error from setont, control outut ncrement and ntegrator gan was set to 5 Q, R 5 () 4 5 [kg/cm ] P [MW] L [m] [s] g. Performance of redctve controller over a wde oeratng range ISBN:

6 Recent Researches n Crcuts and Systems u u u [s] g. 4 Control actons me [s] g.5 Probablty of each model beng the correct one durng the exerment 7 Concluson In the aer, a multle-model redctve control methodology s aled to a boler-turbne coordnated system. he correct model at the current samlng ont s estmated usng the resduals rovded by a bank of Kalman flters. he obtaned state s n the form of a mxture of states rovded by the flters. he arameter of the lnearzed model and the states are then used wthn the redctve control aroach for redcton of the future lant behavor. he smulaton shows that t can be aled to the boler-turbne coordnated system effectvely and the erformance can be mroved by ncrease the number of the lnear models. Acknowledgement he authors kndly arecate the fnancal suort whch was rovded by the uroean Regonal Develoment und under the roject CBIA-ech No. CZ..5/../.89. References: [] X. Wu, J. Shen, Y. L, Control of Bolerturbne Coordnated System Usng Predctve Control, Proceedng of the 8 th I Internatonal Conference on Control and Automaton,, [] P.-C. Chen, J.S. Shamma, Gan scheduled l - otmal control for boler-turbne dynamcs wth actuator saturaton, Journal of Process Control, Vol. 4, 4, [] K. Wu,. Zhang, J. Lv, W. Xang, Model Predctve Control for Nonlnear Boler- urbne System Based on uzzy Gan Schedulng, Proceedngs of the I Internatonal Conference on Automaton and Logstcs, 8,. 5-. [4] X. Lu, J. Lu, P. Guan, Neuro-fuzzy generalzed redctve control of boler steam temerature, Journal of Control heory and Alcatons, Vol. 5, 7, [5] U.-C. Moon and Y.L. K.Y. Lee, A Bolerturbne system control usng a fuzzy autoregressve movng average (ARMA) model, I ransactons on nergy Converson, vol.8,, [6] D. Peng, H. Zhang, P. Yang, Modelng of Boler-urbne Nonlnear Coordnated Control System Based on RB Neural Network, Proceedng of the I Internatonal Conference on Control and Automaton, 7, [7] H. Habb, M. Zelmat, B. O. Bouamama, A dynamc fuzzy model for a drum boler turbne system, Automatca, Vol. 9,,. -9. [8] R. Garduno-Ramrez, K.Y.Lee, Wde Range Oeraton of a Power Unt va eedforward uzzy Control, I ransactons on nergy Converson, Vol. 5,, [9] K.J. Astrom, R.D. Bell, Drum boler dynamcs. Automatca, Vol. 6,, [] V. Havlena, J. Pekař, J. Štecha, Predctve Controller for MIxture of Outut rror Models, Proceedngs of Modellng, Identfcaton and Control, 4, [] J. Novak, P. Chalua, V. Bobal, Automatc Cost-otmal Power Balance Control, I ransactons on Power Systems, Vol. 6,, ISBN: