Fuzzy Modeling and Control of Boiler-Turbine Unit U sing Clustering and Subspace Method

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1 Fuzzy Mdeling and Cntrl f Biler-Turbine Unit U sing Clustering and Subspace Methd Xia Wu, Jing Shen and Yigu i Key abratry f Energy Thermal Cnversin and Cntrl f Ministry f Educatin Sutheast University Nanjing, China wux@seu.edu.cn K wang Y. ee, Fellw IEEE Department f Electrical and Cmputer Engineering Baylr University Wac, TX 76798, USA Kwan! Y _ee@baylr.edu Ahstract- This paper develps a fuzzy mdeling methd fr nnlinear biler-turbine unit using nly the input utput data. The structure f the fuzzy mdel is determined by clustering t achieve an apprpriate divisin f the lcal mdels. Then, the subspace identificatin is extended t develp lcal state-space mdels, which are dependent n the fuzzy membership functins. Owing t the advantages f the bth methds, the resulting fuzzy mdel can represent the biler-turbine unit very clsely, and a fuzzy mdel predictive cntrller is designed based n this mdel. The effectiveness f the prpsed fuzzy mdeling methd is demnstrated thrugh the simulatin. Index Terms--Biler-turbine unit, clustering, fuzzy mdel, predictive cntrl, subspace identificatin. I. INTRODUCTION Biler-turbine unit is an essential part f mdern fssilfuel-fired pwer plant which cnvert the chemical energy in fuel int mechanical energy and then int electrical energy. The central task f a typical biler-turbine cntrl system is t regulate the pwer utput t meet the demand f the grid while maintaining the pressure and water level in drum within given tlerances. As the pwer plants increase in size and participate in grid pwer regulatin mre frequently, cntrl f biler-turbine unit becmes a challenge due t multi-variable, time varying behavir and the severe nnlinearity ver a wide peratin range. Fr this reasn, varius cntrl strategies, such as advanced prprtinal-integral-derivative (PID) cntrl [], [2], mdel predictive cntrl (MPC) [2]-[8], H cntrl [8] [], and gain scheduling cntrl [2], have been develped. In general, mdeling is the first and fremst step in advanced cntrller design, because the cntrllers ' perfrmance is greatly relying n the structure, accuracy and cmplexity f the mdel. In [3] and [9], linear mdel arund a typical perating pint is used. The designed cntrllers shw gd system respnse ver a well-defined range, but they cannt satisfy the requirement ver a wide perating range. In [2] and [4], nnlinear mdels such as the neural-netwrk mdel and neural-fuzzy netwrk are emplyed in cntrller design. Althugh the wide range lad fllwing perfrmance is imprved in these papers, the nnlinear ptimizatin is time cnsuming. T vercme these issues, the fuzzy mdel technique has been widely used in biler-turbine cntrller design recently, which utilizes a cmbinatin f several linear mdels t apprximate the nnlinear system [5]-[7], [0]. The mdeling f fuzzy system is generally divided int the premise part (structure) and cnclusin (lcal mdel parameters) part. Fr the premise part design, clustering is widely accepted as an effective methd which classifies the data accrding t similarities amng them and rganizes the data int grups. Since the number f the clusters is crrespnd t the number f lcal mdels and the clustering centers can be used t calculate the membership functins, a satisfactry clustering can prvide an apprpriate structure design withut much experience n r nnlinear analysis f the plant [3], [4]. Fr the cnclusin part, it is interesting t nte that, a state-space type f lcal linear mdel is adpted in mst f the fuzzy cntrllers because f its advances in multi-variable systems and cntrl thery fr linear systems [5]-[8], [0]. In these wrks, an apprximatin r transfrmatin f the nnlinear system has been used in rder t btain the linear state-space mdel. Hwever, fr cmplex systems such as biler-turbine unit, it is a challenge t develp an accurate mathematical mdel withut the knwledge f thermdynamics and design specificatins f many cmpnents, which has becme ne f the main limitatins fr designing cntrllers fr real pwer plants. Subspace identificatin (SID) ffers an alternative way t develp state-space mdel directly frm the input-utput data f the plant [5]-[7]. The SID is a nn-iterative rbust identificatin methd which can avid lcal minima and cnvergence prblems; therefre, it vercmes the prblems f cnventinal predictin-errr methd (PEM) and is intrinsically suitable fr multi-variable systems [5], [6]. Hwever, due t the limitatin f the SID being nly fr linear This wrk was supprted in parts by the Natinal Natural Science Fundatin f China (NSFC) under Grant and Grant , the Specialized Research Fund fr the Dctral Prgram f Higher Educatin under Grant , the Scientific Research Fundatin f Graduate Schl f Sutheast University under grant yb.ij 20, and the U.S. Natinal Science Fundatin under grant ECCS /3/$ IEEE

2 system mdeling, mst f its existmg applicatins are n linear systems r n a small perating regin f the plant, and few papers can be fund n its applicatin t highly nnlinear biler-turbine unit. Given these reasns, we present this paper t address the mdeling prblems f biler-turbine unit when nly the inpututput data are available. The clustering is first used t develp the structure f the fuzzy system, and then by cmbining the data with the membership functins, the standard SID is extended t develp the lcal state-space mdels efficiently. The resulting fuzzy mdel is accurate and suitable fr varius advanced cntrller design. Mrever, due t its data-driven nature, the prpsed mdeling methd is flexible and can easily be adapted t ther types f systems withut knwing mathematical mdels f the plant. This paper is an extensin t a previus wrk [8], in which the whle perating regin is first divided by nnlinear analysis, and the crrespnding data fr each regin are cllected and used t identify the lcal mdel separately. Then, these mdels are transfrmed int a cmmn basis t frm the integrated multi-mdel system. Cmpared with the methd in [8], the prpsed methd has the fllwing advantages: i) the divisin f the whle peratin range is determined by the clustering, thus less human experience and interventin is needed; ii) it is mre efficient that all the lcal mdels can be identified tgether; iii) the resulting fuzzy mdel has smth transitin between lcal mdels, thus better suited fr cntrller design invlving switchings. II. SYSTEM DESCRIPTION The biler-turbine system used in this paper represents the behavir f a 60MW drum-type il-fired pwer plant. The dynamics f this particular pwer plant were recrded and frmulated int mathematical mdel by Bell and Astrm [9] using bth physical and empirical methds, as shwn belw: - dp 0.9ul u2P 9/8 -u3 () de (( 0.73u2-0.6)P9/8-E)/O (2) dp (3) _I ( l4u3 -(l.lu2-0.9)p)/85 where P dentes drum steam pressure (kg/cm 2 ), E dentes pwer utput (MW), and PI dentes steam-water density (kg/cm\ Cntrl inputs int the system are valve actuatr psitins that cntrl: the mass flw f fuel, represented as Ul; steam t the turbine, U2; and feed water t the drum, U3' The three cntrl inputs are subject t magnitude and rate cnstraints which represent the physical limitatins f the actuatrs. : ul,u2,u3 ; ul 0.007; (4) -2 u2 0.02; u Using the steady-state slutin fr Pfi the drum water level (m) can be calculated using the fllwing equatins: q e (0.854u2-0.47)P ul -2.54u (5) a s ( p I )(0.8P -25.6) Pj( P) 0.05( PI + 00as + q e ) (7) where as is the steam quality and qe is the evapratin rate in kg/s III. DATA-DRIVEN MODEING OF BOIER-TURBINE UNIT Suppse the fllwing discrete fuzzy mdel can be used t present the biler-turbine unit with bth fuzzy inference rules and lcal state-space mdels: R i : IF (jjk E Mi' THEN: {Xk+l Aixk + Biuk +Kiek Yk CiXk + Diuk +ek, i, 2... (8) where R i dentes the i-th fuzzy inference rule, the number f fuzzy rules, Mi the fuzzy sets, xk E SRn the state variables, Uk E SRm the cntrl input variables, Yk E SRP the utput variables, ek E SRP the zer mean white innvatin prcess. The matrices A, Bi, Ci, Di, Ki are lcal systems and bserver matrices. The vectr (jjk is the clustering input t the fuzzy system, which is cmpsed by current and past measurable variables f the plant. et r4 be the nrmalized membership functin f the fuzzy set Mi, then the fuzzy mdel can be expressed by the glbal frm: {Xk+l A",xk + B",uk +K",ek Yk C",xk + D",uk + ek where the matrices A", I OJ Ai' ther matrices are described in the same way. (9) OJ E [0,], Ir4, and The bjectives f the fuzzy system mdeling are: i) using the clustering t determine the number f the lcal mdels, and the membership functins d; ii) determine the mdel parameters { Ai' Bi, Ci, Di, Ki} thrugh SID using the input/utput data f the plant and their crrespnding membership functins. A. Premise Part Design Using Clustering Befre perfrming the clustering, it is imprtant t select the clustering input (jjk Because the dynamics f the plant are greatly relying n the pwer level, drum pressure and water level as well as the valve psitins, the clustering input is chsen as: (jjk [ -l ' Ek-l ' k-l ' UI,k- ' U2.k- ' U3.k- ] in this paper. Then, the data set f clustering, X, can be cnstructed as:

3 N X {Xk Ik,2... N} with the sample X k [ CfJk ', Ek ' k r ' where N is the number f the samples. A mdified Gaustafsn-Kessel (G-K) clustering algrithm [4] is emplyed t find clustering centers V;,, 2... and the partitin matrix U [,u;k ] E [0, ] XN frm the data set X,such that the fllwing bjective functin is minimal: ; k J(X;U,Y, A) (,u;kyn D; (0) where me [, c] is a scalar parameter which determines the fuzziness f the resulting clusters, generally being set as m2, and D;kA, dentes the distance between sample Xk and cluster center V;, which als determines the gemetrical shapes f the clustering: in which the psitive definite matrix A; is btained by: A; [p;det(f)t N ';- N (,u;k)m (Xk - V; )(Xk - V;)T F k t N (,u;kyn k () (2) As a nnlinear ptimizatin prblem, the analytic slutins f G-K clustering is difficult t btain, thus the iterative methd is widely used t minimize the bjective functin. The detailed algrithm can be fund in [4]. Once the clustering centers V;,, 2... are btained, we extract the input centers V;q:>frm them, which are set as the centers f the fuzzy set M;. Then, fr a given input vectr CfJk ' a Gaussian-type membership functin can be calculated thrugh V;q:>:. IICfJk -v/'ii 2 w exp[-(. )] at where (j ; is the wih f the membership functin:. j a' -(--: II V ; - V; II) f3 } (3) (4) with VI,,2,... J being the j clsest centers t the center V;. We set j and f3 4 in this paper and the nrmalized membership functin can be calculated by: ; ; /" ; OJk wk... wk ; (5) Therefre, we have successfully develped the premise part f the fuzzy mdel. B. Cnsequent Part Design Using the SID Cnsidering the glbal fuzzy mdel (9), the prblem left can nw be frmulated as: given the input sequence Uk, utput sequence Yk and their crrespnding membership functins UJk ver a time k,2,...,n, find the system matrices A;, B;, C;, D;, K;. Hwever, since mdel (9) is dependent n the membership functins, the glbal system matrices will be different fr each time step. This will make the data matrices invlved in standard SID grw expnentially with the size f the predictin time k [20], [2], and make it a challenge t apply. T slve this prblem, a simplified frm f mdel (9) is emplyed by making the matrices A, C, K independent f the membership functins (i.e., A;A j A, i, j, 2,..., ), and the new fuzzy mdel is as fllwing: {Xk+ AXk + BwUk +Kek Yk Cxk + DwUk +ek (6) Nte that Bw OJB; B[OJk uk] and Dw OJD; D[OJk Uk], where B [BI Bz... B;] ; D [DI Dz... D;], the membership functin vectr OJk [OJk I OJk z... {]. T, and presents the krnecker prduct. Mdel (6) can be rewritten as: ; { Xk+ Axk + B[ OJk Uk ]+Kek Yk CXk + D[OJk Uk ]Uk + ek (7) Therefre, by cmbining the input data with their crrespnding membership functins, the SID can be extended t find the cnsequent part f the fuzzy mdel. The algrithm f the SID can be fund in [5]-[ 7] and is nt repeated here. Remark 3.: This simplificatin brings great cnvenience fr utilizing the SID at the expense f nnlinear apprximatin ability f the fuzzy mdel. Hwever, since i) the structure f matrices A and D are independent fr each ther; ii)the accuracy f the mdel can be imprved by increasing the number f the clusters, this simplificatin is reasnable and the resulting mdel can still attain a satisfactry accuracy. IV. Fuzzy PREDICTIVE CONTRO OF BOIER -TURB INE UNIT Due t its lcal state-space structure, the prpsed fuzzy mdel is suitable fr varius advanced cntrller design. Therefre, a typical fuzzy MPC is develped in this sectin t achieve a wide range lad fllwing while dealing with the cnstraints. Cnsider the bjective functin: J (:)) f - rf) T Q f (-y f - rf) + D.uR fd.u f (8) where Q f Q > 0, R f R > are weighting matrices f

4 utput and input, rj [r::-l utput trajectry, )) j [));+l ));+2 r::- N y r the desired Y;+Ny r the estimated predictive utput, '.u j ['.U;+ '.U;+2 '.U;+N" r the future cntrl input variatins, N y and Nil, N y ;? Nil are, respectively, the predictin hrizn and the cntrl hrizn. The basic wrking principle f the fuzzy MPC is belw: Step. Get the dependent vectr rpk ' calculate the current nrmalized membership functins 0/ thrugh (3) and (5). Bm Step 2. Calculate the current glbal fuzzy system matrices mbi, Dm Step 3. A, Bm, C, Dm,K. mdi ' Cnstruct the predictr using system matrices Step 4. Calculate the ptimal input variatin sequence '.uj thrugh minimizing the bjective functin (8) subject t the input magnitude and rate cnstraints (4). Step 5. Implement the first input Uk+ uk + '.uk+ t the plant. V. SIMUATION RESUTS This sectin demnstrates the prpsed mdeling strategy fr biler-turbine unit using fuzzy clustering and subspace identificatin. The accuracy f the fuzzy mdel is demnstrated first, and then the prpsed fuzzy MPC is tested and cmpared with ther types f predictive cntrllers. A. Verificatin f Multi-Mdel System The input signals we used t generate data are shwn in Fig.. Since the pwer utput has a fast respnse t the variatin f steam cntrl valve, the sampling time is selected as secnd. Althugh increasing the number f the clusters will imprve the accuracy f the mdel, fr the sake f simplicity, we set 6. The identified mdel utputs are shwn in Fig. 2. Frm the cmparisn with plant utputs, the effectiveness f the prpsed identificatin strategy is clearly demnstrated. A single linear mdel develped by the SID methd using the same data is als tried fr cmparisn; hwever, due t the high nnlinearity f the biler-turbine unit, it leads t a nncnvergent result. B. Testing f Predictive Cntrller T test the perfrmance f the predictive cntrller, a wide range perating pint change frm (P, E, ) (75.6, 5.27,0) t (35.4, 27, 0) is cnsidered. The cntrl missin is tracking the expected perating pints f drum pressure and utput pwer while maintaining the drum water level cnstant. Three different MPCs are tested fr cmparisn: a) Fuzzy mdel predictive cntrller based n the mdel develped using the clustering and subspace methd in Sectin III (FMPC_C&S). b) Multi-mdel predictive cntrller based n the multiple state-space mdels derived frm the Taylr series apprximatin f the nnlinear mdel (MMPC_T) at the three perating pints: (86.4, 36.65,0), (8.8, 85.06,0) and (35.4, 27,0). c) Dynamic matrix cntrller (DMC) using the step respnse mdel develped based n the test data arund the perating pint (08,66.65,0) [3]. Fuel Flw Valve Steam Cntrl Valve Feedwater Flw Valve Fig.. Input signals used in the multi-mdel subspace identificatin methd I Sample Number 40r G -' 80 I ': S " 0 ' 2 i 0,I,,,.. ', E-0. 2 ", " B _ Sarrple I\tJmber Fig. 2. Estimated and real utputs f biler-turbine system (slid line: estimated utputs; ted line: real utputs). Fr FMPC_C&S and MMPC_T, the sampling time is set as 5 and a predictin hrizn N y 05 and cntrl hrizn NulOs are adpted. Fr the DMC, the predictive and cntrl hrizn are set as N y Nil 6005 with sampling time being set as 55. The weighting diagnal matrices Qj' Rj fr all three cntrllers are given as:

5 with diagnal elements: in which i,2, Q'{...,Ny, j,2,... NI/' The simulatin results in Fig. 3 and Fig. 4 shw that the tw multi-mdel based predictive cntrllers have satisfactry perfrmance while the DMC has failed in the wide range peratin. The reasn is that the step respnse mdel build arund ne pint is nt enugh t capture the nnlinear behavir f the unit in the full perating range. r::::::l rzf- : : : : : ---:-- s !::. / : : J Time (Secnd) Fig. 3. Perfrmance f the biler-turbine unit: Output Variables I :- (slid: FMPC_C&S; ted: MMPC_T; dashed: DMC; ted-dashed: reference) ,, : l \&6, u J _m, m_, Time (Secnd) Fig. 4. Perfrmance f the biler-turbine unit: Manipulated Variable (slid: FMPC_C&S; ted: MMPC_T; dashed: DMC). The tw multi-mdel based cntrllers have almst the same perfrmance. Hwever, in MMPC_T, the exact mathematical mdel f pwer plant is required first t build a Taylr series apprximatin mdel, which greatly limits its applicatin; mrever, frm the cntrl pint f view, affine terms exist in the state-space mdel [7], [8], increasing the cntrller design cmplexity and cmputatinal burden. l REFERENCES [II S. Zhang. C. W. Taft,. Bentsman. A. Hussey, and B. Petrus, "Simultaneus gains tuning in biler/turbine PID-based cntrller clusters using iterative feedback tuning methdlgy," ISA Transactins, vl. 5, pp , Sep [2 K. Y. ee,. H. Van Sickel,. A. Hffman, W-H lung, and S-H Kim, "Cntrller design fr a large-scale ultrasupercritical nce-thrugh biler pwer plant," IEEE Transactins n Energy Cnversin, vl. 25, pp , Dec [3] U. Mn and K.Y. ee, "Step-respnse mdel develpment fr dynamic matrix cntrl f a drum-type biler-turbine system," IEEE Transactins n Energy Cnversin, vl. 24 pp , Jun [4] X. iu, P. Guan, and C. W. Chan, "Nnlinear multi variable pwer plant crdinate cntrl by cnstrained predictive scheme," IEEE Transactins n Cntrl System Technlgy, vl.8, pp.6-25, Sep [5] K. Wu, T. Zhang,. v, and W. Xiang, "Mdel predictive cntrl fr nnlinear biler-turbine system based n fuzzy gain scheduling," in Prc IEEE Internatinal COllf"erence n Autmatin and gistics, pp. 5-20, Sep [6] X. Wu,. Shen, Y. i, and K. Y. ee, "Stable mdel predictive cntrl based n TS fuzzy mdel with applicatin t biler-turbine crdinated system," in Prc. 50th IEEE Cnference n Decisin and Cntrl and Cntrl and Eurpean Cntrl, pp , Dec [7] Y. i, J. Shen, K. Y. ee, and X. iu, "Offset-free fuzzy mdel predictive cntrl f a biler-turbine system based n genetic algrithm," Simulatin Mdeing Practive and Thery, vl. 26, pp , Aug [8] M. Keshavarz, M. Barkhrdari Yazdi, and M. R. Jahed-Mtlagh, "Piecewise affine mdeling and cntrl f a biler-turbine unit," Applied Thermal Engineering, vl. 30, 78-79, Jun [9] W. Tan, H. J. Marquez, T. Chen, and J. iu, "Analysis and cntrl f a nnlinear biler-turbine unit," Jurnal f Prcess Cntrl, vl. 5, pp , Dec [0] J. Wu, S. K. Nguang,. Shen, G. iu, and Y. i, "Rbust H tracking cntrl f biler-turbine systems," ISA Transactins, vl. 49, pp , Jul [] K. Zheng, J. Bentsman, and C. W. Taft, "Full perating range rbust hybrid cntrl f a cal-fired biler/turbine unit," Jurnal f Dynamic Systems, Measurement, and Cntrl, vl. 30, pp , Jul [2] P. C. Chen and. S. Shamma, "Gain-scheduledl' -ptimal cntrl fr biler-turbine dynamics with actuatr saturatin," Jurnal f Prcess Cntrl, vl 4, pp , Apr [3] T. Takagi and M. Sugen, "Fuzzy identificatin f systems and its applicatin t mdeling and cntrl," IEEE Transactins n Systems,Man and Cybernetics" vl. 5, pp. 6-32, Jan.-Feb [4] R. Babuska, P.. van de Veen, and U. Kaymak, "Imprved cvariance estimatin fr Gustafsn-Kessel Clustering," in Prc IEEE Internatinal COt!f"erence n Fuzzy System, pp , May [5] P. V. Overschee and B. D. Mr, "A unifying therem fr three subspace system identificatin algrithms," Autmtica, Vl. 3, N. 2, pp , Dec [6] W. Favreel, B. D. Mr, and P. V. Overschee, "Subspace state space system identificatin fr industrial prcesses," Jurnal f Prcess Cntrl, vl. 0, pp , Apr [7] S.. Qin, "An verview f subspace identificatin," Cmputers and Chemical Engineering, vl. 30, pp , Sep [8] X. Wu, J. Shen, Y. i, and K. Y. ee, "Data-driven mdeling and cntrl fr biler-turbine unit," IEEE Transactins n Energy Cnversin, submitted and revised. [9] R. D. Bell and K. J. Astrbm, Dynamic Mdels fr Biler Turbine Alternatr Units: Data gs and Paramter Estimatin fr 60 MW Unit. TRFT-392. und Institute f Technlgy, und, Sweden, 987. [20] J. W. van Wingerden and M. Verhaegen, "Subspace identificatin f multi variable PV systems: the PBSID apprach," in Prc. 47th IEEE Cnference n Decisin and Cntrl, pp , Dec [2] F. Felici, J. W. van Wingerden, and M. Verhaegen, "Subspace identificatin f MlMO PV systems using a peridic scheduling sequence," Autmtica, Vl. 43, N. 0, pp , Dec