Journal of Power Sources

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1 Journal of Power Source 196 (2011) Content lit available at ScienceDirect Journal of Power Source journal homepage: wwweleviercom/locate/jpowour Modeling and control of tubular olid-oxide fuel cell ytem I: Phyical model and linear model reduction Andrew M Colclaure, Borhan M Sanandaji, Tyrone L Vincent, Robert J Kee Engineering Diviion, Colorado School of Mine, 1600 Illinoi St, Golden, CO 80401, USA article info abtract Article hitory: Received 12 May 2010 Received in revied form 22 June 2010 Accepted 22 June 2010 Available online 30 June 2010 Keyword: Tubular SOFC Phyical model Sytem identification Model reduction Thi paper decribe the development of a tranient model of an anode-upported, tubular olid-oxide fuel cell (SOFC) Phyically baed conervation equation predict the coupled effect of fuel channel flow, porou-media tranport, heat tranfer, thermal chemitry, and electrochemitry on cell performance The model output include patial and temporal profile of chemical compoition, temperature, velocity, and current denity Mathematically the model form a ytem of differential-algebraic equation (DAE), which i olved computationally The model i deigned with proce-control application in mind, although it can certainly be applied more widely Although the phyical model i computationally efficient, it i till too cotly for incorporation directly into real-time proce control Therefore, ytem-identification technique are ued to develop reduced-order, locally linear model that can be incorporated directly into advanced control methodologie, uch a model predictive control (MPC) The paper illutrate the phyical model and the reduced-order linear tate-pace model with example 2010 Elevier BV All right reerved 1 Introduction Thi paper i the firt part of a two-paper equence that dicue the incorporation of phyical model into advanced proce control The preent paper develop tranient phyical model for tubular olid-oxide fuel cell (SOFC), and it dicue linear modelreduction trategie near certain teady-tate operating point (OP) The companion paper [1] concentrate on nonlinear reduced model and model predictive control (MPC) deign The deign and control of fuel-cell ytem can benefit greatly from phyically baed model The preent paper i particularly concerned with model that can be applied to develop model-predictive proce-control algorithm In thi application, high-fidelity tranient repone i required The model conider flow and chemitry within a ingle anode-upported SOFC tube, a well a cathode air that flow over the external urface of a tubular tack The approach i baed generally upon earlier work [2 5] However, to facilitate the real-time control application, the model mut be computationally fat and deliver accurate tranient repone To achieve computational efficiency, relatively coare patial dicretization i ued and heterogeneou reforming chemitry i modeled with global reaction The charge-tranfer chemitry i repreented in a Butler Volmer form that i derived from elementary reaction [6] Correponding author Tel: ; fax: addre: rjkee@mineedu (RJ Kee) For proce-control application a pragmatic balance can be made between phyical fidelity and computational efficiency The primary objective of the model i to ait enor interpretation and ubequent actuation To achieve thi objective, the model mut incorporate all the relevant time cale aociated with the phyical and chemical procee It mut alo repreent the relationhip between actuation and repone However, in practice the control actuation depend on feedback from enor, albeit aited by model-baed interpretation In other word, the control i not baed upon model prediction directly A dicued later in thi paper, complex nonlinear phyical model are further reduced to ytem of linear tate-pace model that are incorporated into the control algorithm Becaue the model that are incorporated into the control algorithm can be approximate, the underpinning phyical model can approximate ome phyical attribute in return for computational peed 2 Model development Fig 1 illutrate a prototype SOFC tack that i deigned for a kilowatt-cale ytem Thi tack contain 36 anode-upported tubular cell Fuel enter the SOFC tube from below and exit at the top of the tack Cathode air enter radially at the bottom of the tack and leave at the top through mall clearance between the tube and a top end plate The phyical model concentrate epecially on the flow, chemitry, and thermal behavior within the tube The air flow over the outide of the tube i modeled a a perfectly tirred reactor The model accommodate heat /$ ee front matter 2010 Elevier BV All right reerved doi:101016/jjpowour

2 AM Colclaure et al / Journal of Power Source 196 (2011) Nomenclature E eq c A f,c cro ectional area of fuel channel, m 2 A cro ectional area of anode upport layer, m 2 A f cro ectional area of anode functional layer, m 2 A f, cro ectional area of anode upport and functional layer, m 2 a pecific urface area of anode upport layer, m 1 a f pecific urface area of anode functional layer, m 1 B g permeability, m 2 c p heat capacity of the fuel channel flow, J kg 1 K 1 c p,m heat capacity of the MEA, J kg 1 K 1 d p particle diameter, m D e k,kn effective Knuden diffuion coefficient, m 2 1 D kl binary diffuion coefficient, m 2 1 D e kl effective binary diffuion coefficient, m 2 1 E eq a reverible potential between the anode and electrolyte, V reverible potential between the cathode and electrolyte, V E H2 activation energy for H 2 oxidation in the Butler Volmer equation, J kmol 1 E O2 activation energy for O 2 reduction in the Butler Volmer equation, J kmol 1 E cell cell potential, V E rev reverible cell potential, V F Faraday contant, C kmol 1 G tandard tate Gibb free energy, J kmol 1 h enthalpy of the ga-phae mixture, J kg 1 h k pecie heat enthalpy, J kg 1 h q heat tranfer coefficient between fuel channel flow and MEA, W m 2 K 1 h q,c heat tranfer coefficient between air tream and MEA, W m 2 K 1 i cell local current denity, A m 2 i 0 exchange current denity for H 2 oxidation, A m 2 i H 2 nominal exchange current denity of H 2 oxidation, Am 2 i ref,h 2 parameter i at the reference temperature T H 2 ref,a m 2 i 0,c exchange current denity for O 2 reduction, A m 2 i O 2 nominal exchange current denity of O 2 reduction, Am 2 i ref,o 2 parameter i at the reference temperature T O 2 ref,a m 2 j k ga-phae pecie ma flux, kg m 2 1 J k ga-phae pecie mole flux, kmol m 2 1 j z,k axial diffuive ma flux of k th pecie within the fuel channel flow, kg m 2 1 j r,k radial ma flux of k th pecie from the fuel flow into the porou anode, kg m 2 1 j c r,k radial ma flux of k th pecie from the MEA into the air tream, kg m 2 1 j e r,k radial ma flux of k th pecie from the anode functional layer into the electrolyte, kg m 2 1 j r,k radial ma flux of k th pecie from the anode upport into the functional layer, kg m 2 1 K number of ga-phae pecie n e number of charge tranferred in the overall charge tranfer reaction Nu Nuelt number of fuel channel flow P f perimeter between the fuel channel and anode upport layer, m P perimeter between the anode upport layer and functional layer, m P e,p c perimeter between the MEA and air tream, m p preure, Pa p f,h2 partial preure of H 2 within the anode functional layer, atm p f,h2 O partial preure of H 2 O within the anode functional layer, atm p a,o2 partial preure of O 2 within the air tream, atm p parameter in the expreion of i H 0, atm 2 p parameter in the expreion of i O 0,c, atm 2 q cond axial conductive heat flux within fuel channel flow, Wm 2 q diff axial heat flux within fuel channel flow from diffuion, W m 2 q a radial heat flux from the fuel channel flow into the diff anode from ma tranport, W m 2 q c radial heat flux from the MEA into the air tream diff from ma tranport, W m 2 q mea axial conductive heat flux within the MEA, W m 2 R univeral ga contant, J kmol 1 K 1 r e outer radiu of anode functional layer, m r f outer radiu of fuel channel, m r p pore radiu, m r outer radiu of anode upport layer, m ṡ k molar production rate by urface reaction, kmol m 2 1 t time, T temperature of the fuel channel flow, K T a temperature of the air tream, K T m temperature of the compoite MEA, K UA a overall heat tranfer coefficient between air tream and encloure, W m 2 K 1 V volume of fuel channel flow control volume, m 3 W k pecie molecular weight, kg kmol 1 W mean molecular weight, kg kmol 1 X k ga-phae pecie mole fraction [X k ] ga-phae pecie molar concentration, kmol m 3 [X T ] total ga-phae molar concentration, kmol m 3 Y k ga-phae pecie ma fraction within the fuel channel flow Y a,k ga-phae pecie ma fraction within the air tream flow Y f,k ga-phae pecie ma fraction within the anode functional layer Y,k ga-phae pecie ma fraction within the anode upport layer z axial coordinate, m Greek letter a anodic ymmetric factor in the Butler Volmer equation c cathodic ymmetric factor in the Butler Volmer equation ız axial length of fuel channel control volume, m heat conductivity of fuel channel flow, W m 1 K 1 m heat conductivity of compoite MEA, W m 1 K 1 ga-phae vicoity, kg m 1 1 k reaction toichiometry of pecie k in overall charge tranfer reaction

3 198 AM Colclaure et al / Journal of Power Source 196 (2011) f g g f act act,a act,c ohm ga-phae ma denity of fuel channel flow, kg m 3 ga-phae ma denity within anode functional layer, kg m 3 ga-phae ma denity within anode upport layer, kg m 3 tortuoity of the ga-phae poroity poroity of anode functional layer poroity of anode upport layer local overpotential, V local activation overpotential, V local activation overpotential within the anode, V local activation overpotential within the cathode, V local ohmic overpotential from ion conduction, V Fig 2 Control volume annotated for the overall fuel-flow continuity equation not familiar with uch derivation may wih to conult a text on chemically reacting flow for detail [7] 21 Fuel flow, overall ma continuity Fig 2 illutrate a control volume for a hort tube ection The ma-conervation equation for the flowing gae within tube i written a Fig 1 A tubular SOFC tack coniting of three hexagonal ring of anode-upported cell and ma tranfer between the tube exterior and the cathode air The fuel-cell model i written in term of mall finite volume along the length of an individual tube Fuel flow within the tube (anode ide) and air circulate outide the tube (cathode ide) The fuel-flow model are baed upon a plug-flow approximation in which only axial variation are modeled That i, perfect mixing i aumed acro the radiu of the tube When hydrocarbon fuel are ued, reforming can proceed via catalytic chemitry within the anode upport tructure Charge-tranfer chemitry proceed at the interface between the dene electrolyte and the compoite electrode tructure The model accommodate heat tranfer within the tubular tructure and heat exchange between the tube and gae The following ection preent derivation of the underpinning conervation equation for ma and energy The derivation follow tandard practice for deriving uch equation baed upon appropriate emi-differential control volume However, reader d dt = ṁ ṁ Ṁ, (1) V where i the ma denity, ṁ i the ma flow rate entering the control volume, ṁ i the ma-flow rate leaving the control volume, and Ṁ i the ma-flow rate leaving the control volume and entering the porou-anode tructure, and V i the volume of the control volume Becaue the tube i egmented into mall ection of length ız, the entering ma flow rate ṁ i equal to the ma flow rate ṁ, which i leaving the adjacent uptream egment The ma exchange between the fuel flow and the anode tructure i repreented a Ṁ = j r,k P f ız, (2) where j r,k repreent the radial ma flux of pecie k from the fuel flow into the porou anode tructure The fuel-tube perimeter i P f = 2r f and K i the total number of pecie A perfect-ga equation of tate, = p W RT = p 1 RT, (3) Y k /W k relate the denity, preure p, temperature T, and the mean molecular weight W The pecie ma fraction are Y k and the molecular weight are W k By differentiating the equation of tate, the ma continuity equation can be rewritten to eliminate the ma-denity derivative in favor of temperature and ma-fraction derivative (preure i aumed to be contant) a ṁ = ṁ Ṁ + V T T t + V W K 1 ( 1 W k 1 W K ) Yk t (4)

4 AM Colclaure et al / Journal of Power Source 196 (2011) Fig 3 Control volume annotated for the fuel-flow pecie continuity equation Fig 4 Control volume annotated for the fuel-flow thermal energy equation The ummation to K 1, not K, i becaue exact pecie conervation i enforced by Y K = 1 K 1 Y k Becaue all the pecie ma fraction are not independent, the molecular weight W K for pecie K appear in the lat term 22 Fuel flow, pecie continuity equation Fig 3 illutrate pecie fluxe into and out of a fuel channel control volume The pecie continuity equation for the fuel flow within the tube may be written a V (Y k) t = ṁ k ṁ k Ṁ k + A f,c ( j u z,k jd z,k) (5) In thi equation, Y k repreent the ma fraction within the control volume, which are the ame a the ma fraction leaving the control volume The entering and exiting pecie ma flow rate are ṁ k = ṁ Y k, ṁ k = ṁy k, (6) where Y are the entering ma fraction The rate of pecie ma k exchange between the anode tructure and the fuel channel i Ṁ k = j r,k P f ız (7) The axial diffuive ma fluxe acro the uptream and downtream control urface are repreented a j u z,k and jd z,k, repectively Thee axial fluxe are calculated uing mixture averaged formulation [7] The control-urface area i A f,c = r 2 f By ubtituting the overall ma-continuity equation (Eq (1)) and ome further algebraic manipulation, Eq (5) can be rewritten a an explicit differential equation for the pecie ma fraction a Y k t = ṁ ( ) Y V k Y 1 k V + A f,c V ( j u z,k jd z,k (Ṁk Y k Ṁ ) ), (k = 1,,K 1) (8) The ma fraction of the K th pecie i found a Y K = 1 K 1 Y k 23 Fuel flow, thermal energy A illutrated in Fig 4, the energy equation ha everal contributing term In differential equation form, E t = (ṁ ( h ṁh) + q u cond cond) ( qd Af,c + q u diff diff) qd Af,c q a diff P fız h q (T T m) P f ız In thi equation h i the pecific enthalpy and H = Vh i the total enthalpy The internal energy time derivative E/ t i equal to the enthalpy time derivative becaue the preure of the fuel channel flow i contant E/ t = H/ t The firt term on the right-hand ide repreent the energy convected into and out of the control volume The econd term repreent thermal conduction Uing Fourier law, the heat flux i repreented a q cond = T z, (10) where i the thermal conductivity With the tube dicretized into a erie of control volume, the temperature in each volume i repreented a T The uptream and downtream conduction are evaluated a q u cond = T T u, q d ız cond = T d T, (11) ız where T u and T d are the temperature in the adjacent uptream and downtream control volume The third term in Eq (9) repreent the energy that i carried with the axial pecie diffuive fluxe acro the control urface in the flow direction Evaluating thee term depend upon the direction of the pecie fluxe a j u z,k h k(t u ), j u z,k > 0 q u diff = (12) j u z,k h k(t), j u z,k < 0 If the uptream diffuion flux i into the control volume (ie, j u > 0), then the pecie enthalpy i evaluated at the temperature of the uptream control volume [ie, h k (T u )] If the uptream z,k diffuion flux i out of the control volume (ie, j u < 0), then the z,k pecie enthalpy i evaluated at the temperature of the target control volume [ie, h k (T)] An analogou evaluation pertain to the (9)

5 200 AM Colclaure et al / Journal of Power Source 196 (2011) downtream diffuion flux, q d diff = d z,k h k(t) + k,j d z,k >0 j d j k,j d z,k <0 z,k h k(t d ) (13) The fourth term in Eq (9) repreent the thermal energy that i tranferred between the porou anode and the fuel flow a a reult of ma exchange That i, j r,k h k (T), j r,k > 0 q a diff = (14) j r,k h k (T m ), j r,k < 0 Becaue when j r,k > 0 pecie k i tranported from the fuel flow into the anode tructure, the pecie enthalpy i evaluated at the fuel-flow temperature [ie, h k (T)] When j r,k < 0 pecie k i tranported from the anode tructure into the fuel flow and pecie enthalpy i evaluated at the membrane-electrode aembly (MEA) temperature [ie, h k (T m )] The fifth term in Eq (9) repreent the thermal energy that i tranferred by fluid convection between the anode urface and the fuel flow Thi term i evaluated uing Newton law of cooling, where the heat-tranfer coefficient h q i evaluated from a Nuelt-number correlation a h q = Nu (15) 2r f For the laminar flow that are typical in mall tubular fuel cell, the approximate Nuelt number i Nu = 366 After ubtituting the ma-continuity equation and ome algebraic manipulation, the fuel-flow energy equation can be written a a differential equation for the ga temperature a c p T t = ṁ V (h h) + Ṁ V h ( ) Af,c + q u cond qd ( cond V ) Af,c + q u diff qd diff V q a P f diff K 1 V ız + h q (T m T) P f V ız (h k h K ) Y k t (16) Becaue all pecie ma fraction are not independent, plitting the time derivative of pecific energy into temperature and compoition contribution caue the pecific enthalpy h K of pecie K to appear in the lat term 24 Anode bi-layer model Fig 5 illutrate the tructure and notation for ma balance within the porou compoite anode tructure The model divide the anode tructure into two layer, the upport layer and the functional layer Both layer are typically the ame compoition (eg, Ni-YSZ) The upport layer i relatively thick (eg, m) and ha relatively open poroity There can be coniderable catalytic reforming chemitry within the upport layer, but eentially no charge-tranfer chemitry The functional layer i relatively thin (eg, m), with large three-phae-boundary (TPB) length and relatively mall particle and pore pace The primary role of the functional layer i to facilitate charge-tranfer chemitry near the dene electrolyte The radiu of the interface between the upport and functional layer i r and the radiu of the interface between the functional Fig 5 Control volume annotated for the bi-layer compoite anode tructure layer and the dene electrolyte i r e The average ga-phae compoition within the pore pace of the upport and functional layer are repreented by ma fraction a Y,k and Y f,k, repectively Poroity of the anode upport layer i repreented a and the pecific catalyt urface area (ie, effective are per unit volume) i repreented a a Radial ma fluxe of ga-phae pecie at the interface between the upport and fuel channel are repreented a j r,k, radial ga-phae fluxe at the interface between the upport and functional layer are repreented a j, and radial ga-phae r,k fluxe at the interface between the functional layer and the dene electrolyte are repreented a j e Axial tranport within the anode r,k tructure i negligible becaue the radial gradient are ignificantly greater The pecie ma-continuity equation within the anode upport layer are written a ( t ( Y,k ) = a ṡ k + P f j A r,k P ) j W A r,k k, (17) where ṡ k are the molar production rate of pecie via heterogeneou catalytic chemitry The average ga-phae denity within the pore of the anode upport i The perimeter P f = 2r f i the perimeter between the fuel channel and the upport layer The perimeter P = 2r i the perimeter between the upport layer and the functional layer The axial cro-ectional area of the anode upport i A = (r 2 r2 ) The overall ma continuity equation for the f upport layer, which i found by umming the pecie continuity equation over all pecie, i written a t ( ) = ( Pf j A r,k P ) j W f A r,k k (18) Thi continuity equation i formed auming no urface pecie are involved in the heterogeneou reaction The pecie ma-continuity equation within the anode functional layer are analogou to thoe for the upport layer That i, ( t ( f f Y f,k ) = a f ṡ k + P j A r,k P ) e j e W f A r,k k (19) f The pecific urface area a f and poroity f of the functional layer may be different from that in the upport layer The perimeter at the dene-electrolyte interface i P e = 2r e and the axial croectional area i A f = (re 2 r2 ) The overall continuity equation for

6 AM Colclaure et al / Journal of Power Source 196 (2011) gae within the functional layer i ( t ( P f f ) = j A r,k P ) e j e W f A r,k k (20) f The heterogeneou reforming chemitry i baed upon global reaction for team reforming of methane, water-ga-hift (WGS), and partial oxidation of methane CH 4 + H 2 O CO + 3H 2, (21) CO + H 2 O CO 2 + H 2, (22) CH O 2 CO + 2H 2 (23) The rate of thee global reaction were approximated by fitting to experimental data for methane reforming a meaured in a eparated anode experiment [8] Three radial ma fluxe mut be evaluated within the anode tructure The flux between the upport and the fuel flow j r,k and the flux between the functional layer and the upport layer j r,k are evaluated uing the duty-ga model (DGM) [9,2] The DGM provide an implicit relationhip among the ga-phae pecie molar fluxe through the porou matrix J k, molar concentration [X k ], concentration gradient, and the preure p gradient a, [X l ]J k [X k ]J l [X T ]D e kl + J k D e k,kn l /= k = [X k ] [X k] B g D e p, k,kn (24) where [X T ] = p/rt i the total molar concentration, B g i the permeability, and i the mixture vicoity The ma fluxe j k are related imply to the molar fluxe J k a j k = W k J k Knuden diffuion repreent ma tranport aited by ga-wall colliion The Knuden diffuion coefficient depend upon the porou-media microtructure, including poroity, average pore radiu r p, and tortuoity g The effective binary and Knuden diffuion coefficient D e kl and De can be evaluated a k,kn D e kl = g D kl, D e k,kn = 2 r p g g 3 g 8RT W k (25) The binary diffuion coefficient D kl and the mixture vicoitie are determined from kinetic theory [7] The permeability can be evaluated from the Kozeny Carman relationhip a g 3 B g = d2 p 72 g (1 g ) 2, (26) where d p i the particle diameter Further detail of the DGM and it numerical implementation can be found in Zhu, et al [2] The two fluxe j r,k and j are evaluated baed on the tranport propertie r,k of the anode upport The ma fluxe at the interface between the anode functional layer and the dene electrolyte j e are determined by the chargetranfer r,k chemitry j e r,k = ki cell n e F W k, (27) where i cell i the current denity and F i Faraday contant The variable k and n e are the toichiometric coefficient for pecie k and number of electron tranferred in the overall charge tranfer reaction 25 MEA energy balance The entire thickne of the MEA i repreented by a ingle temperature T m, which varie temporally and axially A illutrated in Fig 6 Control volume annotated for the thermal balance in the MEA, which i the entire tube wall Fig 6, heat enter and exit the MEA through conduction, ma tranport, and convection The energy balance for the MEA i written a ( m c p,m T m ) ( ) 1 (1 m ) = q u mea t qd mea ız + P f A f, j r,k h k P e A f, j c r,o 2 h O2 (28) + P f h q (T T m ) P e h q,c (T m T a ) A f, A f, P e E A cell i cell, f, where A f, i the combined cro ectional area of the anode upport and functional layer The firt term on the right-hand ide repreent axial thermal conduction through the olid material that comprie the MEA The uptream and downtream conduction term are evaluated a q u mea = T m Tm u m, q d mea ız = Tm d T m m, (29) ız where Tm u and T m d are the MEA temperature in the adjacent uptream and downtream control volume and m i the effective thermal conductivity of the MEA The MEA poroity m i an area average of anode-upport and anode-functional layer poroity The econd term on the right-hand ide of Eq (28) repreent the energy that i tranported via pecie flux from the fuel tream to the anode upport layer The third term repreent the energy that i tranported via pecie flux from the cathode to the exterior A dicued in the context of the fuel-flow energy balance (eg, Eq (14)), the temperature at which the enthalpy i evaluated depend upon the direction of the pecie flux In a imilar way, energy exchange between the cathode and the exterior i evaluated a j c r,k h k(t m ), j c r,k > 0 q c diff = (30) j c z,k h k(t a ), j c r,k < 0 Becaue when j c > 0 pecie k i tranported from the cathode r,k into the urrounding air tream, the pecie enthalpy i evaluated

7 202 AM Colclaure et al / Journal of Power Source 196 (2011) at the MEA temperature [ie, h k (T m )] When j c < 0 pecie k i r,k tranported from the urrounding air tream and pecie enthalpy i evaluated at the air tream temperature [ie, h k (T a )] The fourth and fifth term in Eq (28) repreent convective heat tranfer between the MEA and adjacent ga flow The heat-tranfer coefficient on the exterior cathode ide h q,c i an empirical parameter The final term in Eq (28) repreent the electrical energy produced by the cell, which doe not contribute to the thermal energy balance Auming that the dene electrolyte and cathode are very thin, the cathode perimeter i P c = P e = 2r e 26 Cathode air conervation equation The model i baed upon an aumption that the cathode air that circulate around the outide of the tube tack behave a a perfectly tirred reactor That i, the air temperature and compoition may vary temporally, but are uniform patially The rate at which oxygen i tranferred from the circulating air to the SOFC cathode depend upon the cell operating condition Heat i exchanged between the tube and the air by convection and by the energy aociated with the oxygen ma tranfer The pecie continuity equation for the cathode air i written a d(y a,k ) V a = ṁ a dt Y a,k ṁ ay a,k, k /= O 2, (31) where ṁ a i the ma flow rate of the inlet air and Y are the inlet a,k pecie ma fraction (uually only O 2 and N 2 ) The mixture within the volume V a of cathode pace and the exhaut flow are aumed to have the ame compoition, Y a,k The oxygen ma fraction i determined from Y a,o2 = 1 Y K k /= O 2 a,k The overall ma-continuity equation i d V a dt = ṁ a ṁ a Ṁ a, (32) where Ṁ a i the oxygen ma-conumption rate by the SOFC tack The cathode-air energy equation i d(e) V a = ṁ a dt h a ṁ ah a UA a (T a T hell ) L +P e h q,c (T m T a )dz (33) 0 L +P e j r,o2 h O2 dz 0 The firt two term on the right-hand ide repreent the energy aociated with the air flow into and out of the cathode-air compartment The third term, which repreent heat tranfer from the cathode air to the urrounding encloure, ue an overall heat tranfer coefficient UA a The fourth term i an integral that repreent the convective heat tranfer along the length L of an SOFC tube to the urrounding air The final term repreent the energy aociated with oxygen ma exchange between the SOFC cathode and the urrounding air 27 Electrochemitry The firt tep in modeling the electrochemitry i to determine the cell voltage E cell The cell voltage i equal to the reverible voltage minu the um of variou overpotential E cell = E rev act,a + act,c ohm, (34) where E rev i the reverible voltage (cell voltage when no net current i produced) The ohmic overpotential ohm i mainly from ion conduction through the electrolyte There i an activation overpotential to drive charge-tranfer chemitry at the cathode act,c and the anode act,a Becaue each of the overpotential depend upon current denity, Eq (34) can be ued to determine the local current denity for each dicrete control volume along the length of the SOFC tube Becaue the model determine the compoition in the anode functional layer, there i no need to include concentration overpotential The reverible voltage i calculated baed on the global chargetranfer reaction at the anode and cathode ide being in local equilibrium On the cathode, oxygen from the air tream i reduced by reaction with electron, producing an oxygen ion in the electrolyte a 1 2 O 2(g) + 2e (c) O 2 (el) (35) The oxygen ion i tranported acro the dene electrolyte toward the anode functional layer Within the anode functional layer, hydrogen from the fuel tream react with the oxygen ion and deliver electron into the anode phae a H 2 (g) + O 2 (el) H 2 O(g) + 2e (a) (36) The overall reverible potential can be calculated by etting the electrochemical energy change for reaction (35) and (36) to zero The reult i E rev = Go 2F + RT m 2F ln ( ) p f,h2 p 1/2 a,o 2, (37) p f,h2 O G = G f,h 2 O ( G f,h G a,o 2 ), (38) where p k,f and p k,a are the partial preure (evaluated in atm) of pecie k in the anode functional layer and air tream, repectively The tandard tate Gibb free energy change G i only a function of temperature and not concentration Thu, the firt term in Eq (37) account for temperature effect The lat term account for compoition and preure effect Butler Volmer equation are ued to determine the local current denity i at the anode and cathode ide for a given activation overpotential i = i 0 (exp ( af ) act exp RT m ( cf )) act (39) RT m The exchange current denity i a function of the local concentration and temperature at the TPB The global anodic and cathodic ymmetry factor a and c do not necearily um to one For a given charge-tranfer reaction mechanim, the functional form of the exchange current denity and global ymmetry factor are derived by auming one elementary reaction i rate limiting and that all other reaction are in partial equilibrium The aumed hydrogen oxidation mechanim and rate limiting tep are taken from Zhu et al [2] By auming the ymmetry factor for the elementary rate limiting tep are both a = c = 05, the functional form of the exchange current denity for the electrochemical oxidation of hydrogen i i 0 = i H 2 (p f,h2 /p H 2 ) 1/4 (pf,h2 O) 3/4 1 + (p f,h2 /p H 2 ) 1/2, (40) where i and p are only function of temperature Alo, the H 2 H 2 global ymmetry factor for hydrogen oxidation are a = 15 and c = 05 For the cathode, the mechanim for the electrochemical reduction of oxygen i alo taken from Zhu et al [2] Baed on the aumed elementary rate limiting tep having ymmetry factor equal to 05, the exchange current denity for the cathode ide i (p i 0,c = i a,o2 /p ) 1/4 O 2 O (41) (p a,o2 /p 1/2 ) O 2

8 AM Colclaure et al / Journal of Power Source 196 (2011) The global ymmetry factor for the electrochemical reduction of oxygen are a = 05 and c = 05 The nominal exchange current denitie i depend on temperature in the following way i e = i ref,e exp ( E e R ( 1 T 1 T ref )) (42) where e repreent the electrochemical oxidation of oxygen or hydrogen The parameter i i the nominal exchange current ref,e denity at the reference temperature T ref It i important to note that all partial preure in Eq (40) and (41) mut be evaluated in atmophere 3 Implementation Spatial derivative are evaluated uing the finite volume method The reulting governing equation from a et of differential-algebraic equation (DAE) [10] The model i implemented in C++ and linked to Sundial[11] and Cantera[12] The dicretized equation are olved uing the DAE olver IDA contained within Sundial Baed on meh tudie, an axial meh of 100 point accurately capture patial variation along the tube For each axial meh point, there i an anode upport layer, anode functional layer, and fuel channel control volume The chemically reacting flow oftware Cantera i utilized to calculate fluxe within the anode according to the DGM Cantera i alo ued to evaluate thermodynamic propertie, tranport propertie, and net rate of production from thermal chemitry 4 Example model reult A pecific anode-upported SOFC tube i ued to illutrate the model The 15 cm long tube with an outide diameter of 1 cm, i typical of a tube that may be ued in a ub-kilowatt tubular tack Table 1 lit other model parameter, decribing phyical and electrochemical characteritic Mot of thee parameter are Table 1 Model parameter for a ingle tube Many of the parameter are from Zhu et al [4] Parameter Value Thermal propertie MEA thermal conductivity, m (W m 1 K 1 ) 105 MEA poroity, m 035 MEA heat capacity, c p,m (J kg 1 K 1 ) 533 MEA denity, m (kg m 3 ) 7000 Nominal heat tranfer coefficient, h q,c (W m 2 K 1 ) 120 Nominal overall heat tranfer coefficient, UA a (W K 1 ) 085 Nuelt number, Nu D = h qp f /() 366 taken from Zhu et al [4] The effective thermal conductivity of the compoite MEA (ie, tube wall) i approximated a m = 105 W m 1 K 1 [13] The nominal exchange current denitie i = 207 A H 2 cm 2 and i = 068 A cm 2 were adjuted uch that model predicted power denitie match thoe of typical mall cale ytem O 2 (around 03 W cm 2 for fuel utilization of around 85%) For purpoe of illutrating the model, the inlet fuel compoition i 38% H 2,3%H 2 O, 1% CH 4, 19% CO, 03% CO 2, and 38% N 2 Thi i the equilibrium mixture for methane and air at 800 C with a tochiometric ratio to partial oxidation (ie, oxygen to carbon ratio of 05) A mall amount of team (3%) ha been added to the equilbrium mixture Fuel enter the tube at 800 C and atmopheric preure The cathode air, which flow around the outide of the tube, enter the tack at 550 C Both end of the tube conduct heat to manifold at a fixed temperature of 800 C 41 Steady-tate reult Fig 7 illutrate teady-tate patial profile for compoition, current denity, temperature and fuel velocity with the cell operating at E cell = 072 V The fuel inlet velocity i 38 cm 1, and the air flow rate (per tube) i 335 mg 1 The total current and power produced by a ingle tube are 231 A and 166 W, repectively, which correpond to current and power denitie of 049 A cm 2 and 035 Wcm 2 Under thee condition, the cell achieve an efficiency of 499% and fuel utilization of 899% The fuel compoition profile (Fig 7) are primarily driven by the electrochemical conumption of hydrogen The ratio of hydrogen to team decreae along the length of the cell due to current production The heterogeneou WGS proce (Eq (22)) remain near equilibrium Thu, carbon monoxide and team react rapidly, replenihing the electrochemically conumed hydrogen Overall, thee procee conume hydrogen and carbon monoxide, producing team and carbon dioxide The mall amount of methane (1%) entering the tube i quickly reformed by team (Eq (21)) to produce CO and H 2 Becaue thi reaction increae mole, the nitrogen mole fraction decreae lightly The methane team reforming alo produce a light maximum in the velocity The peak in the temperature profile i the reult of internal heat generation and fixing the end temperature Fig 7 illutrate ome intereting effect of temperature and fuel compoition on local current denity Near the tube inlet, the cur- Anode upport layer propertie Thickne, L (m) 950 Poroity, 035 Tortuoity, 35 Pore radiu, r p (m) 05 Mean particle diameter, d p (m) 25 Specific catalyt area, a (m 1 ) 108e5 Anode functional layer propertie Thickne, L f (m) 50 Poroity, f 035 Specific catalyt area, a f (m 1 ) 208e5 Nominal exchange current denity, i (A m 2 ) H Activation energy, E H2 (kj mol 1 ) 120 Reference temperature, T ref ( C) 800 Electrolyte ohmic reitance R = R ot exp(e ion /(RT)) Activation energy, E ion (kj mol 1 ) 80 Reitance prefactor, R o (m 2 K 1 ) 555e-13 Cathode propertie Nominal exchange current denity, i (A m 2 ) O Activation energy, E O2 (kj mol 1 ) 130 Reference temperature, T ref ( C) 800 Fig 7 Steady-tate profile for temperature, fuel channel compoition, current denity, and fuel velocity along the length of the cell The cell voltage, fuel inlet velocity, and air ma flow rate are 072 V, 38 cm 1 and 335 mg 1, repectively

9 204 AM Colclaure et al / Journal of Power Source 196 (2011) rent denity increae The current denity reache a maximum of 074 A cm 2 about 2 cm downtream of the inlet Following the maximum, the current denity decreae monotonically to around 017 A cm 2 at the fuel outlet The reverible potential decreae along the tube a hydrogen i electrochemically conumed and team i produced, tending to reduce the local current denity However, the current denity alo depend upon the local MEA temperature, which i maximum at around 35 cm from the tube entrance Thee competing effect lead to the maximum current denity occurring between the fuel inlet and maximum MEA temperature 42 Tranient imulation The model i deigned to run imulation with time-varying input For intance, the cell voltage, fuel inlet velocity, and air flow rate can be adjuted to meet varying power demand Conequently, the patial profile (compoition, velocity, current denity, etc) are tranient However, the enor are aumed to be poitioned to meaure tube-exhaut condition Thu, the controller i more concerned with the temporal variation of the outlet flow, average MEA temperature, and total current production than the patial variation within the tube Conider the tranient behavior that reult in going from a low current demand to a high current demand One econd into the imulation, the current demand goe from 145 A to 30 A At the tart of the imulation, the cell i operating in teady tate with a cell voltage of E cell = 078 V, producing 113 W The inlet fuel velocity i 25 cm 1 and air flow rate i 335 mg 1 Under thee condition, the cell i operating at 523% efficiency and a fuel utilization of 872% Fig 8 and 9 illutrate tranient repone on hort (econd) and long (minute) time cale, repectively One econd after the tart of the imulation, the current demand i increaed from 145 A to 30 A In an attempt to achieve the new current demand, the cell voltage i reduced from E cell = 078 V to E cell = 069 V Lowering the cell voltage further could permanently degrade the tube The voltage drop caue a harp current increae from 145 A to around 25 A The electrochemical charge-tranfer rate repond intantly to the change in cell voltage However, the uddenly increaed current production increae fuel conumption, cauing a ubequent decreae in current (Fig 8) Thu, the new current demand of 30 A i not met with only a change in cell voltage Alo, the fuel utilization approache 100% (ie, very low H 2 in the exhaut), which i an undeirable condition To meet the current demand of 30 A and offet fuel depletion, the fuel inlet velocity i increaed from 247 cm 1 to 514 cm 1 over a half-econd ramp beginning at 12 The increaed fuel flow caue the current to increae to around 28 A and the H 2 in the exhaut increae to around 7% by around 2 Baed upon the characteritic flow and diffuion time, the repone time for rapid change in operating condition i around 1 Beginning at 2 after the tart of imulation, the current increae on a hallow linear ramp toward and above the deired value of 30 A With the voltage and inlet velocity held fixed, the fuel conumption increae leading to a decreae in the H 2 exhaut mole fraction (Fig 8) During the firt 10, the tube temperature, fuel-exhaut temperature, and cathode air temperature all increae lightly The temperature rie i the reult of increaing polarization loe a the cell power output increae Fig 9 how cell behavior on a longer time cale extending to 160 After 60, the current continue to rie above the demanded value of 30 A due to a continued rie in temperature The increaed current and increaed temperature caue increaed fuel conumption If the tube temperature become ufficiently high for a long period of time, then the tube might be permanently damaged To Fig 8 Total current, exhaut H 2 mole fraction, and temperature a function of time, following a tranient from a relatively low power to high power demand The change i effected by reducing the cell voltage from 078 V to 069 V one econd after the tart of imulation The velocity of the fuel inlet i increaed from 247 cm 1 to 514 cm 1 over a half-econd ramp beginning at 12 after the tart of imulation The repone on the hort time cale are characterized by the fluid-flow dynamic that are on the order of econd tabilize the current production at 30 A and lower the average tube temperature, at around 80 the cathode air flow rate i increaed The cooling effect of the cathode air erve to decreae tube and flow temperature, although on relatively long time cale The average tube temperature decreae and tabilize at 830 C The tube current tabilize at 299 A A the temperature decreae, the fuel-conumption rate decreae (H 2 in exhaut increae) and the total current decreae lightly At the end of imulation (160 ), the cell i operating with an efficiency of 459% with a fuel utilization of 859% Thi imulation illutrate the need for control of fuel cell ytem to quickly meet varying current demand and not violate contraint The deired higher current demand wa only met and tabilized after 100, which i not acceptable Alo, the current wa tabilized at a value 01 A below the deired value of 30 A After the cell voltage decreae, the fuel outlet became almot completely tarved of hydrogen and carbon monoxide Thi could lead to damage of the tube Often the fuel channel outlet tream i combuted in a tail ga combutor to provide required heat for the fuel cell ytem The lean outlet condition experienced during the imulation could caue the flame in tail ga combutor to extinguih 5 Linear identification of the SOFC tack The phyical model contain the coupled effect of fuel channel flow, porou-media tranport, heat tranfer, reforming chemitry, and electrochemitry Although already implified compared to even-more-complex phyical model, the required computational reource are till too great for direct incorporation into an MPC implementation [14] Although the phyical model i highly non-

10 AM Colclaure et al / Journal of Power Source 196 (2011) into a ingle multiple-input multiple-output (MIMO) ytem, and reduced uing balanced model-reduction method [15], reulting in a ingle (MIMO) reduced-order linear model at a particular OP A explained briefly in the following ection, the ubpace ytem identification i ued to capture the mall-ignal model at a particular OP 51 Subpace ytem identification The approach to fit a low-order model to data (here, output from the phyical model) i baed upon ubpace-identification method Although thi proce i well known and documented (eg, [16]), it i mathematically complex Only a brief ummary i provided here An input output time equence (ıu k,ıy k ), (k = 1,,N) obtained and recorded by exerciing the phyical model at a ampling rate T The objective i to find a linear time-invariant (LTI) ytem in tate-pace form, { x k+1 = x k + ıu k ıy k = Cx k + Dıu k, (43) Fig 9 Total current, exhaut H 2 mole fraction, and temperature with repect to time in going from a low power demand to a high power demand The plot capture tube condition change due to thermal dynamic which are on the order of hundred of econd The reult hown for the firt 10 of imulation are exactly the ame a thoe illutrated in Fig 8 The voltage i dropped from 078 V to 069 V one econd after the tart of imulation The velocity of the fuel inlet i increaed from 247 cm 1 to 514 cm 1 over half a econd tarting at time equal to 12 Over the coure of half a econd, the air flow i raied from 335 mg 1 to 435 mg 1 tarting 80 into the imulation linear, it i poible to extract low-order locally linear model that capture the dominant dynamic of the high-order phyical model at particular OP The companion paper [1] extend the method to nonlinear ytem identification over a range of OP Model reduction i accomplihed uing a data-baed approach that i baed on the ubpace cla of ytem identification method In thi approach, the phyical model take the role of an experiment, albeit one without meaurement noie Thi approach, which i an alternative to the direct mathematical linearization of the phyical model, ha everal advantage Firt, there i no need to for the phyical model to be repreented a ordinary differential equation (ODE) in tandard form [ie, y = f (t, y)] The phyical model ued here i expreed a DAE, not ODE Second, a data-baed model provide the bet linear approximation of the nonlinear ytem a meaured over the amplitude and frequency range of the experimental input, wherea a linearized model ha no correponding approximation qualitie The linear identification of the (SOFC) tack proceed a follow Firt, a nominal OP for the tack i elected, (ū, ȳ), where ū are the nominal input and ȳ are the nominal output of the tack A mall-ignal equence ıu i deigned with frequency content that matche the expected ytem bandwidth One input i perturbed around it OP, while fixing other input at their nominal value The mall-ignal repone ıy = y ȳ i recorded Thi procedure i repeated for all ytem input From thi data, a ingle-input ingle-output (SISO) reduced-order model of each input output pair i identified Thee SISO model are combined that can cloely reproduce the data The tate-pace vector i x k R n and all other vector/matrice are aumed to have compatible dimenion The LTI model (Eq (43)) expree the relationhip among x k+1, x k, u k, and y k over a ingle ampling interval The ubpace identification proceed by collecting together larger ampling interval (window) of the input output data The window length i a free parameter that hould be choen to be larger than the expected order of the identified ytem, n Conidering an -length egment of data, Eq (43) can be rewritten a ıy k C D 0 0 ıu k ıy k+1 C = x C D ıu k+1 k + (44) 0 ıy k+ C C 1 C D ıu k+ Becaue the model to be identified i time-invariant (ie, the,, C, and D matrice are time-independent), any hifted window of the data can be expreed a Eq (44) The hifted-window input output data can be expreed in matrix form a Y = X + H U, (45) where ıy k ıy k+1 ıy k+n ıy k+1 ıy k+2 ıy k+1+n Y =, (46) ıy k+ ıy k++1 ıy k++n ıu k ıu k+1 ıu k+n ıu k+1 ıu k+2 ıu k+1+n U =, (47) ıu k+ ıu k++1 ıu k++n X = [ x k x k+1 x k+n ], (48) and D 0 0 C C D C H =, = (49) 0 C 1 C C D

11 206 AM Colclaure et al / Journal of Power Source 196 (2011) The ingular value decompoition (SVD) of Y U T can be repreented a Y U T = S V T, (52) where S and V are orthonormal matrice and i a diagonal matrix containing the ingular value Thee matrice can be further partitioned in two part a [ ] S V T = [S 1 S 2 ] 1 0 [V 0 1 V 2 ] T, (53) 2 where 1 contain the firt n dominant ingular value Under appropriate condition (ie, peritently exciting input), the range pace of S 1 i the ame a Thu, an etimate for i taken to be the n firt column of S ( i defined only within a coordinate tranformation of x k ) With known, the matrix C (Eq (43))ithe firt p row of, where p i the number of ytem output The matrix can be determined through the hift pattern of With and C in hand, and D, which are linear in Eq (45), can be found via olving a linear leat quare problem Fig 10 Small-ignal perturbation with fuel flow rate changing with ampling time of T = 0125 () The matrix i called the extended obervability matrix where >nguarantie be full column rank (obervability condition) Particular ytem parameter are contained in H and, and can be extracted when thee term are known Following the ubpace ytem-identification approach, etimate of,, C, and D matrice can be evaluated [17] Thee tep are briefly explained here The matrice Y and U (Eq (45)) are known from the meaurement obtained by exerciing the phyical model with mall-ignal input ıu and recording the output ıy The H U term can be removed by multiplying Eq (45) from right by U T = I U T (U U T ) 1 U, (50) where U T i the orthogonal complement of U A a reult, Eq (45) become Y U T = X U T (51) 52 Illutration of linear ytem identification Thi ection illutrate the reult of linear identification for the (SOFC) tack at a particular OP The (SOFC) tack variable are partitioned into three input and four output The input variable are cell voltage, fuel ma flow rate, and air ma flow rate The output variable are cell current, hydrogen concentration in exhaut, average MEA temperature, and cathode-exhaut air temperature A decribed earlier, the phyical model repreent a widely diparate range of characteritic time cale Electrical repone i eentially intantaneou (ie, becaue electrochemical double-layer charging i neglected, a change in operating voltage caue an intantaneou change in current) Characteritic repone time for fluid flow and diffuion i of the order of one econd or le The characteritic time for thermal repone i much longer, on the order of minute For the purpoe of illutrating the ytem identification, air flow rate and cell temperature are conidered contant Thu, the illutration here focue upon identifying cell current and hydrogen concentration from cell voltage and fuel flow rate In the companion paper [1], a ytem i identified for the full range of time cale and an MPC controller i deigned Aume a nominal OP for a tack that i operating at a voltage of E cell = 074 V, fuel ma flow rate of 52 mg 1, and air ma flow rate of 335 mg 1 Under thee condition, the phyical model Fig 11 Comparion between low-order and high-order model for validation imulation

12 AM Colclaure et al / Journal of Power Source 196 (2011) (ingle tube) predict a net current of 2496 A with 1024% H 2 mole fraction in fuel outlet The tack i excited by a perturbation equence that i uitable for proce model identification In thi example, a peudo random binary equence (PRBS) with zero mean value i ued a the perturbation equence [18,19] A equence of 500 ample i ued, with each input perturbed eparately Fig 10 how a egment of a imulation at the elected OP In thi perturbation, the fuel flow rate i perturbed with a ampling time of T = 0125, while the other input are held contant at their nominal operating value Fig 10a how the PRBS that i applied a the fuel flow variation Fig 10b how the correponding change in the H 2 mole fraction in fuel exhaut The ubpace identification at thi OP reult in a 12th-order mall-ignal model To validate the mall-ignal model, an input equence i deigned in which all the input vary imultaneouly The identified mall-ignal model hould be able to predict the output of thi imulation Fig 11a and b how the input variation Fig 11c how the predicted cell current, comparing the high-order phyical model and the identified low-order linear model In thi example, the two curve are nearly inditinguihable Fig 11d how another validation reult comparing the hydrogen mole fraction in the anode exhaut Again, the comparion i excellent 6 Concluion A phyically baed, tranient model for a tubular anodeupported (SOFC) i developed a the bai for implementing an MPC controller Conidering the coupled interaction of fuel flow, porou-media tranport, heat tranfer, reforming chemitry, and electrochemitry, the model i implemented with error-controlled DAE oftware that enable the accurate prediction of accurate tranient repone Accounting for widely diparate time cale, the model reolve patial and temporal profile of compoition, temperature, current denity, and velocity throughout the cell Even with approximation to reduce the complexity of the phyical model, it i till to large to be incorporated directly into an MPC implementation Thu, linear model reduction i ued to reduce the high-order phyical model to a low-order, locally linear model An example i ued to illutrate the validity of the reduced model at a particular OP A dicued in a companion paper [1], the reducedorder model form the bai for implementing an MPC controller Reference [1] BM Sanandaji, TL Vincent, AM Colclaure, RJ Kee, J Power Source, ubmitted for publication [2] H Zhu, RJ Kee, VM Janardhanan, O Deutchmann, DG Goodwin, J Electrochem Soc 152 (2005) A2427 A2440 [3] H Zhu, RJ Kee, J Electrochem Soc 153 (2006) A1765 A1772 [4] H Zhu, RJ Kee, J Power Source 169 (2007) [5] H Zhu, RJ Kee, J Electrochem Soc 155 (2008) B715 B729 [6] DG Goodwin, H Zhu, AM Colclaure, RJ Kee, J Electrochem Soc 156 (2009) B1004 B1021 [7] RJ Kee, ME Coltrin, P Glarborg, Chemically Reacting Flow: Theory and Practice, Wiley-Intercience, 2003 [8] ES Hecht, GK Gupta, H Zhu, AM Dean, RJ Kee, L Maier, O Deutchmann, Appl Catal, A 295 (2005) [9] EA Maon, AP Malinauka, Ga Tranport in Porou Media: The Duty-Ga Model, American Elevier, New York, 1893 [10] UM Acher, LR Petzold, Computer Method for Ordinary Differential Equation and Differential-Algebraic Equation, SIAM, Philadelphia, PA, 1998 [11] AC Hindmarh, PN Brown, KE Grant, SL Lee, R Serban, DE Shumaker, CS Woodward, ACM Tranaction on Mathematical Software (TOMS) 31 (3) (2005) 396 [12] DG Goodwin An open-ource, extenible oftware uite for CVD proce imulation, in: M Allendorf, F Maury, F Teyandier, (Ed), Chemical Vapor Depoition XVI and EUROCVD 14, volume PV , page Electrochemical Society, 2003 ee alo [13] T Nihino, H Iwai, K Suzuki, J Fuel Cell Sci Technol 3 (2006) [14] TL Vincent, BM Sanandaji, AM Colclaure, H Zhu, RJ Kee, ECS Tran 25 (2009) [15] B Moore, IEEE Tranaction on Automatic Control 26 (1) (1981) [16] L Ljung, Sytem Identification: Theory for the Uer, 2nd edition, Prentice-Hall, 1999 [17] BM Sanandaji, TL Vincent, AM Colclaure, RJ Kee, In Proceeding of ASME Dynamic Sytem and Control Conference, 2009 [18] T Sodertrom, P Stoica, Sytem Identification, Prentice-Hall, 1988 [19] Y Zhu, Multivariable Sytem Identification for Proce Control, Elevier Science, 2001