Infinite-Dimensional Boundary Observer for Lithium-Ion Battery State Estimation

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1 Syddank Univeritet Infinite-Dimenional Boundary Oberver for Lithium-Ion Battery State Etimation Haan, Agu Imail; Jouffroy, Jerome Publihed in: Energy Procedia DOI: 0.06/j.egypro Publication date: 07 Document verion Publiher' PDF, alo known a Verion of record Document licene CC BY-NC-ND Citation for pulihed verion (APA: Haan, A., & Jouffroy, J. (07. Infinite-Dimenional Boundary Oberver for Lithium-Ion Battery State Etimation. Energy Procedia, 4, 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 providing detail, and we will remove acce to the work immediately and invetigate your claim. Download date: 0. jan.. 09

2 Available online at ScienceDirect Energy Procedia 4 ( th International Conference on Power and Energy Sytem Engineering, CPESE 07, 5-9 September 07, Berlin, Germany Infinite-Dimenional Boundary Oberver for Lithium-Ion Battery State Etimation Agu Haan* and Jerome Jouffroy Center for Unmanned Aircraft Sytem Mærk McKinney Møller Intitute, Univerity of Southern Denmark, 530 Odene, Denmark Abtract Thi paper preent boundary oberver deign for tate-of-charge (SOC etimation of lithium-ion batterie. The lithium-ion battery dynamic are governed by thermal-electrochemical principle, which mathematically modeled by partial differential equation (PDE. In general, the model i a reaction-diffuion equation with time-dependent coefficient. A Luenberger oberver i developed uing infinite-dimenional backtepping method and ue only a ingle meaurement at the boundary of the battery. The oberver gain are computed by olving the oberver kernel equation. A numerical example i performed to how the applicability of the deign. 07 The Author. Publihed by Elevier Ltd. Peer-review under reponibility of the cientific committee of the 4th International Conference on Power and Energy Sytem Engineering. Keyword: Ditributed parameter ytem; oberver deign; lithium-ion battery; battery management ytem.. Introduction Lithium-ion batterie are one the bet commercially available power ource for energy torage ytem for renewable ource, e.g., computer, electrified tranportation, and unmanned aerial vehicle (UAV. The reaon i partially becaue lithium-ion batterie offer the highet power and energy denity, which make them better than nickel cadmium or lead acid cell. However, their reliability i impacted by dicharge rate, temperature, and amount of uage. Furthermore, they have hort cycle live, ignificant degradation with age, and can become unafe if * Correponding author. Tel.: ; fax: addre: agha@mmmi.du.dk The Author. Publihed by Elevier Ltd. Peer-review under reponibility of the cientific committee of the 4th International Conference on Power and Energy Sytem Engineering. 0.06/j.egypro

3 Agu Haan et al. / Energy Procedia 4 ( overheated. A mart battery management ytem baed on tate-of-health (SOH and tate-of-charge (SOC can recognize thee factor, enable uer and operator the information they need for afer operation. Thi paper focue on lithium-ion battery SOC etimation. SOC i defined a the ratio of intantaneou remaining battery charge to it maximum capacity. Lithium-ion battery SOC etimation i a challenging problem ince the battery dynamic are governed by thermal-electrochemical principle, which mathematically modeled by partial differential equation (PDE. A thermal-electrochemical model conit of a et of PDE, which i a Single Particle Model (SPM to model the anode, cathode, and electrolyte dynamic, and an ordinary differential equation (ODE to model the average temperature of a lithium-ion battery. In general, the thermal-electrochemical model i a coupled of time-varying reaction-diffuion equation and an ODE... Literature review Control and etimation of PDE have gained much attention lately, due to the introduction of the backtepping technique []. The method ue an integral tranformation to tranform the ytem into a table target ytem. The method ha been uccefully ued to olve control and etimation problem of many type of PDE including peudo-parabolic equation [,3], KdV equation [4], and PDE-ODE cacade ytem [5]. More recently, the method ha been ued to deign a tabilizing controller for a reaction-diffuion equation with patially-varying coefficient [6]. In the petroleum indutry, the backtepping method ha found everal application, uch a in ga coning control [7], flow control in porou media [8,9], lugging control [0], lot circulation and kick control [,,3], and heave attenuation problem [4,5]. Another emerging application i in battery management ytem (BMS, where the backtepping method i ued to deign a Luenberger oberver for tate etimation of a lithium-ion battery [6,7]. In [8], a backtepping oberver i ued for tate etimation of a ingle particle model with electrolyte dynamic. In thi work, the diffuion coefficient are aumed contant. An improved model uing a thermal-electrochemical model i conidered in [9,0]. Here, the diffuion coefficient are conidered to be of an Arrheniu-like dependency on temperature... Contribution of thi paper The novelty of thi paper i a tate oberver deign for a reaction-diffuion equation with pace and time dependent coefficient uing a ingle boundary meaurement. Furthermore, we implement the boundary oberver deign for SOC etimation in lithium-ion batterie from a thermal-electrochemical model..3. Organization of thi paper In ection, the etimation problem for a reaction-diffuion equation i formulated. The main contribution of thi paper i preented in ection 3. Here, we deign a tate oberver for the reaction-diffuion equation with pace and time dependent uing the backtepping method with only a ingle boundary meaurement. In ection 4, an oberver deign for a thermal-electrochemical model ued in the lithium-ion battery SOC etimation i preented. A imulation i preented to demontrate the uefulne of the oberver deign. Finally, ection 5 contain concluion.. The Space and Time Dependent Reaction-Diffuion Equation We conider the following a pace and time dependent reaction-diffuion equation with mixed boundary condition ( rt, = D( rt, ( rt, + λ ( rt, c( rt, ( t c 0, = 0 c (, t = H( t c(, t + M( t (

4 496 Agu Haan et al. / Energy Procedia 4 ( where c( rt, i the ytem tate, r [ 0,] i the patial variable, and t [ 0, i the time variable. Ht ( M( t are aumed to be mooth function. The diffuion and reaction coefficient are denoted by D( rt, λ ( rt,, repectively. The objective i to deign a tate oberver to etimate the tate c( rt, boundary meaurement c(, t. and and uing only one 3. Infinite-Dimenional Boundary Oberver for State and Time Dependent Reaction-Diffuion Equation We deign the tate oberver for ( a the copy of the plant plu an output injection term a follow ˆ ( rt, = D( rt, ( rt, + λ ( rt, cˆ( rt, + p( rt, ( c(, t cˆ(, t ( t cˆ 0, = 0 ˆ cˆ (, t = H( t cˆ(, t + M( t + p ( t ( c(, t cˆ(, t 0 where p( rt, and p ( t are oberver gain to be determined later. If we define c ( rt, = c( rt, cˆ ( rt, the error ytem i given by 0 (, then ( rt, = D( rt, ( rt, + λ ( rt, c ( rt, p( rt, c (, t ( t c 0, = 0 c (, t = H( t c (, t + M( t p ( t c (, t We employ a Volterra integral tranformation 0 ( ( crt (, = wrt (, p rt,, w t, d r to tranform the error ytem (3 into the following target ytem (3 (4 w ( rt, = D( rt, ( rt, + µ w ( rt, ( t w 0, = 0 w (, t = w (, t w where the free parameter µ can be ued to et the deired rate of tability. The error ytem (5 i exponentially table in the ene of hould atify -norm. Subtituting (4 into (3 and conidering the target ytem (5, the kernel prt (,, (5

5 Agu Haan et al. / Energy Procedia 4 ( (,, ( t ( 0 (, t ( τ ( D( t, p( rt,, p p ( rt,, = D( rt, ( rt,, rt, p rt,, p rrt p 0,, = 0 r µ λ τ = D rt, D, t dτ Furthermore, the oberver gain are obtained a follow p( rt, = p( r,, t D(, t p t H t p t 0 ( = + ( (,, ( p( r,, t D(, t ( µ λ( ( (6 (7 4. Lithium-Ion Battery State Etimation from a Thermal-Electrochemical Model In thi ection, the boundary oberver deigned in the previou ection i ued to etimate the SOC of a lithiumion battery. To achieve accurate SOC etimation, we ue a coupled thermal-electrochemical model. The thermalelectrochemical model i derived from the Fick' law in pherical coordinate. 4.. A thermal-electrochemical model A ingle particle model (SPM with thermal dependency of a lithium-ion battery in each electrode i given by ( r, t = D ( T( t (, (, r t + r t + ( r, t = D ( T( t ( r, t + ( r, t c r c r with Neumann boundary condition ( 0, t 0, ( R, t I( t ( ( I( t ( ( = = r D T t Fa AL + + c + ( 0, t = 0, ( R, t = r D T t Fa AL ± Here c ( r, t ( denote the olid phae of lithium-ion concentration. The ( + ign i for poitive electrode and the ign i for negative electrode. The patial variable in pherical coordinate i denoted by r 0, R ±, while [ 0, ± ± t i the time variable. The parameter F, a, A, L denote the Faraday' contant, pecific interfacial urface area, cell cro ectional area, and electrode thicknee, repectively. The internal average temperature i denoted by T ( t and i calculated from the following equation (8 (9

6 498 Agu Haan et al. / Energy Procedia 4 ( dt c t h T t T t I t V t dt ( 0 = T ( 0 ( ( ( ( ( ( avg ρ = + p cell amb T amb (0 where ρ avg, c, h, T p cell amb denote the lumped denity, heat capacity, heat tranfer coefficient, and ambient ± temperature, repectively. The diffuion coefficient D ( T( t temperature ( ( = ( ( 0 D T t D T e A D T( t T( 0 T( t T( 0 T( t T( 0 A are modelled a an Arrheniu-like dependency on D ( ( 0 ( ( ( ( 0 T t T D T t = D T e where A D ± denote contant model parameter. The input for the model i the current I ( t voltage V( t. Remark that the tate for the two PDE (8 are uncoupled. ( and the output i the 4.. Boundary meaurement and model reduction The voltage output i derived from Butler-Volmer kinetic with a combination of electric over-potential and electrode thermodynamic, and i given by RT ( t I ( t RT ( t I( t V( t = inh inh αf a AL i c t αf a AL i c t where ( 0 ( ( ( ( 0 0 ( ( ( ( ( ( f U c t U c t + R I t c t, R, i, U ± ± denote lithium concentration at particle urface, univeral ga contant, exchange current + denity, and reaction potential, repectively. We aume c ( t = αc ( t + β uch that V( t f c ( t, I( t Thu, c ( t ( =. can be obtain from the meaured output voltage V ( t and input current I( t. Furthermore, the cathode dynamic i approximated by it equilibrium, i.e., c + ( rt, c + ( t =. A a reult, the electrochemical model can be reduced only for anode dynamic, i.e., only the firt equation of (8 with boundary condition only the firt equation of ( Feaibility of the deign r Let r = and c( r, t = rc ( r, t. Computing the derivative of c( r, t R with repect to r and t, the thermalelectrochemical model for the anode dynamic of the lithium-ion battery become

7 ( rt ( t ( T( t D c, =, c 0, = 0 ( R ( R ( rt ( ( ( I t (, t = c(, t D T t Fa AL Agu Haan et al. / Energy Procedia 4 ( (3 D ( T( t Thi ytem reemble ( with D( rt, =, λ ( rt, = 0, ( t ( R ( R I t H =, and M ( t =. D T t Fa AL ( ( Since the anode boundary concentration i available from the meaured output voltage and input current, the oberver i given by ˆ ( ( t ( T( t D cˆ rt, = rt, + p rt, c, t cˆ 0, = 0 ( R ( R ( ( ( ( ( ( ˆ I t (, t = cˆ (, t + p t c, t 0 D T t Fa AL where the oberver gain p ( rt and p ( 0 ( t ( T( t ( R ( R ( T( t, 0 t ( ( ( ( D p( rt, = p r,, t + p r,, t p 3 = D µ are obtained from ( (4 ( Simulation We demontrate the propoed infinite-dimenional oberver in a imulation tudy. Here, we apply the oberver for the thermal-electrochemical model (8-(. Some model parameter are taken from the genetic algorithm-baed parameter identification tudy performed on commercial lithium-iron phophate cell in []. The objective i to etimate the bulk anode SOC, which i defined a θ 3 t = r c r t dr c ( (, 0 (6

8 500 Agu Haan et al. / Energy Procedia 4 ( Figure. Input current (top and volume-ummed bulk (bottom in anode. To demontrate uncertainty in the initial condition, the plant and oberver tate are initialized at different value. Figure how the applied input current in term of C-rate and the bulk anode SOC. It can be een that the SOC i accurately etimated uing only boundary meaurement c. 5. Concluion Thi paper preent boundary oberver deign for a reaction-diffuion equation with pace and time dependent coefficient. The oberver i deigned uing only a boundary meaurement. The oberver deign i ued to etimate the SOC of a lithium-ion battery from a thermal-electrochemical model. It wa hown that the oberver deign etimate the bulk anode SOC accurately. Acknowledgement Thi work wa upported by Free the Drone (FreeD project. Reference [] M. Krtic and A. Smyhlyaev, Boundary Control of PDE: A Coure on Backtepping Deign, SIAM, Philadelphia; 008. [] A. Haan, O.M. Aamo, and B. Fo, Boundary Control for a Cla of Peudo-Parabolic Differential Equation, Sytem & Control Letter, vol. 6, pp , 03. [3] A. Haan, O.M. Aamo, and B. Fo, Global boundary feedback tabilization for a cla of peudo-parabolic partial differential equation, American Control Conference, Wahington DC, USA, 03. [4] A. Haan, Output-Feedback Stabilization of the Korteweg de-vrie Equation, Mediterranean Conference on Control and Automation, Athen, Greece, 06. [5] A. Haan, O.M. Aamo, and M. Krtic, Boundary Oberver Deign for Hyperbolic PDE-ODE Cacade Sytem, Automatica, vol. 68, pp , 06. [6] R. Vazquez and M. Krtic, Boundary Control of Coupled Reaction-Advection-Diffuion Sytem with Spatially-Varying Coeffcient, IEEE Tranaction on Automatic Control, vol. 6, pp , 07. [7] A. Haan, B. Fo, and S. Sagatun, Optimization of Oil Production under Ga Coning Condition, Journal of Petroleum Science and Engineering, vol. 05, pp. 6 33, 03. [8] J.L. Vazquez, The Porou Medium Equation, Clarendon Pre, Oxford; 006. [9] A. Haan, B. Fo, and S. Sagatun, Flow Control of Fluid Through Porou Media, Applied Mathematic and Computation, vol. 9, pp , 0.

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