A quick and efficient method for consistent initialization of battery models

Transcription

1 Electrhemistry Communications 9 (2007) A quick and efficient method f consistent initialization of battery models Vijayasekaran Boovaragavan, Venkat R. Subramanian * Department of Chemical Engineering, Tennessee Technological University, Cookeville, TN 38505, USA Received 2 March 2007; received in revised fm 6 April 2007; accepted 11 April 2007 Available online 21 April 2007 Abstract Secondary batteries are usually modeled as a system of coupled nonlinear partial differential equations. These models are typically solved by applying finite differences other discretization techniques in the spatial directions and solving the resulting system of differential algebraic equations (DAEs) numerically in time. These DAEs are very difficult to solve even using popular DAE solvers. The complications arise partly due to the difficulty in obtaining consistent, closely consistent, initial conditions f the DAEs. n this paper, a shooting method is proposed as an effective and rapid technique f the initialization of battery models. This method is built on a regionwise shooting approach with initial guess at one end of the electrode and physics based shooting criterion on the other end that can ultimately satisfy all the required conditions in a battery unit. Notably, the computation time required f the proposed method is only milliseconds in a FORTRAN environment f the case of initializing a standard physics based lithium-ion battery model. Also the initial values obtained are exact and can readily be fed into any DAE solver f achieving accurate solutions without solver failure. This rapid method will help in simulating batteries in hybrid environments in real-time (milliseconds). Ó 2007 Elsevier B.V. All rights reserved. Keywds: Battery models; Differential algebraic system; Shooting method 1. ntroduction Many investigations on electrhemical devices are fused on electric hybrid vehicles in which lithium-ion battery chemistry plays a crucial role. A maj problem encountered in electric hybrid vehicles is the lack of on-board moniting and control over the electrhemical components. This is due to the complexity in electrhemical models that makes their simulation very difficult, when compared to other models time scale present in the hybrid system. The existing physics based electrhemical models are represented as a system of coupled nonlinear multiple partial differential equations (PDEs) in multiple domains that are usually written in differential algebraic fm f simulation and model analysis. To facilitate successful simulation of these models, there is a necessity f * Cresponding auth. Tel.: ; fax: address: vsubramanian@tntech.edu (V.R. Subramanian). an exact, consistent and rapid initialization of DAEs. mproper heuristic initial values can lead to simulation failure even with popular DAE solvers like DASSL [1], GAMD [2], RADAU [3], etc. Sometimes apprimate initialization may lead to quick solutions, but the solutions are not physically meaningful inaccurate. Since the integration is started far away from the consistent initial values, the difference between apprimate values and consistent values will contribute to a lal err right from the very first step of integration. Some of the latest DAE solvers have features allowing a range f initializing each variable, but are unaffdable due to high computation time. There are some specific techniques developed f the successful initialization of lithium-ion and other secondary battery models. Newman [4] has first started investigating this system by using an innovative BAND subroutine. Here, the initial conditions are obtained by solving the steady-state equations of the nonlinear DAE models using BAND. Zhang and Cheh [5] solved a similar model /$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi: /j.elecom

2 V. Boovaragavan, V.R. Subramanian / Electrhemistry Communications 9 (2007) Nomenclature a i specific surface area of electrode i (i = p,n), m 2 / m 3 brugg i Bruggman coefficient of region i (i = p,s,n) c electrolyte concentration, mol/m 3 c 0 initial electrolyte concentration, mol/m 3 c s,i concentration of lithium ions in the intercalation particle of electrode i (i = p,n), mol/m 3 c s,i,0 initial concentration of lithium ions in the intercalation particle of electrode i (i = p,n), mol/m 3 c s,i,max maximum concentration of lithium ions in the intercalation particle of electrode i (i = p,n), mol/m 3 D electrolyte diffusion coefficient, m 2 /s D s,i lithium-ion diffusion coefficient in the intercalation particle of electrode i (i = p,n), m 2 /s F Faraday s constant, C/mol applied current density, A/cm 2 j i wall flux of Li + on the intercalation particle of electrode i (i = n,p), mol/m 2 k i intercalation/deintercalation reaction rate constant of electrode i (i = p,n), mol/(mol/m 3 ) 1.5 l i thickness of region i (i = p,s,n), m n negative electrode p positive electrode r radial codinate, m R universal gas constant, J/(mol K) R i radius of the intercalation particle of electrode i (i = p,n), m s separat t + Li + transference number in the electrolyte T absolute temperature, K U i open circuit potential of electrode i (i = p,n), V U s standard potential of the solvent reduction reaction, V x spatial codinate, m e i posity of region i (i = p,s,n) e f,i volume fraction of fillers of electrode i (i = p,n) h i dimensionless concentration of lithium ions in the intercalation particle of electrode i (h i = c s,i / c s,i,max ) j ionic conductivity of the electrolyte, S/m j eff,i effective ionic conductivity of the electrolyte in region i (i = p,s,n), S/m r i electronic conductivity of the solid-phase of electrode i (i = p, n), S/m r eff,i effective electronic conductivity of the solidphase of electrode i (i = p,n), S/m U 1 solid-phase potential, V electrolyte phase potential, V U 2 governing a Zn MnO 2 system without using the traditional BAND technique. They used a HYBRD1 subroutine based on the Powell hybrid method that has provision to choose a tolerance limit. They claimed that it took less than 5% of the total time f solving the DAEs. Another initialization subroutine f DAE solvers called DAES was developed in the FORTRAN environment [6,7] based on the repeated solution of a set of nonlinear equations. The success of this method is based on the nonlinear equation solver used in the subroutine. This method has also been demonstrated f the model governing galvanostatic charge/discharge presses of a thin film nickel hydride electrode. Other rigous methods developed f the initialization of general DAEs systems can be found in the literature [8,9]. Rigous methods f the initialization of DAEs depend on the efficiency and robustness of nonlinear algebraic routines optimization routines and may fail due to convergence problems. Meover, they require me computation time, and hence are not ideal f applications such as parameter estimation, capacity fade analysis, on-board simulation and real-time optimization where repetitive simulation of DAEs are essential. Hence a simple, rapid and me robust method f DAEs initialization is of considerable interest. The purpose of this paper is to introduce a novel and efficient method f initialization of DAEs experienced in advanced battery models. This simple and effective initialization technique is based on the physical constraints imposed by the battery chemistry and the shooting method to meet these constraints using steady-state model equations. The method has been described and also demonstrated using a lithium-ion battery model in FORTRAN, Matlab Ò and Maple Ò environments. The proposed method can provide initial values quickly and consistently in any environments. 2. Proposed initialization method The initial conditions f variables with time derivatives are set by the physical properties of the electrhemical system, only the variables without time derivatives are initialized. These equations are sometimes referred as Positive electrode l p Separat l s x Negative electrode Fig. 1. Schematic representation of different regions in a lithium-ion cell. l n

3 Table 1 Pseudo-two-dimensional model f lithium-ion battery with boundary and initial conditions Region Governing equations Boundary conditions Positive electrode e p ot ¼ D o 2 c eff;p þ a pð1 t 2 þ Þj p initial condition cj t¼0 ¼ c 0 D eff;p Separat Negative electrode r eff;p j eff;p r eff;p ¼ a p Fj p 2 ot ¼ D s;p r 2 o e s ot ¼ D o 2 c eff;s 2 r 2 ¼ j eff;s þ 2j eff;srt F þ 2j eff;prt F ð1 t þ Þ o ln c ¼ initial condition c s j t¼0 ¼ 0:5c s; max p ð1 t þ Þ o ln c e n ot ¼ D o 2 c eff;n þ a nð1 t 2 þ Þj n initial condition cj t¼0 ¼ c 0 r eff;n j eff;n þ 2j eff;nrt ð1 t þ Þ o ln c F ¼ r eff;n ¼ a 2 n Fj n ot ¼ D s;n o r 2 s initial condition c r 2 s j t¼0 ¼ 0:85c s; max ;n ¼ 0 and D eff;p x¼0 ¼ D eff;s j eff;p ¼ 0 and j eff;p x¼0 ¼ j eff;s ¼ and r eff;p x¼0 r eff;p ¼ 0 x¼lp ¼ 0 and j p ¼ D s;p r¼0 D eff;p j eff;p ¼ D eff;s ¼ j eff;s r¼rp D eff;s ¼ D eff;n j eff;s ¼ j eff;n r eff;n ¼ 0 and x¼lpþls ¼ 0 and j n ¼ D s;n r¼0 and D eff;s and j eff;s and r¼rn D eff;n ¼ D eff;n ¼ j eff;n and U 2 j ¼ 0 ¼ r eff;n ¼ V. Boovaragavan, V.R. Subramanian / Electrhemistry Communications 9 (2007)

4 V. Boovaragavan, V.R. Subramanian / Electrhemistry Communications 9 (2007) Table 2 Steady-state model of lithium-ion battery used f evaluating initial values Region Governing equations Boundary conditions Positive electrode r eff;p j eff;p ¼ r eff;p ¼ a 2 p Fj p c s j r¼rp ¼ j p 3ðN þ 1Þ R p D s;p þ c s; max ;p j eff;p ¼ 0 and j eff;p x¼0 ¼ j eff;s ¼ and r eff;p x¼0 r eff;p ¼ 0 x¼lp ou2 Separat j eff;s ¼ j eff;p ou2 ¼ j eff;s ou2 and j eff;s ou2 ¼ j eff;n ou2 Negative electrode r eff;n j eff;n ¼ j eff;s ¼ j eff;n and U 2 j ¼ 0 r eff;n ¼ a 2 n Fj n r eff;n c s j r¼rp ¼ j ¼ 0 and n R x¼lpþls ¼ r eff;n p þ c s; max ;n 3ðN þ 1Þ D s;n j p ¼ 2k p ðc s;max;p c s j r¼rp Þ 0:5 c s j 0:5 r¼rp c0:5 sinh 0:5F RT ðu 1 U 2 U p Þ and j n ¼ 2k n ðc s;max;n c s j r¼rp Þ 0:5 c s j 0:5 r¼rp c0:5 sinh 0:5F RT ðu 1 U 2 U n Þ, U p ¼ 4:656þ88:669h2 p 401:119h4 p þ342:909h6 p 462:471h8 p þ433:434h10 p ; h 1:0þ18:933h 2 p ¼ c s j p 79:532h4 p þ37:311h6 p 73:083h8 p þ95:96h10 r¼rp =c s;max;p, p U n ¼ 0:7222 þ 0:1387h n þ 0:029h 0:5 n 0:0172 hn þ 0:0019 þ 0:2808 expð0:90 15h hn 1:5 n Þ 0:7984 expð0:4465h n 0:4108Þ; h n ¼ c s j r¼rp =c s;max;n. steady-state equations. The components of a secondary battery such as positive electrode, separat and negative electrode can be arranged as three regions as shown in Fig. 1. Since the separat consists of liquid electrolyte alone, the steady-state model governing this region is only a single dinary differential equation (ODE) in x that can lead to a closed-fm solution. The lithium-ion battery model [4,10] used to demonstrate the proposed rapid initialization method is given in Table 1. This rigous two-dimensional model consists of three independent variables (x, r and t) and four dependent variables (c, U 1, U 2, c s ). t has a total of ten coupled nonlinear PDEs that are to be solved simultaneously. A successful simulation of this model is possible only when the initial conditions are consistent, meaning that f(t,x,x 0,r) = 0. The steady-state fm of these governing equations used f calculating the consistent initial values are given in Table 2. The solid-phase surface concentration is written in algebraic fm using spatial discretization in the radial direction. This variable can also be represented in algebraic fm using an efficient simplification method proposed by Subramanian et al. [10]. This has been tested f an entirely different set of model parameters [4]. The values of the parameters used in the simulation of model equations are given in Table 3 and Ref. [11]. The steps involved are given below: 1. Assume an initial value f unknown variables at the boundary x = 0 and integrate the DAEs governing the first region from x =0t = l p. The known boundary conditions at x = 0 are solid-phase potential and its flux. The unknown conditions (liquid-phase potential and solid-phase concentration) at first boundary, x = 0 are and mol/m and Table 3 Parameters used f the simulation Parameters Values Electrolyte concentration, c 1000 mol/m 3 Electrolyte conductivity, j S/m Solid-phase conductivity, r 100 S/m Radius of the particle, R p m Specific electrode area, a p and a n and m 2 /m 3 Li ion concentration in particle, c s,max,p and c s,max,p Solid-phase diffusion, D s,p and D s,n and m/s 2 Reaction rate constant, k p and k n and mol/(mol/m 3 ) 1.5 Thickness of Electrodes, l p and l n and m Thickness of separat, l s m Discharge current density, A/m 2 Transference number, t Posity of electrodes, e p and e n and Posity of separat, e s Volume fraction of fillers, e f,p and e f,n assumed and then shooting method is perfmed to the first interface, x = l p to find the values at second interface, x = l p + l s. Since the solid-phase potential flux at the first interface is known, it can be used as the shooting criterion. Befe integrating the governing equations f first region, one needs to solve f the unknown solid-phase surface concentration cresponding to the other known and guessed values using a nonlinear algebraic solver. This will ensure the consistency of the converged initial values. The numerical values converged at the first interface is significant as it maintains the continuity among the regions.

5 1776 V. Boovaragavan, V.R. Subramanian / Electrhemistry Communications 9 (2007) Use the converged values at the interface x = l p as initial values and integrate the governing equation in the separat from x = l p to x = l p + l s. This value at the second interface x = l p + l s will serve as fixed and known initial condition f further integration up to x = L (L = l p + l s + l n ). Usually the governing equation in the separat can be solved easily and a closed-fm solution is used. 3. The shooting method similar to the one followed in first region is adopted in the third region to meet the physical constraint imposed by solid-phase potential at the second boundary, x = L, i.e. solid-phase potential flux should match i app /r eff,n. The integration f third region stops when this flux at the second boundary meets the required tolerance (10 4 ). Modify φ2 guess at x = 0 START Read model equations and parameters Assume initial guess f φ2 at x = 0 Solve f cs at x = 0 based on φ2 guess ntegrate the DAE fm of model equations from x = 0 to x = lp NO s shooting criteria met at x = lp? Evaluate analytical fmula f φ2 variation in the separat using boundary values at x = lp and x = lp+ls Assume initial guess f φ1 at x = lp+ls Solve f cs at x = lp+ls based on φ1 guess ntegrate DAE fm of the model equations from x = lp+ls to x = L s shooting criteria met at x = L? END YES YES Fig. 2. Flow chart f the proposed computation scheme. NO Modify φ1 guess at x = lp+ls nitial Cell voltage, V Maple environment Matlab environment FORTRAN environment Discharge current density, A/m 2 Fig. 3. Comparison of different solvers used f predicting potential loss due to overpotential as a function of discharge current density using the proposed initialization method. The above scheme is simple and easy to program on a computer using any simulation environment with less computational cost (Fig. 2). By following this predure, a closed-fm solution is also possible f the model equation governing the separat. This further assists in increasing the computation efficiency of the proposed scheme. The steady-state lithium-ion battery model in Table 2 is solved f initial values using the proposed initialization method in three different simulation environments. The numerical results obtained f the initial cell voltage as a function of discharge current density is shown in Fig. 3 f various environments. The initial condition obtained f solid-phase potential using different simulation environments match on top of each other as expected. The cresponding computation time required f initialization in different simulation environments (1.7 GHz press and 1 GB RAM) are also given in Table 4. t can be seen that the FORTRAN and DASSL combination wks well and can predict the initial condition within a few milliseconds. The computation in other two environments, namely Matlab s Ò built-in solver and Maple s Ò built-in solver, require a few seconds f successful initialization. The high perfmance computing language Matlab Ò out perfms the symbolic language Maple Ò by me than half of the computation cost. However, the proposed method needs only a very few seconds f initialization even while using Table 4 Computation time f initialization using the proposed method Discharge rate FORTRAN DASSL (ms) Matlab Ò in-built solver (s) C/ C/ C/ C C C C C Maple Ò in-built solver (s)

6 V. Boovaragavan, V.R. Subramanian / Electrhemistry Communications 9 (2007) a computer algebra system. The initial values f dependent variables in polynomial fm that best fit the simulation results are expressed below. F the positive electrode: U 1 ðxþ¼4:246 2: ðl p xþ 2 þ 0: ðl p xþ 3 0: ðl p xþ 4 þ 0: ðl p xþ 5 ; U 2 ðxþ¼0:0005 0: x 2 þ 0: x 3 0: x 4 þ 0: x 5 ; c s ðxþ¼25548:5 0: x þ 0: x 2 0: x 3 þ 0: x 4 : F the separat: U 2 ðxþ ¼0:055 þ i app ðl p xþ: ð4þ j eff;s F the negative electrode: U 1 ðxþ¼0:182 þ 1: ðl p þ l s xþ 2 0: ðl p þ l s xþ 3 ð1þ ð2þ ð3þ 0: ðl p þ l s xþ 4 þ 0: ðl p þ l s xþ 5 ; ð5þ U 2 ðxþ¼0:108 0: ðl xþ 2 þ 0: ðl xþ 3 0: ðl xþ 4 ; c s ðxþ¼25305:363 þ 0: x 0: x 2 þ 0: x 3 0: x 4 : ð6þ ð7þ The above equations, Eqs. (1) (7) can be directly plugged into the computer code that uses popular DAE solvers like DASSL, GAMD, RADAU, etc. However, if the solver needs initial values f time derivatives of the algebraic differential variables, they have to be solved separately. t can be noted that the initial profile governing the separat is a closed-fm solution. Since the initial values obtained are exact, it can ensure successful and rapid simulation. The imptant feature of the proposed predure is that it requires only 15 ms f initialization in a FORTRAN environment that includes solving and printing the results. This unique characteristic of the proposed approach puts fward the developed methodology to today s demanding simulation requirements such as parameter estimation, on-board battery moniting, online optimization of charge/discharge cycles, etc. 3. Conclusion n this paper, a simple and rapid method f consistent initialization of secondary battery models is proposed. This technique is based on a shooting method and has been described and demonstrated using an advanced battery model that is represented by coupled nonlinear multiple PDEs. t is found that the method requires only a few milliseconds to initialize a rigous lithium-ion battery model and could provide exact and consistent initial values. Using the proposed method, the potential loss due to overpotential is plotted as a function of discharge rate in various simulation environments. The computation time is also compared to validate the accurate and rapid initialization properties of the proposed method. The proposed method is suitable f emerging applications such as on-line optimization, on-board moniting real time simulation because of its rapidity and consistency with no programming difficulties. Also, the future publications will describe combination of discretization in time (t) and shooting in the x-direction f rapid simulation of discharge curves f batteries. Acknowledgement The auths are thankful f the partial financial suppt of this wk by the US Army communications Electronics Command (CECOM) through the project on Advanced Ptable Power nstitute. References [1] K.E. Brenan, S.L. Campbell, L.R. Petzold, Numerical Solution of nitial Value Problems in Differential Algebraic Equations, second ed., SAM, [2] F. avernaro, F. Mazzia, Appl. Num. Math. 28 (1998) 107. [3] E. Hairer, G. Wanner, Solving Ordinary Differential Equations : Stiff and Differential algebraic Problems, second ed., Springer-Verlag, [4] M. Doyle, J. Newman, A.S. Gozdz, C.N. Schmutz, J-M. Tarascon, J. Electrhem. S. 143 (1996) [5] Y. Zhang, H.Y. Cheh, J. Electrhem. S. 146 (1999) 850. [6] B. Wu, R.E. White, Comput. Chem. Eng. 25 (2001) 301. [7] S. Santhanagopalan, Q. Guo, P. Ramadass, R.E. White, J. Power Sources 156 (2006) 620. [8] E.C. Biscaia Jr., R.C. Vieira, Comput. Chem. Eng. 25 (2001) [9] R. Lamour, Comput. Math. Appl. 50 (2005) [10] V.R. Subramanian, V.D. Diwakar, D. Tapriyal, J. Electrhem. S. 152 (2005) A2002. [11] P. Ramadass, P.M. Gomadam, R.E. White, B.N. Popov, J. Electrhem. S. 151 (2004) A196.