Simplification and Efficient Simulation of ... - CyberLeninka

Available online at www.sciencedirect.com

ScienceDirect Energy Procedia 104 (2016) 68 – 73

CUE2016-Applied Energy Symposium and Forum 2016: Low carbon cities & urban energy systems

Simplification and efficient simulation of electrochemical model for Li-ion battery in EVs Cheng Lina,b, Aihua Tanga,b,c* a

National Engineering Laboratory for Electric Vehicles, School of Mechanical Engineering, Beijing Institute of Technology, Beijing 100081, China b Collaborative Innovation Center of Electric Vehicles in Beijing, School of Mechanical Engineering, Beijing Institute of Technology, Beijing 100081, China c School of Mechanical Engineering, Sichuan University of Science & Engineering , Zigong 643000, China

Abstract At present, lithium-ion (Li-ion) cells are the core of electric vehicles (EVs). The complexity of electrochemical model makes on-line simulation difficult in electric vehicles. Thence, it is necessary to obtain a simplified model instantaneously under all operating conditions of the batteries. In this paper, simplification of electrochemical models of Li-ion battery to improve simulation and computational efficiency in EVs will be proposed. An isothermal pseudotwo-dimensional (P2D) model based on spatiotemporal dynamics of li-ion concentration, electrode potential in each phase, and the Butler-Volmer kinetics is developed. Since using traditional approaches to simulate the P2D model is computationally expensive, it has limited its use in EV’s applications. Some methods can be used to decrease the number of Partial Differential Equations (PDEs) that must be solved simultaneously and enable faster computation while using limited resources. Moreover, an averaged electrode (AE) model and single particle (SP) model which derive from P2D model embodies high precision and fast simulation of battery performance for a range of working conditions. Finally, the simulation results of the AE and SP model are compared with Doyle-Fuller Newman (DFN) model and show that the SP model can reduce computational amount significantly while still retaining the accuracy. © 2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license

© 2016 The Authors. Published by Elsevier Ltd. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Selection peer-review of under responsibility of CUE Peer-reviewand/or under responsibility the scientific committee of the Applied Energy Symposium and Forum, CUE2016: Low carbon cities and urban energy systems. Key words: electrochemical model; electric vehicles (EVs); pseudo-two-dimensional (SP2D); single particle (SP) model; Partial Differential Equations(PDEs)

1.Introduction Nowadays, lithium-ion (Li-ion) battery is one of the potential candidate for electric vehicles (EVs), because of high power and energy density, long service life, non-memory effect, low self-discharge rate, * Corresponding author. Tel.: +86-106-891-3992; fax: +86-106-891-3992. E-mail address: [email protected].

1876-6102 © 2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the scientific committee of the Applied Energy Symposium and Forum, CUE2016: Low carbon cities and urban energy systems. doi:10.1016/j.egypro.2016.12.013

69

Cheng Lin and Aihua Tang / Energy Procedia 104 (2016) 68 – 73

etc.[1]. However, inappropriate work situations such as successive overcharge or overdischarge can promote degradation processes and even cause the failures. Thence, for safe and reliable work of Li-ion batteries, it is essential to monitor battery performances through the adoption of model-based battery management system (BMS) in EVs. A BMS is the core of battery model which identifies the correlation between currents and voltages obtained at battery terminals and characters the battery instantaneous state using state of charge (SOC) or condition of health (SOH) [2]. The models can be classified into three categories, including equivalent circuit models (ECMs) [3], empirical models and electrochemical models. The core benefit of ECMs is the simplicity. However, they have no immediate electrochemical meaning. Based on a large number of battery test data, the empirical models can get a higher precision of the internal state of the battery. The electrochemical models, especially the P2D, is based on the theories of porous electrodes and concentrated solutions [4]. Comparing to other models, the P2D model can capture the electrochemical reaction dynamics and predict the batteries’ behavior under any type of operating conditions with better accuracy. But, the model requres a heavy computational resources due to its complex coupled with numerous nonlinear differential differential equations (PDEs), which restricts its utilization. For instance, the on-line estimation and prediction cannot be realized in it. Thus, it is important to simplify and reduce the order of models to improve the computational efficiency [5] The goal of current work is to establish an approximate Li-ion electrochemical cell model which can be solvable real time with a properly computing requirement and can estimate the behaviors of cells with little cost of accuracy. 2.P2D model based on electrochemical principles As illustrated in Fig.1 [6], a P2D Li-ion batteries model composes of a negative electrode, a separator, a positive electrode and two current collectors at the sides of the two electrodes. Generally, both electrodes consist of a grain size of quasi-spherical active particles in µm scale. The Li-ions travel inside/outside the active particles via diffusion Charge e e and migration inside the active particles along A e e x the r-axis. It is called solid phase diffusion. The Discharge / δ δ δ void between the particles is packed with Negative Positive Electrode Electrode electrolyte. Under the theories of porous Li Charge Li electrodes and concentrated solutions, P2D Aluminum positive Copper negative Li model is mainly composed of ten coupled current collector current collector Li nonlinear PDEs, which represent the solidDischarge phase diffusion within the particles, the Li Li diffusion in the electrolyte, the balance of the Separator solid-phase potential and the electrolyte Electrolyte potential occurring in the three regions in Li-ion ‫ܥ‬ Li batteries. The details of P2D model are given in U U in the following. Li C Li MnO ρ

Sep

n

+

+

+

+

+

+

‫ݏ‬െ

x

+

6

y

2

Fig. 1 Schematic diagram of a P2D Li-ion cell model

2.1. Li-ion diffusion in the solid phase

Li-ions concentration C s inside solid spherical particles for the positive (p) and negative (n) electrode follows the Fick's second law as the Eq. (1) [1].

wCs wt

Ds w § 2 wCs ¨r r 2 wr © wr

· , for r 0, R ¸ s ¹

(1)

70


To solve the diffusion problems (1), the initial conditions (2) and Neumann boundary conditions (3) are introduced :

Cs ,0

, for C ! 0 at t

C

wcs wr

0, r 0

wcs wr

r Rs

0 r 0, Rs

(2)

j Li Rs 3H s F

(3)

Where Ds is the solid phase diffusion coefficient of Li-ions within the solid particles, jLi(x,t) is the local volumetric transfer current density (jLi>0 represent discharge), Rs is the radius of the particle, and H s is active material volume fraction, F is Faraday’s number. 2.2. Li-ion diffusion in the electrolyte phase In general, the electrolyte concentration distribution ce follows the Eq. (4).

He where

wce wt

w 2 ce (1 t 0 ) Li j , for x 0, L F wx 2

Deeff

(4)

Deeff is the effective electrolyte ionic diffusivity, t 0 is transference number, and follows the Eq.

(5)

Deeff

DeH eBrugg

(5)

The initial and boundary conditions following the Eqs.(6)-(8).

Cs e,0

wce wx

Deeff,n Where

wce wx

He

x G n

Deeff,sep

wce wx

x 0

x G n

wce wx

H e is

(6)

0,

(7)

x L

Deeff,sep

wce wx

x G n G sep

is Electrolyte phase volume fraction porosity,

separator, and positive electrode, defined as L

0 x >0, L@

, for C ! 0 at t

C

Deeff, p

G n , G sep , G p

wce wx

x G n G sep

(8)

are the thicknesses of negative,

the porosities for the liquid region. For convenience, here L is

G n G sep G p .

2.3.Potential equation in the solid phase Generally, the potential distribution in the solid phase

V eff

w Is j Li wx 2 2

Is

follows the Ohm's law as shown in Eq. (9).

for x >0, L@

0,

(9)

And the boundary conditions follows the Eqs. (10) and (11).

wIs wx wI s wx x

V eff

V eff x 0

Gn

wI s wx

wIs wx

I (t ) A

x L

0 x G n G sep

(10) (11)

71


V eff

where

is the effective solid phase ionic diffusivity and follows the Eq. (12).

V eff

VH s

(12)

2.4.Potential equation in the electrolyte phase Generally, the potential distribution in the electrolyte phase

N

eff

2 eff 2 RT w ln ce w 2Ie j Li Nd 2 2 F wx wx

Ie

follows the Ohm's law is as follows.

0 , for x 0, L

(13)

And the boundary conditions follows the Eq. (14).

wIe wx

where

N deff

x 0

wIe wx

0

(14)

x L

is the electrolyte ionic diffusional conductivity and follows the Eq. (15).

2 RTk eff (15) t 1 F eff eff Where N ,which can be derived by k (H e )1.5 N ,is effective electrolyte phase ionic conductivity, and N is electrolyte phase ionic conductivity, R is gas constant, T is temperature.

N deff

2.5 BV kinetics The BV kinetic formula is used to describe the rates of the Li-ion intercalation/deintercalation reactions for each electrode as shown in Eq. (16).

j Li

ª

§ DpF · § D F ·º K ¸¸ exp¨ n K ¸» , for x 0, L ¹¼ © RT © RT ¹

D s i0 cs max cse D cse D «exp¨¨

Where

¬

Ds

concentration,

(16)

is active surface area per electrode unit volume, c smax is maximum solid-phase

a ap

0.5 are change transfers coefficients of the negative and positive

an

K is the electrode activation polarization overpotential for the Li ion deintercalation/intercalation reactions, and i0 is the exchange current density

electrode, cse is solid concentration at electrolyte interface, at an interface and is determined by Eq. (17).

i0

K ce cs max cse cse , for x 0, L D

D

D

K is shown in Eq. (18) K Is Ie U cse , for x 0, L

(17)

where K is kinetic rate constant,

(18)

Here, U is corresponds to the electrode material performance of thermodynamic equilibrium potential, the open circuit voltage of the electrodes. Then the battery terminal voltage V could be given as follows

V (t ) Is ( L, t ) Ie (0, t ) R f I (t ) 3.Simplification of P2D Model

(19)

72


Although the P2D model can completely describe the electrochemical reaction process of the Li-ion cell, it is impossible to apply in the EVs due to its complexity. Thence, it is necessary simplify and reduce the model order. 3.1.AE model To further simplify the P2D model, the following aspects are mainly considered in the AE model: Neglecting the spatial distribution inhomogeneity of the BV current density and the Li ion concentration in solid-phase electrode and electrolyte. In other words, an average value of the solid concentration and an averaging BV current are considered, which satisfies the spatial integral Eq. (9). The negative and positive electrodes described by current average values, and imposing the boundary conditions and continuity at the interface and the solution of Eqs. 9–15. Battery voltage can be obtained as a function of battery current and average concentration of solid particles. 3.2.SP model Based on the hypothesis that the local volumetric transfer current density is constant and equals to the average value of each solid particle within a SP. Accordingly, all particles within the electrodes should behave the same way. The current flowing through the electrodes is evenly distributed over all the particles in this assumption. Therefore, each of the electrodes can be represented by a single spherical particle. In the SP, surface concentrations of Li+ and solid phase potential are only the function of time t, and remain constant along the x-axis of electrode .Thus, given known parameters such as the input current and Li-ion surface concentration of the electrode, the battery terminal voltage in the SP model could be easily calculated. 4.Model validation and discussion In order to demonstrate the accuracy of the AE and SP model, two kinds of simulations were carried out: a current profile according to Freedom CAR test procedure [8] and full discharge/charge at constant current. The model validation was performed versus the Doyle-Fuller Newman (DFN) model described in Ref. [1], developed by Newman and his collaborators [9]. This DFN model also adoped as the generator of experimental data to assess the model’s performance. The steady state accuracy was validated by comparing the cell voltage error under constant current value shown in Fig. 2 and Fig. 3. The voltage response of AE model is consistent with DFN model, while the SP model deviated considerably from the others at high current. The charge or discharge current for the test are ±C/4, ±C/2, ±1C and ±2C. The simulation was terminated when it reaches the upper and lower cut-off voltage. The simulation results in the form of constant current voltage error (RMS) were summarized and the voltage response to several constant discharge rates were showed in Fig. 4. The voltage error increases with C-rate increasing especially in the SP model from Fig. 4. It is vital to note the predicted charge capacity is identical between both types which is essential for applications in EVs.

Fig. 2. Current profile according to Freedom CAR test procedure

Fig. 3. Voltage response for the AE, DFN and SPM model.


Fig. 4. Summary of constant current simulations

5.Conclusions This paper simplifies a Li-ion electrochemical mechanism model for the purpose of better application in EVs. An AE model and SP model are proposed by ignoring the spatial distribution inhomogeneity of local volumetric transfer current density and the Li+ concentration in solid-phase electrode and electrolyte. For validation of the simplified model, two kinds of simulations were carried out together with DFN model. The SP model exhibits less accuracy than the AE model. But it is an attractive candidate for application in EVs within an acceptable error range for its simple. However, their accuracy will deteriorate as the charge/discharge current increases. 6.Copyright Authors keep full copyright over papers published in Energy Procedia. Acknowledgements This work was supported by the National Natural Science Foundation of China (51575044). References [1] Xuebing Han, Minggao Ouyang, Languang Lu, Jianqiu. Simplification of physics-based electrochemical model for lithium ion battery on electric vehicle. Part I: Diffusion simplification and single particle model. J. Power Sources 2015; 278: 802–813. [2] Yinyin Zhao, Song-Yul Choe. A highly efficient reduced order electrochemical model for a large format LiMn2O4/Carbon polymer battery for real time applications. Electrochimica Acta 2015; 164: 97–107. [3] F. Sun, R. xiong, H. He. A systematic state-of-charge of estimate framework for multi-cell battery pack in electric vehicles using bias correction technique. Applied Energy 2016; 162: 1399–1409. [4] de Vidts, P., Delgado, J., and White, R. E. Mathematical Modeling for the Discharge of a Metal Hydride Electrode. J. Electrochem. Soc. 1995; 142: pp. 4006–4013. [5] Weilin Luo, Chao Lyu, Lixin Wang, Liqiang Zhang. A new extension of physics-based single particle model for higher chargeedischarge rates . J. Power Sources 2013; 241: 295–310. [6] Ramin Masoudi, Thomas Uchida. John McPhee. Parameter estimation of an electrochemistry-based lithium-ion battery model. J. Power Sources 2015; 291: 215–224. [7] Shriram Santhanagopalan, Qingzhi Guo, Premanand Ramadass, White, R. E. Review of models for predicting the cycling performance of lithium ion batteries. J. Power Sources 2006; 156: 620–628. [8] FreedomCar Battery Test Manual for Power-Assist Hybrid Elecric Vehicles. 2003; DOE/ID-11069. [9] Newman, J. Fortran Programs for the Simulation of Electrochemical Systems. University of California, Berkley, CA; 2008. [10] Di Domenico, Anna Stefanopoulou, Giovanni Fiengo. Lithium-Ion Battery State of Charge and Critical Surface Charge Estimation Using an Electrochemical Model-Based Extended Kalman Filter. Journal of Dynamic Systems, Measurement, and Control.2010; 132: 061302-1–061302-11.

73