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State-of-charge estimation for lithium ion batteries via the simulation of lithium distribution in the electrode particles
Journal of Power Sources 272 (2014) 68e78
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Journal of Power Sources journal homepage: www.elsevier.com/locate/jpowsour
State-of-charge estimation for lithium ion batteries via the simulation of lithium distribution in the electrode particles Naixing Yang, Xiongwen Zhang*, Guojun Li Key Laboratory of Thermal-Fluid Science and Engineering of MOE, School of Energy & Power Engineering, Xi'an Jiaotong University, Xi'an 710049, China
h i g h l i g h t s A new method of SOC estimation based on computing the amount of Li is proposed. The available Li associated with the temperature and current are investigated. Experiments at different operating conditions are conducted for model validation. The discharge time with the method has less than 1% error with the measurements.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 4 June 2014 Received in revised form 9 August 2014 Accepted 13 August 2014 Available online 23 August 2014
State of charge (SOC) estimation is a key function of the battery management system for humanemachine interactions and systems control. This study proposes a new approach for SOC estimation based on computing the amount of Lithium (Li) in the electrode particles. The distribution of the Li concentration in the electrode particles are simulated and dynamically updated by solving the solid phase diffusion equation. By integrating the Li concentration distribution function over the battery volume, the battery SOC is estimated according to the calculated amount of dischargeable Li in the particles. The capacity changes of a LiPFeO4 battery during discharge are measured and calculated using this approach. The calculated capacities agree well with the measured capacities. The maximum difference is approximately 2.4%. The effects of operating temperature and current density on the Li concentration distribution during discharge are investigated. The Li concentration gradient in the particles increases as the operating temperature decreases or as the discharge rate increases. The capacity of dischargeable Li decreases approximately linearly by 52.2% as the operating temperature decreases from 25 C to 20 C, while it increases less than 3.5% when the operating temperature increases from 25 C to 40 C. © 2014 Elsevier B.V. All rights reserved.
Keywords: Lithium ion battery State of charge Capacity estimation Deintercalation/intercalation
1. Introduction Lithium ion batteries are widely used in mobile phones, laptops, backup power sources, automotive parts, etc. They are currently considered as the best alternative energy storage device for electric vehicles (EVs) and hybrid electric vehicles (HEVs) because they have excellent energy density and specific energy [1]. The state of charge (SOC) of a battery is defined as the percentage of the dischargeable capacity to the total capacity. The SOC is introduced for purposes of humanemachine interaction and system control [2,3]. Accurate knowledge of the SOC can prolong the battery lifecycle and allow efficient battery power management to avoid over-
* Corresponding author. Tel.: þ86 29 82665447; fax: þ86 29 82665445. E-mail addresses: [email protected], [email protected] (X. Zhang). http://dx.doi.org/10.1016/j.jpowsour.2014.08.054 0378-7753/© 2014 Elsevier B.V. All rights reserved.
charging, over-discharging and unpredicted system interruptions [4]. The SOC cannot be measured directly; therefore, it must be estimated from other available information, such as the operating current and the cell voltage. The SOC estimation is one of the most important functions of the battery management system (BMS), especially for EV and HEV applications. If the SOC estimation fails or has large errors, the battery performance, the safety, the working life of the battery, etc., may be compromised. However, the accurate estimation of the SOC is a challenge because the capacity of the charge pulled from a battery depends on many factors, including the geometrical structure of the micro-porous electrodes, the charge/discharge current, the temperature, the battery age, the cutoff voltage, the service history, etc. [5]. Furthermore, the microscopic parameters, such as the material composition and the structure of the micro-porous electrodes, change as the battery is used. In EV applications, the battery discharge is highly transient
N. Yang et al. / Journal of Power Sources 272 (2014) 68e78
with infrequent periods of constant current. These factors lead to variations in the maximum effective battery capacity, and thus increase the difficulty and complexity of the SOC estimation. Because EVs and HEVs have recently attracted considerable attention caused by their potential to reduce the oil dependence and the air pollution in the transport sector, a precise SOC estimation method that can adapt to the changing operating conditions in EVs is urgently required. Currently, there are many SOC estimation methods that have been developed over a long period of time. The classification for these methods varies in the literature [3,6e8]. According to the basic methodologies, the common methods are Ah counting [9,10], open circuit voltage (OCV) measurement [4,8,11,12], terminal voltage measurement [13,14], impedance measurement [15,16], artificial neural networks [17e19], and Kalman filters [20e23]. Ah counting uses the charge/discharge current as the input and the SOC is calculated by integrating the charge/discharge current over time. This approach is simple and easy to implement in real time. However, the drawbacks with Ah counting include accumulated measurement errors and an inaccurate determination of the initial SOC. Moreover, several factors, such as temperature, charge/ discharge current and aging, also affect the accuracy of Ah counting [3]. Another easy method for SOC estimation is OCV measurement, because it relates to the SOC through the battery's intrinsic physical properties. However, OCV measurement requires more resting time to obtain a stable value, thus it is not suitable for online dynamic SOC estimation. Additionally, the relation between the OCV and the SOC is weak and further complicated by the presence of hysteresis in some batteries, such as the lithium iron phosphate (LiFePO4) battery; therefore, a small OCV measurement error leads to a significant error in the SOC estimation [24]. Similarly, the theory for the methods of terminal voltage measurement and impedance measurement is also based on the battery's intrinsic characteristics. The prediction accuracy can be improved by considering the effects of temperature, aging, dynamic behavior, etc., in the models [8,12,25]. The artificial neural network and the Kalman filter for SOC estimation are categorized as adaptive systems [3,7], i.e., they can automatically adjust the SOC for different charging/discharging conditions. However, the artificial neural network requires a large amount of training data for prediction model development [26]. The Kalman filter algorithm that considers all feathers (i.e., nonnormalities and nonlinearities) is difficult to implement [7]. It is known that the SOC of a battery depends directly on the amount of deliverable lithium from the particles of electrodes. The amount of lithium that can be intercalated or deintercalated from the particles is strongly related to the Li diffusion coefficient, current density, particle scale, particle shape, etc. The Li diffusion coefficient is a function of the temperature and of the physical properties of the material. The battery aging affects the SOC estimation via the particle scale and the lower and upper limits of the Li concentration in the particles. The SOC of the battery can be estimated by calculating the amount of deliverable lithium in the electrodes. This study aims to develop a model-based deliverablelithium calculation method for the SOC estimation. The effects of the temperature and the discharge rate on the SOC estimation are studied using experiments and simulations. 2. Model development 2.1. Li intercalation and deintercalation A typical cell configuration of a lithium ion battery includes the negative electrode, the separator, the positive electrode, and current collectors (see Fig. 1). Both the negative electrode and the positive electrode are intercalated lithium compounds with the
69
Fig. 1. Cell configuration of the Li ion battery.
ability to absorb or denote lithium ions to the electrolyte. Generally, the negative electrode is graphite, and the positive electrode is a metal oxide such as LiCoO2, LiFePO4, or LiMn2O4 [27]. The lithium ions deintercalate from the particles of a graphite/lithium compound in the negative electrode and intercalate into the particles of the metal oxide compound in the positive electrode during discharge. The above process reverses when the battery is charging. Fig. 2 shows the process of lithium deintercalation during discharge with a constant current. The lithium compound particles in the electrodes are assumed to be spheres with radius Rs. Initially, the particles are fully intercalated and the Li concentration in the particles is at the maximum value. During discharge, the lithium ions deintercalate from the negative electrode and intercalate into the positive electrode. In the lithium deintercalation process, as shown in Fig. 2(b), the gradient of Li concentration between the particle center and the surface gradually increases until it reaches a steady value. For a specific current density and battery temperature, the concentration gradient of Li between the particle center and the surface is fixed. Generally, it is positively related to the current density and negatively related to the temperature. The Li concentration in the particle decreases continuously during deintercalation until the lithium concentration at the particle surface reaches the minimum. As shown in Fig. 2(c), the quantity of Li that cannot be deintercalated from the particle is positively dependent on the gradient of the lithium concentration in the particle. The unavailable Li can be reduced by decreasing the charge or the discharge current. Similarly, the Li intercalation process is the reverse of the deintercalation process. As shown in Fig. 3(a), the Li concentration near the particle surface increases at the initiation of Li intercalation. The gradient of Li concentration along the radial direction dynamically increases to a steady state if the operating current is constant. As shown in Fig. 3(b), a certain volume cannot be filled at the end of the Li intercalation. The discharge/charge current must be continually reduced to make the entire particle volume available to lithium intercalation. It is noted that the Li intercalation/deintercalation process in the positive electrode particles of LiFePO4 may proceed by two-phase reaction. The Li compounds in the positive electrode particles change between LiFePO4 and FePO4 [28e30] as the charging/discharging process proceeds. According to the references [31,32], the amounts of Li transported by two-phase reaction can be assumed to follow the continuous diffusion process in which the relation between the diffusion coefficient and the phase-transfer rate is defined by a shrinking coreeshell model. It is also known that the available capacity for a Li ion battery is linearly related to the
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Fig. 3. Graphic processes of Li intercalation in the electrode particles (a) start of the Li intercalation (b) end of the Li intercalation.
where cs,upper is the upper limit of the Li concentration in the particle and is generally a function of the total capacity of active Li in the battery. The available capacity is calculated differently for different cases. Fig. 4 shows the Li concentration distributions along the particle radial direction for different operating modes. The Li in the region outlined by red lines in Fig. 4 is the available Li. As shown in Fig. 4(a) and (c), the Li concentration at the particle surface is higher than the minimum Li concentration along the particle radius. In this case, the available Li capacity is defined as follows: Fig. 2. Graphic processes of the Li deintercalation in the electrode particles (a) full intercalated Li (b) start of the Li deintercalation (c) end of the Li deintercalation.
amounts of the available Li in the negative electrode particles or the insertable Li in the positive electrode particles. During discharge or charge, the amounts of the Li transport in both electrodes are same. Mostly, the battery capacity is limited by the capacity of negative electrode. In this work, the battery capacity is estimated by the quantity of Li in the negative electrode (i.e., carbon particles). The current density is assumed uniform across the surface of the spherical particle. Then, the volumeeaverage concentration of Li in the carbon particles at any time is calculated as follows:
Z cs;n ðtÞ ¼
Rs;n 0
4pr 2 cs;n ðr; tÞdr 4pR3s;n 3
(1)
Therefore, the discharge capacity of the battery at any time can be evaluated as follows:
Cdischarge ðtÞ ¼ 26:8Ln An εs;n cs;upper N cs;n t
(2)
Cavailable t ¼ 26:8Ln An εs;n Cs;n Rs;n ; t cs;cut T; N whencs;n Rs;n ; t > cs;n r; t min
(3)
In the case shown in Fig. 4(b), the available Li capacity is calculated as follows:
Cavailable t ¼ 26:8Ln An εs;n cs;n t cs;cut T; N whencs;n Rs;n ; t ¼ cs;n r; t min
(4)
where cs,cut is the lithium concentration corresponding to the cutoff voltage, which is also related to the operating temperature and the cycle number. The capacity of a lithium ion battery decreases during cycling. The mechanisms for capacity fade are associated with unwanted side reactions that occur during cycling, causing electrolyte decomposition, passive film formation and active Li dissolution [33]. The available lithium, i.e., cs,upper, decreases with battery aging as more and more active lithium is lost in the cycling. The real-time SOC during battery discharge is estimated as follows:
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density function, the average particle diameter is obtained as follows:
" # dav df dmin df 1
dmin 1 ¼ Rs ¼ 2 dmax 2 df 1
(8)
By inserting Eq. (8) into Eq. (1), the model estimates the SOC considering the effect of the geometry of the electrodes (via the parameters df, dmin and dmax), the temperature and the discharge/ charge rate (via the parameters cs(r, t) and cs,cut(T, N)), and the age of the battery (via the parameters cs,upper(N) and cs,cut(T, N)). In this study, the SOC estimation for a commercial cylindrical lithium iron phosphate battery model #26650 is conducted for various operating temperatures and discharge rates, using Eq. (5). A crosssection of the battery is shown in Fig. 5. The spiral layer consists of the negative current collector (Cu), the negative electrode (LixC6), the separator, the positive electrode (LiyFePO4) and the positive current collector (Al).
2.2. Electrochemical model
Fig. 4. Sketches of Li concentration distributions for different operating modes (a) switch from charging to discharging (b) steady stage of discharging process (c) charging process.
SOCðtÞ ¼
Cdischarge ðtÞ SOC0 Cdischarge ðtÞ þ Cavailable ðtÞ
! 100%
(5)
The electrodes are considered porous materials, which can be described by the theory of fractal geometry. The particle size follows the statistical properties of porous media. The number of particles of size d can then be calculated as follows:
NðL dÞ ¼
dmax df d
(6)
where d is the particle diameter, dmax is the maximum particle diameter and df is the fractal dimension within [0, 2] and [0, 3] for two dimensions and three dimensions, respectively. The density function of the particle size is taken from the literature [34] as follows: f f ðdÞ ¼ df ddmin dðdf þ1Þ
(7)
where dmin is the minimum particle diameter. The average particle diameter is used for the SOC estimation. Based on the probability
The upper limit of the Li concentration, cs,upper, decreases as the number of battery cycles increases because of Li loss during the charge/discharge cycling. The cut-off concentration, cs,cut, depends on changes in the internal resistance of the battery, which relates to the battery lifetime and the operating temperature. The physical structures of the electrodes also change gradually with the battery cycling. Generally, the quantitative relations between the physical properties and the battery aging are empirically determined, using large amounts of experimental data. This study focuses on the particular charge or discharge process, neglecting the aging effects on cs,upper and cs,cut. Therefore, only the distribution of the Li concentration in the radial direction influences the SOC estimation. The distribution of the Li concentration in the radial direction is determined by the current density across the particle surface and the Li diffusion coefficient. The Li diffusion coefficient in the particle depends strongly on the operating temperature. The changes in the current density and in the temperature can be predicted by the electrochemical model described below. The mass balance of active particles of electrodes is described by Fick's 2nd law as follows:
vcs;i Ds;i v vcs;i R2 ¼ 2 vt vR R vR
(9)
where i ¼ n and p. The flux at the center of the particle is equal to zero and is governed by Fick's 1st law on the particle surface as follows:
Ds;i
vcs;i ¼0 vR R¼0
(10)
Ds;i
vcs;i ¼ ji vR R¼Rs;i
(11)
where ji is the average flux of lithium ion deintercalation/intercalation in the electrode components. The material balance for the electrolyte is expressed as follows:
εl;i
vci v2 c ¼ Deff ;i 2i þ ai ð1 tþ Þji vt vx
(12)
where i ¼ n, sp and p, representing the zones of the negative electrode, the separator, and the positive electrode, respectively. The effective diffusion coefficient is determined by Deff,i ¼ Dεbrug 2i .
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Fig. 5. Schematic of the cross-section for a cylindrical Li ion battery.
The specific area of the active materials is defined as ai ¼ 3εs,i/Rs,i. The species flux of the liquid phase equals zero at the electrode/ current collector interfaces (see boundaries 2 and 5 in Fig. 1) and is continuous at the electrode/separator interfaces (see boundaries 3 and 4 in Fig. 1). Therefore, the boundary conditions for the electrolyte are as follows:
Deff;n
vc vc ¼ D eff ;sp vxx¼3 vxx¼3þ
(13)
vc vc ¼ D eff;p vxx¼4 vx x¼4þ
(14)
Deff;sp
vc vc Deff;n ¼ Deff;p ¼0 vx x¼2 vx x¼5
vfs;i vx
(15)
(16)
where i ¼ n and p. The effective conductance of the electrodes is defined as seff.i ¼ siεs. At the electrode/separator interfaces, there is no charge flux as follows:
seff;n
vfs;p vfs;n ¼ s ¼0 eff;p vx x¼3 vx x¼4
(17)
The charge flux at the positive current collector is equal to the current density applied to the cell. The solid phase potential at the negative current collector is equal to zero. The boundary conditions are as follows:
fs;n x¼2 ¼ 0 is;p x¼5 ¼
I Acell
(18)
vfe;i 2keff;i RTð1 tþ Þ d ln f± vln ci þ 1þ vx d ln ci vx F
(21)
where i ¼ n and p. The surface over-potential of the electrodes is defined as follows:
(22)
where i ¼ n and p. The open circuit voltage of the electrode, U, is assumed to be a function of the temperature and of the lithium ion concentration at the particle surface, and can be approximated by the first term in a Taylor's expansion as follows:
dU i Ui ¼ Uref;i þ T Tref dT
(23)
where Uref,i and dU/dT of carbon [35] and of LiFePO4 [36] are functions of the concentration at the surfaces of the particles. The energy balance in the lithium ion battery is as follows:
rCP
vT 1 v vT v2 T ¼ kT;r r þ kT;z 2 þ Qact þ Qrea þ Q ohm vt r vr vr vz
(24)
where kT,r and kT,z are the thermal conductivity coefficients for the battery in the radial and the axial directions, respectively. The terms representing heat generation caused by ohmic resistance, activation polarization and entropy change are calculated as follows:
Qact ¼ ai Fji hi
(25)
(19) Qrea ¼ ai Fji T
The charge balance in the electrolyte is as follows:
ie;i ¼ keff ;i
a a aa Fhi c ji ¼ ki cs;i;max cs;i;surf cas;i;surf cai a exp RT ac Fhi exp RT
hi ¼ fs;i fe;i DfSEI;i Ui
The charge balance of the porous electrodes is determined by Ohm's law as follows:
is;i ¼ seff;i
the electrode/current collector interfaces and is continuous at the electrode/separator interfaces. The reaction rate at the surface of the particles in the electrode is determined by the ButlereVolmer equation as follows:
(20)
where i ¼ n and p. The effective conductivity of the binary electrolyte is defined as keff,i ¼ kiεbrug 2i . The charge flux is equal to zero at
dUi dT
vfs;i 2 vfe;i 2 2keff ;i RT 1 tþ þ keff;i þ Qohm ¼ seff;i F vx vx d ln f± vln ci vfe;i 1þ þ ai Fji DfSEI d ln ci vx vx
(26)
(27)
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Applying the radiation law and Newton's law of cooling, the boundary condition for the energy balance is as follows:
4 kVT ¼ h Tw T þ ε Tw T4
(28)
k ¼ 1 104 c 10:5 þ 0:074T 6:96 105 T 2 þ6:68 104 c 1:78 105 cT þ 2:8 108 cT 2 2 þ4:94 107 c2 8:86 1010 c2 T
(33)
pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n ¼ 0:601 0:24 103 c þ 0:982½1 0:0052ðT 294Þ 109 c3
3. Results and discussion
(34)
3.1. Model validation The model parameters are taken from the manufacturer, extracted from the literature and based on experimental data. The primary parameters of the battery are listed in Table 1. The diffusion coefficient for Li and the reaction rate constant are calculated using the Arrhenius formula [39]. Based on the discharge capacities and the battery voltages under various operating conditions, the reaction rate constant of LiFePO4 and the diffusion coefficient of Li are revised as follows:
!# "
cs;surface;p Eak;p 1 1 exp kp ¼ k0;p 1 þ M 0:5 exp 1:8 Tref T csmax;p R (29) " EDa;n R
1 1 Tref T
Ds;n ¼ lDs0;n exp
!# (30)
where l is a dimensionless corrective coefficient calculated as follows:
l¼
73
0:998 þ 3:985 ln q þ 54:899 ln2 q 862:878 ln3 q 1 þ 3:27 ln q þ 13:308 ln2 q 349:3 ln3 q 1389:078 ln4 q (31)
where q ¼ T/Tref. Data for the salt diffusivity, D, the conductivity, k, and the thermodynamic factor, y, are derived from the literature [40] as follows:
D ¼ 1 104 104:4354=ðT2290:005cÞ2:210
4
c
The model validation experiments are conducted on a lithium ion battery test platform (see Fig. 6). In the experiments, a high-low alternating temperature testing machine was used to precisely control the temperature environment during battery operations. The constant currenteconstant voltage (CCeCV) charging mode (1 C / 3.6 V / 0.02 C) was selected. The batteries were measured with 1 C at different temperatures and at different discharge rates at 25 C. The discharge cut-off voltage is set at 2.5 V. Fig. 7 shows the output voltage from the experiments compared with the output voltage from the simulations for a discharge rate of 1 C for different temperatures (0 C, 10 C, 25 C and 40 C). Fig. 8 shows the comparisons of the output voltages from the experiments and the simulations at 25 C for different current densities (0.5 C, 1 C, 2 C and 4 C). Overall, the simulation results agree well with the experimental data. The errors are slightly larger at the turning point of the curve late in the discharge, particularly when the temperature is higher than 10 C. This effect is mostly from the estimates of diffusivity and conductivity for the electrolyte used in the model. The diffusivity and conductivity for the electrolyte shows a highly nonlinear relationship with respect to the temperature. The discharge capacities of the battery are measured under different operating conditions. Fig. 9 shows the comparisons of the discharge capacity between the measurements and the model calculations. The maximum difference between the measurements and the model is 2.4%. Fig. 10 shows the comparisons between the measured and calculated discharge capacity for different rates at 25 C. The maximum difference in this case is 0.8%. The capacities calculated by the model are slightly lower than the measured capacities.
(32) 3.2. Effect of temperature on the Li concentration distribution
Table 1 Model parameters for the Li ion battery. Negative Li (mm) si (Sm1)
εs,i εl,i Rs,i (mm) csmax,i (mol m3) cs0,i/csmax,i Ds0,i (m2 s1) EDa,i (J mol1) c0,i (mol m3) tþ k0,i (m2.5 mol0.5 s1) Eka,i (J mol1) aa,i/ac,i brug,i ri (kg m3) RSEI,i (U m2) Acell (m2) h (W m2 K1) ε Tref (K)
m
39 100 [37] 0.585m 0.364m 6.5m 31,370 [38] 0.86a 1.0 1014a 59,760a
Separator 20
m
0.54 [37]
Positive
The data in Figs. 7 and 9 show that decreasing the operating temperature results in decreases in both the discharge voltage and the capacity. The lithium ion diffusivity and the electrical
55m 0.5 [37] 0.513m 0.417m 1.4m 22,806 [39] 0.05a 8.0 1016a 48,000a
1200m 0.363 [37] 1.764 1011 39,000a 0.5 2.4a 1400m 0.01a 0.1537 15a 0.8a 298.15
1.5a 1200 [37]
3.9 1012 30,000a 0.5 1.8a 2100m 0
m a
Manufacturer. Estimated by experiment data.
Fig. 6. Li ion battery test facilities.
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Fig. 7. The changes of the voltage for different temperatures with 1 C discharge.
Fig. 9. Discharge capacity with 1 C at different temperatures.
conductivity of the electrolyte both decrease as the temperature decreases, and, as a result, the output voltage is reduced from the increase in ohmic losses. Fig. 9 also shows that the battery discharge capacity decreases as the temperature of the environment decreases. The slope is approximately 0.028425 Ah C1 when the operating temperature is below 20 C. The slope slowly decreases as the operating temperature rises. This is because the distribution characteristics of the Li concentration in the active particles are different for different operating temperatures. Fig. 11 shows the profiles of Li concentration in the electrode particles for various operating temperatures (0 C, 10 C, 25 C and 40 C) at the end of the discharge. The gradient in the concentration of lithium in the particles increases as the temperature decreases. This is because the diffusivity of lithium in the active particles is positively dependent on the temperature. For a lower operating temperature, the difference between the Li concentration at the particle center and the Li concentration at the particle surface rises and thus can provide enough force to keep the mass transfer rate as high as that in the case of a higher operating temperature. As shown in Fig. 11, the amount of unavailable intercalated or unavailable deintercalated lithium rises as the gradient of the Li concentration increases. It is also shown in Fig. 11 that the Li concentration on the surface of a carbon particle is approximately
equal to 0.04csmax,n for different temperatures at the end of the discharge. For the Li intercalation process in the positive electrode, the Li concentration at the particle surface at the end of the discharge is approximately 0.92csmax,p when the operating temperature is over 25 C. Fig. 12 shows the diagrams of the Li concentration based on the capacity calculation for different operating temperatures with the discharge rate of 1 C. The data in Fig. 12 show that the unavailable Li increases as the operating temperature decreases. The changes in the unavailable Li are significant when the temperature is below 25 C. When the operating temperature increases higher than 25 C, the amount of the unavailable Li changes little with the temperature. The discharge capacities for operating temperatures 20 C, 10 C, 0 C, 10 C and 40 C, respectively, equal 47.8%, 62.7%, 75.9%, 88.7%, and 103.5% of the discharge capacity at 25 C.
Fig. 8. The changes of the voltage for different discharge rates at 25 C.
Fig. 10. Discharge capacity with different rates at 25 C.
3.3. Effect of current rate on the Li concentration distribution The capacity of unavailable intercalated or unavailable deintercalated Li increases as the charge/discharge rate increases (see Figs. 8 and 10). There is a turning point at approximately 4 C. The decrease in the chargeable/dischargeable capacity is significant as
N. Yang et al. / Journal of Power Sources 272 (2014) 68e78
75
Fig. 11. Li concentration profiles in the active particles for different operating temperatures with discharge rate 1 C (a) Negative (b) Positive.
the charge/discharge rate increases on the left side (low side) of the turning point. Comparatively, the chargeable/dischargeable capacity more gradually decreases as the charge/discharge rate increases at discharge rates higher than 4 C. The dischargeable capacity is limited by the amount of Li in the negative electrode. The dischargeable capacity of the battery is calculated based on the distribution of the lithium concentration in the negative carbon particles. Fig. 13 shows the concentration profiles of lithium inside the carbon particle for different discharge rates (1 C, 2 C, 4 C and 8 C) at t ¼ tf. As shown in Fig. 13, the gradient of the Li concentration increases markedly as the discharge current increases from 1 C to 4 C. There are only slight changes in the distribution of Li concentration when the discharge rate changes from 4 C to 8 C. It is known that the mass transfer rate of Li increases as the discharge rate increases. At an increased discharge rate, a higher Li concentration gradient is required to guarantee the mass diffusion rate of Li. However, the generated heat increases as the discharge rate increases. In particular, the heat generation rate increases faster when the discharge rate is higher than 4 C. Therefore, the temperature rise is significant as the discharge rate increases from 4 C to 8 C. An increased temperature leads to the rise of the diffusion coefficient of Li, and thus offsets part of the requirement for the increase in the
Fig. 12. The deintercalated Li in the positive electrode particles for different operating temperatures (a) 0 C (b) 10 C (c) 25 C (d) 40 C.
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Fig. 13. Li concentration profiles inside the carbon particles with different discharge rates.
Li concentration gradient. Fig. 14 shows the lithium distribution in the carbon particle for different rates during discharge. The data in Fig. 14 show that the amount of unavailable discharged lithium increases as the discharge current increases, for values below 4 C. 3.4. SOC estimation during discharge with constant current Fig. 15(a) and (b) shows the estimated SOCs during discharge with 1 C at different temperatures (0 C, 10 C, 25 C and 40 C) and with different rates (1 C, 2 C, 4 C, 8 C) at 25 C. The battery is fully charged at the start of discharge and the initial SOC equals 100%. The SOC decreases linearly to 0% as the battery terminal voltage reaches the cut-off voltage. The discharge experiments are also conducted for the cases shown in Fig. 15. The maximum error of the discharge time between the estimated values and the experimental values is less than 1%. 4. Conclusions One of the most important functions of a battery management system is SOC estimation. This study proposes a model-based SOC estimation approach for a LiFePO4 battery that calculates the amount of Li in the electrode particles. The distributions of Li concentration in the particles are calculated and dynamically updated by simulations using Li-diffusion models. Experiments of the discharge process for a LiFePO4 battery are conducted for different operating temperatures and different current densities. The discharged capacities calculated with the proposed method agree well with the measured values. The maximum difference is approximately 2.4%. The effect of operating temperature on the distribution of Li concentration in the electrode particle is studied with a discharge rate of 1 C. The Li concentration gradient in the electrode particle is negatively associated with the operating temperature. The dischargeable (deintercalation process for the negative electrode and intercalation process for the positive electrode) capacity decreases as the operating temperature decreases. There is a turning point at an operating temperature of approximately 25 C. The dischargeable lithium linearly decreases by 52.2% when the operating temperature decreases from 25 C to 20 C, while it only decreases by 3.5% as the operating temperature changes from 40 C to 25 C. The dischargeable capacity of the battery decreases as the discharge rate increases in the same temperature range. The gradient of the Li concentration in the
Fig. 14. Li deintercalation in the negative electrode particles for different discharge rates (a) 1 C (b) 2 C (c) 4 C (d) 8 C.
N. Yang et al. / Journal of Power Sources 272 (2014) 68e78
cs;i cs,cut cs,upper ci Cavailable Cdischarge CP,i D Ds,i Ds0,i EDa,i Eka,i F h is,i ie,i I ji kT,i k0,i Li Qohm Qact Qrea r R Rs RSEI,i SOC0 tf
Fig. 15. The estimated SOC during discharge (a) 1 C at different temperatures, (b) different rates at 25 C.
electrode particles increases as the discharge rate increases. A larger Li concentration gradient leads to a greater volume of Li unavailable for discharge, thus reducing the total discharge capacity of the battery during discharge with a constant current. The discharge time estimated by the proposed method differs from the experimental data by less than 1%. Acknowledgment This work was funded by the Industrial Research Foundation of Shaanxi Province (Grant No. 2014K06-27). Nomenclature ai Acell brug cs,max,i cs,i
specific surface area (m1) unwinding area of the electrode (m2) Bruggeman tortuosity exponent theoretical maximum Li concentration in the active material particles (mol m3) Li concentration in the active material particles (mol m3)
77
average Li concentration in the active material particles (mol m3) Li concentration corresponding to the cut-off voltage (mol m3) upper limit of the Li concentration achievable in the particle (mol m3) electrolyte concentration in the solution phase (mol m3) available capacity of the battery in real time (Ah) discharged capacity of the battery in real time (Ah) heat capacity (J kg1 K1) diffusion coefficient of the electrolyte (m2 s1) diffusion coefficient of Li in the active material particles (m2 s1) diffusion coefficient of Li in the active material particles at the reference temperature (m2 s1) active energy for solid diffusion (J mol1) active energy for the reaction rate (J mol1) Faraday's constant (96485C mol1) heat transfer coefficient (W m2 K1) solid phase current density (A m2) solution phase current density (A m2) current applied to the battery (A) surface reaction rate (mol m2 s1) thermal conductivity (W m1 K1) reaction rate constant at the reference temperature (m2.5 mol0.5 s1) thickness (mm) ohmic heat generation rate (W m3) active polarization heat generation rate (W m3) reaction heat generation rate (W m3) radius variable of the cylindrical battery (m) gas constant (8.314 J mol1 K1) radius of electrode particle (mm) resistance of the film at the solid electrolyte interphase ( U m2 ) SOC at the start of the discharge time at the end of the discharge (s)
Greek letters aa,i transfer efficiency for anodic current ac,i transfer efficiency for cathodic current εs,i volume fraction of the active material εl,i volume fraction of the electrolyte ε emissivity of the battery can n thermodynamic factor of the electrolyte salt k ionic conductivity of the electrolyte (S m1) ri density (kg m3) si electronic conductivity of the solid matrix (S m1) hi over-potential (V) 4s,i solid phase voltage (V) 4e,i solution phase voltage (V) D4SEI,i voltage drop result from the SEI film resistance (V) Subscripts a anode c cathode e electrolyte n negative p positive s solid phase sp separator surf surface of active particles SEI solid electrolyte interphase ref reference temperature for Arrhenius or open circuit voltage formula
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References [1] K.E. Aifantis, S.A. Hackney, R.V. Kumar, High Energy Density Lithium Batteries: Materials, Engineering, Applications, John Wiley & Sons, 2010. [2] F. Trinh, Chalmers University of Technology, 2012. [3] V. Pop, H.J. Bergveld, P. Notten, P.P. Regtien, Meas. Sci. Technol. 16 (2005) R93eR110. [4] K.-S. Ng, C.-S. Moo, Y.-P. Chen, Y.-C. Hsieh, in: Power and Energy Conference, PECon 2008, IEEE 2nd International, IEEE, 2008, pp. 972e976. [5] K. Kutluay, Y. Cadirci, Y.S. Ozkazanc, I. Cadirci, IEEE Trans. Ind. Electron. 52 (2005) 1315e1327. [6] A. Burke, in: Presented at the Society of Automotive Engineers International Congress and Exposition, Detroit, MI, US, Feb. 27eMar. 3, 1989. [7] W.-Y. Chang, ISRN Appl. Math. 2013 (2013) 1e7. [8] Y. Xing, W. He, M. Pecht, K.L. Tsui, Appl. Energy 113 (2014) 106e115. [9] H.J. Bergveld, W.S. Kruijt, P.H. Notten, Battery Management Systems: Design by Modelling, Springer, 2002. [10] K.S. Ng, C.-S. Moo, Y.-P. Chen, Y.-C. Hsieh, Appl. Energy 86 (2009) 1506e1511. [11] S. Lee, J. Kim, J. Lee, B. Cho, J. Power Sources 185 (2008) 1367e1373. [12] H. He, X. Zhang, R. Xiong, Y. Xu, H. Guo, Energy 39 (2012) 310e318. [13] S. Sato, A. Kawamura, in: Proceedings of the Power Conversion Conference, PCC-Osaka 2002, IEEE, 2002, pp. 565e570. [14] M. Coleman, C.K. Lee, C. Zhu, W.G. Hurley, IEEE Trans. Ind. Electron. 54 (2007) 2550e2557. [15] F. Huet, J. Power Sources 70 (1998) 59e69. [16] J. Xu, C.C. Mi, B. Cao, J. Cao, J. Power Sources 233 (2013) 277e284. [17] T. Weigert, Q. Tian, K. Lian, J. Power Sources 196 (2011) 4061e4066. [18] C. Bo, B. Zhifeng, C. Binggang, Energy Convers. Manag. 49 (2008) 2788e2794.
[19] I.-H. Li, W.-Y. Wang, S.-F. Su, Y.-S. Lee, IEEE Trans. Energy Convers. 22 (2007) 697e708. [20] J. Lee, O. Nam, B. Cho, J. Power Sources 174 (2007) 9e15. [21] D. Di Domenico, G. Fiengo, A. Stefanopoulou, in: 17th IEEE International Conference on Control Applications, Part of 2008 IEEE Multi-conference on Systems and Control, IEEE, 2008, pp. 702e707. [22] C. Hu, B.D. Youn, J. Chung, Appl. Energy 92 (2012) 694e704. [23] F. Sun, X. Hu, Y. Zou, S. Li, Energy 36 (2011) 3531e3540. [24] M. Minarcin, E. Rask, SAE Int. J. Altern. Powertrains 1 (2012) 144e159. [25] O. Erdinc, B. Vural, M. Uzunoglu, in: 2009 International Conference on Clean Electrical Power, IEEE, 2009, pp. 383e386. [26] S. Piller, M. Perrin, A. Jossen, J. Power Sources 96 (2001) 113e120. [27] M.M. Thackeray, J.O. Thomas, M.S. Whittingham, MRS Bull. 25 (2000) 39e46. [28] C. Delacourt, P. Poizot, J.-M. Tarascon, C. Masquelier, Nat. Mater. 4 (2005) 254e260. [29] A. Yamada, H. Koizumi, N. Sonoyama, R. Kanno, Electrochem. Solid-State Lett. 8 (2005) A409eA413. [30] G.K. Singh, G. Ceder, M.Z. Bazant, Electrochim. Acta 53 (2008) 7599e7613. [31] V. Srinivasan, J. Newman, J. Electrochem. Soc. 151 (2004) A1517eA1529. [32] A. Andersson, J.O. Thomas, J. Power Sources 97 (2001) 498e502. [33] P. Arora, R.E. White, M. Doyle, J. Electrochem. Soc. 145 (1998) 3647e3667. [34] B. Yu, Appl. Mech. Rev. 61 (2008) 1e19. [35] M. Guo, G. Sikha, R.E. White, J. Electrochem. Soc. 158 (2011) A122eA132. [36] R.E. Gerver, J.P. Meyers, J. Electrochem. Soc. 158 (2011) A835eA843. [37] Y. Ye, Y. Shi, A.A.O. Tay, J. Power Sources 217 (2012) 509e518. [38] M. Safari, C. Delacourt, J. Electrochem. Soc. 158 (2011) A562eA571. [39] D. Bernardi, E. Pawlikowski, J. Newman, J. Electrochem. Soc. 132 (1985) 5e12. [40] L.O. Valøen, J.N. Reimers, J. Electrochem. Soc. 152 (2005) A882eA891.
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Journal of Power Sources journal homepage: www.elsevier.com/locate/jpowsour
State-of-charge estimation for lithium ion batteries via the simulation of lithium distribution in the electrode particles Naixing Yang, Xiongwen Zhang*, Guojun Li Key Laboratory of Thermal-Fluid Science and Engineering of MOE, School of Energy & Power Engineering, Xi'an Jiaotong University, Xi'an 710049, China
h i g h l i g h t s A new method of SOC estimation based on computing the amount of Li is proposed. The available Li associated with the temperature and current are investigated. Experiments at different operating conditions are conducted for model validation. The discharge time with the method has less than 1% error with the measurements.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 4 June 2014 Received in revised form 9 August 2014 Accepted 13 August 2014 Available online 23 August 2014
State of charge (SOC) estimation is a key function of the battery management system for humanemachine interactions and systems control. This study proposes a new approach for SOC estimation based on computing the amount of Lithium (Li) in the electrode particles. The distribution of the Li concentration in the electrode particles are simulated and dynamically updated by solving the solid phase diffusion equation. By integrating the Li concentration distribution function over the battery volume, the battery SOC is estimated according to the calculated amount of dischargeable Li in the particles. The capacity changes of a LiPFeO4 battery during discharge are measured and calculated using this approach. The calculated capacities agree well with the measured capacities. The maximum difference is approximately 2.4%. The effects of operating temperature and current density on the Li concentration distribution during discharge are investigated. The Li concentration gradient in the particles increases as the operating temperature decreases or as the discharge rate increases. The capacity of dischargeable Li decreases approximately linearly by 52.2% as the operating temperature decreases from 25 C to 20 C, while it increases less than 3.5% when the operating temperature increases from 25 C to 40 C. © 2014 Elsevier B.V. All rights reserved.
Keywords: Lithium ion battery State of charge Capacity estimation Deintercalation/intercalation
1. Introduction Lithium ion batteries are widely used in mobile phones, laptops, backup power sources, automotive parts, etc. They are currently considered as the best alternative energy storage device for electric vehicles (EVs) and hybrid electric vehicles (HEVs) because they have excellent energy density and specific energy [1]. The state of charge (SOC) of a battery is defined as the percentage of the dischargeable capacity to the total capacity. The SOC is introduced for purposes of humanemachine interaction and system control [2,3]. Accurate knowledge of the SOC can prolong the battery lifecycle and allow efficient battery power management to avoid over-
* Corresponding author. Tel.: þ86 29 82665447; fax: þ86 29 82665445. E-mail addresses: [email protected], [email protected] (X. Zhang). http://dx.doi.org/10.1016/j.jpowsour.2014.08.054 0378-7753/© 2014 Elsevier B.V. All rights reserved.
charging, over-discharging and unpredicted system interruptions [4]. The SOC cannot be measured directly; therefore, it must be estimated from other available information, such as the operating current and the cell voltage. The SOC estimation is one of the most important functions of the battery management system (BMS), especially for EV and HEV applications. If the SOC estimation fails or has large errors, the battery performance, the safety, the working life of the battery, etc., may be compromised. However, the accurate estimation of the SOC is a challenge because the capacity of the charge pulled from a battery depends on many factors, including the geometrical structure of the micro-porous electrodes, the charge/discharge current, the temperature, the battery age, the cutoff voltage, the service history, etc. [5]. Furthermore, the microscopic parameters, such as the material composition and the structure of the micro-porous electrodes, change as the battery is used. In EV applications, the battery discharge is highly transient
N. Yang et al. / Journal of Power Sources 272 (2014) 68e78
with infrequent periods of constant current. These factors lead to variations in the maximum effective battery capacity, and thus increase the difficulty and complexity of the SOC estimation. Because EVs and HEVs have recently attracted considerable attention caused by their potential to reduce the oil dependence and the air pollution in the transport sector, a precise SOC estimation method that can adapt to the changing operating conditions in EVs is urgently required. Currently, there are many SOC estimation methods that have been developed over a long period of time. The classification for these methods varies in the literature [3,6e8]. According to the basic methodologies, the common methods are Ah counting [9,10], open circuit voltage (OCV) measurement [4,8,11,12], terminal voltage measurement [13,14], impedance measurement [15,16], artificial neural networks [17e19], and Kalman filters [20e23]. Ah counting uses the charge/discharge current as the input and the SOC is calculated by integrating the charge/discharge current over time. This approach is simple and easy to implement in real time. However, the drawbacks with Ah counting include accumulated measurement errors and an inaccurate determination of the initial SOC. Moreover, several factors, such as temperature, charge/ discharge current and aging, also affect the accuracy of Ah counting [3]. Another easy method for SOC estimation is OCV measurement, because it relates to the SOC through the battery's intrinsic physical properties. However, OCV measurement requires more resting time to obtain a stable value, thus it is not suitable for online dynamic SOC estimation. Additionally, the relation between the OCV and the SOC is weak and further complicated by the presence of hysteresis in some batteries, such as the lithium iron phosphate (LiFePO4) battery; therefore, a small OCV measurement error leads to a significant error in the SOC estimation [24]. Similarly, the theory for the methods of terminal voltage measurement and impedance measurement is also based on the battery's intrinsic characteristics. The prediction accuracy can be improved by considering the effects of temperature, aging, dynamic behavior, etc., in the models [8,12,25]. The artificial neural network and the Kalman filter for SOC estimation are categorized as adaptive systems [3,7], i.e., they can automatically adjust the SOC for different charging/discharging conditions. However, the artificial neural network requires a large amount of training data for prediction model development [26]. The Kalman filter algorithm that considers all feathers (i.e., nonnormalities and nonlinearities) is difficult to implement [7]. It is known that the SOC of a battery depends directly on the amount of deliverable lithium from the particles of electrodes. The amount of lithium that can be intercalated or deintercalated from the particles is strongly related to the Li diffusion coefficient, current density, particle scale, particle shape, etc. The Li diffusion coefficient is a function of the temperature and of the physical properties of the material. The battery aging affects the SOC estimation via the particle scale and the lower and upper limits of the Li concentration in the particles. The SOC of the battery can be estimated by calculating the amount of deliverable lithium in the electrodes. This study aims to develop a model-based deliverablelithium calculation method for the SOC estimation. The effects of the temperature and the discharge rate on the SOC estimation are studied using experiments and simulations. 2. Model development 2.1. Li intercalation and deintercalation A typical cell configuration of a lithium ion battery includes the negative electrode, the separator, the positive electrode, and current collectors (see Fig. 1). Both the negative electrode and the positive electrode are intercalated lithium compounds with the
69
Fig. 1. Cell configuration of the Li ion battery.
ability to absorb or denote lithium ions to the electrolyte. Generally, the negative electrode is graphite, and the positive electrode is a metal oxide such as LiCoO2, LiFePO4, or LiMn2O4 [27]. The lithium ions deintercalate from the particles of a graphite/lithium compound in the negative electrode and intercalate into the particles of the metal oxide compound in the positive electrode during discharge. The above process reverses when the battery is charging. Fig. 2 shows the process of lithium deintercalation during discharge with a constant current. The lithium compound particles in the electrodes are assumed to be spheres with radius Rs. Initially, the particles are fully intercalated and the Li concentration in the particles is at the maximum value. During discharge, the lithium ions deintercalate from the negative electrode and intercalate into the positive electrode. In the lithium deintercalation process, as shown in Fig. 2(b), the gradient of Li concentration between the particle center and the surface gradually increases until it reaches a steady value. For a specific current density and battery temperature, the concentration gradient of Li between the particle center and the surface is fixed. Generally, it is positively related to the current density and negatively related to the temperature. The Li concentration in the particle decreases continuously during deintercalation until the lithium concentration at the particle surface reaches the minimum. As shown in Fig. 2(c), the quantity of Li that cannot be deintercalated from the particle is positively dependent on the gradient of the lithium concentration in the particle. The unavailable Li can be reduced by decreasing the charge or the discharge current. Similarly, the Li intercalation process is the reverse of the deintercalation process. As shown in Fig. 3(a), the Li concentration near the particle surface increases at the initiation of Li intercalation. The gradient of Li concentration along the radial direction dynamically increases to a steady state if the operating current is constant. As shown in Fig. 3(b), a certain volume cannot be filled at the end of the Li intercalation. The discharge/charge current must be continually reduced to make the entire particle volume available to lithium intercalation. It is noted that the Li intercalation/deintercalation process in the positive electrode particles of LiFePO4 may proceed by two-phase reaction. The Li compounds in the positive electrode particles change between LiFePO4 and FePO4 [28e30] as the charging/discharging process proceeds. According to the references [31,32], the amounts of Li transported by two-phase reaction can be assumed to follow the continuous diffusion process in which the relation between the diffusion coefficient and the phase-transfer rate is defined by a shrinking coreeshell model. It is also known that the available capacity for a Li ion battery is linearly related to the
70
N. Yang et al. / Journal of Power Sources 272 (2014) 68e78
Fig. 3. Graphic processes of Li intercalation in the electrode particles (a) start of the Li intercalation (b) end of the Li intercalation.
where cs,upper is the upper limit of the Li concentration in the particle and is generally a function of the total capacity of active Li in the battery. The available capacity is calculated differently for different cases. Fig. 4 shows the Li concentration distributions along the particle radial direction for different operating modes. The Li in the region outlined by red lines in Fig. 4 is the available Li. As shown in Fig. 4(a) and (c), the Li concentration at the particle surface is higher than the minimum Li concentration along the particle radius. In this case, the available Li capacity is defined as follows: Fig. 2. Graphic processes of the Li deintercalation in the electrode particles (a) full intercalated Li (b) start of the Li deintercalation (c) end of the Li deintercalation.
amounts of the available Li in the negative electrode particles or the insertable Li in the positive electrode particles. During discharge or charge, the amounts of the Li transport in both electrodes are same. Mostly, the battery capacity is limited by the capacity of negative electrode. In this work, the battery capacity is estimated by the quantity of Li in the negative electrode (i.e., carbon particles). The current density is assumed uniform across the surface of the spherical particle. Then, the volumeeaverage concentration of Li in the carbon particles at any time is calculated as follows:
Z cs;n ðtÞ ¼
Rs;n 0
4pr 2 cs;n ðr; tÞdr 4pR3s;n 3
(1)
Therefore, the discharge capacity of the battery at any time can be evaluated as follows:
Cdischarge ðtÞ ¼ 26:8Ln An εs;n cs;upper N cs;n t
(2)
Cavailable t ¼ 26:8Ln An εs;n Cs;n Rs;n ; t cs;cut T; N whencs;n Rs;n ; t > cs;n r; t min
(3)
In the case shown in Fig. 4(b), the available Li capacity is calculated as follows:
Cavailable t ¼ 26:8Ln An εs;n cs;n t cs;cut T; N whencs;n Rs;n ; t ¼ cs;n r; t min
(4)
where cs,cut is the lithium concentration corresponding to the cutoff voltage, which is also related to the operating temperature and the cycle number. The capacity of a lithium ion battery decreases during cycling. The mechanisms for capacity fade are associated with unwanted side reactions that occur during cycling, causing electrolyte decomposition, passive film formation and active Li dissolution [33]. The available lithium, i.e., cs,upper, decreases with battery aging as more and more active lithium is lost in the cycling. The real-time SOC during battery discharge is estimated as follows:
N. Yang et al. / Journal of Power Sources 272 (2014) 68e78
71
density function, the average particle diameter is obtained as follows:
" # dav df dmin df 1
dmin 1 ¼ Rs ¼ 2 dmax 2 df 1
(8)
By inserting Eq. (8) into Eq. (1), the model estimates the SOC considering the effect of the geometry of the electrodes (via the parameters df, dmin and dmax), the temperature and the discharge/ charge rate (via the parameters cs(r, t) and cs,cut(T, N)), and the age of the battery (via the parameters cs,upper(N) and cs,cut(T, N)). In this study, the SOC estimation for a commercial cylindrical lithium iron phosphate battery model #26650 is conducted for various operating temperatures and discharge rates, using Eq. (5). A crosssection of the battery is shown in Fig. 5. The spiral layer consists of the negative current collector (Cu), the negative electrode (LixC6), the separator, the positive electrode (LiyFePO4) and the positive current collector (Al).
2.2. Electrochemical model
Fig. 4. Sketches of Li concentration distributions for different operating modes (a) switch from charging to discharging (b) steady stage of discharging process (c) charging process.
SOCðtÞ ¼
Cdischarge ðtÞ SOC0 Cdischarge ðtÞ þ Cavailable ðtÞ
! 100%
(5)
The electrodes are considered porous materials, which can be described by the theory of fractal geometry. The particle size follows the statistical properties of porous media. The number of particles of size d can then be calculated as follows:
NðL dÞ ¼
dmax df d
(6)
where d is the particle diameter, dmax is the maximum particle diameter and df is the fractal dimension within [0, 2] and [0, 3] for two dimensions and three dimensions, respectively. The density function of the particle size is taken from the literature [34] as follows: f f ðdÞ ¼ df ddmin dðdf þ1Þ
(7)
where dmin is the minimum particle diameter. The average particle diameter is used for the SOC estimation. Based on the probability
The upper limit of the Li concentration, cs,upper, decreases as the number of battery cycles increases because of Li loss during the charge/discharge cycling. The cut-off concentration, cs,cut, depends on changes in the internal resistance of the battery, which relates to the battery lifetime and the operating temperature. The physical structures of the electrodes also change gradually with the battery cycling. Generally, the quantitative relations between the physical properties and the battery aging are empirically determined, using large amounts of experimental data. This study focuses on the particular charge or discharge process, neglecting the aging effects on cs,upper and cs,cut. Therefore, only the distribution of the Li concentration in the radial direction influences the SOC estimation. The distribution of the Li concentration in the radial direction is determined by the current density across the particle surface and the Li diffusion coefficient. The Li diffusion coefficient in the particle depends strongly on the operating temperature. The changes in the current density and in the temperature can be predicted by the electrochemical model described below. The mass balance of active particles of electrodes is described by Fick's 2nd law as follows:
vcs;i Ds;i v vcs;i R2 ¼ 2 vt vR R vR
(9)
where i ¼ n and p. The flux at the center of the particle is equal to zero and is governed by Fick's 1st law on the particle surface as follows:
Ds;i
vcs;i ¼0 vR R¼0
(10)
Ds;i
vcs;i ¼ ji vR R¼Rs;i
(11)
where ji is the average flux of lithium ion deintercalation/intercalation in the electrode components. The material balance for the electrolyte is expressed as follows:
εl;i
vci v2 c ¼ Deff ;i 2i þ ai ð1 tþ Þji vt vx
(12)
where i ¼ n, sp and p, representing the zones of the negative electrode, the separator, and the positive electrode, respectively. The effective diffusion coefficient is determined by Deff,i ¼ Dεbrug 2i .
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N. Yang et al. / Journal of Power Sources 272 (2014) 68e78
Fig. 5. Schematic of the cross-section for a cylindrical Li ion battery.
The specific area of the active materials is defined as ai ¼ 3εs,i/Rs,i. The species flux of the liquid phase equals zero at the electrode/ current collector interfaces (see boundaries 2 and 5 in Fig. 1) and is continuous at the electrode/separator interfaces (see boundaries 3 and 4 in Fig. 1). Therefore, the boundary conditions for the electrolyte are as follows:
Deff;n
vc vc ¼ D eff ;sp vxx¼3 vxx¼3þ
(13)
vc vc ¼ D eff;p vxx¼4 vx x¼4þ
(14)
Deff;sp
vc vc Deff;n ¼ Deff;p ¼0 vx x¼2 vx x¼5
vfs;i vx
(15)
(16)
where i ¼ n and p. The effective conductance of the electrodes is defined as seff.i ¼ siεs. At the electrode/separator interfaces, there is no charge flux as follows:
seff;n
vfs;p vfs;n ¼ s ¼0 eff;p vx x¼3 vx x¼4
(17)
The charge flux at the positive current collector is equal to the current density applied to the cell. The solid phase potential at the negative current collector is equal to zero. The boundary conditions are as follows:
fs;n x¼2 ¼ 0 is;p x¼5 ¼
I Acell
(18)
vfe;i 2keff;i RTð1 tþ Þ d ln f± vln ci þ 1þ vx d ln ci vx F
(21)
where i ¼ n and p. The surface over-potential of the electrodes is defined as follows:
(22)
where i ¼ n and p. The open circuit voltage of the electrode, U, is assumed to be a function of the temperature and of the lithium ion concentration at the particle surface, and can be approximated by the first term in a Taylor's expansion as follows:
dU i Ui ¼ Uref;i þ T Tref dT
(23)
where Uref,i and dU/dT of carbon [35] and of LiFePO4 [36] are functions of the concentration at the surfaces of the particles. The energy balance in the lithium ion battery is as follows:
rCP
vT 1 v vT v2 T ¼ kT;r r þ kT;z 2 þ Qact þ Qrea þ Q ohm vt r vr vr vz
(24)
where kT,r and kT,z are the thermal conductivity coefficients for the battery in the radial and the axial directions, respectively. The terms representing heat generation caused by ohmic resistance, activation polarization and entropy change are calculated as follows:
Qact ¼ ai Fji hi
(25)
(19) Qrea ¼ ai Fji T
The charge balance in the electrolyte is as follows:
ie;i ¼ keff ;i
a a aa Fhi c ji ¼ ki cs;i;max cs;i;surf cas;i;surf cai a exp RT ac Fhi exp RT
hi ¼ fs;i fe;i DfSEI;i Ui
The charge balance of the porous electrodes is determined by Ohm's law as follows:
is;i ¼ seff;i
the electrode/current collector interfaces and is continuous at the electrode/separator interfaces. The reaction rate at the surface of the particles in the electrode is determined by the ButlereVolmer equation as follows:
(20)
where i ¼ n and p. The effective conductivity of the binary electrolyte is defined as keff,i ¼ kiεbrug 2i . The charge flux is equal to zero at
dUi dT
vfs;i 2 vfe;i 2 2keff ;i RT 1 tþ þ keff;i þ Qohm ¼ seff;i F vx vx d ln f± vln ci vfe;i 1þ þ ai Fji DfSEI d ln ci vx vx
(26)
(27)
N. Yang et al. / Journal of Power Sources 272 (2014) 68e78
Applying the radiation law and Newton's law of cooling, the boundary condition for the energy balance is as follows:
4 kVT ¼ h Tw T þ ε Tw T4
(28)
k ¼ 1 104 c 10:5 þ 0:074T 6:96 105 T 2 þ6:68 104 c 1:78 105 cT þ 2:8 108 cT 2 2 þ4:94 107 c2 8:86 1010 c2 T
(33)
pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n ¼ 0:601 0:24 103 c þ 0:982½1 0:0052ðT 294Þ 109 c3
3. Results and discussion
(34)
3.1. Model validation The model parameters are taken from the manufacturer, extracted from the literature and based on experimental data. The primary parameters of the battery are listed in Table 1. The diffusion coefficient for Li and the reaction rate constant are calculated using the Arrhenius formula [39]. Based on the discharge capacities and the battery voltages under various operating conditions, the reaction rate constant of LiFePO4 and the diffusion coefficient of Li are revised as follows:
!# "
cs;surface;p Eak;p 1 1 exp kp ¼ k0;p 1 þ M 0:5 exp 1:8 Tref T csmax;p R (29) " EDa;n R
1 1 Tref T
Ds;n ¼ lDs0;n exp
!# (30)
where l is a dimensionless corrective coefficient calculated as follows:
l¼
73
0:998 þ 3:985 ln q þ 54:899 ln2 q 862:878 ln3 q 1 þ 3:27 ln q þ 13:308 ln2 q 349:3 ln3 q 1389:078 ln4 q (31)
where q ¼ T/Tref. Data for the salt diffusivity, D, the conductivity, k, and the thermodynamic factor, y, are derived from the literature [40] as follows:
D ¼ 1 104 104:4354=ðT2290:005cÞ2:210
4
c
The model validation experiments are conducted on a lithium ion battery test platform (see Fig. 6). In the experiments, a high-low alternating temperature testing machine was used to precisely control the temperature environment during battery operations. The constant currenteconstant voltage (CCeCV) charging mode (1 C / 3.6 V / 0.02 C) was selected. The batteries were measured with 1 C at different temperatures and at different discharge rates at 25 C. The discharge cut-off voltage is set at 2.5 V. Fig. 7 shows the output voltage from the experiments compared with the output voltage from the simulations for a discharge rate of 1 C for different temperatures (0 C, 10 C, 25 C and 40 C). Fig. 8 shows the comparisons of the output voltages from the experiments and the simulations at 25 C for different current densities (0.5 C, 1 C, 2 C and 4 C). Overall, the simulation results agree well with the experimental data. The errors are slightly larger at the turning point of the curve late in the discharge, particularly when the temperature is higher than 10 C. This effect is mostly from the estimates of diffusivity and conductivity for the electrolyte used in the model. The diffusivity and conductivity for the electrolyte shows a highly nonlinear relationship with respect to the temperature. The discharge capacities of the battery are measured under different operating conditions. Fig. 9 shows the comparisons of the discharge capacity between the measurements and the model calculations. The maximum difference between the measurements and the model is 2.4%. Fig. 10 shows the comparisons between the measured and calculated discharge capacity for different rates at 25 C. The maximum difference in this case is 0.8%. The capacities calculated by the model are slightly lower than the measured capacities.
(32) 3.2. Effect of temperature on the Li concentration distribution
Table 1 Model parameters for the Li ion battery. Negative Li (mm) si (Sm1)
εs,i εl,i Rs,i (mm) csmax,i (mol m3) cs0,i/csmax,i Ds0,i (m2 s1) EDa,i (J mol1) c0,i (mol m3) tþ k0,i (m2.5 mol0.5 s1) Eka,i (J mol1) aa,i/ac,i brug,i ri (kg m3) RSEI,i (U m2) Acell (m2) h (W m2 K1) ε Tref (K)
m
39 100 [37] 0.585m 0.364m 6.5m 31,370 [38] 0.86a 1.0 1014a 59,760a
Separator 20
m
0.54 [37]
Positive
The data in Figs. 7 and 9 show that decreasing the operating temperature results in decreases in both the discharge voltage and the capacity. The lithium ion diffusivity and the electrical
55m 0.5 [37] 0.513m 0.417m 1.4m 22,806 [39] 0.05a 8.0 1016a 48,000a
1200m 0.363 [37] 1.764 1011 39,000a 0.5 2.4a 1400m 0.01a 0.1537 15a 0.8a 298.15
1.5a 1200 [37]
3.9 1012 30,000a 0.5 1.8a 2100m 0
m a
Manufacturer. Estimated by experiment data.
Fig. 6. Li ion battery test facilities.
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Fig. 7. The changes of the voltage for different temperatures with 1 C discharge.
Fig. 9. Discharge capacity with 1 C at different temperatures.
conductivity of the electrolyte both decrease as the temperature decreases, and, as a result, the output voltage is reduced from the increase in ohmic losses. Fig. 9 also shows that the battery discharge capacity decreases as the temperature of the environment decreases. The slope is approximately 0.028425 Ah C1 when the operating temperature is below 20 C. The slope slowly decreases as the operating temperature rises. This is because the distribution characteristics of the Li concentration in the active particles are different for different operating temperatures. Fig. 11 shows the profiles of Li concentration in the electrode particles for various operating temperatures (0 C, 10 C, 25 C and 40 C) at the end of the discharge. The gradient in the concentration of lithium in the particles increases as the temperature decreases. This is because the diffusivity of lithium in the active particles is positively dependent on the temperature. For a lower operating temperature, the difference between the Li concentration at the particle center and the Li concentration at the particle surface rises and thus can provide enough force to keep the mass transfer rate as high as that in the case of a higher operating temperature. As shown in Fig. 11, the amount of unavailable intercalated or unavailable deintercalated lithium rises as the gradient of the Li concentration increases. It is also shown in Fig. 11 that the Li concentration on the surface of a carbon particle is approximately
equal to 0.04csmax,n for different temperatures at the end of the discharge. For the Li intercalation process in the positive electrode, the Li concentration at the particle surface at the end of the discharge is approximately 0.92csmax,p when the operating temperature is over 25 C. Fig. 12 shows the diagrams of the Li concentration based on the capacity calculation for different operating temperatures with the discharge rate of 1 C. The data in Fig. 12 show that the unavailable Li increases as the operating temperature decreases. The changes in the unavailable Li are significant when the temperature is below 25 C. When the operating temperature increases higher than 25 C, the amount of the unavailable Li changes little with the temperature. The discharge capacities for operating temperatures 20 C, 10 C, 0 C, 10 C and 40 C, respectively, equal 47.8%, 62.7%, 75.9%, 88.7%, and 103.5% of the discharge capacity at 25 C.
Fig. 8. The changes of the voltage for different discharge rates at 25 C.
Fig. 10. Discharge capacity with different rates at 25 C.
3.3. Effect of current rate on the Li concentration distribution The capacity of unavailable intercalated or unavailable deintercalated Li increases as the charge/discharge rate increases (see Figs. 8 and 10). There is a turning point at approximately 4 C. The decrease in the chargeable/dischargeable capacity is significant as
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Fig. 11. Li concentration profiles in the active particles for different operating temperatures with discharge rate 1 C (a) Negative (b) Positive.
the charge/discharge rate increases on the left side (low side) of the turning point. Comparatively, the chargeable/dischargeable capacity more gradually decreases as the charge/discharge rate increases at discharge rates higher than 4 C. The dischargeable capacity is limited by the amount of Li in the negative electrode. The dischargeable capacity of the battery is calculated based on the distribution of the lithium concentration in the negative carbon particles. Fig. 13 shows the concentration profiles of lithium inside the carbon particle for different discharge rates (1 C, 2 C, 4 C and 8 C) at t ¼ tf. As shown in Fig. 13, the gradient of the Li concentration increases markedly as the discharge current increases from 1 C to 4 C. There are only slight changes in the distribution of Li concentration when the discharge rate changes from 4 C to 8 C. It is known that the mass transfer rate of Li increases as the discharge rate increases. At an increased discharge rate, a higher Li concentration gradient is required to guarantee the mass diffusion rate of Li. However, the generated heat increases as the discharge rate increases. In particular, the heat generation rate increases faster when the discharge rate is higher than 4 C. Therefore, the temperature rise is significant as the discharge rate increases from 4 C to 8 C. An increased temperature leads to the rise of the diffusion coefficient of Li, and thus offsets part of the requirement for the increase in the
Fig. 12. The deintercalated Li in the positive electrode particles for different operating temperatures (a) 0 C (b) 10 C (c) 25 C (d) 40 C.
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Fig. 13. Li concentration profiles inside the carbon particles with different discharge rates.
Li concentration gradient. Fig. 14 shows the lithium distribution in the carbon particle for different rates during discharge. The data in Fig. 14 show that the amount of unavailable discharged lithium increases as the discharge current increases, for values below 4 C. 3.4. SOC estimation during discharge with constant current Fig. 15(a) and (b) shows the estimated SOCs during discharge with 1 C at different temperatures (0 C, 10 C, 25 C and 40 C) and with different rates (1 C, 2 C, 4 C, 8 C) at 25 C. The battery is fully charged at the start of discharge and the initial SOC equals 100%. The SOC decreases linearly to 0% as the battery terminal voltage reaches the cut-off voltage. The discharge experiments are also conducted for the cases shown in Fig. 15. The maximum error of the discharge time between the estimated values and the experimental values is less than 1%. 4. Conclusions One of the most important functions of a battery management system is SOC estimation. This study proposes a model-based SOC estimation approach for a LiFePO4 battery that calculates the amount of Li in the electrode particles. The distributions of Li concentration in the particles are calculated and dynamically updated by simulations using Li-diffusion models. Experiments of the discharge process for a LiFePO4 battery are conducted for different operating temperatures and different current densities. The discharged capacities calculated with the proposed method agree well with the measured values. The maximum difference is approximately 2.4%. The effect of operating temperature on the distribution of Li concentration in the electrode particle is studied with a discharge rate of 1 C. The Li concentration gradient in the electrode particle is negatively associated with the operating temperature. The dischargeable (deintercalation process for the negative electrode and intercalation process for the positive electrode) capacity decreases as the operating temperature decreases. There is a turning point at an operating temperature of approximately 25 C. The dischargeable lithium linearly decreases by 52.2% when the operating temperature decreases from 25 C to 20 C, while it only decreases by 3.5% as the operating temperature changes from 40 C to 25 C. The dischargeable capacity of the battery decreases as the discharge rate increases in the same temperature range. The gradient of the Li concentration in the
Fig. 14. Li deintercalation in the negative electrode particles for different discharge rates (a) 1 C (b) 2 C (c) 4 C (d) 8 C.
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cs;i cs,cut cs,upper ci Cavailable Cdischarge CP,i D Ds,i Ds0,i EDa,i Eka,i F h is,i ie,i I ji kT,i k0,i Li Qohm Qact Qrea r R Rs RSEI,i SOC0 tf
Fig. 15. The estimated SOC during discharge (a) 1 C at different temperatures, (b) different rates at 25 C.
electrode particles increases as the discharge rate increases. A larger Li concentration gradient leads to a greater volume of Li unavailable for discharge, thus reducing the total discharge capacity of the battery during discharge with a constant current. The discharge time estimated by the proposed method differs from the experimental data by less than 1%. Acknowledgment This work was funded by the Industrial Research Foundation of Shaanxi Province (Grant No. 2014K06-27). Nomenclature ai Acell brug cs,max,i cs,i
specific surface area (m1) unwinding area of the electrode (m2) Bruggeman tortuosity exponent theoretical maximum Li concentration in the active material particles (mol m3) Li concentration in the active material particles (mol m3)
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average Li concentration in the active material particles (mol m3) Li concentration corresponding to the cut-off voltage (mol m3) upper limit of the Li concentration achievable in the particle (mol m3) electrolyte concentration in the solution phase (mol m3) available capacity of the battery in real time (Ah) discharged capacity of the battery in real time (Ah) heat capacity (J kg1 K1) diffusion coefficient of the electrolyte (m2 s1) diffusion coefficient of Li in the active material particles (m2 s1) diffusion coefficient of Li in the active material particles at the reference temperature (m2 s1) active energy for solid diffusion (J mol1) active energy for the reaction rate (J mol1) Faraday's constant (96485C mol1) heat transfer coefficient (W m2 K1) solid phase current density (A m2) solution phase current density (A m2) current applied to the battery (A) surface reaction rate (mol m2 s1) thermal conductivity (W m1 K1) reaction rate constant at the reference temperature (m2.5 mol0.5 s1) thickness (mm) ohmic heat generation rate (W m3) active polarization heat generation rate (W m3) reaction heat generation rate (W m3) radius variable of the cylindrical battery (m) gas constant (8.314 J mol1 K1) radius of electrode particle (mm) resistance of the film at the solid electrolyte interphase ( U m2 ) SOC at the start of the discharge time at the end of the discharge (s)
Greek letters aa,i transfer efficiency for anodic current ac,i transfer efficiency for cathodic current εs,i volume fraction of the active material εl,i volume fraction of the electrolyte ε emissivity of the battery can n thermodynamic factor of the electrolyte salt k ionic conductivity of the electrolyte (S m1) ri density (kg m3) si electronic conductivity of the solid matrix (S m1) hi over-potential (V) 4s,i solid phase voltage (V) 4e,i solution phase voltage (V) D4SEI,i voltage drop result from the SEI film resistance (V) Subscripts a anode c cathode e electrolyte n negative p positive s solid phase sp separator surf surface of active particles SEI solid electrolyte interphase ref reference temperature for Arrhenius or open circuit voltage formula
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