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A method for SOC estimation based on simplified mechanistic model for LiFePO4 battery
Energy 114 (2016) 1266e1276
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Energy journal homepage: www.elsevier.com/locate/energy
A method for SOC estimation based on simplified mechanistic model for LiFePO4 battery Junfu Li a, Qingzhi Lai a, Lixin Wang a, Chao Lyu a, *, Han Wang b a b
School of Electrical Engineering and Automation, Harbin Institute of Technology, Harbin, 150001, China School of Astronautics, Harbin Institute of Technology, Harbin, 150001, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 7 April 2016 Received in revised form 11 August 2016 Accepted 22 August 2016
Accurate battery state of charge (SOC) estimation conduces to establishing an ideal charging and discharging strategy for battery management system (BMS). And it can also prevent serious damage to battery (pack) from over-charging or over-discharging. This paper firstly establishes a simplified mechanistic model for LiFePO4 battery. Parameter identification conditions are originally designed based on excitation response analysis and essential verifications in terms of model accuracy are conducted. The functional relationship between mechanistic parameters and battery SOC is then established. And lastly, SOC estimation method based on the proposed model is presented. The contributions of this paper are listed as follows: (i) the proposed mechanistic model with obtained parameters can accurately describe LiFePO4 battery behaviors and predict discharge capacity under different operating conditions, (ii) the proposed SOC estimation method based on the battery model has certain applicability and robustness. Analysis and assessment of accuracy of the proposed method indicate that SOC estimation accuracy is acceptable. With the improvement of model simulation, SOC estimation accuracy can be further refined. © 2016 Elsevier Ltd. All rights reserved.
Keywords: State of charge Battery management system Simplified mechanistic model Excitation response analysis Applicability and robustness
1. Introduction Li-ion batteries, as primary power suppliers for many devices or systems, have been attractive due to high specific power and energy density, high cycle lifetime and low self-discharge rate [1]. They have been widely adopted in battery management system (BMS) of electric vehicles (EVs). An advanced BMS can not only accurately estimate the state of charge (SOC) of the batteries connected in series/parallel configuration, but avoid the potential safety hazard and ensure efficient operation as well. However, inaccurate SOC estimation and unbalance problem make it hard for BMS to determine an effective management strategy [2]. Battery SOC is a key evaluation index for BMS which can serve as an indicator of remaining capacity [3]. An accurate battery SOC estimation method plays a vital role in avoiding over-charging and overdischarging and addressing unbalance problem. In addition, it can also be used for developing power and energy management strategies [4].
* Corresponding author. P.O. Box 404, Harbin Institute of Technology, No. 92, West Dazhi Street, Nangang District, Harbin, 150001, Heilongjiang, China. E-mail addresses: [email protected] (J. Li), [email protected] (Q. Lai), wlx@ hit.edu.cn (L. Wang), [email protected] (C. Lyu), [email protected] (H. Wang). http://dx.doi.org/10.1016/j.energy.2016.08.080 0360-5442/© 2016 Elsevier Ltd. All rights reserved.
In the past few years, a large amount of researches concentrated on battery SOC estimation, which had their own limits. One of the most commonly used methods is the ampere-hour counting, which is a kind of open-loop calculation methods [2]. Though it has simple formations and high calculation efficiency, initial offset for SOC and accumulated calculation errors can lead to bad SOC estimation performance [5]. To enhance SOC estimation precision, close-loop method was developed. An OCV-based method was adopted to recalibrate the accumulated errors. Apart from the above two commonly used methods, Kalman filter (KF) method [6], extended Kalman filter (EKF) method [7] and adaptive unscented Kalman filter (AUKF) [8] were also adopted to improve the performance of battery SOC estimation. While, there are also some limitations in such methods [3]. Battery models used in their algorithms must be accurate. Otherwise, unreliable estimation result will occur. Furthermore, measurement noises including system noise and observation noise must be submitted to Gaussian distribution. Reference [9] used Monte Carlo sampling methods to represent any probability density function and particle filter (PF) was employed to determine initial SOC. The proposed method showed good ability to correct initial value and track SOC variation. It is revealed that to increase the accuracy and accelerate the determination of the correct SOC, complex impedance models were needed.
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Generally, the performance of SOC estimation can be improved with accurate battery model. And an accurate battery model is able to describe the real states inside the battery. The pseudo 2 dimensional (P2D) model was based on theories of porous electrodes and concentrated solutions and was developed to capture the microcosmic behaviors of Li-ion at different operating conditions [10]. The development of the P2D model set the foundation for further research on electrical-thermal mechanistic models. A pseudo 3 dimensional model based on P2D model was developed by coupling the mass, charge, energy conservation, and cell electrochemical kinetics to study both electrochemical and thermal characteristics of the battery [11]. With a well developed electrochemical-thermal coupled model, thermal phenomenon can be better understood [12]. Much work has been concentrated on thermal analysis and management to enhance battery model accuracy and ensure battery operating safety [13e15]. As mechanistic models can accurately simulate the entire SOC, they are very appealing in BMS [16,17]. Reference [18] combined P2D model with an EKF algorithm to estimate battery SOC. The proposed state estimation algorithm was able to rapidly and accurately recover the model states with current, voltage and temperature measurements. However, the large computational cost of solving partial differential equations limits the practical application of these models and thus many efforts have concentrated on simplifying these models. References [19,20] took temperature into consideration to expand single particle model and the single particle thermal model showed good agreement with experimental voltage data under different discharge C-rates. Reference [21] obtained an approximate solution for electrolyte concentration distribution, which used a parabolic profile approximation assuming that the electrolyte concentration profile in the porous electrode was a parabolic polynomial in lateral direction at any time. Modified single particle model with the heterogeneous treatment given by the non-random two-liquid model was found to be the best in predicting the constant discharge/charge profiles [22]. With reduced or regrouped parameters, simplified models tend to be more suitable for embedded application in terms of battery SOC estimation in BMS, for they have both acceptable accuracy and less computational cost. The purpose of this work is to establish a method for SOC estimation based on a simplified mechanistic model for LiFePO4 battery. The novelty of our work are as follows: (i) the proposed model and model parameter estimation method have certain generality in terms of battery materials and individuals, (ii) a functional relationship between mechanistic parameters and battery SOC is originally established, (iii) a method for battery SOC estimation based on the simplified mechanistic model with low computational cost and high accuracy is proposed. This paper is organized as follows: in Section 2, a simplified mechanistic model together with parameter estimation method based on excitation response analysis is presented and essential verifications are conducted; in Section 3, a functional relationship between mechanistic parameters and battery SOC is established and estimation procedure is given in detail, followed by analysis and assessment of accuracy, applicability and robustness of the proposed method in Section 4. Finally, Section 5 concludes this paper. 2. Simplified mechanistic model and validation
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and acceptable accuracy is employed [21,23]. It is further assumed that internal average temperature of spiral battery is equal to the surface in the case that the cell has small capacity and thermal increment under large current load. Thus, surface temperature is a measureable variable, which is used to simulate battery overpotential and terminal voltage. Such assumption, on one hand, reduces computational cost of the model. There is no need to calculate battery heat generation and internal average temperature with iterative calculation. On the other hand, compared with our previous work [23], it simplifies the process of parameter estimation which will be discussed in detail in Section 2.2. The proposed mechanistic model is described as follows. 2.1.1. Open circuit voltage The solid phase surface concentration of the particle directly determines open circuit voltage (OCV). Therefore, OCV at reference temperature is described as
Eocv ðtÞ ¼ Up yavg Un xavg
(1)
Where Up and Un are open circuit voltage curves of electrodes [24]. yavg and xavg are average Li-ion concentration inside the particle of each electrode, which are given as
yavg ¼ y0 þ It Qp
(2)
xavg ¼ 1 yofs yavg Qp Qn
(3)
Where y0 and x0 are initial values of yavg and xavg, respectively. Qp and Qn theoretically represent the capacities of effective active materials. We define yofs as the offset of relative position which can be given as
yofs ¼ 1 y0 x0
Qn Qp
(4)
The above four parameters give a clear description on basic operating process. 2.1.2. Solid phase diffusion Concentration of Li-ion on the particle surface of each electrode can be described as [21].
ysurf ðtÞ ¼ yavg ðtÞ þ DyðtÞ
(5)
xsurf ðtÞ ¼ xavg ðtÞ DxðtÞ
(6)
The discrete formation ofDy andDx can be found to be 0
0
Dy ðtkþ1 Þ ¼ Dy ðtk Þ þ Dx0 ðtkþ1 Þ ¼ Dx0 ðtk Þ þ
! s 12 tp 0 Iðt Þ Dy ðtk Þ ðtkþ1 tk Þ 7 Qp k
1
tsp 1
tsn
12 tsn Iðtk Þ Dx0 ðtk Þ ðtkþ1 tk Þ 7 Qn
(7)
(8)
Where tsp and tsn are solid phase diffusion time constants of each electrode.
2.1. Simplified mechanistic model To accurately describe battery charge and discharge behaviors under different operating scenarios, it is essential to have a clear understanding on battery internal physicochemical processes. A simplified mechanistic model which has low computational cost
2.1.3. Liquid phase diffusion The effect of liquid phase diffusion can be equivalently seen as the establishment of concentration difference at current collectors of each electrode. The degree of concentration polarization can be expressed by one fused state variableDc. Therefore, we have [21].
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hcon ðtÞ ¼
2RTðtÞ c þ DcðtÞ ð1 tþ Þln 0 F c0 DcðtÞ
2.2. Parameter estimation
(9)
Where c0 is initial liquid phase Li-ion concentration. State equation ofDc is
Dcðtkþ1 Þ ¼ Dcðtk Þ þ
1
te
ðPcon Iðtk Þ Dcðtk ÞÞðtkþ1 tk Þ
(10)
Where Pcon andte are the proportion coefficient and time constant of liquid phase diffusion, respectively. Pcon with respect to temperature is fitted by Ref. [23].
Pcon ¼
ref Pcon
. exp lcon 1 Tref 1=T
(11)
Some parameters such as initial liquid phase Li-ion concentration c0 and effective heat dissipation area Rohm can be obtained directly. And the others need to be estimated. In this work, parameter identification conditions are originally designed and model parameters are obtained through excitation response analysis. As battery internal physicochemical processes have numerical coupling relations, it is essential to make some assumptions and approximations in step-by-step procedure of parameter estimation, which will be mentioned in each part. 2.2.1. Parameter estimation of basic working process According to the definitions in Section 2.1.1, Qp and Qn can be described as
Qp ¼ Qall =Dy; Qn ¼ Qall =Dx 2.1.4. Reaction polarization According to Butler-Volmer kinetics, reaction polarization overpotential hact can be expressed as [21].
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2RTðtÞ ln m2n ðtÞ þ 1 þ mn ðtÞ F qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ln m2p ðtÞ þ 1 þ mp ðtÞ
hact ðtÞ ¼
mp ðtÞ ¼
1 6Qp c0:5 0
1 ysurf
1 mn ðtÞ ¼ 6Qn c0:5 0
1 xsurf
1 0:5
1 0:5
ysurf
xsurf
0:5 Pact IðtÞ
0:5 Pact IðtÞ
(12)
(13)
(14)
(15)
2.1.5. Ohmic polarization Additionally, the ohmic polarization overpotential hohm can be uniformly expressed as [21].
hohm ðtÞ ¼ Rohm IðtÞ
(16)
Where, Rohm is ohmic resistance expressed in the form of a lumped parameter. The variations of ohmic resistance with respect to temperature are fitted by the following formula [23].
. ref Rohm ¼ Rohm exp lohm 1 Tref 1=T
(17)
Then, terminal voltage Uapp is found to be
Uapp ðtÞ ¼ Eocv ðtÞ hcon ðtÞ hact ðtÞ hohm ðtÞ
Where Qall is total discharge capacity measured at rate of 0.02C. Dx and Dy are maximum variation ranges of stoichiometric number x and y, respectively. Using method of ampere-hour counting, SOC of the battery can be defined as [5,8].
socðtÞ ¼ 1 It=Qall
(20)
Neglecting the effects of polarization and diffusion process inside the battery, ideal OCV can be described as
Pact is a combined parameter and comparable to electrochemical reaction rate constant, which reflects the difficulty of electrochemical reaction. As Pact changes with temperatures over a wide range, Arrhenius formula is employed to modify the parameter [23]:
. ref Pact ¼ Pact exp lact 1 Tref 1=T
(19)
(18)
Theoretically, the basic shape of terminal voltage curve is directly affected by open circuit voltage which is determined by electrode characteristic. The degree of Li-ion diffusion and polarization which is affected by ambient temperature has certain effect on terminal voltage plateau.
Eocv ðtÞ ¼ Up y0 þ Dy ð1 socðtÞÞ Un ðx0 Dx ð1 socðtÞÞÞ (21) Measured voltage can be regarded as Eocv. Least square fit is adopted to fit for initial stoichiometric numbers (x0 and y0) and their variation ranges (Dx and Dy). The fitting input variables are Eocv and soc. Then, Qp, Qn, and yofs can be calculated by Eqs. (19) and (4). 2.2.2. Estimation of ohmic resistance Ohmic resistance is measured by a battery resistance testing equipment (ZM3000E, ZEEMOO, China) at different ambient temperatures (0 C, 15 C, 25 C, 35 C, and 55 C) in a thermal incubator (Partner, PTC14003-M, China). Thermal resistance is tightly attached to the surface of the battery to obtain internal temperature as accurately as possible. The variations of ohmic resistance with respect to temperature are fitted by Eq. (17). 2.2.3. Reaction polarization calculation Dynamic operating condition is specially designed based on the response time of different physicochemical processes. Designed current load (25 C ambient temperature) and schematic diagram of voltage response are shown in Fig. 1. And ambient temperature is separately controlled to 0 C, 15 C, 25 C, 35 C and 55 C. When load current changes from zero to a certain value, as Li-ion concentration in the electrodes will not mutate immediately, DU which ref consists of hact and hohm occurs. With obtained Rohm andlohm in Section 2.2.2, hact can be indirectly calculated. ref According to Eqs.(12)e(15), fitting parameters Pact andlact can be acquired through least square fit. The fitting input variables are the sequences of hact, I, T, yavg and xavg. 2.2.4. Estimation of parameters related to diffusion processes Steady concentration deviations between solid phase average and surface stoichiometric numbers will be established after a period of transition time. Similarly, electrolyte concentration deviation between two current collectors of electrodes needs certain transition time to be stable. According to Eqs. (7), (8) and (10), stable deviations can be given as
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Fig. 1. Designed current load (25 C ambient temperature) and schematic diagram of voltage response. DU occurs when current load changes from zero to the certain value. Steady concentration deviations have been built at the selected points.
Dystable ðtÞ ¼ 2
tsp Qp
IðtÞ; Dxstable ðtÞ ¼ 2
tsn Qn
IðtÞ; Dcstable ðtÞ ¼ Pcon IðtÞ (22)
Steady concentration deviations have been built at the selected points. And terminal voltage can be described as
Uapp ðtÞ ¼ Up yavg þ Dystable Un xavg Dxstable hstable con ðtÞ hact ðtÞ Rohm $IðtÞ
. eðhcon FÞ=ð2RTðtÞð1tþ ÞÞ þ 1
DcðtÞ ¼ c0 eðhcon FÞ=ð2RTðtÞð1tþ ÞÞ 1
(26) With calculatedDc(t) and previously estimatedPcon, the discrete formation of te is found to be
te ðtkþ1 Þ ¼
Pcon $Iðtk Þ Dcðtk Þ $ðt t Þ Dcðtkþ1 Þ Dcðtk Þ kþ1 k
(27)
The average value of te is then considered to be the final estimated result.
(23) The sequence of Uapp(t) in Eq. (23) is directly measured and after transformation, we have
UðtÞ ¼ Up yavg ðtÞ þ Dystable ðtÞ Un xavg ðtÞ Dxstable ðtÞ RTðtÞ c þ Dcstable ðtÞ ð1 tþ Þln 0 2 F c0 Dcstable ðtÞ
(24) The sequence of U(t) on the left side of Eq. (24) is a merged fitting input variable, including three known sub parts: measured Uapp(t), calculated hact(t) and Rohm $IðtÞ. Least square fit is then ref employed to obtain tsp , tsn , Pcon and lcon simultaneously. Transforming Eq. (18), hcon is given as
hcon ðtÞ ¼ Eocv ðtÞ Uapp ðtÞ hact ðtÞ hohm ðtÞ
(25)
Then, hcon can be calculated with previous estimation results. Transforming Eq. (9), Dcl(t) is given as
2.3. Model validations In Section 2.1 and 2.2, a simplified mechanistic model for LiFePO4 battery is proposed and parameters of the model are obtained by excitation response analysis. The estimated parameters are listed in Table 1. Simulation and measurement results of battery terminal voltage under different operating scenarios are shown in Fig. 2. And statistic comparative results are shown in Table 2. The maximum absolute errors (MAE) and average absolute errors (AAE) of voltage between simulation and measurement under constant current loads are generally less than 500 mV and 30 mV, respectively. MAE and AAE of voltage under dynamic current load are 425.36 mV and 22.48 mV, respectively. In addition, discharge capacity estimation relative error ranges from 0.95% to 2.97%. When a battery is discharged under larger rate, the model gives relatively higher errors. We speculate that such result is due to the approximation of electrolyte concentration profile, for the effect of uneven reaction in electrolyte on modeling is neglected.
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Table 1 Values of estimated mechanistic parameters.
Table 3 Battery specifications.
Parameter
Value
Parameter
Value
Specification
Value
Dx () Dy () y0 () yofs () Qp (A s) Qn (A s) Qall (A s)
0.8004 0.4268 0.0305 0.5417 1.5450 104 8.2397 103 6.5949 103 0.0097
lohm(K)
744.8761 3.9591 105 1.2497 103 284.9677 5.0351 383.9544 1.4725 103 68.0082
Pack Dimension
Height:65.0 ± 0.2 mm Width:26.0 ± 0.3 mm Approx:90 g 2300 mAh 3.6 V LiFePO4 Graphite
Rref ( U) ohm
ref Pact (m1.5 mol0.5 s) lact(K) tsp (s) tsn (s) ref Pcon (mol m3 A1) lcon(K) te(s)
Weight Rated capacity Cut-off voltage of charging Positive material Negative material
3. SOC estimation method based on simplified mechanistic model When the simulated voltage gets close to the cut-off voltage, or battery actual SOC is very low, MAE of voltage between simulation and measurement mostly occurs. In practical application, when battery SOC is less than 10%, it will be stopped from working for the sake of safety. From this perspective, the proposed mechanistic model can accurately describe battery behaviors under different operating scenarios. Compared with our previous work [25], it has wider applicability with accurate discharge capacity estimation.
3.1. Stoichiometric numbers and battery SOC In simplified mechanistic model, solid phase average stoichiometric numbers of electrodes directly determine open circuit potential. As battery SOC can better describe internal state in a conventional manner, a functional relationship between stoichiometric numbers and battery SOC is established.
Fig. 2. Simulation and measurement results of battery terminal voltage under different operating scenarios.
Table 2 Simulation results at different discharge C-rates.
10 C 25 C 45 C
MAE (mV) AAE (mV) MAE (mV) AAE (mV) MAE (mV) AAE (mV)
0.1C
0.3C
1.0C
1.5C
2.0C
340.73 22.49 403.46 25.60 490.95 24.51
309.70 28.70 391.13 23.09 478.86 24.12
205.68 22.34 285.13 27.23 384.60 22.51
125.26 28.12 253.89 33.84 290.25 38.25
126.02 67.80 165.66 19.70 250.86 17.02
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Combining Eqs. (2), (3) and (20), we get
. Dy soc ¼ 1 yavg y0 soc ¼ 1 x0 xavg Dx
(28)
It is worth mentioning that stoichiometric number xavg can be obtained by yavg and yofs according to Eqs. (2)e(4). The two calculation results of SOC in Eq. (28) are theoretically identical.
3.2. Model-based SOC estimation method Cylindrical batteries manufactured by America labeled A123ANR26650M1A were used. The battery specifications are summarized in Table 3. Battery SOC estimation algorithm is compiled in MATLAB and the performance of SOC estimation is tested under laboratory circumstance. All the simulations are carried out on a PC with Intel Core i3 CPU 530 @ 2.93 GHz, 2.99 GB RAM and Windows operating system. With the knowledge of Section 3.1, it can be deduced that the core issue of battery SOC estimation is stoichiometric number yavg estimation in this work. Initial SOC guess is determined by stoichiometric number yavg. When a battery is fully charged, initial yavg is equal to y0; when a battery is fully discharged, initial yavg is approximately equal to (y0þDy). In addition, if battery state is unknown, according to Section 2.2.1 initial yavg can be found through the method of dichotomy by Eq. (21).
Fig. 3. The framework of model-based SOC estimation method.
Table 4 Procedure of battery SOC estimation based on simplified mechanistic model. Step
Discrete formula of simplified mechanistic model
1.Initialization 0 0 ¼ 0:8; ytlower socðt0 Þ ¼ f ðEocv ðt0 ÞÞ; ytupper
¼ 0:01; Dy0 ðt0 Þ ¼ 0; Dx0 ðt0 Þ ¼ 0; Dcl ðt0 Þ ¼ 0 yavg ðt0 Þ ¼ y0 þ Dy ð1 socðt0 ÞÞ; xavg ðt0 Þ ¼ ð1 yofs yavg ðt0 ÞÞQp =Qn 2.Estimation þ a) Priori estimation of yavgand xavg: time index update (from tk1 to tk where k ¼ 1, 2, …) þ Þ þ Iðtk1 ÞDt=Qp ; xavg ðtk Þ ¼ ð1 yofs yavg ðtk ÞÞQp =Qn yavg ðtk Þ ¼ yavg ðtk1 b) Open circuit voltageEocv(tk): s tsp 0 0 0 1 12 tn Iðt 0 Dy0 ðtk Þ ¼ Dy0 ðtk1 Þ þ t1s ð 12 k1 Þ Dx ðtk1 Þ Dt 7 Qp Iðtk1 Þ Dy ðtk1 ÞÞDt; Dx ðtk Þ ¼ Dx ðtk1 Þ þ ts 7 Qn p
n
ysurf ðtk Þ ¼ yavg ðtk Þ þ Dy0 ðtk Þ; xsurf ðtk Þ ¼ xavg ðtk Þ Dx0 ðtk ÞEocv ðtk Þ ¼ Up ðysurf ðtk ÞÞ Un ðxsurf ðtk ÞÞ Reaction polarization overpotentialhact(tk): Pact ðtk Þ ¼ Pact expðlact ð1=Tref 1=Tðtk ÞÞÞ ref
mp ðtk Þ ¼ 6Q 1cl 0:5 p 0
hact ðtk Þ ¼
1 Pact ðtk ÞIðtk Þ; mn ðtk Þ ð1ysurf ðtk ÞÞ0:5 ðysurf ðtk ÞÞ0:5
1 ¼ 6Q 1cl 0:5 Pact ðtk ÞIðtk Þ ð1xsurf ðtk ÞÞ0:5 ðxsurf ðtk ÞÞ0:5 n 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2n ðtk Þ þ 1 þ mn ðtk ÞÞ þ lnð m2p ðtk Þ þ 1 þ mp ðtk ÞÞÞ
2RTðtk Þ ðlnð F
Concentration polarization overpotentialhcon(tk): ! kÞ hcon ðtk Þ ¼ 2RTðt ð1 tþ Þln F
cl0 þDcl ðtk Þ cl0 Dcl ðtk Þ
ref expðlcon ð1=Tref 1=Tðtk ÞÞÞ; Dcl ðtk Þ ¼ Dcl ðtk1 Þ þ t1e ðPcon ðtk ÞIðtk1 Þ Dcl ðtk1 ÞÞDt Pcon ðtk Þ ¼ Pcon
Ohmic polarization overpotentialhohm(tk):
Rohm ðtk Þ ¼ Rref expðlohm ð1=Tref 1=Tðtk ÞÞÞ; hohm ðtk Þ ¼ Rohm ðtk ÞIðtk Þ ohm c) Posterior estimation of yavgand xavg: measured ðt Þ; U ðt Þ ¼ E error ¼ jU ðt Þ U ðt Þj Let U1 ðtk Þ ¼ Uapp ocv ðtk Þ hact ðtk Þ hcon ðtk Þ hohm ðtk Þ; DUtk 2 k 1 k 2 k k tk tk Þ; y Þ ¼ ðytk If DUterror > V and U >U , then y ¼ y ðt ðt þ y Þ=2 avg avg 2 1 upper threshold k k lower lower k tk tk tk error If DUtk > Vthreshold and U2
t
t
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Fig. 4. SOC estimation and error results under constant current load (0.5C) at 25 C ambient temperature.
The framework of model-based SOC estimation method is shown in Fig. 3 and a detailed SOC estimation procedure is also listed in Table 4. The numerical value of Vthreshold is determined by sensor precision. Such strategy can prevent the measurement noises from affecting SOC estimation performance. The method of dichotomy is
employed to revise yavg(tk) and xavg(tk) during state updating process. If the model output deviates from the measurement, the algorithm will accordingly adjust the values of ylower(tkþ1) and yupper(tkþ1) at current index, of which the previous values vary with estimated yavg(tk) at last index. The computational efficiency of the method will be discussed in the following section.
Fig. 5. SOC estimation and error results under dynamic current load at 25 C ambient temperature.
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Fig. 6. SOC estimation and error results under constant current load (0.5C) at 45 C and 10 C ambient temperatures.
Fig. 7. SOC estimation and error results with random noises under constant current load (0.5C) at 25 C ambient temperature.
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Fig. 8. SOC estimation and error results with random noises under dynamic current load at 25 C ambient temperature.
4. SOC estimation results and discussion To assess the proposed SOC estimation method based on simplified mechanistic model, simulations and measurements are performed. In this section, the accuracy, applicability and robustness of the proposed method will be systematically analyzed and assessed. Battery real SOC is achieved through ampere-hour counting method, which serves as the reference.
4.1. SOC estimation results under various current loads The developed method is validated using constant current load at 25 C ambient temperature. Estimation and error results are shown in Fig. 4. It is revealed that the maximum SOC error is less than 2.5% over the entire SOC range. MAE and AAE between voltage simulations and measurements are 36.8 mV and 18.7 mV, respectively. Fig. 5 shows SOC estimation and error results under dynamic
current loads. It can be seen that the level of estimation error is comparable with that under constant current. MAE and AAE between voltage simulations and measurements are less than 40 mV and 24.5 mV, respectively. SOC estimation error decreases with time. On the whole, the proposed method can accurately estimate SOC and predict the cut-off voltage. In addition, the computational efficiency of the method is investigated in our work. The maximum indexes of k under dynamic and constant current load are 20289 and 600, and time consumption amount to 24.87s and 0.817s, respectively. The average elapsed time of each iterative calculation is about 1.3 ms, which indicates that the proposed approach has low computational cost.
4.2. SOC estimation results under different ambient temperatures SOC estimation and error results at constant current loads at
J. Li et al. / Energy 114 (2016) 1266e1276
45 C and 10 C ambient temperatures are shown in Fig. 6. The MAE between SOC estimation and simulation under two ambient temperature is less than 2.5%. MAE and AAE between voltage simulations and measurements are 39.5 mV, 17.7 mV (45 C) and 40.0 mV, 22.5 mV (10 C), respectively. As is seen from Fig. 6 (c), estimated SOC at 45 C ambient temperature gets closer to the reference gradually. While, estimated SOC under 10 C ambient temperature converges to the reference gradually over the former half range of SOC. And it deviated from the reference over the latter half range of SOC. In order to maintain the accuracy of model simulation, the estimation algorithm automatically revises yavg(tk) and xavg(tk) during state updating process, which can account for such result of SOC estimation. Generally, the proposed method can provide an accurate estimation result under different ambient temperatures.
4.3. Effects of measurement noises on estimation results As the precision of voltage sensor is generally ±5 mV and the error of current sensor is within 1%, measurements of voltage and current are artificially added with random noises of which the amplitudes are ±10 mV and ±100 mA, respectively, to illustrate the robustness of the proposed SOC estimation method. SOC estimation and error results under constant current load (0.5C) at 25 C ambient temperature are shown in Fig. 7. SOC estimation result shows good agreement for MAE of SOC less than 2.50%, and MAE and AAE of terminal voltage are 39.3 mV and 15.7 mV, respectively, which is comparable to that without additional voltage and current noises in Fig. 4. Fig. 8 depicts the SOC estimation and error results under dynamic current load. Similarly, SOC estimation result shows good agreement for MAE of SOC less than 1.50%. On the whole, the proposed method has certain antiinterference ability. If the value of Vthreshold decreases, according to the estimation algorithm in Table 4, the accuracy of model simulation will be further enhanced. However, the effect of measurable noises on the accuracy of model simulation and SOC estimation will be accordingly stronger.
5. Conclusion This paper firstly establishes a simplified mechanistic model and proposes a model parameter estimation method for LiFePO4 battery. Essential verifications of model accuracy are conducted and statistic results illustrate that the proposed mechanistic model can accurately describe battery behaviors under different operating scenarios. A SOC estimation method for LiFePO4 battery which can provide an accurate result under different operating scenarios and has low computational cost is then presented. Measurements of voltage and current are artificially added with random noises to illustrate the robustness of the proposed SOC estimation method. It is revealed that the proposed method has certain anti-interference ability. When the SOC of battery pack is defined as the minimum value of single cell of the pack, the proposed method of SOC estimation can be directly expanded from single cell to battery pack, which indicates that the proposed SOC estimation method has potential application in EVs. As LiFePO4 battery has a relatively flat OCV-SOC curve over the range of 20e80% SOC, how to revise initial SOC error leaves us room for future research. In addition, researches on model parameter variation during battery aging process are also our future work.
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Acknowledgment We thank the National Natural Science Foundation of China (No. 51477037) for financial support. We sincerely appreciate the significant help on translation supported by Miss. Jia Xie. References [1] Farmann A, Waag W, Marongiu A, Sauer DU. Critical review of on-board capacity estimation techniques for lithium-ion batteries in electric and hybrid electric vehicles. J Power Sources 2015;281:114e30. [2] Xiong R, Sun F, Gong X, Gao C. A data-driven based adaptive state of charge estimator of lithium-ion polymer battery used in electric vehicles. Appl Energy 2014;113:1421e33. [3] He Y, Liu X, Zhang C, Chen Z. A new model for State-of-Charge (SOC) estimation for high-power Li-ion batteries. Appl Energy 2013;101:808e14. [4] Zhang C, Li K, Pei L, Zhu C. An integrated approach for real-time model-based state-of-charge estimation of lithium-ion batteries. J Power Sources 2015;283: 24e36. [5] Hu XS, Li SB, Peng H. A comparative study of equivalent circuit models for Liion batteries. J Power Sources 2012;198:359e67. [6] Lee S, Kim J, Lee J, Cho BH. State-of-charge and capacity estimation of lithiumion battery using a new open-circuit voltage versus state-of-charge. J Power Sources 2008;185(2):1367e73. [7] Plett GL. Extended Kalman filtering for battery management systems of LiPBbased HEV battery packs. J Power Sources 2004;134(2):277e92. [8] Sun F, Hu X, Zou Y, Li S. Adaptive unscented Kalman filtering for state of charge estimation of a lithium-ion battery for electric vehicles. Energy 2011;36(5):3531e40. [9] Remmlinger J, Buchholz M, Meiler M, Bernreuter P, Dietmayer K. State-ofhealth monitoring of lithium-ion batteries in electric vehicles by on-board internal resistance estimation. J Power Sources 2011;196(12):5357e63. [10] Doyle M, Fuller TF, Newman J. Modeling of galvanostatic charge and discharge of the lithium polymer insertion cell. J Electrochem Soc 1993;140(6): 1526e33. [11] Xu M, Zhang Z, Wang X, Jia L, Yang L. A pseudo three-dimensional electrochemical-thermal model of a prismatic LiFePO4 battery during discharge process. Energy 2015;80:303e17. [12] Saw LH, Ye Y, Tay AAO. Electrochemical-thermal analysis of 18650 lithium iron phosphate cell. Energy Convers Manag 2013;75:162e74. [13] Eddahech A, Briat O, Vinassa J-M. Thermal characterization of a high-power lithium-ion battery: potentiometric and calorimetric measurement of entropy changes. Energy 2013;61:432e9. [14] Jeon DH, Baek SM. Thermal modeling of cylindrical lithium ion battery during discharge cycle. Energy Convers Manag 2011;52(8e9):2973e81. [15] Dong Hyup J, Seung Man B. Thermal modeling of cylindrical lithium ion battery during discharge cycle. Energy Convers Manag 2011;52(8e9): 2973e81. [16] Ouyang MG, Liu GM, Lu LG, Li JQ, Han XB. Enhancing the estimation accuracy in low state-of-charge area: a novel onboard battery model through surface state of charge determination. J Power Sources 2014;270:221e37. [17] Han XB, Ouyang MG, Lu LG, Li JQ. 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;278:802e13. [18] Bizeray AM, Zhao S, Duncan SR, Howey DA. Lithium-ion battery thermalelectrochemical model-based state estimation using orthogonal collocation and a modified extended Kalman filter. J Power Sources 2015;296:400e12. [19] Meng G, Sikha G, White RE. Single-particle model for a lithium-ion cell: thermal behavior. J Electrochem Soc 2011;158(2):A122e32. [20] Baba N, Yoshida H, Nagaoka M, Okuda C, Kawauchi S. Numerical simulation of thermal behavior of lithium-ion secondary batteries using the enhanced single particle model. J Power Sources 2014;252:214e28. [21] Luo W, Lyu C, Wang L, Zhang L. An approximate solution for electrolyte concentration distribution in physics-based lithium-ion cell models. Microelectron Reliab 2013;53(6):797e804. [22] Tatsukawa E, Tamura K. Activity correction on electrochemical reaction and diffusion in lithium intercalation electrodes for discharge/charge simulation by single particle model. Electrochim. Acta 2014;115:75e85. [23] Li J, Wang L, Lyu C, Wang H, Liu X. New method for parameter estimation of an electrochemical-thermal coupling model for LiCoO2 battery. J Power Sources 2016;307:220e30. [24] Rahimian SK, Rayman S, White RE. Extension of physics-based single particle model for higher charge-discharge rates. J Power Sources 2013;224:180e94. [25] Li J, Wang L, Lyu C, Zhang L, Wang H. Discharge capacity estimation for Li-ion batteries based on particle filter under multi-operating conditions. Energy 2015;86:638e48.
Nomenclature cl0 : initial electrolyte concentration (mol m3) Dx: maximum variation range of stoichiometric number x ()
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Dy: maximum variation range of stoichiometric number y () Eocv: open circuit voltage, OCV (V) F: Faraday's constant (C mol1) Pact: coefficient of anode reaction polarization (m1.5 mol0.5 s) ref Pact : coefficient of anode reaction polarization at reference temperature (m1.5 mol0.5 s) Pcon: proportional coefficient of liquid phase diffusion (mol m3 A1) ref Pcon : proportional coefficient of liquid phase diffusion at reference temperature (mol m3 A1) Qall: total capacity at discharge rate of 0.02C (A s) Qi, i ¼ n, p:: capacities of effective active material in the electrodes (A s) R: ideal gas constant (J mol1 K1) Rohm: ohmic resistance (U) ref Rohm : ohmic resistance at reference temperature (U) soc: state of charge () T: battery internal temperature (K) Tref: reference temperature (K) t: time (s) tþ: transport number () Ui, i ¼ n, p:: open circuit voltage (V) x0: initial stoichiometric number of negative electrode () xavg: solid phase average stoichiometric number of negative electrode () xsurf: solid phase surface stoichiometric number of negative electrode () y0: initial stoichiometric number of positive electrode () yavg: solid phase average stoichiometric number of positive electrode ()
ysurf: solid phase surface stoichiometric number of positive electrode () yofs: offset of relative position of stoichiometric numbers () Dcl : electrolyte concentration deviation between two current collectors (mol m3) Dx0 : deviations between xsurf and xavg () Dy0 : deviations between ysurf and yavg () tsi ; i ¼ n; p: solid phase diffusion time constants of each electrode (s) te: time constant of liquid phase diffusion (s) hact: reaction polarization overpotential (V) hcon: concentration polarization overpotential (V) hohm: ohmic polarization overpotential (V) lact: proportional coefficient of activation energy (K) lcon: proportional coefficient of activation energy (K) lohm: proportional coefficient of activation energy (K) Subscript act: reaction avg: average con: concentration max: maximum n: negative ohm: ohmic p:: positive ref: reference
Contents lists available at ScienceDirect
Energy journal homepage: www.elsevier.com/locate/energy
A method for SOC estimation based on simplified mechanistic model for LiFePO4 battery Junfu Li a, Qingzhi Lai a, Lixin Wang a, Chao Lyu a, *, Han Wang b a b
School of Electrical Engineering and Automation, Harbin Institute of Technology, Harbin, 150001, China School of Astronautics, Harbin Institute of Technology, Harbin, 150001, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 7 April 2016 Received in revised form 11 August 2016 Accepted 22 August 2016
Accurate battery state of charge (SOC) estimation conduces to establishing an ideal charging and discharging strategy for battery management system (BMS). And it can also prevent serious damage to battery (pack) from over-charging or over-discharging. This paper firstly establishes a simplified mechanistic model for LiFePO4 battery. Parameter identification conditions are originally designed based on excitation response analysis and essential verifications in terms of model accuracy are conducted. The functional relationship between mechanistic parameters and battery SOC is then established. And lastly, SOC estimation method based on the proposed model is presented. The contributions of this paper are listed as follows: (i) the proposed mechanistic model with obtained parameters can accurately describe LiFePO4 battery behaviors and predict discharge capacity under different operating conditions, (ii) the proposed SOC estimation method based on the battery model has certain applicability and robustness. Analysis and assessment of accuracy of the proposed method indicate that SOC estimation accuracy is acceptable. With the improvement of model simulation, SOC estimation accuracy can be further refined. © 2016 Elsevier Ltd. All rights reserved.
Keywords: State of charge Battery management system Simplified mechanistic model Excitation response analysis Applicability and robustness
1. Introduction Li-ion batteries, as primary power suppliers for many devices or systems, have been attractive due to high specific power and energy density, high cycle lifetime and low self-discharge rate [1]. They have been widely adopted in battery management system (BMS) of electric vehicles (EVs). An advanced BMS can not only accurately estimate the state of charge (SOC) of the batteries connected in series/parallel configuration, but avoid the potential safety hazard and ensure efficient operation as well. However, inaccurate SOC estimation and unbalance problem make it hard for BMS to determine an effective management strategy [2]. Battery SOC is a key evaluation index for BMS which can serve as an indicator of remaining capacity [3]. An accurate battery SOC estimation method plays a vital role in avoiding over-charging and overdischarging and addressing unbalance problem. In addition, it can also be used for developing power and energy management strategies [4].
* Corresponding author. P.O. Box 404, Harbin Institute of Technology, No. 92, West Dazhi Street, Nangang District, Harbin, 150001, Heilongjiang, China. E-mail addresses: [email protected] (J. Li), [email protected] (Q. Lai), wlx@ hit.edu.cn (L. Wang), [email protected] (C. Lyu), [email protected] (H. Wang). http://dx.doi.org/10.1016/j.energy.2016.08.080 0360-5442/© 2016 Elsevier Ltd. All rights reserved.
In the past few years, a large amount of researches concentrated on battery SOC estimation, which had their own limits. One of the most commonly used methods is the ampere-hour counting, which is a kind of open-loop calculation methods [2]. Though it has simple formations and high calculation efficiency, initial offset for SOC and accumulated calculation errors can lead to bad SOC estimation performance [5]. To enhance SOC estimation precision, close-loop method was developed. An OCV-based method was adopted to recalibrate the accumulated errors. Apart from the above two commonly used methods, Kalman filter (KF) method [6], extended Kalman filter (EKF) method [7] and adaptive unscented Kalman filter (AUKF) [8] were also adopted to improve the performance of battery SOC estimation. While, there are also some limitations in such methods [3]. Battery models used in their algorithms must be accurate. Otherwise, unreliable estimation result will occur. Furthermore, measurement noises including system noise and observation noise must be submitted to Gaussian distribution. Reference [9] used Monte Carlo sampling methods to represent any probability density function and particle filter (PF) was employed to determine initial SOC. The proposed method showed good ability to correct initial value and track SOC variation. It is revealed that to increase the accuracy and accelerate the determination of the correct SOC, complex impedance models were needed.
J. Li et al. / Energy 114 (2016) 1266e1276
Generally, the performance of SOC estimation can be improved with accurate battery model. And an accurate battery model is able to describe the real states inside the battery. The pseudo 2 dimensional (P2D) model was based on theories of porous electrodes and concentrated solutions and was developed to capture the microcosmic behaviors of Li-ion at different operating conditions [10]. The development of the P2D model set the foundation for further research on electrical-thermal mechanistic models. A pseudo 3 dimensional model based on P2D model was developed by coupling the mass, charge, energy conservation, and cell electrochemical kinetics to study both electrochemical and thermal characteristics of the battery [11]. With a well developed electrochemical-thermal coupled model, thermal phenomenon can be better understood [12]. Much work has been concentrated on thermal analysis and management to enhance battery model accuracy and ensure battery operating safety [13e15]. As mechanistic models can accurately simulate the entire SOC, they are very appealing in BMS [16,17]. Reference [18] combined P2D model with an EKF algorithm to estimate battery SOC. The proposed state estimation algorithm was able to rapidly and accurately recover the model states with current, voltage and temperature measurements. However, the large computational cost of solving partial differential equations limits the practical application of these models and thus many efforts have concentrated on simplifying these models. References [19,20] took temperature into consideration to expand single particle model and the single particle thermal model showed good agreement with experimental voltage data under different discharge C-rates. Reference [21] obtained an approximate solution for electrolyte concentration distribution, which used a parabolic profile approximation assuming that the electrolyte concentration profile in the porous electrode was a parabolic polynomial in lateral direction at any time. Modified single particle model with the heterogeneous treatment given by the non-random two-liquid model was found to be the best in predicting the constant discharge/charge profiles [22]. With reduced or regrouped parameters, simplified models tend to be more suitable for embedded application in terms of battery SOC estimation in BMS, for they have both acceptable accuracy and less computational cost. The purpose of this work is to establish a method for SOC estimation based on a simplified mechanistic model for LiFePO4 battery. The novelty of our work are as follows: (i) the proposed model and model parameter estimation method have certain generality in terms of battery materials and individuals, (ii) a functional relationship between mechanistic parameters and battery SOC is originally established, (iii) a method for battery SOC estimation based on the simplified mechanistic model with low computational cost and high accuracy is proposed. This paper is organized as follows: in Section 2, a simplified mechanistic model together with parameter estimation method based on excitation response analysis is presented and essential verifications are conducted; in Section 3, a functional relationship between mechanistic parameters and battery SOC is established and estimation procedure is given in detail, followed by analysis and assessment of accuracy, applicability and robustness of the proposed method in Section 4. Finally, Section 5 concludes this paper. 2. Simplified mechanistic model and validation
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and acceptable accuracy is employed [21,23]. It is further assumed that internal average temperature of spiral battery is equal to the surface in the case that the cell has small capacity and thermal increment under large current load. Thus, surface temperature is a measureable variable, which is used to simulate battery overpotential and terminal voltage. Such assumption, on one hand, reduces computational cost of the model. There is no need to calculate battery heat generation and internal average temperature with iterative calculation. On the other hand, compared with our previous work [23], it simplifies the process of parameter estimation which will be discussed in detail in Section 2.2. The proposed mechanistic model is described as follows. 2.1.1. Open circuit voltage The solid phase surface concentration of the particle directly determines open circuit voltage (OCV). Therefore, OCV at reference temperature is described as
Eocv ðtÞ ¼ Up yavg Un xavg
(1)
Where Up and Un are open circuit voltage curves of electrodes [24]. yavg and xavg are average Li-ion concentration inside the particle of each electrode, which are given as
yavg ¼ y0 þ It Qp
(2)
xavg ¼ 1 yofs yavg Qp Qn
(3)
Where y0 and x0 are initial values of yavg and xavg, respectively. Qp and Qn theoretically represent the capacities of effective active materials. We define yofs as the offset of relative position which can be given as
yofs ¼ 1 y0 x0
Qn Qp
(4)
The above four parameters give a clear description on basic operating process. 2.1.2. Solid phase diffusion Concentration of Li-ion on the particle surface of each electrode can be described as [21].
ysurf ðtÞ ¼ yavg ðtÞ þ DyðtÞ
(5)
xsurf ðtÞ ¼ xavg ðtÞ DxðtÞ
(6)
The discrete formation ofDy andDx can be found to be 0
0
Dy ðtkþ1 Þ ¼ Dy ðtk Þ þ Dx0 ðtkþ1 Þ ¼ Dx0 ðtk Þ þ
! s 12 tp 0 Iðt Þ Dy ðtk Þ ðtkþ1 tk Þ 7 Qp k
1
tsp 1
tsn
12 tsn Iðtk Þ Dx0 ðtk Þ ðtkþ1 tk Þ 7 Qn
(7)
(8)
Where tsp and tsn are solid phase diffusion time constants of each electrode.
2.1. Simplified mechanistic model To accurately describe battery charge and discharge behaviors under different operating scenarios, it is essential to have a clear understanding on battery internal physicochemical processes. A simplified mechanistic model which has low computational cost
2.1.3. Liquid phase diffusion The effect of liquid phase diffusion can be equivalently seen as the establishment of concentration difference at current collectors of each electrode. The degree of concentration polarization can be expressed by one fused state variableDc. Therefore, we have [21].
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hcon ðtÞ ¼
2RTðtÞ c þ DcðtÞ ð1 tþ Þln 0 F c0 DcðtÞ
2.2. Parameter estimation
(9)
Where c0 is initial liquid phase Li-ion concentration. State equation ofDc is
Dcðtkþ1 Þ ¼ Dcðtk Þ þ
1
te
ðPcon Iðtk Þ Dcðtk ÞÞðtkþ1 tk Þ
(10)
Where Pcon andte are the proportion coefficient and time constant of liquid phase diffusion, respectively. Pcon with respect to temperature is fitted by Ref. [23].
Pcon ¼
ref Pcon
. exp lcon 1 Tref 1=T
(11)
Some parameters such as initial liquid phase Li-ion concentration c0 and effective heat dissipation area Rohm can be obtained directly. And the others need to be estimated. In this work, parameter identification conditions are originally designed and model parameters are obtained through excitation response analysis. As battery internal physicochemical processes have numerical coupling relations, it is essential to make some assumptions and approximations in step-by-step procedure of parameter estimation, which will be mentioned in each part. 2.2.1. Parameter estimation of basic working process According to the definitions in Section 2.1.1, Qp and Qn can be described as
Qp ¼ Qall =Dy; Qn ¼ Qall =Dx 2.1.4. Reaction polarization According to Butler-Volmer kinetics, reaction polarization overpotential hact can be expressed as [21].
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2RTðtÞ ln m2n ðtÞ þ 1 þ mn ðtÞ F qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ln m2p ðtÞ þ 1 þ mp ðtÞ
hact ðtÞ ¼
mp ðtÞ ¼
1 6Qp c0:5 0
1 ysurf
1 mn ðtÞ ¼ 6Qn c0:5 0
1 xsurf
1 0:5
1 0:5
ysurf
xsurf
0:5 Pact IðtÞ
0:5 Pact IðtÞ
(12)
(13)
(14)
(15)
2.1.5. Ohmic polarization Additionally, the ohmic polarization overpotential hohm can be uniformly expressed as [21].
hohm ðtÞ ¼ Rohm IðtÞ
(16)
Where, Rohm is ohmic resistance expressed in the form of a lumped parameter. The variations of ohmic resistance with respect to temperature are fitted by the following formula [23].
. ref Rohm ¼ Rohm exp lohm 1 Tref 1=T
(17)
Then, terminal voltage Uapp is found to be
Uapp ðtÞ ¼ Eocv ðtÞ hcon ðtÞ hact ðtÞ hohm ðtÞ
Where Qall is total discharge capacity measured at rate of 0.02C. Dx and Dy are maximum variation ranges of stoichiometric number x and y, respectively. Using method of ampere-hour counting, SOC of the battery can be defined as [5,8].
socðtÞ ¼ 1 It=Qall
(20)
Neglecting the effects of polarization and diffusion process inside the battery, ideal OCV can be described as
Pact is a combined parameter and comparable to electrochemical reaction rate constant, which reflects the difficulty of electrochemical reaction. As Pact changes with temperatures over a wide range, Arrhenius formula is employed to modify the parameter [23]:
. ref Pact ¼ Pact exp lact 1 Tref 1=T
(19)
(18)
Theoretically, the basic shape of terminal voltage curve is directly affected by open circuit voltage which is determined by electrode characteristic. The degree of Li-ion diffusion and polarization which is affected by ambient temperature has certain effect on terminal voltage plateau.
Eocv ðtÞ ¼ Up y0 þ Dy ð1 socðtÞÞ Un ðx0 Dx ð1 socðtÞÞÞ (21) Measured voltage can be regarded as Eocv. Least square fit is adopted to fit for initial stoichiometric numbers (x0 and y0) and their variation ranges (Dx and Dy). The fitting input variables are Eocv and soc. Then, Qp, Qn, and yofs can be calculated by Eqs. (19) and (4). 2.2.2. Estimation of ohmic resistance Ohmic resistance is measured by a battery resistance testing equipment (ZM3000E, ZEEMOO, China) at different ambient temperatures (0 C, 15 C, 25 C, 35 C, and 55 C) in a thermal incubator (Partner, PTC14003-M, China). Thermal resistance is tightly attached to the surface of the battery to obtain internal temperature as accurately as possible. The variations of ohmic resistance with respect to temperature are fitted by Eq. (17). 2.2.3. Reaction polarization calculation Dynamic operating condition is specially designed based on the response time of different physicochemical processes. Designed current load (25 C ambient temperature) and schematic diagram of voltage response are shown in Fig. 1. And ambient temperature is separately controlled to 0 C, 15 C, 25 C, 35 C and 55 C. When load current changes from zero to a certain value, as Li-ion concentration in the electrodes will not mutate immediately, DU which ref consists of hact and hohm occurs. With obtained Rohm andlohm in Section 2.2.2, hact can be indirectly calculated. ref According to Eqs.(12)e(15), fitting parameters Pact andlact can be acquired through least square fit. The fitting input variables are the sequences of hact, I, T, yavg and xavg. 2.2.4. Estimation of parameters related to diffusion processes Steady concentration deviations between solid phase average and surface stoichiometric numbers will be established after a period of transition time. Similarly, electrolyte concentration deviation between two current collectors of electrodes needs certain transition time to be stable. According to Eqs. (7), (8) and (10), stable deviations can be given as
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Fig. 1. Designed current load (25 C ambient temperature) and schematic diagram of voltage response. DU occurs when current load changes from zero to the certain value. Steady concentration deviations have been built at the selected points.
Dystable ðtÞ ¼ 2
tsp Qp
IðtÞ; Dxstable ðtÞ ¼ 2
tsn Qn
IðtÞ; Dcstable ðtÞ ¼ Pcon IðtÞ (22)
Steady concentration deviations have been built at the selected points. And terminal voltage can be described as
Uapp ðtÞ ¼ Up yavg þ Dystable Un xavg Dxstable hstable con ðtÞ hact ðtÞ Rohm $IðtÞ
. eðhcon FÞ=ð2RTðtÞð1tþ ÞÞ þ 1
DcðtÞ ¼ c0 eðhcon FÞ=ð2RTðtÞð1tþ ÞÞ 1
(26) With calculatedDc(t) and previously estimatedPcon, the discrete formation of te is found to be
te ðtkþ1 Þ ¼
Pcon $Iðtk Þ Dcðtk Þ $ðt t Þ Dcðtkþ1 Þ Dcðtk Þ kþ1 k
(27)
The average value of te is then considered to be the final estimated result.
(23) The sequence of Uapp(t) in Eq. (23) is directly measured and after transformation, we have
UðtÞ ¼ Up yavg ðtÞ þ Dystable ðtÞ Un xavg ðtÞ Dxstable ðtÞ RTðtÞ c þ Dcstable ðtÞ ð1 tþ Þln 0 2 F c0 Dcstable ðtÞ
(24) The sequence of U(t) on the left side of Eq. (24) is a merged fitting input variable, including three known sub parts: measured Uapp(t), calculated hact(t) and Rohm $IðtÞ. Least square fit is then ref employed to obtain tsp , tsn , Pcon and lcon simultaneously. Transforming Eq. (18), hcon is given as
hcon ðtÞ ¼ Eocv ðtÞ Uapp ðtÞ hact ðtÞ hohm ðtÞ
(25)
Then, hcon can be calculated with previous estimation results. Transforming Eq. (9), Dcl(t) is given as
2.3. Model validations In Section 2.1 and 2.2, a simplified mechanistic model for LiFePO4 battery is proposed and parameters of the model are obtained by excitation response analysis. The estimated parameters are listed in Table 1. Simulation and measurement results of battery terminal voltage under different operating scenarios are shown in Fig. 2. And statistic comparative results are shown in Table 2. The maximum absolute errors (MAE) and average absolute errors (AAE) of voltage between simulation and measurement under constant current loads are generally less than 500 mV and 30 mV, respectively. MAE and AAE of voltage under dynamic current load are 425.36 mV and 22.48 mV, respectively. In addition, discharge capacity estimation relative error ranges from 0.95% to 2.97%. When a battery is discharged under larger rate, the model gives relatively higher errors. We speculate that such result is due to the approximation of electrolyte concentration profile, for the effect of uneven reaction in electrolyte on modeling is neglected.
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Table 1 Values of estimated mechanistic parameters.
Table 3 Battery specifications.
Parameter
Value
Parameter
Value
Specification
Value
Dx () Dy () y0 () yofs () Qp (A s) Qn (A s) Qall (A s)
0.8004 0.4268 0.0305 0.5417 1.5450 104 8.2397 103 6.5949 103 0.0097
lohm(K)
744.8761 3.9591 105 1.2497 103 284.9677 5.0351 383.9544 1.4725 103 68.0082
Pack Dimension
Height:65.0 ± 0.2 mm Width:26.0 ± 0.3 mm Approx:90 g 2300 mAh 3.6 V LiFePO4 Graphite
Rref ( U) ohm
ref Pact (m1.5 mol0.5 s) lact(K) tsp (s) tsn (s) ref Pcon (mol m3 A1) lcon(K) te(s)
Weight Rated capacity Cut-off voltage of charging Positive material Negative material
3. SOC estimation method based on simplified mechanistic model When the simulated voltage gets close to the cut-off voltage, or battery actual SOC is very low, MAE of voltage between simulation and measurement mostly occurs. In practical application, when battery SOC is less than 10%, it will be stopped from working for the sake of safety. From this perspective, the proposed mechanistic model can accurately describe battery behaviors under different operating scenarios. Compared with our previous work [25], it has wider applicability with accurate discharge capacity estimation.
3.1. Stoichiometric numbers and battery SOC In simplified mechanistic model, solid phase average stoichiometric numbers of electrodes directly determine open circuit potential. As battery SOC can better describe internal state in a conventional manner, a functional relationship between stoichiometric numbers and battery SOC is established.
Fig. 2. Simulation and measurement results of battery terminal voltage under different operating scenarios.
Table 2 Simulation results at different discharge C-rates.
10 C 25 C 45 C
MAE (mV) AAE (mV) MAE (mV) AAE (mV) MAE (mV) AAE (mV)
0.1C
0.3C
1.0C
1.5C
2.0C
340.73 22.49 403.46 25.60 490.95 24.51
309.70 28.70 391.13 23.09 478.86 24.12
205.68 22.34 285.13 27.23 384.60 22.51
125.26 28.12 253.89 33.84 290.25 38.25
126.02 67.80 165.66 19.70 250.86 17.02
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Combining Eqs. (2), (3) and (20), we get
. Dy soc ¼ 1 yavg y0 soc ¼ 1 x0 xavg Dx
(28)
It is worth mentioning that stoichiometric number xavg can be obtained by yavg and yofs according to Eqs. (2)e(4). The two calculation results of SOC in Eq. (28) are theoretically identical.
3.2. Model-based SOC estimation method Cylindrical batteries manufactured by America labeled A123ANR26650M1A were used. The battery specifications are summarized in Table 3. Battery SOC estimation algorithm is compiled in MATLAB and the performance of SOC estimation is tested under laboratory circumstance. All the simulations are carried out on a PC with Intel Core i3 CPU 530 @ 2.93 GHz, 2.99 GB RAM and Windows operating system. With the knowledge of Section 3.1, it can be deduced that the core issue of battery SOC estimation is stoichiometric number yavg estimation in this work. Initial SOC guess is determined by stoichiometric number yavg. When a battery is fully charged, initial yavg is equal to y0; when a battery is fully discharged, initial yavg is approximately equal to (y0þDy). In addition, if battery state is unknown, according to Section 2.2.1 initial yavg can be found through the method of dichotomy by Eq. (21).
Fig. 3. The framework of model-based SOC estimation method.
Table 4 Procedure of battery SOC estimation based on simplified mechanistic model. Step
Discrete formula of simplified mechanistic model
1.Initialization 0 0 ¼ 0:8; ytlower socðt0 Þ ¼ f ðEocv ðt0 ÞÞ; ytupper
¼ 0:01; Dy0 ðt0 Þ ¼ 0; Dx0 ðt0 Þ ¼ 0; Dcl ðt0 Þ ¼ 0 yavg ðt0 Þ ¼ y0 þ Dy ð1 socðt0 ÞÞ; xavg ðt0 Þ ¼ ð1 yofs yavg ðt0 ÞÞQp =Qn 2.Estimation þ a) Priori estimation of yavgand xavg: time index update (from tk1 to tk where k ¼ 1, 2, …) þ Þ þ Iðtk1 ÞDt=Qp ; xavg ðtk Þ ¼ ð1 yofs yavg ðtk ÞÞQp =Qn yavg ðtk Þ ¼ yavg ðtk1 b) Open circuit voltageEocv(tk): s tsp 0 0 0 1 12 tn Iðt 0 Dy0 ðtk Þ ¼ Dy0 ðtk1 Þ þ t1s ð 12 k1 Þ Dx ðtk1 Þ Dt 7 Qp Iðtk1 Þ Dy ðtk1 ÞÞDt; Dx ðtk Þ ¼ Dx ðtk1 Þ þ ts 7 Qn p
n
ysurf ðtk Þ ¼ yavg ðtk Þ þ Dy0 ðtk Þ; xsurf ðtk Þ ¼ xavg ðtk Þ Dx0 ðtk ÞEocv ðtk Þ ¼ Up ðysurf ðtk ÞÞ Un ðxsurf ðtk ÞÞ Reaction polarization overpotentialhact(tk): Pact ðtk Þ ¼ Pact expðlact ð1=Tref 1=Tðtk ÞÞÞ ref
mp ðtk Þ ¼ 6Q 1cl 0:5 p 0
hact ðtk Þ ¼
1 Pact ðtk ÞIðtk Þ; mn ðtk Þ ð1ysurf ðtk ÞÞ0:5 ðysurf ðtk ÞÞ0:5
1 ¼ 6Q 1cl 0:5 Pact ðtk ÞIðtk Þ ð1xsurf ðtk ÞÞ0:5 ðxsurf ðtk ÞÞ0:5 n 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2n ðtk Þ þ 1 þ mn ðtk ÞÞ þ lnð m2p ðtk Þ þ 1 þ mp ðtk ÞÞÞ
2RTðtk Þ ðlnð F
Concentration polarization overpotentialhcon(tk): ! kÞ hcon ðtk Þ ¼ 2RTðt ð1 tþ Þln F
cl0 þDcl ðtk Þ cl0 Dcl ðtk Þ
ref expðlcon ð1=Tref 1=Tðtk ÞÞÞ; Dcl ðtk Þ ¼ Dcl ðtk1 Þ þ t1e ðPcon ðtk ÞIðtk1 Þ Dcl ðtk1 ÞÞDt Pcon ðtk Þ ¼ Pcon
Ohmic polarization overpotentialhohm(tk):
Rohm ðtk Þ ¼ Rref expðlohm ð1=Tref 1=Tðtk ÞÞÞ; hohm ðtk Þ ¼ Rohm ðtk ÞIðtk Þ ohm c) Posterior estimation of yavgand xavg: measured ðt Þ; U ðt Þ ¼ E error ¼ jU ðt Þ U ðt Þj Let U1 ðtk Þ ¼ Uapp ocv ðtk Þ hact ðtk Þ hcon ðtk Þ hohm ðtk Þ; DUtk 2 k 1 k 2 k k tk tk Þ; y Þ ¼ ðytk If DUterror > V and U >U , then y ¼ y ðt ðt þ y Þ=2 avg avg 2 1 upper threshold k k lower lower k tk tk tk error If DUtk > Vthreshold and U2
t
t
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Fig. 4. SOC estimation and error results under constant current load (0.5C) at 25 C ambient temperature.
The framework of model-based SOC estimation method is shown in Fig. 3 and a detailed SOC estimation procedure is also listed in Table 4. The numerical value of Vthreshold is determined by sensor precision. Such strategy can prevent the measurement noises from affecting SOC estimation performance. The method of dichotomy is
employed to revise yavg(tk) and xavg(tk) during state updating process. If the model output deviates from the measurement, the algorithm will accordingly adjust the values of ylower(tkþ1) and yupper(tkþ1) at current index, of which the previous values vary with estimated yavg(tk) at last index. The computational efficiency of the method will be discussed in the following section.
Fig. 5. SOC estimation and error results under dynamic current load at 25 C ambient temperature.
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Fig. 6. SOC estimation and error results under constant current load (0.5C) at 45 C and 10 C ambient temperatures.
Fig. 7. SOC estimation and error results with random noises under constant current load (0.5C) at 25 C ambient temperature.
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Fig. 8. SOC estimation and error results with random noises under dynamic current load at 25 C ambient temperature.
4. SOC estimation results and discussion To assess the proposed SOC estimation method based on simplified mechanistic model, simulations and measurements are performed. In this section, the accuracy, applicability and robustness of the proposed method will be systematically analyzed and assessed. Battery real SOC is achieved through ampere-hour counting method, which serves as the reference.
4.1. SOC estimation results under various current loads The developed method is validated using constant current load at 25 C ambient temperature. Estimation and error results are shown in Fig. 4. It is revealed that the maximum SOC error is less than 2.5% over the entire SOC range. MAE and AAE between voltage simulations and measurements are 36.8 mV and 18.7 mV, respectively. Fig. 5 shows SOC estimation and error results under dynamic
current loads. It can be seen that the level of estimation error is comparable with that under constant current. MAE and AAE between voltage simulations and measurements are less than 40 mV and 24.5 mV, respectively. SOC estimation error decreases with time. On the whole, the proposed method can accurately estimate SOC and predict the cut-off voltage. In addition, the computational efficiency of the method is investigated in our work. The maximum indexes of k under dynamic and constant current load are 20289 and 600, and time consumption amount to 24.87s and 0.817s, respectively. The average elapsed time of each iterative calculation is about 1.3 ms, which indicates that the proposed approach has low computational cost.
4.2. SOC estimation results under different ambient temperatures SOC estimation and error results at constant current loads at
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45 C and 10 C ambient temperatures are shown in Fig. 6. The MAE between SOC estimation and simulation under two ambient temperature is less than 2.5%. MAE and AAE between voltage simulations and measurements are 39.5 mV, 17.7 mV (45 C) and 40.0 mV, 22.5 mV (10 C), respectively. As is seen from Fig. 6 (c), estimated SOC at 45 C ambient temperature gets closer to the reference gradually. While, estimated SOC under 10 C ambient temperature converges to the reference gradually over the former half range of SOC. And it deviated from the reference over the latter half range of SOC. In order to maintain the accuracy of model simulation, the estimation algorithm automatically revises yavg(tk) and xavg(tk) during state updating process, which can account for such result of SOC estimation. Generally, the proposed method can provide an accurate estimation result under different ambient temperatures.
4.3. Effects of measurement noises on estimation results As the precision of voltage sensor is generally ±5 mV and the error of current sensor is within 1%, measurements of voltage and current are artificially added with random noises of which the amplitudes are ±10 mV and ±100 mA, respectively, to illustrate the robustness of the proposed SOC estimation method. SOC estimation and error results under constant current load (0.5C) at 25 C ambient temperature are shown in Fig. 7. SOC estimation result shows good agreement for MAE of SOC less than 2.50%, and MAE and AAE of terminal voltage are 39.3 mV and 15.7 mV, respectively, which is comparable to that without additional voltage and current noises in Fig. 4. Fig. 8 depicts the SOC estimation and error results under dynamic current load. Similarly, SOC estimation result shows good agreement for MAE of SOC less than 1.50%. On the whole, the proposed method has certain antiinterference ability. If the value of Vthreshold decreases, according to the estimation algorithm in Table 4, the accuracy of model simulation will be further enhanced. However, the effect of measurable noises on the accuracy of model simulation and SOC estimation will be accordingly stronger.
5. Conclusion This paper firstly establishes a simplified mechanistic model and proposes a model parameter estimation method for LiFePO4 battery. Essential verifications of model accuracy are conducted and statistic results illustrate that the proposed mechanistic model can accurately describe battery behaviors under different operating scenarios. A SOC estimation method for LiFePO4 battery which can provide an accurate result under different operating scenarios and has low computational cost is then presented. Measurements of voltage and current are artificially added with random noises to illustrate the robustness of the proposed SOC estimation method. It is revealed that the proposed method has certain anti-interference ability. When the SOC of battery pack is defined as the minimum value of single cell of the pack, the proposed method of SOC estimation can be directly expanded from single cell to battery pack, which indicates that the proposed SOC estimation method has potential application in EVs. As LiFePO4 battery has a relatively flat OCV-SOC curve over the range of 20e80% SOC, how to revise initial SOC error leaves us room for future research. In addition, researches on model parameter variation during battery aging process are also our future work.
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Acknowledgment We thank the National Natural Science Foundation of China (No. 51477037) for financial support. We sincerely appreciate the significant help on translation supported by Miss. Jia Xie. References [1] Farmann A, Waag W, Marongiu A, Sauer DU. Critical review of on-board capacity estimation techniques for lithium-ion batteries in electric and hybrid electric vehicles. J Power Sources 2015;281:114e30. [2] Xiong R, Sun F, Gong X, Gao C. A data-driven based adaptive state of charge estimator of lithium-ion polymer battery used in electric vehicles. Appl Energy 2014;113:1421e33. [3] He Y, Liu X, Zhang C, Chen Z. A new model for State-of-Charge (SOC) estimation for high-power Li-ion batteries. Appl Energy 2013;101:808e14. [4] Zhang C, Li K, Pei L, Zhu C. An integrated approach for real-time model-based state-of-charge estimation of lithium-ion batteries. J Power Sources 2015;283: 24e36. [5] Hu XS, Li SB, Peng H. A comparative study of equivalent circuit models for Liion batteries. J Power Sources 2012;198:359e67. [6] Lee S, Kim J, Lee J, Cho BH. State-of-charge and capacity estimation of lithiumion battery using a new open-circuit voltage versus state-of-charge. J Power Sources 2008;185(2):1367e73. [7] Plett GL. Extended Kalman filtering for battery management systems of LiPBbased HEV battery packs. J Power Sources 2004;134(2):277e92. [8] Sun F, Hu X, Zou Y, Li S. Adaptive unscented Kalman filtering for state of charge estimation of a lithium-ion battery for electric vehicles. Energy 2011;36(5):3531e40. [9] Remmlinger J, Buchholz M, Meiler M, Bernreuter P, Dietmayer K. State-ofhealth monitoring of lithium-ion batteries in electric vehicles by on-board internal resistance estimation. J Power Sources 2011;196(12):5357e63. [10] Doyle M, Fuller TF, Newman J. Modeling of galvanostatic charge and discharge of the lithium polymer insertion cell. J Electrochem Soc 1993;140(6): 1526e33. [11] Xu M, Zhang Z, Wang X, Jia L, Yang L. A pseudo three-dimensional electrochemical-thermal model of a prismatic LiFePO4 battery during discharge process. Energy 2015;80:303e17. [12] Saw LH, Ye Y, Tay AAO. Electrochemical-thermal analysis of 18650 lithium iron phosphate cell. Energy Convers Manag 2013;75:162e74. [13] Eddahech A, Briat O, Vinassa J-M. Thermal characterization of a high-power lithium-ion battery: potentiometric and calorimetric measurement of entropy changes. Energy 2013;61:432e9. [14] Jeon DH, Baek SM. Thermal modeling of cylindrical lithium ion battery during discharge cycle. Energy Convers Manag 2011;52(8e9):2973e81. [15] Dong Hyup J, Seung Man B. Thermal modeling of cylindrical lithium ion battery during discharge cycle. Energy Convers Manag 2011;52(8e9): 2973e81. [16] Ouyang MG, Liu GM, Lu LG, Li JQ, Han XB. Enhancing the estimation accuracy in low state-of-charge area: a novel onboard battery model through surface state of charge determination. J Power Sources 2014;270:221e37. [17] Han XB, Ouyang MG, Lu LG, Li JQ. 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;278:802e13. [18] Bizeray AM, Zhao S, Duncan SR, Howey DA. Lithium-ion battery thermalelectrochemical model-based state estimation using orthogonal collocation and a modified extended Kalman filter. J Power Sources 2015;296:400e12. [19] Meng G, Sikha G, White RE. Single-particle model for a lithium-ion cell: thermal behavior. J Electrochem Soc 2011;158(2):A122e32. [20] Baba N, Yoshida H, Nagaoka M, Okuda C, Kawauchi S. Numerical simulation of thermal behavior of lithium-ion secondary batteries using the enhanced single particle model. J Power Sources 2014;252:214e28. [21] Luo W, Lyu C, Wang L, Zhang L. An approximate solution for electrolyte concentration distribution in physics-based lithium-ion cell models. Microelectron Reliab 2013;53(6):797e804. [22] Tatsukawa E, Tamura K. Activity correction on electrochemical reaction and diffusion in lithium intercalation electrodes for discharge/charge simulation by single particle model. Electrochim. Acta 2014;115:75e85. [23] Li J, Wang L, Lyu C, Wang H, Liu X. New method for parameter estimation of an electrochemical-thermal coupling model for LiCoO2 battery. J Power Sources 2016;307:220e30. [24] Rahimian SK, Rayman S, White RE. Extension of physics-based single particle model for higher charge-discharge rates. J Power Sources 2013;224:180e94. [25] Li J, Wang L, Lyu C, Zhang L, Wang H. Discharge capacity estimation for Li-ion batteries based on particle filter under multi-operating conditions. Energy 2015;86:638e48.
Nomenclature cl0 : initial electrolyte concentration (mol m3) Dx: maximum variation range of stoichiometric number x ()
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Dy: maximum variation range of stoichiometric number y () Eocv: open circuit voltage, OCV (V) F: Faraday's constant (C mol1) Pact: coefficient of anode reaction polarization (m1.5 mol0.5 s) ref Pact : coefficient of anode reaction polarization at reference temperature (m1.5 mol0.5 s) Pcon: proportional coefficient of liquid phase diffusion (mol m3 A1) ref Pcon : proportional coefficient of liquid phase diffusion at reference temperature (mol m3 A1) Qall: total capacity at discharge rate of 0.02C (A s) Qi, i ¼ n, p:: capacities of effective active material in the electrodes (A s) R: ideal gas constant (J mol1 K1) Rohm: ohmic resistance (U) ref Rohm : ohmic resistance at reference temperature (U) soc: state of charge () T: battery internal temperature (K) Tref: reference temperature (K) t: time (s) tþ: transport number () Ui, i ¼ n, p:: open circuit voltage (V) x0: initial stoichiometric number of negative electrode () xavg: solid phase average stoichiometric number of negative electrode () xsurf: solid phase surface stoichiometric number of negative electrode () y0: initial stoichiometric number of positive electrode () yavg: solid phase average stoichiometric number of positive electrode ()
ysurf: solid phase surface stoichiometric number of positive electrode () yofs: offset of relative position of stoichiometric numbers () Dcl : electrolyte concentration deviation between two current collectors (mol m3) Dx0 : deviations between xsurf and xavg () Dy0 : deviations between ysurf and yavg () tsi ; i ¼ n; p: solid phase diffusion time constants of each electrode (s) te: time constant of liquid phase diffusion (s) hact: reaction polarization overpotential (V) hcon: concentration polarization overpotential (V) hohm: ohmic polarization overpotential (V) lact: proportional coefficient of activation energy (K) lcon: proportional coefficient of activation energy (K) lohm: proportional coefficient of activation energy (K) Subscript act: reaction avg: average con: concentration max: maximum n: negative ohm: ohmic p:: positive ref: reference