IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 62, NO. 12, DECEMBER 2015 7557 Butler–Volmer-Equation-Based Electrical Model for High-Power Lithium Titanate Batteries Used in Electric Vehicles Sijia Liu, Jiuchun Jiang, Senior Member, IEEE,WeiShi,ZeyuMa, Le Yi Wang, Fellow, IEEE, and Hongyu Guo

Abstract—The lithium titanate battery, which uses f1(·) Coefficient of the simplified form of BV equation. Li4Ti5O12 (LTO) as its anode instead of graphite, is a promis- f2(·) Coefficient of the simplified form of BV equation. ing candidate for fast charging and power assist vehicular h(·) Hysteresis voltage of battery. applications due to its attractive battery performance in rate characteristics and chemical stability. Unfortunately, com- Io Current flowing through battery. monly used battery models, including a large number of Ipc Reference rate by manufacturers. enhanced electrical models, become problematic when de- J Current density. scribing current–voltage characteristics of lithium titanate J0 Exchange current density. batteries. In this paper, a novel Butler–Volmer equation- M Constant in the electrical circuit model with based electric model is employed to outline unique phe- nomena induced by changing rates for high-power lithium hysteresis. titanate batteries. The robustness of the proposed model R Gas constant. for three types of lithium titanate batteries under varying Rp Polarization resistance of battery. loading conditions, including galvanostatic test and Fed- Rpc Polarization resistance measured at Ipc. eral Urban Dynamic Schedule test, is evaluated and com- R Internal resistance of battery. pared against experimental data. The experimental results Ω of three types of lithium titanate batteries with common S Effective area. anode materials but differentiated cathode materials show T Temperature (K). good agreement with the model estimation results with t0 The initial clock when battery begins to discharge. maximum voltage errors below 2%. t1 The clock when battery begins to rest after Index Terms—Battery model, Butler–Volmer (BV) equa- discharging. tion, lithium titanate batteries, rate characteristics. t2 The end clock of battery operation. t3 The initial clock of parameters identification. NOMENCLATURE t4 The next clock after t3. a1 Coefficient of OCV expression. tp The previous clock in the electrical circuit model a2 Coefficient of OCV expression. with hysteresis. b1 Coefficient of internal resistance expression. Uo Battery terminal voltage. b2 Coefficient of internal resistance expression. Uocv Open-circuit voltage (OCV). c1 Coefficient of polarization capacitor expression. Up Voltage of a RC parallel network. c2 Coefficient of polarization capacitor expression. Up1 Voltage of the RC parallel network with the larg- Cmax Maximum available capacity. er TC. Cp Polarization capacitor of battery. Up2 Voltage of the RC parallel network with the small- F Faraday constant. er TC.

Upmax Static value of battery polarization. Manuscript received December 12, 2014; revised March 8, 2015 and α Transfer coefficient. April 22, 2015; accepted May 15, 2015. Date of publication June 24, ΔSOC SOC variation when battery current changes. 2015; date of current version November 6, 2015. This work was sup- Δt Time variation when battery current changes. ported by the National Natural Science Foundation of China under Grant 51277010. (Corresponding author: Jiuchun Jiang.) ε Constant in the electrical circuit model with S. Liu, J. Jiang, W. Shi, and Z. Ma are with the National Active hysteresis. Distribution Network Technology Research Center, Beijing Jiaotong ε1 A finite variable that is near zero. University, Beijing 100044, China (e-mail: [email protected]; jcjiang@ · bjtu.edu.cn; [email protected]; [email protected]). η( ) Battery overpotential. L. Y. Wang is with the Department of Electrical and Computer τ Time constant (TC) of the RC parallel network. Engineering, Wayne State University, Detroit, MI 48202 USA (e-mail: [email protected]). H. Guo is with Beijing E-power Electronic Company Ltd., Beijing 100044, China (e-mail: [email protected]). I. INTRODUCTION Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. ITHIUM-ION batteries have been widely utilized in elec- Digital Object Identifier 10.1109/TIE.2015.2449776 L trical devices and systems such as telephones, electric 0278-0046 © 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. 7558 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 62, NO. 12, DECEMBER 2015 vehicles and renewable generation systems due to their high (SOC) estimation of different rates was achieved by using a power density, high energy density, and excellent reliability normalization method, which was based on the definition of rate [1]–[4]. To satisfy various power and energy demands of dif- factors [25]. More comprehensively, Zhang et al. [26] proposed ferent applications, batteries can be classified into two groups, an organic combination of an electrical circuit model and the i.e., high-power designs and high-energy designs. The lithium Rakhmatov diffusion model, which was sufficient to capture the titanate battery, which uses Li4Ti5O12 (LTO) as its anode recovery effect. Nevertheless, this improved model was difficult instead of graphite, has emerged as a leading candidate for to configure for its complicated structures. To enhance the fast charging and power assist vehicular applications because model adaptability to high rates and model suitability to system of its attractive battery performance in rate characteristics and simulation, a hybrid battery model was presented in [27], which chemical stability [5], [6]. In particular, it is more suitable for utilized a kinetic model to represent the rate capacity effect frequent start–stop applications than lead–acid batteries owing instead of the highly coupled diffusion model. Notably, the to its higher power capability and longer life [7]. Lithium aforementioned methods rarely discussed the change of model titanate batteries have been deployed in larger scale applications parameters depending on the quantity and direction of current, in China, including thousands of battery-powered buses and a although they were accurate enough from the perspective of series of bullet trains. In order to ensure the safety of battery quantitative results. Lam et al. [28] proposed an empirical for- operations and enhance battery management systems, battery mula to describe current dependence of parameters using curve models of high accuracy are urgently needed [8]–[10]. fitting. However, the popularization of this method is deficient Several battery models have been reported to meet critical re- because of a lack of theoretical derivations and the fact that the quirements of diversified circumstances over the past decades. model validation was only actualized in less than the 2-C rate, Commonly used battery models fall into three categories [1], where the “2-C” rate refers to a rate at which the battery will be [2], [11]–[13], i.e., electrochemical models, analytical models, discharged to empty in half an hour after fully charged at room and electrical circuit models. Electrochemical models accu- temperature. Apart from the general circuit model, an improved rately characterize material properties and reaction mechanism nonlinear battery model was presented in [29], which utilized inside the battery, serving as a basis for the optimal design the well-known electrochemical kinetics equation, namely, the of battery systems [14]–[17]. However, electrochemical mod- Butler–Volmer equation (BV equation), to outline the nonlinear els contain complicated nonlinear differential equations with electrode behavior of the battery. Unfortunately, the application many unknown variables, which not only increase the model of the proposed identification methodology to lithium batteries complexity but also are difficult to be employed in power was problematic because voltage responses of discharge current control systems. Analytical models usually are simplified pulses hardly reached steady state, although this model showed forms of electrochemical models and remain too complicated a promising result for lead–acid batteries. for accurately predicting dynamic performance during battery In this paper, a novel BV equation-based electrical model is runtime [18]. employed to capture unique phenomena induced by changing Electrical circuit models can capture battery current–voltage rates for the high-power lithium titanate battery. The evaluation (I–V) characteristics through a combination of electrical com- results of the model accuracy and robustness under varying ponents, such as voltage sources, resistors, and capacitors loading profiles, including the galvanostatic profile and federal [19]–[22]. These models have simpler structures and less un- urban dynamic schedule (FUDS) profile, are discussed in detail. known variables than the other two kinds of models, and also In addition, the robustness analysis of the model for three types can be easily incorporated into control models of battery- of lithium titanate batteries, which is composed of common powered systems. Low et al. [23] presented an improved model anode materials but differentiated cathode materials, show that comprising two resistance–capacitance (RC) parallel networks, the proposed model has the advantages of estimating the ter- which gave a good prediction of behavior with sufficient accu- minal voltage accurately and reliably with maximum voltage racy for lithium ferro phosphate (LFP) batteries. A compari- errors below 2%. The aforementioned analysis is focused on son of 12 electrical circuit models using the same data were the middle SOC range, namely, 10%–90% SOC, since practical presented by Hu et al. [24], which indicated that the model data are available in the middle SOC range. with a single RC parallel network was preferred for lithium This remaining part of this paper is organized as follows. nickel–manganese–cobalt oxide (LiNMC) batteries, whereas In Section II, the related work, including several commonly the first-order RC model with one-state hysteresis was more used electrical circuit models and derivations of the BV equa- appropriate for LFP batteries. However, there is little mention tion, is presented. Section III specifies the experimental setup of battery models designed for lithium titanate batteries. and procedures. In Section IV, the modeling process of the There are two unique phenomena induced by rate variation, proposed model is outlined. Section V displays the steps for namely, the rate capacity effect and the recovery effect. The parameter extraction. Model verification results are discussed rate capacity effect refers to the change of battery capacity due in Section VI. Finally, Section VII gives a conclusion. to changing rates, which is more visible at high rates. After battery discharges with a specific rate, it is still able to discharge II. RELATED WORK for a period of time with a lower rate, which is named as the A. Electrical Circuit Models recovery effect. To improve the model feasibility for batteries with high-power designs, there are many enhanced electrical There are three emblematical electrical circuit models that circuit models developed by researchers. The state-of-charge are basics for the majority of other electrical models. Thevenin LIU et al.: BV-EQUATION-BASED ELECTRICAL MODEL FOR BATTERIES USED IN ELECTRIC VEHICLES 7559

TABLE I COMMONLY USED ELECTRICAL CIRCUIT MODELS

Fig. 2. Polarization distributions of two types of batteries. (a) High- Fig. 1. Structure of the Thevenin model. energy battery. (b) High-power battery. model, which is composed of a single RC parallel network in ization. Electrochemical polarization appears once the electric series with a resistance, is the leading foundation of various field of the battery has been established; whereas concentration circuit models [1], [2]. In order to achieve a much greater ac- polarization, as a result of lithium ions transfer in the electrolyte curacy, the battery is characterized by adding an appended RC phase when it is dominated by the diffusion effect, requires a parallel network with a different TC, namely, the second-order longer time to be built completely. RC model. Generally, one of these two TCs is much larger than Power batteries have several characteristics, such as small the other one [30]. As aforementioned, the hysteresis effect is internal resistance, short recovery time, and attractive rate so crucial for LFP batteries rather than other kinds of batteries. characteristics [33]. To rigorously investigate unique battery These battery models are summarized in Table I [24], [31], [32]. performance caused by different designs, two batteries pro- Fig. 1 illustrates the detailed structure of the Thevenin model duced by the same manufacturer but with disparate designs are for a single battery, which contains a controlled voltage source, selected and analyzed using the second-order RC model, where a series resistance and a RC parallel network used to represent the two RC parallel networks with different TCs are employed the overpotential (alternatively called polarization). The OCV is to describe the polarization voltage. One of these two RC described as a voltage source dominated by SOC of the battery; parallel networks is the polarization with the larger TC, whereas the series resistance is used as a lumped representation of another is the polarization with the smaller TC. Fig. 2 shows resistive forces in electrodes, electrolyte, and other components the different polarization distributions of these two batteries of the battery, describing energy losses under battery charges or from experiment. It obviously indicates that the polarization discharges. To increase the fidelity of this model, all parameters of a high-energy battery is mainly composed of polarization aforementioned are dependent on SOC of the battery and cer- with the larger TC, whereas polarization with the smaller TC tainly different under charge and discharge conditions as well. is further significant when it comes to a high-power battery. However, traditional electrical circuit models are insufficient to The battery runtime decreases regularly in an inverse proportion characterize dynamic performance of the battery under high- way when current flowing through battery increases because rate applications due to a lack of effective representation on of Peukert law, resulting in a changing tendency shown as 1 . the reaction mechanism inside the battery. Enhanced electrical As a consequence of current growth, these two totally distinct circuit models proposed in [25]–[27] give partial solutions to polarizations will be more serious. Polarization with a smaller this problem by using complex electrochemical models, which TC, particularly, will be much likely in an extremely critical are capable of describing the capacity variation induced by condition owing to short recovery time of a power battery. changing rates, ignoring the vital fact that model parameters Variation tendencies of these two polarizations are marked as have already changed. 2 and 3 , respectively. It can be concluded that polarization with a smaller TC will be affected much heavily as the current increases. Moreover, B. BV Equation it should be quite appropriate to regard this short polarization Battery polarization mainly includes two components, as electrochemical polarization since ions diffusion are more namely, electrochemical polarization and concentration polar- inclined to generate polarization with a larger TC. Therefore, 7560 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 62, NO. 12, DECEMBER 2015 methods capable of characterizing electrochemical polarization induced by changing rates are desperately required. After kinetics of ions transfer is completely established, the BV equation is proposed to model an ideal overpotential η(t) coming from electrochemical polarization under the steady state αF −(1 − α)F J(t)=J exp η(t) − exp η(t) . 0 RT RT (1) Considering there are few approaches to measuring current Fig. 3. Experimental setup for battery tests. density externally, the BV equation can be written as (3) according to Their major parameters are shown in Table II.UsingLTO I(t)=J(t) × S (2) as its anode instead of graphite, LTO batteries are more in- − − clined to high-rate applications in electric vehicles than other 1 αF − (1 α)F I(t)= exp η(t) exp η(t) . lithium batteries of graphite anode, resulting in an unusual J0 × S RT RT (3) reference rate for capacity test presented in Table II.Moreover, Takami et al. [5] also mentioned that 1 C was chosen as the However, it seems impossible to get the overpotential caused reference rate for LTO battery capacity test. The test procedure by current variation because equation (3) cannot be solved with- shown in Fig. 4 is performed to adequately excite comprehen- out any assumptions. Fortunately, the transfer coefficient rarely sive behaviors of these batteries. It begins with the capacity changes suddenly and generally argued as α ≈ 0.5 [15]–[17]. test, which is measured by discharging the battery down to Consequently, the relationship is expressed as the lower cutoff voltage with a current of 1 C after constant ⎛ ⎞  current-constant voltage (CC-CV) charging to the lower cutoff 2RT 1 1 2 current, and this test will not be finished until the difference η(t)= ln ⎝ I(t)+ I(t) +1⎠ . (4) F 2J0S 2J0S of results between two adjacent cycles are within 0.1 Ah. We further examine the OCV dependence on directions of current Although there are still a variety of electrochemical variables using a “positive pulse sequence” (PPS) for 20 times, followed cannot obtained externally, it is quite unsophisticated for us to by a “negative pulse sequence” (NPS) repeated 20 times, where acquire a specific change of overpotential excited by diverse a PPS contains 1-C charge for 180 s to gain 5% of the battery SOC, and rest of 1 h on purpose of acquiring magnitude of currents. Meanwhile, J0 and S are closely related to SOC, reflecting the complexity during the polarization establishment. OCV precisely. Similarly, a NPS has the same structure but in The increase of current, whereas other variables available are opposite directions of current. For an internal resistance test, fixed, will result in a sharp deviation from the equilibrium state. we apply a rich pulse sequence, including several steps at each On the contrary, a battery is gradually becoming steady and specific SOC: 1) 1-C charge for 10 s; 2) rest for 40 s; 3) 1-C reaching equilibrium eventually when current approaches zero. discharge for 10 s; and 4) repeat step 1) to step 3) after replacing More meaningfully, the battery overpotential at a specific SOC, 1 C to 2 C, 3 C, 4 C, 5 C, 6 C, 7 C, 8 C, 9 C, 10 C, respectively. which is only induced by changing currents, is affirmatively ac- Then, rate charging and discharging tests are carried out with cessible in accordance with equation (4). It suggests the battery selected typical rates to investigate the relationship between the has completely reached a nonequilibrium state after deviating detailed parameters and rates. On the other hand, batteries are from equilibrium, which requires some time particularly. excited by the FUDS profile, providing the data set for model verification. The typical current profile of FUDS tests is shown in Fig. 5. Regions between two adjacent dotted red lines are III. EXPERIMENTAL SETUP AND PROCEDURES chosen as representative parts for our subsequent discussions. Fig. 3 exhibits the experimental setup for battery tests, which consists of an Arbin BT2000 tester, a thermal chamber provid- IV. PROPOSED MODEL ing controllable temperature, a computer for monitoring, and data storage and tested batteries. The voltage and current are In this paper, the structure of the Thevenin model is selected measured and recorded per second during the whole test. In this as a reference for the proposed model owing to three reasons: paper, we prefer to evaluate battery characteristics depending on changing rates rather than other factors that will be discussed 1) It is mainly used as the basic part of improved models in our subsequent works. Hence, the thermal chamber is main- developed by other authors [1], [2], thus permits compar- tained at 25 ◦C all the time. ison within diverse models. Three types of lithium titanate batteries are selected in our 2) It has been proved preferred for LiNMC batteries in work, and their shapes are totally different. The prior two [24], also appropriate for LFP batteries after adding a batteries are from leading manufacturers in China, and the hysteresis component. Particularly, hysteresis voltages of third one is produced by a well-known Japanese company. lithium titanate batteries are within 6 mV and will reduce LIU et al.: BV-EQUATION-BASED ELECTRICAL MODEL FOR BATTERIES USED IN ELECTRIC VEHICLES 7561

TABLE II POWER BATTERIES USED FOR TESTS

Fig. 6. Schematic of the proposed electrical model.

Fig. 4. Flowchart of the test procedure. particularly lithium titanate batteries applied in high-rate ap- plications. Moreover, its framework is so minimalistic that less computation and storage cost than other enhanced models mentioned in [25]–[28] are needed. Assuming a battery discharges with a current of I1 during t0

t 1 SOC(t)=SOC(t0) − Io(t)dt (5) Cmax t ⎧  0  − t−t0 ⎨ · − τ RpIo(t)  1 e  ,t0

Uo(t)=Uocv(t) − Up(t) − RΩIo(t) (7)

where τ = RpCp. During the constant discharging process, the SOC of the battery reduces linearly according to equation (5). In addition, the initial value of SOC is necessary for SOC estimation when a battery provides the requisite power for electric motors. This value of SOC is always calculated Fig. 5. Current profiles for FUDS test. (a) No. 1 battery. (b) No. 2 and from the specific OCV–SOC curve of the battery. Given cir- No. 3 battery. cuit parameters are likely to change with SOC, all of these parameters are as functions of SOC. Equation (8) expresses or even vanish in high-rate applications. Hence, hysteresis SOC dependence of Uocv,RΩ,Cp,wherea1,a2,b1,b2,c1,c2 is ignored in this paper. are obtained from circuit parameters of typical SOC points by 3) It is more feasible and suitable for high-power batteries linear interpolation method ⎧ than the second-order RC model, given that dynamic ⎨⎪U [SOC(t)] = a SOC(t)+a high-rate characteristics are affected much severely by the ocv 1 2 R [SOC(t)] = b SOC(t)+b (8) polarization with a smaller TC. ⎩⎪ Ω 1 2 C [SOC(t)] = c SOC(t)+c . Fig. 6 displays the proposed electrical model. It takes the p 1 2 same electrical circuit as Thevenin model but uniquely embeds When discharge current suddenly changes from I1 to I2, a simplified form of the BV equation to predict parametric the time interval can be considered as Δt<ε1 Hence, we can variation, particularly the polarization voltage drop caused easily get equation (9) and infer that ΔSOC ≈ 0 by changing currents. Consequently, the proposed model is sufficient for characterizing comprehensive battery behaviors ΔSOC ≤ max{I1Δt, I2Δt}≤max{I1ε1,I2ε1}. (9) 7562 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 62, NO. 12, DECEMBER 2015

As a lumped representation of resistive forces, internal re- sistance RΩ is precisely correlated with SOC of the battery but clearly independent of current amplitude [30], [34]. How- ever, battery polarization, particularly polarization resistance is heavily affected by current variation [35]. Based on equation (4) and (6), the steady-state value of battery polarization at a specific SOC can be expressed as   − t−t0 · − τ Upmax = RpIo(t) 1 e tτ

= RpIo(t)

= f1(SOC) Fig. 7. OCV–SOC curves of selected lithium titanate batteries. · ln f (SOC) · I (t)+ (f (SOC) · I (t))2 +1 2 o 2 o TABLE III (10) HYSTERESIS VOLTAGES OF TESTED BATTERIES where f1(SOC),f2(SOC) are coefficients of the simplified form of the BV equation. More importantly, these two coef- ficients are also as functions of the current direction. Accord- ing to equation (6), battery polarization can be divided into two categories: polarization establishment when battery current is nonzero and polarization recovery when battery is rested. As a dominate part of polarization establishment, polarization resistance is derived from equation (10). On the other hand, polarization resistance of polarization recovery is irrelevant to current amplitude, since no external excitation is applied on the battery. Therefore, polarization resistance is analytically calcu- lated as (11), shown at the bottom of the page, where Rpc rep- resents polarization resistance measured at the current of 1 C. Clearly, model accuracy is more sensitive to Rp rather than Cp,givenCp is solely dominated by the τ and the fact that recovery time keeps short for high-power batteries. To reduce the storage space required, Cp is assumed to be independent Fig. 8. Typical curve of voltage response for a PPS. of current amplitude, which has been proved pretty feasible in Section VI. time to be entirely relaxed. More specifically, it was confirmed that LTO batteries also showed OCV hysteresis and high rates V. M ODEL EXTRACTION affected the macroscopic processes in a way that hysteresis even vanished for Li-ion batteries [37]. Since what we evaluate are A. OCV–SOC Curves high-power lithium titanate batteries for high-rate applications As an indispensable component of the proposed model, OCV and the actual hysteresis voltages of 1 C measured current as a function of SOC can be determined after a battery has been are already within 6 mV, hysteresis voltages will reduce or rested for a long period. Hysteresis mentioned in [31], [32] even disappear when battery current grows. Hence, hysteresis is specifically describes the OCV difference for a certain SOC, ignored in this paper to simplify the model without noticeable which is caused by current directions. Nevertheless, OCV–SOC effect on accuracy. curves of three selected lithium titanate batteries, which are exhibited in Fig. 7, indicate that there is an extremely high B. Parameters Identification consistency between two parts of an OCV–SOC curve, namely, the hysteresis voltages of lithium titanate batteries are within Fig. 8 demonstrates a typical curve of voltage response 6mVasshowninTable III.Liet al. [36] pointed out that for a PPS. The terminal voltage immediately rises and grows hysteresis easily appeared when a battery required a fairly long exponentially later. Moreover, internal resistance of the battery

⎧ √ ∗ 2 ⎨ f1(SOC) ln f2(SOC)·Io(t)+ (f2(SOC)·Io(t)) +1 ,Io(t) =0 Rp = ⎩ Io(t) (11) Rpc,Io(t)=0 LIU et al.: BV-EQUATION-BASED ELECTRICAL MODEL FOR BATTERIES USED IN ELECTRIC VEHICLES 7563

Fig. 10. Polarization voltages of typical rates.

methodology in [29]. In order to outline battery performances, particularly polarization voltage relationships with rates and to enable the model feasibility for system-level simulation, two coefficients of (11), which are closely related to the SOC of Fig. 9. Internal resistances of typical rates. the battery, are determined from the least squares curve fitting is calculated from (12) by using the instantaneous voltage rise tool. In particular, it becomes much earlier for a battery to from t3 to t4 = t3 +Δt,whereΔt is equal to sampling period, enter the CV charging region when rate rises, complicating namely, 1 s in this paper. Then polarization voltage can be the model description. In addition, current obviously reduces obtained, according to (7) and Uocv measured at a fixed SOC, after entering the CV charging region, bringing much less providing data for identification of the RC parallel network. polarization inside the battery. Therefore, coefficients of (11) to Using (13), polarization resistance and polarization capacitor, capture charging behavior are refined calculated in 10%–75% where polarization resistance refers to that measured at the SOC, whereas polarization resistances of the current of 1 C are current of 1 C, particularly, are determined from the least adopted in other SOC intervals. squares curve fitting tool in MATLAB. It is important to note that two groups of parameters are taken to capture characteristic VI. MODEL VERIFICATION differences between charging and discharging behaviors of the battery To verify the proposed model, the simulation results of galvanostatic test at a high rate and FUDS test, which are Uo(t4) − Uo(t3) carried out on three types of lithium titanate batteries, are RΩ = I (t4) − I(t3) respectively compared with the experimental data. Taking Uo(t3 +Δt) − U(t3) No. 1 battery as an example, the superiority of the BV equation = (12) ⎧ Ipc is prominently shown in a digital form. In this paper, the pro- − t posed model is implemented in MATLAB/Simulink, ensuring ⎨⎪Up(t)=A · [1 − e τ ] A model promising popularization in high-powered applications. Rp = . (13) ⎪ Ipc ⎩ τ Cp = Rp A. Galvanostatic Test at A High Rate Fig. 11 compares simulation results of the galvanostatic C. Battery Performances of Typical Rates test at a high rate with experimental data. There is a close In this part, No. 1 battery is chosen to illustrate unordinary accordance between terminal voltage extracted from experi- battery performances of typical rates. Fig. 9 shows internal mental data and simulation results of the proposed electrical resistances of typical rates, which are measured at a specific model, which is available for selected three lithium titanate SOC, are approximately constant when the current direction is batteries. Maximum voltage errors for galvanostatic tests of fixed. However, polarization voltages of typical rates, which are three batteries are found similarly at the ending of discharging, displayed in Fig. 10, maintains a continuously growing trend, whereas the fact those maximum voltage errors are all within despite the increasing speed is clearly not as fast as before. 1.5%. The voltage error of the No. 1 battery for galvanostatic When the battery is tested under rate charging or discharg- test at 10 C (85 A) is less than 0.5% in 240 s (23.3%–90% ing, the polarization voltage of the middle SOC range clearly SOC), and voltage error is less than 1.5% during the discharging achieves the steady-state value. Given the fact that the internal process. There is a same distribution of voltage errors for dis- resistance is independent of current amplitude and the data of charging process at 10 C (160 A) of No. 2 battery. Although the rate charging tests, the battery polarization voltage of varied simulation result of No. 3 battery is slightly different from that charging rates can be acquired according to (7) after obtaining of the prior two batteries, voltage errors of whole discharging the OCV value and resistance at 1 C measured current. Simi- process is still within 1.5%. Therefore, the proposed model can larly, the battery polarization voltage of varied discharging rates capture galvanostatic performances of lithium titanate batteries can be obtained with the same method. In addition, rate charg- precisely. ing and discharging, particularly high-rate applications, will not Furthermore, the simulation results of the galvanostatic cost too much time compared with the proposed identification charging test at a high rate and experimental data are displayed 7564 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 62, NO. 12, DECEMBER 2015

Fig. 12. Comparison between simulation and experimental data of the Fig. 11. Comparison between simulation and experimental data of the galvanostatic discharging test. (a) No. 1 battery. (b) No. 2 battery. galvanostatic charging test. (a) No. 1 battery. (b) No. 2 battery. (c) No. 3 (c) No. 3 battery. battery. in Fig. 12, confirming that the simulation results of the proposed electrical model are highly consistent with terminal voltage pro- files obtained from the experimental data. It can be concluded that maximum voltage errors for galvanostatic charging tests of three batteries are all less than 2% and occur at the end of the charging process. Although the voltage errors of No. 1 battery at 8 C (68 A) grow in the last 60 s, they are less than 1% during the first 240 s. Moreover, the voltage errors of No. 2 battery at 10 C (160 A) and No. 3 battery at 8 C (160 A) during the whole discharging process are within 1%. Therefore, the proposed model can capture both galvanostatic discharging and charging performances of lithium titanate batteries precisely. To evaluate the applicability of the proposed model, voltage errors of No. 1 battery at several rates are presented in Fig. 13, demonstrating how voltage errors vary with pulse durations. Apparently, model error distributions of the battery discharging test are similar, namely, a sudden rise after holding a small value for a long time. During the approximate horizontal part of Fig. 13. Voltage error of the No. 1 battery galvanostatic tests. (a) Charging test. (b) Discharging test. the discharging curve cluster, the voltage errors slightly change between −0.1% and 0.05%. Although voltage errors quickly increase at the end of discharging, they are within 1%. When and terminal voltage obtained from battery simulation, which it comes to charging test, the model error distributions hold a is shown in Fig. 14, indicates that simulation results well match same shape as discussed in the discharging test and are still experimental data. Maximum voltage errors of No. 1 and No. 2 within 1%. Particularly, model errors of both charging and batteries are less than 1.5%, which happen at the time point discharging tests suddenly rise at the end of battery operations. belonging to [900 s, 1800 s]. Despite the fact that the maximum voltage error of the No. 3 battery appears at the ending of discharging, it is a little larger than that of other two batteries B. Verification With FUDS Profile but still within 2%. As mentioned in Section III, No. 3 battery is On the other hand, we further verify the proposed model only investigated by current that is up to 8 C, because its design with FUDS profile. The comparison between experimental data contains a tradeoff between power and energy although it is LIU et al.: BV-EQUATION-BASED ELECTRICAL MODEL FOR BATTERIES USED IN ELECTRIC VEHICLES 7565

Fig. 16. Comparison between the proposed model and the Thevenin model for the FUDS test. (a) Representative part I. (b) Representative part II.

ing battery comprehensive behaviors, meaningfully, enhances electrical model applicability for high-rate operations. In this part, regions chosen in III are selected as representa- tive parts for our subsequent discussions due to three reasons: Fig. 14. Comparison between simulation and experimental data of the FUDS test. (a) No. 1 battery. (b) No. 2 battery. (c) No. 3 battery. 1) It includes three different trails deviated from a high rate, nearly covering all available trajectories after the current has reached a high value, as 10 C (85 A) for No. 1 battery. 2) Corresponding to three typical performances of a battery, these trails are sufficient to compose all possible high- rate battery behaviors. At the time interval of [190 s, 200 s] of representative part I, the current slightly varies near the 10 C, imitating the battery continuously pro- vides power to power equipment or electric vehicles at a substantial high rate. In addition, the second typ- ical performance appears when battery discharges at Fig. 15. Comparison between the proposed model and the Thevenin [200 s, 240 s], outlining a massive current shift from model for the galvanostatic test. 10 C (85 A) but still larger than zero. This current shift accurately simulates dynamic discharging of a battery. Moreover, the third typical performance, which happens more likely to be applied in power-assist fields. Therefore, the at [275 s, 285 s], displays a sudden change of the cur- proposed model error of the No. 3 battery, as discussed earlier, rent direction. The sudden change at the beginning of is the largest among three batteries. 10 C (85 A), obviously, indicates an abrupt charging after battery discharging. 3) It becomes more visible to show how model description C. Comparison Against Thevenin Model is affected by the BV equation using these regions instead To better illustrate the advantage of BV equation, both the of the whole FUDS test. Thevenin model and the proposed model are taken to describe Fig. 16 expresses current excitations and voltage errors of dynamic performances of the No. 1 battery, respectively, which these two representative parts, based on analyses of battery contain a galvanostatic test at 10 C and FUDS test. Fig. 15 ex- characteristics by using the two models aforementioned. The hibits voltage errors of these two models when the battery Thevenin model keeps a value that is larger than 20 mV, when discharges at a current of 10 C. During the discharging process, battery discharges at [190 s, 200 s] of representative part I. the voltage error of the Thevenin model continuously rises However, the voltage error of the proposed model falls rapidly and eventually exceeds 80 mV. However, the proposed model to 5 mV at the same time interval. As for discharging at [200 s, holds a small voltage error and is lower than 30 mV. It is 240 s], the proposed model is clearly superior to the Thevenin the BV equation that produces two entirely different trends of model with a voltage error that is lower than 15 mV. Similarly, model errors. Hence, the BV equation capable of characteriz- there is a same comparison when battery current direction 7566 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 62, NO. 12, DECEMBER 2015

TABLE IV MODEL PARAMETERS OF NO.1BATTERY

TABLE V MODEL PARAMETERS OF NO.2BATTERY

TABLE VI MODEL PARAMETERS OF NO.3BATTERY

changes at [275 s, 285 s]. It is worth to point out there is concentration polarization, particularly for high-power little difference between two representative parts. Therefore, batteries, through comparison between polarization dis- the BV equation accurately outlines battery characteristics of tributions of two batteries with the same manufacturer but three different trails deviated from a high rate, particularly when disparate designs. current slightly varies near the high rate. 2) A simplified form of BV equation, which is related to According to the highly agreements between simulation SOC and can be directly added to the electrical circuit results and experimental data of three groups of lithium ti- model, is carefully derived from the original BV equation, tanate batteries, the proposed model accuracy and reliability is according to the physical meaning more than a math- verified. Based on discussions upon error distributions of two ematical solution and gives an ideal expression of the models, the advantage of the proposed model also has been overpotential coming from electrochemical polarization proved compared with the Thevenin model. under the steady state. 3) Two coefficients of the simplified form of BV equation can be totally determined after rate charging and dis- VII. CONCLUSION charging tests, given the OCV value and resistance at In this paper, a high-fidelity BV equation-based electrical 1 C measured current and the fact internal resistance is model for high-power lithium titanate batteries is employed independent of current amplitude, which would not cost with the following features. too much test time. In addition, hysteresis voltages of selected lithium titanate batteries are found all within 1) Electrochemical polarization will affect dynamic rate 6 mV at 1 C measured current, resulting in the absence characteristics of lithium batteries more seriously than of the hysteresis model to simplify the complexity of LIU et al.: BV-EQUATION-BASED ELECTRICAL MODEL FOR BATTERIES USED IN ELECTRIC VEHICLES 7567

TABLE VII an extra consideration of the model parameters dependence on COEFFICIENTS OF THE SIMPLIFIED BV EQUATION FOR NO.1BATTERY temperature is needed and will be discussed in our subsequent works.

APPENDIX Model parameters and coefficients of the simplified BV equation are shown in Tables IV–IX.

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Ouyang, “Temperature characteristics formation, system identification, robust control, of power LiFePO4 batteries,” Chin.J.Mech.Eng., vol. 47, no. 18, H-infinity optimization, time-varying systems, pp. 115–120, Sep. 2011. adaptive systems, hybrid and nonlinear systems, information processing [37] M. A. Roscher, O. Bohlen, and J. Vetter, “OCV hysteresis in Li-ion and learning, as well as medical, automotive, communications, power batteries including two-phase transition materials,” Int. J. Electrochem., systems, and computer applications of control methodologies. He has vol. 2011, 2011, Art ID. 984320. been a Keynote Speaker at several international conferences. Dr. Wang was an Associate Editor of the IEEE TRANSACTIONS ON AUTOMATIC CONTROL and several other journals, and currently is an Associate Editor of the Journal of System Sciences and Complexity and Journal of Control Theory and Applications.

Sijia Liu was born in Henan, China. He received Hongyu Guo was born in Guangdong, China. the B.S. degree in electrical engineering from He received the B.S. and Ph.D. degrees in Beijing Jiaotong University, Beijing, China, in electrical engineering from Beijing Jiaotong 2013, where he is currently working toward the University, Beijing, China, in 2005 and 2013, Ph.D. degree. respectively. He is currently with the National Active Dis- He is currently with Beijing E-power Elec- tribution Network Technology Research Center, tronic Company Ltd., Beijing. His research inter- Beijing Jiaotong University. His research inter- ests include battery management systems and ests include battery modeling, state estimation, battery grouping technology. and battery management systems.