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