IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 3, NO. 3, JULY 2016 281 Fractional Modeling and SOC Estimation of Lithium-ion Battery Yan Ma, Xiuwen Zhou, Bingsi Li, and Hong Chen, Senior Member, IEEE Abstract—This paper proposes a state of charge (SOC) estima- tor of Lithium-ion battery based on a fractional order impedance spectra model. Firstly, a battery fractional order impedance model is derived on the grounds of the characteristics of Warburg element and constant phase element (CPE) over a wide range of frequency domain. Secondly, a frequency fitting method and parameter identification algorithm based on output error are presented to identify parameters of the fractional order model of Lithium-ion battery. Finally, the fractional order Kalman filter approach is introduced to estimate the SOC of the lithium-ion battery based on the fractional order model. The simulation results show that the fractional-order model can ensure an acceptable accuracy of the SOC estimation, and the error of estimation reaches maximally up to 0.5 % SOC. Index Terms—Lithium-ion battery, fractional order model, electrochemical impedance spectra, fractional Kalman filter. I. I NTRODUCTION G ENERALLY, the electrochemical reactions inside lithium-ion battery are complicated in the running elec- tric vehicle (EV), which is a highly nonlinear dynamic system. State of charge (SOC) [1] is defined as the percentage of the amount of left energy to the rated capacity of a battery, which cannot be measured directly, it only can be estimated by measured variables such as current and terminal voltage. The accurate estimation of SOC is the key problem in the field of power battery. The methods of SOC estimation are categorized into direct experiment measurement methods and estimation methods based on battery models. Coulomb counting method and cur- rent integration method are the most popular experiment mea- surement methods, which are simple to obtain SOC. However, Manuscript received August 31, 2015; accepted December 21, 2015. This work was supported by National Natural Science Foundation of China (615 20106008, U1564207, 61503149), High Technology Research and Develop- ment Program of Jilin (20130204021GX), Specialized Research Fund for Graduate Course Identification System Program (Jilin University) of China (450060523183), and Graduate Innovation Fund of Jilin University (20151 48). Recommended by Associate Editor YangQuan Chen. Citation: Yan Ma, Xiuwen Zhou, Bingsi Li, Hong Chen. Fractional mod- eling and SOC estimation of Lithium-ion battery. IEEE/CAA Journal of Automatica Sinica, 2016, 3(3): 281-287 Yan Ma is with the State Key Laboratory of Automotive Simulation and Control, Jilin University, China, and also with the Department of Control Science and Engineering, Jilin University (Campus Nanling), Changchun 1300 25, China (e-mail: yma@jlu.edu.cn). Xiuwen Zhou and Bingsi Li are with Jilin University, Changchun 130012, China (e-mail: zhou xiuwen@126.com; lbsmichelle@163.com). Hong Chen is with the State Key Laboratory of Automotive Simulation and Control, Jilin University, Changchun 130012, China, and with the Department of Control Science and Engineering, Jilin University (Campus Nanling), Changchun 130025, China (e-mail: chenh98cn@126.com). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. these methods result in high errors caused by the accumulation of errors in numerical integration in current measurement. State observer [2-3] , Kalman filter (KF) [4-5] and particle filter (PF) [6-7] are used to estimate SOC based on the model of Lithium-ion battery. The SOC estimation error of each method is summarized in [8], which shows that the existing integer order battery SOC estimation methods mainly have estimation error larger than 1 %, which may be because the models which are obtained through the external characteristics of the power battery cannot show the precise internal characteristics. The dynamics of the battery is described by a set of integer order calculus equations. But complex electrochemical reactions are described by the fractional order function. The fractional order calculus (FOC) is a natural extension of the classical integral order calculus. References [9-11] have shown that most phenomena, such as damping, friction, mechanical vibration, dynamic backlash, sound diffusion, etc., have fractional order properties. Thus, FOC is widely used in modeling, kinetics estimation, etc. FOC is also used to develop the electrochemical models of the super capacitors and so on. When it comes to FOC battery modeling and SOC estima- tion, [12] uses FOC model obtained by system identification to estimate crankability of battery, [13] proposes lead acid battery state of charge estimation with FOC, and [14] deals with a fractional order state space model for the lithium-ion battery and its time domain system identification method. The existing FOC modeling for battery meets the same problem, the estimation accuracy in not high enough for battery man- agement system. The electrochemical impedance spectroscopy (EIS) method is one of the most accurate methods to model the electrochem- ical Li-ion batteries. There are many studies which have tried to utilize the impedance spectra directly to estimate SOC, but EIS method is too complicated to be used directly. EIS method is mainly used with equivalent circuit model at present [15-16] . The remainder of the paper is organized as follows. Section II discusses the battery fractional-order modeling based on impedance spectra; Section III discusses how to obtain char- acteristic curve between open circuit voltage (OCV) and SOC, states order identification with frequency method and param- eters identification according to the output error identification algorithm; Section IV presents fractional order Kalman filter for SOC estimation; Section V draws conclusions from the preceding work and offers suggestion for further study. II. FRACTIONAL MODELLING OF BATTERY The impedance spectra curve of the Lithium-ion battery can be got through Electrochemical workstation and is shown in