## 1 Introduction  
Lithium-ion batteries have many advantages, such as high working voltage, small volume, light weight, long cycle life, environmental friendliness, weak memory effect, wide working temperature range, low self-discharge rate, and so on [1; 2; 3; 4]. Lithium-ion batteries have been widely used in mobile phones, digital cameras, laptops, other portable consumer electronics, and renewable energy storage [5; 6]. With the continuous development and improvement of science and technology, the application range of lithium-ion batteries has further widened to the electric power, aerospace, military, and other critical fields [7; 8]. Lithium-ion batteries have become core components of energy supply for many critical devices or systems, and are often critical to the reliability and functionality of the overall system [3; 8]. However, consequences of battery failure could range from inconvenience to catastrophic failure; for example, NASA's Mars Global Surveyor stopped running due to battery failure in November 2006. The reliability of lithium-ion battery systems has yet to be improved. Recently, lithium-ion batteries have reached a degree of implementation that enabled their use in stringent automotive applications; for example, the Nissan LEAF rolled off the assembly line in 2010 as well as the Tesla Roadster in 2008 [9]. Automotive lithium-ion batteries usually consist of hundreds or even thousands of single cells connected in series and parallel so they have high power and energy density coupled with such problems as safety, reliability, and uniformity. At present, an efficient and reliable battery management system (BMS) has become the key to guaranteeing the reliable and safe operation of lithium-ion batteries [10; 11]; there are many aspects to the process of developing a battery management system, such as requirement analysis, modeling and simulation, control strategy research, and on-line hardware test, which all need a capable model to identify the characteristics of the lithium-ion batteries. In order to predict the behavior of the battery, many different models have been established, which can be generally divided into two categories: electrochemical models and equivalent circuit models. Although the electrochemical model accounts for the main underlying physics in the battery dynamics in terms of the main electrochemical parameters and variables [12; 13], the on-site accuracy is not high when it is applied in vehicle control systems due to the complexity of the model, and a large number of design and electrochemical kinetic parameters need to be obtained first. Therefore, the electrochemical mechanism model is usually used to understand the reaction process inside the battery, and to guide the design and manufacture of the battery. In an equivalent circuit model (ECM), resistances, capacitances, and voltage sources are used to describe charging and discharging processes, and the model is built in frequency- or time-domain [14; 15], in which each component has clear physical meaning, simple mathematical expressions, is more intuitive, and is easy to handle. It has been widely applied in the estimation of the state-of-charge (SOC), the state-of-health (SOH) and the state-of-function (SOF) for lithium-ion batteries. The literature [16] based on the first-order RC (resistor-capacitor) model of lithium-ion battery short-circuit detection was explored. The literature [17] used a first-order RC equivalent circuit model to estimate the resistance of lithium-ion batteries and the open circuit voltage. Reference [18] used the first-order RC equivalent circuit model for the modeling and simulation of lithium-ion battery packs. A first-order RC equivalent circuit model was created and validated for a particular battery using a MATLAB (the MathWork, Inc., Natick, MA, USA) Simulink in terms of the internal temperature distribution, the open circuit voltage, the heat generation, and the internal resistance in References [19; 20].