**1. Introduction**

Lithium-ion power batteries are widely used in electric vehicles (EVs), owing to their advantages of high energy density, low self-discharge rate, long cycle life, and no memory effect [1]. To ensure the safe, efficient, and stable operation of the power batteries, it is essential to manage the batteries effectively [2]. It is known that the state of charge (SOC) serves as an important indicator to characterize the remaining battery capacity. Therefore, an accurate SOC estimation is the basis for preventing over-charge and over-discharge, and for equilibrium processing. Accurate SOC estimation is the core of an effective battery management system (BMS) [3–5].

An effective battery model is a prerequisite for estimating the SOC of a battery. A model that is not effective directly reduces the accuracy of the SOC estimation algorithm, and could even cause the estimation algorithm to diverge directly in severe cases. There are three main types of working lithium-ion battery models: the black-box models, the electrochemical models, and the equivalent circuit models (ECMs) [6]. These three types of models describe the characteristics of the lithium-ion batteries from different detail levels [7]. The black-box models are similar to a linear or nonlinear mapping function. This function reflects the characteristics of battery voltage response, whereas ignores

the internal mechanism of the battery and has no physical existence. The electrochemical models contain many equations and parameters, but the simulation accuracy of the battery under complex working conditions is low. The ECMs are used to simulate the external operating characteristics of the battery by the matching of electronic components and are widely used in battery SOC estimation [8]. The pseudo-two-dimensional electrochemical mechanism model proposed by Doyle et al. [9] is often used as a full-order reference mechanism model, and it is also used to evaluate and test other simplified mechanism models. Wang et al. [10] established a nonlinear black-box battery model, and the verification under federal urban driving schedule (FUDS) operating conditions showed that the relative error of voltage was within 3.8%. Plett et al. [11] developed the commonly used ECMs in detail, including internal resistance models, models considering hysteresis effects, etc. Kim et al. [12] proposed a hybrid battery model consisting of a KiBaM model and a dual-polarization model (DPM), which can simultaneously describe the external dynamic characteristics and the recovery effect of the battery. Hu et al. [13] used a second-order fractional-order model to predict the terminal voltage under FUDS cycling conditions, and the average relative error does not exceed 0.1%, which proved the high accuracy of the model.

At present, the SOC estimation methods used both locally and internationally include: the ampere-hour (Ah) integration method [14], electrochemical impedance spectroscopy (EIS) method [15], open-circuit voltage (OCV) method [16], internal resistance method [1], particle filter (PF) [17], Kalman filter (KF) [18], fuzzy logic (FL) [19], artificial neural network (ANN) [20], support vector machine (SVM) [21], and relevance vector machine (RVM) [22]. The authors in [23] critically reviewed the existing SOC estimation methods in the past five years and introduced the basic principles and main disadvantages of various methods. Among these methods, the KF is an optimized autoregressive data filtering algorithm [24], which utilizes the principle of minimum mean square error to achieve an optimal state estimation for a complex dynamic system. Not only does the KF correct the initial error of the system but it also effectively suppresses the noise in the actual measurement process [8]. This feature makes the KF stand out among the current SOC estimation models. Therefore, a variety of improved algorithms have also been derived, such as the extended Kalman filter (EKF) [25], the adaptive unscented Kalman filter (AUKF) [26], and the central difference Kalman filter (CDEF) [27]. In 2004, Gregory L Plett [11] employed the EKF algorithm to perform the battery state and parameters' estimation based on the ECM. He proposed an EKF algorithm as the core control method, which was supported and improved by many researchers [28–30]. Xu et al. [31] proposed a fractional-order model (FOM) for SOC estimation. Compared to the integer-order model, the accuracy of the SOC estimation is significantly improved with the FOM. Lai et al. [32] combined the Ah integration method and the EKF to estimate the SOC based on multi-model global identification. The results proved the robustness of the algorithm. Xu et al. [33] employed the double Kalman filter (DKF) algorithm to estimate the SOC of a lithium-ion battery based on the temperature-dependent DPM. After verification, the battery SOC estimation error could be kept within the range of ± 0.004 under different temperature conditions.

At present, most works in the existing literature focus on the SOC estimation algorithm, and the effects of operating conditions on SOC estimation are barely considered. In this study, we studied the influence of different operating conditions on SOC estimation based on different battery models. Simulation results show that the SOC estimation accuracy of the DPM and the FOM is satisfactory, and the errors are within the range of ±0.06. Under any operating condition, the SOC estimation error of the FOM is always less than that of the DPM, but the adaptability of the FOM is not as good as that of the DPM.

This paper is organized as follows: first, we establish a DPM and a FOM. Second, we apply a mixed-swarm-based cooperative particle swarm optimization (MCPSO) algorithm to identify the battery parameters. The accuracy of the model is verified by using dynamic stress test (DST) operating condition. Third, a hybrid Kalman filter (HKF) algorithm is used to estimate the SOC of the battery, comprehensively utilizing the Ah integration method, KF, and EKF. Finally, the SOC estimation results

of the DPM and the FOM under DST, FUDS, and hybrid pulse power characteristic (HPPC) cycling conditions are analyzed by comparing six sets of experiments.
