In Equation (
1):
is the remaining capacity of the battery.
is the capacity of the battery when discharged at a constant current I. The traditional approaches for estimating the SOC of batteries can be classified into three main methods: Coulomb counting, the open-circuit voltage (OCV) method, and the Kalman filtering method [
11,
12]. In the case of the Coulomb counting method, its accuracy largely depends on the precise measurement of the current during battery discharge and the accuracy of the initial measured SOC of the battery. When the initial SOC is estimated accurately, the error in calculating the battery’s SOC using the integration of charge and discharge currents during the cycle will be smaller. However, it is generally challenging to calculate the initial SOC precisely, and there are inevitably errors in measuring the battery current during the cycle. These errors accumulate over time, increasing the level of inaccuracy in estimating the SOC using the Coulomb counting method. When the correct initial SOC is provided, the Mean Square Error (MSE) can be reduced by 1.05% compared to the case where an incorrect initial SOC is used, which results in an MSE of 7.3% [
13,
14]. This method offers the advantages of high accuracy and simplicity in the calculation. However, the OCV method is associated with a relatively long measurement time, making it impractical for meeting the requirements of online estimation of the SOC [
15]. The standard and modified Kalman filter are widely employed. In the field of automated vehicles, Neel P. Bhatt and Xin Xia et al. (2023) proposed an integrated localization method based on the fusion of an inertial dead reckoning model and 3D LiDAR-based map matching [
16]. In their experiments, the Kalman filter is utilized for stability analysis. Xin Xia and Runsheng Xu et al. (2023) utilize the consensus Kalman information filter (CKIF) to merge shared information from connected vehicles, proposing secure cooperative localization for connected automated vehicles (CAVs) [
17]. In the literature on SOC prediction using Kalman filtering, Sepasi and Ghorbani et al. (2014) extended the application of the Kalman filter method to assess the SOC in battery packs [
12]. This approach leveraged the dynamic characteristics of gas and liquid flows within the battery to improve the accuracy and effectiveness of SOC estimation. Ning and Deng et al. (2022) employed the Kalman filter method to jointly estimate the health condition and SOC of a 48 V battery system. The study conducted showed that their method achieved remarkable results, with average absolute errors of less than 0.88% for SOC estimation and 0.64% for capacity estimation [
18]. Poloni and Figueroa-Santos et al. (2018) proposed a method for estimating the SOC of lithium iron phosphate batteries (LiFePO4) by extracting SOC-related features from the impedance spectrum [
19]. Wang and Zhao et al. (2023) proposed an impedance-based algorithm for estimating the SOC of batteries. By employing the impedance spectrum method for calibration, it is possible to reduce the maximum absolute error to less than 5.4% in estimating the SOC of batteries [
20]. Wadi and Abdel-Hafez et al. (2023) proposed a method to enhance the accuracy of SOC estimation by employing an iterated Extended Kalman Filter (EKF) with correction terms that are independent of output errors [
21]. Wang and Wang et al. (2023) conducted tests on lithium-ion batteries under different temperatures and introduced an enhanced EKF model for SOC estimation [
22]. Their proposed model demonstrated remarkable performance, with SOC estimation errors within 3% under variable and 1% under low temperatures. Du and Shao et al. (2022) presented a collaborative algorithm for estimating batteries’ SOC and the state of health (SOH) of batteries. Their approach combined the least squares method and EKF to improve SOC estimation accuracy [
23]. Lin and Li et al. (2023) introduced a model that combines an EKF with an RC equivalent circuit for battery state estimation [
24]. Through experimental analysis, they observed that this model exhibited superior accuracy compared to the commonly used EKF model. Wang and Lu et al. (2019) developed a method to estimate the SOC of lithium-ion batteries. They employed a dual EKF approach to improve the estimation accuracy with the characteristics of the batteries [
25]. Their experiments demonstrated that this method effectively eliminates measurement noise and ensures the accuracy of model estimation. Luan and Qin et al. (2023) presented a novel approach for estimating the SOC of batteries by developing an equivalent circuit model using a particle swarm optimization algorithm [
26]. Qiu and Li et al. (2019) introduced an improved EKF for estimating the SOC of vanadium redox batteries (VRBs) [
27]. Their research demonstrated that this method outperformed the traditional EKF method concerning convergence speed and accuracy.
In addition to conventional approaches, novel artificial intelligence algorithms utilizing extensive datasets have emerged to estimate the battery’s SOC. The data-driven approaches do not need to consider the internal characteristics of the battery and use the previously existing data and experience to predict the results in unknown cases [
28,
29]. Omer Ali and Ishak et al. (2022) introduced an online estimation method for SOC using Gaussian process regression. Their experiments demonstrated that the Gaussian process regression model, optimized with an RBF kernel, achieved an estimation error of less than 2% [
30]. Zhang and Xia et al. (2021) proposed an optimization technique for backpropagation neural networks using time series models [
31]. Mao and Song et al. (2022) introduced a backpropagation neural network enhanced by the Levy flight-optimized particle swarm algorithm [
15]. Anton and Nieto et al. (2013) conducted a study utilizing support vector machine (SVM) models. They employed a single SVM model to assess the SOC and validated the model using untrained battery data. The study demonstrated that the accuracy of the SVM model for SOC estimation could reach as high as 97% [
32]. Zhang and Li et al. (2019) proposed an evaluation model based on sparse least squares SVM. This model addressed the issue of over-fitting encountered in traditional SVM and demonstrated higher accuracy than the cardless trace Erdmann filter model [
33]. Hu and Ma et al. (2022) developed a deep neural network for estimating the SOC of batteries. Their model achieved a maximum error of 2.5% in SOC estimation during charging, with an MSE of 0.8% [
34]. M.S. and M.A. et al. (2020) utilized an enhanced firefly algorithm in conjunction with a time-delay neural network model to forecast SOC. The prediction outcomes yielded a Root Mean Square Error (RMSE) below 1% [
35]. Duan and Song et al. (2020) introduced a gated recurrent unit recurrent neural network model with an activation function layer. The model demonstrated improved accuracy compared to the recurrent neural network model, with an enhancement ranging from 0.1% to 0.4% when the measurement data contained noise. The RMSE of the proposed model remained stable at approximately 1.9% [
36]. Javid and Abdeslam et al. (2021) introduced an online recurrent neural network for long and short-term memory, which was optimized using an online gradient learning method. The proposed model demonstrated superior performance to the Kalman filter model through experimental verification, highlighting its improved accuracy and effectiveness [
37].
The SOC of lithium-ion batteries is a critical parameter for assessing the mileage of new energy vehicles. Most research focused on training and evaluating SOC models has primarily emphasized applying one or two temperature states above 0 C. While these models demonstrate good accuracy in estimating the SOC of lithium-ion batteries at room temperature, there remains a significant gap in evaluating SOC across a broader temperature range, including 0 C and below. Addressing this limitation and improving the applicability of SOC estimation models across various temperature conditions requires further research.
Researchers have achieved relatively high accuracy in predicting the behavior of lithium-ion batteries using the support vector regression(SVR) model. Exploring the use of advanced mathematical techniques to address SOC prediction problems is also a promising avenue. The application of genetic algorithm(GA) to optimize SVR models for prediction tasks is relatively less explored compared to other methods. Additional research is needed to thoroughly investigate the potential benefits and limitations of employing GA in conjunction with SVR for prediction tasks and to evaluate its performance in different contexts.