Jointed SOH Estimation of Electric Bus Batteries Based on Operating Conditions and Multiple Indicators

Abstract

Accurately estimating the battery State of Health (SOH) is crucial for the safe and reliable operation of electric vehicles. Based on the actual operating data of electric buses, this article proposes a battery SOH estimation method that can be applied to multiple operating conditions and indicators. Specifically, the complex operating conditions are simplified into charging and driving conditions through data preprocessing. Under charging conditions, combined with Coulomb counting and incremental capacity analysis methods, a battery SOH estimation model of capacity indicators based on the Bayesian optimization bidirectional gated recursive unit model (BO-BiGRU) is established. Under driving conditions, the adaptive forgetting factor recursive least squares method considering the influence of current is used to identify the battery internal resistance feature. In addition, two separate battery SOH estimation models are established: one for internal resistance indicators based on BO-BiGRU and another for power indicators derived from the actual operational data feature. Finally, a joint battery SOH estimation method considering temperature and different operating conditions is proposed based on the SOH estimation results of the three battery indicators. The verification results show that the average error of the battery SOH estimation method proposed in this article is less than 2%, which has better accuracy for actual vehicles.

1. Introduction

Lithium-ion batteries (LIBs) are electric vehicles’ primary energy storage devices because of their high energy density and long cycle life [1]. However, battery safety accidents frequently occur as electric vehicles develop, resulting in significant property losses and severe threats to personal safety. Safety issues are still substantial barriers to the development of electric vehicles [2,3]. The performance degradation of batteries during use can inevitably bring safety hazards, and the State of Health (SOH) of batteries characterizes the performance degradation of batteries. Consequently, precise evaluation of SOH is essential for electric vehicles’ secure, efficient, and dependable functioning [4,5].

1.1. Literature Review

The SOH of LIBs is typically expressed as the ratio of the actual value to the initial value of a particular characteristic of the battery [6]. According to different focus areas, SOH can be measured through various indicators, including evaluations based on capacity, Ohmic resistance internal resistance (OIR), or power [7,8]. In order to accurately evaluate SOH, existing estimation methods are mainly divided into experimental, model-based, and data-driven methods. However, in practical applications, applying the definition to accurately calculate the dynamic SOH of a battery is difficult. Consequently, numerous scholars have proposed various methods for estimating the SOH of batteries. These methods are primarily categorized into experimental-based methods, model-based methods, and data-driven methods, each with distinct advantages and disadvantages.

The experimental method is the most direct way to obtain battery state parameters. However, the experimental method has a long testing cycle, high requirements for experimental environment and hardware, and is not suitable for SOH estimation of actual operating vehicles [9]. Model-based prediction methods simulate the dynamic characteristics of batteries by establishing mathematical models, including the Equivalent Circuit Models (ECM), Electrochemical, and Empirical models [10]. The ECM model, combined with state observers such as Kalman and particle filters, is widely used for battery capacity estimation [11]. However, the ECM-based method often exhibits lower stability due to uncertainties in battery modeling. The electrochemical model uses partial differential equations to describe the electrochemical reaction process of battery degradation [12]. However, these models exhibit limited robustness due to their high sensitivity to solving constraint conditions. Empirical models describe battery capacity degradation as a function of various factors, such as cumulative runtime, cumulative charging capacity, and the number of equivalent cycles. However, their applicability is limited under different operating conditions [13]. Model-based approaches require a thorough understanding of battery-aging mechanisms and the development of accurate mathematical models. Model-based approaches are often difficult to apply to on-board batteries under complex operating conditions due to their computational complexity and the difficulty of solving them.

Data-driven estimation methods primarily include Support vector machines (SVM) [14], Relevance vector machines (RVM) [15], Gaussian process regression (GPR) [16], and Neural Networks [17,18,19]. Traditional machine learning methods, such as SVM and RVM, are widely used for SOH estimation and prediction due to their advantages in regression tasks [20,21]. However, a significant limitation of these methods is that it is difficult for them to process large-scale training datasets, which can adversely impact their robustness. The GPR models can handle non-linear prediction problems, but their high spatial complexity becomes problematic when dealing with large datasets. In recent years, with the construction of new energy vehicle big data platforms and the improvement of computing power, data-driven SOH-based estimation methods have become an important approach. Guo et al. [22] combined a Gated Recurrent Unit (GRU) neural network with a Savitzky-Golay filter to establish an SOH prediction model for LIBs, which predicts battery capacity under different charging strategies. Chen et al. [23] proposed MGM, MREGM, and MMREGM models based on metabolic grey theory, and validated their battery capacity prediction ability under different operating conditions using the NASA dataset. Yang et al. [24] proposed a GBLSBooster multi task learning model using the NASA dataset to simultaneously evaluate the SOH and Remaining Useful Life (RUL) of lithium batteries. But the results based on experimental data are challenging to apply to real-world vehicles. Hong et al. [25] proposed a real-vehicle battery SOH prediction method to identify the OIR by Kalman filter and recursive least squares (RLS), demonstrating the feasibility of OIR as a health factor under real-vehicle conditions. Yet, the SOH estimation results based on a single OIR feature have large fluctuations, and it is difficult to provide stable and reliable results. Cheng et al. [26] combined empirical mode decomposition with a back-propagation long short-term memory (LSTM) neural network to develop a SOH estimation and remaining useful life prediction model. Notably, the joint algorithm, which combines the two algorithms, greatly multiplies the parameters in the model, making the parameter adjustment process more complicated, increasing the computational cost of the model, and slowing the estimation response. In order to better highlight the differences between different SOH estimation methods,

Table 1. Comparison of SOH estimation methods with the related literature.

Literature	Data Source	Capacity	OIR	Temperature	Power	Model	Methodological Features
[16]	Laboratory	√		√		GPR-based with optimized similarity measurement	Uses charging curve as inputs and improves SOH accuracy with covariance adjustments.
[22]	Laboratory	√			√	GRU + Filter	Uses an SG filter to denoise and GRU to predict SOH and RUL for non-linear trends.
[23]	Laboratory	√		√		Metabolic grey theory-based models	Proposes metabolic grey models for capacity prediction under varying conditions
[24]	Laboratory	√				EMD-GRU-RF	Integrates deep and machine learning for SOH estimation
[24]	Real Vehicle		√			GRU+RLS based OIR prediction	OIR is feasible as a health factor under real-vehicle conditions
[25]	Real Vehicle	√		√		EMD + LSTM-based hybrid model	Combines empirical mode decomposition and LSTM for SOH prediction
This paper	Real Vehicle	√	√	√	√	Coulomb counting + ICA + AFFRLS + BO-BiGRU	A combination of multiple methods that focus on real-vehicle data.

shows the data types used in the relevant literature, the health indicators considered, and the unique methods used in each study.

1.2. Challenges for SOH Estimation of Batteries

As shown in Table 1, these methods have achieved good results in battery SOH estimation, but they still face several issues. On the one hand, these methods predominantly use OIR or capacity as indices to assess health status, which limits their effectiveness due to the reliance on a single metric. On the other hand, SOH estimation methods that incorporate multiple indicators suffer from slow predictive responses and complex tuning processes [27]. In addition, data-driven estimation methods are primarily based on laboratory data, while it is difficult to apply laboratory results to actual operating data due to complex operating conditions and low data accuracy. To address the aforementioned issues, this paper proposes a comprehensive battery health estimation method that fuses multiple operating conditions and indicators using real-vehicle data.

1.3. Contributions of This Work

In response to the above issues, this article proposes a comprehensive battery health estimation method based on the joint consideration of multiple working conditions and multiple indicators under real-vehicle data. This study attempts to improve and contribute to current research as follows:

(1)

Feature parameter extraction of SOH: A capacity feature extraction method based on the Coulomb counting method combined with incremental capacity analysis is proposed for real-vehicle data characteristics. A self-adaptive forgetting factor recursive least squares method (AFFRLS) considering current influence is proposed to identify battery OIR. A proposed power feature extraction method based on the average maximum output power is used as the evaluation index.

(2)

Joint of Multi-Condition and Multi-Indicator Models: In response to the problems of slow prediction response and the complex parameter-tuning process in multi-indicator SOH estimation methods, a joint SOH estimation method based on temperature and charge–discharge operating conditions is proposed, which effectively improves estimation efficiency and accuracy.

(3)

SOH estimation based on BO-BiGRU model: This study establishes a hyperparameter self-optimized SOH estimation model combining Bayesian and BiGRU methods to improve prediction efficiency and accuracy.

1.4. Organization of the Paper

The remainder of the paper is structured as follows: Section 2 outlines the data sources and presents the data preprocessing results. Section 3 and Section 4 derive the state of health estimation results based on two working conditions: charging and driving. Section 5 and Section 6 implement the SOH joint estimation results and conclude the paper.

2. Data Acquisition and Processing

2.1. Data Acquisition

The dataset used in this study comes from the 2023 Digital Vehicle Competition, which is the operational data of 10 pure electric buses in a certain location in China. The data collection items are in accordance with China’s standardized data transmission regulations, including 73 data items such as vehicle and battery status. The time span of the data collection set is 1–2 years. In addition, the batteries in this dataset are ternary polymer lithium-ion batteries with a rated capacity of 150 Ah. The dataset comprises 14.65 million rows of data. In order to conserve computational resources, this paper employs the data from a single vehicle as a case study for analysis and modeling. The total data volume of this vehicle includes 2.03 million rows collected from 1 January 2022 to 31 December 2022. The starting mileage of the data segment is 35,051 km, and the ending mileage is 168,023 km. The partial items of the dataset are shown in

Table 2. Schematic sheet of dataset.

Date	Status	…	Mileage/km	Voltage/V	Current/A	SOC/%
2022-01-01 00:00:04	1	…	35,051.4	393.5	20.6	92
2022-01-01 00:00:14	1	…	35,051.5	395.3	−10.8	92
2022-01-01 00:00:24	1	…	35,051.5	390.9	66.7	92
2022-01-01 00:00:34	1	…	35,051.6	395.6	−23.9	92
…	…	…	…	…	…

.

2.2. Data Processing

Due to the complex operating conditions of vehicles, various factors can affect the quality of collected data, resulting in abnormal and invalid data in the raw dataset. To ensure the data’s completeness, continuity, and accuracy and to obtain trainable datasets, it is necessary to preprocess the obtained raw dataset. These data processing methods mainly include data-cleaning and data-slicing, as shown in Figure 1. The data cleaning method is used to process and handle abnormal data generated during data collection and transmission, including abnormal values, data duplication, and data loss. The data-slicing method is used to reconstruct and extract driving conditions.

During the data cleaning process, the $3-\sigma$ method is used to correct abnormal values in the data range and size within the data segment. The DBSCAN method is used to delete duplicate data with multiple results at the same time point and the interpolation method is used to fill in missing data. Then, according to the method shown in Figure 1, data-cleaning is performed on indicators such as mileage, speed, driving status, charging status, maximum temperature, minimum temperature, and SOC, which are highly correlated with battery life in the dataset. The result of data-cleaning is shown in Figure 2. Data-cleaning eliminates errors of collected data set and improves the reliability of the data.

The working conditions of new energy vehicles mainly include driving, charging, parking, and specific working conditions. The driving conditions are complex and varied, while the charging conditions are relatively stable, mainly affecting the changes in the health status of the battery [25]. Therefore, this article reconstructs and extracts these two working conditions through data-slicing methods. Firstly, based on the charging operating conditions satisfying $i=2$, $j=1$, $v<0$ and the driving operating conditions satisfying $i=1$, $j=3$, $v>0$ the complex driving conditions are analyzed. Subsequently, the complex driving working condition is simplified into charging and driving working conditions. Since the total voltage and total current vary greatly in different traveling segments, direct cleaning will lead to poor results. Then, based on the changes in adjacent SOC values, the charging dataset and driving data are sliced to obtain data segments of a single charging driving condition. Finally, the data segment is processed using the data-cleaning method, and the results are shown in Figure 3a,b.

3. SOH Estimation Based on Charging Data

Currently, most battery capacity calculation methods are based on the development of cell levels through preset load curves and environmental temperatures. However, in this charging dataset, most charging segments adopt a step charging strategy, with relatively small and step-like changes in charging current. In addition, the randomness of the starting and ending points of SOC values during charging is strong, making it difficult to obtain a complete charging segment. When applied directly under actual driving conditions, these methods may reduce the system-level estimation performance [28]. Therefore, this paper adopts a capacity calculation method that combines Coulomb counting with regional capacity to mitigate the impact of incomplete charging segments and temperature on capacity estimation [29]. Moreover, by analyzing the characteristics that affect capacity, a capacity estimation model under charging conditions can be established to estimate SOH. The SOH estimation method based on charging data is shown in Figure 4. Firstly, select charging segments that meet the capacity calculation requirements. Then, a combination of Coulomb counting and regional capacity is used to calculate the capacity of these fragments, and the results are used as capacity labels. Through further analysis, key features that affect capacity calculation are extracted, and a feature model is constructed based on them. Finally, the Bayesian optimization algorithm is utilized to automatically optimize hyperparameters and establish a BIGRU capacity prediction model for predicting the capacity of the remaining charging segments, achieving capacity estimation for the entire time period.

3.1. Capacity Calculation

The traditional incremental capacity analysis (ICA) is to calculate the derivative of charging capacity concerning open circuit voltage to obtain the rate of change of charging capacity at a specific voltage. On this basis, by combining the Coulomb counting method and the capacity calculation method of regional capacity, the battery’s actual capacity can be evaluated. The specific calculation steps are as follows:

Step 1: Extract the eligible charging segments. Extract charging segments with SOC levels ranging from 40% to 85% and charging currents between 0.5 C and 1 C. Ensure that the charging segments have similar charging rates and starting and ending SOC levels to ensure the effectiveness of ICA and Coulomb calculation methods.

Step 2: Derive the incremental capacity (IC) curve and locate the peak regions. The valued of IC can be expressed as:

$$\mathrm{IC}={\displaystyle \frac{\mathrm{d}Q}{\mathrm{d}U}}\approx {\displaystyle \frac{\mathsf{∆}{Q}_k}{\mathsf{∆}U}}={\displaystyle \frac{Q_k-Q_{k-1}}{U_k-U_{k-1}}}$$

(1)

The IC curve is derived from the capacity $Q$ to the terminal voltage $U$ ratio at two consecutive moments, $k-1$ and $k$. Due to the susceptibility of the original IC curves to approximate derivation and measurement errors, the IC curves across different discharge cycles are smoothed using an SG filter.

Figure 5a shows the IC processing curve of the vehicle during different cycles. Evidently, the area under a specific IC curve, i.e., the regional capacity, typically declines with increased accumulated mileage. The IC curve exhibits two peaks, designated as peaks A and B, whose heights and voltages can reveal the intricate electrochemical degradation mechanism [30]. Figure 5b illustrates the correlation between average temperature and peak voltage location. The Pearson correlation coefficients of the two peak voltage locations are 0.86 and 0.79, respectively, indicating a strong temperature dependence [31]. Since peak B is less affected by noise than peak A, peak B is selected as the capacity feature to be extracted.

Step 3: Determine the voltage interval. Selecting a suitable voltage interval is critical to calculating the area capacity. As the voltage interval increases, the linear relationship between regional and overall capacity becomes more evident [32]. However, this also results in a reduction in the number of charging segments that meet the extraction requirements. Thereby, there exists a tradeoff between accuracy and data readiness. The voltage interval selected in this article is 1 V, decided upon through the analysis of multiple voltage interval data.

Step 4: Determine the integration voltage range of the sliding area and calculate the area capacity. The prerequisite for calculating the capacity of a region is to determine the voltage range of that region. The moving voltage range of the IC curve under different cycles can be calculated using Equation (2).

$$\left\{\begin{array}{l}{\tilde{V}}_1=V_1+\left(V_{\mathrm{peak}}-V_{\mathrm{mean}}\right)\\ {\tilde{V}}_2=V_2+\left(V_{\mathrm{peak}}-V_{\mathrm{mean}}\right)\end{array}\right.$$

(2)

where $V_{\mathrm{peak}}$ is the peak voltage of peak B and $V_1$ and $V_2$ are the starting and ending voltage of the regional voltage interval, respectively. $V_{\mathrm{mean}}$ is the median voltage of the regional voltage range. The moving voltage range of peak B is represented as $\left[{\tilde{V}}_1,{\tilde{V}}_2\right]$. Then, the regional capacity can be expressed as:

$$C_{\mathrm{n}}\left(\mathrm{t}\right)={\displaystyle {\int }_{t\left({\tilde{V}}_2\right)}^{\mathrm{t}\left({\tilde{V}}_1\right)}I\left(t\right)\mathrm{d}t}$$

(3)

where $I\left(t\right)$ is the charging current.

In this study, $V_{peak}$ is between [385 V and 393 V], and the sliding voltage range $V_1$, $V_2$, and $V_{\mathrm{mean}}$ is calculated from this range to be 389 V. Then, the regional capacity can be calculated according to Equation (3).

The comparison between the improved regional capacity calculation method and the traditional fixed interval regional capacity calculation method is shown in Figure 6. From Figure 6, it can be seen that the battery capacity calculated by both methods gradually decreases with the increase of mileage. The method based on fixed voltage regions is more susceptible to changes in temperature. In low-temperature environments, at 35,000 km and 75,000 km, the calculated capacity values are relatively small because the fixed region method does not account for the impact of temperature on the peak voltage position of the IC. Conversely, the sensitivity of the capacity calculation method in the sliding voltage region to temperature changes is significantly reduced, and the rebound phenomenon during the capacity decay process more closely aligns with real-world conditions.

3.2. Feature Extraction Affecting Battery Capacity

The capacity feature is influenced by various factors such as charging rate, temperature, number of cycles, and charging depth [33]. However, under actual traffic conditions, some data are difficult to test and obtain. Therefore, extracting features that impact capacity based on existing vehicle driving data is necessary. Firstly, based on the characteristics of actual vehicle data, the charging dataset is subjected to feature extraction. Accumulated mileage is selected as a substitute for cycle times, and the average current during charging represents the charging rate. Seven parameters, including cumulative mileage, average total current, and average total voltage, were selected from the dataset as influencing capacity characteristics, and the Pearson correlation coefficient method was used for correlation analysis. Figure 7 shows a high correlation between battery capacity and cumulative mileage, temperature, peak temperature, and average current. However, a significant correlation exists between peak temperature and temperature, so peak temperature is removed. This article ultimately determines the input parameters for the capacity estimation model: cumulative mileage, average temperature of charging segments, and average current of charging segments.

3.3. SOH Estimation Method

In Section 3.3, the battery capacity of some charging segments is obtained through calculation, but not every charging segment is continuous and not every charging segment meets the requirements for battery capacity extraction. Therefore, to achieve health status estimation throughout the life cycle, this paper establishes a multi-feature prediction model based on charging capacity. To better handle the problem of time recursion and consider the influence of historical data time order, the bidirectional gated recurrent neural network (BiGRU) is applied to estimate the SOH of batteries. The specific structure of the BiGRU is shown in Figure 8 [34].

As an improved recurrent neural network model, GRU possesses a faster convergence speed through a simplified gating mechanism that reduces the model’s training parameters while retaining important information. However, GRU can only infer from past information, limiting its dealing with bidirectionally dependent information. As shown in Figure 8, the BiGRU model consists of two GRU models with opposite directions, which not only retain the high efficiency of GRU but also capture both forward and backward information, further improving the understanding of time series data. Its specific calculation formula is shown as follows.

$$\left\{\begin{array}{l}{Z}_t=\delta (W_z{x}_t+U_z{h}_{t-1}+b_z)\\ R_t=\delta (W_r{x}_t+U_r{h}_{t-1}+b_r)\\ {\tilde{h}}_t=\tan h(W_h{x}_t+U_h(r_t\odot h_{t-1}))\\ h_t=(1-Z_t)\odot h_{t-1}+Z_t\odot h_t\end{array}\right.$$

(4)

where in $Z_t$ and $R_t$ represent the update gate and reset gate vectors, respectively, while $\delta$ denotes a non-linear sigmoid function. $W$, $U$, and $b$ denote the weight matrices in the GRU model, whereas $\odot$ represents the dot product operation. In BiGRU algorithms, hyperparameter tuning typically relies on trial-and-error methods. However, finding optimal hyperparameters through empirical and trial-and-error methods is time-consuming and may lead to suboptimal solutions, especially with large search spaces. Therefore, this paper proposes a Bayesian optimization (BO) algorithm for automatically tuning hyperparameters of the BiGRU method (BO-BiGRU), including the initial learning rate, number of hidden units, L2 regularization, and maximum iterations in BiGRU. BO is a global optimization algorithm. By using the Gaussian process regression model to estimate the objective function [35], the prior knowledge of the optimization problem is established using the probability model order for randomly selected hyperparameters, and then the objective function $f(x)$ is scalarized [36], as shown in Equation (5).

$$x^{*}=\arg {\max }_{x\in X}f(x)$$

(5)

where $x^{*}$ is the hyperparameter of the final optimization and $f(x)$ is the objective function to be optimized. The BO algorithm framework mainly consists of two core parts: the probability proxy model and the collection function. The probability proxy model includes a prior probability model and an observation model, while the collection function is composed of a posterior probability distribution. By maximizing the collection function, it is possible to find the next most optimal evaluation point. After a limited number of iterations, the optimal hyperparameters are finally found.

3.4. SOH Estimation Result Analysis

The optimal hyperparameters obtained through Bayesian optimization are as follows: learning rate: 0.0982, number of hidden units: 30, maximum iteration number: 192, and L2 regularization coefficient: 1.409 × 10⁻¹⁰. These optimal parameters were used to train a capacity model. The training set contained 168 charging segments, and the validation set contained 71 charging segments. To compare the accuracy and time required for capacity modeling of different neural network models, evaluations including root mean square error (RMSE), mean absolute error percentage (MAPE), and coefficient of determination (R²) were used. The results are shown in

Table 3. Results of different algorithms for capacity modeling.

Model Name	RMSE/Ah	MAPE/%	R2
LightGBM	3.95	3.25	0.83
BP	3.14	3.07	0.78
LSTM	2.93	3.03	0.86
GRU	2.49	2.83	0.84
BiGRU	2.43	2.54	0.85
BO-BiGRU	2.22	2.02	0.88

.

Table 3 shows that the RMSE of the Bo-BiGRU method on the test set is 2.22 Ah, and the MAPE is 2.02%. Compared with methods such as LightGBM, BP, LSTM, and GRU, this method performs better in terms of performance. The capacity prediction results of the model are shown in Figure 9.

As shown in Figure 9a, there is a significant fluctuation between the actual and predicted capacities. This fluctuation arises from uncertainties such as sensor measurements and SOC estimation errors. Therefore, in order to more accurately describe the degradation trend of capacity, it is necessary to smooth the capacity curve. Due to the non-linear nature of the decreasing trend in capacity, traditional fitting methods often cannot accurately reflect this actual trend. Hua et al. [37] utilized GAM to non-linearly analyze NOx concentration with satisfactory results. GAM reduces the risk of linear modeling by using the smoothing function S(x) to handle the target variable. The specific functional form is obtained through an inverse fitting algorithm. Therefore, this paper used cumulative mileage as an input, and the capacity was denoised by S(x) of GAM. The results are shown in Figure 9b.

The data of the remaining 634 charging segments were input into the trained model to obtain the capacity values over the entire period. According to Equation (6), the SOH value of each charging segment was calculated by the current capacity $C_i$ ratio to the initial capacity C₀.

$$SO{H}_{\mathrm{c}}={\displaystyle \frac{C_i}{C_0}}\times 100\%$$

(6)

Finally, the trend of SOH variation with mileage during the entire charging period was obtained. Figure 10 shows that the overall SOH of the car battery pack was less affected by temperature and shows a continuous decline trend. At 168,023 km, the SOH of the battery pack decreased from the initial 96.54% to 81.89%. The error between the predicted results of the model and the actual values is less than 2%. In theory, as the accumulated mileage increases, the health status of the battery should gradually decline. However, due to factors such as temperature-induced battery capacity regeneration, data acquisition accuracy deviation, and SOC estimation error, the SOH value may fluctuate in actual operation. Therefore, it is reasonable for SOH values to be higher than the previous time at certain moments.

4. SOH Estimation Based on Driving Data

Under complex driving conditions, the degradation of batteries is mainly manifested by an increase in OIR and a decrease in output power. This study selected OIR and maximum output power as evaluation indicators for battery SOH. However, when the load current fluctuates sharply, the traditional forgetting factor RLS (FFRLS) algorithm makes it difficult to accurately identify the OIR value of the battery. Therefore, this study proposes the AFFRLS algorithm to reduce the impact of current fluctuations on resistance identification and improve identification accuracy. Additionally, by analyzing the influence characteristics of internal resistance, OIR and power prediction models were constructed to achieve accurate estimation of battery SOH under driving conditions.

4.1. Feature Extraction of Driving Conditions

4.1.1. Feature Extraction for OIRs

In practical applications, the cost of measuring battery OIR using sensors is relatively high. Therefore, this study adopted the establishment of an equivalent circuit model to achieve the acquisition of battery internal resistance. Different ECM structures result in varying simulation accuracies. He et al. [38] evaluated and compared different equivalent circuit models through experimental comparisons and found that the DP model and Thevenin model have better dynamic performance and smaller errors. Compared with the Thevenin model, the DP model adds an RC parallel circuit, which improves dynamic performance and simulation accuracy but which increases model complexity and parameter identification computation. Therefore, under the basic conditions of satisfying dynamic performance and accuracy, this study chose the relatively simple Thevenin model as the research object. Since the data provide the total voltage and total current of the battery pack, the entire battery pack was treated as a whole in the modeling process. The state equation of the Thevenin model is as follows:

$$\left\{\begin{array}{l}{U}_t=U_{OC}-U_P-i_L{R}_0\\ {\dot{U}}_t={\displaystyle \frac{i_L}{C_p}}-{\displaystyle \frac{U_P}{C_p{R}_p}}\end{array}\right.$$

(7)

where $C_p$, $R_p$ are the polarization OIR and polarization capacitance, respectively, and $U_{\mathrm{P}}$ is the voltage drop of the RC parallel circuit. In order to facilitate the practical application, it is necessary to discretize the equation of state, and the discretized result is as follows:

$$\left\{\begin{array}{l}{U}_{t,k}={\Phi }_{1,k}{\mathsf{\theta }}_{1,k}\\ \Phi =[1,U_{t,k-1},i_{L,k},i_{L,k-1}]\\ \mathsf{\theta }={[a_1,a_2,a_3,a_4]}^T\end{array}\right.$$

(8)

where $\Phi$ is the system data matrix, and $\theta$ is the system parameter matrix. Based on the discretized model in Equation (9), a method with parameter identification can be used to identify the relevant parameters. The FFRLS algorithm is one of the most widely used methods in parameter identification. The algorithm design flow of FFRLS is as follows:

$$\left\{\begin{array}{l}{K}_k=P_{k-1}{\Phi }_k^{\mathrm{T}}{({\Phi }_k{P}_{k-1}{\Phi }_k^{\mathrm{T}}+\mu )}^{-1}\\ P_k={\displaystyle \frac{(I-K_k{\Phi }_k)P_{k-1}}{\lambda }}\\ {\widehat{\theta }}_k={\widehat{\theta }}_{k-1}+K_k(y_k-{\Phi }_k{\widehat{\theta }}_{k-1})\end{array}\right.$$

(9)

where $K_k$ is the gain matrix, $P_k$ is the covariance error matrix, $I$ is the identity matrix, and $\lambda$ is the forgetting factor. In this method, the range of $\lambda$ is usually [0.95, 1.00]. By multiplying the $\lambda$ with data, the weights of new and old data can be reallocated to avoid data redundancy and alleviate the problem of data saturation [39]. As $\lambda$ increases, the impact of historical data errors on identification results increases, and the algorithm tends towards the overall optimal solution, but the tracking effect weakens. As $\lambda$ decreases, the tracking effect improves, but the volatility of the identification results increases. Due to the reverse and sudden changes in current caused by operations such as kinetic energy recovery and emergency acceleration during actual driving, traditional FFRLS algorithms perform poorly in parameter recognition. Therefore, this study proposes an AFFRLS algorithm that can better consider the impact of current changes on battery parameter identification. This method describes the variation of $\lambda$ by comparing the current changes at time $k$ and $k-1$. If the current change is $\mathsf{∆}I\left(k-1\right)\le \mathsf{∆}I\left(k\right)$, it indicates a significant change in the current, and the value of $\lambda$ should be reduced to better track the current changes. On the contrary, it indicates that the change is not significant, and the value of $\lambda$ is the value at time $k-1$. The formula is defined as follows:

$$\left\{\begin{array}{l}{\rm E}(k-1)=I(k-1)I(k-2)\\ {\rm E}(k)=I(k)I(k-1)\\ \mathsf{∆}I(k-1)=|I(k-1)-I(k-2)|\\ \mathsf{∆}I(k)=|I(k)-I(k-1)|\hfill \end{array}\right.$$

(10)

where $E(k)$ is the product of two current moments. If $E(k)E(k-1)<0$, it indicates a change in current direction, and if $\lambda$ is set to the minimum value within the interval, it mitigates the impact of sudden current changes on parameter identification. If $\mathsf{∆}I\left(k-1\right)\le \mathsf{∆}I\left(k\right)$, then the forgetting factor correction can be expressed as:

$$\mathsf{∆}\lambda =\rho \cdot \mathsf{∆}I\left(k-1\right)$$

(11)

where $\rho$ is correction factor and is set to −0.01. Moreover, if $\mathsf{∆}I\left(k-1\right)>\mathsf{∆}I\left(k\right)$, $\mathsf{∆}\lambda =0$. The flow of the algorithm is shown in Figure 11.

According to the above AFFRLS method, a driving segment of the vehicle is identified and compared with the battery pack voltage error based on the traditional FFRLS method, as shown in Figure 12, where the error limit is drawn based on the total voltage in the original data. As shown in Figure 12, the error fluctuation of the AFRRLS method is significantly smaller than that of the FFRLS method, and the overall identification error remains within 2% of the total voltage. This indicates that the AFRRLS method has high parameter identification accuracy, and the results obtained can be applied to subsequent tasks.

Due to the fluctuation of OIR values with changes in vehicle operating status and time, it is difficult to uniformly screen for outliers in all OIR identification results. According to relevant experiments [40], the rate of change in the OIR of battery increases with decreasing temperature and decreases with increasing temperature. Therefore, box plots were used to eliminate outliers in the resistance values at various temperatures to ensure the accuracy of OIR modeling, as shown in Figure 13a. After removing abnormal values from the OIR identification results through a box plot, the remaining OIR identification results were arranged based on accumulated mileage and temperature, as shown in Figure 13b. It can be seen that the OIR decreases with increasing temperature and increases with increasing mileage. The results show that the OIR of the identification result conforms to the law and is highly reliable.

4.1.2. Feature Extraction for Battery Output Power

Due to the inability to measure the maximum output power of the operating vehicle’s battery, this study calculated the average of the maximum output power of each driving segment within each 1000-km interval, which was calculated as the maximum output power of the battery over a long duration. The calculated results of maximum output power are shown in Figure 14. As shown in Figure 14, with the increase in accumulated mileage, the maximum output power of the selected vehicle’ battery exhibits an overall downward trend.

4.2. SOH Estimation

4.2.1. SOH Estimation Based on OIR Feature

Similar to the SOH estimation method based on battery capacity under charging conditions in Section 3, the BiGRU method was applied to establish a SOH estimation model based on OIR features under driving conditions. Firstly, based on the analysis results of the Pearson correlation coefficient method, temperature, insulation resistance, and cumulative mileage were selected as input parameters for the OIR prediction model. Then, the battery SOH value was calculated based on the predicted OIR value. The analysis results of the Pearson correlation coefficient method are shown in Figure 15.

In the BiGRU prediction model based on the OIR feature under driving conditions, OIR is the output of the model, and features parameters such as temperature, insulation resistance, and cumulative mileage, which were used as input training parameters. Referring to the process in Figure 4, the BiGRU model hyperparameters were optimized using Bayesian optimization to obtain the optimal hyperparameters: a learning rate of 0.0982, a number of hidden units of 30, a maximum iteration count of 210, and am L2 regularization of 1.309 × 10⁻¹⁰. The OIR model was trained with the optimal parameters. The validation datasets include 932 discharge segments, and the test set includes 400 discharge segments. The training results are shown in Figure 16.

To evaluate the effectiveness of the BO-BIGRU method in predicting OIR features, this study compared the prediction results based on LightGBM, BP, LSTM, GRU, and BiGRU models, as shown in

Table 4. Results of different algorithms for OIR modeling.

Model Name	RMSE/Ω	MAPE/%	R2
LightGBM	0.0049	9.45	0.83
BP	0.0059	10.32	0.78
LSTM	0.0046	9.93	0.86
GRU	0.0042	8.76	0.84
BiGRU	0.0390	8.49	0.85
BO-BiGRU	0.0034	8.26	0.86

. The method based on BO-BiGRU has lower RMSE, MAPE, and higher R², indicating that the method proposed in this paper has batter prediction accuracy.

The SOH value based on the predicted OIR of the model was calculated, defined as follows:

$$SO{H}_{\mathrm{r}}={\displaystyle \frac{R_E-R_i}{R_E-R_0}}\times 100\%$$

(12)

where $R_E$ is the cutoff internal resistance, $R_i$ is the current internal resistance, and $R_0$ is the initial internal resistance. The final trend of the health status $SO{H}_{\mathrm{r}}$ based on OIR as a function of driving mileage is shown in Figure 17. As shown in Figure 17, the battery pack SOH generally decreased with the increase in cumulative mileage. However, due to the high sensitivity of OIR to temperature changes, the SOH evaluation curve based on OIR fluctuates significantly. This indicates that it is challenging to estimate SOH by relying only on OIR, and it is necessary to make a comprehensive judgment in combination with power and capacity.

4.2.2. SOH Estimation Based on Output Power Feature of Batteries

The SOH estimation based on power characteristics under driving conditions is expressed as

$$SO{H}_{\mathrm{P}}={\displaystyle \frac{P_{\mathrm{i}}}{P_{\mathrm{m}}}}\times 100\%$$

(13)

where $P_{\mathrm{m}}$ is the vehicle’s initial maximum output power and $P$ is the maximum power of the battery pack of the driving vehicle. The setting of power extraction intervals based on every 1000 km in Section 4.1.2 results in a limited data sample size. In this case, the neural network model was prone to overfitting, i.e., the model may have remembered the noise and details in the training data, which reduced its ability to generalize based on new data. To avoid this problem, this study chose the linear fitting method for the regression prediction of the maximum output power. Linear fitting can provide more stable and robust regression results when dealing with smaller datasets. The average maximum output power before the accumulated mileage of 40,000 km was selected as the initial maximum power, and the battery health status based on power $SO{H}_{\mathrm{P}}$ was calculated using Formula (13). The results are shown in Figure 18.

5. Joint Estimation of the Charge-Driving SOH

5.1. SOH Joint Estimation with Multiple Indicators

In the above study, capacity prediction models, OIR prediction models, and output power prediction models were established based on charge and driving data, and the SOH of the battery was preliminarily evaluated using the three indicators. Due to the significant differences in the usage status of these three health indicators, the SOH values they represent are usually not consistent [41]. This inconsistency will have an impact on the actual health status assessment, so further processing of the predicted results is needed to achieve the fusion evaluation of SOH. In addition, a large amount of the literature has confirmed that temperature is one of the key factors affecting battery performance [7,27]. As the temperature decreases, the OIR gradually increases, and the rate of change of the OIR also increases, indicating that the lower the temperature, the more significant the impact on the OIR. Specifically, the change in OIR is highly correlated with the maximum output power of the battery. In contrast, battery capacity is less affected by temperature. In this dataset, the charging process does not involve low-temperature operation (below 0 °C). Therefore, in low-temperature environments, SOH evaluation should mainly be based on capacity. In high-temperature environments, a comprehensive evaluation that combines capacity, OIR, and power indicators is more accurate for the true SOH. Finally, by analyzing the SOH estimation results under three indicators, a joint estimation of SOH is performed to improve the accuracy of the evaluation. This article proposes a method for joint estimation of SOH, as shown in Formulas (14) and (15), defined as follows:

$$\left\{\begin{array}{l}r={\displaystyle \frac{|SO{H}_{\mathrm{c}}-\frac{SO{H}_{\mathrm{r}}+SO{H}_{\mathrm{P}}}{2}|}{SO{H}_{\mathrm{c}}}}\hfill \\ K=\left|T_{\mathrm{h}}-T\right|\times \xi \hfill \end{array}\right.$$

(14)

$$SOH=\left\{\begin{array}{l}{\displaystyle \frac{SO{H}_{\mathrm{r}}+SO{H}_{\mathrm{c}}+SO{H}_{\mathrm{P}}}{3}}\hspace{1em}r\ge K\hfill \\ SO{H}_{\mathrm{c}}\hspace{1em}r<K\hfill \end{array}\right.$$

(15)

where $r$ is the SOH estimation error under the three methods, $K$ is the temperature difference quantization coefficient, $T_{\mathrm{h}}$ is the average temperature of all charging segments, $T$ is the average temperature of the current segment, and $\xi$ is the temperature correction factor. The average SOH value of the last five charging segments was used as the final SOH reference value.

According to Equation (15), the estimation error value r was calculated, as was the temperature difference coefficient $K$ of SOH under three methods. Then, using the relationship between $r$ and $K$ in Equation (14), the joint estimated SOH value was calculated. When $r<{\rm K}$, it indicates that the ambient temperature is low, and the jointly estimated SOH should be based on the capacity result as a reference. When $r\ge K$, it indicates that the ambient temperature is high. At this time, the weight allocation of each evaluation index and the influence of temperature should be considered. The final value and estimation error of SOH joint estimation for different $\xi$ values in the interval of [0.005, 0.015] with a step size of 0.001 were calculated. The results are shown in Figure 19.

From Figure 19, it can be seen that when $\xi$ is greater than 0.007, the accuracy of the joint estimation of SOH is relatively close. But when $\xi$ is too large, the results of the joint estimation are very close to the results of the capacity estimation and ignore too many OIR and power factors. Therefore, while considering the accuracy of SOH estimation and the combination of various operating condition information, $\xi$ was selected as 0.013, and the final joint estimated SOH result is shown in Figure 20.

From Figure 20, it can be seen that all four types of SOH estimation curves in the figure show a significant downward trend, which is consistent with the gradual decline of battery performance during long-term cyclic use, thus verifying the effectiveness of the SOH estimation method proposed in this study. The estimated final values of SOH based on OIR, power, and capacity are 93.8%, 88.9%, and 83.7%, respectively, while the reference value of SOH (the average SOH value of the last five charging segments) is 84.52%. In contrast, the estimation final value of the comprehensive estimation method is 85.5%, which is more accurate than the SOH estimation based solely on power or internal resistance, and the error is controlled within 2%. Compared with the capacity-based SOH estimation results, the joint estimation method combines multiple SOH indicators, which not only makes the estimation results closer to the true SOH value of the battery, but also shows a smoother downward trend, effectively reducing the impact of single indicator fluctuations on the estimation results. Therefore, the joint estimation method demonstrates stronger robustness in battery SOH assessment, providing a more reliable basis for battery health management.

5.2. Results and Validation

The dataset from the National Aeronautics and Space Administration (NASA) of the United States was selected for validation to verify the universality and effectiveness of the design method [42]. The SOH estimation results of battery B45 in Appendix A are shown in Figure 21.

Figure 21a shows the SOH estimation results of the battery in 70 cycles, where the SOH estimation results based on charging and discharging data were obtained by the method proposed in this paper, demonstrating the effectiveness of SOH estimation-based OIR and capacity modeling. Compared with the estimation results of a single charge and discharge data, the estimation curve of the comprehensive estimation method is between the first two, and the combination of multiple indicators effectively reduces the fluctuation of the results. In addition, the estimation results of all three methods remained within a 5% error range. Figure 21b shows the SOH prediction curves of three algorithms (GRU, BP, LightGBM) based on capacity health factors, as well as the SOH prediction curve of the joint method. (For more detailed parameters, please refer to

Table A2. Results of different algorithms for capacity modeling.

Model Name	RMSE/%	MAPE/%	R2
Joint	0.0157	1.54	0.96
GRU	0.0176	1.97	0.95
LightGBM	0.0452	3.55	0.78
BP	0.0529	4.80	0.77

in Appendix B) Although single factor algorithms can effectively capture the downward trend of SOH, their prediction accuracy varies to some extent and is susceptible to fluctuations. In contrast, the predicted final value of the joint method is closer to the reference value with an error within ±2, and the curve exhibits higher smoothness. Next, the method proposed in the article was applied to other vehicles of the same type. Figure 22 shows the relationship between the joint SOH estimation results of the other three vehicles and the change in mileage increment. The final SOH estimation error was less than ±2%. Overall, the comprehensive estimation method proposed in this study successfully reduces estimation errors by combining multiple indicators, providing more reliable and robust SOH estimation results than single-indicator methods.

6. Conclusions

This study proposes a joint estimation method that combines vehicle charging and discharging data segments to estimate the SOH of batteries based on actual vehicle operating data. Based on partitioned vehicle charging and driving data, considering the difficulty in obtaining battery capacity, internal resistance, and maximum output power under actual vehicle conditions, this paper proposes a multi-index SOH joint estimation method that integrates capacity, internal resistance, and power through feature extraction, correlation analysis, and data-driven model construction. The main conclusions of the study are as follows:

(1)

The feature extraction method for battery capacity and OIR proposed in this study are tailored to the characteristics of real-vehicle data and can more accurately extract key features that affect SOH. Compared with traditional laboratory methods, the capacity feature extraction method based on Coulomb counting and incremental capacity analysis proposed in this paper effectively eliminates the influence of temperature on capacity features. The OIR identification method based on AFFRLS effectively eliminates the influence of current fluctuations and significantly improves the accuracy of feature extraction.

(2)

Compared with the SOH estimation results based on independent OIR and independent power, the final value accuracy estimated by the multi-condition and multi-index combination method proposed in this study was improved by 8.22% and 3.48%, respectively. In addition, compared with the SOH estimation results of independent battery capacity, the method proposed in this paper not only shows a smoother downward trend but is also closer to the true SOH of the battery. Notably, the method proposed in this study has higher stability and accuracy in dealing with fluctuations in single indicators and external disturbances, with a final value error of less than 2%, significantly better than the SOH estimation method for single operating conditions and single indicators.

(3)

Compared with various machine-learning algorithms, such as BP, LSTM, LightGBM, XGBoost, GRU, etc., the capacity and OIR estimation model of BO BiGRU based on hyperparameter self-optimization proposed in this paper has an average absolute error of less than 4% and 9%, respectively, indicating better accuracy in battery SOH estimation.

In future work, probabilistic and statistical methods will be adopted to optimize the weighting parameters in the algorithm to improve the scientific validity and accuracy of the model. In addition, the impact of bus operation scheduling and charging networks on battery health will be considered to further reflect a more realistic system.