Multiphysics Feature-Based State-of-Energy Estimation for LiFePO4 Batteries Using Bidirectional Long Short-Term Memory and Particle Swarm-Optimized Kalman Filter

Abstract

State-of-energy (SOE) estimation helps to enhance the safety of battery operation and predict vehicle range. However, the voltage plateau of the LiFePO4 (LFP) battery presents a significant challenge for SOE estimation. Therefore, this paper introduces a significantly varying mechanical force feature to tackle the flat voltage curve in the mid-SOE region. A fusion model that integrates a bidirectional long short-term memory (BiLSTM) network, particle swarm optimization (PSO), and Kalman filter (KF) algorithm is proposed for SOE estimation. The BiLSTM is applied to fully capture the temporal dependencies from inputs to output over both local and long cycles. Subsequently, PSO is employed to optimize the parameters of KF, which is utilized to smooth the results of the BiLSTM network, thereby achieving highly accurate SOE estimation. Experimental results across different operating conditions and temperatures reveal that the introduction of mechanical force significantly improves SOE estimation accuracy. Compared to models using only traditional electrical and thermal features, the model with the introduction of mechanical force achieves average improvements of 67.06%, 66.38%, and 66.46% for the root mean square error (RMSE), maximum absolute error (MAXE), and mean absolute error (MAE), respectively. Moreover, the generalizability and robustness of the proposed method are further confirmed by the comparison of different models and preload forces.

1. Introduction

Given the dual imperatives of addressing global environmental pollution and transitioning energy structures, electric vehicles powered by lithium-ion battery technology are emerging as a central focus of the energy revolution within the global transportation sector [1,2]. The LiFePO₄ battery is one of the commonly used batteries in electric vehicles due to its advantages of high safety, long cycle life, large energy density, and low cost [3]. The state of energy (SOE) is one of the core monitoring indicators of the battery management system (BMS) and is crucial for thermal management, optimal energy distribution, and effective battery safety warnings. However, direct measurement of SOE remains technically unfeasible, with its estimation being susceptible to multiple influencing factors, including temperature fluctuations, aging mechanisms, and load variations. Additionally, the intricate electrochemical features of LFP batteries coupled with the dynamic operating conditions encountered in real-world vehicle applications substantially augment the complexity of accurate SOE estimation. Therefore, the development of more effective model input features and advanced SOE estimation methods holds significant research importance.

The SOE represents the ratio between the current remaining energy and the maximum available energy of the battery, serving as an indicator of the battery’s remaining mileage. To date, SOE estimation has predominantly been based on three methodologies: the power integration method, the model-based approach, and the data-driven method [4]. The power integration method, which calculates SOE by integrating current and voltage over time, has been widely adopted in practice due to its simplicity and ease of implementation. However, the method bears similarities to the ampere-hour integration approach for state-of-charge estimation. Both techniques are susceptible to cumulative errors in their calculation processes and exhibit a strong dependence on the initial battery state [5].

Model-based approaches typically rely on electrochemical models or equivalent circuit models (ECMs) combined with filtering algorithms for SOE estimation. The KF, introduced by Rudolf E. Kálmán in 1960, is widely used for the optimal estimation of linear systems [6]. Conventional KF algorithms cannot be directly applied to complex coupled and nonlinear battery systems. To overcome this challenge, some researchers have developed the extended Kalman filter (EKF) algorithm. The EKF algorithm employs Taylor series expansion to linearize the nonlinear problem, enabling accurate estimation of the battery state. Shrivastava et al. [7] proposed the dual forgetting factor adaptive extended Kalman filtering algorithm, which employs dual forgetting factors to adaptively update the noise covariance matrices. By utilizing various discharge profiles, this algorithm establishes a quantitative correlation between state of charge (SOC) and SOE. The method achieves superior performance in terms of both convergence rate and estimation accuracy for joint SOC–SOE estimation [7]. Xia et al. proposed an adaptive noise correction–dual extended Kalman filtering algorithm and a new fusion ECM considering the influence of temperature, which achieves highly accurate co-estimation of SOE and SOC [8]. The EKF algorithm effectively addresses nonlinear estimation problems, but it is susceptible to divergence under high noise conditions and requires the computation of Jacobian matrices. Thus, the unscented Kalman filter (UKF) has been proposed. Lai et al. proposed a joint state-of-health and SOE estimation approach by integrating the forgotten factor recursive least squares algorithm with the UKF, which demonstrates superior SOE estimation accuracy under adverse conditions, particularly during battery aging and at extremely low temperatures [9]. The cubature Kalman filter, an enhanced version of the UKF, utilizes a spherical–radial rule strategy for sigma point selection to improve accuracy in high-dimensional estimation. However, the model-based approach is computationally demanding and heavily reliant on the accuracy of the battery model. Moreover, simplified modeling assumptions can result in substantial deviations from actual battery behavior under complex operating conditions, consequently reducing the accuracy of SOE estimation [10,11].

In contrast, data-driven approaches circumvent the need for explicit modeling of complex electrochemical mechanisms within batteries through end-to-end learning frameworks [12]. Additionally, they directly establish a nonlinear mapping of dynamic properties between multiphysics feature inputs and the SOE, which achieves enhanced accuracy estimation along with higher generalization performance. Ma et al. introduced sliding windows in long short-term memory (LSTM) networks for the estimation of SOC–SOE [13]. Mou et al. proposed a fusion model of a convolutional neural network (CNN) and bilayer gated recurrent unit, which utilizes the BGRU to learn the depth features extracted by the CNN to achieve high-precision estimation of SOE across diverse operating conditions, including different operating conditions, temperatures, and battery materials [14]. Although these models show high accuracy in estimating SOE, their performance remains largely contingent upon data quality. During battery tests, data acquisition is susceptible to noise interference that could lead to significant fluctuations in SOE estimates. Some scholars have used a KF algorithm added after the deep learning model to smooth the outputs of the priors [15,16]. Chen et al. [17] proposed a data-driven model that incorporates transfer learning, multi-scale squeeze-and-excitation networks (MS-SENet), gated recurrent units, and UKF. This methodology employs MS-SENet to extract and enhance key features, which are subsequently fed into the GRU for prediction to obtain a preliminary estimate. The results of SOC and SOE estimation are then optimized through UKF to effectively mitigate noise effects [17].

Numerous studies have demonstrated the effectiveness of data-driven approaches for accurately estimating the SOE in various applications. However, most of the existing literature relies on conventional electrical and thermal signals such as voltage, current, and temperature as inputs to the model. With advancements in sensing technology, battery mechanical force has emerged as a noteworthy parameter for characterizing battery state, attracting increasing attention from researchers. The mechanical properties of LFP are tightly related to the state of the battery. Based on traditional features such as voltage, current, and temperature, Jiang et al. [18] innovatively introduced mechanical force as a new input feature. Additionally, a private multi-condition dataset was established to collect multidimensional feature information. High-precision battery SOC estimation was achieved using the LSTM network [18]. Under normal charge and discharge cycling conditions, the changes in battery mechanical force can be observed at both microscopic and macroscopic levels. At the microscopic level, these changes are manifested as the intercalation and deintercalation of lithium ions between the cathode and anode. At the macroscopic level, they are reflected as the relative expansion between the cathode and anode. This expansion is primarily governed by the anode [19,20]. For LFP batteries, the change in force during the discharge test is significant due to contraction at low and high SOE values and expansion at moderate SOE values. The flat open-circuit voltage in traditional methods reduces the accuracy of SOE estimation. This problem can be effectively addressed by integrating force signals with deep learning approaches. Considering the above studies, this study proposes a force-based fusion model. This model integrates the bidirectional long short-term memory (BiLSTM) network, particle swarm optimization (PSO), and Kalman filter (KF) algorithm for the estimation of the SOE of LFP batteries. The main contributions of this study include the following.

• A sliding window is proposed to make full use of past measurements containing voltage, temperature, current, and mechanical force.
• A BiLSTM with four multiphysics feature inputs is used to obtain preliminary estimation of SOE. The PSO algorithm is introduced to optimize the noise covariance of the KF, thereby minimizing manual parameter adjustments. The optimized KF algorithm then smooths the outputs of BiLSTM, achieving enhanced estimation precision and effectively mitigating noise interference.
• A test bench is established to acquire multiphysics features, including current, voltage, temperature, and mechanical force. The force signal is introduced as a model input feature, which significantly improves the estimation accuracy of SOE during the voltage plateau period.
• The validity of the proposed model and mechanical force signals in high-precision SOE prediction is verified by applying the proposed method to different operating conditions and temperatures. In addition, the comparison of BiLSTM, PSO, and KF (BiLSTM–PSO–KF) with other networks and the prediction results under different preload forces further validate the robustness and generalization performance of the proposed model.

The remainder of this paper is organized as follows. Section 2 presents the experimental procedure, the test bench, and the analysis of the measurements. Section 3 explains the fundamental principles of BiLSTM, PSO, and KF. The hyperparameters’ design and evaluation criteria of the model are also described in detail in this section. Section 4 presents the SOE estimation results under various conditions. Section 5 summarizes the content of this study.

2. Experimental Data Analysis

2.1. Text Platform

In this paper, two brand-new pouch LFP batteries with a nominal capacity of 10 Ah were selected. The specific parameters of the batteries are shown in

Table 1. Specific parameters of the pouch LFP batteries.

Item	LFP Battery
Cathode	LiFePO4
Anode	Graphite
Nominal capacity	10 Ah
Nominal voltage	3.2 V
Charge cutoff voltage/discharge voltage	3.65 V/2.5 V
Size (H/W/L)	135/68/13 mm
Brand	Dingshan New Energy Co., Ltd., Guiyang, China
Model code	1368135-10 Ah
Production date	13 January 2024

.

The test platform consists of a thermostat, an ARBIN battery test system, a data acquisition instrument, a temperature sensor, a mechanical stress sensor, and a battery fixture, as shown in Figure 1. The model of the mechanical stress sensor is ZNHBM-IIX manufactured by Bengbu Zhongnuo Sensor Co., Ltd. in Anhui, Bengbu, China, with a range of 0–3000 N and a resolution of 3–6 N. The temperature sensor uses a negative temperature coefficient thermocouple, model T-K-24, with a range of −40 to 150 °C and a resolution of 0.2 °C. The sensor is set at the positive lug of the battery to minimize the effect of its position on the battery preload force. The battery fixture consists of three iron plates and two epoxy resin plates, and the battery to be tested is placed between the top and middle iron plates. In order to avoid short-circuit accidents that may be caused by the contact between the battery and the iron plate, an epoxy resin plate is placed between the contact surfaces of the battery and the iron plate. The mechanical stress sensor is placed between the middle and bottom iron plates, and the preload force of the battery is adjusted by tightening the nuts around the fixture. During the experiment, the ARBIN battery test system can set up different operating conditions to test the battery. The current, voltage, temperature, and mechanical force information is simultaneously collected by a data acquisition system, with a sampling frequency of 1 Hz. All the experiments are carried out in a thermostat to ensure the homogeneity of the experimental environment.

2.2. Data Acquisition

Batteries generate a solid electrolyte interface (SEI) on their electrode surfaces when they undergo charging and discharging processes. This membrane grows rapidly, especially during the charge and discharge cycle of the new batteries [21]. As the number of cycles increases, the main reactants inside the cell are consumed, causing the creation of the SEI membrane to slow and eventually stabilize [22]. It should be noted that the formation of this layer leads to loss of battery capacity and irreversible expansion. Before the dynamic testing profiles, the battery first needs to be activated. To mitigate this initial effect on the experiment, a preload force of 500 N is applied to the new battery for charge and discharge cycling until both the battery’s capacity and pressure profiles reach a state of equilibrium. The constant current and constant voltage (CCCV) profile is applied to the charging, and the LFP battery is discharged using the constant current profile. In charging, the battery is charged at 0.5 C constant current to 3.65 V, followed by constant voltage charging until the current drops below 0.5 A. The cell is then allowed to rest for 2 h. Subsequently, it is discharged at 0.5 C constant current (CC) until the voltage falls below 2.5 V. A 2 h rest period is maintained between each cycle. The battery activation experiment is carried out for a total of 10 cycles.

To simulate EV driving conditions more comprehensively, this study employs experiments under four distinct operating conditions: the Supplemental Federal Test Procedure Driving Schedule (US06), the Federal Urban Driving Schedule (FUDS), the Dynamic Stress Test (DST), and the Beijing Dynamic Stress Test (BJDST). The experiments are conducted with an initial preload force of 500 N at 15 °C, 25 °C, and 35 °C. Before testing at each temperature condition, the battery energy is recalibrated to minimize the interference from cycling-induced aging effects. Figure 2 shows the experimental procedure under a preload force of 500 N.

The generalization performance of the model is further validated through experimental studies conducted at 25 °C, implementing preload forces of 1000 N and 1500 N, with each scenario encompassing four different operating conditions. An energy calibration procedure is also performed before each experimental set to eliminate potential aging-related effects on the data. The same experimental protocol is used for both battery samples in this study. To validate the transfer learning capability of the model, data from battery 1 and battery 2 are designated as the training and test sets, respectively. Furthermore, DST and BJDST profiles under various conditions are utilized for model training, while FUDS and US06 profiles are employed for model validation.

2.3. Data Analysis

The ARBIN battery testing system establishes operating conditions by regulating the current magnitude per second. Excluding the effects of electromagnetic interference and sampling errors, the calculation of SOE using the discrete power integration method does not introduce cumulative errors under ideal conditions.

$$SO{E}_t=SO{E}_{t-1}+{\displaystyle \frac{\Delta t\times U_t\times I_t}{3600\times E}}$$

(1)

Here, SOE_t is the value of SOE at time step t, ∆t is the sampling time, and U_t and I_t represent voltage and current at time step t, respectively. E denotes the battery energy, which is acquired from the energy calibration experiment. The SOE is calibrated based on the average discharge energy measured over three repeated cycling tests.

The relationships between four multiphysics features and SOE are shown in Figure 3, measured at 25 °C with a preload force of 500 N. By comparing the characteristic curves of the different operating conditions, the advantages of the mechanical force feature in SOE prediction can be summarized as follows.

• The voltage and temperature exhibit a strong dependence on the periodically varying current. Both parameters fluctuate significantly with the current within a localized range. In contrast, the mechanical force remains relatively stable in both local variations within a single operating condition and overall variations across different operating conditions.
• Conventional methods utilize voltage, current, and temperature as model inputs to predict the SOE of LFP batteries. However, similar to SOC estimation, these methods suffer from the issue of the voltage plateau, which leads to reduced estimation accuracy. Figure 3(a2–d2) show the overall trend of voltage with a slow decrease in the range of 0.3 to 0.9 SOE, although it fluctuates periodically with the current. In contrast, the curve of mechanical force shows a significant change with an overall decreasing trend, but a reverse peak occurs in the range of 0.3 to 0.6 SOE.

The raw measurements need to be preprocessed to minimize the impact of noise and outliers on the accuracy of model estimation. The multidimensional data are initially smoothed using a fixed window of size 50. For each feature, the precedent 49 data are smoothed using a variable window. The average is placed to the last unit of each window. As shown in Figure 3(a2–d2), mechanical force varies over a range of approximately one hundred times the voltage variation. The min–max method is employed to uniformly normalize the four features to the range of [0, 1], which reduces the scale sensitivity of the model across different features and accelerates the convergence speed. The equation is as follows:

$${x^{\prime }}_i={\displaystyle \frac{x_i-x_{i,min}}{x_{i,max}-x_{i,min}}}$$

(2)

where x_i represents the original feature data, x_i,max and x_i,min denote the maximum and minimum values of the feature, respectively, and ${x^{\prime }}_i$ denotes the normalized value of the feature.

In order to more effectively capture localized temporal dependencies within the time-series data, a sliding window of size n is employed to input data into the deep learning model. The value of SOE at the k-th time step is estimated using (k − n + 1) previous inputs of current, voltage, temperature, and mechanical force. The window slides incrementally to cover the entire dataset. The SOE values for the first (n − 1) time steps are estimated based on data from a single time step. The value of n is set to 55 in this study. Compared to prediction methods using a single time step, the estimation accuracy of SOE is improved with the addition of a window prediction method. The sliding window is shown in Figure 4.

3. Methodology

3.1. BiLSTM

Numerous studies have demonstrated that recurrent neural networks (RNNs) can be used for handling time-series prediction tasks [23,24,25]. However, in tasks involving long time series, it is difficult for an RNN to effectively capture the long-term dependencies among data due to the gradual decay or infinite increase of the error signal during backpropagation. To solve this challenge, LSTM, an improved version of the RNN, introduces memory cell and gating mechanism to significantly alleviate the gradient explosion and explosion gradient vanishing problems of RNNs. Figure 5 provides a detailed view of an LSTM cell.

Here, X_t, C_t, and h_t represent the input, memory cell, and output at time t, respectively. The forward propagation process of each cell at moment t is as follows:

$$f_t=\sigma (W_{xf}{X}_t+W_{hf}{h}_{t-1}+b_f)$$

(3)

$$i_t=\sigma (W_{xi}{X}_t+W_{hi}{h}_{t-1}+b_i)$$

(4)

$$O_t=\sigma (W_{xo}{X}_t+W_{ho}{h}_{t-1}+b_o)$$

(5)

$$C_t=f_t*C_{t-1}+i_t*\tanh (W_{xc}{X}_t+W_{hc}{h}_{t-1}+b_c)$$

(6)

$$h_t=O_t*\tanh (C_t)$$

(7)

where f_t, i_t, and O_t denote the forget gate, input gate, and output gate, respectively; W_f, W_i, and W_o represent the weights of these gates; b is the bias; σ refers to the sigmoid activation function; and tanh serves as the activation function for the memory cell and hidden state. Compared to RNN, LSTM decides whether to retain or update memories using a combination of three gates. The sigmoid function is used to implement the gate selection mechanism with an output range of [0, 1], with 0 representing a closed door and 1 a fully open door. The tanh maps the data into the range of [−1, 1]. Furthermore, this function provides normalization and nonlinear transformation, facilitating smooth regulation of state updates and outputs.

In this study, BiLSTM is proposed for the preliminary estimation of the SOE. Compared to LSTM, the network introduces two independent LSTM layers to realize forward and backward processing of time-series data, respectively. The forward LSTM layer captures dependencies from past data up to the current moment, while the backward LSTM layer extracts information from future information back to the present [26]. The bidirectional recurrent structure of BiLSTM enhances the accuracy of SOE estimation by leveraging the data from the past, the current, and the future. The framework of BiLSTM is shown in Figure 6.

3.2. KF

The KF is a recursive algorithm that can be used to optimize system state estimation as well as effectively minimize the impact of noise on the results. This paper introduces the KF to smooth the output of the BiLSTM and improve the robustness and stability of the fusion model. The state equation and measurement equation are as follows:

$$x_t=A_t{x}_{t-1}+B_t{u}_t+w_t$$

(8)

$$y_t=H_t{x}_t+{\nu }_t$$

(9)

where x_t presents the state vector at moment t; u_t is the control input of the system; y_t is the vector of observations; and A_t, B_t, and H_t stand for state transfer matrix, input matrix, and observation matrix, respectively. w_t is the process noise and v_t is the observation noise, both of which are Gaussian noise. The process noise covariance Q_t and the observation noise covariance R_t are defined as follows:

$$Q_t=E[{w_t{w}_t}^T]$$

(10)

$$R_t=E[{v_t{v}_t}^T]$$

(11)

The recursive process of the KF is divided into three steps:

• Initialization process

The initial value of the state vector $\widehat{\mathrm{x}}$₀ and the covariance P₀ are set as follows:

$${\widehat{x}}_0=E(x_0)$$

(12)

$$P_0=E[(x_0-{\widehat{x}}_0){(x_0-{\widehat{x}}_0)}^T]$$

(13)

2.

Prediction progress

Calculate the priori state ${\widehat{x}}_t^{\prime }$ and the priori error covariance matrix ${\mathrm{P}}_{\mathrm{t}}^{\prime }$:

$${\widehat{x}}_t^{\prime }=A_t{\widehat{x}}_{t-1}+B_t{u}_{t-1}$$

(14)

$$P_t^{\prime }=A_t{P}_{k-1}{A}^T+Q_t$$

(15)

3.

Correction process

The correction process involves the calculation of the Kalman gain K_t and the update of posteriori state $\widehat{\mathbf{x}}$_t and posteriori error covariance matrix P_t:

$$K_t=P_t^{\prime }{H}_t^T{(H_t{P}_t^{\prime }{H}_t^T+R_t)}^{-1}$$

(16)

$${\widehat{x}}_t={\widehat{x}}_t^{\prime }+K_t(z_t-H_t{\widehat{x}}_t^{\prime })$$

(17)

$$P_t=(1-K_t{H}_t)P_t^{\prime }$$

(18)

The values of SOE estimated by BiLSTM are treated as observation values and the discrete power integral equation is employed as the state equation. The specific equations are as follows:

$$SO{E}_{A,t}=SO{E}_{A,t-1}+{\displaystyle \frac{\Delta t\times U_t\times I_t}{3600\times E_N}}+Q_t$$

(19)

$$y_t=SO{E}_{Bi,t}+R_t$$

(20)

where SOE_A,t−1 is computed through discrete power integral equation and SOE_Bi,t is the output of BiLSTM at time t. E_N, the nominal energy, is obtained by multiplying the nominal voltage by the nominal capacity according to the specifications of the LFP battery. I_t is negative for discharging and positive for charging.

3.3. PSO

As an optimization algorithm based on group intelligence, PSO finds the global optimal solution through information sharing and collaboration among particles. Each particle is characterized by a position vector and a velocity vector. During each iteration in the multidimensional search space, particles update their position and velocity based on their optimal solution and the global optimal solution [27]. The updating process adjusts the influence of individual and collective experience on particle behavior by controlling the learning factors. The position and velocity update equations for the particles are as follows:

$$V_{id}^{k+1}=w{V}_{id}^k+c_1{r}_1(P_{id}^k-X_{id}^k)+c_2{r}_2(P_{gd}^k-X_{id}^k)$$

(21)

$$X_{id}^{k+1}=X_{id}^k+V_{id}^{k+1}$$

(22)

where i denotes the i-th particle; d represents the dimensionality of the particle’s position and velocity; k denotes the iteration number; and V and X are the velocity vector and position vector of the particles, respectively. P_id represents the individual optimal solution; P_gd represents the global optimal solution; and w is the inertia weight, which affects the global search ability of the particle. c represents the learning factor, which is used to regulate the learning ability of the particles for individual and global solutions. r is a random number in the interval [0, 1]. The PSO algorithm introduces the random number r to prevent particles from becoming trapped in local optima and to enhance the probability of finding the best global solution.

3.4. Training and Evaluation

3.4.1. Model Setup

As shown in Figure 4, the size of the sliding window is 55. An excessive or insufficient number of BiLSTM layers can lead to model overfitting or underfitting. Two BiLSTM layers are selected in this study. The first layer uses the sequence output mode, while the second layer uses the last output mode. Both layers contain 128 cells. A fully connected layer is used to output SOE. The number of epochs and the batch size are set to 150 and 128, respectively. Additionally, set the BiLSTM network with a dropout rate of 0.1, initial learning rate of 0.001, learning rate drop factor of 0.1, and learning rate drop period of 10. The Adam algorithm is selected to optimize weights and biases. The noise covariance of the KF is optimized by the PSO algorithm. The number of particles is set to 30, and the maximum number of iterations is set to 50. The ranges of Qt and Rt are established from 1 × 10⁻¹² to 0.1. The proposed model in this paper runs on a computer equipped with an AMD 7945HX CPU @ 2.5 GHz (16 cores, 32 threads) and an Nvidia GeForce RTX 4060.

3.4.2. Evaluation Criteria

The root mean square error (RMSE) is utilized as both the loss function of the model and the fitness function of the PSO in the training process. The test set is imported into the well-trained model to obtain the SOE. Three metrics are selected to evaluate the performance of the model: RMSE, maximum absolute error (MAXE), and mean absolute error (MAE). The specific formulae of these metrics are as follows:

$$RMSE=\sqrt{{\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N{(SO{E}_{ip}-SO{E}_{ir})}^2}}$$

(23)

$$MAXE=max\left|SO{E}_{ip}-SO{E}_{ir}\right|$$

(24)

$$MAE={\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N\left|SO{E}_{ip}-SO{E}_{ir}\right|}$$

(25)

where N is the length of time series, SOE_ip represents the estimated SOE, and SOE_ir is the actual SOE.

3.5. The Fusion Model Framework

This paper integrates a BiLSTM network, PSO, and KF algorithm to accurately estimate the SOE of an LFP battery. Figure 7 illustrates the comprehensive structure of the BiLSTM–PSO–KF model.

4. Results and Discussion

4.1. Comparison of SOE Estimation with Different Operating Conditions

EVs operate under complex conditions in practice. Accurate SOE estimation is challenged by the impact of diverse operating conditions on battery performance [28]. We aimed to verify the model’s performance under various operating conditions and evaluate the potential of mechanical force in SOE estimation. Predictions are conducted based on FUDS and US06 at 25 °C with 500 N. Additionally, the performance of the fusion model is compared with or without the use of force as an input feature.

Table 2. SOE estimation results under different operating conditions.

Training Data	Test Data	RMSE%	MAXE%	MAE%
DST, BJDST (without force)	FUDS	2.35	5.34	2.06
	US06	2.02	5.62	1.50
DST, BJDST (with force)	FUDS	0.64	1.64	0.53
	US06	0.56	2.22	0.39

shows the SOE estimation results under different operating conditions. The addition of the force signal reduced the RMSE from 2.35% to 0.64% and the MAE from 2.06% to 0.53% for the FUDS. The MAXE was also significantly reduced from 5.34% to 1.64%. Similarly, for the US06, the RMSE and MAE were reduced to 0.56% and 0.39%, respectively, while the MAXE decreased from 5.62% to 2.22%.

Figure 8 demonstrates that the BiLSTM–PSO–KF model, incorporating the force signal, dramatically improves the SOE estimation accuracy of the LFP battery under both operating conditions, particularly in the SOE range of 0.4 to 0.9. A long voltage plateau exists in this region where voltage and temperature variations are subtle, making it prone to large errors when using only traditional features. The estimation results under FUDS and US06 strongly demonstrate the significant advantage of incorporating the force signal in improving the accuracy of SOE estimation.

4.2. Comparison of SOE Estimation at Different Temperatures

Temperature can also significantly influence battery performance by altering the rate of electrochemical reactions within the battery and modifying key parameters such as internal resistance and depth of discharge [29]. To verify the performance of the proposed model across various temperatures, data from the dataset at 15 °C, 25 °C, and 35 °C are utilized to test and estimate SOE. Figure 5 shows the results at 25 °C. To avoid redundancy, only the results from the remaining two temperatures are presented in this part.

Table 3. SOE estimation results at different temperatures.

Training Data	Test Data	RMSE%	MAXE%	MAE%
DST, BJDST (without force)	US06 (15 °C)	1.85	7.32	1.58
	US06 (35 °C)	2.65	7.25	1.84
DST, BJDST (with force)	US06 (15 °C)	0.89	2.10	0.71
	US06 (35 °C)	0.76	2.58	0.69

illustrates the three evaluation criteria of SOE estimation results at various temperatures. At a temperature of 15 °C, the RMSE and MAE decreased from 1.85% and 1.58% to 0.89% and 0.71% without mechanical force and with mechanical force, respectively, and the maximum absolute error significantly reduced from 7.32% to 2.10%. Similarly, the estimation results with force at a temperature of 35 °C showed RMSE of 0.76% and MAE of 0.69%. The maximum absolute error was reduced from 7.25% to 2.58%. After combining the results at three temperatures and introducing the force signal, the RMSE was found to be less than 0.9%, the MAE less than 0.8%, and the MAXE less than 3%. By comparing the results in Section 4.1 and Section 4.2, it was found that after introducing mechanical force as an input feature, the average improvements in MARE, MAXE, and MAE were 67.06%, 66.38%, and 66.46%, respectively.

Furthermore, the proposed model demonstrates excellent capability in capturing the complex nonlinear mapping relationships among current, voltage, temperature, mechanical force, and SOE, while maintaining high accuracy of SOE estimation and robustness across different temperatures. As shown in Figure 9, the estimation results with force are much better than those without force.

4.3. Comparison of SOE Estimation of Different Models

To verify the performance of the proposed model, we employed a BiLSTM model and LSTM model for comparison analysis. Additionally, the LSTM hyperparameters were configured according to reference [30], taking current, voltage, temperature, and mechanical force as input features [30]. The hyperparameters of BiLSTM are consistent with those of the proposed model. The test dataset was selected from the FUDS at a temperature of 35 °C and with a preload force of 500 N.

The three evaluation criteria of SOE estimation results are shown in

Table 4. SOE estimation results of different model.

Model	Training Data	Test Data	RMSE%	MAXE%	MAE%
LSTM	DST, BJDST (without force)	FUDS	4.20	15.70	3.28
BiLSTM			3.20	15.18	2.51
Proposed			2.29	6.67	1.73
LSTM	DST, BJDST (with force, 500 N)	FUDS	2.34	9.33	1.74
BiLSTM			1.69	6.05	1.27
Proposed			0.62	2.66	0.48

. For different models, the addition of the mechanical force achieves a significant improvement in all three performance metrics, which further demonstrates the potential of the force feature in SOE estimation of LFP batteries. Compared to the criteria of the LSTM network, the RMSE, MAXE, and MAE of BiLSTM with the bidirectional recurrent structure reduce from 2.34%, 9.33%, and 1.74% to 1.69%, 6.05%, and 1.27%, respectively. The integration of PSO-KF with BiLSTM further improves the accuracy of SOE estimation with an RMSE of 0.62%, MAXE of 2.66%, and MAE of 0.48%.

Figure 10 illustrates the estimation results of various models. The BiLSTM–PSO–KF model demonstrates superior performance with estimation results that are the closest to the actual curves, and exhibits better fitting capabilities compared to other models. It should be noted that the estimation accuracy exhibits a slight decrease in the SOE range of 0.4–0.6. This phenomenon can be attributed to two primary factors. One factor is the nonlinear relationship between force and SOE, which features a distinct peak with an opposite trend within this range. Additionally, a comparison of the curves in Figure 3(a1–a3) reveals that the FUDS exhibits a longer cycle duration compared to the DST and the BJDST, with larger fluctuations within each cycle.

4.4. Comparison of SOE Estimation Under Different Preload Forces

The generalization performance of the model is further verified using data from the US06 at 25 °C ambient temperature with preload forces of 500 N, 1000 N, and 1500 N. The results under 500 N have been shown in Section 4.1. To avoid repetition, only the test dataset is selected with preload forces of 1000 N and 1500 N.

The results under preload forces of 1000 N and 1500 N are presented in

Table 5. SOE estimation results under different preload forces.

Model	Training Data	Test Data	RMSE%	MAXE%	MAE%
LSTM	DST, BJDST (1000 N)	US06	1.86	7.78	1.44
BiLSTM			1.16	3.51	0.93
Proposed			0.59	1.09	0.46
LSTM	DST, BJDST (1500 N)	US06	2.38	8.34	1.85
BiLSTM			1.65	6.70	1.28
Proposed			0.95	2.48	0.70

. The RMSE, MAXE, and MAE of the BiLSTM–PSO–KF model are as low as 0.59%, 1.09%, and 0.46% and 0.95%, 2.48%, and 0.70%, respectively, which demonstrates the model’s robustness and stability under various preload conditions.

Figure 11 presents the curves and absolute errors of the three models under different preload forces, further verifying the superior performance of the BiLSTM–PSO–KF model in SOE estimation compared to the LSTM and BiLSTM models.

5. Conclusions

In this paper, a data-driven model incorporating mechanical force for SOE estimation is proposed. A data acquisition platform is established to simultaneously collect current, voltage, temperature, and mechanical force under four operating conditions consisting of DST, BJDST, FUDS, and US06. To reduce the complexity of manual parameter adjustment, the PSO is introduced to optimize the measurement noise covariance and process noise covariance of the KF. BiLSTM is then integrated with the KF, which further increases the accuracy of SOE estimation. Under different operating conditions, the RMSE is less than 0.65%, the MAXE is less than 2.23%, and the MAE is less than 0.54%. Under different temperatures, the RMSE, MAXE, and MAE are less than 0.9%, 2.59%, and 0.72%, respectively. In both of these operating conditions and variable temperatures, the introduction of the mechanical force results in an average improvement of over 66% in the RMSE, MAXE, and MAE compared to the conventional electrothermal signal model, which strongly proves the potential of force signals in SOE estimation. Additionally, the BiLSTM–PSO–KF model is compared with the LSTM and BiLSTM models under conditions of 35 °C, 500 N, and FUDS. The proposed model demonstrates a superior fit to the actual SOE, with RMSE, MAXE, and MAE reduced to 0.62%, 2.66%, and 0.48%, respectively. To further validate the generalization and robustness of the proposed model, SOE estimation is conducted under US06 at 25 °C with different preload forces. The results demonstrate that the RMSE is less than 0.96%, the MAXE is less than 2.49%, and the MAE is less than 0.71%. In summary, the offline training time of the model is less than one hour. The proposed novel framework introducing mechanical force signals into SOE estimation for lithium-ion batteries—a direction rarely explored in existing literature—offers a promising technical pathway for broader application in battery state estimation.

In the future, the existing model will be extended to other types of lithium-ion battery SOE estimation. Moreover, a joint estimation model for SOE, SOC, and SOH will be developed to achieve a more comprehensive characterization of the state of LFP batteries and enhance the engineering application capability of system-level BMS.