Deep Learning-Based State Estimation for Sodium-Ion Batteries Using Long Short-Term Memory Network

Abstract

Sodium-ion batteries (SIBs) have attracted growing attention as an alternative to lithium-ion technologies for electric mobility and stationary energy-storage applications, owing to the wide availability of sodium resources, cost advantages, and comparatively favorable safety characteristics. Accurate state-of-health (SOH) estimation is essential for safe and reliable SIB deployment, yet existing data-driven methods still suffer from limited accuracy and interpretability, as well as a lack of dedicated aging datasets. This study proposes an explainable SOH estimation methodology based on a long short-term memory (LSTM) network combined with model-agnostic KernelSHAP analysis. Thirteen health indicators (HIs) are extracted from charge/discharge data and post-charge relaxation segments, and the most relevant indicators are selected via Pearson correlation screening as model inputs. Built on these HIs, an LSTM-based multi-step framework is developed to take HI sequences as input and forecast the SOH trajectory over the subsequent 20 cycles. Experimental results show that the proposed method achieves high accuracy and robust cross-cell generalization, with mean absolute error (MAE) below 1.0%, root-mean-square error (RMSE) below 1.2% across all cells, and an average RMSE of about 0.75% in the main cross-cell setting. KernelSHAP-based global and temporal analyses further clarify how different HIs and time positions influence SOH estimates, enhancing model transparency and physical interpretability.

1. Introduction

Across both transportation electrification and grid applications, lithium-ion batteries have become the prevailing solution, owing to their favorable energy-density characteristics and mature long-life performance [1,2]. Nevertheless, the growing need for cost-effective energy storage, coupled with concerns regarding the security and stability of critical material supply chains, is accelerating a shift away from conventional LIB technologies. Owing to their lower cost, the abundance and wide geographic distribution of sodium resources, and a more environmentally friendly materials portfolio, sodium-ion batteries (SIBs) have emerged as a viable candidate to complement or replace lithium-ion batteries in certain applications [3,4]. Wu et al. [5] reviewed five years of progress in SIBs, including electrode design, reaction mechanisms, and the roles of electrolytes, binders, separators, and conductive additives, and highlighted key challenges and development directions for large-scale grid and EV applications. Deshmukh et al. [6] presented an end-to-end overview of SIB technology from materials and cell manufacturing to pack integration, emphasizing cost and resource advantages over LIBs alongside current limitations in energy density and reaction kinetics and prospective grid/EV deployment routes. Zhao et al. [7] reviewed recent advances in solid electrolytes for solid-state sodium-ion batteries and outlined challenges associated with ionic conductivity, electrochemical stability, and interfacial compatibility. Commercial deployment of SIBs is now approaching a critical stage [8].

According to previous LIB studies, the State of Health (SOH) is typically defined as the ratio of the battery’s actual capacity to its rated capacity [9]. Accurate SOH estimation enables the prediction of the remaining service life of SIBs and helps extend their usable lifespan [10]. Accurate SOH estimation of SIBs is also important at the system level, for example, in smart-grid and smart-city applications where SIB packs are deployed in large-scale storage, public transport and second-life scenarios. Integrating data-driven SOH monitoring into energy management and smart charging strategies can enhance the safety, reliability and sustainability of such infrastructures. Therefore, accurate and reliable estimation of battery SOH, a critical technique in the Battery Management System (BMS), is used to monitor and control battery aging [11]. However, capacity and SOH cannot be directly measured in operation; they must be inferred from measurable quantities such as voltage and current, or obtained via calibration tests under prescribed conditions, which are unsuitable for online use [12]. In addition to cycling-induced degradation, sodium-ion batteries can experience measurable capacity loss even when held at rest, suggesting that time-dependent aging contributes alongside charge–discharge operation [13,14].

Current research on SIB SOH estimation remains sparse and preliminary, whereas LIB-related methods are relatively mature [15]. Drawing on methodologies developed for LIBs, SOH estimation schemes for SIBs can be classified into direct measurement methods, model-based methods, and data-driven methods [16].

In the first method, aging-related quantities are obtained offline using dedicated test setups, and the measured parameters are then used to characterize battery degradation and performance loss [17]. Representative diagnostics range from capacity measurements to EIS and OCV analysis, which together provide complementary indications of degradation [18]. Baghdadi et al. [19] proposed a simple two-parameter model that links post-charge relaxation voltage to SOH, revealing an almost linear relationship across three chemistries and achieving mean errors below 2%, thereby enabling rapid pack-level screening. In general, model-based strategies rely on electrochemical models (EM) or equivalent-circuit models (ECM) to infer battery states, and they can provide accurate estimates. During operation, external electrical responses that encode internal electrochemical behavior are measured and analyzed; by incorporating load profiles, material properties, and degradation mechanisms, these models describe SOH dynamics and update parameters via adaptive filtering for online tracking [20]. Hosseininasab et al. [21] designed a fractional-order reduced electrochemical model combined with dual adaptive observers to jointly estimate capacity and resistance as SOH-related quantities, achieving a favorable compromise between accuracy and computational cost for BMS implementation and validating the method under dynamic load profiles and multiple aging states.

Data-driven approaches first extract feature values from charge and discharge data, then learn a mapping from these features to SOH, optimizing the model via dedicated training procedures so that its outputs closely replicate experimental measurements. This paradigm avoids detailed physics-based modeling and tedious parameter identification, while implicitly capturing complex internal degradation mechanisms within the cell [20]. Conventional machine-learning algorithms have been widely applied to SOH estimation, including relevance vector machines [22], support vector machines [23,24], Gaussian process regression [25], and random forest regression [26]. For example, Song et al. [27] proposed a hybrid joint-state estimator that combines a least-squares support vector machine with an unscented particle filter, using a voltage-drop time feature extracted from terminal voltage and current. More recently, deep-learning-based approaches have shown excellent performance in battery SOH estimation. The earlier studies mainly employed recurrent architectures without complex auxiliary modules. Guo et al. [28] used a bidirectional long short-term memory (LSTM) network to learn multi-cycle temporal relationships from degradation-aware health indicators (HIs), thereby improving robustness across different load profiles. Zhang et al. [29] proposed an LSTM-driven time-series capacity model for accurate SOH prediction and demonstrated good cross-dataset generalization on the public datasets. Nguyen Van and Quang [30] designed a two-stage LSTM architecture that first estimates SOH and then infers internal resistances R_e and R_ct from the same stream of HIs, thereby outperforming feed-forward baselines without requiring electrochemical impedance spectroscopy. Nasimov et al. [31] further showed that a feed-forward neural network trained with an improved particle swarm optimization scheme can improve convergence and accuracy relative to conventional neural network baselines on benchmark datasets. To further enhance accuracy and generalization, more advanced and hybrid neural-network architectures have been proposed. Shi et al. [32] used Pearson correlation to select HIs and trained a gated recurrent unit network optimized by a bio-inspired search algorithm to estimate SOH and identify the knee point on MIT–Stanford cells, achieving sub-percent values of error, and outperforming LSTM, support vector regression, and extreme learning machines. Qian et al. [33] presented a CNN–LSTM architecture enhanced with a self-attention mechanism and a homoscedastic-uncertainty joint loss to simultaneously estimate state of charge, state of energy, and SOH across three operating conditions, achieving robust performance and good cross-profile generalization. Zhou et al. [34] adopted a one-dimensional CNN followed by a bidirectional LSTM, with network parameters tuned by a swarm-based optimization algorithm. They achieved sub-percent root-mean-square error in predictions on the public dataset. Zhao et al. [35] developed an attention-augmented CNN–BiLSTM architecture for coupled SOH and remaining useful life prediction from multi-parameter cycle data, reporting a coefficient of determination above 0.99 on the public dataset. Building on these advances, Zhang et al. [36] proposed a genetic-algorithm-optimized hybrid model that combines temporal convolutional networks, gated recurrent unit networks, and a wavelet neural network to fuse multi-scale temporal features, reducing the error to below 1% on the public dataset. In parallel with these SOH-focused studies, AI- and ML-based techniques have also advanced rapidly across other battery-optimization tasks, including remaining useful life prediction [37], fast-charging control [38], and anomaly detection [39], reflecting the broader shift toward data-centric battery science.

In summary, data-driven SOH estimation avoids complex battery modeling and parameter identification, and does not need to explicitly resolve the underlying degradation pathways within the cell [20]. Although this approach entails substantial demands for training data and higher computational costs, the rapid advances in computing have made fast data processing feasible. Consequently, data-driven SOH estimation has become a mainstream choice for battery health assessment [40]. Nonetheless, applying data-driven methods to SIBs presents several challenges: (1) heavy reliance on substantial experimental datasets; in the absence of mature public SIBs datasets, training must use lab-collected data, resulting in limited sample size and insufficient diversity of operating conditions; (2) careful selection of HIs tailored to SIBs characteristics; although SIBs share structure and operating principles with LIBs, fundamental Na–Li differences lead to distinct aging behaviors, so the chosen indicators should reflect SIB-specific phenomena; (3) limited interpretability—In many studies, the mapping from battery HIs to SOH is implemented as a black-box predictor, so the underlying decision logic and intermediate computations are not readily interpretable [41], obscuring how different health features influence the SOH estimate.

To tackle these issues, we develop an interpretable LSTM-based framework designed for SOH estimation in SIBs and aligned with their degradation-related characteristics. By analyzing the charge/discharge profiles and post-charge relaxation segments, a set of HIs was developed to capture SIB-specific aging traits. The LSTM model leverages the multi-cycle temporal dependencies of these HIs to produce suitable short-term SOH predictions. To reduce black-box opacity, a model-agnostic KernelSHAP pipeline offers global explanations. Robustness and generalization are tested via leave-one-cell-out validation on the full dataset. The main contributions are as follows:

• A set of SIB-oriented HIs is systematically designed and evaluated through correlation analysis, capturing polarization, hysteresis, relaxation behavior, and dQ/dV shifts within an interpretable feature space for data-driven SOH estimation.
• An LSTM-driven SOH estimation framework is proposed, which fuses degradation-aware HIs with temporal modeling and enables accurate short-term prediction under practical cycling conditions.
• A two-scheme cross-cell validation protocol is rigorously designed; across both schemes, the proposed LSTM-based SOH model maintains MAE and RMSE within approximately 1.0–1.2%, demonstrating robust cross-cell generalization and resilience to limited training data.
• A comprehensive KernelSHAP-based explainability framework is developed that combines global feature-importance analysis with temporal attributions within the input window, clarifying how different HIs and their time positions jointly influence the estimated SOH and providing guidance for HI selection and window design.

The remainder of this study is organized as follows. Section 2 describes the design of the sodium-ion battery aging experiments and the construction of the resulting aging dataset. Section 3 describes the HIs obtained from the experimental dataset and assesses their relevance to aging through Pearson correlation analysis. Section 4 presents the proposed SOH estimation methodology, including the LSTM-based network architecture and hyperparameter choices, followed by explainability analysis techniques. Section 5 evaluates the proposed SOH estimator and discusses the results. Section 6 summarizes the main limitations and outlines future research directions. Finally, Section 7 provides the conclusions.

2. Experimental Design and Battery Dataset

2.1. The Battery Cycling Aging Test

The battery datasets were constructed from an aging campaign conducted from aging experiments involving six pouch-type SIBs. The cells were custom-made by Veken Technology Co., Ltd., Ningbo, Zhejiang, China. The cathode material is NaNi_1/3Fe_1/3Mn_1/3O₂, and the anode material is hard carbon. The nominal capacity of each battery is 4.1 Ah. The cutoff voltages were set to 3.9 V for charging and 1.5 V for discharging. These six cells are named Cell 1, Cell 2, Cell 3, Cell 4, Cell 5, and Cell 6.

Due to the limited availability of commercial SIB datasets, we undertook an experimental aging campaign to develop a specialized sodium-ion battery dataset. Figure 1a provides an overview of the test protocol, which includes periodic capacity-check runs for reference updating and a separate cycling-aging program for degradation tracking. Additionally, Figure 1a presents a representative full charge–discharge profile of the battery system SIBs. In contrast to many lithium-ion chemistries, the voltage curve exhibits a predominantly sloping shape without an extended flat plateau, reflecting the multi-step Na⁺ insertion/extraction and characteristics of the NFM/hard-carbon SIBs.

All six cells were aged under the same cycling regime: charging at 2C in a CC–CV scheme and discharging at 2C under constant-current control, with a 0.5 h rest between charge and discharge, conducted at 25 °C. During the CC–CV charge, the CV phase was terminated when the current decayed to 0.02 A. The tests spanned 1200 cycles, over which all six cells degraded to approximately 80% SOH. The calibration process employed a 0.2C constant current-constant voltage (CC-CV) charging protocol, maintaining the identical CV cutoff current of 0.02 A, and a 0.2C discharging cycle was conducted to track capacity retention during the aging campaign. Additionally, a 0.5 h rest interval was inserted between the charge and discharge steps. Capacity checks/calibration were carried out at 25 °C after every 100 aging cycles.

2.2. The Battery Aging Dataset

The capacity-fade profiles derived from the aging experiments are compiled in Figure 1b. Figure 1b shows that all six cells exhibit a two-stage trajectory: a slow-fade regime over about 0–300 cycles, during which inter-cell SOH differences remain small, followed by an accelerated phase in which trajectories diverge. Cell 3 shows a pronounced knee point at about 1100 cycles, after which SOH rapidly falls below 80%; apart from Cell 3, no clear knee points are observed—only an increased slope beginning around about 300 cycles—consistent with reports that many cells follow two-phase degradation with a subsequent rapid-fade onset. Cell 1 and Cell 2 display slightly faster overall capacity fade than the remaining cells, while Cell 4, Cell 5, and Cell 6 follow relatively similar trajectories without abrupt transitions. These observations indicate measurable cross-cell differences in fade rates within the same batch, supporting the use of this six-cell dataset to assess the generalization of SOH estimation methods.

3. The Battery Health Indicator Extraction

3.1. Feature Engineering

For data-driven SOH estimation, the model ingests a set of features extracted from cycling data; these inputs strongly govern predictive performance [42]. In this study, we refer to these features as HIs. Hence, identifying HIs that effectively characterize aging is critical. Section 2.1 details the aging test, where voltage–current trajectories were continuously logged throughout cycling. In addition to changes observed during charge/discharge, the relaxation portions exhibited repeatable temporal patterns that developed with increasing age. Leveraging these trends, we extract a suite of HIs from the aging data and retain those showing strong correlation with SOH.

Notably, SIBs differ from LIBs in their electrochemical mechanisms and operating characteristics, as reflected in their charge–discharge profiles. Wang H. et al. [43] reported that SIBs with Na-based Ni–Fe–Mn oxide cathodes do not exhibit an extended voltage plateau during cycling. Wu C. et al. [44] showed that Na⁺ insertion/extraction in hard-carbon anodes follows a two-step storage mechanism, resulting in inherently larger voltage hysteresis than graphite and a further amplification of hysteresis with aging. Li K. et al. [45] indicated that conventional ester-based electrolytes tend to form less stable SEI in SIBs, leading to more pronounced polarization than in LIBs. Accordingly, the selection of HIs for SIBs SOH estimation should explicitly account for SIB-specific electrochemical and operational characteristics rather than inheriting LIB-oriented assumptions.

3.1.1. Average Charge/Discharge Voltage Within 2.0–3.5 V and Their Difference

Figure 2a,b shows that as aging progresses, the charge-segment voltage–time curves and the discharge-segment curves shift leftward; accordingly, the selected indicators should capture this alteration. Specifically, given the lack of an extended plateau in SIBs profiles, we adopt the average voltages within 2.0–3.5 V during charge and discharge as proxy “plateau” levels, denoted as HI1 and HI2, respectively. In addition, because SIBs inherently exhibit pronounced voltage hysteresis, it is essential to explicitly represent this behavior; the difference between HI1 and HI2 provides a direct quantification, which we designate as Health HI3. HI3 thus characterizes the SIB-specific hysteresis, while HI1 and HI2 jointly reflect the aging-induced divergence of charge/discharge voltage alteration. The variation in HI1-HI3 of Cell 1 is shown in Figure 2c–e as an example.

3.1.2. Time During Equal Voltage Increase

The aging-dependent change in the charge voltage–time profile is shown in Figure 3a. To quantify this effect, we evaluate the time spent across predefined voltage intervals, referred to as TEVI. Three TEVI indicators corresponding to 3.3–3.48 V, 3.5–3.68 V, and 3.7–3.88 V are extracted and used as HI4–HI6. These three TEVI intervals were selected by examining the charge voltage–time curves at different aging stages. As cells age, SIBs exhibit pronounced polarization, leading to shifts in the charge curves. To avoid regions that are partially truncated or disappear under severe polarization, we chose a voltage band that remains fully present throughout aging and across all cells. The variation in HI4-HI6 of Cell 1 during battery aging is shown in Figure 3b–d as an example.

3.1.3. Capacity Increment During Constant-Voltage Charging

Figure 4a shows the variation in current–time curves during the constant-voltage (CV) charging process. The time integral of the current curve corresponds to the charge throughput delivered in the CV stage. It can be observed that, as the battery ages, the CV current–time curves gradually shift upward and the charging quantity increases. Based on the CV-stage charge calculation, we construct HI7 to characterize the charging quantity delivered during the CV step. The cycle-to-cycle trend of HI7 for Cell 1 is presented in Figure 4b.

3.1.4. Voltage Drop During Rest

Figure 5a presents the post-charge relaxation voltage–time response over aging. With cycling, the relaxation behavior becomes more pronounced, manifested by a faster initial voltage decay and a larger overall drop. These alterations show more pronounced polarization of SIBs relative to LIBs. Accordingly, we extract four HIs from the first 10 min of the rest segment: the voltage at 10 min (HI8), the instantaneous slope at 5 min (HI9), the mean voltage over 0–10 min (HI10), and the voltage variance over the same window (HI11). The variation in HI8-HI11 of Cell 1 is illustrated in Figure 5b–e as an example.

3.1.5. Peak Value and Position of Charging IC Curve

The incremental capacity (IC) curve features peaks with distinct shapes, amplitudes, and voltage positions that encode information about the electrochemical processes occurring during battery charge and discharge [46]. Variations in these peaks are often associated with the loss of active materials within the electrodes [47]. By differentiating capacity with respect to voltage, IC analysis converts a slowly varying voltage plateau—where internal reactions are intense, but the voltage changes only gradually—into a pronounced dQ/dV peak that is easier to resolve. In this manner, the charge distribution across the voltage window is reflected by the peak features observed in the curve. It captures subtle changes that are difficult to discern directly from voltage–time curves [48]. Owing to these more intuitive aging signatures, IC curves have been extensively employed as informative indicators for battery SOH estimation [49]. Figure 6a illustrates the evolution of the incremental-capacity (IC) curves with cycling. As aging progresses, the IC features exhibit an overall shift toward higher voltage and higher amplitude, indicating increases in both peak position and peak magnitude. In the fresh state, the dQ/dV curves of the NFM/hard-carbon full cells exhibit two prominent peaks. The mid-voltage peak mainly corresponds to Na⁺ insertion/extraction in the O3-type layered NaNi_1/3Fe_1/3Mn_1/3O₂ cathode, where an O3 to P3 stacking transition occurs during charge. This transition involves limited layer gliding and relatively small lattice distortion and is therefore largely reversible, such that the corresponding IC peak persists during aging and its position and amplitude evolve monotonically with SOH [50,51]. By contrast, the high-voltage peak is associated with deep Na extraction at elevated potentials, where the cathode is more prone to irreversible structural distortion toward P′3/O′3-type phases, transition-metal migration, and parasitic side reactions, which rapidly suppress the high-voltage capacity contribution and cause the peak to vanish at low SOH [52,53]. For this reason, the IC-based HIs are constructed from the more resilient left peak, which provides a robust indicator of capacity fade in this SIB system. In this work, two left-peak descriptors are selected as HIs: the peak magnitude (PVIC, HI12) and the peak voltage position (PPIC, HI13). Their cycle-dependent evolution for Cell 1 is presented in Figure 6b,c as a representative case.

3.2. The Correlation Analysis Between Battery SOH and His

Based on the above analysis, a total of 13 health indicators are derived from the charging data and are summarized in Figure 7. HIs include the average charge and discharge voltages within 2.0–3.5 V and their difference, the time during equal-voltage increase (TEVI) in three voltage intervals, the capacity increment during the CV segment of CC–CV charging, four descriptors of the first 10 min post-charge relaxation, and PVIC and PPIC extracted from charging IC curves. Their relationships with SOH are quantified using the Pearson correlation coefficient (PCC), as defined in Equation (1).

$$\rho ={\displaystyle \frac{{\displaystyle \sum _{i=1}^n(x_i-\overline{x})(y_i-\overline{y})}}{\sqrt{{\displaystyle \sum _{i=1}^n{(x_i-\overline{x})}^2}{\displaystyle \sum _{i=1}^n{(y_i-\overline{y})}^2}}}}$$

(1)

where $x_i$ and $y_i$ denote the HI and SOH samples, respectively, $\overline{x}$ and $\overline{y}$ are the corresponding mean values of HI and SOH.

The Pearson correlation analysis for the 13 HIs is summarized in Figure 8. HI2, HI8, and HI10 increase with aging, as indicated by their positive PCC values. HI1, HI3, HI7, HI11, and HI13 exhibit a clear negative correlation. For HI4, HI6, and HI9, the correlation coefficients with aging are below 0.5 across all six cells; these indicators are therefore discarded from subsequent modeling. HI5 and HI12 show relatively low correlation coefficients in specific cells, but still maintain acceptable overall correlation with aging. In particular, the TEVI in the 3.5–3.68 V interval can be considered a representative descriptor of TEVI indicators, and PVIC exhibits characteristic aging-related evolution in SIBs; therefore, these two HIs are retained.

4. SOH Estimation Method

In this section, we detail the overall SOH-estimation pipeline. The core predictor is an LSTM-based deep-learning model, after which model explainability is examined using KernelSHAP.

4.1. LSTM for SOH Estimation

LSTM is a class of RNN designed to capture long-range dependencies in sequential data by introducing gated memory units [54]. Owing to their ability to retain and update information over extended horizons, LSTMs have been successfully applied to speech processing, audio analysis, and a wide range of time-series forecasting tasks. Over time, numerous architectural extensions have been proposed, including stacked LSTMs, bidirectional LSTMs, CNN–LSTM hybrids, and GRU [29]. The LSTM employs a gated mechanism to control information retention and removal, enabling the network to maintain and update its internal memory state over time.

In an LSTM cell, sigmoid-activated gates modulate elementwise interactions, thereby controlling how information is propagated through the network. As shown in Figure 9, the LSTM cell employs three gating functions—input, forget, and output—to regulate its memory update process [55].

The LSTM transition equations are as follows:

$$F_t=\sigma \left(W_F·[H_{t-1},X_t] +b_F\right)$$

(2)

$$I_t=\sigma (W_I\cdot [H_{t-1},X_t]+b_I)$$

(3)

$${\tilde{C}}_t=tanh(W_C·[H_{t-1},X_t]+b_C)$$

(4)

$$C_t=F_t\ast C_{t-1}+I_t\ast {\tilde{C}}_t$$

(5)

$$O_t=\sigma (W_O·[H_{t-1},X_t]+b_O)$$

(6)

$$H_t=O_t\ast tanh(C_t)$$

(7)

In the above equations, $C_t$ and $H_t$ denote the cell state and hidden state, respectively; $\sigma$ is the logistic sigmoid function; and $*$ indicates elementwise multiplication. Accordingly, the output at time step is influenced by earlier inputs and the accumulated internal states. In addition, LSTM models may take multi-sequence inputs, in which several lagged replicas of one series, or multiple distinct series, are provided to the network in parallel. In this way, the LSTM can exploit information distributed across multiple temporal scales [56].

The capacity degradation of SIBs takes a long cycle. In this work, an LSTM-based regression network is designed to exploit the degradation information contained in the engineered HIs of SIBs and to realize short-term multi-step SOH prediction. The final LSTM configuration was selected based on preliminary experiments (the input window length (10–40 cycles), the number of stacked LSTM layers (1–3), the hidden-state dimension (64–192 units) and the prediction horizon (10–20 cycles)) on the training/validation splits: a 20-cycle input window and a 20-cycle prediction horizon provided the best trade-off between capturing short-term degradation trends and keeping a sufficient number of samples, while a three-layer LSTM with 192 hidden units and an inter-layer dropout of 0.2 yielded the lowest average validation RMSE without obvious overfitting. Accordingly, the per-cycle HIs were ordered by cycle index, and the data were converted into supervised time-series samples using a 20-cycle input window and a 20-step prediction horizon. As shown in Figure 10, the core of the network consists of a stack of three LSTM layers, followed by a layer normalization module that stabilizes the temporal representations and a fully connected layer that maps the final hidden state to the multi-step SOH predictions. During inference, the sliding window is shifted along the aging trajectory with overlap, which yields multiple SOH predictions for the same cycle. The predictions are subsequently merged by averaging to assign a single SOH value to each cycle, allowing a continuous SOH trajectory to be reconstructed over the entire cycle range.

4.2. Explainability Analysis of Deep Learning Algorithms

In recent years, explainable artificial intelligence (XAI) has been increasingly explored to enhance the interpretability of complex black-box predictors. Post hoc XAI approaches are commonly grouped into model-agnostic methods and model-specific methods, depending on whether the explanation strategy is independent of the underlying model architecture [57]. This study also includes a post hoc XAI analysis of the proposed LSTM-based SOH estimator.

SHapley Additive exPlanation (SHAP) is adopted as a game-theoretic approach to decompose a model’s prediction into additive contributions from individual feature values [58]. For a given input window, the prediction can be written as the sum of a baseline value and the Shapley values for all input features, where each value is obtained by taking the average marginal contribution of that feature across all feature coalitions, as defined by the Shapley value in Equation (8):

$${\mathsf{\Phi }}_i={\displaystyle \sum _{S\subseteq N\backslash \left\{i\right\}}\frac{\left|S\right|!\left(M-\left|S\right|-1\right)!}{M!}}\left[f_x\left(S\cup \left\{i\right\}\right)-f_x\left(S\right)\right]$$

(8)

Two commonly used SHAP variants are KernelSHAP and TreeSHAP. KernelSHAP reformulates the computation of Shapley values as a weighted linear regression problem, thereby significantly reducing computational burden while retaining the fundamental axiomatic properties of Shapley-based explanations [59]. In addition, KernelSHAP is fully model-agnostic and only requires query access to the predictor. Accordingly, KernelSHAP is selected to interpret the SOH estimation model in this study. The attribution analysis is carried out in Python 3.9 using the open-source shap package (version 0.49.1).

4.3. The General Framework of the SOH Estimation Method

Figure 11 provides an overview of the proposed SOH-estimation workflow. Firstly, the aging dataset is first built from cycling experiments on SIB cells, after which 10 degradation-relevant features are selected from the full HI pool based on their correlation with aging. These inputs are then fed into the LSTM model to estimate SOH, and the accuracy is assessed using the root-mean-square error (RMSE) and the mean absolute error (MAE). Finally, KernelSHAP is employed to interpret the model outputs by quantifying the contributions of individual HIs.

5. Results and Discussion

5.1. The Verification of SOH Estimation

Figure 12a illustrates the train/validation/test partitioning scheme used in this study. In each training run, one of the six cells (Cell 1–Cell 6) is held out for testing; the cell with the subsequent index is assigned for validation, and the other four cells are used for training. For example, in the first run, Cell 1 is used as the test set, Cell 2 as the validation set, and Cell 3–Cell 6 as the training set. Every cell has served once as the test set and once as the validation set, resulting in six training–testing rounds in total. During all runs, the LSTM architecture follows the design described in Section 4.1, and only the network weights are retrained on the corresponding training sets to evaluate the cross-cell generalization of the proposed SOH estimation model.

Training was performed using the Adam optimizer, which leverages momentum together with adaptive step-size updates to accelerate convergence. The learning rate was initialized at 0.001, and early stopping was adopted to reduce the risk of overfitting.

Using the HIs defined above together with the adopted train–validation–test partitioning, Figure 13 presents the SOH estimation results of the proposed LSTM model for two representative cells (Cell 4 and Cell 6). In Figure 13a, the predictions for Cell 4 closely match the reference SOH, indicating high estimation accuracy. Both the gradual capacity fade and the stepwise drops are well reproduced. The SOH trajectory of Cell 6 is likewise accurately captured throughout the aging process, and the predicted curve closely matches the measured SOH, with only slight smoothing of local fluctuations.

To assess SOH estimation performance, this study employs two accuracy metrics, namely the MAE and the RMSE, which are defined as follows:

$$\mathrm{MAE}={\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N\left|y_i-{\widehat{y}}_i\right|}$$

(9)

$$\mathrm{RMSE}={\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N\sqrt{{\left(y_i-{\widehat{y}}_i\right)}^2}}$$

(10)

The SOH estimation metrics of all six cells are summarized in

Table 1. SOH estimation error.

Test Cell	RMSE	MAE
Cell 1	0.77%	0.56%
Cell 2	0.84%	0.68%
Cell 3	1.23%	0.99%
Cell 4	0.56%	0.41%
Cell 5	0.63%	0.49%
Cell 6	0.46%	0.36%

. Overall, the MAE remains below 1.0% for every cell, and the RMSE consistently falls below 1.2%. Only Cell 3 shows a slightly higher MAE due to more pronounced late-stage degradation, whereas the remaining cells maintain MAE values below 0.7%. These results demonstrate that the proposed LSTM-based SOH estimation framework achieves accurate, robust cross-cell prediction over long-term cycling.

This study also analyzes the stepwise errors in multi-step predictions. As introduced earlier, the dataset is formatted into supervised sequences using a 20-step input window, and the model is trained to predict the next 20 steps, thereby capturing the capacity-fade evolution over the subsequent 20 cycles. As shown in Figure 14, the MAE shows a mild upward trend as the prediction step increases, and reaches its maximum at step 20. This gradual and limited increase in error is consistent with the accumulation of uncertainty in multi-step forecasting, meanwhile indicating that the proposed LSTM-based model maintains stable accuracy across all prediction horizons.

To examine model robustness under data-scarce conditions, we additionally considered an alternative data-splitting strategy with reduced training samples. As shown in Figure 12b, only one cell is used for model training in each run, another is reserved for validation, and the remaining four cells are used as the test set. For example, Cell 1 is used as the training set, Cell 2 as the validation set, and Cell 3–Cell 6 as the test set in the first run. This stringent setting, where the training set contains data from only a single cell, is designed to verify whether the proposed LSTM-based SOH estimator can preserve acceptable predictive accuracy under a markedly reduced training dataset.

The comparative results of the two partitioning schemes are summarized in

Table 2. The comparative results of the two partitioning schemes.

Round	Scheme 2	Scheme 1
1	0.82% *	0.77%
2	1.06%	0.84%
3	1.18%	1.23%
4	1.14%	0.56%
5	0.87%	0.63%
6	1.04%	0.46%

* RMSE.

. Using the original cross-cell split, the mean RMSE across the six held-out cells is approximately 0.75%, and the cell-wise RMSE ranges from about 0.46% to 1.23%. When the training set is reduced to a single cell per round, the corresponding RMSE ranges from 0.82% to 1.18%, with an average of 1.02%. Such increases in MAE and RMSE are reasonable because training on only one cell provides much less diversity in degradation patterns, thereby amplifying the distribution shift between the training and test cells. Nevertheless, all errors remain well within 1–1.2%, indicating that the proposed LSTM-based SOH estimation model still maintains satisfactory accuracy and maintains strong generalization performance even when the available training data are limited.

To benchmark the proposed method, its SOH estimation accuracy is compared with several representative data-driven approaches for SIBs, as summarized in

Table 3. RMSE comparison with previously reported methods.

Reference	Method	RMSE/%
Proposed method	LSTM	0.75 *
[60]	Gaussian-kernel SVR	1.26
[60]	EIS-SVR	1.14
[4]	SB-LSTM	<1.1

* The average RMSE of all cells.

. Liu et al. [60] constructed temperature-resilient EIS indicators and trained an SVR model, achieving an average RMSE of 1.14% and MAE of 0.96% on test data collected at 10, 25, and 30 °C, while the Gaussian-kernel SVR variant yielded an RMSE of 1.26% and MAE of 0.89%. Wang et al. [4] developed the DI4SHE framework based on SPA/SPIC features and a stacked bidirectional LSTM, achieving MAE and RMSE below 0.9% and 1.1%, respectively, across multiple sodium half-cells and whole cells. In comparison, the proposed LSTM-based HI model achieves lower MAE and RMSE across all test cells within the considered SOH range, indicating that its estimation accuracy is competitive with that of existing sodium-ion battery SOH estimation approaches.

5.2. Explainability Analysis Based on KernelSHAP

The SHAP supports model interpretability at both the dataset level and the individual-sample level, commonly referred to as global and local explanations, respectively. In this study, KernelSHAP is employed as a global interpreter to analyze how each HI contributes to the LSTM-based SOH estimates over all samples. The SHAP summary plots for Cell 1–Cell 6 are shown in Figure 15. In each subplot, every point corresponds to one sample point; the color encodes the value of the corresponding HI (blue for low values and red for high values), while the X-axis corresponds to the SHAP value associated with that HI. Points on the positive side of the axis imply that the corresponding HI increases the predicted SOH, whereas negative values suggest an opposite effect. Within each cell-specific plot, the HIs are ranked by their mean absolute SHAP values, so the most influential features appear at the top.

It can be seen that SIB-related voltage and polarization indicators play a dominant role in SOH estimation. In particular, HI3 (ΔVavg), together with the average charge and discharge voltages HI1 (Vavg_charge) and HI2 (Vavg_discharge), consistently ranks at the top of the SHAP summary plots for all cells, indicating that the model mainly exploits the shift and widening of the quasi-plateau region and the increase in voltage hysteresis to infer SOH. For HI2, samples with larger average voltages (red) tend to exhibit positive SHAP attributions, whereas smaller voltages (blue) are mainly linked to negative SHAP values. This pattern suggests that an increased HI2 value pushes the prediction toward a higher SOH. By contrast, HI3 exhibits the opposite trend: larger ΔVavg (stronger polarization) yields negative SHAP values, indicating that increased polarization lowers the SOH estimate. This is consistent with the SIB-specific mechanisms discussed in Section 2, in which Na-based cathodes, hard-carbon anodes, and less stable SEI formation lead to more pronounced polarization than in LIBs.

Capacity- and relaxation-related HIs provide complementary information. The constant-voltage capacity increment HI7 (Cap_CV) typically appears in the upper–middle of the ranking, with larger values associated with lower predicted SOH, reflecting the additional charge needed to reach the cut-off current in aged, more polarized cells. Relaxation features HI8 (V_relax10min), HI10 (V_mean_relax10min), and HI11 (V_var_relax10min) usually lie lower in the ranking with narrower SHAP spreads, suggesting that they act as auxiliary indicators of sluggish Na⁺ diffusion and interfacial kinetics that fine-tune, rather than dominate, the prediction. IC-curve features HI13 (PPIC) also show clear patterns: peak shifts toward higher voltage tend to produce negative SHAP values, highlighting the model’s sensitivity to SIB-specific redox-potential shifts and loss of active sodium inventory. Overall, the stable importance ordering of HI1–HI3, HI7, HI8/HI10/HI11, and HI12/HI13 across all six cells confirms that the LSTM has captured common degradation patterns tailored to SIB electrochemistry, and the KernelSHAP analysis quantitatively demonstrates that polarization-, relaxation-, and IC-related features are the primary drivers of the SOH predictions.

In addition to feature-wise importance, the joint significance of each health indicator and its temporal position within the input window is further quantified using a feature–time SHAP representation. After computing SHAP values for all samples, each concatenated dimension corresponding to a specific health indicator at a given time index (for example, HI3 at time step t17) is treated as a separate feature–time element. The average absolute SHAP magnitude over the dataset measures its importance. This quantity is consistent with the ranking criterion used in standard SHAP summary plots. It can be interpreted as the typical “horizontal extent” of the corresponding row in a feature–time beeswarm plot. The resulting importance scores are assembled into a health-indicator-by-time matrix and visualized in Figure 16 as a three-dimensional bar chart, with the two horizontal axes representing the time and health-indicator indices. The bar height represents the mean absolute SHAP value, that is, how strongly each feature–time pair contributes to the state-of-health estimate.

From this feature–time map, a clear temporal pattern emerges. For most HIs, the bars associated with early time steps (approximately from t1 to t10) remain close to zero, whereas the importance increases markedly toward the end of the window. The dominant contributions are concentrated at time steps t16–t19 for HI7, HI3, HI1, HI2, HI13 and HI10, which confirms that the model relies primarily on the increase in constant-voltage charging capacity (HI7), the rise in hysteresis (HI3), shifts in charge and discharge voltages (HI1 and HI2), migration of incremental-capacity peaks (HI13), and relaxation behavior (HI10) in the most recent cycles. At the same time, the same indicators at slightly earlier positions (for example, HI1 at t11–t13, HI3 at t13–t14 and HI13 at t13–t15) still show noticeable but smaller SHAP magnitudes, indicating that they provide secondary support. This gradual decay of importance from late to early time steps suggests that the LSTM model exploits both the current state and its short-term evolution: the latest cycles carry the most direct information on the upcoming degradation trajectory, whereas the earlier cycles mainly supply contextual information on the build-up of polarization, the gradual shift in incremental-capacity peaks, and the evolution of relaxation voltage. The fact that this feature–time importance structure is highly consistent across batteries (Cell 1–Cell 6) demonstrates that the proposed LSTM-based model has learned a stable and physically interpretable temporal weighting scheme that agrees with the degradation behavior of SIBs.

6. Limitations and Outlook

Under controlled aging conditions, the proposed LSTM-based SOH estimator with KernelSHAP interpretability delivers accurate short-horizon SOH predictions on six sodium-ion cells from the same production batch. Nevertheless, several limitations remain, which also indicate directions for future research.

• In this study, 13 candidate HIs are initially constructed from charge/discharge profiles and post-charge relaxation segments, and 10 indicators are ultimately retained via correlation and redundancy screening as inputs for LSTM training. It should be emphasized that the present dataset is restricted to six same-batch NFM/hard-carbon pouch cells aged at 25 °C under a 2C CC–CV charge/2C discharge protocol, and public SIB aging datasets remain scarce. Although the framework operates on physically interpretable HIs rather than raw waveforms and is, in principle, transferable by re-extracting HIs and re-training the model, the current HI set is still calibrated under a single chemistry and operating regime. Future work will therefore extend the dataset to multiple chemistries, formats, temperatures, and C-rates, and will assess whether the selected HIs remain robust or should be augmented under more diverse aging trajectories.
• The 10 HIs used for model training are extracted from full or quasi-full cycles with well-defined rest periods. In practical EV and stationary storage applications, however, batteries are typically subjected to dynamic loading, partial cycling, and irregular usage, so that complete charge/discharge curves and clean relaxation segments may not be available. Under such conditions, some of the HIs defined in this work may be missing, truncated, or distorted. Future work will therefore reformulate the proposed polarization, hysteresis, relaxation, and IC indicators in a segment-based manner compatible with drive cycles and partial usage, and combine the LSTM with masking or imputation strategies so that missing or truncated HIs at certain steps can be handled without sacrificing prediction accuracy.
• This study focuses on single sodium-ion cells; however, in practical applications, batteries are typically integrated into modules and pack-level systems. In such configurations, cell-to-cell inconsistency in capacity and dynamic characteristics may degrade the overall estimation performance and hinder direct deployment of the proposed method. Therefore, future work should extend the SOH estimation framework to pack-level scenarios and explicitly account for cell inconsistency and balancing strategies.
• The explainability module in this study quantitatively evaluates the contribution of each HI to the LSTM-based SOH estimates and helps interpret abnormal predictions. Nevertheless, the present analysis alone cannot establish a direct correspondence between the predicted SOH and specific physicochemical aging mechanisms. Future work will extend the explainability assessment to different aging trajectories and degradation modes, so that the extracted attributions can be more closely aligned with mechanism-aware interpretations and, ultimately, provide stronger diagnostic support for BMS applications.

In addition, although the present study relies on laboratory cycling data, the proposed HI-based framework is compatible with real-time deployment in battery management systems. The required indicators can be computed from voltage, current, and temperature signals routinely measured by on-board BMSs, while IoT-enabled platforms can upload compressed HI sequences for cloud- or edge-based LSTM inference and periodic model updating as field data accumulate. Such a data infrastructure also provides a foundation for future integration into battery digital-twin frameworks, in which streaming HIs continuously update a virtual replica of the physical cell and SHAP-derived insights support anomaly diagnosis, predictive maintenance, and scenario-based virtual testing without extensive physical cycling.

7. Conclusions

Reliable SOH estimation is a prerequisite for safe application of sodium-ion batteries. To this end, we introduce an explainable data-driven SOH estimator that integrates an LSTM-based sequence model with KernelSHAP to interpret the contributions of health indicators. Thirteen candidate HIs are first extracted from experiment data, and ten indicators with strong correlation to capacity degradation are selected as model inputs. Utilizing aging data from six identical pouch cells and employing a leave-one-cell-out train–validation–test strategy with overlapping prediction windows, the proposed methodology attains high precision and robustness, evidenced by MAE below 1.0% and RMSE below 1.2% across all individual cells. The average MAE in the original multi-cell training schema is approximately 0.58%. When the training data are limited to only one cell, the MAE remains approximately 0.67%, and the RMSE is about 1.02%, demonstrating satisfactory generalization capabilities under data-limited scenarios. Moreover, the analysis based on KernelSHAP elucidates the contributions of individual HIs and their temporal positions, aiding interpretation of anomalous estimates and enhancing the transparency of the LSTM model. Overall, the proposed framework enables accurate SOH estimation for sodium-ion batteries while improving interpretability, which makes it an available building block for future SIB-oriented battery management systems.