Adaptive Hybrid LSTM-EKF Model for Reliable State of Charge Estimation in Lithium-Ion Batteries Under Noisy Conditions

A nonlinear equivalent circuit model (ECM) is often used to represent the dynamic behavior of lithium-ion batteries. The state-space representation of this model is given as follows:

1SoCk+1=SoCk−ηΔtCbatIk+Wk\[So{C_{k + 1}} = So{C_k} - \frac{{\eta \Delta t}}{{{c_{bat}}}}{I_k} + {w_k}\]

where

Δt: is the discrete-time step,

η: charge efficiency parameter

C_bat: represents the nominal capacity of the battery,

I_k: is the battery current,

w_k: denotes process noise, assumed to be Gaussian with zero mean and a variance Q.

The measurement equation as follows:

2Vk=Voc(SoCk)−RsIk+vk\[{V_k} = {V_{oc}}(So{C_k}) - {R_s}{I_k} + {v_k}\]

where

V_k: is the measured terminal voltage,

V_oc(SoC_k): is the open-circuit voltage as a function of SoC,

R_s: represents the series resistance.

υ_k: is the measurement noise, assumed to be Gaussian with zero mean and a variance R.

Given the nonlinear nature of the system, traditional filtering techniques such as the Extended Kalman Filter (EKF) have been employed to estimate the internal battery states. However, these methods struggle with capturing complex nonlinear dependencies in SoC estimation, necessitating the integration of machine learning techniques like Long Short-Term Memory (LSTM) networks.

The state-space model represents the dynamics of the system, capturing the evolution of the internal states over time based on the system's inputs and noise. In the context of lithium-ion battery modeling, the state-space approach provides an efficient framework for estimating the battery's state of charge (SoC), voltage, and temperature under uncertain conditions. The state-space model for a nonlinear system can be expressed as:

3xk+1=f(xk,uk)+wkyk=h(xk)+vk \[\begin{array}{*{20}{c}}{{x_{k + 1}} = f({x_k},{u_k}) + {w_k}}\\{{y_k} = h({x_k}) + {v_k}}\end{array}\]

where

x_kis the state vector (SoC, temperature),

u_k represents the input vector (current),

y_kis the output vector (measured voltage),

f(.)and h(.)are the nonlinear dynamic and measurement models, respectively.

To overcome these limitations, this paper introduces an adaptive hybrid LSTM-EKF framework that leverages the strengths of both approaches. The proposed method integrates a dynamic fusion factor (α_k), which adjusts the LSTM and EKF contributions based on real-time confidence metrics, ensuring robust SoC estimation under varying operating conditions.

The LSTM network serves as the data-driven component of our hybrid framework, modeling the complex, time-varying dynamics of lithium-ion batteries. At each time step t, the LSTM processes input features (current, voltage, and temperature) to predict the SoC_LSTM. The network's gating mechanisms enable it to selectively retain or discard information, making it particularly suited for sequential data with long-term dependencies. The LSTM's hidden state h_t and cell state C_t are updated according to the following equations:

4ft=σ(Wf×[ht−1,xt]+bf)it=σ(Wi×[ht−1,xt]+bi)C˜t=tanh(Wc×[ht−1,xt]+bc)Ct=ft⊙Ct−1+it⊙C˜tot=σ(Wo×[ht−1,xt]+bo)ht=ot⊙tanh(Ct)\[\begin{array}{l}{f_t} = \sigma ({W_f} \times [{h_{t - 1}},{x_t}] + {b_f})\\\begin{array}{*{20}{c}}{{i_t} = \sigma ({W_i} \times [{h_{t - 1}},{x_t}] + {b_i})}\\{{{\tilde C}_t} = \tanh ({W_c} \times [{h_{t - 1}},{x_t}] + {b_c})}\\{{C_t} = {f_t} \odot {C_{t - 1}} + {i_t} \odot {{\tilde C}_t}}\\{{o_t} = \sigma ({W_o} \times [{h_{t - 1}},{x_t}] + {b_o})}\\{{h_t} = {o_t} \odot \tanh ({C_t})}\end{array}\end{array}\]

where

- f_t, i_t and o_t are the forget, input, and output gates respectively

- σ denotes the sigmoid activation function

The final state prediction SoC_LSTM,k+1 is obtained through a linear projection layer as follows:

5SoCLSTM,k+1=fLSTM(Ik,Vk,Tk)\[So{C_{LSTM,k + 1}} = {f_{LSTM}}({I_k},{V_{k,}}{T_k})\]\]

The LSTM network effectively models the nonlinear dynamics of lithium-ion batteries by capturing long-term dependencies in voltage, current, and temperature data. However, its lack of real-time error correction necessitates integration with the EKF, as described in the following subsection, to achieve robust SoC estimation under noisy conditions.

While the LSTM captures the nonlinear dynamics of the battery, the EKF provides real-time correction by filtering process and measurement noise. Referring to the system equations (3), the Extended Kalman Filter recursively estimates the states by linearizing the system dynamics and measurement equations around the current state estimate using Jacobian matrices:

• –

  State prediction:

6x^k/k−1=f(xk,uk)\[{\hat x_{k/k - 1}} = f({x_k},{u_k})\]

7Pk|k−1=AkPkAkT+Q\[{{P_{k|k - 1}} = {A_k}{P_k}A_k^T + Q}\]

where Ak=∂f∂x|x^k\[{A_k} = \frac{{\partial f}}{{\partial x}}|{{\hat x}_k}\]

• –

  Measurement Update

  8Kk=Pk|k−1HkT(HkPk|k−1HkT+R)−1\[{K_k} = {P_{k|k - 1}}H_k^T{({H_k}{P_{k|k - 1}}H_k^T + R)^{ - 1}}\]

  9x^k|k=x^k|k−1+Kk(yk−h(x^k|k−1))\[{\hat x_{k|k}} = {\hat x_{k|k - 1}} + {K_k}({y_k} - h({\hat x_{k|k - 1}}))\]

  10Pk|k=(I−KkHk)Pk|k−1\[{{P_{k|k}} = (I - {K_k}{H_k}){P_{k|k - 1}}}\]

where Hk=∂h∂x|x^k\[{{H_k} = \frac{{\partial h}}{{\partial x}}|{{\hat x}_k}}\]

The EKF SoC estimation equation is simplified as:

11SoC^k+1|k+1=SoC^k+1|k+Kk(Vk−Voc(SoC^k+1|k)+RsIk)\[{\widehat {SoC}_{k + 1\left| {k + 1} \right.}} = {\widehat {SoC}_{k + 1\left| k \right.}} + {K_k}({V_k} - {V_{oc}}({\widehat {SoC}_{k + 1\left| k \right.}}) + {R_s}{I_k})\]

The final SoC estimation is obtained by fusing the LSTM prediction and EKF correction through a weighted approach:

12SoC^k+1=αkSoCLSTM,k+1+(1−αk)SoC^EKF,k+1\[{\widehat {SoC}_{k + 1}} = {\alpha _k}So{C_{LSTM,k + 1}} + (1 - {\alpha _k}){\widehat {SoC}_{EKF,k + 1}}\]

Where α_k is an adaptive weighting factor that dynamically balances the contributions of EKF and LSTM. By tuningα_k, the system optimally combines the strengths of both approaches, ensuring accurate and stable SoC estimation under varying operating conditions. The adaptive fusion factor, α_k ∈ [0,1], is computed as:

13αk=σEKF,k2σEKF,k2+σLSTM,k2\[{\alpha _k} = \frac{{{\sigma ^2}_{EKF,k}}}{{{\sigma ^2}_{EKF,k} + {\sigma ^2}_{LSTM,k}}}\]

To maintain consistency in error modeling, Gaussian white noise is introduced in both the LSTM and EKF estimations, ensuring fair comparison and robustness. The variances σLSTM2\[\sigma _{LSTM}^2\] and σEKF2\[\sigma _{EKF}^2\] as follows:

14σLSTM,k2=1N∑i=1N(SoCreal,i−SoCLSTM,i)2\[{\sigma ^2}_{LSTM,k} = \frac{1}{N}\sum\nolimits_{i = 1}^N {{{(So{C_{real,i}} - So{C_{LSTM,i}})}^2}} \]

15σEKF2=Pk|k\[{\sigma ^2}_{EKF} = {P_{k|k}}\]

where P_k|k is the covariance matrix of the state from the EKF.

The fusion with the weighting factor α_k is a critical aspect of the proposed LSTM-EKF hybrid model. It balances the contributions of the LSTM's prediction and the EKF's estimation to refine the State of Charge estimation. By tuning α_k, the method dynamically adjusts the trust given to the data-driven model (LSTM) versus the physics-based model (EKF), thereby enhancing robustness against noise and improving overall estimation accuracy.

This synergistic combination addresses the limitations of individual methods, providing accurate, robust, and adaptive SoC estimation under noisy and nonlinear conditions. The next section validates the framework's performance on NASA's battery datasets, comparing it to standalone and fixed hybrid methods.

In this section, we evaluate the performance of the proposed LSTM-EKF hybrid model for State-of-Charge (SoC) estimation using the NASA battery dataset used by Saha and Goebel.[11]. The observed fluctuations highlight the challenges of traditional SoC estimation methods, which may struggle with sensor noise, nonlinear dependencies, and model uncertainties. This justifies the need for a hybrid approach that combines machine learning (LSTM) with model-based filtering (EKF).

The dataset was split into 80% training and 20% testing. The model was trained for 50 epochs with a batch size of 16, using a validation split of 10% to monitor performance. The training process was evaluated by plotting the loss curves, showing a consistent decrease in MSE over epochs, indicating effective learning. Once trained, the model was tested on the entire dataset to generate SoC predictions. The model summary confirmed its suitability for battery SoC estimation, demonstrating its ability to capture time-dependent patterns in dynamic battery conditions.

Figure 3 illustrates the evolution of the training loss and validation loss for the LSTM model over 50 epochs, measured using the Mean Squared Error (MSE). Overall, the results demonstrate that the LSTM model is effectively trained and provides accurate predictions with minimal overfitting.

Fig. 3.

Training and validation loss for LSTM Model

To refine the SoC predictions generated by the LSTM model, the EKF was applied. The SoC estimation is modeled using a discrete-time state-space representation with the following parameters: A_k = 1, C_bat = 2mh, B_k = –1/(3600 * C_bat), Q = 0.02, R = 0.02, x₀ =0and P = 1

Referring to Equation (12), the combination is performed using the adaptive fusion factor α_k represented by Equation (13).

The evolution of over time (Fig. 4) shows that:

• –

  when LSTM predictions are reliable, increases, giving more weight to the neural network model.

• –

  when uncertainty is high, EKF takes over to stabilize the estimation.

• –

  this adaptive mechanism enhances overall SoC estimation accuracy, balancing both methods dynamically.

Fig. 4.

Evolution of the adaptive fusion factor (α_k)

The fusion factor fluctuates over time, demonstrating that the system dynamically adjusts the influence of each method based on estimation confidence and noise levels. This adaptive mechanism enhances robustness by allowing the hybrid model to intelligently switch between data-driven learning and Kalman filtering, depending on real-time conditions.

Figure 5 presents the SoC estimation results obtained using five different approaches: the standalone LSTM model, the BiLSTM model, the LSTM with attention mechanism, the EKF method, and the proposed adaptive hybrid LSTM-EKF model.

• –

  The LSTM-EKF hybrid model significantly reduces estimation errors, particularly in dynamic and nonlinear regions (as shown in the zoomed-in section)

• –

  The standalone LSTM and its variants (BiLSTM and LSTM with attention) exhibit limitations: while they capture temporal dependencies effectively, they lack real-time correction, leading to higher errors in noisy or transient conditions.

• –

  The EKF method improves upon standalone LSTM by providing real-time correction but struggles in highly nonlinear sections due to its reliance on accurate system modeling.

• –

  The hybrid LSTM-EKF approach effectively combines the predictive capabilities of LSTM with the adaptive filtering ability of EKF, resulting in superior performance. By dynamically balancing their contributions through the fusion factor, it overcomes the limitations of both standalone and recent advanced methods.

Fig. 5.

SoC Estimation results using LSTM, BiLSTM, LSTM-Attention, EKF and LSTM-EKF

These findings highlight the advantage of integrating machine learning with traditional filtering techniques for SoC estimation, ensuring higher accuracy and adaptability to nonlinear and noisy conditions.

Figure 6 presents a comparative evaluation of SoC estimation accuracy for three methods. The metrics used are the Mean Absolute Error (MAE) and Root Mean Square Error (RMSE):

Fig. 6.

Comparison of RMSE and MAE for different SoC estimation methods

16MAE=1n∑i=1n|yi−y^i|\[MAE = \frac{1}{n}\sum\nolimits_{i = 1}^n {|{y_i} - {{\hat y}_i}|} \]

17RMSE=1n∑i=1n(yi−y^i)2\[RMSE = \sqrt {\frac{1}{n}\sum\nolimits_{i = 1}^n {{{({y_i} - {{\hat y}_i})}^2}} } \]

Figure 6 provides a visual comparison of the Root Mean Square Error (RMsE) and Mean Absolute Error (MAE) for five different State of Charge (SoC) estimation methods: LStM, BiLSTM, LSTM-Attention, EKF, and LSTM-EKF. The exact error values are displayed above each bar, allowing for a direct comparison of the performance of each method:

• –

  LSTM Only:

  • –

    RMSE (0.0211) and AME (~0.0165) are the highest.

  • –

    This suggests that the standalone deep learning model introduces some lag and lacks real-time error correction.

• –

  BiLSTM:

  • –

    RMSE = 0.0227, MAE = 0.0182

  • –

    The BiLSTM model does not outperform the standalone LSTM, suggesting that bidirectional processing does not provide substantial benefits for this specific application. The errors are slightly higher than those of the LSTM.

• –

  LSTM-Attention:

  • –

    RMSE=0.0217, MAE=0.0169

  • –

    The LSTM with attention mechanism performs similarly to the standalone LSTM. This indicates that attention mechanisms do not significantly improve the accuracy of SoC estimation in this context.

• –

  EKF Only:

  • –

    RMSE (0.0118) and AME (0.0093) are lower than LSTM.

  • –

    The EKF model shows a notable improvement over the LSTM, with a 43.92% reduction in RMSE. This improvement is due to its ability to dynamically correct state estimates in real-time, addressing one of the key limitations of data-driven models.

• –

  LSTM-EKF (Hybrid Model):

  • –

    RMSE = 0.0062, MAE = 0.0048.

  • –

    The proposed adaptive hybrid LSTM-EKF framework achieves the lowest errors among all methods. This represents a 70.77% reduction in RMSE and a 70.80% reduction in MAE compared to the standalone LSTM. The dynamic fusion factor optimally balances the contributions of the LSTM and EKF, resulting in superior accuracy and robustness under noisy and nonlinear conditions.

The results in Tab.1. demonstrate that the standalone LSTM model, while effective at capturing temporal dependencies, suffers from higher estimation errors due to its lack of real-time correction. The BiLSTM and LSTM with attention models do not significantly outperform the baseline LSTM, suggesting that bidirectional processing and attention mechanisms do not provide substantial benefits for this specific application. In contrast, the EKF model shows a notable improvement, with a 43.92% reduction in RMSE, thanks to its ability to dynamically correct state estimates in real time. However, the most significant improvement is achieved by the proposed adaptive hybrid LSTM-EKF framework, which combines the predictive power of the LSTM with the corrective robustness of the EKF. The dynamic fusion factor (α_k) optimally balances their contributions, resulting in a 70.77% reduction in RMSE and a 70.80% reduction in MAE compared to the standalone LSTM. This superior performance underscores the effectiveness of our hybrid approach in achieving accurate and robust SoC estimation under noisy and nonlinear conditions.

The results show that the hybrid approach achieves the lowest error values, demonstrating superior accuracy and robustness. The LSTM-EKF model effectively reduces errors by dynamically adjusting the contribution of each method through the adaptive fusion mechanism (α_k). This confirms that integrating machine learning with model-based filtering significantly improves SoC estimation, making it a reliable solution for battery management in electric vehicles and renewable energy systems.

This work presented an adaptive hybrid LSTM-EKF framework for accurate and robust state-of-charge (SoC) estimation of lithium-ion batteries under nonlinear and noisy conditions. The key innovation is the dynamic fusion mechanism (α_k), which adaptively balances the predictive capability of the LSTM and the corrective feedback of the EKF in real time. A comprehensive comparison with recent approaches BiLSTM, LSTM-Attention, and EKF confirmed the superiority of the proposed method, achieving about 70% improvement in RMSE and MAE. The model effectively captures complex battery behavior while ensuring stability and fast convergence, showing strong potential for integration into advanced batterymanagement systems. Future work will extend the framework to multi-cell configurations and explore integration with physics-informed neural networks (PINNs) for real-time health prediction.