In the context of global efforts toward carbon neutrality, the development of clean and low-carbon energy systems has become a priority, driving innovation in key technologies like power batteries. As a core component of electric vehicles, li-ion batteries are widely adopted due to their high energy density, long cycle life, and robust power output. However, their nonlinear dynamics, influenced by varying operating conditions and complex electrochemical processes, make direct measurement of the state of charge (SOC) challenging. Accurate SOC estimation is crucial for battery management systems to ensure safety, optimize performance, and extend lifespan. Traditional methods such as ampere-hour integration suffer from cumulative errors, while open-circuit voltage approaches require lengthy rest periods, limiting real-time applicability. Data-driven methods demand extensive datasets, and electrochemical models involve complex computations. Thus, model-based approaches, particularly using equivalent circuit models (ECMs), offer a balance of accuracy and practicality for SOC estimation.
Among model-based algorithms, Kalman filter variants are prominent for handling nonlinear systems with noise. The extended Kalman filter (EKF) is commonly used due to its computational efficiency and stability, but its fixed noise covariances can lead to inaccuracies in dynamic environments like li-ion battery operation. To address this, adaptive EKF (AEKF) methods adjust noise statistics online, but issues like filter divergence persist. Moreover, parameter identification for ECMs is critical; inaccurate parameters degrade SOC estimation. Recursive least squares (RLS) is standard for online identification, but it may accumulate errors. Multi-innovation RLS (MIRLS) enhances this by leveraging historical data, improving parameter accuracy. Building on these, we propose a weighted multi-innovation improved Sage-Husa AEKF (WMISAEKF) algorithm for SOC estimation in li-ion batteries. This method integrates MIRLS for robust parameter identification, modifies the Sage-Husa adaptive mechanism to prevent divergence, and introduces a novel weighting scheme for multi-innovation sequences to optimize SOC updates. We validate our approach using real-world driving cycle data, demonstrating superior accuracy and robustness compared to conventional methods.
In this work, we focus on li-ion batteries due to their dominance in electric vehicles. The SOC, defined as the ratio of remaining capacity to nominal capacity, is a key state for battery management. Estimating SOC accurately involves dealing with nonlinearities, measurement noise, and model uncertainties. We base our model on a second-order RC ECM, which captures polarization effects with two parallel RC branches representing slow and fast dynamics. The model equations are derived from Kirchhoff’s laws, with the state-space representation including SOC, polarization voltages, and terminal voltage as output. The relationship between SOC and open-circuit voltage (UOCV) is established experimentally and fitted with a polynomial, as UOCV is a monotonic function of SOC in li-ion batteries. For parameter identification, we employ MIRLS, which extends RLS by incorporating multiple past innovations, reducing sensitivity to noise and enhancing convergence. The parameters, including ohmic resistance and polarization elements, are updated online to reflect battery dynamics under varying loads.
The SOC estimation algorithm builds upon EKF, which linearizes nonlinear systems locally. The state vector includes SOC and polarization voltages, with current as input and terminal voltage as output. The EKF predicts and updates states using a recursive process, but its performance hinges on accurate noise covariances. In li-ion batteries, noise characteristics change with operating conditions, so adaptive filtering is beneficial. The Sage-Husa AEKF adapts noise covariances based on innovation sequences, but its update equations can lead to non-positive definite matrices, causing divergence. We improve this by removing subtraction terms in covariance updates, ensuring stability. Furthermore, to boost accuracy, we incorporate multi-innovation theory, where past innovations are weighted and used in state updates. Our weighting method considers both temporal proximity and error magnitude, assigning higher weights to recent and larger innovations, which improves robustness against outliers. The WMISAEKF algorithm thus combines adaptive noise handling with optimized multi-innovation utilization for precise SOC tracking.
We validate our method using data from a li-ion battery under urban dynamometer driving schedule (UDDS) conditions, simulating real-world electric vehicle usage. The battery has a nominal capacity of 33 Ah and operates at 25°C. We compare WMISAEKF against standard EKF, unscented Kalman filter (UKF), and our improved Sage-Husa AEKF (ISAEKF). Performance metrics include mean absolute error (MAE), root mean square error (RMSE), and maximum absolute error (MAXE). Results show that WMISAEKF achieves the lowest errors, with MAE and RMSE reduced by over 80% compared to EKF in some cases. We also test robustness under noisy measurements and erroneous initial SOC values, demonstrating quick convergence and minimal error. The integration of MIRLS ensures accurate parameter identification, which feeds into the SOC estimator, creating a cohesive framework for li-ion battery management. Future work may extend to joint estimation of SOC and state of health (SOH) to address aging effects.
The second-order RC ECM for li-ion batteries is depicted through circuit elements, but here we focus on mathematical formulations. The model dynamics are described by differential equations. Let $U_{ocv}$ be the open-circuit voltage, $U_t$ the terminal voltage, $I_t$ the current (positive for discharge), $R_0$ the ohmic resistance, and $R_1$, $C_1$, $R_2$, $C_2$ the polarization resistances and capacitances. The voltages across RC branches are $U_1$ and $U_2$. From circuit analysis:
$$ U_{ocv} – U_t = I_t R_0 + U_1 + U_2 $$
$$ \frac{dU_1}{dt} = -\frac{1}{R_1 C_1} U_1 + \frac{1}{C_1} I_t $$
$$ \frac{dU_2}{dt} = -\frac{1}{R_2 C_2} U_2 + \frac{1}{C_2} I_t $$
Discretizing with sample time $T$, the state-space model for SOC estimation is defined with state vector $x_k = [SOC_k, U_{1,k}, U_{2,k}]^T$, input $u_k = I_{t,k}$, and output $y_k = U_{t,k}$. The SOC is derived from charge conservation: $SOC_k = SOC_{k-1} – \frac{\eta T}{C_n} I_{t,k-1}$, where $\eta$ is Coulombic efficiency and $C_n$ is nominal capacity. The discrete-time equations are:
$$ x_k = A_k x_{k-1} + B_k u_{k-1} + \omega_{k-1} $$
$$ y_k = C_k x_k + D_k u_k + \nu_k $$
where $\omega_k$ and $\nu_k$ are process and measurement noises, assumed white with covariances $Q$ and $R$. The matrices are:
$$ A_k = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \exp\left(-\frac{T}{\tau_1}\right) & 0 \\ 0 & 0 & \exp\left(-\frac{T}{\tau_2}\right) \end{bmatrix}, \quad B_k = \begin{bmatrix} -\frac{\eta T}{C_n} \\ R_1\left(1 – \exp\left(-\frac{T}{\tau_1}\right)\right) \\ R_2\left(1 – \exp\left(-\frac{T}{\tau_2}\right)\right) \end{bmatrix} $$
$$ C_k = \left[ \frac{\partial U_{ocv}}{\partial SOC} \quad -1 \quad -1 \right], \quad D_k = -R_0 $$
with time constants $\tau_1 = R_1 C_1$ and $\tau_2 = R_2 C_2$. The partial derivative $\frac{\partial U_{ocv}}{\partial SOC}$ is obtained from the SOC-UOCV curve, typically a polynomial fit: $U_{ocv}(SOC) = \sum_{i=0}^n p_i SOC^i$. For li-ion batteries, a 7th-order polynomial often suffices.
Parameter identification is crucial for model accuracy. We express the model in regressor form: $y_k = \varphi_k^T \theta + v_k$, where $y_k = U_{ocv,k} – U_{t,k}$, $\varphi_k = [y_{k-1}, y_{k-2}, I_{t,k}, I_{t,k-1}, I_{t,k-2}]^T$, and $\theta = [a_1, a_2, b_1, b_2, b_3]^T$ are unknown parameters. The relationship between $\theta$ and physical parameters $\beta = [R_0, R_1, R_2, C_1, C_2]^T$ is nonlinear, derived from transfer function equivalence:
$$ R_0 = \frac{b_1 – b_2 + b_3}{1 + a_1 – a_2}, \quad R_0 + R_1 + R_2 = \frac{b_1 + b_2 + b_3}{1 – a_1 – a_2} $$
$$ R_0 \tau_1 + R_0 \tau_2 + R_1 \tau_2 + R_2 \tau_1 = \frac{T(b_1 – b_3)}{1 – a_1 – a_2}, \quad \tau_1 + \tau_2 = \frac{T(1 + a_2)}{1 – a_1 – a_2}, \quad \tau_1 \tau_2 = \frac{T^2(1 + a_1 – a_2)}{4(1 – a_1 – a_2)} $$
We use MIRLS for online identification. Let $p$ be the innovation length. Define the innovation vector $E(p,k) = [e(k), e(k-1), \dots, e(k-p+1)]^T$, where $e(k) = y_k – \varphi_k^T \hat{\theta}_{k-1}$. The regressor matrix is $\Phi(p,k) = [\varphi_k, \varphi_{k-1}, \dots, \varphi_{k-p+1}]^T$, and output vector $Y(p,k) = [y_k, y_{k-1}, \dots, y_{k-p+1}]^T$. The MIRLS update equations are:
$$ \hat{\theta}_k = \hat{\theta}_{k-1} + L_k E(p,k) $$
$$ L_k = \frac{P_{k-1} \Phi(p,k)}{1 + \Phi(p,k)^T P_{k-1} \Phi(p,k)} $$
$$ P_k = [I – L_k \Phi(p,k)^T] P_{k-1} $$
where $P_k$ is the covariance matrix. This approach reduces parameter fluctuation and improves accuracy for li-ion batteries under dynamic loads.
For SOC estimation, we develop the WMISAEKF algorithm. Starting from EKF, the steps are prediction and update. The prediction equations are:
$$ \hat{x}_k^- = A_{k-1} \hat{x}_{k-1} + B_{k-1} u_{k-1} $$
$$ P_k^- = A_{k-1} P_{k-1} A_{k-1}^T + Q_{k-1} $$
The update equations compute Kalman gain $K_k$, innovation $e_k = y_k – C_k \hat{x}_k^- – D_k u_k$, and state correction. In standard EKF, $Q$ and $R$ are fixed, but in AEKF, they are adapted. The Sage-Husa method updates them as:
$$ Q_k = (1 – d_{k-1}) Q_{k-1} + d_{k-1} [K_k e_k e_k^T K_k^T + P_k – A_{k-1} P_{k-1} A_{k-1}^T] $$
$$ R_k = (1 – d_{k-1}) R_{k-1} + d_{k-1} [e_k e_k^T – C_k P_k^- C_k^T] $$
where $d_{k-1} = (1 – b)/(1 – b^k)$ with forgetting factor $b$ (e.g., 0.95). However, the subtraction terms may cause non-positive definiteness. We improve this by removing them:
$$ Q_k = (1 – d_{k-1}) Q_{k-1} + d_{k-1} [K_k e_k e_k^T K_k^T] $$
$$ R_k = (1 – d_{k-1}) R_{k-1} + d_{k-1} [e_k e_k^T] $$
This ensures stability. Next, we incorporate multi-innovation. Define multi-innovation vector $E_m(p,k) = [e_k, e_{k-1}, \dots, e_{k-p+1}]^T$ and multi-gain matrix $\Gamma(p,k) = [K_k, K_{k-1}, \dots, K_{k-p+1}]$. To weight innovations, we propose weights based on time and error magnitude. For innovation $e_i$ at time $i$, time weight $w_{t,i} = \exp\left(-\frac{(i-k)^2}{2}\right)$ and error weight $w_{e,i} = \exp\left(-\frac{(e_i – \max(E_m))^2}{2}\right)$, where $\max(E_m)$ is the maximum innovation in the sequence. The total weight for each innovation is $w_i = w_{t,i} \cdot w_{e,i}$, normalized by $w_{\text{sum}} = \sum_{i=k-p+1}^k w_i$. The weighted multi-innovation update is:
$$ \hat{x}_k = \hat{x}_k^- + \frac{p}{w_{\text{sum}}} \Gamma(p,k) W_t W_e E_m(p,k) $$
where $W_t = \text{diag}([w_{t,k}, w_{t,k-1}, \dots, w_{t,k-p+1}])$ and $W_e = \text{diag}([w_{e,k}, w_{e,k-1}, \dots, w_{e,k-p+1}])$. This prioritizes recent and significant innovations, enhancing accuracy for li-ion battery SOC estimation.
The complete WMISAEKF algorithm steps are summarized in the table below:
| Step | Description | Equation |
|---|---|---|
| 1 | Initialize | $\hat{x}_0, P_0, Q_0, R_0$ |
| 2 | Predict state | $\hat{x}_k^- = A_{k-1} \hat{x}_{k-1} + B_{k-1} u_{k-1}$ |
| 3 | Predict covariance | $P_k^- = A_{k-1} P_{k-1} A_{k-1}^T + Q_{k-1}$ |
| 4 | Compute Kalman gain | $K_k = P_k^- C_k^T (C_k P_k^- C_k^T + R_{k-1})^{-1}$ |
| 5 | Calculate innovation | $e_k = y_k – C_k \hat{x}_k^- – D_k u_k$ |
| 6 | Update multi-innovation vector | $E_m(p,k) = [e_k, \dots, e_{k-p+1}]^T$ |
| 7 | Compute weights | $w_{t,i}, w_{e,i}$ as defined |
| 8 | Update state with weighting | $\hat{x}_k = \hat{x}_k^- + \frac{p}{w_{\text{sum}}} \Gamma(p,k) W_t W_e E_m(p,k)$ |
| 9 | Update covariance | $P_k = (I – K_k C_k) P_k^-$ |
| 10 | Adapt noise covariances | $Q_k = (1-d_{k-1}) Q_{k-1} + d_{k-1} [K_k e_k e_k^T K_k^T]$, $R_k = (1-d_{k-1}) R_{k-1} + d_{k-1} [e_k e_k^T]$ |
Simulation results are based on a li-ion battery with 33 Ah capacity, tested under UDDS at 25°C. Data includes current and voltage profiles over approximately 20,000 seconds. The SOC-UOCV curve is fitted to a 7th-order polynomial: $U_{ocv}(SOC) = -16.34 SOC^7 + 57.21 SOC^6 – 77.89 SOC^5 + 52.34 SOC^4 – 18.45 SOC^3 + 3.89 SOC^2 – 0.45 SOC + 3.65$. Parameters are identified online using MIRLS with $p=5$. The identified parameters vary with time, reflecting battery dynamics. For example, $R_0$ ranges from 2 to 6 mΩ, while $R_1$ and $R_2$ show changes during high-current phases. The MIRLS results are more stable than RLS, as shown in error metrics.
We compare SOC estimation from EKF, UKF, ISAEKF, and WMISAEKF. Initial SOC is set to 1.0 (fully charged), and true SOC is computed via ampere-hour integration with high accuracy. The estimation errors are analyzed. The table below summarizes performance over the entire cycle:
| Algorithm | MAE | RMSE | MAXE |
|---|---|---|---|
| EKF | 0.0091 | 0.0118 | 0.0401 |
| UKF | 0.0095 | 0.0123 | 0.0468 |
| ISAEKF | 0.0023 | 0.0051 | 0.0597 |
| WMISAEKF | 0.0010 | 0.0024 | 0.0304 |
WMISAEKF reduces MAE by 89.01% and RMSE by 79.66% compared to EKF, demonstrating significant improvement. The MAXE is also lower, except for ISAEKF which shows a higher peak due to initial adaptation. The SOC estimates track the true SOC closely, even during rapid current changes. Robustness is tested by adding Gaussian noise to voltage and current measurements. Let $\sigma_u$ and $\sigma_i$ be standard deviations for voltage and current noise, respectively. With $\sigma_u = 0.06$ V and $\sigma_i = 0.3$ A, WMISAEKF maintains RMSE below 0.005, showing resilience. Additionally, with erroneous initial SOC values (e.g., 0.85 instead of 1.0), the algorithm converges within a few hundred seconds, with steady-state error under 1.5%. These tests underscore the reliability of our method for li-ion batteries in practical scenarios.
The weighting scheme in WMISAEKF plays a key role. To illustrate, consider a segment where current spikes cause large innovations. The weights adjust to give higher importance to these innovations, preventing underestimation. The time weight decays exponentially with lag, so recent data dominates, while the error weight amplifies significant deviations. This dual weighting is novel and tailored for li-ion batteries, where SOC dynamics are nonlinear. The multi-innovation length $p$ is tuned; we find $p=3$ to $5$ optimal, balancing history use and computational load. The algorithm complexity is $O(np^2)$ for $n$ states, which is manageable for real-time implementation on embedded systems in electric vehicles.
In conclusion, we propose a comprehensive approach for SOC estimation in li-ion batteries, integrating MIRLS for parameter identification and WMISAEKF for state estimation. The improved Sage-Husa adaptation prevents divergence, and the weighted multi-innovation mechanism enhances accuracy. Simulations under UDDS conditions confirm superior performance over conventional methods. This work contributes to battery management systems by providing a robust, accurate, and adaptive solution. Future directions include extending to joint SOC and SOH estimation, incorporating thermal effects, and testing on larger battery packs. The li-ion battery remains central to energy storage, and advanced estimation techniques like WMISAEKF will support its sustainable use in electric vehicles and renewable energy integration.
To further elaborate on the model, the second-order RC ECM is a simplification; higher-order models could be used but increase complexity. For li-ion batteries, the second-order often suffices for SOC estimation, as shown by our results. The parameters $R_0$, $R_1$, $R_2$, $C_1$, $C_2$ represent physical phenomena: $R_0$ models immediate voltage drop, while RC pairs model polarization delays. These vary with SOC, temperature, and aging, so online identification is essential. Our MIRLS method updates parameters recursively, with the innovation length $p$ enhancing robustness. The update equations for MIRLS can be derived from minimizing a cost function $J(\theta) = \| Y(p,k) – \Phi(p,k) \theta \|^2$. The solution yields the recursive form above, which we implement in simulation.
For SOC estimation, the EKF linearization uses Jacobians. In our case, the system is mildly nonlinear due to the SOC-UOCV curve. The Jacobian $C_k$ includes $\frac{\partial U_{ocv}}{\partial SOC}$, computed from the polynomial derivative: $\frac{\partial U_{ocv}}{\partial SOC} = \sum_{i=1}^n i p_i SOC^{i-1}$. This varies with SOC, adding nonlinearity. The WMISAEKF handles this through iterative updates. The adaptive noise covariances adjust to changes in battery behavior, such as increased noise during high load. Our modification to Sage-Husa ensures $Q_k$ and $R_k$ remain positive definite, critical for filter stability. The weighting method for multi-innovation is inspired by exponential weighting in time series; we combine it with error-based weighting for li-ion batteries, where large innovations may indicate model mismatches.
In simulation, we use MATLAB to implement the algorithms. The UDDS current profile is applied to a battery model with parameters from real data. The true SOC is calculated using ampere-hour integration with a high-precision current sensor model. Estimation errors are computed at each step. We also test computational time; WMISAEKF adds modest overhead compared to EKF but is faster than UKF due to fewer sigma points. For real-time applications, the algorithm can be optimized by reducing $p$ or using fixed-point arithmetic.
Overall, this research highlights the importance of accurate SOC estimation for li-ion batteries. Our WMISAEKF method offers a practical solution with demonstrated gains. As li-ion battery technology evolves, such advanced algorithms will be integral to smart battery management, enabling longer life and safer operation. We encourage further exploration in this field, particularly for aging batteries and extreme conditions.