Accurate State of Charge (SOC) estimation is fundamental for the safe and efficient operation of lithium-ion batteries, particularly in demanding applications like electric vehicles. Traditional model-based estimation methods often rely on precise parameter identification for the battery’s equivalent circuit model. However, conventional identification techniques, which typically estimate all parameters simultaneously, frequently suffer from issues like parameter coupling and convergence to local optima. This inaccuracy in the foundational model parameters cascades into significant errors for subsequent state estimations, such as SOC and State of Health (SOH). Therefore, enhancing the precision and robustness of model parameter identification is a critical prerequisite for reliable battery management.
This paper proposes a novel methodology for the joint estimation of SOC and SOH for lithium-ion battery systems, built upon a refined three-stage parameter identification strategy. The core innovation lies in decoupling the battery’s internal dynamic characteristics to achieve more accurate and stable parameter identification, which then feeds into a dual time-scale estimator.
1. Proposed Three-Stage Parameter Identification Framework
The proposed framework significantly improves upon traditional single-step parameter identification by strategically decomposing the process according to the electrochemical time constants inherent in a lithium-ion battery. The workflow is illustrated in the following diagram and elaborated in the subsequent sections.
Stage 1: Model Decomposition into Fast and Slow Dynamics
The foundation is a second-order RC equivalent circuit model, a common compromise between complexity and accuracy for lithium-ion battery modeling. The model comprises an ohmic resistor (R0), two RC parallel networks (R1C1 and R2C2) representing polarization dynamics, and a voltage source (UOC(SOC)) representing the open-circuit voltage. The state-space equations are:
$$
\begin{aligned}
U_t &= U_{OC} – U_1 – U_2 – I R_0 \\
\frac{dU_1}{dt} &= -\frac{1}{R_1 C_1}U_1 + \frac{1}{C_1}I \\
\frac{dU_2}{dt} &= -\frac{1}{R_2 C_2}U_2 + \frac{1}{C_2}I
\end{aligned}
$$
where \(U_t\) is the terminal voltage and \(I\) is the load current. The key insight is to separate the model’s response into Fast Dynamics (FD) and Slow Dynamics (SD) modules. The FD module captures the immediate ohmic drop and the faster electrochemical polarization (R0, R1, C1). The SD module captures the slower concentration polarization and the SOC-dependent OCV (R2, C2, UOC). This separation mitigates the negative effects of parameter coupling during identification.
Stage 2: Parameter Identification for the Fast Dynamics Module
For the FD module, a Forgetting Factor Recursive Least Squares (FFRLS) algorithm is employed. FFRLS is well-suited for tracking time-varying parameters in dynamic systems. The discrete-time transfer function for the FD module voltage \(U_{FD} = I R_0 + U_1\) is derived and used to formulate a standard linear regression model. The recursive equations for FFRLS are:
$$
\begin{aligned}
\hat{\boldsymbol{\theta}}(k) &= \hat{\boldsymbol{\theta}}(k-1) + \mathbf{K}(k)\left[y(k) – \boldsymbol{\phi}^T(k)\hat{\boldsymbol{\theta}}(k-1)\right] \\
\mathbf{K}(k) &= \frac{\mathbf{P}(k-1)\boldsymbol{\phi}(k)}{\lambda + \boldsymbol{\phi}^T(k)\mathbf{P}(k-1)\boldsymbol{\phi}(k)} \\
\mathbf{P}(k) &= \frac{1}{\lambda}\left[\mathbf{I} – \mathbf{K}(k)\boldsymbol{\phi}^T(k)\right]\mathbf{P}(k-1)
\end{aligned}
$$
where \(\hat{\boldsymbol{\theta}} = [R_0, R_1, C_1]^T\) is the parameter vector, \(y(k)\) is the measured FD voltage, \(\boldsymbol{\phi}(k)\) is the regression vector composed of past current and voltage data, \(\mathbf{K}(k)\) is the gain matrix, \(\mathbf{P}(k)\) is the covariance matrix, and \(\lambda\) is the forgetting factor (set to 0.99). This stage provides accurate, real-time estimates for R0, R1, and C1.
Stage 3: Parameter Identification for the Slow Dynamics Module with Optimization
The SD module parameters (UOC, R2, C2) are identified using an Adaptive Extended Kalman Filter (AEKF). The AEKF provides a robust framework for nonlinear state estimation and can adapt its process and measurement noise covariance matrices online. The state vector for this filter is \(\mathbf{x}_{SD} = [U_{OC}, U_2]^T\). However, to prevent the AEKF from settling into a local optimum and to further refine the parameters, its outputs are used as the initial population for a metaheuristic optimizer, the Zebra Optimization Algorithm (ZOA).
ZOA mimics the foraging and defense behaviors of zebra herds. It operates in two phases: a “foraging phase” where candidate solutions move towards the historically best solution, and a “defense phase” where they stochastically choose to either escape from or attack a perceived threat (modeled to encourage exploration). The algorithm updates candidate parameter sets \(\mathbf{X}_i = [U_{OC}^{(i)}, R_2^{(i)}, C_2^{(i)}]\) and selects the set that minimizes the voltage prediction error. This hybrid AEKF-ZOA approach ensures highly accurate and globally optimized parameters for the SD module.
2. Multi-Time Scale Joint SOC-SOH Estimation Using DAEKF
The accurately identified parameters from the three-stage process are then fed into a joint estimator. Crucially, SOC and SOH evolve on different time scales: SOC can change rapidly with current flow (micro-scale), while battery capacity degrades very slowly over charge-discharge cycles (macro-scale). A Dual Adaptive Extended Kalman Filter (DAEKF) architecture is adopted to leverage this multi-time scale characteristic, reducing computational burden and improving estimation accuracy.
The joint state-space model for the lithium-ion battery is defined with an augmented state vector \(\mathbf{x} = [SOC, U_1, U_2, Q]^T\), where \(Q\) is the maximum available capacity (inversely related to SOH). The system and measurement equations are:
$$
\begin{aligned}
\mathbf{x}_k &= \mathbf{A}_k \mathbf{x}_{k-1} + \mathbf{B}_k I_{k-1} + \mathbf{w}_{k-1} \\
U_{t,k} &= U_{OC}(SOC_k) – U_{1,k} – U_{2,k} – I_k R_0 + v_k
\end{aligned}
$$
where \(\mathbf{w}\) and \(v\) are process and measurement noise, respectively. The matrix \(\mathbf{A}_k\) is a function of the identified parameters R1, C1, R2, C2, and \(Q\).
The DAEKF runs two AEKF filters in parallel with different execution rates:
Micro-scale AEKF (for SOC): Executes at every sampling time step (e.g., 1s). It updates the SOC, U1, and U2 states using the latest current/voltage measurements and the most recent capacity estimate \(Q\) from the macro-scale filter.
Macro-scale AEKF (for SOH/Capacity): Executes at a much slower rate, defined by a time-scale factor \(L\). It updates the capacity state \(Q\) by assimilating the accumulated Coulomb count and the SOC change estimated by the micro-scale filter over the last \(L\) seconds.
This structure ensures that the rapidly changing SOC is tracked in real-time, while the slowly varying capacity is updated efficiently without unnecessary computation, leading to a stable and accurate joint estimation.
3. Experimental Validation and Results
The proposed algorithm was validated using experimental data from two different lithium-ion battery cells: the INR18650-20R and the ICR18650-26F. Tests were conducted under various dynamic driving profiles (US06, FUDS, DST, NEDC, UDDS, HWFET) and at different ambient temperatures (-20°C, 25°C, 40°C) to evaluate robustness.
3.1 Determination of Optimal Time-Scale Factor (L)
A preliminary study was conducted to determine the optimal update interval \(L\) for the macro-scale capacity estimator. For the INR18650-20R cell at 25°C under US06 cycling, different values of \(L\) were tested. The results, summarized in the table below, show that \(L = 60\) seconds provides the lowest Root Mean Square Error (RMSE) for both SOC and SOH estimation, balancing responsiveness and stability.
| Time Scale L (s) | SOC RMSE | SOH RMSE |
|---|---|---|
| 30 | 0.0077 | 0.0190 |
| 60 | 0.0069 | 0.0148 |
| 90 | 0.0087 | 0.0215 |
| 120 | 0.0105 | 0.0390 |
3.2 Performance Under Different Temperatures and Profiles
The proposed “Joint Algorithm” (Three-stage ID + DAEKF) was compared against a standard DAEKF with conventional parameter identification and a basic DEKF. The tables below present the average and maximum absolute errors for SOC estimation under challenging conditions for the INR18650-20R cell.
Table 2: SOC Estimation Error at Different Temperatures (US06 Profile)
| Temperature | Error Metric | Proposed Joint Algorithm | DAEKF | DEKF |
|---|---|---|---|---|
| -20°C | Avg. Error (%) | 0.32 | 0.91 | 1.25 |
| Max. Error (%) | 0.62 | 1.70 | 2.75 | |
| 25°C | Avg. Error (%) | 0.38 | 0.76 | 1.07 |
| Max. Error (%) | 1.30 | 1.42 | 2.92 | |
| 40°C | Avg. Error (%) | 0.47 | 0.85 | 1.03 |
| Max. Error (%) | 1.29 | 1.58 | 3.21 |
Table 3: SOC Estimation Error for Different Driving Profiles at 25°C
| Driving Profile | Error Metric | Proposed Joint Algorithm | DAEKF | DEKF |
|---|---|---|---|---|
| US06 | Avg. Error (%) | 0.38 | 0.76 | 1.07 |
| Max. Error (%) | 1.30 | 1.42 | 2.92 | |
| FUDS | Avg. Error (%) | 0.26 | 0.91 | 1.13 |
| Max. Error (%) | 0.82 | 1.42 | 3.28 | |
| DST | Avg. Error (%) | 0.41 | 0.94 | 1.06 |
| Max. Error (%) | 1.05 | 1.41 | 2.31 |
The results consistently demonstrate the superiority of the proposed method. Across all temperatures and dynamic profiles, it achieves the lowest average and maximum SOC errors. For instance, at 25°C under the FUDS profile, the proposed algorithm reduced the average SOC error by 0.65% and 0.87% compared to DAEKF and DEKF, respectively. Furthermore, the algorithm provided stable and accurate SOH estimation, converging reliably to the reference capacity value.
3.3 Validation on a Second Battery Type
To demonstrate generalizability, the algorithm was also tested on data from a different lithium-ion battery model, the ICR18650-26F. The optimal time-scale factor was again found to be L=60 seconds. The proposed method maintained superior SOC estimation accuracy under NEDC, UDDS, and HWFET profiles at room temperature, and delivered precise SOH tracking, confirming its robustness across different battery chemistries and usage patterns.
4. Conclusion
This paper presented a comprehensive framework for high-accuracy joint SOC-SOH estimation in lithium-ion battery systems. The key contribution is a three-stage parameter identification strategy that intelligently decouples the fast and slow dynamic processes within the battery. By employing FFRLS for the fast dynamics and a hybrid AEKF-ZOA approach for the slow dynamics, the method achieves precise, globally optimized model parameters while mitigating coupling issues. These refined parameters are then utilized in a multi-time scale DAEKF, which efficiently and accurately estimates the rapidly changing SOC and the slowly degrading SOH in a coupled manner.
Extensive experimental validation under diverse temperatures and dynamic load profiles, using two distinct lithium-ion battery types, confirmed the algorithm’s effectiveness, robustness, and generalizability. The proposed method consistently outperformed conventional DAEKF and DEKF approaches, offering significantly lower estimation errors. This work provides a promising solution for enhancing the reliability and longevity of battery management systems in real-world applications.