With the rapid advancement of the new energy vehicle industry, lithium-ion power batteries, as the core energy supply component of electric vehicles, play a decisive role in vehicle safety, reliability, and service life. In battery management systems (BMS), the state of charge (SOC) reflects the remaining available energy of the battery, while the state of health (SOH) characterizes battery aging and remaining usable capacity. Accurate estimation of SOC and SOH is essential for ensuring the safe and efficient operation of power batteries. This thesis focuses on the joint estimation of SOC and SOH for a selected 18650 lithium-ion power battery and conducts a systematic study on modeling, parameter identification, and advanced state estimation algorithms. The main contents of this work are summarized as follows.
The study begins with an analysis of the electrochemical operating principles of lithium-ion batteries. An experimental test platform was established using high-precision battery testing equipment, including the Neware WO-13329-166 battery test system and the Jiabao GDS-100A temperature chamber. Maximum available capacity tests and hybrid pulse power characteristic (HPPC) tests were conducted to obtain T–SOC–OCV characteristic curves under different health states and temperature conditions. The dynamic driving cycle experiments under the World Light Vehicle Test Procedure (WLTP) and the China Light-duty Vehicle Test Cycle for passenger cars (CLTC-P) were performed, providing experimental data support for model development and algorithm validation. The experimental battery specifications are listed in Table 1.
| Parameter | Specification |
|---|---|
| Battery type | LiNCM18650 |
| Nominal capacity | 2000 mAh |
| Nominal voltage | 3.7 V |
| Upper cut-off voltage | 4.2 ± 0.5 V |
| Lower cut-off voltage | 2.6 V |
| Operating temperature | Charge: 0 °C–45 °C; Discharge: −10 °C–60 °C |
In the capacity characterization experiments, five temperature points (0 °C, 10 °C, 20 °C, 30 °C, 40 °C) were selected for batteries at three health levels (SOH = 100%, 95%, 90%). The results demonstrate that temperature decrease leads to reduced available capacity due to decreased material activity and increased internal resistance. Similarly, battery aging causes capacity reduction and increased voltage drop rate, which directly manifests as SOH degradation.
The HPPC tests were conducted to establish the relationship between open circuit voltage (OCV) and SOC. A sixth-order polynomial was fitted to represent the OCV-SOC relationship. At 20 °C and SOH = 100%, the fitted OCV-SOC function is:
$$OCV = 7449.25 \cdot SOC^6 + 6065.14 \cdot SOC^5 – 2685.09 \cdot SOC^4 + 657.85 \cdot SOC^3 – 96.70 \cdot SOC^2 + 14.02 \cdot SOC + 27.18$$
The T-SOC-OCV curves exhibit a characteristic “two-plateau and one-steep-slope” pattern. In the low SOC region, voltage changes most dramatically due to significant lithium-ion concentration variations. In the 0.2–0.8 SOC range, the curve slope decreases as the battery undergoes a two-phase coexistence transition. At high SOC levels, the slope increases again as electrode materials approach saturation. At lower temperatures, the OCV decreases more rapidly because of increased internal resistance. For aged batteries (SOH = 90%), the OCV-SOC curve becomes flatter as the effective active material decreases, weakening the phase-transition capability of electrode materials.
For the modeling aspect of this research, four types of equivalent circuit models were evaluated: the Rint model, Thevenin model, PNGV model, and n-order RC model. After comprehensive comparison considering computational cost and accuracy requirements, the second-order RC equivalent circuit model was selected as the battery state estimation model. This model consists of an ideal voltage source, an ohmic resistance, and two RC parallel networks that describe the electrochemical polarization and concentration polarization processes respectively. Based on Kirchhoff’s laws, the model state equations are:
$$U_t = U_{OCV} – U_1 – U_2 – I \cdot R_0$$
$$C_1 \frac{dU_1}{dt} = I – \frac{U_1}{R_1}$$
$$C_2 \frac{dU_2}{dt} = I – \frac{U_2}{R_2}$$
To address the unknown parameters in the model, online parameter identification methods based on least squares were studied. The traditional recursive least squares (RLS) algorithm suffers from data saturation, which weakens the tracking capability of new data. The forgetting factor recursive least squares (FFRLS) algorithm mitigates this issue by assigning different weights to new and old data. However, the fixed forgetting factor cannot simultaneously satisfy stability and dynamic response requirements under complex operating conditions. Therefore, an improved adaptive forgetting factor recursive least squares (IAFFRLS) algorithm was adopted for online parameter identification. The adaptive forgetting factor is calculated as:
$$\lambda(k) = \lambda_{max} – (\lambda_{max} – \lambda_{min}) \cdot \frac{2}{\pi} \cdot \arctan\left(\mu(k)^n\right)$$
$$\mu(k) = round\left(\frac{e_L(k)}{e_0}\right)$$
where $\lambda_{max}$ and $\lambda_{min}$ are the expected maximum and minimum values, $e_L(k)$ is the model terminal voltage error, $e_0$ is the reference error, and $n$ is the power exponent (typically 2 or 4). The IAFFRLS algorithm adjusts the forgetting factor dynamically based on the estimation error. When the error is smaller than the reference value, the forgetting factor rapidly approaches its maximum; conversely, when the error is large, it decreases toward the minimum value. This mechanism enhances the convergence ability when the system has not yet reached a stable state.
The effectiveness of the IAFFRLS algorithm was validated under both WLTP and CLTC-P driving cycles. The voltage prediction results were compared against the FFRLS and AFFRLS algorithms. Table 2 presents the terminal voltage prediction errors under different operating conditions.
| Operating Condition | Identification Algorithm | RMSE (V) | MAE (V) |
|---|---|---|---|
| CLTC-P | FFRLS | 0.0040331 | 0.0082508 |
| CLTC-P | AFFRLS | 0.0030431 | 0.0060746 |
| CLTC-P | IAFFRLS | 0.0029942 | 0.0034494 |
| WLTP | FFRLS | 0.0024512 | 0.0112081 |
| WLTP | AFFRLS | 0.0022845 | 0.0081035 |
| WLTP | IAFFRLS | 0.0021782 | 0.0052701 |
The results demonstrate that the IAFFRLS algorithm produces the smallest RMSE and MAE values in both driving cycles, with terminal voltage errors fluctuating within −0.02 V to 0.02 V. This confirms that the proposed algorithm offers superior stability, real-time tracking capability, and better adaptability to complex operating conditions, making the identified parameters more representative of the actual battery characteristics.
For SOC estimation, the study first analyzed the linear Kalman filter and its limitations in nonlinear systems. The Kalman filter operates through a “predict-correct” iterative process, optimally estimating system states under minimum mean square error criteria. However, its applicability to the EV battery pack is limited because lithium-ion batteries exhibit strong nonlinear and time-varying characteristics. The unscented Kalman filter (UKF) was introduced to better handle these nonlinearities. UKF employs an unscented transform to select sigma points that represent the state distribution, mapping these through the nonlinear system to capture the posterior mean and covariance accurately without linearization errors.
The discretized state-space equations of the battery model are:
$$\begin{bmatrix} SOC_k \\ U_{1,k} \\ U_{2,k} \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & e^{-T/\tau_1} & 0 \\ 0 & 0 & e^{-T/\tau_2} \end{bmatrix} \begin{bmatrix} SOC_{k-1} \\ U_{1,k-1} \\ U_{2,k-1} \end{bmatrix} + \begin{bmatrix} -\frac{T}{Q_n} \\ R_1(1-e^{-T/\tau_1}) \\ R_2(1-e^{-T/\tau_2}) \end{bmatrix} I_{k-1}$$
$$U_{t,k} = U_{OCV,k} – U_{1,k} – U_{2,k} – I_k R_0$$
where $Q_n$ is the nominal capacity, $\tau_1 = R_1 C_1$, $\tau_2 = R_2 C_2$, and $T$ is the sampling interval.
To further improve estimation accuracy, the adaptive unscented Kalman filter (AUKF) was developed. This approach addresses the limitation of UKF where process and measurement noise covariances are treated as constants. In the AUKF, these covariances are updated online based on output residuals, allowing the filter to adapt to changing noise characteristics in real-world environments. The innovation is defined as:
$$e_k = Y_k – \hat{Y}_{k|k-1}$$
with the noise covariance update:
$$R_k = L_k^T L_k + \sum_{i=0}^{2n} W^{(i)}_c (Y_{k|k-1}^{(i)} – \hat{Y}_{k|k-1})(Y_{k|k-1}^{(i)} – \hat{Y}_{k|k-1})^T$$
$$Q_k = K_k L_k K_k^T + \frac{1}{L}\sum_{j=k-L+1}^{k} e_j e_j^T$$
Finally, combining the adaptive filtering concept with multi-innovation identification theory, the multi-innovation adaptive unscented Kalman filter (MIAUKF) algorithm was proposed for high-precision SOC estimation. Instead of using only the current innovation, MIAUKF constructs an innovation matrix from historical data:
$$E_{p,k} = \begin{bmatrix} e_k \\ e_{k-1} \\ \vdots \\ e_{k-p+1} \end{bmatrix} = \begin{bmatrix} Y_k – \hat{Y}_{k|k-1} \\ Y_{k-1} – \hat{Y}_{k-1|k-2} \\ \vdots \\ Y_{k-p+1} – \hat{Y}_{k-p+1|k-p} \end{bmatrix}$$
The Kalman gain is correspondingly expanded to a gain matrix:
$$K_{p,k} = [K_k \quad K_{k-1} \quad \cdots \quad K_{k-p+1}]$$
and the state update becomes:
$$\hat{X}_{k|k} = \hat{X}_{k|k-1} + K_{p,k} E_{p,k}$$
$$P_{k|k} = P_{k|k-1} – p \cdot K_{p,k} P_{yy,k|k-1} K_{p,k}^T$$
The verification results under WLTP and CLTC-P conditions are summarized in Table 3. The MIAUKF algorithm consistently outperforms AUKF in both estimation accuracy and error stability. Under WLTP conditions, the RMSE and MAE of MIAUKF are 0.35565% and 0.30702% respectively, representing improvements of 63.90% and 41.42% over AUKF. Under CLTC-P conditions, where frequent start-stop operations and rapid acceleration/deceleration cause significant current transients, the MIAUKF achieves RMSE and MAE of 0.78568% and 0.67732%, improving over AUKF by 19.44% and 17.35% respectively.
| Operating Condition | Algorithm | RMSE (%) | MAE (%) |
|---|---|---|---|
| CLTC-P | AUKF | 0.97534 | 0.81952 |
| CLTC-P | MIAUKF | 0.78568 | 0.67732 |
| WLTP | AUKF | 0.98532 | 0.52412 |
| WLTP | MIAUKF | 0.35565 | 0.30702 |
A comparative study was also conducted to evaluate the second-order RC equivalent circuit model against the Rint and first-order RC models for SOC estimation using the MIAUKF algorithm under WLTP conditions. The results are presented in Table 4. The Rint model, lacking polarization components, shows significant fluctuations in early discharge stages with RMSE of 2.41% and MAE of 1.73%. The first-order RC model improves upon this with RMSE of 1.81% and MAE of 1.56%. The second-order RC model yields the best performance with RMSE of 0.43% and MAE of 0.58%, representing accuracy improvements of 82.2% and 66.47% over the Rint model. The superior performance stems from its two polarization branches that separately describe the fast and slow polarization processes within the battery.
| Model Type | RMSE (%) | MAE (%) |
|---|---|---|
| Rint model | 2.41 | 1.73 |
| First-order RC model | 1.81 | 1.56 |
| Second-order RC model | 0.43 | 0.58 |
Robustness analysis was performed under CLTC-P conditions with different initial SOC values (0.3, 0.5, 0.8) to evaluate the convergence capability of the MIAUKF algorithm. The results demonstrate that regardless of the initial SOC deviation, the algorithm converges to the true value within a short period. When the initial SOC is set to 0.8, the estimation curve approaches the true value rapidly. For larger initial deviations, the algorithm initially shows notable errors but gradually converges to the true SOC as time progresses. This indicates that the initial error primarily affects the convergence process rather than the final estimation result. After convergence, all estimation curves align closely with the true values without oscillation or divergence, demonstrating excellent robustness and practical applicability.
For SOH estimation, the study first analyzed the influencing factors and emphasized that frequent SOH estimation is unnecessary. SOH is defined based on the maximum available capacity:
$$SOH = \frac{C_k}{C_N} \times 100\%$$
where $C_N$ is the nominal capacity and $C_k$ is the maximum available capacity at time k. The battery aging process exhibits significant time-varying and irreversible characteristics, with capacity decay and internal resistance increase accumulating gradually over long-term use. Since SOH changes slowly compared to SOC, it requires less frequent estimation, reducing computational burden while maintaining estimation accuracy.
In the single-time-scale approach, a MIAUKF filter was used to estimate battery capacity directly. The state and observation equations were established as:
$$C_k = C_{k-1} + r_k$$
$$U_k = U_{OCV,k} – U_{1,k} – U_{2,k} – I_k R_0 + e_k$$
where $C_k$ represents the system state variable (capacity), $r_k$ and $e_k$ denote process noise and measurement noise respectively. Through successive iterations and corrections, the SOH estimation gradually converges to the true value.
Considering the coupling relationship between SOC and SOH and their different time-scale characteristics, multi-time-scale theory was introduced to construct a joint estimation framework using the double multi-innovation adaptive unscented Kalman filter (DMIAUKF). The nonlinear discrete system with multi-time-scale can be expressed as:
$$X_{k,l+1} = F(x_{k,l}, u_k, \theta_k, w_{k,l})$$
$$Y_{k,l} = G(x_{k,l}, u_k, \theta_k, v_{k,l})$$
$$\theta_{k+1} = \theta_k + r_k$$
where k and l represent the macro and micro time scales respectively. When the micro scale l reaches the threshold L, the system transitions to the macro scale k. The relationship between scales is:
$$L = \frac{frequency_{SOC}}{frequency_{SOH}}$$
In the proposed DMIAUKF architecture, two MIAUKF filters operate cooperatively: one estimates SOC at the micro time scale in real-time, while the other estimates SOH at the macro time scale. The SOC estimation provides dynamic information for capacity update, while the capacity estimation feeds back to correct the SOC estimation, creating an information interaction mechanism that enhances overall estimation accuracy.
The evaluation results under WLTP and CLTC-P conditions demonstrate the advantages of the joint estimation approach. Table 5 presents the SOC estimation errors comparing AUKF, MIAUKF, and DMIAUKF algorithms.
| Operating Condition | Algorithm | RMSE (%) | MAE (%) |
|---|---|---|---|
| CLTC-P | AUKF | 0.97534 | 0.81952 |
| CLTC-P | MIAUKF | 0.78568 | 0.67732 |
| CLTC-P | DMIAUKF | 0.60527 | 0.50136 |
| WLTP | AUKF | 0.98532 | 0.52412 |
| WLTP | MIAUKF | 0.35565 | 0.30702 |
| WLTP | DMIAUKF | 0.12883 | 0.10793 |
Under CLTC-P conditions, the DMIAUKF algorithm achieves RMSE and MAE of 0.60527% and 0.50136%, improving accuracy by 22.96% and 25.98% compared to the single-time-scale MIAUKF. Under WLTP conditions, the DMIAUKF achieves even more significant improvement with RMSE of 0.12883% and MAE of 0.10793%, representing accuracy gains of 63.78% and 64.85% over MIAUKF, and 86.93% and 79.41% over AUKF. The improved performance stems from the dynamic consideration of battery aging and capacity changes, allowing SOC estimation to adapt to parameter variations.
SOH estimation results were verified for different true SOH values (100%, 95%, 90%) under both WLTP and CLTC-P conditions. The estimation curves track the true values closely across all cases despite small fluctuations during frequent condition switches. This confirms both the rationality of the second-order RC model and the effectiveness of the estimation approach. Notably, as SOH decreases, the fluctuation amplitude of the estimation curves increases slightly, which is attributed to the more significant internal parameter changes in aged batteries making the SOH estimation more sensitive to operating conditions and noise. Minor overshoots beyond 100% in some SOH estimates are considered normal since fresh batteries can show slight measurement variations during initial use.
The necessity of infrequent SOH estimation was examined by evaluating different time scales L. The relationship between the time scale L and SOC estimation accuracy was found to be non-monotonic. When L increases from 20 to 40, the maximum error decreases by approximately 0.3%. However, when L exceeds 60, the error rebounds; for example, increasing L from 80 to 150 causes a negative offset with the maximum error increasing by about 0.2%. This phenomenon occurs because excessively frequent SOH updates may interfere with battery operation, negatively affecting estimation results. Therefore, an optimal time scale of L = 60 was selected, achieving high SOC accuracy while maintaining reasonable computational costs.
In summary, this study systematically investigates the SOC and SOH estimation problem for EV battery packs through a combination of experimental testing, model construction, parameter identification, and advanced filtering algorithm design. The key contributions of this research include: establishing a comprehensive experimental foundation with T-SOC-OCV characteristic curves across different health states and temperatures; developing an improved adaptive forgetting factor recursive least squares algorithm for accurate online parameter identification; proposing the MIAUKF algorithm that integrates multi-innovation theory with adaptive unscented Kalman filtering for high-precision SOC estimation; and constructing a multi-time-scale DMIAUKF framework for the coordinated estimation of SOC and SOH that achieves superior accuracy and robustness under complex operating conditions.
For future research directions, several aspects merit further investigation. First, this study focuses on single battery cells, whereas real automotive applications involve battery packs with multiple cells connected in series and parallel, where cell-to-cell variations must be considered. Extending the proposed algorithm to battery pack level estimation is a significant challenge. Second, the integer-order second-order RC model has limitations in fully capturing temperature gradient effects, cell consistency differences, and localized aging characteristics under complex operating conditions; fractional-order models may provide better accuracy. Third, the robustness analysis can be further enhanced by incorporating experimental data under various temperature conditions and noise levels to systematically validate the algorithm’s anti-interference capability. Finally, this research is based on offline experimental data and simulation; real-time validation on embedded BMS hardware platforms would provide valuable insights for engineering deployment of the proposed joint estimation algorithm.
The findings of this study provide a solid theoretical basis and technical reference for state perception and management of EV battery packs, contributing to the safe, efficient, and reliable operation of electric vehicles.