Accurate estimation of the State of Health (SOH) of lithium ion batteries is critical for ensuring the safety, reliability, and longevity of battery systems in applications such as electric vehicles and energy storage. Traditional data-driven SOH estimation methods often suffer from inadequate uncertainty quantification, overfitting, and dependency on historical data from the same battery, limiting their practical utility. To address these challenges, we propose a novel framework that integrates the Egret Swarm Optimization Algorithm (ESOA) with Gaussian Process Regression (GPR) to achieve high-precision SOH estimation with robust uncertainty intervals.
1. Health Feature Extraction and Selection
The degradation of lithium ion batteries manifests in measurable electrical parameters during charging and relaxation phases. We extracted health features (HFs) from voltage, current, and relaxation voltage profiles to capture aging-related characteristics.
1.1 Charging Process Features
From the Constant Current-Constant Voltage (CC-CV) charging phase, eight features were initially considered:
- Constant Current Charge Time (CCCT)
- Constant Voltage Charge Time (CVCT)
- CCCT/CVCT Ratio
- Total Charge Time (TCT)
- Duration Corresponding to Equal Current Interval (DCECI)
- Current Variation Corresponding to Charging Time (CVCCT)
- Voltage Variation Corresponding to Charging Time (VVCCT)
- Duration Corresponding to Equal Voltage Interval (DCEVI)
1.2 Relaxation Process Features
From the 30-minute relaxation voltage signal, six statistical features were derived:
- Voltage Maximum (Voltage Max)
- Voltage Minimum (Voltage Min)
- Voltage Variance (Voltage Var)
- Voltage Average (Voltage Avg)
- Voltage Kurtosis (Voltage Kur)
- Voltage Skewness (Voltage Ske)
1.3 Pearson Correlation Analysis
The Pearson coefficient PXYPXY quantified the linear relationship between each HF and battery capacity:PXY=cov(X,Y)σXσYPXY=σXσYcov(X,Y)
where XX is the battery capacity, YY is the HF, and σσ denotes standard deviation. Features with ∣PXY∣>0.8∣PXY∣>0.8 were retained.
Table 1: Selected Health Features and Correlation Coefficients
| Feature | Description | Avg. Pearson Coefficient |
|---|---|---|
| HF1 (DCEVI) | Charging duration in [3.5 V, 3.95 V] | 0.997 |
| HF2 (DCECI) | Charging duration in [2000 mA, 1250 mA] | 0.999 |
| HF3 (Voltage Max) | Maximum relaxation voltage | 0.989 |
| HF4 (Voltage Drop) | Voltage drop in [0 s, 120 s] relaxation | 0.987 |
2. ESOA-Optimized Gaussian Process Regression
Gaussian Process Regression (GPR) provides probabilistic predictions, but its performance depends critically on hyperparameters. The Egret Swarm Optimization Algorithm (ESOA) was employed to optimize these parameters, enhancing model accuracy and robustness.
2.1 Gaussian Process Regression
GPR models the relationship between inputs x∈R4x∈R4 (HFs) and outputs yy (SOH) as a Gaussian distribution:f(x)∼GP(m(x),k(x,x′))f(x)∼GP(m(x),k(x,x′))
where m(x)=0m(x)=0 (zero-mean assumption) and k(x,x′)k(x,x′) is the squared exponential kernel:k(xi,xj)=σf2exp(−∥xi−xj∥22l2)+σn2δijk(xi,xj)=σf2exp(−2l2∥xi−xj∥2)+σn2δij
Here, σf2σf2, ll, and σn2σn2 are hyperparameters representing signal variance, length scale, and noise variance, respectively.
2.2 Egret Swarm Optimization Algorithm
ESOA mimics the foraging behavior of egrets, balancing exploration and exploitation through three strategies:
- Sit-and-Wait Strategy: Local optimization using gradient descent.
- Random Walk Strategy: Global exploration via stochastic movements.
- Encirclement Strategy: Aggressive pursuit of promising regions.
The position update for the ii-th egret小队 is governed by:xi(t+1)=xi(t)+Δxi(t)xi(t+1)=xi(t)+Δxi(t)
where Δxi(t)Δxi(t) combines gradient-guided updates and stochastic perturbations.
2.3 ESOA-GPR Implementation
- Hyperparameter Tuning: ESOA optimizes σfσf, ll, and σnσn by minimizing the mean squared error (MSE):
MSE=1n∑i=1n(yi−y^i)2MSE=n1i=1∑n(yi−y^i)2
- Model Training: Source-domain data (NCA/NCM batteries) were used for training, with five-fold cross-validation to prevent overfitting.
- Uncertainty Quantification: The 95% confidence interval for SOH estimates was derived from the posterior variance:
CI=y^±1.96σ∗2CI=y^±1.96σ∗2
3. Experimental Validation
3.1 Dataset and Setup
Experiments utilized NCA (LiNiCoAlO22) and NCM (LiNiCoMnO22) lithium ion batteries from Tongji University:
- NCA Batteries: 4 cells, 3.5 Ah, 2.65–4.2 V range.
- NCM Batteries: 4 cells, 3.5 Ah, 2.5–4.2 V range.
Table 2: Battery Specifications
| Parameter | NCA | NCM |
|---|---|---|
| Cathode Material | LiNiCoAlO22 | LiNiCoMnO22 |
| Anode Material | Graphite/Si | Graphite |
| Voltage Range | 2.65–4.2 V | 2.5–4.2 V |
| Capacity | 3.5 Ah | 3.5 Ah |
3.2 Performance Metrics
- Mean Absolute Error (MAE):
MAE=1n∑i=1n∣yi−y^i∣MAE=n1i=1∑n∣yi−y^i∣
- Root Mean Squared Error (RMSE):
RMSE=1n∑i=1n(yi−y^i)2RMSE=n1i=1∑n(yi−y^i)2
3.3 Results
Table 3: Comparison of SOH Estimation Errors
| Model | NCA Batteries | NCM Batteries | ||
|---|---|---|---|---|
| MAE (%) | RMSE (%) | MAE (%) | RMSE (%) | |
| LSTM | 0.65 | 0.78 | 0.52 | 0.66 |
| GRU | 0.94 | 0.99 | 0.55 | 0.71 |
| BP Neural Network | 0.34 | 0.46 | 0.35 | 0.48 |
| GPR | 0.35 | 0.50 | 0.25 | 0.34 |
| ESOA-GPR | 0.22 | 0.20 | 0.14 | 0.16 |
The ESOA-GPR model achieved a 57.69% reduction in MAE and 58.82% reduction in RMSE for NCA batteries compared to standard GPR. For NCM batteries, improvements reached 76.12% (MAE) and 78.79% (RMSE).
Table 4: Computational Efficiency (Inference Time)
| Model | NCA (ms) | NCM (ms) |
|---|---|---|
| LSTM | 12.5 | 13.2 |
| GRU | 11.8 | 12.9 |
| GPR | 16.4 | 14.3 |
| ESOA-GPR | 16.2 | 15.6 |
4. Conclusion
This study presents a robust framework for lithium ion battery SOH estimation by integrating ESOA-optimized GPR with data-driven health features. Key contributions include:
- Decoupled Training/Testing: Models trained on source-domain batteries generalize effectively to target-domain batteries, enabling full lifecycle prediction.
- Uncertainty Quantification: The 95% confidence intervals prevent safety risks from SOH overestimation.
- Superior Accuracy: ESOA-GPR reduces MAE and RMSE by >50% compared to state-of-the-art models.
Future work will focus on simplifying feature extraction under variable temperatures and enhancing real-time applicability. The proposed method advances lithium ion battery management systems by delivering precise, reliable, and safe SOH estimates.