Lithium-ion batteries have become indispensable components of our modern technological landscape. Their high energy density, low self-discharge rate, and extended cycle life make them the power source of choice for applications ranging from portable electronics and electric vehicles to large-scale renewable energy storage systems. The reliable and safe operation of these systems is intrinsically linked to the health of the lithium-ion battery. However, as a consumable component, every lithium-ion battery undergoes inevitable performance degradation over time, characterized primarily by a gradual decline in its usable capacity. This aging process, if not properly monitored and managed, can lead to reduced operational range, unexpected failures, and even safety hazards.
Therefore, accurately predicting the Remaining Useful Life (RUL) of a lithium-ion battery—the number of charge-discharge cycles or operational time left before it falls below a critical performance threshold—is a paramount task. Precise RUL estimation forms the cornerstone of advanced Battery Management Systems (BMS), enabling condition-based maintenance, optimizing usage strategies to prolong service life, and providing early warnings for battery replacement, thereby enhancing system reliability, safety, and economic efficiency.
Numerous methods have been developed to tackle the challenge of lithium-ion battery prognostics, broadly falling into model-driven, data-driven, and hybrid categories. Model-driven approaches rely on mathematical representations of the degradation process. While electrochemical and equivalent circuit models offer deep physical insight, they are often complex and computationally expensive. Empirical models, derived from observed data patterns, provide a simpler alternative. Common examples include exponential and polynomial models that approximate capacity fade over cycles. Data-driven methods, leveraging machine learning and deep learning algorithms, learn the mapping between operational data (voltage, current, temperature) and health indicators without explicit physical models. Filtering techniques, particularly Particle Filter (PF), are a popular model-driven approach for RUL prediction as they can handle nonlinear state-space models and provide a probabilistic estimate of the remaining life.
Despite progress, significant challenges remain. A primary limitation is the reliance on a single, fixed degradation model. The capacity fade of a lithium-ion battery is a complex, multi-stage process that may exhibit linear, nonlinear, and accelerated degradation behaviors under different operating conditions or at different life stages. A single exponential or polynomial model often lacks the flexibility to generalize across these varied patterns, leading to reduced prediction accuracy and robustness when conditions change. Furthermore, the standard Particle Filter suffers from the inherent problem of particle degeneracy, where most particles carry negligible weight after a few iterations, impairing the quality of the state estimate and, consequently, the RUL prediction.
To address these dual challenges of model inflexibility and particle degeneracy, this article explores a novel prognostic framework. We propose an Interactive Multiple Model (IMM) framework fused with a Particle Flow Filter (PFF) for the state estimation and RUL prediction of lithium-ion battery. The core idea is to leverage multiple empirical models concurrently, each capturing a different characteristic of battery degradation, and to use an advanced filtering technique that avoids traditional resampling pitfalls.
The Limitation of Single-Model Approaches
Traditional model-based prediction often selects one degradation model a priori. For instance, a double exponential model is popular for its ability to fit the typical curvature of capacity fade:
$$C_k = \alpha_1 \cdot \exp(\alpha_2 \cdot k) + \alpha_3 \cdot \exp(\alpha_4 \cdot k)$$
where $C_k$ is the capacity at cycle $k$, and $\alpha_1, \alpha_2, \alpha_3, \alpha_4$ are model parameters. This model excels at capturing nonlinear decay but may overfit or perform poorly when the degradation is more linear. Conversely, a quadratic polynomial model:
$$C_k = \beta_1 k^2 + \beta_2 k + \beta_3$$
is simpler and can effectively model near-linear or parabolic trends but fails to capture strong nonlinear or saturation dynamics. Another model, the Verhulst or “biological” model, often used in grey system theory, is effective for describing processes that have a gradual then accelerating decay trend, common in some lithium-ion battery aging patterns.
$$C_k = \frac{\gamma_1 C_1}{\gamma_2 C_1 + (\gamma_1 – \gamma_2 C_1) e^{\gamma_1 k}}$$
The choice of model is critical, and an inappropriate choice can lead to significant prediction errors. The table below summarizes the fit of these three models to different lithium-ion battery datasets, illustrating that no single model consistently outperforms the others across all batteries or degradation phases.
| Battery Cell | Model | RMSE | R² Score |
|---|---|---|---|
| Cell A1 | Exponential | 0.0476 | 0.9687 |
| Polynomial | 0.0585 | 0.9527 | |
| Verhulst | 0.0608 | 0.9491 | |
| Cell A2 | Exponential | 0.0159 | 0.9766 |
| Polynomial | 0.0382 | 0.8642 | |
| Verhulst | 0.0383 | 0.8638 |
This variability underscores the need for an adaptive, multi-model strategy that can dynamically weight different degradation hypotheses based on real-time data.
The Proposed Framework: Interactive Multiple Model Particle Flow Filter (IMM-PFF)
Our proposed method, the Interactive Multiple Model Particle Flow Filter (IMM-PFF), is designed to overcome the shortcomings of single-model predictors. It combines the adaptability of a multi-model system with the estimation efficiency of a modern filtering algorithm.
1. Interactive Multiple Model (IMM) Framework
The IMM algorithm is a well-established technique for state estimation in systems that may switch between different behavioral modes. In our context, each “mode” corresponds to a specific capacity degradation model (exponential, polynomial, Verhulst). The IMM runs a bank of filters in parallel—one for each model. Crucially, it calculates the probability that each model is the correct representation of the current degradation phase based on measurement history. The overall state estimate and prediction are then computed as a weighted sum of the estimates from all filters, where the weights are these model probabilities.
The model probability $\mu_j(k)$ for model $j$ at time $k$ is updated using the measurement likelihood $\Lambda_j(k)$:
$$\mu_j(k) = \frac{1}{c} \mu_j(k|k-1) \Lambda_j(k)$$
where $c$ is a normalization constant, and $\mu_j(k|k-1)$ is the predicted model probability based on a Markovian model-switching matrix. This allows the algorithm to seamlessly transition its belief from one model to another as the lithium-ion battery ages, effectively adapting to linear, nonlinear, and accelerated degradation phases.
2. Particle Flow Filter (PFF) for Efficient State Estimation
Within each model’s filter in the IMM bank, we employ a Particle Flow Filter instead of a standard Particle Filter. The PFF addresses particle degeneracy through a fundamentally different mechanism. Instead of using importance sampling and resampling, PFF moves particles continuously from the prior distribution to the posterior distribution along a deterministic “flow” defined by a homotopy parameter $\lambda$.
The core idea is to define a log-homotopy between the prior $p(x)$ and the posterior $p(x|z)$:
$$\log p(x, \lambda) = \log p(x) + \lambda \log p(z|x) – \log K(\lambda)$$
where $\lambda$ smoothly varies from 0 (prior) to 1 (posterior). The particle flow is governed by an ordinary differential equation:
$$\frac{dx}{d\lambda} = f(x, \lambda)$$
The flow $f(x, \lambda)$ is derived by forcing the Fokker-Planck equation for the density $p(x, \lambda)$ to hold. For the Exact Daum-Huang (EDH) flow under Gaussian assumptions, the solution takes a simple form:
$$f(x, \lambda) = A(\lambda)x + b(\lambda)$$
where $A(\lambda)$ and $b(\lambda)$ are computed using gradients of the measurement function and covariance matrices. This process smoothly migrates particles to high-likelihood regions without the sample impoverishment caused by traditional resampling, leading to more accurate and stable parameter estimates for each lithium-ion battery degradation model.
3. Integration for RUL Prediction
The integrated IMM-PFF algorithm operates in a recursive cycle. At each new cycle where capacity is measured:
- Interaction/Mixing: The estimates and covariances from each filter are mixed based on the predicted model probabilities from the previous step.
- Model-Conditional Filtering: Each filter (using PFF) processes the new capacity measurement to update its state estimate (which includes the model parameters and the current capacity state) and compute its likelihood.
- Model Probability Update: The likelihoods from all filters are used to update the model probabilities.
- Combination: The final state estimate (current capacity, degradation parameters) is computed as the probability-weighted sum of all filter estimates.
- Prediction: Each model’s state estimate is projected forward in time until the predicted capacity trajectory crosses the predefined failure threshold (e.g., 70% of initial capacity). The overall RUL distribution is the probability-weighted combination of the RUL predictions from all models.
This structure allows the prognostic system to automatically favor the exponential model when the degradation is highly nonlinear, the polynomial model during quasi-linear fade, and the Verhulst model when a specific saturation trend is observed, all while maintaining robust state estimates via particle flow.
Experimental Validation and Results
The performance of the IMM-PFF method was rigorously evaluated using publicly available aging datasets for lithium-ion battery, including those from CALCE and NASA PCoE. Batteries were cycled under various charge/discharge regimes, and their capacity was recorded at regular intervals. The prediction process was initiated at different points in the battery’s life (early, mid, late) to assess the algorithm’s performance throughout the degradation process.
Prediction Accuracy
The proposed IMM-PFF method was compared against single-model PFF predictors. Key performance metrics included Root Mean Square Error (RMSE) for capacity tracking and absolute error in predicting the End-of-Life (EoL) cycle. The results consistently demonstrated the superiority of the multi-model approach. The following table compares the RUL prediction error for a lithium-ion battery from the CALCE dataset.
| Prediction Start Point | Single Exp-PFF Error (Cycles) | IMM-PFF Error (Cycles) | Improvement |
|---|---|---|---|
| Early Life (Cycle 71) | 23 | 18 | 21.7% |
| Mid Life (Cycle 81) | 13 | 11 | 15.4% |
| Late Life (Cycle 91) | 11 | 10 | 9.1% |
Across the tested batteries and prediction starting points, the IMM-PFF method reduced the average capacity prediction RMSE by approximately 24.3% and the absolute RUL prediction error by about 4.5% compared to the best-performing single-model PFF. More importantly, the IMM-PFF showed consistently reliable performance, whereas single-model filters could produce large errors if the chosen model was a poor match for a particular battery’s aging trajectory.
Model Probability Evolution
Analyzing the evolution of model probabilities $\mu_j(k)$ provides insight into the algorithm’s adaptive behavior. For one battery, the Verhulst model probability remained high throughout, indicating its overall good fit for that cell’s specific degradation shape. For another battery, the polynomial model dominated during an extended linear fade phase, but as the battery entered a more accelerated degradation stage, the probability weight smoothly shifted towards the exponential and Verhulst models. This dynamic adaptation is the key to the method’s robust generalization across different lithium-ion battery types and usage patterns.
Comparison with Advanced Data-Driven Methods
The IMM-PFF method was also benchmarked against state-of-the-art data-driven methods reported in recent literature, such as those based on Long Short-Term Memory (LSTM) networks and Gaussian Process Regression (GPR) optimized with various algorithms. On the NASA battery dataset, the performance was highly competitive. The table below shows a comparison of RUL prediction errors for several NASA battery cells.
| Battery ID | VMD-LSTM-GPR Error | IWOA-LSSVM Error | IMM-PFF Error |
|---|---|---|---|
| B05 | 0 cycles | 0 cycles | 0 cycles |
| B06 | 2 cycles | 0 cycles | 1 cycle |
| B18 | 1 cycle | 0 cycles | 0 cycles |
While some advanced machine learning methods can achieve exceptional accuracy, they often require extensive, high-quality training data from similar batteries and operational conditions. The model-based IMM-PFF approach offers a compelling alternative with strong performance, clear interpretability (through model probabilities), and less dependency on large, labeled historical datasets.
Conclusion and Future Perspectives
Accurate remaining useful life prediction is critical for ensuring the reliability, safety, and economic viability of systems powered by lithium-ion battery. The inherent complexity and variability of battery degradation make single-model prediction frameworks prone to accuracy loss and poor generalization. This work has presented the Interactive Multiple Model Particle Flow Filter (IMM-PFF) as a robust solution to these challenges.
By integrating three complementary empirical degradation models within an IMM framework, the algorithm dynamically identifies the most plausible degradation mode, effectively adapting to the multi-phase aging characteristics of a lithium-ion battery. The use of Particle Flow Filter for the state estimation within each model eliminates the particle degeneracy problem, leading to more stable and precise parameter updates. Experimental validation on real battery aging datasets confirms that the IMM-PFF method significantly improves prediction accuracy and robustness compared to single-model filters, achieving performance on par with sophisticated data-driven techniques.
The proposed method provides a principled, probabilistic, and adaptive framework for lithium-ion battery prognostics. Future research will focus on incorporating more physics-informed or semi-empirical models into the IMM model set to further enhance the interpretability and accuracy of predictions, especially in the very early life stage. Furthermore, exploring the integration of real-time operational stress factors (like temperature and charge/discharge rates) into the model state or the mode-switching logic could unlock even more adaptive and precise RUL prediction capabilities for lithium-ion battery operating under highly dynamic real-world conditions.