In modern battery EV cars, lithium-ion batteries serve as the primary power source, forming the core of the electric powertrain. The performance and health state of these batteries inevitably degrade during usage due to various factors, leading to potential failures or severe accidents. Therefore, monitoring and acquiring usage information during operation, predicting the remaining useful life (RUL), and assessing the health state of lithium-ion batteries are crucial for ensuring the safe operation of battery EV cars and protecting the lives and property of passengers. This study focuses on developing a RUL prediction method that accounts for the self-healing phenomenon—a slight recovery in battery capacity during idle periods—to enhance battery performance and extend lifespan in battery EV cars.
The degradation process of lithium-ion batteries in battery EV cars is inherently nonlinear and stochastic. To model this, we define the battery life T and the RUL Lk at time tk based on the first-passage-time concept of stochastic processes. Let X(t) represent the degradation amount at time t, with X(0) as the initial state and ω as a predefined failure threshold. The life T and RUL Lk are given by:
$$ T = \inf\{ t: X(t) \geq \omega \mid X(0) < \omega \} $$
$$ L_k = \inf\{ l_k: X(t_k + l_k) \geq \omega \mid X(t_k) < \omega \} $$
These definitions form the basis for our degradation modeling and RUL prediction in battery EV cars.
Degradation Process Modeling for Lithium-Ion Batteries in Battery EV Cars
We characterize the degradation process using a stochastic process {X(t), t ≥ 0}. Without considering self-healing effects, the degradation model at time t is expressed as:
$$ X(t) = X(0) + \int_0^t \mu(\tau; \theta_\mu) \, d\tau + \sigma B(t) $$
Here, μ(τ; θμ) is a nonlinear drift coefficient—if constant, the degradation becomes linear—σ is the diffusion coefficient, and B(t) is standard Brownian motion with σB(t) ∼ N(0, σ²), capturing random dynamics in battery EV cars. When self-healing is considered, the total degradation Y(t) at time t includes an additive self-healing amount Z:
$$ Y(t) = X(t) + Z $$
where Z represents the cumulative self-healing effect over the battery’s lifecycle in battery EV cars. This model accounts for capacity recovery during rest intervals, which is common in real-world usage of battery EV cars.
Methodology for RUL Prediction in Battery EV Cars
To derive the probability density functions (PDFs) of T and Lk, we propose two assumptions. Assumption 1 states that if the battery is functioning at time t, no failure has occurred before t. Assumption 2 posits that if the degradation process {X(t), t ≥ 0} reaches the threshold ω at time t (i.e., X(t) = ω), the probability of exceeding ω before t is negligible. Based on these, Theorem 1 provides the PDF of the first-passage time for the degradation process in battery EV cars. If μ(τ; θμ) is continuous on [0, ∞), the conditional PDF of T given random parameters is:
$$ f_T(t) = \frac{S_B(t)}{\sqrt{2\pi t^3} \sigma} \exp\left( -\frac{S_B(t)^2}{2t} \right) $$
where $$ S_B(t) = \frac{\omega – \int_0^t \mu(\tau; \theta_\mu) \, d\tau}{\sigma} $$. For battery EV cars without self-healing, the PDFs of T and Lk are:
$$ f_T(t) = \frac{\omega – \int_0^t \mu(\tau; \theta_\mu) \, d\tau}{\sqrt{2\pi t^3} \sigma} \exp\left( -\frac{\left( \omega – \int_0^t \mu(\tau; \theta_\mu) \, d\tau \right)^2}{2\sigma^2 t} \right) $$
$$ f_{L_k}(l_k) = \frac{\omega – X(t_k) – \int_{t_k}^{t_k + l_k} \mu(\tau; \theta_\mu) \, d\tau}{\sqrt{2\pi l_k^3} \sigma} \exp\left( -\frac{\left( \omega – X(t_k) – \int_{t_k}^{t_k + l_k} \mu(\tau; \theta_\mu) \, d\tau \right)^2}{2\sigma^2 l_k} \right) $$
For battery EV cars with self-healing, the PDFs incorporate the self-healing amount Z, assumed to follow a normal distribution Z ∼ N(μ₁, σ₁²). The PDFs become:
$$ f_T(t) = \frac{\omega – \int_0^t \mu(\tau; \theta_\mu) \, d\tau – \mu_1}{\sqrt{2\pi t^3 (\sigma^2 + \sigma_1^2)}} \exp\left( -\frac{\left( \omega – \int_0^t \mu(\tau; \theta_\mu) \, d\tau – \mu_1 \right)^2}{2(\sigma^2 + \sigma_1^2) t} \right) $$
$$ f_{L_k}(l_k) = \frac{\omega – X(t_k) – \int_{t_k}^{t_k + l_k} \mu(\tau; \theta_\mu) \, d\tau – \mu_1}{\sqrt{2\pi l_k^3 (\sigma^2 + \sigma_1^2)}} \exp\left( -\frac{\left( \omega – X(t_k) – \int_{t_k}^{t_k + l_k} \mu(\tau; \theta_\mu) \, d\tau – \mu_1 \right)^2}{2(\sigma^2 + \sigma_1^2) l_k} \right) $$
These equations enable RUL prediction for lithium-ion batteries in battery EV cars under both scenarios.
Parameter Estimation for Degradation Models in Battery EV Cars
To estimate unknown parameters θ, μ₁, σ₁, and σ, we assume N tested lithium-ion batteries from battery EV cars, with the n-th battery sampled at times t₁ⁿ, …, tₘⁿ, where m is the number of measurements. The degradation trajectory at time tₖⁿ is:
$$ Y^n(t_k) = X^n(0) + \theta t_k^n + \sigma B(t_k) + Z $$
By defining Rⁿ(tₖ) = Xⁿ(tₖ) – Xⁿ(0) with realizations rⁿ(tₖⁿ), we rewrite this as:
$$ R^n(t_k) = \theta t_k + \sigma B(t_k) + Z $$
Let Tⁿ = (t₁ⁿ, …, tₘⁿ)’ and Rⁿ = [rⁿ(t₁ⁿ), …, rⁿ(tₘⁿ)]’. The vector Rⁿ follows a multivariate normal distribution with mean μⁿ and covariance Ωⁿ:
$$ \mu^n = \theta T^n + \mu_1 $$
$$ \Omega^n = \sigma^2 Q^n + \sigma_1^2 I_m $$
where Qⁿ is a matrix with elements Qⁿᵢⱼ = min(tᵢⁿ, tⱼⁿ). The log-likelihood function for all test data R is:
$$ L(\theta, \mu_1, \sigma_1, \sigma) = -\frac{mN}{2} \ln(2\pi) – \frac{1}{2} \sum_{n=1}^N \ln |\Omega^n| – \frac{1}{2} \sum_{n=1}^N (R^n – \mu^n)’ (\Omega^n)^{-1} (R^n – \mu^n) $$
Maximizing this function using multidimensional search methods (e.g., in MATLAB) yields the parameter estimates, essential for accurate RUL prediction in battery EV cars.
Experimental Validation with Battery EV Car Data
We validated our method using degradation data from Panasonic 18650 lithium-ion batteries, commonly used in battery EV cars. The dataset was divided into two groups: one for parameter estimation and another for testing. Self-healing times of 2, 5, and 10 minutes were applied during discharge cycles to simulate real-world conditions in battery EV cars. The testing protocol included charging at 3.4 A (1 C) until 4.2 V, followed by constant-voltage charging until 0.34 A (0.1 C). Discharge was at 3.4 A (1 C) until 3 V, then a self-healing period, resuming discharge to 2.5 V. Cycles repeated until capacity faded to 80% of the rated 3.4 Ah (i.e., 2.7 Ah), with data recorded at 10 Hz. The failure threshold ω was set to 0.2 (20% capacity loss). The degradation curves for different self-healing times are summarized below, showing capacity fade over cycles.
The RUL predictions were compared using three methods: Method 1 (ignoring self-healing), Method 2 (a baseline linear model), and Method 3 (our proposed model with self-healing). The actual RUL was determined from the degradation curves as the cycle count until exceeding ω. The results demonstrate the accuracy of our approach for battery EV cars.
| Method | Self-Healing Time (min) | Actual RUL (cycles) | Predicted RUL (cycles) | Absolute Error (cycles) | Relative Error (%) |
|---|---|---|---|---|---|
| Method 1 | 2 | 165 | 177 | 12 | 7.27 |
| Method 2 | 2 | 165 | 144 | 21 | 12.7 |
| Method 3 | 2 | 165 | 162 | 3 | 1.82 |
| Method 1 | 5 | 184 | 199 | 15 | 8.15 |
| Method 2 | 5 | 184 | 166 | 18 | 9.78 |
| Method 3 | 5 | 184 | 175 | 9 | 4.81 |
| Method 1 | 10 | 187 | 161 | 26 | 13.9 |
| Method 2 | 10 | 187 | 167 | 20 | 10.7 |
| Method 3 | 10 | 187 | 191 | 4 | 2.14 |
Our proposed Method 3 consistently outperforms others, with lower errors across all self-healing times, highlighting its effectiveness for RUL prediction in battery EV cars. The integration of self-healing into the degradation model captures real-world behavior, making it suitable for practical applications in battery EV cars.
Discussion on Battery EV Car Applications
The self-healing phenomenon in lithium-ion batteries for battery EV cars is often overlooked but can significantly impact RUL estimation. By incorporating it into stochastic degradation models, we improve prediction accuracy, which is vital for maintenance scheduling and safety management in battery EV cars. For instance, in battery EV cars, self-healing during parking or charging breaks can temporarily restore capacity, delaying failure. Our model accounts for this through the additive term Z, modeled as a normal random variable. The parameter estimation process leverages real degradation data from battery EV cars, ensuring robustness. Future work could explore adaptive thresholds or machine learning integrations to further enhance predictions for diverse battery EV car scenarios.
Conclusion
This study presents a comprehensive method for predicting the remaining useful life of lithium-ion batteries in battery EV cars, considering the self-healing effect. We developed nonlinear degradation models with and without self-healing, derived probability density functions for life and RUL, and provided a parameter estimation framework. Experimental validation using 18650 battery data showed that our method reduces prediction errors compared to alternatives, underscoring its value for health management in battery EV cars. By addressing self-healing, we contribute to safer and more reliable battery EV cars, extending battery lifespan and optimizing performance. As battery EV cars become more prevalent, such advanced prediction tools will be crucial for sustainable transportation.
In summary, the RUL prediction for lithium-ion batteries in battery EV cars is a critical area that benefits from stochastic modeling and real-data validation. Our approach demonstrates how accounting for self-healing can lead to more accurate forecasts, ultimately supporting the advancement of battery EV cars in the automotive industry. Further research could extend this to other battery types or environmental conditions, broadening its applicability for battery EV cars worldwide.