With the rapid advancement of electric vehicles, there is an increasing demand for higher energy density and improved safety in power batteries. All-solid-state batteries, particularly those based on sulfide solid electrolytes, are considered a promising next-generation technology due to their superior safety profile and potential for high energy density. The solid-state battery design eliminates flammable liquid electrolytes, reducing risks of leakage and thermal runaway. Moreover, the use of solid electrolytes enables the integration of high-capacity electrodes, such as lithium metal, further boosting energy density. Accurate state of health (SOH) estimation is crucial for battery management systems to ensure reliability and longevity. Traditional methods for SOH estimation in liquid lithium-ion batteries are well-established, but research on solid-state batteries is still in its infancy. This paper presents a novel approach for SOH estimation in sulfide-based all-solid-state batteries by combining an electrochemical model with the unscented Kalman filter (UKF) algorithm. The electrochemical model captures the internal reaction mechanisms, while UKF handles nonlinearities and noise, providing robust and accurate SOH estimates. Experimental validation on lab-scale solid-state batteries demonstrates the effectiveness of our method, with rapid convergence and high precision.
The core of our approach lies in the electrochemical modeling of all-solid-state batteries. Unlike conventional batteries, solid-state batteries exhibit unique characteristics, such as solid-solid interfaces, double-layer effects, and effective contact area concepts. For instance, in a composite cathode solid-state battery, the cathode consists of active material (e.g., single-crystal LiNi0.8Co0.1Mn0.1O2 or NCM811), solid electrolyte (e.g., Li6PS5Cl), and conductive carbon. The anode, often lithium-silicon based, interfaces with the electrolyte in a two-dimensional manner. Key reactions during discharge include:
$$ \text{Li}{1-x}\text{Ni}{0.8}\text{Co}{0.1}\text{Mn}{0.1}\text{O}2 + x e^- + x \text{Li}^+ \rightleftharpoons \text{LiNi}{0.8}\text{Co}{0.1}\text{Mn}{0.1}\text{O}_2 $$
$$ \text{Li}{4.4}\text{Si} \rightleftharpoons \text{Li}{4.4-x}\text{Si} + x e^- + x \text{Li}^+ $$
The electrochemical model accounts for lithium ion diffusion in the solid electrolyte, potential distributions, and double-layer capacitances at interfaces. The governing equations include mass balance and charge conservation. For example, the lithium concentration in the electrolyte, denoted as ( c_e ), follows Fick’s law in one dimension:
$$ \frac{\partial c_e}{\partial t} = D_e \frac{\partial^2 c_e}{\partial x^2} + \frac{j}{F} $$
where ( D_e ) is the diffusion coefficient, ( j ) is the pore wall flux, and ( F ) is Faraday’s constant. The potential in the solid phase, ( \phi_s ), and electrolyte phase, ( \phi_e ), are described by:
$$ \frac{\partial}{\partial x} \left( \sigma_{\text{eff}} \frac{\partial \phi_s}{\partial x} \right) = -a F j $$
$$ \frac{\partial}{\partial x} \left( \kappa_{\text{eff}} \frac{\partial \phi_e}{\partial x} \right) + \frac{\partial}{\partial x} \left( \kappa_{\text{D}} \frac{\partial \ln c_e}{\partial x} \right) = a F j $$
Here, ( \sigma_{\text{eff}} ) and ( \kappa_{\text{eff}} ) are effective conductivities, ( \kappa_{\text{D}} ) is the diffusional conductivity, and ( a ) is the specific surface area. The model incorporates boundary conditions, such as zero flux at current collectors, and initial conditions based on state of charge. To reduce complexity, we assume uniform current density and use effective contact areas for solid-solid interfaces. The double-layer capacitance at interfaces, represented as ( c_{\text{dl}} ), affects the transient response and is included in the model.
For SOH estimation, we focus on the degradation of the composite cathode, specifically the volume fraction of active material, ( \varepsilon_p ), as the primary state variable. This choice is motivated by aging studies showing that mechanical degradation in single-crystal NCM811 leads to active material loss, which is a dominant failure mode in solid-state batteries with capacity-overloaded anodes. The state equation in our UKF framework is simple:
$$ \varepsilon_{p,k+1} = \varepsilon_{p,k} $$
reflecting that ( \varepsilon_p ) changes slowly over cycles. The observation equation uses the output voltage from the electrochemical model for the first N seconds of a 1C discharge curve:
$$ V_{\text{out},k+1}^{(1:N)} = g_{\text{ECM}}(\varepsilon_{p,k+1}) $$
where ( g_{\text{ECM}} ) represents the electrochemical model simulation. The UKF algorithm handles the nonlinearities in ( g_{\text{ECM}} ) without linearization, using sigma points to approximate the state distribution. The steps of UKF are as follows:
- Initialize state estimate ( \hat{x}_0 ) and covariance ( P_0 ).
- Generate sigma points ( \mathcal{X}_i ) around the current state:
$$ \mathcal{X}_0 = \hat{x}_k $$
$$ \mathcal{X}_i = \hat{x}_k + \left( \sqrt{(n + \lambda) P_k} \right)_i, \quad i = 1, \dots, n $$
$$ \mathcal{X}_i = \hat{x}_k – \left( \sqrt{(n + \lambda) P_k} \right)_i, \quad i = n+1, \dots, 2n $$
where ( n ) is the state dimension, and ( \lambda = \alpha^2 (n + \kappa) – n ) with parameters ( \alpha ) and ( \kappa ). - Assign weights for mean and covariance:
$$ w_m^{(0)} = \frac{\lambda}{n + \lambda}, \quad w_c^{(0)} = w_m^{(0)} + (1 – \alpha^2 + \beta) $$
$$ w_m^{(i)} = w_c^{(i)} = \frac{1}{2(n + \lambda)}, \quad i = 1, \dots, 2n $$
where ( \beta ) is a parameter for higher-order effects. - Propagate sigma points through state function:
$$ \mathcal{X}{i,k+1}^- = f(\mathcal{X}{i,k}) $$ - Compute predicted state and covariance:
$$ \hat{x}{k+1}^- = \sum{i=0}^{2n} w_m^{(i)} \mathcal{X}{i,k+1}^- $$
$$ P{k+1}^- = \sum_{i=0}^{2n} w_c^{(i)} (\mathcal{X}{i,k+1}^- – \hat{x}{k+1}^-) (\mathcal{X}{i,k+1}^- – \hat{x}{k+1}^-)^T + Q $$
where ( Q ) is process noise covariance. - Generate new sigma points from predicted state.
- Propagate through observation function:
$$ \mathcal{Y}{i,k+1} = g(\mathcal{X}{i,k+1}) $$ - Compute predicted observation and covariances:
$$ \hat{y}{k+1} = \sum{i=0}^{2n} w_m^{(i)} \mathcal{Y}{i,k+1} $$
$$ P{yy} = \sum_{i=0}^{2n} w_c^{(i)} (\mathcal{Y}{i,k+1} – \hat{y}{k+1}) (\mathcal{Y}{i,k+1} – \hat{y}{k+1})^T + R $$
$$ P_{xy} = \sum_{i=0}^{2n} w_c^{(i)} (\mathcal{X}{i,k+1} – \hat{x}{k+1}^-) (\mathcal{Y}{i,k+1} – \hat{y}{k+1})^T $$
where ( R ) is measurement noise covariance. - Compute Kalman gain and update state:
$$ K = P_{xy} P_{yy}^{-1} $$
$$ \hat{x}{k+1} = \hat{x}{k+1}^- + K (y_{k+1} – \hat{y}{k+1}) $$
$$ P{k+1} = P_{k+1}^- – K P_{yy} K^T $$
In our implementation, the state ( x ) is ( \varepsilon_p ), and the observation ( y ) is the voltage vector. The UKF parameters are tuned for convergence and accuracy: ( N = 1800 ) (observation length in seconds), ( Q = 10^{-2} ), ( R = 10^{-5} I ), ( P_0 = 10^{-4} ), and ( \alpha = 0.4 ). The SOH is then calculated from the estimated ( \varepsilon_p ) by simulating the full discharge capacity and normalizing to the initial capacity.
To validate our method, we conducted aging experiments on lab-made sulfide-based all-solid-state batteries. The batteries were fabricated in an argon-filled glovebox to prevent moisture degradation. The composite cathode consisted of 56 wt% single-crystal NCM811, 40 wt% Li6PS5Cl solid electrolyte, and 4 wt% VGCF carbon, ball-milled for homogeneity. The anode was a lithium-silicon compound, and the electrolyte layer was pressed into a pellet. Two batteries, labeled A and B, were cycled under 1C constant current charge and discharge until they reached 80% and 90% of initial capacity, respectively. Reference performance tests included static capacity tests at 1C to measure capacity degradation. The electrochemical model was calibrated using fresh battery data, showing high accuracy with root mean square error below 10 mV for 1C discharge curves.
The SOH estimation results demonstrate the effectiveness of our approach. For both batteries, the UKF-based estimator converged rapidly after one iteration, and subsequent SOH estimates had average errors within 1% and maximum errors within 2%. The table below summarizes the estimation accuracy compared to other methods, highlighting the superiority of our UKF-electrochemical model combination.
| Method | Average Error | Maximum Error |
|---|---|---|
| UKF with Electrochemical Model (Proposed) | < 1% | < 2% |
| UKF with Equivalent Circuit Model | < 1% | — |
| Dual Extended Kalman Filter with Electrochemical Model | < 1% | < 3% |
| Particle Filter with Electrochemical Model | — | < 3% |
Key parameters used in the electrochemical model for solid-state batteries are listed in the following table. These parameters were derived from experimental data and literature, ensuring model fidelity.
| Parameter | Symbol | Value | Unit |
|---|---|---|---|
| Initial Active Material Volume Fraction | \( \varepsilon_p \) | 0.56 | — |
| Solid Electrolyte Diffusion Coefficient | \( D_e \) | 1.0 × 10−10 | m²/s |
| Effective Ionic Conductivity | \( \kappa_{\text{eff}} \) | 0.01 | S/m |
| Double-Layer Capacitance | \( c_{\text{dl}} \) | 0.1 | F/m² |
| Anode Capacity | — | Overloaded | mAh/g |
The aging mechanism in these solid-state batteries primarily involves structural degradation of the single-crystal NCM811, leading to a reduction in ( \varepsilon_p ). This is consistent with studies showing lattice rotation and irreversible distortion under high voltages. By tracking ( \varepsilon_p ), our method effectively captures the health degradation. The UKF algorithm’s ability to handle nonlinearities and noise makes it ideal for this application, as the electrochemical model involves complex, nonlinear relationships.
In conclusion, we have developed a robust framework for SOH estimation in sulfide-based all-solid-state batteries by integrating an electrochemical model with the unscented Kalman filter. This approach leverages the physical insights from the model and the statistical robustness of UKF, providing accurate and convergent estimates. Experimental results on lab-scale batteries validate the method, with errors within practical limits. Future work will extend this approach to broader temperature ranges and cycling conditions, enhancing its applicability for real-world battery management systems. The advancement in solid-state battery technology underscores the importance of such estimation techniques for ensuring the reliability and safety of next-generation energy storage systems.
The potential of solid-state batteries to revolutionize energy storage is immense, and accurate health monitoring is a critical enabler. Our method contributes to this field by providing a physics-based, computationally efficient solution. As research progresses, we anticipate further refinements in models and algorithms, ultimately leading to widespread adoption of solid-state batteries in electric vehicles and grid storage.