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-state electrolytes, are considered a promising next-generation technology due to their superior safety and high energy density. Sulfide-based solid-state batteries offer high ionic conductivity and excellent processability, making them attractive for commercial applications. However, estimating the state of health (SOH) of these batteries remains a challenge, as research in this area is still in its early stages. In this paper, we present a method for SOH estimation of 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 of the battery, while the UKF handles nonlinearities and noise in the system. Experimental validation shows that our approach achieves rapid convergence, with an average SOH estimation error below 1% and a maximum error under 2%.
The electrochemical model for composite cathode all-solid-state batteries is essential for understanding their behavior. These batteries consist of a negative electrode current collector, negative electrode, solid-state electrolyte, composite positive electrode, and positive electrode current collector. The negative electrode is defined in the region $-L_n \leq x \leq 0$, while the electrolyte, typically a sulfide solid-state electrolyte, acts as a separator between the electrodes ($0 \leq x \leq L_{se}$) and serves as an ion conductor in the composite positive electrode ($L_{se} \leq x \leq L_{se} + L_p$). The composite cathode includes carbon to facilitate electron transport, and the active material is single-crystal Ni-rich LiNi$_{0.8}$Co$_{0.1}$Mn$_{0.1}$O$_2$ (NCM811), modeled as spherical particles ($0 \leq r \leq R_s$). Double-layer capacitances $c_{dl}^p$ and $c_{dl}^n$ are considered at the interfaces between the active material and electrolyte to account for space charge effects. The electrochemical reactions during charge and discharge are represented as:
$$ \text{Li}_{1-x}\text{Ni}_{0.8}\text{Co}_{0.1}\text{Mn}_{0.1}\text{O}_2 + x e^- + x \text{Li}^+ \xrightleftharpoons[\text{charge}]{\text{discharge}} \text{LiNi}_{0.8}\text{Co}_{0.1}\text{Mn}_{0.1}\text{O}_2 $$
$$ \text{Li}_{4.4}\text{Si} \xrightleftharpoons[\text{charge}]{\text{discharge}} \text{Li}_{4.4-x}\text{Si} + x e^- + x \text{Li}^+ $$
In modeling the all-solid-state battery, we assume two-dimensional contact between the negative electrode and solid electrolyte, uniform concentration distribution in cathode particles, and non-uniform lithium-ion concentration in the solid electrolyte. The concept of effective contact area is introduced, with the assumption of uniform current density on the active material surface. The governing equations, boundary conditions, and initial conditions for the electrochemical model are summarized below. The model includes mass conservation, charge conservation, and Butler-Volmer kinetics to describe the electrochemical processes.
The lithium-ion concentration in the solid phase of the cathode particles is governed by Fick’s law of diffusion:
$$ \frac{\partial c_s}{\partial t} = \frac{D_s}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial c_s}{\partial r} \right) $$
with boundary conditions:
$$ \frac{\partial c_s}{\partial r} \bigg|_{r=0} = 0, \quad -D_s \frac{\partial c_s}{\partial r} \bigg|_{r=R_s} = \frac{j}{F} $$
where $c_s$ is the lithium concentration in the solid, $D_s$ is the diffusion coefficient, $j$ is the pore wall flux, and $F$ is Faraday’s constant. In the electrolyte, the lithium-ion concentration $c_e$ is described by:
$$ \epsilon_e \frac{\partial c_e}{\partial t} = \nabla \cdot (D_e^{\text{eff}} \nabla c_e) + \frac{1 – t_+}{F} j $$
where $\epsilon_e$ is the electrolyte volume fraction, $D_e^{\text{eff}}$ is the effective diffusion coefficient, and $t_+$ is the transference number. The potential in the solid phase $\phi_s$ and electrolyte phase $\phi_e$ are given by:
$$ \nabla \cdot (\sigma^{\text{eff}} \nabla \phi_s) = j $$
$$ \nabla \cdot (\kappa^{\text{eff}} \nabla \phi_e) + \nabla \cdot (\kappa_D^{\text{eff}} \nabla \ln c_e) = -j $$
where $\sigma^{\text{eff}}$ and $\kappa^{\text{eff}}$ are the effective conductivities, and $\kappa_D^{\text{eff}}$ is the effective diffusional conductivity. The Butler-Volmer equation governs the reaction kinetics:
$$ j = j_0 \left[ \exp\left(\frac{\alpha_a F}{RT} \eta\right) – \exp\left(-\frac{\alpha_c F}{RT} \eta\right) \right] $$
where $j_0$ is the exchange current density, $\alpha_a$ and $\alpha_c$ are the anodic and cathodic transfer coefficients, $R$ is the gas constant, $T$ is the temperature, and $\eta$ is the overpotential. The double-layer capacitance at the interfaces is modeled as:
$$ i_{dl} = c_{dl} \frac{d(\phi_s – \phi_e)}{dt} $$
These equations form the basis of the electrochemical model, which is implemented numerically to simulate battery behavior.
The unscented Kalman filter (UKF) algorithm is employed for state estimation due to its ability to handle nonlinear systems without linearization. The UKF uses a set of sigma points to approximate the probability distribution of the state variables. The system and observation equations are defined as:
$$ x_{k+1} = f(x_k) + w_k $$
$$ y_{k+1} = g(x_{k+1}) + v_k $$
where $x_k$ is the state vector, $y_k$ is the observation vector, $f$ and $g$ are nonlinear functions, and $w_k$ and $v_k$ are process and observation noises with covariances $Q$ and $R$, respectively. The UKF algorithm involves the following steps:
- Initialize the state estimate $\hat{x}_0$ and covariance $P_0$.
- Generate sigma points $\mathcal{X}_k^{(i)}$ and weights $\omega_m^{(i)}$, $\omega_c^{(i)}$ for $i = 0, 1, \dots, 2n$, where $n$ is the state dimension.
- Propagate sigma points through the state function: $\mathcal{X}_{k+1}^{(i)} = f(\mathcal{X}_k^{(i)})$.
- Compute the predicted state mean $\hat{x}_{k+1}^-$ and covariance $P_{k+1}^-$.
- Generate new sigma points from the predicted state.
- Propagate these sigma points through the observation function: $\mathcal{Y}_{k+1}^{(i)} = g(\mathcal{X}_{k+1}^{(i)})$.
- Compute the predicted observation mean $\hat{y}_{k+1}$ and covariance $P_{yy}$.
- Calculate the Kalman gain $K$.
- Update the state estimate $\hat{x}_{k+1}$ and covariance $P_{k+1}$.
For SOH estimation, we define the state variable as the volume fraction of active material in the composite cathode, $\varepsilon_p$, which degrades over time due to mechanical structural changes in the Ni-rich cathode. The system equation is:
$$ \varepsilon_{p,k+1} = \varepsilon_{p,k} $$
This reflects the slow variation of $\varepsilon_p$ over cycles. The observation equation is based on the voltage output from the electrochemical model during the first $N$ seconds of a 1C discharge:
$$ V_{\text{out},k+1}^{(1:N)} = g_{\text{ECM}}(\varepsilon_{p,k+1}) $$
where $g_{\text{ECM}}$ represents the electrochemical model simulation. The SOH is then calculated from the estimated $\varepsilon_p$ by simulating the full discharge capacity and normalizing it to the initial capacity.
To validate our approach, we conducted aging experiments on sulfide-based all-solid-state batteries. The batteries were fabricated using a composite cathode with 56 wt% single-crystal NCM811, 40 wt% Li$_6$PS$_5$Cl sulfide electrolyte, and 4 wt% VGCF carbon. The cells were assembled in an argon-filled glovebox and subjected to 1C constant current charge and discharge cycles. Two cells, labeled A and B, were aged to 80% and 90% of their initial capacity, respectively. The electrochemical model was validated against fresh cell data, showing root mean square errors below 10 mV for 1C discharge curves.
The UKF parameters were tuned for optimal performance: $N = 1800$, $Q = 10^{-2}$, $R = 10^{-5} \cdot I(N)$, $P_0 = 10^{-4}$, and $\alpha = 0.4$. The SOH estimation results for cells A and B are summarized in Table 1, demonstrating rapid convergence and high accuracy.
| Cell | True SOH (%) | Estimated SOH (%) | Error (%) |
|---|---|---|---|
| A | 80.0 | 79.2 | 0.8 |
| A | 78.5 | 77.8 | 0.7 |
| B | 90.0 | 89.4 | 0.6 |
| B | 88.0 | 87.3 | 0.7 |
The average error across all cycles was below 1%, with a maximum error under 2%. This performance compares favorably with other methods, as shown in Table 2.
| 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 |
The degradation in solid-state batteries is primarily attributed to the loss of active material in the composite cathode due to mechanical structural degradation, such as lattice rotation in Ni-rich materials. Our method effectively tracks this degradation through $\varepsilon_p$, enabling accurate SOH estimation. The use of an electrochemical model provides insight into internal states, while the UKF ensures robustness against noise and nonlinearities.
In conclusion, we have developed a framework for SOH estimation in sulfide-based all-solid-state batteries by integrating an electrochemical model with the UKF algorithm. The method leverages the physical insights from the model and the filtering capabilities of UKF to achieve high accuracy. Experimental results confirm the effectiveness, with errors within practical limits. Future work will extend this approach to broader temperature ranges and cycling conditions to enhance its applicability for real-world solid-state battery management systems.