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A Novel State of Charge Estimation Method Based on Electrochemical Impedance Spectroscopy for Solid-State Batteries of Next-Generation Space Power Sources under Different States of Health
Sun et al. · Engineering · Space: Science & Technology 2025
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Energy Storage Data Reporting in Perspective—Guidelines for Interpreting the Performance of Electrochemical Energy Storage Systems
Mathis et al. · Materials Science · Advanced Energy Materials
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The Gaussian Process Hilbert Transform (GP-HT): Testing the Consistency of Electrochemical Impedance Spectroscopy Data
Francesco Ciucci · Computer Science · Journal of The Electrochemical Society
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Ultra-high-rate pseudocapacitive energy storage in two-dimensional transition metal carbides
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A Novel State of Charge Estimation Method Based on Electrochemical Impedance Spectroscopy for Solid-State Batteries of Next-Generation Space Power Sources under Different States of Health

EngineeringPeer-reviewedOpen access
Space: Science & Technology · 2025
Bingxiang Sun et al.authors, affiliations & DOI
Bingxiang Sun, Junfeng Pang, Liying Zhu, Xinze Zhao, Weige Zhang, Shichang Ma, Xiaopeng Liu
Beijing Jiaotong University · Beijing Institute of Spacecraft System Engineering
DOI 10.34133/space.0198·View at publisher ↗

Abstract

With the increasing demand for battery energy density and safety, solid-state batteries are expected to become one of the ideal energy storage devices. Aiming at compensating for the research lack of modeling and state estimation of solid-state batteries, this paper proposes a state of charge (SOC) estimation method based on the electrochemical impedance spectra (EIS) of sample solid-state batteries that consider both accuracy and speed. The SOC estimation model is established using Gaussian process regression by extracting the lowand medium-frequency semicircular parameters that vary markedly with SOC. Considering the online application of the method, the fixed-frequency test impedance is selected as the characteristic parameter to achieve the SOC estimation that takes both accuracy and speed into account, and the method is still applicable under different state of health (SOH). Considering the existence of 3 impedance spectral semicircles in this solid-state battery, the RQ (parallel connection of resistors and constant phase element) process is added to establish a fractional-order model and the fractional-order extended Kalman filter (FOEKF) algorithm is developed. What is more, the FOEKF algorithm and SOC-OCV curve are constructed as a control group to analyze the accuracy, advantages, and disadvantages of the proposed SOC estimation method. The results show that the unique EIS performance of this solid-state battery is more conducive to achieving the estimation of the battery SOC. The application of the EIS online implementation method is expected to rely on the fixed-frequency test to achieve online accurate estimation of the SOC of solid-state batteries.

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What does this paper propose or contribute, and how well does it perform?
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Debrief
The paper contributes by addressing the research gap in modeling and parameter estimation for solid-state batteries Conclusion, p.15. It proposes a novel approach that involves extracting impedance parameters at various frequencies using Electrochemical Impedance Spectroscopy (EIS) testing to develop a solid-state battery State of Charge (SOC) estimation model Conclusion, p.15. This model allows for fast and precise SOC estimation for solid-state batteries Conclusion, p.15. The paper also performs a thorough analysis of two other methods: the FOEKF and the SOC-OCV table lookup approach Conclusion, p.15. The results demonstrate that the estimation model based on EIS characteristic parameters allows for fast and accurate SOC estimation, particularly when implemented online using fixed-frequency testing below 2 Hz Introduction, p.2. Furthermore, the method has been shown to be applicable under various State of Health (SOH) conditions Introduction, p.2. The paper also analyzes the estimation error of the SOC-OCV method by considering resting insufficiency and sampling error Conclusion, p.15.
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What methods, models, or experiments does the paper use?
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Debrief
The paper involves the analysis of the estimation error of the SOC-OCV method by examining resting insufficiency and sampling error Conclusion, p.15. Additionally, the paper analyzes the estimation accuracy of different initial values Conclusion, p.15. The structural framework of the study is presented in Figure 1 S6, p.4.

Problem & motivation

This paper addresses the existing research gap in modeling and parameter estimation for solid-state batteries Conclusion, p.15.

Approach

The paper introduces the theory related to EIS and FOEKF algorithms in the Materials and Methods section Introduction, p.3. The study involves feature parameter extraction and SOC estimation based on EIS for solid-state batteries, and it validates SOC estimation of FOEKF for different operating conditions, as well as analyzing the SOC-OCV (open circuit voltage) application error Introduction, p.3.

Key findings

The provided sources do not contain the main quantitative results and findings of the paper. The sources mention the analysis of estimation accuracy of different initial values Conclusion, p.15 and the analysis of the estimation error of the SOC-OCV method regarding resting insufficiency and sampling error Conclusion, p.15, but they do not present the actual results.

A Novel State of Charge Estimation Method Based on Electrochemical Impedance Spectroscopy for Solid-State Batteries of Next-Generation Space Power Sources under Different States of Health

Citation: Sun B, Pang J, Zhu L, Zhao X, Zhang W, Ma S, Liu X. A Novel State of Charge Estimation Method Based on Electrochemical Impedance Spectroscopy for Solid- State Batteries of Next-Generation Space Power Sources under Different States of Health. Space Sci. Technol. 2025;5:Article 0198. https://doi. org/10.34133/space.0198

Bingxiang Sun1,2*, Junfeng Pang1,2, Liying Zhu³, Xinze Zhao1,2*, Weige Zhang1,2, Shichang Ma1,2, and Xiaopeng Liu1,2

1^{1} National Active Distribution Network Technology Research Center (NANTEC), Beijing Jiaotong University, Beijing 100044, China. 2^{2} Key Laboratory of Vehicular Multi-Energy Drive Systems (VMEDS), Ministry of Education, Beijing Jiaotong University, Beijing 100044, China. 3^{3} Beijing Institute of Spacecraft System Engineering, Beijing 100094, China.

Submitted 22 January 2024 Revised 30 May 2024 Accepted 14 July 2024 Published 29 April 2025

Copyright © 2025 Bingxiang Sun et al. Exclusive licensee Beijing Institute of Technology Press. No claim to original U.S. Government Works. Distributed under a Creative Commons Attribution License (CC BY 4.0).

*Address correspondence to: bxsun@bjtu.edu.cn (B.S.), 21117037@bjtu.edu.cn (X.Z.)

With the increasing demand for battery energy density and safety, solid-state batteries are expected to become one of the ideal energy storage devices. Aiming at compensating for the research lack of modeling and state estimation of solid-state batteries, this paper proposes a state of charge (SOC) estimation method based on the electrochemical impedance spectra (EIS) of sample solid-state batteries that consider both accuracy and speed. The SOC estimation model is established using Gaussian process regression by extracting the lowand medium-frequency semicircular parameters that vary markedly with SOC. Considering the online application of the method, the fixed-frequency test impedance is selected as the characteristic parameter to achieve the SOC estimation that takes both accuracy and speed into account, and the method is still applicable under different state of health (SOH). Considering the existence of 3 impedance spectral semicircles in this solid-state battery, the RQ (parallel connection of resistors and constant phase element) process is added to establish a fractional-order model and the fractional-order extended Kalman filter (FOEKF) algorithm is developed. What is more, the FOEKF algorithm and SOC-OCV curve are constructed as a control group to analyze the accuracy, advantages, and disadvantages of the proposed SOC estimation method. The results show that the unique EIS performance of this solid-state battery is more conducive to achieving the estimation of the battery SOC. The application of the EIS online implementation method is expected to rely on the fixed-frequency test to achieve online accurate estimation of the SOC of solid-state batteries.

Introduction

With the continuous promotion of the “double-carbon” strategy, energy storage technology is poised for broad application prospects. Lithium-ion batteries have gained widespread use in electronic products and power energy storage, owing to their high capacity density, high power density, lack of memory effect, low self-discharge rate, and long service life [1,2]. However, the energy density of the existing material system in lithium-ion batteries has reached its limit. With the rapid progress of new energy technologies and the increasing number of power battery applications in new energy transport vehicles, higher demands are being placed on energy density, safety, and battery volume weight [3]. In contrast to conventional lithium-ion batteries, solid-state batteries utilize solid electrodes and solid electrolytes, eliminating the need for flammable and explosive liquid

electrolytes. They offer higher energy density and lower power density, making solid-state batteries much lighter than lithium-ion batteries of the same capacity. As a result, solid-state batteries are considered a promising alternative energy source for electric vehicles and energy storage systems, given their safety and energy density advantages [4–7]. According to South Korean market research firm SNE Research, the market space for solid-state batteries is expected to reach 20 billion Chinese yuan in 2030, surpassing lithium-ion batteries as the mainstream choice for electric vehicle batteries [8]. However, solid-state batteries differ substantially from conventional lithium-ion batteries in their material system. This leads to fundamental changes in physical, chemical, and mechanical processes, such as material transport, interface electrochemistry, and stress evolution within the battery [3]. Therefore, a systematic analysis of the similarities and differences between solid-state batteries and lithium-ion batteries is essential. This analysis should encompass aspects like characterization technology and theoretical mechanisms to comprehensively strengthen the research theory of solid-state batteries and drive their large-scale market applications.

In recent years, the research on solid-state batteries is still in the stage of material research and improvement. Most of the reviews on solid-state batteries also focus on solid-state electrolytes with high ionic conductivity, electrochemical stability, and mechanical properties, as well as structural and interfacial improvement strategies for composite electrodes [9–13]. There are also some literature on the development and research outlook of solid-state batteries [14–16], but the research on the utilization performance and parameter estimation of solid-state batteries has hardly appeared. Currently, there is only one patent in the public information that mentions SOC estimation for solid-state batteries, which is performed by extracting battery voltage, current, temperature, and power as feature parameters. The validation values are output by 3 models and fused with the original features, and the gradient boosting decision tree model is selected for SOC prediction [17]. Therefore, in this paper, we test the solid-state battery from a lithium-ion battery and try to establish the model and parameter estimation method applicable to solid-state batteries.

Equivalent circuit models are divided into integer-order and fractional-order models according to the composition of circuit elements. The integer order model does not consider the internal reaction mechanism of the battery and mainly reacts to the external characteristics of the battery with fewer parameters, which makes it easy to derive the system space equation of state. It is widely used in system simulation and real-time control, and also has a wide range of applications in battery-related systems. The fractional-order model replaces capacitance C with constant phase element (CPE; commonly used Q) and introduces a fractional-order operator, which has a memory effect and is suitable for describing a memory hysteresis system. For example, modeling with electrochemical impedance spectroscopy (EIS) as a data source is often fitted with a fractional-order model. Hu et al. [18] carried out the establishment and parameter identification of a fractional-order model based on a hybrid multi-particle swarm algorithm, and used a second-order RQ model to verify the dynamic working conditions. Sun et al. [19] carried out a study of a variable-order fractional-order model based on the electrochemical impedance spectra of lithium-ion batteries and verified the model’s good accuracy with parameter identification via the FORPRLS algorithm. Fractional-order models with 1 or 2 RQs were used to estimate the SOC, and better results than the same RC (parallel connection of resistors and capacitance) model were achieved [18,20].

The most widely used method for estimating the state of charge (SOC) of a battery is the ampere-time integration method. Currently, most of the battery management systems (BMSs) also use the ampere-time integration method to estimate the SOC of the battery, which does not require modeling, is simple to implement, and is easy to operate. However, the method relies on the accurate acquisition of the initial value, and if the error of the initial value is large, the estimation results become inaccurate and cannot be corrected. At the same time, the accuracy of current acquisition also greatly affects the accuracy of estimation. SOC estimation based on the extended Kalman filter (EKF) algorithm is to expand the battery state-space equations using the Taylor series, and considering the computational complexity, the first order is mostly retained to linearize the nonlinear system. Reshma and Manohar [21] achieved the estimation of battery SOC by a dual adaptive Kalman filtering algorithm based on a first-order RC equivalent circuit using an improved remora optimization algorithm. Zhao et al. [22] also used a fuzzy method-based adaptive Kalman filtering based on an equivalent circuit, which improves the accuracy of the first hundreds of SOC estimation. Xiong et al. [23] used a fractional-order traceless Kalman filter (KF) to estimate SOC, which proves that the method is capable of achieving accurate estimation of battery terminal voltage and SOC under battery operating conditions over a wide range of temperatures and aging levels. Mawonou et al. [24] establishes a fractional-order model based on the EIS and estimates SOC based on the EKF, which provides a higher level of accuracy compared to the classical equivalent circuit. In addition to algorithm-based SOC estimation, Rodrigues et al. [25] and Buller et al. [26] investigated the variation rule of EIS with SOC for lithium cobalt acid batteries, and the semicircular diameter of the mid-frequency region of EIS monotonically decreases with the increase of SOC. This provides a new method for the estimation of battery SOC.

With the in-depth research on EIS, more and more online methods for obtaining impedance spectra have been published. Zhu et al. [27] used a BMS to generate excitation signals to obtain the battery impedance at different frequencies, realizing the online application of impedance spectra. Su et al. [28] achieved the online acquisition of impedance spectra in the low-frequency band below 2 Hz by injecting a stepped wave instead of sinusoidal waveforms. Sihvo et al. [29] used PRBS (pseudo-random binary sequence) as a reference for the current of discharged batteries and applied the discrete Fourier transform to the measurement of the impedance frequency response to achieve online measurement of lithium-ion battery impedance. Lyu et al. [30] achieved a fast measurement of impedance in the frequency range of 1 kHz to 0.01 Hz based on a hardware circuit by generating a sinusoidal excitation signal from an upper computer and performing a fast Fourier transform of the battery voltage response. Therefore, with the deepening of the research, the online acquisition of impedance spectra is no longer a difficult problem, and the engineering application of impedance spectra becomes possible.

Up until now, research on modeling and parameter estimation of lithium-ion batteries has reached a relatively mature stage with numerous methods available. However, the study of emerging solid-state batteries is still in the early stages, mainly focusing on material research and improvement, while performance and parameter estimation research for solid-state batteries remains lacking. The applicability of solid-state batteries was explored by analyzing the research methods of lithium-ion batteries. Circuit models and algorithms applicable to solid-state batteries were established, various methods were improved to make them applicable to solid-state batteries, and the practical application of solid-state batteries was advanced. This paper aims to address this gap by analyzing the third semicircle trend of the EIS concerning the SOC of the battery. Representative characteristic parameters are extracted from the EIS data, and a SOC estimation algorithm based on Gaussian process regression (GPR) is established. Considering online applications, a fixed-frequency point below 2 Hz was selected to establish an estimation model, achieving relatively accurate SOC estimation for solid-state batteries. Additionally, the training model is utilized for different state of health (SOH) batteries to verify the method’s accuracy to ensure its applicability. In the next steps, the existing models and algorithms designed for lithium-ion batteries are evaluated for their suitability with solid-state batteries. A fractional-order model specific to solid-state batteries is developed, and the fractional-order extended Kalman filter (FOEKF) algorithm is used for SOC estimation. The convergence of the FOEKF algorithm is also analyzed for different initial SOC. Finally, a comprehensive comparison and analysis of the various methods are conducted, focusing on their applicability, advantages, and limitations. The results demonstrate that the estimation model based on EIS characteristic parameters enables fast and accurate SOC estimation, especially when considering online implementation using fixed-frequency testing below 2 Hz. Moreover, the method proves to be applicable under various SOH conditions, providing valuable insights for future research and practical applications of solid-state batteries. The structural framework of the manuscript is shown in Fig. 1.

The remainder of this paper is organized as follows. Materials and Methods introduces the theory related to EIS and FOEKF algorithms. Results describes the parameters and experiments related to solid-state batteries. It also describes the feature parameter extraction and SOC estimation based on EIS for solid-state batteries, validates SOC estimation of FOEKF for different operating conditions, and analyzes the SOC-OCV (open circuit voltage) application error. The last section is the conclusion.

Fractional-order modeling theory of EIS

In Fig. 2, a typical EIS graph for a Li-ion battery is displayed. The graph has a horizontal axis for the real part of EIS and a vertical axis for the negative imaginary part of EIS. The Nyquist curve of the EIS decreases in frequency from left to right, and it is usually divided into high, medium-high, medium-low, and low frequencies. Each part of the spectrum corresponds to electrochemical reactions happening at different time intervals and can be modeled with different equivalent components, such as resistors (R), inductors (L), capacitors (C), normal phase elements (Q), and Warburg impedance (W).

where the resistor RR is an element with only a real part, which is generally used to fit the intersection of the EIS and the real axis with the expression shown in Eq. (1):

ZR=ZRre+ZRim=R,ZRim=0(1)Z_{R} = Z_{Rre} + Z_{Rim} = R, \quad Z_{Rim} = 0 \quad(1)

The inductor LL is a component that has no real part and is often utilized to match the portion of the EIS that falls below the real axis. Its equation is displayed as Eq. (2):

ZL=ZLre+ZLim=jωL,ZLre=0(2)Z_{L} = Z_{Lre} + Z_{Lim} = j \omega L, \quad Z_{Lre} = 0 \quad(2)

The inductor CC is another component that only has an imaginary part. It is usually connected in parallel with the resistor RR to fit into the semicircular section of the EIS above the real axis. The expression for the resistor RR is given in Eq. (3):

ZC=ZCre+ZCim=−j(1/ωC),ZCre=0(3)Z_{C} = Z_{Cre} + Z_{Cim} = - j(1 / \omega C), \quad Z_{Cre} = 0 \quad(3)

The EIS medium frequency band's half-circle arc may sometimes be incomplete and offset due to the “dispersion effect”. This results in the center of the circle being below the real axis, making it difficult for the RC to fit the arc well. To address this issue, the CPE Q is introduced. Its expression, shown in Eq. (4), includes a positive constant Y0Y_{0} with capacitive properties and a dispersion coefficient α\alpha, which determines the degree of dispersion. When α=1\alpha = 1, Q is equivalent to capacitance C, and when α=0\alpha = 0, Q is equivalent to R. When α=−1\alpha = -1, Q is equivalent to L. After transforming it in parallel with the resistor R, its expression is shown in Eq. (5), and it can be represented as a circle with its center at (R/2,−(Rtan⁡θ)/2)(R/2, - (R\tan\theta)/2) and a radius of R/2cos⁡θR/2\cos\theta. The composite element RQ can be used to represent the semicircular arc of the band offset in the EIS, and its impedance diagram is shown in Fig. 3.

ZQ=ZQre+jZQim=(jω)−αY0,0<α<1(4)Z_{Q} = Z_{Qre} + jZ_{Qim} = \frac{(j \omega)^{- \alpha}}{Y_{0}}, 0 < \alpha < 1 \quad(4)

(ZRQre−R2)2+(ZRQim−Rtan⁡θ2)2=(R2cos⁡θ)2,θ=π2−απ2(5)\left(Z_{RQ_{re}} - \frac{R}{2}\right)^{2} + \left(Z_{RQ_{im}} - \frac{R \tan \theta}{2}\right)^{2} = \left(\frac{R}{2 \cos \theta}\right)^{2}, \theta = \frac{\pi}{2} - \frac{\alpha \pi}{2} \quad(5)

Definition of fractional-order differentiation

Fractional-order calculus describes the dispersion effect of a cell very well; therefore, there are now a large number of studies using fractional-order models to describe the external properties of cells. Currently, the commonly used fractional-order definitions include the Riemann–Liouville (R-L) definition, the Grunwald–Letnikov (G-L) definition, and the Caputo definition. Among them, the discrete form of the G-L definition facilitates the construction of state-space equations for battery models and is widely used to deal with mathematical expressions of fractional-order components. In the G-L definition, the α\alpha -order derivative of the function x(t)x(t) is shown in Eq. (6) and the discretized form is in Eq. (7).

t0GLΔtαx(t)=lim⁡h→01hα∑j=0[(t−t0)/h](−1)j(αj)x(t−jTs)(6){}_{t_0}^{GL}\Delta_t^{\alpha} x(t)= \lim_{h \to 0} \frac{1}{h^{\alpha}} \sum_{j=0}^{[(t-t_0)/h]} (-1)^j \binom{\alpha}{j} x(t - jT_s)\quad(6)

Δαxk+1=1Tsα∑j=1k+1(−1)j(αj)xk+1−j(7)\Delta^{\alpha} x_{k + 1} = \frac{1 }{T_{s}^{\alpha} } \sum_{j = 1}^{k + 1} ( - 1 )^{j} \binom{\alpha }{j} x_{k + 1 - j} \quad(7)

where Δ\Delta denotes the differential operator, TsT_{s} is the sampling time, and in this paper Ts=1T_{s}=1.

(αj)=α!j!(α−j)!=Γ(α+1)Γ(j+1)Γ(α−j+1)(8)\binom{\alpha}{j} = \frac{\alpha !}{j ! (\alpha - j)!} = \frac{\Gamma(\alpha + 1)}{\Gamma(j + 1)\Gamma(\alpha - j + 1)} \quad(8)

Based on Eq. (8), it is evident that differential operation involves using all historical data, resulting in a higher computational burden. However, the coefficient of historical data gradually approaches 0 as the distance from the current state increases. To improve computing speed and save hardware resources, we can use historical data within a fixed time window N for differential operation. Therefore, we modify Eq. (7) as follows:

Δαxk={1Tsα∑j=0K(−1)j(αj)xk−j,k<N1Tsα∑j=0N(−1)j(αj)xk−j,k≥N(9)\Delta^{\alpha} x_{k} = \begin{cases} \frac{1}{T_{s}^{\alpha}} \sum_{j=0}^{K} (-1)^{j}{\alpha \choose j} x_{k-j}, k < N \\ \frac{1}{T_{s}^{\alpha}} \sum_{j=0}^{N} (-1)^{j}{\alpha \choose j} x_{k-j}, k \ge N \end{cases} \quad(9)

Fig. 1. The structural framework of the study.. The image is a block diagram outlining the structural framework of a study, divided into three main sections. The top section, 'Data acquisition and processing,' shows an electrochemical workstation applying an AC signal to a solid-state battery, producing an impedance (Nyquist) plot. The middle section, 'Feature extraction,' contains a flowchart with steps including sweep frequency test, impedance parameters analysis, fixed frequency test, and Pearson correlation analysis, with associated frequency values and a correlation formula. The bottom section, 'SOC estimation methods based on different models,' is split into three pathways: a Gaussian process regression approach using training and testing data, a simplified fractional order model with state space equations and an FOEKF under dynamic operating conditions, and a low current test producing an SOC-OCV curve, all converging into the final SOC estimate data.

Fig. 1. The structural framework of the study.

Gaussian process regression

GPR is a nonparametric model for regression analysis of data using a Gaussian process prior. It fits the corresponding Gaussian process through a finite amount of high-dimensional data to predict the value of the function under any random variable. In this paper, using health characteristics as input and battery capacity as output, the GPR exponential model is used for lithium-ion battery capacity estimation based on the MATLAB software tool with the kernel function shown in Eq. (10).

where σf\sigma_{f} is the standard deviation, r is the Euclidean distance, and σl\sigma_{l} is the scale parameter.

k(xi,xj∣θ)=σf2exp(−rσl)(10)k{\left(x_{i}, x_{j} | \theta \right)} = \sigma_{f}^{2} \mathrm{exp} \left(- \frac{r }{\sigma_{l} } \right)\quad(10)

FOEKF algorithm

The KF algorithm is a method that uses input and output observations of a system to estimate its state. It is based on the theory of optimal estimation. While the original KF algorithm is only applicable to linear systems, the EKF algorithm can be used for nonlinear systems by linearizing them through the Taylor expansion. The FOEKF algorithm is based on the same concept as the EKF algorithm and combines it with the fractional-order model. This algorithm consists of a prediction step and an update step. The prediction step predicts the state variable at the next moment, while the update step updates the predicted value using the measurement data. The algorithm can be expressed as follows:

Determination of filtering initial conditions:

x^0=E[x0],P0=E[(x0−x^0)(x0−x^0)T](11)\widehat{x}_{0} = E \left[ x_{0} \right], P_{0} = E \left[ \left(x_{0} - \widehat{x}_{0}\right)\left(x_{0} - \widehat{x}_{0}\right)^{\mathrm{T}} \right] \quad(11)

Fig. 2. Typical electrochemical impedance spectroscopy.. A line plot displays electrochemical impedance spectroscopy (EIS) data, with the x-axis labeled Re(Z) / mΩ and the y-axis labeled -Im(Z) / mΩ. The curve, formed by data points connected by a line, starts near the origin, rises to a peak, dips down, and rises again. Dashed vertical lines divide the curve into regions labeled High frequency, Medium-high frequency, Medium-low frequency, and Low frequency, and a marker highlights a specific point near the origin.

Fig. 2. Typical electrochemical impedance spectroscopy.

Fig. 3. Impedance diagram of parallel composite component RQ.. A schematic impedance diagram for a parallel composite component, plotted on Cartesian axes with Re(Z) on the horizontal axis and -Im(Z) on the vertical axis. A semicircular arc extends from the origin into the lower region, with a dashed line from the origin intersecting the arc at a marked point, forming an angle θ with the horizontal axis. The point is annotated with coordinates involving R/2 and -R tan θ/2.

Fig. 3. Impedance diagram of parallel composite component RQ.

Prediction step (state estimate xkx_{k} and error covariance PkP_{k}):

x^k−=[ΔTsαA+diag⁡(α)]x^k−1+ΔTsαBuk−1−∑j=2k(−1)jγjx^k−j+ωk(12)\widehat{x}_{k}^{-} = \left[ \Delta T_{\mathrm{s}}^{\boldsymbol{\alpha}} A + \operatorname{diag} (\boldsymbol{\alpha})\right] \widehat{x}_{k - 1} + \Delta T_{\mathrm{s}}^{\boldsymbol{\alpha}} Bu_{k - 1} - \sum_{j = 2}^{k} (- 1)^{j} \gamma_{j} \widehat{x}_{k - j} + \omega_{k} \quad(12)

P_{k}^{-}=\big[\Delta T_{s}^{\boldsymbol{\alpha}} A+\mathrm{diag}(\boldsymbol{\alpha})\big] P_{k-1} \big[\Delta T_{s}^{\boldsymbol{\alpha}} A+\mathrm{diag}(\boldsymbol{\alpha})\big]^{\mathrm{T}} + \sum_{j=2}^{k} \gamma_{j} P_{k-1} \boldsymbol{\gamma}_{j}^{\mathrm{T}}+Q_{k} \tag{13}

The gain matrix LkL_{k} is updated:

Lk=Pk−C^kT(C^kPk−C^kT+Rk)−1(14)\boldsymbol{L}_{k} = P_{k}^{-} \widehat{\boldsymbol{C}}_{k}^{\mathrm{T}} \left(\widehat{\boldsymbol{C}}_{k} P_{k}^{-} \widehat{\boldsymbol{C}}_{k}^{\mathrm{T}} + \boldsymbol{R}_{k}\right)^{- 1} \quad(14)

Update step (state estimate xkx_{k} and error covariance PkP_{k}):

x^k=x^k−+Lk(Umeas−(Ckxk+Dkuk))(15)\widehat{x}_{k} = \widehat{x}_{k}^{-} + L_{k} \left(U_{\text{meas}} - \left(C_{k} x_{k} + D_{k} u_{k}\right)\right)\quad(15)

Pk=(I−LkC^k)Pk−(16)P_{k} = \left(I - \boldsymbol{L}_{k} \widehat{\boldsymbol{C}}_{k}\right)P_{k}^{-} \quad(16)

where subscript k denotes the kth moment, superscript ∧\wedge denotes the optimal estimate, right superscript ¬\neg denotes the predicted value, and UmeasU_{meas} is the voltage measurement.

Experimental design

For this experiment, 3 LiCoO 2_{2} solid-state batteries were chosen with varying degrees of aging. EIS tests were conducted at different SOCs. The battery related parameters are shown in Table 1.

Experimental platform

The battery capacitance and impedance spectroscopy test platform has been set up and connected, as depicted in Fig. 4. This platform uses German Bio-Logic Company's VMP-300 to perform EIS tests, covering a frequency range of 10 μ\muHz to 7 MHz. U.S. Arbin Instrumentation Company's charging-discharging equipment is utilized to achieve a current range of ±\pm10 A. To ensure that the battery is in a constant temperature environment during the experiments, a highand low-temperature test chamber is used. This chamber can maintain a set temperature range of −40-40 to +150+150 °C with an accuracy of ±\pm0.5 °C. The experimental design is as follows:

  • Fully rested at 25 °C;

  • Charge the battery to 4.2 V at 1,460 mA, then constant voltage until the current is less than or equal to 365 mA;

  • Rest for 1 h;

Table 1. Solid-state battery parameter table

ParameterValueUnit
Nominal capacity7,300mAh
Rated voltage3.75V
Charging cutoff voltage4.2V
Discharge cutoff voltage3.0V
Standard charge-discharge current1,460mA

Fig. 4. The establishment of an experimental platform.. A block diagram of an experimental platform showing several pieces of laboratory equipment, each represented by a photograph with a label: a computer, an electrochemical workstation, a solid-state battery, charge-discharge equipment, and a thermal chamber. Arrows between the components indicate the flow of interactions, labeled with terms such as 'Device control,' 'Data acquisition,' 'AC signal injection,' and 'Condition testing.' The solid-state battery is positioned centrally, connected to the electrochemical workstation, charge-discharge equipment, and thermal chamber, while the computer communicates with the workstation and the charge-discharge equipment.

Fig. 4. The establishment of an experimental platform.

  • Discharges battery to 3.0 V at 1,460 mA;

  • Rest for 1 h;

  • Cycle ② to ⑤ 3 times, take the average of 3 times discharge capacity as the current battery capacity;

  • Charge the battery to 4.2 V at 1,460 mA, then constant voltage until the current is less than or equal to 365 mA;

  • EIS of the battery was performed at 100% SOC, and the battery was tested at 10 mV, with a frequency range from 100 kHz to 10 mHz, and 12 points were taken at each decibel frequency;

  • Discharge the battery 10% SOC at 1,460 mA and rest for 1 h;

  • Cycle 8 to 9 until battery voltage reaches 3.0 V;

  • Charge the battery to 4.2 V at 1,460 mA, then constant voltage until the current is less than or equal to 365 mA;

  • Rest for 1 h;

  • Cycle discharge the battery in DST (dynamic stress test) condition until the battery voltage reaches 3.0 V;

  • Rest for 1 h;

  • Charge the battery to 4.2 V at 1,460 mA, then constant voltage until the current is less than or equal to 365 mA;

  • Rest for 1 h;

  • Cycle discharge the battery in FUDS (full urban driving schedule) condition until the battery voltage reaches 3.0 V;

  • Rest for 1 h, end of experiment.

where ① to ⑥ are battery capacity calibration, ⑦ to ⑩ are the EIS test, and ⑪ to ⑱ are the working condition test.

Experimental results

The capacity of each battery was calibrated for standard working conditions. Battery #1 had a capacity of 6,753.73 mAh, battery #2 had a capacity of 7,014.56 mAh, and battery #3 had a capacity of 6,601.36 mAh. The EIS test results of the 3 batteries with different SOCs are shown in Fig. 5.

Feature parameter extraction and Pearson correlation analysis

In Fig. 6, the variation of the last semicircle of the battery EIS with SOC is displayed. The battery EIS demonstrates regular changes as SOC decreases, with the middle-low frequency semicircle gradually spreading to the upper right. This correlation with battery SOC is strong.

Based on the last semicircle of the EIS, 5 feature parameters are extracted by taking points at equal intervals, which are 8.95-Hz impedance mode value, 3.39-Hz impedance mode value, 1.56-Hz impedance mode value, 0.72-Hz impedance mode value, and the real part of the inflection point Re(inf), and the 5 features are distributed in the semicircle approximately uniformly, which can adequately represent the trend of the semicircle with the change of the SOC. The variation of SOC is shown in Fig. 7. With the increase of SOC, the 5 features show a gradual decreasing trend.

The Pearson correlation coefficient is used as a test of parameter-capacity correlation and is calculated as shown in Eq. (17).

P=∑i=1n(X(i)−X‾)(Y(i)−Y‾){∑i=1n(X(i)−X‾)2∑i=1n(Y(i)−Y‾)2}(17)P=\frac{\sum_{i=1}^{n} \left(X(i )-\overline{{X}} \right)\left(Y(i )-\overline{{Y}} \right)}{\sqrt{\left\{\sum_{i=1}^{n} \left(X(i )-\overline{{X}} \right)^{2} \sum_{i=1}^{n} \left(Y(i )-\overline{{Y}} \right)^{2} \right\}}} \quad(17)

where X and Y are the 2 variables for surrogate correlation, X‾\overline{X} and Y‾\overline{Y} are the average of the 2 variables for surrogate correlation, and n is the number of variables.

When 0.3≤∣P∣<0.50.3 \leq |P| < 0.5, there is a weak correlation between the 2 variables; when 0.5≤∣P∣<0.80.5 \leq |P| < 0.8, there is a moderate correlation between the 2 variables; when 0.8≤∣P∣<10.8 \leq |P| < 1, there is a strong correlation between the 2 variables; and when ∣P∣=1|P| = 1, there is a perfect correlation between the 2 variables. If P is positive, there is a positive correlation; if P is negative, there is a negative correlation.

Fig. 8 analyzes the correlation between the above feature parameters and the battery SOC. As the battery SOC decreases, the battery impedance gradually increases, all 5 features show a negative correlation with the SOC, and the correlation coefficients are greater than 0.95, which shows a strong correlation, and it is suitable for estimating the battery SOC.

Estimated results

In this paper, batteries #1 and #2 are selected to train the SOC estimation model and #3 to validate the model. Five feature parameters are used as inputs for SOC estimation, and the Gaussian regression exponential model is selected for training based on the Regression Learner App of MATLAB. The estimation results of the model on the SOC of #3 battery after training are shown in Fig. 9, the maximum absolute error is not more than 1.5%, and the average absolute error is 0.564%, which can achieve the accurate estimation of SOC. However,

Fig. 5. EIS test results for solid-state batteries (#1 - #3) with different SOCs.

Fig. 5. EIS test results for solid-state batteries (#1 - #3) with different SOCs.

the frequency corresponding to the low-frequency inflection point of the EIS gradually decreases with the decrease of SOC, and to accurately determine the inflection point, it is necessary to carry out a frequency sweep test of the EIS of the middle and low-frequency bands, and then to determine the inflection point with the geometric characteristics. Taking this battery as an example, the EIS band shown in Fig. 6 is [10.866 Hz, 0.103 Hz], and 6 points are taken for each 10-fold frequency, which is tested by the electrochemical workstation in about 122 s, taking a relatively long time, and does not take advantage of the rapid estimation of the SOC of solid-state batteries for practical applications.

Fixed-frequency feature estimation SOC

Observing the semicircle in Fig. 6, it can be seen that the whole semicircle shows an obvious change rule with the

change of SOC, so it can be considered to select the appropriate frequency impedance and apply the online method to realize the EIS test and SOC estimation. Su et al. [28] utilize the step wave instead of a sine wave to achieve online acquisition of low-frequency EIS below 2 Hz without the aid of other equipment. In this paper, the impedance mode values at 5 frequencies of 1.89, 1.56, 1.28, 1.05, and 0.87 Hz are selected as the characteristic parameters. Their correlation coefficients with SOC are all above 0.95, showing strong correlation. They can be applied to estimate the SOC of the battery. The 5 frequency points are all under 2 Hz, and the stepped-wave method can be applied to obtain the EIS and achieve the SOC estimation by the online method. The 5 frequency points are all below 2 Hz, the stepped wave method can be applied online to obtain the EIS and realize the SOC estimation, the electrochemical workstation is applied to carry out the fixed-frequency test in the laboratory, and the data acquisition time is about 25 s, which is fast.

Fig. 6. Medium-low-frequency semicircle of solid-state battery (#1 - #3) with different SOC.

Fig. 6. Medium-low-frequency semicircle of solid-state battery (#1 - #3) with different SOC.

The above Gaussian regression exponential model is applied to train the model and estimate the SOC of battery #3, and the estimation results are shown in Fig. 10. The characteristic parameters show good estimation results, with the maximum absolute error of 1.5989% and the average absolute error of 0.2998%, which achieves the fast estimation of SOC taking into account the accuracy.

Model applicability under different SOHs

Since the SOHs of the 3 batteries used are different, the 3 batteries are used as the training set and validation set to build the SOC estimation model with the abovementioned fixed-frequency extracted features as an example, and to validate the applicability of the method to build the model under different SOHs. The grouping is shown in Fig. 11.

The estimation results are shown in Table 2, with group 1 having the best estimation results and group 2 having the worst. Therefore, when the training model uses a high SOH cell, the estimation of the SOC of the low SOH cell is better, and the method applies to different SOH cells.

Battery SOC estimation based on FOEKF algorithm

The EIS of this solid-state cell in different SOC states is shown in Fig. 5, and as the SOC decreases, the last semicircle in the impedance spectrum in the low frequency gradually comes to the forefront, and the distribution of relaxation time (DRT)

Fig. 7. The trend of characteristic parameters changing with SOC.

Fig. 7. The trend of characteristic parameters changing with SOC.

analysis of the EIS of the solid-state battery is carried out by using the tool from [31]. As shown in Fig. 12, this solid-state cell is decomposed into 5 relaxation time peaks. The semicircle above the real axis of the EIS is broken up into 3 parts, which adds an unknown electrochemical process to the solid-state cell over the lithium-ion cell. As shown in Fig. 5B, there are 3 semicircles above the real axis of the EIS for cell #2. Based on the specificity of the EIS of this solid-state battery, this paper proposes to fit the EIS through the third-order RC, taking into account the offset caused by the “dispersion effect”, and the 2 RCs are replaced by RQ. The EIS is fitted through the fractional-order model shown in Fig. 13. The impedance expression is shown in Eq. (18). The impedance spectral fitting software ZSimpWin is used to verify the fitting of the fractional-order model, as shown in Fig. 14, which is verified to be able to characterize the impedance spectral information under different SOC states.

Z(jw)=jwL+R0+R11+R1C1(jw)+R21+R2C2(jw)α1+R31+R3C3(jw)α2+1WD(jw)αD(18)\begin{array}{l} Z(jw)= jwL + R_{0} + \frac{R_{1}}{1 + R_{1} C_{1} (jw)} + \frac{R_{2}}{1 + R_{2} C_{2} (jw)^{\alpha_{1}}} \\ + \frac{R_{3}}{1 + R_{3} C_{3} (jw)^{\alpha_{2}}} + \frac{1}{W_{D} (jw)^{\alpha_{D}}} \\ \end{array} \quad(18)

To verify the fitting effect of the model, the fitting error is defined as shown in Eq. (19), and the error of fractional-order model fitting EIS of each SOC point of 3 batteries is calculated. The results are shown in Table 3. The model can better fit the EIS of the solid-state battery.

Fig. 8. The correlation between feature parameters and SOC.

Fig. 8. The correlation between feature parameters and SOC.

Err=1N∑i=1n(Rei∗−Rei)2+(Imi∗−Imi)2Err/%=100%N∑i=1n(Rei∗−Rei)2+(Imi∗−Imi)2Rei2+Imi2(19)\begin{aligned} Err = \frac{1}{N} \sum_{i = 1}^{n} \sqrt{\left(Re_{i}^{*} - Re_{i}\right)^{2} + \left(Im_{i}^{*} - Im_{i}\right)^{2}} \\ Err / \% = \frac{100 \%}{N} \sum_{i = 1}^{n} \frac{\sqrt{\left(Re_{i}^{*} - Re_{i}\right)^{2} + \left(Im_{i}^{*} - Im_{i}\right)^{2}}}{\sqrt{Re_{i}^{2} + Im_{i}^{2}}} \end{aligned} \quad(19)

where Re∗Re^{*}, Im∗Im^{*} denotes the impedance real and imaginary part fitted values, and Re, Im denotes the impedance real and imaginary part measured values.

Fractional-order equivalent circuit model state-space equations

To simplify the parameters and reduce the computational effort of the fractional-order extended KF algorithm, the inductance of the high-frequency inductive part and the Warburg impedance of the low-frequency diffusion process are neglected. Therefore, the fractional-order equivalent circuit model of the solid-state battery consists of an integer-order RC network, 2 fractional-order RQ networks, and the ohmic internal resistance R0R_{0} and the OCV. The expression for the terminal voltage of the fractional-order equivalent circuit model is:

{Δ1Vp(t)=−Vp(t)R1C1+I(t)C1Δα1VCPE1(t)=−VCPE1(t)R2C2+I(t)C2Δα2VCPE2(t)=−VCPE2(t)R3C3+I(t)C3Vo(t)=OCV(SOC)+I(t)Ro+VCPE1(t)+VCPE2(t)+Vp(t)(20)\left\{\begin{array}{c} \Delta^{1} V_{\mathrm{p}} (t)= - \frac{V_{\mathrm{p}} (t)}{R_{1} C_{1}} + \frac{I(t)}{C_{1}} \\ \Delta^{\alpha_{1}} V_{\mathrm{CPE}_{1}} (t)= - \frac{V_{\mathrm{CPE}_{1}} (t)}{R_{2} C_{2}} + \frac{I(t)}{C_{2}} \\ \Delta^{\alpha_{2}} V_{\mathrm{CPE}_{2}} (t)= - \frac{V_{\mathrm{CPE}_{2}} (t)}{R_{3} C_{3}} + \frac{I(t)}{C_{3}} \\ V_{o} (t)= OCV(SOC)+ I(t)R_{o} + V_{\mathrm{CPE}_{1}} (t)+ V_{\mathrm{CPE}_{2}} (t)+ V_{\mathrm{p}} (t)\end{array} \right. \quad(20)

Fig. 9. SOC estimation results.. A single-panel line chart plotting SOC estimation results over time. The horizontal axis represents time in seconds (up to about 15,000 s), the left vertical axis shows SOC as a percentage (0–100%), and the right vertical axis shows error as a percentage. Three data series are displayed with a legend in the upper right: a measured SOC line that decreases steadily from 100%, estimated SOC values shown as square markers that closely follow the measured line, and an error series shown with circle markers that fluctuates around zero.

Fig. 9. SOC estimation results.

Fig. 10. Fixed-frequency feature SOC estimation results.. A line plot showing SOC estimation results over time, with time on the x-axis (0 to 15×10³ s), SOC percentage on the left y-axis (0 to 100%), and error percentage on the right y-axis (-2 to 2%). Three data series are plotted: a measured SOC line that decreases steadily from 100%, estimated SOC points that follow the downward trend, and an error series that fluctuates around zero with a noticeable dip in the middle portion. A legend in the upper right identifies the three series.

Fig. 10. Fixed-frequency feature SOC estimation results.

GroupMaxAE/%MAE/%
Group 11.59890.2998
Group 25.09651.5422
Group 32.46040.9927

Fig. 11. Different SOH batteries grouped as training sets.

Table 2. Results of training models with different SOH batteries

Fig. 12. DRT analysis of different batteries.. A single-panel line plot showing γ(τ) in mΩ on the vertical axis (ranging from 0 to 10) versus τ in seconds on a logarithmic horizontal axis (from 1E-5 to 100). Multiple curves are displayed, each corresponding to a different percentage value from 10% to 100% as indicated in the legend. The curves exhibit oscillatory behavior with peaks at intermediate times and a sharp rise at longer times.

Fig. 12. DRT analysis of different batteries.

Fig. 13. Fractional-order model.. A schematic diagram of an electrical circuit arranged in series from left to right. It begins with an inductor labeled L and a resistor labeled R₀, followed by three parallel RC-type branches with resistors R₁, R₂, R₃ paired with elements C₁, Q₁, and Q₂ respectively, and terminates at an element labeled W on the right.

Fig. 13. Fractional-order model.

where α1\alpha1 and α2\alpha2 are the orders; VpV_{p}, VCPE1V_{CPE1}, and VCPE2V_{CPE2} are the polarization voltages of the integer order and fractional-order networks; I is the current through the battery, which is taken to be positive in the direction of charging; and OCV is the OCV, which is a polynomial function of the polynomial curve about the SOC, with the specific expression of Eq. (21), which is only affected by the ambient temperature and battery aging.

OCV(SOC)=a0+a1SOC1+a2SOC2+a3SOC3+a4SOC4+a5SOC5+a6SOC6+a7SOC7+a8SOC8+a9SOC9(21)\begin{aligned} OCV(SOC)= a_{0} + a_{1} SOC^{1} + a_{2} SOC^{2} + a_{3} SOC^{3} + a_{4} SOC^{4} + a_{5} SOC^{5} + a_{6} SOC^{6} + a_{7} SOC^{7} + a_{8} SOC^{8} + a_{9} SOC^{9}\end{aligned} \quad(21)

where a0a0 to a9a9 denote the polynomial fitting coefficients. In addition, the amount of change in cell SOC at moment tt is:

SO˙C(t)=I(t)3,600Q(22)S \dot{O} C(t)= \frac{I(t)}{3 , 600 Q} \quad(22)

where Q is the actual maximum available capacity of the solid-state battery. Taking the state variable x=[VpVCPE1VCPE2SOC]x = [V_{p} V_{CPE1} V_{CPE2} SOC], the state transfer matrix A=diag([−1/R1C1−1/R2C2−1/R3C30])A = \text{diag}([-1/R_{1} C_{1} - 1/R_{2} C_{2} - 1/R_{3} C_{3} 0]) is obtained from Eqs. (21) and (22), and the observation matrix C=[1 1 1 dOCV/dSOC]C = [1\ 1\ 1\ \text{dOCV/dSOC}], which leads to B=[1/C1 1/C2 1/C3 1/3,600Q]TB = [1/C_{1}\ 1/C_{2}\ 1/C_{3}\ 1/3,600Q]T and D=[Ro]D = [R_{o}], with u=[I(t)]u = [I(t)], and the fractional-order differential operator α=[1 α1 α2 1]\alpha = [1\ \alpha_{1}\ \alpha_{2}\ 1]. At this point, Eq. (20) can be written in the following form:

{ Δαx=Ax+Buy=Cx+Du(23)\begin{cases} \ \Delta^{\alpha} x = Ax + Bu & \\ & y = Cx + Du \end{cases} \quad(23)

Combining Eq. (7), then Eq. (24) holds. Where γj=\gamma_{j}= diag([(1j)(α1j)(α2j)(1j)]).\text{diag}\left(\left[\begin{pmatrix}1\\ j\end{pmatrix}\left(\begin{array}{c}\alpha_{1}\\ j\end{array}\right)\left(\begin{array}{c}\alpha_{2}\\ j\end{array}\right)\left(\begin{array}{c}1\\ j\end{array}\right)\right]\right).

Fig. 14. Fractional-order model fitting effect for #2 and #3.. A scatter plot showing impedance data with the horizontal axis labeled 'Re(Z) / mΩ' and the vertical axis labeled '-Im(Z) / mΩ'. Multiple data series are plotted using distinct markers, with a legend distinguishing 'real' and 'fitting' values at several percentage levels (10%, 30%, 50%, 70%, 90%). The curves form a characteristic impedance spectrum shape, with real and fitted series overlapping across the plotted range. A Nyquist plot labeled B (marked #3) showing the negative imaginary part of impedance, -Im(Z), versus the real part, Re(Z), both in mΩ. Multiple data series compare measured ('real') and fitted curves at several state-of-charge percentages (90%, 70%, 50%, 30%, and 10%), each distinguished by different markers and identified in a legend. The curves form an arc-shaped impedance response, with the fitting results closely tracking the real data across the plotted range.

Fig. 14. Fractional-order model fitting effect for #2 and #3.

Table 3. Fractional-order model fitting error

Battery SOC#1#2#3
Err/mΩErr/%Err/mΩErr/%Err/mΩErr/%
100%0.0700.7680.0430.4450.0550.577
90%0.0400.4410.0220.2830.0720.763
80%0.0580.6070.0580.5700.0650.648
70%0.0520.5350.0250.2940.0600.668
60%0.0981.0200.0620.5870.0590.662
50%0.0980.9980.0240.2800.0280.334
40%0.0700.7080.0880.7590.0550.606
30%0.0890.8620.0380.5150.0500.559
20%0.0650.6090.1161.2740.0570.576
10%0.0590.5310.0430.5350.0300.368

xk=[ΔTsαA+diag (α)]xk−1+ΔTsαBuk−1−∑j=2k(−1)jγjxk−j(24)x_k = \left[ \Delta T_s^\alpha A + \text{diag } (\alpha)\right] x_{k-1} + \Delta T_s^\alpha Bu_{k-1} - \sum_{j=2}^k(-1)^j \gamma_j x_{k-j} \quad(24)

Ultimately, the general form of the discrete expression of the state space equation for the fractional-order model of the solid-state battery is obtained as:

{xk=[ΔTsαA+diag⁡(α)]xk−1+ΔTsαBuk−1−∑j=2k(−1)jγjxk−j+ωkyk=Ckxk+Dkuk+vk(25)\left\{\begin{array}{c} x_{k} = \left[ \Delta T_{\mathrm{s}}^{\boldsymbol{\alpha}} A + \operatorname{diag} (\boldsymbol{\alpha})\right] x_{k - 1} + \Delta T_{\mathrm{s}}^{\boldsymbol{\alpha}} Bu_{k - 1} - \sum_{j = 2}^{k} (- 1)^{j} \gamma_{j} x_{k - j} + \omega_{k} \\ y_{k} = C_{k} x_{k} + D_{k} u_{k} + v_{k} \end{array} \right. \quad(25)

where ωk\omega_{k} is the process noise with mean 0 and covariance QkQ_{k}, vkv_{k} is the measurement noise with mean 0 and covariance RkR_{k}, and ωk\omega_{k} and vkv_{k} are uncorrelated.

Validation of SOC estimation results for different operating conditions

The voltage simulation accuracy of the above fractional-order model is verified through different working conditions, and the simulation results are shown in Fig. 15. The RMSE (root mean square error) of the voltage simulation is 8 mV for the DST condition, and the RMSE of the voltage simulation is 7.3 mV for the FUDS condition, which shows that the above fractional-order model can simulate the battery voltage well and can be used for the SOC estimation.

The above EKF algorithm based on the fractional-order model performs the SOC estimation, and the accuracy is verified using DST and FUDS working conditions. First, the initial value is set to 50% SOC, and the SOC estimation results of different working conditions are shown in Fig. 16. The DST condition converges to an error of no more than 5% after 58 s and to an error of no more than 2% after 126 s, the converged estimation results have a slight deviation of more than 2% at around 45% SOC, and the rest of the estimation results have errors of less than 1%. The FUDS condition converges to an error of no more than 5% after 53 s and to an error of no more than 2% after 109 s, and the converged estimation results are good with an error of less than 1%.

Different initial values are set under FUDS conditions to verify the convergence speed and estimation accuracy of FOEKF. The initial SOC is set to be 0%, 30%, 60%, and 90%, respectively. The estimation results are shown in Fig. 17, and the convergence time for different initial values is shown in Table 4. The FOEKF algorithm can be effectively corrected when the initial value of the battery is unknown, but the convergence speed is affected by the deviation of the initial value,

and the larger the deviation of the initial value, the slower the convergence speed.

SOC estimation for solid-state batteries based on SOC-OCV curves

Currently, the SOC estimation of electric vehicles mostly uses the ampere-time integration method, and the determination of the initial value becomes the main factor affecting its accuracy. The acquisition of the initial value can be calculated based on the SOC-OCV curve lookup table, while the acquisition of OCV requires the battery to be stationary for a long time. Therefore, the OCV can be accurately obtained only when the vehicle is parked and restarted for longer. When the quiescence

Fig. 15. Voltage simulation results under different working conditions — DST (A) and FUDS (B).. Two line plots show voltage versus time, each comparing a 'True' curve with a 'Simulation' curve. Both panels display a general downward trend in voltage from approximately 4.2 V to 3.6 V over time, with the simulation closely tracking the true data. One panel is labeled DST with a time axis extending to about 15,000 s, and the other is labeled FUDS with a time axis extending to about 12,000 s.

Fig. 15. Voltage simulation results under different working conditions — DST (A) and FUDS (B).

Fig. 16. Fractional EKF estimation of SOC under different operating conditions — DST (A) and FUDS (B).. The region contains two side-by-side line plots showing State of Charge (SOC, in %) versus time (in seconds) under different operating conditions. Each plot compares a Measured curve against an Estimated curve, with both curves starting near the top of the SOC range and decreasing over time while closely overlapping. Each panel includes a zoomed-in inset focusing on the initial time interval to highlight small fluctuations and divergence between the two curves, with an arrow pointing from the main plot to the inset.

Fig. 16. Fractional EKF estimation of SOC under different operating conditions — DST (A) and FUDS (B).

Fig. 17. Convergence of different initial values under FUDS operating conditions.. A line plot shows State of Charge (SOC, %) versus Time (s), with the x-axis extending to approximately 14,000 s and the y-axis from 0 to 100%. Five lines are plotted: a measured reference curve and four curves corresponding to different initial SOC values (0%, 30%, 60%, and 90%), as identified in a legend. An inset provides a zoomed-in view of the early time period, highlighting the initial divergence and convergence behavior of the curves, with an arrow pointing from the main plot to the inset region.

Fig. 17. Convergence of different initial values under FUDS operating conditions.

Table 4. Initial value convergence under FUDS working condition

Initial valueLess than 5% error in time/sLess than 2% error in time/s
0%329446
30%93157
60%3881
90%Initial error less than 5%12

Table 5. Comparison of SOC estimation methods for solid-state battery

MethodAdvantageDisadvantage
FOEKFConvergence of initial valuesLong convergence time, greatly influenced by initial values. The model needs to obtain complete EIS data.
SOC-OCV curveThe results are accurate and do not require algorithmsThe battery needs to be left standing for a long time. Only applicable to single-point estimation.
EIS parametersAccurate estimation and short-time consumptionDifferent frequency impedance tests need to be conducted on the battery.

is not sufficient or the sampling error precision of the equipment is limited, the OCV will not be accurately obtained, which also leads to the introduction of SOC estimation error, especially when there is a voltage plateau in the battery OCV, and the error will be greatly increased. The SOC-OCV curve of this solid-state battery is shown in Fig. 18, and the SOC of the current battery can be obtained by obtaining the battery OCV and using the mapping relationship between OCV and SOC of this curve.

Due to the effect of insufficient quiescence or sampling error, when the OCV deviation is 5 mV, the absolute errors generated under different SOCs are shown in Fig. 19, with a maximum absolute error of 1.71% and an average absolute error of 0.507%. Under the condition of full resting and guaranteeing the sampling accuracy, the method is accurate in estimation, but it requires the battery to be rested for a long time to make its voltage reach a stable state, and it is generally applied to the initial SOC calibration of restarting after a long time of parking and can only be used for the estimation of a single point of SOC.

Fig. 18 SOC-OCV curve.. A single line plot shows Open Circuit Voltage (OCV) in volts on the vertical axis (ranging from about 3.2 to 4.2 V) versus State of Charge (SOC) as a percentage on the horizontal axis (0 to 100%). The curve rises steeply from around 3.2 V at low SOC, then increases more gradually, reaching approximately 4.15 V near 100% SOC. A legend in the upper right identifies the curve as OCV.

Fig. 18 SOC-OCV curve.

Fig. 19. Absolute error generated by sampling under different SOC.. A single-panel line plot showing error (%) as a function of state of charge (SOC, %), with SOC on the x-axis ranging from 0 to 100 and error on the y-axis ranging from 0 to 1.8. The curve starts near zero, rises sharply to a peak near 20% SOC (around 1.7%), dips to a local minimum, rises again to a second smaller peak near the middle of the SOC range, and then gradually decreases toward 100% SOC. A legend in the upper right labels the curve as 'Error'.

Fig. 19. Absolute error generated by sampling under different SOC.

Method summary

All of the above methods can achieve a more accurate estimation of the SOC of solid-state batteries, but there are certain advantages, disadvantages, and applicability of each method, as shown in Table 5.

Among the above 3 methods, the EIS estimation method needs to test the EIS of the battery, even if the fixed-frequency points below 2 Hz can be measured without other equipment, but it takes 25 s to obtain the parameters. The estimation process mainly consists of importing data into the model and outputting the results, and the estimation process time can be ignored, which can be used for the initial SOC calibration and is almost unaffected by the battery SOH, as long as a good model is trained. The FOEKF method is mainly for online real-time applications, and online estimation can be realized as long as the model is built, but its model needs to be updated according to the change of the SOH of the battery. The SOC-OCV method can be mainly used for the calibration of the initial SOC, which can be almost instantly accomplished by only measuring the battery voltage, but it also needs to update the curve with the change of the SOH of the battery. Therefore, each of the 3 methods has its advantages and disadvantages. The SOC-OCV method and the EIS method can be used for initial SOC calibration. In terms of computational cost, the SOC-OCV method has a lower computational cost, but it depends on the timely updating of the curves with the change of SOH, while the EIS method is able to avoid this problem.

The results show that establishing an estimation model based on EIS extracted features can balance accuracy and speed. If the online implementation method of EIS is combined with the extraction of battery fixed-frequency impedance modulus as a feature parameter, it is expected to achieve the accurate estimation of online SOC for the full cycle of solid-state battery operation.

Conclusion

This paper addresses the existing research gap in modeling and parameter estimation for solid-state batteries. It presents a novel approach by extracting impedance parameters at various frequencies using EIS testing, leading to the development of a solid-state battery SOC estimation model. This model enables fast and precise SOC estimation for solid-state batteries. Additionally, the article performs a thorough analysis of 2 other methods, namely, the FOEKF and the SOC-OCV table lookup approach. By evaluating the strengths and weaknesses of these methods, the article provides valuable insights into their suitability for SOC estimation in the context of solid-state batteries.

  1. Based on the low-frequency characteristic semicircle in the EIS, the feature parameters that are strongly correlated with the SOC are extracted, and good estimation results are achieved. Considering the online implementation of the method, the fixed-frequency impedance modulus below 2 Hz is selected as the feature parameter, and the estimation results are still good. The applicability of the method under different SOHs is verified.

  • Considering the 3 semicircular frequency bands of solid-state battery EIS, the RQ process is added to establish a fractional-order model applicable to solid-state batteries, and the SOC estimation model is established by combining the FOEKF algorithm. The estimation accuracy is verified by DST and FUDS dynamic working conditions, and the convergence time

and estimation accuracy of different initial values are analyzed. Then, the estimation error of the SOC-OCV method is analyzed by the resting insufficiency and sampling error.

  1. Based on the EIS and SOC-OCV methods, the initial SOC of the battery can be calibrated; the full-cycle SOC estimation of the battery is achieved by applying FOEKF in long-time scales and corrected by combining the EIS method. The accurate estimation of the full cycle of SOC can be ensured, which provides a reference for the subsequent research and application of solid-state batteries.

Acknowledgments

Funding: This work was supported by the National Natural Science Foundation of China (grant no. 52177206) and Beijing Nova Program (grant no. 20220484153).

Author contributions: J.P. performed the numerical experiments. X.Z. and J.P. analyzed the data. All the authors contributed to the writing and revising of the manuscript.

Competing interests: The authors declare that they have no competing interests.

Data Availability

The data are available from the authors upon a reasonable request.

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