Adaptive Dead-Zone Dual Sliding Mode Observer for Reliable Electrochemical Model-Based SOC Estimation

TL;DR

This paper introduces an adaptive dead-zone dual sliding mode observer for electrochemical SOC estimation, improving accuracy, robustness, and computational efficiency in battery management systems.

Contribution

It proposes a novel adaptive dead-zone dual SMO based on an improved electrochemical model, enhancing stability, accuracy, and real-time applicability for SOC estimation.

Findings

Achieves SOC estimation errors within 0.2% under correct initialization.

Maintains errors below 1% with 30% initial SOC error.

Reduces computational time compared to conventional dual SMO.

Abstract

Accurate state of charge (SOC) estimation is critical for ensuring the safety, reliability, and efficiency of lithium-ion batteries in electric vehicles and energy storage systems. Electrochemical models provide high fidelity for SOC estimation but introduce challenges due to parameter variations, nonlinearities, and computational complexity. To address these issues, this paper proposes an adaptive dead-zone dual sliding mode observer(SMO) based on an improved electrochemical single-particle model. The algorithm integrates a state observer for SOC estimation and a parameter observer for online parameter adaptation. A Lyapunov-derived adaptive dead-zone is introduced to ensure stability, activating parameter updates only when the terminal voltage error lies within a rigorously defined bound. The proposed method was validated under constant-current and UDDS dynamic conditions. Results demonstrate that the adaptive dead-zone dual SMO achieves superior accuracy compared with conventional dual SMO and equivalent circuit model-based EKF methods, maintaining SOC estimation errors within 0.2% under correct initialization and below 1% under a 30% initial SOC error, with rapid convergence. Computational efficiency analysis further shows that the adaptive dead-zone dual sliding mode observer reduces execution time compared with the conventional dual SMO by limiting unnecessary parameter updates, highlighting its suitability for real-time battery management applications. Moreover, robustness under battery aging was confirmed using a cycle-aging model, where the adaptive dead-zone dual SMO maintained stable SOC estimation despite parameter drift. These findings indicate that the proposed method offers a reliable, accurate, and computationally efficient solution for SOC estimation.

Tables

Table 1. Table 1: Specifications of the Panasonic NCR18650PF battery

Characteristic	Value
Rated capacity	Min. 2700 mAh; Typical 2900 mAh
Nominal voltage	3.6 V
Voltage range	2.5 V – 4.2 V
Charge method	CC–CV, 4.2 V cutoff, 1365 mA standard current
Weight (max.)	46 g
Energy density (volumetric / gravimetric)	577 Wh/L; 207 Wh/kg
Positive electrode (cathode) material	LiNiMnCoO2 (NMC)
Negative electrode (anode) material	Graphite

Table 2. Table 2: Improved extended single-particle model parameters

Parameter	Description	Value	Unit
$D_n$	Capacity-distribution coefficient (negative electrode)	0.6533	–
$D_p$	Capacity-distribution coefficient (positive electrode)	0.7284	–
$F$	Faraday constant	96485.3	C$\cdot$mol-1
$P_{\mathrm{act}}$	Reaction-polarization coefficient (BV auxiliary)	90424	s$\cdot$(mol$\cdot$m-3)1/2
$P_{\mathrm{con},a}$	Electrolyte diffusion gain (positive side)	150	mol$\cdot$m-3$\cdot$A-1
$P_{\mathrm{con},b}$	Electrolyte diffusion gain (negative side)	60	mol$\cdot$m-3$\cdot$A-1
$Q_{\mathrm{all}}$	Total battery capacity	2894.1	mAh
$R$	Ideal gas constant	8.314	J$\cdot$mol-1$\cdot$K-1
$R_{\mathrm{ohm}}$	Ohmic internal resistance	0.045	$\Omega$
$T$	Internal temperature (assumed constant)	298.15	K
$t_{+}$	Li+ transference number	0.363	–
$c_0$	Initial electrolyte concentration	1000	mol$\cdot$m-3
${\tau }_e$	Electrolyte diffusion time constant	80	s
${\tau }_{s,n,1}$	Solid diffusion time constant (negative) 1	1.1	s
${\tau }_{s,n,2}$	Solid diffusion time constant (negative) 2	10	s
${\tau }_{s,n,3}$	Solid diffusion time constant (negative) 3	0.05	s
${\tau }_{s,p}$	Solid diffusion time constant (positive)	1.85	s
$x_{s,n,0}$	Initial average solid stoichiometry (negative)	0.745	–
$x_{s,p,0}$	Initial average solid stoichiometry (positive)	0.68	–
$n$	Peukert exponent	1.021	–

Table 3. Table 3: SOC estimation errors of each algorithm without initial SOC errors

Working condition	Algorithm	RMSE / %
UDDS	Adaptive dead-zone dual SMO	0.14
UDDS	dual SMO	0.20
UDDS	EKF with ECM [40]	1.08
UDDS	Fixed dead-zone dual EKF with ECM [40]	0.23
1C	Adaptive dead-zone dual SMO	0.13
1C	dual SMO	0.18
1C	EKF with ECM [40]	2.79
1C	Fixed dead-zone dual EKF with ECM [40]	0.39

Table 4. Table 4: SOC estimation errors of each algorithm with 30% initial SOC error

Working condition	Algorithm	RMSE / %
UDDS	Adaptive dead-zone dual SMO	0.68
UDDS	Dual SMO	0.95
UDDS	EKF with ECM [40]	1.28
UDDS	Fixed dead-zone dual EKF with ECM [40]	0.74
1C	Adaptive dead-zone dual SMO	0.17
1C	Dual SMO	0.21
1C	EKF with ECM [40]	3.24
1C	Fixed dead-zone dual EKF with ECM [40]	2.05

Table 5. Table 5: Computation time of different SOC estimation algorithms

Working condition	Initial SOC	SMO	Dual SMO	Adaptive dead-zone Dual SMO
		(ms)	(ms)	(ms)
UDDS	No initial error	6.105	15.147	13.255
UDDS	30% initial error	5.986	14.258	13.419
1C	No initial error	6.093	14.226	11.254
1C	30% initial error	5.951	14.037	10.952

Table 6. Table 6: SOC estimation error (RMSE) under UDDS condition at different cumulative cycling levels

Condition	Algorithm	RMSE (%)
100 cycles	Dual SMO	0.432
	Fixed dead-zone dual SMO	0.193
	Adaptive dead-zone dual SMO	0.187
400 cycles	Dual SMO	0.782
	Fixed dead-zone dual SMO	0.236
	Adaptive dead-zone dual SMO	0.205

Equations

$$U(t)=E_{\mathrm{OCV}}(t)-{\eta }_{\mathrm{con}}(t)-{\eta }_{\mathrm{act}}(t)-{\eta }_{\mathrm{ohm}}(t),$$

$$E_{\mathrm{OCV}}(t)=U_p\left[x_{s,p,\mathrm{surf}}(t)\right]-U_n\left[x_{s,n,\mathrm{surf}}(t)\right],$$

$$\frac{\partial x_{s,i}}{\partial t}(r,t)=\frac{D_{s,i}}{R_{s,i}^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial x_{s,i}}{\partial r}(r,t)\right),$$

$${\frac{\partial x_{s,i}}{\partial r}|}_{r=0}$$

$${\frac{\partial x_{s,i}}{\partial r}|}_{r=R_{s,i}}$$

$$x_{s,p,\mathrm{surf}}(t)=x_{s,p,\mathrm{avg}}(t)+\Delta x_{s,p}(t),$$

$$x_{s,n,\mathrm{surf}}(t)=x_{s,n,\mathrm{avg}}(t)+\Delta x_{s,n}(t),$$

$$Q_p=\frac{Q_{\mathrm{all}}}{D_p},$$

$$Q_n=\frac{Q_{\mathrm{all}}}{D_n},$$

$$x_{s,p,\mathrm{avg}}(t)=x_{s,p,0}+{\int }_0^t\frac{D_p}{Q_{\mathrm{all}}}I(\tau )d\tau ,$$

$$x_{s,n,\mathrm{avg}}(t)=x_{s,n,0}-\frac{D_n}{D_p}\left[x_{s,p,\mathrm{avg}}(t)-x_{s,p,0}\right],$$

$${\dot{\Delta x}}_{s,p}(t)=\frac{1}{{\tau }_p^s}\left(\frac{12}{7}\frac{D_p{r}_p^s}{Q_{\mathrm{all}}}I(t)-\Delta x_{s,p}(t)\right),$$

$${\dot{\Delta x}}_{s,n}(t)=\frac{1}{{\tau }_n^s}\left(\frac{12}{7}\frac{D_n{r}_n^s}{Q_{\mathrm{all}}}I(t)-\Delta x_{s,n}(t)\right).$$

$$\Delta x_{s,p}(t_{k+1})=\Delta x_{s,p}(t_k)+\frac{\Delta t}{{\tau }_p^s}\left(\frac{12}{7}\frac{D_p{r}_p^s}{Q_{\mathrm{all}}}I(t_k)-\Delta x_{s,p}(t_k)\right),$$

$$\Delta x_{s,n}(t_{k+1})=\Delta x_{s,n}(t_k)+\frac{\Delta t}{{\tau }_n^s}\left(\frac{12}{7}\frac{D_n{r}_n^s}{Q_{\mathrm{all}}}I(t_k)-\Delta x_{s,n}(t_k)\right).$$

$${\tau }_{s,n}=\{\begin{array}{ll}{\tau }_{n,1}^s,& I<1.5C,\\ {\tau }_{n,2}^s,& 1.5C\le I<2.5C,\\ {\tau }_{n,3}^s,& I\ge 2.5C.\end{array}$$

$$Q_{\mathrm{now}}=Q_{\mathrm{all}}{\left(\frac{C_{\mathrm{ref}}}{C_{\mathrm{now}}}\right)}^{n-1},$$

$${\eta }_{\mathrm{con}}(t)=\frac{2RT}{F}(1-t_{+})\ln \left(\frac{c_0+\Delta c_1(t)}{c_0-\Delta c_2(t)}\right),$$

$$\Delta c_1(t_{k+1})=\Delta c_1(t_k)+\frac{\Delta t}{{\tau }_e}\left[P_{\mathrm{con},a}I(t_k)-\Delta c_1(t_k)\right],$$

$$\Delta c_2(t_{k+1})=\Delta c_2(t_k)+\frac{\Delta t}{{\tau }_e}\left[P_{\mathrm{con},b}I(t_k)-\Delta c_2(t_k)\right],$$

$${\eta }_{\mathrm{act}}(t)=\frac{2RT}{F}\left\{\ln \left[\sqrt{m_n^2(t)+1}+m_n(t)\right]+\ln \left[\sqrt{m_p^2(t)+1}+m_p(t)\right]\right\},$$

$$m_p(t)=\frac{D_p}{6Q_{\mathrm{all}}{c}_0^{0.5}}{\left(1-x_{s,p,\mathrm{surf}}(t)\right)}^{0.5}{\left(x_{s,p,\mathrm{surf}}(t)\right)}^{0.5}{P}_{\mathrm{act}}I(t),$$

$$m_n(t)=\frac{D_n}{6Q_{\mathrm{all}}{c}_0^{0.5}}{\left(1-x_{s,n,\mathrm{surf}}(t)\right)}^{0.5}{\left(x_{s,n,\mathrm{surf}}(t)\right)}^{0.5}{P}_{\mathrm{act}}I(t),$$

$${\eta }_{\mathrm{ohm}}(t)=R_{\mathrm{ohm}}I(t),$$

$$x_k=A{x}_{k-1}+B{i}_{k-1}+w_{k-1},$$

$$A=\left[\begin{array}{ccccc}1& 0& 0& 0& 0\\ 0& 1-\frac{\Delta t}{{\tau }_{s,p}}& 0& 0& 0\\ 0& 0& 1-\frac{\Delta t}{{\tau }_{s,n}}& 0& 0\\ 0& 0& 0& 1-\frac{\Delta t}{{\tau }_e}& 0\\ 0& 0& 0& 0& 1-\frac{\Delta t}{{\tau }_e}\end{array}\right],$$

$$B={\left[\begin{array}{ccccc}\frac{\Delta t}{Q_{\mathrm{all}}}& \frac{12D_p\Delta t}{7Q_{\mathrm{all}}}& \frac{12D_n\Delta t}{7Q_{\mathrm{all}}}& \frac{P_{\mathrm{con},a}}{{\tau }_e}& \frac{P_{\mathrm{con},b}}{{\tau }_e}\end{array}\right]}^T.$$

$$w_k=\Delta A{x}_k+B\Delta i_k+\Delta B(i_k+\Delta i_k)+{\delta }_k,$$

$$y_{k-1}\triangleq U(t_{k-1})=E_{\mathrm{OCV}}\left[x_{s,i,\mathrm{surf}}(t_{k-1})\right]-{\eta }_{\mathrm{con}}(t_{k-1})-{\eta }_{\mathrm{act}}(t_{k-1})-{\eta }_{\mathrm{ohm}}(t_{k-1})+v_{k-1},$$

$$\hat{x}_{k}=A\hat{x}_{k-1}+Bi_{k-1}+K\,\mathrm{sgn}\!\big{(}x_{k-1}-\hat{x}_{k-1}\big{)},$$

$$\tilde{x}_{k}=A\tilde{x}_{k-1}+\Delta w-K\,\mathrm{sgn}\!\big{(}\tilde{x}_{k-1}\big{)},$$

Full text

\affiliation

[inst1]organization=School of Vehicles and Intelligent Transportation,Fuyao University of Science and Technology,city=Fuzhou, postcode=350109, country=China \affiliation[inst2]organization=School of Mechanical Engineering, Southwest Jiaotong University,city=Chengdu, postcode=610031, country=China \affiliation[inst3]organization=VITO,addressline=Boeretang 200, city=Mol, postcode=2400, country=Belgium \affiliation[inst4]organization=EnergyVille,addressline=Thor Park 8310, city=Genk, postcode=3600, country=Belgium

1 Introduction

Lithium-ion batteries, owing to their high energy and power density compared with other rechargeable batteries, have become the dominant choice for pure electric vehicles, hybrid electric vehicles, as well as a variety of energy storage and power supply applications [1, 2, 3, 4]. Despite the maturity of lithium-ion battery materials and chemistry, the complexity of real-world operating conditions requires advanced battery management systems (BMSs) to serve as the “brain” of the battery, ensuring safe, reliable, and efficient performance. Among the core functions of a BMS, State of Charge (SOC) estimation plays a central role. Accurate SOC estimation not only enables optimal energy management strategies but also safeguards the efficiency, reliability, and longevity of lithium-ion batteries. To further enhance estimation accuracy, high-fidelity battery models are indispensable. In particular, electrochemical models, which capture the internal mechanisms of batteries, have attracted increasing attention from researchers for their potential in improving the reliability of SOC estimation [5, 6].

There are three main categories of battery models: equivalent circuit models (ECMs) [7], electrochemical models [8, 9], and data-driven models [10, 11]. ECMs are widely used due to their simple structure, ease of parameter identification, and computational efficiency. They are particularly suitable for real-time BMS implementation. However, ECMs cannot capture the underlying electrochemical processes of batteries and usually rely on parameter fitting under specific operating conditions, which limits their predictive accuracy when the battery experiences varying temperatures, aging, or dynamic load profiles. Data-driven models, empowered by machine learning and deep learning, can achieve high prediction accuracy when trained with large-scale datasets. Nevertheless, their strong dependence on high-quality training data, lack of physical interpretability, and high computational and hardware requirements make them less suitable for real-time onboard BMS applications. Electrochemical models, in contrast, describe the internal electrochemical reactions of batteries based on physical principles, thereby providing high accuracy and generalizability across operating conditions. Their major drawback lies in the high computational burden associated with solving complex partial differential equations, which hinders direct real-time applications. To overcome this, simplified electrochemical models have been developed. Among them, the improved single-particle model (SPM) [12] reduces the order of partial differential equations while considering concentration gradients, offering a balance between accuracy and computational efficiency. Compared with the traditional pseudo-two-dimensional (P2D) model, the improved SPM retains key internal dynamics with lower complexity, making it an attractive choice for observer design in BMS applications [13, 14, 15].

Since SOC cannot be directly measured, numerous estimation methods have been proposed. The simplest and most commonly used is the Coulomb counting method [16], which integrates the current over time to estimate SOC. Although computationally inexpensive, this method accumulates drift errors over long-term operation. The open-circuit voltage (OCV) method [17, 18] infers SOC from the battery’s OCV–SOC relationship, but it requires the battery to rest for a long relaxation time, making it impractical under dynamic driving conditions. Both methods are open-loop and therefore unable to correct accumulated errors. Model-based estimation methods have thus received significant attention, as they combine battery models with observers or filters to achieve closed-loop estimation. The Kalman filter (KF) family, including the extended Kalman filter (EKF) [19, 20, 21] and unscented Kalman filter (UKF) [22], provides optimal recursive state estimation with good real-time capability. However, the accuracy of KF-based methods heavily depends on the model fidelity and noise assumptions, and they can be sensitive to parameter variations and unmodeled nonlinearities. Particle filters (PFs) [23, 24, 25] are more suitable for highly nonlinear and non-Gaussian systems, but they require a large number of particles, leading to high computational cost and potential instability, especially as the battery degrades. In contrast, sliding mode observers (SMOs) [26, 27, 28] offer several distinct advantages. As nonlinear observers, SMOs are inherently robust against model uncertainties, external disturbances, and parameter variations, while maintaining high estimation accuracy. They can effectively handle strong system nonlinearities and unmodeled dynamics without requiring exact noise statistics, which distinguishes them from KF-based methods. Moreover, SMOs are computationally efficient compared to PFs, making them well suited for real-time SOC estimation in practical BMSs. These advantages have motivated extensive research into improving SMO design, such as incorporating adaptive mechanisms and dead-zone strategies to mitigate chattering and enhance estimation reliability.

A common limitation of model-based approaches is that model parameters are usually identified or estimated from off-line data. As battery parameters evolve with aging and the parameters identified offline inevitably contain fitting errors, it becomes necessary to modify model parameters online in order to achieve accurate SOC estimation in practical applications. Building upon state estimation algorithms, researchers have proposed joint state–parameter estimation approaches that incorporate online parameter adaptation. Representative examples include recursive least square (RLS)-based methods [29], dual Kalman filter algorithms [30, 31, 32, 33], dual $H_{\infty}$ filters [34], and dual particle filters [35]. Most of these studies are based on ECMs. In recent years, however, several researchers have attempted to leverage more accurate electrochemical models for joint estimation of battery states and parameters. For instance, Gao et al. applied a dual extended Kalman filter (DEKF) to simultaneously estimate SOC and state of health (SOH) using an electrochemical model [36]; Hosseininasab et al. employed a dual UKF for SOC estimation, achieving improved accuracy [37]; Wang et al. introduced a dual particle filter based on electrochemical models for SOC estimation [38]. The advantage of using electrochemical models lies in their higher fidelity, which improves SOC estimation accuracy, while the estimated parameters retain physical meaning and can further support the estimation of other health indicators such as SOH, state of power (SOP), and state of available energy (SOA), as well as fault diagnosis.

However, simultaneous state and parameter estimation is prone to divergence, i.e., the algorithm may fail to converge to the correct values. This occurs because estimation errors in the state observer can propagate into the parameter estimator, resulting in biased parameter updates and eventual model deviation. Consequently, the application of joint state–parameter estimation has been limited. To address this issue, our previous work proposed a dead-zone strategy [39] to improve the stability of joint estimation algorithms based on ECMs. Specifically, a DEKF with a predefined dead zone for tracking errors was developed and demonstrated effective mitigation of divergence issues under large initial state errors [40]. The core idea was that by designing a parameter estimator with a dead zone, the transfer of state estimation errors into the parameter estimation loop could be reduced, thereby enhancing robustness. However, the dead zone in [40] was defined as a fixed interval. Such a static design cannot completely block state estimation errors from entering the parameter estimation process. Over the entire battery life cycle, residual state errors can still accumulate, eventually causing divergence of the fixed dead-zone algorithm.

To overcome these limitations, this paper proposes a novel joint state–parameter estimation algorithm based on a dual SMO with an analytically derived adaptive dead-zone structure. By leveraging a higher-fidelity electrochemical model, the proposed method dynamically adjusts the dead-zone boundary according to Lyapunov-based stability conditions, ensuring that parameter updates occur only when the system is provably stable. This adaptive mechanism not only eliminates the drawbacks of static dead zones but also provides formal guarantees of convergence and robustness against model uncertainties, operating condition variability, and long-term aging effects. The effectiveness of the algorithm is validated under both constant-current and dynamic Urban Dynamometer Driving Schedule (UDDS) conditions, as well as under simulated aging scenarios, thereby demonstrating its potential for reliable long-term deployment in advanced battery management systems.

The overall structure of this study is shown in Figure 1 The remainder of this paper is organized as follows. Section 2 introduces electrochemical modeling, battery test and parameter identification. Section 3 presents the design of adaptive dual SMO algorithm for joint state and parameter estimation. Section 4 provides validation results of the proposed algorithm. Section 5 concludes the study and discusses potential directions for future research.

2 An improved extended single-particle model

The simplified electrochemical model used in this paper is an improved extended single particle model (IESP). The IESP model is developed based on the extended single particle model(ESP) by making reasonable modifications to the ohmic resistance and capacitance [41, 12].

2.1 Battery terminal voltage

The input of the ESP model is the applied current $I$, and the output is the terminal voltage $U$. The terminal voltage is obtained by subtracting the concentration over-potential, reaction over-potential, and ohmic over-potential from the open-circuit voltage (OCV):

[TABLE]

where

• •

$E_{\mathrm{OCV}}(t)$ is the open-circuit voltage determined by the electrode surface lithium-ion concentration,

• •

$\eta_{\mathrm{con}}(t)$ is the concentration over-potential caused by the lithium-ion diffusion gradient,

• •

$\eta_{\mathrm{act}}(t)$ is the activation over-potential associated with electrochemical reaction kinetics,

• •

$\eta_{\mathrm{ohm}}(t)$ is the ohmic over-potential arising from the internal resistance of the electrolyte, electrodes, and current collectors.

2.2 Open-circuit voltage (OCV) and solid diffusion equations

The OCV is directly determined by the lithium-ion concentration on the solid surface of the positive and negative electrodes. Assuming that the internal temperature of the battery remains constant, the OCV can be expressed as

[TABLE]

where $U_{p}$ and $U_{n}$ are the equilibrium potentials of the positive and negative electrodes, and $x_{s,p,\mathrm{surf}}(t)$ and $x_{s,n,\mathrm{surf}}(t)$ denote the lithium-ion concentrations on the solid-phase surface of the positive and negative electrodes, respectively.

The solid-phase lithium concentration $x_{s,i}(r,t)$ inside a representative spherical particle evolves according to Fick’s second law:

[TABLE]

where $i\in\{p,n\}$ denotes the positive ($p$) and negative ($n$) electrodes, $D_{s,i}$ is the solid-phase diffusion coefficient, and $R_{s,i}$ is the particle radius. A uniform initial concentration $x_{s,i}(r,0)=x_{i,0}$ is assumed.

The boundary conditions accompanying (3) are

[TABLE]

which enforce symmetry at the particle centre and prescribe the intercalation flux $j_{i}(t)$ at the particle surface.

Direct solution of Eq. (3) is computationally expensive. Therefore, spatial discretization or approximate methods are typically employed, including the parabolic approximation method, Padé approximation, spectral methods, and finite-difference methods [13]. Among these, the parabolic approximation method [42] is widely adopted because of its simplicity, computational efficiency, and relatively high accuracy.

Using the parabolic approximation, the surface concentrations can be expressed as

[TABLE]

[TABLE]

where $x_{s,i,\mathrm{avg}}(t)$ is the average solid-phase concentration of electrode $i$ and $\Delta x_{s,i}(t)$ represents the deviation term describing the surface-to-average concentration difference.

$Q_{p}$ and $Q_{n}$ represent the theoretical capacities of the effective active materials in the positive and negative electrodes, respectively [12]. They are defined as:

[TABLE]

[TABLE]

where $Q_{\mathrm{all}}$ denotes the theoretical maximum discharge capacity, and $D_{n}$ and $D_{p}$ are the maximum variation ranges of the stoichiometric numbers $x_{\mathrm{avg}}$ and $y_{\mathrm{avg}}$, respectively. Here, $x_{\mathrm{avg}}$ and $y_{\mathrm{avg}}$ are the normalized average lithium-ion concentrations of the negative and positive electrodes.

The average solid concentrations are given by

[TABLE]

[TABLE]

and the deviation terms follow first-order dynamics

[TABLE]

[TABLE]

By applying the forward Euler method with step size $\Delta t=t_{k+1}-t_{k}$, the discrete-time form is obtained:

[TABLE]

[TABLE]

In the ESP model, $\tau^{s}_{n}$ is usually regarded as a constant. However, in order to improve the model accuracy under high C-rate charge and discharge conditions, $\tau^{s}_{n}$ can be defined as a piecewise function of the current rate:

[TABLE]

$Q_{\mathrm{all}}$ denotes the nominal battery capacity. In the ESP model, $Q_{\mathrm{all}}$ is assumed constant. In practice, however, the effective capacity decreases with cycling and with increasing charge/discharge current [43, 44, 45]. To account for this effect, Peukert’s law is introduced to represent the variation of capacity under different $C$-rates [45]. Although originally developed for lead–acid batteries, Peukert’s law can be extended to LIBs. The effective capacity is then given by [46]:

[TABLE]

where $C_{\mathrm{ref}}$ is the reference current rate (here $C_{\mathrm{ref}}=1C$), $C_{\mathrm{now}}$ is the applied current rate, $n$ is Peukert’s constant, and $Q_{\mathrm{now}}$ is the effective capacity under the current operating condition. In subsequent formulations, $Q_{\mathrm{all}}$ is replaced by $Q_{\mathrm{now}}$.

2.3 Liquid diffusion (concentration polarization)

The change of liquid-phase lithium-ion concentration at the positive and negative collectors is inconsistent. Thus, the liquid-phase diffusion over-potential can be expressed as

[TABLE]

where $R$ is the universal gas constant, $T$ is the absolute temperature, $F$ is Faraday’s constant, $t_{+}$ is the transference number, $c_{0}$ is the initial electrolyte concentration, and $\Delta c_{1}(t)$ and $\Delta c_{2}(t)$ are the concentration variations at the positive and negative electrodes, respectively.

The discrete-time dynamics of $\Delta c_{1}$ and $\Delta c_{2}$ can be written as

[TABLE]

[TABLE]

where $\tau_{e}$ is the electrolyte diffusion time constant, $P_{\mathrm{con},a}$ and $P_{\mathrm{con},b}$ are empirical coefficients, and $\Delta t=t_{k+1}-t_{k}$ is the discrete sampling interval.

2.4 Reaction polarization

From the Butler–Volmer kinetic equation [42], the reaction polarization over-potential is obtained as

[TABLE]

where $R$ is the universal gas constant, $T$ the absolute temperature, $F$ the Faraday constant, and $m_{p}(t)$ and $m_{n}(t)$ are intermediate variables related to electrode kinetics.

The intermediate variables $m_{p}(t)$ and $m_{n}(t)$ can be expressed as

[TABLE]

[TABLE]

where $c_{0}$ is the initial electrolyte concentration, and $P_{\mathrm{act}}$ is reaction polarization coefficient.

2.5 Ohm polarization

Ohmic polarization follows Ohm’s law, and the internal ohmic resistance of the battery can be obtained experimentally. Thus, the over-potential of ohmic polarization is given by

[TABLE]

where $R_{\mathrm{ohm}}$ is the internal ohmic resistance of the battery and $I(t)$ is the applied current.

2.6 Model parameter

The battery used in the experiment is the Panasonic NCR18650PF cell, and its basic specifications are shown in Table 1. The experimental equipment setup is illustrated in Figure 1(a).The experimental setup consists of an Arbin BT-5HC-5V/100A multifunctional battery tester, a Hongyu HY-TH-150DH temperature and humidity chamber, an electronic computer for data acquisition and recording, and the NCR18650PF battery.

Part of the model parameters were obtained from the literature [47, 48], while others were determined using the parameter identification approach described in [49]. For parameter estimation, we employed the parameter estimation toolbox in Simulink, with the Levenberg–Marquardt optimization algorithm adopted to minimize the error between the measured and simulated terminal voltages. The identified parameter values are summarized in Table 2.

The IESP model is implemented in MATLAB/Simulink. At room temperature (25 ∘C), the input conditions include constant current discharges at 0.5C, 1C, 2C, and 3C rates, as well as the UDDS. The UDDS profile represents a typical dynamic charge/discharge scenario of electric vehicles, and is widely used to evaluate the ability of models to capture dynamic behavior. The corresponding simulation results are shown in Figure 2.

As shown in Figure 2(a), the IESP model accurately reproduces the terminal voltage response of the battery under constant current discharge. At high C-rates, the measured voltage exhibits a sharp drop at the beginning of discharge due to the initial non-uniform lithium-ion concentration distribution inside the electrodes, before gradually recovering to the expected profile. Overall, the IESP model demonstrates strong consistency with experimental results in capturing the voltage variation during constant current discharges. Under the UDDS condition (Figure 2(b,c)), the IESP model is also able to track the actual terminal voltage with high accuracy, with the voltage error remaining within $\pm 0.05$ V. These results confirm that the developed IESP model can reliably simulate the terminal voltage behavior of lithium-ion batteries under both static and dynamic operating conditions.

3 Design of Adaptive Dead-Zone dual SMO Algorithm

The EKF algorithm is traditionally applied to dynamic systems under the assumption of Gaussian white noise perturbations. However, for the nonlinear system of lithium-ion batteries, the system perturbations are usually random and irregular, which leads to degraded estimation accuracy and convergence when EKF is applied.

The SMO, in contrast, is characterized by its simple structure and robustness against unmodeled dynamics and uncertainties. To improve the estimation accuracy under such nonlinear conditions, an adaptive dead-zone dual SMO algorithm is proposed. The algorithm consists of two parts: a state SMO for SOC estimation, and a parameter SMO for online model parameter estimation. To guarantee stability, a dead-zone mechanism is introduced based on Lyapunov theory.

3.1 State SMO Design

The state equation of the battery model can be expressed as

[TABLE]

with

[TABLE]

[TABLE]

Unmodeled dynamics and uncertainty are denoted by

[TABLE]

where $\Delta A$ and $\Delta B$ represent parameter disturbances, $\Delta i_{k}$ denotes current fluctuation, and $\delta_{k}$ is an unmodeled nonlinearity; assume $|w_{k}|\leq\Delta w$.

The terminal-voltage output is

[TABLE]

and the estimated output is $\hat{y}_{k-1}\triangleq\hat{U}(\hat{x}_{k-1},i_{k-1})$.

To begin, we introduce the basic SMO structure without output injection. The state estimation equation can be expressed as

[TABLE]

where $K$ is the sliding-mode gain and $\mathrm{sgn}(\cdot)$ represents the sign function.

Defining the state estimation error as $\tilde{x}_{k}\triangleq x_{k}-\hat{x}_{k}$, the error dynamics are obtained by subtracting (30) from the true system model:

[TABLE]

where $\Delta w$ denotes unmodeled dynamics and bounded disturbances.

Selecting the sliding surface as

[TABLE]

the existence and accessibility conditions of the discrete sliding mode can be written as

[TABLE]

where Eq. (33) ensures that the error trajectory makes a crossing motion with decreasing amplitude, i.e., existence of sliding mode; and Eq. (34) guarantees that the system state reaches the switching surface within finite time, i.e., accessibility of the discrete sliding mode.

By substituting Eq. (31) into Eq. (33) and (34), the sufficient and necessary condition for stability of the discrete SMO can be derived as

[TABLE]

Thus, the observer gain $K$ must be chosen within this interval to ensure convergence under bounded disturbances. Although the SMO designed in this way can guarantee stability and robustness under random disturbances, its tracking performance is often limited due to the absence of direct feedback correction from the output residual. This may lead to slow convergence or steady-state errors in practice. To further improve the tracking performance, an output-injection term is introduced:

[TABLE]

where $L\in\mathbb{R}^{n\times 1}$ is the output-injection gain. This enhanced SMO combines the robustness of sliding mode estimation with output feedback, thereby improving accuracy and dynamic response while maintaining stability.

3.2 Parameter SMO Design

The battery parameters with the greatest influence on terminal voltage are defined as

[TABLE]

Since parameters vary slowly and independently at each sampling instant, the parameter estimator is expressed as

[TABLE]

with the output equation

[TABLE]

The parameter SMO is formulated as

[TABLE]

where $K^{\theta}$ and $L^{\theta}$ are observer gains.

3.3 Adaptive Dead-Zone Proof via Lyapunov Stability

To prevent error accumulation in parameter estimation, we introduce an adaptive dead-zone. The central idea is to derive a Lyapunov-based bound on the terminal-voltage error such that, when the error stays within this bound, the closed-loop observer is (practically) stable; otherwise, parameter adaptation is suspended while the state observer keeps running to ensure robustness. Based on the terminal voltage error, an adaptive dead-zone dual SMO is designed. For the system Eq. (36) define the auxiliary variable

[TABLE]

Thus, the system can be simplified as

[TABLE]

Define the discrete Lyapunov function

[TABLE]

where $P$ is a symmetric positive definite matrix. If

[TABLE]

then the system is stable. Substituting into the Lyapunov difference yields

[TABLE]

where $e_{y/k}$ is the difference between the actual and estimated terminal voltage, $\lambda_{m}$ is the minimum eigenvalue of $Q=A^{T}PA-P$, and $I$ is the identity matrix.

Thus, the stability condition can be expressed as

[TABLE]

By substituting the auxiliary variable, the adaptive dead-zone range of the dual SMO can be expressed as

[TABLE]

Further simplification to remove the sign function yields

[TABLE]

The inequality above defines the adaptive dead-zone boundary. If the terminal voltage error satisfies the inequality, i.e. the error is inside the bound, the system is stable and both state and parameter estimators are activated. If the error exceeds the bound, i.e. lies outside the bound, parameter adaptation is suspended while the state SMO continues to operate, thus guaranteeing the stability of the overall system.

3.4 Implementation Steps

The overall process is illustrated in Figure 1(b).The adaptive dead-zone dual SMO algorithm operates as follows:

1. Step 1:

Initialize the algorithm parameters (observer gains, sliding gain $K$, and dead-zone threshold derived from the Lyapunov condition). 2. Step 2:

Update the state SMO to estimate the battery SOC. 3. Step 3:

Compute the terminal voltage error $e_{y/k}$.

• •

If $|e_{y/k}|$ is within the dead-zone bound, both state SMO and parameter SMO are updated.

• •

If $|e_{y/k}|$ is outside the bound, only the state SMO is updated, and parameter adaptation is suspended to ensure stability. 4. Step 4:

Repeat Steps 2–3 over the operating cycle.

4 Results and Discussion

To validate the effectiveness of the proposed adaptive dead-zone SMO, the following sections present the results under four scenarios: (i) correct SOC initialization, (ii) incorrect SOC initialization, (iii) comparison of computational efficiency, and (iv) estimation accuracy under battery aging conditions.

4.1 Correct SOC Initialization

At room temperature ($25^{\circ}\mathrm{C}$), the input conditions considered are a 1C constant current discharge and the UDDS dynamic condition. The estimation accuracy of the proposed algorithm is evaluated under zero initial SOC error.

For comparison and validation, two estimation algorithms are tested: the proposed adaptive dead-zone dual SMO algorithm and the conventional dual SMO algorithm without dead-zone. The observer gains used in the simulations are set as follows:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

As shown in Figure 3, both the adaptive dead-zone dual SMO algorithm and the conventional dual SMO algorithm achieve high SOC estimation accuracy, which can be attributed to the correct SOC initialization and the use of the electrochemical model. From Table 3, the results are also compared with our previous work using ECM-based methods [40], since both studies employ the same battery type. It can be clearly seen that the proposed electrochemical-model-based approach consistently outperforms ECM-based approaches, highlighting the advantages of electrochemical modeling. Furthermore, the adaptive dead-zone dual SMO algorithm significantly improves accuracy compared with the conventional dual SMO, maintaining SOC estimation errors within $0.2\%$ under both constant current and dynamic conditions. These results demonstrate the effectiveness of the adaptive dead-zone mechanism in enhancing estimation accuracy.

4.2 Incorrect SOC Initialization

To further evaluate the convergence capability of the proposed algorithm under incorrect initial conditions, the same operating profiles as in Section 4.1 are adopted, namely a 1C constant current discharge and the UDDS dynamic condition. The observer gains remain identical to those listed in Section 4.1. However, the initial SOC is deliberately set with a $30\%$ error in order to validate the robustness and convergence properties of the proposed algorithm. The corresponding results are shown in Figure 4 and Table 4.

As shown in Figure 4, when the initial SOC error is $30\%$ under both constant discharge and dynamic conditions, the adaptive dead-zone dual SMO algorithm and the dual SMO algorithm exhibit rapid convergence, requiring only about $30$ s to eliminate the initial error. Moreover, both algorithms achieve final SOC estimation errors below $1\%$. In contrast, our previous study [40] showed that EKF with ECM and fixed dead-zone dual EKF with ECM required approximately $100$ s to converge, highlighting the advantages of the improved model accuracy and the robustness of the SMO-based framework.

From Table 4, it is evident that the adaptive dead-zone dual SMO achieves the highest accuracy among all methods, even in the presence of significant initial SOC errors. It is also observed that the ECM-based algorithms (EKF with ECM and fixed dead-zone dual EKF with ECM) perform better under UDDS conditions than under constant current discharge. This is because dynamic profiles provide richer current excitation, which facilitates state estimation. In contrast, under constant current discharge, the estimation relies more heavily on the intrinsic accuracy of the battery model, which explains the poorer performance of ECM-based methods. The electrochemical model-based approach, however, maintains high accuracy even in constant current operation, further demonstrating its superiority over ECM-based methods.

4.3 Computational Efficiency

The computational cost of an algorithm directly reflects its efficiency and potential applicability in real-time battery management systems. To compare the efficiency of different observers, the simulation times of the SMO, dual SMO, and the proposed adaptive dead-zone dual SMO algorithms are evaluated under identical hardware resources. The experiments are conducted on a computer equipped with an Intel® CoreTM i5-9300H (9th generation) CPU, 16 GB DDR4 memory, and MATLAB R2020a. At room temperature ($25^{\circ}$C), two operating conditions are considered: a 1C constant current discharge and the UDDS dynamic condition. For each condition, the computational times are compared under two scenarios: zero initial SOC error and a $30\%$ initial SOC error. The results are summarized in Table 5.

As shown in Table 5, the dual SMO algorithm requires significantly longer computation time than the single SMO algorithm. This increase is expected, since the dual SMO incorporates an additional parameter estimator, which introduces extra computational overhead. By contrast, the proposed adaptive dead-zone dual SMO reduces the computation time compared with the conventional dual SMO. This is because the adaptive dead-zone mechanism effectively limits unnecessary parameter updates, thereby lowering computational burden without sacrificing accuracy.

Combined with the accuracy results discussed in the previous sections, it can be concluded that the proposed adaptive dead-zone dual SMO achieves both higher estimation accuracy and lower computational cost compared with the conventional dual SMO algorithm. This balance between precision and efficiency highlights its suitability for real-time SOC estimation in practical battery management systems.

4.4 SOC Estimation Accuracy under Battery Aging Conditions

Battery aging leads to gradual changes in model parameters, particularly the available capacity. Cycle aging tests are typically time-consuming and are influenced by multiple factors such as temperature, manufacturing variations, and usage conditions. Even cells of the same type may exhibit different degrees of aging under identical test protocols. Since the focus of this work is on SOC estimation rather than SOH prediction, it is necessary to verify whether the proposed algorithm can still maintain high SOC estimation accuracy under different aging levels.

To this end, the cycle aging model reported in [50] is adopted to generate aged battery behavior after 100 and 400 cycles under UDDS driving conditions. The SOC values provided by this model are used as reference ground truth for validation. Two types of dead-zone strategies are compared: a fixed dead-zone method with a range of $(0,0.001)$, following [40], and the proposed adaptive dead-zone method based on the stability condition in Eq. 47. The corresponding SOC estimation results are shown in Figure 5 and Table 6.

From Figure 5 and Table 6, it can be observed that the proposed algorithm remains stable under different levels of aging. Compared with the results in Table 3 (without aging), the error of the conventional dual SMO increases significantly: after 100 cycles, the RMSE nearly doubles, and after 400 cycles, the error increases from about $0.2\%$ to $0.782\%$, i.e., nearly a fourfold increase. This degradation arises because battery aging changes the cell parameters, and the dual SMO continues to update the parameter estimator even when voltage errors are large, which may drive the algorithm outside the stability boundary and amplify parameter noise.

In contrast, the adaptive dead-zone dual SMO maintains a very low SOC estimation error ($\approx 0.2\%$) across both aging levels. This is because parameter updates are only performed within the stability region defined by the adaptive dead-zone, effectively suppressing error accumulation and improving robustness. Although its error is slightly higher than that under fresh-cell conditions (Table 3), the adaptive dead-zone dual SMO consistently achieves accurate and stable SOC estimation even in the presence of aging.

The fixed dead-zone dual SMO also achieves lower error than the conventional dual SMO, demonstrating the benefit of dead-zone theory in suppressing excessive parameter updates. However, as shown in Figure 5(b)(d), its performance is inferior to the adaptive dead-zone method, since the fixed threshold cannot adjust dynamically to system variations. This highlights the advantage of the adaptive strategy in ensuring stability and minimizing SOC estimation error under diverse operating and aging conditions.

5 Conclusions

This paper proposed an adaptive dead–zone dual SMO for reliable SOC estimation using an improved electrochemical single–particle model. The Lyapunov–based adaptive dead–zone mechanism was shown to guarantee stability by suspending parameter updates when the terminal–voltage error exceeds a stability bound, while enabling joint state–parameter estimation within the safe region. Validation under both constant current and UDDS dynamic conditions demonstrated that the proposed method achieves higher accuracy than the conventional dual SMO and significantly outperforms ECM–based EKF methods, with estimation errors consistently within $0.2\%$ when SOC is correctly initialized. Even under a large initial SOC error of 30%, the adaptive dead–zone dual SMO rapidly converged within 30 s and maintained final errors below $1\%$, markedly faster and more robust than ECM–based approaches. In addition, computational efficiency tests confirmed that the adaptive dead–zone dual SMO requires less time than the conventional dual SMO due to its selective parameter updating, indicating suitability for real–time BMS applications.

The algorithm also proved robust under aging conditions. Compared with the conventional dual SMO, whose errors increased nearly fourfold after 400 cycles, the adaptive dead–zone dual SMO maintained stable accuracy around $0.2\%$. Although the validation relied on an aging model rather than real degradation data, the results indicate that the method can effectively suppress error accumulation even when model parameters drift with aging. Future work should extend validation to real aged cells and further exploit the estimated parameters for SOH tracking, SOP prediction, and fault diagnosis. Overall, the proposed ADZ–DSMO combines accuracy, robustness, and efficiency, making it a promising candidate for practical implementation in advanced battery management systems.

Acknowledgments

This work was supported by the Chengdu Technological Innovation Research and Development Project [Grant No.2024-YF08–00031-GX]; the Sichuan Provincial Department of Science and Technology [Grant No.23YYJCYJ0096, Grant No.2024YFHZ0267]; and the Research Foundation – Flanders (FWO) [Grant No. 1252326N].

Conflict of Interest

The authors declare that they have no conflict of interest.

Author Contributions

Guangdi Hu: Conceptualization, Methodology, Writing – original draft. Keyi Liao: Software, Visualization, Writing – original draft. Jian Ye: Supervision, Writing – review & editing. Feng Guo: Conceptualization, Supervision, Writing – original draft, Writing – review & editing.