Joint estimation of SOC and SOH for lithium-ion batteries based on FOAMIUHF-UKF model

Abstract

Accurate and rapid estimation of power battery state of charge (SOC) and state of health (SOH) is crucial for ensuring the safe and reliable operation of electric vehicles. A joint SOC and SOH estimation method proposed in the study based on a fractional-order model incorporates an adaptive multi-innovations unscented H-infinity filter (FOAMIUHF) to address challenges such as insufficient model accuracy and algorithm robustness in SOC and SOH co-estimation for lithium-ion batteries. First, a fractional-order second-order equivalent circuit model (FOM) is established, then the sparrow search algorithm (SSA) is employed in order to identify model parameters and fractional orders. By integrating the FOAMIUHF algorithm, dynamic adjustments of fading and weighting factors is introduced to suppress noise and enhance the accuracy of the SOC estimation process. Meanwhile, the unscented Kalman filter (UKF) is employed for SOH prediction and battery capacity updates, enabling joint SOC/SOH estimation. Experimental results demonstrate that under dynamic stress test (DST), highway fuel economy test (HWFET), and Japan working conditions, the proposed algorithm achieves a maximum mean absolute error (MAE) of 0.53% and a maximum root mean square error (RMSE) of 0.61% for SOC estimation, outperforming existing methods such as FOMIUKF-UKF and FOMIUKF. Additionally, under various initial SOC (50%–90%) and aging levels (capacity decay to 72.2%), the estimation error of SOC remains below 1%, validating the algorithm’s high accuracy and robustness. The proposed method offers a reliable and effective solution battery state estimation under complex operating conditions and aging scenarios.

Introduction

To address growing resource constraints and environmental challenges, the concept of green development coupled with carbon peaking and carbon neutrality objectives has gained global prominence. The automotive manufacturing industry is systematically advancing energy-efficiency gains and the abatement of environmental pollutants by constructing an incremental, integrated framework that embodies the principles of green development, low-carbon transition, and circularity^1,2. As critical energy storage components for electric vehicles (EVs), lithium-ion batteries have become ubiquitous in EV due to their high energy density, extended cycle life, and environmentally friendly characteristics. Within battery management systems (BMS), precise estimation of two pivotal state parameters - SOC and SOH - is a key technical requirement for ensuring operational safety, optimizing energy management strategies, and prolonging battery service life^3–5. SOC provides essential information about remaining available energy, while SOH reflects capacity degradation and performance deterioration, enabling informed decisions regarding battery maintenance and replacement.

However, lithium-ion batteries exhibit complex nonlinear dynamic characteristics during operation, with parameters that vary over time and are subject to measurement noise interference. Particularly under diverse aging conditions and complex operational scenarios, the intrinsic coupling between SOC and SOH presents significant challenges for state estimation. For instance, capacity fade (a primary indicator of SOH) induces initialization errors in SOC due to cumulative capacity mismatch. Structural changes in the electrode materials of aged cells—such as lithium plating and active-material loss—alter the open-circuit-voltage (OCV) versus SOC relationship, introducing errors into SOC estimates that rely on OCV. At the same time, declining SOH is accompanied by rising internal resistance, which intensifies load-voltage sag and further disturbs SOC estimation under dynamic operating conditions^6,7. Conventional decoupled estimation methods often struggle to balance accuracy and robustness in such scenarios, creating an urgent need for innovative theoretical frameworks and methodological advancements. Recent research directions emphasize the importance of integrated estimation approaches. Many studies have incorporated SOH into SOC prediction through capacity-coupling mechanisms, thereby enhancing estimation accuracy through synergistic state evaluation^8,9. This paradigm shift in battery state estimation is particularly significant for developing adaptive BMS architectures capable of maintaining reliable performance throughout the battery lifecycle.

The estimation of SOC has witnessed substantial methodological advancements, with current approaches broadly divided into three principal categories: parameter-based characterization techniques, model-driven methodologies, and data-driven algorithms. Conventional parameter-centric methods primarily encompass coulomb counting and OCV correlation methods^10–12. Data-driven solutions leverage machine learning architectures like fuzzy logic systems, support vector machines, extreme learning machines, and artificial neural networks to establish nonlinear mappings between measurable parameters and SOC^13–16. Among model-based strategies, equivalent circuit models (ECMs) have emerged as the predominant framework for SOC estimation owing to their favorable compromise between model fidelity and computational tractability. While conventional integer-order ECMs employ resistor-capacitor (RC) networks to simulate battery polarization effects, their capacity to accurately represent the memory characteristics and frequency-dependent behaviors inherent in porous electrode diffusion processes and interfacial reactions remains fundamentally limited^17,18.

In recent years, the application of fractional calculus in lithium-ion battery modeling has attracted increasing attention, primarily because it more effectively captures the non-locality, memory effects, and frequency dependence exhibited by the battery’s internal electrochemical processes¹⁹. Compared with traditional integer-order equivalent circuit models (IOM), fractional-order models (FOM) replace ideal capacitors with constant phase elements (CPEs), enabling more accurate simulation of the dispersion effect at the electrode/electrolyte interface and the impedance behavior during diffusion processes²⁰. These models exhibit particular advantages in capturing both frequency-domain polarization responses and time-domain voltage relaxation patterns^21,22. Representative studies illustrate the potential of this approach: Zhang et al.²³ achieved SOC estimation errors below 3% through fractional impedance modeling combined with genetic algorithm parameter optimization; Mu et al.²⁴ developed a variable-order FOM demonstrating significantly enhanced nonlinear voltage response modeling compared to conventional integer-order counterparts. Nevertheless, critical challenges persist in the practical implementation of FOMs, particularly regarding the dynamic parameter identification of time-variant elements such as aging-dependent internal resistance and temperature-sensitive capacitance characteristics. The development of robust parameter identification techniques remains an essential prerequisite for realizing the full potential of fractional-order modeling in battery management applications.

State estimation algorithms have evolved significantly, with the Kalman filter family demonstrating particular promise in battery management applications due to its inherent adaptability to nonlinear system dynamics. The Extended Kalman Filter (EKF) and UKF implementations have gained significant traction in this domain^25,26. Recent methodological advancements address critical limitations through hybrid filtering architectures: Yu et al.²⁷ developed a synergistic H-infinity/UKF framework that enhances robustness against initial SOC uncertainties and thermal variations through online parameter tracking via H-infinity optimization. Sang et al.²⁸ proposed the dual adaptive central difference H-infinity filter, reporting remarkable estimation precision with mean absolute SOC errors below 0.5% and maximum SOH deviations limited to 0.73% during concurrent state estimation.

While these filtering approaches show improved performance, conventional implementations remain fundamentally dependent on a prior knowledge of noise statistics, rendering them susceptible to performance degradation under non-Gaussian disturbances. Furthermore, their limited inherent adaptability to temporal parameter variations persists as a critical challenge, particularly when confronted with evolving battery characteristics during aging cycles and dynamic operational conditions^27–31. This dual limitation of noise sensitivity and parametric rigidity underscores the continued research imperative for maintaining robustness across diverse operational scenarios and battery degradation states.

Building upon the aforementioned challenges and existing research advancements, this study presents a novel integrated estimation framework that synergistically combines multiple methodological innovations. The proposed approach introduces a fractional-order adaptive multi-innovation unscented H-infinity Kalman filter (FOAMIUHF-UKF), which incorporates three principal enhancements to conventional Kalman filtering paradigms: (1) the implementation of a dynamic weighting mechanism to mitigate historical data dominance and prevent filter saturation, (2) the elimination of Gaussian distribution assumptions and linearization requirements through sigma-point transformation, and (3) the integration of fractional calculus principles with real-time parameter adaptation. This multi-faceted optimization significantly enhances estimation robustness against system nonlinearities, measurement uncertainties, and model parameter drift.

To validate the proposed methodology under realistic operating scenarios, comprehensive experimental verification was conducted using standardized automotive testing protocols. The evaluation framework incorporated three distinct dynamic profiles: Dynamic Stress Test (DST), Highway Fuel Economy Test (HWFET), and Japan working conditions. Experimental results demonstrate superior estimation performance across all test conditions, with particular improvements observed in transient response tracking and error convergence characteristics compared to conventional estimation approaches.

The rest of this article is organized as follows: Chap. 2 briefly introduces the integer-order second-order equivalent circuit model and establishment of the fractional-order equivalent circuit model used in this study respectively. Chapter 3 introduces the battery data acquisition process and the steps of the algorithm used in this study, then presents parameter identification results for different models and algorithms. Chapter 4 details joint estimation methods for SOC and SOH using FOAMIUHF and UKF algorithms, respectively. Chapter 5 evaluates the validation of the estimation accuracy of the proposed method under DST, HWFET, and Japan working conditions, and compares it with FOMIUKF-UKF and FOMIUKF algorithms under different testing cycles. Chapter 6 summarizes the key findings, discusses practical implications, point out the shortcomings of this study and outlines directions for future research.

Battery model

An accurate equivalent circuit model is exceedingly essential for precise estimation of the SOC of a battery. Equivalent circuit models commonly used in the state estimation of batteries include the Rint model, first-order RC model, second-order RC model, PNGV model, and GNL model³². Compared to the Rint model and the first-order RC model, the second-order model provides higher accuracy in simulating the dynamic polarization of lithium-ion batteries and estimating the SOC of batteries under high-rate conditions, while the PNGV model neglects the problem of battery voltage hysteresis and its parameters are too difficult to identify. The GNL model is overly complex and its calculation is computationally intensive, making both models impractical. Therefore, this study adopts the second-order model, which has high estimation precision and relatively low operational difficulty, making it easy to identify parameters.

Integer-order model (IOM)

The second-order model includes an internal resistance , two polarization resistors , and two parallel polarization capacitors ,, and is also applicable to fractional order models. The model structure is displayed in Fig. 1. In Fig. 1, E represents the ideal voltage of the battery and is on behalf of the internal resistance, which represents the resistance between the battery plates, electrolyte, and electrodes; and represent the resistances corresponding to the activated polarization effect and the concentrated polarization effect of the battery, respectively. and represent the capacitances corresponding to the activated polarization effect and the concentrated polarization effect of the battery, respectively.

Fig. 1.

Integer Second-order RC Model.

The discretized state space equation of the integer second-order model is shown in Eqs. (1)–(2):

1

2

Where and are the voltages of two RC circuits, representing the state of charge of the battery, and I representing the battery current.

Definition of fractional calculus

The fractional-order theory first appeared in Leibniz’s notes³³. The three definitions of fractional calculus (FOC) commonly used in relevant researches are Riemann Liouville, Caputo definitions, and Grünwald-Letnikov (GL). This study adopts the GL definition, as shown in Eq. (3).

3

Where represents the continuous fractional calculus operator,  is the order of the fractional system, h is the sampling interval time, L is the length of the window, , for the binomial coefficient, , . When   greater than zero, the formula is differential; When  equal to zero, the formula is the original function, and when less than zero, the formula is an integral, as shown in Eq. (4).

4

Fractional-order model (FOM)

Westerlund and Ekstam indicated that under actual dynamic conditions, most capacitors typically exhibit fractional order characteristics³⁴. To represent the fractional-order properties of capacitance, an expression for fractional capacitance was established:

5

Where represents the capacitance coefficient and n is the order of the fractional capacitance.

Based on the IOM shown in Fig. 1, the integer-order capacitances are replaced with fractional-order capacitances to construct fractional second-order model. The ampere-hour integral method, a widely used estimation technique, is employed to obtain :

6

Where is the initial SOC value of battery and is discharging efficiency, which has to do with factors like the operating temperature and charging and discharging rate. For calculation purposes, the value of in this study is set to 1; is the rated capacity of the battery.

The SOH of the battery is estimated by Eq. (7):

7

Where is the maximum available capacity in the current state; is the nominal capacity of the battery, which is usually considered to be the rated capacity.

Based on the above formulas and the discrete state equations of integer order model, the discretized spatial state equations of the fractional-order model are displayed as Eq. (8):

8

Where T is the sampling interval time; m and n represent the orders of two fractional capacitors, respectively.

By converting Eq. (7) into matrix form, we can obtain:

9

Where , , , , and represent the observation noise and measurement noise, respectively, both of which are Gaussian white noise with a mean of zero.

Model parameter identification

Precise battery model parameters are extremely crucial for accurate SOC estimation. Because of the significant measurement errors associated with battery parameters and the inability to directly measure fractional-order parameters, other methods are required for parameter identification. The parameter identification methods used widely are typically divided into two categories: one is based on traditional methods, such as least squares and related extensions. The other involves the use of optimization algorithms, such as particle swarm optimization and genetic algorithms. These parameter identification methods have relatively high estimation accuracy and have certain advantages in robustness compared to traditional methods.

In the second methods, swarm intelligence optimization algorithms are the most widely used due to its advantages of simple implementation and good stability. Swarm intelligence algorithms primarily seek optimal solutions within a certain range by simulating the behavioral patterns and laws of organisms in nature³⁵. Several algorithms have been developed by observing the behavioral patterns of ants, wolves, and birds, including the ant colony algorithm (ACO), grey wolf algorithm (GWO), and sparrow search algorithm (SSA).

Among them, SSA, proposed by Xue, Shen et al. in 2020, is based on the foraging and evasion behaviors of sparrow populations³⁶. Compared to algorithms such as ACO and GWO, SSA exhibits higher precision, greater robustness and faster convergence. Therefore, this study adopts SSA for parameter identification of the established battery.

SSA algorithm principle

In SSA, sparrows are divided into three categories: discoverers, followers, and vigilantes. Discoverers have a higher adaptability and have a larger search range compared to other sparrows, responsible for finding food and guiding the group to move; followers may follow the discoverer to obtain food for better fitness, and may fly to other areas for foraging when their fitness is low; When vigilantes faces danger, they will release danger signals, adjust their search strategy, and promptly approach other sparrows to avoid being attacked by natural enemies.

Assuming n sparrows are foraging in m-dimensional space, the position of each sparrow in space is:

10

Discoverers, comprising 10% to 20% of the population, follow the location update rules as follows:

11

Where T is the maximum number of iterations; is a random number between ; Q is a random number that follow a normal distribution; L is a vector of all ones; is a random alarm value,; is the safety threshold,. When , the group did not detect any danger and the surrounding environment was safe; when time, the vigilantes discovered natural enemies and the group quickly approached the safe zone.

Apart from the discoverer, all other sparrows are followers, whose position update pattern is as follows:

12

Where represents the worst position of the sparrow in the current iteration; is the optimal location for the discoverer; , A is a randomly generated 1 or −1 vector. When , the follower did not receive food and had low adaptability, requiring them to fly to other areas for foraging; when , the follower would randomly search for food near the optimal location of the discoverer.

Vigilantes generally account for 10% to 20% of the group, and their position update pattern is as follows:

13

Where is the step size control parameter and is a normally distributed random number; K is a directional parameter,; e is a minimal constant to prevent the denominator from being zero; is the fitness of the sparrow; is the optimal fitness for the current group; is the worst fitness for the current group. When, it indicates that the sparrow is easily detected by natural enemies and needs to move closer to other sparrows in order to prevent being attacked by natural enemies; When, it indicates that the sparrow is in the middle of the group and randomly searches nearby.

The detailed flowchart of SSA is shown in Fig. 2.

Fig. 2.

SSA Parameter Identification Flowchart.

Parameter identification method

This study uses SSA to identify model parameters and fractional-order parameters of batteries. The specific identification process is as follows:

Step 1:

Set reasonable upper and lower limits for model parameters based on the initial battery information. The initial population values are related to the parameter limits, and these initial limits affect the convergence speed and accuracy of parameter identification.

Step 2:

Identify model parameters , , ,  and fractional-order parameters m, n. SSA relies on the fitness of individuals within the population, iteratively seeking the optimal population fitness. The vigilante updates enable the algorithm to dynamically escape local optima, balance the search direction, and enhance robustness. The objective of this study is to minimize the sum of squared differences between the actual and estimated model outputs. The equation is in Eq. (14)

14

Where M is the length of the test data; is the actual output value; is the output value for the model. Thus, the fitness function of SSA is obtained:

15

Test bench

The battery test bench used in this study includes a battery testing system (NEWARE T-4008-5V6A-S1), a temperature control box, an experimental data storage and control computer, and a Samsung 18,650 lithium-ion battery cell for testing. The structure is shown in Fig. 3. The specific battery specifications are listed in Table 1.

Fig. 3.

Battery Test Bench.

Table 1.

18,650 lithium battery Parameters.

Parameters	Specifications
Nominal Capacity (A•h)	3.4
Charging Method	CC-CV
Max Charging Current (A)	2
Max Discharging Current (A)	13
Charging Cut-off Voltage (V)	4.2
Discharging Cut-off Voltage (V)	2.65

Firstly, under the controlled experimental temperature of 25 ℃, the battery underwent tests for maximum available capacity, open-circuit voltage, and various dynamic conditions, including dynamic stress test (DST), highway fuel economy test (HWFET), and Japan working conditions.

Next, the battery aging test was conducted: the battery was charged at a rate of 0.5 C CC-CV until the voltage reached the cut-off voltage, followed by a 5-minute rest. Subsequently, the battery was discharged at a rate of 0.5 C CC-CV to the cut-off voltage and rested for another 5 min. This procedure constituted one cycle. Every 100 cycles, the battery was tested for characteristics, including maximum capacity, open-circuit voltage, and dynamic condition tests, until the capacity dropped to approximately 80% of the rated capacity, which defines the battery’s lifespan threshold³⁷. The test was then terminated. The initial available capacity measured for the new battery in this experiment was 3.2709 A • h, slightly lower than the rated capacity.

To conduct battery parameter identification, an accurate OCV-SOC curve is required, which is obtained through an open circuit voltage test. The battery was pulse-discharged at a rate of 0.5 C for 6 min, then rested for 2 h, with discharge tests conducted every 5% SOC interval to gather pulse discharge data. This study performed 5th, 6th, 7th, and 8th -order polynomial fitting on the discharge data to accurately fit the OCV-SOC data, with results shown in Fig. 4.

Fig. 4.

Fitting curves of open circuit voltage at different orders.

From Fig. 4, it can be seen that compared with the 5th, 6th, and 7th order polynomial fitting curves, the 8th -order polynomial is closer to the original data. Therefore, this study selects the 8th -order polynomial for the OCV-SOC fitting curve, with the fitting formula shown in Eq. (16).

16

Parameter identification results

In order to demonstrate the superiority of the fractional second-order model and the SSA algorithm over other models and parameter identification algorithms, this study compares the SSA-FOM parameter identification results with those of SSA-IOM and GWO-FOM. To ensure the effectiveness of the verification, the population size for all three algorithms is set to 20, and the maximum number of iterations is set to 30.

From Fig. 5 (a-b), it can be seen that the simulated terminal voltage obtained from the model parameters identified by the three algorithms closely matches the actual voltage, with SSA-FOM exhibiting the best performance in terms of simulation accuracy. As shown in Table 2, the simulated voltage errors MAE of the three algorithms are 4.5mV, 11.2mV, and 5.2mV, respectively, and the RMSE is 6.2mV, 16.4mV, and 8.1mV, respectively. This indicates that the FOM provides higher accuracy compared to the IOM, while the SSA algorithm has the highest accuracy compared to the GWO algorithm, proving its effectiveness in parameter identification. The intense chemical reactions inside the battery at low SOC levels lead to strong nonlinearity, resulting in increased errors in parameter identification. Therefore, this study selects data from the 10% −100% SOC range for parameter identification and performs average processing. The identification results of SSA-FOM are shown in Table 3.

Fig. 5.

Parameter Identification Simulation Results.

Table 2.

Parameter identification simulation Error.

Algorithm	MAE(mV)	RMSE(mV)
SSA-FOM	4.5	6.2
SSA-IOM	11.2	16.4
GWO-FOM	5.2	8.1

Table 3.

SSA-FOM parameter identification Results.

Parameters						m	n

Joint estimation methods

Most SOC estimation algorithms belong to individual estimation, which does not consider the subtle changes in battery parameters caused by operating cycles. While most individual estimation algorithms achieve high accuracy, various factors within the battery can affect SOC estimation. Ignoring these factors will reduce the accuracy of battery SOC estimation³⁸. To address this, and considering the impact of battery cycling on both SOH and SOC, this study selects battery capacity as the mapping parameter for SOH, dynamically adjusts the parameters of SOC estimation algorithm based on SOH changes, and improves prediction accuracy.

Adaptive fading factor based on multi-innovations unscented Kalman filter

At present, there is a lot of research on UKF algorithm, and numerous studies has explored adaptive changes of UKF. Lin et al.⁸ introduced an adaptive covariance matching method to update the covariance of system noise and measurement noise in real time, thereby reducing the error in SOC estimation. Chen et al.³⁹ designed adaptive factors by changing the Q parameter, but did not consider the influence of historical data during the process, which reduced the estimation accuracy of the algorithm. In this study, an adaptive factor is designed to adjust the prior error covariance matrix of the UKF, as shown in Eq. (17).

17

Where is the adaptive fading factor; is the prior error covariance matrix before adaptive weighting.

This study is based on the novel covariance matrix and window function to set an adaptive fading factor. The multi-innovations theory was first proposed by Ding et al.⁴⁰. The innovation is expressed as Eq. (18):

18

Where M is the window size; is the measured value; is the estimated value. The theoretical value of the covariance matrix for innovations is:

19

Where is the Jacobian matrix of the measurement function; R is the measured covariance matrix of noise. Based on historical data of news, the estimated value of the news covariance matrix defined by the window function is:

20

Based on the above equations, the adaptive fading factor is obtained:

21

Where is the operation of finding the trace. Introducing an adaptive fading factor can effectively weaken the influence of historical data on estimation results and improve the estimation accuracy of the algorithm.

Applying the theory of multi-innovations to the UKF algorithm yields the MIUKF algorithm. Because of the cumulative interference caused by historical data in SOC estimation, there may be significant errors in the prediction results. Therefore, weight factors can be employed to diminish the influence of historical data in the recursive process and reduce interference⁴¹.

22

23

Where a is the adjustable coefficient, , it selected in this study .

Both excessively large and small window sizes can reduce prediction accuracy and affect computational speed. Therefore, it is necessary to choose an appropriate window size. As shown in Fig. 6, the estimation error gradually decreases with the increase in window size. However, a larger window size will reduce the computational speed. Thus, this paper selects a window size of 365.

Fig. 6.

The Impact of Window Size on Estimation Accuracy.

Adaptive unscented H-infinity filter

Since the fact that UKF belongs to the framework of standard Kalman filtering, it requires the tested system to be Gaussian to obtain the optimal solution, while H-infinity filtering (HF), although not requiring the system, is affected by system nonlinearity. Linearizing the system function may introduce significant errors, leading to a decrease in estimation accuracy⁴². Therefore, this study combines the advantages of both approaches to develop an unscented H-infinity filter (UHF) that satisfies the HF minimum cost function in the UKF framework.

Similar to UKF, UHF uses sigma points to update the mean, measurement error covariance, and cross covariance. However, for UHF, the update of state error covariance differs from that of UKF, as shown in Eqs. (24)–(25):

24

25

Where is the prior state covariance; is the cross covariance; is the posterior state covariance; is the performance parameter. Similar to HF, the performance of UHF is very important. If is too small, it will cause the filter to diverge, so needs to be satisfied in Eq. (26):

26

Where is the operation for finding eigenvalues.

UHF combines the advantages of UKF and HF, which not only solves the nonlinear transfer problem of mean and covariance through unscented transformation without the need for Taylor expansion linearization of system equations, but also maximizes the limitation of non-Gaussian noise generated by the system and reduces the errors of SOC estimation. Based on the adaptive factor and weight factor discussed earlier, this study constructs the FOAMIUHF algorithm on the basis of the FOM.

Joint estimation method based on FOAMIUHF-UKF

This study proposes a SOC estimation method that considers the dynamic changes in SOH. The UKF algorithm estimates SOH every 60 s and updates the SOC parameters for the next cycle. The algorithm framework is shown in Fig. 6, with detailed principles and steps described in the following text:

Step 1:

Initialization, for ;

Set SOC estimation initial state , state error covariance , process and measurement noise and , SOH estimation initial parameters , parameter error covariance , measurement noise and .

Step 2:

L is the Estimated time interval for SOH and run the FOAMIUHF filter for each cycle:

1. Unscented transformation generates sigma points:

27

Calculate the weight of the sampling points :

28

Where , , ,.

• (b)Update prior state and covariance :

29

30

31

• (c)Unscented transformation resampling generates sigma points, updates measurement equation and Kalman gain:

32

33

34

35

36

• (d)Update posterior state , covariance :

37

, repeat steps (a) to (d).

Step 3:

When the loop updates to , , , ,

Step 4:

Calculate the UKF filter for each cycle.

1. Generate sigma points through unscented transformation, calculate sampling weights ,

38

39

• (b)Update prior state and covariance :

40

41

• (c)Unscented transformation resampling generates sigma points, updates measurement equation and Kalman gain:

42

43

44

45

46

• (d)Update posterior state , covariance :

47

48

Step 5:

The k time loop ends and the begins.

SOC is updated rapidly on a small time scale. As shown in step 3, when the micro-time step updates to l, the macro-time step is updated, and the micro-time step counter is reset to zero, preparing for the next macro-estimation of SOH and SOC estimation in the next cycle. If the value of l is set too small, it will increase the computational burden. Conversely, if it is set too large, it will lead to untimely parameter updates and affect the accuracy of SOC estimation⁴³. Therefore, considering these factors comprehensively, this paper sets l to 60, meaning that the SOH estimation is performed every 60 s to update the SOC parameters, and the two cycles alternate to form the SOC-SOH joint estimation framework in this study. The overall flowchart of the joint algorithm is shown in Fig. 7.

Fig. 7.

Joint Estimation Flowchart.

Results and discussion

To verify the effectiveness of the proposed joint estimation algorithm, this study validates the method using DST, HWFET, and Japan working conditions, with an initial SOC set at 90%.

Estimated results under different operating conditions

DST

Figure 8(a) and (b) show the terminal voltage estimation results and errors under DST conditions. Figure 8(c) and (d) display the SOC estimation results and errors. Figure 8(e) and (f) present the SOH estimation results and errors. Specific SOC estimation errors are listed in Table 4.

Fig. 8.

Simulation Results of DST.

Table 4.

SOC Estimation error under DST.

	MAE	RSME
FOAMIUHF-UKF	0.0034	0.0039
FOMIUKF-UKF	0.0058	0.0079
FOMIUKF	0.0099	0.0108
FOAMIUHF	0.0077	0.0082

HWFET

Figure 9(a) and (b) show the terminal voltage estimation results and errors under HWFET conditions. Figure 9(c) and (d) display the SOC estimation results and errors. Figure 9(e) and (f) present the SOH estimation results and errors. Specific SOC estimation errors are listed in Table 5.

Fig. 9.

Simulation Results of HWFET.

Table 5.

SOC Estimation error under HWFET.

	MAE	RSME
FOAMIUHF-UKF	0.0043	0.0053
FOMIUKF-UKF	0.0060	0.0073
FOMIUKF	0.0105	0.0118
FOAMIUHF	0.0076	0.0083

Japan

Figure 10(a) and (b) show the terminal voltage estimation results and errors under HWFET conditions. Figures 10(c) and (d) display the SOC estimation results and errors. Figures 10(e) and (f) present the SOH estimation results and errors. Specific SOC estimation errors are listed in Table 6.

Fig. 10.

Simulation Results of Japan.

Table 6.

SOC Estimation error under Japan.

	MAE	RSME
FOAMIUHF-UKF	0.0053	0.0061
FOMIUKF-UKF	0.0067	0.0088
FOMIUKF	0.0096	0.0107
FOAMIUHF	0.0067	0.0074

From Figs. 8, 9 and 10(a-b), it is evident that the FOAMIUHF-UKF algorithm proposed in this study closely matches the battery voltage estimation values, with significantly reduced errors compared to other algorithms. Initially, the algorithm exhibited a larger error but quickly converged to a level that accurately matched the actual voltage, thereby demonstrating the effectiveness of the method. In the later stages of estimation, as the battery voltage approached the cut-off voltage, the model struggled to accurately capture the battery’s dynamic behavior at low SOC, resulting in large terminal voltage estimation errors. However, the mean absolute error remained below 6.1 mV, and the battery terminal voltage was still well-estimated.

Figures 8, 9 and 10(c-d) show that the FOAMIUHF-UKF algorithm has the smallest error compared to other algorithms and maintains high estimation accuracy under various operating conditions. Despite the large initial SOC error due to inaccurate initial values, the algorithm quickly converges to the accurate value. The maximum MAE and RMSE under different operating conditions are 0.53% and 0.61%, respectively. Compared to FOMIUKF-UKF and FOMIUKF, the maximum MAE is reduced by 20.9% and 49.5%, and RMSE is reduced by 30.7% and 48.3%, respectively. As shown in the figures and tables, the FOAMIUHF-UKF algorithm maintains the lowest error across all three operating conditions compared with FOAMIUHF, demonstrating its superiority. FOAMIUHF exhibits slightly higher error than FOAMIUHF-UKF but slightly lower error than FOMIUKF, confirming that the introduction of an adaptive fading factor and H-infinity filtering can effectively reduce SOC estimation error, while joint estimation further contributes to error reduction. Specific SOC estimation errors for different algorithms under various working conditions are listed in Tables 4, 5 and 6.

Figures 8, 9 and 10 (e-f) show that the battery’s SOH varies over time. Dynamically updated SOH values are used for SOC estimation, which contributes to enhance SOC accuracy. Under the three working conditions, the SOH estimation accuracy is relatively high, with a maximum average absolute error of no more than 0.0032%.

Analysis of algorithm robustness

Before performing algorithm estimation, it is essential to know the accurate initial SOC value. However, in practical situations, due to factors such as the internal reactions of the battery and BMS, the initial value set by the algorithm may not be consistent with the actual situation of the battery. Therefore, in order to verify the robustness of the FOAMIUHF-UKF algorithm under different initial SOC values, this study set SOC initial values of 90%, 70%, and 50% respectively, and verified them under DST, HWFET, and Japan working conditions. As shown in Fig. 11, the FOAMIUHF-UKF algorithm maintains accurate SOC estimation under large initial SOC deviations, with fast convergence speed and strong adaptability to different initial SOC values. The detailed SOC estimation error is shown in Table 7, and the results show that the maximum error is in the Japan, with maximum MAE and RSME of 0.54% and 0.66%, respectively, indicating that the algorithm maintains high robustness under various operating and initial SOC conditions.

Fig. 11.

Algorithm Estimation Results under Different Initial SOC Values under DST.

Table 7.

Algorithm Estimation errors under different initial SOC Values.

Initial SOC	DST		HWFET		Japan
	MAE	RMSE	MAE	RMSE	MAE	RMSE
90%	0.0034	0.0039	0.0043	0.0053	0.0053	0.0061
70%	0.0035	0.0048	0.0044	0.0059	0.0054	0.0064
50%	0.0034	0.0050	0.0043	0.0061	0.0053	0.0066

SOC Estimation under different degrees of aging

During the process of battery cycling, its internal materials will gradually undergo chemical reactions, leading to irreversible transformation, battery aging, and a decrease in battery capacity⁴⁴. The capacity degradation directly impacts the accuracy of SOC estimation. Figure 12 shows the capacity decline curve of the battery used in this study. After 300th aging cycles, the battery capacity dropped to a minimum of 72.2% of the rated capacity, nearing the battery’s lifespan threshold⁴⁵. Therefore, to evaluate the SOC estimation performance of FOAMIUHF-UKF algorithm for SOC of aging batteries, the model was validated using operating data from 100th, 200th, and 300th aging cycles, corresponding to 89.3%, 82.8%, and 77.0% of the rated capacity, respectively.

Fig. 12.

Battery Capacity Decline Curve.

Figures 13, 14 and 15 show the estimated results of operating conditions under battery aging. As battery aging progresses, the FOAMIUHF-UKF algorithm maintains accurate SOC estimation at various stages. Table 8 shows that the maximum MAE and RMSE of the FOAMIUHF-UKF algorithm are 0.99% and 1.00%, respectively, with average values of 0.38% and 0.47%, which are significantly lower than other algorithms. The results demonstrate that the proposed algorithm sustains accurate SOC estimation under varying degrees of battery aging, confirming its adaptability and effectiveness in mitigating the adverse effects of aging on battery state estimation.

Fig. 13.

SOC Estimation Results under 100th Cycles.

Fig. 14.

SOC Estimation Results under 200th Cycles.

Fig. 15.

SOC Estimation Results under 300th Cycles.

Table 8.

SOC Estimation errors under different aging Cycles.

Condition	Cycle period	FOAMIUHF-UKF	FOMIUKF-UKF
		MAE/RMSE	MAE/RMSE
DST	100 Cycle	0.0015/0.0021	0.0054/0.0058
	200 Cycle	0.0033/0.0042	0.0063/0.0079
	300 Cycle	0.0030/0.0040	0.0448/0.0505
HWFET	100 Cycle	0.0099/0.0100	0.0187/0.0191
	200 Cycle	0.0039/0.0053	0.0061/0.0075
	300 Cycle	0.0019/0.0035	0.0090/0.0101
Japan	100 Cycle	0.0009/0.0015	0.0110/0.0116
	200 Cycle	0.0048/0.0059	0.0086/0.0108
	300 Cycle	0.0048/0.0054	0.0144/0.0159

Conclusion

This study employs a fractional second-order equivalent circuit model and uses SSA for parameter identification. It proposes a joint SOC and SOH estimation method based on the FOAMIUHF-UKF algorithm. The accuracy and stability of the proposed algorithm are verified through experiments under various operating conditions and aging degrees. The research conclusion is as follows:

1. The proposed algorithm significantly improves the SOC estimation accuracy of lithium-ion batteries. Experimental validation shows that it outperforms the other two algorithms (FOMIUKF-UKF, FOMIUKF) under DST, HWFET, and Japan working conditions. The maximum SOC estimation errors (MAE and RMSE) are 0.53% and 0.61%, respectively, indicating high accuracy in SOC estimation.

2. The algorithm exhibits strong adaptability to different initial SOC values. Experimental results show that under different operating conditions, even with large errors in the initial SOC value, the SOC estimation errors (MAE and RMSE) are still less than 0.54% and 0.64%, respectively, demonstrating its robustness.

3. The algorithm achieves joint estimation of SOC and SOH under different degrees of aging. The experimental results show that the SOC estimation errors (MAE and RMSE) of this algorithm are still less than 0.99% and 1%, respectively, under different degrees of aging, with average values of 0.38% and 0.47%, which are significantly lower than other algorithms (FOMIUKF-UKF), verifying the effectiveness of the algorithm.

Although the proposed method has its advantages, it still has certain limitations. Advanced intelligent algorithms such as machine learning and artificial neural networks have been applied to battery state estimation, and appropriate intelligent algorithms can effectively improve the accuracy of SOC estimation. In addition, temperature is also an important factor affecting battery state estimation. Therefore, future work will particularly focus on exploring the application of advanced intelligent algorithms in battery state estimation and considering the impact of factors such as temperature on the accuracy of battery state estimation.

Author contributions

Zheng Ye: Conceptualization, Experiment, Validation, Writing - Original Draft. ZhiHui Deng: Resources, Methodology, Writing - Review & Editing, Supervision. Lianfeng Lai: Methodology, Validation, Resources. YongHong Xu: Validation, Supervision, Writing - Review & Editing. RuiQian Zhang: Methodology, Validation, Supervision. Liang Tong: Resources, Conceptualization. HongGuang Zhang: Experiment, Software, Writing - Review & Editing. YiYang Li: Experiment, Software. MingHui Gong: Writing - Review & Editing. GuoJu Liu: Experiment, Methodology. MengXiang Yan: Validation, Supervision.

Funding

This work was sponsored by the Beijing Natural Science Foundation (Grant 3244039), “R&D Program of Beijing Municipal Education Commission” (Grant KM202411232021), Fujian Provincial Natural Science Foundation of China (2024H6017), Educational Research Project for Middle-aged and Young Teachers of Fujian Province (JAT241166), New Energy Automobile Motor Industry Technology Development Base, Ningde Normal University, Ningde 352000, China,

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Declarations

Competing interests

The authors declare no competing interests.