Red-Billed Blue Magpie Optimizer for Modeling and Estimating the State of Charge of Lithium-Ion Battery

Abstract

The energy generated from renewable sources has an intermittent nature since solar irradiation and wind speed vary continuously. Hence, their energy should be stored to be utilized throughout their shortage. There are various forms of energy storage systems while the most widespread technique is the battery storage system since its cost is low compared to other techniques. Therefore, batteries are employed in several applications like power systems, electric vehicles, and smart grids. Due to the merits of the lithium-ion (Li-ion) battery, it is preferred over other kinds of batteries. However, the accuracy of the Li-ion battery model is essential for estimating the state of charge (SOC). Additionally, it is essential for consistent simulation and operation throughout various loading and charging conditions. Consequently, the determination of real battery model parameters is vital. An innovative application of the red-billed blue magpie optimizer (RBMO) for determining the model parameters and the SOC of the Li-ion battery is presented in this article. The Shepherd model parameters are determined using the suggested optimization algorithm. The RBMO-based modeling approach offers excellent execution in determining the parameters of the battery model. The suggested approach is compared to other programmed algorithms, namely dandelion optimizer, spider wasp optimizer, barnacles mating optimizer, and interior search algorithm. Moreover, the suggested RBMO is statistically evaluated using Kruskal–Wallis, ANOVA tables, Friedman rank, and Wilcoxon rank tests. Additionally, the Li-ion battery model estimated via the RBMO is validated under variable loading conditions. The fetched results revealed that the suggested approach achieved the least errors between the measured and estimated voltages compared to other approaches in two studied cases with values of 1.4951 × 10⁻⁴ and 2.66176 × 10⁻⁴.

1. Introduction

The widespread utilization of fossil fuels pollutes the environment and causes a rise in the atmospheric temperature around the world, necessitating the expansion of the advancement of numerous alternate energy sources. Renewable energies (REs) like solar and wind energies have lately gotten much interest as alternate solutions. The availability of RE sources (RESs) is intermittent and unpredictable, hence RESs have faced a conflict between time of abundance and time of use. These variable characteristics establish substantial issues for the power networks like power quality and instability concerns. Therefore, to continuously guarantee the presence of electrical energy, REs have to be stored [1,2]. There are a variety of energy storage systems (ESSs) which include chemical, electrical, electrochemical, magnetic, thermal, biological, and mechanical ESSs. The choice of appropriate ESS relies greatly on the energy source, the energy needed for particular utilization, funding, and the infrastructure feasibility [3]. The storage of the electrical energy produced through RESs in the electrochemical energy form via rechargeable batteries is mostly executed [4]. In RE utilizations, the battery cycle should have high stability, and its discharge rate density should be high. The lithium-ion (Li-ion) batteries have several advantages such as large energy density, excellent efficiency, and extended lifetime so they are progressively employed for numerous energy storage utilizations such as power systems, microgrids, and electric vehicles (EVs) [5,6,7,8,9]. However, the Li-ion batteries need to be maintained continuously as the most delicate and pricey element [10]. The electrochemical properties of the battery may deteriorate due to irreversible electrolyte dynamics [11]. Additionally, numerous operating situations may quicken their deterioration including overcharging, intense discharge, and interior electrical short-circuit [12]. Several schemes of energy and battery management involve the battery model in their control rules. However, battery deterioration certainly results in parametric variants [13]. Hence, determination of the battery parameters is needed for the sake of a precise model. Furthermore, the determination of the battery model enables the evaluation of the state of charge (SOC) which points out the deterioration ratio [14,15]. The SOC is used in conjunction with other variables, namely the current, voltage, ambient temperature, and thermal properties to estimate the battery temperature which is vital to guarantee the safety of ESSs. Alternatively, battery temperature can be measured physically or estimated using ultrasonic reflection waves [16]. The SOC directly restricts and affects the EV routing choices that guarantee the chosen route is not only the speediest or shortest but also feasible given the current battery SOC [17]. Advanced routing approaches consider the SOC as the restriction and optimization coefficient [18]. However, the direct measurement of the battery equivalent circuit cannot be performed. The model-based approach is the only way to determine the battery equivalent circuit. Numerous models of Li-ion battery have been constructed and explained, such as data-based models which employ the measurements and look-up tables [19], equivalent circuit models [20] including the RC and Thevenin model, and electrical distribution models including current density distribution models [21]. Electrochemical models involve chemical properties such as the pseudo-two-dimensional model [22] and the Shepherd model which is simpler and precisely depicts the non-linearities of the battery [23]. Several approaches were utilized for determining the Li-ion battery model parameters: switched optimizers, iterative least squares and their uncoupled kinds [24,25,26], and least-square deviation as well as absolute deviation techniques [27]. The latter is aimed at the minimization of deviations between estimated and real battery voltages. Other approaches including sliding mode and confidence zone optimization were also employed to determine the Li-ion battery model parameters [28,29]. Numerous kinds of Kalman filter such as extended (EKF), unscented (UKF), and dual filters (DKF), were also widely employed for determining the parameters and the SOC of the Li-ion battery [30,31,32]. The parameters, SOC, and state-of-health of the Li-ion battery were determined using machine learning (ML) [33,34,35,36] and data-driven approaches [37,38,39]. Despite these approaches yielding actual and good findings, they required enormous datasets, a large memory size, and complex processes. Numerous optimization techniques were employed for determining the Li-ion battery model parameters such as particle swarm optimizer (PSO) [40], genetic algorithm (GA) [41], artificial ecosystem optimizer (AEO) [42], modified white shark optimizer (WSO) [43], bald eagle algorithm (BEA) [44], modified BEA [45], and gazelle optimizer [46]. These optimizers were employed to minimize the deviations between estimated and real battery voltages for extracting the optimal parameters of the Li-ion battery model. Metaheuristic algorithms generally have various merits such as independence on initial conditions, accuracy, and reaching global optimal solutions, and bypassing local optima. Accordingly, they can solve various non-linear optimization problems. The limitations of the published approaches employed in estimating the battery model parameters are listed in

Table 1. Limitations of some published approaches employed in estimating battery parameters.

Ref.	Year	Method	Limitations
[24]	2023	Switched optimizers	No generic performance, reliance on switch logic
[25]	2023	Anti-windup evaluation	Complication, possibility of unsuitability for all models
[26]	2018	Uncoupled least squares	Uncoupled method may miss inter-variable exactness
[27]	2021	Deviation minimization structure	Influence of data uncertainty, large computation cost
[28]	2020	Sliding mode differentiator	Requirement of tuning of sliding mode coefficients
[29]	2020	Confidence zone optimization	Lesser complexity trades off some accurateness
[30]	2024	EKF	Reliance on a precise model and noise evaluation
[31]	2023	UKF	Influence of parameter identification quality on results, complexity
[32]	2019	DKF	More complex; higher computational cost
[33]	2024	improved extreme ML	Impact of data quality on ML precision
[34]	2023	PSO + ML	Sensitivity to data, complexity, overfitting risk
[35]	2024	ML + feature engineering	Intensive data, incomplete popularization
[36]	2024	ML	Reliance on learning data; popularization problems
[37]	2024	Incremental capacity study + transformer model	A lot of resources are required for the transformer model
[38]	2024	Data-driven ML	Need of huge datasets, difficult real-time operation
[39]	2024	Real-world data + ML	Requirements of different high-quality operating data
[40]	2022	PSO	Metaheuristic restrictions (convergence velocity, adjusting)
[41]	2020	Feature extraction + GA	Model complexity, coefficient sensitivity
[42]	2023	AEO	Sensitivity to initial conditions, popularization problems
[43]	2023	Modified WSO	Mixed complexity, need of performance adjusting
[44]	2022	BEA	Risk of local optima, performance variation with problem
[45]	2023	Modified BEA	Reliance on tuning, sensitivity to initial conditions
[46]	2023	Gazelle optimizer	Limited exploration, shortage of benchmarks through problems
This work		Red-billed blue magpie optimizer	No tuning required, effective escape from local optima, global exploration

.

However, in accordance with the no-free-lunch theorem, there is no optimizer that can solve all problems with equal effectiveness. This encourages the utilization of new optimizers to improve execution and processes. Hence, the authors suggest employing the red-billed blue magpie optimizer (RBMO) for the determination of the Shepherd model parameters of the Li-ion battery.

The RBMO was designed in 2024 [47] inspired by the hunting behavior of the red-billed blue magpie (RBM). This algorithm was effectively utilized for other non-linear optimization problems, namely thermal-electrical arrangement [48], planning trajectory and tracing control of vehicles [49,50], and modeling fuel cells [51].

The contributions of this article are listed below:

• The RBMO is suggested for the first time for determining the parameters of the Li-ion battery.
• The results attained via the RBMO are compared with those attained via other algorithms.
• The model and the SOC of the Li-ion battery estimated via the RBMO are validated under variable loading conditions.

The article is structured as follows: Section 2 explains the Li-ion battery model. Section 3 discusses the RBMO. Section 4 covers the simulation findings. Section 5 presents the conclusions.

2. Model of Li-Ion Battery

In this article, the Shepherd model described in [52] is employed for the Li-ion battery. This model is chosen since it requires only a few data points from the manufacturer’s datasheet and the battery discharge curve. The chosen model easily reflects macro-level properties for both the current and voltage indicating an essential level of simulation. Figure 1 illustrates the Shepherd model equivalent circuit which includes an internal resistance and regulated voltage source which is determined in both charging and discharging modes.

The battery terminal voltage ($V_B$) can be stated as follows [53]:

$$V_B=E_0-K\left({\displaystyle \frac{Q}{Q-it}}\right)i-R\times i+A\times e^{-B\times it}$$

(1)

where $E_0$ is the open circuit voltage and the coefficient of polarization is denoted by $K$. The term $Q$ represents the battery size, $it$ refers to the removed actual charge, $i$ is the battery current, and $R$ denotes the internal resistance of the battery. The terms $A$ and $B$ are the amplitude and inverse time constant of the exponential zone, respectively.

The modified Shepherd model presented in [52] uses the polarization voltage term and resistance effect for the discharge model as

$$V_B=E_0-K\left({\displaystyle \frac{Q}{Q-it}}\right)i^{\mathrm{*}}-K\left({\displaystyle \frac{Q}{Q-it}}\right)it-R\times i+A\times e^{-B\times it}$$

(2)

where the filtered current is denoted by $i^{*}$.

The battery voltage during charging mode is computed as follows:

$$V_B=E_0-K\left({\displaystyle \frac{Q}{it-0.1Q}}\right)i^{*}-K\left({\displaystyle \frac{Q}{Q-it}}\right)it-R\times i+A.e^{-B\times it}$$

(3)

Also, the battery SOC can be formulated as

$$\mathrm{S}\mathrm{O}\mathrm{C}\left(t\right)={\mathrm{S}\mathrm{O}\mathrm{C}}_0-{\displaystyle \frac{1}{Q}}\int idt$$

(4)

where ${SOC}_0$ is the initial value of the battery SOC. The Li-ion battery discharge characteristics are displayed in Figure 2. The term Vfull represents the battery voltage at rated capacity, while Vnom and Vexp are the normal voltage and the voltage at the end of the exponential zone, respectively.

There are certain missing data points on the manufacturer’s datasheet; they can be identified using metaheuristic techniques with the help of experiential data. In this article, the process of parameter estimation is formulated as an optimization problem aimed at minimizing the error between the experimental and simulated battery voltages as follows:

$$Minimize\hspace{1em}Fobj={\displaystyle \frac{1}{N}}\sqrt{\sum _{i=1}^N{\left(V_s\left(i\right)-V_e\left(i\right)\right)}^2}$$

(5)

where $Fobj$ is the objective function, $N$ symbolizes the quantity of datasets, $V_s\left(i\right)$, and $V_e\left(i\right)$ are the simulated and experimental terminal voltages at instant $i$. In the formulated problem, seven design variables to be determined are [$E_0$, $Q$, $R$, $A$, $K$, $B$, τ]. These parameters should be defined inside the problem search space as follows:

$$\begin{array}{c}{E}_{0,min}\le E_0<E_{0,max}\\ Q_{min}\le Q<Q_{max}\\ R_{min}\le R<R_{max}\\ A_{min}\le A<A_{max}\\ K_{min}\le K<K_{max}\\ B_{min}\le B<B_{max}\\ {\tau }_{min}\le \tau <{\tau }_{max}\end{array}$$

(6)

where $max$ and $min$ are the maximum and minimum boundaries of the search space, respectively.

3. RBMO

The RBMO is an optimizer that replicates the intelligence of the RBM swarm. The RBMO was published in 2024 [47]. The RBMO has been proven to be very successful when employed in various applications such as thermal-electrical arrangement [48], planning trajectory and the tracing control of vehicles [49,50], and modeling fuel cells [51]. The RBMO draws motivation from the chasing performance of the RBMs involving target seeking, attack, and food retention. The primary habitats of the RBM are forests in China, India, and Southeast Asia. The RBM feeds on fruits, insects, and small vertebrates. Through wandering in small sets, the RBMs discover food more effortlessly and cooperate to increase productivity. Additionally, the RBMs store food to be eaten later and obtain nourishment from the trees and the land.

Several critical parts of the RBMO algorithm are expanded by drawing on the natural behavior of RBMs. These birds are recognized for their cooperative foraging behavior, vocal communication, and social intelligence. They usually forage in tiny, coordinated groups that rely on leader-following, alert calls, and food-sharing behaviors. These processes promoted a balance between exploration and exploitation in RBMO, in which agents communicate information about food sources (solutions) and alter their motions as needed. Biologically based concepts like information sharing, emulating the magpies’ use of visual and vocal cues to communicate resource-rich areas, adaptive movement patterns based on their abrupt directional changes and non-linear flight paths, and memory and learning abilities reflected in the RBMO through fitness-based probability weighting and adaptive memory components are all incorporated into the approach.

The mathematical model of the RBMO contains the stages of exploration and exploitation.

3.1. Initialization

The RBMO produces a random initial solution ${(X}_{j,k})$ as follows:

$$X_{j,k}={LL}_k+r_1\left({HL}_k-{LL}_k\right),\forall j \in Popu \mathrm{and} k\in \mathrm{D}\mathrm{i}\mathrm{m}$$

(7)

where ${LL}_k$ and ${HL}_k$ are the lower and upper limits of solution $k$, $r_1$ is a random number between 0 and 1, $Popu$ is the quantity of the population, and $\mathrm{D}\mathrm{i}\mathrm{m}$ is the problem dimension.

3.2. Seeking Food

The RBMs naturally seek in small sets of 2−5 birds or large sets of 10 and above birds for improving search efficacy. Seeking agent locations are updated using Equations (8) and (9) for small and large sets, respectively.

$$X_j\left(m+1\right)=X_j\left(m\right)+r_2\left({\displaystyle \frac{1}{S}}\sum _{p=1}^S{X}_p\left(m\right)-X_{rc}\left(m\right)\right)$$

(8)

$$X_j\left(m+1\right)=X_j\left(m\right)+r_3\left({\displaystyle \frac{1}{L}}\sum _{p=1}^L{X}_p\left(m\right)-X_{rc}\left(m\right)\right)$$

(9)

where $m$ is the current iteration, $X_j\left(m+1\right)$ symbolizes the jth new seeking agent location, $r_2$ and $r_3$ are random numbers between 0 and 1, $S$ symbolizes the quantity of small sets of RMBs randomly chosen from the population, $X_p$ symbolizes the pth randomly chosen bird, $X_j$ symbolizes the jth bird, $X_{rc}$ symbolizes the seeking agent randomly chosen in the current iteration, and $L$ symbolizes the quantity of large sets of RMBs randomly chosen from the population.

3.3. Attacking Prey

The RMBs show great skill and collaboration throughout attacking prey. Small prey is targeted by a small set as modeled mathematically in Equation (10), while big prey is targeted by a large set as presented in Equation (11).

$$X_j\left(m+1\right)=X_F\left(m\right)+CF.{rn}_1\left({\displaystyle \frac{1}{S}}\sum _{m=1}^S{X}_m\left(m\right)-X_{rs}\left(m\right)\right)$$

(10)

$$X_j\left(m+1\right)=X_F\left(m\right)+CF.{rn}_2\left({\displaystyle \frac{1}{L}}\sum _{m=1}^L{X}_m\left(m\right)-X_{rs}\left(m\right)\right)$$

(11)

where $X_F\left(m\right)$ symbolizes the food position, $CF={\left(1-{\displaystyle \frac{m}{M}}\right)}^{{\displaystyle \frac{2m}{M}}}$, $M$ is the maximum number of iterations, and ${rn}_1$ and ${rn}_2$ are random numbers employed for generating standard normal distribution (standard deviation 1, mean 0).

3.4. Storing Food

The RMBs store extra food for consumption later as follows:

$$X_j\left(m+1\right)=\left\{\begin{array}{c}{X}_j\left(m\right)\hspace{1em}\forall {Fobj}_j^{old}>{Fobj}_j^{new}\\ X_j\left(m+1\right)\hspace{1em}Otherwise\end{array}\right.$$

(12)

where ${Fobj}_j^{old}$ and ${Fobj}_j^{new}$ symbolize the values of the objective function before and after the update of the ith RBM position, respectively. The RBMO has an excellent search scheme and fully robust execution. All details of the RBMO and comparison of its characteristic properties with other meta-heuristic optimization algorithms, accompanied by its pseudo-code can be found in [47]. The flowchart of the RBMO is revealed in Figure 3.

4. Results and Analysis

The suggested RBMO is validated through analyzing two Li-ion batteries with the aid of experimental terminal voltage data; the first one has nominal voltage and a rated capacity of 220 V and 120 Ah, respectively, whereas the second battery has 280 V and 1500 Ah.

Table 2. Real data of batteries under study.

	Vnom (V)	${\mathit{E}}_0$ (V)	$\mathit{Q}$ (Ah)	$\mathit{R}$ (Ω)	$\mathit{A}$ (V)	$\mathit{K}$	$\mathit{B}$ (Ah−1)	$\mathit{\tau }$ (s)
Battery 1	220	238.559	120	0.01833	18.475	0.01374	0.509	20
Battery 2	280	303.6205	1500	0.0018667	23.5133	0.0010988	0.040708	20

displays the real data of the batteries under consideration. The suggested RBMO effectiveness in estimating accurate parameters for the battery model is evaluated by comparing it to other ways such as AEO [42], modified WSO [43], dandelion optimizer (DO), spider wasp optimizer (SWO), barnacles mating optimizer (BMO), and interior search algorithm (ISA). All approaches considered are executed with a population size of 25 and 150 iterations, while 10 independent runs are implemented.

The optimal parameters obtained via the suggested approach and the others for the first considered battery are given in

Table 3. Optimal parameters of the first considered battery obtained through the RBMO and the others.

	AEO [42]	Modified WSO [43]	DO	SWO	BMO	ISA	RBMO
$E_0$ (V)	238.6283	238.568	238.5622	238.5499	238.54	238.5371	238.5602
$Q$ (Ah)	120.2189	121.637	121.8207	118.3956	122.0000	121.9998	122.0000
$R$ (Ω)	0.018458	0.01918	0.0188	0.0171	0.017985	0.0178	0.0188
$A$ (V)	18.4210	121.637	18.4789	18.4790	18.4810	18.4810	18.4833
$K$	0.0130508	0.01352	0.0136	0.0133	0.013531	0.0135	0.0135
$B$ (Ah−1)	0.50142	0.51050	0.5093	0.5164	0.51011	0.5101	0.5101
$\tau$ (s)	21.80907	19.5263	20.2573	19.7822	19.586	19.6326	19.6131
Fobj (V)	7.11 × 10−4	5.6118 × 10−4	2.6913 × 10−4	1.6620 × 10−3	1.5106 × 10−4	1.5092 × 10−4	1.4951 × 10−4
Worst	2.139 × 10−3	6.6383 × 10−4	2.0081 × 10−3	5.7082 × 10−3	5.1664 × 10−4	3.2105 × 10−4	1.5136 × 10−4
Average	1.219 × 10−3	3.7127 × 10−4	9.9714 × 10−4	3.6627 × 10−3	2.9090 × 10−4	2.1237 × 10−4	1.5045 × 10−4
SD	4.57 × 10−4	3.2854 × 10−4	5.0166 × 10−4	1.0132 × 10−3	1.3576 × 10−4	7.0697 × 10−5	6.5684 × 10−7
Elapsed time per trial (s)	4912	1745	3026	113	6054	2706	6434

. Moreover, the best, worst, average, standard deviation (SD), and elapsed time per trial are recorded. The best fetched objective value is 1.4951 × 10⁻⁴ through the suggested RBMO, the ISA came second achieving 1.5092 × 10⁻⁴, while the SWO had the worst fitness value of 1.6620 × 10⁻³. The proposed strategy outperformed the others in terms of fitness value. Also, regarding SD value, the proposed RBMO is the best among the other comparable approaches as it achieved the least value of 6.5684 × 10⁻⁷. Also, the elapsed time per trial is recorded; the proposed strategy consumed 6434 s per trial, it is not the fastest among the considered approaches as the SWO consumed the least time of 113 s. This is not a disadvantage of the RBMO given the offline nature of battery parameter estimation. Figure 4 displays the convergence curves of the RBMO and other comparable methods when solving the first considered battery. The curves confirm the preferred technique, which settled in the lowest fitness value after the shortest number of iterations. The measured voltage and that of the established model through the suggested approach are shown in Figure 5. It is obvious that the estimated data strongly converge with the experimental data validating the competency of the suggested approach in predicting the right parameters of the first battery model.

In order to confirm the validity of the suggested approach, a second battery of 1500 Ah capacity is analyzed. The RBMO is implemented in comparison to others; the obtained results are tabulated in

Table 4. Optimal parameters of the second considered battery obtained through the RBMO and the others.

	DO	SWO	BMO	ISA	RBMO
$E_0$ (V)	303.553	303.667	303.577	303.581	303.589
$Q$ (Ah)	1490.00	1494.94	1504.87	1499.71	1499.68
$R$ (Ω)	0.00191901	0.00191005	0.0017	1.7488 × 10−3	0.00199999
$A$ (V)	23.5755	23.4234	23.5462	23.5442	23.5441
$K$	0.00101638	0.00104697	0.0011	1.09943 × 10−3	0.00109985
$B$ (Ah−1)	0.0405665	0.0413123	0.0406678	4.06838 × 10−2	0.0406879
$\tau$ (s)	20.6021	20.8872	19.0000	20.9986	19.4006
Fobj (V)	4.82065 × 10−4	3.29567 × 10−3	2.85252 × 10−4	2.76219 × 10−4	2.66176 × 10−4
Worst	2.77386 × 10−3	1.2734 × 10−2	2.92551 × 10−3	3.42083 × 10−4	2.80305 × 10−4
Average	1.16742 × 10−3	6.7016 × 10−3	6.82816 × 10−4	2.87921 × 10−4	2.74471 × 10−4
SD	7.49439 × 10−4	2.8224 × 10−3	7.94156 × 10−4	1.99556 × 10−5	6.04758 × 10−6
Elapsed time per trial (s)	2087	98	6075	2981	6420

. The suggested RBMO succeeded in achieving the least fitness value of 2.66176 × 10⁻⁴, ISA came second with a value of 2.76219 × 10⁻⁴, while the SWO came last with a fitness value of 3.29567 × 10⁻³. Also, the proposed strategy outperformed all in terms of SD with the least value of 6.04758 × 10⁻⁶. Figure 6 depicts the convergence curves of the RBMO and other comparable approaches, the curves show that the RBMO outperforms the others in determining the optimum parameters in the shortest number of iterations. Also, Figure 7 shows the experimental terminal voltage and that predicted using the suggested method demonstrating their high degree of convergence.

The obtained findings confirmed the robustness and competency of the suggested approach for creating the battery model by calculating its optimal parameters. The suggested RBMO is statistically tested using Kruskal–Wallis, ANOVA table, Friedman rank, and Wilcoxon rank tests. They were carried out on the first battery under consideration. Each optimizer executed ten runs and the run with the best fitness value was chosen as the best solution.

Table 5. Statistical test findings for the RBMO and other comparable methods.

	DO	SWO	BMO	ISA	RBMO
Kruskal–Wallis test
p-value	1.1664 × 10−8
Friedman test
p-value	6.2472 × 10−8
Friedman rank	7.4	9.4	4.8	4.3	1.6
	(4)	(5)	(3)	(2)	(1)
ANOVA test
p-value	1.9130 × 10−20
Wilcoxon rank test
p-value	1.8165 × 10−4	1.8267 × 10−4	4.3964 × 10−4	5.8284 × 10−4	-
h-value	1	1	1	1	-
Null hypothesis rejection	√	√	√	√	-

shows the statistical test findings for the RBMO and the others. The Kruskal–Wallis test yielded a p-value of 1.1664 × 10⁻⁸ indicating that not all data follow the same distribution. The proposed RBMO came first with a p-value of 1.6 according to the Friedman rank test while SWO was ranked last with a value of 9.4. Furthermore, the p-value obtained using the ANOVA table test is 1.9130 × 10⁻²⁰ indicating that there is a significant difference between column means. The final test is the Wilcoxon rank test in which both p-value and h-value are calculated and displayed in Table 4. The Wilcoxon test confirmed that all methods rejected the same null median. This confirms the preference of RBMO performance among the others. Figure 8 shows a boxplot of fitness function based on the ANOVA table. The penguins created by the suggested RBMO had the smallest paddles when compared to the others. This signifies that the suggested strategy rejects the null hypothesis.

It is critical to examine the performance of the created battery model using the suggested RBMO under the disturbed load. To do so, the varied pattern of the battery current shown in Figure 9 is used. The battery terminal voltage and SOC are monitored and compared using experimental data extracted via the real parameters given in Table 2. After using the present pattern shown in Figure 9, the estimated curves are generated using the fetched parameters obtained through the provided approach. Figure 10 depicts the measured and estimated responses of the first considered battery to both terminal voltage and SOC. Both data for voltage and SOC are tightly converged confirming the ability of the built model to deal with load disturbance. Moreover, the derived curves of the second considered battery are shown in Figure 11 demonstrating that the estimated data closely converge to the experimental one. The proposed RBMO is effective in creating a robust battery model capable of responding to abrupt fluctuations in demand.

The obtained findings demonstrate the advantages of the suggested RBMO in developing a trustworthy model of various Li-ion batteries.

5. Conclusions

This article proposes a new methodology incorporating the RBMO to determine the optimal Li-ion battery model parameters. The Shepherd model has been used since it provides a practical physics-informed procedure to relate electrical performance with implicit electrochemical processes. This improves the comprehension of battery dynamics, enables precise SOC estimation, and upholds the increased efficiency of battery management systems, finally enhancing battery implementation, safety, and lifetime in practical applications. The model parameters have been determined with the aid of experimental terminal voltages while the error between them and those estimated from the established model is minimized. Two different batteries have been analyzed while the suggested RBMO has been compared to other programmed methods of BEO, modified WSO, DO, SWO, BMO, and ISA. Moreover, the proposed RBMO has been statistically assessed by the Kruskal–Wallis, ANOVA table, Friedman rank, and Wilcoxon rank tests.

The main findings of the article are stated as follows:

• The best fetched values of the fitness function, 1.4951 × 10⁻⁴ and 2.66176 × 10⁻⁴, have been obtained through the suggested RBMO for the two considered batteries, respectively.
• The RBMO has achieved the best SD values of 6.5684 × 10⁻⁴ and 6.04758 × 10⁻⁴ for the two batteries considered, respectively.
• Statistical tests have yielded the superiority of the suggested approach compared to the comparable methods.
• The performance examination of the RBMO for modeling and estimating the SOC of the Li-ion battery under the disturbed load has confirmed the ability of the built model to handle load disturbance.

The resilience and competency of the RBMO have been confirmed by the obtained findings, so it can be suggested as an efficient way for modeling the Li-ion battery. The suggested future work includes validation under dynamic conditions using hardware deployment test framework, considering the impact of temperature change and dynamic loading, and the correlation analysis between the current SOC estimation and model parameter error.