A Novel Spotted Hyena Optimizer for the Estimation of Equivalent Circuit Model Parameters in Li-Ion Batteries

Abstract

Li-ion batteries possess significant advantages like large energy density, fast recharge, and high reliability; hence, they are widely adopted in electric vehicles, portable electronics, and military and aerospace applications. Albeit having their merits, accurate battery modeling is subjected to problems like prior information on internal chemical reactions, complexity in problem formulation, a large number of unknown parameters, and the need for extensive experimentation. Hence, this article presents a reliable Spotted Hyena Optimizer (SHO) to determine the equivalent circuit parameters of lithium-ion (Li-ion) batteries. The methodology of the SHO is derived from the living and hunting tactics of spotted hyenas, and it is efficiently applied to solve the battery parameter estimation problem. Nine unknown battery model parameters of a Samsung INR 18650-25R are determined using this method. The model parameters estimated are endorsed for five different datasets with various discharge current values. Further, the effect of parameter range and its selection is also emphasized. Secondly, for validation, various performance metrics such as Integral Squared Error, mean best, mean worst, and Standard Deviation are evaluated to authenticate the superiority of the proposed parameter extraction. From the computed results, the SHO algorithm is able to explore the search area up to 89% in the case of larger search ranges. The chosen model and range of the SHO precisely predict the behavior of the proposed Li-ion battery, and the results are in accordance with the catalog data.

1. Introduction

Batteries are electrochemical devices that provide high round-trip efficiency and remain a major energy source for both pure and hybrid electric vehicles [1]. With their fast transient behavior, they are potentially suitable as an intermittent energy storage system for applications like distributed generation in off-grid/on-grid systems and Uninterrupted Power Supply (UPS) systems [2,3]. Among battery types, Li-ion technology is preferred for its high energy and power density along with its longevity. Despite many advancements made in Li-ion technology in recent years, research on Battery Management Systems (BMSs) and controlling charge and discharge is yet to mature.

In particular, battery modeling is not only essential to optimize its performance but also plays a crucial role in energy management and predicting its behavior due to aging. Over the last few decades, various battery models have evolved to describe batteries’ dynamic behavior. They can be categorized into (a) electrochemical, (b) experimental, (c) mathematical, and (d) equivalent circuit models. Though electrochemical models are highly precise, they involve complex nonlinear differential equations and intense computations [4]. Moreover, they also require prior knowledge on electrochemical reactions, physical systems, and battery identification [5]. Meanwhile, experimental models insist on a set of experiments to derive the battery internal parameters [6,7]. Both electrochemical and experimental models are not well matched to characterize cell dynamics and estimate State of Charge (SoC) [8]. Recently, stochastic approaches have been used to predict battery performance and efficiency [9]. The models are less accurate, with the error percentage lying between 5% and 20%. However, this approach is application-specific and can easily be implemented in real time [10]. Lastly, to match the voltage current characteristics of batteries, equivalent circuit-based modeling was proposed in [11] to determine its electrical properties. The model accuracy lies between electrochemical models and mathematical models and provides efficient simulation results. To mimic battery dynamics, this model uses combinations of the voltage source in series with resistors and capacitors. Among the many models, Thevenin’s single RC equivalent circuit model was benchmarked in the field of battery parameter estimation [12]. The unknown parameters were multivariable functions of SoC with sets of linear and polynomial equations describing the charge and discharge curves of the battery.

It is evident from the literature that the accuracy in battery modeling is complicated by 1. its nonlinear behavior, 2. the number of optimization parameters, and 3. challenges posed by the minimization problem. Hence, battery-equivalent circuit parameter estimation is treated as a constrained or unconstrained optimization problem and solved using various analytical methods like Least Squares (LS) methods and its variants such as Recursive Least Squares (RLS) [13,14], Moving Window Least Squares (MWRLS) [15], Decoupled Weighted Recursive Least Squares (DWRLS) [16], Constrained Nonlinear Least Squares (C-NLS), and Regularization-based Nonlinear Least Squares (R-NLS) [17]. These methods have high efficiency and faster sampling rates but are bound by limitations such as the requirement for an initial guess, the longer time taken to converge, and lower data resolution. Therefore, a few proposed adaptive versions have been used, like Kalman filters [18], Extended Kalman filters (EKFs) [19], Unscented Kalman filters (UKFs) [20], and a dual-filter approach combining EKFs and UKFs [21]. However, the adaptive methods’ effectiveness depends upon the accuracy of the battery model and the number of tuning parameters. Furthermore, these LS and adaptive filtering methods require large quantities of experimental and simulation data to predict model parameters in real time. Alternatively, to identify the unknown battery parameters, nature-inspired meta-heuristic methods such as the Genetic Algorithm (GA) [22], Particle Swarm Optimization (PSO) [23], Enhanced Sunflower Optimization (ESFO), Harmony Search Optimization (HSO), the Grey Wolf Optimizer (GWO), Whale Optimization (WO), Water Cycle Optimization (WCO), Golden Eagle Optimization [24,25], modified COOT (mCOOT) [26], and the Bald Eagle Search method by injecting the adaption of parameters [27] were recently applied.

But a lack of exploration capability in the GA and the convergence to local optima in PSO diminish the exactness of the derived battery model; moreover, model complexity increases with a 31-parameter single RC model [22]. Thus, to reduce the number of unknown parameters, the authors of [24] constructed a linear-state space model consisting of fewer parameters, treating the open-circuit voltage equation as a straight line. However, the modifications introduced could not depict the actual voltage characteristics and hence lacked accuracy. Therefore, with the aim of improving precision, a modified model with nine unknown parameters, reducing the open-circuit voltage equation to a fifth-order polynomial, is detailed in [17]. This methodology is robust and simple, as it uses fewer parameters to determine a battery-equivalent circuit model. However, the complexity of parameter estimation can be condensed when an optimization technique is employed.

Thus, this article proposes the application of a novel Spotted Hyena Optimizer (SHO) [28,29] to identify unknown parameters of a modified equivalent circuit model based on the manufacturer data sheet. Further, this work is probably the first attempt to use a modified single RC model of a battery consisting of nine unknown parameters using an optimization technique. Furthermore, to achieve high accuracy and reproduce the actual battery characteristics, the objective function preferred is the Sum of Squared Error (SSE). The software codes were written for various cases in the Matlab 2017a platform. Most importantly, the appropriate range selection of parameters is mandatory for SHO to achieve the best possible optimal values. Hence, parameter range selection and its effects are also studied. The results are validated to match with the manufacturer data sheet of a Samsung INR-18650-25R, 3.64 V, 2.55 Ah Li-ion battery. The computed results are presented for various discharge current cases of 1 A, 5 A, 10 A, 15 A, 20 A, and 25 A. To authenticate the effectiveness of SHO, the experimental and calculated data of the battery mean, Standard Deviation (SD), best, and worst are also explained in detail.

Lastly, this paper is sectioned as follows: Section 2 is about the mathematical modeling of the battery-equivalent circuit, Section 3 details the problem formulation, Section 4 describes the SHO algorithm in detail, and Section 5 deals with model validation using computed results with experimental data from the data sheet.

2. Single RC Model of a Battery

In a battery pack, each cell is assembled in parallel and series to form a stack that delivers required voltage and power for the output. Each cell of a battery can be modeled as a voltage source having an internal resistance ($R_o$) connected in series with a resistor and capacitor (RC) pair. From the literature [17,30], it is clear that open-circuit voltage $V_{ocv}$ can be represented as a polynomial equation of 5th order, as given in Equation (1). The terms $a_j$ for $j=0,1,2,3,4,5$ are unknown polynomial coefficients, where $a_0$ is the cut-off voltage ($V_{cutoff}$) and the fifth-order polynomial term $a_5$ is obtained by Equations (2) and (3).

$$V_{ocv}\left(SoC\right)={\displaystyle \sum _{j=0}^5{a}_j\ast So{C}^j}$$

(1)

$$a_0=V_{cutoff}$$

(2)

$$a_5=(V_{ocv}-V_{cutoff}-{\displaystyle \sum _{j=1}^4{a}_j)}\ast So{C}^5(t)$$

(3)

The voltage source emulates the open-circuit voltage ($V_{ocv}$), and it is further dependent on the $SoC$ of the battery. The $SoC$ under a discharge condition can be determined by the following equation:

$$SoC=So{C}_0-{\displaystyle \frac{1}{3600\ast Q_c}}I$$

(4)

where $I$ is the discharge current in amperes and $Q_c$ is the rated capacity in Ah (ampere-hours). Depending on the material property, the internal resistance causes a voltage drop during discharging and charging. Hence, it is also a function of $SoC$ with an exponentially varying nature [11,17,31].

$$R_o(SoC)=b_0+b_1{e}^{-b_{2}SoC}$$

(5)

The terms $b_0$, $b_1$, and $b_2$ are constants that are unknown. Additionally, the RC pair in series represents the transient response of the battery voltage. This article considers only one RC pair that is widely used as a benchmark model in the literature [11], and its corresponding equation is written as

$$V_{RC}\left(t\right)=I\ast R\left(1+e^{{\displaystyle \frac{-t}{RC}}}\right)$$

(6)

where the ‘$t$’ is the time constant of the RC pair. The terminal voltage ($V_t$) of the depicted circuit in Figure 1 is derived by sum of the voltage drops across each passive element.

$$V_t=V_{ocv}+V_{RC}+R_{o}I$$

(7)

Finally, to obtain the terminal voltage ($V_t$) equation, Equations (1), (5), and (6) are substituted into Equation (7) to obtain Equation (8), which represents the final single RC model of the battery:

$$\begin{array}{l}V\left(t\right)=a_o+a_{1}SoC\left(t\right)+a_{2}So{C}^2\left(t\right)+a_{3}So{C}^3\left(t\right)+a_{4}So{C}^4\left(t\right)+a_{5}So{C}^5\left(t\right)\\ +I\ast R\left(1+e^{{\displaystyle \frac{-t}{RC}}}\right)+I_0\left(b_0+b_1{e}^{-b_2\ast SoC(t)}\right)\end{array}$$

(8)

3. Problem Formulation

From Equation (8), it can be understood that the terms $\left[\begin{array}{ccc}{V}_{cutoff}& V_{ocv}& SoC\end{array}\right]$ are measurable quantities based on operating conditions, while the other physical parameters $\left[\begin{array}{ccccccccc}{a}_1& a_2& a_3& a_4& b_0& b_1& b_2& R& {\left(RC\right)}^{-1}\end{array}\right]$ remain unknown. These unknown parameters have a significant impact on models that are needed to be extracted and to match as closely as possible with the actual discharge characteristics, i.e., the voltage–Ah curve of a battery.

Subsequently, to identify the unknown parameters

$f=\left[\begin{array}{ccccccccc}{a}_1& a_2& a_3& a_4& b_0& b_1& b_2& R& {(RC)}^{-1}\end{array}\right]$ of a battery, an objective function is to be formulated first. From the previous literature, it is apparent that the goal of optimization is to minimize the Sum of Squared Error (SSE) between the voltage reference data ($V_{ref}$) and the computed voltage ($V_{calc}$), and its corresponding equation is given as in Equation (9):

$$\mathrm{minimize}f\left(X\right)={\displaystyle \sum _{i=1}^d{{\displaystyle \sum _{k=1}^N\left(V_{ref}-V_{calc}\right)}}^2}$$

(9)

$$X\in \left[\begin{array}{cc}LB& UB\end{array}\right]\in R^{+}$$

where $f$ is the objective function, $d$ is the number of datasets used for estimation of parameter extraction, $N$ is the number of reference Vvs Ah data points in each dataset, and LB and UB are the lower and upper bounds of unknown model parameters $f$, respectively.

4. Parameter Identification Using Spotted Hyena Optimizer

Applying the meta-heuristic optimization method based on the living and hunting nature of hyenas, the unknown parameters $f=\left[\begin{array}{ccccccccc}{a}_1& a_2& a_3& a_4& b_0& b_1& b_2& R& {(RC)}^{-1}\end{array}\right]$ are identified. The identification process follows the steps of SHO with the objective of minimizing the SSE value. This procedure continues until optimal values are obtained. The schematic of the optimization-based battery parameter estimation is outlined in Figure 2.

The recently evolved nature-inspired meta-heuristic Spotted Hyena Optimization (SHO) method mimics the social stratification and hunting nature of spotted hyenas to find the solution for constrained and unconfined centered design complications [28,29]. The close-knit clusters of SHO assist the well-ordered teamwork of spotted hyena clan members. The steps of SHO originate from the hunting behavior of spotted hyenas, which include searching/tracking, chasing, encircling, and attacking prey. In this work, mathematical modeling of the communal and hunting nature pertaining to spotted hyenas is described to extract the parameters of a battery.

4.1. Inspiration

The hunting mechanisms of the spotted hyena adopts three important strategies: encircling, hunting, and exploitation. During the first phase, hyenas encircle and harass prey by scouting the potential targeted prey from the whole obtainable roster. In the second phase, the chosen prey is strategically moved out from the group via hunting and rundown. This step is crucial to minimize distraction from external factors that can hinder the complete hunting process. Further, in the third phase, exploitation, various candidates from the clan of spotted hyenas distribute the gains of the hunt in a systematic and hierarchal precedence. The steps of trailing, tracking, encircling, and ambushing behavior of the wild hyena mathematical modeling are outlined in the following section.

4.2. Phase 1: Encircling Prey

Spotted hyenas are amicable within location of prey until they proceed to round the prey up. To mathematically model the communal hierarchy of spotted hyenas, the prey is considered as the best target, closest to the ideal, since the search space is not known in advance. Once the prime search candidate (prey) has been identified, the remaining search entities will attempt to update their locations. The mathematically modeled behavior of the hyenas is expressed as follows [32]:

$$\overrightarrow{D_h}=\left|\overrightarrow{B}.\overrightarrow{P_p}\left(x\right)-\overrightarrow{P}\left(x\right)\right|$$

(10)

$$\overrightarrow{P_p}\left(x+1\right)=\overrightarrow{P_p}\left(x\right)-\overrightarrow{E}.\overrightarrow{D_h}$$

(11)

where $\overrightarrow{D_h}$ denotes the distance between the target and hyena, $x$ specifies the present iteration, $\overrightarrow{B}$ and $\overrightarrow{E}$ are coefficient vectors, $\overrightarrow{P_p}$ designates the location vector of the target, and $\overrightarrow{P}$ is the location vector of the spotted hyena. The values of $\overrightarrow{B}$ and $\overrightarrow{E}$ are estimated using the following expression:

$$\overrightarrow{B}=2.r.\overrightarrow{d_1}$$

(12)

To have an equilibrium between exploration and exploitation, the value of the $\overrightarrow{h}$ vector is linearly decreased from 5 to 0 until the maximum iteration is reached. The number of optimal positions in the group or cluster during the hunting phase is recorded based on the coefficient vector $\overrightarrow{E}$ and is given by

$$\overrightarrow{E}=2.\overrightarrow{h.}r\overrightarrow{d_2}-\overrightarrow{h}$$

(13)

$$\overrightarrow{h}=5-(Iteration\ast ({\displaystyle \frac{5}{Ma{x}_{iterations}}}))$$

(14)

For increasing the randomness of random vectors, $r\overrightarrow{d_1}$ & $r\overrightarrow{d_2}$ are varied between 0 and 1. By fine-tuning the parameters $\overrightarrow{E}$ and $\overrightarrow{B}$, a substantial search space from the current location is covered. The relativistic updated locations of a spotted hyena are obtained by utilizing Equations (13) and (14), and the spotted hyena position is moved indiscriminately in and around the target. To improve the exploration and exploitation, h is linearly decreased from 5 to 0 within the duration of the highest no. of iterations. Furthermore, this methodology helps increase exploitation, as there is an increase in the count of the iteration value. However, $r{d}_1$ and $r{d}_2$ are the random vectors in the range of [0, 1]. Figure 3 presents the properties of Equations (10) and (11) in a 2D setting. In this figure, the spotted hyena (A, B) updates its location towards the location of the prey (X*, Y*). Through fine-tuning the value of vectors $E$ and $B$, there are some different places that can be reached from the current location. The relativistic updated locations of a spotted hyena in a setting are shown in Figure 3. By utilizing Equations (10) and (11), a spotted hyena can update its location indiscriminately around the prey. Therefore, this same concept can further be expanded to deal with the n-dimensional search region.

4.3. Phase 2: Hunting Prey

Similar to grey wolves, spotted hyenas also chase in packs or clans. Moreover, hyenas predominantly rely on a reliable network of other hyenas, based on their ability to track the location of the target. To design a model that mimics the behavior of spotted hyenas, the authors presume that the most dominant search entity has the insight to determine the whereabouts of the target [34]. The remaining search entities form a cluster, comprising a loyal group of companions, moving towards the leading search entity and retaining the optimal solution obtained from the updated locations. This behavior is represented by the following equations:

$$\overrightarrow{D_h}=\left|\overrightarrow{B.}\overrightarrow{P_h}-\overrightarrow{P_k}\right|$$

(15)

$$\overrightarrow{P_k}=\overrightarrow{P_h}-\overrightarrow{E}.\overrightarrow{D_h}$$

(16)

$$\overrightarrow{C}=\overrightarrow{P_k}+{\overrightarrow{P}}_{k+1}+....+{\overrightarrow{P}}_{k+N}$$

(17)

where $\overrightarrow{P_h}$ defines the location of the foremost best spotted hyena and $\overrightarrow{P_k}$ indicates the location of the accompanying spotted hyenas. Here, $N$ indicates the number of spotted hyenas, which is enumerated as follows:

$$N=coun{t}_{nos}\left(\overrightarrow{P_h},{\overrightarrow{P}}_{h+1},{\overrightarrow{P}}_{h+2}.....\left(\overrightarrow{P}+\overrightarrow{M}\right)\right)$$

(18)

where $\overrightarrow{M}$ is a vector in the random range of [0.5, 1], which states the no. of solutions that are taken into account for all the candidate solutions in addition to $\overrightarrow{M}$, which is far away from the initial eminent solution for a specified search region, and ${\overrightarrow{C}}_h$ is the cluster or group of optimal solutions $N$.

4.4. Phase 3: Exploiting Prey

To mathematically model the spotted hyena prey attacking, the vector $\overrightarrow{h}$ value is reduced. Further, vector $\overrightarrow{E}$ is reduced in agreement with the change in the vector $\overrightarrow{h}$ value.

Moreover, the change in $\left|\overrightarrow{E}\right|>1$ and $\left|\overrightarrow{E}\right|<1$ services the clan of spotted hyenas to trail either prey search. The formula for attacking the prey is as follows:

$$\overrightarrow{P}\left(x+1\right)={\displaystyle \frac{\overrightarrow{C_h}}{N}}$$

(19)

$\overrightarrow{P}\left(x+1\right)$ detects the initial potential solution and the locations is updated from the other search entities in accordance with the initial value of the search entity [28,29].

4.5. Prey Search

The search for prey by spotted hyenas involves the location of the pack, represented as vector ${\overrightarrow{C}}_h$. Hyenas move away from each other to scout and attack the prey. By randomizing the vector $\overrightarrow{E}$ values to be less than −1 or greater than 1, the search entities are forced to move farther from the prey, allowing the Spotted Hyena Optimization (SHO) algorithm to conduct a global search. To identify the prey more effectively, Figure 4 illustrates how $\left|\overrightarrow{E}\right|>1$ enables hyenas to move further away from the prey. Additionally, SHO has an integral component for prey exploration, represented by vector $\overrightarrow{B}$ that assigns random weights to the prey. To ensure that the SHO algorithm has randomness, the vector $\overrightarrow{B}>1$ precedence over $\overrightarrow{B}<1$, assumed to detail the distance affected, is observed in Equation (12). Moreover, it is very substantial to explore and to avoid the local optima. Based on the spotted hyena location, the prey weight is randomly assigned. Vector $\overrightarrow{B}$ is deliberately needed to assign the random variables not only during the inceptive iterations but also for terminal iterations. The prey’s weight is randomly assigned based on the hyenas’ locations, and this randomization must be applied not only during the initial iterations but also during the final stages. This ensures that the algorithm does not converge prematurely to local optima. As the algorithm approaches its termination, it is guided by the termination criterion [28,29].

5. Results and Discussion

This section discusses the performance evaluation of the proposed SHO method for parameter extraction of battery models that are best fit for cataloguing Ah–voltage characteristic prediction. The technical specifications and operating conditions of the chosen battery are mentioned in

Table 1. Data sheet specifications of INR-18650-25R.

S. No	Parameters	INR-18650-25R
1.	Nominal voltage ($V$)	3.62 V
3.	Nominal capacity ($Q_c$)	2600 mAH
4.	Open-circuit voltage ($V_{oc}$)	4.2 V
5.	Cut-off voltage ($V_{cu}$)	2.5 V
6.	Operating temperature	−40 to 75 °C
7.	Chemistry	Nickel Manganese Cobalt (NMC)

. The reason for using the SAMSUNG INR-18650-25R battery cell selected is that the discharge characteristics are easily available online or can be reproduced from the literature [35]. The accuracy of the identified parameters is highly dependent on the search range for each model parameter, which must be defined before running the proposed SHO algorithm Additionally, the SHO algorithm was implemented in MATLAB 2017a, and experiments were conducted on a desktop computer with an Intel Core i5-4570T @ 2.90 GHz processor running a 64-bit Windows 10 operating system.

5.1. Parameter Selection and Its Effect

For the appropriate identification of unknown battery parameters and the best results covering the entire search area, the range selection is crucial for any optimization algorithm. Further, the parameters obtained having physical meaning should be realizable. In addition, this process helps to understand if the identified parameters are in line with the model requirement. Hence, two possible search ranges (Range 1 and Range 2) with long and short search bands are taken for study, and their details are given in

Table 2. Parameter ranges used for 1 Amp discharge current.

				Range 1
Parameter	$a_1$	$a_2$	$a_3$	$a_4$	$b_0$	$b_1$	$b_2$	$R$	${\left(RC\right)}^{-1}$
Min	0.01	−26	1	−45	0.005	0.001	6.5	0.005	0.001
Max	5.1	−1.5	45	−2	0.085	8	89.5	0.085	0.095
				Range 2
Parameter	$a_1$	$a_2$	$a_3$	$a_4$	$b_0$	$b_1$	$b_2$	$R$	${\left(RC\right)}^{-1}$
Min	0.9	−6	−1.1	−15	0.004	0.02	16.5	0.004	0.001
Max	2	−0.5	15	−2	0.05	1	35	0.05	0.005

. The identified model parameters using Range 1 (first row) and Range 2 (second row) and applying the SHO algorithm for a 1 A current discharge are tabulated in

Table 3. Identified parameters for Range 1 and Range 2 for 1 Amp discharge current.

	Minimum	$a_1$	$a_2$	$a_3$	$a_4$	$b_0$	$b_1$	$b_2$	$R$	${\left(RC\right)}^{-1}$
Range 1	1.21 × 10−6	1.251	−3.17	7.6811	−7.8306	0.045	1.371	30.151	0.045	0.007
Range 2	4.80 × 10−6	1.702	−5.991	14.082	−14.066	0.0455	1	21.33109	0.0455	0.004
Diff %	−74.7	−36.106	−89.249	−83.336	−79.633	0.285	27.076	29.254	0.285	29.099

. A total of five datasets of 1 A, 5 A, 10 A, 15 A, and 20A discharge datasets are considered for this analysis. The term ‘Minimum’ in Table 3 refers to the best SSE value attained after 20 trials. Furthermore, it also indicates the best curve fit and capability of the SHO algorithm. The corresponding equation is given below.

$$Minimum={\displaystyle \sum _{k=1}^{5}Min{(SSE)}_j}$$

(20)

It is understood from Table 3 that the range selection impacts the results considerably. For instance, it also expands the search area to find the optimal values from 0.2 to 89% for the long range (Range 1) and goodness of the curve fit. By comparing Table 2 with Table 3 for the short range (Range 2), the parameters $a_2$, $a_3$, $a_4$, $b_0$, $b_1$, and $R$ obtained by SHO tend to converge to the corresponding lower bound. To access the parametric variation of the SHO algorithm, both long and short ranges are applied for 1 A, 5 A, 10 A, 15 A, and 20 A discharge currents. The results obtained in graphical form are presented in Figure 4 and Figure 5.

Figure 4. Bar chart representation of obtained parameter set using SHO for long range (Range 1).

Figure 4. Bar chart representation of obtained parameter set using SHO for long range (Range 1).

It is clear that the parameters $b_0$, $b_1$, and $R$ are varying with the discharge current and contributing to internal ohmic losses. Further, these parameters are multiplied with factors of 1000, 10, and 1000 in order to differentiate the variability of the parameters. Hence, to find the best optimal values of the algorithm, one should choose a larger search area (range 1) in order to achieve a lower convergence speed. Henceforth, the long range (Range 1) is used for further analysis to study the SHO performance and efficacy.

5.2. Performance Analysis of SHO Algorithm

To extract the unknown parameters of the single RC model, the long range (Range 1) limits mentioned in Table 2 are utilized. The experimental or reference Ah–voltage data of 34 instances are extracted for discharge currents of 1 A, 5 A, 10 A, 15 A, and 20 A, and the curves are fitted with the SHO extracted parameters. In order to identify the accuracy of the modeling method, analytical tools such as Individual Absolute Error (IAE), Individual Squared Error (ISE), accuracy of curve fit, and convergence to obtain the minimum SSE are determined. Further, to prove the effectiveness of SHO, an error analysis based on the ISE between the reference and computed values at each instance is calculated from Equation (21).

$$ISE={\left(V_{ref}-V_{calc}\right)}^2$$

(21)

The ISE values of the 1 A discharge current curves that correspond to the Samsung battery cell identified using the SHO method are tabulated in

Table 4. Assessment of catalogue values and calculated values of single RC model.

N	Reference Data		Computed Data		N	Reference Data		Computed Data
	$\mathit{A}{\mathit{h}}_{\mathit{r}\mathit{e}\mathit{f}}$	${\mathit{V}}_{\mathit{r}\mathit{e}\mathit{f}}\left(\mathit{V}\right)$	${\mathit{V}}_{\mathit{c}\mathit{a}\mathit{l}\mathit{c}}\left(\mathit{V}\right)$	${\left({\mathit{V}}_{\mathit{r}\mathit{e}\mathit{f}}-{\mathit{V}}_{\mathit{c}\mathit{a}\mathit{l}\mathit{c}}\right)}^2$		$\mathit{A}{\mathit{h}}_{\mathit{r}\mathit{e}\mathit{f}}$	${\mathit{V}}_{\mathit{r}\mathit{e}\mathit{f}}\left(\mathit{V}\right)$	${\mathit{V}}_{\mathit{c}\mathit{a}\mathit{l}\mathit{c}}\left(\mathit{V}\right)$	${\left({\mathit{V}}_{\mathit{r}\mathit{e}\mathit{f}}-{\mathit{V}}_{\mathit{c}\mathit{a}\mathit{l}\mathit{c}}\right)}^2$
1	0.0154	4.1064	4.1	4.0960 × 10−5	18	1.7556	3.4730	3.458	2.2500 × 10−4
2	0.0872	4.0448	4.061	2.6244 × 10−4	19	1.8275	3.4466	3.438	7.3960 × 10−5
3	0.2253	3.9920	3.994	4.0000 × 10−6	20	1.8938	3.4290	3.42	8.1000 × 10−5
4	0.4629	3.9129	3.896	2.8561 × 10−4	21	1.9711	3.4027	3.399	1.3690 × 10−5
5	0.5789	3.8513	3.852	4.9000 × 10−7	22	2.0263	3.3939	3.383	1.1881 × 10−4
6	0.6894	3.8073	3.811	1.3690 × 10−5	23	2.0926	3.3675	3.362	3.025 × 10−5
7	0.7888	3.7721	3.775	8.4100 × 10−6	24	2.1534	3.3499	3.338	1.4161 × 10−4
8	0.8938	3.7457	3.737	7.5690 × 10−5	25	2.2142	3.2971	3.308	1.1881 × 10−4
9	0.9932	3.7017	3.702	9.0000 × 10−8	26	2.2639	3.2531	3.272	3.5721 × 10−4
10	1.065	3.6666	3.6770	1.0816 × 10−8	27	2.3191	3.2003	3.211	1.1449 × 10−4
11	1.1369	3.6490	3.652	9.0000 × 10−6	28	2.3578	3.1124	3.146	0.0011
12	1.2363	3.6138	3.618	1.7640 × 10−5	29	2.3854	3.0508	3.082	9.7333 × 10−4
13	1.3358	3.5786	3.585	4.0960 × 10−5	30	2.4186	2.9804	2.975	2.9160 × 10−5
14	1.3965	3.5610	3.565	1.6000 × 10−5	31	2.4407	2.9100	2.88	9.0000 × 10−4
15	1.4684	3.5434	3.542	1.9600 × 10−6	32	2.4628	2.8045	2.759	0.0021
16	1.54022	3.5170	3.52	9.0000 × 10−6	33	2.4894	2.6901	2.646	0.0019
17	1.6175	3.4994	3.490	8.8360 × 10−5	34	2.4849	2.5318	2.603	0.0051

N—no. of instances.

. Further, the reference data representing the actual values extracted from the data sheet are also incorporated in the table for clarity. From the data given in Table 3, it can be inferred that that the single RC model has minimal ISE error in every instance. The lowest and highest bounds of the ISE values are 1.081 × 10⁻⁸ and 0.0011, respectively. Furthermore, to substantiate the error analysis, a bar chart for each instance at discharge rate of 1 Amp between the reference and computed values for the single RC battery model is depicted in Figure 6. From the figure, it can be understood that the single RC model of the battery fits the actual curve with the lowest marginal error.

In addition, an analysis based on the IAE for all 34 instances of the Ah–voltage curve is performed using the formula given below. The IAE values plotted with respect to each instance are shown in Figure 7.

$$IAE=\left|V_{ref}-V_{calc}\right|$$

(22)

In addition, it is important to mention that the performance of the algorithm is purely based on the generation of random values and probability to find the best fitness value, i.e., SSE. Moreover, it is not assured that one sample run of the algorithm can provide an acceptable performance. Therefore, the obtained nine unknown parameters for the remaining discharge rates 5 A, 10 A, 15 A, 20 A, and 25 A are listed in

Table 5. Identified parameters of single RC model of Samsung INR-18650-25R using proposed SHO algorithm.

Parameters	Discharge Currents
	5 A	10 A	15 A	20 A	25 A
$a_1$	1.7360	1.63	0.9131	1.2563	1.3992
$a_2$	−5.7238	−3.9002	−0.5669	−1.5558	−1.5012
$a_3$	12.4628	6.4609	−1.1861	2.1244	1.5592
$a_4$	−11.4326	−4.4570	−1.9933	−2.2075	−2.4326
$b_0$	0.0168	0.0133	0.0107	0.0110	0.0110
$b_1$	0.1723	0.0530	0.0334	0.0252	0.0228
$b_2$	22.3813	17.715	20.4340	21.1505	20.2482
$R$	0.0168	0.0133	0.0107	0.0110	0.0110
${\left(RC\right)}^{-1}$	0.0094	0.001	0.0038	0.0018	0.0015

. As said above, the parameters belong to the best candidate solutions that are found out of 30 independent runs, and it is not guaranteed that the algorithm can repeat the same values. Further, the unknown parameters are different for each discharge rate due the change in the effective series resistance of the battery cell. To validate the robustness of the SHO algorithm, some of the statistical measures such as best, worst, average, median, variance, and Standard Deviation are computed through each independent run by utilizing the best fitness values and are tabulated in

Table 6. Error minimization result during discharge.

Discharge Current	Best	Worst	Average	Median	Standard Deviation	Variance
1 Amp	0.0202	0.0290	0.0218	0.0206	0.0023	5.3217 × 10−6
5 Amps	0.0170	0.0320	0.0197	0.0172	0.0044	1.9145 × 10−5
10 Amps	0.0138	0.0361	0.0179	0.0154	0.0056	3.180 × 10−5
15 Amps	0.0181	0.0357	0.0213	0.0190	0.0058	3.4015 × 10−5
20 Amps	0.0149	0.0403	0.0205	0.0154	0.0090	8.1216 × 10−5
25 Amps	0.0178	0.0507	0.0253	0.0178	0.0107	1.1361 × 10−4

. The best and worst are the minimum and maximum values of the SSE of 30 independent runs. The average shows the mean values of all 30 SSE values obtained. The variance depicts how far the set of SSE values is from the mean value. The Standard Deviation (SD) shows that the scattering of data and its value should be a low value that tends to be closer to the average value.

The computed parameters of the Samsung INR-18650-25R battery are used for the identification and validation of the SHO results. The identified and validated Ah–V curves of the battery are plotted in Figure 8a–f. From the discharge characteristics, it can be inferred that SHO is able to produce a well-matched simulated curve that is in close agreement with the catalogue or reference dataset. Further, to know the convergence speed, the objection function values for the above test case are presented in Figure 9a–f. The convergence characteristics illustrate that the SHO method converges to the lowest objective function value within 800 iterations. More likely, the SHO behavior is similar for all discharge currents, such as 1 A, 5 A, 10 A, 15 A, 20 A, and 25 A. Further, this method requires fewer tuning parameters, which allows it to converge faster for all the discharge rates. The lowest convergence characteristics are obtained for the discharge current rate of 15 Amps, while for the remaining operating conditions, the value remains lower than 0.2.

6. Conclusions

For applications in EVs, energy storage systems, etc., batteries play a major role in meeting intermittent power demands. Due to the significant advantages of Li-ion batteries, understanding the battery model and its associated parameters is becoming vital in predicting its behavior. This article is mainly focused on the accurate extraction of the unknown parameters of a Li-ion battery by applying the SHO method. Further, the exact fit of the Ah–V characteristics at different rates at currents of 1 A, 5 A 10 A, 15 A, 20 A, and 25 A are plotted for matching the catalogue data of the Samsung INR 18650-25R battery and finding the most suitable value. It is also worth mentioning that the SHO method evaluates better with faster convergence and gives a robust performance. The following are the conclusions made from the results:

• The unknown parameters $a_2$, $a_3$, $a_4$, $b_0$, $b_1$, and $R$ are majorly dependent on the range, and a large search area should be used.
• The parameters $b_0$, $b_1$, and $R$ have drastic variations within the prescribed range, which is attributed to internal resistive losses for higher rate of discharge currents.
• The SHO methods proves that with its exploitation and exploration capability, it can cover up to 89% of the defined search area with minimal iterations and find the best suitable value to accurately fit the single RC model of the Li-ion battery.
• The parameter estimation of the SHO method was evaluated based on the IAE, ISE, curve fit accuracy, and convergence. Further, performance metrics such as best, worst, mean, and SD were analyzed.