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 (
) connected in series with a resistor and capacitor (RC) pair. From the literature [
17,
30], it is clear that open-circuit voltage
can be represented as a polynomial equation of 5th order, as given in Equation (1). The terms
for
are unknown polynomial coefficients, where
is the cut-off voltage (
) and the fifth-order polynomial term
is obtained by Equations (2) and (3).
The voltage source emulates the open-circuit voltage (
), and it is further dependent on the
of the battery. The
under a discharge condition can be determined by the following equation:
where
is the discharge current in amperes and
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
with an exponentially varying nature [
11,
17,
31].
The terms
,
, and
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
where the ‘
’ is the time constant of the RC pair. The terminal voltage (
) of the depicted circuit in
Figure 1 is derived by sum of the voltage drops across each passive element.
Finally, to obtain the terminal voltage (
) 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:
3. Problem Formulation
From Equation (8), it can be understood that the terms are measurable quantities based on operating conditions, while the other physical parameters 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
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 (
) and the computed voltage (
), and its corresponding equation is given as in Equation (9):
where
is the objective function,
is the number of datasets used for estimation of parameter extraction,
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
, 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
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]:
where
denotes the distance between the target and hyena,
specifies the present iteration,
and
are coefficient vectors,
designates the location vector of the target, and
is the location vector of the spotted hyena. The values of
and
are estimated using the following expression:
To have an equilibrium between exploration and exploitation, the value of the
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
and is given by
For increasing the randomness of random vectors,
&
are varied between 0 and 1. By fine-tuning the parameters
and
, 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,
and
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
and
, 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:
where
defines the location of the foremost best spotted hyena and
indicates the location of the accompanying spotted hyenas. Here,
indicates the number of spotted hyenas, which is enumerated as follows:
where
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
, which is far away from the initial eminent solution for a specified search region, and
is the cluster or group of optimal solutions
.
4.4. Phase 3: Exploiting Prey
To mathematically model the spotted hyena prey attacking, the vector value is reduced. Further, vector is reduced in agreement with the change in the vector value.
Moreover, the change in
and
services the clan of spotted hyenas to trail either prey search. The formula for attacking the prey is as follows:
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
. Hyenas move away from each other to scout and attack the prey. By randomizing the vector
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
enables hyenas to move further away from the prey. Additionally, SHO has an integral component for prey exploration, represented by vector
that assigns random weights to the prey. To ensure that the SHO algorithm has randomness, the vector
precedence over
, 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
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].