State-of-Charge Estimation on Lithium-Ion 18650 Under Charging and Discharging Conditions: A Statistical and Metaheuristic Approach

Abstract

Battery management systems are essential in electric vehicles and renewable energy applications, especially in terms of ensuring optimal battery health and performance and regarding the state of charge (SOC) in batteries consisting of many cells. The lifetime and efficiency of the battery depend on the accuracy of the SOC parameter estimation. Moreover, systems that apply active balancing technology are able to move cells with high SOC data to cells with low SOC. Many methods have been developed, but their long execution time makes them less optimal when applied. High-speed SOC estimation is required in active balancing technology, in addition to high accuracy. Therefore, this study proposes the estimation of SOC parameters using a statistical and metaheuristic approach from voltage and current input data in each battery cell. The experimental results showed that the metaheuristic-based method (ANFIS) had better RSME and R² values compared with the polynomial and linear regression or even the machine learning-based method (recurrent neural network) for training data.

1. Introduction

Batteries function as energy storage devices; they discharge energy periodically as required [1]. The increasing need for batteries necessitates an increase in the quantity of batteries in the battery industry and research where batteries are the focus. When choosing the right battery, each system’s battery type has its own distinct characteristics. Due to their high energy density, lithium-ion batteries are widely used [1]. This type of battery is often utilized in the automotive industry as an energy source to drive electric motors must be assembled into packs to achieve the desired voltage and current capacity per hour [2]. However, various problems are often encountered when using battery packs, one of which is an imbalance between battery cells [3]. A battery pack requires a battery management system to regulate voltage stability during charge and discharge [4,5,6]. A battery with unbalanced cells results in a decrease in the performance of the battery management system (BMS) in terms of regulating the battery charging and discharging process [4,7]. The performance of a battery management system (BMS) declines when cell balance is disrupted, as reflected in shortened battery life during charging and discharging cycles [7,8]. The problem of cell imbalance within the battery pack should be addressed by the cell balancing circuit integrated into the battery management system. However, in practice, many battery management system products available on the market lack a cell balancing function within their modules [9]. To increase a battery’s longevity, the battery management system must include a balancing technique. Battery cells can be balanced in a variety of ways. The two main cell balancing techniques in battery management systems are active and passive balancing [10,11]. The active cell balancing method includes several system topologies, i.e., cell-to-cell (C2C), pack-to-cell (P2C), and cell-to-pack (C2P) techniques [12,13]. Battery balancing techniques utilize several types of basic components to transfer energy, including resistors, capacitors, and inductors [10,11]. Given the rapid development of electric vehicles, which has increased the demand for BMS and battery balancers, there is a need to develop a method that effectively integrates battery cell balancers with battery management systems to meet the specific needs and capacities of batteries. However, the availability of battery balancing products on the market does not always align with the capacity of battery cells. This issue highlights the need for further research focused on developing a method that combines multiple battery cell balancers within the desired battery pack. One commonly used product on the market employs the flying capacitor balancing method. This method includes active balancing methods using C2C topology; however, this topology requires a significant amount of time to achieve cell balancing. Conversely, one advantage of the C2C method is that it can balance cells by optimizing the capacity of each available cell [14]. In a previous study, the researchers employed the active cell balancing method with a cell-to-cell topology, specifically utilizing the flying capacitor balancing method obtained from the market [12,13].

In one study, the researchers developed a combination of two or more identical balancing modules that had a battery management system (BMS) integrated on a miniature 8S4P battery pack. The system achieved a balanced state within 600 min, with a discharging ratio of 1:3 and a charging ratio of 35:48 before and after applying the combined balancers. Since battery performance depends on the capacity of the tested battery, this study employed the state-of-charge (SOC) parameter to evaluate the charging and discharging conditions as part of the software development process. The SOC represents the remaining energy of a battery as a percentage of its total capacity, which is critical for ensuring optimal performance and safety, especially in electric vehicles (EVs) and energy storage systems (ESSs) [15]. Common SOC estimation techniques, such as Coulomb counting and voltage-based methods, provide real-time insights into charge levels, enabling efficient energy management. The BMS, on the other hand, oversees battery health by monitoring and controlling SOC [16], preventing overcharging, deep discharge, and extreme temperatures while balancing cell voltages to enhance the efficiency and lifespan of the battery [17]. Accurate SOC estimation allows designers to optimize capacity usage, reduce over-engineering, and design smaller and lighter batteries; it also helps users to maintain the battery within safe charge limits, improving performance and extending service life. Considering SOC’s importance, utilizing a statistical approach, this study designed an SOC estimator for the BMS with active balancing based on the flying capacitor method. A battery’s capacity depends on factors such as discharge time, average current, electrolyte temperature, cutoff voltage, storage duration, and aging [18,19,20,21]. Traditionally, the most accurate SOC determination involves fully charging or discharging the battery to recalibrate the estimation algorithm, a process typically carried out in EVs or plug-in hybrid vehicles (PHEVs). For hybrid electric vehicles (HEVs), SOC is commonly estimated using the Coulomb counting method, which integrates current over time. However, this technique is susceptible to measurement errors, meaning that periodic corrections are required. Therefore, in this research, SOC estimation is based not only on current measurements but also on normalized cell voltage data to enhance accuracy [22] and reliability.

Numerous researchers have developed a variety of methods for estimating SOC parameters, including techniques based on measurement [23,24], adaptive filters [25,26], and data-driven algorithms [27,28]. In general, these methods prioritize the accuracy of SOC estimation while often neglecting execution speed. For example, in lithium-ion battery state-of-charge (SOC) estimation, the ANFIS has shown superior performance compared with conventional Kalman filters [20,22] and neural networks, achieving prediction errors below 1% and maintaining accuracy under varying temperatures and load conditions. In EV energy management systems, the ANFIS has been applied to optimize power sharing between the battery and supercapacitor, improving overall efficiency by up to 8% and extending battery lifespan through reduced stress during acceleration and braking. In another grid-connected ESS study, ANFIS-based controllers effectively managed charging and discharging operations, maintaining voltage stability and minimizing energy losses. Furthermore, in vehicle-to-grid (V2G) applications, ANFIS models were employed to predict energy demand and optimize bidirectional power flow, enhancing grid reliability and reducing peak energy loads. Collectively, these applications demonstrate that the ANFIS provides a powerful, flexible, and computationally efficient solution for managing nonlinear and dynamic energy systems, underscoring its potential in advancing intelligent control and optimization strategies for next-generation EV and ESS technologies. Despite the fact that SOC parameter estimation provides accurate results, prolonged execution times can impair the performance of the BMS when implemented in hardware. Additionally, the majority of these methods have been tested on batteries. Therefore, this study proposes SOC parameter estimation in the BMS using a statistical approach, i.e., a linear regression and metaheuristic approach. The regression equation is derived from the current and voltage data collected from the battery pack used for the testing. This equation is subsequently integrated into the Daly BMS software, which serves as the focus of the testing. Since the equation consists of a mere single formula without filtering or looping processes, this method enables better execution speed to be achieved compared with that obtained using other non-statistical approaches. The SOC estimation speed of the BMS is one of the factors determining the BMS’s performance.

The main contributions of this study are summarized as follows:

• This research proposes the estimation of the SOC parameter using the ANFIS approach.
• This research also performs a comparison with other statistical and machine learning methods, i.e., the linear regression, polynomial regression, and recurrent neural network techniques.
• The proposed approach is validated on a real active balancing BMS using commercially available lithium-ion 18650 cells under charging and discharging conditions.

The contribution and novelty of this research article are rooted in how the optimal method for SOC estimator applications is determined. Several approaches are used: the statistical approaches are based on linear and polynomial regression, and the metaheuristic approaches are based on adaptive neuro-fuzzy inference systems and recurrent neural networks. The remainder of this paper is structured as follows: Section 2 provides an explanation of the theories and methods related to the topic of this paper; Section 3 contains a discussion of the utilized dataset; Section 4 outlines the results and analysis of the experiments performed in this study; and, finally, Section 5 provides a summary of our overall work.

2. Materials and Methods

2.1. Lithium-Ion 18650

The Panasonic lithium-ion 18650 battery is a cylindrical lithium-ion cell measuring 18 mm in diameter and 65 mm in length; it is commonly applied in electric vehicle (EV) systems because of its high energy density, consistent performance, and standardized size [29]. It has a nominal operating voltage of around 3.6–3.7 V, with a maximum charging limit of 4.2 V and a capacity typically between 1500 and 3500 mAh [30]. In EV applications, large numbers of 18,650 cells are connected in series and parallel arrangements to build battery modules and packs that can deliver the required voltage and energy levels for vehicle propulsion. A battery management system (BMS) is essential in these packs to continuously monitor voltage, current, and temperature, ensuring safe operation while protecting the cells from overcharging, excessive discharge, and overheating. The ability of certain 18,650 cells to provide high discharge currents supports vehicle acceleration and enhances the efficiency of regenerative braking [31]. Furthermore, their long cycle life and low self-discharge rate make them well-suited for repeated daily charging cycles. Owing to their modular structure, relatively simple thermal management, and proven reliability, Li-ion 18650 batteries are widely used in electric cars, electric motorcycles, and other EV-related energy storage systems, offering an effective balance between performance, safety, and cost.

2.2. Flying Capacitor

In battery management systems (BMSs), one of the commonly used methods to balance the voltage between the different cells in a battery pack is the flying capacitor technique. This technique applies a capacitor connected between two cells to temporarily transfer energy from the cell with the higher voltage level to the cell with the lower voltage level. Its efficient and rapid balancing capability, in comparison with the resistor bleeding method, makes it a suitable choice for electric vehicle (EV) applications. In addition, the charge transfer is not dissipated as heat, which prevents this method from generating high temperatures, which can occur when using other techniques. Still, this technique has certain drawbacks, including a complex circuit that necessitates precise switching and a higher cost compared with that of other methods. This method falls under the category of active balancing techniques, and its topology is illustrated in Figure 1 [32]. B1 to B4 represent the battery cells, S1 to S4 denote the switches, and C1 to C3 indicate the capacitors.

Figure 1. Balancing cell with flying capacitor topology.

2.3. Polynomial Regression

Regression analysis, in which the correlation between the independent (input) variables and the dependent (output) variables is nonlinear but modeled using a polynomial equation, is referred to as polynomial regression. The polynomial regression of degree n is as presented in Equation (1).

$$y={\beta }_o+{\beta }_{1}x+{\beta }_2{x}^2+{\beta }_3{x}^3+\dots +{\beta }_n{x}^n+e$$

(1)

where $y$ represents the output or dependent variable,$x$, ${\beta }_o, {\beta }_{1x},\dots ,{\beta }_n$ denote the independent variables, n determines the polynomial degree, and e represents the error or residual.

2.4. Linear Regression

Linear regression is a statistical method used to model and analyze the correlation between one independent variable (y) and one more dependent variable ($x_n$) with a regression coefficient (${\beta }_n$) and error (e). This method is also adopted to predict the value of input variables based on their output variables. Meanwhile, Equation (2) demonstrates the correlation between the two.

$$y={\beta }_o+{\beta }_1{x}_1+{\beta }_2{x}_2+\dots +{\beta }_n{x}_n+e$$

(2)

2.5. Adaptive Neuro-Fuzzy Inference System (ANFIS)

The adaptive neuro-fuzzy inference system (ANFIS) is a hybrid system that combines artificial neural networks (ANNs) with fuzzy inference systems (FISs). This integration is designed to leverage the learning capability of neural networks while maintaining the interpretability of fuzzy logic systems. The ANFIS is typically based on the Sugeno FIS model, which provides a structured and adaptive approach to function approximation and decision-making. The structure of the ANFIS consists of five layers that each perform a specific function. The first layer is the fuzzification layer, where each node represents a membership function that defines the degree of input belonging to a fuzzy set. Common membership functions include Gaussian and sigmoid functions, mathematically represented as follows:

$$\mu A\left(x\right)= e^{{-\left({\displaystyle \frac{x-e}{\sigma }}\right)}^2}$$

(3)

where c is the center and σ is the width of the Gaussian function. The second layer is the rule layer, where each neuron represents a fuzzy rule and computes the firing strength of that rule using the T-norm operation, often multiplication:

$$wi= \mu Ai\left(x\right)·\mu Bi\left(y\right)$$

(4)

The third layer is the normalization layer, which normalizes the rule strengths to ensure that their sum adds up to one:

$$\overline{wi}= {\displaystyle \frac{wi}{\sum jwj}}$$

(5)

The fourth layer is the consequent layer, where each node computes a linear function of the input variables, typically expressed as follows:

$$fi=\underset{\_}{wi} (pi·x+qi·y+ri)$$

(6)

where$pi$, $qi$, and $ri$ are the parameters to be learned. Finally, the fifth layer is the output layer, which provides the sum of all the outputs of the previous layer to produce the final ANFIS output:

$$Output= \sum \overline{wi}·fi$$

(7)

ANFIS learning is achieved through a hybrid training approach that combines backpropagation for the fuzzy membership function parameters and least squares estimation (LSE) for the consequent parameters. The feedforward phase calculates the output based on the given inputs, while the backpropagation phase updates the parameters to minimize the error between the predicted and actual outputs. The ANFIS offers several advantages, including adaptability, interpretability, and powerful function approximation capabilities. This system is widely applied in fields such as weather prediction, robotics control, financial forecasting, and medical diagnosis. By combining the strengths of neural networks and fuzzy systems, the ANFIS provides a robust framework for modeling complex, nonlinear systems.

2.6. Research Flow

Figure 2 illustrates the research flowchart for estimating the SOC of the battery pack. First, a literature review was conducted to assess the state of the art in SOC estimation research and identify the most appropriate method for processing battery characterization data.

Figure 2. Flow diagram for SOC estimation research using the active balancing system with statistical algorithm.

Since 2021, state-of-charge (SOC) estimation has rapidly advanced from basic methods like Coulomb counting and simple voltage-based approaches to more sophisticated model-based and machine learning techniques to improve accuracy, adaptability, and robustness in real EV battery systems. Researchers have explored Kalman filters, deep neural networks, and hybrid estimators that better handle nonlinear battery behavior, temperature effects, and sensor noise, significantly reducing estimation errors compared with older techniques. Using the adaptive neuro-fuzzy inference system (ANFIS) as an SOC estimator combines neural networks and fuzzy logic, allowing for it to learn complex nonlinear relationships between inputs like voltage, current, and temperature and the battery’s SOC, and to adjust adaptively in real time. The benefit of the ANFIS is its ability to provide high-accuracy SOC estimation with efficient computation and adaptive learning, which helps to improve battery performance monitoring, optimize charging strategies, and enhance safety compared with simpler estimation methods. After establishing a statistical method to determine the unknown SOC value at a given current, the statistical analysis further revealed the trend of the battery’s SOC value in relation to its service life. Once the appropriate model was obtained, system testing could be performed directly using the active balancing system.

3. Results

This section provides a detailed explanation of the dataset utilized in this research, including the data acquisition strategy and the design of the acquisition device.

3.1. System Architecture

The system architecture for collecting current and voltage data from a battery pack composed of four series of Panasonic Li-ion 18650 batteries, with a total voltage of 16.4 V, is illustrated in Figure 3. After passing through the BMS, the current and voltage were measured using the first INA226 sensor (Texas Instrument, Dallas, TX, USA), which then transmitted the measurement data to the ESP32 (Espressif System, Shanghai, China) for monitoring.

Figure 3. Flow diagram for SOC estimation research using the active balancing system.

The measured power was subsequently transmitted to a buck-boost converter, which adjusts the voltage according to the load requirements by either increasing (boosting) or decreasing (bucking) the voltage. The current and voltage following the buck-boost converter were re-measured using a second INA226 sensor before the power was delivered to the final load. Data from the second INA226 sensor was also transmitted to the ESP32 for further monitoring. The ESP32 microcontroller collected and processed data from both INA226 sensors and could regulate the buck-boost converter to ensure that the load would receive a stable and efficient voltage. With this system, the battery condition could be monitored and the power received by the load could be regulated in real time, allowing for overall system performance to be optimized. The data collected by the ESP32 could be utilized for direct monitoring, or be sent to the database. The database would allow for data to be stored in a structured and secure manner.

By storing parameter readings and estimation data, the SOC could access and recover this information as required, utilizing MongoDB 8.0 for database management. The advantages of MongoDB include its flexible, document-oriented data model, which stores data in JSON format. Additionally, MongoDB does not require a fixed schema, allowing for the storage of documents without predefined structures; it offers high flexibility in data modeling and can easily adapt to changes in data structures. Figure 4 presents the design of the data model using MongoDB, which is set to be implemented. The JSON format represents the measurement data collected from four batteries over a specified period. The data encompasses information on voltage, current, temperature, and SOC. This data was recorded in real time. The current value remained constant within the 400 mA range.

Figure 4. SOC data view on MongoDB.

3.2. Dataset

Table 1 demonstrates several data samples that were generated from the data acquisition process, which were utilized as test data in this research at a current of 3 A. The use of voltage variation datasets for battery charge or discharge conditions is to determine the distribution of data during the two conditions. With dynamic data distribution, it can produce a representative SOC estimator modeling according to the actual SOC value, while the actual SOC in this dataset is obtained through calculations applied to the BMS software by considering the initial SOC before the battery charging or discharging process occurs, the maximum battery capacity, and measurements of the incoming and outgoing electric current.

Table 1. Sample of dataset of SOC estimator.

Id	State (Charge = 1; Discharge = 0)
	Time	V1	V2	V3	V4	Tot V	SOC	State
1	0	2.74	2.74	3.19	2.94	11.61	86	1
2	10	2.74	2.74	3.19	2.94	11.61	86	1
3	20	3.23	3.22	3.23	3.21	12.89	96	1
4	30	3.27	3.28	3.249	3.27	13.069	97	1
5	40	3.289	3.293	3.275	3.28	13.137	98	1
6	50	3.328	3.294	3.288	3.301	13.211	98	1
7	60	3.325	3.52	3.31	3.318	13.473	100	1
8	70	3.356	3.446	3.306	3.345	13.453	100	1
9	80	3.354	3.337	3.329	3.343	13.363	99	1
10	90	3.372	3.356	3.351	3.346	13.425	100	1
…	…	…	…	…	…	…	…	…
25	0	3.25	3.24	3.25	3.25	12.99	100	0
26	12	3.2	3.2	3.22	3.21	12.83	99	0
27	24	3.21	3.2	3.21	3.21	12.83	99	0
28	36	3.2	3.2	3.22	3.21	12.83	99	0
29	48	3.2	3.19	3.22	3.2	12.81	99	0
30	60	3.19	3.19	3.22	3.2	12.8	99	0
31	72	3.18	3.18	3.21	3.19	12.76	98	0
32	84	3.18	3.18	3.21	3.18	12.75	98	0
33	96	3.18	3.17	3.2	3.18	12.73	98	0
34	108	3.17	3.17	3.19	3.18	12.71	98	0
35	120	3.16	3.17	3.2	3.17	12.7	98	0
…	…	…	…	…	…	…	…	…
146	720	0.91	0.93	1.01	0.94	3.77	30	0

The dataset used is one that combines charge and discharge conditions. For charging conditions, data collection is carried out every 10 s, with the initial SOC condition of 86%, until the SOC condition reaches 100%, so that 12 data samplings are obtained when the battery is full. The charging condition is carried out twice so that the total sampling for the charging condition is 24 samplings. For the discharge condition, sampling data is taken twice with a 12 s interval using a constant resistive load from the SOC condition of 100% to 30% so that the number of data obtained for the second experiment is 122. The experiment was stopped when the SOC was at 30 percent in order to maintain the reliability of the lithium-ion battery used. In total, the dataset used in this study comprises 146 pieces of data (24 charging condition data and 122 discharge condition data). For training purposes, both statistical and metaheuristic approaches were used at an 80:20 ratio, namely 80% of the dataset (117 data) was used for training purposes and 20% of the dataset (29 data) was used for testing purposes.

4. Discussion

Utilizing the dataset presented in Table 1, the calculations for the polynomial regression and linear regression methods for the SOC estimation could be obtained, as shown in Equations (8) and (9), respectively.

$$\begin{array}{ll}{y}_p=& -3177.9879{x_1}^2-517.5819x_1{x}_2-338.0708x_1{x}_3+7550.5462x_1{x}_4\\ & -1075.5106x_1+197.7149{x_2}^2-678.8878x_2{x}_3\\ & +1618.7552x_2{x}_4-2751.273x_2+237.5377{x_3}^2\\ & +942.5297x_3{x}_4-1243.7176x_3-5834.7809{x_4}^2\\ & +5093.5519x_4+25.9281\end{array}$$

(8)

$$y_l= 55.3164x_1+5.1262x_2+52.3422x_3-82.5314x_4+0.26633$$

(9)

where $x_1\dots x_n$ is the voltage value in cell 1 to cell n, $y_p$ is the polynomial regression, and the linear regression is denoted by$y_l$. A comparison graph of the SOC estimation results from the two methods is shown in Figure 5.

Figure 5. Comparison of the SOC estimation using polynomial regression and linear regression.

Utilizing a statistical approach based on polynomial regression, the root mean square error (RMSE) was calculated to be 0.3945, accompanied by an R² value of 0.9996. Meanwhile, the linear regression-based approach yielded an RMSE of 1.4576 and an R² value of 0.9949. The RMSE and R² values were calculated using the following equations:

$$RMSE=\sqrt{\sum _{i=1}^n{\displaystyle \frac{{(\widehat{y_i}-y_i)}^2}{n}}}$$

(10)

$$R^2=1-{\displaystyle \frac{\sum _{i=1}^n{(\widehat{y_i}-y_i)}^2}{\sum _{i=1}^n{(\overline{y_i}-y_i)}^2}}$$

(11)

where n is the number of data, $\widehat{y_i}$ is the predicted value, $y_i$ represents the actual value, and $\stackrel{\_}{y_i}$is the actual average value. Given that a smaller RMSE value indicates a closer approximation to the actual SOC and a higher R² value signifies a greater representation of the actual data, linear regression-based modeling serves as a reliable reference for statistic-based SOC modeling in this research.

To provide a comparison in this study, the metaheuristic method based on the adaptive neuro-fuzzy inference system is proposed to optimize the design of the SOC estimator. The data used as input parameters are the voltage data of the four cells (V1, V2, V3, and V4), and the output is the actual SOC value. To obtain an optimal ANFIS structure and at the same time provide the novelty of this study, the number of membership functions in the ANFIS structure was tuned. The tuning in question is the appropriate number of MFs, and it was performed using a PSO (particle swarm optimization)-based optimization process. As the ANFIS structure in this study has four input variables, the limitation of the number of MF combinations is between 2 and 3 MFs for each input variable. The selection of the number of MFs is made by considering the resulting computational load. Thus, the lower limit value is MF_min = 2 and MF_max = 3, numParticles = 20, maxIter = 30, inertia weight = 0.7, c1 = 2.0, and c2 = 2.0, with a gaussian membership function (for solving the nonlinearity relation of each input parameter). The objective function used is to find the smallest RMSE value for each ANFIS training result with a random number of MF combinations.

$$F=Min\left(RMSE\right)$$

(12)

The following is the PSO response obtained:

The Figure 6 shows that in the first iteration, the system reached a convergent condition or an optimal condition, so that the optimal ANFIS structure in this study is [3 3 3 3] with a minimum RMSE value of 0.7126. The contour of the relationship between the input and output parameters in this study is presented in the surface relationship (with the blue surface having a small value and the yellow surface having a high value) outlined in Table 2.

Figure 6. PSO result.

Table 2. Surface view relation.

Id	Parameter	Surface View
1	V1, V2, SOC
2	V1, V3, SOC
3	V1, V4, SOC
4	V3, V4, SOC
5	V2, V3, SOC

Using the ANFIS to model the SOC value obtained an RMSE of 0.2 and R² of 0.9999 (representative close to the actual data shown in Figure 7; the ANFIS curve is close to the actual SOC value). In comparison, with different input structures, namely voltage and current, and an ACO (Ant Colony Optimization) based estimator, the RMSE accuracy is not better than the ANFIS proposed in this study. With ACO-based RMSE value of 0.32238 [33].

Figure 7. (a) Training data response and (b) testing data response.

The addition of machine learning methods has been achieved using the RNN (recurrent neural network) method. This method has also been applied in previous SOC estimator research [34]. To obtain optimal RNN structure results, if in ANFIS the best structure is obtained using optimization techniques on RNN experiments were carried out by modifying the number of hidden neurons and training functions. The experimental method was carried out considering the wide variation of hidden neurons and training function variations. Table 3 the experimental results of modifying the number of neurons until the first hidden neuron totals 19 and the second hidden neuron totals 19 selected as the best number of neurons in estimating the SoC value.

Table 3. RNN Structure Comparison.

Id	Structure	Trainfcn	RMSE	R2
1	1-19	Bayesian Regression	0.78885	0.99852
2	3-19	Bayesian Regression	0.79591	0.9985
3	5-19	Bayesian Regression	0.79516	0.9985
4	7-19	Bayesian Regression	0.78885	0.99852
5	9-19	Bayesian Regression	0.78079	0.99855
6	11-19	Bayesian Regression	0.81358	0.99843
7	13-19	Bayesian Regression	0.78228	0.99855
8	15-19	Bayesian Regression	0.71749	0.99878
9	17-19	Bayesian Regression	0.77095	0.99859
10	19-19	Bayesian Regression	0.65186	0.99899
11	21-19	Bayesian Regression	0.80493	0.99846
12	23-19	Bayesian Regression	0.79564	0.9985
13	25-19	Bayesian Regression	0.75247	0.99866
14	27-19	Bayesian Regression	0.78764	0.99853

The RNN structure experiment was conducted incrementally starting with the smallest number of neurons and odd multiples in the first hidden layer, starting from 1 hidden neuron to 27 hidden neurons. The results showed that the 19-19 structure (marked in bold in Table 3) had the smallest RMSE value of 0.65186 and the largest R² of 0.99899. Furthermore, with this optimal structure, the comparison of the training function in Table 4 was performed to obtain the most optimal RMSE and R² values.

Table 4. RNN Training Function Comparison.

ID	Structure	Trainfcn	RMSE	R2
1	19-19	trainbr	0.65186	0.99899
2	19-19	trainlm	0.4084	0.9996
3	19-19	traingda	1.5401	0.99437
4	19-19	traingdx	1.9879	0.99062
5	19-19	trainbfg	1.9064	0.99137
6	19-19	trainrp	1.4226	0.9952
7	19-19	trainscg	1.2809	0.9961
8	19-19	traincgb	1.3238	0.99584
9	19-19	traincgf	1.356	0.99563
10	19-19	traincgp	1.3729	0.99552
11	19-19	trainoss	1.5257	0.99447
12	19-19	traingdm	17.6435	0.26093

The Lavenberg Marquardt-based training function (trainlm) is still the best weight update algorithm for backpropagation-based training scope. This is shown, that of the 12 training function methods that can be applied to the RNN architecture, Lavenberg Marquardt (marked in bold in Table 4) has the smallest RMSE value of 0.4084 and the largest R² of 0.9996. Thus, the best RNN structure in this case is 2 hidden layers with 19 hidden neurons in the first hidden layer, 19 hidden neurons in the second hidden layer and based on Lavenberg Marquardt.; these results are better than those produced by the linear regression method. For training sessions, the ANFIS method has advantages in modeling non-linear curves. This is indicated by the RMSE value reaching 0.2 and R² of 0.9999, so ANFIS-based modeling is highly recommended. In curve patterns that tend to be linear in the testing data session, the RNN algorithm and statistical regression-based algorithms are superior due to the ability to interpret coefficients, and computational efficiency, making it a fairly efficient standard for analyzing relationships between variables with limited data. Table 5 shows the basic configuration of the ANFIS method used in this study.

Table 5. Final structure of the SOC estimator based on the ANFIS.

Type of Fuzzy Inference System	Number of Membership Functions	Type of Membership Function
Sugeno	[3 3 3 3]	Gaussian

The difference in the execution time of all the methods discussed in this study using the ESP32 board can be seen in Table 6. It appears that all algorithms have a fast execution time under 1 s; the linear regression method had the fastest time, with <1 ms, and the RNN had the slowest time, with an execution time of around 380 ms. For the ANFIS method, the execution time is moderate, namely around 38 ms, with the best training accuracy compared with the other approaches. Therefore, for real-time applications on the ESP32 board, this method is highly recommended compared with its competitors.

Table 6. Comparison of the execution times of all algorithms.

Id.	Method	Execution Time (s)
1	ANFIS	0.0382
2	RNN	0.3805
3	Linear Regression	0.0008
4	Polynomial Regression	0.0022

5. Conclusions

This study proposes the use of a statistical and metaheuristic approach utilizing polynomial regression, linear regression, recurrent neural network (RNN), and adaptive neuro-fuzzy inference system (ANFIS) methods to estimate the SOC of cells within Li-ion battery packs in a battery management system. Voltage data was collected using the INA226 module and transmitted via the ESP32 for storage in the database. The battery management system employed was of the active balancing system type using the flying capacitor technique. The training results found that the SOC estimation using the ANFIS technique yielded a smaller value, showing an RMSE of 0.2 when compared with other techniques. Meanwhile, the R² value of the ANFIS method was greatest, with a value of 0.9999 when compared with either the linear or polynomial regression methods or even the RNN technique for training data. In conclusion, the findings suggest that the ANFIS method demonstrates superior performance compared with the linear and polynomial regression methods for estimating the SOC in lithium-ion battery packs. In further work, the ANFIS-based SOC estimator should be applied to BMS software to allow it to produce stable and precise SOC readings for various electric vehicle applications.