Enhancing battery system identification: nonlinear autoregressive modeling for Li-ion batteries

International Journal of Electrical and Computer Engineering (IJECE)
Vol. 14, No. 3, June 2024, pp. 2449~2456
ISSN: 2088-8708, DOI: 10.11591/ijece.v14i3.pp2449-2456  2449
Journal homepage: http://ijece.iaescore.com
Enhancing battery system identification: nonlinear
autoregressive modeling for Li-ion batteries
Meriem Mossaddek, El Mehdi Laadissi, Chouaib Ennawaoui, Sohaib Bouzaid, Abdelowahed Hajjaji
Laboratory of Engineering Sciences for Energy (LabSIPE), National School of Applied Sciences, Chouaib Doukkali University,
El Jadida, Morocco
Article Info ABSTRACT
Article history:
Received Dec 15, 2023
Revised Mar 1, 2024
Accepted Mar 5, 2024
Precisely characterizing Li-ion batteries is essential for optimizing their
performance, enhancing safety, and prolonging their lifespan across various
applications, such as electric vehicles and renewable energy systems. This
article introduces an innovative nonlinear methodology for system
identification of a Li-ion battery, employing a nonlinear autoregressive with
exogenous inputs (NARX) model. The proposed approach integrates the
benefits of nonlinear modeling with the adaptability of the NARX structure,
facilitating a more comprehensive representation of the intricate
electrochemical processes within the battery. Experimental data collected
from a Li-ion battery operating under diverse scenarios are employed to
validate the effectiveness of the proposed methodology. The identified
NARX model exhibits superior accuracy in predicting the battery's behavior
compared to traditional linear models. This study underscores the
importance of accounting for nonlinearities in battery modeling, providing
insights into the intricate relationships between state-of-charge, voltage, and
current under dynamic conditions.
Keywords:
Artificial neural network
Electric vehicle
Lithium-ion
Modeling
Nonlinear autoregressive
models with exogenous inputs
State of charge
This is an open access article under the CC BY-SA license.
Corresponding Author:
Meriem Mossaddek
Laboratory of Engineering Sciences for Energy (LabSIPE), National School of Applied Sciences, Chouaib
Doukkali University
Azemmour road, National No 1, ElHaouzia, El Jadida, Morocco
Email: mossaddekmeryem@gmail.com/mossaddek.m@ucd.ac.ma
1. INTRODUCTION
In response to the growing concern over fossil fuel scarcity and climate change, there has been a
rapid shift from internal combustion engine (ICE) vehicles to electric vehicles (EVs) [1]. This transition
hinges on two pivotal objectives: augmenting and customizing battery capacity, preferably on-board, and
developing high-speed battery chargers [2]. EV batteries and on-board chargers are also emerging as
potential solutions to the mass-energy storage challenges faced by the electric power sector. Integrating EVs
into the smart grid capitalizes on their predominantly parked status, enabling them to accumulate grid energy
during periods of low demand and supply energy to the grid during peak demand [3]. Various battery
technologies, such as those employing lead, nickel, and lithium, are integral to this transition, albeit
characterized by instability and sensitivity to reaction conditions [4]. Consequently, meticulous monitoring
and control of chemical reactions within cells are imperative to safeguard batteries from a spectrum of
damages, ranging from irreversible capacity loss to catastrophic explosions. Particularly, lithium-ion batteries
demand specialized handling to prevent performance deterioration and mitigate scenarios that could lead to
severe damage or explosions [4].
Ensuring the simplicity of the model identification process aligns with specific requirements. In the
context of electric vehicles, an accurate representation and simulation of battery behavior are crucial for

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Int J Elec & Comp Eng, Vol. 14, No. 3, June 2024: 2449-2456
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assessing storage system performance. A model serves as a streamlined mathematical depiction of a battery,
enabling the prediction of its behavior and the observation of phenomena that are often challenging to
measure under real-world electric vehicle usage [5]. For instance, a model facilitates the simulation of several
years of a storage system's life cycle in a matter of minutes, eliminating the need for recurrent construction of
physical prototypes and costly experiments. Effective predictive engineering entails the development of a
model that closely mirrors reality, addressing pertinent engineering inquiries [6].
The study and control of electric vehicles heavily depend on modeling lithium-ion batteries, a task
that numerous researchers are actively engaged in [7]. Their efforts are directed towards enhancing the
accuracy, robustness, and speed of lithium-ion battery models, considering the myriad factors and
uncertainties that influence the complex electrochemical reactions within batteries. Establishing
mathematical battery models is a multifaceted problem, posing challenges in both academic and industrial
realms. Measurable quantities in battery management systems (BMS) include current input, output
observation, terminal voltage, and temperature [8], [9]. BMS, functioning as an electronic guardian, monitors
battery operation, shielding it from damage during charging and discharging. It ensures that the voltage,
temperature, current, and state of charge of each cell remain within safe parameters, thereby enhancing the
battery's lifespan and autonomy [10]. Consequently, BMS research has been pivotal in fostering innovation,
leading to various types with diverse functions and solutions for improving battery operation [11].
One primary task of BMS is to determine the state of charge, representing the residual potential of
the battery. Given that the state of charge is typically not immediately measured, various estimation strategies
based on battery models have been developed [12]. Electrochemical models, describing battery performance
through chemical processes, offer high accuracy but are often deemed impractical due to their complexity.
Alternatively, circuit modeling, demonstrated by the Thevenin battery model and other variations, has proven
effective [13]. However, limitations persist, such as constant parameter values concerning the state of charge
and temperature. Innovative models, including those incorporating high-frequency cycling effects and battery
self-discharge, offer potential improvements [14].
The application of artificial neural networks (ANNs) emerges as a highly efficacious strategy for
modeling intricate and dynamic systems, transcending the constraints imposed by battery models or
mathematical correlations [15]. Nevertheless, the computational intricacies inherent in the ANN algorithm
pose significant challenges, manifesting as slow convergence, susceptibility to overfitting, and vulnerability
to local minima. Mitigating these challenges necessitates a meticulous selection of the learning algorithm,
activation function, number of hidden layers, neuron count, learning rate, spread value, and input and output
specifications [16].
To enhance the computational efficiency of ANN, careful consideration and optimization of these
parameters become imperative. Various sophisticated ANN methodologies have been advanced for the
estimation of state of charge (SOC), encompassing backpropagation neural network (BPNN) [17], radial
basis function neural network (RBFNN) [18], and recurrent neural network (RNN) [19]. Nevertheless, it is
noteworthy that existing ANN approaches often lean on a laborious trial-and-error paradigm for the
identification of optimal parameter values. This approach, unfortunately, proves inefficient and obstructs the
realization of an optimal solution within a reasonable timeframe. Hence, a more strategic and efficient
exploration of parameter spaces is imperative for advancing the effectiveness of ANN applications in this
domain [20].
The Shepherd equation, presenting a generic model with a controlled voltage source in series with a
fixed internal resistance, is a notable advancement. Although literature elaborates on this model by
incorporating temperature and lifecycle effects, it lacks integration of the state of charge effect. Meanwhile,
ANN have gained widespread application as intelligent mathematical tools for data-driven modeling [21].
Their suitability for handling nonlinear and intricate frameworks makes ANN a compelling choice. In this
study, a neural network (NN) approach is employed to assess the parameters of a Li-ion battery. The
selection of this methodology is motivated by its exceptional ability to address intricate problems.
Specifically, recurrent neural networks of the nonlinear autoregressive models with exogenous inputs
(NARX) type are employed for the estimation process. This approach relies not only on the input data but
also takes into account the feedback from the outputs. This paper employs a well-organized structure,
providing a comprehensive understanding of the neural network model's theory and features in section 2. The
experimental setup is thoroughly explained in this section, laying the foundation for the subsequent
discussions. Section 3 delves into the test results, presenting simulations and engaging in a detailed analysis
of the obtained outcomes. The concluding remarks, encapsulating key insights and potential avenues for
future research, are presented in section 4. This structured approach enhances the clarity and coherence of the
paper, ensuring a systematic and logical flow of information.

Int J Elec & Comp Eng ISSN: 2088-8708 
Enhancing battery system identification: nonlinear autoregressive … (Meriem Mossaddek)
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2. RESEARCH METHOD
With the swift progress in computer processing capabilities and the ongoing refinement of learning
techniques, the prevalence of neural networks is undergoing significant expansion, particularly in fields like
image processing and automatic translation [22]. The ANN serves as a cornerstone model for information
processing, drawing inspiration from the intricacies of the human brain. Consequently, the fundamental
structure of an ANN comprises a network of interconnected computing nodes, intricately linked by directed
and weighted connections. These nodes, akin to neurons, symbolize information-processing units, while the
weighted connections signify the strength of synaptic links between neurons. In this model, a neuron can be
envisioned as the accumulation of potentials derived from incoming synaptic signals. This cumulative sum,
in turn, conveys information through a non-linear transfer function [12].
The activation of an ANN occurs by inputting data into some or all of its nodes and subsequently
propagating this information through the weighted links. Following information propagation, the activation
levels of some or all nodes can be collected and employed for system control, prediction, or classification
purposes [23]. ANNs possess the capability to model variations in real data by continuously adjusting the
weights between nodes based on the information flow during the learning phase. They are well-suited for
capturing intricate relationships between inputs and outputs, demonstrating the ability to adapt and refine
their understanding, making them a potent tool for modeling nonlinear statistical data. The foundational
mathematical model of ANNs is depicted in Figure 1. The mathematical expression for a neuron is
formulated as (1):
𝑌 = 𝐹(∑(𝑋𝑖 ∗ 𝑊𝑖 + 𝐵𝑖)) (1)
where 𝑋𝑖 represents the input of the neuron, 𝑊𝑖 is the weight of the interconnection between input 𝑋𝑖 and the
neuron, and 𝐵𝑖 is the bais of the neuron. The determination of all weights and baises take place during the
training phase.
Figure 1. The basic mathematical model of ANN
Artificial neural networks have demonstrated effectiveness in various tasks related to the prediction
and modeling of time-series data, including applications in financial time series prediction [3] and the
forecasting of communication network traffic. Particularly in scenarios characterized by noisy time series and
nonlinear underlying dynamical systems, ANNs consistently outperform traditional linear techniques, such as
the well-known Box-Jenkins models [5]. The enhanced predictive capabilities of ANN models in these
situations can be attributed to their inherent nonlinearity and heightened resilience to noise. Within the domain
of recurrent neural architectures, NARX represent a distinctive class with limited feedback architectures
stemming exclusively from the output neuron rather than hidden neurons [24]. The NARX constitutes a
significant class of discrete-time nonlinear systems, and its mathematical representation is articulated as (2):
𝑦(𝑛 + 1) = 𝑓[𝑦(𝑛), … , 𝑦(𝑛 − 𝑑𝑦 + 1); 𝑢(𝑛 − 𝑘), 𝑢(𝑛 − 𝑘 − 1), … , 𝑢(𝑛 − 𝑘 − 𝑑𝑢 + 1)] (2)
In this context, 𝑢(𝑛) ∈ 𝑅 and 𝑦(𝑛) ∈ 𝑅 represent, respectively, the input and output of the model at discrete
time step 𝑛. The parameters 𝑑𝑢 ≥ 1 and 𝑑𝑦 ≥ 1, with the condition 𝑑𝑢 ≤ 𝑑𝑦, denote the input memory and
output memory orders respectively. The parameter 𝑘(𝑘 ≥ 0) represents a delay term referred to as the
process dead time. For the sake of generality, we consistently assume 𝑘 = 0 throughout this study, resulting
in the following NARX model:

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𝑦(𝑛 + 1) = 𝑓[𝑦(𝑛), … , 𝑦(𝑛 − 𝑑𝑦 + 1); 𝑢(𝑛), 𝑢(𝑛 − 1), … , 𝑢(𝑛 − 𝑑𝑢 + 1)] (3)
𝑦(𝑛 + 1) = 𝑓[𝑦(𝑛); 𝑢(𝑛)] (4)
In this context, the vectors 𝑦(𝑛) and 𝑢(𝑛) represent the output and input regressors, respectively.
Identifying nonlinear relationships poses frequent challenges and can be approximated through
conventional means, exemplified by a standard multilayer perceptron (MLP). This feed-forward neural
network (FFNN) comprises an input layer, one or more hidden layers, and an output layer, with
interconnections established between nodes within each layer and those of the preceding layer [25]. The
resultant interconnected structure is termed the NARX network, representing a robust class of dynamic
models reminiscent of Turing machines in the realm of computer science. The NARX topology employed in
this manuscript is depicted in Figures 2 and 3.
To conduct the experiments, the setup necessitates the utilization of two power supplies and an
active load, specifically the EA Power Control. The EL9000B, functioning as an active load, boasts a
formidable power capacity of 2,400 W and a current rating of 170 A. Complementing this, the PS9000 3 U
power supply, with an impressive power capacity of 10 kW and a current output of 340 A, is employed.
Furthermore, a constant voltage supply is integrated for relay power, maintaining the ambient temperature at
a controlled 23 °C, Figure 4. The EA Power Control plays a pivotal role in the experimentation process,
serving to record current, voltage, and power profiles. MATLAB takes charge of the subsequent data
processing, while LabView acts as the interface for efficient data acquisition. This comprehensive setup
ensures meticulous control and monitoring of the experimental conditions.
Figure 2. Prediction results of the NARX method Figure 3. NARX model
Figure 4. Experimental data

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It is noteworthy that the EL9000B and PS9000 3 U are selected for their substantial power
capabilities, ensuring a robust and versatile platform for experimentation. The constant voltage supply and
controlled ambient temperature contribute to the stability of the experimental setup. Following data
acquisition, MATLAB simulations are conducted, and the results are meticulously compared with the
experimental data, leading to a thorough performance analysis. This meticulous integration of cutting-edge
equipment and sophisticated software underscores the precision and reliability of the experimental
procedures, facilitating a comprehensive evaluation of the proposed model [12].
3. RESULTS AND DISCUSSION
The battery underwent a complete charging cycle from 0% to 100% and discharging from 100% to
0%, resulting in negligible integration errors due to the precise calibration of the current sensor. The
subsequent figures depict a comparison among the proposed models, the measured voltage, and the standard
model. Figures 5 and 6 specifically illustrate this comparison using the training database. As evident in
Figures 5 and 6, a notable similarity is observed between the proposed and measured voltages in contrast to
the voltage derived from the standard model. The uniqueness of our approach lies in the dynamic nature of
the battery model, which accounts for the influence of temperature and state of charge (SOC) on the battery
models.
While the maximum error of the proposed model is limited to 10%, it is crucial to note that certain
error peaks are discernible, particularly during instances of high battery discharge. These peaks, far from
being shortcomings, actually serve as indicators of the robustness of our model. The ability of our model to
quickly converge to the experimental curve, even in the presence of these error peaks, highlights its resilience
and adaptability under challenging conditions. These occasional peaks in error, associated with intense
battery discharge, underscore the realistic and dynamic nature of the proposed model. The fact that our model
effectively captures and responds to such discharge-induced fluctuations reinforces its reliability and
suitability for real-world applications. In essence, these error peaks contribute to validating the robustness of
our model, showcasing its capacity to navigate and accurately represent the complex dynamics inherent in
various battery operating scenarios.
The DST data at 10C serves as a crucial component in validating the proposed model. Figures 5 to 7
vividly present the simulation results during the validation phase. Notably, the rapid convergence of the
proposed model curve to the measured voltage is evident in the figures. Furthermore, a strikingly low error is
observed between the curves of the proposed model and the measured voltage, underscoring its efficacy in
comparison to the standard model. This robust validation process reaffirms the accuracy and reliability of our
proposed model in capturing the intricate dynamics of the battery system under consideration.
Figure 5. Contrast between a conventional battery model and the proposed battery model

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4. CONCLUSION
In conclusion, this study presented a battery model founded on ANN. The ANN underwent thorough
offline training to ascertain the necessary battery model, utilizing experimental data sourced from The
CALCE battery group website. The simulation results demonstrated not only a remarkable accuracy but also
swift convergence to experimental outcomes, irrespective of the charging and discharging conditions. The
adaptability of the proposed model extends its applicability to a wide array of rechargeable batteries.
Nevertheless, certain challenges associated with the model necessitate careful consideration. For the
implementation of this model in a battery pack, the calculation of the SOC for each cell is imperative.
Considering the varied environmental conditions in which batteries operate, the database used for designing
the model should encompass all potential operational scenarios.

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Looking ahead, future endeavors will revolve around leveraging the state space of this model to
estimate SOC through a robust algorithm. This holistic approach will not only enhance the model's predictive
capabilities but also contribute to addressing the challenges inherent in applying such models to real-world
battery systems. Overall, the findings underscore the potential and significance of the proposed ANN-based
battery model in advancing the understanding and practical application of rechargeable batteries.
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BIOGRAPHIES OF AUTHORS
Meriem Mossaddek is a Ph.D. student at the Laboratory of Engineering Sciences
for Energy (LabSIPE), National School of Applied Sciences, Chouaib Doukkali University, El
Jadida, Morocco. She obtained her master’s degree in electronics, electrical engineering,
automation, and industrial computing from the Faculty of Sciences Ain Chock of Casablanca
(FSAC), Morocco, in 2020. Her research interests encompass battery modeling, battery
management systems (BMS), and electric vehicle fast charging. She can be contacted at email:
mossaddekmeryem@gmail.com.
El Mehdi Laadissi is a professor at the National School of Applied Sciences,
Chouaib Doukkali University, El Jadida, Morocco, and a member of the Laboratory of
Engineering Sciences for Energy (LabSIPE). He received his master degree and his Ph.D in
electrical engineering from the Normal School for Technical Education in Rabat (ENSET),
Mohammed V University, Rabat, Morocco, respectively in 2014 and 2017. His research
interests include renewable energies, battery management system (BMS) and battery
modeling. He can be contacted at email: laadissi.e@ucd.ac.ma.
Chouaib Ennawaoui is a professor at the National School of Applied Sciences,
Chouaib Doukkali University, El Jadida, Morocco, and a member of the Laboratory of
Engineering Sciences for Energy (LabSIPE). He received his PhD in mechanics and energy
and from the National School of Applied Sciences of El Jadida (ENSA), Chouaib Doukkali
University, El Jadida, Morocco, in 2014 and 2019 respectively. His research interests include
industrial engineering, mechanical engineering and materials engineering. He can be contacted
at email: chouaib.enna@gmail.com.
Sohaib Bouzaid received the B.Eng. degree in electronic and automatic systems
engineering from National School of Applied Sciences of Tangier, from Abdelmalek Essaadi
University, Morocco, in 2019. Currently, he is a PhD student at the Laboratory of Engineering
Sciences for Energy, Chouaib Doukkali University. His currently working on battery
management systems for electric vehicles applications. His research interests include battery
management systems, battery fast charge, battery protection algorithms and methods, battery
data acquisition. He can be contacted at email: bouzaid.s@ucd.ac.ma.
Abdelowahed Hajjaji is the director of National School of Applied Sciences,
Chouaib Doukkali University, El Jadida, Morocco, and director of the Laboratory of
Engineering Sciences for Energy (LabSIPE) is a prolific researcher with a broad range of
interests. His work spans across several disciplines, including engineering, materials science,
chemistry, piezoelectricity, polymers, and ferroelectric materials. He can be contacted at
email: hajjaji.a@ucd.ac.ma.

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