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Citation: Dini, P.; Colicelli, A.;
Saponara, S. Review on Modeling and
SOC/SOH Estimation of Batteries for
Automotive Applications. Batteries
2024,10, 34. https://doi.org/
10.3390/batteries10010034
Academic Editor: King Jet Tseng
Received: 25 November 2023
Revised: 9 January 2024
Accepted: 16 January 2024
Published: 18 January 2024
Copyright: © 2024 by the authors.
Licensee MDPI, Basel, Switzerland.
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distributed under the terms and
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Attribution (CC BY) license (https://
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4.0/).
batteries
Review
Review on Modeling and SOC/SOH Estimation of Batteries for
Automotive Applications
Pierpaolo Dini *,† , Antonio Colicelli †and Sergio Saponara †
Department of Information Engineering, University of Pisa, Via G. Caruso n. 16, 56122 Pisa, Italy;
antonio.colicelli@unipi.it (A.C.); sergio.saponara@unipi.it (S.S.)
*Correspondence: pierpaolo.dini@ing.unipi.it
†These authors contributed equally to this work.
Abstract: Lithium-ion batteries have revolutionized the portable and stationary energy industry and
are finding widespread application in sectors such as automotive, consumer electronics, renewable
energy, and many others. However, their efficiency and longevity are closely tied to accurately
measuring their SOC and state of health (SOH). The need for precise algorithms to estimate SOC and
SOH has become increasingly critical in light of the widespread adoption of lithium-ion batteries in
industrial and automotive applications. While the benefits of lithium-ion batteries are undeniable, the
challenges related to their efficient and safe management cannot be overlooked. Accurate estimation
of SOC and SOH is crucial for ensuring optimal battery management, maximizing battery lifespan,
optimizing performance, and preventing sudden failures. Consequently, research and development
of reliable algorithms for estimating SOC and SOH have become an area of growing interest for the
scientific and industrial community. This review article aims to provide an in-depth analysis of the
state-of-the-art in SOC and SOH estimation algorithms for lithium-ion batteries. The most recent
and promising theoretical and practical techniques used to address the challenges of accurate SOC
and SOH estimation will be examined and evaluated. Additionally, critical evaluation of different
approaches will be highlighted: emphasizing the advantages, limitations, and potential areas for
improvement. The goal is to provide a clear view of the current landscape and to identify possible
future directions for research and development in this crucial field for technological innovation.
Keywords: power electronics; automotive; mechatronics; identification; SOC; SOH; Li-ion; batteries
1. Introduction
The application of advanced mathematical models emerges as a key pillar in power
electronics for vehicles, robotics, and mechatronic systems. This approach proves to be
essential for meeting competitive challenges and optimizing operational efficiency [
1
,
2
].
The use of mathematical models enables more accurate and predictive design of devices.
Through advanced simulation, potential issues can be anticipated and resolved, signif-
icantly reducing development time and costs [
3
,
4
]. Detailed simulation of dynamic be-
haviors provides a solid foundation for the development of highly efficient control and
monitoring systems. This methodology makes it possible to optimize control algorithms,
improve monitoring accuracy, and ensure rapid and reliable responses to changes in op-
erating conditions [
5
–
8
]. The integration of mathematical models is synonymous with
technological innovation. Leading companies in the industry take this approach to over-
come the limitations of existing technologies in order to position themselves at the forefront
of the market. Virtual simulation makes it possible to identify and solve inefficiencies and
design problems before the production phase. This results in significant resource optimiza-
tion, which reduces associated costs and improves business sustainability [
9
,
10
]. Early
identification of critical issues during the design ensures the production of reliable and safe
products. In addition, the adoption of mathematical models gives agility and adaptability
to companies, enabling them to remain competitive in an ever-changing market.
Batteries 2024,10, 34. https://doi.org/10.3390/batteries10010034 https://www.mdpi.com/journal/batteries
Batteries 2024,10, 34 2 of 37
The strategic application of mathematical models is a distinctive and indispensable
element for companies engaged in power electronics and mechatronic systems and offers
competitive advantages and contributes significantly to success in the modern industrial
environment [
11
–
14
]. The accurate estimation of the SOC and state of health (SOH) of
batteries holds paramount significance in modern battery management systems and is
primarily driven by the increasing demand for robust, efficient, and reliable energy storage
solutions. Recent research endeavors have been dedicated to the development of advanced
algorithms for SOC/SOH estimation owing to several critical factors:
•
Energy Management Optimization: In various battery-powered systems, precise SOC
estimation is indispensable for efficient energy management [
15
–
17
]. By accurately
gauging the available charge, these algorithms enable optimal utilization of the battery
capacity, thus preventing potentially detrimental conditions such as overcharging
or over-discharging [
18
–
20
]. Avoiding these extremes is vital as they can lead to
irreversible damage to the battery’s internal chemistry and structure, significantly
compromising its overall lifespan and performance [
21
–
23
]. Implementing accurate
SOC estimation techniques facilitates intelligent energy utilization strategies, which
ensure prolonged battery life and sustained system performance over extended opera-
tional periods [24–26].
•
Enhanced Safety Measures: The accuracy of SOC estimation is directly linked to
ensuring the safety and integrity of battery-powered systems [
27
–
29
]. Inaccurate SOC
determination can lead to scenarios of overcharging or over-discharging, exacerbating
the risk of critical safety hazards such as thermal runaway [
30
–
32
]. Uncontrolled
thermal runaway has the potential to trigger severe consequences, including battery
failure, fire hazards, and even catastrophic explosions, thereby posing serious threats
to both the equipment and personnel associated with the system [
33
–
35
]. Hence, the
development of precise and reliable SOC estimation algorithms plays a pivotal role in
enhancing the overall safety and risk management strategies within battery-dependent
applications [36–38].
•
Proactive Battery Health Monitoring: Accurate estimation of SOH is instrumental to ef-
fective battery health monitoring and enables the timely detection of degradation and
performance decline [
39
–
41
]. These algorithms facilitate the assessment of critical bat-
tery parameters, such as capacity fade, impedance changes, and chemical degradation,
to provide insight into the battery’s remaining useful life (RUL) [
42
–
44
]. By imple-
menting proactive SOH estimation methodologies, operators can anticipate potential
battery failures, initiate timely maintenance interventions, and prevent costly down-
time and unplanned disruptions in industrial and automotive
operations [45–47]
. The
ability to predict the RUL allows for informed decision-making regarding battery
replacement or reconditioning, thereby optimizing maintenance costs and ensuring
continuous system reliability [48–50].
The continuous evolution and refinement of SOC/SOH estimation algorithms hold the
promise of significant improvements in the accuracy and reliability of battery management
systems. Leveraging these advanced algorithms facilitates optimized energy utilization,
extended battery lifespans, enhanced operational safety, and streamlined maintenance
practices, thereby fostering increased efficiency, reduced operational costs, and improved
overall performance in a diverse range of industrial and automotive applications. Con-
sidering the diverse array of available algorithms, such as coulomb counting, the voltage
method, the Kalman filter method, neural network algorithms, and hybrid algorithms,
the selection of a specific estimation technique is dependent on various factors, including
the battery’s electro–chemical characteristics, the operational requirements of the specific
application, and the desired level of estimation accuracy. There are several algorithms
used for the estimation of SOC and state of health (SOH) of batteries, each with its own
advantages and limitations. Understanding these algorithms is crucial for efficient battery
management and to ensure optimal performance in various applications. Here is a more
comprehensive overview of the most commonly used methods for SOC/SOH estimation:
Batteries 2024,10, 34 3 of 37
•
Coulomb Counting Method: This widely used method estimates SOC by integrating
the current flowing in and out of the battery over time. Despite its simplicity and
ease of implementation, the Coulomb counting method is susceptible to cumulative
errors arising from measurement inaccuracies, parasitic currents, and changes to the
battery capacity caused by aging and temperature fluctuations [
51
,
52
]. Calibration
and frequent updates are often necessary to mitigate the impact of these factors on the
accuracy of SOC estimation, especially over the long term [53,54].
•
Voltage Method: The voltage method estimates SOC by measuring the battery’s
open-circuit voltage (OCV) and comparing it to a lookup table or a mathematical
model. While being noninvasive and relatively simple, this method is affected by
temperature variations and the dynamic nature of battery aging [
55
–
57
], leading
to potential inaccuracies in SOC estimation. Moreover, the nonlinear relationship
between SOC and OCV necessitates careful calibration and temperature compensation
to improve the accuracy of the estimation, particularly in real-world applications
where temperature variations are common [58–60].
•
Kalman Filter Method: This approach leverages a recursive filter algorithm, such as
the Kalman filter, in combination with a detailed mathematical model of the battery.
The Kalman filter method provides a more accurate estimation of SOC compared to
simpler methods such as Coulomb counting and voltage estimation [
61
,
62
]. However,
its efficacy heavily relies on the accuracy of the underlying battery model, which
requires precise knowledge of the battery’s characteristics and behavior under varying
operating conditions. Additionally, the implementation of the Kalman filter method
demands significant computational resources, limiting its practicality in resource-
constrained applications, particularly in the automotive and industrial sectors where
real-time performance is critical [63–65].
•
Neural Network Algorithm: This method involves training a neural network using a
dataset of battery measurements to establish a relationship between input parameters
(e.g., current, voltage, and temperature) and SOC. The trained neural network is then
utilized to predict the SOC based on real-time or historical battery data [
66
,
67
]. The
neural network algorithm offers enhanced accuracy compared to traditional methods,
especially in complex scenarios where the relationships between input parameters
and SOC are nonlinear and are challenging to model analytically. Nevertheless, the
successful implementation of this algorithm relies heavily on the availability of a
substantial and diverse dataset for training the neural network along with signifi-
cant computational resources for training and inference, which could pose practical
challenges in resource-limited applications [68,69].
•
Hybrid Algorithm: As the name suggests, a hybrid algorithm combines the strengths
of two or more estimation methods to improve the overall accuracy of SOC estimation.
For instance, a hybrid algorithm may integrate the coulomb counting method with the
Kalman filter technique to compensate for their respective limitations and enhance the
accuracy and robustness of SOC estimation [
70
,
71
]. By leveraging the complementary
strengths of multiple algorithms, the hybrid approach aims to mitigate the impact of
individual weaknesses, leading to more reliable SOC estimation in diverse operating
conditions and environments [72–74].
In the context of industrial and automotive applications, the choice of a specific
SOC/SOH estimation algorithm is determined by a multitude of factors, including the type
of battery chemistry, the targeted application requirements, the desired accuracy level, and
the available computational resources. Accurate estimation of SOC/SOH is paramount
for effective battery management systems as it not only optimizes the performance and
reliability of the entire energy storage system but also extends the overall lifespan of the
battery, thereby minimizing operational costs and maximizing the return on investment for
various industrial and automotive applications.
Batteries 2024,10, 34 4 of 37
2. Overview of Battery Systems
Battery energy storage systems (BESSs) include batteries, power converter systems,
control electronics, and other parts for system protection, as has been noted in the litera-
ture [75]. Batteries are essential because they are made up of stacks of cells that transform
chemical energy into electrical energy [
76
]. Tailored qualities are made possible by a variety
of cell topologies and chemistry, including series and parallel connections. The dynamism
of BESS is highlighted by the ongoing large investments and research in this subject [
77
].
Selecting a particular chemistry for a given application requires careful study since every
chemistry has unique benefits and downsides [
78
]. This calls for a comprehensive evalua-
tion of the benefits and drawbacks of different cell shape and chemistry trade-offs [
79
]. We
concentrate on the chemistry’s general characteristics in our examination [80].
Nowadays, lithium-ion (Li-ion) batteries are the most popular option, particularly for
commercial use, because of how quickly they can be charged [
81
]. Although they have a
wide operating window, it is advisable to exercise caution when approaching maximum
and minimum charge states for extended periods of time because doing so might increase
the danger of explosions and fires [
82
]. The focus of lithium battery research is on uses
like automation that need big battery packs [
79
]. LiNiO
2
is an example of a nickel–cobalt–
aluminum (NCA) battery, which is a more affordable option than lithium-ion batteries.
Even though cobalt is less expensive than nickel, aluminum reduces volumetric fluctuations
and ensures a longer lifespan that is appropriate for automation applications. But because
of the cathode’s instability and vulnerability to thermal runaway, safety issues surface [
83
].
LiNixCoyMnzO
2
composition in nickel–manganese–cobalt (NMC) batteries allows for
customization while resolving safety problems associated with NCA batteries (see Figure 1).
It is expected that NMC batteries would offer high capacity, strong C-rate capability, and
performance metrics comparable to NCA batteries since manganese improves safety and
nickel enhances cycle life. Battery management systems (BMSs) with balancing systems are
required due to ongoing safety and cost issues [
84
,
85
]. A more contemporary and affordable
cathode material that exhibits stability at higher temperatures is lithium–iron–phosphate
(LFP). Its SOC function curve is LFP flat, which makes it appropriate for motor applications
even if its volumetric energy density is lower than that of NMC. However, safety issues
necessitate a BMS [86,87].
Figure 1. Qualitative comparison in terms of specific chemistry.
Batteries 2024,10, 34 5 of 37
Although nickel batteries are not as energy efficient as Li-ion batteries under normal
circumstances, they are suitable at high current rates, which eliminates the need for balanc-
ing devices and lowers the cost of control electronics. Metal hydride (Ni-MH) cells provide
a respectably high energy density and C rates. The negative electrode of these cells is made
of hydrogen alloys. They have fewer chances of overheating and internal short circuiting,
which makes them a good substitute for lead–acid batteries in some
situations [88,89]
.
Nickel–cadmium (Ni-Cd) batteries are utilized in sealed, maintenance-free cells with a
high cycle life since they are well-known for their good safety and dependability. Calendar
aging is well supported by Ni-Cd batteries despite their lower energy density compared to
Li-ion batteries [
90
,
91
]. Because sodium is abundant in the Earth’s crust, sodium electrode
batteries provide a financially viable solution to the lack of active materials. Despite being
inexpensive, sodium–sulfur (Na-S) batteries present a safety risk due to their high operating
temperature. The development of room-temperature Na-S batteries is still underway and
offers a promising solution to the problem of aging. Reduction batteries, sometimes referred
to as metal–air batteries, produce energy by a redox reaction with atmospheric oxygen. A
higher energy density is facilitated by their open-air cell form [
92
]. In summary, solid-state
batteries are a possible successor to lithium-ion batteries, which are now the industry
standard for energy storage systems in cars and other consumer products. However, the
landscape is changing. A battery management system (BMS) is necessary for overall system
dependability, safety, efficiency, and performance monitoring regardless of the chemistry
used (see Table 1). A battery energy storage system comprises various logical and physical
levels, each necessitating a dedicated control system.
Table 1. Summary of the pros and cons of each type of battery.
Type of Battery Advantages Disadvantages
Lead–Acid
Low expense and simple to manufacture;
Low cost per watt-hour;
Low self-discharge;
High specific power, capable of high discharge;
Good performance at low and high operating
temperatures.
Low specific energy;
Slow charge: typical full charge requires 14–16 h;
Must be stored in charged condition to prevent
degradation;
Limited cycle life: repeated deep-cycling reduces
battery life;
Transportation restrictions on the flooded type;
Not environmentally friendly.
Nickel–Cadmium
Only battery that can be ultra-fast charged with
little stress;
Good load performance;
Can be stored in discharged state;
Simple storage and transportation: not subject to
regulatory control;
Good low-temperature performance.
Relatively low specific energy compared to new
systems;
Memory effect: needs periodic full discharge
(rejuvenated);
Cadmium is toxic metal: cannot be disposed of;
High self-discharge, needs recharging after storage;
Low cell voltage (typically) of 1.2
V
requires many
cells in series to achieve high voltage.
Lithium-Ion
High specific energy and high load capabilities
with power cells;
Long cycle life and extended shelf life;
Maintenance-free;
High capacity, low internal resistance and good
coulombic efficiency;
Simple charge algorithms;
Relatively short charge times.
Requires protection circuit to prevent thermal run-
away if stressed;
Degrades at high temperature and when stored at
high voltage;
No rapid charge possible at freezing temperatures
(<0◦C or <32 ◦F);
Transportation regulations applicable when ship-
ping in larger quantities.
These levels are intricately connected to ensure the safe and efficient operation of the
entire system. Let us delve into a more detailed discussion of these key levels:
1.
Battery Packs and Cells: The foundational level of the BESS consists of multiple battery
packs, each containing interconnected batteries.
Batteries 2024,10, 34 6 of 37
This arrangement is designed to achieve the desired values of current and voltage.
The batteries are organized to collectively contribute to the overall energy storage
capacity [93,94].
2.
Battery Management System (BMS): Operating at the cellular level, the battery man-
agement system (BMS) is responsible for overseeing the individual cells’ performance.
It ensures that each cell operates within safe voltage, current, and temperature ranges,
which promotes both the safety of the system and the optimal functioning of the
batteries. The BMS also undertakes the critical tasks of calibrating and equalizing the
SOC across all cells, which promotes uniform performance [95].
3.
Power Conversion System (PCS): The battery system interfaces with inverters through
a specific power electronic level known as the power conversion system (PCS). Typ-
ically organized into a conversion unit, the PCS handles the conversion of stored
energy into AC power. Additionally, it integrates auxiliary services necessary for
comprehensive monitoring and controlling the BESS [96,97].
4.
Energy Management System (EMS): At a higher level, the energy management system
(EMS) takes charge of monitoring and controlling the energy flow within the BESS. This
system ensures that the power flow aligns with specific applications and operational
requirements. It plays a pivotal role in optimizing the utilization of stored energy
based on real-time demands and conditions [98,99].
5.
Supervisory Control and Data Acquisition (SCADA) System: The broader monitoring
and control aspects are often encapsulated within the supervisory control and data
acquisition (SCADA) system. This system provides a comprehensive overview of
the entire BESS, including its various components and their performance. It acts
as the centralized control hub for monitoring the overall health and status of the
system [100].
6.
Transformer Connections: Finally, the BESS interfaces with transformers to manage the
voltage levels. Depending on the system’s size, there are connections with medium-
voltage/low-voltage transformers and, in larger systems, high-voltage/medium-
voltage transformers located in dedicated substations. These transformers facilitate
the integration of the BESS with the broader electrical infrastructure [101,102].
In essence, the BESS operates as a multi-tiered system, with each level playing a
crucial role in ensuring the reliability, safety, and efficiency of the overall energy storage
and distribution process.
Modeling electric batteries is a crucial aspect in designing battery management systems
(BMSs) and energy management systems in automotive systems for several reasons.
•
Performance Optimization: An accurate battery model allows the BMS to monitor and
control battery performance efficiently. With a precise model, it is possible to optimize
the battery’s charging and discharging, which maximizes its lifespan and ensures safe
and reliable operation [103].
•
Planning and Thermal Control: Modeling helps with monitoring the battery tempera-
ture, which is a critical parameter for safety and durability. Thermal models enable
the BMS to predict and control temperature variations in order to avoid overheating
situations that could damage the battery [104].
•
Wear Management: Battery models help estimate wear over time, which assists the
BMS with efficiently managing the battery’s lifespan. This is particularly important in
automotive systems, where the battery must operate under varying conditions and
last for an extended period [105].
Regarding the generality and scalability of mathematical battery models, there are
several reasons:
•
Similarities in Basic Behaviors: Even though batteries may use different chemical
technologies (e.g., lithium, lead–acid, and nickel–metal hybrid), there are similarities
in basic behaviors such as charging, discharging, temperature, and internal resistance.
Batteries 2024,10, 34 7 of 37
•
General Parameters: Mathematical models rely on general parameters like internal
resistance, capacity, and open-circuit voltage that are commonly present in all bat-
teries. These parameters can be measured or experimentally derived to fit specific
batteries [106].
•
Physical–Mathematical Approach: Models often rely on physical and mathematical
equations reflecting fundamental principles of battery operation. These principles
are applicable to many different technologies and allow for some universality in
models [107].
•
Adaptability through Configurable Parameters: Models can be configured and adapted
using technology-specific parameters. This allows BMS designers to customize models
to fit the specific characteristics of the battery they are managing [108].
In summary, battery modeling is essential for optimizing performance, ensuring
safety, and extending battery lifespan in automotive energy management systems. The
generality and scalability of models is possible due to similarities in basic principles and
parameter adaptability.
3. Overview of Modeling Approaches
3.1. Empirical Modeling
Empirical models are valuable tools in battery research and management due to
their ability to efficiently represent the behavior of batteries based on experimental data.
They offer a practical approach for predicting battery performance and aging, which
enables researchers and engineers to make informed decisions about battery design and
management strategies. Here is an elaboration on the types of empirical models for Li-ion
batteries as mentioned in the description:
•
Semi-empirical model: A semi-empirical approach combines elements of theoretical
understanding with empirical data, which allows for a more accurate representation
of the electrochemical processes within Li-ion batteries [
109
,
110
]. By incorporating
both theoretical and experimental components, this model can provide insights into
the battery’s performance over its lifetime and aid with the prediction of degradation
and capacity loss [111,112].
•
Empirical data hybrid driven approach: This approach combines empirical data with
other predictive methods to assess the remaining useful life of Li-ion batteries. By
considering capacity diving, this model offers insights into the battery’s degradation
patterns and remaining performance, which is crucial for implementing effective
battery management strategies and prolonging the battery’s lifespan [113,114].
•
Empirical aging model: An empirical aging model is specifically designed to evaluate
the impact of different operating strategies, such as vehicle-to-grid (V2G) approaches,
on the lifespan of Li-ion batteries. By simulating the effects of various usage scenarios,
this model helps with understanding how different operational conditions can affect
the long-term performance and aging of the battery and enables the development of
effective management protocols [115–117].
Empirical models, with their simplified structures and fast computational capabilities,
play a vital role in battery management systems (BMSs) by facilitating efficient monitor-
ing and control of battery performance. When appropriately calibrated, these models
serve as powerful tools for optimizing battery designs and management strategies: ul-
timately contributing to the advancement and widespread adoption of Li-ion batteries
in various applications. In the generic empirical method, the objective is to provide an
“instantaneous” estimate based on a limited amount of information and on instantaneous
direct measurements.
Batteries 2024,10, 34 8 of 37
A typical formulation consists of representing the SOC as an explicit function of time,
of the rough estimate of the initial conditions, and of an empirical factor that expresses the
weight that the charge lost/accumulated due to the passage of current as time passes and
also as a function of temperature.
SOC(t) = Vb(t0)−Vmin
Vm ax −Vmin
+sign(Ib(t)) <Ib(t)>N∆TN
Qn
α(t−t0,Θ)(1)
where the term
Vb(t0)−Vmin
Vmax−Vmin
represents the rudimentary estimate of the initial SOC, while the
term
sign(Ib(t)) <Ib(t)>N∆TN
Qn
represents the fraction of charge lost or accumulated due to
the passage of current for a certain number
N
of samples measured over a time horizon
of length
DeltaTN
. The term
α(t−t0
,
Θ(t))
is the time and temperature function, which
must take into account the battery characteristics, usage time, and (measured) temperature.
From a computational complexity point of view, it is certainly the most efficient method.
From the point of view of accuracy, it is certainly among the methods that can be improved,
and it can be said that it serves to give an approximate view of the state of the battery
based on the direct measurements available and connected to each other in a physically
coherent way.
The Shepherd model [
118
–
121
], widely used in the realm of Li-ion battery analysis,
serves as an empirical representation of the open-circuit voltage (
OCV
). Its foundation lies
in a polynomial equation that correlates the OCV with the SOC (
SOC
) of the battery. The
mathematical formulation of the Shepherd model is outlined as follows: Consideration of
the battery cell model within the Shepherd model includes an internal resistance (
R
) and
the open-circuit voltage (
OCV
). The relationship between the voltage (
VBatt(t)
) and the
current (i(t)) for a constant-current discharge is articulated by the following equations:
VBatt(t) = V0−K·Q
Q−i·t−R·i(t)
OCV(t) = V0−K·Q
Q−i·t
(2)
where
V0
represents the initial voltage,
K
denotes a constant,
Q
signifies the battery capacity,
and t represents time. The polynomial equation is often modified to illuminate the bat-
tery’s exponential behavior in more detail. Notably, the Shepherd model, functioning as a
voltage-dependent model, effectively characterizes the Li-ion voltage behavior for different
scenarios. Its parameters are typically derived from the manufacturer’s data-sheets. More-
over, the Shepherd model is implemented in MATLAB/Simulink for testing the accuracies
for and traits of diverse cell types. The Shepherd model proves to be a valuable tool for
predicting the
OCV
of a Li-ion battery based on its
SOC
. With precise calibration, this
model can swiftly and accurately predict real battery performance and thereby help with
finding practical applications for optimizing battery management strategies and estimating
the remaining useful life of the battery.
The Nernst equation, a cornerstone in electrochemistry, offers an essential understand-
ing of the potential difference and concentration gradient of ions across a membrane. In
battery modeling, the Nernst equation finds application in describing the open-circuit
voltage (OCV) as a function of the battery’s SOC and temperature. Here are the elaborated
mathematical formulations for the Nernst model in batteries:
The Nernst model for batteries, derived from the Nernst equation, is defined as:
OCV =E0−RT
nF ln(a)(3)
where
OCV
symbolizes the open-circuit voltage,
E
0 denotes the standard potential,
R
represents the gas constant,
T
signifies the temperature,
n
signifies the number of electrons
transferred in the reaction,
F
is the Faraday constant, and
a
represents the activity of ions in
the electrolyte.
Batteries 2024,10, 34 9 of 37
The Nernst model is often expanded to incorporate additional parameters to account
for temperature and aging effects on the battery’s performance. Combining the Nernst
model with other battery models such as the equivalent circuit model and the Shepherd
model enhances the accuracy of the representation of the battery’s behavior. The Nernst
model [
122
–
125
] serves as a valuable tool for predicting the OCV of a battery based on its
SOC and temperature. When appropriately calibrated, this model can swiftly and accurately
predict real battery performance and help with finding applications for optimizing battery
management strategies and estimating the remaining useful life of a battery.
3.2. Equivalent Circuit Modeling
Equivalent circuit modeling (ECM) is certainly the most complete approach with
regard to the description of the behavior of a single cell, and being “modular”, it also
describes the electro–thermal dynamics (also starting from chemical considerations) of
the entire battery. Many methods for estimating SOC/SOH are in fact “model-based”;
therefore, concrete and exhaustive formal modeling is certainly a fundamental requirement
for what concerns the design of efficient and reliable estimation algorithms. In the case of
ECM, the main objective of using a dynamic model is the description of the response of the
cells to a variation in the input current (or load current). The combination of simulations
and real data acquired directly from tests on the battery is used to estimate some essential
parameters to make the model reliable for the design of SOC/SOH estimation algorithms.
Below, we propose an “incremental” description starting from the basic idea and showing
which types of details it makes sense to add in a dynamic model for the study of cell/battery
charge/discharge transients in order to finally arrive at the formalization of the structure
of the ECM most used in today’s literature [
126
,
127
]. Let us start with the ideal model
depicted in Figure 2(left). This model is called an “Ideal Voltage Source” and is based on
the obviously far-fetched hypothesis that the battery voltage
v(t)
is independent of the
current
i(t)
absorbed/delivered by the battery, independent of past operating conditions,
and constant over time. In this model, therefore,
v(t) = OCV
would result, where OCV
is denominated as “Open-Circuit Voltage”. Although it is an unrealistic model, it serves
to establish some fundamental concepts: (i) the battery imposes voltage on the load;
(ii) when
not connected to a load, the open-circuit voltage is easily predictable; and
(iii) even
sophisticated models include the ideal voltage source.
Figure 2. Ideal circuit with constant OCV (left) and with OCV dependent on SOC z(t)(right).
The first enrichment of details is based on the experimental evidence that when the
cell/battery is fully charged, its
OCV
is higher than when it is in a discharged condition.
Therefore, the dependence of the
OCV
on the
SOC
can be introduced into the model, which,
by convention, is named the time variable
z(t)
. It is assumed that
z(t) =
100% for a fully
charged battery and
z(t) =
0% for a fully discharged battery. We also call
Q
the total
amount of charge that is removed from the cells during the transition from charged to
discharged conditions, which is normally provided by the manufacturer (or is measurable)
in
Ah
. As shown in Figure 2(right), the voltage source is driven with respect to the charge:
v(t) = OCV (z(t)).
Batteries 2024,10, 34 10 of 37
Knowing that
z(t)
decreases if
i(t)>
0 (discharge condition) and, vice versa, increases
if
i(t)<
0 (recharge condition), it is possible to associate the following differential equation
to describe the variation to SOC as a function of current [128].
dz(t)
dt =−i(t)
Q−→ z(t) = z(t0)−1
QZt
t0
i(τ)dτ(4)
From experimental evidence, we have that not all the charge moved by the current
i(t)
is actually used in the charge/discharge process due to the fact that a certain quantity of
charge is dispersed in some regions of the electrodes; therefore, we introduce as a further
level of detail the so-called “Coulomb efficiency”
η(t) = qout (t)
Q
. Note that in general this
efficiency is a function of time in order to take into account the use of the battery (or its
aging). Efficiency in the classical sense is relative to the input/output energy flow
ρ=Eout
Ein
;
therefore, in general,
ρ=η
. In particular, for lithium-ion batteries, experimentally, we have
values of the type: ρ≃95% and η≃99%.
An evolution of the above model is the so-called “
Rint
model”, in which the presence
of a series resistor
R0
(shown in Figure 3) is included to account for energy losses due
to heating, which obviously do not result in charge losses. The important phenomenon
that through
R0
is (at least partially) included in the modeling is “polarization”, which is
understood as the voltage drop across the battery terminals with respect to
OCV
, such as
the intrinsic voltage drop due to the connection with the load. Of course, serious resistance
alone represents a level of abstraction sufficient only for simple electronic system design
but certainly not enough for BMS design and advanced estimation algorithms in HV/EV
applications [129,130].
Figure 3. Equivalent circuit model taking into account internal resistance.
In the case of the
Rint
model, the circuit equilibrium can be expressed as
v(t) = OCV (z(t)) −R0i(t)
, which shows that in the discharge phase,
v(t)<OCV
; con-
versely,
v(t)>OCV
in the charging phase. The limitation of this model is that the dynamic
response with respect to a change in
i(t)
turns out to be instantaneous (because of the
R0i(t)
term), whereas experimentally, there is a settling transient due to internal electrochemi-
cal processes.
The realistic cell/battery response is shown in Figure 4, where at
tstart
, a pulse of
positive current is applied (thus bringing the cell to a discharged condition) until the instant
tsto p
. It can be seen that during the time window for which the current is constant, the
voltage tends to drop transiently, and when the current returns to zero, the voltage slowly
rises again to a steady-state value, which, however, is less than the starting value. This
phenomenon is called “diffusion voltage”: meaning that
v(t)
can never be instantaneous
but is always characterized by settling transients. To represent even this degree of detail,
circuit components for which the fundamental V-I relationship is integral–differential must
be included in the circuit and, in particular, through parallel RC branches as, for example,
shown in Figure 5, which is called the RC–Thevenin model [131,132].
Batteries 2024,10, 34 11 of 37
Figure 4. Schematic representation of the current pulse response of a “realistic” cell/battery model.
Figure 5. RC–Thevenin model.
In this case, the dynamic electric equilibrium is represented by the set of algebraic–
differential equations shown in Equation (5).
dz(t)
dt =−η(t)
Qi(t)
iR1(t)
dt =−1
R1C1iR1(t) + 1
R1C1i(t) = 1
τ1i(t)−iR1(t)
v(t) = OCV (z(t)) −R1iR1(t)−R0i(t)
(5)
Typically, the model is integrated on a microcontroller-based embedded platform,
so a key step in the implementation of model-based SOC/SOH estimation systems is
the discretization technique so that an efficient and numerically accurate SW task can be
defined. In Equation (6) are given the recursive (or finite difference) equations that are
obtained by applying the ZOH (zero-order-holder) numerical approximation; we denote
by ∆tthe appropriate numerical integration step (or sampling time).
z[k+1] = z[k]−η[k]
Q∆ti[k]
iR1[k+1] = e−∆t
τ1iR1[k] + 1−e−∆t
τ1i[k]
v[k] = OCV (z[k]) −R1iR1[k]−R0i[k]
(6)
In the end, it is possible to use this model efficiently to describe cell/battery behav-
ior by correlating experimental evidence with the choice of parameters through a system-
atic procedure. The basic steps for deriving the parameters of the RC–Thevenin model are
described below:
Batteries 2024,10, 34 12 of 37
-
For estimating the total charge
Q
, a simple and effective method is to bring the
cell/battery to the maximum rated voltage (corresponding to the maximum level of
SOC =
100%), apply a load, and directly measure
qdisch =_intidisch (t′)dt′
, i.e., directly
monitor the “amperes per hour”
Ah
, up to the minimum rated voltage (corresponding
to the minimum level of SOC =0%).
-
For estimating the efficiency
η
, by opposite procedure, one starts from the full dis-
charge condition of the battery and slowly brings it up to the maximum nominal
voltage level (
SOC =
100%) from the minimum (
SOC =
0%); one directly monitors
qch =Rich (t′)dt′. This results in an estimated η≃qdisch
qch .
-
Estimation of the circuit parameters is done during the “Pulse Current Test” analysis
and by applying some simple concepts of linear dynamic systems analysis to the
graphical meaning of the step response. In particular, reference is made to the situation
depicted in Figure 6. As can be seen, at
tsto p
, the voltage has a rising transient
characterized by a “vertical” stretch, which can be associated with the term
R0i(i)
,
so from the direct voltage measurement it is possible to derive the value
∆v0
and
consequently to obtain an estimate of the parameter. Similarly, one can associate
the steady-state value
∆v∞
of the voltage with the series of resistors
R0+R1
to
procedurally derive
R1
as well. As for the value of
C1
, we can consider the pseudo-
empirical relationship between the settling time τ∞and characteristic constant.
Figure 6. Schematic representation of dynamic response to current pulse, with meanings of the model
parameter contributions highlighted.
Further refinements are achieved by increasing the number of RC blocks put in series,
which results in a model that is essentially linear but of such an order that it can approximate
even more complex phenomena, such as the effect of Warberg impedance or the effect of
cell hysteresis. For an “nRC–Thevenin” model (as in Figure 7), the set of recursive equations
to be implemented has the form given in Equation (7).
IR[k+1] =
e−∆t
R1C10 0 · · · 0
0e−∆t
R2C20· · · 0
0 0 e−∆t
R3C3· · · 0
.
.
..
.
..
.
.....
.
.
0 0 0 · · · e−∆t
RnCn
iR1[k]
iR2[k]
iR3[k]
.
.
.
iRn[k]
+
1−e−∆t
R1C1
1−e−∆t
R2C2
1−e−∆t
R3C3
.
.
.1 −e−∆t
RnCn
i[k] = ARC [k]IR[k] + BRC[k]i[k]
v[k] = OCV (z[k]) −∑n
j=1RjiRj[k] + R0i[k]
(7)
Batteries 2024,10, 34 13 of 37
Figure 7. Thevenin equivalent model with nRC branches.
3.3. Other Models
The Partnership for a New Generation of Vehicles (PNGV) battery equivalent model
is used for accurate description and SOC estimation of lithium batteries, particularly for
electric vehicles. It is based on the existing charging state estimation method and the
Davidson’s battery model. The model is established using the Kalman filtering algorithm
and a state space expression. It includes parameters related to environmental tempera-
ture, battery charge and discharge, and SOC. The PNGV model is suitable for modeling
monomers and modules of lithium iron phosphate batteries, and it has been shown to
have higher modeling accuracy, especially when electric vehicles are running in a city.
The PNGV model is a conventional battery equivalent circuit model (ECM) [
133
,
134
]. The
PNGV model for batteries offers several advantages and some limitations.
Advantages:
•
Accurate Description: The PNGV model provides an accurate description of the
discharge behavior of lithium batteries.
•
SOC Estimation: It enables precise SOC estimation for Li-ion batteries based on
multi-model switching.
•
Suitability for Electric Vehicles: The model is suitable for modeling the monomers and
modules of lithium–iron–phosphate batteries with higher accuracy, especially when
electric vehicles are running in a city.
Disadvantages:
•
Complexity: The model may involve a relatively complex equivalent circuit and
state space expression, which may require computational resources and expertise to
implement effectively.
•
Parameter Sensitivity: Some parameters of the model are related to environmental
factors and battery charge and discharge, which may introduce sensitivity and require
careful calibration.
In summary, the PNGV model offers an accurate description and SOC estimation for
lithium batteries and is particularly suitable for electric vehicles. However, its complexity and
parameter sensitivity are aspects that need to be carefully considered during implementation.
The dual polarization (DP) model is an equivalent circuit model used for the charac-
terization of lithium batteries, particularly in the context of battery management systems
and SOC estimation. This model is known for its dynamic performance and accurate SOC
estimation, which makes it a valuable tool for optimizing charging profiles and understand-
ing the transient response during power transfer to and from the battery. The DP model
represents the open-circuit voltage of the battery as a function of the SOC. It is considered
one of the most flexible methods for battery management systems. The model is based on
the polarization characteristics of the battery and is used to simulate and understand the
behavior of lithium batteries under different operating conditions [
135
,
136
]. As with the
other models, the DP model has some advantages and some limitations.
Batteries 2024,10, 34 14 of 37
Advantages:
•
Dynamic Performance: The DP model is known for its excellent dynamic performance,
which makes it valuable for understanding the transient response during power
transfer to and from the battery.
•
Accurate SOC Estimation: It provides the most accurate SOC estimation compared
to other models, which is crucial for effective battery management systems and
electric vehicles.
•
Flexibility: The DP model is considered one of the most flexible methods for battery
management systems as it allows for optimized charging profiles based on proper
battery models.
Disadvantages:
•
Complexity: The model’s dynamic performance and accuracy may come with in-
creased complexity, which may require a sophisticated implementation and computa-
tional resources.
•
Parameter Sensitivity: Like many battery models, the DP model may exhibit sensitivity to
different SOC initial values, which could impact its robustness in practical applications.
In summary, the DP model offers excellent dynamic performance, accurate SOC
estimation, and flexibility for battery management systems. However, its complexity and
potential parameter sensitivity are aspects that need to be carefully considered during
practical implementation.
The Warburg impedance concept is an important element in battery modeling, particu-
larly in the context of electrochemical impedance spectroscopy (EIS) and equivalent circuit
models. It represents the diffusion process of ions in the electrodes and the electrolyte
of a battery and is used to capture the battery’s dynamic behavior at low frequencies.
The Warburg impedance is integrated into battery models, such as the dynamic battery
model, to accurately represent the diffusion processes occurring within the battery during
charging and discharging [
137
]. This integration allows for a more comprehensive and
accurate representation of the battery’s electrochemical behavior, especially at different
operating conditions and charging modes, making it a valuable tool for understanding
and optimizing battery performance. The Warburg impedance is often used in conjunction
with other elements, such as RC pairs, in equivalent circuit models to provide a more de-
tailed representation of the battery’s electrochemical processes [
138
]. This comprehensive
modeling approach, which includes the Warburg impedance, enables better understanding
of the battery’s dynamic behavior and facilitates the development of advanced battery
management systems and charging strategies. In summary, the Warburg impedance con-
cept is integrated into battery models to capture the diffusion processes within the battery,
especially at low frequencies, and is a valuable tool for understanding and optimizing bat-
tery performance, making it an essential element in the development of advanced battery
management systems and charging strategies [139].
Pros of integrating Warburg impedance in battery models:
•
Modeling Slow Dynamic Processes: The Warburg impedance allows for the modeling
of slow dynamic processes happening inside the battery, such as diffusion processes,
to provide a more comprehensive representation of the battery’s behavior.
•
Accurate Representation of Battery Response: By integrating the Warburg impedance,
battery models can accurately represent the response of the battery at low frequencies,
which is essential for understanding the battery’s behavior during different operating
conditions and charging modes.
•
Improved Dynamic Model Performance: The integration of the Warburg impedance
has been shown to improve the performance of dynamic battery models, making
them more effective for various charging modes, including those intended for electric
vehicle charging.
Batteries 2024,10, 34 15 of 37
Cons of integrating Warburg impedance in battery models:
•
Increased Model Complexity: The integration of the Warburg impedance may lead to
increased model complexity, which may require additional computational resources
and expertise for implementation.
•
Effect on Low-Frequency Response: The Warburg impedance primarily affects the
response at low frequencies, and its integration may introduce challenges related to
parameter adjustment and the transformation of visual information obtained from
impedance measurements into evolving parameters.
In summary, integrating the Warburg impedance in battery models offers the advantage
of modeling slow dynamic processes, accurately representing the battery’s response, and
improving dynamic model performance. However, it may also lead to increased model com-
plexity and pose challenges related to the low-frequency response and parameter adjustment.
As can be observed from simulation scenario in Figures 8and 9, the
Rint
model
achieves a minimal SOC error during the first 2020 s because of the exact starting SOC and
Ah counting approach. However, because of the model’s inferior precision, an accumulation
error arises and a maximum SOC error is acquired at the conclusion of the computation.
Figure 8. Simulation of SOC estimation of the discussed models.
Figure 9. SOC estimation error obtained within proposed simulation.
Batteries 2024,10, 34 16 of 37
Similar variations and trends are seen for the other four models that take into account
the polarization features. This indicates that the SOC error is highest during the first
stage of the calculation process and then rapidly decreases toward the correct SOC, albeit
with varying degrees of precision. The maximum SOC error for the DP, RC, Thevenin,
and PNGV models appears at the first stage, whereas it appears at the final stage for
the Rint model. This indicates that the Rint model is not appropriate for long-term use
in SOC estimation, with the exception of timely revision of the initial SOC, while the
other four models perform well at SOC estimation, particularly for extended periods of
time. In contrast, the DP model’s SOC error consistently remains low, with the exception
being the initial 2020 s; this further confirms that the DP model has the best accuracy for
SOC estimate.
4. Main SOC/SOH Estimation Algorithms
4.1. Coulomb Counting Method
The estimation of the SOC of a battery is critical for its efficient and reliable utilization
across a wide array of applications. The coulomb counting method represents a funda-
mental approach to SOC estimation that relies on the measurement of the current flowing
in and out of the battery and its integration over time to estimate the transferred charge
amount. The associated mathematical equation for the coulomb counting method can be
expressed as follows:
SOC(t) = SOC(t0) + R(Ib−Iloss)dt
Crated (8)
We denote as
SOC(t0)
the initial SOC at time
t0
;
Ib
represents the battery current;
Iloss
represents the current consumed by loss reactions; and
Crated
represents the nom-
inal capacity of the battery. While the coulomb counting method is straightforward to
implement, it presents several sources of error, including the accuracy of the initial SOC,
current measurement accuracy, current integration error, uncertainty of battery capacity,
and timing error. Furthermore, the method provides a relative change to SOC rather than
an absolute SOC and thus requires calibration to enhance its accuracy. This approach is
suitable for applications that demand low computational load and moderate precision,
such as mobile phones and laptops. However, it may not be suitable for applications
requiring high precision and real-time performance, such as electric vehicles and grid stor-
age systems. Given the significance of accurate SOC estimation, various research studies
have proposed enhancements to the coulomb counting method to augment its accuracy,
including the implementation of a piecewise SOC-OCV relationship, the use of a Kalman
filter, or a neural network algorithm. However, careful consideration of the battery type,
application context, and required precision is crucial for selecting the most suitable SOC
estimation algorithm. The selection or adaptation of a specific algorithm must account
for the specific application requirements and battery characteristics to ensure efficient and
reliable battery management across a wide range of operational settings. The coulomb
counting method, a foundational technique for estimating the SOC in batteries, has recently
been augmented by the introduction of the enhanced coulomb counting algorithm. This
enhanced approach was developed to address the limitations inherent in the traditional
coulomb counting method and to significantly improve its accuracy for estimating both the
SOC and state of health (SOH) parameters in lithium-ion batteries. The enhanced coulomb
counting algorithm incorporates several key elements to refine the SOC estimation process.
It derives the initial SOC from the loaded voltages during charging and discharging or from
the open-circuit voltages. Moreover, it factors in the losses incurred during these processes
by considering the charging and discharging efficiencies. This algorithm dynamically
re-calibrates the maximum releasable capacity of the battery during operation, thereby
allowing for the concurrent evaluation of the battery’s SOH. Consequently, this approach
contributes to a more precise estimation of the SOC and enhances the overall accuracy and
reliability of the estimation process.
Batteries 2024,10, 34 17 of 37
The algorithm’s technical principle is rooted in the concept of the releasable capacity,
denoted as
Creleasable
, which represents the released capacity when the battery is fully
discharged. As a result, the SOC is defined as the percentage of the releasable capacity
relative to the battery’s rated capacity
Crated
as provided by the manufacturer. Addition-
ally, the algorithm accounts for the varying maximal releasable capacity
Cmax
, which can
deviate from the rated capacity and tends to decline over time. This characteristic is instru-
mental for evaluating the battery’s state of health (SOH) and assessing its aging pattern
during operation.
SOC =Crele asab le
Crated 100%
SOH =Cm ax
Crated 100%
DOD =Crel eas ed
Crated 100%
(9)
Furthermore, the algorithm involves comprehensive considerations of the battery’s
depth of discharge (DOD) and the operating efficiency
η
during the charging and discharg-
ing stages. The integration of these parameters refines the calculation of the DOD: thereby
enabling a more accurate estimation of the SOC. This nuanced approach enhances the
precision of the estimation process, leading to an improved understanding of the battery’s
performance characteristics and health status over its operational cycle.
∆DOD =−Rt0+τ
t0Ib(t)dt
Crated 100%
DOD(t) = DOD(t0) + η∆DOD
SOC(t) = 100% −DOD(t)
SOC(t) = SOH(t)−DOD(t)
(10)
The practical implementation of the enhanced coulomb counting algorithm is fa-
cilitated by its straightforward computational framework and its minimal hardware re-
quirements. This user-friendly and efficient algorithm has the potential for widespread
integration across various portable devices and electric vehicles. Its notable feature of
reducing the estimation error to as low as 1% during the operating cycle highlights its
capacity to enhance the overall performance and reliability of battery management systems
and underscores its significance for advancing the realm of battery technology and energy
management applications. See Figure 10 for a schematic representation of the enhanced
coulomb counting method implementation.
Figure 10. Schematic illustration of the main operation in CCM.
Batteries 2024,10, 34 18 of 37
4.2. Open-Circuit Voltage Method
To model the relationship between the open-circuit voltage (OCV) and the state of
charge (SOC) of a battery, commonly used mathematical functions are the sigmoid and
polynomial functions.
SOCsigmoid =1
1+e−(a·OCV+b)
SOCpoly =∑n
k=0αkOCVk(11)
The objective is to obtain the parameters of the function through a regression process
based on data collection measuring the OCV during different charging and discharging
cycles and iterating the fitting process until obtaining functions useful for estimating
the SOC. The operational limit of this methodology is that the coefficients of the chosen
function may not make physical sense; therefore, it then becomes difficult to evaluate
the state of health of the battery. Furthermore, this method is valid only and exclusively
for the type and operational range experimentally acquired. The calibration procedure
involves collecting data, whereby the open-circuit voltage and known state of charge are
measured. Through the regression process, the parameters are adjusted to best fit the
experimental data. It is important to emphasize that the choice of this function depends on
the specific chemistry of the battery and its response to voltage variations. The sigmoid
function is a common choice, but other mathematical functions may be used based on the
battery type and specific system characteristics. Once the model is calibrated, during the
normal operation of the battery, by measuring the open-circuit voltage, this function can
be used to estimate the corresponding state of charge. Proper temperature compensation
and periodic model revision are important to maintain the accuracy of SOC estimates over
time [140–143].
4.3. Kalman-Filter-Based Method
The Kalman filter is an estimation algorithm that combines measurements with a
system prediction to obtain a more accurate estimation of the system state. It is widely
used in various fields, including navigation, electronic engineering, and robotics, to obtain
accurate estimates of a system’s state in the presence of noise or data uncertainty. In the
context of estimating the SOC and SOH for lithium batteries, the Kalman filter plays a
crucial role in providing accurate and reliable estimates of these key parameters. The SOC
represents the amount of remaining energy available in a battery at a given instant, while
the SOH is a measure of the overall integrity and capacity of a battery over time. This
algorithm works in two main phases—the prediction phase and the correction phase—
using a mathematical model that takes into account the specific characteristics of the battery
and the dynamics of the charging and discharging process. The state model for a lithium
battery can be represented by a set of differential equations that describe the charging and
discharging processes along with factors such as battery age, residual capacity, and internal
resistance. In the following, we report the essential formal aspects.
xk=Fxk−1+Buk+wk
We denote as
xk
the internal system state at time
k
;
F
is the state transition matrix
that represents the system dynamics;
B
is the control matrix that represents the influence
of input
uk
on the state; and
wk
is the zero-mean process noise. The observation model
relates to the actual measurements of SOC and SOH of the lithium battery. The relationship
between the state and the observation can be described as following:
zk=Hxk+vk
where
zk
is the actual measurement at time
k
;
H
is the observation matrix that projects the
state into the measurement space; and vkis the zero-mean measurement noise.
In the following, we report the fundamental aspect for integration of the classic Kalman
filter to linear time-invariant system models.
Batteries 2024,10, 34 19 of 37
1 Prediction Phase:
1.a
State prediction:
ˆ
xk|k−1=Fˆ
xk−1|k−1+Buk(12)
1.b
Covariance prediction:
Pk|k−1=FPk−1|k−1FT+Q(13)
where
ˆ
xk|k−1
is the a priori estimate of the state at time
k
;
Pk|k−1
is the a priori
covariance of the estimation error; Qrepresents the process noise.
2 Correction Phase:
2.a
Kalman gain calculation:
Kk=Pk|k−1HT(HPk|k−1HT+R)−1(14)
2.b
Corrected state estimation:
ˆ
xk=ˆ
xk|k−1+Kk(zk−Hˆ
xk|k−1)(15)
2.c
Covariance update:
Pk= (I−KkH)Pk|k−1(16)
where
Kk
is the Kalman gain at time
k
;
R
represents the measurement noise
covariance matrix.
The precise implementation of these equations requires a detailed understanding of
the specific parameters of the battery and the dynamic models involved in the charging and
discharging process. The choice of a suitable set of input data and the sampling frequency
is crucial to ensure reliable and accurate estimation of the SOC and SOH over time. The
Kalman filter offers a powerful and versatile approach to improve the accuracy of SOC and
SOH estimates for lithium batteries, which allows for more precise monitoring of battery
performance and better management of the overall system. See Figure 11 for a schematic
representation of the prediction and correction loop in the classic linear KF.
Figure 11. Schematic illustration of the KF recursive steps.
Alongside the traditional Kalman filter, the extended Kalman filter (EKF) and un-
scented Kalman filter (UKF) are commonly employed for more complex nonlinear systems,
such as those often encountered in battery state estimation. The extended Kalman filter is
an adaptation designed for nonlinear systems.
Batteries 2024,10, 34 20 of 37
In UKF, the model is linearized around the current mean and covariance: effectively
applying the standard Kalman filter to the linearized system. This allows the EKF to
handle some non-linearities, but it has limitations, particularly for systems with high non-
linearities, where linearization may introduce significant errors. The EKF extends the basic
Kalman filter to nonlinear systems. It approximates the system using a first-order Taylor
expansion around the current mean and covariance estimates (see Figure 12 for a schematic
representation of the EKF phases).
1 Prediction Step:
1.a
State prediction:
ˆ
xk|k−1=f(ˆ
xk−1|k−1,uk)
Fk=∂f(ˆ
xk−1|k−1,uk)
∂ˆ
xk−1|k−1
Bk=∂f(ˆ
xk−1|k−1,uk)
∂uk
(17)
1.b
Error covariance prediction:
Pk|k−1=FkPk−1|k−1FT
k+Qk(18)
2 Correction Step:
2.a
Compute the Kalman gain:
Kk=Pk|k−1HT
k(HkPk|k−1HT
k+Rk)−1(19)
2.b
Update the state estimate:
ˆ
xk=ˆ
xk|k−1+Kk(zk−h(ˆ
xk|k−1)) (20)
2.c
Update the error covariance:
Pk= (I−KkHk)Pk|k−1(21)
Figure 12. Schematic illustration of the EKF recursive steps.
These formulations highlight the nonlinear extensions of the standard Kalman filter to
showcase the additional complexity and computational demands of the EKF and UKF.
Careful consideration of these complexities is vital when integrating them into em-
bedded systems for battery SOC/SOH estimation, particularly concerning computational
resources and real-time performance requirements. On the other hand, the unscented
Kalman filter provides an improved solution to the limitations of the EKF.
Batteries 2024,10, 34 21 of 37
It operates by approximating the mean and covariance of the probability distribution
through a set of carefully chosen points, known as sigma points, propagated through the
nonlinear functions. This technique better captures the nonlinearities without the need
for linearization, thus providing more accurate estimations for highly nonlinear systems
compared to the EKF. The UKF avoids linearization by utilizing a set of carefully chosen
sigma points that capture the mean and covariance of the estimated distribution in order to
preserve moments up to the third order.
1 Prediction Step:
1.a
Generate sigma points:
Xk={χ0
k,χ1
k, . . . , χ2n
k}(22)
1.b
Propagate sigma points through the nonlinear process model:
χi
k+1|k=f(χi
k,uk)(23)
1.c
Compute predicted state and covariance:
ˆ
xk|k−1=∑2n
i=0wm
iχi
k|k−1
Pk|k−1=∑2n
i=0wc
i(χi
k|k−1−ˆ
xk|k−1)(χi
k|k−1−ˆ
xk|k−1)T+Qk
(24)
2 Correction Step:
2.a
Propagate sigma points through the observation model:
Zi
k=h(χi
k|k−1)(25)
2.b
Compute the predicted measurement mean and covariance:
ˆ
zk|k−1=∑2n
i=0wm
iZi
k
Sk=∑2n
i=0wc
i(Zi
k−ˆ
zk|k−1)(Zi
k−ˆ
zk|k−1)T+Rk
(26)
2.c
Compute the cross-covariance matrix:
Px,z=
2n
∑
i=0
wc
i(χi
k|k−1−ˆ
xk|k−1)(Zi
k−ˆ
zk|k−1)T(27)
2.d
Compute the Kalman gain:
Kk=Px,zS−1
k(28)
2.e
Update the state estimate:
ˆ
xk=ˆ
xk|k−1+Kk(zk−ˆ
zk|k−1)(29)
2.f
Update the error covariance:
Pk=Pk|k−1−KkSkKT
k(30)
In Figure 13 is reported a schematic representation of the recursive phases required
for UKF algorithm integration.
While the EKF and UKF offer better adaptability to nonlinear systems compared to
the traditional Kalman filter, they are computationally more intensive due to the increased
number of calculations involved to handle the nonlinearities. This can pose challenges for
Batteries 2024,10, 34 22 of 37
embedded systems, especially in real-time applications where computational resources
are limited.
Moreover, the complexity of implementation and the need for careful tuning of pa-
rameters can increase the development effort and time. When integrating these filters
into embedded systems for Li-ion battery SOC/SOH estimation, it is crucial to consider
the trade-off between computational complexity and estimation accuracy. In resource-
constrained environments, simplifying the model or adopting a less computationally
intensive method might be necessary. However, for applications where high accuracy is
critical, the EKF or UKF might be preferred, with optimization efforts focused on efficient
implementation and utilization of available hardware resources. Furthermore, the inte-
gration of any of these filters into embedded systems demands meticulous consideration
of memory usage, processing power, and real-time performance requirements. Efficient
algorithm optimization, careful selection of hardware components, and algorithmic simpli-
fications tailored to the specific application can aid with the successful integration of these
advanced filters for Li-ion battery SOC/SOH estimation in embedded systems.
Figure 13. Schematic illustration of the UKF recursive steps.
4.4. AI-Based Methods
4.4.1. Bayesian Neural Network
A Bayesian neural network (BNN) is a variant of a neural network that incorporates
Bayesian concepts, which allows the capture of the uncertainty associated with SOC
estimates. In a battery SOC estimation context, the BNN can be formulated to model
the probability distribution of the estimated SOC to provide a probabilistic rather than
a deterministic estimate [
144
–
151
]. Let us consider a simple BNN with a single hidden
layer. The output of the BNN will be a probability distribution for the estimated SOC. The
weights of the neural network, traditionally denoted by
W
, become random variables. In a
BNN, we represent the probability distribution of the weights as P(W).
Using Bayes’ theorem, the goal is to obtain the posterior distribution
P(W|D)
, where
Drepresents the training dataset.
P(W|D)∝P(D|W)·P(W)(31)
Batteries 2024,10, 34 23 of 37
where
P(D|W)
is the likelihood (the probability of the data given the weights), and
P(W)
is the prior (the a priori knowledge about the weights). The output of the BNN given an
input Xis a probability distribution for the estimated SOC P(SOC|X,D).
P(SOC|X,D) = ZP(SOC|X,W)·P(W|D)dW (32)
The integration takes into account the uncertainty associated with the weights. During
training, we maximize the likelihood function
P(D|W)
, which measures how well the
weights explain the observed data.
P(D|W) =
N
∏
i=1
P(SOCi|Xi,W)(33)
The loss function can be formulated as the Kullback–Leibler divergence between the
prior distribution and the posterior distribution of the weights.
J(θ) = −EP(W|D)log P(W|D)
P(W)(34)
where
θ
represents the parameters of the BNN. Training involves minimizing the Bayesian
loss function using Bayesian optimization techniques such as Bayes optimization through
Gaussian processes. The BNN captures the uncertainty associated with SOC estimates,
which is essential for critical applications. It provides a natural extension to probabilistic
prediction and provides more detailed information than a deterministic estimate. In
exploring computational complexity, the BNN emerges as a model that may require a more
considerable amount of resources, mainly due to Bayesian inference. When compared
to support vector machines (SVMs), BNN is on a higher scale in terms of computational
intensity but is comparable to traditional neural networks. By analyzing the complexity
of the software (SW) and hardware (HW) tools necessary for training, the BNN presents
itself as a model that requires significant resources: in line with the needs of traditional
neural networks and superior to those of SVMs. In the context of busy memory, the BNN
may be more expensive due to the probability distributions associated with the weights. In
this aspect, it surpasses SVM and confirms itself in line with the greater resource demand
of traditional neural networks. When evaluating the possibility of implementation on
embedded systems, the BNN may present additional challenges due to its computational
complexity and required memory. When compared to traditional neural networks, BNN
may appear less suitable for implementation on embedded systems, although it is more
efficient than more complex neural networks. Finally, regarding the possibility of real-time
implementation, BNN shows similar flexibility to traditional neural networks, but the speed
of implementation depends on the complexity of the model and the available computing
power. In summary, while BNN offers more accurate management of uncertainty in SOC
estimates, its implementation requires significant computational and memory resources,
which influences implementation choices in real-time and on embedded systems. The
decision between AI models depends on the specific needs of the application and the
available resources.
4.4.2. Support Vector Machines
Support vector machines (SVMs) are supervised learning algorithms used for classifi-
cation and regression. For estimating the SOC of batteries, SVMs can be adopted to model
the relationship between the input variables and the SOC [
152
,
153
]. We consider a linear
SVM model for SOC estimation.
f(x) = wTx+b(35)
Batteries 2024,10, 34 24 of 37
where
f(x)
represents the decision function,
w
is the weight vector,
x
are the input variables,
and
b
is the bias term. The goal of SVMs is to maximize the margin between classes. The
margin is the distance between the closest point of each class and the decision plane.
yi(wTxi+b)≥1 (36)
where
yi
is the class label of
xi
. The margin condition ensures that points are correctly
classified outside the margin. The loss function, also called the cost function, penalizes
classification errors.
min 1
2||w||2+C
N
∑
i=1
max(0, 1 −yi(wTxi+b)) (37)
where the parameter
C
regulates the trade-off between maximizing the margin and min-
imizing errors. SVMs can use the “kernel trick” to handle nonlinearly separable data
without having to explicitly transform them into a feature space.
f(x) =
N
∑
i=1
αiyiK(xi,x) + b(38)
where
K(xi
,
x)
is the kernel function that computes the similarity between
xi
and
x
, and
αi
are the Lagrange multipliers. Training of SVMs involves optimizing the parameters
w
and
b
in order to maximize the margin and minimize classification errors. The computational
complexity of SVMs mainly depends on the size of the dataset and the type of kernel
used. The linear case is more efficient than complex kernels. The memory occupied
depends on the number of support vectors, i.e., the most relevant data points for defining
the margin. SVMs stand out for their ability to generalize excellently, even when faced
with complex data and noise. This feature makes them reliable when dealing with real
scenarios and challenges typical of battery SOC estimation applications [
154
–
156
]. Another
strength of SVMs is their versatility at managing input variables. They can successfully
navigate between both continuous and categorical variables, which offers flexibility that
adapts to different application contexts [
157
,
158
]. SVMs, despite their power, present a
certain sensitivity to parameter configuration. Carefully choosing key factors such as the
regularization parameter
C
and the kernel type can significantly influence the model’s
performance in SOC estimation. This requires special attention during the configuration
phase [
159
,
160
]. Another important consideration is the handling of complex kernels.
While these can offer greater modeling flexibility, their implementation can require large
amounts of data to prevent overfitting. This aspect underlines the importance of a robust
data collection strategy adapted to the peculiarities of SVMs when dealing with more
complex models. SVMs offer a robust approach for estimating battery SOC: providing good
generalization and effectively handling complex data. However, choosing appropriate
parameters and handling large datasets are crucial considerations during implementation.
4.4.3. Long Short-Term Memory
Recurrent neural networks (RNNs)—in particular, long short-term memory (LSTM)—
are used to model complex relationships in the context of estimating the SOC of batteries.
LSTMs are particularly well suited to managing temporal sequences and capturing long-
term dependencies [
161
,
162
]. The long-term memory or cell state
Ct
is a key component of
LSTMs and is updated using the formula:
Ct=ft⊙Ct−1+it⊙˜
Ct(39)
Batteries 2024,10, 34 25 of 37
where
ft
is the forgetting gate,
it
is the input gate, and
˜
Ct
is the new candidate informa-
tion. The hidden state
ht
, which represents the information to pass to the next step, is
calculated as:
ht=ot⊙tanh(Ct)(40)
where
ot
is the output value. The decision gates govern the updating and output of the
LSTM cell. The equations for the doors are given by:
ft=σ(Wf·[ht−1,xt] + bf)
it=σ(Wi·[ht−1,xt] + bi)
˜
Ct=tanh(WC·[ht−1,xt] + bC)
ot=σ(Wo·[ht−1,xt] + bo)
(41)
where
Wf
,
Wi
,
WC
,
Wo
are weight matrices,
bf
,
bi
,
bC
,
bo
are the biases,
σ
is the sigmoid
function, and
tanh
is the hyperbolic tangent function. Training LSTMs involves using the
back-propagation through time (BPTT) algorithm to optimize weights and biases. The
computational complexity of LSTMs depends on the length of the time sequence and
the size of the hidden layers. The use of mini-batches during training contributes to
computational efficiency. The occupied memory is influenced by the size of the hidden
layers and the number of network parameters. Training LSTMs involves back-propagation
through time (BPTT), a key process for optimizing model weights and biases. BPTT is a
variant of traditional back-propagation adapted to recurrent neural networks. Referring to
Figure 14, the learning procedure for LSTM-type networks is explained in detail below.
Figure 14. Schematic illustration of the learning process in LSTM models.
The error at time
t
is calculated based on the difference between the desired output
yt
and the actual output ht:
δt=yt−ht(42)
The error is propagated backwards through the state cell using the hyperbolic
tangent tanh
:
δC,t=δt⊙ot⊙(1−tanh2(Ct)) (43)
Batteries 2024,10, 34 26 of 37
The error propagates through the decision gates:
δo,t=δt⊙tanh(Ct)⊙ot⊙(1−ot)
δf,t=δC,t⊙Ct−1⊙ft⊙(1−ft)
δi,t=δC,t⊙˜
Ct⊙it⊙(1−it)
δ˜
C,t=δC,t⊙it⊙(1−˜
C2
t)
(44)
The error propagates in the hidden state:
δh,t= (WT
o·δo,t) + (WT
f·δf,t) + (WT
i·δi,t) + (WT
C·δ˜
C,t)(45)
The weights and biases are updated using the calculated error:
∆Wo=η·δo,t·[ht−1,xt]T
∆bo=η·δo,t
∆Wf=η·δf,t·[ht−1,xt]T
∆bf=η·δf,t
∆Wi=η·δi,t·[ht−1,xt]T
∆bi=η·δi,t
∆WC=η·δ˜
C,t·[ht−1,xt]T
∆bC=η·δ˜
C,t
(46)
The hidden state weights and biases are updated considering the error in the hidden state:
∆Wh=η·δh,t·hT
t−1
∆bh=η·δh,t
(47)
The parameter
η
represents the learning rate. The BPTT process is iteratively repeated
to optimize the weights and biases of the LSTM network. Long short-term memory (LSTM)
emerges as a powerful tool in the context of SOC estimation of batteries by offering a
number of significant advantages. One of the distinguishing characteristics of LSTMs is
their ability to capture long-term dependencies. This property is of particular relevance
when it comes to modeling the dynamic behavior of the SOC over time [
163
,
164
]. The
recurrent structure of LSTMs gives them a natural adaptability to temporal sequences. This
peculiarity makes them particularly suitable for dealing with data from battery systems, for
which changes to SOC over time must be modeled accurately [
165
,
166
]. However, it is also
essential to consider the limitations associated with the use of LSTMs for SOC estimation.
First, model complexity can be a challenge. Complex LSTM models, although capable of
capturing intricate relationships, may require longer training times. In addition, model
complexity can make them prone to overfitting, whereby the model overfits the training
data, which compromises its ability to generalize [
167
,
168
]. Another critical consideration
concerns the need for sufficient data to ensure the effectiveness of LSTMs. These models
depend on the availability of a representative and large training dataset to capture the
inherent complexity of the system.
Batteries 2024,10, 34 27 of 37
In the absence of sufficient data, LSTMs may not be able to learn optimally and,
consequently, may compromise performance at accurately estimating the SOC [
169
–
171
]. In
summary, while offering significant advantages for capturing long-term relationships and
dealing with sequential data, the implementation of LSTMs for SOC estimation requires a
careful balance between model complexity and the availability of training data.
4.5. Method Comparison Discussion
In the intricate landscape of SOC estimation for batteries, diverse models and algo-
rithms have emerged to address the complex dynamics of energy storage systems. This
section aims to compare three prominent approaches: support vector machines (SVMs),
long short-term memory (LSTM) networks, and Bayesian neural networks (BNNs). Bat-
teries, as intricate electrochemical systems, demand accurate SOC estimation for optimal
operation. From traditional support vector machines (SVMs) to sophisticated LSTM net-
works and probabilistic Bayesian neural networks (BNNs), various models and algorithms
have been proposed. SVMs employ mathematical formulations that optimize the margin
between data points, making them effective at capturing intricate relationships. However,
their adaptability to battery dynamics and temporal sequences may be limited compared
to other methods. LSTMs, a type of recurrent neural network, excel at capturing long-term
dependencies within sequential data. Widely adopted for their ability to adapt to temporal
sequences and handle complex, nonlinear relationships, LSTMs prove effective at modeling
the dynamic behavior of batteries over time. Introducing a probabilistic approach, BNNs
incorporate uncertainty into neural networks. This is advantageous in scenarios for which
uncertainty in predictions is critical, such as battery SOC estimation. Offering a balance
between complexity and efficiency, BNNs present an intriguing option. In Table 2, we
comprehensively compare SVM, LSTM, and BNN based on criteria such as mathematical
formulation, computational complexity, memory usage, implementation considerations,
and adaptability to temporal sequences. This comparative analysis aims to guide the
selection of the most suitable method based on specific application requirements.
This comparative analysis provides valuable insights for researchers and practitioners
navigating the strengths and limitations of each method in the context of SOC estimation
for battery systems. In the complex challenge of estimating the state of charge (SOC)
of batteries, several methods are employed, each with unique approaches and specific
challenges (see Table 3). The coulomb counting method is based on measuring the current
flowing into and out of the battery over time. Its conceptual simplicity and economical
implementation make it attractive, but it is subject to the accumulation of errors over time
due to phenomena such as current measurement drift. This method may be particularly
susceptible to inaccurate results in the presence of batteries that exhibit nonlinear behavior
or variations in efficiency under different operating conditions. The open-circuit voltage
(OCV) method, on the other hand, is based on the open-circuit voltage of the battery. It
is a nonintrusive approach and is relatively simple to implement. However, its accuracy
is closely tied to correct calibration, and it can be less accurate during rapid charge and
discharge transitions. The Kalman filter, a more advanced method, offers adaptability to
changes to the operating conditions. This method is capable of integrating multiple data to
reduce uncertainty, but its implementation requires in-depth knowledge of the system and
a careful choice of parameters. Artificial intelligence techniques, such as neural networks,
present a flexible approach that can adapt to complex models. However, they require a
large amount of data for training, can be sensitive to the quality and representativeness
of the data, and can require significant computational resources. In summary, the choice
of SOC estimation method depends on various factors, such as the required accuracy, the
complexity of the system, and the availability of accurate calibration data. A combined
approach that leverages the advantages of multiple methods could be the ideal solution to
address the specific challenges that each SOC estimation application presents.
Batteries 2024,10, 34 28 of 37
Table 2. Comparison between SVM, LSTM, and BNN.
Support Vector Machine (SVM)
[172–175]
Long Short-Term Memory
(LSTM) [176–178]
Bayesian Neural Network
(BNN) [179–182]
Formulation
SVM optimizes the margin
through constraints and
loss function.
LSTM has a recurrent structure
with decision gates, cell state, and
hidden state.
BNN uses probabilistic neurons
with a Bayesian approach.
Complexity
Complexity depends on the
dataset size and the type of
kernel used.
Complexity depends on the
length of the temporal sequence
and the size of hidden layers.
Complexity depends on the
network depth and layer size,
also requires Bayesian inferences.
Memory Memory usage depends on the
number of support vectors.
Memory depends on the size of
hidden layers and the number
of parameters.
BNN has more efficient memory
usage due to probabilistic
weight utilization.
Embedded-oriented Implementation is possible with
linear models and simple kernels.
Higher computational
complexity, requires
case-by-case evaluations.
Implementation is possible but
may require significant
computational resources due to
the Bayesian approach.
Real-time Implementation is possible with
linear models and simple kernels.
Implementation is possible but
requires significant
computational resources.
Implementation is possible but
may have longer computation
times due to Bayesian inference.
Time-series
Less suitable compared to LSTM.
Highly suitable, excellent for
capturing
long-term dependencies.
Suitable, but the ability to capture
temporal dependencies may
be limited.
Training
May require significant data for
good performance and is
sensitive to
parameter configuration.
Effective with sequential data,
but may require a significant
amount of data.
May require less data compared
to more complex models but
requires data for
Bayesian training.
Robustness
SVM provides good
generalization and can handle
noise with parameter tuning.
LSTM offers good generalization
and can handle complex and
noisy data.
BNN is robust to noise due to the
Bayesian approach but may have
potentially lower
modeling capacity.
Table 3. Comparison of SOC estimation methods for batteries.
Method Advantages Disadvantages
Coulomb Counting Method [183–185]• Conceptual simplicity
• Economical implementation
• Accumulation of errors over time
• Less suitable for complex batteries
Open-Circuit Voltage (OCV)
Method [186–188]
• Non-intrusive
• Simple implementation
• Precision tied to calibration
•
Less accurate during rapid transitions
Kalman Filter [189–191]• Adaptability to variations
• Reduction in uncertainty
• Complex implementation
•
Dependence on parameters and sys-
tem knowledge
Artificial Intelligence Methods [192–195]
• Adaptability to complex models
•
Continuous improvement with
new data
• Requires large amounts of data
•
Sensitivity to data quality and com-
putational resources
5. Conclusions and Future Work
In conclusion, this state-of-the-art analysis has outlined a detailed and in-depth picture
of current methodologies and algorithms for SOC estimation in batteries and emphasized
the crucial importance of such approaches in power electronics for vehicles and mecha-
tronic systems.
Batteries 2024,10, 34 29 of 37
The survey went beyond the limitations of conventional research to provide a level of
detail that is currently lacking in the literature. Examination of the methodologies, such as
Coulomb counting, the voltage method, Kalman filters, neural network algorithms, and hy-
brid algorithms, highlighted the strengths and limitations of each approach. This in-depth
understanding is crucial for the practical implementation of advanced energy management
solutions, which can ensure optimal battery utilization and enhanced operational safety.
The detailed comparison of SOC estimation algorithms identified the operational situations
in which each method excels in order to help provide practical guidelines for selecting the
most suitable algorithm based on the specific needs of the electric vehicle or mechatronic
system. This study stands out for its depth of analysis and detail and fills a gap in the
existing literature. The information gathered provides a vital resource for researchers, engi-
neers, and practitioners interested in optimizing energy management strategies. Looking
ahead, promising insights emerge for further research and development in the field of
power electronics for vehicles and mechatronic systems: The integration of cybersecurity
into power electronics systems is a key research direction. With the expansion of electric
and hybrid vehicles, protection from potential cyber attacks becomes crucial to ensure the
integrity and safety of the vehicle and its occupants. Developing SOC estimation algorithms
that are resilient to cyber threats will become a critical aspect of future design. Exploration
of multi-objective optimization approaches for SOC estimation algorithms that balance
estimation accuracy with computational complexity is necessary. This could lead to more
energy- and computationally efficient solutions. As storage technologies, such as solid-state
batteries, evolve, future research should focus on how SOC estimation algorithms can
adapt to and benefit from these new solutions to help further improve battery efficiency
and lifetime. In conclusion, this study has not only expanded the current understanding of
SOC estimation methodologies but also outlined a road map for future investigations to
address emerging challenges in the field of power electronics for vehicles and mechatronic
systems. Further investigation of these issues will not only ensure significant advances in
energy management but will also help make electric vehicles safer and more reliable in a
rapidly changing environment.
Author Contributions: Conceptualization, P.D. and S.S.; methodology, P.D. and S.S.; software, P.D.;
validation, P.D., S.S. and A.C.; formal analysis, P.D. and S.S.; investigation, P.D. and S.S.; resources,
P.D.; data curation, P.D., S.S. and A.C.; writing—original draft preparation, P.D., S.S. and A.C.;
writing—review and editing, P.D., S.S. and A.C.; visualization, P.D., S.S. and A.C.; supervision, P.D.,
S.S. and A.C.; project administration, P.D., S.S. and A.C.; funding acquisition, S.S. All authors have
read and agreed to the published version of the manuscript.
Funding: This research is partially funded by: MIUR FoReLab Project “Dipartimenti di Eccellenza”;
the PNRR project “CN1 Big Data, HPC and Quantum Computing in Spoke 6 Multiscale Modeling
and Engineering Applications”; and by the ECSEL JU project Hiefficient n. 101007281 (EU ECSEL-
2020-2-RIA call).
Data Availability Statement: The data that support the findings of this study are available from the
corresponding author, P.D., upon reasonable request.
Conflicts of Interest: The authors declare no conflicts of interest.
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