Modeling of Lithium-Ion Batteries for Electric Transportation: A Comprehensive Review of Electrical Models and Parameter Dependencies

Abstract

The power and transportation sectors contribute to more than 66% of global carbon emissions. Decarbonizing these sectors is critical for achieving a zero-carbon economy by mid-century and mitigating the most severe impacts of climate change. Battery packs, which enable energy storage in electric vehicles, are a key component of electrified transport systems. The production of these batteries has significantly increased in recent years to meet rising demand, and this trend is expected to continue. However, current traction batteries exhibit lower energy density compared to fossil fuels. As a result, accurate battery models that balance computational complexity and precision are essential for designing high-performance energy storage systems. This paper provides a comprehensive review of the most used electrical models for lithium-ion batteries in traction applications, as reported in the technical literature. By exploring the strengths and limitations of different modeling approaches, this paper aims to offer valuable insights into their practical applicability for the electrification of transportation systems. Additionally, this paper discusses the primary methods employed to derive the values of the electrical components within these models. Finally, it examines the key parameters—such as temperature, state of charge, and aging—that significantly influence the component values. Ultimately, it guides researchers and practitioners in selecting the most suitable modeling approach for their specific needs.

1. Introduction

The European Union’s (EU) international commitments, as part of the 2015 Paris Agreement, led to the adoption of the European Climate Law in 2021 committing the EU to reducing greenhouse gas (GHG) emissions by $55\%$ by 2030, compared to 1990 and to reach ‘net zero emissions’ (i.e., ‘carbon neutrality’) by 2050 [1]. Vehicle decarbonization is a key factor in achieving the $55\%$ GHG reduction target by 2030 in an efficient manner.

This focus on decarbonization is driven by the low energy efficiency of current road transport vehicles, particularly in the conversion of fuel energy into wheel movement. Under typical driving conditions, passenger cars exhibit efficiency between $20\%$ and $25\%$, while long-distance heavy good vehicles (HGVs) achieve a maximum of $30\%$. Replacing these inefficient and polluting vehicles with zero-emission alternatives—primarily, battery–electric passenger cars—can significantly improve energy efficiency [2].

In the last ten years, batteries have evolved from powering small electronics to becoming a critical enabler of clean energy transition. Today, with 8.5 million electric vehicles (EVs) on the road today, batteries are at the forefront of transportation electrification, supporting the shift to zero-emission vehicles. Looking ahead, investors and carmakers are advancing ambitious plans to expand battery manufacturing, confident that the demand for EV and stationary batteries will continue to grow because of increasing electrification and power grid decarbonization. Global battery manufacturing capacity is projected to exceed 9 TWh by 2030, with around 70% of this capacity already operational or committed. Assuming an $85\%$ maximum utilization rate, nearly 8 TWh of batteries could be produced annually by 2030, with over 5.5 TWh coming from plants either currently operational or with firm commitments [3].

Currently, cathode materials limit the energy density of batteries and, as a result, influence the overall cost. Selecting the appropriate cathode material is critical for designing efficient traction batteries. At present, several lithium-ion battery chemistries are suitable for the transport sector, each with distinct characteristics that affect performance, cost, and safety [4,5]. The most widely used lithium-ion battery chemistries in electric vehicles are nickel cobalt-aluminum (NCA), nickel manganese cobalt oxide (NMC), and lithium iron phosphate (LFP), as illustrated in Figure 1.

In detail,

Table 1. Cathode technology.

Symbol	Specific Energy (Wh/kg)	Number of Cycles	Voltage (V)
NCA	200–260	500–600	3.65
NMC	150–220	1000–2000	3.8–4.01
LFP	90–120	1000–2000	2.3–2.5
LMO	100–150	300–700	4.0
LCO	150–200	500–1000	3.7–3.9
LTO	70–80	3000–7000	2.3–2.5

presents the key parameters of lithium-ion batteries utilized in both the transportation and energy sectors. In addition to the previously mentioned chemistries, it is essential to also consider the following: lithium manganese oxide (LMO), lithium cobalt oxide (LCO), and lithium titanate oxide (LTO).

NMC and NCA batteries are widely used in EV energy storage systems. While NCA tends to be more expensive, with moderate durability and overheating risks, NMC batteries instead are generally considered safe when properly managed [7]. LFP chemistry is known for its high electrochemical performance and long lifespan. These batteries have an extensive operating temperature range, although humidity can negatively impact their lifespan [8]. The three-dimensional architecture of LMO chemistry greatly reduces internal resistance and improves design flexibility, long lifetime, and high specific power. While the energy capacity is moderate, it is relatively low in cost and is considered safe compared to other chemistries [9]. LCO chemistry, introduced by Sony in 1991 for electronics, is characterized by good specific energy and fast power delivery. However, its lifespan varies according to operating conditions, and it poses risks of overheating and fire hazards [10]. Finally, LTO chemistry replaces graphite at the anode of LCO with the Lithium- Titanate for improving battery cooling and generating less heat than graphite-based systems. LTO has an exceptionally long lifespan but relatively low energy capacity and moderate energy delivery capacity [11,12]. A comprehensive performance comparison of the listed above lithium-ion chemistries is shown in Figure 2.

Considering the peculiarities of each chemistry is essential for evaluating which types of batteries are best suited for specific applications. For example, LTO chemistry, due to its high safety, specific power output, and long lifespan, is particularly well suited for railway or tramway applications [13]. However, its high cost and low specific energy make it less appropriate for the automotive sector, unlike the LFP and NMC chemistries [14].

Given that multiple modules of cells form the entire grid of a power source, it is crucial to monitor each cell within the module. Each cell operates under varying conditions due to differences in temperature, the state of charge ($SoC$), and the state of health ($SoH$). Safe and efficient management of lithium-ion batteries is the key to maximizing their energy potential and extending the driving range of electric vehicles. In general, battery states are estimated based on models, which rely on sampling working conditions and are used to develop management strategy in the battery management system (BMS). As a result, the models embedded in the BMS must provide high accuracy and real-time performance. Numerous models have been developed and presented in the technical literature, including electrochemical, electrical, spectroscopy-based, mathematical, data-driven. While electrical, electrochemical, and data-driven models can all be used for real-time monitoring, they address different needs. Electrical models provide a simplified representation of the battery’s electrical behavior, making them particularly effective for the real-time prediction of key performance metrics. In contrast, electrochemical models offer a more detailed insight into internal chemical reactions. Data-driven models, on the other hand, utilize historical data to identify patterns and make predictions, providing an alternative approach for monitoring battery states.

However, electrical models appear to be the most promising for monitoring lithium-ion batteries and optimizing battery pack sizing in transport systems [15]. They accurately predict electric quantities like voltage, current, and the SoC, enhancing safety and efficiency in dynamic operating conditions. Their adaptability to various configurations and chemistries makes them suitable for EVs, buses, and rail systems. Moreover, their scalability and low computational costs enable effective battery management, optimizing energy use and extending battery lifespan. In recent years, electrical models for lithium-ion batteries have great relevance in industry applications, as many companies, particularly in the automotive field, are providing significant investment in laboratories aimed at battery cycling to improve the tuning of electrical models used in the BMS and to evaluate data-driven joint approaches.

This review paper provides a comprehensive overview of electrical models for characterizing lithium-ion batteries, based on the latest technical literature. It elucidates the theoretical foundations and practical implementations of these models, focusing on key methodologies for determining essential parameters such as internal resistance, capacity, and the SoH. The discussion includes how these parameters are affected by factors like temperature, charge and discharge cycles, and environmental conditions. The review also critically evaluates the strengths and limitations of various modeling approaches, from empirical methods to complex electrochemical models, assessing their applicability in real-world scenarios, especially for the electrification of transportation systems. By analyzing recent case studies, this paper offers practical insights into advancing battery technology and management strategies, ultimately guiding researchers and practitioners in selecting the most suitable modeling approaches for their specific needs.

The rest of the paper is organized as follows: Section 2 describes different approaches for modeling lithium-ion batteries, highlighting pros and cons of electrical models. Section 3 provides an overview of the existing electrical models, while Section 4 and Section 5 offer insights into parameter identification and the main dependencies of these parameters, respectively. Concluding remarks end the paper.

2. Classification of Battery Models

Battery modeling involves the formulation of mathematical representations to estimate the state and output of a battery in response to a given input. These models, derived from extensive data collection and research, help clarify fundamental battery behaviors and predict performance under various operating conditions [16]. Different applications require slightly varied modeling approaches and parameters. There is no strict rule on how to categorize battery models, and same models can belong to more than one class [17]. These models are essential across a wide range of applications, requiring adaptable approaches and parameters tailored for specific purposes. Battery models are generally classified into the following categories:

• Electrochemical,
• Mathematical,
• Electrical,
• Reduced-order electrochemical model,
• Based on spectroscopy,
• Data-driven.

Further characterization of battery models can be made based on their transparency and comprehensibility, dividing them into the following groups:

• White box models: These models provide a detailed and transparent representation of the system’s internal mechanisms and processes. It is easy to understand how input variables influence output variables through a series of equations, physical laws, or known algorithms. Electrochemical models and mathematical models can be included in this category.
• Black box models: These models offer only the system’s inputs and outputs, without knowing or considering the specific internal mechanisms or causal relationships driving the system’s behavior. The internal workings remain hidden, and the system is treated as a black box, focusing attention exclusively on the observable results. Data-driven models, such as those based on neural networks, are included in this category.
• Gray box models: These models represent a middle ground between black box and white box models. It combines transparency and compressibility of white box models with the flexibility and adaptability of black box models. It typically includes a combination of known and explicitly modeled components along with unknown or more simplified modeled components. This type of model is often used when you do not have a complete understanding of the system but want to incorporate some degree of knowledge or theoretical constraints into the modeling. Electrical models and reduced-order electrochemical models are examples of this group.

2.1. Electrochemical Models

Electrochemical models delve into the intricate aspects of physical design and fundamental mechanisms of power generation within batteries, even if at the cost of increased complexity and longer computation time. These sophisticated mathematical frameworks are employed to understand the underlying chemical processes occurring within batteries. Electrochemical battery models provide detailed insights into the dynamics of electrochemical reactions at the microscopic level. Initially introduced by Fuller, Doyle, and Newman in the mid-1990s [18,19,20], these models use principles such as porous electrode theory to describe phenomena such as the mass transport, diffusion, and distribution of ions within the battery cell. By relating construction parameters (e.g., electrode materials, separator thickness) to electrical parameters (e.g., voltage, current), as well as thermal behavior, they offer a comprehensive understanding of battery operation. However, the use of electrochemical models presents significant challenges. They require solving complex systems of time-coupled spatial partial differential equations, which require high computational resources and complex numerical algorithms. Obtaining all the parameters necessary for accurate modeling can be arduous and expensive and require the in-depth knowledge of cellular chemistry and manufacturing processes. Despite these complexities, electrochemical models provide unprecedented precision in describing battery behavior [21]. Despite the complexity of the relations governing electrochemical models, methodologies have been developed that make them suitable for real-time applications, such as the single particle model (SPM) [22,23]. The SPM is a simplified approach for modeling the electrochemical behavior of lithium-ion cells. It represents each active electrode particle as a single entity, simplifying calculations and simulations. This model describes ion diffusion and chemical reactions during charging and discharging, allowing rapid estimates of electric quantities such as voltage, current, and the $SoC$. Although a simplified approach, the SPM is effective in capturing the behavior of Li-ion cells and is widely used in commercial BMSs.

2.2. Mathematical Model

Mathematical models, which utilize empirical equations or stochastic methods, are employed to predict system-level behavior such as battery runtime, efficiency, and capacity. They can be categorized into empirical and stochastic models.

Empirical models describe specific battery behaviors through simple equations. Although they are suitable for certain applications and may have a margin of error ranging from 5% to 20%, their main advantage lies in low complexity and the ability to identify parameters in real time. A notable example is the kinetic battery model (KiBaM), developed by Manwell and McGowan [16,24,25], which is based on chemical kinetic processes and provides an intuitive representation of battery processes. In the KiBaM model, the battery charge is divided into two compartments: the available and the constrained charge well. They are shown in Figure 3.

A fraction of the total capacity $c$ is allocated to the available charge well $x\left(t\right),$ while the remaining fraction $(1-c)$ is stored in the constrained charge well $y\left(t\right)$.

The available charge well supplies electrons directly to the load $i\left(t\right)$, whereas the constrained charge well can only transfer electrons to the available charge well. Charge flows from the constrained charge well to the available charge well through a “valve” with fixed conductance $k$. In addition to this parameter, the rate at which the charge flows between wells depends on the height difference between the two wells. The heights of the two wells are given by the following:

$$h_1={\displaystyle \frac{x}{c}}$$

(1)

$$h_2={\displaystyle \frac{y}{1-c}}$$

(2)

The equation described above does not represent modern batteries, which exhibit a different discharge profile. However, if the primary focus is on battery life rather than the actual voltage during discharge, the two-well model of the KiBaM can still be utilized, as it effectively captures both the charging capacity and the recovery effect [16]. On the other hand, stochastic models are based on the principle of discrete-time Markov chains. These models can offer greater accuracy than empirical models while maintaining reduced complexity and fast simulation times.

These models focus on the recovery effect and use probabilities to characterize the behavior of the battery based on its physical characteristics. The first stochastic battery models were developed by Chiasserini [26,27]. In the simplest model, the battery is represented by a time-discrete Markov chain with $N+1$ states, numbered from $0$ to $N$, as in Figure 4. The number of states corresponds to the number of units of charge available in the battery, where one unit of charge represents the energy required to transmit a single packet. $N$ is the number of charge units directly available based on continuous use. In this model, at each time step, a unit of charge is either consumed with probability $a_1=q$ or recovered with probability $a_0=1-q$. The battery is considered empty when the state of absorption 0 is reached or when a maximum of $T$ units of charge is consumed. The number of $T$ charge units is equal to the theoretical capacity of the battery ($T>N$) [28]. While mathematical models are valuable for applications with constant operating conditions, they are less effective at predicting the real-time behavior of batteries. These models are described by simple equations and are computationally efficient, but their accuracy is sensitive to the parameters used, requiring careful calibration and validation against real-world data. In addition, their development can be complex and require extensive experimental data, which may not be readily available.

2.3. Electrical Models

Electrical models for lithium-ion batteries simulate cell behavior with component values optimized through laboratory test data to closely match real cells’ current and voltage responses. These models show good performance under tested conditions but may struggle outside these ones. Known for their simplicity and effectiveness, electrical models use resistors, capacitors, and voltage sources to approximate battery behavior, making them highly suitable for real-time applications in the BMS due to their low computational cost and rapid simulation speeds. However, electrical models are limited in predicting output behavior, due to little insight into the battery’s internal electrochemical states. This simplification can cause inaccuracies when applied beyond tested conditions. Additionally, achieving high accuracy may require several resistor–capacitor (RC) networks, increasing model complexity and potentially impacting computational efficiency. One problem of electrical models is the error drift due to their parameter SoC dependency. However, many methods for an accurate estimation of the SoC exist for electrical models (e.g., Coulomb Counting, ANN, Kalman Filter, etc.). Moreover, as highlighted in Section 4, there are several procedures to estimate model parameters considering their dependencies. Further details on various electrical models found in the literature are available in Section 3 of this paper.

2.4. Reduced-Order Electrochemical Model

The complexity and difficulty of finding all the necessary parameters make electrochemical models suitable only in the battery design phases. To reduce the complexity of the electrochemical model, model order reduction techniques can be applied; in such away, reduced-order models (ROMs) are obtained, which are computationally simple and robust. ROMs typically feature a finite-dimensional rational transfer function, differential equations of minor degree and non-dependence on some parameters considered negligible. ROMs play a significant role in the modeling and management of lithium-ion batteries, effectively balancing complexity and computational efficiency. One of their primary advantages is the ability to simplify the underlying electrochemical processes into a more manageable form. This simplification enables faster simulations, which are particularly beneficial for real-time applications. With reduced computational requirements, ROMs allow for more frequent updates and the monitoring of battery states, leading to enhanced operational safety and efficiency, especially under dynamic conditions typical of electric vehicles and other applications [29]. Additionally, ROMs can facilitate the integration of battery models with control systems, improving the responsiveness of the system to varying operational demands. Their adaptability to different battery configurations and chemistries also makes them versatile for a wide range of applications, from electric vehicles to renewable energy storage solutions. This flexibility allows for tailored solutions that can optimize energy management based on specific use cases. However, there are inherent disadvantages to using ROMs. The primary concern is the potential loss of accuracy due to the simplification of complex dynamics. If the reduced-order model does not sufficiently capture the essential behaviors of the battery, such as nonlinear responses to charge and discharge cycles, it may lead to erroneous predictions and suboptimal performance. Indeed, ROMs are not able to capture the performance of cells cycled at high C-rates and/or temperatures and aged cells. Additionally, the effectiveness of a ROM is contingent upon the quality of the full-order model from which it is derived. If the full-order model lacks fidelity, the resulting ROM may fail to provide reliable insights, particularly in edge cases or under extreme operating conditions. Moreover, ROMs require careful calibration and validation against experimental data to ensure their reliability as elec-trical models. These actions can require resource-intensive efforts, and discrepancies between model predictions and actual performance may necessitate iterative adjustments. Finally, while reduced-order models are invaluable for enhancing the efficiency and speed of battery analysis, their application must be approached with caution, ensuring that their limitations are well understood and managed [30].

2.5. Electrochemical Impedance Spectroscopy

Electrochemical impedance spectroscopy (EIS) provides insight into the dynamic behavior of lithium-ion batteries by analyzing impedance across frequency ranges. EIS typically reveals three sections: high, mid, and low frequencies, each indicative of different battery characteristics. The impedance data are often depicted in a Nyquist diagram, where resistance is plotted against reactance. AC input current is applied, and the AC voltage is measured using fast Fourier transform (FFT) analysis. From this voltage, the impedance can be calculated. In lithium-ion battery modeling, a commonly used approach is Randle’s Circuit, which includes components such as bulk resistance ($R_b$), surface film layer resistance and capacitance ($R_{sei}$, $C_{sei}$), charge transfer resistance ($R_{ct}$), double layer capacitance ($C_{dl}$), and Warburg impedance ($Z_W$). These elements correspond to various electrochemical processes within the battery, as illustrated in Figure 5.

In detail, $R_b$ is the bulk resistance of the cell, accounting for the electrical conductivity of the electrolyte, separator, and electrodes. $R_{sei}$ and $C_{sei}$ correspond to the resistance and capacitance of the surface film layer on the electrodes, such as the solid electrolyte interphase layer (SEI), and are associated with high-frequency impedances. R_b (charge transfer resistance) relates to the charge transfer between the electrolyte and the electrode, while $C_{dl}$ (double-layer capacitance) is the capacitance between the electrolyte and the electrode interface.

These two components explain the mid-frequency response. The Warburg impedance ${\mathrm{Z}}_{\mathrm{w}}$ represents the diffusion of lithium ions between the active material and the electrolyte, corresponding to low frequencies, and is depicted as a line with a 45-degree slope at very low frequencies. In some cases, an inductance is added in series with the bulk resistor to account for the positive reactance observed at high frequencies [31]. These parameters are illustrated in Figure 6, which shows a commonly used circuit along with its corresponding Nyquist diagram. The effects of this Warburg impedance can also be approximated by using multiple resistor–capacitor (RC) networks in series [32]. Although an infinite RC pair network would be required for an exact equivalence, a finite number of RC pairs is often sufficient to the circuit accurately over a certain frequency range. Additionally, the double-layer capacitance $C_{dl}$ is frequently omitted, as its impact is predominant only at very high frequencies. When $C_{dl}$ is excluded and Warburg impedance is replaced by a small finite number of RC pairs, the resulting model is referred to as the Thevenin model, detailed in Section 3. By analyzing impedance characteristics, EIS provides valuable insights into battery performance, degradation, and aging. Impedance models are highly useful for diagnosis Li-Ion cell health. For example, identifying the cause of cell aging can be achieved by observing the parameter that exhibits significant variation. $R_b$ is the contact (or ohmic) resistance, and its variation indicates conductivity loss, collector corrosion, or side reactions in electrolytes. An increase in $R_{sei}$ and $C_{sei}$ reflects an expansion of the solid electrolyte interface, while a rise in R_CT suggests a loss of lithium with the cell. Changes in Warburg impedance typically indicate a reduction in active material [33]. However, this model is challenging to implement in real-time applications, as it always requires the use of a signal generator.

2.6. Data-Driven Models

Data-driven models are widely adopted to both predict the lifespan and estimate the health of batteries. Several methods were employed for battery life prediction, including support vector machines, neural network, autoregressive modeling, Bayesian predictions, and Box–Cox transformations [34,35,36,37]. An example is the use of the data-driven model to predict battery parameters according to the remaining useful life (RUL), until the End-of-Life (EoL), as presented in Figure 7. The main advantage of this approach is that it can be applied even under non-constant operating conditions and can efficiently solve nonlinear equations, enabling an accurate RUL prediction. Advantages of this methodology lie in its suitability for real-time modeling and its application in EV systems, where high precision is crucial for effective performance monitoring and management. Data-driven models can leverage large datasets to make accurate predictions, thereby enhancing decision-making processes in dynamic environments. However, a significant disadvantage of data-driven models is the difficulty in interpreting the underlying processes within the network that lead to the output results. This lack of transparency can hinder the understanding of how various factors influence battery performance, making it challenging to diagnose issues or optimize the model further. Additionally, the data requirements for training these networks are substantial; they necessitate large, high-quality datasets to develop a reliable and robust model. The need for extensive data collection can be a barrier, particularly in emerging applications where such datasets may not be readily available.

2.7. Real-World Scenarios for Different Battery Models

The various models used to represent Li-ion batteries find specific applications in different industrial and research contexts, depending on the level of complexity and details considered. The electrochemical model is widely used for the design and optimization of battery cells. This type of model provides a detailed description of the chemical reactions within the battery and is used to simulate complex electrochemical phenomena, such as the degradation of the active material, which is essential for designing new cells with advanced characteristics [38]. In contrast, the mathematical model finds application in constant or standardized operating conditions, but its simplicity makes it less suitable in dynamic and variable scenarios [39]. Electrical models, on the other hand, are commonly used traction and stationary storage applications. These models focus on the electrical response of the battery, such as voltage and current, making them suitable for designing battery behavior in systems that require high operational stability [40]. Low-order models are used in applications requiring fast, high-performance simulations, such as in real-time control systems for battery management. These models together with electrical ones are often implemented in commercially available BMSs. In advanced research applications, such as electrochemical spectroscopy, specific models are used to analyze the dynamic behavior of batteries as a function of frequency [41]. Finally, data-driven models are increasingly used thanks to the development of artificial intelligence and machine learning, applied for battery life prediction and performance monitoring [42]. A comprehensive summary is given in

Table 2. Battery model applications.

Model	Applications
Electrochemical	Used for cell design
Mathematical	Applicable only for constant operating condition
Electrical	Electric vehicles, energy storage systems
Reduced order	Real-time control systems, Battery Management Systems
Spectroscopy	Analyze dynamic behavior as a function of frequency
Data-driven	Battery life prediction, performance monitoring

.

2.8. Future Trends and Limitations

In recent years, innovation in Li-ion battery modeling has led to the emergence of several significant trends, each addressing specific industry needs. Among the emerging models, advanced electrochemical models, such as those that integrate chemical reaction analysis and charge and mass transport, are gaining prominence for their ability to provide accurate predictions of battery performance [43]. Mathematical and simulation models, such as those based on differential equations, continue to be refined to improve their ability to model behavior under extreme conditions. Furthermore, the increasing availability of data has driven the adoption of data-driven models, which use machine learning techniques to analyze and predict battery performance in a more dynamic and responsive manner [44]. However, these new trends also present significant challenges. Advanced electrochemical models require a huge amount of experimental data and a thorough understanding of the underlying mechanisms, which may limit their applicability in industrial settings. On the other hand, mathematical and simulation models can be complex and require significant computational resources, making them less practical for real-time applications. As for data-driven models, although promising, they face inherent limitations. Their effectiveness depends on the quality and quantity of available data; in the absence of representative data, the risk of overfitting increases, leading to inaccurate predictions. Furthermore, the lack of interpretability is a major problem: many models based on complex algorithms, such as neural networks, operate as ‘black boxes’, making it difficult to understand the underlying mechanisms of their predictions. Finally, data-driven models may struggle to adapt to new chemistries and emerging technologies, requiring costly and time-consuming training and validation processes. To address these challenges, research is aiming to develop hybrid approaches that integrate physical and data-driven models, improving both the accuracy of predictions and the comprehensibility of results. These hybrid approaches could represent the future of battery modeling, overcoming the limitations of purely data-driven models and providing a solid basis for innovation in the field of Li-ion batteries [45].

3. Electrical Models of Lithium-Ion Battery Cells

This section presents various equivalent electrical circuit models commonly found in the literature, such as Refs. [45,46,47,48,49,50]. To facilitate understanding, the circuits will be introduced with increasing complexity, starting with the simplest models—those that are fewer faithful representations of battery progression to those that more accurately simulate battery behavior. Simple models reduce the system to essential components, such as resistors, providing a simplified view of electrical behavior. These models are useful when a rough estimate of operation is desired but may not capture all the dynamics of a real battery. As one moves to more complex models, more elements are introduced that more accurately simulate battery behavior. These advanced models include variable parameters, multiple branches, and nonlinear components that can more accurately reflect real operating conditions.

3.1. Thevenin Model

The simplest electrical model used to represent a battery is the $R_{int}$ model (or simplified model). This model consists of an ideal voltage generator representing the open circuit voltage ($OCV$), in series with a resistor ($r_s$), as conducted by Refs. [51,52,53] and as shown in Figure 8a. The voltage generator describes the terminal voltage of the battery under equilibrium conditions, while the series resistor models the internal resistance of the battery. This resistance accounts for the resistance of the electrode material, the electrolyte, and the contact resistance within the battery. It explains the behavior of the terminal voltage, which drops when the battery is under load and rises when the battery is being charged. Additionally, it represents instantaneous bias, referring to the immediate response of the terminal voltage when a current is applied or removed [54,55]. The equation that characterizes the simplified model is as follows:

$$v=OCV-r_s \ast i$$

(3)

However, real batteries exhibit more complex behavior. Voltage bias can develop gradually over time when current is applied to the battery and decrease just as slowly when the battery is allowed to rest. To more accurately simulate the behavior of a real battery, a more detailed model is required—one that accounts for these time-dependent variations.

3.2. Thevenin Model with RC Branches

Thevenin’s model [56,57], building upon the simplified model, introduces $n$ − $r_p-c_p$ branch in series with the resistor $r_s$, to describe the dynamic behavior of the battery, as illustrated in Figure 8b. This dynamic behavior is related to electrochemical and concentration polarization effects, including charge transfer effect, diffusion, and other factors. By increasing the number of $r_p-c_p$ branches, the model becomes more accurate, but also more complex. Consequently, several studies have attempted to identify the optimal number of branches to use. In [58,59], it is demonstrated that using 2 or 3 $r_p-c_p$ branches provide a good balance between accuracy and computational load, as shown in Figure 8c.

The Thevenin model with a single $r_p-c_p$ branch, shown in Figure 9a, is also referred to as the first-order Thevenin model. It can represent the effects of double-layer capacitance and polarization characteristics. However, due to its short-time constant, this model is limited to describing the short-term transient effects and fails to capture the long-term behavior [60,61,62,63]. The system of equations describing this model, under constant current condition, is as follows:

$$v=OCV-r_s \ast i-v_{p_1}$$

(4)

$$\dot{v_{p_1}}={\displaystyle \frac{v_{p_1}}{r_{p_1} \ast c_{p_1}}}+{\displaystyle \frac{1}{c_{p_1}}}$$

(5)

Thevenin’s model with two $r_p-c_p$ branches, shown in Figure 9b, also incorporates a longer time constant modeling the diffusion phenomenon in the electrolyte [64,65]. The system of equations describing, under constant current conditions, this model is as follows:

$$v=OCV-r_s \ast i-v_{p_1}-v_{p_2}$$

(6)

$$\dot{v_{p_1}}={\displaystyle \frac{v_{p_1}}{r_{p_1} \ast c_{p_1}}}+{\displaystyle \frac{1}{c_{p_1}}}$$

(7)

$$\dot{v_{p_2}}={\displaystyle \frac{v_{p_2}}{r_{p_2} \ast c_{p_2}}}+{\displaystyle \frac{1}{c_{p_2}}}$$

(8)

The Thevenin model with three $r_p-c_p$ branches, shown in Figure 9c, offers greater accuracy [66,67]. While less common, this model is often used for advanced applications, such as battery parameter studies and Vehicle-to-Grid (V2G) applications [68,69]. The system of equations describing, under constant current conditions, this model is as follows:

$$v=OCV-r_s \ast i-v_{p_1}-v_{p_2}-v_{p_3}$$

(9)

$$\dot{v_{p_1}}={\displaystyle \frac{v_{p_1}}{r_{p_1} \ast c_{p_1}}}+{\displaystyle \frac{1}{c_{p_1}}}$$

(10)

$$\dot{v_{p_2}}={\displaystyle \frac{v_{p_2}}{r_{p_2} \ast c_{p_2}}}+{\displaystyle \frac{1}{c_{p_2}}}$$

(11)

$$\dot{v_{p_3}}={\displaystyle \frac{v_{p_3}}{r_{p_3} \ast c_{p_3}}}+{\displaystyle \frac{1}{c_{p_3}}}$$

(12)

3.3. Hysteresis Model

Lithium batteries exhibit a phenomenon known as the hysteresis effect, which must be considered in simulation models of their performance. Hysteresis occurs when the parameters measured during battery charging differ from those measured during discharging [70,71,72]. This effect can be visually represented by a curve showing the discrepancy between an open circuit voltage and state of charge ($SoC$) values during charging and discharging, as illustrated in Figure 10a. The hysteresis effect has a significant impact on the performance and accuracy of battery models. If not considered, it can introduce substantial errors into battery performance predictions. This effect is more pronounced in batteries that exhibit a flat curve in the relationship between an open circuit voltage and an $SoC$ and tends to worsen with increasing battery operating temperature, as shown by Ref. [73]. One of the most common strategies for modeling the hysteresis effect is to calculate the parameters separately during the charging phase and during the discharging phase, storing them in separate tables. This approach addresses the discrepancies caused by hysteresis and provides a more accurate representation of battery performance. To obtain the result, models can calculate the average value or use an interpolation method between the values obtained during the charging and discharging [74,75]. At the circuit level, this approach translates into a configuration where two resistors—one for charging and one for discharging—are placed in series with two diodes. The diodes inhibit current flow in the reverse direction, as shown in Figure 10b. The system of equations describing, under constant current conditions, this model is as follows:

$$Charge:v=OCV+R_c \ast i$$

(13)

$$Discharge:v=OCV-R_d \ast i$$

(14)

3.4. Run-Time Model

The circuits presented so far are unable to estimate electrical model parameters in real time due to their reliance on constant voltage generators. These voltage generators, specifically the open circuit voltage sources, are static and do not adapt based on the state of the battery. The circuits presented so far are unable to estimate electrical model parameters in real time due to their reliance on a constant voltage generator. Specifically the open circuit voltage source, is static and does not adapt according to the state of the battery. This property poses a challenge for dynamic systems like Lithium-ion batteries, where the OCV is not constant but varies depending on the battery’s state of charge and other operational conditions, such as temperature. To address this limitation, M. Chen and G. A. Rincon-Mora proposed a model, referred to as the run-time model [76], which provides a more accurate representation of voltage behavior over time. This model introduces a dynamic voltage generator, where the open circuit voltage is not constant but varies with the $SoC$. In their approach, the $OCV$ is modeled as a nonlinear function of $SoC$, typically obtained empirically and described using polynomial, piecewise linear, or exponential equations to represent the battery’s equilibrium voltage as a function of its $SoC$. Mathematically, this model can be expressed as the following:

$$OCV\left(SoC\right)=f\left(SoC\right)$$

where $f\left(SoC\right)$ defines the characteristic voltage response of the battery under no-load conditions. In the run-time model, the $OCV$ is calculated using a sub-circuit composed of a resistor $r_d\left(t\right)$ and a capacitor $c\left(t\right)$ arranged in parallel, with a current source $i\left(t\right)$ as the driving element, as illustrated in Figure 11. This sub-circuit captures the dynamic nature of the $OCV$ by accounting for the time-varying characteristics of internal resistance and capacitance. The inclusion of these time-dependent elements enables the model to adjust the $OCV$ in response to changes in $SoC$ overtime, allowing for real-time parameter estimation.

The system of equations describing this model, for a run-time model composed of a second-order Thevenin model, is provided below.

$$OCV=r_s \ast i-v_{p_1}$$

(15)

$$\dot{OCV}={\displaystyle \frac{OCV}{r_d \ast c_i}}+{\displaystyle \frac{1}{c_i}}$$

(16)

$$\dot{v_{p_1}}={\displaystyle \frac{v_{p_1}}{r_{p_1} \ast c_{p_1}}}+{\displaystyle \frac{1}{c_{p_1}}}$$

(17)

$$\dot{v_{p_2}}={\displaystyle \frac{v_{p_2}}{r_{p_2} \ast c_{p_2}}}+{\displaystyle \frac{1}{c_{p_2}}}$$

(18)

In the first subcircuit, the resistor $r_d\left(t\right)$ represents the self-discharge of the battery, the capacitor c_i indicates the amount of charge stored, and the current source $i\left(t\right)$ is controlled by measuring the current flowing in the second subcircuit. This subcircuit is designed for energy calculations, such as measuring the state of charge or residual capacity.

The second subcircuit is based on a second-order Thevenin model [77,78,79,80] (or third order Thevenin model), with one key difference: the voltage source is controlled and depends on the voltage measured in the first subcircuit, thereby reflecting the level of the $SoC$. This subcircuit is designed to simulate the current voltage $(I-V)$ performance of the battery.

3.5. Dynamic Model

Dynamic battery models are designed to simulate the time-varying electrical behavior of batteries under different operating conditions. These models incorporate the effects of key parameters, including voltage, internal resistance, and capacitance, as functions of the state of charge, temperature, and load dynamics. The primary advantage of dynamic models is their ability to provide accurate real-time predictions of battery performance, which is crucial for optimizing energy management, improving operational efficiency, and extending battery life in applications such as EVs and grid energy storage systems.

The dynamic model in [81], as shown in Figure 12, extracts model parameters from battery datasheets in a straightforward manner. Only three points on the manufacturer’s discharge curve in a steady state are required to obtain the parameters. Finally, this battery model is included in the Matlab/Simulink toolbox SimPowerSystems and used in detailed simulations of EVs equipped with hybrid storage systems. This approach provides approximate values applicable to various battery chemistries.

The equations characterizing the dynamic model shown in Figure 12 are as follows:

$$\mathrm{D}\mathrm{i}\mathrm{s}\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}:V_t=OCV-r_s \ast i-K \ast {\displaystyle \frac{Q}{Q-it}} \ast \left(it+i^{\ast }\right)+A \ast e^{-B\ast it}$$

(19)

$$\mathrm{C}\mathrm{h}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}:V_t=OCV-r_s \ast i-K \ast {\displaystyle \frac{Q}{it-0.1 \ast Q}} \ast \left(i^{\ast }\right)-K \ast {\displaystyle \frac{Q}{Q-it}} \ast (it)+A \ast e^{-B\ast it}$$

(20)

The battery voltage, OCV is the battery constant voltage, r_s is the internal resistance, i is the battery current, k is the polarization constant or polarization resistance, Q is the battery capacity, it is the actual battery charge estimated by $\int i dt$, A is the exponential zone amplitude, B is the exponential zone time constant inverse, and i* is the filtered current.

The model assumes constant internal resistance during charge and discharge cycles, no variation in battery capacity with the current, no temperature influence, and no self-discharge or memory effect. Model parameters are derived from discharge characteristics and considered identical for both charging and discharging. Limitations include a no-load voltage range from $0 V$ to 2*$E_0$ and a maximum capacity of $Q$, restricting the $SoC$ to $100\%$.

In contrast, the dynamic model described in [54] presents a LiFePO4 battery model for use in a battery simulator. The LiFePO4 battery model accurately represents the nonlinear voltage response to transient loads. To achieve this goal, the model, shown in Figure 13, combines the Shepherd model for nonlinear voltage changes with the Thevenin model to simulate the battery’s transient load response. The model is converted into a discrete-time format for use in simulations, with its parameters determined experimentally. The accuracy of the model is validated by comparing simulated voltage waveforms with experimental data.

In Figure 13, the Shepherd model describes a unique nonlinear $OCV$ waveform by the following equations:

$$OCV=E_0-K \ast {\displaystyle \frac{Q}{Q-\int i dt}}+A{\mathit{e}}^{-(\mathit{b}*\int \mathit{i}\mathit{d}\mathit{t})}$$

(21)

$$V_t=OCV-r_s \ast i$$

(22)

where $E$ is the open circuit voltage, $E_0$ is the battery constant voltage, $K$ is the polarization voltage, $Q$ is the battery capacity, $A$ is the exponential zone amplitude, and $b$ is the exponential zone time constant inverse.

The first-order Thevenin model represents the transient characteristics of the battery. However, this model is challenging to implement in an actual battery simulator because the battery simulator is embedded in a controller, meaning that the continuous-time model must be converted into a discrete time model. Transfer functions, influenced by changes in the $OCV$ and battery current that affect the terminal voltage, are derived using the principle of superposition. Additionally, the discrete-time model of transfer function is obtained by applying the $Z$-transform.

4. Parameter Identification

To improve the accuracy of electrical models, it is crucial to correctly estimate their parameters. However, battery manufacturers often provide only a limited set of parameters, which is insufficient for developing highly precise models. As a result, comprehensive characterization becomes essential to accurately identify the parameters and ensure the model’s realism and reliability. Various tests and methodologies for parameter estimation have been discussed in the literature, emphasizing the importance of thorough characterization to compensate for the lack of detailed manufacturer data [82,83,84,85].

Before beginning parameter characterization, any passivation that the battery may have undergone during transport from the manufacturer to the test site must be removed through a preconditioning method. Additionally, it is necessary to verify the accuracy of the information provided by the manufacturer using a capacity test. After these steps, the battery should be allowed to rest for a certain period, after which characterization can be carried out, as illustrated in Figure 14.

4.1. Preconditioning Test

The purpose of the preconditioning test is to eliminate any possible passivation that may have occurred between the time of battery fabrication and the first tests. Additionally, this test aims to stabilize the battery capacity, as the solid electrolyte interface is not fully formed during the fabrication process, and its porosity and structure change significantly after the first few cycles. The preconditioning test consists of three to five consecutive charge–discharge cycles at a standard $C-rate$ and $25 °\mathrm{C}$ [62,84].

The battery is considered preconditioned when the discharge capacity during two consecutive discharges does not deviate by more than $3\%$ from the nominal capacity specified by the manufacturer. The block diagram in Figure 15 shows the different steps and the behavior of this test.

4.2. Capacity Test

The nominal capacitance provided by the manufacturer is based on optimal laboratory conditions. To accurately characterize the battery’s actual capacity, a capacity test must be performed under specific test conditions. The nominal capacity of a battery can vary depending on different conditions, and this test allows for adjusting the value of the capacity according to the battery’s operating conditions. During the test, the battery is charged at a standard current rate, while the discharge current rate varies with each cycle [84,85,86,87,88]. Figure 16 summarizes the capacity test flowchart and behavior.

4.3. Open Circuit Voltage Test

The open circuit voltage test involves fully charging the battery, followed by applying a discharge current equivalent to $5\%$ of the $SoC$ and allowing for a rest period to allow the battery to relax. This same procedure is also used during charging.

Another variant of this test involves using a $2\%$ step in the $SoC$ ranging from $100\%$ to $90\%$ and from $10\%$ to $0\%$. These ranges are the areas of the $OCV$ curve that provide the most information, with a discharge step of $5\%$ in the middle range of $90-10\%$. Relaxation time is crucial in this test because it allows the battery to reach equilibrium at a stable $OCV$ value. Typically, a relaxation time of 30 min is used, after which the active materials reach their equilibrium. At the end of the test, it is possible to extract $OCV$ points at different levels of the $SoC$ to plot the curve connecting the $OCV$ and $SoC$. The discharge test consists of a stepped decreasing voltage, where each step corresponds to the $OCV$ value at the $SoC$ level after battery relaxation. The charge test, conversely, consists of a stepped increasing voltage [85,89,90]. Figure 17 presents the flowchart for the $OCV$ charge and discharge test when a discharge current equivalent to $5\%$ of the $SoC$ is applied.

4.4. Hybrid Pulse Power Characterization

The Hybrid Pulse Power Characterization (HPPC) test allows for the determination of battery parameters based on variations in operating conditions. This test involves applying an alternating set of charge and discharge pulses of varying amplitudes at each $SoC$ level to simulate battery usage.

Various HPPC test variations were proposed in the literature, differing in the applied current rate and $SoC$ levels studied, depending on the desired application [85,91,92,93,94,95]. To conduct the HPPC discharge test, the battery is first fully charged. Next, two charge and discharge pulses are applied for $10 \mathrm{s}$ each with the $C-rate$, with a $30 \mathrm{s}$ rest period in between. This procedure is repeated, varying the current rate $C-rate$. After applying all desired current rates, the battery is discharged by $5\%$ of its capacity to reach a new state of charge level.

The same set of pulses is then applied again at this new $SoC$ level. The HPPC test during charging follows the same principle but starts with a fully discharged battery. Further methodology of implementing the HPPC test is shown in Figure 18. Specifically, the battery starts fully charged and it is followed by five discharge pulse at different C-rates (0.5, 1, 2, 4, 6) C. After completing the pulsed discharges, there is a SoC reduction of 5% if it is in the range (100–90; 30–0)% or 10% if it is in the range (40–80)%.

4.5. Pulse Discharge Test

The Pulse discharge test (PD) consists of a series of charge and discharge pulses applied to the battery until the cut-off voltage is reached. The PD test was developed to evaluate the effectiveness of the battery model under dynamic conditions. Additionally, this test can be used to assess the energy performance efficiency of the model after estimating the parameters of the lithium-ion battery model. In the automotive industry, several validation tests are available, including the PD test. The current profile consists of two successive discharge pulses: one at $1.5 \mathrm{C}$ for $10\mathrm{s}$ and the other at $0.4 \mathrm{C}$ for $20 \mathrm{s}$, followed by a charge pulse of $0.8 \mathrm{C}$ for $5 \mathrm{s}$. A short rest of $30 \mathrm{s}$ is then applied. This current profile is repeated until the battery is fully discharged [96,97,98,99]. The flowchart of the DP test is presented in Figure 19.

5. Model Parameter Dependencies

This section provides an overview of the variations in the equivalent electrical circuit parameters in relation to temperature and current rate. The variation in the $SoH$ and $SoC$ as a function of these parameters is also discussed. The objective is to demonstrate how these factors affect the overall performance of the electrical system and its reliability over time.

5.1. Temperature

For Li-ion batteries, temperature is a critical factor that directly affects all the electrical, chemical, and mechanical parameters of the system. Among these, capacity, internal resistance, electrochemical reaction rate, and aging are highly dependent on operating temperature. The dynamic behavior of a battery is significantly influenced by thermal conditions, highlighting the need for an integrated model that encompasses both electrical and thermal aspects.

An accurate representation of the battery under different conditions required an approach that integrates both thermal and electrical dynamics. This approach allows for the consideration of the effects of heating or cooling resulting from charge and discharge cycles, as well as their subsequent impact on battery lifespan and performance. Therefore, it is essential to consider a thermal–electrical model to gain a deeper understanding of, and to manage, lithium-ion batteries more efficiently in real applications.

An example of a thermal model that can be integrated with the previously mentioned electrical models is presented in [100] and illustrated in Figure 20.

The thermal model is defined by the following set of equations:

$$C \ast \dot{T}=Q-\left(T-T_{amb}\right) \ast {\displaystyle \frac{1}{R4}}$$

(23)

$$Q=\left|I \ast (OCV-V_t)\right|$$

(24)

$$R={\displaystyle \frac{1}{h \ast S}}$$

(25)

where $C_{core}$ is the thermal capacity, $\dot{T}$ is the partial derivative of temperature $\left(T\right)$ with respect to time, $Q$ is the heat generation rate, $T_{amb}$ is the ambient temperature, $R$ is the thermal resistance, $I$ is the cell current, $OCV$ is the cell open circuit voltage, $V_t$ is the cell voltage, $h$ is the equivalent heat transfer coefficient, and $S$ is the cell surface area. The open circuit voltage of Li-ion batteries is affected by both temperature and state of charge. However, in most common operating conditions, the change in open circuit voltage can be considered relatively independent of temperature.

Significant differences in $OCV$ behavior occur only when the temperature drops to well below zero or when the battery’s state of charge is in the range of 0–20% or 80–100%. In these cases, as shown in Figure 21, the $OCV$ decreases as temperature increases [101].

Additionally, for temperatures below zero and in the $SoC$ range of 0–20%, the resistance curve tends to follow a decreasing exponential shape, in contrast to the nearly linear behavior observed at positive temperatures, as shown in the Figure below. For temperatures ranging from $-10 °\mathrm{C}$ to $-25 °\mathrm{C}$, the resistance waveforms are quite similar. The series resistance of Li-ion batteries is influenced by both the temperature and state of charge. Its value decreases as the temperature increases [94,101,102]. The resistance and capacitance of the first r-c branch are influenced by both the temperature and state of charge. Resistance tends to decrease with increasing temperature, while the capacitance increases as the temperature rises, as illustrated in Figure 22b,c. The resistance values obtained for temperatures below zero exhibit more irregular patterns, and, at $-30 °\mathrm{C}$, the resistance is approximately an order of magnitude higher than at positive temperatures. Conversely, the capacitance value follows the opposite trend. For temperatures ranging from $25 °\mathrm{C}$ to $45 °\mathrm{C}$, the resistance waveforms nearly coincide. The resistance and capacitance of the second $r_p-c_p$ branch is dependent on temperature and state of charge. As shown in Figure 22d,e, the resistance decreases with increasing temperature, while the capacitance increases as temperature rises [78,94,102]. For temperatures ranging from $25 °\mathrm{C}$ to $45 °\mathrm{C}$, the resistance waveforms almost coincide. Unlike the capacitance of the first branch, in this case, the capacitance waveforms nearly coincide when the temperature is in the range of $-10 °\mathrm{C}$ to $-30 °\mathrm{C}$. For both parameters, the waveforms are more regular than in the previous case.

5.2. State of Charge

The state of charge represents the amount of charge remaining in a Li-ion battery relative to its total capacity. It expresses the battery’s charge as a percentage, where $100\%$ indicates a fully charged battery and 0% a discharged battery. The state of charge is defined by the following equation:

$$SoC={\displaystyle \frac{q}{q_{nom}}}$$

(26)

where $q$ is the actual battery charge of the battery and $q_{nom}$ is the nominal value of the battery charge. Additionally, an accurate estimation of the $SoC$ is crucial because it directly influences key parameters that define the equivalent electrical circuit model typically used to represent the behavior of a lithium-ion battery. For instance, a reduction in the $SoC$ generally leads to a decrease in the OCV and an increase in the battery’s internal resistance, particularly the series resistance. In the literature, various methods are proposed for the estimation of the state of charge, including lookup table-based methods, integral method methods, filter-based methods, and data-driven methods. The $OCV$ method falls under the category of lookup table-based methods. This approach converts $OCV$ estimations into $SoC$ estimations by utilizing the $SoC-OCV$ relationship. In [103], the $OCV$ estimation is performed using the Recursive-Least-Square (RLS) algorithm with a forgetting factor, where the model parameters are simultaneously estimated. Integral methods are based on recording and integrating the electric current flowing in and out of the battery. In [104], the Coulomb Counting (CC) methodology is employed for the $SoC$ estimation. The algorithm calculates residual capacitance by accumulating the current overtime. However, the accuracy and efficiency of the CC algorithm depend on the sampling time of the current sensor. Therefore, measurement errors in the current sensor can lead to an inaccurate $SoC$ estimation. The $SoC$ of the battery at each time interval ($SoC\left(t\right)$) based on the CC algorithm can be calculated as follows:

$$SoC\left(t\right)=SoC\left(0\right)+{\int }_{t_0}^T{\displaystyle \frac{I\left(t\right) \ast \eta }{C}}dt$$

(27)

where $SoC\left(0\right)$ is the initial $SoC,$ $I\left(t\right)$ is the current flowing through the battery from $t_0$ to $T$, $C$ is the capacity of the battery, and $\eta$ is the coulombic efficiency that shows the charge efficiency by which electrons are transferred in batteries. Filter-based methods use mathematical models that combine real measurements with theoretical predictions to dynamically estimate the $SoC$. Among them, the Kalman filter is one of the most widely used algorithms. In [105], an $SoC$ estimation method based on the multi-model extended Kalman filter (MM-EKF) algorithm for Li-ion batteries is proposed, and its effectiveness is validated under different ambient temperatures. Experimental results demonstrate that the MM-EKF algorithm, which considers the effects of temperature and current rate, can accurately estimate the $SoC$ of Li-ion batteries. It is necessary to use the EKF because the standard Kalman filter works correctly only for linear systems that can be described by linear differential equations. However, Li-ion batteries exhibit nonlinear dynamics. Consequently, the standard Kalman filter is not suitable for an accurate $SoC$ estimation because the models describing the electrical, thermal, and chemical behavior of the battery are often nonlinear. The EKF can handle nonlinear systems by linearizing the model around the current state using a first-order Taylor approximation. Data-driven methods, on the other hand, rely on algorithms that learn from battery operating data—such as voltage, current, temperature, and charge/discharge cycles—to build models able to estimate the state of the battery. As reported in [106], the most studied types of data-driven methods include neural network (NN), deep learning (DL), support vector machine (SVM), and Gaussian process regression (GPR) methods.

5.3. C-Rate

The $OCV$, in addition to being temperature-dependent, is also influenced by $C-rate$. The dependence of the open circuit voltage differs between the charging and the discharging phase: during charging, as the $C-rate$ increases, the $OCV$ increases; whereas, during the discharging, as the $C-rate$ increases, the $OCV$ decreases [65,107]. While charging, for $SoC$ values ranging from 20 to 100%, the curves are very similar up to a $C-rate$ of $2 C$. However, for $SoC$ values from 0 to 20%, the curves show very differently. In the case of discharge, for $C-rates$ values greater than or equal to $2 C$, the $OCV$ curve displays markedly different values across the entire $SoC$ range, while, for lower $C-rates$, the curves tend to coincide for $SoC$ values between 40 and 100%, as shown in Figure 23.

Series resistance also exhibits a pronounced dependence on both $C-rate$ and temperature. Specifically, the value of series resistance tends to decrease as the $C-rate$ increases [95,100,105,108]. However, for $SoC$ values close to $0\%$, the series resistance can be considered relatively invariant to the $C-rate$, as shown in Figure 24a. The resistance and capacitance of the first $r_p-c_p$ branch show a significant dependence on both $C-rate$ and temperature. As shown in Figure 24a,b, the resistance decreases as the $C-rate$ increases, while the capacitance value increases with rising $C-rate$ [70,82,94,95,105]. The resistance and capacitance of the second r-c branch show a significant dependence on both $C-rate$ and temperature. As shown in Figure 24d,e, the resistance decreases as the $C-rate$ increases, while the capacitance increases with a rising $C-rate$ [78,94,95,105]. In this case, for high $SoC$ values, the resistance values remain similar as the $C-rate$ changes, indicating a weaker $C-rate$ dependency for this parameter compared to others.

5.4. Aging

The aging of Li-ion batteries is a complex process that occurs over time and during use, leading to a gradual degradation of performance, capacity, and efficiency. This degradation can be broadly categorized into two main forms: calendar aging and cyclical aging. Although these two mechanisms operate simultaneously, they are driven by different factors and occur under distinct conditions.

Calendar aging refers to the degradation of the battery that occurs even when it is not in active use, driven by the passage of time rather than by charge–discharge cycles [109]. This type of aging is primarily influenced by external factors such as temperature, storage conditions, and the state of charge during periods of inactivity. For instance, elevated temperatures and high $SoC$ levels during storage are known to accelerate chemical reactions within the cell, such as electrolyte decomposition, growth of the SEI layer on the anode, and the oxidation of the cathode materials. These processes contribute to a reduction in capacity and an increase in internal resistance, even if the battery is not subjected to frequent cycling. Calendar is therefore a critical concern for applications where batteries are stored for extended periods or used intermittently, such as in EVs or grid storage systems [110].

Cyclical aging, on the other hand, results from repeated charge–discharge cycles and is highly dependent on the charge rate and the operating conditions during use. Each cycle induces mechanical and chemical stress on the battery’s electrodes, leading to phenomena such as lithium plating, microcracking of the electrode materials, and further growth of the SEI layer. These effects result in the gradual loss of active material, reduced ionic conductivity, and structural degradation of the electrode. Cycling at high charge and discharge rates or under extreme temperatures exacerbates these stresses, accelerating performance decline. Over many cycles, these effects accumulate, leading to reduced usable capacity, lower energy efficiency, and increased internal impedance, ultimately limiting the battery’s lifespan [111,112].

A key parameter for understanding battery aging is the $SoH$, which quantifies the overall condition of the battery compared to its original capacity and performance. The $SoH$ reflects the gradual loss of capacity and increased internal resistance over time—key indicators of aging. Monitoring the $SoH$ is essential for predicting battery life, optimizing usage, and ensuring safety in applications such as electric vehicles and energy storage systems. The $SoH$ is defined as a measure of the remaining capacity of a battery relative to its initial capacity, indicating the degree to which the battery has aged or degraded. The equation describing the behavior of the ${SoH}_c$ is as follows.

$${SoH}_c={\displaystyle \frac{c}{c_{nom}}}$$

(28)

where c is the value of the actual capacitance, and $c_{nom}$ is the nominal capacitance. From this equation, it is evident that, as the $SoH$ decreases, the battery’s capacity also decreases.

In recent years, a close correlation between series resistance and the $SoH$ has also been recognized, where a decrease in the $SoH$ corresponds to an increase in series resistance. This phenomenon is formalized by the following equation:

$${SoH}_r={\displaystyle \frac{r_{s,N}-r_s}{r_{s,N}-r_{s,0}}}$$

(29)

where $r_{s,N}$ is the value of the series resistance at the end of the battery’s life, $r_s$ is the current value of the series resistance, and $r_{s,0}$ is the value of the series resistance at the beginning of the battery’s life. In the technical literature, the $SoH$ estimation approaches are divided into three categories: physical methods, statistical methods, and machine-learning methods. The first category of models includes incremental capacity analysis techniques (ICA) and the internal resistance estimation approach. Among these, the ICA approach is the most widely used for estimating battery capacity degradation. An ICA can assess aging of the battery, which represents the primary cause of its gradual performance reduction. Similarly, the Coulomb counting method is one of the most extensively documented ICA techniques.

The second category consists of statistical methods that employ a system of equations to accurately describe the electrochemical or electrical model of the battery, allowing for real-time $SoH$ estimation. These approaches typically rely on nonlinear statistical observers, with Kalman filters being the most widely used.

The third category, machine learning (ML), is a data-intensive approach that does not require detailed knowledge of the battery’s internal structure. ML uses artificial intelligence algorithms to estimate the $SoH$ in real time. Since the $SoH$ is related to changes in the internal parameters of the battery—particularly those of the equivalent electrical circuit—it is also dependent on temperature and the $C-rate$.The dependence of the $SoH$ on temperature is particularly pronounced at values below $0 °\mathrm{C}$, where faster degradation and a more rapid decline in the $SoH$ are observed. From $0 °\mathrm{C}$ to $27 °\mathrm{C}$ (the typical operating range for batteries), as the temperature increases, the $SoH$ decreases, as shown in Figure 25 [113]. The dependence of the $SoH$ on the $C-rate$ becomes evident as the $C-rate$ increases. For high $C-rate$ values, the $SoH$ decreases more rapidly [114]. This trend is illustrated in Figure 26, where the curves represent the $SoH$ as a function of EoL. It is defined as the point at which the battery can no longer be used for automotive applications [115] and typically corresponds to an $SoH$ of $80\%$.

6. Conclusions

The power and transportation sectors account for more than $66\%$ of the global carbon emissions. Decarbonizing these sectors is a critical challenge in achieving a zero-carbon economy by mid-century and avoiding the most severe impacts of climate change. Battery packs, which store energy on-board vehicles, are the primary component of electrified transport systems. Their production has rapidly increased in recent years to meet growing demand, and this trend is expected to continue. Therefore, the development of suitable battery models is essential for designing high performance energy storage systems.

In the technical literature, battery models are classified into several categories: electrochemical models, mathematical models, and electrical models. Electrochemical models are the most accurate in simulating internal phenomena but require significant computational resources and are slow. On the other hand, mathematical models are useful for certain calculations or predicting parameters, such as statistical cycle life based on experimental data. Data-driven models hold potential for many applications but require large amounts of data to achieve high accuracy. Finally, electrical models are the most suitable for design, real-time control, or emulation purposes, and are the best option for implementation in BMSs, chargers, V2G charging stations, and similar applications. This paper presented an overview of the electrical battery models, starting from the simplest and least accurate to the most complex and complete, though computationally demanding. Relevant references with more detailed descriptions and in-depth considerations have also been provided. Parameter estimation is another key aspect addressed in this work, as battery manufacturers often do not provide sufficient data on their product. As such, existing methods for deriving the value of model parameters were presented, including some models that can be categorized as electrical.

In conclusion, an appropriate battery model for implementation in electrified transport systems should meet the following requirements: accuracy, computational simplicity, support for real-time operation with fast simulation speed, ease of configuration with fewer parameters to identify and define, and physical interpretability, facilitating the identification of the source of any issue that may arise in battery modules.