Review on Modeling and SOC/SOH Estimation of Batteries for Automotive Applications
Abstract
:1. Introduction
 Energy Management Optimization: In various batterypowered systems, precise SOC estimation is indispensable for efficient energy management [15,16,17]. By accurately gauging the available charge, these algorithms enable optimal utilization of the battery capacity, thus preventing potentially detrimental conditions such as overcharging or overdischarging [18,19,20]. 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 [21,22,23]. Implementing accurate SOC estimation techniques facilitates intelligent energy utilization strategies, which ensure prolonged battery life and sustained system performance over extended operational periods [24,25,26].
 Enhanced Safety Measures: The accuracy of SOC estimation is directly linked to ensuring the safety and integrity of batterypowered systems [27,28,29]. Inaccurate SOC determination can lead to scenarios of overcharging or overdischarging, exacerbating the risk of critical safety hazards such as thermal runaway [30,31,32]. 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 [33,34,35]. Hence, the development of precise and reliable SOC estimation algorithms plays a pivotal role in enhancing the overall safety and risk management strategies within batterydependent applications [36,37,38].
 Proactive Battery Health Monitoring: Accurate estimation of SOH is instrumental to effective battery health monitoring and enables the timely detection of degradation and performance decline [39,40,41]. These algorithms facilitate the assessment of critical battery parameters, such as capacity fade, impedance changes, and chemical degradation, to provide insight into the battery’s remaining useful life (RUL) [42,43,44]. By implementing proactive SOH estimation methodologies, operators can anticipate potential battery failures, initiate timely maintenance interventions, and prevent costly downtime and unplanned disruptions in industrial and automotive operations [45,46,47]. The ability to predict the RUL allows for informed decisionmaking regarding battery replacement or reconditioning, thereby optimizing maintenance costs and ensuring continuous system reliability [48,49,50].
 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 opencircuit 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 [55,56,57], 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 realworld applications where temperature variations are common [58,59,60].
 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 resourceconstrained applications, particularly in the automotive and industrial sectors where realtime performance is critical [63,64,65].
 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 realtime 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 significant computational resources for training and inference, which could pose practical challenges in resourcelimited 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 [72,73,74].
2. Overview of Battery Systems
 Battery Packs and Cells: The foundational level of the BESS consists of multiple battery packs, each containing interconnected batteries.
 Battery Management System (BMS): Operating at the cellular level, the battery management 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].
 Power Conversion System (PCS): The battery system interfaces with inverters through a specific power electronic level known as the power conversion system (PCS). Typically 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].
 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 realtime demands and conditions [98,99].
 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].
 Transformer Connections: Finally, the BESS interfaces with transformers to manage the voltage levels. Depending on the system’s size, there are connections with mediumvoltage/lowvoltage transformers and, in larger systems, highvoltage/mediumvoltage transformers located in dedicated substations. These transformers facilitate the integration of the BESS with the broader electrical infrastructure [101,102].
 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 temperature, 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].
 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.
 General Parameters: Mathematical models rely on general parameters like internal resistance, capacity, and opencircuit voltage that are commonly present in all batteries. 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 technologyspecific parameters. This allows BMS designers to customize models to fit the specific characteristics of the battery they are managing [108].
3. Overview of Modeling Approaches
3.1. Empirical Modeling
 Semiempirical model: A semiempirical approach combines elements of theoretical understanding with empirical data, which allows for a more accurate representation of the electrochemical processes within Liion 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 Liion 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 vehicletogrid (V2G) approaches, on the lifespan of Liion batteries. By simulating the effects of various usage scenarios, this model helps with understanding how different operational conditions can affect the longterm performance and aging of the battery and enables the development of effective management protocols [115,116,117].
3.2. Equivalent Circuit Modeling
 
 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 ${q}_{disch}=\_int{i}_{disch}\left({t}^{\prime}\right)d{t}^{\prime}$, 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 $\eta $, by opposite procedure, one starts from the full discharge condition of the battery and slowly brings it up to the maximum nominal voltage level ($SOC=100\%$) from the minimum ($SOC=0\%$); one directly monitors ${q}_{ch}=\int {i}_{ch}\left({t}^{\prime}\right)d{t}^{\prime}$. This results in an estimated $\eta \simeq \frac{{q}_{disch}}{{q}_{ch}}$.
 
 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 ${t}_{stop}$, the voltage has a rising transient characterized by a “vertical” stretch, which can be associated with the term ${R}_{0}i\left(i\right)$, so from the direct voltage measurement it is possible to derive the value $\Delta {v}_{0}$ and consequently to obtain an estimate of the parameter. Similarly, one can associate the steadystate value $\Delta {v}_{\infty}$ of the voltage with the series of resistors ${R}_{0}+{R}_{1}$ to procedurally derive ${R}_{1}$ as well. As for the value of ${C}_{1}$, we can consider the pseudoempirical relationship between the settling time ${\tau}_{\infty}$ and characteristic constant.
3.3. Other Models
 Accurate Description: The PNGV model provides an accurate description of the discharge behavior of lithium batteries.
 SOC Estimation: It enables precise SOC estimation for Liion batteries based on multimodel 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.
 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.
 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.
 Complexity: The model’s dynamic performance and accuracy may come with increased complexity, which may require a sophisticated implementation and computational 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.
 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.
 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 LowFrequency 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.
4. Main SOC/SOH Estimation Algorithms
4.1. Coulomb Counting Method
4.2. OpenCircuit Voltage Method
4.3. KalmanFilterBased Method
 1
 Prediction Phase:
 1.a
 State prediction:$${\widehat{x}}_{kk1}=F{\widehat{x}}_{k1k1}+B{u}_{k}$$
 1.b
 Covariance prediction:$${P}_{kk1}=F{P}_{k1k1}{F}^{T}+Q$$
 2
 Correction Phase:
 2.a
 Kalman gain calculation:$${K}_{k}={P}_{kk1}{H}^{T}{(H{P}_{kk1}{H}^{T}+R)}^{1}$$
 2.b
 Corrected state estimation:$${\widehat{x}}_{k}={\widehat{x}}_{kk1}+{K}_{k}({z}_{k}H{\widehat{x}}_{kk1})$$
 2.c
 Covariance update:$${P}_{k}=(I{K}_{k}H){P}_{kk1}$$
 1
 Prediction Step:
 1.a
 State prediction:$$\begin{array}{c}{\widehat{x}}_{kk1}=f({\widehat{x}}_{k1k1},{u}_{k})\\ \\ {F}_{k}=\frac{\partial f({\widehat{x}}_{k1k1},{u}_{k})}{\partial {\widehat{x}}_{k1k1}}\\ \\ {B}_{k}=\frac{\partial f({\widehat{x}}_{k1k1},{u}_{k})}{\partial {u}_{k}}\end{array}$$
 1.b
 Error covariance prediction:$${P}_{kk1}={F}_{k}{P}_{k1k1}{F}_{k}^{T}+{Q}_{k}$$
 2
 Correction Step:
 2.a
 Compute the Kalman gain:$${K}_{k}={P}_{kk1}{H}_{k}^{T}{({H}_{k}{P}_{kk1}{H}_{k}^{T}+{R}_{k})}^{1}$$
 2.b
 Update the state estimate:$${\widehat{x}}_{k}={\widehat{x}}_{kk1}+{K}_{k}({z}_{k}h\left({\widehat{x}}_{kk1}\right))$$
 2.c
 Update the error covariance:$${P}_{k}=(I{K}_{k}{H}_{k}){P}_{kk1}$$
 1
 Prediction Step:
 1.a
 Generate sigma points:$${\mathcal{X}}_{k}=\{{\chi}_{k}^{0},{\chi}_{k}^{1},\dots ,{\chi}_{k}^{2n}\}$$
 1.b
 Propagate sigma points through the nonlinear process model:$${\chi}_{k+1k}^{i}=f({\chi}_{k}^{i},{u}_{k})$$
 1.c
 Compute predicted state and covariance:$$\begin{array}{c}{\widehat{x}}_{kk1}=\sum _{i=0}^{2n}{w}_{i}^{m}{\chi}_{kk1}^{i}\\ \\ \phantom{\rule{1.em}{0ex}}{P}_{kk1}=\sum _{i=0}^{2n}{w}_{i}^{c}({\chi}_{kk1}^{i}{\widehat{x}}_{kk1}){({\chi}_{kk1}^{i}{\widehat{x}}_{kk1})}^{T}+{Q}_{k}\end{array}$$
 2
 Correction Step:
 2.a
 Propagate sigma points through the observation model:$${\mathcal{Z}}_{k}^{i}=h\left({\chi}_{kk1}^{i}\right)$$
 2.b
 Compute the predicted measurement mean and covariance:$$\begin{array}{c}{\widehat{z}}_{kk1}=\sum _{i=0}^{2n}{w}_{i}^{m}{\mathcal{Z}}_{k}^{i}\\ \\ \phantom{\rule{1.em}{0ex}}{S}_{k}=\sum _{i=0}^{2n}{w}_{i}^{c}({\mathcal{Z}}_{k}^{i}{\widehat{z}}_{kk1}){({\mathcal{Z}}_{k}^{i}{\widehat{z}}_{kk1})}^{T}+{R}_{k}\end{array}$$
 2.c
 Compute the crosscovariance matrix:$${P}_{x,z}=\sum _{i=0}^{2n}{w}_{i}^{c}({\chi}_{kk1}^{i}{\widehat{x}}_{kk1}){({\mathcal{Z}}_{k}^{i}{\widehat{z}}_{kk1})}^{T}$$
 2.d
 Compute the Kalman gain:$${K}_{k}={P}_{x,z}{S}_{k}^{1}$$
 2.e
 Update the state estimate:$${\widehat{x}}_{k}={\widehat{x}}_{kk1}+{K}_{k}({z}_{k}{\widehat{z}}_{kk1})$$
 2.f
 Update the error covariance:$${P}_{k}={P}_{kk1}{K}_{k}{S}_{k}{K}_{k}^{T}$$
4.4. AIBased Methods
4.4.1. Bayesian Neural Network
4.4.2. Support Vector Machines
4.4.3. Long ShortTerm Memory
4.5. Method Comparison Discussion
5. Conclusions and Future Work
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
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Type of Battery  Advantages  Disadvantages 

Lead–Acid  Low expense and simple to manufacture; Low cost per watthour; Low selfdischarge; 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 deepcycling reduces battery life; Transportation restrictions on the flooded type; Not environmentally friendly. 
Nickel–Cadmium  Only battery that can be ultrafast charged with little stress; Good load performance; Can be stored in discharged state; Simple storage and transportation: not subject to regulatory control; Good lowtemperature 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 selfdischarge, needs recharging after storage; Low cell voltage (typically) of $1.2\mathrm{V}$ requires many cells in series to achieve high voltage. 
LithiumIon  High specific energy and high load capabilities with power cells; Long cycle life and extended shelf life; Maintenancefree; High capacity, low internal resistance and good coulombic efficiency; Simple charge algorithms; Relatively short charge times.  Requires protection circuit to prevent thermal runaway 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 shipping in larger quantities. 
Support Vector Machine (SVM) [172,173,174,175]  Long ShortTerm Memory (LSTM) [176,177,178]  Bayesian Neural Network (BNN) [179,180,181,182]  

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. 
Embeddedoriented  Implementation is possible with linear models and simple kernels.  Higher computational complexity, requires casebycase evaluations.  Implementation is possible but may require significant computational resources due to the Bayesian approach. 
Realtime  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. 
Timeseries  Less suitable compared to LSTM.  Highly suitable, excellent for capturing longterm 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. 
Method  Advantages  Disadvantages 

Coulomb Counting Method [183,184,185] 


OpenCircuit Voltage (OCV) Method [186,187,188] 


Kalman Filter [189,190,191] 


Artificial Intelligence Methods [192,193,194,195] 


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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
Dini P, Colicelli A, Saponara S. Review on Modeling and SOC/SOH Estimation of Batteries for Automotive Applications. Batteries. 2024; 10(1):34. https://doi.org/10.3390/batteries10010034
Chicago/Turabian StyleDini, Pierpaolo, Antonio Colicelli, and Sergio Saponara. 2024. "Review on Modeling and SOC/SOH Estimation of Batteries for Automotive Applications" Batteries 10, no. 1: 34. https://doi.org/10.3390/batteries10010034