Review of State of Power Estimation for Li-Ion Batteries: Methods, Issues, and Prospects
Article information
Abstract
This study provides a review of the available techniques for peak power estimation of Li-ion batteries in battery management systems (BMS). While various methods for estimating the state of charge (SOC) and state of health (SOH) of onboard batteries have been extensively discussed in previous research, the topic of state of power (SOP) estimation has yet to be comprehensively explored. Predicting the maximum power that can be applied to the battery through discharging or charging it during acceleration, regenerative braking, and hill climbing is undoubtedly one of the most challenging tasks for BMS. In Li-ion batteries, due to issues such as temperature and aging, efficient and reliable methods are required to estimate battery power. Accurate SOP estimation enables the BMS to more precisely regulate power flow in applications, optimize battery performance, and correspondingly increase its lifespan. To this end, scientific and technical literature sources were researched, and existing methods were reviewed. Finally, some opportunities and prospects were summarized to support the establishment of an accurate and robust multi-constraint-based SOP estimation technology in electric vehicle applications for intelligent BMS in the future.
INTRODUCTION
In recent years, with the increasing severity of environmental pollution and energy crisis, people’s attention to the field of electric vehicles has been growing [1]. In this field, Li-ion batteries are highly regarded as a green energy product. Li-ion batteries have advantages such as high energy density and long cycle life [2], making them one of the most promising energy storage components in electric vehicles [3]. However, under extreme operating conditions, Li-ion batteries also face some challenges, such as battery aging and mechanical damage. To ensure the safe and reliable operation of Li-ion batteries under complex working conditions, an effective BMS is needed to continuously monitor the internal state of the battery. A reliable and robust BMS is the primary task for a safe driving experience. A robust BMS can accurately monitor the state of the battery, enhance the energy output and power transmission efficiency of the battery, and effectively extend the overall life of the battery [4], such as SOC, SOH, State of Energy (SOE), SOP, and so on [5]. As we know, the state cannot be directly measured by sensors, and it only can be indirectly estimated by analyzing some measurable signals such as voltage and current. Therefore, several estimation algorithms have been developed to estimate different states of the battery pack for achieving powerful BMS, where the SOP estimation plays an important role in ensuring stable and reliable power output of electric vehicles and provides great assurance for the safe operation of electric vehicles over a certain period of time. Furthermore, the estimation of SOP is a relatively complex process that requires considering multiple factors including the chemical characteristics of the battery, usage conditions, and environmental temperature [6,7], among others. Thus, SOP refers to the magnitude of power that can be provided within specific constraints over a certain period.
In recent years, methods for estimating the SOP of Li-ion batteries can be categorized into four categories: Look-up Table (LUT) methods [8,9], Model-Driven (MD) methods [10,11], Data-Driven (DD) methods [12,13], and hybrid methods that combine MD and DD methods [14,15].
The LUT methods rely on pre-established tables derived from extensive experimental data to estimate the SOP of batteries based on corresponding data found in the tables. This method, straightforward in nature, depends heavily on a wealth of experimental data and requires thorough testing under various conditions to ensure accuracy.
To overcome the limitations of LUT methods, the MD based methods have been introduced. This method involves constructing mathematical models to infer internal battery states such as SOP from external characteristics like voltage and current. It enhances the transferability and accuracy of SOP estimation across different conditions. However, this approach demands precise modeling of batteries and parameter identification, directly influencing the accuracy and constraints of SOP estimation.
Although MD based methods can improve flexibility and accuracy in SOP estimation compared to LUT methods, they still rely on precise model construction and parameter identification. Therefore, the DD based methods have emerged to estimate SOP by inputting influencing factors such as voltage, current, temperature, and others. They extract implicit patterns from large-scale measured data, effectively predicting battery states without extensive knowledge of internal battery mechanisms. This reduces dependency on prior experimental data and enhances generalization capability. However, DD based methods require high data quality and quantity, lack intuitive physical interpretations, and may exhibit instability in rare conditions or data biases.
To maximize the advantages of both, researchers have begun employing hybrid methods that combine MD and DD based approaches. By integrating deep insights from physical models with the powerful capabilities of machine learning, these methods not only enhance the accuracy and reliability of SOP prediction but also significantly improve the model’s adaptability across diverse operating conditions. However, this approach also introduces challenges such as increased complexity and reduced computational efficiency. Therefore, current research efforts are focused on selecting efficient algorithms and optimizing computational techniques to strike a balance between model complexity and practical usability.
In the past decades, a significant number of researchers have conducted reviews on different state estimation methods [16,17]. In [18], a comprehensive summary of the current SOC estimation techniques is provided, which not only discusses the technical details and application status but also looks ahead to future research directions, particularly focusing on how real-time reliable SOC estimation can be achieved through technological innovation and the integration of advanced sensor technologies. While many researchers have effectively analyzed and compared methods for estimating the SOC of Li-ion batteries in the past, they have seldom reflected the SOC estimation techniques of lithium-ion battery systems from a control-oriented perspective. In order to fill this gap, [19] predominantly reviews various methods for lithium-ion battery SOC estimation from the perspective of control engineering. These methods are widely categorized into open-loop, closed-loop, and hybrid-based approaches. The applications of deep learning in SOC estimation for Li-ion batteries have been reviewed in [20], discussing the efficacy of various deep neural networks, including fully connected neural networks, recurrent neural networks, and convolutional neural networks, and also explores how advanced applications such as transfer learning can enhance estimation performance. After delving into the SOC estimation techniques of batteries and their advancements, the importance and complexity of battery SOH evaluation naturally emerge. In addition, the internal and external factors influencing battery cycle life have been performed extensively analyzed as well as the predictive methods for battery SOH and remaining useful life (RUL) in [21], while elaborating on the advantages and limitations of various life assessment methods. In [22], the definitions of SOH are summarized, and discusses various SOH evaluation methods have also been discussed in detail, along with analyzing the current technical issues and future development trends. Moreover, a comprehensive review of SOH estimation techniques is provided, which covers the battery degradation mechanisms, SOH definition, modeling methods, evaluation standards, and the classification and discussion of various estimation methods. It also proposes a degradation model considering battery relaxation effects, and concludes with a summary of the current status, challenges, and future development directions of SOH estimation. For assessing retired battery health status, a detailed evaluation of the main methods was provided in [24] to discuss their advantages and disadvantages. Finally, the article points out the technical challenges and prospects, emphasizing the potential role of big data analysis, non-intrusive diagnostic techniques, and artificial intelligence in the safe and sustainable management of retired batteries. In addition, several SOH assessment methods were reviewed in [25], including direct measurement, model-based methods, data-driven methods, and hybrid methods. [26] focuses on the SOH assessment methods for second-life Li-ion batteries, discussing direct and indirect assessment techniques, health indicator identification, required equipment, timelines, accuracy of SOH estimation, and the level of effort required to compute SOH. While both SOC and SOH have been widely studied, to the best of our knowledge, there is a relative paucity of systematically reviewing the SOP methods for batteries. Therefore, this article aims to fill this gap in the field by comprehensively analyzing existing literature, thoroughly discussing battery SOP estimation methods, technical challenges, and future research trends, providing a new perspective and insights to ensure the efficient and safe operation of battery systems.
The rest of this article is organized as follows. Section 2 introduces the current definition of SOP. Section 3 summarizes the commonly used equivalent circuit model (ECM) in MD based SOP estimation. Section 4 discusses various outstanding SOP estimation approaches developed in literatures. Finally, the conclusion and future recommendations are summarized in Section 5.
STATE OF POWER DEFINITION
SOP is an estimated value of battery power, defined as the peak power that the battery can provide at a certain moment, which quantifies the short-term peak power capability released or absorbed by the power system of electric vehicles [27,28]. It is important that the SOP of the battery plays a crucial role in extreme situations such as when an electric vehicle needs to climb a steep hill or perform emergency braking within a few seconds, which can determine whether the electric vehicle can successfully complete these tasks and continue to operate without shutting down [29,30]. Therefore, accurate SOP estimation is one of the important evaluation indicators for BMS [31], which helps provide peak power for the battery and provides a good guarantee for the efficient and safe operation of electric vehicles [32]. Since actual driving conditions for vehicles involve acceleration, braking, climbing, etc., and they are not instantaneous values, it is more meaningful to study peak power sustained over a certain time interval [33].
Currently, SOP is defined as the peak power that a battery can provide or absorb over a certain time span, which is called the peak charging/discharging time. The peak power output from the battery is called discharging SOP, while the peak power input to the battery is called charging SOP. For estimating sustained peak power, choosing the appropriate duration is also crucial. Different time spans will lead to various estimations of SOP, affecting the accurate prediction of the battery’s instantaneous peak power. A shorter time span may better capture the battery’s transient response as it is closer to the real-time state of the battery. Such an estimation can more accurately predict the battery’s maximum power output or absorption capacity in a short period, which is crucial for dealing with abrupt changes in loads or driving conditions. Conversely, a longer time span may smooth out the battery’s peak power output as it considers the average battery performance over a period. Such an estimation may be more suitable for analyzing driving behavior over an extended time range and assessing battery health conditions. Therefore, selecting the appropriate duration is essential for accurately estimating the battery’s SOP, and it requires a balance based on specific application scenarios and requirements.
According to our investigation, some expressions used for the definition of SOP in reference papers are summarized in Table 1. It is more meaningful to estimate the power capability based on the current battery conditions without violating the preset design limits for battery voltage, SOC, or current [34]. Accurate power capability prediction can not only ensure safety but also adjust driving behavior and optimize battery energy consumption. The SOP definitions listed in Table 1 are as follows: The SOP definition derived in [35] is based on the ECM, which calculates the peak current under voltage constraints by incorporating the voltage drop due to diffusion resistance, thus improving the accuracy of SOP under voltage constraints. The definitions in [36] and [37] calculate the peak charging and discharging currents under SOC, voltage, and current constraints, with the charging and discharging currents defined as sustaining peak power over a certain period of time. The definition in [12] estimates the total SOP of the battery pack, also calculating peak power under SOC, voltage, and current constraints. The definition in [38] expresses the peak power under voltage constraints for batteries at different levels of aging, laying the foundation for further research on SOP estimation under SOH constraints.
Commonly used definition of power state in literatures
ECM FOR SOP
To accurately estimate the SOP, constructing an accuracy ECM is a prerequisite for the MD and DD based methods. ECM is a method of using circuit elements to simulate the internal characteristics of Li-ion batteries, which helps us to better understand the working principles and characteristics of batteries. Generally, some parameters in ECM, such as the power supply voltage, internal resistance, capacitance, and open circuit voltage of the battery are contained in ECM, which must be identified. When the ECM is constructed by using the estimated parameters, the dynamic response of the battery can be used for estimating SOP to thereby achieve better monitoring and managing the working state of Li-ion batteries. However, establishing ECM is a complex process that requires the selection of a physical model and parameter identification methods. Table 2 lists the currently widely used ECMs.
ECM for Li-ion batteries
There is a close relationship between SOP estimation and ECMs. Accurate ECMs can improve the accuracy of SOC estimation and peak current estimation under voltage constraints. The current methods for ECM parameter estimation include improved least squares method [39], pseudorandom binary sequence Pseudorandom Binary Sequence -Discrete Extended Kalman Filter battery identification technology [40], improved adaptive battery-state estimator [41], etc. Choosing an accurate method for parameter identification is crucial for the estimation of SOP. In article [39], a linear ECM was established to verify the relationship between SOP and duration. The parameter identification of the linear ECM is relatively simple and does not consider polarization phenomena, so the accuracy of SOP estimation needs to be improved. In article [42], a first-order RC ECM was established to estimate SOC using three Kalman filters (KFs), and online estimation of available capacity and internal resistance was performed based on multiple obtained parameters for SOP estimation. Additionally, articles [43] and [44] also used first-order ECMs, with article [44] replacing the ohmic resistance with a charge/discharge parallel resistor in the first-order ECM. In article [45], a second-order ECM was established for SOC estimation and SOP estimation. In article [33], a fractional-order ECM was established, replacing the polarization resistance in the integer-order ECM with a SPE capacitor, demonstrating higher accuracy and robustness of the fractional-order model (FOM). In [33], the FOM is utilized to estimate SOC, and FOM helps explain the battery’s nonlinear dynamics. Compared to the maximum relative error of approximately 10% in [46], the maximum relative error based on the SOP estimation using the fractional-order ECM is 1.34%, indicating that the accuracy of SOP estimation is further improved based on the fractional-order model. However, it introduces a complex computational burden for battery BMS in online SOC and SOP estimation. Therefore, a 1-RC model is chosen for online SOC and SOP estimation because it has a simple structure, fewer parameters, and a relatively balanced trade-off between model accuracy and computational efficiency. The first-order model is used for SOC estimation, and the secondorder model is used for SOC- open circuit voltage (OCV) curve fitting.
METHODS FOR SOP ESTIMATION
Accurate SOP estimation helps to ensure the safety of electric vehicles in emergency situations. Currently, SOP estimation methods can be mainly divided into test table lookup methods, model driven methods, and data-driven methods.
The LUT methods are based on pre-established lookup tables, which are derived from extensive experimental data. In practical applications, the corresponding data from these tables are used to estimate the SOP of batteries. This method is simple and straightforward but relies heavily on a large amount of experimental data and requires thorough testing under various operating conditions to ensure accuracy. The main steps are as follows: 1) Data collection: Record parameters such as voltage and current, along with the corresponding power state. 2) Establishing the lookup table: Include SOP values for different conditions. 3) Real-time lookup: Use the current operating parameters to find the corresponding SOP value from the lookup table.
The MD based methods use the ECM of the battery to simulate its internal characteristics. By establishing a mathematical model, it is possible to infer the internal state of the battery based on external characteristics such as voltage and current, and subsequently estimate the SOP. This method requires modeling the battery, and the accuracy of the model directly affects the precision of SOP estimation. The main steps can be summarized as follows: 1) Establishing the model based on the equivalent circuit characteristics of the battery. 2) Performing model parameter identification using experimental data. 3) Using the model and experimental data to calculate internal states (such as SOC, SOE, etc.). 4) Calculating internal states from the model and combining with the rated parameters of the battery to compute the SOP value.
The DD methods utilize machine learning, deep learning, and other techniques to learn the behavior patterns of batteries from a large amount of historical data. By training models, it is possible to extract patterns directly from the data to estimate the SOP without fully understanding the internal physical mechanisms of the battery. This method can handle complex nonlinear relationships but requires a substantial amount of high-quality training data, which main steps can be summarized as 1) Collecting a large amount of battery operation data, including voltage, current, historical SOP, etc. 2) Preprocessing the data, including cleaning and normalization. 3) Training the model using machine learning algorithms and perform model validation. 4) Inputting real-time data into the trained model to predict the SOP at the current moment. The MD and DD methods involve using data-driven approaches to predict SOP constraint conditions, and then combining them with model-based methods to improve the accuracy of SOP estimation.
In summary, the LUT methods are simple and intuitive, making them easy to implement. However, they cannot dynamically adapt to battery aging or environmental changes. MD methods require the design of complex mathematical derivations, modeling, and parameter identification, making them complex, but they can dynamically adjust to accommodate battery aging. The complexity of DD methods mainly lies in the need for a large amount of high-quality historical operational data, involving long-term data collection. The fusion of DD and MD methods is relatively complex because it requires not only parameter identification but also neural network training, although the accuracy is higher. Each method has its own advantages, disadvantages, and levels of complexity. The appropriate method should be selected based on the actual situation. As shown in Fig. 1, this is a classification of SOP estimation methods.
SOP estimation methods.
Look-up table method
At present, some effective test and lookup methods for SOP estimation have been developed by using different principles, such as the United States Advanced Battery Consortium (USABC) method, Hybrid Pulse Power Characteristic (HPPC) method, JEVS Japan Electric Vehicle Society (JEVS) method, and China’s 863 Plan method. The essence of the USABC method is to determine the battery’s continuous (30 s) discharge power capability at different depths of discharge (DOD) when the battery’s OCV value is at 2/3, with the power value calculated at 80% DOD being a particularly important data set and defined as the USABC power target peak power for the battery. The JEVS method, also known as the power density testing method, uses a “0-10C” series of currents to test the battery’s power capability. In the China 863 Plan method, the battery’s peak power is the product of voltage and current at 0.1 s after each stage of pulse discharge ends, and the battery’s average power is the quotient of pulse discharge energy and discharge time for each stage [47,48]. In all, a characteristic map can be created by utilizing available power and the relationships between SOC, SOH, temperature, and power pulse time scale for the LUT based method.
Among those aforementioned methods, the HPPC method is relatively simple and widely used. HPPC is a characteristic test used to evaluate the pulse charging and discharging performance of a dynamic battery. The HPPC method involves using a test system that includes discharge and feedback pulses within the charging and discharging range of the battery to determine its dynamic power capability [49]. In this test, the battery is first discharged or charged with a constant current, and the voltage variation is observed. Subsequently, a series of current pulses are applied at different SOC values, and the voltage response is recorded. By determining the charge-discharge power value when the battery voltage reaches a specified voltage under different temperature and SOC constraints, a large number of experimental test values are used to create a map [50]. It should be noted that the HPPC test method is mainly used to determine the static peak power in laboratory environments and is not suitable for estimating continuous peak power available for consecutive sampling intervals. Additionally, this method ignores constraints such as current, SOC, battery aging, and polarization phenomena. Therefore, in practical applications, it may be necessary to combine other methods and strategies to further improve the accuracy of SOP estimation for batteries. However, currently obtained SOP values from HPPC testing can still serve as reference values. In [45,46], more accurate reference SOP values under both static and dynamic load conditions can be obtained by designing incremental pulse tests, which helps improve the assessment accuracy of battery performance.
Model-driven methods for SOP estimation
Compared with the LUT method, the MD based SOP estimation methods as the most promising approach have been widely applied studied in several literatures. In general, the SOP estimation via MD method is particularly performed in the context of multiple constraints with a certain ECM. The commonly used constraint types include the current, voltage, and SOC constraints. The voltage constraint can be calculated based on the ECM and represents the peak power under the voltage constraint. Similarly, the SOC constraint is estimated using the ECM to obtain the peak power under the SOC constraint. Fig. 2 illustrates the SOP estimation under multiple constraints using the ECM. In the following subsections, the SOP estimation methods under different constraints are detailed summarized.
Model-driven estimation of SOP.
SOP estimation under SOC constraint
In many applications, the power output of a battery must meet certain constraints or requirements. SOC directly influences the available energy and charge status of the Li-ion battery, thereby affecting the battery’s output power and performance. By accurately estimating the battery’s SOC, it is possible to better predict the level of output power of the battery, ensuring that the battery can stably and efficiently provide the required power output during operation. Currently, for multi-constraint based SOP estimation, the SOC constraint is crucial for SOP. On the one hand, when the battery’s SOC is low, its ability to provide power may be limited. In such cases, to prevent excessive discharge and damage to the battery, it is necessary to restrict the SOP to a lower level. On the other hand, when the battery SOC is high, it can be expected to provide higher power output. In this case, the battery can be ensured to operate at its maximum potential while avoiding exceeding its design range to prevent overheating or damage. By considering the combination of SOC and SOP, the power output of the battery can be optimized.
Setting the SOC constraint is based on the battery’s state of charge, ensuring that the battery operates within a specific SOC range to maintain its performance and lifespan. The SOC of the battery should have maximum and minimum values set during the charging and discharging processes to prevent overcharging and over-discharging. In experiments, the SOC threshold for charging SOCmax is generally set to 0.9, while the SOC threshold for discharging SOCmin is set to 0.1. The method calculates the peak current based on the definition of SOC, and the peak current under SOC constraint can be expressed as
Among them, SOCk represents the SOC at time k, η denotes the Coulombic efficiency, Cn is the rated capacity, and L*Δt represents the peak charge discharge time (where L is the number of samples, Δt represents the sampling interval time).
According to formula (1), one can know that the peak current at the given time can be calculated based on the SOC at that time. In addition, the peak voltage at that time can also be determined by using a certain ECM, then an estimation of the SOP can ultimately be achieved by multiplying the peak current and peak voltage. Therefore, the accuracy of SOC estimation determines the precision of peak power estimation, as can be seen from formula (1). Recently, SOC estimation is commonly performed using Kalman Filter (KF), Extended Kalman Filter (EKF), Unscented Kalman Filter (UKF), particle filter, H-Infinity (H∞) filtering, and other methods [51]. In [52], the dual square root unscented Kalman filter is adopted for SOC state estimation. Dong et al. [53] proposed the use of a KF for SOC estimation in Li-ion batteries, and experiments showed that the KF method can provide accurate SOC estimation and voltage prediction. The accuracy of SOC estimation is crucial for SOP estimation. One reason for the low accuracy of SOP estimation is the inadequate precision of SOC estimation. Traditional SOC estimation methods suffer from error accumulation over prolonged usage, leading to decreased accuracy. Model-driven SOC estimation requires complex parameter calibration, and inaccurate parameter settings can affect the accuracy of SOC estimation. In the future, improving SOC estimation algorithms and optimizing model parameter calibration methods will further enhance the accuracy and stability of SOC estimation for Li-ion batteries, thereby improving SOP estimation accuracy.
SOP estimation under voltage constraint
Voltage and power exhibit a strong correlation. During battery discharge, the battery’s open-circuit voltage varies with SOC, and when the battery delivers high output power, the voltage correspondingly decreases. Therefore, in SOC estimation, it is imperative to account for the influence of voltage and utilize it as a constraint to limit the battery’s output power. To ensure the safe operation of Li-ion batteries, both limits of upper and lower cutoff voltages should be set. Here is an example derivation using a second-order ECM to calculate the peak current under voltage constraint, represented by the following state-space equation formula (2). UL is the terminal voltage, Uoc is the open circuit voltage, I is the current, R0 is the ohmic resistance, U1 is the voltage across R1, and U2 is the voltage across R2.
Data driven estimation method.
At any k+1 moment, the terminal voltage of the Li-ion battery, pack is expressed as: At any moment k+1, the terminal voltage of the Li-ion battery pack can be expressed as formula (3). UL(k) is the terminal voltage at time k, Uoc (SOCk+1)is open circuit voltage corresponding to SOC at time k+1, IT, k+1 is the current at time k+1, under duration T, U1(k) is the voltage across R1 at time k and U2(k) is the voltage across R2 at time k.
The voltage state value at any time can be expressed as the superposition of zero input response and zero state response, as shown in formula (4). The terms τ1 and τ2 represent the time constants of capacitors C1 and C2, respectively.
Taylor expansion of SOC and OCV functions:
Based on formulas (4) and (5), calculate the terminal voltage Ut(k) to obtain formula (6).
The expression for the peak current at a certain moment is given by formula (8).
Due to the importance of continuous peak power, voltage discretization has been carried out, as shown in formula (9).
Due to the fact that discharge current is a continuous function, SOC can be expressed as a function of current (Zk represents the function equation with current IL,k as the independent variable), and the open circuit voltage can be expressed using a Taylor series expansion as in formula (10).
At any time, the terminal voltage is expressed as in formula (11).
The expressions for charge and discharge currents under voltage constraints, as in formula (12).
Formula (12) represents the calculation of peak current under voltage constraint, and formula (13) represents the value of sustained dynamic peak current under multiple constraints (SOC, voltage, current).
Currently, the multi-constraint SOP estimation primarily considers constraints such as SOC constraint, voltage constraint, and current constraint. The following text will also summarize the influence of SOH constraint, SOE constraint, and temperature on SOP.
SOP estimation under SOH constraint
SOH refers to the health or lifespan status of a battery or other rechargeable device. It represents the current state of the battery, including its capacity and available energy. SOH is one of the important parameters in battery management, and aging batteries can certainly affect the accuracy of SOP estimation results. Sun et al. [54] developed a data-driven joint estimator for battery SOC and SOP based on the adaptive extended Kalman filtering algorithm. At the same time, in order to achieve accurate estimation of SOC and SOP throughout the calendar life of the battery, they discussed and studied the necessity of updating the model parameters with the lowest computational burden. The experiment shows that for aging batteries, online parameter updates, capacity and ohmic resistance parameter updates are required, which can significantly improve the accuracy of battery system control and SOP estimation. Chen et al. [37] proposed an online estimation of the power state of Li-ion batteries, for battery aging issues. In order to make parameter identification more accurate, a first-order ECM was used to estimate SOP. The polarization resistance and Ohmic resistance in the first-order ECM come from Li-ion batteries, with different SOH. According to the experimental graph in the text, SOP decreases with the decrease of SOH, Ohmic resistance decreases with the decrease of SOH, τ and polarization resistance remains almost unchanged. Therefore, aging batteries indirectly affect the voltage constraint at the SOP end. M. J. Esfandayari et al. [55] designed a control system based on model free fuzzy logic to compensate for changes in battery aging status and perform SOP estimation. By comparing the SOP estimation of the aging status without compensation and after compensation, it was found that there was a significant violation of rules in the uncompensated aging battery, while the compensated aging battery did not exceed the current and voltage limits. In [42], a first-order ECM was used for SOP estimation, and the battery parameters were identified using a recursive least squares algorithm with variable forgetting factors. Three KFs were used to estimate the battery SOC, and the maximum available capacity and internal resistance were estimated using the forgetting factor recursive least squares algorithm. Then estimate the peak current and power under the composite constraints of estimated capacity and internal resistance. Compared to peak power under SOC and voltage constraints, the emphasis of peak power under SOH constraints lies in the fact that the ohmic resistance parameters of differently aged batteries may change, leading to inaccurate SOP values under voltage constraints. Therefore, online identification of the ohmic resistance parameters in the ECM model is necessary, making the SOP estimation more accurate. By analyzing the impact of SOH on SOP, future SOP estimation should also consider the online parameter identification of battery capacity parameters and Ohmic resistance in ECM. The updated parameters make the SOP estimation of aging batteries more accurate. To understand better how to estimate SOP to optimize a tradeoff performance and life, how to age the power battery use these aging models for some more advanced SOP estimation algorithms that take advantage of will be the topic of the future work. Meanwhile, the thermal and degradation should be also constrained for more reasonable SOP estimation [56,57].
SOP estimation under the constraint of SOE
SOE refers to the SOC of a battery in terms of its available energy relative to its maximum storage capacity. For Li-ion batteries, SOE usually refers to the current charging or discharging state. The current energy state also affects the estimation of SOP. Xu et al. [58] proposed a new method for estimating the energy and power states of Li-ion batteries, based on a multi-time scale filter. They analyzed the maximum available capacity and OCV characteristics of the battery at different working temperatures. Additionally, they applied the battery’s ECM to analyze its dynamic performance, and used Particle Swarm Optimization-Unscented Kalman Filter (PSO-UKF) to identify the battery model parameters and estimate the battery SOE and SOP at each micro time. However, they did not consider the impact of SOE estimation on SOP estimation, nor did they use SOE constraints to estimate SOP. In [59], a new model-based joint estimator for battery energy and power capability states was developed using Augmented Unscented Kalman Filter (AUKF). The paper mentioned that the peak power prediction based on SOE is mainly used to avoid the dangers of overcharging and over-discharging of the battery. Guo et al. [44] proposed an enhanced multi-constraint SOP estimation algorithm for Li-ion batteries, in electric vehicles with a 120-second prediction window. The multiple constraint conditions were introduced based on SOC, voltage, and current, and SOE constraints were added. They also proposed a voltage-constrained SOP estimation algorithm based on regression to enhance the accuracy of peak current estimation under end voltage limitations. Since Li-ion batteries, serve as power and energy storage systems in electric vehicles, SOC [60] is not sufficient to intuitively reflect and restrict the battery’s power capability. Therefore, an SOP estimation algorithm with SOE constraints was derived.
In [61], the expression for peak current under SOE constraint is shown in (14).
Among them,
Under the above constraints, the peak power for the duration can be calculated by using the peak current multiplied by the peak voltage corresponding to this time period. In addition, under SOE constraints, it is more involved in estimating SOP in the low SOC region. However, the SOC constraint still plays a crucial role in SOP estimation and cannot be completely replaced by SOE constraint. The combined effect of these two constraints results in more accurate SOP estimation.
The model-driven SOP estimation method involves building an ECM and then identifying the parameters of the model. Estimation of constrained variables is conducted based on the ECM, thereby estimating peak current under multiple constraints and obtaining the SOP value.
SOP estimation based on data-driven methods
Due to the need for establishing and adjusting the battery model, as well as the potential for significant errors introduced by the model, the MD based methods may be disadvantaged. In contrast, the DD based methods can circumvent these issues and offer greater flexibility, which is a method that treats batteries as black boxes, without considering the internal reaction mechanisms and characteristics of the battery. By using data analysis and machine learning techniques, the estimated SOP is used as the output of the model, and the influencing factors are used as the input to achieve SOP estimation in the testing and model training of a large amount of data.
However, DD based methods heavily rely on the quality and quantity of training data, and directly collecting continuous reference SOP data is often difficult. Therefore, these methods are rarely used in online SOP estimation. In addition, SOP estimation is carried out under multiple constraints, and accurate estimation of SOP requires reference values calculated based on expressions such as SOC estimation, terminal voltage, and current. Therefore, the main method is based on MD methods for SOP estimation, while DD methods are relatively difficult to evaluate in practice. At present, battery SOP estimation based on DD methods mainly includes Backpropagation (BP) neural network method, Adaptive Neuro Fuzzy Inference System (ANFIS) model method [62], Support Vector Machine (SVM) method, and Genetic Algorithm (GA) [12].
The BP neural network is a multi-layer feedforward network trained using the error backpropagation algorithm, consisting of an input layer, a hidden layer, and an output layer. When using the BP neural network for SOP estimation, the input layer is typically set to voltage, temperature, SOC, etc., while the output layer is set to the SOP of the battery. The error is propagated back along the original circuit, and the weights and thresholds of the nodes in each layer are adjusted using gradient descent until the output layer achieves the desired output. The structure of the BP neural network for estimating SOP is shown in Fig. 4.
Estimating SOP using BP neural network.
The BP neural network selects the appropriate number of input, hidden, and output layers according to the modeling requirements, uses hidden layers to connect input and output layers, and establishes node weights between layers. Then, the normalized training samples are sent to the output layer to train the network. Finally, adjust the network weights based on the error until the error reaches the allowable range.
The ANFIS is a new type of fuzzy inference system structure that combines fuzzy logic and neural networks. A learning system of neural networks is added to the fuzzy system to automatically extract rules from input and output sample data, forming an adaptive neural fuzzy controller. The ANFIS network structure of the first-order T-S fuzzy inference system is shown in Fig. 5, and the framework of the ANFIS network based SOP estimation can be seen in Fig. 6.
Structure of Adaptive Neural Fuzzy System.
ANFIS model with 3 inputs and 1 output for SOP prediction.
The overall prediction process of the system includes data acquisition, model training, and simulation verification. Using experimental methods, first obtain the peak power and battery internal resistance at different temperatures and SOC. In the model training stage, grid generation and subtractive clustering methods can be used to partition the fuzzy intervals of variables, and a single BP neural network and comprehensive training method can be used for model training, which can reduce complexity and improve accuracy. Finally, simulation verification was conducted, and the method solved the local optimal problem of a single BP neural network with high prediction accuracy. In [62], Fuzzy Inference System (FIS) and Artificial Neural Networks (ANN) are combined into the framework of ANFIS. During battery aging, the system can not only provide accurate power prediction at room temperature, but also at lower temperatures, which are the most challenging for power prediction.
The SVM is a machine learning algorithm developed based on statistical theory. It has strong learning capabilities and can effectively map nonlinear functions. This method consists of three layers: the input layer, the hidden layer, and the output layer. The input layer stores data such as voltage, current, temperature, SOC, etc. The hidden layer selects an appropriate kernel function before training the data, and the output layer represents the regression prediction function (SOP value). Reference [42] proposed a Grid Search SVM (GS-SVM) model based on data statistical characteristics for analyzing temperature, internal resistance, SOC, etc. It improves the computational efficiency of SVM and ensures the search for global optimal values by introducing the grid search method. This enhances SOP prediction accuracy.
The SVM algorithm performs classification and regression by finding an optimal hyper-plane, avoiding the traditional process of induction to deduction and simplifying the problem more efficiently. The computational difficulty mainly depends on the number of support vectors rather than the dimensionality of the sample space, effectively avoiding the “curse of dimensionality”. However, the SVM algorithm lacks theoretical guidance in selecting kernel functions and constructing parameters, relying on practical experience and methods such as cross-validation to determine suitable parameters. Additionally, when the training sample size is large, the computational complexity of the algorithm significantly increases, making it difficult to implement. In summary, the SVM algorithm is an efficient approach for simplifying classification and regression problems, but it still faces challenges in kernel function and parameter selection, as well as handling large-scale training samples.
GA is an optimization algorithm that simulates natural selection and genetic mechanisms. It gradually searches and optimizes solutions by simulating the process of biological evolution. GA has a wide range of applications, particularly suitable for complex optimization problems and cases where traditional methods cannot be applied. It is employed in fields such as combinatorial optimization, parameter optimization, and machine learning, effectively searching for and finding better solutions.
Lu et al. [12] proposed a GA-based SOP estimation algorithm to handle long-term scale estimation in power management applications. They also analyzed the sensitivity coefficient (δ) of SOC accuracy on SOP estimation and established the correlation with changing SOH, estimation time scale and δ. Compared to the traditional Taylor method, the proposed GA-based estimation method can improve SOP estimation accuracy by up to 7.2% in certain cases. If temperature constraints and SOE constraints are included, the SOP estimation method will be further enhanced.
The DD non-parametric model estimation method does not consider the internal reaction mechanism of the battery, and can estimate the SOP of any battery, with wide applicability. However, it requires a large amount of experimental data as a prerequisite, and the selected samples during training will have a significant impact on the results. Compared to other methods, the DD based method is computationally complex and requires high hardware requirements, with a long training time in the early stages.
Data driven and model driven fusion methods for estimating SOP
The DD and MD fusion method mainly utilizes DD methods for SOC estimation or dynamic parameter variation of resistance, and combines it with MD methods to accurately estimate SOP under constraint conditions. In [14], a model fusion method (MFM) that jointly estimates SOC and SOP is used. Particle swarm genetic algorithm is used to identify a 2-RCCPE FOM model for SOC estimation, while Dual Extended Kalman Filter (DEKF) algorithm is used to identify a 1-RC model. The SOP is estimated online through model fusion. The accuracy of li-ion battery SOP estimation mainly depends on accurately estimating influencing factors such as SOC, terminal voltage, SOE, SOH, temperature, and capacity, and then calculating the SOP value. For SOC estimation, data-driven methods can be used for accurate estimation. In [15], a new M-1 structure Bidirectional Long Short-Term Memory-Recurrent Time Series Smoothing (BiLSTM-RTSS) algorithm is proposed, which uses improved sliding window M-1 technology to effectively fit current, voltage, temperature, and SOC. The RTSS algorithm is used to update the SOC estimation of BiLSTM to improve the accuracy and speed of SOC estimation. Under Beijing Bus Dynamic Stress Test (BBDST) and HPPC conditions, the maximum estimation errors of SOC are 0.0184 and 0.0229, respectively, and the SOP estimation is more accurate and robust. In [63], a long short-term memory (LSTM)-based method is used to predict the SOC of Li-ion batteries, and a multi-parameter constraint method is used to obtain estimated SOP values within a certain period of time. All of the above are based on data-driven methods for accurate SOC estimation, and then SOP estimation is performed under multi-parameter constraints.
The voltage constraint in SOC estimation mainly depends on the establishment of an ECM and parameter identification. In [44], a new regression algorithm called Model Parameter Forward Prediction-Based Algorithm (RBA) is used for voltage-constrained SOP estimation. A reference SOP is obtained through incremental pulse testing, and the accuracy of static and dynamic SOP estimation is validated. In [64], R. Guo et al. establish a feed-forward neural network (FFNN) with SOC, discharge rate, and pulse operation time as inputs. The polarization voltage of the battery is characterized by simulating current-induced polarization resistance. Experimental results validate the effectiveness of the constructed FFNN in reproducing the nonlinearity of battery polarization characteristics at high discharge rates and demonstrate a significant improvement in SOP estimation accuracy. In [40], a DEKF algorithm excited by Pseudo-Random Binary Sequence (PRBS) battery stimulus is used for parameter identification of a second-order ECM to provide accurate SOC estimation. Similarly, in [41], an improved Adaptive Battery State Estimator (ABSE) is used for more accurate identification of battery model parameters. The error source of the ABSE algorithm lies in the fact that Rp identified during battery charging and discharging at high currents is much higher than the actual value, and ABSE does not consider the influence of load current on the parameters of the ECM. In [5], global optimization is performed on the parameters of a fractional-order circuit model to identify two resistor constant-phase element networks representing battery internal dynamics at different time scales. By fixing the parameters of the first resistor constant-phase element network with slow dynamics and allowing online adaptation of the second resistor constant-phase element network, partial adaptive fractional-order modeling is achieved. Online SOC estimation is implemented using the adaptive EKF algorithm, and an iteration approximation algorithm based on UKF is designed for SOP estimation. The identification of model state and parameters is crucial for power state estimation [43].
In SOP estimation, not only ECM can be constructed for SOP estimation but also electrochemical [65] and physical methods can be employed. In [66], a new physico-based battery power estimation method is proposed, which considers internal constraints such as lithium plating and thermal runaway to estimate SOP. In [67], the Coupled Electro-Thermal (CET) model is combined with Linear Parameter-Varying Model Predictive Control (LPV-MPC) to compute the power dynamic SOP limits of Li-ion batteries. Currently, Physics-Informed Neural Network (PINN) is gaining popularity. PINN is a method that combines physical knowledge and neural networks to solve physical problems. The main idea is to embed known physical equations or laws into the neural network model to improve its accuracy and generalization ability. The core idea of PINN is to train a neural network model by minimizing the residual of physical equations. These physical equations can be partial differential equations, integral equations, or other equations that describe physical phenomena. During training, the model automatically adjusts its network parameters to make the prediction results consistent with the observed data and physical equations. In the future, PINN can be used for parameter identification of Li-ion batteries, to more accurately estimate SOC and SOP.
Temperature also has an impact on SOP estimation. The internal chemical reactions of Li-ion batteries will be accelerated when the temperature is too high, and correspondingly the resistance and losses within the battery will be increased, resulting in more heat generation and reducing the peak power of the battery. On the contrary, if the temperature is too low, the internal chemical reactions of Li-ion batteries will slow down, leading to a decrease in the discharge rate of the battery, which also affects the peak power of the battery. In [68], an adaptive method is proposed for online SOC and SOP co-estimation of batteries considering temperature. To adapt to different loads, SOC, and temperatures, a regularized moving level estimation algorithm embedded with temperature is used to adjust the contribution coefficients of each sub-model online. Additionally, a Fractional-Order Multi-Model Proportional Integral Observer (FO-MM-PIO) is designed for SOC estimation, utilizing H-infinity criteria to achieve fast convergence and disturbance rejection. Furthermore, the Butler-Volmer equation is simulated to describe the nonlinear charge transfer dynamics of the battery under heavy load conditions. An iterative approximation algorithm is then derived to estimate the battery SOP with high fidelity. In [36], a fusion model considering voltage, current, and SOC is also developed, incorporating parameter calibration considering the influence of temperature, to achieve online tracking of power estimation state. Experimental results show that under BBDST conditions, the voltage error of this algorithm is less than 0.08 V, and the online power calculation error is less than 6 W. In [69], a model based on the Butler-Volmer (BV) equation is extended, which captures the relationship between charge state and useful charge state, and its implementation in predicting SOP at different temperatures. By precisely controlling the voltage limit, the actual 10s SOP of the battery is obtained using constant current pulses. Experimental results demonstrate the robustness and accuracy of the model. The model indicates that temperature constraints can enhance the accuracy of SOP estimation, and in the future, temperature constraints should be incorporated, and an algorithm with a temperature factor should be designed for accurate SOP estimation. In [70], special attention has been given to the thermal effects of the battery by establishing a control-oriented electro-thermal model to track the battery’s temperature state in real-time, integrating it into the continuous peak power evaluation. This method of evaluating continuous peak power with temperature constraints improves the accuracy of SOP estimation, significantly reducing power decline due to overheating, especially during the charging phase. In the future, it is necessary to consider better modeling to describe battery behavior in low-temperature environments.
The summary above indicates that the estimation of SOP is primarily driven by data and model fusion. Data-driven approaches are mainly used for estimating SOC, SOE, capacity, etc., and algorithms for estimating temperature are incorporated to impose SOP constraints. Subsequently, estimation of SOP is still required using an ECM under various estimation constraints. Fig. 7 represents the SOP estimation using a fusion of DD and MD methods.
Data-driven and model-driven integration method for estimating SOP.
According to aforementioned analysis in subsections 4.1–4.4, the well-known existing SOP estimation methods have been summarized from their methodologies, advantages and disadvantages. Here, in order to more intuitively represent the characteristics of each method, a schematic representation of different methods is presented as shown in Fig. 8.
A schematic representation of different SOP estimation methods.
CONCLUSIONS AND FUTURE RECOMMENDATIONS
The current SOP estimation method is mainly based on an ECM to accurately estimate SOP under multiple constraint conditions, and the reference value for SOP is either an estimated value at the true SOC or an interpolated pulse method for SOP estimation and calibration. To improve the accuracy of SOP in the future, it is necessary to further improve the accuracy of SOC, SOE, and SOH estimation, while considering multiple limiting factors.
With the increasing demand for electrification in the automotive industry, the development of advanced BMS based on SOP estimation has received more attention. However, there has not been a comprehensive review of SOP research alone. Thus, this paper summarizes existing methods for estimating SOP in electric vehicles. Furthermore, it discusses the advantages and limitations of SOP limiting factors and estimation methods. Therefore, this review collects extensive research results on current SOP estimation methods. Through a literature review and method comparison, it has been found that considering multiple factors such as SOC, temperature, voltage, current, SOH, and capacity can increase the accuracy of SOP estimation. The accuracy of the corresponding SOC estimation results directly affects the SOP estimation value. In addition, the impact of SOE and SOH on the estimation results of SOP is also summarized. Moreover, recent research has introduced reduced-order models in BMS applications. Reduced-order models simplify system representations, thereby significantly improving computational speed while maintaining reasonable accuracy. However, these models may face limitations in accuracy, particularly under complex operating conditions. A key challenge lies in balancing the trade-off between speed and accuracy. In practical applications, it is crucial not only to consider the computational speed and accuracy of these models but also to provide estimated values under different scenarios to better assess their practical effectiveness. For the future improvement of SOP estimation accuracy, accurate ECM can be constructed through parameter identification methods, and multiple limiting factors can be considered for estimation. This provides new ideas for the development of battery management.
Thus, future studies should focus not only on the performance of reduced-order models under various conditions but also on providing estimations in different scenarios. This will enhance the understanding and application of these models. Additionally, combining extensive real-world condition tests with comprehensive data collection will help verify and refine the accuracy and reliability of these models. Furthermore, by conducting extensive experimental tests and real-world condition tests to build a more comprehensive database, we can provide more representative data samples for li-ion battery, SOP estimation and contribute to the advancement of big data. As li-ion battery, technology continues to progress, the accuracy of SOP estimation will have a more significant impact on the performance and safety of electric vehicles and energy storage systems. Therefore, by integrating various methods and compensating for their respective shortcomings, it will be possible to achieve accurate and fast SOP estimation for Li-ion batteries, providing more precise references for power matching in electric vehicles, thereby ensuring the safe and reliable operation of electric vehicles and energy storage systems.
Notes
CREDIT AUTHORSHIP CONTRIBUTION STATEMENT
Wentao Ma: Conceptualization, Funding acquisition, Methodology, Supervision, Writing - review & editing, Supervision. Shizhuo Ren: Methodology, Data curation, Formal analysis, Conceptualization, Software, Writing - original draft. Peng Guo: Data curation, Investigation.
DECLARATION OF COMPETING INTEREST
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
The work was supported by the National Key R.D Program of China (2021YFB2401900) and the National Natural Science Foundation of China (62473308).
