Model reduction of fractional impedance spectra for time–frequency analysis of batteries, fuel cells, and supercapacitors

Joint time–frequency analysis is an emerging method for interpreting the underlying physics in fuel cells,batteries,and supercapacitors.To increase the reliability of time–frequency analysis,a theoretical correlation between frequency-domain stationary analysis and time-domain transient analysis is urgently required.The present work formularizes a thorough model reduction of fractional impedance spectra for electrochemical energy devices involving not only the model reduction from fractional-order models to integer-order models and from high-to low-order RC circuits but also insight into the evolution of the characteristic time constants during the whole reduction process.The following work has been carried out:(i)the model-reduction theory is addressed for typical Warburg elements and RC circuits based on the continued fraction expansion theory and the response error minimization technique,respectively;(ii)the order effect on the model reduction of typical Warburg elements is quantitatively evaluated by time–frequency analysis;(iii)the results of time–frequency analysis are confirmed to be useful to determine the reduction order in terms of the kinetic information needed to be captured;and(iv)the results of time–frequency analysis are validated for the model reduction of fractional impedance spectra for lithium-ion batteries,supercapacitors,and solid oxide fuel cells.In turn,the numerical validation has demonstrated the powerful function of the joint time–frequency analysis.The thorough model reduction of fractional impedance spectra addressed in the present work not only clarifies the relationship between time-domain transient analysis and frequency-domain stationary analysis but also enhances the reliability of the joint time–frequency analysis for electrochemical energy devices.

Characterization and identification towards dynamic-based electrical modeling of lithium-ion batteries

Lithium-ion batteries are widely recognized as a crucial enabling technology for the advancement of electric vehicles and energy storage systems in the grid.The design of battery state estimation and control algorithms in battery management systems is usually based on battery models,which interpret crucial battery dynamics through the utilization of mathematical functions.Therefore,the investigation of battery dynamics with the purpose of battery system identification has garnered considerable attention in the realm of battery research.Characterization methods in terms of linear and nonlinear response of lithium-ion batteries have emerged as a prominent area of study in this field.This review has undertaken an analysis and discussion of characterization methods,with a particular focus on the motivation of battery system identification.Specifically,this work encompasses the incorporation of frequency domain nonlinear characterization methods and dynamics-based battery electrical models.The aim of this study is to establish a connection between the characterization and identification of battery systems for researchers and engineers specialized in the field of batteries,with the intention of promoting the advancement of efficient battery technology for real-world applications.

A novel order-reduced thermal-coupling electrochemical model for lithium-ion batteries

Although the single-particle model enhanced with electrolyte dynamics(SPMe)is simplified from the pseudo-twodimensional(P2D)electrochemical model for lithium-ion batteries,it is difficult to solve the partial differential equations of solid–liquid phases in real-time applications.Moreover,working temperatures have a heavy impact on the battery behavior.Hence,a thermal-coupling SPMe is constructed.Herein,a lumped thermal model is established to estimate battery temperatures.The order of the SPMe model is reduced by using both transfer functions and truncation techniques and merged with Arrhenius equations for thermal effects.The polarization voltage drop is then modified through the use of test data because its original model is unreliable theoretically.Finally,the coupling-model parameters are extracted using genetic algorithms.Experimental results demonstrate that the proposed model produces average errors of about 42 mV under 15 constant current conditions and 15 mV under nine dynamic conditions,respectively.This new electrochemicalthermal coupling model is reliable and expected to be used for onboard applications.

Electrothermal Model Based Remaining Charging Time Prediction of Lithium-Ion Batteries against Wide Temperature Range

Battery remaining charging time(RCT)prediction can facilitate charging management and alleviate mileage anxiety for electric vehicles(EVs).Also,it is of great significance to improve EV users’experience.However,the RCT for a lithiumion battery pack in EVs changes with temperature and other battery parameters.This study proposes an electrothermal model-based method to accurately predict battery RCT.Firstly,a characteristic battery cell is adopted to represent the battery pack,thus an equivalent circuit model(ECM)of the characteristic battery cell is established to describe the electrical behaviors of a battery pack.Secondly,an equivalent thermal model(ETM)of the battery pack is developed by considering the influence of ambient temperature,thermal management,and battery connectors in the battery pack to calculate the temperature which is then fed back to the ECM to realize electrothermal coupling.Finally,the RCT prediction method is proposed based on the electrothermal model and validated in the wide temperature range from-20℃to 45℃.The experimental results show that the prediction error of the RCT in the whole temperature range is less than 1.5%.

Growth and inhibition of zinc anode dendrites in Zn-air batteries:Model and experiment

Zinc(Zn)-air batteries are widely used in secondary battery research owing to their high theoretical energy density,good electrochemical reversibility,stable discharge performance,and low cost of the anode active material Zn.However,the Zn anode also leads to many challenges,including dendrite growth,deformation,and hydrogen precipitation self-corrosion.In this context,Zn dendrite growth has a greater impact on the cycle lives.In this dissertation,a dendrite growth model for a Zn-air battery was established based on electrochemical phase field theory,and the effects of the charging time,anisotropy strength,and electrolyte temperature on the morphology and growth height of Zn dendrites were studied.A series of experiments was designed with different gradient influencing factors in subsequent experiments to verify the theoretical simulations,including elevated electrolyte temperatures,flowing electrolytes,and pulsed charging.The simulation results show that the growth of Zn dendrites is controlled mainly by diffusion and mass transfer processes,whereas the electrolyte temperature,flow rate,and interfacial energy anisotropy intensity are the main factors.The experimental results show that an optimal electrolyte temperature of 343.15 K,an optimal electrolyte flow rate of 40 ml·min^(-1),and an effective pulse charging mode.

Key Issues for Modelling, Operation, Management and Diagnosis of Lithium Batteries: Current States and Prospects

1 Introduction.With the continuous growth of the global population,the energy demand continues to increase.However,due to the dominance of fossil fuels in global energy and fossil fuels are non-renewable,it has led to the global energy crisis[1].Besides,the use of fossil fuels will generate a mass of air pollutants(e.g.,carbon dioxide,sulfur dioxide,etc.),which will cause serious environmental pollution,climate change[2],etc.To resolve the aforementioned issues,countries around the world have implemented a variety of measures hoping to fundamentally adjust the global energy structure and achieve sustainable development.Thereinto,“Paris Agreement”reached in 2015 under the framework of“United Nations Framework Convention on Climate Change”aims to control the increase in the average temperature of the globe to within 2°C below preindustrial levels,and thereafter to peak global greenhouse gas emissions as soon as possible,continuously decreasing thereafter[3].United Kingdom plans to reduce the average exhaust emissions of“new cars”to approximately 50–70 g/km by 20230,which is roughly half of what it is now[4].In addition,China proposed a plan at“United Nations General Assembly”in 2020 to peak carbon dioxide emissions by 2030 and strive to achieve carbon neutrality by 2060.It is a fact that the whole world is committed to changing the current energy structure,protecting the Earth’s ecology,and achieving global sustainable development[5].

Towards a business model for second-life batteries: Barriers,opportunities, uncertainties, and technologies

Electric vehicles(EVs) and the recent pandemic outbreak give cities a new trend to primarily private and shared mobility with low noise and less air pollution.Crucial factors for the widespread of EVs are the electrical charging infrastructure,driving range,and the reduction of the cost of battery packets.For this reason,there is a massive effort from manufacturers,governments,and the scientific community to reduce battery costs and boost sustainable electrical production and distribution.Battery reuse is an alternative to reduce batteries’ costs and environmental impacts.Second-life batteries can be used in a wide variety of secondary applications.Second-life batteries can be connected with off-grid or on-grid photovoltaic and wind systems,vehicle charging stations,forklifts,and frequency control.The present work aims to analyze the main challenges imposed on the reuse of batteries,the leading technologies for their reuse,and the different types of batteries in terms of their feasibility for second-life use.The main novelty of this work is the discussion about the barriers,opportunities,uncertainties,and technologies for the second life market.Here we summarize the present state of the art in reusing lithium-ion batteries discussing technical and economic feasibility,environmental impacts,and perspectives.The results show five business models that have been proposed in the literature,three types of markets for trading second-life batteries,and the main opportunities and barriers for each actor in the battery supply chain.

Physics-based battery SOC estimation methods:Recent advances and future perspectives

The reliable prediction of state of charge(SOC)is one of the vital functions of advanced battery management system(BMS),which has great significance towards safe operation of electric vehicles.By far,the empirical model-based and data-driven-based SOC estimation methods of lithium-ion batteries have been comprehensively discussed and reviewed in various literatures.However,few reviews involving SOC estimation focused on electrochemical mechanism,which gives physical explanations to SOC and becomes most attractive candidate for advanced BMS.For this reason,this paper comprehensively surveys on physics-based SOC algorithms applied in advanced BMS.First,the research progresses of physical SOC estimation methods for lithium-ion batteries are thoroughly discussed and corresponding evaluation criteria are carefully elaborated.Second,future perspectives of the current researches on physics-based battery SOC estimation are presented.The insights stated in this paper are expected to catalyze the development and application of the physics-based advanced BMS algorithms.

The initial stages of Li_(2)O_(2) formation during oxygen reduction reaction in Li-O_(2) batteries:The significance of Li_(2)O_(2) in charge-transfer reactions within devices

Lithium-oxygen batteries are a promising technology because they can greatly surpass the energy density of lithium-ion batteries.However,this theoretical characteristic has not yet been converted into a real device with high cyclability.Problems with air contamination,metallic lithium reactivity,and complex discharge and charge reactions are the main issues for this technology.A fast and reversible oxygen reduction reaction(ORR)is crucial for good performance of secondary batteries',but the partial knowledge of its mechanisms,especially when devices are concerned,hinders further development.From this perspective,the present work uses operando Raman experiments and electrochemical impedance spectroscopy(EIS)to assess the first stages of the discharge processes in porous carbon electrodes,following their changes cycle by cycle at initial operation.A growth kinetic formation of the discharge product signal(Li_(2)O_(2))was observed with operando Raman,indicating a first-order reaction and enabling an analysis by a microkinetic model.The solution mechanism in the evaluated system was ascribed for an equivalent circuit with three time constants.While the time constant for the anode interface reveals to remain relatively constant after the first discharge,its surface seemed to be more non-uniform.The model indicated that the reaction occurs at the Li_(2)O_(2) surface,decreasing the associated resistance during the initial discharge phase.Furthermore,the growth of Li_(2)O_(2) forms a hetero-phase between Li_(2)O_(2)/electrolyte,while creating a more compact and homogeneous on the Li_(2)O_(2)/cathode surface.The methodology here described thus offers a way of directly probing changes in surface chemistry evolution during cycling from a device through EIS analysis.

Online Identification of Lithium-ion Battery Model Parameters with Initial Value Uncertainty and Measurement Noise

Online parameter identification is essential for the accuracy of the battery equivalent circuit model(ECM).The traditional recursive least squares(RLS)method is easily biased with the noise disturbances from sensors,which degrades the modeling accuracy in practice.Meanwhile,the recursive total least squares(RTLS)method can deal with the noise interferences,but the parameter slowly converges to the reference with initial value uncertainty.To alleviate the above issues,this paper proposes a co-estimation framework utilizing the advantages of RLS and RTLS for a higher parameter identification performance of the battery ECM.RLS converges quickly by updating the parameters along the gradient of the cost function.RTLS is applied to attenuate the noise effect once the parameters have converged.Both simulation and experimental results prove that the proposed method has good accuracy,a fast convergence rate,and also robustness against noise corruption.

A hierarchical enhanced data-driven battery pack capacity estimation framework for real-world operating conditions with fewer labeled data

Battery pack capacity estimation under real-world operating conditions is important for battery performance optimization and health management,contributing to the reliability and longevity of batterypowered systems.However,complex operating conditions,coupling cell-to-cell inconsistency,and limited labeled data pose great challenges to accurate and robust battery pack capacity estimation.To address these issues,this paper proposes a hierarchical data-driven framework aimed at enhancing the training of machine learning models with fewer labeled data.Unlike traditional data-driven methods that lack interpretability,the hierarchical data-driven framework unveils the“mechanism”of the black box inside the data-driven framework by splitting the final estimation target into cell-level and pack-level intermediate targets.A generalized feature matrix is devised without requiring all cell voltages,significantly reducing the computational cost and memory resources.The generated intermediate target labels and the corresponding features are hierarchically employed to enhance the training of two machine learning models,effectively alleviating the difficulty of learning the relationship from all features due to fewer labeled data and addressing the dilemma of requiring extensive labeled data for accurate estimation.Using only 10%of degradation data,the proposed framework outperforms the state-of-the-art battery pack capacity estimation methods,achieving mean absolute percentage errors of 0.608%,0.601%,and 1.128%for three battery packs whose degradation load profiles represent real-world operating conditions.Its high accuracy,adaptability,and robustness indicate the potential in different application scenarios,which is promising for reducing laborious and expensive aging experiments at the pack level and facilitating the development of battery technology.

Calculation model of edge carbon atoms in graphite particles for anode of lithium-ion batteries

Based on the hexagonal crystallite model of graphite,the electrochemical characteristics of carbon atoms on the edge and basal plane were proposed by analyzing graphite crystal structure and bonds of carbon atoms in different sites.A spherical close-packed model for graphite particle was developed.The fractions of surface carbon atoms（SCA） and edge carbon atoms（ECA） were derived in the expression of crystallographic parameters and particle size,and the effects of ECA on the initial irreversible capacity and the mechanisms of action were analyzed and verified.The results show that the atoms on the edge are more active for electrochemical reactions,such as electrolyte decomposition and tendency to form stable bond with other atoms and groups.For the practical graphite particle,corresponding modifying factors were introduced to revise the difference in calculating results.The revised expression is suitable for the calculation of the fractions of SCA and ECA for carbon materials such as graphite,disordered carbon and modified graphite.

Working Condition Real-Time Monitoring Model of Lithium Ion Batteries Based on Distributed Parameter System and Single Particle Model

Lithium ion batteries are complicated distributed parameter systems that can be described preferably by partial differential equations and a field theory. To reduce the solution difficulty and the calculation amount, if a distributed parameter system is described by ordinary differential equations （ODE） during the analysis and the design of distributed parameter system, the reliability of the system description will be reduced, and the systemic errors will be introduced. Studies on working condition real-time monitoring can improve the security because the rechargeable LIBs are widely used in many electronic systems and electromechanical equipment. Single particle model （SPM） is the simplification of LIB under some approximations, and can estimate the working parameters of a LIB at the faster simulation speed. A LIB modelling algorithm based on PDEs and SPM is proposed to monitor the working condition of LIBs in real time. Although the lithium ion concentration is an unmeasurable distributed parameter in the anode of LIB, the working condition monitoring model can track the real time lithium ion concentration in the anode of LIB, and calculate the residual which is the difference between the ideal data and the measured data. A fault alarm can be triggered when the residual is beyond the preset threshold. A simulation example verifies that the effectiveness and the accuracy of the working condition real-time monitoring model of LIB based on PDEs and SPM.

A Comparative Study on Open Circuit Voltage Models for Lithium-ion Batteries

The current research of state of charge(SoC) online estimation of lithium-ion battery(LiB) in electric vehicles(EVs)mainly focuses on adopting or improving of battery models and estimation filters. However, little attention has been paid to the accuracy of various open circuit voltage(OCV) models for correcting the SoC with aid of the ampere-hour counting method. This paper presents a comprehensive comparison study on eighteen OCV models which cover the majority of models used in literature. The low-current OCV tests are conducted on the typical commercial LiFePO/graphite(LFP) and LiNiMnCoO/graphite(NMC) cells to obtain the experimental OCV-SoC curves at different ambient temperature and aging stages. With selected OCV and SoC points from experimental OCV-SoC curves, the parameters of each OCV model are determined by curve fitting toolbox of MATLAB 2013. Then the fitting OCV-SoC curves based on diversified OCV models are also obtained. The indicator of root-mean-square error(RMSE) between the experimental data and fitted data is selected to evaluate the adaptabilities of these OCV models for their main features, advantages,and limitations. The sensitivities of OCV models to ambient temperatures, aging stages, numbers of data points,and SoC regions are studied for both NMC and LFP cells. Furthermore, the influences of these models on SoC estimation are discussed. Through a comprehensive comparison and analysis on OCV models, some recommendations in selecting OCV models for both NMC and LFP cells are given.

Simulation of second-order RC equivalent circuit model of lithium battery based on variable resistance and capacitance

With the rise of the electric vehicle industry,as the power source of electric vehicles,lithium battery has become a research hotspot.The state of charge(SOC)estimation and modelling of lithium battery are studied in this paper.The ampere-hour(Ah)integration method based on external characteristics is analyzed,and the open-circuit voltage(OCV)method is studied.The two methods are combined to estimate SOC.Considering the accuracy and complexity of the model,the second-order RC equivalent circuit model of lithium battery is selected.Pulse discharge and exponential fitting of lithium battery are used to obtain corresponding parameters.The simulation is carried out by using fixed resistance capacitance and variable resistance capacitor respectively.The accuracy of variable resistance and capacitance model is 2.9%,which verifies the validity of the proposed model.

SOC distribution-based modeling for lithium-ion battery for electric vehicles using numerical optimization

In order to simulate electrical characteristics of a lithium-ion battery used in electric vehicles in a good manner,a three-layer battery model is established.The charge of the lithium-ion battery is assumed to distribute among the three layers and their interaction is used to depict hysteresis and relaxation effect observed in the lithium-ion battery.The model parameters are calibrated and optimized through a numerically nonlinear least squares algorithm in Simulink Parameter Estimation Toolbox for an experimental data set sampled in a hybrid pulse test of the battery.Evaluation results showed that the established model is able to provide an acceptable accuracy in estimating the State of Charge of the lithium-ion battery in an open-loop fashion for a sufficiently long time and to describe the battery voltage behavior more accurately than a commonly used battery model.The battery modeling accuracy can thereby satisfy the requirement for practical electric vehicle applications.

Modeling the dynamic behavior of a lithium-ion battery for electric vehicles using numerical optimization

In order to simulate the dynamical behavior of a lithium ion traction battery used in elec tric vehicles, an equivalent circuit based battery model was established. The methodology in the guide document of the ADVISOR software was used to determine the initial parameters of the model as a function of state of charge （ SoC ） over an experimental data set of the battery. A numerically nonlinear least squares algorithm in SIMULINK design optimization toolbox was applied to further op timize the model parameters. Validation results showed that the battery model could well describe the dynamic behavior of the lithinm ion battery in two different battery loading situations.

Online model identification of lithium-ion battery for electric vehicles

In order to characterize the voltage behavior of a lithium-ion battery for on-board electric vehicle battery management and control applications,a battery model with a moderate complexity was established.The battery open circuit voltage (OCV) as a function of state of charge (SOC) was depicted by the Nernst equation.An equivalent circuit network was adopted to describe the polarization effect of the lithium-ion battery.A linear identifiable formulation of the battery model was derived by discretizing the frequent-domain description of the battery model.The recursive least square algorithm with forgetting was applied to implement the on-line parameter calibration.The validation results show that the on-line calibrated model can accurately predict the dynamic voltage behavior of the lithium-ion battery.The maximum and mean relative errors are 1.666% and 0.01%,respectively,in a hybrid pulse test,while 1.933% and 0.062%,respectively,in a transient power test.The on-line parameter calibration method thereby can ensure that the model possesses an acceptable robustness to varied battery loading profiles.

Design and management of lithium-ion batteries:A perspective from modeling,simulation,and optimization

Although the lithium-ion batteries(LIBs) have been increasingly applied in consumer electronics, electric vehicles,and smart grid, they still face great challenges from the continuously improving requirements of energy density, power density, service life, and safety. To solve these issues, various studies have been conducted surrounding the battery design and management methods in recent decades. In the hope of providing some inspirations to the research in this field, the state of the art of design and management methods for LIBs are reviewed here from the perspective of process systems engineering. First, different types of battery models are summarized extensively, including electrical model and multi-physics coupled model, and the parameter identification methods are introduced correspondingly. Next, the model based battery design methods are reviewed briefly on three different scales, namely, electrode scale, cell scale, and pack scale. Then, the battery model based battery management methods, especially the state estimation methods with different model types are thoroughly compared. The key science and technology challenges for the development of battery systems engineering are clarified finally.

A Comparative Study of Fractional Order Models on State of Charge Estimation for Lithium Ion Batteries

State of charge(SOC)estimation for lithium ion batteries plays a critical role in battery management systems for electric vehicles.Battery fractional order models(FOMs)which come from frequency-domain modelling have provided a distinct insight into SOC estimation.In this article,we compare five state-of-the-art FOMs in terms of SOC estimation.To this end,firstly,characterisation tests on lithium ion batteries are conducted,and the experimental results are used to identify FOM parameters.Parameter identification results show that increasing the complexity of FOMs cannot always improve accuracy.The model R(RQ)W shows superior identification accuracy than the other four FOMs.Secondly,the SOC estimation based on a fractional order unscented Kalman filter is conducted to compare model accuracy and computational burden under different profiles,memory lengths,ambient temperatures,cells and voltage/current drifts.The evaluation results reveal that the SOC estimation accuracy does not necessarily positively correlate to the complexity of FOMs.Although more complex models can have better robustness against temperature variation,R(RQ),the simplest FOM,can overall provide satisfactory accuracy.Validation results on different cells demonstrate the generalisation ability of FOMs,and R(RQ)outperforms other models.Moreover,R(RQ)shows better robustness against truncation error and can maintain high accuracy even under the occurrence of current or voltage sensor drift.