Characterization of high-power lithium-ion batteries by electrochemical impedance spectroscopy. II: Modelling

A common way to model lithium-ion batteries is to apply equivalent circuit (EC) models. In this work two different EC models are build up and parameterized for a commercial 6.5 Ah high-power lithium-ion cell. Measured impedance spectroscopy data depending on temperature and state of charge (SOC) are used for parameter estimation.The first EC model consists of an ohmic resistor (R), an inductor (L) and three RC-elements (a parallel connection of a capacitor (C) and a resistor). The second EC model consists of one R, one L, two Zarc elements and a Warburg element. The estimated parameters were used to develop two empirical electrical cell models which are able to predict the voltage of the cells depending on current, temperature and SOC. Hereby the internal cell resistance Ri is based on the EC models and a Butler-Volmer adjustment. Both approaches were validated by current profiles, which cover typical automotive applications to prove the model performance at low temperatures and high dynamic operation. An accurate voltage prediction could be realized with both EC models. The second, more complex, model is able to predict cell voltage more precisely, but at the expense of up to four times higher computational effort.     

From single cell model to battery pack simulation for Li-ion batteries

A practical universal modeling and simulation approach is presented in this paper to show that accurate battery pack simulation can be achieved if cell-to-cell variations were taken into account. A generic equivalent circuit model was used in the approach with parameters deduced from cell testing with proper protocols, which could come from live cell monitoring in a control circuitry. Using a single cell model, which was validated against experimental data and demonstrated with validity of high accuracy in predicting cell performance, we showed that such a high accuracy in single cell model is essential for a high fidelity pack simulation. It is also important to derive statistical confidence intervals accurately from experiments to characterize intrinsic cell-to-cell variations in capacity and internal resistance, which need to be considered in the pack model. If parameters for each individual cell were correctly approximated and used in the pack model, the accuracy in the prediction of pack performance could be significantly improved.     

Equivalent circuit components of nickel-cadmium battery at different states of charge

Equations that describe the voltage variations with time of a rechargeable battery during charging and discharging were used to determine the component values of the equivalent circuit of nickel-cadmium batteries under different states of charge (SOC). The equivalent circuit of the battery was described as an ideal voltage source in series with a resistor and the parallel combination of a resistor and a capacitor. The battery model used different values of resistance and capacitance, in the parallel combination, during the different phases of the discharge-rest-charge-rest sequence. The results show that the series resistance is approximately constant with variations in the SOC while the resistor in the parallel RC circuit increases as the SOC decreases. For the discharge and charge phases the capacitor value increased and decreased, respectively, as the SOC decreased. The value of the resistor or capacitor in the parallel RC circuit is an indicator of the battery SOC.     

A comparative study of lumped equivalent circuit models of a lithium battery for state of charge prediction

Battery modeling plays an important role in remaining range prediction and battery management system development. An accurate and realistic battery model is essential to design an efficient electric storage system. The goal of this paper is to investigate the performance of different circuit topologies for diffusion process in the equivalent circuit models (ECMs). The theory of diffusion process approximation by using resistive‐capacitor (RC) networks is explained in frequency domain. The terminal voltage predictive capabilities of the ECMs are compared and validated with test data. The numerical simulation results show that model prediction accuracy and computation burdens increase along with the number of RC pairs. The ECM with three RC networks is the best choice in terms of the balance between accuracy and complexity for ternary lithium batteries. In addition, a novel method of combining unscented Kalman filter (UKF) algorithm with initial state of charge (SOC) acceleration convergence for SOC estimation is proposed. The results of urban dynamometer driving schedule (UDDS) show that ECM with three RC networks has the best comprehensive performance on calculation cost and SOC estimation accuracy.     

Equivalent circuit components of nickel-metal hydride battery at different states of charge

Equations that describe the voltage variations with time of rechargeable batteries during charging and discharging were used to determine the component values of the equivalent circuit of nickel-metal hydride batteries under different states of charge (SOC). The equivalent circuit of the battery was described as an ideal voltage source in series with a resistor and the parallel combination of a resistor and a capacitor. The battery model used different values of resistance and capacitance, in the parallel combination, during the different phases of the discharge-rest-charge-rest sequence. The results show that the resistances in the equivalent circuit are approximately constant with variations in the SOC. For the discharge and charge phases the capacitor value increased and decreased, respectively, as the SOC decreased. The value of the capacitor in the parallel RC circuit is an indicator of the battery SOC.     

An equivalent circuit model of a deformed Li-ion battery with parameter identification

A new equivalent circuit model (ECM) of a Li-ion battery is developed in this study. The developed model is utilized to obtain the dynamic electrical response of the battery when it is deformed under external force. Compared with other models, this model is developed based on a modified Thevenin model, and the parameters of the developed model are relevant to state of charge, the battery surface temperature, and the deformation. In this study, to obtain the real electrical response of the battery when it deformed under external force, batteries that are compressed by different deformations from 0 to 5 mm are studied with pulse discharging tests. Then, the parameters of the circuit elements are identified by a differential evolution algorithm based on the data obtained from these tests. Moreover, the data from the pulse discharging tests of batteries compressed by 3.5, 4.25, and 4.5 mm and the data from the pulse charging tests of batteries compressed by 0 and 1 mm are used to verify the parameters. The results illustrate that the battery capacity should drop significantly when the battery is severely deformed, but the battery still can be charged and discharged. Most importantly, the discharging curves of these tested deformed batteries are similar to those of undeformed ones. Moreover, the developed new ECM can predict the dynamic electrical response of a deformed battery accurately.     

Research on parameter identification and state of charge estimation of improved equivalent circuit model of Li-ion battery based on temperature effects for battery thermal management

The performance and parameters of Li-ion battery are greatly affected by temperature. As a significant battery parameter, state of charge (SOC) is affected by temperature during the estimation process. In this paper, an improved equivalent circuit model (IECM) considering the influence of ambient temperatures and battery surface temperature (BST) on battery parameters based on second-order RC model have been proposed. The exponential function fitting (EFF) method was used to identify battery model parameters at 5 ambient temperatures including −10°C, 0°C, 10°C, 25°C and 40°C, fitting the relationship between internal resistance and BST. Then, the SOC of the IECM was estimated based on the extended Kalman filter (EKF) algorithm. Using the result calculated by the Ampere-hour integration method as the standard, the data of battery under open circuit voltage (OCV) test profile and dynamic stress test (DST) profile at different ambient temperatures has been compared with the ordinary second-order RC model, and the advantages of the SOC estimation accuracy with IECM was verified. The numerical results showed that the IECM can improve the estimation accuracy of battery SOC under different operating conditions.     

Novel equivalent circuit model and statistical analysis in parameters identification

Parameter fitting based on the classical equivalent circuit model does not always reach reasonable solutions. It often gave negative series resistance and exaggerated diode ideality factor. Three problems were identified. The first problem was illuminated I–V is not a simple voltage shift by lumped series resistance. The second problem was rounded I–V curves in concentrator cells. The third problem was statistical instability in data fitting algorithm. A new model was proposed to describe the second problem. It was also effective to partly solve the first and the third problems.     

Current density and state of charge inhomogeneities in Li-ion battery cells with LiFePO4 as cathode material due to temperature gradients

Current density distributions and local state of charge (SoC) differences that are caused by temperature gradients inside actively cooled Li-ion battery cells are discussed and quantified. As an example, a cylindrical Li-ion cell with LiFePO₄ as cathode material (LiFePO₄-cell) is analyzed in detail both experimentally and by means of spatial electro-thermal co-simulations. The reason for current density inhomogeneities is found to be the local electrochemical impedance varying with temperature in different regions of the jelly roll. For the investigated cell, high power cycling and the resulting temperature gradient additionally cause SoC-gradients inside the jelly roll. The local SoCs inside one cell diverge firstly because of asymmetric current density distributions during charge and discharge inside the cell and secondly because of the temperature dependence of the local open circuit potential. Even after long relaxation periods, the SoC distribution in cycled LiFePO₄-cells remains inhomogeneous across the jelly roll as a result of hysteresis in the open circuit voltage. The occurring thermal electrical inhomogeneities are expected to influence local aging differences and thus, global cell aging. Additionally the occurrence of inhomogeneous current flow and SoC-development inside non-uniformly cooled battery packs of parallel connected LiFePO₄-cells is measured and discussed.     

Analysis of multicrystalline silicon solar cells by modified 3-diode equivalent circuit model taking leakage current through periphery into consideration

We proposed a modified 3-diode equivalent circuit model for analysis of multicrystalline silicon (Mc-Si) solar cells. By using this equivalent circuit model, we can precisely evaluate the characteristics of Mc-Si solar cells taking the influence of grain boundaries and large leakage current through the peripheries into consideration and extract electrical properties. The calculated value of current-voltage characteristics for small size (3 mm×3 mm) Mc-Si solar cells using this model completely agreed with the measured value at various cell temperatures. Moreover, the calculated open-circuit voltage (V_oc) obtained by extracted parameters and measured V_oc agreed well.     

Thermal stability of LiPF6-based electrolyte and effect of contact with various delithiated cathodes of Li-ion batteries

Thermal stability of LiPF₆-based electrolyte (1 M LiPF₆/EC + DMC) was studied by in-situ FTIR spectroscopy and C80 calorimetry, which indicated that the electrolyte underwent furious polymerization and decomposition reactions and sharp heat flow was generated below 225 °C. The thermal stability of the electrolyte in contact with various delithiated cathodes (Li_xCoO₂, Li_xNi_0.8Co_0.15Al_0.05O₂, Li_xNi_1/3Co_1/3Mn_1/3O₂, Li_xMn₂O₄, Li_xNi_0.5Mn_0.5O₂, Li_xNi_0.5Mn_1.5O₄ and Li_xFePO₄) was also investigated by C80 calorimetry. The results show that the cathode materials except for Li_xFePO₄ usually have an enhancement effect on the decomposition of the electrolyte, but Li_xFePO₄ exhibits a suppression effect on the reactions of the electrolyte. Li_xFePO₄ is found to be with excellent thermal stability. Among the other cathodes, Li_xCoO₂, Li_xNi_0.8Co_0.15Al_0.05O₂, Li_xNi_0.5Mn_0.5O₂ and Li_xNi_0.5Mn_1.5O₄ promote the decomposition of electrolyte by releasing oxygen and thus considered not favorable for safety, but Li_xNi_1/3Co_1/3Mn_1/3O₂ with a lesser reaction heat and Li_xMn₂O₄ with even less heat flow in the low temperature range (50-225 °C) are believed as promising cathodes for better safety. By comparing X-ray diffraction (XRD) patterns of these cathode materials at room temperature and those heated to 300 °C in the presence of the electrolyte, we have found that Li_xFePO₄ only has experienced tiny structure change, which is greatly different from the other cathode materials.     

Multi-parameter battery state estimator based on the adaptive and direct solution of the governing differential equations

We report the development of an adaptive, multi-parameter battery state estimator based on the direct solution of the differential equations that govern an equivalent circuit representation of the battery. The core of the estimator includes two sets of inter-related equations corresponding to discharge and charge events respectively. Simulation results indicate that the estimator gives accurate prediction and numerically stable performance in the regression of model parameters. The estimator is implemented in a vehicle-simulated environment to predict the state of charge (SOC) and the charge and discharge power capabilities (state of power, SOP) of a lithium ion battery. Predictions for the SOC and SOP agree well with experimental measurements, demonstrating the estimator's application in battery management systems. In particular, this new approach appears to be very stable for high-frequency data streams.     

Development of a universal modeling tool for rechargeable lithium batteries

An interesting universal modeling tool for rechargeable lithium batteries is presented in this paper. The generic model is based on an equivalent circuit technique commonly used in electrochemical impedance characterization. Therefore, the parameters used in the model can be easily parameterized from the electrochemical impedance derivations, which provide a convenient integration with experimental cell characterizations. Such integration offers the universality in this modeling approach.     

Active equalization control strategy of Li-ion battery based on state of charge estimation of an electrochemical-thermal coupling model

Aiming at solving the problem of poor battery cell consistency caused by excessive decay of cell capacity or increased internal resistance during the operation of lithium-ion battery packs for vehicles, the paper proposes an active equalization control with 12-V power supply as an equalization energy source, which achieves efficient energy replenishment of individual cells with low power. The electrochemical-thermal coupling model of lithium-ion battery is built, and the order reduction of large-scale system theory ensures that the model had higher accuracy and lower amount of calculation, which is suitable for vehicle battery management system (BMS). Then the extended Kalman filter algorithm is used to calculate the real-time state of charge (SOC) of each cell and set as an equalization variable. The equalization simulation circuit is built with MATLAB/Simulink, the experimental platform of active equalization system for battery packs is constructed, and the battery packs are tested for equalization in static state. The simulation and experimental results show that the proposed active equalization control strategy can rapidly improve the voltage inconsistency between single cells, and the energy transfer efficiency can reach about 85% during the equalization process.     

Estimation of equivalent circuit parameters of PV module and its application to optimal operation of PV system

A method to estimate the equivalent circuit parameters of a PV (photovoltaic) module is presented. The parameters are calculated using a least-squares fitting of the equivalent model current–voltage characteristic with the measured one. For applications of the equivalent circuit model parameters, a quantitative diagnostic method of the PV modules by evaluating the parameters is introduced and examined by simulation. A new maximum peak power tracking (MPPT) method using the model parameters, a solar insolation, and a cell temperature is also shown. Its performance is compared with other MPPT control algorithms by simulations. The performance of the proposed method was better than other MPPT methods.     

Thermal modeling of a cylindrical LiFePO4/graphite lithium-ion battery

A lumped-parameter thermal model of a cylindrical LiFePO₄/graphite lithium-ion battery is developed. Heat transfer coefficients and heat capacity are determined from simultaneous measurements of the surface temperature and the internal temperature of the battery while applying 2 Hz current pulses of different magnitudes. For internal temperature measurements, a thermocouple is introduced into the battery under inert atmosphere. Heat transfer coefficients (thermal resistances in the model) inside and outside the battery are obtained from thermal steady state temperature measurements, whereas the heat capacity (thermal capacitance in the model) is determined from the transient part. The accuracy of the estimation of internal temperature from surface temperature measurements using the model is validated on current-pulse experiments and a complete charge/discharge of the battery and is within 1.5 °C. Furthermore, the model allows for simulating the internal temperature directly from the measured current and voltage of the battery. The model is simple enough to be implemented in battery management systems for electric vehicles.     

Dynamic modeling of a PEM fuel cell with temperature effects

With the aim of dynamic simulation and control, a new non-linear state-space dynamic non-isothermal polymer electrolyte membrane fuel cell (PEMFC) model is developed in this paper. This mathematical model is developed based on mass and energy equation. The present model takes in account subsequent factors as the effects of charge double layer capacitance, the geometrical capacity and the effect of temperature gradient. In this paper, the authors propose a combination of several dynamic equations to study the effect of suddenly variation of some operating parameters like load resistance, gas pressure and gas temperature input. The results are compared to those of an isothermal model. This model will be extremely functional for the best possible design and real-time control of PEMFC systems. The present model is executed in MATHCAD software and the fuel cell is symbolized by an equivalent circuit which incorporates gas diffusion layer, membrane and electrodes. The analysis results show that the main elements that influence the performance of the cell are load resistance and functioning temperature.     

Simple fabrication of porous NiO nanoflowers: Growth mechanism,shape evolution and their application into Li-ion batteries

Tailoring the shape of nanomaterials is a key factor to control their properties. In this presentation, individual porous NiO nanoflowers via α-Ni(OH)₂ were fabricated through a simple solvothermal process without any surfactants or growth templates and their application in lithium battery was investigated. In the method, nickel acetate and urea were used as starting materials in ethanol media at 190 °C for 3 h followed by calcination at 400 °C. Electron microscopy studies revealed that initially fine nanoparticles precipitate during solvothermal treatment which then undergo aggregation and self-assembly resulting in nanoflowers. In prolonged time, each nanoflower gives rise to a solid well-faceted microparticle. The electrochemical performance of the NiO nanoflowers was investigated by cyclic voltammetry and conventional galvanostatic charge–discharge tests. The results showed an initial high discharge capacity of ～1330 mAhg^?1 after 10 cycles at 0.1 C rate and a stable capacity of 630 mAhg^?1 after 40 cycles in the range of 0.01–3.0 V with the excellent columbic efficiency of ～94%, suggesting that they have a very promising potential in the future application for lithium ion battery.     

A semi-empirical method for electrically modeling of fuel cell: Executed on a direct methanol fuel cell

In this paper, a novel semi-empirical modeling method to mathematically derive a nonlinear equivalent circuit from a special group of impedance fuel cell models is proposed. As an example, a 5-cm² direct methanol fuel cell (DMFC) was modeled by this method. The derived equivalent circuit is composed of lumped nonlinear resistors, capacitors and an inductor. The nonlinear circuit has an impedance equivalent to the target fuel cell in various operating conditions and provides a good approximation of the static and transient behaviors of the fuel cell. The equivalent circuit fuel cell model was validated by comparing its numerical simulation results with its polarization curve and the dynamic behavior of the target DMFC. These comparisons were performed while the DMFC was operating under square current pulses with different upper and low current levels.     

Charge and discharge characteristics of a commercial LiCoO2-based 18650 Li-ion battery

We studied the charge and discharge characteristics of commercial LiCoO₂-based 18650 cells by using various electrochemical methods, including discharging at constant power, ac impedance spectroscopy, and dc-voltage pulse. At 20 °C, these cells deliver 8.7–6.8 Wh of energy when discharged at a power range of 1–12 W between 2.5 and 4.2 V. Ragone plots show that the effect of discharge power on the energy is significantly increased with decreasing of the temperature. For example, energy of the cell is entirely lost when the temperature downs to −10 °C and the discharge rate still remains at 10 W. Impedance analyses indicate that the total cell resistance (R_cell) is mainly contributed by the bulk resistance (R_b, including electric contact resistance and electrolytic ionic conductivity), solid electrolyte interface resistance (R_sei), and charge-transfer resistance (R_ct). Individual contribution of these three resistances to the cell resistance is greatly varied with the temperature. Near room temperature, the R_b occupies up to half of the cell resistance, which means that the rate performance of the cell could be improved by modifying cell design such as employing electrolyte with higher ionic conductivity and enhancing electric contact of the active material particles. At low temperature, the R_ct, which is believed to reflect cell reaction kinetics, dominates the cell resistance. In addition, galvanosatic cycling tests indicate that the charge and discharge processes have nearly same kinetics. The performance discrepancy observed during charging and discharging, especially at low temperatures, can be attributed to these two factors of: (1) substantially higher R_ct at the discharged state than at the charged state; (2) asymmetric voltage limits pre-determined for the charge and discharge processes.