CN112731181A - Lithium ion battery impedance model based on electrochemical principle - Google Patents

Lithium ion battery impedance model based on electrochemical principle Download PDF

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CN112731181A
CN112731181A CN202011608797.5A CN202011608797A CN112731181A CN 112731181 A CN112731181 A CN 112731181A CN 202011608797 A CN202011608797 A CN 202011608797A CN 112731181 A CN112731181 A CN 112731181A
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王大方
张齐
金毅
杨博文
董浩崧
郝自伟
张毕
孙旭
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Weihai Tianda Automobile Technology Co Ltd
Harbin Institute of Technology Weihai
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Abstract

The utility model provides a lithium ion battery impedance model based on electrochemistry principle, relates to lithium ion battery impedance model, and its battery total impedance: z (ω) ═ j ω Lw+Rohm+ZSEI，ino+ZSEI，org+ZDL+Zele+Zsld(ii) a And the detailed research discloses models of liquid phase impedance, solid phase impedance, SEI organic layer impedance and SEI inorganic layer impedance in the total impedance of the battery. The method has the advantages of capability of completely describing the reaction process in the battery, small computation amount of the model, high calculation efficiency of the model and the like.

Description

Lithium ion battery impedance model based on electrochemical principle
Technical Field
The invention relates to a lithium ion battery impedance model, in particular to a lithium ion battery impedance model based on an electrochemical principle, which can completely describe the internal reaction process of a battery, has small computation amount of the model and high calculation efficiency of the model
Background
The use of oil-burning vehicles results in the emission of large amounts of carbon dioxide and other harmful gases, thereby causing environmental pollution and greenhouse effect. In recent years, governments have been in continuous motion to drive pure electric vehicles in order to solve a series of environmental problems caused by fuel-oil vehicles. The lithium ion battery is the best choice for the power source of the electric automobile due to the advantages of high energy and power density, long service life and no pollution.
Although the lithium ion battery has many advantages, the lithium ion battery also has specific operating intervals of voltage, current and temperature, and if the battery operates outside the specific intervals, the performance of the battery is reduced and even safety accidents are caused. In order to ensure the safety of the lithium ion battery, the internal reaction process of the lithium ion battery needs to be monitored in real time, so that early warning is timely carried out before the internal failure of the lithium ion battery occurs. During the operation of the battery, various reaction processes, such as solid-liquid phase diffusion, charge transfer, SEI film growth, etc., occur. In order to monitor the reaction process inside the battery comprehensively, a plurality of test methods are proposed one after another. Electrochemical Impedance Spectroscopy (EIS) is one of the most commonly used electrochemical detection methods at present, which obtains the phase and amplitude, or real and imaginary parts, of a measured object by applying a set of sinusoidal voltages or currents of a single frequency to the measured object, by analog circuit or Fast Fourier Transform (FFT) analysis. The EIS has the advantage that the reaction process can be decoupled in the frequency domain according to the time constant of the reaction process inside the battery, and the impedance spectrum is generally divided into five parts according to the frequency interval, as shown in fig. 1, and the reaction process corresponding to the five parts is as follows.
The first part is in the very high frequency region, the EIS behaves as an inductance, due to the wire inductivity, in the nyquist plot as a straight line close to perpendicular to the real axis;
the intersection of the second part EIS with the real axis, representing the ohmic internal resistance of the cell;
the third part is in the high frequency region, the impedance characteristic is related to the lithium ion transport process in the SEI film, and is represented as a semicircle in the Nyquist diagram;
fourth part in the intermediate frequency region, the impedance characteristics are related to charge transfer and charge accumulation in the electric double layer, appearing as a semicircle in the nyquist plot;
the fifth part is in the low frequency region, the impedance characteristic is related to solid-liquid phase diffusion, and is shown as a straight line at an angle to the real axis in the Nyquist diagram.
Based on the above analysis, the variations of each reaction process in the battery due to factors such as SOC, temperature, aging, etc. can be individually analyzed by EIS, and the estimation of the temperature and the state in the battery can be performed by extracting the characteristics of each reaction process. However, the premise of accurate internal temperature estimation and state estimation is to perform accurate fitting on the EIS, and the current models capable of fitting the EIS include a fractional order model and an electrochemical model.
The correlation between the Fractional order model and EIS is detailed in the literature of Fractional order base modification methods and electric field applications of J.Tian et al, scientific advances and fields [ J ], scientific CHINA technical SCIENCEs,2020,63, as shown in FIG. 2, the Fractional order model fits a semicircular part in the Nyquist plot using a parallel form of an equal phase angle element (CPE) and a resistive element, and fits a linear part in the Nyquist plot using a Warburg impedance element. In many studies of EIS fitting, the fractional order model showed good fitting accuracy.
In order to clearly express the internal reaction process of the battery, some scholars establish an electrochemical model, the most classical model of the lithium ion battery is a pseudo two-dimensional model (P2D model), the P2D model is originally proposed by M.Doyle et al, and the model is based on porous electrode and concentrated solution theory and clearly describes the reaction processes of solid-liquid phase diffusion, charge transfer and the like. Some scholars apply electrochemical models equally to EIS spectral fitting. Doyle et al in The literature Computer Simulations of The Impedance Response of The lithium rechargeable Batteries [ J ], Journal of The Electrochemical Society,147(1)99-110(2000) first proposed The P2D model's deficiencies in fitting EIS and made it better describe The frequency domain characteristics of The cell by adding a description of The electric double layer capacitance in The P2D model. Zhang et al, in the documents of Electrochemical model of lithium-ion battery for side frequency amplification application [ J ], Electrochemical Acta 343(2020)136094, established an extended P2D model considering an electric double layer and an SEI film, and performed multi-dimensional verification on the established model in frequency domain and time domain.
To accommodate the fit of EISs for different cells, the fractional order model typically selects the number of CPE-resistor series structures in the model based on the number of semi-circles on the nyquist plot for EIS, but because the time constants of some reaction processes inside the cell are very close, one semi-circle in the nyquist plot often corresponds to more than one reaction process. Therefore, the fractional order model can simultaneously describe a plurality of reaction processes with close time constants by using only one CPE-resistor series structure according to the shape of the Nyquist diagram, and in this case, the fractional order model cannot sufficiently decouple the internal reaction of the battery, so that the description of the internal reaction process of the battery is unclear. Although the fractional order model can fit the EIS, the reaction process inside the battery cannot be completely described, so that it is difficult to realize complete detection of the reaction process inside the battery through the fractional order model.
Compared with a fractional order model, the electrochemical model can completely describe the reaction process in the battery in detail and can also realize accurate fitting of EIS, but the electrochemical model often comprises a large number of differential equations, so that the solution is complex and the calculation efficiency is low, and therefore the electrochemical model is difficult to realize application in engineering.
Disclosure of Invention
The invention aims to solve the defects of the prior art and provide a lithium ion battery impedance model based on an electrochemical principle, which can completely describe the internal reaction process of a battery, has small computation amount of the model and high calculation efficiency of the model.
The technical scheme adopted by the invention for solving the defects of the prior art is as follows:
a lithium ion battery impedance model based on electrochemical principle is characterized in that:
total impedance of the battery: z (ω) ═ j ω Lw+Rohm+ZSEI，ino+ZSEI，org+ZDL+Zele+Zsld；
Liquid phase impedance:
Figure BDA0002872524520000021
Figure BDA0002872524520000022
Figure BDA0002872524520000023
solid phase impedance: zsld，i＝AB(C+Dj)，i＝n，p，
Figure BDA0002872524520000024
Figure BDA0002872524520000025
Electric double layer impedance:
Figure BDA0002872524520000026
SEI organic layer resistance:
Figure BDA0002872524520000027
Figure BDA0002872524520000028
SEI inorganic layer resistance:
Figure BDA0002872524520000029
Figure BDA00028725245200000210
in the lithium ion battery impedance model, j is an imaginary unit, ω is an angular frequency of an excitation applied to the lithium ion battery, LwThe lead is a lead inductance, the lead refers to a lead connected with two poles of a battery during the impedance spectrum test of the lithium ion battery, and the test lead can show the sensitivity under the high-frequency excitation, so the impedance spectrum obtained by the lithium ion battery in the test of a wider frequency range can simultaneously contain the impedance characteristics of the lithium ion battery and the test lead. RohmOhmic internal resistance D of the interior of the lithium ion batteryeleIs the liquid phase diffusion coefficient of lithium ion, Dsld，nIs the negative electrode lithium ion solid phase diffusion coefficient, D of the lithium ion batterysld，pIs the positive electrode lithium ion solid phase diffusion coefficient, D of the lithium ion batteryorgLithium ion diffusion coefficient, D, of organic layer of SEI film for lithium ion batteryinoThe diffusion coefficient of lithium ions in an SEI film inorganic layer of the lithium ion battery, K is an electrochemical reaction constant at an electric double layer in the lithium ion battery,
Figure BDA00028725245200000311
The liquid phase transfer coefficient of lithium ion in the lithium ion battery,
Figure BDA0002872524520000031
Is the equivalent total thickness L of the cathode, the diaphragm and the anode of the lithium ion batterynThickness L of negative electrode plate of lithium ion batterysIs the thickness, L, of the separator of a lithium ion batterypThickness of positive electrode plate of lithium ion batterynIs the porosity of the negative electrode, epsilon, of a lithium ion batterysPorosity of the membrane, epsilon, for lithium ion batteriespPorosity of the positive electrode, a, of a lithium ion batterynIs the specific surface area of the negative electrode of the lithium ion battery, apIs the specific surface area of the positive electrode of the lithium ion battery, AnArea of negative electrode plate of lithium ion battery, ApArea of positive electrode plate r of lithium ion batterynRadius of spherical particles r of negative electrode of lithium ion batterypIs the radius of the positive spherical particle of the lithium ion batteryorgPorosity, l, of an organic layer of an SEI film for a lithium ion batteryorgThickness of organic layer of SEI film for lithium ion batteryinoThickness of SEI film inorganic layer for lithium ion battery, cele，0The initial concentration of liquid-phase lithium ions in the lithium ion battery,
Figure BDA0002872524520000032
Initial concentration of solid-phase lithium ions of the negative electrode of the lithium ion battery,
Figure BDA0002872524520000033
Is the initial concentration of solid-phase lithium ions of the positive electrode of the lithium ion battery,
Figure BDA0002872524520000034
The maximum concentration of solid-phase lithium ions at the negative electrode of the lithium ion battery,
Figure BDA0002872524520000035
Is the maximum concentration of solid-phase lithium ions of the positive electrode of the lithium ion battery, corg，0Initial concentration of lithium ions in organic layer of SEI film of lithium ion battery, cino，0The initial concentration of lithium ions in an SEI film inorganic layer of the lithium ion battery, F is a Faraday constant, and R is a molar gas constant.
The invention establishes an impedance model special for EIS fitting based on an electrochemical mechanism, and the model only comprises conventional operations such as addition, subtraction, multiplication, division and trigonometric functions, thereby avoiding the solution of a differential equation of the electrochemical model and greatly improving the calculation efficiency of the model.
Drawings
FIG. 1 is a typical EIS Nyquist plot.
Fig. 2 is a diagram of a relationship between a fractional order model and a nyquist diagram.
Fig. 3 is a structural correspondence diagram of an impedance model of a lithium ion battery.
Fig. 4 is a schematic diagram illustrating a movement path of lithium ions in a liquid phase in an idealized lithium ion battery.
Fig. 5 is a comparison of nyquist plots of simulated EIS and measured EIS for lithium cobaltate 18650 cells at different SOCs.
Fig. 6 is a graph comparing real part versus frequency plots of simulated EIS and measured EIS for lithium cobaltate 18650 cells at different SOCs.
Fig. 7 is a comparison of imaginary part-frequency plots of simulated EIS and measured EIS for lithium cobaltate 18650 cells at different SOCs.
Fig. 8 is a comparison of nyquist plots for simulated EIS and measured EIS for a nickel-cobalt-manganese ternary 18650 battery at different SOCs.
Fig. 9 is a graph comparing real part versus frequency plots for simulated EIS and measured EIS for a nickel-cobalt-manganese ternary 18650 battery at different SOCs.
Fig. 10 is a comparison of imaginary part-frequency plots of simulated EIS and measured EIS for a nickel-cobalt-manganese ternary 18650 battery at different SOCs.
Fig. 11 is a comparison of nyquist plots of simulated EIS versus measured EIS for a nickel-cobalt-aluminum ternary 21700 battery at different SOCs.
Fig. 12 is a graph of real part versus frequency plots for simulated EIS versus measured EIS for a nickel-cobalt-aluminum ternary 21700 battery at different SOCs.
Fig. 13 is a graph comparing the imaginary part versus frequency plot for a nickel-cobalt-aluminum ternary 21700 battery with the measured EIS at different SOCs.
Detailed Description
A lithium ion battery impedance model based on electrochemical principle is characterized in that:
total impedance of the battery: z (ω) ═ j ω Lw+Rohm+ZSEI，ino+ZSEI，org+ZDL+Zele+Zsld；
Liquid phase impedance:
Figure BDA0002872524520000036
Figure BDA0002872524520000037
Figure BDA0002872524520000038
solid phase impedance: zsld，i＝AB(C+Dj)，i＝n，p，
Figure BDA0002872524520000039
Figure BDA00028725245200000310
Figure BDA0002872524520000041
Electric double layer impedance:
Figure BDA0002872524520000042
SEI organic layer resistance:
Figure BDA0002872524520000043
Figure BDA0002872524520000044
SEI inorganic layer resistance:
Figure BDA0002872524520000045
Figure BDA0002872524520000046
the theoretical basis of the invention is as follows:
the present invention builds a series impedance model, corresponding to the extended P2D model, as shown in fig. 3. The lower half of fig. 3 shows an extended P2D model, which is a multilayer structure composed of positive and negative electrodes, an electrolyte, a separator, a current collector, an SEI film, and an electric double layer. The positive and negative active particles are assumed to be small balls, the diaphragm separates the positive and negative electrodes, the electrolyte is positioned between the gaps of the positive and negative active particles and the pores of the diaphragm, the current collectors are positioned on the two sides of the positive and negative electrodes, the SEI film is attached to the surface of the negative electrode and is divided into an inorganic layer and an organic layer, and the double electric layers are positioned at the interfaces of the active particles and the electrolyte. The upper part of FIG. 3 shows the impedance model, where LwRepresenting the inductance, R, induced by the conductorohmIndicates the ohmic internal resistance, ZSEI，inoRepresents the impedance of lithium ions passing through the inorganic layer of the SEI film, ZSEI，orgRepresents the resistance of the organic layer of the SEI film, ZDLRepresents the electric double layer impedance, ZeleRepresenting the impedance of the liquid phase, ZsldRepresents the impedance of the solid phase. Thus, the total impedance of the battery can be described as:
Z(ω)＝jωLw+Rohm+ZSEI，ino+ZSEI，org+ZDL+Zele+Zsld (1)
except for LwAnd RohmOther impedances are related to the electrochemical reaction process, and the derivation of each impedance is as follows.
Liquid phase impedance ZeleDerivation of (1):
in all diffusion processes, the diffusion of lithium ions in the electrolyte is the most complicated, because the lithium ions in the liquid phase diffuse through the negative electrode, the positive electrode and the diaphragm, which are different in size and structure, and the diffusion formulas of the lithium ions in the three parts described by the model P2D are also different, as shown in formula 2
Figure BDA0002872524520000047
In the formula, epsilonn、εs、εpPorosity of the negative electrode, the diaphragm and the positive electrode respectively, x is the diffusion distance of lithium ions along the thickness direction of the polar plate, celeRespectively the Li ion concentration at x in the liquid phase,
Figure BDA0002872524520000048
effective diffusion coefficients of lithium ion liquid phases in the negative electrode, the separator and the positive electrode, an、apThe specific surface areas of the negative electrode and the positive electrode,
Figure BDA0002872524520000049
is the Li ion liquid phase transfer coefficient, jave，n、jave，pThe lithium ion flux densities in the x direction of the negative electrode and the positive electrode are shown. The calculation formula of the effective diffusion coefficient and the lithium ion flux density is as follows:
Figure BDA00028725245200000410
Figure BDA00028725245200000411
in the formula IextFor an external current, Ln、LpThe thickness of the plate, A, of the negative and positive electrodes respectivelyn、ApThe areas of the anode and cathode plates are shown respectively, and F is the Faraday constant.
Equation 2 describes the liquid phase lithium ion concentration for each infinitesimal x. For positive and negative electrode parts, one micro element is relative to one spherical particle, the number of lithium ions on the surface of the spherical particle is commonly contributed by three aspects, namely lithium ions diffused from adjacent spherical particles, lithium ions diffused to adjacent spherical particles and lithium ions flowing out or in from the surface of the micro element. The first term on the right of the equal sign in equation 2 describes the effect of the first two aspects and the second term on the right describes the effect of the last aspect. For a separator without spherical particles inside, each cell does not consume and generate lithium ions by itself, so the right side of the equation has no second term.
Formula 2 mainly considers the influence of the microelements themselves and the adjacent microelements on the microelements to obtain the lithium ion concentration, and we can consider the liquid phase diffusion in a relatively macroscopic angle by changing the angle. Taking the discharge of a lithium ion battery as an example, in the initial stage of the discharge of the battery, lithium ions are released on the surface of each spherical particle of the negative electrode, resulting in an overall increase in the negative electrode lithium ion concentration, and lithium ions are consumed on the surface of each spherical particle of the positive electrode, resulting in an overall decrease in the positive electrode lithium ion concentration. Due to the influence of the concentration difference, the lithium ions of the negative electrode can diffuse to the positive electrode, and meanwhile, under the influence of the potential difference between two ends of the liquid phase, the lithium ions of the negative electrode can also perform electromigration to the positive electrode. The lithium ions released from the surface of the spherical particles of the negative electrode are influenced by the same action and move to the positive electrode at the same speed, but because the positions of the spherical particles in the x direction are different, the starting points of the movement of the lithium ions are different, so the lithium ions released from the spherical particles close to the position of the diaphragm in the negative electrode firstly reach the positive electrode, and after reaching the positive electrode, the consumption of the lithium ions of the spherical particles close to the position of the diaphragm in the positive electrode is firstly compensated according to the principle of proximity. It is inferred that the lithium ions released from the spherical particles at the middle position in the negative electrode compensate for the lithium ion consumption of the spherical particles at the middle position in the positive electrode, and the lithium ions released from the spherical particles at the position close to the current collector in the negative electrode compensate for the lithium ion consumption of the spherical particles at the position close to the current collector in the positive electrode. Based on the above analysis, the movement path of lithium ions at each position is shown in fig. 4. As can be seen from fig. 4, when considering the liquid phase diffusion process, we do not consider the effect of the SEI film because the time constant of the liquid phase diffusion is much larger than that of the diffusion process in the SEI film, and only under the excitation of lower frequency, the lithium ions have a significant concentration difference on both sides of the liquid phase, and at this time, the concentration difference of the lithium ions on both sides of the SEI film is negligible due to the very small diffusion time constant in the SEI film. On the other hand, the thickness of the SEI film is much smaller than the dimension of the liquid phase in the x direction, and therefore the influence on the path length along which the liquid phase moves can be ignored.
According to the idea shown in fig. 4, if the concentration of the lithium ions in the liquid phase is only determined at both ends of the liquid phase, only the moving path of the lithium ions from the spherical particle n1 to the spherical particle p1 needs to be studied. The formula of the concentration of the liquid-phase lithium ions in the path is as follows:
Figure BDA0002872524520000051
in order to unify the three parts of the negative electrode, the positive electrode, and the separator, equation 5 can be expressed as follows:
Figure BDA0002872524520000052
in the formula, DeleThe liquid phase diffusion coefficients are uniform, and the liquid phase diffusion coefficients are completely the same because the electrolyte components in the negative electrode, the positive electrode, and the separator are completely the same, and l is the equivalent diffusion length in consideration of the difference in the three porosity.
The boundary condition of equation 6 is:
Figure BDA0002872524520000053
in the formula (I), the compound is shown in the specification,
Figure BDA0002872524520000054
is the equivalent total thickness of the negative electrode, the diaphragm and the positive electrode, /)n、ls、lpThe thicknesses of the negative electrode, the separator and the positive electrode are respectively.
Based on the formulas 6 and 7, firstly, deriving an expression of concentration difference between two ends of the liquid phase in a frequency domain, applying Laplace transform, and converting the formula 6 to the frequency domain to obtain:
Figure BDA0002872524520000055
in the formula,. DELTA.celeIn order to change the concentration of the liquid-phase lithium ions after the battery is excited relative to the concentration of the liquid-phase lithium ions before the battery is excited, the general solution of equation 8 is as follows according to the solving principle of the second-order constant coefficient homogeneous differential equation:
Figure BDA0002872524520000056
substitution of formula 7 to obtain
Figure BDA0002872524520000061
Then, the difference in lithium ion concentration between the two ends of the liquid phase, i.e., L ═ 0 and L ═ L, is:
Figure BDA0002872524520000062
substituting s ═ j ω into equation 11 and separating the real and imaginary parts:
Figure BDA0002872524520000063
the solution for the liquid phase potential in the P2D model is as follows:
Figure BDA0002872524520000064
in the formula, κeffIs an effective ionic conductivity of the liquid phase,
Figure BDA0002872524520000065
is a liquid phase internal electromotive force ieleR is the molar gas constant and T is the temperature. The first term on the right of the equation represents the concentration polarization overpotential of the liquid phase and the second term represents the ohmic polarization overpotential of the liquid phase. In the liquid phase diffusion impedance model, we only consider the influence of concentration polarization, and incorporate the liquid phase ohmic polarization overpotential into the ohmic polarization overpotential of the whole cell. Extracting the concentration polarization overpotential separately and adding ln celeLinearization is carried out to obtain:
Figure BDA0002872524520000066
in the formula, κeffIs an effective ionic conductivity of the liquid phase,
Figure BDA0002872524520000067
is a liquid phase internal electromotive force ieleR is the molar gas constant and T is the temperature. The first term on the right of the equation represents the concentration polarization overpotential of the liquid phase and the second term represents the ohmic polarization overpotential of the liquid phase. In the liquid phase diffusion impedance model, we only consider the influence of concentration polarization, and incorporate the liquid phase ohmic polarization overpotential into the ohmic polarization overpotential of the whole cell. Extracting the concentration polarization overpotential separately and adding ln celeLinearization is carried out to obtain:
Figure BDA0002872524520000068
solid phase impedance ZsldDerivation of (1):
the P2D model describes solid phase diffusion as:
Figure BDA0002872524520000071
in the formula, Dsld，n、Dsld，pThe solid phase diffusion coefficients of the negative electrode and the positive electrode, respectively, csld，n、csld，pThe lithium ion concentrations in the negative electrode active particles and the positive electrode active particles are respectively, R is the diffusion distance of the lithium ions in the active particles along the radius direction, and R is the radius of the active particles.
Order to
Figure BDA0002872524520000072
Equation 16 can be converted to:
Figure BDA0002872524520000073
applying the laplacian transform, transform equation 17 into the frequency domain:
Figure BDA0002872524520000074
according to the solution principle of the second-order constant coefficient homogeneous differential equation, the general solution of equation 18 is:
Figure BDA0002872524520000075
then
Figure BDA0002872524520000076
Introducing boundary conditions
Figure BDA0002872524520000077
To obtain
Figure BDA0002872524520000078
Then, the amount of change in the lithium ion concentration on the surface of the spherical particle from the initial time is:
Figure BDA0002872524520000079
substituting s ═ j ω into equation 23 and separating the real and imaginary parts:
Figure BDA00028725245200000710
the P2D model usually uses a relation between the open-circuit potential (OCV) of the positive and negative electrode materials and the amount of embedded lithium to determine the solid-phase potential of the electrode, but the relation is often complex and difficult to linearize, and a theoretical derivation form of OCV is adopted here:
Figure BDA00028725245200000711
in the formula, csldIs the solid-phase lithium ion concentration, cVConcentration of holes generated for removal of solid-phase lithium ions, csld，maxThe maximum lithium ion concentration in the solid phase. Equation 25 is linearized and the change in OCV is found:
Figure BDA00028725245200000712
in the formula, csld，0Is the solid phase initial lithium ion concentration.
Combining the formula 24 and the formula 26, the impedance expression of solid phase diffusion is given as:
Figure BDA0002872524520000081
impedance of electric double layer ZDLDerivation of (1):
the classical P2D model only has a Butler-Volmer equation for describing the electric double layer and cannot accurately fit the lithium ion battery EIS, so that the double-electric-layer theory is introduced to accurately describe the charge accumulation of the electric double layer. Here, to simplify the calculation, the compact layer and the dispersed layer of the electric double layer are described as a plate capacitor, and the formula of the electric double layer capacitor is as follows:
CDL，i＝qsurf，i/ηi，i＝n，p (28)
in the formula, qsurf，n、qsurf，pThe residual charge density on the surface of the spherical particles of the cathode and the anode respectivelyn、ηpRespectively, the reaction polarization overpotential of the negative electrode and the positive electrode.
qsurf，i＝F∫(jave，i-jf，i)dt，i＝n，p (29)
In the formula, jf，n、jf，pAre respectively a negative electrodeAnd the positive electrode Faraday current flow density which can be obtained by a Butler-Volmer equation:
Figure BDA0002872524520000082
in the formula i0，n、i0，pExchange current density, alpha, of the cathode and anode, respectivelya、αcThe transfer coefficients of the oxidation reaction and the reduction reaction are respectively, and the transfer coefficients are 0.5. By combining equations 4, 27, 28, and 29, and using laplace transform, the following equations are obtained:
Figure BDA0002872524520000083
will be provided with
Figure BDA0002872524520000084
Linearized, and converted to:
Figure BDA0002872524520000085
for simplicity of calculation, the electric double layer capacitance C of the positive and negative electrodes is usedDL，nAnd CDL，pIs unified as CDLIn the above formula:
Figure BDA0002872524520000086
in the formula, Kn、KpThe electrochemical reaction constants of the positive and negative electrodes, also denoted as K, cele，n、cele，pThe lithium ion concentrations of the double electric layers of the negative electrode and the positive electrode close to one end of the electrolyte respectively,
Figure BDA0002872524520000087
the lithium ion concentrations on the surfaces of the spherical particles of the negative electrode and the positive electrode are respectively. As the lithium ion battery is used for ensuring the linearity of a tested object during EIS testAct, the current excitation applied to the lithium ion battery is all in the milliampere level, in this case, the lithium ion concentration at both ends of the liquid phase and the lithium ion concentration on the surface of the solid phase spherical particles are all not much different from their initial concentrations, so cele，n、cele，pCan be taken asele，0，
Figure BDA0002872524520000088
Can be respectively taken as
Figure BDA0002872524520000089
Equation 33, can be converted to:
Figure BDA00028725245200000810
SEI film organic layer impedance ZSEI，orgDerivation of (1):
the organic layer of the SEI film has a porous structure, and research shows that lithium ions diffuse through pores inside the organic layer, and the diffusion expression is as follows:
Figure BDA0002872524520000091
in the formula, corgIs the lithium ion concentration of the organic layer of the SEI film at x, DorgIs the lithium ion diffusion coefficient, ε, of the organic layer of an SEI filmorgIs the porosity of the organic layer of the SEI film.
The boundary conditions are as follows:
Figure BDA0002872524520000092
in the formula IorgIs the thickness of the organic layer of the SEI film. Applying laplacian transform to transform equation 35 to the frequency domain to obtain:
Figure BDA0002872524520000093
according to the solution principle of the second-order constant coefficient homogeneous differential equation, the general solution of equation 37 is:
Figure BDA0002872524520000094
substitution of formula 36 to obtain
Figure BDA0002872524520000095
Then, the organic layers of the SEI film are divided into two ends, i.e., l-0 and l-lorgThe difference in lithium ion concentration between sites is:
Figure BDA0002872524520000096
substituting s ═ j ω into equation 40 and separating the real and imaginary parts yields:
Figure BDA0002872524520000097
the solution of the concentration polarization overpotential of the SEI organic layer is as follows:
Figure BDA0002872524520000098
will ln corgLinearization is carried out to obtain:
Figure BDA0002872524520000099
where c isorg，0The initial liquid phase lithium ion concentration is taken. From equations 41 and 43, the impedance formula of the organic layer of the SEI film is as follows:
Figure BDA0002872524520000101
SEI film inorganic layer impedance ZSEI，inoDerivation of (1):
unlike the diffusion of lithium ions through pores in the organic layer of the SEI film, the diffusion of lithium ions between the inorganic layers of the SEI film occurs through the structure of the solid phase thereof, and the diffusion coefficient of lithium ions in the solid phase can be described as:
Figure BDA0002872524520000102
in the formula, v*For lattice vibration frequency,. DELTA.x is the distance of each step of ion migration, EmThe size of the potential barrier, k, that the lithium ions need to overcome for each step of migrationBBoltzmann constant. The components in the SEI inorganic layer are complex and the diffusion mechanism is not conclusive, but for different inorganic components and diffusion mechanisms, Δ x and E aremAre different from each other and thus the diffusion coefficient distribution is not uniform for complex inorganic layers. In order to simplify the calculation, the diffusion coefficient of the inorganic layer is processed uniformly, and then the diffusion expression of the inorganic layer of the SEI film is as follows:
Figure BDA0002872524520000103
in the formula, cinoIs the lithium ion concentration of the inorganic layer of the SEI film in the x direction, DinoIs the lithium ion diffusion coefficient of the inorganic layer of the SEI film.
The boundary conditions are as follows:
Figure BDA0002872524520000104
in the formula IinoIs the thickness of the inorganic layer of the SEI film. According to the same method for solving the impedance of the SEI organic layer, the impedance formula of the inorganic layer of the SEI film can be obtained as follows:
Figure BDA0002872524520000105
where c isino，0The initial liquid phase lithium ion concentration was also taken.
The invention is characterized in that an impedance model which is suitable for EIS fitting of a lithium ion battery and describes the reaction process in the battery in detail is established, and the specific model is described as follows:
Figure BDA0002872524520000106
Figure BDA0002872524520000111
Figure BDA0002872524520000121
before the impedance model is fitted with the EIS, model parameters need to be obtained first, and the model parameters of the impedance model and the obtaining method thereof are as follows:
Figure BDA0002872524520000122
Figure BDA0002872524520000131
the impedance model can use optimization algorithms such as simulated annealing algorithm, genetic algorithm and the like to identify the parameters to be identified in the model, and the specific identification process is as follows: firstly, determining an initial range of each parameter based on a relevant document, then using the error of an actually-measured EIS and a model simulation EIS as an evaluation index, using an optimization algorithm to find a parameter set which minimizes the error in the range, further reducing the range of each parameter by using each parameter value of the parameter set as a center, and using the optimization algorithm again to find the parameter set which minimizes the error in a new range. After many rounds, the parameter set meeting the error requirement can be found and determined as the final parameter set.
In order to prove that the impedance model established by the invention has higher simulation accuracy, model simulation impedance spectrums of three different lithium ion batteries under different SOC (state of charge) are compared with actually measured impedance spectrums, the parameters of the three batteries are shown in table one, the impedance spectrum comparison results are shown in fig. 5 to 13, and in order to clearly show the comparison of Nyquist diagrams under different SOC (state of charge), images are sequentially moved upwards in the directions of negative virtual axes in fig. 5, 8 and 11. From the comparison results of the impedance spectra of three different batteries, the impedance model established by the invention has higher simulation precision.
Table one: measured battery parameters
Figure BDA0002872524520000132
To further quantify the simulation accuracy of the present invention, we used Mean Absolute Error (MAE), Mean Relative Error (MRE), and Relative Root Mean Square Error (rrme) as evaluation indices. The formula for the three evaluation indices is as follows:
Figure BDA0002872524520000133
Figure BDA0002872524520000134
Figure BDA0002872524520000135
in the formula (I), the compound is shown in the specification,
Figure BDA0002872524520000136
is a frequency of fnThe impedance mode is actually measured when the voltage is measured,
Figure BDA0002872524520000137
is a frequency of fnThe impedance mode is simulated. The evaluation index sizes of three different batteries under different SOCs are shown in table two, table three and table four.
Table two: mean absolute error between simulated impedance spectrum and measured impedance spectrum
Figure BDA0002872524520000141
Table three: average relative error between simulated impedance spectrum and measured impedance spectrum
Figure BDA0002872524520000142
Table four: relative root mean square error between simulated impedance spectrum and measured impedance spectrum
Figure BDA0002872524520000143
From three evaluation indexes, the impedance model provided by the invention can meet the high-precision fitting requirement of EIS. In the conventional method, documents of Electrochemical model of lithium-ion battery for side frequency application [ J ], Electrochemical Acta 343(2020)136094 propose an Electrochemical model for EIS fitting, and the relative root mean square error between a measured impedance spectrum and a simulated impedance spectrum of the model is calculated to be 1.1% when the SOC of a measured battery is 40%, which shows that the fitting accuracy of the impedance spectrum model provided by the invention to EIS is equivalent to that of the conventional Electrochemical model. However, the solution of the traditional electrochemical model needs to solve a large number of differential equations, and the impedance of a single frequency in the impedance spectrum needs to be obtained by applying sinusoidal excitation of a corresponding frequency to the model, which means that the complete impedance spectrum needs to be obtained by applying excitation to the model for multiple times, and the obtaining time of the impedance spectrum is greatly prolonged. However, the impedance spectrum model provided by the invention does not need to solve a differential equation, can directly output the impedance at any frequency after parameter identification is completed, and does not need to separately apply excitation to the model to obtain the impedance at a single frequency, so that the acquisition time of the impedance spectrum is greatly saved. Therefore, the impedance spectrum model provided by the invention has great advantages in EIS fitting compared with the traditional electrochemical model.
The specific meanings of the terms in the present invention are as follows:
P2D model: the Pseudo-two-dimensional (Pseudo-2-dimension) model is a model which is used for describing processes such as solid-liquid phase diffusion, material conservation, charge conservation and the like in a lithium ion battery by using an electrochemical principle according to the basic structure of the lithium ion battery and simulating the characteristics such as voltage, capacity and the like of the lithium ion battery.
Diffusion: the phenomenon that a certain component in different regions moves from a region with a high concentration to a region with a low concentration.
Electric double layer: any two different phases in contact will create an electrical potential between the two phases due to charge separation. Two phases have excessive charges respectively, the electric quantity is equal, the signs are opposite, and the two phases attract each other to form an electric double layer.
SEI film: in the first charge and discharge process of the liquid lithium ion battery, the electrode material and the electrolyte react on a solid-liquid phase interface to form a passivation layer covering the surface of the electrode material. This passivation layer is an interfacial layer, which has the characteristics of a solid electrolyte, is an electronic insulator but is an excellent conductor of Li +, and Li + can be freely inserted and extracted through the passivation layer, so this passivation film is called a "solid electrolyte interface film" (SEI film for short). The SEI film may be divided into an organic layer and an inorganic layer according to the composition of the SEI film, wherein one layer adjacent to the electrode is an inorganic layer, referred to as the inorganic layer of the SEI film, and the other layer is composed of an organic layer, referred to as the organic layer of the SEI film.
Electrochemical Impedance Spectroscopy (EIS for short): applying an alternating current signal with different frequencies and small amplitude to an electrochemical system, and measuring the change of the ratio of the voltage to the current of the alternating current signal (the ratio is the impedance of the system) along with the frequency omega of a sine wave or the change of the phase angle phi of the impedance along with omega. Further, electrode process kinetics, electric double layers, diffusion and the like are analyzed, and mechanisms such as electrode materials, solid electrolytes, conductive polymers, corrosion protection and the like are researched.
Nyquist plot: a graph in which a frequency response is represented in polar coordinates by its amplitude-frequency characteristic and phase-frequency characteristic is called a phase-amplitude diagram or Nyquist diagram.
SOC: statedischarge, is the ratio of the current capacity of the battery to the maximum capacity.
SOH: stateful, which is the ratio of the current maximum available capacity of the battery to the rated capacity.

Claims (1)

1. A lithium ion battery impedance model based on electrochemical principle is characterized in that:
total impedance of the battery: z (ω) ═ j ω Lw+Rohm+ZSEI，ino+ZSEI，org+ZDL+Zele+Zsld；
Liquid phase impedance:
Figure FDA0002872524510000011
Figure FDA0002872524510000012
solid phase impedance: zsld，i＝AB(C+Dj)，i＝n，p，
Figure FDA0002872524510000013
Figure FDA0002872524510000014
Electric double layer impedance:
Figure FDA0002872524510000015
Figure FDA0002872524510000016
SEI organic layer resistance:
ZSEI，org＝A(B+Cj)，
Figure FDA0002872524510000017
Figure FDA0002872524510000018
SEI inorganic layer resistance:
ZSEI，ino＝A(B+Cj)，
Figure FDA0002872524510000019
Figure FDA00028725245100000110
in the lithium ion battery impedance model, j is an imaginary unit, ω is an angular frequency of an excitation applied to the lithium ion battery, LwThe lead inductance and the lead refer to leads connected with two poles of the battery in the impedance spectrum test of the lithium ion battery, RohmOhmic internal resistance, D, inside the lithium ion batteryeleIs the liquid phase diffusion coefficient of lithium ion, Dsld，nIs the negative electrode lithium ion solid phase diffusion coefficient, D, of the lithium ion batterysld，pIs the positive electrode lithium ion solid phase diffusion coefficient, D, of the lithium ion batteryorgLithium ion diffusion coefficient, D, of organic layer of SEI film for lithium ion batteryinoIs the diffusion coefficient of lithium ions in an inorganic layer of an SEI film of the lithium ion battery, K is the electrochemical reaction constant at an electric double layer inside the lithium ion battery,
Figure FDA00028725245100000111
is the liquid phase transfer coefficient of lithium ion in the lithium ion battery,
Figure FDA00028725245100000112
is the equivalent total thickness L of the cathode, the diaphragm and the anode of the lithium ion batterynThickness L of negative electrode plate of lithium ion batterysIs the thickness of the diaphragm of a lithium ion battery、LpThickness of positive electrode plate of lithium ion batterynIs the porosity of the negative electrode, epsilon, of a lithium ion batterysPorosity of the membrane, epsilon, for lithium ion batteriespPorosity of the positive electrode, a, of a lithium ion batterynIs the specific surface area of the negative electrode of the lithium ion battery, apIs the specific surface area of the positive electrode of the lithium ion battery, AnArea of negative electrode plate of lithium ion battery, ApArea of positive electrode plate r of lithium ion batterynRadius of spherical particles r of negative electrode of lithium ion batterypIs the radius of the positive spherical particle of the lithium ion batteryorgPorosity, l, of an organic layer of an SEI film for a lithium ion batteryorgThickness of organic layer of SEI film for lithium ion batteryinoThickness of SEI film inorganic layer for lithium ion battery, cele，0The initial concentration of liquid-phase lithium ions in the lithium ion battery,
Figure FDA0002872524510000021
Initial concentration of solid-phase lithium ions of the negative electrode of the lithium ion battery,
Figure FDA0002872524510000022
Is the initial concentration of solid-phase lithium ions of the positive electrode of the lithium ion battery,
Figure FDA0002872524510000023
The maximum concentration of solid-phase lithium ions at the negative electrode of the lithium ion battery,
Figure FDA0002872524510000024
Is the maximum concentration of solid-phase lithium ions of the positive electrode of the lithium ion battery, corg，0Initial concentration of lithium ions in organic layer of SEI film of lithium ion battery, cino，0The initial concentration of lithium ions in an SEI film inorganic layer of the lithium ion battery, F is a Faraday constant, and R is a molar gas constant.
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