Distribution order equivalent circuit modeling method for representing aging state of lithium battery

Abstract

The invention discloses a distribution order equivalent circuit modeling method for representing the aging state of a lithium battery, and relates to the technical field of lithium battery modeling. The method comprises the following steps: firstly introducing a distribution order calculus theory into equivalent circuit modeling of the lithium ion battery, defining a distribution order weight function in a continuous order interval, then introducing a double-Beta distribution hybrid model to carry out parametric modeling on the distribution order weight function, and further providing an identification strategy for mutual decoupling of an intensity parameter and a normalized weight function. And meanwhile, in combination with weighted nonlinear least square and physical constraint optimization, stable and interpretable identification of continuous order distribution is realized. Therefore, a characterization index system of the aging state of the lithium battery is constructed, the mapping relation between the characterization index system and a specific aging mechanism is determined, and quantitative expression of hidden aging information in a continuous time scale from the impedance spectrum is achieved. The problem that traditional integer order and single fractional order equivalent circuit models are limited in the aspect of describing the multi-time-scale electrochemical dynamic behavior of the lithium ion battery is solved.

Description

Technical Field

[0001] This invention relates to the field of lithium battery modeling technology, and in particular to a distributed-order equivalent circuit modeling method for characterizing the aging state of lithium batteries. Background Technology

[0002] With the rapid development of the new energy industry, lithium-ion batteries are widely used in electric vehicles, energy storage systems, and portable electronic devices. Their performance degradation has a significant impact on the safety, reliability, and economy of these systems. Therefore, accurately characterizing battery aging and establishing high-precision models has become a key requirement for battery management systems (BMS) and equipment health management technologies (EHM).

[0003] During long-term cycling and storage, lithium-ion batteries are affected by cycling stress, temperature gradients, and electrochemical side reactions, resulting in aging phenomena such as capacity decay, increased internal impedance, and accelerated temperature rise. Although traditional integer-order equivalent circuit models (such as Rint, Thevenin, and multi-relaxed RC networks) have advantages such as simple structure and high real-time performance, their parameter dimensions are limited, making it difficult to effectively characterize the complex electrochemical kinetic processes under aging conditions that are multi-scale, nonlinear, and diffusion-dominated.

[0004] In recent years, numerous electrochemical impedance spectroscopy (EIS) studies have shown that the impedance spectrum of batteries exhibits a distinct "distribution order" characteristic after aging. That is, its frequency response is no longer described by a single fixed fractional-order element, but rather shows a continuous distribution of orders within a certain range. This reflects the coupling effects of multiple physical processes, such as the elongation of the solid-phase diffusion path, the growth of the solid-electrolyte interphase (SEI), and the increase in charge transfer impedance. Therefore, establishing a distribution order model that can reflect the aging state of batteries is of great significance for accurately tracking the state of health (SOH) and remaining lifetime. The fractional-order equivalent circuit model (FO-ECM) uses constant-phase elements (CPEs) or Warburg diffusion impedance elements to replace traditional RC branches, and can reproduce EIS curves with fewer parameters over a wider frequency domain. It has been used for online impedance estimation and SOH diagnosis. However, most existing models use a single or a few fixed orders, which makes it difficult to capture the distributed frequency response characteristics that change significantly with aging, resulting in a decrease in model accuracy as the aging state deepens.

[0005] To address the aforementioned shortcomings, Chinese invention patent CN104392080B discloses a fractional-order variable-order equivalent circuit model for lithium batteries and its identification method. This model extends the second-order RC circuit model to non-integer orders and identifies model parameters and fractional orders at different State of Charge (SOC) using the least squares method, thus obtaining a fractional-order equivalent circuit model that varies according to SOC. Based on the FO-ECM, this patent introduces a "variable-order" CPE element that can automatically adjust with the operating point (SOC, temperature, cycle number), and establishes an explicit mapping between the order and aging factor through piecewise linear or polynomial fitting, thereby improving the model's ability to track aging states to some extent. However, this method still uses a finite number of discrete-order fractional-order elements to equivalently describe the complex electrochemical processes inside the battery. The fractional orders are usually assumed to be constant, making it difficult to characterize the continuous time-scale distribution characteristics of lithium-ion batteries at different frequency ranges and aging stages. This results in low-precision and inconsistent characterization of the complex impedance characteristics of the battery under different SOC and cyclic aging states. Summary of the Invention

[0006] To overcome the limitations of traditional integer-order and single-fractional-order equivalent circuit models in describing the multi-timescale electrochemical kinetics of lithium-ion batteries, this invention proposes a distributed-order equivalent circuit modeling method for characterizing the aging state of lithium batteries. This method achieves a unified and high-precision characterization of the complex impedance characteristics of batteries under different SOC and cyclic aging states. The provided technical solution is as follows: On the one hand, a distributed-order equivalent circuit modeling method for characterizing the aging state of lithium batteries is provided, the method comprising: S1, Obtain electrochemical impedance spectroscopy data of lithium batteries at different aging stages: Under preset ambient temperature and state of charge conditions, apply sinusoidal AC excitation to lithium batteries at different cycle aging stages to obtain broadband electrochemical impedance spectroscopy data covering the low-frequency range to the high-frequency range. S2, Establish a distributed-order equivalent circuit model: Based on the distributed-order calculus theory, a distributed-order equivalent circuit model of lithium-ion battery is constructed. In the distributed-order equivalent circuit model, an order weight function is introduced to describe the weight distribution of the dynamic processes corresponding to different orders in the overall polarization response. The order weight function is parameterized using the Beta distribution function. S3, Model Parameter Identification and Inversion: The distributed-order equivalent circuit model constructed in S2 is matched with the broadband electrochemical impedance spectroscopy data obtained in S1, and the model parameters are jointly identified using a weighted nonlinear least squares optimization method.

[0007] On the other hand, a lithium battery aging state analysis method based on the modeling method is provided. The method includes: constructing a characterization index system for the aging state of lithium batteries based on the distributed-order equivalent circuit model obtained by the modeling method and the identified model parameters.

[0008] The beneficial effects of the technical solutions provided by the embodiments of the present invention include at least the following: 1. Compared with traditional integer-order or single fractional-order equivalent circuit models, this invention introduces a distributed-order equivalent circuit model, which realizes a unified modeling of the wide-band electrochemical impedance characteristics of lithium-ion batteries under different SOC and different cycle aging conditions. It can maintain high fitting accuracy in the low-frequency to high-frequency range and effectively overcome the problem of insufficient applicability of existing models in multi-timescale dynamic characterization.

[0009] 2. The distributed-order equivalent circuit model constructed in this invention has a parameter system that can establish a correspondence with various electrochemical processes inside the battery, such as ohmic conduction, interface polarization, and diffusion restriction, at the physical level. This makes the model parameters and their evolution trends with SOC and cycle aging have clear mechanistic orientation, thereby improving the physical interpretability of battery aging state analysis.

[0010] 3. By jointly analyzing the distributed order capacitance, the order weight function and its derived characteristic parameters (including the distributed order centroid, the order distribution width and the weights of high and low orders), this invention constructs a parameterized feature system that can characterize the aging process of lithium-ion batteries from multiple time scales and multiple physical mechanisms. This provides high-quality feature inputs with intrinsic physical meaning for SOC / SOH joint estimation, aging state identification and lifetime prediction methods.

[0011] It should be understood that the description in this section is not intended to identify key or essential features of the embodiments of this disclosure, nor is it intended to limit the scope of this disclosure. Other features of this disclosure will become readily apparent from the following description. Attached Figure Description

[0012] The accompanying drawings are provided for a better understanding of this solution and do not constitute a limitation of this application. Wherein: Figure 1 This is a flowchart of a distributed-order equivalent circuit modeling method for characterizing the aging state of lithium batteries, provided by an embodiment of the present invention. Figure 2 This is a schematic diagram of the distributed-order equivalent circuit model structure of a lithium battery provided in an embodiment of the present invention; Figure 3 This is a schematic diagram of EIS fitting under different SOCs with different number of cycles provided in the embodiments of the present invention; Figure 4This is a schematic diagram of EIS fitting accuracy under different SOCs with different number of cycles provided in the embodiments of the present invention; Figure 5 This is a schematic diagram of the distribution weight function under different SOCs with different number of cycles provided in the embodiments of the present invention; Figure 6 This is a schematic diagram of the distributed capacitance under different SOCs at different cycle numbers provided in the embodiments of the present invention; Figure 7 This is a schematic diagram of R0 under different SOCs with different number of cycles provided in the embodiments of the present invention; Figure 8 This is a schematic diagram of R1 under different SOCs for different number of cycles provided in the embodiments of the present invention; Figure 9 This is a schematic diagram of the distribution order intensity A under different SOCs with different number of cycles provided in this embodiment of the invention; Figure 10 This is a schematic diagram of the centroid of the distribution order under different SOCs with different number of cycles provided in the embodiments of the present invention; Figure 11 This is a schematic diagram of the order distribution width under different SOCs with different number of cycles provided in the embodiments of the present invention; Figure 12 This is a schematic diagram of different SOCs under different number of cycles provided in the embodiments of the present invention; Figure 13 This is a schematic diagram of different SOCs under different number of cycles provided in the embodiments of the present invention; Detailed Implementation

[0013] The following description, in conjunction with the accompanying drawings, illustrates exemplary embodiments of this application, including various details to aid understanding. These should be considered merely exemplary. Therefore, those skilled in the art will recognize that various changes and modifications can be made to the embodiments described herein without departing from the scope and spirit of this application. Similarly, for clarity and brevity, descriptions of well-known functions and structures are omitted in the following description.

[0014] First, we will explain the distributed order calculus (DOC). As a further generalization of integer and fractional order calculus, DOC allows the order of the differential or integral to be distributed over a continuous interval, thus enabling the description of more complex multi-scale and multi-rate processes. In electrochemical impedance spectroscopy (EIS), the distributed order model treats polarization behavior as a weighted superposition of responses of different orders (corresponding to different time scales), which is more consistent with the physical characteristics of actual electrochemical processes compared to the traditional fixed-order model. The general form of the distributed order differential operator can be expressed by equation (1): (1) In the formula, Let be the order of the differential. The order of the differential The range of values ​​for is generally defined as follows: , Let be the order weight function, describing the distribution density of different orders in the overall dynamics, and satisfying the normalization condition. , The order of the differential Fractional differential operators, For definition in The time function.

[0015] In frequency domain analysis, the frequency domain operator corresponding to the distributed derivative is shown in equation (2): (2) In the formula, Indicates Fourier transform, Represents the multiplicative operators in the frequency domain. Let represent the function after the Fourier transform, and The mathematical expression shown in equation (3) defines: (3) Since the actual battery interface exhibits non-ideal capacitive behavior, a constant phase angle element (CPE) is usually used to replace the ideal capacitor, and the impedance of the CPE is shown in equation (4): (4) In the formula, The impedance form of the CPE, For generalized capacitance, This represents the fractional differential operator in the frequency domain. Let be the angular frequency, and , For fractional order, For frequency.

[0016] However, a single CPE can only describe a single diffusion process, making it difficult to capture the complex behavior of multiple overlapping relaxation timescales, and its parameters have relatively insufficient physical interpretability. In contrast, the distributed order method provides a new modeling approach for characterizing the dynamics of complex interfaces.

[0017] Based on this theory, embodiments of the present invention provide a distributed-order equivalent circuit modeling method for characterizing the aging state of lithium batteries. For example... Figure 1 The flowchart shown is a distributed-order equivalent circuit modeling method for characterizing the aging state of lithium batteries. The processing flow of this method may include the following steps: S1, Obtain electrochemical impedance spectroscopy data of lithium batteries at different aging stages: Under preset ambient temperature and state of charge conditions, apply a small-amplitude sinusoidal AC excitation to lithium batteries at different cycle aging stages to obtain broadband electrochemical impedance spectroscopy data covering the low-frequency to high-frequency range.

[0018] Specifically, the broadband electrochemical impedance spectroscopy data must cover at least the low- to high-frequency range that makes the ohmic resistance, interfacial polarization, and diffusion impedance distinguishable in the frequency domain, ensuring the identifiability of each sub-parameter in the distributed-order equivalent circuit model. In other words, it must include at least the low-frequency impedance response dominated by ion diffusion processes, and the mid- to high-frequency impedance response dominated by interfacial polarization and charge transfer processes. The high-frequency range supports the identification of the ohmic resistance parameter R0, while the mid-frequency range supports the order weighting function in the distributed-order equivalent circuit. The inversion of its Beta distribution parameters, identification of the Warburg diffusion coefficient and low-order distribution characteristics in the low-frequency region.

[0019] For example, a nickel-cobalt-manganese (NCM) ternary lithium-ion battery with a nominal capacity of 3.2 Ah was selected as the test object. Electrochemical impedance spectroscopy (EIS) was performed at an ambient temperature of 35℃ and a discharge rate of 1C. To systematically analyze the effects of cycle aging and state of charge (SOC) on the battery's dynamic characteristics, EIS measurements were conducted after 300, 400, 500, and 600 cycles at SOCs of 0.1, 0.2, 0.4, 0.6, 0.8, and 1.0, respectively.

[0020] S2: Establishing a Distributed-Order Equivalent Circuit Model: Based on distributed-order calculus theory, a distributed-order equivalent circuit model of a lithium-ion battery is constructed. By introducing a continuous-order variable β into the equivalent circuit, traditional fixed-order or single-fractional-order capacitor elements are replaced with distributed-order capacitor elements, and a frequency-domain integral expression for the distributed-order impedance is established.

[0021] The equivalent circuit model of a lithium battery based on a distributed-order first-order RC network is shown below. Figure 2 As shown. By Figure 2 As can be seen, in the Distributed Order Equivalent Circuit Model (DO-ECM), R0 is the ohmic internal resistance of the lithium battery, mainly reflecting the intrinsic resistance characteristics of components such as the electrolyte, electrode current collector, and electrode contact interface; R1 is the polarization internal resistance of the lithium battery, used to describe the impedance characteristics when charge transfer occurs at the electrode interface, i.e., charge transfer resistance; W is the Warburg diffusion impedance element of the lithium battery, used to characterize the diffusion impedance caused by the diffusion process of lithium ions inside the electrode material; C eq This refers to the distributed-step capacitance of a lithium battery, used to represent the multi-scale transport process of the electrodes and its distributed-step characteristics.

[0022] Depend on Figure 2It is known that the ohmic resistor is connected in series with one end of the distributed polarization branch, and the other end of the distributed polarization branch is connected in series with the Warburg diffusion impedance element. Specifically, at the high frequency end, the ohmic internal resistance R0 is first connected in series to capture the intrinsic resistance of the electrolyte, current collector, and contact interface; at the mid frequency end, the polarization internal resistance R1 is connected in parallel with the distributed voltage, and then connected in series with the aforementioned R0. This parallel branch forms a flattened Nyquist arc, thereby simultaneously reflecting the charge transfer process and the diffusion effect across multiple time scales; at the low frequency end, the Warburg diffusion impedance element is further connected in series, with its real and imaginary impedance parts being equal and a phase angle of 45°, used to characterize the semi-infinite diffusion behavior of lithium ions in the electrode material.

[0023] Specifically, based on the definition of distributed-order calculus, by introducing a continuous distribution function within the order interval to describe the polarization and relaxation processes at different time scales, distributed-order capacitances of all orders can be superimposed in a weighted form, extending the discrete model of multiple CPEs in series to a continuously distributed form. Assuming that... The CPE components are connected in series, and their total impedance is shown in equation (5): (5) In the formula, for The total impedance of each CPE component Generally, 101 is chosen. CPE component index ( ), For the first The admittance coefficient of a CPE For the first The fractional index of each CPE satisfies .

[0024] To extend the discrete series form to continuous order intervals, the interval is... If divided, then satisfy: (6) According to equation (6) in the first Representative points are taken from the small interval. By approximating the discrete terms, the discrete summation in equation (5) can be rewritten in terms of order. Riemann and. Meanwhile, continuous functions are introduced. Representing the scale of each order point (in the discrete case) (continuous extension), and with Representing an interval The order distribution density on the [aspect] allows for a continuous description of the discrete terms. Then, on the [aspect]... The interval is used to approximate the corresponding discrete terms as: (7) Substitute equation (7) into equation (5) and then... Summing yields the Riemann sum approximation: (8) If function In the interval It is Riemann integrable and satisfies the following conditions during the partitioning refinement process. (i.e., taking the Riemann limit), and with the support of the dominance convergence theorem or other equivalent conditions, the order of the limit and the summation (or the integral and the Fourier / Laplace transform) can be interchanged. Then, the aforementioned Riemann sum converges to the corresponding integral expression when the partition is sufficiently fine. These other equivalent adjustments refer to certain mathematical conditions that need to be satisfied when interchanged with the summation or integral to ensure the legality of the operation, such as the dominance convergence theorem, uniform convergence, and absolute convergence.

[0025] Furthermore, in parameter identification, only the product can usually be obtained. (or identify separately under normalization conditions) and after normalization For powers To maintain phase definition consistency, principal value support must be used: when At that time, .

[0026] When all the above conditions are met, Then, the impedance of the distributed-order capacitance in continuous order form (also known as the distributed-order impedance) can be obtained, as shown in equation (9): (9) remember And let , The impedance forms of a standard distributed step capacitor are: (10) Furthermore, in the distributed-order equivalent circuit model, an order weight function is introduced to describe the weight distribution of the dynamic processes corresponding to different orders in the overall polarization response. The order weight function is parameterized using a Beta distribution function, so that the degrees of freedom of the distributed-order model are concentrated on a small number of physically meaningful shape parameters, thereby avoiding overfitting and improving the stability of aging characteristics. To reduce the model's degrees of freedom and improve the stability of parameter identification, it is assumed in equation (11) that... It is approximately a constant within the order interval, that is, it satisfies: (11) At this point, the impedance expression for the distributed-step capacitor simplifies to: (12) In the formula, The impedance of the distributed step capacitor, The intensity parameter of the distribution order (dimension: ), The weight function for the distributed impedance reflects multi-scale dynamics. It is a fractional exponent, and , fractional exponent The impedance expression for CPE, The excitation angular frequency.

[0027] As can be seen from equation (12), when performing complex logarithmic transformation and impedance decomposition, the complex number can be... Represented in polar coordinates, that is: (13) Taking the natural logarithm of both sides of equation (13), we get: (14) In equation (14), the principal value branch is used to calculate the complex phase angle, so that the phase angle is always maintained at the same level. Within this range, to ensure the consistency and continuity of phase representation in the Nyquist plane. This is achieved using the complex exponential identity of the power function, i.e.: (15) Substituting equation (15) into equation (12), the expression for the impedance of the distributed step capacitor is: (16) Expanding the complex logarithm further, we get: (17) Therefore, the impedance of the distributed step capacitance can be further expressed as: (18) Expanding equation (18) using Euler's formula, the complex exponential term can be decomposed into real and imaginary parts, thus allowing for the analysis of the frequency domain characteristics of the impedance expression. Euler's formula is shown in equation (19). (19) Decompose equation (18) into real and imaginary parts, i.e.: (20) Therefore, the real and imaginary parts of the impedance of the distributed-order capacitor are: (twenty one) In the formula, This represents the real part of the impedance of the distributed-order capacitance, which reflects the energy dissipation behavior of the system and is related to loss mechanisms such as charge transfer and interface polarization. This represents the imaginary part of the impedance of the distributed-order capacitance, characterizing its energy storage behavior and corresponding to charge accumulation or diffusion control features. The integral kernel reflects the power-law dependence of different orders on frequency, representing the frequency domain characteristics of the multi-scale relaxation process, and includes trigonometric function terms. and The phase shift characteristics determine the phase ratio of different orders, reflecting the projection of multi-scale relaxation characteristics in the frequency domain.

[0028] Specifically, in the parameterized representation of weight functions, the Beta distribution is often used to characterize the order exponent in the interval [missing information]. The probability density function of a continuously varying weight distribution can be expressed as: (twenty two) In the formula, For Beta functions, and For shape parameters.

[0029] By adjusting the shape parameters and The Beta distribution allows for flexible control of the weight distribution across different order intervals, effectively characterizing the relative influence of each relaxation scale. Due to its high plasticity, the Beta distribution can effectively describe the weight structure at different relaxation scales in electrochemical systems, providing a physically meaningful parameterized expression for distribution-order models.

[0030] It is also important to understand that, given the coexistence of multiple relaxation mechanisms during the cycling aging process of lithium-ion batteries, a composite order weighting function is constructed using a weighted mixture of two Beta distribution functions. By introducing a mixing ratio η, the weights of the relaxation processes corresponding to the two types of Beta distributions are adjusted to achieve a unified characterization of multi-peak, wide-distribution, or overlapping relaxation behaviors.

[0031] To characterize the multiple relaxation mechanisms that may coexist at the electrochemical interface, a weighting function is constructed using a weighted mixture of two Beta distributions. That is, the composite order weight function: (twenty three) In the formula, This is the mixing ratio, used to adjust the relative weights of the two processes. The shape parameter of the first Beta distribution is used to characterize the mid-frequency relaxation process of charge transfer. The shape parameter of the second Beta distribution characterizes low-frequency diffusion or high-frequency double-layer effect.

[0032] This mixing model can accurately characterize common multi-peak, broad-distribution, or overlapping relaxation behaviors in electrochemical systems. Specifically, it makes preliminary assumptions about the range of values ​​for the mixing ratio based on two well-defined physical processes (such as charge transfer and diffusion, or double-layer polarization and charge transfer), and their respective frequency ranges or distribution characteristics corresponding to different shape parameters of the Beta distribution.

[0033] In the distributed order model, the weight function must satisfy the normalization condition, specifically as follows: (twenty four) In numerical implementation, the discretized weight function is usually normalized using a trapezoidal integral, as follows: (25) In the formula, Indicates will Normalization is performed to the form of a probability density function. This formula eliminates the overall scale factor, ensuring that the composite order weight function represents only the "relative weights / relative probability distribution," rather than a dimensional intensity function. Through this normalization, the scale parameter can be... Effective decoupling from the weight function improves the physical interpretability and numerical stability of the model parameters.

[0034] In numerical implementation, trapezoidal integrals are typically used to calculate the normalization factor on discrete-order grids or at discrete frequency points to ensure numerical stability and feasibility of the calculation process. When the distributed-order capacitance impedance... With resistance When connected in parallel, the total impedance can be obtained from the parallel admittance: (26) Taking the reciprocal gives the total parallel impedance: (27) Equation (27) will serve as the core combination structure in subsequent frequency domain calculations. To further obtain the real and imaginary parts that can be directly used for numerical calculations, let the distributed order impedance be: (28) The imaginary and real parts are as follows: (29) Substituting into the parallel formula (27), we get: (30) Rationalize the denominator, that is: (31) Expanding and rearranging the real and imaginary parts, we get: (32) Therefore, the real and imaginary parts of the parallel impedance are: (33) In characterizing diffusion impedance, a Warburg diffusion impedance element is typically introduced. For a one-dimensional semi-infinite diffusion system, a small-signal perturbation at the interface causes a concentration change, which follows Fick's second law. Its evolution is described by the diffusion equation, namely: (34) In the formula, Indicates the concentration distribution of diffusing substances. This represents the diffusion coefficient, used to characterize the diffusion rate; This represents the rate of change of concentration at a certain point over time. This represents the diffusion driving force determined by the second derivative of the spatial concentration gradient.

[0035] Among them, the boundary The normal flux at a point is determined by the current density, and the expression is: (35) In the formula, It represents current density, which is the flux of electrons participating in electrochemical reactions per unit area. Indicates the number of electrons transferred in the reaction. represents the Faraday constant. This formula reveals the diffusion-controlled characteristics at the electrode interface. After performing a Laplace or Fourier transform on this diffusion equation, the expression for the Warburg impedance in the frequency domain can be obtained, i.e.: (36) The expression for the Warburg coefficient is as follows: (37) This impedance reflects the typical effect caused by semi-infinite diffusion. Frequency-dependent behavior. The frequency-dependent behavior in equation (36) is... Complex number factorization is performed as follows: (38) Substituting equation (38) into equation (36) and rationalizing the denominator, we obtain the expression for the Warburg impedance in the frequency domain after rationalizing the denominator: (39) Use again conjugate The conversion is performed, including: (40) Therefore, the Warburg diffusion impedance in the frequency domain has the following form: (41) The real part and the imaginary part are: (42) Therefore, the real and imaginary parts of the Warburg impedance have the same magnitude, and the phase angle is constant. And its amplitude varies with frequency. The regular decay reflects the typical electrochemical impedance characteristics of a semi-infinite diffusion process.

[0036] according to Figure 2 As shown, the total impedance of the DO-ECM consists of three parts connected in series: an ohmic resistor, a distributed-step capacitor-resistor parallel branch, and a Warburg diffusion impedance. That is, the total impedance of the distributed-step equivalent circuit model includes: the impedance of the ohmic resistor, the impedance of the distributed-step polarization branch, and the impedance of the diffusion impedance element. The impedance of the distributed-step polarization branch includes the impedance of the distributed-step capacitor and the polarization internal resistance. Its total impedance is shown by equation (43): (43) In the formula, It is the ohmic resistance (the contact resistance of the electrolyte, current collector, and electrode interface). The impedance of the distributed-order RC parallel branch, Let be the Warburg diffusion impedance. Expanding each part, the real and imaginary parts of the total impedance are: (44) In the formula, , It is defined by equations (28) and (29).

[0037] To facilitate numerical solutions, the distribution order interval is... Perform uniform discretization and divide into There are equidistant sampling points, as follows: (45) Usually selected A good balance is achieved between computational accuracy and computational load.

[0038] For each frequency point M represents the number of frequency points, the specific number of which is determined by the measured EIS, and its corresponding kernel matrix elements. Defined as: (46) Specifically, the calculation steps for the kernel matrix elements are as follows: 1) Calculate the complex logarithm, specifically: (47) 2) Calculate the exponent term, specifically: (48) 3) Perform complex exponentiation to obtain .

[0039] The kernel matrix elements describe different orders. For a given frequency The response characteristics under these conditions are the core of distributed order impedance reconstruction. Based on the trapezoidal integral method, the distributed order integral can be approximated as: (49) in The trapezoidal integral weighting coefficients are expressed as follows: (50) It is important to understand that the truncation error of the trapezoidal integral described above is of order [order missing]. When taking At that time, the grid step size is The corresponding integral error is approximately The order of magnitude is sufficient to meet the accuracy requirements of distributed order models in frequency domain solutions.

[0040] Specifically, formula (12) gives the continuous form of the distributed-order impedance, i.e., the "infinite-dimensional" function. The integral of formula (49) is truncated into a finite (101) segment using the trapezoidal integral, transforming the integral into a weighted sum. This approach transforms the distributed-order equivalent circuit model from a theoretical formula into an algorithm core that can be identified in real time and embedded in engineering. Otherwise, each iteration requires re-integration, increasing the computational load.

[0041] S3: Model Parameter Identification and Inversion: The distributed-order equivalent circuit model constructed in S2 is matched with the broadband electrochemical impedance spectroscopy data obtained in S1. A weighted nonlinear least squares optimization method is used to jointly identify the model parameters. By minimizing the weighted error between the model impedance and the experimental impedance, the set of model parameters, including ohmic internal resistance, polarization internal resistance, distributed-order intensity parameter, mixing ratio η, Beta distribution shape parameter, and Warburg diffusion coefficient, is obtained through inversion.

[0042] Specifically, in the parameter identification process, a weighted nonlinear least squares optimization method is used to solve for the jointly identified model parameters, denoted as the parameter vector to be identified. Defined as: (51) In the formula, each parameter corresponds to the ohmic resistance, the parallel branch resistance, the weighting function amplitude, the mixing coefficient, the shape parameters of the two sets of Beta distributions, and the Warburg coefficient.

[0043] Parameter identification uses the weighted least squares criterion, and the objective function is in the form of: (52) In the formula, The impedance value was measured experimentally at the following frequency point. , The impedance value predicted by the model, corresponding to the parameter vector. and frequency points are , These are the model parameters to be identified; This represents the total number of frequency sampling points. The weighting coefficient is usually taken as the reciprocal of the impedance amplitude, as shown in equation (53), to balance the influence at different frequency points, specifically: (53) The weighting coefficients are used to balance the influence of different frequency ranges on the objective function, prevent low-frequency high-impedance points from dominating the optimization, and enhance the sensitivity of high-frequency bands to parameter changes, preventing high-frequency low impedance from being ignored due to its low magnitude, thereby enhancing the overall stability of the fitting.

[0044] To ensure the physical rationality and numerical stability of the model solution process, the parameter constraints are set as follows: (54) It should be noted that the parameter constraints have clear physical and numerical meanings, specifically: Resistance and intensity parameters must be kept non-negative to conform to the basic physical laws of electrochemical systems.

[0045] Mixing ratio Restricted to Within the interval, to ensure that the weight function remains an effective convex combination.

[0046] The shape parameter of the Beta distribution is constrained by... Within this range, the setting is mainly based on considerations of physical interpretability and numerical stability. The upper limit can avoid the weight function from having an excessively sharp extreme distribution, and the lower limit can avoid the weight function from having an abnormally flat extreme distribution. At the same time, this range covers the continuous or asymmetric relaxation spectrum characteristics commonly found at electrochemical interfaces, thereby avoiding numerical oscillations or integral instability and improving the overall robustness of parameter identification.

[0047] Regarding the initial value estimation strategy, the high-frequency band mainly exhibits ohmic resistance characteristics, therefore the constraint expression is: (55) The low-frequency band includes polarization impedance, which can be used to estimate... The restricted expression is: (56) In the initial setting of the distribution order parameters, the amplitude intensity The average magnitude of the experimental impedance modulus is taken to provide a reasonable initial scale that matches the actual frequency domain response. The initial magnitude of the distributed order parameter is calculated as shown in equation (57).

[0048] (57) Considering that the importance of the two types of distributions is difficult to predict in the early stages of optimization, the initial value of the mixing ratio can be an empirical compromise value to avoid biasing the initial value towards a particular distribution shape, i.e., satisfying: (58) To improve the convergence of the optimization, the Beta distribution parameters are given physically meaningful prior settings, including two sets of settings.

[0049] The first group uses a right-skewed distribution parameter initial setting, specifically: (59) The second group uses a symmetrically distributed parameter initial setting, specifically: (60) The initial value of the Warburg coefficient is the amplitude intensity ratio, and the constraint expression is: (61) In the aforementioned electrochemical impedance spectroscopy (EIS) tests, the EIS fitting results of the proposed distributed-order equivalent circuit model under different SOC and cycle number conditions are as follows: Figure 3 As shown, and Figure 3(a)-(d) show the EIS fitting results for 300, 400, 500, and 600 cycles, respectively. The Nyquist plots demonstrate that the model can accurately describe the impedance characteristics of the high-frequency ohmic region, mid-frequency polarization region, and low-frequency diffusion region simultaneously over a wide frequency range, indicating that the distributed-order modeling method has a significant advantage in characterizing electrochemical processes across multiple timescales. Notably, when the SOC approaches 1, the battery is in a critical condition with high potential and low reversible buffering. Interface charge transfer is limited, diffusion driving force is weakened, and side reactions are significantly enhanced. Multiple non-ideal processes are activated simultaneously, causing the system to deviate from the weakly linear steady-state conditions assumed by the electrochemical impedance spectroscopy. Under this state, the kinetics shift from mid-to-high-order polarization processes to low-order diffusion and side reactions, manifested as amplified polarization impedance, a shift and significant broadening of the order distribution to lower orders, and discontinuous or "abnormal" changes in related parameters. Therefore, the parameter anomalies at SOC=1 are not model failures or fitting errors, but rather a true physical reflection of battery kinetic degradation and a sudden increase in heterogeneity under high SOC conditions. They are more suitable for aging scale-up and mechanism analysis than for direct use as quantitative indicators of steady-state SOH.

[0050] Based on the distributed-order equivalent circuit model obtained by the aforementioned modeling method, and the identified distributed-order model parameters, a characterization index system for the aging state of lithium batteries is constructed, including: ohmic internal resistance and polarization internal resistance parameters; equivalent value of distributed-order capacitance; centroid of the order weight function; distributed-order intensity; order distribution width; and weight ratios corresponding to low-order and high-order intervals. By analyzing the systematic evolution of the above characteristic parameters with the number of cycles and SOC, a quantitative correlation between battery aging state and dynamic structural changes is established, providing multi-dimensional and interpretable characteristic evidence for aging mechanism analysis and health status assessment.

[0051] Specifically, to extract key indicators characterizing the evolution of electrochemical processes from the frequency domain distribution-order model, a statistical feature extraction and aging analysis method was constructed. The first moment (distribution centroid) is used to measure the overall shift trend of the distribution-order weight function. The distribution centroid is defined as the distribution-order weight function... The mathematical expectation of . The specific expression is: (62) In numerical implementation, the centroid of the distribution can be calculated using a discrete weighted summation, specifically: (63) It is important to understand that during the aging process, the centroid of distribution... The offset can provide crucial information indicating degradation: when When the capacitance shifts to the left (decreases), it indicates a relative increase in the capacitive characteristics of the system, often caused by the continuous thickening of the SEI film leading to an increase in interfacial capacitance, typically reflecting SEI growth-type aging; conversely, when... When the movement to the right (increases), the resistive behavior becomes more pronounced, which is mostly related to the increase in charge transfer resistance caused by the loss of active material, corresponding to the aging mechanism of active material failure.

[0052] To further reveal the intrinsic relationship between aging mechanisms and distribution width, the physical significance of the second-order central moment (i.e., distribution width) as a key aging indicator is analyzed in depth. Distribution width characterizes the distribution order weight function within the interval... The degree of dispersion on the distribution width is defined as follows: (64) Understandably, the distribution width It can directly reflect the concentration of the relaxation process of the system: when When the relaxation time is small, the relaxation time distribution is narrow, indicating that the electrode material and interface properties are relatively uniform and the dominant mechanism is singular; while when When the value is large, the relaxation time distribution broadens significantly, indicating that there is stronger material heterogeneity and multi-scale coupling effect in the system, and multiple relaxation processes participate simultaneously and dominate the electrochemical behavior.

[0053] Furthermore, in the distribution order model, the interval can be divided into three functional regions based on its physical meaning: low values ​​correspond to the diffusion region, mainly reflecting the diffusion behavior of ions in the electrode bulk phase at low frequencies; intermediate values ​​correspond to the mixing region, representing the mid-frequency range, embodying electrochemical reaction kinetics and charge transfer processes; and high values ​​belong to the capacitance region, characterizing the interfacial capacitance characteristics dominated by double-layer charging under high-frequency conditions. To quantify the relative roles of different physical processes in the overall distribution order response, the distribution order weighting function can be adjusted. Integrate over each functional region to obtain the corresponding region weights. The specific definitions are as follows: Low area( (Diffusion-dominated) is defined as: (65) middle area( (Diffusion and capacitance mixing) is defined as: (66) high area( (Capacitor-dominated) is defined as: (67) because After normalization, the weights of the three regions sum to 1, that is: (68) In numerical computation, these integrals are typically achieved through weighted summation over a discrete grid, i.e., accumulating the sampling points on the 𝛽 grid. And combined with integral weighting coefficients This allows us to obtain the weighting ratio of each region. By using the region weights, we can intuitively reflect the relative proportion of different dominant mechanisms (diffusion, charge transfer, capacitive behavior) in the overall impedance response, which is an important quantitative indicator for diagnosing aging modes and determining changes in material properties.

[0054] In the distributed-order equivalent circuit model, each order Each of these can be viewed as corresponding to a generalized fractional branch, and its admittance characteristics reflect the dynamic mechanism dominated by that order. Then used to depict different The contribution ratio in the overall polarization process forms a continuous capacitance spectrum (ECS).

[0055] The complex admittance corresponding to the distributed branch is: (69) The equivalent capacitance is defined as: (70) At the physical level, the equivalent capacitance spectrum can reflect the charge storage capacity of the interface, demonstrate the characteristics of the combined effect of double-layer capacitance and pseudocapacitance, and reveal the distribution and evolution of multi-scale relaxation processes inside the material through its frequency dependence.

[0056] To extract representative interface characteristics and suppress the influence of local noise, the mid-frequency average capacitance can be calculated, which is located in the characteristic frequency range. Internal definition: (71) The reason for using logarithmic averaging is that EIS data are usually sampled uniformly at logarithmic frequencies in experimental design. Therefore, averaging on the logarithmic axis can more accurately reflect the overall electrochemical characteristics in the mid-frequency range and avoid the bias of linear averaging on high-frequency or low-frequency data.

[0057] To verify the proposed distributed-order equivalent circuit model, the asymptotic characteristics of the model under high-frequency and low-frequency conditions are analyzed by examining the limiting frequency behavior. Specifically, the high-frequency limit is used to characterize the transient response of the system under rapid perturbations, and is defined as follows: (72) The total impedance, the limiting expression is: (73) Under high-frequency conditions (i.e., the high-frequency range extracted from S1), the impedance of all capacitive processes decays rapidly with 𝜇(𝛽) and tends to short-circuit, so the system is only controlled by the ohmic internal resistance independent of frequency; correspondingly, the high-frequency end of the Nyquist plot eventually converges to the ohmic internal resistance point on the real axis.

[0058] The low-frequency limit is used to characterize the steady-state polarization behavior of a system under slow perturbations and is an important criterion for judging whether a model can correctly describe slow processes such as diffusion and interface hysteresis. It is defined as follows: (74) when From time to time Therefore, the convergence of the low-frequency limit (i.e., the low-frequency range corresponding to S1) depends entirely on exist Distribution characteristics of the vicinity: If If the integral remains finite at low frequencies, then the integral remains finite at high frequencies; conversely, if... If so, the impedance may diverge at low frequencies.

[0059] Compared to traditional models, when the distribution function satisfies (that is, in) It degenerates into Dirac. When the function is called, that is: (75) Degenerates into a single CPE model, where the CPE parameters satisfy , .

[0060] When taking When, the corresponding expression is: (76) Degenerates into an ideal RC model, where It is an ideal capacitor.

[0061] In the distributed order model It describes the power-law characteristic of impedance as a function of frequency, and its microscopic meaning can be given by the logarithmic slope of the impedance magnitude. The microscopic interpretation of the value is: (77) For power-law impedance, the following constraints apply: (78) therefore, Essentially, it reflects the local frequency response slope of the impedance spectrum, characterizing the non-uniformity and multi-scale nature of the interface process. The actual DOC model can be viewed as different... The superposition of value processes exhibits a frequency behavior that is a combination of multiple power-law characteristics.

[0062] In this embodiment, the inherent limitations of traditional integer-order and single fractional-order models in terms of time-scale representation are abandoned. For the first time, distributed-order calculus theory is systematically introduced into the equivalent circuit modeling of lithium-ion batteries. By defining distributed-order weight functions within continuous-order intervals, a unified physical description of multi-scale, multi-rate relaxation and polarization processes at the electrode interface is achieved, elevating the modeling from discrete-order to continuous-scale. This approach overcomes the shortcomings of traditional CPEs or fractional-order elements, which can only characterize single or finite dispersion processes and struggle to analyze multiple overlapping relaxation mechanisms. It provides a more fundamental and unified mathematical framework for characterizing complex, non-uniform electrochemical interface dynamics, theoretically breaking through the expressive power boundaries of existing equivalent circuit models. Secondly, addressing the problem of high degrees of freedom and difficulty in direct identification of distributed-order models, a novel double-Beta distribution hybrid model is introduced to parameterize the distributed-order weight functions. Furthermore, an identification strategy that decouples the intensity parameters from the normalized weight functions is proposed. Combining weighted nonlinear least squares and physical constraint optimization, stable and interpretable identification of continuous-order distributions is achieved. This approach effectively solves key challenges in practical applications of distributed-order models, such as overfitting, unclear physical meaning of parameters, and instability in identification under limited EIS data. The hybrid Beta distribution can flexibly characterize multi-peak relaxation behavior, while the decoupling normalization strategy allows the weight function to focus on reflecting the mechanistic weight distribution, thus significantly improving the model's engineering usability and mechanistic relevance. This method is not a simple replacement or splicing of existing least-squares or global optimization algorithms. Finally, starting from the identified distributed-order weight function, a novel quantitative diagnostic index system for lithium-ion battery aging status is systematically derived and defined, and its mapping relationship with specific aging mechanisms such as SEI growth, active material loss, and electrode heterogeneity evolution is established. This approach overcomes the limitation of traditional equivalent circuit models, which can only provide a few macroscopic parameters (such as ohmic internal resistance and equivalent capacitance), enabling in-depth mining and quantitative expression of hidden aging information over continuous time scales from impedance spectra. This provides a new technical path and analytical tool for precise diagnosis of battery health status and tracing the origins of aging mechanisms.

[0063] Furthermore, to verify the accuracy of the proposed model, the coefficient of determination is used to measure the model's ability to interpret experimental impedance, defined as: (79) in, (80) In the formula, This is the mean of the data obtained in S1. The closer the value is to 1, the higher the model fitting accuracy.

[0064] Root mean square error (RMSE) is used to measure the overall magnitude of the fitting error, and it is defined as follows: (81) It's important to understand that the smaller the RMSE, the closer the model output is to the experimental data.

[0065] In this embodiment, based on the established distributed-order equivalent circuit model, a weighted nonlinear least squares optimization method is used to identify the model parameters. By introducing frequency-related weights into the objective function, the influence of impedance data from different frequency bands on the parameter estimation results is effectively balanced, thereby improving the model's fitting stability and robustness across the entire frequency domain.

[0066] Figure 4 Statistical results of fitting accuracy under different SOC and different number of iterations are presented. (See attached document.) Figure 4 It can be observed that the root mean square error (RMSE) of the distributed-order equivalent circuit model is less than 0.0009 under all SOC and cycle number conditions, while the coefficient of determination (R²) is higher than 0.97, indicating a high degree of consistency between the model's calculated impedance and the experimentally measured impedance. The low RMSE reflects the model's high-precision fitting ability at the amplitude level, while the high R² indicates that the model can accurately capture the overall trend of impedance variation with frequency. Furthermore, under conditions such as high SOC and deep aging, the model still maintains high fitting accuracy without significant systematic bias, indicating that the distributed-order equivalent circuit model has good adaptability and generalization ability under complex conditions. This verifies the effectiveness of the proposed model in describing the aging evolution and multi-scale dynamic behavior of lithium-ion batteries.

[0067] The fitting results of some parameters of the distributed-order equivalent circuit model are shown in Table 1.

[0068] Table 1. Fitting results of some parameters of the distributed-order equivalent circuit model. Table 1 shows that the mixing ratio η remains relatively stable between 0.48 and 0.51 within the low to medium SOC range, with little change with the number of cycles. This indicates that the interfacial polarization process in the low to medium SOC region is relatively constant, and the model is mainly dominated by a single relaxation process. Under high SOC conditions, η fluctuates significantly: it is approximately 0.84 after 300 cycles, decreases to 0.11 after 400 cycles, and then rises again to 0.85–0.86 after 500–600 cycles. This shows that the interfacial dynamics under high SOC are significantly affected by cycle aging, possibly related to increased lithium-ion concentration, changes in interfacial structure, and SEI growth. Therefore, to ensure the robustness of the model fitting under different SOC and cycling conditions, the initial value of the mixing ratio can be set to a middle compromise value (e.g., 0.7). This balances the stability in the low SOC region with providing a buffer for fluctuations in the high SOC region, thus avoiding excessive bias of the initial value on the optimization results. This verifies the rationality of setting the mixing ratio.

[0069] Table 1 also shows that for the Beta distribution shape parameters p1, q1, p2, and q2, in the low to medium SOC region, the first Beta distribution parameter p1 is mostly in the range of 6–20, and q1 is relatively small, indicating that the mid-frequency charge transfer relaxation process is relatively sharp and concentrated. As the number of cycles increases, p1 and q1 are more stable, fluctuating significantly only under high SOC conditions. For example, at 300 cycles, with SOC=0.8, q1=10.63, significantly higher than other SOCs, indicating that the mid-frequency relaxation distribution in this range is widened. The second Beta distribution parameters p2 and q2 are usually close to 20 and 3–4 under low SOC conditions, indicating that the low-frequency diffusion or high-frequency double-layer relaxation process is relatively dispersed. Under high SOC or deep cycling conditions, the parameter changes are more significant. For example, at 600 cycles with SOC=1, q2 reaches 7.32, indicating that the distribution of low-frequency / high-frequency effects in this range is widened, which may be related to the increase in interfacial impedance and SEI film growth.

[0070] Table 1 shows the coefficients of the Warburg diffusion impedance. The variation with SOC and number of iterations shows a certain pattern. At low iteration counts, The minimum value is observed in the low SOC range, indicating that the diffusion resistance of the battery is low and the diffusion process is smooth; while in the high SOC range... Increasing to 0.0032–0.00346 indicates an increased degree of low-frequency diffusion limitation under high SOC conditions. As the number of cycles increases to 400–600, A slight increase is observed in the low SOC range, indicating that cyclic aging has a more significant impact on diffusion impedance in the medium-to-high SOC range, while the overall change is smaller in the low SOC region. Overall, The changing trend indicates that the low-frequency diffusion process in the battery is relatively stable at low and medium SOC, while diffusion is increasingly hindered under high SOC or long-term cycling conditions. This is closely related to SEI growth and the degradation of electrode active materials. Table 1 shows that the constraint conditions set for the model parameters are consistent with practical application scenarios.

[0071] Figure 5 The distribution order weight functions are presented under different cycle numbers and SOC conditions. Several representative patterns can be analyzed from the figures: First, the distribution order weight functions exhibit a clear non-uniform distribution across the order range, rather than being concentrated in a single order. This indicates that the internal polarization and energy storage processes of the battery are not dominated by a fixed timescale, but rather are the result of multiple kinetic mechanisms acting together at different orders. The weight functions typically have higher amplitudes in the low to medium order range, corresponding to diffusion polarization, limited charge transfer, and mass transfer processes within porous electrodes; while the weights are relatively smaller in the high order range, mainly reflecting rapid interfacial polarization and transient response behavior. This distribution characteristic essentially reveals the "diffuse" kinetic mechanism inside lithium-ion batteries. Second, the shape and centroid position of the order weight functions differ significantly under different SOC conditions. As the State of Charge (SOC) changes, the distribution of the weight function along the order axis shifts and is reconstructed: in the low to medium SOC range, the weight function tends to concentrate on lower orders, indicating that diffusion-limited effects and slow dynamic processes dominate; while in the high SOC range, the weights of higher orders are enhanced, indicating increased interfacial reactivity and a relatively faster polarization process. This demonstrates that the distributed order weight function has high sensitivity to SOC changes and can effectively reflect changes in the internal reaction state of the battery. Furthermore, from the perspective of cycle count, as cycle aging deepens, the order weight function generally exhibits a "lower order" trend, that is, the weights in the lower order range gradually increase, while the weights in the higher order range relatively decrease. This phenomenon is highly consistent with the physical mechanisms such as electrode structure degradation, SEI film thickening, and longer effective diffusion paths during battery aging, reflecting the evolution of the battery's dynamic response from a fast to a slow process. It is worth noting that the distributed order model can still maintain a continuous and smooth weight function distribution at different cycle stages, demonstrating its good modeling stability and physical consistency. In summary, the distributed order weight function not only intuitively depicts the variation law of the dynamic characteristics of lithium-ion batteries under different SOC and cycle aging conditions at multiple time scales, but also provides an important basis for understanding the battery polarization mechanism and aging evolution process from the perspective of parameter distribution, further verifying the effectiveness and advantages of the distributed order modeling method in battery aging state characterization and health assessment.

[0072] like Figure 6 The figure shows the fitting results of the distributed-order capacitance of the equivalent circuit model. From the figure, the following patterns and characteristics can be analyzed: First, the equivalent value of the distributed-order capacitance C... eqIt exhibits a significant nonlinear growth characteristic as the frequency decreases. In the high-frequency range, C eq The values ​​are relatively small and the changes are relatively gradual, mainly reflecting the rapid polarization and ohmic behavior near the electrode / electrolyte interface; while in the mid-to-low frequency and even low frequency range, C eq The rapid increase with decreasing frequency reflects the combined effects of charge transfer, diffusion polarization, and multi-timescale dynamics. This continuous variation across frequency bands is difficult to accurately describe using traditional integer-order or single-fractional-order capacitance models, while the distributed-order model can effectively characterize the complex dispersion dynamics within the battery through its order distribution. Secondly, the overall level of the distributed-order capacitance curves differs significantly under different SOC conditions. With changes in SOC, especially in the low-to-medium SOC range, C... eq The increase is more significant in the low-to-mid frequency range, indicating enhanced non-uniformity of the internal reaction, and more pronounced diffusion-limited effects and polarization accumulation. This demonstrates that the distributed-step capacitance is highly sensitive to changes in SOC and can serve as an important characteristic parameter reflecting the battery's operating state. Furthermore, from the perspective of cycle count, the distributed-step capacitance generally increases in the low-frequency range with increasing cycle aging, while the change is relatively small in the high-frequency range. This phenomenon indicates that the aging process mainly affects slow-dynamic processes, such as electrode structure degradation, loss of active materials, and thickening of the SEI film, resulting in significant changes in effective energy storage capacity and diffusion-related "apparent capacitance" in the low-frequency region. The distributed-step model maintains good fit continuity across different cycling stages, verifying its robustness in aging modeling. In summary, Figure 6 The distributed-order capacitance fitting results shown not only achieve high-precision characterization of experimental data across the entire frequency band, but also clearly reveal the influence of SOC and cycle aging on the multi-timescale dynamic characteristics of the battery, further demonstrating the advantages and applicability of the distributed-order equivalent circuit model in lithium-ion battery mechanism modeling and state perception.

[0073] like Figure 7The figure shows the fitting results of the ohmic internal resistance R0 of the distributed-order equivalent circuit model. The following analytical conclusions can be drawn from the figure: First, under the same cycle number conditions, R0 generally decreases with increasing SOC, especially in the high SOC range (close to full charge). This phenomenon indicates that at high SOC, the effective area of ​​the electrode active material participating in the reaction increases, and the electrolyte conductivity and interfacial contact state are relatively good, thus reducing the ohmic polarization of the battery. In contrast, in the low SOC range, ion transport channels are limited, and interfacial reaction activity decreases, resulting in a relatively high ohmic internal resistance level. Second, from the perspective of cycle number, in the range of 300–600 cycles, the overall change in R0 is small, showing good stability, indicating that the current collector, electrolyte, and main conductive paths of the battery have not yet undergone significant degradation during this aging stage. Only under high SOC and a large number of cycles does R0 show a slight increase in certain operating conditions, reflecting that with the accumulation of cycle aging, the electrode / electrolyte interfacial impedance and contact resistance begin to gradually increase. Furthermore, comparing the cyclic evolution patterns under different SOC conditions reveals that R0 in the low SOC range is slightly more sensitive to the number of cycles than in the medium-to-high SOC range. This indicates that aging has a more significant impact on conductivity under low charge conditions. This characteristic is consistent with aging mechanisms such as SEI film thickening and changes in electrode pore structure. In summary, the R0 fitting results not only reflect the reasonable response of ohmic internal resistance to SOC and cyclic aging, but also maintain the continuity and physical consistency of parameter changes under different operating conditions, further verifying the effectiveness and engineering applicability of the distributed-order equivalent circuit model in ohmic polarization modeling.

[0074] Figure 8The fitting results of the polarization resistance R1 in the distributed-order equivalent circuit model are presented. The figure shows that R1 exhibits a significant upward trend with increasing cycle count, and this trend varies under different SOC conditions. From a mechanistic perspective, during cyclic aging, the SEI / CEI film continuously grows and gradually densifies, while active lithium loss (LLI) accumulates, increasing the charge transfer impedance at the electrode / electrolyte interface and significantly suppressing interfacial reaction kinetics, thus leading to a continuous increase in polarization resistance R1. This indicates that R1 effectively characterizes the degree of degradation of the battery interface. Further comparison of the variation patterns under different SOC conditions reveals that in the high SOC range, the value of R1 is significantly higher than that in the medium and low SOC range, and the increase with increasing cycle count is more pronounced. This is mainly because under high SOC conditions, the electrode is in a highly reactive state, the interfacial reaction rate is accelerated, side reactions (such as electrolyte decomposition and film thickening) are more active, and the polarization effect is enhanced, thereby amplifying the impact of aging on R1. In contrast, the interface reaction is relatively mild under low to medium SOC conditions. Although the increasing trend of R1 still exists, the change is relatively small. In summary, the fitting results of polarization resistance R1 not only reflect the significant impact of cyclic aging on the interface polarization process, but also reveal its high sensitivity to SOC conditions. This further verifies the physical rationality and application value of R1 as a key parameter characterizing interface aging and polarization characteristics in the distributed-order equivalent circuit model.

[0075] Figure 9The variation of the distribution step intensity parameter A under different cycle numbers and SOC conditions is shown. Several physically significant conclusions can be drawn from the figure. First, from the perspective of cycle evolution, as the number of cycles increases from 300 to 600, A shows a continuous downward trend under various SOC conditions, and the decline is relatively smooth. This indicates that as the battery ages, the overall contribution of the multi-timescale polarization and diffusion processes represented by the distribution step element gradually weakens. The fundamental reason is that during cycling, the active material gradually fails, the pore structure deteriorates, and the effective reaction area shrinks, resulting in a continuous decrease in the number of current-carrying and mass-transfer channels that can participate in the distribution step kinetic process. Second, there are significant differences in the numerical level and decay rate of A under different SOC conditions. In the low to medium SOC range (e.g., SOC = 0.1–0.4), the initial value of A is relatively high, and it shows a relatively slow decay trend with increasing cycle number, indicating that the internal structure and reaction environment of the battery are relatively stable under this condition, and the distribution step behavior still has strong overall intensity. In contrast, under high SOC conditions, the value of A is significantly lower, and the decrease is more pronounced, reflecting that factors such as material volume changes, interfacial stress accumulation, and intensified side reactions under high charge conditions exert a stronger inhibitory effect on the distribution-order polarization and diffusion processes. Furthermore, under near-full SOC conditions, the decrease in A is particularly pronounced in some cycling stages, indicating a certain degree of accelerated aging effect under high SOC conditions. This further illustrates that the distribution-order intensity parameter A is highly sensitive to operating stress and aging degree, and can serve as an important indicator for characterizing the multi-scale dynamic degradation and health evolution of the battery.

[0076] Based on the aforementioned distribution order weight function, the distribution order centroid can be further calculated. and order distribution width The results are as follows: Figure 10 and Figure 11 As shown. First, from Figure 10 The centroid of the distribution order shown The pattern of change shows that as the number of cycles increases, The overall trend shows a slow but clear decline, which is more pronounced under high SOC conditions. This indicates that with the continuous accumulation of cycle aging, the dominant kinetic behavior inside the battery gradually evolves from a high-order response approaching ideal capacitance characteristics to a low-order behavior with stronger resistivity and diffusion-limited features. From a physical mechanism perspective, this change is closely related to the thickening of the SEI / CEI film, the extension of the effective diffusion path, and the limitation of interfacial reaction kinetics, reflecting the "slowing down" and dispersion trend of the polarization process inside the battery. Therefore, the distribution order centroid... It can serve as an important indicator for characterizing the evolution of the dominant dynamic mechanism inside the battery and has good characterization ability for aging state. Secondly, Figure 11The given order distribution width The overall trend increases with the number of cycles, with a more pronounced increase in the high SOC range. This indicates that as aging progresses, the polarization and diffusion processes within the battery at different timescales are no longer concentrated within a finite order range, but rather exhibit a more dispersed distribution. The fundamental reason for this lies in the gradual non-homogenization of the material structure, the deterioration of electrode pore distribution, and the expansion of differences in interfacial reaction regions during cycling, leading to the parallel existence and superposition of multiple kinetic mechanisms. (Order distribution width) The increase in the centroid of the distribution order directly reflects the increased complexity of the battery's internal dynamics and the diversification of electrochemical reaction pathways. Comprehensive analysis reveals that the centroid of the distribution order... The decrease and the order distribution width The increase in the magnitude of the distribution order reveals the kinetic evolution of lithium-ion batteries during cycle aging from two dimensions: "dominant order position" and "order dispersion." These two aspects complement each other, enhancing not only the physical interpretability of the distributed order model parameters but also further validating the significant advantages of the distributed order modeling method in characterizing battery aging mechanisms and revealing dynamic changes across multiple time scales.

[0077] Figure 12 and Figure 13 The low-order weights are given under different number of iterations and different SOC conditions. With higher-order weights The changing pattern. From the overall cyclical evolution trend, as the number of cycles increases, Overall, it shows a gradual upward trend, while This manifests as a continuous decline. This contrast clearly shows that during battery aging, the proportion of low-order fractional kinetics in the overall kinetic response continuously increases, while high-order kinetic processes, approaching ideal capacitance characteristics, gradually weaken. The increase in low-order weights typically corresponds to diffusion limitation, increased resistivity, and the intensification of slow-timescale processes, reflecting the fact that internal transport paths are blocked and polarization deepens. From the perspective of SOC dependence, under high SOC conditions, The absolute value and the rate of increase with increasing cycles are both more significant, while The decrease is also more pronounced. This indicates that under high SOC conditions, interfacial reactivity is higher, side reactions and structural stresses are more severe, and aging has a stronger remodeling effect on the kinetic structure, causing the previously dominant high-order quasi-capacitive behavior to shift to low-order dispersion-type kinetics more quickly. A comparison of different SOC ranges reveals that in the low to medium SOC range, although... It still increases with the number of cycles, but the change is relatively mild. The battery remains at a high level, indicating that it retains a certain proportion of rapid response and quasi-capacitive behavior under this operating condition, and the overall dynamic structure is relatively stable. In summary, Figure 12 and Figure 13 The revealed synergistic change between the increase in lower-order weights and the decrease in higher-order weights further confirms, from the perspective of weight allocation, that the internal dynamic mechanism of the battery evolves from a concentrated, rapid higher-order response to a dispersed, slow lower-order fractional dynamic behavior during cyclic aging. This result is consistent with the aforementioned analytical conclusions regarding the decrease in the centroid of the distributed order and the increase in the width of the order distribution, further enhancing the physical consistency and explanatory power of the distributed order model parameters in characterizing the battery aging mechanism and multi-timescale dynamic evolution.

[0078] In summary, the distributed-order equivalent circuit model not only effectively fits the electrochemical responses under different SOCs and cycling conditions, but its parameter evolution also exhibits a clear and consistent aging orientation and interpretable mechanism at the physical level. Through the coordinated changes of parameters such as distributed-order capacitance, ohmic internal resistance, and polarization internal resistance, as well as the comprehensive analysis of derived features such as order weight functions, distributed-order centroids, order distribution width, and high- and low-order weights, the intrinsic laws governing the evolution of lithium-ion batteries from "rapid and concentrated" kinetic responses to "slow and diffuse" fractional-order kinetic behaviors under different SOCs and cycling aging conditions can be systematically revealed. Furthermore, compared with traditional integer-order or single fractional-order models, the proposed distributed-order equivalent circuit model not only demonstrates a significant advantage in fitting accuracy, but its parameter system also establishes a physical correspondence with the multi-timescale electrochemical processes and aging mechanisms within the battery. The parameters of each distribution order and their derived features (such as the centroid of the distribution order, the width of the order distribution, and the weights of high and low orders) characterize the evolution path of the battery dynamic structure from different perspectives, making the model not only "well-fitted" but also "interpretable." Therefore, the proposed distribution order equivalent circuit model not only achieves high-precision fitting of EIS data, but more importantly, it constructs a parameterized characterization framework that can characterize the battery aging process from multiple time scales and multiple physical mechanisms. This model and its parameter system provide a new analytical perspective for a deeper understanding of the aging mechanism of lithium-ion batteries, and also provide richer and more intrinsically physical support for the establishment of subsequent SOC / SOH joint estimation, aging state identification, and lifetime prediction methods, thus possessing high theoretical research value.

[0079] This invention not only improves the fitting accuracy of electrochemical impedance modeling for lithium-ion batteries, but also achieves interpretable characterization of the battery's dynamic behavior and aging mechanism across multiple time scales through a distributed-order parameter system, thus possessing both engineering practical value and theoretical research significance.

[0080] It should be noted that the order of the embodiments described above is merely for descriptive purposes and does not represent the superiority or inferiority of the embodiments. Furthermore, the above description focuses on specific embodiments of this application. Other embodiments are within the scope of the appended claims. In some cases, the actions or steps described in the claims can be performed in a different order than that shown in the embodiments and still achieve the desired results. Additionally, the processes depicted in the drawings do not necessarily require a specific or sequential order to achieve the desired results. In some implementations, multitasking and parallel processing are also possible or may be advantageous.

[0081] The various embodiments in this application are described in a progressive manner. Similar or identical parts between embodiments can be referred to mutually. Each embodiment focuses on describing the differences from other embodiments. In particular, the device, equipment, and storage medium embodiments are basically similar to the method embodiments, so the descriptions are relatively simple; relevant parts can be referred to the descriptions of the method embodiments.

[0082] Those skilled in the art will understand that all or part of the steps of the above embodiments can be implemented by hardware or by a program instructing related hardware. The program can be stored in a computer-readable storage medium, such as a read-only memory, a disk, or an optical disk.

[0083] The specific embodiments described above do not constitute a limitation on the scope of protection of this application. Those skilled in the art should understand that various modifications, combinations, sub-combinations, and substitutions can be made according to design requirements and other factors. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of this application should be included within the scope of protection of this application.

Claims

1. A distributed-order equivalent circuit modeling method for characterizing the aging state of lithium batteries, characterized in that, The method includes: S1, Obtain electrochemical impedance spectroscopy data of lithium batteries at different aging stages: Under preset ambient temperature and state of charge conditions, apply sinusoidal AC excitation to lithium batteries at different cycle aging stages to obtain broadband electrochemical impedance spectroscopy data covering the low-frequency range to the high-frequency range. S2, Establish a distributed-order equivalent circuit model: Based on the distributed-order calculus theory, a distributed-order equivalent circuit model of lithium-ion battery is constructed. In the distributed-order equivalent circuit model, an order weight function is introduced to describe the weight distribution of the dynamic processes corresponding to different orders in the overall polarization response. The order weight function is parameterized using the Beta distribution function. S3, Model Parameter Identification and Inversion: The distributed-order equivalent circuit model constructed in S2 is matched with the broadband electrochemical impedance spectroscopy data obtained in S1, and the model parameters are jointly identified using a weighted nonlinear least squares optimization method.

2. The distributed-order equivalent circuit modeling method for characterizing the aging state of lithium batteries as described in claim 1, characterized in that, The impedance spectral data includes at least the low-frequency impedance response dominated by ion diffusion processes, and the mid-to-high-frequency impedance response dominated by interfacial polarization and charge transfer processes.

3. The distributed-order equivalent circuit modeling method for characterizing the aging state of lithium batteries as described in claim 1, characterized in that, The circuit structure of the distributed-order equivalent circuit model includes: an ohmic internal resistance for characterizing ohmic polarization; a distributed-order polarization branch composed of a distributed-order capacitor and a polarization internal resistance connected in parallel; and a Warburg diffusion impedance element for describing the diffusion process. The ohmic resistor is connected in series with one end of the distributed polarization branch, and the other end of the distributed polarization branch is connected in series with the Warburg diffusion impedance element.

4. The distributed-order equivalent circuit modeling method for characterizing the aging state of lithium batteries as described in claim 3, characterized in that, The total impedance of the distributed-order equivalent circuit model includes: the impedance of the ohmic resistor, the impedance of the distributed-order polarization branch, and the impedance of the diffused impedance element. The impedance of the distributed-step capacitor is specifically expressed as follows: In the formula, The impedance of the distributed step capacitor, For the distribution order intensity parameter, The composite weight function of the distributed impedance is given by the following order. It is a fractional exponent. For the integration kernel, trigonometric function terms and Phase shift characteristics; The real and imaginary parts of the impedance of the distributed-order capacitor satisfy the following: In the formula, Let be the real part of the impedance of the distributed-order capacitor. This represents the imaginary part of the impedance of the distributed-order capacitor.

5. The distributed-order equivalent circuit modeling method for characterizing the aging state of lithium batteries as described in claim 1, characterized in that, Also includes: To address the coexistence of multiple relaxation mechanisms in lithium-ion batteries during cycle aging, a composite order weighting function is constructed using a weighted mixture of two Beta distribution functions. The specific expression for the composite order weight function is as follows: In the formula, For mixing ratio, Let be the shape parameter of the first Beta distribution. The shape parameter of the second Beta distribution; The composite order weight function satisfies the normalization condition; The first Beta distribution is used to characterize the mid-frequency relaxation process of charge transfer type; The second Beta distribution is used to characterize low-frequency diffusion or high-frequency double-layer effect.

6. The distributed-order equivalent circuit modeling method for characterizing the aging state of lithium batteries as described in claim 1, characterized in that, The model parameters for joint identification include ohmic internal resistance, polarization internal resistance, distribution order intensity parameter, mixing ratio parameter, Beta distribution shape parameter, and Warburg diffusion coefficient. The weighted least squares criterion used for parameter identification has the following objective function form: In the formula, Broadband electrochemical impedance spectroscopy data, For frequency points, The impedance value is predicted by the distributed-order equivalent circuit model. For the model parameters of joint identification, This represents the total number of frequency sampling points. These are weighting coefficients; The constraints on the model parameters are as follows: the ohmic resistance, parallel branch resistance, weight function magnitude, and Warburg coefficient are all non-negative, the mixing coefficient ranges from [0,1], and the shape parameters of the two sets of Beta distributions are initially set using right-skewed distribution parameters for the first set and symmetrical distribution parameters for the second set.

7. The distributed-order equivalent circuit modeling method for characterizing the aging state of lithium batteries as described in claim 1, characterized in that, It also includes verifying the accuracy of the distributed-order equivalent circuit model using the coefficient of determination and root mean square error; The coefficient of determination is used to measure the interpretability of the distributed-order equivalent circuit model fitting for broadband electrochemical impedance spectroscopy data. The root mean square error is used to measure the overall magnitude of the fitting error of the distributed order equivalent circuit model.

8. A lithium battery aging state analysis method based on the modeling method of claim 1, characterized in that, Based on the distributed-order equivalent circuit model obtained by the aforementioned modeling method, the identified model parameters are used to construct a characterization index system for the aging state of lithium batteries.

9. The lithium battery aging state analysis method as described in claim 8, characterized in that, The characterization index system includes: ohmic internal resistance and polarization internal resistance parameters, equivalent value of distributed order capacitance, centroid of the order weight function, intensity of the distributed order, width of the order distribution, and weight ratios corresponding to the low-order interval to the high-order interval. The centroid of the distribution is used to measure the overall offset trend of the distribution order weight function; The order distribution width is used to reflect the degree of dispersion of the distribution order weight function in a preset interval; The sum of the weight ratios corresponding to the lower-order intervals and the higher-order intervals is 1.

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