A lithium battery electro-thermal hysteresis coupling modeling method based on fractional order memristor

Abstract

The application discloses a lithium battery electro-thermal hysteresis coupling modeling method based on a fractional order memristor, and belongs to the technical field of battery modeling. The method introduces a Caputo fractional order operator, replaces a fixed polarization resistance in a traditional integer order RC model with a fractional order memristor, establishes a polarization state evolution equation and an improved type memristor mapping function, and realizes physical endogenous coupling of electricity, heat, hysteresis and memory effect in combination with an Arrhenius temperature correction, an SOC-temperature two-dimensional OCV mapping and a state-entropy thermal coupling mechanism. Further, L1 discrete format and short memory principle are adopted for state recursion, a two-stage joint closed loop framework is constructed, offline parameter identification based on an anti-degradation physical boundary and online closed loop filter correction with adaptive slope are fused, and the accuracy, robustness and real-time application capability of the model under a wide temperature range and complex working conditions are improved.

Description

Technical Field

[0001] This invention relates to the field of battery modeling technology, and in particular to a lithium battery electrothermal hysteresis coupling modeling method based on fractional memristors. Background Technology

[0002] As the global energy structure accelerates its transformation towards low-carbon and sustainable development, lithium-ion batteries, with their high energy density, excellent cycle stability, and good environmental adaptability, have been widely used in electric vehicles, large-scale energy storage, and renewable energy grid integration. In battery management systems (BMS), high-fidelity battery modeling is the core theoretical foundation for accurate state estimation, optimized energy scheduling, and fault diagnosis. However, lithium-ion batteries are inherently complex electrochemical systems that are highly nonlinear, time-varying, and strongly coupled with multiple physics fields. Their terminal voltage response involves the synergistic evolution of ion diffusion, charge transfer, interfacial polarization, and thermal effects, making accurate characterization difficult through a single physical mechanism.

[0003] Current mainstream battery modeling methods mainly include electrochemical mechanism models, data-driven models, and equivalent circuit models (ECM). While electrochemical mechanism models offer high mechanistic interpretability, they suffer from high complexity, long solution times, and difficulty in parameter acquisition, making direct deployment in real-time embedded BMS difficult. Data-driven models demonstrate strong fitting capabilities under complex operating conditions, but are highly dependent on the quality of training samples, exhibiting limited generalization ability and physical interpretability across operating conditions, and are prone to significant accuracy degradation in extreme temperature or small sample scenarios. In contrast, ECM uses lumped parameter elements such as resistance and capacitance to equivalently characterize the external characteristics of the battery, achieving a more balanced trade-off between model accuracy, computational complexity, and engineering feasibility.

[0004] While traditional ECMs have achieved widespread application in engineering, their characterization limitations under complex dynamic conditions and wide temperature ranges are becoming increasingly apparent as demands for battery range and safety continue to rise. Firstly, in terms of voltage hysteresis modeling, lithium iron phosphate (LFP) batteries exhibit significant thermodynamic hysteresis due to their two-phase coexistence mechanism. Existing research typically employs empirical mathematical add-ons (such as Preisach models or additional nonlinear networks) for external compensation. These methods lack a clear physical connection to the internal electrochemical mechanisms and are prone to enhanced parameter coupling and decreased numerical stability under dynamic current excitation.

[0005] Secondly, in characterizing polarization dynamics and long-memory characteristics, traditional integer-order RC models exhibit exponential "short-memory" polarization decay, failing to accurately depict the power-law relaxation and long-term tailing characteristics caused by distributed diffusion of porous electrodes in real battery systems. In recent years, although some studies have introduced fractional-order equivalent circuit models (FO-ECM) to capture polarization dynamics across multiple timescales, most existing models separate polarization dynamics from hysteresis behavior, lacking a unified physical description of their coupling mechanism. More importantly, traditional FO-ECM struggles to establish a quantitative correlation between the polarization history state and the battery's non-equilibrium thermodynamic behavior (such as interfacial entropy change and the Gibbs free energy difference of phase transition), leading to a systematic decline in model accuracy in the voltage plateau region and under wide temperature range conditions.

[0006] In summary, the core challenge in current high-precision battery modeling lies in the fact that traditional compensation strategies typically involve exogenously splicing hysteresis, polarization, and temperature correction terms, leading to model redundancy, physical inconsistencies, and difficulties in parameter decoupling. Therefore, proposing a physically consistent multiphysics coupling modeling method that can intrinsically and uniformly describe polarization kinetics, thermodynamic hysteresis, and temperature effects, overcoming the limitations caused by exogenous splicing, has become a key requirement for improving the robustness and generalization performance of models under wide temperature ranges and complex dynamic conditions. Summary of the Invention

[0007] The purpose of this invention is to provide a lithium battery electrothermal hysteresis coupling modeling method based on fractional-order memristors, comprising:

[0008] S1. Introduce the Caputo fractional operator to replace the fixed polarization resistor in the integer RC model with a fractional memristor and construct the equivalent circuit model of the fractional memristor.

[0009] S2. Polarization state evolution equations established based on Caputo fractional operators;

[0010] S3. Construct an improved fractional memristor mapping function;

[0011] S4. Establish a state-entropy thermal coupling correction mechanism based on non-equilibrium thermodynamics to achieve physical endogenous coupling of electricity, heat and memory.

[0012] As a further aspect of the present invention: replacing the fixed polarization resistor in the integer-order RC model with a fractional-order memristor includes:

[0013] Introducing the Caputo fractional operator to replace the traditional integer derivative, for orders... The Caputo fractional operator is defined as follows:

[0014]

[0015] In the formula, Denotes the Gamma function, satisfying , The Caputo fractional operator contains an integral kernel from 0 to t.

[0016] As a further aspect of the present invention: the establishment of the polarization state evolution equation includes:

[0017] The integer-order state equations for the polarization voltage are derived from Kirchhoff's current law:

[0018]

[0019] Where t represents time; Indicates the lithium battery current; Indicates polarization resistance; Indicates polarization capacitance; Indicates polarization voltage;

[0020] The polarization voltage is defined as a state variable. The prototype of the integer-order state equation is obtained:

[0021]

[0022] In the formula, Indicates polarization state; This represents the driving term, reflecting the energy injected by the current into the polarization state; the driving coefficient. ; This represents the dissipation term, describing the spontaneous dissipation of polarization through the resistor, and the dissipation coefficient. Determines the dissipation rate, time constant ;

[0023] Replacing the integer derivative in the prototype of the integer-order state equation with the Caputo fractional operator yields the fractional-order internal state evolution equation:

[0024]

[0025] In the formula, Represents polarization state variables. The fractional order describes the memory depth of the system. For driving coefficients, dimensions ; Describes the driving strength of current injection on the internal polarization state; The dissipation coefficient has dimensions of 1. ; Describes the spontaneous decay rate of the polarization state within the system.

[0026] As a further embodiment of the present invention: the improved fractional memristor mapping function is:

[0027]

[0028] In the formula, Indicates polarization state, This is the physical lower bound of polarization resistance; This refers to the range of resistance variation. The sensitivity coefficient controls the steepness of the tanh curve; The center offset is used to correct the charging and discharging asymmetry.

[0029] As a further aspect of the present invention: the state-entropy thermal coupling correction mechanism includes: performing temperature dynamic correction on the internal resistance, driving coefficient, and dissipation coefficient using Arrhenius law;

[0030]

[0031] In the formula, Indicates the corresponding parameter activation energy, Represents the molar gas constant; These are parameter values ​​at the reference temperature.

[0032] As a further aspect of the present invention: the state-entropy thermal coupling correction mechanism further includes:

[0033] Constructing a two-dimensional temperature-SOC mapping:

[0034]

[0035] This two-dimensional surface is generated by bidirectional PCHIP interpolation in the SOC-temperature domain using quasi-static experimental data at different temperatures.

[0036] The effective open-circuit voltage observation equation for FO-MECM is obtained as follows:

[0037]

[0038] In the formula, A two-dimensional OCV-SOC-T surface constructed from measured data; The coupling coefficient of SC-OCV; The term will define the fractional-order internal polarization state. This is mapped to a correction amount for the OCV benchmark; Temperature modulation factor;

[0039] The complete terminal voltage observation equation for FO-MECM is:

[0040]

[0041] In the formula, Indicates a static thermodynamic base; Indicates dynamic polarization evolution; T represents temperature. Indicates resistance.

[0042] As a further aspect of the present invention, it also includes solving the state of the model based on the L1 scheme and the short memory principle.

[0043] For the Caputo operator in the fractional-order state equation, the L1 approximation scheme is used for discretization:

[0044] sampling step size Next, the The fractional derivative at time t is expressed as:

[0045]

[0046] In the formula, the step size weighting coefficient of the L1 scheme is:

[0047]

[0048] The weighting coefficients satisfy and ,make , We can obtain the L1 recurrence equation:

[0049]

[0050] In the formula, and These represent the driving and dissipation coefficients, respectively, which are updated in real time with temperature.

[0051] As a further embodiment of the present invention: the

[0052] The time complexity of the standard L1 format is Utilizing the power-law decay property of the coefficient:

[0053]

[0054] Introducing a fixed memory length This reduces the computational complexity to The recurrence relation for the state after that is:

[0055]

[0056] Take the initial conditions This corresponds to the assumption that there is no pre-polarization at the start of the battery test; if initial polarization exists, then... Set it to a non-zero initial value.

[0057] As a further aspect of the present invention, it also includes a two-stage robust parameter identification and global state estimation joint closed-loop strategy, including an offline identification stage:

[0058] In the offline identification stage, a covariance matrix is ​​constructed and the adaptive evolution strategy of the covariance matrix is ​​combined with the physical boundary to optimize the parameters.

[0059] As a further aspect of the present invention: the joint closed-loop strategy of two-stage robust parameter identification and global state estimation also includes an online estimation stage:

[0060] Online estimation stage: To address the long-term integral drift generated by open-loop identification in the nonlinear end region, a slope adaptive extended Kalman filter is introduced. The slope adaptive extended Kalman filter dynamically adjusts the measurement noise covariance exponentially based on the real-time OCV slope: actively reducing feedback in the flat region to suppress oscillations caused by disturbances, and enhancing feedback in the steep end region to forcibly correct state deviations.

[0061] The beneficial effects of this invention are:

[0062] (1) By constructing an equivalent circuit model based on fractional-order memristors, this invention integrates polarization dynamics, thermodynamic hysteresis, and temperature effects into the same modeling framework, avoiding parameter redundancy, physical inconsistency, and parameter coupling problems caused by the exogenous splicing of hysteresis compensation, polarization correction, and temperature correction in traditional models, thereby improving the physical consistency and interpretability of the model structure.

[0063] (2) This invention introduces the Caputo fractional-order operator and an improved fractional-order memristor mapping function, which can effectively describe the multi-timescale dynamic response of lithium batteries caused by porous electrode diffusion, interface polarization, and phase transition processes. Compared with the exponential short memory decay of the traditional integer-order RC model, this method can more accurately characterize the power-law relaxation, long-term tailing, and voltage hysteresis phenomenon of the battery polarization voltage, and is especially suitable for modeling the plateau region of lithium iron phosphate batteries.

[0064] (3) This invention combines Arrhenius temperature correction, SOC-temperature two-dimensional OCV mapping, state-entropy thermal coupling mechanism, and fractional-order state solving method based on L1 scheme and short memory principle to reduce computational complexity while ensuring high model accuracy. Furthermore, by combining offline robust parameter identification with online slope adaptive extended Kalman filter estimation strategy, integral drift and end-region estimation errors can be effectively suppressed, improving the model's generalization ability and real-time deployment ability in wide temperature range, dynamic current conditions, and practical BMS applications. Attached Figure Description

[0065] Figure 1 This is a flowchart of the method of the present invention;

[0066] Figure 2 This is a schematic diagram of the characteristic curve of the memristor kneading hysteresis loop provided in an embodiment of the present invention;

[0067] Figure 3 This is a schematic diagram comparing exponential decay and power-law decay provided in an embodiment of the present invention;

[0068] Figure 4 This is a schematic diagram of the equivalent circuit model of a lithium battery provided in an embodiment of the present invention;

[0069] Figure 5 Provided for embodiments of the present invention Schematic diagram of the function curve;

[0070] Figure 6 A schematic diagram illustrating the intrinsic hysteresis generation mechanism based on a fractional-order memristor provided in an embodiment of the present invention;

[0071] Figure 7 This is a schematic diagram of OCV characteristics at different temperatures provided in an embodiment of the present invention;

[0072] Figure 8 This is a schematic diagram of TS-PISE convergence curves at different temperatures provided in an embodiment of the present invention;

[0073] Figure 9 A schematic diagram comparing the terminal voltage and error of each model at different temperatures, provided for embodiments of the present invention;

[0074] Figure 10 This is a schematic diagram of the constitutive characteristic curves of a memristor at different temperatures provided by an embodiment of the present invention;

[0075] Figure 11 This is a schematic diagram of the internal state and dynamic memristor evolution curves under different temperatures in the DST operating condition provided in the embodiments of the present invention;

[0076] Figure 12 A schematic diagram comparing the static average open-circuit voltage and effective open-circuit voltage at different temperatures under DST conditions, provided for an embodiment of the present invention.

[0077] Figure 13 A schematic diagram of the evolution of dynamic hysteresis voltage correction at different temperatures under DST conditions as a function of SOC, provided for embodiments of the present invention.

[0078] Figure 14 This is a schematic diagram of the time-series evolution curves of internal irreversible transient heat generation power and cumulative dissipated energy at different temperatures under DST conditions provided in an embodiment of the present invention. Detailed Implementation

[0079] Embodiments of the present invention are described in detail below. Examples of these embodiments are illustrated in the accompanying drawings, wherein the same or similar symbols denote the same or similar elements or elements having the same or similar functions throughout. The embodiments described below with reference to the accompanying drawings are exemplary and are only used to explain the present invention, and should not be construed as limiting the present invention.

[0080] The present invention will now be further described with reference to the accompanying drawings. The following embodiments are intended to illustrate the present invention in more detail and provide detailed implementation methods. However, the scope of protection of the present invention is not limited to the following embodiments.

[0081] like Figure 1 As shown, the core contradiction in current high-precision battery modeling lies in the fact that traditional compensation strategies typically involve exogenously splicing hysteresis, polarization, and temperature correction terms, leading to model structural redundancy, physical inconsistencies, and difficulties in parameter decoupling. This invention proposes a lithium battery electrothermal hysteresis coupling modeling method based on fractional-order memristors, as detailed below:

[0082] 1. Introduce the Caputo fractional operator

[0083] The memristor is the fourth basic passive component in circuit theory. It combines memory and resistance, and its basic mathematical definition is derived from magnetic flux. With charge Definition of nonlinear relationships:

[0084] （1）

[0085] In the formula, Indicates magnetic flux; It represents the integral of the historical charge (i.e., the integral of current over time).

[0086] In macroscopic circuits, the equivalent performance is as follows:

[0087] （2）

[0088] In the formula, Indicates voltage; Indicates current; The memristor value is directly dependent on the historical integer integral of the current ( The core characteristic of memristors is their... The curve exhibits classic pinch-hysteresis loop characteristics under sinusoidal excitation: the curve must pass through the origin, and forms closed loops along different paths during charging and discharging, such as... Figure 2 As shown.

[0089] from Figure 2It can be seen that this path-dependent memory characteristic is highly isomorphic to the polarization behavior of lithium iron phosphate batteries: during lithium-ion insertion and extraction, the polarization resistance of the electrode interface dynamically changes with the accumulation of historical current, resulting in a large hysteresis difference in the charge and discharge open-circuit voltage, which in turn... A distinct hysteresis loop forms on the plane. This physical phenomenon cannot be mathematically generated by the traditional fixed-resistance RC model.

[0090] Traditional memristors typically represent the state of a simple summation of currents (integer-order integrals), implicitly suggesting that the influence of historical information on the current state is linear or decays rapidly in an exponential manner. However, the memristors of real-world physicochemical systems, such as lithium-ion battery polarization and complex ion diffusion, exhibit long-lasting tailing effects, with polarization relaxation following power-law decay rather than exponential decay. Therefore, a Caputo fractional operator is introduced to replace the traditional integer-order derivative. For the order... The Caputo fractional operator is defined as follows:

[0091] （3）

[0092] In the formula, Denotes the Gamma function, satisfying , The Caputo fractional operator contains an integral kernel from 0 to t, meaning that the current state depends not only on the previous state but on all historical states, and the decay of this historical influence follows a power-law decay (long memory) rather than an exponential decay (short memory), as shown below. Figure 3 As shown.

[0093] The advantage of the Caputo operator over the Riemann-Liouville operator is that its Laplace transform directly utilizes integer-order initial conditions and has a clear physical meaning.

[0094] （4）

[0095] In the formula, Represents the Laplace transform operator; Representation function Caputo fractional operators; Represents the complex frequency variable in the Laplace transform; Representing time-domain functions After Laplace transform Image function of the domain; The use of integer-order initial physical conditions allows fractional-order models to seamlessly integrate with the initial conditions of conventional integer-order ECMs. This is because the Laplace transform of the Caputo fractional operator produces ordinary integer-order initial values, such as f(0+), rather than abstract "fractional-order initial values." This allows fractional-order models to directly use the initial state values ​​of conventional integer-order ECMs during initialization, without introducing difficult-to-interpret fractional-order initial conditions. Therefore, they can seamlessly integrate with the initial conditions of conventional integer-order ECMs.

[0096] The introduction of fractional calculus theory has substantially expanded the physical concept of traditional memristors. Fractional memristors no longer simply track the charge that accumulates over time. Instead, it tracks an internal state variable that is strictly controlled by fractional-order dynamics. First, at the level of macroscopic constitutive relations, its mathematical description is:

[0097] （5）

[0098] In the formula, Indicates the macroscopic memristor value; Represents the internal state variable, which represents the equivalent polarization accumulation on the electrode surface. A positive value usually indicates accumulated polarization during charging, while a negative value indicates accumulated polarization during discharging.

[0099] 2. Fractional-order state evolution equations

[0100] Therefore, this paper introduces fractional-order memristor theory into the traditional equivalent circuit model to construct an equivalent circuit model of a lithium battery fractional-order memristor based on hysteresis, temperature, and memory properties, such as... Figure 4 As shown.

[0101] Figure 4 (a) is a first-order RC equivalent circuit model. Wherein, Indicates the terminal voltage of the lithium battery; Indicates the open-circuit voltage of the lithium battery; Indicates the lithium battery current; Indicates polarization resistance; This indicates a polarized capacitor. Figure 4 (b) is the equivalent circuit model of the fractional-order memristor constructed in this paper. Wherein, Indicates the terminal voltage of the lithium battery; Indicates the open-circuit voltage of the lithium battery; Indicates the lithium battery current; This indicates the dynamic resistance value of a fractional memristor; This indicates a polarized capacitor.

[0102] For traditional first-order RC equivalent circuits, such as Figure 4As shown in (a), the polarization voltage can be obtained from Kirchhoff's current law (KCL). Integer-order state equations:

[0103] （6）

[0104] Polarization voltage is defined as a state variable. The prototype of the integer-order state equation is obtained:

[0105] （7）

[0106] In the formula, This represents the driving term, reflecting the energy injected by the current into the polarization state; the driving coefficient. The smaller the capacitance, the more sensitive the current is to the driving force of polarization accumulation; This represents the dissipation term, describing the spontaneous dissipation of polarization through the resistor, and the dissipation coefficient. Determines the dissipation rate, time constant .

[0107] Under constant current excitation, the time-domain solution of equation (7) is an exponential response (RC charging curve), and the polarization decays exponentially—that is, short-memory behavior. Solving equation (7) under zero initial conditions, let… For constants:

[0108] （8）

[0109] Equation (8) is the time-domain solution of the integer-order state equation under constant current excitation, in the form of an RC charging curve. When the current is cut off ( The polarized free relaxation response is:

[0110] （9）

[0111] The core drawback of integer-order models lies in their fixed time constant. The exponential polarization decay. This "short memory" characteristic causes the effects of historical states to disappear rapidly at an exponential rate, making it impossible to accurately characterize the long-lasting tail memory effect in real battery systems.

[0112] However, electrochemical impedance spectroscopy (EIS) experiments on lithium batteries show that the polarization relaxation at the electrode interface exhibits constant phase element (CPE) characteristics, appearing as a flattened semicircle on the Nyquist plot, with the impedance being:

[0113] （10）

[0114] In the formula, This represents the complex impedance of the CPE; Represents the imaginary unit; Indicates angular frequency; Represents the generalized capacitance constant; Indicates the order of a fraction. (From...) Figure 4 It can be seen that the corresponding time-domain relaxation is power-law decay. It decays much more slowly than exponentially. The smaller the value, the longer the tail of the power-law decay, and the longer the memory retention time. This indicates that the integer derivative in formula (7) cannot capture the long memory behavior of true polarization relaxation.

[0115] To compensate for the above deficiencies, this paper replaces the integer derivative in formula (7) with the Caputo fractional operator, resulting in the fractional internal state evolution equation:

[0116] （11）

[0117] In the formula, The fractional order characterizes the depth of the system's memory. Driving coefficient (dimensions) ), which describes the driving strength of current injection on the internal polarization state; Dissipation coefficient (dimensions) ), describing the spontaneous decay rate of the polarization state within the system; both are parameters to be identified, determined through time-domain charge-discharge data; when hour, Equation (11) degenerates into integer order equation (7), which is in complete agreement with the traditional RC polarization model.

[0118] The structure on the right-hand side of the equation (11) remains unchanged because of the driving term. and dissipation terms It describes the transient force balance relationship in the current state, and its functional form depends only on and The current value is used, without explicitly including historical trajectory information. Therefore, the instantaneous rate of change on the left is used. Extended to a fractional rate of change carrying all historical weights This transforms the system's time response from exponential decay to power-law decay with long memory characteristics. Expand:

[0119] （12）

[0120] In the formula, the integrator kernel right Weighted average throughout – the closer to the current moment, the greater the weight; the farther away, the greater the weight, which decays according to a power law but never becomes zero. This is the mathematical expression of long memory.

[0121] 3. Fractional memristor mapping function

[0122] Before constructing the memristor mapping hysteresis function, the polarization resistance Four physical constraints must be satisfied: (1) Bounded positive definiteness ( (1) constant positive); (2) monotonicity (the deeper the polarization, the greater the resistance); (3) saturation (when it approaches the extreme, it tends to the boundary value and cannot grow indefinitely); (4) asymmetry (the charging and discharging characteristics are asymmetrical). The simplest family of functions that satisfy the above four conditions is the sigmoid function. This paper selects the hyperbolic tangent function, whose original mathematical definition is:

[0123] （13）

[0124] when hour, ;when hour, .in, Function curves as follows Figure 5 As shown.

[0125] from Figure 5 It can be seen that, The value range is stuck Between these points, symmetrical about the origin, with a midpoint value of 0, it cannot be directly used as a resistor. Therefore, the original... The value range is strictly controlled within First, Shift the whole thing upwards by 1, and it becomes Then divide by 2 to compress into Construct the normalized mapping factor:

[0126] （14）

[0127] To satisfy asymmetry, the input function is... Perform a linear transformation, let Substituting this into formula (14), we get:

[0128] （15）

[0129] In the formula, Indicates sensitivity. Controlling the steepness of the S-curve, The larger the value, the more the polarization state accumulates resistance, and the resistance quickly tends to saturate. This indicates the offset, moving the curve inflection point from... Translate to It is used to correct charge-discharge asymmetry. When hour, ;when hour, It meets the boundedness requirement.

[0130] To scale the value range to the physical range, let and will Linear mapping to the target resistance range:

[0131] （16）

[0132] Formula (16) is a linear scaling. When hour, (Physical lower bound); when hour, (Physical upper limit). This determines the maximum range of change in polarization resistance.

[0133] Combining formulas (15) and (16), we obtain the improved memristor mapping function:

[0134] （17）

[0135] In the formula, Indicates polarization state; This is the physical lower bound of polarization resistance; The range of resistance values ​​indicates the width of the hysteresis loop; a larger range indicates a wider hysteresis loop. The sensitivity coefficient controls the steepness of the S-curve. The larger the value, the more the polarization state accumulates resistance, and the resistance quickly tends to saturate. The center offset is used to correct charging and discharging asymmetry. When hour, ;when hour, ;when At the neutral point, ; any Down It is strictly valid and guarantees physical rationality.

[0136] 4. Construction of FO-MECM circuit model

[0137] Based on the above fractional-order memristor theory, this paper will change the fixed polarization resistance of the traditional first-order RC model. Replace with fractional memristor Polarized capacitor It is still retained, forming a parallel dynamic polarization network of memristors and capacitors, such as Figure 4 As shown in (b).

[0138] The terminal voltage observation equation for FO-MECM is:

[0139] （18）

[0140] The relationship between the polarization voltage of the FO-MECM and the memristor is as follows:

[0141] （19）

[0142] To meet the requirements of real-time discrete solution of the model, a first-order backward difference approximation is adopted for the time derivative term. Substitute into equation (19) and consider the polarization voltage at the current moment. By performing an explicit solution (when calculating the unknown quantity at the current moment, the known quantity can be directly substituted into the formula to calculate it, without needing to solve the equation or iterate), the polarization voltage can be obtained:

[0143] （20）

[0144] In the formula, Indicates the sampling time; This represents the polarization voltage at the current moment; The dynamic memristor value at the current moment is obtained in advance by the subsequent L1 recursive formula (50).

[0145] 5. Temporal Analytical Solution (Mittag-Leffler Convolution)

[0146] Taking the Laplace transform of both sides of the state evolution equation (11), and using formula (4), let the initial conditions be zero. ,have:

[0147] （21）

[0148] In the formula, Indicates the system output; This represents the system's input.

[0149] The transfer function is obtained by rearranging:

[0150] （22）

[0151] right Perform the inverse Laplace transform, using the standard conclusions of the Mittag-Leffler function:

[0152] （23）

[0153] The two-parameter Mittag-Leffler function is defined as follows:

[0154] （24）

[0155] make ,but:

[0156] （25）

[0157] By the convolution theorem, the time-domain analytical solution of equation (11) is obtained as follows:

[0158] （26）

[0159] In the formula, the convolution kernel Let the system's impulse response function be determined by applying data to the entire historical interval. Integration fully characterizes the long-memory properties of a system. In particular, when... hour, The degeneracy into the exponential convolution response of the standard RC demonstrates the consistent inclusion of the fractional-order model for integer-order components.

[0160] In addition, when and At that time, it can be known from the complete monotonicity of the Mittag-Leffler function that right This holds true. Simultaneously, the power function... It is always positive on the positive real axis. Therefore, this fractional-order convolution kernel is strictly positive throughout the entire integration interval. This mathematical property guarantees the internal state solution. Always maintain the same sign as the external excitation current, i.e., during charging ( There must be polarization accumulation. During discharge ( There must be a polarization reversal. This constant positivity will serve as a fundamental lemma, directly supporting a rigorous mathematical proof regarding the mechanism of hysteresis generation.

[0161] 6. Rigorous mathematical proof of hysteresis

[0162] In the kinetic characteristics of lithium-ion batteries, the hysteresis effect manifests as the uncertainty in the mapping relationship between SOC and OCV. For LFP batteries, the difference in Gibbs free energy and micro-stress effects during the phase transition process cause a significant deviation in the charging and discharging paths on the voltage platform, forming a closed hysteresis loop.

[0163] Suppose At that moment, the current undergoes an instantaneous step change, from the positive charging current. The sudden change to negative discharge current ,like Figure 6 As shown in (a).

[0164] During charging ( The excitation signal for the system is a constant charging current. According to the output response formula of the fractional-order model, the polarization residual value at the moment of commutation is expressed as:

[0165] （27）

[0166] In the formula, This represents the memory variables accumulated within the system at the instant of current switching.

[0167] In the subsequent discharge phase ( A key difference emerges between integer-order (IO) and fractional-order (FO) dynamics. In the traditional IO model (… In ), state parameters Experiencing rapid exponential decay ( ), rapidly dissipating its historical state. In contrast, such as Figure 6 As shown in (b), the FO model ( The fractional polarization exhibits a significant long-tailed "power-law decay." Therefore, the dissipation rate of fractional polarization is much slower than that of exponential polarization, and it retains historical memory over a longer period. The polarization response of a fractional polarization system can be expressed as:

[0168] （28）

[0169] The prolonged retention of this polarization state directly determines the electrical response of the nonlinear impedance element. According to the hysteresis impedance mapping function, the system impedance depends on the current polarization state. Because... , If the condition is always true, substituting into equation (17) yields:

[0170] （29）

[0171] In the formula, This is the zero-polarization reference value. When... hour, The reference resistance value is exactly located at the midpoint between the physical upper and lower bounds; when hour, Asymmetric deviation occurs, parameters The physical meaning is to adjust the reference resistance value when there is no polarization, so that the model can accurately fit the hysteresis characteristics of the LFP battery's asymmetrical charge and discharge. Parameter identification must be constrained. Reasonable to ensure physical rationality.

[0172] The macroscopic hysteresis fundamentally stems from this microscopic memory retention. As the current reverses, the IO memristor (memristor value) rapidly decreases to its reference value, while the FO memristor, due to the long-tail effect, slowly decays from a higher positive value, resulting in a significant memristor difference in the early stages of discharge. ),like Figure 6 As shown in (c). This persistent additional polarization resistance, through an intrinsic mechanism, forces the terminal voltage at the same SOC to be lower than the terminal voltage during the charging process, thereby causing a significant voltage difference ( Therefore, as Figure 6 As shown in (d), the discharge trajectory of FO naturally separates from the charging path on the OCV-SOC phase plane. This seamlessly reconstructs the thermodynamic hysteresis loop through an endogenous physical mechanism, completely eliminating the dependence on any exogenous mathematical splicing.

[0173] 7. State-entropy coupling correction based on non-equilibrium thermodynamics

[0174] Since OCV is a core parameter describing the thermodynamic state of lithium-ion batteries, defined as the equilibrium potential difference between two electrodes without external current excitation, it essentially reflects the chemical potential difference of the electrode materials at a given SOC and is directly controlled by the change in the system's Gibbs free energy. Therefore, the OCV-SOC relationship not only establishes a thermodynamic reference potential for ECM but is also a key foundation for accurate SOC estimation and battery state analysis.

[0175] For LFP batteries, the charge-discharge process is primarily dominated by the two-phase transition between the lithium-rich and lithium-poor phases. Since the chemical potential remains almost constant within this coexisting region, the open voltage velocity (OCV) changes extremely slowly with state of charge (SOC), resulting in a characteristic flat voltage plateau spanning approximately 0.1 to 0.95 SOC. In addition to this broad plateau, LFP batteries also exhibit a significant thermodynamic hysteresis effect. At any given SOC, the charging OCV is consistently higher than the discharging OCV. This phenomenon mainly stems from the lattice mismatch stress at the phase interface and the kinetic barrier during the phase transition, forcing the charge-discharge process to evolve along distinctly different thermodynamic paths, ultimately forming a closed hysteresis loop on the OCV-SOC phase plane.

[0176] The key point is that ambient temperature has a significant coupling and modulating effect on these electrochemical properties. For example... Figure 7 As shown, temperature not only determines the utilization rate and available capacity of active substances, but also fundamentally alters the evolutionary profile of OCV. Specifically, Figure 7 (b) shows that the hysteresis voltage is highly sensitive to thermodynamic conditions. As the temperature decreases from 50°C to 10°C, the hysteresis width of the voltage plateau region increases significantly, from about 0.05V to over 0.1V. This significant temperature dependence indicates that the phase transition kinetics, interfacial charge transfer processes, and solid-phase lithium-ion diffusion within the LFP cell become increasingly sluggish at lower temperatures.

[0177] To integrate these complex characteristics into the ECM, the average value of the charge and discharge OCV is typically used as a thermodynamic reference.

[0178] （30）

[0179] The average OCV was obtained through low-rate, quasi-static charge-discharge experiments. At each SOC sampling point, the battery was allowed to rest for a sufficiently long time to reach near thermodynamic equilibrium, and a smooth lookup table was constructed using piecewise cubic Hermite interpolation polynomials (PCHIP).

[0180] Furthermore, to accurately capture the temperature dependence of OCV, a two-dimensional temperature-SOC mapping was constructed:

[0181] （31）

[0182] This two-dimensional surface was generated by bidirectional PCHIP interpolation in the SOC-temperature domain using quasi-static experimental data at multiple temperatures. This method provides an absolute thermodynamic basis for the model's OCV benchmark, thereby completely eliminating the static background error caused by temperature fluctuations.

[0183] To achieve accurate initialization in SOC estimation, the monotonicity of the OCV lookup table must be guaranteed. Since the discharge OCV is strictly monotonically increasing over a wide SOC range, it can be used for reverse lookup to circumvent the non-uniqueness problem in the plateau region. Real-time SOC tracking employs the ampere-hour integration method.

[0184] （32）

[0185] In the formula, This represents the actual usable capacity of the battery at the current temperature. Let be the instantaneous current at step k. This is the sampling step size.

[0186] According to the transition state theory of statistical mechanics and Arrhenius's law, the internal dynamic parameters of the system are controlled by the activation energy. Its ohmic resistance... Driving coefficient and dissipation coefficient All satisfy the general correction formula:

[0187] （33）

[0188] In the formula, Indicates the corresponding parameter activation energy, =8.314 J / (mol·K) represents the molar gas constant; Reference temperature ( The parameter values ​​at K = 293.15. Specifically, the dissipation coefficient. The temperature dependence quantitatively describes the physical process by which temperature rise accelerates polarization dissipation: high temperature enhances lattice vibration and reduces the frictional resistance of phase boundary movement, and polarization residues are more easily relaxed under thermal excitation, thus causing the hysteresis width to narrow as the temperature increases.

[0189] To characterize the sensitivity coefficient Based on the evolution of temperature, a linear approximation correction model is introduced:

[0190] （34）

[0191] In the formula, Temperature compensation coefficient (unit: This characterizes the sensitivity of the hysteresis phenomenon to temperature fluctuations.

[0192] Furthermore, to establish a fully coupled electrothermal framework, the internal heat generation and temperature evolution must be quantitatively determined. In the proposed model, the total internal heat generation rate... It mainly consists of ohmic internal resistance and dynamic polarization resistance (composed of memristor). The irreversible Joule heating composition generated on the surface (characterized by):

[0193] （35）

[0194] Based on the lumped parameter thermal model, the average internal temperature of the battery The dynamic evolution is controlled by the energy balance between internal heat generation and heat dissipation to the external environment:

[0195] （36）

[0196] In the formula, For ambient temperature, For temperature difference, This represents the equivalent convective thermal resistance between the battery surface and the environment. This represents the lumped heat capacity of the battery. Through the above heat generation and temperature evolution equations, the electrical state (current and polarization) continuously drives the evolution of the thermal state, and the updated temperature in turn modulates the electrical parameters, thus forming a strict closed-loop endogenous electro-thermal coupling mechanism.

[0197] To fundamentally characterize the modulation of battery thermodynamic properties by polarization states, the DeDonder-Prigogine internal variable theory is introduced. Let the system be in non-equilibrium internal variable... Perform a Taylor series expansion of the entropy function at the point (corresponding to the thermodynamic equilibrium state):

[0198] （37）

[0199] In the formula, The non-equilibrium entropy represents the entropy value of the battery during the actual charging and discharging dynamic process. The equilibrium entropy represents the ideal entropy value of a battery when it is completely at rest and without current, and is determined by the state of charge (SOC) and temperature (T). Represents a higher-order infinitesimal; This indicates absolute temperature.

[0200] According to the second law of thermodynamics, the entropy of an isolated system reaches a maximum in equilibrium, and the first variation satisfies the extremum condition:

[0201] （38）

[0202] Meanwhile, to ensure the stability of the equilibrium state, its second derivative must satisfy negative definiteness:

[0203] （39）

[0204] Let the coupling coefficient Ignoring higher-order terms, we can obtain a simplified expression for the non-equilibrium entropy:

[0205] （40）

[0206] The physical meaning of equation (40) is: polarization state The non-uniform charge distribution or local disorder at the electrode interface represents the deviation from equilibrium. This deviation from equilibrium leads to the effective entropy of the system being lower than the theoretical equilibrium value. The entropy reduction is proportional to the square of the polarization intensity, ensuring the thermodynamic stability of the system—that is, any polarization deviation will lead to the degradation of the system's energy quality.

[0207] In an electrochemical system, the Gibbs free energy is related to the equilibrium open-circuit voltage ( Satisfying basic relations Using the Gibbs-Helmholtz relation, partial derivatives with respect to temperature are obtained:

[0208] （41）

[0209] In the formula, Indicates the number of electrons transferred in the reaction; Denotes Faraday's constant, approximately ; It represents the potential difference between the two electrodes of a battery when it is in thermodynamic equilibrium (an ideal state of no polarization and hysteresis); The entropy-heat coefficient at equilibrium; It is the Gibbs free energy, which represents the maximum useful work that a reaction can do under constant temperature and pressure. It is the reaction entropy change, which represents the change in the system's disorder during the electrode reaction process and is directly related to the reversible heat of the battery.

[0210] Therefore, the entropy-heat coefficient under equilibrium state can be obtained as follows:

[0211] （42）

[0212] In non-equilibrium dynamic processes, the corrected Substituting into formula (42), we obtain the dynamic apparent entropy heat coefficient:

[0213] （43）

[0214] In the formula, , The contribution to the overall polarization state at the two electrode interface; This represents the entropy difference between the positive and negative poles; This is the defined state-entropy coupling constant. Because... The square term ensures that polarization will lead to an apparent decrease in reversible thermal energy during both charging and discharging processes, which is completely consistent with the thermodynamic law that polarization is always accompanied by irreversible entropy increase.

[0215] Combining the above Arrhenius kinetic corrections with non-equilibrium thermodynamic coupling, the observation equation for the effective open-circuit voltage (SC-OCV) of the FO-MECM is obtained:

[0216] （44）

[0217] In the formula, The two-dimensional OCV-SOC-T surface is constructed from measured data (Equation (31)); V represents the SC-OCV coupling coefficient. The term utilizes the boundedness of the hyperbolic tangent function to define the fractional-order internal polarization states. The mapping to the OCV benchmark enables an endogenous coupling from electrochemical polarization kinetics to thermodynamic potential. It is a temperature modulation factor that makes the hysteresis effect increase as the temperature decreases and decrease as the temperature increases.

[0218] Will (Pure static OCV) equations and A comparison of the complete equations clearly shows the nonlinear modulation effect of the state-entropy coupling term on the OCV. The complete terminal voltage observation equation for FO-MECM is:

[0219] （45）

[0220] In the formula, the static thermodynamic base ( ) and dynamic polarization evolution ( Complete decoupling, memristor through Simultaneously modulating the polarization voltage and OCV correction term achieves an endogenous physical coupling of electro-thermal-memory properties, without requiring any exogenous mathematical splicing.

[0221] 8. State solving and complexity optimization based on L1 discretization

[0222] For the Caputo operator in the fractional-order state equation, an L1 approximation scheme is used for discretization. At the sampling step size... Next, the The fractional derivative at time t can be expressed as:

[0223] （46）

[0224] In the formula, the step size weighting coefficient of the L1 scheme is:

[0225] （47）

[0226] The weighting coefficients satisfy and This ensures the numerical stability of the discrete iterative process. Substituting equation (47) into equation (11) and rearranging, let... , We can obtain the L1 recurrence equation:

[0227] （48）

[0228] In the formula, and These represent the driving and dissipation coefficients, respectively, which are updated in real time with temperature.

[0229] The time complexity of the standard L1 format is The computational load accumulates significantly over time. The power-law decay characteristic of the coefficients is utilized:

[0230] （49）

[0231] Introducing a fixed memory length This reduces the computational complexity to The optimized state recurrence relation is:

[0232] （50）

[0233] Take the initial conditions This corresponds to the assumption that there is no pre-polarization at the start of the battery test; if initial polarization exists, it can be... Set it to a non-zero initial value.

[0234] To balance computational accuracy and real-time performance, the truncation length must be precisely determined. Using the Telescoping property, the upper bound of the truncation error is:

[0235] （51）

[0236] According to the preset accuracy requirements To deduce the minimum memory length required:

[0237] （52）

[0238] In engineering practice, Lmem≈400 steps (corresponding to a historical window of about 400 seconds) can be taken to meet the computational constraints of embedded real-time deployment while ensuring the integrity of the hysteresis characteristics.

[0239] To quantitatively evaluate the accuracy of the proposed model, this paper selects root mean square error (RMSE) and mean absolute error (MAE) as the core evaluation metrics:

[0240] （53）

[0241] 9. Two-stage robust parameter identification and global state estimation strategy

[0242] Because the proposed model couples multiple parameters to its endogenous mechanism, the topological structure of its objective function solution space is highly non-convex and riddled with local optimum traps. When exploring such high-dimensional spaces, traditional heuristic algorithms are prone to getting trapped in local optima and experiencing population degradation. Meanwhile, the extremely flat OCV platform of LFP batteries often leads to severe divergence and numerical oscillations in traditional state estimation algorithms. To address these pain points, this paper proposes a joint closed-loop framework of two-stage parameter identification and state estimation (TS-PISE):

[0243] For each temperature point, parameter identification is performed sequentially: first, the ohmic internal resistance is initialized using current-voltage regression, and relevant hysteresis parameters are initialized based on the OCV hysteresis characteristics; then, the parameters to be identified are mapped to the range of 0 to 1 according to their lower and upper bounds, with an initial step size of 0.3 and a population size of 150. During the non-convergence period of optimization, 150 candidate individuals are generated based on the current mean, step size, and covariance matrix; for each candidate individual, the FO-MECM model is simulated using the L1 recursive scheme, and the corresponding objective function value is calculated; then, the top λ / 2 elite individuals are selected to update the distribution mean, the global step size is updated through a cumulative step size adaptive mechanism, and the covariance matrix is ​​corrected using rank-1 and rank-μ updates. After convergence, the optimal 0-to-1 scale parameters are restored to the original physical parameters and stored in the row corresponding to the temperature point in the parameter matrix. After identification, terminal voltage prediction is performed for the three operating conditions (DST, FUDS, and US06) and the case where the SOC is in the range of 0.02 to 1.0: The SA-EKF observer is initialized with the initial state set to the initial SOC plus 0.02, the covariance matrix is ​​set to a diagonal form consisting of the initial covariance and 10^-4, and the initial system bias is set to 0. Then, state estimation is performed cyclically from the second sampling time to the Nth sampling time. The current SOC is predicted based on the previous SOC, current, sampling interval, battery capacity, and a time conversion factor of 3600, and the prior terminal voltage is calculated using FO-MECM. Then, the measurement noise is adaptively calculated based on the endpoint noise, plateau noise, observation sensitivity, and exponential decay relationship. Finally, Kalman filtering is used to correct and update the SOC and system bias, and the results are physically bounded. After processing all temperature points, operating conditions, and sampling times, the identification parameter matrix and predicted terminal voltage sequence are returned.

[0244] In the offline parameter identification stage, TS-PISE combines the Covariance Matrix Adaptive Evolution Strategy (CMA-ES) with strict "anti-degradation" physical boundary constraints to ensure the physical self-consistency of the extracted intrinsic parameters. Although this open-loop identification can efficiently extract parameters within the intermediate flat plateau region, the pure open-loop model is highly susceptible to long-term drift of the current integral and initial state deviation across the entire SOC range (0% to 100%), especially in the highly nonlinear end regions.

[0245] To achieve high-precision state tracking across the entire domain, a closed-loop observer—the slope-adaptive extended Kalman filter (SA-EKF)—is introduced during the online evaluation phase. This observer constructs a two-dimensional augmented state vector incorporating the state of charge (SOC) and the real-time voltage system bias to simultaneously compensate for model state errors and system residuals. Crucially, its measurement noise covariance... It will be based on the slope of the real-time OCV curve It performs continuous, exponential, adaptive adjustments. In extremely flat plateau regions, the algorithm automatically increases... This reduces the weight of voltage feedback, thus preventing severe oscillations caused by small voltage disturbances and making the system more reliant on current integrals. Conversely, in steep regions at both ends, It will decrease smoothly to make full use of highly sensitive voltage information for forced correction.

[0246] To ensure accuracy and robustness, this paper implements rigorous parameter configuration. During the CMA-ES calibration phase, the population size is set to [value missing]. The maximum number of iterations is The initial search step size is Nine core physical parameters are mapped to a unified normalized space. Global optimization is performed within the system, and its strict physical boundaries are detailed in Table 1. The identification data is specifically extracted from the intermediate flat region (SOC 0.1 to 0.95) of the dynamic operating condition to isolate the interference of nonlinearity at both ends on the extraction of intrinsic parameters.

[0247] Table 1. Boundaries of TS-PISE Identification Parameters

[0248]

[0249] During the global closed-loop validation phase, an initial SOC error of +2% was artificially injected into the SA-EKF to assess its corrective capability. The state covariance matrix was initialized as follows: Enhanced voltage bias Depending on the SOC interval, it is dynamically constrained to ~mV to Between ~mV. Furthermore, to strike a balance between physical consistency and numerical stability, a strict step-size limiter was specifically implemented in the low SOC range, which is highly susceptible to numerical drift, tightening it to... Finally, to quantitatively analyze internal heat generation, the first-order lumped thermal model sets the battery heat capacity to 80 J / K and the convective thermal resistance to 15 K / W.

[0250] To comprehensively evaluate the model's tracking accuracy under complex operating conditions and ensure the physical validity of the parameters, this paper constructs the following comprehensive objective function:

[0251] （54）

[0252] In the formula, This represents the overall weighting coefficient assigned to different dynamic cyclic operating conditions.

[0253] 10. Model Validation and Analysis

[0254] To verify the effectiveness and accuracy of the proposed model, an LFP battery with a nominal capacity of 1100mAh was used. The verification environment was selected at 10℃, 30℃, and 50℃, and parameter identification and verification were performed using dynamic stress testing (DST). The convergence curve based on TS-PISE is shown below. Figure 8 As shown, the proposed FO-MECM model integrates multiple highly coupled electrochemical and thermodynamic parameters, resulting in an extremely complex objective function surface with numerous local optima. The evolution trend of the convergence curve reveals that the objective function value decreases rapidly in the early stages of optimization. This is primarily due to the TS-PISE algorithm's ability to continuously and adaptively update the covariance matrix of the population distribution, effectively capturing the strong correlations between high-dimensional parameters. As the number of iterations increases, the algorithm smoothly transitions to a local fine-search phase; therefore, the fitness curve gradually flattens and eventually achieves stable convergence. This convergence characteristic not only overcomes the inherent defects of traditional optimization algorithms in dealing with high-dimensional, multi-field coupled parameter spaces, which are prone to getting trapped in local optima and population degradation, but also demonstrates the algorithm's superior performance in achieving stable and efficient global optimization in complex parameter spaces. The final steady-state minimum indicates that the identified parameter combination gives the model extremely high terminal voltage tracking accuracy.

[0255] Table 2 shows the identification results of the core parameters of the model at different temperatures (10℃, 30℃, and 50℃). Overall, the evolution trends of each parameter are highly consistent with the electrochemical and thermodynamic mechanisms inside the battery. As the temperature increases from 10℃ to 50℃, the changes in ohmic internal resistance, reference polarization internal resistance, and polarization impedance all show a significant monotonically decreasing trend. This intuitively reflects the reduction of the ion diffusion barrier and the enhancement of charge transfer activation kinetics inside the battery under high-temperature conditions. Simultaneously, the hysteresis potential coefficient decreases from 0.0442 to 0.0305, accurately characterizing the physical phenomenon of the narrowing of the thermodynamic phase transition hysteresis loop at high temperatures. Regarding the fractional order and memristor dynamic characteristics, the fractional order steadily increases with temperature (from 0.6868 to 0.8381). This indicates that high temperatures accelerate the polarization relaxation process and weaken the long-term memory tail effect. However, this parameter is strictly limited to the effective physical range of non-integer orders, effectively preventing model degradation. Furthermore, the memristor mapping center offset, which characterizes the charge-discharge polarization asymmetry, remains stable at all temperatures, thus ensuring physical consistency in accurately calibrating the true polarization center location.

[0256] Table 2 FO-MECM parameter identification results

[0257]

[0258] To fully verify the accuracy and robustness of the proposed FO-MECM, this paper conducts a comprehensive comparison with the traditional first-order RC equivalent circuit model (ECM), the first-order fractional-order model (FO-ECM), the first-order model with hysteresis correction (H-ECM), and the reference fractional-order memristor model without thermal coupling correction (FO-MECM-NT). Figure 9 The terminal voltage tracking curves and error distributions of each model under DST conditions are shown. Furthermore, Table 3 quantitatively summarizes the terminal voltage RMSE and MAE of each model at different temperatures.

[0259] from Figure 9 The terminal voltages and their error curves shown indicate that, under the DST dynamic conditions at three different temperatures (10℃, 30℃, and 50℃), all evaluated models can roughly follow the overall trend of the battery reference voltage. However, through the error distribution subplot ( Figure 9(b), (d), and (f) clearly demonstrate significant differences in model accuracy. Traditional ECM and FO-ECM models exhibit large error fluctuations throughout the entire discharge cycle, especially towards the end of the discharge period (when the time exceeds 6400 s). Due to the intensified polarization effect and enhanced nonlinear characteristics within the battery, traditional models show obvious voltage prediction deviations and error divergence. In contrast, the voltage prediction error of the proposed FO-MECM model is strictly limited to an extremely narrow range near the zero mark. This not only effectively smooths voltage fluctuations but also maintains high tracking stability even at the end of the discharge period, intuitively demonstrating its superior dynamic response capability.

[0260] The quantitative statistics in Table 3 further validate the qualitative observations above. Under all test temperature conditions, FO-MECM exhibited the best predictive performance, with the lowest RMSE and MAE among the five models. Taking a normal temperature environment of 30℃ as an example, the RMSE and MAE of FO-MECM were only 5.66mV and 4.44mV, respectively, which significantly reduced the errors by approximately 83.3% and 72.1% compared to the traditional ECM model (33.82mV and 15.89mV). Meanwhile, compared to the H-ECM model with hysteresis correction (RMSE of 20.73mV), the accuracy advantage of FO-MECM remains significant, fully demonstrating the superiority of combining fractional calculus with memristor theory in characterizing the complex history memory effect and nonlinear polarization process of batteries.

[0261] Table 3. Terminal voltage RMSE and MAE of each model at different temperatures (unit: mV)

[0262]

[0263] Furthermore, comparing models with and without thermal coupling correction (FO-MECM-NT and FO-MECM) reveals that introducing thermal coupling correction is crucial for improving the model's robustness over a wide temperature range. At low temperatures of 10°C and high temperatures of 50°C, the impact of temperature on the internal electrochemical reaction kinetics of the battery is more pronounced. Under these conditions, the RMSE of the FO-MECM model decreases by 26.7% (from 16.30mV to 11.94mV) and 29.0% (from 7.83mV to 5.56mV), respectively, compared to the FO-MECM-NT model without thermal coupling correction. This comparison strongly demonstrates that the proposed FO-MECM model, by comprehensively considering thermodynamic coupling effects, successfully overcomes the parameter inaccuracies of traditional models at extreme temperatures, exhibiting extremely high modeling accuracy and excellent robustness across different environmental temperatures.

[0264] Figure 10 To determine the internal state variables of a fractional-order memristor under different ambient temperatures. With dynamic polarization internal resistance The figure shows a nonlinear constitutive mapping relationship between the two. It can be directly observed from the figure that, regardless of temperature, the dynamic memristor value is strictly confined between the lower and upper physical bounds, exhibiting a smooth and asymmetric S-shaped evolution trajectory. This bounded and smooth mapping mechanism not only perfectly matches the path-dependent characteristics of the polarization state driven by historical current accumulation in lithium iron phosphate batteries, but also ensures the numerical stability of the model during long-term dynamic integration.

[0265] The internal state and dynamic memristor evolution curves at different temperatures under DST conditions are as follows: Figure 11 As shown, from Figure 11 It can be clearly observed that, under the DST condition, the internal state variables of the FO-MECM model... and dynamic memristor value All exhibit periodic oscillation characteristics highly correlated with the charging and discharging current. Figure 11 (a) It can be seen that as the ambient temperature decreases from 50℃ to 10℃, the state variable... The peak values ​​and fluctuation amplitudes increased significantly. This reflects the more intense accumulation and memory effect of polarization states caused by the sluggish electrochemical reaction kinetics within the battery at low temperatures. This accumulation of states directly maps to... Figure 11 (b) Dynamic polarization internal resistance Above: The overall memristor resistance level shows a clear monotonic increase as temperature decreases (the local extreme value at 10℃ approaches 100mΩ). Conversely, at a high temperature of 50℃, The overall resistance decreased significantly and the fluctuation range narrowed considerably, which is highly consistent with the physical mechanism that high temperature promotes ion diffusion and reduces the charge transfer polarization barrier. Overall, both the state variables and the memristor value remained strictly within reasonable physical boundaries during long-term complex dynamic cycles without numerical divergence. This fully demonstrates that the proposed model has excellent dynamic tracking ability and numerical stability when characterizing the evolution of memory polarization at different temperatures.

[0266] Figure 12 The static average open-circuit voltage at three ambient temperatures of 10°C, 30°C, and 50°C under DST conditions is shown. ) and the effective open-circuit voltage calculated by the model ( The relationship between SOC and its evolution. Overall, within the entire SOC range, All can closely follow The global trend of change is accurately mapped to the typical wide and flat voltage plateau characteristics of lithium iron phosphate batteries. However, comparing the local details at different temperatures reveals that at a low temperature of 10℃, The curve exhibits a distinct step-like local fluctuation in the middle region of SOC. This profoundly reflects the intensified thermodynamic phase transition hysteresis effect inside the battery under low-temperature conditions, leading to a wider hysteresis loop around the static equilibrium potential in the dynamic evolution of the effective potential. As the ambient temperature rises to 30℃ and 50℃, the polarization and ion diffusion kinetics performance inside the battery are significantly improved, and the thermodynamic hysteresis effect is greatly reduced, resulting in… The step-like fluctuations in the curve have largely subsided, and The overlap is extremely high. The above evolution trend not only intuitively reveals the significant impact of temperature on the dynamic hysteresis characteristics inside the battery, but also fully verifies the ability of the established model to dynamically track and accurately reconstruct the effective open-circuit voltage under complex operating conditions.

[0267] Figure 13 The figure shows the evolution trajectory of the hysteresis voltage correction term with SOC at different ambient temperatures. It can be intuitively observed from the figure that within the main SOC operating range, the hysteresis voltage exhibits a periodic fluctuation characteristic highly synchronized with the DST dynamic current excitation. Notably, temperature has a decisive influence on the amplitude of this hysteresis correction term: the hysteresis voltage fluctuation is most severe at a low temperature of 10℃ (with locally higher peak values); while as the temperature increases to 30℃ and 50℃, the overall level and amplitude of the hysteresis correction term show a significant monotonic decay trend. This evolution law profoundly reflects the thermodynamic characteristics of the internal phase transition mechanism of lithium iron phosphate batteries. Specifically, low temperatures exacerbate the phase transition hysteresis effect of the lithium insertion / extraction process within the material, resulting in a larger thermodynamic potential difference between the dynamic charge and discharge paths; high temperatures effectively promote phase transition kinetics, significantly narrowing the hysteresis loop. Furthermore, in the fully charged boundary region where SOC approaches 1.0, extremely sharp abrupt changes in hysteresis voltage occur at all temperatures. This is mainly attributed to the rapid nonlinear changes in the internal polarization state and phase transition barrier of the battery when it is at the high SOC boundary. Overall, the evolution curve fully demonstrates that, under complex dynamic conditions, deeply coupling temperature variables into the hysteresis correction model is indispensable for accurately tracking the true nonlinear thermodynamic state inside the battery and improving the accuracy of the full-condition model.

[0268] Figure 14 This study demonstrates the temporal evolution characteristics of the transient power and cumulative energy consumption of irreversible heat generation within the battery under DST dynamic conditions and a wide temperature range of 10℃, 30℃, and 50℃. From the perspective of transient power, the fluctuations in ohmic and polarization heating power are not only highly synchronized with the pulse amplitude of the external DST excitation current but also deeply modulated by the ambient temperature. At the low-temperature environment of 10℃, due to the sharp increase in electrolyte viscosity and the obstruction of interfacial ion migration, both the ohmic internal resistance and dynamic polarization impedance of the battery remain at a high level, resulting in a significantly higher peak transient heating power than under the high-temperature conditions of 30℃ and 50℃. Further analysis from the perspective of energy-time integration... Figure 14(a) and Figure 14 (c) The alternating fluctuations of transient power directly constitute the driving force. Figure 14 (b) and Figure 14 (d) The core driving force behind the cumulative evolution of total thermal energy. Throughout the entire test cycle, the cumulative energy of both ohms and polarization exhibits a monotonically increasing stepwise trend, and its step slope closely matches the timing of the external high-current pulse. As the ambient temperature rises from 10℃ to 50℃, the enhanced electrochemical activation kinetics significantly reduces the transient impedance, macroscopically manifested as an overall downward shift in the cumulative energy curve and a significant reduction in total heat dissipation. This spatiotemporal coupling mechanism of "transient power modulates energy accumulation, and temperature effect reconstructs energy bandwidth" fully demonstrates that the established model, while realistically simulating the real-time evolution of internal impedance with temperature, achieves deep intrinsic integration and high physical consistency in electro-thermal-memory multi-field coupling.

[0269] In summary, this paper comprehensively validates the proposed FO-MECM model under wide temperature range and dynamic operating conditions. Parameter identification using the TS-PISE algorithm yields highly physically consistent results, accurately reflecting the temperature-dependent characteristics of internal electrochemical kinetics and phase transition thermodynamics. Comparative analysis shows that FO-MECM significantly outperforms other models. Even under the most challenging low-temperature and highly dynamic operating conditions, the proposed model still improves the terminal voltage tracking accuracy by more than 60% compared to the traditional ECM, with its minimum RMSE suppressed to only 4.66mV. The fundamental advantage of FO-MECM lies in its state-entropy coupling mechanism, which intrinsically correlates the internal state variables of the memristor with thermodynamic characteristics to reconstruct the dynamic effective open-circuit voltage. This method successfully captures multi-timescale polarization relaxation, path-dependent thermodynamic hysteresis, and irreversible heat dissipation, without relying on any empirically based external mathematical joins. Therefore, FO-MECM establishes a robust, high-precision, and physically self-consistent theoretical framework, demonstrating excellent generalization ability for advanced battery management systems under low temperature and strong dynamic excitation.

[0270] The above description is only a preferred embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any equivalent substitutions or modifications made by those skilled in the art within the scope of the technology disclosed in the present invention, based on the technical solution and inventive concept of the present invention, should be covered within the scope of protection of the present invention.

[0271] It should be understood that the terms "center," "longitudinal," "lateral," "length," "width," "thickness," "upper," "lower," "front," "rear," "left," "right," "vertical," "horizontal," "top," "bottom," "inner," "outer," "clockwise," "counterclockwise," "axial," "radial," and "circumferential" indicate the orientation or positional relationship based on the orientation or positional relationship shown in the accompanying drawings. They are used only for the convenience of describing the present invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation. Therefore, they should not be construed as limitations on the present invention.

[0272] Furthermore, the terms "first" and "second" are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the number of technical features indicated. Thus, a feature defined as "first" or "second" may explicitly or implicitly include at least one of that feature. In the description of this invention, "a plurality of" means at least two, such as two, three, etc., unless otherwise explicitly specified.

[0273] In this invention, unless otherwise explicitly specified and limited, the terms "installation," "connection," "linking," and "fixing," etc., should be interpreted broadly. For example, they can refer to a fixed connection, a detachable connection, or an integral part; they can refer to a mechanical connection, an electrical connection, or a connection that allows communication between them; they can refer to a direct connection or an indirect connection through an intermediate medium; they can refer to the internal communication of two components or the interaction between two components, unless otherwise explicitly limited. Those skilled in the art can understand the specific meaning of the above terms in this invention according to the specific circumstances.

[0274] In this invention, unless otherwise explicitly specified and limited, "above" or "below" the second feature can mean that the first feature is in direct contact with the second feature, or that the first feature is in indirect contact with the second feature through an intermediate medium. Furthermore, "above," "over," and "on top" of the second feature can mean that the first feature is directly above or diagonally above the second feature, or simply that the first feature is at a higher horizontal level than the second feature. "Below," "below," and "under" the second feature can mean that the first feature is directly below or diagonally below the second feature, or simply that the first feature is at a lower horizontal level than the second feature.

Claims

1. A lithium battery electrothermal hysteresis coupling modeling method based on fractional-order memristors, characterized in that, include: S1. Introduce the Caputo fractional operator to replace the fixed polarization resistor in the integer RC model with a fractional memristor and construct the equivalent circuit model of the fractional memristor. S2. Polarization state evolution equations established based on Caputo fractional operators; S3. Construct an improved fractional memristor mapping function; S4. Establish a state-entropy thermal coupling correction mechanism based on non-equilibrium thermodynamics to achieve physical endogenous coupling of electricity, heat and memory.

2. The lithium battery electrothermal hysteresis coupling modeling method according to claim 1, characterized in that, The step of replacing the fixed polarization resistor in the integer-order RC model with a fractional-order memristor includes: Introducing the Caputo fractional operator to replace the traditional integer derivative, for orders... The Caputo fractional operator is defined as follows: In the formula, Denotes the Gamma function, satisfying , The Caputo fractional operator contains an integral kernel from 0 to t.

3. The lithium battery electrothermal hysteresis coupling modeling method according to claim 1, characterized in that, The establishment of the polarization state evolution equation includes: The integer-order state equations for the polarization voltage are derived from Kirchhoff's current law: Where t represents time; Indicates the lithium battery current; Indicates polarization resistance; Indicates polarization capacitance; Indicates polarization voltage; The polarization voltage is defined as a state variable. The prototype of the integer-order state equation is obtained: In the formula, Indicates polarization state; This represents the driving term, reflecting the energy injected by the current into the polarization state; the driving coefficient. ; This represents the dissipation term, describing the spontaneous dissipation of polarization through the resistor, and the dissipation coefficient. Determines the dissipation rate, time constant ; Replacing the integer derivative in the prototype of the integer-order state equation with the Caputo fractional operator yields the fractional-order internal state evolution equation: In the formula, Represents polarization state variables, The fractional order describes the memory depth of the system. For driving coefficients, dimensions ; Describes the driving strength of current injection on the internal polarization state; The dissipation coefficient has dimensions of 1. ; Describes the spontaneous decay rate of the polarization state within the system.

4. The lithium battery electrothermal hysteresis coupling modeling method according to claim 1, characterized in that, The improved fractional memristor mapping function is as follows: In the formula, Indicates polarization state, This is the physical lower bound of polarization resistance; This refers to the range of resistance variation. The sensitivity coefficient controls the steepness of the tanh curve; The center offset is used to correct the charging and discharging asymmetry.

5. The lithium battery electrothermal hysteresis coupling modeling method according to claim 1, characterized in that, The state-entropy thermal coupling correction mechanism includes: applying temperature dynamics correction to the internal resistance, driving coefficient, and dissipation coefficient using Arrhenius's law. In the formula, Indicates the corresponding parameter activation energy, Represents the molar gas constant; These are parameter values ​​at the reference temperature.

6. The lithium battery electrothermal hysteresis coupling modeling method according to claim 5, characterized in that, The state-entropy thermal coupling correction mechanism also includes: Constructing a two-dimensional temperature-SOC mapping: This two-dimensional surface is generated by bidirectional PCHIP interpolation in the SOC-temperature domain using quasi-static experimental data at different temperatures. The effective open-circuit voltage observation equation for FO-MECM is obtained as follows: In the formula, A two-dimensional OCV-SOC-T surface constructed from measured data; The coupling coefficient of SC-OCV; The term will define the fractional-order internal polarization state. This is mapped to a correction amount for the OCV benchmark; Temperature modulation factor; The complete terminal voltage observation equation for FO-MECM is: In the formula, Indicates a static thermodynamic base; Indicates dynamic polarization evolution; T represents temperature. Indicates resistance.

7. The lithium battery electrothermal hysteresis coupling modeling method according to claim 1, characterized in that, It also includes solving the state of the model based on the L1 scheme and the short memory principle: For the Caputo operator in the fractional-order state equation, the L1 approximation scheme is used for discretization: sampling step size Next, the The fractional derivative at time t is expressed as: In the formula, the step size weighting coefficient of the L1 scheme is: The weighting coefficients satisfy and ,make , We can obtain the L1 recurrence equation: In the formula, and These represent the driving and dissipation coefficients, respectively, which are updated in real time with temperature.

8. The lithium battery electrothermal hysteresis coupling modeling method according to claim 7, characterized in that, The The time complexity of the standard L1 format is Utilizing the power-law decay property of the coefficient: Introducing a fixed memory length This reduces the computational complexity to The recurrence relation for the state after that is: Take the initial conditions This corresponds to the assumption that there is no pre-polarization at the start of the battery test; If initial polarization exists, then... Set it to a non-zero initial value.

9. The lithium battery electrothermal hysteresis coupling modeling method according to claim 1, characterized in that, It also includes a two-stage robust parameter identification and global state estimation joint closed-loop strategy, including an offline identification stage: In the offline identification stage, a covariance matrix is ​​constructed and the adaptive evolution strategy of the covariance matrix is ​​combined with the physical boundary to optimize the parameters.

10. The lithium battery electrothermal hysteresis coupling modeling method according to claim 9, characterized in that, The two-stage robust parameter identification and global state estimation joint closed-loop strategy also includes an online estimation stage: Online estimation stage: To address the long-term integral drift generated by open-loop identification in the nonlinear end region, a slope adaptive extended Kalman filter is introduced. The slope adaptive extended Kalman filter dynamically adjusts the measurement noise covariance exponentially based on the real-time OCV slope: actively reducing feedback in the flat region to suppress oscillations caused by disturbances, and enhancing feedback in the steep end region to forcibly correct state deviations.

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