Systems and methods for optimizing current excitation to identify battery electrochemical parameters based on analytical sensitivity expressions

Abstract

The present invention provides an analytical sensitivity formula related to parameters of a battery electrochemical model based on reconstruction techniques such as model reduction, single particle assumption, Pade approximation and Laplace transform. The analytical sensitivity formula has a simple form related to the current input and allows direct optimization of input-related sensitivities. The present invention enables current optimization that is very sensitive to several battery electrochemical parameters that are very important (lithium solid phase diffusion coefficient, volume fraction of electrode active material, and reaction rate constant). In addition, electrochemical parameters can be reliably identified by using optimal current profiles.

Description

Technical Field

[0001] This disclosure relates to systems and methods for optimizing current excitation to identify battery electrochemical parameters.

[0002] This application claims priority to Korean Patent Application No. 10-2020-0097631, filed in Korea on August 4, 2020, the disclosure of which is incorporated herein by reference. Background Technology

[0003] Parameter identification is a crucial topic in battery modeling and control research. This is because parameter accuracy determines model fidelity, which in turn determines the performance of model-based battery state estimation and control. Since many parameters used in electrochemical battery models cannot be directly measured, it is common practice to use algorithms to fit model parameters to measured input and output data (e.g., current, voltage, and temperature). In this case, data quality significantly affects the accuracy of parameter identification and estimation, depending on the data's sensitivity to the target parameters. Traditionally, parameter estimation has been performed using undesigned or heuristic input current profiles and output voltage responses. Examples include constant current (CC) curves, constant voltage (CV) curves, pulsed currents, sinusoidal currents, and driving cycle currents. However, the resulting data is often insensitive to most target parameters, leading to poor accuracy in parameter identification or estimation.

[0004] In recent years, research on data analysis and optimization has become a trend in order to improve the accuracy of parameter estimation. Indicators including sensitivity and sensitivity-based Fisher information matrix (FIM) and Cramer-Rao boundary have been used to quantify the quality of data or the accuracy of estimation results.

[0005] For example, the optimal current curve for estimating parameters of a li-ion equivalent-circuit battery model can be designed by maximizing the determinant of the Fisher information matrix. This method is disclosed in the paper "Genetic optimization and experimental validation of a testcycle that maximizes parameter identifiability for a li-ion equivalent-circuit battery model" by MJ Rothenberger, DJ Docimo, M. Ghanaatpishe, and HK Fathy, Journal of Energy Storage, Vol. 4, pp. 156-166, 2015. These works show promising results in improving the quality of parameter estimation because the designed curves produce better estimation accuracy than the undesigned baseline.

[0006] However, existing work on data / input optimization for battery parameter estimation has significant limitations. Specifically, the aforementioned work requires first imposing a heuristic pattern on the input curve and then optimizing the pattern. For example, a dual sinusoidal current pattern is considered, optimizing the coefficients of the sinusoidal current (such as frequency, amplitude, and phase angle). In another example, a CC-CV current pattern is considered, and current and voltage constraints are optimized. In yet another example, various current patterns, including pulses, sine waves, and drive cycles, are selected from a pre-built library, and the current patterns are combined to form the optimal curve. It is worth noting that the curve obtained in this way is optimal only with respect to the specific pattern considered, and is not necessarily the final global optimum.

[0007] There is great interest in finding the ultimate optimal curve without any pre-defined pattern and exploring the characteristics of the optimal data to estimate different parameters. A major challenge in directly optimizing inputs without imposed patterns is the complexity of computational sensitivity and sensitivity-based metrics (such as Fisher information). This computational complexity is particularly pronounced for first-principles electrochemical models based on PDEs. A commonly used method for sensitivity calculation is the perturbation method. The perturbation method involves perturbing the target parameter and simulating the model to quantify the change in output. A more precise method is to solve the sensitivity differential equation (SDE). The SDE is obtained by taking the partial derivatives of the original model equations with respect to the target variable. For both methods, the computational load is intractable for optimization. This is because most algorithms require iterations over a large search space to find the optimum. Summary of the Invention

[0008] Technical issues

[0009] This disclosure aims to provide a system and method for determining the optimal current excitation to estimate various electrochemical parameters.

[0010] Furthermore, this disclosure also aims to provide a system and method for reliably identifying electrochemical parameters by using optimal current profiles that exhibit sensitivity to electrochemical parameters.

[0011] Technical solution

[0012] In one aspect of this disclosure, a system for optimizing current excitation to identify battery electrochemical parameters is provided, the system comprising: a current application unit coupled to the battery; a voltage measurement unit configured to measure the voltage of the battery; and a control unit operatively coupled to the current application unit and the voltage measurement unit.

[0013] Preferably, the control unit can be configured to: (i) determine a sensitivity transfer function corresponding to the partial derivative of the transfer function from the battery current to the particle surface concentration of the electrode with respect to the electrochemical parameters using an electrochemical model of the battery; (ii) determine an overpotential slope corresponding to the partial derivative of the overpotential of the electrode with respect to the particle surface concentration or the partial derivative of the overpotential of the electrode with respect to the electrochemical parameters using the particle surface concentration of the electrode and the Butler-Volmere equation defining the correlation between the electrochemical parameters and the overpotential of the electrode; (iii) determine an OCP (open circuit potential) slope corresponding to the partial derivative of the OCP function of the electrode with respect to the particle surface concentration; (iv) determine a sensitivity curve of the electrochemical parameters for the battery voltage of the electrochemical model in the time domain using the sensitivity transfer function, the overpotential slope, and the OCP slope; and (v) change the battery current in the time domain and determine an optimal current curve such that the square integral of the sensitivity curve in the time domain according to the change of the battery current is maximized.

[0014] Preferably, the control unit can be configured to change the battery current in the time domain without deviating from preset current boundary conditions.

[0015] Furthermore, the control unit can be configured to change the battery current in the time domain so that the battery voltage of the electrochemical model does not deviate from the preset voltage boundary conditions.

[0016] Preferably, the control unit is configured to determine the optimal current curve using a pseudospectral method, such that the square integral of the sensitivity curve in the time domain according to the change in battery current is maximized.

[0017] According to one implementation, the pseudospectral method is the Legendre-Gauss-Radau (LGR) pseudospectral method with adaptive multi-mesh-interval collocation.

[0018] In this disclosure, the electrochemical model of the battery may employ the single-particle assumption, and the battery voltage (V) is expressed by the following equation.

[0019] V=φ s,p -φ s,n =(U p (c se,p )-U n (c se,n ))+(φ e,p -φ e,n )

[0020] +(η p -η n )-IR l ,

[0021] (Φ s，i Electrode potential, Φ e，i : Electrolyte potential at the electrode boundary, U: Predefined OCP function (V), c se，i Lithium ion particle surface concentration (mol·m⁻¹) -3 ), η i R1: Overpotential at the electrode-electrolyte interface; R2: Lumped ohmic resistance of the battery (Ω·m) 2 (where i = p represents the positive electrode and i = n represents the negative electrode)

[0022] Preferably, the transfer function from the battery current to the particle surface concentration of the electrode is represented by the following equation.

[0023]

[0024] (c se,i : Lithium particle surface concentration in the intercalated electrode (mol·m -3 I: Battery current (A), R s,i Radius (m) of the electrode particle, D s,i Solid-phase diffusion coefficient of electrode particles (m) 2 s -1 A: Electrode area (m²) 2 ), δ i Electrode thickness (m), ε s,i F: Volume fraction of active material at the electrode (unitless), F: Faraday constant (C·mol⁻¹) -1), i: an index representing the type of electrode, which is p for the positive electrode and n for the negative electrode, s: a Laplace transform variable)

[0025] According to one embodiment, the electrochemical parameter may be the solid-phase diffusion coefficient D of the electrode. s，i Furthermore, the control unit can be configured to determine the solid-phase diffusion coefficient D of the electrode in the time domain using the following equation. s，i Sensitivity curve to the battery voltage V.

[0026]

[0027] ( It is the overpotential η of the electrode i For particle surface concentration c se，i The slope of the overpotential corresponding to the partial derivative. It is the OCP function U of the electrode i For particle surface concentration c se，i The partial derivatives of correspond to the slope of OCP, and It is related to the particle surface concentration c from the battery current to the electrode. se，i The transfer function of the electrode and the solid-phase diffusion coefficient D s，i (The partial derivatives of the sensitivity transfer function)

[0028] According to another embodiment, the electrochemical parameter can be the volume fraction ε of the active material of the electrode. s，i The control unit can be configured to determine the volume fraction ε of the active material of the electrode in the time domain using the following equation. s，i Sensitivity curve to the battery voltage V

[0029]

[0030] ( It is the overpotential η of the electrode i For particle surface concentration c se，i The slope of the overpotential corresponding to the partial derivative. It is the overpotential η of the electrode i For the volume fraction ε of the active material s，i The slope of the overpotential corresponding to the partial derivative. It is the OCP function U of the electrode i For particle surface concentration c se，i The partial derivatives of correspond to the slope of OCP, and It is related to the particle surface concentration c from the battery current to the electrode. se，i The transfer function of the electrode's active material volume fraction ε s，i (The partial derivatives of the sensitivity transfer function)

[0031] In another aspect of this disclosure, a system for identifying battery electrochemical parameters using the above-described system for optimizing current excitation is also provided, and the control unit can be configured to: (i) generate a measured battery voltage curve by simultaneously measuring the battery voltage while applying the optimal current curve to the battery during a time period corresponding to the time domain; (ii) generate a predicted battery voltage curve by predicting the battery voltage from the battery current curve during the time period corresponding to the time domain using the electrochemical model; (iii) reduce the difference between the predicted battery voltage curve and the measured battery voltage curve to a preset reference value by adjusting the electrochemical parameters when the difference between the predicted battery voltage curve and the measured battery voltage curve is greater than a threshold; and (iv) identify the adjusted electrochemical parameters as the current electrochemical parameters of the battery.

[0032] In another aspect of this disclosure, a system for optimizing current excitation to identify battery electrochemical parameters is also provided, the system comprising: a current applying device coupled to the battery; a voltage measuring device configured to measure the voltage of the battery; and a control unit operatively coupled to the current applying device and the voltage measuring device.

[0033] Preferably, the control unit can be configured to: (i) determine an overpotential slope corresponding to the partial derivative of the overpotential of the electrode with respect to the reaction rate constant of the electrode using the particle surface concentration of the electrode and the Butler-Volmer equation defining the correlation between the reaction rate constant of the electrode and the overpotential of the electrode; (ii) determine a sensitivity curve of the reaction rate constant for the cell voltage of an electrochemical model in the time domain using the overpotential slope; and (iii) change the cell current in the time domain and determine an optimal current curve such that the square integral of the sensitivity curve in the time domain according to the change of the cell current is maximized.

[0034] Preferably, the control unit is configured to change the battery current in the time domain without deviating from a preset current boundary condition. Furthermore, the control unit can be configured to change the battery current in the time domain such that the battery voltage of the electrochemical model does not deviate from a preset voltage boundary condition.

[0035] Preferably, the control unit can be configured to determine the optimal current curve using a pseudospectral method, such that the square integral of the sensitivity curve in the time domain according to the change of the battery current is maximized.

[0036] According to one implementation, the pseudospectral method is the Legendre-Gauss-Radow LGR pseudospectral method with an adaptive multi-grid spacing configuration.

[0037] In this disclosure, the control unit can be configured to determine the reaction rate constant k of the electrode in the time domain using the following equation. i Sensitivity curve to the battery voltage V.

[0038]

[0039] (I: Battery current, R: Universal gas constant (J·mol⁻¹)) -1 ·K -1 T: Battery temperature (K), F: Faraday constant (C·mol⁻¹) -1 ), ε s，i j: Volume fraction (unitless) of active material that is active at the electrode. 0，i Exchange current density (A·m) -2 ), α: charge transfer coefficient, A: effective electrode area (m²) 2 ), R s，i : Radius of the electrode particle (m), δ i Electrode thickness (m)

[0040] In another aspect of this disclosure, a system for identifying battery electrochemical parameters using the above-described system for optimizing current excitation is also provided, and the control unit can be configured to: (i) generate a measured battery voltage curve by simultaneously measuring the battery voltage while applying the battery current curve to the battery during a time period corresponding to the time domain; (ii) generate a predicted battery voltage curve by predicting the battery voltage based on the battery current curve during a time period corresponding to the time domain using an electrochemical model of the battery; (iii) reduce the difference between the predicted battery voltage curve and the measured battery voltage curve to a preset reference value by adjusting the reaction rate constant of the electrode when the difference between the predicted battery voltage curve and the measured battery voltage curve is greater than a threshold; and (iv) identify the adjusted reaction rate constant of the electrode as the current reaction rate constant.

[0041] In another aspect of this disclosure, a method for optimizing current excitation to identify battery electrochemical parameters is also provided, the method comprising: (a) determining a sensitivity transfer function corresponding to the partial derivative of the transfer function from battery current to particle surface concentration of an electrode with respect to the electrochemical parameters using an electrochemical model of the battery applying a single-particle assumption; (b) determining an overpotential slope corresponding to the partial derivative of the overpotential of the electrode with respect to the particle surface concentration or the partial derivative of the overpotential of the electrode with respect to the electrochemical parameters using the particle surface concentration of the electrode and the Butler-Wolmer equation defining the correlation between the electrochemical parameters and the overpotential of the electrode; and (c) determining an OCP slope corresponding to the partial derivative of the OCP function of the electrode with respect to the particle surface concentration; (d) determining a sensitivity curve for the electrochemical parameters of the battery voltage for the electrochemical model in the time domain using the sensitivity transfer function, the overpotential slope, and the OCP slope; and (e) varying the battery current in the time domain and determining an optimal current curve such that the square integral of the sensitivity curve in the time domain according to the variation of the battery current is maximized.

[0042] In another aspect of this disclosure, a method for identifying battery electrochemical parameters is also provided, the method comprising: generating a measured battery voltage curve by simultaneously measuring the battery voltage while applying the optimal current curve to the battery during a time period corresponding to the time domain; generating a predicted battery voltage curve by predicting the battery voltage based on the battery current curve during the time period corresponding to the time domain using the electrochemical model; reducing the difference between the predicted battery voltage curve and the measured battery voltage curve to a preset reference value by adjusting the electrochemical parameters when the difference between the predicted battery voltage curve and the measured battery voltage curve is greater than a threshold value; and identifying the adjusted electrochemical parameters as the current electrochemical parameters.

[0043] In another aspect of this disclosure, a method for optimizing current excitation to identify battery electrochemical parameters is also provided, the method comprising: (a) determining an overpotential slope corresponding to the partial derivative of the overpotential of the electrode with respect to the reaction rate constant of the electrode using the particle surface concentration of the electrode and the Butler-Wolmer equation defining the correlation between the reaction rate constant of the electrode and the overpotential of the electrode; (b) determining, in the time domain, a sensitivity curve of the reaction rate constant for a battery voltage for an electrochemical model using the overpotential slope; and (c) varying the battery current in the time domain and determining an optimal current curve such that the square integral of the sensitivity curve in the time domain according to the variation of the battery current is maximized.

[0044] In another aspect of this disclosure, a method for identifying battery electrochemical parameters is also provided, the method comprising: generating a measured battery voltage curve by simultaneously measuring the battery voltage while applying the battery current curve to the battery during a time period corresponding to the time domain; generating a predicted battery voltage curve by predicting the battery voltage based on the battery current curve during a time period corresponding to the time domain using the electrochemical model; reducing the difference between the predicted battery voltage curve and the measured battery voltage curve to a preset reference value by adjusting the reaction rate constant of the electrode when the difference between the predicted battery voltage curve and the measured battery voltage curve is greater than a threshold value; and determining the adjusted reaction rate constant of the electrode as the current reaction rate constant.

[0045] Beneficial effects

[0046] This disclosure provides optimization of the current excitation for estimating battery electrochemical parameters. A method for designing the optimal current profile within a given time domain has been formulated based on analytical sensitivity expressions. In one embodiment of this disclosure, the solid-phase diffusion coefficient D is shown. s Volume fraction ε of electrode active material s The results for the three parameters—the reaction rate constant k, the current curve, and the current curve. By correlating these results with analytical expressions related to parameter sensitivity, optimal patterns and potential mechanisms for different parameters were discovered. Interestingly, it was noted that the optimal patterns for different parameters are fundamentally different. Numerical results may depend on the specific battery chemistry and the set of parameters considered. However, the basic patterns and characteristics considered in this disclosure are considered generalizable. In future work, the obtained optimized current curves will be used to estimate individual parameters with the goal of significantly improving estimation accuracy. Attached Figure Description

[0047] The accompanying drawings illustrate preferred embodiments of the present disclosure and, together with the foregoing disclosure, serve to provide a further understanding of the technical features of the present disclosure; therefore, the present disclosure should not be construed as limited to the drawings.

[0048] Figure 1 It is a graph showing the slope of the positive electrode's OCP (open circuit potential) relative to the battery's overall SOC (state of charge).

[0049] Figure 2 This indicates the volume fraction (ε) of the positive electrode active material. s，p The optimized current curve and the corresponding voltage and SOC response characteristics are plotted.

[0050] Figure 3 It is a graph that independently shows the characteristics of two factors (i.e., the semi-linear dynamic term and the nonlinear dynamic term) that will be compared to analyze the optimization of the current curve.

[0051] Figure 4 This shows the solid-phase diffusion coefficient D for the positive electrode. s，p The graph shows the optimized current excitation and the corresponding terminal voltage and SOC response characteristics.

[0052] Figure 5 This shows the reaction rate constant k. p Optimized current curve and corresponding reaction rate constant k p A graph showing the voltage response characteristics.

[0053] Figure 6 This is a schematic diagram of a system for optimizing current excitation to identify battery electrochemical parameters and for identifying electrochemical parameters, according to embodiments of the present disclosure. Detailed Implementation

[0054] Preferred embodiments of the present disclosure will be described in detail below with reference to the accompanying drawings. Before the description, it should be understood that the terminology used in the specification and appended claims should not be construed as limited to its general and dictionary meaning, but rather is interpreted based on the principle of allowing the inventors to appropriately define the terminology for best interpretation, and on the meanings and concepts corresponding to various technical aspects of the present disclosure. Therefore, the description presented herein is merely a preferred example for illustrative purposes only and is not intended to limit the scope of the invention; thus, it should be understood that other equivalents and modifications can be made thereto without departing from the scope of the present disclosure.

[0055] In this disclosure, analytical sensitivity expressions for the parameters of the battery electrochemical model will be derived based on model simplification and reconstruction techniques, such as the single-particle hypothesis, Pad approximation, and Laplace transform.

[0056] The analytical expression for parameter sensitivity obtained in this disclosure is a compact form that has a clear relationship with the current input, which can directly optimize the input-related sensitivity.

[0057] In this disclosure, current optimization is performed for three key battery electrochemical parameters (i.e., solid-phase lithium diffusion coefficient, volume fraction of electrode active material, and reaction rate constant) by using derived analytical expressions.

[0058] In this disclosure, patterns or characteristics of optimized current curves will also be observed, and the underlying mechanisms behind these patterns will be explored by relating them to analytical expressions of parameter sensitivity.

[0059] In this disclosure, a Cramer-Rao boundary analysis will also be performed to quantify the expected estimation accuracy by using designed curves.

[0060] First, we provide analytical sensitivity expressions for the battery's electrochemical parameters. As one implementation method, we will provide the solid-phase diffusion coefficient D. s Volume fraction ε of electrode active material s The analytical sensitivity equations for the reaction rate constant k are presented. These parameters reflect key battery electrochemical characteristics related to critical battery performance and are difficult to measure directly. Therefore, these parameters are chosen as target variables to be identified from the data. Without loss of generality, the methods disclosed herein can be applied to other battery parameters of interest. Based on the analytical sensitivity expressions, an optimization problem is then formulated, aiming to find the optimal current profile for estimating the target parameters. Furthermore, the Cramer-Rao boundary analysis is briefly reviewed, which will be used to quantify the expected estimation accuracy under the optimized current profile.

[0061] First, the various symbols used in the embodiments of this disclosure are defined. If a symbol used in the formulas of this disclosure is not defined, reference may be made to the following definitions.

[0062] c se : Particle surface concentration of lithium-embedded solid particles [mol·m -3 ]

[0063] c e Lithium concentration in electrolyte [mol·m -3 ]

[0064] Φ s Potential [V] of solid-phase particles

[0065] Φ e Electrolyte potential [V]

[0066] J i Li Lithium-ion current density in the electrode [A·m] -2 ]

[0067] i0: Exchange current density [A·m] -2 ]

[0068] η: Overpotential [V]

[0069] k: Dynamic reaction rate [s] -1 ·mol -0.5 ·K 2.5 ]

[0070] R: Universal gas constant [J·mol⁻¹] -1 ·K -1 ]

[0071] F: Faraday constant [C·mol] -1 ]

[0072] T: Temperature [K]

[0073] α a Charge transfer coefficient at the negative electrode [unitless]

[0074] α c Charge transfer coefficient at the positive electrode [unitless]

[0075] c s，max The maximum concentration of lithium in solid particles [mol·m] -3 ]

[0076] δ: Thickness of the predetermined region [m]

[0077] I: Battery current [A], where the charging current is negative and the discharging current is positive.

[0078] V: Battery terminal voltage [V]

[0079] A: Effective electrode area [m] 2 ]

[0080] Vol: Electrode volume [m] 3 ]

[0081] D s Solid-phase diffusion coefficient [m] 2 ·s -1 ]

[0082] D e Electrolyte diffusion coefficient [m] 2 ·s -1 ]

[0083] a s Effective surface area per unit volume of electrode (m²) 2 ·m -3 , corresponding to 3*ε s / R s )

[0084] ε s Volume fraction of active material in the electrode [unitless]

[0085] Rs: Radius of solid-phase active material particles [m]

[0086] U: Open-circuit potential [V] of solid-phase active materials

[0087] R f Solid electrolyte interfacial membrane resistance [Ω·m] 2 ]

[0088] R lump The lumped resistance of the battery [Ω·m] 2 ]

[0089] t0 + Lithium-ion migration [unitless]

[0090] Subscript eff: valid

[0091] Subscript s: solid phase

[0092] Subscript e: Electrolyte phase

[0093] Subscript p: positive electrode

[0094] Subscript n: negative pole

[0095] [Analysis of parameter sensitivity expressions]

[0096] Analytical expressions for the sensitivity of battery electrochemical parameters are derived based on the single-particle hypothesis, electrochemical model simplification, and reconstruction techniques. A brief overview is provided here for reference.

[0097] First, in the single-event model (SPM), the battery voltage V can be expressed as the following equation (1-1).

[0098] Equation 1-1

[0099] V=φ s,p -φ s,n =(U p (c se,p )-U n (c se,n ))+(φ e,p -φ e,m )

[0100] +(η p -η n )-IR l ,

[0101] V: Battery voltage (V), Φ s,i Electrode potential, Φ e,i Electrolyte potential at the electrode boundary, U: Open circuit potential (OCP) function (V), c se,i Lithium particle surface concentration (mol·m -3 ), η i R1: Overpotential at the electrode-electrolyte interface; R2: Lumped ohmic resistance of the battery (Ω·m) 2 (i=p represents the positive electrode, i=n represents the negative electrode)

[0102] Surface concentration c se,i The evolution of lithium diffusion is governed by Fick's diffusion law in spherical coordinates. For electrode particles, the boundary condition for lithium diffusion can be expressed as equation (1-2), which captures the lithium solid phase concentration c. s,iThe change in time and space along the particle radius direction (r). The symbol i is a symbol indicating the type of electrode. If the symbol i is p, it represents the positive electrode; if the symbol i is n, it represents the negative electrode.

[0103] Equation 1-2

[0104]

[0105] D s,i The solid-phase diffusion coefficient of lithium (m) 2 ·s -1 ), c s,i Solid concentration of lithium (mol·m -3 ), R s,i Electrode particle radius (m), ε s,i : Volume fraction of active material at the electrode (unitless), F: Faraday constant (C / mol), r: variable in spherical coordinates.

[0106] In the single-event model (SPM), the current density J in equation (1-2) is assumed to be... i Li The cross-electrode current is constant, and therefore it can be calculated by dividing the total current I by the electrode volume, as in the following equation (1-3).

[0107] Equation 1-3

[0108]

[0109] J i Li Ion current density at the electrode (A·m) -2 A: Electrode area (m²) 2 ), δ i Electrode thickness (m)

[0110] Preferably, the partial differential equation (PDE) represented by equation (1-2) can be discretized before solving it. In one embodiment of this disclosure, the Laplace transform and the Pad approximation are used for the discretization of equation (1-2).

[0111] The Pad approximation is a theory of approximations that uses rational functions of a given order to approximate a function. Specifically, the Pad approximation approximates a predetermined function as a rational function with an nth-order polynomial as the denominator and an mth-order polynomial as the numerator.

[0112] Specifically, for the discretization of equation (1-2), the Laplace transform of equation (1-2) yields the following equation (1-4).

[0113] Equation 1-4

[0114]

[0115] Then, by matching the boundary conditions of equation (1-2), the lithium concentration c on the particle surface is obtained from the input current I. se，i The transcendental transfer function can be obtained as shown in the following equation (1-5).

[0116] Equation 1-5

[0117]

[0118] C se，i : Lithium particle surface concentration in the intercalated electrode (mol·m -3 I: Battery current (A), R s，i Radius (m) of the electrode particle, D s，i Solid-phase diffusion coefficient of electrode particles (m) 2 ·s -1 A: Electrode area (m²) 2 ), δ i Electrode thickness (m), F: Faraday constant (C / mol), ε s，i : Volume fraction of active material at the electrode (unitless), s: Laplace transform variable, e: natural constant.

[0119] The transcendental transfer function represented by equation (1-5) cannot be directly solved in the time domain. Therefore, it is approximated using a low-order rational transfer function based on moment matching.

[0120] As a result, the lithium concentration c on the particle surface se，i The three-dimensional Pad approximation equation can be obtained as follows (2), and the Pad approximation can be transformed to the time domain using the state-space expression.

[0121] That is, in the embodiments of this disclosure, by using the single-event hypothesis, Laplace transform, and Pad approximation to solve the control equation, the equation from current I to c can be obtained as shown in equation (2) below. se，i The transfer function.

[0122] Equation 2

[0123]

[0124] c se，i : Lithium particle surface concentration in the intercalated electrode (mol·m -3 I: Battery current (A), R s，i Radius (m) of the electrode particle, D s，i Solid-phase diffusion coefficient of electrode particles (m) 2 s -1 A: Electrode area (m²)2 ), δ i Electrode thickness (m), ε s，i F: Volume fraction of active material at the electrode (unitless), F: Faraday constant (C·mol⁻¹) -1 ), where i = p represents the positive pole and i = n represents the negative pole.

[0125] Lithium concentration c on the particle surface se，i The Pad approximation is disclosed in the papers “Design and parametrization analysis of a reduced-order electrochemical model of graphite / LiFePO4 cells for SOC / SOH estimation” by J. Marckcki, M. Cannova, A.T. Conlisk, and G. Rizzoni, Journal of Power Sources, vol. 237, pp. 310-324, 2013, and “Reduction of an Electrochemistry-Based Li-Ion Battery Model vis Quasi-Liniearization and Pade Approximation” by J.C. Forman, S. Bashash, J.L. Stein, and HK. Fathy, Journal of The Electrochemical Society, vol. 158, no. 2, p. A93, 2011. The contents of these papers are incorporated herein by reference.

[0126] According to equation (2), the target parameter (i.e., the volume fraction of active material ε) s，i and solid diffusion coefficient D s，i Controlling the particle surface concentration c se，i The dynamics. Furthermore, according to equation (1-1), the particle surface concentration c se，i The battery voltage V is affected by the open circuit potential U.

[0127] The overpotential η controlling the insertion / extraction of lithium ions into electrode particles / from electrode particles i It can be calculated using equation (3) and equation (4) derived by reversing the Butler-Wolmer equation.

[0128] Equation 3

[0129]

[0130] Equation 4

[0131]

[0132] J i Li Based on the current density (A·m) of lithium ion movement -2 ), j 0，i Exchange current density (A·m) -2 R: Universal gas constant (J·mol⁻¹) -1 ·K -1 T: Battery temperature (K), α: Charge transfer coefficient (unitless), F: Faraday constant (C·mol⁻¹) -1 ), ε s，i Volume fraction of active material that is active at the electrode (unitless).

[0133] In equation (4), j 0，I It is the exchange current density (A·m) -2 It is calculated using the following equation (5).

[0134] Equation 5

[0135]

[0136] c s，i max Maximum solid-phase lithium concentration (mol·m -3 ), c se，i Particle surface concentration (mol·m -3 ), α: charge transfer coefficient (unitless).

[0137] Under the single-event assumption, the current density j i Li It can be approximated as the average current on the electrode, as shown in equation (6) below.

[0138] Equation 6

[0139]

[0140] A: Electrode area (m²) 2 I: Battery current (A), δ i Electrode thickness (m)

[0141] According to equations (3) to (5), the reaction rate constant k i and the volume fraction of active materials ε i respectively through j 0，i 、 and ξ i Influence of overpotential η i This affects the battery voltage. Furthermore, due to the overpotential η...i Depending on the surface concentration c in equation (5) se，i The relevant exchange current density j 0，i Therefore, the solid-phase diffusion coefficient D s，i and the volume fraction of active materials ε i Also through particle surface concentration c se，i It has an indirect effect on overpotential.

[0142] Based on the above relationship between the parameters and the terminal voltage, by taking the partial derivatives with respect to the voltage, we can obtain analytical expressions representing the sensitivity of each parameter.

[0143] In one example, the solid-phase diffusion coefficient D s，I The sensitivity can be expressed as shown in the following equation (7).

[0144] Equation 7

[0145]

[0146] V: Battery voltage (V), η i Overpotential (V) of the electrode, c se，i Particle surface concentration (mol·m -3 ), U i : OCP(V), D of the electrode s,i Solid-phase diffusion coefficient of the electrode (m) 2 ·s -1 ), t: time (s)

[0147] here, It is the slope of OCP.

[0148] In equation (7), By taking the right c se，i The partial derivatives are derived from equation (3) and can be expressed as equation (8) below.

[0149] Equation 8

[0150]

[0151] α: Charge transfer coefficient, c e Lithium electrolyte phase concentration (mol·m -3 ), c se，i Lithium particle surface concentration (mol / m³) -3 ), c s，max，i Maximum solid-phase ion concentration (mol·m -3 ), ε s，i R: Volume fraction of active material at the electrode (unitless), universal gas constant (J·mol⁻¹) -1 ·K-1 F: Faraday constant (C·mol⁻¹) -1 ), R s，i : Radius of the electrode particle (m), J i Li Based on the current density (A·m) of lithium ion movement -2 ), j 0，1 Exchange current density (A·m) -2 ).

[0152] It is worth noting that, Usually in It dominates. Characterized by the sensitivity transfer function (STF), which is obtained by taking the values ​​of the solid-phase diffusion coefficient D. s，i The partial derivatives are derived from equation (2). The solid-phase diffusion coefficient D s，I The sensitivity transfer function (STF) can be expressed as the following equation (9).

[0153] Equation 9

[0154]

[0155] According to one implementation method, the solid-phase diffusion coefficient D s，i The sensitivity transfer function can be easily implemented in the time domain using a state-space model in a canonical format as shown in equation (10) for sensitivity calculation or optimization.

[0156] In equation (10), the initial conditions of states x1, x2, x3 and x4 can be set to 0, but this disclosure is not limited thereto.

[0157] Equation 10

[0158]

[0159] also, The predefined OCP function U of the electrode can be used. i (c se,i ) Perform calculations. (Compare with the OCP function U) i The input corresponds to the particle surface concentration c se,i It can be calculated using equation (2), which corresponds to c. se,i The Padre approximation equation. That is, the particle surface concentration c. se，i Equation (2) can be readily computed in the time domain by converting it into a state-space model in a canonical format such as Equation (11) below. In Equation (11), the initial conditions of states X1, X2, and X3 can be set using the initial SOC of the battery.

[0160] Equation 11

[0161]

[0162] Similarly, the volume fraction ε of the active material can be derived. s，i The analytical sensitivity expression.

[0163] First, as shown in equation (12) below, by taking equation (1-1) to adjust the volume fraction ε of the active material s，i The partial derivatives can be used to obtain the volume fraction ε of the active material. s，i Analytical sensitivity

[0164] Equation 12

[0165]

[0166] V: Battery voltage (V), η i Overpotential of the electrode (V), U i : OCP(V), c of the electrode se，i Particle surface concentration (mol·m -3 ), ε s，i Volume fraction of active material that is active at the electrode (unitless).

[0167] In equation (12), as shown in equation (13) below, by taking equation (3) for the volume fraction ε of the active material s，i The partial derivatives can be obtained

[0168] Equation 13

[0169]

[0170] In equation (12), as shown in equation (14) below, by taking equation (2) for the volume fraction ε of the active material s，i The partial derivatives can be obtained

[0171] Equation 14

[0172]

[0173] According to one embodiment, the volume fraction of the active material ε s，i The sensitivity transfer function can be readily computed in the time domain using a state-space model in canonical form as shown in equation (15) for sensitivity calculation or optimization. In equation (15), the initial conditions of states X1, X2, X3, and X4 can be set to 0, but this disclosure is not limited thereto.

[0174] Equation 15

[0175]

[0176] In equation (12), calculate The method is basically the same as described above.

[0177] Finally, by taking equation (1-1) as shown in equation (16), the reaction rate constant k is... i The partial derivatives of the partial derivatives can be used to obtain the reaction rate constant k. i The analytical sensitivity expression.

[0178] In electrochemical models applying the single-particle (SPM) model, the reaction rate constant k i The analytical sensitivity expression corresponds to the overpotential η of the electrode. i For the reaction rate constant k i The partial derivative, i.e., the slope of the overpotential.

[0179] Equation 16

[0180]

[0181] The sensitivity expressions obtained above are all low-order functions with battery current as the input variable.

[0182] The analytical expressions according to embodiments of this disclosure can directly optimize the current curve to maximize sensitivity. Furthermore, the analytical sensitivity expressions also provide theoretical insights into the fundamental mechanisms underlying the electrochemical model.

[0183] The derived expressions have been validated for their accuracy and sensitivity obtained from numerical simulations of full-order electrochemical models.

[0184] [Optimize the settings]

[0185] The goal of optimization is to obtain the optimal current profile for identifying electrochemical parameters. The optimal current profile maximizes the sensitivity or sensitivity-related indices (e.g., Fisher information) of certain parameters subject to operational constraints. Under voltage constraint c1 and current constraint c2, the optimization problem of the current profile can be expressed as the following equation (17).

[0186] Equation 17

[0187]

[0188] c1:V min ≤V(t)≤V max

[0189] c2:I min ≤I(t)≤I max

[0190] In the optimization problem of equation (17), the objective function is The integral of the square (i.e., the square integral) represents the expression of a parameter θ from t0 to t0. f The normalized sensitivity over a specified time interval. The solution to the optimization problem is in the time domain (t0 to t). f The current curve in the equation allows for maximizing the magnitude of the objective function. Under the assumption of independent and identically distributed Gaussian measurement noise, the objective function is equivalent to Fisher information estimated using a single parameter. Normalization sensitivity. The ratio of output and parameter variation is quantified, and is the sensitivity of the parameter multiplied by its nominal value, as shown in equation (18) below.

[0191] Equation 18

[0192]

[0193] In the battery model, the nominal values ​​of different parameters vary considerably. Therefore, normalized sensitivity can better represent the influence of these parameters. The sensitivity can be calculated in the time domain using the analytical expressions described earlier. Inequality constraints c1 and c2 are employed to keep the battery within a reasonable operating range. The constraint on the input current considers the validity of the single-event hypothesis in deriving the analytical sensitivity and battery health. Furthermore, the constraint on the voltage is adapted to the recommended voltage window for battery chemistry.

[0194] Because optimization problems are highly nonlinear and nonconvex, local optima conditions are a major challenge. The pseudospectral method is an effective tool for solving complex nonlinear optimization problems and has been widely applied to real-world engineering problems. The pseudospectral method is a numerical analysis method used in applied mathematics and engineering computation to calculate solutions to partial differential equations. It is also known as discrete variable representation.

[0195] In embodiments of this disclosure, the Legendre-Gauss-Radau (LGR) pseudospectral method with an adaptive multi-grid spacing configuration is employed. The LGR pseudospectral method is implemented using general-purpose optimal control software (GPOPS).

[0196] GPOPS is a commercially available general-purpose MATLAB software that can obtain numerical solutions to optimization problems. GPOPS can also implement new types of variable-order Gaussian quadrature methods.

[0197] The variable-order Gaussian quadrature method approximates the continuous-time optimization problem as a sparse nonlinear programming problem (NLP) to facilitate solving. Then, solvers such as the interior-point optimizer (IPOPT) and the sparse nonlinear optimizer (SNOPT) are used to solve the NLP.

[0198] [Kramer-Romagna border]

[0199] After obtaining the optimal current curve using the LGR pseudospectral method, the expected estimation accuracy achievable using the corresponding curve based on Cramer-Rhodes boundary analysis can be evaluated. Given a current curve from 0 to t... f The output data curve y(t) measured within a certain time period, under the assumption of independent and identically distributed Gaussian measurement noise, the Fisher information of a certain model parameter θ can be expressed as the following equation (19). In equation (19), σ 2 y It is the variance of the noise.

[0200] Equation 19

[0201]

[0202] The Fisher information, expressed as equation (19), is equal to the objective function expressed in equation (17). By inverting the Fisher information matrix, the Cramer-Rhodes boundary in equation (20) can be obtained.

[0203] Equation 20

[0204]

[0205] Equation (20) represents the lower bound of the covariance of the estimation error of the unbiased estimator. The Cramer-Rhodes boundary is often used as an algorithm-independent metric to assess the quality of data.

[0206] [Optimization Results and Discussion]

[0207] To demonstrate the optimization method according to this disclosure, the solid diffusion coefficient D will be considered separately. s，p Volume fraction of active material ε s，p and reaction rate coefficient k p The optimized current curves are disclosed within an 1800-second time domain. The results are followed by a characteristic analysis of the optimized curves. The battery used in this embodiment is a lithium cobalt oxide battery. The parameters are taken from S. Moura's paper "Singleparticle model with electrolyte and temperature: An electrochemical battery model" (https: / / github.com / scott-moura / SPMeT, published 2019-08-20). The OCP slope is shown for the entire state-of-charge (SOC) range of the battery. like Figure 1 As shown.

[0208] OCP slope It plays an important role in determining the sensitivity of target parameters and the optimal current mode. Through the following equation (21), the state of charge (SOC) of the battery is related to the particle surface concentration c. se，p There is a correlation between them.

[0209] Equation 21

[0210]

[0211] In equation (21), β is c se，p / c s，max，p That is, β is the lithium particle surface concentration (c se，p ) and the maximum solid concentration of lithium that can be contained in electrode particles (c s，max，p The ratio of β to 1. 100％ This is the β value when SOC is 100%. 0％ This is the β value when SOC is 0%.

[0212] During the optimization of the current curve, the GPOPS software can maximize the sensitivity of the parameters within a given time of 1800 seconds. The initial SOC is set to 50%, and V is used as a boundary condition. max It was set to 4.2V, V min It was set to 3.105V, I max It was set to 72A (during discharge), and I max It is set to -72A (during charging).

[0213] [Volume fraction of active material ε] s，p [Optimized current curve]

[0214] Volume fraction ε of cathode active material s，p The optimized current curve and the resulting voltage and SOC response are shown in Figure 2 As shown in the image.

[0215] According to the sensitivity expression in equation (12), the volume fraction ε of the active material in the positive electrode s，p It includes two items. The first item is... It can be called a nonlinear, non-dynamic term because it is primarily a static nonlinear function as shown in equation (13). The second term is... This can be called a semi-linear dynamic term because it relates to the kinetics of solid-phase lithium diffusion. It is important to note that... Usually more Much smaller, therefore ignored in subsequent analysis. To analyze and optimize the current curve, the two items (i.e., and ) can be like Figure 3 It is shown as a separate chart.

[0216] Depend on Figure 3 It can be seen that the volume fraction ε of the active material s，p Sensitivity to semilinear dynamic terms Dominated by. Therefore, the pattern of the optimal current curve may be largely determined by. kinetics and OCP slope Determined. Specifically, state sensitivity. The dynamics can be captured by the sensitivity transfer function in equation (14), which characterizes the pole at s = 0. Essentially, it is the integral of current over time, which increases / decreases under non-zero current and remains constant under zero current. Therefore, the optimal curve tends to push the battery SOC to the point with the maximum OCP slope (i.e., the point where SOC is 84.5%) and hold at the corresponding point to reach the maximum value. To achieve this, the optimal current profile begins at the maximum current (MC) - maximum voltage (MV) charging phase to approach the desired state of charge (SOC) as quickly as possible. Figure 2 As shown. Then the current is gradually cut off to maintain the ideal SOC. The latter half of the curve characterizes a series of SOC-maintaining current pulses. The purpose is to utilize the current-related nonlinear term. Further enhance sensitivity, but its contribution is as follows: Figure 3 What is shown is only a minor detail.

[0217] When using the optimal current profile to estimate parameters, the Cramer-Rao boundary can be calculated to assess the expected estimation accuracy. Under the optimized current profile, ε s，p The normalized Fisher information can be calculated as shown in equation (22) below.

[0218] Equation 22

[0219]

[0220] Furthermore, equation (22) gives the normalized Cramer-Robben boundary as shown in equation (23) below.

[0221] Equation 23

[0222]

[0223] Equation (23) indicates that if the standard deviation σ of the voltage measurement noise v =0:1V, then σ(ε) s，p The boundary will be the volume fraction ε of the active material. s，p The nominal value is 0.04 x 0.1 x 100%, which is 0.4%.

[0224] [Solid diffusion coefficient D] s，p [Optimized current curve]

[0225] Figure 4 The solid-phase diffusion coefficient D of the positive electrode is shown. s，p Optimized current excitation and corresponding voltage and SOC response.

[0226] According to the sensitivity expression in equation (7), D s，p The voltage sensitivity includes a semi-linear dynamic term. But it does not include ε s The sensitivity to nonlinear, non-dynamic terms. Note. and The value is the smallest and is ignored in subsequent analysis. Therefore, D s，p Sensitivity is affected by state sensitivity And the OCP slope of the positive electrode The combined effects of control. State sensitivity. Characterized by the sensitivity transfer function in equation (9), there is no pole at s = 0, which is consistent with the volume fraction ε of the active material. s，p Their sensitivities differ. Therefore, continuous current excitation is required to exhibit this. It is a non-zero value. Regarding the slope of OCP, such as... Figure 1 As shown, three peaks were observed at SOC = 61.5%, SOC = 76.6%, and SOC = 84.5%, respectively. To obtain maximum sensitivity, it is desirable to... and It also has a large value. Because and The linkage effect causes the optimized curve to oscillate between the peak positions of the OCP slope, such as... Figure 4 As shown in (c), during the transition between peak values, the current excitation of maximum current-maximum voltage will establish / maintain a significant [value / sustain]. It is connected to a large OCP slope at the peak position. When the SOC reaches its peak, the current curve switches to a short pulse period. This temporarily holds the SOC at the peak of the OCP slope to maximize the effect of the large OCP slope. However, due to the stability property of the sensitivity transfer function, the current curve is limited during the SOC maintenance pulse period. As the current decreases, the optimal curve must quickly switch back to the maximum current-maximum voltage mode to drive the SOC toward the next OCP slope peak position.

[0227] Under the optimized current curve, D s，p Fisher information can be calculated according to the following equation (24).

[0228] Equation 24

[0229]

[0230] Equation (24) gives the normalized Cramer-Robben boundary, which can be expressed as equation (25).

[0231] Equation 25

[0232]

[0233] Equation (25) indicates that if the standard deviation of the voltage noise σ v =0:1V, then regarding σ(D) s，p The Kramer-Rhodes boundary will be D s，p The nominal value is 0.858 x 0.1 x 100% = 8.56%, or 8.58%. If desired, optimizing the current curve over a longer time domain can reduce the error boundary.

[0234] [Reaction rate constant k] p [Optimized current curve]

[0235] Figure 5 The reaction rate constant k is provided in the middle. p The optimized current curve and corresponding voltage response.

[0236] The current pattern is essentially maximum current (MC) - maximum voltage (MV) charging, followed by maximum current discharging, occasionally alternating between maximum current (MC) charging and maximum current (MC) discharging. This is achieved by relating k in equation (16). p The analytical sensitivity expression makes this feature easy to understand. According to equation (16), k p The sensitivity is a monotonically increasing function of the current I, therefore, maximum current charging or maximum current discharging is required to maximize the sensitivity. By exchanging current density j 0，i For c se The slight dependence on / SOC has only a negligible effect on sensitivity.

[0237] Under the optimized current curve, Fisher information is obtained from the following equation (26).

[0238] Equation 26

[0239]

[0240] Equation (26) gives the normalized Cramer-Robben boundary, which can be expressed by the following equation (27).

[0241] Equation 27

[0242]

[0243] Equation (27) indicates that if the standard deviation σ of the voltage measurement noise v =0:1V, then regarding σ(k) p The Kramer-Rhodes boundary will be k p The nominal value is 0.46 x 0.1 x 100% = 4.6%, that is, 4.6%.

[0244] This disclosure provides an optimization of the current excitation for estimating battery electrochemical parameters. Based on the obtained analytical sensitivity equation, a method for designing the optimal current profile within a given time domain is formulated.

[0245] In one embodiment of the present invention, the solid-phase diffusion coefficient D is shown. s Volume fraction ε of electrode active material s The results for the three parameters—the reaction rate constant k, the current curve, and the current curve. By correlating these results with analytical expressions for parameter sensitivity, optimal patterns for different parameters and underlying mechanisms were discovered. Interestingly, it was noted that the optimal patterns for different parameters are fundamentally different. Numerical results can depend on the specific battery chemistry and the set of parameters considered. However, the basic patterns and characteristics considered in this disclosure are considered generalizable. In future work, the obtained optimized current curves will be used to estimate the individual parameters, with the goal of significantly improving estimation accuracy.

[0246] Preferably, the excitation of the optimal current curve for identifying the battery electrochemical parameters and the use of the excitation to identify the parameters can be achieved through a computer system.

[0247] Figure 6 This is a block diagram schematically illustrating a system 10 for generating an optimal current profile for identifying electrochemical parameters and estimating electrochemical parameters using the optimal current profile, according to an embodiment of the present disclosure.

[0248] Reference Figure 6 The system 10 may include: a current application unit 12 for applying a charging current and / or a discharging current to the battery 11; a voltage measurement unit 13 and a temperature measurement unit 14 for measuring the voltage and temperature of the battery 11 respectively when current flows through the battery 11; a storage unit 15 for storing programs and data required to implement this disclosure; a communication unit 16 for sending and receiving data with external devices; and a control unit 17 for controlling the operation of the overall system.

[0249] The voltage measuring unit 13 includes a voltage measuring circuit capable of measuring the voltage of the battery 11, and the temperature measuring unit 14 includes a thermocouple capable of measuring the temperature of the battery 11.

[0250] There are no particular restrictions on the type of storage unit 15, as long as it is a storage medium capable of recording and erasing information. For example, storage unit 15 can be a hard disk, RAM, ROM, EEPROM, register, or flash memory. Storage unit 15 stores and / or updates and / or erases and / or transmits programs including various control logics executed by control unit 17 and / or data generated during the execution of control logic, lookup tables, predefined functions and parameters, chemical / physical / electrical constants, etc.

[0251] The communication unit 16 includes a known communication interface that supports communication between two different communication media. In one example, the communication interface may support CAN communication, daisy-chain communication, RS232 communication, etc.

[0252] The control unit 17 may optionally include processors, application-specific integrated circuits (ASICs), another chipset, logic circuits, registers, communication modems, data processing devices, etc., known in the art, to perform various control logics.

[0253] The control unit 17 can determine the optimal current profile that maximizes sensitivity for the electrochemical parameters included in the electrochemical model of the battery 11. Equation (1-1) above is an example of an electrochemical model.

[0254] Preferably, the control unit 17 can combine the above equations as needed to determine the optimal current curve that has the greatest sensitivity to electrochemical parameters within a preset time domain (e.g., 1800 seconds), and record the optimal current curve in the storage unit 15.

[0255] Preferably, the optimal current profile can vary depending on the type of electrochemical parameters. In other words, the optimal current profile depends on the electrochemical parameters.

[0256] Control unit 17 can determine the particle surface concentration (c) from the battery current (I) to the electrode using an electrochemical model of a battery that applies the single-particle assumption. se，i The transfer function of the transfer function is used to determine the optimal current profile and the partial derivatives of the corresponding transfer function with respect to the electrochemical parameter (θ). The corresponding sensitivity transfer function.

[0257] Preferably, the transfer function from the battery current to the particle surface concentration of the electrode can be determined by solving the control equation of the electrochemical model using the single-particle assumption, Laplace transform, and Pad approximation, as in equation (2).

[0258] In one example, when the electrochemical parameter is the solid-phase diffusion coefficient D of the electrode... s，i When the sensitivity transfer function is given by equation (9), the sensitivity transfer function can be expressed as equation (9). In another example, when the electrochemical parameter is the volume fraction ε of the active material of the electrode...s，i When the sensitivity transfer function is expressed as equation (14), the sensitivity transfer function can be expressed as equation (14).

[0259] Control unit 17 can also be used by defining the surface concentration of electrode particles (c se，i ) and electrode overpotential (η) i The correlation between the Butler-Volmer equation (3) and the electrode overpotential (η) is determined by the Butler-Volmer equation (as in Equation 8 or Equation 13). i ) on particle surface concentration (c se，i The partial derivative of ) or the overpotential of the electrode (η) i The electrochemical parameters (θ, e.g., the volume fraction of the active material ε) affect the following: s，i The slope of the overpotential corresponding to the partial derivative of ).

[0260] The control unit 17 can also determine the OCP slope corresponding to the partial derivative of the OCP function of the electrode with respect to the particle surface concentration. Preferably, the control unit 17 can determine this slope by using a predefined OCP function U. i (c se，i To determine the slope of the OCP.

[0261] The control unit 17 can determine the sensitivity curve of the electrochemical parameters for the battery voltage of the electrochemical model within a preset time domain by using the sensitivity transfer function and / or the overpotential slope and / or the OCP slope.

[0262] The control unit 17 can also change the battery current and determine the current curve in a preset time domain, thereby maximizing the square integral of the change in the sensitivity curve in the time domain. Here, the current curve is the optimal current curve that has the greatest sensitivity to electrochemical parameters.

[0263] In one example, the control unit 17 can determine the solid diffusion coefficient D of the counter electrode by using equations (7) and (17). s，i The optimal current profile exhibits maximum sensitivity. Furthermore, the control unit 17 can determine the volume fraction ε of the active material in the counter electrode using equations (12) and (17). s，i The optimal current curve exhibiting maximum sensitivity. Furthermore, the control unit 17 can determine the reaction rate constant k against the electrode using equations (16) and (17). i The optimal current curve that provides the greatest sensitivity.

[0264] Preferably, the control unit 17 can determine an optimal current profile so as not to deviate from preset current boundary conditions. Furthermore, the control unit 17 can determine an optimal current profile such that the battery voltage determined from the electrochemical model does not deviate from preset voltage boundary conditions.

[0265] Specifically, when determining the optimal current curve under voltage and current boundary conditions using equation (17), control unit 17 can use a pseudospectral method (preferably, a Legendre-Gauss-Radow (LGR) pseudospectral method with an adaptive multi-grid spacing configuration). That is, control unit 17 uses the pseudospectral method to adaptively determine the optimal current curve, such that the normalized sensitivity equation of equation (18) is... The square integral in the target time domain (t0~t) f The sensitivity equation is maximized in ( ). This corresponds to the sensitivity curve, which represents the change in sensitivity within a preset time domain. Due to the sensitivity equation... Since the battery current is included as a variable, the sensitivity curve depends on the current curve. In one example, control unit 17 can implement the LGR pseudospectral method using General Purpose Optimal Software (GPOPS). GPOPS can accept normalized sensitivity equations related to the time domain, voltage boundary conditions, current boundary conditions, and sensitivity. Furthermore, it generates the variable curve (optimal current curve) of the sensitivity equation in the time domain, thereby maximizing the square integral of the normalized sensitivity equation.

[0266] After determining the optimal current profile for the electrochemical parameters, the control unit 17 can apply the optimal current profile to the battery 11 using the current application unit 12 for a duration corresponding to a preset time range (e.g., 1800 seconds).

[0267] Applying the optimal current profile to the battery means adjusting the current magnitude according to the optimal current profile while charging and / or discharging the battery 11 within the time corresponding to the time domain.

[0268] The control unit 17 can use the voltage measurement unit 13 to measure the battery voltage, and simultaneously apply an optimal current curve to the battery 11 within a preset time domain to generate a measured battery voltage curve, and record the measured battery voltage curve in the storage unit 15. Here, the measured battery voltage curve is a curve representing the change of battery voltage over time.

[0269] The control unit 17 can also use the temperature measurement unit 14 to measure the battery temperature, and simultaneously apply an optimal current curve to the battery 11 within a preset time domain, generating a battery temperature measurement curve and recording the battery temperature measurement curve in the storage unit 15. Here, the battery temperature measurement curve is a curve representing the change of battery temperature over time.

[0270] The control unit 17 can also generate a predicted battery voltage curve by predicting the battery voltage from the battery current curve and the measured battery temperature curve within a time corresponding to the time domain using the electrochemical model of the battery 11, and can store the predicted battery voltage curve in the storage unit 15.

[0271] In this disclosure, the electrochemical model of battery 11 can be predefined by the above equation (1-1). The parameters and functions of the electrochemical model, U... i (c se，i ), Φ e，i and η i The specific parameters and functions of the electrochemical model can vary depending on the chemical properties of battery 11. Furthermore, the functions can be simplified (reduced) using techniques such as the single-particle hypothesis, Laplace transform, and Pad approximation. Additionally, the parameters and functions of the electrochemical model can be pre-recorded in storage unit 15 or included as variables or functions in a program executed by control unit 17.

[0272] Various electrochemical models are known in the art. Obviously, any known electrochemical model can be used to implement this disclosure. As one example, the electrochemical model disclosed in the paper “Design and parametrization analysis of aduced-order electrochemical model of graphite / LiFePO4 cells for SOC / SOHestimation” (Journal of Power Sources. Vol. 237, pp. 310-324, 2013) may be referenced, but this disclosure is not limited thereto.

[0273] If the difference between the predicted battery voltage curve and the measured battery voltage curve is greater than a threshold, the control unit 17 can also adaptively adjust the electrochemical parameters to reduce the difference between the predicted battery voltage curve and the measured battery voltage curve to a preset reference value. The adjusted electrochemical parameters are then identified as the current electrochemical parameters of the battery and recorded in the storage unit 15.

[0274] The difference between the predicted battery voltage curve and the measured battery voltage curve can be determined by calculating the integral value of each curve in the time domain and determining the ratio of the difference between the two integral values ​​to the integral value of either curve, but this disclosure is not limited thereto.

[0275] The control unit 17 can also calculate the change in the electrochemical parameters identified at the current time and the change in the electrochemical parameters of the battery 11 in the BOL (Start of Life) state, quantitatively determine the SOH (State of Health) of the battery 11 based on the change, and record it in the storage unit 15.

[0276] In one example, when the solid-phase diffusion coefficient D of the positive electrode of battery 11... s，p When the SOH of the positive electrode of battery 11 is reduced by 10% compared to the BOL state, it can be determined that it is 90%. In another example, when the volume fraction ε of the active material of the positive electrode... s，p When the SOH at the positive electrode of battery 11 is reduced by 10% compared to the BOL state, it can be determined to be 90%. In another example, when the reaction rate constant k at the positive electrode... i When the SOH of the positive electrode of battery 11 is reduced by 10% compared to the BOL state, it can be determined to be 90%.

[0277] The State of Harmony (SOH) determined according to this disclosure can be used to modify the charging / discharging control logic of battery 11. In one example, as the SOH increases, the charging cutoff voltage can be decreased or the discharging cutoff voltage can be increased. In another example, as the SOH increases, the amplitude of the maximum charging / discharging current may decrease. In yet another example, the width of the State of Charge (SOC) portion performing charging / discharging can be reduced. Furthermore, it will be apparent to those skilled in the art that factors affecting the safety of battery 11 can be adjusted according to the increase of SOH.

[0278] The control unit 17 can transmit the optimal current curve or identified electrochemical parameters to an external device via the communication unit 16. Furthermore, the control unit 17 can receive programs, parameters, functions, chemical / physical / electrical constants, etc., necessary for executing this disclosure from an external device and record them in the storage unit 15.

[0279] The aforementioned system 10 can be included in the control system of a device including the battery 11. This device can be an electric vehicle, a hybrid electric vehicle, a plug-in hybrid vehicle, an energy storage system, etc., but the invention is not limited thereto. The control system can, for example, be a BMS (Battery Management System).

[0280] In another example, the system 10 described above can be included in a device for diagnosing the performance of the battery 11. In this case, information related to parameters, functions, chemical / physical / electrical constants, etc., included in the electrochemical model for each model of the battery 11 can be recorded in the storage unit 15. Additionally, the control unit 17 can receive model information about the battery 11 before generating the optimal current profile. The model information of the battery 11 can be directly input via a program interface or transmitted from the control system of the device including the battery 11 via the communication unit 16. The control unit 17 can then determine the optimal current profile corresponding to the target electrochemical parameters by referring to the parameters, functions, and chemical / physical / electrical constants included in the electrochemical model matching the model of the battery 11.

[0281] It will be apparent to those skilled in the art that the combination of control logic performed by the control unit 17 described above may include steps in a method for optimizing current excitation to identify battery electrochemical parameters, and / or steps in a method for using the method to identify electrochemical parameters.

[0282] Furthermore, one or more of the various control logics of the control unit 17 can be combined, and the combined control logic can be written into a computer-readable code system and recorded in a computer-readable recording medium. There are no particular limitations on the recording medium, as long as it can be accessed by a processor included in a computer. As an example, the storage medium includes at least one selected from ROM, RAM, registers, CD-ROM, magnetic tape, hard disk, floppy disk, and optical data recording devices.

[0283] The code scheme can be assigned to a networked computer for storage and execution. Furthermore, the functional programs, code, and code segments used to implement the combinational control logic can be readily deduced by programmers in the art to which this disclosure pertains.

[0284] In the description of the various exemplary embodiments of this disclosure, it should be understood that the elements referred to as "units" are distinguished functionally rather than physically. Therefore, each element can be selectively integrated with other elements, or each element can be divided into sub-elements to efficiently implement control logic. However, it will be apparent to those skilled in the art that integrated or divided elements fall within the scope of this invention if their functional identity can be confirmed.

[0285] This disclosure has been described in detail. However, it should be understood that while the detailed description and specific examples indicate preferred embodiments of this disclosure, they are given by way of illustration only, as various variations and modifications within the scope of this disclosure will become apparent to those skilled in the art from this detailed description.

Claims

1. A system for optimizing current excitation to identify a battery electrochemical parameter, the system comprising: a current application unit coupled to a battery; a voltage measurement unit configured to measure a voltage of the battery; and a control unit operably coupled to the current application unit and the voltage measurement unit, wherein the control unit is configured to: determine a sensitivity transfer function corresponding to a partial derivative of a transfer function from a battery current to a particle surface concentration of an electrode with respect to an electrochemical parameter by using an electrochemical model of the battery; determine an overpotential slope corresponding to a partial derivative of an overpotential of the electrode with respect to the particle surface concentration or a partial derivative of the overpotential of the electrode with respect to the electrochemical parameter by using the particle surface concentration of the electrode and a Butler-Volmer equation defining a correlation between the electrochemical parameter and the overpotential of the electrode; determine an OCP slope corresponding to a partial derivative of an open-circuit potential (OCP) function of the electrode with respect to the particle surface concentration; determine a sensitivity profile of the electrochemical parameter for a battery voltage of the electrochemical model in a time domain by using the sensitivity transfer function, the overpotential slope, and the OCP slope; and vary a battery current in the time domain and determine an optimal current profile such that a squared integral of the sensitivity profile with respect to the battery current variation in the time domain is maximized.

2. The system for optimizing current excitation to identify a battery electrochemical parameter according to claim 1, the control unit is configured to vary the battery current in the time domain without deviating from a pre-set current boundary condition.

3. The system for optimizing current excitation to identify a battery electrochemical parameter according to claim 1, wherein the control unit is configured to vary the battery current in the time domain such that a battery voltage of the electrochemical model does not deviate from a pre-set voltage boundary condition.

4. The system for optimizing current excitation to identify a battery electrochemical parameter according to claim 1, wherein the control unit is configured to determine the optimal current profile using a pseudospectral method such that a squared integral of the sensitivity profile with respect to the battery current variation in the time domain is maximized.

5. The system for optimizing current excitation to identify a battery electrochemical parameter according to claim 4, wherein the pseudospectral method is a Legendre-Gauss-Radau (LGR) pseudospectral method with an adaptive multi-grid spacing configuration.

6. The system for optimizing current excitation to identify a battery electrochemical parameter according to claim 1, wherein the electrochemical model of the battery employs a single particle assumption and the battery voltage V is expressed by the following equation:

7. The system for optimizing current excitation to identify a battery electrochemical parameter according to claim 1, wherein a transfer function from a battery current to a particle surface concentration of an electrode is represented by the following equation: V = φ s,p - φ s,n = (U p (c se,p )- U n (c se,n )) + (φ e,p - φ e,n ) + (η p - η n )- IR l , where Φ s，i : electrode potential, Φ e，i : electrolyte potential at the electrode boundary, U: predefined OCP function V, c se，i : particle surface concentration of lithium ions, η i : overpotential at the electrode-electrolyte interface, R l : lumped ohmic resistance of the battery, p denotes the positive electrode, n denotes the negative electrode, I: battery current.

8. The system for optimizing current excitation to identify a battery electrochemical parameter according to claim 1, wherein 9. The system for optimizing current excitation to identify a battery electrochemical parameter according to claim 1, wherein c se,i : particle surface concentration of lithium embedded in the electrode, I: battery current, R s，i : radius of the electrode particle, D s，i : solid phase diffusion coefficient of the electrode particle, A: electrode area, δ i : thickness of the electrode, ε s，i : volume fraction of active material active at the electrode, F: Faraday constant, i: index indicating the type of electrode, which is p and n for the positive and negative electrode, respectively, s: Laplace transform variable. ​ wherein, The electrochemical parameter is the solid phase diffusion coefficient D of the electrode s，i and The control unit is configured to determine the solid phase diffusion coefficient D of the electrode in the time domain by using the following equation s，i sensitivity curve to the battery voltage V wherein is the overpotential η of the electrode i the partial derivative of the particle surface concentration c se，i corresponds to the overpotential slope, is the OCP function U of the electrode i the partial derivative of the particle surface concentration c se，i corresponds to the OCP slope, and is the sensitivity transfer function of the solid-state diffusion coefficient D of the electrode se，i the partial derivative of the particle surface concentration c s，i of the electrode with respect to the battery current. ​ wherein, The electrochemical parameter is the active material volume fraction ε of the electrode s，i and The control unit is configured to determine the active material volume fraction ε of the electrode in the time domain by using the following equation s，i sensitivity curve to the battery voltage V wherein is the overpotential of the electrode i is the partial derivative of the overpotential of the electrode se，i corresponding to the overpotential slope, is the overpotential of the electrode i is the partial derivative of the overpotential of the electrode s，i corresponding to the overpotential slope, is the OCP of the electrode i is the partial derivative of the OCP of the electrode se，i corresponding to the OCP slope, and is the sensitivity transfer function of the electrode se，i corresponding to the partial derivative of the active material volume fraction of the electrode s,i of the transfer function from the battery current to the particle surface concentration of the electrode 10. A system for identifying battery electrochemical parameters using the system of claim 1, wherein the control unit is configured to: generate a measured battery voltage curve by measuring battery voltage while applying the optimal current profile to the battery during a time corresponding to a time domain; generate a predicted battery voltage curve by predicting battery voltage from the battery current profile during the time corresponding to the time domain using the electrochemical model; reduce a difference between the predicted battery voltage curve and the measured battery voltage curve to a pre-set reference value by adjusting the electrochemical parameters when the difference between the predicted battery voltage curve and the measured battery voltage curve is greater than a threshold value; and identify the adjusted electrochemical parameters as current electrochemical parameters of the battery.

11. A system for optimizing current excitation to identify battery electrochemical parameters, the system comprising: a current applying device coupled to a battery; a voltage measuring device configured to measure voltage of the battery; and a control unit operably coupled to the current applying device and the voltage measuring device, wherein the control unit is configured to: determine an overpotential slope corresponding to a partial derivative of an overpotential of an electrode with respect to a reaction rate constant of the electrode by using a Butler-Volmer equation defining a correlation between the overpotential of the electrode and the reaction rate constant of the electrode and a particle surface concentration of the electrode; determine a sensitivity curve of the reaction rate constant of a battery voltage for an electrochemical model in a time domain by using the overpotential slope; and vary a battery current in the time domain and determine an optimal current profile such that a squared integral of the sensitivity curve with respect to the battery current variation in the time domain is maximized.

12. The system for optimizing current excitation to identify battery electrochemical parameters of claim 11, the control unit is configured to vary the battery current in the time domain without deviating from a pre-set current boundary condition. wherein, 13. The system for optimizing current excitation to identify battery electrochemical parameters of claim 11, the control unit is configured to vary the battery current in the time domain such that a battery voltage of the electrochemical model does not deviate from a pre-set voltage boundary condition. wherein 14. The system for optimizing current excitation to identify battery electrochemical parameters of claim 11, the control unit is configured to determine the optimal current profile using a pseudospectral method such that a squared integral of the sensitivity curve with respect to the battery current variation in the time domain is maximized. wherein 15. The system for optimizing current excitation to identify battery electrochemical parameters of claim 14, the pseudospectral method is a Legendre-Gauss-Radau (LGR) pseudospectral method with an adaptive multi-grid spacing configuration. wherein 16. The system for optimizing current excitation to identify battery electrochemical parameters of claim 11, 17. A system for identifying battery electrochemical parameters using the system of claim 11, wherein The control unit is configured to determine the reaction rate constant k of the electrode in the time domain by using the following equation i sensitivity curve to the battery voltage V: where I: battery current, R: universal gas constant, T: battery temperature, F: Faraday constant, ε s，i : volume fraction of active material having activity at the electrode, j 0，i : exchange current density, α: charge transfer coefficient, A: effective electrode area, R s，i : radius of the electrode particle, δ i : thickness of the electrode, η i : overpotential at the electrode-electrolyte interface, i: index indicating the type of electrode. the control unit is configured to: wherein, ​ generating a measured battery voltage profile by measuring battery voltage while the battery current profile is applied to the battery during a time corresponding to a time domain; generating a predicted battery voltage profile by predicting battery voltage from the battery current profile during a time corresponding to the time domain using an electrochemical model of the battery; when a difference between the predicted battery voltage profile and the measured battery voltage profile is greater than a threshold value, reducing the difference between the predicted battery voltage profile and the measured battery voltage profile to a preset reference value by adjusting a reaction rate constant of the electrode; and identifying the adjusted reaction rate constant of the electrode as a current reaction rate constant.

18. A method for optimizing current excitation to identify a battery electrochemical parameter, the method comprising the steps of: (a) determining a sensitivity transfer function corresponding to a partial derivative of an electrochemical parameter with respect to a transfer function from a battery current to a particle surface concentration of an electrode by using an electrochemical model of the battery applying a single particle assumption; (b) determining an overpotential slope corresponding to a partial derivative of an overpotential of the electrode with respect to the particle surface concentration of the electrode or a partial derivative of an electrochemical parameter with respect to an overpotential of the electrode by using the particle surface concentration of the electrode and a Butler-Volmer equation defining a correlation between the electrochemical parameter and the overpotential of the electrode; (c) determining an OCP slope corresponding to a partial derivative of an OCP function of the electrode with respect to the particle surface concentration; (d) determining a sensitivity profile of the electrochemical parameter for a battery voltage of the electrochemical model in a time domain by using the sensitivity transfer function, the overpotential slope, and the OCP slope; and (e) varying a battery current in the time domain and determining an optimal current profile such that a squared integral of the sensitivity profile with respect to the battery current variation in the time domain is maximized.

19. A method for identifying a battery electrochemical parameter using the method of claim 18, the method comprising the steps of: generating a measured battery voltage profile by measuring battery voltage while the optimal current profile is applied to the battery during a time corresponding to a time domain; generating a predicted battery voltage profile by predicting battery voltage from the battery current profile during a time corresponding to the time domain using the electrochemical model; when a difference between the predicted battery voltage profile and the measured battery voltage profile is greater than a threshold value, reducing the difference between the predicted battery voltage profile and the measured battery voltage profile to a preset reference value by adjusting the electrochemical parameter; and identifying the adjusted electrochemical parameter as a current electrochemical parameter.

20. A method for optimizing current excitation to identify a battery electrochemical parameter, the method comprising the steps of: (a) determining an overpotential slope corresponding to a partial derivative of an overpotential of an electrode with respect to a reaction rate constant of the electrode by using a particle surface concentration of the electrode and a Butler-Volmer equation defining a correlation between the reaction rate constant of the electrode and an overpotential of the electrode; (b) determining a sensitivity curve for a reaction rate constant of a battery voltage for an electrochemical model in a time domain by using the overpotential slope; and (c) changing a battery current in the time domain and determining an optimal current curve such that a squared integral of the sensitivity curve according to the battery current change in the time domain is maximized.

21. A method of identifying an electrochemical parameter of a battery using the method of claim 20, the method comprising the steps of: generating a measured battery voltage curve by measuring a battery voltage while the battery current curve is applied to the battery during a time corresponding to a time domain; generating a predicted battery voltage curve by predicting a battery voltage from the battery current curve during the time corresponding to the time domain using the electrochemical model; when a difference between the predicted battery voltage curve and the measured battery voltage curve is greater than a threshold value, reducing the difference between the predicted battery voltage curve and the measured battery voltage curve to a preset reference value by adjusting the reaction rate constant of the electrode; and identifying the adjusted reaction rate constant of the electrode as a current reaction rate constant.

22. A method of identifying an electrochemical parameter of a battery using the method of claim 20, the method comprising the steps of: generating a measured battery voltage curve by measuring a battery voltage while the battery current curve is applied to the battery during a time corresponding to a time domain; generating a predicted battery voltage curve by predicting a battery voltage from the battery current curve during the time corresponding to the time domain using the electrochemical model; when a difference between the predicted battery voltage curve and the measured battery voltage curve is greater than a threshold value, reducing the difference between the predicted battery voltage curve and the measured battery voltage curve to a preset reference value by adjusting the reaction rate constant of the electrode; and identifying the adjusted reaction rate constant of the electrode as a current reaction rate constant.

23. A method of identifying an electrochemical parameter of a battery using the method of claim 20, the method comprising the steps of: generating a measured battery voltage curve by measuring a battery voltage while the battery current curve is applied to the battery during a time corresponding to a time domain; generating a predicted battery voltage curve by predicting a battery voltage from the battery current curve during the time corresponding to the time domain using the electrochemical model; when a difference between the predicted battery voltage curve and the measured battery voltage curve is greater than a threshold value, reducing the difference between the predicted battery voltage curve and the measured battery voltage curve to a preset reference value by adjusting the reaction rate constant of the electrode; and identifying the adjusted reaction rate constant of the electrode as a current reaction rate constant.

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