Method for determining a battery state variable and battery management system

Abstract

The invention relates to a method (100) for determining a battery state variable (111) of a battery cell (120), comprising the following steps: Obtaining (101) a cell-specific parameter set (110) which is characteristic of the battery cell (120); Obtaining (102) a rest voltage (U OCV ) of the battery cell (120); and determining (103) the battery state variable (111) of the battery cell (120) based on a battery cell model (130) which shows a relationship between a state of charge (SoC) and the open-circuit voltage (U OCV ) of the battery cell (120) based on an analytical function using the cell-specific parameter set (110) of the battery cell (120); wherein the analytical function is based on two incomplete Eulerian beta functions depending on the state of charge (SoC) of the battery cell (120) and the cell-specific parameter set.

Description

Technical field

[0001] The present invention relates to methods for determining a battery state variable of a battery cell, for example, for a battery-powered vehicle, and to a battery management system, for example, of a battery-powered vehicle. In particular, the invention relates to techniques for determining an agnostic, adaptable SOC-OCV (state of charge open circuit voltage) model for battery cells for highly accurate calculation of battery state variables SOx and use in filters. State of the art

[0002] The BMS (Battery Management System) and the software it runs are required, for example, to calculate the current state of the battery (SOC, "State of Charge") and to predict its performance (SOP, "State of Power"). The accuracy of the SOx functions is particularly important here. They enable, for example, optimal and safe operation of the battery until it is recharged by accurately determining the range and prevent the vehicle from breaking down, for example. Previous methods for determining the SOC, such as the Open Circuit Voltage (OCV) method, have their limitations, particularly when it comes to accurately mapping the hysteresis between charging and discharging curves. This leads to inaccuracies in the determination of the SOC, which can also affect battery performance and service life. Description of the invention

[0003] An object of the invention is therefore to provide a concept for a more accurate determination of the state of charge (SOC) of the battery cell, in particular based on an OCV model, which does not result in any significant impairment of the battery performance and the service life of the battery cell.

[0004] In particular, an object of the invention is to provide a method for a more accurate determination of the state of charge (SOC) of the battery cell, in which the hysteresis between charging and discharging curves can also be accurately mapped.

[0005] The object is achieved by the subject matter of the independent claims. Advantageous developments of the invention are specified in the dependent claims, the description, and the accompanying figures.

[0006] The invention is based on the idea of overcoming the limitations described above when using an OCV model for the battery cell model, as shown below.

[0007] The disclosure introduces a more precise function for describing the OCV curves, which, by taking hysteresis into account, creates a more accurate SOC-OCV model. More accurate SOC estimation can be achieved, for example, using a Kalman filter (or an extended Kalman filter). The improved accuracy of this new function allows users to more accurately monitor the actual battery state of charge, thus ensuring optimal energy management, increasing electric vehicle performance, and extending battery life.

[0008] The new OCV model presented here can be used, for example, in the following areas: 1. SOC determination (OCV) and calculation of other battery state variables such as capacity, SOHQ, SOP and active balances (SOS). 2. Integration of different filters (Kalman filter, DDR filter) in a battery condition detection system. 3. Analysis of the OCV characteristic (curve discussion) for cell / battery characterization (hysteresis, aging, etc.) 4. Modeling and simulation.

[0009] The new OCV model and method for determining a battery state variable presented here is based on an agnostic SOC OCV model, which describes the open circuit voltage (OCV) of a large class of batteries of different types, materials, etc. with a high degree of accuracy. By using the new battery cell model presented here, the manufacturer can overhaul, replace, or decommission the battery over time without compromising the accuracy of the determination of the battery state variable. The new OCV model is also suitable for real-time updating of the parameter values, e.g., when replacing the battery cell. The method presented here can be executed on a battery management system and thus perform adaptive battery management, taking into account hysteresis, aging, and tolerances in SOx monitoring.

[0010] In contrast to previous OCV models, which, according to the applicant's experience, do not meet or only partially meet the requirements for agnosticism, adaptability, and accuracy, these requirements are fully met with the new OCV model presented here. The new OCV model presented here is ideally suited for application in a Kalman filter or an extended Kalman filter.

[0011] Embodiments of the invention achieve the above-described problem by transforming the underlying Nernst equation into an integral equation, which is then generalized and solved analytically. It can be shown mathematically that the original Nernst equation is reproduced by a specific set of parameters. Initial measurements with two different cathode materials (NCM, LFP) show that the achieved accuracies are of the same order of magnitude, regardless of battery types and electrode materials.

[0012] According to a first aspect of the invention, the object is achieved by a method for determining a battery state variable of a battery cell, comprising the following steps: Obtaining a cell-specific parameter set which is characteristic of the battery cell; Obtaining a rest voltage (U OCV ) of the battery cell; and determining the battery state variable of the battery cell based on a battery cell model which establishes a relationship between a state of charge (SoC) and the open-circuit voltage (U OCV ) of the battery cell based on an analytical function using the cell-specific parameter set of the battery cell; where the analytical function is based on two incomplete Eulerian beta functions depending on the state of charge (SoC) of the battery cell and the cell-specific parameter set.

[0013] The battery cell can be used in many applications. For example, it can be a battery cell for a battery-powered vehicle or a battery storage system, such as a photovoltaic system or other electrical system. The battery cell can be used, for example, in a starter battery or in other types of energy sources.

[0014] Such a method allows a more precise description of the OCV curves and determines the battery state parameter, e.g., the state of charge of the battery cell based on a more accurate SOC-OCV model taking hysteresis into account. A more precise estimation of the SOC can be achieved, for example, using a Kalman filter (or an extended Kalman filter). The improved accuracy of this new method allows users to more precisely monitor the actual state of charge of the battery or battery cell, thus ensuring optimal energy management (load balancing), increasing the performance of electric vehicles or other devices in which the battery cell is used, and extending the battery's service life.

[0015] The cell-specific parameter set can, for example, be determined from offline measurement data of the specific battery cell and stored in a memory. This determination is usually non-linear due to the incomplete Euler beta functions. A nonlinear optimization algorithm can be used for the determination, for example, a gradient-based iterative search method that iteratively works on the offline measurement data and determines a solution for the cell-specific parameter set.

[0016] According to an exemplary embodiment of the method, the analytical function is based on a sum of the two incomplete Eulerian beta functions weighted using the cell-specific parameter set and a linear function of the state of charge (SoC) of the battery cell.

[0017] This has the technical advantage that the analytical function can be precisely determined with knowledge of the open-circuit voltage and the cell-specific parameter set of the battery cell, and thus precise statements can be made about the state of charge of the vehicle or the device in which the battery cell is used, or other battery state variables.

[0018] According to an exemplary embodiment of the method, the analytical function gives the relationship between the state of charge (SoC) and the rest voltage (U OCV ) of the battery cell regardless of the technology or cell chemistry of the battery cell.

[0019] This has the technical advantage that the battery state variable depends only on the open-circuit voltage and the corresponding cell-specific parameter set and can be determined independently of the technology or cell chemistry of the battery cell.

[0020] According to an exemplary embodiment of the method, the analytical function is based on a Nernst equation generalized in the form of an integral equation.

[0021] This allows the battery state variable to be determined advantageously independently of the cell chemistry and the technology of the battery cell.

[0022] According to an exemplary embodiment of the method, the Nernst equation defines the relationship between the state of charge (SoC) and the rest voltage (U OCV ) of the battery cell based on a logarithmic function of the state of charge (SoC) and a logarithmic function of the inverse state of charge.

[0023] This model is a good starting point for a generalization to precisely describe cell behavior.

[0024] According to an exemplary embodiment of the method, the Nernst equation defines the relationship between the state of charge (SoC) and the rest voltage (U OCV ) of the battery cell as follows: U OCV (θ) = E0 + µ1ln (θ) + µ2ln (1 - θ), where θ is the state of charge of the battery cell and E0,µ1,µ2 are cell-specific parameters of the battery cell for the Nernst equation.

[0025] This form of the Nernst equation can be seen below, for example, Fig. 1, especially with regard to equation (3). This form of the Nernst equation is well suited as a starting point for a generalization to precisely describe cell behavior.

[0026] According to an exemplary embodiment of the method, the analytical function is based on the following integral equation: UOCV(θ)=M0+∫M1(1−θ)θ−∫(M1+M2)θ(1−θ)θ, where θ is the state of charge of the battery cell and M0, M1, M2 are cell-specific parameters of the battery cell for the integral equation.

[0027] This integral equation is, for example, below Fig. 2, especially regarding equation (10). This generalized representation of the Nernst model as an integral equation allows the behavior of the battery cell to be described independently of the technology.

[0028] According to an exemplary embodiment of the method, the integral equation is further generalized by two additional exponents in the two respective integrals of the integral equation as follows: UOCV(θ)=M0+∫M1(1−θ)α1θσ1−∫(M1+M2)θ(1−θ)α2θσ2, where θ is the state of charge of the battery cell and M0, M1,M2, α1,α2, σ1, σ2 are cell-specific parameters of the battery cell for the generalized integral equation.

[0029] This generalized integral equation is, for example, below Fig. 2, especially for equation (11). This generalized representation of the integral equation allows for a precise description of the hysteresis effects.

[0030] According to an exemplary embodiment of the method, the analytical function is as follows: U OCV,VGL (θ) = L0 + L1θ + L2Beta(θ,1 - κ, σ) + L3Beta(1 - θ, σ, 2 - κ), where θ is the state of charge of the battery cell and L0,L1,L2,L3,σ,κ are cell-specific parameters of the battery cell for the analytical function.

[0031] This analytical function can be seen below, for example, Fig.2, particularly Equation (19). This equation provides a sufficiently accurate mathematical description of the OCV characteristic, which precisely describes all materials (cathode, anode) with different hysteresis characteristics over the entire SOC and temperature range.

[0032] According to a second aspect of the invention, the object is achieved by a battery management system with one or more battery cells, wherein the battery management system comprises: a processor which is designed to carry out the method according to one of the preceding claims in order to determine a battery state variable, in particular a state of charge (SoC) or a state of power (SoP), of the one or more battery cells.

[0033] The battery management system can be used in many applications. For example, it can be used in a battery-powered vehicle, especially during vehicle operation, or in a battery storage system, such as a photovoltaic system or other electrical system. The battery management system can also be used in a starter battery or other types of energy sources.

[0034] Such a battery management system allows for precise determination of battery state variables, such as state of charge (SoC), state of power (SoP), etc., in order to actively balance the battery cells and increase the range and service life of the battery.

[0035] According to an exemplary embodiment of the battery management system, the processor is configured to determine the state of charge (SoC) of the one or more battery cells based on a Kalman filter or an extended Kalman filter configured to iteratively evaluate the battery cell model.

[0036] This offers the advantage that a very accurate prediction of the battery condition is possible via the Kalman filter or the extended Kalman filter when using the battery cell model presented here (SOC OCV model), which is based on the evaluation of an analytical function, so that the use of the battery can be optimally controlled in order to optimize the range and service life.

[0037] According to an exemplary embodiment of the battery management system, the processor is configured to initiate a charge equalization of the battery cells based on the determined state of charge (SoC) of the respective battery cells.

[0038] This has the advantage of providing precise information about the charge level of the individual battery cells, which can be used to set an optimal charge balance across all battery cells.

[0039] With the battery cell model presented here and the corresponding method for determining a battery state variable, gentle and safe operation of the battery can be ensured over the battery's lifetime.

[0040] The agnostic OCV model presented here can be optimally and sufficiently accurately parameterized for each battery system for a fixed number of parameters (e.g., up to 9 parameters, but also more). This is advantageous compared to the state of the art, as the manufacturer could overhaul, replace, or decommission the battery over time. The cells can differ in format and material. This includes not only lithium-ion cells, but also other cell chemistries such as those of lead-acid starter batteries. The applicant is currently not aware of any mathematical model for the OCV characteristic curve that can function under different environments, i.e., is interoperable. According to the new EU Battery Regulation, it will be required in the future that the BMS must be reset for a "second life." For example, ifIf the BMS is then used for a different battery system, the OCV model does not need to be changed during SOC determination or directly in the Kalman filter; only the parameter values need to be updated. This typically requires cell characterization in the laboratory. However, the parameter values can also be updated online using the current, voltage, and temperature measurement signals, either via a cloud or directly on the BMS.

[0041] For filters such as Kalman filters, a SOC OCV model is required, which increases the accuracy of the filter and reduces the convergence time of the filter. Due to the significantly higher accuracy of almost 50% compared to the state of the art with the same number of parameters, it is possible to describe inaccuracies such as hysteresis more precisely with the SOC OCV model and then correct them using a Kalman filter. This is directly related to the available hardware resources of the BMS, especially when individual cells or modules are to be monitored. Filters are generally very computationally intensive, so an accurate, adaptable SOC OCV model is a major advantage over the state of the art. Short character description

[0042] An advantageous embodiment of the invention is explained below with reference to the accompanying figures. They show: Fig.1 an OCV-SOC representation 10 of open-circuit voltage characteristics 11, 12, 13 in the charging and discharging direction and according to the Plett model; Fig. 2 shows a schematic representation of a method 100 according to the invention for determining a battery state variable according to the disclosure; Fig. 3 is a schematic representation of a battery management system 200 according to the disclosure; Fig. 4 a comparative representation 400 of the measurement data for the open-circuit voltage of the NCM pouch cell in the charging and discharging direction with the results of the model according to the invention according to equation (19); Fig. 5 a comparative representation 500 of the measurement data for the hysteresis of the NCM pouch cell with the results of the model according to the invention according to equation (19); Fig.6 a comparative representation 600 of the measurement data for the open-circuit voltage (OCV) of the LFP cell in the discharge direction with the results of the model according to the invention according to equation (15); Fig. 7 a comparative representation 700 of the normalized deviation of the SOC error of an NCM pouch cell for the HPPC (“Hybrid Pulse Power Characterization”) cycle with extended Kalman filter; and Fig. 8 a comparative illustration 800a, 800b of the convergence time of the extended Kalman filter (EKF) at a low and high initial charge state for a fault injection test.

[0043] The figures are merely schematic representations and serve only to illustrate the invention. Identical or equivalent elements are provided with the same reference numerals throughout.

[0044] In the following detailed description, reference is made to the accompanying drawings, which form a part hereof, and in which is shown by way of illustration specific embodiments in which the invention may be practiced. It is understood that other embodiments may be utilized and structural or logical changes may be made without departing from the scope of the present invention. The following detailed description, therefore, is not to be taken in a limiting sense. Further, it is to be understood that the features of the various embodiments described herein may be combined with one another unless specifically indicated otherwise.

[0045] The aspects and embodiments are described with reference to the drawings, wherein like reference numerals generally refer to like elements. In the following description, for purposes of explanation, numerous specific details are set forth in order to provide a thorough understanding of one or more aspects of the invention. However, it may be apparent to one skilled in the art that one or more aspects or embodiments may be practiced with a lesser level of specific detail. In other instances, well-known structures and elements are shown in schematic form to facilitate describing one or more aspects or embodiments. It is to be understood that other embodiments may be utilized and structural or logical changes may be made without departing from the concept of the present invention.

[0046] This disclosure describes models or battery cell models for the open circuit voltage of battery cells, in particular SOC OCV models, ie state of charge open circuit voltage models.

[0047] SOC OCV models are required to describe the (quasi-) open-circuit voltage of a cell / battery. The description should consider all known static and dynamic dependencies with the highest possible accuracy. These include, for example, the SOC, the charge / discharge direction (hysteresis), temperature, aging, and tolerances. Ideally, the description should be possible across the entire SOC (0 ... 100%) and temperature range (-30°C ... 60°C) using only a fixed number of different parameters (no model changes).

[0048] The first model was proposed by Plett: "[1] Plett, GL Extended Kalman filtering for battery management systems of LiPB-based HEV battery packs: Part 2. Modeling and identification. J. Power Sources 2004, 134, 262-276". This is ultimately a combined model, which is derived from the Shepherd model UOCV(θ)=Eo−μθ , the Unnewehr model UOCV(θ)=E0−μ θ and the Nernst model UOCV(θ)=E0+μ1 ln(θ)+μ2ln(1−θ) consists of: UOCV,Plett(θ)=K0−K1θ−K2θ+K3 ln(θ)+K4 ln(1−θ)

[0049] The description only covers the SOC dependency. For the temperature dependency, either an additional Arrhenius factor with parameters is used or the five parameters (K0(T), ... , K4(T)) are temperature-dependent. The accuracy of the model is not sufficient to significantly describe other dependencies such as charge-discharge direction (hysteresis), aging, and tolerances. The accuracy also depends heavily on the technology, the SOC, and the temperature range. This model provides particularly good agreement with the open circuit voltage (OCV) characteristic of lithium iron phosphate (LFP) batteries, which is expressed in the low value for the (Root Mean Square (RMS) and maximum error) (see Table 1). Using the SOC-OCV model, UCell,k=UOCV(θk)−R ik The SOC or SOP is determined using a Kalman filter. The second model is discussed in "[2] Zhang, C.; Jiang, J.; Zhang, L.; Liu, S.; Wang, L.; Loh, PC A Generalized SOC-OCV Model for Lithium-Ion Batteries and the SOC Estimation for LNMCO Battery. Energies 2016, 9, 900": UOCV(θ)=K0+K1e−α(1−θ)−K2θ

[0050] Here, the study reveals low values for the above-mentioned errors, especially for LNMCO batteries (see Table 1). A combined model, which includes an additional parameter m as an exponent of the log function, i.e., m ≠ 1, and in which the term proportional to log(1 - θ) is neglected, is proposed in [2]: UOCV(θ)=a+b(−ln(θ))m+c θ+d en(θ−1)

[0051] Limiting the number of freely selectable parameters to 6 certainly also plays a role here. The nonlinearity of the OCV curve is only taken into account by the exponent m. The models (Eqs. 4, 6, 7) were tested for different materials; the results are summarized in Table 1. OCV model RMS error for LNMCO battery in mV Max. error for LNMCO battery in mV RMS error for LFP battery in mV Max. error for LFP battery in mV 1 16.6 36.5 6.2 14.8 2 9.7 21.8 34.9 141 3 13.0 20.6 15.3 27.3

[0052] OCV Model 1: VOC(s)=K0−K1 / s−K2s+K3ln(s)+K4ln(1−s)

[0053] OCV Model 2: VOC(s)=K0+K1e−α(1−s)−K2 / s

[0054] OCV Model 3: VOC=a+b⋅(−lns)m+c⋅s+d⋅en(s−1)

[0055] Table 1: OCV models (state of the art) with accuracies for different technologies (LNMCO, LFP) according to [2].

[0056] Regarding the RootMeanSquare (RMS) and maximum error values for the model according to equation (7) and Table 1, OCV model 3, it is clearly noticeable that the errors lie between the values for equation (4) and equation (6). Therefore, the model equation (7) cannot be considered a technology-independent, general description of the open-circuit voltage UOCV (θ) of a cell / battery. The achieved accuracy is insufficient to adequately account for the charge / discharge direction (hysteresis), aging, and tolerances across the entire SOC and temperature range.

[0057] The respective OCV models (equations (1) to (7)) are required for a variety of different Kalman filters, either in their own right or in the form of Jacobian matrices. The advantages of using a parameter set model defined by an analysis function instead of a numerical representation of the OCV curve with tables are the following: 1. SOC accuracy: Increased accuracy due to better OCV characteristics—represented in Kalman filter state-space matrices. In the case of a numerical representation of the OCV curve using tables, linear interpolation is required between the sampling points. 2. Reusability: easier reusability and adaptability to different cell configurations and types. 3. Reduction of electronic platform resources, especially in multi-cell configurations and OCV adjustment with age. Instead of lookup tables, a fixed number of parameters per cell can be stored in long-term memory and adjusted with age.

[0058] The hysteresis is either only determined by the mean OCV curve UOCV,AVG(θ)=UOCV,DISC(θ)+UOCV,CH(θ)2 or using various models, the so-called “zero- / one-state hysteresis model” is taken into account. The Plett model U OCV,Plett(θ) (equation (4)), which cannot describe the hysteresis (see Figure 1), is used e.g. in an EKF “The one-state hysteresis model”, see [1].

[0059] It turns out that part of the error is compensated for by the Kalman filter itself, but this also affects the convergence time of the Kalman filter. This is understandable because the tolerances and hysteresis, for example, can also be interpreted as measurement inaccuracies of the Kalman filter. While the Kalman filter demonstrably improves the SOC accuracy, this cannot be clearly attributed to the dependencies considered in the model. Due to this issue, recent publications, particularly for the hysteresis of LFP, instead use a higher-order interpolation function for the SOC OCV for the EKF. According to the theorems of analysis, any function can be approximated with arbitrary precision using a simple n-th-order interpolation function. UOCV(θ)=∑k=0Nakθk described. Since the hysteresis characteristic and other dependencies (aging, tolerances) are only inadequately described, the interpolation function (equation (9)) is used as the SOC-OCV model in recent publications. The major disadvantage is that the optimal number of parameters must be determined individually for each technology, e.g. LFP (characteristic parameters) and cells from different manufacturers (design parameters). There are studies that distinguish between N = 4, ... , 9 parameters. Ideally, the open-circuit voltage should only be possible across the entire SOC and temperature range using a fixed number of different parameters (no model change). For these reasons, equation (9) does not allow a technology-independent, general description of the open-circuit voltage. The second disadvantage is that numerical inaccuracies occur during the calculation, since the summands can become very small at higher orders.

[0060] The applicant does not have access to the cells and measurement data used in [2], therefore, only own measurement data from an NCM (pouch) and LFP cell (18650) were used. NCM batteries consist of a lithium nickel manganese cobalt oxide cathode and are abbreviated differently in the literature. It is assumed that the so-called LNMCO batteries (see Table 1) are made of the same material as the NCM batteries used here (LiNi1 / 3Mn1 / 3Co1 / 3O2).

[0061] Fig. 1 shows an OCV-SOC representation 10 of open-circuit voltage characteristics 11, 12, 13 in the charging and discharging direction as well as according to the Plett model according to “[1] Plett, GL Extended Kalman filtering for battery management systems of LiPB-based HEV battery packs: Part 2. Modeling and identification. J. Power Sources 2004, 134, 262-276”.

[0062] The dependence of the open-circuit voltage characteristic on the charging and discharging direction is shown in Fig. 1. This effect is called hysteresis. The solid line 12 is the solution of the so-called Plett model, here for the discharge direction. This model is most commonly used in publications and in BMS software.

[0063] In Fig. 1 is the comparison between the OCV curve in the charging direction (13) U OCV,CH (θ), discharge direction (11) U OCV,DISC (θ) and the Plett model (12) U OCV,Plett (θ). It is clearly evident that the OCV in the charge and discharge directions is only inadequately described by the SOC OCV model (equation (4)). The hysteresis characteristics of the materials (anode, cathode) are completely lost here. Of course, the SOC OCV model according to equation (7) results in U OCV,Plett (θ) better results (see Table 1, e.g. LNMCO), but instead of four parameters, six parameters are now required.

[0064] In summary, it can be stated: 1. According to the state of the art, there is no sufficiently accurate mathematical description of the OCV characteristic with a general mathematical model that includes all materials (cathode, anode) with different hysteresis characteristics and the entire SOC and temperature range. In the known models [equations (1) to (7)], the number of parameters is fixed. 2. There are conditions (temperature) or materials (cathode, anode) for which a sufficiently accurate description is possible at least with the interpolation function (equation (8)), but the necessary number of parameters is not determined, ie in this mathematical model the number of parameters is not fixed. 3. For these reasons, the applicant considers the known models unsuitable for agnostic BMS software, as they depend, among other things, on the cells used, which the manufacturer may overhaul or decommission over time. The applicant is currently unaware of any mathematical model for the OCV characteristic curve that can function under different environments, i.e., is interoperable.

[0065] Fig. 2 shows a schematic representation of an inventive method 100 for determining a battery state variable according to the disclosure.

[0066] The method 100 is used to determine a battery state variable 111 of a battery cell 120. The battery cell 120 may, for example, be a battery cell 120 for a battery-powered vehicle, for example, a vehicle 220 as in Fig. 3. The method 100 includes the following steps: Obtaining 101 a cell-specific parameter set 110 which is characteristic of the battery cell 120; Obtaining 102 of a rest voltage (U OCV ) of the battery cell 120; and Determining 103 the battery state variable 111 of the battery cell 120 based on a battery cell model 130, which shows a relationship between a state of charge SoC and the rest voltage (U OCV ) of the battery cell 120 based on an analytical function using the cell-specific parameter set 110 of the battery cell 120; wherein the analytical function is based on two incomplete Eulerian beta functions depending on the state of charge (SoC) of the battery cell 120 and the cell-specific parameter set, as described below, for example, Fig. 4 described in more detail.

[0067] The cell-specific parameter set can be determined, for example, from offline measurement data of the specific battery cell. This determination is usually nonlinear due to the two incomplete Euler beta functions mentioned above. For example, a nonlinear optimization algorithm, which is usually executed iteratively, can be used for the determination. One possibility is the use of a gradient-based iterative search method that operates on the offline measurement data.

[0068] The analytical function may be based on a sum of the two incomplete Eulerian beta functions weighted using the cell-specific parameter set 110 and a linear function of the state of charge (SoC) of the battery cell 120, as further illustrated below in equations (15) and (19).

[0069] The analytical function can determine the relationship between the state of charge (SoC) and the open-circuit voltage (U OCV) of the battery cell 120 regardless of a technology or cell chemistry of the battery cell 120.

[0070] The analytical function can be based on a Nernst equation generalized in the form of an integral equation, as further illustrated below in equation (10).

[0071] The Nernst equation can describe the relationship between the state of charge (SoC) and the rest voltage (U OCV ) of the battery cell 120 based on a logarithm function of the state of charge (SoC) and a logarithm function of the inverse state of charge (SoC), as further illustrated above in equation (3).

[0072] The Nernst equation can describe the relationship between the state of charge (SoC) and the rest voltage (U OCV ) of the battery cell 120 as follows (see equation (3) above): U OCV(θ) = E0 + µ1ln (θ) + µ2ln (1 - θ), where θ is the state of charge of the battery cell 120 and E0,µ1,µ2 are cell-specific parameters of the battery cell 120 for the Nernst equation.

[0073] The analytical function can be based on the following integral equation (see equation (10) below): UOCV(θ)=M0+∫M1(1−θ)θ−∫(M1+M2)θ(1−θ)θ, where θ is the state of charge of the battery cell 120 and M0,M1,M2 are cell-specific parameters of the battery cell 120 for the integral equation.

[0074] The integral equation can be further generalized by two additional exponents in the two respective integrals of the integral equation as follows (see equation (11) below): UOCV(θ)=M0+∫M1(1−θ)α1θσ1−∫(M1+M2)θ(1−θ)α2θσ2, where θ is the state of charge of the battery cell 120 and M0,M1,M2,α1,α2,σ1,σ2 are cell-specific parameters 110 of the battery cell 120 for the generalized integral equation.

[0075] The analytical function can be written as follows (see equation (19) below): U OCV,VGL (θ) = L0 + L1θ + L2Beta(θ,1 - κ,σ) + L3Beta(1 - θ,σ,2 - κ), where θ is the state of charge of the battery cell 120 and L0,L1,L2,L3,σ,κ are cell-specific parameters 110 of the battery cell 120 for the analytical function.

[0076] Fig. 3 shows a schematic representation of a battery management system 200 according to the disclosure. The battery management system 200 can be used, for example, in a battery-powered vehicle 220, in particular during operation of the vehicle, as schematically shown in Fig. 3 shown.

[0077] The battery management system 200 may, for example, be arranged in a battery-powered vehicle 220 with one or more battery cells 120.

[0078] The battery management system 200 can also be used in many other applications, such as battery storage systems for photovoltaic systems or other electrical systems. The battery management system 200 can be used, for example, in a starter battery or in other types of energy sources.

[0079] The battery management system 200 comprises a processor 210 configured to execute the method 100 as described above. Fig. 2 and below to the Fig. 4 to 8, for example during operation of the vehicle 220 or the device in which the battery management system 200 is used, in order to determine a battery state variable 111, for example a state of charge (SoC) or a state of power (SoP), of one or more battery cells 120.

[0080] The processor 210 may be configured to determine the state of charge (SoC) of the one or more battery cells 120 based on a Kalman filter or an extended Kalman filter. The battery cell model 130 may be evaluated iteratively using the Kalman filter or the extended Kalman filter.

[0081] The processor 210 may be configured to initiate a charge equalization of the battery cells 120 based on the determined state of charge (SoC) of the respective battery cells 120.

[0082] In the following, a derivation of the novel battery cell model according to the invention and the corresponding method 100 for determining a battery state variable is shown.

[0083] Agnostic battery state detection can be used for any battery system. Dependence on other systems or components, such as the HV battery or the cells, can be applied using a specific, optimized parameter set adapted to the target system. Agnostic battery state detection does not require model adaptations that require extensive verification. Ideally, a technology-independent, general description of the (quasi) open-circuit voltage of a cell / battery is required for the SOx functions.

[0084] The description should consider all known static and dynamic dependencies with the greatest possible accuracy. These include, for example, the SOC, the charge / discharge direction (hysteresis), temperature, aging, and tolerances. Ideally, the description should be possible across the entire SOC (0 ... 100%) and temperature range (-30°C ... 60°C) using only a fixed number of different parameters (no model changes).

[0085] To derive the new method, the Nernst model (equation (3)) is used and the model is written in the form of an integral equation: UOCV(θ)=M0+∫M1(1−θ)θ−∫(M1+M2)θ(1−θ)θ

[0086] The denominator of the respective integrals is now generalized with two additional exponents: UOCV(θ)=M0+∫M1(1−θ)α1θσ1−∫(M1+M2)θ(1−θ)α2θσ2

[0087] The integrals can be calculated analytically and yield two so-called incomplete Eulerian beta functions B θ (α, σ), which are available by default in the “MathWorks Function Set” (The MathWorks Inc. (2022)) as “Regularized Beta Function (betainc)”: Iθ(α,σ)=1B(α,σ)Bθ(α,σ) where B(α, σ) = B1(α, σ) is the complete beta function. Furthermore, the inverse function "inverse of the incomplete beta function I θ -1 (α, σ)" is required. This function is also included as standard and can be implemented, for example, using Simulink® Coder™. Of course, both functions can also be implemented individually in C++ if required. Furthermore, there is an important connection with Euler's beta function: Beta(θ,2−κ2,1−α2)=Beta(2−κ2,1−α2)−Beta(1−θ,1−α2,2−κ2)

[0088] With L1=M1, L2=(M1+M2), σ1=1−α1, σ2=1−α2, L0=M0−L2Beta(2−κ2,σ2) in combination with the Shepherd and Unnewehr model analogous to equation (4) the new SOC OCV model results in: UOCV,DRX(θ)=L0+L1θ+L2θ+L3 Beta(θ,1−κ1,σ1)+L4 Beta(1−θ,σ2,2−κ2) U OCV,DRX (θ) can of course also be approximated by the interpolation function according to equation (9), but more parameters are then required to achieve the same accuracy. Due to the properties of the beta function Beta(θ,1,0)=−ln(1−θ) or Beta(1−θ,1,0)=−ln(θ) applies to a specific parameter set L3=−K3,L4=−K4 κ1=0, σ1=0, κ2=2, σ2=1 UOCV,DRX(θ)=UOCV,Plett(θ) resulting in the SOC-OCV model of equation (4). In other words, the new SOC-OCV model according to equation (15) is strictly mathematically a generalization of equation (4).

[0089] The applicant has no access to the cells and measurement data used in [2], but with equation (18) the results from [2], e.g. for LFP, would also be applicable for U OCV,DRX (θ) is reproducible. For example, an RMS error of 6.2 mV was reported for LFP (see Table 1). This is the lowest RMS error for the specific parameter set according to equation (17) and the four freely selectable parameters. However, this does not necessarily mean that a smaller RMS error can be found for equation (15) for the nine parameters.

[0090] To compare the accuracies of the different SOC OCV models, the number of parameters must be taken into account. The goal is to make a statement about whether the model U OCV,DRX (θ) represents a technology-independent, general description of the OCV; for this purpose, the measurement data of the NCM pouch cell were examined in more detail.

[0091] To ensure the accuracy of U OCV,DRXTo compare (θ) with equation (7), the Shepherd model according to equation (1) is omitted in equation (15) and (κ = κ1 = κ2) and (σ = σ1 = σ2) are set, resulting in the same number of parameters as in equation (7): UOCV,VGL(θ)=L0+L1θ+L2 Beta(θ,1−κ,σ)+L3 Beta(1−θ,σ,2−κ)

[0092] Fig. 4 shows a comparative representation 400 of the measurement data 401, 402 for the rest voltage of the NCM pouch cell in the charging (402) and discharging direction (401) with the results of the model according to the invention according to equation (19).

[0093] The fit result in charge and discharge direction is in the Fig. 4. The comparison of RMS and Max errors of the new reduced SOC OCV model according to equation (19) and the state-of-the-art according to equation (7) is shown for the discharge direction in Table 2.

[0094] Accordingly, the battery cell model according to equation (19) also achieves error values for NCM in the same order of magnitude (RMS: 7.1 mV, Max Error: 16.15 mV) as for LFP [2] in Table 1.

[0095] However, it must be expressly pointed out that, in general, not all cells, e.g., those with LFP material, will have exactly the same error values. Therefore, we also used our own measurement data for an 18650 LFP cell, and determined values that are twice as accurate (columns 5 and 6). In summary, we can say: 1. The following applies to our own measurement data: Compared to the state of the art with the same number of parameters, the new reduced model according to the invention (U OCV,VGL (θ)) achieves higher accuracy. For the same number of parameters, the RMS error improves by 56% for the NCM cell and by 44% for the LFP cell compared to the state-of-the-art SOC OCV model according to equation (7). 2. With the reduced model (U OCV,VGL (θ)), an error of the same order of magnitude can be achieved for the LFP cell from [2] (see Table 1, row 2, last two columns) and the NCM cell (see Table 2, row 3, columns 3 and 4). SOC-OCV model (discharge) RMS error for NCM cells in mV Max. error for NCM cells in mV RMS error for LFP cells in mV Max. error for LFP cells in mV 1 (according to [2]) 16.17 28.85 7.03 12.42 2 (according to Eq. (19)) 7.13 16.15 3.91 8.5

[0096] SOC-OCV Model 1: VOC=a+b⋅(−lns)m+c⋅s+d⋅en(s−1)

[0097] SOC-OCV Model 2: UOCV,VGL(θ)=L0+L1θ+L2 Beta(θ,1−κ,σ) +L3 Beta(1−θ,σ,2−κ)

[0098] Table 2: Comparison of accuracies of OCV models (6 parameters) for technology (NCM, LFP) based on different measurement data (DRX, [2]).

[0099] The Model U OCV,VGL(θ) according to equation (19) yields a significantly smaller error than the model according to equation (7). The values for the two technologies, NCM and LFP, are almost identical, thus demonstrably generalizing the model with respect to different cell chemistries.

[0100] The RMS error for the OCV in the charging direction is better (approx. 0.0054 mV) and slightly worse in the discharging direction (0.0069 mV).

[0101] Fig. Figure 5 shows a comparative representation 500 of the measurement data 501, 502, 503 for the hysteresis of the NCM pouch cell with the results of the inventive model according to equation (19). (A) is an excerpt. Due to the RMS error, the hysteresis would no longer be detectable until 90% SOC.

[0102] In Fig.Figure 5 illustrates the hysteresis using the OCV model according to equation (19). The accuracy naturally also varies with temperature; this is only shown as an example for 25 °C. At a maximum model error (dashed black line 402, 7.13 mV), the hysteresis would no longer be detectable until 90% SOC.

[0103] The number of parameters plays a major role in the computational effort, for example if the parameters of each individual cell connected in parallel have to be regularly re-determined. In addition, a plausibility check must also be carried out. Measurements for different temperatures are required for initialization. After that, it must be noted that the parameters for the OCV model must be regularly corrected online using various methods due to aging, tolerances, etc. In some cases, either ECM or OCV parameters are identified online using a DEKF or FFRLS method. In addition, different architectures are used for battery state detection. The most complex case arises when the SOx functions of the individual cells are to be calculated for an HV battery with serial and parallel connections of cells.This has a direct and significant impact on the memory capacity and computing speed and thus on the choice of microcontroller. Table 3: Comparison of accuracies of the general (equation (15)) and reduced OCV model (equation (19)) for LFP cell (own measurement data). Model comment Number of parameters RMS error for LFP cells in mV, discharge Max. error LFP cells in mV, discharge RMS error for LFP cells in mV, charge Max. error LFP cells in mV, charge U OCV,DRX (i) Unnewehr + Shepherd + κ1 ≠ κ2, σ1 ≠ σ2 9 1.97 4.71 1.72 4.1 U OCV,VGL (i) Unnewehr κ1 = κ2, σ1 = σ2 6 3.91 8.5 3.79 8.52

[0104] The comparison of the accuracies of U OCV,DRX (θ) according to equation (15) and U OCV,VGL (θ) according to equation (19) is shown in Table 3. The RMS and maximum error for the LFP cell (own measurement data) is for the complete model U OCV,DRX (θ) twice as accurate as the reduced model U OCV,VGL (θ). For both SOC-OCV models U OCV,VGL (θ) and U OCV,DRX(θ), the parameters can be updated online using a recursive non-linear least-square method to account for hysteresis and cell aging. According to publication DE 10 2021 110 384 A1, an analytical determination of the parameters is only possible for equation (4) for (K1 = K4 = 0).

[0105] Fig. 6 shows a comparative representation 600 of the measurement data 601, 602 for the open-circuit voltage (OCV) of the LFP cell in the discharge direction with the results of the model according to the invention according to equation (15).

[0106] Fig. 6 shows the comparison between the measured data of an LFP cell and the complete model U OCV,DRX(θ) according to equation (15) in the charging direction. In this case, the RMS error is only 1.72 mV. What is special here is that the specific plateau (the flat region of the OCV) is described very precisely in terms of location (SOC value) and height (maximum gradient of the OCV). This enables a precise analysis of the OCV characteristics (curve analysis) for cell / battery characterization (hysteresis, aging, etc.).

[0107] Fig.7 shows a comparative representation 700 of the normalized deviation of the SOC error of an NCM pouch cell for the HPPC ("Hybrid Pulse Power Characterization") cycle with an extended Kalman filter (dotted line) 703 and without an EKF (dashed line) 701. For comparison, the normalized hysteresis of the cell (solid line) 702 is also shown. The correlation between the SOC error (dashed line) 701 and the hysteresis (solid line) 702 and the error correction of the EKF (dotted line) 703 is clearly visible, also enlarged in detail.

[0108] For the NCM pouch cell, measurement data (load profile) from a so-called HPPC cycle was used to demonstrate the effect of hysteresis 702 on SOC accuracy when only the open-circuit voltage (OCV) is used to track the state of charge. At each point, two SOC values are calculated and compared: the SOC from the Ah throughput and the SOC taken from the open-circuit voltage table at the resting points of the cycle. A correlation between hysteresis 702 and the SOC deviation of the OCV-SOC 701 and the SOC determined by counting the Ah throughput ("real" SOC) 703 is visible. This suggests that the deviations (dashed line 701) are most likely indeed caused by hysteresis. However, conclusive proof would require a precise error analysis, which would require further measurements.

[0109] An extended Kalman filter (EKF) is parameterized to track the SOC on the same load profile. The dotted line 703 in Fig. Figure 7 shows the result of the extended Kalman filter (EKF). Due to the EKF correction, the error was significantly reduced, to only 40%, and the characteristic curve with the three extrema was corrected without an explicit hysteresis model. The precise description of the open-circuit voltage characteristic by the new SOC OCV model according to Equation (15) plays a crucial role here.

[0110] This can be clearly demonstrated by the convergence time of the extended Kalman filter. The convergence time and the accuracy of the extended Kalman filter depend significantly on the Jacobian matrices (∂U OCV / ∂ SOC ) depends on the accuracy of the SOC OCV model (or its derivative (see Fig. 6, section (A)).

[0111] Fig.8 shows a comparative illustration 800a, 800b of the convergence time of the extended Kalman filter (EKF) at a low and high initial charge level for a fault injection test.

[0112] In the Fig. Figure 8 shows the convergence time of the extended Kalman filter for a fault injection test and a low and high initial SOC value.

[0113] With a low initial SOC value, the initial error decreases very quickly (see lower figure 800b). This is different with a high initial SOC value. Here, the convergence time is much longer. This depends on the slope of the OCV (∂U OCV / ∂ SOC ) (stronger curvature of the OCV characteristic). If the mathematical description by the SOC OCV model takes this behavior into account, ie, if it reproduces the SOC-dependent behavior with high accuracy, instead of just averaging the values over the SOC range as in equation (7), then the convergence time will also be shorter.

[0114] This would also be the case with a numerical calculation of the Jacobian matrix. However, it should be noted that, depending on the requirements / architecture, each individual cell of the HV battery may need to be monitored. The parameters of the SOC OCV model must be readjusted from time to time due to temperature, aging, and tolerances. Since the calculations for the Kalman filter are very memory- and computationally intensive, the goal here is to find an SOC OCV model with an optimal number of parameters that is also technology-independent, meaning that it can be adapted to new cells by changing the parameters. The same relationship naturally applies to the accuracy of the Kalman filter.

[0115] The result (dotted line 703 in Fig. 7) suggests that if the SOC OCV model is so accurate that it correctly describes the hysteresis, the filter is more likely to filter out the deviations. The hysteresis (solid line 702 in Fig.7) is SOC dependent, the deviation, i.e. the SOC value calculated by the Kalman filter, is almost constant over the entire SOC (dotted line 703 in Fig. 7), the absolute value no longer depends on the hysteresis. LIST OF REFERENCE SYMBOLS 100 Method for determining a battery state variable 101 first step: Obtain parameter set 102 second step: Maintain resting voltage 103 Third step: Determine battery state variable 110 cell-specific parameter set 111 Battery state variable 120 battery cells 130 battery cell model U OCV Open circuit voltage or rest voltage 200 Battery management system, BMS 210 processor 220 battery-powered vehicle, electric vehicle 10 OCV-SOC representation 11, 12, 13 Open-circuit voltage characteristics 400 comparative representation of measurement data for the resting voltage 401, 402 Measurement data for the resting voltage 500 comparative representation of measurement data for the hysteresis of the NCM Pouch cell 501, 502, 503 Measurement data for the hysteresis of the NCM Pouch cell 600 comparative representation of the measurement data for the resting voltage of the LFP cell 601, 602 Measurement data for the resting voltage of the LFP cell 700 comparative representation of the normalized deviation of the SOC error of an NCM pouch cell 701, 702, 703 Measurement data 800a, 800b Representation of the convergence time of the extended Kalman filter 801, 802, 803 measurement data 804, 805 measurement data QUOTES CONTAINED IN THE DESCRIPTION

[0000] This list of documents submitted by the applicant was generated automatically and is included solely for the convenience of the reader. This list is not part of the German patent or utility model application. The DPMA assumes no liability for any errors or omissions. Cited patent literature

[0000] DE 10 2021 110 384 A1

[0104] Cited non-patent literature

[0000] Plett, GL Extended Kalman filtering for battery management systems of LiPB-based HEV battery packs: Part 2. Modeling and identification. J. Power Sources 2004, 134, 262-276 [0048, 0061] Zhang, C.; Jiang, J.; Zhang, L.; Liu, S.; Wang, L.; Loh, PC A Generalized SOC-OCV Model for Lithium-Ion Batteries and the SOC Estimation for LNMCO Battery. Energies 2016, 9, 900

[0049]

Claims

[1] Method (100) for determining a battery state variable (111) of a battery cell (120) with the following steps: Obtaining (101) a cell-specific parameter set (110) which is characteristic of the battery cell (120); Obtaining (102) a rest voltage (U OCV ) of the battery cell (120); and Determining (103) the battery state variable (111) of the battery cell (120) based on a battery cell model (130) which shows a relationship between a state of charge (SoC) and the open-circuit voltage (U OCV ) of the battery cell (120) based on an analytical function using the cell-specific parameter set (110) of the battery cell (120); where the analytical function is based on two incomplete Eulerian beta functions depending on the state of charge (SoC) of the battery cell (120) and the cell-specific parameter set. [2] The method (100) of claim 1, wherein the analytical function is based on a sum of the two incomplete Eulerian beta functions weighted using the cell-specific parameter set (110) and a linear function of the state of charge (SoC) of the battery cell (120). [3] Method (100) according to one of the preceding claims, wherein the analytical function determines the relationship between the state of charge (SoC) and the rest voltage (U OCV ) of the battery cell (120) regardless of a technology or cell chemistry of the battery cell (120). [4] Method (100) according to one of the preceding claims, wherein the analytical function is based on a Nernst equation generalized in the form of an integral equation. [5] Method (100) according to claim 4, wherein the Nernst equation describes the relationship between the state of charge (SoC) and the rest voltage (U OCV) of the battery cell (120) based on a logarithmic function of the state of charge (SoC) and a logarithmic function of the inverse state of charge (SoC). [6] Method (100) according to claim 4 or 5, wherein the Nernst equation describes the relationship between the state of charge (SoC) and the rest voltage (U OCV ) of the battery cell (120) is defined as follows: UOCV(θ)=E0+μ1ln(θ)+μ2ln(1−θ), where θ is the state of charge of the battery cell (120) and E0,µ1,µ2, are cell-specific parameters of the battery cell (120) for the Nernst equation. [7] Method (100) according to one of the preceding claims, wherein the analytical function is based on the following integral equation: UOCV(θ)=M0+∫M1(1−θ)θ−∫(M1+M2)θ(1−θ)θ, where θ is the state of charge of the battery cell (120) and M0, M1, M2 are cell-specific parameters of the battery cell (120) for the integral equation. [8] The method (100) of claim 7, wherein the integral equation is further generalized by two additional exponents in the two respective integrals of the integral equation as follows: UOCV(θ)=M0+∫M1(1−θ)α1θσ1−∫(M1+M2)θ(1−θ)α2θσ2, where θ is the state of charge of the battery cell (120) and M0, M1, M2, α1, α2, σ1, σ2 are cell-specific parameters (110) of the battery cell (120) for the generalized integral equation. [9] Method (100) according to one of the preceding claims, wherein the analytical function is as follows: UOCV,VGL(θ)=L0+L1θ+L2Beta(θ,1−κ,σ)+L3Beta(1−θ,σ,2−κ), where θ is the state of charge of the battery cell (120) and L0,L1,L2,L3,σ,κ are cell-specific parameters (110) of the battery cell (120) for the analytical function. [10] Battery management system (200) with one or more battery cells, the battery management system (200) comprising: a processor (210) configured to execute the method (100) according to any one of the preceding claims in order to determine a battery state variable (111), in particular a state of charge (SoC) or a power state (SoP), of the one or more battery cells (120). [11] Battery management system (200) according to claim 10, wherein the processor (210) is configured to determine the state of charge (SoC) of the one or more battery cells (120) based on a Kalman filter or an extended Kalman filter configured to iteratively evaluate the battery cell model (130). [12] Battery management system (200) according to claim 10 or 11, wherein the processor (210) is configured to initiate a charge equalization of the battery cells (120) based on the determined state of charge (SoC) of the respective battery cells (120).

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