DE102021110384A1 - Electronic circuit for determining the state of charge of a battery cell - Google Patents

Abstract

The invention relates to an electronic circuit (100) for determining a state of charge (114) of a battery cell of a battery system, in particular for an electric vehicle, the electronic circuit (100) being designed to have a plurality of measured values (101) of an open-circuit voltage of the battery cell with associated values time values (102) at which the measured values (101) of the open-circuit voltage were recorded, an equilibrium voltage (112) of the battery cell based on a linear regression (110) of the plurality of measured values (101) of the open-circuit voltage with respect to a predetermined mathematical function ( 111) of the time values (102) associated with the measured values (101) of the no-load voltage, and to determine the state of charge (114) of the battery cell based on a predetermined characteristic curve (113) which shows a relationship between the equilibrium voltage (112) and a state of charge ( 114) of the battery cell indicates.

Description

The invention relates to an electronic circuit for determining a state of charge of a battery cell of a battery system, for example for an electric vehicle, and a corresponding method. In particular, the invention relates to the aging-dependent online prediction of the relaxation of battery cells such as lithium-ion cells using a computationally and memory-saving linear regression method.

Battery management systems (BMS) can be found in a wide variety of interconnections of accumulator cells, for example traction batteries in electric cars, uninterruptible power supplies, mobile phones or notebooks. One of the major challenges is determining the state of charge, which for many types of accumulator can only be determined imprecisely based on the cell voltage and the internal resistance. The determination and display of the battery condition or the remaining operating time is usually not reliable.

The BMS or BMU (Battery Management Unit) is woken up at regular intervals and uses the CMCs (Charge Module Controller) to determine the off-load voltage of the battery. Depending on the state of charge and the temperature, the open-circuit voltage changes to a quasi-equilibrium voltage after a certain time (the relaxation time). With the help of the equilibrium voltage, the state of charge of the battery can be reliably determined (OCV recalibration). However, the relaxation time increases significantly with decreasing state of charge and temperature. After a short time, e.g. half an hour, e.g. when parking for a short time, the open-circuit voltage still deviates considerably from the equilibrium voltage, since the cells of the battery are not yet in a state of equilibrium.

An SOC determination using the open-circuit voltage is only possible with a certain inaccuracy. In addition, the relaxation and thus the accuracy of the SOC (state of charge) determination changes with the aging of the battery.

It is an object of the invention to create a concept for improved determination of the state of charge of the battery cells of a battery system, for example for an electric vehicle.

In particular, it is an object of the invention to create a state of charge determination in which an SOC determination with greater accuracy is possible after a short time, taking aging into account, using simple, computationally efficient methods.

This object is solved by the objects with the features according to the independent claims. Advantageous embodiments are the subject matter of the dependent claims, the description and the drawings.

The invention is based on the idea of adapting (fitting) the time profile of the open-circuit voltage to the equilibrium voltage of the cells in the battery using a new mathematical function, the parameters of which can already be precisely determined online after just a few data points. This takes into account cell tolerances, measurement errors and aging effects that affect the values of the parameters. Because of the type of function, the parameters can be determined using a computationally economical regression method.

With the help of the invention, an SOC determination can thus be carried out with greater accuracy after a short time, taking aging into account, using simple, computationally efficient methods.

For the prediction of the equilibrium voltage after a short time, especially at low temperatures and states of charge, only 3 parameters are required for the first time instead of 4, which minimizes the storage effort. Furthermore, only linear instead of non-linear regression methods are required for the online determination of the parameters, so that the computing time can be further optimized here. The comparison of the results from the previous and new method also shows a much more accurate result than before in certain SOC and temperature ranges.

The solution presented here is therefore superior to the previously used method, in which a mathematical function consisting of two nested exponential functions and a constant part as a fit function are used for the technical solution to the problem. To determine the online parameters are on Due to the function type, however, non-linear methods that require a great deal of computation and memory are required for the method used up to now, which can be omitted with the new method presented here. While with the previously used method an exact prediction of the equilibrium value is only possible after times in the order of magnitude of the relaxation time, the new method presented here already results in a good prediction of the equilibrium value before the relaxation time is reached. In the previous method, aging effects are usually taken into account by measurements on cells that have aged in a defined manner, while in the new method presented here, aging in the solution is also taken into account.

The solution presented here is suitable for customer-specific and oriented solutions for battery status detection in electric vehicles, but also for other battery-powered systems, such as notebooks, mobile communication devices, uninterruptible power supplies, etc. With the help of the solution according to the invention, the battery can be operated more gently and safely guaranteed over the entire lifetime of the battery.

According to a first aspect of the invention, the object is achieved by an electronic circuit for determining a state of charge of a battery cell of a battery system, in particular for an electric vehicle, wherein the electronic circuit is designed to use a plurality of measured values of an open-circuit voltage of the battery cell with associated time values, for which to obtain the measured values of the open circuit voltage have been taken; determine an equilibrium voltage of the battery cell based on a linear regression of the plurality of open circuit voltage measurements with respect to a predetermined mathematical function of the time values associated with the open circuit voltage measurements; and to determine the state of charge of the battery cell based on a predetermined characteristic curve, which indicates a relationship between the equilibrium voltage and a state of charge of the battery cell.

Such an electronic circuit ensures an improved determination of the state of charge of the battery cells of a battery system, for example an electric vehicle. With the electronic circuit, an SOC determination with higher accuracy is possible after a short time, taking aging into account, using simple, low-calculation methods.

According to an exemplary embodiment of the electronic circuit, the predetermined mathematical function specifies a time profile of the open-circuit voltage of the battery cell.

By specifying the time profile of the no-load voltage via the mathematical function, the electronic circuit can efficiently determine an estimated value of the no-load voltage based on a linear regression of the measured values, with the estimated value predicting the time profile of the no-load voltage very precisely. The electronic circuit can then very precisely determine the state of charge of the battery cell.

This achieves the technical advantage that the mathematical function accurately represents the course of the open-circuit voltage of the battery cell over time in order to enable rapid regression and approximation.

According to an exemplary embodiment of the electronic circuit, the predetermined mathematical function is a linear function.

If the mathematical function is a linear function, the electronic circuit can determine the estimated value of the open-circuit voltage quickly and in a resource-saving manner, i.e. the calculation time is short and no complex calculation steps are required.

According to an exemplary embodiment of the electronic circuit, the predetermined mathematical function is a linear function that depends on: a first parameter, or a first parameter and a second parameter, or a first parameter, a second parameter and a third parameter.

This achieves the technical advantage that fewer parameters are required than in the previous method, which requires four parameters. The computing complexity with three or fewer parameters is therefore lower than in the previous method.

According to an exemplary embodiment of the electronic circuit, the predefined mathematical function specifies a time profile of the open-circuit voltage of the battery cell as a function of a relaxation time of the battery cell.

This achieves the technical advantage that the estimated value of the no-load voltage can be specified as a function of the relaxation time of the battery cell, which is a characteristic time constant of the battery cell.

wobei U_OCV(θ) einen zeitlichen Verlauf der Ruhespannung der Batteriezelle darstellt, welche abhängig ist von der Variablen θ, welche die Zeit bezogen auf die Relaxationszeit der Batteriezelle darstellt, und wobei die vorgegebene mathematische Funktion von dem ersten Parameter a, dem zweiten Parameter b und dem dritten Parameter c abhängt.According to an exemplary embodiment of the electronic circuit, the predefined mathematical function is determined as follows: $$u_{OCV}\left(\theta \right)=ƒ\left(a,b,c\right)=a+b\times \theta +c\times \text{log}\left(\theta \right),$$

where U _OCV (θ) represents a time course of the open-circuit voltage of the battery cell, which depends on the variable θ, which represents the time related to the relaxation time of the battery cell, and wherein the predetermined mathematical function depends on the first parameter a, the second parameter b and the third parameter c.

This achieves the technical advantage that such a mathematical function can very precisely approximate the exponential course of the open-circuit voltage of the battery cell. The tests to those presented below 3 until 5 have shown, for example, that there is a very small deviation from the actual course of the no-load voltage.

According to an exemplary embodiment of the electronic circuit, the electronic circuit is designed to determine at least one of the three parameters based on a statistical evaluation of the measured values of the no-load voltage and the associated time values at which the measured values of the no-load voltage were recorded.

This achieves the technical advantage that a statistical evaluation can be determined using standard statistical functions such as mean value, variance, correlation, which are standard functions that can be determined efficiently and quickly.

$$s_{lnx}={\displaystyle \sum _{i=1}^n\text{log}\left(x_i\right)};s_{ln{x}^2}={\displaystyle \sum _{i=1}^n{\left(\text{log}\left(x_i\right)\right)}^2};s_{xlnx}={\displaystyle \sum _{i=1}^n{x}_i\text{log}\left(x_i\right)};s_{ylnx}={\displaystyle \sum _{i=1}^n{y}_i\text{log}\left(x_i\right)}$$

wobei y_i die Messwerte der Ruhespannung bezeichnen und x_i die zugehörigen Zeitwerte, zu denen die Messwerte der Ruhespannung erfasst wurden. Die Variable n bezeichnet die Anzahl der Messwerte.According to an exemplary embodiment of the electronic circuit, the electronic circuit is designed to base the statistical evaluation on the determination of the following functions: $$s_x={\displaystyle \sum _{i=1}^n{x}_i};s_y={\displaystyle \sum _{i=1}^n{y}_i};s_{x^2}={\displaystyle \sum _{i=1}^n{x}_{\mathrm{<figure-callout\; id="2"\; label="i"\; filenames="DE102021110384A1\_0029.png"\; state="\{\{state\}\}">i</figure-callout>}}^2};s_{xy}={\displaystyle \sum _{i=1}^n{x}_i{y}_i}$$

$$s_{lnx}={\displaystyle \sum _{i=1}^n\text{log}\left(x_i\right)};s_{ln{x}^2}={\displaystyle \sum _{i=1}^n{\left(\text{log}\left(x_i\right)\right)}^2};s_{xlnx}={\displaystyle \sum _{i=1}^n{x}_i\text{log}\left(x_i\right)};s_{ylnx}={\displaystyle \sum _{i=1}^n{y}_i\text{log}\left(x_i\right)}$$

where y _i denote the measured values of the no-load voltage and x _i the associated time values at which the measured values of the no-load voltage were recorded. The variable n designates the number of measured values.

This achieves the technical advantage that these functions can easily be determined from the available measured values and time values. The accuracy of the prediction increases with the number n of measured values. As the number n of measured values increases, the predicted value becomes more and more accurate.

According to an exemplary embodiment of the electronic circuit, the electronic circuit is designed to provide a first estimated value of the equilibrium voltage of the battery cell by determining the first parameter even before the relaxation time of the battery cell is reached.

This achieves the technical advantage that when the first estimated value is determined by only evaluating the first parameter a, there is already a very good approximation of the no-load voltage, as can be seen from tests, see for example 6 and related description below.

According to an exemplary embodiment of the electronic circuit, the electronic circuit is designed to provide a second estimated value of the equilibrium voltage of the battery cell before the relaxation time of the battery cell is reached by determining the second parameter, wherein the second estimated value is determined as a sum of the first parameter and the second parameter.

This achieves the technical advantage that when the second estimated value is determined by evaluating the first parameter a and the second parameter b, a very precise approximation of the no-load voltage results. Such an approximation also saves resources and can be determined quickly with the electronic circuit.

According to an exemplary embodiment of the electronic circuit, the electronic circuit is designed to store a time profile of at least the first parameter in a memory, and is also designed to determine the equilibrium voltage of the battery cell based on the stored time profile of at least the first parameter in a further determination of the state of charge of the battery cell to determine parameters.

This achieves the technical advantage that the electronic circuit can make an online prediction, which can be based on the results of earlier estimates of the course over time, which enables an even more accurate and faster estimate. With increasing consideration of past determinations of the state of charge, the time from which an exact prediction of the state of charge is possible therefore decreases. With an increasing number of measured values n»0, a more and more accurate result is achieved. The term "fast" in this context means with the same accuracy, but at an earlier point in time.

According to an exemplary embodiment of the electronic circuit, the time course extends from the start of the determination of the state of charge until the relaxation time of the battery cell is reached.

This achieves the technical advantage that one cycle of the parameter a is stored from the start of the measurement until the relaxation time of the battery cell is reached and can be taken into account in future determinations of the state of charge.

According to an exemplary embodiment of the electronic circuit, the electronic circuit is designed to store a plurality of time curves of at least the first parameter based on successive determinations of the state of charge of the battery cell in the memory, and is also designed to detect an aging of the battery cell based on the plurality of chronological To determine the state of charge of the battery cell.

This achieves the technical advantage that aging is also taken into account in the online prediction of the relaxation of the battery cell and that even aging can be explicitly determined from the recorded time curves.

According to a second aspect of the invention, the object is achieved by a battery management system with: at least one charging module controller for detecting a plurality of measured values of an open-circuit voltage in at least one battery cell of a battery system, in particular for an electric vehicle, and of associated time values at which the measured values the open-circuit voltage can be detected; and an electronic circuit according to the first aspect for determining the state of charge of the at least one battery cell of the battery system.

As with the electronic circuit according to the first aspect, the advantage of an improved determination of the state of charge of the battery cells of a battery system, for example an electric vehicle, also results here. With such a battery management system, an SOC determination with greater accuracy is possible after just a short time, taking aging into account, using simple, computationally efficient methods.

According to a third aspect of the invention, the object is achieved by a method for determining a state of charge of a battery cell of a battery system, in particular for an electric vehicle, the method having the following steps: Obtaining a plurality of measured values of an open-circuit voltage of the battery cell with associated time values which the measured values of the open-circuit voltage were taken; determining an equilibrium voltage of the battery cell based on a linear regression of the plurality of open circuit voltage measurements with respect to a predetermined mathematical function of the time values associated with the open circuit voltage measurements; and determining the load State of the battery cell based on a predetermined characteristic curve, which indicates a relationship between the equilibrium voltage and a state of charge of the battery cell.

As with the electronic circuit according to the first aspect, the method according to the third aspect also has the advantage of improved determination of the state of charge of the battery cells of a battery system, for example an electric vehicle. With such a method, an SOC determination with higher accuracy is possible after a short time, taking aging into account, using simple, computationally efficient methods.

According to a fourth aspect of the invention, the object is achieved by a computer program with a program code for executing the method according to the third aspect on an electronic circuit according to the first aspect or a battery management system according to the second aspect.

This achieves the technical advantage that the computer program can be easily executed on a vehicle control unit, which can implement the real-time requirement with no unit costs, without the need for additional external hardware components.

• 1 eine schematische Darstellung einer elektronischen Schaltung 100 zur Bestimmung eines Ladezustands einer Batteriezelle gemäß der Offenbarung;
• 2 eine Darstellung eines beispielhaften zeitlichen Verlaufs 200 der Ruhespannung einer Batteriezelle;
• 3 eine Darstellung der Abweichung 300 zwischen Messwert und Schätzwert der Ruhespannung einer Batteriezelle bei einer Temperatur von -10 Grad Celsius gemäß einem Beispiel;
• 4 eine Darstellung der Abweichung 400 zwischen Messwert und Schätzwert der Ruhespannung einer Batteriezelle bei einer Temperatur von +10 Grad Celsius gemäß einem Beispiel;
• 5 eine Darstellung der Abweichung 500 zwischen Messwert und Schätzwert der Ruhespannung einer Batteriezelle bei einer Temperatur von +25 Grad Celsius gemäß einem Beispiel;
• 6 eine Darstellung einer Kennlinie 600, welche den Zusammenhang zwischen Gleichgewichtsspannung und Ladezustand einer Batteriezelle beschreibt, sowie des Parameters a bei einer Temperatur von +25 Grad Celsius gemäß einem Beispiel; und
• 7 eine schematische Darstellung eines Verfahrens 700 zur Bestimmung eines Ladezustands einer Batteriezelle gemäß der Offenbarung.

The invention is described in more detail below using exemplary embodiments and the figures. In the figures show:

• 1 a schematic representation of an electronic circuit 100 for determining a state of charge of a battery cell according to the disclosure;
• 2 an illustration of an exemplary time profile 200 of the open-circuit voltage of a battery cell;
• 3 an illustration of the deviation 300 between the measured value and the estimated value of the open-circuit voltage of a battery cell at a temperature of -10 degrees Celsius according to an example;
• 4 an illustration of the deviation 400 between the measured value and the estimated value of the open-circuit voltage of a battery cell at a temperature of +10 degrees Celsius according to an example;
• 5 an illustration of the deviation 500 between the measured value and the estimated value of the open-circuit voltage of a battery cell at a temperature of +25 degrees Celsius according to an example;
• 6 a representation of a characteristic curve 600, which describes the relationship between the equilibrium voltage and the state of charge of a battery cell, as well as the parameter a at a temperature of +25 degrees Celsius according to an example; and
• 7 a schematic representation of a method 700 for determining a state of charge of a battery cell according to the disclosure.

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 concept of the present invention. The following detailed description is, therefore, not to be taken in a limiting sense. Furthermore, it is understood that the features of the various exemplary embodiments described herein can be combined with one another, unless specifically stated otherwise.

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

In the following description, reference is made to battery cells and the battery management system (BMS), as well as charge module controllers (CMCs). So that battery cells, such as lithium For ion cells to operate safely, cell voltages and temperatures must be continuously monitored. This is done via monitoring electronics, the Cell Supervision Electronics or CSE for short. The individual cells in a battery are combined into cell modules. Each cell module has monitoring electronics, the CSE, to which the battery cells are connected. The CSE measures the voltage and temperature of the cells.

The BMS is the central control unit of a battery. The information from the individual cell monitors, the CSEs, comes together at the BMS. The BMS uses the cell voltages to determine the current state of charge (SOC = State of Charge), issues the command for balancing and takes over communication with the vehicle. It also ensures that the battery cells are not overcharged or over-discharged.

The behavior of battery cells as a function of the relaxation time is described below. The relaxation time describes the characteristic time in which a system, i.e. here the battery cell, approaches (usually exponentially) the stationary state. The stationary state here corresponds to the equilibrium voltage on the battery cell. Clearly, after a relaxation time constant has elapsed, the system has noticeably moved towards its state of equilibrium; after a period of three to six relaxation time constants, one can usually assume that the relaxation is largely complete.

In the following description reference is made to electronic circuits. An electronic circuit is a combination of electrical and, in particular, electronic components (e.g. diodes and transistors) to form a functioning arrangement. Electronic circuits can perform very simple functions such as B. the flashing of a lamp or the control of an automatic door. But also many complex technical devices, such. B. computers or electric vehicles are based on electronic circuits, often in the form of integrated circuits (ICs). An IC typically contains a combination of numerous electronic semiconductor components which are electrically connected to one another, such as transistors, diodes and/or other active and passive components.

1 shows a schematic representation of an electronic circuit 100 for determining a state of charge of a battery cell of a battery system according to the disclosure. The battery system can be a battery system of an electric vehicle, for example.

The electronic circuit 100 is designed to receive a plurality of measured values 101 of an open-circuit voltage of the battery cell with associated time values 102 at which the measured values 101 of the open-circuit voltage were recorded. The electronic circuit 100 is also designed to determine an equilibrium voltage 112 of the battery cell based on a linear regression 110 of the plurality of measured values 101 of the no-load voltage with respect to a predefined mathematical function 111 of the time values 102 associated with the measured values 101 of the no-load voltage. The electronic circuit 100 is also designed to determine the state of charge 114 of the battery cell based on a predetermined characteristic curve 113, which indicates a relationship between the equilibrium voltage 112 and a state of charge 114 of the battery cell.

With the help of the electronic circuit 100, an SOC determination can already be carried out with greater accuracy after a short time, taking into account aging using simple, computationally efficient methods.

The time course of the off-load voltage towards the equilibrium voltage of the cells in the battery is fitted with a new mathematical function 111, the parameters of which can be precisely determined online after just a few data points. This takes into account cell tolerances, measurement errors and aging effects that affect the values of the parameters. Because of the function type, the parameters can be determined with the aid of a computationally economical regression method 110 .

The electronic circuit 100 can be constructed in hardware and/or software.

In the electronic circuit 100, the predefined mathematical function 111 can specify a time profile 200 of the open-circuit voltage of the battery cell, as for example in 2 shown.

The predetermined mathematical function 111 can be a linear function.

For example, the predetermined mathematical function 111 can be a linear function that depends on: a first parameter (hereinafter referred to as parameter a), or a first parameter and a second parameter (hereinafter referred to as parameter b), or a first parameter , a second parameter and a third parameter (hereinafter referred to as parameter c).

The specified mathematical function 111 can be a time profile 200 of the open-circuit voltage of the battery cell, such as in 2 shown, specify as a function of a relaxation time of the battery cell.

Hierbei stellt U_OCV(θ) einen zeitlichen Verlauf der Ruhespannung der Batteriezelle dar, welche abhängig ist von der Variablen θ, welche die Zeit bezogen auf die Relaxationszeit der Batteriezelle darstellt, und wobei die vorgegebene mathematische Funktion von dem ersten Parameter a, dem zweiten Parameter b und dem dritten Parameter c abhängt, wie unten näher beschrieben.For example, the predetermined mathematical function 111 can be determined as follows: $$u_{OCV}\left(\theta \right)=ƒ\left(a,b,c\right)=a+b\times \theta +c\times \text{log}\left(\theta \right),$$

Here, U _OCV (θ) represents a time course of the open-circuit voltage of the battery cell, which is dependent on the variable θ, which represents the time related to the relaxation time of the battery cell, and the predetermined mathematical function of the first parameter a, the second parameter b and the third parameter c, as described in more detail below.

The electronic circuit 100 can be designed to determine at least one of the three parameters based on a statistical evaluation of the measured values 101 of the no-load voltage and the associated time values 102 at which the measured values 101 of the no-load voltage were recorded.

$$s_{lnx}={\displaystyle \sum _{i=1}^n\text{log}\left(x_i\right)};s_{ln{x}^2}={\displaystyle \sum _{i=1}^n{\left(\text{log}\left(x_i\right)\right)}^2};s_{xlnx}={\displaystyle \sum _{i=1}^n{x}_i\text{log}\left(x_i\right)};s_{ylnx}={\displaystyle \sum _{i=1}^n{y}_i\text{log}\left(x_i\right)}$$

The electronic circuit 100 can be designed to base the statistical evaluation on the determination of the following functions: $$s_x={\displaystyle \sum _{i=1}^n{x}_i};s_y={\displaystyle \sum _{i=1}^n{y}_i};s_{x^2}={\displaystyle \sum _{i=1}^{\mathrm{<figure-callout\; id="2"\; label="n\; x\; i"\; filenames="DE102021110384A1\_0029.png"\; state="\{\{state\}\}">n</figure-callout>}}{x}_i^{}{}_2^{}};s_{xy}={\displaystyle \sum _{i=1}^n{x}_i{y}_i}$$

$$s_{lnx}={\displaystyle \sum _{i=1}^n\text{log}\left(x_i\right)};s_{ln{x}^2}={\displaystyle \sum _{i=1}^n{\left(\text{log}\left(x_i\right)\right)}^2};s_{xlnx}={\displaystyle \sum _{i=1}^n{x}_i\text{log}\left(x_i\right)};s_{ylnx}={\displaystyle \sum _{i=1}^n{y}_i\text{log}\left(x_i\right)}$$

In this case, y _i designates the measured values 101 of the no-load voltage and x _i the associated time values 102 at which the measured values 101 of the no-load voltage were recorded, as described in more detail below.

The electronic circuit 100 can be designed to provide a first estimated value of the equilibrium voltage 112 of the battery cell before the relaxation time of the battery cell is reached by determining the first parameter, as described in more detail below.

The electronic circuit 100 can be designed to provide a second estimated value of the equilibrium voltage 112 of the battery cell before the relaxation time of the battery cell is reached by determining the second parameter, the second estimated value being determined as a sum of the first parameter and the second parameter, as described in more detail below shown.

The electronic circuit 100 can be designed to store a time profile 600 of at least the first parameter 602 in a memory. The electronic circuit 100 can also be designed to determine the equilibrium voltage 112 of the battery cell based on the stored time curve 600 of at least the first parameter 602 in a further determination of the state of charge 114 of the battery cell, as below 6 shown in more detail.

The time course 600 can extend from the start of the determination of the state of charge 114 until the relaxation time of the battery cell is reached, as described in more detail below.

The electronic circuit 100 can be designed to store a plurality of time curves 600 of at least the first parameter 602 based on successive determinations of the state of charge 114 of the battery cell in the memory, as below 6 described in more detail and to determine an aging of the battery cell based on the plurality of time curves 600 of the state of charge 114 of the battery cell.

The electronic circuit 100 can be used in a battery management system (BMS), eg an electric vehicle. Such a battery management system comprises at least one charging module controller for acquiring a plurality of measured values 101 of an open-circuit voltage in at least one battery cell of a battery system, e.g. an electric vehicle, and associated time values 102 at which the measured values 101 of the open-circuit voltage are recorded. The battery management system includes the electronic circuit 100 described above for determining the state of charge 114 of the at least one battery cell of the battery system.

The functioning of the electronic circuit 100 is described in more detail below using the example of a lithium-ion cell.

erfolgt, sondern mit der neuen mathematischen Funktion: $$U_{OCV}\left(\theta \right)=ƒ\left(a,b,c\right)=a+b\times \theta +c\times \text{log}\left(\theta \right)$$

wobei $$\theta =\frac{t}{RT}.$$

The innovation is that the relaxation of a lithium-ion cell, described by U _cell (t) the cell voltage [V], no longer uses the exponential approach commonly used in the literature: $$u_{cell}=u_{OCV}+I_c\left(R_{ƒ}{e}^{-\frac{t-{\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="DE102021110384A1\_0028.png,DE102021110384A1\_0029.png"\; state="\{\{state\}\}">t</figure-callout>}}_0}{{\tau }_{ƒ}}}+R_{ct}{e}^{-\frac{t-{\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="DE102021110384A1\_0028.png,DE102021110384A1\_0029.png"\; state="\{\{state\}\}">t</figure-callout>}}_0}{{\tau }_{ct}}}\right)$$

takes place, but with the new mathematical function: $$u_{OCV}\left(\theta \right)=ƒ\left(a,b,c\right)=a+b\times \theta +c\times \text{log}\left(\theta \right)$$

whereby $$\theta =\frac{t}{RT}.$$

Here τ _ƒ = R _ƒ C _ƒ and τ _ct = R _ct C _{dl denote} different time constants with R _ƒ polarization resistance and C _ƒ capacitance, and RT the relaxation time. In the new mathematical function 111, a, b, c are parameters that are determined online using the well-known "regression" method as a function of the aging, the state of charge and the temperature of the cell. In this application (relaxation), the variable t is the time, which is renamed “x” in the following.

What is special about the new function 111 is that it is a linear equation in which the parameters can be determined mathematically using least square fitting, just like the equation for a straight line.

For example, vertical least squares fitting is done by finding the sum of the squares of the vertical deviations R ² of a set of n data points: $${\mathrm{<figure-callout\; id="2"\; label="R"\; filenames="DE102021110384A1\_0029.png"\; state="\{\{state\}\}">R</figure-callout>}}^2\equiv {\displaystyle \sum {\left[y_i-ƒ\left(x_i,a_1,a_2,...,a_n\right)\right]}^2}$$

für i = 1,...,n. Daraus ergeben sich folgende Gleichungen: $$\boxed{\begin{array}{c}-2{\displaystyle {\sum }_{i=1}^n\left(y_i-\left(a+b\ast x_i+c\ast log\left(x_i\right)\right)\right)=0}\\ -2{\displaystyle {\sum }_{i=1}^n\left(y_i-\left(a+b\ast x_i+c\ast log\left(x_i\right)\right)\right)x_i=0}\\ -2{\displaystyle \sum _{i=1}^n\left(y_i-\left(a+b\ast x_i+c\ast log\left(x_i\right)\right)\right)log\left(x_i\right)=0}\end{array}}$$

For the function f above, the necessary condition for R ² to be a minimum is the following: $$\frac{\partial \left(R^2\right)}{\partial a_i}=0$$

for i = 1,...,n. This results in the following equations: $$\boxed{\begin{array}{c}-2{\displaystyle {\sum }_{i=1}^n\left(y_i-\left(a+b\ast x_i+c\ast lOG\left(x_i\right)\right)\right)=0}\\ -2{\displaystyle {\sum }_{i=1}^n\left(y_i-\left(a+b\ast x_i+c\ast lOG\left(x_i\right)\right)\right)x_i=0}\\ -2{\displaystyle \sum _{i=1}^n\left(y_i-\left(a+b\ast x_i+c\ast lOG\left(x_i\right)\right)\right)lOG\left(x_i\right)=0}\end{array}}$$

$$s_{lnx}={\displaystyle \sum _{i=1}^n\text{log}\left(x_i\right)};s_{ln{x}^2}={\displaystyle \sum _{i=1}^n{\left(\text{log}\left(x_i\right)\right)}^2};s_{xlnx}={\displaystyle \sum _{i=1}^n{x}_i\text{log}\left(x_i\right)};s_{ylnx}={\displaystyle \sum _{i=1}^n{y}_i\text{log}\left(x_i\right)}$$

$$nenner=-2\times \left(s_{lnx}\right)\times \left(s_x\right)\times \left(s_{xlnx}\right)+n\times {\left(s_{xlnx}\right)}^2+{\left(s_{lnx}\right)}^2\times \left(s_{x^2}\right)+s_{ln{x}^2}\times \left({\left(s_x\right)}^2-n\times s_{x^2}\right)$$

ergibt sich: $$\begin{array}{l}a=\frac{1}{nenner}[\left({\left(s_{xlnx}\right)}^2-\left(s_{ln{x}^2}\right)\times s_{x^2}\right)\times s_y+\left(\left(-s_x\right)\times s_{xlnx}+\left(s_{lnx}\right)\times s_{x^2}\right)\times s_{ylnx}\\ +\left(\left(s_{ln{x}^2}\right)\times s_x-\left(s_{lnx}\right)\times s_{xlnx}\right)\times s_{xy}]\end{array}$$

$$\begin{array}{l}b=\frac{1}{nenner}[\left(\left(s_{ln{x}^2}\right)\times s_x-\left(s_{lnx}\right)\times s_{xlnx}\right)\times s_y+\left(\left(-s_{lnx}\right)\times s_x+n\times s_{xlnx}\right)\times s_{ylnx}\\ +\left({\left(s_{lnx}\right)}^2-n\times s_{ln{x}^2}\right)\times s_{xy}]\end{array}$$

$$\begin{array}{l}c=\frac{1}{nenner}[\left(\left(-s_x\right)\times s_{xlnx}+\left(s_{lnx}\right)\times s_{x^2}\right)\times s_y+\left({\left(s_x\right)}^2-n\times s_{x^2}\right)\times s_{ylnx}\\ +\left(\left(-s_{lnx}\right)\times s_x+n\times s_{xlnx}\right)\times s_{xy}]\end{array}$$

These equations can be solved for the parameters a, b, c. With the abbreviations: $$s_x={\displaystyle \sum _{i=1}^n{x}_i};s_y={\displaystyle \sum _{i=1}^n{y}_i};s_{x^2}={\displaystyle \sum _{i=1}^{\mathrm{<figure-callout\; id="2"\; label="n\; x\; i"\; filenames="DE102021110384A1\_0029.png"\; state="\{\{state\}\}">n</figure-callout>}}{x}_i^{}{}_2^{}};s_{xy}={\displaystyle \sum _{i=1}^n{x}_i{y}_i}$$

$$s_{lnx}={\displaystyle \sum _{i=1}^n\text{log}\left(x_i\right)};s_{ln{x}^2}={\displaystyle \sum _{i=1}^n{\left(\text{log}\left(x_i\right)\right)}^2};s_{xlnx}={\displaystyle \sum _{i=1}^n{x}_i\text{log}\left(x_i\right)};s_{ylnx}={\displaystyle \sum _{i=1}^n{y}_i\text{log}\left(x_i\right)}$$

$$nenne\mathrm{right}=-2\times \left(s_{lnx}\right)\times \left(s_x\right)\times \left(s_{xlnx}\right)+n\times {\left(s_{xlnx}\right)}^2+{\left(s_{lnx}\right)}^2\times \left(s_{x^2}\right)+s_{ln{x}^2}\times \left({\left(s_x\right)}^2-n\times s_{x^2}\right)$$

surrendered: $$\begin{array}{l}a=\frac{1}{nenne\mathrm{right}}[\left({\left(s_{xlnx}\right)}^2-\left(s_{ln{x}^2}\right)\times s_{x^2}\right)\times s_y+\left(\left(-s_x\right)\times s_{xlnx}+\left(s_{lnx}\right)\times s_{x^2}\right)\times s_{ylnx}\\ +\left(\left(s_{ln{x}^2}\right)\times s_x-\left(s_{lnx}\right)\times s_{xlnx}\right)\times s_{xy}]\end{array}$$

$$\begin{array}{l}b=\frac{1}{nenne\mathrm{right}}[\left(\left(s_{ln{x}^2}\right)\times s_x-\left(s_{lnx}\right)\times s_{xlnx}\right)\times s_y+\left(\left(-s_{lnx}\right)\times s_x+n\times s_{xlnx}\right)\times s_{ylnx}\\ +\left({\left(s_{lnx}\right)}^2-n\times s_{ln{x}^2}\right)\times s_{xy}]\end{array}$$

$$\begin{array}{l}c=\frac{1}{nenne\mathrm{right}}[\left(\left(-s_x\right)\times s_{xlnx}+\left(s_{lnx}\right)\times s_{x^2}\right)\times s_y+\left({\left(s_x\right)}^2-n\times s_{x^2}\right)\times s_{ylnx}\\ +\left(\left(-s_{lnx}\right)\times s_x+n\times s_{xlnx}\right)\times s_{xy}]\end{array}$$

The n data points from time and voltage value are therefore used to determine the parameters a, b and c. As before, the following applies: the larger n is, the more precisely the equilibrium voltage can be predicted with the function.

According to the formula, the equilibrium voltage of the cell after the relaxation time RT is as follows: $$u_{OCV}\left(\theta =1\right)=a+b$$

Verschiedene Experimente haben außerdem ergeben, dass der Parameter b sehr klein ist. Dies ist bei dem Vergleich zwischen dem Parameter a und der Gleichgewichtsspannung zu sehen. Hier konnte bereits nach 10 Minuten der Wert a in dieser Genauigkeit bestimmt werden.On the one hand there are measured values for which the time t is less than RT. Second, b is very small. So if a is taken for the equilibrium voltage, then the nearly constant value of the function profile is taken. Even if t is still smaller than RT, i.e. θ < 1, here it means that $$b\times \theta \approx c\times \text{log}\left(\theta \right)$$

Various experiments have also shown that the parameter b is very small. This can be seen in the comparison between the parameter a and the equilibrium stress. Here the value a could already be determined with this accuracy after 10 minutes.

In the following, the algorithm is explained, with which an exact determination of the state of charge is possible with the help of the open-circuit voltage after a short time. The values for the parameter “a” of the function are recorded in a characteristic map after 15 minutes, for example, and are saved if the equilibrium voltage with SOC calculation is close to present, for example after the relaxation time RT. As in the illustration of 6 shown, the value "a" can of course also be used directly as an approximation for the equilibrium voltage. A function results that describes the dependency of the parameter "a" on the SOC with values between 0 ... 100.

If this algorithm has been trained sufficiently, the end value can be determined as follows after 15 minutes, for example: The SOC range is divided into four equal ranges and the pairs of values {a, SOC} are fitted again with the same fit function. It is advisable to save new parameters under a different name. e.g. alpha, beta, gamma. During the next rest phase, the expected state of charge can then be calculated after a short time. The values in the table are then updated at regular intervals to take aging into account.

2 FIG. 2 shows an illustration of an exemplary time curve 200 of the no-load voltage of a battery cell.

The open-circuit voltage starts at time t=0 at a voltage value of about 3.505 V and then transitions approximately exponentially to the equilibrium state, which is reached after about 8000 seconds. An equilibrium voltage of about 3.53 V is then established. The course 200 over time was measured at a temperature of -10 degrees Celsius.

With the help of the electronic circuit described above, an SOC determination with greater accuracy is possible after a short time, taking aging into account, using simple, computationally efficient methods. This is in the illustration of the 2 illustrated. The parameters of the fit function used here are determined after 3600 seconds and extrapolated up to 5400 seconds.

The course of the rest voltage is extrapolated so well that in the representation of the 2 no difference is recognizable between the measured open-circuit voltage and the estimated open-circuit voltage.

3 FIG. 3 shows a representation of the deviation 300 between the measured value and the estimated value of the open-circuit voltage of a battery cell at a temperature of -10 degrees Celsius according to an example.

gegenüber dem hier beschriebenen neuen Verfahren 302, welches auf der neuen mathematischen Funktion, wie bereits oben zu 1 beschrieben, basiert: $$U_{OCV}\left(\theta \right)=ƒ\left(a,b,c\right)=a+b\times \theta +c\times \text{log}\left(\theta \right)$$

zeigt darüber hinaus in bestimmten SOC- und Temperaturbereichen ein wesentlich genaueres Ergebnis als bisher.For the prediction of the equilibrium voltage after a short time, especially at low temperatures and states of charge, only 3 parameters are required for the first time instead of 4, which minimizes the storage effort. Furthermore, only linear instead of non-linear regression methods are required for the online determination of the parameters, so that the computing time can be further optimized here. The comparison of the results from the previous method 301, ie with the usual exponential approach: $$u_{cell}=u_{OCV}+I_c\left(R_{ƒ}{e}^{-\frac{t-{\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="DE102021110384A1\_0028.png,DE102021110384A1\_0029.png"\; state="\{\{state\}\}">t</figure-callout>}}_0}{{\tau }_{ƒ}}}+R_{ct}{e}^{-\frac{t-{\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="DE102021110384A1\_0028.png,DE102021110384A1\_0029.png"\; state="\{\{state\}\}">t</figure-callout>}}_0}{{\tau }_{ct}}}\right)$$

compared to the new method 302 described here, which is based on the new mathematical function, as already mentioned above 1 described, based: $$u_{OCV}\left(\theta \right)=ƒ\left(a,b,c\right)=a+b\times \theta +c\times \text{log}\left(\theta \right)$$

also shows a much more accurate result than before in certain SOC and temperature ranges.

3 shows the comparison of the square mean between the measured value and the value of the approximation x 1000 via SOC in % for the new approach 302 and the previously used approach 301 for a temperature of -10°C. It is noteworthy that the new approach 302 with only one SOC value shows a significantly worse result here, but otherwise a similar or better result. Presumably, this also depends on the cell specifically examined, so that even better results can be expected when measuring with other cells.

The method presented here is suitable for all energy cells, in particular polymer cells such as silicon polymer cells as well as lead battery cells. It has been shown that particularly good results can be achieved with conventional lead starter batteries, i.e. battery cells which are used as starter batteries in almost all vehicles today. The electronic circuit 100 can thus advantageously also be used in connection with starter batteries which are made, for example, on a lead basis, for example in a BMS system for monitoring the starter battery.

4 shows a representation of the deviation 400 between measured value and estimated value of the open circuit voltage of a battery cell at a temperature of +10 degrees Celsius according to an example.

4 shows the comparison of the root mean square between measured value and value of the approximation x 1000 over SOC in % for the new approach 402, ie using the mathematical function 111 as above 1 described, and the previously used approach 401, which is based on the exponential functions described above, for a temperature of +10° C.

Here, too, the new approach 402 shows a significantly worse result with only one SOC value, but otherwise a similar or better result.

5 shows a representation of the deviation 500 between the measured value and the estimated value of the open circuit voltage of a battery cell at a temperature of +25 degrees Celsius according to an example.

5 shows the comparison of the root mean square between measured value and value of the approximation x 1000 over SOC in % for the new approach 502, ie using the mathematical function 111 as above 1 described, and the previously used approach 501, which is based on the exponential functions described above, for a temperature of +25° C.

Here, too, the new approach 502 shows a significantly worse result with only one SOC value, but otherwise a similar or better result.

In summary, it can be stated: The advantage is that the available memory (RAM) and the performance of the CPU can be improved with the inventive approach presented here, i.e. additional and more complex algorithms can be implemented on the BMU with the same hardware. With the same function of a battery core ("BatteryCore"), a cheaper chip can be used under certain circumstances.

6 shows a representation of a characteristic curve 600, which describes the relationship between the equilibrium voltage and the state of charge of a battery cell, as well as the parameter a at a temperature of +25 degrees Celsius according to an example. A measured value 601 of the equilibrium voltage and an estimated value 602 of the equilibrium voltage are plotted, here represented by the parameter a, as described above.

The parameter a results from the new mathematical function, as already mentioned above 1 described, which reads as follows: $$u_{OCV}\left(\theta \right)=ƒ\left(a,b,c\right)=a+b\times \theta +c\times \text{log}\left(\theta \right)$$

In the characteristic curve, an equilibrium voltage of 3.4 V results in a state of charge of approximately 10 percent. With an equilibrium voltage of 3.6 V, the state of charge is approximately 35 percent. With an equilibrium voltage of 4 V, the state of charge is 80 percent. With an equilibrium voltage of 4.2 V, the state of charge is approximately 98 percent. Almost no difference can be seen between the measured value 601 of the equilibrium voltage and the estimated value 602 of the equilibrium voltage, represented here by the parameter a, except perhaps at very low SOC values.

Based on the in 6 In the representation shown, an algorithm can be set up with which an exact determination of the state of charge is possible with the help of the open-circuit voltage after a short time. The values for the parameter “a” 602 of the mathematical function 111 are recorded in a characteristic map and stored if the equilibrium voltage with SOC calculation is almost present, for example after the relaxation time RT. Like in the 6 shown, the deviation between the measured value 601 and the estimated value of the equilibrium stress is only minimal and almost negligible. Therefore, the value "a" can be used directly as an approximation for the equilibrium voltage. A function results that describes the dependency of the parameter "a" on the SOC with values between 0 ... 100, as in 6 shown.

If this algorithm has been trained sufficiently, the end value can be determined as follows after a short time, i.e. well before the relaxation time: The SOC range is divided into four equal ranges and the value pairs {a, SOC} are again used with the same fit function fitted. It is advisable to save new parameters under a different name. e.g. alpha, beta, gamma. During the next rest phase, the expected state of charge can then be calculated after a short time. The values in the table are then updated at regular intervals to take aging into account.

7 FIG. 7 shows a schematic representation of a method 700 for determining a state of charge of a battery cell of a battery system according to the disclosure. The battery system can be a battery system of an electric vehicle, for example.

The method 700 comprises the following steps: Obtaining 701 a plurality of measured values 101 of an open-circuit voltage of the battery cell with associated time values 102 at which the measured values 101 of the open-circuit voltage were recorded, as, for example, above 1 described; determining 702 an equilibrium voltage 112 of the battery cell based on a linear regression 110 of the plurality of measured values 101 of the no-load voltage with respect to a predefined mathematical function 111 of the time values 102 associated with the measured values 101 of the no-load voltage; and determining 703 the state of charge 114 of the battery cell based on a predetermined characteristic curve 113, which indicates a relationship between the equilibrium voltage 112 and a state of charge 114 of the battery cell.

Furthermore, a computer program with a program code for executing the method 700 on an electronic circuit 100 or a battery management system, as above 1 described, are provided.

reference list

100

electronic switch

101

Measurements of the open-circuit voltage of the battery cell

102

Associated time values at which the measured values of the no-load voltage were recorded

110

Linear regression

111

predetermined mathematical function

112

equilibrium voltage

113

Specified characteristic, which indicates the relationship between the equilibrium voltage and the state of charge of the battery cell

114

state of charge, SoC

200

time course of the open-circuit voltage

300

Deviation between the measured value and the estimated value of the open-circuit voltage

301

Deviation with exponential approach

302

Deviation in the approach according to the invention

400

Deviation between the measured value and the estimated value of the open-circuit voltage

401

Deviation with exponential approach

402

Deviation in the approach according to the invention

500

Deviation between the measured value and the estimated value of the open-circuit voltage

501

Deviation with exponential approach

502

Deviation in the approach according to the invention

600

Characteristic that describes the relationship between the equilibrium voltage and the state of charge of a battery cell

700

Method for determining the state of charge of a battery cell

701

first step in the process

702

second process step

703

third step

Claims (15)

Electronic circuit (100) for determining a state of charge (114) of a battery cell of a battery system, in particular for an electric vehicle, the electronic circuit (100) being designed to have a plurality of measured values (101) of an open-circuit voltage of the battery cell with associated time values (102) , for which the measured values (101) of the open-circuit voltage were recorded; an equilibrium voltage (112) of the battery cell based on a linear regression (110) of to determine a plurality of measured values (101) of the no-load voltage with respect to a predetermined mathematical function (111) of the time values (102) associated with the measured values (101) of the no-load voltage; and to determine the state of charge (114) of the battery cell based on a predetermined characteristic curve (113), which indicates a relationship between the equilibrium voltage (112) and a state of charge (114) of the battery cell. Electronic circuit (100) after claim 1 , wherein the predetermined mathematical function (111) specifies a time profile (200) of the open-circuit voltage of the battery cell. Electronic circuit (100) after claim 1 or 2 , where the predetermined mathematical function (111) is a linear function. Electronic circuit (100) according to any one of the preceding claims, where the given mathematical function (111) is a linear function dependent on: a first parameter, or a first parameter and a second parameter, or a first parameter, a second parameter and a third parameter. Electronic circuit (100) after claim 4 , wherein the predetermined mathematical function (111) specifies a time profile (200) of the open-circuit voltage of the battery cell as a function of a relaxation time of the battery cell. Electronic circuit (100) after claim 5 , where the given mathematical function (111) is determined as follows: $$u_{OCV}\left(\theta \right)=ƒ\left(a,b,c\right)=a+b\times \theta +c\times \text{log}\left(\theta \right),$$

where U _OCV (θ) represents a time course of the open-circuit voltage of the battery cell, which depends on the variable θ, which represents the time related to the relaxation time of the battery cell, and wherein the predetermined mathematical function depends on the first parameter a, the second parameter b and the third parameter c. Electronic circuit (100) after claim 6 , wherein the electronic circuit (100) is designed to determine at least one of the three parameters based on a statistical evaluation of the measured values (101) of the no-load voltage and the associated time values (102) at which the measured values (101) of the no-load voltage were recorded . Electronic circuit (100) after claim 7 , wherein the electronic circuit (100) is designed to base the statistical evaluation on the determination of the following functions: $$s_x={\displaystyle \sum _{i=1}^n{x}_i};s_y={\displaystyle \sum _{i=1}^n{y}_i};s_{x^2}={\displaystyle \sum _{i=1}^n{x}_i^2};s_{xy}={\displaystyle \sum _{i=1}^n{x}_i{y}_i}$$

$$s_{lnx}={\displaystyle \sum _{i=1}^n\text{log}\left(x_i\right)};s_{ln{x}^2}={\displaystyle \sum _{i=1}^n{\left(\text{log}\left(x_i\right)\right)}^2};s_{xlnx}={\displaystyle \sum _{i=1}^n{x}_i\text{log}\left(x_i\right)};s_{ylnx}={\displaystyle \sum _{i=1}^n{y}_i\text{log}\left(x_i\right)}$$

where y _i denote the measured values (101) of the no-load voltage and x _i the associated time values (102) at which the measured values (101) of the no-load voltage were recorded. Electronic circuit (100) according to any one of Claims 5 until 8th , wherein the electronic circuit (100) is designed to provide a first estimated value of the equilibrium voltage (112) of the battery cell before the relaxation time of the battery cell is reached by determining the first parameter. Electronic circuit (100) after claim 9 , wherein the electronic circuit (100) is designed to provide a second estimated value of the equilibrium voltage (112) of the battery cell before the relaxation time of the battery cell is reached by determining the second parameter, the second estimated value being a sum of the first parameter and the second parameter definitely. Electronic circuit (100) according to any one of Claims 5 until 10 , wherein the electronic circuit (100) is designed to store a time profile (600) of at least the first parameter (602) in a memory, and wherein the electronic circuit (100) is designed, in a further determination of the state of charge (114) of Battery cell to determine the equilibrium voltage (112) of the battery cell based on the stored time history (600) of at least the first parameter (602). Electronic circuit (100) after claim 11 , wherein the time profile (600) extends from a start of the determination of the state of charge (114) until the relaxation time of the battery cell is reached. Electronic circuit (100) after claim 12 , wherein the electronic circuit (100) is designed to store a plurality of time curves (600) of at least the first parameter (602) based on successive determinations of the state of charge (114) of the battery cell in the memory, and wherein the electronic circuit (100 ) is designed to determine an aging of the battery cell based on the plurality of time curves (600) of the state of charge (114) of the battery cell. Battery management system with: at least one charging module controller for detecting a plurality of measured values (101) of an open-circuit voltage on at least one battery cell of a battery system, in particular for an electric vehicle, and of associated time values (102) at which the measured values (101) of the open-circuit voltage are recorded; and an electronic circuit (100) according to any one of the preceding claims for determining the state of charge (114) of the at least one battery cell of the battery system. Method (700) for determining a state of charge (114) of a battery cell of a battery system, in particular for an electric vehicle, the method (700) having the following steps: Obtaining (701) a plurality of measured values (101) of an open-circuit voltage of the battery cell with associated time values (102) at which the measured values (101) of the open-circuit voltage were recorded; determining (702) an equilibrium voltage (112) of the battery cell based on a linear regression (110) of the plurality of measured values (101) of the open-circuit voltage with respect to a predetermined mathematical function (111) of the measured values (101) of the open-circuit voltage associated time values (102); and Determining (703) the state of charge (114) of the battery cell based on a predetermined characteristic curve (113) which indicates a relationship between the equilibrium voltage (112) and a state of charge (114) of the battery cell.

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