Method and device for determining the state of charge and health of a rechargeable battery

Description

[0001] The invention relates to a method for determining the state of charge of a rechargeable battery of a given battery type or a physically related parameter, in particular a residual charge amount contained in the battery, as well as a method for determining the state of health of a rechargeable battery of a given battery type or a physically related parameter, in particular a current capacity of the battery. Furthermore, the invention relates to a device for carrying out these methods.

[0002] The state of charge (SOC) of a rechargeable battery is defined as SOC = Q C .

[0003] The State Of Health (SOH) of a rechargeable battery is defined as SOH = C C N .

[0004] In these equations, Q the remaining charge in the battery, with Cthe capacity, ie the amount of charge that can be extracted from a fully charged battery, and C N is the nominal capacity, i.e., the capacity of a new battery. SOC and SOH are defined as dimensionless quantities between 0 and 1. In practice, SOC and SOH are often expressed as percentages.

[0005] In many battery systems today, SOC determination is performed by a so-called battery management system (BMS), which makes this information available to the user, for example, via a display. A variety of different methods are known for the approximate determination of SOC, including model-based methods. These operate exclusively on the basis of current-controlled battery models, so measured values ​​for the battery current are required as an input signal. However, current measurements are not possible with the desired accuracy, so the accuracy of the known methods for determining SOC is often not good enough.

[0006] For example, US 2014 / 0203813 A1 and US 2011 / 0093223 A1 disclose methods and devices for determining the SOC of a battery based on such (at least partly) current-controlled models. The SOC is determined using an observer, which can be implemented as a Kalmann filter.

[0007] Methods for determining SOH often rely on measuring battery current during a cycle between a fully discharged and fully charged state. However, this is not possible during normal battery use.

[0008] US 2019 / 0170827 proposes a method for determining the SOH, which is also based on the use of an observer configured as a Kalmann filter. The use of an observer requires considerable computational effort during implementation. Furthermore, EP 3 174 175 A1 discloses a device for the safe use of a battery, which determines the SOH of the battery as a function of a quotient formed from the difference between the battery voltage values ​​measured at the beginning and end of an observation period and the energy supplied or absorbed by the battery recorded during the observation period.

[0009] Based on this prior art, the invention is based on the object of creating a method for the approximate determination of the state of charge of a rechargeable battery of a given battery type or of a parameter physically related thereto, in particular a residual charge quantity contained in the battery, which can be implemented in a battery management system in a simple manner and with little computational effort.Furthermore, the invention is based on the object of creating a method for determining the state of health of a rechargeable battery of a given battery type or a parameter physically related thereto, in particular a current capacity of the battery, which method enables the determination of an approximate value for the state of health of the battery during normal use of the battery and which can be implemented in a battery management system simply and with little computational effort. Finally, the invention is based on the object of creating a device which enables the implementation of one or both of the aforementioned methods.

[0010] The invention is defined by the independent claims. Advantageous embodiments are defined in the dependent claims.

[0011] The invention is based on the finding that the use of a voltage-controlled dynamic mathematical battery model, which is described by an implicit system of equations, enables the simple determination of an approximate value for the SOC. This merely requires measuring the battery voltage (over a specific period of time). Depending on the specifically selected battery model, the battery temperature or the ambient temperature can also be used as input variables for the battery model. The model can also include certain time dependencies, for example, a hysteresis behavior of the battery or a time dependency of double layers.

[0012] At the same time, the voltage-controlled battery model provides the (model) battery current as an output variable. This is used, according to the invention, to determine the battery's SOH.

[0013] The method according to the invention uses a dynamic mathematical battery model for the battery or the predetermined battery type (ie a plurality of similar batteries), which links a state of charge of the battery SOC mod or a parameter physically related thereto, in particular the residual charge quantity Q contained in the battery, with a battery current I mod and which defines a voltage U mess measured between two terminals (poles) of the battery as a function of a sum of an open circuit voltage U 0< of the battery and an overvoltage η, wherein the open circuit voltage U 0< depends at least on the residual charge quantity Q or the parameter physically related thereto and the overvoltage η depends at least on the battery current I mod.

[0014] This dynamic mathematical battery model can then be parameterized for the specific battery or a specific battery type, i.e. the parameters for the battery model are chosen such that the parameterized model describes the specific battery or the specific battery type with sufficient accuracy.

[0015] Thus, the method according to the invention enables the determination of an approximate value SOC mod for the actual state of charge SOC of the battery or the parameter physically related thereto by measuring the battery voltage U mess and calculating the approximate value SOC mod using the parameterized dynamic mathematical battery model.

[0016] A device for implementing this method therefore only needs to be configured to measure the battery voltage U mess, for example, using a unit for measuring the battery voltage U mess, and to calculate the approximate value SOC mod for the SOC of the battery from the measured battery voltage U mess. For this purpose, the device can comprise a computing unit for calculating the SOC. The parameterized battery model can also be stored in the computing unit or supplied to it from a higher-level unit.

[0017] The battery model also enables the calculation of the (model) battery current I mod . According to the invention, this can also be used in a further step to determine an approximate value SOH mod for the actual SOH of the battery (see below). It is not absolutely necessary that the SOC of the battery is also determined when using the method according to the invention to determine an approximate value for the SOH of a battery. Of course, the use of a voltage-controlled dynamic mathematical battery model allows only the (model) battery current I mod to be calculated, without the SOC of the battery having to be output or used beforehand or simultaneously.

[0018] In its full form, however, the method according to the invention creates an algorithm (a mathematical rule) that allows the state of charge SOC and the state of health SOH to be determined from measured values ​​for the battery voltage U mess and the current I mess of the battery – and optionally from measured values ​​for the temperature T mess of the battery or the ambient temperature. This fully developed method is described in Fig. 1 shown schematically. It consists of an overall algorithm 100, which comprises two components, namely a first component 102 with a battery model and an algorithm for determining the SOC and a second component 104 with an SOH algorithm for determining the SOH. The measured battery voltage U mess and, optionally, the measured temperature T mess of the battery are fed to the first component 102 as a mandatory input variable. Instead of the measured temperature T mess, the ambient temperature T ambient< can also be fed to the battery model if the battery model includes a thermal sub-model. The first algorithm component 102 with the dynamic mathematical battery model supplies a value SOC mod as an output variable in the full embodiment of the method according to the invention, which value is used as an approximate value for the actual SOC of the battery 106.The second algorithm component 104 receives as input variables, on the one hand, the value for the model battery current I mod calculated using the battery model and, on the other hand, the measured battery current I mess . As an output variable, the second algorithm component provides a value SOH mod , which is used as an approximate value for the actual SOH of the battery 106.

[0019] According to one embodiment of the invention, the dynamic mathematical battery model consists of an implicit system of equations or can be developed from it, which includes the following equations: dSOC mod d t = − I mod C N U mess = U 0 SOC mod T mess t + η I mod SOC mod T mess t where CN is a given nominal capacity of the battery, T mess is a measured temperature of the battery and t is the time, where the open circuit voltage U 0< necessarily shows a dependence on the state of charge SOC mod, while the dependence on the measured temperature T mess and the time t is optional, and where the overvoltage η necessarily shows a dependence on the battery current I mod, while the dependence on the state of charge SOC mod, the measured temperature T mess and the time t is optional.

[0020] These two very general relationships implicitly describe the temporal behavior of the two unknowns SOC mod and Imod . While the first equation merely describes that the change in the residual charge contained in the battery corresponds to the battery current and can be adopted in this way for all further, special designs or formulations of the mathematical battery model, the second equation can be adapted to a specifically selected battery model and, if necessary, replaced by a complex system of equations that describes the selected battery model mathematically and physically.

[0021] For example, according to one embodiment of the invention, the open-circuit voltage U 0< can be assumed to be exclusively dependent on the state of charge SOC mod, and a constant internal resistance R i of the battery can be assumed. Furthermore, it can be assumed that the battery model is temperature-independent. In this case, the above general equations for the battery model can be replaced by the following equations: dSOC mod d t = − I mod C N U mess = U 0 SOC mod − R i ⋅ I mod , where the dependence of the open-circuit voltage U 0< on the state of charge SOC mod is determined, in particular, by measurement, particularly as measured discrete values ​​or as an analytical function. In this case, a very easy-to-solve system of equations results, which can be implemented into a battery management system with little effort.

[0022] In the case of this battery model, the two above equations can be solved by analytical inversion, where the state of charge SOC mod is determined from the equations dSOC mod d t = − 1 R i ⋅ C N ⋅ U 0 SOC mod − U mess I mod = 1 R i U 0 SOC mod − U mess can be calculated. The dependency U 0< (SOC mod ) can be given either in tabular form, i.e. in the form of measured values, or in the form of an analytical function. It should be noted at this point that the model current I mod can of course also be calculated from these equations if the SOH is to be determined in addition to the SOC. Depending on the method chosen to solve the two aforementioned equations, only the first equation may be sufficient to calculate the SOC. If the modeled battery current I mod is also to be calculated, the second equation must also be used in any case.

[0023] In practice, the aforementioned pairs of equations for calculating the SOC mod will be solved using numerical methods, since the input variables supplied to the algorithm for determining the SOH mod are also recorded in the form of discrete measured values, preferably at an equidistant time interval Δt (sampling interval). In particular, the voltage U mess is recorded in the form of discrete voltage measured values. U mess i recorded at a specified time interval Δt. The specified time interval can also vary. The measured values ​​can also be recorded at specified times.

[0024] To numerically solve the above equations for the assumed very simple battery model, the explicit forward Euler method can be used for discretization. This discretization method leads to the relations SOC mod i + 1 = SOC mod i − Δ t R i ⋅ C N ⋅ U 0 SOC mod i − U mess i I mod i = 1 R i U 0 SOC mod i − U mess i where the respective quantities are provided with an index i, which denotes a time step of Δt. Of course, a suitable starting value for SOC mod i = 0 be elected.

[0025] Instead of the explicit forward Euler method, other explicit or implicit methods can be used to discretize the above equations. However, with the implicit backward Euler method, a closed-form solution is no longer possible. In this case, additional numerical methods, such as the Newton method, must be used.

[0026] However, a special case arises if, according to an embodiment of the invention, a linear relationship between the open-circuit voltage and the SOC of the battery is assumed. This further simplifies the equations for the simple battery model given above to dSOC mod d t = − 1 R i ⋅ C N U L − U E ⋅ SOC mod + U E − U mess and I mod = 1 R i U L − U E ⋅ SOC mod + U E − U mess where UL denotes a final charging voltage and UE a final discharging voltage.

[0027] From these equations, the approximate value for the state of charge SOC, i.e., the value SOC mod , can be calculated using a mathematical discretization method and, if necessary, further numerical methods. Using the implicit backward Euler method, this leads to the equations I mod i + 1 = 1 U L − U E C N Δ t + R i U L − U E ⋅ SOC mod i + U E − U mess i + 1 and SOC mod i + 1 = SOC mod i − I mod i + 1 C N Δ t , from which SOC mod can also be easily calculated. As already mentioned, determining SOH mod does not necessarily require calculating a value for the battery current I mod, but in this particular case, this value is automatically determined due to the coupling of the two equations.

[0028] According to a further embodiment of the invention, the method for determining the SOC or SOH of the battery can also be carried out by means of a relatively complex equivalent circuit model for the battery according to Fig. 3 This battery model can be described by the following equations: dSOC mod d t = − I mod C N U mess = U 0 SOC mess T mod + ΔU hys I mod − U RC , an − U RC , ca − I mod ⋅ R s T mod d U RC , an d t = 1 C DL , an T mod I mod − U RC , an R CT , an T mod d U R C , ca d t = 1 C DL , ca T mod I mod − U RC , ca R CT , ca T mod I mod d T mod d t = 1 C th I mod 2 R s T mod + U RC , an 2 R CT , an T mod + U RC , ca 2 R CT , ca T mod I mod − T mod − T mess Umg R th + I mod ⋅ T mod − T 0 ⋅ d U 0 SOC mod d T wobei mit U RC is a voltage drop across an RC element, with R CT is a charge transfer resistor, with C DL a double-shift capacity, with C th a heat capacity, with R th is a heat transfer resistance at the battery surface, with d U 0< (SOC mod ) / d T a temperature dependence of the open circuit voltage, with T 0< an associated reference temperature and with T mess Um a measured ambient temperature of the battery, where the indices "an" and "ca" refer to an anode and a cathode of the battery and where an asymmetry of the open circuit voltage U 0< is given by Δ U hys and an asymmetry of the cathode resistance is caused by a current dependence of R CT,ca . From this system of equations, the value for SOC mod , i.e. an approximate value for the actual state of charge SOC, can be calculated, preferably using one or more numerical mathematical methods.

[0029] The method according to the invention for determining the state of health of a rechargeable battery of a given battery type or a parameter physically related thereto, in particular a current capacity C of the battery, comprises the following steps: Creating a voltage-controlled dynamic mathematical battery model parameterized for the battery or the specified battery type with the features explained above, determining a charge quantity Q in,mess absorbed by the battery during a first observation period and a charge quantity Q out,mess delivered by the battery during a second observation period by measuring and integrating a battery current I mess delivered or absorbed by the battery, wherein the second observation period is preferably selected to be identical or at least overlapping with the first observation period;Calculation of a charge quantity Q in,mod absorbed by the battery during the first observation period and a charge quantity Q out,mod released by the battery during the second observation period using a voltage-controlled dynamic mathematical battery model parameterised for the battery or the specified battery type, determination of an approximate value SOH mod for the actual state of health SOH by calculating a charge state of health SOH in as the quotient of the charge quantity Q in,mess and the charge quantity Q in,mod and / or a discharge state of health SOH out as the quotient of the charge quantity Q out,mess and the charge quantity Q out,mod and using the charge state of health SOH in or the discharge state of health SOH out or an average value calculated therefrom as an approximate value SOH mod for the actual state of health SOH. ;

[0030] It should be clarified that in the step of calculating the charge quantities Q in,mod or Q out,mod using the voltage-controlled battery model, measurement data for the battery voltage are of course also used.

[0031] According to one embodiment, the first and second time periods can be selected such that the charge quantities Q in,mess charged and / or the charge quantities Q out,mess discharged during the respective time period and / or the sum of these amounts are greater than a predefined value, wherein the predefined value is preferably greater than the nominal capacity CN of the battery. This ensures sufficient accuracy of the determined value SOH mod.

[0032] According to a further embodiment, the end times of the first and second time periods can be selected such that the same state of charge SOC ref exists at the end times as at the beginning times and / or that the same current direction of the measured battery current I mess prevails at the end times as at the beginning times. This can reduce the influence of hysteresis effects or model deviations.

[0033] The variants of battery models explained above and the methods presented for solving the systems of equations enable the calculation of the battery current in a simple manner Imod . According to the invention, the charge quantities Q in,mod and Q out,mod can be easily calculated by analytical or numerical integration of the battery current I mod calculated from the dynamic mathematical battery model. Thus, the approximate value SOH mod for the actual state of health of the battery can also be determined very easily.

[0034] Further embodiments of the invention emerge from the subclaims.

[0035] The invention is explained in more detail below with reference to exemplary embodiments illustrated in the drawings. In the drawings: Fig. 1 a schematic diagram of the method according to the invention; Fig. 2 a schematic block diagram of a battery operated under load with a device according to the invention for carrying out the method; Fig. 3 a special, complex equivalent circuit model for a lithium iron phosphate-based lithium-ion cell, which consists of an electrical submodel ( Fig. 3a ) and a thermal submodel ( Fig. 3b ) consists; Fig. 4 a U 0< (SOC) diagram of a lithium-ion battery with nickel-manganese-cobalt oxide / graphite chemistry (NMC-graphite) from the manufacturer Kokam, type SLPB533459H4, with a nominal capacity of 0.74 Ah and a nominal voltage of 3.7 V; Fig. 5 simulated discharge-charge characteristics (battery voltage versus charge quantity) for the battery according to Fig. 3 at three temperatures ( Fig. 5a at 5°C; Fig. 5b at 20°C and Fig. 5c at 35°C) and three current intensities (0.06C, 0.28C, 0.93C), whereby model B was parameterized for this battery for simulation; Fig. 6 a section of the 100 consecutive charging cycles according to Birkl for the battery according to Fig. 4 , where Fig. 6a shows the measured voltage U mess as input value for the battery model and Fig. 6b the measured current (curve (a) and the current calculated using the battery model (model A) (curve (b)); Fig. 7 Results of the new procedure (according to Model A) for the battery according to Fig. 4 , where Fig. 7a represents the state of charge SOC during the first hours of cycling according to Birkl and Fig. 7b the state of charge (SOC) during the last few hours; the results of the method (curves (b)) are compared with results obtained using a conventional method (curves (a)); Fig. 8 Results of the new method for the health state SOH over the entire experimental period (curve (b)) using model A; these results are compared with an SOH curve obtained by charge counting (curve (v)). Fig. 9 Results of an experiment with the new process for a lithium-ion battery for stationary storage with LFP graphite chemistry with a nominal capacity of 158 Ah for a wide range of discharge and charge cycles; Fig. 9a shows the measured voltage and Fig. 9b shows the measured (curve (a)) and the modeled (curve (b)) current (using battery model B); and Fig. 10 Results of this procedure for the SOC ( Fig. 10a ; curves (b)) and the SOH ( Fig. 10b ); the results of the procedure are compared with a precise comparison measurement (curves (v)).

[0036] The Fig. 1 und 2 show a schematic representation of the device and method according to the invention for determining the SOH and SOC of a battery 106 which is operated on a load RL. As can be seen from Fig. 1 As can be seen, the method can be implemented by means of an overall algorithm 100, for example, by integration into an existing battery management system. The overall algorithm 100 comprises a first component 102 for SOC determination, which also comprises a dynamic mathematical battery model selected for the specific battery or the specific battery type. The measured battery voltage U mess and the measured temperature T mess of the battery are fed to this first component as input variables. Instead of the measured temperature of the battery, the measured ambient temperature can also be fed to this first component of the overall algorithm 100. T mess umg This first component of the overall algorithm is able to calculate the measured battery voltage U measure (input variable) the state of charge SOC mod and the current Imod (output variables) of the battery. The type of battery model itself is irrelevant for the procedure as long as it meets these requirements. Empirical models, equivalent circuit models, multiphysics models, or other models are conceivable. Only the output variable SOC mod is required to determine the SOC.

[0037] To determine the SOH, the overall algorithm 100 has a second component 104, to which the current I mod calculated by means of the first component 102 and the measured current I mess are supplied as input variables.

[0038] The essential feature of the invention is that the dynamic model operates voltage-controlled, ie with the measured voltage U mess as the input variable. The charge level to be transmitted to the user is derived directly from the model. This can be done, for example, via a display unit.

[0039] The accuracy of the method depends heavily on how accurately the model can reproduce the actual SOC and current I mess of the real battery from the given voltage, ie how well SOC mod and SOC or I mess and I mod match. To increase the accuracy, a measured temperature can also be added to the model. T measurement of the cell or the environment T mess umg or other measured values ​​are passed on.

[0040] As in Fig. 2 As shown, this overall algorithm 100 can be easily integrated into an existing battery management system. For this purpose, the battery management system (not shown) only needs to contain a device 108 for carrying out the method. The device 108 comprises a unit 110 for measuring the battery voltage U mess , which is connected to the connections (poles) of the battery 106. Furthermore, the device 108 comprises a unit 112 for measuring the battery current I mess , which can be designed in any desired way. For example, the unit 112 can comprise a shunt resistor that lies in the current path between the battery poles and any load RL, which is also designated by the reference numeral 114. The unit 112 can be designed to measure the voltage across the shunt resistor and to calculate the current from the measured voltage drop and the resistance value of the shunt resistor.

[0041] The device 108 can also include a display unit 116 on which the determined values ​​for the SOC or SOH are displayed. The device 108 includes a computing unit 118, which can be embodied, for example, as a microprocessor unit, to perform the calculations required to implement the method. The microprocessor unit can also have an analog / digital converter that samples the analog variables U mess and I mess supplied to it over time and converts them into digital values.

[0042] The basic principle of the method according to the invention as well as special variants are explained in more detail below.

[0043] The battery model has two minimum requirements. First, it must be dynamic, meaning there must be at least one time-dependent state variable. This is typically the remaining charge. Q or the charge level SOC = Q C N , which change over time due to an applied current. Second, the voltage curve must be described as a function of state of charge and current.

[0044] In a general representation, the model consists of two equations, dSOC mod d t = − I mod C N U mess = U 0 SOC mod T mess t + η I mod SOC mod T mess t

[0045] The two equations implicitly describe the temporal behavior of the two unknowns (SOC mod , I mod ) as a function of the measured input variable U measurement with given parameters and battery properties ( C N, U 0< , η ).

[0046] The two terms on the right side of the second equation represent the open circuit voltage U 0< and the overvoltage η, ie voltage drops due to slow internal processes such as reactions and transport. The open circuit voltage U0< depends primarily on the SOC, with further possible dependencies on the temperature and the temporal charge / discharge history (e.g., for battery materials with hysteresis such as lithium iron phosphate). The overvoltage depends on the SOC, the current I mod and the temperature T mess and also has a pronounced dynamic (temporal) characteristic due to electrochemical double layers. To describe the behavior of U 0< and η, further model equations can be used depending on the model complexity. Here and in the following, the sign of the battery current (measured and calculated) in the case of battery discharge is chosen to be positive, i.e. I > 0, and in the case of battery charge negative, ie I < 0.

[0047] The solution of the implicit system of equations (1) and (2) according to SOC mod and Imod requires an inversion. This can be done analytically or numerically, depending on the model or implementation. The solution is illustrated in the examples below, but other methods are also conceivable.

[0048] A possible implementation of the method is the use of a real (e.g. measured) relationship between open circuit voltage U 0< and SOC, U 0< = U 0< (SOC), assuming a constant internal resistance R i , from which the overvoltage to η = - R i · I mod, and assuming temperature independence. This simplifies the system of equations (1) and (2) to dSOC mod d t = − I mod C N U mess = U 0 SOC mod − R i ⋅ I mod .

[0049] The connection U0< (SOC mod ) can be specified either in tabular form (e.g., measured values) or in the form of an analytical function. This model is referred to as "Model A" or "Simple Model" below. The system of equations (3) and (4) is sufficiently simple that it can be analytically inverted. For this purpose, equation (4) is I mod and the resulting relationship is inserted into equation (3). dSOC mod d t = − 1 R i ⋅ C N U 0 SOC mod − U mess I mod = 1 R i U 0 SOC mod − U mess with U mess as independent (given) variable, SOC mod and I mod as dependent (searched) variables, and U 0< (SOC), R i and C N as model parameters.

[0050] In a real system, voltage measurements are usually recorded as discrete values U mess i at a given time interval Δ t measured. Here is iis the index for a time step. Therefore, a time discretization is necessary to solve the system of equations (5) and (6). A discretization using the explicit forward Euler method yields the following solution: SOC mod i + 1 = SOC mod i − Δ t R i ⋅ C N U 0 SOC mod i − U mess i I mod i = 1 R i U 0 SOC mod i − U mess i

[0051] Equation (7) provides a concrete calculation rule for the new method for SOC determination. The only input variable is the current voltage measurement value. U mess i . The only value saved is the one calculated in the previous step. SOC mod i necessary. The calculated new value SOC mod i + 1 is passed on to the user as the current SOC of the battery. This determination of the SOC from equation (7) is therefore extremely simple and can be performed quickly with low computing power. It can be easily implemented on a microcontroller, as only simple calculation steps are required. The measurement effort is also low; only the voltage U mess needs to be measured in the form of time-discrete voltage values. U mess i measured. Unlike conventional methods for determining SOC, measuring the current is not required.

[0052] Equation (8) provides the corresponding current I mod i This is not required for the SOC determination, but is required for the SOH determination (see below).

[0053] Equations (7) and (8) were derived from equations (5) and (6) by an explicit forward Euler discretization. There are also alternative discretization methods. An implicit backward Euler discretization is, however, not suitable due to the nonlinear relationship U 0< (SOC mod ) is not possible in a closed form, further numerical methods would be necessary here (e.g. Newton's method). However, a special case is the assumption of a linear relationship between voltage and SOC according to the relationship U 0< = ( U L - U E ) · SOC + U E , where with U L the final charging voltage and with U E which is the final discharge voltage. This simplifies equations (5) and (6) further to dSOC mod d t = − 1 R i ⋅ C N U L − U E ⋅ SOC mod + U E − U mess I mod = 1 R i U L − U E ⋅ SOC mod + U E − U mess

[0054] The backward Euler discretization yields the following result: I mod i + 1 = 1 U L − U E C N Δ t + R i U L − U E ⋅ SOC mod i + U E − U mess i + 1 SOC mod i + 1 = SOC mod i − I mod i + 1 C N Δ t

[0055] In this case, too, the calculation procedure is very simple. This form is also numerically more stable. However, the results would be very different from the highly simplified and unrealistic linear U 0< (SOC mod ) relationship. For sufficiently small time steps Δ t However, for periods ≤ 10 s, calculations have shown no significant difference between explicit and implicit discretization.

[0056] Of course, other analytically specified U 0< (SOC) relationships are conceivable. Depending on the analytical form, the system of equations (5) and (6) can be solved analytically or using suitable numerical methods.

[0057] The following example illustrates a complex battery model, also referred to as Model B, and its use for determining SOC and SOH. Real batteries exhibit complex dynamic current-voltage-temperature behavior that cannot be fully represented by the simple model of equations (5) and (6). Therefore, more complex models can be used to increase the reliability of the method.

[0058] The following describes the Fig. 3 The equivalent circuit model shown is considered. It is an electrical-thermal model. The electrical model consists of the open-circuit voltage source U 0< (SOC), a serial resistance R s and two resistor-capacitor (RC) elements ( R CT and C DL , one each for the two electrodes, anode and cathode). Elements are also added that describe an asymmetry of the open circuit voltage (ΔU hys ) and a charge / discharge asymmetry of the cathode resistance. This makes the model suitable for simulating a lithium iron phosphate (LFP) / graphite lithium-ion cell (see H. Kim, "Modeling an parameterization of a commercial LFP / Graphite Lithium-ion cell considering charge and discharge characteristics," Master's thesis, Offenburg University of Applied Sciences, 2018), which exhibits these asymmetries. The thermal model consists of heat sources and heat transfer to the environment. All parameters are also assumed to be temperature-dependent.

[0059] The equivalence circuit model can be represented in the following differential-algebraic system of equations: dSOC mod d t = − I mod C N U mess = U 0 SOC mod T mod + Δ U hys I mod − U RC , an − U RC , ca − I mod ⋅ R s T mod d U RC , an d t = 1 C DL , an T mod I mod − U RC , an R CT , an T mod d U RC , ca d t = 1 C DL , ca T mod I mod − U RC , ca R CT , ca T mod I mod d T mod d t = 1 C th I mod 2 R s T mod + U RC , an 2 R CT , an T mod + U RC , ca 2 R CT , ca T mod I mod − T mod − T mess umg R th + I mod ⋅ T mod − T 0 ⋅ d U 0 SOC mod d T

[0060] Hier sind U RC is the voltage drop across the RC element, R CT is the charge transfer resistance, C DL is the double-layer capacity, Cth is the heat capacity, R th is the heat transfer resistance at the battery surface, d U 0< (SOC model) / d T the temperature dependence of the open circuit voltage, T 0< the corresponding reference temperature, and the indices an and ca stand for "anode" and "cathode," i.e., the two electrodes. The asymmetry of the open-circuit voltage is determined by Δ U hys, and the asymmetry of the cathode resistance is determined by the current dependence of R CT,ca . By comparing equations (13) and (14) with equations (1) and (2), it can be seen that this is a further form of the basic model. The additional equations (15) to (17) are required for its description. As before, equations (13) to (17) implicitly describe the temporal behavior of the two unknowns (SOC mod , I mod ) as a function of the measured input variables, namely the voltage Umeasurement and the ambient temperature T mess umg .

[0061] Due to the coupling of the equations, an analytical solution is not possible. For the simulation results presented below, an implicit numerical solver provided in the MATLAB software package was used. However, the solution is also possible using other methods.

[0062] It should be expressly emphasized again that the two presented models A and B serve only to demonstrate the new procedure, which is in no way limited to these two specific models. Rather, any simpler or even more complex models are conceivable.

[0063] The models presented above (A: "Simple Model" and B: "Equivalence Circuit Model") can describe different batteries or battery types depending on their parameterization. In the following, two models are parameterized for specific lithium-ion battery cells.

[0064] As an example of a battery used in the field of electromobility, we consider a lithium-ion battery with nickel-manganese-cobalt oxide / graphite chemistry (NMC-graphite), specifically a cell from the manufacturer Kokam, model SLPB533459H4, with a nominal capacity of 0.74 Ah and a nominal voltage of 3.7 V. Birkl and Howey provide detailed datasets for this cell for free use (Christoph Birkl, David Howey, "Oxford Battery Degradation Dataset 1", University of Oxford, DOI: 10.5287 / bodleian:KO2kdmYGg, website: https: / / ora.ox.ac.uk / objects / uuid:03ba4b01-cfed-46d3-9b1a-7d4a7bdf6fac (2017)), which are ideally suited for demonstrating the present process.

[0065] Model A ("Simple Model") is parameterized for this cell as follows: The open circuit voltage curve U0< (SOC) was determined by averaging a charge and discharge characteristic curve, each recorded at 40 mA ("quasi-open circuit voltage"). The capacity was normalized to the maximum. The resulting U 0< (SOC) relationship is in Fig. 4 The nominal capacity C N = 0.74 Ah was taken directly from the manufacturer's specifications. The internal resistance was determined by reading the cell charge voltages at 50% SOC for discharges of 40 mA and 740 mA from the experiments. R i = − U 40 mA − U 740 mA 40 mA − 740 mA = 0 , 0464 Ω The parameters are assumed to be temperature-independent. Complete parameterization therefore requires only two experimental charge / discharge curves (at 40 mA and 740 mA).

[0066] As an exemplary representative of a cell chemistry used in the field of stationary power storage (home storage, commercial storage, storage for grid applications, uninterruptible power supply), a lithium-ion battery with lithium iron phosphate / graphite chemistry (LFP-graphite) is considered below, specifically a cell from the manufacturer Sinopoly, model SP-LFP180AHA with a nominal capacity of 180 Ah and a nominal voltage of 3.2 V. This cell was characterized at Offenburg University of Applied Sciences (see H. Kim, ibid.).

[0067] For this example, model B (equivalence circuit model) was parameterized to this specific cell. Fig. 5 shows simulated discharge-charge curves (cell voltage versus charge quantity) after successful parameterization at three temperatures (5°C, 20°C, and 35°C) and three current intensities. The current intensities are specified with the unit of the C-rate (i.e., in this case, 1C=180 A). The arrows indicate the direction in which the curves are traversed, namely the discharge direction (arrow pointing right) and the charge direction (arrow pointing left). The upper of the associated curves is the charge curve, and the lower one is the discharge curve. The darkest curves (top and bottom) are the charge and discharge curves for a current of 0.93 C, and the upper and lower innermost curves (lightest) are the charge and discharge curves for a current of 0.06 C. The curves between these curves are the charge and discharge curves for a current of 0.28 C.The points on the curves represent measured values, and the solid lines represent simulation values. As can be seen from . Fig. 5 As can be seen, the measured values, including the asymmetry and hysteresis of the experiments, can be very well predicted by the model over the entire range.

[0068] The following explains in more detail how to determine the SOH using the SOH algorithm. Its specific application also requires the parameterization of the dynamic mathematical battery model used, as explained above.

[0069] The battery models shown above are able to measure the current in addition to the state of charge SOC mod I mod as the output value. In addition to the current according to the model, the measured current I required for SOH determination. The SOH algorithm is able to I mod and I measure to calculate the health status SOH.

[0070] The algorithm is based on the battery model, as a "digital twin," performing the same cycling as the real battery, since (as explained above) it is operated at the same voltage as the real battery. However, unlike the real battery, the battery model does not age. The difference between I mod (without loss of capacity) and I mess (with loss of capacity), the SOH can be determined. For this purpose, charge meters are used, ie the measured current is integrated I measurement and the current determined by the model I mod . This calculation is shown below.

[0071] As explained above, a voltage-controlled battery model is a prerequisite for the algorithm, because only then is I mod an output variable that is IThe new method for determining SOH is therefore closely linked to the new method for determining SOC.

[0072] For a complete battery discharge (full to empty) starting at time t 0 and completed at time tl, an SOH out can be defined by the relationship: SOH out = ∫ 0 t l I bat , akt d t ∫ 0 t l I bat , n d t where I bat,n is the battery current of a new (not aged battery) and I bat,akt is the battery current of the aged battery.

[0073] Assuming that I bat,act corresponds to the measured current of the real battery (with capacity loss) and I bat,n the simulated current from the battery model (without capacity loss), the SOH out can be represented as: SOH out = ∫ 0 t l I mess d t ∫ 0 t l I mod d t

[0074] Similarly, a SOH can also be defined during a full battery charge if the battery is assumed to be empty: SOH in = ∫ 0 t v I mess d t ∫ 0 t v I mod d t where it is assumed that the charging process starts at time t 0 and is completed at time t V .

[0075] In practice, complete charging and discharging processes rarely occur (e.g., the battery of an electric car will never be completely discharged, as it would then no longer be functional). Therefore, the algorithm must be able to determine the SOH even with partial charges and discharges. We define an arbitrary period [ t 1 ; t 2 ] without a priori knowledge as to whether this period involves a charge, a discharge, or both (e.g., one or more full cycles or partial cycles). The period [ t 1 ; t 2 ] is typically in the range of hours; the more precise requirements are given below. Within this period, we can calculate the SOH as SOH out = ∫ t 1 t 2 I mess , out d t ∫ t 1 t 2 I mod , out d t und SOH in = ∫ t 1 t 2 I mess , in d t ∫ t 1 t 2 I mod , in d t , where the indices "out" and "in" denote a discharge (out) and a charge (in) of the battery, respectively. In these equations, I mess , in = I mess für I mess < 0 0 für I mess ≥ 0 I mess , out = 0 für I mess < 0 I mess für I mess ≥ 0 bzw . I mod , in = I mod für I mod < 0 0 für I mod ≥ 0 I mod , out = 0 für I mod < 0 I mod für I mod ≥ 0

[0076] In equation (21), integration is only performed during discharge, while in equation (22) integration is performed only during charge. In principle, each of these SOH values, ie, SOH out or SOH in , could already serve as an approximate value for the actual SOH. However, an increase in accuracy can be achieved by averaging according to the relationship SOH = SOH out + SOH in 2

[0077] The period [ t 1 ; t 2 ] is initially arbitrary; however, the choice of the period influences the accuracy of the method. In a concrete implementation of the method, this period can preferably be chosen taking the following conditions into account: The quantities of cargo discharged and loaded during the period must be greater than a predefined threshold Q s (e.g. C N or a multiple of C N ). At the beginning and end of the period, the same state of charge (SOC ref) prevails (e.g., 50%). At the beginning and end of the period, the same current direction prevails (e.g., the battery is charging). Different, overlapping time periods can also be used for the two variables SOH out and SOH in. This is used, for example, in the implementation of the method shown below.

[0078] This method has the advantage that, unlike conventional methods, neither the experimental execution of full cycles nor an algorithm for counting cycles is necessary.

[0079] In a concrete implementation as program code running on a microcontroller, the algorithm can be designed in such a way that it has four charge counters ( Q out,mess , Q out,mod , Q in,mod , Q in,mod ) and two SOH values ​​(SOH out , SOH in ) are determined and stored. The algorithm is periodically executed after a period Δ t This period is ideally the same as in the SOC calculation. The value for I mod is obtained from the SOC algorithm. The following logic then runs: 1. Charge counter (a) Actual I mod < 0 Entladung ? Yes Q out , mod = Q out , mod − I mod ⋅ Δ t (are I mod > 0 Ladung ? Ja : Q in , mod = Q in , mod + I mod ⋅ Δ t (c) Is I mess < 0 Entladung ? Yes: Q out , mess = Q out , mess − I mess ⋅ Δ t (d) Is I mess > 0 Ladung ? Yes: Q in , mess = Q in , mess + I mess ⋅ Δ t 2. SOH calculation (a) Are Q out,mod > Q s and Q out,mess > Q s and SOC mod = SOC ref and is the battery discharging? Yes: SOH out = Q out , mess Q out , mod Set Qout,mess and Q out,mod returns to zero. (b) If Q in,mod > Q s and Q in,mess > Q s and SOC mod = SOC ref and is the battery charging? Yes: SOH in = Q in , mess Q in , mod Set Q in,mess and Q in,mod returns to zero. (c) SOH = SOH out + SOH in 2

[0080] The last calculated value SOH is returned by the algorithm and can be displayed to the user, for example. The parameters Q s and SOC ref influence the performance of the SOH diagnostics; in the examples shown below, Q s = C N and SOC ref = 50%.

[0081] The process is demonstrated below using a lithium-ion battery cell with NMC graphite chemistry (representative of the electromobility application area). Openly available experimental data from Birkl (loc. cit.) are used. Model A ("Simple Model") is used.

[0082] Birkl's experiments (loc. cit.) were conducted as follows: The battery under test was subjected to a large number of consecutive cycles until the battery's service life was reached. In each cycle, the battery was charged using the CCCV (Constant Current Constant Voltage) method and discharged using a dynamic load profile simulating a city driving cycle. After every 100 cycles, a measurement cycle was performed to characterize the battery properties. The battery or a battery cell was discharged with a constant current of 0.74 A to a final voltage of 2.7 V and then recharged to a final voltage of 4.2 V. This characterization process was repeated approximately 80 times. However, only the full discharge / charge cycles of the characterization cycles have been published. These were combined for the present demonstration and thus represent accelerated aging behavior.At the same time, this demonstrates another strength of the new process, namely the ability to switch to battery operation at any time and also to handle incomplete data.

[0083] Fig. 6 shows the measured or calculated quantities for the procedure using the simple model (Model A), namely in Fig. 6a the measured voltage and in Fig. 6b the measured I meas current (curve (a)) or the calculated I mod current (model output, input for the SOH algorithm; curve (b)) of the battery. The noticeable difference between the model and the measurement is a result of remaining deficiencies in the model used. Nevertheless, meaningful results can be obtained, as shown below.

[0084] Fig. 7 shows the state of charge (SOC) determined for this battery using the new method (in the specific embodiment according to equation (7)) (curves (b)). Also shown is a SOC value determined using conventional methods based on charge counting (normalized to the fully discharged cell) (curves (a)). Fig. 7a shows the first few hours of cycling. The new method can determine the SOC with very good accuracy (compared to the conventional method). Fig. 7b shows the final hours of cycling. Here, the cell has already aged significantly, meaning it has lost capacity. The new method can reliably represent full cycling. However, the conventional method fails due to cell aging: Despite charging to the end-of-charge voltage, the SOC in the conventional method only reaches an (incorrect) value of approximately 75%. This comparison demonstrates the robustness of the new method for determining SOC against capacity loss due to cell aging.

[0085] The result of the SOH determination is in Fig. 8 (curve (b)). Furthermore, a comparison with values ​​from a simple charge count according to Equation (19) is shown (curve (v)). The agreement is very good. These results indicate that the SOH can be reliably determined using the new method.

[0086] In the following, the process is demonstrated using a lithium-ion battery cell with LFP graphite chemistry (representative for the stationary storage application area), using model B ("equivalent circuit model").

[0087] For the experiments, over 670 consecutive charge / discharge cycles were conducted, with discharging at a constant current of 150 A and charging using the CCCV charging method. The final discharge voltage was 2.85 V, and the final charge voltages were 3.8 V. The test duration was approximately 1,500 h. The capacity of the new CN cell was 158 Ah. This dataset can be used to demonstrate both SOC determination (during any single cycle) and SOH determination (over the entire duration of the test). In addition, highly precise SOC and SOH values ​​were determined using suitable measurement technology using the conventional charge counting method, which serve as a comparison with the new method.

[0088] Fig. 9 shows the input variables for the procedure, ie the measured voltage U mess ( Fig. 9a ) and the measured current I mess ( Fig. 9b ; curve (a)) for this battery. Fig. 9b also shows the modeled current I mod (output of the model and input for the SOH algorithm; curve (b)). The noticeable difference between the model and the measurement is a result of remaining deficiencies in the model used.

[0089] The result of the procedure, the state of charge SOC and the state of health SOH of the battery, is in Fig. 10 shown (curves (b)). For comparison, values ​​resulting from the precise comparative measurement are also shown (curves (v)). Fig. 10a shows the SOC. The new method can reliably represent the battery's cycling between 0% and 100% SOC, albeit with a certain small error compared to precise measurements. Fig. 10b shows the SOH. The new method is able to reliably reproduce the battery's capacity loss over the test period of approximately 1,500 hours. Compared to the precise measurement, only increased noise is observed.

[0090] These results demonstrate the effectiveness of the method described above for determining the state of charge and health of batteries.

[0091] List of essential names of variables and parameters C Battery capacity CN Nominal battery capacity I mess Measured battery current I mod Model battery current Q Remaining charge in the battery SOC actual state of charge SOC mod State of charge according to the battery model SOH actual health status SOH mod Health status according to the battery model t time T measure measured battery temperature T measurement measured ambient temperature U measure measured battery voltage U mod Model battery voltage U 0 Open circuit voltage Δtsampling interval List of reference symbols

[0092] 100Overall algorithm 102First component of the overall algorithm 104Second component of the overall algorithm 106Battery 108Device for carrying out the method 110Voltage measurement unit 112Current measurement unit 114Load 116Display unit 118Calculation unit

Claims

1. Method for determining the state of charge of a rechargeable battery or a parameter physically related thereto, in particular a residual charge quantity Q contained in the battery, wherein the method comprises the following steps: (a) creating a voltage-controlled dynamic mathematical battery model for the battery (106) or a specified battery type for the battery (106), (i) which links a state of charge of the battery SOCmod or the parameter physically related thereto to a battery current Imod, and (ii) which defines a voltage Umess measured between the two poles of the battery (106) as a function of a sum of an open circuit voltage U0 of the battery (106) and an overvoltage η, (iii) wherein the open circuit voltage U0 is dependent at least on the residual charge quantity Q or the parameter physically related thereto and the overvoltage η is dependent at least on the battery current Imod; and (iv) wherein the voltage-controlled dynamic mathematical battery model consists of or is developed from an implicit system of equations comprising the following equations: (1) dSOC mod d t = I mod C N (2) U mess = U 0 SOC mod T mess t + η I mod SOC mod T mess t where CN denotes a specified nominal capacity of the battery (106), Tmess denotes a measured temperature of the battery (106) and t denotes the time, wherein the open circuit voltage U0 necessarily exhibits a dependence on the state of charge SOCmod, while the dependence on the measured temperature Tmess and the time t is optional, and wherein the overvoltage η necessarily exhibits a dependence on the battery current Imod, while the dependence on the state of charge SOCmod and the time t is optional; (b) parameterizing the voltage-controlled dynamic mathematical battery model for the battery (106) or the specified battery type; and (c) determining an approximate value SOCmod for the actual state of charge SOC of the battery (106) or the parameter physically related thereto by measuring the battery voltage Umess and calculating the approximate value SOCmod using the parameterized voltage-controlled dynamic mathematical battery model.

2. Method according to claim 1, characterized in that the open circuit voltage U0 is assumed to be exclusively dependent on the state of charge SOCmod, in that a constant internal resistance Ri of the battery (106) is assumed, and in that temperature independence of the battery model is assumed, such that the battery model is defined by the following equations: (a) dSOC mod d t = − I mod C N (b) U mess = U 0 SOC mod − R i ⋅ I mod , the dependence of the open circuit voltage U0 on the state of charge SOCmod being ascertained in particular by measurement, in particular as measured discrete values or as an analytical function.

3. Method according to claim 2, characterized in that the two equations are solved by analytical inversion, the state of charge SOCmod being calculated from the equations dSOC mod d t = − 1 R i ⋅ C N ⋅ U 0 SOC mod − U mess I mod = 1 R i U 0 SOC mod − U mess 4. Method according to claim 3, characterized in that the voltage Umess is captured in the form of discrete measured voltage values U mess i at a specified time interval Δt or at specified points in time, and in that the approximate value SOCmod for the state of charge is calculated by discretization of the equation according to claim 3 using a mathematical discretization method, in particular using the explicit forward Euler method, from the equation SOC mod i + 1 = SOC mod i − Δ t R i ⋅ C N ⋅ U 0 SOC mod i − U mess i where i denotes the index for a time step.

5. Method according to claim 3, characterized in that in addition a linear correlation is assumed between the open circuit voltage U0 and the state of charge SOCmod, and in that the state of charge is calculated from the relationships dSOC mod d t = 1 R i ⋅ C N U L − U E ⋅ SOC mod + U E − U mess and I mod = 1 R i U L − U E ⋅ SOC mod + U E − U mess where UL denotes an end-of-charge voltage and UE denotes an end-of-discharge voltage.

6. Method according to claim 5, characterized in that the voltage Umess is captured in the form of discrete measured voltage values U mess i at a specified time interval Δt, and in that the approximate value SOCmod for the state of charge is calculated by discretization of the equations according to claim 5 using a mathematical discretization method, in particular using the implicit backward Euler method, from the equations I Modell i + 1 = 1 U L − U E C N Δ t + R i U L − U E ⋅ SOC Modell i + U E − U Messung i + 1 and SOC Modell i + 1 = SOC Modell i − I Modell i + 1 C N Δ t where i denotes the index for a time step.

7. Method according to claim 1, characterized (a) in that, as the mathematical battery model, an equivalent circuit model is used, which is described by the equations according to features (a) (iv) (1) and (a) (iv) (2) of claim 1, the equation according to feature (a) (iv) (2) of claim 1 being replaced by the system of equations: U mess = U 0 SOC mess T mod + Δ U hys I mod − U RC , an − U RC , ca − I mod ⋅ R s T mod d U RC , an d t = 1 C DL , an T mod I mod − U RC , an R CT , an T mod d U RC , ca d t = 1 C DL , ca T mod I mod − U RC , ca R CT , ca T mod I mod d T mod d t = 1 C th I mod 2 R s T mod + U RC , an 2 R CT , an T mod + U RC , ca 2 R CT , ca T mod I mod − T mod − T mess Umg R th + I mod ⋅ T mod − T 0 ⋅ d U 0 SOC mod d T where URC denotes a drop in voltage across an RC element, RCT denotes a charge transfer resistance, CDL denotes a double-layer capacitance, Cth denotes a heat capacity, Rth denotes a heat transfer resistance on the battery surface, dU0(SOCmod) / dT denotes a temperature dependence of the open circuit voltage, T0 denotes an associated reference temperature, and T mess Umg denotes a measured ambient temperature of the battery (106), where the indices "an" and "ca" refer to an anode and a cathode of the battery and where an asymmetry of the open circuit voltage U0 is described by ΔUhys and an asymmetry of the cathode resistance is described by a current dependence on RCT,ca, and (b) in that the value for the state of charge SOCmod is calculated from this system of equations, preferably by means of one or more numerical mathematical methods.

8. Method for determining the state of health of a rechargeable battery of a specified battery type or a parameter physically related thereto, in particular a present capacity C of the battery, comprising the following steps: (a) creating a voltage-controlled dynamic mathematical battery model for the battery (106) or a specified battery type for the battery, (i) which links a state of charge of the battery SOCmod or the parameter physically related thereto to a battery current Imod, and (ii) which defines a voltage Umess measured between the two poles of the battery (106) as a function of a sum of an open circuit voltage U0 of the battery and an overvoltage η, (iii) wherein the open circuit voltage U0 is dependent at least on the residual charge quantity Q or the parameter physically related thereto and the overvoltage η is dependent at least on the battery current Imod, (iv) wherein the voltage-controlled dynamic mathematical battery model consists of or is developed from an implicit system of equations comprising the following equations: (1) dSOC mod d t = − I mod C N (2) U mess = U 0 SOC mod T mess t + η I mod SOC mod T mess t where CN denotes a specified nominal capacity of the battery (106), Tmess denotes a measured temperature of the battery (106) and t denotes the time, wherein the open circuit voltage U0 necessarily exhibits a dependence on the state of charge SOCmod, while the dependence on the measured temperature Tmess and the time t is optional, and wherein the overvoltage η necessarily exhibits a dependence on the battery current Imod, during parameterization of the voltage-controlled dynamic mathematical battery model for the battery (106) or the specified battery type; (b) determining a charge quantity Qin,mess received by the battery during a first observation period and / or a charge quantity Qout,mess provided by the battery (106) during a second observation period, by measuring and integrating a battery current Imess supplied or received by the battery (106), wherein the second observation period is preferably selected to be identical to or at least overlapping with the first observation period; (c) calculating a charge quantity Qin,mod received by the battery during the first observation period and a charge quantity Qout,mod supplied by the battery (106) during the second observation period using the voltage-controlled dynamic mathematical battery model parameterized for the battery (106) or the specified battery type; and (d) determining an approximate value SOHmod for the actual state of health SOH, by calculating a charge state of health SOHin as a quotient of the charge quantity Qin,mess and the charge quantity Qin,mod and / or a discharge state of health SOHout as a quotient of the charge quantity Qout,mess and the charge quantity Qout,mod and using the charge state of health SOHin or the discharge state of health SOHout or an average value calculated therefrom as an approximate value SOHmod for the actual state of health SOH.

9. Method according to claim 8, characterized in that the first and the second period are selected such that the charge quantities Qin,mess charged and / or the charge quantities Qout,mess discharged during the relevant period and / or the sum of these quantities are greater than a value predefined in each case, the predefined value being preferably greater than the nominal capacity CN of the battery (106).

10. Method according to any of claims 8 or 9, characterized in that the end points in time of the first and the second period are selected such that the same state of charge SOCref exists at the end points in time as at the start points in time and / or that the same current direction of the measured battery current Imess prevails at the end points in time as at the start points in time.

11. Method according to any of claims 8 to 10, characterized in that the charge quantities Qin,mod and Qout,mod are calculated by integrating the battery current Imod calculated from the dynamic mathematical battery model.

12. Method according to any of claims 8 to 11, characterized in that the open circuit voltage U0 is assumed to be exclusively dependent on the state of charge SOCmod, in that a constant internal resistance Ri of the battery (106) is assumed, in that temperature independence of the battery model is assumed, and in that the battery current Imod is calculated from the equation obtained by analytical inversion: I mod = 1 R i U 0 SOC mod − U mess the dependence of the open circuit voltage U0 on the state of charge SOCmod being ascertained in particular by measurement, in particular as measured discrete values or as an analytical function.

13. Method according to claim 12, characterized in that the voltage Umess is captured in the form of discrete measured voltage values U mess i at a specified time interval Δt, and in that the battery current Imess is calculated by discretization of the equation according to claim 14 using a mathematical discretization method, in particular using the explicit forward Euler method, from the equation I mod i = 1 R i U 0 SOC mod i − U mess i where i denotes the index for a time step.

14. Method according to claim 12, characterized in that in addition a linear correlation is assumed between the open circuit voltage U0 and the state of charge SOCmod, and in that the battery current is calculated from the relationships dSOC mod d t = − 1 R i ⋅ C N U L − U E ⋅ SOC mod + U E − U mess and I mod = 1 R i U L − U E ⋅ SOC mod + U E − U mess where UL denotes an end-of-charge voltage and UE denotes an end-of-discharge voltage.

15. Method according to claim 14, characterized in that the voltage Umess is captured in the form of discrete measured voltage values U mess i at a specified time interval Δt or at specified points in time, and in that the approximate value SOCmod for the state of charge is calculated by discretization of the equation according to claim 16 using a mathematical discretization method, in particular using the implicit backward Euler method, from the equations I mod i + 1 = 1 U L − U E C N Δ t + R i U L − U E ⋅ SOC mod i + U E − U mess i + 1 and SOC mod i + 1 = SOC mod i − I mod i + 1 C N Δ t where i denotes the index for a time step.

16. Method according to claims 8 to 11, characterized (a) in that, as the mathematical battery model, an equivalent circuit model is used, which is described by the equations according to features (a) (iv) (1) and (a) (iv) (2) of claim 8, the equation according to feature (a) (iv) (2) of claim 8 being replaced by the system of equations: d U RC , an d t = 1 C DL , an T mod I mod − U RC , an R CT , an T mod d U RC , ca d t = 1 C DL , ca T mod I mod − U RC , ca R CT , ca T mod I mod d T mod d t = 1 C th I mod 2 R s T mod + U RC , an 2 R CT , an T mod + U RC , ca 2 R CT , ca T mod I mod − T mod − T mess Umg R th + I mod ⋅ T mod − T 0 ⋅ d U 0 SOC mod d T where URC denotes a drop in voltage across an RC element, RCT denotes a charge transfer resistance, CDL denotes a double-layer capacitance, Cth denotes a heat capacity, Rth denotes a heat transfer resistance on the battery surface, dU0(SOCmod) / dT denotes a temperature dependence of the open circuit voltage, T0 denotes an associated reference temperature, and T mess Um denotes a measured ambient temperature of the battery, where the indices "an" and "ca" refer to an anode and a cathode of the battery (106) and where an asymmetry of the open circuit voltage U0 is described by ΔUhys and an asymmetry of the cathode resistance is defined by a current dependence on RCT,ca, (b) in that the battery current Imod is calculated from this system of equations, preferably by means of one or more numerical mathematical methods and (c) in that the charge quantities Qin,mod and Qout,mod are calculated by integrating the battery current Imod calculated from the dynamic mathematical battery model.

17. Device for carrying out the method according to any of claims 1 to 7, comprising a unit (110) for measuring the battery voltage Umess, preferably at equidistant time intervals Δt or at specified points in time, and comprising a unit (118) for calculating an approximate value SOCmod for the actual state of charge SOC of the battery (106), which device is designed to carry out the method according to any of claims 1 to 7.

18. Device for carrying out the method according to any of claims 8 to 16, comprising a unit (110) for measuring the battery voltage Umess, preferably at equidistant time intervals Δt, comprising a unit (112) for measuring the battery current Imess and comprising a unit (118) for calculating an approximate value SOHmod for the actual state of health SOH of the battery, which device is designed to carry out the method according to any of claims 8 to 16.