Method and device for determining capacity, internal resistance and open-circuit voltage curve of a battery

Abstract

The invention relates to a method, a device and a computer program for carrying out a method for determining the capacity, internal resistance and open-circuit voltage curve of a chargeable battery. The method is based on measuring battery voltage (V mess) and battery current (I mess) over a time period. The measured voltage is applied to a voltage-regulated battery model that calculates a current intensity (I mod). The battery model is provided with arbitrarily assumed values for the parameters to be determined (capacity and / or internal resistance and / or open-circuit voltage curve). Since these do not as a rule correspond to the values of the real battery, there is a difference between simulated current intensity (I mod) and measured current intensity (I mess). From this difference, the deviation between assumed and real values for capacity, internal resistance and open-circuit voltage curve is determined by means of suitable calculation rules. From this, and from the assumed values, follow the real values for capacity, internal resistance and / or open-circuit voltage characteristic, which can be stored or displayed to a user. If one or more of the parameters are already known from the beginning, the known values are used in the model, and only the unknown values are determined. The determined values can also be used to update the model. Further measurement data, or a repetition using the same measurement data, then permit the measuring accuracy to be increased. A continuous measurement of capacity, internal resistance and / or open-circuit voltage curve over longer periods is also possible, which thus enables the ageing of the battery to be assessed.

Description

[0001] Method and device for determining capacity, internal resistance and open circuit voltage curve of a battery The invention relates to a method for determining the internal resistance and / or the open circuit voltage curve and / or the capacity of a rechargeable battery, wherein the method comprises various steps.Furthermore, the present invention relates to a device for determining the internal resistance and / or the open-circuit voltage curve and / or the capacity of a rechargeable battery, comprising a detection device for detecting measured values ​​for the battery current ^^exp( ^^) and the battery voltage ^^exp( ^^) of the rechargeable battery, preferably at equidistant time intervals Δt or at predetermined times, and an evaluation and control device to which the detected measured values ​​can be fed, wherein the evaluation and control device is designed to carry out the method for determining the internal resistance and / or the open-circuit voltage curve and / or the capacity of a rechargeable battery. An evaluation and control device is understood to mean any suitable device that provides this functionality, regardless of whether in the individual case a control in the narrower sense (iea control of an output variable without feedback) or a control (i.e. a control of an output variable using feedback). Furthermore, the present invention relates to a computer program for determining the internal resistance and / or the open circuit voltage curve and / or the capacity of a rechargeable battery for a device, wherein the computer program is designed such that when the computer program is executed in the evaluation and control unit of a device for determining the internal resistance and / or the open circuit voltage curve and / or the capacity of a rechargeable battery, the method for determining the internal resistance and / or the open circuit voltage curve and / or the capacity of a rechargeable battery is carried out. The internal resistance is one of the important characteristic properties of a battery. It causes a drop in the battery terminal voltage when the battery is loaded with current.There are various methods for measuring internal resistance, such as pulse testing or electrical impedance spectroscopy. In a pulse test, current ^^1 and voltage ^^1 are measured, then the current or voltage is changed rapidly (< 1 ms), and the current ^^2 and voltage ^^2 are measured again a few seconds after the change. The internal resistance ^^ (usually expressed in Ω) is then calculated as: (1) The internal resistance is responsible for voltage drops and heat generation during battery operation; it typically increases over time (battery aging). Thus, the battery's performance decreases. Knowledge of the internal resistance over the battery's service life is therefore of great importance for quantifying aging and performance. The open-circuit voltage curve is another characteristic of a battery. It describes the course of the open-circuit voltage. 0(^^^^^^) (also known as open-circuit voltage (OCV)) as a function of depth of discharge (DOD). There are various methods for measuring the open-circuit voltage curve. In a "quasi-OCV" measurement, the battery is fully charged and discharged using very low currents; the open-circuit voltage curve ^^ 0(DOD) is the mean value of the measured voltage curves for charge and discharge. The open-circuit voltage curve allows statements to be made about the battery chemistry, i.e., which electrode materials are used in the battery, as well as about the aging state and aging mechanisms. Knowledge of the open-circuit voltage curve is therefore of great importance when characterizing unknown batteries (e.g., used batteries for second-life applications) or when assessing the aging state. The capacity ^^ of a rechargeable battery indicates the amount of charge (typically in ampere-hours, Ah) that can be drawn from a fully charged battery. It is another key characteristic of a battery. Capacity is typically measured in a laboratory test by fully discharging a full battery with a constant current ^^ and measuring the capacity after ^^ = ^^ ∙ ^^ Entladung(2). In DE 102019127828 A1, a method for determining the relative capacity ^^ of an aged cell based on the capacity ^^ ^^of a fresh cell ("nominal capacity"), referred to there as the "state of health" (SOH). This method is further developed here to determine the capacity of an unknown battery. The above-mentioned measurement methods (pulse test, quasi-OCV measurement, constant current discharge) require measurements in a laboratory environment using precise measuring instruments. This is generally not possible for batteries in practical applications, as they are an integral part of a device (e.g., smartphones, electric cars, home storage devices) and cannot be removed and transferred to a laboratory, or only with great effort.Based on this prior art, the invention is based on the object of creating a method for the approximate determination of the capacity and / or the internal resistance and / or the open-circuit voltage curve of a rechargeable battery during normal use of the battery, which method has improved accuracy and is also easy to implement in a battery management system. Furthermore, the invention is based on the object of creating a device that enables the aforementioned method to be carried out. Finally, the invention is based on the object of creating a computer program that enables the aforementioned method to be carried out. The technical object is achieved by the present invention by a method for determining the internal resistance and / or the open-circuit voltage curve and / or the capacity of a rechargeable battery with the following steps.A dynamic, voltage-controlled, mathematical battery model is created, where the internal resistance ^^ and / or the open circuit voltage curve ^^. 0 and / or the capacity C given initial values ​​^^ 0 mod , ^^ mod , ^^ mod The initial values ​​can be chosen arbitrarily. Alternatively, known values ​​can be used if one or more of these parameter values ​​are known and should not be determined. The model describes the dependence of the current on the voltage, ie, it has an internal resistance ^^ mod Depending on the model complexity, the internal resistance results from a single model equation with a single parameter (e.g., Ohm's law) or a combination of model equations and multiple parameters. The model is voltage-controlled. Accordingly, the measured voltage is ^^ mess the input variable and the predicted current ^^ modthe output variable. According to one embodiment of the invention, the dynamic mathematical battery model may consist of or be developed from a system of equations that includes, but is not limited to, the following equations: d DOD d ^^ = 1 ^ ^ ∙ ( ^^ 0 (DOD) − ^^ mess ) s ^^ where the model has three parameters serial resistance ^^ s , battery capacity ^^ and open circuit voltage curve ^^ 0 (DOD). The depth of discharge (DOD) takes values ​​between 0 and 1, where DOD = 0 represents a fully charged battery and DOD = 1 represents a fully discharged battery. This system of equations allows the calculation of the output variable ^^ mod based on the input variable ^^ mess. This is a voltage-controlled model (voltage as input variable). Alternatively, more complex models can be used for the new method, e.g. extended equivalent circuit models. Alternatively, the model equations and model parameters can also be specified as a function of the state of charge (SOC), where SOC = 1 - DOD, SOC = 1 is a fully charged battery and SOC = 0 is a fully discharged battery, or as a function of another related battery property. Measured values ​​for the battery current ^^mess( ^^) and the battery voltage ^^mess( ^^) of the rechargeable battery are recorded as a function of time over a predetermined period of time ^^. For some embodiments of the invention, measured values ​​of the battery temperature ϑ(t) are additionally recorded as a function of time over the period T.The type of charging and discharging (constant or varying current intensity, interruptions, temporary change in current direction) in the period T is fundamentally irrelevant for the method. The method of the present invention is therefore also applicable to measured values ​​from practical battery operation. The period ^^ preferably comprises at least one full cycle of the battery, i.e. a complete charge from approximately 0% state of charge to approximately 100% state of charge and a complete discharge from approximately 100% to approximately 0% state of charge. Alternatively, the period T can also comprise only one full charge cycle, i.e. a complete charge from approximately 0% state of charge to approximately 100% state of charge. Furthermore, the period ^^ can also comprise only partial cycles, under the condition that the current intensity is not zero over the entire period (no resting battery).The time period ^^ can also cover longer or shorter periods of time, whereby the general rule is that the longer ^^, the more accurate the determined values. The specified time period ^^ can also be determined during the measurement, for example by counting the cumulative charge throughput and counting the achievement of a specified charge throughput as reaching a specified time period ^^. The specified charge throughput can, for example, correspond to the equivalent charge throughput of a full cycle, in this case the time period T corresponds to a so-called equivalent full cycle. For an equivalent full cycle, it is irrelevant between which charge states or with which cycle depth the battery is operated, only the cumulative charge throughput is relevant. Measured values ​​for the battery voltage ^^mess( ^^) are used as input variables for the dynamic, voltage-controlled, mathematical battery model, and values ​​for a simulated current ^^.mod ( ^^) are calculated as the output of the battery model. Typically, both the input values, which are the measured values ​​for the battery voltage, and the output values ​​of the battery model, which are the simulated values ​​for the current, take the form of a set of time-discrete measured or output values. This means that the input values ​​are a time-discrete series of measurements of the battery voltage, and the output values ​​are a time-discrete series of values ​​of the simulated current. Using the values ​​for the simulated current ^^mod( ^^) and the recorded measured values ​​for the current ^^mess( ^^), values ​​for the internal resistance ^^ are calculated using a predefined calculation rule. cal and / or the open circuit voltage curve ^^ 0cal and / or the capacitance Ccal. A separate calculation rule is used for each of the determined quantities. The values ​​for the simulated current ^^mod( ^^) and the recorded measured values ​​for the current ^^mess( ^^) are used in such a way that a deviation of the respective corresponding values ​​from each other is used in the calculation rule. The deviation can preferably be a difference between the values ​​for the simulated current ^^ mod ( ^^) and the recorded measured values ​​for the current ^^mess( ^^) or a quotient of these. The calculation rules only require the simulated current ^^mod( ^^) and the recorded measured values ​​for the current ^^ mess( ^^) for a precise determination of the values ​​for the internal resistance ^^cal and / or the open-circuit voltage curve ^^0cal and / or the capacity Ccal. The method of the present invention is thus not only feasible under laboratory conditions, but also during any everyday use of the battery. According to a further embodiment of the invention, the method can comprise at least two iteration steps, wherein each iteration step comprises the implementation of the complete method according to the first embodiment. That is, creating a dynamic, voltage-controlled, mathematical battery model, wherein for the internal resistance ^^ and / or the open-circuit voltage curve ^^ 0 and / or the capacity C given initial values ​​^^ 0 mod , ^^ mod , ^^ modused, collecting measured values ​​for the battery current ^^mess( ^^) and the battery voltage ^^mess( ^^) of the rechargeable battery over a specified period of time ^^, using the measured values ​​for the battery voltage ^^mess( ^^) as input for the dynamic, voltage-controlled, mathematical battery model, calculating values ​​for a simulated current ^^mod( ^^) as output of the battery model and determining calculated values ​​for the internal resistance ^^cal and / or the open circuit voltage curve ^^ 0 cal and / or the capacity C caleach using the values ​​for the simulated current ^^mod( ^^) and the recorded measured values ​​for the current ^^mess( ^^), each with a predefined calculation rule. Preferably, in each iteration step except the first, i.e., during each complete execution of all steps of the method, when creating a dynamic, voltage-controlled, mathematical battery model for the specified initial values ​​^^ mod , ^^ 0 mod and C mod for the internal resistance ^^and / or the open circuit voltage curve ^^ 0 and / or the capacitance C the calculated values ​​for the internal resistance ^^cal and / or the open circuit voltage curve ^^ determined in the previous iteration step 0 cal and / or the capacity ^^ calused. Such an adaptation of the initial conditions for the subsequent iteration step is also called a "model update." A measurement period T is associated with an iteration step. Alternatively, already known values ​​can be used as initial values, or only individual values ​​can be updated. According to a further embodiment of the invention, the method can comprise at least two iteration steps, wherein each iteration step comprises the implementation of steps (a), (c) and (d) of the method according to claim 1. In each iteration step, except for the first, at the location of the initial values ​​specified in step (a), ^^ mod , ^^ 0 mod and C mod for the internal resistance ^^ and / or the open circuit voltage curve ^^ 0 and / or the capacitance C the calculated values ​​for the internal resistance ^^cal and / or the open circuit voltage curve ^^ determined in step (d) of the previous iteration step 0 caland / or the capacity ^^ calused. In this embodiment, a data set of measured values ​​is repeatedly evaluated without re-measuring. This allows available data sets measured in the past to be evaluated without a physically present battery. The iterations are repeated until the determined values ​​converge. Convergence is achieved, for example, when the determined values ​​from an iteration step differ by less than a predetermined percentage, e.g., 1%, from the determined values ​​from the previous iteration step. According to a further embodiment of the invention, the methods are combined such that measured values ​​are acquired over a period T1, then evaluated with several, but at least two, iteration steps until the desired values ​​converge, and this is repeated with a further period T2. The second period can directly follow the first. The second period can also have a time interval, e.g.,one day. In this case, a battery would be measured once a day, and the data set would be evaluated iteratively several times. This would monitor the battery's aging state. According to a further embodiment of the invention, the method can be implemented such that, using a deviation between the values ​​for the simulated current ^^mod( ^^) and the measured values ​​for the battery current ^^mess( ^^), deviations Δ ^^, Δ ^^ are calculated. 0 , ΔC between the given initial values ​​^^mod, ^^ 0 mod , C mod and the respective calculated values ​​^^ cal , ^^ 0 cal , C cal From the deviations Δ ^^, Δ ^^ 0 , ΔC and the given initial values ​​^^mod, ^^ 0 mod, Cmod for internal resistance ^^ and / or open circuit voltage curve ^^ 0 and / or capacity C, the calculated values ​​for internal resistance ^^ caland / or open circuit voltage curve ^^ 0 cal and / or capacity Ccal. The deviations Δ ^^, Δ ^^ 0 , ΔC between the given initial values ​​^^mod, ^^ 0 mod, Cmod and the calculated values ​​^^ cal , ^^ 0 cal , C cal The deviation between the simulated current values ​​^^mod( ^^) and the measured battery current values ​​^^mess( ^^) can be differences. Quotients or other methods of calculating the deviations are also possible. The use of the deviation between the simulated current values ​​^^mod( ^^) and the measured battery current values ​​^^ mess ( ^^) to determine the deviations Δ ^^, Δ ^^ 0 , ΔC between the given initial values ​​^^mod, ^^ 0 mod, Cmod and the calculated values ​​^^cal, ^^ 0 cal, Ccal allows the determination of the calculated values ​​^^ cal , ^^ 0cal , C cal with completely arbitrary initial values ​​^^ mod , ^^ 0 mod , C mod This allows for very simple application of the method, even without any prior knowledge of the battery's parameters. If certain values, such as the battery's capacity, open-circuit voltage curve, or internal resistance, are known, these can be used as initial values ​​^^mod, ^^ 0 mod, Cmod can be used to accelerate the determination of the remaining values. According to a further embodiment of the invention, the following calculation rule can be used to determine the internal resistance ^^: , where Δ ^^ is the difference between the internal resistance of the battery model ^^ mod and the calculated value for the internal resistance ^^ cal , ^^ = d ^^ 0 / dDOD the slope of the open circuit voltage curve, ^^ mod the current of the battery model and ^^ messthe measured current of the battery. According to a further embodiment of the invention, for the determination of the open circuit voltage curve ^^ 0 the following calculation rule can be used: , where Δ ^^ 0 the difference between the open circuit voltage curve of the battery model ^^ m 0 O d and the calculated value for the open circuit voltage curve ^^ c 0 a l , ^^ the slope of the open circuit voltage curve, ^^ mod the current of the battery model and ^^ mess is the measured current of the battery. According to a further embodiment of the invention, the following calculation rule can be used to determine the capacity C: ^ ^ ^ ^ , where Δ ^^ is the quotient of the capacity of the battery model ^^ mod and the calculated value for the battery capacity ^^ cal , ^^ modthe current of the battery model and ^^ mess the detected current of the battery. According to a further embodiment of the invention, for the simultaneous determination of the internal resistance ^^ and the open circuit voltage curve ^^ 0 the following calculation rule can be used: with Δ ^^ tot = Δ ^^ 0 − Δ ^^ ∙ ^^ mess , where Δ ^^ tot the total voltage difference, Δ ^^ 0 the difference between the open circuit voltage curve of the battery model ^^ m 0 O d and the calculated value for the open circuit voltage curve ^^ c 0 a l , Δ ^^ the difference between the internal resistance of the battery model ^^ mod and the calculated value for the internal resistance ^^ cal , ^^ the slope of the open circuit voltage curve, ^^ mod the current of the battery model and ^^ messthe measured current of the battery. The voltage difference due to the internal resistance Δ ^^ and the voltage difference due to the open circuit voltage curve Δ ^^ 0 combine to form a total difference Δ ^^ tot (the subscript "tot" stands for total). With this method, the determination of the internal resistance and the open-circuit voltage curve run simultaneously, so that only a few cycles are necessary to determine both parameters very accurately. According to a further embodiment of the invention, for the calculation of the time-discrete values ​​of the difference in the internal resistance Δ ^^ n and / or the time-discrete values ​​of the difference of the open-circuit voltage curve Δ ^^ ^^ 0and / or for the discrete values ​​of the total difference ^^ ^^^^ ^^ ^^, ^^a numerical solution method is used. Here, n stands as an index for the time-discrete values ​​of the measured quantities ^^mess( ^^) and Vmess( ^^) at specific discrete points in time ^^ ^^. Preferably, an implicit Euler method is used to solve the equations. However, the method is not limited to a solution using the implicit Euler method, but can also be solved using other numerical solution methods. The calculation of the discrete values ​​of the difference in internal resistance Δ ^^n is carried out in this embodiment according to: ^ , where ^^ ^^ the value of the slope of the open circuit voltage curve at time ^^, ^^ ^^−1 the value of the slope of the open circuit voltage curve at time ^^ − 1, ^^ mess, ^^ the recorded value of the battery current at time ^^, ^^mess, ^^−1the recorded value of the battery current at time ^^ − 1, ^^mod, ^^ the current of the battery model at time ^^, Δ ^^ the time interval between two consecutive measurements and Δ ^^ ^^−1 denotes the discrete value of the difference in internal resistance at time ^^ − 1. The calculation of the discrete values ​​of the difference in the open circuit voltage curve (Δ ^^ ^^ 0 ) is carried out in this embodiment according to: ^ ^ , where Δ ^^ ^^ 0 the discrete value of the difference of the open circuit voltage curve at time n ^ 0 ^ −1 denotes the discrete value of the difference in the open-circuit voltage curve at time n-1. The calculation of the discrete values ​​of the total difference (^^ ^^^^ ^^ ^^, ^^) is carried out in this embodiment as follows: ^ ^ , where ^^ ^^^^ ^^ ^^, ^^denotes the discrete value of the difference of the total difference at time ^^ and Δ ^^^^ ^^ ^^, ^^−1denotes the discrete value of the difference of the total difference at time ^^ − 1. According to a further embodiment of the invention, the following calculation rule can be used to calculate the capacity with time-discretely recorded values ​​of the battery current: ^ ^ , Δ ^^ where ^^ is the number of time steps, Δ ^^ is the time between two consecutive measurements, Δ ^^ is the quotient of the capacity of the battery model ^^ mod and the calculated value for the battery capacity ^^ cal , ^^ mess, ^^ the recorded value of the battery current at time ^^ and ^^ mod, ^^ denotes the simulated current of the battery model at time ^^. According to a further embodiment of the invention, at least two time-discrete values ​​of the difference in the internal resistance Δ ^^^^ to an average value of the difference in internal resistance ത Δ തത ^ ത ^ and / or at least two time-discrete values ​​of the difference of the open-circuit voltage curve Δ ^^ ^^ 0 to an average value of the difference of the open circuit voltage curve ത Δ ത ^ ത ^ ത0ത averaged over a measurement period ^^, where in the measurement period ^^ n discrete measurements of current ^^ mess and excitement ^^ mess According to a further embodiment of the invention, the internal resistance is determined as a function of depth of discharge and / or current and / or temperature ^^(DOD, ^^, ^^), wherein several values, e.g. mean values, of the difference of the internal resistance ത Δ തത ^ ത ^ over the measurement period ^^ are used for the determination, and the mean values ​​of the difference in internal resistance ത Δ തത ^ ത^ be determined by measuring sectionally over different ranges of depth of discharge DOD and / or current ^^ mess and / or temperature ^^. This allows the determination of a state-of-charge, current, and / or temperature-dependent internal resistance ^^(DOD, ^^, ^^). For this purpose, N measured values ​​are taken in the period T for the current ^^ mess, ^^ and the tension ^^ mess, ^^ of the battery. Each measured value is assigned the temperature ^^ and depth of discharge DOD of the battery corresponding to that time. From the measured values, time-discrete values ​​of the difference in internal resistance Δ ^^n are calculated with the same assignment. This means, for example, a time-discrete value of the difference in internal resistance Δ ^^ n the same assigned value for the temperature ^^ and depth of discharge DOD of the battery as the measured values ​​for the current ^^ mess, ^^ and the tension ^^ mess, ^^The time-discrete values ​​of the difference in internal resistance Δ ^^n can be averaged in so-called bins for identical temperatures (e.g., in specified steps of 1 K), for identical depths of discharge (e.g., in specified steps of 1%), or for identical currents (e.g., in specified steps of 1% of a specified nominal current, as specified, for example, in a data sheet). These averaged values ​​are used to determine a state-of-charge, current-intensity, and / or temperature-dependent internal resistance ^^(DOD, ^^, ^^). This helps in identifying possible causes of faults or aging conditions in the battery. According to a further embodiment of the invention, the open-circuit voltage curve is plotted as a function of the depth of discharge and / or the temperature, ^^ 0 (DOD, ^^) was determined, whereby several average values ​​of the open circuit voltage curve ത Δ ത ^ ത ^ ത0തover the measurement period ^^ are used for the determination, and the mean values ​​of the open circuit voltage curve ത Δ ത ^ ത ^ ത0ത be determined by averaging sectionally over different ranges of the depth of discharge (DOD) and / or the temperature ^^. For this purpose, N measured values ​​are taken in the period T for the current ^^ mess, ^^ and the tension ^^ mess, ^^ of the battery. Each measured value is assigned the temperature ^^ and depth of discharge DOD of the battery at that time. From the measured values, time-discrete values ​​of the difference of the open-circuit voltage curve Δ ^^ are calculated. 0 n , calculated with the same assignment. That is, for example, a time-discrete value of the difference of the open-circuit voltage curve Δ ^^ 0 n the same assigned value for the temperature ^^ and depth of discharge DOD of the battery as the measured values ​​for the current ^^ mess, ^^ and the tension ^^ mess, ^^. The time-discrete values ​​of the difference of the open-circuit voltage curve Δ ^^ 0 n can be averaged in so-called bins for equal temperatures (e.g., in 1-°C increments) or depths of discharge (e.g., in 1-% increments). These averaged values ​​are used to create a state-of-charge and temperature-dependent open-circuit voltage curve ^^ 0 (DOD, ^^) is determined. This helps in the search for possible causes of faults or aging of the battery components. Furthermore, the technical object of the present invention is achieved by a device for determining the internal resistance and / or the open-circuit voltage curve and / or the capacity of a rechargeable battery with a detection device for detecting measured values ​​for the battery current ^^mess( ^^) and the battery voltage ^^ mess( ^^) and, for some embodiments of the invention, for the battery temperature ^^( ^^) of the rechargeable battery, preferably at equidistant time intervals Δt or at predetermined times, and an evaluation and control device to which the recorded measured values ​​can be fed. The device is characterized in that the evaluation and control device is designed to carry out the described method for determining the internal resistance and / or the open circuit voltage curve and / or the capacity of a rechargeable battery. The evaluation and control device can in particular be integrated into the battery management system (BMS) used today in many battery systems, which makes this information available to the user, for example by means of a display.Furthermore, the technical object of the present invention is achieved by a computer program for determining the internal resistance and / or the open-circuit voltage curve and / or the capacity of a rechargeable battery. The computer program is designed such that when the computer program is executed in the evaluation and control device, the method for determining the internal resistance and / or the open-circuit voltage curve and / or the capacity of a rechargeable battery is carried out. The invention is explained in more detail below with reference to exemplary embodiments illustrated in the drawing. The drawing shows: Fig. 1 a schematic diagram of the method for determining internal resistance ^^, open-circuit voltage curve ^^. 0and / or capacity C of a rechargeable battery; Fig. 2 shows a schematic block diagram of a battery operated under load with a device according to the invention for carrying out the method; Fig. 3a) a simple equivalent circuit model of a battery; Fig. 3b) a more complex equivalent circuit model of a battery; Fig. 4 shows an experimentally determined open-circuit voltage curve ^^ 0 (DOD) and its derivative d ^^ 0 / dDOD; Fig.5a) the measured voltage ^^ mess ( ^^), plotted against time for four consecutive full cycles, starting with a fully discharged battery; Fig.5b) the measured current ^^ mess ( ^^), plotted against time for four consecutive full cycles, starting with a fully discharged battery; Fig.6 the open circuit voltage curve ^^ 0 , the voltage of the real battery ^^ mess and the voltage of the battery model ^^ mod, plotted against the depth of discharge DOD; Fig.7 the results of the new method for determining the internal resistance ^^, exemplified by four consecutive experimental full cycles (T1 to T4); Fig.7a) the measured voltage ^^ mess as input variable; Fig.7b) the measured current ^^ mess and simulated current ^^ mod of the voltage-controlled model, plotted over time t; Fig. 7c) the difference between simulated and experimental resistance ΔR determined according to Eq. (17); Fig. 7d) the calculated internal resistance ^^cal of the battery according to Eq. (19) after each period T; the dashed line is the independently determined reference value for the internal resistance; Fig. 8 the demonstration of the new method for determining the internal resistance ^^, exemplified using experimental partial cycles (between 25% and 75% state of charge); Fig. 8a) the measured voltage ^^ mess , plotted against time t; Fig.8b) the measured current ^^mess , plotted against time t; Fig.8c) the calculated internal resistance ^^cal of the battery, starting from an arbitrary starting value (here 9 mΩ), over a total of 10 consecutive model updates; Fig.9 the demonstration of the new method for determining the internal resistance ^^, exemplified by experimental driving cycles in the "Worldwide Harmonised Light-Duty Vehicles Test Procedure" (WLTP) protocol; Fig.9a) the measured voltage ^^ mess , plotted against time t; Fig.9b) the measured, dynamically strongly varying current ^^ mess , plotted over time t; Fig.9c) the calculated internal resistance ^^cal of the battery, starting from an arbitrary starting value (here 9 mΩ), over a total of 40 consecutive model updates; Fig.10 the real open-circuit voltage curve ^^ e 0 xpof a lithium-ion battery cell, plotted as voltage versus depth of discharge DOD, and the initial assumption of the open circuit voltage curve in ; Fig.11 the results of the new method for determining the open circuit voltage curve ^^ 0 (DOD), exemplified by four consecutive experimental full cycles (T1 to T4); Fig.11a) the measured voltage ^^ mess , plotted against time t; Fig.11b) the measured current ^^ mess and the simulated current ^^ mod of the voltage-controlled model, plotted against time t; Fig.11c) the difference between simulated and experimental open-circuit voltage Δ ^^ determined according to Eq. (29) 0 ; Fig.11d) the open circuit voltage curve determined according to equation (30) ^^ c 0 a l(DOD) of the battery after each period ^^; the dashed line is the independently determined reference curve; Fig.12 shows the results of the new method for determining the capacity ^^, exemplified by four consecutive experimental full cycles ( ^^1 to ^^4); Fig.12a) the measured voltage ^^ mess , plotted against time t; Fig.12b) the measured current ^^ mess and the simulated current ^^ mod of the voltage-controlled model, plotted against time t; Fig.12c) the difference between simulated and experimental open-circuit voltage Δ ^^ determined according to Eq. (37); Fig.12d) the determined capacitance ^^ cal of the battery, starting from an arbitrary starting value of 30 Ah; the dashed line is the independently determined reference value; Fig. 13 shows the results of the new method for determining the capacity ^^, exemplified by experimental partial cycles; Fig. 13a) the measured voltage ^^ mess, plotted against time t for eight consecutive partial cycles; Fig.13b) the measured current ^^ mess and the simulated current ^^ mod of the voltage-controlled model, plotted against time t; Fig.13c) the determined capacity ^^ cal of the battery, starting from an arbitrary starting value (here ^^ mod = 2Ah); the dashed line is the independently determined reference value. Fig. 14 shows the results of the new method for determining capacity ^^, exemplified by experimental driving cycles in the "Worldwide Harmonised Light-Duty Vehicles Test Procedure" (WLTP) protocol; Fig. 14a) the measured voltage ^^ mess , plotted against time t for several discharges in the WLTP driving cycle and subsequent constant current charging; Fig.14b) the measured current ^^ mess and the simulated current ^^ mod of the voltage-controlled model, plotted against time t; Fig.14c) the determined capacity ^^ calof the battery, starting from an arbitrary starting value (here ^^ mod = 10 Ah); the dashed line is the independently determined reference value; Fig.15 shows the results of the new method for the simultaneous determination of the internal resistance ^^ and the open-circuit voltage curve ^^ 0 (DOD), exemplified by four consecutive experimental full cycles ( ^^1 to ^^4); Fig.15a) the measured voltage ^^ mess , plotted against time t for four consecutive full cycles; Fig.15b) the measured current ^^ mess and the simulated current ^^ mod of the stress-controlled model, plotted against time t; Fig.15c) the difference between simulated and experimental stress Δ ^^ determined according to Eq. (42) tot ; Fig.15d) the calculated internal resistance ^^ calof the battery according to Eq. (19) after each period T; the dashed line is the independently determined reference value for the internal resistance; Fig.15e) the open circuit voltage curve determined according to Eq. (30) ^^ c 0 a l (DOD) of the battery after each period ^^; the dashed line is the independently determined reference curve; Fig. 16 shows the results of the new method for the simultaneous determination of the internal resistance ^^ and the capacity ^^, exemplified by four consecutive experimental full cycles ( ^^1 to ^^4); Fig. 16a) the measured voltage ^^ mess , plotted against time t for four consecutive full cycles; Fig.16b) the measured current ^^ mess and the simulated current ^^ mod of the voltage-controlled model, plotted against time t; Fig.16c) the determined capacity ^^ calof the battery after each period T; the dashed line is the independently determined reference value for the capacity; Fig.16d) the calculated internal resistance ^^ cal of the battery according to Eq. (19) after each period T; the dashed line is the independently determined reference value for the internal resistance; Fig. 17 shows the results of the new method for the simultaneous determination of the internal resistance ^^ and the capacity ^^, exemplified by twelve consecutive experimental sub-cycles; Fig. 17a) the measured voltage ^^ mess , plotted against time t for twelve consecutive partial cycles; Fig.17b) the measured current ^^ mess and the simulated current ^^ mod of the voltage-controlled model, plotted against time t; Fig.17c) the determined capacity ^^ calof the battery after each period T; the dashed line is the independently determined reference value for the capacity; Fig. 17d) the calculated internal resistance ^^cal of the battery according to Eq. (19) after each period T; the dashed line is the independently determined reference value for the internal resistance; Fig. 18 the results of the new method for the simultaneous determination of capacity ^^ and open circuit voltage curve ^^ 0 , exemplified by an experimental full cycle; Fig.18a) the measured voltage ^^ mess , plotted against time t for one full cycle; Fig.18b) the measured current ^^ mess , plotted against time t; Fig.18c) the determined capacity ^^ cal of the battery, starting from an arbitrary starting value (here ^^ mod = 10Ah) over 9 consecutive model updates; the dashed line is the independently determined reference value for the capacity; Fig.18d) the determined open circuit voltage curve ^^ c0 a l (DOD) of the battery over 9 consecutive model updates; the dashed line is the independently determined reference curve; Fig. 19 Results of the new method for the simultaneous determination of capacity ^^, internal resistance ^^ and open circuit voltage curve ^^ 0 , exemplified by an experimental full cycle; Fig.19a) the measured voltage ^^ mess , plotted against time t for one full cycle; Fig.19b) the measured current ^^ mess , plotted against time t; Fig.19c) the determined capacity ^^ cal of the battery, starting from an arbitrary starting value (here ^^ mod = 10Ah) over 19 consecutive model updates; the dashed line is the independently determined reference value for the capacity; Fig.19d) the calculated internal resistance ^^ calof the battery, starting from an arbitrary starting value of 9 mΩ over 19 consecutive model updates; the dashed line is the independently determined reference value for the internal resistance; Fig.19e) the determined open circuit voltage curve ^^ c 0 a l (DOD) of the battery over 19 consecutive model updates; the dashed line is the independently determined reference curve. Fig. 1 shows a schematic representation of the method. It shows the rechargeable battery 106, the first and second components of the overall algorithm 102, 104, and the overall algorithm 100 itself. Recorded voltage measurements ^^ mess (t) of the rechargeable battery 106 are transferred to the first part of the overall algorithm 102. The first part of the overall algorithm 102 comprises the voltage-controlled battery model. In the battery model, arbitrarily assumed initial values ​​for the quantities to be determined, internal resistance ^^ mod and / or the open circuit voltage curve ^^0 mod and / or the capacity ^^ mod used. From the measured voltage ^^ mess (t) and the arbitrary quantities, values ​​for a simulated current ^^mod( ^^) are calculated as the output of the battery model. The values ​​of the simulated current ^^mod( ^^) and recorded measured values ​​for the battery current ^^ mess ( ^^) of the rechargeable battery 106 are transferred to the second part of the overall algorithm 104. Using a given calculation rule, this part calculates values ​​for the internal resistance ^^cal and / or the open circuit voltage curve ^^ 0 cal and / or the capacity Ccal using the values ​​for the simulated current ^^ mod ( ^^) and the recorded measured values ​​for the current ^^mess( ^^) are determined using a predefined calculation rule. The determined values ​​^^cal, ^^ 0cal and Ccal are transferred as an update to the voltage-controlled model and replace the assumed values ​​^^ mod , ^^ 0 mod and ^^ mod . The determined values ​​^^cal, ^^ 0cal and Ccal are output as the result of the overall algorithm 100. As shown in Fig. 2, 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 an evaluation and control unit 120 for carrying out the method. The evaluation and control unit 120 comprises a unit 110 for measuring the battery voltage Umess, which is connected to the connections (poles) of the rechargeable battery 106. Furthermore, the evaluation and control unit 120 comprises a unit 112 for measuring the battery current Imess, which can be designed in any desired manner. For example, the unit 112 can comprise a shunt resistor located 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 of the shunt resistor. In a further embodiment of the invention, the evaluation and control unit 120 can also comprise a device for detecting the temperature of the battery (not shown). The evaluation and control unit 120 can also comprise a display unit 116 on which the determined values ​​are displayed. The evaluation and control unit 120 comprises a computing unit 118, which can be designed, for example, as a microprocessor unit, to carry out the calculations required to implement the method. The microprocessor unit can also have an analog / digital converter which converts analog variables U supplied to it. mess and I messtemporally samples and converts them into digital values. The battery model used in the process must be able to predict the temporal course of the current for a given voltage curve. To do so, the model must have the following properties. The model describes the dependence of the voltage on the state of charge (SOC) or a related variable such as the depth of discharge (DOD), the remaining charge, or the remaining energy. A necessary model parameter for this is the capacity ^^ of the battery. Another necessary model parameter is the open-circuit voltage curve ^^ 0 (DOD). The model describes the dependence of voltage on current, ie it has an internal resistance ^^ modDepending on the model complexity, the internal resistance results from a single model equation with a single parameter (e.g., Ohm's law) or a combination of model equations and multiple parameters. The internal resistance could be determined by a pulse test applied to the model according to Equation (1). The model is voltage-controlled. Accordingly, the measured voltage is ^^ mess the input variable and the predicted current ^^ mod the output variable. There are many different modeling approaches that meet these requirements, e.g., equivalent circuit models or physical-chemical models. A simple equivalent circuit model, sufficient for demonstrating the method, is shown in Fig. 3 a). It consists of a voltage source ^^ 0 and a serial resistor ^^ s This model is mathematically described by a differential-algebraic system of equations: (3) (4) The model has the three parameters serial resistance ^^ s , battery capacity ^^ and open circuit voltage curve ^^ 0 (DOD). The depth of discharge (DOD) takes values ​​between 0 and 1, where DOD = 0 represents a fully charged battery and DOD = 1 represents a fully discharged battery. The DOD is directly related to the state of charge (SOC): SOC = 1 − DOD . (5) The state of charge (SOC) is a commonly used parameter to indicate how full the battery is. The system of equations (3) and (4) allows the calculation of the output variable ^^ mod based on the input variable ^^ mess. This is a voltage-controlled model (voltage as the input variable). Other, more complex models are also suitable for use in the new method, e.g., extended equivalent circuit models as in Fig. 3b). By using prior knowledge about the battery in more complex models, e.g., the assumption of voltage hysteresis ^^ hys , the accuracy of the method can be increased. The equivalent circuit in Fig.3b) is an example of a model in which the internal resistance ^^ modfollows from several model elements, here from the Rs-(RC)1-(RC)2 chain. To demonstrate the present method, experiments were conducted with commercial lithium-ion pouch cells with a nominal voltage of 3.75 V and a nominal capacity of 20 Ah. The cells have a negative electrode made of graphite and a positive electrode made of a mixture of lithium nickel manganese cobalt oxide (NMC) and lithium manganese oxide (LMO). The cells were measured at an ambient temperature of 25 °C. Three different measurement protocols were carried out. The data from these measurements form the basis for all methods presented here. 1. Full cycles: CCCV discharge to 3.0 V, CCCV charge to 4.2 V, 1 C rate, C / 10 cut-off current, no pause) for several cycles, starting with a fully discharged battery. This measurement data is shown in Fig. 5. 2. Partial cycles: CC discharge and charge between 25% and 75% state of charge for several partial cycles 3.Driving cycles: Starting with a fully charged battery, a dynamic load profile was performed based on the Worldwide Harmonized Light-Duty Vehicles Test Procedure (WLTP). This profile contains rapidly successive discharge and charge phases resulting from the acceleration and braking processes of an electric vehicle. Furthermore, a quasi-OCV measurement was performed at a 0.05C rate. The resulting open-circuit voltage curve ^^. 0 (DOD) and its derivative d ^^ 0 / dDOD are shown in Fig. 4 and serve as a reference for the new method. The open-circuit voltage curve is shown. ^^ 0(DOD) is plotted in a dotted line as a voltage versus the depth of discharge DOD. The curve shows an almost linear discharge of the battery until shortly before complete discharge. The derivative of the open circuit voltage curve d ^^0 / dDOD is shown as a solid line and plotted as a voltage versus the depth of discharge DOD. The voltage for the derivative can be read off the axis on the right side of the diagram. The curve shows an almost constant course until shortly before complete discharge. The internal resistance was determined independently from the full cycles at 1C according to (6) with ^ ത ^ chg as average charging voltage between 25% and 75% SOC, ^ ത ^ disthe average discharge voltage between 75% and 25% SOC and ^^ = 20 A. A value of ^^ = 4.579 mΩ was determined. The battery capacity was also determined from the full cycles to be ^^ = 19.96 Ah, which corresponds almost exactly to the nominal capacity of 20 Ah. These values ​​for ^^ and ^^ also serve as a reference for the new method. The new method is applied below, in particular, to measured values ​​from four consecutive full cycles. These are shown in Fig. 5. The curve of the measured voltage ^^ can be seen. mess( ^^), plotted against time in Figure a), and the curve of the measured current ^^mess( ^^), plotted against time in Figure b). The four full charge cycles are clearly visible. The procedure began with a completely discharged battery. Furthermore, the procedure is also demonstrated using the partial cycles and the WLTP load profile. Determination of the internal resistance The real battery has a real internal resistance, which we denote by R. A representative value R is determined using the procedure. cal which is very close to the real internal resistance. The model has an assumed internal resistance, which we denote by ^^ mod We denote the difference as Δ ^^ with Δ ^^ = ^^ mod − ^^ cal . (7) Due to this difference, the voltage-controlled battery model will have a different current ^^ mod than the real battery. We denote the measured current of the real battery by ^^ mess. From the difference between ^^ mod and ^^ mess can therefore be inferred to Δ ^^. This relationship is derived below. The derivation is based on Fig. 6. This shows the voltage behavior during a battery discharge with a constant current. First, the open-circuit voltage curve ^^ is shown. 0 (DOD), here an example curve of a lithium-ion battery cell with a final charge voltage of 4.2 V and a final discharge voltage of 3.0 V. We assume that the real battery is at an arbitrary operating point, marked in the figure as "operation point exp", which corresponds to a certain depth of discharge DOD exp When discharging with the current ^^ mess is the voltage of the real battery ^^ mess (DOD exp ) due to the internal resistance R lower than the open circuit voltage, namely on the curve shown in Fig.6 ^^ mess(DOD). We assume that the internal resistance of the battery model is greater than that of the real battery, i.e. Δ ^^ > 0. The battery model therefore has the same depth of discharge DOD exp an even lower voltage ^^ mod (DOD exp ) on the curve shown in Fig.6 ^^ mod (DOD). We denote the voltage difference between the two curves as Δ ^^ R = ^^ mod (DOD exp ) − ^^ mess (DOD exp ) . In the example of Fig.6, Δ ^^ R < 0. We define the current as positive for battery discharge. According to Ohm's law, the relationship Δ ^^ follows. R = −Δ ^^ ∙ ^^ mess(8) The method presented here uses a voltage-controlled battery model. Therefore, the model has, by definition, the same voltage as the real battery at any given time. Shifting the characteristic curves by Δ ^^R relative to each other (for the same DODexp) results in the model having a different depth of discharge DODmod compared to the real battery (at the same voltage ^^ mess ). The model is therefore at the operating point marked in Fig.6, "operation point (mod)". We refer to the difference in the depths of discharge as ΔDOD with ΔDOD = DOD mod − DOD exp . (9) In our example, ΔDOD < 0. Fig.6 clearly shows that Δ ^^ R and ΔDOD form a slope triangle. The slope −Δ ^^ R / ΔDOD corresponds to the slope of the characteristic curve d ^^ / dDOD and, because this is shifted parallel to the open-circuit voltage, the slope of the open-circuit voltage curve d ^^ 0 / dDOD, which we will refer to as ^^ in the following: The negative sign is necessary because Δ ^^ < 0 and ΔDOD < 0, but also ^^ < 0. Substituting Eq. (8) into Eq. (10) provides a relationship between ΔDOD and the unknown quantity Δ ^^, ΔDOD =Δ ^^∙ ^^mess^^ . (11) We next develop an expression for ΔDOD. The depth of discharge changes over time due to an applied current. This can be described with a simple differential equation, which we apply to both the real battery and the model: d DOD exp d ^^ = ^^mess ^^ , (12) d DOD mod ^^mo d ^^ = d ^^ . (13) We subtract Eq. (12) from Eq. (13) and substitute Eq. (9) to This equation describes the temporal evolution of ΔDOD for a difference between simulated and experimental current. We insert Eq. (11) and obtain This equation describes the relationship between the desired quantity Δ ^^, the measured quantity ^^ mess, the output of the voltage-controlled model ^^ mod and the model parameters ^^ and ^^. To calculate Δ ^^, the time derivative of the left-hand side of Eq. (15) must be integrated. In practical battery operation, the measured quantity ^^ mess at certain discrete times ^^ ^^ Therefore, an implicit Euler method is suitable for the solution of Eq. (15): Here ^^ is the current time of the measurement and Δ ^^ = ^^ ^^ − ^^ ^^−1 the time interval from the previous measurement point. This equation can be solved for the desired value Δ ^^: This equation is the central result of this analysis. It allows the calculation of Δ ^^ from discrete time series of ^^ mess and ^^ mod . For each time step, a value of Δ ^^ is obtained. This can be averaged over several time steps ^^ if necessary according to Δ തതത ^ ത ^ = 1 ^^ ∑ ^ ^^ ^ =1Δ ^^ ^^ . (18) This averaging can also be performed over specific DOD sections or over sections of current or temperature, so that the DOD, current, or temperature dependence of Δ ^^ is obtained. The quantity to be determined for the internal resistance of the real battery ^^ cal results in a final step according to Eq. (7) to Up to this point, the derivation is completely independent of the type of battery model used. This only becomes relevant when calculating Equation (19). The internal resistance of the model required for this formula ^^ mod is calculated from the model parameters. For the simple equivalent circuit model in Fig.3a), ^^ mod = ^^ m . For the exemplary complex equivalent circuit model in Fig.3b) we get ^^ mod = ^^ s+ ^^1+ ^^2. To increase the accuracy of the method and / or to use the model for additional measurement data, the model parameters can be subsequently adjusted (“updated”). For the simple equivalent circuit model in Fig. 3a), this is done analogously to Eq. (19) according to For more complex models, the determined value ത Δ തത ^ ത ^ be distributed appropriately among the model parameters. For the exemplary complex equivalent circuit model in Fig.3b), for example, ത Δ തത ^ ത ^ 1 / 3 of each of the three parameters ^^ s, ^^1 and ^^2 are subtracted. The resulting equation (15) shown above was derived using Fig. 6 under the assumption of a constant current. This assumption was merely for plausibility reasons and is not a prerequisite for the present method. Battery operation with a current that varies arbitrarily over time (discharge or charge) can be conceptually divided into short sections of constant current - one such section is, for example, the distance Δ ^^ between two measuring points, as used in the discretized form Eq. (17). For infinitesimally short time periods, the gradient triangle shown in Fig. 6 changes from the difference quotient Δ ^^ / ΔDOD to the differential quotient d ^^ / dDOD. The resulting equation (15) is therefore exactly valid, independent of the dynamics and sign of the current. Using the derived equations, the internal resistance is determined in practice in the following steps.First, a voltage-controlled battery model with known and / or predetermined parameter values ​​for the capacity ^^ is created. mod and the provided. Arbitrary starting values ​​are provided for the parameter(s) related to the internal resistance ^^ of the model (e.g. ^^ ^^ for the simple equivalent circuit model in Fig.3a). The battery is charged over a period of time ^^ with measurement of the current ^^ mess and the tension ^^ mess operated. The simulated current ^^ mod over the period ^^ using the voltage-controlled model. This is followed by the calculation of Δ ^^ according to Eq. (17). The values ​​Δ ^^ ^^ are averaged over the period ^^ ത Δ തത ^ ത ^ averaged. Then the approximate value ^^ calfor the real internal resistance according to Eq. (19). Optionally, the procedure is repeated, whereby the one or more with the internal resistance ^^ mod related parameters in the battery model are set to the determined values ​​("model update"). This results in an iterative approximation of the model's internal resistance to the true internal resistance. The procedure is demonstrated below using the previously mentioned experimental data, specifically all three data sets (full cycles, partial cycles, driving cycles). The simple equivalent circuit model from Fig. 3a is used. The capacity is set to the reference value of ^^ = 19.96 Ah, and the open-circuit voltage curve is set to the reference curve shown in Fig. 4. The parameter for the serial resistance is set to an arbitrary starting value, here, for example, ^^ ^^= 9 mΩ. The results for experimental full cycles are shown in Fig. 7. The period ^^ is chosen as a charge / discharge cycle (approx. 2.1 h). Fig. 7a) shows the measured voltage ^^ mess as input for the voltage-controlled model. Fig.7b) shows both the measured current ^^ mess as well as the simulated current ^^ mod from the voltage-controlled model. During the period ^^1 (between 0 and 2 h) there is a significant deviation ^^ mod − ^^ mess of the two curves. Fig. 7c) shows the difference between simulated and experimental resistance Δ ^^ determined according to Eq. (17) using the data shown in Fig. 7b). The value Δ ^^ varies over time. In the first cycle (between 0 and 2 h) it assumes values ​​around 4 mΩ, with clear peaks particularly at the end of the charge and discharge. The value averaged over the first cycle duration ^^1 is ത Δ തത ^ ത^ = 4.32 mΩ. From this, the calculated internal resistance ^^ is obtained according to Eq. (19). cal of the battery to ^^cal = ^^mod − Δ തതത ^ ത ^ = ^^s − Δ തതത ^ ത ^ = 9 mΩ − 4.32 mΩ = 4.68 mΩ . This value is very close to the reference value of 4.58 mΩ. Thus, the internal resistance can be determined using the new method after just the first full cycle. This successfully demonstrates the method. The model's series resistance is now set to the new value according to Equation (20), ^^ s,neu = ^^ s − ത Δ തത ^ ത^, before continuing with the second iteration step in period ^^2. The procedure is analogous to ^^2, ^^3, and ^^4. The determined internal resistances are shown in Fig. 7d), starting from the assumed starting value. The procedure stabilizes near the reference value. At the same time, the prediction quality of the voltage-controlled model improves (cf. Fig. 7b) for periods > 2 h) and thus Δ ^^ becomes smaller (cf. Fig. 7c) for periods > 2 h). These results use full cycles. To demonstrate the flexibility of the method, it was further applied to partial cycles (25% to 75% state of charge) and to driving cycles (load on the battery in the electric vehicle). The results are shown in Fig. 8 (partial cycles) and Fig. 9 (driving cycles). The experimental data sets consist of 2-3 hours of battery operation each. We follow the described procedure. The starting value for the series resistance is ^^ s= 9 mΩ. For the period ^^, we choose the duration of the entire data set (2.1 h for the partial cycles, 3.2 h for the driving cycles). According to Equation (19), we obtain a new value for ^^ cal . We then update the model according to Eq. (20) and repeat this with the same experimental data from the period ^^, until ^^ cal converged to a constant value. For the partial cycles, this is the case after approximately 5 updates, for the driving cycles after approximately 25. The converged values ​​are at ^^ cal = 4.20 mΩ (partial cycles, Fig.8d) or ^^ cal= 4.14 mΩ (driving cycles, Fig. 9d), these values ​​are close to the reference value of ^^ = 4.58 mΩ. Using each of the data sets shown, the new method for determining the internal resistance of a rechargeable battery was successfully demonstrated and its high flexibility with regard to input data was demonstrated. Determination of the open-circuit voltage curve. The real battery has a real open-circuit voltage curve, which we denote by ^^ 0 (DOD). The procedure uses a value ^^ c 0 a l which is very close to the real internal resistance. The model has an assumed open-circuit voltage curve, which we denote by ^^ m 0 O d We denote the difference as Δ ^^ 0 with All three parameters Δ ^^ 0 , ^^ m 0 O d and ^^ c 0 a ldepend on the depth of discharge (DOD). Due to this difference, the voltage-controlled battery model will generally have a different current ^^ mod than the real battery ^^ mess . From the difference between ^^ mod and ^^ mess can therefore be set to Δ ^^ 0 This relationship is derived below. Fig. 10 shows two open-circuit voltage curves. As a real curve ^^ e 0 xp (DOD) is shown as an example of a lithium-ion battery cell with a final charge voltage of 4.2 V and a final discharge voltage of 3.0 V. For the a linear relationship between the final voltages is assumed. For a given operating point DOD exp (in Fig.10 marked as “operation point exp.”) this leads to the difference Δ ^^ 0 ; in the example of Fig.10, Δ ^^ 0< 0. The method presented here uses a voltage-controlled battery model. Therefore, by definition, the model has the same voltage as the real battery at any given time. Shifting the characteristic curves by Δ ^^0 against each other (for the same DODexp) results in the model having a different depth of discharge DODmod compared to the real battery (at the same voltage ^^ mess ). The model is therefore at the operating point "operation point mod" marked in Fig. 10. We refer to the difference in the depths of discharge as ΔDOD with ΔDOD = DOD mod − DOD mess . (22) In our example, ΔDOD < 0. Fig.10 clearly shows that Δ ^^ 0 and ΔDOD form a slope triangle. The slope −Δ ^^ 0 / ΔDOD corresponds to the slope of the characteristic curve d ^^ m 0 O d / dDOD, which we call ^^: Next, we develop an expression for ΔDOD. The depth of discharge changes over time due to an applied current. This can be described with a simple differential equation, which we apply to both the real battery and the model: dDOD exp = ^^mess d ^^ ^^ , (24) d DOD mod d ^^ = ^^mod ^^ . (25) We subtract Eq. (24) from Eq. (25) and substitute Eq. (22) to get d(ΔDOD) ^^mo − ^^ d ^^ = d mess ^^ . (26) This equation describes the temporal evolution of ΔDOD for a difference between simulated and experimental current. We insert Eq. (23) and obtain This equation describes the relationship between the desired quantity Δ ^^ 0 , the measured value ^^ mess , the output of the voltage-controlled model ^^ modand the model parameters ^^ and ^^. The solution requires integrating the time derivative of the left-hand side of Eq. (27). In practical battery operation, the measured quantity ^^ mess at certain discrete times ^^ ^^ Therefore, an implicit Euler method is suitable for the solution of Eq. (27): Here ^^ is the current time of measurement and Δ ^^ = ^^ ^^ − ^^ ^^−1 the time interval to the previous measurement point. This equation can be expanded to the desired value Δ ^^ 0 be dissolved: This equation is the central result of this analysis. It allows the calculation of Δ ^^ 0 from discrete time series of ^^ mess and ^^ mod For each time step, a value of Δ ^^ 0 obtained. Since Δ ^^ 0depends on DOD, averages must be calculated section by section (e.g. every 1-DOD percentage point). The open circuit voltage curve of the real battery to be determined is obtained in a final step according to Eq. (21) as where ^^ m 0 O d (DOD) is the parameter used in the model. To increase the accuracy of the method and / or to use the model for additional measurement data, the model parameter can be subsequently adjusted ("updated") according to ^^ n 0 eu = ^^ m 0 O d − Δ ^^ 0 . (31) The determination of the open-circuit voltage curve is carried out in practice in the following steps. First, a voltage-controlled battery model with known parameter values ​​for the capacity ^^ mod and for the one with the internal resistance ^^ mod related parameters (e.g. ^^ ^^for the simple equivalent circuit model in Fig.3a). An arbitrary starting value is used for the course of the open-circuit voltage curve ^^ m 0 O d ( ^^ ^^ ^^) is assumed, preferably a linear curve between the final charge and discharge voltage. The battery is charged over a period of time ^^ with measurement of current ^^ mess and excitement ^^ mess operated. The simulated current ^^ mod over the period ^^ using the voltage-controlled model. This is followed by the calculation of Δ ^^ 0 according to Eq. (29). The values ​​Δ ^^ ^^ 0 are calculated section by section for DOD areas to the mean ത Δ ത ^ ത ^ ത0ത(DOD) is averaged over the period ^^. The approximate value for the real open-circuit voltage curve is then calculated according to Equation (30). Optionally, the procedure can be repeated, whereby the parameter for the open-circuit voltage curve in the battery model is set to the determined value ("model update"). This results in an iterative approximation of the model's open-circuit voltage curve to the real open-circuit voltage curve. The averaging of Δ ^^ ^^ 0 can also be carried out section by section for different ranges of measured temperatures. This allows a temperature-dependent open-circuit voltage curve to be obtained. 0 (DOD, ^^) can be determined. The open circuit voltage curve ^^ 0depends on the depth of discharge. For complete recording, the period ^^ must therefore be selected so that the battery has cycled through all states of charge between 0% and 100% at least once. If only a partial range is cycled through within ^^, the open-circuit voltage curve can only be determined in this partial range. The described procedure is demonstrated below using the previously mentioned experimental data, specifically the full cycles (Fig. 5). The simple equivalent circuit model of Fig. 3a) is used. The capacity is set to the reference value of ^^ = 19.96 Ah, the series resistance to the reference value of ^^ ^^ = 4.579 mΩ. The open circuit voltage curve ^^ 0(DOD) is set to an arbitrary starting value, namely a linear progression between the final voltages. A charge / discharge cycle (approximately 2.1 h) is chosen as the period ^^. The results are shown in Fig. 11. Fig. 11a) shows the measured voltage ^^ mess as input for the voltage-controlled model, plotted over time. Fig.11b) shows both the measured current ^^ mess as well as the simulated current ^^ mod from the voltage-controlled model, plotted over time. During the period ^^1 (between 0 and 2 h) a significant deviation ^^ is observed. mod − ^^ mess of the two curves, which becomes smaller and smaller in the following time periods. Fig.11c) shows the difference between simulated and experimental open-circuit voltage Δ ^^ determined according to Eq. (29) 0 using the data shown in Fig. 11b). The value Δ ^^ 0varies over time during the period ^^1 (between 0 and 2 h): The values ​​are symmetrical with respect to charge and discharge and exhibit fluctuations down to -0.35 V. Over the first cycle duration ^^1, these values ​​are averaged section by section for each DOD percentage point. From this, the approximate value for the real open-circuit voltage curve ^^ is calculated according to Equation (30). 0(DOD). Fig. 11d) shows the initially assumed linear curve as a thick solid line and the curve determined according to ^^1 as a thin solid line. This is already close to the reference curve, which is also shown as a dashed line in Fig. 11d). After just two full cycles, the open-circuit voltage curve can be determined using the new method. This successfully demonstrates the method. After period ^^1, the model is updated with the determined curve before continuing with the second cycle in period ^^2. The procedure is analogous to ^^2, ^^3, and ^^4. The determined curves are also shown in Fig. 11d). The method stabilizes close to the reference curve. At the same time, the prediction quality of the voltage-controlled model improves (cf. Fig. 11b) for periods > 2 h) and thus Δ ^^ 0smaller (see Fig. 11c) for periods > 2 h). With these results, the new method for determining the open-circuit voltage curve of a rechargeable battery was successfully demonstrated. Determination of capacity The real battery has a real capacity, which we denote by ^^. A value ^^ is determined as a representative value using the method. cal which is very close to the real capacity. The model has an assumed capacity, which we call ^^ mod We denote the difference as Δ ^^ with Since the capacity assumed in the model usually does not correspond to the real capacity, the voltage-controlled battery model will generally have a different current ^^ mod than the real battery ^^ mess . From the difference between ^^ mod and ^^ messcan therefore be inferred to Δ ^^. This relationship is derived below. The battery is operated for a period of time ^^. The amount of charge ^^ cal results from integration according to We choose the current value to be independent of the type of operation (charging, discharging, or a combination of both)—only the absolute amount of charge passed through is relevant. The voltage-controlled model is subjected to the experimentally measured voltage over the same period. The amount of charge passed through the model is ^^ mod is obtained analogously by integration according to The quotient of ^^ mod and ^^ cal corresponds to the quotient of ^^ mod and ^^ cal , i.e. The combination of equations (32) to (35) gives This equation describes the relationship between the desired quantity Δ ^^, the measured quantity ^^ messand the output of the voltage-controlled model ^^ mod For practical application, the integrals in Eq. (36) must be calculated. In practical battery operation, the measured quantity ^^ mess at certain discrete times ^^ ^^ This gives Eq. (32) as with ^^ as the number of measurement points in the period ^^ and Δ ^^ as the time step size. This equation is the central result of this analysis. It allows the determination of Δ ^^ from discrete time series of ^^ mess and ^^ mod The capacity of the real battery to be determined is determined in a final step according to Eq. (32) as To increase the accuracy of the method and / or to use the model for additional measurement data, the model parameters can be subsequently adjusted (“updated”). For the simple equivalent circuit model in Fig. 3a), this is done analogously to Eq. (38) according to In practice, the capacity is determined in the following steps. First, a voltage-controlled battery model with known parameter values ​​for the internal resistance ^^ mod related parameters (e.g. ^^ ^^ for the simple equivalent circuit model in Fig.3a) and for the open circuit voltage curve ^^ m 0 O d (DOD). An arbitrary starting value is used for the capacity ^^ mod The battery is charged over a period of time ^^ with measurement of current ^^ mess and excitement ^^ mess operated. The simulated current ^^ modover the period ^^ using the voltage-controlled model. This is followed by the calculation of Δ ^^ according to Eq. (37). Then, the approximate value for the real capacity is calculated according to Eq. (38). Optionally, the procedure can be repeated, whereby the parameter for the capacity in the battery model is set to the determined value ("model update"). This results in an iterative approximation to the real value of the capacity. In the following, the described procedure is demonstrated using the experimental data already mentioned, specifically using all three data sets (full cycles, partial cycles, driving cycles). The simple equivalent circuit model from Fig. 3a) is used. The series resistance is set to the reference value of ^^ ^^ = 4.579 mΩ, and the open-circuit voltage curve is set to the reference curve shown in Fig. 4. Any starting values ​​for the capacity are chosen; here, as an example, different values ​​are used for the three data sets examined.Results for full cycles are shown in Fig. 12. ^^ = 30 Ah is used as the starting value for the capacity assumed in the model. A charge / discharge cycle (approximately 2.1 h) is chosen as the period ^^, and the algorithm is applied after four periods ^^1 to ^^4 as an example of continuous use of the experimental time series. Fig. 12a) shows the experimentally measured voltage. This data serves as the input for the voltage-controlled model. Fig. 12b) shows the measured current ^^. mess and the current simulated with the model ^^ modIn period ^^1 (between 0 and 2 h), this deviates significantly from the experimental measured value. This is a sign that the capacity assumed in the model of ^^ = 30 Ah is incorrect. Fig. 12d) shows the capacity values ​​determined using the method, starting from the assumed initial capacity, here as a function of the updates performed. After a period ^^1 of approximately 2 hours, the capacity was determined for the first time; the value is already very close to the reference value. In the second period ^^2, the deviation between simulated and experimental current shown in Fig. 12b) is further reduced, while at the same time the quotient Δ ^^ in Fig. 12c) approaches one. At the end of the data set, after a good eight hours of measurement with four full cycles and four updates, the reference value is reached. Fig. 13 shows results using the same procedure, but based on experimental partial cycles; here the battery was cycled between 25% and 75% state of charge.Equal time periods ^^1 to ^^4 of approximately 2 hours each are selected, corresponding to 2 partial cycles. ^^ = 2 Ah is used as the starting value for the capacity assumed in the model. Here, too, the reference value for the capacity is reached at the end of the eight-hour measurement series. Fig. 14 shows results for experimental driving cycles. The entire data set shown, measuring just over eight hours, is used as the time period ^^. ^^ = 10 Ah is used as the starting value for the capacity assumed in the model. The algorithm is applied to this data several times, and the model parameter is updated each time. The reference value is reached after four such updates. Using each of the data sets shown, the new method for determining the capacity of a rechargeable battery was successfully demonstrated, and its high flexibility with regard to input data and starting values ​​was shown.Simultaneous determination of internal resistance and open-circuit voltage curve. The internal resistance and open-circuit voltage curve can be determined simultaneously. To do this, we combine the approaches for determining the internal resistance and the open-circuit voltage curve. The voltage difference due to the internal resistance Δ ^^. R according to equation (8) (Fig.6) and the voltage difference due to the open circuit voltage curve Δ ^^ 0 according to Eq. (21) (Fig.10) combine to form a total difference Δ ^^ tot (the index “tot” for total) according to Δ ^^ tot = Δ ^^ 0 − Δ ^^ ∙ ^^ mess (40) Analogous to equations (15) and (27), the following expression can be derived: The discretization gives ^ ^ ^^ . (42) This equation allows the calculation of Δ ^^ tot from discrete time series of ^^ mess and ^^ mod For each time step, a value of Δ ^^ totUsing Eq. (40) Δ ^^ can be calculated in a subsequent step. 0 and Δ ^^ are calculated. For this, Δ ^^ tot sectionally over a matrix of DOD and ^^ mess For each DOD section, a linear fit of Δ ^^ is calculated according to Eq. (40). tot against ^^ mess The y-intercept results in Δ ^^ 0( DOD ) , from the slope Δ ^^(DOD). The latter value can be averaged over all DODs if necessary. The battery properties to be determined are then cal and ^^ c 0 a l determined analogously to Eqs. (19) and (30). Finally, the model parameters can be updated analogously to Eqs. (20) and (31). The simultaneous determination of internal resistance and open-circuit voltage curve is carried out in practice in the following steps. First, a voltage-controlled battery model with a known parameter value for the capacity ^^ is created. modprovided. An arbitrary starting value is set for the one or more with the internal resistance ^^ mod related parameters (e.g. ^^ ^^ for the simple equivalent circuit model in Fig.3a) and for the course of the assumed (it makes sense to assume a linear curve between the final charge and discharge voltage). The battery is charged over a period of time ^^ with measurement of current ^^ mess and excitement ^^ mess operated. The simulated current ^^ mod over the period ^^ using the voltage-controlled model. This is followed by the calculation of Δ ^^ tot according to Eq. (42). The values ​​Δ ^^ tot are sectionally in a matrix of DOD and ^^ mess - sections to the mean ത Δ തത ^ ത ^ t ത o തത t (DOD, ^^ exp ) are averaged over the period ^^. Subsequently, Δ ^^ 0( DOD) and Δ ^^(DOD) according to Eq. (40) by linear regression of ത Δ തത ^ ത ^ t ത o തത t (DOD, ^^ mess ) against ^^ mess for each DOD section. Subsequently, Δ ^^(DOD) over all DOD is ത Δ തത ^ ത ^ averaged. Then the approximate value for the real internal resistance ^^ cal according to Eq. (19) and the approximate value for the real open circuit voltage curve ^^ c 0 a l (DOD) is calculated according to Eq. (30). Optionally, the procedure is repeated, whereby the one or more resistors with the internal resistance ^^ mod related parameters and the open circuit voltage curve ^^ m 0 O din the battery model are set to the determined values ​​("model update"). This results in an iterative approximation to the true values ​​of internal resistance and open-circuit voltage curve. The averaging of Δ ^^ tot can also be performed section by section for different measured temperatures. The value Δ ^^(DOD) does not necessarily have to be averaged over all DODs. This allows state-of-charge, current, and / or temperature-dependent values ​​for the internal resistance ^^(DOD, ^^, ^^) and the open-circuit voltage curve ^^ to be determined. 0 (DOD, ^^). The procedure described is demonstrated below using the experimental data already mentioned, specifically the full cycles (Fig. 5). The simple equivalent circuit model shown in Fig. 3a is used. The battery model is given an arbitrary starting value for the series resistance, here ^^ s = 9 mΩ. The open circuit voltage curve ^^ 0The discharge voltage (DOD) is also set to an arbitrary starting value, namely a linear curve between the two final voltages. The capacity is set to the reference value of ^^ = 19.96 Ah. The period ^^ is chosen as a charge / discharge cycle (approximately 2.1 h). The results are shown in Fig. 15. Fig. 15a) shows the measured voltage ^^ mess as input for the voltage-controlled model. Fig.15b) shows both the measured current ^^ mess as well as the simulated current ^^ mod from the voltage-controlled model. In the period ^^1 (between 0 and 2 h) there is a significant deviation ^^ mod − ^^ mess of the two curves. Fig.15c) shows the difference between simulated and experimental stress Δ ^^ determined according to Eq. (2) tot using the data shown in Fig.15b). The value Δ ^^ totvaries over time during the period ^^1 (between 0 and 2 h). Over the first cycle duration ^^1, these values ​​are plotted section by section in a matrix for each DOD percentage point and each current ^^ mess (to whole amperes). For each individual DOD section, a linear regression of the curve Δ ^^ tot against ^^ mess and from this, according to Eq. (40), the values ​​Δ ^^ 0 (DOD) and Δ ^^(DOD). The latter value is averaged over the entire DOD range to ത Δ തത ^ ത^. Finally, the model parameters are set to the new values ​​according to Eqs. (20) and (31). The procedure is repeated for the periods ^^2 to ^^4. The internal resistances determined in this way are shown in Fig. 15d), starting from the assumed starting value. The procedure stabilizes after three cycles near the reference value. The determined open-circuit voltage curves are shown in Fig. 15e. Here, too, the procedure stabilizes after three cycles near the reference curve. At the same time, the prediction quality of the voltage-controlled model improves (cf. Fig. 15b) for periods > 2 h) and thus Δ ^^ totsmaller (cf. Fig. 15c) for periods > 2 h). With these results, the new method for the simultaneous determination of internal resistance and open circuit voltage curve of a rechargeable battery was successfully demonstrated. Simultaneous determination of internal resistance and capacity The methods for determining capacity and internal resistance presented above can be combined. This allows the simultaneous determination of these two parameters. No new theoretical development is necessary for this, merely a combination of practical implementations. The simultaneous determination of capacity and internal resistance is carried out in practice in the following steps. First, a voltage-controlled battery model with known parameter values ​​for the provided. Any starting values ​​are used for the capacity ^^ mod and the one with the internal resistance ^^ mod related parameters are assumed (e.g. ^^ ^^for the simple equivalent circuit model in Fig.3a). The battery is charged over a period of time ^^ with measurement of current ^^ mess and excitement ^^ mess operated. The simulated current ^^ mod over the period ^^ using the voltage-controlled model. This is followed by the calculation of Δ ^^ according to Eq. (17). The values ​​for Δ ^^ are averaged over the period ^^ ത Δ തത ^ ത ^ averaged. Δ ^^ is then calculated according to Eq. (37). The approximate value for the real internal resistance ^^ is then cal according to Eq. (19) and the approximate value for the real capacity according to Eq. (38). Optionally, the procedure is repeated, whereby the capacity ^^ mod and the one with the internal resistance ^^ modThe related parameters in the battery model are set to the determined values ​​("model update"). This results in an iterative approximation to the true values ​​of capacity and internal resistance. The averaging of Δ ^^ can also be performed section by section for different depth of discharge (DOD) ranges, different measured temperatures, or different currents ^^ mess This allows the determination of a state-of-charge, current-, and / or temperature-dependent internal resistance ^^(DOD, ^^, ^^). The described method is demonstrated below using the previously mentioned experimental data, namely full cycles (Fig. 5) and partial cycles. The simple equivalent circuit model of Fig. 3a) is used. The battery model receives the measured open-circuit voltage curve ^^ 0(DOD) as shown in Fig. 4. The capacity is set to an arbitrary starting value, for example, ^^ = 30 Ah for the full cycles and ^^ = 2 Ah for the partial cycles. The parameter for the serial resistance is also set to an arbitrary starting value, for example, ^^ ^^= 1 mΩ. Fig. 16 shows results for the full cycles. A charge / discharge cycle (approx. 2.1 h) is chosen as the period ^^. From Fig. 16c) and Fig. 16d) it can be seen that the capacitance and internal resistance converge towards the reference value after just three model updates (i.e. after three full cycles). Fig. 17 shows similar results for the partial cycles. A partial charge / discharge cycle (approx. 1 h) is chosen as the period ^^. From Fig. 17c) and Fig. 17d) it can be seen that the capacitance and internal resistance converge towards the reference value after around ten model updates (i.e. after ten partial cycles). These results demonstrate the successful use of the new method for the simultaneous determination of internal resistance and capacitance. Simultaneous determination of the open-circuit voltage curve and capacitance The previously described methods for determining the capacitance and open-circuit voltage curve can be combined. This allows the simultaneous determination of these two parameters.This doesn't require any new theoretical development, merely a combination of practical implementations. The simultaneous determination of capacity and open-circuit voltage curves is carried out in practice in the following steps. First, a voltage-controlled battery model with known parameter values ​​for the internal resistance ^^ is created. mod related parameters (e.g. ^^ ^^ for the simple equivalent circuit model in Fig.3a). Arbitrary starting values ​​are provided for the capacitance ^^ mod and for the course of the assumed (it makes sense to assume a linear curve between the charging and discharging voltages). The battery is charged over a period of time ^^ with measurement of current ^^ mess and excitement ^^ mess operated. The simulated current ^^ mod over the period ^^ using the voltage-controlled model. This is followed by the calculation of Δ ^^ 0according to Eq. (29). The values ​​for Δ ^^ 0 are calculated section by section for DOD areas to the mean ത Δ ത ^ ത ^ ത0ത (DOD) is averaged over the period ^^. Δ ^^ is then calculated according to Eq. (37). The approximate values ​​for the real open-circuit voltage curve ^^ are then c 0 a l (DOD) according to Eq. (30) and for the real capacity ^^ cal calculated according to Eq. (38). Optionally, the procedure is repeated, whereby the capacity ^^ mod and the open circuit voltage curve ^^ m 0 O din the battery model are set to the determined values ​​("model update"). This results in an iterative approximation to the true values ​​of capacity and open-circuit voltage curve. The described procedure is demonstrated below using the previously mentioned experimental data, specifically a full cycle. The simple equivalent circuit model from Fig. 3a) is used. The capacity is set to an arbitrary starting value, here as an example ^^ = 10 Ah. The open-circuit voltage curve ^^ 0(DOD) is also set to an arbitrary starting value, namely a linear curve between the two final voltages. The parameter for the series resistance is set to the reference value of ^^ ^^ = 4.579 mΩ. A charge / discharge cycle (approx. 2.1 h) is selected as the period ^^. The method is applied a total of nine times to this period, and a model update is performed each time. The results are shown in Fig. 18. From Fig. 18c), it can be seen that the capacity converges to the reference value after just three model updates. Fig. 18d) shows that the determination of the open-circuit voltage curve requires further updates; here, the reference value is reached after nine updates. These results demonstrate the successful use of the new method for the simultaneous determination of the open-circuit voltage curve and capacity of a rechargeable battery.Simultaneous Determination of Capacity, Internal Resistance, and Open-Circuit Voltage Curve The methods described above for determining capacity, internal resistance, and open-circuit voltage curves can be combined. This allows the simultaneous, complete characterization of all relevant parameters of an unknown battery. No new theoretical development is required for this, merely a combination of practical implementations. The simultaneous determination of capacity, internal resistance, and open-circuit voltage curves is carried out in practice in the following steps. First, a voltage-controlled battery model is provided. Arbitrary starting values ​​for the capacity ^^ are used. mod , the one with the internal resistance ^^ mod related parameters (e.g. ^^ ^^ for the simple equivalent circuit model in Fig.3a) and for the course of the assumed (it makes sense to assume a linear curve between the charging and discharging voltages). The battery is charged over a period of time ^^ with measurement of current ^^ mess and excitement ^^ mess operated. The simulated current ^^ mod over the period ^^ using the voltage-controlled model. This is followed by the calculation of Δ ^^ tot according to Eq. (42). The values ​​for Δ ^^ tot are in a matrix of DOD and ^^ mess -sections to the mean ത Δ തത ^ ത ^ t ത o തത t (DOD, ^^ mess ) averaged over the period ^^. Δ ^^ 0( DOD ) and Δ ^^(DOD) are calculated according to Eq. (40) by linear regression of ത Δ തത ^ ത ^ t ത o തത t (DOD, ^^ mess ) against ^^ messfor each DOD section. The values ​​Δ ^^(DOD) are calculated over all DOD to ത Δ തത ^ ത ^ averaged. Δ ^^ is calculated according to Eq. (37). Subsequently, the approximate values ​​for the real internal resistance are calculated according to Eq. (19), for the real open-circuit voltage curve according to Eq. (30), and for the real capacitance according to Eq. (38). Optionally, the procedure is repeated, whereby the capacitance ^^ mod and the one with the internal resistance ^^ mod related parameters and the open circuit voltage curve ^^ m 0 O din the battery model are set to the determined values ​​("model update"). This results in an iterative approximation to the true values ​​of capacity, internal resistance, and open-circuit voltage curve. The described procedure is demonstrated below using experimental data, specifically a full cycle. The simple equivalent circuit model shown in Fig. 3a) is used. The battery model receives arbitrary starting values ​​for the capacity of ^^ = 10 Ah and for the parameter related to the internal resistance of ^^ s = 9 mΩ. The open circuit voltage curve ^^ 0(DOD) is also set to an arbitrary starting value, namely a linear curve between the two final voltages. A charge / discharge cycle (approx. 2.1 h) is selected as the period ^^. The procedure is applied a total of 19 times to this period, and a model update is carried out each time. The results are shown in Fig. 19. Fig. 19a) and Fig. 19b) show the voltage and current of the battery over the period ^^. The determined values ​​of capacity, internal resistance, and open-circuit voltage curve as a function of the continuous model updates are shown in Fig. 19c), Fig. 19d) and Fig. 19e), each starting from the starting values. The dashed lines are the independently determined reference values. After 19 model updates, all three parameters (capacity, internal resistance, and open-circuit voltage curve) converge towards the reference values.These results successfully demonstrate the new method for the simultaneous determination of internal resistance, open-circuit voltage curve, and capacity of a rechargeable battery. The method thus allows the complete characterization of all relevant parameters of an unknown battery.

[0002] List of essential names of variables and parameters C Capacity of the battery^^mod Initial value in the battery model for the capacityCcal Calculated value for the capacityΔC Deviation between initial value of the capacity and calculated value of the capacity ^^ mess ( ^^) measured battery current ^^mess,n recorded value of the battery current at time ^^ ^^mod( ^^) model battery current ^^ mod,n simulated current of the battery model at time ^^ ^^ mess ( ^^) measured battery voltage ^^mess,n recorded value of the battery voltage at time ^^ ^^ 0 Open circuit voltage curve ^^ 0mess measured open circuit voltage curve ^^ 0 mod initial value in the battery model for the open circuit voltage curve ^^ 0 cal calculated value for the open circuit voltage curve Δ ^^ 0 Deviation between initial value of the open circuit voltage curve and calculated value of the open circuit voltage curve Δ ^^ 0 n time-discrete values ​​of the difference of the open-circuit voltage curve Δ ^^ tot Total voltage difference Δ ^^tot,n discrete values ​​of the total difference ത Δ ത ^ ത ^ ത0ത Average of the difference of the open circuit voltage curve ^^ 0 (DOD, ^^) Open circuit voltage curve as a function of depth of discharge and / or temperature ^^ Slope of the open circuit voltage curve ^^ Internal resistance ^^mod Initial value in the battery model for the internal resistance ^^ calcalculated value for the internal resistance Δ ^^ Deviation between initial value of the internal resistance and calculated value of the internal resistance Δ ^^n time-discrete values ​​of the difference in the internal resistance ത Δ തത ^ ത ^ Mean value of the difference in internal resistance ^^(DOD, ^^, ^^) Internal resistance as a function of depth of discharge and / or current and / or temperature SOC State of charge DOD Depth of discharge t TimeΔt Sampling interval T (Measuring) period^^ Temperature BMS Battery management system List of reference symbols 100 Overall algorithm 102 First component of the overall algorithm 104 Second component of the overall algorithm 106 Battery 110 Unit for voltage measurement 112 Unit for current measurement 114 Load 116 Display unit 118 Computing unit 120 Evaluation and control unit

Claims

Claims 1. A method for determining the internal resistance and / or the open-circuit voltage curve and / or the capacity of a rechargeable battery (106), the method comprising the following steps: (a) creating a dynamic, voltage-controlled, mathematical battery model, wherein for the internal resistance ^^ and / or the open-circuit voltage curve ^^ 0 and / or the capacity C given initial values ​​^^ 0 mod , ^^ mod , ^^ modused; (b) acquiring time-discrete measured values ​​for the battery current ^^mess( ^^) and the battery voltage ^^mess( ^^) of the rechargeable battery over a specified period of time ^^; (c) using the measured values ​​for the battery voltage ^^mess( ^^) as input to the dynamic, voltage-controlled, mathematical battery model and calculating values ​​for a simulated current ^^mod( ^^) as output of the battery model; and (d) determining calculated values ​​for the internal resistance ^^ cal and / or the open circuit voltage curve ^^ 0 cal and / or the capacity Ccal using the values ​​for the simulated current ^^ mod ( ^^) and the recorded measured values ​​for the current ^^ mess(^^) each with a predetermined calculation rule.

2. Method according to claim 1, characterized in that (a) the method comprises at least two iteration steps; (b) each iteration step comprises the implementation of steps (a) to (d) of the method according to claim 1; and (c) where in each iteration step, except the first, at the location of the initial values ​​^^mod, ^^ given in step (a) 0 mod and Cmod for the internal resistance ^^and / or the open circuit voltage curve ^^ 0 and / or the capacitance C the calculated values ​​for the internal resistance ^^cal and / or the open circuit voltage curve ^^ determined in step (d) of the previous iteration step 0 cal and / or the capacity ^^ cal3. Method according to claim 1 or 2, characterized in that (a) the method comprises at least two iteration steps; (b) wherein each iteration step comprises carrying out steps (a), (c) and (d) of the method according to claim 1; and (c) wherein in each iteration step, except for the first, at the location of the initial values ​​predetermined in step (a), ^^ mod , ^^ 0 mod and C mod for the internal resistance ^^and / or the open circuit voltage curve ^^ 0 and / or the capacitance C the calculated values ​​for the internal resistance determined in step (d) of the previous iteration step ^^ cal and / or the open circuit voltage curve ^^ 0 cal and / or the capacity ^^ cal 4. A method according to any one of claims 1 to 3, wherein (a) using a deviation between the values ​​for the simulated current ^^mod( ^^) and the measured values ​​for the battery current ^^ mess( ^^) deviations Δ ^^, Δ ^^ 0 , ΔC between the given initial values ​​^^mod, ^^ 0 mod, Cmod and the calculated values ​​^^cal, ^^ 0 cal, Ccal can be calculated; and (b) from the deviations Δ ^^, Δ ^^ thus determined 0 , ΔC and the specified initial values ​​^^ mod , ^^ 0 mod , C mod for internal resistance ^^ and / or open circuit voltage curve ^^ 0 and / or capacity C the calculated values ​​for internal resistance ^^cal and / or open circuit voltage curve ^^ 0 cal and / or capacitance Ccal are calculated.

5. Method according to one of claims 1 to 4, characterized in that the following calculation rule is used to determine the internal resistance ^^: , where Δ ^^ is the difference between the internal resistance of the battery model ^^ mod and the calculated value for the internal resistance ^^ cal, ^^ the slope of the open circuit voltage curve, ^^ mod the current of the battery model and ^^ mess the detected current of the battery.

6. Method according to one of claims 1 to 4, characterized in that for the determination of the open circuit voltage curve ^^ 0 the following calculation rule is used: , where Δ ^^ 0 the difference between the open circuit voltage curve of the battery model ^^ m 0 O d and the calculated value for the open circuit voltage curve ^^ c 0 a l , ^^ the slope of the open circuit voltage curve, ^^ mod the current of the battery model and ^^ mess denotes the measured current of the battery.

7. Method according to one of claims 1 to 4, characterized in that the following calculation rule is used to determine the capacity C: ^^ ^ ^, where Δ ^^ is the quotient of the capacity of the battery model ^^ mod and the calculated value for the battery capacity ^^ cal , ^^ mod the current of the battery model and ^^ mess the detected current intensity of the battery.

8. Method according to one of claims 1 to 4, characterized in that for the simultaneous determination of the internal resistance ^^ and the open-circuit voltage curve ^^ 0 the following calculation rule is used: with Δ ^^ tot = Δ ^^ 0 − Δ ^^ ∙ ^^ mess , where Δ ^^ tot the total voltage difference, Δ ^^ 0 the difference between the open circuit voltage curve of the battery model ^^ m 0 O d and the calculated value for the open circuit voltage curve ^^ c 0 a l , Δ ^^ the difference between the internal resistance of the battery model ^^ modand the calculated value for the internal resistance ^^ cal , ^^ the slope of the open circuit voltage curve, ^^ mod the current of the battery model and ^^ mess the detected current of the battery.

9. Method according to claims 5 to 8, characterized in that for the calculation of the time-discrete values ​​of the difference in the internal resistance Δ ^^ n and / or the time-discrete values ​​of the difference of the open-circuit voltage curve Δ ^^ ^^ 0 and / or for the time-discrete values ​​of the total voltage difference ^^ ^^ tot, ^^ and / or for the value of the quotient ΔC from the capacity of the battery model ^^ mod and the calculated value for the battery capacity ^^ cal a numerical solution method is used.

10. Method according to claims 4 to 9, characterized in that at least two time-discrete values ​​of the difference of the internal resistance Δ ^^ ^^to an average value of the difference in internal resistance ത Δ തത ^ ത ^ and / or at least two time-discrete values ​​of the difference of the open-circuit voltage curve Δ ^^ ^^ 0 to an average value of the difference of the open circuit voltage curve ത Δ ത ^ ത ^ ത0ത over a measurement period ^^ are averaged.

11. Method according to claims 1 to 10, characterized in that the internal resistance is determined as a function of depth of discharge and / or current and / or temperature ^^(DOD, ^^, ^^), wherein several average values ​​of the difference of the internal resistance ത Δ തത ^ ത ^ are used for the determination, and the mean values ​​of the difference in internal resistance ത Δ തത ^ ത ^ can be determined by taking values ​​of the difference in internal resistance Δ ^^ ^^over the measurement period ^^ sectionally over different ranges of the depth of discharge DOD and / or current ^^ and / or temperature ^^ are averaged.

12. Method according to claims 1 to 9, characterized in that the open circuit voltage curve as a function of the depth of discharge and / or the temperature, ^^ 0 (DOD, ^^) is determined, whereby several average values ​​of the open circuit voltage curve ത Δ ത ^ ത ^ ത0ത used for the determination, and the mean values ​​of the open circuit voltage curve ത Δ ത ^ ത ^ ത0ത be determined by taking values ​​of the open circuit voltage curve Δ ^^ ^^ 0over the measurement period ^^ sectionally over different ranges of the depth of discharge (DOD) and / or the temperature ^^ are averaged.

13. Device for determining the internal resistance and / or the open circuit voltage curve and / or the capacity of a rechargeable battery (106) with an evaluation and control unit (120) which has a detection device for Recording measured values ​​for the battery current ^^mess( ^^) and the battery voltage ^^ mess ( ^^) and / or the temperature ^^ mess( ^^) (110, 112) of the rechargeable battery, preferably at equidistant time intervals Δt or at predetermined times, and a computing unit (118) to which the recorded measured values ​​can be fed, characterized in that the evaluation and control device (120) is designed to carry out the method according to one of claims 1 to 12.

14. Computer program for determining the internal resistance and / or the open circuit voltage curve and / or the capacity of a rechargeable battery (106), characterized in that the computer program is designed such that when the computer program is executed in a data processing unit, in particular the evaluation and control device according to claim 13, the method according to one of claims 1 to 12 is carried out.

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