DE102020005169A1 - Method for estimating a current distribution within an on-board electrical system of a motor vehicle using a battery management system and battery management system - Google Patents

Abstract

The invention relates to a method for estimating a current distribution within an on-board electrical system of a motor vehicle, at least one first battery cell and one second battery cell connected in parallel to the first battery cell being provided in the on-board electrical system, and using a mathematical model of an electronic computing device of the battery management system on the basis at least one first characterizing parameter of the first battery cell and on the basis of at least one second characterizing parameter of the second battery cell, the current distribution is estimated by means of the electronic computing device, with the on-board electrical system being converted into a simplified equivalent circuit diagram in the mathematical model and by determining a current at an infinite point in time as a function of the at least two characterizing parameters, by determining a current at a zero instant as a function of of the at least two characterizing parameters and by determining a time constant as a function of the at least two characterizing parameters, the estimation of the current distribution is carried out using the mathematical model. The invention also relates to a battery management system (12).

Description

The invention relates to a method for estimating a current distribution within an on-board electrical system of a motor vehicle using a battery management system according to the preamble of patent claim 1. The invention also relates to a battery management system.

Electrical energy stores, which can also be referred to as batteries, in motor vehicles contain in particular a large number of battery cells connected in parallel and in series in order to cover the energy and power requirements of motor vehicles, in particular of at least partially electrically operated motor vehicles. Due to current/state of charge and temperature differences between the battery cells, which can be configured as lithium-ion cells, for example, caused by cell production and battery design tolerances, the performance of the battery is limited to the most heavily loaded cell. In order to achieve the best possible battery performance in terms of rapid charging, recuperation and acceleration of the vehicle, the current distribution within the cells connected in parallel on board must be estimated.

In the scientific publication by A. Fill, S. Koch, K.P. Birke: "Analytical model of the current distribution of parallel-connected battery cells and strings", Journal of Energy Storage 23 (2019), pages 37-43, a simplified equivalent circuit diagram for the on-board electrical system is shown in particular.

The object of the present invention is to provide a method and a battery management system by means of which an efficient estimation of a current distribution within an on-board electrical system of a motor vehicle can be implemented.

This object is achieved by a method and by a battery management system according to the independent patent claims. Advantageous embodiments are specified in the dependent claims.

One aspect of the invention relates to a method for estimating a current distribution within an electrical system of a motor vehicle using a battery management system, wherein in the electrical system at least a first battery cell and a second battery cell connected in parallel to the first battery cell are provided, and using a mathematical model of an electronic Computing device of the battery management system based on at least one first characterizing parameter of the first battery cell and on the basis of at least one second characterizing parameter of the second battery cell, the current distribution is estimated by means of the electronic computing device.

It is provided that in the mathematical model the on-board electrical system is converted into a simplified equivalent circuit diagram and by determining a current at an infinite point in time as a function of the at least two characterizing parameters, by determining a current at a zero point in time as a function of the at least two characterizing parameters and by determining a time constant as a function of the at least two characterizing parameters, the current distribution is estimated using the mathematical model.

This enables a more efficient estimation of the current distribution. In particular, the corresponding current distribution can be determined with little computing effort using the electronic computing device. This makes it possible for the corresponding current distribution to be estimated in real time, in particular inside the motor vehicle. In particular, a semi-analytical mathematical model for estimating the current distribution is thus disclosed, which can be implemented with little parameterization and computing effort for implementation in the battery management system inside the motor vehicle.

According to an advantageous embodiment, the current in the first battery cell is determined by subtracting the current from the second battery cell from the current in the vehicle electrical system and vice versa. In other words, it can be provided that the current in the second battery cell is carried out by subtracting the current from the first battery cell from the current in the vehicle electrical system. The current distribution within the on-board electrical system can thus be implemented with reduced effort and efficiently with little computing effort.

bestimmt wird, wobei x_Ri einem jeweiligen Batteriezellewiderstand im Verhältnis zu einem jeweiligen erwarteten Batteriezellenwiderstand, x_Ci einer Batteriezellenkapazität im Verhältnis zu einer jeweilig erwarteten Batteriezellenkapazität, f einem Verhältnis der jeweiligen Spannungssteilheiten der Batteriezellen, I_batt dem Strom im elektrischen Bordnetz, t einem Zeitpunkt und τ der Zeitkonstante entsprechen. Insbesondere kann mittels der Formel vereinfacht die Stromverteilung innerhalb des elektrischen Bordnetzes bestimmt werden.It is also advantageous if a first current in the first battery cell using the formula: $$I_1\left(t\right)=\frac{x_{C_1}}{x_{{\mathrm{<figure-callout\; id="1"\; label="C"\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">C</figure-callout>}}_1}+\xi x_{C_2}}{I}_{batt}\left[\left(\frac{x_{R_2}}{x_{{\mathrm{<figure-callout\; id="1"\; label="R"\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">R</figure-callout>}}_1}+x_{R_2}}\frac{x_{{\mathrm{<figure-callout\; id="1"\; label="C"\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">C</figure-callout>}}_1}+\xi x_{C_2}}{x_{{\mathrm{<figure-callout\; id="1"\; label="C"\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">C</figure-callout>}}_1}}-1\right)e^{-\frac{t}{\tau }}+1\right]$$

is determined, where x _Ri a respective battery cell resistance in relation to a respective expected battery cell resistance, x _Ci a battery cell capacity in relation to a respective expected battery cell capacity, f a ratio of the respective voltage gradients of the battery cells, I _batt the current in the electrical system, t a point in time and τ correspond to the time constant. In particular, the current distribution within the on-board electrical system can be determined in a simplified manner using the formula.

durchgeführt wird, wobei x_Ci einer Batteriezellenkapazität im Verhältnis zu einer jeweilig erwarteten Batteriezellenkapazität, ξ einem Verhältnis der jeweiligen Spannungssteilheiten der Batteriezellen und I_batt dem Strom im elektrischen Bordnetzentsprechen.It is also advantageous to determine a first current in the first battery cell at an infinite point in time using the formula: $${\lambda }_{\infty }=I_1=\frac{x_{C_1}}{x_{{\mathrm{<figure-callout\; id="1"\; label="C"\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">C</figure-callout>}}_1}+\xi x_{C_2}}{I}_{batt}$$

is carried out, where x _Ci corresponds to a battery cell capacity in relation to a respectively expected battery cell capacity, ξ to a ratio of the respective voltage gradients of the battery cells and I _batt to the current in the vehicle electrical system.

durchgeführt wird, wobei x_Ri einem jeweiligen Batteriezellenwiderstand im Verhältnis zu einem jeweiligen erwarteten Batteriezellenwiderstand und I_batt dem Strom im elektrischen Bordnetz entsprechen. Somit kann aufwandsreduziert und insbesondere kraftfahrzeugintern die Stromverteilung bestimmt werden.In a further advantageous embodiment, a first current in the first battery cell is determined at a zero point in time using the formula: $${\lambda }_0=I_1=\frac{x_{R_2}}{x_{R_1}+x_{R_2}}{I}_{batt}$$

is carried out, where x _Ri corresponds to a respective battery cell resistance in relation to a respective expected battery cell resistance and I _{batt corresponds} to the current in the vehicle electrical system. In this way, the effort can be reduced and, in particular, the current distribution can be determined inside the motor vehicle.

durchgeführt, wobei x_Ri einem Batteriezellenwiderstand im Verhältnis zu einem jeweiligen erwarteten Batteriezellenwiderstand, x_Ci einer Batteriezellenkapazität im Verhältnis zu einer jeweilig erwarteten Batteriezellenkapazität, ξ einem Verhältnis der jeweiligen Spannungssteilheiten der Batteriezellen, m_U,2 der Spannungssteilheit der zweiten Batteriezelle, µ_R dem erwarteten Batteriezellenwiderstand und µ_c der erwarteten Batteriezellenkapazität entsprechen.In a further advantageous embodiment, the time constant is determined using the formula: $$\tau =\frac{\left(x_{{\mathrm{<figure-callout\; id="1"\; label="R"\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">R</figure-callout>}}_1}+x_{R_2}\right){\mu }_R{\mu }_C{x}_{C_1}{x}_{C_2}}{m_{u\mathrm{,2}}\left(x_{{\mathrm{<figure-callout\; id="1"\; label="C"\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">C</figure-callout>}}_1}+\xi x_{C_2}\right)}$$

carried out, where x _Ri a battery cell resistance in relation to a respective expected battery cell resistance, x _Ci a battery cell capacity in relation to a respective expected battery cell capacity, ξ a ratio of the respective voltage gradients of the battery cells, m _U,2 the voltage gradient of the second battery cell, µ _R the expected Battery cell resistance and µ _c correspond to the expected battery cell capacity.

According to a further advantageous embodiment, the equivalent circuit diagram is simplified by specifying that the first battery cell and the second battery cell are each described by a series resistor and a voltage source. The on-board electrical system can thus be described in a simple manner, with the determination of the current distribution being able to be carried out with reduced computation.

Provision can furthermore be made for the equivalent circuit diagram to be simplified by specifying that a respective series resistance is invariant. In other words, the series resistance is set to be constant. The determination of the current distribution can thus be carried out in a simple manner and with reduced computation.

Furthermore, it has proven to be advantageous if the equivalent circuit diagram is simplified by specifying that a respective battery cell capacity of a respective battery cell is invariant. In other words, the respective battery cell capacity is specified as constant. As a result, the current distribution within the on-board electrical system can be determined in a simple manner and with reduced computation.

It can preferably also be provided that it is also specified that the temperature of the respective battery cell is specified as constant. In particular, it is specified that the first battery cell and the second battery cell are connected in parallel to one another. This also makes it possible for the current distribution within the on-board electrical system to be determined in a simple manner and with reduced computing power.

A further aspect of the invention relates to a battery management system for a motor vehicle with at least one electronic computing device, the battery management system being designed to carry out a method according to the preceding aspect. In particular, the method is carried out using the battery management system.

Yet another aspect of the invention relates to a motor vehicle with a battery management system according to the preceding aspect. The motor vehicle is at least partially electrically operated. In particular, the motor vehicle is operated fully electrically.

Further advantages, features and details of the invention result from the following description of preferred exemplary embodiments and from the drawings. The features and combinations of features mentioned above in the description and the features and combinations of features mentioned below in the description of the figures and/or shown alone in the figures can be used not only in the combination specified in each case, but also in other combinations or on their own, without going beyond the scope of the leave invention.

• 1 ein schematisches Blockschaltbild einer Ausführungsform eines elektrischen Bordnetzes mit einer Ausführungsform eines Batteriemanagementsystems; und
• 2 ein schematisches Schaltbild zur Darstellung eines mathematischen Modells.

show:

• 1 a schematic block diagram of an embodiment of an electrical system with an embodiment of a battery management system; and
• 2 a schematic circuit diagram to represent a mathematical model.

Elements that are the same or have the same function are provided with the same reference symbols in the figures.

1 shows a schematic block diagram of an embodiment of an electrical system 10. The electrical system 10 has a battery management system 12 on. Furthermore, the on-board electrical system 10 has a first battery cell 14 and a second battery cell 16 . The first battery cell 14 and the second battery cell 16 are connected to one another in parallel. In particular, a simplified equivalent circuit diagram for the on-board electrical system 10 is thus shown. The simplified equivalent circuit diagram assumes in particular that a respective battery cell 14, 16 can be described by means of a voltage source Usc and a serial resistor R _SER . In particular, therefore, the first battery cell 14 has a first voltage source U _SC1 and a first serial resistor R _SER1 . The second battery cell 16 has a second voltage source U _SC2 and a second series resistor R _SER2 . A first current I ₁ flows within the first battery cell 14 and a second current I ₂ flows within the second battery cell 16 . The entire vehicle electrical system has a current of I _batt .

In a method for estimating a current distribution 18 of the electrical system 10 of a motor vehicle 20, shown purely schematically, using the battery management system 12, the electrical system 10 is provided with at least the first battery cell 14 and the second battery cell 16 connected in parallel to the first battery cell 14, and it is a mathematical model 22 of an electronic computing device 24 of the battery management system 12 based on at least one first characterizing parameter of the first battery cell 14 and based on at least one second characterizing parameter of the second battery cell 16, the current distribution 18 is estimated by means of the electronic computing device 24.

Provision is made for the on-board electrical system 10 to be represented in the mathematical model 22 in a simplified equivalent circuit diagram, as in 1 shown, is converted, and by determining a current λ _∞ at an infinite point in time as a function of the at least two characterizing parameters, by determining a current λ ₀ at a zero point in time as a function of the at least two characterizing parameters and by determining a time constant τ as a function the estimation of the current distribution 18 is carried out by means of the mathematical model 22 from the at least two characterizing parameters.

In particular, it is therefore provided that the counting behavior is transferred to an equivalent circuit in order to demonstrate the non-linearity of the current distribution 18 and the interaction of the various cell parameters and states. For this purpose, the equivalent circuit with two RC pairs was used in particular. Based on Kirchhoff's mesh equation, the respective currents I ₁ , I ₂ can be described with the polarization and diffusion overpotentials, the series resistances R _SER and the battery current I _batt . The fact that the cell currents interact with the current distribution 18 as a function of the SOC change (state of charge) and are also dependent on a cell temperature results in a non-linear system of equations for the current distribution 18.

By assuming as in 1 shown that the cell behavior of a battery cell 14, 16 is limited to the voltage source Usc and the serial resistance R _SER , this can be simplified. Furthermore, it is assumed in particular that the battery cells 14, 16 are in a parallel-connected state. In particular, it is specified that the series resistances R _SER are invariant and therefore constant. Furthermore, it is specified in particular that the battery cell capacities C are likewise invariant and constant. Furthermore, a constant cell temperature is assumed. This results in the simplified equivalent circuit diagram, as shown in 1 is shown. Furthermore, it is specified that the following formulas apply within each time contactor Δt: $$\begin{array}{c}{I}_{batt}=kOnstant;\\ \frac{\delta u_{SC,i}}{\delta SO{C}_i}-m_{u,i}=kOnstant\end{array}$$

$$\frac{d{I}_1}{dt}=\frac{1}{{\mu }_R\left(x_{R_1}+x_{R_2}\right)}\frac{d{\Delta }_{2-1}{U}_{SC}}{dt},\frac{d{I}_2}{dt}=\frac{d{I}_1}{dt}$$

unter der Annahme der abgeleiteten Zeit eines jeden Ladungszustands (SOC) einer jeden Zelle ergibt sich $$\frac{dSO{C}_1}{dt}=\frac{1}{x_{C_1}{\mu }_C}{I}_1,\frac{d\delta SO{C}_2}{dt}=\frac{1}{x_{C_2}{\mu }_C}{I}_2$$

wobei die Zeitableitung von Usc mittels der Formel $$\frac{d{U}_{SC\mathrm{,1}}}{dt}=\frac{m_{U\mathrm{,1}}}{x_{C_1}{\mu }_C}{I}_1,\frac{d{U}_{SC\mathrm{,2}}}{dt}=\frac{m_{U\mathrm{,2}}}{x_{C_2}{\mu }_C}{I}_2,$$

bestimmt werden kann, wobei hier eine individuelle Spannungssteilheit mit m_U,1 und m_U,2 einer jeden Zelle bereitgestellt ist. Die Abweichung einer jeweiligen Spannungsquellendifferenz ist mit den Formeln $$\begin{array}{c}\frac{d{\Delta }_{2-1}{U}_{SC}}{dt}=\frac{m_{U\mathrm{,2}}}{x_{C_2}{\mu }_C{x}_{C_1}}\left[x_{C_1}{I}_{batt}-\left(x_{C_1}+\xi x_{C_2}\right)I_1\right]\\ \xi =\frac{m_{U\mathrm{,1}}}{m_{U\mathrm{,2}}}\end{array}$$

bestimmt. Dadurch kann das lineare Differential für die Zellspannungen bestimmt werden: $$\frac{d{I}_1}{dt}=\frac{m_{U\mathrm{,2}}}{{\mu }_R\left(x_{R_1}+x_{R_2}\right)x_{C_2}{\mu }_C{x}_{C_1}}\left[x_{C_1}{I}_{batt}-\left(x_{C_1}+\xi x_{C_2}\right)I_1\right]$$

In this case, a linear correlation is carried out between the change in the voltage source at the SOC m _U,i of each cell. The slope and the battery voltage are updated within each time step. With this mathematical model 22, the combination of the discretization and the analytical solution requires an exact simulation as well as little computation effort and little storage space. The respective battery cell resistance R _SER,i and the capacity C _i are related to the expected values of the cell resistance μ _R and μ _c . Due to the assumptions made previously, the total difference in cell currents is reduced to $$x_{R_i}=\frac{R_{SER,i}}{{\mu }_R},x_{C_i}=\frac{C_i}{{\mu }_C}$$

$$\frac{\mathrm{i.e}{I}_1}{\mathrm{i.e}t}=\frac{1}{{\mu }_R\left({\mathrm{<figure-callout\; id="1"\; label="x\; R"\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">x</figure-callout>}}_{R_{}}+x_{R_2}\right)}\frac{\mathrm{<figure-callout\; id="2"\; label="i.e\; \Delta "\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">i.e</figure-callout>}{\Delta }_{}{}_{2-1}{u}_{SC}}{\mathrm{i.e}t},\frac{\mathrm{i.e}{I}_2}{\mathrm{i.e}t}=\frac{\mathrm{i.e}{I}_1}{\mathrm{i.e}t}$$

assuming the derived time of each state of charge (SOC) of each cell $$\frac{\mathrm{i.e}\mathrm{<figure-callout\; id="1"\; label="S\; O\; C"\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">S</figure-callout>}O{C}_{}{}_1}{\mathrm{i.e}t}=\frac{1}{{\mathrm{<figure-callout\; id="1"\; label="x\; C"\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">x</figure-callout>}}_{C_{}}{\mathrm{<figure-callout\; id="1"\; label="\mu \; C\; I"\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">\mu </figure-callout>}}_C}{I}_{}{}_1,\frac{\mathrm{i.e}\delta \mathrm{<figure-callout\; id="2"\; label="S\; O\; C"\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">S</figure-callout>}O{C}_{}{}_2}{\mathrm{i.e}t}=\frac{1}{{\mathrm{<figure-callout\; id="2"\; label="x\; C"\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">x</figure-callout>}}_{C_{}}{\mathrm{<figure-callout\; id="2"\; label="\mu \; C\; I"\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">\mu </figure-callout>}}_C}{I}_{}{}_2$$

where the time derivative of Usc is given by the formula $$\frac{\mathrm{i.e}{u}_{SC\mathrm{,1}}}{\mathrm{i.e}t}=\frac{m_{u\mathrm{,1}}}{x_{C_1}{\mu }_C}{I}_1,\frac{\mathrm{i.e}{u}_{SC\mathrm{,2}}}{\mathrm{i.e}t}=\frac{m_{u\mathrm{,2}}}{x_{C_2}{\mu }_C}{I}_2,$$

can be determined, with an individual voltage rate of rise with m _U,1 and m _U,2 being provided for each cell. The deviation of a respective voltage source difference is with the formulas $$\begin{array}{c}\frac{\mathrm{i.e}{\mathrm{<figure-callout\; id="2"\; label="\Delta "\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">\Delta </figure-callout>}}_{2-1}{u}_{SC}}{\mathrm{i.e}t}=\frac{m_{u\mathrm{,2}}}{{\mathrm{<figure-callout\; id="2"\; label="x\; C"\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">x</figure-callout>}}_{C_{}}{\mu }_C{x}_{C_1}}\left[x_{C_1}{I}_{batt}-\left({\mathrm{<figure-callout\; id="1"\; label="x\; C"\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">x</figure-callout>}}_{C_{}}+\xi x_{C_2}\right)I_1\right]\\ \xi =\frac{m_{u\mathrm{,1}}}{m_{u\mathrm{,2}}}\end{array}$$

certainly. This allows the linear differential for the cell voltages to be determined: $$\frac{\mathrm{i.e}{I}_1}{\mathrm{i.e}t}=\frac{m_{u\mathrm{,2}}}{{\mu }_R\left({\mathrm{<figure-callout\; id="1"\; label="x\; R"\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">x</figure-callout>}}_{R_{}}+x_{R_2}\right){\mathrm{<figure-callout\; id="2"\; label="x\; C"\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">x</figure-callout>}}_{C_{}}{\mu }_C{x}_{C_1}}\left[x_{C_1}{I}_{batt}-\left({\mathrm{<figure-callout\; id="1"\; label="x\; C"\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">x</figure-callout>}}_{C_{}}+\xi x_{C_2}\right)I_1\right]$$

welche teilweise integriert werden kann, so dass die nachfolgende Formel: $$-\frac{1}{\left(x_{C_1}+\xi x_{C_2}\right)}\text{ln}{x}_{C_1}{I}_{batt}-I_1\left(x_{C_1}+\xi x_{C_2}\right)=\frac{m_{U\mathrm{,2}}}{{\mu }_R\left(x_{R_1}+x_{R_2}\right)x_{C_2}{\mu }_C{x}_{C_1}}t+C_0$$

formuliert werden kann, wobei C₀ als konstant integriert ist. Die Formel kann dann weiterhin transformiert werden zu: $$x_{C_1}{I}_{batt}-I_1\left(x_{C_1}+\xi x_{C_2}\right)=\widetilde{C_0}{e}^{-\frac{m_{U\mathrm{,2}}\left(x_{C_1}+\xi x_{C_2}\right)}{{\mu }_R\left(x_{R_1}+x_{R_2}\right)x_{C_2}{\mu }_C{x}_{C_1}}t}$$

This equation can be determined by separating the variables and defining limits accordingly. With balanced battery cells 14, 16, the state of charge of the first battery cell 14 corresponds to the state of charge of the second battery cell at a point in time t=0. This leads to the following formula $$\left[x_{C_1}{I}_{batt}-\left({\mathrm{<figure-callout\; id="1"\; label="x\; C"\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">x</figure-callout>}}_{C_{}}+\xi x_{C_2}\right)I_1\right]\mathrm{i.e}{I}_1=\frac{m_{u\mathrm{,2}}}{{\mu }_R\left({\mathrm{<figure-callout\; id="1"\; label="x\; R"\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">x</figure-callout>}}_{R_{}}+x_{R_2}\right){\mathrm{<figure-callout\; id="2"\; label="x\; C"\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">x</figure-callout>}}_{C_{}}{\mu }_{\mathrm{<figure-callout\; id="1"\; label="C\; x\; C"\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">C</figure-callout>}}{x}_{C_{}}{{}_1}_{}}\mathrm{i.e}t$$

which can be partially integrated so that the following formula: $$-\frac{1}{\left({\mathrm{<figure-callout\; id="1"\; label="x\; C"\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">x</figure-callout>}}_{C_{}}+\xi x_{C_2}\right)}\text{ln}{x}_{C_1}{I}_{batt}-I_1\left({\mathrm{<figure-callout\; id="1"\; label="x\; C"\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">x</figure-callout>}}_{C_{}}+\xi x_{C_2}\right)=\frac{m_{u\mathrm{,2}}}{{\mu }_R\left({\mathrm{<figure-callout\; id="1"\; label="x\; R"\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">x</figure-callout>}}_{R_{}}+x_{R_2}\right){\mathrm{<figure-callout\; id="2"\; label="x\; C"\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">x</figure-callout>}}_{C_{}}{\mu }_C{x}_{C_1}}t+{\mathrm{<figure-callout\; id="0"\; label="C"\; filenames="DE102020005169A1\_0042.png"\; state="\{\{state\}\}">C</figure-callout>}}_0$$

can be formulated with C ₀ integrated as a constant. The formula can then be further transformed to: $$x_{C_1}{I}_{batt}-I_1\left({\mathrm{<figure-callout\; id="1"\; label="x\; C"\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">x</figure-callout>}}_{C_{}}+\xi x_{C_2}\right)=\tilde{{\mathrm{<figure-callout\; id="0"\; label="C"\; filenames="DE102020005169A1\_0042.png"\; state="\{\{state\}\}">C</figure-callout>}}_0}{e}^{-\frac{m_{u\mathrm{,2}}\left({\mathrm{<figure-callout\; id="1"\; label="x\; C"\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">x</figure-callout>}}_{C_{}}+\xi x_{C_2}\right)}{{\mu }_R\left({\mathrm{<figure-callout\; id="1"\; label="x\; R"\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">x</figure-callout>}}_{R_{}}+x_{R_2}\right){\mathrm{<figure-callout\; id="2"\; label="x\; C"\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">x</figure-callout>}}_{C_{}}{\mu }_C{x}_{C_1}}t}$$

wobei dann die Integrationskonstante bestimmt werden kann, und es ergibt sich die Formel $$\widetilde{C_0}=-\frac{x_{R_2}}{x_{R_1}+x_{R_2}}{I}_{batt}+\frac{x_{C_1}}{x_{C_1}+\xi x_{C_2}}{I}_{batt}$$

The formula is based on the boundary conditions of the balanced cells and Kirchhoff's mesh equation $$I_1\left(t=0\right)=\frac{x_{R_2}}{{\mathrm{<figure-callout\; id="1"\; label="x\; R"\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">x</figure-callout>}}_{R_{}}+x_{R_2}}{I}_{batt}$$

where the constant of integration can then be determined, and the formula results $$\tilde{{\mathrm{<figure-callout\; id="0"\; label="C"\; filenames="DE102020005169A1\_0042.png"\; state="\{\{state\}\}">C</figure-callout>}}_0}=-\frac{x_{R_2}}{{\mathrm{<figure-callout\; id="1"\; label="x\; R"\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">x</figure-callout>}}_{R_{}}+x_{R_2}}{I}_{batt}+\frac{x_{C_1}}{{\mathrm{<figure-callout\; id="1"\; label="x\; C"\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">x</figure-callout>}}_{C_{}}+\xi x_{C_2}}{I}_{batt}$$

was wiederum in der Stromverteilung 18 resultiert.By setting the constant integration and a further transformation, the $$I_1\left(t\right)=\frac{x_{C_1}}{{\mathrm{<figure-callout\; id="1"\; label="x\; C"\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">x</figure-callout>}}_{C_{}}+\xi x_{C_2}}{I}_{batt}\left[\left(\frac{x_{R_2}}{{\mathrm{<figure-callout\; id="1"\; label="x\; R"\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">x</figure-callout>}}_{R_{}}+x_{R_2}}\frac{{\mathrm{<figure-callout\; id="1"\; label="x\; C"\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">x</figure-callout>}}_{C_{}}+\xi x_{C_2}}{{\mathrm{<figure-callout\; id="1"\; label="x\; C"\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">x</figure-callout>}}_{C_{}}}-1\right)e^{-\frac{t}{\tau }}+1\right]$$

which in turn results in the current distribution 18 .

mit dem ersten Strom λ₀ in der ersten Batteriezelle 14 zu dem Nullzeitpunkt: $${\lambda }_0=I_1=\frac{x_{R_2}}{x_{R_1}+x_{R_2}}{I}_{batt}$$

und der Zeitkonstante τ: $$\tau =\frac{\left(x_{R_1}+x_{R_2}\right){\mu }_R{\mu }_C{x}_{C_1}{x}_{C_2}}{m_{U\mathrm{,2}}\left(x_{C_1}+\xi x_{C_2}\right)}$$

gebildet ist.The current distribution 18 thus has 3 significant distribution values, as a result of which the first current λ _∞ in the first battery cell 14 is determined at an infinite point in time: $${\lambda }_{\infty }=I_1=\frac{x_{C_1}}{{\mathrm{<figure-callout\; id="1"\; label="x\; C"\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">x</figure-callout>}}_{C_{}}+\xi x_{C_2}}{I}_{batt}$$

with the first current λ ₀ in the first battery cell 14 at the zero time: $${\mathrm{<figure-callout\; id="0"\; label="\lambda "\; filenames="DE102020005169A1\_0042.png"\; state="\{\{state\}\}">\lambda </figure-callout>}}_0=I_1=\frac{x_{R_2}}{{\mathrm{<figure-callout\; id="1"\; label="x\; R"\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">x</figure-callout>}}_{R_{}}+x_{R_2}}{I}_{batt}$$

and the time constant τ: $$\tau =\frac{\left(x_{{\mathrm{<figure-callout\; id="1"\; label="R"\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">R</figure-callout>}}_1}+x_{R_2}\right){\mu }_R{\mu }_C{x}_{C_1}{x}_{C_2}}{m_{u\mathrm{,2}}\left({\mathrm{<figure-callout\; id="1"\; label="x\; C"\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">x</figure-callout>}}_{C_{}}+\xi x_{C_2}\right)}$$

is formed.

2 12 shows a schematic diagram for estimating the current distribution 18. In particular, two diagrams are shown. A discharging curve is shown in the upper diagram and a charging curve is shown in the lower diagram. In particular, the current I is plotted on the ordinate and the charging flow in Colombs per ampere-hour is plotted on the abscissa. Furthermore, the ratio of the respective voltage gradients ξ is also plotted on the ordinate. the 2 12 shows in particular a first curve 26 which corresponds to an actually measured curve of the first current I ₁ . A second curve 28 corresponds in particular to the simulated first current I ₁ . A third curve 30 corresponds in particular to a measured current I ₂ . A fourth curve 32 corresponds to the simulated second current I ₂ . Furthermore, the voltage gradients f are shown in each case in a fifth curve 34 . To determine an effect of an open circuit voltage (OCV) and an overpotential on mu, a differential voltage analysis (DVA) is useful and discharging at different C values is performed. $$DVA=\frac{\Delta u}{\Delta SOC}$$

The parameterization of the mathematical model 22 can then be carried out using the data from the DVA, with a number of curves being able to be carried out depending on a simulation accuracy and a memory capacity.

$$SO{C}_{\mathrm{1,1}}=SO{C}_{\mathrm{2,1}}=SO{C}_0$$

$${\Delta }_{2-1}{U}_{SC\mathrm{,1}}=0$$

$$m_{U\mathrm{,1}}=m_{U\mathrm{,0}}$$

$$I_{\mathrm{1,1}}=\frac{x_{R_1}}{x_{R_1}+x_{R_2}}{I}_{batt\mathrm{,1}},I_{\mathrm{2,1}}=I_{batt\mathrm{,1}}-I_{\mathrm{1,1}}$$

Berechnung: k ≥ 2 $$\begin{array}{l}{t}_k=t_{k-1}+\Delta t\\ SO{C}_{\mathrm{1,}k}=SO{C}_{\mathrm{1,}k-1}\ast \frac{I_{\mathrm{1,}k-1}\Delta t}{{\mu }_C{x}_{C_1}},SO{C}_{\mathrm{2,}k}=SO{C}_{\mathrm{2,}k-1}\ast \frac{I_{\mathrm{2,}k-1}\Delta t}{{\mu }_C{x}_{C_2}}\end{array}$$

$${\Delta }_{2-1}{U}_{SC,k}=I_{\mathrm{1,}k-1}{\mu }_R\left(x_{R_1}+x_{R_2}\right)-I_{batt,k}{x}_{R_2}{\mu }_R$$

$$m_{U,i,k}^{*}=DVA\left(SOC,I_{batt}\right)$$

$$m_{U,i,k}=\{\begin{array}{c}{m}_{U,i,k}^{*},wenn{t}_k>5{\tau }_{RC}\\ \left(m_{U,i,k-1}-m_{U,i,k}^{*}\right)e^{-\frac{\Delta t}{{\tau }_{RC}}}+m_{U,i,k}^{*},\end{array}$$

oder $${\xi }_k=\frac{m_{U\mathrm{,1,}k}}{m_{U\mathrm{,2,}k}}$$

$${\lambda }_{\mathrm{0,}k}=\frac{{\Delta }_{2-1}{U}_k}{{\mu }_R\left(x_{R_1}+x_{R_2}\right)}+\frac{x_{C_1}}{x_{C_1}+\xi x_{C_2}}{I}_{batt,k}$$

$${\lambda }_{\infty ,k}=\frac{x_{C_1}}{x_{C_1}+{\xi }_k{x}_{C_2}}{I}_{batt,k}$$

$${\tau }_k=\frac{\left(x_{R_1}+x_{R_2}\right){\mu }_R{\mu }_C{x}_{C_1}{x}_{C_2}}{m_{U\mathrm{,2,}k}\left(x_{C_1}+{\xi }_k{x}_{C_2}\right)}$$

$$I_{\mathrm{1,}k}=\left({\lambda }_{\mathrm{0,}k}-{\lambda }_{\infty ,k}\right)e^{-\frac{\Delta t}{{\tau }_{RC}}}+{\lambda }_{\infty ,k}$$

$$I_{\mathrm{2,}k}=I_{batt,k}-I_{\mathrm{1,}k}$$

In order to use the semi-analytical model described for an in-vehicle evaluation of the cell currents, the corresponding calculation steps of the algorithm must be carried out. Since the overpotentials arise during the current load, the calculation of mu is divided into a time-dependent and a time-independent part, whereby this is carried out depending on the time constant τ of the polarization and diffusion process τ _RC and the duration t of the current load. In addition to estimating the current distribution 18 on the vehicle side, the mathematical model 22 offers a method for estimating the cell currents from the voltage data.
Initialization: k = 1 $${\mathrm{<figure-callout\; id="1"\; label="t"\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">t</figure-callout>}}_1=0$$

$$SO{C}_{1.1}=SO{C}_{2.1}=\mathrm{<figure-callout\; id="0"\; label="S\; O\; C"\; filenames="DE102020005169A1\_0042.png"\; state="\{\{state\}\}">S</figure-callout>}O{C}_{}{}_0$$

$${\mathrm{<figure-callout\; id="2"\; label="\Delta "\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">\Delta </figure-callout>}}_{2-1}{u}_{SC\mathrm{,1}}=0$$

$$m_{u\mathrm{,1}}=m_{u\mathrm{,0}}$$

$$I_{1.1}=\frac{x_{R_1}}{{\mathrm{<figure-callout\; id="1"\; label="x\; R"\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">x</figure-callout>}}_{R_{}}+x_{R_2}}{I}_{batt\mathrm{,1}},I_{2.1}=I_{batt\mathrm{,1}}-I_{1.1}$$

Calculation: k ≥ 2 $$\begin{array}{l}{t}_k=t_{k-1}+\Delta t\\ \mathrm{<figure-callout\; id="1"\; label="S\; O\; C"\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">S</figure-callout>}O{C}_{}{}_{\mathrm{1,}k}=\mathrm{<figure-callout\; id="1"\; label="S\; O\; C"\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">S</figure-callout>}O{C}_{}{}_{\mathrm{1,}k-1}\ast \frac{I_{\mathrm{1,}k-1}\Delta t}{{\mu }_C{\mathrm{<figure-callout\; id="1"\; label="x\; C"\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">x</figure-callout>}}_{C_{}}},\mathrm{<figure-callout\; id="2"\; label="S\; O\; C"\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">S</figure-callout>}O{C}_{}{}_{\mathrm{2,}k}=\mathrm{<figure-callout\; id="2"\; label="S\; O\; C"\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">S</figure-callout>}O{C}_{}{}_{\mathrm{2,}k-1}\ast \frac{I_{\mathrm{2,}k-1}\Delta t}{{\mu }_C{\mathrm{<figure-callout\; id="2"\; label="x\; C"\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">x</figure-callout>}}_{C_{}}}\end{array}$$

$${\mathrm{<figure-callout\; id="2"\; label="\Delta "\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">\Delta </figure-callout>}}_{2-1}{u}_{SC,k}=I_{\mathrm{1,}k-1}{\mu }_R\left({\mathrm{<figure-callout\; id="1"\; label="x\; R"\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">x</figure-callout>}}_{R_{}}+x_{R_2}\right)-I_{batt,k}{\mathrm{<figure-callout\; id="2"\; label="x\; R"\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">x</figure-callout>}}_{R_{}}{\mu }_R$$

$$m_{u,i,k}^{*}=DVA\left(SOC,I_{batt}\right)$$

$$m_{u,i,k}=\{\begin{array}{c}{m}_{u,i,k}^{*},wenn{t}_k>5{\tau }_{RC}\\ \left(m_{u,i,k-1}-m_{u,i,k}^{*}\right)e^{-\frac{\Delta t}{{\tau }_{RC}}}+m_{u,i,k}^{*},\end{array}$$

or $${\xi }_k=\frac{m_{u\mathrm{,1,}k}}{m_{u\mathrm{,2,}k}}$$

$${\mathrm{<figure-callout\; id="0"\; label="\lambda "\; filenames="DE102020005169A1\_0042.png"\; state="\{\{state\}\}">\lambda </figure-callout>}}_{\mathrm{0,}k}=\frac{{\mathrm{<figure-callout\; id="2"\; label="\Delta "\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">\Delta </figure-callout>}}_{2-1}{u}_k}{{\mu }_R\left({\mathrm{<figure-callout\; id="1"\; label="x\; R"\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">x</figure-callout>}}_{R_{}}+x_{R_2}\right)}+\frac{x_{C_1}}{{\mathrm{<figure-callout\; id="1"\; label="x\; C"\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">x</figure-callout>}}_{C_{}}+\xi x_{C_2}}{I}_{batt,k}$$

$${\lambda }_{\infty ,k}=\frac{x_{C_1}}{{\mathrm{<figure-callout\; id="1"\; label="x\; C"\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">x</figure-callout>}}_{C_{}}+{\xi }_k{x}_{C_2}}{I}_{batt,k}$$

$${\tau }_k=\frac{\left(x_{{\mathrm{<figure-callout\; id="1"\; label="R"\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">R</figure-callout>}}_1}+x_{R_2}\right){\mu }_R{\mu }_C{x}_{C_1}{x}_{C_2}}{m_{u\mathrm{,2,}k}\left({\mathrm{<figure-callout\; id="1"\; label="x\; C"\; filenames="DE102020005169A1\_0041.png,DE102020005169A1\_0042.png"\; state="\{\{state\}\}">x</figure-callout>}}_{C_{}}+{\xi }_k{x}_{C_2}\right)}$$

$$I_{\mathrm{1,}k}=\left({\mathrm{<figure-callout\; id="0"\; label="\lambda "\; filenames="DE102020005169A1\_0042.png"\; state="\{\{state\}\}">\lambda </figure-callout>}}_{\mathrm{0,}k}-{\lambda }_{\infty ,k}\right)e^{-\frac{\Delta t}{{\tau }_{RC}}}+{\lambda }_{\infty ,k}$$

$$I_{\mathrm{2,}k}=I_{batt,k}-I_{\mathrm{1,}k}$$

2 shows in particular the results of a complete discharge and charge of the two battery cells 14, 16 connected in parallel, these being in particular in the form of lithium-ion cells. It is shown that the simulated currents closely match the actual measurement results. This accuracy is sufficient to be able to check the corresponding current distribution 18 inside the motor vehicle, and this can be carried out in a very computationally efficient manner, as shown. In particular, the computing and parameterization effort is reduced. A characteristic crossing point 36 of the cell currents I ₁ results after the minimum of the DVA for the discharge. Because of the increase in the DVA, f then exceeds the value 1. Based on the characteristic value of the cell current _λ∞ , the capacity of the second battery cell 16 C ₂ is weighted with ξ. Hence the condition f=1 is a tipping point of the current distribution 18. This influence is also shown in the figure, where f=1 is reached just before the crossing of the cell currents, which is represented by the corresponding point 36 in the figure. The same effect of ξ causes current passage during charging. In addition, the correlation of the time constant τ with the slope m _U affects the current distribution 18, which leads to rapid convergence of the cell currents at the beginning of charging.

A semi-analytical mathematical model 22 for the current current distribution 18 within battery cells 14, 16 connected in parallel is thus shown. Due to the analytical solution, the current distribution 18 can be described by the three characteristic values. In this case, the computing and parameterization effort of the mathematical model 22 for the battery management system 12 is reduced. The mathematical model 22 is in particular, as in 2 shown, validated accordingly.

reference list

10

On-board electrical system

12

battery management system

14

First battery cell

16

Second battery cell

18

power distribution

20

motor vehicle

22

Mathematical Model

24

electronic computing device

26

First curve

28

Second curve

30

Third curve

32

Fourth curve

34

Fifth curve

35

crossing point

Ibatt

current

I1

First stream

I2

second stream

USC,1

First voltage source

USC,2

Second power source

RSER,1

First serial resistor

RSER,2

Second serial resistor

Claims (10)

Method for estimating a current distribution (18) within an on-board electrical system (10) of a motor vehicle (20) using a battery management system (12), wherein in the on-board electrical system (10) at least one first battery cell (14) and one for the first battery cell (14) second battery cell (16) connected in parallel are provided, and using a mathematical model (22) of an electronic computing device (24) of the battery management system (12) on the basis of at least one first characterizing parameter of the first battery cell (14) and on the basis of at least a second characterizing parameter Parameters of the second battery cell (16), the current distribution (18) is estimated by means of the electronic computing device (24), characterized in that in the mathematical model (22) the electrical system (10) is converted into a simplified equivalent circuit diagram and by determining a current ( λ _∞ ) at an infinite point in time in Dep Dependence on the at least two characterizing parameters, by determining a current (λ ₀ ) at a zero point in time depending on the at least two characterizing parameters and by determining a time constant (τ) depending on the at least two characterizing parameters, estimating the current distribution (18) is carried out by means of the mathematical model (22). procedure after claim 1 , characterized in that a first current (I ₁ ) in the first battery cell (14) by subtracting the current (I ₂ ) from the second battery cell (16) from the current (I _batt ) in the electrical system (10) and vice versa is determined. procedure after claim 1 or 2 , characterized in that a first current (I ₁ ) in the first battery cell (14) using the formula: $$I_1\left(t\right)=\frac{x_{C_1}}{x_{C_1}+\xi x_{C_2}}{I}_{batt}\left[\left(\frac{x_{R_2}}{x_{R_1}+x_{R_2}}\frac{x_{C_1}+\xi x_{C_2}}{x_{C_1}}-1\right)e^{-\frac{t}{\tau }}+1\right]$$

is determined, where x _{R i} a respective battery cell resistance in relation to a respective expected battery cell resistance, x _{C i} a battery cell capacity in relation to a respectively expected battery cell capacity, f a ratio of the respective voltage gradients of the battery cells (14, 16), I _{batt correspond} to the current in the vehicle electrical system (10), t correspond to a point in time and τ to the time constant. Method according to one of the preceding claims, characterized in that the determination of a first current (λ _∞ ) in the first battery cell (14) at an infinite time using the formula: $${\lambda }_{\infty }=I_1=\frac{x_{C_1}}{x_{C_1}+\xi x_{C_2}}{I}_{batt}$$

is performed, where x _{C i} a battery cell capacity in relation to a respectively expected battery cell capacity, ξ a ratio of the respective voltage gradients of the battery cells (14, 16) and I _{batt correspond} to the current in the vehicle electrical system (10). Method according to one of the preceding claims, characterized in that the determination of a first current (λ ₀ ) in the first battery cell (14) at a zero point in time using the formula: $${\lambda }_0=I_1=\frac{x_{R_1}}{x_{R_1}+x_{R_2}}{I}_{batt}$$

is performed, where x _{R i} a respective battery cell resistance in relation to a respective expected battery cell resistance and I _{batt correspond} to the current in the vehicle electrical system (10). Method according to one of the preceding claims, characterized in that the determination of the time constant (τ) by means of the formula: $$\tau =\frac{\left(x_{R_1}+x_{R_2}\right){\mu }_R{\mu }_C{x}_{C_1}{x}_{C_2}}{m_{u\mathrm{,2}}\left(x_{C_1}+\xi x_{C_2}\right)}$$

is performed, where x _{R i} a respective battery cell resistance in relation to a respective expected battery cell resistance, x _{c i} a battery cell capacity in relation to a respectively expected battery cell capacity, f a ratio of the respective voltage gradients of the battery cells (14, 16), m _U,2 the voltage gradient of the second battery cell (16), µ _R the expected battery cell resistance and µ _c the expected battery cell capacity. Method according to one of the preceding claims, characterized in that the equivalent circuit diagram is simplified by specifying that the first battery cell (14) and the second battery cell (16) are each replaced by a series resistor (R _ser ) and a voltage source (Usc) are described. procedure after claim 7 , characterized in that the equivalent circuit diagram is simplified by specifying that a respective serial resistance (R _ser ) is invariant. procedure after claim 7 or 8th , characterized in that the equivalent circuit diagram is simplified by specifying that a respective battery cell capacity of a respective battery cell (14, 16) is invariant. Battery management system (12) for a motor vehicle (20), with at least one electronic computing device (24), wherein the battery management system (12) for performing a method according to one of Claims 1 until 9 is trained.

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