Techniques for estimating operating parameters of a battery cell using a battery cell model with a number of N RC elements

Abstract

The invention relates to a method (100) for determining at least one estimated operating parameter (111) of a battery cell of a battery-powered vehicle, comprising the following steps: receiving (101) at least one measured operating parameter (110) of the battery cell; determining (102) at least one estimated operating parameter (111) of the battery cell from the at least one measured operating parameter (110) of the battery cell using a battery cell model (120) to describe a dynamic behavior of the battery cell, wherein the battery cell model (120) is based on an equivalent circuit (121) of the battery cell comprising a number of N RC elements (122), wherein the battery cell model (120) represents a cell voltage (U) applied to the battery cell. Cell ) depending on model parameters (R i , C i ) of the battery cell model (120) is defined, where the model parameters (R i , C i) on a plurality of first coefficients (b i ) are based on, where a respective first coefficient (b i ) is a sum of second coefficients (a i ) determined over the number of N RC terms (122), where a respective second coefficient (a i ) on a Taylor series expansion of a resistor (R i ) of a corresponding RC element of the N RC elements (122) based on a state of charge of the battery cell around an operating point of the battery cell.

Description

Technical area

[0001] The present invention relates to techniques for estimating operating parameters of a battery cell, such as a lithium-ion cell, using a battery cell model with a number of N RC elements. Several such battery cells form a battery that can be used in a battery-powered vehicle. In particular, the invention relates to a general method for parameterizing N RC ECM models. State of the art

[0002] In the automotive sector, so-called ECM models ("Equivalent Circuit Models") are used for a wide variety of tasks. Single and double RC models are predominantly used to simulate the electrical behavior of lithium-ion cells. Models with multiple RC elements are also known. The choice of model – the number N of RC elements – and the accuracy of the parameters play a major role in ensuring the accuracy of the simulation and thus its application for the diverse tasks. This includes, for example, the design of the battery system in the vehicle according to the requirements of the OEM ("Original Equipment Manufacturer"), the development and testing of control units, in particular the "Battery Management System" (BMS) and the software running on it.

[0003] The literature generally states that the nonlinear behavior of the cell can be simulated more accurately using more complex models (e.g. electrochemical model) or by ECM models with additional RC elements. Usually, this only involves a comparison of 1- or 2-RC models with the mean square error between measured and simulated values, and a decrease in the RMS (Root Mean Square) error for 2-RC ECM models is numerically proven. The decrease in the RMS error with the number of RC elements is demonstrated using various driving cycles. It is unclear whether the RMS value converges to the value "0" for N >> 0. It is unclear whether an even remotely complete description of cell behavior can be achieved with N-RC elements. Usually, reference is made to electrochemical models, which enable more precise modeling.However, this fact is not mathematically proven in the literature and the relationship with current amplitude, pulse length and SOC dependence, especially for multiple RC elements, is not systematically investigated.

[0004] The determination of operating parameters, such as the state of charge, is known from DE 10 2022 105 481 A1, from DE 10 2022 134 261 A1 or also from WO 2019 / 020 620 A1. Description of the invention

[0005] An object of the invention is therefore to create a concept for an improved battery cell model that can be used, for example, for performance prediction in the battery management system of battery-powered vehicles.

[0006] In particular, it is an object of the invention to provide an improved identification of the parameters of an N-RC-ECM model, ie a battery cell model with an equivalent circuit based on a number of N RC elements.

[0007] The object is achieved by the subject matter of the independent claims. Advantageous developments of the invention are specified in the dependent claims, the description, and the accompanying figures.

[0008] The invention is based on the idea of ​​overcoming the limitations made so far regarding the dependencies of the parameters in the battery cell model, as shown below.

[0009] The 1-RC-ECM model is described by a system of two first-order differential equations (DELs). A special analytical solution to the DEL exists, which is used to find a fitting function. A comparison of the values ​​of this function with measured values ​​then yields a set of ECM parameters. For the special analytical solution, certain assumptions are made regarding the dependencies of the parameters. In order to be able to compare the values ​​of the analytical solution with the measured values, certain restrictions regarding the magnitude and duration of the current loads also apply to the measurement. The validity range of the fully parameterized model at a given accuracy is therefore ultimately also limited. Expanding the validity range regarding the magnitude and duration of the current loads results in lower accuracy.

[0010] This problem is now solved by the invention in that the assumptions previously required for deriving the analytical solution are no longer completely necessary. This allows the evaluation of data in which the state of charge of the lithium-ion cell changes due to the magnitude and duration of the current loads during a charge or discharge pulse. The removal of assumptions can be transferred to any N-RC-ECM model, thus eliminating the previous limitations in the parameterization of static offline models for simulating a cell / pack / entire battery system or for a vehicle simulation and an adaptive state observer or parameter estimator with a BMS2Cloud connection (digital twin).

[0011] At this point, two further areas of application of the invention should be mentioned: 1. The invention is suitable for determining a limit current at a given state of charge, state of health, temperature and limit voltage or equivalently for the highly accurate adaptive power prediction in the electric vehicle (SOP, "state of power" or SOF, "state of function" function). 2. For the implementation of an extended Kalman filter (EKF).

[0012] With an adaptive state observer / parameter estimator directly on the control unit, the issue remains that resources—available memory (RAM) and performance (CPU)—are limited. By more precisely determining the model parameters of the battery cell model, as described in this disclosure, advantages can also be achieved with regard to the aforementioned applications.

[0013] The invention offers the technical advantages described below over solutions according to the current state of the art.

[0014] According to the current state of the art, cell characterization and parameter identification of an ECM model, for example, for a digital twin, requires specific current and voltage data series to be recorded in the laboratory to ensure the required accuracy of the ECM model (e.g., 2%) after parameterization. Ideally, a synthetic measurement profile must be available that contains current pulses that each contribute only a slight change to the battery's SOC. This means that the pulses must not exceed a certain pulse length and amplitude to achieve the required accuracy. For an online parameter estimator that is intended to account for cell aging, a balance must be struck between the accuracy and frequency of parameter determination.

[0015] The solution according to the invention offers the following technical advantages: 1. Any current-voltage data series can be used for parameter identification of RC-ECM models. 2. With an adaptive state observer / parameter estimator, parameter determination can be carried out regularly with the same accuracy. 3. The new method provides additional information about the necessary number of RC elements, such as below for the Fig. 6 and Fig. 9 described. 4. This method allows a systematic investigation and optimized determination of the number of RC elements (or number of necessary parameters) for a specified accuracy. 5. Trial and error - investigations that risk not achieving the required accuracy or over-dimensioning the system without improving accuracy - as is the case with the current state of the art - are avoided. 6. The new redundant method can be used alongside other methods (such as AI, neural networks) to validate the old method, to further secure the values ​​(diagnosis) and to significantly increase the accuracy / reliability / robustness of a parameter estimator (BMS2Cloud connection). 7. The new method can also explicitly determine a parameterization of a new RC element, which specifically allows deviations up to order O[t] 2 taken into account, such as below for Fig. 8 described.

[0016] The invention can be used in customized and customer-oriented solutions for battery condition detection in electric vehicles. This ensures the goal of the most gentle and safe battery operation possible over its lifetime. The invention significantly improves existing solutions and is compatible with such existing solutions. The invention can be used in many other areas of application and is not limited to battery condition detection. One such area of ​​application is, for example, the so-called digital service for companies, which involves monitoring the service life of EV batteries in electric vehicles in the field using a digital twin and offering services in the areas of diagnostics, error analysis, and warranty.

[0017] The essential difference between the new procedure for parameterizing ECM models presented here and the standard procedures is easily recognized by comparing the parameterized ECM model with the measurement. In the following description of Fig. Figure 8, for example, shows a comparison between the measurement signal, the solution for a 2-RC model according to the standard method, and the new method according to the invention. While the previous method searches for a parameter set using the RMS (Root Mean Square) for the second RC element, the new method according to the invention explicitly finds parameters that specifically describe deviations up to the order O[t]. 2 This is easy to see. The curve according to the inventive method lies on the curve according to the standard method in the first 150 s. Only after that, when higher terms in O[t] 3play a role, deviations arise between the measurement and simulation curves.

[0018] According to a first aspect of the invention, the object is achieved by a method for determining at least one estimated operating parameter of a battery cell of a battery-powered vehicle, comprising the following steps: receiving at least one measured operating parameter of the battery cell;Determining at least one estimated operating parameter of the battery cell from the at least one measured operating parameter of the battery cell using a battery cell model to describe a dynamic behavior of the battery cell, wherein the battery cell model is based on an equivalent circuit of the battery cell comprising a number N of RC elements, wherein the battery cell model defines a cell voltage applied to the battery cell as a function of model parameters of the battery cell model, wherein the model parameters are based on a plurality of first coefficients, wherein a respective first coefficient is determined from a sum of second coefficients over the number N of RC elements, wherein a respective second coefficient is based on a Taylor series expansion of a resistance of a corresponding RC element of the N RC elements according to a state of charge of the battery cell around an operating point of the battery cell;

[0019] Such a dependence of the first coefficients b i (i=1... M) of the second coefficients a i,1 (i=1... N) is described in more detail below in the description of Fig. 2, in particular equations (17) and (18) define such a relationship. An example of such a Taylor series expansion of the resistance of the respective RC element is shown below. Fig. 2, in particular defined by equations (7) and (8).

[0020] Such a method creates improved modeling of the battery cell using a battery cell model. It can be used, for example, to more accurately predict performance in the battery management system of battery-powered vehicles. The method presented here enables improved identification of the parameters of the N-RC-ECM model. The N-RC-ECM is a battery cell model with an equivalent circuit based on N RC elements. N is generally greater than one, but the method can also be used for a 1-RC-ECM model, i.e., a battery cell model with an equivalent circuit that includes only one RC element.

[0021] According to an exemplary embodiment of the method, the first coefficients define a temporal dependence of the cell voltage applied to the battery cell and are in the form of parameters of a polynomial representation of the cell voltage over time.

[0022] Such a polynomial representation of the cell voltage with time is shown below, for example, Fig. 2, in particular by equation (17), in which the time terms t, t 2 , t 3 , ... t M with the corresponding first coefficients b i (i=1... M) are defined.

[0023] According to an exemplary embodiment of the method, the first coefficients associated with odd orders of the cell voltage versus time polynomials have positive signs; and the first coefficients associated with even orders of the cell voltage versus time polynomials have negative signs.

[0024] An example of such first coefficients is shown below. Fig. 2, for example represented by equation (23), where b1, b3 and b5 have positive signs and where b2, b4 and b6 have negative signs.

[0025] According to an exemplary embodiment of the method, a capacitance of a first RC element of the number N RC elements decreases over time with increasing order of the polynomials of the cell voltage and tends towards a limit value; and the capacitance abruptly leaves the limit value from a certain order.

[0026] An example of such a dependence on order is in Fig. 6, where the capacitance value of C1, ie the capacitance of the first RC element, decreases as a function of the order M=1 to M=6 from an amount of 120,000F (M=0) to 40,000F (M=6), which represents the limit value, and then decreases for M=7 to a value close to 0F (= jump to a very low value).

[0027] According to an exemplary embodiment of the method, an optimal number of first coefficients is determined based on the order of the polynomials of the cell voltage with the time which lies before the order at which the capacity abruptly leaves the limit value.

[0028] Here in the example of Fig. 6 would be the optimal number of first coefficients b i thus a number of 6, because the limit value (40,000F) is abruptly left after order 6 and jumps towards zero.

[0029] According to an exemplary embodiment of the method, the model parameters of the battery cell model depend on the state of charge of the battery cell and a temperature of the battery cell; and the operating point of the battery cell is defined for a specific state of charge and a specific temperature of the battery cell.

[0030] This dependence of the operating point of the battery cell on the temperature and the state of charge SOC of the battery cell can be seen below, for example: Fig. 2, especially equation (6) is described in more detail.

[0031] According to an exemplary embodiment of the method, the method comprises: detecting the state of charge of the battery cell and the temperature of the battery cell; transmitting the state of charge and the temperature of the battery cell to a controller for determining the model parameters of the battery cell model based on the detected state of charge and the detected temperature; and receiving the model parameters of the battery cell model determined by the controller.

[0032] This realizes the technical advantage that the computationally complex calculations do not have to be performed in the battery cell or battery management system (BMS), but can be executed by an external controller or a processor in the cloud. The relevant data simply needs to be sent from the battery cell or BMS to the external controller, and the results are received by the external controller.

[0033] According to an exemplary embodiment of the method, the at least one measured operating parameter of the battery cell defines a discharge pulse or a charge pulse of the battery cell depending on the time.

[0034] The SOC ("state of charge") dependence of the cell's internal resistance is particularly evident during prolonged, continuous charging or discharging at low current or during a prolonged charging / discharging pulse with very high current. This disclosure demonstrates that the SOC dependence of a battery cell's internal resistance can be fully described by an N-RC-ECM model, using a lithium-ion cell as an example. The method presented here can therefore be used for a wide variety of charging and discharging pulses and leads to a precise determination of the model parameters of the battery cell model, regardless of the specific type of charging or discharging pulse selected.

[0035] According to an exemplary embodiment of the method, the model parameters of the battery cell include capacitances and resistances of the N RC elements; and the model parameters of the battery cell also include an internal resistance and an open-circuit voltage of the battery cell.

[0036] The N-RC-ECM model presented here and the determination of the model parameters of the battery cell are based on an equivalent circuit of the battery cell with a general number of N RC elements. N is generally greater than one. However, the model also delivers suitable results with only one RC element (N=1). Furthermore, the equivalent circuit includes an internal resistance R0 of the battery cell. The equivalent circuit is shown for the example N=1 in Fig. 2. The corresponding time constants τ i = R i C i from the resistance and capacitance values ​​of the RC elements as well as the open circuit voltage U 0CVThe battery cell model defines a relationship between the battery cell voltage U Cell and the cell current I flowing through the battery cell Cell depending on the model parameters.

[0037] According to an exemplary embodiment of the method, the equivalent circuit of the battery cell comprises a series circuit of an internal resistor and the N RC elements.

[0038] The mathematical analysis presented below shows that with such an equivalent circuit, suitable model parameters can be determined which very accurately simulate the properties of the battery cell, in particular its dynamic behavior.

[0039] According to a second aspect of the invention, the object is achieved by a computer program having a program code for executing the method according to the first aspect on an electronic circuit or on a computer or on a battery management system or on a processor in a cloud.

[0040] This provides the technical advantage that the computer program can easily be executed on different hardware Short character description

[0041] An advantageous embodiment of the invention is explained below with reference to the accompanying figures. They show: Fig. 1 is a schematic representation of a method for determining at least one estimated operating parameter of a battery cell of a battery-powered vehicle according to the disclosure; Fig. 2 is a diagram showing a general equivalent circuit of a 1-RC-ECM model according to an example; Fig. 3 a comparative representation of a measurement and a standard function for the 1-RC ECM model at a short discharge pulse of 108 seconds according to an example; Fig. 4 a comparative illustration of a measurement and a standard function for the 1-RC ECM model at a long discharge pulse of 1080 seconds according to an example; Fig. 5 a representation of a SOC-dependent comparison of the parameters b1 and ψ; Fig. 6 a representation of the parameter C1 for different orders O[t] M ; Fig. 7 a representation of the difference between measurement and fit function according to equation (17); Fig. 8 a comparative representation of a measurement and a result of the method according to the invention for a 2-RC-ECM model according to an example; and Fig. 9 a comparative representation of a measurement and a result of the method according to the invention for a 3-RC-ECM model according to an example.

[0042] The figures are merely schematic representations and serve only to illustrate the invention. Identical or equivalent elements are provided with the same reference numerals throughout.

[0043] In the following detailed description, reference is made to the accompanying drawings, which form a part hereof, and in which is shown by way of illustration specific embodiments in which the invention may be practiced. It is understood that other embodiments may be utilized and structural or logical changes may be made without departing from the scope of the present invention. The following detailed description, therefore, is not to be taken in a limiting sense. Further, it is to be understood that the features of the various embodiments described herein may be combined with one another unless specifically indicated otherwise.

[0044] The aspects and embodiments are described with reference to the drawings, where like reference numerals generally refer to like elements. In the following description, for purposes of explanation, numerous specific details are set forth in order to provide a thorough understanding of one or more aspects of the invention. However, it may be apparent to one skilled in the art that one or more aspects or embodiments may be practiced with a lesser level of specific detail. In other instances, well-known structures and elements are shown in schematic form to facilitate describing one or more aspects or embodiments. It is to be understood that other embodiments may be utilized and structural or logical changes may be made without departing from the concept of the present invention.

[0045] Fig. 1 shows a schematic representation of a method 100 for determining at least one estimated operating parameter 111 of a battery cell of a battery-powered vehicle according to the disclosure.

[0046] The method 100 comprises the steps of: receiving 101 at least one measured operating parameter 110 of the battery cell; and determining 102 at least one estimated operating parameter 111 of the battery cell from the at least one measured operating parameter 110 of the battery cell using a battery cell model 120 for describing a dynamic behavior of the battery cell.

[0047] Here, the battery cell model 120 is based on an equivalent circuit 121 of the battery cell, which comprises a number of N RC elements 122, as shown by way of example in Fig. 2 for a number of N=1.

[0048] The battery cell model 120 defines a cell voltage U applied to the battery cell Celldepending on model parameters R i , C i of the battery cell model 120, as exemplified in Fig. 2 shown.

[0049] The model parameters R i , C i on a plurality of first coefficients b i , where a respective first coefficient b i consists of a sum of second coefficients a i by the number of N RC elements 122, and where a respective second coefficient a i on a Taylor series expansion of a resistor R i a corresponding RC element 122 according to a state of charge of the battery cell around an operating point of the battery cell, as described below Fig. 2 described in more detail.

[0050] The first coefficients b i can be a temporal dependence of the cell voltage U applied to the battery cell Celldefine and in the form of parameters of a polynomial representation of the cell voltage U Cell over time, as shown below Fig. 2 described in more detail.

[0051] The first coefficients b i , which odd orders of the polynomials of the cell voltage U Cell associated with time can have positive signs; and the first coefficients b i , which even orders of the polynomials of the cell voltage U Cell associated with time can have negative signs.

[0052] A capacity 600 of a first RC element of the number N RC elements 122 can be increased with increasing order of the polynomials of the cell voltage U Cell decrease over time and tend towards a limit value of 601, as for example in Fig. 6. The capacity can leave the limit value 601 abruptly 603 from a certain order, as also shown in Fig. 6 shown.

[0053] An optimal number of first coefficients b i is determined, for example, based on the order of the polynomials of the cell voltage U Cell with the time which lies before the order at which the capacity leaves the limit 601 abruptly 603, as in Fig. 6 shown.

[0054] The model parameters R i , C i The values ​​of the battery cell model 120 depend, for example, on the battery cell's state of charge and the battery cell's temperature. The operating point of the battery cell is defined, for example, for a specific state of charge and a specific battery cell temperature.

[0055] The method 100 may, for example, comprise the further steps of: detecting the state of charge of the battery cell and the temperature of the battery cell; transmitting the state of charge and the temperature of the battery cell to a controller for determining the model parameters R i , C iof the battery cell model 120 based on the detected state of charge and the detected temperature; and receiving the model parameters R determined by the controller i , C i of the battery cell model 120.

[0056] Thus, the battery cell model 120 does not have to be determined by the method 100 itself, but can be determined by an external control computer or a processor in the cloud based on the states of the battery cell detected by the method. The results determined by the controller, i.e., the external control computer or the processor in the cloud, can then be used by the method. The method 100 can, for example, be a computer-implemented method that is implemented on a computer, a control processor, or an electronic circuit. The computer on which the method 100 is executed then does not need to have particularly high hardware complexity, since the particularly computationally intensive tasks are executed on the external control computer or in the cloud.

[0057] For example, the at least one measured operating parameter 110 of the battery cell can define a discharge pulse or a charge pulse of the battery cell depending on time.

[0058] For example, the model parameters R i , C i of the battery cell capacitances and resistances of the N RC elements 122. The model parameters R i , C i The battery cell can also have an internal resistance R0 and an open circuit voltage U OCV of the battery cell, as in Fig. 2 is shown as an example.

[0059] The equivalent circuit 121 of the battery cell may comprise a series circuit of an internal resistance R0 and the N RC elements 122, as shown in Fig. 2 is shown as an example for a single RC element, i.e., N=1. It is understood that N is generally greater than one, so that the equivalent circuit 121 of the battery cell comprises several RC elements 122.

[0060] Fig. 2 shows a representation of a general equivalent circuit 121 of a 1-RC-ECM model according to an example.

[0061] The equivalent circuit 121, which describes the dynamic behavior of the battery cell, consists of an internal resistance R0, to which an RC element 122 is connected in series. The internal resistance R0 is connected to an open-circuit voltage U OCV switched, which causes a current flow I Cell caused by the battery cell. At the output of the battery cell, a cell voltage U Cell The RC element 122 consists of a parallel circuit of an electrical resistor R1 with a capacitance C1. When flowing through the RC element 122, the cell current I Cell into a resistive component I R1 and a capacitive component I C1 A voltage of U1 drops across the RC element 122.

[0062] In an equivalent circuit 121 comprising a number of N RC elements, several such RC elements 122 are connected together, preferably in series with each other and with the internal resistance R0. The voltages U1 to U across the N RC elements N are added together with the voltage U0 dropping across the internal resistance R0 to give the cell voltage U Cell to generate.

[0063] That is, the cell voltage U Cell The battery cell consists of the various components (open-circuit voltage, etc.) and the voltages of the RC elements.

[0064] This relationship can also be represented as follows: U(t)≈UOCV+U0+∑i=1NUi, where U0=ICellROhm

[0065] Where R Ohm the ohmic component of the total resistance.

[0066] The SOC (“State Of Charge”) dependence of the cell’s internal resistance is particularly evident during longer, continuous charging or discharging with low

[0067] Current or a longer charge / discharge pulse (e.g., 30 s) with very high current. In the following sections, it is shown mathematically that the SOC dependence of the internal resistance of a lithium-ion cell can be fully described by an N-RC-ECM model.

[0068] For simple illustration and technical description, a 1-RC-ECM model is initially chosen according to the representation in Fig. 2 and the derivatives are then transferred to an N-RC-ECM model, i.e. a battery cell model with a number of N RC elements 122. The N-RC-ECM model (e.g. N = 3) extends the 1-RC-ECM model by adding (e.g. two) further RC elements 122.

[0069] To describe the dynamic behavior of a lithium-ion cell as an example of a battery cell, the Fig. 2 shown cell model with the voltage source U OCV and an equivalent circuit model (ECM). This model consists of an ohmic resistor and an RC element with a fixed time constant τ1 = R1C1.

[0070] The U OCV , R0, R1 and C1 values ​​depend on the temperature and the state of charge (SOC). If the notation is according to Fig. 2 is used, the simulated voltage response of the battery cell is: UCell(SOC(t),T(t))=UOCV(SOC(t),T(t))+R0(SOC(t),T(t))ICell(t)+U1(SOC(t),T(t))

[0071] The equivalent circuit diagram 121 according to Fig. 2 can be converted into a differential equation for the voltage U1(t): τ1(SOC(t),T(t))ddtU1(t)+U1(t)=R1(SOC(t),T(t))ICell(t), ddtSOC(t)=ICell(t)36 CCell, τ1=R1(SOC(t),T(t))C1(SOC(t),T(t)) where C Cell is the capacity of the cell. The solution of the differential equation for U1(t) is possible under certain conditions and yields a fitting function for two cases. In the first case, the solution U Cell (t) the voltage behavior of the cell during a discharge pulse: UCell(t)=UOCV+R0ICell+R1ICell(1−e−tτ1), τ1=R1C1

[0072] In the second case, the behavior of the cell when the current is switched off, i.e. during the relaxation process, is reproduced: UCell(t)=UOCV+U1e−tτ1, τ1=R1C1

[0073] These fitting functions are used by default to identify the R1 and C1 values ​​at different cell states of charge and temperatures. The following assumptions are made in the literature for the analytical solution of the ODE: 1. The open circuit voltage is a function of U 0CV(SOC, T) is already well determined and independent of the current direction. 2. For the time being, all parameters are assumed to be current independent, both in direction and amplitude. 3. Near each operating point (SOC, T) the values ​​R0, R1, C1 and τ1 are assumed to be constant.

[0074] Because the parameters are dependent on SOC and temperature, according to Equation (2), they are also explicitly dependent on current and time. Therefore, points 2 and 3 are obviously not independent of each other. The larger the current amplitude, the shorter the time must be chosen for which the values ​​R0, R1, C1 can be assumed to be constant. These assumptions, which of course only apply in the limiting case "pulse time → 0", provide a set of parameters for a specific operating point (e.g., SOC = 50%, T = 25°C). Once the parameters for all operating points are fixed (SOC in steps of, for example, 10%), the simulation results of the parameterized ECM model are usually compared with another measurement (e.g., a driving cycle with longer pulse times), and the parameters are optimized manually or with another optimization algorithm. This method usually finds a parameter set that mathematically exhibits the smallest squared error (local minimum).

[0075] The general procedure for parameterizing N - RC ECM models is described below.

[0076] For this purpose, the case is considered where the characteristic time constant τ is approximately constant for each pulse: T0τ1(SO(t),T(t),SOC(t0))=τ1(SOC(t0))

[0077] Here, T0 denotes the 0th term of a Taylor series expansion and means that higher-order terms T1... can be neglected. This assumption has already been confirmed by many measurements with moderate pulse times and current amplitudes. The time constant plays a crucial role in the relaxation of the voltage to the equilibrium state. During the relaxation time, however, τ does not change over time because the cell is not under load in this state. The SOC dependence of τ essentially becomes apparent after the cell has been conditioned to different states of charge. Mathematically, this means that the Taylor series with the expansion point (SOC, T) is truncated after the first term at a constant temperature T, for example. In contrast, the change in R1 (SOC(t), T(t)) during the pulse should be taken into account first. This is always necessary when the pulse times or current amplitudes play a role.

[0078] In the Fig. 3 and Fig. 4 is, for example, a discharge of the cell with a low current (C / 3) and a pulse duration of 108 s (1% SOC, Fig. 3) and 1080 s (10 % SOC, Fig. 4). The cell voltage 300, 400 is shown over time, with the curve 301 in Fig. 3 and curve 401 in Fig. 4 shows a standard function for a 1-RC-ECM model according to equation (4). The curve 302 in Fig. Figure 3 shows a measurement with a short pulse of 108 s at T=25°C and SOC=100%. Curve 402 in Fig. Figure 4 shows a measurement with the long pulse of 1080 s at T=25°C and SOC=100%.

[0079] As an example, the first term in the Taylor series of R1 is considered: T1R1(SOC(t),SOC(t0))=R1(SOC(t0))+ddSOCR1(SOC)|SOC→SOC(t0)(SOC(t)−SOC(t0))

[0080] As can easily be seen, the SOC dependence of R4(SOC(t), T(t)) can be determined using the respective derivatives of the polarization resistance (or internal resistance) R1 with respect to the state of charge SOC, so that: τ1(SOC(t),T(t))ddtU1(t)+U1(t)=∑k=0Ma1,kSOC(t)k a1,0=R1(SOC(t0))ICell

[0081] The 1st index of the parameter α 1,k expresses that for further RC elements further parameters α 2,k , ... α N,k must be taken into account. With the help of the solution for the SOC SOC(t)=SOC(t0)+tICell36 CCell the SOC dependence can also be equivalently expressed in terms of time t: τ1(SOC(t),T(t))ddtU1(t)+U1(t)=∑k=0Ma1,ktk

[0082] This differential equation can now be solved in a series representation. For the sake of clarity, it is sufficient to demonstrate this here: U1(t)=e−tτ∑k=0M(−1)1−ka1,kτk(Γ[1+k]−Γ[1+kr−tτ1])

[0083] The complete and incomplete gamma functions Γ [α] and Γ [a, x], respectively, are special mathematical functions. The case M = 0 yields the well-known solution (see equation (4)), which is used as the standard solution for case 1: U1(t)=e−tτ1(1−+etτ1)a1,0 a1,0=R1(SOC(t0))ICell

[0084] For M = 1 we get: U1(t)=e−tτ1(−1+etτ1)a1,0+a1,1t+τa1,1(−1+etτ1)

[0085] Since the sum is defined here as an expansion of the terms in O[t] M is to be understood as follows: limτ1>>tτa1,1(−1+etτ1)=0

[0086] Therefore, it can be written approximately: U1(t)≈e−tτ1(−1+etτ1)a1,0+a1,1t

[0087] In the following, instead of approximately “≈”, “=” is used. According to equation (15), an additional linear term is obtained. In the publication DE 10 2022 105 481 A1 it was shown that the linear part ψ = a 1,1 , which has to be taken into account in a 1-RC model due to the SOC change during a pulse, can be written as follows: ψ=(ddSOCR1(SOC)|SOC→SOC(t0)+ddSOCR0(SOC)|SOC→SOC(t0) +1ICellddSOCUOCV(SOC)|SOC→SOC(t0))ICell36CCell

[0088] If the 1-RC-ECM model is extended by adding further RC elements, the respective voltages (U1(t)... U N (t)) of the RC elements can be described analogously to U1(t). However, if higher-order terms in time are considered for the SOC dependence at high currents and pulse durations according to equation (11) and solution (1), the following applies: U(t)=UOCV+U0+e−tτ0(−1+etτ0)b0+b1t+b2t2+b3t3+…+bMtM

[0089] The total voltage of the cell is described here, which, unlike U1, is available for measurement and, of course, is also used exclusively in the literature for parameterizing RC elements. Analogous to the voltages, the coefficient b1, which increases linearly with time, is composed of the respective coefficients a. i,1 together: b1≈∑i=1Nai,1=ψ+∑i=2Nai,1

[0090] Here, the parameter ψ has already been explicitly calculated in equation (16). The equation explicitly states that the coefficient b1 is the sum of the coefficients a i,1 of the respective voltages or in other words the t-dependence of the measuring signal (e.g. as shown in Fig. 4) is calculated additively from the t-dependence of the respective voltages (U1(t)... U N (t)) of the RC elements. According to equation (18), for N > 2 b1 ≠ ψ.

[0091] In other words, if a pulse such as in Fig. 4 the SOC changes significantly, then the t-dependence of other RC elements also plays a role. To prove this assumption, the coefficient proportional to "t" of several long pulses (1080 s) at different SOC values ​​was evaluated (see illustration in Fig. 5, curve 502) and with the parameter ψ(SOC(t)) ( Fig. 5, curve 501, see also equation (16)). Since the two curves 502, 501 have a different course, it follows that the coefficient ψ does not completely describe the t-dependence, but that further t-dependent terms a 2,1 ... a N,1 play a role.

[0092] In Fig. Figure 3 shows a discharge pulse at an initial state of charge of SOC = 100% of 108 s with C / 3 at 25°C. This discharge pulse was fitted with the standard fit function 301 for a 1-RC ECM model (M = 0). The fit yields the following ECM parameters and RMS values: R0=0.0011 R1=0.00198; C1=36378 τ1=R1C1=72.2 RMS=0.0011

[0093] In comparison, Fig. 4 a longer discharge pulse of 1080 s. All other values ​​and the fit function are identical: R0=0.0017 R1=0.0118; C1=118285 τ1=R1C1=1407.2 RMS=0.0021

[0094] It is immediately noticeable that all parameters are larger with the longer discharge pulse. In particular, the characteristic time τ has increased by a factor of 20. The reason for this is that the implemented solution methods, which are based on standard software for evaluating mathematical formulas (e.g., Mathematica from Wolfram Research or Matlab from The MathWorks Inc.), in the course of minimizing the mean square error RMS, the linear behavior of the measurement (see Fig. 4) by larger parameters R1 and C1 or by a larger τ= R1C1.

[0095] The linear part is also included in the characteristic part of the 1-RC model, as the Taylor series expansion shows: R1(1−etR1C1)=tC1−t22C12R1+…+O[t]3

[0096] The other reason is that the measured total voltage at the terminals is used to determine the ECM parameters, and the voltage U1 of the 1-RC model is, of course, unknown. In other words, when using a 1-RC model to describe the voltage behavior of the cell, U≈UOCV+U0+U1,where U0=IcellROhm

[0097] If a cell measurement is used for parameter identification in which the synthetic measurement profile consists of short and long pulses, the standard procedure provides different values ​​for the ECM parameters depending on whether short or long pulses are evaluated.

[0098] According to equation (17), the question now arises which order M in t should be chosen for the fitting function if the RMS error changes only minimally with M, combined with different values ​​for R1 and C1. Fig. Figure 6 shows an example of the values ​​for C1 for different orders in “t”.

[0099] For M = 0, the value for C1 is not plausible for the reasons mentioned above. As can be seen from Fig. 6, the value of C1 decreases with increasing order M and reaches approximately the value of a short pulse at M = 6. For M = 7, a significant jump is observed and here one obviously finds the parameters of a completely new RC element, which determines the voltage behavior (see Fig. 4) with the t-dependence with M = 7. Further investigations at the short pulse times (see Fig. 3) also show the significant jump, but as expected already at M = 2.

[0100] Regression models for ECM models are usually created using NLS ("Nonlinear Least Square"). The inaccuracy between measured and simulated values ​​in 1-RC models arises from the fact that NLS, based on RMS, finds parameters that are only local (pulse) but not global (cycle), and are also non-physical, especially for long pulse durations and currents.

[0101] In Fig. Figure 7 shows the difference between measurement and simulation for the case where equation (17) is used instead of the standard model (equation 3). After that, a deviation between measurement and simulation is only visible for the first few seconds.

[0102] The long pulse, which is accompanied by a 10% SOC change, is completely described by the fitting function (17). This leads to an interesting observation: When fitting the measurement curve with the fitting function equation (17) and determining the parameters, it is observed that the coefficients of the polynomial change their signs, i.e., the odd orders (M = 1, 3, 5, ...) always have positive signs, while the even orders (M = 2, 4, 6, ...) always have negative signs: b1=1.22 10−5 b2=−2.14 10−8 b3=4.03 10−11 b4=−4.63 10−14 b5=2.88 10−17 b6=−7.34 10−21

[0103] The expansion of an N-RC-ECM model according to order in t yields exactly equation (17) with the characteristic signs of the polynomial coefficients. From this, it can be concluded that equation (17) can also be written as follows: U(t)=UOCV+U0+etτ1(−1+etτ1)b0+etτ2(−1+etτ2)b1+…+etτN(−1+etτN)bN

[0104] This mathematically demonstrated that the voltage behavior of the cell can be described by an N-RC-ECM model. Expanding equation (24) again for t up to terms of order O[t] 4 : U(t)−UOCV+U0+etτ1(−1+etτ1)b0=(1C2+1C3)t−(12C22R2+12C32R3)t2+(16C23R22+16C33R32)t3−(124C24R23+124C34R33)t4+O[t]5 This immediately provides the possibility of determining the parameters of the RC elements by solving a system of equations. For a 2-RC model, this results in: 1C2=b1&&−12C22R3=b2

[0105] The system of equations can be solved algebraically and yields the solutions: C2=1b1and R2=−b122b2

[0106] The first additional RC element can now be parameterized from the numerical values ​​for b1 and b2. For the long pulse, this results in: R0=0.00103 R1=0.00054; C1=37422.7; τ1=R1C1=20.07 R2=0.0035; C2=82061.4; τ2=R2C2=284.5

[0107] The challenge, especially for models with multiple RC elements, is to specify initial values ​​for these RC elements so that the numerical method can find physically meaningful parameters. There are usually several ways to select the parameters so that the difference between the measured value and the function value of the regression model is minimized. With this method, N RC elements can be found numerically by solving a polynomial equation. Up to order 4, this is also possible algebraically; from order 5 onwards, it is possible to use numerical methods.

[0108] Fig. 8 shows three different curves, 801, 802, 803. The first curve 801 is the measurement signal from Fig. 4, where U Ocv(resting voltage), U0 and U1 according to equation (12) with the parameters (R0, R1, C1) equation (28) have been subtracted. Therefore, curve 801 starts at (t= 0) with a value of 0. The value deviating from 0, e.g., at (t= 400), means that the pulse cannot be described solely by a 1-RC model due to the SOC dependence, since higher terms in t must be taken into account here due to the SOC dependence. The third curve 803 is the result of the calculation of the second RC element using the new method according to the invention. In the first approximately 100 seconds, the course of the third curve 803 is identical to that of the first curve 801.

[0109] A Taylor expansion provides the explanation: R2(1−etR2C2)=1.22 10−5t−2.14 10−8t2+2.51×10−11t3+O[t]4

[0110] The second curve 802 provides the parameterization of a total 2-RC model, in which the parameters are determined using the standard method via the RMS. This results in parameter values ​​for (R2, C2) that, due to the higher terms in t, are of order O[t] 3 using RMS as in Fig. 4 are adapted.

[0111] The method used here finds exactly those parameters (R0, R2, C2) according to Eq. (26), which in the expansion again yield b1 and b2 according to Eq. (26). In other words, in this case (N = M). Only with the parameter b3 - compare Eq. (29) and Eq. (23) - do differences arise, which are clearly evident in Fig. 8 (first curve 801 and third curve 803 at (t~150s) ). Here, terms of order O[t] play a role. 3 a role.

[0112] When determining two additional RC elements, i.e., in a 3-RC-ECM model, an algebraic solution is found using standard software for evaluating mathematical formulas (e.g., Mathematica from Wolfram Research). However, this is no longer practical for the concrete calculation of the required parameters, as this solution requires a large number of individual terms to be taken into account. To demonstrate the practical applicability to a generalization of N-RC-ECM models, the system of equations was solved using the @mathworks optimization toolbox. The two curves (first curve 801 (abs|Umess - (UOCV + U0 + U1)|) and third curve 803 (+ 2 RC elements according to the new inventive parameterization method) are shown in Fig. 9 exactly on top of each other.

[0113] This shows that if pulses are to be considered in a profile (synthetic or driving profile) where the SOC changes by 10% during pulse discharge, this can only be correctly simulated by a 3-RC ECM model (N = (M - 1)). Finally, the values ​​according to the standard procedure (values ​​marked with *) R2*=0.0011; C2*=147787 R3*=0.075; C3*=186406 and according to the new inventive method R2=0.0011; C2=146991 R3=0.075; C3=185775

[0114] The values ​​from both methods are almost identical. They also no longer differ in the Taylor coefficients. This means that, taking into account terms of order O[t] 4 the behavior of the cell is fully described at a longer pulse.

[0115] In contrast to the knowledge of known solutions, the new inventive method presented here provides additional information (see illustrations in the Fig. 6 and Fig. 9) about the relevant t-dependence (size M) and about the number of RC elements (size N) that are necessary to describe the behavior of the cell with sufficient accuracy during a longer, continuous charge or discharge with low current or a longer charge or discharge pulse (e.g. 30s) with very high current.

[0116] This disclosure presents a novel method for determining model parameters according to the invention, and in particular, provides a theoretical description of the new method. The method presented here can be implemented in existing software and thus leads to a quantitative improvement in the computational effort and accuracy of the complete ECM parameter set for specific synthetic and real measurement profiles. LIST OF REFERENCE SYMBOLS 100 Method for determining at least one estimated operating parameter of a battery cell 101 first step: receiving 102 second step: Determine 110 at least one measured operating parameter of the battery cell 111 at least one estimated operating parameter of the battery cell 120 battery cell model or battery model 121 Equivalent circuit of the battery cell 122 RC element UCell Cell voltage applied to the battery cell I Cell Cell current flowing through the battery cell U OCV Open circuit voltage R i , C i Model parameters of the battery cell model, for i=1, 2, 3, ... b i Plurality of first coefficients, for i=1, 2, 3, ... a i or a i,k Plurality of second coefficients, for i=1, 2, 3, ... k=1, 2, 3, ... U i Voltage drop across the respective RC element, for i=1, 2, 3, ... R0 internal resistance of the cell 300 cell voltage with short pulse 301 with standard function 302 with measurement of short pulse 400 cell voltage with long pulse 401 with standard function 402 with long pulse measurement 500 SOC dependent course of the parameter b1 501 Course of ψ according to equation (16) 502 Course of b1 according to equation (18) 600 History of parameter C1[F] 601 Course of the parameter C1[F] for short pulse 602 Course of the parameter C1[F] as a function of the order O[t] M 700 Course of the difference between measurement and fit function according to equation (17) 800 Course of the difference between measurement and fit function according to equations (12) and (28) 801 Measurement progress 802 Course of the standard function for 1-RC-ECM according to equation (4) 803 Course of the new procedure for 2-RC-ECM according to equation (27) 900 Course of the difference between measurement and fit function according to equations (12) and (31) 901 Measurement progress 903 Course of the new procedure for 3-RC-ECM according to equation (27)

Claims

[1] Method (100) for determining at least one estimated operating parameter (111) of a battery cell of a battery-powered vehicle, comprising the following steps: Receiving (101) at least one measured operating parameter (110) of the battery cell; Determining (102) at least one estimated operating parameter (111) of the battery cell from the at least one measured operating parameter (110) of the battery cell using a battery cell model (120) for describing a dynamic behavior of the battery cell, wherein the battery cell model (120) is based on an equivalent circuit diagram (121) of the battery cell, which comprises a number N of RC elements (122), wherein the battery cell model (120) has a cell voltage (U Cell ) depending on model parameters (R i , C i ) of the battery cell model (120), where the model parameters (R i , Ci ) on a plurality of first coefficients (b i ), where a respective first coefficient (b i ) is a sum of second coefficients (a i ) is determined by the number of N RC elements (122), where a respective second coefficient (a i ) on a Taylor series expansion of a resistor (R i ) of a corresponding RC element of the N RC elements (122) according to a state of charge of the battery cell around an operating point of the battery cell. [2] Method (100) according to claim 1, wherein the first coefficients (b i ) a time dependence of the cell voltage applied to the battery cell (U Cell ) and in the form of parameters of a polynomial representation of the cell voltage (U Cell ) over time. [3] Method (100) according to claim 2, where the first coefficients (b i), which odd orders of the polynomials of the cell voltage (U Cell ) are associated with time, have positive signs; and where the first coefficients (b i ), which even orders of the polynomials of the cell voltage (U Cell ) associated with time have negative signs. [4] Method (100) according to claim 2 or 3, wherein a capacity (600) of a first RC element of the number N RC elements (122) with increasing order of the polynomials of the cell voltage (U Cell ) decreases with time and tends towards a limit value (601); and where the capacity leaves the limit value (601) abruptly (603) from a certain order. [5] Method (100) according to claim 4, wherein an optimal number of first coefficients (b i ) is based on the order of the polynomials of the cell voltage (U Cell) is determined by the time which lies before the order at which the capacity leaves the limit value (601) abruptly (603). [6] Method (100) according to one of the preceding claims, where the model parameters (R i , C i ) of the battery cell model (120) are dependent on the state of charge of the battery cell and a temperature of the battery cell; and where the operating point of the battery cell is defined for a specific state of charge and a specific temperature of the battery cell. [7] Method (100) according to one of the preceding claims, comprising: Recording the battery cell charge level and battery cell temperature; Transferring the state of charge and temperature of the battery cell to a controller to determine the model parameters (R i , C i ) of the battery cell model (120) based on the detected state of charge and the detected temperature; and Receiving the model parameters determined by the controller (R i , C i ) of the battery cell model (120). [8] Method (100) according to one of the preceding claims, wherein the at least one measured operating parameter (110) of the battery cell defines a discharge pulse or a charge pulse of the battery cell depending on time. [9] Method (100) according to one of the preceding claims, where the model parameters (R i , C i ) of the battery cell capacitances and resistances of the N RC elements (122); and where the model parameters (R i , C i ) of the battery cell also has an internal resistance (R0) and an open circuit voltage (U OCV ) of the battery cell. [10] Method (100) according to one of the preceding claims, wherein the equivalent circuit (121) of the battery cell comprises a series circuit of an internal resistance (R0) and the N RC elements (122).

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