Method for estimating the state of charge of a battery cell from a Doyle-Fuller-Newman model of the cell and a nonlinear Bayesian reduced state vector filter

Abstract

The proposed method combines a two-dimensional, multi-particle Doyle-Fuller-Newman model of a cell with a nonlinear Bayesian filter. Advantageously, the Bayesian filter is a UKF or S3-UKF Kalman filter. An additional advantage of using an S3-UKF filter is the lower dimensionality of its state matrix compared to the UKF filter, which reduces the number of operations required at each filter run. The number of operations is further reduced by incorporating the estimated lithiation rates for a single electrode into the state vector. The predicted lithiation rates for the other electrode are determined from those predicted for the same electrode in the previous iteration, from those predicted for the first electrode, and from a total lithium quantity on both electrodes, assumed to be constant over a given time interval or provided by a cell aging model.Figure 2.

Description

Title of the invention: Method for estimating the state of charge of a battery cell from a Doyle-Fuller-Newman model of the cell and a nonlinear Bayesian reduced state vector filter

[0001] The invention relates to the field of electric batteries and more specifically to that of algorithms for estimating the state of charge of a battery cell.

[0002] The invention is particularly applicable for estimating the state of charge of electric vehicle batteries used for electric mobility applications such as electric vehicles or for stationary electrical energy storage solutions.

[0003] An electric battery is composed of several rechargeable elementary cells connected in series and / or in parallel between two main voltage supply terminals.

[0004] A battery is generally associated with a management device connected to the terminals of the elementary cells of the battery and / or to the main terminals of the battery and which can implement functions of balancing the charge of the cells in particular.

[0005] For this purpose, the management device implements an algorithm enabling it to estimate at any time the state of charge, also called SoC (from the English "State of Charge") of each elementary cell of the battery or of the complete battery.

[0006] State of charge refers here to the ratio between the charge contained in the cell or battery and the total capacity of the cell or battery at the given moment. The management device determines the state of charge using predefined algorithms that take as input measurements taken by sensors connected to the battery cells.

[0007] Indeed, the state of charge of a cell cannot be directly measured online using sensors; it is generally estimated from models and / or estimation algorithms which take current and / or voltage measurements as input.

[0008] The algorithm for estimating the state of charge of a cell must be both sufficiently accurate and of low complexity to be implemented in real time.

[0009] There are various state-of-the-art methods for estimating the state of charge of a cell or battery. Some methods are based on a current measurement, but these are sensitive to measurement accuracy and can drift over time. Other methods are based on a voltage measurement, but these can be inaccurate and lead to high state of charge (SoC) estimation errors. Still other methods exploit the lithiation rate or lithium concentration. in the electrodes of a cell. One drawback is that there are no sensors to measure these quantities, which must therefore be estimated using a model.

[0010] In the literature, various algorithms exist for estimating the state of charge of a battery or an elementary cell based on equivalent circuit models of the cell or on physical models. Reference [1] describes an example of such a two-pseudo-2-dimensional (P2D) physical model, also known as the Doyle-Fuller-Newman model. This model provides information on the internal states of a cell and its physical behavior.

[0011] One drawback of these methods lies in the inaccuracies inherent in the model, as well as in the measurements provided as input to the model. Furthermore, the numerous parameters of the model are identified through an optimization process performed using a set of measurements from a characterization campaign. This parameter identification step results from a global optimization, and therefore a compromise. Consequently, the raw accuracy of the model at certain temperatures, over certain ranges of state-of-charge (SoC) values, and for certain current profiles may be more or less high.

[0012] There is therefore a need to improve the accuracy of methods for estimating the state of charge of a battery from a Doyle-Fuller-Newman model of a cell.

[0013] Moreover, some of the state-of-the-art methods use Kalman filters or more generally Bayesian filters to estimate the state of charge from measurements.

[0014] Linear Kalman filters are limited to linear systems which are not suitable for modeling the behavior of a battery.

[0015] Extended Kalman filters are compatible with nonlinear problems but require analytical equations involving differentiable functions describing the battery behavior. For this reason, this type of filter is generally associated with simple electrical models or single-particle physical models.

[0016] Reference [2] gives an example of such a method.

[0017] One objective of the invention is to propose a new method for estimating the state of charge of a battery that is both sufficiently accurate and low in complexity to be compatible with a real-time implementation.

[0018] The proposed method combines a two-dimensional multi-particle Doyle-Fuller-Newman model of a cell with a nonlinear Bayesian filter. Advantageously, the Bayesian filter is a UKF or S3-UKF Kalman filter. An additional advantage of using an S3-UKF filter is a higher dimensionality. weak state matrix compared to UKF filter, which reduces the number of operations to be performed at each execution of the filter.

[0019] The number of operations to be performed is further reduced by taking into account, in the state vector, the lithiation rates estimated for a single electrode. The predicted lithiation rates for the second electrode are determined from those predicted for the same electrode in the previous iteration, those predicted for the first electrode, and a total quantity of lithium on both electrodes, assumed to be constant over a given time interval or provided by an aging model that takes into account cyclable lithium losses.

[0020] The invention relates to a method for estimating the state of charge of a battery cell, the cell being equipped with two electrodes, each electrode being spatially discretized according to a predefined spatial mesh, the method comprising the iterative steps of: - Receive a current measurement, a temperature measurement, and a voltage measurement taken on the cell, - Execute multiple instances of a two-dimensional multi-particle model of the cell to estimate the voltage across the cell and a first lithiation rate for each point of the spatial mesh of each electrode from current measurement, temperature measurement, and a prediction made at a previous iteration of a lithium concentration on each electrode, said model acting as a state-space model of a nonlinear Bayesian filter, - Execute a correction step of the nonlinear Bayesian filter to predict a state vector including a prediction of a second corrected lithiation rate for a first electrode among the two electrodes and for each point of the spatial mesh, from the voltage measurement, the voltage estimate and the first estimated lithiation rate for the first electrode obtained for all execution instances of the model, - Determine, from the state vector, a state matrix comprising several modified vectors obtained by transformations of the state vector, and provide the modified vectors as input to the model at a subsequent iteration. - Determine, from the state matrix, the first lithiation rate of the second electrode at the model output, and a total quantity of lithium distributed between the two electrodes, several vectors, each containing a different prediction of the second corrected lithiation rate for the second electrode, to be provided as input to the model at the next iteration. - Determine a first estimate of the state of charge from the state vector.

[0021] According to a particular aspect of the invention, the model is a Doyle-Fuller-Newman model.

[0022] In one embodiment, the method according to the invention further includes the conversion of each corrected lithiation rate into a lithium concentration prediction to be provided as input to the model.

[0023] According to a particular aspect of the invention, the step of determining several vectors, each containing a different prediction of the second corrected lithiation rate for the second electrode, is carried out by means of the substeps of: - Determine the average value of the lithiasis rate of each modified vector, - Determine, for each modified vector, an average value of the second corrected lithiation rate for the second electrode, based on the average value of the modified vector and the total amount of lithium on the electrodes. - Determine the prediction vectors of the second corrected lithiation rate for the second electrode from the first estimated lithiation rates for the second electrode at the output of the model (200) multiplied by the ratio between the average value of the second corrected lithiation rate for the second electrode and the average value of the first estimated lithiation rate for the second electrode.

[0024] According to a particular aspect of the invention, the total quantity of lithium on the electrodes is determined from the respective average values ​​of the a priori estimates of the lithiation rates of each electrode obtained at the output of the model.

[0025] According to a particular aspect of the invention, the model further delivers at output an estimate of an electrical potential difference for each electrode which are supplied at the input of the Bayesian filter.

[0026] According to a particular aspect of the invention, the nonlinear Bayesian filter is a UKF or S3-UKF type Kalman filter.

[0027] According to a particular aspect of the invention, the state matrix is ​​of dimension (N,N+2) when the Kalman filter is of type S3-UKF, with N an integer equal to the number of points in the spatial mesh of the first electrode.

[0028] In one embodiment, the method according to the invention comprises the steps of: - Calculate an average lithiation rate of the first electrode from the state vector, - Calculate the first estimate of the state of charge from the average lithiation rate of the first electrode.

[0029] In one embodiment, the method according to the invention comprises the steps of: - Calculate the average lithiation rate of the second electrode from the average lithiation rate of the first electrode and the total amount of lithium on the electrodes. - Calculate an estimated second state of charge from the average lithiation rate of the second electrode.

[0030] The invention also relates to a device for managing the state of charge of a battery comprising a processing unit configured to implement the steps of the method according to the invention.

[0031] Other features and advantages of the present invention will become more apparent from the following description in relation to the following accompanying drawings.

[0032] [Fig-1] represents a diagram of an electric battery management system adapted to implement the invention.

[0033] [Fig.2] represents a synoptic diagram of the method for estimating the state of charge of a cell of an electric battery according to an embodiment of the invention.

[0034] Fig. 1 schematically represents a system comprising an electric battery BAT and an electronic management device GES configured to implement a method for estimating the state of charge of the battery according to the invention.

[0035] The battery BAT may be a single battery cell or comprise several individual cells connected in series and / or parallel. The GES management device includes a SENS measuring device with one or more sensors adapted to measure one or more physical quantities of the battery, for example, the current flowing through the battery and / or the voltage across the battery terminals and / or the battery temperature. The GES management device further includes a PROC processing unit, for example, a microprocessor, which receives the data measured by the SENS measuring device and calculates an estimate of the state of charge.

[0036] Figure [Fig.2] represents a diagram of a method for estimating the state of charge of a battery according to an embodiment of the invention.

[0037] The method works iteratively. It takes as input current measurements I, temperature T and voltage Ucen taken on a cell at a predefined input frequency and provides as output an estimate of the cell's state of charge SoC(t) as a function of time at a predefined output frequency which may be the same as or different from the input frequency.

[0038] The method is based on the execution of a two-dimensional, multi-particle model of the cell, for example a Doyle-Fuller-Newman model, which is executed from the measurement of current I, the measurement of temperature T and Predictions of lithium concentration values ​​at each electrode, Clposet Clneg, and in electrolyte C2 are provided. The model outputs an estimate of the cell voltage U, the respective lithiation rates at each electrode xLipOs, xLineg, and a lithium concentration in electrolyte C2, which is fed back to the model input. The lithiation rates are expressed for several points within the volume of an electrode using a predefined 2D spatial mesh. The lithium concentrations in the electrolyte are also expressed for several points within the electrolyte mesh, but using a different mesh than that of the electrodes. Typically, an electrode is modeled by Nx particles along an axis normal to the cell's principal plane. Each particle is discretized into Nr points along a radial direction.The lithiation rates and lithium concentrations are therefore, for each electrode, matrices of dimension Nx * Nr. Without departing from the scope of the invention, the spatial discretization parameters of an electrode may be identical or different between the two electrodes of the same cell.

[0039] Model 200 also provides as output an average lithiation rate on each electrode xLipm -, xLineg which can be calculated from the lithiation rates of all points of the spatial mesh or calculated directly by the model.

[0040] Optionally, the model also provides as output an estimate of an electrical potential difference of each electrode, ddppos, ddpneg.

[0041] Other EST estimates of physical quantities characterizing the cell can be produced as output of the model.

[0042] Model 200 is executed as a state model of a Bayesian filter to provide a first a priori prediction of lithiation rates.

[0043] The Doyle-Fuller-Newman model used is, for example, that described in reference [1]. Other models may be considered insofar as they take into account a 2D spatial discretization of the electrode volume and provide an estimate of the same physical quantities.

[0044] The method is further based on the execution of a correction step using a nonlinear Bayesian filter 203, for example, a UKF (Unscented Kalman Filter) or S3-UKF (Scaled Spherical Simplex Unscented Kalman Filter) type filter. A complete description of an S3-UKF (also called S3F) Kalman filter is given in reference [3]. The correction step 203 provides a second a posterior prediction of the lithiation rates, which is a corrected version of the first a priori prediction.

[0045] The correction step 203 takes as input the estimated voltage across the cell U and the estimated lithiation rate of a single electrode xIJpos, for example the positive electrode in the example of [Fig.2].

[0046] Optionally, the correction step 203 also takes as input the estimates of the electrical potential differences of the two electrodes.

[0047] The correction step 203 allows the calculation of a new corrected a posteriori prediction of the lithiation rates on the first electrode in the form of a state vector x.

[0048] Thus, the non-linear Bayesian filter consists of the sequence of step 200 of a priori prediction using a cell state model, step 203 of correction of the a priori predictions to provide a posteriori predictions and step 204 of generation of the state matrix X.

[0049] More specifically, the correction step 203 aims to determine the state vector x from several sets of input data obtained by running multiple instances of the same model 200. For each execution instance, model 200 receives the same current and temperature measurements as input, but different versions of the lithium concentrations on the two electrodes Clposet Clneg. The lithium concentrations are determined from the lithiation rates by multiplying by a gain Cipos >max or Cineg,max for the positive and negative electrodes, respectively.

[0050] For the first electrode (the positive electrode in the example in [Fig. 2]), the different versions of the lithiation rates of this first electrode are determined from the state vector x by constructing a state matrix X, each column of which corresponds to a modified version of the state vector with a slight modification obtained by applying a predetermined mathematical transformation to the state vector. The determination of the state matrix X is carried out using equations specific to the type of Kalman filter used. For a UKF type Kalman filter, the state matrix has dimension N by 2N+1, where N is the size of the state vector, which is equal to the number of points in the spatial mesh of an electrode.

[0051] For a Kalman filter of type S3-UKF, the state matrix is ​​of dimension N by N +2. Thus, the S3-UKF filter has the advantage of requiring fewer executions of model 200.

[0052] The state matrix X, like the state vector x, contains assumptions about lithiation rates. A conversion step can be applied to convert the lithiation rates into lithium concentrations. This step consists of applying a Cipos >max gain to the model input.

[0053] The corrected lithiation rates for the second electrode (the negative electrode in the example of [Fig.2]) are obtained from the state matrix X by means of the following steps.

[0054] In step 205, the average of each modified version of the state vector corresponding to a column of the state matrix is ​​first determined. This average is denoted vjf „„ with SP an index varying over all columns of the pos.s matrix / state X.

[0055] Next, in step 206, the corresponding averages of the corrected lithiation rates on the second electrode x1*ttegçp (the negative electrode in the example) are determined from the averages obtained for the first electrode xL^p^p and a total quantity of lithium on both electrodes nLi. This total quantity is, for example, assumed to be constant over a time step regardless of the respective proportions on each electrode. It is, for example, a mass of lithium.

[0056] The total quantity of lithium nLi is obtained in step 202 from the averages of the lithiation rates of the two electrodes determined at the output of model 200. The total mass of lithium is considered constant if the aging phenomena of the cell are ignored or this phenomenon can be taken into account in which case, a cell aging model makes it possible to take into account the evolution over time of the total mass of lithium.

[0057] Then in step 207, the corrected lithiation rates for the second electrode are determined from the first lithiation rates estimated for the second electrode at the output of the Doyle-Fuller-Newman model 200 multiplied by the ratio between the average value of the second corrected lithiation rate for the second electrode and the average value of the first estimated lithiation rate for the second electrode.

[0058] This calculation is represented in two sub-steps in the diagram of Figure 2. The lithiation rate on the second electrode obtained at the output of the model is first divided

[0059] by its average at step 201. Then the normalized vector obtained is multiplied by the averages xlï\te^p obtained at the output of step 206. The corrected lithiation rates on the second electrode xLi^^g^p thus obtained are fed back into the input of model 200 in the form of lithium concentration on the second electrode after application of the Cineg gain >max.

[0060] Thus, several instances of the 200 model are executed in parallel (2N+1 for a UKF filter, N+2 for an S3-UKF filter).

[0061] The set of outputs of the model 200 obtained for all execution instances are provided as input to the correction step 203 of the filter which therefore determines a corrected state vector x from a voltage measurement Ucen and several sets of quantities estimated by the model (voltage, lithiation rate of the first electrode).

[0062] At a given frequency, an estimate of the state of charge SoC is calculated for at least one of the two electrodes by means of the following steps.

[0063] In step 208, an average of the lithiation rates of the state vector is calculated to obtain a corrected average lithiation rate xu*pos on the first electrode.

[0064] In step 209, the state of charge SoC is finally calculated using the following relationship:

[0065] Soc(t) = xLipos max”xLip <js [°066] xlïpot „ is the average lithiation rate calculated for the positive electrode at step 208,

[0067] xLipos max and xLi^ç rmn are respectively the maximum and minimum values ​​of the lithiation rates. These quantities are inputs to the method that depend on the type of cell. They are typically constant values ​​that depend on the minimum and maximum voltages across the cell corresponding respectively to a state of charge of 0% and 100%.

[0068] The state of charge SoC is expressed as a percentage: 0% corresponding to a totally discharged state and 100% corresponding to a totally charged state.

[0069] Optionally, a second estimate of the state of charge SoC is calculated for the negative electrode.

[0070] For this, in step 210, an average of the corrected lithiation rates xLl neg on the second electrode is calculated from the average of the corrected lithiation rates xLfpos on the first electrode and the total quantity of lithium nLi.

[0071] In step 211, the estimated second state of charge (SoC) is calculated using the following relationship:

[0072] n xL ïfieg max^Li^eÿ. j xLi»^ max“XLÎ / K?£ min

[0073] , is the average lithiation rate calculated for the negative electrode at the stage 210,

[0074] xLLeg max and xL΄e„ mjn are respectively the max and min values ​​of the lithiation rates corresponding to the maximum and minimum voltages allowed for the cell.

[0075] The filter is driven by a set of input parameters PAR and involves the calculation of several internal variables VAR calculated iteratively such as innovation, Kalman filter gain or state estimation error covariance matrix S, which can be used as output, for example to characterize the convergence or uncertainty of the filter.

[0076] The filter also produces an output observation vector y which includes the a posteriori estimate of the voltage. Optionally, the filter can also provide predictions of the electrical potential differences of the two electrodes: ddp*pos and ddp*neg.

[0077] The principle of the UKF and S3-UKF Kalman filters is based on the UT (Unscented Transform) described in reference [4].

[0078] Its objective is to estimate the statistics of a random Gaussian variable that is subjected to a nonlinear transformation. To this end, a set of points, called sigma points, is generated in order to access the statistical properties of the distribution. A UT transform is applied to the distribution.

[0079] A Kalman filter requires a description of the system to be modeled through space-time equations. These equations are of the type

[0080] X^^X^^U^^w)

[0081] yk=g(xk,uk,v)

[0082] The state vector x contains the unobservable variables. In this case, it is the lithiation rate at each point of the spatial mesh of a single electrode. The observation vector y can contain values ​​known from measurements such as the voltage across the cell.

[0083] One advantage of using an S3-UKF filter is the reduced number of sigma points compared to the UKF filter, due to their modified spatial distribution. This has a direct impact on the number of instances of model 200 to be executed in parallel, the computational load, and therefore the execution time of the method. Indeed, the number of sigma points corresponds to one dimension of the state matrix X (the second dimension, given that the first dimension is that of the state vector).

[0084] In the case of the S3-UKF filter, geometric properties are used to reduce the number of sigma points to be generated. The distribution of the sigma points is optimized compared to that of the UKF filter. This modification of the distribution makes it possible to reduce the number of sigma points from 2N+1 to N+2 with virtually no loss of precision.

[0085] Furthermore, an additional advantage of the invention is that the dimension of the state vector of the Kalman filter is limited to the number of points in the spatial mesh of a single electrode. The lithiation rates of the second electrode are predicted from those of the first electrode and the assumption of a constant amount of lithium on both electrodes over a time step.

[0086] In one embodiment, the total mass of lithium can vary over time according to a variation model that takes into account the aging phenomena of the cell such as a cyclable physical lithium loss model or a lithium loss estimation algorithm.

[0087] Similarly, the number of execution instances of model 200 is also reduced because it is equal to the number of sigma points generated by the filter which depends on the dimension of the state vector.

[0088] In particular, this number is divided by two compared to a method which would exploit the lithiation rates of the two electrodes within the same state vector.

[0089] The method according to the invention can be implemented as a computer program comprising instructions for its execution. The computer program can be stored on a storage medium readable by a processor. It can be executed by the PROC processing unit of a GES battery charge state management device.

[0090] Although the invention has been described for estimating the state of charge of a cell in a battery, it can be extended to estimate the state of charge of a group of cells in a similar way. References

[0091] [1] Doyle, M., Fuller, T. & Newman, J. (1993), Modeling of Galvanostatic Charge and Discharge of the Lithium / Polymer / Insertion Cell. Journal of The Electrochemical Society, Volume 140, Number 6, DOI 10.1149 / 1.2221597

[0092] [2] Oehler, Nürnberger, Sturm, & Jossen. (2022). Embedded real-time State observer implémentation for lithium-ion cells using an electrochemical model and extended Kalman filter. Journal of Power Sources.

[0093] [3] Papakonstantinou, K., Amir, M., & Warn, G. (2022). A Scaled Spherical Simplex Filter (S3F) with a decreased n + 2 sigma points set size and équivalent 2n + 1 Unscented Kalman Filter (UKF) accuracy. Mechanical Systems and Signal Processing, 163,107433.

[0094] [4] van der Merwe, R., & Wan, E. (2001). The square-root unscented Kalman filter for State and parameter-estimation. 2001 IEEE International Conférence on Acoustics, Speech, and Signal Processing, (pp. 3461-3464). Sait Lake City, UT, USA.

Claims

1. Demands Method for estimating the state of charge of a battery cell, the cell being equipped with two electrodes, each electrode being spatially discretized according to a predefined spatial mesh, the method comprising the iterative steps of: - Receive a current measurement, a temperature measurement, and a voltage measurement taken on the cell, - Execute (200) several instances of a two-dimensional multi-particle model of the cell to estimate the voltage across the cell terminals and a first lithiation rate for each point of the spatial mesh of each electrode from the current measurement, the temperature measurement and a prediction made at a previous iteration of a lithium concentration on each electrode, said model acting as a state model of a nonlinear Bayesian filter, - Execute (203) a correction step of the nonlinear Bayesian filter to predict a state vector including a prediction of a second corrected lithiation rate for a first electrode among the two electrodes and for each point of the spatial mesh, from the voltage measurement, the voltage estimate and the first estimated lithiation rate for the first electrode obtained for all execution instances of the model, - Determine (204), from the state vector, a state matrix comprising several modified vectors obtained by transformations of the state vector and provide the modified vectors as input to the model at a subsequent iteration, - Determine (205,206,207), from the state matrix, the first lithiation rate of the second electrode at the output of the model and a total quantity of lithium distributed between the two electrodes, several vectors each containing a different prediction of the second corrected lithiation rate for the second electrode, to be provided as input to the model at the next iteration, - Determine (208,209) a first estimate of the charge state from the state vector.

2. Method for estimating the state of charge of a battery cell according to claim 1 wherein the model is a Doyle-Fuller-Newman model.

3. Method for estimating the state of charge of a battery cell according to any one of the preceding claims, further comprising converting each corrected lithiation rate into a prediction of lithium concentration to be provided as input to the model.

4. A method for estimating the state of charge of a battery cell according to any one of the preceding claims, wherein the step of determining several vectors, each containing a different prediction of the second corrected lithiation rate for the second electrode, is carried out by means of the substeps of: - Determining (205) the average value of each modified vector, - Determining (206), for each modified vector, an average value of the second corrected lithiation rate for the second electrode from the average value of the modified vector and the total amount of lithium on the electrodes,- Determine (207) the prediction vectors of the second corrected lithiation rate for the second electrode from the first estimated lithiation rates for the second electrode at the output of model (200) multiplied by the ratio between the average value of the second corrected lithiation rate for the second electrode and the average value of the first estimated lithiation rate for the second electrode.

5. Method for estimating the state of charge of a battery cell according to any one of the preceding claims wherein the total amount of lithium on the electrodes is determined (202) from the respective average values ​​of the estimated lithiation rates of each electrode obtained at the output of the model (200).

6. A method for estimating the state of charge of a battery cell according to any one of the preceding claims, wherein the model (200) further outputs an estimated state of charge of a electrical potential difference for each electrode which are supplied as input to the Bayesian filter correction stage.

7. Method for estimating the state of charge of a battery cell according to any one of the preceding claims wherein the nonlinear Bayesian filter is a UKF or S3-UKF type Kalman filter.

8. Method for estimating the state of charge of a battery cell according to claim 7 wherein the state matrix is ​​of dimension (N,N+2) when the Kalman filter is of type S3-UKF, with N an integer equal to the number of points in the spatial mesh of the first electrode.

9. Method for estimating the state of charge of a battery cell according to any one of the preceding claims comprising the steps of: - Calculating (208) an average lithiation rate of the first electrode from the state vector, - Calculating (209) the first estimate of the state of charge from the average lithiation rate of the first electrode.

10. Method for estimating the state of charge of a battery cell according to claim 9 comprising the steps of: - Calculating (210) an average lithiation rate of the second electrode from the average lithiation rate of the first electrode and the total amount of lithium on the electrodes, - Calculating (211) an estimated second state of charge from the average lithiation rate of the second electrode.

11. Battery state of charge management (GES) device comprising a processing unit (PROC) configured to implement the steps of the method according to any one of the preceding claims.