METHOD FOR ESTIMATING THE STATE OF CHARGE OF A BATTERY CELL USING A DOYLE-FULLER-NEWMAN MODEL OF THE CELL AND A NONLINEAR BAYESIAN FILTER

Abstract

Method for estimating the state of charge of a battery cell, comprising the iterative steps of: Receiving a current measurement, a temperature measurement, and a voltage measurement taken on the cell; Running (200) several instances of a cell model to estimate the voltage across the cell and a first lithiation rate of each electrode; Running (203) a correction using the nonlinear Bayesian filter to predict a state vector including a prediction of a second corrected lithiation rate; Determining (204), from the state vector, a state matrix comprising several modified vectors obtained by transformations of the state vector; Providing each modified vector as input to the model at a subsequent iteration; Determining (205, 206) an estimate of the state of charge from the state vector. 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 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 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] To this end, the management system implements an algorithm enabling it to estimate at any given moment 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] By state of charge, we mean here the ratio between the charge contained in the cell or in the 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. More recent methods seek to exploit the lithiation rate or lithium concentration in the electrodes of a cell. One drawback is that it There are no sensors available 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 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 nonlinear Bayesian filter correction step to predict a state vector including a prediction of a second corrected lithiation rate for each point of the spatial mesh and for each electrode from the voltage measurement and the outputs of the model executed for all instances, - Determine, from the state vector, a state matrix comprising several modified vectors obtained by transformations of the state vector, - Provide each modified vector as input to the model at a subsequent iteration, - Determine an estimate of the state of charge from the state vector.

[0020] According to a particular aspect of the invention, the model is a Doyle-Fuller-Newman model.

[0021] In one embodiment, the method further includes a step of converting each modified vector into a lithium concentration prediction to be provided as input to the model at a subsequent iteration.

[0022] According to a particular aspect of the invention, the model further provides, as output, an estimate of an electrical potential difference for each electrode.

[0023] According to a particular aspect of the invention, the model further provides as output an average lithiation rate of each electrode, the method further comprising the calculation of a total quantity of lithium distributed between the two electrodes which is provided as input to the nonlinear Bayesian filter as an observation.

[0024] According to a particular aspect of the invention, the total quantity of lithium is constant over time or follows a predefined variation pattern depending on the aging state of the cell.

[0025] According to a particular aspect of the invention, the nonlinear Bayesian filter is a UKF or S3-UKF type Kalman filter.

[0026] 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 sum of the points of the spatial mesh of the two electrodes.

[0027] In one embodiment, the method comprises the steps of: - Calculate an average lithiation rate for each electrode from the state vector, - Calculate an estimate of the state of charge from the average lithiation rate of one of the electrodes.

[0028] 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.

[0029] Other features and advantages of the present invention will become more apparent from the following description in relation to the following accompanying drawings.

[0030] [Fig.1] represents a diagram of an electric battery management system adapted to implement the invention.

[0031] [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.

[0032] 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.

[0033] 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.

[0034] Figure 2 shows a schematic of a method for estimating the state of charge of a battery according to an embodiment of the invention.

[0035] The method operates iteratively. It takes as input current I, temperature T and voltage Ucen measurements taken on a cell at a frequency predefined input and provides as output an estimate of the cell charge state SoC(t) as a function of time at a predefined output frequency which may be the same as or different from the input frequency.

[0036] The method is based on the execution of a two-dimensional meshed multi-particle model of the cell, for example, a Doyle-Fuller-Newman model, which is run using current measurements I, temperature measurements T, and predictions of lithium concentration values ​​at each electrode, Clposet Clneg, and in the electrolyte, C2. The model outputs an estimate of the cell voltage U, the respective lithiation rates of each electrode, and a lithium concentration in the electrolyte C2, which is fed back to the model input. The lithiation rates and lithium concentrations are expressed for several points in the volume of an electrode according to a predefined 2D spatial mesh. The lithium concentrations in the electrolyte are also expressed for all points in the electrolyte mesh, but according to a different mesh than that of the electrodes.Typically, a porous electrode is meshed in one dimension with a number Nx of cells along an axis normal to the principal plane of this electrode. Furthermore, a particle is associated with each cell of the porous electrode. Each particle is discretized in one dimension, with a number Nr of cells, along the radial direction.

[0037] The lithiation rates and lithium ion 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.

[0038] The model also provides as output an average lithiation rate within each electrode xLipos -, xLineg which can be calculated from the lithiation rates of all points of the spatial mesh or calculated directly by the model.

[0039] Optionally, the model also provides as output an estimate of an electrical potential difference of each electrode, ddppos, ddpneg.

[0040] Other EST estimates of physical quantities characterizing the cell can be produced as output of the model.

[0041] Model 200 is executed as a state model of a Bayesian filter to provide a first a priori prediction of lithiation rates.

[0042] 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 volumetric discretization of the electrode volume, in other words a 2D spatial discretization of the electrodes and provide an estimate of the same physical quantities.

[0043] 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. 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 posteriori prediction of the lithiation rates, which is a corrected version of the first prediction.

[0044] The correction step 203 takes as input certain outputs of the model 200, in particular the voltage predictions U, the lithiation rates of each electrode, as well as a measurement of the voltage across the terminals of the cell Ucen.

[0045] Optionally, a prediction of the electrical potential differences of the two electrodes can be provided as input to step 203.

[0046] 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.

[0047] The correction step 203 also takes as input an estimate of the total amount of lithium nLi, which is calculated 202 from the average lithiation rates on each electrode xLipos ■> xllngg ■. The total amount of lithium is, for example, expressed as a total lithium mass. This total amount is provided as input to the filter as a constraint in the observation vector y. This is a virtual measurement. The total lithium mass is considered constant if cell aging phenomena are ignored, or this phenomenon can be taken into account, in which case a cell aging model allows the evolution of the total lithium mass over time to be considered. This constraint is respected by the filter 203 when calculating its predictions.

[0048] The correction step 203 allows the calculation of a new corrected a posteriori prediction of the lithiation rates on the two electrodes in the form of a state vector x.

[0049] More specifically, the correction step 203 aims to determine the state vector x from several sets of input data which are obtained by executing several instances of the same model 200. For each execution instance, the model 200 receives as input the same current and temperature measurements but different versions of the lithium concentrations (from the lithiation rates).

[0050] These different versions are determined from the state vector x by constructing a state matrix X, each column of which corresponds to a slight modification of the state vector, obtained via a predefined mathematical transformation. 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 dimensions N by 2N+1, where N is the size of the state vector, which is equal to the sum of all the points in the mesh of the two electrodes.

[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 contains, like the state vector x, assumptions about lithiation rates. A conversion step 207 can be applied to convert the lithiation rates into lithium concentration. This step 207 consists of applying a CjpOS îmax or Cinegmax.

[0053] Thus, several instances of the 200 model are executed in parallel (2N+1 for a UKF filter, N+2 for an S3-UKF filter).

[0054] The set of outputs of the model 200 obtained for all execution instances are provided as input to the correction step 203 which therefore determines a corrected state vector x from a voltage measurement Uceii, the virtual measurement of total quantity of lithium nLi and several sets of quantities estimated by the model (voltage and lithiation rate).

[0055] 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.

[0056] In step 205, an average of the lithiation rates of the state vector is calculated for each respective electrode, to obtain corrected average lithiation rates xLl pos ' XLl neg

[0057] In step 206, the state of charge SoC is finally calculated using the following relationship calculated for at least one of the two electrodes.

[0058] / \ xLiam-xLr(t) - xLimax-xLin!in

[0059] xLi*(t) is the average lithiation rate calculated for one of the electrodes in step 205,

[0060] xLimax and xLi^ are respectively the max and min values ​​of the lithiation rates. These quantities are inputs to the method and depend on the cell type. They are typically constant values ​​that depend on the minimum and maximum voltages across the cell, corresponding to a charge state of 0% and 100%, respectively.

[0061] 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.

[0062] The Kalman 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 the state covariance matrix S which can be used as output.

[0063] The filter also produces an output observation vector y containing the estimated voltage. Optionally, the filter can also provide a posteriori predictions of the electrical potential differences between the two electrodes: ddp*pos and ddp* neg*

[0064] The principle of the UKF and S3-UKF Kalman filters is based on the UT (Unscented Transform) described in reference [4].

[0065] 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.

[0066] A Kalman filter requires a description of the system to be modeled through space-time equations. These equations are of the type

[0067] x^fCx^DU^Dw)

[0068] yk=g(xk,uk,v)

[0069] The state vector x contains the unobservable variables. In this case, these are the lithiation rates. The observation vector y can contain values ​​known from measurements such as the voltage across the cell or from virtual measurements such as the total quantity of lithium.

[0070] 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).

[0071] 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.

[0072] 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.

[0073] 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

[0074] [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

[0075] [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.

[0076] [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.

[0077] [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

Demands

1. A method for estimating the state of charge of a battery cell, the cell having two electrodes, each electrode being spatially discretized according to a predefined spatial mesh, the method comprising the iterative steps of: - Receiving a current measurement, a temperature measurement and a voltage measurement taken on the cell, - Executing (200) several 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 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 each point of the spatial mesh and for each electrode from the voltage measurement and the outputs of the model executed for all instances, - Determine (204), from the state vector, a state matrix comprising several modified vectors obtained by transformations of the state vector, - Provide each modified vector as input to the model at a subsequent iteration, - Determine (205,206) an 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 (200) is a Doyle-Fuller-Newman model.

3. A method for estimating the state of charge of a battery cell according to any one of the preceding claims, further comprising a step of: - Convert (207) each modified vector into a lithium concentration prediction to be provided as input to the model at a subsequent iteration.

4. Method for estimating the state of charge of a battery cell according to any one of the preceding claims wherein the model (200) further provides at output, an estimate of an electrical potential difference for each electrode.

5. Method for estimating the state of charge of a battery cell according to any one of the preceding claims wherein the model (200) further provides as output an average lithiation rate of each electrode, the method further comprising the calculation (202) of a total amount of lithium distributed between the two electrodes which is provided as input to the nonlinear Bayesian filter as an observation.

6. Method for estimating the state of charge of a battery cell according to claim 5 wherein the total amount of lithium is constant over time or follows a predefined variation pattern depending on the aging state of the cell.

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 sum of the points of the spatial mesh of the two electrodes.

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 (205) an average lithiation rate of each electrode from the state vector, - Calculating (206) an estimate of the state of charge from the average lithiation rate of one of the electrodes.

10. Battery state of charge management (SCM) device comprising a processing unit (PCU) configured to put implement the steps of the method according to any one of the preceding claims.

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