Method for determining the state of charge of a battery

Abstract

A method for determining the state of charge of a battery comprising the following steps: – producing an expression for the state of charge of the battery and the voltage across the battery terminals, and expressing the corresponding components and the maximum capacity of the battery by a continuous analytical expression dependent on the state of charge, current, and temperature; – producing a prediction of the state of charge of the battery; – producing a simulation of the voltage across the battery terminals; – measuring the voltage across the battery terminals; – correcting said prediction of the state of charge by applying a Kalman filter, adding a value depending on the error between the predicted voltage across the battery terminals and the measured voltage across the battery terminals. Figure for the abstract: Fig. 4A

Description

Title of the invention: Method for determining the state of charge of a battery technical field

[0001] The invention relates to the field of electrochemical batteries and means of managing these batteries. The invention relates to techniques for measuring and estimating essential battery parameters, and in particular the state of charge.

[0002] Batteries are essential components of many devices, machines, and means of transport. These batteries are used, for example, in stationary or automotive applications, such as electric vehicles or electric lifting equipment (pallet trucks, forklifts, aerial work platforms, etc.). For this type of use, controlling and optimizing battery performance is a considerable challenge, as the batteries generate the power necessary for the vehicle or machine's movements. The charge level indicator directly reflects the remaining operating range of the equipment. Accurate and reliable information provides the end user with important data regarding remaining operating capacity or the distance that can be traveled, allowing the equipment to adjust its operation to maximize operating range without compromising battery life.

[0003] The real-time determination of the state of charge of a battery is a complex operation given the difficulties in accurately knowing the actual operating conditions of the battery, current and voltage measurement errors, and the limitations of battery management systems (BMS for "Battery Management System"). PREVIOUS ART

[0004] Patent application CN106054085 describes a method for estimating the state of charge of a battery as a function of temperature. Battery parameters are measured under different temperature conditions to create a database for fitting the battery model according to the temperature conditions. During estimation, the battery temperature is detected, and the corresponding temperature data from the database is retrieved. A Kalman filter is used to fit the battery model and estimate the state of charge.

[0005] This model can be improved with regard to its real-time accuracy and its resource consumption. Description of the invention

[0006] The invention aims to improve prior art methods for determining the state of charge of a battery.

[0007] To this end, the invention relates to a method for determining the state of charge of a battery, from measurements of current, voltage and temperature of the battery, comprising a step of modeling the battery and a step of applying a Kalman filter, this method being characterized in that it comprises the following steps: - establish a model of the battery by a circuit comprising at least: an open circuit voltage of the battery, an ohmic resistance of the battery, a diffusion resistance of the battery, a diffusion capacitor of the battery, and a loss resistance of the battery; - from the battery model, produce an expression for the state of charge of the battery based on the integration of current measurements taking into account the loss resistance of the battery; - from the battery model, produce an expression for the voltage across the battery terminals as a function of at least the following components: the battery loss resistance, the battery ohmic resistance, the battery diffusion capacitor, the battery diffusion resistance, and the battery open circuit voltage; - express each of said components, as well as the maximum capacity of the battery, by a continuous analytical expression that is dependent on both the state of charge of the battery, the current and the temperature of the battery; - from said expression of the state of charge of the battery and said continuous analytical expressions of said components, produce a prediction of the state of charge of the battery; - from the said expression of the voltage across the battery terminals and the said continuous analytical expressions of the said components, produce a simulation of the voltage across the battery terminals; - to measure the voltage across the battery terminals; - determine the error between said prediction of the voltage across the battery terminals and said measurement of the voltage across the battery terminals; - correct said prediction of the battery state of charge by applying a Kalman filter, by adding to said prediction of the battery state of charge a value depending on said error between the prediction of the voltage across the battery terminals and the measurement of the voltage across the battery terminals.

[0008] The batteries covered by the invention can be any electrochemical device enabling the storage of electrical energy, regardless of their chemistry. Examples include lead-acid batteries, lithium or nickel batteries, etc.

[0009] The term battery refers both to elementary components and to more complex assemblies comprising several of these interconnected components. These may be battery cells, modules comprising several of these cells, or batteries or battery packs comprising several modules or several batteries.

[0010] The invention offers, through a model, the possibility of predicting the state of charge (SOC) of a battery in real time, under all conditions of use (charging, discharging and self-discharging), taking into consideration the ambient temperature, which has an important impact on the evolution of the state of charge.

[0011] From the sensors associated with the battery, the process is based on measurements of current, voltage and temperature, to accurately determine the state of charge.

[0012] The invention differs from prior art by taking into account, in particular, two phenomena that have a significant impact on the battery's state of charge: temperature and self-discharge. These phenomena are generally underestimated, or roughly estimated, in the prior art, when they are not simply neglected.

[0013] Indeed, ambient temperature directly impacts the internal functioning of the battery. More specifically, during discharge, a high temperature provides more energy to the system, facilitating agitation and thus the movement of chemical species within the cell. This allows a large portion of the available active material within the electrodes to be accessed, thereby increasing the battery's autonomy. Conversely, a low temperature slows the circulation of chemical species, which form thicker crystals and struggle to utilize the active material. This reduces the battery's autonomy. This well-known but often overlooked physical phenomenon is centrally considered here.

[0014] With regard to self-discharge, a battery never truly reaches a perfect state of equilibrium. Even at rest, very localized electrochemical reactions persist within the electrodes, resulting in the circulation of a very low current and thus a slow loss of capacity, which constitutes the self-discharge phenomenon. This phenomenon is often neglected due to its limited short-term impact. However, in certain applications, or during long-distance transport, batteries are sometimes required to remain at rest for several months. In these situations, batteries can become deeply discharged, which not only causes irreversible damage but also poses logistical problems. The invention takes advantage of the self-discharge formulation to optimize the use of the battery model.

[0015] The method according to the invention allows the taking into account of self-discharge and temperature for an estimation of the state of charge of the battery, in real time, continuously, without any recourse to pre-established databases which discretize the phenomena and make the estimates less precise.

[0016] Accurate determination of the battery's state of charge, estimated during discharge, charging, and self-discharge, regardless of the battery's internal temperature, represents a considerable advantage, particularly for industrial equipment. This solution allows the machine operator to work under optimal conditions, eliminating the risk of sudden battery failure resulting from an inaccurate indicator that overestimates the remaining battery life. The invention also enables the manufacturer of such machines to fine-tune the performance of their product according to the battery's state of charge, thereby preserving the batteries and reducing maintenance costs, which the invention positively impacts in the long term.

[0017] The method according to the invention may include the following additional features, alone or in combination:

[0018] - said continuous analytical expression expressing each of said components is a combination of exponential and polynomial forms;

[0019] - said continuous analytical expression expressing each of said components presents the following generic form: 4 V +ta«h k-iJO-ts (¾.:5+4 s — SOC': SOC being the state of charge, and Qnom the nominal capacity of the battery;

[0020] - each of the sub-components Cq depends on the temperature T of the battery: with ViJ.k' € B. ;

[0021] - each of the sub-components Cjj depends on the temperature T of the battery and of Battery health status (SOH): 7“ 4- Ci^T 4' Cj.ys.scw avec GM;

[0022] - the factor is written in the form: — (5OH ~ 4- ^., / .8,5 4- tanh ^,^.§,7 (SO.H -- c^^.a) | ) SOH being the value of the battery's health status, and the parameters and C / jg / being constants;

[0023] - said first estimate of the battery's state of charge is based on the following expression: socri) - soc(to) + 7r ~...... / k-) - Dr. Jty LJ

[0024] - during the step of correcting said prediction of the battery state of charge by application of a Kalman filter, the corrected state of charge of the battery is expressed by the corrected state vector of the Kalman filter;

[0025] - during the step of correcting said prediction of the battery state of charge by By applying a Kalman filter, the correction of the battery state of charge prediction is performed based on the error ÿk between the simulated voltage HsimÀ and its measured value umesk, for each iteration k of the process:

[0026] - during the step of correcting said prediction of the battery state of charge by Applying a Kalman filter, the error yk is multiplied by a covariance matrix Kk of the Kalman filter:

[0027] - said covariance matrix Kk is defined by the following expression: / < (7¾ 4- 7¾ 1¾ “1 with : ; observation matrix evaluated at iteration k of the process; Pk: covariance matrix of the state estimated at iteration k of the process : covariance matrix of the measurement noise estimated at iteration k of the process;

[0028] - said expression of the voltage across the battery terminals is as follows: with : M: the voltage across the battery terminals Ra: the battery loss resistance Rs: the ohmic resistance of the battery E: the battery's open-circuit voltage i: the current flowing through the battery Qc: the capacitance of the capacitor Cd CD: the battery capacitor;

[0029] - said capacitance Qc of capacitor Cd, is defined by the differential equation next: dQc = ^iE Qc ( 1 1 \ df R^ + IQ Cd (¾ ! 1Q + R») with : Ra: the battery loss resistance Rs: the ohmic resistance of the battery E: the battery's open-circuit voltage i: the current flowing through the battery CD: the battery capacitor Rd: diffusion resistance;

[0030] - said battery model is a circuit comprising: - a perfect voltage generator delivering a voltage E (which corresponds to the open-circuit voltage of the battery); - a resistance Rs (which corresponds to the ohmic resistance of the battery) in series with the voltage source E; - an RC circuit in series with the voltage source E, this RC circuit comprising a resistance Rd (which corresponds to the diffusion resistance of the battery) and a capacitor Cd (which corresponds to the diffusion capacitance of the battery); - a resistance Ra (corresponding to the loss resistance of the battery) in parallel with the voltage source E;

[0031] - the steps of: - produce a prediction of the battery's state of charge; - produce a simulation of the voltage across the battery terminals; - to measure the voltage across the battery terminals; - determine the error between said prediction of the voltage across the battery terminals and said measurement of the voltage across the battery terminals; - correct said prediction of the battery state of charge by applying a Kalman filter; are repeated cyclically following iterations & of the process;

[0032] - the iterations of the process are carried out over a period of approximately 1 millisecond to 1 second;

[0033] - the process includes an initial calibration step, in which the process is implemented according to a first iteration with the battery fully charged, and in this first iteration: - said prediction of the battery state of charge corresponds to a value of 100%; - said simulation of the voltage across the battery terminals corresponds to a voltage measurement. PRESENTATION OF THE FIGURES

[0034] Other features and advantages of the invention will become apparent from the following non-limiting description, with reference to the accompanying drawings in which:

[0035] - [Fig. 1] illustrates an example of an equivalent electrical circuit of a battery;

[0036] - [Fig.2A] and [Fig.2B] are graphs representing the evolution of the state of charge as a function of time, respectively in discharge phase and in charge phase;

[0037] - [Fig. 3A] and [Fig. 3B] are graphs representing the evolution of the voltage depending on the time, respectively in discharge phase and in charge phase, under real conditions;

[0038] - Fig. 4A, Fig. 4B, and Fig. 4C are graphs representing the evolution of the state of charge, respectively at -20 °C, at 0 °C, and at 40 °C.

[0039] Similar and common elements in the various embodiments bear the same reference numbers to the figures. DETAILED DESCRIPTION

[0040] The invention relates to a method for estimating the state of charge of a battery based on a model of the electrical operation of an electrochemical accumulator. It allows for the precise determination of the real-time state of charge of a battery. As previously stated, the term "battery" here refers to any cell, module, pack, or any other assembly of electrochemical batteries, regardless of the battery's chemistry.

[0041] This state-of-charge estimation method can be implemented during both battery discharge and charging, and also allows for the simulation of self-discharge occurring during periods of rest. Furthermore, the state-of-charge estimation method can adapt to the battery's internal temperature, which significantly impacts its autonomy. This method will be described in general terms and is easily adaptable to different battery models.

[0042] The method for determining the state of charge of a battery comprises the following general steps: - establish an electrical model of the battery; - produce an expression of the battery's state of charge; - produce an expression for the voltage across the battery terminals; - express each of the components of the expressions for the state of charge of the battery and the voltage across the battery terminals, by a continuous analytical expression that is dependent on both the state of charge of the battery, the current and the temperature of the battery; - determine a first estimate of the state of charge of the battery (called "state of charge prediction") and of the voltage across the battery terminals (called "simulated voltage"); - correct said prediction of the state of charge using a Kalman filter, by the error between the simulated voltage and its measured value.

[0043] The voltage simulation allows the correction of a first estimate of the state of charge, via the Kalman filter.

[0044] To implement this method, the battery is first modeled by an equivalent electrical circuit which includes at least the following elements, each representing a characteristic of the battery: - the perfect voltage generator E of the battery (corresponding to the open circuit voltage, or no-load voltage, of the battery); - the ohmic resistance Rs of the battery; - the battery's Cd capacitor; - the diffusion resistance Rd of the battery; - the loss resistance Ra of the battery.

[0045] The ideal voltage generator E maintains a constant voltage across its terminals, regardless of the current flowing through it. Thus, it illustrates the evolution of the open-circuit voltage ("OCV") as a function of the state of charge and temperature. However, this component is insufficient to model the voltage variations resulting from the difference in electrical potentials associated with the reactions of each battery electrode.

[0046] The ohmic resistance Rs of the battery represents the direct resistive losses in the battery due to the passage of current through internal conductive materials, such as the electrodes, connectors, and electrolyte. The ohmic resistance Rs is an instantaneous resistance that appears as soon as a current flows through the battery, resulting in a voltage drop correlated to the current intensity. The ohmic resistance Rs allows for the simulation of immediate energy losses, which affect battery efficiency and reduce the energy actually available for the application.

[0047] The battery capacitor Cd is typically representative of the electrical double layer. The first layer, commonly called the "Stern layer", is The first layer is formed by ions absorbed onto the oppositely charged electrode. Furthermore, due to electrostatic forces, a second, more dispersed region of ions, the diffuse layer, appears near the first. The ions in this second layer are distributed according to electrical forces and thermal agitation, similarly to a capacitor.

[0048] Combining this with a diffusion resistor Rd connected in parallel allows for controlled discharge of the capacitor Cd when it is no longer powered, while reducing rapid voltage fluctuations. Thus, the diffusion resistor Rd reflects the diffusion phenomena of chemical species within the battery and adds a dynamic character to the model.

[0049] The battery loss resistance Ra is a dynamic component that allows modeling of self-discharge effects during discharge and outgassing phenomena at the end of charging. Thus, it ensures a more accurate simulation of the state of charge and voltage.

[0050] The chosen model can be any known model incorporating these elements.

[0051] Figure 1 illustrates an example of such a simple equivalent electrical circuit, based on A Thévenin equivalent circuit consists of a perfect voltage source delivering a voltage E (corresponding to the battery's open-circuit voltage) and a resistor Rs (corresponding to the ohmic resistance) in series, with a voltage Ur across its terminals. The model is completed by an RC circuit connected in series with the resistor Rs. This RC circuit includes a resistor Rd (corresponding to the diffusion resistance) and a capacitor Cd (corresponding to the diffusion capacitance), with the voltage Ud across this RC circuit.

[0052] The model also includes the resistance Ra (corresponding therefore to the loss resistance) placed between the terminals of the equivalent circuit.

[0053] The voltage across the battery terminals is denoted “ , and the battery current is denoted h. The other currents flowing through each cell are denoted *A, *P, *R, C-

[0054] The aim of the method is to obtain a corrected version of the state of charge, which is closer to reality. The method is based on a first estimate of the battery's state of charge, which includes a first term obtained, in a known manner, by integrating the current flowing through the battery over a given time (from t₀ to t). However, this integral is performed on this current, from which a second term corresponding to the modeling of water electrolysis or losses is subtracted, according to the following equation (here, the "receiver" convention is adopted, according to which the current is negative during discharge and positive during charge): [Math 1] SOC(t) = SOCfto! + tA™

[0055]

[0056]

[0057] This leads to a differential equation, since the resistance Ra itself depends on the state of charge of the battery: [Math 2] dSOC 1 / . u \ dt Qmax \ / Qmax corresponds to the maximum capacity of the battery, * to the current flowing through the battery, etu to the voltage across the battery terminals. Regarding the voltage u, this battery model allows us to produce an expression for this voltage “across the battery terminals, by expanding the sum of the three potential differences of the circuit in the model: [Math 3]

[0058]

[0059]

[0060] with : - u: the voltage across the battery terminals - Ra: the battery loss resistance - Rs: the ohmic resistance of the battery - E: the open-circuit voltage of the battery - * : the current flowing through the battery - Qc: the capacitance of the capacitor Cd - Cd: the battery capacitor. The capacitance Qc of the capacitor Cd in the model is defined by the following differential equation: [Math 4] dQc = IQi-E Qc ( 1 1 \ df X+X c (i vx ! sq + rJ with : - Ra: the battery loss resistance - Rs: the ohmic resistance of the battery - E: the open-circuit voltage of the battery - ' : the current flowing through the battery - CD: 1st capacitor - Rd: diffusion resistance

[0061] Following this, each of the components involved, namely Ra, E, Rs, Cd, Rd, and Qmax, was determined based on conventional experimental tests. For example, these tests were conducted at various temperatures (ranging from -20 °C to 40 °C) on several 6-volt lead-acid batteries. During discharge, the tests were performed using a square wave signal with a duty cycle of unity, i.e., a test in which the current is applied constantly for a certain duration and then switched off for an equivalent duration. During charging, a conventional charging protocol was followed.

[0062] Any other type of test may be carried out, such as the experimental tests described in document CN106054085.

[0063] However, these experimental tests do not lead to the creation of databases or lookup tables, but rather to the production of continuous analytical expressions. Indeed, following the experimental tests used to approximate the components Ra, E, Rs, Cd, Rd, and Qmax, the process involves the design of a general continuous analytical expression that depends on the state of charge, the current in the battery, and the battery temperature.

[0064] Advantageously, each of the components Ra, E, Rs, Cd, Rd, as well as Qmax, can be written in the form of a combination of exponential and polynomial forms. An example of such an expression is given below in the notation C{ which generically denotes each component Ra, E, Rs, Cd, Rd, Qmax): [Math 5] r 52 [Cf.144-4« “ SOC) i )

[0065] SOC being the state of charge, and Qnom the nominal capacity of the battery.

[0066] Each of the sub-components Cîj allows this generic expression Ci to be adapted for each component Ra, E, Rs, Cd, Rd, Qmax. All the components Ra, E, Rs, Cd, Rd, Qmax are then expressed by a single generic continuous analytical expression, of which only the sub-components Cq are adapted for each particular case of component Ra, E, Rs, Cd, Rd, Qmax.

[0067] Each of the subcomponents C'y depends on the temperature T: [Math 6] h "t U.,; / ,? F Mj8 with >

[0068] Optionally, if the battery's state of health (SOH) value is known, the model offers the possibility of adapting its robustness and accuracy. It is then also Alternatively, it is possible to express these sub-components C(j) in a way that depends on the battery temperature T, and also on the battery's state of health SOH: 67 7' -i- with jajk € jR and with jgSO.ff which is written in the form: SOH + bmh 4, / ,8,7 (50 / 7 -

[0069] These parameters and Cf jgj are constants that can be easily determined experimentally by a person skilled in the art, for a given battery. These parameters are therefore specific to each component Ra, E, Rs, Cd, Rd, Qmax within the generic analytical expression.

[0070] The method then includes a step of applying a Kalman filter to lead to a better evaluation of the state of charge, by performing a real-time correction.

[0071] Advantageously, the Kalman filter used is an extended Kalman filter, adapted to the present model, which is nonlinear. Thus, the state and observation transition functions do not need to be linear; their differentiability is sufficient.

[0072] In a classic manner for a Kalman filter, in this example, we consider the following system: [Math 7] ^(0 = f(X(t), &(t)) + w(7) where X represents the state vector, U the measurement prediction vector, w the process noise, and r the measurement noise. In this example, the noises w and r are assumed to be Gaussian, centered, and have covariance matrices Q and R, respectively. The function f calculates a state prediction based on the previous estimation. Similarly, the function h determines a measurement prediction using the previously estimated values. At each time interval (the process is, in fact, preferably repeated cyclically), the Jacobian matrices of the transition and observation states are evaluated, a process that essentially linearizes the system. Considering g and $, the solution algorithm is written as: [Math 8] Prediction: f X(t) - f(X(tp U(t)) ( Xk ^X(tk) Resolution << (pm-nmv) - puwt । (a = p(M Update : - pk ni (¾ h- ip pk ai)} Pk^(J-KbHk)Ph ÿk ™ btnes}k “ bk Xk = ..¥fe P Kk ÿk

[0073] with: [Math 9] Meanings: ....... Z\P- M,ru< < of p good evaluated at time t Hk: Matrix, observation evaluated at iteration k — ÿk ■ Error between the measured voltage and the simulated voltage at iteration k - Ak: State vector estimated at iteration k ...... JC* : State vector corrected at iteration k — Pk: Matrix, with covariance of Tet estimated at iteration k Q(t): Covariance matrix of the noise tic process estimated at time l — Rk: Covariance matrix of the nusure noise estimated at iteration k I: Identity Matrix

[0074] The transition and observation matrices are defined as the following Jacobians: [Math 10]

[0075] If we take the general formulation and apply it to our model, we obtain the following system: [Math 11] fy,,, kh(X(t},O(t'}}\ (wt\ (SÔC(i\\ / wA ( Uk — h(Xk) + t? — Usim,fc -r V

[0076] for which it is necessary to calculate the Jacobian matrices associated respectively with the state vector and the observation vector: FU) as oc ■ aaa / o.

[0077]

[0078]

[0079]

[0080]

[0081]

[0082]

[0083]

[0084]

[0085] Furthermore, it is possible to make this filter adaptive by allowing the covariance matrices of process noise and measurement noise to evolve over time. This improves the convergence of the algorithm: [Math 12] “* ^ai(ur£R-3^ Then, for each iteration, the error yk between the simulated voltage MsimA and its measured value umesJc allows the value of the state of charge SOC, expressed by the variable x, to be corrected by implementing the covariance matrix. [Math 13] ÿk “ Wræs.fe [Math. 14] [Math. 15] Kk = PkHk (Rk + HkPk Hk) 1 By considering the model equations mentioned above, it is then possible to determine the state of charge of the battery in real time, from the current, voltage and temperature measurements of the battery. For this so-called real-time updating, it is possible to cyclically repeat the process following k iterations of the prediction and correction steps. Periods on the order of 1 ms to 1 s allow for a preferred updating rate, and even more preferably with a period on the order of 100 ms. The application of the Kalman filter, which has just been described as an example, can of course be implemented in any other known way, with any suitable variation of the Kalman filter, which leads to the same result: correcting the value of the state of charge SOC of the battery (relative to the first estimate based on the integration of the battery current), thanks to the error between the simulation of the voltage across the battery terminals and its measured value. APPLICATION EXAMPLES The invention has been implemented on various batteries of FLA (“Flooded Lead-Acid”) and AGM (“Absorbed Glass Mat”) technologies and nominal capacities ranging from 150 Ah to 400 Ah.

[0086] A preliminary study made it possible to define a set of parameters specific to each of the batteries used, based on classical experimental tests described previously.

[0087] Next, in order to measure the performance of the state of charge determination according to the invention, the batteries were subjected to a comparison between a state of charge determined by the invention and the actual state of charge measured by complete battery discharge. Thus, discharge profiles were applied to the batteries cyclically until complete discharge. These tests were repeated at different temperatures (-20°C, 0°C, 20°C, and 40°C).

[0088] Load tests were then carried out according to conventional load protocols, under the same temperature conditions.

[0089] In these experiments, it is assumed that the battery discharges and recharges completely, that is to say that the initial state of charge is fixed at 100% in discharge and 0% in charge, and that a perfect determination of the state of charge (with the method of the invention) is supposed to indicate 0% at the end of discharge and 100% at the end of charge.

[0090] Figures 2A and 2B illustrate the results of these tests by representing the evolution of the load states estimated by the method according to the invention (on the ordinate of Figures 2A and 2B, as a percentage of the maximum load) as a function of time (on the abscissa, in hours).

[0091] Figure 2A illustrates the evolution of the state of charge determined according to the invention for batteries in discharge at different temperatures (-20°C, 0°C, 20°C, and 40°C). The measurements according to the invention make it possible to consistently demonstrate the impact of temperature on the battery's state of charge (and therefore its potential operating time) during discharge. The operating time is effectively doubled between the test at -20°C and the test at 20°C. Methods for estimating the state of charge of a battery that do not take temperature into account are, in fact, very limited in accuracy.

[0092] On the other hand, the trend of the discharge curves is consistent with physical reality: the curves are cyclically linear. Indeed, the same power profile (for discharge) is applied cyclically.

[0093] Figure 2B illustrates the evolution of the state of charge determined according to the invention for batteries during charging at different temperatures (-20°C, 0°C, 20°C and 40°C). The trends are also consistent with physics.

[0094] Furthermore, the performance of the method for determining the state of charge is measured by completely discharging the battery to compare the value given by the determination according to the invention and the actual measured value of the state of charge, as described above.

[0095] These measurements indicate an average error in the estimation of the state of charge across all the batteries tested of less than 5%.

[0096] However, in the case of longer tests, including several successive discharges and charges, prior art models produce an estimate of the state of charge that degrades rapidly, with an accumulation of errors. The determination of the state of charge according to the invention, however, remains just as accurate.

[0097] Figures 3A and 3B illustrate the evolution of the simulated voltage according to the invention SIM and the measured voltage EXP (on the y-axis, in volts) as a function of time (on the x-axis, in minutes during the discharge phase for [Fig. 3A] and in hours during the charging phase for [Fig. 3B]). The solid lines represent the measured experimental data EXP and the dashed lines the simulated SIM values ​​of the battery voltage.

[0098] Fig. 3A illustrates the voltage simulations for a partial discharge, corresponding to a real and demanding use of the battery in an industrial environment, with many voltage variations.

[0099] Fig. 3B illustrates the voltage variation for a full battery charge.

[0100] The observed results are sufficiently accurate to apply the correction in the best conditions, with an average error across all batteries studied of less than 40 mV, representing an absolute error of less than 0.6%.

[0101] Figures 4A, 4B, 4C illustrate the results of the determination of the state of charge, before correction (SIM, solid lines) and after correction by the invention (AEKF, dashed lines), during a test comprising three complete discharges, each followed by a complete charge, at -20 °C ([Fig.4A]), at 0 °C ([Fig.4B]), and at 40 °C ([Fig.4C]).

[0102] The curves clearly show the accumulation of simulation errors of the state of charge over time, but these are partially corrected according to the invention (dotted curves) which improves the estimation, and thus provides long-term robustness to the determination of the state of charge.

[0103] Overall, the error in determining the state of charge is reduced by 0.6% in discharge and by 3.9% in charge, which makes it possible to obtain, thanks to the invention, an average error in determining the state of charge of less than 3% on all the batteries tested.

[0104] Various embodiments can be implemented. In particular, the simple model given as an example, as the first step of the process, can be supplemented by any other circuit element to make it more realistic. For example, it is possible to add an additional RC circuit to the electrical model of the battery. This component can allow for a more realistic representation of the different diffusion phenomena of species within the battery, whose time constants can vary considerably.

[0105] Moreover, the process can be applied to any electrochemical battery technology, other than the lead-acid battery examples mentioned, and in particular to lithium or nickel type batteries.

[0106] Alternatively, it is possible to integrate SOH health status data into the process, as mentioned above, for batteries where this indicator is available. Thus, the process according to the invention would be able to adapt to a decrease in battery performance due to a decline in its health status.

Claims

1. Demands A method for determining the state of charge of a battery, from measurements of current, voltage and temperature of the battery, comprising a step of modeling the battery and a step of applying a Kalman filter, this method being characterized in that it comprises the following steps: - establish a model of the battery by a circuit comprising at least: an open circuit voltage (E) of the battery, an ohmic resistance (Rs) of the battery, a diffusion resistance (Rd) of the battery, a diffusion capacitor (Cd) of the battery, and a loss resistance (Ra) of the battery; - from the battery model, produce an expression for the state of charge of the battery based on the integration of current measurements taking into account the loss resistance (Ra) of the battery; - from the battery model, produce an expression for the voltage across the battery terminals as a function of at least the following components: the loss resistance (Ra) of the battery, the ohmic resistance (Rs) of the battery, the diffusion capacitor (Cd) of the battery, the diffusion resistance (Rd) of the battery, and the open circuit voltage (E) of the battery; - express each of said components (Ra, E, Rs, Cd, Rd), as well as the maximum capacity Qmax of the battery, by a continuous analytical expression dependent on both the state of charge of the battery, the current and the temperature of the battery; - from said expression of the state of charge of the battery and said continuous analytical expressions of said components, produce a prediction of the state of charge of the battery; - from the said expression of the voltage across the battery terminals and the said continuous analytical expressions of the said components, produce a simulation of the voltage across the battery terminals; - to measure the voltage across the battery terminals; - determine the error between the predicted battery voltage and the measured battery voltage; - correct the predicted battery state of charge by applying a Kalman filter, adding to the predicted battery state of charge a value dependent on the error. between the prediction of the battery voltage and the measurement of the battery voltage.

2. A method according to claim 1, characterized in that said continuous analytical expression expressing each of said components (Ra, E, Rs, Cd, Rd, Qmax) is a combination of exponential and polynomial forms.

3. A method according to claim 2, characterized in that said continuous analytical expression expressing each of said components (Ra, E, Rs, Cd, Rd, Qmax) has the following generic form: Ci = Ci.i exp - 2 --- | -raj h- <;i 5 (WG - cW -rd - racp (SOC - r <S1 )c' 4- 22 -r taiih (c$,U+4* -SOC) : ) SOC étant l’état de charge, et Qnom la capacité nominale de la batterie. ^Revendication 4] Procédé selon la revendication 3, caractérisé en ce que chacun des sous-composants Ci j dépend de la température T de la batterie : Q exp [c,.,2 x / f + y., J— ] + c T2 + Cî..

5. A method according to claim 3, characterized in that each of the subcomponents Cjj depends on the temperature T of the battery and the state of health SOH of the battery:

6. A method according to claim 5, characterized in that the factor Ci.ys.SOJZ is written in the form: Cij.îxSOH (SOff ~~ Ci^) + 4- -j-tanh (SOi? -• cyyy) Q SOH being the value of the state of health of the battery, and the parameters Ciet Ci j^j being constants.

7. A method according to any one of the preceding claims, characterized in that said first estimation of the state of charge of the battery is based on the following expression: SOC(t) — SOC(fo) -F ......... / p(r) — | dr Wm&x JL J.

8. A method according to any one of the preceding claims, characterized in that, during the step of correcting said prediction of the battery state of charge by applying a Kalman filter, the corrected state of charge of the battery is expressed by the corrected state vector of the Kalman filter.

9. A method according to any one of the preceding claims, characterized in that, during the step of correcting said prediction of the battery state of charge by applying a Kalman filter, the correction of the prediction of the battery state of charge is carried out on the basis of the error ÿ, between the simulation of the voltage and its measured value umesjt, for each iteration & of the method:

10. A method according to claim 9, characterized in that, during the step of correcting said prediction of the battery state of charge by applying a Kalman filter, said error is multiplied by a covariance matrix Kk of the Kalman filter:

10. ■X7 ~ A& *i“ K & I / fe

11. A method according to claim 10, characterized in that said covariance matrix Kk is defined by the following expression: kj. - a (¾ + A- n hJ ) 1 with: Hk; observation matrix evaluated at iteration & of the method P^: covariance matrix of the state estimated at iteration k of the method Æk: covariance matrix of the measurement noise estimated at iteration k of the method.

12. A method according to any one of the preceding claims, characterized in that said expression for the voltage across the battery terminals is as follows: & p . , QgJ U = .......................... h + K., î -b............ ' < + BV ° J with:

12. u: the voltage across the battery terminals Ra: the loss resistance of the battery Rs: the ohmic resistance of the battery E: the open-circuit voltage of the battery » : the current flowing through the battery Qc: the capacitance of the capacitor Cd Cd: the capacitor of the battery.

13. The method according to claim 12, characterized in that said capacitance Qc of the capacitor Cd, is defined by the following differential equation: dQc - jRg1 - E - Qc / 1 t 1 \ m “ JV+B G1 5 + ¾ J with: Ra: the loss resistance of the battery Rs: the ohmic resistance of the battery E: the open-circuit voltage of the battery * : the current flowing through the battery Cd: the capacitor of the battery Rd: the diffusion resistance

14. A method according to any one of the preceding claims, characterized in that said battery model is a circuit comprising: - a perfect voltage generator delivering a voltage E (which corresponds to the open-circuit voltage of the battery); - a resistance Rs (which corresponds to the ohmic resistance of the battery) in series with the voltage source E; - an RC circuit in series with the voltage source E, this RC circuit comprising a resistance Rd (which corresponds to the diffusion resistance of the battery) and a capacitor Cd (which corresponds to the diffusion capacitance of the battery); - a resistance Ra (corresponding to the loss resistance of the battery) in parallel with the voltage source E.

15. A method according to any one of the preceding claims, characterized in that the steps of: - producing a prediction of the state of charge of the battery; - producing a simulation of the voltage across the battery terminals; - carrying out a measurement of the voltage across the battery terminals; - determining the error between said prediction of the voltage across the battery terminals and said measurement of the voltage across the battery terminals; - correct said prediction of the battery state of charge by applying a Kalman filter; are repeated cyclically following iterations & of the process.

16. A method according to claim 15, characterized in that the iterations of the method are carried out over a period of approximately 1 millisecond to 1 second.

17. A method according to any one of the preceding claims, characterized in that it comprises an initial calibration step, in which the method is implemented according to a first iteration with the battery fully charged, and in this first iteration: - said prediction of the state of charge of the battery corresponds to a value of 100%; - said simulation of the voltage across the battery terminals corresponds to a measurement of the voltage.

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