Method for determining the water content of a membrane of a proton exchange polymer membrane fuel cell

Abstract

The present invention relates to a method for determining the water content of a membrane of a fuel cell (1), the method using a dynamic model of the membrane (MOD DYN), an electrochemical model of the fuel cell (MOD ELC), and a mass balance (BIL MAS) at the anode and at the cathode. For this method, the voltage and the current at the terminals of the fuel cell, as well as the temperature of the fuel cell, are measured; the flow rates of reagents at the anode and the cathode are acquired (ACQ), and an adaptive extended Kalman filter (EKF) is applied to the models by means of the voltage measurement, the current measurement, the temperature measurement and the acquired flow rates in order to determine the water content within the membrane.

Description

[0001]METHOD FOR DETERMINING THE WATER CONTENT OF A MEMBRANE OF A FUEL CELL WITH A PROTON EXCHANGE POLYMER MEMBRANE Technical field The present invention relates to the field of monitoring, control and diagnosis of a fuel cell, in particular a fuel cell with a proton exchange polymer membrane. The invention relates in particular to the determination of the water content of the membrane of such a fuel cell. Much work is currently devoted to the development of fuel cells as sources of electrical energy, both for stationary applications and for on-board applications, in particular for vehicles driven by electrical machines.Indeed, the advantages of fuel cells (no greenhouse gas emissions, higher energy density than batteries) are assets that are becoming important, particularly for the transport sector, especially for heavy transport. A fuel cell allows chemical energy to be transformed into electrical energy. In the case of a hydrogen / oxygen fuel cell, the chemical reaction implemented is a redox equation of dihydrogen with dioxygen, the reaction of which can be written by the equation: [Chem 1]. In a fuel cell, the electrochemical oxidation of hydrogen is carried out at an anode made of a conductive catalytic material, while the electrochemical reduction of oxygen occurs at a cathode, generally made of the same catalytic material. In addition, the anode and cathode compartments are separated by an electrolyte allowing the exchange of protons or ions. For vehicles, the fuel cells that currently prove most promising are so-called proton exchange membrane cells, operating from a source of hydrogen coming either from a bottle on board the vehicle or from a unit producing hydrogen directly in the vehicle. Thus, hydrogen can be produced directly using a reformer operating with a suitable fuel, such as methanol, gasoline, diesel, etc. Advantageously,the fuel cell can be a low-temperature fuel cell, for example a proton exchange polymer membrane (PEM) fuel cell. This type of fuel cell is currently the most suitable for mobile applications, particularly due to its cost. For economic purposes, their operation must respect a maximization of the lifetime while involving an acceptable energy efficiency. To understand, a proton exchange polymer membrane fuel cell comprises several subsystems that can be grouped into three parts: - A "cathode" volume in which the incoming air (wet or dry) comes out charged with water following the oxidation-reduction equation involved; the air arrives at the level of channels of a bipolar plate,and the oxygen transport is done by diffusion in the gas diffusion layer GDL (from the English "Gas Diffusion Layer") to the place of the reduction reaction near the membrane / catalytic layer CL interface (from the English "Catalytic Layer"), at the platinum catalytic sites; - An "anode" volume in which the incoming H2 (wet or dry) crosses the third part which is the membrane, to react with the dioxygen O2 on the cathode side; the dihydrogen H2 arrives at the level of the channels of the bipolar plate to then migrate by diffusion towards the membrane / catalytic layer CL interface (from the English "Catalytic Layer"), in which the oxidation reaction takes place, near the catalytic sites; and - A membrane which will allow the protons H, +to cross from the anode to the cathode so that the reduction reaction takes place at the membrane interface on the cathode side; the membrane is also the location of water transfers between cathode and anode; depending on its water content, it gives a greater or lesser resistance to the fuel cell. Figure 1 schematically illustrates a proton exchange membrane fuel cell. The fuel cell 1 comprises an anode 2, a cathode 3 and a polymer membrane 4 arranged between the anode and the cathode. The anode 2 comprises a bipolar plate 5 provided with channels 6 for supplying dihydrogen H2, a gas diffusion layer 7 and a catalytic layer 8. The cathode 3 comprises a bipolar plate 12 provided with channels 11 for supplying air, a gas diffusion layer 10 and a catalytic layer 9. The chemical reactions implemented result in the movement of H ions +in the membrane 4 as well as the movement of electrons e- ensuring the generation of electricity, symbolized here by the curved arrows and the symbol V. In addition, the water present and / or generated at the anode and the cathode crosses the membrane. This exchange is represented by a double arrow in Figure 1. Prior art Water management at the heart of the fuel cell, particularly at the heart of the membrane, is essential. On the one hand, if the core of the fuel cell is not wet enough, it loses efficiency and the drying of the membrane causes irreversible damage. On the other hand, if the core of the fuel cell is too wet, liquid water can appear, generally on the cathode side, and clog the catalytic sites (which may be composed of platinum). In this case, the fuel cell loses efficiency, and the catalytic sites degrade by dissolution, possibly by dissolution of the platinum.To do this, it is necessary to measure or estimate this water content in the membrane upstream. It seems relevant to choose to locate the measurement within the cell membrane itself. The measurement is certainly more difficult to access there, but it provides a much more precise idea of ​​the water behavior. Different measuring devices exist and provide an estimate of the membrane's proton resistance which, via an empirical equation depending on the type of membrane considered, makes it possible to trace this water content. Despite everything, these tools can be expensive. In addition, the sensitivity of the measurement to other quantities such as temperature does not allow a direct conclusion on a reliable membrane water content. For example, patent applications CN109841879, US2019348694 describe such solutions based on proton resistance or impedance to deduce the water content.Additionally, the paper “Lukas Bohler, Daniel Ritzberger, Christoph Hametner, and Stefan Jakubek. Constrained extended Kalman filter design and application for on-line state estimation of high-order polymer electrolyte membrane fuel cell systems. International Journal of Hydrogen Energy, 46(35):18604–18614, 2021” describes the application of an extended Kalman filter to reconstruct different fuel cell states, including the water activity at the membrane. However, this paper uses a simple model of the membrane that does not allow for accurate determination of the water content. Furthermore, this paper concerns water activity at the membrane level and not the water content within the membrane. Thus, the water activity determined by this method is not representative of what is happening at the center of the fuel cell.Summary of the invention The aim of the invention is to accurately determine the water content within the membrane of a polymer membrane fuel cell, without expensive and complex instrumentation, this determination being able to be carried out in real time. For this purpose, the present invention relates to a method for determining the water content of a membrane of a fuel cell, implementing a dynamic model of the membrane, an electrochemical model of the fuel cell, and a mass balance at the anode and cathode.For this method, a measurement of the voltage and current at the terminals of the fuel cell is implemented, as well as the temperature of the fuel cell, the flow rates of reactants at the anode and cathode are acquired, (possibly the pressures at the inlets of the anode and cathode, and the relative humidity entering the cathode) and an adaptive extended Kalman filter is applied to the models by means of the voltage measurement, the current measurement, the temperature measurement and the flow rates acquired to determine the water content within the membrane (and possibly the anode and cathode pressures, the relative humidity at the cathode). The models allow an accurate representation of the operation of the fuel cell, therefore an accurate determination of the water content within the membrane.Voltage measurements and mass flow acquisition do not require complex and expensive instrumentation: these can be quantities already measured within the fuel cell. In any case, the invention does not implement a direct measurement in the membrane. The adaptive extended Kalman filter allows adaptation of the filter gain, which allows for speed gains, and a fortiori makes real-time application possible. The invention further relates to a control method implementing such a method for determining the water content within the membrane.The invention relates to a method for determining the water content within a membrane of a proton exchange polymer membrane fuel cell, said fuel cell comprising an anode, a cathode, and a polymer membrane arranged between said anode and said cathode, said anode being supplied with reactants such as dihydrogen and optionally water to form products such as H ions. +, said cathode being supplied with reactants such as oxygen and nitrogen to form products such as water. The method implements a dynamic model of said membrane which relates the water content of the membrane to the surface current and to the mole fractions of the reactants at the cathode and the anode, an electrochemical model of said fuel cell which relates the surface current to the voltage of said fuel cell and to the water content of said membrane, and a mass balance at the anode and the cathode which relates said surface current to the mole fractions of the reactants and products at said anode and cathode and to the mass flow rates of the reactants at the anode and cathode, and in that the following steps are implemented: a. A voltage and a current are measured at the terminals of said fuel cell, as well as the temperature of the fuel cell; b.Mass flow rates of said reactants are acquired at said anode and cathode; and c. The water content within said membrane, preferably at the center of said membrane, is determined by means of an adaptive extended Kalman filter, applied to said dynamic model of said membrane, to said electrochemical model of said fuel cell and to said mass balance, and by means of said measured voltage, to the measured current, to the measured temperature, and to said acquired mass flow rates of reactants. According to one embodiment, pressures of the reactants are further acquired at said anodes and cathodes, said electrochemical model depending on said pressures of the reactants. According to one implementation, said electrochemical model is constructed by means of the following equation: ^. ^^^^ = ^ ^^^^^^ − ^ ^^^ − ^ ^^^^ − ^ ^^^ with ^ ^^^^ the voltage of a fuel cell, ^ ^^^activation losses as a function of surface current, ^ ^^^^ the losses by diffusion of oxygen at the cathode as a function of the surface current, ^ ^^^ ohmic losses due to the proton resistance of the membrane as a function of the surface current and the water content of said membrane, ^ ^^^^^^ the potential available under the operating conditions. Advantageously, the water content of said membrane at the interface with said anode and / or at the interface with said cathode is further determined. According to one aspect, said dynamic model of the membrane takes into account the transport of water, and / or the electroosmosis reaction, and / or the absorption or desorption phenomena at said anode and cathode. Advantageously, said dynamic model of the membrane is constructed using content in the center of the membrane, ^ ^ the water content of the membrane at the interface with the anode, ^ ^the water content of the membrane at the interface with the cathode, 78 the diffusivity coefficient, j the surface current, 9 ^ the number of water molecules carried by a proton in the membrane, 1 ^^,^ ^^^ the absorption or desorption coefficient of the anode, 1 ^^,^ ^^^ the absorption or desorption coefficient of the cathode, ^ ^ ^ 4 the equivalent water content of the anode as a function of the mole fraction of water at the anode, ^ ^ ^ 4the water content of the cathode as a function of the molar fraction of water at the cathode, :; the thickness of the membrane, :=> the thickness of the catalytic layer, ?^ the fixed charge concentration of the membrane, F the Faraday constant. According to one embodiment, said mass balance is constructed at said cathode, taking into account oxygen consumption, water production as well as water absorption and desorption. According to one embodiment, said mass balance is constructed at said anode, taking into account hydrogen consumption, as well as water absorption and desorption. According to one implementation, said extended Kalman filter is written: @A = B^@, C, D^,0 = E ^ @, C, D ^, F = GC, with x the membrane water content, @A the time derivative of x, u the measured or acquired values ​​including the fuel cell voltage and reactant flow rates, y the measured fuel cell current, C the observation matrix, B the second membrane dynamic equation, E an algebraic constraint function and z the algebraic fuel cell states, preferably, the dynamic equation of the system is discretized as follows, with k the discrete time step:@HI% − @H − J:B^@HI%, CHI%, DH^ = 0E ^ @ HI% , C HI% , D H ^ = 0. Advantageously, said water content within the membrane is determined by applying an adaptive extended Kalman filter using the following steps, given the estimates ^@L HM% , Ĉ HM% ^, the covariance matrices O HM% , P HM% , the estimated covariance matrix Q HM%at the previous time step k-1:i. We calculate the Jacobian identity matrix, ii. We calculate the predicted state by solving the dynamic equation of the system and calculating the Jacobian iii. We calculate the predicted estimate of the covariance iv. We calculate the innovation [H = FH − GĈH|HM%, the covariance of the innovation and the Kalman gain] = Z M% H QH|HM%XH\H .v. We find the updated state @L H = @L H|HM% + ] H [ H , and we deduce the water content of the membrane, vi. We calculate the algebraic state Ĉ H solution of the dynamic equation of the system and the updated innovationεH = FH − GĈH .vii. We then update the covariance matrix identity matrix, and the covariance matricesO H P= _P + ^1 − _^ + ^aa Z + ZH HM% HH XHQH|HM%XH^.Furthermore, the invention relates to a method for controlling a proton exchange polymer membrane fuel cell, said fuel cell comprising an anode, a cathode, and a polymer membrane arranged between said anode and said cathode, said anode being supplied with reactants such as dihydrogen and optionally water to form products such as H ions +, said cathode being supplied with reactant such as oxygen, nitrogen to form products such as water. For this method, the following steps are implemented: a. The water content of the membrane is determined by means of the method for determining the water content of the membrane of a fuel cell according to one of the preceding characteristics; and b. The flow rate of at least one reactant of said anode and cathode is adapted as a function of said determined water content of the membrane and / or the pressure at the inlet of the cathode is adapted as a function of said determined water content of the membrane and / or the temperature of the fuel cell is adapted as a function of said determined water content and / or the humidity of the air at the inlet of the cathode is adapted as a function of said determined water content.Other characteristics and advantages of the method according to the invention will appear on reading the following description of non-limiting examples of embodiments, with reference to the appended figures described below. List of figures Figure 1, already described, illustrates a constitution of a fuel cell with a proton exchange polymer membrane. Figure 2 illustrates the steps of the method according to a first embodiment of the invention. Figure 3 illustrates the steps of the method according to a second embodiment of the invention. Figure 4 illustrates, for an example, the water content of a membrane of a fuel cell, with a reference curve REF, a curve corresponding to a method of the prior art AA, and a curve corresponding to the implementation of the invention INV.Description of embodiments The present invention relates to a method for determining the water content of a membrane of a proton exchange polymer membrane fuel cell. The water content of the membrane can be seen as the ratio of the water concentration of the membrane to the concentration of ion exchange sites (SO3- for nafion for example) of the membrane, it is expressed without dimension.A proton exchange polymer membrane fuel cell comprises: - An anode, generally comprising a bipolar plate (for distributing dihydrogen), a gas diffusion layer GDL and a catalytic layer CL, for example made of platinum, or platinum alloy, or with platinum-doped catalysts or alternative platinum deposition materials, - A cathode, generally comprising a bipolar plate (for distributing dioxygen and discharging water), a gas diffusion layer GDL and a catalytic layer CL, for example made of platinum, or platinum alloy, or platinum-doped catalysts or alternative platinum deposition materials, and - A polymer electrolyte membrane arranged between the anode and the cathode, particularly between the catalytic layer of the anode and the catalytic layer of the cathode, for example the membrane may be made of fluoropolymer or based on polybenzimidazole (PBI) doped with phosphoric acid.The cathode is supplied with reactants: moist or dry air (therefore the cathode reactants include oxygen and nitrogen). During fuel cell operation, the cathode produces water as a result of the oxidation-reduction equation involved. The anode is supplied with reactants: hydrogen, therefore the anode reactants include hydrogen. During fuel cell operation, the anode produces H ions. +following the oxidation equation involved. The H+ ions pass through the polymer membrane, as well as the water. Depending on the humidification (humidity level) of the membrane, linked to its water content which changes over time, it gives a greater or lesser resistance to the fuel cell. The fuel cell implemented in the method according to the invention may conform to the constitution of Figure 1, or to any constitution of a fuel cell with a proton exchange polymer membrane. The method according to the invention implements the following models: - A dynamic model of said membrane which links the water content of the membrane to the surface current (current per unit area within the fuel cell) and to the molar fractions of the reactants at the cathode and anode,the dynamic model of the membrane reporting the water behaviors within the membrane itself as a function of the humidity conditions at the interfaces with the anode and the cathode, - An electrochemical model of said fuel cell which links the surface current to the voltage of said fuel cell and to the water content of said membrane, the electrochemical model characterizing the operation of the fuel cell as a function of the operating conditions, this model being able to be linked to the polarization curve of the cell, and - A mass balance at the anode and the cathode which links said surface current to the molar fractions of the reactants and products at said anode and cathode and to the mass flow rates of the reactants at the anode and cathode, the mass balance making it possible to take into account the consumption of the reactants and the production of water. The invention makes it possible to determine the water content within the membrane of the fuel cell,preferably in the center of the fuel cell membrane. According to one embodiment of the invention, the method can further determine the water content of the membrane at the interface with the anode and / or at the interface with the cathode. These two water contents make it possible to model the water exchanges with the external cathode and anode volumes. In this way, the physical phenomena within the membrane can be determined more precisely, which makes it possible in particular to adapt the control, monitoring or diagnosis of the fuel cell. The method according to the invention implements the following steps: 1. Measurement of the voltage, current and temperature 2. Acquisition of the mass flow rates of the reactants 3. Determination of the water content of the membrane. Steps 1 and 2 can be implemented simultaneously. Step 3 can be implemented by computer means,in particular a computer or a fuel cell controller. The steps will be detailed in the rest of the description. For the implementation of the method in real time, all the steps are implemented in real time. Figure 2 illustrates, schematically and in a non-limiting manner, the method according to a first embodiment of the invention. The voltage and current (MES) are measured at the terminals of the fuel cell, as well as the temperature of the fuel cell. The mass flow rates (ACQ) of the reactants at the anode and cathode are simultaneously acquired, and possibly the pressures at the inlet of the anode and the cathode. Then, an adaptive extended Kalman filter is applied to the three models: dynamic membrane model MOD DYN, electrochemical model MOD ELC, and mass balance BIL MAS, and by means of the voltage and current measurement as well as the measurement of the temperature and the acquired mass flow rates,to obtain the water content λ within the membrane. According to one embodiment, the method may comprise a step of controlling the fuel cell according to the determined water content. The control makes it possible to avoid operating the fuel cell when the membrane is too dry or too wet, which avoids losses of efficiency and avoids degradation of the fuel cell. Thus, for this embodiment, the method comprises the following steps: 1. Measurement of the voltage, current and temperature 2. Acquisition of the mass flow rates of the reactants 3. Determination of the water content of the membrane 4. Control of the fuel cell Steps 1 and 2 can be implemented simultaneously. Steps 3 and 4 can be implemented by computer means, in particular a computer or a controller of the fuel cell. The steps will be detailed in the rest of the description. For the implementation of the method in real time,all steps are implemented in real time. Figure 3 illustrates, schematically and in a non-limiting manner, the steps of the method according to this second embodiment. The steps already described in relation to Figure 2 are not detailed again. The method further comprises a step of controlling CON of the fuel cell as a function of the determined water content λ. According to an implementation of the invention, the method may comprise preliminary steps of constructing the models. In this case, the method may comprise the following steps: A. Construction of the dynamic model B. Construction of the electrochemical model C. Construction of the mass balance 1. Measurement of the voltage, current and temperature 2. Acquisition of the mass flow rates of the reactants 3. Determination of the content at the top of the membrane 4. Control of the fuel cell Steps A, B and C can be carried out previously only once offline,whereas steps 1 to 4 can be performed in real time. Steps A, B and C can be performed in any order. Steps 1 and 2 can be implemented simultaneously. Steps 3 and 4 can be implemented by computer means, in particular a computer or a controller of the fuel cell. The steps will be detailed in the remainder of the description. A. Construction of the dynamic model During this optional step, the dynamic model of the membrane is constructed which links the water content of the membrane to the surface current and to the mole fractions of the reactants at the cathode and the anode. For the embodiment for which the water content at the interface of the membrane with the anode and / or at the interface of the membrane with the cathode is further determined, the dynamic model of the membrane can depend on these different water contents. According to one aspect of the invention,the dynamic model of the membrane can take into account the transport of water, and / or the electroosmosis reaction, and / or the absorption or desorption phenomena at the level of said anode and cathode. Thus, the different physical phenomena within the membrane are modeled, which contributes to the precision of the determination of the water content. According to an implementation of the invention, the dynamic model of the membrane can be constructed using the following equation:, ^ ^ the content in the center of the membrane, ^ ^ the water content of the membrane at the interface with the anode, ^ ^ the water content of the membrane at the interface with the cathode, 78 the diffusivity coefficient, j the surface current (in other words the current per unit area within the fuel cell), 9 ^the number of water molecules carried by a proton in the membrane, 1^^,^^^^ the absorption or desorption coefficient of the anode,1 ^^^ the absorption or desorption coefficient ^4^^,^ nt of the cathode, ^^ the equivalent water content of the anode as a function of the molar fraction of water at the anode, ^ ^ ^ 4 the water content of the cathode as a function of the molar fraction of water at the cathode: ; the thickness of the membrane, : => the thickness of the catalytic layer, ? ^ the concentration of ion exchange sites in the membrane, F the Faraday constant. The first term of this equation corresponds to the transport phenomenon following a concentration gradient, generally from the cathode to the anode where water is least present; the diffusivity coefficient 78 ^ ^ ^ depends on the water content ^ in the membrane and its equation can be written: With R the ideal gas constant, T the fuel cell temperature, T ref the reference temperature taken at 298K, and ^ ^^^ the activation energy of the reaction. The second term of this equation corresponds to the phenomenon of electroosmosis; during use, H protons + will migrate from the anode to the cathode and will carry water molecules with them via H3O ions + ; the number of water molecules carried by an H proton + is given by the electroosmotic coefficient 9 ^ ^λ^ : With λ ^ the water content at the center of the fuel cell. The last term in this equation only concerns the water contents at the interfaces ^ ^ , ^ ^ and highlights the exchanges of water with the cathode and anode volumes by absorption or desorption phenomenon; coefficient 1 ^^^^^ defines the absorption or desorption coefficient while ^ ^ ^ 4 {: ^ ^ ^ 4 are the equivalent water contents of the cathode and anode volumes and depend explicitly on the molar fractions of water in these two volumes: +17.81~ − 39.85~^ + 36~! if 0 ≤ ~ < 1,λ^ = |14 + 1.4^~ − 1^ if 1 ≤ ~ ≤ 3,s i ~ ≥ 3. With i denoting the anode or cathode, ^ ^ ^ ^ ^ the mass fraction of water at the anode or cathode, Pi the pressure of the reactants at the anode or cathode, Q ^^^the saturated vapor pressure. This model can be adapted if the method only determines the water content at the center of the membrane. B. Construction of the electrochemical model During this optional step, an electrochemical model of said fuel cell is constructed which links the surface current to the voltage of said fuel cell and to the water content of said membrane. The electrochemical model can be a voltage-current equation which characterizes the operation of the fuel cell as a function of the operating conditions; this model can be linked to the polarization curve of the fuel cell. According to one embodiment of the invention, the electrochemical model can also depend on the pressures of the reactants at the anode and at the cathode. For this embodiment, the method can comprise an additional step of acquiring the pressures of hydrogen and oxygen respectively at the anode and at the cathode.This embodiment promotes the accuracy of the model, and therefore the accuracy of the water content determination. According to one implementation, the electrochemical model can be constructed using the following equation: ^. ^^^^ = ^ ^^^^^^ − ^ ^^^ − ^ ^^^^ − ^ ^^^ with ^ ^^^^ the voltage of a fuel cell, ^ ^^^ activation losses as a function of surface current, ^ ^^^^ the losses by diffusion of oxygen at the cathode as a function of the surface current, ^ ^^^ ohmic losses due to the proton resistance of the membrane as a function of the surface current and the water content of said membrane, ^ ^^^^^^ the potential available under the operating conditions. According to a non-limiting example, the terms of this equation can be determined as follows: which defines the potential available under the required operating conditions, with JX the enthalpy of the reaction under standard conditions, J\ the entropy of the reaction under standard conditions, F the Faraday constant, T the fuel cell temperature, R the ideal gas constant, Q ^^ the hydrogen pressure, Q ^^ the oxygen pressure, Q ^^^ the reference pressure; ^ ^^^ = ln ¢ £ £ ¤ ¥, which defines the losses by activation of the chemical reaction, with R the ideal gas constant, T the fuel cell temperature, F the Faraday constant, j the surface current, . n the surface exchange current under the reference pressure and temperature conditions taken respectively at 1 bar (0.1 Mpa) and 353K, _ a calibration coefficient of the model so that the resulting polarization curve approaches the actual polarization curve of the fuel cell; - ^ ^^^^= which defines the losses by diffusion of dioxygen at the cathode, with B a calibration coefficient of the model so that the resulting polarization curve approaches the real polarization curve of the fuel cell, j the surface current, . ^ a model calibration coefficient so that the resulting polarization curve approaches the actual polarization curve of the fuel cell; - ^ ^^^ = ©\P^^ ^ ^., which characterizes the ohmic losses due to the proton resistance of the membrane, with ©\P^^ ^ ^ the resistance of the fuel cell membrane, j the surface current. For example, the resistance can be defined by a formula of the form: ©\P^^ ^ ^ = ^^ ^n.nn^%!ª^ Mn.nn ^ ° °with ^ ^the «!^®¯^%^¬&^^±²M³^^water content in the middle of the membrane, T the temperature of the fuel cell,:; the thickness of the membrane.C. Mass balance During this optional step, a mass balance is constructed at the anode and cathode which links said surface current to the mole fractions of the reactants and products at the anode and cathode and to the mass flow rates of the reactants at the anode and cathode, the mass balance makes it possible to take into account the consumption of the reactants and the production of water. According to one embodiment of the invention, the mass balance on the cathode side can take into account the consumption of oxygen, the production of water as well as the absorption and desorption of water (i.e. the quantity of water which is entering or leaving the membrane at the interface with the cathode). For example, the oxygen consumption at the cathode can be written: 9 ^ £´µ¶·^¶ ^ ,^^^^ = − $, with 9 ^^,^^^^the quantity of moles of oxygen consumed, j the surface current, © ^^^ the active surface of the membrane, ^¸ the number of cells in the fuel cell, F the Faraday constant. For example, water production can mass flow rate of produced water, j the surface current, © ^^^ the active surface of the membrane, ^¸ the number of cells in the fuel cell, F the Faraday constant. In addition, the absorption and desorption of water at the cathode can be written as [9^^^,^^^¹,^ = with [9^^^,^^^¹,^ the mass flow rate of water absorbed or desorbed at the cathode, © ^^^ the active surface of the membrane, ^¸ the number of fuel cell cells, 1 ^^,^ ^^^ the absorption or desorption coefficient of the cathode, ^ ^ ^ 4 the water content of the cathode as a function of the molar fraction of water at the cathode, ^ ^the water content of the cathode. The mass balance of the cathode can be summarized by the following equation: With ^ ^ ^ ^ the mass fraction of oxygen at the cathode, ^ ^ ^ ^ the mass fraction of nitrogen at the cathode, the mass fraction of water at the cathode, [9 the mass flow rate ^ ^^^ ^ ^ 0 ^ ^ ^ ^6the mass fractions at the inlet, j the surface current, © ^^^ the surface ^ ^ ^^^ active membrane, ^¸ the number of cells in the fuel cell, F the Faraday constant, 1 ^^,^ ^^^ the absorption or desorption coefficient of the cathode, ^ ^ ^ 4 the water content of the cathode as a function of the molar fraction of water at the cathode, ^ ^the water content of the cathode. According to one implementation of the invention, the mass balance on the anode side can take into account hydrogen consumption, as well as water absorption and desorption (i.e., the amount of water that is entering or leaving the membrane at the interface with the anode). For example, the hydrogen consumption at the anode can be written as: [9^^,^^^^ = − £´µ¶·^¶ ^ , avec [9 ^^,^^^^ the mass flow rate of hydrogen consumed, j the surface current,©^^^ the active surface of the membrane, ^¸ the number of cells in the fuel cell, F the Faraday constant. As an example, the absorption and desorption of water at the anode can be written: 9^^^,^^^¹,^ = with [9^^ the mass flow rate of water absorbed or at the anode, © ^^^ the active surface of the membrane, ^¸ the number of fuel cell cells, 1 ^^,^ ^ ^ ^the absorption or desorption coefficient of the anode, ^ ^ ^ 4 the water content of the anode as a function of the molar fraction of water at the cathode, ^ ^ the water content of the anode. The mass balance of the cathode can be synthesized by the following equation: With ^ ^ ^ ^ the mass fraction of dihydrogen at the anode, ^ ^ ^ ^ the mass fraction of nitrogen at the anode, ^ ^ ^ ^ ^ the mass fraction of water at the anode, [9 ^^ the mass flow rate of dihydrogen, ^ ^^^ ^ ^ 0 ^ ^ ^ ^6the mass fractions at the inlet, j the surface current, © ^^^ the surface ^ ^ ^^^ active membrane, ^¸ the number of cells in the fuel cell, F the Faraday constant, 1 ^^,^ ^^^ the absorption or desorption coefficient of the anode, ^ ^ ^ 4the water content of the cathode as a function of the molar fraction of water at the cathode, ^ ^the water content of the cathode. 1. Measurement of voltage, current and temperature During this step, the voltage and current are measured at the terminals of the fuel cell. This provides a simple and inexpensive way to obtain an operating variable of the fuel cell. For example, the measurement can be carried out by a voltage sensor and a current sensor, in particular a V sensor as illustrated in Figure 1. In addition, during this step, the temperature of the fuel cell is measured. For example, the measurement can be carried out by a temperature sensor. The fuel cell therefore does not require specific instrumentation for determining the water content within the membrane, in particular the invention does not require instrumentation of the membrane. 2. Acquisition of flow rates During this step, the flow rates of reactants at the anode and cathode are acquired.In this way, the chemical reactions within the fuel cell can be quantified. Advantageously, the acquisition of reactant flow rates can be carried out by measurement, in particular by means of a flow meter. Thus, approximations can be limited. Alternatively, the acquisition of reactant flow rates can be determined by a controller, which controls the intake of air and hydrogen to the fuel cell. This solution does not require any specific sensor. Alternatively, the acquisition of reactant flow rates can be an estimation of these flow rates. According to one embodiment of the invention, the pressures of the reactants at the anode and at the cathode can also be acquired. Advantageously, this acquisition of pressures can be carried out by at least one pressure sensor. Thus, approximations can be limited.Alternatively, the acquisition of reactant pressures can be determined by a controller, which controls the air and hydrogen intake of the fuel cell. This solution does not require any specific sensors. Alternatively, the acquisition of reactant flow rates can be an estimation of these flow rates. 3. Determination of the water content within the membrane In this step, the water content within the membrane, preferably at the center of the membrane, is determined by means of an adaptive extended Kalman filter applied to the dynamic model (possibly constructed in step A), the electrochemical model (possibly constructed in step B), the mass balance at the anode and cathode (possibly constructed in step C), by means of measurements from step 1, and acquisitions from step 2.In other words, at the input of the adaptive extended Kalman filter we have the measured voltage, the measured current, the measured temperature and the acquired mass flow rates, and at the output the water content within the membrane, and possibly at the interfaces between the membrane and the anode and between the membrane and the cathode. According to one embodiment of the invention, the adaptive extended Kalman filter can also take into account pressure acquisitions of the reactants at the anode and at the cathode. These pressures are then inputs of the adaptive extended Kalman filter. According to one implementation of the invention, the adaptive extended Kalman filter can also take into account the relative humidities at the cathode and at the anode, so as to fully calculate the molar fractions of the gas mixture at the inlet. In other words, the relative humidities at the cathode and at the anode can be inputs of the adaptive extended Kalman filter.Applying a Kalman filter allows us to obtain a state observer. We recall that a state observer, or a state estimator, is, in automatic control and systems theory, an extension of a model represented in the form of a state representation. When the state of the system is not measurable, we construct an observer that allows us to reconstruct the state from a model. The extended Kalman filter allows us to locally linearize nonlinear systems. Furthermore, it is recognized that the covariance matrices Q0 and R0 have a significant impact on the performance of the observer. If these are not well adjusted, the filter is less efficient. The adaptive Kalman filter allows the implementation of a calculation of covariance matrices based on the innovation [. Hand the residue to improve the accuracy of the filter. In addition, the adaptive extended Kalman filter is robust. In the following, a non-limiting embodiment of the application of such an adaptive extended Kalman filter is described. According to one aspect of the invention, the extended Kalman filter can be written: @A = B^@, C, D^, 0 = E^@, C, D^, F = GC, with x the water content of the membrane, @A the time derivative of x, u the measured or acquired values ​​including the fuel cell voltage and the reactant flow rates, y the measured fuel cell current, C the observation matrix, B the second membrane of the dynamic equation, E an algebraic constraint function, and z the algebraic fuel cell states, preferably, the dynamic equation of the system can be discretized as follows, with k the discrete time step: @HI% − @H − J:B^@HI%, CHI%, DH^ = 0E^@ HI% , C HI% , D H^ = 0. Depending on the optional measurements (pressures, temperature, relative humidities at the anode and cathode) and the optional outputs (water content at the interfaces), we can write: @= ^^^ ^^ ^^^Z , G = ^1 0 0 0 0 0^ With ^ ^ the water content within the membrane at the interface with the anode, ^ ^ the water content in the center of the membrane, ^ ^ the water content within the membrane at the interface with the cathode, j the surface current, ^^^^ the mass fraction of oxygen at the cathode, ^^ la f ^ ^^ mass reaction of nitrogen at the cathode, ^^^ mass fraction of dihydrogen at the anode, ^ ^ ^ ^ the mass fraction of nitrogen at the anode, ^ ^^^^ the measured fuel cell voltage, [9 ^^^ the air flow rate at the cathode inlet, [9 ^^ the flow rate of dihydrogen at the anode inlet, Q ^^ the pressure at the cathode inlet, Q ^^the anode inlet pressure, XP ^^ cathode relative humidity, XP ^^ the relative humidity of the anode, T the temperature of the fuel cell. We can write the discrete dynamic equation of the system in the form of a more classical non-linear explicit system in order to write a classical adaptive extended Kalman filter: @ HI% = U^@ H , D H ^, F HI% = ℎ ^ @ HI% , D H ^ . With U^@, D^ = 3V − J:B^@, C^@, D^, D^5 M% and ℎ^@, D^ = GC^@, D^ where C^@, D^ is the solution of the discrete equation for @ {: D fixed. According to an implementation of the invention, said water content of the membrane can be determined by means of the following steps, given the estimates ^@L HM% , Ĉ HM% ^, the covariance matrices O HM% , P HM% , the estimated covariance matrix Q HM% : i. We calculate the Jacobian identity matrix, ii. We calculate the predicted state by solving the dynamic equation of the system and calculating the Jacobian iii. We calculate the predicted estimate of the covariance iv. We calculate the innovation [H = FH − GĈH|HM%, the covariance of the innovation and the Kalman gain] = Q XZ M% HH|HM% H\H .v. We find the updated state @L H = @L H|HM% + ] H [ H , and we deduce the water content of the membrane, vi. We finally calculate the algebraic state Ĉ H solution of the dynamic equation of the system and the updated innovation vii. We then update the covariance matrix identity matrix, and covariance matrices In the covariance matrix update, the coefficient _ allows to switch from an extended Kalman filter to an adaptive extended Kalman filter, so that the gain matrices Q à and R Ãvary during the trajectory. 4. Fuel cell control In this optional step, at least one flow rate of a reactant of the anode and cathode is controlled according to the water content within the membrane determined in step 3. In this way, the control of the fuel cell can be adapted so that the membrane operates in a suitable water content range, limiting its degradation and limiting efficiency losses.For example, the control may comprise at least one of the following operations: - The air flow rate at the cathode inlet is adjusted (reduced or increased) if the water content is not within a predefined water content range, - The pressure at the cathode inlet is adjusted (reduced or increased) if the water content is not within a predefined water content range, - The temperature of the fuel cell is adjusted (reduced or increased) if the water content is not within a predefined water content range. - The humidity of the air at the cathode inlet is adjusted (reduced or increased) if the water content is not within a predefined water content range.The invention further relates to a method for monitoring and / or diagnosing a proton exchange polymer membrane fuel cell, in which the following steps are implemented: - The water content within the membrane is determined by means of the method according to any one of the variants described above, - The operation of the fuel cell is monitored and / or diagnosed as a function of the water content. For example, if the membrane is too wet compared to a first predefined threshold or too dry compared to a second predefined threshold, a malfunction of the fuel cell can be diagnosed and / or a user can be alerted to this malfunction and / or maintenance of the fuel cell can be provided. Example The characteristics and advantages of the method according to the invention will become more clearly apparent upon reading the comparative example below.The example concerns the simulation of a proton exchange polymer membrane fuel cell using the Simcenter Amesim simulation software. TM(SIEMENS, Germany) suitable for predicting the multidisciplinary performance of multi-domain systems. We consider a fuel cell with a power of 90kW with 330 cells, an active surface area of ​​320cm². The membrane thickness is taken at 15 µm, as well as that of the catalytic layer. The membrane is assumed to be made of Nafion type ionomer. We assume that the temperature of the fuel cell is well regulated around 80°C. The fuel cell is simulated from the dedicated brick available in AMESIM. This is then nested in a vehicle simulation platform that operates with an input current demand. The simulation platform is divided into three blocks: - The air loop with an air filter, a compressor, a passive humidifier at the intake and a backpressure valve at the exhaust.- The hydrogen loop with an intake tank and a recirculation device to reintegrate the H2 in the exhaust that has not been consumed in the intake loop - The fuel cell block. The platform is co-simulated with Simulink software. TM(The Mathworks, USA) with previously established control strategies to control the operating conditions with optimized efficiency giving the current requested at the input with the dedicated controllers (compressor, backpressure valve, H2 injector). When the initial water content is known, the simulation allows to determine the water content at the center of the membrane, this value is considered as the reference. The simulation allows to know the voltage of the fuel cell, the gas inlet flow rates, the gas inlet pressures, and the temperature of the fuel cell. From these quantities, the method according to the invention is applied (the adaptive extended Kalman filter and the three models) to determine the water content at the center of the membrane, when the initial water content of the membrane is not known. Figure 4 illustrates for this simulation the water content at the center of the membrane λ as a function of time T in s.In this graph, REF indicates the reference curve resulting from the simulation, AA indicates a curve obtained by a prior art method in open loop (starting from an unknown initial condition), and INV indicates the curve obtained by the method according to the invention starting from the same unknown initial condition in AA. We note that the INV curves are quickly superimposed on the REF curve, which is why we can conclude that the invention makes it possible to determine the water content precisely and quickly without knowing its initial value beforehand, which allows its use in real time.

Claims

Claims 1. Method for determining the water content within a membrane of a fuel cell (1) with a proton exchange polymer membrane, said fuel cell comprising an anode (2), a cathode (3), and a polymer membrane (4) arranged between said anode (2) and said cathode (3), said anode (2) being supplied with reactants such as dihydrogen and optionally water to form products such as H ions +, said cathode (3) being supplied with reactants such as oxygen and nitrogen to form products such as water, characterized in that the method implements a dynamic model (MOD DYN) of said membrane (4) which relates the water content of the membrane to the surface current and to the mole fractions of the reactants at the cathode and the anode, an electrochemical model (MOD ELC) of said fuel cell (1) which relates the surface current to the voltage of said fuel cell and to the water content of said membrane, and a mass balance (BIL MAS) at the anode (2) and the cathode (3) which relates said surface current to the mole fractions of the reactants and products at said anode and cathode and to the mass flow rates of the reactants at the anode (2) and cathode (3), and in that the following steps are implemented: a.A voltage and a current are measured (MES) at the terminals of said fuel cell (1), as well as the temperature of the fuel cell (1); b. Mass flow rates of said reactants are acquired (ACQ) at said anode (2) and cathode (3); and c. The water content within said membrane (4), preferably at the center of said membrane (4), is determined by means of an adaptive extended Kalman filter (EKF), applied to said dynamic model of said membrane (MOD DYN), to said electrochemical model (MOD ELC) of said fuel cell and to said mass balance (BIL MAS), and by means of said measured voltage, to the measured current, to the measured temperature, and to said acquired mass flow rates of reactants.

2. The method of claim 1, wherein pressures of the reactants are further acquired (ACQ) at said anodes and cathodes, said electrochemical model depending on said pressures of the reactants. 3.Method according to one of the preceding claims, in which said m is constructed. odèle électrochimique (MOD ELC) au moyen de l’équation suivante : ^^^^^ = ^ ^^^^^^ − ^^^^ − ^^^^^ − ^^^^ avec ^^^^^ la tension d’une cellule de la pile à fuel, ^ ^^^ activation losses as a function of surface current, ^ ^^^^ the losses by diffusion of oxygen at the cathode as a function of the surface current, ^ ^^^ ohmic losses due to the proton resistance of the membrane as a function of the surface current and the water content of said membrane, ^ ^^^^^^the potential available under the operating conditions.

4. Method according to one of the preceding claims, in which the water content of said membrane (4) at the interface with said anode (2) and / or at the interface with said cathode (3) is further determined.

5. Method according to one of the preceding claims, in which said dynamic model (MOD DYN) of the membrane takes into account the transport of water, and / or the electroosmosis reaction, and / or the absorption or desorption phenomena at said anode and cathode.

6. Method according to claims 4 and 5, in which said dynamic model (MOD DYN) of the membrane is constructed using the following equation: with ^ ^ the content in the center of the membrane, ^ ^ the water content of the membrane at the interface with the anode, ^ ^ the water content of the membrane at the interface with the cathode, 78 the diffusivity coefficient, j the surface current, 9 ^the number of water molecules carried by a proton in the membrane, 1 ^^,^ ^ ^ ^ the absorption or desorption coefficient of the anode, 1 ^^,^ ^ ^ ^ the absorption or desorption coefficient of the cathode, ^ ^ ^ 4 the equivalent water content of the anode as a function of the mole fraction of water at the anode, ^ ^ ^ 4 the water content of the cathode as a function of the molar fraction of water at the cathode: ; the thickness of the membrane, : => the thickness of the catalytic layer, ? ^the fixed charge concentration of the membrane, F the Faraday constant.

7. Method according to one of the preceding claims, in which said mass balance (BIL MAS) is constructed at said cathode (3), taking into account oxygen consumption, water production and water absorption and desorption.

8. Method according to one of the preceding claims, in which said mass balance (BIL MAS) is constructed at said anode (2), taking into account hydrogen consumption, water absorption and desorption.

9. Method according to one of the preceding claims, in which said K filter alman étendu s’écrit : @ A = B^@, C, D^, 0 = E^@, C, D^, F = GC, with x the membrane water content, @A the time derivative of x, u the measured or acquired values ​​including fuel cell voltage and reactant flow rates, y the measured fuel cell current, C the observation matrix, B the second membrane dynamic equation, E a function of algebraic constraints and z the algebraic fuel cell states, preferably the dynamic equation of the system is discretized as s uivante, avec k le pas de temps discret : @ HI% − @H − J:B^@HI%, CHI%, DH^ = 0 E^@ HI% , C HI% , D H ^ = 0.

10. A method according to claim 9, wherein said water content within the membrane is determined by applying an adaptive extended Kalman filter by means of the following steps, given the estimates ^ @L HM% , Ĉ HM% ^ , the covariance matrices the estimated covariance matrix Q HM%at the previous time step k-1: i . On calcule le Jacobien identity matrix, ii. We calculate the predicted state by solving the dynamic equation of the system and calculating the Jacobian i ii. On calcule l’estimation prédite de la covariance i v. On calcule l’innovation [ H = FH − GĈH|HM%, L a covariance de l’innovation e t le gain de Kalman ] = Q XZ M% H H|HM% H\H . v. We find the updated state @L H = @L H|HM% + ] H [ H , and we deduce the water content of the membrane, vi. We calculate the algebraic state Ĉ H solution of the dynamic equation of the system and the updated innovation ε H = F H − GĈ H . v ii. On met ensuite à jour la matrice de covariance matrice d’identité, e t les matrices de covariances 11. Method for controlling a fuel cell (1) with a proton exchange polymer membrane, said fuel cell comprising an anode (2), a cathode (3), and a polymer membrane (4) arranged between said anode (2) and said cathode (3), said anode (2) being supplied with reactants such as dihydrogen and optionally water to form products such as H ions +, said cathode (3) being supplied with reactant such as oxygen, nitrogen to form products such as water, characterized in that the following steps are implemented: a. The water content of the membrane (4) is determined by means of the method for determining the water content of the membrane of a fuel cell according to one of the preceding claims; and b. The flow rate of at least one reactant of said anode (2) and cathode (3) is adapted as a function of said determined water content of the membrane and / or the pressure at the inlet of the cathode (3) is adapted as a function of said determined water content of the membrane and / or the temperature of the fuel cell (1) is adapted as a function of said determined water content and / or the humidity of the air at the inlet of the cathode (3) is adapted as a function of said determined water content.

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