Fuel cell system and operating method for operating the fuel cell system

Abstract

The presented invention relates to an operating method (100) for operating a fuel cell system (200), wherein the operating method (100) comprises: - measuring (101) an oxygen content in the exhaust gas of the fuel cell system (200) by an oxygen concentration sensor (205) arranged in the exhaust tract (203) of the fuel cell system (200), - determining (103) a system fuel utilization factor of the fuel cell system (200) based on the measured oxygen content, - Adjusting (105) the fuel cell system (200) depending on the determined system fuel utilization rate.

Description

[0001] The presented invention relates to an operating method for operating a fuel cell system, a fuel cell system and a program product according to the appended claims. State of the art

[0002] Known solid oxide fuel cell (SOFC) systems are based on feedforward control of characteristic parameters such as gas utilization, since these cannot be determined online. Disclosure of the invention

[0003] Within the scope of the invention presented, a fuel cell system and an operating method for operating the fuel cell system, as well as a program product, are presented. Further features and details of the invention emerge from the respective subclaims, the description, and the drawings. Features and details described in connection with the operating method according to the invention naturally also apply in connection with the fuel cell system according to the invention or the program product according to the invention, and vice versa, so that with regard to the disclosure of the individual aspects of the invention, reference is always made to each other.

[0004] The invention presented serves in particular to provide a possibility for fuel-efficient and robust operation of a fuel cell system.

[0005] Thus, according to a first aspect of the invention presented, an operating method for operating a fuel cell system is presented.

[0006] The presented operating method comprises measuring an oxygen content in the exhaust gas of the fuel cell system by an oxygen concentration sensor arranged in the exhaust tract of the fuel cell system, determining a system fuel utilization factor of the fuel cell system based on the measured oxygen content and adjusting the fuel cell system depending on the determined system fuel utilization factor.

[0007] The presented operating procedure is based on measuring the oxygen content in the exhaust gas to determine the system fuel efficiency as a control variable.

[0008] A system fuel efficiency (FU sys) is to be understood in the context of the presented invention as a ratio of an electrochemically oxidized gas portion to a supplied oxidizable gas stream. The relationship according to equation system (1) applies. TGB: tailgas burner; OG: offgas; and NG: natural gas. FUsys=el.−chem.oxidized gas fractionsupplied oxidizable gas flow=Δn˙O2,StkΔn˙O2,Stk+Δn˙O2,tgb=Δn˙O2,StkΔn˙O2,Sys

[0009] It can further be provided that the system fuel efficiency is determined as a function of an air molar flow determined by means of a first flow sensor and an exhaust gas molar flow determined by means of a second flow sensor.

[0010] With the current / , the number of cells N cell and the Farady constant, as well as an applied air mole flow ṅ air and an exhaust gas mole flow ṅ og the relationship follows according to the system of equations (2) Δn˙O2,stk=INcell4FΔn˙O2,sys=n˙airxO2,air−n˙ogxO2,air,og

[0011] Where x 02,air,og a value determined by the oxygen concentration sensor.

[0012] It may be provided that a lambda probe, in particular a broadband lambda probe, is selected as the oxygen concentration sensor.

[0013] Lambda sensors, such as those known from the automotive sector, are designed and constructed specifically for use in exhaust gases, which means that direct use of such a lambda sensor in the fuel cell system presented is possible without any adaptation.

[0014] It may also be provided that the fuel cell system is a solid oxide fuel cell system and is preferably operated with natural gas. Alternatively, other fuel types such as LNG, NH3, or diesel are also possible.

[0015] If the change in the number of moles ξ of the overall reaction is defined, ṅog with the air mole flow ṅ air and the fuel gas mole flow ṅ og can be expressed using the following equation (3). ξ=n˙ogn˙air+n˙ng=1︸100%CH4≈1︸Natural gas

[0016] This applies to natural gas in particular under the condition of SOFC system operation with a high excess of air, which is usually the case since the SOFC is cooled via the air flow.

[0017] If an oxygen concentration x 02,αir,og from the oxygen concentration sensor and the air mole flow ṅ air and the fuel gas mole flow ṅ ng determined by flow sensors, the system fuel efficiency FU sys can be determined using equation (4). FUsys=INcell4F(n˙airxO2,air(1−n˙air+n˙ngn˙airξxO2,ogxO2,air))−1

[0018] Based on the system fuel efficiency, a corresponding control or feedforward control can be set up that adjusts the fuel cell system to be fuel-efficient and robust.

[0019] The measurement of the oxygen concentration in the exhaust gas provided according to the invention is particularly advantageous because the boundary conditions for the oxygen concentration sensor in the exhaust gas are less demanding than, for example, in the intake tract, and installation and maintenance of the oxygen concentration sensor are particularly easy to carry out.

[0020] It can further be provided that a fuel cell stack fuel utilization rate is inferred from the system fuel utilization rate and that the fuel cell system is adjusted depending on the fuel cell stack fuel utilization rate.

[0021] Furthermore, it can be provided that the fuel cell stack fuel utilization rate is determined as a function of the system fuel utilization rate in combination with a currently set recirculation rate.

[0022] In the case of a fuel cell system with anode exhaust gas recirculation, knowledge of the system fuel utilization rate can be used to determine the recirculation rate and, in combination, the fuel cell stack fuel utilization rate.

[0023] In fuel cell systems without anode exhaust gas recirculation, even a direct determination of the fuel cell stack fuel efficiency is possible, since FU sys = FU stk applies.

[0024] It can further be provided that a target value for a natural gas mole flow that is supplied to a fuel cell stack of the fuel cell system is determined on the basis of a predetermined target value for the system fuel utilization factor in combination with the oxygen content in the exhaust gas and an air mole flow determined by means of a first flow sensor.

[0025] If the oxygen concentration sensor detects an oxygen concentration x 02,air,og and the air mole flow ṅ air determined by means of a flow sensor and an operating point FU sys,sp specified, a setpoint for a natural gas mole flow ṅ which is supplied to a fuel cell stack of the fuel cell system can be ng,SP can be determined according to equation (5). n˙ng,SP=n˙air(xO2,airxO2,ogξ−1)−INcell4F1ξxO2,ogFUsys,sp

[0026] In this approach, the setpoint for the natural gas mole flow ṅ ng,SPa relative size, since this is always based on a specific natural gas composition.

[0027] It may also be provided that the setpoint for the natural gas mole flow is adjusted iteratively.

[0028] In a natural gas mass flow controller, the natural gas mole flow can be determined based on a correlation to the thermal capacity, and in a simple adjustable valve based on a correlation to the valve position, via a characteristic curve, through an iterative adjustment of the relative natural gas mole flow. For this purpose, after setting the specified operating point, an oxygen concentration in the exhaust gas is measured. Using the relative natural gas mole flow, the system fuel efficiency based on this is calculated according to equation (4) using the relative natural gas mole flow. The fuel flow is then corrected accordingly.

[0029] It can further be provided that the operating method further comprises determining a natural gas composition, a natural gas mole flow supplied to the fuel cell system, on the basis of an iterative adjustment of the natural gas mole flow, in order to determine an electron coefficient of the natural gas mole flow.

[0030] It can further be provided that a feedforward control of the fuel cell system comprises calculating a recirculation rate based on a predetermined target value for a fuel cell stack fuel utilization factor and the determined system fuel utilization factor, determining a recirculating anode exhaust gas mole flow based on the calculated recirculation rate and adapting an operating point of a recirculation blower of the fuel cell system to the determined natural gas composition.

[0031] An example of a procedure for iteratively adjusting the natural gas volume flow in a fuel cell system with anode exhaust gas recirculation in the case of unknown natural gas composition or quality and simultaneous measurement of the oxygen concentration in the exhaust gas can be as follows: 1. Specification of the operating parameters power to be provided / , fuel cell stack fuel utilization rate and oxygen to carbon ratio. 2. Estimation of a natural gas composition with an electron coefficient K e-,Init or specification of a reference value. 3. Calculation of the recirculation rate r as a function of the current I, the fuel cell stack fuel efficiency, the oxygen to carbon ratio and a natural gas quality. 4. Calculation of a theoretical system fuel efficiency FU sys,calc using equation (6) FUsys,calc=FUstk,sp1−r+rFUstk,sp 5. Calculation of the theoretically required natural gas volume flow ṅ ng,calc with the assumed natural gas quality according to equation (7) n˙ng,calc=INcellKe−,init,FUsys,calcF 6. Adjust the natural gas volume flow calculated under 5 by, for example, opening a valve. 7. After the system has stabilized: Calculate the theoretical oxygen concentration in the exhaust gas according to equation (8) xO2,og,calc=n˙airxO2,airFUsys,calc−INcell / 4F(n˙air+n˙ng,calc)FUsys,calc 8. Comparison of the measured oxygen concentration in the exhaust gas with the theoretically calculated oxygen concentration in the exhaust gas by: a) x O2,og,sensor < x O2,og,calc → Reduction of the natural gas flow rate by closing the valve, or b) x O2,og,sensor > x O2,og,calc → Increase the natural gas flow rate by opening the valve. 9. Repeat steps 6-8 until the expected oxygen concentration in the exhaust gas is reached. This can be implemented, for example, by a simple PI controller, whereby only the oxygen mole flow changes in equation (8), since the theoretically required natural gas volume flow is not updated. 10. Optionally, if knowledge of the exact natural gas composition is of interest, the calculation can also be restarted by updating the estimate under 2.

[0032] It can further be provided that a fuel cell stack fuel efficiency is determined as a function of an oxygen molar flow at the outlet of an afterburner of the fuel cell system, wherein the oxygen molar flow is determined based on an oxygen concentration determined by a further oxygen concentration sensor at the outlet of the afterburner as well as a water molar flow and a carbon dioxide molar flow at the inlet of the afterburner, wherein the water molar flow and the carbon dioxide molar flow at the inlet of the afterburner are determined by at least one calculation method from the following list of calculation methods: Calculation via a fuel cell stack current provided by a fuel cell stack of the fuel cell system, Calculation via an equilibrium calculation,Measuring the water mole flow and the carbon dioxide mole flow at the inlet of the afterburner via a changed operating point of a further oxygen concentration sensor arranged at the inlet of the afterburner.

[0033] By monitoring the oxygen concentration in this way, a sudden change in natural gas quality during operation can also be detected and reacted to accordingly. In a fuel cell system without anode exhaust gas recirculation, steps 3-5 are omitted, since FU sys = FU stk applies.

[0034] The above procedure can be carried out until the result is FU sys,calc and FU sys,realcorrespond to each other. It is important that the natural gas mole flow is not determined directly, but rather a relative value is adjusted, which is possible, for example, by adjusting the opening width of a low-cost valve as a replacement for a cost-intensive natural gas mass flow controller.

[0035] From this, the electron molar flow in natural gas can ultimately be calculated according to equation (9). This describes the molar flow of potentially releasable electrons in natural gas during a complete electrochemical conversion. n˙ng,e−=4n˙O2,trFUsys=INcellFFUsys

[0036] From this, the electron molar current at the afterburner can be determined according to equation (10). n˙tgb,e−=(1−FUsys)⋅n˙ng,e−=(1−FUsys)⋅INcellFFUsys

[0037] With the oxygen molar flow of the air at the afterburner inlet, which is known (see below), the oxygen molar flow at the afterburner outlet can now be determined according to equation (11) n˙O2,tgb−out=n˙O2,air−tgb−in−n˙e−,fuel−tgb−in4

[0038] From this, the total molar flow at the afterburner outlet or system outlet can be determined according to equation (12) and, if necessary, the change in the number of moles of the overall reaction can be updated. n˙og=n˙O2,tgb−outxO2,tgb−out

[0039] By updating the mole number change of the overall reaction, conclusions can be drawn about the natural gas composition.

[0040] Typically, when controlling an SOFC fuel cell system, a setpoint for the system fuel efficiency is specified to comply with limits for the fuel cell stack. Consequently, the following procedure can be run as feedforward control: 1. Definition of a target value for a system fuel efficiency 2. Calculation of the recirculation rate based on the specified target value for the system fuel utilization rate and a determined fuel utilization rate 3. This allows the recirculating mole flow to be calculated, or in particular an operating point for a recirculation fan to be set. 4. An adjustment of the recirculation blower, e.g. its characteristic curve to the current natural gas mixture, can be carried out using the calculated change in the number of moles of the overall reaction or the updated natural gas composition with electron coefficient, e.g. as part of a calibration.

[0041] The air volume flow at the afterburner inlet can usually be clearly calculated based on the air supplied to the fuel cell system and the power supplied.

[0042] Using the standard oxygen concentration of 21%, the oxygen content at the afterburner inlet as well as the nitrogen concentration (and thus also the molar flow rate ṅ) can be determined. Alternatively, an oxygen measuring device, such as a broadband lambda sensor, can be used at the afterburner inlet.

[0043] The nitrogen molar flow at the afterburner outlet corresponds to a first approximation to the nitrogen volume flow at the inlet, since it does not participate in the electrochemical conversion.

[0044] The oxygen concentration at the outlet can be measured with an appropriate oxygen measuring device, such as a broadband lambda sensor.

[0045] The following formulas and relationships can be established: xO2,out=n˙O2,outn˙N2+n˙O2,out+n˙CO2,out+n˙H2O,outn˙CO2,out+n˙H2O,out=n˙CO2,out+n˙H2O,in+n˙CO,in+n˙H2,inn˙CO2,out+n˙H2O,out=n˙CO2,in+n˙H2O,in+Δn˙O2⋅0.5Δn˙O2⋅0.5=n˙CO,in+n˙H2,inΔn˙O2=n˙O2,in−n˙O2,out

[0046] The first equation gives: 1=(1−xO2,out)n˙O2,outxO2,out⋅(n˙N2+n˙CO2,out+n˙H2O,out) n˙O2,out=xO2,out⋅(n˙N2+n˙CO2,out+n˙H2O,out)(1−xO2,out)

[0047] By substituting we get: n˙O2,out=xO2,out⋅(n˙N2+n˙CO2,in+n˙H2O,in+Δn˙O2⋅0.5)(1−xO2,out)n˙O2,out=xO2,out⋅(n˙N2+n˙CO2,in+n˙H2O,in+(n˙O2,in−n˙O2,out)⋅0.5)(1−xO2,out)nO2,out⋅(1−xO2,out)=xO2,out⋅(n˙N2+n˙CO2,in+ n˙H2O,in+n˙O2,in⋅0.5)−n˙O2,out⋅0.5⋅xO2,outn˙O2,out⋅(1−0.5⋅xO2,out)=xO2,out⋅(n˙N2+n˙CO2,in+n˙H2O,in+n˙O2,in⋅0.5)n˙O2,out=xO2,out⋅(n˙N2+n˙CO2,in+n˙H2O,in+n˙O2,in⋅0.5)(1−0.5⋅xO2,out)

[0048] The determination of the oxygen mole flow at the afterburner outlet therefore depends only on the water and carbon dioxide mole flow at the afterburner inlet. Four approaches can be pursued for this: a) Determination / estimation of the stack current

[0049] In the stack, H2 and CO are converted to CO2 and H2O. Consequently, the following relationship can be established according to equation (13): n˙CO2,ein+n˙H2O,ein=IStack2F⋅(1−r)

[0050] The recirculation rate is assumed to be known. However, CH4 conversion is neglected. Consequently, a correction term must be incorporated, which depends on system variables such as stack current and stack temperatures, or can be described with them according to Equation (14): n˙CO2,ein+n˙H2O,ein=IStack2F⋅(1−r)+ε b) Determination via equilibrium calculation

[0051] The equilibrium calculation approach uses the temperatures at the inlet and outlet. These represent easily measurable quantities.

[0052] At the inlet, the air composition (N2, O2) and the temperature are given and thus completely determined. For the fuel gas flow, the temperature and the relationships listed above are given.

[0053] The oxygen concentration, the nitrogen molar flow and the temperature are given for the outlet.

[0054] Assuming a thermodynamic equilibrium at the afterburner outlet, there must then be a unique molar flow of hydrogen and carbon monoxide, each of which requires an oxygen atom for combustion, which leads to a specific oxygen concentration and a specific temperature at the afterburner outlet. c) Changed operating point of a broadband lambda sensor

[0055] With a broadband lambda sensor, H2O and CO2 can be measured quantitatively by first measuring the oxygen content, whereby the lambda sensor regulates to UN = 450 mV and then setting higher target values of UN or Up, at which the H2O and CO2 decomposition then also takes place.

[0056] Consequently, further information is obtained as the concentration of the sum of CO2 and H2O.

[0057] Furthermore, the combination of the above-mentioned methods is possible.

[0058] From this, a difference in oxygen concentration across the afterburner can ultimately be calculated. The following relationship applies to this according to equation (15): Δn˙O2=n˙O2,in−n˙O2,out=n˙e-,gas,afterburner,in=f(n˙H2,n˙CO)

[0059] The electron mole flow of natural gas can therefore be calculated according to formulas (16): n˙ng,e−=n˙e−,gas,afterburner,on+I⋅NcellF

[0060] This allows the system fuel efficiency to be calculated according to formula (17): FUSystem=n˙e-,Gas,Afterburner,einn˙ng,e-

[0061] From this, the stack-side gas utilization factor can then be calculated for known r according to equation (18): n˙e-,anode,out=n˙e-,gas,afterburner,on(1−r)

[0062] It may further be provided that a machine learner or a machine learning method is used to determine an error between a real value and a value determined by a sensor of the fuel cell system, where the value determined by the sensor is corrected based on the value determined by the machine learner.

[0063] In order to achieve an improvement of the presented operating procedure with the aim of achieving higher accuracy, the presented operating procedure can be combined with a machine learning method in the sense of a hybrid system.

[0064] The machine learning method or the machine learner can be used in particular to calculate an error between the real value h s and a value h measured with the sensor concepts shown s,Sensor and thus improve the latter values.

[0065] To do this, the machine learner only needs to be trained in advance with training data to estimate the error at various operating points. The machine learner can be set up as a function of the measured variables of the oxygen concentration sensor, such as pump current, pump voltage, temperature, and Nernst voltage, as well as other variables (temperatures, pressures, volume flows, etc.) from the fuel cell system in which the oxygen concentration sensor is used. The sensor error then represents the output variable. In addition to multivariate linear regression, the application of a neural network and, in particular, the application of a Gaussian process represent suitable methods for implementing the machine learner.

[0066] According to a second aspect, the presented invention relates to a fuel cell system for converting energy, wherein the fuel cell system is a solid oxide fuel cell system and wherein the fuel cell system comprises a number of fuel cell stacks, an oxygen concentration sensor arranged in the exhaust tract of the fuel cell system and a computing unit, wherein the computing unit is configured to carry out a possible embodiment of the presented operating method.

[0067] In the context of the invention presented, a computing unit is understood to mean a computer, a processor, a control unit or any other programmable circuit.

[0068] According to a third aspect, the presented invention relates to a program product, wherein the program product comprises program code means which, when the program product is executed on a computing unit, configure the computing unit to execute a possible embodiment of the presented operating method.

[0069] Advantages described in detail with respect to the operating method for operating a fuel cell system according to the first aspect of the invention apply equally to the fuel cell system for converting energy according to the second aspect of the invention and the program product according to the third aspect of the invention.

[0070] Further advantages, features, and details of the invention will become apparent from the following description, in which exemplary embodiments of the invention are described in detail with reference to the drawings. The features mentioned in the claims and in the description may be essential to the invention individually or in any combination.

[0071] They show: Fig. 1 a schematic representation of a possible design of the presented operating procedure, Fig. 2 a possible design of the presented fuel cell system, and Fig. 3 a possible design of an algorithm according to the presented program product.

[0072] In Fig. 1 shows an operating method 100 for operating a fuel cell system.

[0073] The operating method 100 comprises a measuring step 101 in which an oxygen content in the exhaust gas of the fuel cell system is measured by an oxygen concentration sensor arranged in the exhaust tract of the fuel cell system, a determining step 103 in which a system fuel utilization factor of the fuel cell system is determined based on the measured oxygen content, and an adjusting step 105 in which the fuel cell system is adjusted depending on the determined system fuel utilization factor.

[0074] In Fig. 2 shows a fuel cell system 200 for converting energy.

[0075] The fuel cell system 200 is a solid oxide fuel cell system and comprises a fuel cell stack 201, an oxygen concentration sensor 205 arranged in the exhaust tract 203 of the fuel cell system 200, a computing unit 207 and a metering valve 209 for adjusting a fuel flow supplied to the fuel cell stack 201.

[0076] The fuel cell stack is supplied with fresh air via an air supply system 211.

[0077] The computing unit 207 is configured to execute the operating method 100 according to Fig. 1 and, for example, to set an opening degree of the metering valve 209 based on the system fuel utilization rate.

[0078] In Fig. Figure 3 shows an algorithm 300 for operating a fuel cell system. The algorithm 300 starts with a specification step 301 in which operating parameters, such as a target value for the system fuel utilization rate, are specified.

[0079] In an estimation step 303, a fuel quality of fuel supplied to the fuel cell system is estimated.

[0080] In a first calculation step 305, system variables are calculated and in a setting step 307, the fuel volume flow supplied to the fuel cell system is set.

[0081] In a second calculation step 309, after a transient phase of the fuel cell system, an oxygen concentration in the exhaust gas of the fuel cell system is calculated

[0082] In a comparison step 311, the oxygen concentration in the exhaust gas calculated in the second calculation step 309 is compared with a value measured by an oxygen concentration sensor in the exhaust tract of the fuel cell system.

[0083] If the calculated oxygen concentration deviates from the measured value by more than a predetermined threshold value, the setting step 307 can be adjusted in an iteration step in order to adjust the fuel volume flow supplied to the fuel cell system such that the calculated oxygen concentration deviates from the measured value by less than the predetermined threshold value.

[0084] Alternatively or additionally, the fuel quality estimated in the estimation step 303 can be adjusted so that in the first calculation step 305 the calculated system variables for adjusting the fuel cell system change accordingly.

[0085] In the event of a deviation of the calculated oxygen concentration from the measured value by less than the specified threshold value, it can be assumed that a calculated system fuel utilization rate corresponds to the actual system fuel utilization rate and that the fuel cell system is optimally adjusted and a corresponding validation signal is output in a validation step 313.

Claims

[1] Operating method (100) for operating a fuel cell system (200), wherein the operating method (100) comprises - measuring (101) an oxygen content in the exhaust gas of the fuel cell system (200) by an oxygen concentration sensor (205) arranged in the exhaust tract (203) of the fuel cell system (200), - determining (103) a system fuel utilization factor of the fuel cell system (200) based on the measured oxygen content, - Adjusting (105) the fuel cell system (200) depending on the determined system fuel utilization rate. [2] Operating method (100) according to claim 1, characterized by that a lambda probe is selected as the oxygen concentration sensor (205). [3] Operating method (100) according to claim 1 or 2, characterized by that the fuel cell system (200) is a solid oxide fuel cell system and is operated with natural gas. [4] Operating method (100) according to one of the preceding claims, characterized by that the system fuel efficiency is determined as a function of an air mole flow determined by means of a first flow sensor and an exhaust gas mole flow determined by a second flow sensor. [5] Operating method (100) according to one of the preceding claims, characterized by that the fuel cell stack fuel utilization factor is used to determine the fuel cell stack fuel utilization factor and that the fuel cell system is adjusted depending on the fuel cell stack fuel utilization factor. [6] Operating method (100) according to claim 5, characterized by that the fuel cell stack fuel efficiency is determined as a function of the system fuel efficiency, in combination with a currently set recirculation rate. [7] Operating method (100) according to one of the preceding claims, characterized bythat a target value for a natural gas mole flow that is supplied to a fuel cell stack (201) of the fuel cell system (200) is determined on the basis of a predetermined target value for the system fuel utilization factor, in combination with the oxygen content in the exhaust gas and an air mole flow determined by means of a first flow sensor. [8] Operating method (100) according to claim 7, characterized by that the setpoint for the natural gas mole flow is adjusted iteratively. [9] Operating method (100) according to one of the preceding claims, characterized by that the operating procedure (100) further includes: - Determining a natural gas composition of a natural gas mole flow supplied to the fuel cell system (200) based on an iterative adjustment of the natural gas mole flow to determine an electron coefficient of the natural gas mole flow. [10] Operating method (100) according to claim 9, characterized by, a pre-control of the fuel cell system (200) comprises: - Calculating a recirculation rate based on a specified target value for a fuel cell stack fuel efficiency and the determined system fuel efficiency, - Determination of a recirculating natural gas mole flow based on the calculated recirculation rate, - adapting an operating point of a recirculation fan of the fuel cell system (200) to the determined natural gas composition. [11] Operating method (200) according to one of the preceding claims, characterized by , that a fuel cell stack fuel efficiency is determined as a function of an oxygen mole flow at the outlet of an afterburner of the fuel cell system (200), wherein the oxygen mole flow is determined based on an oxygen concentration at the outlet of the afterburner determined by a further oxygen concentration sensor and a water mole flow and a carbon dioxide mole flow at the inlet of the afterburner, wherein the water mole flow and the carbon dioxide mole flow at the inlet of the afterburner are determined by at least one calculation method from the following list of calculation methods: calculation via a fuel cell stack current provided by a fuel cell stack (201) of the fuel cell system (200), calculation via an equilibrium calculation, measuring the water mole flow and the carbon dioxide mole flow at the inlet of the afterburner via a changed operating point of a yet further oxygen concentration sensor arranged at the inlet of the afterburner. [12] Operating method (100) according to one of the preceding claims, characterized by , that a machine learner is used to determine an error between a real value and a value determined by a sensor of the fuel cell system (200), where the value determined by the sensor is corrected based on the value determined by the machine learner. [13] Fuel cell system (200) for converting energy, wherein the fuel cell system (200) is a solid oxide fuel cell system, wherein the fuel cell system (200) comprises: - a number of fuel cell stacks (201), - an oxygen concentration sensor (205) arranged in the exhaust tract (203) of the fuel cell system (200), - a computing unit (207), wherein the computing unit (207) is configured to execute an operating method (100) according to one of claims 1 to 12. [14] A program product, wherein the program product comprises program code means which, when the program product is executed on a computing unit, configure the computing unit to execute an operating method (100) according to any one of claims 1 to 12.

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