Soft-Sensor based Operation of a SOFC System with Anode Exhaust Gas Reciculation

A solid oxide fuel cell (SOFC) converts chemical energy of a fuel gas directly into electrical energy representing an auspicious technology for electricity generation with high efficiency. A SOFC system with anode exhaust gas recirculation (AEGR) provides the highest potential of net electrical DC-efficiency of up to 65%. For efficient and durable operation, it is of crucial importance to monitor relevant characteristic parameters and keep them within safe and durable operating limits. The oxygen-to-carbon-ratio and the fuel utilization are such characteristic parameters and must not exceed stackand reformer-specific thresholds. Monitoring and control of these characteristic parameters is not trivial due to the enhanced complexity of a SOFC system with AEGR and dependence on fluctuating natural gas (NG) quality. In this paper, a soft-sensor concept is presented to determine the H/C-ratio of NG as measure of its quality with high accuracy. It is composed of an energy balance by means of enthalpy flow rates and Gaussian process regression models for estimation of unknown amount of inert gas species xN2, xCO2 in NG as well as the enthalpy error as a corrector term of the ideal energy balance.


Introduction
Limited (fossil) resources, growth of worldwide energy demand and progressive global warming are currently one of the major challenges, where a reduction of greenhouse gas emissions is needed. An enhanced use of renewable energies as well as replacement of conventional energy converters with high carbon footprint by technologies with higher efficiency and thus lower carbon footprint is a way to reduce primary energy consumption [1,2]. One of the major promising technologies that has the potential to overcome these challenges is the fuel cell [2][3][4]. Fuel cells are electrochemical devices, which convert chemical energy directly into electrical energy by oxidizing a fuel gas, leading to an efficient electricity generation.
One of the most promising fuel cell technologies represents the high temperature solid oxide fuel cell (SOFC). It uses nonprecious metals as electrocatalyst material of the electrodes and an ion-conducting ceramic as electrolyte, which simplifies the cell construction and eliminates a complex electrolyte management in comparison to lowtemperature fuel cell types [5]. The resource-efficient use of nonprecious and abundant metals as electrocatalyst leads to lower costs and the potential of internal reformation of natural gas (NG) within the cell, offers a great fuel flexibility and a reduced complexity for fuel pre-processing [2,[5][6][7]. Moreover, SOFC technology is enabling a high energy utilization with a net electrical efficiency of up to 65% concerning direct current (DC) output, which is comparable to efficiencies reached by large power plants [5,6,[8][9][10][11]. This can be enhanced to an overall efficiency of around 90%, if the SOFC system is operated as a combined heat and power generator by utilizing the available heat in the exhaust gas with temperatures around 200°C [2,4,5,7,[9][10][11][12][13][14].
When using NG as fuel, the system configuration with increased efficiency can be obtained by anode exhaust-gas recirculation (AEGR), where a part of the depleted anode exhaust gas is recirculated and mixed with fresh NG prior to entering the reformer [15].
This increases electrical efficiency and supplies steam for the reforming process [14,15].
Next to cost reduction, reduced start-up time and transient operating conditions caused by a high share of renewable energies in the future, improvement of durability with less degradation and enhanced system efficiency are main challenges for the SOFC technology [5,9,[16][17][18][19][20][21]. This includes robustness to variations in NG quality supplied to the SOFC system, which can result from fluctuating NG quality itself or the feed-in of synthetic hydrogen or methane produced by Power-to-X technologies.
Characteristic parameters such as fuel utilization or the oxygen-to-carbon-ratio to avoid fuel depletion or carbon formation, respectively, are of key importance for optimal operation in this regard. They have to be kept within safe and durable operating ranges defined by stack-and reformer-specific limits [22]. For high system efficiency, the parameters need to be kept as narrow as possible to the limiting values though. The definition of the system and stack specific fuel utilization (FU) as well as the oxygen-tocarbon-ratio at the reformer inlet can be seen in equation (1), (2) and (3). A common control is a feed-forward control, which is based on uncertain parameters recirculation ratio , NG coefficients − , and as well as molar flow rate of NG ̇. The cell number and current are known or measured.
The uncertain parameters cannot be measured directly by a sensor and rely on NG composition, which quality is changing over time and normally is not constantly tracked [1,23,24]. Figure 1 lists the fluctuation range of major gas species contained in NG for some locations in Europe taken from Hering [23]. The major species contained in NG is methane with a share of ~80-90%. Additional species in NG are ethane, carbon dioxide, propane, nitrogen and butane. Sulfur is not included in the list, because it is only present in a neglectable low range of some parts per million (ppm) and is not included as measured variable in the NG data. In some studies it is stated that a sulfur content above 10 or even 2 ppm is sufficient to start catalyst deactivation [24][25][26]. More related information on sulfur tolerance and poisening is given in the related literature [27][28][29][30]. For more detailed information on the variation of NG compositions and the related standardization, the reader is referred to the relevant literature [23,[31][32][33][34][35][36].
Installing concentration measurement devices at the system inlet is a possible and easy way to determine NG quality. However, it comes along with in most cases not acceptable additional costs and increases the size of a SOFC system. Accurate concentration measurements imply also long measurement times, which is partly due to the extractive measuring methods, hence being an unsuitable method for commercial use [18,[37][38][39][40]. The most common remedy represents a safety buffer of up to 10%-points on the characteristic parameters to ensure safe operation, which means a great amount of unused fuel gas in the stack and a loss in efficiency. A different method is thus needed to determine NG quality, where a respective concept is presented in this paper.
The remainder of this paper is structured as follows. Firstly, fundamentals regarding SOFC systems, gas coefficients and respective correlations as well as Gaussian process regression are outlined. Subsequently, the basic concept to use models of SOFC components (tail-gas burner, hotbox) as calorimeter is illustrated. The experimental procedure is followed by the results, which are divided into inert gas estimation, estimation for enthalpy errors and the essential results for determination of H/C-ratio. A short summary and an outlook on further work is given in the last part of this paper.

SOFC system
A SOFC system comprises in general the process steps gas treatment, electrochemical conversion and afterburning [5,9,12,41]. Gas treatment consists of gas reforming and desulfurization [5,24,41]. The electrochemical conversion takes place in the stacks.
Additionally, a SOFC system consists of some balance of plants components such as pumps, blowers, heat exchangers (HEX) and a tail-gas burner (TGB).
A detailed description of the SOFC system used for this study has been presented by Weeber [42] and in the work of Hering [23]. The SOFC system is intended for NG operation with a net electric power output of 10 kW and a net electric system efficiency of ηel,sys ≈ 60%. A simplified piping and instrumentation diagram is shown in figure 2. The SOFC system owns an AEGR, where a part of the depleted fuel gas leaving the anode is recirculated with a blower and mixed with fresh NG. The recirculation ratio is defined as the ratio of recirculated to total anode exhaust gas molar flow and has two purposes [1,15,43]. First, since operation of a SOFC stack is typically limited to a in the range of 60 -80% to avoid fuel depletion, part of the unused fuel is recirculated leading to a higher fuel utilization on system level and higher efficiency [1,21,41,[44][45][46][47]. Second, the needed water vapor for steam reforming is supplied without an external device [48]. Too much of recirculation leads to a fuel gas dilution though resulting in a drop of Nernst voltage and lower power output [11,47,49].
Sulfur contained in NG is removed by an integrated desulfurization unit prior to the reformer [41,48]. NG contains long-chain hydrocarbons, which need to be cracked in a reforming process in presence of water vapor or oxygen to primarily hydrogen and carbon monoxide for the electrochemical conversion in the stack [5,11,24,50]. The reformate is fed to the anode side of the SOFC module comprising one or more SOFC stacks with multiple cells. The reformate reacts electrochemically at the anode with oxygen anions [41,50].
Anode exhaust gas, which is not recirculated, and the oxygen depleted cathode exhaust air are supplied to the TGB ensuring a complete combustion [41]. The heat in the exhaust gas is used in an internal heat recovery system to preheat the air and NG. The hot components, being the SOFC module, TGB, reformer and heat exchangers, are assembled together in a common enclosure, named hotbox, to reduce heat losses. Figure 2: Basic structure of SOFC system by Bosch [23].
The SOFC itself is based on the planar metal supported cell (MSC) technology by Ceres 1 allowing stack operating temperatures of 500-600°C. The cells are composed of gadolinia-doped ceria (CGO) as electrolyte material, Lanthanum Perovskite for the cathode, Ceria-Nickel-Cermets as anode material as well as a laser drilled substrate made of ferritic stainless-steel foil [46]. The cells are assembled to 5 kW stacks, offering a of up to 75%, an internal conversion of hydrocarbons such as CH4 of up to 100% and an electrical efficiency ηel,FC of > 63% based on the lower heating value of CH4 [46].
SOFC system control is divided into a control for the air-supply and control for the supply of fuel gas. The supplied air is used to cool the SOFC module, which results in an air utilization AU << 100%. The air supply is controlled via a closed loop control to meet inlet and outlet temperature of the SOFC module. The fuel gas supply is based on a feedforward control of the NG flow rate and the recirculation flow rate , as shown in equation (4) and (5) [15,23]. Therein and need to be defined. NG composition needs to be known to calculate the respective gas coefficients , and − .
In a SOFC system with AEGR FU is defined with respect to the fuel gas flow rate at the anode inlet as or NG at the system inlet as , see equation (1) and (2) [ 1,7,11,15,17,23,37,43]. Without AEGR ( = 0) both and are equal, as it is stated in equation (2) for the relation of and via recirculation-ratio [11,15,37,43].

Gas coefficients and correlations
Gas coefficients are used to describe NG composition. These are the gas coefficients for carbon, oxygen, hydrogen and nitrogen as well as an electron gas coefficient in mol per mol. Since each NG type owns its proper coefficients, e.g. pure CH4 has − = 8, the coefficients are used to classify gas mixtures. Two NG compositions with identical gas coefficients lead to identical behavior of the SOFC system [23]. The gas coefficients according to equation (6) to (9) for carbon, oxygen, hydrogen and nitrogen are calculated on the basis of the individual concentrations , and the corresponding number of atoms contained in the respective molecule as weighting factor [23]. Each coefficient thus defines the number of atoms of a particular element (C,O,H,N) [23]. The electron gas coefficient according to equation (10) describes the amount of potentially releasable electrons in a complete electrochemical conversion in dependence of the number of electrons − , for each gas species i [11,23] = ∑ , • , = 4 , 4 + 6 , 2 6 + 8 , 3 An important parameter to describe NG composition is the ratio of H-atoms to C-atoms (H/C). The H/C-ratio correlates to − , , as well as other gas properties such as cp (specific molar heat capacity of gas in J K -1 mol -1 ) or specific enthalpy h (in J mol -1 ).
New correlations for − , , are derived within this work with H/C-ratio as descriptive variable under the prerequisite of constant or known amount of inert gas species 2 , 2 . These correlations for H/C ≤ 4 are shown in equation (11) to (14). The index "no" states that inert gas species are not considered for the respective H/C-ratio. The derivation of the correlations is given for the interested reader in appendix Fehler! Verweisquelle konnte nicht gefunden werden..
The relationship between H/C-ratio and − , , for H/C > 4 is specified using regressions. NG data for H/C > 4 are based on the limit gas G222 [52] and some arbitrary gas mixtures of CH4 and H2 up to a H/C-ratio of 12. Good correlations are defined without considering inert gas species initially ("no") and a subsequent respective conversion as shown in equation (15) to (17 Heat capacity and enthalpy h for NG are calculated based on a reference NG temperature by regression functions in equation (18) and (19) for H/C ≤ 4 and in equation (20) and (21) for H/C > 4 without considering inert species being part of NG composition.
NG data for some locations in Europe is taken for H/C ≤ 4 from the work of Hering [23].
Inert gas species are taken into account by a respective conversion as shown in equation (22) to (23) A smooth transition between H/C ≤ 4 and H/C > 4 by equation (24) is used to ensure a continuous calculation of a variable needed for a stable solving process, where the function for H/C ≤ 4 is multiplied with and the function for H/C > 4 is multiplied 3 Soft-sensor models of SOFC components

Basic concept
The soft-sensor concept proposed within this work represents a hybrid model combining The GP regression model is used for correction of the ideal energy balance to achieve a good fit with the component's real behavior. Modelling of the enthalpy error is done by supplying a fuel gas with known composition such as CH4 or by supplying NG in conjunction with a high precision gas analyzer (e.g. gas chromatograph). Based on Dubourg et al. [57], the kernel for all the Gaussian process regression models used within this work is the product of a constant kernel and an RBF kernel. Multiplying the RBF by a constant kernel represents a scaling of the RBF kernel, which leads to a significant improvement in the accuracy of the enthalpy error estimation in terms of mean absolute error (MAE) with respect to validation data (see below).
The concept of the hybrid model can thus be generalized as stated in equation (25).
Therein ̇ is the enthalpy flux i coming or leaving a component, and describe measured and calculated variables used in the energy balance. It shall be noted that the enthalpy error term ̇ comprises both the physical present heat loss of the specific component as well as the error of the ideal energy balance since it cannot be differentiated.
Based on the H/C-ratio the gas coefficient − is calculated by means of the correlation shown above. Subsequently can be calculated by equation (1) and can be calculated by equation (2) for known recirculation ratio. The method also allows to calculate the heat capacity cp or density of NG. Using NG heat capacity, it is possible to correct the NG flow measured with a thermal flowmeter. Using NG density, the density at a recirculation blower can be determined to enhance recirculation ratio calculation.

Tailgasburner
The H/C-ratio in the TGB exhaust gas is calculated by performing an energy balance at its boundaries. Assuming a gas-tight system, the H/C-ratio in the TGB exhaust gas is equal to the H/C-ratio in the NG. The general energy balance is given in equation (26) and , , Assuming no gas leakage in the SOFC system, the molar flow rate of carbon atoms is constant and is used to describe the molar flow rate of fuel gas entering the TGB by equation (30). The gas composition at TGB inlet is calculated as mentioned above.
The enthalpy flux of air ̇, , is calculated by equation (31). The specific enthalpy for nitrogen and oxygen is calculated by means of NASA polynoms [54] in dependence of the respective temperature. The cathode outlet oxygen molar flow rate is calculated according to equation (32) in dependence of the oxygen amount entering the SOFC system and the cell current. Since N2 is an inert species, its amount stays constant and its molar flow rate is calculated by equation (33) in dependence of the oxygen amount in ambient air, which is typically around 21%.
̇2 , , , =̇, (1 − 2 , ) + 2 , •̇ The enthalpy error ̇, is modeled with a GP regression model to achieve a good fit with the real TGB behavior, using available measurement data from the SOFC system at different operating points. It is modelled by some training data supplying CH4 as fuel gas, which has a constant H/C-ratio of 4, and NG in conjunction with measurements by a high precision gas chromatograph system (Agilent 7890 [65]).
By inserting all the equations in the main equation (26) for the TGB energy balance, all input variables are measured or calculated values, except the H/C-ratio, which is the unknown of interest and is calculated using a numerical process.

Hotbox
The Hotbox is the second component used within this study as calorimeter to estimate H/C-ratio in the exhaust gas by performing an energy balance at its boundaries. Assuming a gas-tight system, it is equal to the H/C-ratio in the NG feed line.
The general energy balance is given in equation (39) The electric DC power of the stack , is calculated as product of the measured stack voltage and current. The enthalpy flux of NG ̇ is calculated according to equation (40) in dependence of the respective molar flow rate ̇ and specific enthalpy ℎ , where the latter is calculated by equation (23). The molar flow rate ̇ is defined by equation (41) in dependence of the volume flow rate and the molar volume of ideal gas. NG volume flow rate is controlled with a thermal mass flow meter calibrated with CH4. The measured volume flow rate is thus adjusted by NG temperature as well as the specific heat capacity of CH4 and NG, which is expressed as a function of the H/C-ratio as shown above.
The enthalpy flux of air ̇, entering the hotbox is calculated by equation (42). The specific enthalpy for nitrogen and oxygen is calculated by means of NASA polynoms [54].
The molar flow rate of O2 and N2 are calculated by equation (43) and (44) ̇2 , , =̇, (1 − 2 , ) With the sole difference in gas mixture temperature, the enthalpy flux of the offgas ̇, , is calculated by equation (45) analogously to the enthalpy flux of the TGB offgas in equation (34). After the complete conversion in the TGB, there is no further change in gas composition, but a change in temperature due to internal heat recovery in the SOFC system. The respective specific enthalpies are calculated by means of NASA polynoms [54] and the molar flow rates by equation (35) to (38).
The enthalpy error ̇ is modeled with a GP regression model, using available measurement data from the SOFC system supplying CH4 and NG as fuel gas.
By inserting all equations in the main equation (39), all input variables are measured or calculated values, except the H/C-ratio, which is the unknown of interest and is calculated using a numerical process to solve the energy balance.

Experimental procedure
Real data of a SOFC system is required for the implementation of the concepts. It

Inert gas species estimation
The presented concepts are based on the assumption that the concentrations of CO2 and N2 are known or constant. The latter is not true for NG from the grid. Since it is aimed to determine the H/C-ratio without NG analysis, these variables can also not be assumed  The accuracy of the GP regression models with respect to training and validation data are shown in figure 4 (a) and figure 4 (b). A negligible deviation is achieved for the training data in figure 4 (a), which is attributed to its use for training of the GP regression models.
A good accuracy of the overall estimation on validation data is shown in figure 4 (b).
Small deviations in the estimation in figure 4 (b) are present for data point 1 and 24 for both CO2 and N2, where the latter is accompanied by some negligible low deviations for the neighboring data samples. Data point 1 and data point 24 are both operated with an of 65% and a cell current of 22.2 A, which are similar to other operating conditions used as data points within the study though. A respective influence of a unique operating point is therefore not present for data point 24, which means that the deviations cannot be attributed to it. At data point 1 the SOFC system has been operated with a slightly smaller recirculation ratio compared to the other data points, to which the deviations can accordingly be attributed.
The analysis with the Pearson correlation coefficients also showed a mutual interference of CO2 and N2. Its concentration is therefore added as mutual input variable for the GP regression models. Figure 5 (a) shows the respective results for estimation of CO2 and N2 concentrations on the relevant validation data.  neighboring data points has been improved significantly by the mutual integration. The mutual integration of inert gas species concentrations is therefore an improvement in the overall estimation quality.
Since no concentration value is available at the beginning of the calculation process for either CO2 or N2, the following procedure is applied. In a first step, the CO2 fraction is estimated without N2 as input variable as shown in Figure 4. In a second step, the N2 fraction is estimated, where the estimated CO2 concentration from the first step is included as input variable. In a third step, the CO2 fraction is determined with the N2 concentration of the second step as input variable. The N2 fraction from the second step and the CO2 fraction from the third step are then used in the soft-sensor models.

Estimation results for enthalpy errors
The   Figure 7 shows the results for the estimation of the H/C-ratio on the training data for both the TGB and the hotbox. Figure 8 shows the results on the validation data for the TGB and the hotbox separately as well as its respective relative errors.

Summary and conclusion
For safe and durable operation of a SOFC system it is of crucial importance to monitor the oxygen-to-carbon-ratio and fuel utilization and keep them within operating limits.
Monitoring and control of these characteristic parameters is not trivial to both, the correlations within a SOFC system with AEGR and on fluctuating NG quality. Further studies, which need to be done subsequently include the following tasks: -Derive a broader base of NG data with different compositions and variation of operating parameters.
-Extend offline tests to SOFC systems with different system layouts, analyze the differences and evaluate the potential to generalize the presented approach by a small calibration procedure to account for variations in system behavior.
-Performing of experimental tests directly on a real SOFC system for verification of the concept.

B. Gaussian process regression
Machine learning algorithms are a common tool for estimation, regression and modelling and has been used in a variety of studies in general [67,68] and in the fuel cell field [69][70][71][72][73][74][75][76][77][78]. One of the best-known variants is the Gaussian process (GP). GP modelling is a supervised machine-learning algorithm based on known training data for model generation in classification and regression problems, where the latter is used in this work A GP is defined by its mean function m(x) and its covariance function ( , ) for two input vectors , given in equation (B.1) and (B.2) [67,[79][80][81]83]. A common assumption is zero as mean of the GP [79,80,82]. The covariance function or kernel defines a priori the covariance between functional values (f(xi), f(xj)) and contains the prior knowledge of a GP [67,[79][80][81][82]. A typical covariance function is the squared exponential covariance function defined in equation (B.3), which is also called radial-basis function (RBF) [79][80][81][82]84]. Therein 2 is the signal variance (vertical scale) as maximum value of allowable covariance (for ≈ ) and the characteristic lengthscale (horizontal scale) as hyperparameters, which need to be optimized [67,79,80,82,83]. The characteristic lengthscale defines the distance of two correlated functional values describing the smoothness of the underlying function [67,[80][81][82]. A new functional value f* with the input vector x* is estimated based on a GP model, which has been trained with known data [81]. Its output is a Gaussian distribution with its predicted mean μ* as expected value for f* at point x* and a predicted variance of μ*, which captures the uncertainty or confidence of the estimation [67,79,[81][82][83].