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Publication# Control of a Solid Oxide Fuel Cell System with Biased Gas Temperature Measurements

Résumé

An efficient control system is paramount for the operability of Fuel Cell systems since, in ideal cases, it allows the regulation of power output, temperatures and economic performance under a dynamic working environment where they need to operate. Although several control strategies, scenarios and methodologies have been broadly investigated, it is usually taken for granted that the measurements from the system, necessary for the control feedback, are correct. Nonetheless, when simple thermocouples are used to measure gas temperatures there is a significant danger of systematic errors due to radiation effects between the surroundings and the thermocouple. The discrepancy between a real gas temperature and the one measured depends mainly on the temperature difference between the gas and the solids around as well as the gas velocity, radiation factors etc. The phenomenon has been described in our past publications. In this work we simulate an SOFC system and apply control scenarios in order to investigate potential problems arising from such systematic errors. The results show that important dysfunctions may occur and caution should be applied in design of both control and of the systems themselves.

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Solid Oxide Fuel Cells (SOFCs) are electrochemical devices that convert chemical energy from fuel directly into electrical and thermal energy. They present high electrical efficiency and because of this feature they are considered to be an important power source alternative. However, due to their high temperature of operation, SOFCs are subject to degradation, that is, their ability to produce electricity deteriorates in time. The reasons and the conditions that contribute to degradation are multiple. Among the studied factors, operating temperature is an important one. It has been found that a stable and controllable thermal environment can mitigate this undesirable phenomenon. Therefore, the pursuit of appropriate thermal conditions during their operation is of paramount importance for their lifetime and consequently for their commercialisation. Studying and obtaining such conditions is also known as thermal management. Another important point regarding the study of SOFCs is the development and use of adequate mathematical models that the engineer and generally the researcher will use as tools to analyse their behaviour and will determine conditions for safe and efficient operation. For practical applications, these models must be validated against experimental data. The result of such validation is the determination, or calibration, of the model parameters so that the output of the model fits measurements taken in the laboratory to the best possible extent. This process is called parameter estimation or system identification. This thesis tackles both aspects. In its first part it works on the thermal management of a specific product, an SOFC autonomous unit fed with methane, producing electrical power and liberating hot off-gases that can be used as a source of thermal power. A dynamic model of the physical system is developed and validated against experimental data. During this process, it was found that measurement of gas temperatures using thermocouples may be severely biased due to radiation effects of surrounding solids to the thermocouples. In order to overcome this hurdle, the phenomena that take place around the thermocouple were incorporated into the system's mathematical model. Such a systematic error may have important consequences on the thermal management and generally on the control of the SOFC system. Indeed, an investigation effectuated in this thesis revealed how incorrect gas temperature measurements affect the system's control to such an extent that, depending on the conditions, may lead to system failure. The validation of the system model proved to be particularly challenging. This fact incited the author to investigate the factors that contribute to reliable parameter estimations. This is the subject of the second part of this thesis. A first study was performed on model-based Design of Experiments (DoE) for SOFC button cells, i.e. on how measurements on small cells may be optimised. To our knowledge the study treats for the first time the issue of repetitive measurements and their impact on the quality of parameter estimation systematically. It distinguishes the notion of measurements, which can be repetitive, from that of measurement points (or design points), which are defined once for an experiment. Introducing a new type of graph that provide information on quality criteria as functions of the number of measurement points for constant number of measurements, it stresses the importance of repetitions of measurements during experiments, which up to now has been neglected. A mathematical formula is also given that helps the experimenter define the necessary number of repetitions for a specific degree of precision of the information matrix. The theoretical findings on Design of Experiments are validated experimentally with measurements on button cells. The thesis introduces a novel method with which repetitive parameter estimations are obtained using data from polarisation curves. This method allows the calculation of histograms for the model parameters approximating this way their stochastic behaviour. Standard deviations, covariance and correlation matrices are calculated directly from the available data, avoiding the approximate calculation based on sensitivity (Jacobian) matrices. The experimental results showed again the importance of repetitions on the quality of parameter estimation. A rule-of-thumb is introduced suggesting that the number of repetitions of measurements needs to be at least equal to the number of calculated parameters plus three in order to avoid high correlations. However, repetitions do not influence the values of parameters or the fit to the experimental data, but their precision. These values are influenced by the number of measurement points. Another point of importance shown in this thesis is that there is a counterbalance between the parameters' variances and covariances and their correlations, that is, if the one improves, the other deteriorates. This is demonstrated to be in accordance with the Cramèr-Rao theorem in statistics. A consequence of this is the conclusion that parameter estimation cannot be obtained to an arbitrarily high precision. Another consequence is that assessment of system identification based on only one of the widely known criteria may not be adequate, a result that concurs with the findings of other authors in the field of DoE. Finally the available results from the afore-mentioned method revealed a potential relation between correlations of parameters and the non-uniformity of their histograms. Among the investigated examples, a case of fast degradation exhibits how the introduced parameter estimation method may be used as a diagnostics tool. However such a use requires the employment of models that describe the phenomena adequately, especially after degradation. The thesis ends with the calculation of the histograms of the parameters of the SOFC unit's heat exchanger providing a stochastic perspective of parameter estimation at an SOFC system level.

Joel Albrektsson, Daniel Favrat, Leonidas Tsikonis, Jan Van Herle

In this paper the effect of systematic errors in gas temperature measurements on the thermal control of a Solid Oxide Fuel Cell system is demonstrated. Three control schemes were tested under two different conditions. First, the real gas temperature at the point of interest was used as input for the controller. Second, the reading of the thermocouple that takes the measurement at the same point was used as input. The second approach included the bias on the measurement due to radiation and heat conduction on the thermocouple. The results showed an overestimation of temperatures with important influence on the system control. Furthermore, in one case the system could not follow the required power demand and it failed. A successful solution to the problem was obtained by increasing the dimension of the system's heat exchanger. (C) 2013 Elsevier B.V. All rights reserved.

This report is intended to summarize the work of a semester project conducted on a gPROMS model (developed in LENI) of an existing stand-alone 1kW solid oxide fuel cell (SOFC) system. Different means of controlling the cathode gas temperature with single-input-single-output PID controllers were investigated. Particular attention was directed to the issue of erroneous thermocouple readings due to radiation and how this error can be minimized. It was shown that the discrepancy between thermocouple reading and real temperature depends strongly on the method used to control the temperature. In some cases, the incorrect thermocouple reading can be fatal for the system even in regular operation. With this basic type of PID controller, lambda control was shown to be the scenario with the smallest discrepancy. However, it is still non-negligible and a more advanced control approach should be investigated before an implementation on the physical system.

2011