System Identification: A New Modelling Approach for SOFC Single Cells

Due to the great number and complex nature of physico-chemical processes, a comprehensive model describing SOFC single cell operation is at present not available. System identification, a modelling approach from control theory is proposed. Instead of infering models from physical laws, the method is based on experimental data to build a mathematical model. Model parameters may subsequently be related to physical processes in the cell. The method is carried to the point of pre-identification of SOFC electrochemical impedance spectroscopy data. A new technique has been developed for calculating the distribution of relaxation times directly from the data. No equivalent circuit has to be assumed. In addition the method yields a higher resolution of dynamical processes than non-linear least squares curve fit. Kramers-Kronig transformation of impedance data was used to calculate the ohmic resistance of the electrolyte and the total polarization resistance of the cell. INTRODUCTION In order to optimize the SOFC single cell with respect to maximum electrical efficiency and long term stability the interaction between the cell parameters, i.e. its material properties and operating conditions, and its electrical performance ought to be well understood. Modelling is usually done by splitting up the system into sub-systems, whose properties are well established from physical laws. The sub-models are combined to obtain a model of the whole system. However, comprehensive models describing SOFC single cell operation are at present not available because of the high complexity of concurring physico-chemical processes. In this paper, an alternative to theoretical modelling based on physical laws is presented: system identification. This approach originates from control theory. It deals with the problem of infering mathematical models of dynamical systems based on observed data from the systems (1). With this approach, there is in principle no need for a priori knowledge of the physical processes in the cell. In other words, the cell is regarded as a technical process in a ”black box” (Fig. 1). The input-output data of SOFC operation are recorded by electrical measurement. Wellknown examples of such input-output data are electrochemical impedance spectra, I/Vcharacteristics and long term measurements. Possible input signals are current and impedance distortion applied to the system, measured output signals are cell voltage, voltage losses and impedance data. Cell parameters are all kinds of properties that influence cell performance: materials characteristics (e.g. composition and porosity of electrode layers), Solid Oxide Fuel Cells VI , Ed.: Singhal, S. C., Dokiya, M., Electrochem. Soc. Proc. Ser., Pennington NJ, 1069-1077, 1999. 2 changes in microstructure • interfaces • porosity • layer thicknesses • cation diffusion / demixing output signals • cell voltage • voltage losses • impedance spectra • I/V-characteristics • long term measurements materials characteristics • composition • grain size distribution • porosity / layer thickness • thermal / electrical properties input signals • impedance distortion • current load operational parameters • temperature • gas composition • fuel utilization • thermal cycling production parameters • powder preparation • screen printing • sintering process SOFC Figure 1: SOFC operation as a technical process production parameters, working conditions like temperature, gas composition, etc., and material parameters obtained from post-test analysis (Fig. 1). System identification is an iterative method that consists of three steps: pre-identification, model estimation and model validation (Fig. 2). During pre-identification, the data from the system under investigation are subjected to numerical analysis in order to make it suitable for modelling and a corresponding model structure is chosen. In the next step, model estimation, the parameters of the model are determined by suitable parameters estimation methods. It then remains to test whether this model is ”good enough”. Such tests are known as model validation. If the model does not relate to observed data or if the dynamical range of the model is to be extended, we must go back and revise the three steps of system identification. The obtained model represents information about the underlying physical processes as long as they can be distinguished in terms of time constants. If an adequate model can be found, changes in the physical parameters of the cell can be related to changes in the formal parameters of our model. Once a relationship is established, the effect of cell parameter changes on cell performance can be predicted. The scope of the present paper is an initial step into system identification of SOFC single cell operation. Fig. 3 shows the wide range of time constants at which physical processes are observed together with some examples of these processes. Different measurement techniques are employed to cover this time scale. At present, electrochemical impedance