An empirical mixed-effect regression model for comparative analysis of long-term SOFC degradation tests

Solid oxide fuel cell (SOFC) is an electrochemical device, which directly converts the chemical energy into electricity with high efficiency and low pollutant emissions. Thus, SOFC systems are expected to play a significant role in many scenarios of distributed and stationary power generation. Since high reliability levels are required, a big effort was devoted to extend their durability. In particular, degradation at the cell/stack level is an issue of major concern, and a number of features influencing cell/stack degradation have been identified. However, methods to directly observe degradation modes and to measure their evolution over time are difficult to implement, and hence indirect cell/stack performance indicators are adopted, typically related to the cell/stack voltage measurements in long-term tests. These tests show that different shapes of the cell/stack voltage as a function of time can be observed, as a consequence of technological improvements and/or operating conditions. Also, a variability (non reproducibility) of the voltage degradation path is often observed across cells in the same stack, as a consequence of lack of uniformity in the manufacturing process and/or of different operating conditions across cells. Furthermore, voltage measures are affected by an experimental noise, whose primary sources are experimental temperature, pressure and reactant concentration. Thus, it appears that, in order to describe long-term degradation tests, (at least) three components of the voltage measurements should be modelled: the smooth variability of voltage over time for each single unit; the variability of voltage behaviour among units; and the variability due to experimental noise. To this aim, in this paper we propose an empirical mixed-effect regression model of polynomial type, enabling to evaluate separately these three types of variability. Point and interval estimates are also derived for some quantities of interest, such as the mean voltage degradation over time and the prediction interval on future observations. The proposed methodology is then applied to real case-studies of long-term degradation tests. The results appear to confirm the ability of the proposed method to enable accurate comparisons between long-term durability tests. This need nowadays arises because rapid technology advances tend to reduce degradation rates, thus requiring tools with increased sensitivity.