## 2 Methods and Modeling
### The Degradation Model  
Degradation mechanisms of SOFC have been studied experimentally [24; 25] and theoretically [11] to investigate how the microstructure evolves over long-term operation under different conditions, such as temperature, current density [24; 25], and initial composition of the porous anode, and the average size of Ni and YSZ grains within the microstructure [11]. Since a real SOFC operates at high temperatures (at \(T\) = 800-1000 \({}^{\circ}\)C), the microstructure evolution, such as grain growth or particle coarsening, is significant [11]. The microstructure becomes increasingly coarser and denser with increasing testing temperature, current density, and testing time [24]. At 750 \({}^{\circ}\)C an increase in grain size occurs mainly in the bulk and interface compared with a non-degraded cell. At 850 \({}^{\circ}\)C and above, the grain coarsening in the bulk becomes more pronounced than at 750 \({}^{\circ}\)C. The coarsened interface region is about 1-2 \(\upmu\)m in width at 850 \({}^{\circ}\)C and up to 5 \(\upmu\)m at 950 \({}^{\circ}\)C. It was also seen that at 850 \({}^{\circ}\)C the process of grain coarsening and interface densification continues with increasing testing time up to 17,500 h [24]. From these results one can note that microstructure evolution generally progresses with increasing temperature and testing time. Ostwald-ripening is an important mechanism of Ni coarsening in SOFC anode [26]. It occurs due to the transport of volatile Ni species via the gas phase and diffusion of vacancies, driven by different grain sizes. Despite the detailed studies, only a few papers [27; 28] deal with the problem of how the microstructural degradation actually affects the electric performance of SOFC. In this paper the electric performance of a degraded SOFC was calculated by adjusting fewer parameters compared to the number of parameters that need to be set in the phase field model [11]. Moreover, the validation of such a model would require detailed analysis with three-dimensional (3-D) tomography techniques. These techniques have shown that the Ni (YSZ) grains grow exponentially with time [9]. Therefore, the degradation of SOFC due to grain growth was modeled by considering the power law [9]:  
\[d^{\kappa}=d_{0}{}^{\kappa}+k\cdot t \tag{22}\]  
Time-dependent diameter of Ni grains is denoted with \(d\), initial diameter with \(d_{0}\), growth coefficient with \(k\) and time with \(t\). The exponential coefficient is denoted with \(\kappa\). It equals 2 for grain growth and 3 for particle coarsening due to Ostwald-ripening [11]. However, Equation (22) has to be slightly adapted for numerical implementation. It is assumed that \(d_{0}\) is determined by the minimum distance between the two adjacent points where active TPBs are located. This simplification is done due to problems with detecting grain boundaries of the Ni (YSZ) particles and their actual diameter by processing a grey-scale image. Moreover, reconstruction of the porous anode according to the phase field model would be necessary if actual grain growth was  
Figure 5: (a) Schematic explanation of the detection of active triple phase boundary (TPB) and (b) percolated gas phases. considered rigorously [11]. That is beyond the scope of the present paper, so the following algorithm was proposed. First, the active TPBs are resolved as described in Section 2.6. Second, two vectors are defined for storing two distances (\(d_{1,\min}\), \(d_{2,\min}\)) from one to the closest two TPB points. Third, \(d_{1,\min}\) and \(d_{2,\min}\) are found. Fourth, the SOFC model is evaluated at each iteration (q) using a fixed time step (\(\Delta t\)).