## 1 Introduction  
Solid oxide fuel cell (SOFC) systems are a promising technology for stationary applications since they convert hydrogen (H\({}_{2}\)) directly into electricity at high conversion efficiency (\(\eta\)) [1]. The \(\eta\) of chemical to electrical energy is usually higher compared to the gas turbines or engines with internal combustion, which are limited by the Carnot cycle [2]. SOFCs can be fueled with some other hydrocarbon gases (e.g., methane) that need to be reformed externally in a reformer [3] or internally within the cell [3, 4]. When the cell is fueled with hydrocarbon gas, the exhaust gas also contains carbon dioxide (CO\({}_{2}\)). The major drawbacks of using SOFC systems are relatively high manufacturing costs and a fast degradation rate. The overall degradation of SOFC is commonly a combination of electrochemical degradation, structural degradation, and mechanical failure [5]. One of the most critical issues is structural degradation of the porous anode due to nickel (Ni) agglomeration. Ni agglomeration increases the polarization resistance of SOFC since it decreases the density of electrochemical reaction sites within the anode active layer. Numerous experimental and theoretical studies have been conducted to elucidate the degradation processes. Electric performance losses were studied when the SOFC was fed with fuel containing poisonous gas [6]. It was found that sulfur (S) adsorbs on the nickel (Ni) surface within the porous anode layer and inhibits the reaction sites at triple-phase boundaries (TPBs). Consequently, an obvious drop in SOFC output voltage was observed in the initial few hours of the test. The degradation mechanisms occurring within a planar, nickel-yttria-stabilized zirconia (Ni-YSZ), anode-supported, SOFC with direct internal reforming of dry CH\({}_{4}\)-CO\({}_{2}\) mixtures were investigated during the ageing tests [7]. One of the tests was conducted for about 300 h at the temperature of 770 \({}^{\circ}\)C, current of 15 A (current density of 0.3 A cm\({}^{-2}\)), and the fuel utilization of 60%. The CH\({}_{4}\) and CO\({}_{2}\) flow rates were both 43 cm\({}^{3}\) min\({}^{-1}\), whereas the airflow rate was 750 cm\({}^{3}\) min\({}^{-1}\). The SOFC output voltage decreasedby about 20% in less than 300 h due to the deactivation of Ni particles as a consequence of carbon formation and deposition [7]. Industrial size (10 cm \(\times\) 10 cm), planar SOFCs were tested under various polarization conditions to show their degradation rate [8]. Post-mortem analysis of the microstructure revealed that the average size of Ni particles increased within the anode. Small Ni agglomeration occurred in the cell that operated under activation polarization (low current density), whereas large Ni agglomeration was observed in the cell operated under concentration polarization (high current density) [8]. This implies that the agglomeration of Ni particles has detrimental effects on electro-catalytic performance. The degradation reflects an increased polarization resistance of the anode that is attributed to the reduced density of TPBs within the anode due to the Ni grain coarsening [9]. Another paper [10] aims to predict the degradation of SOFC due to the growth of Ni particles. A theoretical approach to assess the agglomeration of Ni particles was adopted. A simple power law was employed to model the growth of average particle diameter over time. An excellent agreement with experimentally observed changes of the microstructure during the initial 1000 h of the test was accomplished. The change of TPB density was accounted for accordingly. An improved phase field model was proposed recently [11] to simulate the microstructure evolution of the SOFC electrode. Three different sets of parameters for the YSZ/Ni anode, YSZ/Ianthanum strontium manganite (LSM) cathode, and reference electrode were used to tune the interfacial energies in the model. The model was validated by simulating the grain growth and particle coarsening.