## 1 Introduction  
Solid oxide fuel cells (SOFCs) have become the focus of the research community due to the relatively high energy efficiency and power density, all-solid state, cleanliness, and capability of working with various fuels [1; 2; 3]. In the past decade, extensive efforts have been devoted to lowering the working temperature of SOFC to 350-650 \({}^{\circ}\)C [4], because the low operating temperature would make it possible to apply these cells to portable devices [5], reduce material costs [6; 7; 8], and extend the lifetime of fuel cells [6]. The ionic conductivity of traditional electrolyte materials (i.e., yttria-stabilized zirconia: YSZ) decreases exponentially as the operation temperature drops [9; 10]. Proton-conductive electrolyte materials have been used to replace oxygen ionic electrolyte materials to overcome SOFC difficulties, while the operation temperature is reduced [11]. Compared with oxygen ionic electrolyte materials, proton-conductive electrolyte materials (i.e., BaZr\({}_{0.1}\)Ce\({}_{0.7}\)Y\({}_{0.2}\)O\({}_{3-\delta}\) (BZCY) [12], BaZr\({}_{0.1}\)Ce\({}_{0.7}\)Y\({}_{0.1}\)Yb\({}_{0.1}\)O\({}_{3}\) (BZCYYb) [13; 14]) have higher proton conductivity around the intermediate temperature zone [15; 16]. Furthermore,unlike the O\({}^{2-}\)-SOFCs, protonic ceramic fuel cells (PCFCs) have the following advantages: (a) low energy loss during H\({}^{+}\) transport [17; 18; 19], (b) the cathode produces water to avoid dilution of the fuel in anode side [20], (c) faster fuel shipments lead to low concentration at anode [21; 22], and (d) transference of H\({}^{+}\) increases with the decreasing temperature [23]. The choice of cathode material is crucially important to the performance of PCFC [24; 25]. The cathode material, La\({}_{0.8}\)Sr\({}_{0.2}\)MnO\({}_{3}\) (LSM), which is widely used at high temperature is restricted, while the temperature drops below 800 \({}^{\circ}\)C due to the lack of oxygen vacancies [26; 27; 28]. Therefore, ion/e\({}^{-}\) mixed conductive cathode materials (i.e., Ba\({}_{0.5}\)Sr\({}_{0.5}\)Co\({}_{0.8}\)Fe\({}_{0.2}\)O\({}_{3}\)-delta (BSCF) [29], La\({}_{0.6}\)Sr\({}_{0.4}\)Co\({}_{0.2}\)Fe\({}_{0.8}\)O\({}_{3-5}\) (LSCF) [30; 31]) are widely used in intermediate temperature fuel cells. [PERSON] et al. [32] pointed out that the poor performance of most PCFCs is attributed to having no suitable cathode material. H\({}^{+}\)/e\({}^{-}\)/O\({}^{2-}\) mixed conducting material (i.e., BaCo\({}_{0.4}\)Fe\({}_{0.4}\)Zr\({}_{0.1}\)Y\({}_{0.1}\)O\({}_{3-.8}\): BCFZY\({}_{0.1}\)[33]) helps to improve the oxygen reduction reaction kinetics at intermediate temperature PCFC. [PERSON] et al. [34] proposed a H\({}^{+}\)/e\({}^{-}\)/O\({}^{2-}\) mixed conducting composite cathode BZCY-LSCF in PCFC. Many H\({}^{+}\)/e\({}^{-}\)/O\({}^{2-}\) mixed conductive materials [35; 36; 37] (i.e., Pr\({}_{0.5}\)Ba\({}_{0.5}\)Co\({}_{0.7}\)Fe\({}_{0.3}\)O\({}_{3-d}\)) have also been invented as the PCFC cathode materials and good performances have been achieved. Compared with the conventional composite electrodes, the reaction sites and reaction process in the mixed conductive cathode of PCFCs are very different. The reaction in H\({}^{+}\)/e\({}^{-}\)/O\({}^{2-}\) mixed conducting cathodes is divided into two parts. The electrochemically active sites (EAS) for oxygen reaction (O\({}_{2}\)\(\rightarrow\)O\({}^{2-}\)) are the percolated double-phase boundary sites (DPBs) (i.e., LSCF-pore), while the EAS for vapor formation reaction is the percolated LSCF-BZCY-pore three-phase boundary sites (TPBs). With the development of PCFC electrode manufacturing technology, in order to improve the performance of PCFC, a variety of electrode materials have been invented, and the structures/components are also more complex [38; 39]. The microstructure of electrodes has a great influence on the PCFC performance [40; 41]. There is an increasing need to predict the effect of the microstructure parameters on the PCFC working details.