Generating power from renewable biogas in solid oxide fuel cells (SOFCs) is an environment-friendly, efficient, and promising energy conversion process. Biogas can be used in SOFCs via a reforming process for which dry reforming is more suitable as the reforming agent exists in the biogas mixture. Biogas can be directly reformed to $H_2$ -rich fuel stream in the anode chamber of a SOFC by the heat released during power generation. Exploiting the heat and water produced in the SOFC for internal reforming of biogas makes the energy conversion process very efficient; however, various challenges are reported. Thus, indirect internal reforming is opted for which a separate reforming domain is required. In an indirect internal reformer operating at usual conditions, dry reforming rate is quite high in the inlet and it decreases steeply toward the fuel outlet. Great temperature gradients develop over the reformer, since the dry reforming reaction is strongly endothermic. The abruptly varying rate of the reforming reaction affects the temperature fields in the adjacent components of SOFC and hence intolerable thermal stresses emerge on the SOFC components. In our preceding study, we graded the reforming domain, homogenized the temperature profile over the reforming domain, and executed performance and durability experiments. However, most of the experiments failed due to fracturing SOFC components hinting at existence of thermal stresses. In that study, we focused on minimizing the temperature gradients within the reforming domain; namely, we neglected the other processes. To eliminate the thermal stresses, we modeled the entire module of SOFC equipped with a reformer featuring a graded reforming domain. We found that the mismatch between the thermal conductivities of the adjacent module components is the major reason for the thermal stresses. When the mismatch is eliminated, thermal stresses disappear even if the reforming domain is not graded.