Utilizing renewable energy sources (RESs) to their full potential provides an opportunity for lowering carbon emissions and reaching a state of carbon neutrality. DC transmission lines have considerable potential for the integration of RESs. However, faults in DC transmission lines are challenging due to the lack of zero-crossing, large fault current magnitudes and a short rise time. This research proposes using a superconducting fault current limiter (SFCL) for effective current limitation in PV-based DC systems. To properly design an SFCL, the present work investigates the effect of copper stabilizer thickness on SFCL performance by using an accurate multilayer thermoelectric model. In the MATLAB/Simulink platform, the SFCL has been modeled and tested using different copper stabilizer thicknesses to demonstrate the effectiveness of the SFCL model in limiting the fault current and the impact of the copper stabilizer thickness on the SFCL’s performance. In total, four different thicknesses of the copper stabilizer were considered, ranging from 10 μm to 80 μm. The current limitation and voltage profile for each thickness were evaluated and compared with that without an SFCL. The developed resistance and temperature profiles were obtained for various thicknesses to clarify the mechanisms behind the stabilizer-thickness impact. An SFCL with an 80 µm copper stabilizer can reduce the fault current to 5.48 kA, representing 71.16% of the prospective current. In contrast, the fault current was reduced to 27.4% of the prospective current (2.11 kA) when using a 10 µm copper stabilizer.