Background: The electricity production market benefits from combined cycles with their high efficiency. In the lower power range, Fuel Cells are a good candidate to combine with gas turbines. Such systems can achieve efficiencies of 60% and more. High temperature Solid Oxide Fuel Cells (SOFC) offer good opportunities for this coupling. A systematic method to select a design according to user specifications is presented. Method of approach: The most attracting configurations of this technology coupling are identified using a thermo-economic multi-objective optimization approach. The SOFC model includes detailed computation of losses of the electrodes and thermal management. The system is integrated using pinch based methods. A thermo-economic method is then used to compute the integrated system performances, size and cost. This allows to perform the optimization of the system with regard to two objectives: minimize the specific cost and maximize the efficiency. Results: Optimization results prove the existence of designs with costs from 2400 $/kW for a 44% efficiency to 6700 $/kW for a 70% efficiency. Several design options are analysed regarding fuel processing, pressure ratio or turbine inlet temperature among others. Conclusions: A pressurized SOFC-$\mu$GT hybrid cycle has been modeled for design purposes. Combining state-of-the-art planar SOFC with high speed micro gas turbine sustained by air bearings. The proposed optimization method allows to identify the most interesting design configurations to reach high efficiencies for a given cost.