A hybrid system including the solid oxide fuel cell (SOFC), Carnot cycle, and Brayton cycle is proposed, where the high-grade waste heat and residual fuel from the SOFC is utilized and transformed into power. Based on electrochemistry and thermodynamics, the electric power, flow rate of waste heat, and composition of residual fuel from the SOFC are calculated, respectively, under a given flow rate of natural gas into the anode of SOFC. The working substance of Carnot cycle absorbs heat in the cooling-tube of SOFC stack and the efficiency of the cycle is derived by considering the finite-rate heat transfer and internal irreversible losses. The working substance of Brayton cycle absorbs high-temperature heat from the combustion products of the residual hydrogen and the efficiency of Brayton cycle is obtained by considering the finite-rate heat transfer and irreversible compression and expansion processes. To achieve optimal power of SOFC or the maximum power of the hybrid system, the favorable working temperature, mole flow rate of natural gas, and hydrogen utilization factor in SOFC are calculated by using MATLAB software. (C) 2017 Elsevier Ltd. All rights reserved.