An optimal performance analysis of a combined Carnot cycle (two single Carnot cycles in a cascade), including internal irreversibility for steady state operation, has been carried out. It is shown that the effects of internal irreversibility on maximum power may be characterized by two cycle-irreversibility parameters representing the ratios of the entropy differences for each of the cycles. The effects of these two irreversibility parameters on power and thermal efficiency are studied. When there is no internal irreversibility, the analysis reduces to an endo-reversible combined cycle. The results for the endo-reversible case are compared with those of Wu et al. It is shown that the maximum power cannot exceed that of a single endo-reversible cycle operating between the same heat-source and heat-sink temperatures. It is also shown that the efficiency at maximum power for an endo-reversible combined cycle is the same as the efficiency of a single cycle given by Curzon and Ahlborn.