The expansion of renewable energy generation must go hand in hand with measures for reliable energy supply and energy storage. A combination of hydrogen and oxygen as storing media provided from electrolysis at high pressure and zero emission power plants is a very promising option. The Graz cycle is an oxy-fuel combined power cycle that can operate with internal H2/O2 combustion and steam as working fluid. It offers thermal efficiencies up to 68.5% (lower heating value - LHV). This work applies a second law analysis to the Graz cycle and determines its exergetic efficiency. Exergy destruction is broken down to the cycle's components, thus providing insights on the location and magnitude of the cycle's inefficiencies. A sensitivity analysis identifies the cycle's exergetic and energetic efficiency as a function of representative parameters, offering an approach for future improvements. The combination of the cycle with an electrolysis plant is subsequently analyzed as an electric energy storage system. The round trip efficiency of the storage and back conversion system is computed by taking into account the additional compression of the reactants. As part of this analysis, the effect of the electrolyzer's operational pressure is studied by comparing several commercial electrolyzers.