Exergy losses represent true losses of potential to generate a desired product, exergy efficiencies always provide a measure of approach to ideality, and the links between exergy and both economics and environmental impact can help develop improvements. In this study, PV-coupled Solid Oxide Fuel Cell (SOFC) and Gas Turbine (GT)-electrolyzer hybrid power generation system is considered to determine the contribution of different hybrid system components in the total exergy loss. The number of panels, the power of SOFC–GT, and the power of electrolyzer can have different values. Therefore, to obtain the optimum combination from ecological, economical, and reliability points of view, a multi-objective optimization algorithm (PESA) is considered. This optimization method chooses a set of optimum solutions that is known as Pareto frontier. The exergy loss of some of these optimum solutions is compared with each other. The effect of panel angle and SOFC–GT fuel type on the hybrid system exergy loss is considered in this study. Also, the hybrid system exergy loss is determined in different months of the year to obtain the worst month from exergy loss view.