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This study delineates the development and evaluation of a power generation cycle characterized by the absence of carbon dioxide emissions, achieved through the direct combustion of oxygen and natural gas, a system commonly known as the Graz cycle. The analysis incorporates both thermodynamic and economic dimensions. In this system, hydrogen (H2) was initially produced from two separate sources to enable the conversion of carbon dioxide (CO2), sourced from the Graz cycle, into methane. The predominant method for hydrogen production is through the biomass gasification system, complemented by the use of hydrogen separation membranes. The solid oxide electrolyzer cell serves as the secondary source, wherein the necessary electrical energy is supplied by photovoltaic collectors. This research employs a robust methodological framework to undertake a comprehensive analysis of economic variables, with the objective of providing an accurate depiction of empirical conditions in the real world. This study conducts a comprehensive analysis of all relevant costs within the specified framework. A series of ANN-centered optimization analyses was performed to ascertain conditions that concurrently optimize the economic and thermodynamic dimensions of the system.