The increasing demand for energy in both commercial and residential sectors poses a pressing challenge, driving the ongoing quest for sustainable and virtually limitless energy resources. As nations become increasingly reliant on energy, the need to enhance energy system efficiency while curbing costs and reducing greenhouse gas emissions, particularly CO2 from fossil fuel combustion, becomes imperative. This paper presents a thermodynamics modeling, economic assessment, and optimization of a multigenerational module that utilizes compressed-air energy storing and polymer electrolyte membrane electrolysis with methane and hydrogen fuels, with a case study focused on Munich, Germany. Using EES, a system comprising of a modified organic Rankine cycle, gas turbine, PEM electrolyzer, and CAES unit was modeled. The combustion chamber was used to exploit methane and a combination of methane and hydrogen. The performance of the suggested module under these two scenarios was evaluated using multi-objective optimization to select the optimal scenario based on TOPSIS. In this study, we conducted 700 runs and obtained data on seven inputs. To derive a mathematical function from this data, we employed an Artificial Neural Network (ANN). This function was subsequently integrated into a multi-objective optimization system, allowing us to optimize the desired objectives. It was determined that Scenario 1 was the best choice for lowering the cost rate, reducing CO2 emissions, and optimizing energy round-trip effectiveness (ERTE). The highest exergy destruction rates were observed in the gas turbine, combustion chamber, and recuperator of the suggested system. The cost rates for ORC, CAES, and PEM electrolyzer were the highest according to the economic evaluation of the proposed module. Moreover, the gas turbine had the highest price. The study assessed the power demand of a building and evaluated the efficiency of the proposed power supply system. Additionally, an investigation was conducted in Munich, Germany, to examine how the system's performance is influenced by daily changes the annual variation in the environment temperature. The results revealed that the exergy destruction rates for the charging and discharging units were identified as 1706.38 kWh and 3019.31 kWh, respectively. additionally, based on the economic analysis, the total cost rate for the proposed energy production system was projected to be 68.073 USD per hour. The subsystems were ranked in the following order of costs: CAES > ORC > PEM electrolyzer. Additionally, within the specified system equipment, the gas turbine equipment exhibited the highest cost rate.