Power-to-X-to-Power energy systems can serve as storage and sector-coupling solutions for the transition from fossil fuels to renewable energy. The process uses green hydrogen, which can be produced from wind and solar power via water electrolysis. Within a chemical reactor, hydrogen reacts with carbon dioxide under high pressure and temperature to form methane or methanol. The methane or methanol are stored and can be converted into heat and power when needed. Such a system can be used for seasonal energy storage and mastering dark doldrums. This work introduces a Power-to-X-to-Power energy system demonstrator, which has been developed to optimize the operation of the individual components and the overall system. The demonstrator is based on a methane synthesis reactor and an oxyfuel combustion process. The oxyfuel combustion is realized in a 20-kW combined heat and power gas engine with a CO2 separation unit. The CH4/O2 mixture is diluted with 50 mol-% of CO2 to keep the temperature in the combustion chamber below 2000°C. The methane synthesis reactor uses Ni/Al2O3 catalysts and operates at 350°C and up to 20 bar of pressure. A maximum of 80 l/min at STP of methane can be produced by the reactor. In parallel, a digital twin of the energy system demonstrator is being developed. The Python, ESTECO SpA and LOGEsoft software packages are utilized for the development. Low-fidelity and high-fidelity simulations based on detailed chemistry are performed for the methane synthesis reactor, the oxyfuel gas engine, and the CO2 separation unit. The results are used to generate metamodels, which are incorporated into the digital twin. Subsequently, the digital twin is used to perform a genetic optimization of the energy system. Round-trip efficiencies of 42 - 48 % could be achieved by optimizing the operating parameters, recycling O2 and CO2 and reusing waste heat.