Interlink Converter (ILC) is the backbone of the hybrid AC/DC microgrid (HMG), and any failure in its operation may lead to financial and technical losses. The reliability enhancement of the ILC is paramount, as it is responsible for power exchange between sub-grids to maintain system stability and facilitate optimal energy transactions in economic scheduling. Besides, battery energy storage system (BESS) play a pivotal role in coping with the intermittent behaviour of renewable energy sources and the economic operation of grid-connected HMGs by charging at low energy prices and discharging to the grid at high energy prices. Given that BESS represent one of the most expensive components in electricity networks, it is imperative to account for their capacity fading constraints in the optimal energy management system. This paper introduces a reliability index for the ILC based on the Arrhenius equation, while also exploring the energy throughput constraint to mitigate the cyclic aging of the BESS. Through three comprehensive case studies, the simulation results signify the remarkable effectiveness of the proposed energy management system in reducing the failure rate of the ILC and mitigating the cyclic aging of the BESS.