Optimal sizing of the power system can drastically reduce the total cost, which is challenging due to the fluctuation in output power of RE (primarily wind and solar) and pollution from thermal generators. The main purpose of this study is to cope with this output power uncertainty of renewables by considering ADLC, residential PV, and BESS at the lowest cost and with the least amount of carbon emission, while putting less burden on consumers by minimizing the IL. This paper optimizes the cost and carbon emission function of a hybrid energy system comprising PV, WG, BESS, and DG at Aguni Island, Japan, using a multi-objective optimization model. To solve the proposed problem in the presence of ADLC, the ϵ-constraint method and MILP are utilized. After obtaining all possible solutions, the FSM selects the best possible solution among all solutions. The result shows that while case 1 has a lower energy cost than the other cases, the quantity of IL is quite significant, putting customers in a burden. In case 2 and case 3, the total energy cost is 11.23% and 10% higher than case 1, respectively, but the sum of the IL is 99% and 95.96% lower than case 1 as the ADLC is applied only for the consumers who have residential PV and BESS, which can reflect the importance of residential PV and BESS. The total cost of case 3 is 1.72% lower than case 2, but IL is higher because sometimes home PV power will be used to charge the home BESS.