There is a global push to develop renewable energy to further a low-carbon society. However, the nature of variable renewable energy (VRE) sources such as wind power and solar photovoltaic (PV) systems may create problems because electricity grids require a stable power supply to match demand. To evaluate the potential capacity of VREs that may be installed, we develop an optimized model that balances power supply and demand and also considers grid balancing by battery storage and load frequency control. The model was applied to a case study of an isolated grid on a remote Japanese island. When set to optimize the grid in terms of lowest cost, the model suggested that, compared with the base case, the capacity of wind power should be increased by a factor of 1.7 and 15.8 for situations without and with battery storage, respectively. Since it was always considered to be more expensive than wind power, no change in solar PV capacity was observed. These approaches resulted in a decrease in the total power generation cost of 2% and 24%, respectively, while total CO2 emissions fell by 3% and 52%, primarily driven by decreased used of the existing fossil-fueled thermal plant.