Net zero energy buildings and positive energy buildings are gaining more and more interest. This paper evaluates the impact of the integration of a battery in a positive energy building used to increase its self-consumption of electricity. Parametric studies are carried out by varying the building envelope characteristics, the power supply system, the climate, the lighting and appliances profiles, the roof tilt angle, the battery size and the electricity tariffs, leading to 3200 cases. The analysis is performed on an annual basis in terms of self-consumption and self-production rate and payback period. It is shown that the battery size leading to the minimum payback period within the input range is comprised between 2.6 and 4.5 kWh. The lowest payback periods, (~7 years), are reached with a well-insulated building envelope, a high lightning and appliance consumption, a low feed-in tariff and a 3.7 kWh battery. Finally, simple correlations (based on the feed-in tariff, the annual electrical consumption and production) to predict the optimal size of battery and the lowest payback period are proposed.