The energy continues to cover the home’s loads. If there is surplus production and feed-in is permitted, the surplus may be directed to the grid, in accordance with the connection terms and the system settings.
When feed-in is zero or limited, the installation’s controller reduces production to stay within the permitted limits. The energy the panels could produce in addition at that moment is not stored for later. You do not need to run appliances unnecessarily just to consume it.
In backup mode, with the property isolated from the grid, production and charging are controlled by the compatible equipment. The behavior depends on the configuration and is not the same in every system.
If the battery often fills early, review the production and consumption curves before deciding whether to expand the system. Scheduling existing needs during surplus hours may help. The additional cost of greater storage capacity should be assessed separately.