NOVELTY - The apparatus (1,11) has output terminal (14) that is connected to system having engine (2), and electric storage device having monitoring apparatus and relay (12). Controller (22) is configured to determine whether variation value detected by detector is lower than opening threshold larger than engine activation low-threshold that is lowest level of amount of charge at which engine is started by preset value, and to execute opening process to switch state of relay from closed state to open state if variation value is equal or lower than opening threshold.    USE - Electric storage apparatus e.g. battery e.g. starter battery, iron phosphate lithium ion battery and assembled battery used in vehicle e.g. engine vehicle, aircraft and hybrid vehicle. Can also be used in manganese lithium ion battery, secondary battery of cell, lead electric storage battery and nickel metal hydride battery.    ADVANTAGE - The less power is consumed by monitoring apparatus of the electric storage device, since the controller determines whether variation value detected by detector is equal or lower than opening threshold and executes opening process to switch state of relay from closed state to open state if variation value is equal to or lower than opening threshold. The power provided by the electric storage device is less likely to be continuously consumed by the electric load because the relay does not remain in the closed state for long time while the engine is not started after the relay is switched back to the closed state. The relay can be used to prevent overcharge. The electric storage apparatus can itself control the amount of charge independently from system outside the electric storage apparatus so as not to decrease to the level at which the engine cannot be started. Thus the power consumption in electric storage apparatus is reduced while power supply to electric load is halted.    DESCRIPTION OF DRAWING(S) - The drawing shows a block diagram of the battery, on-vehicle device, and devices.    Battery (1)    Engine (2)    Assemble battery (11)    Relay (12)    Output terminal (14)    Controller (22)