While wind farms have no inherent storage to supply power in calm conditions, this paper demonstrates that large tidal turbine farms in channels have short-term energy storage. This storage lies in the inertia of the oscillating flow and can be used to exceed the previously published upper limit for power production by currents in a tidal channel, while simultaneously maintaining stronger currents. Inertial storage exploits the ability of large farms to manipulate the phase of the oscillating currents by varying the farm's drag coefficient. This work shows that by optimizing how a large farm's drag coefficient varies during the tidal cycle it is possible to have some flexibility about when power is produced. This flexibility can be used in many ways, e.g. producing more power, or to better meet short predictable peaks in demand. This flexibility also allows trading total power production off against meeting peak demand, or mitigating the flow speed reduction owing to power extraction. The effectiveness of inertial storage is governed by the frictional time scale relative to either the duration of a half tidal cycle or the duration of a peak in power demand, thus has greater benefits in larger channels.