  Low-carbon societies will need to store vast amounts of electricity to
balance intermittent generation from wind and solar energy, for example,
through frequency regulation. Here, we derive an analytical solution to the
decision-making problem of storage operators who sell frequency regulation
power to grid operators and trade electricity on day-ahead markets.
Mathematically, we treat future frequency deviation trajectories as functional
uncertainties in a receding horizon robust optimization problem. We constrain
the expected terminal state-of-charge to be equal to some target to allow
storage operators to make good decisions not only for the present but also the
future. Thanks to this constraint, the amount of electricity traded on
day-ahead markets is an implicit function of the regulation power sold to grid
operators. The implicit function quantifies the amount of power that needs to
be purchased to cover the expected energy loss that results from providing
frequency regulation. We show how the marginal cost associated with the
expected energy loss decreases with roundtrip efficiency and increases with
frequency deviation dispersion. We find that the profits from frequency
regulation over the lifetime of energy-constrained storage devices are roughly
inversely proportional to the length of time for which regulation power must be
committed.
