  In power systems, one wishes to regulate the aggregate demand of an ensemble
of distributed energy resources (DERs), such as controllable loads and battery
energy storage systems. We suggest a notion of predictability and fairness,
which suggests that the long-term averages of prices or incentives offered
should be independent of the initial states of the operators of the DER, the
aggregator, and the power grid. We show that this notion cannot be guaranteed
with many traditional controllers used by the load aggregator, including the
usual proportional-integral (PI) controller. We show that even considering the
non-linearity of the alternating-current model, this notion of predictability
and fairness can be guaranteed for incrementally input-to-state stable (iISS)
controllers, under mild assumptions.
