  Virtual power plants and load aggregation are becoming increasingly common.
There, one regulates the aggregate power output of an ensemble of distributed
energy resources (DERs). Marecek et al. [Automatica, Volume 147, January 2023,
110743, arXiv:2110.03001] recently suggested that long-term averages of prices
or incentives offered should exist and be independent of the initial states of
the operators of the DER, the aggregator, and the power grid. This can be seen
as predictability, which underlies fairness. Unfortunately, the existence of
such averages cannot be guaranteed with many traditional regulators, including
the proportional-integral (PI) regulator with or without deadband. Here, we
consider the effects of losses in the alternating current model and the
deadband in the controller. This yields a non-linear dynamical system (due to
the non-linear losses) exhibiting discontinuities (due to the deadband). We
show that Filippov invariant measures enable reasoning about predictability and
fairness while considering non-linearity of the alternating-current model and
deadband.
