  A significant portion of electricity consumed worldwide is used to power
thermostatically controlled loads (TCLs) such as air conditioners,
refrigerators, and water heaters. Because the short-term timing of operation of
such systems is inconsequential as long as their long-run average power
consumption is maintained, they are increasingly used in demand response (DR)
programs to balance supply and demand on the power grid. Here, we present an
\textit{ab initio} phase model for general TCLs, and use the concept to develop
a continuous oscillator model of a TCL and compute its phase response to
changes in temperature and applied power. This yields a simple control system
model that can be used to evaluate control policies for modulating the power
consumption of aggregated loads with parameter heterogeneity and stochastic
drift. We demonstrate this concept by comparing simulations of ensembles of
heterogeneous loads using the continuous state model and an established hybrid
state model. The developed phase model approach is a novel means of evaluating
DR provision using TCLs, and is instrumental in estimating the capacity of
ancillary services or DR on different time scales. We further propose a novel
phase response based open-loop control policy that effectively modulates the
aggregate power of a heterogeneous TCL population while maintaining load
diversity and minimizing power overshoots. This is demonstrated by low-error
tracking of a regulation signal by filtering it into frequency bands and using
TCL sub-ensembles with duty cycles in corresponding ranges. Control policies
that can maintain a uniform distribution of power consumption by aggregated
heterogeneous loads will enable distribution system management (DSM) approaches
that maintain stability as well as power quality, and further allow more
integration of renewable energy sources.
