  Coordinated aggregation of a large population of thermostatically controlled
loads (TCLs) presents a great potential to provide various ancillary services
to the grid. One of the key challenges of integrating TCLs into system level
operation and control is developing a simple and portable model to accurately
capture their aggregate flexibility. In this paper, we propose a geometric
approach to model the aggregate flexibility of TCLs. We show that the set of
admissible power profiles of an individual TCL is a polytope, and their
aggregate flexibility is the Minkowski sum of the individual polytopes. In
order to represent their aggregate flexibility in an intuitive way and achieve
a tractable approximation, we develop optimization-based algorithms to
approximate the polytopes by the homothets of a given convex set. As a special
application, this set is chosen as a \emph{virtual battery model} and the
corresponding optimal approximations are solved efficiently by equivalent
linear programming problems. Numerical results show that our algorithms yield
significant improvement in characterizing the aggregate flexibility over
existing modeling methods. We also conduct case studies to demonstrate the
efficacy of our approaches by coordinating TCLs to track a frequency regulation
signal from the Pennsylvania-New Jersey-Maryland (PJM) Interconnection.
