A Geometric Approach to Virtual Battery Modeling of Thermostatically Controlled Loads

A large population of thermostatically controlled loads (TCLs) can be coordinated to provide various ancillary services to the grid. Effective operation and coordination of TCLs requires an accurate and simple model to capture their aggregate power flexibility. One appealing approach is the virtual battery method, which models the aggregate flexibility offered by a collection of TCLs as a simple scalar dynamical system that resembles the dynamic behavior of a battery with limits on the energy capacity and the output power. In this paper, we propose a novel geometric approach to design a virtual battery model for a given TCL population. We adopt a discrete time formulation of the individual TCL dynamics over a finite time-horizon. This allows for a clear geometric interpretation of the individual flexibility, which is shown to be a polytope. The aggregate flexibility can be represented by the Minkowski sum of the individual polytopes. In view of the special structure of these polytopes, we further propose two design methods to optimally extract each TCL's flexibilities, both via approximating the polytope with respect to the scaling and translation of a given polytope. The optimal scaling and translation factors are solved efficiently from equivalent linear programming problems. The aggregate flexibility is approximated by a virtual battery model obtained from an easy calculation of the Minkowski sum of those obtained similar polytopes. Simulation results show significant improvement and superior performance over the existing methods.