A Model for Aggregate Demand Response of an Ensemble of Variable-Speed Air Conditioners

A substantial amount of research in recent years has investigated the potential of air conditioners (ACs) in providing demand-side services in the electricity grid. ACs with a variable-speed compressor, like various other thermostatically controlled loads (TCLs), are rapidly becoming widespread in many markets today due to their high efficiency and flexibility. However, existing methods for aggregate TCL prediction and control are still largely developed around models for air conditioners with fixed-speed (on/off) compressors. Building on a recent simple dynamic model for a variable-output compressor air conditioner (VOCAC), this paper develops a mean-field approach for analysis and control of the aggregate demand of ensembles of such TCLs. The main contribution of the paper is to rigorously show that the aggregate demand response of a heterogeneous ensemble of VOCACs with log-normally distributed thermal capacitance can be approximated by that of a single VOCAC model with a capacitance equal to the mean value of the distribution, under the assumption that all other parameters are homogeneously distributed. Analytic bounds for the approximation error are derived based on a sharp version of Jensen's inequality. Numerical simulation results illustrate and validate the proposed model performance against that of the aggregate demand response of a numerically simulated heterogeneous ensemble of 1000 VOCACs to broadcast step changes in thermostat set-points and enforced capping of electricity demand.