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Aggregate load modeling is of fundamental importance for systematic analysisand design of various demand response strategies. Instead of keeping track ofthe trajectories of individual loads, the aggregate modeling problem focuses oncharacterizing the density evolution of the load population. Most existingmodels are only applicable to Thermostatically Controlled Loads (TCL) withfirst-order linear dynamics. This paper develops a unified aggregate modelingapproach that can be used for general TCLs as well as deferrable loads. Wepropose a deterministic hybrid system model to describe individual loaddynamics under demand response rules, and develop a general stochastic hybridsystem (SHS) model to capture the population dynamics. We also derive a set ofpartial differential equations (PDE) that governs the probability densityevolution of the SHS. Our results cannot be obtained using the exiting SHStools in the literature as the proposed SHS model involves both random anddeterministic switchings with general switching surfaces in multi-dimensionaldomains. The derived PDE model includes many existing aggregate load modelingresults as special cases and can be used in many other realistic modelingscenarios that have not been studied in the literature.

A Unified Stochastic Hybrid System Approach to Aggregate Modeling of Responsive Loads