This dissertation delivers new theoretical and computational frameworks for systematically modeling the aggregate dynamics of large-scale cyber-physical systems. Particularly focused on the hierarchical demand response management system in smart grid, we develop both control-oriented and optimization-oriented aggregate models for coordinating a large population of responsive loads, including both thermostatically controlled loads (TCLs) and deferrable loads.

For control-oriented modeling, we develop a unified stochastic hybrid system (SHS) framework to derive the partial differential equations (PDE) that characterize the dynamical evolution of the load distribution. A deterministic hybrid system is proposed for modeling general individual responsive load. An SHS is proposed for modeling the population dynamics after accounting for different uncertainties.Existing literature usually derives the PDE based on the physical principles and specifies the associated boundary conditions heuristically. Our method is based on the adjoint relation between the differential operator associated with the PDE and the extended generator of the SHS process. In particular, it enables us to determine the PDE boundary conditions directly from the boundary condition satisfied by the SHS generator. The obtained PDE model systematically generalizes many existing aggregate models. It is fundamentally important for designing various aggregate control strategies.

The optimization-oriented modeling is to characterize the constraint sets satisfied by the aggregate load power, also known as the aggregate flexibility. We show that the individual power flexibility can be modeled by a polytope and the aggregate flexibility is the Minkowski sum of the individual flexibility polytopes. Exact Minkowski sum of these polytopes is computationally prohibitive. Therefore, we develop optimization-based algorithms to approximate the aggregate flexibility. For TCLs, we propose to approximate individual flexibility polytopes using homothetic polytopes. This leads to efficient algorithms to approximate the aggregate flexibility as the Minkowski sum of homothetic polytopes reduces to just the ordinary vector sum. We also provide extensive simulation to show the significant performance improvement over the existing modeling methods.

Aggregate flexibility modeling of deferrable loads is more challenging than that of TCLs due to the additional timing constraints. A sufficient characterization (i.e., inner approximation) of the aggregate flexibility with controllable model complexity has not been proposed in the literature before. We interpret the Minkowski sum as a projection operation and develop an algorithm to inner approximate the projection optimally using homothetic polytopes. The algorithm can be executed efficiently in a distributed way. An energy arbitrage example is further employed to demonstrate the application of the proposed algorithm.