Aggregate load modeling for Demand Response via the Minkowski sum

Aggregations of flexible loads can provide several power system services through demand response programs, for example load shifting and curtailment. The capabilities of demand response should therefore be represented in system operators' planning and operational routines. However, incorporating models of every load in an aggregation into these routines could compromise their tractability by adding exorbitant numbers of new variables and constraints. We propose a novel approximation for concisely representing the capabilities of a heterogeneous aggregation of flexible loads. We assume that each load is mathematically described by a convex polytope, i.e., a set of linear constraints. We discuss the polytopic formulation of many classes of loads including deferrable loads, thermostatically controlled loads, and generic energy storage. The set-wise sum of the loads is the Minkowski sum, which is in general computationally intractable. In this thesis, we develop a new outer approximation of the Minkowski sum. The new approximation is applicable for linear constraints, is easily computable, and only uses one variable per time period corresponding to the aggregation's net power usage. We prove that the approximation is exact when applied to deferrable loads without power constraints and loads modelled for two time periods only. Additionally, numerical results indicate that the approximation is accurate for broad classes of loads. We also develop a tightening procedure to further improve upon the accuracy of the approximation. Following, we extend the above approximation to semidefinite constraints and second-order cone constraints. The approximation is extended to loads that do not have matching constraints via the use of the Gershgorin circle theorem. This extension allows for the modelling of loads with apparent power constraints or for stochastic loads modelled with chance constraints. Numerical results are shown for the case where load charging efficiency and total energy demand are taken as stochastic quantities. Finally, we consider the problem of finding inner approximation to load models. We use an ellipsoidal projection technique to find the maximum inscribed ellipsoid for the Minkowski sum of several loads. We perform numerical simulations to illustrate the technique.Ph.D