Exact constraint aggregation with applications to smart grids and resource distribution

Abstract

As hierarchical predictive control of large-scale distributed systems grow in complexity, it eventually becomes necessary to consider aggregation of lower-level units into larger groups of units that can be handled efficiently at higher levels in the hierarchy. When aggregating similar units in this manner, it is advantageous if the aggregation maintains a certain degree of genericity, since the higher-level algorithms can then be designed with a higher degree of modularity. To achieve this goal, however, it is not only necessary to examine aggregation of models of the underlying units, but also the accompanying constraints. Constraint sets for rate-and storage volumeconstrained units can often be represented as polytopes in high-dimensional Euclidean space; unfortunately, adding such polytopic sets in higher dimension than 2 has so far been considered a combinatorial problem. In this paper, we present a novel method for computing such polytopic constraint sets for integrating units, which achieves a much lower computational complexity than previous results. The concept is demonstrated via simulations of a smart grid control scenario.

A Geometric Approach to Aggregate Flexibility Modeling of Thermostatically Controlled Loads

“…Besides, the outer approximation cannot guarantee the feasibility of the aggregate power profile. The references [24]- [26] deals with the so-called resource polytopes, which arise from the flexibility modeling of deferrable loads, such as plugin electric vehicles (PEVs), dishwashers, among many others. In particular, the exact Minkowski sum of resource polytopes was considered in [24].…”

An Overview of Modeling Approaches Applied to Aggregation-Based Fleet Management and Integration of Plug-in Electric Vehicles †

“…Another approach that can be used for representing the aggregated flexibility more precisely is Minkowski summation [48][49][50]. Constraint sets for the charging rate and the battery storage volume of PEVs are represented as polytopes in high-dimensional Euclidean space.…”

“…Minkowski summation, formed by adding together the vector of each polytope, is derived to represent the aggregated flexibility in a more generic, mathematical manner. As shown in Figure 5, the flexibility of the two PEV batteries (IC1 and IC2), represented by their constraints, can be summed together to form the representation of the aggregated flexibility [49]. This modular approach supports both visualization of aggregated flexibility of PEVs and applications that use the aggregated flexibility to power system operations through charging management [51].…”

“…However, compared to using the lumped battery approach, the computational cost of Minkowski summation can be very high. Minkowski summation of two resource polytopes for a two-sample horizon (P and E refer to power and energy limitations, respectively) (adapted from [49]). …”

“…Minkowski summation of two resource polytopes for a two-sample horizon (P and E refer to power and energy limitations, respectively) (adapted from [49]). …”