Extracting flexibility of heterogeneous deferrable loads via polytopic projection approximation

This paper develops a novel approach to extract the aggregate flexibility of deferrable loads with heterogeneous parameters via polytopic projection approximation. First, an exact characterization of their aggregate flexibility is derived analytically, which in general contains exponentially many inequality constraints with respect to the number of loads. In order to have a tractable solution, we develop a numerical algorithm that gives a sufficient approximation of the exact aggregate flexibility. Geometrically, the flexibility of each individual load is a polytope and their aggregation is the Minkowski sum of these polytopes. Our method is motivated by an alternative interpretation of the Minkowski sum as a projection operation. The aggregate flexibility can be viewed as the projection of a high-dimensional polytope onto the subspace representing the aggregate power. We formulate a robust optimization problem to optimally approximate the polytopic projection with respect to the homothets of a given polytope. To enable efficient and parallel computation of the aggregate flexibility for a large number of loads, a muti-stage aggregation strategy is proposed. Finally, an energy arbitrage problem is solved to demonstrate the effectiveness of the proposed method.