New Insights From the Shapley-Folkman Lemma on Dispatchable Demand in Energy Markets

In this paper we consider the aggregation of common convex and non-convex individual Demand Response (DR) models for responsive loads, and apply the Shapley-Folkman (SF) lemma to show that such an aggregate is approximately convex in its action space and cost, and strictly convex under mild conditions. We then discuss how reduced order convex aggregate models can be passed on to System Operators for inclusion in economic and operational models, including a novel polytope approximation as well as an ellipsoidal model derived in a distributed fashion. Numerical simulations confirm our application of the Shapley-Folkman lemma and show that the new reduced order models outperform conventional virtual generator approximations.