In this paper, we present a real-time trading framework for distribution networks where a rational aggregator is identified as a broker by contracting with individual demands and dealing with the distribution company. Demand response capability is characterized by the coexistence of elastic and inelastic demand components. A one-leader multi-follower bi-level model is proposed to derive the procurement strategies, i.e., the upper-level problem intends to maximize the profit of the proactive distribution company, while the lower-level expresses the profit maximization per rational aggregator. The proposed model is then transformed into a solvable mathematical program with equilibrium constraints through a primal-dual approach. A modified 33-bus distribution network is utilized to demonstrate the effectiveness of the proposed model.
We study the equilibria reached by strategic producers in a pool-based network-constrained electricity market. The behavior of each producer is modeled by a mathematical program with equilibrium constraints (MPEC) whose objective is maximizing profit and whose complementarity constraints describe market clearing. The joint solution of all these MPECs constitutes an equilibrium problem with equilibrium constraints (EPEC). The equilibria associated with the EPEC are analyzed by solving the strong stationarity conditions of all MPECs, which can be linearized without approximation by mixed-integer linear programming (MILP) techniques. The resulting mixed-integer linear conditions can be reformulated as an optimization problem that allows establishing diverse objectives to differentiate among alternative equilibria.
We prove that groups acting geometrically on delta-quasiconvex spaces contain no essential Baumslag-Solitar quotients as subgroups. This implies that they are translation discrete, meaning that the translation numbers of their nontorsion elements are bounded away from zero.