Generalized Convex Hull Pricing for the AC Optimal Power Flow Problem

We study an abstract myopic electricity market subject to general nonconvex power flow models and convex generator models (without discrete commitment variables) in a setting where there may be no market equilibrium that guarantees revenue adequacy of the independent system operator. A generalized version of convex hull prices (CHPs) are proposed and are defined to solve a novel multiobjective minimum uplift problem that captures the tradeoff between generator side-payments and potential congestion revenue shortfall (PCRS). A convex primal counterpart of this multiobjective minimum uplift problem, called the primal CHP problem, is formulated in terms of the convex hull of the set of feasible net power injections. Indeed the term convex hull price derives from the result that the CHPs are equivalent to the optimal Lagrange multipliers of the primal CHP problem. However, depending on the chosen model of the transmission network, the convex hull of the feasible set of net power injections may be intractable to evaluate. In this case, CHPs are approximated using state-of-the-art convex relaxations that are efficiently solvable. This is the first proposed method of approximating CHPs in polynomial-time that is general enough to accommodate the nonlinear transmission constraints in the AC optimal power flow (OPF) problem. In our abstract myopic market setting, we show that tight relaxations of the AC OPF problem can be used to effectively approximate CHPs that decrease potential congestion revenue shortfall significantly with little effect to side-payments.