Network-Constrained Stackelberg Game for Pricing Demand Flexibility in Power Distribution Systems

We propose a network-constrained Stackelberg game framework to set energy prices for flexible consumers in a distribution grid. In this set-up, an aggregator acts as the leader, setting energy prices for each node, and price-responsive consumers are the followers, adjusting their demand according to the price charged. We show that this problem has an equilibrium in which the optimal demands can be written as a function of the Lagrange multipliers of the problem. For each node, voltage and current shadow costs have a cumulative effect that depends both on the upstream path to the substation and on the downstream demand level. We compare the Stackelberg solution to a centralized approach which maximizes social welfare. Our analysis reveals that, although the system-level optimal demand is higher in the centralized case, some individual nodes have higher consumption in the Stackelberg game. This counter-intuitive result cannot be observed in network-free formulations commonly adopted in game-theoretic works on demand-side management, where a centralized approach benefits every individual consumer. Numerical studies on an IEEE 123-bus feeder provide a system-level and a node-level analysis of this problem, highlighting the effect of network constraints on the optimal demands, and comparing the Stackelberg and the centralized solutions.