Two-stage stochastic optimization for distribution networks considering demand response uncertainties

Abstract

High penetration of distributed energy resources and uncertain loads creates financial and operational challenges for distribution networks. To address these challenges, this paper proposes a risk-averse, two-stage stochastic coordination framework for a distribution network interacting with the main grid. The day-ahead stage determines the procurement schedule, whereas the real-time stage corrects deviations through real-time market transactions and demand response (DR) mobilization. DR availability is explicitly modeled as scenario-dependent uncertainty. The physical network is formulated using a second-order cone programming (SOCP)-relaxed branch-flow model, which enables an exact mathematical decomposition of distribution locational marginal prices (DLMPs). A Conditional Value-at-Risk (CVaR) metric is integrated to manage extreme tail-risk penalties. Simulations on the IEEE 33-bus system show that the proposed framework reduces the expected daily power shortage by over 10% compared with the risk-neutral baseline. Stronger risk aversion leads to more conservative day-ahead procurement. This forward over-procurement reduces expected real-time marginal scarcity, decreases the expected reliance on local DR during most hours, and dampens absolute DLMP levels at high-priced downstream nodes over a large portion of the operating horizon. These results indicate that tail-risk management reshapes both real-time flexibility utilization and localized economic signals in distribution networks.