Distributionally robust risk-constrained optimal power flow using moment and unimodality information

As we incorporate more random renewable energy into the power grid, power system operators need to ensure physical constraints, such as transmission line limits, are not violated despite uncertainty. Risk-constrained optimal power flow (RCOPF) based on the Conditional Value-at-Risk (CVaR) is a convenient modeling tool, ensuring that these constraints are satisfied with a high probability (e.g., 95%). However, in practice, it is often difficult to perfectly estimate the joint probability distribution of all uncertain variables, including renewable energy production and load consumption. In this paper, we propose a distributionally robust RCOPF approach by considering all possible probability distributions that share the same moment (e.g., mean and covariance) and unimodality properties. Moment and unimodality information can be estimated based on historical data, and so the proposed approach can be applied in a data-driven manner. In view of the computational challenges, we derive a conservative and a relaxed approximation of the problem. We reformulate these approximations as semidefinite programs (SDPs) facilitating the use of highly efficient off-the-shelf optimization solvers (e.g., CVX). We demonstrate the proposed approach based on a modified IEEE 9-bus power network.