A scenario-based robust distribution expansion planning under ellipsoidal uncertainty set using second-order cone programming

This paper intends to propose a new mathematical framework for distribution expansion planning (DEP) based on which the uncertainties associated with electric demand and wind production are modeled through several plausible ellipsoidal uncertainty sets. In this regard, a hybrid model combining stochastic programming and robust optimization is constructed. Data related to the demand-wind scenarios are first divided into clusters of similar elements using an existing clustering technique. Then, an ellipsoidal uncertainty set is created corre-sponding to each of these generated clusters based on the theory of minimum volume covering ellipsoid (MVCE) using the Khachiyan algorithm (KA). The hybrid robust/stochastic scheme is formulated as a two-stage tri-level min-max-min optimization problem in which the convex conic relaxation of AC power flow is used to represent the electric distribution network. The multi-uncertainty-set-based model is then solved by employing the classic column-and-constraint-generation (C&CG) technique which guarantees the convergence to the optimal solution in a limited number of iterations. In this regard, a master problem and several subproblems related to each scenario will be solved, both of which are second-order cone programs. Numerical simulations reveal the su-periority of the hybrid model compared to the existing ones.