Three Solution Approaches to Stochastic Multi-Period AC Optimal Power Flow in Active Distribution Systems

This paper proposes three novel solution approaches (A1, A2, A3) to solve stochastic multi-period AC optimal power flow (S-MP-OPF) for day-ahead flexibility procurement from distributed energy resources (DER) in active distribution systems (ADSs). The S-MP-OPF is a mixed-integer nonlinear programming (MINLP) problem due to the binary variables modeling the operation of storage devices and flexible loads. The proposed three approaches have a shared common first step, which resorts to a new mixed-integer linear programming (MILP) model approximation of the S-MP-OPF problem. The MILP model employs second-order Taylor series expansion of trigonometric terms and formulates the linear approximations relying on variables such as square of voltage magnitude and voltage angle difference. This first step serves also the purpose of fixing the binary variables to the values computed by the MILP problem. Then, the approach A1 only checks the AC feasibility of MILP solution while approaches A2 and A3 further optimize continuous variables. Specifically, the sophisticated heuristic approach A2 employs sequential linear programming and AC power flow while the approach A3 models and solves directly the remaining nonlinear programming (NLP) problem. The performances of these approaches, a benchmark MINLP solver, and a state-of-the-art method are thoroughly compared in three radial or weakly meshed ADSs of 34, 31, and 191 nodes, respectively. The numerical results indicate that, albeit the approach A2 performs best overall, these approaches present distinct accuracy vs speed trade-offs, which make them suitable to problems of different sizes and diverse accuracy or speed requirements.