Optimally Managing the Impacts of Convergence Tolerance for Distributed Optimal Power Flow

The future power grid may rely on distributed optimization to determine the set-points for huge numbers of distributed energy resources. There has been significant work on applying distributed algorithms to optimal power flow (OPF) problems, which require separate controllers to agree on shared boundary variable values. Although controllers find feasible solutions to subproblems which optimize power flow within their region of the network, they must reach consensus on shared variables at tie-lines between regions to find a system-wide feasible solution. While running the distributed algorithm, controllers iteratively compute local subproblem solutions and communicate shared variable information until the mismatches between shared variables fall below a user-selected primal convergence tolerance. Looser tolerances for the mismatches in these shared variables generally yield faster convergence at the expense of exacerbating constraint violations, but there is little quantitative understanding of how the convergence tolerance affects solution quality. To address this gap, we first quantify how convergence tolerance impacts constraint violations when the distributed OPF solution is applied to the power system. Using insights from this analysis, we then develop a bound tightening algorithm which guarantees that operating points from distributed OPF algorithms will not result in violations despite the possibility of shared variable mismatches within the convergence tolerance. We also explore how bounding the cumulative shared variable mismatches can prevent unnecessary conservativeness in the bound tightening. While ensuring feasibility, the proposed approach enables control of the trade-off between computational speed, which improves with looser convergence tolerance, and distributed OPF solution cost, which degrades with looser convergence tolerance due to tightened constraints.