Solving Min-Cost Concave Generalized Dynamic Flows and Approximating Dynamic Optimal Power Flows

Abstract: Assuming power travels instantaneously, can be steered by us, and is lost quadratically in each power line, the dynamic optimal power flow problem simplifies to a min-cost dynamic generalized flow with quadratic losses (MCDGFWQL) problem. As this is a special case of the min-cost concave dynamic generalized flow (MCCDGF) problem, we derive both general results for the MCCDGF problem and stronger results for the MCDGFWQL problem. Our main contribution is a fully polynomial-time approximation scheme for a mild restriction of the MCDGFWQL problem. We also implement and benchmark a slight modification of this algorithm, finding it to be efficient in practice, with a slightly superlinear and possibly subquadratic dependence of execution time on the edge count of the input graph. Our secondary contributions are the derivation of a reduction from the MCCDGF problem to a convex program and a corresponding sensitivity analysis of the MCCDGF problem. We also provide a brief comparison of our dynamic optimal power flow simplification and the classic direct current simplification and point out some interesting directions for future research.