Smoothing Property of Load Variation Promotes Finding Global Solutions of Time-Varying Optimal Power Flow

Abstract

This article analyzes solution trajectories for optimal power flow (OPF) with time-varying load. Despite its nonlinearity, time-varying OPF is commonly solved every 5–15 min using local-search algorithms. Failing to obtain the globally optimal solution of power optimization problems jeopardizes the grid's reliability and causes financial and environmental issues. The objective of this article is to address this problem by understanding the optimality behavior of OPF solution trajectories. An empirical study on California data shows that, with enough variation in the data, local search methods can solve OPF to global optimality, even if the problem has many local minima. To explain this phenomenon, we introduce a backward mapping that relates the time-varying OPF's global solution at a given time to a set of desirable initial points. We show that this mapping could act as a stochastic gradient ascent algorithm on an implicitly convexified formulation of OPF, justifying the escape of poor solutions over time. This work is the first to mathematically explain how temporal data variation affects the complexity of solving power operational problems.