Multi-stage Stochastic Alternating Current Optimal Power Flow with Storage: Bounding the Relaxation Gap

TL;DR

This paper develops a multistage stochastic model for AC optimal power flow in radial distribution networks with renewable energy and storage, providing conditions for convex relaxation to be tight and demonstrating its effectiveness through realistic case studies.

Contribution

It introduces a priori conditions for zero relaxation gap in multistage stochastic AC OPF with storage, enabling convex optimization solutions for complex power systems.

Findings

Relaxation gap can be null with light reverse power flows or sufficient storage.

A posteriori bounds on the relaxation gap are provided.

Results are validated on realistic distribution network scenarios.

Abstract

We propose a generic multistage stochastic model for the Alternating Current Optimal Power Flow (AC OPF) problem for radial distribution networks, to account for the random electricity production of renewable energy sources and dynamic constraints of storage systems. We consider single-phase radial networks. Radial three-phase balanced networks (medium-voltage distribution networks typically have this structure) reduce to the former case. This induces a large scale optimization problem, which, given the non-convex nature of the AC OPF, is generally challenging to solve to global optimality. We derive a priori conditions guaranteeing a vanishing relaxation gap for the multi-stage AC OPF problem, which can thus be solved using convex optimization algorithms. We also give an a posteriori upper bound on the relaxation gap. In particular, we show that a null or low relaxation gap may be expected for applications with light reverse power flows or if sufficient storage capacities with low cost are available. Then, we discuss the validity of our results when incorporating voltage regulation devices. Finally, we illustrate our results on problems of planning of a realistic distribution feeder with distributed solar production and storage systems. Scenario trees for solar production are constructed from a stochastic model, by a quantile-based algorithm.