# The Price of Uncertainty: Chance-constrained OPF vs. In-hindsight OPF

## TL;DR

This paper compares chance-constrained optimal power flow (ccOPF) with in-hindsight OPF (hOPF), analyzing how uncertainty impacts their costs and constraints, and introduces measures to quantify their performance gap.

## Contribution

It provides theoretical conditions under which ccOPF matches hOPF and proposes using total variational distance to measure the performance gap.

## Key findings

- ccOPF can be equivalent to hOPF under certain conditions
- Total variational distance quantifies the performance gap
- Illustrative example demonstrates the theoretical results

## Abstract

The operation of power systems has become more challenging due to feed-in of volatile renewable energy sources. Chance-constrained optimal power flow (ccOPF) is one possibility to explicitly consider volatility via probabilistic uncertainties resulting in mean-optimal feedback policies. These policies are computed before knowledge of the realization of the uncertainty is available. On the other hand, the hypothetical case of computing the power injections knowing every realization beforehand---called in-hindsight OPF(hOPF)---cannot be outperformed w.r.t. costs and constraint satisfaction. In this paper, we investigate how ccOPF feedback relates to the full-information hOPF. To this end, we introduce different dimensions of the price of uncertainty. Using mild assumptions on the uncertainty we present sufficient conditions when ccOPF is identical to hOPF. We suggest using the total variational distance of probability densities to quantify the performance gap of hOPF and ccOPF. Finally, we draw upon a tutorial example to illustrate our results.