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Distributionally Robust Optimization for a Resilient Transmission Grid During Geomagnetic Disturbances

Mowen Lu, Sandra D. Eksioglu, Scott J. Mason, Russell Bent, Harsha Nagarajan

TL;DR

A two-stage, distributionally robust (DR) optimization formulation that models uncertain GMDs and mitigates the effects of GICs on power systems through existing system controls and develops a decomposition framework for solving the problem.

Abstract

In recent years, there have been increasing concerns about the impacts of geomagnetic disturbances (GMDs) on electrical power systems. Geomagnetically-induced currents (GICs) can saturate transformers, induce hot-spot heating and increase reactive power losses. Unpredictable GMDs caused by solar storms can significantly increase the risk of transformer failure. In this paper, we develop a two-stage, distributionally robust (DR) optimization formulation that models uncertain GMDs and mitigates the effects of GICs on power systems through existing system controls (e.g., line switching, generator re-dispatch, and load shedding). This model assumes an ambiguity set of probability distributions for induced geo-electric fields which capture uncertain magnitudes and orientations of a GMD event. We employ state-of-the-art linear relaxation methods and reformulate the problem as a two-stage DR model. We use this formulation to develop a decomposition framework for solving the problem. We demonstrate the approach on the modified Epri21 system and show that the DR optimization method effectively handles prediction errors of GMD events.

Distributionally Robust Optimization for a Resilient Transmission Grid During Geomagnetic Disturbances

TL;DR

A two-stage, distributionally robust (DR) optimization formulation that models uncertain GMDs and mitigates the effects of GICs on power systems through existing system controls and develops a decomposition framework for solving the problem.

Abstract

In recent years, there have been increasing concerns about the impacts of geomagnetic disturbances (GMDs) on electrical power systems. Geomagnetically-induced currents (GICs) can saturate transformers, induce hot-spot heating and increase reactive power losses. Unpredictable GMDs caused by solar storms can significantly increase the risk of transformer failure. In this paper, we develop a two-stage, distributionally robust (DR) optimization formulation that models uncertain GMDs and mitigates the effects of GICs on power systems through existing system controls (e.g., line switching, generator re-dispatch, and load shedding). This model assumes an ambiguity set of probability distributions for induced geo-electric fields which capture uncertain magnitudes and orientations of a GMD event. We employ state-of-the-art linear relaxation methods and reformulate the problem as a two-stage DR model. We use this formulation to develop a decomposition framework for solving the problem. We demonstrate the approach on the modified Epri21 system and show that the DR optimization method effectively handles prediction errors of GMD events.

Paper Structure

This paper contains 30 sections, 3 theorems, 53 equations, 4 figures, 9 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Electric Power Model--AC
  4. Electric Power Model--GIC
  5. $\widetilde{\nu}_e^d$ calculation
  6. Transformer modeling
  7. GIC Modeling
  8. Electric Power Model--Coupled GIC and AC
  9. A Two-stage OTSGMD Model
  10. A Two-stage DR-OTSGMD Model with Uncertainty
  11. Convex Relaxations
  12. Quadratic Fuel Cost Function.
  13. AC Power Flow Physics.
  14. Line Thermal Limits.
  15. GIC-Associated Effects.
  16. ...and 15 more sections

Key Result

Lemma 2.1

Given that $\Omega'$ is a linear relaxation of $\Omega$ and $\Omega' \in \Omega$. Then, for any given first-stage decisions $\boldsymbol{y}$ and $\boldsymbol{\lambda}$, there exists an extreme point of $\Omega'$ which is an optimal (worst-case) solution of $\boldsymbol{\widetilde{\omega}} \in \Omega

Figures (4)

  • Figure 1: DC equivalent circuits for different types of transformers. $h$, $l$, and $n$ (blue font) represent high-side bus, low-side bus, and neutral bus, respectively.
  • Figure 2: Examples of feasible regions of $\widetilde{\nu}^E$ and $\widetilde{\nu}^N$. The geo-electric field $\vec{E}$ is formed by $\widetilde{\nu}^E$ and $\widetilde{\nu}^N$. In Fig. \ref{['fig:vd_WN']}, the magnitude of $\vec{E}$ equals $\overline{\nu}^M$ and angle relative to east is larger than $\frac{\pi}{2}$. In Fig. \ref{['fig:vd_EN']}, the magnitude of $\vec{E}$ is less than $\overline{\nu}^M$ and the angle is smaller than $\frac{\pi}{2}$.
  • Figure 3: A linear relaxation of support set $\Omega$. The blue shadow represents $\Omega'$ and $\boldsymbol{\omega}^*_i$ for $i \in \{1,2,..,5\}$ denote the vertices of $\Omega'$.
  • Figure 4: Cost comparisons among all cases.

Theorems & Definitions (6)

  • Lemma 2.1
  • proof
  • Proposition 1
  • proof
  • Proposition 2
  • proof