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Adjustable Robust Nonlinear Network Design Without Controllable Elements under Load Scenario Uncertainties

Johannes Thürauf, Julia Grübel, Martin Schmidt

TL;DR

The work tackles robust network design under load scenario uncertainties for nonlinear potential-based flows by employing adjustable robust optimization. It proves that robust feasibility for a given expansion decision can be certified by solving a finite set of nonlinear subproblems that identify worst-case load scenarios, enabling an exact adversarial algorithm that terminates in finite iterations. Computational experiments on GasLib networks (GasLib-40, GasLib-60, and GasLib-135-derived) show that only a small number of worst-case scenarios are needed in practice and that enhanced relaxations and acyclic inequalities substantially speed up solution times. The approach generalizes to various utility networks (gas, hydrogen, water) and provides a practical, controllable-element-free framework for robust expansion planning.

Abstract

We study network design problems for nonlinear and nonconvex flow models without controllable elements under load scenario uncertainties, i.e., under uncertain injections and withdrawals. To this end, we apply the concept of adjustable robust optimization to compute a network design that admits a feasible transport for all, possibly infinitely many, load scenarios within a given uncertainty set. For solving the corresponding adjustable robust mixed-integer nonlinear optimization problem, we show that a given network design is robust feasible, i.e., it admits a feasible transport for all load scenario uncertainties, if and only if a finite number of worst-case load scenarios can be routed through the network. We compute these worst-case scenarios by solving polynomially many nonlinear optimization problems. Embedding this result for robust feasibility in an adversarial approach leads to an exact algorithm that computes an optimal robust network design in a finite number of iterations. Since all of the results are valid for general potential-based flows, the approach can be applied to different utility networks such as gas, hydrogen, or water networks. We finally demonstrate the applicability of the method by computing robust gas networks that are protected from future load fluctuations.

Adjustable Robust Nonlinear Network Design Without Controllable Elements under Load Scenario Uncertainties

TL;DR

The work tackles robust network design under load scenario uncertainties for nonlinear potential-based flows by employing adjustable robust optimization. It proves that robust feasibility for a given expansion decision can be certified by solving a finite set of nonlinear subproblems that identify worst-case load scenarios, enabling an exact adversarial algorithm that terminates in finite iterations. Computational experiments on GasLib networks (GasLib-40, GasLib-60, and GasLib-135-derived) show that only a small number of worst-case scenarios are needed in practice and that enhanced relaxations and acyclic inequalities substantially speed up solution times. The approach generalizes to various utility networks (gas, hydrogen, water) and provides a practical, controllable-element-free framework for robust expansion planning.

Abstract

We study network design problems for nonlinear and nonconvex flow models without controllable elements under load scenario uncertainties, i.e., under uncertain injections and withdrawals. To this end, we apply the concept of adjustable robust optimization to compute a network design that admits a feasible transport for all, possibly infinitely many, load scenarios within a given uncertainty set. For solving the corresponding adjustable robust mixed-integer nonlinear optimization problem, we show that a given network design is robust feasible, i.e., it admits a feasible transport for all load scenario uncertainties, if and only if a finite number of worst-case load scenarios can be routed through the network. We compute these worst-case scenarios by solving polynomially many nonlinear optimization problems. Embedding this result for robust feasibility in an adversarial approach leads to an exact algorithm that computes an optimal robust network design in a finite number of iterations. Since all of the results are valid for general potential-based flows, the approach can be applied to different utility networks such as gas, hydrogen, or water networks. We finally demonstrate the applicability of the method by computing robust gas networks that are protected from future load fluctuations.
Paper Structure (20 sections, 6 theorems, 49 equations, 2 figures, 7 tables, 1 algorithm)

This paper contains 20 sections, 6 theorems, 49 equations, 2 figures, 7 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Potential-Based Networks
  4. Robust Network Design
  5. Exact Adversarial Approach
  6. Enhanced Solution Techniques
  7. Reducing the Number of Adversarial Problems
  8. Computing Lower Bounds
  9. Acyclic Inequalities
  10. How Many Scenarios Do We Need?
  11. Computational Study
  12. Gas Networks
  13. Uncertainty Modeling
  14. Algorithmic and Computational Setup
  15. Numerical Results
  16. ...and 5 more sections

Key Result

Lemma 1

Let $x \in X$ be fixed and let $G\xspace'(x)=(V\xspace, A\xspace(x))$ be the expanded graph. Further, we assume that $G\xspace'(x)$ is weakly connected. For a fixed load scenario $d \in U$, there are potentials $\pi'$ and unique flows $q\xspace$ such that the set of feasible points that satisfies Co where $\mathds{1}\xspace$ is a vector of ones in appropriate dimension.

Figures (2)

  • Figure 1: Academic network: existing arcs in solid black, expanded arcs in dashed red.
  • Figure 2: Performance profile for the runtimes regarding all instances that are solved by at least one approach.

Theorems & Definitions (16)

  • Definition 1
  • Remark 1
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Lemma 2
  • proof
  • Theorem 2
  • proof
  • ...and 6 more