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Global Optimization of Gas Transportation and Storage: Convex Hull Characterizations and Relaxations

Bahar Cennet Okumusoglu, Burak Kocuk

TL;DR

This paper tackles the global optimization of gas transportation and storage, a nonconvex MINLP arising from gas physics and discrete control decisions. It develops a conic-relaxation framework built on exact convex hulls for pipes (SOC-representable) and a complete convex hull characterization for compressors (power-cone representable), complemented by a practical SOC outer-approximation. An algorithmic framework using these convexifications yields near-global solutions and strong warm-starts, demonstrated via extensive GasLib computational experiments that outperform a state-of-the-art global solver in many cases. The work provides scalable, high-accuracy tools for operators to optimize gas networks with storage and active elements, with potential extensions to uncertainty and integrated energy systems.

Abstract

Gas transportation and storage has become one of the most relevant and important optimization problems in energy systems. This problem inherently includes highly nonlinear and nonconvex aspects due to gas physics, and discrete aspects due to the control decisions of active network elements. Obtaining even locally optimal solutions for this problem presents significant mathematical and computational challenges for system operators. In this paper, we formulate the gas transportation and storage problem as a nonconvex mixed-integer nonlinear program (MINLP) through disjunctions on the flow directions. Moreover, we study the nonconvex sets induced by gas physics and propose mixed-integer second-order cone programming relaxations for the nonconvex MINLP problem. The proposed relaxations are based on the convex hull representations of two nonconvex sets: Firstly, we give the convex hull representation of the nonconvex set for pipes and show that it is second-order cone representable. Secondly, we also give a complete characterization of the extreme points of the nonconvex set for compressors and show that the convex hull of the extreme points is power cone representable. Moreover, for practical applications, we propose a second-order cone outer-approximation for the nonconvex set for compressors. To obtain (near) globally optimal solutions, we develop an algorithmic framework based on our convex hull results. We evaluate our framework through extensive computational experiments on various GasLib networks in comparison with the convex relaxations from the literature and a state-of-the-art global solver. Our results highlight the computational efficiency and convergence performance of our convex relaxation method compared to other methods. Moreover, our method also consistently provides (near) global solutions as well as high-quality warm-starting points for local solvers.

Global Optimization of Gas Transportation and Storage: Convex Hull Characterizations and Relaxations

TL;DR

This paper tackles the global optimization of gas transportation and storage, a nonconvex MINLP arising from gas physics and discrete control decisions. It develops a conic-relaxation framework built on exact convex hulls for pipes (SOC-representable) and a complete convex hull characterization for compressors (power-cone representable), complemented by a practical SOC outer-approximation. An algorithmic framework using these convexifications yields near-global solutions and strong warm-starts, demonstrated via extensive GasLib computational experiments that outperform a state-of-the-art global solver in many cases. The work provides scalable, high-accuracy tools for operators to optimize gas networks with storage and active elements, with potential extensions to uncertainty and integrated energy systems.

Abstract

Gas transportation and storage has become one of the most relevant and important optimization problems in energy systems. This problem inherently includes highly nonlinear and nonconvex aspects due to gas physics, and discrete aspects due to the control decisions of active network elements. Obtaining even locally optimal solutions for this problem presents significant mathematical and computational challenges for system operators. In this paper, we formulate the gas transportation and storage problem as a nonconvex mixed-integer nonlinear program (MINLP) through disjunctions on the flow directions. Moreover, we study the nonconvex sets induced by gas physics and propose mixed-integer second-order cone programming relaxations for the nonconvex MINLP problem. The proposed relaxations are based on the convex hull representations of two nonconvex sets: Firstly, we give the convex hull representation of the nonconvex set for pipes and show that it is second-order cone representable. Secondly, we also give a complete characterization of the extreme points of the nonconvex set for compressors and show that the convex hull of the extreme points is power cone representable. Moreover, for practical applications, we propose a second-order cone outer-approximation for the nonconvex set for compressors. To obtain (near) globally optimal solutions, we develop an algorithmic framework based on our convex hull results. We evaluate our framework through extensive computational experiments on various GasLib networks in comparison with the convex relaxations from the literature and a state-of-the-art global solver. Our results highlight the computational efficiency and convergence performance of our convex relaxation method compared to other methods. Moreover, our method also consistently provides (near) global solutions as well as high-quality warm-starting points for local solvers.

Paper Structure

This paper contains 23 sections, 19 theorems, 38 equations, 3 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Our Contributions
  4. Problem Formulation
  5. Problem Setting
  6. Gas Network Modeling
  7. MINLP Formulation
  8. Main Results
  9. Notation
  10. Convexification of the feasible set of a pipe
  11. Second-order cone representable convex hull
  12. Outer-approximations via different convex sets
  13. Polyhedral outer-approximation
  14. Second-order cone outer-approximation
  15. Convexification of the feasible set of a compressor
  16. ...and 8 more sections

Key Result

Theorem 1

$\textup{conv}(\mathcal{X}_{ij})$ is SOCr.

Figures (3)

  • Figure 1: The GasLib-11 instance (figure is taken from burlacu2019). Here, $\mathcal{N}_{source} = \{\text{S1, S2, S3}\}$, $\mathcal{N}_{sink} = \{\text{T1, T2, T3}\}$, $\mathcal{N}_{inner} = \{\text{N1, N2, N3, N4, N5}\}$, $\mathcal{C} = \{\text{Cm1, Cm2}\}$, $\mathcal{R} = \{\text{Vl1}\}$, $\mathcal{P}$ is the set of unlabeled arcs.
  • Figure 2: Polyhedral outer-approximation of $\mathcal{X}_{ij}$ in the $(f_{ij}, \pi_i - \pi_j)$-space.
  • Figure 3: The illustrations of the second-order cone representable outer-approximations.

Theorems & Definitions (27)

  • Theorem 1
  • Proposition 1
  • proof
  • Proposition 2
  • proof : Proof of Theorem \ref{['theorem:convexHull_pipes']}
  • Remark 1
  • Theorem 2
  • Lemma 1
  • proof
  • Proposition 3
  • ...and 17 more