Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Tight MIP Formulations for Optimal Operation and Investment of Storage Including Reserves

Maaike B. Elgersma, Germán Morales-España, Karen I. Aardal, Niina Helistö, Juha Kiviluoma, Mathijs M. de Weerdt

TL;DR

The paper addresses the scalability and accuracy challenges of optimizing energy storage investments and operations with reserves in large-scale power systems. It derives the convex hull of feasible solutions for storage operation in a single time period and develops tight MIP formulations—TO-MIP for operation, TOR-MIP for operation with reserves, and TIR-MIP for investment with reserves—whose LP-relaxations coincide with this convex hull, ensuring no tighter MIP or better LP approximation exists for one period. The authors validate the approach through unit-commitment and transmission-expansion planning case studies, showing that the tightened LP relaxations more effectively prevent simultaneous charging and discharging and can reduce solver time in multi-period problems. Overall, the work enhances the reliability and speed of large-scale energy system models that include storage and reserves, while highlighting open questions for extending convex-hull results to multi-period settings.

Abstract

Fast and accurate large-scale energy system models are needed to investigate the potential of storage to complement the fluctuating energy production of renewable energy systems. However, standard Mixed-Integer Programming (MIP) models that describe optimal investment and operation of these storage units, including the optional capacity to provide up/down reserves, do not scale well. To improve scalability, the integrality constraints are often relaxed, resulting in Linear Programming (LP) relaxations that allow simultaneous charging and discharging, while this is not feasible in practice. To address this, we derive the convex hull of the solutions for the optimal operation of storage for one time period, as well as for problems including investments and reserves, guaranteeing that no tighter MIP formulation or better LP approximation exists for one time period. When incorporating this convex hull into a multi-period formulation and including it in large-scale energy system models, the improved LP relaxations can better prevent simultaneous charging and discharging, and the tighter MIP could positively affect the solving time. We demonstrate this with illustrative case studies of a unit commitment problem and a transmission expansion planning problem.

Tight MIP Formulations for Optimal Operation and Investment of Storage Including Reserves

TL;DR

The paper addresses the scalability and accuracy challenges of optimizing energy storage investments and operations with reserves in large-scale power systems. It derives the convex hull of feasible solutions for storage operation in a single time period and develops tight MIP formulations—TO-MIP for operation, TOR-MIP for operation with reserves, and TIR-MIP for investment with reserves—whose LP-relaxations coincide with this convex hull, ensuring no tighter MIP or better LP approximation exists for one period. The authors validate the approach through unit-commitment and transmission-expansion planning case studies, showing that the tightened LP relaxations more effectively prevent simultaneous charging and discharging and can reduce solver time in multi-period problems. Overall, the work enhances the reliability and speed of large-scale energy system models that include storage and reserves, while highlighting open questions for extending convex-hull results to multi-period settings.

Abstract

Fast and accurate large-scale energy system models are needed to investigate the potential of storage to complement the fluctuating energy production of renewable energy systems. However, standard Mixed-Integer Programming (MIP) models that describe optimal investment and operation of these storage units, including the optional capacity to provide up/down reserves, do not scale well. To improve scalability, the integrality constraints are often relaxed, resulting in Linear Programming (LP) relaxations that allow simultaneous charging and discharging, while this is not feasible in practice. To address this, we derive the convex hull of the solutions for the optimal operation of storage for one time period, as well as for problems including investments and reserves, guaranteeing that no tighter MIP formulation or better LP approximation exists for one time period. When incorporating this convex hull into a multi-period formulation and including it in large-scale energy system models, the improved LP relaxations can better prevent simultaneous charging and discharging, and the tighter MIP could positively affect the solving time. We demonstrate this with illustrative case studies of a unit commitment problem and a transmission expansion planning problem.

Paper Structure

This paper contains 24 sections, 3 theorems, 15 equations, 2 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Motivation
  3. Background & Literature Review
  4. Contributions
  5. Background: common storage models
  6. Modeling Storage Operation Problems
  7. Modeling Storage Operation Problems Including Reserves
  8. Modeling Storage Investment Problems Including Reserves
  9. Tight storage models
  10. Method: Obtaining the Convex Hull
  11. Tight MIP Formulation of Storage Operation
  12. Tight MIP Formulation of Storage Including Reserves
  13. Tight MIP Formulation of Storage Investment Including Reserves
  14. Illustrative case studies
  15. LP Storage Operation incl. Reserves in Unit Commitment Case Study
  16. ...and 9 more sections

Key Result

Theorem 1

The LP-relaxation of the eq:chso model (eq:chso) describes the convex hull of the solutions to the eq:bso model for one time period.

Figures (2)

  • Figure 1: The up and down reserve capacities when discharging, bounded by the charging and discharging capacities, as modeled by different MIP formulations.
  • Figure 2: The up and down reserve capacities when discharging, bounded by the total storage capacity, as modeled by different MIP formulations.

Theorems & Definitions (5)

  • Example 1
  • Theorem 1
  • Example 2
  • Theorem 2
  • Theorem 3