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Scalable Multi-Level Optimization for Sequentially Cleared Energy Markets with a Case Study on Gas and Carbon Aware Unit Commitment

Yuxin Xia, Iacopo Savelli, Thomas Morstyn

TL;DR

The paper tackles the challenge of scalable optimization for Mixed-Integer Multi-Level problems with Sequential Followers (MIMLSF) in energy markets, where upper-level decisions must anticipate outcomes of lower-level sequential market clears. It introduces a lexicographic-weighted-sum single-level (SLP) approximation that asymptotically converges to the original MIMLSF as the scaling parameter $\gamma \to 1$, and develops a dedicated multi-level Benders decomposition that splits the complex Benders subproblem into two tractable pieces and further into $n$ subproblems. The authors apply this framework to a four-level Unit Commitment with Gas and Carbon Awareness (UCGCA) model in the Northeastern US, integrating electricity, natural gas, and carbon markets and enforcing bid-validity constraints to prevent unprofitable GFPP commitments. Computational results show that the specialized Benders approach reduces computing time by about 32.23% and shrinks optimality gaps by about 94.23% versus direct solution, with $\gamma=0.9999$ balancing accuracy and numerical stability. The work demonstrates meaningful operational benefits, including reduced gas price surges and prevention of GFPP losses, highlighting the practical value of anticipatory, multi-market coupling in system operations and guiding future extensions to stochastic and chance-constrained settings.

Abstract

This paper examines Mixed-Integer Multi-Level problems with Sequential Followers (MIMLSF), a specialized optimization model aimed at enhancing upper-level decision-making by incorporating anticipated outcomes from lower-level sequential market-clearing processes. We introduce a novel approach that combines lexicographic optimization with a weighted-sum method to asymptotically approximate the MIMLSF as a single-level problem, capable of managing multi-level problems exceeding three levels. To enhance computational efficiency and scalability, we propose a dedicated Benders decomposition method with multi-level subproblem separability. To demonstrate the practical application of our MIMLSF solution technique, we tackle a unit commitment problem (UC) within an integrated electricity, gas, and carbon market clearing framework in the Northeastern United States, enabling the incorporation of anticipated costs and revenues from gas and carbon markets into UC decisions. This ensures that only profitable gas-fired power plants (GFPPs) are committed, allowing system operators to make informed decisions that prevent GFPP economic losses and reduce total operational costs under stressed electricity and gas systems. The case study not only demonstrates the applicability of the MIMLSF model but also highlights the computational benefits of the dedicated Benders decomposition technique, achieving average reductions of 32.23% in computing time and 94.23% in optimality gaps compared to state-of-the-art methods.

Scalable Multi-Level Optimization for Sequentially Cleared Energy Markets with a Case Study on Gas and Carbon Aware Unit Commitment

TL;DR

The paper tackles the challenge of scalable optimization for Mixed-Integer Multi-Level problems with Sequential Followers (MIMLSF) in energy markets, where upper-level decisions must anticipate outcomes of lower-level sequential market clears. It introduces a lexicographic-weighted-sum single-level (SLP) approximation that asymptotically converges to the original MIMLSF as the scaling parameter , and develops a dedicated multi-level Benders decomposition that splits the complex Benders subproblem into two tractable pieces and further into subproblems. The authors apply this framework to a four-level Unit Commitment with Gas and Carbon Awareness (UCGCA) model in the Northeastern US, integrating electricity, natural gas, and carbon markets and enforcing bid-validity constraints to prevent unprofitable GFPP commitments. Computational results show that the specialized Benders approach reduces computing time by about 32.23% and shrinks optimality gaps by about 94.23% versus direct solution, with balancing accuracy and numerical stability. The work demonstrates meaningful operational benefits, including reduced gas price surges and prevention of GFPP losses, highlighting the practical value of anticipatory, multi-market coupling in system operations and guiding future extensions to stochastic and chance-constrained settings.

Abstract

This paper examines Mixed-Integer Multi-Level problems with Sequential Followers (MIMLSF), a specialized optimization model aimed at enhancing upper-level decision-making by incorporating anticipated outcomes from lower-level sequential market-clearing processes. We introduce a novel approach that combines lexicographic optimization with a weighted-sum method to asymptotically approximate the MIMLSF as a single-level problem, capable of managing multi-level problems exceeding three levels. To enhance computational efficiency and scalability, we propose a dedicated Benders decomposition method with multi-level subproblem separability. To demonstrate the practical application of our MIMLSF solution technique, we tackle a unit commitment problem (UC) within an integrated electricity, gas, and carbon market clearing framework in the Northeastern United States, enabling the incorporation of anticipated costs and revenues from gas and carbon markets into UC decisions. This ensures that only profitable gas-fired power plants (GFPPs) are committed, allowing system operators to make informed decisions that prevent GFPP economic losses and reduce total operational costs under stressed electricity and gas systems. The case study not only demonstrates the applicability of the MIMLSF model but also highlights the computational benefits of the dedicated Benders decomposition technique, achieving average reductions of 32.23% in computing time and 94.23% in optimality gaps compared to state-of-the-art methods.

Paper Structure

This paper contains 14 sections, 4 theorems, 17 equations, 9 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Single-Level Approximation of the MIMLSF Problem
  3. Dedicated Benders Decomposition with Multi-level Subproblem Separability
  4. Benders Subproblem Decomposition for Case I
  5. Benders Subproblem Decomposition for Case II
  6. Benders Decomposed Subproblems with Further Multi-level Separation for Case III
  7. The Four-level Unit Commitment with Gas and Carbon Awareness Model
  8. The Physical and Economic Coupling of Three Markets and the Bid Validity Constraints
  9. Relationship between the four-level UCGCA model and the MIMLSF problem
  10. Real-World Implementation of MIMLSF: Four-Level UCGCA Model in the Northeastern United States
  11. Effectiveness of the Bid-Validity Constraints
  12. Effectiveness of the Dedicated Benders Decomposition Algorithm
  13. Strong Duality Analysis and Numerical Performance with Different Scaling Values
  14. Conclusion

Key Result

Theorem 2.1

The MIMLSF problem eq:1 can be asymptotically approximated by the following single-level problem (SLP) eq:2 for some $\gamma \in (0,1)$. Moreover, when $\gamma \rightarrow 1$, the optimal solution of problem eq:2 converges to the optimal solution of problem eq:1. In SLP eq:2, the scaling factor appl

Figures (9)

  • Figure 1: Different models of hierarchical decision making process. UL: Upper-Level, ML: Middle-Level, LL: Lower-Level. (a) classic bi-level problem; (b) classic tri-level problem; (c) tri-level problem with sequential ML and LL problems; (d) multi-level problem with $n$ sequential LL problems.
  • Figure 2: Theoretical Contributions. The equation numbers in parentheses denote the complete mathematical formulation of the optimization model. Case I: general formulation; Case II: coefficients in Problem \ref{['eq:1']} satisfy $c_{x1}^T = c_1^T, c_{xi}^T = 0, \forall i \in [n]^+$; Case III: Case II without the upper dual complicating constraints\ref{['eq:1:u4']}. RMP: Relaxed Master Problem; BSP: Benders Subproblem.
  • Figure 3: Four-level UCGCA model within an integrated electricity, gas, and carbon market clearing framework.
  • Figure 4: Electric and gas transmission networks in the Northeastern United States, depicted with simplified point-to-point connections. Node markers are sized proportionally to maximum supply and load capacities at $(\eta_e,\eta_g)=(1,1)$, with gas system markers shown on a logarithmic scale. GFPPs in the electric network are connected to their nearest natural gas receipt points in the gas network.
  • Figure 5: Three-stage sequential Benchmark Model (BM) without GFPPs' bid validity constraints.
  • ...and 4 more figures

Theorems & Definitions (8)

  • Theorem 2.1
  • proof
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof