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A computationally efficient Benders decomposition for energy systems planning problems with detailed operations and time-coupling constraints

Anna Jacobson, Filippo Pecci, Nestor Sepulveda, Qingyu Xu, Jesse Jenkins

TL;DR

The paper tackles macro-energy systems planning with detailed hourly operations and time-coupled constraints, proposing a budgeting-variable Benders decomposition that separates investments from operations and decouples temporal subperiods for parallel processing. The master-subproblem scheme enables solving many small operational problems in parallel, with multiple cuts per iteration improving convergence and allowing near-linear scaling with temporal resolution. Across 2–52 week horizons and up to 19 zones, the method substantially outperforms monolithic MILP solvers, reducing runtime and enabling high-resolution planning under policy constraints such as RPS and CO2. This approach mitigates structural uncertainty by enabling finer temporal detail in investments and operations, with applicability to other infrastructure domains and potential openness for open-source solver use.

Abstract

Energy systems planning models identify least-cost strategies for expansion and operation of energy systems and provide decision support for investment, planning, regulation, and policy. Most are formulated as linear programming (LP) or mixed integer linear programming (MILP) problems. Despite the relative efficiency and maturity of LP and MILP solvers, large scale problems are often intractable without abstractions that impact quality of results and generalizability of findings. We consider a macro-energy systems planning problem with detailed operations and policy constraints and formulate a computationally efficient Benders decomposition separating investments from operations and decoupling operational timesteps using budgeting variables in the master model. This novel approach enables parallelization of operational subproblems and permits modeling of relevant constraints coupling decisions across time periods (e.g. policy constraints) within a decomposed framework. Runtime scales linearly with temporal resolution; tests demonstrate substantial runtime improvement for all MILP formulations and for some LP formulations depending on problem size relative to analagous monolithic models solved with state-of-the-art commercial solvers. Our algorithm is applicable to planning problems in other domains (e.g. water, transportation networks, production processes) and can solve large-scale problems otherwise intractable. We show that the increased resolution enabled by this algorithm mitigates structural uncertainty, improving recommendation accuracy.

A computationally efficient Benders decomposition for energy systems planning problems with detailed operations and time-coupling constraints

TL;DR

The paper tackles macro-energy systems planning with detailed hourly operations and time-coupled constraints, proposing a budgeting-variable Benders decomposition that separates investments from operations and decouples temporal subperiods for parallel processing. The master-subproblem scheme enables solving many small operational problems in parallel, with multiple cuts per iteration improving convergence and allowing near-linear scaling with temporal resolution. Across 2–52 week horizons and up to 19 zones, the method substantially outperforms monolithic MILP solvers, reducing runtime and enabling high-resolution planning under policy constraints such as RPS and CO2. This approach mitigates structural uncertainty by enabling finer temporal detail in investments and operations, with applicability to other infrastructure domains and potential openness for open-source solver use.

Abstract

Energy systems planning models identify least-cost strategies for expansion and operation of energy systems and provide decision support for investment, planning, regulation, and policy. Most are formulated as linear programming (LP) or mixed integer linear programming (MILP) problems. Despite the relative efficiency and maturity of LP and MILP solvers, large scale problems are often intractable without abstractions that impact quality of results and generalizability of findings. We consider a macro-energy systems planning problem with detailed operations and policy constraints and formulate a computationally efficient Benders decomposition separating investments from operations and decoupling operational timesteps using budgeting variables in the master model. This novel approach enables parallelization of operational subproblems and permits modeling of relevant constraints coupling decisions across time periods (e.g. policy constraints) within a decomposed framework. Runtime scales linearly with temporal resolution; tests demonstrate substantial runtime improvement for all MILP formulations and for some LP formulations depending on problem size relative to analagous monolithic models solved with state-of-the-art commercial solvers. Our algorithm is applicable to planning problems in other domains (e.g. water, transportation networks, production processes) and can solve large-scale problems otherwise intractable. We show that the increased resolution enabled by this algorithm mitigates structural uncertainty, improving recommendation accuracy.
Paper Structure (18 sections, 1 theorem, 34 equations, 8 figures, 8 tables)

This paper contains 18 sections, 1 theorem, 34 equations, 8 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Overall problem formulation
  4. Solution method
  5. Benders decomposition algorithm
  6. Numerical experiments
  7. Computational setup
  8. Results
  9. Conclusions
  10. Future Directions and Novel Capabilities
  11. Notation
  12. Problem Description
  13. Investment constraints
  14. Operational constraints
  15. Policy constraints
  16. ...and 3 more sections

Key Result

Theorem 1

Problem eq:prob is equivalent to:

Figures (8)

  • Figure 1: Block structure of Problem \ref{['eq:prob']} with both complicating constraints and variables, where $n_W=|W|$. Investment-only constraints \ref{['eq:probinv']} are not pictured.
  • Figure 2: IPM regions used in our numerical experiments.
  • Figure 3: Runtime by weeks (left) and by zone (right) obtained by applying Benders decomposition to solve the MILP problem in the CO$_2$-constrained case. Runtime grows linearly with the number of weeks, while it increases quadratically with the number of zones.
  • Figure 4: Runtime per iteration by week (left) and number of iterations (right), shown for different numbers of zones in the reference case. Runtime per iteration increases both with the number of weeks and the number of zones modeled. Number of iterations increases with the number of zones but decreases with the number of weeks modeled.
  • Figure 5: Optimality gap vs. runtime for our Benders algorithm with decomposed subproblem \ref{['eq:subprob']} and a standard Benders implementation with a full operational subproblem \ref{['eq:subprob_classic']}. Plotted on logarithmic x- and y-axes.
  • ...and 3 more figures

Theorems & Definitions (1)

  • Theorem 1