The Unit Commitment Model with Concave Emissions Costs: A Hybrid Benders' Decomposition with Nonconvex Master Problems

Jennifer van Dinter, Steffen Rebennack, Josef Kallrath, Paul Denholm, Alexandra Newman

Research output: Contribution to journalArticlepeer-review

9 Scopus Citations


We present a unit commitment model which determines generator schedules, associated production and storage quantities, and spinning reserve requirements. Our model minimizes fixed costs, fuel costs, shortage costs, and emissions costs. A constraint set balances the load, imposes requirements on the way in which generators and storage devices operate, and tracks reserve requirements. We capture cost functions with piecewise-linear and (concave) nonlinear constructs. We strengthen the formulation via cut addition. We then describe an underestimation approach to obtain an initial feasible solution to our model. Finally, we constitute a Benders' master problem from the scheduling variables and a subset of those variables associated with the nonlinear constructs; the subproblem contains the storage and reserve requirement quantities, and power from generators with convex (linear) emissions curves. We demonstrate that our strengthening techniques and Benders' Decomposition approach solve our mixed integer, nonlinear version of the unit commitment model more quickly than standard global optimization algorithms. We present numerical results based on a subset of the Colorado power system that provide insights regarding storage, renewable generators, and emissions.

Original languageAmerican English
Pages (from-to)361-386
Number of pages26
JournalAnnals of Operations Research
Issue number1
StatePublished - 2013

NREL Publication Number

  • NREL/JA-6A20-53534


  • Benders' Decomposition
  • Convex underestimators
  • Integer programming applications
  • Mixed integer nonlinear programming
  • Power systems
  • Renewables
  • Spinning reserves
  • Storage
  • Unit commitment model


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