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Data-Driven Continuous-Time Framework for Frequency-Constrained Unit Commitment

Mohammad Rajabdorri, Enrique Lobato, Lukas Sigrist, Jamshid Aghaei

TL;DR

This work addresses the challenge of frequency-constrained unit commitment in low-inertia systems by transitioning from hourly discrete schedules to a continuous-time formulation based on Bernstein polynomials. The authors develop a MILP-compatible CUC framework that represents generation, ramping, and trajectories with Bernstein coefficients, and they introduce a data-driven linear constraint to approximate the nonlinear frequency nadir constraint. The approach also integrates RoCoF and steady-state frequency considerations, enabling more accurate intra-hour frequency management. Validation on the La Palma island demonstrates timely solution times and improved frequency response, with the learned nadir constraint effectively preventing critical nadir violations while signaling the trade-off with operation costs. This methodology offers a practical, scalable path for robust operation of island grids under rising sub-hourly variability and low system inertia.

Abstract

The conventional approach to solving the unit commitment problem involves discrete intervals at an hourly scale, particularly when integrating frequency dynamics to formulate a frequency-constrained unit commitment. To overcome this limitation, a novel continuous-time frequency-constrained unit commitment framework is proposed in this paper. In this approach, Bernstein polynomials represent continuous variables in the unit commitment problem and enable the calculation of frequency response-related metrics such as the rate of change of frequency, quasi-steady-state frequency, and frequency nadir. Notably, startup and shut-down trajectories are meticulously considered, transforming the formulation into a fully continuous-time model and simplifying constraints related to variable continuity. To address the complexities associated with integrating the obtained non-linear frequency nadir constraint into a mixed-integer linear problem, an alternative data-driven frequency nadir constraint is proposed, which accurately constrains frequency nadir deviations throughout the time interval. To validate the proposed model, it is applied to the real-life network of the Spanish Island of La Palma. The results demonstrate the effectiveness of the proposed formulation, indicating that the model is solved timely while mitigating the impact of intra-hour real-time power fluctuations on system frequency.

Data-Driven Continuous-Time Framework for Frequency-Constrained Unit Commitment

TL;DR

This work addresses the challenge of frequency-constrained unit commitment in low-inertia systems by transitioning from hourly discrete schedules to a continuous-time formulation based on Bernstein polynomials. The authors develop a MILP-compatible CUC framework that represents generation, ramping, and trajectories with Bernstein coefficients, and they introduce a data-driven linear constraint to approximate the nonlinear frequency nadir constraint. The approach also integrates RoCoF and steady-state frequency considerations, enabling more accurate intra-hour frequency management. Validation on the La Palma island demonstrates timely solution times and improved frequency response, with the learned nadir constraint effectively preventing critical nadir violations while signaling the trade-off with operation costs. This methodology offers a practical, scalable path for robust operation of island grids under rising sub-hourly variability and low system inertia.

Abstract

The conventional approach to solving the unit commitment problem involves discrete intervals at an hourly scale, particularly when integrating frequency dynamics to formulate a frequency-constrained unit commitment. To overcome this limitation, a novel continuous-time frequency-constrained unit commitment framework is proposed in this paper. In this approach, Bernstein polynomials represent continuous variables in the unit commitment problem and enable the calculation of frequency response-related metrics such as the rate of change of frequency, quasi-steady-state frequency, and frequency nadir. Notably, startup and shut-down trajectories are meticulously considered, transforming the formulation into a fully continuous-time model and simplifying constraints related to variable continuity. To address the complexities associated with integrating the obtained non-linear frequency nadir constraint into a mixed-integer linear problem, an alternative data-driven frequency nadir constraint is proposed, which accurately constrains frequency nadir deviations throughout the time interval. To validate the proposed model, it is applied to the real-life network of the Spanish Island of La Palma. The results demonstrate the effectiveness of the proposed formulation, indicating that the model is solved timely while mitigating the impact of intra-hour real-time power fluctuations on system frequency.
Paper Structure (25 sections, 22 equations, 7 figures, 1 table)

This paper contains 25 sections, 22 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Methodology
  3. Bernstein Polynomials
  4. Function Approximation
  5. Differential of a Polynomial
  6. Convex Hull Property
  7. Sum and Difference polynomials
  8. Order Elevation
  9. Multiplying Polynomials
  10. CUC Formulation
  11. MILP formulation of CUC
  12. Load Approximation
  13. Bernstein Zero-order Continuity
  14. Bernstein First-order Continuity
  15. Power Balance
  16. ...and 10 more sections

Figures (7)

  • Figure 1: Summary of the cited studies.
  • Figure 2: Bernstein basis functions of order 3.
  • Figure 3: Hourly discrete, 5 min discrete, and Bernstein approximation of load.
  • Figure 4: An illustrative sequence of binary variables, SU, and SD trajectories for unit $i$.
  • Figure 5: The exact calculation of \ref{['eq:nadir']} is compared with the approximation in \ref{['eq:nadir_approx']}
  • ...and 2 more figures