  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.
