Modern power systems throughout the world, particularly in islands such as Great Britain, face a major problem: the significantly reduced level of system inertia due to integration of Renewable Energy Sources (RES). Inertia, which refers to the physical inertia of the rotating masses in thermal generators, acts as a buffer of kinetic energy, which helps prevent blackouts in the event of an unexpected generator outage. Given that most RES such as wind and solar are decoupled from the power grid through power electronics converters, they do not naturally contribute to system inertia, therefore reducing this valuable energy buffer. In order to avoid blackouts in low-carbon systems, a higher amount of alternative ancillary services such as Frequency Response becomes necessary.

This thesis focuses on studying mathematical methods to schedule ancillary services for frequency security in a cost-effective manner. To do so, we have obtained novel conditions for post-fault frequency security that can be implemented as constraints into optimisation problems such as the Unit Commitment, in order to co-optimise the provision of energy and frequency ancillary services. The key challenge lies in incorporating the differential-equation-driven frequency dynamics into an algebraic-equation-constrained optimisation problem. Furthermore, the constraints for frequency security must be computationally efficient, since they may be used in computationally intense problems like Stochastic Unit Commitment: while Mixed-Integer Linear Programming has been extensively used in power system scheduling and markets, this thesis exploits the recent development of convex optimisation algorithms to also propose Mixed-Integer Second-Order Cone constraints for a safe post-fault frequency evolution.

The developed frequency-secured optimisation framework has been applied to several relevant case studies, which allow to inform sensible designs for ancillary-services markets and planning decisions that would lead to an optimal operation of a low-carbon power system, achieving significant economic savings and reduction in carbon emissions. An optimal procurement of these services is of uttermost practical relevance in modern power grids, as highlighted by recent events such as the Great Britain blackout of August 9th, 2019.