  A reduced level of system inertia due to renewable integration increases the
need for cost-effective provision of ancillary services, such as Frequency
Response (FR). In this paper we propose a closed-form solution to the
differential equation describing frequency dynamics, which allows to obtain
frequency-security algebraic constraints to be implemented in optimisation
routines. This is done while considering any finite number of FR services with
distinguished characteristics, such as different delivery times and activation
delays. The problem defined by these frequency-security constraints can be
formulated as a Mixed-Integer Second-Order Cone Program (MISOCP), which can be
efficiently handled by off-the-shelf conic optimisation solvers. This paper
also takes into account the uncertainty in inertia contribution from the demand
side by formulating the frequency-security conditions as chance constraints,
for which an exact convex reformulation is provided. Finally, case studies
highlighting the effectiveness of this frequency-secured formulation are
presented.
