## 1 Introduction  
In recent decades, decarbonization targets have been set in order to face and limit global warming. On a timeline, the last milestone of this challenge has been set with the Paris Agreement in 2015, which established the goal of keeping global warming well below 2 \({}^{\circ}\)C, and possibly under 1.5 \({}^{\circ}\)C, with respect to the pre-industrial level [1, 2]. In this scenario, one of the most important layers of intervention is the energy sector, which in 2019 accounted for 41.8% of the CO\({}_{2}\) emissions in the world, consequently representing their major contributor [3]. An outlined solution to reduce its impact is to increase renewable energies utilization. In this sense, for example, the EU set the target of covering 32% of its energy consumptions with renewable energies by 2030 [4]. These ambitious targets are inducing deep changes in the energy mix, leading to equally difficult technological challenges. In particular, renewable generators are displacing conventional fossil-fueled ones, leading to a variety of issues mainly related to their non-dispatchability and inability in providing ancillary services. A potential future scenario is presented in report [5]. It offers an immediate idea of the modification that will hit the electric system in less than thirty years: if the world will adopt measures to comply with the Paris agreement (\"REmap Case\"), the share of renewable energies in the power generation sector could reach 86%, while the sector of variable renewables such as solar and wind would be about 60%, with an annual addition of capacity, respectively, up to 360 GW/yr for photovoltaic plants and up to 240 GW/yr for wind plants in 2050. Starting from this estimation, it is clear how important it will be in the future to create renewable generators able to provide the services that nowadays are the responsibility of traditional ones. According to this, regulators and System Operators (SO) are starting to ask for some of these services also to renewable plants in the updated versions of grid codes to face the system developing and keep ensuring its safety and reliability [6]. For variable renewable sources, the implementation of techniques to meet these requirements is asking huge research efforts due to their nature. In fact, they are used to be connected with the grid through power electronic devices, which have no rotational parts and are, therefore, unable to contribute to the total inertia of the power system. Inertia is a fundamental concept while analyzing a power system. Indeed, transient stability and frequency stability are strongly related to the kinetic energy present in the system, whose amount in turn is correlated to the inertia constant. A reduction in the total inertia constant is reflected in a stability decrease [7]. Focusing on the frequency stability problem, inertia constant impact can be appreciated in Equation (1) [7]:  
\[df/dt=0.5f_{0}\;(P_{gen}-P_{load})/(S_{tot}\;H_{tot}), \tag{1}\]  
where \(f\) is the frequency of the system, \(f_{0}\) is the nominal frequency, \(P_{gen}\) is the total generated power in a certain time instant and \(P_{load}\) the total consumed one, \(S_{tot}\) is the sum of the rated power of all the generators, and \(H_{tot}\) is the equivalent inertia constant of the system as a whole. It is evident that if \(H_{tot}\) decreases, given a certain active power unbalance between generation and load, the frequency variation will be larger. To contain the frequency variations and keep the frequency as close as possible to its nominal value, it is thus necessary to create certain levels of reserve available at different timescales. The spread of variable renewable energies (VRE) mentioned above impacts on the reserve problem in more than one way: the uncertainty of the primary source availability, and so of the power production, requires additional reserve in the system; and the participation in the system of the conventional units that are able to provide reserves reduces, and so the stability of the system.