## 2 Methodology
### Fundamentals of Power System Inertia  
In AC electrical networks, the generation and consumption of power have to be balanced at any given point in time [6]. The indicator for such an equilibrium is the grid frequency, \(f_{grid}\), the representation of the rotational speed of all synchronously connected rotating masses [6]. The grid frequency deviates from its nominal value in the event of an imbalance of power generation and consumption, \(\Delta P\)[6]. The ROCCF, \(\delta f/\delta t\), which represents the speed with which the grid frequency changes, is determined by the overall provided inertia from all, synchronously to the system-connected rotating masses, \(J_{sys}\), and depicted in Equation (1) [1]. \[\frac{\delta f}{\delta t}=\frac{\Delta P}{4\cdot\pi^{2}\cdot f_{grid}\cdot J_{ sys}} \tag{1}\]  
Kinetic energy stored in the rotational motion of the synchronously connected machine is described by Equation (2), where \(E_{g,kin}\) is the stored kinetic energy of the unit \(g\). \[E_{g,kin}=\frac{1}{2}\cdot J_{g}\cdot 4\cdot\pi^{2}\cdot f_{grid}^{2} \tag{2}\]  
Commonly, the provided inertia of a single rotating mass connected synchronously to the power system is expressed via the inertia constant, \(H_{g}\)[6]. The inertia constant is the proportional ratio of the machines' stored kinetic energy, \(E_{g,kin}\), with respect to the units of 3. apparent power, \(S_{g}\)[1,6]. \[H_{g}=\frac{E_{g,kin}}{S_{g}} \tag{3}\]  
The inertia constant describes the theoretical time a synchronously connected rotating mass, e.g., synchronously connected generator, is able to supply nominal power only by the generators' stored kinetic energy [6]. In the same manner, the robustness of electrical systems in case of deviations between power generation and consumption can be expressed with the power system inertia constant, \(H_{sys}\)[1]. Generation and energy storage units which are connected to the power system via grid frequency converters are electronically decoupled from the power system. Hence, even if a rotating mass exists as in the case of wind turbines, their rotational speed is isolated from the electrical grid and stored kinetic energy is hidden [1]. However, such units are capable of emulating the inertial response of a synchronously connected rotating machine in the event of a power imbalance known as Synthetic Inertia (SI) [1; 3]. The provided inertial power, \(P_{SI}\) is described by Equation (4) and determined by the emulated moment of inertia, \(J_{SI}\), as well as the grid frequency and the ROCOF. \[P_{SI}=J_{SI}\cdot 4\cdot\pi^{2}\cdot f_{grid}\cdot\frac{\delta f}{\delta t} \tag{4}\]  
Inertia provided by synchronously connected rotating masses cannot directly be subsituted by SI [5]. The emulated inertial response is delayed due to the measurement of the grid frequency and precise detection of events which demand an inertial response [25]. Thus, to sustain controllability of the grid frequency in power systems with a high share of non-synchronous penetration, sufficient synchronous and non-synchronous inertia has to be provided. For synchronous inertia which responds instantaneously to limit the ROCOF, and non-synchronous inertia which is activated with a time delay after the power imbalance occurrence and synchronous inertia in combination, both provide time to units supplying grid frequency control services and thus limiting the grid frequency nadir [1; 25; 26]. This is depicted in Equation (5) and further illustrated in Figure 1, in which \(H_{sync}\) is the provided synchronous inertia, \(H_{SI}\) is the provided non-synchronous inertia, \(H_{sys,sync,min}\) the minimum required synchronous system inertia and \(H_{sys,min}\) the minimum required system inertia comprising a synchronous and a non-synchronous share. Thus, both requirements being met prevents ROCOF relays from triggering and the disconnection of loads or generation units due to the violation of grid frequency thresholds.