# Potential of Onshore Wind Turbine Inertia in Decarbonising the Future Irish Energy System
## 2 Methodology
### Synchronous and Synthetic Inertia in Power Systems  
In AC-power systems, power generation and power consumption have to be balanced at all times [2]. The indicator for power balance is the grid frequency, \(f_{grid}\)[2]. A power imbalance leads to a deviation of the grid frequency from its nominal value [2]. The speed with which the grid frequency changes is dominated by power system inertia [1]. Equation (1) illustrates this relationship where \(\delta f/\delta t\) is the ROCOF, \(P_{gen}\) the accumulated power generation, \(P_{load}\) the accumulated power consumption and \(j_{sys}\) is the overall power system inertia, i.e., the sum of the moment of inertia of all synchronously connected machines [1].  
\[\frac{\delta f}{\delta t}=\frac{P_{gen}-P_{load}}{4\cdot\pi^{2}\cdot f_{grid} \cdot J_{sys}} \tag{1}\]  
It is state-of-the-art to describe the inertia of a synchronously connected machine with the inertia constant, \(H_{g}\). The inertia constant represents the proportional expression of the units stored kinetic energy, \(E_{g,kin}\), with respect to its apparent power rating, \(S_{g}\), as depicted in Equation (2) [1,3]. The inertia constant represents the theoretical time, e.g., a synchronous generator, is able to provide its rated power solely using its stored kinetic energy [1]. Likewise, the resilience of a power system against power imbalances can be described with the power system inertia constant [1].  
\[H_{g}=\frac{E_{g,kin}}{S_{g}} \tag{2}\]  
Frequency converter connected generation sources or energy storage units are able to emulate the behaviour of a synchronous generator in the event of a power imbalance known as synthetic inertia (SI) [3; 4]. A common strategy to substitute inertia with WTs is to change the electric power output based on the ROCOF and the frequency [3]. Due to the modified power output, the rotational speed of the WT changes [18]. That might cause the WT to operate at non-optimal rotational speed [18]. In the worst case, the WT disconnects from the system due to over- or underspeed protection while providing SI [19]. Thus, the SI support as well as the non-inertia related power feed-in of the WT is lost [19]. Hence, from the TSO's point of view, the situation changes for the worse. To counteract such problems, a novel control methodology to emulate inertia with WTs is introduced by [PERSON] et al. [19]. The authors propose to scale the inertia constant with the actual operating point of the WT, considering the cut-in speed and the rated speed of the WT as supporting points [19]. Thereby, the risk of disconnecting the WT while providing inertia is diminished. The variable inertia constant is normalised such that the inertia constant demanded by the TSO, \(H_{dem}\), is provided at rated speed. \(H_{dem}\) is determined by the TSO and based on power system requirements. Equation (3) depicts how the provided SI of a WT is calculated were \(\omega_{actual,WT}\) is the actual rotational speed of the WT, \(\omega_{rated,WT}\) is the rated speed of the WT and \(\omega_{cut\_in,WT}\) the cut-in wind speed of the WT.  
\[\frac{H_{var}}{H_{dem}}=\frac{0.5\cdot J_{WT}\cdot(\omega_{actual,WT}^{2}- \omega_{cut-in,WT}^{2})}{0.5\cdot J_{WT}\cdot(\omega_{rated,WT}^{2}-\omega_{ cut-in,WT}^{2})} \tag{3}\]  
SI is not a direct substitution of synchronous inertia since it is not an inherent inertial response [3]. Hence, in future power systems, the overall system inertia constant comprises an inherent and and non-inherent term as depicted in Equation (4).  
\[H_{sys}=\frac{\frac{\sum\limits_{g}H_{g,sync}\cdot S_{g,sync}}{\sum\limits_{g} S_{g}}}{\sum\limits_{g}S_{g}}+\frac{\sum\limits_{g}H_{g,synt}\cdot S_{g, synt}}{\sum\limits_{g}S_{g}} \tag{4}\]