## 5 Results
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Scenario**} & \multirow{2}{*}{**Inertia Constraints**} & \multicolumn{4}{c}{**Parameter Combination**} \\ \cline{3-5}  & & **1 Hz/s, 30 s** & **1 Hz/s, 2 s** & **2 Hz/s, 2 s** \\ \hline \multirow{3}{*}{**NT**} & \(\mathbf{J_{min,sys,sync}}\) [\(\mathbf{kg\cdot m^{2}}\)] & 668,066 & 668,066 & 334,033 \\  & \(\mathbf{J_{min,sys}}\) [\(\mathbf{kg\cdot m^{2}}\)] & 4,811,009 & 1,074,668 & 1,074,668 \\ \cline{1-1} \cline{2-5}  & \(\mathbf{J_{min,sys,sync}}\) [\(\mathbf{kg\cdot m^{2}}\)] & 577,437 & 577,437 & 288,718 \\ \cline{1-1} \cline{2-5}  & \(\mathbf{J_{min,sys}}\) [\(\mathbf{kg\cdot m^{2}}\)] & 4,158,352 & 928,879 & 928,879 \\ \cline{1-1} \cline{2-5}  & \(\mathbf{J_{min,sys,sync}}\) [\(\mathbf{kg\cdot m^{2}}\)] & 832,540 & 832,540 & 412,305 \\ \cline{1-1} \cline{2-5}  & \(\mathbf{J_{min,sys}}\) [\(\mathbf{kg\cdot m^{2}}\)] & 6,615,523 & 1,341,097 & 1,341,097 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Overview of the required system synchronous inertia and overall system inertia for Germany per scenario and parameter combination. the first three hours of the depicted day. Thereafter, more units of these categories get connected to the system, which results in more inertia provided. Until 17:00, pump storage and power to gas units get connected to the system for a single hour, explaining the inertia peaks. As soon as gas-fired power plants get connected to the electrical grid, supplied inertia increases significantly. Figure 6 depicts the results of the NT scenario and the 1 Hz/s, 2 s parameter combination. Again, feed-in from onshore and offshore wind turbines plus photovoltaic system feed-in during the midday could potentially satisfy power demand. However, due to the minimum synchronous inertia constraint and minimum system inertia constraint, more hydro and pump storage units are connected to the electrical grid until 17:00 of the day to meet the inertia constraints. This is visible when comparing the bottom subplots of Figures 5 and 6. The inertia constraints are satisfied by hydro and pump storage units. Although the provided SI from wind turbines is high, pump storage and hydro units are both connected to the system to provide synchronous inertia, since the single connection of either of the units themselves would not be high enough to meet the system synchronous inertia constraint. Thus, both inertia constraints are met by the units providing an inherent inertial response, and SI by wind turbine generators would not be needed. However, due to the fact that inertia provision by wind turbines is not associated with costs and due to the model's logic of the binary variable, \(x_{g}^{source\_inertia}\), wind turbines provide full SI. By the end of the day, when natural-gas-driven generation units are connected to the electrical system to compensate decreasing feed-in from RES, synchronous inertia provided by such units increases. Figure 5: Visualisation of a full day of the NT scenario without considering inertia constraints. The top subplot illustrates results of the power output flows, and the bottom subplot shows the supplied moment of inertia inertia. Table 7 provides an overview of the optimisation results for each scenario and parameter combination. Shown are results of overall CO2 emission, the number of synchronous condensers, the overall system costs and the costs for the provided inertia. The results of costs for the provided annual inertia are calculated by taking additional system costs due to the provision of inertia with respect to the additionally provided inertia to meet the inertia constraints and maintain grid frequency controllability. With respect to the base scenarios, carbon dioxide emissions for the 1 Hz/s, 30 s parameter combination of the NT as well as of the DE increase. In the NT scenario and parameter combination 1 Hz/s, 30 s, additional synchronous condensers provide a moment of inertia of overall 12 \(\times\) 10\({}^{9}\) kg\(\cdot\)m\({}^{2}\) for the modelled time period.