## 2 Methodology
### Unit Commitment and Dispatch Model Formulation
#### 2.2.4 Inertia by Wind Turbines  
A source with a high potential for SI is wind turbines, which are thus integrated [1; 8]. A common method to provide a synthetic inertial response with wind turbines is to adapt the electric power output with respect the power systems need, i.e., ROCOF and the grid frequency [1]. As a result, the wind turbines' rotational speed changes, and stored kinetic energy is exchanged with the power system [44]. The wind turbine might then be operated at non-optimal rotational speed which could lead to undesired behaviour of the wind turbine and in worst case to the disconnection of the wind turbine due to over- or underspeed safety measures while providing a synthetic inertial response [8]. Thus, not only the SI support is lost, but the wind turbines' regular non-inertia-related feed-in is lost as well [8]. To prevent such undesired consequences, the variable H controller is introduced by [PERSON] et al. [8]. The solution suggested by the authors is to scale the provided emulated moment of inertia with the actual operation point of the wind turbine. Therefore, the emulated inertial response is scaled with cut-in wind speed and the nominal speed of the wind turbine generator [8]. Hence, the risk of the wind turbine being disconnected from the grid is reduced while SI is provided. Therefore, the power systems' demand for inertia is expressed by the inertia constant, \(H_{dem}\)[8]. It is defined by the respective TSO and based on the power systems' specific requirements. The wind turbines' variable inertia constant is normalised such that it is supplied at nominal speed [8].  
A simplified approach to depict the potential provided SI with wind turbines based on the control concept of the variable H controller is implemented in OpInMod [27]. The method is illustrated in Figure 3. Since the simplified method to model wind inertia is based on the actual operation point of the wind turbine, normalised specifications of the open-source NREL 5 MW wind turbine generator are used [45; 46]. The wind turbine's operation point is used to determine corresponding normalised rotational speed. This is illustrated in the Figure 3 top subplot. Afterwards, the determined normalised rotational speed is used to specify the variable inertia constant, \(H_{var}\), with with regard to the demanded inertia constant, \(H_{dem}\). This is visualised in the Figure 3 bottom subplot.  
2.2.5. Inertia by Non-Synchronously Connected Energy Storage Units  
The fourth source for inertia implemented is non-synchronously connected energy storage units. Different types of non-synchronously connected energy storage units can be applied to provide SI-like battery storage units [3; 47]. Since most non-synchronously connected energy storage units do not have a physical rotating mass, the emulated moment of inertia, \(J_{SI}\), is almost freely selectable, thus resulting in the SI power provision, \(P_{SI}\), depicted in Equation (4). However, the maximum provided SI power provision is limited by the energy storage units' rated power, \(P_{rated}\), and the power output to balance power demand, \(P_{power\_balance}\), as depicted in Equation (11).  
\[P_{SI}=P_{rated}-P_{power\_balance} \tag{11}\]  
Combining Equation (4) and Equation (11) leads to Equation (12), which determines the emulated moment of inertia, \(J_{SI}\), of a non-synchronously connected energy storage unit.  
\[\frac{P_{rated}-P_{power\_balance}}{4\cdot\pi^{2}\cdot f_{grid}\cdot\frac{gf}{ \delta t}}=J_{SI} \tag{12}\]