## 2 Methodology
### Unit Commitment and Dispatch Model Formulation  
To evaluate system costs in the future German power system due to the provision of synchronous and non-synchronous inertia, the open-source modelling tool Open Inertia Modelling (OpInMod) is used. OpInMod is an open-source modelling framework able create unit commitment and economic inertia dispatch models [27]. It is built upon the basic functionalities of the Open Energy Modelling Framework (oemof) package solph [28; 29]. OpInMod is implemented in python and inherits oemof's design philosophy and core functionalities. Hence, scientific principles such as reproducibility, transparency and easy accessibility are met [30; 31; 32; 33]. Like oemof.solph, OpInMod creates Mixed Integer Linear Programming (MILP) optimisation problems [29] applying python's pyomo package [34]. An external solver is used to solve the optimisation problem. A simple validation and verification of OpInMod is presented in [27]. Therin, a stepwise presentation of OpInMods's capabilites is shown. The presentation builds upon oemof's examples which are provided alongside the model via Github [35]. Further, oemof itself has been reviewed and discussed in [36; 37]. OpInMod is published under MIT License and can be accessed via Github [38] and Zenodo [39]. A simplistic summary of the modelling procedure is illustrated in Figure 2.  
Figure 1: Schematic illustration of inertia in present power system and the system inertia composition in future power system. In order to create a model of the German power system and to assess costs due to the provision of synchronous inertia and SI, the scenarios have to be defined first. Since the modelling tool OpInMod does not provide data by source, information such as feed-in series from wind turbines or solar photovoltaic system have to be provided. The same applies, e.g., for power demand series or generator characteristics such as the inertia constant or the generators' efficiency. The unit dispatch and economic inertia dispatch model is created using OpInMod's functionalities. This includes oemof's core functions to represent energy systems. The created optimisation function of the unit dispatch and economic inertia dispatch model is solved by an external solver. The solver is freely selectable and has to be installed separately. Solver output is processed and transferred into a tabular data format for further analysis. The optimisation function of the unit commitment and economic inertia dispatch model depicted in Equation (6) minimise the total system costs. The first term of the equation represents the costs of the energy flows by the variable costs of the unit, \(c_{g}^{wc}\), and the flow variable, \(x_{g}^{flow}\), and is inherented from oemof.solph [29]. The second part of the optimisation function describes inertia-related costs. The generators' moment of inertia is depicted by \(J_{g}\) and the costs of the unit providing inertia by \(c_{g}^{ic}\). A synchronously connected rotating machine provides its total inertia to the power system while it is connected with the grid. The binary variable \(x_{g}^{source\_inertia}\) expresses this logic. Investment costs of storage units providing inertia, \(c_{g}^{inrest\_inertia}\), are defined as the annualised inertia costs including fixed operation and maintenance costs. \[ \begin{split}\min:\sum_{t}\sum_{g}\overbrace{c_{g}^{wc}\cdot x_{g }^{flow}}^{\text{costs flow}}(t)+\overbrace{x_{g}^{source\_inertia}\cdot x_{g}^{ ic}\cdot J_{g}(t)}^{\text{costs inertia}}+\\ \sum_{g}\overbrace{c_{g}^{inrest\_inertia}\cdot J_{g}}^{\text{ costs inertia}}\cdot J_{g}\end{split}  \tag{6}\]  
Figure 2: Schematic representation of the three-step modelling process. Scenarios are defined and data such as feed-in series of variable RES, generator characteristics and power demand series are used to feed the unit dispatch and economic inertia dispatch model. To complete the initialisation part of the modelling process, OpInMod including oemof’s core functionality is used to build an energy system.