## 4 Methodology
### Gas System Model  
Similar to the power system network, the gas network is described by a directed graph \(G=(V,E)\) composed of nodes \(V\) and branches \(E\). Facilities with an inlet, outlet and flow direction are modeled as branches, while connection points between these branches as well as entry and exit stations are represented by nodes. Branches, in turn, can be distinguished between active and passive branches. Active branches represent controlled facilities, which can change their state or control during simulation, such as compressor stations, regulator stations and valves, while passive branches, such as pipelines and resistors represent facilities or components whose state is fully described by the physical equations, derived from the conservation laws. A description of the different branch types is given in Table 1. Nodes can also be differentiated according to their function into supply, demand, storage and junctions. A description of the different node types and their corresponding node facilities is given in Table 2.  
The gas model proposed in this study includes sub-models of all important facilities comprising a gas transport system, such as pipelines, compressor stations (CS), production fields (PRO), cross-border import (CBI) and export stations (CBE), city gate stations (CGS), stations of direct served customers (GFPPs, IND), liquefied natural gas (LNG) regasification terminals and underground gas storage (UGS) facilities. The model is able to capture appropriately [52; 53] the reaction of gas transport systems to load variations (i.e., daily and seasonal changes of gas demands at offtake points) and disruption events (e.g., loss of supply from an entry point, failure in a compressor station, etc.) with moderate computational cost, taking into account the physical laws governing the dynamic behavior of gas transport systems. The accuracy of the proposed gas model has been confirmed in [52; 53], where it is benchmarked against a commercial software package. In the following we give a brief description of the physical equations used fro describing the gas system. We refer to previous publications, for more details on the gas model implemented in SAInt [52; 54; 53]. The dynamic behavior of a gas transport system is predominantly determined by the gas flow in pipelines. The set of non-linear hyperbolic partial differential equations (PDE) describing the transient flow of natural gas in pipelines are derived from the law of conservation of mass, momentum and energy and the real gas law. Applying these laws on an infinitesimal control volume (\(CV\)) of a general pipeline with a constant cross-sectional area \(A\) and an infinitesimal length \(dx\) (see Figure 2) and assuming the parameters describing the gas flow dynamics along the pipe coordinate \(x\) are averaged over \(A\), yields the following set of fundamental Partial Differential Equations (PDEs) describing the gas flow through pipelines (the assumption of averaging the flow parameters over the cross-sectional area can be justified as long as the pipe length \(L\) is much greater than the pipe diameter \(D\) which is the case in transmission networks where \(\frac{D}{L}\) is of order \(O(10^{-5})\) or lower):