## 4 Methodology
### Gas System Model
\begin{table}
\begin{tabular}{c c c} \hline
**Element Types** & **Description** & **Facilities** \\ \hline \hline \multirow{2}{*}{\begin{tabular}{c} pipe \\ \end{tabular} } & \multicolumn{2}{c}{**Passive Elements**} \\ \cline{2-3}  & models a section of a pipeline, basic properties are length, diameter, roughness and pipe efficiency & \\ \hline \hline \multirow{2}{*}{\begin{tabular}{c} resistor \\ \end{tabular} } & models passive devices that cause a local pressure drop (e.g., meters, inlet piping, \\  & coolers, heaters, scrubbers etc.) & \\ \hline \multicolumn{3}{c}{**Active Elements**} \\ \hline \hline compressor & \multirow{2}{*}{\
\begin{tabular}{c} models a compressor station with generic constraints, allows the specification of a control mode of the station (e.g., outlet pressure control, inlet pressure control, flow rate control etc.) \\ \end{tabular} } \\ \cline{2-3}  & & models a pressure reduction and metering station located at the interface of two neighboring networks with different maximum operating pressures, allows the specification of a control mode of the station (e.g., outlet pressure control, inlet pressure control, flow rate control etc.) \\ \hline \hline \multirow{2}{*}{
\begin{tabular}{c} valve \\ \end{tabular} } & models a valve station, which is is either opened or closed \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Basic elements comprising gas transport networks. \begin{table}
\begin{tabular}{c c c} \hline
**Node Type** & **Description** & **Facilities** \\ \hline \hline \multirow{2}{*}{\begin{tabular}{c} demand \\ \(L>0\) \\ \end{tabular} } & point, where gas is extracted from the network, connected & \multirow{2}{*}{\begin{tabular}{c} CGS, CBE, GFPP, \\ IND \\ \end{tabular} } \\ \cline{2-3}  & & facilities are typically flow or pressure controlled & \\ \hline \multirow{2}{*}{\begin{tabular}{c} supply \\ \end{tabular} } & point, where gas is injected into the network; connected facilities & \multirow{2}{*}{\begin{tabular}{c} PRO, CBI, LNG \\ PRO, CBI, LNG \\ \end{tabular} } \\ \cline{2-3}  & & are typically flow or pressure controlled; for LNG regasification & \\ \cline{2-3}  & & terminals the working gas inventory is monitored and the flow rate is reduced in case of low inventory & \\ \hline \hline \multirow{2}{*}{\begin{tabular}{c} storage \\ \end{tabular} } & point, where gas is injected or extracted from the network and & \multirow{2}{*}{\
\begin{tabular}{c} UGS \\ \end{tabular} } \\ \cline{2-3}  & where the maximum supply/loads depend on the working gas & & \\ \cline{2-3}  & inventory, which is monitored along the transient simulation; & \\ \cline{2-3}  & connected facilities are typically flow or pressure controlled & \\ \hline \hline \multirow{2}{*}{
\begin{tabular}{c} junction \\ \end{tabular} } & point, where a topological change or a change in pipe properties & \multirow{2}{*}{-} \\ \cline{2-3}  & occurs (e.g., diameter, inclination); no specific control & \\ \hline \end{tabular}
\end{table} 
Table 2: Classification and characteristics of nodes in the network. Law of Conservation of Mass--Continuity Equation:  
\[\frac{\partial\rho}{\partial t}+\frac{\partial(\rho v)}{\partial x}=0 \tag{17}\]  
_Newton's Second Law of Motion--Momentum Equation:_  
\[\underbrace{\frac{\partial(\rho v)}{\partial t}}_{\text{inertia}}+\underbrace{ \frac{\partial(\rho v^{2})}{\partial x}}_{\text{convective term}}+ \underbrace{\frac{\partial p}{\partial x}}_{\text{pressure force}}+\underbrace{ \frac{\lambda\rho v|v|}{2D}}_{\text{shear force}}+\underbrace{\frac{\rho gsina}{ \text{force of gravity}}}_{\text{force of gravity}}=0 \tag{18}\]  
_First Law of Thermodynamics - Energy Equation:_  
\[\frac{\partial}{\partial t}\left[\left(c_{v}T+\frac{1}{2}v^{2}\right)\rho A \right]+\frac{\partial}{\partial x}\left[\left(c_{v}T+\frac{p}{\rho}+\frac{1}{ 2}v^{2}\right)\rho uA\right]+\rho uAgsina=\Omega \tag{19}\]  
_Real Gas Law - State Equation:_  
\[\frac{p}{\rho}=Z\,R\,T \tag{20}\]  
The fundamental equations are typically simplified by adapting them to the prevailing conditions in transport pipelines.