## 2 Literature Review  
Various methods have been published to model a gas network and solve the governing equations of gas flow through the pipelines, with the vast majority applying numerical methods due to the non-linearity of the equations. In the literature, a Steady-State Model (SSM) is frequently used where the pressure (the driving force) and flow rate at the inlet and outlet of the pipeline are assumed constant. The steady-state equations of gas flow are non-linear algebraic, and in most cases, for solving this type of equation, numerical solutions are employed. Some valuable studies of SSM have been published. The principles and fundamentals of graph theory can be used to model a network along with the empirical flow equations for low-, medium-, and high-pressure flow. Various different empirical flow equations have been proposed such as [PERSON]'s equation, the Polyflo equation, the Panhandle equation, and the Weymouth equation (for high-pressure networks). [PERSON] (2019) applied the Newton-Raphson method to deal with the non-linear equations. In this work, a step-by-step, Newton-Raphson method using a Jacobian matrix in graph theory was presented to model a horizontal gas pipe network. The essence of the work is the presentation of a computational method based on physical and mathematical laws for simplification of fundamental equations. While this work is one of the leading references in the modelling of gas networks, no specified example for a pipeline with inclination was presented. Many authors for simplification neglect the inclination term; however, [PERSON], [PERSON], and [PERSON] et al. (2011); [PERSON] (2012); [PERSON] (2013) are some authors who have considered inclination in the pipe flow equation, because of its importance for the validation of models with real-world data. Simulation of a natural gas network focusing on gas quality using a numerical-iteration method was carried out by [PERSON] (2016). An important point of this literature is the inclusion of gases with different density in the network. Varying gas composition will impact gas pipeline system characteristics such as pressure and flow rate. With a move to decarbonise the NG network, this is vital in the evaluation of the networks' ability to store synthetic natural gas or hydrogen. [PERSON] (2018), after describing three different gas networks, low, middle, and high pressure, indicated two methods to run the gas system model, which are the nodal method and loop method. In the nodal method, the flow rate is the dependent variable, and the pressure drop (differential pressures between two nodes of a pipe/branch) through each branch is the independent variable \(Q_{ij}=f(dP_{ij})\), while the loop method describes that pressure is a function of flow rate (Equation (1)). The article investigates the impact of ambient temperature on the flow rate. \[dP_{ij}=f(Q_{ij}) \tag{1}\][PERSON]'s [27] paper focused on simulation of transmission pipelines in a gas network focusing on compressor stations. The equations are mainly based on the mass balance, flow principles, and compression characteristics of gas in pipes. The Newton-Raphson method has been applied to solve the equation using C++. The model evaluated the energy consumption for various configurations in order to optimize the transmission pipeline flow and pressure. There are a number of papers that have implemented the MATLAB-Simulink tool to simulate pipelines: [PERSON] and [PERSON] [28] and [PERSON] in [24] applied the Simulink toolbox in MATLAB to model the gas network. [PERSON] and [PERSON] [28] have manipulated the Simulink model of a gas system, and in order to verify the accuracy, the results have been compared with outputs from the finite difference scheme (FSM) and experimental test cases. The proposed model can predict the transient response of the dependent variable (pressure) with a good correlation with the nonlinear finite difference models.