Natural gas suppliers and recipients are usually connected through a large-scale, high pressure and integrated transmission pipeline network. These transmission systems are the most cost effective ways to transmit fluid products over long distances , and are usually supplied from multiple sources and use inline compression units to deliver gas to the end users. Adaptation to the varying demands of customers plays an important role, and it is the responsibility of the dispatchers in gas transmission systems to supply adequate amounts of natural gas to meet consumers’ requirements and maintain the pressure level above the minimum required values. Safe operation of transmission pipeline ...view middle of the document...
The quantity of gas contained in a given pipeline segment is defined as the line–pack , , which is a very important parameter in gas transmission systems , considering its use in the compensation of abrupt load changes.
As a result, state estimation techniques play a key role in several problems related to the gas industry such as dynamic data reconciliation, determining the line-pack of the pipelines, the appropriate calculation corresponding to natural gas transactions , leak detection, demand estimation, optimal sensor placement  and as an aid to control or optimum design of the system.
Line-pack and demand estimation estimates are essential in the evaluation of the current status of a network and can be helpful in reliable planning for the future.
Increasing and decreasing pipeline inventory, called line packing and drifting, respectively, is one of the key tools available for the gas dispatchers to balance the time-varying demands with supplies.
The available measurements are mostly pressure and mass flow at the inlet and outlet of the pipeline sections. In compression units, apart from pressure and mass flow, the inlet and outlet temperatures are measured as compression of the gas increases its temperature.
For a gas dispatcher, it is desirable to have an accurate measurement of the customer demand flows as the basic information required in the operation of the network. However, mass flow meters are much more expensive than pressure sensors to install, and so a method capable of estimating the flow demands from pressure measurements is an economically attractive alternative.
There are several standard numerical methods that can be used to simulate the dynamic behavior of a gas transmission system such as spectral method –, the method of characteristic , , explicit and implicit finite difference –, the finite volume , , finite element ,  and high order approaches such as third order Runge-Kutta discontinuous Galerkin method .
Some of the main characteristics of most common methods used in transient analysis of GTN are discussed in . A literature review reveals that although many methods have been proposed for numerical simulation of dynamic behavior of GTN’s, but they mostly focus on isothermal models and transient behavior of fluid in a single pipe. Furthermore, flow reversal arising either from inverse pressure gradient or sudden valve opening or closing, is not addressed in them.
Including energy equation in the dynamic model of a GTN, introduces a difficulty in the modeling as the governing equations of the boundary points depend on the flow direction or network connectivity. The direction of flow in a specific equipment might change during flow reversals. On the other hand, opening or closing a valve might also alter the connectivity of the network, which leads to connecting or isolating different compartments of a GTN.
Here, we are motivated by discontinuities...