Network: System Transient Calculations by Implicit Method
Authors: Wylie E. Benjamin , Stoner Michael A. , Streeter Victor L. 
Published: December 2007

Abstract
The calculation of transient flows and pressures in a gas network by the characteristics pressures in a gas network by the characteristics method has a time interval restricted to the shortest pipe length divided by the isothermal wave speed. To avoid this restriction, the centered implicit-difference method is developed. It requires two equations for each pipe and one equation for each junction. The Newton-Raphson method of handling the nonlinear equations is utilized and sparse matrix algebra reduces the time of solution of the simultaneous equations. An example is presented which compares results obtained with the implicit scheme with results obtained from the characteristics method of solving for conditions at each node in turn. The new method is then applied to a simplified grid of the Consumers Power Co. and a comparison is made with field Power Co. and a comparison is made with field measurements.

Introduction
Steady-state analysis and design of natural gas transmission and distribution systems has progressed to a fairly sophisticated state in recent years. When time variations of parameters are added to an analysis, an entirely new independent dimension is added to the mathematical model. The analyst must be concerned with the variation of pressure with time as well as position and, additionally, must be concerned with the space and time variation of mass flow rate. From the system operator's point of view the latter modeling represents the real physical operation of a system. Some progress has physical operation of a system. Some progress has been made in this area of dynamic simulation of system operation. This paper is intended to further the knowledge of analysis and design techniques concerned with unsteady flow in natural gas systems. The primary complication in a treatment of transient-flow phenomena is in the handling of the variables in their true distributed character. The parameters of concern include frictional, inertial, parameters of concern include frictional, inertial, pressure, and gravitational forces, and system pressure, and gravitational forces, and system storage and compliance. To accomplish a solution of the partial-differential equations that relate these parameters, normally one of three different approaches is followed: an implicit procedure, an explicit procedure, or the method of characteristics. Computer programs that have been developed include PIPETRAN, developed by Electronic Associates, Inc., under contract to the American Gas Assn.; the G. E. Simulator, developed by General Electric Co.; CAP, developed at the Engineering Research Station of the Gas Council; and SATAN, developed at the London Research Station of the Gas Council. The G.E. Simulator and CAP bob use an implicit-solution procedure for the dynamic equations; PIPETRAN and SATAN use an explicit modeling. The method of characteristics has also been used successfully. The method of characteristics provides a stable, accurate solution of the equations, but has the disadvantage of costly computations for long-duration transients in a complicated system. For stability reasons the time increment in the model cannot exceed the reach length divided by the isothermal wave speed. The explicit approach is limited in the same manner; however, users of the method have successfully relaxed this restriction on the time step by using certain empirical restraints to achieve stability. The implicit finite-difference representation of the equations offers the advantage of guaranteed stability for a large time step, but has the disadvantage of requiring the solution of a set of nonlinear simultaneous equations at each time step. For a complicated gas network, the matrix becomes quite large, the computer storage requirements become very large, and the solution time again becomes excessive.
SPEJ
P. 356