Sensitivity Analysis Applied to a Steady-State Model of Natural Gas Transportation Systems

Abstract

Abstract A nodal formulation for the mathematical model of a gas transportation system is presented and the Newton-Raphson method is proposed as a means of solving the resulting nonlinear equations. Once a solution has been obtained, a linearized form of the model can be used to assess the sensitivity or interaction of system variables. The equations necessary to perform this sensitivity analysis are developed and demonstrated on a sample system. A scheme for performing Gaussian elimination on a general set of simultaneous linear equations having a sparse coefficient matrix is presented. Introduction This paper espouses the use of a nodal formulation for the steady-state model of a gas transportation system and attempts to show how sensitivity analysis applied to this model can be used to aid in the design and operation of the prototype systems. the mathematical model used here has been described in a previous paper, which was based on earlier work. The basic model is recapped here. Solutions of this model are obtained using the n-dimensional Newton-Raphson (N-R) technique with an acceleration scheme, the main purpose of which is to prohibit divergence. The N-R method requires the solution of a set of simultaneous linear equations whose coefficient matrix for this application is very sparse. In preparing digital computer programs using this model, this sparsity has been exploited to great advantage in core storage savings and execution time efficiency. The basic aspects of a sparse matrix solution scheme for a general set of simultaneous linear equations and rudimentary programming steps are covered in the Appendix. Once a solution has been obtained for the model, the user often poses questions such as, "How different would the solution be if I had specified a different value for this load?" Or "How much more horsepower would be needed here if we had increased this pressure 15 psi? The user could make the specified changes and resolve his model to obtain the answers. However, if he is willing to accept approximate answers to these questions, then an appropriately executed sensitivity analysis can provide the answers with considerable saving in computer time and associated expense. The development of the equations necessary to perform a general sensitivity analysis using the proposed model is presented. Computational shortcuts using the sparse matrix solution of a set of simultaneous linear equations are demonstrated. Finally, an example is hypothesized to demonstrate one way of using the proposed sensitivity analysis scheme.