Numerical Issues and Approximated Models for the Diagnosis of Transmission Pipelines

Abstract

The chapter concerns numerical issues encountered when the pipeline flow process is modeled as a discrete-time state-space model . In particular, issues related to computational complexity and computability are discussed, i.e., simulation feasibility which is connected to the notions of singularity and stability of the model. These properties are critical if a diagnostic system is based on a discrete mathematical model of the flow process. The starting point of the study is determined by the partial differential equations obtained from the momentum and mass conservation laws by using physical principles. A realizable computational model is developed by approximation of the principal equations using the finite difference method. This model is expressed in terms of the recombination matrix A which is the key of the analysis by taking into account its possible singularity and stability . The nonsingularity of the matrix A for nonzero and finite, time and spatial steps is proven by the Lower–Upper decomposition. A feature of the discrete model allows the derivation of a nonsingular aggregated model, whose stability can be analyzed. By considering the Courant–Friedrichs–Lewy condition and data from experimental studies, numerical stability conditions are derived and limitations for the feasible discretized grid are obtained. Moreover, the optimal relationship between the time and space steps which ensures a maximum stability margin is derived. Because the inverse of matrix A, composed of four tridiagonal matrices, is required for the main diagnosis methods, two analytical methods for the inversion are discussed which reduce the system’s initialization time and allow designing an accurate and fast diagnosis algorithm. By considering that each inversion method generates its particular structure, two different flow models are generated: one based on auxiliary variables and the other suitable if the stability condition of A is satisfied. The applicability of the two models is shown by considering the norm of the difference between their behaviors for a finer discretization grid . A similarity measure is proposed which considers the number of pipeline segments as well as the ratio between the time and spatial steps . Thus, the system’s computational efficiency is improved and satisfactory results are shown with respect to the base model, if a highly dimensional model with the approximated diagonal matrix is considered.