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We derive a linear system of ordinary differential equations (ODEs) toapproximate the dynamics of natural gas in pipeline networks. Although aclosed-form expression of the eigenvalues of the state matrix does notgenerally exist, the poles of an irrational transfer function corresponding tothe linearized partial differential equations are used to approximate theeigenvalues of the ODE system. Our analysis qualitatively demonstrates that theeigenvalues of the state matrix of the entire network system are "pipelineseparable" in the sense that the eigenvalues are dominated by the individualpipeline parameters and not the incidence connectivity of the network graph.The linear system is used as the dynamic constraints of a linear optimalcontrol problem (OCP) to design the control actions of compressor units tominimize the energy that they expend. The motivation of this work is to reducethe computational complexity of optimizing gas dynamics in large networks tomeet the unpredictable and highly variable demand from electric generators. Thelinear and corresponding nonlinear OCPs are discretized in time to obtainlinear and nonlinear optimization problems, which are demonstrated on a testnetwork to illustrate the validity of linear programming. Moreover, ananalytical bound on the error between the solutions of the linear and nonlinearflow dynamics is presented using Lyapunov functions and verifiedcomputationally by plotting the error against the size of the flow variationaround the steady-state solution.

Linear System Analysis and Optimal Control of Natural Gas Dynamics in Pipeline Networks