The fully implicit finite-difference method is used to solve the continuity, momentum, and energy equations for flow within a gas pipeline. This methodology (1) incorporates the convective inertia term in the conservation of momentum equation, (2) treats the compressibility factor as a function of temperature and pressure, and (3) considers the friction factor as a function of the Reynolds number and pipe roughness. The fully implicit method representation of the equations offers the advantage of guaranteed stability for a large time step, which is very useful for gas pipeline industry. The results show that the effect of treating the gas in a nonisothermal manner is extremely necessary for pipeline flow calculation accuracies, especially for rapid transient process. It also indicates that the convective inertia term plays an important role in the gas flow analysis and cannot be neglected from the calculation.
A detailed mathematical model of compressor stations and pipes is essential for optimizing the performance of the gas pipeline system. Most of the available literature on compressor station modeling is based on isothermal solutions for pipe flow, which is inadequate for our purposes. In the present work, the pipe flow is treated as nonisothermal unsteady one-dimensional compressible flow. This is accomplished by treating the compressibility factor as a function of pressure and temperature, and the friction factor as a function of Reynolds number. The solution method is the fully implicit finite difference method that provides solution stability, even for relatively large time steps. The compressors within the compressor station are modeled using centrifugal compressor map-based polynomial equations. This modeling technique permits the designation of different models of compressors in the compressor station. The method can be easily extended to include other types of compressors. Using this mathematical model as a basis, a nonlinear programing problem (NLP) is set up wherein the design variables are the compressor speeds and the objective function to be minimized is the total fuel consumption. The minimum acceptable throughput is imposed as a constraint. This NLP is solved numerically by a sequential unconstrained minimization technique, using the mathematical model of the system for the required function evaluations. The results show that this approach is very effective in reducing fuel consumption. An application of this methodology for selecting the number of compressors to be shut down for the most fuel-efficient operation is also presented. Our results further indicate that station-level optimization produces results comparable to those obtained by network-level optimization. This is very significant because it implies that the optimization can be done locally at the station level, which is computationally much easier.
One of the key factors in the operation of a natural gas pipeline network is the linepack in the network. The desired operation of the network as derived from estimated receipts and deliveries is expressed in terms of the desired linepack profile that must be maintained. The compressor stations in the pipeline network are then operated in a manner that generates this linepack profile. Generally, the operating points selected for the units in the compressor stations are based on experience and experimentation and are therefore not optimal. In this paper, we present a systematic approach for operating the units of a compressor station to meet a specified linepack profile. The first step in developing this approach is the derivation of a numerical method for analyzing the flow through the pipeline under transient nonisothermal conditions. We have developed and verified a fully implicit finite difference formulation that provides this analysis capability. Next, the optimization of the compressor stations is formulated as a standard nonlinear programing problem in the following form: Find the values in the design variable vector denoted by b=[b1,b2,…,bn]T, to minimize a given objective function F(b), subject to the constraints gj(b)⩽0, j=1,…,m. Here, n is the number of operational parameters whose optimal value is to be determined, while m is the number of operational constraints that must be enforced. In our formulation, the design variables are chosen to be the operating speeds of the units in the compressor stations, while the objective function is taken to be the average fuel consumption rate over the interval of interest, summed over all units. The constraint functions gj(b) are formulated suitably to ensure that operational limits are met at the final solution that is obtained. The optimization problem is then solved using a sequential unconstrained minimization technique (SUMT), in conjunction with a directed grid search method for solving the unconstrained subproblems that are encountered in the SUMT formulation. The evaluation of the objective function and constraint functions at each step of the optimization is done by using the fully implicit analysis method mentioned above. A representative numerical example has been solved by the proposed approach. The results obtained indicate that the method is very effective in finding operating points that are optimal with respect to fuel consumption. The optimization can be done at the level of a single unit, a single compressor station, a set of compressor stations, or an entire network. It should also be noted that the proposed solution approach is fully automated and requires no user involvement in the solution process.
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