Minimizing fuel consumption of a gas pipeline in transient states by dynamic programming

Gas pipelines are often operated in transient states because of time-varying consumer demands for natural gas. For a certain operating period of a gas pipeline, given the delivery rates at the pipeline's delivery points and the flow rates or pressures of its gas sources as functions of time, pipeline dispatchers face a transient optimization problem with the objective of minimizing pipeline fuel consumption for the period. To address this problem, a two-level model is constructed. The model includes a pipeline level that optimizes the discharge pressure of each compressor station and a station level that optimizes the operating scheme of each compressor station. A key assumption, that the discharge pressures of each compressor station on the pipeline are time-independent, is made to simplify the problem. Based on this assumption, a dynamic programming (DP)-based algorithm is proposed for the pipeline level in which the discharge pressures are chosen as the state variables. The algorithm is essentially a recursion from the pipeline destination to the origin. The i-th stage of the recursion covers the i-th and the (i + 1)-th compressor stations together with the pipeline segment between them, and the state transition of the stage reflects the relationship between the discharge pressures of the two compressor stations. Unlike in steady-state operation, determining this relationship involves the partial differential equations governing transient gas flow in the pipeline segment between two adjacent stations. Because these equations must be solved by numerical simulation, it is much more difficult to determine the relationship for transient operation than for steady-state operation. To compress the gas going through a compressor station to a compression ratio assigned at the pipeline level, the operating scheme of the station is solved at the station level. A preprocessing method is devised to reduce the computational time of the DP approach. The method approximates the transient optimization problem as a series of steady-state optimization problems. The results provide information that can narrow the discharge pressure region of each compressor station. A simple example shows that the approximate minimum fuel consumption given by the proposed algorithm is 6.2% higher than the true optimum. Typical cases for different operating periods are tested on a gas pipeline. The proposed approach is also tested on a network studied in the literature. For most of the case studies, optimal solutions can be computed by the proposed approach within 15 min. In addition, the computational complexity grows modestly as the period becomes longer.