Model relaxations for the fuel cost minimization of steady-state gas pipeline networks

Natural gas, driven by pressure, is transported through pipeline network systems. As the gas flows through the network, energy and pressure are lost due to both friction between the gas and the pipesʹ inner wall, and heat transfer between the gas and its environment. The lost energy of the gas is periodically restored at the compressor stations which are installed in the network. These compressor stations typically consume about 3–5% of the transported gas. This transportation cost is significant because the amount of gas being transported worldwide is huge. These facts make the problem of how to optimally operate the compressors driving the gas in a pipeline network important. s paper, we address the problem of minimizing the fuel cost incurred by the compressor stations driving the gas in a transmission network under steady-state assumptions. In particular, the decision variables include pressure drops at each node of the network, mass flow rate at each pipeline leg, and the number of units operating within each compressor station. We present a mathematical model of this problem and an in-depth study of the underlying mathematical structure of the compressor stations. Then, based on this study, we propose two model relaxations (one in the compressor domain and another in the fuel cost function) and derive a lower bounding scheme. We also present empirical evidence that shows the effectiveness of the lower bounding scheme. For the small problems, where we were able to find optimal solutions, the proposed lower bound yields a relative optimality gap of around 15–20%. For a larger, more complex instance, it was not possible to find optimal solutions, but we were able to compute lower and upper bounds, finding a large relative gap between the two. We show this wide gap is mainly due to the presence of nonconvexity in the set of feasible solutions, since the proposed relaxations do a very good job of approximating the problem within each individual compressor station. hasize that this is, to the best of our knowledge, the first time such a procedure (lower bound) has been proposed in over 30 years of research in the natural gas pipeline area