Solving power-constrained gas transportation problems using an MIP-based alternating direction method

We present a solution algorithm for problems from steady-state gas transport optimization. Due to nonlinear and nonconvex physics and engineering models as well as discrete controllability of active network devices, these problems lead to hard nonconvex mixed-integer nonlinear optimization models. The proposed method is based on mixed-integer linear techniques using piecewise linear relaxations of the nonlinearities and a tailored alternating direction method. In addition to most other publications in the field of gas transport optimization, we do not only consider pressure and flow as main physical quantities but further incorporate heat power supplies and demands as well as a mixing model for different gas qualities. We demonstrate the capabilities of our method on Germany’s largest transport networks and hereby present numerical results on the largest instances that were ever reported in the literature for this problem class. The optimization of real-world gas transport networks is a mathematically challenging task. The combination of highly nonlinear and nonconvex models of gas dynamics and engineering together with the discrete nature of controllable network devices leads to large-scale nonconvex mixed-integer nonlinear optimization or feasibility problems (MINLPs). The size of the networks result in instances that are far beyond of being solvable by general-purpose MINLP solvers and thus call for tailored optimization methods. The main contribution of this article is a tailored alternating direction method (ADM) that is based on mixed-integer linear (MIP) models. This combination enables us to solve MINLP problems on gas transport networks of sizes that were never reported before in the literature. Furthermore, we extend standard models of gas transport such that mixing of different gas qualities is also considered. In this way, we can satisfy heat power constraints that are of crucial importance in practice. The standard setting of gas network optimization is to prescribe bounds on the pressure and the mass flow at supply and demand nodes. Using mass flow is the natural setting from the physical point of view as mass flow is a conserved quantity. From the point of view of a network planner, however, the more natural quantity is heat power as it is the quantity that is used in contracts with the customers. Heat power is simply the product of mass flow and the calorific value of the gas. At supply nodes, the calorific value is given so that there is no distinction between these two notions. For demand nodes, however, the calorific value depends on the mixture of the gas, which is highly dependent on the specific flow situation and thus cannot be computed a priori. The gas quality of different sources in typical European networks vary such that the effects of mixing are substantial. Thus, for the practical application of automated methods in network planning it is important Date: November 27, 2014. 2010 Mathematics Subject Classification. 65K10, 90-08, 90B10, 90C06, 90C11, 90C35, 90C90.