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

Abstract 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 difficult 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. Most other publications in the field of gas transport optimization only consider pressure and flow as main physical quantities. In this work, we additionally 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.

[1]  Miloslav Feistauer,et al.  Mathematical Methods in Fluid Dynamics , 1993 .

[2]  Alexander Martin,et al.  Mixed integer linear models for the optimization of dynamical transport networks , 2011, Math. Methods Oper. Res..

[3]  C. Lawson.  Characteristic properties of the segmented rational minmax approximation problem , 1964 .

[4]  Mohammed Alfaki,et al.  Strong formulations for the pooling problem , 2013, J. Glob. Optim..

[5]  Andrzej J. Osiadacz,et al.  Comparison of isothermal and non-isothermal pipeline gas flow models , 2001 .

[6]  R. Glowinski,et al.  Sur l'approximation, par éléments finis d'ordre un, et la résolution, par pénalisation-dualité d'une classe de problèmes de Dirichlet non linéaires , 1975 .

[7]  Claude Le Pape,et al.  Exploring relaxation induced neighborhoods to improve MIP solutions , 2005, Math. Program..

[8]  M. Powell,et al.  Approximation theory and methods , 1984 .

[9]  T. Pavlidis,et al.  Uniform piecewise polynomial approximation with variable joints , 1974 .

[10]  Christodoulos A. Floudas,et al.  ADVANCES FOR THE POOLING PROBLEM: MODELING, GLOBAL OPTIMIZATION, AND COMPUTATIONAL STUDIES , 2009 .

[11]  P. I. Barton,et al.  Stochastic pooling problem for natural gas production network design and operation under uncertainty , 2011 .

[12]  Björn Geißler,et al.  A New Algorithm for MINLP Applied to Gas Transport Energy Cost Minimization , 2013 .

[13]  Stephen J. Wright,et al.  Numerical Optimization (Springer Series in Operations Research and Financial Engineering) , 2000 .

[14]  Alexander Martin,et al.  Combination of Nonlinear and Linear Optimization of Transient Gas Networks , 2011, INFORMS J. Comput..

[15]  Stephen P. Boyd,et al.  Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers , 2011, Found. Trends Mach. Learn..

[16]  James Williamson,et al.  THE LAWS OF FLOW IN ROUGH PIPES , 1951 .

[17]  Mikhail V. Lurie,et al.  Modeling of Oil Product and Gas Pipeline Transportation , 2008 .

[18]  Martin Schmidt,et al.  High detail stationary optimization models for gas networks , 2015 .

[19]  F. Clarke.  Optimization And Nonsmooth Analysis , 1983 .

[20]  Martin Schmidt,et al.  High detail stationary optimization models for gas networks: validation and results , 2016 .

[21]  Björn Geißler,et al.  Using Piecewise Linear Functions for Solving MINLP s , 2012 .

[22]  John N. Tsitsiklis,et al.  Parallel and distributed computation , 1989 .

[23]  B. Mercier,et al.  A dual algorithm for the solution of nonlinear variational problems via finite element approximation , 1976 .

[24]  Martin Schmidt,et al.  A Primal Heuristic for Nonsmooth Mixed Integer Nonlinear Optimization , 2013 .

[25]  Joseph B. Franzini,et al.  Fluid Mechanics with Engineering Applications. 6th Ed. By R. L.DAUGHERTY and J. B. FRANZINI. McGraw-Hill. 1965. 574 pp. $9.95 or 80s. Fluid Dynamics. By J. W. DAILY and D. R. F. HARLEMAN. Addison Wesley. 1966. 454 pp. $12.50 or 94s. , 1967, Journal of Fluid Mechanics.

[26]  Armin Fügenschuh,et al.  Mathematical optimization for challenging network planning problems in unbundled liberalized gas markets , 2014 .

[27]  Michael Jünger,et al.  Facets of Combinatorial Optimization , 2013 .

[28]  Frode Rømo,et al.  Optimization Models for the Natural Gas Value Chain , 2007, Geometric Modelling, Numerical Simulation, and Optimization.

[29]  Jamal Saleh,et al.  Fluid Flow Handbook , 2002 .

[30]  Tom van der Hoeven,et al.  Math in gas and the art of linearization , 2004 .

[31]  Armin Fügenschuh,et al.  Validation of nominations in gas network optimization: models, methods, and solutions , 2015, Optim. Methods Softw..

[32]  Richard E. Wendell,et al.  Minimization of a Non-Separable Objective Function Subject to Disjoint Constraints , 1976, Oper. Res..

[33]  Kathrin Klamroth,et al.  Biconvex sets and optimization with biconvex functions: a survey and extensions , 2007, Math. Methods Oper. Res..

[34]  Thorsten Koch,et al.  Evaluating Gas Network Capacities , 2015, MOS-SIAM Series on Optimization.