Solving Highly Detailed Gas Transport MINLPs: Block Separability and Penalty Alternating Direction Methods

Detailed modeling of gas transport problems leads to nonlinear and nonconvex mixed-integer optimization or feasibility models (MINLPs) because both the incorporation of discrete controls of the network and accurate physical and technical modeling are required to achieve practical solutions. Hence, ignoring certain parts of the physics model is not valid for practice. In the present contribution we extend an approach based on linear relaxations of the underlying nonlinearities by tailored model reformulation techniques yielding block-separable MINLPs. This combination of techniques allows us to apply a penalty alternating direction method and thus to solve highly detailed MINLPs for large-scale, real-world instances. The practical strength of the proposed method is demonstrated by a computational study in which we apply the method to instances from steady-state gas transport including both pooling effects with respect to the mixing of gases of different composition and a highly detailed compressor station ...

[1]  Lorenz T. Biegler,et al.  MPEC problem formulations and solution strategies with chemical engineering applications , 2008, Comput. Chem. Eng..

[2]  Jon W. Tolle,et al.  Exact penalty functions in nonlinear programming , 1973, Math. Program..

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

[4]  Roger Z. Ríos-Mercado,et al.  Optimization problems in natural gas transportation systems. A state-of-the-art review , 2015 .

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

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

[7]  Björn Geißler,et al.  Solving power-constrained gas transportation problems using an MIP-based alternating direction method , 2015, Comput. Chem. Eng..

[8]  Martin Schmidt,et al.  A generic interior-point framework for nonsmooth and complementarity constrained nonlinear optimization , 2013 .

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

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

[11]  Olvi L. Mangasarian,et al.  Exact penalty functions in nonlinear programming , 1979, Math. Program..

[12]  Nikolaos V. Sahinidis,et al.  A polyhedral branch-and-cut approach to global optimization , 2005, Math. Program..

[13]  Björn Geißler,et al.  Penalty Alternating Direction Methods for Mixed-Integer Optimization: A New View on Feasibility Pumps , 2017, SIAM J. Optim..

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

[15]  I. Nowak.  Relaxation and Decomposition Methods for Mixed Integer Nonlinear Programming , 2005 .

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

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

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

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

[20]  Alexander Martin,et al.  A mixed integer approach for time-dependent gas network optimization , 2010, Optim. Methods Softw..

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

[22]  Thorsten Koch,et al.  Mathematical optimization for evaluating gas network capacities , 2015 .

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

[24]  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 .

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

[26]  Armin Fügenschuh,et al.  Chapter 2: Physical and technical fundamentals of gas networks , 2015, Evaluating Gas Network Capacities.

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

[28]  Arne Stolbjerg Drud,et al.  CONOPT - A Large-Scale GRG Code , 1994, INFORMS J. Comput..

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

[30]  Alexander Martin,et al.  Mixed Integer Models for the Stationary Case of Gas Network Optimization , 2006, Math. Program..

[31]  Marc C. Steinbach,et al.  Computational optimization of gas compressor stations: MINLP models versus continuous reformulations , 2016, Mathematical Methods of Operations Research.

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