Control Strategies for Transport Networks under Demand Uncertainty

In this article, we consider transport networks with uncertain demands. Network dynamics are given by linear hyperbolic partial differential equations and suitable coupling conditions, while demands are incorporated as solutions to stochastic differential equations. For the demand satisfaction, we solve a constrained optimal control problem. Controls in terms of network inputs are then calculated explicitly for different assumptions. Numerical simulations are performed to underline the theoretical results.

[1]  Achim Klenke,et al.  Probability theory - a comprehensive course , 2008, Universitext.

[2]  Peter Benner,et al.  Gas Network Benchmark Models , 2018 .

[3]  Ravindra K. Ahuja,et al.  Network Flows , 2011 .

[4]  H. T. Papadopoulos,et al.  Queueing theory in manufacturing systems analysis and design: A classification of models for production and transfer lines , 1996 .

[5]  D. M. Miller,et al.  Application of discrete event simulation in production scheduling , 1998, 1998 Winter Simulation Conference. Proceedings (Cat. No.98CH36274).

[6]  Michele D. Simoni,et al.  A Fast Lax-Hopf Algorithm to Solve the Lighthill-Whitham-Richards Traffic Flow Model on Networks , 2020, Transp. Sci..

[7]  M. Herty,et al.  Network models for supply chains , 2005 .

[8]  Christian A. Ringhofer,et al.  A Model for the Dynamics of large Queuing Networks and Supply Chains , 2006, SIAM J. Appl. Math..

[9]  Mauro Garavello,et al.  Flows on networks: recent results and perspectives , 2014 .

[10]  M. Sørensen,et al.  The Pearson Diffusions: A Class of Statistically Tractable Diffusion Processes , 2007 .

[11]  Simone Göttlich,et al.  A model for a network of conveyor belts with discontinuous speed and capacity , 2018, Networks Heterog. Media.

[12]  Alexander Brodsky,et al.  SC-CoJava: A Service Composition Language to Unify Simulation and Optimization of Supply Chains , 2012 .

[13]  T. Ware,et al.  Polynomial Processes for Power Prices , 2017, Applied Mathematical Finance.

[14]  B. Øksendal.  Stochastic differential equations : an introduction with applications , 1987 .

[15]  Michael Herty,et al.  Electric transmission lines: Control and numerical discretization , 2016 .

[16]  R. Korn,et al.  Optimal Inflow Control Penalizing Undersupply in Transport Systems with Uncertain Demands , 2019, Progress in Industrial Mathematics at ECMI 2018.

[17]  Mauro Garavello,et al.  Traffic Flow on Networks , 2006 .

[18]  Axel Klar,et al.  Gas flow in pipeline networks , 2006, Networks Heterog. Media.

[19]  Eric V. Denardo,et al.  Flows in Networks , 2011 .

[20]  Tadashi Dohi,et al.  Optimal production planning under diffusion demand pattern , 1995 .

[21]  Guenter Leugering,et al.  On the Modelling and Stabilization of Flows in Networks of Open Canals , 2002, SIAM J. Control. Optim..

[22]  Mathematical Modeling, Simulation and Optimization for Power Engineering and Management , 2021, Mathematics in Industry.

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

[24]  Freddy Delbaen,et al.  An Interest Rate Model with Upper and Lower Bounds , 2002 .

[25]  R. LeVeque.  Finite Volume Methods for Hyperbolic Problems: Characteristics and Riemann Problems for Linear Hyperbolic Equations , 2002 .

[26]  Simone Göttlich,et al.  Chance‐constrained optimal inflow control in hyperbolic supply systems with uncertain demand , 2020, Optimal Control Applications and Methods.

[27]  Simone Göttlich,et al.  Optimal control of electricity input given an uncertain demand , 2018, Mathematical Methods of Operations Research.

[28]  C. Gouriéroux,et al.  Multivariate Jacobi process with application to smooth transitions , 2006 .

[29]  R. Korn,et al.  Modeling the Intraday Electricity Demand in Germany , 2021 .

[30]  Ali Gharbi,et al.  Optimal Production Control of Hybrid Manufacturing/Remanufacturing Failure-Prone Systems under Diffusion-Type Demand , 2013 .