# American Institute of Mathematical Sciences

June  2020, 15(2): 171-195. doi: 10.3934/nhm.2020008

## Convexity and starshapedness of feasible sets in stationary flow networks

 1 Department Mathematik, Friedrich-Alexander Universität Erlangen-Nürnberg, Cauerstraße 11, 91058 Erlangen, Germany 2 Fakultät für Mathematik, Universitäet Duisburg-Essen, Thea-Leymann-Strasse 9, 45127 Essen, Germany

* Corresponding author: Michael Schuster

Received  January 2019 Revised  October 2019

Fund Project: The authors are supported by DFG grant CRC/Transregio 154

In this paper, we consider a stationary model for the flow through a network. The flow is determined by the values at the boundary nodes of the network. We call these values the loads of the network. In the applications, the feasible loads must satisfy some box constraints. We analyze the structure of the set of feasible loads. Our analysis is motivated by gas pipeline flows, where the box constraints are pressure bounds.

We present sufficient conditions that imply that the feasible set is star-shaped with respect to special points. Under stronger conditions, we prove the convexity of the set of feasible loads. All the results are given for passive networks with and without compressor stations.

This analysis is motivated by the aim to use the spheric-radial decomposition for stochastic boundary data in this model. This paper can be used for simplifying the algorithmic use of the spheric-radial decomposition.

Citation: Martin Gugat, Rüdiger Schultz, Michael Schuster. Convexity and starshapedness of feasible sets in stationary flow networks. Networks & Heterogeneous Media, 2020, 15 (2) : 171-195. doi: 10.3934/nhm.2020008
##### References:
 [1] M. K. Banda, M. Herty and A. Klar, Coupling conditions for gas networks governed by the isothermal Euler equations, Netw. Heterog. Media, 1 (2006), 295-314.  doi: 10.3934/nhm.2006.1.295.  Google Scholar [2] M. K. Banda, M. Herty and A. Klar, Gas flow in pipeline networks, Netw. Heterog. Media, 1 (2006), 41-56.  doi: 10.3934/nhm.2006.1.41.  Google Scholar [3] A. Bermúdez, J. González-Díaz, F. J. González-Diéguez, M. González-Rueda and M. P. Fernández de Córdoba, Simulation and optimization models of steady-state gas transmission networks, Energy Procedia, 64 (2015), 130-139.   Google Scholar [4] R. M. Colombo and C. Mauri, Euler systems for compressible fluids at a junction, J. Hyperbolic Differ. Equ., 5 (2008), 547-568.  doi: 10.1142/S0219891608001593.  Google Scholar [5] P. Domschke, B. Hiller, J. Lang and C. Tischendorf, Modellierung von Gasnetzwerken: Eine Übersicht, (2019), Preprint on Webpage http://tubiblio.ulb.tu-darmstadt.de/106763/. Google Scholar [6] M. H. Farshbaf-Shaker, R. Henrion and D. Hömberg, Properties of chance constraints in infinite dimensions with an application to PDE constrained optimization, Set-Valued Var. Anal., 26 (2018), 821-841.  doi: 10.1007/s11228-017-0452-5.  Google Scholar [7] A. Genz and F. Bretz, Computation of Multivariate Normal and $t$ Probabilities, Lecture Notes in Statistics, 195. Springer, Dordrecht, 2009. doi: 10.1007/978-3-642-01689-9.  Google Scholar [8] T. González Gradón, H. Heitsch and R. Henrion, A joint model of probabilistic/robust constraints for gas transport management in stationary networks, Comput. Manag. Sci., 14 (2017), 443-460.  doi: 10.1007/s10287-017-0284-7.  Google Scholar [9] C. Gotzes, H. Heitsch, R. Henrion and R. Schultz, On the quantification of nomination feasibility in stationary gas networks with random load, Math. Methods Oper. Res., 84 (2016), 427-457.  doi: 10.1007/s00186-016-0564-y.  Google Scholar [10] M. Gugat, F. M. Hante, M. Hirsch-Dick and G. Leugering, Stationary states in gas networks, Netw. Heterog. Media, 10 (2015), 295-320.  doi: 10.3934/nhm.2015.10.295.  Google Scholar [11] M. Gugat, R. Schultz and D. Wintergerst, Networks of pipelines for gas with nonconstant compressibility factor: Stationary states, J. Comput. Appl. Math., 37 (2018), 1066-1097.  doi: 10.1007/s40314-016-0383-z.  Google Scholar [12] M. Gugat and M. Schuster, Stationary gas networks with compressor control and random loads: Optimization with probabilistic constraints, Math. Probl. Eng., 2018 (2018), Art. ID 7984079, 17 pp. doi: 10.1155/2018/7984079.  Google Scholar [13] M. Gugat and S. Ulbrich, The isothermal Euler equations for ideal gas with source term: Product solutions, flow reversal and no blow up, J. Math. Anal. Appl., 454 (2017), 439-452.  doi: 10.1016/j.jmaa.2017.04.064.  Google Scholar [14] M. Gugat and D. Wintergerst, Transient flow in gas networks: Traveling waves, Int. J. Appl. Math. Comput. Sci., 28 (2018), 341-348.  doi: 10.2478/amcs-2018-0025.  Google Scholar [15] H. Heitsch, On probabilistic capacity maximization in a stationary gas network, Optimization, 69 (2020), 575–604, Preprint on Webpage http://wias-berlin.de/publications/wias-publ/run.jsp?template=abstract&type=Preprint&year=2018&number=2540. doi: 10.1080/02331934.2019.1625353.  Google Scholar [16] T. Koch, B. Hiller, M. E. Pfetsch and L. Schewe, Evaluating Gas Network Capacities, MOS-SIAM Series on Optimization, 21. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2015. doi: 978-1-611973-68-6.  Google Scholar [17] A. Prékopa, Stochastic Programming, Mathematics and its Applications, 324. Kluwer Academic Publishers Group, Dordrecht, 1995. doi: 10.1007/978-94-017-3087-7.  Google Scholar [18] D. Rose, M. Schmidt, M. C. Steinbach and B. M. Willert, Computational optimization of gas compressor stations: MINLP models versus continuous reformulations, Math. Methods Oper. Res., 83 (2016), 409-444.  doi: 10.1007/s00186-016-0533-5.  Google Scholar [19] J. O. Royset and E. Polak, Extensions of stochastic optimization results to problems with system failure probability functions, J. Optim. Theory Appl., 133 (2007), 1-18.  doi: 10.1007/s10957-007-9178-0.  Google Scholar [20] L. Schewe, M. Schmidt and J. Thürauf, Structural properties of feasible bookings in the European entry-exit gas market system, 4OR, (2019). doi: 10.1007/s10288-019-00411-3.  Google Scholar [21] W. van Ackooij, I. Aleksovska and M. Munoz-Zuniga, (Sub-)Differentiability of probabilistic functions with elliptical distributions, Set-Valued Var. Anal., 26 (2018), 887-910.  doi: 10.1007/s11228-017-0454-3.  Google Scholar [22] W. van Ackooij and R. Henrion, Gradient formulae for nonlinear probabilistic constraints with Gaussian and Gaussian-like distribution, SIAM J. Optim., 24 (2014), 1864-1889.  doi: 10.1137/130922689.  Google Scholar [23] D. Wintergerst, Application of chance constrained optimization to gas networks, (2019), Preprint on Webpage https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/start/5/rows/10/sortfield/score/sortorder/desc/searchtype/simple/query/wintergerst/docId/158. Google Scholar

show all references

##### References:
 [1] M. K. Banda, M. Herty and A. Klar, Coupling conditions for gas networks governed by the isothermal Euler equations, Netw. Heterog. Media, 1 (2006), 295-314.  doi: 10.3934/nhm.2006.1.295.  Google Scholar [2] M. K. Banda, M. Herty and A. Klar, Gas flow in pipeline networks, Netw. Heterog. Media, 1 (2006), 41-56.  doi: 10.3934/nhm.2006.1.41.  Google Scholar [3] A. Bermúdez, J. González-Díaz, F. J. González-Diéguez, M. González-Rueda and M. P. Fernández de Córdoba, Simulation and optimization models of steady-state gas transmission networks, Energy Procedia, 64 (2015), 130-139.   Google Scholar [4] R. M. Colombo and C. Mauri, Euler systems for compressible fluids at a junction, J. Hyperbolic Differ. Equ., 5 (2008), 547-568.  doi: 10.1142/S0219891608001593.  Google Scholar [5] P. Domschke, B. Hiller, J. Lang and C. Tischendorf, Modellierung von Gasnetzwerken: Eine Übersicht, (2019), Preprint on Webpage http://tubiblio.ulb.tu-darmstadt.de/106763/. Google Scholar [6] M. H. Farshbaf-Shaker, R. Henrion and D. Hömberg, Properties of chance constraints in infinite dimensions with an application to PDE constrained optimization, Set-Valued Var. Anal., 26 (2018), 821-841.  doi: 10.1007/s11228-017-0452-5.  Google Scholar [7] A. Genz and F. Bretz, Computation of Multivariate Normal and $t$ Probabilities, Lecture Notes in Statistics, 195. Springer, Dordrecht, 2009. doi: 10.1007/978-3-642-01689-9.  Google Scholar [8] T. González Gradón, H. Heitsch and R. Henrion, A joint model of probabilistic/robust constraints for gas transport management in stationary networks, Comput. Manag. Sci., 14 (2017), 443-460.  doi: 10.1007/s10287-017-0284-7.  Google Scholar [9] C. Gotzes, H. Heitsch, R. Henrion and R. Schultz, On the quantification of nomination feasibility in stationary gas networks with random load, Math. Methods Oper. Res., 84 (2016), 427-457.  doi: 10.1007/s00186-016-0564-y.  Google Scholar [10] M. Gugat, F. M. Hante, M. Hirsch-Dick and G. Leugering, Stationary states in gas networks, Netw. Heterog. Media, 10 (2015), 295-320.  doi: 10.3934/nhm.2015.10.295.  Google Scholar [11] M. Gugat, R. Schultz and D. Wintergerst, Networks of pipelines for gas with nonconstant compressibility factor: Stationary states, J. Comput. Appl. Math., 37 (2018), 1066-1097.  doi: 10.1007/s40314-016-0383-z.  Google Scholar [12] M. Gugat and M. Schuster, Stationary gas networks with compressor control and random loads: Optimization with probabilistic constraints, Math. Probl. Eng., 2018 (2018), Art. ID 7984079, 17 pp. doi: 10.1155/2018/7984079.  Google Scholar [13] M. Gugat and S. Ulbrich, The isothermal Euler equations for ideal gas with source term: Product solutions, flow reversal and no blow up, J. Math. Anal. Appl., 454 (2017), 439-452.  doi: 10.1016/j.jmaa.2017.04.064.  Google Scholar [14] M. Gugat and D. Wintergerst, Transient flow in gas networks: Traveling waves, Int. J. Appl. Math. Comput. Sci., 28 (2018), 341-348.  doi: 10.2478/amcs-2018-0025.  Google Scholar [15] H. Heitsch, On probabilistic capacity maximization in a stationary gas network, Optimization, 69 (2020), 575–604, Preprint on Webpage http://wias-berlin.de/publications/wias-publ/run.jsp?template=abstract&type=Preprint&year=2018&number=2540. doi: 10.1080/02331934.2019.1625353.  Google Scholar [16] T. Koch, B. Hiller, M. E. Pfetsch and L. Schewe, Evaluating Gas Network Capacities, MOS-SIAM Series on Optimization, 21. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2015. doi: 978-1-611973-68-6.  Google Scholar [17] A. Prékopa, Stochastic Programming, Mathematics and its Applications, 324. Kluwer Academic Publishers Group, Dordrecht, 1995. doi: 10.1007/978-94-017-3087-7.  Google Scholar [18] D. Rose, M. Schmidt, M. C. Steinbach and B. M. Willert, Computational optimization of gas compressor stations: MINLP models versus continuous reformulations, Math. Methods Oper. Res., 83 (2016), 409-444.  doi: 10.1007/s00186-016-0533-5.  Google Scholar [19] J. O. Royset and E. Polak, Extensions of stochastic optimization results to problems with system failure probability functions, J. Optim. Theory Appl., 133 (2007), 1-18.  doi: 10.1007/s10957-007-9178-0.  Google Scholar [20] L. Schewe, M. Schmidt and J. Thürauf, Structural properties of feasible bookings in the European entry-exit gas market system, 4OR, (2019). doi: 10.1007/s10288-019-00411-3.  Google Scholar [21] W. van Ackooij, I. Aleksovska and M. Munoz-Zuniga, (Sub-)Differentiability of probabilistic functions with elliptical distributions, Set-Valued Var. Anal., 26 (2018), 887-910.  doi: 10.1007/s11228-017-0454-3.  Google Scholar [22] W. van Ackooij and R. Henrion, Gradient formulae for nonlinear probabilistic constraints with Gaussian and Gaussian-like distribution, SIAM J. Optim., 24 (2014), 1864-1889.  doi: 10.1137/130922689.  Google Scholar [23] D. Wintergerst, Application of chance constrained optimization to gas networks, (2019), Preprint on Webpage https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/start/5/rows/10/sortfield/score/sortorder/desc/searchtype/simple/query/wintergerst/docId/158. Google Scholar
Example of differently structured graphs
Example for illustrating the notation (graph numbered by breadth-first search)
Example for illustrating the triple and quadruple indices on the path from $G_1$ to $G_5$
Linear graph of example 1
Feasible sets for different pressure bounds
Linear graph of example 2
Set $M_{a}$ for $(p^{+, \min})_{a} = [1, 1, 1, 1]^T$ and $(p^{+, \max})_{a} = [3, 3, 3, 3]^T$
Set $M_{b}$ for $(p^{+, \min})_{b} = [2.5, 2, 1.5, 1]^T$ and $(p^{+, \max})_{b} = [3, 2.5, 2, 1.5]^T$
Linear graph with one compressor edge of example 3
Set $M_{a}$ for $(p^{+, \min})_{a} = [1, 1, 1, 1]^T$ and $(p^{+, \max})_{a} = [3, 3, 3, 3]^T$
Set $M_{b}$ for $(p^{+, \min})_{b} = [2.5, 2, 1.5, 1]^T$ and $(p^{+, \max})_{b} = [3, 2.5, 2, 1.5]^T$
Graph of example 2
Feasible sets for different pressure bounds
Graph of example 5
Set $M_{a}$ for $(p^{+, \min})_{a} = [1, 1, 1, 1]^T$ and $(p^{+, \max})_{a} = [3, 2, 2, 2]^T$
Set $M_{a}$ for $(p^{+, \min})_{b} = [2, 1, 1, 1]^T$ and $(p^{+, \max})_{b} = [3, 2, 2, 1.5]^T$
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