Modeling and optimizing a road de-icing device by a nonlinear heating

Bernardin, Frederic; Munch, Arnaud

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 53 (2019) no. 3, pp. 775-803.

Suite au passage du modèle économique de la revue en S20, le texte intégral des articles des années 2019 et 2020 est accessible uniquement sur le site de l'éditeur et est réservé aux abonnés.

In order to design a road de-icing device by heating, we consider in the one dimensional setting the optimal control of a parabolic equation with a nonlinear boundary condition of the Stefan–Boltzmann type. Both the punctual control and the corresponding state are subjected to a unilateral constraint. This control problem models the heating of a road during a winter period to keep the road surface temperature above a given threshold. The one-dimensional modeling used in this work is a first step of the modeling of a road heating device through the circulation of a coolant in a porous layer of the road. We first prove, under realistic physical assumptions, the well-posedness of the direct problem and the optimal control problem. We then perform some numerical experiments using real data obtained from experimental measurements. This model and the corresponding numerical results allow to quantify the minimal energy to be provided to keep the road surface without frost or snow.

[1] F. Ammar-Khodja, S. Micu and A. Münch, Controllability of a string submitted to unilateral constraint. Ann. Inst. Henri Poincaré Anal. Non Linéaire 27 (2010) 1097–1119. | DOI | Numdam | MR | Zbl

[2] S. Asfour, F. Bernardin and E. Toussaint, Experimental validation of 2d hydrothermal modelling of porous pavement for heating and solar energy retrieving applications. To appear in: Road Mater. Pavement Des. J. doi: 2018). | DOI

[3] S. Asfour, F. Bernardin, E. Toussaint and J.-M. Piau, Hydrothermal modeling of porous pavement for its surface de-freezing. Appl. Therm. Eng. 107 (2016) 493–500. | DOI

[4] B. Azmi and K. Kunisch, Receding horizon control for the stabilization of the wave equation. Discrete Contin. Dyn. Syst. 38 (2018) 449–484. | DOI | MR | Zbl

[5] L. Bouilloud and E. Martin, A coupled model to simulate snow behavior on roads. J. Appl. Meteorol. Climatol. 45 (2006) 500–516. | DOI

[6] E. Casas, C. Clason and K. Kunisch, Parabolic control problems in measure spaces with sparse solutions. SIAM J. Control Optim. 51 (2013) 28–63. | DOI | MR | Zbl

[7] M. Chen, S. Wu, H. Wang and J. Zhang, Study of ice and snow melting process on conductive asphalt solar collector. Solar Energy Mater. Solar Cells 95 (2011) 3241–3250. | DOI

[8] L.-P. Crevier and Y. Delage, Metro: a new model for road-condition forecasting in canada. J. Appl. Meteorol. 40 (2001) 2026–2037. | DOI

[9] G. Duvaut and J.-L. Lions, Inequalities in mechanics and physics. Translated from the French by C.W. John, Grundlehren der Mathematischen Wissenschaften 219. Springer-Verlag, Berlin-New York (1976). | DOI | MR | Zbl

[10] W. Eugster, Road and bridge heating using geothermal energy. overview and examples. In: Proceedings European Geothermal Congress 2007 Unterhaching (2007).

[11] R. Mallick, B. Chen, S. Bhowmick and M. Huken, Capturing solar energy from asphalt pavements. In: International ISAP Symposium on Asphalt Pavements and Environment. Zurich, Switzerland (2008).

[12] A. Mammeri, L. Ulmet, C. Petit and A. Mokhtari, Temperature modelling in pavements: the effect of long- and short-wave radiation. Int. J. Pavement Eng. 16 (2015) 198–213. | DOI

[13] J. Martínez-Frutos, M. Kessler, A. Münch and F. Periago, Robust optimal Robin boundary control for the transient heat equation with random input data. Int. J. Numer. Methods Eng. 108 (2016) 116–135. | DOI | MR

[14] X. Min Zhou, Z.J. Yang, C. Chang and G. Song, Numerical assessment of electric roadway deicing system utilizing emerging carbon nanofiber paper. J. Cold Regions Eng. 26 (2012) 1–15. | DOI

[15] K. Morita and M. Tago, Operational characteristics of the gaia snow-melting system in Ninohe, Iwate, Japan. In: World Geothermal Congress 2000. Beppu–Morioka, Japan (2000) 3511–3516.

[16] A. Münch and F. Periago, Numerical approximation of bang–bang controls for the heat equation: an optimal design approach. Syst. Control Lett. 62 (2013) 643–655. | DOI | MR | Zbl

[17] E. Sachs, A parabolic control problem with a boundary condition of the Stefan-Boltzmann type. Z. Angew. Math. Mech. 58 (1978) 443–449. | DOI | MR | Zbl

[18] A. Scacht, M. Munk, C. Busen, M. Osser and B. Steinauer, Application of a porous interlayer for road temperature control. In: XIV International Winter Road Congress. Institute for Road and Traffic Engineering, RWTH Aachen university, Deutschland, PIARC, Andoraa la Vella (2014).

[19] T. Scharf, La route thermorégulée. In: ARTE Futuremag, Les routes de demain. Paris, France (2016).

[20] Z. Tang, Z. Li, J. Qian and K. Wang, Experimental study on deicing performance of carbon fiber reinforced conductive concrete. J. Mater. Sci. Technol. 21 (2005) 113–117.

[21] W. Yu, X. Yi, M. Guo and L. Chen, State of the art and practice of pavement anti-icing and de-icing techniques. Sci. Cold Arid Reg. 6 (2014) 14–21.

[22] J. Zhang, D. Das and R. Peterson, Selection of effective and efficient snow removal and ice control technologies for cold-region bridges. J. Civil Environ. Architectural Eng. 3 (2009) 1–14.

[23] H. Zhao, Z. Wu, S. Wang, J. Zheng and G. Che, Concrete pavement deicing with carbon fiber heating wires. Cold Regions Sci. Technol. 65 (2011) 413–420. | DOI