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Designing optimal trajectories for a skimmer ship to clean, recover and prevent the oil spilled on the sea from reaching the coast

Susana Gomez

Susana Gomez

Instituto de Matemáticas Aplicadas y Sistemas, Universidad Nacional A. de MéxicoMéxico, America
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Gomez, Susana
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Benjamin Ivorra

Benjamin Ivorra

Departamento de Análisis Matemático y Matemática Aplicada, Instituto de Matemática Interdisciplinar, Universidad Complutense de MadridMadrid, España
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Ivorra, Benjamin
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Angel M. Ramos

Angel M. Ramos

Departamento de Análisis Matemático y Matemática Aplicada, Instituto de Matemática Interdisciplinar, Universidad Complutense de MadridMadrid, España
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Ramos, Angel M.
  
31 déc. 2018
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Publié en ligne: 31 déc. 2018
Pages: 553 - 570
Reçu: 18 sept. 2018
Accepté: 18 déc. 2018
DOI: https://doi.org/10.2478/AMNS.2018.2.00043
Mots clés
© 2018 Susana Gomez et al., published by Sciendo
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Fig. 1

Graphical representation of the function coef(.) with λ1 = 20, λ2 = 4 and λ3 = 1.
Graphical representation of the function coef(.) with λ1 = 20, λ2 = 4 and λ3 = 1.

Fig. 2

(Left) Satellite image of the Prestige oil spill situation taken by the Envisat ASAR satellite (European Spatial Agency) on 17 November 2002. (Right) Oil concentration simulated by the model presented in this work for the same date. The coast is also represented in green.
(Left) Satellite image of the Prestige oil spill situation taken by the Envisat ASAR satellite (European Spatial Agency) on 17 November 2002. (Right) Oil concentration simulated by the model presented in this work for the same date. The coast is also represented in green.

Fig. 3

(Left) NASA satellite image presenting the Oleg Naydenov oil spill situation on 21 April 2015. The zone of interest is inside the red rectangle. The ship position is represented by a blue circle and the oil spill with a black line. (Right) Oil concentration simulated by the model presented in this work at the same date, considering the computational domain Ωz. The land is represented in green and the position of the Oleg Naydenov ship by a blue star.
(Left) NASA satellite image presenting the Oleg Naydenov oil spill situation on 21 April 2015. The zone of interest is inside the red rectangle. The ship position is represented by a blue circle and the oil spill with a black line. (Right) Oil concentration simulated by the model presented in this work at the same date, considering the computational domain Ωz. The land is represented in green and the position of the Oleg Naydenov ship by a blue star.

Fig. 4

(Left) Top view and (Right) 3D view of the final oil concentration distribution on 19 November 2002. Furthermore, the trajectory of the Prestige ship (continuous line) and the final position of the Prestige ship (o) are also presented.
(Left) Top view and (Right) 3D view of the final oil concentration distribution on 19 November 2002. Furthermore, the trajectory of the Prestige ship (continuous line) and the final position of the Prestige ship (o) are also presented.

Fig. 5

Top view of the final oil concentration distribution at 19 November 2002, skimmer ship trajectory (continuous red line), final position of the skimmer ship (x) and final position of the Prestige ship (o) for the experiments: (Top-Left) trajectory of the Prestige ship (continuous blue line) with no pumping process, (Top-Right) OPT-0.3-(3), (Bottom-Left) OPT-4-(3) and (Bottom Right) OPT-8-(3). The trajectory of the Prestige ship is presented in dotted blue line.
Top view of the final oil concentration distribution at 19 November 2002, skimmer ship trajectory (continuous red line), final position of the skimmer ship (x) and final position of the Prestige ship (o) for the experiments: (Top-Left) trajectory of the Prestige ship (continuous blue line) with no pumping process, (Top-Right) OPT-0.3-(3), (Bottom-Left) OPT-4-(3) and (Bottom Right) OPT-8-(3). The trajectory of the Prestige ship is presented in dotted blue line.

Fig. 6

3D view of the final oil concentration distribution on the 19th of November 2002 for the experiments: (Top-Left) case with no pumping process, (Top-Right) OPT-0.3-(3), (Bottom-Left) OPT-4-(3) and (Bottom Right) OPT-8-(3). A plane at the same height as the maximum value of the oil concentration is also presented. In this plane the skimmer ship trajectory (continuous red line), the Prestige ship trajectory when there is no pumping process (continuous blue line), final position of the skimmer ship (x) and final position of the Prestige ship (o) are depicted.
3D view of the final oil concentration distribution on the 19th of November 2002 for the experiments: (Top-Left) case with no pumping process, (Top-Right) OPT-0.3-(3), (Bottom-Left) OPT-4-(3) and (Bottom Right) OPT-8-(3). A plane at the same height as the maximum value of the oil concentration is also presented. In this plane the skimmer ship trajectory (continuous red line), the Prestige ship trajectory when there is no pumping process (continuous blue line), final position of the skimmer ship (x) and final position of the Prestige ship (o) are depicted.

Fig. 7

Top view of the distribution cw, defined in Section 3.2.2, on 19 2002, skimmer ship trajectory (continuous red line), final position of the skimmer ship (x); final position of the Prestige ship (o) for the experiments: (Top-Left) trajectory of the Prestige ship (continuous blue line) with no pumping process, (Top-Right) OPT-0.3-(4), (Bottom-Left) OPT-4-(4) and (Bottom Right) OPT-8-(4). The trajectory of the Prestige ship is also presented in dotted blue line.
Top view of the distribution cw, defined in Section 3.2.2, on 19 2002, skimmer ship trajectory (continuous red line), final position of the skimmer ship (x); final position of the Prestige ship (o) for the experiments: (Top-Left) trajectory of the Prestige ship (continuous blue line) with no pumping process, (Top-Right) OPT-0.3-(4), (Bottom-Left) OPT-4-(4) and (Bottom Right) OPT-8-(4). The trajectory of the Prestige ship is also presented in dotted blue line.

Fig. 8

3D view of the distribution cw, defined in Section 3.2.2, on 19 November 2002 for the experiments: (Top-Left) no pumping process, (Top-Right) OPT-0.3-(4), (Bottom-Left) OPT-4-(4) and (Bottom Right) OPT-8-(4). A plane at the same height as the maximum value of the oil concentration is also presented. In this plane the skimmer ship trajectory (continuous red line), the Prestige ship trajectory with no pumping process (continuous blue line), final position of the skimmer ship (x) and final position of the Prestige ship (o) are depicted.
3D view of the distribution cw, defined in Section 3.2.2, on 19 November 2002 for the experiments: (Top-Left) no pumping process, (Top-Right) OPT-0.3-(4), (Bottom-Left) OPT-4-(4) and (Bottom Right) OPT-8-(4). A plane at the same height as the maximum value of the oil concentration is also presented. In this plane the skimmer ship trajectory (continuous red line), the Prestige ship trajectory with no pumping process (continuous blue line), final position of the skimmer ship (x) and final position of the Prestige ship (o) are depicted.

Fig. 9

Top view of the final oil concentration distribution on 19 November 2002, skimmer ship trajectory (continuous red line), final position of the skimmer ship (x) and final position of the Prestige ship (o) for the experiments: (Top-Left) trajectory of the Prestige ship (continuous blue line) with no pumping process, (Top-Right) OPT-0.3-(4), (Bottom-Left) OPT-4-(4) and (Bottom Right) OPT-8-(4). The trajectory of the Prestige ship is also presented in dotted blue line.
Top view of the final oil concentration distribution on 19 November 2002, skimmer ship trajectory (continuous red line), final position of the skimmer ship (x) and final position of the Prestige ship (o) for the experiments: (Top-Left) trajectory of the Prestige ship (continuous blue line) with no pumping process, (Top-Right) OPT-0.3-(4), (Bottom-Left) OPT-4-(4) and (Bottom Right) OPT-8-(4). The trajectory of the Prestige ship is also presented in dotted blue line.

Fig. 10

3D view of the final oil concentration distribution on 19 November 2002 for the experiments: (Top-Left) no pumping process, (Top-Right) OPT-0.3-(4), (Bottom-Left) OPT-4-(4) and (Bottom Right) OPT-8-(4). A plane at the same height as the maximum value of the oil concentration is also presented. In this plane the skimmer ship trajectory (continuous red line), the Prestige ship trajectory with no pumping process (continuous blue line), final position of the skimmer ship (x) and final position of the Prestige ship (o) are depicted.
3D view of the final oil concentration distribution on 19 November 2002 for the experiments: (Top-Left) no pumping process, (Top-Right) OPT-0.3-(4), (Bottom-Left) OPT-4-(4) and (Bottom Right) OPT-8-(4). A plane at the same height as the maximum value of the oil concentration is also presented. In this plane the skimmer ship trajectory (continuous red line), the Prestige ship trajectory with no pumping process (continuous blue line), final position of the skimmer ship (x) and final position of the Prestige ship (o) are depicted.

Fig. 11

(Left) Top view and (Right) 3D view of the final oil concentration distribution on 19 November 2002, skimmer ship trajectory (continuous line), final position of the skimmer ship (x) and final position of the Prestige ship (o) for the experiments with two skimmer ships, one following the trajectory generated by OPT-8-(4) and another one following the Prestige ship trajectory. The trajectory of the prestige ship is also presented in dotted line. In the 3D view case, a plane at the same height as the maximum value of the oil concentration is also presented. In this plane the skimmer ship trajectory (continuous line), Prestige ship trajectory (dotted line), final position of the skimmer ship (x) and final position of the Prestige ship (o) are depicted.
(Left) Top view and (Right) 3D view of the final oil concentration distribution on 19 November 2002, skimmer ship trajectory (continuous line), final position of the skimmer ship (x) and final position of the Prestige ship (o) for the experiments with two skimmer ships, one following the trajectory generated by OPT-8-(4) and another one following the Prestige ship trajectory. The trajectory of the prestige ship is also presented in dotted line. In the 3D view case, a plane at the same height as the maximum value of the oil concentration is also presented. In this plane the skimmer ship trajectory (continuous line), Prestige ship trajectory (dotted line), final position of the skimmer ship (x) and final position of the Prestige ship (o) are depicted.

Objective function (4) value (O_ Func_) and percentage reduction (Reduc_) of this value regarding the scenario Q=0 given by the experiments (Exp_): OPT-0_3-(4), OPT-4-(4), OPT-8-(4), PT-0_3-(4), PT-4-(4) and PT-8-(4)_

Exp.OPT-0.3-(4)OPT-4-(4)OPT-8-(4)
O. Func.2.144e101.366e101.105e10
Reduc.114355
Exp.PT-0.3-(4)PT-4-(4)PT-8-(4)
O. Func.2.3632e102.2057e101.330e+10
Reduc.2945

Objective function (3) value (O_ Func_) expressed in tons of oil and the percentage of oil pumped (Reduc_) given by the experiments (Exp_): OPT-0_3-(3), OPT-4-(3), OPT-8-(3), PT-0_3-(3), PT-4-(3) and PT-8-(3)

Exp.OPT-0.3-(3)PT-4-(3)OPT-8-(3)
O. Func.50,91532,24131,627
Reduc.64042
Exp.PT-0.3-(3)PT-4-(3)PT-8-(3)
O. Func.52,71236,96823,574
Reduc.23156

Objective function (4) value (O_ Func_) expressed in tons of oil and the percent reduction (Reduc_) of this value regarding the scenario Q=0 given by the experiments (Exp_): OPT-0_3-(4), OPT-4-(4), OPT-8-(4), PT-0_3-(4), PT-4-(4) and PT-8-(4)_

Exp.OPT-0.3-(4)OPT-4-(4)OPT-8-(4)
O. Func.50,97341,23340,136
Reduc.62426
Exp.PT-0.3-(4)PT-4-(4)PT-8-(4)
O. Func.52,71236,96823,574
Reduc.23156
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